Extraction of beam-spin asymmetries from the hard exclusive π + channel off protons in a wide range of kinematics
S. Diehl, K. Joo, A. Kim, H. Avakian, P. Kroll, K. Park, D. Riser, K. Semenov-Tian-Shansky, K. Tezgin, K.P. Adhikari, S. Adhikari, M.J. Amaryan, G. Angelini, G. Asryan, H. Atac, L. Barion, M. Battaglieri, I. Bedlinskiy, F. Benmokhtar, A. Bianconi, A.S. Biselli, F. Boss`u, S. Boiarinov, W.J. Briscoe, W.K. Brooks, D. Bulumulla, V.D. Burkert, D.S. Carman, J.C. Carvajal, A. Celentano, P. Chatagnon, T. Chetry, G. Ciullo, L. Clark, P.L. Cole, M. Contalbrigo, V. Crede, A. D'Angelo, N. Dashyan, R. De Vita, M. Defurne, A. Deur, C. Dilks, C. Djalali, R. Dupre, H. Egiyan, M. Ehrhart, A. El Alaoui, L. El Fassi, P. Eugenio, A. Filippi, T.A. Forest, Y. Ghandilyan, G.P. Gilfoyle, K.L. Giovanetti, F.X. Girod, D.I. Glazier, E. Golovatch, R.W. Gothe, K.A. Griffioen, M. Guidal, L. Guo, H. Hakobyan, N. Harrison, M. Hattawy, T.B. Hayward, D. Heddle, K. Hicks, M. Holtrop, Y. Ilieva, D.G. Ireland, B.S. Ishkhanov, E.L. Isupov, D. Jenkins, H.S. Jo, S. Joosten, D. Keller, M. Khachatryan, A. Khanal, M. Khandaker, C.W. Kim, W. Kim, V. Kubarovsky, S.E. Kuhn, L. Lanza, M. Leali, P. Lenisa, K. Livingston, I.J.D. MacGregor, D. Marchand, N. Markov, L. Marsicano, V. Mascagna, B. McKinnon, Z.E. Meziani, T. Mineeva, M. Mirazita, V. Mokeev, C. Munoz Camacho, P. Nadel-Turonski, et al. (38 additional authors not shown)
EExtraction of beam-spin asymmetries from the hard exclusive π + channel off protonsin a wide range of kinematics S. Diehl,
7, 24
K. Joo, A. Kim, H. Avakian, P. Kroll, K. Park, D. Riser, K. Semenov-Tian-Shansky, K. Tezgin, K.P. Adhikari, S. Adhikari, M.J. Amaryan, G. Angelini, G. Asryan, H. Atac, L. Barion, M. Battaglieri,
40, 18
I. Bedlinskiy, F. Benmokhtar, A. Bianconi,
43, 21
A.S. Biselli, F. Boss`u, S. Boiarinov, W.J. Briscoe, W.K. Brooks,
41, 40
D. Bulumulla, V.D. Burkert, D.S. Carman, J.C. Carvajal, A. Celentano, P. Chatagnon, T. Chetry, G. Ciullo,
16, 11
L. Clark, P.L. Cole, M. Contalbrigo, V. Crede, A. D (cid:48)
Angelo,
36, 19
N. Dashyan, R. De Vita, M. Defurne, A. Deur, C. Dilks, C. Djalali,
32, 38
R. Dupre, H. Egiyan, M. Ehrhart, A. El Alaoui, L. El Fassi, P. Eugenio, A. Filippi, T.A. Forest, Y. Ghandilyan, G.P. Gilfoyle, K.L. Giovanetti, F.X. Girod, D.I. Glazier, E. Golovatch, R.W. Gothe, K.A. Griffioen, M. Guidal, L. Guo, H. Hakobyan,
41, 50
N. Harrison, M. Hattawy, T.B. Hayward, D. Heddle,
6, 40
K. Hicks, M. Holtrop, Y. Ilieva,
38, 14
D.G. Ireland, B.S. Ishkhanov, E.L. Isupov, D. Jenkins, H.S. Jo, S. Joosten, D. Keller, M. Khachatryan, A. Khanal, M. Khandaker, ∗ C.W. Kim, W. Kim, V. Kubarovsky,
40, 34
S.E. Kuhn, L. Lanza, M. Leali,
43, 21
P. Lenisa, K. Livingston, I.J.D. MacGregor, D. Marchand, N. Markov, L. Marsicano, V. Mascagna,
42, 21, † B. McKinnon, Z.E. Meziani, T. Mineeva, M. Mirazita, V. Mokeev, C. Munoz Camacho, P. Nadel-Turonski, G. Niculescu, M. Osipenko, M. Paolone, L.L. Pappalardo,
16, 11
E. Pasyuk, W. Phelps,
6, 40
O. Pogorelko, J.W. Price, Y. Prok,
33, 47
B.A. Raue,
12, 40
M. Ripani, A. Rizzo,
19, 36
P. Rossi,
40, 17
J. Rowley, F. Sabati´e, C. Salgado, A. Schmidt, R.A. Schumacher, Y.G. Sharabian, U. Shrestha, O. Soto, N. Sparveris, S. Stepanyan, P. Stoler, I.I. Strakovsky, S. Strauch,
38, 14
J.A. Tan, N. Tyler, M. Ungaro,
40, 34
L. Venturelli,
43, 21
H. Voskanyan, E. Voutier, D.P. Watts, X. Wei, M.H. Wood,
3, 38
