Final state interactions and the extraction of neutron single spin asymmetries from SIDIS by a transversely polarized $^3$He target

TL;DR

This paper develops a realistic model incorporating final state interactions to improve the extraction of neutron single spin asymmetries from SIDIS experiments using a transversely polarized helium-3 target.

Contribution

It introduces a distorted spin-dependent spectral function using a generalized eikonal approximation to account for nuclear effects beyond the plane wave impulse approximation.

Findings

Enhanced accuracy in neutron Sivers and Collins asymmetry extraction.

Quantitative assessment of nuclear effects on asymmetry measurements.

Applicability to Jefferson Lab and Electron Ion Collider kinematics.

Abstract

The semi-inclusive deep inelastic electron scattering off transversely polarized $^3$He, i.e. the process, $e + \vec{^3 {\rm He}} \to e' + h+X$, with $h$ a detected fast hadron, is studied beyond the plane wave impulse approximation. To this end, a distorted spin-dependent spectral function of a nucleon inside an A=3 nucleus is actually evaluated through a generalized eikonal approximation, in order to take into account the final state interactions between the hadronizing system and the (A-1) nucleon spectator one. Our realistic description of both nuclear target and final state is a substantial step forward for achieving a reliable extraction of the Sivers and Collins single spin asymmetries of the free neutron. To illustrate how and to what extent the model dependence due to the treatment of the nuclear effects is under control, we apply our approach to the extraction procedure of the neutron single spin asymmetries from those measured for $^3$He for values of the kinematical variables relevant both for forthcoming experiments at Jefferson Lab and, with an exploratory purpose, for the future Electron Ion Collider.

Tables

Table 1. Table 1: The PWIA values of the dilution factors d n ​ ( p ) ​ ( x B ​ j , z ) subscript 𝑑 𝑛 𝑝 subscript 𝑥 𝐵 𝑗 𝑧 d_{n(p)}(x_{Bj},z) and their product with the corresponding effective polarizations, in PWIA, for the kinematical conditions of the planned experiments at JLab, with scattering angle θ e = 30 o subscript 𝜃 𝑒 superscript 30 𝑜 \theta_{e}=30^{o} and detected pion angle θ π = 14 o subscript 𝜃 𝜋 superscript 14 𝑜 \theta_{\pi}=14^{o} . The effective polarizations are evaluated with extrema of integrations depending upon the kinematics. At Q 2 = 5.73 superscript 𝑄 2 5.73 Q^{2}=5.73 (GeV/c) 2 , i.e. the central Q 2 superscript 𝑄 2 Q^{2} value for an energy beam ℰ = 8.8 ℰ 8.8 {\cal{E}}=8.8 GeV, one obtains p n subscript 𝑝 𝑛 p_{n} = 0.876, p p subscript 𝑝 𝑝 p_{p} = -0.024 (cf Eq. ( 78 )), very close to the corresponding asymptotic values 0.878 and -0.024 (i.e. in the Bjorken limit).

$ℰ$,		$x_{Bj}$		$\nu$		$P_{\pi }$		$d_n(x_{Bj},z)$		$p_n{d}_n$		$d_p(x_{Bj},z)$		$p_p{d}_p$
GeV				GeV		GeV/c
8.8		0.21		7.55		3.40		0.304		0.266		0.348		-${8.410}^{-3}$
8.8		0.29		7.15		3.19		0.286		0.251		0.357		-${8.510}^{-3}$
8.8		0.48		6.36		2.77		0.257		0.225		0.372		-${8.910}^{-3}$
11		0.21		9.68		4.29		0.302		0.265		0.349		-${8.310}^{-3}$
11		0.29		9.28		4.11		0.285		0.250		0.357		-${8.510}^{-3}$

Table 2. Table 2: The same as in Table I, but taking into account FSI within GEA framework. For all the presented kinematical conditions, one gets distorted polarizations , evaluated by using Eq. ( 78 ), with distorted distributions. They amount to p n ≃ 0.756 similar-to-or-equals subscript 𝑝 𝑛 0.756 p_{n}\simeq 0.756 , p p ≃ − 0.0265 similar-to-or-equals subscript 𝑝 𝑝 0.0265 p_{p}\simeq-0.0265 for Q 2 = 5.73 superscript 𝑄 2 5.73 Q^{2}=5.73 (GeV/c) 2 , i.e. the central Q 2 superscript 𝑄 2 Q^{2} value for an energy beam ℰ = 8.8 ℰ 8.8 {\cal{E}}=8.8 GeV. In these conditions one gets, for the quantities < N n > expectation subscript 𝑁 𝑛 <N_{n}> and < N p > expectation subscript 𝑁 𝑝 <N_{p}> , Eq. ( 76 ), the values 0.85 and 0.87, respectively.

$ℰ$,		$x_{Bj}$		$\nu$		$P_{\pi }$		$d_n(x_{Bj},z)$		$p_n{d}_n$		$d_p(x_{Bj},z)$		$p_p{d}_p$
GeV				GeV		GeV/c
8.8		0.21		7.55		3.40		0.353		0.267		0.405		-$1.1\cdot {10}^{-2}$
8.8		0.29		7.15		3.19		0.332		0.251		0.415		-$1.1\cdot {10}^{-2}$
8.8		0.48		6.36		2.77		0.298		0.225		0.432		-$1.2\cdot {10}^{-2}$
11		0.21		9.68		4.29		0.351		0.266		0.405		-$1.0\cdot {10}^{-2}$
11		0.29		9.28		4.11		0.331		0.250		0.415		-$1.1\cdot {10}^{-2}$

Equations

$$\frac{d\sigma }{d{\phi }_{\ell }d{x}_{Bj}dyd{\mathbf{P}}_h}=\frac{{\alpha }_{em}^2{m}_N}{Q^4}y\frac{1}{2E_h}{L}^{\mu \nu }{W}_{\mu \nu }^{s.i.}({\mathbf{S}}_A,Q^2,P_h),$$

$$L_{\mu \nu }=2\left[k_{\mu }{k}_{\nu }^{\prime }+k_{\mu }^{\prime }{k}_{\nu }-(k\cdot k^{\prime })g_{\mu \nu }\right].$$

$$W_{\mu \nu }^{s.i.}({\mathbf{S}}_A,Q^2,P_h)$$

$$|P_h,X{⟩}^{PWIA}=|P_{A-1}⟩\otimes |P_h⟩\otimes |X^{\prime }⟩,$$

$$\sum _{\lambda }\int \frac{d{\mathbf{p}}_N}{2E_N(2\pi {)}^3}|\lambda ,p_N⟩⟨\lambda ,p_N|=1,$$

$$\sum _{f_{A-1}}\sum {\int }_{{ϵ}_{A-1}^{\ast }}\rho \left({ϵ}_{A-1}^{\ast }\right)\int \frac{d{\mathbf{P}}_{A-1}}{2E_{A-1}(2\pi {)}^3}|{\Phi }_{{ϵ}_{A-1}^{\ast }}^{f_{A-1}},{\mathbf{P}}_{A-1}⟩⟨{\Phi }_{{ϵ}_{A-1}^{\ast }}^{f_{A-1}},{\mathbf{P}}_{A-1}|=1,$$

$${\rho }_{2bbu}=\frac{1}{(2\pi {)}^3},{\rho }_{3bbu}=\frac{1}{(2\pi {)}^6}\frac{m_N\sqrt{m_N{ϵ}_2^{\ast }}}{2},$$

$$\sum _{X}d{\tau }_X\to \sum _{X^{\prime }}d{\tau }_{X^{\prime }}\sum _{f_{A-1}}\sum {\int }_{{ϵ}_{A-1}^{\ast }}\rho \left({ϵ}_{A-1}^{\ast }\right)\int \frac{d{\mathbf{P}}_{A-1}}{2E_{A-1}(2\pi {)}^3},$$

$$W_{\mu \nu }^{s.i.;IA}({\mathbf{S}}_A,Q^2,P_h)=\sum _{X^{\prime },\lambda {\lambda }^{\prime }}\sum _N\int dE{P}_{\lambda {\lambda }^{\prime }}^{N{\mathbf{S}}_A}(E,{\mathbf{p}}_N)\frac{1}{2E_N}⟨{\lambda }^{\prime },p_N|{\hat{j}}_{\mu }^N|P_h,X^{\prime }⟩⟨P_h,X^{\prime }|{\hat{j}}_{\nu }^N|\lambda ,p_N⟩$$

