High Precision Measurement of the Proton Elastic Form Factor Ratio at Low Q 2
HHigh Precision Measurement of the Proton ElasticForm Factor Ratio at Low Q by Xiaohui Zhan
Submitted to the Department of Physicsin partial fulfillment of the requirements for the degree ofDoctor of Philosophyat theMASSACHUSETTS INSTITUTE OF TECHNOLOGYJanuary 2010c (cid:13)
Massachusetts Institute of Technology 2010. All rights reserved.Author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Department of PhysicsJanuary 25, 2010Certified by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .William BertozziProfessor of PhysicsThesis SupervisorCertified by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Shalev GiladPrinciple Research ScientistThesis SupervisorAccepted by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Thomas J. GreytakAssociate Department Head for Education a r X i v : . [ nu c l - e x ] A ug igh Precision Measurement of the Proton Elastic FormFactor Ratio at Low Q byXiaohui Zhan Submitted to the Department of Physicson January 25, 2010, in partial fulfillment of therequirements for the degree ofDoctor of Philosophy
Abstract
Experiment E08-007 measured the proton elastic form factor ratio µ p G E /G M in therange of Q = 0 . − . /c ) by recoil polarimetry. Data were taken in 2008 at theThomas Jefferson National Accelerator Facility in Virginia, USA. A 1.2 GeV polarizedelectron beam was scattered off a cryogenic hydrogen target. The recoil proton wasdetected in the left HRS in coincidence with the elasticly scattered electrons taggedby the BigBite spectrometer. The proton polarization was measured by the focalplane polarimeter (FPP).In this low Q region, previous measurement from Jefferson Lab Hall A (LEDEX)along with various fits and calculations indicate substantial deviations of the ratiofrom unity. For this new measurement, the proposed statistical uncertainty ( < G Ep at this region. Beyond the intrinsicinterest in nucleon structure, the new results also have implications in determiningthe proton Zemach radius and the strangeness form factors from parity violationexperiments.Thesis Supervisor: William BertozziTitle: Professor of PhysicsThesis Supervisor: Shalev GiladTitle: Principle Research Scientist 3 cknowledgments This work could not have be completed without the people who have supportedand helped me along this long journey. I am extremely grateful for their care andconsiderations along these years for which I don’t have many opportunities to showmy gratitude in front of them.First, I would like to thank my advisor, Prof. William Bertozzi, for giving methe opportunity to start the graduate study at MIT, and for his continuing guidance,attention and support throughout these years. I won’t forget the “hard moments” hegave me during the preparation of the part III exam the same as the encouragementwhen I was frustrated. He helped me to understand how to become a physicist and atthe mean time a happy person in life. I also would like to thank my another advisorDr. Shalev Gilad for his valuable advices and suggestions during the whole analysisand the encouragement throughout my study and research at Jefferson Lab. Withouttheir support, I would not complete the thesis experiment and finish the degree.I would like to thank my academic advisor Prof. Bernd Surrow for his carefulguidance in my graduate courses and the discussions for the future career. Manythanks to my thesis committee members: Prof. William Donnelly and Prof. IainStewart for their valuables comments and suggestions to this thesis.Although it’s only been less than two years since I join the E08-007 collaboration,I had a wonderful experience and learned a lot in working with the spokespersons,post-docs and former graduate students. Individually, I sincerely appreciate Prof.Ronald Gilman for his guidance and support before and during the running of theexperiment, and his helpful suggestions and discussions for the analysis afterwards. Iwould like to thank Dr. Douglas Higinbotham for being my mentor at Jefferson Laband giving valuable advices in resolving different problems I encountered along theway. I would like to thank Dr. Guy Ron, for providing the first hand experimentalrunning and analysis experience, the experiment would not run so smoothly withouthis effort. I would like to thank Dr. John Arrington for the valuable comments anddiscussions on the analysis and providing the form factors global fits. I also would5ike to thank Prof. Steffen Strauch, Prof. Eliazer Piasetzky, Prof. Adam Sarty, Dr.Jackie Glister and Dr. Mike Paolone for their guidance and inspiring discussionsthrough the whole analysis process. In addition, I would like to thank the groupformer post-docs Dr. Nikos Sparveris and Dr. Bryan Moffit for their generous helpon the experimental setup and assistance through the experiment. This work couldnot be done without the contribution from any one of them.I would like to thank the Hall A staff members and the entire the Hall A collab-oration for their commitment and shift efforts for this experiment. I would also liketo thank the Jefferson Lab accelerator crew for delivering high quality beam for thisexperiment.In the earlier days at Jefferson Lab, I worked with the saGDH/polarized Hegroup. It was very special to me since that’s when I completed my first analysisassignment and learned quite some knowledge about the target system. I wouldlike to thank Dr. Jian-Ping Chen for his supervision and guidance when I started theresearch in Jefferson Lab without any experience, his passion and rigorous attitude forphysics have served as a model for me. I also would like to thank the former graduatestudents Vince Sulkosky, Jaideep Singh, Ameya Kolarkar, Patricia Solvignon andAidan Kelleher for their patience and generous support on various things. I wouldlike to thank the He lab/transversity fellow students: Chiranjib Dutta, Joe Katich,and Huan Yao for the great experience we had worked together.Although I started my graduate life at MIT, I spent the last four years at JeffersonLab. I am lucky to have friendships at both cities which made my graduate study andresearch an enjoyable experience. I would like to thank them for their support andencouragement: Bryan Moffit, Vince Sulkosky, Bo Zhao, Kalyan Allada, Lulin Yuan,Fatiha Benmokhtar, Ya Li, Jianxun Yan, Linyan Zhu, Ameya Kolarkar, Xin Qian,Zhihong Ye, Andrew Puckett, Peter Monaghan, Jin Huang, Navaphon Muangma,Kai Pan, Wen Feng, Wei Li, Feng Zhou,. Especially I would like to thank the grouppost-doc Vince Sulkosky for his effort in reading and correcting my thesis draft.And Finally, I would like to show my deepest appreciation to my parents. I wouldnot be anywhere without them and there is no words could ever match the love and6upport they gave since I was born. I also want to thank my fiance Yi Qiang, for theendless support over the past 8 years, and loving me for who I am.7 ontents Q . . . . . . . . . . . . . . . . . . . . . . . . . 77 Data Analysis II 161
A Kinematics in the Breit Frame 223B Algorithm for Chamber Alignment 227C Extraction of Polarization Observables 229
C.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22911.2 Azimuthal asymmetry at the focal plane . . . . . . . . . . . . . . . . 229C.3 Weighted-sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230C.4 Maximum likelihood . . . . . . . . . . . . . . . . . . . . . . . . . . . 231C.5 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232C.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 D pC Analyzing Power Parameterizations 241E Neutral Pion Photoproduction Estimation 245
E.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245E.2 Phase Space Simulation . . . . . . . . . . . . . . . . . . . . . . . . . 246E.3 Photon flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249E.4 Cross Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249E.5 Pion Electroproduction . . . . . . . . . . . . . . . . . . . . . . . . . . 252E.6 Rate Estimation and Polarization corrections . . . . . . . . . . . . . . 252E.6.1 BigBite Acceptance . . . . . . . . . . . . . . . . . . . . . . . . 253E.6.2 Hall C Inclusive Data . . . . . . . . . . . . . . . . . . . . . . . 254E.6.3 Corrected Proton Polarizations . . . . . . . . . . . . . . . . . 254E.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
F Cross Section Data 257 ist of Figures ep elastic scattering. . . . . . . . . . . . 341-2 World data of G Ep from unpolarized measurements [1, 2, 3, 4, 5, 6, 7,8, 9, 10, 11, 12, 13], using the Rosenbluth method, normalized to thedipole parameterization. . . . . . . . . . . . . . . . . . . . . . . . . . 481-3 World data of G Mp from unpolarized measurements [1, 3, 14, 15, 5,7, 9, 10, 11, 12, 13], using the Rosenbluth method, normalized to thedipole parameterization. . . . . . . . . . . . . . . . . . . . . . . . . . 491-4 World data of the ratio µ p G Ep /G Mp from unpolarized measurements(black symbols) using the Rosenbluth method and from polarizationexperiments (colored symbols) [16, 17, 18, 19, 20, 21, 22, 23]. . . . . 501-5 Ratio µ p G Ep /G Mp extracted from polarization transfer (filled diamonds)and Rosenbluth method (open circles). The top (bottom) figures showRosenbluth method data without (with) TPE corrections applied tothe cross sections. Figures from [24]. . . . . . . . . . . . . . . . . . . 511-6 Pertubative QCD picture for the nucleon EM form factors. . . . . . . 5313-7 The scaled proton Dirac and Pauli form factor ratio: Q F F (upperpanel) and QF F (lower panel) as a function of Q in GeV . The dataare from [17, 18]. Shown with statistical uncertainties only. The dash-dotted curve is a new fit based on vector meson dominance model(VMD) by Lomon [25]. The thin long dashed curve is a point-formspectator approximation (PFSA) prediction of the Goldstone bosonexchange constituent quark model (CQM) [26]. The solid and the dot-ted curves are the CQM calculations by Cardarelli and Simula [27] in-cluding SU(6) symmetry breaking with and without constituent quarkform factors, repectively. The long dashed curve is a relativistic chiralsoliton model calculation [28]. The dashed curve is a relativistic CQMby Frank, Jennings, and Miller [29]. Figure from [30]. . . . . . . . . . 561-8 Diagrams illustrating the two topologically different contributions whencalculating nucleon EM form factors in lattice QCD [31]. . . . . . . . 571-9 Lattice QCD results from the Nicosia-MIT group [32] for the isovectorform factors F V (upper left) and F V (lower left) as a function of Q .Both the quenched results ( N F = 0) and unquenched lattice resultswith two dynamical Wilson fermions ( N F = 2) are shown for threedifferent pion mass values. The right panels show the results for G VE (upper right) and G VM (lower right), divided by the standard dipoleform factor, as a function of Q in the chiral limit. The filled trianglesshow the experimental results for the isovector form factors extractedfrom the experimental data for the proton and neutron form factors.Figure from [32]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 581-10 Isovector form factor F V ( Q ) lattice data with best fit small scale ex-pansion (SSE) at m π = 292 .
99 MeV (left panel). The line in theright-hand panel shows the resulting Dirac radii, (cid:104) r (cid:105) . Also shown asthe data points are the Dirac radii obtained from dipole fits to the formfactors at different pion masses. Figure from [33]. . . . . . . . . . . . 591-11 Photon-nucleon coupling in the VMD picture. . . . . . . . . . . . . . 6014-12 The proton form factor ratio µ p G Ep /G Mp from Jefferson Lab Hall Atogether with calculations from various VMD models. . . . . . . . . . 611-13 The proton form factor ratio µ p G Ep /G Mp from Jefferson Lab Hall Atogether with calculations from dispersion theory fits. Figure from [30] 621-14 The nucleon electromagnetic form factors for space-like momentumtransfer with the explicit pQCD continuum. The solid line gives thefit [34] together with the world data (circles) including the JLab/CLASdata for G Mn (triangles), while the dashed lines indicate the error band.Figure from [34]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 631-15 Comparison of various relativistic CQM calculations with the data for µ p G Ep /G Mp . Dotted curve: front form calculation of Chung and Co-ester [35] with point-like constituent quarks; thick solid curve: frontform calculation of Frank et al. [29]; dashed curve: point form cal-culation of Boffi et al. [36] in the Goldstone boson exchange modelwith point-like constituent quarks; thin solid curve: covariant specta-tor model of Gross and Agbakpe [37]. Figure from [38]. . . . . . . . . 661-16 Result for the proton form factor ratio µ p G pE /G pM computed with fourdifferent diquark radii, r . Figure from [39]. . . . . . . . . . . . . . . 671-17 The proton form factors in the relativistic baryon χ PT of [40] (IRscheme) and [41] (EOMS scheme). The results of [40] including vectormesons are shown to third (dashed curves) and fourth (solid curves)orders. The results of [41] to fourth order are displayed both with-out vector mesons (dotted curves) and when including vector mesons(dashed-dotted curves). Figure from [38]. . . . . . . . . . . . . . . . . 691-18 Comparison between charge and magnetization densities for the protonand neutron. Figure from [38] . . . . . . . . . . . . . . . . . . . . . . 721-19 Kelly’s fits [42] to nucleon electromagnetic form factors. The errorbands were of the fits. Figure from [42]. . . . . . . . . . . . . . . . . . 741-20 The parametrization of Bradford et al. compared with Kelly’s, togetherwith world data. Figure is from [43]. . . . . . . . . . . . . . . . . . . 7515-21 Extracted values of G E and G M from the global analysis. The opencircles are the results of the combined analysis of the cross section dataand polarization measurements. The solid lines are the fits to TPE-corrected cross section and polarization data. The dotted curves showthe results of taking G E and G M from a fit to the TPE-uncorrectedreduced cross section. Figure from [24]. . . . . . . . . . . . . . . . . . 761-22 The difference between the measure nucleon form factors and the 2-components phenomenological fit of [44] for all four form factors. . . . 771-23 The world data from polarization measurements. Data plotted arefrom [23, 45, 46, 21, 22, 16, 19, 20] . . . . . . . . . . . . . . . . . . . 791-24 Recent world high precision polarization data [16, 19, 20] compared toseveral fits [47, 24, 48, 44] and parameterizations [49, 36, 50, 51]. . . . 802-1 Layout of the CEBAF facility. The electron beam is produced at theinjector and further accelerated in each of two superconduction linacs.The beam can be extracted simultaneously to each of the three exper-imental halls. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 822-2 Hall A floor plan during E08-007. . . . . . . . . . . . . . . . . . . . . 842-3 Schematic of beam current monitors. . . . . . . . . . . . . . . . . . . 852-4 Beam spot at target. . . . . . . . . . . . . . . . . . . . . . . . . . . . 882-5 Layout of the Møller polarimeter. . . . . . . . . . . . . . . . . . . . . 902-6 Beam helicity sequence used during experiment E08-007. . . . . . . . 922-7 Target ladder. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 932-8 Schematic of Hall A High Resolution Spectrometer and the detector hut. 952-9 Left HRS detector stack during E08007. . . . . . . . . . . . . . . . . 962-10 Schematic diagram and side view of VDCs. . . . . . . . . . . . . . . . 972-11 Configuration of wire chambers. . . . . . . . . . . . . . . . . . . . . . 982-12 Layout of scintillator counters. . . . . . . . . . . . . . . . . . . . . . . 992-13 Layout of the Focal Plane Polarimeter. . . . . . . . . . . . . . . . . . 10016-14 The simulated FPP figure of merit with different carbon door thick-nesses [52]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1012-15 FPP coordinate system. . . . . . . . . . . . . . . . . . . . . . . . . . 1022-16 Straws in two different planes of a FPP straw chamber. . . . . . . . . 1032-17 Block diagram for the logic of the FPP signal. (l.e. = leading edge,t.e. = trailing edge). . . . . . . . . . . . . . . . . . . . . . . . . . . . 1042-18 A side view (left) and top view (right) of the BigBite magnet showingthe magnetic field boundary and the large pole face gap. . . . . . . . 1052-19 A side view (left) and top view (right) of the BigBite spectrometerduring this experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . 1052-20 A side view of the BigBite detector package during this experiment. . 1062-21 The BigBite shower counter hit pattern for kinematics K8, δ p = -2%.The hot region corresponds to the elastic electrons. For productiondata taking, only the shower blocks inside the ellipse were on. . . . . 1072-22 Left HRS single arm triggers diagram during E08-007. . . . . . . . . . 1092-23 The BigBite trigger diagram during E08-007. . . . . . . . . . . . . . . 1112-24 Coincidence trigger diagram during E08-007. . . . . . . . . . . . . . . 1123-1 The flow-chart of the E08-007 analysis procedure. . . . . . . . . . . . 1143-2 Hall coordinate System (top view). . . . . . . . . . . . . . . . . . . . 1153-3 Target coordinate system (top and side views). . . . . . . . . . . . . . 1163-4 Detector coordinate system (top and side views). . . . . . . . . . . . 1183-5 Transport coordinate system. . . . . . . . . . . . . . . . . . . . . . . 1193-6 Rotated focal plane coordinate system. . . . . . . . . . . . . . . . . . 1203-7 The TDC width of the u1 wire group and the demultiplexing cut. . . 1223-8 Illustration of the procedure to find clusters in a FPP chamber. Thethree layers represent the three planes, and the circles are cross-sectionalcuts of the straws. The filled circles represent the fired straws. . . . . 12417-9 4 possible tracks for two given fired straws with given drift distances d and d . The good track is the one with the lowest χ when takinginto account all planes of all chambers. . . . . . . . . . . . . . . . . . 1253-10 The difference between the VDC track and the FPP front track before(in black) and after (in red) the chamber alignment. The difference iscentered at 0 after the alignment. . . . . . . . . . . . . . . . . . . . . 1273-11 φ fpp versus zclose before and after the FPP chamber alignment. . . . 1283-12 Cartesian angles for tracks in the transport coordinates system. . . . 1283-13 Spherical angles of the scattering in the FPP. . . . . . . . . . . . . . 1293-14 Left HRS VDC track number distribution. . . . . . . . . . . . . . . . 1313-15 HRS acceptance cuts for kinematic setting K5 δ p = 0%. . . . . . . . . 1323-16 Elastic cut on dpkin (left), and the corresponding 2D cut on the protonangle θ p versus momentum δ p . . . . . . . . . . . . . . . . . . . . . . . 1333-17 The BigBite pre-shower ADC sum versus shower ADC sum with (rightpanel) and without (left panel) the coincidence trigger cut (T5). Thelow energy background were highly suppressed with the coincidenceconfiguration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1343-18 BigBite shower counter hit pattern in the upper panel and the profileson x (vertical) and y (horizontal) in the left and right panels, respectively.1353-19 Proton acceptance (angle versus momentum) with BigBite shower y > y <
0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1353-20 The distribution of the FPP polar scattering angle θ fpp and the appliedcut. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1363-21 Cut applied to zclose after the manual correction for setting K2 δ p = 0%.1373-22 sclose distribution and cut applied to it for setting K2 δ p = 0%. . . . 1373-23 The cone-test in the FPP. The cone of angle θ fpp around track 1 isentirely within the rear chambers acceptance, while the one aroundtrack 2 is not. Track 2 fails the cone-test and is rejected. . . . . . . . 1383-24 Polarimetry principle: via a spin-orbit coupling, a left-right asymmetryis observed if the proton is vertically polarized. . . . . . . . . . . . . . 14018-25 The dipole approximation of the spin transport in the spectrometer:only a perfect dipole with sharp edges and a uniform field. The protonspin only processes along the out-of-plane direction. . . . . . . . . . . 1423-26 Asymmetry difference distribution along the azimuthal scattering an-gle φ fpp at kinematics K6 ( Q = 0 . ). The black solid curverepresents the sinusoidal fit to the data ( χ /ndf = 0 . µ p G Ep /G Mp = 1 in dipole approximation. . . . . . . . . . . . . . . . . 1443-27 Close up view of Fig. 3-26. The black solid curve represents the sinu-soidal fit to the data, while the dashed light blue curve correspondsto a hypothetical distribution assuming µ p G Ep /G Mp = 1 in dipole ap-proximation. There is ∼ ◦ shift between these two curves at the zerocrossing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1453-28 The target scattering coordinate system (solid lines) is the frame wherethe polarization is expressed while the TCS (dashed lines) is the onein which COSY does the calculation. . . . . . . . . . . . . . . . . . . 1473-29 Histograms of the four spin transport matrix elements, S xy (upper left), S xz (upper right), S yy (lower left) and S yz (lower right) at Q = 0 . for the elastic events. The ones plotted in black are from dipoleapproximation, and the ones in red are from the full spin transportmatrix generated by COSY. For the dipole approximation, S xy and S yz are exactly zero, and S xy = 1 by ignoring the transverse componentsof the field. The full spin precession matrix gives broad distributionsfor these elements which represent the effect from the quadrupoles andthe dipole fringe field. . . . . . . . . . . . . . . . . . . . . . . . . . . 14919-30 Analyzing power fit part 1: A y plotted with different parameterizationin the low energy region ( T p <
170 MeV). The error bars shown arestatistical only. The dashed lines are from the LEDEX [53] parameteri-zation, the dashed dotted lines are from the “low energy” McNaughtonparameterization [54], and the solid lines are from the new parameter-ization for experiment E08-007. . . . . . . . . . . . . . . . . . . . . . 1563-31 Analyzing power fit part 2: A y plotted with different parameterizationin the high energy region ( T p >
170 MeV). The error bars shown arestatistical only. The dashed lines are from the LEDEX [53] parameteri-zation, the dashed dotted lines are from the “low energy” McNaughtonparameterization [54], and the solid lines are from the new parameter-ization for experiment E08-007. . . . . . . . . . . . . . . . . . . . . . 1573-32 Weighted average analyzing power (cid:104) A y (cid:105) with respect to T p for scatter-ing angles 5 ◦ ≤ θ fpp ≤ ◦ . . . . . . . . . . . . . . . . . . . . . . . . . 1594-1 The y tg spectrum for LH and Al dummy data with the cut shown bythe vertical solid lines. . . . . . . . . . . . . . . . . . . . . . . . . . . 1644-2 The normalized dpkin spectrum for LH and Al dummy at setting K2 δ p = − y -axis is the dif-ference between the results, the x -axis which was manually shifted fordifferent cuts for a better view. . . . . . . . . . . . . . . . . . . . . . 1674-5 Coordinates for electrons scattering from a thin foil target. L is thedistance from Hall center to the floor mark, and D is the horizontaldisplacement of the spectrometer axis from its ideal position. Thespectrometer set angle is θ and the true angle is denoted by θ s whenthe spectrometer offset is considered. . . . . . . . . . . . . . . . . . . 1714-6 Carbon pointing y tg for kinematics K8 ( Q = 0 . ). . . . . . . . 17120-7 NMR reading with probe D versus the central momentum setting (leftpanel), and the deviation between the value from the linear fit functionand the set value. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1754-8 Proton scattering angle θ p versus the momentum δ p for kinematics K8 δ p = 0%. The anticipated elastic peak position is plotted as the blackdash line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1774-9 Dependence of µ p G Ep /G Mp on the proton target quantities for kine-matics K7 ( Q = 0 . ). The full precession matrix calculated byCOSY (solid quare) is compared to the dipole approximation (opensquare) and a constant fit. The data points are shown with statisticalerror bars only. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1794-10 Alternative ways to calculate the spin precession matrix S ij . . . . . . 1804-11 Fit of the out-of-plane angle difference between the target and the focalplane. θ tr = θ det − ◦ (K6 δ p = 0%). The peak at zero corresponds toa 45 ◦ bending angle in the spectrometer. . . . . . . . . . . . . . . . . 1814-12 FPP chamber rotation along z and the shift of the azimuthal angle φ . 1834-13 FPP chamber rotation along x ( y ) and the change of φ distribution. . 1844-14 The non-zero y component in the rotated frame. . . . . . . . . . . . . 1854-15 The track difference in y versus x before and after the software alignment.1864-16 The track difference ( y ) and its profile versus x after the software align-ment. The solid line is a linear fit to the profile with a slope of 1 × − .1864-17 The form factor ratio binning on the FPP polar scattering angle θ fpp for kinematic setting K6 ( Q = 0 . ) and K7 ( Q = 0 . ). . 1874-18 Comparison of the major contributions to the systematic uncertaintiesand the statistical uncertainty for each kinematics. . . . . . . . . . . 1894-19 Radiative corrections to the recoil polarization. The solid and dashedlines correspond to the longitudinal and transverse components with s = 8 GeV . Figure from [55]. . . . . . . . . . . . . . . . . . . . . . . 19121-20 Radiative corrections to the ratio of the recoil proton polarization inthe region where the invariant mass of the unobserved state is close tothe pion mass and s = 8 GeV . Figure from [55]. . . . . . . . . . . . 1924-21 The 2 γ exchange correction to the recoil proton longitudinal polar-ization components P l and the ratio of the transverse to longitudinalcomponent for elastic ep scattering at Q = 5 GeV . Figure from [56]. 1924-22 The relative correction to the proton form factor ratio from 2 γ exchangeas a function of ε for 5 different Q [57, 58]. . . . . . . . . . . . . . . 1935-1 The proton form factor ratio µ p G E /G M as a function of Q with worldhigh precision data [16, 19, 20] ( σ tot < µ p G E /G M as a function of Q shown withworld high precision data [16, 19, 20] ( σ tot < µ p G E /G M as a function of Q shown withworld high precision polarization data [16, 19, 20, 18, 60]. . . . . . . . 1995-4 Rosenbluth separation of G E and G M constrained by R = µ p G E /G M .For each Q , the reduced cross section σ R is plotted against ε . Thesolid blue line is the standard Rosenluth separation fit without anyconstraint on R . The dotted red line is fit with an exact ratio constraint.2015-5 The new extraction of G E and G M plotted together with the worldunpolarized data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20222-6 The global fit for the proton form factor ratio with world high precisiondata. The red points are the new results (E08-007 I and E03-104), theother points are from previous polarization measurements [16, 19, 20].The black line is the AMT fit to the world 2 γ exchange corrected crosssection and polarization data. The red line is the new fit by includingthe new data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2045-7 The global fit for the proton electric form factor G E . The black lineis the AMT fit to the world 2 γ exchange corrected cross section andpolarization data. The red line is the new fit by including the new data.2055-8 The global fit for the proton magnetic form factor G M . The black lineis the AMT fit to the world 2 γ exchange corrected cross section andpolarization data. The red line is the new fit by including the new data.2055-9 The uncertainty of the Zemach radius as a function of Q . The greenband shows the coverage of the new data. . . . . . . . . . . . . . . . . 2105-10 A linear fit to previous world polarization data, shown by the solid(blue) line and error band. The fit was done up to the region of Q =0 .
35 GeV where the linear expansion is valid for the transverse radiidifference. The shaded area indicates (cid:104) b (cid:105) Ch > (cid:104) b (cid:105) M . The dashed(red) line shows the critical slope when (cid:104) b (cid:105) M = (cid:104) b (cid:105) Ch . Figure from [61]2115-11 New fit with the E08-007 data, shown by the solid (blue) line and errorband. The shaded area indicates (cid:104) b (cid:105) Ch > (cid:104) b (cid:105) M . The dashed (red)line shows the critical slope when (cid:104) b (cid:105) M = (cid:104) b (cid:105) Ch . . . . . . . . . . . . 21223-12 The accessible kinematic region in ε/Q space. The black dots repre-sent the chosen settings (centers of the respective acceptance). Thedotted curves correspond to constant incident beam energies in stepsof 135 MeV (”horizontal” curves) and to constant scattering angles in5 ◦ steps (”vertical” curves). Also shown are the limits of the facility:the red line represents the current accelerator limit of 855 MeV, withthe upgrade, it will be possible to measure up to the light green curve.The dark green area is excluded by the minimal beam energy of 180MeV. The maximum (minimum) spectrometer angle excludes the dark(light) blue area. The gray shaded region is excluded by the uppermomentum of spectrometer A (630 MeV/ c ). Figure from [62]. . . . . 2155-13 Spin-dependent ep elastic scattering in Born appromixation. . . . . . 2175-14 The kinematics for the two simultaneous measurements. The scatteredelectrons e (cid:48) and e (cid:48) are detected in left and right HRS, respectively. Therecoil protons p and p point in the direction of the q-vector (cid:126)q and (cid:126)q , respectively. (cid:126)S denotes the target spin direction. . . . . . . . . . . 2185-15 The proposed Q points and projected total uncertainties for the secondpart of E08-007. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2185-16 The uncertainty of the Zemach radius as a function of Q . The greenband shows the coverage of the new data from this work, and the yellowband shows the proposed coverage of the second part of E08-007. . . 2195-17 Projection of E08-007 part II measurements on the new fit by assumingthe same slope as Q decreases. . . . . . . . . . . . . . . . . . . . . . 220A-1 Elastic scattering in the Breit frame. . . . . . . . . . . . . . . . . . . 224C-1 False asymmetry Fourier series coefficients vs. δ p for kinematics K6 δ p = 2%. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23424-2 Histograms of the extracted ratio P y /P z by weighted-sum method withno false asymmetry ( s = s = 0) in the simulation. N is the samplesize of each trial in the simulation. At large statistics, the extractedratio is in good agreement with the set ratio in the simulation. . . . . 235C-3 Extracted ratio mean value by weighted-sum method vs. different sam-ple size N with false asymmetry s = 0 (left) and s = 0 . P y = 0 . , P z = 0 .
1, lower panel with set polarization P y = 0 . , P z = 0 .
2, showing that the results of the tests do not dependon the value of the set ratio P y /P z . . . . . . . . . . . . . . . . . . . . 236C-4 Extracted ratio mean value deviation from the set value divided bythe sample standard deviation (RMS) vs. different sample size N with false asymmetry s = 0 (left) and s = 0 . P y = 0 . , P z = 0 .
1, lower panel is with set polarization P y = 0 . , P z = 0 .
2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237C-5 Proton induced polarization component, as a function of the electron θ cm scattering angle for different beam energies. The dash (solid) lineshows the total (elastic only) 2 γ exchange effect. The y-axis P y isactually P tgx for the convention used here. . . . . . . . . . . . . . . . 238C-6 Extracted ratio mean value and relative deviation vs. different samplesize N with false asymmetry s = 0 .
1, and different combinations ofset polarization P , P . . . . . . . . . . . . . . . . . . . . . . . . . . . 239C-7 Extracted ratio mean value and relative offset from the set value vs. dif-ferent sample size N with difference false asymmetries: s = 0 . , c =0 . , s = 0 . , c = 0 .
01, and set polarizations: P = P = 0 . , P y = P z = 0 .
1, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . 240E-1 Data and simulated spectrum on δ p − δ p ( φ ). . . . . . . . . . . . . . . 24725-2 Simulated proton kinematics for π p at E γ = 500 MeV and elastic. P p is the proton momentum and θ p is the scattering angle. . . . . . . . . 247E-3 Proton elastic cut on δ p − δ p ( φ ) spectrum for kinematics K2. . . . . . 248E-4 Simulated ep and π p spectrum for kinematics K2 and K8. The bluelines are the corresponding elastic cut applied to the data. . . . . . . 249E-5 World data and calculations for π p differential cross section at E γ =1185 MeV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251E-6 Phase space simulation for ep , π p and epπ with E γ = 1190 MeV. . . 252E-7 The proton singles spectra and the full background simulation fromHall C Super-Rosenbluth experiment with beam energy 849 MeV (leftpanel) and 985 MeV (right panel). The spectra in red in the protonelastic peak, and the one in magenta is the simulated pion production. 255E-8 Calculations for the π p polarization observable at E γ = 1185 MeV. . 25626 ist of Tables ◦ < θ fpp < ◦ . T p is the protonaverage kinetic energy at the center of the carbon door. . . . . . . . . 1594.1 Aluminum foil thickness. . . . . . . . . . . . . . . . . . . . . . . . . . 1634.2 The upper limit of the Al background fraction R max for each kinemat-ics. The numbers listed are the average over all δ p settings. . . . . . . 1634.3 Polarization P y ( z ) of LH , Al dummy and corrected values for kinemat-ics K1 ( Q = 0 .
35 GeV ). . . . . . . . . . . . . . . . . . . . . . . . . . 1654.4 Polarization P y ( z ) of LH inside, outside the coincidence timing cut andthe corrected values for kinematics K8 ( Q = 0 . ). . . . . . . . 1664.5 Shifts of the form factor ratio associated with shifts of the individualtarget quantities for each kinematic setting. . . . . . . . . . . . . . . 1694.6 Spectrometer nominal ( θ ) and real ( θ s ) central angle for each kine-matic setting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1724.7 Target materials in the beam energy loss calculation. . . . . . . . . . 1724.8 Converted uncertainty in φ tg with ∆( E e ) = 0 . B in kG with probe D for each momentumsetting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1744.10 Converted uncertainty in φ tg from P . . . . . . . . . . . . . . . . . . . 1744.11 Target materials that the proton passed through before entering thespectrometer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1754.12 Proton momentum loss [MeV/c] for each kinematics. . . . . . . . . . 1764.13 Uncertainty of φ tg with ∆ δ p = 0 . φ tg from the external parameters. . . . . . . . . . 1774.15 φ tg uncertainty for each kinematics. . . . . . . . . . . . . . . . . . . . 1784.16 Systematic uncertainty in R = µ p G E /G M for each kinematics associ-ated with left HRS optics. . . . . . . . . . . . . . . . . . . . . . . . . 1784.17 Systematic error in µ p G E /G M associated with COSY. . . . . . . . . . 1824.18 Errors in the FPP scattering angles and the associated systematic errorin µ p G E /G M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1874.19 Errors of the VDC angles and associated systematic error in µ p G E /G M .1884.20 Errors of the kinematic factors and the resulting uncertainty in theform factor ratio R for kinematics K7 ( Q = 0 . ). . . . . . . . . 1884.21 Final results with statistical and systematic uncertainties for each kine-matics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1905.1 The extracted values of G E and G M , with and without the constraintof µ p G E /G M from the new measurements. The errors are indicated inparentheses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2035.2 Proton charge rms-radius from different parameterizations. . . . . . . 2075.3 Summary of corrections for electronic hydrogen. . . . . . . . . . . . . 2095.4 Zemach radii, ∆ Z for different parameterizations. . . . . . . . . . . . 2095.5 The absolute asymmetry difference (∆ A P V ), the normalized differenceby the experimental uncertainty (∆ A P V /σ ) and the relative asymmetrydifference (∆ A P V /A P V ) between using the AMT [24] parameterizationand the new one. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21428.1 Electron scattering angle θ cm for each kinematics ( δ p = 0%). . . . . . 237C.2 Deviation from the set value ∆ R with different combinations of P and P . The set transferred polarization is P y = P z = 0 .
1. Simulation withsample size N = 10 and number of trial N trial = 10 . The standarddeviation for extracted values is ∼ . θ fpp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242D.2 Binning on T p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242D.3 Coefficients of different parameterizations for the pC analyzing power A y . The reduced χ of the new fit is 0.74 with a χ of 272.5 and 368degrees of freedom. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243E.1 HRS acceptance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246E.2 Simulated π p to ep phase space ratio at kinematics K2. . . . . . . . 248E.3 π p to ep phase space ratio for different kinematics with E γ = 1180,1185, and 1190 MeV. . . . . . . . . . . . . . . . . . . . . . . . . . . . 250E.4 Real photon flux at different energies with 1.192 GeV electron beam. 250E.5 ep and π p differential cross sections in the lab frame and the ratio R XS for different kinematics. . . . . . . . . . . . . . . . . . . . . . . . 251E.6 Estimated ratio of π p to ep for kinematics K2. . . . . . . . . . . . . 253E.7 ep and π p differential cross sections in the lab frame and the ratio R XS for different kinematics. . . . . . . . . . . . . . . . . . . . . . . . 254E.8 Polarization observable . . . . . . . . . . . . . . . . . . . . . . . . . . 255290 hapter 1Introduction When the proton and the neutron were discovered in 1919 and 1931 respectively, theywere believed to be Dirac particles, just like the electron. They were expected to bepoint-like and to have a Dirac magnetic moment, expressed by: µ D = qmc | (cid:126)s | (1.1)where q , m , and s are the electric charge, mass and spin of the particle respectively.However, later measurements of these nucleons magnetic moments revealed the exis-tence of the nucleon substructure. The first direct evidence that the proton has aninternal structure came from the measurement of its anomalous magnetic moment 70years ago by O. Stern [63], µ p = 2 . µ B , (1.2)where µ B is the Bohr magneton. The first measurement of the charge radius of theproton by Hofstadter et al. [64, 65] yielded a value of 0.8fm, which is quite close tothe modern value.Starting from 1950s, electron scattering experiments were used to unravel the nu-cleon internal structure. Through the measurements of electromagnetic form factorsand nucleon structure functions in elastic and deep inelastic lepton-nucleon scattering,it’s commonly accepted that in a simplistic picture, a nucleon is composed of threevalence quarks interacting with each other through the strong force. The strong in-31eraction theory, Quantum Chromodynamics (QCD) can make rigorous predictionswhen the four-momentum transfer squared, Q , is very large and the quarks becomeasymptotically free. However, predicting nucleon form factors in the non-perturbativeregime is difficult due to the dominance of the soft scattering processes. As a conse-quence there are several phenomenological models which attempt to explain the datain this domain, and precise measurements of the nucleon form factors are desired toconstrain and test these models.In the one-photon-exchange (OPE) approximation, the ep elastic scattering crosssection formalism is well known and can be parameterized by two form factors, G E and G M which are functions of Q . At low momentum transfer, the form factors canbe interpreted as the fourier transform of the nucleon charge and magnetic densities.Earlier experiments measured the cross section of the ep elastic scattering whichcontains information about the internal structure responsible for the deviation fromthe scattering off point-like particles. However, after four decades of effort, therewere still large kinematic regions where only very limited measurements of the formfactors were possible, since the cross section of the unpolarized electron scatteringis only sensitive to a specific combination of the form factors and the lack of a freeneutron target.In the last two decades, advances in the technology of intense polarized electronbeams, polarized targets and polarimetry have ushered in a new generation of electronscattering experiments which rely on spin degrees of freedom. Compared to the crosssection measurement, the polarization techniques have several distinct advantages.First, they have increased sensitivity to a small amplitude of interest by measuringan interference term. Second, spin-dependent experiments involve the measurementof polarizations or helicity asymmetries, and these quantities are independent of thecross section normalization, since most of the helicity independent systematic uncer-tainties can be canceled by measuring a ratio of polarization observable.The first experiment to measure the recoil proton polarization observable in ep elastic scattering was done at SLAC by Alguard et al. [66], but the impact of theresults was severely limited by the low statistics. Followed by that, the proton form32actor measurements using recoil polarimetry were carried out at MIT-Bates [45, 67]and MAMI [68, 69]. Due to limited statistics and kinematics coverage, the ratiovalues were in agreement within uncertainties with the unpolarized measurements.More recent measurements of the proton form factor ratio µ p G E /G M using recoilpolarimetry at Jefferson Lab [16, 17, 18], which have much better precision at high Q , deviated dramatically from the unpolarized data. This has prompted intensetheoretical and experimental activities to resolve the discrepancy. The validity ofanalyzing data in the OPE approximation has been questioned, and two-photon-exchange (TWE) processes are now considered as an significant correction to theunpolarized data and mostly account for the discrepancy at high Q [24].While extending our knowledge at higher momentum transfer region is an ongoingendeavor, the proton form factor ratio behavior at low Q has also become the subjectof considerable interest, especially, when potential discrepancy was observed from themost recent high precision measurements for Q < . BLAST [19] did the firstproton form factor ratio measurement via beam-target asymmetry at Q values of 0.15to 0.65 Gev , and the results are consistent with 1 in this region. LEDEX [20], whichused the recoil polarimetry technique, observed a substantial deviation from unity at Q ∼ .