N. Zachariou, J. Zhang, and Z.W. Zhao (The CLAS Collaboration) Argonne National Laboratory, Argonne, Illinois 60439 California State University, Dominguez Hills, Carson, CA 90747 Canisius College, Buffalo, NY Carnegie Mellon University, Pittsburgh, Pennsylvania 15213 IRFU, CEA, Universit´e Paris-Saclay, F-91191 Gif-sur-Yvette, France Christopher Newport University, Newport News, Virginia 23606 University of Connecticut, Storrs, Connecticut 06269 Duke University, Durham, North Carolina 27708-0305 Duquesne University, 600 Forbes Avenue, Pittsburgh, PA 15282 Fairfield University, Fairfield CT 06824 Universita’ di Ferrara , 44121 Ferrara, Italy Florida International University, Miami, Florida 33199 Florida State University, Tallahassee, Florida 32306 The George Washington University, Washington, DC 20052 Idaho State University, Pocatello, Idaho 83209 INFN, Sezione di Ferrara, 44100 Ferrara, Italy INFN, Laboratori Nazionali di Frascati, 00044 Frascati, Italy INFN, Sezione di Genova, 16146 Genova, Italy INFN, Sezione di Roma Tor Vergata, 00133 Rome, Italy INFN, Sezione di Torino, 10125 Torino, Italy INFN, Sezione di Pavia, 27100 Pavia, Italy Universit’e Paris-Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay, France James Madison University, Harrisonburg, Virginia 22807 Justus Liebig University Giessen, 35392 Giessen, Germany Kyungpook National University, Daegu 41566, Republic of Korea Lamar University, Beaumont, Texas 77705 Mississippi State University, Mississippi State, MS 39762-5167 National Research Centre Kurchatov Institute - ITEP, Moscow, 117259, Russia University of New Hampshire, Durham, New Hampshire 03824-3568 Norfolk State University, Norfolk, Virginia 23504 National Research Centre Kurchatov Institute, Petersburg Nuclear Physics Institute, RU-188300 Gatchina, Russia Ohio University, Athens, Ohio 45701 Old Dominion University, Norfolk, Virginia 23529 Rensselaer Polytechnic Institute, Troy, New York 12180-3590 University of Richmond, Richmond, Virginia 23173 a r X i v : . [ nu c l - e x ] J u l Universita’ di Roma Tor Vergata, 00133 Rome Italy Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University, 119234 Moscow, Russia University of South Carolina, Columbia, South Carolina 29208 Temple University, Philadelphia, PA 19122 Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606 Universidad T´ecnica Federico Santa Mar´ıa, Casilla 110-V Valpara´ıso, Chile Universit`a degli Studi dell’Insubria, 22100 Como, Italy Universit`a degli Studi di Brescia, 25123 Brescia, Italy University of Glasgow, Glasgow G12 8QQ, United Kingdom University of York, York YO10 5DD, United Kingdom Virginia Tech, Blacksburg, Virginia 24061-0435 University of Virginia, Charlottesville, Virginia 22901 College of William and Mary, Williamsburg, Virginia 23187-8795 Fachbereich Physik, Universitat Wuppertal, D-42097 Wuppertal, Germany Yerevan Physics Institute, 375036 Yerevan, Armenia
We have measured beam-spin asymmetries to extract the sin φ moment A sin φLU from the hardexclusive (cid:126)ep → e (cid:48) nπ + reaction above the resonance region, for the first time with nearly full coveragefrom forward to backward angles in the center-of-mass. The A sin φLU moment has been measured upto 6.6 GeV in − t , covering the kinematic regimes of Generalized Parton Distributions (GPD) andbaryon-to-meson Transition Distribution Amplitudes (TDA) at the same time. The experimentalresults in very forward kinematics demonstrate the sensitivity to chiral-odd and chiral-even GPDs.In very backward kinematics where the TDA framework is applicable, we found A sin φLU to be negative,while a sign change was observed near 90 ◦ in the center-of-mass. The unique results presented inthis paper will provide critical constraints to establish reaction mechanisms that can help to furtherdevelop the GPD and TDA frameworks. PACS numbers: 13.60.Le, 14.20.Dh, 14.40.Be, 24.85.+p
