Storage Ring to Search for Electric Dipole Moments of Charged Particles -- Feasibility Study
F. Abusaif, A. Aggarwal, A. Aksentev, B. Alberdi-Esuain, A. Atanasov, L. Barion, S. Basile, M. Berz, M. Beyß, C. Böhme, J. Böker, J. Borburgh, C. Carli, I. Ciepał, G. Ciullo, M. Contalbrigo, J.-M. De Conto, S. Dymov, O. Felden, M. Gagoshidze, M. Gaisser, R. Gebel, N. Giese, K. Grigoryev, D. Grzonka, M. Haj Tahar, T. Hahnraths, D. Heberling, V. Hejny, J. Hetzel, D. Hölscher, O. Javakhishvili, L. Jorat, A. Kacharava, V. Kamerdzhiev, S. Karanth, C. Käseberg, I. Keshelashvili, I. Koop, A. Kulikov, K. Laihem, M. Lamont, A. Lehrach, P. Lenisa, N. Lomidze, B. Lorentz, G. Macharashvili, A. Magiera, K. Makino, S. Martin, D. Mchedlishvili, U.-G. Meißner, Z. Metreveli, J. Michaud, F. Müller, A. Nass, G. Natour, N. Nikolaev, A. Nogga, A. Pesce, V. Poncza, D. Prasuhn, J. Pretz, F. Rathmann, J. Ritman, M. Rosenthal, A. Saleev, M. Schott, T. Sefzick, Y. Senichev, D. Shergelashvili, V. Shmakova, S. Siddique, A. Silenko, M. Simon, J. Slim, H. Soltner, A. Stahl, R. Stassen, E. Stephenson, H. Straatmann, H. Ströher, M. Tabidze, G. Tagliente, R. Talman, Yu. Uzikov, Yu. Valdau, E. Valetov, T. Wagner, C. Weidemann, A. Wirzba, A. Wrońska, P. Wüstner, P. Zupranski, M. Żurek
SStorage Ring to Searchfor Electric Dipole Moments of Charged Particles
Feasibility Study
F. Abusaif, A. Aggarwal, A. Aksentev, B. Alberdi-Esuain, A. Atanasov, L. Barion, S. Basile, M. Berz, M. Beyß, C. Böhme, J. Böker, J. Borburgh, C. Carli, I. Ciepał, G. Ciullo, M. Contalbrigo, J.-M. De Conto, S. Dymov, O. Felden, M. Gagoshidze, M. Gaisser, R. Gebel, N. Giese, K. Grigoryev, D. Grzonka, M. Haj Tahar, T. Hahnraths, D. Heberling, V. Hejny, J. Hetzel, D. Hölscher, O. Javakhishvili, L. Jorat, A. Kacharava, V. Kamerdzhiev, S. Karanth, C. Käseberg, I. Keshelashvili, I. Koop, A. Kulikov, K. Laihem, M. Lamont, A. Lehrach, P. Lenisa, N. Lomidze, B. Lorentz, G. Macharashvili, A. Magiera, K. Makino, S. Martin, D. Mchedlishvili, U.-G Meißner, ,b, , Z. Metreveli, J. Michaud, F. Müller, A. Nass, G. Natour, N. Nikolaev, A. Nogga, ,b A. Pesce, V. Poncza, D. Prasuhn, J. Pretz, F. Rathmann, J. Ritman, M. Rosenthal, A. Saleev, M. Schott, T. Sefzick, Y. Senichev, D. Shergelashvili, V. Shmakova, S. Siddique, A. Silenko, M. Simon, J. Slim, H. Soltner, A. Stahl, R. Stassen, E. Stephenson, H. Straatmann, H. Ströher, M. Tabidze, G. Tagliente, R. Talman, Yu. Uzikov, Yu. Valdau, E. Valetov, T. Wagner, C. Weidemann, A. Wirzba, ,b A. Wro ´nska, P. Wüstner, P. Zupranski, M. ˙Zurek. Abstract:
The proposed method exploits charged particles confined as a storage ringbeam (proton, deuteron, possibly helium-3) to search for an intrinsic electric dipole mo-ment (EDM) aligned along the particle spin axis. Statistical sensitivities could approach − e.cm. The challenge will be to reduce systematic errors to similar levels. The ringwill be adjusted to preserve the spin polarisation, initially parallel to the particle velocity, fortimes in excess of 15 minutes. Large radial electric fields, acting through the EDM, will ro-tate the polarisation. The slow rise in the vertical polarisation component, detected throughscattering from a target, signals the EDM.The project strategy is outlined. It foresees a step-wise plan, starting with ongoing COSYactivities that demonstrate technical feasibility. Achievements to date include reduced polar-isation measurement errors, long horizontal-plane polarisation lifetimes, and control of thepolarisation direction through feedback from the scattering measurements. The project con-tinues with a proof-of-capability measurement (precursor experiment; first direct deuteronEDM measurement), an intermediate prototype ring (proof-of-principle; demonstrator forkey technologies), and finally the high precision electric-field storage ring.CERN-PBC-REPORT-2019-002CPEDM CollaborationDecember 2019 a r X i v : . [ h e p - e x ] D ec Institut für Kernphysik, Forschungszentrum Jülich, 52425 Jülich, Germany b Institute for Advanced Simulation, Forschungszentrum Jülich, 52425 Jülich, Germany Institute of Physics, Jagiellonian University, 30348 Cracow, Poland III. Physikalisches Institut B, RWTH Aachen University, 52056 Aachen, Germany University of Ferrara and INFN, 44100 Ferrara, Italy Department of Physics and Astronomy, Michigan State University, East Lansing, Michigan 48824, USA European Organization for Nuclear Research, CERN CH-1211 Genéve 23, Switzerland Institute of Nuclear Physics PAS, 31-342 Cracow, Poland CNRS/IN2P3 - UGA, LPSC , Grenoble, France Laboratory of Nuclear Problems, Joint Institute for Nuclear Research, 141980 Dubna, Russia Dept. of Electrical and Computer Engineering, Agricultural University of Georgia, 0159 Tbilisi, Georgia Institut für Hochfrequenztechnik, RWTH Aachen University, 52056 Aachen, Germany Budker Institute of Nuclear Physics, 630090 Novosibirsk, Russia High Energy Physics Institute, Tbilisi State University, 0186 Tbilisi, Georgia Bethe Center for Theoretical Physics, Universität Bonn, 53115 Bonn, Germany Zentralinstitut für Engineering, Elektronik und Analytik, Forschungszentrum Jülich, 52425 Jülich, Germany L.D. Landau Institute for Theoretical Physics, 142432 Chernogolovka, Russia Samara National Research University, 443086 Samara, Russia Institute of Physics, Johannes Gutenberg University Mainz, 55128 Mainz, Germany Institute for Nuclear Research, Russian Academy of Sciences, 117312 Moscow, Russia Research Institute for Nuclear Problems, Belarusian State University, 220030 Minsk, Belarus Indiana University Center for Spacetime Symmetries, Bloomington, Indiana 47405, USA INFN, 70125 Bari, Italy Cornell University, Ithaca, New York 14850, USA Andrzej Soltan Institute for Nuclear Studies, Warsaw, Poland Helmholtz-Institut für Strahlen und Kernphysik, Universität Bonn, 53115 Bonn, Germany Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA cknowledgements
The authors would like to acknowledge important discussions with, and significant contributions by,Yannis K. Semertzidis and his colleagues from the Center for Axion and Precision Physics (CAPP) of theKorean Advanced Institute of Science and Technology (KAIST, South Korea) to this study; appendicesB and F were authored by them.This report is supported by an ERC Advanced Grant of the European Commission (srEDM, ontents
Acronyms and abbreviations viiiExecutive Summary 11 Introduction 13 CP -violating QCD ¯ θ parameter . . . . . . . . . . . . . . . . . . 232.3 EDM analysis based on non-perturbative methods . . . . . . . . . . . . . . . . . . . . . . 232.3.1 Determination of the ¯ θ induced nucleon EDM . . . . . . . . . . . . . . . . . . . . . . . 232.3.2 Estimates of the nucleon EDM terms in the BSM scenario . . . . . . . . . . . . . . . . . 242.3.3 Estimates of the nuclear EDM matrix elements for light nuclei . . . . . . . . . . . . . . 252.4 Option for oscillating EDM searches at storage rings . . . . . . . . . . . . . . . . . . . . . 26 iii.1 Introduction – BNL design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 778.2 Preparedness for the full-scale ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 808.3 New ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
10 Sensitivity and Systematics 93
11 Polarimetry 106
12 Spin Tracking 126
13 Roadmap and Timeline 137
Appendices 139A Results and achievements at Forschungszentrum Jülich 140
A.1 Results and achievements at COSY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140A.1.1 High precision spin tune measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . 140A.1.2 Long horizontal polarisation lifetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141A.1.3 Feedback and control of polarisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141A.1.4 Invariant spin axis measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142A.1.5 Radio-Frequency Wien filter for spin manipulation . . . . . . . . . . . . . . . . . . . . 142A.1.6 Measurements of deuteron carbon and proton carbon analysing powers . . . . . . . . . . 144A.1.7 Orbit control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144A.1.8 Beam Based Alignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144A.1.9 Beam Position Monitor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145A.1.10 Electrostatic and combined Deflector development . . . . . . . . . . . . . . . . . . . . . 145A.1.11 "Spin-Offs" . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148A.2 Results and achievements from the Jülich/Bonn theory group . . . . . . . . . . . . . . . . 149
B Mitigation of background magnetic fields 154
B.1 Static magnetic field configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154B.1.1 Static radial magnetic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154B.1.2 Static longitudinal magnetic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157B.1.3 Static vertical magnetic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158B.2 Effect of alternating magnetic fields and the geometric phases . . . . . . . . . . . . . . . . 158B.3 Magnetic shielding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160B.3.1 Residual field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161B.3.2 Time stability of the residual field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161B.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
C Statistical Sensitivity 165
C.1 Statistical error on EDM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165C.2 Precursor Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168C.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
D Gravity and General Relativity as a ‘Standard Candle’ 170E Additional Science Option: Axion Search 173
E.1 Concept of Search for Axion-like Particles . . . . . . . . . . . . . . . . . . . . . . . . . . 173E.2 Technical Considerations for an Axion Search . . . . . . . . . . . . . . . . . . . . . . . . 174v.3 Initial Tests with Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175E.4 Immediate Plans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
F New ideas: Hybrid Scheme 180
F.1 Experimental Method using a hybrid ring lattice . . . . . . . . . . . . . . . . . . . . . . . 180F.2 Systematic errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183F.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
G New ideas: Doubly Magic EDM Measurement Method 185
G.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185G.1.1 Major previous EDM advances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185G.1.2 Koop spin wheel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186G.1.3 Proposed EDM measurement technique . . . . . . . . . . . . . . . . . . . . . . . . . . 186G.1.4 Polarimetry assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186G.2 Orbit and spin tune calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186G.2.1 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186G.2.2 Fractional bending coefficients η E and η M . . . . . . . . . . . . . . . . . . . . . . . . . 187G.2.3 Spin tune expressed in terms of η E and η M . . . . . . . . . . . . . . . . . . . . . . . . . 187G.2.4 The “magic energy” condition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187G.2.5 Wien filter spin-tune adjustment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188G.3 “MDM comparator trap” operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188G.3.1 Dual beams in a single ring. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188G.3.2 Sensitivity to imperfections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188G.3.3 Spin tune invariance and spin tune comparator trap precision . . . . . . . . . . . . . . . 189G.4 Secondary beam solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190G.4.1 Analytic formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190G.5 Three practical doubly-magic solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191G.5.1 Promising doubly-magic solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191G.5.2 Perturbative variant of all-electric (original) holy grail ring. . . . . . . . . . . . . . . . . 192G.5.3 Proton-positron solution—the (new) holy grail. . . . . . . . . . . . . . . . . . . . . . . 192G.5.4 Helion-proton solution, JEDI-capable option. . . . . . . . . . . . . . . . . . . . . . . . 193G.5.5 Stability of the 233 MeV all-electric fixed point . . . . . . . . . . . . . . . . . . . . . . 193G.6 Gravitational effect EDM calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195G.7 The need for non-destructive resonant polarimetry . . . . . . . . . . . . . . . . . . . . . . 195G.8 The EDM measurement campaign . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 H New ideas: Spin Tune Mapping for EDM Searches 199
H.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199H.2 The mixing of EDM signal with systematic background from MDM . . . . . . . . . . . . 199H.3 Advantage of electrostatic rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200H.4 The effect of the Wien filter on beam and spin . . . . . . . . . . . . . . . . . . . . . . . . 200H.4.1 Spin tune shift by Wien filter and EDM . . . . . . . . . . . . . . . . . . . . . . . . . . . 201vi.5 Spin wheel at the beam energy of frozen spin . . . . . . . . . . . . . . . . . . . . . . . . . 203H.6 Other possibilities for EDM measurements with counter-circulating beams . . . . . . . . . 204H.6.1 An option with two RF Wien filters in the prototype EDM ring . . . . . . . . . . . . . . 204H.6.2 An option with static Wien filters in the prototype EDM ring . . . . . . . . . . . . . . . 204H.7 Summary and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
I New ideas: Deuteron EDM Frequency Domain Determination 206
I.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206I.2 Universal SR EDM measurement problems . . . . . . . . . . . . . . . . . . . . . . . . . . 207I.2.1 Spin motion perturbation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207I.2.2 Expected machine imperfection SW roll rate . . . . . . . . . . . . . . . . . . . . . . . . 208I.2.3 Spin decoherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209I.2.4 Machine imperfections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209I.3 Main methodology features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210I.4 EDM estimator statistic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210I.5 Effective Lorentz factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211I.6 Guide field flipping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212I.7 Statistical precision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
J New ideas: Distinguishing the effects of EDM and magnet misalignment by Fourier anal-ysis 215K New ideas: External Polarimetry 219
K.1 Pellet-extracted beam sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219K.1.1 Pellet-extracted beam sampling; qualitative . . . . . . . . . . . . . . . . . . . . . . . . 219K.1.2 Experimental confirmation of wire and pellet beam extraction at COSY . . . . . . . . . 222K.1.3 Re-interpretation and revision of COSY moving wire beam experiments . . . . . . . . . 223K.1.4 Quantitative formulation of pellet beam sampling . . . . . . . . . . . . . . . . . . . . . 225K.1.5 Derivations of required formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226vii cronyms and abbreviations
A AC Alternating CurrentALP Axion-like ParticleANKE Name of detector at COSYB BMT Bargmann-Michel-Telegdi (equation)BNL Brookhaven National LaboratoryBPM Beam Position MonitorBSM Beyond Standard Model (of elementary particle physics)C C Charge (symmetry)CAPP Center for Axion and Precision Physics (Research) (Daejeon, South Korea)CCW Counter Clock-WiseCDR Conceptual Design ReportCeNTREX Name of experiment to search for proton EDM in Tl-nucleiCERN Conseil Européan pour la Recherche NucléaireCESR Cornell Electron-Positron Storage RingChPT Chiral Perturbation TheoryCKM Cabibbo-Kobayashi-Maskawa (matrix)COSY Cooler Synchrotron (storage ring) (Forschungszentrum Jülich, Germany)CP Charge-parity (invariance)CPEDM Charged Particle Electric Dipole Moment (collaboration)CPT Charge-parity-time reversal (symmetry)CSR Cryogenic Storage Ring (Max-Planck Institute, Heidelberg, Germany)CW Clock-WiseD DC Direct CurrentDESY Deutsches Elektronen Synchrotron (Hamburg, Germany)DORIS Name of a detector at DESYE EDM Electric Dipole MomentELENA Extra Low Energy Antiproton (ring) (CERN)F FNAL Fermi National Accelerator Laboratory (Chicago, USA)FRM-II Forschungsreaktor München (Heinz Maier-Leibnitz, München, Germany)G G Magnetic anomalyGR General RelativityH HGF Helmholtz-Gemeinschaft Deutscher ForschungszentrenI IBS Institute for Basic Science (South Korea)IKP Institut für Kernphysik (Institute for Nuclear Physics of FZJ) (Jülich, Germany)ILL Institut Laue-Langevin (Grenoble, France)ISOLDE Isotope Separator On Line Device (CERN)J JEDI Jülich Electric Dipole moment Investigations (collaboration)J-PARC Japan Proton Accelerator Research Complex (Tokai, Japan)JULIC Jülich Light Ion Cyclotron (FZJ, Germany)K KAIST Korea Advanced Institute of Science and Technology (South Korea)KM Kobayashi-Maskawa (mixing matrix)KVI KVI - Center for Advanced Radiation Technology (Groningen, The Netherlands)L LANL Los Alamos National Laboratory (Los Alamos, USA)LC Inductance-CapacitorM MDM Magnetic Dipole Moment viiiV Megavolt (10 V)N NEG Non Evaporable Getter (pumps)P PAC Program Advisory CommitteePNPI Petersburg Nuclear Physics Institute (Gatchina, Russia)PSI Paul Scherrer Institute (Villigen, Switzerland)PTR Prototype RingQ QCD Quantum ChromodynamicsR R&D Research and DevelopmentRF RadiofrequencyRLC Resistance-Conductance-CapacitorROI Region of InterestRWTH Rheinisch-Westfälische Technische Hochschule (RWTH Aachen University, Germany)S SCT Spin Coherence TimeSM Standard Model (of elementary particle physics)SNS Spallation Neutron Source (Oak Ridge, USA)SQUID Superconducting Quantum Interference DevicesrEDM Storage Ring Electric Dipole MomentSUSY SupersymmetryT T Tesla (unit), Time-reversal (symmetry)TDR Technical Design ReportTlF Thallium FluorideTRIUMF Canada’s particle accelerator center (Vancouver, Canada)U US DOE United States Department of Energyix xecutive Summary Science context and objectives
Symmetry considerations and symmetry-breaking patterns have played an important role in the develop-ment of physics in the last 100 years. Experimental tests of discrete symmetries ( e.g. , parity P , charge-conjugation C , their product CP , time-reversal invariance T , the product CP T , baryon- and/or leptonnumber) have been essential for the development of the Standard Model (SM) of particle physics.Subatomic particles with nonzero spin (regardless whether of elementary or composite nature) canonly support a nonzero permanent electric dipole moment (EDM) if both time-reversal ( T ) and parity ( P )symmetries are violated explicitly while the charge symmetry ( C ) can be maintained (see e.g. [2]). As-suming the conservation of the combined CP T symmetry, T -violation also implies CP -violation. The CP -violation generated by the Kobayashi-Maskawa (KM) mechanism of weak interactions contributesa very small EDM that is several orders of magnitude below current experimental limits. However, manymodels beyond the Standard Model predict EDM values near the current experimental limits. Findinga non-zero EDM value of any subatomic particle would be a signal that there exists a new source of CP violation, either induced by the strong CP violation via the θ QCD angle or by genuine physics be-yond the SM (BSM). In fact, the best upper limit on θ QCD follows from the experimental bound on theEDM of the neutron. CP violation beyond the SM is also essential for explaining the mystery of theobserved baryon-antibaryon asymmetry of our universe, one of the outstanding problems in contempo-rary elementary particle physics and cosmology. A measurement of a single EDM will not be sufficientto establish the sources of any new CP -violation. Complementary observations of EDMs in multiplesystems will thus prove essential. Up to now measurements have focused on neutral systems (neutron,atoms, molecules). We propose to use a storage ring to measure the EDM of charged particles.The storage ring method would provide a direct measurement of the EDM of a charged particlecomparable to or better than present investigations on ultra-cold neutrons. The neutron investigationsmeasure the precession frequency jumps in traps containing magnetic and electric fields as the sign ofthe electric field is changed. These experiments are now approaching sensitivities of − e · cm [3] andpromise improvements of another order of magnitude within the next decade. Because proton beams trapsignificantly more particles, statistical sensitivities may reach the order of − e · cm [4] with a new,all-electric, high-precision storage ring. Indirect determinations for the proton produce model-dependentEDM limits near × − e · cm using Hg [5]. Thus storage rings could take the lead as the mostsensitive method for the discovery of an EDM.It should be noted that the rotating spin-polarised beam used in the EDM search is also sensitiveto the presence of an oscillating EDM resulting from axions or axion-like fields, which correspond to thedark-matter candidates of a pseudo-scalar nature. These may be detected through a time series analysisof EDM search data or by scanning the beam’s spin-rotation frequency in search of a resonance with anaxion-like mass in the range from µ eV down to − eV [6, 7]. Methodology
The electric dipole must be aligned with the particle spin since it provides the only axis in its rest frame.The EDM signal is based on the rotation of the electric dipole in the presence of an external electricfield that is perpendicular to the particle spin. The particles are formed into a spin-polarised beam.Measurements are made on the beam as it circulates in the ring, confined by the ring electromagneticfields that always generate an electric field in the particle frame pointing to the centre of the ring.For a particle propagating in generic magnetic (cid:126)B and electric (cid:126)E fields the spin motion is described Update of the version submitted to the European Strategy for Particle Physics [1]
1y the Thomas-Bargmann-Michel-Telegdi (Thomas-BMT) equation and its extension for the EDM [8] : d (cid:126)S d t = = (cid:16) (cid:126) Ω MDM + (cid:126) Ω EDM (cid:17) × (cid:126)S, (1) (cid:126) Ω MDM = − qm (cid:34)(cid:18) G + 1 γ (cid:19) (cid:126)B − γGγ + 1 (cid:126)β (cid:16) (cid:126)β · (cid:126)B (cid:17) − (cid:18) G + 1 γ + 1 (cid:19) (cid:126)β × (cid:126)Ec (cid:35) ,(cid:126) Ω EDM = − ηq mc (cid:20) (cid:126)E − γγ + 1 (cid:126)β (cid:16) (cid:126)β · (cid:126)E (cid:17) + c(cid:126)β × (cid:126)B (cid:21) . The angular velocities, (cid:126) Ω MDM and (cid:126) Ω EDM , act through the magnetic dipole moment (MDM) and electricdipole moment (EDM) respectively. (cid:126)S in this equation denotes the spin vector in the particle rest frame, t the time in the laboratory system, (cid:126)β and γ the relativistic Lorentz factors. The magnetic anomaly G and the electric dipole factor η are dimensionless and introduced via the magnetic dipole moment (cid:126)µ and electric dipole moment (cid:126)d , which are both pointing in the same direction and are proportional to theparticle’s spin (cid:126)S : (cid:126)µ = g q (cid:126) m (cid:126)S = (1 + G ) q (cid:126) m (cid:126)S, (cid:126)d = η q (cid:126) mc (cid:126)S, (2)where q and m are the charge and the mass of the particle, respectively.The effect of the torque for a positively charged particle is illustrated in Fig. 1. In this examplethe magnetic and electric fields are purely vertical and purely radial, respectively. A particle is confinedon an ideal planar, closed orbit in the ring. Its velocity (cid:126)v = c(cid:126)β is along the orbit. The spin axis isgiven by the purple arrow that rotates in a plane perpendicular to (cid:126)E . If the initial condition begins withthe spin parallel to the velocity, then the rotation caused by the EDM will make the vertical componentof the beam polarisation change. This rotation receives a contribution from both the external field (cid:126)E and the motional electric field c(cid:126)β × (cid:126)B , and becomes the signal observed by a polarimeter located in thering. This device allows beam particles to scatter from nuclei in a fixed bulk material target (black). Thedifference in the scattering rate between the left and right directions (into the blue detectors) is sensitiveto the vertical polarisation component of the beam. Continuous monitoring will show a change in therelative left-right rate difference during the time of the beam storage if a measurable EDM is present. Ev Bp
POLARIMETER
Fig. 1:
Diagram showing a particle travelling around the storage ring confined by purely vertical magnetic andpurely radial electric fields. The polarisation, initially along the velocity, precesses slowly upward in responseto the radial electric field acting on the EDM. The vertical component of this polarisation is observed throughscattering in the polarimeter.
The angular velocities ( (cid:126) Ω ) in Eq. (1) describe the rotation of the spin vector of the particle as ittravels around the ring. Because the magnetic moments of all particles carry an anomalous part, thepolarisation will in general rotate in the plane of the storage ring relative to the beam path. This rotation More details on the application of the Thomas-BMT equation for circular accelerators and the inclusion of gravity effectsare discussed in Chap. 4. (cid:126) Ω MDM to the cyclotron frequency (cid:126) Ω cycl = − qγm (cid:32) (cid:126)B ⊥ − (cid:126)β × (cid:126)E r β c (cid:33) , (3) i.e. (cid:126) Ω MDM = (cid:126) Ω cycl , a condition called “frozen spin”. Under this condition, the vertical polarisation canbuild up. In a magnetic ring, this condition requires that (since (cid:126)β · (cid:126)B = 0 ) a radial electric field is addedto the ring bending elements with E r = GBcβγ − Gβ γ . (4)For particles such as the proton where G > , it is also possible to build an all-electric ring( (cid:126)B = 0 ) provided that one can choose γ = (cid:112) /G . For the proton, this gives p = 0 . GeV/c.The kinetic energy of T = 232 . MeV fortuitously comes at a point where the spin sensitivity of thepolarimeter is near its maximum ( e.g. , carbon target), creating an advantageous experimental situation.The statistical error for one single machine cycle is given by [4] σ stat ≈ (cid:126) √ N f τ P AE . (5)Assuming the parameters given in Table 2, the statistical error for one year of running ( i.e. , 10000 cyclesof 1000 s length) is σ stat (1 year) = 2 . × − e · cm . (6)The challenge is to suppress the systematic error to the same level. Table 2:
Parameters relevant for the statistical error in the proton experiment. beam intensity N = 4 · per fillpolarisation P = 0 . spin coherence time τ = 1000 selectric fields E = 8 MV/mpolarimeter analyzing power A = 0 . polarimeter efficiency f = 0 . Many of the systematic errors in the EDM search may be eliminated by looking at the differencebetween two experiments run with clockwise (CW) and counter-clockwise (CCW) beams in the ring.One beam represents the time-reverse of the other, and the difference will show only time-odd effectssuch as the EDM. For the proton, the choice of an all-electric ring allows the two beams to be present inthe ring at the same, an advantage when suppressing systematic effects. Figure 2 illustrates two featuresof the all-electric proton experiment, the counter-rotating beams and the alternating direction of thepolarisation (along or against the velocity) in separate beam bunches, which is important for geometricerror cancellation in the polarimeter.In general any phenomena other than an EDM generating a vertical component of the spin limitsthe sensitivity ( i.e. the smallest detectable EDM) of the proposed experiment. Such systematic effectsmay be caused by unwanted electric fields due to imperfections of the focusing structure (such as themisalignment of components) or by magnetic fields penetrating the magnetic shielding or produced in-side the shield by the beam itself, the RF cavity, or gravity. A combination of several such phenomena,or a combination of an average horizontal spin and one of these phenomena, may as well lead to suchsystematic effects. 3 − + − + − + − + − + − + − + − + − + − + − + − ~p~s Fig. 2:
Electric storage ring with simultaneously clockwise and counter clockwise circulating beams (dark andlight blue arrows), each with two helicity states (green and red arrows for each beam). The gray circles representelectric field plates.
In many cases, as for example effects due to gravity, the resulting rotations of the spin into thevertical plane do not mimic an EDM because the observations for the two counter-rotating are not com-patible with a time-odd effect. In this case, the contributions from the two counter-rotating beams tendto cancel, provided the forward and reverse polarimeters can be calibrated with sufficient precision. Insome cases, for example, magnetic fields from the RF cavity, the resulting spin rotations into the verticalplane can be large.The most important mechanism dominating systematic effects is an average static radial magneticfield that mimics an EDM signal. For a 500 m circumference frozen-spin EDM ring, an average magneticfield of about − T generates the same vertical spin precession as the EDM of − e · cm the finalexperiment aims at being able to identify. In order to mitigate systematic effects, the proposed ringwill be installed in state-of-the-art magnetic shielding that reduces residual fields to the nT level [4].The vertical position difference between the two counter-rotating beams that is caused by the remainingradial field will be measured with special pick-ups that must be installed at very regular locations aroundthe circumference to measure the varying radial magnetic field component created by the bunched beamseparation. A complete, thorough study of systematic errors in the EDM experiment is very delicateand not yet available. Studies of systematic effects have been carried out and are underway by severalteams in the CPEDM collaboration to further improve the understanding of basic phenomena to be takeninto account and to estimate the achievable sensitivity. The preliminary conclusion is that the intendedsensitivity is very challenging. Meeting this challenge requires that we proceed in a series of stages (seeFig. 3) where each one depends on the knowledge gained from the preceding stage’s experience. Readiness and Expected Challenges
The JEDI (Jülich Electric Dipole moment Investigations) Collaboration has worked with COSY (COolerSYnchrotron at the Forschungszentrum Jülich in Germany) for the last decade to demonstrate the fea-sibility of critical EDM technologies for the storage ring. Historically, these studies were begun withdeuterons, and the switch has not been made yet to protons in order to preserve and build on the deuteronexperience. These studies are briefly itemised below.– The beam may be slowly brought to thick ( ∼ ν S = Gγ , polarisation revolutions per turn)to a part in in a single cycle of 100 s length [11]. The polarimeter signals permit feedbackstabilisation of the phase [12] of the in-plane precession to better than one part per billion ( )over the time of the machine store. This is necessary to maintain frozen spin.– By using bunched beam, electron cooling, and trimming of the ring fields to sextupole order, thepolarisation decoherence with time may be reduced, yielding a lifetime in excess of 1000 s [13].Observation of the spin tune variations allows for the measurement of the direction of the invariantpolarisation axis with a precision of about 1 mrad [14].With deuterons in the COSY ring at 970 MeV/c with non-frozen spin, the polarisation precessesin the horizontal plane at 121 kHz relative to the velocity. The EDM is associated with a tilt of theinvariant spin axis away from the vertical direction. This tilt of the spin rotation axis generates anoscillation of the vertical spin component. This effect is too small for a measurement with a reasonablesensitivity. However, using an RF Wien filter in the ring with fields oscillating synchronously with thespin precession in the horizontal plane, a vertical spin component builds up over the whole duration ofthe fill. This has become the basis for the precursor experiment (see Fig. 3, stage 1). Initial runningreveals that EDM-like signals arise from systematic perturbations of the deuteron spin as it goes aroundCOSY. These resemble the effects of small rotational misalignments of the Wien filter about the beamline and longitudinal polarisation changes induced by a solenoid located across the ring from the Wienfilter. Using these two effects for reference, measurements can lead to the location of the invariant spinaxis. In parallel, studies and preparations for the proton EDM measurement in a fully-electric frozen-spin ring are on-going. Some of the key technologies are currently under development for the final ring.These include:– electrostatic deflector design that requires testing full scale prototypes in a magnetic field withbeam to levels of at least 8 MV/m;– beam position monitors are needed to operate at a precision of 10 nm for a measurement time of1000 s;– the ring must be shielded to provide isolation from systematic radial magnetic fields to the nTlevel [4].Spin tracking calculations are needed to verify the level of precision needed in the ring constructionand the handling of systematic errors. For a detailed study during beam storage and buildup of the EDMsignal, one needs to track a large sample of particles for billions of turns. The COSY-Infinity [15] andBmad [16] simulation programs are utilised for this purpose. Given the complexity of the tasks, particleand spin tracking programs have been benchmarked and simulation results compared to beam and spinexperiments at COSY to ensure the required accuracy of the results.Finally, a strategy will be needed to verify any signal produced by the experiment after the CW-CCW subtraction through a series of critical tests and independent analyses.When constructed, the proton EDM experiment will be the largest electrostatic ring ever built. Itwill have unique features, such as counter-rotating beams and strenuous alignment and stability require-ments. It may also require stochastic cooling and weak magnetic focusing consistent with dual beamoperation. Intense discussions within the CPEDM collaboration have concluded that the final ring can-not be designed and built in one step; instead, a smaller-scale prototype ring (see Fig. 3, stage 2) mustbe constructed to confirm and refine the following critical features.5 The ring stores high beam intensities for a sufficiently long time.– Beam injection must allow for multiple polarisation states (longitudinal fore and aft, sideways forpolarisation coherence monitoring) in both CW and CCW beams.– The ring must circulate CW and CCW beams simultaneously, both horizontally polarised.– The ring must support frozen spin.– Magnetic shielding must operate to reduce the ambient magnetic fields (esp. radial) to suitablelevels while allowing full operation of the ring high voltage, vacuum, monitoring, and control.– Polarimeter measurements must be made for both CW and CCW beams using the same target. Asecond polarimeter is needed with independent beam extraction onto its target.– Beam cooling (electron-cooling before injection, or stochastic) is required to reduce the beamphase space. CPEDM Strategy
As emphasised above this challenging project needs to proceed in stages that are also outlined in Fig. 3.1. COSY will continue to be used as long as possible for the continuation of critical R&D associatedwith the final experiment design. An important requirement is to test as many of the results aspossible with protons where the larger anomalous magnetic moment leads to more rapid spinmanipulation speeds.2. The precursor experiment will be completed and analysed. Some data will be taken with an im-proved version of the Wien filter with better electric and magnetic field matching.3. The next stage is to design, fund, and build a prototype ring (discussed in detail below) to addresscritical questions concerning the features of the EDM ring design. At 30 MeV, the ring with onlyan electric field can store counter-rotating beams, but they are not frozen spin. At 45 MeV withan additional magnetic field, the frozen spin condition can be met. But the magnetic fields alsoprevent the CW and CCW beams from being stored at the same time. Even so, an EDM experimentmay be done with these two beams used on alternating fills.4. Following step 3, the focus will be to create the final ring design, then fund and construct it.5. Once the ring is ready, the longer term activity will be to commission and operate the final ring, im-proving it with new versions as the systematic errors and other experimental issues are understoodand improved. Precursor Experiment Prototype Ring All-electric Ring dEDM proof-of-capability (orbit and polarization control;first dEDM measurement) pEDM proof-of-principle (key technologies,first direct pEDM measurement) pEDM precision experiment (sensitivity goal: 10 -29 e cm)- Magnetic storage ring- Polarized deuterons- d-Carbon polarimetry- Radiofrequency (RF) Wien-filter - High-current all-electric ring- Simultaneous CW/CCW op.- Frozen spin control (withcombined E/B-field ring)- Phase-space beam cooling - Frozen spin all-electric(at p = 0.7 GeV/c)- Simultaneous CW/CCW op.- B-shielding, high E-fields- Design: cryogenic, hybrid,…Ongoing at COSY (Jülich)2014 (cid:1) (cid:1) (cid:1)
Fig. 3:
Summary of the important features of the proposed stages in the storage ring EDM strategy.
Future scientific goals may include conversion of the ring to crossed electric and magnetic field6peration so that other species besides the proton could be examined for the presence of an EDM. Anal-ysis of the data may be made for signs of axions using a frequency decomposition and investigation ofcounter-rotating beams with different species used in novel EDM comparisons.The prototype ring and the CPEDM stages 2 and 3 are host-independent. If the prototype is builtat COSY, it would take advantage of the existing facility for the production of polarised proton (anddeuteron) beams, beam bunching, and spin manipulation. COSY itself could be used for producingelectron-cooled beams. It may also be built at another site ( e.g. , CERN) provided that a comparablebeam preparation infrastructure is made available. In either case, the lattice design will mimic that of thehigh-precision ring in order to test as many features as possible on a smaller scale.
Details of the Prototype EDM ring
The prototype ring (PTR) will be small (circumference of 100 m) and operate in two modes (see stage3 in Fig. 3 and Table 3). The ring will be as inexpensive as possible, consistent with being capable ofachieving its goals. The first mode would operate with all-electric bending (at 30 MeV), a demonstrationthat such a concept works and may be used to demonstrate feasibility of the ring with simultaneouscounter-rotation beams. The second would extend the operating range to 45 MeV with the addition ofmagnetic bending (air core). With this combination, frozen spins could be demonstrated for a protonbeam, other spin manipulation tools developed, and a reduced-precision proton EDM value measured.Alternating fills in counter-rotating directions would allow cancellation of the average radial magneticfield (cid:104) B r (cid:105) that is the leading cause of systematic error (though with a large systematic error associatedwith the needed magnetic field reversal).This section describes a starting-point lattice in terms of geometry, type and strength of the el-ements. The ring is square with 8 m long straight sections. The basic beam parameters are given inTable 3. Table 3:
Basic beam parameters for the Prototype ring E only E, B unitkinetic energy 30 45 MeV β = v/c γ (kinetic) 1.032 1.048momentum 239 294 MeV/cmagnetic rigidity Bρ · mElectric field only 6.67 MV/mElectric field E (frozen spin) 7.00 MV/mMagnetic field B (frozen spin) 0.0327 T Prototype ring requirements and goals
The foremost goal of the prototype ring is to demonstrate the ability to store enough protons ( ∼ )to be able to perform proton EDM measurements in an electric storage ring, recognising that somesuperimposed magnetic bending is likely to be necessary to meet this goal.Since ultimate EDM precision will require simultaneously counter-circulating beams a prototypering has to demonstrate the ability to store and control simultaneously two such beams.Cost-saving measures in the prototype, such as room temperature operation, minimal magneticshielding, and avoidance of excessively tight manufacturing and field-shape matching tolerances, are ex-pected to limit the precision of any prototype ring EDM measurement. Nonetheless, data for reliable costestimation and extrapolation of the systematic error evaluation to the full scale ring has to be obtained.7 rototype ring design The lattice has fourfold symmetry, as shown in Fig. 4. The basic parameters for the prototype ringare given in Tables 4 and 5. The bending, for example for 45 MeV protons, is done by eight ◦ electric/magnetic bending elements. The acceptance of the ring is to be 10 mm · mrad for particles.The lattice is designed to allow a variable tune between 1.0 and 2.0 in the radial plane and between 1.6and 0.1 respectively in the vertical plane. Table 4:
Geometry units ◦ ) (cid:15) x = (cid:15) y π mm · mradacceptance a x = a y π mm · mrad Table 5:
Bend elements, 45 MeV units
Electric electric field 7.00 MV/mgap between plates 60 mmplate length 6.959 mtotal bending length 55.673 mtotal straight length 44.800 mbend angle per unit ◦ m Magnetic magnetic field 0.0327 Tcurrent density 5.000 A/mm windings/element 60 The injector:
Injection into the prototype ring will closely resemble injection into a nominal all-electricring. In particular there will be an even number of bunches in each beam, with alternating sign polarisa-tions, whether in single beam or counter-circulating beam operation. The injector for the prototype ringcould be the electron-cooled beam from COSY or make use of equipment at CERN. The beams will beprotons in the 30 to 45 MeV range, in a cooled phase space of 1 π mm · mrad, with the beams bunchedinto 2, 4, 6 or 8 bunches to be fed into counter-circulating beams in the prototype ring.Injection into the prototype ring will be done using switching magnets distributing the beams intoclockwise (CW) or counter clockwise (CCW) direction as sketched in Fig. 4. All beam bunches aretransferred with vertical polarisation, either up or down. Electric bends:
The electrostatic deflectors consist of two cylindrically-shaped parallel metal plates withequal potential and opposite sign. With the zero voltage contour of electric potential defined to be thecentre line of the deflector, the ideal orbit of the design particle stays on the centre line. The electricalpotential vanishes on the centre line of the bends, as well as in drift sections well outside the bends.So the electric potential vanishes everywhere on the ideal particle orbit. With the electric potential seenby the ideal particle continuous at the entrance and exit of the deflector, its total momentum is constanteverywhere (even through the RF cavity).The designed ring lattice requires electric gradients in the range from 5 to 10 MV/m. This ismore than the standard values for most accelerator deflectors separated by a few centimetres. Assuming60 mm distance between the plates, to achieve such high electric fields we have to use high voltage powersupplies. At present, two 200 kV power converters have been ordered for testing deflector prototypes.The field emission, field breakdown, dark current, electrode surface and conditioning will be studiedusing two flat electrostatic deflector plates, mounted on the movable support with the possibility ofchanging the separation from 20 to 120 mm. The residual ripple of power converters is expected to bein the order of ± − at a maximum of 200 kV. This will lead to particle displacement on the order ofmillimetres. A smaller ripple or stability control of the system will be a dedicated task for investigationsplanned at the test ring facility. Magnetic bends:
The experiments require periodic reversals of the magnetic bending field to use sym-8
Fig. 4:
The basic layout of the prototype ring, consisting of 8 dual, superimposed electric and magnetic bends;3 families of quadrupoles (Focusing, Defocusing, and Straight-section); and four 8-m long straight sections. Thetotal circumference is about 100 m. Injection lines for injecting counter-circulating beam are represented just asstubs. Costs given in the Addendum are restricted to just the prototype ring, which is truly site-independent. Thepossibly greater infrastructure costs associated with producing appropriately polarised beams are neither given norsite-independent. metry to suppress systematic deviations. The reversal of the magnetic field should be done with bestpossible reproducibility. This is why the magnetic field production will iron-free (see Fig. 5).
Other components:
All quadrupoles will be electrostatic. Their design will follow the principles of theHeidelberg CSR ring [17]. Both DC and AC Wien filters and solenoids will be required for spin control.The RF cavity design is under study.The requirement for the vacuum is mainly given by the minimum beam lifetime requirement ofabout 1000 s. The emittance growth in the ring caused by multiple scattering from the residual gas is0.005 mm · mrad/s. At − Torr vacuum, the emittance at the beginning, assumed to be 1 mm · mrad,will have increased to 5 mm · mrad within 1000 s, assuming a nitrogen (N ) partial pressure. This is aboutthe cooling rate expected for stochastic cooling. (One notes in passing that stochastic cooling becomesimpractical for very low tunes.) For such an ultra-high vacuum only cryogenic or NEG pumping systemscan be used. Bake-out must be foreseen for either cryogenic or NEG systems.The choice of NEG requires a beam pipe with a diameter of 300 mm over the full circumferenceof 100 m. This can easily be plated with the NEG material. We will then have an active area of ≈ m for the whole ring. The roughing speed will be about 5000 liter/s per meter of length of vacuum pipe.There are beam position monitors located around the ring. A BPM is placed at the entrance and theexit of each bending unit. One BPM will be placed additionally in close connection to the quadrupolesin the straight sections. A new type of BPM, of Rogowski coil design [19], has been developed at theIKP of the FZ-Jülich. These pick-ups are presently in a development stage. The position resolution ismeasured to be 10 µ m over an area with a diameter of about 90 mm. These BPMs require only a shortbeam insertion length of 60 mm and an offset-bias free response to counter-circulating beams. All-electric storage ring
This document describes the vision of CPEDM culminating in the design, construction, and operation ofa dedicated, high-precision storage ring for protons. Operating at the all-electric, frozen-spin momentum9 agnetic coilconductorsElectric fieldelectrodes
Beam
300 mm
Fig. 5:
Shown on the left is a cutaway drawing of the prototype ring in the (cid:126)E × (cid:126)B version. A side view of the lowerhalf of a ◦ bend element is shown. The electrodes have a gap of 60 mm. The magnetic coil conductors (single, × mm copper bars) produce a highly uniform “cosine-theta” dipole field. Shown on the right is a transversesection displaying an end view of the (inner legs of the) magnet coil, as well as a field map of the good magneticfield region. of 0.7 GeV/c, the signals from counter-rotating beams aim to measure the proton electric dipole momentwith a sensitivity of − e · cm. The major challenge is the handling of all systematic errors to obtainan overall sensitivity of a similar size. The main source of systematic uncertainties will be due to anyunknown or unidentified radial magnetic field acting through the much larger magnetic dipole momentand leading to a false EDM signal. The level at which this can be mitigated remains to be determined.Invaluable results and experiences are expected from the intermediate step, the construction of asmaller, prototype ring. The attempts to examine the control of counter-rotating beams and study directlythe conditions for frozen spin will have a huge impact on the detailed outline of the high-precision ringdesign.The concept of an all-electric storage ring with extremely well-fabricated and aligned elementsrunning two longitudinally polarised proton beams in opposite directions in the absence of significantmagnetic fields serves as the current starting point. There are new ideas under development that offer theprospect of further mitigation of the systematic issues:– A hybrid electric/magnetic ring [20] with magnetic focusing (in addition to electric deflector con-tributions) will change the electromagnetic environment in significant ways. Even in the presenceof uncontrolled radial magnetic fields, this geometry offers at least one point at which the magneticfield vanishes. Beam-based alignment techniques will tend to find these points and place the beamthere. This substantially relaxes the requirement that radial magnetic fields be made to nearlyvanish. The magnetic focusing, however, does not produce counter-rotating beams with the samephase space profile. So periodic reversal of the magnetic focusing would be required to provide aset of signals that must be averaged to obtain an EDM value.– It is possible to find pairs of unlike polarised beams for which the same superimposed electricand magnetic bending yields a frozen spin condition for both (e.g. protons and He) [21]. Sincethe two beams would not have the same revolution frequency, to circulate simultaneously theywould run with appropriately different RF harmonic numbers. Though not yielding either EDMvalue directly, the resulting EDM difference will be independent of the (otherwise dominant) radialmagnetic field systematic error. Any EDM signal differences would be interpreted as the presenceof an EDM on at least one of the two beams.10ork on these concepts can proceed using the prototype ring with the possibility of yielding new physicsresults.
Prototype ring costs
Preliminary prototype ring cost estimates are given in Table 6. Many items are currently receiving R&Dfunding. The bend element high voltage supplies are presently under development. Neither building norinjection line costs are included. The accuracy of this cost estimation is preliminary. The magnetic bendequipment for the frozen spin experiments in a 2 nd stage will require additional costs for the magnets anda Wien Filter of about 7000 k C. component cost [k C]bends 9200electric-quads 1700vacuum 1800pick-ups 400control 2000polarimeter 1200RF equipment 300sum machine 16600 Table 6:
Summary of preliminary cost estimates for the prototype ring first stage.
Roadmap and Timeline
FeasibilityPrecursorpEDM Dev.Other (axion)
ReportsPrototype
CERN CDR TDR Propose Procure / Build Commission
EventsRunning: pEDM Dev.PrototypeOther
30 MeV ‘26 ‘27 ‘28 ‘29 ‘30 ‘31 ‘32
45 MeV
Events 5 4All-electric ring
CDR TDR Propose Procure / Build Run in ‘34+
Event Key:1. Strategic program evaluation Helmholtz Association (HGF)2. Start of HGF funding period3. End of “srEDM” Grant of European Research Council4. HGF Mid-term Review5. Start of next HGF funding period
Y/N Y/N
HGF HGF
Fig. 6:
GANTT chart of major activities and events for CPEDM.
A staged approach to the CPEDM project (outlined in Fig. 3 and expanded in detail in Fig. 6) is cur-rently ongoing with work on the precursor experiment and feasibility studies. This is partially funded byan ERC Advanced Grant that runs until September 2021 (event 3). Meanwhile, this longer “Yellow Re-port” having been published will be followed by an Helmholtz-Gemeinschaft Deutscher Forschungszen-tren (HGF) evaluation (event 1) and preparation for the start of the new HGF funding period (event 2).11ince 2017, preparation has been underway on the design for a prototype electric and mixed field stor-age ring to verify CPEDM concepts that will appear in a CDR/TDR available for funding considerationby 2021. With approval, construction and commissioning of the prototype ring will begin. In parallel,experimental work at COSY would refocus on feasibility studies for proton beams. By the beginning ofsubsequent funding period, the first prototype results should show the best techniques for the all-electricfull-energy ring. These will be the subject of another CDR/TDR study. If approved, efforts will switchto the construction and running of this new ring.The storage ring EDM feasibility studies made so far show encouraging results. Handling system-atic errors is the main challenge. The path to addressing this lies through the construction and operationof a small-scale prototype ring from which will come the design for the high-precision ring with the bestsensitivity to new physics.
References [1] F. Abusaif et al. , arXiv:1812.08535 (2018).[2] J. Engel, M. J. Ramsey-Musolf, and U. van Kolck, Prog. Part. Nucl. Phys. , 21 (2013).[3] J. M. Pendlebury et al. , Phys. Rev. D , 115116 (2015).[4] V. Anastassopoulos et al. , Rev. Sci. Instrum. , 092003 (2015).[5] B. Graner et al. , Phys. Rev. Lett. , 161601 (2016), Erratum: Phys. Rev. Lett. , 119901(2017).[6] S. P. Chang et al. , arXiv 1710.05271 (2017); S. Park et al. , Proc. Sci. PSTP2017.[7] C. Abel et al. Phys. Rev. X , 1350147 (2013).[9] N. P. M. Brantjes et al. , Nucl. Instrum. Methods A , 49 (2012).[10] Z. Bagdasarian et al. , Phys. Rev. ST, Accel. Beams , 052003 (2014).[11] D. Eversmann et al. , Phys. Rev. Lett. , 094801 (2015).[12] N. Hempelmann et al. , Phys. Rev. Lett. , 014801 (2017).[13] G. Guidoboni et al. , Phys. Rev. Lett. , 054801 (2016).[14] A. Saleev et al. , Phys. Rev. ST, Accel. Beams , 072801 (2017).[15] For COSY-Infinity, see http://bt.pa.msu.edu .[16] For Bmad, see .[17] R. von Hahn et al. , Rev. Sci. Instrum. , 063115 (2016).[18] Y. Orlov, E. Flanagan, and Y. Semertzidis, Phys. Lett. A , 2822 (2012).[19] F. Trinkel, Ph. D. thesis, Dec. 17, RWTH Aachen (see JEDI Web site).[20] S. Haciomeroglu and Y. Semertzidis, Phys. Rev. Accel. Beams , 034001 (2019).[21] R. Talman, arXiv 1812.05949 (2018). 12 hapter 1Introduction Project Scope
An experiment is described to detect a permanent electric dipole moment (EDM) of the proton with asensitivity of − e · cm by using counter-rotating polarised proton beams at the “magic” momentum of0.7007 GeV/c in an all-electric precision storage ring.The science case for such a project is based on the fact that measurements of EDMs of funda-mental particles provide “a unique, extraordinarily sensitive way to probe for a physical phenomenon ofprofound significance, [the] violation of microscopic time-reversal invariance” (F. Wilczek). Assuming CP T -symmetry conservation, T -violation implies violation of the combined CP -symmetry, one of theingredients required for explaining the matter-antimatter asymmetry of our Universe. EDM searches aresensitive to new physics beyond the Standard Model of elementary particle physics at a scale of the orderof 1000 TeV. Moreover, the storage ring technology will also allow a search for oscillating EDMs, whichmay be connected with axions or axion-like particles. The physics motivation is thus evident and wellsupported by the community.The storage ring concept has been well developed over the years, including a detailed examina-tion of the experimental method, required technologies and involved systematics. R&D has progressedin parallel on essential storage ring components such as electrostatic deflectors, beam instrumentation,magnetic shielding and polarimetry.A good understanding of the key systematic errors has been achieved, and their potential con-straints on the ultimate sensitivity of the storage ring approach have been quantified. The leading sys-tematic uncertainty is due to a residual radial magnetic field interacting with the magnetic moment tomimic the EDM signal. A radial magnetic field will lead to a vertical separation of the counter-rotatingbeams. Measurement of this separation will provide a handle to mitigate this systematic.The ultimate goal is to design, build, and operate an all-electric storage ring for protons at theirmagic momentum (0.7 GeV/c) with clockwise (CW) and counter-clockwise (CCW) longitudinally po-larised beams to achieve a sensitivity of the order of − e · cm. To this end a number of ring latticeoptions have been developed. These options make reasonable assumptions about the achievable electricfield, deflector size, instrumentation requirements etc., and have led to the adoption of a baseline ringdesign of some 500 m in circumference.To fully confirm the validity of the approach, a small all-electric prototype ring is proposed. Thiswould allow:(i) to deploy and test key hardware components of the all-electric ring;(ii) to verify that an intense proton beam can be stored for at least 1000 s;(iii) to deploy and use beam instrumentation, like the polarimeter;(iv) to demonstrate the ability to master key systematics via the use of counter-clockwise beams.The prototype is seen as a key step in demonstrating the credibility of the full ring proposal. A baselineproposal for this prototype has been developed and foresees two phases (see chapter 7 “Prototype Ring”): • Phase 1: All-electric ring for 30 MeV proton beams (CW and CCW) • Phase 2: Combined E- and B-fields for 45 MeV proton beams (frozen spin) to allow a first pEDMmeasurement 13t is expected that the prototype ring will provide invaluable information for the outline and designof the final ring.
Key Accomplishments
Significant insight into the storage ring EDM project and accompanying technological advances havebeen achieved in the past few years. These include early findings of the srEDM collaboration at BNL(USA) and contributions from KAIST (South Korea). Recently a new level of measurement-technologyachievements has been reached by the JEDI collaboration working at the Cooler Synchrotron COSY atthe Forschungszentrum Jülich (Germany). These achievements are enumerated below.1. A deuteron beam polarisation lifetime near 1000 s in the horizontal plane of the magnetic storagering COSY has been observed [1]. This long “spin coherence time” (SCT) was obtained through acombination of beam bunching, electron cooling, sextupole field corrections, and the suppressionof collective effects through beam current limits. This record lifetime is required for a storage ringsearch for an intrinsic electric dipole moment on the deuteron and paves the way for similarly largeSCT for protons.2. A new method to determine the spin tune was established and tested [2]. In an ideal planar mag-netic storage ring, the “spin tune” − defined as the number of spin precessions per turn − is givenby ν S = γG ( γ is the Lorentz factor, G the gyromagnetic anomaly). At 0.97 GeV/c, the deuteronspins coherently precess at a frequency of about 121 kHz in COSY. The spin tune was deducedfrom the up-down asymmetry of deuteron-carbon scattering. In a time interval of 2.6 s, the spintune was determined with a precision of the order − , and to − for a continuous 100 s accel-erator cycle. This renders the new method a precision tool for accelerator physics; observing andcontrolling the spin motion of particles to high precision is again mandatory for the measurementof electric dipole moments of charged particles in a storage ring.3. The successful use of feedback from a spin polarisation measurement to the revolution frequencyof a 0.97 GeV/c bunched and polarised deuteron beam in COSY has been realised in order tocontrol both the precession rate ( ≈ kHz) and the phase of the horizontal polarisation com-ponent [3]. Real time synchronisation with a radio frequency (RF) solenoid made possible therotation of the polarisation out of the horizontal plane, yielding a demonstration of the feedbackmethod to manipulate the polarisation. In particular, the rotation rate shows a sinusoidal functionof the horizontal polarisation phase (relative to the RF solenoid), controlled to within a one stan-dard deviation range of σ = 0 . rad. The minimum possible adjustment was 3.7 mHz out of arevolution frequency of 751 kHz, which changes the precession rate by 26 mrad/s. Such capabilitymeets the requirement for the use of storage rings to look for an intrinsic electric dipole momentof charged particles.4. Procedures have been developed and tested that allow for systematic errors in the measurement ofthe vertical polarisation component (that carries the EDM signal) to be corrected to a level belowone part in [4]. This requires a prior calibration of the polarimeter for rate and geometric erroreffects and the use of two opposite polarisation states in the measurement. The extra polarisationstate allows for an independent estimate of the size of the systematic error. Such corrections may bemade in real time. This meets the sensitivity requirement to measure the small vertical componentpolarisation changes expected in the EDM search.Further details will be given in later chapters of this report. European and global context
Permanent electric dipole moments are sought in various elementary and complex systems; the mostrecent experimental limits are given in the “Physics Motivation” Chapter 2 below. A rather complete listof the international EDM efforts can be found in [5].14eutron EDM searches are conducted/under development/proposed at nuclear fission reactor fa-cilities (ILL Grenoble, FRM-2 Munich, PNPI Gatchina) and spallation neutron sources (PSI Villigen,ESS Lund) in Europe as well as in the US and Canada (SNS Oak Ridge, LANL Los Alamos, TRIUMF,Vancouver). Molecular and atomic EDMs are sought by a numerous groups worldwide including projectsat radioactive beam facilities such as ISOLDE (CERN). The muon EDM is measured as a by-product of ( g − µ experiments at FNAL (Batavia, USA) and J-PARC (Tokai, Japan).A new experiment, called CeNTREX, has recently been launched at Yale University (USA) tosearch for a deformation in the shape (nuclear Schiff moment) of the atomic nucleus Tl inside athallium fluoride (TlF) molecule [6]. The experiment will be complementary to
Hg and primarilysensitive to the proton EDM and θ QCD . It is expected that it will improve the indirect proton EDM limitof × − e · cm by more than one order of magnitude in the coming years.Storage ring EDM searches for the proton and other light nuclei (deuteron, He) have been dis-cussed for a few years (see “Background” Chapter 4). It is our strong belief that eventually a result forthe pure proton system will be required to complement the free neutron EDM. This is currently pursuedby the JEDI and CPEDM collaborations and constitutes the motivation for the present document.
Contents of the report by chapter – The
Executive Summary was prepared and submitted in December 2018 for the European Strat-egy for Particle Physics (ESPP) update to consider in their review of the storage ring EDM searchalong with other experimental programs in nuclear and high energy physics research. The Sum-mary described the concept of the experiment, the strategic path forward that includes a prototypering for further feasibility testing, and an outline of plans for the final EDM ring.[2] The
Physics Motivation begins with a summary of the status of other major searches for an EDMon various systems and discusses both measured and derived upper bounds. These are comparedwith what one would expect for an intrinsic EDM on the basis of a naïve dimensional analysis. Butan EDM appears to be suppressed in nature, as illustrated by the small size of the CP -violating θ QCD parameter. The size of this parameter may be estimated from the current limit on the EDMin the neutron. A brief mention is made of the possibility to search for axion-like particles througha search for an oscillating EDM with a frequency related to the axion mass.[3] The
Background chapter gives a summary of the history of the storage ring EDM search from theoriginal ideas developed at the Brookhaven National Laboratory in the USA. The story is followedas it moved into feasibility testing at the COSY storage ring located at the ForschungszentrumJülich in Germany. This led to the first direct measurement of an EDM upper limit for the deuteronusing a Wien filter located on the storage ring. The section also explores the experience of thecollaboration members and work being done on supporting technologies for the final EDM ring.[4] The
Experimental Method chapter describes the storage ring EDM search beginning with themost basic concept of the experiment and developing all of the essential ideas needed for this ex-periment. Various experimental possibilities are presented and considered. Essential formalism isshown for both the ring and the polarisation measurement. There is also a discussion of systematiceffects and the challenge of managing all aspects of the experiment. This section is intended forthe novice to EDM searches.[5] The
Strategy describes briefly the idea to build a prototype ring featuring both electric and mixedfield designs so that more of the critical technologies for the EDM search may be demonstrated.This will lead to a design for the final ring.[6] The
Precursor Experiment at COSY adds an RF Wien filter to the existing COSY storage ring sothat the cancellation of the EDM signal due to the in-plane rotation of the polarisation is broken.This allows in principle for a sensitivity to the deuteron EDM, which is the beam currently in usefor EDM feasibility studies. A series of first measurements were completed in the fall of 2018 andthe preliminary results are presented here. 157] The
Prototype Ring chapter presents the detailed design considerations for the small ring to bebuilt as a site for the continuation of feasibility studies for the final EDM experiment. The proto-type will operate in two modes: (i) an all-electric setup that allows for the simultaneous storage ofboth clockwise and counter-clockwise travelling beams, and (ii) a combined electric and magneticfield ring that creates the conditions for frozen spin operation. In both cases, systematic errors maybe studied in an environment that uses the same electric field structures that are proposed for thefinal EDM ring.[8] The
All-electric Proton EDM Ring is described more fully in this chapter, based on the design forthe lattice in the prototype ring. Ring specifications are provided. A table is included that showsthe status and preparedness of various aspects of the projects.[9] The
Electric Fields chapter reviews the status of various accelerator systems that will be neededfor the EDM ring. Electric fields pose a particular challenge since the best experiment is associatedwith large field strength. This then generates requirements on voltage holding capability and theability to suppress dark currents. Focusing and beam injection elements will also be needed, andthe prototype ring offers a chance to develop various designs.[10] The
Sensitivity and Systematics chapter describes in detail the considerations that lead to anexpected statistical sensitivity reach of − e · cm for charged particle EDMs in a storage ring.The main part of this section is devoted to an assessment of the size and nature of systematic effectsthat can mimic an EDM signal and means, such as simultaneous clockwise and counter-clockwisemeasurements, that may be used to cancel these errors.[11] The Polarimetry chapter begins with the items needed during the beam preparation phase of theexperiment to verify the polarisation of the beam. For the main EDM ring, the polarimeter target(most likely carbon) and the detectors needed for making online polarisation measurements andcancelling the systematic errors associated with this process. The requirements of the polarimeterare explained and details given for the choice of detector acceptance in order to maximise thefigure of merit of the device. Examples are provided for both the prototype and final EDM ringdesigns. This section also details the work so far accomplished in fashioning the calorimeterdetectors and other event tracking hardware that will be needed. Details are provided on the use ofthe polarimeter as a device for maintaining the frozen spin operating condition in the EDM ring.[12]
Spin Tracking consists of those calculations needed to describe the history of the polarised beamas it circulates in the EDM ring. It is also a testing laboratory where we can explore varioussources of systematic error ( e.g. magnet misalignment) and ways to mitigate it. This calls forreliably calibrated programs using well-understood techniques for treating electric and magneticfield effects.[13] The last formal chapter of the report covers a
Roadmap and Timeline . Special Appendices [A] The
Results and Achievements at Forschungszentrum Jülich covers (i) polarimetry, (ii) highprecision tune measurements, (iii) long horizontal polarisation lifetime, (iv) feedback control ofpolarisation, (v) invariant spin axis measurements, (vi) RF Wien filter construction, (vii) referencedata bases for deuteron and proton-induced reactions on carbon, (viii) progress in orbit measure-ment and control (ix) electrostatic deflector development, (x) EDM and axion theory, and (xi) spintracking simulations.[B] The
Mitigation of Background Magnetic Fields chapter describes the influence of magneticfields on the EDM experiment and why they need to be small. Stray magnetic fields need tobe managed with shielding and perhaps some active elements. And the effects of any residualfield need to be well understood. For injection and perhaps for spin manipulation, time-varyingmagnetic fields may be needed, so their effects on the experiment need to be explored.[C] The appendix on
Statistical Sensitivity gathers the derivations for the contribution of event col-16ection statistics to the final EDM result. Connections are shown to critical ring and experimentalrequirements.[D] The appendix on
Gravity and General Relativity as a ’Standard Candle’ contains the inclusionof gravity as an explicit item in the Thomas-BMT equation. From this the level of the signal fromgravity acting on the beam may be estimated, opening the possibility of using it as a marker ofsensitivity.[E] The
Axion Search appendix contains preliminary plans to use the rotating polarisation of theCOSY beam to search for an oscillating EDM that is a possible signature of an axion-like particlein nature. The first experiment to develop this techniques was scheduled to run in April 2019.
Appendices Describing New Ideas [F] The
Hybrid Scheme addresses the problems of minimising the residual horizontal magnetic fieldin the all-electric storage ring by imposing a magnetic focusing system. This system along withbeam-based alignment techniques draws the beam toward the point in each quadrupole wherethe field vanishes. This reduces the requirements on the elimination of the residual backgroundfield by orders of magnitude. This does break the symmetry between the CW and CCW rotatingbeams. Symmetry may be restored by operating with both focusing field polarities and averagingthe results. Independent confirmation of this scheme is underway.[G] The appendix on a
Doubly Magic EDM Measurement Method describes the possibility for hav-ing CW and CCW beams being different polarised particle species ( e.g. protons and He) circulat-ing at different energies under “frozen spin” conditions for both beams. This would enable precisecomparisons of EDM properties and magnetic moments across the two species.[H] The
Spin Tune Mapping for EDM Searches appendix explores a more generalised method ofmaking EDM searches by replacing the requirement of “frozen spin” with corrections applied bya Wien filter mounted in the storage ring. This method may be generalised to allow comparison ofmultiple particle species.[I] The
Frequency Domain appendix introduces the notion of utilising “spin wheel” rotation of thepolarisation about the horizontal transverse axis to obtain sensitivity to the magnetic and electricdipole contributions together. By measuring the frequency of the resulting rotation, a precisesubtraction to obtain the EDM contribution becomes possible.[J] The
EDM from Fourier Analysis appendix explores the idea of separating EDM effects fromsystematic due to machine errors through the use of a Fourier analysis of the experimental signals.[K] The
External Polarimetry appendix addresses the problem that a block target located at the edgeof the beam does not necessarily sample the polarisation across the full beam. This allows theeffects of a polarisation distribution across the beam to become a systematic error in the results.The scheme presented in this appendix uses pellets dropped through the beam to extract a fractionof the beam into a channel branching from the main beam line where it strikes a large and thickpolarimeter target that spans the entire beam profile. The efficiency for this scheme is expected tobe comparable to the block target scheme used at COSY.
Final Comments
We would like to emphasise that this write-up is a status report of what has been achieved and what isknown at the time of the editorial deadline (December 2019) – work is ongoing at COSY, CERN andother places towards the realisation of the storage ring EDM project.
References [1] G. Guidoboni et al. , Phys. Rev. Lett. , 054801 (2016).[2] D. Eversmann et al. , Phys. Rev. Lett. , 094801 (2015).173] N. Hempelmann et al. , Phys. Rev. Lett. , 014801 (2017).[4] N. P. M. Brantjes et al.
