A realistic study of the SiDIS process 3He(e  , e

π)X in the Bjorken

limit is briefly reviewed, showing that the nuclear effects, present in the extraction

of the neutron information, are largely under theoretical control, within an ImpulseApproximation approach. In view of the forthcoming experimental data, we shortly

present a novel Poincar´e covariant description of the nuclear target, implementing a

Light-Front analysis at finite Q2, within the Bakamijan-Thomas construction of the

Poincar´e generators. Furthermore, as a by-product of the introduction of a LightFront spin-dependent spectral function for a J = 1/2 system, we straightforwardly

extend our analysis to the quark-quark correlator, obtaining three new exact relations between the six leading-twist Transverse-Momentum–Dependent distributions

Del Dotto, A.

Pace, E.

Salmè, G.

Scopetta, S.

English

arXiv

A realistic study of the SiDIS process He(e, e π)X in the Bjorken
limit is briefly reviewed, showing that the nuclear effects, present in the extraction of the neutron information, are largely under theoretical control, within an Impulse- Approximation approach. In view of the forthcoming experimental data, we shortly present a novel Poincar ́e covariant description of the nuclear target, implementing a Light-Front analysis at finite Q2, within the Bakamijan-Thomas construction of the Poincar ́e generators. Furthermore, as a by-product of the introduction of a Light- Front spin-dependent spectral function for a J = 1/2 system, we straightforwardly extend our analysis to the quark-quark correlator, obtaining three new exact rela- tions between the six leading-twist Transverse-Momentum–Dependent distributions

Scopetta, S

Del Dotto, A

Salme`, G.

