Incoherent deeply virtual Compton scattering off 4 He
aa r X i v : . [ nu c l - t h ] A ug Incoherent deeply virtual Compton scattering off He Sara Fucini, Sergio Scopetta, and Michele Viviani Dipartimento di Fisica e Geologia, Universit`a degli Studi di Perugia and Istituto Nazionale di Fisica Nucleare,Sezione di Perugia, via A. Pascoli, I - 06123 Perugia, Italy INFN-Pisa, 56127 Pisa, Italy (Dated: August 27, 2020)Very recently, for the first time, the two channels of nuclear deeply virtual Compton scattering,the coherent and incoherent ones, have been separated by the CLAS collaboration at the JeffersonLaboratory, using a He target. The incoherent channel, which can provide a tomographic view ofthe bound proton and shed light on its elusive parton structure, is thoroughly analyzed here in theImpulse Approximation. A convolution formula for the relevant nuclear cross sections in terms ofthose for the bound proton is derived. Novel scattering amplitudes for a bound moving nucleonhave been obtained and used. A state-of-the-art nuclear spectral function, based on the Argonne 18potential, exact in the two-body part, with the recoiling system in its ground state, and modelled inthe remaining contribution, with the recoiling system in an excited state, has been used. Differentparametrizations of the generalized parton distributions of the struck proton have been tested. Agood overall agreement with the data for the beam spin asymmetry is obtained. It is found thatthe conventional nuclear effects predicted by the present approach are relevant in deeply virtualCompton scattering and in the competing Bethe-Heitler mechanism, but they cancel each other to alarge extent in their ratio, to which the measured asymmetry is proportional. Besides, the calculatedratio of the beam spin asymmetry of the bound proton to that of the free one does not describethat estimated by the experimental collaboration. This points to possible interesting effects beyondthe Impulse Approximation analysis presented here. It is therefore clearly demonstrated that thecomparison of the results of a conventional realistic approach, as the one presented here, with futureprecise data, has the potential to expose quark and gluon effects in nuclei. Interesting perspectivesfor the next measurements at high luminosity facilities, such as JLab at 12 GeV and the futureElectron Ion Collider, are addressed.
PACS numbers: 13.60.Hb,14.20.Dh,27.10.+h
I. INTRODUCTION
A quantitative understanding of the European Muoncollaboration (EMC) effect in inclusive deep inelasticscattering (DIS) off nuclear targets [1] is still missingafter several decades. Since then, it is clear that theparton structure of bound nucleons is modified by thenuclear medium (see Ref. [2] for a recent report), butso far it has not been possible to distinguish betweenseveral different explanations, proposed using differentdescriptions of the structure of the bound nucleons. It iswidely understood that measurements beyond DIS, suchas semi-inclusive DIS (SIDIS) and nuclear deeply virtualCompton scattering (DVCS), the hard exclusive lepto-production of a real photon on a nuclear target, will playa fundamental role in shedding light on this long-standingproblem of hadronic Physics [3, 4]. Crucial steps forwardare expected from a new generation of planned measure-ments at high energy and high luminosity facilities inthe next years, including the Jefferson laboratory (JLab)at 12 GeV [5] and the future electron-ion collider (EIC)[6, 7]. From the theoretical point of view, this programmeimplies the challenging description of complicated pro-cesses. One of them, incoherent DVCS off He nuclei,for which the first data have been collected and recentlypublished [8], is the subject of this work. In DVCS, the parton structure is encoded in the socalled Compton Form factors (CFFs), defined in termsof the generalized parton distributions (GPDs) [9], nonperturbative quantities providing a wealth of novel infor-mation (for exhaustive reports see, e.g., Ref. [10–12]).In particular, nuclear DVCS could unveil the presence ofnon-nucleonic degrees of freedom in nuclei [13], or mayallow to better understand the spatial distribution of nu-clear forces [14, 15] (to develope this latter program, theuse of positron beams, presently under discussion at JLab[16], would be of great help). Besides, the tomography ofthe target, i.e., the distribution of partons with a givenlongitudinal momentum in the transverse plane, is cer-tainly one of the most exciting information accessiblein DVCS through the GPDs formalism [17]. In nuclei,DVCS can occur through two different mechanisms, i.e.,the coherent one A ( e, e ′ γ ) A , where the target A recoilselastically and its tomography can be ultimately studied,and the incoherent one A ( e, e ′ γp ) X , where the nucleusbreaks up and the struck proton is detected, so that itstomography could be obtained. The comparison betweenthis information and that obtained for the free protoncould provide ultimately a pictorial view of the realiza-tion of the EMC effect. From an experimental point ofview, the study of nuclear DVCS requires the very diffi-cult coincidence detection of fast photons and electronstogether with slow, intact recoiling protons or nuclei. Forthis reason, in the first measurement of nuclear DVCS atHERMES [18], a clear separation between the two dif-ferent DVCS channels was not achieved. Recently, forthe first time, such a separation has been performed bythe EG6 experiment of the CLAS collaboration [19], withthe 6 GeV electron beam at Jefferson Lab (JLab). Thefirst data for coherent and incoherent DVCS off He havebeen published in Refs. [20] and [8], respectively. Amongfew nucleon systems, for which a realistic evaluation ofconventional nuclear effects is possible in principle, Heis deeply bound and represents the prototype of a typicalfinite nucleus. Realistic approaches allow to distinguishconventional nuclear effects from exotic ones, which couldbe responsible of the observed EMC behaviour. Withoutrealistic benchmark calculations, the interpretation of thedata will be hardly conclusive. Indeed, in Refs. [8, 20],the importance of new calculations has been addressed,for a successful interpretation of the collected data and ofthose planned at JLab in the next years [21, 22]. In factsavailable estimates, proposed long time ago, correspondin some cases to different kinematical regions [23, 24].New refined calculations are certainly important, aboveall, for the next generation of accurate measurements. Inthis sense, the use of heavier targets, due to the difficultyof the corresponding realistic many-body calculations, isless promising. Among few-body nuclear systems, H isvery interesting, for the extraction of the neutron infor-mation and for its rich spin structure [13, 25, 26, 28].In between H and He, He could allow to study the A dependence of nuclear effects and it could give an easyaccess to neutron polarization properties, due to its spe-cific spin structure. Besides, being not isoscalar, flavordependence of nuclear effects could be studied, in par-ticular if parallel measurements on H targets were pos-sible. A complete impulse approximation (IA) analysis,using the Argonne 18 (AV18) nucleon-nucleon potential[29] and the UIX three nucleon force model of Ref. [30],is available and nuclear effects on GPDs are found to besensitive to details of the used nucleon-nucleon interac-tion [31–35]. Measurements for He have been addressed,planned in some cases but they have not been performedyet. We have therefore analyzed successfully, in impulseapproximation (IA), coherent DVCS off He [36], obtain-ing an overall good agreement with the data [20]. In arecent rapid communication [37], we have proposed ananalogous analysis for the incoherent channel, to see towhat extent a conventional description can describe therecent data [8], which have the tomography of the boundproton as the ultimate goal. In that analysis, the inco-herent DVCS beam spin asymmetry has been evaluatedin IA framework, in terms of a diagonal spectral function[38] based on the AV18+UIX nuclear interactions and theGPDs model by Goloskokov and Kroll [39], obtaining anoverall good description of the available data.We retake here the subject in detail. The expressionsfor all the relevant scattering amplitudes for a bound,moving proton are fully derived and explicitly given. In terms of them, the relevant cross sections are calculated,showing the effects of the use of different descriptions ofthe nuclear structure and of the nucleon GPDs. Resultsare shown for the differential cross sections and the beamspin asymmetry, investigating carefully the source of nu-clear effects on both of these observables.The paper is structured as follows. The framework andthe main formalism are presented in the next section,while details are collected in two extended appendices.In the third section, the ingredients of the calculationare described, while numerical results are presented anddiscussed in the following one. Conclusions and perspec-tives are eventually given in the last section.
