Search for a muon EDM using the frozen-spin technique
A. Adelmann, M. Backhaus, C. Chavez Barajas, N. Berger, T. Bowcock, C. Calzolaio, G. Cavoto, R. Chislett, A. Crivellin, M. Daum, M. Fertl, M. Giovannozzi, G. Hesketh, M. Hildebrandt, I. Keshelashvili, A. Keshavarzi, K.S. Khaw, K. Kirch, A. Kozlinskiy, A. Knecht, M. Lancaster, B. Märkisch, F. Meier Aeschbacher, F. Méot, A. Nass, A. Papa, J. Pretz, J. Price, F. Rathmann, F. Renga, M. Sakurai, P. Schmidt-Wellenburg, A. Schöning, M. Schott, C. Voena, J. Vossebeld, F. Wauters, P. Winter
SSearch for a muon EDM using the frozen-spin technique
A. Adelmann,
1, 2
M. Backhaus, C. Chavez Barajas, N. Berger, T. Bowcock, C. Calzolaio, G. Cavoto,
5, 6
R. Chislett, A. Crivellin,
2, 8, 9
M. Daum, M. Fertl, M. Giovannozzi, G. Hesketh, M. Hildebrandt, I. Keshelashvili, A. Keshavarzi, K.S. Khaw,
13, 14
K. Kirch,
1, 2
A. Kozlinskiy, A. Knecht, M. Lancaster, B. M¨arkisch, F. Meier Aeschbacher, F. M´eot, A. Nass, A. Papa,
2, 17
J. Pretz,
11, 18
J. Price, F. Rathmann, F. Renga, M. Sakurai, P. Schmidt-Wellenburg, ∗ A. Sch¨oning, M. Schott, C. Voena, J. Vossebeld, F. Wauters, and P. Winter ETH Z¨urich, 8093 Z¨urich, Switzerland Paul Scherrer Institut, 5232 Villigen PSI, Switzerland University of Liverpool, Liverpool, UK PRISMA + Cluster of Excellence and Institute of Nuclear Physics,Johannes Gutenberg University Mainz, Mainz, Germany Sapienza Universit`a di Roma, Dip. di Fisica, P.le A. Moro 2, 00185 Roma, Italy Istituto Nazionale di Fisica Nucleare, Sez. di Roma, P.le A. Moro 2, 00185 Roma, Italy University College London, London, UK CERN, 1211 Geneva, Switzerland University of Z¨urich, Z¨urich, Switzerland PRISMA + Cluster of Excellence and Institute of Physics,Johannes Gutenberg University Mainz, Mainz, Germany Institut f¨ur Kernphysik, Forschungszentrum J¨ulich, J¨ulich, Germany Department of Physics and Astronomy, University of Manchester, Manchester, UK Tsung-Dao Lee Institute, Shanghai Jiao Tong University, Shanghai, China School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai, China Physik-Department, Technische Universit¨at M¨unchen, Garching, Germany Brookhaven National Laboratory, USA University of Pisa and INFN, Pisa, Italy RWTH Aachen, Aachen, Germany Physics Institute, Heidelberg University, Heidelberg, Germany Argonne National Laboratory, Lemont, USA
This letter of intent proposes an experiment to search for an electric dipole moment of the muonbased on the frozen-spin technique. We intend to exploit the high electric field, E = 1 GV / m,experienced in the rest frame of the muon with a momentum of p = 125 MeV /c when passingthrough a large magnetic field of | (cid:126)B | = 3 T. Current muon fluxes at the µ E1 beam line permitan improved search with a sensitivity of σ ( d µ ) ≤ × − e · cm, about three orders of magnitudemore sensitivity than for the current upper limit of | d µ | ≤ . × − e · cm (C.L. 95%). With theadvent of the new high intensity muon beam, HIMB, and the cold muon source, muCool, at PSI thesensitivity of the search could be further improved by tailoring a re-acceleration scheme to match theexperiments injection phase space. While a null result would set a significantly improved upper limiton an otherwise un-constrained Wilson coefficient, the discovery of a muon EDM would corroboratethe existence of physics beyond the Standard Model. Keywords: ∗ Electronic address: [email protected] a r X i v : . [ h e p - e x ] F e b I. SUMMARY
This letter of intent proposes a search for the electric dipole moment (EDM) of the muon. We plan todesign, mount, and operate an experiment to search for an EDM of the muon (muEDM), using the frozen-spintechnique [1] at the High Intensity Proton Accelerator (HIPA) of the Paul Scherrer Institute (PSI).Particle EDMs are generally considered as excellent probes of physics Beyond the StandardModel (BSM) [2], indicating the violation of the combined symmetry of charge and parity (CPV). IndeedCPV is one of three necessary conditions to explain the creation of a matter-dominated Universe from aninitially symmetric condition [3]. The observed Baryon Asymmetry of the Universe (BAU) [4] cannot beexplained by the otherwise successful Standard Model (SM) of particle physics [5] as, among others, theexisting CPV in the weak sector of the SM is insufficient.The EDM limit of the muon d µ ≤ . × − e · cm (95% C.L.) [6] is the only EDM of a fundamental particleprobed directly on the bare particle. Assuming simple scaling in mass, by the ratio ( m µ /m e and lepton uni-versality the electron EDM, d e ≤ . × − e · cm (95% C.L.) [7], measured using thorium monoxide (ThO)molecules provides a much tighter indirect limit, assuming the electron is the only source of CPV. As wewill argue in section II, current B -meson decay anomalies at LHC [8] and the persistent 3 . σ discrepancyof the muon g − × − e · cm per year of data-taking. The baseline concept plans to use muons with a momentumof p = 125 MeV /c ( β = 0 .
77) and an average polarization of 90% from the µ E1 beam line at PSI with aparticle flux of up to 2 × µ + / s. Two concepts are currently under evaluation and discussed in thisletter of intent. In section III C we discuss a storage ring sketched in Figure 1a, with a magnetic field of | (cid:126)B | =1 . g −
2) group [10] and is discussed in section III D. (a) (b)
FIG. 1: Sketch of storage ring with lateral injection (a), and the helix muEDM search (b) using a vertical injectioninto a uniform solenoid field.
In both cases a nested electrode system provides a radial electric field E f for the frozen-spin technique,discussed in section III B. Positive muons will be stored one at a time on a stable orbit inside the frozen-spinregion. The muon sees a large electric field (cid:126)E ≈ γ (cid:126)β × (cid:126)B , about 1 GV / m for | (cid:126)B | = 3 T, which leads toa precession of the spin in the presence of a muEDM, while the oscillation from the anomalous magneticmoment is suppressed. The muons will decay after an average lifetime of γτ µ = 3 . µ s in the lab systeminto a positron and two neutrinos. Due to the parity violating decay, the positron is preferentially emittedalong the spin of the muon, with an average asymmetry of A = 0 .
3. By detecting the vertical asymmetry ofpositrons ejected upwards or downwards with a tracker placed inside the helix/orbit the build-up in time ofan asymmetry due to an EDM will be measured.
FIG. 2: Historical overview of EDM limits (90% C.L.). The labels in the plot next to the date (Cs, Tl, TlF, ThO,Xe, and YbF) refer to the measured system from which the limit was derived. So far, all EDM measurements werein agreement with a null result and were therefore interpreted as upper limits.