Nucl. Instrum. Methods Phys. Res. Sect. A , 49 (2012).[5] See (PSI Group)[6] See https://demillegroup.yale.edu/research (Demille Group, search for the electricdipole moment of the electron) 18 hapter 2Physics Case for CPEDM
Introduction
Both continuous and discrete symmetries combined with possible breaking patterns have been decisivefor the development of physics in the last 100 years. This was exemplarily demonstrated by the construc-tion of the Standard Model (SM) of particle physics. Measurements of sizes or limits with which discretesymmetries (as, e.g. , parity P , charge-conjugation C , their product CP , time-reflection invariance T , theproduct CP T , baryon- and/or lepton number) are respectively broken or conserved have been essentialfor this task in the second part of the last century. These tests currently play, and will continue to play,an essential role for constructing and identifying physics beyond the SM (BSM).As it is the case for all stationary states of finite and parity–non-degenerate quantum systems, theground-state of any of the known non-selfconjugate subatomic particles with nonzero spin (regardless ofelementary or composite nature) can only support a nonzero permanent electric dipole moment (EDM),if both time-reflection ( T ) and parity ( P ) symmetries are violated explicitly, while the charge symmetry( C ) can be maintained. Assuming the conservation of the combined CP T symmetry, T violation alsoimplies CP violation.The CP violation generated by the Kobayashi-Maskawa (KM) mechanism of weak interactionsinduces a very small EDM that is several orders of magnitude below current experimental limits. How-ever, many models beyond the standard model predict EDM values near these limits. Hence, there isa window in which the search for nonzero EDMs corresponds to a search for CP violation beyond theweak interaction one. In fact, finding a non-zero EDM value for any subatomic particle (above the KMlimit of the SM which experimentally is out of reach for the foreseeable future) will be a signal that thereexists a new source of CP violation, either induced by the strong CP violation via the QCD ¯ θ angle orby genuine physics beyond the SM. The latter is essential for explaining – within the framework of theBig Bang and inflation – the mystery of the observed baryon-antibaryon asymmetry of our universe, oneof the outstanding problems in contemporary elementary particle physics and cosmology. Over the years, the quest in improving the bounds of the permanent EDM of the neutron, d n , pioneeredmore than 60 years ago by the work of Purcell, Ramsey, and Smith [2], has served to rule out or, at least,to severely constrain many models of CP violation, demonstrating the power of sensitive null results.The current bound of the neutron EDM resulting from these efforts is | d n | < . × − e · cm (90% C.L.) [3, 4] which corresponds to | d n | < . × − e · cm at a 95% confidence upper limit [4]. As reported below,the prediction of the CKM matrix is at least four orders of magnitude smaller: | d SM n | (cid:46) − e · cm , seeChap. 2.2.1 for more details.There are complimentary constraints from atomic and molecular physics experiments. Especially,the EDM bounds on paramagnetic atoms, e.g. , (cid:12)(cid:12) d Cs (cid:12)(cid:12) < . × − e · cm (95% C.L.) [5] , E.g. the ρ and ω vector mesons are particles with nonzero spin. But as they are selfconjugate, i.e. their own antiparticles,they cannot possess an electric dipole moment, while the ρ ± or the K ∗ as non-selfconjugate vector mesons have this possibility. Actually the best upper limit on this parameter of Quantum Chromodynamics follows from the experimental bound on theEDM of the neutron. (cid:12) d Tl (cid:12)(cid:12) < . × − e · cm (95% C.L.) [6, 7] , and the constraints from dipolar molecules and molecular ions indirectly lead to the following upperlimits on the electron EDM: (cid:12)(cid:12)(cid:12) d ↓ YbF e (cid:12)(cid:12)(cid:12) < . × − e · cm (90% C.L.) [9] , (cid:12)(cid:12)(cid:12) d ↓ ThO e (cid:12)(cid:12)(cid:12) < . × − e · cm (90% C.L.) [10–12] , (cid:12)(cid:12)(cid:12) d ↓ HfF + e (cid:12)(cid:12)(cid:12) < . × − e · cm (90% C.L.) [13] . These bounds should be put into perspective since they are quite large compared to the prediction of theCKM mechanism in the SM: | d SM e | ∼ − e · cm , see, e.g. , [14].In contrast to the paramagnetic cases which are sensitive to the their electron clouds, in diamag-netic atoms the EDM-defining spin is carried by the pertinent nucleus. Corresponding upper limits onthe EDMs of diamagnetic atoms are, e.g. , (cid:12)(cid:12) d Xe (cid:12)(cid:12) < . × − e · cm (95% C.L.) [15] , (cid:12)(cid:12) d Ra (cid:12)(cid:12) < . × − e · cm (95% C.L.) [16] , (cid:12)(cid:12) d Hg (cid:12)(cid:12) < . × − e · cm (95% C.L.) [1] . Because of Schiff screening, the indirect bounds on the neutron and proton EDM obtained by applyingnuclear physics methods [17] are much weaker than their parent atom bounds. From the currently bestcase, Hg , the following indirect bounds on the neutron and proton EDM could be derived [1]: | d ↓ Hgn | < . × − e · cm (95% C.L.) , | d ↓ Hgp | < . × − e · cm (95% C.L.) . The indirect bound on | d p | is by a factor of 13 or 6 weaker than the indirect or direct | d n | counterparts,respectively, and therefore not really competitive.The current status of the already excluded EDM regions derived from the experimental upperlimits of the various particles mentioned above are summarised in Figure 2.1. In this proposal, we discuss an experimental opportunity, provided by the storage ring technology, to pusha direct measurement of the proton EDM to − e · cm sensitivity, corresponding to an improvement bynearly 5 orders of magnitude. Such dramatic improvement is made possible by new ideas and techniquesdescribed in this document. Several new neutron EDM experiments involving ultra-cold neutrons (UCN)have already been started worldwide with the aim to eventually approach | d n | ∼ − e · cm sensitivity.Compared to that, the storage ring studies target a | d p | sensitivity more than an order of magnitudebeyond | d n | expectations which are primarily limited by the achievable number of trapped UCNs. Suchan improved sensitivity might be crucial in reaching the forefront of the underlying mechanisms behindbaryogenesis and BSM-induced CP violation. In view of the entirely unknown isospin properties ofthe latter, even at the lower − (27 — e · cm sensitivity the proton EDM studies are complementary Note that the EDMs of paramagnetic atoms and the P - and T -violating observables in polar molecules are dominatedby system-dependent linear combinations of the electron EDM and the nuclear-spin–independent electron-nucleon interaction,which couples to the scalar current components of the pertinent nuclei. An extraction of an electron EDM value d e cannotindependently be performed from the extraction of this semi-leptonic four-fermion interaction C S , while the quoted | d e | boundsassume that the measured paramagnetic systems are saturated by the electron EDM alone. For more details on this issue, onfurther EDM bounds, and also on the analogous extractions of the | d n | and | d p | bounds of valence and core nucleons fordiamagnetic atoms see, e.g. , the reviews [7, 8] and quoted references therein. - - - - - - - c m • ed m / e Cs Tl Xe Ra Hg n nHg) ( pHg) ( e(ThO) m paramagnetic atomsdiamagnetic atoms hadronsleptons indirect measurements Fig. 2.1:
Current status of excluded regions of electric dipole moments. Shown are direct and/or derived EDMbounds of the particles and a selection of atoms discussed in Chap. 2.1.1, and, additionally, the upper limit on theEDM of the muon, | d µ | < . · − e · cm (95% C.L.), by the muon ( g − collaboration [18]. to the neutron ones and will be essential in discriminating between – or at least constraining – variousmechanisms for baryogenesis or competing models of CP violation, e.g. , variants of supersymmetric(SUSY) models, multi-Higgs models, left-right symmetric ones, or the strong CP violation from theQCD ¯ θ term. Note that a priori the results for d n and d p are independent and could have significantlydifferent values. Only when interpreted within the context of a specific theoretical framework do theirvalues become related and a comparison is meaningful. Even if d n is found to differ from zero, themeasurement of d p (and perhaps additional measured EDMs of light nuclei, e.g. , deuteron or helion,which might also be studied in a storage ring experiments) will prove crucial in unfolding the new sourceof CP violation. Dimensional analysis
Because of its inherent P and CP violation, the upper limit on the permanent EDM ( d N ) of the nucleon( i.e. neutron or proton) can be estimated [19] from the product of the P - and CP -conserving nuclearmagnetic moment (approximated by the nuclear magneton µ N = e m N ∼ − e · cm ) times a suppressionscale counting the P violation ( ∼ G F f π ∼ − in terms of the Fermi constant G F and the axial decayconstant of the pion f π ) times the additional CP -violating scale ( ∼ − derived from the absolute ratioof the CP -forbidden and CP -allowed amplitudes A ( K L → ππ ) and A ( K S → ππ ) , respectively). Thusthe absolute value of the nucleon EDM cannot be much larger than the natural scale | d N | (cid:46) µ N × − × − ∼ − e · cm (2.1)21ithout getting into conflict with known physics constraints – on top of the experimental neutron EDMbound which nowadays is even more restrictive, see above.In the absence of the QCD ¯ θ term, the SM only possesses a nonzero CP -violating phase if theCKM-matrix involves at least three generations, such that in this case the above estimate inherentlyimplies a flavour change. The EDM, however, is flavour-neutral. Therefore, the upper bound for thenucleon EDM in the SM with zero ¯ θ term necessarily involves a further G F f π ∼ − suppressionfactor to undo the flavour change: | d SM N | (cid:46) − × − e · cm ∼ − e · cm , (2.2)This simple estimate agrees in magnitude with the three-loop calculations of Refs. [20, 21] and also withthe two-loop calculations of Refs. [22, 23] that include a strong penguin diagram and the long-distanceeffect of a pion loop. It is even consistent with modern loop-less calculations involving charm-quarkpropagators [24,25]. From the bounds (2.1) and (2.2) one can infer that an EDM of the nucleon measuredin the window − e · cm > | d N | (cid:38) − e · cm (2.3)will be a clear sign for new physics beyond the KM mechanism of the SM: either strong CP violationby a sufficiently large QCD ¯ θ term or CP violation by BSM physics, as, e.g. , supersymmetric models,multi-Higgs models or left-right symmetric models. A rough estimate of the scale of BSM physics probed by EDM experiments can be derived from anexpression of a subatomic EDM d i that is solely based on dimensional considerations and that holdsfor dimension-six extensions of the SM, since the SM symmetries and the pertinent chirality constraintspreclude any contribution from dimension-five operators: d i ≈ π m i Λ e i sin φ . (2.4)Here e i and m i are the charge and mass, respectively, of the relevant quark or lepton, sin φ results fromthe CP -violating BSM phases, and Λ BSM is the mass scale of the underlying BSM physics. In general,the coupling of BSM physics to subatomic particles induces at least one quantum loop and therefore a g / (16 π ) ∼ − suppression factor (assuming g ∼ ) is included in addition.For current quark masses of the order m q ∼ , we might therefore expect | d N | ∼ − (cid:18) BSM (cid:19) | sin φ | e · cm . (2.5)If Λ BSM (cid:38) and sin φ ∼ , this result is compatible with the upper limit (2.1) derived from thenaive estimate, i.e. it is compatible with all the known physics, except the constraints from direct orindirect EDM measurements. The projected sensitivity for | d p | ∼ − e · cm would in turn allow oneto test the CP -violating phase φ of a theory of mass scale M ∼ down to values of φ (cid:38) − ,while for natural values of the CP -violating phases, φ ∼ , a mass scale range up to M ∼
300 TeV canbe probed. Strictly speaking, the quark masses are scale- and scheme-dependent. These numbers refer to one-loop processes as, e.g. , supersymmetric extensions. They are suppressed by about two ordersof magnitude for two-loop (so-called Barr-Zee [26]) processes as, e.g. , in multi-Higgs scenarios, while they are enhanced bythe same factor for loop-free particle exchanges as, e.g. , for leptoquarks. .2.3 Estimate of the strong CP -violating QCD ¯ θ parameter Even if a natural-sized ¯ θ parameter (which is given by the sum of the original θ that couples to theproduct of the gluon and dual gluon field strength tensors and the phase of the determinant of the quarkmass matrix) is removed by the Peccei-Quinn mechanism [27], it can not be excluded that a fine-tuned ¯ θ , compatible with the | d n | bound [3, 4], will reemerge from Planck-scale physics upon UV completion.The scale of the nucleon EDM induced by the ¯ θ parameter can be estimated, in a similar way tothe expression given in (2.1), by [28–30] | d ¯ θ N | ∼ | ¯ θ | · m ∗ q Λ QCD · e m N ∼ | ¯ θ | · − e · cm , (2.6)where m ∗ q = m u m d m s / ( m u m d + m s m u + m s m d ) ∼ m u m d / ( m u + m d ) is the reduced quark mass. Theadditional suppression factor given by the ratio of the reduced quark mass to the QCD scale parameter Λ QCD ∼
200 MeV takes into account that the ¯ θ induced EDM would have to vanish if any quark masswere vanishing, since in that (chiral) limit the complete ¯ θ term could be rotated away by an axial U (1) transformation acting on the quark with zero mass [30]. Applying the above estimate (2.6) and utilisingthe empirical bound on the neutron EDM [3, 4], one finds the following upper limit for ¯ θ : | ¯ θ | (cid:46) − . Taking into account the limit (2.2) of the Kobayashi-Maskawa induced nucleon EDM, the accessiblewindow for determining ¯ θ by nucleon EDM measurements is therefore − (cid:38) | ¯ θ | (cid:38) − , while the projected sensitivity for | d p | ∼ − e · cm would allow a measurement of the value of, or thebound on, the parameter ¯ θ down to the order − . EDM analysis based on non-perturbative methods
EDM measurements are of low-energy in nature and therefore all predictions of EDM values of sub-atomic particles, especially nucleons belong to the realm of non-perturbative QCD. ¯ θ induced nucleon EDM The QCD ¯ θ -term is manifestly a flavour-neutral, isoscalar source of CP violation. It is instructive thatthe underlying non-perturbative physics nonetheless entails d p (cid:54) = d n .The best way to predict the ratios d p / ¯ θ or d n / ¯ θ in the ¯ θ term scenario would be the applicationof lattice QCD methods. Unfortunately, all current high-precision lattice calculations dedicated to thesepredictions have been based on the computation of the T - and P -violating F form factors of the neutronor proton and have not taken into account that, in a finite volume, the Dirac states of the nucleon acquirean axial rotation in the mass term, such that there is sizeable admixture of the large F (Pauli) in thesmall F (EDM) form factor, as first pointed out in Ref. [31]. When the reported numbers of d n / ¯ θ or d p / ¯ θ had been reanalysed, the corrected values turned out to be compatible with zero within one standarddeviation [31, 32]. It is expected that lattice estimates with better accuracy will become achievable inthe next 1-2 years, though. Note that no general consensus has been reached about the need to perform this correction to the form factors. The authorsof Ref. [33] assert that in their method, which differs from other attempts by a purely imaginary value of the vacuum angle, theexpansion is about a topologically non-trivial vacuum and that therefore the mixing-angle dependence has been included. A step in that direction has been made in Ref. [34]. Using the gradient-flow method with proper axial rotation and byextrapolating from dynamical quark masses corresponding to admittedly large pion masses of (700, 570, 410) MeV this paperpredicts the following results: d n / ¯ θ = − . . · − e · cm and d p / ¯ θ = 1 . . · − e · cm which in turn imply | ¯ θ | < . · − as an upper bound on the QCD theta angle from the experimental bound on the EDM of the neutron.
23n the meantime, chiral perturbation theory (ChPT) can be applied to estimate the contribution ofthe pion one-loop terms to the ¯ θ -induced neutron and proton EDMs [35] – note that the leading termwhich involves a CP -violating but isospin-conserving pion-nucleon vertex was already estimated nearlyforty years ago [29], while the loop diagram with the isospin-breaking counter part is subleading. Bothdiagrams are divergent and have logarithmic scale dependence, which in principle can be cured by theaddition of two independent CP -violating photon-nucleon contact terms [36, 37]. The signs and sizesof the latter, however, cannot be determined in ChPT and need external input, either from experiment orfrom lattice QCD which currently, as mentioned above, produces inconclusive results. The leading pionloop term predicts a contribution (at the mass scale of the nucleon) of ∆ d p / ¯ θ = − ∆ d n / ¯ θ = (1 . ± . · − e · cm (2.7)that is of isovector nature – see [35] with input parameters from [38]. Note, however, that here thesubleading isoscalar loop-term is neglected and that the sizes and signs of the two missing contact termsare not known, such that ChPT itself cannot predict the ratio of the proton to the neutron EDM.For a real test or falsification of the ¯ θ hypothesis as the leading ( i.e. dimension-four) CP -violatingmechanism in case d n and d p are measured, one needs the anticipated results of lattice QCD. However,even without lattice QCD calculations, additional measurement of the deuteron or helion or both EDMswould enable independent tests, since ChPT and chiral effective field theory methods can be used to getan estimate of the genuine nuclear contributions of these light nuclei (including triton) [38], i.e. ( d H − . d p − . d n ) / ¯ θ = (0 . ± . · − e · cm , (2.8) ( d He − . d n + 0 . d p ) / ¯ θ = − (1 . ± . · − e · cm , (2.9) ( d H − . d p + 0 . d n ) / ¯ θ = (2 . ± . · − e · cm . (2.10)These numbers are comparable with the predictions for the single-nucleon EDM case— cf. Eq. (2.7) andfootnote 7. Therefore, they can equally well be used to test or constrain the value of the ¯ θ term to ∼ − level, assuming that the above listed EDMs can be measured to − e · cm sensitivity. Again lattice QCD is the first choice for an estimate of the EDM contributions of the dimension-six CP -violating operators, which can be grouped [39–41] into quark operator terms ( CP -violatingphoton–quark vertex terms), the quark-chromo operator terms ( CP -violating gluon–quark vertex term),the isoscalar Weinberg three-gluon term [42], isospin-conserving CP -violating four-quark terms, andisospin-breaking four quark terms which can be traced back to left-right symmetric models. While theredo not exist lattice QCD calculations for any of the four-quark operators, exploratory studies have juststarted in the Weinberg three-gluon case. In the quark-chromo scenario there already exist promisingsignals for the connected contributions, but results with (quark-)disconnected diagrams, non-perturbativemixing and renormalisation are still missing – see [32,43] and reference therein for further details. In thequark EDM case, however, lattice QCD has delivered since the pertinent weight factors of the u-, d-, ands-quark EDMs follow via a chiral rotation from the corresponding flavour-diagonal quark tensor charges, g u , d , sT , i.e. d n = d γ u g dT + d γ d g uT + d γ s g sT , (2.11) d p = d γ u g uT + d γ d g dT + d γ s g sT , (2.12)where the predictions of the tensor charges improved considerably from 2015 to 2018: g uT = 0 . , g dT = − . , g sT = − . [45]; (2.13) Here and in the following, the signs of the EDMs always refer to the convention e > . Note that the flavour assignments of the tensor charges refer to the proton case, while for the neutron the role of u and d have to be interchanged. The cited values refer to the MS scheme at 2 GeV [44]. uT = 0 . , g dT = − . , g sT = − . [46]; (2.14) g uT = 0 . , g dT = − . , g sT = − . [47]. (2.15)While the ratio g uT /g dT ≈ − is compatible with the estimate from the naive non-relativistic quark modeland the one from QCD sum rules [39], the absolute values of g uT and g dT are smaller, approximatelyreduced by a factor / relative to the naive quark model estimate and the central values in the QCD sumrule case (see also below).The above predictions of the tensor quark charges allow for stringent tests of the split SUSYscenario with gaugino mass unification [48–50], since in this case there is a strong correlation betweenthe electron and neutron (or proton) EDMs [51], the latter governed by the quark EDM operators, whileall other CP -violating operators are highly suppressed. In particular, the results (2.15) and the indirectexperimental bound | d e | < . · − e · cm [12] imply | d n | < . · − e · cm as upper bound in thesplit-SUSY scenario [47]. This limit is still in the range of sensitivity of a dedicated proton EDM storagering experiment.With the exception of the quark EDM case mentioned above, currently there do not exist anypredictions of lattice QCD or ChPT for any of the other CP -violating BSM operators. In the latter cases,only QCD sum rule estimates of the quark and quark-chromo contributions to the nucleon EDMs areavailable [39], d n (cid:39) (1 ± . × (cid:8) . (cid:0) d γ d − . d γ u (cid:1) + 3 . e d d cd − . e u d cu ) (cid:9) ± (0 .
02 GeV) e d W , (2.16) d p (cid:39) (1 ± . × (cid:8) . (cid:0) d γ u − . d γ d (cid:1) + 3 . e u d cu − . e d d cd ) (cid:9) ± (0 .
02 GeV) e d W , (2.17)where d γ u , d and d cu , d denote the u-flavour and d-flavour quark and quark-chromo EDMs, respectively with e u,d the corresponding quark charges, while d W (of dimension mass − ) stands for the prefactor of theWeinberg term. Taking these large uncertainties into account, currently we have no reliable prediction ofthe ratio of the proton to neutron EDM for any of the BSM extensions (SUSY, multi-Higgs models, left-right symmetric models), with the notable exception of the above discussed split-SUSY case (assumingthat quark EDM ratios follow the quark mass (times quark charge) ratios). If, however, storage ring experiments are planned to measure the deuteron and/or helion EDMs, theseresults would determine the genuine nuclear EDM contributions. The relevant, i.e. leading, CP -violatingnuclear matrix elements are governed by tree-level operators and are predicted in the framework of chiraleffective field theory (chEFT) with reasonable uncertainties [38, 52, 53]: d H − . d p + d n ) = { . g − . π } e · fm , (2.18) d He − . d n + 0 . d p = { . g + 0 . g + 0 . π − . C / fm + 0 . C / fm (cid:9) e · fm , (2.19) d H − . d p + 0 . d n = {− . g + 0 . g − . π + 0 . C / fm − . C / fm (cid:9) e · fm . (2.20)Here g and g are the dimensionless low-energy constants of the isospin–conserving and isospin–breaking CP -violating pion-nucleon vertices, respectively, while ∆ π · m N is the prefactor of the CP -violating three-pion term and C and C are the coefficients of the two leading CP -violating four-nucleon terms. The values of the three hadronic low-energy constants g , g and ∆ π can be predictedfrom the coefficients of the CP -violating terms of the underlying theory at the quark-gluon level, e.g. ,from ¯ θ in the case of QCD [38, 54] or from the prefactors of the quark-chromo [55] or the left-right-model-induced four-quark terms – see [56] and references wherein. While the ¯ θ mechanism assigns adominant role to g , the quark-chromo mechanism predicts g and g of about equal magnitude, whereas25 dominates in the left-right scenario. There do not exist analogous predictions for the hadronic coef-ficients C and C . The order of their contributions can so far only be estimated by naive dimensionalanalysis and thus has to be included in the theoretical uncertainties. Note that the role and magnitudeof the CP -violating four-nucleon and three-pion terms have not been investigated for A > nuclei –see [7, 8] for more information on EDM calculations for heavy nuclei. Option for oscillating EDM searches at storage rings
The storage ring technology also allows to search for time-varying (oscillating) components of the EDMin addition to the static (permanent) one [57, 58] and therefore to test the hypothesis that the dark mat-ter content in our galaxy is (at least partially) saturated by a classical oscillating field of axionsor axion-like particles (ALPs), even if the axion/ALP mass m a were in the range from − eV to − eV [59, 60]. This mass range is very challenging for any other technique to reach, since, e.g. , theresonance cavities of the microwave (haloscope) method would have to be unwieldy large in size [62].There are, though, some astrophysical constraints from the bounds of supernova energy losses, Big Bangnucleosynthesis, and the spatial extent of dwarf galaxies [63]. For instance, the latter give an upper boundon the de Broglie wavelength and therefore the lower bound of − eV on the mass of a non-relativisticbosonic particle trapped in the halo of a dwarf galaxy.All interactions of the axions/ALPs are either suppressed by the very large axion/ALP decay con-stant f a or are just of gravitational nature. Thus, in the so-called pre-inflationary Peccei-Quinn symmetrybreaking scenario [61], the initial displacement (misalignment) of axion/ALP field a from the minimumof its potential energy density, given by m a a / , leads to a coherent oscillation of the classical ax-ion/ALPs field at a Compton frequency ω a = m a c / (cid:126) . The idea is to equate the energy density inthese oscillations with the mass-energy associated with dark matter [59, 60]. The axions/ALPs trappedin the halo of our galaxy and to be observed in terrestrial experiments acquire in addition a velocity v of the size of the virial velocity of our solar system relative to the centre of our galaxy ∼ − c . Thustheir frequency is second-order Doppler-shifted, ω (cid:48) (cid:39) ω a (cid:0) v / c (cid:1) . This implies a finite coherencetime of order τ a ≈ (cid:126) /m a v , thus a Q -value of the size ( c/v ) ∼ , and a coherence length of order (cid:126) / ( m a v ) . For any terrestrial experiment smaller than this coherence length, which is at least . for m a c (cid:46) . µ eV , the oscillating axion field corresponds to [57, 58] a ( t ) = a cos( ω (cid:48) ( t − t ) + φ ) ≈ a cos( (cid:126) m a c ( t − t ) + φ ) , (2.21)where the undetermined local phase φ , which is approximately constant as long as the measurementperiod | t − t | is smaller than the coherence time τ a , is correlated with the choice of the start point t ofthe measurement cycle. The amplitude a of this classical field oscillating at the frequency ω (cid:48) ≈ ω a can be estimated by saturating the local dark matter density in our galaxy, ρ LDM ≈ . / cm [61],with the total energy density of the oscillating axion/ALPs field, i.e. ρ LDM ≈ m a a / . Assuming theQCD-axion coupling to the gluons and therefore an effective theta angle θ a = a f a ≈ √ ρ LDM m a f a ≈ √ ρ LDM . m π f π ∼ × − , (2.22) The mode occupation numbers of dark matter bosons of mass < suffice for the formation of a classical field. This assumes that the initial misalignment angle of the axion or ALP field in this light-mass scenario is tuned so small thatthe resulting ‘dark matter’ particles do not overclose the universe – see, e.g. , [61] for more details. That means that a resonance in an RF cavity in a strong magnetic field is excited by the inverse Primakoff effect. Starting with the QCD epoch ( ∼ − s after the Big Bang), the axion mass m a is constrained as m a ≈ . m π f π /f a ,where m π and f π are the pion mass and decay constant, respectively – see, e.g. , Ref. [61] for more details. The to-be-measured value of the phase φ ensures that at the beginning of a measurement period, t = t , all spectral ω (cid:48) components of the axion field, irrespective of their velocity | (cid:126)v | < v esc (= the escape velocity from our galaxy), start coherentlywith the common phase cos( φ ) and stay approximately coherent as long as | t − t | < τ a .
26e would get from the naive formula (2.6) for the ¯ θ -induced nucleon EDM the following estimate of theaxion-induced oscillating component of the nucleon EDM: d osc N ( t ) ∼ − · a ( t ) f a ∼ · − cos( (cid:126) m a c ( t − t ) + φ ) e · cm . (2.23)The detection of an oscillating EDM of such an amplitude would be very demanding. In the case of anALP, however, there is no strict relation between its mass m a and its decay constant f a , such that massregions with m a < . m π f π /f a and therefore effective ALPs angles with θ a > × − become acces-sible. In fact, first exclusion bounds in the domain of axion/ALPs mass (frequency) versus axion/ALP-gluon coupling strength have already been extracted from the recent neutron EDM measurements [63]and dedicated experiments applying nuclear magnetic resonance techniques or superconducting toroidalmagnets are currently realised [62, 64–67].In complete analogy to the neutron EDM experiment, the measurement/bounds of the proton (ordeuteron) EDM by the frozen spin method in storage ring experiments can of course be analysed for slowoscillations, such that the neutron ALP-bounds can potentially be improved by the ratio of the projectedsensitivity of the proton EDM measurement to the current neutron EDM limit, · − e · cm . But theadvantage of the storage ring technique is really the search/scan for an oscillating EDM at the resonanceconditions between the axion/ALP frequency and the g − precession frequency of the storage ring.Such a resonance enhancement would allow to investigate an axion/ALP frequency range of ∼ to ∼
100 MHz , where the lower limit is just due to the current bound on the spin coherence time while theupper bound is due to the spin-rotation frequency. Furthermore, the resonance method should by fiat beless affected by systematical uncertainties than the frozen spin one. And moreover, in a combined electricand magnetic storage ring (which is needed in the case of the deuteron or helion and maybe realised inthe prototype scenario) effective radial electric fields in the centre-of-mass frame of the rotating particlecan be achieved that are one or even two-orders of magnitude bigger than the presently realisable E fields in the laboratory. In this way, the projected sensitivity for oscillating EDM measurements by theresonance method may even reach the − e · cm level. Synopsis
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Beginnings at Brookhaven National Laboratory (BNL, USA)
The idea for using a storage ring to confine a charged-particle beam while testing it for the presenceof an EDM grew out of the Brookhaven g − experimental effort. Even at low sensitivity, the datafrom this experiment may be checked for effects that arise from an EDM. The results from BNL [1]and an even earlier CERN experiment [2] reported upper limits for the muon EDM in the − e · cmrange. Discussions in the late 1990’s centred mostly on the muon experiment [3], but also considered thedeuteron which has a similar magnetic anomaly to mass ratio.A regular pattern of BNL meetings for discussion and planning developed. In 2004, a proposalfor a storage ring search on the deuteron at the − e · cm level was submitted to the BNL ProgramAdvisory Committee as Experiment 970. In light of the discrepancy between theory and experiment forthe muon value of g − [4], it was considered possible that contributions from triangle graphs associatedwith meson exchange in the deuteron would lead to an enhancement in the EDM of the deuteron andmore favourable prospects for a search. However, the BNL PAC did not find the proposal sufficientlycompetitive with other smaller scale EDM searches to warrant the cost of constructing a new storagering. For a while, ring designs shifted to the development of resonant techniques to amplify and thusidentify systematic errors [5]. But eventually these schemes were discarded as unworkable at the greatersensitivities needed, and attention returned to a more standard storage ring design.Beginning in 2005, feasibility experiments were run at the KVI cyclotron facility in Groningen tomeasure broad range spin sensitivities for deuteron scattering on carbon near 100 MeV. These showedlarge analysing powers but also sensitivities to beam alignment errors that could not be cancelled withstandard first-order analysis techniques [6]. In 2007, more definitive experiments were proposed for theCOSY storage ring (Experiment 170) and approved for running. Tests began in 2008 leading to a finalconfirmation run in 2009 to demonstrate that, with a calibration of the sensitivity of the polarimeter tosystematics, errors could be corrected to levels below one part in [6]. This was the first of whatwould become a series of beam studies to develop techniques needed for the EDM search.In 2008, a second deuteron proposal was submitted to the BNL PAC. This time several improve-ments led to an anticipated sensitivity of − e · cm with up to a year of data collection [7]. This ledto a technical review that was held in 2009 (see [8] for a Web site for the review). In the meantime,it was realised that a first experiment on the proton offered some technical advantages, including theability to have counter-rotating beams travelling the same path in the same ring. This would optimise thecancellation of a large class of time-reversal conserving systematic errors. From this point on, proposalsfeatured the proton rather than the deuteron. Work at COSY continued with the deuteron because ofthe investment already made in deuteron operation and the sense that any conclusions would apply toeither proton or deuteron beams. Development continued at BNL and a second technical review washeld in 2011, again with encouraging results (see BNL EDM site). In October of 2011 a full proposalwas forwarded to the US DOE, but no formal evaluation was ever initiated. In collaboration with the USNSF, the two funding agencies decided to terminate all further work along these lines. Continuation at the Forschungszentrum Jülich (FZJ, Germany)
First contributions to the storage ring EDM effort were made at FZJ in 2008-2009, when members ofthe BNL srEDM collaboration and scientists from the Groningen KVI started experiments together with31cientists from the Institute for Nuclear Physics (IKP) to investigate polarimetry issues at the coolersynchrotron COSY. It soon was realised that COSY [9] with its polarised proton and deuteron beamsin the energy range required for srEDM establishes a unique and ideal facility to perform the requiredR&D. The COoler SYnchrotron (COSY) is a worldwide unique facility for polarised and phase-spacecooled hadron beams, which was utilised for hadron physics experiments until the end of 2014. Sincethen it has been used as a test and exploration facility for accelerator and detector development as well asfor the preparation and execution of precision experiments (JEDI (Jülich Electric Dipole moment Investi-gations), TRIC (Time Reversal Invariance at COSY)). The COSY facility comprises sources for polarisedand polarised protons and deuterons, the injector cyclotron JULIC (Jülich Light Ion Cyclotron), the syn-chrotron to accelerate, store and cool beams, and internal and external target stations for experimentalset-ups.H − (D − )-ions are pre-accelerated up to 0.3 (0.55) GeV/c in JULIC, injected into COSY via strip-ping injection and subsequently accelerated to the desired momentum below 3.7 GeV/c. Three installa-tions for phase-space cooling can be used: (i) a low-energy electron cooler (between 0.3 and 0.6 GeV/c),installed in one of the straight sections, (ii) stochastic cooling above 1.5 GeV/c, and (iii) a new high-energy electron cooler in the opposite straight section, which can be operated between 0.3 and 3.7 GeV/c.Well-established methods are used to preserve polarisation during acceleration. A fast tune jump-ing system, consisting of one air-core quadrupole, has been developed to overcome depolarising reso-nances. Preservation of polarisation across imperfection resonances is achieved by the excitation of thevertical orbit using correcting dipoles to induce total spin flips. The polarisation can be continuouslymonitored by an internal polarimeter (EDDA); an additional polarimeter, making use of the WASA for-ward detectors, has recently been set up, and a further new polarimeter, based on LYSO-scintillators,is under development and will be installed in the ring in early 2019. For protons, a beam polarisationof 75% up to the highest momentum has been achieved. Vector and tensor polarised deuterons are alsoroutinely accelerated with a degree of polarisation of up to 60%. Dedicated tools have been developedto manipulate the stored polarised beam and to precisely determine the beam energy.In 2011, the JEDI (Jülich Electric Dipole moment Investigations) collaboration [10] was created,aiming to exploit COSY not only for the development of the key technologies for srEDM but also forperforming a first direct EDM measurement for deuterons (“precursor experiment”). Since COSY is aconventional storage ring with magnetic bending, a dedicated insertion (“Radio-frequency (RF) Wien-filter”) must be used to be sensitive to an EDM. This latter project towards a proof-of-principle for srEDMis supported by an “Advanced Grant” of the “European Research Council” (2016 – 2021) [11].Meanwhile, significant experimental progress has been made at COSY and elsewhere (see Ap-pendix A, “Results and Achievements”). However, it has also become clear that between now (COSY,precursor experiment) and then (final clockwise, counter-clockwise all-electric EDM ring), an interme-diate step (prototype, demonstrator) is required to test/demonstrate key issues, such as:– Storage time of the beam (stochastic cooling)– Spin coherence time– Polarimetry– Clock-wise (CW) and counter clock-wise (CCW) operation– Effects of the magnetic moment(see Chapter 13, “Road-map and Timeline”).The prototype may, if equipped with magnetic elements in addition to electric deflectors, be used inthe frozen-spin condition to determine an EDM limit for the proton (see Chapter 7, “The EDM PrototypeRing”). 32 .3 Ongoing activity: the precursor experiment at COSY
During autumn 2018, the JEDI collaboration performed a first measurement of the deuteron EDM atCOSY, with the analysis of the results in progress. In a pure magnetic storage ring such as COSY, anEDM will generate an oscillation of the vertical polarisation component. For a 970 MeV/c deuteronbeam with the spin precession frequency of 120 kHz, a tiny amplitude is expected, e.g. × − for anEDM of d = 10 − e · cm. To allow for a build-up of the vertical polarisation proportional to the EDM,a radio-frequency (RF) Wien-filter [12] has to be operated. A prototype Wien-filter was successfullyinstalled and operated in COSY in 2014. A new device with a stronger magnetic field (0.05 T · mm) wasdeveloped and constructed together with the Institut für Hochfrequenztechnik (IHF) at RWTH AachenUniversity and ZEA-1 in Jülich. This new RF Wien-filter was installed in COSY in May 2017 and a firstcommissioning run was successfully conducted in June 2017. Charged-particle EDM Initiative and experience of the collaboration
In connection with the “Physics Beyond Colliders” (PBC) initiative of CERN and the “European Strat-egy for Particle Physics” (ESPP) update, a cooperation under the name “Charged Particle Electric DipoleMoment” (CPEDM) was formed in early 2017, comprising members of the srEDM and JEDI collabora-tions as well as new interested scientists from CERN in order to prepare the science case for a storagering EDM search for the proton (deuteron and He) and the technical design study – in other words thecurrent document.The JEDI members of the Institut für Kernphysik (IKP) of the Forschungszentrum Jülich havea decade-long experience to design, to build and operate as well as to further develop accelerators:foremost JULIC and COSY, but also the polarised and unpolarised ion sources for protons and deuterons.IKP has also contributed significantly to the various versions of linear accelerators for spallation neutronsources and it has designed a superconducting linac, which was planned to replace JULIC as the injectorfor COSY. Recently, it has delivered the proton source for commissioning of the ELENA antiproton ringat CERN.Unique experience is available to produce and accelerate polarised beams without polarisation lossand to manipulate them in COSY, to select polarisation states and to determine the degree of polarisationby the use of nuclear reactions with polarimeters, based on scintillator detectors. A huge expertise hasbeen accumulated over the years to cool and store beams, to accelerate and decelerate them and to usethem during energy ramping or at a fixed energy at internal target stations with thin solid, gas or pellettargets. It is also possible to provide (slow (resonant and stochastic) or fast) extracted beams to externaltarget stations – this option was previously used for the TOF-spectrometer and is now exploited for allkinds of detector tests.Electron cooling at low momenta (up to 600 MeV/c) has been used in COSY early on; morerecently a high energy electron cooler (E e < 2 MeV) has been installed and commissioned in the ring.Stochastic cooling is also used routinely in COSY (momentum range from 1.5 to 3.3 GeV/c); here newpick-up and kicker-devices have been developed at and implemented in COSY.A group working at KAIST in South Korea (IBS Center for Axion and Precision Physics research,CAPP) has developed a large expertise in the use of SQUID magnetometers. A prototype EDM ringsection has been constructed to investigate the cryogenic environment and magnetic sensitivity. Thiseffort is in conjunction with the building of a magnetically shielded chamber to simulate conditions inan EDM beam line.A group of scientists from CERN with enormous experience in accelerator design has joined theCPEDM project from the start. They already have made essential contributions to the study of electricdeflection and to various kinds of systematic effects. Limiting the effects of systematics is the centralissue in the success of the EDM storage ring project.33 .5 Further developments
Work is underway at COSY to develop electrostatic plates usable in a final EDM ring. An initial seriesof tests with half spheres demonstrated fields of 17 MV/m for stainless steel separated by 1 mm and30 MV/m for aluminium separated by 0.1 mm. The next phase of the project will test a prototype electricfield section 1-m long located in an existing dipole magnet with a large gap (ANKE, dipole 2) outsidethe COSY ring.
Summary
Summarising, it must be emphasised that in contrast to other EDM projects, e.g. for the neutron, theelectron, the muon and others, which are pursued in many different places worldwide, for CPEDM,Europe will be in a unique position to design, construct and host such a project.
References [1] G.W. Bennett et al. , Phys. Rev. D , 052008 (2009).[2] J. Bailey et al. , J. Phys. G , 345 (1978).[3] F.J.M. Farley et al. , Phys. Rev. Lett. , 052001 (2004).[4] G.W. Bennett et al. , Phys. Rev. D , 072003.[5] Yuri F. Orlov et al. , Phys. Rev. Lett. , 214802 (2006).[6] N.P.M. Brantjes et al. , Nucl. Instrum. Methods Phys. Res. A , 49 (2012).[7] [8] [9] R. Maier, Nucl. Instrum. Methods Phys. Res. A , 1 (1997).[10] http://collaborations.fz-juelich.de/ikp/jedi/ [11] ERC Advanced Grant (srEDM, 694340); see [12] J. Slim et al. , Nucl. Instr. Methods Phys. Res. A , 116 (2016); ibid. , 52 (2017).34 hapter 4Experimental Method Introduction
The existence of a permanent Electric Dipole Moment (EDM) for fundamental particles or subatomicsystems is still an open question in physics since such a quantity has never been detected. The EDM isa vector-like intrinsic property which measures the asymmetric charge distribution along its spin axis .Hence, an experiment to measure the latter often relies on the spin precession rate in an electric field.However, for charged particles, such a measurement cannot be made while maintaining the particle atrest since any applied electric field leads to acceleration. Instead, those fields can be provided as a partof a particle trap. For the experiment considered here, the trap is a storage ring with crossed vertical (cid:126)B y and radial (cid:126)E r fields that confine a beam of spin-polarised particles ( e.g. , protons, deuterons, etc.)into a design orbit (see Fig. 4.1). The Electric Dipole Moment ( (cid:126)d ) couples to the electric fields whilethe Magnetic Dipole Moment ( (cid:126)µ ) couples to the magnetic fields so that, for a particle at rest, a spinprecession will occur which is given by: d (cid:126)S d t = (cid:126)d × (cid:126)E + (cid:126)µ × (cid:126)B , (4.1)In general, the MDM of subatomic particles is known to high precision and the aim of the proposedexperiment is to determine the EDM part which leads only to much smaller spin rotations. Nevertheless,since the charged particle is subject to combined electromagnetic fields and therefore is not at rest, oneneeds to account for the kinematical effect that may alter its spin precession. For that reason, one shallinvoke the Thomas-BMT equation [1] which gives the precession rate of the angle between the spin andmomentum vectors in the inertial rest frame of the particle. Spin evolution in electric and magnetic fields
In Chapter the T-BMT equation was introduced for generic (cid:126)B and (cid:126)E - fields. The latter are defined in thelaboratory frame while the spin is defined in the inertial rest frame of the particle. In a storage ring wherethe particle is being continuously deflected by the guiding electromagnetic fields to perform a closedorbit trajectory, it is convenient to rewrite the equation of motion in the non-inertial frame rotating withthe velocity vector of the particle. A natural way to describe the rotation of the coordinate system is touse the Frenet-Serret frame attached to the reference orbit [2, 3] and therefore lying in the mid-plane ofthe accelerator as illustrated in Fig. 4.1. In that case, the angular velocity describing the rotation of thecoordinate system is given by: (cid:126) Ω cycl = − βcρ (cid:126)u y (4.2)where ρ is the bending radius, β the Lorentz factor and (cid:126)u y is the unit vector perpendicular to the mid-plane of the ring. Writing the relativistic form of Newton’s second law for the reference particle in aperfect machine without any imperfections, and projecting it into the horizontal plane, it can be easilyshown that: ρ = − qmγβ c E r + qmγβc B y (4.3) The dipole moment must be aligned with the only other vector quantity as a consequence of the Wigner-Eckart theorem. Only the field components acting on a particle on the reference orbit in a perfect machine are taken into account to explainthe basic idea of the measurement method: (cid:126)E r = E r (cid:126)u r and (cid:126)B y = B y (cid:126)u y where (cid:126)u r is the unit vector pointing radially outwards, (cid:126)u z is the unit vector co-linear with the velocity vector of the particle and (cid:126)u y is the unit vector defined such that (cid:126)u y = (cid:126)u z × (cid:126)u r .Note that for the electric field to point inwards, E r < . E r ~B y vertical y longitudinal z radial r r e f e r e n c e o r b i t Fig. 4.1:
Coordinate system used in eqs. 4.5 and 4.6 for the case that the beam runs clockwise. Note that theelectromagnetic fields refer to the laboratory frame.
Now, making use of Eqs. (4.2) and (4.3), it results that the spin motion of the reference particle is givenby the subtracted T-BMT equation: d (cid:126)S d t = (cid:104)(cid:16) (cid:126) Ω MDM + (cid:126) Ω EDM (cid:17) − (cid:126) Ω cycl (cid:105) × (cid:126)S , (4.4)where (cid:126) Ω MDM − (cid:126) Ω cycl = − qm (cid:34) G (cid:126)B y − (cid:18) G − γ − (cid:19) (cid:126)β × (cid:126)E r c (cid:35) (4.5) (cid:126) Ω EDM = − ηq mc (cid:104) (cid:126)E r + c(cid:126)β × (cid:126)B y (cid:105) . (4.6)In Eq. 4.4, (cid:126)S is the spin unit vector in units of (cid:126) / (for fermions) defined in the Frenet-Serret frame ofthe reference particle, and t is the time in the laboratory frame of reference.The dimensionless EDM parameter η is related to the electric dipole moment (cid:126)d through (cid:126)d = η q (cid:126) mc (cid:126)S , (4.7)In addition, it is important to note that the form of the Thomas-BMT equation shown in Eqs. (4.4)–(4.6)does not include the effects of gravity. However, this will be described in the appendix D, Gravity andGeneral Relativity as a ’Standard Candle’ and has been studied by several authors [4–7]. The storage ring EDM search
The signal of an electric dipole moment (EDM) using the storage ring method relies on the direct ob-servation of the rotation of the electric dipole and thus, the spin in the presence of an external electricfield that is perpendicular to the axis of the particle spin [8]. The particles being studied are formed intoa beam that is spin polarised, and the changes in the polarisation components are measured on the beamas a whole while it is confined in the ring. However, the MDM can also contribute to the polarization36uildup in the same way that EDM does. Thus, the main idea of the storage ring EDM search (in a per-fect machine) is to maintain the spin frozen along the momentum direction in order to nullify the MDMcontribution and maximize the EDM signal buildup, hence the frozen spin concept that we discuss in thenext section.
To simplify the discussion, one shall assume that the particle is moving on the reference orbit in aperfect machine such that the only fields acting on it are the bending fields, (cid:126)B y and (cid:126)E r as illustrated inFig. 4.1. Then, from Eq. (4.5), a general relationship between the fields can be established that sets thespin precession frequency due to the MDM (or g-2 precession) to zero in the Frenet-Serret frame of theparticle: G (cid:126)B y − (cid:18) G − γ − (cid:19) (cid:126)β × (cid:126)E r c = 0 (4.8)and the radial E-field that is sensed by the EDM is given by: E r = cβγ GB y β γ G − (4.9)In other words, for each energy, there exits ( B y , E r ) combinations such that the spin precession frequencydue to the MDM equals the particle angular velocity. Thus, if the EDM contribution is disregarded andthe initial condition begins with the spin parallel to the velocity, the spin will remain frozen in thehorizontal plane along the momentum direction. However, in the presence of EDM, the spin will precessaround the radial axis leading to a vertical spin component as sketched in Fig. 4.2.Furthermore, if the anomalous magnetic moment G of the particle is positive, then from Eq. (4.8), thefrozen spin condition can be satisfied for an all electric ring and for one specific momentum that onegenerally refers to as the magic momentum p m : p m = mc √ G (4.10)For the proton, this corresponds to a momentum p m = 700 . MeV/c i.e. to a particle kinetic energyof 232.8 MeV.For particles with a negative anomalous magnetic moment such as deuterons for instance, there is nomagic momentum and a combination of radial electric and vertical magnetic fields is necessary to achievethe frozen spin condition. In this case, the electric field must be pointing away from the centre of the ring( E r > ), thus reducing the bending of the beam from magnetic fields alone. This yields an increase ofthe ring circumference.In Fig. 4.2, the frozen spin concept is illustrated where (cid:126)v is the particle velocity along the orbit, (cid:126)B and (cid:126)E are possible external fields (acting on a positively charged particle), and the spin axis is givenby the purple arrow that rotates in a plane perpendicular to (cid:126)E . If the initial condition begins with thespin parallel to the velocity, then the rotation caused by the EDM will make the vertical component ofthe beam polarisation change. This becomes the signal observed by a polarimeter located in the ring.This device allows beam particles to scatter from nuclei in a fixed target. The difference in the scatteringrate towards the left and right sides of the beam is sensitive to the vertical polarisation component of thebeam. Continuous monitoring by a pair of detectors, illustrated in blue in Fig. 4.2, will show a changein the relative left-right rate difference during the time of the beam storage if a measurable vertical spincomponent due to an ED, (or perturbations described in the next paragraph) is generated. A practicalconsideration is the need for an optimum polarimeter efficiency, which is the case for magic energyprotons (see Chap. 12, Polarimetry).Under realistic conditions, beam particles will execute transverse "betatron" and longitudinal "syn-chrotron" oscillations in an imperfect machine constructed with finite mechanical tolerances, positioning37 v Bp POLARIMETER
Fig. 4.2:
A diagram of the EDM experimental effect. A circulating beam particle (yellow) travelling counter-clockwise in a storage ring has its initial polarisation parallel to the velocity. If the orbit is constrained magneticallywith a (cid:126)B field down, then in the particle frame a radial inward electric field (cid:126)E is produced. The orbit may also beconstrained by a radial (cid:126)E field created using electric field plates. If the EDM lies along the spin axis and is thusperpendicular to the electric field, then a precession as shown will be induced. This creates a vertical component tothe polarisation that may be observed in the left-right asymmetry of scattering from a target (black) into detectors(blue) in the lower right part of the orbit. errors of elements and stray fields from surrounding structures. Various effects can rotate the spin fromthe longitudinal into the vertical direction even without an EDM and may lead to systematic errors of themeasurement. An example for the "frozen spin" fully electric proton EDM are residual magnetic fields.To mitigate the effect, the proposal includes installation of the ring in a state-of-the-art magnetic shield-ing reducing the residual field to about 1 nT. Even with such a shielding the residual radial magneticfield couples to the MDM and is expected to limit the sensitivity of the experiment to values well above − e.cm. Measures to further mitigate the effect due to the average radial magnetic field are describedin section 4.3.2. A more thorough analysis of systematic effects is given in chapter 11.The kinematic diagrams shown in Figs. 4.3 and 4.4 show the momentum and ring radius respec-tively as a function of the electric and magnetic fields available to fulfill the frozen spin condition forprotons and deuteron beams. Pure electric rings lie along the horizontal axis. For the case of deuterons,no purely electric "frozen spin" solution exists which is consistent with the observation in Fig. 4.4 thatnone of the curves crosses the horizontal axis. The red dots in Fig. 4.3 labelled "pure electric ring" arefor a realistic electric field of 8 MV/m corresponding to a bending radius of about 52 m. The red dotslabelled prototype ring in Figs. 4.3 and 4.4 are motivated by the prototype described in chapter 8 andwith a bending radius of 8.9 m. The energy is limited by the electric field around 7 MV/m; for protons,the "frozen" spin condition is fulfilled with 45.2 MeV kinetic energy and a magnetic field of 0.0326 T(see Fig. 4.3, both electric and magnetic field deflect in the same direction). For deuterons, the "frozenspin" condition would be fulfilled reversing the electric field, adding a magnetic field of 0.360 T and forkinetic energy of 164.4 MeV (indicated as red point in Fig. 4.4).Figure 4.4 includes only the mixed-field prototype ring operating point for the deuteron at a much highermagnetic field than is required for the proton. There is no pure electric solution for the deuteron. The large size of MDM effects compared to EDM effects also means that any storage ring experimentis sensitive to problems that might arise from issues such as fringe fields, component alignment, strayEM interference, etc., with the design and construction of the physical machine. One strategy to dealwith these problems in general is based on the realisation that the EDM is time-reversal violating whilethe majority of the problems are time-reversal conserving. The experiment could be changed to a timereverse of itself by inverting the direction of all velocities, reversing all spins, and reversing all magneticfields while maintaining the electric fields as is. In this case where the time-reversed beam travels inside38 E r /(MV/m)0.000.020.040.060.080.10 B v / T prototype ringpure electric ring p /(GeV/ c ) . . . . . . . E r /(MV/m)0.000.020.040.060.080.10 B v / T prototype ring pure electric ring R /m . . . . . . . . . . . . Fig. 4.3:
Proton momentum p (left) and storage ring bending radius r (right), for different frozen spin combinationsof electric and magnetic fields (the absolute value of the field is shown). For the pure electric ring the momentumis fixed to 0.7007 Gev/c. E r /(MV/m)0.00.10.20.30.40.5 B v / T B v = E r / c prototype ring p /(GeV/ c ) . . . . . . . . E r /(MV/m)0.00.10.20.30.40.5 B v / T R /m prototype ring B v = E r / c . . . . . . . . . . Fig. 4.4:
Deuteron momentum p (left) and storage ring radius r (right), for different frozen spin combinations ofelectric and magnetic fields. the same machine as the initial experiment and is subject to all of the same imperfections as the originalexperiment, the results could be compared directly. In other words, addition of the measured rotationsof the two counter-rotating beams will cancel all machine-related systematic imperfections such that theremaining part will correspond to the EDM signal (twice).Nevertheless, if a residual radial magnetic field does not reverse between the two counter-rotating beams,this will yield a signal mimicking the EDM one. For the all-electric proton storage ring concept witha ring circumference C = 500 m, an average radial magnetic field as low as B r = 9.3 aT will generatethe same vertical spin component as the EDM signal of − e.cm. This is probably the most serious39ystematic imperfection that needs to be corrected to reach the high sensitivity goal of the experiment.The first line of defence against such magnetic fields is shielding. State-of-the-art multilayer shieldingwith degaussing procedures can reduce the ambient field to the 1 nT level. Noting in addition that sucha residual radial magnetic field does separate the orbits of the counter-rotating beams vertically, thenthe idea to remediate such an imperfection is to operate the machine with low vertical tune, i.e. withweak vertical focusing in order to maximize the separation between the two beams. The latter willbe measured with ultra-sensitive SQUID magnetometers. For instance, with a vertical tune Q y = 0.1,the same radial magnetic field of 9.3 aT leads to an average orbit separation of 5 pm. The measuredvertical separation of the two counter-rotating beams will be reduced by an additional radial magneticfield to compensate. This method and, in particular the limitations, is further discussed in Chapter 10,"Sensitivity and Systematics". Various categories of EDM storage rings are shown in Table 4.1. Of these, only the proton cases areseriously analysed in the present report. The deuteron and electron cases have been mentioned earlierin the report, but are not described in any further detail. The all-magnetic case is exploited to the extentpossible in the COSY precursor experiments. But frozen spin is not possible with only magnetic bendingand an effect is possible only because an RF Wien filter synchronised to the polarisation precession ratebreaks the cancellation that prevents an EDM signal accumulation.
Table 4.1:
General possibilities according to BMT equation.
Field Particle G-factor Kinetic Beams commentconfiguration type energy (MeV) CW/CCWall-electric proton +1.79285 232.8 concurrent final ring, prototype requiredelectron +0.00160 14.5 concurrent challenging polarimetrymuon +0.00165 2991 concurrent impractically short lifetimeE/B proton +1.793 45 consecutive compromised EDM precisioncombined deuteron -0.143 variable consecutive E/B technological challengehelium-3 -4.191 39 consecutive must develop polarimetryall-magnetic used for precursorno frozen spin possibilityDetails of the ring design may be found in other chapters of this report: Chap. 7 describing theCOSY precursor experiment based on deuterons, Chap. 8 for the proton EDM prototype ring and Chap. 9,for the all-electric proton EDM ring. The route toward the final ring, i.e. the all-electric proton EDMring will be explained in the next chapter. In what follows, we discuss the experimental observable andthe basic measurement sequence.
As described in Ref. [9], the 232.8 MeV proton ring has a 500 m circumference and a confining elec-tric field of 8 MV/m. The accumulation rate for a signal corresponds to a rotation of the polarisationaccording to Ω EDM = 2
E d (cid:126) . (4.11)For an EDM of d = 10 − e · cm, the rate would be about . × − rad/s.The plan for an EDM-sensitive polarisation measurement is to record the horizontal asymmetry inthe scattering of sampled protons from a carbon target at forward angles. At the energies where the EDM40earch would be made, the interaction between the polarised protons and the carbon nucleus containsa large spin-orbit term. This gives rise in elastic scattering to an asymmetry between left and right-going particles when there is a vertical polarisation component present. For a complete description ofpolarisation observables and effects, see Tanifugi [10].For spin-1/2 particles, this effect is described by the differential scattering cross secion given inEq. (4.12) with the angles defined in Fig. 4.5. The polarisation along any given axis is given in termsof the fraction of the particles in the ensemble whose spins, through some experiment, are shown to lieeither parallel or anti-parallel to that axis. If these fractions are f + and f − with f + + f − = 1 for the twoprojections of the proton’s spin-1/2, the polarisation becomes p = f + − f − and ranges between 1 and − . The scattering cross section σ POL may be written in terms of the unpolarised cross section σ UNP as σ POL ( θ ) = σ UNP ( θ ) (cid:16) pA Y ( θ ) cos φ sin β (cid:17) (4.12)with the vertical component given by p Y = p cos φ sin β . (4.13)The angles are defined with respect to a coordinate system shown in Fig. 4.5 in which a particle fromthe beam, travelling in the + z direction, is scattered by an absorber into the + x or “left” side of the xz plane. The scattering angle is θ . The polarisation direction, shown as the red arrow, is defined bythe two polar coordinate angles β and ϕ . The polarisation effect reverses if the particles are detected atthe same θ on the − x or “right” side of the beam due to the cos ϕ dependence in Eq. (4.12). Thus thisleft-right asymmetry measures the vertical polarisation component p Y . The size of the signal is governedby the strength of the spin-orbit interaction, which gives rise to the asymmetry scaling coefficient A Y ( θ ) ,otherwise known as the analysing power. ⚫ Fig. 4.5:
The coordinate system for polarization experiments where the beam defines the z axis. The detectorposition in the xz plane corresponds to the scattering angle θ which is used to determine the spin cross section asshown in Eq. (4.12). The angles defining the orientation of the positive polarization direction (see Eq. (4.12) arelabelled in this diagram. In the case of the deuteron, which is spin-1, there are three fractions that describe the magneticsub-state population, f + , f , and f − where f + + f + f − = 1 . The two polarisations are vector,41 V = f + − f − , and tensor p T = 1 − f , which can range from 1 to −
2. If we are interested onlyin the EDM, then the vector polarisation suffices as a marker and the deuteron polarised cross section(Cartesian coordinates following the Madison Convention [10]) becomes σ POL ( θ ) = σ UNP ( θ ) (cid:16) p V A Y ( θ ) cos ϕ sin β (cid:17) . (4.14)Tensor polarisation is usually present to a small degree in polarised deuteron beams. There are threeindependent tensor analysing powers that each add another “ p T A ” term to the equation above. Theireffects may prove useful in polarisation monitoring or checking for systematic effects. Because thisreport explores the possibility of a proton storage ring, the deuteron spin dependence will not be furtherelaborated here.In the energy range where we would like to run the EDM search, it happens that the spin-orbitinteraction between light particles such as the proton and deuteron and the carbon nucleus provides alarge analysing power A Y ( ∼ A typical single measurement sequence is outlined below with the aim of giving some notions of theoverall approach. There are still many open questions, and it is clear that experience of operating, firstlya prototype, and subsequently the full ring will be required to firmly establish the procedures. Details ofthe beam preparation process and data taking may be found in the Polarimetry chapter 11.– Several bunches with vertically polarized protons are injected CW and CCW into the storage ring.– Beams must be injected into the ring in both directions in reasonably rapid succession. The polar-isation begins perpendicular to the ring plane.– Using an RF solenoid, the spins of the particles are rotated into the horizontal plane.– Subsequently the beams are continuously extracted onto the target for ≈ times per year of operation. Note that for a singlestore, statistical effects will be over two orders of magnitude larger than any EDM effect at the expectedlevel of sensitivity. References [1] V. Bargman, L. Michel, and V. L. Telegdi, “Precession of the polarization of particles moving in ahomogeneous electromagnetic field,”
Phys. Rev. Lett. , vol. 2, p. 435, 1959.[2] E. D. Courant and R. D. Ruth, “The Acceleration of Polarized Protons in Circular Accelerators,”
BNL-51270 , 1980.[3] A. J. Silenko, “Comparison of spin dynamics in the cylindrical and Frenet-Serret coordinate sys-tems,”
Phys. Part. Nucl. Lett. , vol. 12, no. 1, pp. 8–10, 2015, 1502.05970.424] Y. F. Orlov, E. Flanagan, and Y. K. Semertzidis, “Spin Rotation by Earth’s Gravitational Field in a"Frozen-Spin" Ring,”
Phys. Lett. , vol. A376, pp. 2822–2829, 2012, 1904.00339.[5] Y. N. Obukhov, A. J. Silenko, and O. V. Teryaev, “Manifestations of the rotation and gravityof the Earth in high-energy physics experiments,”
Phys. Rev. , vol. D94, no. 4, p. 044019, 2016,1608.03808.[6] A. J. Silenko and O. V. Teryaev, “Equivalence principle and experimental tests of gravitational spineffects,”
Phys. Rev. , vol. D76, p. 061101, 2007, gr-qc/0612103.[7] N. N. Nikoalev, F. Rathmann, A. Saleev, and A. J. Silenko, “Spin Dynamics in Storage Rings inApplication to Searches for EDM,” in , 2019.[8] F. J. M. Farley, K. Jungmann, J. P. Miller, W. M. Morse, Y. F. Orlov, B. L. Roberts, Y. K. Se-mertzidis, A. Silenko, and E. J. Stephenson, “New method of measuring electric dipole momentsin storage rings,”
Phys. Rev. Lett. , vol. 93, p. 052001, Jul 2004.[9] V. Anastassopoulos et al. , “A Storage Ring Experiment to Detect a Proton Electric Dipole Mo-ment,”
Rev. Sci. Instrum. , vol. 87, no. 11, p. 115116, 2016, 1502.04317.[10] M. Tanifugi,
Polarization Phenomena in Physics, Applications to Nuclear Reactions . Singapore:World Scientific, 2018. 43 hapter 5Strategy
Introduction
The project to search for charged-particle electric dipole moments in storage rings has a strong sciencecase, but at the same time it is facing demanding technological and metrological challenges. Moreover, itis obvious that such high-precision measurements will require commitments for a long period of time. Inorder to justify the significant expenditures for the ring(s), it will be inevitable to outline a clear plan (seeChapter 13, Road Map and Timeline) for moving towards the ultimate goal of an all-electric polarisedproton EDM-facility with clockwise and counter-clockwise beams operating at the magic momentum:this must include not only the verification of all key technologies, but also the demonstration that theaimed-for sensitivities are feasible. This has already started with several polarised beam techniquesmeeting the EDM experimental requirements. It is now clear that the only viable way to continue this isto pursue a staged approach with a prototype ring as the essential demonstration milestone.
Starting point of the staged approach
The charged-particle EDM project is in an excellent position to start with, since a conventional ( i.e. ,using magnetic deflection) storage-ring facility exists that provides all the required elements for R&Dand will even allow a “proof-of-capability” measurement. COSY, the cooler synchrotron at the Institutefor Nuclear Physics (IKP) of Forschungszentrum Jülich (FZJ) in Germany, is a storage ring for polarisedproton and deuteron beams between 0.3 (0.55) and 3.7 GeV/c. Besides phase-space cooling (electron,stochastic), well-established methods are used to provide, manipulate and investigate stored polarisedbeams. Over the past decade, the JEDI (Jülich Electric Dipole moment Investigations) Collaboration hasmade significant progress using COSY as an EDM test facility (see: Appendix A, Results and achieve-ments at Forschungszentrum Jülich). Currently, JEDI is conducting a precursor experiment (see belowand Chapter 6, Precursor Experiment) to obtain a first directly measured EDM limit for the deuteron byexploiting a radiofrequency (RF) Wien filter in the ring. The experiment is sensitive to the EDM throughits effect on the direction of the invariant spin axis of the ring. A first measurement has been conductedand is currently being analysed. Additional measurements are planned for the second half of 2019 and2020.
Route toward the final ring
A prototype ring (see Chapter 7) offers new capabilities, in a small testing environment, that can addressEDM design issues not accessible otherwise. These include electric field beam transport with the pos-sibility to store two counter-circulating beams using ring lattice construction suitable for the final EDMexperiment. With the addition of air-core magnetic bending, it becomes possible to “freeze” the beampolarisation along the direction of the beam velocity, thus making possible a more sensitive search foran EDM compared to the precursor experiment. Tests will show the limits on beam storage and theprecision of beam monitoring and control. Most importantly, systematic effects that limit the sensitivityin EDM experiments may be studied directly along with efforts to mitigate them.It is the large number of uncertainties, the most fundamental of which is coping with inevitableresidual magnetic fields, that currently prevents a realistic full-scale ring design going beyond the pre-viously published report [Ref, RSI paper]. (See Chapter 8, “All-electric proton EDM ring”). The finalfull-scale design will be an (essentially) all-electric, 233 MeV ring with simultaneously counter-rotating,frozen spin proton beams. 44
Precursor Experiment Prototype Ring All-electric Ring dEDM proof-of-capability (orbit and polarization control;first dEDM measurement) pEDM proof-of-principle (key technologies,first direct pEDM measurement) pEDM precision experiment (sensitivity goal: 10 -29 e cm)- Magnetic storage ring- Polarized deuterons- d-Carbon polarimetry- Radiofrequency (RF) Wien-filter - High-current all-electric ring- Simultaneous CW/CCW op.- Frozen spin control (withcombined E/B-field ring)- Phase-space beam cooling - Frozen spin all-electric(at p = 0.7 GeV/c)- Simultaneous CW/CCW op.- B-shielding, high E-fields- Design: cryogenic, hybrid,…Ongoing at COSY (Jülich)2014 (cid:1) (cid:1) (cid:1)
Fig. 5.1:
Summary of the important features of the proposed stages in the storage ring EDM strategy.
With experience gained from the prototype ring, the information needed for a detailed design ofthe all-electric proton EDM ring (see Chapter 8) should be available. From prototype test results weexpect to be able to justify the technology used and the sensitivity level to be achieved. Finally, detailedand realistic cost estimates will then be possible.
Science case beyond EDM
The rotation of the polarisation (precession of the spin vector) involved in an EDM search may alsocouple to any oscillating EDM associated with a surrounding axion field. Data from the EDM searchmay be scanned as has been shown in neutron and atomic EDM searches to be possible for any evidenceof an axion. In addition, moving the EDM ring parameters away from the frozen spin condition enablesa broader range search to be conducted.It may also be possible to find conditions where the counter-rotating beams obey frozen spinrequirements for different particle species, thus allowing a class of high precision comparisons of relativemagnetic moments and EDMs, if they are observable. Thus the EDM ring will become a facility fordifferent experimental programs with discovery potential at the frontier of new science.45 hapter 6Precursor Experiment
Introduction
The first step in the staged approach is a set of "proof-of-capability" tests referred to as the "precursorexperiment". It is performed at the Cooler Synchrotron COSY at Forschungszentrum Jülich in Germany,which is a magnetic storage ring providing polarised protons and deuterons in the momentum range 0.3to 3.7 GeV/c, see Fig. 6.1.
Fig. 6.1:
The Cooler Synchrotron COSY at Forschungszentrum Jülich in Germany.
Principle of the Measurement
In a magnetic storage ring the precession of the polarisation vector in the horizontal plane prevents abuild-up of a vertical polarisation due to the EDM. The EDM just causes an oscillation of the verticalpolarisation component with amplitude ξ = βη/ (2 G ) . This signature is used in the muon g − exper-iment to measure the muon EDM. For hadrons this method is less sensitive because | G hadron | (cid:29) G µ .The precursor experiment is performed with deuterons at a momentum of p = 970 MeV /c . In this casethe amplitude is only · − for an EDM of d = 10 − e.cm. ‘To allow for a build-up of the vertical polarisation proportional to the EDM, a radio-frequency(RF) Wien-filter can be utilised [1, 2]. Such a device was developed and constructed, see Refs. [3–5]. Itwas installed in COSY in May 2017. Fig. A.5 shows a drawing of the Wien filter.In order to obtain a build-up, the Wien filter has to be operated in resonance with the spin preces-sion frequency f spin . The resonance condition is given by f WF = f rev | k + ν s | , k integer , (6.1) The dimensionless factor η is related to the EDM d via the relation d = η e (cid:126) mc S . ν s = f spin /f rev is the spin tune, defined as the number of spin revolutions per turn. For theexperiments at COSY, the Wien filter was operated at a frequency of f WF ≈
871 kHz which correspondsto k = − . The revolution frequency is f rev ≈
751 kHz . The integral magnetic field of the Wienfilter is .
019 Tmm and the corresponding integral electric field amounts to . . A build-up is onlyobserved if the relative phase Φ between the fields of the Wien filter fields and the horizontal polarisationcomponent match. The polarisation vector has to be parallel to the momentum vector in the Wien filterwhen the E and B fields are at their maximum.The precursor experiment requires several additional prerequisites:1. a long spin coherence time, [6]2. a precise monitoring of the precession in the horizontal plane, [7]3. a feedback system controlling the relative phase of the polarisation vector and the Wien filter fields,[8]which have all been achieved. More details are given in appendix A. As one example, we discuss thespin coherence time. Fig. 6.2 shows the normalised polarisation in the horizontal plane as a function oftime or turn number. Even after 1000 s approximately 50% of the initial polarisation is left. f rev f spin ~B ( a ) ( b ) no r m a li z ed po l . i n ho r i z on t a l p l ane · nb. of turns ( a ) ( b ) Fig. 6.2: left: Initially all spins point in the same direction (a). Difference in the precession frequency f spin leadto decoherence (b). Right: After optimisation a spin coherence time of the order of 1000 s was reached at COSY. Current Status
With all these tools available, a first precursor test run was performed. The main operating parametersof COSY for the precursor experiment are shown in Tab. 6.1. The COSY ring indicating the maincomponents used in the experiment is shown in Fig. 6.3.COSY circumference 183 mdeuteron momentum 0.970 GeV/ cβ ( γ ) G ≈ − . revolution frequency f rev ≈ Table 6.1:
Values of the COSY operating parameters for the deuteron precursor EDM experiment.