II. FORMALISM
In this section, we present the relevant formalism forthe IA description of the handbag approximation to theincoherent DVCS process He ( e, e ′ γp ′ ) X , shown in Fig.1. In such a description of the process, the protonchanges its momentum from p to p ′ after the interac-tion of the virtual photon with one quark belonging toone nucleon, i.e., only nucleonic degrees of freedom areincluded and coherent effects, such as shadowing, are ne-glected. The other IA assumption is that any furtherscattering between the proton and the remnant system X is disregarded in the final state. The factorizationproperty can be applied to this process when the ini-tial photon virtuality, Q = − q = −( k − k ′ ) , is muchlarger than the momentum transferred at hadronic level, t = ∆ = ( p − p ′ ) . We note also that, in the presentIA approach, ∆ = ( q − q ) , that is, the momentumtransferred to the system coincides with that transferredto the struck proton. For high enough values of Q , IAusually describes the bulk of nuclear effects in a hard elec-tron scattering process (see, e.g., Ref. [40] for an experi-mental study of the onset of the validity of IA). Similarexpectations hold in this study, although only the com-parison with data can establish the validity of the chosenframework. In this way, the hard vertex of the diagramillustrated in Fig. 1 can be calculated using perturba-tive methods while the soft part can be parametrizedthrough the GPDs of the bound proton. Such non per-turbative objects, namely the GPDs, are functions of ∆ ,of the so-called skewness ξ = − ∆ + / P + , i.e., the differencein plus momentum fraction between the initial and the fi-nal states, and of x , the average plus momentum fractionof the struck parton with respect to the total momentum.(the notation a ± = ( a ± a )/√ q = ( q + q )/ P = p + p ′ ). Actually GPDs, asany other parton dostribution, depend on the momentumscale Q according to QCD evolution equations. Such anobvious dependence is omitted in the rest of the paperto avoid a too heavy notation. We adopted the referenceframe proposed in Ref. [41], with the target at rest, thevirtual photon with energy ν moving opposite to the ˆ z axis and the leptonic and hadronic planes of the reactiondefining the angle φ . Using energy-momentum conser-vation, one gets for the azimuthal angle of the detectedproton the relation φ p ′ = φ + φ e and, since in the chosenframe one has, for the electron azimuthal angle, φ e = φ p ′ coincides with φ .Since x cannot be experimentally accessed, GPDs can-not be directly measured. Some help comes from the factthat the leptoproduction of a real photon always occursthrough two different mechanisms leading to the same fi-nal state ( e ′ γp ′ ) : the DVCS process, discussed above andrelated to the parton content of the target, and the elec-tromagnetic Bethe-Heitler (BH) process, shown in Fig.2. In facts, the complete squared amplitude for the lep-toproduction process has to be read as A = T DV CS + T BH + I . (1)In particular, in the kinematical region tested at JLaband of interest here, the BH mechanism is dominatingthe DVCS one. For this reason, a key handle to accessthe GPDs is the interference between these two compet-ing processes, i.e. I = R e ( T DV CS T ∗ BH ) . This term, con-taining T DV CS is sensitive to the parton content of thetarget through the GPDs. Such information is encapsu-lated in the Compton Form Factors (CFFs) F related tothe generic GPDs F by: F ( ξ, ∆ ) = ∫ dx F ( x, ξ, ∆ ) x ± ξ + iǫ . (2)Since in the CFFs the dependence on x is integrated out,they can be measured. Also for the CFFs the obvious Q dependence is omitted here and in the following. Wenote in passing that the possibility that the final photonis emitted by the initial nucleus, or by the final nuclearsystem X, has been neglected, being the BH cross sectionapproximately proportional to the inverse squared massof the emitter. Therefore, with respect to the emissionfrom the electrons, this contribution is negligibly small.In facts, the experimental collaboration EG6 has not con-sidered this occurrence in its analysis. From a theoreticalpoint of view, if these contributions are neglected, gaugeinvariance is not respected. Nonetheless, we have to pointout that in the present IA analysis gauge invariance is inany case not fulfilled and it could be restored only imple-menting many-body currents at the nuclear level. Thesecorrections have not been included in the calculation yetand they could be more relevant than photon emissionfrom nuclear systems in the initial and final state.The clearest way to experimentally access the relevantinterference term is the measurement of the beam-spinasymmetry (BSA) for the process where the unpolarizedtarget (U), He in this case, is hit by a longitudinallypolarized (L) electron beam with different helicities ( λ =± ). So, the observable under scrutiny reads A LU = dσ + − dσ − dσ + + dσ − . (3) Since the interference term is directly proportional tothe helicity of the beam, the difference of cross sectionsfor different beam helicities in the numerator of Eq.(3),up to a phase space factor, gives a direct access to suchterm. We will show in the following that the quantities dσ ± in Eq. (3) are actually 4-times differential cross sec-tions.Our aim is thus the evaluation of the complete expres-sion for the leptoproduction cross section at LO in IA inorder to study the theoretical behaviour of the BSA andcompare it with the data. The details of the calculationare described in the following. γ γ (q )(q ) P p p’ = p + ∆ He GPDs (x, ξ , ∆ ) X Factorization
A−1 (p ) fe e’(k) (k’) ∆ = q − q A * FIG. 1: (color online) Incoherent DVCS process off He in theIA to the handbag approximation.
FF ( ∆ )Pe(k) e’(k’) ( ∆ ) γ (q ) γ * p’p X He k’+q A FF ( ∆ ) γ γ e(k) e’(k’)P He X ( ∆ ) * (q )p’p k−q + A FIG. 2: (color online) The Bethe Heitler process in IA.
In our IA approach, we account only for the kinemat-ical off-shellness of the initial bound proton so that theenergy of the struck proton is obtained from energy con-servation and reads p = M A − √ M ∗ A − + ⃗ p ≃ M − E − T rec , (4)where we define the removal energy E = M ∗ A − + M − M A = ǫ ∗ A − + ∣ E A ∣ − ∣ E A − ∣ in terms of the binding energy (mass)of He and of the 3-body system, E A ( M A ) and E A − ( M ∗ A − ), respectively, and of the excitation energy of therecoiling system, ǫ ∗ A − . Finally, T rec is the kinetic en-ergy of the recoiling 3 − body system and M is the protonmass. A straightforward but lengthy analysis, detailed inappendix A, leads to a complicated convolution formulafor the cross section, which can be cast in the followingform dσ ± Inc = ∫ exp dE d ⃗ p p ⋅ kp ∣⃗ k ∣ P He (⃗ p, E ) dσ ± b (⃗ p, E, K ) , (5)where the main ingredients are the nuclear spectral func-tion P He (⃗ p, E ) and the cross section for a DVCS pro-cess off a bound proton, dσ ± b . As thoroughly describedin Appendix A, the integral on the removal energy refersto the full spectrum of He, both discrete and continu-ous. In Eq. (5), K is the set of kinematical variables { x B = Q /( M ν ) , Q , t, φ } . The range of K accessed inthe experiment fixes the proper energy and momentumintegration space, denoted as exp and described in ap-pendix A. From Eq. (5) we get the measured differentialcross sections, appearing in Eq. (3), dσ ± ≡ dσ ± Inc dx B dQ d ∆ dφ = ∫ exp dE d ⃗ p P He (⃗ p, E ) (6) ×∣ A ± (⃗ p, E, K )∣ g (⃗ p, E, K ) , where g (⃗ p, E, K ) is a complicated function which arises,as explicitely detailed in Appendix A, from the integra-tion over the phase space and includes also the flux factor p ⋅ k /( p ∣⃗ k ∣) of Eq. (5). This latter term comes from thefact that one has at disposal only non-relativistic nuclearwave functions to evaluate the spectral function. In thepresent approach this implies that the number of parti-cle sum rule is respected, but the momentum sum ruleis slightly violated. Such a problem could be solved ul-timately within a Light Front approach, along the linesproposed in Ref. [42] for a 3 − body system.The BSA (3), written in terms of the above cross sec-tions, yields the schematic form A IncohLU ( K ) = I He ( K ) T HeBH ( K ) , (7)where I He ( K ) = ∫ exp dE d ⃗ p P He (⃗ p, E ) g (⃗ p, E, K ) I (⃗ p, E, K ) ,T HeBH ( K ) = ∫ exp dE d ⃗ p P He (⃗ p, E ) g (⃗ p, E, K )× T BH (⃗ p, E, K ) , (8)refer to a moving bound nucleon and generalize theFourier decomposition of the DVCS cross section off aproton at rest, at leading twist, derived in Ref. [41].Without going into technical details, that are presentedin appendix B, we summarize the structure of the differ-ent contributions.For the BH part, we considered the full sum of azimuthalharmonics, i.e T BH = c bound + c bound cos φ + c bound cos ( φ ) , (9)where the coefficients c boundi contain the Dirac and Pauliform factors (FFs). The azimuthal dependence of the -0.4-0.2 0 0.2 0.4 0 50 100 150 200 250 300 350 A L U ( φ ) φ [deg] t =-0.28 GeV , x B =0.25 , Q =1.95 GeV no ∆ /Q ∆ /Q PRL 100, 162002 (2008)
FIG. 3: Beam spin asymmetry for a proton at rest considering(full curve) and ignoring (dot-dashed curve) term of order∆ / Q in the interference part. In this kinematics, ∆ / Q ≃ . amplitudes is due to the expression of the BH propaga-tor as reported in Appendix B. We stress that in thepresent IA approach no nuclear modifications occur forthe FFs of the bound proton. Concerning the interferencepart in the numerator of Eq. (3), terms proportional to∆ / Q have been considered as well as corrections pro-portional to ǫ = M x B / Q , accounting for target masscorrections. The latter terms are fundamental in orderto obtain a fully consistent comparison with the BSA fora proton at rest, which will be shown in the next sec-tion. The main reason is that in the amplitudes for abound proton it is not always possible to isolate suchterms, since the obtained expressions are function of the4-momentum of the bound, off-shell proton. In our ap-proach the parton content of the bound proton plays arole only in the imaginary part of the CFF H . In the kine-matics of interest and in the present model, this quantitycan be expressed in terms of only one GPD of the boundproton, H ( x, ξ, ∆ ) , selected in the slice x = ± ξ , i.e. I m H ( ξ ′ , t ) = H ( ξ ′ , ξ ′ , t ) − H (− ξ ′ , ξ ′ , t ) , (10)where H ( ξ ′ , ξ ′ , t ) is summed over the u, d, s flavours ofthe quarks. We notice that the off-shellness of the boundnucleon enters the proton parton structure through thedependence of the GPDs on ξ ′ = − q /( P ⋅ q ) . In this way,the modification at partonic level is due to this rescal-ing of the skewness that, for a proton at rest, becomes ξ = x B ( + ∆ / Q )/( − x B + x B ∆ / Q ) , keeping termsproportional to ∆ / Q . III. INGREDIENTS OF THE CALCULATION