35 GeV . However, the data quality of LEDEX was compromised due to thelow beam polarization and background issues [70]. Hence, it was necessary to carryout a new high precision measurement to either confirm to refute the deviations atlow momentum transfers. Beyond the intrinsic interest in the nucleon structure, animproved proton form factor ratio also impacts other high precision measurementssuch as parity violation experiments (HAPPEX) [71, 72], deeply virtual Comptonscattering (DVCS) [73, 74], and also determination of other physical quantities suchas the proton Zemach radius.This thesis presents the analysis and results of experiment E08-007, which wasconducted in 2008 at Jefferson Lab Hall A. In this experiment, the proton formfactor ratio µ p G E /G M was measured at Q = 0 . − . using recoil polarimetry.33igure 1-1: The leading order diagram of ep elastic scattering. When scattered off a nuclear target, the electron exchanges virtual photons with thenucleus, which probes the electromagnetic structure of the nucleus. The electromag-netic coupling is small enough ( α = 1 / e ( k ) + P ( p ) → e ( k (cid:48) ) + P ( p (cid:48) ),the leading order diagram is shown in Fig. 1-1. Initial and final electrons have four-momenta k = ( E, (cid:126)k ) and k (cid:48) = ( E (cid:48) , (cid:126)k (cid:48) ) respectively, and the initial and final protons p = ( E p , (cid:126)p ) and p (cid:48) = ( E (cid:48) p , (cid:126)p (cid:48) ). The virtual photon has four-momentum q = ( ω, (cid:126)q ), andthe Lorentz-invariant four-momentum transfer squared Q is defined as: Q = − q = − ( ω − (cid:126)q ) = − ( k − k (cid:48) ) ∼ EE (cid:48) sin θ e , (1.3)where the last expression is valid in the Lab frame by neglecting the electron mass.The amplitude of Q is associated with the scale that the electromagnetic probe issensitive to.For exclusive elastic scattering, the recoil proton is also detected, so that Q can34e defined from the proton momenta: Q = − ( p (cid:48) − p ) = − [( E (cid:48) p − E p ) − ( (cid:126)p (cid:48) − (cid:126)p ) ] . (1.4)In the Lab frame,the initial proton is at rest, and Eq. 1.4 becomes: Q = − ( E (cid:48) + m p − E (cid:48) p m p − (cid:126)p ) = − (2 m p − m p E (cid:48) p ) = 2 m p T p , (1.5)where m p is the proton mass and T p = E (cid:48) p − m p is the kinetic energy of the finalproton in the Lab frame. One of the advantages of the electromagnetic probe lies in the fact that the leptonicvertex e ( k ) → e ( k (cid:48) ) + γ ∗ ( q ) is fully described by the theory of the electromagneticinteraction, Quantum ElectroDynamics (QED), and the information related to theunknown electromagnetic properties of the nucleon are contained by only the hadronicvertex γ ∗ ( q ) + P ( p ) → P ( p (cid:48) ). From the Feynman diagram in Fig. 1-1, the amplitudefor ep elastic scattering can be written as: i M = [ ie ¯ v ( p (cid:48) )Γ µ ( p (cid:48) , p ) v ( p )] − ig µν q [ ie ¯ u ( k (cid:48) ) γ ν u ( k )] (1.6)= − iq [ ie ¯ v ( p (cid:48) )Γ µ ( p (cid:48) , p ) v ( p )][ ie ¯ u ( k (cid:48) ) γ µ u ( k )] , (1.7)where γ µ , µ = 0 , , , × γ = , (cid:126)γ = (cid:126)σ − (cid:126)σ (1.8)35 and (cid:126)σ is the set of standard Pauli matrices: σ = , σ = − ii , σ = − . (1.9) u ( k ) and ¯ u ( k (cid:48) ) are the Dirac spinors for the initial and final electron, and v ( p ),¯ v ( p (cid:48) ) are the Dirac four-spinors for the initial and recoil proton respectively. Inparticular, the proton spinors enter in the plane-wave solution for a spin 1/2 particle ψ ( x ) = v ( p ) e − ip · x which satisfies the Dirac equation:( − iγ µ ∂ µ − m ) ψ ( x ) = 0 , (1.10)and one can write: v ( p ) = √ p · σχ √ p · ¯ σχ (1.11)with σ µ ≡ (1 , (cid:126)σ ) , ¯ σ ≡ (1 , − (cid:126)σ ) and χ is a normalized two-spinor, such that χ † χ = 1 . (1.12)While the leptonic current j µ = ie ¯ µ ( k (cid:48) ) γ µ u ( k ) is fully described by QED, the hadroniccurrent J µ = ie ¯ v ( p (cid:48) )Γ µ v ( p ) involves the factor Γ µ , which contains the informationabout the internal electromagnetic structure of the proton. In general Γ µ is someexpression that involves p, p (cid:48) , γ µ and constants such as the proton mass m , the electriccharge e . Since the hadronic current transforms as a vector, Γ µ must be a linearcombination of these vectors, where the coefficients can only be function of Q . It isconvenient to write the current in the following way: J µ = ie ¯ v ( p (cid:48) )Γ µ v ( p ) = ie ¯ v ( p (cid:48) )[ γ µ F ( q ) + iσ µν q ν m κF ( q )] v ( p ) , (1.13)where σ µν = i [ γ µ , γ ν ], κ (cid:39) .
793 is the proton anomalous magnetic moment and F , ( Q ) are the proton elastic form factors. They contain the information about theelectromagnetic structure of the proton. 36 .1.3 Nucleon Form Factors F ( Q ) and F ( Q ) are distinguished according to their helicity ( (cid:126)σ · (cid:126)p/ | (cid:126)p | ) character-istics, the projection of electron intrinsic spin (cid:126)σ along its direction of motion (cid:126)p/ | (cid:126)p | . F ( Q ) is the Dirac form factor; it represents the helicity-preserving part of the scat-tering. On the other hand, the Pauli form factor F ( Q ) represents the helicity-flipping part. F and F are defined in a similar way for the neutron. The formfactors are normalized according to their static properties at Q = 0. For the proton: F p (0) = 1 , F p (0) = 1 , (1.14)and for the neutron: F n (0) = 0 , F n (0) = 1 . (1.15)For reasons that will soon become obvious, it is more convenient to use the Sachsform factors [75]: G E ( Q ) and G M ( Q ), which are defined as: G E = F − τ κF G M = F + κF , (1.16)where τ = Q m is a kinematic factor. The Sachs form factors also have particularvalues at Q = 0 according to the static properties of the corresponding nucleon: G Ep (0) = 1 , G Mp (0) = µ p (1.17) G En (0) = 0 , G Mn (0) = µ n , (1.18)where µ p = 2 .
79 and µ n = − .
91 in units of nuclear magneton.
In the Breit frame, which is defined as the frame where the initial and final nucleonmomenta are equal and opposite, the hadronic current has a simplified interpretation.37 definition of variables in the Breit frame, which are noted with a subscript B , iselaborated in Appendix A. Using the Gordon identity [76]¯ v ( p (cid:48) ) γ µ v ( p ) = ¯ v ( p (cid:48) )[ p (cid:48) µ + p µ m + iσ µν q ν m ] v ( p ) (1.19)similarly, we can write:¯ v ( p (cid:48) )Γ µ v ( p ) = ¯ v ( p (cid:48) )[( F + κF ) γ µ − ( p + p (cid:48) ) µ m κF ] v ( p ) . (1.20)In the Breit frame, the explicit expression of the hadronic current J = ( J , (cid:126) J ) issimplified: J = ie ¯ v ( p (cid:48) )[( F + κF ) γ − E pB m κF ] v ( p ) (1.21) (cid:126) J = ie ( F + κF )¯ v ( p (cid:48) ) (cid:126)γv ( p ) , (1.22)where E pB is the Using ¯ v ( p (cid:48) ) = v † ( p (cid:48) ) γ , the time component J can be expressedby: J = ie [( F + κF ) v † ( p (cid:48) ) v ( p ) − κF E pB m v † ( p (cid:48) ) γ v ( p )] . (1.23)By the definition of v ( p ) and γ in Eqs. 1.11 and 1.8, we now have: J = ie ( F + κF ) χ (cid:48)† (cid:18)(cid:113) p (cid:48) · σ, (cid:113) p (cid:48) · ¯ σ (cid:19) √ p · σ √ p · ¯ σ χ − ieκF E pB m χ (cid:48) (cid:18)(cid:113) p (cid:48) · σ, (cid:113) p (cid:48) · ¯ σ (cid:19) √ p · σ √ p · ¯ σ χ. (1.24)Then, by the expressions: (cid:113) p (cid:48) · σ √ p · σ = (cid:113) p (cid:48) · ¯ σ √ p · ¯ σ = m (1.25) (cid:113) p (cid:48) · σ √ p · ¯ σ + (cid:113) p (cid:48) · ¯ σ √ p · σ = 2 E pB (1.26) τ = Q m = (cid:126)q B m = E pB − m m , (1.27)38e can finally get the simple relation: J = ie mχ (cid:48)† χ ( F − τ κF ) = ie mχ (cid:48)† χG E . (1.28)The vector current (cid:126) J can also be expressed in a similar way in the Breit frame: (cid:126) J = − eχ (cid:48)† ( (cid:126)σ × (cid:126)q B ) χ ( F + κF ) = ieχ (cid:48)† ( (cid:126)σ × (cid:126)q B ) χG M . (1.29)Therefore, in the Breit frame, the electric form factor G E is directly related to theelectric part of the interaction between the virtual photon and the nucleon, and themagnetic form factor G M is related to the magnetic part of this interaction. Theelectric and magnetic form factors can be associated with the Fourier transforms ofthe charge and magnetic current densities in this frame in the non-relativistic limit.The Fourier transforms can be expanded in powers of q : G E,M ( Q ) = (cid:90) ρ ( (cid:126)r ) e i(cid:126)q · (cid:126)r d (cid:126)r (1.30)= (cid:90) ρ ( (cid:126)r ) d (cid:126)r − (cid:126)q (cid:90) ρ ( (cid:126)r ) (cid:126)r d (cid:126)r + . . . (1.31)Notice that the first integral gives the total electric or magnetic charge, and thesecond integral defines the RMS electric or magnetic radii of the nucleon. However,the Breit frame is a mathematical abstraction, and for different Q value, the Breitframe experiences relativistic effect which is essentially a Lorentz contraction of thenucleon along the direction of motion. This results in a non-spherical distributionfor the charge densities, and complicates the Fourier transform interpretation of theform factors in the rest frame. 39 .2 Form Factor Measurements The differential cross section for ep scattering in the lab frame can be written as: dσ = (2 π |M| )4( k · p ) δ ( k + p − k (cid:48) − p (cid:48) ) d (cid:126)k (cid:48) (2 π )2 E (cid:48) d (cid:126)p (cid:48) (2 π )2 E (cid:48) p , (1.32)where we have neglected the electron mass, and M is the amplitude defined in Eq. 1.7.Integrating over (cid:126)k (cid:48) and (cid:126)p (cid:48) gives: dσd Ω e = |M| π m (cid:32) E (cid:48) p E p (cid:33) , (1.33)where Ω e is the solid angle in which the electron is scattered, and |M| has the form: |M| = [ J µ − iq j µ ][ J ν − iq j ν ] ∗ = (cid:32) q (cid:33) [ J µ J ν ∗ ][ j µ j ∗ ν ] = (cid:32) e q (cid:33) W µν L µν . (1.34)The hadronic and leptonic tensors are defined respectively as: W µν = 1 e J µ J ν ∗ (1.35) L µν = 1 e j µ j ∗ ν . (1.36)For unpolarized cross section, W µν and L µν are averaged over the incident particlespin states, and summed over the final particle spin states. Since the contraction ofthese two tensors is a Lorentz invariant, they can be calculated in any frame, as longas they are both calculated in the same frame.In the single-photon exchange (Born) approximation, the formula for the differ-ential cross section of electron scattering off nucleons is given by [77]: dσd Ω e = (cid:32) dσd Ω (cid:33) Mott E (cid:48) E { F ( Q ) + 2( F ( Q ) + F ( Q )) tan θ e } , (1.37)40here (cid:32) dσd Ω (cid:33) Mott = (cid:32) e E (cid:33) (cid:32) cos θ e sin θ e (cid:33) (1.38)is the Mott cross section for the scattering of a spin-1/2 electron from a spinless,point-like target, and EE (cid:48) is the recoil factor. This is the most general form for theelectron elastic scattering cross section. Using Eq. 1.16, we can rewrite Eq. 1.37without the interference term: dσd Ω e = α Q (cid:32) E (cid:48) E (cid:33) [2 τ G M + cot θ τ ( G E + τ G M )] , (1.39)where α = e / π ∼ /
137 is the electromagnetic fine structure constant, and thisexpression is known as the
Rosenbluth formula . Rosenbluth Separation
The Rosenbluth cross section has two contributions: the electric term G E and themagnetic term G M . As noted earlier, there is no interference term, so that the twocontributions can be separated. We define the reduced cross section as: σ red = dσd Ω ε (1 + τ ) dσd Ω Mott = τ G M + εG E , (1.40)where ε = (1 + 2(1 + τ ) tan ( θ e / − is the virtual photon polarization parameter.The quantity ε can be changed at a given Q , by changing the incident electronbeam energy and the scattering angle. Therefore, at a fixed Q by varying (cid:15) , one canmeasure the elastic scattering cross section and separate the two form factors usinga linear fit to the cross section. The slope is equal to G E and the intercept is equalto τ G M .This method has been extensively used in the past 40 years to measure the elasticform factors and proved to be a very powerful method to measure the proton andthe neutron magnetic form factors up to large Q . However, there are practicallimitations. First, the neutron electric form factor is normalized to the static electriccharge of the neutron, which is 0, and the cross section is completely dominated by41he magnetic form factor. For the proton, the magnetic term will also dominate atlarge Q , since the factor τ = Q m increases quadratically as Q increases. As anexample, at Q = 2GeV the magnetic term contributes about 95% of the total crosssection. On the other hand, in the low Q region, the magnetic term is suppressed forthe same reason and the electric term becomes dominant. Besides the difficulties fromthe physics side, the precision of Rosenbluth separation is also limited by the crosssection measurements due to a widely different kinematic settings in order to covera wide range of ε . Systematic errors are introduced by the inconsistent acceptance,luminosity, detector efficiency between different kinematics. In 1974, Akhiezer and Rekalo [78] discussed the interest of measuring an interferenceterm of the form G E G M by measuring the transverse component of the recoilingproton polarization in the reaction (cid:126)e + p → e (cid:48) + (cid:126)p (cid:48) . Thus, one could obtain G E in thepresence of a dominating G M at large Q . Instead of directly measuring the separateform factors, the ratio G E /G M can be accessed by measuring the polarization of therecoil nucleon. The virtue of the polarization transfer technique is that it is sensitiveonly to the ratio G E /G M and does not suffer from the dramatically reduced sensitivityto a small component. Another way of measuring the interference term would be tomeasure the asymmetries in the scattering of a polarized beam off a polarized target.By measuring the polarization P ˆ u of the recoil nucleon along a unit vector ˆ u , wemeasure a preferential orientation of the spin along ˆ u . In this case, when we averageover initial proton spin states and sum over final proton spin states, the completenessrelation (cid:88) s =1 , χ s χ † s = 1 (1.41)no longer holds. Instead we have to use: (cid:88) s =1 , χ (cid:48) s χ (cid:48)† s = 1 + (cid:126)σ · ˆ u (1.42)42o that the hadronic tensor becomes: W µν = 12 T r [ F µ (1 + (cid:126)σ · ˆ u ) F ν † ] = W µνu + W µνp , (1.43)where W µνu is the unpolarized hadronic tensor and W µνp is the polarized one: W µνp = 12 T r [ F µ F ν † (cid:126)σ · ˆ u ] . (1.44)For recoil proton polarization measurements, a longitudinally polarized beam is re-quired. The polarization of the beam is defined as: h = N + − N − N + + N − , (1.45)where N + and N − are the number of electrons with their spin parallel and anti-parallel to their momentum respectively. Therefore, with a polarized electron beam,the leptonic tensor is modified and a new γ matrix is introduced: γ = iγ γ γ γ = − (1.46)The operator: 1 − γ . (1.47)projects the spin along the momentum in a preferential direction. If the beam polar-ization is h , and by further neglecting the electron mass, the leptonic tensor can bewritten as: L µν = 12 T r [( γ · k (cid:48) − m e ) γ µ (1 − hγ )( γ · k − m e ) γ ν ]= 2 k µ k (cid:48) ν + 2 k ν k (cid:48) µ − g µν k · k (cid:48) + 2 ih(cid:15) µναβ k α k (cid:48) β = L uµν + L pµν , (1.48)where (cid:15) µναβ is the Levi-Civita symbol. It is 0 if any two indices are identical, -1 under43n even number of permutations and +1 under an odd number of permutations. Notethat L pµν is anti-symmetrical.In order to get the polarized amplitude, we contract the leptonic and the hadronictensors: W µν L µν = W µνu L uµν + W µνu L pµν + W µνp L uµν + W µνp L pµν (1.49)where • W µνu L uµν is the amplitude squared of the unpolarized process. • W µνu L pµν = 0 because it is the product of a symmetrical and an anti-symmetricaltensors. • W µνp L uµν is the induced polarization , it represents the polarization state of therecoil proton after scattering with an unpolarized beam off an unpolarized tar-get. • W µνp L pµν is the transferred polarization , it represents the polarization state ofthe recoil proton after scattering with a polarized beam.The recoil polarization along the vector ˆ u are given by: P ind ˆ u = W µνp L uµν W µνu L uµν hP transf ˆ u = W µνp L pµν W µνu L uµν . (1.50)With the equations above, we can write the amplitude as: W µν L µν = W µνu L uµν (1 + P ind ˆ u + hP transf ˆ u ) , (1.51)where h is the polarization of the beam.First, assume we measure the polarization along the 1-direction, and we can deriveeach term of the hadronic tensor: W µνp, = 12 T r [ F µ F ν † σ ] . (1.52)44sing σ σ = iσ , σ σ = iσ and σ σ = iσ , we have: F † σ = 2 mG E σ (1.53) F † σ = − (cid:113) Q G M σ (1.54) F † σ = i (cid:113) Q G M (1.55) F † σ = 0 . (1.56)The (cid:126)σ matrices have the trace properties: T r [ γ µ γ ν ] = 4 g µν T r [ γ µ γ ν γ ρ γ σ ] = 4( g µν g ρσ − g µρ g νσ + g µσ g νρ ) , (1.57)where g µν is the Minkowski metric. The only non-zero terms arising are: W p, = i (cid:113) Q mG E G M W p, = − i (cid:113) Q mG E G M . (1.58)We note here that the polarized tensor is anti-symmetrical, hence, when it multipliedby the unpolarized leptonic tensor, the terms will vanish, which applies for all thecomponents.The corresponding polarized terms of the leptonic tensor in the Breit frame areanti-symmetrical, and obeys: L p = − L p . (1.59)According to Eq. 1.48, L p = 2 ih(cid:15) αβ k α k (cid:48) β = 2 ih ( k B k (cid:48) B − k B k (cid:48) B ) = − ihQ cot θ B . (1.60)By contracting the hadronic tensor and the leptonic tensor, we get the transferred45olarization amplitude: W µνp, L pµν = 4 hmQ (cid:113) Q cot θ B G E G M . (1.61)Therefore, measuring the 1-component, or transverse component of the recoil protonpolarization, gives access to the interference term G E G M , which is inaccessible froman unpolarized cross section measurement.The derivation for the 2-component is exactly identical. It involves the terms W p, and L p of the tensors, in particular: L p = 2 ih(cid:15) αβ k αB k (cid:48) βB = 2 ih ( k B k (cid:48) B − k B k (cid:48) B ) = 0 (1.62)since k B = k (cid:48) B = 0. Therefore, in the Born approximation, there is no normalcomponent to the transferred polarization in elastic scattering .The same derivation applies to the longitudinal, 3-component, and we can obtain: W µνp, L pµν = − hQ (cid:113) Q G M sin θ B , (1.63)hence, the measurement of the longitudinal component of the recoil proton polariza-tion is a measurement of the magnetic form factor G M .We can now change the notation of the transferred polarization components by1 → y, → x, → z . By applying the transformation from the Breit frame to theLab frame as defined in Appendix A, we have: σ red P x = 0 σ red P y = − ε (cid:113) τ (1 + τ ) tan θ e G E G M σ red P z = ε E + E (cid:48) m (cid:113) τ (1 + τ ) tan θ e G M , (1.64)where σ red = εG E + τ G M is the reduced cross section as defined in Eq. 1.40. Fromthis equation, we can see that the ratio of the form factors G E /G M can be extractedby a simultaneous measurement of the transverse and longitudinal components of the46olarization of the recoil proton: G E G M = − P y P z E + E (cid:48) m tan θ e . (1.65)Compared to the cross section measurement, this method offers several experimentaladvantages: • Only a single measurement is required at each Q , and this greatly reducesthe systematic error associated with the settings of the spectrometer and beamenergy changes. • Since it’s a polarization ratio measurement, it is not sensitive to the knowledgeof helicity independent factors, such as the detector efficiency, beam polarizationand the analyzing power of the polarimeter. • The measurement of the interference term G E G M provides a much more accu-rate characterization of the electric form factor. • There is no need to measure the absolute cross section, therefore, the associatedsystematic uncertainties are usually much smaller.With so many advantages, the polarization measurements cannot provide absolutemeasurements of either form factor by themselves. However, when coupled with crosssection measurements, they allow for a precise extraction of both form factors, evenin regions where the cross section is sensitive to only one of the form factors.
Proton and neutron form factors have been measured for over 50 years at differentelectron accelerator facilities around the world. Earlier cross section measurements(Rosenbluth separation) at low Q found that the form factors, except G En , were inapproximate agreement with the diploe form: G Mp µ p (cid:39) G Ep (cid:39) G Mn µ n (cid:39) G D (1.66)47 [GeV/c] Q -2 -1
10 1 D / G P E G Qattan et al.Andivahis et al.Walker et al.Simon et al.Borkowski et al. Murphy et al.Bartel et al.Hanson et al.Price et al.Berger et al. Litt et al.Janssens et al.Christy et al.
Figure 1-2: World data of G Ep from unpolarized measurements [1, 2, 3, 4, 5, 6, 7, 8,9, 10, 11, 12, 13], using the Rosenbluth method, normalized to the dipole parameter-ization.where: G D = (1 + Q . ) − . (1.67)This implies that the charge and magnetization distributions would be well describedby an exponential distribution, corresponding to the Fourier transform of the dipoleform.Figs. 1-2 and 1-3 give a summary of the world data on the separated proton formfactors for unpolarized measurement using the Rosenbluth separation method. It isclear that the extractions of G Ep from Rosenbluth separation are of limited precisionat high Q , and for G Mp , the data follow the dipole shape reasonably well up to Q ∼ but show a large deviation from this behavior at higher Q . Fig. 1-4shows the ratio µ p G Ep /G Mp from Rosenbluth separation. Earlier results generallyindicated that the form factor ratio stays around 1 but with poor quality.The polarization transfer technique was used for the first time by Milbrath etal. [45] at the MIT-Bates facility at Q values of 0.38 and 0.50 GeV . A follow-48 [GeV/c] Q -1
10 1 10 D / G P M G Qattan et al.Andivahis et al.Walker et al.Bosted et al.Sill et al. Borkowski et al.Bartel et al.Price et al.Berger et al.Litt et al. Janssens et al.Christy et al.
Figure 1-3: World data of G Mp from unpolarized measurements [1, 3, 14, 15, 5, 7, 9, 10,11, 12, 13], using the Rosenbluth method, normalized to the dipole parameterization.up measurement was performed at the MAMI facility [68], for Q = 0 . . Agreater impact of the polarization transfer measurement was made by two recentexperiments [17, 16, 18], at Jefferson Lab Hall A as shown in Fig. 1-4. The moststriking feature of the data is the sharp, practically linear decline as Q increases.In order to resolve the discrepancy between the results of the form factor ratio fromthe two experimental techniques, an ε dependent modification of the cross section isnecessary. More recently, two-photon-exchange (TPE) contribution is considered asthe main origin of this discrepancy. A number of recent theoretical studies of TPEin elastic scattering have been performed [79, 57, 80, 56, 58, 81, 82, 83, 84]. Theseindicate that TPE effects give rise to a strong angular-dependent correction to theelastic cross section, which can lead to large corrections to the extracted ratio. Infact, the results of quantitative calculations based both on hadronic intermediatestates and on generalized parton distributions, provide strong evidence that TPEeffects can account for most of the difference between the polarized and unpolarizeddata sets. Fig. 1-5 shows a comparison of the Rosenbluth data and the polarization49 [GeV Q -1
10 1 P M / G P E G μ Punjabi et al. (GEp-I)Gayou et al. (GEp-II)Crawford et al. (BLAST)Ron et al. (LEDEX)Hu et al. Gayou et al. (2001)Milbrath et al.Pospischil et al.Dieterich et al.Jones et al. (RSS)Qattan et al.Christy et al.Andivahis et al.Walker et al. Bartel et al.Berger et al.Litt et al.Janssens et al.
Figure 1-4: World data of the ratio µ p G Ep /G Mp from unpolarized measurements(black symbols) using the Rosenbluth method and from polarization experiments(colored symbols) [16, 17, 18, 19, 20, 21, 22, 23].50igure 1-5: Ratio µ p G Ep /G Mp extracted from polarization transfer (filled diamonds)and Rosenbluth method (open circles). The top (bottom) figures show Rosenbluthmethod data without (with) TPE corrections applied to the cross sections. Figuresfrom [24].data from the global analysis [24]. The TPE correction brings the high Q µ p G E /G M points from unpolarized measurements into decent agreement with the polarizationtransfer measurement data. While the world experimental data have been quite fruitful for the nucleon electro-magnetic form factors, significant theoretical progress has also been made in recentyears in understanding the nucleon electromagnetic structure from the underlyingtheory of QCD. As the theory of the strong interaction, QCD has been extremely51ell tested in the high-energy region, i.e., in the perturbative QCD (pQCD) regime.Ideally, one should be able to calculate the nucleon electromagnetic form factors di-rectly in pQCD regime to confront the data. Unfortunately, it’s impossible to solveQCD analytically in the confinement regime where the available world experimentaldata are located. Lattice QCD calculations based on first principles of QCD, on theother hand, have shown much promise in this field, and is developing rapidly. WhilepQCD give prediction for the nucleon form factors in the perturbative region, QCDeffective theories such as the chiral perturbation theory can in principle provide reli-able prediction in the very low energy region. In between the low energy region andthe pQCD regime, various QCD-inspired models and other phenomenology modelsexist. Thus, precision data in all experimentally accessible regions is crucial in test-ing these predictions. There are some recent reviews [30, 85, 38] that provide a nicesummary on these models and predictions.The newly developed Generalized Parton distributions (GPDs) [73, 74, 86, 87,88], which can be accessed through deeply virtual Compton scattering and deeplyvirtual meson production, connect the nucleon form factors and the nucleon structurefunctions probed in the deep inelastic scattering experiments. The GPDs provide newinsights into the structure of the nucleon, and possibly provide a complete map of thenucleon wave-function.The rest of the section will give a brief discussion of various theoretical approachesused to calculate the nucleon electromagnetic form factors.
Scaling and pQCD
In contract to the QED dynamics of the leptonic probe, the QCD running couplingconstant at 1-loop order is: α s ( Q ) = α s (0)1 + α s (0)16 π (11 − N f )ln( Q Λ ) , (1.68)where the string coupling constant α s → →
0. Thus, onecan solve QCD using the perturbation method in the limit of Q → ∞ . As illustrated52igure 1-6: Pertubative QCD picture for the nucleon EM form factors.in Fig. 1-7, in pQCD picture, the large momentum of the virtual photon resolves thethree leading quarks of the nucleon, and the momentum is transferred between thequarks through two successive gluon exchanges. Brodsky and Farrar [89] proposedthe following scaling law for the proton Dirac ( F ) and Pauli form factor ( F ) at largemomentum transfers based on dimensional analysis: F ∝ ( Q ) − , F ∼ F Q (1.69)This prediction is a natural consequence of hadron helicity conservation. Hadronhelicity conservation arises from the vector coupling nature of the quark-gluon inter-action, the quark helicity conservation at high energies, and assumption of zero quarkorbital angular momentum state in the nucleon. In terms of the Sach’s form factors G Ep and G Mp , the scaling result predicts: G Ep G Mp → constant at large Q . Such scalingresults were confirmed in a short-distance pQCD analysis carried out by Brodskyand Lepage [90]. Considering the proton magnetic form factor at large Q in theBreit frame, the initial proton is moving in the z direction and is struck by a highlyvirtual photon carrying a large transverse momentum, q ⊥ = Q . The form factorcorresponds to the amplitude that the composite proton absorbs the virtual photonand stays intact. Thus, the form factor becomes the product of the following threeprobability amplitudes: • the amplitude for finding the valence | qqq > state in the incoming proton. • the amplitude for this quark state to scatter from the incoming photon produc-53ng the final three-quark state with colinear momenta. • the amplitude for the final three-quark state to reform a proton.Based on this picture, Brodsky and Lepage obtained the following result within theirshort-distance pQCD analysis [90]: G M ( Q ) = 32 π α s ( Q ) Q (cid:88) n,m b nm (ln Q Λ ) − γ n − γ m [1 + O ( α s ( Q ) , m /Q )] → π C α s ( Q ) Q (ln Q Λ ) − / β ( − e (cid:107) ) , (1.70)where α s ( Q ) and Λ are the QCD strong coupling constant and scale parameterrespectively, b nm and γ m,n are QCD anomalous dimensions and constants, and e (cid:107) ( − e (cid:107) )is the mean total charge of quarks with helicity parallel (anti-parallel) to the nucleon’shelicity. For protons and neutrons, the mean total charge is given by: e p (cid:107) = 1 , − e p (cid:107) = 0 , e n (cid:107) = − e n (cid:107) = − / , (1.71)and based on the fully symmetric flavor-helicity wave function. For the proton electricform factor, one obtains similar results for the Q dependence in the Q → ∞ limit,and the short-distance pQCD analysis predicts the same scaling law as the dimen-sional analysis for the proton form factors: G Ep G Mp → constant and Q F F → constant.Recently, Belitsky, Ji and Yuan [91] performed a pQCD analysis of the nucleon’sPauli form factor F in the asymptotically large Q limit. They found that the leadingcontribution to F goes like 1 /Q , which is consistent with the scaling result obtainedby Brodsky and Farrar [89]. Fig. 1-7 shows data on the scaled proton Dirac andPauli form factor ratio Q F F from Jefferson Lab as a function of Q together withvarious predictions. While the short-distance pQCD analysis [90] predicts a constantbehavior for the Q F F in the Q → ∞ , the data are in better agreement with the QF F scaling behavior. The data could imply that the asymptotic pQCD scaling region hasnot been reached so far or that hadron helicity is not conserved in the experimentallytested regime. However, Brodsky, Hwang and Hill [92] were able to fit the Jefferson54ab data using a form consistent with pQCD analysis and hadron helicity conserva-tion by taking into account higher twist contributions. Ralston and Jain [93] arguethat the QF F scaling behavior is expected from pQCD when one takes into accountcontributions to the proton quark wave function from states with non-zero orbitalangular momentum. Miller [49] recently used light front dynamics in modeling thenucleon as a relativistic system of three bound constituent quarks surrounded by acloud of pions. While the pion cloud is important for understanding the nucleonstructure at low momentum transfer, particularly in understanding the neutron elec-tric form factor, quark effects are expected to dominate at large momentum transfers.The model was able to reproduce the observed constant behavior of QF F as a functionof Q and the QF F is predicted to be a constant up to a Q value of 20 GeV . Lattice QCD Calculations
An analytical approach in solving QCD at low momentum transfers is preventeddue to the non-perturbative nature of QCD at large distance. However, importantconceptual and technical progress has been made over the last decade in solving QCDon the lattice. In general, lattice QCD calculations are a discretized version of QCDformulated in terms of path integrals on a space-time lattice [94] with the bare quarkmasses and the coupling constant as the only parameters. The parameters commonlydefined in lattice calculations are: • lattice spacing a : separate calculation at several values of a is required in orderto extrapolate results at finite lattice spacing a to a = 0 by continuum theory. • spatial length of the box L : as lattice calculations are performed for a finitelattice size, one must define a box size large enough to fit the hadrons inside,and this requires to increase the number of sites as one decreases a . • pion mass m π : to keep finite volume effects small, one must have a box sizemuch larger than the Compton wavelength of the pion. Present lattice QCDcalculations take Lm π ≥
5. 55 Q (GeV ) Q F / F Jones et al.Gayou et al. Q F / F SU(6) + CQ ffSU(6)CQMSolitonPFSAVMD (a)(b)
Figure 1-7: The scaled proton Dirac and Pauli form factor ratio: Q F F (upper panel)and QF F (lower panel) as a function of Q in GeV . The data are from [17, 18].Shown with statistical uncertainties only. The dash-dotted curve is a new fit basedon vector meson dominance model (VMD) by Lomon [25]. The thin long dashedcurve is a point-form spectator approximation (PFSA) prediction of the Goldstoneboson exchange constituent quark model (CQM) [26]. The solid and the dotted curvesare the CQM calculations by Cardarelli and Simula [27] including SU(6) symmetrybreaking with and without constituent quark form factors, repectively. The longdashed curve is a relativistic chiral soliton model calculation [28]. The dashed curveis a relativistic CQM by Frank, Jennings, and Miller [29]. Figure from [30].56igure 1-8: Diagrams illustrating the two topologically different contributions whencalculating nucleon EM form factors in lattice QCD [31].State-of-the-art lattice calculations for nucleon structure studies use a ≤ . L ∼ m q isproportional to m π for small quark masses). As the computational costs of suchcalculations increase like m − π , it was only until very recently that pion mass valuesbelow 350 MeV [95, 96] have been reached.Also, most of the lattice results obtained so far were carried out in the so-calledquenched approximation in which the quark loop contributions, i.e. the sea quark con-tributions, are suppressed. As illustrated in Fig. 1-8, the disconnected diagram (rightpanel) involves a coupling to a q ¯ q loop, thus, it requires a numerically more intensivecalculation and is neglected in most lattice studies. The Nicosia-MIT group [32] hasperformed a high-statistics calculation of nucleon isovector EM form factors, both inthe quenched approximation and in full QCD, using two dynamical Wilson fermions.The largest Q value accessible is around Q (cid:39) . When comparing with exper-iments, the Nicosia-MIT group uses a linear fit in m π . As shown in Fig. 1-9, one cansee that both the quenched and unquenched lattice results of [32] largely overestimatethe data for F V . For F V , one observes a stronger quark mass dependence, bringingthe lattice results closer to experiment with decreasing m π .The lattice calculations at present are still severely limited by available computingpower. Hence, the uncertainties in extrapolating lattice results to the physical quark57igure 1-9: Lattice QCD results from the Nicosia-MIT group [32] for the isovectorform factors F V (upper left) and F V (lower left) as a function of Q . Both thequenched results ( N F = 0) and unquenched lattice results with two dynamical Wilsonfermions ( N F = 2) are shown for three different pion mass values. The right panelsshow the results for G VE (upper right) and G VM (lower right), divided by the standarddipole form factor, as a function of Q in the chiral limit. The filled triangles show theexperimental results for the isovector form factors extracted from the experimentaldata for the proton and neutron form factors. Figure from [32].58igure 1-10: Isovector form factor F V ( Q ) lattice data with best fit small scale ex-pansion (SSE) at m π = 292 .
