Two-Photon Exchange in Electron-Proton Elastic Scattering: Theory Update
aa r X i v : . [ h e p - ph ] N ov October 29, 2018 17:18 WSPC - Proceedings Trim Size: 9in x 6in afanasev˙2photon TWO-PHOTON EXCHANGE IN ELECTRON-PROTONELASTIC SCATTERING: THEORY UPDATE
Andrei V. AFANASEV
Department of Physics, Hampton University, Hampton, VA 23668, USAandTheory Center, Jefferson Lab, Newport News, VA 23606, USA
Recent theoretical developments in the studies of two-photon exchange effectsin elastic electron-proton scattering are reviewed. Two-photon exchange mech-anism is considered a likely source of discrepancy between polarized and un-polarized experimental measurements of the proton electric form factor at mo-mentum transfers of several GeV . This mechanism predicts measurable effectsthat are currently studied experimentally. Keywords : Electron scattering; nucleon form factors; two–photon exchange;QED radiative corrections
Measurements of elastic nucleon form factors reached a new level ofaccuracy, with separation of electric and magnetic contributions made pos-sible at high transferred momenta. At Jefferson Lab, due to a 100% dutyfactor of the electron beam and implementation of nucleon spin polarizationtechniques, electric nucleon form factors were measured up to 4-momentumtransfers Q = 5.6 GeV for the proton [1] and Q = 1.45 GeV for the neu-tron [2]. Extension of the measurements up to Q = 9 GeV via recoil protonpolarimetry is underway [3].Polarization-based results, however, appeared to be in conflict withearlier unpolarized cross section measurements at SLAC [4]. In high- Q kinematics, the difference between the measured values of the proton elec-tric/magetic form factor ratio, G Ep /G Mp , was as large as a factor of five,resulting in important theoretical and phenomenological implications, c.f. Ref. [5]. The observed discrepancy between unpolarized and polarized ex-perimental techniques prompted new cross section measurements at Jeffer-son Lab [6]. These later measurements also appeared to be in conflict withpolarization data, confirming a systematic difference between the data from ctober 29, 2018 17:18 WSPC - Proceedings Trim Size: 9in x 6in afanasev˙2photon the different experimental techniques.The resolution of this conflict was suggested [9] to be due to a higher-order electromagnetic effect of two-photon exchange not accounted for inthe experimental analysis. Model calculations [10] and [11] lead to similarconclusions, attributing over a half of experimental discrepancy to two-photon exchange corrections. A detailed account of the status of theoryand experiment in two-photon exchange can be found in the recent re-views, Refs. [7,8]. Here, I present an update on the status of the two-photonexchange problem and its implications, and highlight related theoretical is-sues.Let us start with reviewing a full set of higher-order electromagneticcorrections to electron-proton scattering Fig.1. Contributions of the dia-grams Fig.1a,c,d can be calculated using standard QED techniques. Theyare enhanced by large logarithmic factors log Q m e , resulting in radiativecorrections of the order of tens per cent that in addition depend on detailsof experimental cuts in the phase space of the radiated photon and elec-tron scattering angles (at fixed Q ). Vacuum polarization, Fig.1b, albeithas model uncertainties due to hadronic loop contributions, does not alterangular dependence of cross section at fixed momentum transfer Q , and ctober 29, 2018 17:18 WSPC - Proceedings Trim Size: 9in x 6in afanasev˙2photon hence it has no impact on Rosenbluth separation. Subprocesses with an ad-ditional photon coupling only to the proton Fig.1g,f show negligible angulardependence when constrained by kinematic cuts of elastic scattering.The bremsstrahlung correction of Fig.1c,d was calculated in Ref. [13]in soft-photon approximation, and this result was applied in data analy-sis in Ref. [4]. If this contribution is calculated fully including also hard-photon emission, for example, according to Ref. [14] or Ref. [15], it leads toabout 1 per cent additional absolute correction [16] to the experimental [4]Rosenbluth slope at Q =6 GeV . This additional correction accounts forabout one fifth of the discrepancy between Rosenbluth [4] and polariza-tion data [16] when missing mass cuts on the radiated photon are chosen tomatch experimental ones. The choice of kinematic cuts is essential since themagnitude of bremsstrahlung correction strongly depends on them. For ex-ample, if one uses a generic energy cut parameter for all electron scatteringangles ( e.g. , c=0.97 as in Ref. [17]) the extracted Rosenbluth slope reducesby about 5% at Q =6 GeV , thereby seemingly ‘resolving’ disagreement be-tween Rosenbluth and polarization data. It is therefore very important thatall