In order to actually evaluate Eq. (7), we need an inputfor the proton GPD and for the proton spectral functionin He. Concerning the nuclear part, only old attemptsexist of obtaining a complete spectral function of He [44,45]. The unpolarized spectral function, whose emergence d σ [ nb / G e V ] φ [deg] t =-0.15 GeV , x B =0.16, Q = 1.39 GeV Free protonBound protonPRC 98, 045203 (2018)
FIG. 4: The cross section for the BH process on the freeproton (dashed line) and on a proton bound in He (full line),according to the present treatment, in the kinematics reportedon the top of the frame, corresponding to data presented inRef. [55], as a function of the azimuthal angle φ . The preciseposition of the data and their errors are taken from [58]. d σ [ nb / G e V ] φ [deg] t =-0.15 GeV , x B =0.16, Q = 1.39 GeV BH+InterferenceBHPRC 98, 045203 (2018)
FIG. 5: The cross section for the BH process (full line) andthe one obtained including the interference between the BHand DVCS processes (dot-dashed line), for a proton boundin He, according to the present treatment, in the kinematicsreported on the top of the frame, corresponding to data pre-sented in Ref. [55], as a function of the azimuthal angle φ .The precise position of the experimental data and their errorsare taken from [58]. in this process is thoroughly described in appendix A,can be cast in the form P He (⃗ p, E ) = ∑ f A − ⟨ He ∣ f A − ; N ⃗ p ⟩⟨ f A − ; N ⃗ p ∣ He ⟩× δ ( E − E min − ǫ ∗ A − ) . (11)It is therefore clear that its realistic evaluation wouldrequire the knowledge, at the same time, of exact solu-tions of the Schr¨odinger equation with realistic nucleon-nucleon potentials and three-body forces for the Henucleus and for the three-body recoiling system f A − .This system can be either in its ground state, when E = E min = ∣ E He ∣ − ∣ E H ∣ , or unbound with an exci-tation energy ǫ ∗ A − . The description of this latter part d σ [ nb / G e V ] φ [deg] t =-0.15 GeV , x B =0.16, Q = 1.39 GeV Free protonBound protonPRC 98, 045203 (2018)
FIG. 6: The cross section for the bound proton (full line)and for the free proton (dot-dashed line) in the kinematicsreported on the top of the frame, corresponding to data pre-sented in Ref. [55], as a function of the azimuthal angle φ .The precise position of the data and their errors are takenfrom [58]. -0.3-0.2-0.1 0 0.1 0.2 0.3 0 50 100 150 200 250 300 350 A L U I n c o h ( φ ) φ [deg] t = -0.15 GeV , x B = 0.22, Q = 1.82 GeV FIG. 7: The BSA A IncohLU , Eq. (7), as a function of the az-imuthal ange φ , compared to data corresponding to the anal-ysis leading to Ref [8] represents a challenging few-body problem, whose solu-tion is presently unknown. A full realistic calculation ofthe He spectral function is planned and has started but,in this work, for P He (⃗ p, E ) use is made of the modelpresented in Ref. [38, 46]. In that approach, when therecoiling system is in its ground state and E = E min ,an exact description is used in terms of variational wavefunctions for the 4-body [47] and 3-body [48] systems,obtained through the hyperspherical harmonics method[49], within the Av18 NN interaction [29], including UIXthree-body forces [30]. The cumbersome part of the spec-tral function, with the recoiling system excited, is basedon the Av18+UIX interaction, proposed in Ref. [38, 46],an update of the two-nucleon correlation model of Ref.[50]. We note that realistic calculations of GPDs for He,for which an exact spectral function is available, haveestablished the importance of properly considering the E -dependence of the spectral function [32]. To have an ] [GeV Q ) ° ( I n c o h L U A B x ] -t [GeV He, A IncohLU ( K ) , for φ = o : results of this approach(red dots are obtained using the diagonal spectral function as described along the text, blue stars using the momentumdistribution in the so called ”closure approximation” ) compared with data (black squares) [20]. From left to right, the quantityis shown in the experimental Q , x B and t bins, respectively. Shaded areas represent systematic errors. ] [GeV Q ) ° ( I n c o h L U A B x ] -t [GeV FIG. 9: (Color online) Azimuthal beam-spin asymmetry for the proton in He, A IncohLU ( K ) , for φ = o : results of this approach(red dots are obtained using the GK GPD model [39], blue triangles using the MMS model [52] ) compared with data (blacksquares) [20]. From left to right, the quantity is shown in the experimental Q , x B and t bins, respectively. Shaded areasrepresent systematic errors. idea of the importance of a proper treatment of the E -dependence in this process, and, in general, of the draw-back of the use of a less refined nuclear description, in thenext section we will show also results obtained using theso called ”closure” approximation. It consists in evalu-ating the spectral function considering, in the argumentof the delta function in Eq. (11), an average value of theremoval energy, so that the closure of the f A − states canbe used, yielding P Heclosure ( p, E ) = n gr ( p ) δ ( E − E min )+ n ex ( p ) δ ( E − ¯ E ) , (12)where the momentum distribution for the proton withthe recoiling system in its ground or excited state, n gr ( k ) and n ex ( k ) , respectively, have been introduced, with ¯ E the average excitation energy of the recoiling system.A similar approach has been used to model the non-diagonal He spectral function in the description of co-herent DVCS off He, in Ref. [36]. We note that, whenthis approximation is used, also the off-shellness of thestruck proton, governed by Eq. (12), has to be changedaccordingly, i.e. p = M A − √ M ∗ A − + ⃗ p Ð → M − ¯ E − T rec . (13)As we will see in the following, this produces importanteffects in the cross section, due to the fact that the com-ponents of the four momentum of the proton enter scalarproducts present in the relevant scattering amplitudes.For the nucleonic GPD, two models have been used.One is the model of Goloskokov and Kroll (GK) [39], al-ready successfully exploited in the coherent case [36]. Itis worth to remind that the model is valid in principle at Q values larger than those of interest here, in particu-lar at Q ≥ . Nonetheless we have checked thatthe GK model can reasonably describe free proton datacollected in similar kinematical ranges, for example theones in Ref. [51], as it is discussed in the next section(see Fig. 3).The other model is taken from Ref. [52]. It is basedon an original compact version of the double distribu-tion prescription. It is developed at leading twist andat leading order in α s (of course NLO corrections maybe sizable also in the valence region, at moderate energy,see, e.g., the discussion in Ref. [53]). With respect tothe GK model, only the valence region is modified andthe momentum scale evolution is the same. The modelis expected to work in the region − t / Q ≤ .
1, wherefactorization is supposed to work. To obtain the relevantnumbers for that model, use has been made of the virtualaccess infrastructure ”3DPARTONS” [54].
IV. NUMERICAL RESULTS
We can now evaluate the beam spin symmetry (BSA),Eq. (7) , and compare it with the recently published data[8].First of all, let us check if the GK model we used, forvalues of Q smaller than those for which it is supposed towork, Q ≥ , is still describing the available datareasonably well. To this aim, we show in Fig. 3 that, inone of the kinematics presented in Ref. [51] for DVCSoff the free proton, not far from the ones of interest here,a reasonable description of the BSA data, is obtainedcalculating this quantity for the free proton with the GKmodel. We notice that the azimuthal angle φ , used by theexperimental collaboration and exploited here, is relatedto the one previously defined and used in this paper bythe relation φ = π − φ . The relevance of terms of order t / Q , discussed in the previous section, is also shown.In general, the BSA is rather sensitive to changes of thekinematics, to t especially. Data for the free proton arenot available for the kinematics of the experiment underscrutiny so that we have to compare with results of otherexperiments.Then, let us show the results of our model for the dif-ferential cross sections (6) which are used later to cal-culate the BSA. All the cross sections shown here beloware obtained considering a positive electron helicity, asan example.To have a first glimpse at the nuclear effects on therelevant processes, the cross section for the BH processon the free proton (dashed) and on a proton bound in He (full), according to the present treatment, is shownin Fig. 4, as a function of the azimuthal angle φ , in one ofthe kinematical ranges of the data presented in Ref. [55]. The data, corresponding to the full DVCS process offthe free proton, are presented here for illustration only.Relevant nuclear effects are clearly seen. To our knowl-edge, this figure and the next two are the first ones inthe literature where the comparison of cross sections forfree and bound nucleons, with a difference arising from amicroscopic calculation, is presented.In Fig. 5, the cross section for the BH process is com-pared with that obtained including also the only relevantterm, as discussed in Appendix B, of the the interfer-ence between the BH and DVCS processes, for a protonbound in He according to the present treatment, againin the kinematics of Ref. [55], as a function of the az-imuthal angle φ (see Appendix B for the discussion of therelevant term included). It is clearly seen as a relevant φ asymmetry is generated including the DVCS mecha-nism. The data for the free proton are again reportedfor illustration. It is seen that a reasonable descriptionis obtained.In Fig. 6, in the same kinematics of the previous two,the full cross section is shown, for a bound and for a freeproton, to expose the role of the nuclear effects on theproton DVCS cross-section, found to be overall sizable.Let us now present results for the BSA A IncohLU , Eq.