99 MeV (left panel). The line in the right-hand panelshows the resulting Dirac radii, (cid:104) r (cid:105) . Also shown as the data points are the Diracradii obtained from dipole fits to the form factors at different pion masses. Figurefrom [33].mass are rather large, particularly with the naive linear extrapolation in quark mass.Thus, the challenge is to find an accurate and reliable way of extracting the latticeresults to the physical quark mass. The extrapolation methods which incorporatethe model independent constraints of chiral symmetry [97, 98], especially the leadingnon-analytic (LNA) behavior of chiral perturbation theory [99] and the heavy quarklimit [100] are exciting development in these years. Recently, the LHPC collabora-tion [33] calculated new high-statistics results using a mixed action of domain wallvalence quarks on an improved staggered sea, and performed chiral fits to both vec-tor and axial form factors. Through the comparison with the experimental data (seeFig. 1-10), they found that a combination of chiral fits and lattice data is promisingwith the current generation of lattice calculations. Vector Menson Dominance (VMD) Model
In the low Q region, several effective models have been developed to describe thenucleon properties. Most of them are semi-phenomenological, which means that theyrequire experimental data as inputs and thus have little predictive power. Usuallyeach model is valid in a limited Q range. One of the earlier attempts to describe theproton form factors is a semiphenomenological fit introduced by Iachello et al. [101].59 e (cid:255) V C (cid:74) (cid:90)(cid:85) ,
22 2
Qm m (cid:14) jV F N N (cid:255) * (cid:74) Figure 1-11: Photon-nucleon coupling in the VMD picture.It is based on a model that the scattering amplitude is written as an intrinsic formfactor of a bare nucleon multiplied by an amplitude derived from the interaction withthe virtual photon via vector meson dominance (VMD). As shown in Fig. 1-11, thenucleon form factors are expressed in terms of photon-meson coupling strengths C γV and meson-nucleon vertex form factors F jV : F is,ivj ( Q ) = (cid:88) i m i C γV i m i + Q F jV i ( Q ) , (1.72)where the sum is over vector mesons of mass m i and is and iv correspond to theisoscalar and isovector electromagnetic currents respectively. The form factors arethen given by: 2 F jp = F isj + F ivj ; 2 F jn = F isj − F ivj , (1.73)where j = 1 , p and n denote the proton and neutron respectively.Various forms of the intrinsic bare nucleon form factor have been used: dipole,monopole, eikonal. However, since this function is multiplicative, it cancels out inthe ratio G E /G M . The VMD amplitude was written in terms of parameters fit todata. Gari and Kr¨umpelmann [102] extended the basic VMD model with an addi-tional term to include quark dynamics at large Q via pQCD. Lomon updated this60 (GeV/c) Q10 -2 -1 p M / G p E G p (cid:996) Jones et al.Gayou et al.Lomon mpelmannuGari and KrIachello
Figure 1-12: The proton form factor ratio µ p G Ep /G Mp from Jefferson Lab Hall Atogether with calculations from various VMD models.model [25] by including the width of the ρ meson and additional higher mass vectormeson exchanges. The model has been further extended [103] to include the ω (cid:48) (1419)isoscalar vector meson pole in order to describe the Jefferson Lab proton form factorratio data at high Q . Fig. 1-12 shows the proton form factor ratio data as a functionof Q together with predictions from various VMD models discussed above. Whilethese models have limited predictive power due to the tunable parameters, once thehigh Q data have fixed the parameters, the approach to low Q can be constrained.However, one can obviously see that these calculations are still different in the low Q range. H¨ohler [104] fit the e − N scattering data with a dispersion ansatz, and thecontributions from ρ, ω, φ, ρ (cid:48) and ω (cid:48) were included and parameterized. The protonform factor ratio is obtained and is in good agreement with the Jefferson Lab dataup to Q ≈ as shown in Fig. 1-13.In recent years, these VMD relation approaches have been extended to includechiral perturbation theory [105, 106, 107, 108, 109]. Mergell et al. [105] obtained abest fit that gave an rms proton radius near 0.85 fm, which is close to the acceptedvalue of 0.86 fm. However, simultaneously fitting the neutron data did not yield betterresults. Hammer et al. [106] included the available data in the time-like region in thefit to determine the model parameters. The later work by Kubis [109] was restricted61 (GeV/c) Q10 -2 -1 p M / G p E G p (cid:996) Jones et al.Gayou et al.KubisHammerMergellhleroH
Figure 1-13: The proton form factor ratio µ p G Ep /G Mp from Jefferson Lab Hall Atogether with calculations from dispersion theory fits. Figure from [30]to the low Q domain of 0 − . and used the accepted proton RMS radius of0.86 fm as a constraint. The comparison between data and the different models areshown in Fig. 1-13. It is not a surprise to find that these models failed to describethe high Q data when their region of validity was claimed to be for Q ≤ . .Recently, an updated dispersion-theoretical analysis [110] describes the nucleonform factors through the inclusion of additional unphysical isovector and isoscalarpoles whose masses and widths are fit parameters to the form factors. The parametriza-tion of the spectral functions includes constraints from unitarity, pQCD, and recentmeasurements of the neutron charge radius. Belushkin et al. [34] updated the analysisby including contributions from the ρπ and K ¯ K isoscalar continua as independentinputs, in addition to the 2 π continuum. The 2 π continuum is evaluated using thelatest experimental data for the pion time-like form factor [34]. The K ¯ K continuum isobtained from an analytic continuation of KN scattering data [111]. World data wereanalyzed in both space-like and time-like regions, and the fits were in general agree-ment with the data. Fig. 1-14 shows the results for space-like momentum transferscompared to the published world data, which includes preliminary CLAS data.62 G E n G M n / ( (cid:996) n G D ) [GeV ]0.40.60.81 G E p / G D [GeV ]0.60.70.80.911.11.2 G M p / ( (cid:996) p G D ) Figure 1-14: The nucleon electromagnetic form factors for space-like momentumtransfer with the explicit pQCD continuum. The solid line gives the fit [34] togetherwith the world data (circles) including the JLab/CLAS data for G Mn (triangles),while the dashed lines indicate the error band. Figure from [34].63 onstituent Quark Models In the constituent quark model (CQM), the nucleon is described as the ground stateof a three-quark system in a confining potential. In this picture, the ground statebaryon, which is composed of the three lightest quarks ( u , d , s ), is described by SU (6) flavor wave functions and an antisymmetric color wave function. This non-relativistic model, despite its simplicity, gives a relatively good description of baryonstatic properties, such as nucleon magnetic moments and the charge and magneticradii.However, to calculate electromagnetic form factors in the high- Q (1 −
10 GeV )region, relativistic effects need to be considered. Relativistic constituent quark models(RCQM) are based on relativistic quantum mechanics as opposed to quantum fieldtheory. The goal is to formulate a mechanics where the Hamiltonian acts on a suitableHilbert space, similar to the non-relativistic case. For any relativistic quantum theory,it must respect Poincar´e invariance. There are three classes of hamiltonian quantumdynamics which satisfy Poincar´ e invariance [112]: the instant form, light-front form,and point form.In the instant form, the Einstein mass relation p µ p µ = m takes the form: p = ± (cid:113) (cid:126)p + m (1.74)which has two solutions for p , thus allowing quark-antiquark pair creation and an-nihilation in the vacuum, and it makes the theory complicated. In this case, thegenerators of the Poincar´e group are the energy of the system, whereas, the rotationsdo not contain interactions. This allows states of good angular momentum to beeasily constructed.In the point form, where the dynamical variables refer to the physical conditionson some three-dimensional surface rather than an instant, boosts and rotations aredynamical. It has the angular momenta and Lorentz boosts the same as the free case,but has complications in dealing with all four momentum components.In the light-front dynamics, the space-time variables x and t are transformed64o x ± = √ ( t ± x ) with corresponding canonical momenta p ± . This system hasthe advantages of a simple Hamiltonian without negative energies, the ability toseparate the center of mass from the relative motion of particles, and boosts whichare independent of the interactions.Several theorists have calculated the proton electric and magnetic form factors us-ing various versions of CQM. Chung and Coester [35], Aznauryan [113], and Schlumpf [114]all used RCQM to calculate nucleon form factors in the Q range of 0 − . Bothgroups were able to reproduce the available data on F p and F p between Q from 2to 4 GeV . The calculation by Schlumpf is in good agreement with the unpolarizeddata, showing a rise in the ratio µ p G Ep /G Mp , but fails to reproduce the polarizeddata from Jefferson Lab.More recent calculations have been made using the CQM in light front dynamics(LFCQM) [27, 115]. This approach uses a one-body current operator with phe-nomenological form factors for the CQMs and light-front wave functions which areeigenvectors of a mass operator . The SU (6) symmetry breaking effects with andwithout the constituent quark form factor are also included. These calculations areable to describe the trend of the high- Q polarized data.Previously, Frank, Jennings and Miller [29] considered medium modifications inreal nuclei and calculated the proton form factors in CQM. Their results for the freeproton are in reasonable agreement with the polarization data and predict a changein sign of G Ep at slightly higher Q .A relativistic quark model (RQM) calculated by Li [116] requires symmetry in thecenter-of-mass frame. By adding additional terms to the baryon wave function, whichare generated by the SU (6) symmetry requirements, it represents the inclusion of thesea quarks. The result of this calculation originally preceded the publication of thepolarized data from Jefferson Lab, and the model has good agreement with the data.A variant of the CQM model is the diquark model of Kroll et al. . Two of the con-stituent quarks are tightly-bound into a spin-0 or 1 diquark with a phenomenologicalform factor which allows the diquark to behave as free quarks at high Q . When anelectron scatters from the spin-1 diquark, helicity-flip amplitudes are generated. Ma,65igure 1-15: Comparison of various relativistic CQM calculations with the data for µ p G Ep /G Mp . Dotted curve: front form calculation of Chung and Coester [35] withpoint-like constituent quarks; thick solid curve: front form calculation of Frank etal. [29]; dashed curve: point form calculation of Boffi et al. [36] in the Goldstoneboson exchange model with point-like constituent quarks; thin solid curve: covariantspectator model of Gross and Agbakpe [37]. Figure from [38].66igure 1-16: Result for the proton form factor ratio µ p G pE /G pM computed with fourdifferent diquark radii, r . Figure from [39].Qing, and Schmidet [117, 118] performed calculations of a quark spectato-diquarkmodel using the light-cone formalism. They also describe the available data well.Recently, Wagenbrunn Boffi et al. [26] calculated the neutron and proton electromag-netic form factors for the first time using the Goldstone-boson-exchange constituentquark model. The calculations are performed in a covariant frame work using thepoint-form approach to relativistic quantum mechanics, and is in good agreementwith the form factors from polarized data. The comparison between various CQMmodels and the data are shown in Fig. 1-15.Recently, Cl¨et et al. [39] calculated the form factors contributed by a dressed-quark core. It is defined by the solution of a Pioncar´a covariant Faddeev equation, inwhich dressed-quarks provide the elementary degree of freedom and the correlationsbetween them are expressed via diquarks. The nucleon-photon vertex only has thediquark charge radius as the free parameter. The calculation of the proton Sach’sform factor ratio through this model is compared with the experimental data as shownin Fig. 1-16.Irrespective of the diquark radius, however, the proton’s electric form factor pos-67esses a zero and the magnetic form factor is positive definite. For Q < , theresult of the calculation lies below experiment, which can likely be attributed to theomission of pseudoscalar-meson-cloud contributions. Pion Cloud Models
As the lightest hadrons, pions dominate the long-distance behavior of hadron wavefunctions and yield characteristic signatures in the low momentum transfer behaviorof hadronic form factors. Therefore, a natural way to qualitatively improve the CQMsis to include the pionic degrees of freedom [119].In the early MIT Bag Model, the nucleon is described as three quark fields confinedin a potential that maintains them within a finite sphere of radius R . The introductionof the pion cloud [120, 121] improves the static properties of the nucleon by restoringchiral symmetry and also provides a convenient connection to πN and N N scattering.To extend the calculation to larger Q , Miller performed a light-front cloudy bagmodel calculation [49], which give a relatively good global account of the data bothat low and larger Q . Chiral Perturbation Theory
At low momentum region, the nucleon form factors can also be studied within chiralperturbation theory ( χ PT) expansions based on chiral Lagrangians with pion andnucleon fields. In χ PT, the short-distance physics is parameterized in terms of low-energy-constants (LECs), which ideally can be determined by matching to QCD; butin practice, they are fit to experimental data or estimated using resonance satura-tion. In the calculation of the nucleon form factors, the LECs can be obtained fromthe nucleon static properties, such as the charge radii and the anomalous magneticmoments. Once these LECs are determined, the Q -dependence of the form factorscan be predicted.The calculation of the nucleon EM form factors involves a simultaneous expan-sion in soft scales: Q and m π , which are understood to be small relative to thechiral symmetry breaking scale Λ χSB ∼ χ PT of [40] (IRscheme) and [41] (EOMS scheme). The results of [40] including vector mesons areshown to third (dashed curves) and fourth (solid curves) orders. The results of [41]to fourth order are displayed both without vector mesons (dotted curves) and whenincluding vector mesons (dashed-dotted curves). Figure from [38].developed in the literature. Early calculations of the nucleon from factors in thesmall scale expansion (SSE) [122] have been performed in [123]. Since such an ap-proach is based on a heavy baryon expansion it is limited to Q values much below0.2 GeV . Subsequently, several calculations of the nucleon form factors have beenperformed in manifestly Lorentz invariant χ PT. Kubis and Meissner [40] performeda calculation in relativistic baryon χ PT, employing the infrared regularization (IR)scheme. Schindler [41] also performed a calculation employing the extended on-mass-shell (EOMS) renormalization scheme. Both groups found that when only pion andnucleon degrees of freedom are included, one cannot well describe the data over asignificant range of Q . On the other hand, it was found that the vector meson polediagrams play an important role, which also confirms the findings of VMD modelsand dispersion theory mentioned earlier. The corresponding results in both IR andEOMS schemes are shown in Fig. 1-17. 69 ucleon Charge and Magnetization Densities Although models of nucleon structure can calculate the form factor directly, it isdesirable to relate form factors to spatial densities because our intuition tends to begrounded more firmly in space than momentum transfer. The interpretation of theform factors G E and G M has the simplest interpretation in the nucleon Breit framewhere the energy transfer vanishes, and the charge and magnetization densities canbe written as: ρ NRch ( r ) = 2 π (cid:90) ∞ dQQ j ( Qr ) G E Q (1.75) µρ NRm ( r ) = 2 π (cid:90) ∞ dQQ j ( Qr ) G M Q . (1.76)However, this naive inversion is only valid when it ignores the variation of the Breitframe with Q , also known as the non-relativistic (NR) limit. For the nucleon, whenthe form factors are measured for Q values much larger than M , one needs to takethe effect of relativity into account. Kelly [42] provided a relativistic prescriptionto relate the Sachs form factors to the nucleon charge and magnetization densities,which accounts for the Lorentz contraction of the densities in the Breit frame relativeto the rest frame.If we start from the spherical charge ρ ch ( r ) and magnetization densities ρ m ( r ) inthe nucleon rest frame which are normalized according to the static properties: (cid:90) ∞ drr ρ ch ( r ) = Z (1.77) (cid:90) ∞ drr ρ m ( r ) = 1 , (1.78)the Fourier-Bessel transforms of the intrinsic densities are defined as:˜ ρ ( k ) = (cid:90) ∞ drr j ( kr ) ρ ( r ) , (1.79)where k is the spatial frequency (or wave number), and ˜ ρ ( k ) is described as theintrinsic form factor. If one can find the connection between the Sachs form factor70nd the intrinsic form factors, the intrinsic density is obtained simply by invertingthe Fourier transform: ρ ( r ) = 2 π (cid:90) ∞ dkk j ( kr ) ˜ ρ ( k ) . (1.80)In the non-relativistic limit, k → Q and ˜ ρ ( Q ) → G ( Q ), where G ( Q ) is the ap-propriate Sachs form factor. However, this naive inversion causes unphysical cuspsat the origin for the common dipole and Galster parameterizations. Licht and Pag-namenta [124] attributed these failures to the replacement of the intrinsic spatialfrequency k with the momentum transfer Q and demonstrated that the density soft-ens, when a Lorentz boost from the the Breit frame with momentum q B = Q to therest frame is applied. Consequently, the spacial frequency is replaced by: k = Q τ , (1.81)where τ = Q / M N , and a measurement with Breit-frame momentum transfer q B = Q probes a reduced spatial frequency k in the rest frame.Unfortunately, unique relativistic relationships between the Sachs form factorsmeasured by finite Q and the static charge and magnetization densities in the restframe do not exist. The fundamental problem is that electron scattering measurestransition matrix elements between states of a composite system that have differentmomenta, and the transition densities between such states are different from thestatic densities in the rest frame. Several models have employed similar relativisticprescriptions, which can be written in the following form:˜ ρ ch ( k ) = G E ( Q )(1 + τ ) λ E (1.82) µ ˜ ρ m ( k ) = G M ( Q )(1 + τ ) λ M (1.83)where k and Q are related as in Eq. 1.81 and λ is a model-dependent constant. Onecan see that the accessible spatial frequency is limited to k ≤ M N determined bythe nucleon Compton wavelength. To account for an asymptotic 1 /Q form factorbehavior, Kelly followed the choice λ E = λ M = 2, and he employed linear expansions71igure 1-18: Comparison between charge and magnetization densities for the protonand neutron. Figure from [38]in complete sets of basis functions to minimize the model dependence. Fig. 1-18 showsthe charge and magnetization densities for neutron and proton from his analysis asdetermined from fits of the world data.The low Q behavior of the form factors also play an important role in definingthe transition radii obtained from integral moments of the underlying density. Theintegral moments are defined by: M α = (cid:90) ∞ drr α ρ ( r ) . (1.84)While the lowest nonvanishing moment is free of discrete ambiguities, the highermoments depend upon λ . For example, the proton radius retains a small dependenceupon λ , < r > λ,p = − dG (0) dQ | Q → − λ m p , (1.85)Recently, Miller [125] proposed a model independent analysis in the infinite-momentum-frame (IMF). In this frame, the charge density ρ ( b ) in the transverse72lane is in fact a two-dimensional Fourier transform of the F form factor: ρ ( b ) ≡ (cid:88) q e q (cid:90) dxq ( x, b ) = (cid:90) d q (2 π ) F ( Q = q ) e iq · b . (1.86)In contrast with earlier expectations, from this analysis, the neutron charge densityis negative at the center, and the proton’s central d quark charge density is largerthan that of the u quark by about 30%. Global Fits
As the most basic quantities, nucleon electromagnetic form factors are needed forvarious calculations in nuclear physics. Hence, a simple parametrization which accu-rately represents the data over a wide range of Q and has reasonable behavior forboth Q → Q → ∞ would be convenient.For reasonable behavior at low Q , the power-series representation should involveonly even powers of Q . At high Q , dimensional scaling rules require G ∝ Q − .However, at present the most common parameterizations violate one or both of theseconditions. Often the reciprocal of a polynomial in Q [126, 127, 128] is used, butthis method has difficulty in determining the RMS radius due to the unphysical oddpowers of Q . Recently, Kelly [42] proposed a much simpler parametrization whichtakes the form: G ( Q ) ∝ (cid:80) nk =0 a k τ k (cid:80) n +2 k =1 b k τ k , (1.87)where both numerator and denominator are polynomials in τ = Q / m p and thedegree of the denominator is larger than that of the numerator to ensure the ∝ Q − for large Q . Good fits by this form require only four parameters each for G Ep , G Mp and G Mn , and only two for G En . Fig. 1-19 shows the results of the parametrization.Bradford et al. [43] did another parametrization that uses the same functionalbut with two additional constraints. The first constraint comes from local duality,and a second constraint is based on QCD sum rules including a further applicationof duality. The constraints were implemented by scaling the high Q data of G Mp and then adding these scaled points to the data sets for G En and G Mn during the73igure 1-19: Kelly’s fits [42] to nucleon electromagnetic form factors. The error bandswere of the fits. Figure from [42].fits. Fig. 1-20 shows the new parameterizations. Arrington and Sick [48] performeda fit of the world data at very low momentum transfer by a Continued Fraction (CF)expansion: G CF ( Q ) = 11 + b Q b Q ··· . (1.88)This expansion is suitable for the lower momentum transfers, and extends up to Q = √ Q ≈ . /c . The analysis included the effect of Coulomb distortionand the Two-Photon-Exchange (TPE) exchange beyond Coulomb distortion, whichincludes only the exchange of an additional soft photon. Later on, Arrington et al. [24]performed a global analysis of the world data. The analysis combined the correctedRosenbluth cross section and polarized data, and this is the first extraction of G E and G M including the explicit TPE correction. Fig. 1-21 shows this global analysiscompared with the world data.In 2003, Friedrich and Walcher performed various phenomenological fits [44] atlow Q with the “bump and dip” structure on the base of the smooth 2-dipole form.Shown in Fig. 1-22 are the difference between the measured nucleon form factors at74igure 1-20: The parametrization of Bradford et al. compared with Kelly’s, togetherwith world data. Figure is from [43]. 75igure 1-21: Extracted values of G E and G M from the global analysis. The opencircles are the results of the combined analysis of the cross section data and polar-ization measurements. The solid lines are the fits to TPE-corrected cross section andpolarization data. The dotted curves show the results of taking G E and G M from afit to the TPE-uncorrected reduced cross section. Figure from [24].76igure 1-22: The difference between the measure nucleon form factors and the 2-components phenomenological fit of [44] for all four form factors.that time and the smooth part of their phenomenological ansatz. It is found that allfour form factors exhibit similar structure at low momentum transfer region, whichthey interpreted as an effect of the pion cloud around a bare nucleon. They found avery long-range contribution to the charge distribution in the Breit frame extendingout to about 2 fm which could arise from the pion cloud. With the hint of theexistence of the “bump and dip” structure, their analysis reinvigorated the interestin investigating the form factor behavior in the low Q region. Q Q , it is generally accepted that the proton form factor ratio µ p G Ep /G Mp decreases smoothly with increasing Q . In the low Q ( < ) region, the worldexisting data appear to be less conclusive about where this deviation starts. On theother hand, from the fits performed by Friedrich and Walcher, the data somehow77ndicate the existence of structure.Fig. 1-23 presents all world polarization data for Q < and Fig. 1-24presents only the high precision ones ( σ tot < . Later on, BLAST(MIT-Bates) performed the first measurement of µ p G Ep /G Mp from (cid:126)H ( (cid:126)e, e (cid:48) p ) in the Q region between 0.15 and 0.65 GeV [19]. The extracted ratio from these data isconsistent with unity. In 2006, Jefferson Lab [20] performed another recoil polariza-tion measurement focusing at low Q , which overlaps the region covered by BLAST.While both results gave similar behavior over the whole range, a strong deviation fromunity is observed at Q ∼ . in LEDEX. However, due to limited statisticsduring the experiment and the background issue [70], such a deviation is not conclu-sive at that moment. Interestingly, both data sets are inconsistent with Friedrich andWalcher fit.The experiment reported in this thesis aimed to provide a high precision survey ofthe proton form factor ratio ( σ stat. < Q = 0 . − . . Withthe proposed accuracy, we will be able to either confirm or refute the existence of anydeviation from unity and local “structure” in this low momentum transfer region.In addition, the range that we cover is particularly important for tests of effectivefield theory predictions, future precision results from lattice QCD and also helps toquantify the pion cloud effect in nucleon structure. Besides, improved form factormeasurements also have implications in the extraction of other physics quantities,such as the ultra-high precision test of QED from the hydrogen hyperfine splittingand the strange quark content of the nucleon through parity violation experiments.78 [GeV Q0 0.2 0.4 0.6 0.8 1 P M / G P E G μ Pospischil et al.Milbrath et al.Gayou et al.Hu et al.Dieterich et al.Punjabi et al. (GEp-I)Crawford et al. (BLAST)Ron et al. (LEDEX)
Figure 1-23: The world data from polarization measurements. Data plotted arefrom [23, 45, 46, 21, 22, 16, 19, 20] 79 [GeV Q0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 P M / G P E G μ Punjabi et al. (GEp-I)Crawford et al. (BLAST)Ron et al. (LEDEX)
Kelly FitAMTArrington & Sick FitFriedrich & Walcher FitMiller LFCBMBoffi et al. PFCCQMBelushkin, Hammer & Meiner VMDFaessler et al. LCQM
Figure 1-24: Recent world high precision polarization data [16, 19, 20] compared toseveral fits [47, 24, 48, 44] and parameterizations [49, 36, 50, 51].80 hapter 2Experimental Setup
Experiment E08-007 was completed in the summer of 2008 at Thomas Jefferson Na-tional Accelerator Facility in Hall A. The polarized electron beam was produced andaccelerated by the Continuous Electron Beam Accelerator Facility (CEBAF). Witha 1.19 GeV beam on a liquid hydrogen target, the elasticly scattered electrons weredetected by the BigBite spectrometer in coincidence with the recoil proton detectedby the left High Resolution Spectrometer (HRS). The transferred proton polarizationwas measured in the focal plane polarimeter (FPP). The proton form factor ratio weremeasured at 8 kinematics, which are listed in Table 2.1. This chapter will describethe experimental setup and instrumentation used for this experiment.Kine. Q [GeV ] θ e [deg] θ p [deg] ε K1 0.35 31.3 57.5 0.85K2 0.30 28.5 60.0 0.88K3 0.45 36.7 53.0 0.80K4 0.40 34.2 55.0 0.82K5 0.55 41.9 49.0 0.75K6 0.50 39.2 51.0 0.78K7 0.60 44.6 47.0 0.72K8 0.70 49.8 50.0 0.66Table 2.1: E08-007 kinematics.81 orth Linac South LinacHeliumRefrigeratorEnd Stations
A B C
Injector ExtractionElements45 MeVAccelerator RecirculationArcs (magnets)
Figure 2-1: Layout of the CEBAF facility. The electron beam is produced at theinjector and further accelerated in each of two superconduction linacs. The beam canbe extracted simultaneously to each of the three experimental halls.
CEBAF (see Fig. 2-1) accelerates electrons up to 5.7 GeV by recirculating the beamup to five times through two superconducting linacs. Each linac contains 20 cryo-modules with a design accelerating gradient of 5 MeV/m, producing a nominal energygain of 400 MeV per pass, and this gain can be tuned up to about 500 MeV per passif required by the experimental halls. Ongoing insitu processing has already resultedin an average gradient in excess of 7 MeV/m, which has made it possible to accelerateup to about 5.7 GeV.Electrons can be injected into the accelerator from either a thermionic or a polar-ized gun. With the polarized gun a strained GaAs cathode is illuminated by a 1497MHz gain-switched diode laser, operated at 780 nm. The absorption of a right orleft circularly polarized laser light preferentially produces electrons with a spin down82r up respectively in the conduction band, thus longitudinally polarizing the beam,up to 85%. The laser light is circularly polarized using a Pockels cell. The electronbeam polarization is measured at the injector with a 5 MeV Mott polarimeter [129]and the polarization vector can be oriented with a Wien filter [130]. The sign of thebeam helicity is flipped pseudo-randomly at a rate of 30 Hz by switching the circularpolarization of the laser, which is achieved by changing the voltage of the Pockelscell. The current sent to the three Halls A, B and C can be controlled independently.The design maximum current is 200 µ A in CW (continuous wave) mode, which canbe split arbitrarily between three interleaved 499 MHz bunch trains. One such bunchtrain can be peeled off after each linac pass to any one of the Halls using RF sep-arators and septa. CEBAF can deliver 100 µ A beam to one or both of the Hall Aand Hall C, while maintaining high polarization low current (1 nA) to Hall B. Hall Chas been operational since November 1995, Hall A since May 1997 and Hall B sinceDecember 1997.For this experiment (E08-007), a 1.19 GeV CW beam was delivered into Hall A,with current 4 − µ A for production data taking for various kinematics. The averagebeam polarization during the experiment was ∼ All three experimental halls have their bulk volumes underground with a shield ofconcrete and a thick layer of earth. Hall A is the largest one with a diameter of 53 m.The layout of Hall A during E08-007 is shown in Fig. 2-2. The key elements includethe beamline, cryogenic target in the scattering chamber, the left High ResolutionSpectrometers (LHRS) and the BigBite spectrometer.83
Beam Dump
BPMMollerARC Compton BCMRaster eP 6 cm LH2 Target
Figure 2-2: Hall A floor plan during E08-007.
The beam energy during the experiment was monitored by “Tiefenbach” energy [131].The value is calculated by the current values of Hall A arc B · dl and Hall A arc beamposition monitors (BPM). This number is continuously recorded in the data streamand is calibrated against the Arc energy of the 9th dipole regularly. The accuracyfrom this measurement is about 0.5 MeV. For this experiment, the results are notsensitive to the absolute beam energy; therefore, there were no invasive measurementsperformed during the experiment. The beam current is measured by the beam current monitors (BCMs) [131] in HallA, which provides a stable, low-noise, no-invasive measurement. It consists of an84
HzDownConverter
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HzDownConverter 50 kHzBWFilter50 kHzBWFilter 1X1X3X VToFVToFVToFVToFVToFVToFRMS-DCRMS-DC
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DownstreamBCMUpstreamBCMUNSER
Sampled DataIntegratedDataUnser Measurement DataStreamScaler toData StreamDataStream
Beam
Figure 2-3: Schematic of beam current monitors.Unser monitor, two RF cavities, associated electronics and a data-acquisition system.The cavities and the Unser monitor are enclosed in a temperature-stabilized magneticshielding box which is located 25 m upstream of the target.Fig. 2-3 shows the schematics of BCMs. The Unser monitor is a Parametric Cur-rent Transformer which provides an absolute measurement [132]. The monitor iscalibrated by passing a known current through a wire inside the beam pipe and hasa nominal output of 4 mV/ µ A. As the Unser monitor’s output signal drifts signifi-cantly on a time scale of several minutes, it is not suitable for continuous monitoring.However, the drift can be measured during the calibration runs and the net measuredvalue is used to calibrate the two RF BCMs. The two resonant RF cavity monitorson either side of the Unser monitor are stainless steel cylindrical high-Q ( ∼ − µ A to 200 µ A.A set of amplifiers has been introduced with gain factors of 1, 3, and 10 in order toallow for lower currents at the expense of saturation at high currents. Hence, there isa set of three signals coming from each RF BCM. These six signals are fed to scalerinputs of each spectrometer, providing redundant beam charge information.The beam charge can be derived from BCM scaler reading by Q BCM × A,H = N BCM × A,H clock H − offset × A,H constant × A clock H , (2.1)where A = 1, 3 or 10 is the gain factor, H =plus, minus or 0 (ungated) is the beamhelicity state, and clock H is the total clock time of corresponding helicity gate. TheBCM calibration is typically performed every 2 − ± .
5% down to a current of 1 µ A. The position and direction of the beam at the target location is determined by twoBeam Position Monitors (BPMA and BMPB) which are located at 7.345 m and 2.214m upstream of the Hall A center respectively.The standard difference-over-sum technique is used to determine the relative po-sition of the beam to within 100 µ m for currents above 1 µ A [131, 133]. The absoluteposition of the beam can be determined from the BPMs by calibrating them withrespect to wire scanners (superharps) which are located adjacent to each BPM. The86ire scanners are regularly surveyed with respect to the Hall A coordinates; the re-sults are reproducible at the level of 200 µ m. The position information from theBPMs are recorded in the raw data stream by two ways: average value and event-by-event. The real beam position and direction at the target can be reconstructed usingthe BPM positions calculated from 8 BPM antennas’ readout (2 × x, y target = x, y BPMa · ∆ z BPMb − x, y BPMb · ∆ z BPMa z BPMb − z BPMa (2.2) (cid:126)x beam = (cid:126)x BPMb − (cid:126)x BPMa | (cid:126)x BPMb − (cid:126)x BPMa | , (2.3)where ∆ z = z BPM − z target .For liquid or gas targets, high current beam ( > µ A) may damage the target cellby overheating it. To prevent this, the beam is rastered by two pairs of horizontal (X)and vertical (Y) air-core dipoles located 23 m upstream of the target, and the size ofrastered beam is typically several millimeters. The raster can be used in two modes,sinusoidal or amplitude modulated. In the sinusoidal mode both the X and Y magnetpairs are driven by pure sine waves with relative 90 ◦ phase and frequencies ∼ ◦ phase between X andY, producing a circular pattern. The radius of this pattern is changed by amplitudemodulation at 1 kHz.During the experiment, a new triangular raster was used, which copied the HallC design [134]. The new raster provides a major improvement over the sinusoidalraster by reducing dwell time at the peaks. A uniform density distribution of beamon the target is achieved by moving the beam position with a time-varying dipolemagnetic field with a triangular waveform. The raster contains two dipole magnets,one vertical and one horizontal, which are located 23 m upstream from the target.In the electronics design, an “H-bridge” is used that allows one pair of switchesto open and another pair to close simultaneously and rapidly at 25 kHz. the current87 eam Position Y [m]0 0.001 0.002 0.003 0.004 0.005 B ea m P o s i t i on X [ m ] Target
Figure 2-4: Beam spot at target.is drawn from HV supplies and rises according to I ( t ) = (cid:15)R (1 − e − t/τ ) (2.4)where τ = L/R is the time constant with resistance R and inductance L of thecontrolling electronics. The time and applied voltage are t and (cid:15) , respectively. Fig. 2-4 is a sample beam spot at target with raster on. In this experiment, a 1.5 mm × There are three methods to measure the electron beam polarization: • Mott method. • Møller polarimetry. • Compton polarimetry.The Mott measurement [129] is performed at the polarized electron source, and theother two polarimetries are performed in the experimental Hall. During this exper-iment, since the beam polarization is canceled in the result, continuous monitoring88f the polarization is not required, so only the Møller measurement was performedduring this experiment.
Møller Polarimetry
The Møller polarimetry [135] measures the process of Møller scattering of the polar-ized beam electrons off polarized atomic electrons in a magnetized foil (cid:126)e − + (cid:126)e − → e − + e + . The cross section of the Møller scattering depends on the beam and targetpolarization P b and P t as σ ∝ (1 + (cid:88) i = X,Y,Z ( A ii P b,i P t,i )) , (2.5)where i = X, Y, Z defines the projections of the polarizations. A ii is the analyzingpower, which depends on the scattering angle in the center of mass (CM) frame θ CM .Assuming that the beam direction is along the Z -axis and that the scattering happensin the ZX plane, we have A ZZ = − sin θ CM · (7 + cos θ CM )(3 + cos θ CM ) A XX = − A Y Y = − sin θ CM (3 + cos θ CM ) . (2.6)The analyzing power does not depend on the beam energy. At θ CM = 90 ◦ , theanalyzing power has its maximum A ZZ,max = 7/9. The Møller polarimeter of HallA detects pairs of scattered electrons in a range of 75 ◦ < θ CM < ◦ . The averageanalyzing power is about < A ZZ > = 0 .
