Polarization Transfer Observables in Elastic Electron Proton Scattering at Q^2 = 2.5, 5.2, 6.8, and 8.5 GeV^2
A. J. R. Puckett, E. J. Brash, M. K. Jones, W. Luo, M. Meziane, L. Pentchev, C. F. Perdrisat, V. Punjabi, F. R. Wesselmann, A. Afanasev, A. Ahmidouch, I. Albayrak, K. A. Aniol, J. Arrington, A. Asaturyan, H. Baghdasaryan, F. Benmokhtar, W. Bertozzi, L. Bimbot, P. Bosted, W. Boeglin, C. Butuceanu, P. Carter, S. Chernenko, E. Christy, M. Commisso, J. C. Cornejo, S. Covrig, S. Danagoulian, A. Daniel, A. Davidenko, D. Day, S. Dhamija, D. Dutta, R. Ent, S. Frullani, H. Fenker, E. Frlez, F. Garibaldi, D. Gaskell, S. Gilad, R. Gilman, Y. Goncharenko, K. Hafidi, D. Hamilton, D. W. Higinbotham, W. Hinton, T. Horn, B. Hu, J. Huang, G. M. Huber, E. Jensen, C. Keppel, M. Khandaker, P. King, D. Kirillov, M. Kohl, V. Kravtsov, G. Kumbartzki, Y. Li, V. Mamyan, D. J. Margaziotis, A. Marsh, Y. Matulenko, J. Maxwell, G. Mbianda, D. Meekins, Y. Melnik, J. Miller, A. Mkrtchyan, H. Mkrtchyan, B. Moffit, O. Moreno, J. Mulholland, A. Narayan, S. Nedev, Nuruzzaman, E. Piasetzky, W. Pierce, N. M. Piskunov, Y. Prok, R. D. Ransome, D. S. Razin, P. Reimer, J. Reinhold, O. Rondon, M. Shabestari, A. Shahinyan, K. Shestermanov, S. Sirca, I. Sitnik, L. Smykov, G. Smith, L. Solovyev, P. Solvignon, R. Subedi, E. Tomasi-Gustafsson, A. Vasiliev, M. Veilleux, B. B. Wojtsekhowski, et al. (7 additional authors not shown)
PPolarization Transfer Observables in Elastic Electron-Proton Scattering at Q = . A. J. R. Puckett, ∗ E. J. Brash,
2, 3
M. K. Jones, W. Luo, M. Meziane, L. Pentchev, C. F. Perdrisat, V.Punjabi, F. R. Wesselmann, A. Afanasev, A. Ahmidouch, I. Albayrak, K. A. Aniol, J. Arrington, A. Asaturyan, H. Baghdasaryan, F. Benmokhtar, W. Bertozzi, L. Bimbot, P. Bosted, W.Boeglin, C. Butuceanu, P. Carter, S. Chernenko, M. E. Christy, M. Commisso, J. C. Cornejo, S. Covrig, S. Danagoulian, A. Daniel, A. Davidenko, D. Day, S. Dhamija, D. Dutta, R.Ent, S. Frullani, † H. Fenker, E. Frlez, F. Garibaldi, D. Gaskell, S. Gilad, R. Gilman,
3, 24
Y. Goncharenko, K. Hafidi, D. Hamilton, D. W. Higinbotham, W. Hinton, T. Horn, B. Hu, J.Huang, G. M. Huber, E. Jensen, C. Keppel, M. Khandaker, P. King, D. Kirillov, M. Kohl, V.Kravtsov, G. Kumbartzki, Y. Li, V. Mamyan, D. J. Margaziotis, A. Marsh, Y. Matulenko, J.Maxwell, G. Mbianda, D. Meekins, Y. Melnik, J. Miller, A. Mkrtchyan, H. Mkrtchyan, B. Moffit, O. Moreno, J. Mulholland, A. Narayan, S. Nedev, Nuruzzaman, E. Piasetzky, W. Pierce, N.M. Piskunov, Y. Prok, R. D. Ransome, D. S. Razin, P. Reimer, J. Reinhold, O. Rondon, M.Shabestari, A. Shahinyan, K. Shestermanov, † S. ˇSirca,
31, 32
I. Sitnik, L. Smykov, † G. Smith, L. Solovyev, P. Solvignon, † R. Subedi, E. Tomasi-Gustafsson,
16, 33
A. Vasiliev, M. Veilleux, B.B. Wojtsekhowski, S. Wood, Z. Ye, Y. Zanevsky, X. Zhang, Y. Zhang, X. Zheng, and L. Zhu University of Connecticut, Storrs, CT 06269 Christopher Newport University, Newport News, VA 23606 Thomas Jefferson National Accelerator Facility, Newport News, VA 23606 Lanzhou University, Lanzhou 730000, Gansu, Peoples Republic of China College of William and Mary, Williamsburg, VA 23187 Norfolk State University, Norfolk, VA 23504 The George Washington University, Washington, DC 20052 North Carolina A&T State University, Greensboro, NC 27411 Hampton University, Hampton, VA 23668 California State University Los Angeles, Los Angeles, CA 90032 Argonne National Laboratory, Argonne, IL, 60439 Yerevan Physics Institute, Yerevan 375036, Armenia University of Virginia, Charlottesville, VA 22904 Duquesne University, Pittsburgh PA, 15282 Massachusetts Institute of Technology, Cambridge, MA 02139 Institut de Physique Nucl´eaire, CNRS/IN2P3 and Universit´e Paris-Sud, France Florida International University, Miami, FL 33199 University of Regina, Regina, SK S4S OA2, Canada JINR-LHE, Dubna, Moscow Region, Russia 141980 Ohio University, Athens, Ohio 45701 IHEP, Protvino, Moscow Region, Russia 142284 Mississippi State University, Mississippi, MS 39762 INFN, Sezione Sanit`a and Istituto Superiore di Sanit`a, 00161 Rome, Italy Rutgers, The State University of New Jersey, Piscataway, NJ 08855 University of Glasgow, Glasgow G12 8QQ, Scotland UK University of Witwatersrand, Johannesburg, South Africa University of Maryland, College Park, MD 20742 SLAC National Accelerator Laboratory, Menlo Park, CA 94025 University of Chemical Technology and Metallurgy, Sofia, Bulgaria University of Tel Aviv, Tel Aviv, Israel Faculty of Mathematics and Physics, University of Ljubljana, SI-1000 Ljubljana, Slovenia Joˇzef Stefan Institute, SI-1000 Ljubljana, Slovenia DSM, IRFU, SPhN, Saclay, 91191 Gif-sur-Yvette, France (Dated: August 14, 2018)
Background:
Interest in the behavior of nucleon electromagnetic form factors at large momentum transfers hassteadily increased since the discovery, using polarization observables, of the rapid decrease of the ratio G pE /G pM of the proton’s electric and magnetic form factors for momentum transfers Q (cid:38) , in strong disagreementwith previous extractions of this ratio using the traditional Rosenbluth separation technique. Purpose:
The GEp-III and GEp-2 γ experiments were carried out in Jefferson Lab’s (JLab’s) Hall C from 2007-2008, to extend the knowledge of G pE /G pM to the highest practically achievable Q given the maximum beamenergy of 6 GeV, and to search for effects beyond the Born approximation in polarization transfer observables of a r X i v : . [ nu c l - e x ] A ug elastic (cid:126)ep scattering. This article provides an expanded description of the common experimental apparatus anddata analysis procedures, and reports the results of a final reanalysis of the data from both experiments, includingthe previously unpublished results of the full-acceptance dataset of the GEp-2 γ experiment. Methods:
Polarization transfer observables in elastic (cid:126)ep → e(cid:126)p scattering were measured at central Q values of2.5, 5.2, 6.8, and 8.54 GeV . At Q = 2 . , data were obtained for central values of the virtual photonpolarization parameter (cid:15) of 0.149, 0.632, and 0.783. The Hall C High Momentum Spectrometer detected andmeasured the polarization of protons recoiling elastically from collisions of JLab’s polarized electron beam with aliquid hydrogen target. A large-acceptance electromagnetic calorimeter detected the elastically scattered electronsin coincidence to suppress inelastic backgrounds. Results:
The final GEp-III data are largely unchanged relative to the originally published results. The statisticaluncertainties of the final GEp-2 γ data are significantly reduced at (cid:15) = 0 .
632 and 0 .
783 relative to the originalpublication.
Conclusions:
The final GEp-III results show that the decrease with Q of G pE /G pM continues to Q = 8 . ,but at a slowing rate relative to the approximately linear decrease observed in earlier Hall A measurements. At Q = 8 . , G pE /G pM remains positive, but is consistent with zero. At Q = 2 . , G pE /G pM derivedfrom the polarization component ratio R ∝ P t /P (cid:96) shows no statistically significant (cid:15) -dependence, as expectedin the Born approximation. On the other hand, the ratio P (cid:96) /P Born(cid:96) of the longitudinal polarization transfercomponent to its Born value shows an enhancement of roughly 1.4% at (cid:15) = 0 .
783 relative to (cid:15) = 0 . ≈ . σ significance based on the total uncertainty, implying a similar effect in the transverse component P t thatcancels in the ratio R . I. INTRODUCTION
Electron scattering is of central importance to the char-acterization of nucleon and nuclear structure, because ofthe relative weakness of the electromagnetic interaction(compared to a strongly interacting probe), the struc-tureless character of the leptonic probe, and the avail-ability of electron beams of high intensity, duty cycle,energy, and polarization. The field of elastic electron-nucleus scattering started with the availability of elec-tron beams with energies up to 550 MeV at the HighEnergy Physics Laboratory (HEPL) in Stanford in themid-1950s. One notable result of these early experimentswas the first determination of a proton radius [1], which,together with the anomalous magnetic moment of theproton, discovered in 1933 by Otto Stern [2], completedthe picture of the proton as a finite-size object with aninternal structure.The utility of electron-nucleon scattering as a probeof nucleon structure derives from the validity of the sin-gle virtual photon exchange (Born) approximation, upto radiative corrections that are modest in size com-pared to the leading (Born) term, and precisely calcu-lable in low-order QED perturbation theory, due to thesmall value of the fine structure constant α = e π(cid:15) (cid:126) c ≈ / . ep → ep scattering am-plitude is completely specified by two form factors (FFs), ∗ Corresponding author:[email protected] † Deceased. which encode the interaction of the pointlike electromag-netic current of the electron with the proton’s charge andmagnetic moment distributions. The “Dirac” form fac-tor F describes the charge and Dirac magnetic momentinteractions, while the “Pauli” form factor F describesthe anomalous magnetic moment interaction. F and F are real-valued functions of the Lorentz-invariant four-momentum transfer squared between the electron and thenucleon, defined as Q ≡ − q = − ( k − k (cid:48) ) , with k and k (cid:48) the four-momenta of the incident and scattered electron.In fixed-target electron scattering, q is a spacelike invari-ant that is always negative. The reaction kinematics andphysical observables are thus typically discussed in termsof the positive-definite quantity Q . A detailed overviewof the theoretical formalism of the Born approximationfor elastic ep scattering is given in Ref. [4].An equivalent description of the nucleon electromag-netic form factors (EMFFs) is provided by the so-called“Sachs” form factors [5, 6] G E (electric) and G M (mag-netic), defined as the following experimentally convenientindependent linear combinations of F and F , G E ≡ F − τ F (1) G M ≡ F + F , (2)in which τ ≡ Q M p , with M p the mass of the proton. Interms of the Sachs form factors, the differential cross sec-tion for elastic ep scattering in the Born approximation isgiven in the nucleon rest frame (which coincides with thelab frame in fixed-target experiments) by the Rosenbluthformula [7]: dσd Ω e = (cid:18) dσd Ω e (cid:19) Mott (cid:15)G E + τ G M (cid:15) (1 + τ ) , (3) (cid:18) dσd Ω e (cid:19) Mott = α cos θ e E e sin θ e E (cid:48) e E e , (4) FIG. 1. G pE /G D extracted from cross section measurementsversus Q . The data from before 1980 are: open triangle[6], multiplication sign [8], open circle [9], filled diamond [10],filled square [11], crossed diamond [12], crossed square [13]and open square [14]. The SLAC data from the 1990’s arefilled star [15] and open diamond [16]. The JLab data areasterisk [17] and filled triangle [18]. Figure adapted from Fig.(3) of Ref. [4]. in which (cid:16) dσd Ω e (cid:17) Mott represents the theoretical Born crosssection for electron scattering from a pointlike, spinlesstarget of charge e , E e is the beam energy, E (cid:48) e is thescattered electron energy, θ e is the electron scatteringangle, and (cid:15) ≡ (cid:0) τ ) tan θ e (cid:1) − is the longitu-dinal polarization of the virtual photon. The expres-sion (3) provides a simple technique for the extractionof G E and G M known as Rosenbluth or L/T (for longi-tudinal/transverse) separation, in which the differentialcross section is measured at fixed Q while varying theparameter (cid:15) . A plot of the (cid:15) dependence of the “reduced”cross section, obtained by dividing the measured, radia-tively corrected cross section by the Mott cross sectionand the kinematic factor in the denominator of Eq. (3),yields a straight line with a slope (intercept) equal to G E ( τ G M ).Until the late 1990s all (or most) form factor measure-ments suggested that both G pE and G pM decreased like Q at large Q , and that the ratio µ p G pE /G pM was approxi-mately equal to one, regardless of Q . It also appearedthat the dipole form G D ≡ (cid:16) Q Λ (cid:17) − , with Λ = 0 . , provided a reasonable description of G pE , G pM /µ p and G nM /µ n , as illustrated in Figs. 1 and 2 (for G pE and G pM ). G nE was expected to have an entirely different Q dependence, given the zero net charge of the neutron,which imposes G nE = 0 at Q = 0.The helicity structure of the single-photon-exchange FIG. 2. G pM /µ p G D extracted from cross section measure-ments versus Q . The symbols are the same as in Fig. 1. Ad-ditional data points at the highest Q , open square [19] andopen star [20], were extracted from cross sections assuming µ p G pE / G pM = 1. The solid (dashed) line is a fit by Ref. [21](Ref. [22]). Figure adapted from Fig. (4) of Ref. [4]. amplitude also gives rise to significant double-polarization asymmetries, with different sensitivities tothe form factors compared to the spin-averaged crosssection. Non-zero asymmetries occur in the case wherethe electron beam is longitudinally polarized , andeither the target nucleon is also polarized or the polar-ization transferred to the recoiling nucleon is measured.The polarization transferred to the recoil proton inthe scattering of longitudinally polarized electrons byunpolarized protons has only two non-zero components,longitudinal, P (cid:96) , and transverse, P t , with respect tothe momentum transfer and parallel to the scatteringplane [23, 24]: P t = − hP e (cid:114) (cid:15) (1 − (cid:15) ) τ G E G M G M + (cid:15)τ G E P (cid:96) = hP e (cid:112) − (cid:15) G M G M + (cid:15)τ G E (5) G E G M = − P t P (cid:96) (cid:114) τ (1 + (cid:15) )2 (cid:15) . The effects of transverse polarization of the electron beam aresuppressed by factors of m e /E e , leading to asymmetries of order10 − in experiments with ultra-relativistic electrons at GeV-scaleenergies. In the context of electromagnetic form factor measure-ments in the Q regime of this work, these effects are negligiblecompared to the asymmetries for longitudinally polarized elec-trons and the precision with which they are measured. Here h denotes the sign of the electron beam helicity, and P e is the electron beam polarization. The observablesfor scattering on a polarized proton target are relatedto those for polarization transfer by time-reversal sym-metry [25–27]. Specifically, the transverse asymmetry A t = P t , while the longitudinal asymmetry A (cid:96) = − P (cid:96) .The sign change between A (cid:96) and P (cid:96) is caused by theproton spin flip required to absorb transversely polarizedvirtual photons.The interest in measuring these double-polarizationobservables is multi-faceted. First, the ratio G E /G M is directly and linearly proportional to the ratio P t /P (cid:96) in the recoil polarization case or, equivalently, the ra-tio A t /A (cid:96) of the beam-target double-spin asymmetries inthe polarized target case. Compared to the Rosenbluthmethod, polarization observables provide enhanced sen-sitivity to G E ( G M ) at large (small) values of Q . More-over, polarization observables provide an unambiguousdetermination of the relative sign of G E and G M , whereasthe Rosenbluth method is only sensitive to the squares of the form factors. Finally, because of the ratio na-ture of the asymmetries, radiative corrections tend to benegligible, whereas they can and do affect the cross sec-tion measurements and Rosenbluth separations signifi-cantly, especially in kinematics where the relative con-tribution of either the (cid:15)G E or the τ G M term to theBorn cross section (3) is small. The polarization transfermethod in particular is highly attractive, as a simultane-ous measurement of both recoil polarization componentsin a polarimeter facilitates a very precise measurement of G E /G M in a single kinematic setting, with small system-atic uncertainties resulting from cancellations of quanti-ties such as the beam polarization, the polarimeter ana-lyzing power, and the polarimeter instrumental asymme-try.In recent years the nucleon’s elastic form factors haveattracted steadily increasing attention, due in part to theunexpected results of the first polarization transfer mea-surement of the ratio G pE /G pM at JLab. This increasingattention is evident in the number of reviews of the sub-ject published in the last 15 years [4, 33–40]. The firstmeasurement of G pE /G pM by recoil polarization took placein 1994, at the MIT-Bates laboratory, at Q values of 0.38and 0.50 GeV , with 5% statistical uncertainties [41].The first two polarization transfer experiments at JLab,hereafter denoted GEp-I [28, 29] and GEp-II [30, 42],consisted of measurements of the ratio R ≡ µ p G pE /G pM for 0 . ≤ Q (GeV ) ≤ .
6. Together, the results of GEp-I and GEp-II, shown in Fig. 3, established conclusivelythat the concept of scaling of the proton form factor ra-tio had to be abandoned. There is a clear discrepancybetween the values of G pE /G pM extracted from doublepolarization experiments, and those obtained from crosssection measurements. Among possible explanations forthis discrepancy, the most thoroughly investigated is thehard two-photon exchange (TPEX) process, the ampli- tude for which does not “factorize” from the underly-ing nucleon structure information, cannot presently be FIG. 3. The ratio µ p G pE /G pM from the first two JLab experi-ments filled circle [28, 29], and filled square [30, 31], comparedto Rosenbluth separation results, open diamond [16], opencircle [17], filled diamond [18], and open square [6, 9–15, 32].The curve shows the linear fit to the polarization data fromRef. [30]. Figure adapted from Fig. (9) of Ref. [4]. calculated model-independently, and is neglected in the“standard” radiative corrections to experimental data. Arecent overview of the theory, phenomenology and experi-mental knowledge of TPEX effects in elastic ep scatteringis given in Ref. [43].In the general case, elastic eN scattering can be de-scribed in terms of three complex amplitudes [44–46],which can be written as ˜ G M , ˜ G E , and ˜ F , the first twochosen as generalizations of the Sachs electric and mag-netic form factors, G E and G M , and the last one, ˜ F , be-ing O ( α ) relative to the Born terms and vanishing in theBorn approximation. The “generalized form factors” ˜ G M and ˜ G E can be decomposed into sums of the real-valuedSachs form factors appearing in the Born amplitudes anddepending only on Q , plus O ( α ) complex-valued correc-tions that vanish in the Born approximation and dependon both Q and (cid:15) as follows:˜ G M ( Q , (cid:15) ) ≡ G M ( Q ) + δ ˜ G M ( Q , (cid:15) ) (6)˜ G E ( Q , (cid:15) ) ≡ G E ( Q ) + δ ˜ G E ( Q , (cid:15) ) . (7)In terms of the generalized complex amplitudes, the re-duced cross section σ R ≡ (cid:15) (1+ τ ) τ σ/σ Mott and polarizationobservables are given at next-to-leading order in α by: σ R = G M + (cid:15)τ G E + 2 G M (cid:60) (cid:16) δ ˜ G M + (cid:15)νM ˜ F (cid:17) + 2 (cid:15)τ G E (cid:60) (cid:16) δ ˜ G E + νM ˜ F (cid:17) , (8) P t = − hP e σ R (cid:114) (cid:15) (1 − (cid:15) ) τ (cid:104) G E G M + G M (cid:60) (cid:16) δ ˜ G E + νM ˜ F (cid:17) + G E (cid:60) (cid:16) δ ˜ G M (cid:17)(cid:105) , (9) P (cid:96) = hP e σ R (cid:112) − (cid:15) (cid:20) G M + 2 G M (cid:60) (cid:18) δ ˜ G M + (cid:15) (cid:15) νM ˜ F (cid:19)(cid:21) , (10) P n = (cid:114) (cid:15) (1 + (cid:15) ) τ σ R (cid:20) − G M (cid:61) (cid:16) δ ˜ G E + νM ˜ F (cid:17) + G E (cid:61) (cid:18) δ ˜ G M + 2 (cid:15) (cid:15) νM ˜ F (cid:19)(cid:21) , (11) R ≡ − µ p (cid:114) τ (1 + (cid:15) )2 (cid:15) P t P (cid:96) = µ p G E G M (cid:60) (cid:34) − δ ˜ G M G M + δ ˜ G E G E + ν ˜ F M (cid:18) (1 + (cid:15) ) G M − (cid:15)G E (1 + (cid:15) ) G E G M (cid:19)(cid:35) , (12)in which (cid:15) and τ are defined as above, the symbols (cid:60) and (cid:61) denote real and imaginary parts of the amplitudes, and νM ≡ (cid:114) τ (1 + τ ) 1 + (cid:15) − (cid:15) . (13)The reduced cross section and the polarization transfercomponents P t and P (cid:96) are defined only by the real partsof the two-photon amplitudes. The normal polarizationtransfer component, P n , which is zero in the Born ap-proximation, is defined by the imaginary parts of thetwo-photon exchange amplitudes.There are several noteworthy features of Eqs. (8)-(12).The corrections to the reduced cross section beyond theBorn approximation are additive with the Born terms,implying that even a small TPEX correction can seriouslyobscure the extraction of G E ( G M ) at large (small) Q when the relative contribution of either Born term to σ BornR is small enough to be comparable to the TPEXcorrection. The ratio R defined in Eq. (12), on the otherhand, is directly proportional to its Born value: R = µG E /G M (1 + O ( α )), and is subject only to relative O ( α )TPEX corrections, in principle. In the limit G E → R ; the limit of Eq. (12) as G E → R → R Born + (cid:60) (cid:104) µ δ ˜ G E G M + µ νM ˜ F G M (cid:105) , assuming δ ˜ G M /G M (cid:28) R measured in polarization trans-fer experiments only becomes significantly sensitive toTPEX corrections when R Born is comparable to α , thereduced cross section becomes sensitive to TPEX correc-tions at relatively low Q even for R Born (cid:29) α . Giventhe superior sensitivity to G E at large Q of the ra-tio P t /P (cid:96) and its relative robustness against radiativeand TPEX corrections as compared to the Rosenbluthmethod, a general consensus has emerged that the po-larization transfer data provide the most reliable deter-mination of G pE in the Q range where cross section andpolarization data disagree. Nevertheless, a large amountof experimental and theoretical effort is ongoing to under-stand the source of the discrepancy and develop a maxi-mally model-independent prescription for TPEX correc-tions to elastic ep scattering observables.The subject of this article is the third dedicated seriesof polarization transfer measurements in elastic (cid:126)ep scat- tering at large Q , carried out in Jefferson Lab’s (JLab’s)Hall C from October, 2007 to June, 2008. ExperimentsE04-108 (GEp-III) and E04-019 (GEp-2 γ ) used the sameapparatus and method to address two complementaryphysics goals. The goal of GEp-III was to extend thekinematic reach of the polarization transfer data for G pE /G pM to the highest practically achievable Q , giventhe maximum electron beam energy available at the time.The goal of GEp-2 γ was to measure the (cid:15) -dependence of G pE /G pM at the fixed Q of 2.5 GeV with small statis-tical and systematic uncertainties, in order to test thepolarization method and search for signatures of TPEXeffects in two polarization observables.The results of GEp-III [31] and GEp-2 γ [47] have al-ready been published in short-form articles. The purposeof this article is to provide a detailed description of theapparatus and analysis methods common to both exper-iments and report the results of a full reanalysis of thedata, carried out with the aim of reducing the system-atic and, in the GEp-2 γ case, statistical uncertainties.Our reanalysis of the GEp-2 γ data includes the previ-ously unpublished results of the full-acceptance analysisat (cid:15) = 0 .
