Generalized parton distributions of 3He and the neutron orbital structure
aa r X i v : . [ nu c l - t h ] M a r Noname manuscript No. (will be inserted by the editor)
Generalized parton distributions of He and the neutronorbital structure
M. Rinaldi · S. Scopetta
Received: date / Accepted: date
Abstract
The two leading twist, quark helicity conserving generalized parton distributions(GPDs) of He, accessible, for example, in coherent deeply virtual Compton scattering(DVCS), are calculated in impulse approximation (IA). Their sum, at low momentum trans-fer, is found to be largely dominated by the neutron contribution, so that He is very promis-ing for the extraction of the neutron information. Anyway, such an extraction could be nottrivial. A technique, able to take into account the nuclear effects included in the IA analysisin the extraction procedure, even at moderate values of the momentum transfer, is proposed.Coherent DVCS arises therefore as a crucial experiment to access, for the first time, theneutron GPDs and the orbital angular momentum of the partons in the neutron.
Keywords
Three body systems · Generalized parton distributionsGeneralized Parton Distributions (GPDs) [1] parameterize the non-perturbative hadronstructure in hard exclusive processes, allowing to access unique information such as, forexample, the parton total angular momentum [2]. By subtracting from the latter the helicityquark contribution, measured in other hard processes, the parton orbital angular momentum(OAM), contributing to the nucleon spin, could be then estimated, a crucial step towards thesolution of the so called “Spin Crisis”.The cleanest process to access GPDs is Deeply Virtual Compton Scattering (DVCS),i.e. eH e ′ H ′ g when Q ≫ M ( Q = − q · q is the momentum transfer between theleptons e and e ′ , D the one between hadrons H and H ′ with momenta P and P ′ , and M isthe nucleon mass. Another relevant kinematical variable is the so called skewedness, x = − D + / ( P + + P ′ + ) ). Despite severe difficulties to extract GPDs from experiments, datafor proton and nuclear targets are being analyzed, see, i.e., Refs. [3,4]. The measurement ofGPDs for nuclei could be crucial to distinguish between different models of nuclear medium M. RinaldiDipartimento di Fisica, Universit`a degli studi di Perugia and INFN sezione di Perugia, Via A. Pascoli 06100Perugia, ItalyE-mail: [email protected]. ScopettaDipartimento di Fisica, Universit`a degli studi di Perugia and INFN sezione di Perugia, Via A. Pascoli 06100Perugia, ItalyE-mail: [email protected] In this paper, a ± = ( a ± a ) / √ -4 -3 -2 -1 Δ μ [fm -1 ] |G (Δ )| x G ~ M3 (x, D , x ) x (b) Fig.1: (a): The magnetic ff of He, G M ( D ) , with D m = √− D . Full line: the present IA calculation, obtainedas the x-integral of (cid:229) q ˜ G , qM (see text). Dashed line: experimental data [14]. (b): The quantity x ˜ G M ( x , D , x ) ,where x = M / M x and x = M / M x , shown at D = − . and x = .