Hard exclusive pseudoscalar meson electroproductionprocesses offer a unique opportunity to study the struc-ture of the nucleon. They allow one to vary the size ofboth the probe (i.e. the photon virtuality Q ) and thetarget (the four-momentum transfer to the nucleon (me-son) t ( u )). These reactions reveal rich information aboutthe structure of the nucleon and the reaction dynamics.At very forward kinematics ( − t/Q (cid:28)
1) where theBjorken limit is applicable, hard exclusive pseudoscalarmeson electroproduction can be factorized into a pertur-batively calculable hard sub-process at the quark level, γ ∗ q → πq , and the hadronic matrix elements which areexpressed via the leading twist Generalized Parton Distri-butions (GPDs) of the nucleon and the pion DistributionAmplitude (DA) [1–3] as shown in Fig. 1 (a). GPDs arelight-cone matrix elements that can be expressed as non-local bilinear quark and gluon operators that describe thetransition from the initial to the final nucleon and revealthe 3-dimensional structure of the nucleon at the par-ton level by correlating the internal transverse positionof the partons to their longitudinal momentum [4–6]. Afirst experimental confirmation of the applicability of theleading twist GPD framework was provided by deeplyvirtual Compton scattering (DVCS) experiments at Jef-ferson Lab (JLab), DESY and CERN (see, e.g., [7–12]).While the DVCS process gives access to all chiral-evenGPDs H , (cid:101) H , E and (cid:101) E , pseudoscalar meson productionis especially helpful in probing the polarized GPDs ( (cid:101) H and (cid:101) E ), which contain information about the spatial dis-tribution of the quark spin [13, 14]. However, extensive experimental [15–29] and theoretical [2, 3, 30–32] inves-tigations of hard exclusive pseudoscalar meson electro-production in recent years have shown that the asymp-totic leading-twist approximation is not readily applica-ble in the range of kinematics accessible to current ex-periments. In fact, there are strong contributions fromtransversely polarized virtual photons that are asymp-totically suppressed by 1 /Q in the cross sections andhave to be considered by introducing chiral-odd GPDs( H T , (cid:101) H T , E T , and (cid:101) E T ) into the framework. For in-stance for π and η electroproduction, the contributionsfrom transversely polarized virtual photons are signifi-cant and the introduction of chiral-odd GPDs is neededto reproduce the measured cross sections as well as largebeam- and target-spin asymmetries with GPD models[2, 3, 21, 23, 24, 28, 33, 34].A further generalization of the GPD concept has beenintroduced for non-diagonal transitions where the ini-tial and final states are hadronic states of differentbaryon number [35–38]. In very backward kinematics( − u/Q (cid:28)
1) the collinear factorized description canbe applied in terms of a convolution of a hard partcalculable in perturbative QCD, and the soft parts ex-pressed in terms of nucleon-to-pion baryonic TransitionDistribution Amplitudes (TDAs) and the nucleon DAas shown in Fig. 1 (b). Like GPDs, nucleon-to-mesonTDAs are defined through hadronic matrix elements ofnon-local three-quark light-cone operators. Nucleon-to-meson TDAs are universal functions that parameter-ize the non-perturbative structure of hadrons. Withinthe reaction mechanism involving TDAs, the three-quarkcore of the target nucleon absorbs most of the virtual pho-ton momentum and recoils forward, while a pion from themesonic cloud of the nucleon remains with a low momen-tum heading backward. Therefore, the process brings abulk of new information on hadronic structure and canbe used e.g. to probe the non-minimal Fock componentsof hadronic light-cone wave functions. In contrast to thevery