II. MOTIVATION
A non-zero EDM of a fundamental particle violates time-reversal symmetry, and by invoking the CPT-theorem of quantum field theories [11], also the combined symmetry of charge conjugation and parity in-version (CP). Many BSM theories have new complex parameters which are sources of CP violation as theseparameters are naturally expected to have a generic phase of order one. In fact, the only complex parameterwithin the SM (disregarding the vanishingly small QCD theta term), the phase of the Cabibbo KobayashiMaskawa (CKM) matrix [12], is close to maximal [13, 14]. Furthermore, CP violation is also one of threenecessary conditions to explain the observed BAU [3]. However, even though the CKM phase is close to max-imal, CP violation within the SM is by far not sufficient to explain the observed BAU [15–20]. This stronglymotivates theories with additional complex parameters as extensions of the SM, providing additional sourcesof CP violation. Clearly, such sources of CP violation are expected to generate at some level non-vanishingelectric dipole moments of fundamental particles, which can significantly exceed the tiny values within theSM [21].Therefore, many experiments searching for non-vanishing electric dipole moments have been performedover the last decades, as summarized in Figure 2, and the current status can be found in [22]. As we can see,the limits on the muon EDM are particularly weak compared to the other constraints. Therefore, a searchfor a permanent EDM of the muon gives access to one of the least tested areas of the SM of particle physicsand is hence an important piece of this comprehensive and complementary experimental strategy to unveilBSM physics [23].One reason why in the past the focus of EDM searches was obviously not on the muon EDM is that theimpressive limits on the electron EDM from measurements using atoms or molecules, e.g. thorium oxidemolecules d e < . × − e · cm [7], were commonly rescaled, assuming MFV [24–27] (by the ratio m µ /m e )resulting in d µ < . × − e · cm. However, MFV is, to some extent, an ad hoc symmetry invented toallow light particle spectra, in particular within the MSSM where this reduces the degree of fine-tuning inthe Higgs sector while respecting at the same time flavor constraints. Since the LHC did not discover anynew particles directly [28, 29] the whole concept of naturalness is challenged. Furthermore, LHCb, Belle andBaBar discovered significant tensions in semi-leptonic B decays [30–37] implying a 5 σ level discrepancy whenanalyzed together [38–40]. These remarkable hints for new physics point towards the violation of LeptonFlavor Universality (LFU) and are therefore not compatible with MFV in the lepton sector [41].Furthermore, there is the longstanding 3.7 σ tension between the measured value of the anomalous magneticmoment (AMM) of the muon [42] and its SM prediction [43]. The AMM is directly related to the EDMsince the former measures the real part of the same Wilson coefficient whose imaginary part gives rise tothe non-vanishing EDM. While the measurement of the AMM of the muon is by itself consistent with theassumption of MFV, in general any TeV scale explanation of the AMM of the muon requires a chirally FIG. 3: Contours of d µ as a function of the anomalous momentum ∆ a µ and the phase of the associated Wilsoncoefficient [55]. enhanced effect that automatically provides an a priori free phase. For example, the B anomalies motivatethe introduction of leptoquarks, which can account not only for them, but at the same time for the AMMof the muon [44] via a m t /m µ enhanced effect [45, 46] whose phase is completely unconstrained.Therefore, it is well-motivated that New Physics (NP) has a flavor structure beyond MFV. A notionoften contested on grounds of naturalness arguments. However, note that in the limit of vanishing neutrinomasses, which is an excellent approximation taking into account their smallness, lepton flavor is conserved.Thus it possible to completely disentangle the muon from the electron EDM via a symmetry, meaning thatno fine-tuning is necessary. This could for example be achieved via a L µ − L τ symmetry [47–49] which cannaturally give rise to the observed PMNS matrix [50–52], and, even after its breaking, protects the electronEDM and AMM from NP effects [53]. Also from an EFT point of view [54], it is clear, that the muon EDMcan be large and that a measurement of it is in practice the only way of determining the imaginary part ofthe associated Wilson coefficient. In summary, this clearly demonstrates that a more sensitive measurementof the muon EDM has the potential to discover CP violation and further corroborates the existing hints forthe violation of LFU [55]. This can be clearly seen from Figure 3 which shows the potential reach for thecomplex phase of the Wilson coefficients of a future muon EDM search at PSI. In fact, a discovery of anon-vanishing muon EDM would consolidate the existence of physics beyond the SM and lead to a paradigmshift in our understanding of nature. III. EXPERIMENTAL SEARCH FOR A MUON EDMA. Spin motion of muons in electric and magnetic field in the presence of an EDM
The spin dynamics of a muon at rest in a magnetic field (cid:126)B is described by d (cid:126)s/ d t = (cid:126)µ × (cid:126)B = (cid:126)ω L × (cid:126)s where (cid:126)µ = ge/ (2 m ) (cid:126)s is the magnetic dipole moment with | (cid:126)s | = (cid:126) / (cid:126)ω L = − µ (cid:126)B/ (cid:126) the Lamor precessionfrequency. Similarly, a hypothetical electric dipole moment (cid:126)d = ηe/ (2 mc ) (cid:126)s results in a spin precession ofthe muon (cid:126)ω d = − d (cid:126)E/ (cid:126) in an electric field (cid:126)E .The first search for a muEDM resulted in an upper limit of 2 . × − e · cm (95% C.L.) [56, 57] and waspublished in 1958. Half a century later, the current best upper limit of d µ < . × − e · cm (95% C.L.) [6]was deduced using the spin precession data from the ( g −
2) storage ring experiment E821 at BNL [42].For the further discussion of the spin dynamics of a moving muon with momentum (cid:126)p , (cid:126)β = (cid:126)v/c and γ = (1 − β ) − / in magnetic, (cid:126)B , and electric, (cid:126)E , fields it is useful to change to the unit polarization threevector (cid:126) Π = (cid:126)s/ | (cid:126)s | . Then the change in polarization with time is given byd (cid:126) Πd t = (cid:126) Ω × (cid:126) Π , (1)where (cid:126) Ω = − emγ (cid:34) (1 + γa ) (cid:126)B − aγ ( γ + 1) (cid:16) (cid:126)β · (cid:126)B (cid:17) (cid:126)β − γ (cid:18) a + 1 γ + 1 (cid:19) (cid:126)β × (cid:126)Ec (cid:35) (2)is the Thomas precession [58], when replacing the anomalous moment of the muon a [42] with ( g − / λ in [58] by ge/ (2 mc ).In the case that no electric field is applied parallel to the momentum, the acceleration of the muon ispurely transverse to its motion d (cid:126)β d t = eγmc (cid:16) (cid:126)E + (cid:126)βc × (cid:126)B (cid:17) , (3)which is equivalent to d (cid:126)β d t = (cid:126) Ω c × (cid:126)β, (4)where (cid:126) Ω c = − emγ (cid:32) (cid:126)B − γ γ − (cid:126)β × (cid:126)Ec (cid:33) (5)is the cyclotron frequency. The relative spin precession (cid:126) Ω of a muon in a storage ring with an electric field (cid:126)E and magnetic field (cid:126)B is then given by: (cid:126)
Ω = (cid:126) Ω − (cid:126) Ω c = qm (cid:34) a (cid:126)B − aγ ( γ + 1) (cid:16) (cid:126)β · (cid:126)B (cid:17) (cid:126)β − (cid:18) a + 11 − γ (cid:19) (cid:126)β × (cid:126)Ec (cid:35) . (6)which is the known T-BMT equation [59] when replacing q = − e . The presence of the EDM adds a secondterm (cid:126) Ω = (cid:126) Ω − (cid:126) Ω c = qm (cid:34) a (cid:126)B − aγ ( γ + 1) (cid:16) (cid:126)β · (cid:126)B (cid:17) (cid:126)β − (cid:18) a + 11 − γ (cid:19) (cid:126)β × (cid:126)Ec (cid:35) + ηq m (cid:34) (cid:126)β × (cid:126)B + (cid:126)Ec − γc ( γ + 1) (cid:16) (cid:126)β · (cid:126)E (cid:17) (cid:126)β (cid:35) . (7)The first line of equation (7), is the anomalous precession frequency ω a , the difference of the Larmorprecession and the cyclotron precession. The second line is the precession ω e due to an EDM coupling to therelativistic electric field of the muon moving in the magnetic field (cid:126)B , oriented perpendicular to (cid:126)B . In the casethat momentum, magnetic field, and electric field form an orthogonal basis, the scalar products of momentumwith fields, (cid:126)β · (cid:126)B = (cid:126)β · (cid:126)E = 0, drop out. A special configuration was chosen for the E821 experiment; muonswith a so-called “magic” momentum of p magic = m/ √ a = 3 .