Fig. 6.4 shows the build-up rate ˙ α with α = ˙ P vertical /P horizontal of the vertical polarisation com-ponent as a function of the relative phase Φ . The expected sinusoidal shape is observed. To obtain this47 ig. 6.3: The COSY ring with the main components used in the precursor experiment. data requires about 4 hours of beam time. For every single data point the relative phase Φ was set usingthe feedback system. Systematic effects, like misalignments of ring elements, deviation from the designorbit cause at this stage fake EDM-signals orders of magnitude larger than real EDM effect. These ef-fects are under investigation. To get an idea about the statistical sensitivity the hypothetical signal of anEDM of d = 10 − e.cm is indicated by the gray line. The statistical error of the measurement is of theorder of the symbol size, indicating that statistically one is sensitive to EDMs well below the − e.cmlevel. [rad] φ − − [ / s ] α Fig. 6.4:
Build-up of the vertical polarisation ˙ α with α = P vertical /P horizontal as function of the relative phase Φ between the RF Wien filter fields and the horizontal polarisation component. The WF was at its nominal position( Φ WF = 0 ) and snake was switched off ( χ sol = 0 ). The effect of an hypothetical EDM of d = 10 − e.cm isindicated by the grey line. The statistical errors of the measurement are of the order of the symbol size. To get a deeper understanding of systematic effects, the invariant spin axis was varied. The in-variant spin axis, ˆ n , is defined as the rotation axis of the polarisation vector. In an ideal ring with noEDM ˆ n points in the vertical y -direction. An EDM adds a radial x -component such that ∠ (ˆ n, (cid:126)e x ) = ξ = βη/ (2 G ) . A Wien filter rotation around the longitudinal beam axis acts in the same direction,48 (ˆ n, e x ) = Φ WF . The solenoidal field in the snake causes a tilt in the beam z-direction such that ∠ (ˆ n, e z ) = χ sol / (2 sin( πν s )) ( χ sol depends on the snake current). Thus, physically rotating the Wienfilter by an angle Φ WF around the beam axis and introducing a longitudinal magnetic field using thesnake will introduce deliberate perturbations that cause changes in the build-up of a vertical polarisation.At a setting where the introduced perturbations cancel the imperfections of the COSY ring and the EDMeffect, the build-up should vanish.Fig. 6.4 shows the build-up for Φ WF = 0 and χ sol = 0 . In Fig. 6.5 the so called resonance strengthwhich is proportional to the amplitude of the observed oscillation in Fig. 6.4 is shown for various valuesof Φ WF and χ sol in a range between − . ° and . °. The surface is a fit to the data using an analyticexpression for the build-up. The minimum of this graph gives the location of the invariant spin axis. Inan ideal ring its location is ( ξ = ηβ/ (2 G ) , in radial and longitudinal direction.It should be noted that in total three "maps" were taken indicated by the different symbols in theplot. The fact that they give consistent results although taken several days apart indicates that the stabilityof COSY is sufficient to perform this kind of precision studies down to sensitivities corresponding toEDM values well below − e cm . Of course at this stage the deviation of the minimum from ( ξ, is mostly attributed to systematic effects (e.g. misalignment of magnets and beam position monitorscausing deviations from the design orbit). Work is going on to minimise these effects using beam basedalignment and quantify them with the help of simulations. - ] o [ W F f - ] o [ s o l c - · e - · e P R E L I M I N A R Y Fig. 6.5:
The resonance strength (cid:15) which is proportional to the amplitude of the observed oscillation in Fig. 6.4( (cid:15) = ˙ α/ (2 πf rev )) , for various values of Φ WF and χ sol . The surface is a fit to the data using an analytic expressionfor the build-up. The minimum of this graph gives the gives the location of the invariant spin axis. .4 Outlook
In the first half of 2020 a second EDM run is planned by the JEDI collaboration. Prior to this run, beambased alignment procedures are performed in order to calibrate beam position monitors which in turn willlead to an improved orbit. This will likely reduce the shift of the invariant spin axis due to systematiceffects.In the same time simulations tools are developed (see Chapter 12) in order to estimate the contri-bution of systematic effects on the invariant spin axis. The goal is to perform with COSY a first EDMmeasurement with a precision similar to the one of the muon, i.e. − e.cm.It should also be clear that gaining further orders of magnitude in precision is only possible witha dedicated storage ring using counter rotating beams where many systematic effects mentioned abovecancel. References [1] Frank Rathmann, Artem Saleev, and N. N. Nikolaev. The search for electric dipole moments of lightions in storage rings.
J. Phys. Conf. Ser. , 447:012011, 2013.[2] William M. Morse, Yuri F. Orlov, and Yannis K. Semertzidis. rf wien filter in an electric dipolemoment storage ring: The “partially frozen spin” effect.
Phys. Rev. ST Accel. Beams , 16:114001,Nov 2013.[3] J. Slim, F. Rathmann, A. Nass, H. Soltner, R. Gebel, J. Pretz, and D. Heberling. Polynomial ChaosExpansion method as a tool to evaluate and quantify field homogeneities of a novel waveguide RFWien Filter.
Nucl. Instrum. Meth. , A859:52–62, 2017.[4] J. Slim et al. Electromagnetic Simulation and Design of a Novel Waveguide RF Wien Filter forElectric Dipole Moment Measurements of Protons and Deuterons.
Nucl. Instrum. Meth. , A828:116–124, 2016.[5] C. Weidemann et al. Toward polarized antiprotons: Machine development for spin-filtering experi-ments.
Phys. Rev. ST Accel. Beams , 18:020101, Feb 2015.[6] G. Guidoboni et al. How to Reach a Thousand-Second in-Plane Polarization Lifetime with 0.97-GeV/c Deuterons in a Storage Ring.
Phys. Rev. Lett. , 117(5):054801, 2016.[7] D. Eversmann et al. New method for a continuous determination of the spin tune in storage ringsand implications for precision experiments.
Phys. Rev. Lett. , 115(9):094801, 2015.[8] N. Hempelmann et al. Phase locking the spin precession in a storage ring.
Phys. Rev. Lett. ,119(1):014801, 2017. 50 hapter 7The EDM Prototype Ring (PTR)
Introduction
Intense discussions within the CPEDM collaboration have concluded that the final ring cannot be de-signed and built in one step (see Chapter 5, “Strategy”). Instead, a smaller-scale prototype ring PTRmust be constructed and operated as the next step.In the following Chapter 8, “All-Electric Proton EDM Ring”, the state of preparedness for a full-scale all-electric proton EDM of approximately 500 m circumference is discussed. Ideally containingonly electric fields and no magnetic fields, this ring needs to be capable of storing 232.8 MeV frozenspin protons circulating in either clockwise (CW) or counter-clockwise (CCW) directions. Initially thesebeams will circulate consecutively. But, for best possible suppression of systematic EDM errors, thebeams will later have to circulate concurrently.As part of the preparation of the present report, the level of preparedness for constructing wasstudied in considerable detail, with the results distilled down to Table 8.2 in Chapter 8. There the “lacksof preparedness” are sorted by perceived “degrees of severity”.Naturally, such a sorting cannot be arithmetically precise, but a kind of “triage” sorting is possible.Some all-electric EDM storage ring problems can be judged to be “show stoppers” which would defini-tively prevent EDM measurement from being accomplished. It is only because no relativistic all-electricstorage ring has ever been designed and built that problems of this degree of severity cannot be ruled outby experience. Such problems have been colour-coded "red".Other problems, though obviously still in need of further refinement, have been colour-coded“green” to indicate that, based on wide experience that has been gained with polarized beams in magneticstorage rings, there is no need to be concerned about their performance in a full scale all-electric ring,neither in impairing beam performance, nor in limiting the precision of the EDM determination.Finally there are cautionary problems, clearly between red and green in seriousness. These arecoded “yellow”, for caution, in Table 8.2. The potential importance of these problems, in essentially allcases, is that they are capable of restricting the precision with which the proton EDM can be determined.The primary basis for the conclusion that a prototype ring is needed is the presence of show-stopper entries in Table 8.2. These flags must certainly be cleared before serious full-scale planning canbegin responsibly. Furthermore, the lack of experimental experience with electric rings prevents evenany extrapolation from established experience.Once all of the red flags have been cleared, serious design of a full-scale ring can be contemplated.Even then, a complete full-scale design will require that most, or perhaps all, of the yellow flags becleared as well. These were the main predicates from which the proposed PTR program has been derived.
Goals for the PTR prototype ring have been constructed to correlate sensibly with these preparednessdesignations. In particular, two stages are planned. The goals of stage 1, after re-confirming beamcontrol procedures that have already been developed at COSY, will be to turn all red flags in Table 8.2at least to G(-) or Y(+). The goals of stage 2 will be more diverse. But their common thread will be togain the experience needed to complete the design of the full-scale ring. This has to include acquiringinformation needed to predict the potential precision with which the proton EDM can be measured.51ertainly, as a prototype, the ring should be small and simple, and as inexpensive as possible. Yetthe ring has to be designed to be capable of achieving its claimed goals. The primary goal of stage 1 isto demonstrate that performance routinely obtained in magnetic rings can be replicated in an all-electricring. The goals for stage 2 mainly require frozen spin protons. (Except at the 232.8 MeV “magic” kineticenergy for which proton spins can be frozen in an all-electric ring) proton frozen spins require verticalmagnetic field B z to be superimposed on the radial electric field E r .Several considerations went into the determination of kinetic energies for stages 1 and 2. Toavoid new building costs, the ring circumference was constrained to not exceed 100 m. After allowingfor adequate drift space for needed equipment, this led to a bending radius less than 9 m. For technicalreasons the power supply voltages have been constrained to not exceed ± kV. A consequence of theserequirements was that the proton kinetic energy should not exceed 30 MeV.The proton polarimeter figure of merit is satisfactory at 30 MeV, but decreases with decreasingenergy. As a result of these considerations, the 30 MeV nominal all-electric, stage 1, proton beam energywas adopted. (As it happens, electrons circulating under identical ring conditions will be approximately“magic”, meaning that their spins will be “frozen”. Except for the quite low efficiency of currentlyavailable electron polarimetry, this means that, in principle, the electron EDM can also be measured inthe PTR.)From 30 MeV proton energy for stage 1, the choice of 45 MeV for stage 2, frozen spin operation,followed almost automatically. In order to achieve the frozen spin condition for protons near this energy,approximately 1/3 of the bending shall be provided by magnetic elements. Since the magnet needs to beiron free (to avoid hysteresis and obtain the required reproducibility) air core magnets must be used. Therequired magnetic field is sufficiently low that this is not a serious constraint.Up to this point in PTR design studies there has been no differentiation between all-electric,30 MeV, stage 1 optics and 45 MeV, stage 2 frozen spin optics. The basic design has sufficient flexi-bility to meet both goals. In detail, of course, the working points and other details will be essentiallydifferent. Detailed lattice design and performance is described later, in Section 7.6. During task force studies the ring evolved from a race-track to a square shape of various sizes. Thepresent report describes the adopted “square ring”, having 8 m long straight sections. The basic protonkinematic data and field strengths are given in Table 7.1, and the ring layout is shown in Figure 7.1.
Table 7.1:
Basic beam parameters for the PTR ring E only E, B unitKinetic energy 30 45 MeV β = v/c γ (kinetic) 1.032 1.048Momentum 239 294 MeV/cMagnetic rigidity Bρ · cmElectric field only 6.67 MV/mElectric field E (frozen spin) 7.00 MV/mMagnetic field B (frozen spin) 0.0327 T Goals for the 30 MeV all-electric PTR
The two primary goals for the 30 MeV stage can be expressed quantitatively. They are1. to demonstrate the ability to store the polarized protons thought to be the minimum number52 Fig. 7.1:
The basic layout of the prototype ring consists of 8 dual, superimposed electric and magnetic bends; 2families of quadrupoles – focusing (F) and de-focusing (D); with an optional skew quadrupole family at mid-pointsof the four 8 m long straight sections. The total circumference is about 100 m. needed to be able to perform proton EDM measurements in a predominantly electric storage ring,and2. as needed for reducing systematic error, the ability to produce two polarized beams, each with thesame proton intensity, simultaneously counter-circulating in the same ring.Technically, it would be sufficient for these goals to be achieved with unpolarised beams, since there is noreason to suppose that the storage capability depends in any way on the state of beam polarization. Theproton intensity goal has been set conservatively low to avoid distractions associated with preserving po-larization through the injection process—this can be perfected later, using well-understood experimentaltechniques.Polarimetry already demonstrated in COSY will be sufficient to complete these goals. As inCOSY, the spins will not be frozen, nevertheless the spin coherence time (SCT) can be determined. Alsophase-locked spin control can be reconfirmed. Secondary , qualitative goals for stage 1 therefore include replicating spin-control abilities in an all-electric ring, such as phase-locked loop stabilization of the beam polarization. This capability is requiredto provide input signals to the external correction circuits needed to manipulate the beam polarization. Asin COSY, this capability does not require frozen-spin protons—it is enough for the spins to be “pseudo-frozen”; i.e. with spin tune equal to the ratio of two small numbers so that, viewed at fixed location, thepolarization vector appears frozen.Certain tertiary goals for stage 1 will also be needed to steer the upgrading of PTR for a moreadvanced second stage. But any such upgrades need to preserve the gross geometry of the ring. (Mainlyto reduce cost, and speed progress) it seems prudent initially, to economize, with flexibility for laterupgrades. Investigations in the first stage, can produce PTR modification possibilities needed to producea more productive second stage. Some examples follow.It is currently uncertain whether a completely cryogenic vacuum will be necessary. Connectedto this issue is whether or not the beam emittance can be adequately controlled by stochastic cooling,and whether stochastic cooling adversely affects EDM experiments. Also connected with vacuum un-certainty is the possibility of a regenerative breakdown mechanism that could limit the proton beamcurrent. Such a breakdown could commence with a temporarily free electron being accelerated toward53he positive electrode. Secondary electrons created on impact, would be immediately re-captured, butphotons produced could strike the other electrode, producing secondary electron emission that couldlead to regenerative failure. No such phenomenon has ever been observed in magnetic rings—but thisis irrelevant, since there is no corresponding electron acceleration present. Some proton intensity limi-tations in non-relativistic rings seem consistent with such as interpretation. But no such limitation hasbeen observed in electrostatic separators in either electron or proton high energy storage rings. Any suchbreakdown mechanism would presumably tend to be moderated by a superimposed magnetic field. Butweak magnetic fields could be ineffective.Magnetic shielding is another uncertain issue. There are well-understood (but expensive) passivemagnetic shielding methods known, improving the shielding by at least one or two orders of magnitude.But they require detailed understanding of the apparatus, that can, realistically, be studied experimentallyonly in situ. Certainly magnetic shielding could be upgraded in the interval between stages. No activefield control based on magnetic measurement is planned for stage 1, but could, optionally, be developedfor stage 2.The possibility of significant upgrading of positioning and alignment is also anticipated betweenstages 1 and 2. Ferrite kickers, assumed for stage 1, may need to be replaced by air core or electrostatickickers for stage 2.Greatly improved critical analysis of beam position monitor (BPM) performance is also expectedof stage 1, for possible inclusion in stage 2. Similar investigations in the stability of basic mechanicaland electrical parameters will be performed.
Goals for the 45 MeV combined E/B PTR
Stage 2 will be largely dedicated to the development of operational capabilities and identification ofissues that need to be resolved before an eventual full scale ring design can be committed. The followinggoals are essential:1. To lend confidence to an eventual full-scale EDM ring proposal, experimental methods are to bedeveloped and demonstrated for measuring the proton EDM in a ring with superimposed electricand magnetic bending. Cost-saving measures in the prototype, such as room temperature oper-ation, minimal magnetic shielding, avoidance of obsessively tight manufacturing and field-shapematching tolerances, are expected to limit the precision of the prototype ring EDM measurement.But data needed for extrapolation to the full scale ring has to be obtained from the PTR.2. Demonstrate frequency domain control and measurement capability; for example a phase-lockedspin wheel frozen spin beam control. (See Section 7.9)3. Investigate, understand, and measure the general relativistic (GR) correction to the proton EDMmeasurement, which is expected to produce a “fake” EDM measured value of approximately × − e · cm. This quantity can be calculated to better than one percent accuracy. This processis useful as a “standard candle”, whose measurement can provide a major physics motivation forstage 2. The factor of 15 by which the GR effect exceeds the nominal − e · cm effect providesa factor of 15 “cushion” in isolating systematic effects. The fundamental physical significanceof this gravitation-dependent measurement is debatable. But any credible deviation from generalrelativity at atomic scales would be as alarming as any measurably large EDM. The fundamentalexperimental value for calibration purposes is also of debatable value; a fake EDM of far greater(more convenient) magnitude can be reliably mimicked using a Wien filter.4. Finally, a first precise storage ring proton EDM measurement can be made. For various reasons,mainly due to cost-saving measures in PTR design, the achieved precision cannot, however, beexpected to provide a significant test of the standard model. But information gained from thisprototype measurement can be expected to produce specifications the nominal all-electric ringneeds to meet to reach that goal. 54 .4 Relation between PTR and the nominal all-electric ring
This section provides fine-grained technical details concerning the relation of the proposed prototype tothe full scale ring. This is detail the average reader expects to exist for subsequent design refinement, butwithout the immediate need for such detail. Such readers may prefer to glance through this section onlyperfunctorily.The details describe a four parameter lattice design for a complete family of stable all-electricstorage rings, ranging from the PTR ring at the small radius, low energy end, to a full scale, large radius,high energy end. Especially for measuring the EDMs of particles other than protons, there are validreasons for considering electric rings everywhere in this range. And, for the proton rings emphasizedin this report, when comparing the results of different particle tracking programs, it is important for allassumed lattice parameter to be identical, even down to the fine-grained detail given here.The structure of the prototype ring (PTR) has been obtained from the full-scale ring by down-scaling from the full scale Anastassopoulos et al. [1] design to 30/45 MeV, trying to keep the two designsas close as possible. After the down-scaling, mainly to make element lengths sensible for a low energyring, small changes were then made to the PTR design before scaling back up to the full-scale ring. Inthis way, physical properties of the scaled-back-up full-scale ring and the Anastassopoulos et al. ring canbe compared as in Table 9.1, in the full scale ring chapter. (As expected) agreement is quite good forall parameters, well within the ranges of parameter values of the various 2016 ring designs. The skeletalPTR prototype lattice design is shown in Figure 7.2. q F q F D q F q F D q F q F D q F q F D q \ ss q \ ss q F q \ ss . . . . . .
34 0 . . . . . . . .
34 0 . . F u ll ce ll o f m c i r c u m f e r e n ce , M e V p r o t on E D M p r o t o t yp e r i ng Θ Fig. 7.2:
Lattice layouts for proposed lattice half-cell (left) and full ring (right). The accumulated drift lengthis not enough for the ring to operate “below transition”. When scaling up to the eventual, full energy, all-electricring, from four-fold to sixteen-fold symmetry, with drift lengths and bend lengths preserved (but bend angles fourtimes less) the total circumference is to be about 500 m and operation will be well below transition.
55n both ring designs, for flexibility, focusing is provided both by separated-function electric quad-rupoles, and by (very weak) alternating-gradient, combined-function, electrode-shape focusing. (Currentdesigns have favoured electric-quadrupole-only focusing).It was decided that the scaling relation between prototype and full-scale ring would be performedby relating the ring super-periodicities in the ratio of 4 to 16, while leaving all lengths (except for straightsection lengthening to be explained) within each super-period constant. Expressed as ring “shapes”, thisscaling gives the prototype ring the appearance of a square with rounded corners (see Figure 7.2), whilethe full ring appears very nearly circular; (see Figure 8.1 of the nominal all-electric chapter). In thisprocess the bend per super-period was reduced by an (integer) factor of 4. The values of the four mainscaling parameters are shown as (upper-case) parameter values in Table 7.2.
Table 7.2:
The four major parameters for scaling between full scale ring and prototype ring. They describea continuum of stable, all-electric storage rings ranging from small, low energy to large, high energy. Theiruppercase names make these parameters easily recognizable in lattice description files. The remaining (minor)parameters are given in Table 7.3. parameter E_30MeV EM_35MeV EM_45MeV EM_55MeV E_233MeVR_NOMINAL [m] 9.0 9.0 9.0 9.0 40.0L_LONG_STRAIGHT [m] 6.0 6.0 6.0 6.0 14.8N_SUPER 4 4 4 4 16M_NOMINAL 0.1 0.1 0.1 0.1 0.1
The adopted scaling relations follow: the field index scales inversely with super-periodicity N_SU-PER; with m = ± M_NOMINAL being the field indices of the prototype ring, the scaling relation is m = ± M_NOMINAL*4/N_SUPER. Lattice names are in the column headings. Minor scalings areindicated in Table 7.3.
Table 7.3:
Minor geometric parameters: Theta, r , leh , lss = 0 . m, and llsh are, respectively, bend/half-period,bend radius, bend-half-length, short-straight-length, and long-straight-half-length. K is proton kinetic energyand ± m in are alternating field index values. Minor kinetic parameters: lq is quad length, qF and qD are quadstrengths, gBy is half-gap width, Q x and Q y are tunes. lattice name K0 m_in Theta r0 leh llsh lq qF/qD circ. gBy2 Q_x/Q_y[MeV] [r] [m] [m] [m] [m] [1/m] [m] [m]E_30MeV 0.0300 0.100 0.785 9 3.53 2.60 0.2000 ∓ . ∓ . ∓ .
501 0.015 1.815/0.145
Detailed lattice descriptions (needed for computer processing) are contained in the following files,and identified in the tables by the first column entries (are available on request).
EM_45MeV-con_xml : “ .xml ” file containing all parameters (both symbols and their values) for a small(85 m circumference) proton EDM prototype ring, including (symbolic) parameters for scaling tothe large (500 m circumference) all-electric proton EDM ring.
EM_45MeV-nocon_xml : Symbolic “ .xml ” file describing idealized lattice design.
EM_45MeV.adxf : Numerical “ .adxf ” file describing idealized lattice design.
EM_45MeV.sxf2 : Numerical “ .sxf ” file describing fully-instantiated lattice design (though withoutdifferentiated (i.e. individualized) parameter values.)Initially, for both prototype and full-scale ring, the horizontal tune was expected to be just below 2.0and the vertical tune less than 1.0, and tuneable to a value as low as 0.02. This ultra-low vertical tunewas needed to reduce the vertical restoring force, to enhance the beam “self-magnetometry” sensitivityto beam displacement caused by radial magnetic field.(As an aside, it can now be mentioned, that the doubly-magic EDM measurement method possi-bility avoids the need for ultraweak vertical focusing, allowing the focusing to be much stronger than56as initially anticipated. A very thorough and valuable 2015 study by V. Lebedev [3] analysed twofrozen-spin all-electric designs, one very weak-focusing, the other stronger focusing. With ultra-lowvertical tune no longer necessary, the scaled down PTR can be said to more nearly correspond to thestronger-focusing ring favoured there.)For the full-scale ring the correspondingly smaller tune advance per super-period causes the fo-cusing to be weaker. This is what permits the long straight sections of the full scale ring to be morethan doubled, compared to the prototype (from 6 m to 14.8 m). This has the beneficial (perhaps evenobligatory) effect, for the full-scale ring, of operating “below transition”. This ameliorates intrabeamscattering, as can be explained in connection with stochastic cooling. (Conversely, this is one respectin which the prototype ring optics is a not-quite-faithful prototype.) This choice was made to reducethe prototype size. Also, with the COSY ring as a candidate low energy injector ring, for reasons ofbeam bunch-to-bunch separation, the EDM prototype ring circumference of 91 m, exactly one-half ofthe COSY circumference, would be a natural choice.
Electric and magnetic bends
The electrostatic deflectors consist (ideally) of two cylindrically-shaped parallel metal plates with equalpotential and opposite sign. With the zero voltage contour of electric potential defined to be the centerline of the deflector, the “ideal orbit” of the design particle stays on the center line. The electrical potentialis defined to vanish on the center line of the bends, as well as in drift sections well outside the bends.So the electric potential vanishes everywhere on the ideal particle orbit. With the electric potential seenby the ideal particle continuous at the entrance and exit of the deflector, its total momentum is constanteverywhere (even through the RF cavity, except during very weak acceleration needed to keep the beamcentroid on the design axis).There are restrictions on the minimum distance between deflectors. Recent candidate ring latticestudies have limited the horizontal good-field-region for stored particles to be 50 mm. This requires theminimum distance between electric deflector plates to be about 60 mm. The vertical beam size is severaltimes larger than the horizontal. This imposes restrictions on the vertical dimensions of the flat part ofthe deflector too. Minimum vertical dimensions of the bending elements will be more than 100 mm. Inorder to minimize breakdown probability between the flat regions of the deflectors and move them tothe edge, the shape of deflectors should follow Rogowski profiles at both vertical edges. The ends ofindividual deflectors need to be shaped to match the stray fields with subsequent deflectors.The designed ring lattice requires electric gradients in the range 5-10 MV/m. This is more than thestandard values for most accelerator deflectors separated by a few centimeters. Assuming 60 mm distancebetween the plates, to achieve such high electric fields we have to use high voltage power supplies. Atpresent, two 200 kV power converters have been ordered for testing deflector prototypes. The fieldemission, field breakdown, dark current, electrode surface and conditioning are to be studied using twoflat electrostatic deflector plates, mounted on the movable support with the possibility of changing theseparation from 20 to 120 mm. The residual ripple of power converters is expected to be in the orderof ± − at maximum 200 kV. This will lead to particle displacement on the order of millimeters. Asmaller ripple or stability control of the system will be a dedicated task for investigations planned at thetest ring facility. The electric part of the ring can be considered a plate capacitor, whose distance parameter was determinedfrom beam optics considerations. The 2D cross section is shown in Figure 7.3. The good-field region,or the region of interest (ROI), was specified to have dimensions 20 mm ×
60 mm. The contours of theupper and lower edges of the plates were rounded according the Rogowski shape principle. Due to the57 ig. 7.3:
Cross section of the capacitor (in red) inside the beam tube (outer circle). The distance between the platesis 60.7 mm and their height (straight part) is 151.5 mm. The region of interest (ROI) is represented by the twocentral rectangles. finite radius of curvature of the plates of about 8 m a field gradient is generated. Its magnitude can beestimated in the case of infinitely high capacitor plates, because in this case, the electric potential ispurely logarithmic, and its gradient - the electric field - can be obtained analytically. For finitely highcapacitor plates, this should still provide a good approximation. U ( ρ ) = U i + ( U o + U i ) . ln( ρ/ρ i )ln( ρ /ρ i ) (7.1)The corresponding electrical field in radial direction is given by: E ρ ( ρ ) = − ∂∂ρ U ( ρ ) = − U o − U i ρ . ρ /ρ i ) (7.2) U i and U o are the potential values on the inner and the outer capacitor plates, respectively, with thecorresponding values of the radii ρ i = 8.831 m and ρ o = 8.891 m. Here U i = - U o = 210.2 kV.Figure 7.4 shows the potential values and the electric field strength between the capacitor platescalculated with these parameters. There are two ways of dealing with the field strength gradient depictedin Figure 7.4. The accumulated effect may be compensated by electric quadrupoles outside the bendingsection. This solution is valid, and the required quadrupole strengths can be estimated from the figuresgiven above. On the other hand, the gradient can be compensated locally by shaping the contour ofthe electrodes, giving the inner and the outer plate convex and concave shapes, with radii of curvature8 m, respectively. Figure 7.5 suggests how the plates should be manufactured in order to provide thiscompensation.The homogeneity profile of the electric field in the ROI is shown in Figure 7.6. The averagevalue is about 7 MV/m, the same as predicted by the theoretical considerations leading to the results ofFigure 7.4. The maximum relative difference of the electric field in the ROI is about 2.1 × −3 .58 ig. 7.4: Potential values and corresponding electric field strengths between the capacitor plates in the case ofinfinitely high capacitor plates. The average field strength is about 6.998 MV/m.
The geometry is not yet fixed but may be altered according to engineering needs. A guide to theexpected homogeneity values on changing the distance and the height of the capacitor plates is depictedin Figure 7.6. The figure caption gives an example about how to read the graphics.
The nominal magnetic bending field is vertical, B y = B · ˆy . For the combined E/B-prototype ring a firstdesign has been made based on the requirements on integrated electric and magnetic fields. Specificallythe magnetic flux density of the magnet should be B =32.65 mT and the corresponding electric field is E The required vertical flux density of 32.65 mT is small enough to envisage a solution with normal con-ducting, even air-cooled magnets. The magnets are designed according to the cos θ -scheme to ensurea high level of homogeneity of the magnetic field. In order to avoid detrimental magnetic fields fromthe return paths of the cables in the cos θ -dipole, even these have been distributed in a cos θ -fashion.This reduces the effective field in the region of interest, but the flux density is not very high anyway( B y =32.65 mT). The cross section of the cos θ magnet looks as depicted in Figure 7.8.In this design the conductors have a cross section of 50 mm × ×
60 mm. Theaverage flux density in the ROI is 32.65 mT. Figure 7.9 shows the deviation from this value. It is lessthan 1 µ T—so small that in reality the homogeneity will be dominated by manufacturing tolerances. This59 ig. 7.5:
Introduction of concave and convex shapes of the electrodes as one possibility to reduce the gradient fielddue to the curvature. The inner (left) electrode has a concave shape depicted in the inset, whereas the outer onehas a convex one. The corresponding arc can barely be distinguished from the straight line coming down from theends of the curved sections, because the radius of curvature is about 8 m. The corresponding sagitta is only about0.33 mm. contour plot is slightly asymmetric, because the magnet is not straight but follows a radius of about 8.8m. This curvature introduces a gradient in the magnetic flux density, leading to a left-right asymmetry.This asymmetry has been reduced by the introduction of a slight rotation of the upper conductors and areverse rotation of the lower ones by about 0.16° around the center of the arrangement, which cannot beperceived in Figure 7.8, because of the smallness of this angle.The current density in the conductors is about 2.6 A/mm . For the present design the generatedpower amounts to about 43 kW at a current of 1053 A, corresponding to a voltage drop of about 41.0 V.This may be too high a value to rely on air cooling alone for the removal of the generated heat, but designstudies have been carried out which show that the length of the conductors can be enlarged from 8.1 mm,thus reducing the current density and the heat load without compromising the field homogeneity. Atpresent it seems reasonable to assume a water-cooled magnet. The mass of the copper conductors for asingle magnet amounts to about 3000 kg. The magnet can be accommodated outside the vacuum tube. A staged approach was agreed on to match electric and magnetic fields. A global matching of the electricand magnetic fields based on field integrals will suffice in the first stage. This requirement can easily befulfilled from an engineering point of view and the design would already be well described at this point,but it will in the end not be sufficient to lower the EDM level to the anticipated values. For this purposeit will be necessary to ensure local matching of the magnetic and the electric field in a second stage.Due to the fact that the global matching does not represent a major obstacle we are here concerned60 ig. 7.6:
Variation (on a logarithmic scale) of the homogeneity of the electrical field strength as given by thedifference of the maximum and minimum values on the circumference of the region of interest for a straightcapacitor. Example: Without any change to the geometry the homogeneity is close to 10 −2.7 = 2.1 × −3 . Theenlargement of the (nearly) straight section of the plates by about 20 mm improves this value to 10 −3.2 = 6.3 × −4 (see horizontal arrow). A subsequent increase of the plate half distance by 12 mm deteriorates this value again toabout 10 −2.7 (vertical arrow). with the task of locally matching electrical and magnetic field. Inside the magnet and inside the capacitorthe fields are quite constant in amplitude, and their ratio can be chosen according to the requirements. Inthe stray field regions both fields reveal different decay lengths, because the magnetic field componentis much larger in size than the electric one. For this reason, the magnetic stray field has a much largerdecay length and the geometry of the electric capacitor has in some way to be adapted to the decay ofthe magnetic field. The decay of the magnetic field can hardly be changed, because the way the innerconductors are to be connected to the outer returning counterparts is more or less determined by the crosssection shown in Figure 7.8.Figure 7.10 shows how the magnetic field behaves along the central trajectory between adjacentmagnets.It is well known from electrostatics, that the electric field of a plate capacitor is inversely propor-tional to the distance of the plates for fixed potential difference. Several simulations for this study haveshown that also locally the electric field follows this rule. More specifically, as long as the field plates aremuch higher than the gap distance the local electric field is indirectly proportional to the plate distanceat this location. For this reason a flux density distribution like that shown in Figure 7.10 can be regardedas the inverse gap distance of a capacitor providing the same field behavior. From this considerationwe can already draw the conclusion that it will be difficult to fulfill the requirement of locally match-ing the two fields at all locations on the trajectory because the magnetic field drops to very low valuesoutside the magnet pairs, which would correspond to a very large capacitor gap. It may still be possiblebetween the two magnets because the field reduction is given by a factor of 27 in Figure 7.10, which61 ig. 7.7: Overview of one quarter of the combined E/M prototype ring. Two cos θ -dipoles surround the beam tube,in which the capacitor plates are accommodated (not visible). would correspond to a gap distance of about 27 ×
60 mm = 1620 mm. There may be a concern of how toaccommodate such a large capacitor inside the vacuum tube, but it must be kept in mind that, where thegap increases considerably, there is no magnet to restrict this expansion and the beam tube may locallybe much larger.From Figure 7.10 it becomes clear that this solution is not viable if the magnetic field decreaseseven further, as in the region between two neighboring magnets in different quarters, as shown in thisfigure for the outermost distance values. In this case the question arises whether several such capacitorswith step-wise decreasing potentials can be stacked along the trajectory to approximate the magneticfield decay in a step-wise fashion.Figure 7.11 shows an example of this stacking principle for the field decay between magnets indifferent quarters. This figure shows the normalized electric and magnetic field obtained with numericalsimulations. These normalized field values cannot be distinguished on this scale but the difference values(red curve) show small features in the overlap region (red curve) where two neighboring capacitors meet.In all there are 5 capacitors with decreasing potential differences, which require the same number ofpower supplies unless a solution with voltage dividers is chosen. The number of capacitors is dictated bythe maximum expansion factor accepted, which for the example in this figure is about 1.9. This translatesinto a local distance of the capacitor of 60 mm × ig. 7.8: Upper part of the cross section of the cos θ dipole, showing inner conductors (that dominate the field) aswell as the return paths of the conductors (that return the currents without degrading the uniformity, at the smallcost of reducing the field strength).. The current direction is represented by the colour of the conductors. Thebeam tube is represented by the two concentric circles with an inner diameter of 300 mm. The outer diameterof the conductor circumference is 1148 mm. The ROI can be seen as a rectangle in the center, surround by tworectangles representing the electrodes. The field homogeneity in the ROI is shown in more detail in the next figure. Fig. 7.9:
Deviation of the flux density in vertical direction from the average specified value of 32.65 mT in theROI. Enlarged view from Figure 7.8. ig. 7.10: Flux density between the two magnets in Figure 7.7 along the central trajectory within the ROI. Inthe center of the magnets a flux density of 32.65 mT is obtained whereas, midway between adjacent magnets, at . × mm, the flux density drops to a value of about 1 mT. Fig. 7.11:
Normalized electric and magnetic field in the stray field region. The electrical field has been obtainedby stacking 5 expanding capacitors at different potential, each starting again with a gap distance of 60 mm whereits expanded predecessor ends up to a gap nearly double as large. ig. 7.12: Top view of the region between adjacent bend elements, showing capacitor plate separations expandingup to a maximum distance, before starting with a new capacitor at reduced potential. Matching Figure 7.11, theoverall length is 2 m. The minimum separation values are approximately equal to the main bending field electrodeseparation. The various potential levels of the plates are indicated. There may be space to accommodate additionaldevices in the gap of the capacitor with the smallest potential difference. This example shown requires 5 capacitors,with over-all length of about two meters.
Fig. 7.13:
Same as Figure 7.11, but with a larger expansion factor of about 2.95, yielding a larger space at alongitudinal coordinate of 1800 mm with a diameter of 60 mm × .6 Ring design
The basic PTR geometric ring parameters have been given earlier in Table 7.1. As mentioned previously,to this point the ring optics for 30 MeV and 45 MeV have been taken to be identical. The bending,for example for 45 MeV protons, is done by eight 45-degree electric/magnetic bending elements. Theacceptance of the ring is to be 10 π · mm · mrad for particles. The lattice is based on fourfold symmetry,as shown in Figure 7.1.The bend elements consist of electric and magnetic bending. Pure electric bending can be used for30 MeV protons but, for a (nominal maximum) proton energy of 45 MeV, superimposed magnetic andelectric bending will be applied. The magnetic part of the bending has to be provided by a pure air coilmagnet, to avoid hysteresis effects caused by using iron for the magnet. It will be possible to store bothCW and CCW beams consecutively, but not concurrently.The present design for the prototype is a “square” ring with four 8 m long straight sections. Thishas been the result of lattice studies using different shapes such as round or race-track shaped. The ringis shown in Figure 7.1. It consists of 4 unit cells each of them bending ◦ . Each unit cell consists of afocusing structure F-B-D-B-F, where F is a focusing quadrupole, D is a defocusing quadrupole, and B isan electric/magnetic bending unit. The lattice is designed to allow a variable tune between 1.0 and 2.0 inthe radial plane and between 1.6 and 0.1 in the vertical plane, as shown in Figure 7.14.The straight sections have to house separate injection regions for clockwise (CW) and counter-clockwise (CCW) beam operation. There will also be a quadrupole of type QSS in the centre of each ofthe straights, to provide additional tuning possibilities. The voltages on the electric bending plates willbe limited to ± KV for technical reasons. The horizontal gap is determined by the horizontal beamsize, which is determined by the maximum acceptance and the maximum beta function to be x max =2 (cid:112) a x, max β x, max = 2 √ × ≈ mm. With a safety factor of 1.2, the gap between the plates isthen 60 mm, and the maximum electric field in the gap is E max = 2 × / .
06 = 6 . MV/m. Themaximum vertical beta function determines the vertical beam size to be y max = 2 (cid:112) a y,max β y, max =2 √ × ≈ mm. In the good field region x max × y max = 50 ×
90 mm , the field relativefield homogeneity is specified to be better than − . Ring element counts, geometry and other bendparameters are given in Tables 7.4 and 7.5.Lattice flexibility is a goal for the design. The betatron working points can be varied over a largerange as shown in Figure 7.14. A typical plot of the beta functions is given in Figure 7.15. Table 7.4:
Geometry units ◦ ) (cid:15) x = (cid:15) y π mm · mradacceptance a x = a y π mm · mrad Table 7.5:
Bend elements, 45 MeV units
Electric electric field 7.00 MV/mgap between plates 60 mmplate length 6.959 mtotal bending length 55.673 mtotal straight length 44.800 mbend angle per unit ◦ m Magnetic magnetic field 0.0327 Tcurrent density 5.000 A/mm windings/element 6066 Figure 2b: the horizontal betatron tune 𝑄 𝑥 and the vertical betatron tune 𝑄 𝑦 is plotted versus the strength of the Quadrupol family QSS. The other Quadrupole families QF and QD are kept constant as it has been marked in figure 2a. A typical plot of the beta functions is given in figure 3. Figure 3: beta functions for a typical working point: k QF=0.05, k QD = 0.3, k QSS=0, 𝑄 𝑥 = 1.73,𝑄 𝑦 = 1.20 . k QSSl Tune vs. Strength K QSS, K QF=0.05, K QD=-0.3
Q x Q y b e t a / m Distance along optic axis /m beta functions beta x ext beta y ext k QF=0.05, k QD=-0.3, k QSS=0
Fig. 7.14:
On the left the horizontal betatron tune Q x and the vertical betatron tune Q y are plotted versus thestrength of the QF quadrupole family; the quadrupole families QD and QSS are constant, while the QD quadrupolefamily QD is varied. The marked points are continued in the figure on the right. s (m)EDM-Lattice (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) β x β y D x KQf=0.05, KQd=-0.3,KQss=0.0S(m)
Fig. 7.15:
Beta functions and dispersion for a typical working point: k QF =0.05, k QD =0.3, k QSS =0, Q x =1.73, Q y =1.20. .7 Components
There are about 20 beam position monitors located around the ring, as shown in Figure 7.16. A BPM isplaced at the entrance and the exit of each bending unit. One BPM will be placed additionally close tothe quadrupoles in the straight sections. The BPMs have to be mounted precisely and rigidly, as close aspossible to the quadrupoles, to which they are accurately and rigidly attached.A new type of BPM has been developed at the IKP of the Forschungszentrum Juelich. Thesepick-ups are presently in a development stage. The position resolution is measured to be µ m overan active volume of × mm [11]. These Rogowski design BPMs are very attractive because oftheir short length of only 60 mm, and their anticipated accurate absence of systematic relative transversedisplacement of forward and backward beams. Figure 11.21.1: Drawing of the Rogowski Pick-up module. The inner diameter is 100mm. The schematic view on the right side is from F. Trinkl ’ s thesis (Tri-17-12). Figure 11.21.2: 20 Beam Position Monitors are located around the ring. Figure 11.21.1: Drawing of the Rogowski Pick-up module. The inner diameter is 100mm. Figure 11.21.2: 20 Beam Position Monitors are located around the ring.
Fig. 7.16:
On the left are pictures of the Rogowski pick-up module. The inner diameter is 100 mm. Beam positionmonitor locations around the ring are shown on the right.
The quadrupoles for PTR are characterized by aperture diameter 80 mm, powered at +/- 20 kV. We havesimulated a design with a vacuum chamber of diameter 400 mm (Figure 7.17) on the left). The maximumpole tip potential is 30 kV to allow some margin for conditioning. A 3D design has been carried out. Thecalculated sextupole, octupole and higher harmonics of the integrated field seem very reasonable. The3D integration model (Figure 7.17) on the right) suggests that the device can be built within the allocated800 mm longitudinal length, but the (quite large) radial diameter is 620 mm.
Table 7.6:
Calculated multipole content of electric quadrupoles. Index value 2 designates the fundamentalquadrupole content. index strength index strength index strength2 1145.915 9 1.36256e-06 15 9.79269e-073 1.14093e-06 10 1.63810e-06 16 9.85316e-084 7.20433e-09 11 5.69516e-075 1.18116e-06 12 1.07131e-076 1.63343e-06 13 1.10359e-067 1.31927e-06 14 1.52276e-0668 ig. 7.17:
The electrostatic quadrupole design figure is shown on the left. Calculated central field quality is shownon the right. Integrated 2D multipole amplitudes are given in Table 7.6.
The vertical polarization of a stored beam can be rotated into the horizontal plane by the longitudinal fieldof an rf solenoid. As shown in Figure 7.18, the RF solenoid at COSY is a 25-turn air-core water-cooledcopper coil with a length of 57.5 cm and an average diameter of 21 cm. It has an inductance of about41 µ H, and produces a maximum longitudinal RF magnetic field of about 1.17 mT rms at its center. Thesolenoid is a part of an RLC resonant circuit, which typically operates near 917 kHz at an RF voltage ofabout 5.7 kV rms , producing a longitudinal RF field integral of 0.67 T-mm. Typical ramp-up times, fromvertical to horizontal polarization, are about 200 ms. Fig (cid:883)(cid:882) (cid:2879)(cid:2869)(cid:2870) (cid:1840) (cid:2870) (cid:883)(cid:882) (cid:2879)(cid:2869)(cid:2870) (cid:448)(cid:286)(cid:374)(cid:410)(cid:349)(cid:374)(cid:336)(cid:859)(cid:400)(cid:1007)(cid:856) T(cid:346)(cid:349)(cid:400) (cid:400)(cid:410)(cid:381)(cid:396)(cid:258)(cid:336)(cid:286) (cid:396)(cid:349)(cid:374)(cid:336) (cid:349)(cid:400) (cid:258) (cid:393)(cid:396)(cid:381)(cid:410)(cid:381)(cid:410)(cid:455)(cid:393)(cid:286) (cid:258)(cid:374)(cid:282) (cid:258) (cid:400)(cid:349)(cid:336)(cid:374)(cid:349)(cid:296)(cid:349)(cid:272)(cid:258)(cid:374)(cid:410) (cid:374)(cid:437)(cid:373)(cid:271)(cid:286)(cid:396) (cid:381)(cid:296) (cid:448)(cid:286)(cid:374)(cid:410)(cid:349)(cid:374)(cid:336)(cid:859)(cid:400) (cid:349)(cid:400) (cid:393)(cid:258)(cid:396)(cid:410) (cid:381)(cid:296) (cid:410)(cid:346)(cid:286) (cid:282)(cid:286)(cid:448)(cid:286)(cid:367)(cid:381)(cid:393)(cid:373)(cid:286)(cid:374)(cid:410)
Fig. 7.18:
COSY LC-resonant RF solenoid. In COSY this element precesses the polarization vectors of all particlebunches identically. Its role in the PTR ring will be the same.
Outer part (left) and inner part (right) of the COSY waveguide RF-Wien-Filter are shown in Fig 7.19.Beam tuning manoeuvres described earlier in this chapter have employed small radial magnetic fieldsfor applying small controlled torque to the beam polarization to control the spin wheel (explained furtheris Section 7.9). Such a radial magnetic field also causes an undesirable beam orbit perturbation. Insome cases the applied radial magnetic field causes an acceptably small orbit perturbation. But, whenthis is not the case, the RF-Wien filter has to be used instead. One way of expressing the Wien filter69 ig. 7.19:
Outer part (left) and inner part (right) of the COSY transmission line RF-Wien-Filter. In COSY thisdevice emulates the spin precession caused by a deuteron EDM. In the PRT ring it can act identically on allbunches, or for precessing individual bunch spins, without influencing the other bunches [5]. "strength" is to give the spin wheel angular velocity caused per watt of power applied to the RF-Wienfilter. In a COSY precursor RF Wien filter experiment a Wien filter magnetic field times length integralof × − T-m, caused a 0.16 Hz spin-wheel frequency f spin − wheel = Ω spin − wheel / (2 π ) = 0 . Hz.The power conversion was such that an RF power level of 1 kW provided a magnetic field times lengthintegral equal to . × − T-m. This calibration factor was deduced from an experiment using 0.97GeV/c deuterons stored in COSY.
The requirement for the vacuum is mainly given by the minimum beam lifetime requirement of about1000 s. The emittance growth in the ring caused by multiple scattering from the residual gas needs to beless than 0.005 mm · mrad/s. With the initial emittance assumed to be 1 mm · mrad the pressure will haveincreased to 5 mm · mrad within 1000 s. This requires partial pressures of less than − Torr for N and · − Torr for H . The cooling rate for stochastic cooling should be better than · − mm mrad/s.For such an ultra-high vacuum only cryogenic or NEG pumping systems can be used [10]. Bake-out must be foreseen for either cryogenic or NEG systems.The usage of NEG systems introduces some problems: 1. The NEG material becomes saturatedafter several pump-downs; 2. The aging NEG material becomes brittle and leaves some dust in thevacuum vessel; 3. This storage ring is a prototype and a significant number of pump-downs will be partof the development program; 4. The high voltage system requires excellent vacuum.A cryogenic vacuum system has also been considered for the PTR ring. The beam pipe wouldhave to be a system of three concentric pipes. The inner shell would carry the liquid He. Next, in theoutwards direction, is the 70 K pipe, while the outer shell would house super-insulation (and heatingdevices). To avoid these complications and expenses, it might be recommended to use the NEG basedvacuum system. A system of NEG cartouches is presently under discussion. Beam preparation
Before describing proposed injection sequences, it is useful to establish some principles common to allor most schemes, whether for prototype ring or full-scale ring, single beam or dual beam injection.Assuming the harmonic number is 80 for the full-scale ring, one can have a lop-sided fill with asmany as 60 consecutive stable buckets filled and the other 20 empty. This allows the injection kicker70o be pulsed on for half a microsecond or so, which is comfortably long. For filling the other beam,the CCW one, the bunch train and kicker duration can be the same. Similar considerations apply to theprototype ring.The stored bunches would then be too close to be acted on individually, so maybe one would preferto have just 30 filled buckets, alternating with empty buckets. The spacing between bunches would betoo close for single bunch injection or extraction, but it could be amply long for ”tweaking” bunchesindividually.A useful principle recognizes that the final ring is the ”experiment” and the injection system is not.Any time spent in the final ring adjusting the bunch structure is time taken away from the experiment. Sotime taken to trim the spins after injection should be minimized. The responsibility for best arranging thebunch pattern is therefore delegated to the injection system. Minimum injection time would be achievedby injecting just two trains of prepared bunches, which could reduce the set-up time to as little as tenseconds or so.Most of the following principles are intended to ensure the uniformity of all polarized bunchproperties, at least to the extent possible, by assuring that all bunches are subject to identical injectiontreatment:1. All spin flips should be performed in the low energy injection ring, where (at COSY) essentially100 percent efficiency has been persuasively demonstrated [9].2. During any single data collection sequence, there should be no change in the low energy sourceregion. (Except for test purposes) this includes keeping identical bunch polarization. The basisfor this constraint is to best maintain identical parameters for all bunches. (This constraint is notactually imposed from the point of view of minimizing the duration of the entire injection process.In fact, the time needed to change parameters for a subsequent train is expected to be only about 5seconds.)3. All injected bunches will have been pre-cooled in the low energy injection ring. In all cases, onlyvertically polarized bunches (all up, or all down) will be injected into the EDM ring.4. Injection as close as possible to the magic frozen spin energy will be desirable, but the injectedbeam energy will always be off-energy by an amount great enough for the loss of beam polariza-tion (after betatron and synchrotron equilibration, either by filamentation or by active damping ofcoherent oscillations) to be negligible.5. Finally, and most importantly (not counting special polarimetry investigations) after all buncheshave been populated with vertically polarized bunches, identical external fields will be applied toevery bunch to bring all polarization orientations into their desired final injection state— i.e. theinitial EDM measurement configuration state.
The polarized bunch filling sequence can be described in general terms without having frozen the RFfrequency or harmonic numbers. The same discussion can also apply to either a small prototype ring orthe eventual full scale ring. In both cases preliminary commissioning will use just a single, say clockwise(CW) beam. However, since the sequential injection of simultaneously circulating beams does not greatlycomplicate the process, only dual beam injection and bunch polarization manipulation will be describedhere. It will be obvious below, which steps are to be skipped for single beam injection.The longitudinal bunch patterns of counter-circulating beams in a predominantly electric EDMring will be quite similar to the bunching pattern of first generation, single ring, electron-positron col-liders like the Cornell Electron-Positron Storage Ring (CESR) or the DESY Doppel-Ring (DORIS). Inall cases the RF timing has to be arranged so that all bunches, both CW and CCW, pass through the RFcavity (or cavities) at stable phases.Assuming a single RF cavity, there will be a number of stable RF buckets, both CW and CCW,71qual to the harmonic number of the radio-frequency. Not all stable buckets will be filled. Single turn(or “kick”) injection will require the presence of pulsed kickers in the ring, whose turn-on and turn-offpulse-edge durations will have to be restricted to time intervals during gaps in the charge distributionsof both CW and CCW beams. The length of each of these gaps has to be at least one (or higher integermultiple) of RF bucket lengths. We assume gaps and filled buckets alternate more or less uniformlyaround the ring.Ideally, every bunch will have the same number of particles and be maximally polarized. But, forreasons of polarimetry, it is optimal for the polarization signs of adjacent bunches to alternate. When theinjection phase has been (almost) completed in each of the beams the fill pattern will consist of regularrepetitions in a single sequence: “up-polarized-bunch, gap, down-polarized-bunch, gap”. For the smallPTR, two such sequences are planned—for a larger ring probably more.In a final injection phase the bunch polarisations will be rotated, but, until this final injectionphase, all bunch polarisations will be up or down, and bunches will be referred to as “up bunches” in “upbuckets” or “down bunches” in “down buckets”. One could contemplate an “up bunch” being parkedtemporarily, for example, into a “down bucket” but, by an injection principle, this would not be favoured.
Injection will proceed in the following steps (for some of which there are optional procedures):1. At some point a beam (cooled and at full energy) in the injection ring is selected for one injectionpath or the other. It consists of a train of uniformly-spaced, identical, vertically-polarized protonbunches—say “up bunches”.2. All CW “up buckets” in the EDM ring are then filled by kick injection of a single train of appro-priately spaced, timed, and cooled “up bunches” from the injection ring. For this injection phasethe EDM ring energy will be slightly different, say higher, than the magic energy—just enough toprevent decoherence.3. Next, with no change in ring energy, all CCW “up” buckets in the EDM ring are filled by kick in-jection of a single train of appropriately spaced, timed, and cooled “up bunches” from the injectionring.4. For the next two steps, bunches identical to the previous train except for having been flipped into“down bunches”, and therefore, having all other properties the same (to the extent possible).5. The previous two steps are then repeated, injecting “down bunches” from the injection ring.6. After this sequence all “up buckets” and “down buckets” will have been properly populated. Upto this point, all bunch polarisations have been vertical, either up or down.7. Then, by ramping the RF frequency down to the magic energy, uniformly and adiabatically, allbunch energies will have been tuned onto the magic energy. (Though all spins are still vertical,this no longer provides protection against decoherence).8. Then, for a time interval that is an integral number of synchrotron oscillation periods, by applyingan adjustable, uniformly distributed, radial, magnetic, B r trim field, all spin orientations will havebeen rolled through π/ around the radial axis, ending up with longitudinally polarized buncheswith alternating signs. Alternatively, this maneuver could be performed using a waveguide RFWien filter.9. Especially towards the ends of the previous two steps, both horizontal and vertical polarimetry willprobably be needed to control the orientations of all bunches as intended. Figure 7.20 contains a John Jowett space-time beam bunch evolution diagram. There are two CW andtwo CCW bunches. The space-time trajectories are helical, with time advancing to the right for the redbunches (with solid arrows); viewed from the left, these bunches are receding along CW orbits, with72 t cw1 2ccw cw 2cw 2CCW2intervalkicker intervalCW1kicker intervalCCW1kicker intervalCW2kicker previous bunchcrossing point next bunchcrossing point either beaminjection kicker gap empty bucket(Modified) John Jowett space−time bunch fill and kicker timing diagram0 empty bucket060 120 180 240 300 360time spaceCW CCWbeam viewingpoint Fig. 7.20:
A modified “John Jowett beam bunch space-time plot” illustrating a patterns of counter-circulatingbunches. For this example there are six stable buckets (for each beam direction) with two CCW blue bunches(separated by an empty bucket), two CW red bunches (separated by an empty bucket), and two empty gaps,available for single turn injection of a train of two (up or down) polarized bunches into one or the other beams.(This example is directly applicable to the EDM prototype ring bunch filling scheme explained in the text.) time advancing from left to right. Blue bunches (with broken-line arrows) when viewed from the left,are approaching along CCW orbits, with time advancing from right to left. Though representing toroids,the plot is rendered in two dimensional by projection onto the plane of the paper. With advancing time,red and blue bunches “collide” (or, rather, pass harmlessly through each other) at the points indicated.A kicker placed anywhere in PTR, pulsed with proper polarity, for a time conservatively shorterthan 1/8 of a revolution period, can deflect any bunch without affecting any of the other three storedbunches.
Fundamental physics opportunities for PTR
To explain the essential differences between stages 1 and 2 it is useful to expand language that is currentlyin common use.Stage 1 discusses spin effects that are implicitly understood to imply in-plane precession, where“in-plane” implies precession in the (horizontal) plane of the accelerator. Eventually, if the beam polar-ization is frozen in this plane (where “frozen” implies “frozen relative to the particle trajectory) the EDMeffect will accumulate monotonically. But, in stage 1, the proton spins cannot be frozen.Stage 2 concentrates on “out-of-plane” precession, where “out-of-plane” refers to spin vector pre-cession in the “vertical plane instantaneously tangent to the particle orbit”. It is precession into thisplane, that is driven by a symmetry-violating effect such as a proton EDM, and that is the subject of theEDM measurement. In a paper discussing spin decoherence, Koop [4] introduced the “spin wheel” asa picturesque way of describing precession of the beam polarization vector in this “out-of-plane” plane.Though grammatically-dubious, this language is very helpful for visualizing the experimental investi-gations intended for stage 2. That this metaphorical language is due to Koop is of little importance,compared to Koop’s probably correct contention that, to the extent his “spin wheel” executes multiplerevolutions, there is a strong suppression of spin decoherence, with a corresponding increase in the spincoherence time (SCT).Regrettably, the magnitudes of the out-of-plane precessions due to the proton EDM, or the earth’sgravity, have to be expressed in units of nanoradian per second (nrad/s). The length of a run observinga single full turn of the spin wheel might take ten years. This means that, from an experimental point73f view, the Koop decoherence argument simply does not apply to any experiment in which the beampolarization is frozen in all degrees of freedom. However the Koop wheel picture remains as valuable asever, as the rest of this introduction is intended to explain.Expanding the imagery, if in-plane precession of the beam polarization vector is visualized as apropeller blade of a helicopter, and out-of-plane precession as a blade of a wind farm propeller, then,the remaining possible precession direction (azimuthal around the beam axis) can be visualized as thepropeller of a propeller-powered airplane.This remaining freedom, precession of the beam polarization around the beam axis, is driven bysolenoidal ( i.e. along the orbit) magnetic field acting on the proton MDM. Except for one phenomenon,this precession might seem to have negligible influence on the EDM experiment. (With further abuse oflanguage) the airplane-propeller and helicopter-propeller precessions “do not commute”. This failure ofcommutation produces a “wind-farm propeller-like precession” which produces a spurious EDM signal; i.e. systematic error. (Important though this source of sytematic error is, there is currently no plan(other than avoiding solenoidal fields) to study this effect in stage 2—the importance of this issue has tobe addressed theoretically, for example by simulation. This is commonly referred to as “the geometricphase problem”.)The main thrust of stage 2 is to study “out-of-plane” beam polarization precessions which, asjust explained, can usefully be visualized as the rolling motion of a “spin wheel”. After this cartoonishdescription, a more technically informative discussion can be based on the matched-pair of graphs inFigure 7.21.In each graph the horizontal axis is magnetic field (but note the huge difference in the scales) andthe vertical axis describes the Koop wheel response (but note that angular velocity is plotted in (a) whilethe angular advance after a given time is plotted in (b)). The main point of these two graphs is that, inspite their vastly different scales, the slopes are determined by a single, truly constant, physical constantof nature—the magnetic dipole moment of the proton. (That this quantity is, itself, known to 10 decimalpoint accuracy, though interesting, is not the point here. It is the constancy.)With the aid of a transmission line Wien filter, such as shown in Figure 7.19, and a frozen spinpolarized proton beam, from data implied by the figure on the left of Figure 7.21, it will be possible to“calibrate” this slope, as a function of a single, externally-imposed current, which is, itself, experimen-tally reproducible to better than parts per million accuracy [6–8]. The ultimate EDM precision dependson either improving this accuracy, or on scheming to exploit it most effectively. Appropriately trans-formed to match the parameters of the experiment implied on the right, this calibration can be appliedto determine the slopes on the right, in spite of the nine orders of magnitude difference of horizontalscales. The accuracy with which data implied by the figure on the right can be used to determine theproton EDM depends, primarily, on the precision with which the data points are determined, as indicatedby their error bars and point locations—which have been chosen arbitrarily for the figure. Though just acartoon, the fact that the two parallel lines on the right do not quite coincide, is intended to suggest thepresence of errors in the extrapolations on the right. And there is another significant ambiguity in thefigure on the right; the ∆ β ranges may, or may not include the critical β = 0 point, at which the beamsare truly frozen in all degrees of freedom. Though it is not obvious from the figures, the vast differencein horizontal scales “amplifies” this ambiguity.One example of the “scheming most effectively” mentioned above, that can be developed usingPTR, would be to exploit the wave-guide Wien filter to isolate just one of the bunches for phase-lockingits spin wheel angle β . This would, of course, destroy any EDM information contained in this particularbunch. But the phase-locking would, to high precision, have no effect on the other bunch polarizations;they would still freely respond to the EDM torques (including spurious EDM-mimicking torques). With the exception of vertical positioning, which needs to be controlled to micro-meter accuracy, it is element positioningrigidity, current resettability, and time-indepence of all parameters, more than absolute accuracy, that needs to be achieved. B r B r ∆ "identical"experimentallycalibrated slopes T µ multiple complete spin wheel rotations ∆β (fT)spin wheelangular velocity (Hz) r B ( ) ∆β spin wheel angle advance (mr)(b)( a ) MICROSCOPIC (MEASUREMENT) VIEWapplicability condition: applicability condition: total spin wheel rotation << π TELESCOPIC (CALIBRATION)VIEW run duration long enough for run duration so short that
Fig. 7.21:
Graphs indicating dependence of spin wheel orientation angle β on radial magnetic field B r . The left-hand graph (a) is especially appropriate for spin wheel calibration, with large radial torques intentionally appliedby stripline Wien filter (for example for calibration purposes). The right-hand graph (b) is especially appropriatefor representing the dependence of change ∆ β = β end − β beg . (for example as the result of measuring unknownphysically interesting torques during a long EDM or GR measurement run). For the graphs to be intelligible, thescales have to be unambiguously shown and the error bars need to be shown—here they are just order of magnitudeestimates. The prototype ring PTR can be used as a “dry run” prototype for investigating operational issueslike this, to be applied later in the full-scale EDM ring. This is an example of the sort of investigationsto be performed in stage 2.For various technical reasons (mainly connected with minimizing the PTR cost) the EDM mea-surement just described is expected to provide a proton EDM upper limit only of about − e · cm. Onthe other hand, the out-of-plane precession induced by earth gravity, is calculated to mimic a protonEDM of × − e · cm. This “standard candle” data rate will therefore be 15 times greater than thenominal EDM rate, and the run duration needed to collect the same number of counts 15 times shorter.Because of its stochastic nature, the statistics-dominated “standard candle” run duration is correspond-ingly reduced, compared to the year-long data collection period assumed in the nominal proton EDMmeasurement plan. This vastly improved true and known data rate can be expected to permit systematicerror investigations that would be impractical for an actual EDM measurement. Furthermore, the GR-predicted precession reversal upon magnetic field and beam direction reversal provides a further factorof two effective data rate improvement. (Depending on the precision with which magnetic field reversalcan be controlled) this makes the goal of measuring the gravitational effect well worth pursuing in PTR. Summary and Outlook
The concept of, and the need for, the prototype storage ring has been outlined in the previous sections tothe best of our current understanding. The next stage is to produce a detailed design report which should75emonstrate the technical feasibility of a prototype storage ring. The plan of the CPEDM collaborationis to finalize this TDR in 2022 (see Chapter 13).
References [1] V. Anastassopoulos et al.,
A storage ring experiment to detect a proton electric dipole moment,
Review of Scientific Instruments, , 115116 (2016)[2] V. Anastassopoulos et al., A proposal to measure the proton electric dipole moment with − e · cmsensitivity, Limitations on an EDM ring design,
Report to the EDM collaboration meeting,Forschungscentrum, Juelich, Nov. 11, 2015[4] I.A. Koop,
Asymmetric energy colliding ion beams in the EDM sorage ring,
Paper TUPWO040, inProceedings of IPAC2013, Shanghai, China, 2013[5] J. Slim et al.,
Electromagnetic simulation and design of a novel waveguide RF Wien filter for electricdipole moment measurements of protons and deuterons,
Nucl. Instrum. Meth., vol. A828, pp. 116-124, 1603.01567, 2016[6] G. Fernquist, E.Halvarsson, and J. Pen,
The CERN Current Calibrator - a New Type of Instrument,
IEEE Trans. , 2, 2003[7] G. Fernquist et al., Design and Evaluation of a 10-mA DC Current Reference Standard,
IEEE Trans.Instr. and Meas, , 2, 2003[8] G. Fernquist, High precision measurement,
CERN Power Point presentation, CAS 2004[9] C. Weidemann et al.,
Toward polarized antiprotons: Machine development for spin-filtering exper-iments,
PRST-AB , 020101, 2015[10] H. Jagdfeld, F. Klehr, private communication, FZ-Juelich, 2018[11] F. Abusaif, Beam position monitors, report at the JEDI Collaboration meeting, Cracow, Oct-01-2018[12] R. Stassen, private communication, FZ-Juelich, Aug-2018[13] F. Trinkl, Development of a Rogowski coil beam position monitor for electric dipole moment mea-surements at storage rings, Thesis, Dec. 17, RWTH Aachen[14] A. Zelenski,
The RHIC Polarized Ion Source Upgrade,
POS (PSTP2013) 048, 201376 hapter 8All Electric Proton EDM Ring
Introduction – BNL design
It has usually been assumed and, as far as we know it is still true, that the most sensitive proton EDMmeasurement will be made in a dedicated, precision, all-electric storage ring, in which clock-wise (CW)and counter clock-wise (CCW) beams are circulating concurrently at the “magic” kinetic energy of232.8 MeV, for which the proton spins are “frozen”, for example pointing in the forward direction ev-erywhere in the ring. Most recently, a design for this ring has been outlined in a publication of Anas-tassopoulos et al. [1], as the design had evolved from the more detailed earlier proposal [2]. Substantialanalysis of (a quite similar earlier version of) this design has been produced by Lebedev [3]. Parametersfor the Anastassopoulos et al. design are given in Table 8.1 (column “full scale”), and one quadrant ofthe full ring is shown in Figure 8.1. The present report does not attempt to replicate material in thatpublication in any substantive way. The purpose for any material copied is only for ease of comparison.Planning for the prototype ring PTR began by down-scaling from the Anastassopoulos et al. de-sign, by approximate factors of five in both lengths and kinetic energy. The down-scaling prescriptionis described in detail in the “EDM prototype” chapter. A result of the down-scaling is that, though thefull-scale ring shown in Figure 8.1 looks “round”, the PTR ring shown in Figure 7.2 looks “square”. Thisis an artifact resulting from the scaling of lattice functions rather than the scaling of appearances.After minor changes to match element lengths at the reduced beam energy, the adopted PTR di-mensions were up-scaled, back to the full-scale size. Recalculated lattice parameters for the up-scaledring are shown in Table 8.1. Agreement is quite good for all parameters, well within the ranges of pa-rameter values of the various 2016 ring designs. For transverse optical properties this agreement followsmore or less automatically from the scaling. For longitudinal dynamics the scaling is less transparent,since cavity frequencies and harmonic numbers do not scale automatically. However the well-establishedsynchrotron oscillation formalism is expected to apply quite directly to both PTR and full-scale ring.The only significant defect of the down-scaling has to do with sensitivity to intrabeam scattering.(For emphasis concerning this deficiency) the following paragraph is copied verbatim from Chapter 7.“For the full-scale ring the correspondingly smaller tune advance per super-period causes the focusing tobe weaker. This is what permits the long straight sections of the full scale ring to be more than doubled,compared to the prototype (from 6 m to 14.8 m). This has the beneficial (perhaps even obligatory) effect,for the full-scale ring, of operating “below transition”. This ameliorates intrabeam scattering, as canbe explained in connection with stochastic cooling. (Conversely, this is one respect in which PTR is anot-quite-faithful prototype.) This choice was made to reduce the prototype size.”77 able 8.1:
Lattice parameter comparison between a lattice up-scaled from the prototype PTR lattice (in the lastcolumn) to the the same parameters for full-scale all-electric EDM lattice (in the second to last column). Anydifferences between entries in these two columns lie well within the ranges of values for existing full-scale all-electric proton EDM rings. parameter symbol unit full scale PTR-scaledbending radius r m 52.3 electrode spacing g cm 3 3electrode height d cm 20 20deflector shape cylindrical ≈ cylindricalelectrode index m E Mv/m 8.0 8.92number long straights 4 16long strght sec. leng. lss m 20.8 12.0polarimeter sections 2 2injection sections 2 2total circumference C h
100 100RF frequency 35.878 35.878number of bunches 100 25particles per bunch 2.5e8 5e8mom. spread(not/cooled) ± ± β x, max m 47 48max. vert. beta func. β y, max m 216 183dispersion D m 29.5 46.1horizontal tune Q x Q y (cid:15) x mm-mr 3.2/3 3/3vert. emit.(not/cooled) (cid:15) y mm-mr 17/3 17/3slip-factor η = α − /γ -0.19278 ig. 8.1: One quadrant of a full-scale, all-electric, frozen spin EDM storage ring. The total circumference is 500 m.Copying from the original caption, the deflector radius is 52.3 m and the plate pacing is 3 cm. The electric field isdirected inward between the plates. The spin and momentum vectors are kept aligned for the duration of storage. Arealistic lattice will include 40 bending sections separated by 36 straight sections 2.7 m long each, with electrostaticquadrupoles in an alternating gradient configuration, and four 20.8 long straight sections for polarimetry and beaminjection. It will also include SQUID-based magnetometers distributed around the ring. .2 Preparedness for the full-scale ring
Table 8.2 gives a long (but surely still incomplete) list of requirements that need to be satisfied beforeserious construction of a full scale EDM ring can begin. Each of these topics has been discussed inpreparing this report, at least to the level of formulating criteria for assigning the “preparedness rankings”shown in this table. Though highly abbreviated in this table, most of the issues are expanded uponelsewhere in the report. The assignment of colour-coded scores is explained in the table caption. Thesescores are loosely correlated with the PTR prototype ring staging described in Chapter 8.Operations Rank Comment Referencespin control feed-back G COSY R&D App. A.1.3spin coherence time G(-) COSY R&D App. A.1.2polarimetry Y polarimetry is destructive Chap. 11beam current limit R enough protons for EDM Sect. 7.2CW/CCW operation R systematic EDM error reduction Ref. [1]TheoryGR gravity effect G(+) this paper, standard candle bonus App. Dfrozen spin fixed point stable? G stable, this paper App. G.5.5intrabeam scattering Y may limit run duration Ref. [3]geometric/Berry phase theory Y needs further study Ref. [4]Componentsquads G e.g. CSR design Chap. 9polarimeter G COSY R&D Chap. 11waveguide Wien filter G COSY R&D precursor App. A.1.5electric bends R(+) sparking/cost compromise App. A.1.10Physics & Engineeringcryogenic vacuum Y required?—cost issue only Ref. [5]stochastic cooling Y ultraweak focusing issue Ref. [6]power supply stability Y(-) may prevent phase lock Chap. 7regenerative breakdown R(+) specific to mainly-electric,not seen in E-separatorsEDM systematicspolarimetry G(-) COSY R&D Chap. 11CW/CCW beam shape matching Y Chap. 10beam sample extraction Y systematic error? Chap. 11, App. Kcontrol current resettability Y Ref. [7]BPM precision Y(-) Rogowski? Squids? Chap. 7, Chap. 10element positioning & rigidity Y(-) must match light source stability Ref. [8]theoretical analysis Chap. 10 and refs.Radial B-field B r R assumed to be dominant Ref. [1]
Table 8.2:
Status of colour-coded preparedness levels for the full-scale, all-electric ring. Green means “readyto break ground”, Yellow means “promising only”, Red means “critical challenge”. Plus (+) and (-) are to beinterpreted as for college course grades. So the ranking, with most prepared first, is: +G-+Y-+R . Success inmeeting prototype ring goals could amount, for example, to upgrading all scores to Y(+) or better.