PACE, EMANUELE

Transversity studies with a polarized 3He target

Scientific Open-access Literature Archive and Repository

DOI 10.1393/ncc/i2012-11184-xColloquia: Transversity 2011IL NUOVO CIMENTO Vol. 35 C, N. 2 Marzo-Aprile 2012Transversity studies with a polarized 3He targetS. Scopetta(1), A. Del Dotto(2), E. Pace(3) and G. Salme`(2)(1) Dipartimento di Fisica, Universita` degli Studi di Perugia and INFN, Sezione di Perugiavia A. Pascoli, I-06100 Perugia, Italy(2) INFN, Sezione di Roma - P.le A. Moro 2, 00185 Rome, Italy(3) Physics Department, University of Rome “Tor Vergata” and INFNSezione di “Tor Vergata” - Via della Ricerca Scientifica 1, I-00133 Rome, Italyricevuto il 31 Ottobre 2011; approvato il 6 Dicembre 2011pubblicato online il 2 Marzo 2012Summary. — A realistic study of the SiDIS process 3He(e, e′π)X in the Bjorkenlimit is briefly reviewed, showing that the nuclear effects, present in the extractionof the neutron information, are largely under theoretical control, within an Impulse-Approximation approach. In view of the forthcoming experimental data, we shortlypresent a novel Poincare´ covariant description of the nuclear target, implementing aLight-Front analysis at finite Q2, within the Bakamijan-Thomas construction of thePoincare´ generators. Furthermore, as a by-product of the introduction of a Light-Front spin-dependent spectral function for a J = 1/2 system, we straightforwardlyextend our analysis to the quark-quark correlator, obtaining three new exact rela-tions between the six leading-twist Transverse-Momentum–Dependent distributions.PACS 13.60.Hb – Total and inclusive cross sections(including deep-inelastic processes).PACS 13.85.Hd – Inelastic scattering: many-particle final states.PACS 13.85.Ni – Inclusive production with identified hadrons.1. – IntroductionDue to ongoing and forthcoming challenging measurements in various Laboratories,the partonic structure of transversely polarized hadrons, not accessible in inclusive ex-periments, is widely studied by the hadronic Physics Community. In particular, semi-inclusive deep inelastic scattering (SiDIS) is one of the processes proposed to accesstransversity observables. The theoretical description of SiDIS implies a complicatedformalism, accounting for the transverse motion of the quarks in the target [1-4]. Inparticular, the non-perturbative effects of the intrinsic transverse momentum kT of thequarks inside the nucleon may induce significant hadron azimuthal asymmetries [5,6]. Inc© Societa` Italiana di Fisica 101102 S. SCOPETTA, A. DEL DOTTO, E. PACE and G. SALME`order to describe such quantities, a central role is played by the Transverse-Momentum–Dependent parton distributions (TMDs), that depend, besides on the fraction x of lon-gitudinal momentum of the target carried by the parton, as the standard parton distri-butions (PDs), also on its intrinsic transverse momentum. The number of TMDs, six atleading twist, is fixed by counting the scalar quantities allowed by hermiticity, parity andtime-reversal invariance. However, the existence of leading twist final state interactions(FSI) allows for two additional time-reversal odd functions [7], namely, the Sivers andthe Boer-Mulders functions [8, 9].The experimental scenario which arises from the analysis of SiDIS off transverselypolarized proton and deuteron targets is puzzling, showing a strong flavor dependence [10,11]. With the aim of extracting the neutron information and clarifying the situation, ameasurement of SiDIS off transversely polarized 3He has been proposed [12], and anexperiment, planned to measure azimuthal asymmetries in the production of leading π±from transversely polarized 3He, has been just completed at Jefferson Lab (JLab), witha beam energy of 6GeV [13]. Another experiment will be performed after upgrading theJLab beam to 12GeV [14,15]. In this contribution, a brief review of the realistic analysisof SiDIS off transversely polarized 3He, as presented in ref. [16], is given and it is extendedin order to implement both the Poincare´ covariant treatment of the nuclear target and thefinite values of the actual four-momentum transfer Q2. In particular, the Light-front (LF)spin-dependent spectral function for a generic J = 1/2 target is introduced, exploitingthe Bakamijan-Thomas (BT) construction of the Poincare´ generators (see, e.g. [17] for adetailed review) and it is applied to the quark-quark correlator, obtaining new relationsbetween the six leading-twist TMDs.2. – A realistic description of 3He Single Spin AsymmetriesIn