(7). This quantity, evaluated using the GK model forthe GPD entering the DVCS part, is shown in Fig. 7,as a function of the azimuthal ange φ , compared to datacorresponding to the analysis leading to Ref [8]. A con-vincing agreement is found, in particular at φ = φ = o ,the fixed value at which the BSA has been extracted andat which it will be shown in the following.The BSA is a function of the azimuthal angle φ andof the kinematical variables Q , x B and t . Due to lim-ited statistics, in the experimental analysis these lat-ter variables have been studied separately with a two-dimensional data binning. The same procedure has beenused in our calculation. For example, each point at agiven x B has been obtained using for t and Q the cor-responding average experimental values, which are re-ported for definiteness in Tables I-III, together with thenumerical values of the calculated theoretical asymme-tries discussed in the following. x B < Q > [GeV ] < t > [GeV ] A GKLU A MMSLU Q and t in x B bins. In Fig. 8 it is seen that, overall, the calculation repro-duces the data rather well in all of these bins. For thisobservable, in most of the cases the present accuracy ofthe data does not allow to distinguish between the fullcalculation and that performed using the closure approx-imation, Eq. (12). In any case, whenever the disagree- Q [GeV ] < x B > < t > [GeV ] A GKLU A MMSLU x B and t in Q bins. t [GeV ] < x B > < Q > [GeV ] A GKLU A MMSLU -0.145 0.213 1.82 0.145 0.094-0.282 0.255 2.13 0.164 0.118-0.490 0.284 2.31 0.190 0.144-1.11 0.308 2.41 0.173 0.140TABLE III: The BSA, obtained using the GK [39] or MMS[52] models, using the nuclear spectral function, for the aver-age values of x B and Q in t bins. ment with the data is sizable, the proper treatment ofthe excitation energy within the spectral function helpsin describing the data. Besides, we note that the agree-ment is not satisfactory only when the GK model is usedin the region of low Q . Indeed, this is evident only in theexperimental points corresponding to the lowest values of Q , x B and t . One should notice that the average valueof Q grows with increasing x B and t (cf. tables I-III),so that a not satisfactory description at low Q affectsalso the first x B and t bins. Actually, the GK model isdesigned to describe the available data for Q ≤ ,e.g at values higher than the typical ones accessed by theCLAS collaboration in the experiment under scrutiny.The problems found using the GK parametrization aretherefore somehow expected. We have therefore repeatedthe calculation using as a nucleonic partonic input themodel MMS, introduced in Ref. [52], briefly described inthe previous section. The comparison of the two resultsis presented in Fig 9, where it is seen that the data favorthe MMS model with respect to the GK one. The successof the MMS model, with parameters chosen precisely tobe realistic in the Q range typical at JLab, is remarkableand points to a solid predictivity of the IA, emphasizing,at the same time, the dependence of the results on thechoice of the nucleonic model. In any case, the resid-ual disagreement, or the problems found using the GKmodel, could be also due to some final state interaction(FSI) effects that in the present IA are not considered.For this reason, a careful analysis of the interplay be-tween the t and Q dependence of the data is required toestablish whether FSI play a relevant role. The presentaccuracy of the data does not allow such an analysis, butthe data expected from the planned future measurementscertainly will. In the light of this discussion, we can con-clude that a careful use of basic conventional ingredientsis able to reproduce the available data. In order to betterunderstand our results, addressing nuclear modificationsof the parton strucure, possibly related therefore to the EMC effect, as an illustration we perform a specific anal-ysis, detailed in what follows.Let us define, in each experimental bin, specific ra-tios to expose the nature of nuclear effects, namely, theratio between the BH-DVCS interference cross sectionfor the proton bound in He and the free one at rest, R I ( K ) , the corresponding quantity for the pure BH pro-cess, R BH ( K ) , and the ratio of the two, R ALU ( k ) , pro-viding the ratio of the bound proton to the free proton BSA in our calculation scheme. These quantities read,respectively R I ( K ) = N I He ( K ) I p ( K ) , (14) R BH ( K ) = N T HeBH ( K ) T pBH ( K ) , (15) R ALU ( K ) = R I ( K ) R BH ( K ) = A IncohLU ( K ) A pLU ( K ) . (16)In the equations above the factor N =∫ exp dE d ⃗ pP He (⃗ p, E ) accounts for the fact that onlya part of the spectral function is selected in a givenexperimental bin. The meaning of the integration space exp is clarified in appendix A . The ratios (14)-(16) at φ = o , using the GK model for the nucleon GPD, areshown in Fig. 10. It is clearly seen that the nucleareffects obtained within the present IA scheme in theratios (14) and (15) are rather sizable, while the effectsare dramatically reduced in the ”super-ratio” (16). Thisfact points to relevant conventional nuclear effects in thepure BH and pure DVCS processes, which are anyhow ofa similar origin, so that they cancel out to a large extentin the ratio.Something similar happens when the closure approx-imation is applied to estimate the nuclear effects. InFigs. 11 and 12 it is seen that, in some cases, the differ-ence between the results of the full calculation, performedconsidering the distribution of the removal energy withinthe spectral function, and of the one obtained with theclosure approximation, is rather sizable in the ratio (14)and (16). In Fig. 14 is seen instead that the effect is dra-matically reduced in the ratio of these two quantities, thesuper-ratio (16), showing that the effects in the numer-ator and in the denominator basically compensate eachother.The dots shown in this latter figure are related to an-other intriguing observation, obtained following a proce-dure used by the experimental collaboration to exposenuclear effects [8]. Our BSA for the proton bound in He has been divided by the corresponding quantity fora free proton at rest, using the GK model, and plotted asa function of x B . It is seen that the results underestimatethose obtained in the analysis of the experimental collab-oration. This points to interesting effects not included inthe present IA scheme, either at the parton level (mediummodifications of the parton structure due exotic effects,such as dynamical off-shellness) or of conventional origin, ] [GeV Q R a t i o s B x ] -t [GeV BH /R Int R Int R BH R FIG. 10: (Color online) The ratios (14) (red dots), (15) (blue triangles), (16) (black squares), at φ = o and using the GKmodel for the nucleon GPD. From left to right, the quantity is shown in the experimental Q , x B and t bins, respectively. ] [GeV Q I n t R B x ] -t [GeV ClosureSpectral function
FIG. 11: (Color online) The ratio (14) (blue triangles), obtained using either the spectral function (red dots) or the closureapproximation (black stars), at φ = o and using the GK model for the nucleon GPD. From left to right, the quantity is shownin the experimental Q , x B and t bins, respectively. such as FSI, not yet included in the calculation. In Fig.14 we show the results obtained with the spectral func-tion and with either the GK or the MMS model, almostindistinguishable between themselves. Clearly, while inthe result for A LU the difference between the differentmodels was in some cases sizable, in this specific quan-tity, which can be built in principle from data taken forprotons in He and for the free proton at the same kine-matics, this ratio seems to be be essentially independenton the model used for the nucleon. In general nuclear ef-fects are found to be rather small in IA for this quantity,which seems therefore very promising to expose exoticnuclear effects.To dig further into this interesting result and to real-ize to what extent a medium modification of the partonstructure is predicted by our calculation, we observe that the ratio (16) can be sketched as follows A IncohLU A pLU = I He I p T pBH T HeBH ∝ ( nucl.ef f. ) I ( nucl.ef f. ) BH , (17)i.e., it is proportional to the ratio of the nuclear effects onthe BH-DVCS interference to the nuclear effects on thepure BH cross section. If the nuclear dynamics modifies I and the T BH in a different way, the effect can be bigeven if the parton structure of the bound proton doesnot change appreciably. We analyze this occurrence inFig. 15, where, together with the ratio (16), we showtwo others quantities, as functions of x B . One of them,labelled ”pointlike”, is obtained considering in the ratiopointlike protons. It is seen that, at low x B , where sizableeffects are found within our IA approach, the big effect isstill there. Besides, in the same figure we show an ”EMC-like” quantity, i.e., a ratio of a nuclear parton observable,0 ] [GeV Q B H R B x ] -t [GeV ClosureSpectral function
FIG. 12: (Color online) The ratio (15), evaluated using the spectral function (blue triangles) and the closure approximation(black stars) for φ = o and using the GK model for the proton GPD. From left to right, the quantity is shown in theexperimental Q , x B and t bins, respectively. ] [GeV Q p L U / A I n c o h L U A B x ] -t [GeV Spectral FunctionClosure
FIG. 13: (Color online) The ratio (16), evaluated using the spectral function (blue triangles) and the closure approximation(black stars) for φ = o and using the GK model for the proton GPD. From left to right, the quantity is shown in theexperimental Q , x B and t bins, respectively. The results is compared with the same ratio estimated by the EG6 collaboration(black squares). [56]. the imaginary part of the CFF, to the same observablefor the free proton: R EMC − like = N ∫ exp dE d ⃗ p P He (⃗ p, E ) I m H ( ξ ′ , ∆ ) I m H ( ξ, ∆ ) . (18)One should notice that this ratio would be one if nu-clear effects in the parton structure were negligible. Asseen in Fig. 15, this ratio is close to one and it resemblesthe EMC ratio, for He, at low x B (cf the data in Ref.[57]). Since in our analysis the inner structure of thebound proton is entirely contained in the CFF and thisproduces a mild modification, the sizable effect found forthe ratio (17) for the first x B bin, shown in Fig. 15, haslittle to do with the modifications of the parton content driven by the IA and analyzed here. Rather, the effect isdue to a different dependence on the 4-momentum com-ponents, affected by nuclear effects, of the interferenceand BH terms for the bound proton.It will be very interesting to study the ratio (16) whenconsistently collected data will be available for the protonand for He, to look for effects to be ascribed to exoticmodifications of the parton content or to a complicatedconventional behaviour, beyond IA.