Almost from its beginning in early 2018, an EDM feasibility task force realized that a smallscale, inexpensive, prototype EDM ring would be needed, in order to investigate, experimentally, issuesessential for an eventual, full scale EDM ring. At that time a unique full-scale lattice design had not yetbeen adopted. Different investigations were based on different lattice design files, but the differences80ere slight enough to be insignificant, as regards scaling down to a prototype ring. It was further decidedthat the ring designs for the prototype ring and the final ring should be as closely identical as possible.With the frozen-spin proton kinetic energy being 233 MeV, prototype proton kinetic energies of 35, 45,and 55 MeV were considered. This scaling is described in full detail in Chapter 7.A detailed list of needs and goals for the prototype ring is given in Chapter 7. In particular theprocedure employed in scaling down the full scale ring (by a factor of roughly five in circumference)to the prototype ring is described and justified there in full detail. To check this scaling, the entries inthe final column of Table 8.1 were calculated, for comparison with the Anastassopoulos et al. valuesgiven in the second-to-last column. Since the focusing is very weak in both cases, there is little reasonto question the reliability of the scaling, as regards transverse optics. Of course, because of the differentbeam energies, there are substantial differences in the longitudinal dynamics. But, since this formalismis very well established in both cases, there is little reason to doubt this aspect of the scaling.The most obvious need for building a prototype ring was the lack of significant experience withrelativistic all-electric accelerators and especially storage rings. Of course this is due to the much morepowerful bending that is possible with magnetic, rather than electric fields, and the resulting absence ofall-electric role models. To make up for this deficiency, the electric fields have to be increased to a levelthat is limited by electrical breakdown. Experience in this area is largely based on high energy particleelectrical separators. Just one, of many, but perhaps the most important, technical uncertainty has to dowith the highest field, smallest bending radius, and hence the least expensive, all-electric ring that can beconservatively constructed and guaranteed to store 233 MeV protons.The PTR staging can, to some extent, be correlated with the color-coded entries in Table 8.2.Red entries in that table represent critical challenges that would necessarily delay commencement of thefull scale ring.. They are also referred to as “quantitative goals” for PTR stage 1. The main goal ofstage 1 is to remove the “R” flags from the table. This includes the “R(+)” associated with the electricbend/sparking compromise. This score was increased from “R” to “R(+)” only to acknowledge that, byincreasing the ring radius sufficiently, sparking can be sufficiently suppressed. But this could lead tounacceptable cost increase.Especially with the EDM precision being roughly proportional to proton beam current, the exper-imental determination of achievable beam intensity will be needed for any eventual full scale EDM ringdesign. Operational experience with electric rings has been very limited. There has been a significantlygreat accumulation of polarized beam experience, but all in magnetic rings, none in electric rings. In anycase, another important goal for stage 1 is to remove the “R” flag associated with beam current limit.Most of the entries in the table with yellow “Y” flags are to be studied in PTR stage 2. In Chapter 8they are characterized as “qualitative goals”, at least partly to acknowledge their indefinite nature. Forexample there is one weakness in the PTR down-scaling, that will limit the extent to which prototyperesults can be reliably extrapolated to the full scale ring. (Along with well-understood residual vacuumgrowth) intrabeam scattering is expected to be a significant source of beam emittance growth, which, aswell as increasing spin decoherence, can cause beam loss and limit run length. Full 3D equilibration ofthis growth source is only possible for ”below transition” ring operation. This condition is met in the fullscale Anastassopoulos et al. design, but not in the PTR design. (It would have required an approximatedoubling of the ring circumference.) Investigating this issue will be one of the goals of the prototypering. The bottom (2-line) entry in Table 8.2 requires special explanation. This entry is not assigned acolour; it applies to the inevitable residual radial magnetic field average (cid:104) B r (cid:105) after all efforts have beenexhausted to trim it to its ideally zero value. This average (cid:104) B r (cid:105) value is expected to dominate the protonEDM systematic error. But, because its value depends on the uncertain values of all other entries in thetable, the (cid:104) B r (cid:105) uncertainty cannot be compared directly to the other entries in the table—it depends onthe accumulated effect of all the other values, and on their theoretical systematic error calculation.81 .3 New ideas
Of course one also expects investigations with a prototype ring to give rise to new ideas. In fact, theplanning phase itself, now well begun, can motivate the development of new ideas. This study was noexception. By and large, though, to reduce the proliferation of speculative descriptions, the body of thepresent report concentrates mainly on fleshing out ring design and experimental methods as establishedin the first several months of the study.Some of the main new ideas obtained during the preparation of this feasibility study, are describedin appendices to this report. The titles of these appendices are prefixed with “New ideas:” to distinguishthem from the preceding, more conventional, appendices. Also these appendices are introduced by briefabstracts.
References [1] V. Anastassopoulos et al.,
A storage ring experiment to detect a proton electric dipole moment,
Review of Scientific Instruments, , 115116 (2016)[2] V. Anastassopoulos et al., A proposal to measure the proton electric dipole moment with − e · cmsensitivity, Limitations on an EDM ring design,
Report to the EDM collaboration meeting,Forschungszentrum, Juelich, Nov. 11, 2015[4] M. H. Tahar and C. Carli,
On solving the Thomas Bargman-Michel-Telegdi equation using theBogoliubov Krylov method of averages and the calculation of the Berry phases , arXiv:1904.07722[physics.acc-ph], 2019.[5] R. von Hahn, et al.
The Cryogenic Storage Ring, arXiv:1606.01525v1 [physics.atom-ph], 2016[6] D. Mohl,
Stochastic Cooling,
CERN 87-03, p. 453, 1987[7] G. Fernquist et al.,
Design and Evaluation of a 10-mA DC Current Reference Standard,
IEEE Trans.Instr. and Meas, , 2, 2003[8] G. Decker, Beam stability in synchrotron light sources,
ITWM01, Proceedings of DIPAC, LyonFrance, 2005 82 hapter 9Electric Fields
Assumptions and boundary conditions
One proposal for the nominal all-electric EDM ring [1] is a fully electric strong focusing lattice, toobtain a 500 m circumference storage ring. It consists of 4 Long Straight Sections (LSS), to be usedfor injection of each beam (clockwise and counter clockwise) and two polarimeters. The long straightsections are linked with 10 cells, each containing 3 bending sections (bends, see Fig. 9.1), 2 Short StraightSections (SSS) housing magnetometers and separate quadrupoles (quads). This lattice produces the mostchallenging requirements for the quadrupoles compared to a ’soft focusing’ lattice, where at least someof the focusing is included in the bending elements.An alternative soft focusing lattice [2] may use bends with soft focusing in the vertical plane.This will lead to bends with a field index between m = 0 . and m = 0 . , which further increases thechallenge for the bend design and manufacturing tolerance, while the quadrupole requirements would beless demanding with respect to the quads of the strong focusing lattice.The fully electric machine imposes stringent requirements for the background magnetic field ofless than − nT [1]. This requirement has an immediate impact on the construction of the ringelements and the materials that can be used. Austenitic stainless steels show a paramagnetic behaviourat room temperature and the relative magnetic permeability is typically in the order of . − . for the fully annealed, fully austenitic grades. One must avoid the use of work hardened and/or weldedcomponents, where magnetic susceptibility could be higher as a function of the grade used. One couldconsider fully austenitic grades such as 316LN to avoid non-linear behaviour due to traces of ferromag-netic phases. Alternatively, titanium alloys could be used at likely higher cost, but the consequences oftheir relatively poor heat conduction is still to be studied in further detail. Depending on what approachwill be retained to achieve the required vacuum level, poor heat conduction may complicate bake-outs oralternatively operation at cryogenic temperatures.The required vacuum level is in the 10 -11 mbar range. This implies that the equipment should becompatible with either bake-out at 200 ◦ C or 300 ◦ C, if the ring is to be operated at room temperature.Alternatively, it should be compatible with cryogenics cool down, in case that the ring is to be operatedat cryogenic temperatures to avoid too many cold/warm transitions [3]. Running the electric devices atlow temperatures may lead to a reduced voltage breakdown rate, but it requires the need for (a not yetexisting design for) a cryogenic >200 kV feedthrough and possibly a bus bar at cryogenic temperatures.Both are substantial challenges to design and operate reliably.The aim is to keep machine’s cross section below 1 × including the magnetic shielding thatis needed to shield the background magnetic field (earth magnetic field and stray fields), and, as such,this has a direct impact on the design of the beam elements. All elements studied have a smaller crosssection, but depending on the space needed for the magnetic shielding or the cryostats, this requirementmay have to be revised. Electrode material
The electrodes of the bending dipoles, as well as the quadrupoles, are large sized objects, and need toprovide significant fields to achieve the required deflection. The high fields assumed for the ring needto be produced reliably with large electrode surfaces as well as 30 mm gaps. The High Voltage (HV)breakdown rate is expected to be in the order of 1/day for the entire machine, which is very hard to achievewith conventional electrode materials. The choice of the electrode material is also strongly influenced83 ig. 9.1:
Mechanical dipole concept. by the vacuum requirements and the constraints that these impose on the materials. For example, coatedaluminium is commonly used for large septa electrodes at CERN to operate up to 15 MV/m, but isincompatible with bake-out or cryogenics cool-down due to crack forming in the oxide coating of theelectrode.Stainless steel and titanium are compatible with the required vacuum conditions. Older literaturedemonstrated that titanium has a better Voltage Holding Capacity (VHC) than stainless steel [5, 6]. Op-erational experience with larger electrodes (of about 1 m) and similar gaps (30 mm) appears howeverlimited to around 8 MV/m [7, 8]. Alternative electrode materials may be needed to achieve improvedperformance on similarly sized electrodes using similarly sized electrode gaps. In this respect, the workdone on niobium electrodes [9] and TiN coated aluminium electrodes [10] using small electrodes, as wellas TiN coated stainless steel electrodes [11,12] using small electrodes and small gaps, are very encourag-ing. At CERN a campaign of breakdown conditioning and breakdown rates for various metals and alloyswas done [13,14] and demonstrates that there is a difference of more than an order of magnitude betweenthe performance achieved in these small scale laboratory tests and the reported performance of large DCdevices. This is a basis to expect that an increase of operational fields in the electric field devices to beused in the EDM ring may be possible compared to what is used for large DC electric field devices inaccelerators so far.One should not lose sight of the scaling laws for the voltage effect and, more importantly in ourapplication, the area effect [15], where the VHC scales with the surface as:VHC = √ E · U (9.1)VHC = U √ d ∝ A − µ , (9.2)where d is the distance between parallel plates, U the applied voltage, A the surface of the electrodesand µ can be determined empirically. Ring elements
In the strong focusing lattice, the focusing is entirely left over to the quadrupoles, and the dipoles donot focus in the vertical plane. The main dipoles of the strong focusing lattice use cylindrical electrodes,and an integrated field quality 1 ppm in (cid:31)
20 mm central Good Field Region (GFR) is requested [17].Table 9.1 shows the principal requirements assumed for the dipoles.84 able 9.1:
Main dipole parameters as assumed for the final the strong focusing lattice and as proposed for thestrong focusing concept dipole.
Strong focusing Alternative proposedlattice assumption concept designPhysical length [m] 2.739 4.16Equivalent length [m] 2.739 3.80Required deflection [mrad] 52.36 78.54Gap width [mm] 30 30Electrode height [mm] 200 280Beam aperture ( a x × a y ) [mm ] 30 ×
200 30 × (cid:31)
20 mm [ppm] 1 700Main field [MV/m] 8 8.67Voltage per polarity [kV] ± ± × −4 , whilefor the full aperture of 30 ×
200 mm the integrated field homogeneity is 4 × −3 [19]. The main elec-tric field is 8.67 MV/m, but the peak fields are around 10 MV/m. In Fig. 9.3 the fields at the horizontaland vertical mid-plane of the dipole are shown. Electrodes using Rogowski profile edges [18] shouldbe explored to reduce the peak fields further. These simulation results seem to indicate that to reach therequired homogeneity, the electrodes should be increased further in (vertical) size (see present design inFig: 9.2 left), making the cross section requirement for the machine (to keep the cross section below 1 × R = R R R R R R R R R R R R R R R R (9.3) XX (cid:48) YY (cid:48) = horizontal displacementhorizontal angular displacementvertical displacementvertical angular displacement (9.4) Additionally to the concept without vertical focusing, a concept for a bend with soft focusing in the ver-tical plane is being studied. Using quasi-spherical electrodes, this bends with a field index of m = 0 . ,85 ig. 9.2:
2D field plots of the optimised electrode cross section (left) and the top view of the end field (right).
Fig. 9.3:
The electric field as computed for the horizontal and vertical midplanes of the main dipole. i.e. with a radius of 48.4 meter in the horizontal plane and 250 m in the vertical plane. The electrodecurvature amount to just 24 µ m at the top and bottom of the 200 mm tall electrodes. This highlights theneed for very high manufacturing tolerances, both for the electrodes themselves, as well as the electrodefixation inside the vacuum vessel. Electrode manufacturing for both variants will be challenging. To avoid very heavy electrodes (80 kg ifmade of solid titanium), hollow electrodes manufacturing techniques, respecting the required tolerances,are to be developed. The mechanical strength of these electrodes needs to be designed taking into accountthe non-negligible force applied on them by the electric field to obtain the required field precision.A mechanical concept was also developed for the dipole (see Fig. 9.1). The electrode supportsare located close to the end of the central tank section. At this location the vacuum vessel is reinforcedwith external webs, to optimise its stability and to guarantee the electrode position is not affected bytank deformation due to vacuum forces. Three support feet will be mounted onto these support webs toallow precise alignment of the tank. To make sure the requested field quality of − could be reached86 ig. 9.4: Plot of the relevant transfer matrix correlations. The other elements of k (Eq. 9.3) are small. in the GFR of (cid:31)
20 mm, the electrodes need to be aligned parallel in the vertical plane with a precisionbetter 0.3 µ m, corresponding to 1.5 µrad . Therefore, the electrode supports should be adjustable (radialposition, angle and height) to facilitate the electrode alignment during assembly, but this will be verysubstantial challenge. Upstream, the electrodes are longitudinally fixed to the electrode supports, whileon the downstream end the fixation allows for longitudinal movement to limit stress on the ceramicinsulators during bake-out or cool-down.In principle, all dipoles will be powered in parallel to reduce the impact of errors provoked bythe power converter stability. Conditioning may become challenging if it is to be done with all devicesin parallel. This will be even more challenging, if the electrode materials used only have little marginwith respect to the required electric field. Therefore, it is planned to disconnect each device and con-dition it individually. Since the electrode position is fixed, a two stage conditioning process could beenvisaged. First, each polarity will be separately conditioned, mainly to condition the deflectors on theelectrode supports and feedthrough. This would then be followed by bi-polar conditioning, to conditionthe principal electrode surfaces. The principal design assumptions [21] for the strong focusing lattice quadrupoles are shown in Table9.2. The present baseline lattice assumes quads of 400 mm physical length. Our studies however,have shown that the required field quality cannot be met with 400 mm long quads, partly due to theunrealistically high field gradient and unachievable VHC, as well as due to the effect of the end fieldson the field quality. The field requirements can potentially be met with a 1 metre device (Fig. 9.5). Firstsimulations [17] have shown the integrated field error of the 1 metre long (flange to flange) to be ofthe order of × − . The maximum field on the electrodes should still be optimised, but it appearsvery difficult to keep this below 10 MV/m. Two quadrupole variants have been studied in further detail.The first is a fully symmetric variant that uses simplified round electrodes to facilitate manufacturing.The integrated field precision required seems however difficult to reach with cylindrical electrodes [3].Therefore the second variant uses asymmetric hyperbolic poles: narrow gap poles in the horizontal planewill allow powering with a lower voltage of this pair, ultimately requiring only one large HV feedthrough.This facilitates integration as well as lowers the cost. The principal performance parameters of both87 able 9.2: Principal quadrupole parameters.
Lattice Simplified Asymmetricassumption 3D design 3D designPhysical length [m] 0.4 1.0 1.0Equivalent length [mm] 400 730 750Beam aperture ( a x × a y ) [mm ] 30 ×
200 30 ×
200 30 × ] 50 27.4 26.66Electrode voltage [kV] ± ± ± ∼ ∼ ∼ (cid:31) mm 1 × − . × − × − variants are also shown in Table 9.2. The 2D field plots [19] for the quadrupole using the 3 differentpole shapes (hyperbolic, round, and asymmetric) are shown in Fig. 9.6. The asymmetric quadrupole’sleft/right electrode length is longer than the top/bottom electrode length, with the aim to approach as wellas possible the corresponding iso-potential surface of an ideal symmetric quadrupole, (see Fig. 9.7). Theneed for at least 2 HV feedthroughs makes the requirement to keep the cross section below 1 × challenging, in particular when using a perfectly symmetric (horizontal vs. vertical) electrode design,where 2 large feedthroughs will be needed. Fig. 9.5:
Two long straight sections are dedicated to the injection of the two beams. To inject, the beam is deflectedby an electrostatic septum followed by a fast-pulsed separator (fast deflector). The principal parametersof these devices are shown in Table 9.3 [21].
The septum and its (anode) support need to be curved to limit the gap width to 30 mm, while displacingthe beam by 86 mm. The septum can be made of bent or segmented 1 mm thick titanium sheet. Bykeeping the gap limited to 30 mm, the operational voltage required is approximately 240 kV. The vertical88 ig. 9.6:
Electric field of ideal symmetric quad (left), simplified quad (middle) and asymmetric quad (right). acceptance is reduced with respect to the vertical acceptance of the ring to make sure the septum remainsvertically straight when subject to the mechanical force induced by the electric field. The solid cathodecould be made of titanium.
The fast deflector gap width is taking into account the beam sagitta using straight electrodes. The externalelectrode is installed so that at the exit the gap is 30 mm wide and the entrance gap width is 42.5 mm. Thisallows the operating voltage to be limited to 30.4 kV. The HV feedthroughs will have to be developed,since these are not commercially available. The feasibility of the fast deflector pulse generator still needsdetailed study. In particular, the required rise and fall time feasibility are still to be confirmed. The pulse
Fig. 9.7:
Rendering of the asymmetric quadrupole. The low voltage electrodes are longer than the high voltagepair (top/bottom). able 9.3: Principal injection element parameters.
Septum Fast deflectorPhysical length [m] 3.5 3.0Equivalent length [m] 4.0 2.5Deflection angle [mrad] 57.34 10Gap width ( a x × a y ) [mm ] 30 ×
200 42.5 × − ± ∞ T rise and T fall (0.2 % - 99.8 %) [ µ s] ∞ ∼ < generator can use semi-conductor switch stacks (MOSFETs most likely), but no commercially availableswitches have been identified. Required R&D
To make sure that all requirements of the electric field elements are feasible, the following topics forfurther research are identified so far:– Electrode material performance and their compatibility with bake-out or cryo cooling.– Feasibility of required electrode alignment hence field precision for ring elements.– Stable electrode fixation, allowing adjusting of the electrode position sufficiently precisely to ob-tain the required field quality.– Feasibility of electrode manufacturing precision.– Feasibility of the fast deflector pulse generator, in particular with respect to the required rise andfall times.
Summary
The present strong focusing lattice bend concept design achieves 700 ppm field homogeneity in thecentral GFR of (cid:31)
20 mm, which is worse than required. The electric field levels on the bend electrodesmight be achievable with titanium, but alternative materials such as niobium or coated aluminium mayprovide more margin, and should reduce the spark rate. Further study of the voltage holding capacity(VHC) of large electrode materials is essential to make sure that the proposed elements can be operatedat the required fields for extended periods while respecting the desired spark rate.A design for the quad is under development, albeit with a physical length of 1 meter instead of the400 mm assumed in the lattice. The asymmetric variant is supplied with 133/20 kV, with the maximumvoltage close to the voltage used for the dipoles. The achievable integrated field gradient homogeneityin the GFR is for the time being insufficient, but by further optimising the electrode extremities it isexpected that the requirement of 10 − is reachable. The electric fields on the electrodes are compatiblewith the choice of titanium as electrode material. The cross-section of this asymmetric quad is somewhatsmaller than a symmetric variant, facilitating integration within the planned cross section of the ring.To allow the technical design of the electric field elements and in particular to determine the re-quired mechanical tolerances for the dipoles and quadrupoles, such as electrode profile, electrode align-ment etc., an analysis of these tolerances on the performance reach of the EDM storage ring should beperformed. Since this is a problem with many input variables, one could use the Polynomial Chaos Ex-pansion (PCE) method that was already successfully applied to determine the tolerances of an RF Wienfilter [22]. 90or the injection of the beams into the storage rings, the feasibility of a curved septum followedby a fast deflector was studied. Both elements operate with conservative fields and voltages, althoughthe feasibility of the fast deflector pulse generator is still to be studied further in detail with respect to therise and fall time requirements. References [1] V. Anastassopoulos et al. et al. , "The cryogenic storage ring CSR", Rev. Sci. Instrum. 87, 063115 (2016)[4] G.W. Bennett et al. , "Improved limit on the muon electric dipole moment", Phys. Rev. D ,052008 (2009).[5] C.K. Sinclair et al., "Dramatic reduction of DC field emission from large area electrodes by plasmasource ion implantation", PAC 01, Chicago, USA, (2001).[6] B.M. Dunham et al., "Performance of a very high voltage photoemission electron gun for ahigh brightness, high average current ERL injector", PAC 07, Alberquerque, New Mexico, USA,(2007).[7] L. Sermeus et al. , "The design of the special magnets for PIMMS/TERA", Proc. EPAC 2004,Lucerne, Switzerland (2004).[8] J. Borburgh et al. , "Design of electrostatic septa and fast deflector for MEDAUSTRON", Proc.IPAC2011, San Sebastian, Spain (2011).[9] M. BastaniNejad et al. , “Evaluation of niobium as candidate material for dc high votage photo-electron guns", Phys. Rev. Special topics – acc. and beams 15, 083502 (2012).[10] A. Mamun et al. , "TiN coated aluminum electrodes for DC high voltage electron guns", Journalof Vacuum Science & Technology A 33, 031604 (2015).[11] A. Mamun, "Thin film studies toward improving the performance of accelerator electron sources",Doctor of Philosophy (PhD), dissertation, Mechanical & Aerospace Engineering, Old DominionUniversity, https://digitalcommons.odu.edu/mae_etds/6, (2016).[12] Kirill Grigoriev et al. , "Electrostatic deflector studies using small prototypes", Review of ScientificInstruments 90, 045124, https://doi.org/10.1063/1.5086862, (2019).[13] A. Descoeudres et al. , "DC breakdown conditioning and breakdown rate of metals and metallicalloys under ultrahigh vacuum", Phys. Rev. Speical topics - Accelerators and Beams 12, 032991(2009).[14] A. Descoeudres et al. , "DC breakdown experiment with Cobalt electrodes", CERN CLIC-Note-875, CERN-OPEN-2011-029, (2009).[15] A. de Lorenzi et al. , "HV holding in vacuum, a key issue for the ITER neutral beam injector",proceedings ISDEIV 2018, Greiswald (D).[16] Y. Semertzidis, "Storage ring EDM experiments: Proton and Deuteron", EDM kickoff meeting,March 2017, http://indico.cern.ch/event/609422[17] J. Borburgh et al. ,“Challenges for the Electric Field Devices for a CERN Proton EDM StorageRing", proceedings ISDEIV 2018, Greifswald (D).[18] Rogowski, W. Archiv f. Elektrotechnik (1923) 12: 1. https://doi.org/10.1007/BF01656573[19] M. Atanasov et al. et al. , "Polynomial Chaos Expansion method as a tool to evaluate and quantify fieldhomogeneities of a novel waveguide RF Wien filter", NIM in Phys. A, July 2017, pages 52-62,http://dx.doi.org/10.1016/j.nima.2017.03.0492 hapter 10Sensitivity and Systematics Statistical Sensitivity
The statistical error on the EDM d is given by σ EDM = √ β pr s (cid:126) √ N f AP Erτ (10.1)with β pr = (cid:40) = Gγ G +1 precursor experiment = 1 prototype & final ringThe parameters of Eq. (10.1) are described in Table 10.1. Eq. (10.1) assumes that the beam isconstantly extracted on a target in order to measure the polarisation. It also assumes that the beampolarisation does not decohere during the measurement time τ . Details how to arrive at Eq. (10.1) aregiven in Appendix C. The statistical error for the different stages are given in Tab. 10.1.pure magnetic ring & combined ring all electric ringWien filter (precursor)polarisation P N · · fraction of particles detected f A τ E, B . / m , µ T 7 . / m , .
03 T 8 MV / m , − fraction of ring where fields are present r / σ EDM (1fill) /e cm 8 . · − . · − . · − σ EDM (1year) /e cm 8 . · − . · − . · − Table 10.1:
The statistical uncertainty for the three different stages proposed. Note that in the executive summaryfactor r was omitted. Systematic Effects
Any phenomena other than an EDM generating a vertical component of the spin limit the sensitivity, i.e.the smallest detectable EDM, of the proposed experiment. Such systematic effects may be generated byunwanted electric fields due to imperfections of the focusing structure as misalignments of components,by magnetic fields penetrating the magnetic shielding or generated inside the shield e.g. by the beamitself or the RF cavity, or gravity. A combination of several such phenomena or a combination of anaverage horizontal spin and one of these phenomena may as well lead to such systematic effects. Thischapter describes the present stage of the understanding of systematic effects limiting the sensitivity ofthe experiment concentrating on the measurement of the EDM in an electrostatic "frozen spin" ring [1,2],which is considered the present baseline proposal. Nevertheless, many of the mechanisms described arerelevant for other proposals as a hybrid ring with electric bending magnetic focusing [3] and the "doublemagic" ring [4]. 93tudies on systematic effects have been carried out and are underway by several teams of theCPEDM collaboration to further improve the understanding of basic phenomena to be taken into accountand estimate the achievable sensitivity. Note that these studies are still underway and the present report isa snapshot aiming at describing the present understanding. The preliminary conclusion is that achievingthe sensitivity target of − e · cm is very challenging and will probably not be possible with the presentbaseline fully electrostatic " frozen spin" synchrotron. Fig. 10.1:
Sketch of the proposal to measure the proton EDM in a frozen spin "magic energy" electrostatic ring
The basic idea of the proposal to measure the proton EDM in an electrostatic ring [1, 2] is de-picted in Fig. 10.1. Bunches, represented by red and blue arrows, of protons polarized in longitudinaldirection are circulating in an electrostatic ring. The bending electric field pointing towards the inside isrepresented by green arrows. Bunches circulating clockwise (CW) are represented by blue arrows andbunches circulating counter-clockwise (CCW) by red arrows. The direction of the arrows indicates thepolarization. For the case sketched in Fig. 10.1, both the CW and the CCW beam have bunches polarizedparallel to the direction of movement and opposite to the direction of movement. Such a bunch structureis advantageous to reduce some of the systematic effects reducing the sensitivity of the experiment, buton the other hand is difficult to generate . The signature of an electric dipole moment (cid:126)d (aligned with thespin of the particles and, for the rotation indicated in the sketch parallel to the spin), is a rotation of thespin into the vertical direction.The basic equation, used for most of the consideration presented, is the "subtracted form of theThomas-BMT equation" and is the difference between the angular frequency (cid:126) Ω s of the spin rotation andan angular frequency (cid:126) Ω p of the rotation of the direction of movement of the particle. With the choice ofa vanishing longitudinal component of the angular frequency describing the rotation of the direction of Some proposals [5] foresee as well bunches polarized in radial direction circulating simultaneously with the bunchespolarized parallel or anti-parallel to the direction of movement. Such bunches allow to quantify and, possibly with appropriatefeedback systems, to reduce some systematic effects. , this quantity is given by: ∆ (cid:126) Ω = (cid:126) Ω s − (cid:126) Ω p = − qm (cid:34) G (cid:126)B ⊥ + ( G + 1) (cid:126)B (cid:107) γ − (cid:18) G − β γ (cid:19) (cid:126)β × (cid:126)Ec + η (cid:32) (cid:126)E ⊥ c + 1 γ (cid:126)E (cid:107) c + (cid:126)β × (cid:126)B (cid:33)(cid:35) (10.2)where q and m are the charge and mass of the particle, (cid:126)B and (cid:126)E the magnetic and electric field, β and γ the relativistic factors and (cid:126)β a vector with length β and a direction parallel to the velocity. (cid:126)B (cid:107) = (cid:16) (cid:126)β · (cid:126)B (cid:17) (cid:126)β/β and (cid:126)E (cid:107) = (cid:16) (cid:126)β · (cid:126)E (cid:17) (cid:126)β/β denote the longitudinal components in direction of the velocityof the magnetic and electric fields. (cid:126)B ⊥ = (cid:126)B − (cid:126)B (cid:107) and (cid:126)E ⊥ = (cid:126)E − (cid:126)E (cid:107) are the components of the magneticand electric field perpendicular to the direction of movement . The quantities G and η describe themagnetic dipole moment, which is in general well known and the electric dipole moment to be measured.For the case of protons G = 1 . . Note that for a proton EDM of d s = 10 −29 e · cm, which is oftenquoted as expected sensitivity of the proposed facility, η is as low as η s = 1.9 × −15 .In a fully electrostatic machine installed inside a perfect magnetic shielding to reach (cid:126)B = 0 andwithout EDM, a particle spin aligned with the direction of the movement will rotate together with theparticle velocity, i.e. fulfil the "frozen spin" condition, if: βγ = β m γ m = 1 √ G (10.3)where β m and γ m denote the "magic" relativistic factors. For protons β m = 0 . and γ m = 1 . .With the "magic" relativistic factors one obtains the magic proton momentum p m = β m cγ m m = 700 . MeV/c and the "magic" kinetic energy ( γ m − mc = 232 . MeV. Notethat real "magic" relativistic factors are obtained only for positive values of the quantity G , as is the casefor protons. Thus, a purely electrostatic ring fulfilling the "frozen spin" condition is not possible forparticles with negative G such as deuterons.An electric dipole moment described by a non-zero η generates a rotation of the spin from the lon-gitudinal direction into the vertical direction. In an electrostatic ring with circumference of C = 500 m,which is about the minimum required with the given beam energy and in order keep feasible maximumelectric field strength, the angular frequency for η = η s is about 1.6 nrad/s. This small vertical spinrotation has to be detected by precise polarimetry in order to identify the particle EDM.Additional ingredients to the "magic energy" proton EDM measurement concept are (i) to simul-taneously circulate polarized beams in both clockwise (CW) and counter-clockwise (CCW) direction,(ii) to operate the synchrotron with a very weak vertical tune Q V (proposals varying between Q V < . and up to Q V = 0 . with some variants even foreseeing to periodically vary Q V by say about ± %)and (iii) to use a measurement of the vertical separation of the two counter-rotating beams to estimatethe average radial magnetic field, which causes the most important systematic measurement error. Fur-thermore, the average horizontal spin will be continuously monitored with the polarimeters. A feedbackloop will be implemented to bring the measured horizontal spin back to zero. Note that this feedback The transverse component of the angular frequency describing the rotation of the particle direction (cid:126)t = (cid:126)v/ | (cid:126)v | , with (cid:126)v theparticle velocity, is given by (cid:126) Ω p, ⊥ = (cid:126)t × (cid:0) d(cid:126)t/dt (cid:1) . An arbitrary longitudinal component (cid:126) Ω p, (cid:107) = κ(cid:126)t can be added such that theangular frequency describing the rotation of the particle direction is given by (cid:126) Ω p = (cid:126)t × (cid:0) d(cid:126)t/dt (cid:1) + κ(cid:126)t . It is easy to show that d(cid:126)t/dt = (cid:126) Ω p × (cid:126)t for any value of the free parameter κ . Here (cid:126) Ω p, (cid:107) = 0 is assumed. Nevertheless, the longitudinal componentsof Ω p and of ∆Ω have to be interpreted with care and is the topic of discussions on-going at present.Considerations presented in this chapter implicitly assume a coordinate system attached to the trajectory (or rather the closedorbit) of the particle. The rotation of this coordinate system can be described by a unique angular frequency Ω p with anappropriate choice of a (small) longitudinal component. This is somewhat inconsistent with the choice (cid:126) Ω p, (cid:107) = 0 made here.Studies are on-going and will be publishes soon. Using these definitions for the longitudinal and perpendicular field components and (cid:126) Ω p =( q/γm ) (cid:16)(cid:16) (cid:126)β × (cid:126)E/c (cid:17) /β − (cid:126)B ⊥ (cid:17) , it is simple to show the this equation is consistent with the Thomas-BMT Eq. 1 inchapter . Any phenomenon other than an EDM of the particle generating a rotation of the spin into the verticaldirection generates signals, which can be misinterpreted and lead to systematic errors of the measure-ment. Effects considered so far and contributing either alone or in combination with other effects to sucha rotation are:– Magnetic fields: Even small magnetic fields around ∆ B = 1 nT penetrating state-of-art multilayershielding after degaussing procedures may lead to spin rotations into the vertical plane, which areorders of magnitude larger than the ones due to smallest EDM, one would like to be able to detect.In particular, an average radial magnetic field as low as B s = 9 . aT for C = 500 m circumferencemachine generates the same vertical spin component as an EDM of 10 −29 e · cm.– Imperfections of the electrostatic machine: Typical imperfections of electrostatic synchrotrons aremisalignments of bends and quadrupoles or mechanical imperfections of components (e.g. smallerrors of the spacing between electrodes of bends alter the electric field and as consequence thedeflection), which lead to deformations of the so-called closed orbit, i.e. the average transverseoffset of the circulating beam and, in consequence, to local shifts of the kinetic energy of theparticle directly impacting spin rotations as described in Eq. (10.2). Combination of several suchimperfections can lead to a a rotation of the spin from the longitudinal into the vertical direction.– Gravity: gravity leads to a spin rotation from the longitudinal direction into the vertical directionof 44 nrad/s for protons [6–9]. Nevertheless, the phenomenon does not mimic an EDM in the sensethat the spin rotations due to gravity correspond to an EDM of opposite sign for the CW and CCWbeam. This effect is unrelated to other sources of systematic effects and will not be treated anymore.– Average horizontal spin component: an average horizontal component of the spin, which may notbe seen by the polarimeter due to an asymmetry or even the result of a feedback loop aiming atrotating the spin in the horizontal plane into the longitudinal direction may lead to a generation ofvertical spin combination with vertical closed orbit perturbations.– Cavity misalignment and closed orbit perturbation (offset of the transverse beam position) at thelocation of the cavity: The azimuthal magnetic field of the cavity is a special case of magneticfields, which (i) creates strong effects already with small offsets between the beam position andthe center of the cavity and (ii) has a strong gradient.Phenomena possibly generating systematic measurements errors compromising the sensitivity ofthe experiments, which have not yet been studied in detail, are:– Betatron oscillations and different beam emittances of the two counter-rotating beams: studies96escribed here are for particles following the "closed orbit" and betatron oscillation are not yettaken into account.– Inhomogeneous beam distributions: a small vertical polarisation, which is different for particlesat the center of the bunch and particles executing large synchrotron and/or betatron oscillations,could be generated by the beam preparation process. If particles with large oscillation amplitudestend to be intercepted by the polarimeter earlier than particles from the center of the bunches, theaverage observed vertical spin changes over a measurement cycle even in the absence of an EDM.– Electromagnetic field generated by other particles in the same bunch or by particles of the beamrotating in opposite direction.– Electromagnetic field generated by the interaction of the circulating beams with the surroundingvacuum chambers (image currents etc.).For numerical evaluations, the C = 500 m circumference strong focusing lattice [10] will be used.This lattice has been optimised to obtain beam-lifetimes close to requirements with Intra Beam Scatter-ing (IBS) and foreseen intensities and, amongst all proposals, is the closest to a ring, which could beconstructed. Horizontal or "radial" magnetic fields are the only perturbation generating a rotation of the spin from thelongitudinal into the vertical direction, which is directly proportional to the perturbation . There are twomajor sources for magnetic fields not generated by the beam itself and acting on the beam, which haveas well different impacts on the measurement: residual static magnetic fields penetrating the shieldingand magnetic fields from the RF cavity. Typical residual magnetic fields inside state-of-the-art multi-layer shielding with degaussing proceduresare around ∆ B = 1 nT, which is about eight orders of magnitude larger than horizontal magnetic fields B s = 9.3 aT generating the same effect than the EDM sensitivity aimed for in typical proposals [1,2]. Theaverage of the radial magnetic field around the ring circumference will be somewhat smaller than ∆ B ,but still orders of magnitude larger than B s (The radial magnet field will vary strongly over a distancecomparable to the transverse size of the shielding, which is expected to be around 1 m.) Assumingoptimistically that the circumference C = 500 m can be divided into 500 sections with a length of 1 mwith about constant field and no correlation of the field between different sections, one comes to an RMSvalue of the transverse field of about ∆ B s / √ ≈ pT. Note that static average horizontal fieldscoupling to the known proton magnetic moment mimic an EDM in the sense that the contributions fromthe two counter-rotating beams do not cancel for the final result.An essential ingredient of the proton EDM measurement proposals in an electrostatic ring is tooperate the machine with weak vertical focusing, such that horizontal magnetic fields lead to a verticalseparation of the two counter-rotating beams, which is measured with ultra-sensitive SQUID based pick-ups. Note that for the strong focusing lattice proposal [2,10] with a vertical tune of Q V = 0.44, an averagehorizontal magnetic field B s leads to an average vertical separation of the two counter-rotating beamsof ∆ y s ≈ . pm . The measured beam separation will be compensated by additional magnetic fieldsgenerated by electrical currents inside the shielding as much as possible with the achievable measurement Disregarding gravity [6–9]mentioned already and well understood and not a concern for the EDM measurement. Other proposals foresee weaker vertical focusing and lower tunes [1]. An average horizontal magnetic field B s with avertical tune of Q V = 0 . would give a vertical separation of the beams of ∆ y s ≈ pm. However, with the foreseen intensitiesthey feature IBS growth rates not compatible with typical assumptions on the machine cycle length of around 1000 s; optimisinga machine with such a low vertical to obtain IBS rates compatible with expected cycle lengths and intensities leads to excessivevertical beam sizes. d s = 10 −29 e · cm. The following effects may compromise the sensitivity of the experiment:– Limited accuracy of orbit difference measurements even with averaging over many pick-ups andover long durations.– Observation of orbit difference only at discrete positions around the circumference: even withthe assumption that the focusing is perfectly constant around the circumference, the average of theorbit difference measured by a finite number of equally spaced pick-ups is in general slightly differ-ent from the average [10]. A rough estimate for the strong focusing ring proposed [1, 2, 10], wherethe pick-ups not perfectly spaced , leads to the conclusion that this effect limits the uncertainty ofthe final result to an EDM value about four orders of magnitude higher than d s ≈ −29 e · cm. Theeffect can be mitigated in theory by an optimized spacing of the orbit difference pick-ups and amodulation of the vertical tune [2, 11]. The feasibility of the latter implies that the working pointhas to regularly cross betatron resonances, which is delicate and may lead to unacceptable beamlosses.– Wanted and unwanted variations of the Twiss betatron functions around the circumference: Ingeneral, the transverse focusing is not homogeneous around the circumference. Even the "smoothfocusing" lattices feature field free straight sections without focusing at all. In consequence, theso-called Twiss betatron functions vary around the circumference. Thus, the effect of a local hori-zontal magnetic field on the average orbit separation, which depends on the local betatron function,will depend on the position. A rough estimate based on the strong focusing lattice proposal leadsto the conclusion that this effect limits the uncertainty of the final result to an EDM value aboutfive orders of magnitude higher than d s ≈ −29 e · cm. The effects can be mitigated by designinga lattice with small variations of the vertical betatron functions. Note that these considerationstriggered the proposal of a hybrid ring [3] with electric fields bending the beam and magneticquadrupoles for focusing. The situation is even more delicate for a realistic ring with "beta beat-ing", i.e. unwanted and unknown variations of the betatron function w.r.t. the lattice design dueto unknown focusing errors. Careful studies assuming realistic focusing errors and realistic proce-dures to quantify and correct the resulting betatron beating are required to assess the effect and theimplication on the achievable sensitivity.– Coupling of the betatron motion between the two transverse planes: Unavoidable skew quadrupo-lar components due to mechanical imperfections (for example rotation of quadrupoles around thelongitudinal axis, electrodes of bendings not being perfectly parallel ..) couple the betatron oscilla-tion in the two transverse planes. A horizontal separation between CW and the CCW beam due toresidual vertical magnetic fields at the location of such skew quadrupolar components will generatedifferent vertical deflections for the two counter-rotating beams. The resulting vertical separationbetween the two counter-rotating beams is misinterpreted as the signature of a horizontal "radial"magnetic field and leads to a systematic measurement error. Typical azimuthal magnetic field of RF cavities are orders of magnitude higher than the ones relevantfor a study of systematic errors of a proton EDM measurement. Even in case of perfectly aligned cav-ity, individual particles will "see" horizontal magnetic fields and spin rotation into the vertical (and thehorizontal) direction. However, the effect to the final result of the EDM measurement will be strongly This lattice with four-fold symmetry and, each quarter consisting out of 5 arc cell and one straight section cell, has 36 orbitdifference pick-ups adjacent to quadrupoles in arcs only. ∆ y = 100 µm. The integrated horizontal field seen by the CW beamin ring operated below transition due to a cavity operated with harmonic h and peak RF voltage V RF is ∆ Bdl = π β m c C h V RF ∆ y . Inserting parameters V RF = 6 kV and h = 100 for the strong focusingEDM ring proposal one obtains ∆ Bdl = 0 . nT m, which has to be compared with the integrated fieldaround the circumference B s C = 4 . fT m generating the same rotation of the spin into the verticaldirection than an EDM of d s = 10 −29 e · cm. Thus, an offset of ∆ y = 100 µm between the electricalcenter of the cavity and the vertical closed orbit leads to a rotation of the spin into the vertical directiona factor 1.6 × larger than the effect for a proton EDM d s . As the direction of the magnetic field isinverse for CW and CCW beams, the effect does not mimic EDM in the sense that contributions fromthe two counter-rotating beams to the final result cancel each other in a perfect measurement set-up.Nevertheless, this cancellation relies on a measurement of the vertical spin build up with high precisionfor both beams, which may be very challenging . Several cases of effects, where two different perturbations as e.g. residual vertical and longitudinalmagnetic fields penetrating the shielding generate a vertical spin component, will be described in thissection. These phenomena are second order effects in the sense that the resulting vertical spin for smallperturbations is proportional to the square of the perturbation (if both the vertical and the longitudinalmagnetic field in the example are increased by a factor k , the resulting vertical spin is increased by afactor k . All consideration reported apply strictly speaking for beam particles following the closed orbitof the ring and not executing any betatron oscillations.Several but not all of the effects described below have been reported and interpreted in terms ofgeometric phase effects. A geometrical interpretation of the effect rotating the spin from the horizontal into the vertical direction,which has been described in the past [12], is sketched in Fig. 10.2. If the "frozen spin" condition isfulfilled, the rotation of the spin and the direction of the trajectory are described by the same angularfrequency vector (cid:126)ω s = (cid:126)ω p , which is pointing downwards with a small longitudinal component ω s,s = β m cρ y (cid:48) co with y (cid:48) co = d y co ds the slope of the vertical orbit and ρ the curvature radius. This yields, even if the"frozen spin" condition is fulfilled, to a build up of the vertical spin of d s y dt = ω s,s s x = β m cρ y (cid:48) co s x . Thevertical spin generated over one turn given by: ∆ s y = C (cid:90) ds y (cid:48) co ( s ) ρ ( s ) s x ( s ) . (10.4) Another mitigation measure in case of imperfections of the polarity measurement to be discussed is a feedback loopdetecting spin rotations of the CW and CCW not compatible with EDM (not "mimicking" EDM) and correcting them forexample acting on the vertical closed orbit at the location of the RF cavity. Note that there are other effects described in thenext section generating as well spin rotations of the two counter-rotating beams which are not compatible with an EDM andwould be corrected by such a feedback loop. ig. 10.2: Mechanism rotating the spin from the horizontal into the vertical direction by a slope of the verticalclosed orbit inside a bend.
The average horizontal spin of the particles will be monitored continuously using the polarimeter and afeedback loop mentioned in section 10.2.1. A small asymmetry of the polarimeter may lead to a non-zero horizontal component of the spin ∆¯ s x . Spin rotations in the horizontal plane, which cancel overone turn, may as well lead to non-zero average horizontal spin even if the horizontal spin vanishes at thelocation of the polarimeter. With the help of Eq. (10.4) and using a nomenclature different from the onein reference [12] giving the same result, the average rate of change of the vertical spin becomes: d s y dt = 2 π β m cC < y (cid:48) co > s x (10.5)with N b and L b the number and length of bends and < y (cid:48) co > the average slope of the vertical closedorbit inside bends.An average of the horizontal spin ∆¯ s x = 0 . mrad and a vertical misalignment of one verticallydefocusing quadrupole at the transition from an arc to a straight section of "strong focusing" lattice( N b = 120 and L b = 2 . ) by ∆ y = 0 . mm leads to an average slope of < y (cid:48) co > = − . · − rad anda vertical spin build up of − µ rad/s. As the average horizontal spin of the two counter-rotating beamsmay not be correlated (independent polarimeter for CW and CCW beam with different uncorrelatedasymmetries), the systematic EDM measurement error due the effect considered here cannot be reducedby counter-rotating beams. However, even an imperfect polarimeter together with a feedback acting onbunches polarized parallel to the movement and opposite to the movement of the same (say CW) beamwould generate the same horizontal residual spin; the effect on the final EDM result from these buncheswith opposite polarization cancels. Note that residual horizontal magnetic fields will generate smallervertical orbit distortions and, thus, generate a smaller effect than typical misalignments of quadrupolesor tilts of electric bends.A thorough investigation of the effect requires an simulation of a machine with realistic imperfec-tions and a correction scheme based on (imperfect) position pick-ups and correctors.Another mitigation measure proposed for some of the schemes is to foresee bunches with hori-zontal polarization [5] in addition to the bunches with longitudinal polarization to measure and, possibly,correct a rotation from the horizontal into the vertical direction100 Fig. 10.3:
Misalignment in both transverse planes of two quadrupoles at opposite positions in the ring with oppositeoffset.
A case considered in the past [13] are simultaneous transverse offsets of electrostatic quadrupolesin vertical and horizontal direction. To better understand the mechanism generating a vertical spin com-ponent, a special case with two quadrupoles located at opposite positions in the ring and misaligned withtransverse offset in both transverse planes. The sign of the transverse offsets for the two quadrupoles areopposite. Such transverse offsets by ∆ x = ∆ y = ± . mm of two quadrupoles located in the centerof (opposite) straight section of the strong focusing ring proposal results in the closed orbits (1 st ordercontributions taken into account only) shown in Fig. 10.3. The energy of a particle following the closedorbit x co inside a bend is in general different from the magic energy due to the non-zero electric potential.This energy offset leads to a rotation of the spin around a vertical axis. The resulting small horizontalspin of a proton polarized parallel to its momentum circulating in CW direction is given by: s x = s (cid:90) s ds γ m x co ( s ) ρ ( s ) . (10.6)Using Eq. (10.4), the vertical spin obtained over one revolution is given by: ∆ s y = C (cid:90) ds y (cid:48) co ( s ) ρ ( s ) s x ( s ) . (10.7)The functions required to compute the resulting vertical spin build up are plotted in the lowerimage of Fig. 10.3. The result can be interpreted in terms of a geometric phase effect as it is the result oftwo rotation, one around a vertical axis and the other around a longitudinal axis, which are out of phase.For the case, based on the strong focusing lattice, described, one obtains an average build up of thevertical spin of ds y dt = −4.5 µrad/s. The effect to be expected in a realistic machine can only be estimatedby thorough simulations with realistic assumptions for misalignments of components and closed orbitcorrection. Note that this effect does not mimic an EDM in the sense that the contributions from the CW101nd the CCW beam to the final result cancel. Nevertheless, a fast rotation from the longitudinal into thevertical plane may be challenging for the polarimeter as the build-up of vertical spin has to be measuredwith very high relative precision. One may as well consider a feedback system to correct spin rotations(from this and other effects that do not mimic EDM) into the vertical plane not compatible with an EDMfor example by acting on the vertical position at the location of the RF cavity.Note that the effect can not be cured by using a "weak focusing" lattice as a horizontal offset androtations around the longitudinal axis result in the same phenomena, but in addition a more direct rotationfrom the longitudinal into the vertical direction, which will be described in the next section. Fig. 10.4:
Misalignment of two pairs of electric bends around the center of opposite arcs.
An electric bend with a horizontal offset ∆ x induces an electric potential at the location of thereference orbit and, in consequence, moves the kinetic energy of a beam particle from the "frozen spin"condition. An additional rotation of the same bend around the longitudinal axis by an angle α leads to avertical electric field component. The consequence is a non-zero radial component of ∆ (cid:126) Ω and a rotationof the spin from the longitudinal direction into the vertical direction, which differs from the rotation ofthe direction of movement. In reality, the situation is slightly more complicated as the misalignment ofthe bends affect as well the closed orbits in both planes such that (i) the closed orbit gives an additionalcontribution to the shift of the kinetic energy from the "frozen spin" condition and (ii) the effect describedin the previous section 10.2.4.3 has to be taken into account as well. For numerical evaluations weconsider again the strong focusing lattice with (for symmetry reasons) two electric bends on either sideof the center of opposite arcs misaligned by ∆ x = ± . mm and α = ± . mrad. The net rotation ofthe spin from the longitudinal into the vertical direction taking both effects into account is given by: s x ( s ) = s (cid:90) s d ˆ s γ m x co (ˆ s ) − ∆ x (ˆ s ) ρ (ˆ s )∆ s y = C (cid:90) ds γ m x co ( s ) − ∆ x ( s ) ρ ( s ) α ( s ) + C (cid:90) y (cid:48) co ( s ) ρ ( s ) s x ( s ) . (10.8)102ne finally obtains ds y dt = −5.45 µrad/s. Again, this effect does not mimic EDM as contributions from theCW and the CCW beam to the final result cancel. Fig. 10.5:
Vertical and longitudinal magnetic fields generating build-up of a vertical spin component.
Either a vertical magnetic and electric field or a horizontal magnetic and electric field lead to orbitdeformations in both transverse planes and, in turn, to a build-up of a vertical spin component in a waysimilar to the mechanism described in section 10.2.4.3. However, the cases presented here mimic EDMin the sense that the contributions from the CW and CCW to the final EDM value do not cancel.For the case with vertical electric and electric field components, the vertical magnetic field con-tribute as well to the rotation of the spin in the horizontal plane. Thus, stronger vertical spin build isexpected than for horizontal magnetic field components and this case is treated here. An integrated ver-tical magnetic field of ± ∆ B y = 1 nTm at the location of quadrupoles located in the center of oppositestraight sections in the strong focusing lattice is assumed. These quadrupoles are vertically misalignedby ± . mm. The resulting orbit distortions as well the spin rotation s x in the horizontal plane and thederivative of the vertical closed orbit are plotted in Fig. 10.5. The vertical spin generated over one turnis given by Eq. (10.4) and evaluates to ∆ s y = 8 . · − rad. The resulting vertical spin build-up ratefor this probably optimistic case is . nrad/s, which is almost a factor two larger than the one due to anEDM of d s = 10 −29 e · cm. The phenomena considered as potential limitations to reaching the target sensitivity of d s = 10 −29 e · cmare summarized in Tab. 10.2. Static (not due to cavity with a vertical offset) horizontal (radial) magneticfields are expected to be the main source of systematic errors and to limit the achievable sensitivity to avalue larger than this target. In addition, several second order effects, where two distinct imperfections ofthe real machine w.r.t. the perfect design case lead to a spin rotation from the longitudinal to the verticaldirection, have been considered. Higher order effects as well as betatron and synchrotron oscillationshave not been taken into account and are expected to give smaller contributions to systematic effects.For most second order phenomena, only simple special cases with sometimes optimistic assump-tions have been considered aiming at understanding the underlying mechanisms. This has to followedby more realistic studies with positioning errors of all elements, realistic orbit correction scenarios and103 able 10.2: Main systematic effects considered as possibly limitation of the achievable sensitivity
First order effects
Static horizontal magnetic field Mimics EDM (no cancellations between contributions from CWand CCW beam on final result), Effect due to typical magneticfields inside state-of-the-art shielding about eight orders of mag-nitude larger than effect due to smallest EDM to be detected; ex-pected to be the main limitation to achievable sensitivity evenwith orbit separation measurement to estimate (and correct) av-erage horizontal magnetic field.Horizontal magnetic field due tocavity offset Does not mimic EDM, but fast rotation of spin requiring highprecision polarimetry and/or feedback.Gravity Effect about factor 30 larger than the one due to d s = 10 − e.cm.Does not mimic EDM (no cancellations between contributionsfrom CW and CCW beam on final result) and, thus, not expectedto limit the sensitivity of the experiment. Second order effects
Horizontal spin and non-zero av-erage slope of vertical orbit in-side bends Depends on polarimeter properties of the two rings, contributionmimicking EDM and incompatibility with sensitivity goal d s =10 − e.cm likely. Mitigation by bunches polarized in forwardand backward direction? Mitigation by additional bunches withhorizontal polarization?Horizontal and vertical offsets ofelectric quadrupoles Does not mimic EDM, large effects expected, high precision po-larimeter and/or feedback required.Electric bends with simultane-ous horizontal offset and rota-tion around longitudinal axis Does not mimic EDM, large effects expected, high precision po-larimeter and/or feedback required.Static vertical and longitudinalmagnetic fields Does not mimic EDM, moderate effect probably not limiting sen-sitivity.Vertical magnetic field from cav-ity and static longitudinal mag-netic field Mimics EDM, effect small and not expected to limit sensitivityMagnetic and electric fields gen-erating orbit deformations inboth planes Mimics EDM, worst case with vertical magnetic and verticalelectric field probably rules out to reach the sensitivity goal of d s = 10 − e.cm.cross-checked with simulations. Some effects do not mimic EDM in the sense that the contributions fromthe CW and CCW beam on the final results cancel. Nevertheless, there are cases leading to a vertical spinbuild-up several orders of magnitude faster than the smallest EDM to be measured. This requires eitherto measure the vertical polarization with high accuracy or to implement a feedback system reducing theeffect (could be based on any of the effects generating such a spin rotation as for example bends withhorizontal offsets and rotations around the longitudinal axis).The optimum operational scenarios depend on the main source for systematic errors of the ex-periments. In case, the main contribution comes from the average horizontal magnetic field (after theimplementation of mitigation measures), operation with simultaneous CW and CCW beams is impor-tant, but the filling patterns of the two rings is not critical. If second order effects generate a significantcontribution to systematic effects, the filling pattern of the two beams becomes important. The optimumoperational scenario to control systematic effects would be to have both the CW and the CCW beam withpart of the bunches polarized in forward direction, part of the bunches polarized in backward direction104nd some of the bunches with horizontal polarization. However, a filling scenario to generate such abunch pattern is not available. References [1] V. Anastassopoulos et. al., A Proposal to Measure the Proton Electric Dipole Momentwith 10 −29 e · cm Sensitivity, .[2] V. Anastassopoulos et. al., A Storage Ring Experiment to Detect a Proton Electric Dipole Moment,Rev. Sci. Instrum. 87, 115116 (2016).[3] S. Haciomeroglu and Y.K. Semertzidis, A hybrid ring design in the storage-ring proton electricdipole moment experiment, arXiv:1806.09319.[4] R. Talman, A doubly-magic storage ring EDM measurement method, arXiv:1812.05949.[5] Y.K. Semertzidis, people from the Jülich team, private communication.[6] Y. Orlov, E. Flanagan and Y. Semertzidis, Spin rotation by Earth’s gravitational field in a "frozen-spin" ring, Phys. Lett. A 376 (2012) 2822-2829.[7] Y. N. Obukhov, A. J. Silenko, and O. V. Teryaev, Manifestations of the rotation and gravity of theEarth in high-energy physics experiments, Phys. Rev. D94 044019 (2016).[8] A. J. Silenko and O. V. Teryaev, Equivalence principle and experimental tests of gravitational spineffects, Phys. Rev. D76, 061101 (2007).[9] A. Laszlo and Z. Zimboras, Quantification of GR effects in muon g-2, EDM and other spin preces-sion experiments, Class. Quantum Grav. 35 (2018) 175003.[10] V. Lebedev, Accelerator Physics Limitations on an EDM ring Design - Comments to the BNLproposal from 2011, http://collaborations.fz-juelich.de/ikp/jedi/public_files/usual_event/AccPhysLimitationsOnEDMring.pdf [11] Y. Semertzidis, Storage Ring EDM Experiments, EPJ Web of Conferences 118, 01032 (2016).[12] S. Haciomeroglu, Quadrupole Misplacement Studies for the pEDM Rings, presentation at theCPEDM Meeting in Jülich on 8th and 9th March 2018.[13] S. Haciomeroglu and Y. K. Semertzidis, Systematic Errors related to Quadrupole Misplacement inan all-electric Storage Ring for Proton EDM Experiment, arXiv 1709.01208.[14] S. Haciomeroglu, Y. Orlov and Y. Semertzidis, Magnetic field Effects on the Proton EDM in acontinuous all-electric Storage Ring, arXiv:1812.02381.105 hapter 11Polarimetry Introduction to Polarimetry
The quantum nature of the nucleon or nucleus requires that any electric dipole moment (EDM) be alignedwith the spin axis. Thus the experimental connection with the EDM is found through the preparationof beams with spin polarisation and the measurement of small spin polarisation changes that may beinterpreted as evidence for interactions that are signatures of an EDM. Fortunately, polarised beamsand the measurement of nuclear spin polarisation through strong interaction processes are both maturetechnologies and capable of high precision. It is also fortunate that polarimeters for spin polarisationmeasurements are at their most sensitive in the range of beam energies where storage ring technologyfor spin manipulation also works well. This chapter describes the polarisation measurements planned forthe EDM search in detail.The chapter begins with a short review of polarisation terminology as codified in the MadisonConvention for protons (spin-1/2) and deuterons (spin-1). It then moves to a summary of the polarisationmeasurements of the beam as it proceeds through the preparation and acceleration process. The rest ofthe chapter deals with the polarimetry planned for the EDM storage ring itself, showing new technologydeveloped for the calorimeter detectors and arrangements for making polarisation measurements withhigh efficiency and precision. Much of this chapter is devoted to ways to handle systematic error prob-lems and limits on the use of counter-rotating beams for identifying the time-reversal violating EDM.
Polarimeter Spin Formalism
The plan for an EDM-sensitive polarisation measurement is to record the horizontal asymmetry in thescattering of protons or deuterons from a carbon target at forward angles. At the energies where the EDMsearch would be made, the interaction between the polarised particles and the carbon nucleus contains alarge spin-orbit term. This gives rise in elastic scattering to an asymmetry between left and right-goingparticles when there is a vertical polarisation component present.For spin-1/2, the polarisation along any given axis is given in terms of the fraction of the particlesin the ensemble whose spins, through some experiment, are shown to lie either parallel or anti-parallel tothat axis. If these fractions are f + and f − for the two projections of the proton’s spin-1/2, the polarisationbecomes p = f + − f − which ranges between 1 and − with f + + f − = 1 . The scattering cross section σ P OL may be written in terms of the unpolarised cross section as σ P OL ( θ ) = σ UNP OL ( θ )[1 + pA Y ( θ ) cos ( φ ) sin ( β )] (11.1)with the vertical polarisation component p Y = p cos ( φ ) sin ( β ) (11.2)and where the angles are defined with respect to a coordinate system shown in Fig. 11.1 below.The left-right asymmetry measures the vertical polarisation component p Y . The size of the signalis governed by the strength of the spin-orbit interaction, which gives rise to the asymmetry scaling coef-ficient A Y ( θ ) , otherwise known as the analysing power. The left-right asymmetry arises from the cos( φ )dependence of the cross section on the azimuthal angle of the polarisation. If two identical detectors areplaced symmetrically about the z axis and their rates are L and R , then the asymmetry is given by106 ig. 11.1: The coordinate system for polarisation experiments where the beam defines the z axis and the detectorestablishes both the xz plane and the positive x direction. The scattering angle is given by θ . The polar angle β and azimuthal angle φ define the orientation of the positive polarisation direction. (cid:15) = pA Y ( θ ) = L − RL + R (11.3)In the case of the deuteron, which is spin-1, there are three fractions that describe the magneticsub-state population, f + , f , and f − and f + + f + f = 1 . The two polarisations are vector, p V = f + − f − ,and tensor p T = 1 − f which can range from 1 to − . If we are interested only in the EDM, then thevector polarisation suffices as a marker and the deuteron polarised cross section (Cartesian coordinates)becomes σ POL ( θ ) = σ UNPOL ( θ ) (cid:20) p V A Y ( θ ) cos ( φ ) sin ( β ) (cid:21) (11.4)Tensor polarisation is usually present to a small degree in polarised deuteron beams. There arethree independent tensor analysing powers that each add another “ p T A ” term to the equation above.Their effects may prove useful in polarisation monitoring or checking for systematic errors. Beam Preparation
The essential spin-related parts of the EDM storage ring injector beam line are shown in Fig. 11.2. Thesecomponents are site-independent in the description below. The diagram contains a polarised protonsource with its associated low energy polarimeter, spin rotation and proton acceleration equipment, atrip through the a storage ring for a phase space reduction through electron cooling and bunching, andsuitably located polarimeters that confirm that all this works and that calibrate the polarisation of theproton beam. This section summarises the polarisation features.High intensity, pulsed polarised proton sources suitable for use on colliders have reached a maturestate with adequate beam ( /pulse) for a storage ring EDM search [1]. The version currently inoperation at RHIC may be considered as a model. Brookhaven has a crew of a few members whose jobis to maintain, operate, and improve the ion source.The source creates a high-brightness proton beam in a high-efficiency extraction system before itis neutralised in hydrogen gas to make a well-collimated atomic beam. From there it converges into apulsed He ionizer that makes a low-emittance proton beam within a strong axial magnetic field. Inside107 O L A R I Z E D I O N S O U R C E A T O M I C B E A M P O L A R I M E T E R SPINROTATORE-field
PRE-ACCELERATION Polarimeter(~60 MeV)
STORAGERING(e-cooling)
Polarimeter(232.8 MeV)
Fig. 11.2:
Block diagram showing the main components of the injector beam line that are related to spin manipu-lation and measurement. the high-field region, polarised electrons are added to the protons from an optically pumped rubidiumvapour. The neutral atoms proceed to a Sona transition that transfers the electron polarisation to theprotons [2]. The atoms are given an additional negative charge in a sodium vapour and extracted at 35keV to form a beam for subsequent acceleration. Either state of polarisation along the magnetic fieldaxis is possible. The polarisation is in excess of 80%. As an alternative to ionisation, an atomic beampolarimeter is present that is capable of measuring the atomic polarisation. This allows tuning of thesource parameters without requiring acceleration of the beam to higher energy.For transport through the storage ring, the polarisation direction must be perpendicular to thering plane (aligned with the ring magnetic fields). To accomplish this prior to acceleration, electrostaticplates bend the proton beam without spin precession. This produces a sideways polarisation that passesthrough a solenoid where it rotates into the vertical direction. Initial acceleration is then provided by alinear accelerator such as an radio-frequency quadrupole or a drift tube linac. Once the protons reachan energy where nuclear scattering can yield high spin sensitivities, a carbon-target polarimeter becomesfeasible. One should be installed along the beam line so as to verify that the ionisation, spin rotation,and first acceleration have not altered the polarisation. Measurements of the proton-carbon analysingpower [3], shown in Fig. 11.3 over a range of angles and energies between 60 and 70 MeV, indicate largevalues near one at angles less than 60 ◦ that are practical for mounting monitor detectors (usually plasticscintillators). This involves the construction of a small scattering chamber. The target may be a thin foilof carbon mounted on a movable ladder.The passage of the beam through the storage ring is critical for two reasons. First, the in-planepolarisation in the EDM storage ring has a polarisation lifetime that is improved if the phase space of thebeam is made as small as possible [4]. This may be achieved while also using this ring for the secondpurpose, to accelerate the proton beam to the energy of 232.8 MeV before injection into the final storagering. During the acceleration process, the proton energy will pass through the Gγ = 2 imperfectionresonance. This resonance, which is driven by magnetic field errors in the ring, is often strong enough todepolarise the beam. Usually, the remedy is to make the resonance even stronger by introducing brieflyan imperfection in the form of a vertical steering bump that causes the polarisation to completely flip signas it passes through the resonance. A crucial step in the setup of the beam is to tune the bump so that themaximum polarisation survives acceleration. For this, another polarimeter is needed just past extraction108 ig. 11.3: Contour map of the analysing power for proton scattering from carbon in the energy range from 20 to84 MeV [3]. The ridge in the 40 ◦ to 60 ◦ range is particularly well suited for monitoring beam polarisation. from the storage ring. Again, scattering from carbon, shown here in Fig. 11.4 at 250 MeV [5], providesscattering angles with very large analysing powers that may, in fact, be used as the polarisation standardfor this experiment. A simple foil target and scintillation detectors are again appropriate. Fig. 11.4:
Measurements of the angular distribution of the analysing power for proton scattering from carbon ata beam energy of 250 MeV [4]. Note that both the first and second interference peaks show very large analysingpower values. The second peak is close enough to one that it may serve as a calibration standard for the subsequentuse of the beam.