forward kinematic regime in the Bjorken limit, thecontribution of the transversely polarized virtual photonexchange is expected to dominate the process to leadingtwist-3 accuracy in very backward kinematics. Recentpublications on exclusive π + electroproduction by theCLAS collaboration [39] and on ω electroproduction fromJLab Hall C [40] in very backward kinematics have showna first indication of the applicability of the TDA modelto predict the magnitude and the scaling behavior of thecross section, as well as the dominance of the transverseover the longitudinal cross section at sufficiently large Q in the backward regime. FIG. 1: (a) Exclusive electroproduction of a pion on theproton in very forward kinematics ( − t/Q (cid:28) − u/Q (cid:28) The GPD and TDA approaches describe complemen-tary kinematic domains. While GPDs are applicable forsmall − t , TDAs can be applied for small − u , correspond-ing to large − t . Although these two approaches deal withdomains that are well distinct at asymptotic energies,they are not well separated in the kinematic range acces- sible to current experiments. Therefore, it is importantto investigate in detail the phenomenological differencesof the two approaches over a large range of momentumtransfer t . In previous publications, e.g. [27, 39], onlyvery limited kinematic regions covering either the GPDor the TDA regime exclusively have been investigated. Inthis letter, we present a measurement of the beam-spinasymmetries (BSA) for the hard exclusive electroproduc-tion ep → e (cid:48) nπ + for full π + center-of-mass (CM) angularcoverage with a large range of t or u .GPDs and TDAs can be accessed through different ob-servables in exclusive meson production, such as differ-ential cross sections and beam and target polarizationasymmetries [41, 42]. The focus of this work is on the ex-traction of the A sin φLU moment from the beam-spin asym-metry. The beam-spin asymmetry in the one-photon ex-change approximation is defined as follows [41]: BSA ( t, φ, x B , Q ) = dσ + − dσ − dσ + + dσ − = A sin φLU sin φ A cos φUU cos φ + A cos 2 φUU cos 2 φ , (1)where dσ ± is the differential cross section for each beamhelicity state ( ± ). For the positive / negative helicity thespin is parallel / anti-parallel to the beam direction. Thesubscripts ij represent the longitudinal (L) or unpolar-ized (U) state of the beam and the target, respectively. φ is the azimuthal angle between the electron scatteringplane and the hadronic reaction plane.Due to the interference between the amplitudes for lon-gitudinal ( γ ∗ L ) and transverse ( γ ∗ T ) virtual photon polar-izations, the moment A sin φLU is proportional to the polar-ized structure function σ LT (cid:48) [41]: A sin φLU = (cid:112) (cid:15) (1 − (cid:15) ) σ LT (cid:48) σ T + (cid:15)σ L , (2)where the structure functions σ L and σ T correspond tolongitudinal and transverse virtual photons, and (cid:15) de-scribes the ratio of their fluxes.Hard exclusive π + electroproduction was measured atJefferson Lab with the CEBAF Large Acceptance Spec-trometer (CLAS) [43]. Beam-spin asymmetries were ex-tracted over a wide range in Q , t , x B and φ . The inci-dent electron beam was longitudinally polarized and hadan energy of 5.498 GeV. The target was unpolarized liq-uid hydrogen. The CLAS detector consisted of six identi-cal sectors within a toroidal magnetic field. The momen-tum and the charge polarity of the particles were deter-mined by 3 regions of drift chambers from the curvatureof the particle trajectories in the magnetic field. The elec-tron identification was based on a lead-scintillator elec-tromagnetic sampling calorimeter in combination with aCherenkov