A Light-Front approach to the 3 He spectral function
Sergio Scopetta, Alessio Del Dotto, Leonid Kaptari, Emanuele Pace, Matteo Rinaldi, Giovanni Salmè
aa r X i v : . [ nu c l - t h ] N ov Few-Body Systems manuscript No. (will be inserted by the editor)
Sergio Scopetta · Alessio Del Dotto · LeonidKaptari · Emanuele Pace · Matteo Rinaldi · Giovanni Salm`e
A Light-Front approach to the He spectral function
Received: date / Accepted: date
Abstract
The analysis of semi-inclusive deep inelastic electron scattering off polarized He at finitemomentum transfers, aimed at the extraction of the quark transverse-momentum distributions in theneutron, requires the use of a distorted spin-dependent spectral function for He, which takes careof the final state interaction effects. This quantity is introduced in the non-relativistic case, and itsgeneralization in a Poincar´e covariant framework, in plane wave impulse approximation for the momentbeing, is outlined. Studying the light-front spin-dependent spectral function for a J=1/2 system, suchas the nucleon, it is found that, within the light-front dynamics with a fixed number of constituentsand in the valence approximation, only three of the six leading twist T-even transverse-momentumdistributions are independent.
Information on the three-dimensional proton structure can be obtained from the quark transversemomentum distributions (TMDs) [1], which can be accessed through semi-inclusive deep inelasticelectron scattering (SIDIS). In particular the single spin asymmetries (SSAs) allow one to extract theSivers and the Collins contributions, expressed in terms of different TMDs and fragmentation functions(ff) [1]. A large Sivers asymmetry was measured in p ( e, e ′ π ) x [2] and a small one in D ( e, e ′ π ) x [3],showing a strong flavor dependence of TMDs. To clarify the flavour dependence of the quark transversemomentum distributions, high precision experiments, involving both protons and neutrons, are needed[4]. In Ref. [5] an experiment to extract information on the neutron TMDs from experimental measure-ments of the SSAs on He, at JLab at 12 GeV, was proposed. To obtain a reliable information one has
S. ScopettaDipartimento di Fisica e Geologia, Universit`a di Perugia and INFN, Sezione di Perugia, ItalyTel.: +39-075-5852721Fax: +39-075-44666E-mail: [email protected]. Del DottoDipartimento di Fisica, Universit`a di Roma Tre and INFN, Roma 3, ItalyL. KaptariBogoliubov Lab. Theor. Phys., 141980, JINR, Dubna, RussiaE. PaceDipartimento di Fisica, Universit`a di Roma “Tor Vergata” and INFN, Roma 2, ItalyM. RinaldiDipartimento di Fisica e Geologia, Universit`a di Perugia and INFN, Sezione di Perugia, ItalyG. Salm`eINFN Sezione di Roma, Italy to take carefully into account the structure of He, the interaction in the final state (FSI) between theobserved pion and the remnant debris, and the relativistic effects. The present paper reports on ourefforts about these items. He A polarized He is an ideal target to study the polarized neutron since, at a 90% level, a polarized He is equivalent to a polarized neutron. Dynamical nuclear effects in inclusive deep inelastic electronscattering (DIS) He ( e, e ′ ) X (DIS) were evaluated [6] with a realistic He spin-dependent spectralfunction, P τσ,σ ′ ( p , E, S He ), with p the initial nucleon momentum in the laboratory frame and E themissing energy [7]. It was found that the formula A n ≃ p n f n (cid:0) A exp − p p f p A expp (cid:1) (1)can be safely adopted to extract the neutron information from He and proton data. This formulais actually widely used by experimental collaborations. The nuclear effects are hidden in the protonand neutron ”effective polarizations”, p p ( n ) . With the AV18 nucleon-nucleon interaction [8] the values p p = − . p n = 0 .
878 were obtained [9]. The quantities f p ( n ) in Eq. (1) are the dilution factors. Toinvestigate if an analogous formula can be used to extract the SSAs, in [9] the processes He ( e, e ′ π ± ) X ,with He transversely polarized, were evaluated in the Bjorken limit and in PWIA, i.e. the final stateinteraction (FSI) was considered only within the two-nucleon spectator pair which recoils. In such aframework, SSAs for He involve convolutions of P τσ,σ ′ ( p , E, S He ), with TMDs and ffs. Ingredients ofthe calculations were: i) a realistic P τσ,σ ′ ( p , E, S He ) for He, obtained using the AV18 interaction; ii)parametrizations of data for TMDs and ff, whenever available; iii) models for the unknown TMDs andff. It was found that, in the Bjorken limit, the extraction procedure through the formula successfulin DIS works nicely for the Sivers and Collins SSAs [9]. The generalization of Eq. (1) to extractSivers and Collins asymmetries from He and proton asymmetries was recently used by experimentalcollaborations [10].In SIDIS experiments off He, the relative energy between the spectator ( A −
1) system and thesystem composed by the detected pion and the remnant debris (see Fig. 1) is a few GeV and FSI canbe treated through a generalized eikonal approximation (GEA) [11]. The GEA was already succesfullyapplied to unpolarized SIDIS in Ref. [12]. The FSI effects to be considered are due to the propagationof the debris, formed after the γ ∗ absorption by a target quark, and the subsequent hadronization,both of them influenced by the presence of a fully interacting ( A −
1) spectator system (see Fig. 1). h~A~l l ′ Q X ′ A − N Fig. 1
Interaction between the ( A −
1) spectator system and the debris produced by the absorption of a virtualphoton by a nucleon in the nucleus.