There need to be two paths by which the injection of the beam is made into the EDM ring in orderto have both clockwise (CW) and counter-clockwise (CCW) beam circulating during the measurement.The switch between these two paths should be relatively rapid so that the requirement that the two beamsbe as identical as possible is easier to The critical elements are shown in Fig. 11.5.109
Bendingmagnets
EDMRING40.22°
Fig. 11.5:
Diagram of elements essential for spin handling during injection. Beams must be injected into the ring inboth directions in reasonably rapid succession. It should be possible to have polarisation along any direction. Thepolarisation begins perpendicular to the ring plane. A solenoid (up to 2 T · m strength and likely superconducting)is capable of rotating the polarisation by at least 90 ◦ in either direction. This is followed for the beam line to eitherside of the EDM ring by a bending magnet whose angle is 40.22 ◦ . This makes the plane of the polarisation parallelto the beam direction so that the changes are forward and back. A second solenoid rotates this polarisation intothe horizontal plane. Some time will be needed for ramping these solenoids if it is desired to have a variety ofdirections within one beam store. It is assumed that a complete spin flip will be made at the ion source. Main Ring Polarimeter Design Goals
The goal for the EDM search makes certain requirements on the polarimeter system, including bothtarget and detectors along with the associated data acquisition system. • The system must make efficient use of the beam particles. A polarisation sensitivity at the level ofone part per million requires capturing usable polarimeter events, a process that may requiremany months of data taking. For this, we have explored the use of thick targets located at the edgeof the beam at COSY. Particles that enter the front face will be lost from the beam, but the thickness(17 mm in tests) enhances the probability of scattering into one of the polarimeter detectors. Thegoal would be to achieve an efficiency near 1% . Eventually, most of the beam is used up hittingthis target. • At the same time the analysing power A Y should be as large as possible. Values in excess of 0.5are available for optimal choices of the detector acceptance for either protons or deuterons. • The method of choosing which events to include in the data set should be relatively insensitiveto the choice of cuts so that small changes have a minimal effect on the measured asymmetry. Inthe case of deuterons, it may be important to insert a range absorber ahead of the trigger detectorso that most of the breakup proton flux is removed before being processed in the data acquisitionsystem. Data acquisition firmware that digitises pulse shape and has a high throughput may makethis requirement less stringent. To ensure a proper early-to-late asymmetry difference, the triggerthreshold must be stable over time. • In the EDM search, the left-right asymmetry carries the information on the EDM. At the same timea monitor is needed for the magnitude of the beam polarisation. Such a measurement requires thatwe rotate the polarisation periodically from its frozen spin orientation into the sideways direction.Alternatively, some bunches may be loaded into the EDM ring with a sideways orientation. In this110onfiguration, the polarisation is measured through the down-up asymmetry in the scattering. Thusa full azimuthal acceptance is needed in the polarimeter detectors. If these detectors are segmented,some elements near 45 ◦ lines may be used for both left-right and down-up measurements, thusincreasing the useful efficiency of the device. • For the counter-rotating beams (CW and CCW, see below), one block target may serve for themeasurement of polarisation for both beams. This implies that a set of detectors be located in bothdirections from the target. Backscatter from the target is not expected to be a problem, even witha − sensitivity requirement on the measured asymmetry. Proton-carbon elastic scattering dataindicates such cross sections are down by eight orders of magnitude [5]. • Extraction of beam onto the block target at COSY has usually been achieved by heating the beamto enlarge the phase space in the plane where the target is located. Horizontal and vertical heatingmay be operated independently, creating the opportunity for two independent polarimeter locationson the EDM ring. More than one polarimeter is useful as a check against systematic errors. • Studies undertaken in 2008-2009 demonstrated that the sensitivity of the polarimeter to systematicerrors (rate and geometry changes) may be calibrated. With the use of positive and negative polar-isation states, such a calibration can be used to remove the effects of the systematic errors. Such atechnique thus becomes an important requirement.
Implementation of the Polarimeter
While in principle, the polarisation may be deduced from an absolute measurement of the cross sectionalone, experience with polarisation measurements strongly favours the use of both a left and right detectorsimultaneously. In addition, polarised ion sources can provide beam in either a positive or a negativepolarisation state, and the use of both states for the experiment is recommended. In the EDM storagering, the beam injection plan is to fill the ring with both CW and CCW beams, allow them to come toequilibrium in a coasting state without bunching, then impose bunching. The beam is vertically polarisedand both CW and CCW beams are filled using a single polarisation state from the ion source. Oncebunched into the final pattern, an RF solenoid with multiple harmonics of the bunched beam frequencywill be used to precess the bunch polarisations into the ring plane with alternate bunches polarised inopposite directions. The higher harmonic portions of the RF solenoidal field will be optimised so thatall parts of each bunch are polarised in the same direction following the rotation. Once in plane, theorientation of the polarisation in the bunches is maintained using feedback. The feedback system alsorotates the polarisations so that the spin alignment axis is parallel to the beam velocity, thus creatingthe frozen spin condition. The orientation is then maintained by nulling the down-up asymmetry from acontinuous polarisation measurement.For the rotation of the polarisation into the ring plane, the solenoid must carry a complex waveformwith several harmonics. The goal is to have the amount of rotation, given by the solenoid strength, be thesame across the length of each bunch. Outside of the bunch, the solenoid is not constrained. Figure 11.6below shows fits with 3 and 4 harmonics that have been adjusted to best reproduce a level equal to oneover half of the time ( − π/ to π/ ) as indicated by the faint red line. The lower plot is an enlargementof this critical region. Variations are less than 3% for 3 harmonics and less than 0.5% for 4. Theseries converges rapidly. Particles within the bunch will sample various areas of this plot as they undergosynchrotron oscillations of different amplitudes, so the size of the differences with unity tell us somethingabout the residual vertical polarisation in the beam at the end of this process. The expansion used herecontains only odd harmonics and in this way provides two flat-tops. This rotates alternate bunches inopposite directions into the ring plane. So with an even number of bunches we obtain a condition withhalf of the bunches having one polarisation and the other half with the opposite. Thus the RF solenoidruns on a harmonic that is half the bunch number. (Note that in this scheme each bunch sees the sameincremental rotation every time it passes the solenoid because the field is the same. This works becauseat the frozen spin condition with a zero spin tune, there is no rotation of the in-plane component of the111olarisation as the bunch goes around the ring. It is clearly important to inject the beam at the propermomentum.) Fig. 11.6:
Two curves showing the ability to achieve a flat function over an interval where the beam pulse existsand whose polarisation must be rotated uniformly into the ring plane. The upper panel shows the whole curve, thelower panel shows an enlargement of the top of the curve. Inclusion up to harmonics 3 and 4 yields larger andsmaller variations.
If the polarisation states are + and − , and the detectors L and R, then the EDM asymmetry, whichis the product of the polarisation and the analysing power, may be given by the “cross ratio” formula (cid:15) = r − r + 1 , where r = σ L + σ R − σ L − σ R + (11.5)This formula has the advantage that it cancels to first order common errors that depend on differ-ences in the acceptance of the left and right detector systems and differences in the integrated luminositiesfor the plus and minus polarisation states. At the precision required for an EDM search, higher ordererrors still affect this formula for the asymmetry and must be removed, as will be discussed later.Detectors above and below the beam (“up” and “down”) are sensitive the horizontal x -componentof the polarisation. With frozen spin, this should vanish. Thus any non-zero value implies that thematch between the polarisation and velocity rotation rates is not perfect. As had been demonstrated atCOSY [6], such information may be fed back to a suitably sensitive adjustment, such as the rf cavityfrequency, to correct the misalignment. This needs to be done continuously during the EDM experiment.In addition, at regular intervals the polarisation should be rotated into the sideways direction (or allowedto precess through a full circle) to provide a monitor of the polarisation magnitude. The accumulation ofthe EDM signal goes as the time integral of this magnitude.For tensor polarised deuterons, it is possible to utilise a comparison between the scattering rates indifferent polar angle ranges as a beam polarisation monitor, if the tensor polarisation is made intentionallylarge. This eliminates the necessity for periodic rotations of the polarisation away from the direction ofthe beam. However, a tensor polarisation may also generate a left-right (EDM-like) asymmetry if thereis a misalignment between the polarisation axis and the velocity. This systematic error will be discussedlater. 112he necessity to monitor continuously both the vertical and horizontal (sideways) polarisationcomponents during the experiment places a premium on polarimeter efficiency, the fraction of particlesthat scatter usefully into the detectors divided by the number that are removed from the beam. A sensi-tivity requirement for the EDM asymmetry of − implies recorded and useful events, which maynecessitate as much as a year of running time. Polarimeters used in double scattering experiments [7, 8]show that for proton and deuteron beams of a few hundred MeV, efficiencies of 1-3% have already beenachieved using thick (few centimetre of solid material) targets. This makes these energies an ideal choicefor the EDM ring. (Higher energies imply larger storage rings and additional construction costs.) Choice of analysing reaction
Work with highly-efficient double-scattering polarimeters at these energies has concentrated on the elas-tic scattering channel at forward angles ( ◦ to ◦ ) as the best choice for polarimetry. Essentially allpolarimeters have employed carbon as the target material. The angular distribution represents an in-terference pattern created by scattering from opposite sides of the nucleus. This angle range typicallyencompasses one full analysing power oscillation for deuterons and half of an oscillation for protons.At angles less than ◦ , Coulomb scattering takes over from the nuclear and the analysing power quicklygoes to zero. This region should be avoided. Fig. 11.7:
Measurements of the cross section, analysing power, and special figure of merit for proton elasticscattering on C at 250 MeV bombarding energy. The top panel shows the differential cross section and the middlepanel the analysing power. In the bottom panel is the figure of merit calculated using
F oM = σ ( θ ) A Y ( θ ) sin ( θ ) In Fig. 11.7 is the cross section and analysing power angular distribution for proton scatteringfrom carbon at 250 MeV beam energy [5]. The Coulomb-nuclear interference region lies inside ◦ . Be-yond this angle, the cross section arises almost exclusively from nuclear scattering. The elastic scatteringchannel shown here dominates all other reactions inside about ◦ . With a positive spin-orbit interactionand an attractive nuclear field at the surface of the nucleus, there is a strong sensitivity to the polari-sation of the incident proton that results in a positive analysing power. Both the cross section and theanalysing power show an oscillation pattern that reflects interference from opposite sides of the nucleus.113he relative merits of various parts of the angular distribution for polarimetry purposes is usually eval-uated through the use of a figure of merit: F oM = σ A . This is shown in the bottom panel with afactor of sin ( θ ) included to adjust for the falling solid angle near ◦ . This leads to a clear peak in the F oM . Beyond about ◦ the analysing power is falling through zero and little additional information isavailable. A similar peak characterises deuteron scattering, but it is a few degrees more narrow.Because the forces leading to the large positive analysing power are a property of the nuclearsurface and have nothing directly to do with any reactions that might take place (so long as the energytransfer is much smaller than the beam energy), similar features exist also for a large number of otherdirect reaction channels. So there is no particular requirement that the detector be capable of resolvingthe elastic scattering group exclusively, which requires high resolution in the measurement of the elasticscattering (or other charged particle) energy. This simplifies polarimeter design. The critical feature thenbecomes the choice of an acceptance that maximises the figure of merit, and how stable this acceptance(and trigger threshold) is over time. Fig. 11.8:
A collection of operating point analysing powers for proton-carbon polarimeters at intermediate energies[9]. The curve is a guide to the eye. The red line marks the magic energy of the EDM search.
Efficient double-scattering polarimeters for protons have been built and used successfully between100 and 800 MeV with a carbon target and simple polarimeter detectors consisting of thin plastic scin-tillators [9]. McNaughton’s summary plot shown below in Fig. 11.8 illustrates that the best analysingpower falls almost exactly at 232.8 MeV where the proton EDM experiment may run with an all-electricring. Most proton polarimeters have used carbon as the target because of its ease of handling, wide inter-ference pattern period, and large forward cross section. All targets in this mass region of the period tabletend to give similar results. The all-electric frozen spin energy is marked in red.These considerations lead to a simple conceptual design for the EDM polarimeter that is shownschematically here in Fig. 11.9.The main detectors consist of an energy loss detector that identifies the particle followed by atotal energy calorimeter. In many cases, proton polarimeters have used only the dE/dx detector. Forthe deuteron, the main background will be protons from deuteron breakup. These protons have almostno sensitivity to beam polarisation, and every effort should be made to eliminate them from the triggerrather than relying on post-detector processing. Various groups [7, 8] have successfully employed ironabsorbers ahead of the scintillator system. If the absorber is appropriately designed, the event triggerfrom the scintillators may be optimised for large figure of merit and small sensitivity to scintillator gaindrifts. In order to make more precise models of any EDM polarimeter, data base runs have been made at114 ig. 11.9:
Schematic layout showing the important components of an EDM polarimeter. The beam goes from rightto left, and passes through a thick carbon target. Scattered particles first encounter a tracking detector that tracesrays back to the target. Next an absorber removes unwanted events. Lastly, a ∆ E and E detector pair identify theenergy of the particles of interest along with the particle type.
COSY for deuterons at a variety of energies between 170 and 380 MeV. A similar run for protons wascompleted in the fall of 2018. The analysis of these data is in progress. Figure 11.10 shows the resultsof a deuteron data base run at the KVI Groningen at 110 MeV.The upper left panel shows a 2-D representation of the events recorded at ◦ . Clear bans forprotons, deuterons, and tritons appear. The coloured regions indicate places to include in the polarimetry(green) and places to avoid (magenta). The proton band shows a large contributions from deuteronbreakup that has almost no spin dependence. On the right are two panels for deuterons and protonsindividually. The regions are marked there as well. The proton distribution from breakup is large. Thisshould be mostly eliminated if absorber material were installed ahead of the detector system. Target operation in a storage ring
Before our COSY investigation, there was no information on highly efficient polarimeter operation in astorage ring. So we undertook tests to see if a thick target could be operated in this environment whilestill allowing the beam to circulate. What worked was placing a square-cornered block about 3 mm fromthe beam centre line. Various schemes were tested to bring the beam to the target slowly, extractingthe beam over an extended period of time. We found that it is better to move the beam than the target,since the beam moves smoothly. A steering bump changes the beam path length, creating a problemfor maintaining the spin tune (needed for frozen spin). So most of the COSY runs have made use ofwhite noise heating applied through a set of strip-line plates that enlarges the beam through phase spacegrowth. The white noise is applied over a narrow frequency range around one of the betatron oscillationharmonics, and this couples well to the beam.Extraction of the beam using white noise appears to be a two-step process, based on tilted beamstudies [10]. The mean distance from the surface closest to the beam near the point where the particlespenetrate the leading face of the target block is about 0.2 mm. This is much larger than the change perturn in the position of the beam. The first step in the extraction process is an encounter with a microscopicridge on the close face of the target block. This induces a betatron oscillation in the particle, which often115 ig. 11.10:
Panels showing as sample from a broad range data base run taken at the KVI-Groningen. The deuteronenergy was 110 MeV. Scattering from a carbon target was observed at ◦ . Particle type is distinguished in the ∆ E by E upper-left panel. Energy for particles emerging as protons or deuterons are shown in the two right-sidepanels. Areas outlined in green have significant analysing power and could be used for polarimetry. Areas markedin purple have a low spin sensitivity and should be avoided. survives to continue around the storage ring. On some subsequent pass, that oscillation takes it far enoughfrom the beam centre that it impacts the front face of the block. With a 0.2 mm typical distance from theleading corner to the point of impact, many of the perturbed particles penetrate the entire carbon blockand therefore have a maximal probability of undergoing a scattering event. This model is confirmedby the observation that the efficiency of the COSY carbon block target is consistent with Monte Carlocalculations that assume a full interaction with the target [10].The main disadvantage of this target arrangement is that it favours particles that are in the haloof the beam. Below deuterons/fill at COSY, there appears to be no issue that is associated with thisas the polarisation lifetime measurements show a smooth depolarisation curve, as expected. At higherbeam currents, structures appear in the time dependence of the polarisation that indicate more complexhistories in bringing the beam to the target. Modelling of the time dependence confirms this.Carbon block target thicknesses at COSY were typically 17 mm with a density of approximately1.7 g/cm . As the target is made thicker, the energy loss of particles in the target increases. Modellingof the response must therefore consider changes in the cross section and analysing power angular distri-butions with changing particle energy. These changes, plus considerations of beam alignment, probablyrestrict carbon block thicknesses to less than 5 cm. This thickness is enough, however, to achieve effi-ciencies on the order of 1%. Development of calorimeter detectors for an EDM polarimeter
For some time, Irakli Keshelashvili and his group at COSY have been utilising dense LYSO crystals asthe EDM polarimeter’s calorimeter detector. They have developed × cm modules that include asilicon photo-multiplier as a readout (see Fig. 11.11). The resolution for stopped 270-MeV deuterons is116pproximately 1%. Fig. 11.11:
A sample LYSO crystal showing the parts with labels.
These modules will ultimately go into a larger volume array (see Fig. 11.12) that surrounds thebeamline.
OpenDegrader ClosedDegrader2D tracking + dE Pl. Scintillators LYSOModules
Fig. 11.12:
Model view of the detector system inside the LYSO-based polarimeter. The scattered beam expandingfrom the target is shown in red. The LYSO-crystal calorimeter detectors appear in light blue in the segmentedarrangement likely to be used for left-right and up-down asymmetry measurements. Just in front of the calorimeteris an outline sketch of two layers of triangular-shaped scintillation detectors. SiPM light collectors located on theends of the bars are not shown. All particle tracks penetrate both vertical and horizontal layers. Energy sharingbetween neighbouring scintillators allows for a more precise position determination. The shutter assembly in frontallows for an absorbing layer to be imposed in front of the detectors to remove unpolarised background events. Amechanical system will open and close the shutter leaves.
A 48-module mock-up has been tested and will then be moved to the ANKE target area at COSYfor installation on the beam line and further testing at the beginning of 2019. A block target will be apart of the installation.Initial tests of the LYSO modules have been made at 93, 196, 231, and 267 MeV with a deuteronbeam in the external beam “Big Karl” area at COSY. An overlay of a preliminary series of spectrais shown in Fig. 11.13. The energies listed in the figure take into account losses from windows andupstream trigger detectors. The four energies were measured in different setups and relative gains havenot been reconciled. 117 ig. 11.13:
Spectra showing the energy deposited in the LYSO crystal for deuteron beams of different energies, asindicated in the box insert.
Use of the polarimeter to maintain frozen spin
For EDM data runs of about 1000 s, the precision with which the polarisation must be maintained parallelto the proton velocity is about one part in over the course of a beam store. Prior to storing the beam,the value of the spin tune cannot be known to this level of precision, so a feedback mechanism mustbe used to maintain alignment. One such mechanism was tested at COSY [6]. The analysis of thepolarimeter data for in-plane polarisation yields a magnitude and phase for each time interval (1-4 s induration, for example). A scheme was developed to provide very precise changes to the frequency of theRF cavity controlling the beam based on a running analysis of the polarimeter data as it was acquired.Figure 11.14 shows an initial situation in which the spin tune is not matched to the rotation of thebeam. This result is a slope with time for the phase data. At two times, a signal and its opposite weresent to the RF signal generator requesting a change in frequency. This was immediately reflected in achange in the slope of the phase, which is a measure of the spin tune relative to an assumed value. time (s) ph a s e ( r a d ) Fig. 11.14:
Phase of the rotating in-plane polarisation relative to a standard clock reference as a function of time inthe store. Measurements were made by observing the oscillating down-up asymmetry, or sideways polarisation, asa function of time. A slope in the line indicates that the spin tune frequency is not matched to the reference clock.A small change, and then a change back, in the beam revolution frequency changes the spin tune frequency andhence the slope of the phase with time.
In another test, shown in Fig. 11.15, a change was sent to the RF generator and then quickly118eversed, so that the spin tune remained the same after as before. But the pulse caused the phase itself toshift. With the changes calibrated, the figure shows steps of about 1 rad resulting from a series of suchpulses sent to the RF generator. ph a s e ( r a d ) time (s) Fig. 11.15:
Phase of the rotating in-plane polarisation relative to a standard clock reference as a function of timein the store. Periodically a pulse is sent to the COSY rf generator that makes a step in the revolution frequencyand then quickly reverses it. This pulse makes a step in the phase. Once calibrated, these steps can be tuned to beabout 1 rad.
Figure 11.16 is an example of how this might look in a realistic situation. The upper curve isthe corrected phase and the lower curve shows the time sequences of changes made to the the rf cavityfrequency in order to maintain that level of phase reproducibility. The average deviation, indicated bythe grey band, is ± . rad. This level of control is adequate for the EDM experiment. Fig. 11.16: (top) Measurements of the phase of the rotating in-plane polarisation with reference to an externalclock as a function of time. These measurements are being used to correct the COSY revolution frequency in realtime so that the phase remains stable at zero (arbitrarily chosen), beginning at 89 s. The grey band indicates theRMS deviations in the phase. (bottom) Depiction of the actual changes generated by the feedback system and sentto the RF frequency generator as a function of time.
This technology is essential for maintaining the frozen spin condition needed to observe an EDM.In must be in place and operating as soon as the beam is injected into the main storage ring.119
Correction of rate and geometry errors in the polarimeter
A cross-ratio analysis of data from a polarimeter, as described in the requirements section above, cancelsmost first-order errors. In a storage ring, the beam itself is continuously changing with time in bothintensity and geometric placement, so higher-order effects need to be addressed. This is particularly trueif sensitivities approaching − need to be probed. Fig. 11.17:
Measurements with the EDDA detector of the left-right asymmetry as a function of the angle errorin mrad (down triangles) or the position error (up triangles). Various features of the measurements are marked,including some interpretations through model parameters (usually logarithmic derivatives) from the fits throughthe data.
In 2008 and 2009, the EDDA detector system, then used as a polarimeter for COSY, was calibratedfor geometric and rate error sensitivity. The beam was scanned horizontally in both angle and position.The effects of rate were also present in the data as the rate changed with the time in the store. An exampleof a piece of the geometric data for the left-right asymmetry is seen in Fig. 11.17. Measurements of anumber of polarisation observables were made with five different polarisation states. Angular deviations(down triangles in mrad) and position variations (up triangles in mm) were recorded. As seen in thefigure, the effects are large and clear. In the same set of data, changes due to the data acquisition rate werealso recorded. A model of all of the error effects was constructed in terms of the logarithmic derivativesof the cross section and analysing power as geometric parameters, and these parameters as well as otherfactors including rate changes were used to reproduce the data, as shown. The free geometric variable inthe model was taken to be the angle deviation from a straight beam. The model was sufficiently robustthat it could predict effects for any of the measured polarisation observables within the errors in theobservable measurements.In the geometric case, Fig. 11.17 shows different effects for angle and position changes. Thesecould be reconciled provided an effective distance to the detector was assumed, and this became oneof the fitting parameters. If this substitution works well, then it can become the basis for reducing thegeometry effects to a single parameter. The quality of this result is shown in Fig. 11.18. Measurements120f the left-right asymmetry correction are overlaid for both angle and position, and shown to lie along asimilar slope.
Fig. 11.18:
Changes to the left-right asymmetry as a function of an index parameter (defined below). The indexparameters is tied to either position or angle variations of the beam on target. The overlap of these two sets of datainto one universal line indicates that a single index parameter is capable of correcting both types of errors.
We chose to recast this relationship in terms of an index parameter φ : φ = s − s + 1 where s = σ L + σ L − σ R + σ R − (11.6)This quantity is available experimentally in real time. Thus independent of the cross ratio or any otherpolarisation observable, a correction may be applied. The model is used to calculate the correction,such as a change in position along the sloped line in Fig. 11.18. This can be applied to any polarisationobservable. A term, W = (cid:88) σ i , (11.7)is also available for the counting rate. An example is shown in Fig. 11.19. The measurements of a beamwith a constant polarisation is given by the red data. The time dependence is an error that depends onthe data rate as it creates pile-up effects in the detectors. Correction of that error yields the blue data.But these data are still not right because of a geometric misalignment. The final correction leads to theblack data, with are constant in time to better than one part in , which is statistically limited. If thecalibrations are known in advance, such corrections may be made in real time during the experiment, afeature that will be essential in maintaining the polarisation pointing along the velocity through feedback.121 ig. 11.19: Three versions of a set of left-right asymmetry measurements as a function of time during a storein COSY, during which the rate first rose slightly and then started to fall as a declining rate. The red points areuncorrected; the blue points are corrected for rate effects; the black points are corrected for both rate and geometryeffects. A line through the black points is indicated and the error in its slope (consistent with zero) is shown.
The example in Fig. 11.17 of calibration data for five polarisation states is linear in the case ofthe left-right asymmetry. Higher-order effects appear as curvatures of various ranks, which may also beparameterised using powers of the logarithmic derivatives. The combination of all of these properties ofthe model-based calibration and driver-term corrections makes it possible to extract a signal as small as δ(cid:15) = 10 − reliably from a series of time-dependent asymmetry measurements. Polarimeter rotations, energy loss, and deuteron tensor polarisation effects
The polarimeter must be set up so that the coupling between horizontal asymmetries and vertical asym-metries, as established by the location of the ring plane, is as small as possible. Such a correlation ismeasured by stepping the polarisation direction (registered as a phase in the feedback circuit) around thein-plane circle and comparing vertical and horizontal asymmetries as shown in Fig. 11.20 for t = 0 s.Imagined data points for the correlation with an incomplete cancellation are indicated by plus signs. Astime proceeds through the beam store, any EDM effect will cause the left-right asymmetry ( (cid:15) L − R ) to risewith time, taking the correlation with it. This allows in principle for a separation of these effects, but itmust be remembered that for a single store, the statistics on each of the data points will be over two ordersof magnitude larger than any EDM effect at the expected level of sensitivity. The correlation cancellationis likely to be incomplete because of the large running time needed to establish the correlation.For an off-centre block carbon target, the simple down-up raw asymmetry may be of the order of0.2. Since left-right sensitivities as small as − may be involved in the EDM signal itself, cancellationof these polarimeter rotation effects to a similar degree must also be arranged, either by a mechanicaladjustment in the polarimeter detector acceptance or cancellation through terms in the systematic errorcalibration described earlier. The risk, for example, is that energy loss due to collisions with backgroundgas (or the polarimeter target) will populate lower energy particle orbits, causing on average a drift awayfrom the frozen spin condition that increases with time. While continuous polarisation measurementsare used to correct the frozen spin condition, errors that tend in one direction may produce a bias inthe result, and would appear in the figure as data points no longer symmetrically distributed about the122 ig. 11.20: Mock data showing a correlation between vertical and horizontal asymmetries as variations in thevertical asymmetry are used (and corrected) in order to maintain frozen spin through feedback. The figure alsosuggests that this correlation plot rises to include a larger horizontal asymmetry (perhaps from and EDM signal)as time progresses during the store. vertical axis.A correlation plot similar to the figure has also been suggested for the elimination of other effects,such as the slow vertical polarisation growth associated with a residual sideways magnetic field or verticalelectric field in the EDM ring. In that case, the horizontal axis of the plot will be some measurement ofthe sideways field, such as that indicated by the SQUID readout proposed for the proton ring. Becauseall of these effects operate at the same level or higher than any EDM signal, a comprehensive analysismust be done at the end for all of the data. Magnetic field errors are apt to appear as changes from storeto store while polarimeter rotation effects always appear within a store and are not as likely to changeover time.A similar effect arises in addition for the deuteron beam since, in most polarised ion sources, it isimpossible to eliminate completely a tensor polarisation component in the beam. One is often present atthe percent level. (Indeed, there may be arguments for having a large tensor polarisation since it may bemonitored as a continuous measure of the beam polarisation through the T analysing power withouthaving to periodically rotate the polarisation axis of the beam into the sideways direction.) A rotation ofthe tensor polarisation axis to either the left or right will directly generate a left-right asymmetry throughthe T analysing power. This analysing power is not directly driven by the spin-orbit force, so its valuesfor most forward angle polarimeter geometries are typically less than a few percent.Systematic errors of due to polarimeter rotation and T may be detected in a running experimentby looking for a correlation between the down-up asymmetry driving the feedback circuit to hold thefrozen spin condition and the left-right asymmetry that in principle carries the EDM signal. In thedeuteron case, left-right asymmetry sensitivity through either an effective polarimeter rotation or T sensitivity may be separated in a calibration using a series of in-plane polarisation directions and lookingat the nature of the correlation with rotation angles up to π/ . A linear relationship indicates a polarimeterrotation effect while a dependence that goes as the sine of twice the in-plane rotation angle indicates a T sensitivity. These two effects will in general have different sizes or slopes for small rotation angles. Time-reversed experiment
The EDM violates the symmetries of parity conservation and time reversal. In the case of time reversal,the direction of rotation of the beam around the ring would be changed and all polarisations and magneticfields would have the opposite sign. Since this is a physically realisable experimental condition, it hasbeen suggested that it be a part of the protocol for the EDM search. In the case of the proton with a123ositive anomalous magnetic moment, the condition of frozen spin may be realised with only an electricfield. This field remains the same under time reversal, thus it should be possible to operate the storagering with both beams (CW and CCW) at the same time. This offers the chance to compare beam loca-tions, profiles, intensities, and polarisations in order to verify that they are, in fact, identical. A secondpolarimeter would need to be installed in the ring in order to capture measurements of the reversed-direction polarisation. Some economies of construction and the use of only one extraction mechanismfavour a design in which the two polarimeter detector schemes are located on either side of a single blocktarget. Measurements made to large scattering angles of elastic proton scattering from carbon [5] showa drop of eight orders of magnitude of the cross section between the forward scattering angles used forpolarisation measurements and similar backward scattering angles. This should be enough to suppressany interference with small changes being measured through the forward scattering asymmetry.Essentially all systematic error effects that give rise to an EDM-like signal (changing vertical com-ponent of the polarisation over time) are time-reversal conserving. This would appear as a rising signalfor both CW and CCW cases while the EDM signal would rise in one instance and fall (go negative) inthe other. So any unsuppressed systematic error could be cancelled by subtracting the CW and CCWmeasurements.Since the measurement (for small angles of vertical rotation of the polarisation) is one of a contin-uously rising effect, let us denote scattering to the left as: σ P OL = σ UNP OL [1 + ( S + E ) pA ] (11.8)where S is the rate of rise due to remaining systematics and E is the rate of rise due to the EDM. Thesimple left-right asymmetries for CW and CCW become: (cid:15) CW = L − RL + R = ( S + E ) pA and (cid:15) CCW = ( S − E ) pA (11.9)so
12 ( (cid:15) CW − (cid:15) CCW ) =
EpA . (11.10)This subtraction works only to the extent that pA values for both CW and CCW are well calibrated.If we define p and A to be the average values for CW and CCW, and we define δp and δA to be thedifference between the calibrated and the actual values of the polarisations and analysing powers, then,when expanded to first order:
12 ( (cid:15) CW − (cid:15) CCW ) =
EpA + SpA (cid:18) dp + aA (cid:19) . (11.11)This means that the systematic contribution to the EDM signal can be suppressed only to the extent thatthe unknown fractional errors in the CW − CCW polarisation and analysing power differences are smallenough to render the systematic error negligible compared to the EDM signal.In the case of the beam polarisation, this introduces the requirement that the CW and CCW beamsin the experiment be filled using the same polarisation state from the ion source. Likewise, care mustbe taken in the construction of the polarimeters and the setup of their detector readout to ensure thatthe effective analysing powers are also as identical as possible. This puts a premium on other efforts toreduce the systematic error contribution initially.
References [1] A. Zelenski,
The RHIC polarised Ion Source Upgrade,
POS (PSTP2013) 048, 2013[2] P.G. Sona, Energ. Nucl. 14, 295, 1967 1243] M. Ieiri et al.,
A multifoil carbon polarimeter for protons between 20 and 84 MeV,
Nucl. Instrum.Methods A 257, 253, 1987[4] G. Guidoboni et al.,
How to Reach a Thousand-Second in-Plane polarisation Lifetime with 0.97GeV/c Deuterons in a Storage Ring,
Phys. Rev. Lett. 117, 054801, 2016[5] H.O. Meyer et al.,
Proton elastic scattering from 12C at 250 MeV and energy dependent potentialsbetween 200 and 300 MeV,
Phys. Rev. C37, 544, 1988[6] N. Hempelmann et al. , “Phase locking the spin precession in a storage ring,” Phys. Rev. Lett. ,014801 (2017).[7] V.P. Ladygin et al. , “POMME: A medium energy deuteron polarimeter based on semi-inclusived-carbon scattering,” Nucl. Instrum. Methods, Nucl. Phys. A , 129 (1998).[8] B. Bonin et al. , “analysing powers for the inclusive reaction of deuterons on carbon at energiesbetween 0.175 and 1.6 GeV,” Nucl. Instrum. Methods, Nucl. Phys. A , 389 (1990).[9] M.W. McNaughton et al. , “The p-C analysing power between 100 and 750 MeV,” Nucl. Instrum.Methods, Nucl. Phys. A , 435 (1985).[10] N.P.M. Brantjes et al. , “Correcting systematic errors in high-sensitivity deuteron polarisation mea-surements,” Nucl. Instrum. Methods Phys. Res. A , 49 (2012).125 hapter 12Spin Tracking
Introduction
Spin tracking simulations of the complete EDM experiment are crucial to explore the sensitivity of theplanned storage ring EDM searches and to investigate the systematic limitations. Existing spin trackingprograms have been extended to properly simulate spin motion in presence of an electric dipole moment.The appropriate EDM kicks and electric field elements (static and RF) have been implemented andbench-marked. Furthermore, a symplectic description of fringe fields, field errors, and misalignments ofmagnets has been adapted and verified. For a detailed study during particle storage and spin build-up ofan EDM signal, a large sample of particles must be tracked for billions of turns. This is a challengingtask because it requires beam and spin tracking for about turns . Simulation Programs
Given the complexity of the task, and in order to ensure the credibility of the results, various simula-tion programs using different algorithm are upgraded and bench-marked with the required accuracy andefficiency:– COSY Infinity [1], based on map generation using differential algebra and the subsequent calcula-tion of the spin-orbital motion for an arbitrary particle including fringe fields of elements. COSYInfinity and its updates are used including higher-order nonlinearities, normal form analysis, andsymplectic tracking. COSY Infinity contains elements to simulate E × B elements (static and RF).– COTOBO (COSY Toolbox) [2] has been developed to perform the simulations, based on a C++interface for COSY Infinity. The usability of ROOT [3] enables a fast and easy way to analyse thesimulation results.– MODE (Matrix integration of Ordinary Differential Equations) [4, 6] is a software package thatprovides nonlinear matrix maps building for spin-orbit beam dynamics simulation. MODE mathe-matical model is based on Lorentz and Thomas-BMT equations that are expanded to Taylor seriesup to the necessary order of nonlinearity including fringe fields of elements. The numerical algo-rithm is based on matrix presentation of Lie propagator.– Bmad [7] has various tracking algorithms, including Runge-Kutta and symplectic (Lie algebraic)integration. Routines for calculating transfer matrices, emittances, Twiss parameters, dispersion,coupling, and fringe field contributions are also included. Bmad, by interfacing with the PTCtracking code [8], can, for example, compute Taylor maps to arbitrary order and do normal formanalysis.– Homemade integrating program [11], solving equations of particle and spin motion in electric andmagnetic fields using Runge-Kutta integration. The programs models spin-orbital motion includ-ing fringe fields in elements. The algorithm used in the code is by several orders of magnitudeslower than codes based map generation using differential algebra. Therefore, the program waspredominantly used to investigate short-time phenomena and for bench-marking the other codes.– Simulation code for numerical Integration of beam and spin motion [12] is a very simple butgeneral approach and integrate the equation of motion as well as the T-BMT equation numerically.Standard algorithms like the fourth order Runge-Kutta algorithm are compared to newer ones andgreat emphasis is placed on the modular implementation in C++ for maximal flexibility. Corresponds to measurement time of 15 minutes on the circumference of the COSY lattice.
Status and Plans
Different possible scenarios for EDM measurements have been investigated to explore the sensitivity. Ina first step the resonant method [14, 15] has been developed to be able to perform an EDM measurementat COSY. In parallel detailed studies have been carried out to explore the sensitivity of the deuteronprecursor experiment at COSY [16–18]. In this context two different approaches have been investigatedto perform deuteron EDM measurements in dedicated storage rings:– The frozen-spin method [19], where the electrostatic and magnetic bending fields in a storage ringare adjusted according to the particle momentum in such a way that the longitudinally polarisedspins of the particle beam are kept aligned (frozen) with their momenta.– The quasi-frozen-spin method [21,22], where the anomalous magnetic moment of the particles hasto have a small negative value like for deuterons. In this case electric and magnetic field deflectorscan be spatially separated. The spins oscillate around the momentum direction in the horizontalplane back and forth every time the particle passes through a magnetic or an electrostatic field.The spin oscillations of individual particles compensate each other with respect to the momentumvector in the magnetic and electrostatic part of the ring.Different examples of spin tracking results for deuteron EDM storage rings utilising various lattice con-figurations are published in [21–25].The CPEDM consortium started a new initiative to design a prototype EDM (Electric DipoleMoment) storage ring [26], with predominantly electric bending. Operated at proton beam energiesbetween 30 and 50 MeV, the main purpose of this ring will be to carry out R & D work related to afinal 233 MeV frozen-spin proton EDM ring. Recently spin tracking simulation started to support thisdevelopment for dedicated proton EDM rings [28–30]. Spin tracking Simulations for Deuteron and Proton EDM Measurements
As described before, several simulation programs are utilised to simulate the vertical polarisation build-up for Deuteron and Proton EDM measurements induced by field and alignment errors of magnets andcompared in detail to the polarisation build up assuming different EDM magnitudes.
To be able to simulate the polarisation build up for the precursor experiment applying the resonantmethod [14, 15], time-dependent transfer maps have been implemented in COSY Infinity [2] to investi-gate the sensitivity of the precursor experiment using an RF Wien filter. This device provides superim-posed electric and magnetic RF field such that they do not influence the particles’ trajectory but lead toan additional rotation of the spin around the magnetic field of the device. Thus, this so called
Wien Filter will change the invariant spin axis. In order to determine the polarisation build up due to the electricdipole moment, it is necessary to know the orientation of the invariant spin axis. Since the particle andspin motion is perturbed by imperfections of the storage ring magnets, shifts, tilts and rotations can besuperimposed to study randomised sets of magnet misalignments [2, 5]. The resulting closed orbits canbe corrected by the orbit correction system to suppress false spin rotations via the magnetic moment.127
Fig. 12.1:
Maximum vertical spin buildup per turn for different EDM magnitudes and Gaussian distributedquadrupole shifts with different standard deviations. The RMS value of the vertical orbit displacement is used as ameasure for misalignments.
Different magnitudes of the standard deviation of the Gaussian distributed quadrupole shifts between1 µ m and 1 mm have been simulated. For each of these misalignments a tracking simulation has beenperformed using different EDM magnitudes. The Wien filter field’s phase has been locked to the situationof maximum buildup. This results in the shown buildup for different RMS values of the vertical orbitdisplacements at the quadrupoles in 12.1 [2].As long as the EDM contribution to the polarisation buildup is significantly larger than the buildupintroduced by misalignments of magnets, both effects can be experimentally separated. For a randomisederror with a standard deviation of 0.1 mm, the RMS value of the displacements is around 1 mm. In thiscase, the contribution to build up from misalignments of magnets and EDM is in the same order for η = 10 − . This corresponds to an EDM of roughly · − e cm.Results of benchmarking concerning changes in tune and chromaticity as well as driven oscilla-tions of the polarisation vector can be found in Ref. [9]. In order to determine the polarisation build up due to the electric dipole moment, it is necessary to knowthe orientation of the invariant spin axis. One current challenge for the precursor experiment is the lackof knowledge of the radial component of the invariant spin axis (cid:126)n that cannot be measured. A possiblesolution is its determination by simulating the COSY lattice and performing spin tracking. The EDMas well as misalignments of lattice elements are affecting the particles trajectory and therefore the spin128otion and tilt the invariant spin axis.
Fig. 12.2:
Due to a permanent EDM the invariant spin axis tilts into horizontal direction the angle ξ EDM . In case of an ideal ring and a vanishing EDM the invariant spin axis always points in verticaldirection as the spin precesses in the horizontal plane. In the presence of an EDM the invariant spin axisis tilted in the horizontal direction by the angle ξ EDM as sketched in figure 12.2. This angle is directlyproportional to the magnitude of the EDM and can be written as tan( ξ EDM ) = ηβ G . (12.1)In order to determine the invariant spin axis, the spin of the reference particle is tracked for N turns resulting in an ensemble of spin vectors (cid:126)s j where j ∈ N and j ∈ [ 1 , N ] [20]. For each possibleconfiguration of three spin vectors ( (cid:126)s , (cid:126)s , (cid:126)s ) an invariant spin axis (cid:126)n i is calculated as follows. (cid:126)u i = (cid:126)s ,i − (cid:126)s ,i (12.2) (cid:126)v i = (cid:126)s ,i − (cid:126)s ,i (12.3) (cid:126)n i = (cid:126)v i × (cid:126)u i | (cid:126)v i × (cid:126)u i | (12.4)Figure 12.3(left) shows a schematic description of the calculation. The invariant spin axis is thencalculated as the average of all (cid:126)n i vectors. Spin tracking is done using the software library Bmad .Figure 12.3(right) shows the spin distribution after tracking through the COSY lattice including themisalignments of dipoles and quadrupoles as well as an illustration of individual spin vectors (cid:126)s j , theinvariant spin axes (cid:126)n i and the average invariant spin axis (cid:104) (cid:126)n (cid:105) .129 After starting to design a prototype EDM storage ring [26], operated at proton beam energies between 30and 50 MeV, spin tracking simulation are performed to investigate the sensitivity of such a ring for EDMmeasurements. Spin tracking simulation are also carried out for several groups to simulate spin motionfor dedicated EDM rings for 233 MeV frozen-spin protons [10, 27–30].
The spin is determined from the T-BMT equation describing the precession rate of the angle betweenthe spin and momentum vectors of a relativistic particle in the presence of electromagnetic fields. Thelatter have to be evaluated at each location of the particle. Thus, the T-BMT equation is coupled tothe equation of motion for which, in general, a closed form solution cannot be obtained. Given thehigh sensitivity aimed, precise numerical simulations are necessary and bench-marking with analyticalestimates can help understand the major systematic effects. For instance, an average radial magneticfield of a few aT yields a vertical spin buildup similar to an EDM signal level of − e.cm . Thus, aprecise knowledge of the field at each integration step is crucial in order to determine its impact on thespin dynamics. In addition, due to the coupling between the different spin components which inducesadditional phases, the rapid oscillatory behaviour of the spin has to be finely resolved. In what follows,one discusses several examples where the considered ring lattice is based on the strong focusing onethat was proposed by V. Lebedev to achieve the beam requirements suitable for EDM: the simulatedring consists of 4 superperiods each containing 5 FODO cells. There are 6 electric bends per FODOcell characterised by 8 MV/m radial electric field for 3 cm plate separation. In the interface between thebending and the straight sections, the hard edge model was assumed, which means that the electric fieldsare constant everywhere within the element and drop abruptly to zero at the edges. Nevertheless, theenergy change of the particle was taken into account. This is a particularly useful model to simplify theanalysis. In what follows, one discusses some selected cases of lattice imperfections yielding a verticalspin buildup. Further details regarding some of the numerical simulations and their comparison with theanalytical estimates can be found in [30]. In particular, it appears that the established formalism which isbased on the Bogoliubov-Krylov-Mitrpolski method of averages [31] is very useful to calculate the spinprecession rates at the observation point, i.e. at the location of the polarimeter. In addition, it enabled thecalculation of the geometric phases as discussed below. In the case of misplacement of lattice elements, such as electric quadrupoles, orbit distortions can occurleading to a vertical spin build-up [28]. The latter can exhibit a linear and/or quadratic increase with time,depending on the amplitude of the perturbation. Example of tracking simulations for the all-electric ring,in the case where one defocusing quadrupole was misaligned by several micrometers are shown in fig12.4. A particle with an energy spread of ∆ p/p = 10 − was tracked on the perturbed closed orbit and itsspin is recorded after each turn completion. Good agreement was achieved between the tracking simula-tions and the analytical estimates and it is shown that the quadratic increase is due to two contributions:a longitudinal spin precession mainly caused by the vertical slope in the electrostatic deflectors and alinear radial spin buildup due to the displacement from the magic energy so that s y ∝ y (cid:48) ∆ p/p . In thelimit where ∆ p/p = 0 , the quadratic increase vanishes and one obtains a linear increase due to higherorder terms. The next tracking simulation example considered is that of the geometric phases, also referred to as theBerry phases [32]: In the case where the parameters of the system are varied slowly such that the value ofthe particle coordinates end the same as they started and if the average perturbations are balanced within130ne revolution, then the non trivial phase picked-up by such perturbations is called the Berry phase. Sucheffects, due to the non-commutation of spin rotations around different axes, can dominate if the beamenergy is very close to the magic one. Let’s assume that the beam is injected at the magic energy andthat the lattice has alternating magnetic field imperfections. Such an imperfection is represented by thepresence of both vertical and longitudinal magnetic fields which are 90 degrees out of phase as illustratedin fig 12.5. In particular, it can be seen that the radial spin is rotated into the vertical plane by means ofthe longitudinal magnetic fields such that the leading term of the vertical spin buildup is given by [30]: ds y dt ≈ cβ l C (cid:16) em (cid:17) (cid:18) G + 1 γ (cid:19) Gγ B y L y B z L z (12.5) ≈ . ∗ B y L y B z L z (12.6)Thus, assuming integrated field perturbations such that B y L y = B z L z = 1 nT.m, this yields a spinprecession rate of ≈ . ∗ − nrad/s which is well below the EDM signal level. Fig 12.6 shows acomparison of the tracking simulations with the first order analytical estimate of the spin buildup due tothe Berry phases where one can see a good agreement of both estimates. In addition, it is important tonote that such an effect can be cancelled by using two counter-rotating beams and taking the differenceof the signals. 131 ig. 12.3: General schematic illustration to calculate the invariant spin axis and spin tracking results. Left: Methodfor calculating the invariant spin axis from three spin vectors. Right: Spin distribution resulting from misalignedmagnets (blue) and average invariant spin axis (red). .0*10 -5 -4 -4 -4 -4
0 0.002 0.004 0.006 0.008 0.01 S p i n v e r ti ca l [r a d ] Time [s] ∆ y = 10 µ m ∆ y = 50 µ m ∆ y = 100 µ m Fig. 12.4:
Vertical spin versus time for a lattice with different misalignment errors and a momentum offset ∆ p/p = 10 − . The analytical results are shown in solid lines while the dashed lines display the tracking results. -80 -60 -40 -20 0 20 40 60 80 100 120-180-160-140-120-100-80-60-40-20 0 20-1.6e-17-1.4e-17-1.2e-17-1e-17-8e-18-6e-18-4e-18-2e-18 0 2e-18Y [m] Azx y -By+By -Bz+Bz Ideal closed orbitPerturbed closed orbitProjected perturbed orbitX [m] Z [m] Fig. 12.5:
Spin and orbit evolution for a lattice with alternating magnetic field imperfections: a vertical magneticfield B y yields a horizontal spin component which is rotated into the vertical plane by means of a longitudinal fieldcomponent B z . The closed orbit of the perturbed motion is shown in blue and the particle motion is clockwisestarting from Point A. S p i n v e r ti ca l [r a d ] Time [s]TrackingAnalytical
Fig. 12.6:
Vertical spin buildup from tracking simulations and comparison with the analytical estimate given byEq. (12.6). eferences https://indico.cern.ch/event/609422/contributions/2463327/attachments/1426326/2188381/spin_tracking_selcuk.pdf [29] Julien Michaud (Laboratoire de Physique Subatomique & Cosmologie IN2P3 (CNRS), France,PhD thesis at Universite Grenoble Alpes, to be published (2019).[30] M. Haj Tahar and C. Carli, On solving the Thomas Bargman-Michel-Telegdi equation us-ing the Bogoliubov Krylov method of averages and the calculation of the Berry phases, seehttps://arxiv.org/pdf/1904.07722.pdf[31] N.N. Bogoliubov and Y.A. Mitropolskii, Asymptotic methods in the theory of nonlinear oscilla-tions, Gordon and Breach, New York (1961).[32] M. Berry. Quantal Phase Factors Accompanying Adiabatic Changes. Proc. Roy. Soc. A 392, 45(1984). 136 hapter 13Roadmap and Timeline
CPEDM Strategy
As emphasised above this challenging project needs to proceed in stages that are also outlined in Fig. 13.1:1. COSY will continue to be used as long as possible for the continuation of critical R&D associatedwith the final experiment design. An important requirement is to test as many of the results aspossible with protons where the larger anomalous magnetic moment leads to more rapid spinmanipulation speeds.2. The precursor experiment will be completed and analysed. Some data will be taken with an im-proved version of the Wien filter with better electric and magnetic field matching.3. The next stage is to design, fund, and build a prototype ring to address critical questions concerningthe features of the EDM ring design. At 30 MeV, the ring with only an electric field can storecounter-rotating beams, but they are not frozen spin. At 45 MeV with an additional magnetic field,the frozen spin condition can be met. But the magnetic fields also prevent the CW and CCW beamsfrom being stored at the same time. Even so, an EDM experiment may be done with these twobeams used on alternating fills.4. Following step 3, the focus will be to create the final ring design, then fund and construct it.5. Once the ring is ready, the longer term activity will be to commission and operate the final ring, im-proving it with new versions as the systematic errors and other experimental issues are understoodand improved. Precursor Experiment Prototype Ring All-electric Ring dEDM proof-of-capability (orbit and polarization control;first dEDM measurement) pEDM proof-of-principle (key technologies,first direct pEDM measurement) pEDM precision experiment (sensitivity goal: 10 -29 e cm)- Magnetic storage ring- Polarized deuterons- d-Carbon polarimetry- Radiofrequency (RF) Wien-filter - High-current all-electric ring- Simultaneous CW/CCW op.- Frozen spin control (withcombined E/B-field ring)- Phase-space beam cooling - Frozen spin all-electric(at p = 0.7 GeV/c)- Simultaneous CW/CCW op.- B-shielding, high E-fields- Design: cryogenic, hybrid,…Ongoing at COSY (Jülich)2014 (cid:1) (cid:1) (cid:1)
Fig. 13.1:
Summary of the important features of the proposed stages in the storage ring EDM strategy.
Future scientific goals may include conversion of the ring to crossed electric and magnetic fieldoperation so that other species besides the proton could be examined for the presence of an EDM. Anal-ysis of the data may be made for signs of axions using a frequency decomposition and investigation ofcounter-rotating beams with different species used in novel EDM comparisons.The prototype ring and the CPEDM stages are host-independent. If the prototype is built at COSY,it would take advantage of the existing facility for the production of polarised proton (and deuteron)137eams, beam bunching, and spin manipulation. COSY itself could be used for producing electron-cooledbeams. It may also be built at another site ( e.g. , CERN) provided that a comparable beam preparationinfrastructure is made available. In either case, the lattice design will mimic that of the high-precisionring in order to test as many features as possible on a smaller scale.
Timeline
As shown , a staged approach is pursued with step-1 (“Precursor Experiment”) currently ongoing. Thisis partially funded by an ERC Advanced Grant, which runs until September 2021. The next stage (step-2,“Prototype Ring”) has started last year (2017) and a CPEDM task force is working on the “ConceptualDesign Report” (CDR, due in 2020) and will subsequently finalise the “Technical Design Report” (TDR,ready in 2022). If funding can be secured, construction could start beyond 2022. Currently, about 5years are foreseen for building and operation of the prototype. Only after that does it seem conceivableto design, build and operate the final ring (step-3, “All-electric Ring”).A more detailed timeline is presented in Fig. 13.2. See the caption for details.
FeasibilityPrecursorpEDM Dev.Other (axion)
ReportsPrototype
CERN CDR TDR Propose Procure / Build Commission
EventsRunning: pEDM Dev.PrototypeOther
30 MeV ‘26 ‘27 ‘28 ‘29 ‘30 ‘31 ‘32
45 MeV
Events 5 4All-electric ring
CDR TDR Propose Procure / Build Run in ‘34+
Event Key:1. Strategic program evaluation Helmholtz Association (HGF)2. Start of HGF funding period3. End of “srEDM” Grant of European Research Council4. HGF Mid-term Review5. Start of next HGF funding period
Y/N Y/N
HGF HGF
Fig. 13.2:
This “Timeline” follows the anticipated evolution of the storage ring EDM project through several events(numbered) and stages. At present (2019), experimental work will continue with COSY to look into feasibilityissues regarding electron-cooling, begin development of a search for axion-like particles, and continue to refinethe precursor experiment as a first measurement of the deuteron EDM. Meanwhile, a long “Yellow report” isbeing prepared in CERN format to outline the plans for a prototype ring and the eventual construction of a fullscale all-electric ring to measure the proton EDM. Later in the year, the Helmholtz Research Association (HGF)will begin the strategic evaluation process for the research program “Matter and the Univers” (MU) for the next“Program-oriented Funding” (PoF) period that will start at the beginning of 2021. Also in parallel, work will beginon a Conceptual Design Report (CDR), followed by a Technical Design Report (TDR), for the creation of a smallelectric (and later electromagnetic) storage ring to answer feasibility questions about the design and use of suchrings for EDM searches. If support continues during PoF 4, then the efforts with the COSY machine will switchto a development with polarised proton beams that duplicates what has been achieved for deuterons (red band).Other types of research in related symmetries also continues (green band). Work will also start for the constructionof the electric version of the prototype ring (orange). Commissioning with first beam at 30 MeV starts in 2025 todemonstrate high intensities and counter-rotating two-beam operation. A second version with magnetic bendingadded to enable frozen spin operation begins in 2028 at 45 MeV. As new feasibility studies with the prototypecome to fruition, work starts with a CDR/TDR for the proton EDM experiment. This project will be commissionedin the mid-2030s. ppendices ppendix AResults and achievements at Forschungszentrum Jülich
This appendix describes results and achievements obtained up to now. It comprises results obtained theCOoler SYnchrotron COSY at Forschungszentrum Jülich and of the Jülich Theory group. Activities andachievements like polarimetry and spin tracking are described in dedicated chapters.
A.1
Results and achievements at COSY
For most of the studies the parameters listed in Tab. A.1 were used.COSY circumference 183 mdeuteron momentum 0.970 GeV/ cβ ( γ ) G ≈ − . revolution frequency f rev ≈ Table A.1:
Values of the COSY operating parameters for most of the studies reported in this appendix.
A.1.1 High precision spin tune measurements
Although not directly connected to the EDM measurement in a dedicated storage ring using the frozenspin method, the measurement of the the fast 120 kHz precision of the polarisation vector around themagnetic guiding field in the horizontal plane is an import step in understanding and controlling spinprecession in a storage ring.In an ideal planar magnetic storage ring, the spin tune − defined as the number of spin precessionsper turn − is given by ν s = γG ≈ − . ( γ is the Lorentz factor, G the gyromagnetic anomaly). At p = 970 MeV /c , the deuteron spins coherently precess at a frequency of about 120 kHz in COSY. Thespin tune was deduced from the up-down asymmetry of deuteron-carbon scattering. In a time intervalof 2.6 s, the spin tune was determined with a precision of the order 10 − , and to − for a continuous100 s accelerator cycle [1], see Fig. A.1.To appreciate this high relative precision of σ ν s /ν s ≈ − in a 100 s cycle, a comparison tothe equivalent quantity in the muon g − measurement is helpful. Here the precision reached is about σ ν s /ν s ≈ − per year, i.e. a ppm measurement of a = ( g − / in one year. The three order ofmagnitude higher precision in a much shorter time is mainly explained by the fact that the cycle lengthis much larger (100 s compared to 600 µ s).Note that a spin rotation due to an electric dipole moment of d = 10 − e cm for one turn isgiven by ν s = vmγdes = 5 · − . This means that with the statistical precision σ ν s = 10 − reachedin a single cycle of 100 s one is statistically sensitive to EDM of the order of − e cm even with anaccelerator not constructed for this purpose. Of course additional rotations due to the magnet momentdue to imperfections of the storage ring are orders of magnitude larger and have to be understood.140
10 20 30 40 50 60 70 [ r ad ] ϕ ∼ time [s] ] number of particle turns [10 ] [ s ν ∆ ] [ s ν ∆ Fig. A.1:
Upper plot: The phase of the polarisation vector in the horizontal plane evaluated close to spin revolutionfrequency of the polarisation vector using a Fourier analysis over turns. Middle: spin tune change fromobtained from two consecutive phase measurements. Lower: spin tune change obtained from a parabolic fit in theupper plot. A.1.2 Long horizontal polarisation lifetime
To reach the desired statistical precision, a spin coherence time of the order of 1000 s has to be reached.A rough estimate shows that this is not a simple task. A momentum spread of ∆ p/p ≈ − correspondsto ∆ γ/γ ≈ · − . Since the spin tune is given by γG , after ≈ turns ( i.e. ≈ ∆ p/p can be cancelled and thespin coherence time reaches a few seconds.Using a combination of beam bunching, electron cooling, sextupole field corrections, and thesuppression of collective effects through beam current limits a deuteron beam polarisation lifetime of1000 seconds in the horizontal plane of the magnetic storage ring COSY could be reached [2]. The resultis displayed in Fig. A.2. A.1.3 Feedback and control of polarisation
The precise measurement of the horizontal spin precession together with long spin coherence timesallowed us to set up a polarisation feedback system. In a dedicated ring its role is to maintain thepolarisation vector always (anti-) parallel to the momentum vector to maximise the statistical sensitivity.The use of feedback from a spin polarisation measurement to the revolution frequency of a 0.97GeV/c bunched and polarised deuteron beam in the Cooler Synchrotron (COSY) storage ring has beenrealised in order to control both the precession rate ( ≈
120 kHz) and the phase of the horizontal polar-isation component. Real time synchronisation with a radio frequency (rf) solenoid made possible the141 ig. A.2:
Polarisation in the horizontal plane as a function of time. The line shows a comparison to a model (seeRef. [2]), the lower plot the deviation to the model rotation of the polarisation out of the horizontal plane, yielding a demonstration of the feedback methodto manipulate the polarisation, see Fig. A.3. In particular, the rotation rate shows a sinusoidal functionof the horizontal polarisation phase (relative to the rf solenoid), which was controlled to within a onestandard deviation range of σ = 0 . rad, see Fig. A.4. The minimum possible adjustment was 3.7 mHzout of a revolution frequency of 753 kHz, which changes the precession rate by 26 mrad/s [3]. Such acapability meets a requirement for the use of a dedicated storage rings for EDM measurements. A.1.4 Invariant spin axis measurements
Another application of the precise spin tune measurement is the the measurement of the invariant spinaxis. An extended paper entitled "Spin tune mapping as a novel tool to probe the spin dynamics in storagerings" describes this in detail [4]. It is motivated by the fact that precision experiments, such as thesearch for electric dipole moments of charged particles using storage rings, demand for an understandingof the spin dynamics with unprecedented accuracy. New methods based on the spin tune response ofa machine to artificially applied longitudinal magnetic fields, which is called "spin tune mapping", hasbeen developed. The technique was experimentally tested in 2014 at COSY and, for the first time, theangular orientation of the stable spin axis at two different locations in the ring has been determined to anunprecedented accuracy of better than 2.8 µ rad. A.1.5 Radio-Frequency Wien filter for spin manipulation
In a pure magnetic storage ring like COSY, an EDM will generate an oscillation of the vertical polarisa-tion component. For a 970 MeV/c deuteron beam with the spin precession frequency of 120 KHz, a tinyamplitude is expected, e.g. , · − for an EDM of d = 10 − e cm. To allow for a build-up of the ver-tical polarisation proportional to the EDM, a radio-frequency (RF) Wien-filter has to be operated. Sucha device with was developed and constructed, see Fig. A.5. This RF Wien-filter was installed in COSYin May 2017. A first commissioning run was successfully conducted in June 2017. [5, 6]. Fig. A.5shows a drawing of the Wien filter. During the 2018 test run it was operated with magnetic (electric)142 ig. A.3: The left right asymmetry, proportional to the vertical polarisation as a function of time. Initially thepolarisation is pointing upwards (red points) or downwards (black points)depending on injection. At t ≈ s thepolarisation is flipped into the horizontal plane with the help of the rf-solenoid. The polarisation vector starts toprecess in the horizontal plane. At t ≈ s the solenoid is switched on again. The fact that vertical polarisationraises is a proof that the feedback systems is working.
90 100 110 120 130 140 150 160 170 180 [ r ad ] s e t φ φ P ha s e − − − feedback onfeedback off time after injection [s]90 100 110 120 130 140 150 160 170 180 [ H z ] C O SY f ∆ − − frequencycorrections 2 × Fig. A.4:
Upper: The phase as a function of time with feedback off (blue) and on (red). The read points stay withina RMS of 0.21 rad (grey band). Lower: Adjustments of the COSY frequency.
23 4 56789 x y z
Fig. A.5:
Left: Design model of the RF Wien filter showing the parallel-plates waveguide and the support structure.1: beam position monitor (BPM); 2: copper electrodes; 3: vacuum vessel; 4: clamps to hold the ferrite cage; 5:belt drive for ° rotation, with a precision of . ° ( .
17 mrad ); 6: ferrite cage; 7: CF160 rotatable flange; 8:support structure of the electrodes; 9: inner support tube. The axis of the waveguide points along the z -direction,the plates are separated along x , and the plate width extends along y . During the EDM studies, the main fieldcomponent E x points radially outwards and H y upwards with respect to the stored beam. Right: Photograph ofWien filter installed in COSY. field integrals of . . . First results obtained with this device will be discussed in the nextsubsection. A.1.6 Measurements of deuteron carbon and proton carbon analysing powers
The way to measure the vertical polarisation proportional to the EDM is to scatter deuterons or protonselastically off a carbon target. To achieve high accuracy the analysing should be large and should beknown with small uncertainties. A series of measurements were performed. Fig. A.6 shows the analysingpower of deuteron carbon scattering for various beam energies as a function of the polar angle of thedeuteron in the lab system. Data using a polarised proton beam were also taken. The analysis is goingon.
A.1.7 Orbit control
Systematic errors for EDM measurement occur for example due to magnet misalignments and orbitoffsets. At COSY many new devices and procedures could be tested and implemented to improve theorbit. First of all, an automatised orbit control system was implemented which allows one to correct theorbit in real time. This system reduces the orbit correction procedure from about 10 hours to less than onhour. As an example Fig. A.7 shows the result of an orbit after correction. The RMS of the horizontal(vertical) orbit is 1.46 (0.90) mm.
A.1.8 Beam Based Alignment
Beam based alignment is a procedure to verify that the beam passes through the centre of a quadrupole.A off-centre path through a quadrupole results in a deviation of the beam. Modifying the quadrupolestrength, this deviation can be measured. From a surveying procedure the quadrupole position is knownto approximately . . Using the beam based alignment procedure the position of the BPM relativeto the quadrupoles could be determined. Fig. A.8 shows preliminary results. For 12 (of the total 56)quadrupoles offset of the BPMs of a few millimetres were found. These offset can now be corrected.This should result in a orbit closer to the design orbit and will reduce the systematic error of the precursorexperiment. 144 [deg] Lab Q - - - - - a A na l yz i ng P o w e r A y + = 0.0 a
380 MeV = -0.4 a
340 MeV = -0.8 a
300 MeV = -1.2 a
270 MeV = -1.6 a
235 MeV = -2.0 a
200 MeV = -2.4 a
170 MeV
Vector Analyzing Power for Elastic dC Scattering
Fig. A.6:
Reconstructed vector analysing power for deuteron beam energies of (from top to bottom) 380 MeV, 340MeV, 300 MeV, 270 MeV, 235 MeV, 200 MeV and 170 MeV. The curves are subsequently offset by 0.4 for betterreadability. The statistical errors are indicated by the black error bars on the data points. The systematic error isshown as red regions.
A.1.9 Beam Position Monitor
New devices, so called Rogowski coils were built and tested in COSY. The Rogowski coils consist foursegments (up-right, down-right, down-left, up-right). A time varying beam induces a voltage in the fourcoils. Combining the four voltages the beam position can be determined. Fig A.9 shows a photograph ofa coil installed in COSY and the principle setup of the coils. First calibration measurements show thatthe accuracy is less than 100 µ m can be reached, see Fig. A.10. A.1.10 Electrostatic and combined Deflector development
The future measurements of the EDM at COSY storage ring require development of a prototype ofelectrostatic or combined electromagnetic beam bending element. In case of proton beam and magicmomentum of 701 MeV/ c all elements of such ring can be electric, but in all other cases existence ofthe magnetic fields is obligatory. The electrostatic deflectors should consist of two parallel metal plates145 ig. A.7: COSY orbit measurement. The upper plot shows the vertical (red) and horizontal (blue) orbit as afunction of the longitudinal position in COSY. The desired orbits are shown in gold and green (both coincide withthe x = y = 0 line) for the vertical and horizontal orbit, respectively. The RMS of the horizontal (vertical) orbit is1.46 (0.90) mm. The plot in the centre shows the steering magnet currents applied for the correction. QT1 QT4 QT5 QT8 QT9 QT12 QT17 QT18 QT21 QT22 QT28 QT32101234 O ff s e t / mm P r e li m i n a r y HorizontalVertical
Fig. A.8:
Offset of the beam at the position of various quadrupoles. Since the quadrupoles are aligned to . these values can be used to calibrate the beam position monitors (BPMs). under equal potential of a different sign. Equal electric potential seen by the particle at the entrance andat the exit of the deflector will not change the total momentum of the particle. This puts restrictions on theminimum distance between deflectors. Recent possible ring lattices studies limit the good-field-regionfor stored particles to be 40 mm. It leads to the minimum distance between electric deflector plates tobe about 60 mm. The vertical beam size is several times larger than horizontal one impose restrictionson the vertical dimensions of the flat part of the deflector too. Minimum transverse dimensions of thebending elements will be more than 100 mm. In order to minimise breakdown probability between theflat regions of the deflectors and move it to the edge, the shape of deflectors should follow Rogowskiprofile on both vertical ends. The end caps of individual deflector should be made to couple the strayfields with subsequent deflectors. Designed ring lattice require electric gradients in the order of 5-10 MV / m . This is more than the standard values for many accelerator deflectors located at a distances ofa few centimetres apart. Assuming 60 mm distance between the plates, to achieve such high electric146 ig. A.9: Photograph of a Rogowski coil installed in COSY and schematic of the Rogowski coil setup.