$$\times {\delta }^4\left(P_A+q-P_{A-1}-P_h-P_{X^{\prime }}\right)d{\tau }_{X^{\prime }}\frac{d{\mathbf{P}}_{A-1}}{(2\pi {)}^3},$$

$$⟨{\Phi }_{{ϵ}_{A-1}}^{\ast f_{A-1}},{\mathbf{P}}_{A-1};\lambda ,p_N|{\mathbf{S}}_A,P_A⟩=\sqrt{2E_{N}2E_{A-1}2M_A}(2\pi {)}^3\times$$

$$\delta \left({\mathbf{P}}_A-{\mathbf{P}}_{A-1}-{\mathbf{p}}_N\right)⟨{\Phi }_{{ϵ}_{A-1}}^{\ast f_{A-1}};\lambda ,{\mathbf{p}}_N|{\mathbf{S}}_A,{\Phi }_A⟩.$$

$$P_{\lambda {\lambda }^{\prime }}^{N{\mathbf{S}}_A}(E,{\mathbf{p}}_N)=\sum _{f_{A-1}}\sum {\int }_{{ϵ}_{A-1}^{\ast }}\rho \left({ϵ}_{A-1}^{\ast }\right){\mathcal{O}}_{\lambda {\lambda }^{\prime }}^{N{\mathbf{S}}_A}({ϵ}_{A-1}^{\ast },{\mathbf{p}}_N)\delta \left(E+M_A-m_N-M_{A-1}^{\ast }\right),$$

$${\mathcal{O}}_{\lambda {\lambda }^{\prime }}^{N{\mathbf{S}}_A}({ϵ}_{A-1}^{\ast },{\mathbf{p}}_N)=⟨{\Phi }_{{ϵ}_{A-1}^{\ast }}^{f_{A-1}},\lambda ,{\mathbf{p}}_N|{\mathbf{S}}_A,{\Phi }_A⟩⟨{\mathbf{S}}_A,{\Phi }_A|{\Phi }_{{ϵ}_{A-1}^{\ast }}^{f_{A-1}},{\lambda }^{\prime },{\mathbf{p}}_N⟩.$$

$$\frac{1}{2}\sum _{\lambda N}\int dE\int d{\mathbf{p}}_N{P}_{\lambda \lambda }^{N{\mathbf{S}}_A}(E,{\mathbf{p}}_N)=1.$$

$$W_{\mu \nu }^{s.i.;IA}({\mathbf{S}}_A,Q^2,P_h)=\sum _{\lambda {\lambda }^{\prime }}\sum _N\int d{\mathbf{p}}_N\int dE\frac{m_N}{E_N}{w}_{\mu \nu }^{Ns.i.}({\tilde{p}}_N,P_h,{\lambda }^{\prime }\lambda )P_{\lambda {\lambda }^{\prime }}^{N{\mathbf{S}}_A}(E,{\mathbf{p}}_N),$$

$$w_{\mu \nu }^{Ns.i.}({\tilde{p}}_N,P_h,{\lambda }^{\prime }\lambda )=\frac{1}{2m_N}\sum _{X^{\prime }}⟨p_N,{\lambda }^{\prime }|{\hat{j}}_{\mu }^N|P_h,X^{\prime }⟩⟨P_h,X^{\prime }|{\hat{j}}_{\nu }^N|p_N,\lambda ⟩{\delta }^4\left({\tilde{p}}_N+q-P_h-P_{X^{\prime }}\right)d{\tau }_{X^{\prime }},$$

$${\sigma }^{A;IA}({\mathbf{S}}_A)=\sum _{\lambda {\lambda }^{\prime }}\sum _N\int d{\mathbf{p}}_N\int dE\frac{\tilde{\alpha }{m}_N}{E_N}{\sigma }_{\lambda {\lambda }^{\prime }}^N{P}_{\lambda {\lambda }^{\prime }}^{N{\mathbf{S}}_A}(E,{\mathbf{p}}_N),$$

$${\sigma }_{\lambda {\lambda }^{\prime }}^N\equiv \frac{d{\sigma }_{\lambda {\lambda }^{\prime }}^N}{d{\phi }_{\ell }d{x}_{Bj}dyd{\mathbf{P}}_h}=\frac{{\alpha }_{em}^2{m}_N}{Q^4}\frac{m_N\nu }{(p_N\cdot k)}\frac{1}{2E_h}{L}^{\mu \nu }{w}_{\mu \nu }^{Ns.i.}({\tilde{p}}_N,P_h,{\lambda }^{\prime }\lambda )$$

$$\tilde{\alpha }\equiv \frac{(p_N\cdot k)}{\mathcal{E}{m}_N}$$

$$\underset{\to Bj}{\lim }\tilde{\alpha }=\frac{A(p_N\cdot q)}{(P_A\cdot q)}.$$

$$W_{\mu \nu }^{s.i.}({\mathbf{S}}_A,Q^2,P_h)=\frac{1}{2M_A}\sum _X⟨S_A,P_A|{\hat{J}}_{\mu }({\hat{\mathbf{r}}}_i)|P_h,X⟩⟨P_h,X|{\hat{J}}_{\nu }({\hat{\mathbf{r}}}_i)|S_A,P_A⟩\delta \left(M_A+\nu -E_X-E_h\right)d{\tau }_X,$$

$${\hat{J}}_{\mu }({\hat{\mathbf{r}}}_1,{\hat{\mathbf{r}}}_2,{\hat{\mathbf{r}}}_3)\approx \sum _i{\hat{j}}_{\mu }({\mathbf{r}}_{\mathbf{i}}),$$

$$⟨P_h,X|{\hat{J}}_{\mu }({\hat{\mathbf{r}}}_1,{\hat{\mathbf{r}}}_2,{\hat{\mathbf{r}}}_3)|{\mathbf{S}}_3,P_3⟩=⟨{\Psi }^f(1,2,3)|{\hat{J}}_{\mu }({\hat{\mathbf{r}}}_1,{\hat{\mathbf{r}}}_2,{\hat{\mathbf{r}}}_3)|{\Psi }_3^{{\mathbf{S}}_3}(1,2,3)⟩\approx \sum _i⟨{\Psi }^f(1,2,3)|{\hat{j}}_{\mu }({\hat{\mathbf{r}}}_i)|{\Psi }_3^{{\mathbf{S}}_3}(1,2,3)⟩=$$

$$=3⟨{\Psi }^f(1,2,3)|{\hat{j}}_{\mu }({\hat{\mathbf{r}}}_1)|{\Psi }_3^{{\mathbf{S}}_3}(1,2,3)⟩.$$

$$⟨{\mathbf{r}}_1{\mathbf{r}}_2{\mathbf{r}}_3|{\Psi }^f(1,2,3)⟩\approx \frac{\mathcal{A}}{\sqrt{V}\sqrt{3}}{\Psi }_{23}^f({\mathbf{r}}_2,{\mathbf{r}}_3){\chi }_{{\lambda }_Y}\varphi ({\xi }_Y)\sqrt{2E_Y}{e}^{i{\mathbf{p}}_Y{\mathbf{r}}_1}\mathcal{G}({\mathbf{r}}_1,{\mathbf{r}}_2,{\mathbf{r}}_3),$$

$$⟨{\mathbf{r}}_1^{\prime },{\mathbf{r}}_2^{\prime },{\mathbf{r}}_3^{\prime }|\hat{\mathcal{G}}|{\mathbf{r}}_1,{\mathbf{r}}_2,{\mathbf{r}}_3⟩=\delta ({\mathbf{r}}_1^{\prime }-{\mathbf{r}}_1)\delta ({\mathbf{r}}_2^{\prime }-{\mathbf{r}}_2)\delta ({\mathbf{r}}_3^{\prime }-{\mathbf{r}}_3)\mathcal{G}({\mathbf{r}}_1,{\mathbf{r}}_2,{\mathbf{r}}_3).$$

$$\displaystyle{\bf r}_{1}=\frac{2}{3}{\mbox{\boldmath$\rho$}}+{\bf R}~{},$$

$$\displaystyle{\bf r}_{2}=-\frac{1}{3}{\mbox{\boldmath$\rho$}}+\frac{1}{2}{\bf r}+{\bf R}~{},$$

$$\displaystyle{\bf r}_{3}=-\frac{1}{3}{\mbox{\boldmath$\rho$}}-\frac{1}{2}{\bf r}+{\bf R}~{},$$

$$\displaystyle{\cal G}({\bf r}_{1},{\bf r}_{2},{\bf r}_{3})~{}\to~{}{\cal G}({\bf r},{\mbox{\boldmath$\rho$}})~{}.$$

$$\displaystyle\langle\,{\bf r}_{1}{\bf r}_{2}{\bf r}_{3}|\Psi^{f}(1,2,3)\,\rangle\approx{{\cal A}\over\sqrt{V}~{}\sqrt{3}}~{}{{\sqrt{2E_{Y}}}}~{}e^{i{\bf p}_{Y}(2{\mbox{\boldmath$\rho$}}/3+{\bf R})}~{}\chi_{\lambda_{Y}}\phi(\xi_{Y})~{}\sqrt{2E_{23}}e^{i{\bf{P}}_{23}({\bf R}-{\mbox{\boldmath$\rho$}}/3)}~{}\phi_{\epsilon^{*}_{23}}^{f_{23}}({\bf r})~{}{\cal G}({\bf r},{\mbox{\boldmath$\rho$}})~{},$$

$$⟨{\Psi }^f(1,2,3)|{\hat{j}}_{\mu }({\hat{\mathbf{r}}}_1)|{\Psi }_3^{{\mathbf{S}}_3}(1,2,3)⟩\approx \frac{1}{\sqrt{3}\sqrt{V}}\sqrt{2E_Y}$$