76. A transverse polarization also produces anasymmetry, though the analyzing power is lower: A XX,max = A ZZ,max /
7. The mainpurpose of the polarimeter is to measure the longitudinal component of the beampolarization.The polarized electron target consists of a thin magnetically saturated ferromag-netic foil. An average electron polarization of about 8% [135] can be obtained. Thefoil is magnetized along its plane and can be tilted at angles from 20 ◦ to 160 ◦ to thebeam. 89 o r i z on t a l L e f t R i gh t TargetHelmholtz coils Quad1 Quad2 Quad3 Dipole Detector
Top View V e r ti ca l up Side View
Target foil
Figure 2-5: Layout of the Møller polarimeter.The scattered electrons are detected by a magnetic spectrometer (see Fig. 2-5).The spectrometer consists of a sequence of three quadrupole magnets and a dipolemagnet. The detector consists of scintillators and lead-glass calorimeter modules, andsplit into two arms in order to detect the two scattered electrons in coincidence. Thebeam longitudinal polarization is measured as: P b,Z = N + − N − N + + N − · P t · cos θ t · < A ZZ > , (2.7)where N + and N − are the measured counting rates with two opposite mutual ori-entation of the beam and target polarization. While < A ZZ > is obtained usingMonte-Carlo calculation of the Møller spectrometer acceptance, P t is derived fromspecial magnetization measurements of the foil samples, θ t is measured using a scalewhich is engraved on the target holder and seen with an TV camera, and also usingthe counting rates measured at different target angles.The Møller polarimeter can be used at beam energies from 0.8 to 6 GeV. Themeasurement is invasive, since the beam needs to be tuned through the Møller chicane,and the measurement is performed with low current ( ∼ . µ A). One measurement90ate Wien P b ± ∆ P b ( stat. )2008/05/16 − . ◦ − . ± . − . ◦ − . ± . . − . ◦ and − . ◦ respectively. Theresults are reported in Table 2.2. The experiment were mostly running with the lattersetting. For experiment E08-007, the “G0 helicity scheme” [136] was used. The schematicsis shown in Fig. 2-6. There are three relevant signals: macro-pulse trigger (MPS),quartet trigger (QRT), and Helicity. The characteristics of this scheme are: • MPS is the master pulse at 30 Hz which is used as a gate to define periods whenthe helicity is valid. • The helicity sequence has a quartet structure (+ − − + or − + + − ). The helicityof the first MPS gate is chosen pseudorandomly. • Quartet trigger (QRT) denotes when a new random sequence of four helicitystates has begun.There is a blank-off period of about 0.5 µ s for each 33.3 ms gate period. This blank-offis the time during which the Pockel cell at the source is changing and settling. Thequartet sequence provides for exact cancelation of linear drifts over the sequence’stimescale. All three bits (helicity, QRT, gate) are read in the datastream for eachevent, and the copies are sent to scalers which have input registers. The delay of thehelicity reporting breaks any correlations with the helicity of the event by suppressingcrosstalk. For this experiment, we used the configuration with no delay.91 .5 ms33.33 ms - + + - QRTMPSHelicity
Figure 2-6: Beam helicity sequence used during experiment E08-007.
The scattering vacuum chamber [137] consists of several rings, and is supported ona 607 mm diameter central pivot post. The stainless steel base ring has one vacuumpump-out port and other ports for viewing and electrical feed-throughs. The middlering is made out of aluminum and located at beam height with 152 mm verticalcutouts on each side of the beam over the full angular range (12 . ◦ ≤ θ ≤ . ◦ ).The cutouts are covered with a pair of flanges with thin aluminum foils. It also hasentrance and exit beam ports. The upper ring is used to house the cryotarget. Thechamber vacuum is maintained at 10 − Torr to insulate the target and to reduce theeffect of multiple scattering.
A 6 cm liquid hydrogen cryogenic target was used for this experiment. The targetsystem was mounted inside the scattering chamber along with sub-systems for cooling,gas handling, temperature and pressure monitoring, target control and motion, andan attached calibration and solid target ladder (see Fig. 2-7).92 oop1He 20cmLoop2D 20cm Loop3H 6cm H 15cm Optics 10cmOptics 24cmEmptyBe thinBe thickCarbonBeOLithumBeam
Figure 2-7: Target ladder.The target system had three independent target loops: a liquid hydrogen (LH )loop, a liquid deuterium (LD ) loop and a gaseous helium loop. The LH loop hadtwo aluminum cylindrical target cells, 15 cm and 6 cm length, mounted on the verticalstack which could be moved from one position to another by remote control. Boththe LD and gaseous helium loops had only single 20 cm aluminum cell. All the liquidtarget cells had diameter φ = 63 . µ m thick, widthentrance and exit windows approximately 71 and 102 µ m thick, respectively. Theupstream window consisted of a thick ring holder with an inner diameter of 19 mm,large enough for the beam to pass through.Below the cryogenic targets were two sets of carbon foil optics targets constructedof two thin pieces of carbon foils spaced by 10 or 24 cm. A solid target, attached atthe bottom, had six target positions: an empty target, two Be targets with differentthickness, a single carbon foil (can also be used for optics data taking), a BeO foil(typically used for direct beam observation), and a lithium target.The LH (LD ) target were cooled at 19 K (22 K) with pressure of 0.17 MPa (0.1593Pa), about 3 K below their boiling temperature. Under these conditions, they havea density of 0.0723 g/cm and 0.167 g/cm . The nominal operating condition for He( He) was 6.3 K at 1.4 MPa (1.1 MPa). The coolant (helium) was supplied bythe End Station Refrigerator (ESR). The helium from ESR is available at 15 K witha maximum cooling power of 1 kW, and at 4.5 K with a lower maximum coolingcapacity near 600 W. Typically 15 K coolant is used for liquid cells while 4.5 K forgaseous cells. At the full 1 kW load of 15 K coolant, up to 130 µ A beam currentmay be incident on the liquid target with temperature slightly over 20 K. In thisconfiguration the beam heating alone deposits 700 W in the target where the rest ofpower arises from circuiting fans and small heaters required to stabilize the target’stemperature. The coolant supply is controlled with Joule-Thompson (JT) valves,which can be adjusted either remotely or locally.
One of the key pieces of equipments for this experiment is the left High ResolutionSpectrometers (HRS), which was used to detect the recoil proton. A schematic viewof the HRS is shown in Fig. 2-8, and the main design characteristics are provided inTable 2.3. The vertically bending design includes a pair of superconducting cos(2 θ )quadrupoles followed by a 6.6 m long dipole magnet with focusing entrance and exitpolefaces, including additional focusing from a field gradient, n, in the dipole. Follow-ing the dipole is a third superconducting cos(2 θ ) quadrupole. The first quadrupoleQ1 is convergent in the dispersive (vertical) plane. Q2 and Q3 are identical and bothprovide transverse focusing. In this configuration, the spectrometer can provide amomentum resolution better than 2 × − with a 9% momentum acceptance. The detector packages of the spectrometer were designed to provide various infor-mation in the characterization of charged particles passing through the spectrometer.These include: a trigger to readout the data-acquisition electronics, tracking informa-94 . m . m . m o Septum Q1 Q2 Dipole Q31st VDCPlane DetectorHutCentral Trajectory
HRS Design Layout
Figure 2-8: Schematic of Hall A High Resolution Spectrometer and the detector hut.Configuration QQDQ vertical bendBending angle 45 ◦ Optical lengh 24.2 mMomentum range 0.3-4.0 GeV /c Momentum acceptance ± . δp/p )Momentum resolution 2 × − Dispersion at the focus (D) 12.4 mRadial linear magnification (M) -2.5
D/M ±
30 mradVertical angular acceptance ±
60 mradHorizontal resolution 1.5 mradVertical resolution 4.0 mradSolid angle at δp/p = 0 , y = 0 6 msrTransverse length acceptance ± (cid:20) (cid:54)(cid:21) (cid:73)(cid:85)(cid:82)(cid:81)(cid:87)(cid:3)(cid:86)(cid:87)(cid:85)(cid:68)(cid:90)(cid:3)(cid:70)(cid:75)(cid:68)(cid:80)(cid:69)(cid:72)(cid:85)(cid:86)(cid:74)(cid:68)(cid:86)(cid:3)(cid:70)(cid:72)(cid:85)(cid:72)(cid:81)(cid:78)(cid:82)(cid:89) (cid:70)(cid:68)(cid:85)(cid:69)(cid:82)(cid:81)(cid:3)(cid:71)(cid:82)(cid:82)(cid:85)(cid:85)(cid:72)(cid:68)(cid:85)(cid:3)(cid:86)(cid:87)(cid:85)(cid:68)(cid:90)(cid:3)(cid:70)(cid:75)(cid:68)(cid:80)(cid:69)(cid:72)(cid:85)(cid:86) (cid:83)(cid:85)(cid:72)(cid:86)(cid:75)(cid:82)(cid:90)(cid:72)(cid:85)(cid:57)(cid:39)(cid:38) Figure 2-9: Left HRS detector stack during E08007.tion (position and direction), coincidence determination, and particle identification.The configuration of the detectors on the left spectrometer for this experiment isshown in Fig. 2-9. The detector package includes: • a set of two vertical drift chambers (VDCs) which provide tracking information. • two scintillator planes which provide basic triggers. • a CO gas Cerenkov detector for particle identification. • the focal plane polarimeter (FPP) measure the recoil proton polarization. • a pair of lead glass pion rejectors for PID.For this experiment, the key instruments are the scintillator planes, VDCs and theFPP. 96 o Upper VDC Lower VDC
V2U2V1U1 d =26mm uv2 d =26mm uv1 d =335mm v Norminal Particle Trajectory Norminal Particle Trajectory45 o Side ViewTop View
Figure 2-10: Schematic diagram and side view of VDCs.
The vertical drift chamber (VDCs) [138, 139], provides a precise measurement of theincident position and angle of the charged particles at the spectrometer focal plane .The tracking information from the VDC measurement is combined with the knowledgeof the spectrometer optics to reconstruct the position, angle and momentum of theparticles in the target coordinate system.The pair of VDC chambers are laid horizontally. The top VDC is placed 33.5 cmabove the bottom VDC and shifted by another 33.5 cm in the dispersive directionto account for the 45 ◦ central trajectory (see Fig. 2-10). Each VDC consists of twoplanes of wires in a standard UV configuration: the wires of each successive plane areoriented at 90 ◦ to one another. There are a total of 368 sense wires in each plane,spaced 4.24 mm apart.During operation, the VDC chambers have their cathode plane at about − /
38% argon-ethane(C H ) mixture, with a flow rate of 10 liter/hour [131]. When a charged particletravels through the chamber, it ionizes the gas inside the chamber and leaves a track The focal plane is a plane associated with the lower VDC of each spectrometer. A detaileddescription and the definition of related coordinate systems can be found in [140] athode Plane 26 mmGeodeticPath ofIonizationelectrons ParticleTrajectoryAnodeWire FieldLinesPerpendicularDistance Figure 2-11: Configuration of wire chambers.of electrons and ions along its trajectory behind. The ionized electrons acceleratetoward the wires along the path of least time (geodetic path). The Hall A VDCsfeature a five cell design, i.e a typical 45 ◦ track will fire five wires as shown in Fig. 2-11. The fired wires are read out with time-to-digital converters (TDCs) operatingin common stop mode. In this configuration, a smaller TDC signal correspondsto a larger drift time. With a 50 µ m/ns drift velocity and time shift constants, thedistances of the track to each fired wires are precisely reconstructed. The position anddirection of the track is then determined. In the focal plane, the position resolution σ x ( y ) ∼ µ m, and the angular resolution σ θ ( φ ) ∼ . There are two planes of trigger scintillators S1 and S2 in the left HRS, separatedby a distance of about 2 m. Each plane is composed of six overlapping paddlesmade of thin plastic scintillator (5 mm BC408) to minimize hadron absorption (seeFig. 2-12). The active area for the scintillator paddles are 29 . × . (S1) and37 . × . (S2), and are viewed by two photomultiplier tubes (PMTs) (Burle98 ransverse (Y) D i s p e r s i v e X ) ( Active Area Light GuideLeft PMTs Right PMTsOverlap of planes
Figure 2-12: Layout of scintillator counters.8575). The scintillators were used to generate triggers for the data acquisition system.The time resolution of each plane is about 0.30 ns. The scintillators can also be usedfor particle identification by measuring the Time-of-flight (TOF) between the S1 andS2 planes.Additionally, the S0 scintillator counter is usually used for trigger efficiency analy-sis. It was removed for this experiment to reduce the energy loss of the low momentumprotons ( ∼
550 MeV/ c ). The Focal Plane Polarimeter (FPP) measured the polarization of protons in thehadron spectrometer [141]. It was developed by the College of William & Mary,Rutgers University, Norfolk State University and the University of Georgia.The FPP is located between the VDCs and the lead glass counter, it consists of 4straw chambers and a carbon analyzer (see Fig. 2-13). When the polarized protonspass through the carbon analyzer, the nuclear spin-orbit force leads to an azimuthal99 (cid:84)
Front chambers Rear chambersCarbon door
Figure 2-13: Layout of the Focal Plane Polarimeter.asymmetry due to the scattering from carbon nuclei. The particle trajectories, inparticular the scattering angles in the carbon analyzer, are determined by the frontand rear chambers.The front straw chambers are separated by about 114 cm, and are located beforeand after the gas Cerenkov detector. The second chamber is followed by S2, which isin turn followed by the FPP carbon analyzer. The rear chambers, chamber 3 and 4are separated by 38 cm and are immediately behind the carbon analyzer.The carbon analyzer consists of 5 carbon blocks. Each block is split in the middleso that it can be moved in or out of the proton paths. The total thickness of thecarbon analyzer can be adjusted accounting for different proton momentum. Theblock thicknesses, from front to rear are 9”, 6”, 3”, 1.5” and 0.75”. The block positionsare controlled through EPICS [142]. For this experiment, the proton momentum wasbetween 550 MeV /c and 930 MeV /c . We adjusted the carbon door thicknesses basedon a Monte Carlo simulation (see Fig. 2-14). The thicknesses of the carbon door usedfor different kinematics are listed in Table 2.4.The straw chambers include X, U, and V planes. The central ray defines the z -axis. X wires are along the horizontal direction and measure position along thedispersive direction. As illustrated in Fig. 2-15, the UV planes are oriented at 45 ◦ J Brash’s MC Simulations of FPP Performance For E05-103: KEEP ¾” Carbon analyzer in all the time(SOLID LINES)
Last Updated: 22 July 2006
Proton Momentum (GeV/c) F PP F i gu r e - o f - M e r i t: < (cid:72) A c > Region of¾” C door ONLY Region of¾” C PLUS 1.5 “ C Region of¾” C PLUS 3.0 “ C
Figure 2-14: The simulated FPP figure of merit with different carbon door thick-nesses [52]. Kine. Q [(GeV /c ) ] P p [GeV /c ] Carbon thickness [inch]K1 0.35 0.616 2.25K2 0.30 0.565 2.25K3 0.45 0.710 3.75K4 0.40 0.668 3.75K5 0.55 0.794 3.75K6 0.50 0.752 3.75K7 0.60 0.836 3.75K8 0.70 0.913 3.75Table 2.4: Carbon thickness along the proton momentum at each kinematics.101 yU V Figure 2-15: FPP coordinate system.Chamber Ch.1 Ch.2 Ch.3 Ch.4Active legnth (cm) 209.0 209.0 267.5 292.2Active width (cm) 60.0 60.0 122.5 140.6Wire spacing (cm) 1.095 1.095 10.795 1.0795Configuration 3U + 3V 3U + 3V 2U+ 2V + 2X 3U + 3VStraws per plane 170 170 249 292Table 2.5: Dimensions of the FPP straw chambers.with respect to the transverse plane of the XY coordinate system, with +U betweenthe +X and +Y axes, and +V between the +Y and -X axes. The configurations foreach chamber are listed in Table 2.5.4. The FPP has angular resolution better than1 mrad and accepts second scattering angles of at least 20 ◦ .The straw chambers are a set of cylindrical tubes of radius 0.5 cm, with a thin wirerunning along a central axis of each tube (straw), as shown in Fig. 2-16. The wire isat positive high voltage ( ∼ µ m/s. When the electrons get within about 100 µ m of the102 roton trajecoryStrawwire Figure 2-16: Straws in two different planes of a FPP straw chamber.wire, the increase in electric field strength is larger enough so that addition atomsionize; this leads to an avalanche effect and produces a gain of about 10 per primaryionization. The movement of the positive and negative ions leads to a voltage drop onthe wire and produces a negative electrical signal. The analog signal is then sent tothe read out board, where it is pre-amplified and discriminated to give a logic pulse(see Fig. 2-17).Because of the straw around each wire forms a physical ground, a proton trackleaves a signal only in one wire of a plane. Multiplexing the signal in groups ofeight neighboring wires, and reading out the entire group by the same multiplexingchip, it significantly reduces the amount of electronics required for the FPP. Thismultiplexing chip is setup to give a logic pulse whose width depends on which wirefired. This 45 mV signal is converted to a 800 mV signal in the level shifter andis sent to the FastBus TDC modules, whose output is readout to the data stream.The multi-hit TDCs records the arrival of the leading edge and the trailing edge ofthe logic signal. The time difference between the leading edge and the common stopgiven by the trigger gives the drift time. 103 isc.Disc.Disc. Multiplex Levelshifter TDCStopStart l.e.t.e.
HRS trigger
Analog signalfrom sense wire t.e.l.e.
Preamp.
Figure 2-17: Block diagram for the logic of the FPP signal. (l.e. = leading edge, t.e.= trailing edge).
Due to the constraints from the preceding experiments, the BigBite spectrometer wasused to detect the electrons instead of the originally proposed right HRS. Comparedwith the standard HRS, the BigBite spectrometer has larger angular and momentumacceptance. Recently, the spectrometer has been upgraded to detect electrons withadequate momentum and angular resolution for a series of experiments [143, 144].The central component of the spectrometer is a large acceptance, non-focusingdipole magnet. The magnet was originally designed and built for use at NiKHEFin the Netherlands [145, 146]. The large pole-face gap (25 cm in the horizontal and84 cm in the vertical directions) allows for a larger bite of scattered particles in theangular acceptance (see Fig. 2-18).In this experiment, the magnet was located ∼ . ∼ /c , and the solid angle acceptance is ∼
96 msr, roughly sixteen timeslarger than the nominal HRS acceptance.As shown in Fig. 2-20, the BigBite electron package consists of: • • a gas Cerenkov counter. 104igure 2-18: A side view (left) and top view (right) of the BigBite magnet showingthe magnetic field boundary and the large pole face gap.Figure 2-19: A side view (left) and top view (right) of the BigBite spectrometerduring this experiment. 105 ront wire chamberRear wire chambers Gas CerenkovPreshower Scintillator ShowerBigBite magnet Figure 2-20: A side view of the BigBite detector package during this experiment. • a pre-shower counter. • a scintillator plane. • a shower counter.The scintillator plane consists of 13 scintillator paddles with PMTs on both sides, andeach paddle has a size of 17 × × ×
27 lead glassblocks (8 . × . ×
37 cm), and each block is oriented perpendicular to the particletracks. The shower counter has 7 ×
27 lead glass blocks and are aligned parallel tothe tracks. The signal detected by lead glass blocks is linearly proportional to theenergy deposited by the incoming particle [147]. Electromagnetic showers develop inthe counter, whereas hadronic showers do not due to the longer hadronic mean freepath. Therefore, the longitudinal distribution of the energy deposited in the countercan be used to identify the incident particles.The HV for both the pre-shower and shower counters were calibrated by cosmicsbefore the experiment. Since the kinematics can be well determined from the hadronarm for the elastic events, trajectory information is not required on the BigBite side.106 olumn0 1 2 3 4 5 6 7 R o w Figure 2-21: The BigBite shower counter hit pattern for kinematics K8, δ p = -2%.The hot region corresponds to the elastic electrons. For production data taking, onlythe shower blocks inside the ellipse were on.Therefore, only the shower counters were turned on during the production data takingto tag the electrons and form the coincidence trigger. The pre-shower counter wasturned off to further reduce the background. Fig. 2-21 is an example of the showerrate pattern for one of the kinematics settings. The hot region corresponds to theelastic peak on the left HRS. The Hall A data acquisition (DAQ) system used CODA (CEBAF On-line Data Ac-quisition) [148] developed by the Jefferson Lab Data Acquisition Group.CODA is a tool kit composed of a set of software and hardware packages fromwhich a data acquisition system can be constructed which will manage the acquisition,monitoring and storage of data of nuclear physics experiments. The DAQ includesfront-end Fastbus and VME digitization devices (ADCs, TDCs and scalers), the VMEinterface to Fastbus, single-board VME computers running VxWorks operating sys-tem, Ethernet networks, Unix or Linux workstations, and a mass storage tape silo107MSS) for long-term data storage. The custom software components of CODA are: • a readout controller (ROC) which runs on the front-end crates to facilitate thecommunication between CODA and the detectors. • an event builder (EB) which caches incoming buffers of events from the differentcontrollers then merges the data streams in such a way that data which wastaken concurrently in time appears together. • an event recorder (ER) to write the data built by EB to the disk. • an event transfer (ET) system which allows distributed access to the data streamfrom user processes and inserts additional data into the data stream every a fewseconds from the control system. • a graphical user interface (Run Control) to set experimental configuration, con-trol runs, and monitor CODA components.A recorded CODA file consists the following major components: • Header file including a time stamp and other run information like run number,pre-scale factors and event number. • CODA physics events from the detectors. • CODA scaler events: the DAQ reads the scaler values every 1 − • EPICS [142] data from the slow control software used at JLab, e.g., the spec-trometer magnet settings and angles, target temperature and pressure, etc.
In this experiment, six different types of triggers were generated and used in thedata acquisition. T1 and T3 are singles triggers from the electron arm (BigBite)108
80 nsecanalog delayto ADCNIM toTriggerNIM toTDC viaNIM/ECLDISCR.DISCR. “AND” Logic
Left and Right PMTsare AND’d. Have 16outputs. (16 )(16)
CAMAC (1)
R. Michaels (Aug 2003) S m P M T S i gn a l s P/S 75716 in “OR”d to 1 out (16) delay S2mSignal LE F TLE F T R I GH T R I GH T S ! P M T S i gn a l s DISCR.DISCR.
Single Arm Triggers in Each Spectrometer
LeCroy 4516“AND” Logic On back of 4516 is output corresp.to the Logic result (L.and.R) “OR”d.So, this is S1.
Cherenkovor Other Detector “AND” LogicLeCroy 4516
MainScint.Trigger TriggerSupervisor
Coinc. Trig(see other diagram)LogicFan In“OR” S2mS1
GateRetiming
MLUS1 S2mCher. strobe“AND”Logic T1StrobeandRetiming(S2m leads) 2/3 Trig.Cher RTT1..and.Cher. L1ATM
Gates forADCs, TDCs (6)(6) (1) T1
Notes: S2m defines timing for T1, strobe, and RT.MLU defines 2 out of 3 trigger. EDTM signal addedto S1 with EDTM modules, and added to S2m via pulserinput to discriminator.P/S 758
Figure 2-22: Left HRS single arm triggers diagram during E08-007.and the hadron arm (L-HRS) respectively. T4 is the left HRS scintillator triggerused for trigger efficiency. T5 is the coincidence trigger of T1 and T3. T7 is theBigBite cosmic trigger for testing. T8 is the EDTM pulser trigger used to measurethe trigger efficiency. The trigger system was built from commercial CAMAC andNIM discriminators, delay units, logic units and memory lookup units (MLU).
T3 was formed by requiring that both scintillator planes S1 and S2 have at least onefired scintillator bars (both phototubes fired) and they are close enough to form avalid track. Thus, this main trigger requires four fired PMTs. The T3 trigger diagramis illustrated in Fig. 2-22. 1091 was formed by the BigBite shower total sum as illustrated in Fig. 2-23. Thetotal sum (TS) was defined as the sum of all the pre-shower (PS) and shower (SH)ADCs of the two adjacent rows, e.g.,
P S sum = P S L + P S R + P S L + P S R, (2.8) SH sum = SH + SH + · · · + SH + SH + SH + · · · + SH , (2.9) T S
P S sum + SH sum . (2.10)The electron trigger was given by the “OR” of the total sum signals. The diagram of coincidence triggers is shown in Fig. 2-24. Coincidence trigger T5 issimply an “AND” of T1 and T3 triggers.
A summary of triggers used in E08-007 is listed in Table 2.6. After generated, all typesof triggers have their copies sent to a scaler unit for counting and a trigger supervisor(TS) unit to trigger data acquisition. The TS unit has a pre-scale function. If thepre-scale factor for a specific trigger type is N , then only 1 out of N triggers of thattype is recorded in the data stream. This function is very useful to decrease thecomputer dead time caused by frequent data recording while keeping all the eventswith useful physics information. Therefore, during the production data taking, all thesingle arm triggers were highly pre-scaled, and all the T5 (coincidence) trigger eventswere kept in the data stream. The rates of each trigger after the pre-scale factors arealso listed in Table 2.6. 110 igBite Trigger Logic for E08-007 Preshower
PS1LPS1R APS776APS776 ADC (cid:256)
P01L (cid:257)
ADC (cid:256)
P01R (cid:257)
SPL0.5/0.5SPL0.5/0.5 SH1/1SH1/4SH1/3SH1/2SH1/5SH1/6SH1/7SH2/1SH2/4SH2/3SH2/2SH2/5SH2/6SH2/7APS776APS776PS2LPS2R SPL0.5/0.5SPL0.5/0.5ADC (cid:256)
P02L (cid:257)
ADC (cid:256)
P02R (cid:257)
FI/OL428F S8 ADC (cid:256)
S1.1-S1.7 (cid:257)
FI/OL428F DPS706DPS706 ANDPS758S8FI/OL428FAPS776APS776 SPL0.5/0.5SPL0.5/0.5 S8 FI/OL428F DPS706DPS706 ANDPS758 OR ORADC (cid:256)
S2.1-S2.7 (cid:257)
ADC (cid:256)
S3.1-S3.7 (cid:257)
ADC (cid:256)
P201A (cid:257)
ADC (cid:256)
PS02A (cid:257)
ADC (cid:256)
P03L (cid:257)
ADC (cid:256)
P03R (cid:257)
ADC (cid:256)
TS01A (cid:257)
TDC (cid:256)
TS01T (cid:257)
ADC (cid:256)
TS02A (cid:257)
TDC (cid:256)
TS02T (cid:257) (cid:256) electron (cid:257)
PS3LPS3R
Shower
AmpFI/FODiscrLogicLogicLogicORSum8Splitter PS776LRS428FPS706PS758PS755LRS622PS757
SH3/1SH3/4SH3/3SH3/2SH3/5SH3/6SH3/7
Figure 2-23: The BigBite trigger diagram during E08-007.Trigger Definition Rate after pre-scaleT1 electron arm singles (total shower sum) ∼
20 HzT3 hadron arm singles (S1 AND S2) ∼
20 HzT4 hadron arm efficiency (S1 OR S2) ∼
10 HzT5 coincidence (T1 AND T3) ∼ oincidence Trigger
250 nsdelay 226 nsdelay210 nsdelay 85 nsdelay
BigBite singlestrigger
Front-end to DAQweldment Cable delay (cid:256)
AND Logic (cid:257)
LeCroy 4516
Left HRS singlestrigger
Gates for ADCs,TCDs
Figure 2-24: Coincidence trigger diagram during E08-007.112 hapter 3Data Analysis I
The Hall A C++ Analyzer [149] was used to replay the raw data and generate theprocessed data files for this experiment. The Analyzer was developed by Hall A soft-ware group and is based on ROOT [150], a powerful object-oriented framework thathas been developed at CERN by and for the nuclear and particle physics commu-nity. From the replayed data files, the proton form factor ratio was extracted by theweighted sums technique [151].The flow-chart of the E08-007 analysis procedure is illustrated in Fig. 3-1. The rawdata recorded from the detectors were first transformed into ntuples by the Analyzerafter calibration. The recoil proton’s second scattering angle was extracted from theFPP reconstruction. The spin transport matrix were generated by COSY (a modelsimulating the spectrometer transport system). With these inputs, the recoil protonpolarization and hence the form factor ratios were extracted by the main analysiscode PALM [152].
The particle trajectory at the focal plane of the left HRS is determined by raw wirehits and drift times in the VDCs. These trajectories are transported from the focal113 aw DataAnalyzer replay
FPP c a li b r a t i o n a nd a li g n m e n t HRS cuts FPP cuts V D C v a r i ab l e s F PP v a r i ab l e s PALM C O S Y S P m a t r i x General cutsAnalyzingpower fit Target polarization& form factor ratio
Figure 3-1: The flow-chart of the E08-007 analysis procedure.114 eamdump H R S L H R S R x z y Figure 3-2: Hall coordinate System (top view).plane to the target using a calibrated “optics” matrix of the spectrometer. Thereconstructed target quantities (momentum and angles) allow for the determinationof the kinematics of each event. For this experiment, these target quantities areimportant in another way as the inputs for the spin transport matrix calculation,which determines the recoil proton polarization at the target.
In this section, a short overview of Hall A coordinate conventions is presented. Moredetails can be found in reference [140].
Hall Coordinate System (HCS)
The origin of the HCS is defined by the intersection of the electron beam and thevertical symmetry axis of the target system. (cid:126)z is along the beam line and points inthe direction of the beam dump, and (cid:126)y is vertically up, see Fig. 3-2.
Target Coordinate System (TCS)
The TCS is defined with respect to the central axis of the spectrometer. A lineperpendicular to the sieve slit surface of the spectrometer and going through the115 tg L (cid:3) (cid:3) tg x tg D y Z react x beam y beam D x y tg x sieve y sieve Z ^ Top ViewSide View
Electron BeamScattered ParticleScattered ParticleOrigin of HCSOriginof TCSTarget SievePlaneSpectrometerCentral Ray Z ^ Y ^ X ^ Figure 3-3: Target coordinate system (top and side views).midpoint of the central sieve slit hole define the (cid:126)z tg -axis. The (cid:126)y tg -axis points to theright facing the spectrometer, and (cid:126)x tg -axis is vertically down as illustrated in Fig. 3-3.In the ideal case where the spectrometer is pointing directly at the hall center andthe sieve slit is perfectly centered on the spectrometer, the TCS has the same originas HCS. However, it typically deviates from HCS center by D x and D y in the verticaland horizontal directions in TCS, respectively, and the offsets are given by surveys.The distance of the midpoint of the collimator from the TCS origin is defined to bethe length L for the spectrometer. The out-of-plane angel θ tg and the in-plane angle φ tg are given by the tangent of the real angle, dx sieve /L and dy sieve /L .The TCS variables are used to calculate the scattering angle and the reaction pointalong the beam line for each event. Combined with the beam positions (measured inthe Hall coordinate system), the scattering angle and reaction point are given by: θ scat = arc cos cos( θ ) − φ tg sin( θ ) (cid:113) θ tg + φ tg (3.1)116 react = − ( y tg + D y ) + x beam (cos( θ ) − sin( θ ))cos( θ ) φ tg + sin( θ ) , (3.2)where θ denotes the spectrometer central angle. The in-plane and out-of-plane anglescan be determined using sieve hole positions: φ tg = y sieve + D y − x beam cos( θ ) + z react sin( θ ) L − z react cos( θ ) − x beam sin( θ ) (3.3) θ tg = x sieve + D x + y beam L − z react cos( θ ) − x beam sin( θ ) (3.4)and the position at the target is given by: y tg = y sieve − Lφ tg (3.5) x tg = x sieve − Lθ tg . (3.6) Detector Coordinate System (DCS)
The Detector Coordinate System (DCS) is defined by the positions of the VDC planes.The intersection of wire 184 of the VDC1 U1 plane and the perpendicular projectionof wire 184 in the VDC1 V1 plane onto the VDC U1 plane defines the origin of theDCS. (cid:126)z is perpendicular to the VDC planes pointing vertically up, (cid:126)x is along the longsymmetry axis of the lower VDC pointing away from the hall center (see Fig. 3-4).Using the trajectory intersection points p n (where n = U1, V1, U2, V2) with thefour VDC planes, the coordinates of the detector vertex can be calculated from thefollowing expressions: tan( η ) = p U2 − p U1 d (3.7)tan( η ) = p V2 − p V1 d (3.8) θ det = 1 √ η ) + tan( η )) (3.9) φ det = 1 √ − tan( η ) + tan( η )) (3.10) x det = 1 √ p U1 + p V1 − d tan( η )) (3.11)117 op ViewSide View V2U2V1U1 VDC1 VDC2YZ ^^ X ^ X ^ U - V - Figure 3-4: Detector coordinate system (top and side views). y det = 1 √ p U1 + p V1 − d tan( η )) (3.12)where d = 0 .
115 m is the distance between the U and V planes in both chambers,and d = 0 .
335 m is the distance between the two planes.
Transport Coordinate System (TRCS)
The TRCS at the focal plane is generated by rotating the DCS clockwise around its y -axis by 45 ◦ . It’s typically used as a intermediate position state from DCS to theFCS (focal plane coordinate system), which will be described in the next section; thebending angle related to the spin transport can also be calculated from the differenceof the out-of-plane angles ( θ tg − θ tr ) between the TCS and TRCS. The transportcoordinates can be expressed in terms of the detector coordinates as follows: θ tr = θ det + tan( ρ )1 − θ det tan( ρ ) (3.13)118 ^ ^ X ^45 o Figure 3-5: Transport coordinate system. φ tr = φ det cos( ρ ) − θ det sin( ρ ) (3.14) x tr = x det cos( ρ )(1 + θ tr tan( ρ )) (3.15) y tr = y det + sin( ρ ) φ tr x det , (3.16)where ρ = − ◦ is the rotation angle, see Fig. 3-5. Focal Plane Coordinate System (FCS)
The focal plane coordinate system (FCS) chosen for the HRS analysis is a rotatedcoordinate system. Because of the focusing of the HRS magnet system, particles fromdifferent scattering angles with the same momentum will be focused at the focal plane.Therefore, the relative momentum from the central momentum of the spectrometer,which is selected by the HRS dipole magnet field setting, δ = ∆ pp = p − p p , (3.17)is approximately only a function of x tr , and p in the formula stands for the centralmomentum setting of the HRS. The FCS is obtained by rotating the DCS aroundits y -axis by an varying angle ρ ( x tr ) to have the new z -axis parallel to the localcentral ray, which has the scattering angle θ tg = φ tg = 0 for the corresponding δ atposition x tr (see Fig. 3-6). In this rotated coordinate system, the dispersive angle θ fp is small for all the points across the focal plane, and the distribute is approximately119 ZX ^^ Y ^ Trajectories with = =0 (cid:3) (cid:4) tg tg
Figure 3-6: Rotated focal plane coordinate system.symmetric with respect to θ fp = 0. This symmetry greatly simplifies further opticsoptimization.With proper systematic offsets added, the coordinates of focal plane vertex canbe written as follows: x fp = x tr (3.18)tan( ρ ) = (cid:88) t i x i fp (3.19) y fp = y tra − (cid:88) y i x i fp (3.20) θ fp = x det + tan( ρ )1 − θ det tan( ρ ) (3.21) φ fp = φ det − (cid:80) p i x i fp cos( ρ ) − θ det sin( ρ ) . (3.22)The coordinate transformation is not unitary and we have x fp equal to x tr for sim-plicity. For each event, two angular coordinates ( θ det and φ det ) and two spatial coordinates( x det and y det ) are measured at the focal plane detectors. The position of the particleand the tangent of the angle made by its trajectory along the dispersive directionare given by x det and θ det , while y det and φ det give the position and tangent of theangle perpendicular to the dispersive direction. These variables are corrected for any120etector offsets from the ideal central ray of the spectrometer to obtain the focal planecoordinates x fp , θ fp , y fp and φ fp . The focal plane observables are used to reconstructthe variables in the target system by matrix inversion.The first order optics matrix can be expressed as, δθyφ tg = < δ | x > < δ | θ > < θ | x > < δ | θ > < y | y > < y | φ > < φ | y > < φ | φ > · xθyφ fp . (3.23)The null tensor elements result from the mid-plane symmetry of the spectrometer.In practice, the expansion of the focal plane coordinates is performed up to the fifthorder. A set of tensors D jkl , T jkl , Y jkl and P jkl relates the focal plane coordinates tothe target coordinates according to [153] δ = (cid:88) jkl D jkl θ j fp y k fp φ l fp (3.24) θ tg = (cid:88) jkl T jkl θ j fp y k fp φ l fp (3.25) y tg = (cid:88) jkl Y jkl θ j fp y k fp φ l fp (3.26) φ tg = (cid:88) jkl P jkl θ j fp y k fp φ l fp , (3.27)where the tensors D jkl , T jkl and P jkl are polynomials in x fp . For example, D jkl = m (cid:88) i =0 C Dijkl x i fp . (3.28)The optics matrix used in this experiment was optimized for the Transversity [144] ex-periment. The core of the optimization program is the TMinuit package of ROOT [150].This package varies the optics matrix parameters to minimize the variance σ of thereconstructed data from their actual values.121
20 40 60 80 100 120 140020040060080010001200
L.fpp.u1.width
Figure 3-7: The TDC width of the u1 wire group and the demultiplexing cut.
As the key instrument to measure the recoil proton polarization, the FPP recon-structs the second scattering angles of the proton in the analyzer. There are basicallyfour steps: identifying the wires that have fired, calculating the drift distances, re-constructing the tracks in the front and rear chambers, and determining the secondscattering angles. All the steps are done in the Analyzer program by incorporatingthe FPP tracking library.