Fig. A.10:
The left figure shows the difference of the measured positions and the prediction from a model fordifferent regions in the x − y plane as indicated by the right figure. fields we have to use high voltage power supplies. At present, two 200 kV power converters are orderedfor testing deflector prototypes. The field emission, field breakdown, dark current, electrode surfaceand conditioning should be studied using two flat electrostatic deflector plates, mounted on the movablesupport with possibility to change distance between 20 and 120 mm. The residual ripple of powerconverters in the order of 1e-5pp at maximum 200kV will lead to particle displacement on the order ofmillimetres. A smaller ripple or stability control of the system will be a dedicated task for investigationsplanned at the test ring facility. 147 .1.11 "Spin-Offs" This subsection just lists a number of publications that were initiated by the studies for an storage ringEDM measurement but also have application in other areas.1. Polynomial Chaos Expansion method as a tool to evaluate and quantify field homogeneity of anovel waveguide RF Wien filter [5]A full-wave calculations demonstrated that the waveguide RF Wien filter is able to generate high-quality RF electric and magnetic fields. In reality, mechanical tolerances and misalignments de-crease the simulated field quality, and it is therefore important to consider them in the simulations.In particular, for the electric dipole moment measurement, it is important to quantify the field errorssystematically. Since Monte-Carlo simulations are computationally very expensive, we discusshere an efficient surrogate modelling scheme based on the Polynomial Chaos Expansion methodto compute the field quality in the presence of tolerances and misalignments and subsequently toperform the sensitivity analysis at zero additional computational cost.2. Computational framework for particle and spin simulations based on the stochastic Galerkin method [7]An implementation of the polynomial chaos expansion is introduced as a fast solver of the equa-tions of beam and spin motion of charged particles in electromagnetic fields. We show that, basedon the stochastic Galerkin method, our computational framework substantially reduces the re-quired number of tracking calculations compared to the widely used Monte Carlo method.3. Control of systematic uncertainties in the storage ring search for an electric dipole moment bymeasuring the electric quadrupole moment [8]Measurements of electric dipole moment (EDM) for light hadrons with use of a storage ring havebeen proposed. The expected effect is very small, therefore various subtle effects need to beconsidered. In particular, interaction of particle’s magnetic dipole moment and electric quadrupolemoment with electromagnetic field gradients can produce an effect of a similar order of magnitudeas that expected for EDM. This paper describes a very promising method employing an RF Wienfilter, allowing to disentangle that contribution from the genuine EDM effect. It is shown that boththese effects could be separated by the proper setting of the RF Wien filter frequency and phase. Inthe EDM measurement the magnitude of systematic uncertainties plays a key role and they shouldbe under strict control. It is shown that particles’ interaction with field gradients offers also thepossibility to estimate global systematic uncertainties with the precision necessary for an EDMmeasurement with the planned accuracy.4. Extraction of Azimuthal Asymmetries using Optimal Observables [9]Azimuthal asymmetries play an important role in scattering processes with polarised particles.This paper introduces a new procedure using event weighting to extract these asymmetries. Itis shown that the resulting estimator has several advantages in terms of statistical accuracy, bias,assumptions on acceptance and luminosities com- pared to other estimators discussed in the liter-ature.5. Amplitude estimation of a sine function based on confidence intervals and Bayes’ theorem [10]This paper discusses the amplitude estimation using data originating from a sine-like function asprobability density function. If a simple least squares fit is used, a significant bias is observedif the amplitude is small compared to its error. It is shown that a proper treatment using theFeldman-Cousins algorithm of likelihood ratios allows one to construct improved confidence in-tervals. Using Bayes’ theorem a probability density function is derived for the amplitude. It isused in an application to show that it leads to better estimates compared to a simple least squaresfit.6. General dynamics of tensor polarisation of particles and nuclei in external fields [11]The tensor polarisation of particles and nuclei becomes constant in a coordinate system rotatingwith the same angular velocity as the spin, and it rotates in the laboratory frame with the aboveangular velocity. The general equation defining the time dependence of the tensor polarisation is148erived. An explicit form of the dynamics of this polarisation is found in the case when the initialpolarisation is axially symmetric.
A.2
Results and achievements from the Jülich/Bonn theory group
The IKP-3/IAS-4 at the Forschungszentrum Jülich together with the theory group at the Helmholtz-Institut für Strahlen- und Kernphysik (HISKP) at the University of Bonn – both headed by Ulf Meißner- have performed a number of benchmark calculations for the EDMs of proton, neutron and light nucleiusing chiral effective nuclear field theory (chiral perturbation theory and its extension to few-baryonsystems) and lattice QCD simulations.This project on hadronic electric dipole moments started with the diploma thesis of KonstantinOttnad (HISKP) on electric dipole form factors of the neutron in chiral perturbation theory in the year2009 [12]. His work culminated in a publication [13] that analysed the QCD ¯ θ -angle induced EDMs ofthe neutron and proton to third order in U(3) L × U(3) R baryon chiral perturbation theory, in a covariantand by the number of colours ( N c ) extended version. A new upper bound on the vacuum angle, | ¯ θ | (cid:46) . · − was given and the matching relations for the three-flavor representation to the SU(2) case wasderived. These relations still comprise today’s ¯ θ -induced EDM predictions for the neutron and proton inchiral perturbation theory.In 2012 IAS-4/IKP-3 extended the above work to the QCD ¯ θ -term-induced electric dipole mo-ment (EDM) of the deuteron, where the genuine two-nucleon contributions of the P - and T -violatingform factor F of the deuteron was calculated in the Breit frame of this nucleus using chiral effectivefield theory up to and including next-to-next-to-leading order [14]. In particular, it was found that thedifference between the deuteron EDM and the sum of proton and neutron EDMs corresponds to a valueof (0 . ± .
39) ¯ θ × − e.cm. Both the nucleon-nucleon potential and the transition current contri-butions were calculated, where the CP - and isospin-violating πN N coupling constant g θ was identifiedas the source of the dominating contribution to the uncertainty. The role that the vacuum alignmentplays for the generation of g θ was outlined and an estimate of the additional and previously unknowncontribution to g θ was derived from a resonance saturation mechanism involving the odd-parity nucleonresonance S (1535) .In the same year Guo (HISKP) and Meißner calculated the electric dipole form factors and mo-ments of the ground state baryons in chiral perturbation theory at next-to-leading order [15]. It wasshown that the baryon electric dipole form factors at this order depend only on two combinations oflow-energy constants. This was used to derive various relations for the baryon EDMs that are free ofunknown low-energy constants which can be used to cross-check future lattice QCD results. Thus fora precision extraction from lattice QCD data, the next-to-leading order terms have to be accounted for.Akan (HISKP), Guo and Meißner revisited in 2014 the above work by investigating finite volume cor-rections to the CP -odd nucleon matrix elements of the electromagnetic current, which can be related tothe electric dipole moments originating from strong CP violation in the continuum, in the framework ofchiral perturbation theory up to next-to-leading order taking into account the breaking of Lorentz sym-metry [16]. A chiral extrapolation of the recent lattice results of both the neutron and proton electricdipole moments was performed in addition.In 2014 Jan Bsaisou (IKP-3/IAS-4) finished his PhD thesis at the university of Bonn on electricdipole moments of light nuclei in chiral effective field theory [17]. Starting from the QCD ¯ θ -term and theset of P - and T -violating effective dimension-six operators, he presented a scheme to derive the inducedeffective Lagrangians at energies below Λ QCD ∼
200 MeV within the framework of chiral perturbationtheory (ChPT) for two quark flavors – applying the formulation of Gasser and Leutwyler. It was shown The estimate is modulo the unknown contributions of the contact interactions needed to removed the infinities of theone-loop calculations. Part of this work was documented in the prior publication [14]. P and T violation manifest themselves in specific hierarchiesof coupling constants of P - and T -violating vertices. He computed the relevant coupling constants of P-and T-violating hadronic vertices which are induced by the QCD ¯ θ -term with well-defined uncertaintiesas functions of the parameter ¯ θ . The relevant coupling constants induced by the effective dimension-six operators were given as functions of so far unknown Low Energy Constants (LECs) which can notbe determined by ChPT. Estimates of the coupling constants from Naive Dimensional Analysis (NDA)proved to be sufficient to reveal certain hierarchies of coupling constants. The different hierarchies ofcoupling constants translated into different hierarchies of the nuclear contributions to the EDMs of lightnuclei. In this way he could calculate within the framework of ChPT the two-nucleon contributions to theEDM of the deuteron up to and including next-to-next-to leading order and the two-nucleon contributionsto the EDMs of the helion ( He nucleus) and the triton ( H nucleus) up to and including next-to-leadingorder. These computations comprised thorough investigations of the uncertainties of the results from boththe P - and T -violating and conserving components of the nuclear potential. Quantitative predictions ofthe nuclear contributions to the EDMs of the deuteron, the helion and the triton induced by the QCD ¯ θ -term as functions of ¯ θ with well-defined uncertainties were presented, while the EDM predictions forthe effective dimension-six sources were given as function of the unknown LECs with NDA estimates.Several strategies to falsify the QCD ¯ θ -term as a relevant source of P and T violation were presented,whereby a suitable combination of measurements of several light nuclei and, if needed, supplementarylattice QCD input could be used. He demonstrated how particular effective dimension-six sources canbe tested by EDM measurements of light nuclei – with supplementary Lattice QCD input in the future.While the above thesis discussed strategies to separate the various dimension-six EDM operatorsindividually, the IAS4-/IKP-3 publication by Dekens et al. [18], using information from this thesis andfrom the paper by Dekens and de Vries [19] on the renormalisation group running of the dimension-sixsources for P and T violation, showed that the proposed measurements of the electric dipole moments oflight nuclei in storage rings would put strong constraints on models of flavor-diagonal CP violation [18].This analysis was exemplified by a comparison of the Standard Model including the QCD theta term,the minimal left-right symmetric model, a specific version of the so-called aligned two-Higgs doubletmodel, and, “en passant”, a minimal supersymmetric extension of the Standard Model. Again by usingeffective field theory techniques it was demonstrated to what extent measurements of the electric dipolemoments of the nucleons, the deuteron, and helion could discriminate between these scenarios and howmeasurements of electric dipole moments of other systems relate to light-nuclear measurements. Inparticular, the focus was on the most important P -, T -violating hadronic interactions that appear in eachof the scenarios, especially on the P -, T -violating pion-nucleon interactions and the nucleon EDMs. Itwas demonstrated that chiral effective field theory is a powerful tool to study the observables of lightnuclei and that measurements of light-nuclear EDMs can be used to disentangle different underlyingscenarios of CP violation.The EDM predictions of IAS-4/IKP-3 up to the year 2014 were summarised in Ref. [20], and aconsistent and complete calculation of the electric dipole moments of the deuteron, helion, and triton bychiral effective field theory was given in Ref. [21]. The CP -conserving and CP -violating interactionswere treated on equal footing and the CP -violating one-, two-, and three-nucleon operators were con-sidered up to next-to-leading-order in the chiral power counting. In particular, for the first time EDMcontributions induced by the CP -violating three-pion operator were calculated. It was found that ef-fects of CP -violating nucleon-nucleon contact interactions are larger than those predicted in previousstudies involving phenomenological models for the CP -conserving nucleon-nucleon interactions. Theresults which apply to any model of CP violation in the hadronic sector can be used to test variousscenarios of CP violation. In particular, the implications on the QCD ¯ θ -term and the minimal left-rightsymmetric model were demonstrated. Furthermore, in Ref. [22] the underlying scheme was presentedto derive – within the framework of chiral effective field theory - the list of parity- and time-reversal-symmetry-violating hadronic interactions that are relevant for the computation of nuclear contributions150o the electric dipole moments of the hydrogen- 2, helium-3 and hydrogen-3 nuclei. The scatteringand Faddeev equations required to compute electromagnetic form factors in general and electric dipolemoments in particular were documented there in addition.In 2015 Shindler, Luu and de Vries (IAS-4/IKP-3) proposed a new method to calculate electricdipole moments induced by the strong QCD ¯ θ -term [23]. The authors based their method on the gra-dient flow for gauge fields which is free from renormalisation ambiguities. The method was tested bycomputing the nucleon electric dipole moments in pure Yang-Mills theory at several lattice spacings,enabling a first-of-its-kind continuum extrapolation that is theoretically sound.In the same year Guo et al. (2015) [25] presented an entirely dynamical calculation of the electricdipole moment of the neutron on the lattice. They computed the electric dipole moment d n of the neutronfrom a fully dynamical simulation of lattice QCD with 2 + 1 flavors of clover fermions and nonvanishing θ -term. The latter was rotated into a pseudoscalar density in the fermionic action using the axial anomaly.To make the action real, the vacuum angle θ was taken to be purely imaginary. The physical value of d n was obtained by analytic continuation ( d n = − . × − θ e.cm) and an upper bound on theQCD theta angle ( | θ | (cid:46) . × − ) was presented.In 2016 Meißner and de Vries reviewed the progress in the theoretical description of the violationof discrete space-time symmetries in hadronic and nuclear systems [26]. They focused on parity-violatingand time-reversal-conserving interactions which are induced by the weak interaction of the StandardModel, and on parity- and time-reversal-violating interactions which can be caused by a nonzero QCDtheta term or by beyond-the-Standard Model physics. Especially, they reviewed the development of thechiral effective field theory extension that includes discrete symmetry violations and discussed the con-struction of symmetry-violating chiral Lagrangians and nucleon-nucleon potentials and their applicationsin few-body systems. In their review of the parity- and time-reversal violation, of course informationfrom the above mentioned HISKP and IAS-4/IKP3 publications was used, but also results of three recentpublications coauthored by the IAS-4 member de Vries were integrated: the first on the constraint ofthe neutron EDM on the value of the CP -and isospin-violating pion-nucleon coupling constant g in thecase of dimension-6 interactions [27], the second on the extension to SU(3) chiral perturbation theory andthe update on the determination of the CP -violating isospin-conserving pion-nucleon coupling constant g θ [28], and the third on direct and indirect constraints on the complete set of anomalous CP -violatingHiggs couplings to quarks and gluons originating from dimension-6 operators [29].In 2017 Wirzba, Bsaisou and Nogga [30] gave an update on the predictions of Refs. [21, 22],especially by extending the computation of the relevant matrix elements of the nuclear EDM operatorsin the deuteron case to the N4LO level of chiral effective field theory. Furthermore, they incorporated areview about the underlying principle that the existence of a nonzero EDM of an elementary or compositeparticle (in fact, of any finite system) necessarily involves the breaking of a symmetry, either by thepresence of external fields ( i.e. electric fields leading to the case of induced EDMs) or explicitly by thebreaking of the discrete parity and time-reflection symmetries in the case of permanent EDMs.In a series of publications, a collaboration including a current and two former members of IAS-4/IKP-3 refined the method of Ref. [23] by extending it from the calculation of EDMs induced by thestrong QCD ¯ θ -term [31] to include the dimension-6 Weinberg term [32], and the quark-chromo EDMoperator [33]. This work accumulated in Ref. [34] where the electric dipole moment of the nucleoninduced by the QCD theta term was calculated in the gradient flow method with N f = 2 + 1 flavorsof dynamical quarks corresponding to pion masses of (700, 570, and 410) MeV which are used byperforming an extrapolation to the physical point based on chiral perturbation theory. The calculationsapplied 3 different lattice spacings in the range of .
07 fm < a < .
11 fm at a single value of thepion mass, to enable control on discretisation effects. Also finite size effects were investigated using 2different volumes. A novel technique was applied to improve the signal-to-noise ratio in the form factor In fact, their method was already documented in the publication [24] in a more broader context. d n / ¯ θ = − . · − e.cm and d p / ¯ θ = 1 . . · − e.cm. Assuming the theta term is the only source of CP violation, the experimental bound on the neutron electric dipole moment limits was predicted as | ¯ θ | < . · − . References [1] D. Eversmann et al. , “New method for a continuous determination of the spin tune in storage ringsand implications for precision experiments,”
Phys. Rev. Lett. , vol. 115, no. 9, p. 094801, 2015,1504.00635.[2] G. Guidoboni et al. , “How to Reach a Thousand-Second in-Plane Polarization Lifetime with 0.97-GeV/c Deuterons in a Storage Ring,”
Phys. Rev. Lett. , vol. 117, no. 5, p. 054801, 2016.[3] N. Hempelmann et al. , “Phase locking the spin precession in a storage ring,”
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Nucl. Instrum. Meth. , vol. A859, pp. 52–62, 2017, 1612.09235.[6] J. Slim et al. , “Electromagnetic Simulation and Design of a Novel Waveguide RF Wien Fil-ter for Electric Dipole Moment Measurements of Protons and Deuterons,”
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Phys. Lett. , vol. B687, pp. 42–47, 2010, 0911.3981.[14] J. Bsaisou, C. Hanhart, S. Liebig, U.-G. Meißner, A. Nogga, and A. Wirzba, “The electric dipolemoment of the deuteron from the QCD θ -term,” Eur. Phys. J. , vol. A49, p. 31, 2013, 1209.6306.[15] F.-K. Guo and U.-G. Meißner, “Baryon electric dipole moments from strong CP violation,”
JHEP ,vol. 12, p. 097, 2012, 1210.5887.[16] T. Akan, F.-K. Guo, and U.-G. Meißner, “Finite-volume corrections to the CP-odd nucleon matrixelements of the electromagnetic current from the QCD vacuum angle,”
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Electric Dipole Moments of Light Nuclei in Chiral Effective Field Theory . PhD thesis,152onn University, April 2014.[18] W. Dekens, J. de Vries, J. Bsaisou, W. Bernreuther, C. Hanhart, U.-G. Meißner, A. Nogga, andA. Wirzba, “Unraveling models of CP violation through electric dipole moments of light nuclei,”
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The EDM signal can be mimicked by magnetic fields in several different ways. The most critical effectcomes from a static radial magnetic field, requiring a cancellation down to attoTesla level. Static longi-tudinal magnetic field has a similar effect, but with a few orders of magnitude more flexible restriction.Moreover, several configurations of alternating magnetic fields result in EDM-like spin precession too.We studied each of these scenarios and proposed solutions to cancel the effect. Throughout this section, static and alternating fields refer to the particle’s rest frame. As an exam-ple, the earth’s field is alternating in the particle’s rest frame even if it is purely static in the lab frame.
B.1
Static magnetic field configurations
B.1.1 Static radial magnetic field
As Figure B.1 shows, the static radial magnetic field should be kept at attoTesla level to avoid the sys-tematic error (assuming d p = 10 − e · cm). This is obviously not possible with magnetic shielding alone.It should be measured and compensated actively. Our proposal is to measure the relative position of thecounter-rotating beams, proportional to the average radial magnetic field. For the all-electric baselinering, an attoTesla level field splits the counter-rotating beams vertically by picometers. The split beamsinduce a magnetic field in the horizontal direction. The magnitude of this field ( B x ) can be measured bya magnetometer/gradiometer at a few cm horizontal distance (Figure B.2). Fig. B.1:
Assuming d p = 10 − e · cm for the baseline ring, 10 MV/m electric field and 17 aT magnetic field resultin similar spin precession in the vertical plane. We are planning to use SQUID-based beam position monitors (BPM) to measure B x . In orderto suppress the environmental noise, the vertical motion of the beams will be modulated at 1-10 kHzby means of the quadrupoles. The typical white noise of the DC SQUIDs at that range is less than 1 This appendix was authored by Y.K. Semertzidis and S. Haciomeroglu of the Center for Axion and Precision PhysicsResearch, KAIST, South Korea. √ Hz. B x due to the split beams in such a case is given as B x ( t ) = µ I ∆ yπr A cos( ω m t ) (B.1)with the beam current I , vertical split ∆ y , horizontal distance between the pickup loop and the beams r ,and the modulation amplitude and frequency A and ω m respectively. Putting I = 10 mA, ∆ y = 0 . pm, A = 0 . and r = 2 cm into Eq B.1 gives B x ≈ aT cos( ω m t ) .As a reference, with an array of 8 SQUIDs of − T sensitivity at 1 Hz bandwidth (1 fT / √ Hz),it requires × seconds of averaging to achieve SNR > as B = 10 − fT / √ × × = 0 . aT. Fig. B.2:
A magentometer can pick up the magnetic field of horizontal direction, that is induced by the verticallysplit counter-rotating beams.
B.1.1.1 Preliminary tests with SQUID-based BPM
SQUID-based magnetometers can measure magnetic field variations with unprecedented noise level be-low fT / √ Hz. This is why they became the best candidates for the beam position monitors in the pEDMexperiment. In addition to high resolution, the SQUID-based magnetometers have sufficient bandwidthand compact size that allows using multi sensor arrays placed along the beam trajectory inside a super-conductive shielding structure.Figure B.3 shows the 3D drawing of the BPM. It will operate in vacuum at 4 Kelvin. They arepositioned on the horizontal plane to measure the vertical split.The BPM works inside a magnetically shielded room (MSR). The data transfer between theSQUIDs and the computer is done via fiber lines to minimise electromagnetic noise. Figure B.4 showsthe picture of the first prototype.The operation frequency was chosen to be around 1-10 kHz for minimising the external noise.This will be achieved by modulating the vertical tune Q y as mentioned above.We conducted preliminary tests with a setup having a similar SQUID electronics but a differentdesign of pickup loop geometry and Dewar. The Dewar and the eight SQUID gradiometers are shown155 ig. B.3: The beam (white arrow) will pass between two arrays of SQUID gradiometers. The SQUID sensor andthe pickup loops (dark blue) will be kept cold to LHe level. Above the SQUIDs is the LHe tank, shown withturquoise layer. The whole setup can fit in a cube of 1m volume. in Figure B.5. They were originally designed at KRISS/Korea for biomagnetic applications. It has 8axial wire-wound first-order gradiometers positioned along a bottom line inside a fiberglass Dewar. Eachgradiometer has 20 mm diameter and 50 mm baseline and bonded to the double relaxation oscillation(DROS) SQUID current sensor. DROS SQUIDs have a large flux-to-voltage transfer coefficient thatminimises the contribution of the direct read-out electronics noise. The white noise of the gradiometersis about fT / √ Hz at frequencies above 1 Hz.For these measurements with long time averaging, the magnetic field was generated by two paralleltraces of a µ m separation on a PCB, and carrying opposite currents of µ A. The applied currentwas a 300 Hz sinusoidal AC, corresponding to around 200 fT amplitude field at the pickup loop location.The measurements showed more than two orders of magnitude suppression of the white noise fromthe gradiometers and the SQUID read-out electronics (Figure B.6). This corresponds to ≈ aT / √ Hzwith 5-hour averaging. This indicates very high long-time stability and low intrinsic fluctuation level inthe instrument that includes all cryogenic and semiconductor, both analog and digital electronics.The real design proposed for the experiment (Figure B.4) includes 16 magnetometers with two-turn 17 mm diameter pick-up coils bonded to DROS SQUIDs. It allows us to expect more than 3 timeslower white noise floor, i.e. about aT / √ Hz, after 5 hour averaging. For the further noise decrease weexpect using single-chip integrated magnetometers similar to ML12 reported in [2] but with chip size × mm . Such magnetometers have white noise below . fT / √ Hz at frequency above 1 kHz.In the hybrid ring design, the compensation of the radial magnetic field does not have to be so strict,because the magnetic focusing mechanism makes a partial cancellation. According to the simulations,156 ig. B.4:
The first prototype of the BPM. The three layers of the dewar and the LHe tank are covered withaluminized mylar. The partially inserted half cylinder below the LHe tank has housings for the SQUIDs. The otherhalf was not inserted for easier visibility. the restriction releases by five orders of magnitude.
B.1.2 Static longitudinal magnetic field
A static longitudinal magnetic field can appear in the presence of an electric current passing through thehorizontal plane at the inner side of the ring. For instance, 25 mA current passing through the centre ofthe ring induces B L ≈ nT. In such a case, the spin precession s V on the vertical plane becomes s V ( t ) = egB L mγω a (cid:104) cos( ω a t ) − (cid:105) . (B.2)where g , e and m are the magnetic anomaly, electric charge and mass of the proton, γ is the relativisticLorentz factor and ω a is spin precession rate on the vertical plane. Note that s V becomes quadratic if ω a has a constant nonzero value. Figure B.7 shows the spin components on the horizontal ( s r ) and vertical( s V ) directions in the presence of 50 pT longitudinal and vertical magnetic fields. While the verticalmagnetic field does not affect s V directly, it has an indirect effect via ω a . As seen in the plot, s V growsmuch faster compared to the EDM signal. B.1.2.1 Eliminating the effect of the longitudinal magnetic field
One needs to have a 1 fT level average magnetic field in the vertical and longitudinal directions to reducethe effect to the level of the EDM signal (nrad/s).The spin precession rate on the vertical plane is ω R = em g γ B L s R . (B.3)with s R , the horizontal spin component. As the equation shows, the effect of B L enhances proportionallywith s R . This effect can be exploited by using a radially polarised test bunch. According to EquationB.3, the spin precession rate from B L = 1 fT is ω R = 2 . × × − = 220 nrad/s without anycontribution from the EDM as (cid:126)s × (cid:126)E = 0 Monitoring that bunch with the polarimeter, its ω R can befrozen by applying an inverse longitudinal magnetic field with 1 fT resolution.157 ig. B.5: Time-averaging measurements were done with a setup having the same electronics but different Dewar(Left) and the gradiometer (Right) designs. -3 -2 -1 fT / Hz PS D ( f T / H z / ) f ( kHz ) Δ t = sec Δ t = min Δ t = hr Fig. B.6: µ A current was applied through the parallel traces, resulting 200 fT on the pickup. The noise at 1s isa few fT / √ Hz, consistent with the 3 fT / √ Hz sensitivity of the SQUIDs. The noise decreases down to 25 aT / √ Hzafter 5 hours of averaging.
B.1.3 Static vertical magnetic field
As seen in the above section, the static vertical magnetic field does not have a direct effect on s R . But itenhances the effect of the longitudinal field. It can be cancelled similar to the static radial magnetic fieldcase. But this time the field requirement is much more flexible. B.2
Effect of alternating magnetic fields and the geometric phases
We have studied the major configurations of the magnetic field in a continuous ring. In each case, wehave simulated pairs of 1 nT fields at perpendicular directions with different phases. In some cases158 s r ( µ r a d ) t (ms) -80-60-40-200 0 0.3 0.6 0.9 s V ( p i c o r a d ) t (ms) simulationanalytical Fig. B.7:
The spin components as simulated for 1 ms storage time with a magic particle in an electric ring. Becauseof the short storage time compared to one cycle of ω a , s R changes linearly, and s V approximates to a quadraticfunction (See Equation B.2). Left : 50pT vertical magnetic field causes ω a ≈ . mrad/s on the horizontal plane. Right : Having linear dependence on ω a , s V has quadratic dependence on time. Combination of 50pT longitudinaland vertical static magnetic fields grows the vertical spin component up to 67 prad , matching well with theanalytical estimation. Table B.1:
Summary of the major independent magnetic field configurations. (cid:104) ω r (cid:105) is the average spin precessionrate on the vertical plane. Each simulation was done with 1 nT magnetic field strength. Field AC Phase (cid:104) ω r (cid:105) [rad/s] SolutionDC B R n/a 0.18 Measurement and active cancellationwith BPMsDC B L n/a < . × − , proportional to ω a Current to be limited to < mA andDC B V to be avoidedDC B V n/a 0 Can be avoided with BPM similar to B R case B V & B L ◦ × − CW/CCW cancel B R & B V ◦ . × − CW/CCW average out B R & B L , ◦ < − CW/CCW average out B R & B V ◦ < − CW/CCW average out B V & B L ◦ Negligiblewe have seen the spin growing much faster than the EDM signal, like in the case of longitudinal andvertical magnetic fields with ◦ phase difference ( B V & B L , ◦ of Table B.1). Some configurations areharmless as they average out themselves. Some of them cancel out thanks to the counter-rotating beamdesign.Table B.1 summarises all of the studied cases, including static (DC) and alternating field config-urations. According to our studies, the effect of the magnetic field can be kept under control by meansof – SQUID-based BPMs for static radial magnetic field– less sensitive BPMs for static vertical magnetic field– a radially polarised test bunch for the static longitudinal magnetic field– counter-rotating beams. 159hile the coupling between the magnetic fields of perpendicular directions is harmless in a continuousring, the coupling between the beta function and some multipoles of an alternating radial magnetic fieldsplits the beams the same way as a static radial magnetic field. Our simulations show that the magneticfield must be smooth down to 1 pT level to avoid this systematic error. As will be seen in the next section,we have shown that the magnetic field along the shielding prototype is smooth at the level 10 pT withinthe storage time. Another one or two orders of magnitude can be gained by flipping the quadrupole signsbetween runs.We have also proposed the hybrid ring design to avoid this problem. Magnetic focusing in thebaseline ring compensates the external fields effectively, suppressing the above-mentioned systematicerror significantly, B.3
Magnetic shielding
We are considering the magnetic shielding for keeping the beam more stable in the presence of largetransient fields. We have designed a prototype, collaborating with P. Fierlinger’s group at TUM/Germany(Figure B.8). It contains two layers of magnifer, a high permeability material for low frequency shielding.High frequency shielding requires a material with high conductivity, like aluminium. The shielding factorof the system is approximately 500 at low frequencies.
Fig. B.8:
The magnetic shielding prototype was developed in collaboration with Fierlinger Magnetics, a Germany-based company. It contains two layers of high permeability material, separated by ≈ cm. The thickness of eachlayer is 1mm. The length is approximately 2.5 m. The working principle of the magnifer relies on the domain structure inside it. The direction ofmagnetisation is uniform in these small regions, separated with the so-called domain walls. Externalmagnetic field can move the domain walls, changing the total magnetisation of the material. The shield-ing structure gets magnetised over time because of this effect. Demagnetisation (or degaussing) is acommonly used method to avoid it. It is basically conducted by application of alternating field with adecreasing amplitude. This has an effect similar to shaking, which randomises the domain magnetisationover the material. The red cable in Figure B.8 is used for applying a current for degaussing. Our studiesshowed that the uniformity of the cables along the material matters for the degaussing performance at theinner layer, but not at the outer. Therefore unlike the outer one, the inner magnifer layer has uniformlydistributed degaussing cables. 160 .3.1 Residual field
There are several key factors for the performance of degaussing. First of all, the amplitude of the appliedmagnetic field should be large enough to saturate the material. The cycles should be slow enough to letthe domain movement. ( ≈ Hz for this prototype). The last steps of degaussing should be smoothenough for more evenly distributed domain configuration. At the end, the material would still have anonzero magnetisation which results in the so-called “residual field” inside the shielded volume. FigureB.9 shows the residual field measurement inside the prototype after degaussing. As seen, 1 nT field canbe easily achieved with two-layer shielding after degaussing.
Fig. B.9:
Residual field measurement inside the prototype. The x-axis is the longitudinal position of the fluxgatesensors. The field is larger at the edges due to the caps of the prototype, which will not be used when installed atthe ring.
B.3.2 Time stability of the residual field
Time stability of the residual field becomes critical especially when the beta function of the beam is notuniform. Coupling between the varying beta function and the magnetic field moves the beam vertically,mimicking a DC radial magnetic field. We proposed to change the polarity of the quadrupoles to cancelthis effect. According to our simulations, this requires a stable residual field along the ring to < pTlevel. We tested the prototype inside our magnetically shielded room (MSR) as seen in Figure B.10.In the tests we used only the outer layer of the prototype. Then, after degaussing it, we measured themagnetic field inside the prototype. Figure B.11 shows the field at three locations along the axis separatedby 70 cm. The measurement lasted almost 25 hours. The variation of the field is mainly related to thetemperature. It decreases over night and increases after the sunrise. Of course the stability during thewhole day is irrelevant in the pEDM experiment. Rather, we are interested with the stability within onestorage or two.Figure B.12 zooms the morning period of Figure B.11, where the temperature changes the mostrapidly. According to the plot, the change in 20 minutes is around 100pT. For the effect mentioned above,the beta function varies at different locations in the ring. Therefore one needs to look at the correlationbetween different points. If the field changes the same way, then the beta function will not vary the sameway with the magnetic field and the effect will be smaller.161igure B.13 shows the difference between two sensors at the same region with Figure B.12. Thedistance between those two points is 1.4m The residual field changes by 10 pT over the period of 20minutes. Fig. B.10:
Time stability measurements were done inside the MSR.
To sum up, we have a prototype to prevent the effects from transient magnetic fields in the pEDMexperiment. Its residual field is as low as 1 nT with good time stability and field uniformity along thecylindrical axis. The time stability within 1.4 meter is measured to be ≈ pT within 20 minutes. B.4
Summary
Our studies show that we can keep the static magnetic field under control in an alternating gradient,all-electric ring. The active and passive cancellation of the magnetic field requires several componentsincluding counter-rotating beams, beam position monitors, a test bunch with horizontal polarisation andmagnetic shielding. The tests that we have made with the SQUID-based BPMs and the magnetic shield-ing prototype yielded promising results.Alternating magnetic fields are harmless in a continuous all-electric ring. But the coupling be-tween the beta function and the alternating radial magnetic field causes a vertical split similar to thestatic radial magnetic field. This can be suppressed by flipping the quadrupoles at every run and keepingthe residual field uniformity at 10 pT level. But, the hybrid ring is a more efficient solution to this prob-lem. Overall, according to our simulations, the hybrid ring has more flexible requirement for the fieldcancellation in all scenarios. 162 ������� � � � � � � � � � � � � � � � �� � �� � �� � ���� � ������� � ������ � ��� � � ��� � � � � � ��� ������ � �� � �� � Fig. B.11:
Magnetic field is measured over the course of 25 hours. The sensors were located at several locationsalong the prototype. The dominant reason for the change of the field is the temperature. ������������������������������ � ���� � ���� � �� � ���� � ���� � ���� � ���� � ���� � � ��� � � � � � ��� ������ � Fig. B.12:
The protons will be stored for 20 minutes in the ring. Therefore the stability of the field at that periodis important. Zooming in the depicted 20 minute region of Figure B.11, one sees that the field change is ≈ pT.Note that this is the worst period of the measurement, where the temperature changes rapidly. �������������������������������� � ���� � ���� � �� � ���� � ���� � ���� � ���� � ���� � � ��� � � � � � ��� ������ � ��� � Fig. B.13:
The field at different locations around the ring are quite correlated. The difference between the field140cm apart change together in the measurements. The difference is ≈ pT level. eferences [1] I. Altarev et al. , A large-scale magnetic shield with damping at millihertz frequencies , Journalof Applied Physics , 183903 (2015) DOI: https://doi.org/10.1063/1.4919366[2] M. Schmelz, R Stolz, V Zakosarenko, T Schonau, S Anders, L Fritzsch, M Muck and H-GMeyer, Field-stable SQUID magnetometer with sub-fT / √ Hz resolution based on sub-micrometercross-type Josephson tunnel junctions , Supercond. Sci. Technol. (2011) 065009(5pp) DOI:10.1088/0953-2048/24/6/065009 164 ppendix CStatistical Sensitivity C.1
Statistical error on EDM
The spin motion, relative to the momentum vector, is governed by the BMT-equation d (cid:126)S d t = (cid:16) (cid:126) Ω MDM + (cid:126) Ω EDM (cid:17) × (cid:126)S (C.1) Ω MDM = − qm (cid:34) G (cid:126)B + (cid:18) G − γ − (cid:19) (cid:126)v × (cid:126)Ec (cid:35) (C.2) Ω EDM = − ηq mc (cid:104) (cid:126)E + (cid:126)v × (cid:126)B (cid:105) (C.3)with (cid:126)d = η q mc (cid:126)s , and (cid:126)µ = 2( G + 1) q m (cid:126)s . For this discussion (cid:126)B and (cid:126)E denote a vertical magnetic anda radial electric field, respectively.For a pure electric ring the angular precession frequency due to the EDM is given by (cid:126) Ω EDM = ηq mc (cid:126)E . (C.4)One finds Ω EDM = dEs (cid:126) (C.5)Using the numerical values of Tab. C.1 for protons (i.e. s = 1 / ) one arrives at Ω EDM = 2 . · − s − . (C.6) N · particles per fill · CW, · CCW E T cyc P A f s d − e cm EDM Table C.1:
Parameters used to evaluate the statistical error.
To evaluate the statistical error we discuss three different scenarios:1. There is only a precession due to the EDM, i.e. one observes only the initial linear rise of thepolarisation vector because Ω EDM τ (cid:28) The polarisation is continuously measured indicated bythe points in the graph below: 165 time/a.u. v e r t i c a l po l / a . u .
2. As for scenario 1), but half of the beam is extracted at t = 0 , the other half is extracted at t = τ : time/a.u. v e r t i c a l po l / a . u .
3. In this scenario, the precession is dominated by systematic effects, one observes thus many oscil-lations during the cycle of length T cyc : time/a.u. - - v e r t i c a l po l / a . u . In all three cases it is assumed that the EDM is extracted form the difference and sums of the polarisationsof the CW and CCW measurements.1) Assuming a polarisation vector initially along the momentum vector, we get ˙ P ⊥ = Ω EDM P = dEs (cid:126) P , resulting in d = s (cid:126) ˙ P ⊥ EP . (C.7)166n general the variance on the slope parameter s of a straight line is V ( s ) = σ N points V ( t ) where σ is the error on each individual point. N points is the number of points entering the fit and V ( t ) is the variance of the points along the horizontal axis (i.e. time in the EDM case). For evenly distributedvalues in time, one has V ( t ) = T cyc / . The slope in the EDM measurement is just ˙ P . The error on onepolarisation measurement, determined from the azimuthal distribution of events, is σ P = 2( N f /N points ) A . (C.8)The variance on the slope ˙ P is, thus σ P ⊥ = 2(( N f ) /N points ) A N points T cyc = 24( N f )( AT cyc ) From eq. C.7 we find σ EDM = s (cid:126) EP σ ˙ P ⊥ (C.9)which results in σ EDM = s (cid:126) EP √ √ N f AT cyc (C.10) = √ s (cid:126) √ N f AP ET cyc (C.11)Here it is assumed that there is no decoherence during T cyc .2) Taking only two measurements at t = 0 and t = τ , the slope is determined by ˙ P ⊥ = P ( T cyc ) − P (0) T cyc . (C.12)Using σ P (0) = σ P ( T cyc ) = 4( N f ) A one finds σ P ⊥ = 2 T cyc N f ) A = 8( N f )( AT cyc ) resulting in σ d = s (cid:126) EP σ ˙ P ⊥ (C.13) = s (cid:126) EP √ √ N f AT cyc (C.14) = √ s (cid:126) √ N f AP ET cyc (C.15)3) According to Ref. [1] the error on the frequency is given by σ = 2 24( N f )( AP T cyc ) Ω EDM = dEs (cid:126) one finds for the error on d σ d = √ s (cid:126) √ N f AP ET cyc
C.2
Precursor Experiment
For the precursor experiment the build-up of the vertical polarisation is given by P ⊥ = n ηβ G emc (cid:18) G + 1 γ β (cid:19) EL P . (C.16)One finds the following expression for the error on the EDM: σ EDM = (cid:12)(cid:12)(cid:12)(cid:12) (cid:126) s (cid:18) Gγ G + 1 (cid:19) ULE (cid:12)(cid:12)(cid:12)(cid:12) √ √ N f AP τ (C.17)To arrive at eq. C.7 the number of turns n were replaced by the time of the measurement τ times therevolution frequency f rev , n = τ f rev . The revolution frequency can be expressed as f rev = βc/U ,where U is the circumference of the ring.Using the following parameters G − . deuterons s γ p = 1 GeV/ cE · L . kV corresponds to B · L = 0 . Tmm τ P A N particles per fill f U σ EDM (1 fill ) = 8 . · − e · cm , per fill of 1000s. C.3
Summary
The statistical error on the EDM d is given by σ EDM = α β pr s (cid:126) √ N f AP rET cyc with β pr = (cid:40) = Gγ G +1 precursor experiment , = 1 prototype & final ringThe factor α depends on the way the polarisation is measured and on the spin coherence time τ . Thefactor r added is the fraction of the ring equipped with E -field (or Wien filter, in the case of the precursor168.) 2.) 3.) time/a.u. v e r t i c a l po l / a . u . time/a.u. v e r t i c a l po l / a . u . time/a.u. - - v e r t i c a l po l / a . u . pol. const. during T cyc √ ≈ . √ ≈ . √ ≈ . ( τ = ∞ ) P = P e − t/τ , T cyc ≈ τ Table C.2:
Factor α for various cases. experiment). In section C.1 it was assumed that the polarisation is constant over T cyc . If τ ≈ T cyc , theaverage polarisation is smaller by the factor (cid:82) T cyc e − t/τ dtT cyc = 1 − e − ≈ . assuming an exponential decrease. The error is increased accordingly. The factor α is listed in Tab. C.2for the various cases.For the best (worst) case in the table the errors on d in units of e cm, using the values in Tab. C.1for the final ring are: one cycle one year ( cycles) . × − . × − × − × − References [1] J. Pretz, “Statistical uncertainties of frequency measurements.” JEDI Internal Note 01/2014.169 ppendix DGravity and General Relativity as a ‘Standard Candle’
The Thomas-BMT equation including General Relativity corrections reads [1–3]: d (cid:126)S dt = (cid:16) (cid:126) Ω MDM + (cid:126) Ω EDM + (cid:126) Ω GRgeo (cid:17) × (cid:126)S , (D.1)where, in the Frenet-Serret coordinate system whose axis orientation is determined from the local particlemotion [4], (cid:126) Ω MDM refers to the angular velocity from the magnetic dipole moment minus the cyclotronangular velocity, (cid:126) Ω EDM to the one from the electric dipole moment, and (cid:126) Ω GRgeo to the angular velocityof the geodetic (de Sitter) minus the corresponding angular velocity for the particle revolution: (cid:126) Ω MDM = − qm (cid:34) G (cid:126)B − γGγ + 1 (cid:126)β (cid:16) (cid:126)β · (cid:126)B (cid:17) − (cid:18) G − γ − (cid:19) (cid:126)β × (cid:126)Ec (cid:35) , (D.2) (cid:126) Ω EDM = − ηq mc (cid:20) (cid:126)E − γγ + 1 (cid:126)β (cid:16) (cid:126)β · (cid:126)E (cid:17) + c(cid:126)β × (cid:126)B (cid:21) , (D.3) (cid:126) Ω GRgeo = − γγ − (cid:126)β × (cid:126)gc . (D.4)Here (cid:126)g is the gravitational acceleration at the Earth’s surface – for further definitions see Ref. [1] andChap. 4. Furthermore, according to [1], (cid:126)E and (cid:126)B in Eqs. (D.2) and (D.3) have to be replaced by (cid:126)E + (cid:126)E (cid:126)g and/or (cid:126)B + (cid:126)B (cid:126)g , respectively, where (cid:126)E (cid:126)g and/or (cid:126)B (cid:126)g are focusing fields compensating the gravitationaldownwards pull on the beam particles of mass m and velocity c(cid:126)β , (cid:126)F (cid:126)g = γ (cid:16) | (cid:126)β | (cid:17) m(cid:126)g = 2 γ − γ m(cid:126)g . (D.5)This follows from the storage-ring lattice condition for the closed orbit: γ − γ m(cid:126)g + q (cid:16) (cid:126)E (cid:126)g + c(cid:126)β × (cid:126)B (cid:126)g (cid:17) ≡ , (D.6) e.g. the upwards pointing vertical electric field for a pure electric ring reads (cid:126)E (cid:126)g = ( (cid:126)E (cid:126)g · ˆ y )ˆ y = 2 γ − γ mq ( − (cid:126)g ) , (D.7)while the gravity-compensating radially inwards/outwards pointing magnetic field for a counterclock-wise/clockwise beam would be (cid:126)B (cid:126)g = ( (cid:126)B (cid:126)g · ˆ x )ˆ x = (2 γ − γγ − mq (cid:126)β × (cid:126)gc = 2 γ − (cid:112) γ − m | (cid:126)g | cq ˆ β × ˆ g (D.8) Deviating from the local coordinate system used in Chap. 4, here the right-handed, beam-comoving coordinate system ( x, y, z ) is defined by the unit vectors ˆ z = (cid:126)β/ | (cid:126)β | ≡ ˆ β , ˆ y = − (cid:126)g/ | (cid:126)g | ≡ − ˆ g and ˆ x = − ˆ z × ˆ y = ˆ β × ˆ g , i.e. the unit vector ˆ y is always pointing opposite to the gravitational acceleration (cid:126)g . Thus for a clockwise beam we have ˆ x = ˆ r , while for acounterclockwise beam ˆ x = − ˆ r , where ˆ r is the outside-pointing radial unit vector inside the storage-ring plane. (cid:126) Ω GRgeo is calculated from the difference between the gravity-induced ‘spin-orbit’ precession around a radial axis in theEarth gravitational field, (cid:126) Ω LS (the de Sitter precession aka the geodetic effect) [5–7], and the particle revolution around thesame axis in the Earth’s gravitational field, (cid:126) Ω rev , cf. [1]: (cid:126) Ω LS = 2 γ + 1 γ + 1 (cid:126)β × (cid:126)gc , (cid:126) Ω rev = 1 + β β (cid:126)β × (cid:126)gc = 2 γ − γ − (cid:126)β × (cid:126)gc , (cid:126) Ω GRgeo = (cid:126) Ω LS − (cid:126) Ω rev . c(cid:126)β × (cid:126)B (cid:126)g = (2 γ − γγ − mq (cid:16) (cid:126)β ( (cid:126)β · (cid:126)g ) − (cid:126)g ( (cid:126)β · (cid:126)β ) (cid:17) = 2 γ − γ mq ( − (cid:126)g ) . (D.9)1. Note that for the case of the frozen-spin (fs) condition, / ( γ −
1) := G , in an all-electric ring wehave [7] (cid:126) Ω B =0 , fsGR = − γγ − (cid:126)β × (cid:126)gc (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) fs = − ˆ β × (cid:126)gc G (cid:114) GG √ G = − | (cid:126)g |√ Gc ˆ β × ˆ g , (D.10)which agrees with the earlier result of Orlov, Flanagan, and Semertzidis [8]. Thus the geodeticeffect of general relativity would induce a ‘fake’ proton EDM value of, e.g. , d GR p ≈ . · − e cm ( i.e. η GR p ≈ . · − ) corresponding to E r = 10 MV / m ,d GR p ≈ . · − e cm ( i.e. η GR p ≈ . · − ) corresponding to E r = 1 MV / m , where E r is the mean radial component of the electric field, and could therefore serve as a standardsource or ‘standard candle’ for EDM measurements in frozen-spin all-electric storage rings, whilethe gravity-compensating fields just correspond to E (cid:126)g ≈ . µ V / m or B (cid:126)g ≈ .
967 fT . Suchtiny focusing fields are automatically generated by a minuscule orbit displacement by the Earthgravity pull.2. If the radial component B x = ˆ x · (cid:126)B of the magnetic field is identically zero, the (cid:126)F (cid:126)g compensatingfield only arises from the vertical electric field E y = ˆ y · (cid:126)E and therefore we would have as thegravity-induced contribution to the angular velocity [1, 7] (cid:126) Ω B x =0GR = (cid:126) Ω GRgeo − qm (cid:18) G − γ − (cid:19) (cid:126)β × (cid:126)E (cid:126)g c = 1 − G (2 γ − γ (cid:126)β × (cid:126)gc . (D.11)Obviously, in the frozen-spin case, / ( γ −
1) = G , this result will become the one of Eq. (D.10)and of Ref. [8].3. If the vertical electric field E y = ˆ y · (cid:126)E is identically zero, the (cid:126)F (cid:126)g compensating field only arisesfrom the radial magnetic field B x = ˆ x · (cid:126)B and therefore we would find as the gravity-inducedcontribution to the angular velocity [1, 4, 7] (cid:126) Ω E y =0GR = (cid:126) Ω GRgeo − qm G (cid:126)B (cid:126)g = − γγ − (cid:16) G (2 γ − (cid:17) (cid:126)β × (cid:126)gc . (D.12)If the frozen-spin condition / ( γ −
1) = G of the all-electric ring is inserted, the result ofEq. (D.12) is enhanced by a factor (3 + G ) in comparison to Eq. (D.10), i.e. (cid:126) Ω E y =0 , fsGR = − | (cid:126)g | c (3 + G ) √ G ˆ β × ˆ g . (D.13)4. In a mixed ring with E y (cid:54) = 0 (cid:54) = B x using κ ≡ cβB x E y ≈ const . , (D.14)we can derive from the storage-ring lattice condition (D.6), m | (cid:126)g | q γ − γ = E y + cβB x = E y (1 + κ ) , (D.15) Assuming here and in the following that the storage ring plane is normal to (cid:126)g . (cid:126) Ω E B GR ( κ ) = (cid:26) − γγ − β | (cid:126)g | c − qm Gcβ (cid:18) cβB x + (cid:18) − Gγ − (cid:19) β E y (cid:19)(cid:27) ˆ β × ˆ g = (cid:26) − γγ − − qm | (cid:126)g | Gβ (cid:18) cβB x + E y − γ (cid:18) G (cid:19) E y (cid:19)(cid:27) (cid:126)β × (cid:126)gc = − γγ − (cid:26) γ − (cid:18) G − G + 1 γ (1 + κ ) (cid:19)(cid:27) (cid:126)β × (cid:126)gc = 11 + κ (cid:16) (cid:126) Ω B x =0GR + κ(cid:126) Ω E y =0GR (cid:17) . (D.16)Of course, one recovers the expressions (D.11) and (D.12) from (D.16) if one simply inserts κ → or κ → ∞ , respectively, while by applying the “frozen-spin value", / ( γ −
1) = G , we wouldget the general form − ˆ β × ( (cid:126)g/c ) √ G (1 + (3 + G ) κ ) / (1 + κ ) .Note that the contributions (D.10)–(D.13) and (D.16) switch sign if a counterclockwise beam is replacedby a clockwise one. This clearly separates these contributions from any (MDM-term induced) fake -EDMsignal when a radial magnetic field points, for both beams, in the same direction – either in the outward( ˆ r ) or in the inward radial ( − ˆ r ) direction. In fact, if the scenario E y = 0 can be realised (or the valueof κ can be determined in the general case (D.16) by some means), the lattice orbit condition (D.6)ensures that B x of each of the beams is determined, in the average, by Eq. (D.15), respectively. Thus theextraction of the gravity-induced spin rotation from the half-sum/half-difference of counterclockwise andclockwise beams – assuming that the horizontal spins of the beams point in the opposite/same direction – would determine the orbit-averaged value of the effective radial magnetic field which then could beused to correct the EDM signal. References [1] Yuri N. Obukhov, Alexander J. Silenko, and Oleg V. Teryaev. Manifestations of the rotation andgravity of the Earth in high-energy physics experiments.
Phys. Rev. , D94(4):044019, 2016.[2] Alexander J. Silenko and Oleg V. Teryaev. Semiclassical limit for Dirac particles interaction with agravitational field.
Phys. Rev. , D71:064016, 2005.[3] Alexander J. Silenko and Oleg V. Teryaev. Equivalence principle and experimental tests of gravita-tional spin effects.
Phys. Rev. , D76:061101, 2007.[4] Alexander J. Silenko. Comparison of spin dynamics in the cylindrical and Frenet-Serret coordinatesystems.
Phys. Part. Nucl. Lett. , 12(1):8–10, 2015.[5] I. B. Khriplovich and A. A. Pomeransky. Equations of motion of spinning relativistic particle inexternal fields.
J. Exp. Theor. Phys. , 86:839–849, 1998. [Zh. Eksp. Teor. Fiz. 113, 1537 (1998)].[6] A. A. Pomeransky, R. A. Senkov, and I. B. Khriplovich. Spinning relativistic particles in externalfields.
Phys. Usp. , 43:1055–1066, 2000. [Usp. Fiz. Nauk 43, 1129 (2000)].[7] Nikolai N. Nikoalev, Frank Rathmann, Artem Saleev, and Alexander J. Silenko. Spin Dynam-ics in Storage Rings in Application to Searches for EDM. In , 2019.[8] Yuri F. Orlov, Eanna Flanagan, and Yannis K. Semertzidis. Spin Rotation by Earth’s GravitationalField in a "Frozen-Spin" Ring.
Phys. Lett. , A376:2822–2829, 2012. Here, the qualifier “opposite/same spin direction” in the frozen-spin scenario refers to the setting that the components ofthe horizontal spin in beam direction of the clockwise and counterclockwise case agree/differ in sign. ppendix EAdditional Science Option: Axion Search
E.1
Concept of Search for Axion-like Particles
The theoretical prediction of a neutron electric dipole moment based on QCD is given by | d θn | = θ QCD · − e · cm [1]. However, the most sensitive experimental result [2], d n = 2 . × − e · cm (90% C.L.)for the neutron, sets a very strict upper limit on the parameter . Since there is no natural explanation forthe extremely small value of θ QCD , this is sometimes referred to as the strong CP problem.The axion originated from a new symmetry postulated by Roberto Peccei and Helen Quinn tosolve the strong CP problem in QCD physics [3]. The small parameter is explained by a dynamic scalarfield that maintains the symmetry. The particle associated with this field is the axion. Furthermore, ifthe axion is very light, it interacts so weakly that it would be nearly impossible to detect in conventionalexperiments. But it would be an ideal dark matter candidate as it interacts gravitationally with the matteraround it.The axion couples to gluons, fermions, nucleons, etc. This coupling induces an oscillating electricdipole moment (EDM) in nucleons [4, 5]. This may be expressed as: d n = 1 . × − a ( t ) f a e · cm = 1 . × − a cos( m a t + φ a ) f a e · cm (E.1)where a ( t ) is the axion/dark matter field, m a is the axion mass, and f a is the axion decay constant. φ a isan unknown local phase that we will need to consider later.Three conditions must be met in order to consider using the horizontally polarised deuteron beamat COSY to search for axions. First, the effects of the axion must be coherent across a large spatial rangeso that, as the beam circulates, it remains under the influence of a single axion. This will also mean thatall of the deuterons in the beam will show the oscillating EDM property simultaneously. Thus, an electricdipole moment parallel to the polarisation (average orientation of the deuteron spins) may be used to testfor the presence of an axion. Second, this interaction must remain present in the COSY experimentalhall for a time long enough for the beam to respond. Crossing an axion resonance in a scanning searchwould likely require a few seconds. Any axions in the neighbourhood of earth are likely bound to theMilky Way galaxy and thus there is a lower limit on how quickly they will vanish from view. Third, thedensity of axions must be high enough that the chance of observing one is substantial during the timethat the store is under way. Estimates [4, 5] based on the confinement of the axion to our region of theMilky Way galaxy suggest that these coherence requirements are met at the frequency where we wouldmake a feasibility study ( ∼ kHz) with a quality factor Q for the axion’s oscillation exceeding .The experiment to search for an axion would consist of a series of runs in which the revolutionfrequency of the machine is changed continuously in a slow ramp [6]. Measurements of the polarisationcomponents would be made during the ramp. If the in-plane polarisation precession frequency happensto match the axion frequency, then a resonance between the two will cause the vertical component of thepolarisation to undergo a jump proportional to the ratio of the size of the oscillating EDM to the speed ofthe ramp. A comparison of polarisation asymmetries collected during non-ramped times at the beginningand ending of the scan would suffice to quantify the size of any suspected change.Experimental signals based on a sub-atomic EDM depend on a torque about an electric field alongthe radial direction in a storage ring that lifts the polarisation direction out of the ring plane, givingit a small and rising vertical component. Despite the large electric field that exists in the beam framefrom the magnets that confine the deuteron beam to the COSY ring, the continuous rotation of the in-173lane polarisation relative to the beam velocity makes it impossible for any static EDM signal to becomelarge enough through a (cid:126)d × (cid:126)E torque to observe directly. Progress is cancelled by retreat whenever theprojection of the EDM on the tangential direction reverses. But if there is an oscillating EDM that variesat the same frequency as the rotating polarisation, then a vertical component of the polarisation will startto grow. A proposal was accepted by the COSY Program Advisory Committee to develop and describetechniques that could be used in such an axion search and to quantify the sensitivity for reasonableoperating conditions. The plan is to start with the deuteron momentum of p = 0 . GeV/c wherethere is COSY experience with the preparation of a horizontally polarised beam with a long polarisationlifetime [7].
E.2
Technical Considerations for an Axion Search
Accumulation of the vertical component polarisation signal depends on the alignment of the polarisationalong the direction of the beam velocity with the maximum of the value of the oscillating EDM. Thisalignment is controlled by the axion phase φ a . If these two oscillations are out of phase by π/ , then noaccumulation will occur. The plan to overcome this difficulty is to operate COSY on the fourth harmonic(for which hardware already exists), producing four circulating bunches in the ring at the same time. Ifan RF solenoid is used to precess the polarisation from the vertical direction (as it is upon injection intothe ring) into the horizontal plane, then the resulting laboratory-frame polarisation pattern of the fourbeams in the ring is shown in Fig. E.1 for f SOL = f REV (1 + Gγ ) where G is the deuteron’s magneticanomaly and γ is the usual relativistic parameter. AB CD
Fig. E.1:
Laboratory-frame polarisation directions of the four beam bunches as seen from above the plane of thestorage ring. The labels show the order (A, B, C, D) in which they were generated by the RF solenoid operatingon the Gγ harmonic. This pattern features two directions (A and D) that are nearly orthogonal. This means that the experi-ment carries sensitivity to both components of the phase of the oscillating EDM (sine and cosine). Theremaining two polarisation directions may be used to verify that the amplitude of any prospective axionsignal varies in a sinusoidal pattern around the circle in Fig. E.1 in a manner consistent with the twophase components present in the A and D directions. In addition, there are pairs of polarisation direc-tions that are nearly opposite. This provides an opportunity to use them as opposite polarisation states ina “cross ratio” which would serve to reduce or eliminate first-order errors in the scanning process due togeometric or rate-dependent systematic errors that can develop during the beam store [8]. Bunch B maybe compared with the average of A and C, and bunch C may be compared with the average of B and D.One way to search for an axion-like particle is to vary the polarisation rotation rate continuously174hile monitoring the vertical polarisation. If the frequency of rotation happens to match the axion fre-quency at some time during the scan, the resonance condition will create a jump in the polarisation, asshown in Fig. E.2.
Fig. E.2:
A calculation of the resonance crossing with a scan rate of 0.5 Hz/s. The strength of the oscillating EDMis . × e · cm. Within the span of less than one second, this causes a jump of − . in the p Y component ofthe beam polarisation (assumed to initially be completely polarised in the ring plane). A practical scheme for producing such scans would require that the range of the scan is not beso large that it passes outside the acceptance of the storage ring. In addition, the frequency of the RFsolenoid that initially precesses the polarisation from the vertical to the horizontal direction must beadjusted to match the Gγ spin tune resonance. The easiest way to organise the scan is to vary therevolution frequency of the beam. Critical magnetic ring components, such as the dipole magnets, wouldbe programmed to follow. E.3
Initial Tests with Beam
In December, 2018, there was an opportunity to switch COSY to operate on the h = 4 harmonic. Atthat time, the RF solenoid was running on the − Gγ harmonic and the sextupole magnets along withelectron cooling had been set for long in-plane polarisation lifetime. In Fig. E.3 there is a representationof the count rate in the WASA detector as a function of time in the store (horizontal) and position aroundthe ring (vertical).The four beam bunches show clearly after 80 s following a period of electron cooling. At this timethe RF solenoid frequency was associated with the − Gγ harmonic. This yields a different pattern ofpolarisation directions compared to Fig. E.1. In the laboratory frame we have:Like the pattern shown in Fig. E.1, this pattern also presents bunches A and D with polarisationdirections that are nearly perpendicular. So this pattern also suffices to detect the axion for any valueof the axion phase. But the other two polarisation directions, B and C, lie in the same quadrant. Theirpolarisation directions are similar, and any axion signal will tend to have a similar signature as A andD. Thus we cannot use these signals in a cross ratio treatment to eliminate systematic errors in themeasurements of the asymmetry.Experimental verification of these polarisation directions depends on measurements made with apolarimeter located at one spot on the COSY storage ring. It will see the four bunches sequentially atdifferent time. Given that the polarisation continues to rotate in the ring plane, this leads to a differentset of directions measured at the polarimeter. Since the polarisation is rotating at about 630 kHz, it ismost useful to consider expressing this polarisation as a magnitude and a phase with respect to a starting175 ig. E.3: Count rate in the polarimeter as a function of time in the store (horizontal) and position around thecircumference of COSY (0 to π ). Extraction of the beam onto the WASA polarimeter begins at 90 s. Prior to 80 sthe beam is being electron cooled. There are four horizontal ridges corresponding to the four beam bunches. Fig. E.4:
Directions of the in-plane polarisations in the laboratory frame for the case of an RF solenoid operatedon the Gγ harmonic. The labels follow the scheme of Fig. E.1. The opening angle for adjacent pairs is shownas . ◦ . time that is the beginning of data acquisition. Normally, trimming the fields in the ring, especially thesextupole components, is very useful in maintaining the size of the IPP. Then the important question is ameasurement of the phases for the four beam bunches. An example is shown in Fig. E.5.The match (red lines) with a prediction consistent with Fig. E.4 is good (see caption). The patternon phases shows three angular separations of 1.822 rad and a final separation of 0.817 rad. This set ofunequal gaps indicates that phase A is uniquely identifiable as the bunch synchronised with the maximain the 871 kHz RF solenoid pattern at t = 0 (start of solenoid operation). Like the phase pattern inFig. E.5, the angular separations in Fig. E.4 are also the same ( . ◦ ) except for the separation betweenbunches D and A, which is much larger.In the case of the Gγ harmonic recommended for this process, the pattern of three wideand one narrow angular separation in the polarimeter measurements changes to three narrow and onewide different in the phase pattern. The narrow angle is . ◦ and the wide angle is . ◦ . This leadsto a separation angle for the polarimeter measurement of . ◦ between successive beam bunches inFig. E.1. 176 ig. E.5: Measurements of the polarisation phases for the four beam bunches in a test run made in December, 2018.The phase, measured along the horizontal axis, is shown as a function of time in the store. The phase is relative toa calculation of the polarisation direction based on an assumed value for the spin tune frequency ( f REV Gγ ) thatyields a prediction of the phase at any moment in time during the store. A perfect match between the prediction andthe measurements yields phase values that remain constant with time. The numbers on the curve correspond to thefour bunches (A through D). Along the left-hand axis, a diagram using red lines shows the predicted relative phaseseparations that corresponds to the polarisation pattern shown in Fig. 4. Given the value of Gγ , the separationof the phase lines should be either 1.822 rad (for pairs A-B, B-C, and C-D including wrapping through π ) or0.817 rad (for pair D-A). This diagram gives a good account of the phase separations as measured. In the scan for the axion, different axion phases are distributed with a sinusoidal dependence onthe axion phase, as shown in Fig. E.6.The different polarisation jumps is one of the features that distinguishes the detection of an axionfrom the observation of a machine resonance. In the case of the machine resonance, there is no phase andall four bunches should observe the same polarisation jump. The distribution of polarisation directionsas shown in Fig. E.1 ensures the signals will appear with opposite signs for some pairs of directions.Machine resonances must also appear at frequencies related the value of Gγ through Gγ = (cid:96) + mν X + nν Y + kν SY NC (E.2)where (cid:96) , m , n , and k are integers and the tunes ( ν ) are connected to horizontal (X) and vertical (Y)betatron oscillations as well as synchrotron oscillations [9]. Smaller integers generally indicate strongerresonances. These check should allow for the separation of axion signals from other effects. E.4
Immediate Plans
A running period started 1 April 2019 with COSY for the purpose of testing the feasibility of creatinga 4-beam setup and a frequency ramp with properties appropriate for conducting an axion search. Thesetup includes previously developed conditions for long IPP lifetime, which requires electron coolingas well as trimming the ring fields with sextupole components such that the X and Y chromaticities aresimultaneously set to zero [7].The new features begin with the four-bunch setup. The bunches must be well separated spatiallyso that there is no significant transfer of beam particles from one bunch to the next. This would tend177 ig. E.6:
The polarisation jump for four different choices of the axion phase. The phase is marked for each curve. to depolarise the bunches, as the pattern in Fig. E.1 requires nearly opposite polarisation directions forneighbouring bunches. polarisation measurements would ensue to check that the understanding of therelative phases between bunches is correct. This would constitute a confirmation of the patterns shownin the previous section.The next step in the preparation would be the creation of conditions for ramping the machinerevolution frequency to make frequency scans possible. Speeds would be slow, perhaps 0.1 Hz/s. Storagetimes of 150 s during the ramp means a frequency step of 15 Hz per scan. Since there are nearlyopposite polarisation directions represented in the laboratory polarisation pattern, only one polarisationstate is needed from the ion source. The data from one scan cannot be directly combined with anothersince the relative phase may change, even if the same axion is present. This value depends as wellon the start time for the RF solenoid, and this cannot be synchronised with the axion phase. Multiplescans of the same frequency range are advisable since at any given time, an axion may not be present.Ramping the magnetic field of the COSY ring along with the frequency is required in order to maintainthe circumference of the orbit. This allows the spin tune ( Gγ ) to be known from the revolution frequency.However, the development of the software for dealing with IPP also provides for a direct measurementof the spin tune at any time during the process [10], and this will act as a confirmation that the machineconditions are being maintained.With each scan, a comparison of the vertical polarisation component difference between the be-ginning and the end of the run is needed to determine whether or not there is evidence for a polarisationjump during the scan. The statistics of this comparison may be improved if there are times of no rampingbefore and after the actual ramp. Some threshold (such as 2 or 3 standard deviations) must be chosen. Ifpassed, the scan should be repeated to determine whether or not it was an outlier. Once identified, addi-tional scans are needed in order to have the statistics to determine the time location of the polarisationjump with precision.Initial results from this development period are expected to be modest in terms of both the sensi-tivity and frequency range covered. 178 eferences [1] E. Mereghetti, J. de Vries, W. H. Hockings, C. M. Maekawa, and U. van Kolck, Phys. Lett. B ,(2011).[2] C. A. Baker at al. , Phys. Rev. Lett. , 131801 (2006).[3] R. D. Peccei and H. R. Quinn, Phys. Rev. Lett. , 1440 (1977).[4] P. W. Graham and S. Rajendran, Phys. Rev. D , 055013 (2011).[5] P. W. Graham and S. Rajendran, Phys. Rev. D . 035023 (2013).[6] Seung Pyo Chang, Selçuk Haciömero˘glu, On Kim, Soohyung Lee, Seongtae Park, and Yannis K.Semertzidis, Phys. Rev. D , 083002 (2019).[7] G. Guidoboni et al. , Phys. Rev. Lett. , 054801 (2016).[8] N. P. M. Brantjes et al. , Nucl. Instrum. Methods Phys. Res. A , 49 (2012).[9] S. Y. Lee, Spin Dynamics and Snakes in Synchrotrons (World Scientific, Singapore, 1997) p. 26.[10] D. Eversmann et al. , Phys. Rev. Lett. , 094801 (2015).179 ppendix FNew ideas: Hybrid Scheme
Abstract
This appendix examines the possible replacement of electrical quadrupoles bymagnetic quadrupoles for performing the focusing in the full scale ring, whichis then referred to as a “hybrid ring”. Because alternating gradient magneticfocusing is used, simultaneous CW and CCW storage continues to be possible,while still allowing for moderately strong vertical focusing, along with thesimultaneous CW and CCW storage needed for canceling important systematicerrors. This promises to greatly reduce the contribution of radial magnetic fielduncertainty to the EDM systematic error. F.1
Experimental Method using a hybrid ring lattice
Simultaneous storage in clock-wise (CW) and counter-clock-wise (CCW) allows for the cancellation ofimportant systematic errors [1, 3]. In combined electric and magnetic fields, e.g., the deuteron ring [2],it is not possible to store the beams CW and CCW simultaneously and much of the systematic errorwork was geared towards fixing potential problems due to that fact. The all-electric ring allows forit, however the main potential systematic error is large (a consequence of the large sensitivity on theproton EDM) and the required level to know the radial B-field around the ring is at the 10 aT level.High precision SQUID-based BPMs have been developed to be able to detect the required signal causedby the splitting of the counter-rotating beams [3, 4]. In order for the method to have high sensitivityto the potential systematic error, the vertical focusing strength is kept low, making it rather difficult tohandle. A hybrid ring, in which alternating magnetic focusing is used, allowing simultaneous CW andCCW storage, allows for strong vertical focusing, and simultaneous CW and CCW storage for cancelingimportant systematic errors [5].The counter-rotating beams do not actually go through the same places everywhere, due to the factthat the vertical focusing includes magnetic focusing. Therefore, those beams may not exactly cancelthose systematic errors at all places. However, we have shown that it is possible to use the same magneticquads with flipped field directions (opposite sign currents) and on average the particles do follow thesame trajectories. This idea seems to work very well, eliminating completely the radial B-field issue. Inaddition, the vertical dipole E-field effect is cancelled completely in CW and CCW injections as is theeffect of gravity. The suggested working lattice is shown in Figure F.1, which is a modification of thelattice shown in the paper [4] describing the all-electric storage ring method, but this time the electricquadrupoles are replaced with corresponding magnetic ones. Figure F.2 shows the vertical beta-functionof the CW and CCW stored beams, and Figure F.3 the corresponding for the horizontal. Flipping thesign of the currents in the magnetic quadrupoles will produce symmetric beta-functions for the CW andCCW beams.However, it is always possible that some electric focusing will be present somewhere in the ring.This focusing and/or defocusing could originate from the bending electric field plates, which producethe required radial E-field. One or both plates could be misaligned, readily producing a vertical dipole,but also a quadrupole or even higher multipole E-fields. There could also exist induced charges (imagecharges) from any horizontally placed metals around the lattice, the tune shift and tune spread effects This appendix was authored by Y.K. Semertzidis and S. Haciomeroglu of the Center for Axion and Precision PhysicsResearch, KAIST, South Korea. k2k2 k3 k3 k3 k3 k3k1 k4 k4 k4 k4k1 c m c m c m O O F . m k2 k3k3k3k3k3 k1k4k4k4k4k1 k3 k3 k3 k3 k3k1 k4 k4 k4 k4k1k2k3k3k3k3k3 k1k4k4k4k4k1 Fig. F.1:
A detail of the storage ring lattice is shown here with focusing and defocusing quadrupoles (shown as k and k ). The bending sections, including the short straight sections, have a length of 10.417 m, three sectionsassembled as one unit. The long straight sections are 20.834 m long with a quadrupole (shown as k ) in themiddle and two half-length quads (shown as k ) at both ends. The values of the magnetic quadrupole strengthare: k = 0 . T/m, k = − . T/m, k = − . T/m, k = 0 . T/m. The vertical tune, when running with thesequadrupole strengths, is Q y = 0 . , while the horizontal tune is Q x = 1 . . The total effect, i.e. the vertical spin precession rate, is going to be in a functional form: R V = R EDM + R B r × Q + ...ζ × Q + Q + ... where R V is referring to the total vertical spin precession rate, R EDM refers to the portion due to the par-ticle EDM, Q = f ( Q , Q , Q , ... ) corresponds to the square of the tun-ing due to non-magnetic effects, Q is the square of the tune due to the magnetic quads, Q , Q , Q are the square of the tunes due to the electric quads, the forces due to in-duced charges, and the forces due to the beam intensity, correspondingly. R Br refers to the vertical spinprecession rate due to the radial B-field. The point is that a net radial B-field can create a vertical spinprecession, which can only be canceled exactly by another B-field; in this case we assumed it to be themagnetic focusing. Magnetic focusing can essentially eliminate this systematic error provided that it isthe only source focusing the beam. Figure F.4 shows the average vertical offset of the stored beam asa function of the radial B-field multipole whose amplitude is always kept at 1 pT. Figure F.5 shows thevertical spin precession rate under the same conditions. A genuine EDM signal for − e · cm is largerthan 1 nrad/s, and therefore much larger than the above background signal. However, if on one of themagnetic quadrupoles we add an overlapping electrical quadrupole with a strength of 1 kV/m , then weget the much larger spin precession rate of 0.4 nrad/s, for N = 4 harmonic case of the radial B-field. Thiseffect will be further and effectively suppressed by applying varying levels of magnetic field focusing, asdescribed in the section below. 181 (cid:1) ( m ) s (m)polarity - - Fig. F.2:
The vertical beta-function values around the ring for CW and CCW operations. They flip sign when themagnetic quadrupoles are running with the opposite sign and therefore the counter-rotating particles on averagetrace the same paths. (cid:1) ( m ) s (m)polarity - - Fig. F.3:
The horizontal beta-function values around the ring for CW and CCW operations. -5-4-3-2-1 < y > ( n m ) N : B-feld multipoles
Fig. F.4:
The average vertical beam offset when only magnetic focusing is used, as a function of the radial B-fieldmultipoles ( N -values). The amplitude of the background radial B-field is always kept at 1 pT, while the quadrupolestrength is kept at ± Experimental Approach.