refined calculations of bremsstrahlung corrections are also as accuratein the choice of kinematic cuts when compared with specific experimentalanalysis.Let us take a closer look at the two-photon exchange process, Fig.1e,f. Inthe approach developed by Tsai [13], these contributions were calculated ina limit when one of the exchanged photons carries a negligible 4-momentum.This contribution to the scattering amplitude is (infra-red) divergent, andthe divergence is cancelled at the cross-section level by adding interfer-ence between the bremsstrahhlung diagrams Fig.1c,d and f. It thereforealso depends on the details of experimental cuts on the radiated photonkinematics. The good news is that the calculation with soft second photonexchange does not require additional knowledge on the nucleon structure:It can be done in terms of one-photon exchange contribution times a ‘soft’factor that is independent on nucleon structure [13].As opposed to bremsstrahlung and vertex corrections, two-photon ex-change is not enhanced by large logarithms. It is instructive to see theeffect of the soft-photon-exchange portion on the Rosenbluth plot. In Fig.2,its effect on the cross section is shown for the kinematics of SLAC exper-iment [4] at Q = 6 GeV . The correction is angular-dependent, varyingbetween about -5% for backward scattering and 0 for forward scatteringangles. The Rosenbluth slope measured at SLAC [4] at Q = 6 GeV wasclose to 5% with the above correction included . It emphasizes importance of ctober 29, 2018 17:18 WSPC - Proceedings Trim Size: 9in x 6in afanasev˙2photon Ε -0.05-0.04-0.03-0.02-0.01 ∆ H Γ L soft Fig. 2. ‘Soft’ two-photon exchange correction combined with interference of electronand proton bremsstrahlung according to Ref. [13]. The relative radiative correction tothe cross section is plotted aganist the standard kinematic variable ǫ for Q = 6 GeV . two-photon exchange: Without this correction included, the extracted valueof electric proton form factor [4] would be about a factor of √ m q and a charge e q : δ γ = − e q α π log sm q , (1)where s is a Mandelstam variable, and α is a fine structure constant. Thecorresponding correction is zero for forward electron scattering. Note thatthe correction is negative for positive-charge quarks, and it grows logarith-mically with beam energy; numerically, it is a few per cent for relevantkinematics, if a constituent quark mass of m q = 300 MeV is taken for theestimate. Therefore this correction has the proper sign, magnitude and an-gular dependence to mimic a contribution of electric form factor to the crosssection of electron-proton scattering.It motivates more detailed studies of two-photon exchange, especially ata partonic level. Such a partonic approach was developed in Refs. [11,12],where two-photon exchange amplitude was factorized into a hard subpro-cess of electron-quark scattering and a soft subprocess described by gen-eralized parton distributions (GPD). A representative result is shown inFig.3, where it can be seen that linear ǫ -dependence of the cross section ismodified by a non-linear contribution from two-photon exchange. A dot-ted line labeled ‘1 γ ’ is an expectation from a pure one-photon exchangemechanism with a proton electric form factor taken from polarization mea- ctober 29, 2018 17:18 WSPC - Proceedings Trim Size: 9in x 6in afanasev˙2photon Cross section for ep elastic scattering , m.Reg. GPD, G
MBrash · , gauss. GPD, G MBrash · data s R / ( m p G d i po l e ) e Q = 6 GeV Fig. 3. Reduced ep-scattering cross section at Q = 6 GeV . Data points are from Ref.[4]. The dotted line shows an expected result from one-photon exchange using G Ep fit topolarization data [1]; solid and dashed curves have the two-photon exchange mechanismincluded within partonic approach [11,12] using different models of GPD. surements [1]. Noticing that at the same time two-photon exchange doesnot significantly alter interpretation of polarization data, we conclude thatwithin the considered model, this mechanism partially reconciles results ofexperimental techniques using polarized and unpolarized scattering.In a different approach [10,19], the virtual Compton amplitude enter-ing the two-photon exchange mechanism was approximated with nucleonpole diagrams with on-shell form factors substituted in photon-nucleon ver-tices. Despite of different dynamical models for the nucleon Compton am-plitudes, the conclusions of Refs. [10,19] and Refs. [11,12] are in qualitativeagreement. Addition of ∆-excitation mechanism [20] to the approach ofRefs. [10,19] somewhat reduced the predicted magnitude of the two-photoneffect. Higher nucleon resonances are estimated Ref. [21] to contribute aboutan order of magnitude less than nucleon and ∆.Clearly, the problem of two-photon exchange, especially the real part