Demultiplexing
As noted in Section 2.5.4, the signals from the sense wires are multiplexed in groupsof eight to decrease the number of TDCs. By assigning a different pulse width to eachstraw of the group, one can make a cut to identify which wire fired. Fig. 3-7 is anexample of the raw pulse width spectrum from one wire group. Then the signal has tobe demultiplexed in the analysis. The straw group, the leading edge and the trailingedge of the TDC signal are fed into Analyzer, which calculates two time differences:the difference between the trigger signal (common stop) and the leading edge givesthe drift time, while the difference between the leading edge and the trailing edgeidentifies which straw fired in the group.Once the drift time for each wire that fired has been determined, one can convert122t into the drift distance; and hence, the tracks can be reconstructed in the chambers.First, an offset is applied to the drift time spectrum, to correct for various delays inthe electronics. Except when the event passes very close to the anode wire, the driftdistance is proportional to the drift time.When the particle approaches the anode wire, the electric field becomes strongenough for secondary ionization, which starts an avalanche. In this region, the driftvelocity increases near the sense wire, and the drift distance d is obtained from afifth-order polynomial in drift time t : d = (cid:88) n =0 T ( j, n ) t n , (3.29)where T ( j, n ) are obtained from fitting the integrated drift time spectra for a plane j . These coefficients were all re-calibrated for this experiment. More details of theFPP calibration can be found in [154]. Track Reconstruction
Using the FPP library in the Analyzer, the raw data were replayed and the trackswere reconstructed in the straw chambers. The front and rear chambers were analyzedseparately to produce both a rear and front track. For each set of chambers, the u and v directions are also analyzed separately. The x planes in chamber 3 were notused .The first step is to identify hit clusters in the sets of u planes of each chamber. Inthis set, a cluster can have at most one hit per plane. The code searches for a trackby looking at the adjacent straws first. In Fig. 3-8, the colored circles stands for thefired straws. The code looks at the top plane and finds a hit in S , then it looks inthe second plane at the straws adjacent to S . It finds that S fired, then S and S start to form a cluster. When it looks further to the third plane, at straws thatare adjacent to S or S which are both adjacent to S even though S didn’t fire.It finds S and forms the first cluster ( S → S → S ), S also fired and forms The original design of the x plane is to provide additional information of the out-of-plane posi-tion, but it was found later that the u and v planes were sufficient. Figure 3-8: Illustration of the procedure to find clusters in a FPP chamber. Thethree layers represent the three planes, and the circles are cross-sectional cuts of thestraws. The filled circles represent the fired straws.another cluster ( S → S → S ). The area around S is now all scanned, so thecode starts looking at the rest of the first plane. It finds S , and finds nothing elsein this cluster on the next planes. When the entire first plane has been scanned, itgoes to the second plane. A hit is found at S , which forms a cluster with S . Whenlooking at the third plane, no hit is found that is not already included in a cluster sothe procedure is complete. As a results, the code has found a total of four clusters:( S → S → S ), ( S → S → S ), ( S ), ( S → S ).The same procedures are applied to the second chamber. All combinations of pairsof clusters in both chambers are considered. For each combination, several tracks arereconstructed. From the drift distance, the track can be passing left or right of thesense wire of every fired straw, therefore, there are 4 track possibilities with two givendrift distance, as illustrated in Fig. 3-9. Straight lines are then fitted, and a χ foreach possible trajectory is calculated. Since it is easier for a cluster with very fewhits to give a very good χ , a weight is give to the χ corresponding to the number ofhits for the track. The track with the lowest χ is then considered as the good track.The procedure is repeated for the v direction.124 straws4 possible tracks Figure 3-9: 4 possible tracks for two given fired straws with given drift distances d and d . The good track is the one with the lowest χ when taking into account allplanes of all chambers. Chamber Alignment
In order to determine the proton scattering angle in the carbon analyzer, the positionsof the chambers have to be well known so that the second scattering angles φ fpp and θ fpp are correctly reconstructed. To achieve the precision of ∆ φ fpp ∼ ∆ θ fpp ∼ φ fpp , therefore, anyrotation between the front chambers and the rear chambers will directly shift φ fpp .Second, what we really care about is the proton polarization at the target; therefore,the FPP front and rear chambers have to be aligned with respect to a well knowncoordinate system so that the second scattering angle is calibrated referring to thatcoordinates system and can easily be related to the target frame. As described inSection 3.2.1, the transport coordinate system (TRCS) defined by the VDCs is aconvenient choice. By taking “straight-through” data with the carbon door open, thetrajectory determined by the FPP should coincide with the trajectory reconstructed125y the VDCs after alignment has been completed.There are two different methods to do the alignment. In the first approach, asoftware procedure is applied to fit the alignment offset parameters u , v , z , andthe rotation angles θ zu , θ zv , θ uv , by minimizing the trajectory difference between theVDCs and the FPP. The advantage of this method is the direct link between thealignment parameters and the physical offsets of the chambers.For this experiment, the alignment procedure was done by the second approach,which was developed for experiment E93-049 [155]. Compared to the chamber align-ment approach mentioned above, this method directly applies a correction to thereconstructed track instead of the individual chambers. It first aligns the front cham-ber track with respect to the VDC track, and then the rear chamber track is alignedwith respect to the well aligned front chamber track. The alignment parameters ob-tained with this method are not easily related to the physical offsets or rotations, butthe extension to higher order corrections is straight forward. The detailed alignmentalgorithm is presented in Appendix B. For high precision measurements, the previ-ous experiment analysis [155] showed that using the second method by extending thecorrections with higher order terms can achieve better results.The “straight-through” data (electron) was taken during experiment E04-007 [156],which ran just before this experiment . The histograms of the track difference ( x diff , y diff , θ diff , φ diff ) between the VDCs and the FPP front chambers before (black)and after (red) the software alignment are shown in Fig. 3-10. As one can see, thedifferences are well centered at 0 after the alignment.Another way to see the alignment quality is by looking at zclose , which is thelocation along the spectrometer axis of closest approach between the front and rearFPP tracks and stands for the second scattering vertex in the carbon analyzer. Forthe ideal alignment, the reaction vertex should not depends on the azimuthal angle φ fpp , so the plot of zclose versus φ fpp should be “straight” in the zclose dimension,with sharply defined edges centered at the physical position of the carbon analyzer. The FPP chambers were installed before experiment E04-007 took data and were not toucheduntil this experiment was finished. _diff_f [cm]-10 -8 -6 -4 -2 0 2 4 6 8 100200040006000800010000120001400016000180002000022000
L.fpp.x_diff_f y_diff_f [cm]-10 -8 -6 -4 -2 0 2 4 6 8 10020004000600080001000012000140001600018000200002200024000
L.fpp.y_diff_f th_diff_f [deg]-3 -2 -1 0 1 2 3020004000600080001000012000140001600018000
L.fpp.th_diff_f ph_diff_f [deg]-3 -2 -1 0 1 2 3020004000600080001000012000140001600018000
L.fpp.ph_diff_f
Figure 3-10: The difference between the VDC track and the FPP front track before(in black) and after (in red) the chamber alignment. The difference is centered at 0after the alignment.Fig. 3-11 shows a plot before and after the alignment. One can obviously see the“snake” shape is gone after the alignment.
Scattering Angle Calculation
For the determination of the polar and azimuthal angles of the second scattering, onefirst needs to rotate the coordinates system so that its z -axis is along the momentumof the incident track, and then express the scattered track in this new coordinatesystem.As shown in Fig. 3-12, for the incident track (cid:126)f in the transport coordinates system, (cid:126)z is along the spectrometer axis at the focal plane, (cid:126)x is perpendicular to (cid:126)z andvertically down, and (cid:126)y = (cid:126)z × (cid:126)x . θ f and φ f are the Cartesian angles: θ f is the anglebetween the projection of the track on the x − z plane and the z -axis, and φ f is theangle between the projection on the y − z plane and the z -axis. For convenience, wedefine ψ f as the angle between the track and its projection on the y − z plane, and127 close [cm]300 320 340 360 380 400 420 440 460 480 [ deg ] f pp φ Before zclose [cm]300 320 340 360 380 400 420 440 460 480 [ deg ] f pp φ After
Figure 3-11: φ fpp versus zclose before and after the FPP chamber alignment. z (cid:38) x (cid:38) y (cid:38) f (cid:73) f (cid:92) f (cid:38) f (cid:84) Projection on (xz) Projection on (yz)
Figure 3-12: Cartesian angles for tracks in the transport coordinates system.128 ncident track (cid:84) (cid:73) r (cid:38) r (cid:38) ' z (cid:38) ' x (cid:38) ' y (cid:38) Figure 3-13: Spherical angles of the scattering in the FPP.the relation between the angles is:tan ψ f = tan θ f cos φ f , (3.30)Therefore, the rotation can be decomposed into two rotations: first, a rotation of the y − z plane around the x -axis by an angle φ f , and followed by a second rotation byangle ψ f so that the new z (cid:48) -axis lies along the incident track. The new projection ofthe incident track (cid:126)f is given by: f (cid:48) x f (cid:48) y f (cid:48) x = cos ψ f − sin ψ f ψ f ψ f φ f − sin φ f φ f cos φ f f x f y f z . (3.31)Similarly, the new projection of the scattered track (cid:126)r is now: r (cid:48) x r (cid:48) y r (cid:48) x = cos ψ f − sin ψ f ψ f ψ f φ f − sin φ f φ f cos φ f r x r y r z . (3.32)We can now define the scattering angles ( θ fpp , φ fpp ) as the spherical angles of thescattered track in this new coordinate system as illustrated in Fig. 3-13. If (cid:126)r is the129rojection of (cid:126)r on the x (cid:48) − y (cid:48) plane, we have: r = r (cid:48) x + r (cid:48) y , (3.33) θ fpp = tan − ( r r (cid:48) z ) , (3.34) φ fpp = tan − ( r (cid:48) x r (cid:48) y ) . (3.35) Before we extracts the physics asymmetries, a series of cuts were applied to select theelastic events and to minimize the experimental systematic uncertainties.
One-Track-Only Cut
First, the one-track-only cut was applied to the events reconstructed from VDC clus-ters. The drift times range from 0 to 360 ns. In the Analyzer, a software cut of 400 nsis applied after the first wire fires to ensure the completeness of the track searching.If only one track is observed in an event, the track reconstruction will be accurate. Ifmultiple tracks for an event are found in the analysis, the first track reconstructionmay be distorted due to the interference of a nearby second track. This cut removes ∼ .
5% of the total events (see Fig. 3-14). η ( ON E ) = N ( n track = 1) N ( n track ) > ∼ .
5% (3.36)
HRS Acceptance Cut
As mentioned in Section 2.5, the HRS has a finite momentum and angular acceptance.Events with the target coordinates reconstructed outside the physical acceptance needto be cut out. On the other hand, since this experiment measures the helicity depen-130 umber of track-1 0 1 2 3 4 510 Figure 3-14: Left HRS VDC track number distribution.dent asymmetry difference at the focal plane, precise knowledge of the acceptance isnot required compared to an absolute cross section measurement. In order to avoidpotential problems arising from the spin transport at the edge of the acceptance,relatively tight cuts were applied compared to the HRS nominal acceptance. Thereaction vertex cut was also applied ( y tg ) to reduce the number of events from thequasi-elastic scattering off the aluminum end cap. Typical cuts on different targetvariables are shown in Fig. 3-15. Elastic Cut for the Hadron Arm
Since during the production of this experiment, only part of the BigBite showerblocks were turned on, we cannot reconstruct the electron kinematics. However, dueto the small acceptance and high resolution of the HRS, the elastic kinematics arewell determined by the hadron arm. The beam energy ( ∼ . δ p . The resolution of “dpkin” represents the momentum and angular resolution of131 δ -0.05 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05050010001500 tg φ -0.05 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.0505001000150020002500 tg θ -0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.10500100015002000 tg y-0.05 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.050500100015002000 Figure 3-15: HRS acceptance cuts for kinematic setting K5 δ p = 0%.the hadron spectrometer. With sufficient statistics, we applied a tight cut on theproton elastic peak to keep ∼
80% of the elastic events. This cut corresponds to a 2dimensional cut on the proton angle versus momentum (see Fig. 3-16).This elastic cut minimizes the contributions from the radiative tail, inelastic eventsand the background due to the proton re-scattering inside the spectrometer withoutsacrificing too many of the elastic events. Although most the singles triggers were pre-scaled away during the experiment, therewere still ∼
20 Hz T1 and T3 events left in the data stream. An event-type cut wasapplied to select the coincidence trigger T5.A coincidence timing cut was applied to the TDC spectrum of T3. The accidental This becomes crucial for low momentum protons, since the re-scattering can change the mo-mentum and direction of the proton at the focal plane while the reconstruction is still within theacceptance. The spin transport is totally different for this type of events. pkin-0.05 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.050200040006000 p δ -0.05 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05 [ deg ] p θ Figure 3-16: Elastic cut on dpkin (left), and the corresponding 2D cut on the protonangle θ p versus momentum δ p .background under the elastic cut is ∼ . During this experiment, the BigBite shower counter was used to tag the scatteredelectrons and form the coincidence trigger. The entire pre-shower counter and part ofthe shower blocks outside the elastic peak were turned off during the production datataking to reduce the background. To ensure we turned on the right shower counterregion, we did test runs after every kinematic setting change in which both the pre-shower and shower counters were turned on to locate the electron elastic peak (seeFig. 3-18).From these test runs, the electron energy deposited in the pre-shower and showercounters were reconstructed. Fig. 3-17 shows the BigBite shower ADC sum versusthe pre-shower ADC sum when both of them were turned on, with and without thecoincidence cut (T5). Clearly, the coincidence trigger can effectively suppress thepions and low energy electrons. Additionally, the plots of the proton acceptance withBigBite shower y > y <
0) (see Fig. 3-19), directly demonstrate the correlation133 igBite pre-shower ADC [chn]0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 B i g B i t e s ho w e r A DC [ c hn ] B i g B i t e s ho w e r A DC [ c hn ] Figure 3-17: The BigBite pre-shower ADC sum versus shower ADC sum with (rightpanel) and without (left panel) the coincidence trigger cut (T5). The low energybackground were highly suppressed with the coincidence configuration.between the electrons and the protons.
Scattering Angle Cut
In order to select the correct reconstructions of the second scattering in the FPP,several cuts were applied on the FPP variables. First, a cut was applied on the polarscattering angle: 5 ◦ < θ fpp < ◦ . This cut removes the small scattering angle events,which are dominated by Coulomb scattering with little analyzing power, and thelarger scattering angle events, which have large instrumental asymmetry and smalleranalyzing power. Fig. 3-20 shows an example of the θ fpp distribution and the appliedcut. Scattering Vertex Cut
In order to ensure that the scattering originated from within the carbon block, a tightcut on the reaction vertex was applied. Due to the imperfect alignment of the FPPchambers, a manual correction was applied to zclose along the azimuthal scatteringangle φ fpp . In this procedure, a set of coefficients were generated along φ fpp by theprofile of the 2D plot of φ fpp versus zclose . After this correction, a straight line cutwas applied to the corrected zclose . Fig. 3-21 shows the plot of φ fpp versus zclose igBite shower Y [m]-0.3 -0.2 -0.1 0 0.1 0.2 0.3 B i g B i t e s ho w e r X [ m ] -1-0.500.51 02004006008001000 BigBite shower X [m]-1 -0.5 0 0.5 1 C oun t s C oun t s Figure 3-18: BigBite shower counter hit pattern in the upper panel and the profileson x (vertical) and y (horizontal) in the left and right panels, respectively. p δ -0.05 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 t g φ -0.04-0.0200.020.04 BigBite shower Y >0 tg δ -0.05 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 t g φ -0.04-0.0200.020.04 BigBite shower Y <0
Figure 3-19: Proton acceptance (angle versus momentum) with BigBite shower y > y <
0. 135 pp θ C oun t s Figure 3-20: The distribution of the FPP polar scattering angle θ fpp and the appliedcut.after the correction, and the applied cut .The correlation between the FPP front track and rear track is represented by sclose , which is the distance of the closest approach between these two tracks. Toensure the quality of the FPP tracking, a cut on sclose of 2 cm or less was applied(see Fig. 3-22). Cone-test Cut
To avoid large non-physical asymmetries arising at the edges of the rear chambersdue to the limited size, a cone-test was applied. For a scattering angle θ fpp , if theentire cone of angle θ fpp around the incoming track is within the acceptance of therear chamber, this event passes the cone-test. As illustrated in Fig. 3-23, track 1passes the cone test, while track 2 fails and is rejected. This test eliminated ∼ ◦ . The purpose of this correction is to make the cut simpler, and it doesn’t change the FPPalignment. close [cm]300 320 340 360 380 400 420 440 460 480 [ deg ] f pp φ Figure 3-21: Cut applied to zclose after the manual correction for setting K2 δ p = 0%. sclose [cm]0 1 2 3 4 5 6 7 8 9 1005000100001500020000 Figure 3-22: sclose distribution and cut applied to it for setting K2 δ p = 0%.137 ront Chambers Rear ChambersTarck 1 Track 2 Figure 3-23: The cone-test in the FPP. The cone of angle θ fpp around track 1 isentirely within the rear chambers acceptance, while the one around track 2 is not.Track 2 fails the cone-test and is rejected.138 .4 Recoil Polarization Extraction For the events passing all the cuts described in the previous section, the recoil polar-ization and the form factor ratio was extracted. In this section, the distribution of thescattering angle of the recoil proton in the FPP is analyzed. With the reconstructionof the spin precession through the spectrometer, the proton polarization is extracted.In addition, the discussion of the carbon analyzing power is presented.
For the polarization measurement of the recoil proton, the events of interest are thosethat have scattered in the carbon analyzer via the strong interaction with a carbonnucleus. As illustrated in Fig. 3-24, the interaction between the polarized proton andan analyzer nucleus is sensitive to the direction of the incident proton’s spin througha spin-orbit coupling. A left-right asymmetry in the scattering will be occurred ifthe proton spin is preferentially up or down. The sign of the force is determined bythe sign of (cid:126)L · (cid:126)S scalar product, where (cid:126)L is the orbital angular momentum of theproton with respect to the analyzer nucleus, and (cid:126)S is the proton spin. Protons arescattered to the left with spins up and to the right with spin down (correspondingto the polarization of the incident proton). Hence, an asymmetry in the horizontaldirection will be observed. Similarly, an vertical asymmetry will be observed when thepolarization is along the horizontal direction. However, the longitudinal componentdoes not result in an asymmetry.In general, the angular distribution for a large sample of incident polarized pro-tons is expressed by a sinusoidal function of the vertical P fppx and horizontal P fppy polarization components: f ± ( θ, φ ) = 12 π (cid:15) ( θ, φ )(1 ± A y ( θ, T p )( P fppx cos φ − P fppy sin φ )) , (3.37)where ± refers to the sign of the beam helicity. In this expression, (cid:15) ( θ, φ ) is thenormalized efficiency, which describes the non-uniformities in the acceptance due to139 (cid:38) y (cid:38) (cid:73) L (cid:38) L (cid:38) S (cid:38) S (cid:38) (cid:33)(cid:16)(cid:152) (cid:31)(cid:14)(cid:152) SL SL (cid:38)(cid:38) (cid:38)(cid:38) (cid:31)(cid:16)(cid:152) (cid:33)(cid:14)(cid:152)
SL SL (cid:38)(cid:38) (cid:38)(cid:38) C Figure 3-24: Polarimetry principle: via a spin-orbit coupling, a left-right asymmetryis observed if the proton is vertically polarized.140hamber misalignment and detector inefficiency. A y ( θ, T p ) is the analyzing power ofthe reaction A ( p, N ) X , which represents the strength of the spin-orbit coupling ofthe nuclear scattering, thus the sensitivity to the incident particle polarization. Theanalyzing power depends on the scattering polar angle θ and the proton kinetic energy T p . For Coulomb scattering, there is no analyzing power, since there is no spin-orbitcoupling.
The FPP measures the proton polarization at the focal plane, however, the formfactor ratio G Ep /G Mp is obtained from the polarization in the target frame; hence,the measured polarization at the FPP has to be transported to the one at the target.The relation between the polarization components in these two frames is complicateddue to the proton spin precession through the spectrometer magnets. Dipole Approximation
Before we try to fully describe the spin transport through the spectrometer, a simpleapproximation can be used by considering a single perfect dipole, as illustrated inFig. 3-25. With only a transverse field with respect to the particle momentum, thespin rotates along the y -axis. In this case, the spin precession angle is a simplefunction of the trajectory bending angle Θ bend : χ = γ ( µ p − bend , (3.38)where γ = 1 / √ − β . The HRS dipole central bending angle is ∼ ◦ ; in thisapproximation, the relation between the polarization components at the target and The details of the analyzing power analysis are presented in Section 3.4.4. XY B ZXY
P P fpp χ Figure 3-25: The dipole approximation of the spin transport in the spectrometer:only a perfect dipole with sharp edges and a uniform field. The proton spin onlyprocesses along the out-of-plane direction.at the focal plane is: P fppx P fppy P fppz = cos χ χ − sin χ χ P x P y P z . (3.39)Note that the transverse component P y does not precess, since it is parallel to themagnetic field. As mentioned earlier, in the one-photon-exchange approximation,the ep elastic scattering process has no induced polarization, which means that thenormal part of the polarization is: P x = 0 . (3.40)Since the FPP can measure only the two perpendicular components to the momentumat the focal plane, the relation in Eq. 3.39 is further simplified: P fppx P fppy = χ P y P z . (3.41)142sing the angular distribution function from Eq. 3.37, the polarization componentsat the focal plane can be extracted. By taking the difference of the distributions withrespect to two beam helicities, the efficiency term cancels in the first order. Assumingthe efficiency is fairly uniform over the FPP so that the higher order terms can beignored, the asymmetry difference distribution has the simple form: f diff = f + − f − ≈ π [ A y ( P fppx cos φ − P fppy sin φ )] . (3.42)This expression can be written equivalently as: f diff = C cos( φ + δ ) , (3.43)where: C = 1 π (cid:113) ( P fppx ) + ( P fppy ) tan δ = P fppy P fppx . (3.44)In the simple dipole approximation (Eq. 3.41), P fppy is equal to the transverse com-ponent at the target frame P y , which is proportional to the product G Ep G Mp , and P fppx is related to the longitudinal component which is proportional to G Mp , via P fppx = sin χP z . Therefore, the phase shift of the helicity difference distribution is adirect measure of G Ep /G Mp : G Ep G Mp = K P y P z ≈ K sin χ (cid:32) P fppy P fppx (cid:33) , (3.45)where K = E + E (cid:48) m tan ( θ e / f diff and a fit to the data. The blacksolid curve is a sinusoidal fit to the data (K6, δ p = 0%), with a χ of 0.94 per degreeof freedom. The dashed light blue curve is a hypothetical distribution assuming µ p G Ep /G Mp = 1, as predicted by the dipole model. By zooming in this figure, one cansee a small but clear deviation between these two curves in Fig. 3-27, which is a direct143 [deg] fpp φ - / N - - N + / N + N -0.3-0.2-0.100.10.20.3 / ndf χ ± -0.1102 p1 0.0009414 ± -0.1287 / ndf χ ± -0.1102 p1 0.0009414 ± -0.1287 Figure 3-26: Asymmetry difference distribution along the azimuthal scattering angle φ fpp at kinematics K6 ( Q = 0 . ). The black solid curve represents the sinu-soidal fit to the data ( χ /ndf = 0 . µ p G Ep /G Mp = 1 in dipole approximation.indication that the form factor ratio deviates from unity in dipole approximation. Full Spin Precession Matrix and COSY
In reality, the spectrometer magnets are more complicated than just a simple perfectdipole. First, the field is not uniform inside the dipole, it is distorted by the fringefields at the entrance and exit apertures. In addition, there are three quadrupolesthat have field components in both x and y directions; hence, the matrix that relatesthe two polarizations measured in the FPP and in the target frame takes the generalform: P fppx P fppy P fppz = S xx S xy S xz S yx S yy S yz S zx S zy S zz P x P y P z . (3.46)The coefficients S ij depend on the trajectory of the proton as it passes through thespectrometer. Within the HRS acceptance, the protons recoiling with different angles144 [deg] fpp φ
100 110 120 130 140 150 160 170 180 - / N - - N + / N + N -0.1-0.0500.050.1 Figure 3-27: Close up view of Fig. 3-26. The black solid curve represents the sinu-soidal fit to the data, while the dashed light blue curve corresponds to a hypotheticaldistribution assuming µ p G Ep /G Mp = 1 in dipole approximation. There is ∼ ◦ shiftbetween these two curves at the zero crossing.and momenta at the target frame have different trajectories inside the spectrometer,they experience different magnetic fields along their trajectories, and hence, theirspin precession is different. Therefore, the coefficients are calculated event by eventto account for this difference in path length.The COSY model was used to calculate the spin precession matrices. It is a differ-ential algebra-based code written by M. Berz of the Michigan State University [157].This model is originally developed for the simulation, analysis and design of particleoptics systems. COSY takes the dimensions and positions of the magnetic elements,such as the diameter and the path length of the magnet, the central momentumof the particle, etc. as the inputs. The fringe fields are also taking into accountby a set of coefficients that were determined from measurements when Hall A wascommissioned. With all these ingredients, COSY calculates a table of the expansioncoefficients C klmnpij of the rotation matrix. This matrix is calculated event by eventbased on the particle trajectory variables located at the target coordinate system145TCS), which is defined in Section 3.2.1: S ij = (cid:88) k,l,m,n,p C klmnpij r k r l r m r n r p (3.47)where: r = x (3.48) r = p x /p (3.49) r = y (3.50) r = p y /p (3.51) r = δ K = ( K − K ) /K . (3.52) x and y are the positions, p and K are the particle momentum and kinetic energy respectively. From Eq. 3.46, the transverse polarization component at the focal planeis P fppy = S yy P y + S yz P z . Compared to the dipole approximation, the non-zero term S yz brings the contribution from the longitudinal target component P z ; this termis mainly due to the precession of the spin in the non-dispersive direction from thequadrupoles, which is neglected in the dipole approximation.The spin rotation matrix given by COSY only relates the polarization at the targetcoordinate system (TCS) to the transport coordinate system (TRCS). Therefore, twoaddition rotations, from the target scattering frame to the target coordinate system(TCS) and from the transport coordinate system (TRCS) to the focal plane frame atthe FPP, are needed.First, we need to express the proton track in the TCS. As illustrated in Fig. 3-28,the target scattering frame is defined as: (cid:126)x = (cid:126)k i × (cid:126)k f | (cid:126)k i × (cid:126)k f | (cid:126)y = (cid:126)z × (cid:126)x The particle mass is assigned in the code so that the matrix is calculated according to the correctmomentum. rans z (cid:38) trans x (cid:38) trans y (cid:38) y (cid:38) x (cid:38) qz (cid:38)(cid:38) (cid:32) (cid:73) (cid:92) (cid:92) spec (cid:52) Beam direction
Spectrometer axis
TCSTarget frame
Figure 3-28: The target scattering coordinate system (solid lines) is the frame wherethe polarization is expressed while the TCS (dashed lines) is the one in which COSYdoes the calculation. (cid:126)z = (cid:126)k i − (cid:126)k f | (cid:126)k i − (cid:126)k f | , (3.53)where (cid:126)k i and (cid:126)k f are vectors along the incident and scattered electron momenta, re-spectively. In the elastic case, (cid:126)q is the vector along the momentum of the recoilproton: (cid:126)q = (cid:126)k i − (cid:126)k f , (3.54)so that: (cid:126)k i × (cid:126)k f = (cid:126)k i × (cid:126)k i − (cid:126)k i × (cid:126)q = (cid:126)q × (cid:126)k i . (3.55)Eq.3.53 becomes: (cid:126)x = (cid:126)q × (cid:126)k i | (cid:126)q × (cid:126)k i | (cid:126)y = (cid:126)z × (cid:126)x(cid:126)z = (cid:126)q | (cid:126)q | . (3.56)In the lab frame, (cid:126)k i is the beam direction, which is along the z -axis. The momentum147ransfer (cid:126)q is in the direction of the outgoing proton, they both can be expressed inthe TCS: (cid:126)k i = − sin Θ spec cos Θ spec , (cid:126)q = sin ψ cos ψ sin φ cos ψ cos φ . (3.57)Finally, the matrix of the transformation from the target frame to the TCS, T canbe obtained by Eq. 3.53 and Eq. 3.57.Second, we need to perform a rotation from the TRCS to the FPP local frame,whose z -axis is along the proton momentum. In a similar way as defined in Fig. 3-12, the transformation can be done by a rotation around the x -axis by an angle φ f ,which is then followed by a rotation by an angle ψ f around the new y -axis. For thistransformation, the coordinates are related by the matrix T : P fppx P fppy P fppz = cos φ f − sin ψ f sin φ f − sin ψ f cos φ f φ f − sin φ f sin ψ f cos ψ f sin φ f cos ψ f cos ψ f (cid:124) (cid:123)(cid:122) (cid:125) T P trx P try P trz . (3.58)Therefore, the total rotation matrix S consists of T , T and the spin rotation matrix S sp given by COSY. The measured polarization at the focal plane can be expressedas P fpp = T S sp T (cid:124) (cid:123)(cid:122) (cid:125) S P tg . (3.59)As an example, Fig. 3-29 shows the four major elements of the full spin transportmatrix for one of the kinematic settings. With the scattering angles reconstructed by the FPP and the rotation matrix cal-culated by COSY, we are able to extract the polarization components at the target.There are 3 different methods to extract the polarization observables, as discussedin [151]. For the transferred polarization analysis, the weighted-sum method is used.148 y S-0.25 -0.2 -0.15 -0.1-0.05 0 0.05 0.1 0.15 0.2 0.2505001000150020002500300035004000 xz S-1 -0.8 -0.6 -0.4 -0.2 00200400600800100012001400 yy S0.97 0.975 0.98 0.985 0.99 0.995 1 1.0050200040006000800010000 yz S-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.205001000150020002500300035004000
Figure 3-29: Histograms of the four spin transport matrix elements, S xy (upper left), S xz (upper right), S yy (lower left) and S yz (lower right) at Q = 0 . for theelastic events. The ones plotted in black are from dipole approximation, and the onesin red are from the full spin transport matrix generated by COSY. For the dipoleapproximation, S xy and S yz are exactly zero, and S xy = 1 by ignoring the transversecomponents of the field. The full spin precession matrix gives broad distributions forthese elements which represent the effect from the quadrupoles and the dipole fringefield. 149he advantage with this technique is that by using different beam helicities, we canignore the efficiency of the acceptance in extracting the transferred polarization. Thedetailed formalism of the weighted-sum method will be presented in this section. Weighted-sum
As noted earlier, the probability that a proton scatters in the analyzer with angles( θ, φ ) with a polarization ( P fppx , P fppy ) is given by Eq. 3.37: f ± ( φ ) = 12 π (cid:15) (1 ± A y ( P fppy sin φ − P fppx cos φ )) , (3.60)where (cid:15) is the normalized instrumental efficiency (acceptance): (cid:15) ( φ i ) = f + + f − π . (3.61)By considering the spin transport, the probability function can be written in termsof the polarization components at the target frame: f ( φ ) = 12 π (cid:15) (1 + λ x P tgx + λ y hP tgy + λ z hP tgz ) , (3.62)where λ x = A y ( S yx sin φ − S xx cos φ ) λ y = ηhA y ( S yy sin φ − S xy cos φ ) λ z = ηhA y ( S yz sin φ − S xz cos φ ) , (3.63)where η is the sign for the beam helicity, and h is the beam polarization. Notethat the contribution from the induced (normal) polarization P tgx is beam helicityindependent. In the Born approximation, P tgx = 0; hence, Eq. 3.62 reduces to: f ( φ ) = 12 π (cid:15) ( φ )(1 + λ y hP tgy + λ z hP tgz ) . (3.64)150s derived in [151], for Eq. 3.64, with different beam helicities we can alwaysconstruct an effective acceptance that has a symmetry period of π in φ so that theacceptance (cid:15) cancels in the integral. We can obtain the equations: (cid:90) π f ( φ ) λ y dφ = hP tgy (cid:90) π f ( φ ) λ y dφ + hP tgz (cid:90) π f ( φ ) λ y λ z dφ (3.65) (cid:90) π f ( φ ) λ z dφ = hP tgy (cid:90) π f ( φ ) λ y λ z dφ + hP tgz (cid:90) π f ( φ ) λ z dφ, (3.66)since for n + m odd, (cid:90) π (cid:15) ( φ ) sin m φ cos n φdφ = 0 . (3.67)By replacing the integrals in Eqs. 3.66 with corresponding sums over the observedevents, we have (cid:80) i λ y,i (cid:80) i λ z,i = (cid:80) i λ y,i λ y,i (cid:80) i λ z,i λ y,i (cid:80) i λ y,i λ z,i (cid:80) i λ z,i λ z,i P tgy P tgz . (3.68)With the accumulation of a large event sample, Eq. 3.68 can be solved to obtain P tgy and P tgz . Eq. 3.68 is rewritten as: B = M · PP = M − · B . (3.69)The statistical error is given by:∆( P i ) = (cid:113) ( M − ) ii (3.70)with i = y, z , and the correlation factor between the two is ρ ij = ( M − ) ij (cid:113) ( M − ) ii ( M − ) jj . (3.71)151hen the form factor ratio is given by: µ p G Ep G Mp = Kr (3.72)where a = P y , and b = P z , and r = a/b . K is the kinematic factor: K = − µ p E e + E e (cid:48) M p tan θ e . (3.73)The statistical errors are calculated by:∆( G Ep G Mp ) = (cid:115) ( drda ) (∆ a ) + ( drdb ) (∆ b ) + 2 ρ drda ∆ a drdb ∆ b (3.74)where: drda = K bdrdb = − K ab . (3.75)The weighted-sum technique is valid under the condition that there is no inducedpolarization in the physics asymmetry, since this helicity independent term breaksthe symmetry period of (cid:15) . In reality, non-zero P tgx may arise from the 2 γ exchangeprocess. From a detailed study which considered the non-zero induced polarizationin Appendix C, we have concluded that the weighted-sum method is valid given therequired precision. From Eq. 3.72, one can see that since the polarization components are measured si-multaneously, for the ratio of P y and P z , the knowledge of the beam polarization h and the analyzing power A y , which cancel out in the ratio is not necessary. However,certain properties of the analyzing power are useful in giving the correct statisticaluncertainty. As noted earlier, the analyzing power A y depends only on the scattering152ngle θ fpp and the proton kinematic energy T p . For example, by taking the θ fpp depen-dence of the analyzing power into account provides more weight to events scattered atangles corresponding to high analyzing power and less weight to events scattered atsmaller angles with low analyzing power, which is dominated by Coulomb scattering.Although the absolute value of the analyzing power is irrelevant in the extractionof the form factor ratio, it is a byproduct of this measurement. In the first pass ofthe data analysis, the analyzing power A y was ignored by setting it to be 1. Since thebeam polarization is well know from the Møller measurements and is included in theanalysis, the solutions of Eq. 3.68 become A y P y and A y P z . We can rewrite the protonpolarization components as a function of the form factor ratio only, independent ofthe beam polarization and the the analyzing power: P y = − (cid:113) τ (1 + τ ) tan θ e G E G M G E + ( τ /(cid:15) ) G M = − (cid:113) τ (1 + τ ) tan θ e G E G M ( G E G M ) + ( τ /(cid:15) ) (3.76) P z = E + E (cid:48) m (cid:113) τ (1 + τ ) tan θ e G M G E + ( τ /(cid:15) ) G M = E + E (cid:48) m (cid:113) τ (1 + τ ) tan θ e ( G E G M ) ( τ /(cid:15) ) . (3.77)With the measured ratio G E /G M , we can calculate P y and P z . By comparing themwith the measured A y P y and A y P z , we can extract the analyzing power: A y = α a b + βb (3.78)∆ A y = (cid:118)(cid:117)(cid:117)(cid:116)(cid:32) dA y da (cid:33) (∆ a ) + (cid:32) dA y db (cid:33) (∆ b ) + 2 ρ dA y da ∆ a dA y db ∆ b, (3.79)with: dA y da = 2 α ab (3.80) dA y db = − α (cid:18) ab (cid:19) + β, (3.81)153here α = E e + E e (cid:48) m (cid:113) τ (1 + τ ) (3.82) β = τ [1 + 2(1 + τ ) tan ( θ e )] E e E e (cid:48) m (cid:113) τ (1 + τ ) tan ( θ e ) . (3.83)In the above expression, ρ is the correlation factor as defined in Eq. 3.71, and a = A y P y and b = A y P z are the output of the first pass analysis with A y = 1.A parameterization of the analyzing power for large solid angle spectrometers wasfirst suggested by Ransome et al. [158] and was later expanded by McNaughton etal. [54] for inclusive p C experiments at Los Alamos. The parameterization is dividedat T p = 450 MeV into a “low energy region” and a “high energy region”, where T p isthe proton kinetic energy at the center of the carbon analyzer.For the low energy fit, the suggested fitting function in [54, 158] is: A y = ar br + cr , (3.84)where r = p p sin( θ fpp ) and p p is the proton momentum in GeV/ c at the center of thecarbon analyzer. The coefficients a , b , c are polynomials of the momentum. In 2006,the LEDEX [53] experiment extracted the carbon analyzing power for proton energiesfrom 82 to 217 MeV. A similar functional form as shown in Eq. 3.84 was used: A y = ar br + cr + dr , (3.85)where the dr term was added in order to improve the quality of fit. The coefficientsare expanded as follows: a = (cid:88) i =0 a i ( p p − p ) i (3.86) b = (cid:88) i =0 b i ( p p − p ) i (3.87)154 = (cid:88) i =0 c i ( p p − p ) i (3.88) d = (cid:88) i =0 d i ( p p − p ) i , (3.89)where p , a i , b i , c i and d i are the parameters of the fit.For this experiment, we have much better statistics and much larger proton energycoverage (90 to 360 MeV). We used the same functional form in Eq. 3.85. Theanalyzing power was extracted by binning the data with respect to θ fpp and T p , andthe average values of each bin were used to fit the parameters. The parameterizationbased on the new data is provided in Appendix D.As illustrated in Figs. 3-30 and 3-31, the analyzing power in the low energy region( T p <
130 MeV) rises slowly with respect to the scattering angle θ fpp . For T p > ◦ and decreases rapidly at very smallangles and angles larger than 25 ◦ . In the final analysis, events with angle below 5 ◦ and above 25 ◦ were rejected.The new parameterization based on this experiment is in good agreement withboth the McNaughton [54] and LEDEX [53] parameterizations in the energy/angleregimes for which they were intended, considering all fits were done for differentpolarimeters and for different carbon block thicknesses. Compared to the older fits,the new parameterization extends the kinematics coverage and provides a smoothtransition from the low energy to the high energy region.The statistical uncertainty of the ratio µ p G Ep /G Mp depends on the uncertaintyof the asymmetries’ amplitudes at the focal plane hA y P fppx , hA y P fppy , which is pro-portional to the number of events N that contribute to the amplitude via the stronginteraction in the analyzer: ∆( hA y P fppx ( y ) ) ∝ (cid:115) N . (3.90)First we define the efficiency of the polarimeter (cid:15) ( θ ) = N eff ( θ ) N . (3.91)155 [deg] fpp θ y A Tp=0.1060 GeV [deg] fpp θ y A Tp=0.1151 GeV [deg] fpp θ y A Tp=0.1265 GeV [deg] fpp θ y A Tp=0.1352 GeV [deg] fpp θ y A Tp=0.1454 GeV [deg] fpp θ y A Tp=0.1545 GeV [deg] fpp θ y A Tp=0.1547 GeV [deg] fpp θ y A Tp=0.1695 GeV
Figure 3-30: Analyzing power fit part 1: A y plotted with different parameterizationin the low energy region ( T p <
170 MeV). The error bars shown are statistical only.The dashed lines are from the LEDEX [53] parameterization, the dashed dotted linesare from the “low energy” McNaughton parameterization [54], and the solid lines arefrom the new parameterization for experiment E08-007.156 [deg] fpp θ y A Tp=0.1913 GeV [deg] fpp θ y A Tp=0.2070 GeV [deg] fpp θ y A Tp=0.2122 GeV [deg] fpp θ y A Tp=0.2290 GeV [deg] fpp θ y A Tp=0.2508 GeV [deg] fpp θ y A Tp=0.2642 GeV [deg] fpp θ y A Tp=0.3312 GeV [deg] fpp θ y A Tp=0.3476 GeV
Figure 3-31: Analyzing power fit part 2: A y plotted with different parameterizationin the high energy region ( T p >
170 MeV). The error bars shown are statistical only.The dashed lines are from the LEDEX [53] parameterization, the dashed dotted linesare from the “low energy” McNaughton parameterization [54], and the solid lines arefrom the new parameterization for experiment E08-007.157 is the number of incoming protons, and N eff is the number of valid outgoing tracksthat passed a series of the FPP cuts (cone-test, zclose , sclose , etc.) and scatteredwith a polar angle θ . In other words, N eff ( θ ) is the effective number of events whichparticipated in the measurement of the asymmetry.Since the analyzing power A y has a dependence on the scattering angle θ , fromEq. 3.90 the effective number of events has to be multiplied by a weight, A y ( θ ); hence,the weighted effective number of events N ( θ ) is N ( θ ) = N (cid:15) ( θ ) A y ( θ ) . (3.92)The total effective number of events N is obtained by integrating over the scatteringangle θ : N = (cid:90) N ( θ ) dθ = N (cid:90) θ max θ min (cid:15) ( θ ) A y ( θ ) dθ = N · F OM, (3.93)where
F OM = (cid:90) θ max θ min (cid:15) ( θ ) A y ( θ ) dθ (3.94)is the Figure of Merit (FOM) and is an intrinsic characteristic of the polarimeter.Then, Eq. 3.90 can be expressed as:∆( hA y P fppx ( y ) ) ∝ (cid:115) N = (cid:115) N · F OM (3.95)The weighted average analyzing power (cid:104) A y (cid:105) for T p = 90 to 360 MeV is shown inFig. 3-32, and the FOM for each kinematics is summarized in Table 3.1158 p [GeV]0.1 0.15 0.2 0.25 0.3 0.35 > y < A Figure 3-32: Weighted average analyzing power (cid:104) A y (cid:105) with respect to T p for scatteringangles 5 ◦ ≤ θ fpp ≤ ◦ .Table 3.1: FPP performance for E08-007 with 5 ◦ < θ fpp < ◦ . T p is the protonaverage kinetic energy at the center of the carbon door.Kinematics Q [(GeV /c ) ] T p [MeV] (cid:104) A y (cid:105) (cid:15) fpp [%] FOM [%]K1 0.35 141.2 0.3938 3.67 0.57K2 0.30 109.8 0.2191 5.30 0.25K3 0.45 195.4 0.4876 4.09 0.97K4 0.40 165.3 0.4662 4.36 0.95K5 0.55 252.5 0.4305 4.34 0.81K6 0.50 221.4 0.4659 3.81 0.83K7 0.60 282.2 0.3923 4.41 0.68K8 0.70 335.6 0.3343 4.74 0.5315960 hapter 4Data Analysis II In this chapter, the inelastic background, systematic errors, and the radiative effectswill be discussed in detail.