632 and (cid:15) = 0 . Q nucleon FF data is given in Section V A, while theimplications of the GEp-2 γ data for the understandingof TPEX contributions in elastic ep scattering and thediscrepancy between cross section and polarization datafor G pE /G pM are discussed in Section V B. Our conclusionsare summarized in Section VI. TABLE I. Central kinematics of the GEp-III and GEp-2 γ experiments. Q denotes the central or nominal Q value, definedby the central momentum setting of the High Momentum Spectrometer (HMS) in which the proton was detected. (cid:15) is the valueof the kinematic parameter defined in equation (3) computed from the incident beam energy (not corrected for energy loss inthe target prior to scattering), and the central Q . E e is the incident beam energy, averaged over the duration of each runningperiod. E (cid:48) e is the scattered electron energy at the nominal Q . The central angle of BigCal is denoted θ e , and can differ slightlyfrom the electron scattering angle at the central Q . p p is the HMS central momentum setting. θ p is the HMS central angle. χ is the central spin precession angle in the HMS, P e is the average beam polarization, and D cal is the distance from the originto the surface of BigCal.Dates (mm/dd-mm/dd, yyyy) Q (GeV ) (cid:15) E e (GeV) E (cid:48) e (GeV) θ e ( ◦ ) p p (GeV) θ p ( ◦ ) χ ( ◦ ) P e (%) D cal (m)11/27-12/08, 2007 2.50 0.154 1.873 0.541 105.2 2.0676 14.5 108.5 85.9 4.9301/17-01/25, 2008 2.50 0.150 1.868 0.536 105.1 2.0676 14.5 108.5 85.5 4.9412/09-12/16, 2007 2.50 0.633 2.847 1.515 44.9 2.0676 31.0 108.5 84.0 12.0012/17-12/20, 2007 2.50 0.772 3.548 2.216 32.6 2.0676 35.4 108.5 85.8 11.1601/05-01/11, 2008 2.50 0.789 3.680 2.348 30.8 2.0676 36.1 108.5 85.2 11.0311/07-11/20, 2007 5.20 0.377 4.052 1.281 60.3 3.5887 17.9 177.2 79.5 6.0505/27-06/09, 2008 6.80 0.506 5.711 2.087 44.2 4.4644 19.1 217.9 79.5 6.0804/04-05/27, 2008 8.54 0.235 5.712 1.161 69.0 5.4070 11.6 262.2 80.9 4.30 II. EXPERIMENT DESCRIPTION
Longitudinally polarized electrons with energies upto 5.717 GeV produced by JLab’s Continuous ElectronBeam Accelerator Facility (CEBAF) were directed ontoa liquid hydrogen target in experimental Hall C. Elasti-cally scattered protons were detected by the High Mo-mentum Spectrometer (HMS), equipped with a doubleFocal Plane Polarimeter (FPP) to measure their polar-ization. Elastically scattered electrons were detected by alarge-solid-angle electromagnetic calorimeter (BigCal) incoincidence with the scattered protons. The main trig-ger for the event data acquisition (DAQ) was a coinci-dence between the single-arm triggers of the HMS andBigCal within a 50-ns window. Details of the coincidencetrigger logic and the experiment data acquisition can befound in Ref. [48]. Table I shows the central kinematicsand running periods of the GEp-III and GEp-2 γ exper-iments. The two running periods at E e ≈ .
87 GeVwere combined and analyzed together as a single kine-matic setting. The same is true of the running periodsat E e = 3 .
548 GeV and E e = 3 .
680 GeV. In both cases,the near-total overlap of the Q and (cid:15) acceptances of twodistinct measurements differing only slightly in beam en-ergy and HMS central angle justifies combining the twosettings into a single measurement . The beam energyfor each running period quoted in Table I represents theaverage incident beam energy during that period, andis not corrected for energy loss in the LH target. The In this context, combining the data from two distinct measure-ments means combining all events from each of the two kinemat-ically similar settings in a single unbinned maximum-likelihoodextraction of P t and P (cid:96) , in which the small differences in cen-tral kinematics are accounted for event-by-event. This amountsto the assumption that P t and P (cid:96) are the same for both set-tings. The data were also analyzed separately and found to beconsistent with this assumption. (cid:15) value quoted in Tab. I is computed from the averageincident beam energy and central Q value, and differsslightly from the acceptance-averaged value, hereafter re-ferred to as (cid:104) (cid:15) (cid:105) , and the “central” value (cid:15) c quoted withthe final GEp-2 γ results, which is computed from thecentral Q value and the average beam energy, correctedevent-by-event for energy loss in the LH target materi-als upstream of the reconstructed scattering vertex (seeTab. XI and XII).CEBAF consists of two antiparallel superconductingradio-frequency (SRF) linear accelerators (linacs), eachcapable (ca. 2007-2008) of approximately 600 MeV ofacceleration, connected by nine recirculating magneticarcs, with five at the north end and four at the south end.With this “racetrack” design, the electron beam can beaccelerated in up to five passes through both linacs, for amaximum energy of approximately 6 GeV before extrac-tion and delivery to the three experimental halls. Po-larized electrons are excited from a “superlattice” GaAsphotocathode using circularly polarized laser light. De-tails of the CEBAF accelerator design and operationalparameters are described in Refs. [49, 50], while moredetails specific to the running period of the GEp-III andGEp-2 γ experiments can be found in Ref. [48]. The typ-ical beam current on target during the experiment was60-100 µ A, while the typical beam polarization was 80-86%. The beam helicity was flipped pseudorandomly [51]at a frequency of 30 Hz throughout the experiment.During normal operations, the Hall C arc magnets,which steer the beam extracted from the CEBAF accel-erator to Hall C, are operated in an achromatic tune. Fora measurement of the beam energy, the arc magnets areoperated in a dispersive tune. The central bend angle of Where data from kinematically similar settings have been com-bined, the “central” (cid:15) value quoted with the final result repre-sents a weighted average of the “central” values from each of thecombined settings.
TABLE II. Arc measurements of the beam energy ( E arc )taken during the GEp-III and GEp-2 γ experiments. No ded-icated Hall C arc measurement was performed during the pe-riod from December 17-20, 2007, during which the nominalbeam energy was 3.548 GeV. The data at a central Q of 6.8GeV were collected at the same nominal beam energy as the Q = 8 . data during April-June, 2008.Date Q (GeV ) Number of passes E arc (MeV)11/19/2007 5.2 5 4052.34 ± ± ± ± ± ± the arc is 34.3 ◦ . The field integral of the arc magnets hasbeen measured as a function of the power supply current.The beam position and arc magnet current setting infor-mation are used in the feedback system which stabilizesthe beam energy and position. This system has been cal-ibrated using dedicated arc beam energy measurementsfrom Halls A and C, and is used for continuous monitor-ing of the beam momentum.Table II shows the Hall C arc measurements of thebeam energy performed during the GEp-III and GEp-2 γ experiments. The arc energy of E e = 5 .
717 GeV mea-sured at the beginning of the Q = 8 . running inApril 2008 differs slightly from the average beam energyfor this run period and the subsequent 6.8 GeV running,shown in Table I. During the 8.5 GeV running, a numberof slight changes in accelerator tune to optimize the per-formance of CEBAF in the context of simultaneous deliv-ery of longitudinally polarized beam to Halls A and C atdifferent passes resulted in several slight changes in beamenergy at the 1-2 MeV level. While no additional arc en-ergy measurements were performed, the small, occasionalchanges in beam energy were detected by the online beamenergy monitoring system, and also confirmed by shifts inthe elastic peak position in the variables used for elasticevent selection in the offline analysis (see section III A).These small changes were included in the final beam en-ergy database for the offline analysis. Except for the firstfew days at 8.5 GeV , during which the beam energy was5.717 GeV, the actual incident beam energy varied be-tween 5.710 and 5.714 GeV during most of the 8.5 GeV running, averaging 5.712 GeV. The incident beam energywas stable at 5.711 GeV during the 6.8 GeV running.As discussed in section III A and Ref. [52], the contri-bution of the systematic uncertainty in the beam energyto the total systematic uncertainties in the polarizationtransfer observables is a small fraction of the total.The target system used for this experiment consistsof several different solid targets and a three-loop cryo-genic target system for liquid hydrogen (LH ). The solidtargets include thin foils of Carbon and/or Aluminumused for spectrometer optics calibrations and to measure the contribution of the walls of the cryotarget cell to theexperiment background. The spectrometer optics cali-brations and systematic studies are described in detail inRef. [52], while details of the solid targets are describedin Ref. [48]. For the first kinematic point taken fromNov. 7-20, 2007, a 15-cm LH cryotarget cell was used.For all of the other production kinematics of both exper-iments, a 20-cm cryotarget cell was used. The center ofthe 20-cm cell was offset 3.84 cm downstream of the ori-gin along the beamline to allow electrons scattered by upto 120 degrees to exit through the thin scattering cham-ber exit window and be detected by the calorimeter. Theliquid hydrogen targets were operated at a constant tem-perature of 19 K and nominal density of ρ ≈ .
072 g/cm throughout the experiment. The size of the beam spoton target was enlarged to a transverse size of typically2 × by the Hall C fast raster magnet system, tominimize localized heating and boiling of the liquid hy-drogen and resulting fluctuations in target density andluminosity. More details of the cryogenic target systemcan be found in Ref. [48]. A. Hall C HMS
The High Momentum Spectrometer (HMS) is part ofthe standard experimental equipment in JLab’s Hall C.It is a superconducting magnetic spectrometer with threequadrupoles and one dipole arranged in a QQQD layout.The HMS has a 25-degree central vertical bend angle andpoint-to-point focusing in both the dispersive and non-dispersive planes when operated in its “standard” tune.The HMS dipole field is regulated by an NMR probe andis stable at the 10 − level, while the quadrupole magnetpower supplies are regulated by current and are stableat the 10 − level. The HMS solid angle acceptance isapproximately 6.74 msr when used with the larger of itstwo retractable, acceptance-defining octagonal collima-tors, as it was in this experiment. The HMS momentumacceptance is approximately ±
9% relative to the centralmomentum setting. The maximum central momentumsetting is 7.4 GeV/c. The HMS detector package and su-perconducting magnets are supported on a common car-riage that rotates on concentric rails about the centralpivot of Hall C. The detector package is located inside aconcrete shield hut supported on a separate carriage fromthe detector and magnet supports. With the exceptionof small air gaps between the scattering chamber exitwindow and the HMS entrance window and between theHMS dipole exit window and the first HMS drift cham-ber, the entire flight path of charged particles throughthe HMS is under vacuum, minimizing energy loss andmultiple scattering prior to the measurement of chargedparticle trajectories.As shown in Fig. 4, the HMS detector package wasmodified by removing the gas Cherenkov counter andthe two rearmost planes of scintillator hodoscopes fromthe standard HMS detector package to accommodate theFocal Plane Polarimeter (FPP), leaving only the two up-stream planes of scintillators (“S1X” and “S1Y”) to forma fast trigger. The HMS calorimeter was not removed,and its signals were recorded to the data stream, but itwas not used either in the trigger or in the offline analy-sis, except for crude pion rejection in the analysis of theHMS optics calibration data, for which the HMS was setwith negative polarity for electron detection. The stan-dard HMS drift chambers, described in detail in Ref. [53],were used to measure the trajectories of elastically scat-tered protons. The measured proton tracks were thenused to reconstruct the event kinematics at the targetand to define the incident trajectory for the secondarypolarization-analyzing scattering in the CH analyzers ofthe FPP. Because the two rear planes of scintillators hadbeen removed, the “S1X” and “S1Y” planes could not,by themselves, provide an adequately selective trigger formost kinematic settings of the experiment. To overcomethis challenge, two additional 1 cm-thick plastic scintil-lator paddles were installed between the exit window ofthe HMS vacuum and the first HMS drift chamber, withsufficient area to cover the envelope of elastically scat-tered protons for all kinematic settings. These two pad-dles were collectively referred to as “S0”. The S0 planereduced the trigger rate to a manageable level by restrict-ing the acceptance to the region populated by elasticallyscattered protons and suppressing triggers due to inelas-tic processes that occur at a much higher rate for large Q values. During most of the experiment, the HMS triggerrequired at least one paddle to fire in each of the “S1X”,“S1Y” and “S0” planes. During part of the measurementat E e = 2 .
847 GeV and the entire duration of the mea-surements at E e = 3 .
548 GeV and E e = 3 .
680 GeV, forwhich the HMS was located at relatively large scatteringangles, the trigger was based on “S1X” and “S1Y” only,as the rates were low enough to use this less-selectivetrigger in coincidence with the electron calorimeter. Theprice to pay for installing the S0 trigger plane upstreamof the drift chambers is that the angular resolution ofthe HMS was significantly degraded due to the additionalmultiple scattering in S0 [48]. More details of the customHMS trigger logic used for these experiments are givenin Ref. [48].
B. Focal Plane Polarimeter
A new focal plane polarimeter (FPP) was designed,built and installed in the HMS to measure the polariza-tion of the recoiling protons. It consists of two CH ana-lyzer blocks arranged in series to increase the efficiency,each followed by a pair of drift chambers. A design draw-ing of the HMS detector package with the FPP, the HMSdrift chambers and the trigger scintillator planes is shownin Figure 4. CH analyzer blocksFPP drift chamber pairsS1X+S1Y trigger planeHMS drift chambersS0 trigger plane FIG. 4. Design drawing of the FPP installed in the HMS de-tector package, with the HMS drift chambers and the triggerplanes.
1. FPP Analyzer
The FPP analyzer is made of polyethylene (CH ).It consists of two retractable doors, each made of twoblocks, allowing for the collection of “straight-through”trajectories for calibration and alignment studies. Eachpair is 145 cm (tall) ×
111 cm (wide) ×
55 cm (thick) andmade of several layers of CH held together by an outeraluminum frame. To reduce the occurrence of leakagethrough the seam when the doors are inserted, an over-lapping step was designed into the edge of both doors.Given their substantial weight, the CH blocks were sup-ported on a different frame than the detector and at-tached directly to the floor of the shield hut, ensuringthat the other detectors did not move while inserting orretracting the doors.The choice of CH as the analyzer material was drivenby a compromise among the analyzing power and opti-mal thickness of the material on the one hand, and thecost and space constraints within the HMS hut on theother. Measurements of the analyzing power of the reac-tion (cid:126)p +CH → X at Dubna [54] showed that the overallfigure of merit of the polarimeter does not increase whenthe analyzer thickness is increased beyond the nuclearcollision length λ T of CH . With this result in mind, theHMS FPP was designed as a double polarimeter withtwo analyzers, each approximately one λ T thick and fol-lowed by pairs of drift chambers to measure the angu-lar distribution of scattered protons. The analyzers andthe drift chambers were designed to be large enough tohave 2 π azimuthal angular acceptance for transverse mo-menta p T ≡ p sin ϑ up to 0.7 GeV/c, beyond which thepolarimeter figure of merit essentially saturates. TABLE III. Characteristics of the wires used in the FPP driftchambers. The sense wires are gold-plated tungsten, while thecathode and field wires are made of a beryllium-bronze alloy.Type Diameter ( µ m) Tension (g)Sense 30 70Field 100 150Cathode 80 120
2. FPP drift chambers
The tracking system of the FPP consists of two driftchamber pairs, one after each analyzer block. All fourchambers are identical in design and construction. Theactive area of each chamber is 164 cm (tall) ×
132 cm(wide). Each chamber contains three detection planessandwiched between cathode layers. Each detection layerconsists of alternating sense wires and field wires with aspacing of 2 cm between adjacent sense wires (1 cm be-tween a sense wire and its neighboring field wires). Thewire spacing in the cathode layers, located 0.8 cm aboveand below the detection layers, is 3 mm. The character-istics of the different wires are given in Table III. Thesense wire planes have three different orientations, de-noted “U”, “V”, and “X”. The stacking order along the z axis of the planes in each chamber is VXU. The “V”wires are strung along the +45 ◦ line relative to the x axisand thus measure the coordinate along the − ◦ -line; i.e., v ≡ x − y √ . The “X” wires are strung perpendicular to the x axis and thus measure the x coordinate. The “U” wiresare strung along the − ◦ line relative to the x axis andthus measure the coordinate u ≡ x + y √ . The U and Vlayers have 104 sense wires each, while the X layers have83 sense wires. Each layer within each chamber has asense wire passing through the point ( x, y ) = (0 , .Each drift chamber is enclosed by 30 µ m-thick alu-minized mylar gas windows and a rigid aluminum frame.Each pair of chambers is attached to a common set ofrigid spacer blocks (two on each side of the chamberframe) by a set of two aligning bolts per block penetrat-ing each chamber. Each of the two spacer blocks alongboth the top and bottom sides of the chamber frameis also attached to a third threaded steel rod that goesthrough both chambers in the pair. The chamber pair isthen mounted to the FPP support frame via C-shaped The symmetry created by this common intersection point andthe relative lack of redundancy of coordinate measurements, withonly six coordinate measurements along each track, creates anessentially unresolvable left-right ambiguity for a small fractionof tracks passing through the region near the center of the cham-bers at close to normal incidence, for which two mirror-imagesolutions of the left-right ambiguity exist with identical combi-nations of drift distances that are basically indistinguishable interms of χ . channels machined into the top spacer blocks that matewith a cylindrical Thomson rail attached to the top of thesupport frame, and via protrusions of the bottom spacerblocks with guide wheels that slide into a “U” channel onthe bottom of the FPP support frame. After installation,each chamber pair was bolted to a hard mechanical stopbuilt into the support frame. The design ensures that therelative positioning of the two chambers within a pair isfixed and reproducible.The FPP drift chambers used the same 50%/50% ar-gon/ethane gas mixture as the HMS drift chambers. Thebasic drift cell in the FPP drift chambers has the sameaspect ratio as the HMS drift cell, but the dimensions aretwice as large. The cathode and field wires were main-tained at a constant high voltage of -2400 V, while thesense wires were at ground potential. This operationalconfiguration gives the FPP drift chambers similar, butnot identical, electric field and drift velocity character-istics to the HMS drift chambers. The main differenceis that the HMS drift chambers were operated with adifferent electric field configuration in which three differ-ent high voltage settings were applied to the field andcathode wires according to their distance from the near-est sense wire, leading to nearly cylindrical equipotentialsurfaces surrounding each sense wire. This in turn meansthat the drift time measured by the HMS chambers is afunction of the distance of closest approach of the trackto the wire, rather than the in-plane track-wire distance.Since the tracks of interest in the HMS drift chambersare very nearly perpendicular to the wire planes, the dif-ference between these two distances is small in any case.The FPP wire signals are processed by front-end am-plifier/discriminator (A/D) cards attached directly to thechambers. Each A/D card processes the signals fromeight sense wires. The amplified, discriminated FPP sig-nals are digitized by TDCs located close to the chamberswithin the HMS shield hut. A significant advantage ofthe Hall C FPP DAQ system compared to previous ex-periments using the Hall A FPP [29, 42] is that each sensewire was read out individually by a dedicated multi-hitTDC channel, whereas the straw chamber signals in theHall A FPP were multiplexed in groups of eight wires bythe front-end electronics to reduce the number of read-out channels required, effectively preventing the resolu-tion of multi-track events in which two or more trackscreate simultaneous signals on straws located within thesame group of eight. As discussed in Sec. III B 7, theability to isolate true single-track events significantly in-creased the effective analyzing power of the Hall C FPPrelative to the Hall A FPP for equivalent analyzer ma-terial and thickness. From the start of the experimentin October 2007 to February 2008, VME-based F1 TDCmodules [55] housed in a pair of VME crates in the HMSshield hut were used to read out the FPP signals. For thehigh- Q data collection from April to early June of 2008,the FPP signals were read out using LeCroy 1877-modelFastbus TDCs. The FPP data acquisition was changedfrom VME to Fastbus TDCs due to relatively frequent0malfunctions of the VME DAQ system encountered dur-ing the GEp-2 γ production running, especially for thedata taken at the relatively forward HMS central angleof 14.5 degrees, for which the detector hut was fairlyclose to the beam dump and the hit rates in the FPPchambers were relatively high. Since no such problemswere observed with the Fastbus TDCs used concurrentlyto read out the HMS drift chambers, a second Fastbuscrate equipped with LeCroy 1877 TDC modules was in-stalled in the HMS shield hut during the planned two-month accelerator shutdown in February and March of2008 in preparation for the high- Q running at an HMSangle of 11.6 degrees. As expected based on the expe-rience with the HMS drift chamber readout, the Fast-bus TDC readout for the FPP drift chambers functionedfairly smoothly throughout the 2008 high- Q running. C. Electron Calorimeter
Elastically scattered electrons were detected by an elec-tromagnetic calorimeter, named BigCal, built specificallyfor this experiment. The calorimeter was made of 1,744lead-glass blocks (TF1-0 type) stacked with a frontal areaof 122 ×
218 cm . The array was constructed from blocksof two different sizes. The bottom part of the calorime-ter consisted of a 32 ×
32 array of blocks with dimen-sions of 3 . × . ×
45 cm originating from the IHEP inProtvino, Russia, while the top part of the calorimeterconsisted of a 30 ×
24 array of blocks with dimensionsof 4 × ×
40 cm from the Yerevan Physics Institute inYerevan, Armenia, used previously in a Compton scat-tering measurement in Hall A [56]. The 45-cm (40-cm)depth of the Protvino (Yerevan) blocks corresponds to16.4 (14.6) radiation lengths, sufficient to absorb the totalenergy of elastically scattered electrons. The Cherenkovlight created in the glass by relativistic particles fromthe electromagnetic cascade was registered by photomul-tiplier tubes (PMTs) of type FEU-84, coupled opticallyto the end of each block with a 5 mm-thick transparentsilicon ”cookie” to compensate for a possible misalign-ment between the two elements. The blocks were opti-cally isolated from each other via an aluminized mylarwrapping. For each kinematic setting, the calorimeterwas positioned at an angle corresponding to the central Q value and beam energy. The distance from the ori-gin to the surface of BigCal was chosen to be as largeas possible, consistent with matching between the solidangle acceptance of BigCal for elastically scattered elec-trons and the fixed solid angle of the HMS for elasticallyscattered protons. For the kinematics at E e = 3 .