ref. [16], within an impulse approximation (IA) framework, the formal expressionsof the Collins [18] and Sivers [8] contributions to the azimuthal Single Spin Asymmetry(SSA) for the production of leading pions off 3He have been derived. In particular, theinitial transverse momentum of the struck quark has been included, and, in order to give arealistic description of the nuclear dynamics, 3He SSAs have been calculated by using thenucleon spin-dependent spectral function (SF), that yields the probability distributionto find a nucleon with given three-momentum and missing energy inside a target nucleuswith polarization SA. The adopted SF was the one obtained in ref. [19] within a non-relativistic framework and corresponding to the AV18 nucleon-nucleon interaction [20].Moreover, the nucleonic structure, i.e. PDs and fragmentation functions, has been de-scribed by proper parametrizations of data [21] or suitable model calculations [22]. Inref. [16], the crucial issue of extracting the neutron information from 3 He data has beenalso discussed and it was showing that, even for SiDIS, where fragmentation functionsare present beside PD’s, one can adopt the same extraction scheme already successfullyapplied to DIS. In this case, in order to take effectively into account the momentum andenergy distributions of the polarized bound nucleons in 3He, it was adopted a modelindependent procedure, based on the realistic evaluation of the proton and neutron po-larizations in 3He [23], pp and pn, respectively. According to this extraction procedure,the neutron SSA, Ain, is obtained from the nuclear one, Aexp,i3 , as follows:(1) Ain 1pndn(Aexp,i3 − 2ppdpAexp,ip),TRANSVERSITY STUDIES WITH A POLARIZED 3He TARGET 103where dn(dp) is the neutron (proton) dilution factor. For a neutron n (proton p) in 3He,the diluition factor, experimentally known, is given by(2) dn(p)(x, z) =∑q e2qfq,n(p)(x)Dq,h(z)∑N=p,n∑q e2qfq,N (x)Dq,h(z),where fq,N (x) are the standard parton distributions and Dq,h(z) the fragmentation func-tions with z = Eh/ν (see [16] for details). In eq. (1), the nucleon effective polarizations,pp and pn take properly care of all the effects in 3He, like Fermi motion and binding,that produce a depolarization of the bound neutron. In ref. [16], it has been shown that,for any experimentally relevant x and z, the extraction scheme of eq. (1) is quite effec-tive. This important result is due to the peculiar kinematics of the JLab experiments,which helps in two ways. Firstly, it favors the emission of fast pions from the current-quark fragmentation, (z has been chosen in the range 0.45 < z < 0.6, which means thatonly high-energy pions are observed). Secondly, the pions are detected in a narrow conearound the direction of the momentum transfer, making nuclear effects in the fragmen-tation functions rather small [16]. Then, the leading nuclear effects result to be the onesaffecting the parton distributions, as already found in inclusive DIS [23], and they can betaken into account in the usual way, i.e. by using eq. (1) for the extraction of the neutroninformation. In conclusion, eq. (1) represents a valuable tool for the experimental dataanalysis [13,14].While the analysis in [16] was performed assuming the experimental set-up of theexperiment described in ref. [13], but using DIS kinematics, an analysis at the actual,finite values of the momentum and energy transfers, Q2 and ν, is in progress [24], andsome of the results are anticipated in the next section. Moreover, in order to improvethe relativized framework where the non relativistic SF of ref. [19] was embedded, thenew analysis is based on a fully Poincare´ covariant description of the nuclear effects, byintroducing a LF spin-dependent SF (see below).3. – A Light-Front description of 3 HeIn the previous Section, it has been reported that the extraction procedure of theneutron information, successfully applied in DIS, nicely works also in SiDIS, though thecalculation has been performed [16] in the Bjorken limit, using a non relativistic spin-dependent SF. Therefore, it is natural to test the extraction procedure in the actual JLabkinematics, rather far from the asymptotic one, and to investigate relativistic effects atnuclear level, in a more consistent way. In order to achieve this, we adopt the RelativisticHamiltonian Dynamics (RHD) framework [25, 17], suitable for describing an interactingsystem in a Poincare´ covariant way.The RHD, introduced by Dirac [25], can be combined with the Bakamijan-Thomasexplicit construction of the