09 GeV /c were used, simplifying equation (7)on the reference orbit to (cid:126) Ω = qm (cid:34) a (cid:126)B + η (cid:32) (cid:126)β × (cid:126)B + (cid:126)Ec (cid:33)(cid:35) , (8)making the anomalous precession frequency independent of electric fields needed for steering the beam. Inthe presence of a muEDM the precession plane is tilted out of the orbital plane defined by the movement ofthe muon in this “magic” configuration. Hence, a vertical precession ( (cid:126)ω e ⊥ (cid:126)B ) with an amplitude proportionalto the EDM with a frequency (cid:126)ω e phase shifted by π/ (cid:112) ω + ω . (9) B. The frozen-spin technique
The experimental setup proposed for this dedicated search for an EDM of the muon is based on ideas andconcepts discussed in [1, 9].The salient feature of the proposed search for this hypothetical muon EDM is the exploitation of the largeelectric field (cid:126)E ∗ = γc(cid:126)β × (cid:126)B ≈ / m in the rest frame of the muon, while canceling the effect of theanomalous moment by a meticulously-chosen electric field. Here, as in the remainder of the document,fields in the rest frame of the particle will be indicated by an ∗ while all other notation indicate fields in alaboratory frame. The anomalous precession term in equation (7) can be set to zero by applying an electricfield such that a (cid:126)B = (cid:18) a − γ − (cid:19) (cid:126)β × (cid:126)Ec . (10)In the idealized case of (cid:126)β · (cid:126)B = (cid:126)β · (cid:126)E = 0, and (cid:126)B · (cid:126)E = 0 we find that E f ≈ aBcβγ . By selecting the exactfield condition of equation (10), the cyclotron precession frequency is modified such that the relative anglebetween momentum vector and spin remains unchanged if η = 0, hence it is “frozen”. In the presence of anelectric dipole moment the change in polarization is described byd (cid:126) Πd t = (cid:126)ω e × (cid:126) Π , (11)where (cid:126)ω e = ηq m (cid:34) (cid:126)β × (cid:126)B + (cid:126)E f c (cid:35) = 2 d µ (cid:126) (cid:16) (cid:126)βc × (cid:126)B + (cid:126)E f (cid:17) (12)is the precession frequency due to the electric dipole moment of the muon. For the idealized case, see above,this results in a vertical build-up of the polarization | (cid:126) Π( t ) | = P ( t ) = P sin ( ω e t ) (13) ≈ P ω e t ≈ P d µ (cid:126) E f aγ t. (14)From the slope d P d d µ = 2 P E f ta (cid:126) γ (15)multiplied by the mean analysis power of the final polarization, A , we calculate the sensitivity as σ ( d µ ) = a (cid:126) γ P E f √ N τ µ A , (16)for a search of the muon EDM by replacing t with the mean free laboratory lifetime of the muon in thedetector γτ µ and scaling by 1 / √ N for the Poisson statistics of N observed muons. The initial polarization P > .
93 of a beam of muons from backward decaying free pions was measured for a momentum of125 MeV /c muons on µ E1 beam line, see Sec. III F. For the mean decay asymmetry we take A = 0 .
3. Hencethe EDM sensitivity for a single muon is σ ( d µ ) ≈ × − e · cm, assuming a magnetic field of B = 1 . E f = 0 .
96 MV / m. At µ E1 beam linea total muon flux of 2 × µ + / s was reported before [9]. Injection simulations indicate a 0.14% efficiencyfor lateral injection without material. With thin aluminum electrodes this reduces further by a factor 14to 1 × − , c.f. Sec. III C 1. These numbers indicate that one could store one muon at a time at a rate of1 / ( γτ + (cid:104) t d (cid:105) ) = 18 kHz, where (cid:104) t d (cid:105) = 50 µ s is the mean waiting time between two successive measurements.Assuming 200 days per year for data taking, this results in a total of 3 . × detected positrons per yearwhich in turn yields a sensitivity of σ ( d µ ) ≈ × − e · cm.We also investigate the option of a vertical injection as described in [10], see Sec. III D. The clearadvantage is that the muons do not have to pass several times through electrodes, as the lateral injection.Further, the deployment of a magnetic field of up to 3 T seems better feasible, as it results in a largerelectric field of E f ≈ / m, which can be deployed more easily in this scheme as the injection channel ismoved far away from the electric field.To avoid a triggered magnetic field kick, we also investigated a scenario deploying the vertical 3D-injectionsand avoiding the storage of the muons on a stable orbit altogether. Instead we let them drift through thefrozen field configuration on a helix. In this case, all muons which can be injected also contribute to thefinal sensitivity, as we do not have to wait for the decay of each muon before the next is admitted tothe experiment. However, the larger the drift angle, this means the velocity along the solenoid axis, theshorter the mean time required to pass through the frozen-field region. As a consequence, the fraction ofdecays within the frozen field region is reduced, although the rate of injected muons increases to 234 kHz,resulting in an effective mean storage time of about 40 ns. This in turn results in an annual sensitivity of σ ( d µ ) ≈ × − e · cm in the case of a 3 T magnetic field.The most sensitive scenario would be to apply a vertical magnetic kick to the injected muon to store it on astable orbit as in the classical storage-ring concept. On the one hand, losses due to multiple scattering on theelectrodes and injection channel vanish compared to the lateral injection scheme. While on the other hand,the requirement of triggering the injection procedure relaxes to about 50 ns. Combined with a magnetic fieldof 3 T this results in a sensitivity of σ ( d µ ) ≈ × − e · cm , (17)as detailed in Sec. III D. C. Compact storage ring with lateral injection
A straw-man idea for the lateral injection approach of the experiment is shown in Figure 1a. A muonbeam from PSI’s µ E1 (or π E1) beam line is first collimated to limit the vertical divergence of the beam. It isthen guided to the central region of a weak-focusing magnet through a magnetic channel (injection channel).Upon entering the magnet, the muon is displaced from its storage orbit by a few centimeters. Without anymagnetic/electric steering, it will come back to the same place, hit the magnetic channel and scatter out ofthe storage ring.As the period of the muon cyclotron motion is around 10 ns, conventional beam steering techniques, forexample the one-turn-pulsed-magnetic kicker utilized by the Muon g − n = 0 .
72 ( Q x = √ − n = 0 .
5, condition for half-integerresonance). It is then damped over 20 turns ( ≈
200 ns) until the muon is relaxing onto its storage orbit.The effective magnetic field index of the system is reduced from 0.72 to 0.25 over the same time period.Once the muon is stored, the required radial electric field E r ∼ aBcβγ that can be applied by means of twoconcentric cylinder electrodes will “freeze” the muon spin relative to the momentum as the muon circulatesin the storage ring.For the detection of decay positrons, an EDM detector can be installed at the top and bottom of the muonorbital plane. As the muon beam is circulating in the storage ring, the spin will follow the direction of themuon momentum if the muon has no EDM. If an EDM exists for the muon, the spin will slowly precessout of the muon orbital plane. Thus an observable up-down asymmetry that oscillates with time, with afrequency directly proportional to the muon EDM, can be observed. In the case of a muon EDM smallerthan the current limit, a slow increase in the up-down asymmetry in Figure 4b is expected, as the amountof spin precession out of the orbital plane is limited by the muon lifetime. As most of the positrons willcurl into the center of the storage ring as shown in Figure 5, a positron tracker made of scintillating fibersand depleted monolithic active pixel sensors (DMAPS) can be installed in the inner part of the storage ringto track the positrons. It can be used to measure the residual anomalous precession signal and fine-tuningthe radial E-field to reach the “frozen-spin” condition, and to discriminate up and downward tracks for theEDM analysis. (a) μ Time [1 − − ( t )) ↓ ( t ) + N ↑ ( t )) / ( N ↓ ( t )- N ↑ A sy mm e t r y A ( t ) = ( N / ndf χ e A 0.0010 ± e ω ± e φ ± − × = 100%, N = 5.0 μ P cm ⋅ e -17 × = 1.8 μ d (b) FIG. 4: (a) A detection concept for the muon EDM. (b) Simulated ideal up-down asymmetry plot assuming a largemuon EDM of d µ = 1 . × − e · cm. The red line is the fit to simulated data points. (a) (b) FIG. 5: (a) Top view and (b) side view of the ideal simulation with 200 muons at 125 MeV/ c . The stored muonorbits are shown in red and the decay positron trajectories are in blue. The dark rings, changing their width are aperspective view of the electrodes.