In addition to the ep elastic events, there are three major types of background thatcan potentially contaminate the measurement. The first background is the scatteringoff the aluminum (Al) end cap of the liquid hydrogen (LH ) cell through the reaction Al( (cid:126)e, e (cid:48) (cid:126)p ); the second is the accidental background under the coincidence timingpeak, and the final one is from the photoproduction of pions. In this section, thebackground analysis and the impact to the final results are discussed.
To estimate the Al background from the target end cap, we took Al dummy runsfor every kinematic setting. The elastic polarization results need to be corrected ifthere is a significant amount of Al events passing the cuts, which can have a differentproton polarization. The corrected target polarization P y ( z ) is calculated by using: Y el. = Y H − Y Al , (4.1) Y el. P y,el. = Y H P y,H − Y Al P y,Al , (4.2)161 el. P z,el. = Y H P z,H − Y Al P z,Al , (4.3)where Y is the normalized yield. First, we first need to estimate the fraction of Alevents in the elastic data to obtain the corrected proton polarization. In order tobe consistent with the elastic proton polarization extraction, the same relevant cutswere applied: • HRS acceptance cut ( φ tg , θ tg , δ p ). • Coincidence event type cut (T5). • Coincidence timing cut. • Elastic proton peak on dpkin;The fraction of Al in LH data was estimated by using the charge normalizationmethod . By assuming the running conditions (beam energy, position and size, trig-ger setup, etc.) were the same between the LH and the Al dummy run and thepolarization of the background polarization is independent of the reaction location( y tg ), the fraction of Al in LH can be extracted by: R = Y Al /Y H = f · N Al × C H × (1 − DT H ) N H × C Al × (1 − DT Al ) , (4.4)where N H ( Al ) is the number of events in the LH (Al) run after applying the samecuts , C H ( Al ) is the charge, and DT H ( Al ) is the DAQ dead time. In the expression, f is the ratio of the Al foil thickness for the LH and the Al dummy target. From theAl foil thicknesses reported in Table. 4.1, f = 0 . , R , for each kinematic setting is summarized in Table 4.2 . These are the Due to the small acceptance of the HRS, it’s difficult to select a pure Al sample spectra in LH data; hence, the normalization factors obtained from comparing the Al and LH spectra could highlyoverestimate the Al contamination. These include the HRS acceptance cut, coincidence trigger and timing cut, but no target vertexcut was applied to avoid the inconsistency due to the position shift between the LH target cell andthe Al dummy target. The first two δ p settings of kinematics K1 were with the entire BigBite shower counter on; hence,more Al background was included for these data compared to the other kinematic settings, whichhad only a limited set of shower blocks turned on. (6cm) 0.0113Al dummy (6cm) 0.100Table 4.2: The upper limit of the Al background fraction R max for each kinematics.The numbers listed are the average over all δ p settings.Kinematics Q [(GeV /c ) ] R max K1 0.35 0.0021K2 0.30 0.0001K3 0.45 0.0001K4 0.40 0.0001K5 0.55 0.0001K6 0.50 0.0001K7 0.60 0.0001K8 0.70 0.0001upper limits of R , since in the elastic analysis a cut on the target reaction vertexwas applied ( y tg ), and the events from the target end caps were further suppressed.Fig. 4-1 illustrates the spectrum of y tg for the LH and Al dummy runs respectivelywith the location of the target vertex cut indicated by the vertical lines. Fig. 4-2gives an example of the normalized LH and Al spectra after applying all the cuts(including the target vertex cut).The recoil proton polarization of the LH and Al dummy targets for each kinemat-ics were extracted. As an example, the results of kinematics K1 ( Q = 0 .
35 GeV )are reported in Table 4.3. As can be seen that the correction to the elastic form fac-tor ratio µ p G E /G M is less than 0.001, which is negligible compared to the statisticalerror. The corrections for the other kinematic settings are at the same level. In this experiment, the coincidence trigger helped to significantly reduce the inelasticbackground. The accidental background can be estimated by using the same method163 [m] tg y-0.05 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05 C oun t s cut tg y LH [m] tg y-0.05 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05 C oun t s Al Dummy
Figure 4-1: The y tg spectrum for LH and Al dummy data with the cut shown by thevertical solid lines. 164 pkin-0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 C oun t s LH Al proton dpkin cut
Figure 4-2: The normalized dpkin spectrum for LH and Al dummy at setting K2 δ p = − P y ( z ) of LH , Al dummy and corrected values for kinematicsK1 ( Q = 0 .
35 GeV ).Pol. LH Al LH corrected P y -0.2624 ± . ± ± . P z ± . ± ± . ≤ . ≤ . Q = 0 . ), the results are reported in Table 4.4.165able 4.4: Polarization P y ( z ) of LH inside, outside the coincidence timing cut andthe corrected values for kinematics K8 ( Q = 0 . ).Pol. LH Accidental LH corrected P y -0.3636 ± . ± ± . P z ± . ± ± . Due to the reduced detector configuration of the BigBite spectrometer, events cannotbe easily distinguished between an electron or a photon that decayed from a π ,which fired the shower counter, since the coincidence trigger could be formed bypion photoproduction via γ + p → p + π . A study was made to estimate the pioncontamination which is elaborated in Appendix E. Due to the small acceptance andhigh resolution of the HRS combined with the tight elastic cut applied on the protonkinematics, we have concluded that the contribution from pion photoproduction isless than 10 − , and the correction to the proton polarization is also at < − level,which is negligible.As a simple demonstration to test whether the results are sensitive to the elasticcut applied in this work, 3 different cuts were applied on the peak of the proton dpkinas shown in Fig. 4-3: ± . σ , ± . σ and ± . σ . As shown in Fig. 4-4, the resultswith different elastic cuts are consistent within the statistical uncertainty. In the finalanalysis, a ± . σ cut was applied. For this experiment, the proposed statistical uncertainties were achieved ( ≤ pkin-0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 C oun t s = 0% p δ K6 = 0.5 GeV Q Figure 4-3: Different elastic cuts on proton dpkin. ] [GeV Q0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 R δ -0.02-0.015-0.01-0.00500.0050.010.0150.02 σ ± dpkin cut σ ± dpkin cut σ ± dpkin cut Figure 4-4: The ratio difference with different elastic cuts. The y -axis is the differencebetween the results, the x -axis which was manually shifted for different cuts for abetter view. 167 Spin precession: HRS optics and the COSY model. • Scattering angle reconstruction in the FPP. • Beam energy and HRS mis-pointing. • Charge asymmetry.The helicity independent factors such as the acceptance, beam current, target den-sity etc., cancel in the polarization ratio. The beam energy and the spectrometersetting are used to calculate the kinematic factors; however, the form factor ratio isless sensitive to these parameters with the current experimental precision. The spinprecession and the FPP reconstruction are directly related to the extraction of theproton polarization at the target, and therefore, they are the most important compo-nents for this type of measurement. In this section, the analysis for all the significantsystematic uncertainties will be discussed.
What we measured in this experiment is the proton polarization detected at the focalplane: P fppx , P fppy . However, the polarization at the target is directly related to thephysics of interests. In reality, the magnetic structure of the spectrometer is morecomplicated than just a simple perfect dipole; COSY [157] was used to calculatethe the full precession matrix S ij to relate the polarization at the target to the onedetected at the focal plane by Eq. 3.46.To calculate the matrix S ij , two inputs are required. The first input is a table ofthe expansion coefficients C klmnpij which is generated by COSY, and the second is thetarget coordinates of each event which are reconstructed by the HRS optics matrix.Hence, it is natural to separate the spin precession systematics error into two parts:HRS optics and COSY. 168able 4.5: Shifts of the form factor ratio associated with shifts of the individual targetquantities for each kinematic setting.Kinematics δ p (+0.001) φ tg (+1 mrad) θ tg (+1 mrad) y tg (+1 mm)K1 0.0015 0.0064 -0.0004 0.0011K2 0.0018 0.0064 -0.0002 0.0011K3 0.0005 0.0064 -0.0006 0.0015K4 0.0013 0.0066 -0.0005 0.0010K5 0.0004 0.0064 -0.0009 0.0019K6 0.0006 0.0064 -0.0008 0.0015K7 0.0007 0.0064 -0.0010 0.0022K8 0.0005 0.0064 -0.0014 0.0027 HRS Optics
The optics database used for this experiment was optimized for experiment E06-010 [144] . We used two steps to estimate the systematic uncertainty due to theoptics. First, the uncertainties in the central deviation of each target quantity(∆ δ p , ∆ φ tg , ∆ θ tg , ∆ y tg ) were estimated. Then they were shifted separately by theamount of the estimated uncertainties to determine the impact on the form factorratio µ p G E /G M . The sensitivities of the ratio µ p G E /G M to each target quantityare summarized in Table 4.5. Clearly φ tg is the most important quantity and hencerequires additional attention.To evaluate the quality of the optics, especially the uncertainty in φ tg , we takeadvantage of the proton elastic kinematics, since the angle is well constrained when thebeam energy and the proton momentum are fixed. Beforehand, we need to evaluate allthe parameters which are relevant in determining φ tg and convert their uncertaintiesinto ∆ φ tg ( x ). Then, the offset between the anticipated proton elastic peak positionand the reconstructed proton spectrum is quoted as ∆ φ tg ( of f ). The total error in Experiment E06-010 took the optics data for the left HRS at a similar momentum setting ( p =1.2 GeV). We also have the optics acquired in 2000 during experiment E89-044 [159], which was alsocarefully optimized. Both sets of optics were utilized and produced similar results, which indicatesthat the spectrometer optics reconstruction is fairly stable over the past ten years. tg is quoted conservatively as:∆ φ tg = (cid:113) ∆ φ tg ( x ) + ∆ φ tg ( of f ) . (4.5)The relevant parameters which would affect the anticipated proton elastic peakposition are: • Spectrometer central angle θ s . • Beam energy E e . • Proton central momentum P . • δ p reconstruction.Each one of them is discussed in the following subsections.1. Spectrometer Central AngleDue to the misplacement between the front and the end of the spectrometer duringmovement, the HRS central angle can be off by a small amount as illustrated in Fig 4-5. During the experiment, we took carbon foil data at each kinematics to determinethe spectrometer central angle. With the target position survey and ignoring thehigher order terms introduced by φ tg , we can determine the spectrometer horizontaloffset D from its ideal position by: z = − ( y tg + D ) / sin θ + x beam cot θ , (4.6)where x beam is the horizontal beam position. The y tg is the peak value, which is fitas shown in Fig. 4-6. The actual spectrometer angle θ s is corrected by D in the firstorder: θ s ≈ θ − DL , (4.7)where L is the distance between the hall center and the floor marks where the anglesare scripted (8.458 m). By considering the uncertainty of the survey ( ± (cid:84) L (cid:84) D (cid:84)(cid:39) z Hall center Spectrometercentral ray Beam tg y tg y Scattered electron
Figure 4-5: Coordinates for electrons scattering from a thin foil target. L is thedistance from Hall center to the floor mark, and D is the horizontal displacement ofthe spectrometer axis from its ideal position. The spectrometer set angle is θ andthe true angle is denoted by θ s when the spectrometer offset is considered. / ndf χ ± ± -0.0005287 Sigma 0.000037 ± [m] tg y-0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 C oun t s / ndf χ ± ± -0.0005287 Sigma 0.000037 ± Figure 4-6: Carbon pointing y tg for kinematics K8 ( Q = 0 . ).171able 4.6: Spectrometer nominal ( θ ) and real ( θ s ) central angle for each kinematicsetting. Kinematics θ [deg] θ s [deg] ∆ φ tg ( θ s ) [mrad]K1 57.5 57.478 ± ± ± ± ± ± ± ± ± ± y tg ( ± D is derived by:∆ D = (cid:113) ∆ y tg + sin θ ∆ z (4.8)The spectrometer central angle for each kinematics was corrected using the pointingmethod, and the results are reported in Table 4.6. One can see that the angle mis-pointing is small which is consistent with previous records [160]. This observationwas anticipated due to the large value of L .2. Beam EnergyDuring the experiment, the beam energy was given by the Tiefenbach value. Ac-cording to [131], the uncertainty for this non-invasive measurement is 0 . φ tg with ∆( E e ) = 0 . φ tg ( E e ) [mrad]K1 0.11K2 0.11K3 0.14K4 0.14K5 0.16K6 0.14K7 0.18K8 0.18for a 1.19 GeV beam is 1 . E e = ( E tiefenbach − . ± . . (4.9)Table 4.8 gives the converted uncertainty of φ tg for each kinematics due to the uncer-tainty of E e .3. Proton central momentum P The momentum we reconstructed is the relative momentum δ p , which refers tothe central momentum P . At the beginning of the experiment, we switched to NMRprobe D instead of probe A, which is typically used. From the calibration study at 1GeV/c, the offset between probe A and D is 1 . × − [161]. The NMR values foreach momentum setting are listed in Table 4.9.From a previous calibration study [162] with NMR probe A, we know that thecentral momentum P is fairly linear with the central magnetic field B . The relationbetween P and B is given by: P = Γ B + Γ B , (4.10) There is also some uncertainty in the value of the beam energy loss due to the possible non-uniformity of the material thicknesses; however, this uncertainty is much less than 0.5 MeV giventhe precision of the survey. B in kG with probe D for each momentumsetting. Kinematics δ p = − δ p = 0% δ p = 2%K1 2.647 2.595 2.543K2 2.426 2.378 2.330K3 3.049 2.989 2.929K4 2.869 2.813 2.757K5 3.410 3.342 3.276K6 3.229 3.166 3.102K7 3.591 3.521 3.450K8 3.923 3.846 3.769Table 4.10: Converted uncertainty in φ tg from P .Kinematics ∆ φ tg ( P ) [mrad]K1 0.12K2 0.13K3 0.13K4 0.13K5 0.12K6 0.12K7 0.13K8 0.12where B is measured in kG. For the left HRS, Γ = 270 . ± .
15, and Γ = − . × − ± . × − , which is much smaller than Γ . With probe D, a linear fit yields: P = Γ B d . (4.11)As shown in Fig. 4-7, the linearity was well preserved when the probed was switched.Based on the differences between the set values and the ones derived from the newfit, we conservatively estimate ± .
15 MeV/c as the uncertainty on the proton centralmomentum. The converted uncertainty in φ tg for each kinematics is summarized inTable 4.10.4. Proton momentum loss in the targetThe recoil protons passed through a few materials before they entered the spec-174 [kG] B2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4 M e V / c P P500 550 600 650 700 750 800 850 900 950 M e V / c P Δ -0.1-0.0500.050.1 Figure 4-7: NMR reading with probe D versus the central momentum setting (leftpanel), and the deviation between the value from the linear fit function and the setvalue.Table 4.11: Target materials that the proton passed through before entering thespectrometer. Material Thickness [cm]LH ± .
01 (radius)Al wall 0.0127 ± . ± . ± . ± . P loss for each momentum setting is summarized in Table 4.12. We conservativelyquote ± . /c as the uncertainty in the average proton momentum loss in thematerials by considering the uncertainty in the material thicknesses.5. δ p reconstructionThe last parameter we need to consider is the uncertainty of the reconstructedmomentum δ tg . From the optimization results [163], we conservatively quote ± × − as the uncertainty of δ p , and convert it to an uncertainty in φ tg . The results for eachkinematics are listed in Table 4.13.In Table 4.14, the uncertainty in φ tg converted from the uncertainties of the ex-175able 4.12: Proton momentum loss [MeV/c] for each kinematics.Kinematics δ p = − δ p = 0% δ p = 2%K1 3.69 3.83 3.98K2 4.30 4.48 4.67K3 2.95 3.05 3.16K4 3.23 3.35 3.48K5 2.54 2.62 2.70K6 2.72 2.81 2.91K7 2.39 2.46 2.54K8 2.19 2.25 2.31Table 4.13: Uncertainty of φ tg with ∆ δ p = 0 . φ ( δ p ) [mrad]K1 0.25K2 0.25K3 0.30K4 0.30K5 0.33K6 0.30K7 0.35K8 0.35ternal parameters as discussed above are given. ∆ φ ( x ) is defined as:∆ φ ( x ) = (cid:118)(cid:117)(cid:117)(cid:116) N (cid:88) i =0 ∆ φ ( x i ) , (4.12)where ∆ φ ( x i ) are the converted uncertainties in φ tg from the related parameters.The next step is to quote ∆ φ ( of f ), which is the average deviation of the replayedproton kienmatics from the anticipated elastic peak position. From Figure 4-8, we seethat the slope of the elastic strips generally matches the predicted slopes. The averageoffset across the acceptance between the predicted peak position and the center ofthe data ∆ φ ( of f ) is a combined effect of the optics and external parameters (beamenergy, proton momentum, etc.). The systematic uncertainty from the optics is given176able 4.14: Total uncertainty in φ tg from the external parameters.Kinematics ∆ φ ( E e ) ∆ φ ( θ s ) ∆ φ ( δ p ) ∆ φ ( P ) ∆ φ ( P loss ) ∆ φ ( x ) [mrad]K1 0.11 0.14 0.25 0.12 0.08 0.34K2 0.11 0.14 0.25 0.13 0.09 0.34K3 0.14 0.14 0.30 0.13 0.09 0.39K4 0.16 0.14 0.30 0.13 0.08 0.40K5 0.16 0.14 0.33 0.12 0.08 0.42K6 0.14 0.14 0.30 0.12 0.08 0.39K7 0.18 0.14 0.35 0.13 0.09 0.45K8 0.18 0.13 0.35 0.12 0.08 0.44by: ∆ φ tg = (cid:113) ∆ φ ( of f ) + ∆ φ ( x ) . (4.13)The final uncertainty in φ tg is summarized in Table 4.15. The uncertainties of the p δ -0.05 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05 ( deg ) p θ Figure 4-8: Proton scattering angle θ p versus the momentum δ p for kinematics K8 δ p = 0%. The anticipated elastic peak position is plotted as the black dash line.other target quantities ( θ tg , δ p , y tg ) are quoted according to their difference when thedata were replayed by using different HRS optics:∆ θ tg = 2 mrad , ∆ δ p = 0 . , ∆ y tg = 1 mm (4.14)177able 4.15: φ tg uncertainty for each kinematics.Kinematics ∆ φ off [mrad] ∆ φ ( x ) [mrad] ∆ φ tot [mrad]K1 1.30 0.34 1.34K2 0.76 0.34 0.83K3 0.87 0.39 0.95K4 0.87 0.40 0.96K5 0.70 0.42 0.67K6 0.87 0.39 0.95K7 1.22 0.45 1.30K8 1.05 0.44 1.14Table 4.16: Systematic uncertainty in R = µ p G E /G M for each kinematics associatedwith left HRS optics. Kinematics ∆ R (optics)K1 0.0087K2 0.0057K3 0.0062K4 0.0068K5 0.0051K6 0.0063K7 0.0090K8 0.0084Combining the results in Table 4.15, Eq. 4.14 and Table 4.5, the total systematicuncertainty from the left HRS optics is summarized in Table 4.16. COSY
Another source of systematic error, which is related to the spin precession, is COSY. Ifthe precession matrix determined by COSY is correct, the form factor ratio µ p G Ep /G Mp should not depend on any target quantities. As illustrated in Fig. 4-9, while the resultswith dipole approximation show a strong dependence on δ p and φ tg , COSY providesa nice correction to these quantities and gives a reasonable χ with a constant fit. Toestimate the systematic error of COSY, more detailed studies were carried out. TheCOSY systematic error was separated into two parts:178 δ -0.04 -0.02 0 0.02 0.04 M / G E G p μ χ ± χ ± COSYDIPOLE tg y-0.025 -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015 M / G E G p μ / ndf χ ± χ ± COSYDIPOLE tg θ -0.06 -0.04 -0.02 0 0.02 0.04 0.06 M / G E G p μ / ndf χ ± χ ± COSYDIPOLE tg φ -0.03 -0.02 -0.01 0 0.01 0.02 0.03 M / G E G p μ χ ± χ ± COSYDIPOLE
Figure 4-9: Dependence of µ p G Ep /G Mp on the proton target quantities for kinematicsK7 ( Q = 0 . ). The full precession matrix calculated by COSY (solid quare)is compared to the dipole approximation (open square) and a constant fit. The datapoints are shown with statistical error bars only. • The first one is associated with the spectrometer configuration and settingsdefined in the COSY input file. Through a series of tests, the most sensitiveparameters were identified. Then, those parameters were changed and the spinprocession was calculated in different ways to see the variation in the form factorratio. • The other part was determined from the COSY optics map, which also reflectsthe quality of the model. We used the target quantities reconstructed by COSYinstead of the ones from the ANALYZER to calculate the spin precession matrix S ij . 179ig. 4-10 demonstrates the alternative ways to calculate the spin precession and esti-mate the model’s systematics uncertainty. Focal Plane Coordinates
Target Coordinates I Target Coordinates II
ANALYZERoptics COSYopticsCOSY inputsSpin Precession Matrix S klmnpij C tg y ),,,( (cid:71)(cid:73)(cid:84) tg y )',',','( (cid:71)(cid:73)(cid:84) Figure 4-10: Alternative ways to calculate the spin precession matrix S ij .1. Configuration InputsIn the COSY input file, geometries and settings of the magnets were defined.Many of the parameters were determined by comparing them with the field maps. Wefocused on the ones that are either intuitive or examined in the previous study [164].The tested parameters included: • Dipole bending angle Θ . • Dipole radius. • Drift distances between magnets. • Quadrupole alignment coefficients. 180he dipole bending angle was found to be the most important parameter in thismeasurement; on the other hand, the impact from the other parameters was negligible.The default setting for the dipole bending angle is 45 ◦ . Ideally we should be able tocheck the central bending angle from the trajectory determined by the VDCs, whencombined with the VDC position survey [165]. However, it’s very difficult to definethe spectrometer central trajectory . To minimize bias, we cut on a very small regionof the central part of the HRS acceptance and treated the events in this region as thecentral trajectories. By fitting the out-of-plane angle difference ( θ tg − θ tr ) betweenthe target frame and the focal plane, the dipole central bending angle was verified.The cuts applied to select the central trajectories were: • − . < δ p < . • − . < y tg < . • − . < θ tg < . / ndf χ ± ± -5.917e-05 Sigma 0.00011 ± tr θ - tg θ -0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1 C oun t s / ndf χ ± ± -5.917e-05 Sigma 0.00011 ± Figure 4-11: Fit of the out-of-plane angle difference between the target and the focalplane. θ tr = θ det − ◦ (K6 δ p = 0%). The peak at zero corresponds to a 45 ◦ bendingangle in the spectrometer.the mean value of the peak is very close to 0, which corresponds to a 45 ◦ bending anglebetween the target and the focal plane. In a previous analysis [166], a ± Usually the central sieve hole position was used to define the spectrometer central trajectory. µ p G E /G M associated with COSY.Kinematics Bending angle (+5.5 mrad) COSY opticsK1 -0.0018 0.0012K2 -0.0012 0.0018K3 -0.0029 0.0011K4 -0.0022 0.0002K5 -0.0043 0.0005K6 -0.0035 0.0006K7 -0.0048 0.0004K8 -0.0062 0.0002uncertainty in the dipole bending angle Θ was quoted from a fit to the 180 ◦ rotationdata, and this uncertainty was also used in this analysis. Therefore, the central dipolebending angle Θ was changed by 5.5 mrad in the COSY input file and another setof spin precession matrices were generated to extract the ratio. The difference in theresulting form factor ratios was quoted as the systematic error associated with thecentral bending angle. The results are provided in Table 4.17.2. COSY optics mapCOSY not only generates the spin precession matrix but also produces the opticsmap. With the quantities measured at the VDCs, we could use the COSY optics mapto reconstruct the target quantities. The COSY reconstructed target quantities arein general agreement with the target quantities determined via the ANALYZER. Thecentral peak differences are a couple of mrad for the angles ( φ tg , θ tg ) and a couple ofmm for the position ( y tg ). By using the COSY reconstructed target quantities in thespin precession matrix calculation, the difference in the form factor ratio was quotedas another part of the systematic error from COSY. The results from this study arereported in Table 4.17. As previously mentioned, the second scattering angles at the FPP are directly relatedto the proton polarizations measured in the focal plane; to make sure the angles at182he FPP were determined correctly, the software alignment was completed, which iselaborated in Section 3.2.3. Ideally, the straight-through data should uniformly coverthe FPP chamber’s full acceptance. However, full coverage is difficult to achievefor the rear chambers due to their larger area, which was designed for the secondscattering; the lack of uniform coverage inevitably changes the weight of the dataover the acceptance and can affect the fits of the alignment coefficients.The misalignment of the chambers involves both offsets and rotations. For offsets,the effect is equivalent to a non-uniform acceptance A ( φ ), whose effect can be canceledby flipping the beam helicity. For rotations, we can separate them into two types asillustrated in Figs. 4-12 and 4-13. The chamber rotations along x and y -axis inducean elliptical acceptance which can also be absorbed in the non-uniform acceptance;hence, they are not our primary concern. The rotation along the z -axis, which is theparticle’s incident direction, will shift φ by an additional offset and cannot be canceled.This type of rotation will directly change the result of the ratio µ p G Ep /G Mp . x yx (cid:255) y (cid:255) (cid:73) ' (cid:73) FPP chamber (cid:73)(cid:39) xx (cid:255) y (cid:255) y Figure 4-12: FPP chamber rotation along z and the shift of the azimuthal angle φ .The events at the FPP are mostly dispersed in x -direction (vertical), whereas theyare close to zero in y . To make the estimation simpler, the reasonable assumption of y = 0 is made. As illustrated in Fig. 4-14, if there is a small rotation along z or x ,183 x ( y ) x (cid:255) ( y (cid:255) ) xy ' (cid:73)(cid:73) Figure 4-13: FPP chamber rotation along x ( y ) and the change of φ distribution.the difference in y between the VDC track and the FPP track will depend on x by: dy ≈ ∆ φ × x. (4.15)Before applying the software alignment, there is an obvious slope between dy and x as shown in Fig. 4-15, which indicates a rotation around z . After the softwarealignment, the slope is gone. If we zoom in and fit the spectrum after alignment, theresidual slope is at the 1 × − level as shown in Fig. 4-16. The same fit was appliedto the rear track, and the slope is at the same order of magnitude ( ∼ − × − ).Since in the first order this slope can only be caused by a rotation along z , weconservatively quote twice the residual slope value to be the uncertainty in the angleof rotation along z . The slopes of the front and rear alignment were added togetheras the final uncertainty of φ fpp , which is ∼ θ fpp with a constantfit, and there is no indication of any systematic dependence on this variable with thecurrent precision. 184 y (cid:73)(cid:39) x (cid:255) y (cid:255) dy Figure 4-14: The non-zero y component in the rotated frame. The VDC quantities θ tr and φ tr were used to calculate the spin rotation matrixbetween the transport frame and the FPP local frame. By manually shifting thesevariables, the systematic error on the ratio was obtained and reported in Table. 4.19. Charge Asymmetry
In the analysis code, we randomly throw out a small fraction of events with onebeam helicity state to test the sensitivity to the charge asymmetry. With the chargeasymmetry ( < ≤ . (cm) at z=0 cm-100 -80 -60 -40 -20 0 20 40 60 80 y _d i ff _ f ( c m ) -4-3-2-101234 050100150200250300350 Before alignment x (cm) at z=350 cm-100 -80 -60 -40 -20 0 20 40 60 80 y _d i ff _ f ( c m ) -4-3-2-101234 050100150200250300350 After alignment
Figure 4-15: The track difference in y versus x before and after the software alignment. x (cm) at z=350 cm-50 -40 -30 -20 -10 0 10 20 30 40 y _d i ff _ f ( c m ) -1.5-1-0.500.511.5 050100150200250300350 Figure 4-16: The track difference ( y ) and its profile versus x after the software align-ment. The solid line is a linear fit to the profile with a slope of 1 × − . Kinematics factors
From the form factor ratio formula: R = µ p G E /G M = − µ p P y P z E + E (cid:48) m p tan( θ e − µ p K P y P z , (4.16)knowledge of the kinematic factor K is also required. In the analysis code, theinitial inputs are the beam energy and the proton scattering angle ; therefore, thekinematic factor K is a function of E and θ p in this analysis. Based on the systematicstudies mentioned earlier, we quote ± . The reconstruction of the electron kinematics is not available due to reduced configuration ofthe BigBite spectrometer. µ p G E /G M . Kinematics θ fpp (+1 mrad) φ fpp (+1 mrad)K1 -0.0003 0.0018K2 0.0002 0.0018K3 0.0001 0.0018K4 -0.0002 0.0019K5 -0.0001 0.0018K6 -0.0002 0.0018K7 -0.0001 0.0019K8 -0.0001 0.0019 fpp θ P M / G P E G μ / ndf χ ± χ ± fpp θ P M / G P E G μ / ndf χ ± χ ± Figure 4-17: The form factor ratio binning on the FPP polar scattering angle θ fpp forkinematic setting K6 ( Q = 0 . ) and K7 ( Q = 0 . ).0 . ◦ as the proton scattering angle uncertainty. As an example, Table 4.20 lists theuncertainty of each factor and the resulting uncertainty in the ratio for one of ourkinematics (K7). Clearly, the change of the form factor ratio is negligible ( < . As a summary, Fig 4-18 shows the major uncertainties for each Q point. All thesecontributions are added quadratically to obtain the total systematic error for thisexperiment. Table. 4.21 presents the final results with both the statistical and sys-187able 4.19: Errors of the VDC angles and associated systematic error in µ p G E /G M .Kinematics θ tr (+1 mrad) φ tr (+1 mrad)K1 -0.0002 0.0002K2 -0.001 -0.0002K3 0.0003 -0.0002K4 -0.0004 -0.0002K5 -0.0006 -0.0004K6 -0.0005 -0.0003K7 -0.0002 -0.0004K8 -0.0001 -0.0007Table 4.20: Errors of the kinematic factors and the resulting uncertainty in the formfactor ratio R for kinematics K7 ( Q = 0 . ). δ R ( E )( ± . R ( θ )( ± . ◦ ) ∆ R Q increases, the systematic error starts to dominate the totaluncertainty. For electron scattering, the radiative process is inevitably involved. This includes theelectron initial and final state Bremsstrahlung, loop correction, as well as 2 γ exchangeeffects. The radiative correction to this experiment is discussed by providing theresults from recent theoretical calculations.Afanasev et al. [55] performed a numerical analysis for the radiative correctionsin elastic ep scattering when the kinematic variables are only reconstructed from therecoil proton. This study calculated the radiative correction to the cross sections andasymmetries differential in Q . Fig. 4-19 shows the correction to the longitudinaland transverse polarization components as a function of the inelasticity u m = ( k + p − p ) − m , where m is the electron mass, k is the electron initial momentum,and p is the initial (final) proton momentum at s = 8 GeV . The magnitude188 [GeV/c] Q0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 P M / G P E G μ Δ -0.015-0.01-0.00500.0050.010.0150.020.025 Stat.HRS OpticsCOSYFPP alignmentTotal Syst.