Poincare´ generators [26] for obtaining a description of SiDISoff 3He which: i) is fully Poincare´ covariant; ii) has a fixed number of on-mass-shellconstituents; iii) allows one to use the Clebsch-Gordan coefficients to decompose the wavefunction (a benefit from the BT construction). Obviously, this approach is not explicitlycovariant, since locality is lost. Between the three possible forms of RHD, the Light-Frontone has several advantages: i) seven Kinematical generators: three LF-boosts (at variancewith the dynamical nature of the Instant-form boosts), three components of the LF-momentum, P˜ ≡ {P+,P⊥}, and the rotation around the z-axis; ii) the LF-boosts have asubgroup structure, so that the separation of the intrinsic motion is trivially achieved (as104 S. SCOPETTA, A. DEL DOTTO, E. PACE and G. SALME`in the NR case); iii) P+ ≥ 0 leads to a meaningful Fock expansion, in presence of massiveboson exchanges; iv) there are no square roots in the operator P−, propagating the statein the LF-time; v) the Infinite Momentum frame description of DIS is easily included.The main drawback is that the transverse LF-rotations are dynamical, but within theBT construction, one can define a kinematical, intrinsic angular momentum, reobtainingthe rotational invariance even in presence of a truncated Fock space. This is a peculiarfeature of RHD, at variance with Quantum Field Theory where infinite degrees of freedomare present due to the locality. The LF Hamiltonian Dynamics (LFHD) framework hasbeen proven to be very suitable for phenomenological studies (see, e.g. [27, 28]).For SiDIS, the nuclear hadronic tensor plays a central role, and in IA it readsWμν(Q2, xB , z, τhf , hˆ, SHe) ∝∑σ,σ′∑τhf∫∑ maxSminSdS∫ (MX−MS)2M2NdM2f∫ ξupξlodξ(2π)3(3)× 1ξ2(1− ξ)∫ Pmax⊥Pmin⊥dP⊥sin θ(P+ + q+ − h+)wμνσσ′(τhf , q˜, h˜, P˜)Pτhfσ′σ (k, S , SHe).All the formal steps for obtaining eq. (3) and the explicit expression of the integrationlimits will be reported elsewhere [24]. Let us describe here only the two crucial termsappearing in the above equation. The first one is wμνσσ′(τhf , q˜, h˜, P˜), the SiDIS nucleonictensor, that depends not only on its spins σ, σ′ and its LF-momentum, P˜, but alsoon the isospin τhf and LF-momentum h˜ of the produced pseudo-scalar meson. Thesecond one is Pτhfσ′σ (k, S , SHe), i.e., the LF spin-dependent SF, describing a nucleon withCartesian momentum k inside a 3He with polarization SHe, when the spectator pair hasan excitation energy S . The LF spectral function is defined as(4) Pτσ′σ(k˜, S , SHe)∝∑σ1σ′1D12[R†M (k˜)]σ′σ′1Sτσ′1σ1(k˜, S , SHe)D12[RM (k˜)]σ1σ,with D12 [RM (k˜)] = m + k+ − ıσ · (zˆ × k⊥)/√(m + k+)2 + |k⊥|2 the unitary MeloshRotations and the instant-form SF given bySτσ′1σ1(k˜, S , SHe) =∑JSJzSα∑TSτS〈TS , τS , α, SJSJzS ;σ′1; τ,k|Ψ0SHe〉(5)×〈SHe,Ψ0|kσ1τ;JSJzSS , α, TS , τS〉 =[Bτ0,SHe(|k|, E) + σ · fτSHe(k, E)]σ′1σ1,where fτSHe(k, E) = SABτ1,SHe(|k|, E)+ kˆ(kˆ ·SA)Bτ2,SHe(|k|, E). It is tempting to approxi-mate the instant-form SF by the one evaluated in a non relativistic framework, since theBT construction imposes the same constraints adopted for the non relativistic Hamilto-nian [17].From eqs. (4) and (5), it follows that, remarkably, the constituent SF for a J = 1/2system is given in terms of only three independent functions (Bi, with i = 0, 1, 2), withina RHD framework.From the phenomenological side, it will be very important to use the LF nucleartensor, eq. (3), to evaluate the SSAs and to figure out whether or not the proposedextraction procedure still holds in the LF analysis. The preliminary calculations clearlyindicate that the effect of considering the integration limits evaluated in the actual JLabkinematics at 6GeV, and not in the Bjorken limit, as previously done, is negligible. Fromthis point of view, the situation will become even better in the experiments planned at12GeV.TRANSVERSITY STUDIES WITH A POLARIZED 3He TARGET 1054. – The Light-Front quark-quark correlatorIn general, the six leading-twist TMDs for a J = 1/2 system with 4-momentum Pand polarization S are introduced as a proper parametrization of the so called quark-quark correlator, Φ(k, P, S) for a quark of 4-momentum k, in order to fulfill all thegeneral symmetries (see, e.g., ref. [1]). In particular, one can starts with the followingcombinations of Dirac structures and scalar functions, Ai and A˜i:Φ(k, P, S) =12{A1/P + A2SLγ5/P + A3/Pγ5/S⊥(6)+1MA˜1k⊥ · S⊥γ5/P + A˜1SLM/Pγ5/k⊥ +1M2A˜1k⊥ · S⊥/Pγ5/k⊥}.Then, the six leading-twist TMDs, f1, g1L, g1T , h1, h1L, h1T , are identified as follows:12P+Tr(γ+Φ) = f1,(7a)12P+Tr(γ+γ5Φ) = SLg1L +1Mk⊥ · S⊥g1T ,(7b)12P+Tr(iσi+γ5Φ) = Si⊥h1 +SLMki⊥h⊥1L −1M2(ki⊥kj⊥ +12k2⊥gij⊥)S⊥,jh⊥1T .