1. Simulation of lateral injection into a compact storage ring
The lateral injection of muons into a weakly-focusing magnetic field of 1 . G4beamline [63, 64]. Figure 6a shows the implemented geometry, the weakly-focusing magnetic fieldwas modeled, using the formalism defined in [65], as (cid:126)B ( r, z ) = B z + G − rz z − r / , (18)where B z = − .
509 T and G = − . / T / mm , which results in B z = − . r = 280 mm and z = 0. The radial electric field of E f = 0 .
962 MV / m is applied within acylindrical volume of 243 mm < r <
313 mm and − . < z < . d = 20 µ m thickness. The muons for the injection simulations are created at ϕ = 0 (see Figure 6a), (a) (b) FIG. 6: Geometry of the storage ring (a) used for the simulation study of lateral injection. The origin of the coordinatesystem is the center of the reference trajectory in the z = 0 plane with radius r = 280 mm. In the simulation themuons are created in the injection zone which extends for 20 mm radially and 10 mm vertically just outside of thenegative charged high-voltage electrode. A second injection channel is required for counterclockwise measurements.Along the nominal orbit two perturbation fields (b) are applied during injection and ramped to zero within 150 ns.Note that for counterclockwise injection a second pair of perturbators is required, but not shown here. (a) (b) FIG. 7: Radial phase-space evolution during injection (a) for the case that all materials are set to vacuum in thesimulation. The track color changes with time from dark blue at t = 0 to brown t = 158 ns. Note, the nominal orbit r = 280 mm for muons with p = 125 MeV /c is indicated by the black vertical line. The two blue vertical lines indicatethe position of the ground and high voltage electrode. Red points depict the initial creation of the simulated muon.The injection ratio (b) illustrates the effect once materials are included into the simulation. Without material 93% ofthe muons decay to a positron and two neutrinos within a volume defined by the electrodes and −
55 mm ≤ z ≤
55 mm.The dramatic loss of muons, only 11% decay within the frozen-spin volume, can be traced back to multiple scatteringwithin the thin aluminum electrode (thickness d = 20 µ m) during many passages through matter. A reduction ofthese losses is possible by using even thinner electrodes, however, the addition of a required low-field region from theinjection channel in the injection zone reduces the injection efficiency further. In the illustrated case this result in atotal injection efficiency of just above 7%.
313 mm < r <
333 mm and −
10 mm < z <
10 mm, with a divergence of −
10 mrad < r (cid:48) <
10 mrad and −
10 mrad < z (cid:48) <
10 mrad. The perturbation fields, shown in Figure 6b, expand over ∆ φ = ± ° at φ = 110 ° and φ = 200 ° and are ramped down with a delay of 4 ns after creation within 150 ns. The field shape is a copyof the field published in [61]. In addition to the electrodes generating the frozen-field region, an additionalground electrode will be needed outside of the charged electrode. For the simulation, this was modeled ashalf cylinder made of copper in the range φ =0 ° to 180 ° at r = 348 mm and − . < z < . φ =180 ° to 360 ° two injection channels made of magnet iron are positioned.The choice for the reference injection phase space was driven by a series of simulations varying the lateraland vertical phase-space parameters. In the case when all materials are set to vacuum, losses only occur0due to a too large vertical divergence, which could be counteracted by an even stronger weakly-focusingcomponent G . Figure 7a shows the radial phase space evolution using the injection phase space above-mentioned and a kick field which is linearly ramped down within 150 ns. Without vertical divergence, itwas possible to inject nearly 99% of all muons from a lateral phase space of 28 ×
17 mm · mrad as wasalso demonstrated in [9]. However, as simulations quickly showed, most losses in a realistic configurationoccur due to multiple scattering in the many passages of the muon through the high voltage electrode or byhitting the entrance channel. Hence, a further refinement of the vertical divergence and adaptation of G seemed superfluous. Figure 7b illustrates, nearly 90% of all muons are lost during injection due to multiplescattering once material properties are turned on in the simulation. A change of the electrode design couldmost probably reduce these losses by using electrodes made of low- Z material, e.g., a thin Kapton foil coatedwith an even thinner layer of aluminum. However, losses also occur due to a return of the muons into theinjection region. In the case of the simulation presented here, this leads to another 30% loss due to passagesthrough the low field area from the injection channel at every turn. In total, we observed a loss of 93% ofall created muons in the injection zone, which reduces the injection efficiency by a factor 13.6. D. Stored or continuous measurement using a vertical helix injection
An alternative injection, originally proposed and pioneered for the Japanese ( g −
2) project at J-PARC [10],is the injection of muons outside the central and highly uniform magnetic field under a vertical angle ζ = (cid:126)p (cid:107) /(cid:126)p ⊥ into a field produced by a solenoid-like coil package. Here (cid:107) indicates the momentum componentparallel to the magnetic field (cid:126)B ( r, z ) on the symmetry axis r = 0 ∀ z . Figure 8a shows a possible coilpackage producing the field shown in Figure 8b. This method circumvents the large losses due to multiplepassages through material. In combination with a trigger/tagger system upstream, see Sec. III H 2 it lendsitself well for single-muon storage measurement by applying a vertical magnetic kick as described in [10].The entrance trigger will also set a veto in order to inhibit a second magnetic field pulse during the storageperiod of the muon. Muons which still enter into the solenoid, will quickly pass through the central regionand are stopped far away from the positron detection system. The veto is removed by the detection of adecay positron by rapid scintillating tiles next to the positron tracker, or latest after four laboratory lifetimes of about 14 µ s. The spectrometer is again ready to accept the next muon for storage.A second option is to operate this configuration continuously without magnetic kick and let the muonsdrift through the entire field. In this case an event-by-event reconstruction will be implemented using a muontagger, see Sec. III H 2, at the entrance providing the injection angle ζ and start time t for each muon. Incombination with the information of the central positron tracker, the decay vertex and the vertical decayasymmetry can be reconstructed.
1. Stored muons from vertical injection
As in the lateral injection case, the vertical injection needs a triggered magnetic field to kick the muonsonto a stable orbit in the magnetic-field plane at z = 0. Also, in this case, a fast trigger/tagger system isrequired to start the magnetic kick. However, as demonstrated below in the simulation, Sec. III D 2, the kickerneeds to be triggered only after 50 ns, which is considerably longer than in the lateral case. One could usea combination of machine frequency and anti-coincidence between an entrance and veto scintillators insidethe injection channel to produce the trigger for the magnetic kick power supply. As multiple transitionsthrough material could be avoided, we expect a significant gain in injection efficiency once the magneticweakly-focusing field and the magnetic kick are optimized. First, simulations for the vertical injection wereperformed and are described in Sec. III D 2. More details will be studied soon to optimize the design. Fornow, it seems to be sensible to start with an injection phase space of about the same as in the lateralinjection case: 20 ×
20 mm · mrad horizontal and 20 ×
20 mm · mrad vertical (both FWHM). Together withthe measurements presented in section III F and the preliminary injection efficiency deduced in simulations,this results in an injection efficiency of 0 . × − and a positron detection rate of about 60 kHz. Which inturn translates into the statistical sensitivity given in equation (17), as the losses due to multiple scatteringcan be eliminated, and the dominant factor remains the mean storage time of about 3 . µ s.