ctober 29, 2018 17:18 WSPC - Proceedings Trim Size: 9in x 6in afanasev˙2photon of the amplitude, is challenging because 4-dimensional momentum integra-tion in the box (and cross-box) diagrams Fig.1e,f requires knowledge of thenucleon Compton amplitude over a broad (infinite, to be exact!) range ofkinematic variables not available from experiment. On the other hand, theo-retical models applied so far are valid only within certain kinematic regions.One may also try a dispersive approach that takes advantage of analyticityand unitarity of the two-photon amplitude. In the kinematics of forwardelectron scattering, it is possible to reduce model uncertainties by using in-elastic electroproduction structure functions measured experimentally [22].A different category of papers attempt to evaluate the two-photon exchangeeffect using the experimentally observed difference between Rosenbluth andpolarization data, [9,23–25].The two-photon exchange mechanism also contributes to parity-violation studies of electron scattering through interference with Z -bosonexchange, as was pointed out in Ref. [26]. The effect was evaluated inRef. [26] in GPD framework at about 2% for backward angles and large Q .For smaller momentum transfers the two-photon effect is less significant,but Q dependence of a γZ box contribution was found to be essential [27]for extraction of strange-quark effects in the proton neutral weak current.Authors of Ref. [27] used a hadronic model [10,19] and found that a com-bined effect of 2 γ and γZ exchange on the values of G sE + βG sM extracted inrecent experiments can be as large as -40% in certain kinematics. A similarcalculation was also presented in Ref. [28].A comprehensive series of experiments are either underway or in prepa-ration at Jefferson Lab with a purpose to study two-photon exchange effectsin electron-proton scattering. Since two-photon exchange correction to elec-tron scattering observables is proportional to an odd power of the electroncharge, it can be measured directly by comparing electron and positronscattering. This method will be used in JLab experiment E-07-005, witha tertiary beam obtained from photoproduction of electron-positron pairs.Another JLab experiment, E-05-017, analyzes non-linearity of Rosenbluthplot caused two-photon exchange. Angular dependence of double-spin ob-servables is also affected by two-photon exchange at a few per cent level [12],and it is being looked for in polarization-transfer measurements (JLab ex-periment E-04-019). A single-spin target asymmetry caused by two-photonexchange will be studied in JLab experiment E-05-015 on a polarized Hetarget.So far, the only definitive experimental observation of two-photon ex-change effects came from the measurements of normal beam asymmetry ctober 29, 2018 17:18 WSPC - Proceedings Trim Size: 9in x 6in afanasev˙2photon , GeV -20-15-10-50 N o r m a l A s y mm e t r y [ - ] HAPPEX
Normal beam asymmetry for elastic e- He scattering
Unitarity-based model predictions E e = 3 GeV Fig. 4. Single-spin normal beam asymmetry on a He target in units of parts per million.The curve is a prediction of a unitarity-based model [35] extended to a nuclear target,with total photoproduction cross section and Compton t -slope on He used for input.Experimental data point is from Ref. [32]. Contribution of Coulomb distortion is belowa few parts per billion in the shown kinematics. at MIT-Bates [29], MAMI [30], and JLab [31,32]. The observations ap-pear to be in reasonable agreement with theoretical calculations at lowerenergies [33,34], nucleon resonance region [34] and above the resonance re-gion [22,35,36]. Note that a unitarity-based approach [35] applied for anuclear He target agrees both in sign and magnitude with recent measu-ments from JLab HAPPEX collaboration [32], see Fig.4; while the predic-tion in the kinematics of upcoming PREX measurement (JLAB E06-002)on Pb-target is about -5 ppm. At the same time, this asymmetry appearsto be several orders of magnitude larger than predictions from a knownmechanism of Coulomb distortion for small-angle electron scattering kine-matics [37].We conclude that the two-photon exchange effect stands as a possiblesource of the difference between Rosenbluth and polarization techniques forproton electric form factor measurements. It has to be included in the anal-ysis of other precision experiments. The two-photon exchange mechanismleads to new effects that can be studied experimentally. ctober 29, 2018 17:18 WSPC - Proceedings Trim Size: 9in x 6in afanasev˙2photon Notice: Authored by Jefferson Science Associates, LLC under U.S. DOEContract No. DE-AC05-06OR23177. The U.S. Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproducethis manuscript for U.S. Government purposes.
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