548 GeVand 3.680 GeV (see Tab. I), BigCal was placed closer tothe target than the acceptance-matching distance due to The purpose of this accelerator down was to install refurbishedcryomodules in CEBAF to reach the maximum beam energy of5.7 GeV needed for the high- Q running of GEp-III. FIG. 5. BigCal calorimeter with its front aluminum shieldingplates removed, exposing the stack of 1744 lead glass blocks. limitations imposed by the signal cable length and thelocation of the BigCal readout electronics, as well as theavailable space in Hall C. At Q = 8 . , the elec-tron solid angle for acceptance matching was 143 msr,or about twenty times the solid angle acceptance of theHMS. Fig. 5 shows BigCal with the front shielding platesremoved, revealing the array of lead-glass blocks.The analog signals from the PMTs were sent to spe-cialized NIM modules for amplification and summing,with eight input channels each. The outputs includedcopies of the individual input signals amplified by a fac-tor of 4.2, and several copies of the analog sum of theeight input signals. The amplified analog signals fromthe individual PMTs were sent to LeCroy model 1881Mcharge-integrating Fastbus ADCs for readout. One copyof each “first level” sum of eight blocks was sent to afixed-threshold discriminator, the output of which wasthen sent to a TDC for timing readout. Additional copiesof each sum of eight were combined with other sums-of-eight into “second-level” sums of up to 64 blocks us-ing identical analog summing modules. These “level 2”sums, of which there are a total of 38, were also sentto fixed-threshold discriminators, and a global “OR” ofall the second-level discriminator outputs was used todefine the trigger for BigCal. The groupings of blocksfor the “level 2” sums were organized with partial over-lap to avoid regions of trigger inefficiency, as detailed inRef. [48]. Because there was no overlap in the triggerlogic between the left and right halves of the calorime-ter, the trigger threshold was limited to slightly less than1half of the average elastically scattered electron energy.A higher threshold would have resulted in significant ef-ficiency losses at the boundary between the left and righthalves of the calorimeter.Four one-inch thick aluminum plates (for a total ofabout one radiation length) were installed in front of theglass to absorb low-energy photons and mitigate radia-tion damage to the glass. This additional material de-grades the energy resolution, but does not significantlyaffect the position resolution. All four aluminum plateswere used for all kinematics except the lowest (cid:15) pointof the GEp-2 γ experiment, for which only one plate wasused. For this setting, the calorimeter was placed at thebackward angle of θ e ≈ ◦ , for which the elasticallyscattered electron energy was only E (cid:48) e ≈ .
54 GeV, theradiation dose rate in the lead-glass was low enough thatthe additional shielding was not needed, and the betterenergy resolution afforded by removing three of the fourplates was needed to maintain high trigger efficiency atthe operating threshold.The glass transparency gradually deterioratedthroughout the experiment due to accumulated radia-tion damage. The effective gain/signal strength in theBigCal blocks was monitored in situ throughout the ex-periment using the known energy of elastically scatteredelectrons, reconstructed precisely from the measuredproton kinematics. The PMT high voltages were period-ically increased to compensate for the gradual decreasein light yield and maintain a roughly constant absolutesignal size, in order to avoid drifts in the effective triggerthreshold and other deleterious effects. However, asdiscussed in Ref. [52], the reduced photoelectron yieldcaused the energy resolution to deteriorate. With thefour-inch-thick aluminum absorber in place, the energyresolution worsened from about 10 . / √ E following theinitial calibration to roughly 22% / √ E at the end of theexperiment. During the early 2008 accelerator shutdown,the glass was partially annealed using a UV lamp systembut it did not fully recover to its initial transparencyand energy resolution prior to the start of the high- Q running in April 2008, at which point the transparencyresumed its gradual deterioration. The achieved energyresolution, while relatively poor for this type of detectorand dramatically worsened by radiation damage, wasnonetheless adequate for triggering with the thresholdset at half the elastically scattered electron energy orless. In contrast to the energy resolution, the positionresolution of BigCal, estimated to be roughly 6 mmusing the Q = 6 . data collected at the end of theexperiment [48, 52], did not change noticeably duringthe experiment. The achieved coordinate resolution ofBigCal was significantly better than needed given theexperimentally realized angular, momentum and vertexresolution of the HMS, and proved essential for thesuppression of the inelastic background, especially athigh Q , as discussed in section III A. More details ofthe calibration and event reconstruction procedures forBigCal can be found in Refs. [48, 52]. III. DATA ANALYSIS
The analysis of the data proceeds in three phases:1. Decoding of the raw data and the reconstruction ofevents2. The selection of elastic ep events and the estimationof the residual contamination of the final sample byinelastic backgrounds and accidental coincidences3. The extraction of the polarization transfer observ-ables from the measured angular distributions ofprotons scattered in the FPP.The raw data decoding and the event reconstruction pro-cedure, including detector calibrations and reconstruc-tion algorithms, are described in the technical supple-ment to this article [52] as well as the Ph.D. thesis [48].The elastic event selection and background estimationprocedure are discussed in Sec. III A. The extraction ofpolarization observables is presented in Sec. III B. Thedetailed evaluation of systematic uncertainties is pre-sented in Refs. [48, 52]. A. Elastic event selection
Elastic events were selected using the two-body kine-matic correlations between the electron and the proton.Accidental coincidences were suppressed by applying aloose, ±
10 ns cut to the time-of-flight-corrected differ-ence ∆ t between the timing signals associated with theelectron shower in BigCal and the proton trigger in thefast scintillator hodoscopes of the HMS. The resolutionof the coincidence time difference ∆ t is dominated by thetiming resolution of BigCal, which varied from 1 . − ±
10 nscut region was less than 10% before applying the exclu-sivity cuts described below, and negligible after applyingthe cuts. The transferred polarization components forthe accidental coincidence events were found to be simi-lar to those of the real coincidence events for the inelasticbackground [57], such that the accidental contaminationof the inelastic background sample at the level of 10% orless did not noticeably affect the corrections to the elastic ep signal polarizations, which were essentially negligibleexcept at Q = 8 . .The beam energy is known with an absolute accu-racy ∆ E/E (cid:46) × − from the standard Hall C “arc”measurement technique. The “per-bunch” beam energyspread under normal accelerator operating conditions istypically less than 3 × − and is continuously moni-tored using synchrotron light interferometry [58], whilethe CEBAF fast energy feedback system maintains the“long term” stability of the central beam energy at the10 − level [59]. The spread and systematic uncertaintyin the electron beam energy is significantly smaller than2 - - p p d · E ve n t s = 5.2 GeV Q - e p d · E ve n t s = 5.2 GeV Q - fd · E ve n t s = 5.2 GeV Q - - p p d · E ve n t s = 6.8 GeV Q - e p d · E ve n t s = 6.8 GeV Q - fd · E ve n t s = 6.8 GeV Q - p p d · E ve n t s = 8.5 GeV Q - e p d · E ve n t s = 8.5 GeV Q - fd · E ve n t s = 8.5 GeV Q FIG. 6. Simplified illustration of elastic event selection for the GEp-III kinematics: Q = 5 . (top row), Q = 6 . (middle row) and Q = 8 . (bottom row). Exclusivity cut variables are δp p ≡ × p p − p p ( θ p ) p (left column), δp e ≡ × p p − p p ( θ e ) p (middle column), and δφ ≡ φ e − φ p − π (right column). The distribution of each variable is shown for allevents (red empty circles), events selected by applying ± σ cuts of fixed width to both of the other two variables (black filledsquares), and events rejected by these cuts (blue empty triangles). Vertical dotted lines indicate the ± σ cut applied to eachvariable. Similar plots for the GEp-2 γ kinematics can be found in Ref. [52]. Note that the horizontal axis range in each plot isa fixed multiple of the elastic peak width, which varies with Q and E e . the HMS momentum resolution of σ p /p ≈ − , and itscontribution to the systematic uncertainty in the deter-mination of the reaction kinematics is small.The scattering angles and energies/momenta of bothoutgoing particles are measured in each event. Becausethe energy resolution of BigCal was too poor to pro-vide meaningful separation between elastic and inelas-tic events for any cut with a high efficiency for elasticevents, no cuts were applied to the measured energy ofthe electron, beyond the hardware threshold imposed bythe BigCal trigger and the software threshold imposedby the clustering algorithm. This leaves the proton mo-mentum and the polar and azimuthal scattering anglesof the electron and proton as useful kinematic quantitiesfor the identification of elastic events. Figure 6 shows a simplified version of the procedurefor isolating elastic ep events in the GEp-III data usingthe two-body kinematic correlations between the electrondetected in BigCal and the proton detected in the HMS.Similar plots for the GEp-2 γ kinematics can be found inRef. [52]. The proton momentum p p and scattering angle θ p in elastic scattering are related by: p p ( θ p ) = 2 M p E e ( M p + E e ) cos( θ p ) M p + 2 M p E e + E e sin ( θ p ) . (14)The difference δp p ≡ × p p − p p ( θ p ) p , where p is thecentral momentum of the HMS, provides a measure of“inelasticity” for the detected proton independent of anymeasurement of the electron kinematics. The δp p spectra3exhibit significant inelastic backgrounds before applyingcuts based on the measured electron scattering angles,especially at Q = 8 . .The scattered electron’s trajectory is defined by thestraight line from the reconstructed interaction vertex tothe measured electron impact coordinates at the surfaceof BigCal. The correlation between the electron polarscattering angle θ e and the proton momentum p p was ex-pressed in terms of the difference δp e ≡ × p p − p p ( θ e ) p ,where p p ( θ e ) is calculated from elastic kinematics as fol-lows: E (cid:48) e ( θ e ) = E e E e M p (1 − cos θ e ) ,Q ( θ e ) = 2 E e E (cid:48) e ( θ e )(1 − cos θ e ) ,p p ( θ e ) = (cid:112) Q ( θ e ) (1 + τ ( θ e )) , (15)with τ ( θ e ) ≡ Q ( θ e )4 M p . Finally, coplanarity of the outgoingelectron and proton is enforced by applying a cut to δφ ≡ φ e − φ p − π . The azimuthal angles of the detected particlesare defined in a global coordinate system in which thedistribution of φ e ( φ p ) is centered at + π/ − π/ φ e = φ p + π for all elastic ep events within the detector acceptances.The simplified elastic event selection procedure shownin Fig. 6 corresponds to fixed-width, ± σ cuts centeredat zero for all variables. It should be noted, however, thatfor the final analysis, cuts of variable width (mean) wereapplied to δp p ( δφ ) to account for observed variations ofthe width (position) of the elastic peak within the HMSacceptance (for details, see [52]). While the differences instatistics and analysis results between the full procedureand the simple procedure of Fig. 6 are small for suffi-ciently wide cuts, the full procedure optimizes the effec-tive signal-to-background ratio and efficiency of the elas-tic event selection procedure, and suppresses cut-inducedsystematic bias in the reconstructed proton kinematics.In contrast to δp p and δφ , the resolution of δp e is ap-proximately constant within the acceptance, and mostlydominated by the HMS momentum resolution. In gen-eral, the observed correlations of δp e with the recon-structed proton kinematics are small compared to ex-perimental resolution. Moreover, the extracted polariza-tion transfer observables are generally less sensitive tothe systematic error in the reconstructed proton momen-tum than to the errors in the reconstructed proton an-gles, which dominate the experimental resolution of δp p and δφ . The results are thus less susceptible to system-atic bias induced by the δp e cut than that induced bythe δp p and δφ cuts, given the experimentally realizedangular and momentum resolution of the HMS. There-fore, a fixed-width, ± σ cut centered at zero was appliedto δp e for all kinematics, which has the added benefitof simplifying the estimation of the residual backgroundcontamination of the final elastic event sample, as shownin Fig. 7 and discussed below.For electron scattering from hydrogen, elastically scat- tered protons have the highest kinematically allowed mo-menta for positively charged particles at a given θ p .Events at δp p < π photoproduction ( γp → π p )near the Bremsstrahlung end point ( E γ → E e ), with oneor both π decay photons detected by BigCal, and, toa lesser extent, π electroproduction ( ep → e (cid:48) π p ) nearthreshold, with the scattered electron detected in BigCal.At the multi-GeV energies characteristic of these experi-ments, the kinematic separation between the ep and π p reactions in terms of δp p is comparable to the experi-mental resolution, such that there is significant overlapbetween the π p and ep reactions in the vicinity of theelastic peak. The 20-cm liquid hydrogen target is it-self a ∼ .
2% radiator, creating a significant “external”Bremsstrahlung flux along the target length in additionto the real and virtual photon flux present in the electronbeam independent of the target thickness .Events at positive δp p (the so-called “super-elastic” re-gion) originate from quasi-elastic and inelastic scatteringin the aluminum entry and exit windows of the liquidhydrogen target cell, and from non-Gaussian tails of theHMS angular and/or momentum resolution. Because thealuminum window thickness is only ∼
5% of the total tar-get thickness by mass (12% by radiation length), and theexclusivity cut variables are smeared by Fermi motion ofthe nucleons in aluminum, the contribution of scatteringfrom the target end windows to the total event yield isessentially negligible ( (cid:46) − ) after the cuts.The residual peaks at zero in the δp e and δφ spectraof rejected events result from radiative effects and non-Gaussian tails of the experimental resolution. In particu-lar, the remnant peaks in the δφ distributions of rejectedevents contain significant contributions from the elasticradiative tail, because events affected by radiation fromthe incident electron beam (coherent or incoherent withthe hard scattering amplitude) are strongly suppressedby both the δp e and δp p cuts without affecting the co-planarity of the outgoing particles.Figure 7 illustrates the procedure for estimating theresidual background contamination in the final sample ofelastic events. By far the worst case for background con-tamination after applying exclusivity cuts is Q = 8 . , for which the contamination approaches 5% for ± σ cuts. The δp e distribution of the background in thevicinity of the elastic peak after applying cuts to δp p and δφ is well approximated by a Gaussian distribution, aswas confirmed by examining the events rejected by the δp p and/or δφ cuts, as well as by Monte Carlo simula-tions of the main background processes. The shape ofthe elastic ep radiative tail in the δp e distribution wasalso well-reproduced by Monte Carlo simulations with For example, at Q = 8 . , the observed fractional con-tamination by inelastic backgrounds of the final sample of eventsselected as elastic increases by a factor of 1.6 from the upstreamend of the target to its downstream end. (%) e p d - - - E ve n t s = 8.5 GeV Q DataBackground
FIG. 7. Example of the Gaussian sideband fit of the δp e distribution used to estimate the residual background con-tamination of the final elastic event selection cuts at Q = 8 . . Data (black filled circles) are shown after applying ± σ cuts to both δp p and δφ . In this example, the estimated frac-tional background contamination, integrated within the ± σ cut region (black vertical lines), is f ≡ BS + B = (4 . ± . S and B refer to the signal and the background, respec-tively, and the quoted uncertainty is statistical only. See textfor details.TABLE IV. Estimated fractional background contamination f ≡ BS + B (where B and S refer to the background and thesignal, respectively) within the final, ± σ cut region of the δp e distribution, for all the kinematics of the GEp-III andGEp-2 γ experiments. The estimates shown are obtained af-ter applying ± σ cuts to δp p and δφ . The quoted uncertain-ties are statistical only. The quoted beam energy E e is thevalue from Table I, which is averaged over the duration of therunning period, and not corrected for energy loss in the LH target. Q (GeV ) E e (GeV) ( f ± ∆ f stat ) (%)2.5 1.873 0 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . radiative corrections to the unpolarized cross section fol-lowing the formalism described in Ref. [60]. In Fig. 7,the δp e distribution of the background was fitted with aGaussian by excluding the region − . ≤ δp e ≤ . ± σ (%) p p d - - - f ( % ) = 8.5 GeV Q FIG. 8. Fractional background contamination f as a functionof δp p at Q = 8 . , with ± σ cuts applied to δp e and δφ . The horizontal “error bars” represent the rms deviationfrom the mean of the δp p values of all events in each bin. Theuncertainties ∆ f are statistical only and are smaller than thedata points. cuts used for all six kinematics. The inelastic contamina-tion estimates shown in Table IV are determined directlyfrom the data, but are not used directly in the final analy-sis, because the background contamination and the trans-ferred polarization components of the background bothvary strongly as a function of δp p within the final cut re-gion, as the dominant background process evolves from π p photo/electro-production to quasielastic Al( e, e (cid:48) p ).The π p contribution rises rapidly for negative δp p val-ues as the kinematic threshold is crossed, whereas the δp p distribution of the (very small) target endcap contami-nation is relatively uniform within the cut region. Therecoil proton polarization for the inelastic π p reactionon hydrogen generally differs strongly from that of theelastic ep process, while the proton polarization in quasi-elastic Al( e, e (cid:48) p ) is generally similar to elastic ep , since itis basically the same process embedded in a nucleus (seeFigs. 16 and 17). Fig. 8 shows the δp p dependence ofthe fractional background contamination f for Q = 8 . , the setting with (by far) the greatest residualbackground contamination. Details of the backgroundsubtraction procedure are given in section III B and thesystematic uncertainties associated with the backgroundsubtraction are presented in Ref. [52].The stability of the transferred polarization compo-nents with respect to the width of the elastic eventselection cuts and the amount of background includedin the final event sample was checked by varying thewidth of the δp p , δφ , and δp e cuts independently be-tween ± . σ and ± . σ and observing the variations inthe background-corrected results. The observed varia-tions of P t , P (cid:96) , and the ratio P t /P (cid:96) were compatible withpurely statistical fluctuations for all kinematics. There-fore, no additional systematic uncertainty contributionswere assigned. The cut sensitivity study also confirmed5that the application of cuts that were adequately looseand carefully centered with respect to the elastic peakeliminated any cut-induced systematic bias of the re-constructed proton kinematics. This was a non-trivialconcern for this analysis given the exaggerated effect ofmultiple-scattering in “S ” on the event-by-event errorsin the reconstructed proton angles and the very high sen-sitivity of the spin transport calculation to systematic er-rors in these angles, particularly the non-dispersive-planeangle φ tar (see Ref. [52] for a detailed discussion). B. Extraction of Polarization Transfer Observables
1. FPP Angular Distribution
An expression for the general angular distribution inthe polarimeter is given by: N ± ( p, ϑ, ϕ ) = N ± ε ( p, ϑ ) E ( ϑ, ϕ )2 π × (cid:2) ± A y ( P F P Py,tr cos ϕ − P F P Px,tr sin ϕ )+ A y ( P F P Py,ind cos ϕ − P F P Px,ind sin ϕ ) (cid:3) , (16)where N ± is the number of incident protons correspond-ing to a ± ε ( p, ϑ ) is the fractionof protons of momentum p scattered at a polar angle ϑ and producing one single track, E ( ϑ, ϕ ) representsthe angular dependence of the combined effective po-larimeter acceptance/detection efficiency, which factor-izes from the differential nuclear scattering cross sec-tion, A y = A y ( p, ϑ ) represents the analyzing power of (cid:126)p + CH → one charged particle + X scattering, and P F P Px,tr/ind and P F P Py,tr/ind are the transverse componentsof the proton polarization at the focal plane, with P tr ( P ind ) denoting transferred (induced) polarization. Asexplained below, only the ϕ dependence of the detectoracceptance/efficiency is relevant for polarimetry.Note that for all kinematics of the GEp-III and GEp-2 γ experiments, N +0 = N − = N total / (cid:126)ep scattering cross sectionfor an unpolarized target (in the one-photon-exchangeapproximation), and the rapid (30 Hz) helicity reversal,which cancels the effects of slow drifts in experimentalconditions such as luminosity and detection efficiency.As described in [48], the azimuthal scattering angle ϕ was defined in a coordinate system that is comoving withthe incident proton, in which the HMS track defines the z axis, the y axis is chosen to be perpendicular to the HMStrack, but parallel to the yz plane of the fixed TRANS-PORT coordinate system (see Ref. [52]), and the x axis isdefined by ˆ x = ˆ y × ˆ z . In this coordinate system, ϕ is theazimuthal angle of the scattered proton trajectory mea-sured clockwise from the x axis toward the y axis. Notethat this convention for the definition of ϕ differs from theconvention used in the analysis of the GEp-I and GEp-IIexperiments [29, 42]. In the GEp-I and GEp-II analyses, ϕ was defined such that ϕ = 0 for scattering along the+ y axis, and ϕ was measured counterclockwise from the y axis toward the x axis (see Eq. (4) of Ref. [42]). With ϕ defined as in the GEp-III/GEp-2 γ analysis, the sin( ϕ )asymmetry is dominant, whereas the cos( ϕ ) asymmetryis dominant using the GEp-I/GEp-II convention.In the one-photon-exchange approximation in elastic ep scattering, the induced polarization terms are iden-tically zero due to time reversal invariance. When twophotons are exchanged, a non-zero induced polarizationof elastically scattered protons can occur at subleadingorder in α due to the interference between the one-photonand two-photon-exchange amplitudes. Because it is sub-leading order in α , it is not expected to exceed (cid:39) ep scatter-ing plane due to parity invariance of the electromagneticinteraction. The helicity-independent azimuthal asym-metry resulting from a small induced polarization at thislevel is smaller yet as the analyzing power does not ex-ceed roughly 20% at any ( p, ϑ ) in these experiments.The “false” or instrumental asymmetry resulting fromthe effective acceptance/efficiency function E ( ϕ ) can beexpressed in terms of its Fourier expansion: E ( ϕ ) = C (cid:34) ∞ (cid:88) m =1 ( c m cos( mϕ ) + s m sin( mϕ )) (cid:35) ≡ C [1 + µ ( ϕ )] , (17)with an overall multiplicative constant C that is ulti-mately absorbed into the overall normalization of thedistributions when integrating over the dependence onkinematic variables other than ϕ . A clean extractionof the transferred polarization components is obtainedfrom the difference and/or the difference/sum ratio be-tween the angular distributions for positive and negativebeam helicities, integrated over all momenta within theHMS acceptance and a limited ϑ range chosen to excludesmall-angle Coulomb scattering and large-angle scatter-ings for which A y ≈
0. The helicity difference and sumdistributions are given by: f + − f − ≡ π ∆ ϕ (cid:20) N + ( ϕ ) N +0 − N − ( ϕ ) N − (cid:21) = ¯ A y (cid:2) P F P Py,tr cos ϕ − P F P Px,tr sin ϕ (cid:3) × [1 + µ ( ϕ )] ≈ ¯ A y (cid:2) P F P Py,tr cos ϕ − P F P Px,tr sin ϕ (cid:3) (18) f + + f − ≡ π ∆ ϕ (cid:20) N + ( ϕ ) N +0 + N − ( ϕ ) N − (cid:21) = [1 + µ ( ϕ )] × (cid:2) A y ( P F P Py,ind cos ϕ − P F P Px,ind sin ϕ ) (cid:3) ≈ µ ( ϕ ) (19)where ∆ ϕ is the bin width in ϕ and ¯ A y is the averageanalyzing power within the range of ϑ considered . Note also that in the context of Eqs. (18)- (19), N ± is the to- f + − f − f + + f − = ¯ A y (cid:0) P F P Py,tr cos ϕ − P F P Px,tr sin ϕ (cid:1) A y (cid:16) P F P Py,ind cos ϕ − P F P Px,ind sin ϕ (cid:17) ≈ ¯ A y (cid:0) P F P Py,tr cos ϕ − P F P Px,tr sin ϕ (cid:1) (20)2 f ± f + + f − = 1 ± ¯ A y (cid:0) P F P Py,tr cos ϕ − P F P Px,tr sin ϕ (cid:1) A y (cid:16) P F P Py,ind cos ϕ − P F P Px,ind sin ϕ (cid:17) ≈ ± ¯ A y (cid:0) P F P Py,tr cos ϕ − P F P Px,tr sin ϕ (cid:1) (21)where in Eqs. (20)-(21), the induced polarization termsin the denominator are neglected. Equations (18)-(21)show that the false asymmetries and/or the induced po-larization terms are cancelled by the beam helicity rever-sal in the different asymmetry observables. The helicity-difference distribution cancels the induced polarizationterms but is sensitive at second order to the false asym-metry µ , while the difference-sum ratio cancels the falseasymmetry terms, but is sensitive at second order toany induced polarization terms. The helicity-sum dis-tribution cancels the transferred polarization terms, butincludes contributions from false asymmetries and anyinduced polarization terms, if they exist. The trans-ferred polarizations, the induced polarizations, and thefalse asymmetry terms can all be rigorously separated, inprinciple, via Fourier analysis of the distributions (18)-(21), assuming infinite statistical precision. In practice,however, it is very statistically and systematically chal-lenging to separate the induced polarization terms fromthe false asymmetry terms when both are “small”, asis the case in this experiment, especially for the inducedpolarization terms. For the transferred polarization com-ponents, on the other hand, it can be shown [61] thatthe false asymmetry effects are cancelled exactly to allorders by the beam helicity reversal in the linearizedmaximum-likelihood estimators for P t and P (cid:96) defined insection III B 6 below, given sufficient statistical precisionthat the sums over all events entering the maximum-likelihood estimators are a good approximation to thecorresponding weighted integrals over the azimuthal an-gular distribution discussed in [61].