2. Simulation of vertical injection
The magnetic field and the trajectory of a muon are intimately linked to each other, and essentially itneeds many iterations to arrive at an optimized magnetic-field configuration to permit an efficient injectionand a stable and well-defined central orbit within the region with frozen-spin condition. The following1 (a) (b)
FIG. 8: (a) Computer-rendered image of the coil geometry deployed to calculate the magnetic field. The light yellow-grey, and red cylinders are individual coils, the coil package, used in finite element simulation to create a highly-uniform magnetic field shown in (b). The currents shown in the the inlay were deduced by manual adjustments, withthe goal to produce a field that is less than 1% smaller in the injection area 760 > | z | >
740 mm than in the center z = 0, and to create a weak-focusing field at z = 0. (b) Magnitude of the magnetic field for several radii, R ; theweak-focusing field is clearly visible in the range z = − . . considerations define the starting point of the initial magnetic field and trajectory simulations presentedbelow.1. Magnetic adiabatic collimation indicates that an initial beam divergence ζ inj in the injection area(where B inj ), will be increased to ζ c = arccos (cid:32)(cid:114) B inj B c cos( ζ inj ) (cid:33) , (19)where ζ c is the divergence in the central plane with magnetic field B c , see also Figure 9.2. For the storage of the muon in the central plane a weak-focusing field is required with a large verticalacceptance,3. and the pulsed magnetic field should efficiently “stop” the vertical drift of muons with an as large aspossible vertical divergence.For the finite element calculation of the uniform solenoid field shown in Figure 8 we used Agros2D [66]while for post processing
Matlab was used to create magnetic-field maps. These field maps were then usedin
G4beamline to simulate trajectories of muons and decay positrons. Figure 10 shows the side and topview of an injected muon into the central field region including the frozen-spin electric field, E f ≈ / m,and the decay positron that leaves the solenoid to the top.In an initial simulation, using the field shown in Figure 8b, we looked at the time-reversed process bygenerating a positron at z = 0 and r = 14 cm with a momentum of p = 125 MeV /c . As it had no verticalmomentum component, it stayed at z = 0 until the magnetic kick at t = 48 ns was started lasting for T = 100 ns with a half sinusoidal period, B kick ( t ) = B A sin( πδt/T ). The magnetic field along the positrontrajectory is shown in Figure 11a, while Figure 11b shows the amplitude of the magnetic kick B kick . Thereverse of the vertical ejection angle, ζ eject = 89 . z = 750 mm, where B rminj = B ( z = 750 mm) is defined, was used to fix the nominal injection angle for muons incident through theinjection channel. Note that the injection channel was not yet included in the simulation, instead we defineda 20 ×
20 mm region inclined by the injection angle in which muons with at horizontal and vertical divergenceof ±
10 mrad were generated.2
FIG. 9: Plot of injection angle offset δζ inj versus drift angle ζ c in the center of the magnet. Each data pair indicatesthe ratio B c /B inj , and the nominal injection angle ζ inj , which results in ζ c = 0. In this first simulation attempt, about 20% of all muons were finally stored in the central part of the frozen-spin region and decayed to positrons. Figure 12a shows the vertical phase-space acceptance for injection inthe injection zone, while Figure 12b shows the vertical phase-space trajectories of muons that are stored,closed circles, bypass the central zone, or are reflected. For this first simulation we have chosen a field indexof n = 2 . E −
4, a further optimisation will need to balance potential systematic effects due to a verticalbetatron oscillation and the increase injection efficiency due to a larger vertical phase space. A coupling ofthe vertical and horizontal phase space prior injection will significantly reduce the vertical divergence in thecentral plane and hence further improve the injection efficiency [10, 67].3 (a) (b)
FIG. 10: (a) Simulation images of a single muon injected into the magnetic field shown in Fig. 8b, including electrodesystem (dark brown cylinders) and electric fields for the frozen-spin configuration. The muon (blue, large radiushelix) enters from the top and is kicked when entering the region of the magnetic kicker, −
300 mm < z <
300 mm. Asinusoidal kick stops the vertical drift and the muon is stored in the central region, here −
20 mm < z <
20 mm, untilit decays to a positron (dark purple, small radius helix). Note that no detection system is present and the positronescapes the system. (b) Top view of (a). The electric-field region is defined by the purple zone. (a) (b)
FIG. 11: (a) Magnetic field along the reference trajectory, top vertical, bottom radial. The trajectory is from atime-reversed simulated using a positrons. Within the first 48 ns the positron remained on a stable orbit at z = 0.The distinctive feature at z ≈
320 mm is the pulsed magnetic kick δt ≈
70 ns after the pulse was initiated. (b) Radial-symmetric magnetic-field amplitude B A of the kick used to stop an injected muon for three different radii. The pulseis a half-sine with B kick ( t ) = B A sin( πδt/T ), where T = 100 ns is the pulse duration.
3. Continuous drift measurement
As alternative to a single muon storage ring we also investigated the concept of a continuous injectionwithout triggered magnetic field perturbations. In this case all muons which have passed through the injectionchannel enter the detector system continuously. A combination of accelerator radio frequency, muon entrancetagger, and positron tracker permits the reconstruction of each event. The muons will drift parallel to themagnetic field through the frozen-spin field region with v (cid:107) = βc sin( ζ ) as function of the effective injectionangle ζ . The average time of a muon with velocity v (cid:107) ( ζ ) within the central frozen-spin region of length l is4 (a) (b) FIG. 12: (a) Vertical phase-space acceptance of muons injected into the uniform solenoid field. Blue dots indicatethe initial vertical position and vertical momentum of all generated muons. Red dots indicate the initial conditions ofmuons which decay into a positron within the frozen-spin region. (b) Vertical phase-space plot of muon trajectories.The color code heat map indicates the time after injection. Dark blue to dark red spans a duration of 148 ns whilebrown indicates later times. The magnetic kick is applied at 48 ns for a duration of 100 ns with half a period of asinusoidal shape. The peculiar shape at z ≈
320 mm coincides with the maximum of the kick amplitude in Figure 11b.Apparently, some muons are reflected by the pulse. (cid:104) t ( ζ ) (cid:105) = (cid:82) l/v (cid:107) ( ζ )0 t exp ( − t/ ( γτ )) (cid:82) ∞ exp ( − t/ ( γτ )) . (20)By averaging over the drift angle ζ = arccos (cid:16)(cid:112) ( B inj /B c ) cos( ζ inj ) (cid:17) , a smooth function of the injectionangle ζ inj and the magnetic fields B inj and B c , one obtains the average time, (cid:104) t (cid:105) = (cid:82) ζ (cid:104) t ( ζ ) (cid:105)F ( ζ ) (cid:82) ζ F ( ζ ) , (21)a muon is within the frozen-spin region. Here F ( ζ ) is the vertical divergence of the injected beam, typicallymodelled using a Gaussian distribution. Figure 13a shows the mean-passage times in the case of themeasured vertical phase space, see section III F, and a central-field region of l = 1 m. Figure 13b shows theexpected annual sensitivity for an experiment coupled to µ E1 as a function of width of the distribution ofthe injection angle ζ inj and a magnetic-field ratio of B inj /B c = 0 . v θ along the principalmagnetic-field direction. For the sake of calculations, we will use a magnetic field of 3 T, muons with amomentum of 125 MeV/ c ( γ = 1 . β = 0 . l = 1 m.This results in a radius of about r = 0 .