Full text

I Introduction

In recent years, special efforts on both experimental and theoretical sides have been focused on semi-inclusive deep inelastic scattering (SIDIS), i.e. the process $A(l,l^{\prime}h)X$ where, in the final state, a scattered lepton $l^{\prime}$ and a hadron $h$ are detected in coincidence, after the interaction of a lepton $l$ with a hadronic system $A$. Nowadays, it is clear that inclusive deep inelastic scattering (DIS), i.e. the process $A(l,l^{\prime})X$, despite of intense experimental investigations in the last decades, cannot answer to a few crucial questions on hadron structure. Indeed, at least three long standing problems cannot be explained through DIS measurements, namely: (i) the fully quantitative description of the so-called EMC effect (i.e. the modification of the nucleon partonic structure due to the nuclear medium EMC ); (ii) the solution of the so-called “spin crisis”, i.e. the fact that the nucleon spin does not originate from only the spins of its valence quarks polEMC ; (iii) the measurement of the chiral-odd parton distribution function (PDF) called transversity (see, e.g., Refs. barone ; barone1 and references therein quoted) that complements the leading-twist collinear description of a polarized nucleon. As it is well-known, transversity is related to the amount of transversely-polarized quark inside a transversely-polarized nucleon and it is not measurable in DIS, where a flip of the quark chirality cannot take place. Through DIS processes, on both proton (see, e.g., Refs. unpol ; polarized ) and nuclear targets (see, e.g., Refs. Arneodo ; Piller ; emclast ), it is possible to investigate only partonic distributions of longitudinal momentum (i.e. parallel to the direction of the incoming lepton) and helicity. Therefore, in order to access information on the transverse structure of the target, either in coordinate or momentum space, one necessarily has to go beyond DIS measurements (see, e.g. Refs. markus and 3Dmio for recent reviews on nucleon and nuclear targets, respectively).

SIDIS processes are an important tool for increasing our knowledge on hadron dynamics. Indeed, if the detected hadron is fast, it likely originates from the fragmentation of the active quark, after absorbing the virtual photon. Hence, the detected hadron opens a valuable window on the motion of quarks inside the parent nucleon, before the interaction with the photon occurs. In particular their transverse motion, not seen in the collinear case, represents the subject of intense experimental efforts in the study of SIDIS reactions, through which one can access the so-called transverse-momentum-dependent parton distributions (TMDs) (see, e.g., Ref. barone1 ). Those distributions provide a wealth of information on the partonic dynamics, eventually shedding light on the challenging three issues listed above. Beside the main topic represented by TMDs, one should remind that the detected hadron carries also information on the hadronization mechanism itself. The SIDIS cross sections can be parametrized, at leading twist, by six TMDs; this number reduces to three in the collinear case (with only two TMDs measurable in DIS barone1 ) and increases to eight once the so-called time-reversal odd TMDs (i.e. the Sivers sivers and Boer-Mulders boemu functions) are considered barone1 .

It should be emphasized that, in order to experimentally investigate the wide field of TMDs, one should measure cross-section asymmetries, using different combinations of beam and target polarizations (see, e.g., Ref. d'alesio ). Moreover, for completing the study of TMDs, one should achieve a sound flavor decomposition, possible only by collecting a detailed knowledge of the neutron TMDs. The present investigation moves from the observation that free neutron targets are not available and nuclei have to be used as effective neutron targets. In particular, the study of the neutron spin structure is highly favored by choosing a polarized 3He target, as it has been done extensively in DIS studies. In the 90’s, procedures to extract the neutron spin-dependent structure functions from 3He data in the DIS regime, taking properly into account Fermi-motion and binding effects, were proposed antico and successfully applied (see, e.g. appl ). Such a detailed description of the target nucleus was obtained in plane wave impulse approximation (PWIA) by using the so-called spin-dependent spectral function, whose diagonal elements yield the probability distribution to find a nucleon with a given momentum, missing energy and polarization inside the nucleus. It is worth noting that, within PWIA, accurate 3He spin-dependent spectral functions, based on realistic calculations of both the target nucleus and the spectator pair in the final state (fully interacting through the $NN$ interaction adopted for 3He), have been built and used in the last twenty years cda ; SS ; cda1 ; pskv ; umnikovkapt ; plbold .

The question whether similar procedures can be extended to SIDIS is of great relevance, due to the several experiments that exploit a polarized 3He target (see, e.g., Ref. He3exp ), for accessing the transverse momentum and spin of the partons inside the neutron. For instance, a wide interest has arisen about the possibility to use a transversely polarized 3He target for measuring azimuthal single-spin asymmetries (SSAs) of the neutron, which are sensitive to time-reversal odd TMDs and to the Collins fragmentation functions (FF) collins generated by leading twist final state interactions Brodsky:2002cx . In the first measurements of SSAs, through SIDIS off transversely polarized proton and deuteron targets, the proton SSAs were found to be sizable hermes , while those of deuteron were found to be negligible compass , pointing to a large cancellation between the proton and neutron contributions. A high luminosity environment coupled to a suitable neutron target, as a polarized 3He (at level of 90 % an effective neutron target), allows one first to better assess the flavor separation and then accurately test its sensitivity to quark angular momenta. It became clear the need of increasing the experimental knowledge on neutron TMDs through an independent measurement, and an experiment of SIDIS off transversely polarized 3He was soon proposed bro . As it is well-known, some significant steps have been already carried out along the suggested path, since azimuthal asymmetries in the production of leading $\pi^{\pm}$ $(K^{\pm})$ from transversely polarized 3He have been already measured at Jefferson Lab (JLab), with a beam energy of 6 GeV prljlab and new experiments will be soon performed after completing the 12 GeV upgrade jlab12 .

In view of those experimental efforts, a realistic PWIA analysis of SIDIS off transversely polarized 3He has been performed mio . A realistic spin-dependent spectral function, corresponding to the nucleon-nucleon AV18 interaction av18 , has been used for the description of nuclear dynamics and the issue of the extraction of the neutron information from 3He data has been addressed. According to Ref. mio , one can safely extend to SIDIS, where both PDFs and FFs are involved, the model-independent extraction procedure based on the realistic evaluation of the proton and neutron polarizations in 3He and widely used in inclusive DIS appl . As a matter of fact, such an extraction procedure is able to take into account effectively the momentum and energy distributions of the polarized bound nucleons in 3He.

In general, SIDIS off nuclear targets can happen through at least two, rather different, sets of processes:

the standard reaction (most familiar), where a fast hadron is detected mainly in the forward direction, implying that the hadron has been produced by the leading quark. Therefore, this reaction, representing the dominant mechanism in the kinematics of the Jlab experiments of Refs. prljlab ; jlab12 , can be used to investigate TMDs inside the hit nucleon; 2. 2.

the spectator SIDIS, where a slow $(A-1)$ nucleon system, acting as a spectator of the photon-nucleon interaction, is detected, while the produced fast hadron is not.