1, together with the neutron(dashed) and the proton (dot-dashed) contribution. −0.6−0.4−0.20 X X Fig.2: (a): The quantity x ˜ G n , qM ( x , D , x ) for the neutron at D = − . and x = . u , d and u + d contributions (full lines), compared with the approximation x ˜ G n , q , extrM ( x , D , x ) , Eq. (6), (dashed). (b): Theratio r n ( x , D , x ) = ˜ G n , extrM ( x , D , x ) / ˜ G nM ( x , D , x ) , in the forward limit (full), at D = − . and x = D = − . and x = . modifications of the nucleon structure, an impossible task in the analysis of DIS experimentsonly. Moreover, the neutron measurement, which requires nuclear targets, is a very relevantinformation because it permits, together with the proton one, a flavor decomposition ofGPDs. In studies of the neutron polarization, He plays a special role, due its spin structure(see, e.g., Ref. [5]). This is true in particular for GPDs. In fact, among the latters, the ones ofinterest here are H q ( x , D , x ) and E q ( x , D , x ) . He, among the light nuclei, is the only onefor which the combination ˜ G , qM ( x , D , x ) = H q ( x , D , x ) + E q ( x , D , x ) of its GPDs couldbe dominated by the neutron, being H and He not suitable to this aim, as discussed in Ref.[6]. To what extent this fact can be used to extract the neutron information, is shown in Refs.[6,7], and summarized here.The formal treatment of He GPDs in Impulse Approximation (IA) can be found inRefs. [8], where, for the GPD H of He, H q , a convolution-like equation in terms of thecorresponding nucleon quantity is found. Very recently, the treatment has been extended to˜ G , qM (see Refs. [6,7] for details), yielding˜ G , qM ( x , D , x ) = (cid:229) N Z dE Z d p ˜ P N ( p , p ′ , E ) x ′ x ˜ G N , qM ( x ′ , D , x ′ ) , (1)where x ′ and x ′ are the variables for the bound nucleon GPDs and p ( p ′ = p + D ) is its4-momentum in the initial (final) state. Besides, ˜ P N ( p , p ′ , E ) is a proper combination ofcomponents of the spin dependent, one body off diagonal spectral function: eneralized parton distributions of He and the neutron orbital structure 3 X Fig.3: r n ( x , D , x ) = ˜ G n , extrM ( x , D , x ) / ˜ G nM ( x , D , x ) , at D = 0.1 GeV and x =
0, using the model of Ref. [12] forthe nucleon GPDs (dashed) and the one of Ref. [15] (full). P NSS ′ , ss ′ ( p , p ′ , E ) = ( p ) M √ ME Z d W t (cid:229) s t h P ′ S ′ | p ′ s ′ , t s t i N h p s , t s t | P S i N , (2)where S , S ′ ( s , s ′ ) are the nuclear (nucleon) spin projections in the initial (final) state, respec-tively, and E = E min + E ∗ R , being E ∗ R the excitation energy of the two-body recoiling system.The main quantity appearing in the definition Eq. (2) is the intrinsic overlap integral h p s , t s t | P S i N = Z d y e i p · y h c sN , Y s t t ( x ) | Y S ( x , y ) i (3)between the wave function of He, Y S , with the final state, described by two wave func-tions: i) the eigenfunction Y s t t , with eigenvalue E = E min + E ∗ R , of the state s t of the intrinsicHamiltonian pertaining to the system of two interacting nucleons with relative momentum t , which can be either a bound or a scattering state, and ii) the plane wave representing thenucleon N in IA. For a numerical evaluation of Eq. (1), the overlaps, Eq. (3), appearingin Eq. (2) and corresponding to the analysis of Ref. [9] in terms of Av18 [10] wave func-tions [11], have been used, together with a simple nucleonic model for ˜ G N , qM [12] (see Ref.[7] for details). Since there are no He data available, it is possible to verify only a fewgeneral GPDs properties, i.e., the forward limit and the first moments. In particular the cal-culation of H q ( x , D , x ) fulfills these constraints [8]. In the ˜ G , qM ( x , D , x ) case, since thereis no observable forward limit for E q ( x , D , x ) , the only possible check is the first moment: (cid:229) q R dx ˜ G , qM ( x , D , x ) = G M ( D ) ; where G M ( D ) is the magnetic form factor (ff) of He.The result obtained is in perfect agreement with the one-body part of the AV18 calculationpresented in Ref. [13] (see Fig.1a). Moreover, for the values of D which are relevant for thecoherent process under investigation here, i.e., − D ≪ .