Figure 4-18: Comparison of the major contributions to the systematic uncertaintiesand the statistical uncertainty for each kinematics.of the correction does not exceed 1 . Q andinelasticity cut. Fig. 4-20 gives the correction to the measured ratio of final protonpolarization; the correction is negative and does not exceed 1%.Afanasev et al. [56] also estimated the 2 γ exchange contribution to elastic ep scattering at large momentum transfer by using a quark-parton representation ofvirtual Compton scattering. While the correction is significant for cross-section mea-surements, the impact upon the recoil polarization measurement is small. Fig. 4-21shows the calculated transferred proton polarization with and without the 2 γ ex-change terms, for 100% right-handed electron polarization and with a fixed Q of 5GeV .Blunden et al. [57, 58] performed an explicit calculation of the 2 γ exchange dia-gram in which nucleon structure effects were fully incorporated. They also appliedit to systematically calculate the effects in a number of electron-nucleon scatterings.Fig. 4-22 shows the relative correction of the proton form factors ratio µ p G E /G M asa function of ε at different Q .For the kinematic condition of this experiment ( Q < , 0 . < ε < . s = 3 .
12 GeV ), we have concluded that the radiative corrections to the form factor189able 4.21: Final results with statistical and systematic uncertainties for each kine-matics. Kinematics (cid:104) Q (cid:105) [(GeV /c ) ] R ∆ R sys. ∆ R stat. K1 0.3458 0.9433 0.0093 0.0088K2 0.2985 0.9272 0.0071 0.0114K3 0.4487 0.9314 0.0073 0.0060K4 0.4017 0.9318 0.0076 0.0066K5 0.5468 0.9274 0.0071 0.0055K6 0.4937 0.9264 0.0076 0.0056K7 0.5991 0.9084 0.0104 0.0053K8 0.6951 0.9122 0.0107 0.0045ratio is less than 0 .
3% based on the current theoretical calculations.190igure 4-19: Radiative corrections to the recoil polarization. The solid and dashedlines correspond to the longitudinal and transverse components with s = 8 GeV .Figure from [55]. 191igure 4-20: Radiative corrections to the ratio of the recoil proton polarization in theregion where the invariant mass of the unobserved state is close to the pion mass and s = 8 GeV . Figure from [55].Figure 4-21: The 2 γ exchange correction to the recoil proton longitudinal polarizationcomponents P l and the ratio of the transverse to longitudinal component for elastic ep scattering at Q = 5 GeV . Figure from [56].192 R Q = Figure 4-22: The relative correction to the proton form factor ratio from 2 γ exchangeas a function of ε for 5 different Q [57, 58].19394 hapter 5Discussion and Conclusion In this chapter, the experimental data are compared with the world data and variousmodels and fits. In addition, the impacts of the new results to other physics quantitiesare discussed, and the future outlook to access lower Q is also presented. Fig. 5-1 and 5-3 show the new results of this work, µ p G E /G M as a function of Q together with previous high precision measurements ( σ tot < Q = 0 . which will also be published soon is from one of the LEDEXexperiments E03-104 [167]. The new data have the following features: • The new results are in good agreement with the high precision point at Q = 0 . , which was taken in 2006 with a different configuration and analyzedindependently. • The whole data set slowly decrease along Q in the region of Q = 0 . ∼ . ; no obvious indication of any “narrow structure”. • The new data strongly deviate from unity by several percent which is unex-pected from the previous measurements. E03-104 used right HRS to detect the electron and the electron kinematics was well known. [GeV/c] Q0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 P M / G P E G μ Punjabi et al. (GEp-I)Crawford et al. (BLAST)Ron et al. (LEDEX)Paolone et al. (E03-104)E08-007 I
Kelly FitAMTArrington & Sick FitFriedrich & Walcher Fit
Figure 5-1: The proton form factor ratio µ p G E /G M as a function of Q with worldhigh precision data [16, 19, 20] ( σ tot < [GeV/c] Q0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 P M / G P E G μ Punjabi et al. (GEp-I)Crawford et al. (BLAST)Ron et al. (LEDEX)Paolone et al. (E03-104)E08-007 I
Miller LFCBMBoffi et al. PFCCQMBelushkin, Hammer & Meissner VMDLomon GK(05)Faessler et al. LCQM
Figure 5-2: The proton form factor ratio µ p G E /G M as a function of Q shown withworld high precision data [16, 19, 20] ( σ tot < et al. ) overlap with the new data in thevicinity of Q = 0 .
36 GeV , the highest Q point is ∼ σ above the new data. Toinvestigate this potential discrepancy, we reanalyzed the LEDEX data and found thatthe Al background was overestimated in the original analysis [70]; hence, the datawere overcorrected for dilution effect from the Al end cap. The preliminary results ofthe LEDEX reanalysis are in good agreement with the new data, and we expect topublish the erratum soon.The other two data points contributed by JLab in this region are from GEp-Imeasurement [16], which was performed in 1998. The point at Q = 0 . is ∼ . σ higher than the new results, and the point at Q = 0 . is ∼ . σ higher than the E03-104 result (Paolone et al. ). The investigation of the originalGEp-I analysis is still underway, which includes the consistency check of differentanalysis codes, the accuracy of the kinematic parameters , the discussion of the cutsand the systematic error analysis .Another discrepancy is in the comparison with the BLAST [19] results. The newdata are systematically lower by 2 to 3 σ , which is hard to explain by statisticalfluctuations. Since BLAST used the beam-target asymmetry technique, the origin ofthe systematic uncertainty is different. While the investigation of this discrepancyis needed, a third measurement by using the beam-target asymmetry technique inthis region is strongly recommended to uncover any unknown systematic errors in therecent measurements.In summary, Fig. 5-3 shows the new results plotted with a different scale togetherwith the world polarization data, which includes the preliminary results of the GEp-III measurement [60]. In the very early days of the Hall A running, the beam energy and spectrometer momentumwere not very well known. GEp-I used the right HRS to detect the recoil proton instead of the left HRS; therefore, theoptics and spin transport were different. [GeV Q -1
10 1 10 P M / G P E G μ Punjabi et al. (GEp-I)Gayou et al. (GEp-II)Crawford et al. (BLAST)LEDEX reanalysis (Preliminary)Paolone et al. (E03-104)GEP III (Preliminary)E08-007 I
Figure 5-3: The proton form factor ratio µ p G E /G M as a function of Q shown withworld high precision polarization data [16, 19, 20, 18, 60]. In Fig. 5-1 and Fig. 5-3, the data are shown together with a representative set of theexisting theoretical models and fits. Analytical fits from Kelly [42] and AMT [24]are based on the data over all Q , while the fits from Arrington and Sick [48] andFriedrich and Walcher [44] concentrate on the lower Q data. Due to the absence ofphysical interpretation and the dominance of the old data, it is plausible to expectthat the global fits are substantially above our new data. On the other hand, thenew results cannot completely rule out the existence of the structure given by thephenomenological fit of Friedrich and Walcher [44]; however, the average value of thestructure would be much lower then what they predicted if there is any.The existing theoretical models also cannot accurately predicts the results. Achiral constituent quark model by Boffi et al. [36], and a Lorentz covariant chiralquark model by Faessler et al. [51] are both above the new data. A Light-front cloudybag model by Miller [49], which includes the pion cloud effect, generally reproduces199he large deviation from unity in this region; however, this calculation decreases toorapidly compared to the data. The VMD calculations by Belushkin et al. [50] andLomon [59] are also above the new data. Although this type of calculation is knownto be very successful in representing the existing world data, the large number oftunable parameters in these models inevitably weaken the predictive power; therefore,the current disagreement is not surprising. To extract the individual form factors, the data must be combined with cross sectionmeasurements to determine the absolute magnitudes of G E and G M . From Eq. 1.40 σ red = ε (1 + τ ) dσ/d Ω( dσ/d Ω) Mott = εG E + τ G M , (5.1)if the ratio R = µ p G E /G M is completely fixed, there is only degree of freedom left inthe linear fit of the reduced cross section. A new set of G E and G M were extracted byforcing the ratio µ p G E /G M to be the experimental value of the new results (E08-007I and E03-104). The cross sections used in this extraction are listed in AppendixF. Fig. 5-4 shows the fits of the reduced cross sections at 9 different Q s, which arein the vicinity of the new ratio measurements. Table 5.1 provides the results of theextractions of G E and G M by the standard Rosenbluth separation and the constrainedfit. The new extracted G E and G M are plotted with the world data in Fig. 5-5.With the constraint of the new ratio results, the uncertainty of the individualform factors are significantly improved. While the new G E obviously deviates fromunity by a few percent, G M is slightly higher than the world unpolarized data, andboth of them show a relatively smooth evolution along Q in this region.However, forcing the fit to match the ratio results gives too much weight to thepolarization data. To avoid this issue, a global combined fit [168] was performed byJohn Arrington. This new fit followed the same procedure as in [24, 128, 169] with atreatment for the TPE effect in the cross section data. The χ of the combined fit is200 r ed σ = 0.292 GeV Q ε r ed σ = 0.350 GeV Q ε r ed σ = 0.389 GeV Q ε r ed σ = 0.467 GeV Q ε r ed σ = 0.506 GeV Q ε r ed σ = 0.545 GeV Q ε r ed σ = 0.584 GeV Q ε r ed σ = 0.701 GeV Q ε r ed σ = 0.779 GeV Q Figure 5-4: Rosenbluth separation of G E and G M constrained by R = µ p G E /G M .For each Q , the reduced cross section σ R is plotted against ε . The solid blue line isthe standard Rosenluth separation fit without any constraint on R . The dotted redline is fit with an exact ratio constraint. 201 [GeV/c] Q -2 -1
10 1 D / G P E G Qattan et al.Andivahis et al.Walker et al.Simon et al.Borkowski et al. Murphy et al.Bartel et al.Hanson et al.Price et al.Berger et al. Litt et al.Janssens et al.Christy et al.E08007 I + E03-104 [GeV/c] Q -1
10 1 10 D / G P M G Qattan et al.Andivahis et al.Walker et al.Bosted et al.Sill et al. Borkowski et al.Bartel et al.Price et al.Berger et al.Litt et al. Janssens et al.Christy et al.E08007 I + E03-104
Figure 5-5: The new extraction of G E and G M plotted together with the worldunpolarized data. 202able 5.1: The extracted values of G E and G M , with and without the constraint of µ p G E /G M from the new measurements. The errors are indicated in parentheses. Q Unconstrained LT Separation Constrained Fit[(GeV /c ) ] G E /G D G M /µ p G D χ /ndf G E /G D G M /µ p G D χ /ndf0.292 1.003(44) 0.936(22) 0.17 0.906(13) 0.977(11) 1.270.350 0.935(61) 0.971(25) 0.05 0.921(14) 0.976(12) 0.050.389 0.972(16) 0.965(11) 1.14 0.928(07) 0.995(05) 1.600.467 0.993(54) 0.972(20) 0.19 0.925(12) 0.993(10) 0.520.506 0.999(40) 0.957(25) 1.15 0.926(09) 0.999(08) 1.870.545 0.982(69) 0.983(20) 1.60 0.924(13) 0.997(11) 1.350.584 0.971(18) 0.984(08) 0.50 0.915(08) 1.007(05) 1.050.701 1.078(10) 0.981(21) 0.47 0.919(14) 1.007(11) 0.900.779 0.949(41) 1.004(12) 0.55 0.921(10) 1.012(06) 0.52the contribution from the cross section measurements plus the additional contributionfrom the polarization ratio measurements: χ = χ σ + N R (cid:88) i =1 ( R i − R fit ) ( dR stat ) + N exp (cid:88) i =1 (∆ j ) ( dR sys ) , (5.2)where R = µ p G E /G M , dR stat and dR sys are the statistical and systematics uncertain-ties in R , and R fit is the new ratio parameterization by including the new results. N R is the total number of polarization measurements of R , ∆ j is the offset for eachdata set and N exp is the number of the polarization data sets.The form factors are fit to the following functional form: G E ( Q ) , G M ( Q ) /µ p = 1 + (cid:80) ni =1 a i τ i (cid:80) n +2 i =1 b i τ i , (5.3)where τ = Q / M . The first pass of the new fit [168] was performed by removingthe lowest Q Punjabi et al. (GEp-I) point and highest Q Ron et al. (LEDEX)point, since the reanalysis is still underway. The other data points from the worlddata sets kept the same so that we have a conservative estimate of how much thefit changed. The new fit has a slightly increased χ compared to the previous AMTfit [24], which is mainly due to the change in the polarization data set. Fig. 5-6 shows203 [GeV/c] Q0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 P M / G P E G μ JLab (old)Bates BLASTJLab (new)
J. Arrington (new)AMT
Figure 5-6: The global fit for the proton form factor ratio with world high precisiondata. The red points are the new results (E08-007 I and E03-104), the other pointsare from previous polarization measurements [16, 19, 20]. The black line is the AMTfit to the world 2 γ exchange corrected cross section and polarization data. The redline is the new fit by including the new data.the high precision world data with the previous AMT fit and the new fit. As one cansee, the new fit is still slightly above the new data set. The fit to the individual formfactors are shown in Fig. 5-7 and Fig. 5-8, respectively. While G M stays almost thesame, the new fit indicates a ∼
2% decrease in G E in this low Q region. In the past, the proton root-mean-square (rms) radius in general has been determinedfrom the low Q form factor measurements. In the non-relativistic limit, the protoncharge radius is related to the electric form factor as: r p = (cid:32) − dG E ( Q ) dQ (cid:33) Q → . (5.4)204 [GeV/c] Q0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 D / G E G J. Arrington (new)AMT
Figure 5-7: The global fit for the proton electric form factor G E . The black line isthe AMT fit to the world 2 γ exchange corrected cross section and polarization data.The red line is the new fit by including the new data. [GeV/c] Q0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 D / G M G J. Arrington (new)AMT
Figure 5-8: The global fit for the proton magnetic form factor G M . The black line isthe AMT fit to the world 2 γ exchange corrected cross section and polarization data.The red line is the new fit by including the new data.205he most cited value is from the analysis of Simon et al. [4], which gives r p =0 . ± .
012 fm by using the unpolarized data up to Q < − . Occasionally,fits with 2- or 4-pole expressions [170] were performed, and significantly bigger values(0.88 ± ± (cid:104) r (cid:105) term. In parallel, fits based ondispersion relations and VMD [171, 105] models give 0.854 ± ρ ( r ) as thedensity in the nucleon rest frame, and the moment is defined by M α = (cid:90) ∞ drr α ρ ( r ) , (5.5)where α is an even integer. For a charge density, these moments are related to theelectric form factor by M = G E (0) , (5.6) M = (cid:32) − dG E ( Q ) dQ (cid:33) Q → − λ m G E (0) . (5.7)While the definition for the intrinsic charge radius depends upon the choice of λ E employed to fit the form factor, the radius parameter ξ p = (cid:32) − d ln G ( Q ) dQ (cid:33) / Q → = (cid:32) M M + 3 λ m (cid:33) / (5.8)is a model-independent quantity to be compared with the Lamb shift results and otherform factor fits. This approach yields that ξ p = 0 . ± .
01 fm, which represents amodel-independent property of the data even if its interpretation as a charge radiusdepends upon the choice of λ E . Kelly [42] also provided a simple fit with a rationalfunction of Q , which is consistent with dimensional scaling at high Q . It providesexcellent fits to the existing data, and the rms radii are consistent with those in [47].Recently, Sick [172] used the Continued-fraction (CF) expansions to deal properlywith the higher moments after accounting for the Coulomb distortion, and this leads206able 5.2: Proton charge rms-radius from different parameterizations.Form factor r p [fm] yearDipole 0.851 -FW [44] 0.808 2003Kelly [42] 0.878 2004AS [48] 0.879 2007AMT [24] 0.885 2007BS [173] 0.897 2008New (pre.) 0.868 2009to a radius of 0.895 ± .
018 fm, which is significantly larger than the radii used inthe past. Later on, Blunden and Sick [173] investigated the effect of 2 γ exchangeprocesses in the analysis; they found that the change in the radius by removing thecontribution of 2 γ exchange is small (+0.0052 fm). With the new fit presented in theprevious section, we give an updated proton charge rms-radius and compare it withrecent representative parameterizations in Table 5.2. High-precision measurements and calculations of the hydrogen hyperfine-splitting(hfs) provide very high precision tests of QED [174, 175, 176, 177]. Experimentally,the hfs of the hydrogen ground state is known to 13 significant figures in frequencyunits [178], E hfs ( e − p ) = 1420 . . (5.9)One the theoretical side, the QED corrections have reached a level of a ppm accuracy.The major theoretical uncertainty comes from nuclear structure-dependent contribu-tions, which are determined exclusively by the spatial distribution of the charge andmagnetic moment of the proton.The calculated hfs can be given as [179, 180] E hfs ( e − p ) = (1 + ∆ QED + ∆ p hvp + ∆ pµ vp + ∆ p weak + ∆ S ) E pF , (5.10)207here E pF is the Fermi energy E pF = 8 α m r π µ B µ p = 16 α µ p µ B R ∞ (1 + m l /m p ) . (5.11)The mass m r = m l m p / ( m p + m l ) is the reduced mass, and R ∞ is the Rydberg constant(in frequency units).The first four corrections are due to QED, hadronic vacuum polarization, muonicvacuum polarization, and weak interactions ( Z exchange), which are all well known.The proton structure dependent corrections are∆ S = ∆ Z + ∆ pR + ∆ pol , (5.12)where the individual terms stand for “Zemach”, “recoil”, and “polarizability”. TheZemach correction is given by [181]∆ Z = − αm r r Z (1 + δ radZ ) , (5.13)where r Z is the Zemach radius r Z = − π (cid:90) ∞ dQQ (cid:32) G E ( Q ) G M ( Q )1 + κ p − (cid:33) . (5.14)Note that this term depends on the knowledge of the elastic form factors. Due to the1 /Q term in the integral, the form factors at low Q dominate the contribution.Carlson et al. [84] performed an analysis by including the most recent publisheddata on proton spin-dependent structure functions. Table 5.3 shows the results ofthis study. Note that the uncertainty of the polarizalibity term is now comparablewith the uncertainty of the Zemach term.We calculated the Zemach term with the new fit [168], and compared it with otherparameterizaions in Table 5.4. The new fit gives a slightly larger Zemach term (+0.22ppm) which shifts the total calculation in the “right” direction. The deficit is nowreduced to 0.63 and is within one standard deviation. Fig. 5-9 shows the uncertainty208able 5.3: Summary of corrections for electronic hydrogen.Quantity value [ppm] uncertainty [ppm]( E hfs ( e − p ) /E pF ) − QED pµ vp + ∆ p hvp + ∆ p weak Z ( using [24]) -41.43 0.44∆ pR ( using [24]) 5.85 0.07∆ pol ( using [24]) 1.88 0.64Total 1102.63 0.78Deficit 0.85 0.78Table 5.4: Zemach radii, ∆ Z for different parameterizations.Form factor r Z [fm] ∆ Z [ppm] yearDipole 1.025 -39.29 -FW [44] 1.049 -40.22 2003Kelly [47] 1.069 -40.99 2004AS [48] 1.091 -41.85 2007AMT [24] 1.080 -41.43 2007New fit (pre.) 1.075 -41.21 2009of the Zemach radius integrand as a function of Q . The new results ( Q = 0 . ∼ . ) contributed ∼
11% of the uncertainty with an optimistic approach as Q goesto zero . As noted in Section 1.4, unique relativistic relationships between the Sachs formfactors measured at finite Q and the nucleon densities in the rest frame do not exist.Miller [125] showed that the form factor F can be interpreted as a two dimensionalFourier transform of charge density in transverse space in the infinite-momentum- This is by assuming a smooth behavior of G M in the region where it is not well measured( Q < . ), and the uncertainty goes to zero as Q → [GeV ] r i n t e g r a nd Z E08-007 I
Figure 5-9: The uncertainty of the Zemach radius as a function of Q . The greenband shows the coverage of the new data.frame (IMF) ρ Ch ( b ) ≡ (cid:88) q e q (cid:90) dxq ( x, b ) = (cid:90) d q (2 π ) F ( Q = q ) e iq · . b (5.15)Recently, Miller et al. [61] extended the analysis and showed that the form factor F may be interpreted as the two dimensional Fourier transform of the magnetizationdensity by ρ M ( b ) = (cid:90) d q (2 π ) F ( Q ) e i q · b . (5.16)For small values of Q it is possible to make the following expansion: F ( Q ) ≈ − Q (cid:104) b (cid:105) Ch , (5.17) F ( Q ) ≈ κ (cid:32) − Q (cid:104) b (cid:105) M (cid:33) , (5.18)where (cid:104) b (cid:105) Ch ( M ) is the second moment of ρ Ch ( M ) ( b ). The effective ( ∗ ) square radiivia the small Q expansion of the Sachs form factors are defined as G E ( Q ) ≈ − Q R ∗ E , (5.19)210igure 5-10: A linear fit to previous world polarization data, shown by the solid (blue)line and error band. The fit was done up to the region of Q = 0 .
35 GeV where thelinear expansion is valid for the transverse radii difference. The shaded area indicates (cid:104) b (cid:105) Ch > (cid:104) b (cid:105) M . The dashed (red) line shows the critical slope when (cid:104) b (cid:105) M = (cid:104) b (cid:105) Ch .Figure from [61] G M ( Q ) ≈ − Q R ∗ M . (5.20)Then the form factor ratio can be expanded as R = µ p G E /G M ≈ Q R ∗ M − R ∗ E ) , (5.21)and the charge and magnetization transverse densities can be related to the ratio R by: (cid:104) b (cid:105) M − (cid:104) b (cid:105) Ch = µ p κ
23 ( R ∗ M − R ∗ E ) + µ p M p , (5.22)where u p M p ≈ . represents the relativistic correction. It is a consequence ofthe Foldy term [182], which arises from the interaction of the anomalous magneticmoment of the nucleon with the external magnetic field of the electron.Fig. 5-10 shows the results of a linear fit to the previous world data, and Fig. 5-11211 R Q P M / G P E G μ /6 R Q P M / G P E G μ E08007 - I LEDEX ReanalysisCrawford et al.Pospischil et al.Gayou et al.Dietrich et al.Milbrath et al. ) -R (R6 Q 1 + ch > = - (cid:104) b (cid:105) M . The dashed (red) line shows thecritical slope when (cid:104) b (cid:105) M = (cid:104) b (cid:105) Ch .shows the fit with the new results from experiment E08-007 I and the preliminaryresults of LEDEX reanalysis [183]. The charge and magnetization second momentsdifference changed from (cid:104) b (cid:105) M − (cid:104) b (cid:105) Ch = 0 . ± . (5.23)to (cid:104) b (cid:105) M − (cid:104) b (cid:105) Ch = 0 . ± . (5.24)Note that the new fit improves the uncertainty by a factor of ∼
2, and the mag-netic density still extends further than the electric density in the transverse space.This result can be related to the failure of quarks spin to account for the total angu-lar momentum of the proton and the expected importance of quark orbital angularmomentum [184]. 212 .7 Strangeness Form Factors
The parity-violating (PV) asymmetry in elastic ep scattering can be used to extractthe strangeness form factors [185, 186, 187]. The PV asymmetry arises due to in-terference between photon exchange and Z -boson exchange. The asymmetry in theBorn approximation is given by [188]: A P V = − G F Q πα √ A E + A M + A A τ G Mp + εG Ep , (5.25)where G F is the Fermi constant, and α is the fine structure constant. The individualasymmetry terms can be written in terms of the proton form factors G Ep and G Mp and the proton neutral weak vector and axial form factors G ZEp , G ZMp and G ZA : A E = εG Ep G ZEp , (5.26) A M = τ G Mp G ZMp , (5.27) A A = (1 − θ W ) ε (cid:48) G Mp G ZA , (5.28)where θ W is the weak mixing angle, and ε (cid:48) = (cid:113) τ (1 + τ )(1 − ε ). With the assumptionof isospin symmetry, the weak vector form factors can be expressed in terms of theproton and neutron form factors together with the strangeness form factors: G Es and G Ms . Neglecting the contributions from heavier quarks [187], A P V is given by: A P V = − G F Q πα √ (cid:20) (1 − θ W ) − εG Ep ( G En + G Es ) + τ G Mp ( G Mn + G Ms ) ε ( G Ep ) + τ ( G Mp ) − (1 − θ W ) ε (cid:48) G Mp G ZA ε ( G Ep ) + τ ( G Mp ) (cid:21) . (5.29)Clearly, the measurements of the strangeness form factors require the knowledge ofthe nucleon form factors.From our data, we estimated the impact of the new fit to the existing strangenessform factor measurements by comparing them with the AMT parameterization [24].The difference in the extracted physics asymmetry is summarized in Table 5.5.213able 5.5: The absolute asymmetry difference (∆ A P V ), the normalized differenceby the experimental uncertainty (∆ A P V /σ ) and the relative asymmetry difference(∆ A P V /A P V ) between using the AMT [24] parameterization and the new one. Q [GeV ] ∆ A P V [ppm] ∆ A P V /σ ∆ A P V /A P V
Experiment0.38 -0.178 0.42 1 .
6% G0 FWD [189]0.56 -0.347 0.50 1 .
6% G0 FWD1.00 -0.414 0.30 0 .
8% G0 FWD0.23 +0.038 0.12 0 .
2% G0 BCK [190]0.65 +0.014 0.14 0 .
3% G0 BCK0.50 -0.299 0.50 1 .
7% HAPPEX III [191]
The A1 collaboration [192] at Mainz Microtron (MAMI) completed a very high preci-sion elastic ep cross section measurement in the range of Q = 0 . − [62]. Theexperiment aimed to measure the cross section at a fixed Q for several settings of ε to perform the Rosenbluth separation of the individual form factors. The accessibleregion is determined by the accelerator and the properties of the detector system.Fig. 5-12 shows the accessible kinematic region for the experiment.Due to the large cross section in the low Q region, a very small statistical un-certainty can be achieved. The collaboration estimated a < .
5% statistical errorplus a 0 .
5% systematics uncertainty, leading to a total error of ∼
1% or less for ev-ery cross section measurement, which is an unprecedentedly small for cross sectionmeasurements.The Mainz experiment plans to extract the individual form factors using twomethods. The first way is by using the standard Rosenbluth separation which utilizesa linear fit to the cross section at constant Q but different ε . This works in acompletely model independent way except for the larger Q where the two photonexchange contribution becomes larger. A second approach is to fit the global ansatzfor the form factors directly to the cross sections. With a flexible ansatz, this is quasi214igure 5-12: The accessible kinematic region in ε/Q space. The black dots representthe chosen settings (centers of the respective acceptance). The dotted curves corre-spond to constant incident beam energies in steps of 135 MeV (”horizontal” curves)and to constant scattering angles in 5 ◦ steps (”vertical” curves). Also shown are thelimits of the facility: the red line represents the current accelerator limit of 855 MeV,with the upgrade, it will be possible to measure up to the light green curve. Thedark green area is excluded by the minimal beam energy of 180 MeV. The maximum(minimum) spectrometer angle excludes the dark (light) blue area. The gray shadedregion is excluded by the upper momentum of spectrometer A (630 MeV/ c ). Figurefrom [62]. 215odel-independent and is an even more powerful method to directly test availablemodels. The second part of experiment E08-007 is tentatively scheduled in 2012. This partwill measure the proton form factor ratio in the range of Q = 0 . − . usingthe beam target asymmetry technique.For longitudinally polarized electrons scattering from a polarized proton target,the differential cross section can be written as [193]: dσd Ω = Σ + h ∆ , (5.30)where Σ is the unpolarized differential cross section, h is the electron helicity and ∆is the spin-dependent differential cross section given by:∆ = (cid:32) dσd Ω (cid:33) Mott f − recoil (cid:20) τ v T (cid:48) cos θ ∗ G M − (cid:113) τ (1 + τ ) v T L (cid:48) sin θ ∗ cos φ ∗ G M G E (cid:21) , (5.31)where θ ∗ and φ ∗ are the polar and azimuthal proton spin angles defined with respectto the three-momentum transfer vector (cid:126)q and the scattering plane (see Fig. 5-13), and v T (cid:48) and v T L (cid:48) are kinematic factors [193].The spin-dependent asymmetry A is defined as: A = σ + − σ − σ + + σ − , (5.32)where σ +( − ) is the differential cross section for the two different helicities of thepolarized electron beam. The spin-dependent asymmetry A can be written in termsof the polarized and unpolarized differential cross-sections as: A = ∆Σ = − τ v T (cid:48) cos θ ∗ G M − (cid:113) τ (1 + τ ) v T L (cid:48) sin θ ∗ cos φ ∗ G M G E (1 + τ ) v L G E + 2 τ v T G M . (5.33)The experimental asymmetry A exp is related to the spin-dependent asymmetry by the216igure 5-13: Spin-dependent ep elastic scattering in Born appromixation.relation: A exp = P b P t A, (5.34)where P b and P t are the beam and target polarizations, respectively. By measuringthe asymmetry simultaneously in two spectrometers with different angles between themomentum transfer and the target spin as illustrated in Fig. 5-14, the following superratio is directly related to the ratio G E /G M : R = A A = τ v T (cid:48) cos θ ∗ − (cid:113) τ (1 + τ ) v T L (cid:48) sin θ ∗ cos φ ∗ G E G M τ v T (cid:48) cos θ ∗ − (cid:113) τ (1 + τ ) v T L (cid:48) sin θ ∗ cos φ ∗ G E G M , (5.35)which is independent of the knowledge of the beam and target polarization.The solid polarized proton target developed by UVa will be used. In this tar-get, NH is polarized by Dynamic Nuclear Polarization (DNP) [194] in a strongmagnetic field (5 T) at very low temperature ( ∼ Q points and projected total errors are shown inFig. 5-15. This future measurement will overlap with the part I points and the lowerrange of the BLAST [19] measurement. This will provide a direct comparison withthe BLAST results by using the same technique and allow an examination of anyunknown systematic uncertainties of the recent measurements.Beyond the curiosity in the form factor behavior in the extremely low momentum217 (cid:38) )( qp (cid:38) )( qp (cid:38) S (cid:38) ' e ' e * (cid:84) * (cid:84) Figure 5-14: The kinematics for the two simultaneous measurements. The scatteredelectrons e (cid:48) and e (cid:48) are detected in left and right HRS, respectively. The recoil protons p and p point in the direction of the q-vector (cid:126)q and (cid:126)q , respectively. (cid:126)S denotes thetarget spin direction. [GeV/c] Q0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 P M / G P E G μ Punjabi et al. (GEp-I)Crawford et al. (BLAST)Ron et al. (LEDEX)Paolone et al. (E03-104)E08-007 IE08-007 II projection
Miller LFCBMBoffi et al. PFCCQMBelushkin, Hammer & Meissner VMDLomon GK(05)Faessler et al. LCQM
Figure 5-15: The proposed Q points and projected total uncertainties for the secondpart of E08-007. 218 [GeV ] r i n t e g r a nd Z E08-007 II E08-007 I
Figure 5-16: The uncertainty of the Zemach radius as a function of Q . The greenband shows the coverage of the new data from this work, and the yellow band showsthe proposed coverage of the second part of E08-007.transfer region, the motivation for the running of the experiment E08-007 part IIalso comes from the determination of the proton Zemach radius. As illustrated inFig. 5-16, the second part of this measurement will cover the peak region where theexisting data contribute ∼
60% of the total uncertainty in r Z . By assuming the formfactor ratio follows in a similar trend as the part I data, a conservative change of ∼ . Z is expected.As mentioned in Section 5.6, the low Q data will also greatly improve our knowl-edge of the proton transverse densities in the impact parameter space. The expectedresults from the part II measurement are shown in Fig. 5-17 and will allow us to makea definitive fit to this quantity. In conclusion, this thesis presents the details of the proton electric to magnetic formfactor ratios measurements at Q = 0 . − . . This experiment used the stan-dard Hall A experimental system with one of the two high resolution spectrometers.To reduce the inelastic background, the BigBite calorimeter was used to tag the elec-219 R Q P M / G P E G μ /6 R Q P M / G P E G μ E08007 - II Projected UncertaintiesE08007 - I LEDEX ReanalysisCrawford et al.Gayou et al.Dietrich et al.Milbrath et al. ) -R (R6 Q 1 + ch > = -
Figure 5-17: Projection of E08-007 part II measurements on the new fit by assumingthe same slope as Q decreases.trons and form the coincidence trigger. The central experimental equipment was theFocal Plane Polarimeter (FPP), which measured the polarization of the recoil protonin the elastic scattering of polarized electrons from an unpolarized liquid hydrogentarget. The statistical uncertainty in this experiment is determined by the polariza-tion of the electron beam and the figure of merit of the FPP. The main source of thesystematic uncertainty in this measurement comes from the spin precession of theproton in the magnetic field of the spectrometer. With an 85% beam polarizationand 21 days of running, we have achieved the best statistics to date. For the most ofthe Q kinematic points, the systematic error dominates the total uncertainty.The results of this measurement together with a high precision point at Q = 0 . (from experiment E03-104) strongly deviate from unity, and are systematicallybelow the world polarization data. The preliminary reanalysis of the LEDEX data isin agreement with the new data, but the discrepancy between the BLAST results andthe new data still needs to be investigated. The new results do not favor any narrow220tructure in this region as suggested by the phenomenological fit [44]. At the Q = 0limit, the ratio is forced to unity by definition, and the current slope of the data inthis region appears to be too smooth to meet this condition, which might indicate achange in the slope as Q approaches 0.The low Q range measured in this experiment does not allow for a pQCD calcu-lation, which necessitates the development of low energy effective field theories andthe use of fits to the data in order to describe the form factors. None of the currenttheories accurately predicts the entire data set, which is mainly due to the “free”parameters that had been tuned to the older data in those calculations. On theother hand, fast developments of computational capabilities may allow theories suchas Lattice QCD to offer a complete and model-independent description in the nearfuture.In the mean time, the new results from this experiment have been used in global fitto extract the individual form factors. The preliminary fit suggests a smaller G Ep inthis region, while the change in G Mp is relatively small. The new data also changed theresults of the proton transverse density as proposed in [61]; the difference between thetransverse RMS magnetic and electric radius is smaller with improved precision. Theimproved knowledge of the individual form factors also has a significant impact in theultra-high precision test of QED in the hydrogen hyperfine splitting calculations andin the extraction of the strangeness form factors from parity-violation experiments.The second part of this experiment, which will access the region of Q = 0 . − . is tentatively scheduled in 2012. In addition to resolving the potential datadiscrepancy, this part will be the first polarization measurement in the extremely low Q region and will offer a great opportunity to vastly improve our knowledge of thenucleon structure. 22122 ppendix AKinematics in the Breit Frame The Breit frame, also called the “brickwall frame”, is the frame where the momentaof the initial and final nucleon are equal and opposite: (cid:126)p B = − (cid:126)p (cid:48) B = − (cid:126)q B , (A.1)so there is no energy transfer in the elastic scattering in this frame: E pB = E (cid:48) pB (A.2) ω B = E pB − E (cid:48) pB = 0 . (A.3)The four-momentum transfer in the Breit frame is: Q = − q B = (cid:126)q B (A.4)For the electron kinematics, Eq. A.3 imposes E B = E (cid:48) B (A.5) (cid:126)k B = (cid:126)k (cid:48) B (A.6) (cid:126)k B = (cid:126)q B + (cid:126)k (cid:48) B . (A.7)As illustrated in Fig. A-1, the three-momentum of the electron have:223 B k (cid:38) B k ' (cid:38) B q (cid:38) B (cid:84) )(* B q (cid:38) (cid:74) B qp (cid:38)(cid:38) (cid:32) B qp (cid:38)(cid:38) (cid:16)(cid:32) Figure A-1: Elastic scattering in the Breit frame. k B = k (cid:48) B = | (cid:126)q B | θ B √ Q θ B k B = k (cid:48) B = 0 (A.9) k B = − k (cid:48) B = | (cid:126)q B | √ Q θ B in the Lab frame. The Breit frame ismoving along the 3-axis, so that the 1 and 2 components of the electron momentumare left unchanged by the Lorentz transformation: k = k B = k (cid:48) = k (cid:48) B = √ Q θ B k = k B = k (cid:48) = k (cid:48) B = 0 . (A.12)The (cid:126)q is along the 3-axis, so we can write k = ( (cid:126)k · (cid:126)q ) (cid:126)q = (cid:126)k · (cid:126)k − (cid:126)k · (cid:126)k (cid:48) (cid:126)q = ( E ) + ( EE (cid:48) cos θ e ) − E EE (cid:48) cos θ e (cid:126)q . (A.13)By using the relation Q = 4 EE (cid:48) sin θ e k = (cid:126)k − k = (cid:126)k (cid:126)q − ( (cid:126)k · (cid:126)q ) (cid:126)q = [( E ) + E E (cid:48) − E EE (cid:48) cos θ e ] − [( E ) + ( EE (cid:48) cos θ e ) − E EE (cid:48) cos θ e ] (cid:126)q = E E (cid:48) sin θ e (cid:126)q = Q (cid:126)q cot θ e , (A.15)where the electron mass is neglected. Since q = p (cid:48) − p (A.16) p = p (cid:48) = m p , (A.17)we can write p (cid:48) = ( q + p ) = q + 2 p · q + p (A.18) q = − q · p = − ωm p (A.19) ω = − q m p = Q m p . (A.20)Using Q = − ( ω − (cid:126)q ), we can express (cid:126)q as (cid:126)q = Q (1 + Q m p ) = Q (1 + τ ) . (A.21)Now Eq. A.15 can be replaced by k = Q τ ) cot θ e , (A.22)and the angle θ B can be expressed ascot θ B θ e τ . (A.23)22526 ppendix BAlgorithm for Chamber Alignment The FPP chamber alignment by matrix expansion used in this analysis was devel-oped in experiment E93-049. Instead of doing the physical alignment for each FPPchamber, the correction is directly applied to the track reconstructed by the FPPreferring to the VDC track. The alignment algorithm is described as the following.For the “straight-through” events, the VDCs have the reconstructed track T0( x , y , θ , φ ), and the FPP chambers have the reconstructed track T1 before thealignment. The difference between the two tracks is ∆T = T1 − T0. The goal is toapply the correction terms ∆T for each track of the FPP. Intuitively, this correctiondepends on where the track is hitting at, so it’s convenient to expend the correctionin terms of the polynomial of the track position x , y at the focal plane, which are1 , x , y , x , y , x · y .The vector V is defined as: V = x y x y x · y ∆T = x − x y − y x − x y − y . The matrix A is constructed by: A = V · V (cid:48) . (B.1)The matrix B is constructed by: B = ∆ T · V (cid:48) . (B.2)The correction matrix M is defined by: M = A − · B (cid:48) = ( V · V (cid:48) ) − · ( ∆ T · V (cid:48) ) (cid:48) (B.3)= ( V (cid:48) ) − · ∆ T . (B.4)The simple matrix calculation leads to: T1 = T0 + V (cid:48) · M (B.5)From Eq. B.5, the FPP track is corrected by the matrix M . In the real procedure,the front chamber track is first aligned with respect to the VDC tracks, and the rearchamber track is aligned by the same way with respect to the aligned front track.228 ppendix CExtraction of PolarizationObservables C.1 Introduction
For experiment E08-007 we measured the recoil proton polarization in the elastic re-action H ( (cid:126)e, e (cid:48) (cid:126)p ). With the scattering angles reconstructed by the FPP and the spinrotation matrix generated by COSY, we are able to extract the polarization compo-nents at the target. Three different methods to extract the polarization observablesare presented in [151]. In this work, the weighted-sum and maximum likelihoodmethod were discussed. Since we are dealing with the ≤
1% statistical uncertaintyin this measurement, the validity of the approximations used in the formalism wascarefully examined.