The spectator SIDIS process has been proven very useful to investigate the unpolarized DIS functions $F_{1,2}(x)$ of a bound nucleon, and therefore to clarify the origin of the EMC effect (see, e.g., ourlast ; scopetta ; ckk ; smss ; veronica ). At the same time, this process can provide also useful information on quark hadronization in medium, complementary to that obtained so far by the standard SIDIS process. Noteworthy, the polarization degrees of freedom of the target substantially enrich the wealth of information one can gather, as shown in Ref. nostro , where a spectator SIDIS, with a detected deuteron, off a polarized 3He target was studied. Through such a polarized SIDIS, one can obtain fresh information on the spin-dependent structure functions $g_{1,2}(x)$ for bound nucleons and, ultimately, on the origin of the polarized EMC effect.

In polarized (as well as unpolarized) SIDIS processes, the effects of the final state interaction (FSI) that occur among the hadronizing system (produced after the quark-photon knock-out) and the $(A-1)$ spectator system has to be carefully analyzed. For the case of a polarized 3He, this study started in Ref. nostro , where the trinucleon distorted spin-dependent spectral function has been introduced, but restricted to the deuteron spectator system. In order to realistically take into account the above mentioned FSI, it has been adopted a generalized eikonal approximation (GEA), i.e. a framework successfully introduced for describing unpolarized SIDIS off nuclei ourlast . To apply such a distorted spin-dependent spectral function to the standard polarized SIDIS by ${}^{3}\vec{\rm He}$, one has to consider all the possible states of the two-nucleon spectator system. But due to FSI between the spectator system and the quark debris, produced after DIS off an internal nucleon with given polarization, this novel distribution function is remarkably more complicated than the PWIA spin-dependent spectral function, adopted in the description of both DIS by unpolarized 3He antico and SIDIS mio . However, efforts for evaluating a realistic distorted spin-dependent spectral function are worth attempting since its thorough knowledge represents a fundamental help for reliably disentangling TMDs from the nuclear structure, in the experimental cross sections. In perspective, an experimental check of the robustness of the description of the nuclear effects could be in principle carried out by exploiting the isodoublet nature of the trinucleon bound states. In the case of a polarized 3H, one could extract (i) the proton polarized structure functions when a spectator SIDIS is considered or (ii) the relevant TMDs when a standard SIDIS is investigated. The proton information extracted from 3H could be compared with the ones gathered using free proton targets, shedding light on the relevance and nature of nuclear effects. Nowadays, the use of a polarized 3H target seems too challenging, but it is worth mentioning that important achievements have been obtained in the last decade in handling such a problematic target, as demonstrated by the final approval (with scientific rating A), at JLab, of an experiment dedicated to DIS by a 3H target marathon .

As a concluding remark, it should be pointed out that, at the present stage, the needed relativistic description of SIDIS is restricted to the kinematics and the elementary cross-section, as discussed in the following Sections. Indeed, in order to embed the very successful non relativistic phenomenology of the nuclear structure, developed over the past decades, in a fully Poincaré covariant approach, one could exploit the Light-front framework, that originates from the seminal work by P.A.M. Dirac on the forms of relativistic Hamiltonian dynamics Dirac . A thorough formal investigation of a Light-front spin-dependent spectral function for a $J=1/2$ target, in impulse approximation, has been recently presented in Ref. tobe (see also lussino ; Pacelc16 for preliminary results). Obviously, this novel distribution function is the first step for constructing a Poincaré covariant description of SIDIS reactions, since in analogy with the transition from the PWIA spectral function to the distorted one, FSI effects have to be taken into account also in the Poincaré covariant approach.

Aims of the present paper are first to extend the calculation of the distorted spin-dependent spectral function of 3He performed in Ref. nostro , in order to include the excited states of the two-nucleon spectator system (recall that in Ref. nostro only the deuteron state was retained). As a second step, we apply our formalism to the standard SIDIS process, with kinematical conditions typical of experiments to be performed in the next years at JLab and in the future (possibly near) at the electron ion collider (EIC), focusing on the extraction of quark TMDs inside the neutron, i.e. the needed ingredients for making complete the flavor decomposition. One can easily realize that, since in standard SIDIS the final fast hadronic state can re-interact with a two-nucleon scattering state, this process is much more involved than spectator SIDIS, where FSIs occur between the final hadronic state and the detected deuteron.

The paper is organized as follows. In Section II we present the basic formalism for the cross section, valid for the standard SIDIS process, where a hadron $h$ is detected in coincidence with the scattered charged lepton. The main quantities relevant for the calculations are presented and the PWIA framework is reviewed, to better appreciate the difference with the FSI case, discussed in the next Sections. In Section III, the SIDIS reaction $\vec{{}^{3}{\rm He}}(e,e^{\prime}h)X$ is investigated in detail, introducing the distorted spin-dependent spectral function, that represents the main ingredient of our method for implementing FSI effects, through a generalized eikonal approximation. In Section IV, the dependence of the nuclear hadronic tensor upon the target polarization is studied. In Section V the expressions to be used for evaluating the nuclear SSAs, both in PWIA and with FSI taken into account, are presented and a strategy for the extraction of the neutron information is discussed. In Section VI, the results for the distorted spectral functions and light-cone momentum distributions are presented and compared with the corresponding PWIA calculations; furthermore the finite values of the momentum and energy transfers corresponding to the actually proposed experiments are adopted for the evaluation of the 3He Collins and Sivers asymmetries and for the extraction of neutron asymmetries with FSI effects taken into account and implementing the comparison with the PWIA calculations. Eventually, in the last Section, conclusions are drawn and perspectives presented. Important formal details are collected in two appendixes.

II The SIDIS cross section

The differential cross section for the generic SIDIS process off a polarized target $A$, i.e. $l+\vec{A}=l^{\prime}+h+X$ when the final pseudoscalar hadron $h$ is detected, can be written in the laboratory frame and in one-photon exchange approximation as follows (cf, e.g., Refs. barone1 ; scopetta ; nostro ),

[TABLE]

where, for incoming and outgoing charged leptons with 4-momentum $k^{\mu}=({\cal E},\vec{k})$ and $k^{\prime\mu}=({\cal E^{\prime}},\vec{k}^{\prime})$, one has $Q^{2}=-q^{2}=-(k-k^{\prime})^{2}=\vec{q}^{\,\,2}-\nu^{2}=4{\cal E}{\cal E}^{\prime}\sin^{2}(\theta_{\ell}/2)$, i.e. the square 4-momentum transfer in ultrarelativistic approximation (with $\vec{q}=\vec{k}-\vec{k}^{\prime}$, $\nu={\cal E}-{\cal E}^{\prime}$ and $\theta_{\ell}\equiv\theta_{\widehat{\vec{k}\vec{k}^{\prime}}}$). Moreover, $x_{Bj}=Q^{2}/(2m_{N}\nu)$ is the Bjorken scaling variable, $y=\nu/{\cal E}$, $m_{N}$ the nucleon mass, $\alpha_{em}$ the electromagnetic fine structure constant, $\varphi_{\ell}$ the azimuthal angle of the detected charged lepton, $P_{h}=(E_{h},{\bf{P_{h}}})$ the 4-momentum of the detected-hadron $h$, with mass $m_{h}$ and ${\bf{S}}_{A}$ the polarization vector of the target nucleus.

The unpolarized leptonic tensor $L_{\mu\nu}$ is an exactly calculable quantity in QED. In the ultrarelativistic limit it gets the form

[TABLE]

The semi inclusive (s.i.) hadronic tensor of the target with polarization four-vector $S_{A}$ and mass $M^{2}_{A}=P^{2}_{A}$ is defined as

[TABLE]

where the covariant normalization $\langle\,p|p^{\prime}\,\rangle=2E(2\pi)^{3}\delta\left({\bf p-p}^{\prime}\right)$ has been assumed and $d\tau_{X}$ is the suitable phase-space factor for the undetected state $X$, given in turn by a state $X^{\prime}$ with baryon number 1 and an $A-1$ recoiling nuclear system. One should notice that, in Eq. (3), the integration over the phase-space volume of the detected hadron, $h$, does not have to be performed.

In the following, the cross section for SIDIS off transversely polarized 3He will be worked out, taking into account final state interaction effects. To this aim, it is necessary first to recall the results obtained in PWIA.