14 m and v θ = 1 . × m / s. We will use a radial electric field E r = aB z /v θ (cid:0) /c (cid:0) a − / ( γ − (cid:1)(cid:1) − to establish the frozen-spin condition. In the case that v (cid:107) = v z (cid:54) = 0equation (6) changes to (cid:126) ω a = q/m a B r B θ B z − aγ ( v θ B θ + v z B z )( γ + 1) c v θ v z + 1 c (cid:18) a − γ − (cid:19) v z E r − v θ E r . (22)In the case of applying the frozen-spin electric field in combination with a uniform magnetic field along the5 (a) (b) FIG. 13: (a) The differential, equation (20), and integral mean lifetime, equation (21), of a muon within the frozen-spin region. (b) Expected annual sensitivity of the continuous helix muon EDM search connect to the beam line µ E1as a function of width of the vertical injection angle ζ inj . solenoid axis with strength | (cid:126)B | and assuming B r = 0, we get (cid:126) ω a = q/m a B z − aγ ( v z B z )( γ + 1) c v θ v z + 1 c (cid:18) a − γ − (cid:19) v z E r − v θ E r (23)= q/m a B z − aγ ( v z B z )( γ + 1) c v θ v z + a v z /v θ B z − B z (24)= − q/m a γζβ B z ( γ + 1) − γ ζ , (25)where ζ = v z /v θ is the drift angle in the central part of the solenoid. In the next section we show that thecontinuous drift with ζ <
55 mrad does not generate a systematic effect larger than d µ ≤ × − e · cm.However, as Figure 13b shows, the continuous muon helix concept is less sensitive than both storage conceptsand will only be of interest in an initial phase, if the vertical magnetic kicker is not yet implemented. E. Systematic effects
A excellent starting point for a discussion of systematic effects is provided by the seminal publication byFarley and colleagues [1]. Any non-uniformity or misalignment of the magnetic and electric field and thepositron detection system might cause a spin precession or appear as one. As rotations do not commute,particular care has to be taken if several of these effects are combined. • Radial magnetic fields B r • Azimuthal magnetic field B θ • Vertical electric field E V , i.e. (cid:126)E · (cid:126)B (cid:54) = 0 on orbit • Misalignment in positron detector • Early to late change in detector responseThe muons are only stored on orbits where (cid:104) B r (cid:105) = 0, where (cid:104) . (cid:105) denotes the orbit average. In the caseswhere the orbit average of the magnetic and electric field components are zero, (cid:104) B θ (cid:105) = 0 and (cid:104) E V (cid:105) = 0, nosystematic effect occur without a remanent ( g −
2) precession which would lead to non-commutative rotationsof the spin. An exact specification to which precision the electric field for the frozen-spin technique needsto be controlled will be derived and cross checked by simulation. This in turn will indicate the required6precision for a measurement of the anomalous precession as a function of the applied electric field. Theprecise knowledge of a µ [42] permits us to measure (cid:104) B (cid:105) on the orbit for E = 0.Most effects and combination of effects will cancel when combining clockwise and counter clockwise in-jection of muons into the spectrometer and averaging data over multiples of orbit periods, T <
10 ns. Asimulation, supporting analytical derivations will specify to which degree the inverse magnetic field forcounter clockwise injection needs to be identical to the field for clockwise injection, but in sign.A specific systematic effect may occur in the less sensitive case were the muons continuously drift throughthe central part of the spectrometer. In order to make sure that the drift does not generate a relevantsystematic effect the vertical precession ω ζ must be kept much smaller than the sensitivity to an EDM of d µ ≤ × − e · cm. Hence, | ω ζ | = aqm γζ β B z ( γ + 1) ≤ d µ E ∗ (cid:126) (26) ζ ≤ maq γ ( γ − B z d µ E ∗ (cid:126) (27) ζ ≤ (cid:115) md µ caq (cid:126) (cid:115) γ (cid:112) γ − γ −
1) (28) ζ ≤
55 mrad (cid:114) d µ × − e · cm (29)where the minimum is reached for γ = 1 .
62, corresponding to p = 135 MeV /c .In turn this indicates that any drift angle in the central plane up to 55 mrad is acceptable. However, asFigure 9 illustrates the larger the ratio between the magnetic field in the central plane B c and the injectionplane B inj , the stronger the drift-angle distribution will spread in the central plane. F. µ E1 beam line revisited
The precision measurement of the muon EDM requires a high-flux polarized muon beam with a small beamemittance as the sensitivity scales with √ N and P . As E f is proportional to γ , equation (16) indicatesthat the sensitivity increases with higher γ values. Therefore, this demands the use of a fairly high muonmomentum and the µ E1 beam line at PSI is considered to be a potential beam line to host the muon EDMexperiment. Note that even higher momenta would result in higher values of the laboratory electric fieldsneeded for the frozen-spin condition, which are more difficult to realise.Figure 14 shows the layout of the µ E1 area at PSI: pions produced at target E are extracted, selected inmomentum by the dipole magnet ASX 81, and then transported through a 5 T superconducting solenoid,where muons are collected from pion decays followed by the selection of backward decay muons by a secondmomentum selection performed by the dipole magnet ASK 81.In 2019, a characterization of the µ E1 beam line was performed and the muon-beam rate, transverse phasespace (emittance), and polarization level were studied up to the muon-beam momentum of 125 MeV/ c withtwo different beam line settings, the so-called ‘new tune’ and ‘ µ SR-tune’. Note, that for both settings allconfigurations of the proton accelerator up to the µ -channel in Figure 14, were not changed.A scintillating fiber (SciFi) beam monitoring detector mounted 526 mm downstream of the quadrupoleQSE83 (see Figure 14),was used to measure the muon-beam rate and transverse beam size. Then thetransverse phase space was explored by employing a quadrupole-scan technique, which uses the quadraticrelationship between the magnetic-field strength of the final focusing quadrupole in the beam line upstreamof the beam monitoring detector and the transverse beam size to extract the phase-space parameters, namelyTwiss parameters and emittance. Note that such a technique relies on the independent knowledge of thedispersion function for each strength value of the quadrupole used for the scan in order to disentangle thebetatronic from the dispersive part of the measured beam size.A maximum muon-beam rate of 1 . × µ + /s at 2 . c asshown in Figure 15. Figure 16 presents the corresponding horizontal and vertical phase-space ellipses withemittances of 945 mm · mrad and 716 mm · mrad (1 σ ), respectively, and Figure 17 summarizes the horizontaland vertical emittances for two beam line settings as a function of the muon-beam momentum.While we made sure, using Transport -simulations, that for the new beam line setting the beam di-vergence is large and symmetric around the position ‘FS81’ and close to zero at the position of the beammonitor, this was not the case for the ‘ µ SR-tune’ used for solid-state research. Hence, Twiss parametersand emittance for the horizontal planes of the ‘ µ SR-tune’ are not completely correct as they still retain adependence on the dispersion function.7
FIG. 14: Layout of the µ E1 area (see [68]).
This first characterization serves as starting point to optimize the transfer beam line between the muondecay channel and the muon EDM experiment. If calculations and simulations indicate that an alternativebeam layout would further increase the rate and better match the injection phase space, the modificationof the beam line shown in Figure 14 is in principle possible. This probably also requires a second test withbeam.The polarization measurement was performed with a copper stopping target inside the existing µ SRdetector of the GPD instruments and an example of the measured up-down counting asymmetry A ( t ) at125 MeV/ c with the new tune is shown in Figure 18: A ( t ) = α ( N ↑ ( t ) − B ↑ ) − ( N ↓ ( t ) − B ↓ ) α ( N ↑ ( t ) − B ↑ ) + ( N ↓ ( t ) − B ↓ ) , (30)where α accounts for the different detector efficiencies and solid angles, N ↑ and N ↓ are the number of positroncounts in up and down detectors, respectively, and B ↑ and B ↓ represent the constant backgrounds in thecorresponding detectors.Since the oscillation amplitude of A ( t ) is proportional to the initial muon-beam polarization, the compar-ison of the amplitude determined from the measurement and the Geant4 simulation assuming 100% beampolarization results in an absolute value of the muon-beam polarization. The absolute muon-beam polariza-tion at the µ E1 beam line for both beam tunes as a function of the muon-beam momentum is summarizedin Figure 19 and confirms that the initial polarization is above 93%.8
60 70 80 90 100 110 120 130 Muon momentum [MeV/c]020406080100120 · / s ] + m R a t e @ . m A [ n ew tune @ m E1E1 m SR tune @ m FIG. 15: Muon-beam rate at the µ E1 beam line for two beam line settings as a function of the muon-beam momentum. (a) Horizontal (b) Vertical
FIG. 16: Horizontal (a) and vertical (b) phase-space ellipses (1 σ ) at the SciFi beam monitoring detector positionwith the new tune at a muon-beam momentum of 125 MeV/ c . G. Injection channel
All proposed schemes require an injection channel to transport muons from the exit of the beam linethrough the cryostat, coil package, and vacuum tank into the injection zone inside the magnetic field. Forthis purpose, injection using a superconducting magnetic shield, pioneered more than fifty years ago by Firthand coworkers for a 1 .
75 T bubble chamber at CERN [69, 70], and also used for the BNL/FNAL ( g − T c the9
60 70 80 90 100 110 120 130 Muon momentum [MeV/c]0200400600800100012001400 m r ad ] (cid:215) E m i tt an c e [ mm E1 m New tune @ X Y E1 m SR tune @ m X Y
FIG. 17: Horizontal and vertical emittances (1 σ ) at µ E1 beam line for two beam line settings as a function of themuon-beam momentum.