V. CONCLUSIONS
An impulse approximation analysis, based on state-of-the-art models for the proton and nuclear structure, using1 ] [GeV Q p L U / A I n c o h L U A B x ] -t [GeV GKMMS13
FIG. 14: (Color online) The ratio (16) of the azimuthal beam-spin asymmetry for the proton in He, A Inco,hLU ( K ) , to thecorresponding quantity for the free proton at rest, for φ = o , using for the proton GPD the GK model [39] (red dots), andthe MMS model [52] (blue triangles) compared with the ratio estimated by the EG6 collaboration (black squares) [56]. Fromleft to right, the quantity is shown in the experimental Q , x B and t bins, respectively. B x ) ° (90 pLU /A IncohLU
A ) ° (90 pLU /A IncohLU
Pointlike A
EMC-like R FIG. 15: (color online) The ratio A IncohLU / A pLU , Eq. (17) (reddots), compared to the result obtained with pointlike protons(black diamonds) and to the EMC-like ratio Eq. (18) (bluecrosses). a conventional description in terms of nucleon degrees offreedom, has been thoroughly described. Recent data onincoherent DVCS off He are overall well reproduced.The results can be summarized as follows:i) the main experimental observable, the only onemeasured so far, the BSA, turns out to be sensitive tothe nucleonic model used, in particular at low values of Q ; parametrizations for generalized parton distributionsbased on high Q data seem to have limited predictivepower in the low Q sector;ii) given the present accuracy of the data, the beamspin asymmetry is mildly sensitive to the details of thenuclear model used in the calculation, as it can be argued using a spectral function or its closure approximation.Results obtained within the spectral function are anywaycloser to a good description of the data;iii) the behaviour at low Q could point also to possibleFSI effects, to be investigated, or to other quark andgluon effects. The present accuracy of the data does notallow a further analysis towards this direction;iv) a careful study of nuclear effects in the differentprocesses contributing to the BSA, the BH in the denom-inator and the DVCS-BH-Interference in the numerator,has exposed sizable effects; besides, a clear difference isfound, in some kinematical points, if the spectral functionor the closure approximation are used. The separatedmeasurements of these contributions, which correspondto those of the differential cross sections and not only totheir ratio, would be very interesting and deserve to beattempted in the future experiments;v) all these effetcs actually basically disappear in theratio of the interference to the BH contributions. In ourIA approach, the latter ratio represents that between theBSA for incoherent DVCS off He and coherent DVCSoff the free proton. Its stability against different nuclearand nucleon models, found in this study, demonstratesthat it can be used to expose interesting exotic effectsbeyond the ones included in IA. We can preliminarly as-sert that our calculation of this quantity overestimatesthe estimate of the experimental collaboration.We would conlcude that, given the present accuracy ofthe data, there is no point in going beyond the exhaustiveanalysis presented here. New tagged measurements withdetection of residual nuclear final states, planned at JLab[22] and under study for the future EIC, will shed morelight to this respect. The presence of specific nuclearfinal states in these processes will also make possible aprecise evaluation of FSI in terms of few-body realisticwave functions, allowing for a conclusive comparison with2data.While a benchmark calculation in the kinematics of thenext generation of precise measurements will require animproved treatment of both the nucleonic and the nuclearparts of the calculation, such as a realistic evaluation ofthe diagonal spectral function of He, the straightfor-ward approach proposed here can be used as a workableframework for the planning of future measurements. Pos-sible exotic quark and gluon effects in nuclei, not clearlyseen within the present experimental accuracy, will beexposed by comparing forthcoming data with our con-ventional results. To this aim, a novel Montecarlo eventgenerator [59], tested so far with our model of the coher-ent process, will be used to simulate incoherent DVCSoff He, described within the approach presented here,to plan the next generation of experiments at JLab andat the future EIC.
Acknowledgements
We warmly thank R. Dupr´e and M. Hattawy for manyhelpful discussions and technical information on the EG6experiment. S.F. thanks P. Sznajder and C. Mezragfor some tuition on the use of the virtual access infras-tructure 3DPARTONS, funded by the European Union’sHorizon 2020 research and innovation programme un-der grant agreement No 824093. This work was sup-ported in part by the STRONG-2020 project of the Eu-ropean Union’s Horizon 2020 research and innovationprogramme under grant agreement No 824093, WorkingPackage 23, ”GPDS-ACT” and by the project “DeeplyVirtual Compton Scattering off He”, in the programmeFRB of the University of Perugia.
Appendix A: The convolution formula
Let us start considering the cross section dσ ± appearing in Eq. (3). It can be written in a generic frame, for theincoherent channel of the DVCS process under scrutiny, namely e ( k ) A ( P A ) → e ( k ′ ) N ( p N ) γ ( q ) X ( p X ) off a nucleartarget A , in the following way ( dσ ± ) Inc = ( π ) P A ⋅ k ∑ σ ∑ N ∑ X ∣ A ± ∣ δ ( P A + k − k ′ − p X − p N − q ) d ˜ p X d ˜ k ′ d ˜ q d ˜ p N (A1)where the dynamical information is encoded in the squared amplitude. The latter is given by three different contri-butions, namely ∣ A ∣ = ∣ A DV CS ∣ + ∣ A BH ∣ + I BH − DV CS . A generic phase-space integration volume reads d ˜ k ≡ d k ( π ) k . (A2)In Eq. (A1), the sums are extended to the inner nucleons of type N in the target, to the polarization σ of the finaldetected proton and to the undetected nuclear system X . The status f of the latter is identified by a set { α f } ofdiscrete quantum numbers and by the excitation energy E f , for which discrete and continuous values are possible.One has therefore, in Eq. (A1), ∑ X d ˜ p X → ∑ f ∑ { α } f ⨋ E f ρ ( E f ) d ˜ p f , (A3)where ρ ( E f ) is the density of final states. The amplitudes A BH and A DV CS appearing in Eq.(A1) are given by thecontraction of a leptonic tensor ( L νDV CS / Q and L µνBH / ∆ for DV CS and BH , respectively) with the appropriatehadronic tensor. For a generic DVCS process of a target A with initial(final) polarization S ( S ′ ) reads T DV CSµν ( P A , ∆ , q, S, S ′ ) = ∫ dre iq ⋅ r ⟨ P ′ A S ′ ∣ T { ˆ J µ ( r ) ˆ J ν ( )}∣ P A S ⟩ . (A4)Since a convolution formula with the same structure can be obtained for any of the DV CS , BH and interferenceterms exploiting the same steps, to fix the ideas in what follows we specify our treatment to the DV CS part. Letus consider therefore the scattering amplitude of the incoherent DVCS process off an He target, i.e e ( k ) He ( P A ) → e ( k ′ ) N ( p N ) γ ( q ) X ( p X ) A A,N,fDV CS = − ie ∑ λ ′ ¯ u ( k ′ , λ ′ ) γ µ u ( k, λ ) Q T µνA,N,f ǫ ∗ ν ( q ) = ǫ ∗ ν ( q ) Q L DV CSµ ( λ ) T µνA,N,f , (A5)where it appears the hadronic tensor T A,Nµν , defined in terms of T A,Nµν = ∫ d re − iq ⋅ r H A,Nµν , (A6)3being H A,Nµν the matrix element of ˆ O N = T { ˆ J Nµ ( x ) ˆ J Nν ( )} properly evaluated between the states describing the initialand the final nucleon N in the nucleus A , respectively. Here and in the following, we are assuming that the interactiongoes through the nucleons in the nucleus, which are the only degrees of freedom in the present Impulse Approximation(IA). Disregarding for the moment the integration on x , let us focus on the matrix element H A,N,fµν .We will use in the following the standard covariant normalization of the states ⟨ pσ ∣ p ′ σ ′ ⟩ = ( π ) p δ (⃗ p ′ − ⃗ p ) δ σ,σ ′ (A7)and the notation ∑ p = ∫ d ˜ p is used. The matrix element in Eq. (A6) is therefore H A,N,fµν = ⟨ p N σ, p f { α f } E f ∣ ˆ O N ∣ P A ⟩ , (A8)where the final state contains the detected nucleon with momentum p N and polarization σ and the A −