(7c)The LF SF introduced in the previous section for obtaining a Poincare´ covariant de-scription of the nucleon inside a polarized 3He, can be formally adopted for describinga quark inside a polarized nucleon. Namely, ceteris paribus, the LF nucleon spectralfunction, PN , is the analogous of Φ(k, P, S), within a Poincare´ covariant description ofthe nucleon. Then, one is able to perform the following identification:12Tr(PNI) = cB0 = fLF1 ,(8a)12Tr(PNσz) = Sz[a(B1 + B2 cos2 θ) + b cos θ|k⊥|2kB2](8b)+S⊥ · k⊥[aB2cos θk+ b(B1 + B2 sin2 θ)]= SLgLF1L +1MS⊥ · k⊥gLF1T ,12Tr(PNσy) = Sy[(a +d2|k⊥|2)B1 +12(a− bk2⊥ cos θk)B2](8c)+Szky[acos θkB2 − b(B1 + B2 cos2 θ)]+(kxkySx − 12k2⊥Sy)[(ak2− bcos θk)B2 − dB1]= SyhLF1 +SLMkyh⊥LF1L +1M2(kxkySx − 12k2⊥Sy)h⊥LF1T .Therefore, the six TMDs depend actually upon three independent functions, within aLFHD framework. The quantities a, b, c and d, appearing in eqs. (8a)-(8c), are kine-matical factors, predicted by the LF procedure (details will be given elsewhere [29]).106 S. SCOPETTA, A. DEL DOTTO, E. PACE and G. SALME`5. – ConclusionsA realistic study of 3He(e, e′π)X in the Bjorken limit has been summarized, and thepreliminary results of its generalization within a Light-Front analysis at finite Q2 havebeen shortly discussed. The spin-dependent LF spectral function for a J = 1/2 systemhas been presented and, using the BT construction of the Poincare´ generators, an intrigu-ing simplification in the theoretical description of SiDIS is found. Three exact relationsare established between the six T-even TMDs. As a future step, it will be investigatedif similar relations can be found in other theoretical frameworks and eventually in theanalysis of the experimental results.A detailed analysis of these new relations will be given elsewhere [29].∗ ∗ ∗This work has been partially supported by the Italian MUR through the PRIN 2008.One of us, SS, thanks the organizers of the Conference for the kind invitation.REFERENCES[1] Barone V., Drago A. and Ratcliffe P., Phys. Rep., 359 (2002) 1.[2] Bacchetta A., Diehl M., Goeke K., Metz A., Mulders P. J. and Schlegel M.,JHEP, 0702 (2007) 093.[3] D’Alesio U. and Murgia F., Prog. Part. Nucl. Phys., 61 (2008) 394.[4] Pasquini B., Cazzaniga S. and Boffi S., Phys. Rev. D, 78 (2008) 034025.[5] Mulders P. J. and Tangerman R. D., Nucl. Phys. B, 461 (1996) 197 (484 (1997)538(E)).[6] Cahn R. N., Phys. Lett. B, 78 (1978) 269.[7] Brodsky S. J., Hwang D. S. and Schmidt I., Phys. Lett. B, 530 (2002) 99.[8] Sivers D. W., Phys. Rev. D, 41 (1992) 83; 43 (1991) 261.[9] Boer D. and Mulders P. J., Phys. Rev. D, 57 (1998) 5780.[10] Airapetian A. et al. (HERMES Collaboration), Phys. Rev. Lett., 94 (2005) 012002.[11] Alexakhin V. Y. et al. (COMPASS Collaboration), Phys. Rev. Lett., 94 (2005)202002.[12] Brodsky S. J. and Gardner S., Phys. Lett. B, 643 (2006) 22.[13] Qian X. et al., Phys. Rev. Lett., 107 (2011) 072003.[14] Gao H. et al., Eur. Phys. J. Plus, 126 (2011) 2.[15] Cates G. et al., E12-09-018, JLAB approved experiment, hallaweb.jlab.org/collab/PAC/PAC38/E12-09-018-SIDIS.pdf; Del Dotto A., Master Thesis, Universita`di Roma “La Sapienza” (2011) (unpublished).[16] Scopetta S., Phys. Rev. D, 75 (2007) 054005.[17] Keister B. D. and Polyzou W. N., Adv. Nucl. Phys., 21 (1991) 225.[18] Collins J. C., Nucl. Phys. B, 396 (1993) 161.[19] Kievsky A., Pace E., Salme` G. and Viviani M., Phys. Rev. C, 56 (1997) 64.[20] Wiringa R. B., Stocks V. G. J. and Schiavilla R., Phys. Rev. C, 51 (1995) 38.[21] Anselmino M., Boglione M., D’Alesio U., Kotzinian A., Murgia F. and ProkudinA., Phys. Rev. D, 71 (2005) 074006; 72 (2005) 094007 (72 (2005) 099903(E)).[22] Amrath D., Bacchetta A. and Metz A., Phys. Rev. D, 71 (2005) 114018.[23] Ciofi degli Atti C., Scopetta S., Pace E. and Salme` G., Phys. Rev. C, 48 (1993)968.[24] Del Dotto A., Salme` G. and Scopetta S., in preparation.[25] Dirac P. A. M., Rev. Mod. Phys., 21 (1949) 392.[26] Bakamijan B. and Thomas L. H., Phys. Rev., 92 (1953) 1300.[27] Lev F. M., Pace E. and Salme` G., Phys. Rev. C, 62 (2000) 064004.[28] Cardarelli F., Pace E., Salme` G. and Simula S., Phys. Lett. B, 357 (1995) 267.[29] Pace E., Salme` G. and Scopetta S., in preparation.

http://eprints.bice.rm.cnr.it/17741/1/ncc10197.pdf

Transversity studies with a polarized 3He target

Abstract

Similar works

Full text

Available Versions

ART

Scientific Open-access Literature Archive and Repository