Up-down counting asymmetry @125 MeV/c, new tune m Time (0.3 - - - A sy mm e t r y
1 Alpha 0.8631 0.00122 Phase 85.92 0.313 Asy_Tot 0.2664 0.00114 field 97.599 0.0265 lambda 0.0491 0.0072 asymmetry 3TFieldCos 2 fun1simpleGss 5fun1 = par4 * gamma_mu musrfit: 2019-08-12, 12:30:07, chisq = 366.80000000000001 , NDF = 315 , chisq/NDF = 1.1644444444444444
FIG. 18: Measured up-down counting asymmetry plot with the new tune at a muon-beam momentum of 125 MeV/ c . field within the injection shield is “frozen” even when the outside field is ramped to its nominal strength.Essentially, by ramping the outside field persistent currents will be induced inside the superconductorcounteracting to the outside field. This effect is maintained if the shield thickness is sufficiently large for agiven outside field and the mean lifetime of the shielding current is long enough. Once the field starts topenetrate, the outside field has to be ramped down and the superconducting shield can be reset by heatcycling.The group of D. Barna at the Wigner Research Center for Physics in Budapest has investigated threeoptions for a superconducting shield intended for the use in a septum magnet of the CERN FCC [72–74]. Figure 20b on top shows all three variants: multi-layer Nb-Ti/Nb/Cu sheets embedded in a copperblock, superconducting HTS tapes wound in a helical coil onto a copper tube and bulk MgB sintered intothe desired shape. Experimental tests demonstrate excellent performance of the multi-layer Nb-Ti/Nb/Cusheets [73] while the sintered variant made of MgB showed flux jumps at low fields after a successful initial0
60 70 80 90 100 110 120 130 Muon momentum [MeV/c]90919293949596979899100 P o l a r i s a t i on [ % ] E1 m New tune @ E1 m SR tune @ m FIG. 19: Initial muon-beam polarization at the µ E1 beam line for two beam tunes as a function of the muon-beammomentum. field test [74]. The third version using HTS ribbons could not shield the external field, it fully penetratedthrough the shield above 0 .
25 T. The reason for this might be that the soft-soldering process which was usedto attach the ribbons to the copper tube damaged the superconductor. The group of D. Barna decided notto further investigate the HTS version, but instead started to produce Nb-Ti/Nb/Cu sheets, as the originalproducer, Nippon Steel Ltd. in Tokyo, discontinued production. First sheets will be available by the end of2020.For our purpose a HTS version would be favorable as this would not require liquid helium temperatures.However, using the Nb-Ti/Nb/Cu sheets within a sturdy copper structure could be cooled to about 4 Kusing a pulse tube cooler. This might also be attractive for a thermal reset in the case that the external fieldstarts to penetrate.
H. Muon identification and trigger
The concepts to use a magnetic field pulse to kick the muons onto a stable orbit for storage requiresan entrance muon detector to trigger the magnetic field. Further, we consider a muon tagger, providinginformation on the trajectory of the muon, as helpful for a reconstruction of the decay vertex.
1. Entrance trigger
A quick and reliable detection method for an incident muon within the acceptance phase space is requiredto trigger the magnetic field pulse. A possible scenario is depicted in Figure 21, combining a thin scintillatorat the entrance of the injection channel and scintillators on the wall. A robust trigger can be made byconstructing an anti-coincidence between wall scintillators and entrance trigger and including the protonaccelerator frequency of 50 MHz. In this situation only multiple scattering can occur at the entrance of theinjection channel and is controlled by the anti-coincidence. Note, that in average we do not expect morethan one muon per accelerator pulse of 20 ns.
2. Muon tagger
The muon tagger will provide a measurement of time and trajectory of the muon at the injection in themagnetic field. Combined with the time and trajectory of the positron in the central tracker, it will provide1 (a) (b)
FIG. 20: (a) Possible layout for an injection channel based on a shield made of superconductor. For clockwise andcounterclockwise injection two superconducting channels will be required. Images (b) show three different options forthe construction of an injection channel. From left to right: multi-layer Nb-Ti/Nb/Cu sheets embedded in a copperblock, high-temperature superconducting (HTS) tapes wound in a helical coil onto a copper tube and bulk MgB sintered into the desired shape. Below an image of the copper tube with HTS superconducting tape. Courtesy ofD. Barna for all images [72].FIG. 21: Sketch of the scintillator inside the injection channel, not to scale. Muons enter from the far side, if theypass the channel with out hitting the walls, the magnetic field kick is triggered. Possible dimensions are a crosssection of 1 × and a length of 100 cm. the time of flight and positron emission angle to be used in the spin precession analysis. It requires a timeresolution below 10 ns and a resolution on the vertical angle of the muon of O (1 mrad). For 125 MeV /c muons the multiple Coulomb scattering (MS) in the detector materials is likely to dominate the trackingresolution of the detector. Moreover, if muons are excessively deflected from their nominal trajectory, theinjection efficiency would significantly drop. It means that, in order to provide useful track information andnot to compromise the injection efficiency, the material budget of the muon tagger needs to be kept to anextremely low level. For the case of using a different beam-line than µ E1, the muon tagger should also beable to identify electrons and pions possibly contaminating the muon beam.These requirements favor the use of gaseous detectors with a very light, helium-based gas mixture likehelium/isobutane in 90/10 concentration, combined with a thin and possibly segmented fast scintillator infront of the tagger, although some gaseous detectors could also provide timing with the required resolution.We envisaged a couple of possible configurations for a muon tagger based on gaseous detectors, but wealso made a comparison of performances with a design based on solid state detectors and scintillating fibers,that could have the advantage of sharing the detector technology with the positron tracker. The guiding2
FIG. 22: (Left) Top view of a straw tube tracker with five detector stations (gray) and a scintillator (dark blue)in front of the first station. The trajectory of a positive muon (solid line) within the magnetic field is also shown.(Right) a sketch of a detector station made of two arrays of straw tubes with a small vertical angle among them. principle is to have all detection elements inside the main magnet just after exit of the injection channel.The sensitive elements of the detector should be distributed along the circular muon trajectory and covera vertical extent that, depending on the nominal injection angle, can go from O (few cm) to O (few 10 cm).For the determination of the injection angle, the best resolutions should be on the ( φ , Z ) plane.If gaseous wire detectors are considered, it implies that radial wires should be used. It makes a traditional,cylindrical drift chamber with longitudinal wires unsuitable. While one could adopt planar drift chamberswith very light cathode walls (a small version of the MEG drift chambers [75], but radially oriented),the necessity of operating the detector in vacuum makes thin-wall straw tubes the most natural choice.They have been already developed to work in vacuum, for instance for the upcoming µ → e conversionexperiments [76, 77], and there is a continuous R&D effort to further reduce the wall thickness and improvetheir gas tightness. A set of detector stations, each made of two vertical arrays of radially-oriented tubes,would provide a very good resolution in the ( φ , Z ) plane. With a small vertical angle between the twoarrays, each station would also provide a measurement of the radial coordinate. An example of such anarrangement is sketched in Figure 22. In this configuration, the scintillator before the first detector stationalso provides the track timing that is necessary for the precise determination of the drift time (and hencethe drift distance) of ionization electrons in the tubes. As a drawback, straw tubes would give very poorcapabilities of separating tracks crossing the detector within < < µ s.An alternative configuration could make use of a time projection chamber (TPC) shaped as a cylindricalshell sector (see Figure 23). In this case, a radial drift field, with electron amplification and readout placed onthe outer cylindrical surface, would provide the required ( φ , Z ) resolution and would reduce the average driftdistance with respect to the usual longitudinal configuration, so to avoid or at least reduce the concentrationof heavy additives like CO that could be needed to reduce the electron diffusion, in particular when heliumis used as a base for the mixture. Moreover, the drift field could be matched to the electric field used inthe spin precession region in order to minimize systematic uncertainties. Cylindrical gas electron multipli-ers (GEM)s [78] could be used for the amplification and readout. Thin walls will need to be used to containthe gas, but this requirement is limited to the surfaces crossed by the muons. The replacement of traditionalGEMs with high-granularity detectors like GEMPix [79] or GridPix [80] could provide an extremely highcapability of resolving multiple tracks crossing the detector within ∆ t < µ s, so that the same technologycould be used for