15 GeV , our results compare wellalso with the data [14]. With the comfort of this succesfull check, results for GPDs of Heare now discussed. In the forward limit, necessary to measure OAM, the neutron contribu-tion strongly dominates the He quantity, but increasing D the proton contribution growsup (see Fig.1b), in particular for the u flavor [6,7]. It is therefore necessary to introducea procedure to safely extract the neutron information from He data. This can be done byobserving that Eq. (1) can be written as˜ G , qM ( x , D , x ) = (cid:229) N Z MAM x dzz g N ( z , D , x ) ˜ G N , qM (cid:18) xz , D , x z , (cid:19) , (4)where g N ( z , D , x ) is a “light cone off-forward momentum distribution” which, close to theforward limit, is strongly peaked around z =
1. Therefore, for x = ( M A / M ) x < M. Rinaldi, S. Scopetta ˜ G , qM ( x , D , x ) ≃ low D ≃ (cid:229) N ˜ G N , qM (cid:0) x , D , x (cid:1) Z MAM dzg N ( z , D , x )= G , p , pointM ( D ) ˜ G pM ( x , D , x ) + G , n , pointM ( D ) ˜ G nM ( x , D , x ) . (5)Here, the magnetic point like ff, G , N , pointM ( D ) = R MAM dz g N ( z , D , x ) , which would givethe nucleus ff if the proton and the neutron were point-like particles with their physicalmagnetic moments, are introduced. These quantities are very well known theoretically anddepend weakly on the potential used in the calculation [7].Eq. (5) can now be used to extract the neutron contribution:˜ G n , extrM ( x , D , x ) ≃ G , n , pointM ( D ) n ˜ G M ( x , D , x ) − G , p , pointM ( D ) ˜ G pM ( x , D , x ) o . (6)In Fig. 2a, the comparison between the free neutron GPDs, used as input in the calcu-lation, and the ones extracted using our calculation for ˜ G M and the proton model for ˜ G pM ,shows that the procedure works nicely even beyond the forward limit. The only theoreticalingredients are the magnetic point like ffs, which are completely under control. This is evenclearer in Fig. 2b, where the ratio r n ( x , D , x ) = ˜ G n , extrM ( x , D , x ) ˜ G nM ( x , D , x ) is shown in a few kinematicalregions. The procedure works for x < .
7, where data are expected from JLab. Moreover,the extraction procedure depends weakly on the used nucleonic model (see Fig. 3 and Ref.[7]).In closing, we have shown that coherent DVCS off He at low momentum transfer D is an ideal process to access the neutron GPDs; if data were taken at higher D , a relativistictreatment [16] and/or the inclusion of many body currents, beyond the present IA scheme,should be implemented. References
1. Mueller, D. et al., Fortsch. Phys. , 101 (1994); Radyushkin, A. V. Phys. Lett. B , 417 (1996);Ji,X. -D. Phys. Rev. Lett. , 610 (1997).2. Diehl, M. Phys. Rept. 388, 41 (2003); Belitsky, A. V. and Radyushkin, A. V., Phys. Rept. 418, 1 (2005);Boffi, S. and Pasquini, B. Riv. Nuovo Cim. 30, 387 (2007).3. Airapetian, A. et al. [HERMES Collaboration], Nucl. Phys. B (2010) 1; Phys. Rev. C (2010)035202; Mazouz, M. et al. [Jefferson Lab Hall A Collaboration], Phys. Rev. Lett. , 17 (2010).5. Friar, J. L. et al. , Phys. Rev. C et al. , Phys. Rev. C
968 (1993).6. Rinaldi, M. and Scopetta, S. Phys. Rev. C 85, 062201(R) (2012).7. Rinaldi, M. and Scopetta, S. arXiv:1208.2831 [nucl-th].8. Scopetta, S. Phys. Rev. C , 64 (1997).10. Wiringa, R. B., Stoks V. G. J. and Schiavilla, R. Phys. Rev. C
38 (1995).11. Kievsky, A., Viviani, M. and Rosati, S. Nucl. Phys. A
511 (1994).12. Musatov, I. V. and Radyushkin, A. V. Phys. Rev. D , 3069 (1998).14. Amroun, A. et al. , Nucl. Phys. A , 596 (1994).15. Scopetta, S. and Vento, V. Eur. Phys. J. A16