Within PWIA, the nuclear tensor Eq. (3) is approximated using the following assumptions: (i) the nuclear current operator is written as the sum of single nucleon operators $\hat{j}_{\mu}^{N}$; (ii) the FSI between the debris originating by the struck nucleon and the fully interacting (A-1) nuclear system is disregarded, as suggested by the kinematics of the process under investigation; (iii) the coupling of the virtual photon with the $(A-1)$ system is disregarded, due to the large 4-momentum transferred in the process; (iv) the effect of boosts is not considered (they will be properly taken into account in a Light-front framework elsewhere, following the procedure addressed in Refs. tobe ; lussino ; Pacelc16 ). In this way, the complicated final baryon states $|P_{h},X\rangle$ are approximated by a tensor product of hadronic states, viz

[TABLE]

where $|P_{A-1}\,\rangle$ indicates the state (properly antisymmetrized) of the fully-interacting $(A-1)$-nucleon system, which acts merely as a spectator, $|X^{\prime}\,\rangle$ describes the baryonic state, that originates together with $|P_{h}\,\rangle$ from the hadronization of both the quark which has absorbed the virtual photon and the other colored remnants. The nuclear tensor $W_{\mu\nu}^{s.i.}({\bf{S}}_{A},Q^{2},P_{h})$ can be related therefore to the one of a single nucleon. This is obtained inserting in Eq. (3) complete sets of nucleon plane waves and $(A-1)$-nucleon interacting states, given by

[TABLE]

[TABLE]

where $p_{N}\equiv\{{{E}}_{N}=\sqrt{m^{2}_{N}+|{\bf p}_{N}|^{2}},{\bf p}_{N}\}$ is the on-shell four-momentum of a nucleon, $\Phi_{\epsilon^{*}_{A-1}}^{f_{A-1}}$ is the intrinsic part of the $(A-1)$-nucleon state with quantum numbers ${f_{A-1}}$ and energy eigenvalue $\epsilon^{*}_{A-1}$. Moreover, $E_{A-1}=\sqrt{(M^{*}_{A-1})^{2}+|{\bf P}_{A-1}|^{2}}$ with $M^{*}_{A-1}=Z_{A-1}m_{p}+(A-1-Z_{A-1})m_{n}+\epsilon^{*}_{A-1}$. The symbol with the sum overlapping the integral indicates that the $(A-1)$ system has both discrete and continuum energy spectra: this corresponds to negative and positive values of the eigenvalue $\epsilon^{*}_{A-1}$. In Eq. (6), $\rho\left(\epsilon^{*}_{A-1}\right)$ is the proper state density, that for $A=3$ reads

[TABLE]

with the labels $2bbu$ and $3bbu$ indicating the two-body and three-body break-up channels, respectively. Furthermore, recalling that Eq. (4) implies

[TABLE]

one obtains the following expression for the nuclear tensor in PWIA

[TABLE]

where, w.r.t. Eq. (3), $P_{X^{\prime}}+P_{A-1}$ is in place of $P_{X}$, and the nucleon three-momentum, ${\bf p}_{N}={\bf P}_{A}-{\bf P}_{A-1}$, is fixed by the translation invariance of the initial nuclear vertex, viz

[TABLE]

In Eq. (10), $\Phi_{A}$ is the intrinsic wave function of the target nucleus, with mass $M_{A}$ and $(P_{A}-P_{A-1})^{2}\neq m^{2}_{N}$.

The matrix elements ${P}^{N\,{\bf S}_{A}}_{\lambda\lambda^{\prime}}(E,{\bf p}_{N})$ in Eq. (9) contain the description of the nuclear structure and are given in PWIA by

[TABLE]

where $E$ is the usual missing or removal energy, $E=M^{*}_{A-1}+m_{N}-M_{A}=\epsilon^{*}_{A-1}+B_{A}$, with $B_{A}$ the binding energy of the target nucleus. The quantity $m_{N}-E$ is the off-shell mass of a nucleon inside the target nucleus, when the $(A-1)$ system acts as a spectator. In Eq. (11), ${\cal O}^{N\,{\bf S}_{A}}_{\lambda\lambda^{\prime}}(\epsilon^{*}_{A-1},{\bf p}_{N})$ is the following product of PWIA overlaps

[TABLE]

The quantities ${P}^{N\,{\bf S}_{A}}_{\lambda\lambda^{\prime}}(E,{\bf p}_{N})$, Eq. (11), are the matrix elements of the $2\otimes 2$ spin-dependent spectral function of a nucleon inside the nucleus $A$, with polarization ${\bf S}_{A}$ cda1 . The trace of the spectral function yields the probability distribution to find a nucleon in the nucleus $A$ with three-momentum ${\bf p}_{N}$, removal energy $E$ and spin projection equal to $\lambda$. The suitable normalization is

[TABLE]

Assuming the polarized target in a pure state, the nuclear wave function has definite spin projections on the spin quantization axis, chosen as usual along the polarization vector ${\bf S}_{A}$. In agreement with the definition of the spin-dependent spectral function given in Refs. SS ; cda1 , in the complete set of the nucleon plane waves, the spin projections $\lambda$ and $\lambda^{\prime}$ are defined with respect to the $z$ axis.

As for the Cartesian coordinates, we adopt the DIS convention, i.e. the $z$ axis is directed along the three-momentum transfer ${\bf q}$ and the plane $(x,z)$ is the scattering plane. Notice that, in the DIS limit, the direction of the three-momentum transfer coincides with that of the lepton beam, i.e. ${\bf q}\ ||\ {\bf k}_{e}$.

The nuclear tensor Eq. (9) can be written

[TABLE]

where the integration over ${\bf P}_{A-1}$ has been changed to the one over ${\bf p}_{N}={\bf P}_{A}-{\bf P}_{A-1}$, and the semi-inclusive nucleon tensor (cf. Eq. (3)) is given by

[TABLE]

where $\tilde{p}_{N}=P_{A}-P_{A-1}$ is such that $\tilde{p}^{2}_{N}\neq m^{2}_{N}={p}^{2}_{N}$.

Eventually, for the nuclear cross section given in Eq. (1), $\sigma_{A}({\bf S}_{A})\equiv~{}\displaystyle\frac{d\sigma({\bf S}_{A})}{d\varphi_{\ell}dx_{Bj}dyd{\bf P}_{h}}$, one gets the following expression in PWIA

[TABLE]

where

[TABLE]

represents the corresponding cross section for the scattering of a charged lepton from a polarized moving nucleon. In Eq. (16), $\tilde{\alpha}$ is given by

[TABLE]

and it is usually called the ”flux factor”. When energies are close to the Bjorken limit $\tilde{\alpha}$ coincides with the light-cone momentum fraction of the nucleon inside the nucleus, i.e.

[TABLE]

III The distorted spin-dependent spectral function

In order to go beyond PWIA (cf. Eq. (4)), it is necessary to deal with the FSI between the debris, originating from the struck nucleon, and the fully interacting (A-1) nuclear system. In view of this, the dependence upon the space coordinates in the current operator is kept, since we will focus on the action of the current onto the final state in coordinate space.

The starting point is the hadronic tensor written as follows

[TABLE]

For a 3He target, the matrix element of the current operator $\hat{J}_{\mu}(\hat{\bf r}_{1},\hat{\bf r}_{2},\hat{\bf r}_{3})$ between the nuclear ground state, $|\Psi_{3}^{{\bf S}_{A}}(1,2,3)\,\rangle$, and a generic final state, $|\Psi^{f}(1,2,3)\,\rangle$, is evaluated by introducing the following approximation

[TABLE]

where $\hat{j}_{\mu}(\bf{r}_{i})$ is the one-body transition current operator, that describes the electromagnetic response of the single nucleon inside the target. In this way the matrix element becomes

[TABLE]

In what follows, for the sake of concreteness, the active nucleon is labeled ”i=1” and the spectator indexes are ”23”.

For constructing a realistic approximation of FSI, it is useful to consider that, in SIDIS processes, we aim at investigating, the momentum transfer ${\bf q}$ is rather large, and therefore $h$, the leading pseudoscalar meson to be detected, and $X^{\prime}$, that has baryon number equal to $1$ (cf Eq. (4)), move throughout the $A-1$ remnants with high velocity. This observation motivates the introduction of the generalized eikonal approximation (see, e.g., Refs. nostro ; ourlast and references quoted therein) for estimating the rest of FSI not taken into account through PWIA (cf Eq. (4)). Then, the final state can be approximated in coordinate space as

[TABLE]

where ${\cal A}$ is the antisymmetrization operator that acts on the final state, given by a recoiling two-nucleon system and a debris $Y$ originated by the struck nucleon (see below), $\Psi_{23}^{f}({\bf r}_{2},{\bf r}_{3})$ is the properly antisymmetrized wave function of the recoiling two-nucleon system, $V$ is the normalization volume of the global motion of the final state, and the amplitude ${\cal G}({\bf r}_{1},{\bf r}_{2},{\bf r}_{3})$, identically equal to 1 in PWIA, is the non singular part of the matrix elements of the Glauber operator, i.e.