counter. For the selection of deeply inelasticscattered electrons, cuts on Q > and on theinvariant mass of the hadronic final state W > e (cid:48) π + n finalstate, events with exactly one electron and one π + weredetected, and a cut around the neutron peak in the miss-ing mass spectrum was performed. The mean signal-to-background ratio in the forward region is 15.3, while itdecreases to 4.9 in the backward region.Beam-spin asymmetries (BSA) were measured in the Q range from 1 to 4.6 GeV , − t up to 6.6 GeV and x B from 0.1 - 0.6. The BSA and its statistical uncer-tainty were determined experimentally from the numberof counts with positive and negative helicity ( N ± i ), in aspecific bin i as: BSA = 1 P b N + i − N − i N + i + N − i , σ BSA = 2 P b (cid:115) N + i N − i ( N + i + N − i ) , (3)where P b is the average magnitude of the beam po-larization. P b was measured with a Møller polarime-ter upstream of CLAS and was 74.9 ± ± φ moment A sin φLU , the beam-spinasymmetry was measured as a function of the azimuthalangle φ . Then a fit of the data with the functionalform shown in Eq. (1) was applied. Figure 2 shows thebeam-spin asymmetry as a function of φ for events in theforward and backward regions, integrated over all otherkinematic variables. Experimentally the forward regionis defined as cos θ CM > − t < , while thebackward region is defined by a cut on cos θ CM < − u < , where θ CM is the polar angle of thepion in the frame boosted along the momentum trans-fer (cid:126)q direction. As expected the φ -dependence can be FIG. 2: Beam-spin asymmetry as a function of φ for π + emit-ted in the forward (left) and backward (right) regions, inte-grated over all other kinematic variables. The vertical errorbars show the statistical uncertainty of each point, while thehorizontal bars correspond to the bin width. The red lineshows the fit with the functional form of Eq. (1). well described by Eq. (1). The asymmetry of the back-ground has been extracted with the side-band methodby selecting a missing-mass interval on the right side ofthe missing neutron peak. These events represent the background under the region of interest and thereforeits asymmetry has to be subtracted. The amplitude ofthe background asymmetry has been determined in thesame way as for the exclusive events, with a sin φ fit ofthe φ -dependence of the BSA. The sin φ amplitude of thebackground is 0 . ± .
006 in the forward region and de-creases to 0 . ± .
01 in the backward region. Based onthe signal-to-background ratio determined from a fit ofthe missing mass spectrum in each kinematic bin, a bin-by-bin background subtraction has been performed forthe extracted A sin φLU values.Several sources of systematic uncertainty were investi-gated, including particle identification, background sub-traction, beam polarization, and the influence of the A cos φUU and A cos 2 φUU moments. The correlation between theunpolarized moments and A sin φLU was found to be verysmall. The systematic uncertainty for each contribu-tion was determined by a variation of the contributingsource around its nominal value. To estimate the impactof acceptance effects, a Monte Carlo simulation whichincluded a parametrization of the kinematic behaviourfollowing that of the actual data was performed. Theimpact of acceptance effects turned out to be small andis included in the systematic uncertainty. The total sys-tematic uncertainty in each bin is defined as the square-root of the quadratic sum of the uncertainties from allsources. It has been found to be comparable to the sta-tistical uncertainty.Figure 3 shows the results for A sin φLU in the region of − t up to 0.75 GeV ( − t/Q ≈ .
