Deeply virtual Compton Scattering off 4 He
aa r X i v : . [ nu c l - t h ] O c t Deeply virtual Compton Scattering off He Sara Fucini, Sergio Scopetta
Dipartimento di Fisica e Geologia, University of Perugia and INFN, Perugia,I-06100 Perugia, ItalyE-mail: [email protected], [email protected]
Michele Viviani
INFN-Pisa, I-06100 Pisa, ItalyE-mail: [email protected]
Abstract.
Deeply virtual Compton scattering is a fascinating process which can provide a tomographicview of nuclei and bound nucleons. The first experimental results for He targets, recentlyreleased at Jefferson Lab, have been analyzed here in a rigorous Impulse Approximation scenario.For both the coherent and incoherent channels of the process, the main experimental observableshave been written in terms of state-of-the-art models of the nuclear spectral function and ofthe parton structure of the bound proton. A good overall agreement with the data is obtained.The calculation shows that a comparison of our conventional results with future precise datacan expose novel quark and gluon effects in nuclei.
1. Introduction
It is nowadays clear that inclusive Deep Inelastic Scattering measurements do not allow aquantitative understanding of the origin of the EMC effect [1], i.e. the nuclear mediummodification to the parton structure of the bound nucleon. Nevertheless, a new generationof semi-inclusive and exclusive experiments, performed in particular at Jefferson Lab (JLab),are expected to give new insights into the problem [2, 3]. A powerful tool in this sense isdeeply Virtual Compton Scattering (DVCS). In DVCS, the inner parton content of the target isparametrized through non-perturbative functions, the so-called generalized parton distributions(GPDs), which provide a wealth of novel information (for an exhaustive report, see, e.g., Ref.[4]). The one directly linked to this talk is the possiblity to obtain a parton tomography of thetarget [5]. In a nucleus, such a process can occur in two different channels: the coherent one,where the nucleus remains intact and the tomography of the whole nucleus can be accessed, andthe incoherent one, where the nucleus breaks up, one nucleon is detected and its structure canbe studied. As a target, He is very convenient, being the lightest system showing the dynamicalfeatures of a typical atomic nucleus. Moreover, it is scalar and isoscalar and its description interms of GPDs is easy. Recently, DVCS data for this target have become available at JLabwhere the coherent and incoherent channels have been successfully disentangled, for the firsttime [6, 7].A rigorous theoretical description of the process, whose results could be compared with thedata in a conclusive way, requires a proper evaluation of conventional nuclear physics effects actorizatione e’ (k) (k’) x+ x x− x gg * (q ) (q ) GPDs (x, x ,t) A A A (p) (p’) D = p’ −p Figure 1.
The handbag approximation to the coherent DVCS of He.in terms of wave functions corresponding to realistic nucleon-nucleon potentials. This kind ofrealistic calculations, although very challenging, are possible for a few-body system (e.g. seeRef. [8] for H and Ref. [9] for He) as the target under scrutiny. Previous calculations for Hehave been performed long time ago [10, 11], in some cases in kinematical regions different fromthose probed at JLab. In this talk, a review of our main results obtained from the study of thehandbag contribution to both DVCS channels, in impulse approximation, is presented.
2. DVCS formalism
In this section, the general formalism for both DVCS channels, whose handbag approximationwill be studied in Impulse Approximation (IA), is presented. In this scenario, we assume thatthe process occurs off one quark in one nucleon in He, that only nucleonic degrees of freedomare considered, and that further possible rescattering of the struck proton with the remnantsystems is not relevant. As reference frame, we choose the target at rest, with an azimuthalangle φ between the electron scattering plane and the hadronic production plane. The processcan be described in terms of four independent variables, usually chosen as x B = Q / (2 M ν ), Q = − q = − ( k − k ′ ) , ∆ = ( p ′ − p ) = ( q − q ) and φ . In such a process, if the initialphoton virtuality Q is much larger than the momentum transferred to the hadronic system withinitial, the factorization property allows to distinguish the hard vertex, which can be studiedperturbatively, from the soft part, given by the blob in Figs 1 and 3, that is parametrized interms of GPDs. Besides Q and t , GPDs are also a function of the so-called skewness ξ = − ∆ + P + i.e., the difference in plus momentum fraction between the initial and the final states, and on x ,the average plus momentum fraction of the struck parton with respect to the total momentum,not experimentally accessible. Because of this latter dependence, GPDs cannot be directlymeasured. For this reason, Compton Form Factors (CFFs), where GPDs ( H q ) are hidden, aredefined in the following way ( e q being the quark electric charge): ℑ m H ( ξ, t ) = X q e q ( H q ( ξ, ξ, ∆ ) − H q ( − ξ, ξ, ∆ )) , (1) ℜ e H ( ξ, t ) = Pr X q e q Z (cid:18) ξ − x − ξ + x (cid:19) ( H q ( x, ξ, t ) − H q ( − x, ξ, t )) (2)The experimental observable which gives access to these quantities is the beam spin asymmetry(BSA), that for the target under scrutiny is given by A LU = dσ + − dσ − dσ + + dσ − , (3) A L U H e ( (cid:176) ) Q [GeV ] -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.1 0.15 0.2 0.25 0.3 A L U H e ( (cid:176) ) x B -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.06 0.08 0.1 0.12 0.14 0.16 A L U H e ( (cid:176) ) -t [GeV ] Figure 2. He azimuthal beam-spin asymmetry A LU ( φ = 90 o ): results of Ref. [18] (redstars) compared with data (black squares) [6]. From left to right, the quantity is shown in theexperimental Q , x B and t = ∆ bins, respectively.where the differential cross section for the different beam polarization ( ± ) appears. Arealistic calculation of conventional effects for the BSA corresponds to a plane wave impulseapproximation analysis, presented in the following.
3. Coherent DVCS channel
The most general coherent DVCS process A ( e, e ′ γ ) A allows to study the partonic structure ofthe recoiling whole nucleus A through the formalism of GPDs. In the IA scenario presentedabove, a workable expression for H Heq ( x, ξ, ∆ ), the GPD of the quark of flavor q in the Henucleus, is obtained as a convolution between the GPDs H Nq of the quark of flavor q in the boundnucleon N and the off-diagonal light-cone momentum distribution of N in He and reads H Heq ( x, ξ, ∆ ) = X N Z | x | dzz h HeN ( z, ξ, ∆ ) H Nq (cid:18) xζ , ξζ , ∆ (cid:19) . (4)The light cone momentum distribution in the previous equation is defined as h HeN ( z, ∆ , ξ ) = Z dE Z d~p P HeN ( ~p, ~p + ~ ∆ , E ) δ (cid:18) z − ¯ p + ¯ P + (cid:19) , (5)where the off diagonal spectral function P HeN ( ~p, ~p + ~ ∆ , E ) governs the size and relevance ofnuclear effects. It represents the probability amplitude to have a nucleon leaving the nucleus withmomentum ~p and leaving the recoiling system with an excitation energy E ∗ = E − | E A | + | E A − | ,with | E A | and | E A − | the nuclear binding energies, and going back to the nucleus with amomentum transfer ~ ∆. The full realistic evaluation of P HeN requires an exact description ofall the He spectrum, including three-body scattering states; for this reason, it represents achallenging, presently unsolved few body problem. So, while the complete evaluation of thisobject has just begun, as an intermediate step in the present calculation a model of the nuclearnon-diagonal spectral function [12], based on the momentum distribution corresponding to theAv18 NN interaction Ref.[13] and including 3-body forces [14], has been used when excited 3-and 4- body states are considered. For the ground state, exact wave functions of 3- and 4-bodysystems, evaluated along the scheme of Ref. [15], have been used. Concerning the nucleonicGPD appearing in Eq. (4), the well known GPD model of Ref. [16] has been used. With theseingredients at hand, as an encouraging check, typical results are found, in the proper limits, forthe nuclear charge form factor and for nuclear parton distributions. In this way, our model for H Heq allowed us to have a numerical evaluation of Eqs. (1) and (2), which define quantities also g (q )(q ) P p p’ = p + D He GPDs (x, x , D ) X Factorization
A−1 (p ) fe e’(k) (k’) D = q − q A * FF ( D ) g g e(k) e’(k’)P He X ( D ) * (q )p’p k−q + A FF ( D )Pe (k) e’(k’) ( D ) g (q ) g * p’p X He k’+q A + Figure 3.
Incoherent DVCS off He in IA. To the left, pure DVCS contribution; to the rightthe two Bethe Heitler terms.appearing in the explicit form of the BSA of the coherent DVCS channel that reads: A LU ( φ ) = α ( φ ) ℑ m ( H A ) α ( φ ) + α ( φ ) ℜ e ( H A ) + α ( φ ) (cid:18) ℜ e ( H A ) + ℑ m ( H A ) (cid:19) . (6)Here above, α i ( φ ) are kinematical coefficients defined in Ref. [17]. As shown in Fig. 2, avery good agreement is found with the data [18]. One can conclude that a careful analysis ofthe reaction mechanism in terms of basic conventional ingredients is successful and that thepresent experimental accuracy does not require the use of exotic arguments, such as dynamicaloff-shellness.