[TABLE]

The Glauber amplitude depends only upon intrinsic coordinates, {\bf r},{\mbox{\boldmath\rho}}, related to ${\bf r}_{i}$ through

[TABLE]

and therefore

[TABLE]

In Eq. (23), $Y$ is the final debris produced by the nucleon after the absorption of the virtual photon. In the process under consideration, it coincides with a leading pseudoscalar meson to be detected and a baryonic remnant $X^{\prime}$ (cf. Fig. 1). The function $\phi(\xi_{Y})$ characterizes the internal structure of the debris that will hadronize in $h+X^{\prime}$, $\chi_{\lambda_{Y}}$ its spin state, while $e^{i{\bf p}_{Y}{\bf r}_{1}}$ is the plane wave describing the propagation of the c.m. of the debris.

By using intrinsic coordinates, the final state in Eq. (23) becomes

[TABLE]

where ${\bf{P}}_{23}$ is the total momentum of the $(2,3)$ system and the intrinsic part of the two-nucleon state, $\phi_{\epsilon^{*}_{23}}^{f_{23}}({\bf r})$, has quantum numbers ${f_{23}}$ and energy eigenvalue $\epsilon^{*}_{23}$.

Disregarding the photon coupling to the spectator pair, one can apply the familiar approximation

[TABLE]

with

[TABLE]

where $\psi_{3}^{{\bf{S}}_{3}}({\bf{r}},{\bf{\rho}})$ is the intrinsic nuclear wave function and the total momentum of the nucleus is ${\bf{P}}_{3}=0$.

Moreover, if ${\cal G}({\bf r}_{1},{\bf r}_{2},{\bf r}_{3})$ is such that: (i) it does not depend upon spins and (ii) it commutes with $\hat{j}_{\mu}({\bf r}_{1})$ (as it does in PWIA, since ${\cal G}({\bf r}_{1},{\bf r}_{2},{\bf r}_{3})\equiv 1$ ), one can write

[TABLE]

This is the main assumption of our approach, that is exact when the one-body operator $\hat{j}_{\mu}$ does not contain the momentum $\hat{\bf p}$. Otherwise one can have a non-zero commutator $[\hat{j}_{\mu},{\cal G}]$. In the present SIDIS case, the explicit expression of the transition current operator $\hat{j}_{\mu}$ is unknown and we cannot compute the commutator, but we assume a vanishing result, namely [\hat{\bf p},{\cal G}(1,2,3)]\sim\partial/\partial{{\mbox{\boldmath\rho}}}~{}{\cal G}({\bf r},{{\mbox{\boldmath\rho}}})\sim 0. It is worth noting that if only the longitudinal part of the current operator is relevant and the dependence on the coordinates in the Glauber operator is mainly given by the transverse components, one can largely justify our assumption. As a matter of fact, we adopt in the following the same approach used in Ref. nostro , where the distorted spectral function was evaluated only in the 2bbu channel. This amounts to consider GEA (see, e.g., Ref. ourlast and references therein). In this scheme, the Glauber amplitude reads

[TABLE]

where the parallel and perpendicular components of the vectors ${\bf r}_{i}$ are determined with respect to ${\bf{p}}_{Y}$, i.e. to the direction of propagation of the debris. In DIS, when $|{\bf q}|^{2}\gg|{\bf p}_{N}|^{2}$, this direction coincides with the direction of ${\bf q}$. The profile function $\Gamma\left({\bf r}_{i\perp}-{\bf r}_{1\perp},{\bf r}_{i||}-{\bf r}_{1||}\right)$ in Eq. (31), unlike in the standard Glauber approach, depends not only upon the transverse relative separation but also upon the longitudinal one. The Heavyside function $\theta({\bf r}_{i||}-{\bf r}_{1||})$ assures causality in the re-scattering process. In the following we adopt for $\Gamma\left({\bf r}_{i\perp}-{\bf r}_{1\perp},{\bf r}_{i||}-{\bf r}_{1||}\right)$ the expression already used in Refs. nostro ; ourlast , based on the hadronization model of Ref. ckk to evaluate the total cross section of the debris-nucleon interaction, depending on the kinematics of the process, viz

[TABLE]

In this approach, the resulting Glauber operator turns out to be mildly dependent on the longitudinal distance, so that the assumption of a vanishing commutator between the operator and the current is qualitatively justified in the present scheme. Details on the model and on the corresponding parameters can be found in Refs. nostro ; ourlast .

An important issue has now to be addressed. The effective cross section, $\sigma_{eff}$, in Eq. (32), models the hadronization of the debris interacting with the recoiling nuclear system. The debris consists of one nucleon and radiated mesons and gluons. The number of radiated gluons depends on the momentum scale of the process, given by $Q^{2}$. Besides, the emission of mesons and gluons will stop when a maximum longitudinal distance is reached, which increases with the invariant mass, $W_{Y}$, of the debris. As a consequence, $\sigma_{eff}$ depends also on $W_{Y}$. Therefore, in Eq. (32) one should write $\sigma_{eff}({\bf r}_{i||}-{\bf r}_{1||},Q^{2},W_{Y})$ and not simply $\sigma_{eff}({\bf r}_{i||}-{\bf r}_{1||})$. Nevertheless, in the kinematics we are going to discuss in this paper it occurs that: i) for a given value of ${\cal{E}}$, the range of variation of $Q^{2}$ is not wide enough to produce important changes in the gluon radiation rate; ii) $\sigma_{eff}$ depends weakly on the maximum longitudinal distance. In other words, in the kinematics we are going to analyze, for a given ${\cal{E}}$, the dependence of $\sigma_{eff}$ on $Q^{2}$ and $W_{Y}$ is weak. As a matter of facts, in Refs. ourlast ; nostro , $\sigma_{eff}({\bf r}_{i||}-{\bf r}_{1||},Q^{2},W_{Y})\simeq\sigma_{eff}({\bf r}_{i||}-{\bf r}_{1||})$ was assumed in actual calculations. In Ref. ourlast , the model of $\sigma_{eff}$ with this assumption was proven to be able to reasonably describe data of Ref. sigmaeff for unpolarized spectator SIDIS processes, in a kinematics which is close to the one we are discussing. Therefore, to avoid a too heavy notation, throughout the paper we drop the dependence of $\sigma_{eff}$ on $Q^{2}$ and $W_{Y}$ in the relevant expressions.

For completeness we mention that, in the actual form for {\cal G}({\bf r},{{\mbox{\boldmath\rho}}}), Eq. (31), there is a theta-function that generates a contribution to the commutator proportional to \delta^{3}({\mbox{\boldmath\rho}}). Obviously, such a contribution is vanishing if not too much severe singularities are present in both target and spectator wave functions. It is worth noticing that in the quasi-elastic case, where an explicit form of the current operator is commonly accepted, the above assumption, called the factorized form of FSI has been discussed against the unfactorized one in Ref. unfact .

Coming back to Eq. (30) and following the spirit of the standard procedure adopted in PWIA, one can insert the one-nucleon completeness (cf Eq. (5))

[TABLE]

where $I$ is the identity, and the free nucleon states $|k,\lambda\,\rangle$ are normalized according to $\langle\,k,\lambda|k^{\prime},\lambda^{\prime}\,\rangle=2E_{k}\delta_{\lambda\lambda^{\prime}}~{}{(2\pi)^{3}\delta({\bf k}-{\bf k}^{\prime})}$. Then, one can obtain from Eq. (30) the following expression

[TABLE]

By changing coordinates, (see Eq. (27)), and exploiting the translation invariance of the initial vertex in Eq. (10), one gets for Eq. (34)

[TABLE]

where $\langle\,p_{Y},\lambda_{Y}\phi(\xi_{Y})|\hat{j}_{\mu}(0)|k,\lambda\,\rangle$ is the matrix element of the unknown transition current operator involved in the quark-photon vertex (notice that the factor $\sqrt{2E_{Y}2E_{k}}$ is put inside the matrix element), and $p_{mis}\equiv\{E_{mis}=\sqrt{M^{2}_{N}+|{\bf p}_{mis}|^{2}},~{}-{\bf p}_{mis}\}$, with ${\bf p}_{mis}={\bf q}-{\bf p}_{Y}={\bf P}_{23}$ the three-momentum of the system ”23” in the final state, $\Psi_{23}^{f}({\bf r}_{2},{\bf r}_{3})$. Indeed, ${\bf p}_{mis}$ is also the three-momentum of the initial spectator system and eventually of the nucleon (with opposite sign) before absorbing the virtual photon. This is a consequence of the assumed commutativity between the one-body current and the Glauber amplitude. It should be pointed out that the matrix element $\langle\,p_{Y},\lambda_{Y}\phi(\xi_{Y})|\hat{j}_{\mu}(0)|p_{mis}\lambda\,\rangle$ describes SIDIS off a free nucleon, within our approach.