C.2 Azimuthal asymmetry at the focal plane
The detection probability for a proton scattered by the analyzer with polar angle θ and azimuthal angle φ is given by [16]: f ± ( θ, φ ) = 12 π (cid:15) ( θ, φ )(1 ± A y ( P fppy sin φ − P fppx cos φ )) , (C.1)229here ± refers to the sign of the beam helicity, P fppx and P fppy are the transverse andnormal polarization components at the analyzer with plus beam helicity, respectively; P fppz is not measured because it does not result in an asymmetry. (cid:15) ( θ, φ ) is the nor-malized efficiency (acceptance) which describes the non-uniformities in the detectorresponse that results from misalignments and inhomogeneities in detector efficiency. A y is the analyzing power. Based on Eq. C.1 the efficiency can be extracted by: (cid:15) ( θ, φ ) = f + ( θ, φ ) + f − ( θ, φ ) π . (C.2) C.3 Weighted-sum
The spin transport matrix is defined by: P fppx P fppy = S xx S xy S xz S yx S yy S yz P tgx ηhP tgy ηhP tgz , (C.3)where P tgx , P tgy , P tgz is the polarization component at the target. By writing Eq. C.1in terms of the polarization components at the target, we now have: f ( φ ) = 12 π (cid:15) (1 + λ x P tgx + λ y hP tgy + λ z hP tgz ) , (C.4)where λ x = A y ( S yx sin φ − S xx cos φ ) λ y = ηA y ( S yy sin φ − S xy cos φ ) λ z = ηA y ( S yz sin φ − S xz cos φ ) . (C.5) η is the sign for the beam helicity, and h is the beam polarization. Note that thecontribution from the induced (normal) polarization P tgx is independent of the beamhelicity. In the Born approximation, the induced polarization P tgx = 0. As noted230n [151], with different beam helicities, we can always construct an effective acceptancewhich has a symmetry period of π in φ . The integrals can be expressed as: (cid:90) π f ( φ ) λ y dφ = hP tgy (cid:90) π f ( φ ) λ y dφ + hP tgz (cid:90) π f ( φ ) λ y λ z dφ + (cid:90) π f ( φ ) λ z dφ = hP tgy (cid:90) π f ( φ ) λ y λ z dφ + hP tgz (cid:90) π f ( φ ) λ z dφ. (C.6)By replacing the integrals in Eqs. C.6 with corresponding sums over the observedevents, we have (cid:80) i λ y,i (cid:80) i λ z,i = (cid:80) i λ y,i λ y,i (cid:80) i λ z,i λ y,i (cid:80) i λ y,i λ z,i (cid:80) i λ z,i λ z,i hP tgy hP tgz . (C.7)So P tgy and P tgz can be solved from the equation above. Problems may arise if P tgx isnon-zero from the 2 γ exchange, since an acceptance with symmetry period of π in φ cannot be constructed. C.4 Maximum likelihood
The individual polarization components can also be extracted by the maximum-likelihood (ML) technique. Based on Eq. C.1, we can express the probability forthe experimental angular distribution as the product of all the individual probabili-ties: F = N (cid:89) i =1 π (cid:15) [1 + A y ( P fppy sin φ i − P fppx cos φ i )] . (C.8)The likelihood function is given by: L ( P tgx , P tgy , P tgz ) = N (cid:89) i =1 π (cid:15) (1 + λ y hP tgy + λ z hP tgz ) , (C.9)231here λ y , λ z are the same as defined in Eq. C.5. By maximizing the probabilityfunction: ∂ ln L∂P tgy = 0 ∂ ln L∂P tgz = 0 , (C.10)we can extract P tgy and P tgz . The normalized efficiency term (cid:15) is eliminated in thederivative since it dose not depend on P ( ∂ ln (cid:15)∂P = 0). To linearize the equations, anapproximation is applied: ln(1 + x ) ≈ x − x o ( x ) , (C.11)where x = λ y hP tgy + λ z hP tgz . By omitting the o ( x ) term, the equations is simplifiedas: (cid:80) i λ y,i (cid:80) i λ z,i = (cid:80) i λ y,i λ y,i (cid:80) i λ y,i λ z,i (cid:80) i λ z,i λ y,i (cid:80) i λ z,i λ z,i hP tgy hP tgz , (C.12)which is the same as Eq. C.7, and the weighted-sum and ML methods converge atthis point. Here we still assume the induced polarization P tgx = 0 to simplify thecontext, since determining the induced polarization P tgx which is sensitive to the falseasymmetry is not the intent of this experiment. C.5 Simulation
Although it is clearly derived from the above sections and also in [151] that falseasymmetry can be canceled by flipping the beam helicity, it is straight forward toconfirm the results within a certain precision and test the statistical sensitivity of theweighted-sum method by simulation.For simplicity, we use the dipole approximation for the spin transport and assumethat there is no induced polarization. Then, for each trial the simulation generates a232ample of events by the probability: f ± ( φ ) = (cid:15) ( φ )(1 ± P y sin χ cos φ ± P z sin φ ) (C.13)Here P y , P z represent the transferred polarization components at the target. To besimilar to the real case, we choose the spin rotation angle χ = 90 ◦ ∼ ◦ . We setthe pseudo efficiency (cid:15) : (cid:15) = 1 + s sin φ. (C.14)False asymmetries with higher order terms ( c , s , c ) were also tested, and the resultsare similar. With the simulated sample events, we extracted the pseudo ratio P y /P z by Eq. C.7. We also varied the event sample size N used for each trial to test thestatistical sensitivity. The extracted ratio distributions are shown in Fig. C-2 with5000 trials per plot, and the sample sizes for each trial N is from 100 to 50000.The mean value for the extracted ratio versus the sample size is plotted in Fig. C-3,and the deviation from the set value ∆ R divided by the standard deviation of thesimulated distribution versus the sample size is plotted in Fig. C-4.Results from the simulation with two different set ratios P y /P z = 0 . P y /P z =1 are shown in Fig. C-3 and Fig. C-4. From these results we can see that the weighted-sum method can extract the ratio without any problem even with a significant sizeof the false asymmetry. The comparison shown here is between s = 0 and s = 0 . N > P tgx = 0). The problemwith non-zero induced polarization is that we cannot exactly construct an acceptance233 p-0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04-0.1-0.08-0.06-0.04-0.0200.020.040.060.080.1 S1 dp-0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04-0.1-0.08-0.06-0.04-0.0200.020.040.060.080.1 C1 dp-0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04-0.1-0.08-0.06-0.04-0.0200.020.040.060.080.1 S2 dp-0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04-0.1-0.08-0.06-0.04-0.0200.020.040.060.080.1 C2 Figure C-1: False asymmetry Fourier series coefficients vs. δ p for kinematics K6 δ p = 2%.with symmetry period of π in φ as mentioned earlier, so Eq. C.6 is not exactlytrue. However, through simulation, we can give an estimate with a small amount ofinduced polarization P tgx as predicted by [196]. The predicted induced polarizationis shown in Fig. C-5. The electron scattering angle θ cm for each kinematic setting islisted in Table C.1. To simulate the case with non-zero induced polarization, the setprobability at the focal plane Eq. C.13 becomes: f ± ( φ ) = (cid:15) ( φ )(1 ± P y sin χ cos φ ± P z sin φ + P cos φ + P sin φ ) , (C.15)where P , P represent the polarization components raised from the induced polariza-tion at the focal plane. To test the extreme case, we set them to be comparable to thephysics asymmetry P , P = 0 . s = 0 . P , P were tested and corresponding results are listed in Table C.2.The simulations for set polarization P y = 0 . , P z = 0 . ath Entries 5000Mean 0.3453RMS 1.897 -6 -4 -2 0 2 4 6 8010203040506070 rath
Entries 5000Mean 0.3453RMS 1.897=100 N rath Entries 5000Mean 0.9542RMS 1.693 -6 -4 -2 0 2 4 6 8020406080100 rath
Entries 5000Mean 0.9542RMS 1.693=500 N rath Entries 5000Mean 1.145RMS 1.248 -6 -4 -2 0 2 4 6 8020406080100120 rath
Entries 5000Mean 1.145RMS 1.248=1000 N rath Entries 5000Mean 1.178RMS 0.889 -6 -4 -2 0 2 4 6 8020406080100120140160 rath
Entries 5000Mean 1.178RMS 0.889 =2000 N rath Entries 5000Mean 1.071RMS 0.4292 -6 -4 -2 0 2 4 6 8020406080100120140160180200220 rath
Entries 5000Mean 1.071RMS 0.4292=5000 N rath Entries 5000Mean 1.037RMS 0.3075 -6 -4 -2 0 2 4 6 8050100150200250300 rath
Entries 5000Mean 1.037RMS 0.3075 =8000 N rath Entries 5000Mean 1.029RMS 0.2761 -6 -4 -2 0 2 4 6 8050100150200250300 rath
Entries 5000Mean 1.029RMS 0.2761 =10000 N rath Entries 5000Mean 1.018RMS 0.1828 -6 -4 -2 0 2 4 6 8050100150200250300350400 rath
Entries 5000Mean 1.018RMS 0.1828=20000 N rath Entries 5000Mean 1.007RMS 0.1122 -6 -4 -2 0 2 4 6 80100200300400500600 rath
Entries 5000Mean 1.007RMS 0.1122=50000 N Figure C-2: Histograms of the extracted ratio P y /P z by weighted-sum method withno false asymmetry ( s = s = 0) in the simulation. N is the sample size of eachtrial in the simulation. At large statistics, the extracted ratio is in good agreementwith the set ratio in the simulation.results show that the deviation is much less within one standard deviation. For thereal case, the induced polarization at the target P tgx ∼ − . To estimate the effectclose to the real case, we used the similar size of P , and P as predicted, and alsowith the “full” false asymmetry: (cid:15) = 1 + s sin φ + c cos φ + s sin 2 φ + c cos 2 φ (C.16)where s = 0 . , c = 0 . , s = 0 . , c = 0 .
01 which is assigned according to themaximum of their real sizes in this experiment. We assume that P = P = 0 . By considering the spin rotation matrix elements S yx , S xx (cid:28)
1, the contribution from P at thefocal plane is actually (cid:28) − N F a l s e r a t i o -10123 N F a l s e r a t i o -1.5-1-0.500.511.522.5 N F a l s e r a t i o -1-0.500.511.52 N F a l s e r a t i o -1-0.500.511.52 Figure C-3: Extracted ratio mean value by weighted-sum method vs. different samplesize N with false asymmetry s = 0 (left) and s = 0 . P y = 0 . , P z = 0 . P y = 0 . , P z = 0 .
2, showing that the results of thetests do not depend on the value of the set ratio P y /P z .which is very conservative compared to the real case (10 − ) after considering the spinrotation. The simulation results are shown in Fig. C-7.From the simulation (Fig. C-7) we can see that the deviation of the ratio ∆ R is ∼ . P , P are even smaller and the statistics aremuch better in the real case, we do not expect any noticeable effect from the inducedpolarization. C.6 Summary
Through this study, we have confirmed the results in [151]. The approximations andassumptions used were carefully examined. From the simulation, we have confirmedthat the weighted-sum method is valid and false asymmetry plays a negligible role inextracting the transferred polarization and thus the form factor ratio.236 N / R M S R Δ -0.3-0.2-0.100.10.2 N / R M S R Δ -0.3-0.2-0.100.10.2 N / R M S R Δ -0.0500.050.10.15 N / R M S R Δ -0.0500.050.10.15 Figure C-4: Extracted ratio mean value deviation from the set value divided by thesample standard deviation (RMS) vs. different sample size N with false asymmetry s = 0 (left) and s = 0 . P y = 0 . , P z = 0 .
1, lower panel is with setpolarization P y = 0 . , P z = 0 . θ cm for each kinematics ( δ p = 0%).Kinematics Q [(GeV /c ) ] θ cm [deg]K1 0.35 55.5K2 0.30 50.9K3 0.45 63.9K4 0.40 60.2K5 0.55 71.5K6 0.50 67.7K7 0.60 75.4K8 0.70 82.5237igure C-5: Proton induced polarization component, as a function of the electron θ cm scattering angle for different beam energies. The dash (solid) line shows the total(elastic only) 2 γ exchange effect. The y-axis P y is actually P tgx for the conventionused here.Table C.2: Deviation from the set value ∆ R with different combinations of P and P . The set transferred polarization is P y = P z = 0 .
1. Simulation with sample size N = 10 and number of trial N trial = 10 . The standard deviation for extractedvalues is ∼ . P = 0 P = 0 . P = 0 0.0015 0.002 P = 0 . N F a l s e r a t i o -2-1012 =0 =P P =0.1 z =0.1, P y P N / R M S R Δ -0.4-0.3-0.2-0.100.10.2 =0 =P P =0.1 z =0.1, P y P N F a l s e r a t i o -2-10123 =0 =0.2, P P =0.1 z =0.1, P y P N / R M S R Δ -0.4-0.3-0.2-0.100.10.2 =0 =0.2, P P =0.1 z =0.1, P y P N F a l s e r a t i o =0.2 =0, P P =0.1 z P_y=0.1, P N / R M S R Δ -0.4-0.3-0.2-0.100.10.2 =0.2 =0, P P =0.1 z =0.1, P y P N F a l s e r a t i o -2-1012 =0.2 =P P =0.1 z =0.1, P y P N / R M S R Δ -0.4-0.3-0.2-0.100.10.2 =0.2 =P P =0.1 z =0.1, P y P Figure C-6: Extracted ratio mean value and relative deviation vs. different samplesize N with false asymmetry s = 0 .
1, and different combinations of set polarization P , P . 239 N F a l s e r a t i o -2-1012 N / R M S R Δ -0.4-0.3-0.2-0.100.10.2 Figure C-7: Extracted ratio mean value and relative offset from the set value vs.different sample size N with difference false asymmetries: s = 0 . , c = 0 . , s =0 . , c = 0 .
01, and set polarizations: P = P = 0 . , P y = P z = 0 .
1, respectively.240 ppendix D pC Analyzing PowerParameterizations
The carbon analyzing power A y was extracted for this measurement. We appliedtwo-dimensional binning in the analysis to extract the dependence on θ fpp and T p .The mean values of the two variables of each bin were used for the fit. The θ fpp binning is the same for all the kinematics settings as listed in Table D.1, and the T p binning is summarized in Table D.2. The “low energy” McNaughton, the LEDEXand the new parameterizations are summarized in Table D.3.241able D.1: Binning on θ fpp .Bin θ low [deg] θ high [deg]1 4 52 5 63 6 84 8 105 10 126 12 157 15 188 18 219 21 2410 24 2811 28 36Table D.2: Binning on T p .Kinematics carbon thickness [inch] T low [MeV] T high [MeV] Bin size [MeV]K1 3 120 150 10K2 3 90 120 10K3 3.75 160 220 20K4 3.75 140 200 20K5 3.75 220 280 20K6 3.75 200 260 20K7 3.75 240 300 20K8 3.75 300 360 20242able D.3: Coefficients of different parameterizations for the pC analyzing power A y .The reduced χ of the new fit is 0.74 with a χ of 272.5 and 368 degrees of freedom.LEDEX McNaughton (low) NewEnergy Range 82 ∼
127 MeV 95 ∼
483 MeV 90 ∼
360 MeV p a a a a a b b b -5847.9 1333.5 -2207.81 b -21750 -3713.5 6089.94 b c c -8639.4 949.50 102.862 c c × c -2.1720 × -118830.0 595619 d d × × d × -1.99475 × d -1.1201 × × d -1.9356 × -4.41281 × ppendix ENeutral Pion PhotoproductionEstimation E.1 Introduction
For experiment E08007, we measured the recoil proton polarization in the elasticreaction H( (cid:126)e, e (cid:48) (cid:126)p ). For the production data taking we required a coincidence betweena proton detected in the left HRS and a signal in a limited set of BigBite shower blockswhich a coincident elastic ep electron would be expected to hit. Since the particleidentification is limited in the BigBite shower counter due to the configuration, itmay allow contamination by background reactions. Therefore, in the data analysis,an elastic cut was applied to the proton kinematics (angle vs. momentum).This study is to investigate whether there is a significant contribution from pionphotoproduction γ + p → p + π with the current event selection. In this work,we do not consider possible backgrounds from virtual Compton scattering which areexpected to be much smaller than the backgrounds from pion production. Since thegoal is to give an estimate for the order of magnitude, some approximations wereapplied to simplify the simulation. Based on this study we will see if a full simulationis needed.For the pion photoproduction estimation, it includes the following inputs:245 tg ±
30 mrad θ tg ±
60 mrad δ p ± . • phase space simulation for both e + p → e + p and γ + p → p + π . • real photon flux estimation. • BigBite calorimeter acceptance. • elastic cross section, pion photoproduction cross section and polarization ob-servable from the world database and calculations.We took the lowest momentum kinematics setting K2 as an example. The pro-cedure described below was applied to every kinematics, and the results are reportedin the end. E.2 Phase Space Simulation
The proton was detected in the left HRS which has a small acceptance and highresolution. To simplify the phase space simulation, we first put constraints on pro-tons only, assuming that all the pions (decayed photons) could be detected in theBigBite shower counter. By applying the same elastic cut on the simulated protonspectrum, we can get the π p to ep phase space ratio. The momentum resolutionwas manually adjusted in the simulation to match the resolution of the data. Theresolution comparison was made on δ p − δ p ( φ ), which is the difference between themeasured momentum and the one reconstructed from the scattering angle via theelastic kinematics. The acceptance cuts were applied according to the HRS defaultacceptance as reported in Table E.1. The comparison between the real data and thesimulated spectrum are shown in Fig. E-1. The spectrometer resolution becomes worse with lower proton momentum, hence, more difficultto separate the pion background via the elastic cut. / ndf χ ± ± ± ) φ ( p δ - p δ -0.05 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.0502000400060008000100001200014000 / ndf χ ± ± ± Data = 0.3 GeV = -2% Q δ K2 / ndf χ ± ± -0.0009603 Sigma 0.00001 ± ) φ ( p δ - p δ -0.05 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05020004000600080001000012000 / ndf χ ± ± -0.0009603 Sigma 0.00001 ± MC = 0.3 GeV = -2% Q δ K2 Figure E-1: Data and simulated spectrum on δ p − δ p ( φ ).The simulation also generated the phase space for γ + p → p + π . The range ofthe photon energy is from 0 to 1192 MeV (beam energy). However, the proton frompion production will only be detected in the HRS acceptance when the photon carriesalmost all the beam energy. As an example, for kinematics K2, the HRS central angleis 60 ◦ , for the δ p = 0% setting, the central momentum is 565 MeV. As demonstratedin Fig. E-2, the proton kinematics for π p at E γ = 500 MeV is far away from theHRS setting. To estimate the π p to ep phase space ratio, a cut was applied to thesimulated spectrum on δ p − δ p ( φ ) according to the elastic cut applied to the data asillustrated in Fig. E-3. (deg) p θ ( M e V ) p p ElasticPhotopion = 500 MeV γ E Figure E-2: Simulated proton kinematics for π p at E γ = 500 MeV and elastic. P p isthe proton momentum and θ p is the scattering angle.247 φ ( p δ - p δ -0.05 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.0501000200030004000500060007000 proton elastic cut Figure E-3: Proton elastic cut on δ p − δ p ( φ ) spectrum for kinematics K2. E γ [MeV] R phase . × − × − . × − . × − . × − . × − Table E.2: Simulated π p to ep phase space ratio at kinematics K2.It is not surprising to find that only when E γ > π p phase spacebecomes noticeable. The procedure is repeated at several photon energy intervalsfrom 1150 MeV to 1192 MeV. Table E.2 gives the π p to ep phase space ratio.For higher Q settings, although the π p kinematics is getting closer to the ep kinematics, the proton momentum resolution improves and π p can be more clearlyseparated by the elastic cut. Fig. E-4 shows the phase space simulation for kinematicsK2 ( Q = 0.3 GeV ) and K8 ( Q =0.7 GeV ). It is clear to see that at K8, π p aremostly cut away. The same procedure was applied to every kinematics with differentphoton energies, the π p to ep phase space ratios for different kinematics for thesimulated π p are listed in Table. E.3. 248 φ ( p δ - p δ -0.05 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.050200040006000800010000 = 0% p δ K2 = 0.3 GeV Q = 1190 MeV γ E elasticpion ) φ ( p δ - p δ -0.05 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.050100002000030000400005000060000 = 1190 MeV γ E = 0% p δ K8 =0.7 GeV Q elasticpion Figure E-4: Simulated ep and π p spectrum for kinematics K2 and K8. The bluelines are the corresponding elastic cut applied to the data. E.3 Photon flux
The real photon flux from bremsstrahlung were calculated using [197], with 3 cmliquid hydrogen target. The results are listed in Table. E.4
E.4 Cross Sections
In order to compare the rate, the cross sections for ep and π p are required. Theelastic cross section in the lab can be directly estimated from the rate during the249ine. Q [GeV] R phase [1180 MeV] R phase [1185 MeV] R phase [1190 MeV]K1 0.35 1 . × − . × − . × − K2 0.3 9 . × − . × − . × − K3 0.45 8 . × − . × − . × − K4 0.40 1 . × − . × − . × − K5 0.55 8 . × − . × − . × − K6 0.50 4 . × − . × − . × − K7 0.6 3 . × − . × − . × − K8 0.7 0 . . × − . × − Table E.3: π p to ep phase space ratio for different kinematics with E γ = 1180, 1185,and 1190 MeV. E γ range [MeV] Γ γ × − × − × − × − . × − . × − Table E.4: Real photon flux at different energies with 1.192 GeV electron beam.experiment. For K2, with 4 µ A beam, 6 cm target, the coincidence rate is around 3kHz. The maximum HRS acceptance (6msr) is used for d Ω. The elastic differentialcross section in the lab frame can be estimated by: L = 6 cm · . g/cm · . × /g · × − A · . × /C (E.1)= 16 × /cm · s (E.2) dσ el d Ω = 3 × /s (16 × /cm · s ) · × − sr (E.3)= 3 . × − µb/sr. (E.4)The π p differential cross section for E γ ∼ ∼ . × µ b/sr in250igure E-5: World data and calculations for π p differential cross section at E γ =1185 MeV.the C.M. frame, Jacobian J = 1 . ∼ . × . µ b/sr. The cross section ratio can be obtained: R XS = σ π p /σ ep = 2 / . × − ∼ . (E.5)The ep and π p differential cross sections in the lab frame for different kinematics arelisted in Table E.7. Kine. dσ ep d Ω ( µb/sr ) dσ π p d Ω ( µb/sr ) R XS K1 6 . × − . × − . × − . × − . × − . × − . × − . × − ep and π p differential cross sections in the lab frame and the ratio R XS for different kinematics. 251 .5 Pion Electroproduction The electroproduction reaction e + p → e + p + π was checked as well. Althoughthe virtual photon flux is ∼ ) φ ( p δ - p δ -0.05 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.050200040006000800010000 = 0% p δ K2 elasticpion photopion electro Figure E-6: Phase space simulation for ep , π p and epπ with E γ = 1190 MeV. E.6 Rate Estimation and Polarization corrections
With all the information above, we can estimate the π p to ep rate ratio assuming allthe decayed photons were detected by: r = N π p /N ep = (cid:88) E γ R phase × R XS × Γ . (E.6)The results are listed in Table. E.6. So the total π p to ep ratio at K2 is ∼ × − if allthe decayed photons can be detected in the BigBite. Actually, during the experiment,only part of the BigBite shower counter was turned on, if the BigBite acceptance is252 γ range [MeV] r . × − . × − . × − . × − . × − . × − r ∼ × − Table E.6: Estimated ratio of π p to ep for kinematics K2.taking into account, the π p rate will be further reduced. E.6.1 BigBite Acceptance
For K2, a section of 3x3 shower blocks were on. The area of each shower block is 8 . × . π aimed at the center of the 9 shower blocks, sothe in-plane and out-of-plane acceptance is about ± . ◦ . This corresponds to ± ◦ in the C. M. frame where the photons are uniformly distributed. The π p rate wouldbe further suppressed by: f BB = (cos 0 ◦ − cos 35 ◦ ) / cos 0 ◦ = 0 .
18 (E.7)After multiplying the factor above, the π p to ep ratio is: r = r × f BB = 3 × − × .
18 = 0 . × − . (E.8)For different kinematics, the pion momentum is different as well as the acceptanceof BigBite. f BB for each kinematics is listed in Table E.7. Together with the otherinformation mentioned earlier, the π p to ep ratio r for each kinematics are obtainedin Table E.7. This is a very conservative estimation, since the central angle of π wasactually 0 . ∼ . < × − for all the kinematics we havetaken. Kine. shower blocks acceptance in C.M. [deg] f BB K1 3x3 ±
33 0.16K2 3x3 ±
35 0.18K3 3x3 ±
31 0.14K4 3x3 ±
32 0.15K5 3x3 ±
30 0.13K6 4x4 ±
40 0.23K7 5x5 ±
47 0.32K8 5x5 ±
44 0.28Table E.7: ep and π p differential cross sections in the lab frame and the ratio R XS for different kinematics. E.6.2 Hall C Inclusive Data
The Hall C Super-Rosenbluth experiment [199] took the singles elastic data at similar Q with a bit lower beam energies. Fig. E-7 shows the full simulation of the protonsingles spectra at 2 different beam energies. One can clearly see that the higher energymoves the pion production closer to the elastic peak, but the pion contamination isstill much less than 1% if tight elastic cuts are applied. In our experiment, thecoincidence trigger and the limited BigBite acceptance greatly suppressed the inelasticbackground. In addition, compared to the HMS, the Hall A HRS has much betterresolution which makes it much easier to cut out protons from the pion production.In another word, the smallness of the pion contamination in our data is also expectedfrom the Hall C data and simulation. E.6.3 Corrected Proton Polarizations
Now we can look at the the possible correction to the proton polarizations with1 × − π p contamination after we applied the elastic cut. For kinematics K2, asshown in Fig. E-8, the proton polarization for π p from [198] are listed in Table. E.8.254igure E-7: The proton singles spectra and the full background simulation from HallC Super-Rosenbluth experiment with beam energy 849 MeV (left panel) and 985 MeV(right panel). The spectra in red in the proton elastic peak, and the one in magentais the simulated pion production.The corrected C x and C z for the elastic events are: E γ (MeV) C x C z C ep = C raw − r · C π p − r (E.9)where r = 1 × − is the estimated π p to ep ratio. The corrected form factor ratiowould shift by ∼ . π p polarization observable at E γ = 1185 MeV. E.7 Summary
With the procedure presented above, we conservatively estimated the contributionfrom π p to be < − level. The resulting correction to the proton polarization isalso at 10 − level which is negligible. 256 ppendix FCross Section Data E E p θ e ε σ δ σ Ref.[(GeV /c ) ] [GeV] [GeV] [ ◦ ] [1] [nb/sr] [nb/sr]0.2922 0.6240 0.4683 59.997 0.58070 55.56 2.278 [12]0.2916 0.5280 0.3726 74.996 0.43960 32.16 1.319 [12]0.2923 0.4680 0.3123 89.995 0.31590 20.55 1.048 [12]0.2916 0.3990 0.2436 119.993 0.13340 10.50 0.5251 [12]0.2915 0.3800 0.2247 134.993 0.07340 8.812 0.3525 [12]0.3498 0.6920 0.5056 59.997 0.57710 36.94 1.884 [12]0.3500 0.5880 0.4015 74.996 0.43580 22.06 0.8823 [12]0.3503 0.4270 0.2403 134.993 0.07240 6.458 0.2583 [12]0.3894 0.9000 0.6925 46.557 0.70860 59.11 2.896 [12]0.3891 0.7360 0.5287 59.997 0.57460 30.66 1.226 [12]0.3897 0.6270 0.4193 74.996 0.43330 17.65 0.7235 [12]0.3894 0.5570 0.3495 89.995 0.31050 11.82 0.5792 [12]0.3898 0.4790 0.2713 119.993 0.13050 6.218 0.3109 [12]0.3893 0.4570 0.2495 134.993 0.07170 5.123 0.2049 [12]0.3894 0.4470 0.2395 144.992 0.04280 4.690 0.2345 [12]0.3894 1.9035 1.6960 19.999 0.93540 408.9 8.996 [200]0.3903 1.5370 1.3290 25.249 0.89970 226.5 4.559 [10]0.3891 1.2490 1.0416 31.738 0.84780 131.6 2.577 [10]0.3892 1.2310 1.0236 32.268 0.84330 130.0 2.580 [10]0.3892 1.1420 0.9346 35.148 0.81780 107.4 2.188 [10]0.3890 0.8480 0.6407 50.057 0.67380 45.62 0.9317 [10]0.3895 0.6960 0.4884 64.716 0.52860 25.14 0.4075 [10]0.3894 0.5560 0.3485 90.265 0.30840 11.71 0.2287 [10]0.4671 0.9500 0.7011 49.507 0.67490 33.12 1.689 [12]0.4672 0.9000 0.6510 53.037 0.63930 27.62 1.381 [12]0.4677 0.7000 0.4508 74.996 0.42850 11.67 0.4668 [12]0.4675 0.5150 0.2659 134.993 0.07040 3.529 0.1765 [12]0.4674 0.5040 0.2549 144.992 0.04200 3.240 0.1620 [12]0.5061 0.9500 0.6803 52.517 0.64240 24.26 1.189 [12]0.5066 0.7350 0.4650 74.996 0.42610 9.320 0.3821 [12]0.5064 0.5430 0.2732 134.993 0.06980 2.882 0.1441 [12]0.5072 1.7700 1.4997 25.249 0.89700 127.3 2.574 [10]258 E E p θ e ε σ δ σ Ref.[(GeV /c ) ] [GeV] [GeV] [ ◦ ] [1] [nb/sr] [nb/sr]0.5451 0.9500 0.6595 55.597 0.60900 18.17 0.9084 [12]0.5452 0.9000 0.6095 59.797 0.56700 14.20 0.7383 [12]0.5453 0.7690 0.4784 74.996 0.42380 7.793 0.3195 [12]0.5445 0.5700 0.2798 134.993 0.06920 2.454 0.1227 [12]0.5456 0.5590 0.2683 144.992 0.04130 2.347 0.1173 [12]0.5840 0.9500 0.6388 58.747 0.57510 13.27 0.6504 [12]0.5837 0.8020 0.4910 74.996 0.42150 6.573 0.3352 [12]0.5833 0.5970 0.2862 134.993 0.06860 2.126 0.1063 [12]0.5841 2.3617 2.0505 19.999 0.93240 155.0 4.031 [200]0.5840 1.0720 0.7608 50.057 0.66300 17.69 0.3563 [10]0.5843 1.0420 0.7306 51.957 0.64360 16.64 0.3373 [10]0.5844 0.8920 0.5806 64.166 0.52180 9.945 0.1983 [10]0.5837 0.8860 0.5749 64.716 0.51650 9.656 0.1985 [10]0.5845 0.7180 0.4065 90.075 0.29950 4.517 0.9927E-01 [10]0.5844 0.7170 0.4056 90.265 0.29820 4.504 0.8936E-01 [10]0.5846 0.6470 0.3354 110.294 0.17210 2.926 0.6754E-01 [10]0.5834 0.6450 0.3341 110.714 0.17000 2.969 0.6057E-01 [10]0.5847 1.9120 1.6004 25.249 0.89530 89.17 1.781 [10]0.5842 1.6290 1.3177 30.238 0.85450 60.86 1.190 [10]0.5844 1.5400 1.2286 32.268 0.83670 51.12 0.9907 [10]0.5841 1.5220 1.2107 32.698 0.83290 48.94 0.9816 [10]0.5843 1.4310 1.1196 35.148 0.81040 42.28 0.8441 [10]0.7009 0.9500 0.5765 68.886 0.46990 5.588 0.2738 [12]0.7005 0.8990 0.5257 74.996 0.41460 4.392 0.2196 [12]0.7006 0.8640 0.4907 79.996 0.37200 3.609 0.1732 [12]0.7012 0.6770 0.3034 134.993 0.06680 1.280 0.6402E-01 [12]0.7013 0.6640 0.2903 144.992 0.03980 1.177 0.5886E-01 [12]0.7790 1.7890 1.3739 32.698 0.82630 23.13 0.4753 [10]0.7784 1.6830 1.2682 35.148 0.80320 19.65 0.5852 [10]0.7791 1.3920 0.9768 44.478 0.71000 11.25 0.2274 [10]0.7791 1.0640 0.6488 64.166 0.51020 4.457 0.1089 [10]0.7792 0.8650 0.4498 90.075 0.28990 2.118 0.4660E-01 [10]0.7783 0.7840 0.3693 110.124 0.16650 1.384 0.2976E-01 [10]25960 ibliography [1] I.A. Qattan et al. Phys. Rev. Lett. , 94:142301, 2005.[2] L. Andivahis et al.
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