2. FPP event selection criteria
Useful scattering events for polarimetry were selectedaccording to several criteria, detailed in Ref. [52]. First,only single-track events were included in the analysis ofeach polarimeter, as the analyzing power for events withtwo or more reconstructed tracks in either polarimeterwas found to be much lower than that of the single-track tal number of incident protons corresponding to beam helicity ± ϑ range. events, such that even a separate analysis of the multi-track events did not meaningfully improve the polarime-ter figure-of-merit in a weighted average with the single-track events. Secondly, cuts were applied to the parame-ters s close , defined as the distance of closest approach be-tween incident and scattered tracks, and z close , defined asthe z -coordinate of the point of closest approach betweenincident and scattered tracks. A loose, ∼ σ upper limitfor s close was chosen to optimize the statistical precisionof the analysis, by excluding events at large s close valueswith low analyzing power. The z close ranges consideredfor FPP1 and FPP2 events correspond to the physical ex-tent of the CH analyzers ( L CH = 55 cm) plus a smalladditional tolerance (∆ z = ± . z close while excluding the “unphysical” regionclose to (and including) the drift chambers themselves.A “cone test” was applied to each candidate scatter-ing event, to minimize instrumental asymmetries in the ϕ distribution arising from the geometrical acceptance ofthe FPP, and to guarantee full 2 π azimuthal acceptanceover the full range of ( ϑ, z close ) values included in theanalysis. Simply defined, the cone test requires that theprojection of the cone of opening angle ϑ from the recon-structed interaction vertex z close to the rearmost wireplane of the FPP drift chamber pair that detected thetrack lie entirely within the active area of the chamberfor all possible azimuthal scattering angles ϕ . This inturn guarantees that the effective range of ϑ integrationis the same for all ϕ values, such that the average ana-lyzing power is ϕ -independent. As a result, the analyz-ing power, which depends strongly on ϑ , cancels reliablyin the ratio of polarization components P F P Py /P F P Px atthe focal plane and P t /P (cid:96) at the target, regardless of therange of ϑ included in the analysis. Due to the largeactive area of the FPP drift chambers, the efficiency ofthe cone test is close to 100% for scattering angles up toabout 30 degrees. The details of the cone test calculationare given in [48].The useful range of ϑ varies with Q , because the widthof the multiple-Coulomb-scattering peak at small ϑ andthe angular distributions of both the scattering probabil-ity and the analyzing power are observed to scale approx-imately as 1 /p p . The useful range of ϑ was selected foreach Q by applying a cut to the “transverse momentum” p T ≡ p p sin ϑ , where ϑ is the proton’s polar scattering an-gle in the FPP, and p p is the incident proton momentum.The value of p p used in the definition of p T is correctedfor the mean energy loss along the path length in CH traversed by the incident proton prior to the scattering.For all three (cid:15) values at Q = 2 . , the range of p T included in the analysis was 0.06 GeV ≤ p T ≤ ≤ p T ≤ p T cutoff is large compared to theintrinsic angular resolution of the FPP drift chambers,which is about 1.9 (2.1) mrad in the x ( y ) direction. Inthe worst case, at 8.5 GeV , the 0.05 GeV minimum p T - - - - fpp j - - ) - + f + ) / (f - - f + (f FPP1> = 0.153 e <0 1 2 3 4 5 6 (rad) fpp j - - ) - + f + ) / (f - - f + (f > = 0.638 e < fpp j - - ) - + f + ) / (f - - f + (f > = 0.790 e < fpp j - - ) - + f + ) / (f - - f + (f FPP20 1 2 3 4 5 6 (rad) fpp j - - ) - + f + ) / (f - - f + (f fpp j - - FIG. 9. Focal-plane helicity difference/sum ratio asymmetry( f + − f − ) / ( f + + f − ), defined as in Eq. (20), for the GEp-2 γ ( Q = 2 . ) kinematics, for single-track events selectedaccording to the criteria discussed in Sec. III B 2. (cid:104) (cid:15) (cid:105) is theacceptance-averaged value of (cid:15) . The left (right) column showsthe asymmetries for events scattering in the first (second)analyzer. Asymmetries are shown for (cid:104) (cid:15) (cid:105) = 0 .
153 (top), (cid:104) (cid:15) (cid:105) =0 .
638 (middle) and (cid:104) (cid:15) (cid:105) = 0 .
790 (bottom). Red curves are fitsusing ( f + − f − ) / ( f + + f − ) = c cos( ϕ ) − s sin( ϕ ). Asymmetryfit results are shown in Table V. corresponds to a minimum ϑ of about 9 mrad or 4 . σ .More details of the FPP event selection criteria, p T dis-tributions, track multiplicities per event, and closest ap-proach parameters can be found in Ref. [52].
3. Focal plane azimuthal asymmetries
Figure 9 shows the ratio of the helicity-differenceand helicity-sum azimuthal distributions A ≡ ( f + ( ϕ ) − f − ( ϕ )) / ( f + ( ϕ ) + f − ( ϕ )), defined in Eq. (20), for each ofthe GEp-2 γ kinematics, for each polarimeter separately,fitted with a function A = c cos ϕ − s sin ϕ . The fit resultsare shown in Table V. The asymmetries are consistentwith a pure sinusoidal ϕ dependence, and Fourier anal-ysis including a constant term and higher harmonics upto 8 ϕ showed no statistically significant evidence for thepresence of terms other than cos ϕ and sin ϕ , as expected (rad) fpp j ) - + f + ) / (f - - f + (f - = 5.2 GeV Q p < c p ‡ c all (rad) fpp j ) - + f + ) / (f - - f + (f - = 6.8 GeV Q (rad) fpp j ) - + f + ) / (f - - f + (f - = 8.5 GeV Q FIG. 10. Focal plane helicity difference/sum ratio asym-metry ( f + − f − ) / ( f + + f − ), defined as in Eq. (20), for theGEp-III kinematics, for FPP1 and FPP2 data combined, forsingle-track events selected according to the criteria discussedin Sec. III B 2. Asymmetry fit results are shown in Table V.The asymmetry at Q = 5 . is also shown separately forevents with precession angles χ < π and χ ≥ π , illustratingthe expected sign change of the sin( ϕ ) term. from Eq. (20). This suggests that the beam helicity rever-sal does an excellent job of suppressing the instrumentalasymmetries, which are significant at certain values of ϑ and z close . The FPP1 and FPP2 asymmetries are mostlyconsistent with each other, and are always consistent interms of the ratio c/s = P F P Py /P F P Px , or equivalently, interms of the phase of the asymmetry, since the analyzingpower cancels in this ratio. For the GEp-2 γ kinematics,the use of identical event selection criteria for all three (cid:15) values eliminates, in principle, point-to-point systematicvariations of the effective average analyzing power aris-ing from the cuts on the scattering parameters ϑ, s close ,and z close .Figure 10 shows the difference/sum ratio asymmetry( f + − f − ) / ( f + + f − ) for the GEp-III kinematics, for bothpolarimeters combined. For the GEp-III kinematics, thecombined asymmetries are also compatible with a purelysinusoidal ϕ dependence, albeit with much lower statis-tical precision. The asymmetry amplitude at Q = 8 . is larger than for the other two kinematics despite8 fpp j e <0 2 4 6 (rad) fpp j e < fpp j e < fpp j fpp j fpp j + f - f) - +f + /(f +
2f ) - +f + /(f - - + f + (f FIG. 11. Azimuthal angular distributions for FPP1 (leftcolumn) and FPP2 (right column) for the GEp-2 γ kinemat-ics, for events selected according to the criteria discussed inSec. III B 2. The helicity-sum distribution (black filled trian-gles) cancels the asymmetry due to the proton’s transferredpolarization. The raw ϕ distributions for the + (pink emptycircles) and − (green empty squares) helicity states includecontributions from the transferred polarization and the falseasymmetry. The corrected ϕ distributions 2 f + / ( f + + f − ) (redfilled circles) and 2 f − / ( f + + f − ) (blue filled squares) exhibitpure sinusoidal behavior, and include only contributions fromthe transferred polarization terms, assuming the induced po-larization terms are small. the lower analyzing power, because of the precession ofthe proton spin in the HMS. The central precession angleat Q = 8 . is close to 270 degrees, and the asym-metry magnitude is maximal at sin χ = ±
1. In contrast,the central precession angle for Q = 5 . is closeto 180 degrees, such that the acceptance-averaged asym-metry is close to zero. However, as shown in Fig. 12 anddiscussed below, the χ acceptance of the HMS for each Q point is wide enough to provide sufficient sensitivity to P (cid:96) , and the precision of the form factor ratio extractionis not dramatically affected by the unfavorable preces-sion angle, since P (cid:96) is quite large (58%-98%) in all thekinematics of these experiments. Table V summarizesthe focal-plane helicity-difference asymmetry fit results.For each of the Q = 2 . kinematics, the FPP1and FPP2 asymmetries are fitted separately, while the results shown for the GEp-III kinematics are for FPP1and FPP2 combined. For Q = 5 . , the asymme-try results are also fitted separately for precession angles χ < π and χ ≥ π , illustrating the expected sign change of s , the − sin( ϕ ) coefficient of the asymmetry. If Q werechosen such that the HMS acceptance were centered ex-actly at χ = π , and if the effects of quadrupole precessionwere absent, we would expect the values of s for χ < π and χ ≥ π to be equal and opposite. However, the cen-tral value of χ for Q = 5 . is 177.2 ◦ , such thatthe HMS acceptance extends to slightly greater | sin( χ ) | for χ < π than for χ ≥ π (see also Fig. 12). Moreover,as discussed in Ref. [52], the mixing of P t and P (cid:96) due toquadrupole precession shifts the “expected” location ofthe zero crossing of the − sin( ϕ ) coefficient of the asym-metry to about 180.4 degrees instead of the nominal 180degrees. Both of these effects lead to the expectation ofa slightly larger sin( ϕ ) asymmetry for χ < π than for χ ≥ π , as observed.Figure 11 shows the raw ϕ distributions f + , f − , f + + f − and 2 f ± / ( f + + f − ) for the GEp-2 γ kinematics. Similarresults with lower statistical precision are obtained forthe GEp-III kinematics. The normalized distributions2 f ± / ( f + + f − ) are consistent with the pure sinusoidal be-havior predicted by Eq. 21 for all kinematics and for bothpolarimeters separately. The helicity sum distribution f + + f − , which cancels the asymmetry due to the trans-ferred polarization, exhibits a characteristic instrumen-tal asymmetry with several notable features common toall kinematics. The dominant feature of the false asym-metry is a cos(2 ϕ ) term that is roughly independent ofkinematics, negative, and about 2-3% in magnitude whenaveraged over the useful ϑ acceptance at Q = 2 . .This asymmetry appears at small ϑ as a consequence ofthe x/y resolution asymmetry of the FPP drift chambersand at large ϑ due to acceptance/edge effects, and is gen-erally small at intermediate ϑ values near the maximumof the analyzing power distribution (see Sec. III B 7). Al-though the “cone test” (see Section III B 2) is designed toeliminate acceptance-related false asymmetries, it cannotdo so completely because it is applied based on the recon-structed parameters of the incident and scattered tracks,which are affected in a ϕ -dependent way by the FPP x/y resolution asymmetry.The other prominent feature of the false asymmetryis the presence of small peaks at 45-degree intervals cor-responding to the FPP drift chamber wire orientations.The peaks are absent at ϕ = 0 deg., 180 deg., and 360deg., angles corresponding to scattering along the dis-persive ( x ) direction. These artificial peaks are causedby incorrect solutions of the left-right ambiguity due tothe irreducible ambiguity in the drift chambers’ design,resulting from the symmetry of the wire layout and thelack of redundancy of coordinate measurements. Theseincorrect solutions occur primarily for small-angle trackstraversing the chambers at close to normal incidence nearthe center of the drift chambers, where the x , u , and v wires share a common intersection point in the xy plane.9 TABLE V. Focal plane helicity difference/sum ratio asymmetry fit results for GEp-2 γ ( Q = 2 . , top) and GEp-IIIkinematics (bottom). The fit function is f + − f − f + + f − = c cos( ϕ ) − s sin( ϕ ). To first order, c = ¯ A y P FPPy and s = ¯ A y P FPPx . FPP1and FPP2 asymmetries are shown separately for GEp-2 γ , while the combined asymmetries are shown for GEp-III. (cid:104) (cid:15) (cid:105) is theacceptance-averaged value of (cid:15) .Nominal Q (cid:104) (cid:15) (cid:105) c ± ∆ c stat − . ± . − . ± . − . ± . s ± ∆ s stat − . ± . − . ± . − . ± . χ /ndf /
178 188 /
178 137 / c ± ∆ c stat − . ± . − . ± . − . ± . s ± ∆ s stat − . ± . − . ± . − . ± . χ /ndf /
178 145 /
178 167 / Q (GeV ) (cid:104) (cid:15) (cid:105) Combined c ± ∆ c stat Combined s ± ∆ s stat Combined χ /ndf − . ± . − . ± . . / χ < π ) 0.382 − . ± . − . ± . . / χ ≥ π ) 0.382 − . ± . . ± . . / − . ± . . ± . . / − . ± . . ± . . / When an incorrect left-right assignment occurs for eventsin the Coulomb peak of the ϑ distribution, the recon-structed track position at one or both sets of drift cham-bers is incorrectly placed on the opposite side of all threewires that fired in that drift chamber. If the left-right as-signment of the hits in one chamber (but not the other)in a pair is incorrect, the reconstructed point of closestapproach “collapses” to the location of the chamber forwhich the left-right combination was correctly assigned,and the value of ϕ “collapses” to one of the three dif-ferent wire orientations depending on the topology ofthe event and the measured drift distances of the in-correctly assigned hits. The overwhelming majority ofthese mistracked events are rejected by the z close cut,which excludes the unphysical region corresponding tothe drift chambers themselves. However, for z close valueswithin the analyzer region but close to the chambers,some of these mistracked events leak into the “good”event sample due to detector resolution, producing thepattern of small, residual artificial peaks observed in( f + + f − )( ϕ ). These “mistracked” events have low/zeroanalyzing power and tend to dilute the asymmetry inthe z close region closest to the drift chambers. In prin-ciple, they can be further suppressed by excluding thepart of the analyzer region closest to the drift chambers.In practice, this is unnecessary, because the instrumentalasymmetry they generate is cancelled by the beam he-licity reversal, and the resulting dilution of the effectiveaverage analyzing power cancels in the ratio of polar-ization components, such that they cause no systematiceffect whatsoever on the extraction of R . The effect ofthe mistracked events on the average analyzing power,which is important for the extraction of the (cid:15) dependenceof P (cid:96) /P Born(cid:96) , is measured and accounted for, and is thesame for all three (cid:15) values at 2.5 GeV . The sensitivityof the measured P (cid:96) /P Born(cid:96) ratio to the range of z close and p T included in the analysis was examined and found to be small compared to the statistical and systematicuncertainties in this observable.
4. FPP efficiency
Table VI summarizes the total elastic ep statistics col-lected and the effective “efficiency” of the FPP, defined asthe fraction of incident protons producing a useful sec-ondary scattering for polarimetry. The raw FPP wireefficiencies and angular distributions were examined ona run-by-run basis, and runs with data quality issues ineither FPP1 or FPP2 (or both) were rejected for the po-larimeter in question. The total number of elastic eventsshown in Table VI, which serves as the denominator forthe efficiency determination, is not corrected for runs re-jected from the analysis because of FPP data quality is-sues. In other words, FPP-specific data losses due totransient malfunctioning of the data acquisition systemfor either set (or both sets) of FPP drift chambers dur-ing runs of otherwise good data quality are included inthe effective efficiencies shown . These data losses areresponsible for reducing the experimentally realized effi-ciency for FPP1 by several percent for the Q = 2 . data at (cid:104) (cid:15) (cid:105) = 0 . Q = 8 . Since the FPP1 and FPP2 drift chambers were read out by differ-ent VME crates during most of the experiment prior to the switchto Fastbus DAQ during the high- Q running from April-June of2008, a somewhat common occurrence was a data acquisition runwith only one of the two sets of drift chambers providing usabledata. TABLE VI. Experimentally realized effective global FPP efficiencies. “Total elastic events” is the number of events passing theelastic event selection cuts, including the requirement that a definite beam helicity state was recorded for the event. The FPP1(FPP2) efficiency is the fraction of the total number of elastic events passing all the event selection criteria from Section III B 2for FPP1 (FPP2). Note that the efficiencies quoted here do not include single-track events in FPP2 reconstructed as havingscattered in the first analyzer, that failed the event selection criteria for FPP1. These events were included in the GEp-IIIanalysis, but excluded from the GEp-2 γ analysis. Note also that the “efficiencies” are not corrected for data runs that wererejected due to data quality issues in either FPP1, FPP2 or both. See text for details. Q (GeV ) (cid:104) (cid:15) (cid:105) Total elastic events ( × ) FPP1 efficiency (%) FPP2 efficiency (%) Combined efficiency (%)2.5 0.153 99.2 20.5 11.5 32.02.5 0.638 96.8 23.8 10.6 34.42.5 0.790 161.2 26.1 12.8 38.95.2 0.382 9.15 16.8 8.6 25.46.8 0.519 4.96 17.1 8.0 25.18.5 0.243 5.01 15.0 7.0 22.0 GeV to nearly 39% at ( Q , (cid:104) (cid:15) (cid:105) ) = (2 . , . λ T , leads to anefficiency gain of approximately 50% relative to the useof a single polarimeter with one λ T analyzer thickness,regardless of p p .
5. HMS Spin Transport
The asymmetries measured by the FPP are propor-tional to the transverse components of the proton po-larization at the HMS focal plane (see equation (20)),which are related to the reaction-plane transferred po-larization components P t and P (cid:96) by a rotation due to theprecession of the proton polarization in the HMS mag-netic field. The HMS is a focusing spectrometer charac-terized by its 25-degree central vertical bend angle andrelatively small angular acceptance in both the disper-sive and non-dispersive directions. The precession of thepolarization of charged particles with anomalous mag-netic moments moving relativistically in a magnetic fieldis described by the Thomas-BMT equation [62]. Thespin transport for protons (anomalous magnetic moment κ p ≈ .
79) through the HMS is dominated by a rota-tion in the dispersive plane by an angle χ ≡ γκ p θ bend relative to the proton trajectory, where γ ≡ E p /M p isthe usual relativistic γ factor and θ bend is the trajectorybend angle in the dispersive plane. In this so-called “idealdipole” approximation, the dispersive plane componentof the proton polarization precesses by an angle χ , andthe non-dispersive plane component does not rotate, suchthat P F P Px ≈ − sin χP e P (cid:96) and P F P Py = P e P t .Figure 12 illustrates the dominant dipole precession ofthe proton spin using the ratio A FPPx (cid:104) A y (cid:105) P e P (cid:96) , where A F P Px = A y P F P Px is the − sin ϕ coefficient of the asymmetry( f + − f − ) / ( f + + f − ) (see Eq. (20)), (cid:104) A y (cid:105) is the averageanalyzing power within the accepted range of scatteringangles, and P e is the average beam polarization. In theideal dipole approximation, A F P Px = − (cid:104) A y (cid:105) P e P (cid:96) sin χ .The ideal dipole approximation accounts for most of the c - - L P e > P y < A F PP x A = 2.5 GeV Q > = 0.153 e < > = 0.638 e < > = 0.790 e < = 5.2 GeV Q = 6.8 GeV Q = 8.5 GeV Q FIG. 12. Normalized focal-plane asymmetry A FPPx (cid:104) A y (cid:105) P e P (cid:96) , in-tegrated over the accepted ϑ range, vs. precession angle χ ≡ γκ p θ bend , for all six kinematics. The black solid curve is − sin χ , which is the expected value of the normalized asym-metry in the ideal dipole approximation to the spin transportin the HMS. observed χ dependence of the asymmetry. The Q = 5 . data show how it is possible to achieve good pre-cision on the ratio P t /P (cid:96) even when the acceptance-averaged asymmetry is close to zero; due to the largevalue of P (cid:96) and the relatively large χ acceptance of theHMS, the relative statistical uncertainty ∆ P (cid:96) /P (cid:96) is al-most a factor of four smaller than ∆ P t /P t .Deviations from the ideal dipole approximation arisedue to the quadrupole magnets and the finite angular ac-ceptance of the HMS. The forward spin transport matrixof the HMS depends on all the parameters of the protontrajectory at the target and must be calculated for eachevent. Owing to the relatively simple magnetic field lay-out and small angular and momentum acceptance of theHMS, it was not necessary to perform a computationallyexpensive numerical integration of the BMT equation foreach proton trajectory. Instead, a general fifth-order ex-pansion of the forward spin transport matrix in terms ofall the trajectory parameters at the target was fitted to1a sample of random test trajectories populating the fullacceptance of the HMS that were propagated through adetailed COSY [63] model of the HMS including fringefields. Unlike the parameters describing charged parti-cle transport through the HMS, which are independentof the central momentum setting for the standard tune,the spin transport coefficients had to be computed sepa-rately for each central momentum setting, since the spinprecession frequency relative to the proton trajectory isproportional to γ . The use of the same central momen-tum setting for all three kinematics at Q = 2 . ensures that the magnetic field and the spin transportmatrix are the same for all three kinematics. This in turnminimizes the point-to-point systematic uncertainties inthe polarization transfer observables, which is essentialin the accurate determination of their (cid:15) dependence.The fitted expansion coefficients of the COSY spintransport model express the matrix elements of the abso-lute total rotation of the proton spin in the fixed TRANS-PORT coordinate system. The total rotation of the pro-ton spin relevant to the extraction of the polarizationtransfer observables also includes a rotation from the re-action plane coordinate system in which P t and P (cid:96) aredefined to the TRANSPORT coordinate system that isfixed with respect to the HMS optical axis at the tar-get, and a rotation from the fixed TRANSPORT coordi-nate system at the focal plane to the coordinate systemcomoving with the proton trajectory, in which the po-lar and azimuthal scattering angles ϑ and ϕ are defined.Details of the calculation of the total rotation matrix foreach event are given in Ref. [48].