Summarizing the above results and recalling that $X\to(A-1)\otimes Y\to(A-1)\otimes h\otimes X^{\prime}$, one can write the hadron tensor for a polarized 3He target as follows

[TABLE]

where ${\bf p}_{Y}={\bf p}_{h}+{\bf X^{\prime}}$ has been inserted and the following phase space of the spectator system has been adopted

[TABLE]

In conclusion, the nuclear hadronic tensor reads

[TABLE]

where the semi-inclusive nucleon tensor (cf. Eq. (15)) is given by

[TABLE]

with $\tilde{p}_{mis}\equiv\{E-m_{N},{\bf p}_{mis}\}$ and the isospin formalism has been released (i.e. $3\to\sum_{N}$). In Eq. (38), it has been introduced the distorted spin-dependent spectral function given by the following expression for a polarized 3He target

[TABLE]

with the product of distorted overlaps defined by

[TABLE]

with an obvious meaning of the adopted notation (see the Appendix A for the detailed expression of the overlaps).

One shoud notice that the distorted spectral function depends, through the profile function Eq. (32), on the effective cross section $\sigma_{eff}({\bf r}_{i||}-{\bf r}_{1||})$. As discussed above, below Eq. (32), this quantity depends, in principle, also on $Q^{2}$ and $W_{Y}$. As a consequence, the distorted spectral function is a process dependent quantity, at variance with the spectral function evaluated in PWIA. In principle, at any kinematical point (given by ${\cal E},\theta_{e},x_{Bj}$, and $\theta_{p_{mis}q}$) one should evaluate a different distorted spectral function. Nevertheless, for the reasons discussed below Eq. (32), in the kinematics we are going to study, for a fixed initial electron energy ${\cal E}$ and scattering angle $\theta_{e}$ the dependence of $\sigma_{eff}$ on $Q^{2}$ and $W_{Y}$ is rather mild and can be disregarded. As a consequence, also the spectral function, for fixed ${\cal E}$ and $\theta_{e}$, can be considered independent on $x_{Bj}$ and $\theta_{p_{mis}q}$. To avoid a too heavy notation, this dependence is not shown throughout the paper.

The generalization of the above formalism to a polarized nuclear target with $A$ nucleon is straightforward. In particular, for the nuclear cross section $\sigma^{A}({\bf S}_{A})\equiv~{}\displaystyle\frac{d\sigma({\bf S}_{A})}{d\varphi_{e}dx_{Bj}dyd{\bf P}_{h}}$ one has

[TABLE]

One should notice that, formally, Eq. (38) coincides with Eq. (14), relative to the PWIA case, if the distorted spectral function is substituted by the PWIA one. This is a consequence of the assumption made between Eqs. (28) and (30), concerning the commutation property of the Glauber operator with the nucleon current. The FSI described in this manner, called factorized FSI in the literature (see, e.g. Ref. unfact and references therein), lead to convolution-like formulas, as the ones obtained in the PWIA case, where the distorted spectral function appears instead of the PWIA one. The latter can be recovered just putting the Glauber operator identically equal to 1. This observation has crucial consequences in the following sections of the present paper.

IV The dependence of the nuclear hadronic tensor

upon the target nucleus polarization

As a matter of facts, the whole formalism developed in the PWIA case in Ref. mio can be exploited now in the present scenario, once the distorted overlaps are properly evaluated and inserted in the relevant equations.

Notice that, in PWIA, the spectral function $P^{{\bf S}_{3}}_{\lambda\lambda}(E,{\bf p}_{mis})$ in (40) defines the probability to remove from a polarized 3He with polarization ${\bf S}_{3}$ a polarized nucleon with momentum $-{\bf p}_{mis}$ and polarization ${\bf s}_{N}$ (characterized by spin projection $\lambda$ on the quantization axis) leaving the remnant $(A-1)$ system with removal energy $E$. Once the full FSI is taken into account, even through GEA, the probabilistic interpretation of the distorted spectral function is somehow lost.

A further issue is represented by the fact that the direction of the target polarization-axis, ${\bf S}_{3}$, may not always be parallel to the direction which determines the eikonal ${\cal G}$-matrix, i.e. the direction of ${\bf p}_{Y}$ (or, in DIS, the direction of q). In particular, in the SIDIS process of interest here, the target nucleus is transversely polarized, i.e. ${\bf S}_{3}\perp{\bf q}$. To reconcile the polarization axis and the eikonal approximation, one needs to rotate the quantization axis of the target wave function from the direction of ${\bf q}$ to the direction of the polarization ${\bf S}_{3}$, namely

[TABLE]

where the subscript indicates the direction of the quantization axis, $\cos\beta={\hat{\bf S}_{3}}\cdot\hat{\bf q}$ and the polarization vector ${\bf S}_{3}$ is supposed to be in the $(x,z)$ plane. In Eq. (43),

$D^{1/2}_{\sigma^{\prime}\sigma}$ are the suitable Wigner D-functions varsha . Therefore in the general case, the nuclear tensor in Eq. (38) is modified and reads

[TABLE]

[TABLE]

In the above equations, we have defined

[TABLE]

where the third components $M$ and $M^{\prime}$ are defined with respect to the direction $\hat{\bf q}$. In Eq. (46) one has (cf Eq. (40))

[TABLE]

with $\tilde{\cal O}_{\lambda\lambda^{\prime}}^{N\,M\,M^{\prime}\,f_{23}}$, a natural non-diagonal generalization of Eq. (41), viz

[TABLE]

It is worth noticing that, in Eq. (44), the upper scripts $\frac{1}{2}\frac{1}{2}$ $\left(-\frac{1}{2}-\frac{1}{2}\right)$ denote a nucleus polarized along (opposite) the quantization-axis, while $\pm\frac{1}{2}\,\mp\frac{1}{2}$ indicate a nucleus polarized in the perpendicular (wrt the quantization-axis) plane, i.e., in our case, along the $x$-axis.

Let us consider first a longitudinally polarized nucleus; in this case, we have to consider in Eq. (44) only the terms with $M=M^{\prime}=\pm 1/2$. One gets the following longitudinal contribution to the hadronic tensor

[TABLE]

In Eq. (49), $w_{\mu\nu}^{N\,\lambda\lambda^{\prime}}$ is a short-hand notation for $w_{\mu\nu}^{N\,s.i.}(p_{mis},P_{h},\lambda^{\prime}\lambda)$, previously used. In the SIDIS process under investigation, since leptons are unpolarized, the leptonic tensor is symmetric and, as a consequence, only the symmetric part of the hadronic spin-dependent tensor, $w_{\mu\nu}^{sN\lambda\lambda^{\prime}}$, is involved. For the diagonal terms of the symmetric part of the nucleon tensor (see, e.g., Ref. barone for its general structure), one gets

[TABLE]

while for the off-diagonal terms one has

[TABLE]

Then, making use of the properties under complex conjugation of the quantities (48), defined with respect to the quantization axis, namely

[TABLE]

one obtains

[TABLE]

In Eq. (54) the first term in square brackets represents the parallel spin-dependent spectral function.

We are interested in single spin asymmetries measured with transversely polarized targets. The relevant hadronic tensor is therefore

[TABLE]

where we choose ${\bf S}_{\perp}$ along the $x$ axis, i.e. $\beta=90^{o}$. Then, using Eqs. (44) and (45), the quantity relevant to describe the JLAB experiments turns out to be

[TABLE]

Therefore, we have to evaluate

[TABLE]

Therefore one obtains, for the term in the last line of Eq. (57),

[TABLE]

where the relations (51) and (53) have been used.