In this approach, the key quantity to introduce the FSI effects is the distorted spin-dependentspectral function, whose relevant part in the evaluation of SSAs is: P P W IA ( F SI ) || = O IA ( F SI )
12 12 − O IA ( F SI ) − − (2)with: O IAλλ ′ ( p , E ) = XZ dǫ ∗ A − ρ (cid:0) ǫ ∗ A − (cid:1) h S A , P A | Φ ǫ ∗ A − , λ ′ , p ih Φ ǫ ∗ A − , λ, p | S A , P A i δ (cid:0) E − B A − ǫ ∗ A − (cid:1) , (3)and O F SIλλ ′ ( p , E ) = XZ dǫ ∗ A − ρ (cid:0) ǫ ∗ A − (cid:1) h S A , P A | ( ˆ S Gl ) { Φ ǫ ∗ A − , λ ′ , p }ih ( ˆ S Gl ) { Φ ǫ ∗ A − , λ, p }| S A , P A i× δ (cid:0) E − B A − ǫ ∗ A − (cid:1) , (4)where ρ (cid:0) ǫ ∗ A − (cid:1) is the density of the ( A − ǫ ∗ A − , and | S A , P A i isthe ground state of the A -nucleons nucleus with polarization S A . ˆ S Gl is the Glauber operator:ˆ S Gl ( r , r , r ) = Y i =2 , (cid:2) − θ ( z i − z ) Γ ( b − b i , z − z i ) (cid:3) (5)and Γ ( b i , z i ) the generalized profile function: Γ ( b i , z i ) = (1 − i α ) σ eff ( z i )4 π b exp (cid:20) − b i b (cid:21) , (6)where r i = { b i , z i } with z i = z − z i and b i = b − b i . The models for the profile function, Γ ( b i , z i ), and for the effective cross section, σ eff ( z i ), as well as the values of the parameters α and b are the ones exploited in Ref. [12] to nicely describe the JLab data corresponding to the unpolarizedspectator SIDIS off the deuteron.It occurs that P P W IA || and P F SI || can be very different, but the relevant observables for the SSAsinvolve integrals, dominated by the low momentum region, where the FSI effects on P || are minimizedand the spectral function is large [11]. As a consequence the effective nucleon polarizations changefrom p p = − . p n = 0 .
878 to p F SIp = − . p F SIn = 0 .
76, where p F SIp ( n ) = Z dǫ S Z d p T r [ S He ∗ σ P p ( n ) F SI ( p , E, S He )] , (7)with P p ( n ) F SI ( p , E, S He ) the distorted spin-dependent spectral function, defined in terms of the overlapsof Eq. (4) [11]. Then p p ( n ) with and without the FSI differ by 10-15% . Actually, one has also toconsider the effect of the FSI on dilution factors. We have found, in a wide range of kinematics typicalfor the experiments at JLAB [5; 10], that the product of polarizations and dilution factors changesvery little [13]. Indeed the effects of FSI in the dilution factors and in the effective polarizations arefound to compensate each other to a large extent and the usual extraction appears to be safe : A n ≃ p F SIn f F SIn (cid:0) A exp − p F SIp f F SIp A expp (cid:1) ≃ p n f n (cid:0) A exp − p p f p A expp (cid:1) (8)In [9] the calculation was performed within a non relativistic approach for the spectral function, butwith the correct relativistic kinematics in the Bjorken limit. For an accurate description of SIDISprocesses, the role played by relativity has to be fully investigated: it will become even more importantwith the upgrade of JLab @ 12 GeV. To study relativistic effects in the actual experimental kinematics,we adopted [14] the Relativistic Hamiltonian Dynamics (RHD) introduced by Dirac [15]. Indeed theRHD of an interacting system with a fixed number of on-mass-shell constituents, with the Bakamijan-Thomas construction of the Poincar´e generators [16], is fully Poincar´e covariant. The Light-Front (LF)form of RHD has a subgroup structure of the LF boosts, allows a separation of the intrinsic motionfrom the global one, which is very important for the description of DIS, SIDIS and deeply virtualCompton scattering (DVCS) processes, and allows a meaningful Fock expansion. The key quantity toconsider in SIDIS processes is the LF relativistic spectral function, P τσ ′ σ (˜ κ, ǫ S , S He ), with ˜ κ an intrinsic Table 1