6. Maximum-Likelihood Extraction of P t , P (cid:96) , and R The transferred polarization components P t and P (cid:96) areextracted from the measured FPP angular distributionsusing an unbinned maximum-likelihood estimator. Ne-glecting induced polarization terms, the likelihood func-tion is defined up to an overall normalization constantindependent of P t and P (cid:96) as L ( P t , P (cid:96) ) = N event (cid:89) i =1 E ( ϕ i )2 π × (cid:26) h i P e A ( i ) y (cid:104) (cid:16) S ( i ) yt P t + S ( i ) y(cid:96) P (cid:96) (cid:17) cos ϕ i − (cid:16) S ( i ) xt P t + S ( i ) x(cid:96) P (cid:96) (cid:17) sin ϕ i (cid:105)(cid:27) , (22)where E ( ϕ i ) ∝ (cid:80) n [ c n cos( nϕ i ) + s n sin( nϕ i )] is thefalse/instrumental asymmetry for the i -th event , h i = ± i -th event, P e is In principle, the false asymmetry Fourier coefficients can dependon ϑ , p and any other parameters of the event such as s close , z close the beam polarization, A ( i ) y ≡ A y ( p i , ϑ i ) is the analyzingpower of (cid:126)p +CH scattering, which depends on the pro-ton momentum p i and scattering angle ϑ i , and the S ( i ) jk ’sare the forward spin transport matrix elements relatingpolarization component P k in reaction-plane coordinatesto component P j in the comoving coordinate system ofthe secondary analyzing reaction measured by the FPP.The product over all events in Eq. (22) was convertedto a sum by taking the logarithm, and then the prob-lem of maximizing ln L as a function of the parameters P t and P (cid:96) was linearized by truncating the expansionof ln(1 + x ) at quadratic order in x ; i.e., ln(1 + x ) = x − x + O x . In this context, “ x ” represents the sumof all the ϕ -dependent terms in Eq. (22). The largestacceptance-averaged helicity-dependent “raw” asymme-try observed in either experiment was about 0.12 (seeFig. 9), while the largest raw asymmetry observed atany ϑ was about 0.16. The acceptance-averaged helicity-independent false/instrumental asymmetries are at thefew-percent level . It is therefore estimated that themaximum truncation error in ∆(ln L ) / ln L due to thelinearization procedure is (cid:12)(cid:12)(cid:12) x − x / x ) − (cid:12)(cid:12)(cid:12) (cid:46) .
82% at any ϑ , and smaller when averaged over the full ϑ acceptance.The linearized maximum-likelihood estimators for P t and P (cid:96) are given by the solution of the following linearsystem of equations: (cid:88) i (cid:16) λ ( i ) t (cid:17) λ ( i ) t λ ( i ) (cid:96) λ ( i ) t λ ( i ) (cid:96) (cid:16) λ ( i ) (cid:96) (cid:17) (cid:20) ˆ P t ˆ P (cid:96) (cid:21) = (cid:88) i (cid:34) λ ( i ) t λ ( i ) (cid:96) (cid:35) , (23)where λ ( i ) t and λ ( i ) (cid:96) given by λ ( i ) t ≡ h i P e A ( i ) y (cid:16) S ( i ) yt cos ϕ i − S ( i ) xt sin ϕ i (cid:17) λ ( i ) (cid:96) ≡ h i P e A ( i ) y (cid:16) S ( i ) y(cid:96) cos ϕ i − S ( i ) x(cid:96) sin ϕ i (cid:17) (24)are the coefficients of P t and P (cid:96) in the equation for thelikelihood function (22).Note that the false asymmetry E ( ϕ ) does not enter thedefinition of the estimators. Up to the effects of spin pre-cession, the estimators defined by Eq. (23) are equivalentto the “weighted sum” estimators of Ref. [61]. In Ref. [61]it was shown that these estimators are unbiased and ef-ficient, and in particular that the instrumental asymme-tries are cancelled to all orders by the beam helicity rever-sal, which provides an effective detection efficiency thatis 180-degree symmetric; i.e., E ( ϕ ) = E ( ϕ + π ). The The magnitude of the cos(2 ϕ ) false asymmetry arising from the x/y resolution asymmetry and the acceptance of the FPP driftchambers rises to the ∼
10% level at the extremes of the accepted ϑ range. A P = b , the solution ofwhich is P = A − b . The symmetric 2 × A − ,with A defined by the 2 × P . The stan-dard statistical variances in P t and P (cid:96) are given by thediagonal elements of A − , while the covariance of P t and P (cid:96) is given by the off-diagonal element:∆ P t = (cid:113) ( A − ) tt (25)∆ P (cid:96) = (cid:113) ( A − ) (cid:96)(cid:96) (26)cov( P t , P (cid:96) ) = (cid:0) A − (cid:1) t(cid:96) = (cid:0) A − (cid:1) (cid:96)t (27)The ratio R ≡ − µ p (cid:113) τ (1+ (cid:15) )2 (cid:15) P t P (cid:96) ≡ − K P t P (cid:96) , which equals µ p G pE G pM in the one-photon-exchange approximation, iscomputed from the results of the maximum-likelihoodanalysis for P t and P (cid:96) . The uncertainty in R is computedusing the standard prescription for error propagation us-ing the covariance matrix A − discussed above: (cid:18) ∆ RR (cid:19) = (cid:18) ∆ P t P t (cid:19) + (cid:18) ∆ P (cid:96) P (cid:96) (cid:19) − P t , P (cid:96) ) P t P (cid:96) (28)Although it is not immediately obvious from Eq. (23),both the beam polarization and the analyzing power can-cel in the ratio P t /P (cid:96) . All of the matrix elements on theLHS of (23) are proportional to ( P e A y ) , while the com-ponents of the vector on the RHS of (23) are proportionalto P e A y . The estimators ˆ P t and ˆ P (cid:96) are thus proportionalto ( P e A y ) − , and the statistical variances in P t and P (cid:96) areproportional to ( P e A y ) − . Strictly speaking, the cancel-lation of A y in the ratio P t /P (cid:96) requires that the effectiverange of integration in ϑ (or equivalently p T ) be indepen-dent of ϕ , which is guaranteed in principle by the appli-cation of the cone test. According to the χ of a constantfit, the extracted ratio R showed no statistically signifi-cant p T dependence for any of the six kinematic settings,as detailed in Ref. [52].Figure 13 shows the Q dependence of P t , P (cid:96) and R within the HMS acceptance for the GEp-2 γ kinematics.As discussed below, P t and P (cid:96) are equal to their Bornapproximation values for (cid:104) (cid:15) (cid:105) = 0 .
153 by definition, sincethis point is used for the analyzing power calibration at2.5 GeV . At a fixed central Q of 2.5 GeV , the fixed an-gular acceptance of the HMS corresponds to a Q accep-tance that is roughly three times greater at (cid:104) (cid:15) (cid:105) = 0 . (cid:104) (cid:15) (cid:105) = 0 . Q dependence of R within the acceptance of each kinematic setting is sta-tistically compatible with both the expected R ( Q ) anda constant R value. The observed Q dependences of P t and P (cid:96) are also similar to those of P Bornt and P Born(cid:96) , pro-viding important added confirmation that both the HMSspin transport and the momentum dependence of A y (seeSec. III B 7) are accounted for correctly. The P (cid:96) values at (cid:104) (cid:15) (cid:105) = 0 .
790 show a clear excess over P Born(cid:96) . The curve (GeV Q0.51.0 R > = 0.153 e < > = 0.638 e <> = 0.790 e < ), global fit R(Q (a) (GeV Q0.20 - - - T P (b) (GeV Q0.60.81.0 L P (c) BornT,L
P> = 0.153 e < > = 0.638 e < > = 0.790 e < FIG. 13. Q dependence of R (panel (a)), P t (panel (b))and P (cid:96) (panel (c)) for the GEp-2 γ kinematics. Uncertain-ties shown are statistical only. R is compared to R ( Q ) from“Global Fit II” described in appendix A, which includes theGEp-2 γ results reported in this work. Transferred polariza-tion components are compared to their Born approximationvalues, which are also computed in each bin from “Global FitII”. Note that for (cid:104) (cid:15) (cid:105) = 0 . P t and P (cid:96) are equal to theirBorn approximation values by definition, because this settingis used to determine the analyzing power A y . See text fordetails. for R ( Q ) shown in Fig. 13 is obtained from the globalfit to proton form factor data described in appendix A.The kinematic factors τ , (cid:15) and K are computed fromthe beam energy E e and the proton momentum p p foreach event. The value of K is averaged over all eventsin computing the acceptance-averaged R value from theacceptance-averaged unbinned maximum-likelihood esti-mators for P t and P (cid:96) . Q and (cid:15) are one-to-one correlatedwithin the acceptance at each setting due to the fixedbeam energy. P t and P (cid:96) depend on both Q and (cid:15) , andcan vary significantly within the acceptance of a singlemeasurement. R depends only on Q (in the Born ap-proximation), and its expected variation within the ac-ceptance is generally smaller than that of P t or P (cid:96) . Thecorrelated ( (cid:15), Q ) acceptances of all kinematic settingsare small enough that, to within experimental precision, P t , P (cid:96) , and R vary linearly with Q within the accep-tance, and the acceptance-averaged values of P t , P (cid:96) and3 R equal their values at the acceptance-averaged kinemat-ics.The choice to use the measured quantities E e and p p to compute Q and (cid:15) is not unique; the reaction kine-matics in ep → ep are fixed by choosing any two of E e , E (cid:48) e , θ e , θ p , p p . The choice of any two of these fivevariables gives equivalent results; radiative effects on theaverage kinematics of the final elastic ep sample are sup-pressed to a negligible level by the tight exclusivity cuts.The use of the beam energy E e and the proton momen-tum p p to compute the event kinematics is optimal be-cause the beam energy is known with a high degree ofcertainty, Q depends only on the proton momentum inelastic ep scattering, and the systematic uncertainty in p p is easily quantifiable and independent of (cid:15) for a fixedHMS central momentum setting/nominal Q value.
7. Analyzing Power Calibration
The analyzing power A y is not a priori known. How-ever, the elastic ep process is “self-calibrating” with re-spect to the analyzing power, as it can be extracteddirectly from the measured asymmetries, provided thebeam polarization is known. The Møller measurementof the electron beam polarization is subject to a globaluncertainty of approximately 1% and point-to-point un-certainties of (∆ P e /P e ) ptp = 0 . R = − K P t P (cid:96) does not depend on the beam polarization or the analyz-ing power because both of these quantities cancel in theratio P t /P (cid:96) . Moreover, in the one-photon-exchange ap-proximation, the values of P t and P (cid:96) depend only on theratio r ≡ G E /G M ≡ R/µ p , and not on G E or G M sepa-rately. In terms of r , Eq. (5) becomes (for P e = 1): P Bornt = − (cid:114) (cid:15) (1 − (cid:15) ) τ r (cid:15)τ r P Born(cid:96) = √ − (cid:15) (cid:15)τ r (29)The average analyzing power in a particular bin of ϑ and/or p p is determined by computing the maximum-likelihood estimators ˆ P t and ˆ P (cid:96) assuming A y = 1, with P e taken from the Møller measurements, and forming theratios to P Bornt and P Born(cid:96) :ˆ P ( A y =1) t = ¯ A y P t (30)ˆ P ( Ay =1) (cid:96) = ¯ A y P (cid:96) (31)¯ A y = ˆ P ( A y =1) t P Bornt = ˆ P ( A y =1) (cid:96) P Born(cid:96) (32)The value of A y in any kinematic bin is computed froma weighted average of A y ( ˆ P t ) ≡ ˆ P t /P Bornt and A y ( ˆ P (cid:96) ) ≡ ˆ P (cid:96) /P Born(cid:96) , that is usually dominated by A y ( ˆ P (cid:96) ). Al-though P t and P (cid:96) are determined with comparable ab-solute precision, P (cid:96) is determined with a much better J sin( p p ” T p0.00.10.2 y A = 2.5 GeV Q = 5.2 GeV Q = 6.8 GeV Q = 8.5 GeV Q FIG. 14. Analyzing power A y vs. p T ≡ p p sin ϑ for the fourdifferent Q values from GEp-III/2 γ . Data are from bothpolarimeters combined. Curves are the fits to the data, usedto estimate the position and value of the maximum in A y ( p T ).See text for details. relative precision than P t , because the magnitude of P (cid:96) is several times greater than that of P t for all kinematics.Figure 14 shows the angular dependence of A y , ex-pressed in terms of the “transverse momentum” p T , forall four Q values. The shape of A y ( p T ) is qualitativelysimilar for all four HMS central momentum settings. Themaximum A maxy ( p T = p maxT ) was estimated by fittingeach A y ( p T ) curve with the following simple parametriza-tion: A y ( p T ) = (cid:26) N ( p T − p T ) α e − b ( p T − p T ) β , p T ≥ p T , p T < p T (cid:27) , (33)which is positive-definite, vanishes at asymptoticallylarge p T and p T < p T , and is sufficiently flexible to de-scribe the data with reasonable accuracy. An adequatedescription of the GEp-III data is achieved by fixing theexponents ( α, β ) = (1 ,
2) and the zero-offset p T = 0and varying only the normalization constant N and the“slope” b of the exponential. The data at 2.5 GeV areprecise enough that all five parameters had to be variedto achieve a good description, and in contrast to the GEp-III data, strongly favor a “zero offset” p T ≈ .
05 GeV.Table VII shows the best-fit parameters and their un-certainties, and the resulting values for A maxy and p maxT ,defined, respectively, as the maximum value of the ana-lyzing power and the p T value at which it occurs. The p maxT values exhibit some variation with Q but are sta-tistically compatible with a constant value p maxT ≈ . A maxy and ¯ A y , the average analyzing power within the ac-cepted p T range, compared to selected existing measure-ments in (cid:126)p +CH scattering in the few-GeV momentumrange, including GEp-II [42] in JLab’s Hall A and ded-icated measurements performed at the JINR in Dubna,4 TABLE VII. A y ( p T ) fit results. Fit parametrization is as in Eq. (33). Uncertainties in fit parameters are statistical only.Parameters with no uncertainties are fixed at the quoted values. A maxy is the maximum value of A y occuring at p T = p maxT .Uncertainties in derived quantities A maxy and p maxT are computed from the full covariance matrix of the fit result and thegradient of the fit function with respect to the parameters evaluated at the maximum. ¯ A y values are for 0 . ≤ p T (GeV) ≤ . Q (GeV ) 2.5 ( (cid:104) (cid:15) (cid:105) = 0 . N . ± .
02 0 . ± .
03 0 . ± .
02 0 . ± . α . ± .
02 1 1 1 β . ± .
05 2 2 2 b . ± .
03 2 . ± . . ± . . ± . p T (GeV) 0 . ± .
002 0 0 0 A maxy . ± . . ± .
005 0 . ± .
004 0 . ± . A y . ± . . ± .
003 0 . ± .
003 0 . ± . p maxT (GeV) 0 . ± .
03 0 . ± .
09 0 . ± .
08 0 . ± . -1 (GeV -1p p0.00.10.20.3 y A (This work) maxy A (GEp-II) maxy
A (Dubna05) maxy
A (This work) y A (GEp-II) y A FIG. 15. Maximum ( A maxy ) and average ( ¯ A y ) analyzingpower as a function of (cid:104) p p (cid:105) − , compared to existing data fromRefs. [42] (GEp-II) and [54] (Azhgirey 2005). The average A y values for GEp-III/GEp-2 γ and GEp-II are computed for0 . ≤ p T (GeV) ≤ .
2. Curves are linear fits to the data. Seetext for details.
Russia [54]. It is worth remarking that the GEp-II datawere obtained with two different analyzer thicknesses; thelowest- Q (largest p − p ) measurement used a CH thick-ness of 58 cm, which is similar to the thickness used inHall C for each of the two FPPs, while the three measure-ments at higher Q used a thickness of 100 cm, leading toan apparent reduction in A y that was at least partiallyoffset by an increase in the efficiency. The linear fits tothe GEp-II data shown in Fig. 15 only include the threehighest- Q points, which used the same analyzer thick-ness. It is also worth noting that the Dubna measure-ments [54] do not correspond to constant analyzer thick-ness; the Dubna measurement at p p = 1 .
75 GeV used aCH thickness of 37.5 g/cm , significantly less than thethickness used in either the Hall C or Hall A polarimeters.The Dubna measurements at higher proton momenta cor-respond to a range of analyzer thicknesses generally lyingbetween the ∼
50 g/cm thickness used for each of the twoanalyzers in the Hall C double-FPP and the ∼
90 g/cm thickness of the GEp-II polarimeter. While the Dubna data appear to have a significantly greater slope than theJLab data, the difference is not statistically significant,given the large uncertainty of the Dubna measurementat p p = 1 .
75 GeV, and the fact that this measurementcorresponds to a CH thickness of approximately half thethickness used for the other measurements at higher p p .For the GEp-III/2 γ experiments, A maxy and ¯ A y de-pend linearly on p − p . Notably, the extrapolated values of A maxy and ¯ A y at asymptotically large proton momentum(1 /p p →
0) are non-zero and positive for the conditionsof GEp-III/2 γ , although in the case of ¯ A y , the asymp-totic value is only ∼ σ different from zero. The exper-imentally realized effective analyzing power for the HallC double-FPP is substantially greater than that of theGEp-II or Dubna polarimeters at similar p p . The differ-ence is attributable to the Hall C drift chambers’ ability,given their overall performance characteristics and thetrigger and DAQ conditions specific to Hall C, to sepa-rate true single-track events from multiple-track events,revealing the significantly higher analyzing power for truesingle-track events compared to the totally inclusive sam-ple. In the straw chambers of the Hall A FPP, for ex-ample, groups of eight adjacent wires in a plane weremultiplexed into a single readout channel by the front-end electronics [29], preventing the resolution of multiple-track events in which two or more tracks pass through thesame group of eight straws within a plane simultaneously.The effective analyzing power of a given sample of (cid:126)p +CH scattering events is clearly sensitive to experi-mental details such as the analyzer thickness, the mo-mentum distribution of incident protons, the trackingresolution/efficiency, the background rate/occupancy ofthe detectors, the trigger and data acquisition conditions,and the cuts applied to select events. For this reason, it isgenerally not possible to predict A y using previous mea-surements such as [29, 42, 54] with sufficient accuracy foran absolute determination of P (cid:96) commensurate with thestatistical precision of the GEp-2 γ data.Nonetheless, the relative variation of P (cid:96) /P Born(cid:96) with (cid:15) can be precisely extracted from the GEp-2 γ data by ex-ploiting the fact that the experimental conditions whichinfluence the effective average analyzing power are the5same across all three kinematics measured at Q = 2 . . In particular, the application of identical cutson the FPP scattering parameters ensures that the ef-fective average analyzing power is the same for all three (cid:15) values, up to differences in the momentum distribu-tion of incident protons. As shown in Fig. 15, the av-erage analyzing power for a given p T range is inverselyproportional to the proton momentum p p , while the p T distribution of the analyzing power is approximately in-dependent of p p . Given these experimental realities, themomentum dependence of A y can be accounted for onan event-by-event basis in the maximum-likelihood anal-ysis by assuming that the overall momentum dependencefactorizes from the ϑ and/or p T dependence: A y ( p p , p T ) = A y ( p T ) ¯ p p p p , (34)where A y ( p T ) and ¯ p p are, respectively, the acceptance-averaged values of A y ( p T ) and p p . For the extraction ofthe ratio R , the analyzing power calibration is only rele-vant insofar as it optimizes the statistical figure-of-meritof the maximum-likelihood analysis by properly weight-ing events according to A y . The extraction of the (cid:15) de-pendence of P (cid:96) /P Born(cid:96) in the GEp-2 γ analysis relies onthe assumption that A y is the same for all three mea-surements, up to a global p − p scaling that factorizes fromthe p T dependence according to Eq. (34). The lowest- (cid:15) data ( (cid:104) (cid:15) (cid:105) = 0 . A y ( p T ) for the GEp-2 γ analysis for several reasons. First,the value of P Born(cid:96) approaches one as (cid:15) → r at (cid:104) (cid:15) (cid:105) = 0 . P Born(cid:96) /P Born(cid:96) due to theuncertainty in r is more than three times smaller at thelowest (cid:15) than at either of the two higher (cid:15) values, andnegligibly small compared to the statistical uncertaintyin P (cid:96) itself (see Tab. XI). Moreover, despite the fact thatthe measurement at (cid:104) (cid:15) (cid:105) = 0 .
153 has the worst relativestatistical precision for the ratio P t /P (cid:96) , it has the bestrelative precision for P (cid:96) due to the large magnitude of P (cid:96) .
8. Background Subtraction
The maximum-likelihood estimators are modified bythe residual inelastic contamination of the elastic ep sam-ple as follows: (cid:88) i (cid:16) λ ( i ) t (cid:17) λ ( i ) t λ ( i ) (cid:96) λ ( i ) t λ ( i ) (cid:96) (cid:16) λ ( i ) (cid:96) (cid:17) (cid:20) ˆ P t ˆ P (cid:96) (cid:21) = (cid:88) i (cid:34) λ ( i ) t (1 − λ ( i ) bg ) λ ( i ) (cid:96) (1 − λ ( i ) bg ) (cid:35) . (35) The coefficients λ ( i ) t , λ ( i ) (cid:96) defined by Eq. (24) become: λ ( i ) t → (1 − f ( i ) bg ) λ ( i ) t λ ( i ) (cid:96) → (1 − f ( i ) bg ) λ ( i ) (cid:96) , (36)with f ( i ) bg denoting the fractional background contamina-tion evaluated at the reconstructed kinematics of the i -thevent, estimated according to the procedure discussed insection III A. The background asymmetry term appear-ing on the RHS of Eq. (35) is defined as: λ ( i ) bg ≡ f ( i ) bg h i P e A ( i ) y (cid:104)(cid:16) S ( i ) yt cos ϕ i − S ( i ) xt sin ϕ i (cid:17) P inelt + (cid:16) S ( i ) y(cid:96) cos ϕ i − S ( i ) x(cid:96) sin ϕ i (cid:17) P inel(cid:96) (cid:105) , (37)where P inelt and P inel(cid:96) are the transferred polarizationcomponents of the inelastic background, which are mea-sured using the inelastic events as described below.The residual inelastic contamination of the final selec-tion of elastic ep events is estimated directly from thedata using the procedure described in Sec. III A. Aver-aged over the acceptance of the final cuts, the fractionalcontamination f ranges from 0 .