1- body systemdescribed by a set of quantum numbers { α f } , whose constituents are moving with momenta p f . Let us insert to theleft and to the right-hand sides of the hadronic operator two complete sets of states; the first set corresponds to thenucleon N , supposed free, interacting with the virtual photon, whose completeness reads ∑ p ′ N σ ′ ∣ p ′ N σ ′ ⟩⟨ p ′ N σ ′ ∣ = , (A9)while the completeness of the second set of states, describing the hadronic undetected system, is given by: ∑ { α f } ⨋ E f ρ ( E f ) ∑ p f ∣ p f { α f } E f ⟩⟨ p f { α f } E f ∣ = . (A10)Now let us use the IA. This means that the interaction goes only through the nucleons, as already said, and that thefinal state can be written as a tensor product ∣ p N σ, p f { α f } E f ⟩ = ∣ p N σ ⟩ ⊗ ∣ p f { α f } E f ⟩ , (A11)i.e., the interactions between the particles in the final state (FSI) have been neglected. At the light of these facts, wearrive to the following formula H A,N,fµν = ∑ { α ′ f } ⨋ E ′ f ρ ( E ′ f ) ∑ p ′ f ∑ p ′ N σ ′ ⟨ p N σ ∣⟨ p f { α f } E f ∣ ˆ O N ∣ p ′ f { α ′ f } E ′ f ⟩∣ p ′ N σ ′ ⟩⟨ p ′ N σ ′ ∣⟨ p ′ f { α ′ f } E ′ f ∣ P A ⟩ . (A12)Now, assuming in IA that the one-body operator ˆ O N acts only on the nucleonic states, we can consider the normal-ization (A7) to perform trivially some integrals, obtaining the following form: H A,Nµν = ∑ p ′ N σ ′ ⟨ p N σ ∣ ˆ O N ∣ p ′ N σ ′ ⟩⟨ p f { α f } E f ∣⟨ p ′ N σ ′ ∣ P A ⟩ . (A13)A relevant issue has to be discussed at this point. Since relativistic nuclear wave functions for three and four bodysystems are not at hand, in the following we will be forced to use non relativistic wave functions in the overlaps ofthe above equation. Therefore, we will use for the states in the overlap a non relativistic normalization ⟨⃗ ps ∣⃗ p ′ s ′ ⟩ = δ (⃗ p − ⃗ p ′ ) δ ss ′ . (A14)For the same reason, in the overlap we can disentangle the global motion from the intrinsic one ∣ p f { α f } E f ⟩ = ∣ Φ { α f } E f ( p f ′ , σ f ′ ) ; p x s x ⟩ , (A15)where Φ { α f } E f represents the intrinsic motion of the final system, described by A − A − p f ′ and intrinsic quantum numbers σ f ′ , while p x and s x specify the state of the centerof mass of the A − ∣ Φ { α f } E f ⟩ instead of ∣ Φ { α f } E f ( p f ′ , σ f ′ )⟩ ). In this way the overlap becomes {⟨ Φ { α f } E f p x s x ∣}⟨⃗ p N ′ σ ′ ∣ ⃗ P A ⟩ = [( π ) / ] √ M A √ p ′ N √ p x √ p f ⟨⃗ p ′ N σ ′ , Φ { α f } E f ∣ Φ A ⟩ δ ( ⃗ P A − ⃗ p x − ⃗ p ′ N ) δ σ ′ , − σ f − s x , (A16)4where the momentum delta function accounts for the center of mass free motion and Φ A is the intrinsic wave functionof the target nucleus. The other delta function yields a formal condition to be fulfilled between the discrete quantumnumbers appearing in the overlap. The terms at the beginning of the r.h.s. account for the chosen non relativisticnormalization of the states Eq. (A14). In this way, from Eq. (A13) we get H A,N,fµν = ∑ σ ′ ∑ p ′ N [( π ) / ] √ M A √ p ′ N √ p x √ p f ⟨ p N σ ∣ ˆ O N ∣ p ′ N σ ′ ⟩⟨⃗ p ′ N σ ′ , Φ { α f } E f ∣ Φ A ⟩ δ ( ⃗ P A − ⃗ p x − ⃗ p ′ N ) δ σ ′ , − σ f − s x , so that the complete expression for the hadronic tensor in the incoherent DVCS channel becomes: T A,Nµν = ∑ σ ′ ∑ p ′ N ∫ dre iq ⋅ r [( π ) / ] √ M A √ p ′ N √ p x √ p f ⟨ p N σ ∣ ˆ O N ∣ p ′ N σ ′ ⟩⟨⃗ p ′ N σ ′ , Φ { α f } E f ∣ Φ A ⟩ δ ( ⃗ P A − ⃗ p x − ⃗ p ′ N ) δ σ ′ , − σ f − s x , which can be inserted in the DVCS amplitude Eq. (A5) obtaining A A,N,f,λDV CS = − ie ∑ λ ′ ¯ u ( k ′ , λ ′ ) γ µ u ( k, λ ) Q ∑ σ ′ ∫ dre iq ⋅ r ∑ p ′ N ⟨ p N σ ∣ T ( ˆ J µN ( r ) ˆ J νN ( ))∣ p ′ N σ ′ ⟩ (A17) × [( π ) / ] √ M A √ p ′ N √ p x √ p f ⟨ p N σ ∣ ˆ O N ∣ p ′ N σ ′ ⟩⟨⃗ p ′ N σ ′ , Φ { α f } E f ∣ Φ A ⟩ δ ( ⃗ P A − ⃗ p x − ⃗ p ′ N ) δ σ ′ , − σ f − s x ǫ ∗ ν . Now, let us consider the squared amplitude appearing in the expression of the cross section, Eq. (A1) ∣ A A,N,f,λDV CS ∣ = ( π ) M A ∑ σ ′′ ∑ σ ′ ∑ p ′ N ∑ p ′′ N p x p f √ p ′ N √ p ′′ N ∣ A N,λDV CS ( p N , p ′ N , σ, σ ′ )∣ ⟨⃗ p ′ N σ ′ , Φ { α f } E f ∣ Φ A ⟩ (A18) × ⟨ Φ A ∣ ⃗ p ′′ N σ ′′ , Φ { α f } E f ⟩ δ ( ⃗ P A − ⃗ p x − ⃗ p ′ N ) δ ( ⃗ P A − ⃗ p x − ⃗ p ′′ N ) δ σ ′ , (− σ f − s x ) δ σ ′′ , (− σ f − s x ) , where the squared DVCS amplitude off a nucleon is given by ∣ A N λDV CS ( p N , p ′ N , σ ′ )∣ = ∑ σ ∣ A N λDV CS ( p N , p ′ N , σ, σ ′ )∣ (A19) = − g µν Q ∑ σ ∫ dr ′ e − iq ⋅ r ′ ∫ dre iq ⋅ r L ρDV CS ( λ ) L DV CS ( λ ) σ † ⟨ p N σ ∣ ˆ O Nµν ∣ p ′ N σ ′ ⟩⟨ p ′ N σ ′ ∣ ˆ O N † ρσ ∣ p N σ ⟩ . In this way, substituting the obtained expression in the cross section (A1), taking into account that, due to theseparation of the global motion from the intrinsic one in the A − ∑ X dp X → ∑ x ∑ f ′ ∑ { α } f ⨋ E f ρ ( E f ) d ˜ p x d ˜ p f ′ , (A20)and using the delta functions we arrive to ( dσ A ) Inc = ( π ) P A ⋅ k ∑ N ∑ σ ′ ∑ x ∑ f ′ d ⃗ p x d ⃗ p f ′ ∑ { α f } ⨋ E f ρ ( E f )∣ A N,λDV CS ( p N , p N ′ , σ ′ )∣ M A p ′ N (A21) × ⟨⃗ p ′ N σ ′ , Φ { α f } E f ∣ Φ A ⟩⟨ Φ A ∣⃗ p ′ N σ ′ , Φ { α f } E f ⟩ δ ( P A + k − k ′ − p X − p N − q ) d ˜ k ′ d ˜ q d ˜ p N , where one has to read σ ′ ≡ − ( σ f + s x ) . Finally, defining the diagonal spectral function as P HeN (⃗ p N , E ) = ∑ { α f } ∫ d ⃗ p f ′ ρ ( E )⟨⃗ p ′ N σ ′ , Φ { α f } E ∣ Φ A ⟩⟨ Φ A ∣⃗ p ′ N σ ′ , Φ { α f } E ⟩ , (A22)where the standard removal energy definition E ≡ E f = ∣ E A ∣ − ∣ E A − ∣ + E ∗ f has been adopted, the cross section (A21)can be rewritten in the following compact way dσ λInc = P A ⋅ k ∑ σ ′ ∑ N ⨋ E ∫ d ⃗ pP HeN (⃗ p, E ) M A p ∣ A N,λDV CS ( p, p N , σ ′ )∣ ( π ) δ ( P A + q − p N − q − p X ) d ˜ k ′ d ˜ p N d ˜ q = P A ⋅ k ∑ σ ′ ∑ N ⨋ E ∫ d ⃗ pP HeN (⃗ p, E ) M A p ∣ A N,λDV CS ( p, p N , σ ′ )∣ ( π ) δ ( p + q − p N − q ) d ˜ k ′ d ˜ p N d ˜ q (A23)5where we used that ⃗ p X = ⃗ p f + ⃗ p x and that ⃗ p f = ∑ f ′ ⃗ p f ′ =
0. Besides, we also made use of the condition given by (A16),i.e ⃗ p N = ⃗ P A − ⃗ p x ; in addition to this, in the spirit of the IA, we have energy conservation at the nuclear vertex, sothat p N = P A − p x . In the last step we changed the name of the integration variables defining a four momentum of anoff-shell nucleon, p = ( p , ⃗ p ) .Now, keeping in mind that for a coherent DVCS process off a single nucleon the analogous cross section reads dσ λ,NCoh = p ⋅ k ∣ A N,λDV CS ( p, p N , σ ′ )∣ ( π ) δ ( p + q − p N − q ) d ˜ k ′ d ˜ p N d ˜ q (A24)we can rewrite Eq. (A23) as a clear convolution formula between the spectral function P HeN of the inner nucleonsand the cross section for a DVCS process off an off-shell nucleon, namely dσ λInc = ∑ σ ∑ N ⨋ E ∫ d ⃗ p p ⋅ kP A ⋅ k M A p P HeN (⃗ p, E ) dσ λ,NCoh . (A25)If the above equation is evaluated in the target rest frame, it becomes dσ λInc = ∑ σ ∑ N ⨋ E ∫ d ⃗ p p ⋅ kp E k P HeN (⃗ p, E ) dσ λ,NCoh . (A26)We have now to obtain a workable expression for the differential cross section to be used in the actual calculationand to be related to experimental data for the beam spin asymmetry. To this aim, let us rewrite the invariant phasespace ( LIP S ) for the coherent cross section for a moving nucleon, Eq. (A24), that reads explicitly