We apply a series of B-field focusing strengths, from weak to strongerones to probe the EDM effect. With magnetic focusing the main systematic error is the out-of-planedipole electric field, which is cancelled by CW and CCW beam storage as in the deuteron storage ringEDM experiment. [2] Since simultaneous CW and CCW storage is possible in the current configuration,then most of the issues related to E-field direction stability go away. In addition, any focusing effect fromthe electric field plates or any other sources is sorted out by running the experiment at different alternatingmagnetic focusing strengths as shown in Figure F.6. Here, an additional electric focusing exists togetherwith a DC ( N = 0 ) radial magnetic field around the ring with strength of 1 pT. The electric focusing isoriginated by shaping all the bending plates, producing a vertical focusing with a field index of m = 0 . .182 -11 -11 -11 -11 -11 -11 d s y / d t > (r ad / s ) N : B-feld multipoles
Fig. F.5:
The vertical spin precession of the counter-rotating beams when only magnetic focusing is used, fordifferent radial B-field multipoles ( N -values). The amplitude of the radial B-field is always kept at 1 pT, whilethe quadrupole strength is kept at ± − e · cm is larger than 1 nrad/s, andtherefore much larger than the background signal. The spin precession rate equation, when expanded, can be written as R V = R EDM + R B r Q P m − R B r Q P m + ... with P m = 1 / ( ζ × Q ) , showing clearly that for a large magnetic focusing tune, i.e., P m → ,the spin precession rate corresponds to the EDM signal. Hence, the DC offset in Figure F.6 correspondsto the EDM signal and the obtained value is consistent with the simulations. In Figure F.6, the spinprecession rate corresponds to − e · cm EDM level to prove the principle of the method. It will beadvantageous to keep the spin precession rate lower by adding much stronger magnetic focusing casesand keep the electric focusing below the m = 0 . level. The method will work best, requiring lessleverage, when the magnetic focusing is dominating all other focusing effects. In a similar way, we canprove that the sextupole vertical electric field cancels with CW and CCW storage, etc., provided that thebeam emittances are the same to an adequate level. From our simulations we infer that the SQUID-basedBPMs resolution requirements are relaxed by several orders of magnitude over the lattice where electricfocusing is used, which is a major breakthrough. The new requirements are a well-shaped quadrupolemagnetic field in the ring, so that the center of the CW and CCW beams overlap within 100 nm at allmagnetic quadrupole strengths, using the SQUID-based BPM signals. In addition, the ring needs to beflat (absence of corrugation) to 100 nrad, which we achieve by a combination of mechanical alignment,beam-based alignment and by using bunches polarized in the radial direction. A summary of the mainsystematic errors in the experiment with hybrid fields (electric bending and magnetic focusing) and theircurrent remediation plan is given in Table F.1. F.2
Systematic errors
F.3
Conclusions
The hybrid ring, where the radial E-field bends the stored beam, and an alternating B-field providesfocusing allows for simultaneous CW and CCW storage eliminating the most important systematic errorsource. The experiment will also run at various magnetic focusing strengths to eliminate possible electricfocusing sources, etc. In addition, the counter-rotating beams will sense any quad misalignment to betteraccuracy than needed as well as the spin precession of a beam with a radial spin direction. The methodneeds to be studied by an independent group, which should take less than six months to complete.183 (cid:1) r ( r ad / s ) P m Simulation resultFit result
Fig. F.6:
The vertical spin precession rate as a function of the P m = 1 /Q y when the background effect is due toa combination of a DC ( N = 0 ) radial magnetic field around the ring with strength of 1 pT and a large electricfocusing effect of the bending plates. The bending plate focusing corresponds to an (electric) vertical focusingfield index of m = 0 . . The fit result is from a first order polynomial. The DC-offset corresponds to the EDMprecession rate, which in this case is − . × − rad/s, consistent within the estimated errors to the input EDMvalue corresponding to − . × − rad/s. Table F.1:
Main systematic errors and their remediation when hybrid fields (electric bending and magnetic focus-ing) are used.
Effect Remediation
Radial B-field. Magnetic focusing.Radial B-field when other then Varying magnetic focusing and fit for themagnetic focusing is present. DC offset in the vertical precession rate.Dipole vertical E-field. CW and CCW beam storage.Corrugated (non-planar) orbit. Observe CW vs. CCW beam split with magnetometers, e.g.,SQUID-based BPMs [3].Probe with stored beams with their spins frozen in the radial di-rection [2].RF cavity misalignment Vary the longitudinal lattice impedance to probe the effect of thecavity’s vertical angular misalignment. CW and CCW beamscancel the effect of a vertically misplaced cavity. [4]
References [1] F.J.M. Farley et al. , “New method of measuring electric dipole moments in storage rings.”
PhysicalReview Letters , 052001 (2004).[2] D. Anastassopoulos et al. , “AGS proposal: search for a permanent electric dipole moment ofthe deuteron nucleus at the − e · cm level,” Storage Ring EDM Collaboration, October 2008.Available from .[3] V. Anastassopoulos et al. , “A proposal to measure the proton electric dipole moment with − e · cm sensitivity,” Storage Ring EDM Collaboration, October 2011. Available from .[4] V. Anastassopoulos et al. , “A storage ring experiment to detect a proton electric dipole moment,”Storage Ring EDM Collaboration, Reviews of Scientific Instruments , 115116 (2016).[5] S. Haciomeroglu and Y.K. Semertzidis, “A hybrid ring design in the storage-ring proton electricdipole moment experiment,” arXiv:1806.09319 (2018).184 ppendix GNew ideas: Doubly Magic EDM Measurement Method Abstract
This appendix discusses “doubly-magic trap” operation of storage rings withsuperimposed electric and magnetic bending, allowing spins in two beams tobe frozen (at the same time, if necessary), and their application to electricdipole moment (EDM) measurement . Especially novel is the possibility ofsimultaneous storage in the same ring of frozen spin beams of two differ-ent particle types. A few doubly-magic cases have been found: One has an86.62990502 MeV frozen spin proton beam and a 30.09255159 MeV frozenspin positron beam (with accuracy matching their known magnetic moments)counter-circulating in the same storage ring. (Assuming the positron EDM tobe negligibly small) the positron beam can be used to null the worst source ofsystematic EDM error – namely, the existence of unintentional and unknownaverage radial magnetic field < B r > which, acting on the MDM, causesspurious background spin precession indistinguishable from foreground EDM-induced precession. The resulting measured proton minus positron EDM dif-ference is then independent of < B r > . This amounts to being a measurementof the proton EDM.Most doubly-magic features can be tested in one or more “small” EDM pro-totype rings. One promising example is a doubly-magic proton-helion com-bination, which would measure the difference between helion (i.e. helium-3)and proton EDM’s. This combination can be used in the near future for EDMmeasurement in a small, 10 m bending radius ring, using only already well-understood and proven technology. In the standard model both EDM’s arenegligibly small. Any measurably large difference between these EDM valueswould represent “physics beyond the standard model”. G.1
Introduction
G.1.1 Major previous EDM advances
Comparably important EDM advances that have been made in the recent past can be listed: The storagering “frozen spin concept” according to which, for a given particle type, there can be a kinetic energyfor which the beam spins are “frozen” in a storage ring—for example always pointing along the lineof flight, Farley et al. [1]; The recognition of all-electric rings with “magic” frozen spin kinetic ener-gies (14.5 MeV for electrons, 233 MeV for protons) as especially appropriate for EDM measurement,Semertzidis et al. [2]; The “Koop spin wheel” mechanism, in which a small radial magnetic field B r applied to an otherwise frozen spin beam causes the beam polarisation to “roll” around a locally-radialaxis [3] (systematic precession around any axis other than this would cancel any accumulating EDMeffect); Spin coherence times long enough for EDM-induced precession to be measurably large, Ev-ersmann et al. [4]; “Phase-locking” the beam polarisation, which allows the beam polarisation to beprecisely manipulated externally, Hempelmann et al. [5]. This appendix, very slightly changed here, was originally distributed by Richard Talman as arXiv article 1812.05949,[physics.acc-ph], submitted 14 Dec 2018. .1.2 Koop spin wheel
By design, the only field components in the proposed ring would be the radial electric component E x , andideally-superimposed magnetic bending would be provided by a vertical magnetic field component B y .There also needs to be a tuneable radial magnetic field B r ≡ B x , both to compensate any unintentionaland unknown radial magnetic field and to control the roll-rate of the Koop spin wheel.For a “Koop spin wheel” rolling around the radial x -axis, unpublished notes from a Juelich lectureby I. Koop [6] provide formulas for the roll frequencies (expressed here in SI units, with Bρ in T.m), Ω B x x = − Bρ Gγ cB x , and Ω EDM x = − η Bρ (cid:18) E x c + βB y (cid:19) . (G.1) G is the anomalous magnetic moment, β , γ are relativistic factors. Ω EDM x is the foreground, EDM-induced roll frequency. Ω B x x is a roll frequency around the same radial axis, caused by a radially magneticfield B x acting on the MDM. cBρ = pc/ ( qe ) ≡ pc/ ( Ze ) is the standard accelerator physics specificationof storage ring momentum. The factor η expresses the electric dipole moment d = ηµ in terms of themagnetic moment µ of the beam particles. G.1.3 Proposed EDM measurement technique
The proposed EDM measurement technique starts by measuring and nulling Ω Koop x = Ω EDM x + Ω B x x −→ (G.2)for the spin wheel of a secondary beam. The secondary beam is then dumped and, with no change ofring conditions whatsoever , the matching frozen spin primary beam is stored. Since the primary beam issubject to the same radial magnetic fields as the secondary beam, its Ω EDM x roll rate will then provide adirect measurement of the primary beam EDM d .Previously one will, of course, also have followed Koop in minimising (cid:104) B x (cid:105) , by measuring thedifferential vertical separation of the two beams, which is similarly proportional to (cid:104) B x (cid:105) . G.1.4 Polarimetry assumptions
Ultimate EDM precision may depend on resonant polarimetry, probably based on the Stern-Gerlachinteraction [14] [15] [16]. Meanwhile, impressive beam polarisation control has been achieved usingpolarimetry based on left-right scattering asymmetry of protons or deuterons from carbon [5], and muchmore progress will undoubtedly be made with this method. Any prototype EDM ring to be built in thenear future will need to rely initially on this form of scattering asymmetry polarimetry.
G.2
Orbit and spin tune calculation
G.2.1 Terminology
Fields are “cylindrical” electric E = − E ˆx r /r and, superimposed, uniform magnetic B = B ˆy . Thebend radius is r > . Terminology is needed to specify the relation between electric and magneticbending: Cases in which both forces cause bending in the same sense will be called “constructive” or“frugal”; Cases in which the electric and magnetic forces subtract will be referred to as “destructive” or“extravagant”. There is a reason for the “frugal/extravagant” terminology to be favoured. Electric bend-ing is notoriously weak (compared to magnetic bending) and iron-free (required to eliminate hysteresis)magnetic bending is also notoriously weak. As a result an otherwise-satisfactory configuration can betoo “extravagant” to be experimentally feasible.The design particle has mass m > and charge qe , with electron charge e > and q = ± (orsome other integer). These values produce circular motion with radius r > , and velocity v = v ˆz ,186here the motion is CW (clockwise) for v > or CCW for v < . With < θ < π being thecylindrical particle position coordinate, the angular velocity is dθ/dt = v/r .To limit cases we consider only electrons (including positrons) protons, deuterons, tritons, andhelions; that is e-, e+, p, d, t, and h. The circulation direction of the so-called “master beam” (of whatevercharge q ) is assumed to be CW or, equivalently, p > . The secondary beam charge q is allowed tohave either sign, and either CW or CCW circulation direction. G.2.2 Fractional bending coefficients η E and η M (In MKS units) qeE and qeβcB are commensurate forces, with the magnetic force relatively weakenedby a factor β = v/c because the magnetic Lorentz force is qe v × B . Newton’s formula for radius r circular motion can be expressed using the total force per unit charge in the form F tot . e = β pc/er = qE + qβcB , (G.3)Coming from the cross-product Lorentz magnetic force, the term qβcB is negative for backward-travelling orbits because the β factor is negative. The “master” beam travels in the “forward”, CW direc-tion. For the secondary beam, the β factor can have either sign. For q = 1 and E = 0 , formula (G.3)reduces to the standard accelerator physics “cB-rho=pc/e”. For E (cid:54) = 0 the formula incorporates therelative “effectiveness” of E /β and cB .Fractional bending coefficients η E and η m are then defined by η E = r pc/e E β , and η M = r pc/e cB , (G.4)neither of which is necessarily positive. They satisfy η E + η M = 1 . G.2.3 Spin tune expressed in terms of η E and η M With α being the angle between the in-plane component of beam polarisation and the beam direction,the “spin tune” is defined to be the variation rate per turn of α , expressed as a fraction of π . Spin tunesin purely electric or purely magnetic rings are given by Q E = (cid:16) G − γ − (cid:17) γβ = Gγ − G + 1 γ , Q M = Gγ, (G.5)With superimposed fields, the spin tune can be expressed in terms of the fractional bending coefficients, Q S = dαdθ = Q E η E + Q M η M . (G.6) G.2.4 The “magic energy” condition.
Superimposed electric and magnetic bending permits beam spins to be frozen “frugally”; i.e. with a ringsmaller than would be required for all-electric bending. The magic requirement is for spin tune Q S tovanish; Q S = η E Q E + (1 − η E ) Q M = 0 . Solving for η E , η E = GG + 1 γ , η M = 1 − GG + 1 γ . (G.7)For example, with proton anomalous moment G p = 1 . , trying γ = 1 . , we obtain η E = 1 . which agrees with the known proton 233 Mev kinetic energy value in an all-electric ring. For protons inthe non-relativistic limit, γ ≈ and η NR E ≈ / . The magic electric/magnetic field ratio is EcB = βη E η M = βGγ G (1 − γ ) = Gβγ − Gβ γ . (G.8)187 .2.5 Wien filter spin-tune adjustment Superimposed electric and magnetic bending fields allow small correlated changes of E and B to alterthe spin tune without affecting the orbit. Being uniformly-distributed, appropriately matched electricand magnetic field components added to pre-existing bend fields can act as a (mono-directional) “globalWien filter” that adjusts the spin tune without changing the closed orbit. Replacing the requirement that η E and η M sum to 1, we require ∆ η M = − ∆ η E , and obtain, using the same fractional bend formalism,for a Wien filter of length L W the spin tune shift caused by a Wien filter of length-strength product EL W is given by ∆ Q WS = − π Gβ γ EL W mc /e . (G.9)For “global” Wien filter action, L W is to be replaced by πr . G.3 “MDM comparator trap” operation
G.3.1 Dual beams in a single ring.
This section digresses temporarily to describe the functioning of dual beams in the same ring as a “spintune comparator trap”. A “trap” is usually visualised as a “table-top apparatus”. For this appendix “table-top radii” of , , or , meters (or rather curved sectors of these radii, expanded by straight sectionsof comparable length) are considered.Gabrielse [8] has (with excellent justification) boasted about the measurement of the electronmagnetic moment (with 13 decimal point accuracy) as “the standard model’s greatest triumph”, basedon the combination of its measurement to such high accuracy and on its agreement with theory to almostthe same accuracy. Though other magnetic moments are also known to high accuracy, compared to theelectron their accuracies are inferior by three orders of magnitude or more. One purpose for a spin-tune-comparator trap would be to “transfer” some of the electron’s precision to the measurement of othermagnetic dipole moments (MDM’s). For example, the proton’s MDM could perhaps be determined toalmost the current accuracy of the electron’s.Different (but not necessarily disjoint) co- or counter-circulating beam categories include differentparticle type, opposite sign, dual speed, and nearly pure-electric or pure-magnetic bending. Cases inwhich the bending is nearly pure-electric are easily visualised. The magnetic bending ingredient can betreated perturbatively. This is especially practical for the 14.5 MeV electron-electron and the 233 Mevproton-proton counter-circulating combinations.Storage of different beam types in the same ring is illustrated in Figure G.1. As explained inthe caption, the bending can be either frugal or extravagant (i.e. constrictive or destructive). For agiven particle type, if the clockwise (CW) bending is frugal, the counter-clockwise (CCW) bending isextravagant. For stable orbits the net radial force has to be centripetal. For the three cases described inthis appendix, the electric force magnitude exceeds the magnetic force magnitude. This means that onlypositive particle beams can be stable.Eversmann et al. [4] have demonstrated the capability of measuring spin tunes with high accuracy.By measuring the spin tunes of beams circulating in the same ring (not necessarily simultaneously) theMDM’s of the two beams can be accurately compared. G.3.2 Sensitivity to imperfections
So far only perfect apparatus has been considered. Here we comment on imperfections. The mainattribute to be claimed for the spin tune comparator will be its relative insensitivity to imperfections.Whatever validity is claimed will come from a combination of (1) basing parameter determinations onlyon frequency measurement, (2) accurate knowledge of the MDM’s, and (3) on the degree to which thespatial orbits of co- or counter-circulating beams are constrained to be identical to high accuracy. Alsoimportant will be the degree to which the ratio of electric to magnetic field is constant around the ring.188 + p F M F E vvv + e+e vv BB EE (VR)(VR) (d) (e) F E F E F M F E F M F E F M < ~ β ~~ c β ~~ c β< ~ β < ~ β M BBB EE pp (NVR)(NVR)(NVR) (a) (b) (c) E master beam−1 Master beam and potential secondary beams on the same design orbit "extravagant" "extravagant" "frugal" "extravagant" "frugal"
Fig. G.1:
Examples of “secondary beams” designed to have the same design orbits as a (shaded) beam 1“masterbeam”. Electric and magnetic force strengths are crudely represented by the lengths of their (bold-face) vectors.This figure is limited to very-relativistic (VR) electrons (of either sign) and not-very-relativistic (NVR) protons (ofeither sign). CW and CCW orbits are identical, except for traversal direction. For stable beam circulation the sumof electric and magnetic forces has to be centripetal. This condition is violated in case (a); the centrifugal electricforce exceeds the centripetal magnetic force. (To be shown shortly) radial positioning errors are not a serious concern but requiring the designorbits to be accurately planar (i.e. lying in a single horizontal plane) markedly improves the MDM (andlater the EDM) measurement accuracy.The reason for controlling vertical orbit excursions to better accuracy than horizontal has to do withspin precession control. Let us assume that element positions are established initially to ±
100 micrometeraccuracy horizontally, and ±
10 micrometer accuracy vertically. Corresponding angular precision toler-ances of about one-tenth milliradian horizontally and one-hundredth milliradian vertically will also beassumed.Quoting G. Decker from 2005 [9] “Submicron beam stability is being achieved routinely at manyof these light sources in terms of both AC (rms 0.1 - 200 Hz) and DC (one week drift) motion.” For fairly-smooth orbits, if the orbits are that close at all BPM locations, they will be almost that close everywhere.With both spin tunes accurately measured, and their MDM’s known, the average circumference uncer-tainty will be dominated by spin tune measurement inaccuracy, which could correspond to 11 decimalpoint circumference accuracy.In any case it is the circumference differences rather than the individual circumferences that willgovern the accuracy of the spin tune comparator. After nulling all BPM differences, the CW and CCWcircumferences will then be equal to about 13 decimal places.With revolution period known “perfectly” from RF frequency measurement, and average velocityknown “perfectly” from frozen spin and accurately known MDM, even the absolute circumference valuewill be known to high accuracy.
G.3.3 Spin tune invariance and spin tune comparator trap precision
By Eqs. (G.5) spin tunes Q E and Q M depend only on G and γ but not on bend radius r . This implies, forplanar orbits, that spin tunes are conserved constants of the motion, independent of horizontal steering189rrors—assuming, of course, that components stay rigidly fixed in place.But (because of commutativity failure for rotations around non-parallel axes) vertical steeringerrors prevent the spin tune formulas from being universally valid conservation laws. Even so, from up-down symmetry, one expects the change ∆ Q S in spin tune caused by a vertical deflection angle ∆ y (cid:48) tobe proportional to ∆ y (cid:48) . By limiting the magnitudes of vertical deflection angles ∆ y (cid:48) to be less than, say − , one can expect the spin tunes Q E and Q M to be independent of lattice errors to, e.g. 14 decimalplace accuracy. Knowing the spin tunes and γ values of both beams precisely, and knowing the MDMof the particles in one of the beams, allows the MDM of particles in the other beam to be determined tohigh accuracy.This is how a “spin tune comparator trap” can compare MDM’s precisely. Parameter tolerancesfor EDM measurement will be comparable to those discussed in the previous section. G.4
Secondary beam solutions
G.4.1 Analytic formulation
Assume the parameters of a frozen spin master beam have already been established. As well as fixingthe bend radius r , this fixes the electric and magnetic bend field values E and B . A further constraintthat needs to be satisfied for secondary beam operation is implicit in the equations already derived. Tosimplify the formulas we make some replacements and alterations, starting with pc/e → p, and mc /e → m, (G.10)The mass parameter m will be replaced later by, m p , m d , m tritium , m e , etc., as appropriate for theparticular particle type. These changes amount to switching the energy units from joules to electronvolts and setting c = 1 .The number of ring and beam parameters can be reduced by forming the combinations E = qE r , and B = qcB r . (G.11)After these changes, the closed orbit condition has become p − B p + ( B − E ) p − E m = 0 , (G.12)an equation to be solved for secondary beam momentum p . Any solution meets the requirement for spintune comparator functionality, but not yet, in general, the doubly-magic, vanishing-spin-tune condition.Any stable secondary beam orbit has to satisfy this equation but, because the electric and magneticfield values have been squared, not every solution of the equation has electric and magnetic field valuesthat match the signs or magnitudes of the field values E and B constrained by the primary beam. Sosolutions of Eq. (G.12) have to be culled for consistency. The bending force has to be centripetal andconsistent with bending in a circle of radius r .By construction the already-established existence of a stable master beam implies the existenceof a real, CW (i.e. p > ) solution of the equation, say with mass m = m . We look for otherstable solutions, say with mass m = m and momentum p , for which there are no parameter changeswhatsoever, neither in E nor B , nor in the sign or magnitude of the bend radius of curvature.For spin tune comparator functionality, satisfying Eq. (G.12) is sufficient for finding compati-ble dual beam parameters, including determining their spin tunes to the high precision with which theanomalous magnetic moments are known.If anti-protons, anti-deuterons, or other anti-baryons were experimentally available, the flexibilityprovided by Eq. (G.12) would be especially useful. The TCP combination of time, charge, and paritysymmetry transformations would then provide TCP-matched solutions of the equation. But the only190vailable negative particle is the negative electron, so TCP invariance applies usefully only to beamcombinations containing an electron or a positron beam.Limiting particle types to positron, proton, deuteron, tritium, and helions, a fairly comprehensivelist of promising “doubly-magic candidate” solutions has been produced, satisfying these requirements,including the requirement that the master beam satisfy the magic beam condition.For EDM measurement functionality the further constraints to be met are severe. With parame-ters established and set such that the “master beam” is magic, the only remaining free parameter is thesecondary beam energy. Doubly magic solutions are sought by varying this energy (always constrainingthe primary beam to satisfy the spin condition G.7). As well as meeting the vanishing spin tune con-dition, the energy also has to be such that beam production and handling is practical, and high qualitypolarimetry is available. G.5
Three practical doubly-magic solutions
G.5.1 Promising doubly-magic solutions
Several doubly magic beam pairs have been discovered. For this appendix just three cases are consid-ered. Their parameters are given in Table G.1. Details are given in the figure caption and case by caseexplanations are given in the sequel..Eq. (G.12) has been solved with MAPLE to produce Table G.1. (Intended only for checkingderived results, and otherwise unreliable) the numerical anomalous magnetic moment values used havebeen: G [positron, e+] = 0 . G [proton, p] = 1 . G [helion, h] = − . (G.13) r0 beam1 KE E0 B0 η E beam2 KE2 pc2 QS2m GeV V/m T GeV GeV(b) PERTURBED DOUBLY-MAGIC PROTON-PROTON (original) HOLY GRAIL option50 CW p 0.2328 8.386e+06 1.6e-08 1 CCW p 0.2328 -0.7007 -2.144e-06CW p 0.2328 0.7007 -1.024e-15(c1) DOUBLY-MAGIC PROTON-POSITRON (new) HOLY GRAIL option20 CW p 0.08663 6.355e+06 0.016 0.766 CCW e+ 0.03009 -0.0306 5.000e-06(c2) DOUBLY-MAGIC POSITRON-PROTON (inverse of (c1))20 CW e+ 0.03009 6.355e+06 -0.016 4.155 CCW p 0.08664 -0.4124 5.842e-05(q1) DOUBLY-MAGIC HELION-PROTON (JEDI “currently”-capable option)10 CW h 0.03924 5.265e+06 -0.028 1.351 CCW p 0.03859 -0.2719 -6.173e-06(q2) DOUBLY-MAGIC PROTON-HELION (inverse of (q1))10 CW p 0.03859 5.265e+06 0.028 0.6958 CCW h 0.03924 -0.4711 1.245e-05 Table G.1:
Beam-pair combinations for the three EDM experiments discussed in this appendix; master beamentries on the left, secondary beam on the right. “(b)”, “(c1)”, etc. are case labels. Dual rows allow either particletype to be designated “master beam”. Candidate beam particle types are ”e+”,“p”, “d”, “t”, “h” labelling positron,proton, deuteron, triton, and helion rows. Bend radii, particle type, and kinetic energies are given in the first threecolumns. There is no fundamental dependence of spin tune Q s on r , but r values have been chosen to limit | E | to realistic values. Bend radii choices of 10 m, 20 m, and 50 m result from the compromise between reducingring size and limiting electric field magnitude. r can be increased beneficially except for cost in all cases, but notnecessarily decreased . Master beam spin tunes are always exactly zero. Spin tunes of secondary beams are givenin the final column. In all cases they are close enough to guarantee they can be tuned exactly to zero. Further, caseby case, explanations are given in the text. G.5.2 Perturbative variant of all-electric (original) holy grail ring.
Case (b) in Table G.1 represents the all-electric frozen-spin proton ring which, up to now, has beenimplicitly anticipated to be the ultimate apparatus for measuring the proton EDM. With its detailedfeatures not yet understood this ring has been christened as the “holy grail” ring. Not intentionallypejorative, this language has been intended to acknowledge the significant uncertainties concerning thedetailed properties of such a ring. In the table this name has been changed to “(original) HOLY GRAIL”ring, so as to leave available the name “(new) HOLY GRAIL” ring, for the ultimate EDM ring proposedin case (c1).In fact, case (b) is already a more realistic representation of the all-electric ring in the sense thatsome residual non-vanishing vertical magnetic field will be inevitable, even in an all-electric ring. Thiswill require simultaneously-frozen-spin beam energies to have slightly different energies in all cases.With distributed electric and magnetic fields, using Eq. (G.9) to describe the performance of theentire ring as a Wien filter, it will not be difficult to meet the doubly-magic condition, even in the presenceof extraneous weak vertical magnetic field. In itself, this would not justify distributed magnetic field,however, as the same trimming could be done with a short local Wien filter.However the “perturbative” solutions (available also for all-electric electron, triton, and carbon 13frozen spin rings) are very robust in the sense that the superimposed magnetic field can be varied over alarge range while preserving the doubly-magic capability. This opens up the possibility of investigatingsystematic EDM errors by varying the magnetic bending fraction by a large factor.This robust property applies uniquely to perturbations away from an all-electric ring. (In this caseonly) the structure of Eq. (G.12) guarantees that there is a continuum of doubly-magic solutions in thevicinity of the all-electric condition. With counter-circulating beams of the same particle type, if thebending is frugal for one beam it is necessarily extravagant for the other. But, since the sign of η M reverses at the all-electric point, the continuity of solutions of Eq. (G.12) guarantees the existence of acontinuum of doubly-magic solutions in this vicinity. This is the justification for attaching “perturbed”to the name of case (b). This opens the possibility of reducing EDM systematic errors by acquiring datain configurations with substantially different magnetic fields.There is a complication concerning RF frequency, in that slightly different beam velocities willcause either slightly different orbits or slightly different revolution periods. For slow particles, such asprotons, this may require running on different harmonics of a single RF cavity. For positrons, becausethey are fully relativistic, this would probably be impractical, and the orbits would have to differ slightly.This RF issue is addressed explicitly below in the discussion of proton-helion case (q1). G.5.3 Proton-positron solution—the (new) holy grail.
From the point of view of greatest promise for ultimate fundamental physics discovery , case (c1) (withequivalent case (c2)), for proton and positron beams, seems to be the most promising case. It enablesmeasurement of the difference between a master beam containing protons and a secondary beam con-taining positrons.Cancelling the Koop wheel roll rate of the secondary beam containing positrons cancels the radialmagnetic field (under the assumption that the positron EDM is negligibly small). This allows the primarybeam Koop wheel roll rate to serve as a measurement of the proton EDM.As well as providing a clean, frequency difference measurement of the proton EDM, the beams192an circulate simultaneously. Because positron and baryon velocities differ by an order of magnitude, itis probably impractical for the acceleration to be provided by harmonics of a single RF cavity; dual RFsystems will be needed.A major impediment in this case is the low analysing power of existing polarimetry methodsfor electrons (of either sign). To remove the “holy grail” qualification in this case will require thedevelopment of resonant electron polarimeter. This limitation is discussed further below. Achievingnon-destructive, high analysing power electron polarimetry seems likely to be the only remaining majorimpediment to using EDM measurement to test the “standard model” of particle physics.
G.5.4 Helion-proton solution, JEDI-capable option.
Like the doubly-magic baryon-positron pair solutions, doubly-magic, different-type baryon-baryon pairscan be used to obtain EDM differences. A doubly-magic triton/proton solution has been found, but itrequires electric fields that are probably unachievable, even in the largest ring currently under consider-ation. However, by fortuitous accident of their anomalous magnetic moments, there is a doubly-magichelion/proton solution (q1) (with equivalent (q2)) that needs only a small ring. (The development of apolarised helion beam at BNL is described by Huang et al. [18].) For this case radius r has been takenin the table, in round numbers, to be 10 m. But (with electric field increased by 10 percent) this case isintended to match the 9 m bending radius EDM prototype ring described elsewhere in this present CERNyellow report.The (q1) case has a CW, frugal bending solution for protons as master beam, with a CCW, extrava-gant bending helion beam as secondary beam. Carbon scattering asymmetry polarimetry will presumablybe used for both beams.With a single RF cavity, to account for the different proton and helion velocities, the RF harmonicnumbers can be 107 and 180, resulting in revolution period fractional difference of × − .What makes this doubly-magic proton-helion option exciting is that, in the near future, using onlycurrently-established experimental techniques, an upper limit for the EDM of baryons can be substan-tially reduced from current limits, possibly even to a level capable of demonstrating “physics beyond thestandard model”. G.5.5 Stability of the 233 MeV all-electric fixed point
An important motivation for building an EDM prototype ring followed from the observation that in-evitable magnetic field contamination will cause an all-electric ring to be an unrealisable idealisation.Stated more succinctly, it is impossible to construct a ring with exact time reversal symmetry. (Especiallyconsidering the extreme narrowness of the frozen spin condition) this observation has the important con-sequence that beams, identical except for direction, cannot counter-circulate simultaneously. This threat-ens to make a truly all-electric ring unsatisfactory. Note, though, that the word identical was italicisedin the previous sentence. This leaves open the possibility of non-identical beams counter-circulatingsimultaneously. The following discussion expands upon this possibility.This section defines “frozen-spin, fixed point stability” to be general enough to assure that counter-circulating proton beams can be simultaneously frozen, even if some fraction of the bending is magneticrather than electric. With vanishing magnetic field this is assured by time reversal symmetry. Theproblem is that any magnetic field present in the ring, no matter how small, will destroy the time-reversalsymmetry property that a truly all-electric ring guarantees. For brevity the treatment of only the mostserious failure of time reversal symmetry will be described. Formally speaking, all that is known is that,for protons, the ring has a fixed point near 233 MeV. But it is not automatically known whether this fixedpoint is stable or unstable. 193t was shown theoretically in an appendix to the original Courant and Snyder alternating gradientpaper [19], that magnetic ring orbits are guaranteed to have a stable periodic closed orbit fixed point,with the consequence that there is an ellipsoidal region of six dimensional phase space in which particlescan circulate indefinitely. The following argument, essentially a continuation of the earlier “perturbeddoubly-magic proton-proton” sub-section, shows that the all-electric ring is similarly stable, even in thepresence of (sufficiently weak) superimposed magnetic field.Assume, for example, the presence of a small, unintentional, vertical magnetic bending field per-turbation ∆ B y , superimposed on an otherwise-ideal all-electric ring. Such a perturbing field wouldprovide constructive bending for one beam and destructive bending for the other. This would violate thetime reversal symmetry of the apparatus, which might seem to prevent both beams from having frozenspins at the same time. It needs to be shown that this is not the case.The leading effect of the ∆ B y perturbation would be, after RF capture and bunching, to reducethe average momentum of one beam and increase that of the other. Would neither beam, therefore, havethe “magic” momentum needed for frozen spins? In fact, as in the prototype ring, the spin tune canvanish even in the presence of some magnetic bending. So at least one of the beams, let us say the lowermomentum beam, can be adjusted to have frozen spins.What about the other beam, the beam with higher momentum? Most succinctly stated, the frozenspin condition is for the spin tune to vanish. Plotted as a function of beam momentum, the spin tunetherefore has to change sign as the beam momentum passes smoothly through the magic condition. For-tunately the electric/magnetic frozen spin bending requirement has the same dependence on momentum;if the electric and magnetic bending fractions have to be constructive (as they do for protons for mo-mentum below the all-electric magic value), then they have to be destructive for momentum above theall-electric value. As just explained, for the higher momentum, oppositely directed, proton beam, theelectric and magnetic forces do, indeed, combine destructively—as needed. As a result, with the avail-able degrees of freedom, if one beam satisfies the frozen spin condition, parameters can be varied suchthat the spins of the other beam are frozen as well. In fact, in the small B y limit, if one beam is magic,then, automatically, the other beam is magic also.The conclusion is that the “all-electric” ring does not, in fact, have to be all-electric for both beampolarisations to be frozen. In the present language, the ring is “doubly magic”.Of course, a price has to be paid. With the two beams having different (average) momenta, yetbunched longitudinally by the same RF cavity, the two beam circumferences will have to be (slightly)different. A correction for the effect of this imperfection would then have to be dead-reckoned theoreti-cally, or handled some other way. Such corrections need to be applied in every precise experiment. Andthis correction will surely be small compared to other anticipated errors.Continuing the fixed-point stability proof, the prototype ring accepts the existence of a B y =0 . T magnetic bending perturbation that is many orders of magnitude greater than will be likely in thefull-scale ring, yet freezes the beam spin nevertheless. This is not at all ideal for EDM determination,because an inevitable consequence of intentionally applying a strong B y bending field in the prototypering will be an unintentional ∆ B r radial magnetic field, that can be expected to be much greater thanwhatever radial magnetic field will be present in the nominal, all-electric ring. By being “all-electric”the full-scale ring will be optimal, at least in this respect.A useful way of understanding the effect of such an added ∆ B y magnetic field, is to realise thatthe entire ring can be regarded as a distributed Wien filter. For protons of velocity v p there is a matchingradial electric field ∆ E r = v p ∆ B y field which, applied to the bending electrodes, exactly restores oneor the other of the circulating beam design orbits, while doubling the perturbing effect on the other.This procedure effects all bunches identically. As explained in the prototype chapter, to tune individualbunches requires a local, stripline-based Wien-filter.Once one has accepted the existence of an "unintentional" B y magnetic field (perhaps at the micro-194esla level) one is tempted to consider the intentional inclusion of a B y magnetic field (perhaps at themilli-Tesla level, or even at the 0.03 T level of the prototype ring.) (While the deuteron combined E/Bring was still in serious contention, the inevitable presence of a strong magnetic field was occasionallypitched as a “feature”, rather than as a “bug”, because it provided another variable degree of freedomthat could be used to refine the EDM determination.)But the price of intentionally stronger B y is correspondingly greater. Greater momentum differ-ence between counter-rotating beams could lead to unacceptably large orbit differences that could onlybe avoided by dual RF cavities, or running on adjacent harmonic numbers of the single cavity. Neitherapproach is attractive, but either could, perhaps be implemented. G.6
Gravitational effect EDM calibration
Various authors [11] [12] [13] have pointed out that general relativity (GR) introduces effects that couldbe measurably large in proposed EDM rings. László and Zimborás [10] calculate the GR influenceon storage rings designed for EDM measurement. The GR effect mimics the EDM effect. Mistakenattribution to proton EDM produces a spurious proton EDM value of approximately × − e-cm.This is about thirty times greater than the precision anticipated for the (original) holy grail ring andnot inconsistent with an Orlov, Flanagan, Semertzidis [13] estimate. It is an accuracy that should beachievable with a small EDM prototype ring.The GR effect has two main ingredients, both essentially classical.– One is a “toy-top-like precession” caused by the earth’s (uniform) gravitation field applying torqueto the particle angular momentum. More accurately, the torque causing out-of-plane precession isapplied by ring focusing electric fields acting on the particle MDM. Long time beam survivalguarantees the absence of average vertical force.– The other is a secular “Foucault-pendulum-like” precession of the angular momentum during re-peated transits of a closed circular path.Once under control, the GR signal will serve as a valuable calibrator of the EDM detection apparatus.The absolute level of this calibration signal will be at the optimistic (i.e. large EDM value) end of therange of plausible “physics beyond the standard model”. G.7
The need for non-destructive resonant polarimetry
Arthur Schawlow, co-inventor of the laser, is credited with the advice to “Never measure anything butfrequency”. Though not emphasised up to this point, this principle is implicit in the present paper.Though this advice is often accepted, its basis is rarely explained.In our case the EDM signal at the end of an hour-long run may be an EDM-induced beam polari-sation angular difference of, say, a milliradian, between initial and final beam polarisation orientations.Expressed as a fraction of a complete revolution of the beam polarisation, this is − / (2 π ) . For anysingle run this angular shift is likely to be comparable with the difference uncertainty of destructive po-larimetry initial and final orientation measurements. (Then by averaging over, say, one thousand runs,the statistical error can be reduced by a factor of thirty or so. )Consider the same hour-long run with non-destructive resonant polarimetry, assuming, for themoment, the polarimeter natural resonant frequency to be the same as the beam revolution frequency.When sensed instantaneously, the resonator phasor angular advance from run beginning to run end islikely to approximately match the − / (2 π ) difference of the previous paragraph, with “phase noise”having yielded approximately the same uncertainty. But (absent other sources of low frequency noise)after non-destructive averaging the resonator phase for few-minute intervals at both beginning and end,the per-run phasor angular advance can be determined with far less uncertainty than is possible withdestructive scattering asymmetry. 195his has not yet included two other factors that favour resonant polarimetry. One of these factorsis that the whole beam is measured at both beginning and end. With destructive polarimetry, at best,orientation of only half of the beam is measured at run beginning; the other half of the beam is measuredat the end.The other advantage of resonant polarimetry would be that, in practice, the resonant polarimeterfrequency will be in the GHz range, 1000 times higher than the revolution frequency. Generally speaking,absolute precision seem to increase inexorably as technological advances allow processing at ever higherfrequencies. But it would not be legitimate to therefore claim a 1000 times higher precision, withouthaving acquired a deeper understanding of the issues. In our case, for example, at every instant of timethere will be a significantly large spread of particle revolution frequencies, more or less centred on afrequency that is known with exquisite accuracy from the known beam magnetic moments. Withouthaving a clear understanding of the fluctuations and averaging it is hard to refine the determination of thephase precision of resonant polarimetry.Regrettably, the entire discussion of resonant polarimetry up to this point has been “countingchickens before they’ve hatched”. Resonant polarimetry has never, in fact, been demonstrated to bepractical. However, theoretical calculations (admittedly due largely to the present author) based on theStern-Gerlach interaction, have shown that the regular passage of bunches of polarised electrons througha cavity should produce detectably-large cavity excitation [14] [15] [16]. The latter two of these refer-ences describe, in considerable detail, experiments being planned to test both transverse and longitudinalpolarimetry, using a polarised electron linac beam in the CEBAF injection line at the Jefferson Labo-ratory in Newport News, Virginia. Within a few years tests like these should have resolved the issueconcerning the practicality of Stern-Gerlach polarimetry for electrons.The proton’s magnetic dipole moment is three orders of magnitude smaller than the electron’s.In the absence of noise background a proton Stern-Gerlach signal reduced by this factor, would still bedetectably large but, without extremely narrow band lock-in detection, the proton polarimetry signal islikely to be swamped by noise. This makes phase-locked-loop proton beam polarisation control basedon resonant polarimetry likely to fail, even if resonant electron polarimetry has been demonstrated tosucceed. This is my expectation.It is this expectation that makes the doubly-magic proton-positron combination for measuringbaryon EDM’s seem especially important. With a positron beam phase-locked to resonant Stern-Gerlachpolarimeters (both transverse and longitudinal) the Koop wheel manipulations, so optimistically assumedin the present paper, should, indeed be extremely precise for the positron beam.By exploiting the known relation between positron and proton MDM’s, it should then be possibleto freeze the co-rotating proton spins just by controlling the positron beam spin tune and phase. With thefrequency and phase of the proton beam magnetisation then known to such high precision, the frequencyfiltering of a proton beam Stern-Gerlach resonator can be selective to reject the noise which would,otherwise, prevent the accurate resonant determination of the magnetisation signal.Only when non-destructive positron polarimetry has been successfully demonstrated will it belegitimate to remove the “holy grail” designation from the case (c1) positron-proton doubly-magic EDMring design, to make the discovery of physics beyond the standard model likely. G.8
The EDM measurement campaign
The majority of my work in the storage ring EDM area for the last several years has been performedduring, and in connection with, my stays at the IKP Institute for Nuclear Physics of Forschungszentrum,Juelich.During 2018, in response to a CERN invitation, an EDM task force at the IKP laboratory has beenperforming a feasibility study of measuring electric dipole moments, especially of the proton. A fullreport is due by the end of the year. The initial motivation for building a small prototype EDM ring was196o demonstrate the ability to store enough protons to enable an EDM measurement in a storage ring withpredominantly electric bending. A preliminary report was issued after the first quarter of 2018 [17]. Thepresent appendix has been coordinated with this task force planning.As well as developing long term planning, an important thrust of the task force has been to advo-cate the immediate development of designs for a “small” EDM prototype storage ring. The doubly-magicdesign should have a major impact on motivation. This design eliminates the need to use the vertical sep-aration of counter-revolving beam orbits to suppress radial magnetic field. Previous EDM designs haverequired excruciatingly small vertical betatron tune in order to enhance this “self-magnetometry” sensi-tivity to vertical beam separation of counter-circulating beams. The correspondingly weak focusing wasexpected to set a small limit on the proton beam intensity.The doubly-magic EDM ring design transfers this self-magnetometry responsibility to a secondaryfrozen spin beam (with the admitted cost of measuring EDM differences rather than absolute EDMvalues). Elimination of the need for ultraweak focusing should enable the beam current intensities tobe limited only by previously-encountered understood effects. This will permit the storage ring to havemuch stronger, alternating gradient focusing, which can be expected to increase the achievable protonbeam current substantially.Another motivation for building a small prototype EDM ring has been to develop and demonstratethe performance of instrumentation and procedures that will be needed for a subsequent larger ring.These applications are implicit in the examples of Table G.1. Especially relevant is the doubly-magiccombination of case (q1), which can be used to measure the difference of proton and helion EDM’s, Thiscan be done using carbon scattering polarimetry of the type that has been developed, and is already inservice, in the Juelich COSY ring. As already stated, any miserably large difference between proton andhelion EDM’s would constitute physics beyond the standard model.Important contributions by my EDM collaborators need to be acknowledged, especially to SigMartin and Helmut Soltner for detailed discussions of implementation practicalities. Acknowledgementsare also due to Maxime Perlstein for insisting on a less confusing treatment of the orbitry, to EannaFlanagan and Andras Laszlo for communications concerning general relativistic effects, and to AndreasWirzba for conveying and explaining a GR analysis by Kolya Nikolaev.
References [1] J. Farley et al.,
New Method of Measuring Electric Dipole Moments in Storage Rings,
Phys. Rev.Lett., 052001, 2004[2] Y. Semertzidis for the Storage Ring EDM Collaboration,
A Proposal to Measure the Proton ElectricDipole Moment with − e-cm Sensitivity, October, 2011[3] I.A. Koop,
Asymmetric energy colliding ion beams in the EDM sorage ring,
Paper TUPWO040, inProceedings of IPAC2013, Shanghai, China, 2013. When beginning to prepare the present paper,though aware of the “Koop spin wheel”, we had not realised that the shared beam EDM approach,not including the doubly-magic possibility, nor the possibility of electron or positron secondarybeam, nor the spin tune comparator functionality, had been proposed by Koop five years earlier.[4] D. Eversmann et al.,
New method for a continuous determination of the spin tune in storage ringsand implications for precision experiments,
Phys. Rev. Lett.
Phase-locking the spin precession in a storage ring,
P.R.L. , 119401,2017[6] I. Koop,
Spin wheel – a new method of suppression of spin decoherence in the EDM storage rings, http://collaborations.fz-juelich.de/ikp/jedi/public-files/student_seminar/SpinWheel-2012.pdf[7] A. Wolski,
Beam Dynamics in High Energy Particle Accelerators,
Eq. (3.175), Imperial CollegePress, 2014[8] G. Gabrielse,
The standard model’s greatest triumph,
Physics Today, p. 24, December, 20131979] G. Decker,
Beam stability in synchrotron light sources,
Proceedings of Lyon, France, DIPAC 2005[10] A. László and Z. Zimborás,
Quantification of GR effects in muon g-2, EDM and other spin preces-sion experiments,
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Manifestations of the rotation and gravity of the Earth inhigh-energy physics experiments,
Phys. Rev. D94, 2016[12] Y. Obukov, A. Silenko, and O. Terayev,
Manifestations of the rotation and gravity of the Earth inspin physics experiments,
Int. J. Mod. Phys. A31, 1645030, 2016[13] Y. Orlov, E. Flanagan, and Y. Semertzidis,
Spin rotation by Earth’s gravitational field in a “frozen-spin" ring,
Phys. Lett. A376, 2822, 2012[14] R. Talman,
The Electric Dipole Moment Challenge,
IOP Publishing, 2017[15] R. Talman, B. Roberts, J. Grames, A. Hofler, R. Kazimi, M. Poelker, and R. Suleiman,
Resonant(Longitudinal and Transverse) Electron Polarimetry,
XVii International Conference on PolarizedSources, Targets, and Polarimetry, Daejeon, Republic of Korea, 2017[16] R. Talman,
Prospects for Electric Dipole Moment Measurement Using Electrostatic Accelerators,
Reviews of Accelerator Science and Technology, A. Chao and W. Chou, editors, Volume 10, 2018,not yet in print[17] JEDI EDM task force “Easter Report”, unpublished internal report, Institute for Nuclear Physics ofForschungszentrum, Juelich, Germany, April 30, 2018[18] H, Huang et al.,
Optics setup in the AGS and AGS booster for polarized helion beam,
PaperWEPRO071 in the Proceedings of Dresden, Germany, IPAC2014[19] E. Courant and H. Snyder,
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Annals of Physics, , 1,1958 198 ppendix HNew ideas: Spin Tune Mapping for EDM Searches Abstract
The appendix describes an EDM measurement method that uses the Wien filterto produce spin phase advance in the same plane in which the MDM-inducedspin precession occurs. For protons at 30 MeV (non-frozen spin) this planeis the horizontal plane of the ring. For protons at 45 MeV with “frozen spin”this plane is orthogonal to the radius of the ring, where the MDM-inducedspin precession is produced by horizontal magnetic and vertical electric fieldsof the ring lattice imperfections. The EDM can then be extracted by ultra-precise determination of the shift of the spin precession frequency when thesign of the MDM component is reversed between running the beam clockwiseand counter-clockwise in the ring. An important virtue of the method is that,in the case of a pure electric ring (not necessarily frozen spin), it is free ofthe background from imperfect magnetic fields of the ring lattice and also al-lows to protect against the presence of external magnetic fields. Also, becausefrozen spin operation is not required, the method can be used to measure thedeuteron EDM. For searches of proton EDM in the prototype EDM storagering, sensitivity ≈ . · − e · cm at beam energy 30 MeV can be reached. H.1
Introduction
Interaction of MDM with vertical electric imperfection fields in the pure electric storage ring creates thetilt of invariant spin axis (cid:126)c = (cid:126)c y + (cid:126)c MDM xz away from vertical direction (cid:126)e y , where (cid:126)c MDM xz ⊥ (cid:126)e y and (cid:126)c y (cid:107) (cid:126)e y ,also c y ≈ . Projection (cid:126)c MDM xz is a function of azimuthal angle – it depends on which point in the ringthe invariant spin axis is viewed at. The reason for that is non-commutativity of spin rotations in theimperfection fields. Reduction of imperfection fields implies that all elements of the ring are preciselyaligned relative to common vertical axis which becomes a normal vector to the planar beam orbit. Then (cid:126)c MDM xz → at every point of the ring.Interaction of EDM with electric field in the ring tilts the invariant spin axis (cid:126)c towards X-axis(X-axis is pointing against the radius of the ring). This tilt is an indication of EDM signal. For pureelectrostatic storage ring the tilt angle ξ EDM due to EDM is defined as tan ξ EDM = ηβ − β (1 + G )) (H.1)where G is anomalous magnetic moment of particle, η is related to EDM. For protons with kinetic energyT=30 MeV, ξ EDM ≈ . η . H.2
The mixing of EDM signal with systematic background from MDM
In non-ideal storage ring, the tilt due to EDM adds up to the tilt induced by MDM (up to a first orderexpansion in small ξ EDM ) and EDM signal mixes with systematic effects of MDM spin rotation inimperfection fields: (cid:126)c = (cid:126)c y + (cid:126)c MDM xz + ξ EDM (cid:126)e x . (H.2)Experimental determination of the orientation of the invariant spin axis was performed at COSY[1]. The method was based on the observation of the most precise quantity measured presently at COSY199t − level for 100 seconds of the beam cycle – a spin tune [2]. Two static solenoids, one in eachstraight section of the ring, were acting as artificial imperfections which induced the change of the spintune when powered on. The change of the spin tune was predicted by the theoretical model. The unknownparameters were the tilts of the invariant spin axis towards Z-axis (which points along the momentum), c z , at the spots where the solenoids were located. Sensitivity to the angular direction of the invariant spinaxis . · − rad was achieved.Determination of c x projections with this method requires the use of static Wien filters with trans-verse horizontal magnetic fields ( (cid:126)B = (cid:126)e x B ). Such Wien filter rotates the spin around X-axis by constantangle each turn and changes the spin tune. Presently at COSY, there are two Wien filters that can workwith horizontal orientation of B-field, but both of them are radio-frequency devices. Running such RFWien filter on beam revolution frequency allows it to perform as a static one. The time when RF fieldreaches its maximum should be synchronized with the time when the bunch is passing through the Wienfilter. However, the measurement of ξ EDM (cid:126)e x separately from the direction of (cid:126)c MDM xz would still be notpossible for COSY (see [6]). The use of two Wien filters would provide information about azimuthaldependence of the sum (cid:126)c MDM xz + ξ EDM (cid:126)e x . This will give an input to the model of the ring which shouldbe based on the precise knowledge of the fields and beam orbit. Then variations of (cid:126)c MDM xz from one pointto another can be predicted and compared with measured ones, at the same time ξ EDM will be unknownparameter which needs to be determined.
H.3
Advantage of electrostatic rings
The advantage of pure electrostatic machine is that two counter-circulating beams can be stored si-multaneously. It allows to control the unwanted magnetic fields in the ring by observing the relativeseparation of closed orbits for clockwise (CW) and counterclockwise (CCW) beams. Then if unwanted,non-reversible magnetic fields are removed, closed orbits become equal and following relations are true: (cid:126)c cw = (cid:126)c y + (cid:126)c MDM xz + ξ EDM (cid:126)e x (H.3) (cid:126)c ccw = − (cid:126)c y − (cid:126)c MDM xz + ξ EDM (cid:126)e x (H.4)As it was already explained in previous section, (cid:126)c MDM xz is a function of azimuthal angle, therefore thisproperty depends on where in the ring the (cid:126)c cw and (cid:126)c ccw are viewed at – it should be the same point forboth CW and CCW bunches.Eqs. H. − H. are also true for any storage ring operating at a non-frozen spin, be it puremagnetic, pure electric or hybrid electric and magnetic ring, assuming correct expression in Eq. H. for ξ EDM .If the condition (cid:126)c cw ( ξ EDM = 0) = − (cid:126)c ccw ( ξ EDM = 0) can be guaranteed by making CW andCCW beams equal, then Eqs. H. − H. would allow an extraction of the EDM signal from the sumof measured (cid:126)e x -projections of (cid:126)c cw and (cid:126)c ccw . In the sum the systematic effects of MDM spin rotationsrelated to the imperfections of electrostatic ring lattice are cancelled. Hence the prototype electrostaticEDM ring opens a unique opportunity to test the principle of separating the EDM signal from the MDMsystematic effect using simultaneously counter-circulating beams with non-frozen spin. H.4
The effect of the Wien filter on beam and spin
Measurement of (cid:126)e x -projection of invariant spin axis by observation of spin tune perturbations demandsthe use of static Wien filter with horizontal transverse spin rotation axis (cid:126)w = (cid:126)e x . But zero Lorentz forcecondition for the fields of the Wien filter can only be fulfilled for one direction of the beam: (cid:126)E + (cid:126)β × (cid:126)B = 0 . (H.5)200 ig. H.1: A schematic drawing of the ring. Two crosses define the location where CW and CCW bunches alwaysintersect with each other. The points where CW and CCW bunches are always located diametrically opposite toeach other in the ring are marked as circles. At one of such points the Wien filter can be installed.
In order to fulfill zero Lorentz force condition for opposite beam direction, magnetic field in the Wienfilter should change the sign: (cid:126)E + ( − (cid:126)β ) × ( − (cid:126)B ) = 0 . (H.6)The change of magnetic field direction can be achieved by making it an RF field that oscillates at thebeam revolution frequency, in a similar way as proposed in section H. . Electric field should remainconstant for every turn.There are two points in the ring where CW and CCW bunches are always diametrically opposite toeach other and where they intersect, as shown on Fig. H. . Azimuthal position of this points is controlledby RF cavity. Then the Wien filter should be installed at the point where the CW and CCW bunches arediametrically opposite to each other on every turn, so that after half of the revolution period, either CWor CCW bunch enters the Wien filter (see Fig. H. ).The ideal Wien filter has exactly crossed E and B fields matched to a zero Lorentz force and rotatesthe spin around X-axis. Consider now the case of the imperfect Wien filter with horizontal magnetic andvertical electric fields which are not strictly orthogonal to each other and the ratio between E and B doesnot exactly match the zero Lorentz force condition. Such Wien filter will steer the beam vertically andit can have other components of spin rotation axis, (cid:126)w = (cid:126)e x w x + (cid:126)e y w y + (cid:126)e z w z , besides w x ≈ . TheWien filter changes closed orbit, which leads to the change in the direction of the invariant spin axisat the point where the Wien filter is installed. This change adds up to the effect of systematic MDMspin rotations (cid:126)c MDM xz . For both counter-circulating beams these additions are equivalent if the changesof closed orbits are also equal. This will be the case if magnetic field precisely reverses the directionbetween the appearances of CW and CCW beams at the Wien filter. Then Eqs. H. − H. remain valid.The axis of spin rotation in the Wien filter for CW beam in comparison to that of CCW beam isexactly opposite because of B-field reversal: (cid:126)w cw = − (cid:126)w ccw . (H.7) H.4.1 Spin tune shift by Wien filter and EDM
The analysis presented here is assuming that the beam in the storage ring has the energy away of the“frozen spin” condition. It is bunched and polarization of the bunch is in horizontal plane. Continuousmeasurement of time-dependent horizontal polarization P x = (cid:80) i = N S ix allows to determine the spintunes of CW and CCW bunch ( N -number of particles). The sextupole fields are set up to provide at least τ = 1000 seconds of spin coherence time. 201 ig. H.2: A time-line of Wien filter operation.
The change of the spin tune ∆ ν s produced by the spin kick ψ in the Wien filter, is given by: cos π ( ν s + ∆ ν s ) = cos πν s cos ψ − (cid:126)c · (cid:126)w sin πν s sin ψ (H.8)The difference of scalar products (cid:126)c · (cid:126)w for CW and CCW beams gives: (cid:126)c cw · (cid:126)w cw − (cid:126)c ccw · (cid:126)w ccw = 2 w x sin ξ EDM (H.9)Then the difference of the spin tunes for CW ( ν cws = ν s + ∆ ν cws ) and CCW ( ν ccws = ν s + ∆ ν ccws )bunch is proportional to EDM tilt angle and spin kick of Wien filter, while the effects of MDM spinrotations cancel: ν cws − ν ccws = 1 π ξ EDM ψ (H.10)Time dependence of transverse horizontal projection of polarization is measured (see Fig. H. ).The spin tunes of CW and CCW bunches are determined, each one should depend quadratically on ψ .In order to control the time-dependent systematic effects within a beam cycle, the phase shift ∆ Q · t = 2 π ( ν cws − ν ccws ) f rev t between spin oscillations of CW and CCW beam can be monitored (here f rev = βc/U - a revolution frequency). Statistical sensitivity to the EDM is given by: σ ( | (cid:126)d | ) = (cid:126) γ | (cid:126)s | | − β ( G + 1) | G + 1 U √ EL √ N f AP τ (H.11)Sensitivity is inversely proportional to the electric field integral EL in the Wien filter.202 ig. H.3: Time dependence of horizontal spin projection S x for CW (red) and CCW (blue) particle. Black curvedenotes projection S x for either CW or CCW particle in case EDM is zero or Wien filter is switched off. Systematic effects that are coming from external magnetic fields (such as magnetic field of Earth)lead to (cid:126)c cw ( ξ EDM = 0) (cid:54) = − (cid:126)c ccw ( ξ EDM = 0) for both points in the ring (see Fig. H. ) where theWien filters could be installed. Moreover, due to different direction of the external magnetic field atevery element of the ring in the particle rest frame and because of non-commutativity of spin rotations, (cid:126)c cw1 ( ξ EDM = 0) + (cid:126)c ccw1 ( ξ EDM = 0) (cid:54) = (cid:126)c cw2 ( ξ EDM = 0) + (cid:126)c ccw2 ( ξ EDM = 0) , where indices 1 and 2distinguish between two points in the ring for location of Wien filters. If the external magnetic field isweak and orbit separation between CW and CCW beams is not measurable, then the crosscheck of themeasurement ν cws − ν ccws with second Wien filter for the same beam can provide another control of sucheffects. Inequality ν cws − ν ccws (cid:54) = ν cws − ν ccws indicates that systematic effects are present, while in theideal case, the r.h.s. of Eq. H. should be the same for the measurement with every Wien filter andindependent of where in the ring it is installed.The EDM limit of σ p ≈ . · − e · cm can be achieved over the year of measurements, if thenew technique is applied at the prototype EDM storage ring for protons at 30 MeV. Integral field 0.0005Tm at the maximum peak of B-field is required in the Wien filter together with 37.5 KV of constantintegral electric field. Polarization P=0.8 and beam intensity particles per fill is assumed. Detectionefficiency f = 0 . and analyzing power A = 0 . can be achieved with multi-foil carbon polarimeterfor protons [4].In order to minimize the effects of synchrotron oscillations which lead to non-compensated Lorentzforce for head and tail particles in long bunches, it is advisable to have a flat-top pulsed B-field. H.5
Spin wheel at the beam energy of frozen spin
In a special case when proton energy is such that "frozen spin" condition is met, vertical component ofinvariant spin axis c y (cid:126)e y vanishes in Eq. H. and evolution of vertical polarization should be measured.If no imperfection fields are present in the ring, EDM aligns the invariant spin axis with X-axis: ξ EDM = π/ and (cid:126)c = (cid:126)e x , while the spin tune becomes ν s = ηγβ .When the Wien filter that is described in section H. works in the ring at "frozen spin", it allowsto align the invariant spin axis with X-axis in the presence of imperfection fields. That leads to the "SpinWheel" (see [3] and section . ) for both CW and CCW bunches that has frequency proportional to thespin kick of the Wien filter. Then the difference of the "spin wheel" tunes is: ν cws − ν ccws = ηβγ (H.12)203here η is directly proportional to EDM, β and γ -Lorentz factors. The wheel frequency is 10 Hz for thementioned E- and B-field integrals at proton kinetic energy of frozen spin T (cid:39)
232 MeV in the "nominalall electric storage ring" (see chapter ) and 153 Hz for protons at T (cid:39)
45 MeV in the prototype EDM ring(PTR) where combined E- and B-fields in the deflectors are used to freeze the spin. In the latter case,either CW or CCW beam is stored in consecutive beam cycles.