16% for Q = 2 . , (cid:15) = 0 .
638 to 4 .
89% for Q = 8 . (see Tab. IV). Themeasured polarizations P obst,(cid:96) are related to the signal andbackground polarizations by P obst,(cid:96) = (1 − f ) P elt,(cid:96) + f P inelt,(cid:96) , (38)where P elt,(cid:96) and P inelt,(cid:96) are, respectively, the transferred po-larizations of the elastic “signal” and the inelastic “back-ground”.Figures 16 and 17 show the δp p dependence of P inelt,(cid:96) forthe GEp-2 γ and GEp-III kinematics, respectively. Thebackground polarizations are extracted directly from thedata by applying the maximum-likelihood method de-scribed above to the inelastic events, using the analyz-ing power resulting from the elastic events. Backgroundevents were selected by excluding a two-dimensional re-gion of ( δp e , δφ ) in which the elastic peak and radiativetail contributions are significant. The background po-larizations exhibit a strong δp p dependence in the re-gion of the elastic peak, a behavior explained by thedifferent background processes involved and their rela-tive contributions. In the inelastic region ( δp p < π p events, the background po-larizations are approximately constant and differ stronglyfrom the signal polarizations. The polarization transferobservables for (cid:126)γp → π (cid:126)p , measured rather precisely asa byproduct of this experiment, are interesting in theirown right, and were already the subject of a dedicatedpublication [57], which also addressed the induced polar-ization, which is non-negligible for the (cid:126)γp → π (cid:126)p pro-cess. The induced polarization of the π p background isignored here, as its effect on the extraction of the trans-ferred polarization of the elastic signal is negligible. Inthe region of overlap with the elastic peak, the back-ground polarizations evolve rapidly toward values that6 - - - - - p p d - - i n e l T P (a) - - - - - p p d - - i n e l L P (b) >=0.153 e < >=0.638 e < >=0.790 e < FIG. 16. Transferred polarization components of the inelas-tic background vs. δp p for the GEp-2 γ kinematics. Panel (a)shows the transverse component P inelt , while panel (b) showsthe longitudinal component P inel(cid:96) . Vertical lines illustratethe approximate final cut regions for (cid:15) = 0 .
153 (black dot-ted), (cid:15) = 0 .
638 (red dot-dashed) and (cid:15) = 0 .
790 (blue doubledot-dashed). are similar (but not identical) to the signal polarizations.This transition reflects the sharp kinematic cutoff for π p production and the transition to a regime in which thedominant background process is quasi-elastic Al( e, e (cid:48) p )scattering in the end windows of the cryotarget. The δp p -dependences of the contamination f and the back-ground polarizations P inelt,(cid:96) are accounted for in the fi-nal, background-subtracted maximum-likelihood analy-sis. The total corrections to R , P t and P (cid:96) are dominatedby the lowest δp p bins within the final cut region, andare slightly smaller than would be implied by correct-ing the acceptance-averaged results using the acceptance-averaged values of f and P inelt,(cid:96) using Eq. (38). Table VIIIshows the effect of the background subtraction on P t , P (cid:96) and R . The uncertainties associated with the backgroundsubtraction procedure are discussed in Ref. [52]. In allcases, the correction to P t ( P (cid:96) ) is negative (positive), andthe resulting correction to R is always positive. In gen-eral, the corrections to R and P (cid:96) are very small, exceptin the case of Q = 8 . , for which the size of thecorrection to R is comparable to the total systematic un- - - - - p p d - - i n e l T P (a) - - - - p p d - - i n e l L P (b) = 5.2 GeV Q = 6.8 GeV Q = 8.5 GeV Q FIG. 17. Transferred polarization components of the inelas-tic background vs. δp p for the GEp-III kinematics. Panel (a)shows the transverse component P inelt , while panel (b) showsthe longitudinal component P inel(cid:96) . Vertical lines illustrate theapproximate final cut regions for Q = 5 . (black dot-ted), Q = 6 . (red dot-dashed) and Q = 8 . (blue double dot-dashed).TABLE VIII. Inelastic background corrections to P t , P (cid:96) , and R . Systematic uncertainties associated with the backgroundcorrection are discussed in Ref. [52]. Q (GeV ) (cid:104) (cid:15) (cid:105) ∆ P t ∆ P (cid:96) ∆ R certainty. Despite the similar levels of inelastic contami-nation between (cid:104) (cid:15) (cid:105) = 0 .
638 and (cid:104) (cid:15) (cid:105) = 0 .
790 at 2.5 GeV ,the corrections at (cid:104) (cid:15) (cid:105) = 0 .
790 are significantly smaller,because of the smaller differences between the signal andbackground polarizations.7
TABLE IX. Estimated model-independent relative radiativecorrections to R = µ p G pE /G pM and the longitudinal trans-ferred polarization component P (cid:96) , calculated using the ap-proach described in Ref. [65]. Note that a negative (positive)value for the radiative correction as presented below impliesa positive (negative) correction to obtain the Born value fromthe measured value for the observable in question. These cor-rections have not been applied to the final results shown inTables X and XI. See text for details. Q (GeV ) E e (GeV) u max (GeV ) R obs R Born − P obs(cid:96) P Born(cid:96) − − . × − . × − − . × − . × − − . × − . × − − . × − . × − − . × − . × − − . × − . × − − . × − . × − C. Radiative Corrections
The “standard”, model-independent O ( α ) radiativecorrections (RC) to polarized elastic (cid:126)ep scattering havebeen discussed extensively in Refs. [64–67], and in-clude standard virtual RC such as the vacuum polar-ization and vertex corrections, and emission of real pho-tons (Bremsstrahlung). Radiative corrections to double-polarization observables, such as the beam-target double-spin asymmetry in scattering on a polarized target, orpolarization transfer as in this experiment, tend to besmaller than the RC to the unpolarized cross sections,because polarization asymmetries are ratios of polarizedand unpolarized cross sections, for which the factorized,virtual parts of the RC tend to partially or wholly cancelin the expression for the relative RC to the asymmetry.Moreover, the effect of Bremsstrahlung corrections canbe suppressed by the exclusivity cuts used to select elas-tic events. The ratio of transferred polarization compo-nents P t /P (cid:96) , which is directly proportional to G pE /G pM in the Born approximation, is a ratio of ratios of crosssections, and is subject to RC that are typically as smallas or smaller than the RC to the individual asymmetries,depending on the kinematics and cuts involved.The model-independent RC to the ratio R were esti-mated using the formulas described in Ref. [65]. Theresults for the relative RC to R and P (cid:96) /P Born(cid:96) are shownin Table IX. The corrections are very small in all cases.For the ratio R , the correction is negative for everykinematic. The corrections to P (cid:96) are also negligible inmagnitude, and do not exceed 10 − for any kinematic.The upper limit on the Lorentz-invariant “inelasticity” u ≡ ( k + p − p ) , with k , p , and p denoting the four-momenta of incident electron, target proton, and recoilproton, respectively, was chosen according to the effectiveexperimental resolution of u by plotting the distributionof u for events selected by the exclusivity cuts describedin Sec. III A. It is assumed in the calculations that only the outgoing proton is observed, and the kinematics ofthe unobserved scattered electron and/or the radiatedhard Bremsstrahlung photon are integrated over. In re-ality, the tight exclusivity cuts applied to the kinematicsof both the electron and proton angles and the protonmomentum are such that Bremsstrahlung corrections areeven more strongly suppressed than in the case of a sim-ple cut on u reconstructed from the measured protonkinematics. The “true” model-independent RC to theratio could be expected to be even smaller than thosereported in Tab. IX, which can be regarded as conser-vative upper limits. No radiative corrections have beenapplied to the final results for R and P (cid:96) /P Born(cid:96) reportedin Sec. IV below, as the estimated values of the RC areessentially negligible compared to the statistical and sys-tematic uncertainties of the data. Note also that no hardTPEX corrections are applied to the results, as there ispresently no model-independent theoretical prescriptionfor these corrections. Existing calculations give a widevariety of results, varying both in sign and magnitude,but are in general agreement that these corrections aresmall.
IV. RESULTSA. Summary of the data
The final results of the GEp-III and GEp-2 γ exper-iments are shown in Fig. 18 and reported in Tables Xand XI. The acceptance-averaged values of the relevantobservables can be considered valid at the acceptance-averaged kinematics ( Q and (cid:15) ). The final results ofthe GEp-III experiment for R = µ p G pE /G pM are essen-tially unchanged relative to the original publication [31],showing small, statistically and systematically insignifi-cant increases for all three Q points, despite non-trivialmodifications to event reconstruction and elastic eventselection in the final analysis. The statistical uncertain-ties of the GEp-III data are also slightly modified, asit was discovered during the reanalysis of the data thatthe effect of the covariance term expressing the correla-tion between P t and P (cid:96) was not included in the originallypublished statistical uncertainties, whereas it is includedin this work. The effect of the covariance term on ∆ R stat is only significant for Q = 5 . , for which the cor-relation coefficient is ρ ( P t , P (cid:96) ) ≈ − .
17. Because P t and P (cid:96) are opposite in sign at this Q , a negative correlationcoefficient tends to reduce the magnitude of the statis-tical error (see Eq. (28)). The larger correlation coef-ficient observed at 5.2 GeV compared to all the otherkinematics is related to the unfavorable precession an-gle centered near 180 degrees and the reduced sensitivityof the measured asymmetry to P (cid:96) . The final systematicuncertainties of the GEp-III data are also smaller thanthose originally published, as a result of more thoroughanalysis of the data from the study of the non-dispersiveplane optics of the HMS [52], which reduced the uncer-8 TABLE X. Final results of the GEp-III experiment. These results supersede the originally published results from [31]. Thecentral Q value is defined by the HMS central momentum setting. The average beam energy (cid:104) E beam (cid:105) is the result of correctingthe incident beam energy event-by-event for the mean energy loss in the target materials upstream of the reconstructedinteraction vertex. The kinematics of each setting are described by the average, RMS deviation from the mean and totalaccepted range of Q and (cid:15) . The ratio R = µ p G pE /G pM is quoted with its statistical and total systematic uncertainty. Thepolarization transfer components P t and P (cid:96) are quoted with their statistical uncertainties to illustrate the relative statisticalprecision with which the two components are simultaneously measured a . The quoted values of P t and P (cid:96) are the maximum-likelihood estimators obtained after calibrating the analyzing power at each Q as in Sec. III B 7. The value of P Born(cid:96) is quotedwith its statistical uncertainty, which is due solely to the uncertainty in R . ρ ( P t , P (cid:96) ) is the correlation coefficient between P t and P (cid:96) resulting from the maximum-likelihood analysis. See text for details.Central Q (GeV ) 5.200 6.800 8.537 (cid:104) E beam (cid:105) (GeV) 4.049 5.708 5.710 (cid:10) Q (cid:11) ± ∆ Q rms (GeV ) 5 . ± .
12 6 . ± .
19 8 . ± . Q min , Q max ) (GeV ) (4 . , .
47) (6 . , .
21) (8 . , . (cid:104) (cid:15) (cid:105) ± ∆ (cid:15) rms . ± .
026 0 . ± .
027 0 . ± . (cid:15) min , (cid:15) max ) (0 . , .
44) (0 . , .
59) (0 . , . R ± ∆ R stat ± ∆ R syst (final) 0 . ± . ± .
006 0 . ± . ± .
010 0 . ± . ± . P t ± ∆ stat P t − . ± . − . ± . − . ± . P (cid:96) ± ∆ stat P (cid:96) . ± .
034 0 . ± .
027 0 . ± . P Born(cid:96) ± ∆ stat P Born(cid:96) . ± .
002 0 . ± .
002 0 . ± . ρ ( P t , P (cid:96) ) -0.167 -0.076 0.052 a The difference between the absolute statistical errors ∆ P t and ∆ P (cid:96) is entirely explained by spin precession. TABLE XI. Final results of the GEp-2 γ experiment. These results supersede the originally published results from [47]. Averagekinematics and ranges are as in Tab. X. The central (cid:15) value corresponds to the average beam energy and the central Q of 2.5GeV . The results at (cid:104) (cid:15) (cid:105) = 0 .
790 are obtained by combining the data collected at E e = 3 .
549 GeV and E e = 3 .
680 GeV (seeTab. I) and analyzing them together as a single setting, which is justified by the very similar acceptance-averaged values of Q and (cid:15) at these two energies. The acceptance-averaged values of the ratio R ≡ − µ p P t P (cid:96) (cid:113) τ (1+ (cid:15) )2 (cid:15) and the longitudinal polarizationtransfer component P (cid:96) are quoted with statistical and total systematic uncertainties. R bcc is the “bin-centering-corrected” valueof R at the central Q of 2.5 GeV (see Tab. XII and discussion in Sec. IV B). P t is quoted with its statistical uncertainty only a .The total systematic uncertainty in P (cid:96) is dominated by the beam polarization measurement. The point-to-point systematicuncertainties are defined relative to (cid:15) = 0 . . R ( P (cid:96) /P Born(cid:96) ). ρ ( P t , P (cid:96) ) is the correlation coefficient between P t and P (cid:96) resulting from the maximum-likelihood analysis. See text for details.Central Q (GeV ) 2.500 2.500 2.500Central (cid:15) (cid:104) E beam (cid:105) (GeV) 1.867 2.844 3.632 (cid:10) Q (cid:11) ± ∆ Q rms (GeV ) 2 . ± .
032 2 . ± .
074 2 . ± . Q min , Q max ) (GeV ) (2 . , .
58) (2 . , .
68) (2 . , . (cid:104) (cid:15) (cid:105) ± ∆ (cid:15) rms . ± .
015 0 . ± .
018 0 . ± . (cid:15) min , (cid:15) max ) (0 . , .
19) (0 . , .
67) (0 . , . R ± ∆ R stat ± ∆ R totalsyst (final) 0 . ± . ± . . ± . ± . . ± . ± . R ptpsyst (cf. (cid:104) (cid:15) (cid:105) = 0 . . . . R bcc ± ∆ stat R bcc . ± . . ± . . ± . P t ± ∆ stat P t − . ± . − . ± . − . ± . P (cid:96) ± ∆ stat P (cid:96) ± ∆ totalsyst P (cid:96) . ± . ± . . ± . ± . . ± . ± . P Born(cid:96) ± ∆ stat P Born(cid:96) . ± . . ± . . ± . P (cid:96) P Born(cid:96) ± ∆ stat (cid:16) P (cid:96) P Born(cid:96) (cid:17) ± ∆ totalsyst (cid:16) P (cid:96) P Born(cid:96) (cid:17)
N/A 1 . ± . ± . . ± . ± . ptpsyst (cid:16) P (cid:96) P Born(cid:96) (cid:17) (cf. (cid:104) (cid:15) (cid:105) = 0 . ρ ( P t , P (cid:96) ) 0.019 0.009 0.006 a As in Tab. X, the quoted values of P t and P (cid:96) correspond to the maximum-likelihood estimators obtained using the results of theanalyzing power calibration of Sec. III B 7, performed at (cid:104) (cid:15) (cid:105) = 0 .
153 under the assumption P (cid:96) = P Born(cid:96) and applied to all threekinematic settings. FIG. 18. Final results of GEp-III (black filled triangles)for µ p G pE /G pM , with selected existing data from cross sectionand polarization measurements. The error bars shown are sta-tistical. The band below the data shows the final, one-sidedsystematic uncertainties for GEp-III. The originally publishedresults [31] (black empty triangles) are shown for comparison,offset slightly in Q for clarity. The final weighted-averageresult of GEp-2 γ for R at Q = 2 . is shown as thepink empty star. Existing polarization transfer data are fromRefs. [29] (blue filled circles) and [30, 42] (red filled squares).Rosenbluth separation data are from Refs. [17] (green emptycircles), [16] (green empty diamonds), and [18] (green filleddiamonds). tainty in the total bend angle in the non-dispersive planeto ∆ syst φ bend ≈ .
14 mrad.The values of P Born(cid:96) quoted in Tables X and XI areacceptance-averaged values, computed event-by-eventfrom Eq. (29) using the parametrized global Q depen-dence of R resulting from “Global Fit II” of appendix A,which includes the final results of GEp-III and GEp-2 γ reported in this work. The statistical uncertainty∆ P Born(cid:96) is computed at each kinematic by propagatingthe statistical uncertainty in R through Eq. (29), and isbasically negligible compared to the uncertainty in P (cid:96) it-self. The use of a global parametrization of R ( Q ) to cal-culate P Born(cid:96) is necessary for a self-consistent extractionof the (cid:15) dependence of P (cid:96) /P Born(cid:96) at 2.5 GeV . For theGEp-III kinematics and the lowest (cid:15) measurement fromGEp-2 γ , which is used for the analyzing power calibra-tion, the differences between P Born(cid:96) computed from theglobal parametrization of R ( Q ) and P Born(cid:96) computeddirectly from the measurement result for R are negligi-ble.The results in Tables X and XI are the product of athorough reanalysis of the data, aimed at reducing thesystematic and statistical uncertainties of the final re-sults. The most significant difference between the anal-ysis reported here and that of the original publications is that this work uses the full dataset of the GEp-2 γ experiment to achieve a significant reduction in the sta-tistical uncertainties. The original analysis, publishedin Ref. [47], applied acceptance-matching cuts to thedata at (cid:104) (cid:15) (cid:105) = 0 .
638 and (cid:104) (cid:15) (cid:105) = 0 .
790 to match the en-velope of events at the HMS focal plane populated bythe (cid:104) (cid:15) (cid:105) = 0 .
153 data, and further restricted the protonmomentum to | δ | ≤
2% for all three settings. Thesecuts selected subsamples of the data with essentially thesame average Q , and thus the same average analyzingpower, and suppressed possible (cid:15) -dependent systematiceffects resulting from the different phase space regionspopulated by elastically scattered protons, including themomentum dependence of the analyzing power, “bin cen-tering” effects, and the quality of the reconstruction ofthe proton kinematics and the calculation of the spintransport matrix elements.The acceptance-matching and δ cuts applied in theoriginal analysis [47] reduced the total number of eventsby a factor of approximately 2.5(3.4) at (cid:15) = 0 . . p − p scaling of Eq. (34), and that the HMS optics andspin transport are well-calibrated within the wider phasespace regions populated by the two higher- (cid:15) settings (seeFig. 13 and additional discussion in Ref. [52]). As aresult, the statistical uncertainties in R and P (cid:96) /P Born(cid:96) are significantly reduced relative to Ref. [47], without in-creasing the systematic uncertainty. Other changes in thefinal analysis common to both experiments are mainly re-lated to event reconstruction and elastic event selection.Details of the improvements in event reconstruction andelastic event selection, and the final evaluation of system-atic uncertainties, can be found in Ref. [52].Fig. 19 shows the final results for the (cid:15) -dependence of R and P (cid:96) /P Born(cid:96) . The data collected at E e = 3 .
548 GeV( (cid:104) (cid:15) (cid:105) = 0 . E e = 3 .
680 GeV ( (cid:104) (cid:15) (cid:105) = 0 . Q and (cid:15) rangesjustifies combining these two measurements into the sin-gle result reported in Tab. XI and shown in Fig. 19. Forboth observables, the final results are consistent with theoriginally published results, but with significantly smallerstatistical uncertainties at the two highest (cid:15) values. No-tably, the enhancement of P (cid:96) /P Born(cid:96) at (cid:104) (cid:15) (cid:105) = 0 .
790 rel-ative to (cid:104) (cid:15) (cid:105) = 0 .
153 persists in the full-acceptance anal-ysis and is consistent with the ∼
2% enhancement seenin the original publication. The deviation from unity ofthe final result is 5.3 times the statistical uncertainty, 2.3times the point-to-point systematic uncertainty, and 1.9times the “total” uncertainty defined as the quadraturesum of the statistical and total systematic uncertainties.The ∼ .
6% enhancement at (cid:15) = 0 .
638 is roughly a 2 σ effect statistically, but also consistent with no enhance-ment within the point-to-point systematic uncertainty.0 e R Meziane11This workPunjabi05 (a) e B o r n L / P L P (R) totalsyst D (R) ptpsyst D ) BornL /P L (P ptpsyst D (b) FIG. 19. Final, acceptance-averaged results of the GEp-2 γ experiment, without bin-centering corrections, as a func-tion of (cid:15) , for the ratio R ≡ − µ p P t P (cid:96) (cid:113) τ (1+ (cid:15) )2 (cid:15) (panel (a)),and the ratio P (cid:96) /P Born(cid:96) (panel (b)), compared to the origi-nally published results [47] (Meziane11), and the GEp-I re-sult [29] (Punjabi05) at Q = 2 .
47 GeV . Error bars onthe data points are statistical only. For R , the (one-sided)total and point-to-point (relative to (cid:15) = 0 .
79) systematicuncertainty bands are shown, while only the point-to-point(relative to (cid:104) (cid:15) (cid:105) = 0 . P (cid:96) /P Born(cid:96) (also one-sided). The originally published pointsfrom Ref. [47] have been offset by -0.03 in (cid:15) for clarity. Notethat P (cid:96) /P Born(cid:96) ≡ (cid:104) (cid:15) (cid:105) = 0 . The total and point-to-point systematic uncertainties in P (cid:96) /P Born(cid:96) are dominated by the point-to-point uncer-tainty ∆ P e /P e = ± .
5% in the beam polarization. Itis worth noting that the global ±
1% uncertainty of theMøller measurement of the beam polarization is irrele-vant to the determination of the relative (cid:15) dependence of P (cid:96) /P Born(cid:96) , because a global overestimation (underesti-mation) of the beam polarization is exactly compensatedby an equal and opposite underestimation (overestima-tion) of the analyzing power at (cid:104) (cid:15) (cid:105) = 0 . TABLE XII. Summary of bin-centering corrections to R at Q = 2 . . (cid:10) Q (cid:11) and (cid:104) (cid:15) (cid:105) are the acceptance-averagedkinematics. (cid:15) c is the central (cid:15) value computed from the cen-tral Q value and the average beam energy. R bcc is the bin-centering-corrected value of R with statistical uncertainty. R bcc − R avg is the bin-centering correction relative to the re-sults for the average kinematics reported in Tab. XI. (cid:10) Q (cid:11) (GeV ) (cid:104) (cid:15) (cid:105) (cid:15) c R bcc ± ∆ stat R bcc R bcc − R avg . ± . . ± . . ± . B. “Bin centering” effects in R at Q = 2 . GeV In contrast with the original publication [47], theacceptance-averaged results of the full-acceptance anal-ysis of the GEp-2 γ data are quoted at significantly dif-ferent average Q values (see Tab. XI), such that theexpected variation of R with Q can noticeably affectits apparent (cid:15) -dependence, even in the absence of sig-nificant two-photon-exchange effects in this observable.The expected variation of R with Q within the accep-tance of each point is much larger than its expected (cid:15) de-pendence, which is zero in the Born approximation andsmall in most model calculations of the hard TPEX cor-rections widely thought to be responsible for the crosssection-polarization transfer discrepancy. For example, R ( Q ) from the global fit described in appendix A variesby approximately seven times the statistical uncertaintyof the acceptance-averaged result for R within the Q acceptance of the measurement at (cid:15) = 0 .