Proton and neutron effective polarizations within the non relativistic appproach (NR) and preliminaryresults within the light-front relativistic dynamics approach (LF). First line : longitudinal effective polarizations;second line : transverse effective polarizations. proton NR proton LF neutron NR neutron LF R dEd p T r ( P σ z ) S A = b z -0.02263 -0.02231 0.87805 0.87248 R dEd p T r ( P σ y ) S A = b y -0.02263 -0.02268 0.87805 0.87494 nucleon momentum and ǫ S the energy of the two-nucleon spectator system. The LF spectral functionwill be very useful also for other studies (e.g., for nuclear generalized parton distributions (GPDs),where final states have to be properly boosted, studied so far only within a non-relativistic spectralfunction [17; 18]). The LF nuclear spectral function, P τσ ′ σ (˜ κ, ǫ S , S He ), is defined in terms of LF overlaps[19] between the ground state of a polarized He and the cartesian product of an interacting state oftwo nucleons with energy ǫ S and a plane wave for the third nucleon. Within a reliable approximation[19] it can be given in terms of the unitary Melosh Rotations, D [ R M ( ˜ κ )], and the usual instant-formspectral function P τσ ′ σ : P τσ ′ σ ( ˜ κ, ǫ S , S He ) ∝ X σ σ ′ D [ R † M ( ˜ κ )] σ ′ σ ′ P τσ ′ σ ( p , ǫ S , S He ) D [ R M ( ˜ κ )] σ σ (9)Notice that the instant-form spectral funtion P τσ ′ σ ( p , ǫ S , S He ) is given in terms of three indepen-dent functions, B , B , B [7], once parity and t-reversal are imposed: P τσ ′ σ ( p , ǫ S , S He ) = (cid:2) B τ ,S He ( | p | , E ) + σ · f τS He ( p , E ) (cid:3) σ ′ σ (10)with f τS He ( p , E ) = S He B τ ,S He ( | p | , E ) + ˆp ( ˆp · S He ) B τ ,S He ( | p | , E ) (11)Adding FSI, more terms should be included.We are now evaluating the SSAs using the LF hadronic tensor, at finite values of Q . The pre-liminary results are quite encouraging, since, as shown in Table 1, LF longitudinal and transversepolarizations change little from the ones obtained within the NR spectral function and weakly differfrom each other. Then we find that the extraction procedure works well within the LF approach as itdoes in the non relativistic case.Concerning the FSI, we plan to include in our LF approach the FSI between the jet produced fromthe hadronizing quark and the two-nucleon system through the Glauber approach of Ref. [11]. J = 1 / The TMDs for a nucleon with total momentum P and spin S are introduced through the q-q correlator Φ ( k, P, S ) αβ = Z d z e ik · z h P S | ¯ ψ qβ (0) ψ qα ( z ) | P S i = 12 { A P/ + A S L γ P/ + A P/ γ S ⊥ + 1 M e A k ⊥ · S ⊥ γ P/ + e A S L M P/ γ k ⊥ + 1 M e A k ⊥ · S ⊥ P/ γ k ⊥ / } αβ , (12)where k is the parton momentum in the laboratory frame, so that the six twist-2 T-even functions, A i , e A i ( i = 1 , Φ ( k, P, S ).Indeed, particular combinations of the functions A i , e A i ( i = 1 ,
3) can be obtained by the followingtraces of Φ ( k, P, S ) :12 P + Tr( γ + Φ) = A , P + Tr( γ + γ Φ) = S L A + 1 M k ⊥ · S ⊥ e A , P + Tr(i σ j+ γ Φ) = S j ⊥ A + S L M k j ⊥ e A + 1 M k ⊥ · S ⊥ k j ⊥ e A ( j = x, y ) . (13) The six T-even twist-2 TMDs for the quarks inside a nucleon can be obtained by integration of thefunctions A i , e A i on k + and k − as follows f ( x, | k ⊥ | ) = 12 Z dk + dk − (2 π ) δ [ k + − xP + ] 2 P + A ,∆f ( x, | k ⊥ | ) = 12 Z dk + dk − (2 π ) δ [ k + − xP + ] 2 P + A ,g T ( x, | k ⊥ | ) = 12 Z dk + dk − (2 π ) δ [ k + − xP + ] 2 P + e A ,∆ ′ T f ( x, | k ⊥ | ) = 12 Z dk + dk − (2 π ) δ [ k + − xP + ] 2 P + (cid:18) A + | k ⊥ | M e A (cid:19) ,h ⊥ L ( x, | k ⊥ | ) = 12 Z dk + dk − (2 π ) δ [ k + − xP + ] 2 P + e A ,h ⊥ T ( x, | k ⊥ | ) = 12 Z dk + dk − (2 π ) δ [ k + − xP + ] 2 P + e A . (14)Let us consider the contribution to the correlation function from on-mass-shell fermions Φ p ( k, P, S ) = ( k/ on + m )2 m Φ ( k, P, S ) ( k/ on + m )2 m = (15)= X σσ ′ u LF (˜ k, σ ′ ) ¯ u LF (˜ k, σ ′ ) Φ ( k, P, S ) u LF (˜ k, σ )¯ u LF (˜ k, σ ) , and let us identify ¯ u LF (˜ k, σ ′ ) Φ ( k, P, S ) u LF (˜ k, σ ), up to a kinematical factor K , with the LF nucleonspectral function, P σ ′ σ (˜ κ, ǫ S , S ) [19]:¯ u LF (˜ k, σ ′ ) Φ ( k, P, S ) u LF (˜ k, σ ) = K P σ ′ σ (˜ κ, ǫ S , S ) . (16)In a reference frame where P ⊥ = 0, the following relation holds between the off-mass-shell minuscomponent k − of the momentum of the struck quark and the spectator diquark energy ǫ S : k − = M P + − ( ǫ S + m ) 4 m + | k ⊥ | P + − k + . (17)Let us approximate the full correlation function Φ ( k, P, S ) by its particle contribution Φ p ( k, P, S ).Then, through relations analogous to the ones of Eq. (13), which allow one to obtain the functions A i , e A i from the traces of Φ ( k, P, S ), the valence approximations A Vi , e A Vi ( i = 1 ,
3) for the functions A i , e A i can be obtained by the traces [ γ + Φ p ( k, P, S )], [ γ + γ Φ p ( k, P, S )], and [ k/ ⊥ γ + γ Φ p ( k, P, S )].However these same traces can be also expressed through the LF spectral function, since from Eqs.(15,16) one has T r (cid:2) γ + Φ p ( k, P, S ) (cid:3) = k + m K T r [ P (˜ κ, ǫ S , S )] (18) T r (cid:2) γ + γ Φ p ( k, P, S ) (cid:3) = k + m K T r [ σ z P (˜ κ, ǫ S , S )] (19) T r (cid:2) k/ ⊥ γ + γ Φ p ( k, P, S ) (cid:3) = k + m KT r [ k ⊥ · σ P (˜ κ, ǫ S , S )] (20)In turn, as in the He case, the traces Tr( P I ), Tr( P σ z ), Tr( P σ i ) ( i = x, y ) can be expressed in terms ofthree scalar functions, B , B , B and known kinematical factors. Then, within the LF approach witha fixed number of particles and in the valence approximations, the six leading twist T-even functions A Vi , e A Vi ( i = 1 ,
3) can be expressed in terms of the three independent scalar functions B , B , B .Therefore only three of the six T-even TMDs are independent [19]. We stress however that this result,not valid in general in QCD (see, e.g., Ref. [20]), is a prediction of our peculiar framework, i.e., hamil-tonian light-front dynamics with a fixed number of constituents, finalized to yield a proper Poincar´ecovariant description of the nucleon in the valence approximation. Its experimental check would betherefore a test of the reliability of our scheme to describe SIDIS processes in the valence region. A realistic study of the DIS processes off He and in particular of the SSAs in the reaction He ( e, e ′ π ± ) X beyond the PWIA and the non relativistic approach is under way. The FSI effects have been evaluatedthrough the GEA and the introduction of a distorted spin-dependent spectral function. The relativis-tic effects are studied through the analysis of a LF spectral function (up to now only in PWIA).Preliminary results are encouraging, in view of a sound extraction of the neutron information fromexperimental data. Nuclear effects in the extraction of the neutron information are found to be undercontrol, even when the interaction in the final state is considered, and the relativistic effects appearto be small. An analysis at finite Q with the LF spectral function is in progress. The next step is tocomplete this program. Then we will apply the LF spectral function to other processes (e.g., DVCS) toexploit other possibilities to use He as an effective neutron target and as a laboratory for light-frontstudies.The introduction of a LF spin-dependent spectral function for a nucleon allowed us to find relationsamong the six leading twist T-even TMDs, which are exact within LF dynamics with a fixed numberof degrees of freedom, in the valence approximation. It can be shown that, in this case, only threeof the six T-even TMDs are independent. The above relations, although not true in QCD, could beexperimentally checked to test our LF description of SIDIS in the valence region.
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