H.6
Other possibilities for EDM measurements with counter-circulating beams
H.6.1 An option with two RF Wien filters in the prototype EDM ring
Development and construction of the Wien filter described in section H. requires time and resources.There is another option to perform the EDM measurement at non-frozen spin energies of the beam inelectrostatic ring. It is based on the method [5] described in chapter . Combined effect of EDM andMDM on the spin motion in the ring is given by Eq. H. . Small vertical spin oscillations producedfrom horizontal components of (cid:126)c are resonantly excited by the spin kicks in the radiofrequency Wienfilter (it has vertical spin rotation axis (cid:126)w = (cid:126)e y ). That leads to much greater amplitude of S y oscillationswhich becomes accessible for polarimetry. The frequency of S y oscillations is proportional to (cid:107) (cid:126)c MDM xz + ξ EDM (cid:126)e x (cid:107) and integral electric (or magnetic) field in the Wien filter.The RF Wien filter is designed such that the E-field follows the B-field oscillations. It means thatLorentz force is zero only for one beam direction and the RF signal should be gated out when counter-circulating beam comes. This can be achieved by installation of the RF Wien filters at two points wherethe counter-circulating beams are opposite to each other (see Fig. H. ). Then gating out the RF signalof both Wien filters with beam revolution frequency allows to run CW and CCW beams simultaneously.The outcome is similar to the one discussed in section H. : (cid:107) (cid:126)c MDM xz + ξ EDM (cid:126)e x (cid:107) at two points of thering is determined, and if direction (cid:126)c MDM xz can be predicted from the model assumptions, it allows to find ξ EDM . However, direct extraction of EDM signal is also possible when both RF Wien filters are switchedto make zero Lorentz force for opposite beam directions. In this case the directions of (cid:126)c
MDM xz at the Wienfilter locations change sign. Spin rotations produced by one RF Wien filter for CW and CCW beamsare compared. Additionally, static solenoid is needed to suppress the (cid:126)e z projection in (cid:126)c MDM xz , otherwise (cid:107) (cid:126)c MDM xz + ξ EDM (cid:126)e x (cid:107) ∝ ξ EDM .Another advantage of this option is that RF Wien filter is transparent for the off-momentum parti-cles. The Wien filter RF phase and amplitude of the field can be adjusted such that only a slow build-upof vertical polarization is observed during the whole beam cycle. That allows to increase the statisticalsensitivity of this method in 2.5 times in comparison to the method discussed in section H. , assumingthe same field integrals in the Wien filters. Disadvantage of the method is that direct extraction of EDMsignal from the measured P y polarization build-ups produced with the same RF Wien filter for CW andCCW beams, depends on the equality of the orbits in the consecutive CW - CCW beam injections. H.6.2 An option with static Wien filters in the prototype EDM ring
Instead of RF Wien filters described in section H. . , one or more Wien filters with static vertical electricand static horizontal magnetic fields can be used. The placement of Wien filters is not crucial. For asingle Wien filter, all conclusions are the same as previously stated in section H. . The only differenceis that Eq. H. for ν cws − ν ccws is calculated for CW and CCW beams that are running consecutively,and B-field of the Wien filter(s) is reversed between the injections in CW and CCW direction. This canlead to a systematic error if field reversal is not exact. Assuming that one can achieve two orders ofmagnitude higher field integrals for static fields, such method can have an advantage that it allows toreduce statistical error to EDM by two orders of magnitude compared to the one discussed in section H. . 204 .7 Summary and outlook
Here we propose a new method for measurement of charged particles EDM’s in electrostatic storagerings. One of the advantages of such rings is that CW and CCW bunches could be stored simultaneouslywhich allows to cancel the systematic effects of ring lattice imperfections. The advantage of the methodover the BNL proposal (see chapter ) is that the ring operation mode is not fixed only to the energy of"frozen spin" which means it can be of much smaller size and different particle species could be studied.The disadvantage is that sensitivity to EDM signal is suppressed by 4 orders of magnitude compared tothat at "frozen spin", assuming the electric field integral 37.5 KV in the Wien filter. Because of this themethod seems as an intermediate step towards ultimate EDM precision searches, and it is applicable atthe prototype (PTR) EDM ring. It serves as a complement for BNL proposal when applied at "frozenspin” for protons. It allows to control the systematic effects of unwanted MDM spin rotations producedby external magnetic fields when two Wien filters are used for spin tune mapping. References [1] JEDI Collaboration, proposals available http://collaborations.fz-juelich.de/ikp/jedi/ [2] Eversmann, D., et al.: New method for a continuous determination of the spin tune in storage ringsand implications for precision experiments. Phys. Rev. Lett. , 094801 (2015). DOI 10.1103/PhysRevLett.115.094801[3] I. Koop, “Asymmetric Energy Colliding Ion Beams in the EDM Storage Ring”, in
Proc. 4th Int.Particle Accelerator Conf. (IPAC’13) , Shanghai, China, May 2013, paper TUPWO040, pp. 1961–1963.[4] Ieiri, M., Sakaguchi, H., Nakamura, M., Sakamoto, H., Ogawa, H., Yosol, M., Ichihara, T., Isshiki,N., Takeuchi, Y., Togawa, H., Tsutsumi, T., Hirata, S., Nakano, T., Kobayashi, S., Noro, T., Ikegami,H.: A multifoil carbon polarimeter for protons between 20 and 84 mev. Nuclear Instruments andMethods in Physics Research Section A: Accelerators, Spectrometers, Detectors and AssociatedEquipment (2), 253 – 278 (1987). DOI https://doi.org/10.1016/0168-9002(87)90744-3. URL [5] Rathmann, F., Saleev, A., Nikolaev, N.N.: Search for electric dipole moments of light ions in storagerings. Phys. Part. Nucl. , 229–233 (2014)[6] Saleev, A., et al.: Spin tune mapping as a novel tool to probe the spin dynamics in storage rings.Phys. Rev. Accel. Beams , 072801 (2017) 205 ppendix INew ideas: Deuteron EDM Frequency Domain Determination Abstract
This appendix describes suppressing the geometric phase and machine im-perfection systematic errors, which are encountered in any frozen spin stor-age ring EDM measurement method based on observation of a slow, gradualchange in the beam polarization vector. The geometric phase error is caused bynon-commutating wobbling precessions of the polarization vector, which aresignificant only if the polarization vector precession rate is small. Geometricphase can be suppressed by dispensing with operating on the spin resonance(i.e., 3D frozen spin) state, in favour of operating on the 2D frozen spin state,represented by a rolling spin wheel. To eliminate the machine imperfectionsystematic error, the imperfection fields themselves are utilized as the driversof the spin wheel.The method is intended for a combined storage ring; the bend fields are mag-netic and the frozen spin condition is met using multiple, uniformly-distributed,discrete Wien filters. Reversing the bending field (along with the beam direc-tion) reverses the imperfection fields. The EDM measurement consists of mea-suring the difference of spin wheel roll rates, which is proportional to the EDM.Though motivated by the need to measure the deuteron EDM, the method canbe applied also to the proton.
I.1
Motivation
Storage ring-based methods of search for the electric dipole moments (EDMs) of fundamental particlescan be classified into two major categories, which we will call 1. Space Domain, and 2. FrequencyDomain methods.In the Space Domain paradigm, one measures a change in the spatial orientation of the beampolarisation vector caused by the EDM .The original storage ring, frozen spin-type method, proposed in [1], is a canonical example of amethodology in the space domain: an initially longitudinally-polarized beam is injected into the storagering; the vertical component of its polarisation vector is observed. Under ideal conditions, any tilting ofthe beam polarisation vector from the horizontal plane is attributed to the action of the EDM.Two technical difficulties are readily apparent with this approach:1. it poses a challenging task for polarimetry [2];2. it puts very stringent constraints on the precision of the accelerator optical element alignment.The former is due to the requirement of detecting a change of about · − to the cross sectionasymmetry ε LR in order to get to the EDM sensitivity level of − e · cm . [1, p. 18]The latter is to minimize the magnitude of the vertical plane magnetic dipole moment (MDM)precession frequency: [1, p. 11] ω syst ≈ µ (cid:104) E v (cid:105) βcγ , (I.1)206nduced by machine imperfection fields. According to estimates done by Y. Senichev, if it is to be ful-filled, the geodetic installation precision of accelerator elements must reach − m. Today’s technologyallows only for about − m.At the practically-achievable level of element alignment uncertainty, ω syst (cid:29) ω edm , and changesin the orientation of the polarisation vector are no longer EDM-driven.Another crucial problem one faces in the space domain is geometric phase error. [3, p. 6] Theproblem here lies in the fact that, even if one can somehow make field imperfections (either due tooptical element misalignment or spurious electro-magnetic fields) zero on average , since spin rotationsare non-commutative, the polarisation rotation angle due to them will not be zero.By contrast, the Frequency Domain methodology is founded on measuring the EDM contribution to the total (MDM and EDM together) spin precession angular velocity .The polarisation vector is made to roll about a nearly-constant, definite direction vector ¯ n , with anangular velocity that is high enough for its magnitude to be easily measureable at all times. Apart fromeasier polarimetry, the definiteness of the angular velocity vector is a safeguard against geometric phaseerror. This “Spin Wheel” may be externally applied [4], or otherwise the machine imperfection fieldsmay be utilized for the same purpose (wheel roll rate determined by equation (I.1)). The latter is madepossible by the fact that ω syst changes sign when the beam revolution direction is reversed. [1, p. 11] I.2
Universal SR EDM measurement problems
By way of introduction to the proposed measurement methodology, let us briefly summarize some mea-surement problems encountered by any EDM experiment performed in a storage ring; they can begrouped into two big categories:– Problems solved by a Spin Wheel:– spurious electro-magnetic fields;– betatron motion.– Problems having specific solutions:– spin decoherence;– machine imperfections.
I.2.1 Spin motion perturbation
Problems from the first category are ones introducing geometric phase error. Indeed, both the spuriousand the focusing fields, when acting on a betatron-oscillating particle, perturb the direction and magni-tude of its spin precession angular velocity vector. The effect is a spin kick in the direction defined bythe perturbation.Assume that the EDM provides a spin kick about the radial ( ˆ x -) axis. The magnitude of the angularvelocity vector has a general form ω = (cid:113) ω x + ω y + ω z , where ω y is minimized by fulfilling the frozen spin condition; ω z (the constant part of which is dueto machine imperfections) can be minimized via the installation of a longitudinal solenoid on the opticaxis. In the space domain, one also tries to minimize the ω (cid:104) E v (cid:105) contribution to ω x = ω edm + ω (cid:104) E v (cid:105) .Consequently, spin kicks must be minimized to (significantly) less than ω edm , so as to reduce geometricphase to less than the accumulated EDM phase. − T. ω = (cid:113) ( ω edm + ω SW ) + ω y + ω z ≈ ( ω edm + ω SW ) · (cid:34) ω y + ω z ω SW (cid:35) / ≈ ( ω edm + ω SW ) · (cid:32) ω y + ω z ω SW (cid:33) ≈ ω SW + ω edm + 12 ω y + ω z ω SW (cid:124) (cid:123)(cid:122) (cid:125) (cid:15) . Since our goal is to observe the EDM-related value shift in ω , we need to minimize randomvariable (cid:15) : ω y + ω z ω SW < ω edm . Let’s make some preliminary estimates. Suppose ω SW ≈ rad/sec (the reason for choosing thisvalue will be explained shortly), ω edm ≈ − rad/sec (corresponding to the EDM value − e · cm).Then, ω y + ω z must be reduced to less than − rad/sec, or equivalently, either angular velocity to lessthan · − rad/sec. This is several orders of magnitude greater than the expected standard error on theangular velocity estimate, [5] and hence should not be a problem to achieve.One case left to be considered is MDM spin kicks about the ˆ x -axis. These are not attenuated,and cause the most trouble. They come in three varieties: a) permanent, not caused by optical elementmisalignments; b) semi-permanent, caused by element tilts about the optic axis; c) spurious.Semi-permanent radial spin kicks (be they caused by magnetic or electric fields) change sign whenthe beam revolution direction is reversed from clockwise (CW) to counter-clockwise (CCW). Spuriouskicks can be dealt with by statistical averaging. Permanent, insensitive to either the guide field or thebeam circulation direction, cannot be controlled. On the bright side, their sources should not be presentunder normal circumstances.For more details on spin motion perturbation effects on the measurement of the EDM in frequencydomain, please refer to [6]. I.2.2 Expected machine imperfection SW roll rate
In the estimates above, we used a roll rate ω SW ≈ rad/sec for the spin wheel. This is our expected ω syst caused by machine imperfections.Denote the standard deviation of the imperfection radial magnetic field distribution σ [ B x ] . For thewhole ring, MDM precession will be distributed with a standard deviation [7] σ [ ω MDMx ] = emγ G + 1 γ σ [ B x ] √ n , where n is the number of misaligned elements, G = ( g − / is the anomalous magnetic dipole moment.For deuterons in lattices [8] of n on the order of 100 elements, rotated about the optic axis byangles Θ tilt ∼ N (0 , − ) rad, Y. Senichev estimates [7] ω MDMx between 50 and 100 rad/sec.Our simulations done in COSY INFINITY seem to confirm this result. In Figure I.1 you see theresults of the simulation in which we rotated the 32 E+B spin rotator elements used in the frozen spin208codename BNL) lattice [8] by angles randomly picked from the distribution N ( µ · ( i − , σ ) , where µ = 10 · σ = 10 − rad, i ∈ { , . . . , } .At (cid:104) Θ tilt (cid:105) = 10 − we observe a roll rate of 500 rad/sec. We should keep in mind, however, thatSenichev assumes σ Θ tilt = 10 − rad, which means, for a lattice with n = 100 tilted elements, a standarddeviation of the mean σ (cid:104) Θ tilt (cid:105) = σ Θ tilt / √
100 = 10 − . The dependence of ω MDMx on (cid:104) Θ tilt (cid:105) is linear,which means in an actual lattice we would observe an ω syst ≤ rad/sec with 68% probability, and ω syst ≤ rad/sec with 95% probability, and with 27% probability ≤ ω syst ≤ . Fig. I.1:
Spin precession frequency (radial and vertical components) versus the mean E+B element tilt angle
I.2.3 Spin decoherence
Spin coherence is a measure or quality of preservation of polarisation in an initially fully-polarizedbeam. [9] Spin decoherence refers to the depolarisation caused by the difference in the beam particles’spin precession frequencies.The difference in spin tunes is due to the difference of the particles’ orbit lengths, and hence theirequilibrium energy levels, on which spin tune depends. One way spin decoherence can be suppressed isby utilization of sextupole fields. We consider how this can be accomplished in [10].
I.2.4 Machine imperfections
As we have seen, the problem with machine imperfections is twofold: a) they are practically impossibleto remove at the present level of technology; but what’s even worse, b) their removal leaves one in thespace domain, and opens the measurement up to geometric phase error.209ortunately for us, the imperfection spin kicks they induce change sign when the beam circulationdirection is reversed. Their magnitude is also sufficient for use as a Koop Wheel. The one remainingdifficulty is the accuracy of the Koop wheel roll direction flipping. Hopefully, we can make a persuasiveenough argument as to how this can accomplished.
I.3
Main methodology features
The method we propose is characterized by two main features:1. It is a frequency domain method;2. The fields induced by machine imperfections, instead of being suppressed, are used as a KoopWheel.– The Koop Wheel roll direction is reversed by flipping the direction of the guide field;– its roll rate is controlled through observation of spin precession in the horizontal plane.The advantages of the frequency domain, such as a) ease of polarimetry, and b) immunity togeometric phase error, have been discussed in previous sections. Now we will turn to the description ofhow machine imperfection fields can be used as a Koop Wheel.
I.4
EDM estimator statistic
Since the angular velocity measured in the frequency domain methodology includes contributions dueto both the magnetic and electric dipole moments, the EDM estimator statistic requires two cycles tocompose: one in which the Koop Wheel rolls forward, the other backward.The change in the Koop Wheel roll direction is affected by flipping the direction of the guide field.When this is done: (cid:126)B (cid:55)→ − (cid:126)B , the beam circulation direction changes from clockwise (CW) to counter-clockwise (CCW): (cid:126)β (cid:55)→ − (cid:126)β , while the electrostatic field remains constant: (cid:126)E (cid:55)→ (cid:126)E . According to theT-BMT equation, spin precession frequency components change like: ω CWx = ω MDM,CWx + ω EDMx ,ω CCWx = ω MDM,CCWx + ω EDMx ,ω MDM,CWx = − ω MDM,CCWx , (I.2a)and the EDM estimator ˆ ω EDMx := 12 (cid:0) ω CWx + ω CCWx (cid:1) (I.2b) = ω EDMx + 12 (cid:0) ω MDM,CWx + ω MDM,CCWx (cid:1)(cid:124) (cid:123)(cid:122) (cid:125) ε → . (I.2c)To keep the systematic error term ε below required precision, i.e. ensure that equation (I.2a) holdswith sufficient accuracy, Y. Senichev devised [7] a guide field flipping procedure based on observationof the beam polarisation precession frequency in the horizontal plane.To explain how it works, we need to introduce the concept of the effective Lorentz factor.210 .5 Effective Lorentz factor
Spin dynamics is described by the concepts of spin tune ν s and invariant spin axis ¯ n . Spin tune dependson the particle’s equilibrium-level energy, expressed by the Lorentz factor: ν Bs = γG,ν Es = β γ (cid:16) γ − − G (cid:17) = G +1 γ − Gγ. (I.3)Unfortunately, not all beam particles share the same Lorentz factor. A particle involved in betatronmotion will have a longer orbit, and as a direct consequence of the phase stability principle, in an accel-erating structure utilizing an RF cavity, its equilibrium energy level must increase. Otherwise it cannotremain the bunch. In this section we analyze how the particle Lorentz factor should be modified whenbetatron motion, as well as non-linearities in the momentum compaction factor are accounted for.The longitudinal dynamics of a particle on the reference orbit of a storage ring is described by thesystem of equations: (cid:40) dd t ∆ ϕ = − ω RF ηδ, dd t δ = qV RF ω RF πhβ E (sin ϕ − sin ϕ ) . (I.4)In the equations above, ∆ ϕ = ϕ − ϕ and δ = ( p − p ) /p are the deviations of the particle’s phase andnormalized momentum from those of the reference particle; all other symbols have their usual meanings.The solutions of this system form a family of ellipses in the ( ϕ, δ ) -plane, all centered at the point ( ϕ , δ ) . However, if one considers a particle involved in betatron oscillations, and uses a higher-orderTaylor expansion of the momentum compaction factor α = α + α δ , the first equation of the systemtransforms into: [11, p. 2579] d∆ ϕ d t = − ω RF (cid:34) (cid:18) ∆ LL (cid:19) β + (cid:0) α + γ − (cid:1) δ + (cid:0) α − α γ − + γ − (cid:1) δ (cid:35) , where (cid:0) ∆ LL (cid:1) β = π L [ ε x Q x + ε y Q y ] , is the betatron motion-related orbit lengthening; ε x and ε y are thehorizontal and vertical beam emittances, and Q x , Q y are the horizontal and vertical tunes.The solutions of the transformed system are no longer centered at the same single point. Orbitlengthening and momentum deviation cause an equilibrium-level momentum shift [11, p. 2581] ∆ δ eq = γ γ α − (cid:34) δ m (cid:0) α − α γ − + γ − (cid:1) + (cid:18) ∆ LL (cid:19) β (cid:35) , (I.5)where δ m is the amplitude of synchrotron oscillations.We call the equilibrium energy level associated with the momentum shift (I.5), the effective Lorentzfactor : γ eff = γ + β γ · ∆ δ eq , (I.6)where γ , β are the Lorentz factor and relative velocity factor of the reference particle.Observe, that the effective Lorentz factor enables us to account for variation in the value of spintune due to variation in the particle orbit length. It is crucial in the analysis of spin decoherence [10] andits suppression by means of sextupole fields.It plays a big role, as well, in the successful reproduction of the MDM component to the total spinprecession angular velocity. 211 .6 Guide field flipping
Two aspects of the problem need to be paid attention to:1. What needs to be kept constant from one measurement cycle to the next;2. How it can be observed.The goal of flipping the direction of the guide field is to accurately reproduce the radial componentof the MDM spin precession frequency induced by machine imperfection fields. This point should notbe overlooked: a mere reproduction of the magnetic field strength would not suffice, since the injectionpoint of the beam’s centroid, and hence its orbit length — and, via equations (I.6) and (I.3), spin tune, —is subject to variation. (Apart from that, the accelerating structure might not be symmetrical, in terms ofspin dynamics, with regard to reversal of the beam circulation direction.)What needs to be reproduced, therefore, is not the field strength, but the effective Lorentz factorof the centroid.Regarding the second question, we mentioned earlier that the Koop Wheel roll rate is controlledthrough measurement of the horizontal plane spin precession frequency. This plane was chosen becausethe EDM angular velocity vector points (mainly) in the radial direction; its vertical component is dueto machine imperfection fields, and is small compared to the measured EDM effect. Therefore, in firstapproximation, when we manipulate the vertical component of the combined spin precession angularvelocity, we manipulate the vertical component of the MDM angular velocity vector.Moving on to the effective Lorentz factor calibration procedure. Let T denote the set of all tra-jectories that a particle might follow in the accelerator. T = S (cid:83) F , where S is the set of all stabletrajectories, F are all trajectories such that if a particle gets on one, it will be lost from the bunch.Calibration is done in two phases:1. In the first phase, the guide field value is set so that the beam particles are injected onto trajectories t ∈ S .2. In the second phase, it is fine-tuned further, so as to fulfill the FS condition in the horizontal plane.By doing this, we physically move the beam trajectories into the subset S| ω y =0 ⊂ S of trajectoriesfor which ω y = 0 .Spin tune (and hence precession frequency) is an injective function of the effective Lorentz-factor γ eff , which means ω y ( γ eff ) = ω y ( γ eff ) → γ eff = γ eff . The trajectory space T is partitionedinto equivalence classes according to the value of γ eff : trajectories characterized by the same γ eff areequivalent in terms of their spin dynamics (possess the same spin tune and invariant spin axis direction),and hence belong to the same equivalence class. Since ω y ( γ eff ) is injective, there exists a unique γ eff at which ω y ( γ eff ) = 0 : [ ω y = 0] = [ γ eff ] ≡ S| ω y =0 . If the lattice didn’t use sextupole fields for the suppression of decoherence, S| ω y =0 would be asingleton set. We have shown in [10] that if sextupoles are utilized, then ∃D ⊂ S such that ∀ t , t ∈ D : ν s ( t ) = ν s ( t ) , ¯ n ( t ) = ¯ n ( t ) . By adjusting the guide field strength we equate D = S| ω y =0 , and hence S| ω y =0 contains multiple trajectories. Therefore, once we ensured that the beam polarisation does not precess in the horizontal plane, allof the beam particles have γ eff , equal for the CW and CCW beams.Guide field flipping procedure simulation results can be found in [12]. Strictly speaking, even if sextupoles are used there remains some negligible dependence of spin tune on the particle orbitlength (linear decoherence effects, cf. [10]). Because of that, the equalities for ν s and ¯ n are approximate, and the set S| ω y =0 should be viewed as fuzzy: we will consider trajectories for which | ω y | < δ for some small δ as belonging to [ ω y = 0] . .7 Statistical precision
Members of the JEDI Collaboration have studied the statistical precision of spin precession angular ve-locity estimation from sparse (one detector event per 100 spin revolutions) [13] and dense [5] polarisationdata. According to [13], the maximum likelihood estimator for the spin precession frequency estimatehas a standard error σ ˆ ω = 1 P T (cid:114) N , where N is the total number of recorded detector events, P is the beam polarisation, T is the measure-ment time.Assuming N = 7 . · events, polarisation P = 0 . , and cycle duration T = 1 , seconds(same parameters as in the simulation done in [5]), we have σ ˆ ω ≈ . · − rad/sec at the cycle level.Estimates made in [5] agree with this result.This precision is sufficient to obtain a mean estimate with statistical uncertainty σ (cid:104) ˆ ω (cid:105) ≈ · − rad/sec in one year of measurement, with the accelerator operational 70% of the time. An EDM of − e · cm should induce an ω edm on the level of − rad/sec in storage rings proposed in [8]. Thus,we expect to be able to measure the deuteron EDM at the − e · cm level in one year of measurementtime. References [1] D. Anastassopoulos et al., “AGS Proposal: Search for a permanent electric dipole moment of thedeuteron nucleus at the − e · cm level,” BNL, 2008.[2] S. Mane, “A distillation of Koop’s idea of the Spin Wheel,” arXiv:1509.01167 [physics] http://arxiv.org/abs/1509.01167 .[3] V. Anastassopoulos et al., “A Storage Ring Experiment to Detect a Proton Electric Dipole Moment.”Rev. Sci. Instrum., 87(11), 2016. http://arxiv.org/abs/1502.04317 .[4] I. Koop. “Asymmetric energy colliding ion beams in the EDM storage ring,” Proc. of IPAC13(2013). http://accelconf.web.cern.ch/accelconf/ipac2013/papers/tupwo040.pdf .[5] A. Aksentev, Y. Senichev. “Statistical precision in charged particle EDM search in storage rings.”2017 J. Phys.: Conf. Ser. 941 012083.[6] A. Aksentev, Y. Senichev, “Spin Motion Perturbation Effect on the EDM Statistic in the FrequencyDomain Method,” presented at the 10th International Particle Accelerator Conf. (IPAC’19), Mel-bourne, Australia, May. 2019, paper MOPTS011.[7] Y. Senichev, A. Aksentev, A. Ivanov, E. Valetov, “Frequency domain method of the search forthe deuteron electric dipole moment in a storage ring with imperfections,” arxiv:1711.06512[physics.acc-ph] https://arxiv.org/abs/1711.06512 .[8] Y. Senichev, S. Andrianov, S. Chekmenev, M. Berz, E.Valetov. “Investigation of Lattice forDeuteron EDM Ring,” Proc. of ICAP15 (2015). http://accelconf.web.cern.ch/AccelConf/ICAP2015/papers/modbc4.pdf .[9] E. Valetov, “Field modeling, symplectic tracking, and spin decoherence for the EDM and muong-2 lattices.” PhD tehsis, Michigan State University, Michigan, USA. http://collaborations.fz-juelich.de/ikp/jedi/public_files/theses/valetovphd.pdf .[10] A. Aksentev, Y. Senichev, “Spin decoherence in the Frequency Domain Method for the search ofa particle EDM,” presented at the 10th International Particle Accelerator Conf. (IPAC’19), Mel-bourne, Australia, May. 2019, paper MOPTS012.[11] Y. Senichev et al., “Spin tune decoherence effects in Electro- and Magnetostatic Structures.” Pro-ceedings of IPAC 2013, Shanghai, China, pp. 2579-2581.21312] A. Aksentev, Y. Senichev, “Simulation of the Guide Field Flipping Procedure for the FrequencyDomain Method,” presented at the 10th International Particle Accelerator Conf. (IPAC’19), Mel-bourne, Australia, May 2019, paper MOPTS010.[13] J. Pretz, “Determination of polarization and frequency parameters on sparse data,” JEDI internalnote ppendix JNew ideas: Distinguishing the effects of EDM and magnet misalignmentby Fourier analysis Abstract
This appendix shows that, by measuring vertical polarisation in two properlyseparated positions in the storage ring, it is possible to estimate the magnitudeof the major systematic uncertainty induced by ring imperfections. The im-precise positioning of the magnets causes the creation of a radial field, and theinteraction of the magnetic dipole moment with this field induces the effectmimicking the EDM signal. The ring imperfections are distributed rather ran-domly along the ring, while the dipole magnets form a very regular pattern.Therefore the changes of the vertical polarisation induced by magnets mis-alignment have a non-harmonic pattern. On the other hand, the EDM-inducedvertical polarisation has an almost harmonic pattern, since it results from thevertical field of ring dipoles, which is not strongly affected by their misalign-ments. Within a simple model the vertical polarisation induced by EDM andring imperfections is calculated as a function of time. Then it is shown thatFourier analysis of obtained signals sampled twice per beam revolution allowsto distinguish these two effects. It is done by comparison of the Fourier am-plitudes for revolution frequency and for difference of this frequency and spinprecession frequency. Even for unknown misalignments it is possible to pre-dict, with the given likelihood, the magnitude of the systematic uncertaintyinduced by ring imperfections.A reliable limit on the value of EDM in any experiment can be given only when systematic un-certainty is under control. In an experiment measuring EDM with a storage ring the most importantsystematic effect comes from the radial field arising due to magnets misalignment. For all proposedscenarios for EDM measurements based on the detection of induced vertical polarisation s y ( t ) , this sys-tematic effect mimics the expected EDM signal. One might rely on the simulations of misalignmenteffect on s y ( t ) , but magnets rotations and displacements are in fact unknown. Therefore a direct experi-mental estimation of systematic uncertainty of misalignment effect by means of Fourier analysis of s y ( t ) seems a much better solution.The presented method of misalignment effect calculations is an extension of the formalism pre-sented in [1]. The model is limited to particles moving on the central trajectory, but offers an analyticsolution with a detailed insight into the general features characterising the time dependence of polarisa-tion s y ( t ) . In the following example, for simplicity only, misalignment of COSY dipole magnets due torotation around beam axis are considered. With a known placement of magnets and their individual mis-alignments the distributions of all fields are represented by Fourier series with V , V cj , V sj representingthe Fourier coefficients for vertical field B V ( t ) and R , R cj , R sj for radial field B R ( t ) . Then the solutionof the BMT equation for longitudinal spin component s z ( t ) is expressed by those coefficients and twofrequencies: orbital ω o and that of spin precession ω s . Finally, s y ( t ) is obtained as the integral over timeof the product s z ( t ) B V ( t ) for the EDM effect and of s z ( t ) B R ( t ) for the misalignment effect. For moredetails of derivation see [1].To illustrate the time pattern for s y ( t ) the first leading terms are presented:215 DM misalignment × - - · - - · - · - · - t [ s ] s y ( t ) Fig. J.1:
Time dependence of vertical component of polarisation s y ( t ) induced by EDM (red line) and by magnetsmisalignment (blue line) shown for two periods of spin precession. s y ( t ) = ω X X sin( ω s t ) ω s + ∞ (cid:88) j =1 X cj (cid:20) sin( jω o t − ω s t ) jω o − ω s + sin( jω o t + ω s t ) jω o + ω s (cid:21) + ∞ (cid:88) j =1 X sj (cid:20) cos( jω o t − ω s t ) jω o − ω s + cos( jω o t + ω s t ) jω o + ω s (cid:21) + ... , (J.1)where for the misalignment effect ω X = ω s , X = R , X cj = R cJ , X sj = R sJ and for the EDM effect ω X = DβcB / (cid:126) , X = V , X cj = V cJ , X sj = V sJ with D being the EDM value and B dipole magnetfield. Even though the functional time dependence of both effects is the same, different values of Fouriercoefficients lead to different time histories for the two effects. In COSY as in any storage ring the magnetsform a regular pattern and the vertical field disorders due to magnets misalignment are small since theyscale with cosine of the misalignment angle. Therefore V cj for odd j and all V sj coefficients are small.On the other hand radial field scales with sine of the misalignment angle, than its distribution is quiterandom and all radial field Fourier coefficients have arbitrary values. This causes some differences intime dependence of s y ( t ) for the EDM and the misalignment effects seen in Fig. J.1. The numericalresults presented in this figure and hereafter are obtained for D = 4 . · − e · cm and the measuredCOSY dipoles misalignment angles.The differences in the time dependence of s y ( t ) can be quantified via Fourier analysis of theobserved signals. From Eq. J.1 it is seen that Fourier amplitudes for s y ( t ) should peak at frequencies ω s and ω o ± ω s (in general jω o ± ω s ). These maxima can be determined by sampling (measuring) verticalpolarisation with a proper frequency. In Fig. J.2 the Fourier amplitudes for s y ( t ) sampled with frequency ω o and ω o are shown. The first one corresponds to polarisation measurement at one place on the orbit,while for the second the polarisation needs to be measured in two, reasonably separated places. It is seenthat sampling with ω o is not sufficient to distinguish between the EDM and the misalignment effects.For the parameters chosen for numerical calculations the Fourier amplitudes F ( ω s ) at ω s for both effects216 DM with ω o samplingmisalignment with ω o samplingEDM with 2 ω o samplingmisalignment with 2 ω o sampling · - · - · - · - · - · - ω [ MHz ] F o u r i e r a m p l i t u d e ω s ω o - ω s Fig. J.2:
Fourier amplitude of the s y ( t ) signal for sampling with ω o frequency for EDM (red dashed line) andmisalignment (blue dashed line) and for sampling with ω o frequency for EDM (red solid line) and misalignmenteffect (blue solid line). are almost the same. Sampling s y ( t ) with ω o frequency, however, allows to observe a peak in Fourieramplitude F ( ω o − ω s ) at ω o − ω s frequency. In this case the amplitude for the EDM effect is by two ordersof magnitude smaller the the amplitude for the misalignment effect. Hence, determination of the F ( ω o − ω s ) amplitude for misalignment effect allows to determine also the magnitude of the amplitude F ( ω s ) forthis effect. Since for the EDM measurement at COSY two polarimeters will be available, the presentedmethod will allow to experimentally determine the misalignment-related systematic uncertainty for themeasured limit of the EDM value.The values of real misalignments of all magnets at COSY are known with a rather poor accuracy.In such case the presented method allows to calculate the probability of occurrence of a certain ratio ofFourier amplitudes F ( ω s ) /F ( ω o − ω s ) . Then, setting a confidence level it is possible to determine anupper limit for the systematic effect contributing to the measured F ( ω s ) amplitude. Since the magnitudeof the Fourier amplitudes for the EDM effect depends very weakly on magnets misalignments, it ispossible to determine the limit for the EDM value. An example of such analysis is shown in Fig. J.3. Theprobability distribution of the ratio F ( ω s ) /F ( ω o − ω s ) was obtained assuming that the rotation anglesof COSY dipoles have Gaussian distribution with a standard deviation of . ◦ References [1] A. Magiera,
Control of systematic uncertainties in the storage ring search for an electric dipole mo-ment by measuring the electric quadrupole moment , Phys. Rev. Accel. Beams , 094001, (2017)217 F ( ω s )/ F ( ω o - ω s ) p r o b a b i l i t y Fig. J.3:
The likelihood of the Fourier amplitude ratio F ( ω s ) /F ( ω o − ω s ) determined for the misalignment-induced s y ( t ) . ppendix KNew ideas: External Polarimetry Abstract
This appendix describes a pellet extraction scheme for extracting beam sam-ples from the beam core, rather than from the beam tails (as had been assumedup till now). Though not a new idea, itself, pellet beam extraction has been,until now, very erratic, largely because of the poorly controlled “spray” of pel-let directions. Recent pellet gun developments have made this approach muchmore promising. The appendix is largely didactic, collecting formulas neededfor the design of the pellet beam extraction. For EDM, the merit of pellet beamsampling is the elimination of the need for beam heating to produce the beamtails (with their dubious lattice function dependence and questionable system-atic validity) which enables internal target polarimetry, but cancels stochasticcooling possibilities. Because the pellets pass approximately through the beambunch centres, pellet-produced beam samples will be very representative of thetrue particle distributions (that can be further monitored by optical tracking ofthe pellets).
K.1
Pellet-extracted beam sampling
K.1.1 Pellet-extracted beam sampling; qualitative
K.1.1.1 Successful pellet injector implementation
Sun et al [1] have demonstrated a lithium pellet injector that can be copied more or less unchanged forthe beam sampling requirements of the EDM experiment. The upper part of Figure K.1 (copied directlyfrom their figure) shows the Sun et al pellet injector. The lower part of the same figure shows the extrafocusing (and isolation) stage needed to send pellets, one-by-one through our polarised proton (or otherbaryon) beam. The Sun application requires, fast lithium pellet microspheres, for the application oftriggering an EAST Tokomak). Available pellet speeds range from 30 to 110 m/s, ideal for our pelletbeam-sample extraction. Our application requires pellet material having highest possible charge number Z , for which pellet behaviour is expected to be closely similar. K.1.1.2 General description of pellet-induced beam sample extraction
The ideal polarimetry for an EDM measurement experiment would be non-destructive and continuous forhour-long runs, with no beam extraction sampling required. However, at present, the only practical formof polarimetry—left-right asymmetry proton-carbon scattering—consumes stored particles. One canimagine such scattering polarimetry from an internal carbon target—for example from carbon pellets.It is easy to show that this cannot be practical. A pellet big enough to have satisfactory polarimetryscattering efficiency will kill the entire beam within seconds. Beam sample extraction onto a “thick”carbon target is therefore required—so that the particle can scatter within a thick external polarimetrytarget.As it happens, the ability just mentioned, of a single pellet to destroy an entire beam can actuallybe exploited to produce very clean and efficient extraction of controllable samples from the core of astored proton (or other baryon) beam. Basically one person’s “suddenly destroyed beam” can be anotherperson’s “efficient slow-extracted beam sampling”. This is illustrated in Figure K.2. (Objection to thisconfiguration for polarimetry, based on the obvious left-right asymmetry of the extraction apparatus, is219 o 45 MeV, polarized proton beam(maybe 3 m below)1 cm D (maybe 1 m) impellerrotating paddle vacuumeverywhere focusing reflectorcirculating particle beam Fig. K.1:
The Sun et al lithium pellet launcher adopted for use as the pellet beam sampler of the EDM prototypestorage ring. The ability to switch among four pellet types would be unnecessary but, otherwise, the design canjust be copied. But the pellet sizes needed for the EDM application will be some five times smaller than for theTokomak triggering application. Their apparatus fed more than one at a time too-small pellets (far smaller thanthey needed) but their paper explains how a single gap height could be reduced to repair this behaviour. to be be addressed later.) When a particle in the circulating beam, by chance, passes through a transitorypassing pellet in one straight section, the particle loses enough momentum that, when it gets to the nextstraight section, it has become physically separated from the main beam—i.e. it has been “extracted”.The most important parameter, for the performance of the sampled beam extraction, is ∆ K p e . g . ≈− KeV, the kinetic energy change of a particle (for example, proton) in its centred passage througha pellet. Roughly half of the protons hitting the pellet will suffer very nearly this same energy loss; therest, because of their more glancing incidence, will suffer reduced energy loss, from this value all theway down to zero.For slow protons—for example 45 MeV kinetic energy—the dE/dx stopping power of protons islarge—about 7 times minimum-ionising. See Figure K.6. In virtually all cases the energy loss sufferedby a beam particle passing through any single pellet is far larger than the maximum energy that can be220 F q F D . mm > m < e nd s e g m e n t e d m < m > m s i ng l e p e ll e t s po r a d i c b ea m s a m p l e e x t r ac t o r po l a r i ze d b ea m p a r ti c l e s z through beam p e ll e t − e dg e s t r a gg l e r s L a nd a u s t r a gg l e r ss p i r a ll e d − i n b ea m θ x 6 − 26 +2A = = 0.5 detectorleftrightdetector Fig. K.2:
Top view of the left-right asymmetry of protons scattering through angles θ from the seven graphenefoils of the polarimeter target. (Some polarimeter components are traced from Figure 1(a) of reference [2].) Shorthash marks along the polarimeter centreline actually represent, first, an entry scintillation counter, followed byseven carbon foil polarimeter graphene foil targets and, finally two exit scintillation counters. The figure showshow one quarter of a ring with horizontal betatron tune Q x ≈ can act as a 180 degree spectrometer (even thoughit looks like 90 degrees), with point source at the pellet and the polarimeter at the “focus”. The regular beamfocusing serve to focus the extracted beam as well. recovered in a single passage through an RF cavity (should one be encountered along the path). All suchprotons will therefore have been ejected from their stable RF buckets, but their radial positions have notinstantaneously been altered in the process. The extracted “beam bunch” duration will be, for example,about . ms, which is the transit time for velocity vP pellets from entry to exit of the beam bunch.Meanwhile, because of their far greater velocity v p , the beam bunches will have made perhaps 100circulations of the storage ring, the extracted “bunch” will therefor be made up of 100 “sub-bunches”,each of the same length as the stored bunches, but staggered in time by time intervals equal to the ringcirculation time T ≈ µ s.Apart from this spreading in time, the beam being extracted is still a pencil beam emerging froma point source. But most of these protons have off-momentum values near ∆ p/p = − . . At a pointin the ring with dispersion D = 10 m, these about-to-be extracted protons are initially displaced fromtheir nominal off-momentum closed orbits by about 1 cm. Interpreted as a betatron amplitude, this isalmost twice the nominal beam bunch radius. After a horizontal betatron phase advance of π their radialbetatron displacements will be reversed to -1 cm, relative to a nominal orbit that is, itself, also displacedby -1 cm. As a result, the transverse separation of extracted bunch relative to stored bunch is about 2 cm.The extracted beam particles, though all starting from the same point source, also “remember”their initial betatron slope amplitudes. Downstream, the extracted sub-bunch transverse particle dis-placements (from their appropriately-reduced off-momentum closed orbit) will be approximately the221ame as those of the co-travelling bunch from which they were extracted. The separation of stored beamand extracted beam bunches may be about 4 times the nominal bunch radius. This is what can pass as“clean” slow beam sample extraction. (It is not-unlike ion-stripping injection in which Liouville’s law isfoiled by a sudden change of particle rigidity.)There will also be multiple scattering suffered by each extracted proton in its passage through thepellet—for example θ rms = ± mr for this angular deflection. Though not a small angle, at least the coreof the extracted bunch remains within the radial acceptance of the ring, both horizontally and vertically.The extracted beam will be broadened somewhat, and acquire transverse tails from this source. As ithappens, though, the same horizontal phase advance that doubles the extracted beam separation alsorefocuses multiple-scattered protons back to a point focus at the polarimeter scattering target.In short, when observed at the polarimeter in the next straight section, most of the protons thathave touched the carbon pellet will have been slow-extracted into a bunch of much the same dimensionsas the original bunch, somewhat broadened, but mainly displaced by 2 cm from the circulating beam. Anoticeable exception to this analysis concerns protons that have barely grazed the pellet. Though almostcertainly extracted from their stable buckets, these protons can decohere and form a more-or-less stablecoasting beam of reduced radius, but surely at the percent level, at most. Though not welcome, suchprotons should have acceptably small effect on the EDM measurement—to be worried about later.Suggested starting parameters for an EDM experiment are then: that the pellet material shouldhave the highest atomic number Z available, with radius µ m; the number of stored protons, ;the number of protons extracted by the first pellet, 25 million; and the total number of pellets, 400(irrespective of the run length). However all parameters mentioned so far apply only to the starting beamconditions. As the beam intensity falls, say by a factor of two, to maintain the extracted beam flux willrequire the pellet rate to double. So the total number of pellets will be larger than has been stated so far.By controlling the rate at which pellets are launched the beam attenuation pattern can be made linear, orwhatever is most favourable.Making, for example, the assumption that the very first pellet is launched into a beam of protons, and the (unduly optimistic assumption of 100 percent extraction efficiency) the number of ex-tracted beam protons through the polarimeter from just one pellet will be 25 million. Using detailed crosssection values copied unchanged from the (invaluable) paper of M. Ieiri [2], the polarimeter efficiency iscalculated to be . , with analysing power A pol . = 0 . . From the first pellet we therefore antici-pate 5500 total polarimeter counts, with ± scattering to the right (predictably, since we assumethe proton beam is 100 percent vertical polarised) and ± scattering to the left. This would produce(statistically) a better than 2 percent r.m.s. beam polarisation measurement. K.1.2 Experimental confirmation of wire and pellet beam extraction at COSY
K.1.2.1 Previous moving wire investigations
Though the pellet extraction of small beam samples from the centre of a beam bunch has not yet beendemonstrated, nor the high quality of the extracted beam quantitatively confirmed, the concept has,itself, been confirmed experimentally, as show in Figure K.3. In this test by Keshelashvili and others,a stretched µ m carbon fibre was passed suddenly and repeatedly, 10 times through a stored COSYbeam in order to show that the basic considerations given here are correct. The upper oscilloscopepicture indicates the resulting synchronous counting rate bursts in counters of the EDDA polarimeter.The bottom figure shows the beam intensity being reduced in a staircase-like fashion.By reducing the target dimensionality from 2D to 1D, the concept of beam sampling has beenconfirmed. But, while a µ m carbon fibre may seem hardly intrusive, the beam attenuation per wiretransit is still three orders of magnitude too great for the intended application. The need for furtherdimensionality reduction from 1D to 0D—wire to point—seems inescapable. The proposed pellets, withradii three orders of magnitude less than the circulating beam transverse area will provide this needed222actor. Furthermore, the possible performance degradation by electrostatic charging of an insulator in abeam has been shown to be unimportant, at least for a wire. Fig. K.3:
Results of an experimental investigation, by I. Keshelashvili et al. [3] of the interaction of a µ m radiuscarbon wire with the COSY beam for two consecutive cycles. The top graph shows the rate in a detector; thebottom part shows the stepwise reduction of beam intensity for each beam crossing of the wire. K.1.2.2 Pellet formulation applied to moving wire investigation
Later in this section, Eq. (K.4) is derived, giving the opacity O Bp of a moving pellet. Here “opacity” isthe fraction of the circulating beam particles that touch a single pellet (typically over many beam turns)during a single pellet transit. Here we simply copy this formula, with minor modification, to give O WBp ,which is a crude approximation to the opacity of a single transit of a moving wire. The result is O WBp ≈ (cid:18) rWr ⊥ B (cid:19) r ⊥ B /vW C /v p = r ⊥ B rW O Bp (K.1)By the replacement P → W , pellet radius rP becomes wire radius rW , pellet transit time tP becomeswire transit time tW , pellet material density ρP becomes wire material density ρW , and pellet velocity vP becomes wire velocity vW ; (of these, only rW and vW appear in Eq. (K.1)). Apart from these,purely symbolic, changes, the only change has been to multiply the pellet opacity by a (large) multi-plicative factor r ⊥ B /rW . Inclusion of this factor amounts to visualising the moving wire as being madeup of a (large) number r ⊥ B /rW , of length rW , radius rW cylindrical pellets stacked end to end. For rW = rP = 10 µ m and r ⊥ B = 1 cm, the wire opacity is one thousand times greater than the pelletopacity.As an aside, it can be commented that it is the large factor r ⊥ B /rW that makes pellets so muchmore satisfactory than wires for bunch sample extraction. But this factor does not impede our purposehere, of experimentally confirming the moving pellet formalism using moving wire experimentation. K.1.3 Re-interpretation and revision of COSY moving wire beam experiments
The COSY experience with beam sampling by moving an obstacle rapidly through a circulating beamis summarised in Figure K.3, and can be characterised by two qualitative features: the staircase-likereduction of beam current in equal steps, synchronous with transits of a moving wire, and the furtherdetection of similarly synchronous bursts of radiation in nearby counters of the EDDA polarimeter. The223onstant downward beam current steps prove that beam particles are hitting the moving carbon wire; thelocal EDDA counter radiation bursts suggest that the extracted beam energy is dissipated locally.The former conclusion is incontrovertible, but the latter is not. It is our understanding that theEDDA counters are not sensitive to small angle particles less than ten degrees or so. Yet the dominantcontribution to the total cross section for high energy charged particles incident on very thin targets ismultiple scattering at angles much less than ten degrees. The present note therefore assumes that scatteredbeam particles are not contributing significantly to the EDDA signals. This and other contentions of thepresent note, can be tested experimentally using existing COSY moving wire apparatus, either with orwithout new instrumentation.
K.1.3.0.1 Moving wire investigation without new instrumentation—ready immediately.
The simplest suggested experiment is to replace the 10 µ m carbon wire by a 10 µ m tungsten wire.According to Eq. (K.1), the moving wire opacity O WBp is independent of the wire medium density ρW .In our model, every particle that touches a pellet is extracted, irrespective of the wire medium. Theswitch from carbon to tungsten wires should therefore not significantly affect the step-wise reduction ofbeam current shown in the bottom oscilloscope trace in Figure K.3. On the other hand the local, largeangle radiation should be roughly proportional to the wire medium density. The effect of switching fromcarbon to tungsten should therefore increase the ratio of EDDA counts/pellet to beam current loss/pelletby an order of magnitude. K.1.3.0.2 Moving wire investigation with new instrumentation—ready in a few months.
The proposed test without new instrumentation is a significant consistency test, but it does not confirmour contention that the extracted beam particles can be conveyed with significantly large efficiency ontoa carbon polarimeter scattering target. What is needed, for example, is a downstream phosphor screen,or other radiation sensitive imaging device. Judicially-placed in the lattice, such an imaging device candetermine, at least roughly, the angular distribution of beam particles scattered (at small angles) from themoving wire.The choice of a high Z such as tungsten for moving wire medium helps any such investigationsignificantly. The sudden betatron amplitude discontinuity, ∆ x β , derived later in these notes, is given byEq. (K.16), which needs only the symbol conversion ρP → ρW .The switch from carbon to tungsten increases ∆ x β by an order of magnitude. Though the disper-sion function at the moving wire is, presumably, more or less fixed, the displacement of the extractedbeam is also proportional to D p at the screen location. In the COSY lattice there are natural high disper-sion points (of order m in the straight sections at arc centres). It seems natural to consider putting theextracted beam screen at one or the other of these points. This is still not enough though. It is also mostfavourable for the horizontal phase advance to be an odd multiple of π . To complete even a preliminarydesign the true COSY lattice functions have to be known, and preferably be tuneable, to optimise theextracted beam separation. K.1.3.0.3 Full demonstration and calibration of pellet extraction—2+1years.
A cartoon for a pellet extraction test set-up is shown in Figure K.4. Using a COSY lab cyclotron (orequivalent spectrometer at any lab) a 45 MeV proton beam can be used to confirm, optimise, and cali-brate pellet beam sample extraction. The magnetic spectrometer mimics one quarter of the EDM proto-type ring. Inset phosphor screen images show anticipated charge distributions, with and without pelletcontribution. Charge densities are crudely represented by gray-scale shading. The dark elliptical regionis the image of the main beam. The broken-line rectangle indicates a satisfactory placement region foran external polarimeter target. Because the less-strongly-deflected intensity overlaps the main beam, theon-target extraction efficiency has to be at least somewhat less than 100 percent.224
Cyclotron45 MeVPol. Proton
Big KarlExperimental area J U L I C
2D imaging Pelletsource
Main beam
I II ~ΔE
Fig. K.4:
Cartoon of a practical pellet beam sample extraction test set-up.
K.1.4 Quantitative formulation of pellet beam sampling
It is necessary to establish many parameters for pellet beam sample extraction. Symbol definitions for thevarious parameters and kinematic quantities are given in Table K.1. Fortunately pellets are “everywhere”these days, and there is a large choice of materials from which accurate microspheres can be acquired.Parameters for materials that seem to be especially promising are given in Table K.2. It is not our purposeto determine the parameters with high accuracy. Rather, the initial purpose is to acquire a sufficientlyquantitative understanding of the relative advantages of low-Z versus high-Z materials. (Surprisingly, itseems that high-Z pellets are more favourable for our application.)In spite of the ubiquitous availability of high quality plastic pellets we have ruled out all organicmaterials, because their hydrogen content has the potential to harm the vacuum. This mainly leaves pureelements, metals and ceramics. To simplify the analysis we pretend that ceramics can be approximatedas pure single-element metals, quartz as silicon, sapphire as aluminium, etc. Table K.2 contains physicalproperties of an incomplete list of satisfactory and available pellet materials limited in this way. Thereare many possibilities. The main deficiency in the list is the absence of a really high-Z pellet material, asindicated by question marks in the table. If no such pellets exist it can only be that there has, as yet, beenno commercial application requiring high-Z pellets.As well as being needed for analysing the kinematics of pellet acceleration, which is entirelydescribable by classical and statistical mechanics, physical properties are also needed in the table tocalculate the slowing down by ionisation loss as well as the multiple Coulomb scattering of any beamparticle that happens to find itself within the material of a pellet.We picture our pellet bulk material as being in the condensed liquid state of particles that would“evapourate” to form an ideal gas if only they could be heated to a sufficiently high temperature withoutburning or melting— which is not even close to possible.
The requirement to extract one pellet at atime from a fluid of pellets is the main technical challenge in shooting pellets, one-by-one, through ourparticle beam. Fortunately the Su et al apparatus shown in Figure K.1 shows that it is possible to producea reasonably well controllable pellet gun source with the parameters we need.Ideally we could dial up our pellet gun, on demand, to deliver exactly one pellet with an exactspeed and direction. In practice this is unrealistic since, once the pellet fluid medium has been shaken225 ymbol definition (MKS units in formulas, but MeV energies) units in tablesP pelletsubscript p beam particle (proton, deuteron or helion, not electron)subscript B beam (of particles)
ZP/AP pellet material charge/mass number ρP mass density of pellet material gm/cm nP number density of atoms in pellet material 1/ cm n e electron number density in pellet material 1/ cm XP pellet material radiation “length” (i.e. times density) gm/cm NP number of atoms in a pellet MP ≈ NP AP m p pellet mass gm rP radius of pellet microsphere µ m vP speed of pellet < ∼ m/s tP = 2 rP ρP target “thickness” of pellet microsphere gm/cm N p total number of stored beam particles < ∼ N extr . number of beam particles extracted by a single pellet ∼ r ⊥ B transverse radius of (circular) particle beam cm C circumference of storage ring m v p velocity of beam particle m/s η p ( v p ) slowing-down enhancement factor (relative to minimum ionising) ∼ K p kinetic energy of beam particle m/s cp p beam particle momentum (expressed in energy units) MeV T = C /v p beam revolution period s T P = 2 r ⊥ B /vP pellet transit time through beam s O Bp “Opacity” of one pellet transit to beam particles O WBp “Opacity” of one wire transit to beam particles ∆ K p ionisation energy loss, particle through pellet centre MeV δ p = ∆ p p /p p corresponding fractional momentum loss of particle Table K.1:
Definition of symbols for the various parameters and kinematic quantities. m p is the proton mass(which is approximately equal to the a.m.u.). enough to make ejecting pellets one at a time possible, their momentum vectors will have much the samedistributions and uncertainties as given by the Rayleigh-Maxwell distribution of ideal gas molecules.Fortunately, for our application, the pellet beam requirements are not strict. The required averagepellet rate will be of order 1 Hz, but the arrival times can be Poisson distributed in time. Also the pelletbeam width need only be comparable with particle beam transverse dimensions of order one centimetre. K.1.5 Derivations of required formulas
K.1.5.1 Popcorn analogy
When cooking popcorn on a stove top, the kernels, when they pop, supply enough energy to stir thingsup enough to require the sauce pan lid to be kept on. But this also prevents steam from escaping, whichcan make the popcorn soggy. As a compromise one can leave the top slightly ajar. As a result, everyonce in a while, an unpopped kernel comes flying out through the opening between pan and lid. Voila! asource of fast corn pellets.In our application we do not have popping kernels, and it is not even thinkable to supply enoughheat to stir up the pellets thermally—they are far too heavy. We need a moving “impeller” to “evaporate”the pellets into a “vapor”. This necessarily makes the momentum of each particle uncertain, with aMaxwell-like distribution of velocities. In a laboratory scale enclosed vessel with transverse dimensionsof order (cid:96) , there are enough micropellets to run all EDM experiments for centuries, (if none are wasted).Say, therefore, that the volume of pellet material is less than the vessel volume by a factor ofone thousand, with pellets all sitting, condensed, at the bottom of the vessel. Some sort of agitator can,however, stir up the pellets enough that any individual pellet of mass m , with gravitational acceleration mg , can have acquired a kinetic energy mv / of order mg(cid:96) , enough to have a significant probability of226 lement ZP AP Z/A ln (287/Z) Z(Z+1) ρP XP dE/dx | min C r gm/cm gm/cm MeV/(gm/cm) lithium 3 7 0.428 4.56 12 0.534 82.8 1.639 ∼ ∼ Table K.2:
Material properties of high-quality, available microsphere pellets. They should be hydrogen-free,which rules out plastic. All materials are crudely treated as single element metals; quartz treated as silicon, sapphire(aluminium oxide) as aluminium, stainless steel as iron. Plausible coefficient of restitution ( C r ) values are givenin the final column. It has to be realised, though, that even if treating, for example, sapphire as aluminium may becrudely valid for calculating slowing down and multiple scattering of relativistic particles, it is not at all a sensibleapproximation for determining coefficients of restitution [6]. The value given for tungsten, though the result of anactual experiment [7], applies to bouncing for which the pellet velocity is much less than we require. being, for example, in the top half of the vessel. This establishes, a velocity v ∼ √ g(cid:96) , independent of m ,which the agitator has to apply randomly to the pellets, in order for at least some of the pellets to behavelike a gas.This has set a lower limit requirement for the impeller velocity. But this limit is far lower thanthe pellet velocity we require. We could, as a response, use a much faster impeller. But this would be amistake, since this would introduce large and unmanageable transverse velocities. (As always in acceler-ators) we should start with a low energy injector, before applying exclusively longitudinal acceleration.We therefore need two impellers, one to jiggle pellets free, and another to accelerate individual pellets to“high” speed v P ∼ m/s.In the apparatus of Sun et al. shown in Figure K.1, the initial agitation iS supplied by the oscillatingPZO piezo-electric element, coloured purple in the figure, and the secondary acceleration is provided bythe rotating “paddle impeller”, coloured orange in the figure. (As commented earlier, the capability toswitch pellet sizes—indicated by large red open arrow—is superfluous for our application.) K.1.5.2 Pellet acceleration by rotating paddle impeller
When a micropellet approaches at right angles a (not necessarily made of the pellet material) flat surfaceat rest, with momentum p inc . , the pellet bounces with momentum p refl . . The coefficient of restitution [4]is defined as the ratio of these momenta; C r ( v paddle ) = p refl . p inc . , (K.2)which is a number in the range from 0 to 1, that depends on the pellet velocity, and on the pellet and sur-face materials. (The notation here is a bit garbled; C r ( v paddle ) depends on the paddle speed from which p inc . acquires its value in the paddle rest frame, and p refl . inherits the same velocity in its transformationto the laboratory.) In our case the flat surface is a paddle, far more massive than the pellet, and movingin the laboratory with velocity v paddle . In this case the pellet recoils with velocity vP = v paddle (1 + C r ( v paddle )) , (K.3)which can be as large as v paddle . The pellet will lose some of its speed in the reflection from thespherical focusing “mirror” It will also acquire angular velocity (that will have no significant effect onthe subsequent circulating beam sampling).Figure K.5 shows the velocity dependence of sapphire pellets incident on aluminium. Coefficientof restitution values for a few possible pellet materials are also given in Table K.2.227 ig. K.5: Dependence of coefficient of restitution for aluminium oxide (sapphire) normally incident on aluminium[5]. In these notes paddle and pellet media are taken to be identical. (This is not really legitimate for sapphire onaluminium, since aluminium is less rigid than sapphire.)
K.1.5.3 “Opacity” O Bp of a pellet to beam particles Our storage ring of circumference C has some N p ≈ particles circulating with period T at speed v p , with very small fractional momentum spread δ B ∼ − , in a beam with circular cross section ofradius r ⊥ B . Because a pellet is quite small, and is moving quickly, it is unlikely for any particular beamparticle to come close enough to a pellet to be affected. In fact, this is a very sharp distinction, a particleeither hits a pellet or it does not. Assuming the beam is distributed uniformly in a circle of radius r ⊥ B ,in a single passage the probability is ( rP/r ⊥ B ) . However, because the beam particles are relativisticand a pellet speed is much less, each beam particle has multiple opportunities, given by the pellet transittime multiplied by the beam circulation frequency, each time with the same probability. As a result theopacity, which is the probability that a proton will encounter a single pellet, is given by O Bp ≈ (cid:18) rPr ⊥ B (cid:19) r ⊥ B /vP C /v p = 2 rPr ⊥ B rPvP T , (K.4)where T is the beam revolution period. Later we will introduce ρP as the pellet material density, and tP = 2 rP ρP as the “target thickness” of the pellet, expressed in gm/cm . Here we are anticipatingthe approximation that the particle path lengths though the pellet of a substantial fraction of the pelletsdiffer little from the pellet diameter. In practice a pellet will be struck by many beam particles, but onlya very small number of beam particles will be aware of the passage of the pellet. On the other hand,because the pellet is so massive, its passage will be unaffected, even though it is hit by many beamparticles. Furthermore, a single beam particle passing through the pellet will, at first, scarcely notice theinteraction. But, because the binding of a particle in a stable RF bucket is so weak, such a particle isalmost certainly doomed or, in less gloomy terms, “extracted” from its RF bucket. The number of beamparticles extracted by a single pellet is then given by N extr . = O Bp N p . (K.5)228f course the circulating beam particles will be be reduced by exactly this number, but the circulatingbeam will be otherwise unaffected. This has reduced our task to finding the fate of of the N extr . “ex-tracted” particles. (The quotation marks on “extracted” serve as a reminder that, though the particles areno longer captured in stable buckets, they have not necessarily been extracted from the storage ring anddelivered to a polarimeter. K.1.5.4 Expressing pellet mass in terms of pellet “target thickness”
The role of a pellet is to slow down the beam particles that happen to pass through it. This slowingdown is caused almost entirely by collisions of the beam particle with electrons in the pellet. And yet theelectrons make only a negligible contribution to the pellet mass
M P .The pellet dynamics depends on pellet mass
M P and the beam particle slowing down depends onthe pellet “target thickness” tP = 2 rP ρP . The number of free parameters can be reduced by relatingthese two quantities. M P = ρP π rP = 2 π rP tP. (K.6) K.1.5.5 Slowing down of beam particle passing through pellet
The slowing down of a weakly relativistic elementary particle passing through a medium falls inverselywith its squared-velocity v p , “bottoming out” at a “minimum ionization” value dE/dx as the speed ap-proaches c . This is illustrated graphically in Figure K.6. Minimum ionising values for our promisingpellet media are given in in Table K.2. One sees that these minimum ionisation values are approximatelyindependent of the medium, with approximate value 1.6 MeV/(gm/cm ). It was commented earlier that,since our beam particle velocities are significantly less than the speed of light, our slowing down is en-hanced by some voltage-dependent factor η p ( v p ) ≈ , where the value “7” is specific to our 45 MeV,proton beam energy. This value can be regarded as constant for present purposes, since we are concen-trating only on the determination of pellet parameters. With longitudinal position variable z , we cantherefore use dE p d ( zρ ) = − η p ( v p ) dE p dx (cid:12)(cid:12)(cid:12)(cid:12) min ≈ − × . [MeV/(gm/cm-sq)] . (K.7) K.1.5.6 Longitudinal momentum reduction of “extracted” particles
To track extracted beam particles out of the ring it is particle momentum (in the form δ = δ p /p , ratherthan particle energy, that is needed. It would not be flagrantly wrong, and consistent with other relationsused in these notes, to simply use the non-relativistic relation K = (1 / p /m for this purpose. But,for greater generality, let us use a formula that is more nearly correct relativistically, starting with themass-energy-momentum relationship; γ p m p c = E p = ( m p c + K p ) = m p c + p p c . (K.8)Solving for p p p p c = 2 m p c K p (cid:18) K p m p c (cid:19) . (K.9)Differentiating this equation, and keeping only the leading term in K p / ( m p c ) , yields δ ≡ ∆ p p p p ≈ (cid:18) K p m p c (cid:19) dK p K p . (K.10)Substituting from Eq. (K.7) produces δ ≈ (cid:18) K p m p c (cid:19) η p ( v p ) dE p dx (cid:12)(cid:12)(cid:12)(cid:12) min tP, (K.11)229 ig. K.6: NIST Standard Reference Database 124, dE/dx for protons incident on carbon, with cross adjustedfor 45 MeV. In choosing among different pellet material, the ratio of dE/dx at 45 MeV to the ionisation minimumwill be more or less independent of pellet medium, because the beam particle velocity is being held constant. (which is negative).
K.1.5.7 Transverse displacement of “extracted” beam particles (Neglecting any pre-existing betatron or synchrotron amplitude of a beam particle passing through apellet) let us assume the beam particle is on the design orbit as it enters the pellet. At the location in thering of the pellet injector let the particle, horizontal dispersion function value be D p , and the dispersionfunction slope be zero, meaning that the transverse position of a particle with fractional momentum offset δ is given by x p = D p δ. (K.12)On entry we have assumed x p = x (cid:48) p = 0 . Because the pellet is so “short”, the particle will still beon the design orbit (with any non-zero slope having been caused by multiple scattering which we aretemporarily neglecting) as it exits the pellet. But, on exit, the pellets fractional momentum offset is givenby Eq. (K.11); this means the pellet is not on its off-momentum closed orbit—the particle has acquired a(positive) horizontal betatron displacement given by x out β = − D p δ out , (K.13)just right to cancel its sudden, newly-established (negative) off-momentum closed orbit displacement D p δ out . In the absence of any further disturbance, the particle will continue to oscillate with betatronamplitude given by Eq. (K.12) about this newly-displaced closed orbit. For example, when the betatronphase has increased by π , with D p assumed constant, the particle will be displaced from the true, on-momentum design orbit D p δ , with δ given by Eq. (K.11).In general, the betatron perturbation just calculated will simply be superimposed on any previously-neglected betatron and synchrotron amplitudes. K.1.5.8 Sudden particle translation expressed in terms of pellet opacity
The sudden transverse displacement D p , δ (relative to its off-momentum closed orbit) of a beam particlethat has passed through a pellet causes the particle to be extracted. For clean sample extraction we want230o maximise this displacement (by increasing pellet size or atomic number). But, at the same time, wewant to minimise the pellet opacity O Bp , in order to minimise beam particle consumption per pellet—fora bigger sample one need only send more pellets.To analyse this compromise it is useful to express the sudden displacement in terms of the opacity—that is, to express D p δ in terms of O Bp . Toward this end, we re-arrange Eq. (K.4) into tPρP = (cid:113) O Bp r ⊥ B v P T , (K.14)which, conveniently, depends on pellet parameters only through the opacity. We also combine Eq. (K.11)and Eq. (K.13); ∆ x β = − D p (cid:18) K p m p c (cid:19) η p ( v p ) dE p dx (cid:12)(cid:12)(cid:12)(cid:12) min (cid:18) tPρP (cid:19) ρP, (K.15)and substitute from Eq. (K.14), ∆ x β = − D p (cid:18) K p m p c (cid:19) η p ( v p ) dE p dx (cid:12)(cid:12)(cid:12)(cid:12) min (cid:113) O Bp r ⊥ B v P T ρP, (K.16)which is boxed to emphasise its importance. A striking implication of this equation is that, at fixedopacity, ∆ x β is proportional to the density ρP of the pellet material. The importance of this dependencecan be assessed from the density column of Table K.2. K.1.5.9 Angular spread caused by multiple scattering
As well as the loss of momentum just calculated, each extracted beam particle acquires a multiple scatter-ing angular distribution. The r.m.s. angular spread can be expressed in terms of the particle momentum p p , in conjunction with the radiation length XP and target thickness tP of the pellet material The radia-tion length, expressed in units of gm/cm-sq, is defined [8] by XP = 716 . ZP ( ZP + 1) ln √ ZP . (K.17)Values of XP for promising pellet media are given in Table K.2 The r.m.s. angular spread caused bypassage through the pellet with target thickness tP and momentum p p is given by θ r . m . s . = 21 MeV p p cβ p (cid:114) tPXP = 21 MeV (cid:112) m p c K p (cid:0) K p / (4 m p c ) (cid:1) β p (cid:114) tPXP , (K.18)where p p c has been substituted from Eq. (K.9). K.1.5.10 Pellet radius required for efficient bunch sampling extraction
As explained earlier, with dispersion function D p assumed constant, when the betatron phase has in-creased by π (or any odd multiple of π ), a particle passing through a pellet centre will be displacedfrom its previous off-momentum closed orbit by an amount D p δ , with δ given by Eq. (K.11). Anypolarimeter in the ring is assumed to be located at such a position.Even particles touching a pellet will not, in general, pass through the pellet centre. About 1/2 ofthe pellets will be sufficiently off-centre for their path length through the pellet to be at least 30 percentless than the pellet diameter. These pellets we ignore, under the assumption that their energy loss hasbeen insufficient for them to be differentiated from the surviving main beam, and therefore unlikely toregister in the polarimetry. The path lengths of the remaining particles will all be approximately thesame. They will be treated as if centred on the pellet.231 eferences [1] Z. Sun et al., First results of ELM triggering with a multichamber lithium granule injector intoEAST discharges,
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Juelich ballistic diamond pellet target for storage ring EDM measurement,
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