79 (see Fig. 13).In order to correct the results for R to a common cen-tral Q of 2.5 GeV , a bin-centering correction to R iscomputed for each kinematic under the assumption that R depends only on Q , or, equivalently, under the weakerassumption that the global Q dependence of R factor-izes from any potential (cid:15) dependence of R , at least withinthe acceptance of each kinematic. The corrected value of R is obtained by multiplying the acceptance-averaged re-sult, which corresponds to the average Q and (cid:15) , by theratio R (2 . ) /R ( (cid:10) Q (cid:11) ), where R ( Q ) is evaluatedusing the results of the global proton form factor fit described in appendix A. The corrected results are thenplotted at the value of (cid:15) corresponding to the central Q ,as opposed to the acceptance-averaged value of (cid:15) . Thebin-centering correction to R is always negative, becausethe slope of R ( Q ) is negative and the average Q is lessthan the “central” Q for all three settings (due to the Q dependence of the acceptance-convoluted cross sec-tion). Tab. XII shows the results for R corrected to the“central” kinematics at Q = 2 . . The magni- The corrections shown in Tab. XII are computed using the resultsof “Global Fit II” of appendix A. The corrections obtained using“Global Fit I” are indistinguishable. TABLE XIII. Linear and constant fit results for the (cid:15) de-pendence of R , with and without bin-centering corrections.Quoted uncertainties in fit results are statistical only.No b.c.c. b.c.c.Slope dR/d(cid:15) − . ± . − . ± . χ /ndf p ”-value 0.18 0.31Linear fit R ( (cid:15) = 0) 0 . ± .
011 0 . ± . R . ± . . ± . χ /ndf p ”-value 0.37 0.36 tude of the correction is small but noticeable comparedto the uncertainties for the two higher (cid:15) points, whilebeing essentially negligible for (cid:15) = 0 . (cid:15) values are small.Tab. XIII shows the results of linear and constant fits tothe (cid:15) dependence of R for both the average and centralkinematics. While the corrected and uncorrected databoth favor a slightly negative slope for R as a functionof (cid:15) , the slope is also compatible with zero in both cases.Indeed, the constant fits actually give higher “ p -values”than the linear fits, although the comparison of these val-ues is not particularly meaningful given the small num-ber of degrees of freedom and the dramatically differentshape of the theoretical χ distributions for ν = 1 and ν = 2.Fig. 20 shows the final, bin-centering-corrected valuesof R as a function of (cid:15) at 2.5 GeV . The linear fit quotedin Tab. XIII is also shown in Fig. 20 with its 68% con-fidence band. The full-acceptance data, which are sig-nificantly more precise at the two highest (cid:15) values thanthe originally published data [47], slightly favor a small,negative slope dR/d(cid:15) = − . ± .
017 (see Tab. XIII),after correcting the data to the common central Q of2.5 GeV . The uncertainty in the slope dR/d(cid:15) is domi-nated by the statistical uncertainties of the data, as thepoint-to-point systematic uncertainties are small. Theobserved slope is consistent with zero, but is more likelyto be negative than positive. No bin-centering correc-tions were necessary for the ratio P (cid:96) /P Born(cid:96) , other thanto quote the results at the central kinematics as opposedto the average kinematics. This is because the observed Q dependence of P (cid:96) closely follows the predicted Q de-pendence of P Born(cid:96) (see Fig. 13), such that the Q depen-dence of the ratio P (cid:96) /P Born(cid:96) is consistent with a constantwithin the acceptance of each kinematic.
V. COMPARISON TO THEORETICALPREDICTIONSA. Theoretical interpretation of G pE /G pM at large Q Among the primary motivations for measuring nucleonelastic electromagnetic form factors to larger Q values is e R = 2.5 GeV Q FIG. 20. Bin-centering-corrected results for the (cid:15) dependenceof the ratio R at the common central Q of 2.5 GeV (redfilled squares), with statistical uncertainties only. The redsolid line is the linear fit to the corrected data reported inTab. XIII. The red shaded region indicates the point-wise,1 σ uncertainty band of the linear fit (68% confidence level).The blue dashed horizontal line is the weighted average of thethree measurements assuming no (cid:15) dependence of R . The bluehatched region indicates the 68% confidence interval (1 σ ) forthe weighted average. The results of the constant fit are alsoquoted in Tab. XIII. The GEp-I result [29] (empty triangle),corrected to 2.5 GeV using the same approach as the GEp-2 γ data, is shown for comparison. to observe the transition from strong coupling and con-finement to the regime of perturbative QCD (pQCD)physics. However, the applicability of pQCD to hardexclusive processes such as elastic electron-nucleon scat-tering may require much larger momentum transfers thanthose currently accessible. One fact that the new protondata have revealed beyond a doubt, is the importance ofquark orbital angular momentum to the understandingof nucleon structure. The role of orbital angular momen-tum is also revealed in a global way, by the very fact thatthe nucleon magnetic moment is strongly anomalous, dif-fering from the Dirac magnetic moment by ∼ ± Q range for lattice calcula-tions of nucleon FFs has been limited to Q (cid:46) by computing power and other technical issues. The ex-pectation, given increases in computational power andtechnical innovations in the methodology of the calcu-lations, is that lattice QCD will be applicable up to 10GeV or higher in the near future. At the present timeonly phenomenological models which include some, butnot all of the fundamental characteristics of QCD are pos-sible. Some of the most successful models include Vector2Meson Dominance (VMD), the relativistic ConstituentQuark Models (RCQM), Generalized Parton Distribu-tions (GPD), Dyson-Schwinger QCD, and others. Wediscuss a selection of these approaches in more detail hereand compare them with the data.
1. Vector Meson Dominance
The earliest models explaining the global features ofthe nucleon form factors, such as their apparent and ap-proximate dipole behavior, were vector meson dominance(VMD) models. In this picture the photon couples to thenucleon through the exchange of vector mesons. A sin-gle vector meson exchange with simple couplings gives an m V / ( m V − q ) factor, from its propagator, for the falloffof the form factor. One can obtain a Q − high momen-tum falloff, in accord with observation or with pQCD,from cancellations among two or more vector meson ex-changes with different masses, or by giving the vectormesons themselves a form factor in their coupling to nu-cleons.An early example of a VMD fit to form factor data wasgiven by Iachello, Jackson, and Lande [68] or IJL. Theyhad several fits, but the one most cited is a 5-parameterfit with a more complicated ρ propagator than the formnoted above, to account for the large decay width of the ρ meson. (The ω and φ are narrow enough that modifyingtheir propagators gives no numerical advantage.)The IJL work was improved by Gari andKr¨umpelmann [69, 70] to better match the powerlaw pQCD expectations at high Q , that F ∼ Q − and F ∼ Q − , but also including some ln( Q ) correctionsto the falloffs based on the running behavior of thecoupling α s ( Q ).Further improvement in VMD fits was made by Lomon[71], who included a second ρ as the ρ (cid:48) (1450), and lateralso a second ω as the ω (cid:48) (1419), and obtained a good pa-rameterization for all the nucleon form factors. The firstof the polarization transfer measurements [28] becameavailable in time for Lomon’s 2001 work [71]. Lomonfurther tuned his fits [71] when the second set of polar-ization transfer data became available [30].In addition, the original IJL fits [68] were not as goodfor the neutron as for the proton. Both the spacelikeneutron form factors and timelike nucleon form factorswere addressed in what may be termed IJL updates, byIachello and Wan [72] and Bijker and Iachello [73], bothin 2004. Further, Lomon and Pacetti [74] have updatedand analytically continued the earlier Lomon fits in orderto also give a good account of data in both timelike andspacelike regions. The VMD models are of course fits toexisting data, and they have been regularly updated asnew data appeared. It will be interesting to check the“predictions” for the neutron form factors as new dataappear. A plot of the existing situation for the proton isgiven in Fig. 21.VMD models are a special case of the more general FIG. 21. Several VMD fits compared to the JLab G pE /G pM data. The solid curve (black) is the fit of Lomon [71],the dashed curve (blue) is that of Iachello, Jackson, andLande [68], and the dotted curve (red) is that of Bijker andIachello [73]. Data are from Refs. [29] (blue circles), [42] (redsquares), and the present work (black triangles for GEp-IIIand pink star for GEp-2 γ ). The GEp-2 γ result shown is theweighted average of the three (cid:15) points without bin-centeringcorrections (see Tab. XIII). Figure adapted from Fig. 23 ofRef. [4]. dispersion relation approach which relates the nucleonform factors in the space-like ( q <
0) region accessible infixed-target electron scattering to the time-like ( q > e + e − → p ¯ p (or p ¯ p → e + e − ). The analytic properties of FFs justifya common interpretation of scattering and annihilationexperiments and the precision reachable at colliders re-quires a unified description of form factors for both space-like and time-like q . Although the separation of G E and G M has been challenging in the time-like region due tothe low luminosities of e + e − colliders relative to fixed-target experiments, some data on the form factor ratioin the time-like region do exist, mainly from the studyof the initial-state radiation (ISR) process e + e − → p ¯ pγ .The most recent and precise data in the time-like regioncome from the BABAR collaboration [75, 76].
2. Constituent Quark Models
The early success of the non-relativistic constituentquark model was in explaining static properties, in-cluding magnetic moments and transition amplitudes.Examples are the models of De R´ujula, Georgi, andGlashow [77] and of Isgur and Karl [78]. However, todescribe the data presented here in terms of constituent3
FIG. 22. The JLab G pE /G pM data compared to the resultsof a selection of constituent quark models. The short dashedcurve (blue) is from Boffi et al. [79], the solid (orange) fromde Melo et al. [80], the long dash (magenta) from Gross etal. [81], the dotted (red) from Chung and Coester [82], andthe dash-dot (cyan) from Cardarelli et al. [83]. Data are thesame as in Fig. 21. Figure adapted from Fig. 24 of Ref. [4]. quarks, it is necessary to include relativistic effects be-cause the momentum transfers involved are much largerthan the constituent quark mass.Constituent quark models (CQMs) have been used tounderstand the structure of nucleons, beginning whenquarks were first hypothesized and predating the emer-gence of QCD as the theory of the strong interaction.In the CQM, ground state nucleons (and other baryonsin the lowest-lying spin-1/2 octet and spin-3/2 decuplet)are composed of three valence quarks, selected from thethree lightest flavors up ( u ), down ( d ) and strange ( s ),and described using SU (6) spin-flavor wave functions anda completely antisymmetric color wave function. Fig-ure 22 compares a selection of CQM calculations to thepolarization transfer data for µ p G pE /G pM from the GEp-I,GEp-II, GEp-III and GEp-2 γ experiments.A crucial question for a form factor calculation, sincethe nucleon must be moving after or before the interac-tion or both, is how the wave function in the rest frametransforms to a moving frame. The relative ease of ex-actly transforming states from the frame where the wavefunctions are calculated or otherwise given, to any otherframe, makes the light-front form attractive for form fac-tor calculations. The light-front form in this contextwas introduced by Berestetsky and Terentev [84, 85], andlater developed by Chung and Coester [82]. The light-front form of the wave function is obtained by a Meloshor Wigner rotation of the Dirac spinors for each quark.Chung and Coester [82] used a Gaussian wave func- tion. They did obtain a falling G pE /G pM ratio. This isapparently a feature shared by many relativistic calcu-lations and is caused by the Melosh transformation [86].Frank, Jennings, and Miller [87, 88] used the light-frontnucleonic wave function of Schlumpf [89, 90] and founda decreasing G pE /G pM ratio, obtaining a zero between Q of 5 and 6 GeV . Cardarelli et al. [83, 91] also usedthe light-front formalism which used quark wave func-tions obtained from a potential of Capstick and Isgur [92].They made the point that the one-gluon exchange is cru-cial to obtaining high momentum components in the wavefunction to explain the form factor data.A comparable amount of high-momentum componentsin the nucleon wave function can be obtained in theGoldstone-boson-exchange (GBE) quark model [93, 94].This model relies on constituent quarks and Goldstonebosons, which arise as effective degrees of freedom oflow-energy QCD from the spontaneous breaking of thechiral symmetry. The GBE CQM was used by Boffi etal. [79] to calculate the nucleon electromagnetic formfactors in the point-form. Relativistic CQM calculationsby Wagenbrunn et al. [95] compared using Goldstone-boson-exchange to one-gluon-exchange in the point-formand found little difference between the calculations forproton form factors.De Sanctis et al. [96, 97] have calculated the ratio G pE /G pM within the hypercentral constituent quark modelincluding relativistic corrections. Parameters of the po-tential are fit to the baryon mass spectrum. With theinclusion of form factors for the constituent quarks, goodfits are obtained for the nucleon form factors [97], for thelatest polarization transfer G pE results [31].Another type of covariant CQM calculation was doneby Gross, Ramalho, and Pe˜na [81], partly based on earlierwork of Gross and Agbakpe [98], avoiding questions ofdynamical forms by staying in momentum space. Theyperformed CQM calculations using a covariant spectatormodel, where the photon interacts with one quark andthe other two quarks are treated as an on-shell diquarkwith a definite mass. They modeled the nucleon as asystem of three valence constituent quarks with their ownparameterized form factors, where the CQ form factorsare obtained with parameters that they fit to the data.Their fit from the 9-parameter “model IV” achieves arather good description of the existing data, includingthe recent higher- Q data for G nE from Ref. [99], whichhad not yet been published at the time.
3. Perturbative QCD
In the context of elastic scattering and other hard ex-clusive processes, perturbative QCD (pQCD) is only ex-pected to be applicable at very large momentum trans-fers [100, 101], perhaps one to several tens of GeV in themost optimistic scenario. In this limit, the virtual photonmakes a hard collision with a single valence quark, whichthen shares the large momentum transfer with the other4 FIG. 23. Selected data for Q F p /F p from cross sectionand polarization observables. Polarization transfer data andsymbols are the same as in Fig. 18. Cross section data arefrom Refs. [16] (filled green circles), [18] (empty green circles),and [17] (empty green triangles). The cross section data showflattening starting at Q ≈ . However, the polarizationtransfer data continue to rise up to Q = 8 . . two, nearly collinear quarks through two hard gluon ex-changes. pQCD predicts that Q F and Q F /F shouldbecome constant for asymptotically large Q , where theextra power of Q for F relative to F is a consequenceof helicity conservation at high energies. The predic-tions were given by Brodsky and Farrar [102, 103] andby Matveev et al. [104]. By a simple rearrangement ofEq. (2), the ratio of Dirac and Pauli FFs is given in termsof the Sachs ratio r = G E /G M by F /F = (1 − r ) / ( τ + r ).Figure 23 shows the JLab polarization data together withselected cross section data for Q F p /F p . The cross sec-tion data (without TPEX corrections) show flattening for Q (cid:38) . However, the GEp-I, GEp-II and GEp-IIIdata do not yet show the pQCD scaling behavior.In 2003 Belitsky et al. [105] investigated the assump-tion of quarks moving collinearly with the proton un-derlying the pQCD prediction. They reiterated the factthat the helicity of a massless (or very light) quark can-not be flipped by the virtual photon of the ep reaction.Instead, the leading contribution to F p at large Q re-quires one unit of orbital angular momentum in eitherthe initial or final-state light-cone nucleon wave func-tion, leading to a modified logarithmic scaling behav-ior Q F /F ∝ ln ( Q / Λ ) at large Q , with Λ a non-perturbative mass scale. With Λ = 0 . F p /F p agree qualita- FIG. 24. Same data as Fig. 23, plotted as (cid:0) Q / ln (cid:0) Q / Λ (cid:1)(cid:1) F p /F p as proposed by Belitsky et al. [105],for Λ = 0 . tively with such double-logarithmic enhancement . Ral-ston [106] and Brodsky et al. [107] have also discussedthe role of quark orbital angular momentum in produc-ing a ratio F p /F p which falls more slowly than 1 /Q .While the “precocious” scaling behavior observed in theproton’s F /F ratio is interesting, it is important to notethat the neutron FF data up to 3.4 GeV [99] do not sup-port the logarithmic pQCD scaling behavior for a cutoffparameter Λ similar to that which describes the protondata. The detailed flavor decomposition of the individualquark contributions to the nucleon form factors [108, 109]suggests that the pQCD-like scaling behavior observedfor the proton’s F /F ratio is probably largely acciden-tal, and a consequence of the delicate interplay betweenthe u and d quark contributions to F and F .In 2006 Braun et al. [110] evaluated leading order con-tributions to the nucleon EMFFs within the light-conesum rule (LCSR) approach, using both asymptotic dis-tribution amplitudes (DAs) and DAs with QCD sumrule-based corrections. The LCSR approach with asymp-totic DAs yields values of G pM and G nM which are closeto the data in the range Q ∼ . The elec-tric form factors were found to be much more difficultto describe, with G nE overestimated, and G pE /G pM nearlyconstant. The ratio G pE /G pM was found to be very sensi-tive to the details of the DAs. A qualitative descriptionof the proton and neutron electric form factors was ob- This observation is not particularly sensitive to the choice of Λwithin a range of values comparable to Λ
QCD and/or Λ ≈ (cid:126) cr p ≈ .
235 GeV et al. [111] to in-clude the next-to-leading-order pQCD corrections to thecontributions of both twist-3 and twist-4 operators anda consistent treatment of nucleon mass corrections. InRef. [111], the DAs were extracted using the form factordata and compared to lattice QCD results, leading to aself-consistent description. The LCSR approach is, how-ever, not yet able to describe all four nucleon EMFFs toa degree of accuracy comparable to that of the data.Kivel and Vanderhaeghen [112, 113] investigated thesoft rescattering contribution to nucleon form factors us-ing soft collinear effective theory (SCET). They havebeen able to show that the soft or Feynman process canbe factorized into three subprocesses with different scales:a hard rescattering , a hard-collinear scattering, and softnonperturbative modes. For the Q range of the presentdata, SCET qualitatively predicts that Q F /F shouldnot be a constant, but exhibit a slow rise, as seen in thedata.
4. Generalized Parton Distributions
The elementary hard scattering process in large- Q electron-nucleon scattering is virtual photoabsorption bya single quark, embedded in the target nucleon as partof a complex, many-body, relativistic system of valencequarks, sea quark-antiquark pairs, and gluons, describedby the Generalized Parton Distributions (GPDs). TheGPDs provide a framework to describe the process ofemission and re-absorption of a quark by a hadron inhard exclusive reactions via the “handbag” mechanism.The GPDs are universal non-perturbative objects aris-ing in the QCD factorization of hard exclusive processessuch as deeply virtual Compton scattering (DVCS) anddeeply virtual meson production (DVMP). The form fac-tors F and F are related to the vector ( H ) and tensor( E ) GPDs by model-independent sum rules [114]: (cid:90) +1 − dx H q ( x, ξ, Q ) = F q ( Q ) , (cid:90) +1 − dx E q ( x, ξ, Q ) = F q ( Q ) , (39)where F q ( F q ) represents the contribution of quark flavor q to the Dirac (Pauli) FF of the nucleon. These relationsallow us, if we have complete measurements or good mod-els for the GPDs, to predict the electromagnetic formfactors [115]. Alternatively, the measured form factorsat high Q , when combined with the forward partondistributions measured in deep-inelastic scattering, pro-vide fairly stringent constraints on the GPDs, particu-larly with respect to their behavior at large x and/or − t values [116, 117]. Early theoretical developments inGPDs indicated that measurements of the separated elas-tic form factors of the nucleon to high Q might also shed light on the nucleon spin decomposition, via Ji’s angularmomentum sum rule [114] for the total (spin and orbital)angular momentum J q carried by the parton flavor q :2 J q = (cid:90) − [ H q ( x, ,
0) + E q ( x, , xdx. (40)The model-independent extraction of GPDs from ob-servables of hard exclusive processes is an area of highcurrent activity and interest. Some recent and less-recent reviews of the subject can be found in Refs. [118–123]. The GPDs can be represented in impact-parameterspace via two-dimensional Fourier transforms of the t -dependence of GPDs at zero skewness [124], allow-ing a three-dimensional “tomography” of the nucleonin two transverse spatial dimensions and one longitu-dinal momentum dimension. By forming the charge-squared-weighted sum over quark flavors and integrat-ing the impact-parameter-space GPDs over longitudinalmomentum fractions x , Miller [125, 126] derived model-independent expressions for the impact-parameter-spacecharge and magnetization densities of the nucleon interms of two-dimensional Fourier-Bessel transforms of F and F : ρ ch ( b ) = (cid:90) ∞ Q π J ( Qb ) F ( Q ) dQ (41)˜ ρ M ( b ) = b π sin φ (cid:90) ∞ Q π J ( Qb ) F ( Q ) dQ, (42)in which b is the magnitude of the transverse displace-ment from the center of the nucleon, and φ is the anglebetween the direction of b and the direction of the trans-verse magnetic field or, equivalently, the transverse nu-cleon polarization. Venkat et al. [127] performed a firstextraction with realistic uncertainty estimation of ρ ch ( b )and ˜ ρ M ( b ) for the proton.
5. Lattice QCD
Lattice gauge theory is presently the only knownmethod for calculating static and dynamic properties ofstrongly interacting systems from first-principles, non-perturbative QCD in the regime of strong coupling andconfinement. Practical computations in lattice gaugetheory involve numerical solutions of QCD on a finite-volume lattice of discrete space-time points. In the re-cent past, these calculations have often been performedfor unphysically large quark masses due to computationallimitations, whereas modern calculations often work at ornear the physical pion mass. Calculations are typicallyperformed for several lattice volumes, spacings and quarkmasses and then extrapolated to the infinite-volume, con-tinuum limit and to the physical pion mass. Early calcu-lations of nucleon electromagnetic form factors in latticeQCD emphasized the isovector ( p − n ) form factors, whichare simpler to calculate since contributions from discon-nected diagrams are suppressed [128]. Until quite re-cently, most calculations of nucleon form factors in lattice6 FIG. 25. Lattice QCD results for µ p G pE /G pM obtained using anovel method based on the Feynman-Hellman theorem [134](pink filled circles), compared to polarization transfer datafrom Refs. [28, 29] (blue empty circles), [30, 42] (red emptysquares), the final GEp-III data (black empty triangles), andthe weighted-average of the final GEp-2 γ data (pink emptystar). The solid curve is the fit to the data using Eqn. 44from Ref. [4], and has not been re-fitted using the final resultsreported in this work. QCD [128–132] have been restricted to relatively low mo-mentum transfers Q (cid:46) , because the rapid falloffwith Q of the form factors leads to very small signal-to-noise ratios in the extraction of hadronic three-pointcorrelators, and related systematic uncertainties due toexcited-state contamination, among other issues. Lin etal. [133] employed a novel technique using anisotropic lat-tices with both quenched and dynamical ensembles with m π ≥
450 MeV to reach Q ≈ .The prospects for lattice QCD form factor calcula-tions to reach high Q have recently been improvedby a novel application of the Feynman-Hellman theo-rem [134], through which hadronic matrix elements canbe related to energy shifts. In the context of nucleonform factor calculations on the lattice, the Feynman-Hellman method allows access to the matrix elementsrelevant to form factor calculations via two-point corre-lators as opposed to more complicated three-point func-tions, and exploits strong correlations in the gauge en-sembles to enhance the signal-to-noise ratios for high-momentum states. Figure 25 shows an initial resultfrom the QCDSF/UKQCD/CSSM collaborations [134]for µ p G pE /G pM reaching Q ≈ . with uncertain-ties approaching the precision of the experimental data. FIG. 26. Comparison of polarization transfer data for µ p G pE /G pM with the DSE based calculation of Ref. [135].