LIP S = d ˜ k ′ d ˜ p N d ˜ q = d k ′ E ′ ( π ) d p N E ( π ) d q ν ′ ( π ) . (A27)Let us choose, as everywhere in this paper, the target rest frame where the spacelike virtual photon propagates alongthe negative z-axis,i.e q = ( k − k ′ ) = ( ν, , , − q z ) with Q = − q . In this frame, the kinematical variables are (it isassumed that ⃗ k lies in the xz plane): k = ( E k , E k sin θ e , , E k cos θ e ) (A28) k ′ = ( E ′ , ⃗ k ′ ) (A29) P A = ( M A , ⃗ ) (A30) p N = ( E , ∣⃗ p N ∣ sin θ N cos φ N , ∣⃗ p N ∣ sin θ N sin φ N , ∣⃗ p N ∣ cos θ N ) (A31) q = ( ν ′ , ⃗ q ) (A32)We have to specify the components of the 4-momentum of the bound nucleon. In this framework, the energy conser-vation in the electromagnetic nuclear vertex yields p = M A − p x = M A − √ M ∗ A − + ⃗ p A − ≈ M − E − K R . (A33)The interacting nucleon has 3-momentum ⃗ p ( ϑ is the polar angle of ⃗ p , so that the angle between ⃗ p and ⃗ q is π − ϑ )and K R is the kinetic energy of the recoiling A − x B = Q /( M ν ) , ∆ = ( q − q ) , φ N , Q . In addition to these variables, in the following we will makeuse of the quantity: ǫ = M x B / Q . The LIPS, in terms of these variables, read LIP S = J ( p N → ∆ ) d ∆ d cos θ N dφ N Q ( π ) M E k x B dQ dx B dφ k ′ d q ν ′ ( π ) , (A34)where the term J ( p N → ∆ ) is proportional to the jacobean of the transformation and reads, since the processtakes place on a moving nucleon, J ( p N → ∆ ) = ( π ) ∣ ∣⃗ p N ∣ ∣⃗ p ∣ cos θ ˆ pp N E − p ∣⃗ p N ∣ ∣ , (A35)where6cos θ ˆ pp N = cos θ N cos ϑ + sin θ N sin ϑ cos ( φ N − ϕ ) . (A36)Substituting Eq. (A34) in Eq. (A23), using the delta function on the three-momenta to obtain ⃗ q = ⃗ p + ⃗ q − ⃗ p N ,and using this result in the delta function on the energy variables to integrate on cos θ N , one finally obtains the crosssection in the nuclear rest frame dσ λInc dx B dQ d ∆ dφ N = Q E k M ( π ) x B ∑ σ ∑ N ⨋ E ∫ exp d ⃗ pP HeN (⃗ p, E ) × ∣ A N,λDV CS ( p, p N , σ )∣ G ( p, ∣⃗ p N ∣ , K ) . (A37)In the equation above, we have defined the set of kinematical variables K = { x B = Q /( M ν ) , Q , t, φ } and G ( p, ∣⃗ p N ∣ , K ) = p ∫ ( π ) δ ( p − p N − q + q ) J ( p N → ∆ ) d q ( π ) ν ′ d cos θ N = π ∣ ∣⃗ p N ∣(∣⃗ p ∣( sin ϑ cot ¯ θ N cos ( φ N − ϕ ) − cos ϑ ) − q z ) ∣ J ( cos ¯ θ N ) , (A38)where J ( cos ¯ θ N ) is the expression J ( p N → ∆ ) evaluated for cos ¯ θ N , which is obtained from the energy conservationcondition √∣⃗ p ∣ + ∣⃗ p N ∣ + ∣ q z ∣ − ∣⃗ p ∣∣⃗ p N ∣ cos θ pp N − ∣⃗ p N ∣ q z cos θ N + ∣⃗ p ∣ q z cos ϑ − p + E − ν = , (A39)where Eq. (A36) is exploited. We note that the quantity ∣⃗ p N ∣ can be obtained from the relation∆ = ( p N − p ) = M + p − ∣⃗ p ∣ − p √ M + ∣⃗ p N ∣ + ∣⃗ p N ∣∣⃗ p ∣ cos θ ˆ pp N , (A40)where the expression for the angle between ⃗ p and ⃗ p N is given by Eq. (A36). The values of cos ¯ θ N and ∣⃗ p N ∣ to beconsidered in the following are obtained through the numerical solution of the system of equations (A39) and (A40).In order to have a clear comparison between our cross section and that for a DVCS process off a proton at rest, i.e. dσ λ r est dx B dQ d ∆ dφ N = α x B y πQ √ + ǫ ∣ A DV CS e ∣ , (A41)let us rewrite Eq. (A37) in the following way, corresponding to Eq. (6) dσ λInc dx B dQ d ∆ dφ N = ∑ N ⨋ E ∫ exp d ⃗ p P HeN (⃗ p, E )∣ A N,λDV CS ( p, p N , K )∣ g ( E, ⃗ p, K ) , (A42)where g ( E, ⃗ p, K ) = α Q π E k M x B e G ( p, ∣⃗ p N ∣ , K ) (A43)and the sum over the proton polarization in Eq. (A37) has been absorbed by the squared amplitude. The label exp in the above equation describes the fact that the integration region is restricted to the components of ⃗ p and to thevalues of E fulfilling the conditions (A39) and (A40). Appendix B: Scattering amplitudes for the proton bound in He In this appendix we report the expression to be used for the amplitudes relevant to photon-electroproduction off abound off-shell proton in He. This will be achieved generalizing the result obtained for a free proton at rest. Let usrecall first the main formalism for that case.7
1. Formalism for the proton in the rest frame.
Let us study coherent DVCS (e + p → e’+ γ +p’) off a proton at rest, with 4-momentum p = ( M, ⃗ ) . Using thenotation and the reference frame discussed in the text and in the previous appendix, the general cross section, dσ = p ⋅ k ∣ T ∣ d k ′ E ′ ( π ) d p N E ( π ) d q ν ′ ( π ) δ ( p + k − k ′ − p N − q ) , (B1)with ∣ T ∣ = T BH + T DV CS + I BH − DV CS . Here and in the following, if not differently stated, we take into account termsof order ∆ Q , ǫ with ǫ = Mx B Q , so that the virtual photon and the final photon have 4-momentum components q = ( Qǫ , , , − √ + ǫ Qǫ ) ,q = ( Qǫ + ∆ M )( , − sin ( θ γ ) cos ( φ N ) , − sin ( θ γ ) sin ( φ N ) , cos ( θ γ )) , (B2)respectively, and the struck proton has final momentum (A31) with ∣⃗ p N ∣ = ¿ÁÁÀ − ∆ ( − ∆ M ) , cos θ N = − ǫ Q ( − ∆ / Q )− x B ∆ x B M ∣⃗ p N ∣√ + ǫ . We note that the electron scattering angle is given by cos θ e = − + yǫ / √ + ǫ , and we remindthat P = p + p N , q = q + q .In the following, we will review the computation of the BH and Interference amplitudes for the proton at rest, andtheir decomposition in Fourier harmonics depending on φ N , which turns out to be equal to φ in our framework. Inthe following section of the Appendix, we will generalize these expressions to describe a moving, bound proton. Wedo not treat the pure DVCS process because it is expected to be very small in the JLab kinematics of interest hereand it has been neglected in our analysis. a. Bethe-Heitler term The amplitude corresponding to the diagrams in Fig. 2 can be computed exactly starting from T BH = e ∆ ǫ ∗ µ ( q ) ¯ u ( k ′ , s ′ )( γ µ / k − / ∆ γ ν + γ ν / k ′ + / ∆ γ µ ) u ( k, s ) J ν . (B3)The φ dependence of the amplitude comes from the lepton propagators (cf. Fig. 2) which read: P ( φ ) = ( k ′ + ∆ ) Q = + k ⋅ ∆ Q = − y ( + ǫ ) ( J + K cos ( φ )) , (B4) P ( φ ) = ( k − ∆ ) / Q = ∆ − k ⋅ ∆ Q = + ∆ Q + y ( + ǫ ) ( J + K cos ( φ )) , (B5)where we have rewritten the scalar product k ⋅ ∆ in terms of the following quantities: J = ( − y − yǫ )( + ∆ Q ) − ( − x )( − y ) ∆ Q , (B6)˜ K = − ∆ Q ( − x )( − y − y ǫ )( − Q ∆ ( − x B )( −√ + ǫ )+ ǫ x B ( − x B )+ ǫ ) (B7) [√ + ǫ + x B ( − x B )+ ǫ ( − x B ) ( ∆ Q − ( − x B )( −√ + ǫ )+ ǫ x B ( − x B )+ ǫ ))] . (B8)Ignoring the electron mass, Eq. (B3) yields: ∣ T BH ∣ = e ∆ ∑ s ′ ,S ′ ( − g µµ ′ ) L † µν L µ ′ ν ′ J ν J † ν ′ = e ∆ J BHν ′ ν L νν ′ BH , (B9)8where, in the last step, the hadronic and the leptonic tensors obtained summing over the final proton and electronpolarizations, S ′ ans s ′ , respectively, read J µνBH = [ F ( ∆ ) + ( F ( ∆ ) + F ( ∆ )) − ∆ M F ( ∆ )]( p ν p µN + p µ p νN ) + ∆ ( F ( ∆ ) + F ( ∆ )) g µν + ( F ( ∆ ) − ( F ( ∆ ) + F ( ∆ )) − ∆ M F ( ∆ ))( p µ p ν + p µN p νN ) , (B10)where F and F are the nucleonic Dirac and Pauli form factors, and L µνBH = Q P ( φ ) P ( φ ) { [ k ⋅ ∆ + Q ( − ∆ Q )] ( k ′ ν q µ + k ′ µ q ν ) − Q ( − ∆ Q )( k ν k ′ µ + k µ k ′ ν ) +− g µν (( k ′ ⋅ q ) + ( k ⋅ q ) ) − ∆ Q g µν − k ′ µ k ′ ν ( k ⋅ q ) + k µ k ν ( k ′ ⋅ q ) + ( k ⋅ ∆ )( k ν q µ + k µ q ν )} . (B11)Contracting the above two tensors, one gets T BH = e ( + ǫ ) x B y ∆ P ( φ ) P ( φ ) { c ( ¯ K ) + c ( ¯ K ) cos ( φ ) + c ( ¯ K ) cos ( φ )} . (B12)where ¯ K = { x B , ∆ , Q , M } accounts for the dependence of the coefficients c i upon the kinematical invariants of theprocess, explicitely given, e.g., in Ref. [41]. b. Interference term Since it is linear in the CFFs and allows the experimental extraction of these functions, the interference term I BH − DV CS = R e [ T DV CS T ∗ BH ] (B13)is the most interesting quantity for GPDs phenomenology. The interference amplitude, in terms of leptonic andhadronic tensors, reads I BH − DV CS = e ∆ q ( − g µµ ′ ) ∑ S ′ s ′ ( L DV CSν L BHµ ′ ρ T µν J ρ † + c.c ) = e ∆ q ( − g µµ ′ ) ∑ S ′ ( L νµ ′ ρ T µν J ρ † + c.c ) . (B14)The amplitude of the pure DVCS process, T DV CS , depicted in Fig. 1, is related to the DVCS hadronic tensor T µν given by the time-ordered product of the electromagnetic currents j µ ( z ) = e ∑ q ǫ q ¯ ψ q ( z ) γ µ ψ q ( z ) of quarks with afractional charge ( ǫ q ) sandwiched between hadronic states with different momenta (see, for details, Ref. [41]). Themost general expression for the hadronic tensor T µν , which can be decomposed in a complete basis of CCFs F that,up to twist three, reads F ( ξ, ∆ , Q ) = { H , E , ˜ H , ˜ E , H + , E + , ˜ H + , ˜ E + } , (B15)has been worked out in Ref. [41] and, at leading twist, for an unpolarized target, at JLab kinematics, can beapproximated as T µν ≃ − P µσ g στ P τν q ⋅ V P ⋅ q , (B16)with the projector operator P µν = g µν − q µ q ν q ⋅ q , (B17)which ensures current conservation, since q µ P µν =