future upgrades of the experiment, when pileup could become an issue. Also for a TPC, ascintillator has to provide the starting time for the drift distance measurement.If solid state detectors or scintillating fibers are considered, radial planes of such devices should be used,with one layer of pixels or two crossed layers of fibers. With currently available technologies, the minimumthickness of silicon detectors and scintillating fibers that could suit this application are 50 µ m and 250 µ m,respectively.All these devices would provide a resolution of O (100 µ m) or better on the vertical position, and the mainuncertainty on the muon track extrapolation would come from the resolution on the vertical angle. We per-formed calculations to infer the resolution that could be reached by detectors built with the aforementionedtechnologies, under the assumptions summarized in Table I.We used approximated formulae [81] assuming that a simple χ fit of a straight line to the measured points3 E FIG. 23: Top view of a TPC (gray) with a radial drift field E and GEM readout on the outer surface (dark red). Thetrajectory of a positive muon (solid line) and the drift lines of ionization electrons (dashed lines) within the magneticfield are also shown.Technology Detector materials Single-hit resolution number of 3D hitsTPC He:i-C H (90:10)25 µ m Mylar ® walls 300 µ m 3.3 per cmStraw tubes He:i-C H (90:10)12.5 µ m Mylar ® tubes 100 µ m 1 per station (2 tubes)Silicon pixels 50 µ m Silicon 10 µ m 1 per stationScintillating fibers 250 µ m plastic scintillator 72 µ m 1 per station (2 layers)TABLE I: Overview of different technologies for the muon tagger. is performed on the ( φ , Z ) plane in a uniform magnetic field. We considered the contribution of the single-hitposition resolution, the multiple Coulomb scattering in the active material of the detector and, for the TPC,the MS in the chamber wall that is crossed to exit the detector. It is assumed to be a Kapton ® or Mylar ® foil of 25 µ m thickness, although a dedicated R&D would be required to determine the minimum thicknessneeded to bear the gas pressure against the vacuum outside the detector. The results of our calculationsare shown in Figure 24 as a function of the φ extent of the TPC or the number of detector stations forthe other technologies, with being four the minimum number of 3D points that is needed to fit a helix.As expected, the amount of material in silicon detectors, scintillating fibers, and even straw tubes makesthe contribution of the single-hit resolution subleading with respect to the MS. The best performances areobtained with a TPC, with an expected resolution slightly below 2 mrad. It is important to notice that, insuch a MS-dominated tracking problem, a significant improvement of the resolution with respect to thesecalculations can be obtained with a broken-curve fit, usually solved with the Kalman-Filter technique [82].Nonetheless, the MS in the last detector or in the exit wall of the TPC gives an irreducible contribution tothe resolution, that is about 1 mrad under the assumptions above. I. Pixel tracker for positrons
Silicon pixel detectors became a workhorse in many tracking applications. The advent of DMAPS allowedfor very thin detectors, suitable for the detection of low-momentum particles like in this application. TheMu3e collaboration developed a pixel detector using this technology [83]. A thickness of ≈ . µ m thin silicon pixel chip with an aluminium-polyimide-basedhigh-density interconnect. The chip dimensions are approximately 20 ×
23 mm with an active region ofabout 20 ×
20 mm per chip. With some restrictions due to the electrical connections, detector moduleswith arbitrary shapes can be designed and assembled to cover the desired area. Proven designs for detectorbarrels exist from Mu3e, made up with modules arranging up to 18 chips in a row. The cooling of the pixelchips will be done using gaseous helium, as in Mu3e. This will require a volume separation between thevacuum and the location of the pixel detector. A similar development is currently in progress at PSI for aprototype study to build a detector for muon spin rotation experiments.Figure 25 shows a possible configuration where positrons would be tracked outside of the muon trajectory.The polygonal shapes are two layers of pixel barrels. The indicated example trajectory of a positron demon-strates how the hit information can be reconstructed to determine a) the decay vertex of the muon, b) thedirection of the positron, and c) to determine the momentum of it. The momentum gets reduced in every4
50 100 150 200 250 300 350 ] (cid:176) extent [ f [ m r ad ] qd [ m r ad ] qd [ m r ad ] qd [ m r ad ] qd radial TPC straw tubessilicon pixel detectors scintillating fibers FIG. 24: Total expected resolution (black) on the vertical angle from a muon tagger made of a radial TPC (top left)or a set of detector stations with straw tubes (top right), silicon pixel detectors (bottom left) or scintillating fibers(bottom right), as a function of the angular extent of the TPC or the number of detector stations. The contributionsof the single-hit resolution (red), the MS in the detectors (blue) and the MS in the exit wall of the TPC (green)are also shown. The resolution of straw tubes, silicon pixels and fibers is dominated by the MS and the blue line inhidden behind the black one. µ + Point of decay Pixelse + FIG. 25: A rough sketch of a possible pixel sensor arrangement. Pixel modules are arranged concentric around themuon trajectories inside the storage ring. Upon a decay, positrons escape and follow a circular trajectory until theyget lost. Hits of such a trajectory are marked with asterisk. Note: depending on the choice of magnetic field, anarrangement of pixels inside the muons or on both sides might be chosen. turn due to the material being traversed, and eventually the positron will escape the detector plane. Fromexperience with Mu3e, the expected resolution for the vertex extrapolation would be within O (200 µ m)(dominated by multiple scattering the volume separation) and O (1 %) for the momentum. A scintillatorbased end trigger will supplement the positron tracker to provide for a rapid signal to raise the injectionveto.5 IV. FUTURE PROSPECTS USING HIMB AND MUCOOL
At PSI, the HIMB project is currently being pursued that aims at delivering a surface muon beam of10 µ + /s to two experimental areas for next-generation particle physics experiments and novel methods toperform muSR measurements. The existing 5 mm long TgM is replaced by a slanted, 20 mm long targetTgH that emits around 10 surface- µ + /s to either side. In order to capture a large fraction of this flux,two normal-conducting, radiation-hard solenoids are placed 250 mm away from the target. After this initialcapture and in order to reach a high overall efficiency, the use of large-aperture solenoids and dipolesis continued along the beam line. Preliminary simulations showed that in that way around 10 % overallcapture and transmission efficiency can be reached leading to surface muon rates of around 10 µ + /s in theexperimental areas. The project is currently developing the Conceptual Design Report. The realization isforeseen during the Swiss funding period 2025-2028.In addition to HIMB, the muCool project is also progressing. Within this project the cooling of a positivemuon beam is studied in order to reduce its six-dimensional phase space by a factor 10 . A standardmuon beam is injected into a cryogenic helium gas cell and stops. The combination of strong magnetic andelectric fields together with a position-dependent gas density allows to compress the stopped muon beamboth in transverse and longitudinal directions into a point and extract the muons through a small orifice backinto vacuum [84]. While both the transverse and longitudinal compression stages have been demonstratedexperimentally [85, 86], work is currently ongoing on their combination and the extraction of the muons intovacuum. The extracted muons will be accelerated by a pulsed electrode system to around 10 keV energyand taken out of the strong magnetic field by terminating it non-adiabatically. The final low-energy muonbeam will have a transverse size of around 1 mm and an energy of 10 keV with 10 eV spread. Due tothe expected efficiency of 10 − and the reduced six-dimensional phase space by a factor 10 , the overallbrightness of the beam will increase by seven order of magnitude. Within the science case of the HIMBproject, re-acceleration of the muCool beam to higher energies than the planned 10 keV within the muCoolproject will be studied. While we currently do not see any reason why such a re-acceleration should not bepossible with high efficiency to the momenta required by a muon EDM search, the whole scheme still needsto be worked out. The combination of HIMB, muCool and re-acceleration would deliver a very bright beamof around 10 µ + /s that could be injected into the storage ring or solenoid discussed in the previous sectionswith high efficiency and would allow to push the EDM search to its limit. V. CONCLUSIONS
The search for an electric dipole moment of the muon is a great science opportunity to unveil new sourcesof CP violation and to test lepton universality in one of the least tested domains of particle physics. The pro-posed frozen-spin approach in combination with a three-dimensional injection is a novel concept permittingto search for a muon EDM with an unprecedented sensitivity of better than 6 × − e · cm. VI. ACKNOWLEDGMENTS
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