6. Dyson-Schwinger Equations
In recent years, significant progress has also been re-alized in the explanation and prediction of static anddynamic properties of “simple” hadronic systems suchas the pion, the nucleon and the ∆(1232) in contin-uum non-perturbative QCD, within the framework ofQCD’s Dyson-Schwinger Equations (DSEs) [37]. Wherethe calculation of nucleon electromagnetic form factorsis concerned, the DSE approach requires the solutionof a Poincar´e covariant Faddeev equation. One analyti-cally tractable, symmetry-preserving truncation schemethat has achieved considerable success in describing theobserved behavior of the nucleon FFs involves dressedquarks and non-pointlike scalar and axial vector diquarksas the dominant degrees of freedom.In the DSE framework, the nucleon EMFFs at large Q values are sensitive to the momentum dependenceof the running masses and couplings in the strong in-teraction sector of the Standard Model [136]. In a re-cent study, Segovia et al. [135] achieved simultaneouslygood descriptions of the nucleon and ∆(1232) elastic andtransition form factors using identical propagators andinteraction vertices for the relevant dressed quark anddiquark degrees of freedom. One notable prediction is azero crossing in the ratio G pE /G pM at Q = 9 . andin the neutron FF ratio G nE /G nM at Q ≈
12 GeV . Inthis framework, any change in the quark-quark interac-tion that shifts the location of the zero in G pE to larger Q implies a corresponding shift in the location of a zero in G nE to smaller Q . The location of the zero in G pE is par-ticularly sensitive to the rate of transition of the dressedquark mass function between the non-perturbative andperturbative regimes, with a slower fall-off of G pE /G pM corresponding to a faster transition to the perturbative7regime, consistent with the “dimensional scaling” expec-tation discussed in Sec. V A 3. This prediction will beseverely tested by planned near-future precision mea-surements of G pE ( G nE ) to Q ≈
12 (10) GeV at JLab.Fig. 26 shows the calculation of Segovia et al. [135] for µ p G pE /G pM , compared to the polarization transfer datafrom Halls A and C. B. Implications of GEp-2 γ for TPEX Shortly after the publication of GEp-I and GEp-II, twogroups independently suggested that the difference be-tween cross section and double polarization results mightbe attributable to previously neglected hard TPEX pro-cesses; these were Guichon and Vanderhaeghen [44], andBlunden et al. [137]. Notably, some of the earliest po-larization experiments for elastic ep were done to as-sess the contribution of the two photon exchange pro-cess [77, 138–140]. In general, cross section data requirelarge radiative corrections, whereas double-polarizationratios do not [64, 65]. Several calculations and/or ex-tractions of the hard TPEX contribution involving vari-ous models, assumptions and approximations have beenpublished over the last decade. A partial list of theseefforts includes Refs. [45, 141–145]. Many of the calcu-lations partially resolve the discrepancy, but a model-independent theoretical prescription for TPEX correc-tions constrained directly by data remains elusive. Re-cent reviews of the subject can be found in Refs. [43, 146].In addition to the significant theoretical work to re-solve the discrepancy, major experimental efforts werecarried out over the last decade to search for experi-mental signatures of significant TPEX contributions toelastic eN scattering. These signatures include possi-ble non-linearities of the Rosenbluth plot [18, 150], anon-zero target-normal single-spin asymmetry [151] orinduced normal recoil polarization, and deviations fromthe Born approximation in polarization transfer observ-ables, as in this work and Ref. [47]. The beam-normalsingle spin asymmetries in elastic eN scattering have alsobeen precisely measured as byproducts of a large num-ber of parity violation experiments [152–158], albeit ina Q range well below the region of the discrepancy.These beam-spin asymmetries are typically at the few-ppm level, and are also sensitive to the imaginary partof the TPEX amplitudes. The most direct observable toaccess the real part of the TPEX amplitude is a deviationof the e + p/e − p cross section ratio from unity [141], as thereal part of the interference term between the Born andTPEX diagrams changes sign with the charge of the lep-ton beam. Three major experiments with very differentand complementary technical approaches have recentlymeasured the e + p/e − p cross section ratio [159–162].Figure 27 shows the final, bin-centering-corrected re-sults of GEp-2 γ for the (cid:15) dependence of R , compared toseveral theoretical predictions for the hard TPEX cor-rections to this observable. Blunden et al. [148] recently FIG. 27. Final, bin-centering-corrected results of GEp-2 γ for the ratio R , compared to several theoretical predictionsfor the (cid:15) dependence of R at Q = 2 . due to TPEXcorrections. The blue solid horizontal line is the weightedaverage of the corrected data (see Tab. XIII and Fig. 20).Curves are: Borisyuk et al. [147] (cyan dashed), Blunden etal. [148] (green dot-dashed ( N only) and green dotted ( N +∆)), Bystritskiy et al. [143] (pink dot-long dashed), Afanasev et al. [45] (black solid), and Kivel et al. [149] (red dotted(BLW) and red dashed (COZ)). Note that because the ratio R is proportional to the Born value of µ p G pE /G pM , each curvecan be renormalized, in principle, by an overall multiplicativefactor. See text for details. evaluated the hard TPEX corrections to elastic ep scat-tering within a dispersive approach, which avoids off-shell uncertainties inherent in the direct evaluation ofloop diagrams [163]. The box and crossed diagrams forTPEX corrections involving nucleon and ∆ intermediatehadronic states were evaluated both algebraically and nu-merically within the dispersive approach using empiricalparametrizations of the nucleon elastic and N → ∆ tran-sition form factors. The result including the contribu-tions of both N and ∆ intermediate states is consistentin slope with the final GEp-2 γ data, and also achieves areasonable description of the e + p/e − p cross section ra-tios which, however, are only measured for Q (cid:46) . . At Q = 2 . , it appears that a descriptionin terms of hadronic degrees of freedom with only the nu-cleon elastic and ∆ intermediate states is adequate. Athigher Q values where the discrepancy between crosssection and polarization data is more severe, the effectsof higher-mass resonances, inelastic nonresonant inter-mediate states including the πN continuum, and the fi-nite widths of resonances are expected to increase in im-portance. Borisyuk et al. [147] also used the dispersiveapproach to compute the contribution of the P par-tial wave of the πN channel to the TPEX amplitude,8which effectively includes the ∆ contribution with real-istic shape, width, and nonresonant background “auto-matically”. The prediction of Ref. [147] for the ratio R exhibits similar behavior to the calculation of Blunden etal. , which is not surprising, given its similar physics con-tent. Bystritskiy et al. [143] used the electron structurefunction method to compute the higher-order radiativecorrections to all orders in perturbative QED in the lead-ing logarithm approximation. Their method predicts nonoticeable (cid:15) dependence at the level of precision of theGEp-2 γ data, consistent with our results.Afanasev et al. [45] approached the TPEX correc-tions to elastic ep scattering in a parton-model approachassuming dominance of the “handbag” mechanism, inwhich both hard virtual photons are exchanged with asingle quark, embedded in the nucleon via GPDs. Thisapproach is expected to be valid for simultaneously largevalues of s , − u , and Q . The parton-model evaluationof TPEX corrections predicts a strong, non-linear (cid:15) de-pendence for R that is not observed the in GEp-2 γ data.Kivel et al. [149] computed the hard TPEX correctionto elastic ep in a perturbative QCD approach in whichthe leading contribution for asymptotically large Q in-volves two hard photon exchanges occuring on differ-ent valence quarks, and a single hard gluon exchangeoccurring on the third valence quark. In the pQCDapproach, the TPEX amplitude can be expressed in amodel-independent way in terms of leading-twist nucleondistribution amplitudes (DAs). In Fig. 27, the calcula-tion of Ref. [149] is shown for two different models forthe DAs: that of Braun et al. (BLW [110]), and thatof Chernyak et al. (COZ [164]). The GPD and pQCDmodels for the hard TPEX correction predict a significantpositive slope dR/d(cid:15) which is disfavored by the data. Itmust be noted, however, that the GEp-2 γ measurementat (cid:104) (cid:15) (cid:105) = 0 .
153 in particular lies outside the expected kine-matic range of applicability of a partonic description.The deviation from unity of P (cid:96) /P Born(cid:96) at large (cid:15) , giventhe absence of significant (cid:15) dependence of the ratio R , im-plies a similar deviation from the Born approximation in P t that cancels in the ratio P t /P (cid:96) . This deviation wasnot predicted by any of the TPEX calculations avail-able at the time of the original publication [47], whichgenerally expected small TPEX corrections to this ob-servable. A deviation from unity in P (cid:96) /P Born(cid:96) was sub-sequently predicted within the SCET approach by Kivel et al. [46]. Guttmann et al. [165] performed an extrac-tion of the TPEX amplitudes from a global analysis ofelastic ep scattering data including the original GEp-2 γ results [47] and the Hall A “Super-Rosenbluth” data atthe similar Q of 2.64 GeV [18], using the formalism ofEqs. (8)-(12). Under the assumptions used in their anal-ysis, the observed deviation from unity of P (cid:96) /P Born(cid:96) andthe constant value of R imply that the TPEX amplitudes Y E ≡ (cid:60) (cid:16) δ ˜ G E /G M (cid:17) and Y ≡ (cid:0) ν/M (cid:1) (cid:60) (cid:16) ˜ F /G M (cid:17) (seeEqs. (7)-(13)), which are mainly driven by the originalGEp-2 γ data, are roughly equal in magnitude and oppo-site in sign, and approach the 2-3% level at (cid:15) ≈ . Q = 2 . . VI. CONCLUSIONS
This article has described two proton form factor ex-periments, GEp-III and GEp-2 γ , which utilized the re-coil polarization method in Hall C at Jefferson Lab tomeasure the ratio of the proton’s electric and magneticform factors, R ≡ µ p G pE /G pM . The results of these ex-periments were previously published in two separate ar-ticles [31, 47]. The purpose of this article was to providean expanded description of the apparatus and analysismethod common to both experiments and report theresults of a full reanalysis of the data with significantimprovements in detector calibration, event reconstruc-tion, elastic event selection, and the evaluation of sys-tematic uncertainties. The final results of GEp-III areessentially unchanged relative to the originally publishedresults [31]. The new analysis has resulted in a signif-icant reduction in the systematic uncertainty, due to amore thorough evaluation of the systematic uncertaintyin the total bend angle of the proton trajectory in thenon-dispersive plane of the HMS. The high- Q pointsconfirmed the results of the GEp-I and GEp-II experi-ments from Hall A, namely that R continues to decreasetoward zero, but with clear indication that the rate of thisdecrease is slowing down. The impressive agreement ofthe measurements of R in GEp-III and GEp-2 γ with theprevious Hall A measurements at the same or similar Q (but not necessarily the same (cid:15) ) demonstrates that thesystematic uncertainties of the recoil polarization methodare well understood, and that deviations from the Bornapproximation in the extraction of G pE /G pM from polar-ization transfer observables are not large within the Q range presently accessible to experiment.The GEp-2 γ data, originally published in Ref. [47],consist of measurements for three different (cid:15) values ata fixed Q of 2.5 GeV , obtained by changing the elec-tron beam energy and the detector angles. The relative (cid:15) dependence of the ratio P (cid:96) /P Born(cid:96) was also extractedfrom the GEp-2 γ data with small uncertainties by ex-ploiting the fact that the polarimeter analyzing power,the proton momentum, and the HMS magnetic field werethe same for all three (cid:15) values. The lowest (cid:15) point wasused to calibrate the polarimeter analyzing power, giventhe large value of P (cid:96) and its negligible sensitivity to R at this (cid:15) . The results of the reanalysis of the GEp-2 γ data reported in this work include the previouslyunpublished full-acceptance data for the two highest (cid:15) points, increasing the statistics by a factor of 2.5 (3.4) at (cid:104) (cid:15) (cid:105) = 0 . . γ experiment serves as a precise test ofthe validity of the polarization transfer method. Indeed,as expected from the Born approximation, the GEp-2 γ data demonstrate that R is compatible with a constantfor a wide range of (cid:15) between 0.15 and 0.79. The onlydeviation from the Born approximation is observed in9the longitudinal polarization at (cid:15) = 0 . P (cid:96) /P Born(cid:96) =1 . ± . ± . P t , such thatthe form factor ratio remains constant. In addition, thestatistically improved, simultaneous measurements of theindependent observables P (cid:96) /P Born(cid:96) and R at the samekinematics provide important tools for testing TPEXmodels and constraining the extraction of TPEX formfactors.The accelerator at Jefferson Lab has recently been up-graded to a maximum beam energy of 12 GeV. There areapproved experiments at Jefferson Lab that will extendthe knowledge of G pE /G pM to Q = 12 GeV , G nE /G nM to Q = 10 GeV , and G nM to 14 GeV . Dedicated mea-surements of the elastic ep unpolarized differential crosssection over a wide Q range from 2-16 GeV with (cid:46) G pM within the entire Q range accessible withJLab’s upgraded electron beam. The program of high- Q form factor measurements using the upgraded JLabelectron beam will enable the detailed flavor decomposi-tion of the nucleon EMFFs to Q = 10 GeV , providingsignificant constraints on the predictions of theoreticalmodels, and insight into the important degrees of free-dom in understanding nucleon structure across a broadrange of Q . VII. ACKNOWLEDGMENTS
The collaboration thanks the Hall C technical staffand the Jefferson Lab Accelerator Division for their out-standing support during the experiment. This materialis based upon work supported by the U.S. Department ofEnergy, Office of Science, Office of Nuclear Physics, un-der Award Number DE-SC-0014230 and contract Num-ber(s) DE-AC02-06CH11357 and DE-AC05-06OR23177,the U.S. National Science Foundation, the Italian Insti-tute for Nuclear Research, the French Commissariat `al’Energie Atomique and Centre National de la RechercheScientifique (CNRS), and the Natural Sciences and En-gineering Research Council of Canada.
Appendix A: Global Proton Form Factor Fit(s)Using Kelly Parametrization
Several global fits of the proton form factors to mea-surements of differential cross sections and polarizationobservables in elastic ep scattering were performed forthis analysis using a procedure similar to that describedin Ref. [170]. The results were used for the GEp-2 γ analy-sis to estimate the bin centering effects in the ratio R andto calculate the event-by-event and acceptance-averagedvalues of P Born(cid:96) in the maximum-likelihood analysis. Asin Ref. [170], the “first-order” Kelly [21] parametriza- ) (GeV Q0 2 4 6 8 p M / G p E G p m - Punjabi05Puckett12This workChristy04Andivahis94Global fit I Global fit IICrawford07 Ron11Zhan11 Paolone11Qattan05 ) = 2.5 GeV This work (Q
FIG. 28. Global fit results for the proton form factorratio µ p G pE /G pM , compared to selected data from measure-ments of cross sections and polarization observables, includ-ing the final results of GEp-III (black solid triangles) andGEp-2 γ (pink empty star, weighted average). Other polariza-tion data are from Refs. [28, 29] (Punjabi05), [30, 42] (Puck-ett12), [166] (Crawford07), [167] (Ron11), [168] (Zhan11),and [169] (Paolone11). Rosenbluth separation data arefrom [16] (Andivahis94), [17] (Christy04), and [18] (Qat-tan05). Global fit I includes the data from Refs. [167–169],while excluding the data from Ref. [166] and the two low-est Q points from Ref. [29]. Global fit II excludes the datafrom Refs. [167–169]. Shaded regions indicate 1 σ , pointwiseuncertainty bands. tion was used in which G pE and G pM /µ p are describedas ratios of a polynomial of degree n and a polynomialof degree n + 2 in τ = Q / M p (with n = 1). TheKelly parametrization enforces G pE (0) = G pM (0) /µ p = 1and also enforces the “dimensional scaling” behavior atasymptotically large Q predicted by perturbative QCD: Q F ∝ Q F ∝ constant.Compared to Ref. [170], the fits presented here differin a few key respects. The data selection for differentialcross section measurements is largely the same as before,and includes representative results from twelve differ-ent experiments spanning approximately 0 .
005 GeV ≤ Q ≤
31 GeV (Refs. [8, 9, 11–20, 171]). However, thedatabase of polarization observables is modified substan-tially. First, the final results of GEp-III and GEp-2 γ re-ported in this work are now included in the fit, whereasin the original fit, the GEp-III results from Ref. [31] wereused and the GEp-2 γ results were not included at all, asthey were not yet published at the time. The three high-est Q points from the original GEp-II data [30] havebeen replaced by the results of the reanalysis of thesedata published in Ref. [42]. The data from Ref. [172]have also been replaced by the reanalysis results pub-lished in Ref. [167]. Finally, the high-precision data from0Refs. [168, 169] have been added. Given the apparentinconsistency of the various polarization experiments atlow Q , an inconsistency which is not yet explained, twodifferent fits were performed. In the first fit, hereafterreferred to as “Global fit I”, the recent precise data fromRefs. [167–169] were included, while the polarized targetasymmetry data from Ref. [166] and the two lowest Q points from GEp-I [29] were excluded from the fit. In thesecond fit, referred to as “Global fit II”, the data fromRefs. [167–169] were excluded, while all other R p datafrom polarization observables were included.The prescription for treating the cross section data,particularly in the high- Q region where the inconsis-tency with the polarization transfer data exists, is alsoslightly modified here compared to Ref. [170]. As be-fore, three iterations of the fit are performed, using theresulting parameters and their uncertainties and correla-tions from the previous fit as the starting point for thesubsequent fits. In Ref. [170], the value of G pE ( Q ) wasfixed for Q ≥ using the result of the previ-ous fit, or, on the first iteration, Kelly’s 2004 result [21],when computing the χ contribution of individual crosssection data, effectively forcing G pE to be entirely deter-mined by polarization data for Q ≥ . In the fitsreported here, G E ( G M ) was fixed in the same way whenthe fractional contribution of the (cid:15)G E ( τ G M ) term inthe reduced cross section was less than 10%, regardlessof Q . This prescription removes the influence of indi-vidual cross section measurements on the determinationof G E ( G M ) at high (low) Q when said measurementshave very low sensitivity to the respective form factors.In particular, a cutoff of 10% of the reduced cross sec-tion excludes all cross section data for Q (cid:38) . from participating in the determination of G E , and somelower- Q data, depending on (cid:15) . The other significantdifference between the fits reported here and those ofRef. [170] is that in Ref. [170], the overall normalizationuncertainties in the absolute cross section data were es-sentially ignored in the χ calculation, whereas in the fitspresented here, the overall normalization of each of thetwelve experiments included in the global fit was allowedto float within a range of ± . R p from Refs. [167–169] in the 0.1-1 GeV region on theone hand, and the discrepancy between cross section andpolarization data at large Q on the other. Allowing thecross section normalizations to float leads to a reductionof the χ per degree-of-freedom from 1.78 in Ref. [170] toapproximately 1.54 in the fits reported here.Table XIV summarizes the global fit results. The best-fit values of the parameters describing G pE and G pM andtheir (1 σ ) uncertainties are presented together with the TABLE XIV. Summary of global proton FF fit results. Formfactor parametrization is G ( Q ) = a τ b τ + b τ + b τ , where G ( Q ) = G E ( Q ) or G M ( Q ) /µ p . The uncertainty bandsshown in Fig. 28 represent the pointwise, 1 σ errors computedfrom the full covariance matrix of the fit result. The asymp-totic values of the form factors shown below are normalizedto a dipole form G D = (cid:0) Q / Λ (cid:1) − with scale parameterΛ = 0 .
66 GeV corresponding to an RMS radius r p = 0 . χ and degrees of freedom are shown along withthe breakdown of χ contributions among cross section ( σ R )and polarization ( R polp ) data. The χ contributions of crosssection measurements are also separated into “low” ( Q ≤ ) and “high” ( Q > ) data. The best-fit normal-ization constants of the cross section experiments are omittedfor brevity. Fit Global fit I Global fit II a E − . ± . − . ± . b E . ± .
18 12 . ± . b E . ± . . ± . b E ± ± a M . ± .
022 0 . ± . b M . ± .
073 11 . ± . b M . ± . . ± . b M . ± . . ± . Q →∞ G pE G D ( r p =0 .
84 fm) − . ± . − . ± . Q →∞ G pM µ p G D ( r p =0 .
84 fm) . ± .
09 0 . ± . χ /ndf (all data) 706/460 696/455 χ /n data ( σ R ) 672/427 653/427 χ /n data ( R polp ) 34/53 44/48 χ /n data ( σ R , Q ≤ ) 337.7/275 308.4/275 χ /n data ( σ R , Q > ) 334.5/152 344.1/152 implied asymptotic values of G pE and G pM , normalized toa dipole form factor with a scale parameter Λ = 0 . , corresponding to an RMS radius of 0.84 fm, con-sistent with the proton charge radius extracted from mea-surements of the Lamb shift in muonic hydrogen [173].As pointed out in Ref. [174], a dipole form factor with r p = 0 .
84 fm describes the low- Q G pE data better thanthe “standard” dipole form factor with Λ = 0 .
71 GeV (corresponding to r p = 0 .
81 fm). As measured by χ , theoverall quality of both fits is relatively good, except forthe cross section data in the high Q region, for whichthe χ per datum exceeds two. No attempt was made tocorrect the high- Q cross section data for the effects oftwo-photon-exchange thought to be responsible for thediscrepancy, as these effects are presently only poorlyconstrained experimentally and incompletely understoodtheoretically [175]. Instead, the “excess” (cid:15) -dependence ofthe reduced cross sections observed in the high- Q data(i.e., the “excess” slope in the Rosenbluth plot relative tothe expectation from polarization transfer data) is simplyaveraged over in determining G M , with the ratio G E /G M fixed by the polarization data. While this procedure maybias the determination of G M in principle, the potentialsize of the effect on G M in the high- Q region is mitigated1by the smallness of the fractional contribution of G E tothe reduced cross section. The inconsistency among po-larization experiments in the low- Q region is anotherissue that awaits resolution. While the fits reported hereprovide an adequate representation of the proton FFs inthe Q region in which they are directly constrained bydata, the values and uncertainties in the extrapolation ofthese fits to larger Q should not be taken too seriously.The high-precision polarization data for R in both the 0.1-1 GeV region [29, 166–169] and at 2.5 GeV as re-ported in this work, combine to exert significant influenceon the extrapolation of G E and G M to Q values beyondthe reach of existing data, as is evident from the notice-ably different asymptotic behaviors of the two fits, whichdiffer only in the choice of low- Q polarization data. Thisis a consequence of fitting a smooth, relatively inflexibleparametrization of the form factors, with no specific the-oretical justification other than its asymptotic behavior,to high-precision data at significantly different Q values. [1] R. Hofstadter, Rev. Mod. Phys. , 214 (1956).[2] I. Estermann, R. Frisch, and O. Stern, Nature , 169(1933).[3] C. Patrignani et al. (Particle Data Group), Chin. Phys. C40 , 100001 (2016).[4] V. Punjabi, C. F. Perdrisat, M. K. Jones, E. J. Brash,and C. E. Carlson, Eur. Phys. J.
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