0, and V ρ = P ρ q ⋅ hq ⋅ P H ( ξ, ∆ ) + P ρ q ⋅ eq ⋅ P E ( ξ, ∆ ) . (B18)9The above expression is given in terms of CFFs and Dirac bilinears, defined as follows [41] h ρ = ¯ u ( p N , S ′ ) γ ρ u ( p , S ) , (B19) e ρ = ¯ u ( p N , S ′ ) iσ ρν ∆ ν M u ( p , S ) . (B20)Using (B16) - (B18) , a term appearing in Eq. (B14), after summation over the final proton polarizations, can beeffectively cast in the following way ∑ S ′ q ⋅ VP ⋅ q J ρ † + c.c = P ρ [ C intunp ( F )] + q ρ ∆ Q C int,vecunp ( F ) , (B21)where we introduced the following combination of CFFs C intunp ( F ) = F H ( ξ, ∆ ) − ∆ M F E ( ξ, ∆ ) , (B22) C int,vecunp ( F ) = ξ ( F + F )( H ( ξ, ∆ ) + E ( ξ, ∆ )) . (B23)As everywhere in this paper, the dependence of the CFFs on the scale Q is omitted. After contracting the leptonicand the hadronic tensors, the interference term can be decomposed in harmonics, i.e. I BH − DV CS = e y x B ∆ P ( φ ) P ( φ ) ( c I + ∑ n = c I n cos ( φ ) + s I n sin ( nφ )) . (B24)As it can be read in the expressions explicitely given in Ref. [41], the only terms not suppressed at JLab kinematicsare c I and s I , with the latter clearly dominating the former. Besides, in the BSA, only s I , linear in λ , appears.We therefore consider it as the only relevant contribution to the interference. In particular, it turns out that s I depends only on the combination of CFFs given in (B22), with the term proportinal to H clearly dominating at JLabkinematics. Therefore in the following we consider H as the only relevant CFF. For later convenience, we notice thatthe only part of the leptonic tensor in Eq. (B14) which is ontributing to the s I term is¯ L µνρ = − iλQ ( P ( φ ) g µν ǫ ρkk ′ q − P ( φ ) g νρ ǫ µkk ′ q − P ( φ ) k ρ ǫ µνk ′ q + P ( φ ) q ρ ǫ µνkk ′ − P ( φ ) k ′ µ ǫ νρkq − P ( φ ) k ′ ν ǫ µρkq − P ( φ )( k ⋅ q ) ǫ µνρk ′ + P ( φ ) k ′ ρ ǫ µνkk ′ − P ( φ ) k ′ µ ǫ νρkk ′ − P ( φ ) k ′ ν ǫ ρµkk ′ − P ( φ ) k ′ ν ǫ µρkq − P ( φ ) k ′ ρ ǫ µνkq − P ( φ ) k µ ǫ νρk ′ q + P ( φ ) q µ ǫ νρkk ′ − P ( φ )( k ⋅ q ) ǫ µνρk ′ − P ( φ ) k µ ǫ νρkk ′ − P ( φ ) Q ǫ µνρk ′ ) . (B25)Explicitely, one gets s I = λ ˜ Ky ( − y ) I m ( F ( ∆ ) H ( ξ, ∆ )) and therefore I BH − DV CS = λe ˜ K ( − y ) sin φy x B P ( φ ) P ( φ ) ∆ I m ( F ( ∆ ) H ( ξ, ∆ )) . (B26)If one considers corrections of order ǫ and ∆ / Q , both coming from the leptonic part, it reads I BH − DV CS = λe ˜ K sin φP ( φ ) P ( φ ) ∆ x B y ( + ǫ ) ( J + K cos φ + y ( + ǫ )) I m ( F ( ∆ ) H ( ξ, ∆ )) . (B27)We used this formula for the interference part in the present calculation in order to have a coherent comparisonbetween results for the bound proton and for the free one.
2. Generalization to Deeply Virtual Compton Scattering off a moving off-shell proton
First of all, let us define the components of the bound off-shell proton p = ( p , ∣⃗ p ∣ sin ϑ cos ϕ, ∣⃗ p ∣ sin ϑ sin ϕ, ∣⃗ p ∣ cos ϑ ) (B28)where p ≠ √ M + ∣⃗ p ∣ (see Eq. (4)).0 a. Bethe Heitler term Our goal is to obtain a formula for the BH contribution which generalizes the harmonic decomposition obtainedfor a proton at rest, well known in the literature. So, first, let us consider the general expression for Bethe Heitleramplitude given by Eq. B3. In the square of the above mentioned amplitude, after summation over the final protonpolarizations, the hadronic part reads ∑ S ′ J µ J † ν = [ F ( ∆ ) + ( F ( ∆ ) + F ( ∆ )) − ∆ M F ( ∆ )]( p ν p µN + p µ p νN ) + ∆ ( F ( ∆ ) + F ( ∆ )) g µν + ( F ( ∆ ) − ( F ( ∆ ) + F ( ∆ )) − ∆ M F ( ∆ ))( p µ p ν + p µN p νN ) + ( − ∆ + M − p ⋅ p N )[ g µν ∗ ( F ( ∆ ) − F ( ∆ ) + F ( ∆ ) p ⋅ p N M ) − F ( ∆ ) M (( p µN p ν + p νN p µ ) − ( p µN p νN + p µ p ν ))] . (B29)This expression accounts for the motion of the initial proton and reduces to the one obtained for a proton at restgiven by Eq. (B10) when p → M, ⃗ p → ⃗ P ( φ ) = + ( J ( K b ) − K ( K b ) cos φ ) Q ,P ( φ ) = ∆ − ( J ( K b ) − K ( K b ) cos φ ) Q , (B30)but J and K become functions of the invariant kinematical variables and of the 4-momentum components of theinitially moving bound proton, i.e K b = { M, x B , ∆ , Q , ⃗ p, p } : J ( K b ) = E k ( E − p − cos θ e (∣⃗ p N ∣ cos θ N − ∣⃗ p ∣ cos ϑ ) + ∣⃗ p ∣ sin θ e sin ϑ cos ϕ ) (B31) K ( K b ) = E k sin θ e ∣⃗ p N ∣ sin θ N . (B32)With these ingredients at hand, one can compute the full contraction between the leptonic contribution (B11) andthe hadronic one for the BH process. In this way, a long and complicated analytical expression is obtained [60]. It isnot reported here but the interested readers can obtain either a Mathematica notebook or a Fortran code from theauthors upon request. The scalar products there appearing have to be evaluated considering the motion of the initialnucleon and its off-shellness. If one evaluates instead the scalar products for a proton at rest, the obtained expressionreduces to the one of the previous section for a proton at rest, as expected. b. Interference term The BH-DVCS interference term for a moving proton will be given, as always, by the contraction of a lepton anda hadronic tensor. The leptonic part is the same already obtained for a proton at rest and written in Eq. (B25), butnow the lepton propagators have to evaluated according to Eq. (B30).Concerning the hadronic tensor, we obtain the following result for the contribution Eq.(B21) when the off-sehellproton is moving ∑ S q ⋅ VP ⋅ q J ρ † + c.c = P ρ C intunp ( F ) + q ρ ∆ Q C int ,vecunp ( F ) (B33)where the the combination of CFFs has to be read: C intunp ( F ) = F ( ∆ ) H ( ξ, ∆ ) − F ( ∆ ) E ( ξ, ∆ ) ∆ M [ + ξ ( ∆ − M + p ⋅ p N ∆ )] , (B34) C int ,vecunp ( F ) = ξ [ F ( ∆ ) H ( ξ, ∆ )( + M −∣⃗ p ∣ ∆ ) + F ( ∆ ) E ( ξ, ∆ ) + F ( ∆ ) H ( ξ, ∆ ) + F ( ∆ ) E ( ξ, ∆ )( + p ⋅ p N M − M ∆ + p ⋅ p N ∆ − p ⋅ p N M ( + p ⋅ p N ∆ ))] , ⋅ q ≈ − ξ ( P ⋅ q ) and, for the relevant scalar product, one has p ⋅ p N = p E − ∣⃗ p ∣ ∣⃗ p N ∣ cos ( θ ˆ pp N ) .In order to get the explicit expression for the only term appearing in the interference, the contraction between theleptonic part, given by Eq. (B25), and the hadronic tensor, Eq. (B33), has to be performed. Also here, in the actualcalculation we are considering the dominance of H ( ξ, ∆ ) . The final result reads: I BH − DV CS = λ sin ( φ ) Q ∆ P ( φ ) P ( φ ) yǫ ( ( P ( φ ) − P ( φ )) + P ( φ ) + P ( φ ) )( ∣⃗ p N ∣ Q sin θ N ( p sin θ e √ + ǫ + ∣⃗ p ∣ sin θ e cos ϑ − ∣⃗ p ∣( cos θ e + √ + ǫ ) cos ϕ sin ϑ ) ) I m [ F ( ∆ ) H ( ξ ′ , ∆ )] , (B35)where the propagators P , ( φ ) are again given by Eqs. (B4) with the proper definition of the quantities appearing inthere and given by Eqs. (B30). Nuclear effects on the parton content of the bound proton appears only in the CFF,which has to be evaluated properly using the skewness ξ ′ = [ Q ( + ∆ Q )] /( P ⋅ q ) , accounting for the motion of thebound proton in the nuclear medium.Therefore, using the above interference term and the one discussed in the previous subsection for the squared ofthe BH amplitude, we can evaluate the cross sections (6), for a given kinematic and electron helicity and, in turn, thebeam spin asymmetries and all the results shown in this paper.2 [1] J. J. Aubert et al. [European Muon Collaboration], Phys.Lett.
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