25) where the leading-twist GPD framework is applicable and compares themto the theoretical predictions from the GPD-based modelby Goloskokov and Kroll (GK) [44]. The experimentaldata is binned in − t and integrated over the complete Q distribution ranging from 1.0 to 4.5 GeV and x B ranging from 0.1 to 0.6. The band on the theoreticalprediction represents the range in Q and x B accessiblewith our measurements. The GK model includes chiral-odd GPDs to calculate the contributions from the trans-versely polarized virtual photon amplitudes, with their t -dependence incorporated from Regge phenomenology.The GPDs are constructed from double distributions andconstrained by results from lattice QCD and transversityparton distribution functions [44]. A special emphasis isgiven to the GPDs H T and E T = 2 (cid:101) H T + E T , while contri-butions from other chiral-odd GPDs are neglected in thecalculations, unlike chiral-even GPDs, where some con-tributions are negligible but still included. The pion polecontribution to the amplitudes is taken into account forboth the longitudinally and transversely polarized virtualphotons. However, its contribution to the transverselypolarized virtual photon amplitudes is very small.The magnitude of A sin φLU (see Eq. (2)) is proportional tothe ratio of the interference structure function σ LT (cid:48) andthe unseparated cross section σ = σ T + (cid:15)σ L , where σ FIG. 3: A sin φLU (black rectangles) as a function of − t in theforward kinematic regime and their systematic uncertainty(grey bins). For comparison the theoretical prediction fromthe GPD-based Goloskokov-Kroll model (blue band) is shown.The band of the theoretical prediction corresponds to therange accessible with our measurements in Q and x B . is forward peaked due to the pion pole term contributionand σ LT (cid:48) is constrained to be zero at t = t min ( θ CM = 0)due to angular momentum conservation. σ LT (cid:48) can be ex-pressed through the convolutions of GPDs with subpro-cess amplitudes (twist-2 for the longitudinal and twist-3for the transverse amplitudes) and contains the productsof chiral-odd and chiral-even terms [2]: σ LT (cid:48) ∼ Im (cid:104) (cid:104) E T − eff (cid:105) ∗ (cid:104) (cid:101) H eff (cid:105) + (cid:104) H T − eff (cid:105) ∗ (cid:104) (cid:101) E eff (cid:105) (cid:105) , (4)where all involved GPDs are influenced directly or indi-rectly by the pion pole term, for example: (cid:101) E eff = (cid:101) E + pole term, (5) (cid:101) H eff = (cid:101) H + ξ − ξ (cid:101) E eff , (6)with the skewness ξ ∼ x B / (2 − x B ). For π + the imag-inary part of small chiral-odd GPDs in σ LT (cid:48) is signifi-cantly amplified by the pion pole term, which is real andexactly calculable. This feature increases the sensitivityof polarized observables to chiral-odd GPDs in contrastto the π and η channels where the pole contributionis not present. In the GK model σ LT (cid:48) is dominated by Im [ (cid:104) H T − eff (cid:105) ∗ (cid:104) (cid:101) E eff (cid:105) ] and (cid:101) E eff is dominated by the pionpole term, while other GPD products are considered tobe negligible.The comparison between the experimental results andthe theoretical predictions shows that the magnitude ofthe GK model calculations is overestimated and the t -dependence of the measured A sin φLU values shows a muchflatter slope than the predicted curve. The discrepancy inmagnitude and t -dependence might be due to the inter-play of the pion pole term with the poorly known chiral-odd GPDs H T and E T . Even though previous studies showed that the GPD model can be well applied to pre-dict π and η cross sections [20, 21, 23], the results in Fig.3 show that the GPDs and the model have to be tunedto describe BSA as well. While the beam-spin asymme-try calculations for the π + channel are overestimated bythe GK model, the absence of the pion pole term in caseof the π and η channels leads to a significantly smallerpredicted beam-spin asymmetry by the GK model, whichunderestimates the experimental observation as shown inRef. [28]. The combined analysis of these unique π + datawith the π and η channels [16, 20, 21, 23, 28] can beperformed to significantly constrain these poorly knownGPDs.While the framework of GPDs is only applicable invery forward kinematics, a complete understanding ofthe reaction mechanism requires measurements over thecomplete range of − t . As shown in Fig. 4, we extendedthe kinematic region for the extraction of A sin φLU up to − t = 6.6 GeV , which is close to the maximal accessi-ble − t value. The data are binned in − t and integratedover the complete Q distribution ranging from 1 GeV - 4.5 GeV and x B ranging from 0.1 to 0.6. FIG. 4: A sin φLU as function of − t . The shaded area representsthe systematic uncertainty. The sign of of A sin φLU in forward kinematics is clearlypositive, which is confirmed by the most recent GPDmodels [44], while in backward kinematics a clearly neg-ative sign is observed. Large t corresponds to small u , sothat at backward angles the u channel dominates (Fig. 1(b)). Thus it is expected that the TDA-based frameworkcan be applied in very backward kinematics.Similarly to Eq. (4) for very forward kinematics, σ LT (cid:48) in the backward regime can be expressed through the in-terference between the leading twist transverse amplitudeof the convolution in terms of twist-3 πN TDAs ( H tw3 )and nucleon DAs ( φ tw3 ) and the next leading sub-processlongitudinal amplitude of the convolution involving twist-4 TDAs ( H tw4 ) and DAs ( φ tw4 ) [46–48]: σ LT (cid:48) ∼ Im (cid:104) (cid:104) H tw3 i φ tw3 j (cid:105) (cid:0) (cid:104) H tw4 i φ tw3 j (cid:105) + (cid:104) H tw3 i φ tw4 j (cid:105) (cid:1) ∗ (cid:105) . (7)A complete theoretical study of this twist-4 longitudinalamplitude is not yet available, and it is an open questionwhich particular twist-4 πN TDAs and DAs will con-tribute to the BSA and what kind of phenomenologicalmodels can be implemented for these quantities. Never-theless, our measurement will significantly constrain thenearly unknown TDAs and help to further develop theTDA-based framework.Also, for the intermediate kinematic region around θ CM = 90 ◦ , first models have been introduced [49, 50].However, calculations exist only for wide-angle Comp-ton scattering [49] and the photoproduction of pions [50].Nevertheless, the introduced concepts can also be appliedto electroproduction and will help to connect the GPDand TDA kinematic regimes in the future.As shown in Fig. 4, the t -dependence of A sin φLU makesa clear transition from positive values with a maximumvalue of 0.10 in the forward region to negative valuesdown to a minimum value of -0.06 in the backward re-gion. The sign change occurs around − t = 3 GeV , whichcorresponds to θ CM = 90 ◦ , and marks the transition be-tween the π + emitted in the forward and backward di-rections. Therefore, the sign change may be interpretedas an indication for a transition between the GPD andTDA regimes. The wide range of kinematics presentedin this work will also enable the development of a moreconsistent reaction mechanism for the intermediate kine-matical regime in-between the very forward regime withGPD-based description and the very backward regimewith description in terms of TDAs.Figure 5 shows A sin φLU as a function of Q , integratedover x B in the top plots and as a function of x B , in-tegrated over Q in the bottom plots, for pions goingin the forward (left) and backward (right) regions, asdefined earlier. The figure clearly shows that the signchange between the forward and the backward region ispresent for all Q - and x B -bins. In the forward region,the Q -dependence shows a rather flat behavior, while A sin φLU rises for small x B until it reaches a constant levelfor x B > Q - and x B -dependencies show a rather flat behavior. However, theeffect is not statistically significant.In summary, we have measured for the first timethe sin φ moment A sin φLU of beam-spin asymmetries for (cid:126)ep → e (cid:48) nπ + at large photon virtuality, above the res-onance region over the full range of polar angles θ CM that cover the complete kinematic region of the GPDand TDA frameworks simultaneously. A comparison invery forward kinematics showed that our A sin φLU measure-ment cannot be described in magnitude or t -dependenceby the most advanced GPD-based model [44]. In veryforward kinematics where the GPD framework is appli-cable, we measure clearly positive values of A sin φLU , whilein very backward kinematics where the TDA frameworkis applicable, negative A sin φLU values have been measured.A clear sign change of A sin φLU has been observed around FIG. 5: A sin φLU as function of Q (top) and x B (bottom) forpions going in the forward (left) and backward (right) regions.The shaded area represents the systematic uncertainty. θ CM = 90 ◦ . The presented data provide important con-straints for the development of a reaction mechanism thatdescribes the complete kinematic regime including GPDsand TDAs as well as the intermediate regime. To obtaina deeper understanding, and to reveal more details of thereaction mechanism, measurements with a higher preci-sion and over a larger range of Q will be performed withthe upgraded 12 GeV CEBAF accelerator at JLab and inthe crossed reaction ¯ N N → γ ∗ π , accessible with ¯PANDAat FAIR [51–53] and πN → N γ ∗ or πN → N J/
Ψ at J-PARC [54]. The data-set presented in this work can bedownloaded from Ref. [45].We acknowledge the outstanding efforts of the staffof the Accelerator and the Physics Divisions at Jef-ferson Lab in making this experiment possible. Wealso acknowledge very helpful discussions with L. Szy-manowski and B. Pire. This work was supported inpart by the U.S. Department of Energy, the NationalScience Foundation (NSF), the Italian Istituto Nazionaledi Fisica Nucleare (INFN), the French Centre Nationalde la Recherche Scientifique (CNRS), the French Com-missariat pour l (cid:48)
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