4. Incoherent DVCS channel
In the process A ( e, e ′ γp ) X depicted in Fig. 3, the parton structure of the bound proton can beaccessed. In order to have a complete evaluation of Eq. (3), the cross-section for a DVCS processoccurring off a bound moving proton in He is required. Working within an IA approach, weaccount for the pure kinematical off-shellness of the initial bound proton obtaining a convolutionformula for the cross sections differential in the experimental variables: dσ ± ≡ dσ ± Inc dx B dQ d ∆ dφ = Z exp dE d~p P He ( ~p, E ) |A ± ( ~p, E, K ) | g ( ~p, E, K ) , (7)where K is the set of kinematical variables { x B , Q , t, φ } . The intervals of these variables probedin the experiment select the relevant part of the diagonal spectral function P HeN ( ~p, E ), whichhas therefore to be integrated in the range exp . The quantity g ( p, p N , K ) is a complicatedfunction arising from the integration over the phase space and including also the flux factor p · k/ ( p | ~k | ). In the above equation, the squared amplitude includes three different terms, i.e A = T DV CS + T BH + I DV CS − BH as shown in Fig. 3 and each contribution has to be evaluatedfor an initially moving proton. Our amplitudes generalizes the ones obtained for a proton atrest in Ref. [19] and the main assumptions done are summarized in Ref. [20]. Since in thekinematical region of interest at Jlab the BH part is dominating, the key partonic insights areall hidden in the interference DVCS-BH term entering in the BSA in the following way A IncohLU = R exp dE d~p P He ( ~p, E ) g ( ~p, E, K ) I DV CS − BH R exp dE d~p P He ( ~p, E ) g ( ~p, E, K ) T BH . (8)Since our ultimate goal is to have a comparison with the experimental data, we exploit theazimuthal dependence of Eq. (8) decomposing in φ harmonics the interference and the BH B x ) (cid:176) ( I n c o h L U A Figure 4.
Azimuthal beam-spin asymmetry for the proton in He, A IncohLU , Eq. (8), for φ = 90 o :results of this approach [20](red dots) compared with data (black squares) [7]. B x ) (cid:176) (90 pLU /A IncohLU
A ) (cid:176) (90 pLU /A IncohLU
Pointlike A
Figure 5.
The ratio A IncohLU /A pLU , Eq. (9) (red dots), compared to the result obtained withpointlike protons (black diamonds).part. All the information about the parton content of the bound proton is encapsulated inthe imaginary part of CFF, that accounts for the modification at structure level through therescaling of the skewness, that depends explicitly on the 4-momentum components of the initialproton. In the present calculation we considered only the dominating contribution given by the H q ( x, ξ ′ , t ) GPD, for which use of the GK model has been made [16]. The results are depictedin Fig. 4 [20]. As expected, the agreement with experimental data is good except the region oflowest Q , corresponding to the first x B bin. In this region, in facts, the impulse approximationis not supposed to work well, since final state interaction effects, neglected in IA, could besizable. In order to have an idea about how the nuclear effects affect the results obtained, i.e.if they are related to some medium modification of the inner parton structure described by theGPD, we considered the ratio between the BSA for a bound nucleon, given by Eq. (8) and thatfor a free proton, given in our scheme by the GK model: A IncohLU A pLU ∝ I HeDV CS − BH I pDV CS − BH T pBH T HeBH = ( nucl.ef f. ) Int ( nucl.ef f. ) BH . (9)The above quantity is proportional to the ratio of the nuclear effects on the BH and DV CS interference I DV CS − BH to the nuclear effects on the BH cross section. If the nuclear dynamicsodifies I DV CS − BH and the BH cross sections in a different way, the effect can be big evenif the parton structure of the bound proton does not change appreciably. Indeed, this is whathappens: considering the same ratio for pointlike protons, the big observed effect is still present,as we can see from Fig. 5.
5. Conclusions
We can conclude that for both channels, given the present experimental accuracy, the descriptionof the data does not require the use of exotic arguments, such as dynamical off shellness.Nevertheless, a serious benchmark calculation in the kinematics of the next generation of precisemeasurements at high luminosity [21] will require an improved treatment of both the nucleonicand the nuclear parts of the evaluation. The latter task includes the realistic computation of aone-body non diagonal (for the coherent channel) and diagonal (for the incoherent channel)spectral function of He. Work is in progress towards this challenging direction. In themeantime, the straightforward approach proposed here can be used as a workable framework forthe planning of future measurements.
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