In Appendix B it is shown that the contribution of the last line in Eq. (58) can be safely neglected, being of higher order in ${\bf p}_{\perp}/m_{N}$, where ${\bf p}_{\perp}$ is the nucleon transverse-momentum inside the target, with ${\bf p}=-{\bf p}_{mis}$. Besides, in the remaining expression, only the zero order term in ${\bf p}_{\perp}/m_{N}$ yields a sizable contribution. Hence, ${\bf p}_{\perp}$ does not give relevant contributions to the hadronic tensor, and the expression of the nucleon hadronic tensor obtained in a collinear frame, where $\bf p_{\perp}=0$, for example the one given in Ref. barone for the Collins process (cf section 6.5), can be safely used. As a consequence, the final expression for the nuclear hadronic tensor, suitable for calculations of SSAs, reads:

[TABLE]

where Eqs. (50) and (52) have been used to obtain the last term.

In Eq. (59), the transverse spectral function has been introduced

[TABLE]

and the quantity

[TABLE]

has been defined. Furthermore, in Eq. (59), the transverse-longitudinal spectral function,

[TABLE]

is a real quantity which represents, in PWIA, the probability to find a longitudinally polarized nucleon in a transversely polarized nucleus. It should be pointed out that, in PWIA, the transverse spectral function ${\cal P}^{N\,\perp}(E,{\bf p}_{mis})$ yields the probability to find a transversely polarized nucleon in a transversely polarized nucleus with a polarization vector ${\bf S}_{3}$ along the $x$-axis.

For the nuclear cross section Eq. (42) one gets

[TABLE]

where $\sigma^{N\,\perp}$ and $\sigma^{N\,||}$ are the cross sections Eq. (17) for transversely and longitudinally polarized nucleons, respectively.

Note also that, in PWIA, one has

[TABLE]

where ${P}^{N\,||}(E,p,\theta)$ is the spin-dependent spectral function considered, for example, in Ref. mio . It has to be pointed out that ${P}^{N\,\perp}\neq{P}^{N\,||}$ in the relativistic case (see, e.g., Ref. Pacelc16 ).

V The Collins and Sivers asymmetries for 3He

As discussed in the Introduction, a series of SIDIS experiments are planned at JLab, using a transversely polarized 3He target and an unpolarized electron beam, detecting a fast pion (kaon) in the final state. The Sivers and Collins SSAs of 3He will be therefore measured, with the aim of extracting the corresponding neutron quantities. The formal results of the present approach for the 3He SSAs, and for the extraction of the neutron information, are presented in this Section.

The Sivers and Collins asymmetries are defined through proper moments of the experimental SIDIS cross sections, viz

[TABLE]

where $\phi_{h}$ is the azimuthal angle between the hadron and the lepton planes, $\phi_{S_{3}}$ is the azimuthal angle between the target polarization and the lepton plane, according to the conventions fixed in Ref. trento ; $z=E_{h}/\nu$ is the fraction of energy transfer carried by the detected meson. Inserting the cross section Eq. (63) in the above equation, one gets

[TABLE]

where ${\cal E}$ is the energy of the incoming lepton (see below Eq. (1)) and $f^{\perp,i}_{p(n)}(\alpha,Q^{2},{\cal E})$ are the light-cone momentum distributions of transversely polarized nucleons in a transversely polarized nucleus for $i=$ PWIA or FSI. One defines

[TABLE]

where

[TABLE]

with $W_{Y}$ the invariant mass of the debris $Y$, that hadronizes in a nucleon and, at least, one pseudoscalar meson. For the sake of definiteness, in Eq. (68) and in what follows we consider a $\pi^{-}$ in the final state. Let us recall that in the unpolarized case, the light-cone momentum distributions read

[TABLE]

with

[TABLE]

where ${{\cal P}^{Ni}(E,{\bf p}_{mis})}=\sum_{\lambda}{\cal P}_{\lambda\lambda}^{Ni}$. In Eqs. (68) and (70), the delta function can be eliminated by integrating over the angle between ${\bf{p}}_{mis}$ and ${\bf q}$; the limits of integration on $|{\bf p}_{mis}|$, i.e. $|{\bf p}_{min}|$ and $|{\bf p}_{max}|$, and on $E$, $E_{min}$ and $E_{max}$, are determined from the condition $|\cos\theta_{pq}|\leq 1$ and, from the requirement $W_{Y}^{2}\geq(m_{N}+m_{\pi})^{2}$, since we consider SIDIS with at least one pion in the final state. As a consequence, $|{\bf p}_{min}|$ and $|{\bf p}_{max}|$ are functions of $\alpha,E,Q^{2},{\cal E}$. One should notice that, in the Bjorken limit, they would be functions of $\alpha$ and $E$ only. In Eqs. (67) and (69), one has $E_{min}=B_{3}-B_{2}\sim 5.5~{}MeV$.

Moreover, as explained in the previous section, one can obtain the distributions for the two cases, $i=$PWIA, FSI, just substituting, in the same equations, the corresponding spectral functions ${\cal P}^{N\,i}(E,{\bf p}_{mis})$ and ${\cal P}^{N\perp\,i}(E,{\bf p}_{mis})$. The evaluation of ${\cal P}^{N\,i}(E,{\bf p}_{mis})$, when both the nuclear structure and the effects of FSI are included, is the main technical achievement of this paper. Actual numerical results, based on (i) two and three nucleon wave functions pisa evaluated with the nucleon-nucleon AV18 interaction av18 , and (ii) the GEA mechanism, are discussed in detail in the following Section. In what follows, when the distorted spectral functions will be considered in Eqs. (68) and (70), we will call the distribution functions in Eqs. (67) and (69) distorted light-cone momentum distributions (see Appendix B).

In Eq. (65), one should notice that, after multiplying the nuclear hadronic tensor by $\sin(\phi_{{S}_{3}}\pm\phi_{h})$ and integrating over $\phi_{{S}_{3}}$, the transverse-longitudinal term in Eq. (63) does not contribute to the numerators in the asymmetries above defined, due to the properties of the spin-dependent SIDIS nucleon tensor barone1  .

In Eq. (66), the quantities $\Delta\sigma_{Col(Siv)}^{N}$ and $\sigma^{N}$, related to the structure of the bound nucleon, are defined as follows (see, e.g., barone1 )

[TABLE]

[TABLE]

[TABLE]

In the last three equations, the quantities ${\bf k}_{T}$ and {{\mbox{\boldmath\kappa}}_{T}} are the intrinsic transverse momenta of the parton in the bound nucleon and in the produced hadron, respectively; following the notation of SIDIS, a subscript $T$ means transverse with respect to ${\bf{P}}_{h}$ (the three-momentum of the final pion or kaon), while the subscript $\perp$ means transverse with respect to ${\bf{q}}$. The transverse momentum dependent parton distributions, $h_{1}^{q,N}(x_{Bj},{\bf k}_{T}^{2})$, $f_{1T}^{\perp q,N}(x_{Bj},{\bf k}_{T}^{2})$, $f_{1}^{q,N}(x_{Bj},{\bf k}_{T}^{2})$, and the transverse momentum dependent fragmentation functions, D_{1}^{q,h}(z,(z{{\mbox{\boldmath\kappa}}_{T})^{2}}), H_{1}^{\perp q,h}(z,(z{{\mbox{\boldmath\kappa}}_{T})^{2}}), appearing in Eqs. (71), (72) and (73), have been evaluated using experimental data whenever possible, or using proper model estimates. One should realize that the main goal of the present study is the estimate of nuclear effects in the extraction of the neutron information, rather than obtaining absolute predictions on the SSAs of 3He, which would be affected anyhow by the poor present knowledge of some of the distributions necessary to perform the actual calculation. Any reasonable choice of the distribution functions of the nucleon is therefore suitable for our study. In particular, in the actual calculations we have made use of the same functions adopted in Ref. mio , namely:

for the unpolarized parton distribution, $f_{1}^{q,N}(x_{Bj})$, it has been used the parametrization of Ref. gluu , with a gaussian ansatz for the ${\bf k}_{T}$ dependence; 2. 2.

for the transversity distribution, $h_{1}^{q,N}$, it has been exploited the ansatz $h_{1}=g_{1}$, i.e., the transversity distribution has been taken to be equal to the helicity distribution. This gives certainly the correct order of magnitude. In particular, the parametrization of Ref. glup has been used; 3. 3.

for the Sivers function, $f_{1T}^{\perp q}(x_{Bj},{\bf k}_{T}^{2})$ in Eq. (72), it has been adopted the fit proposed in Ref. ans ; 4. 4.

for the unpolarized fragmentation function $D_{1}^{q,h}(z)$, different models are used for evaluating the Sivers and Collins asymmetries. In particular, for the Sivers asymmetry, the parametrization in Ref. kret has been used while, for the Collins one, the model calculation of Ref. amr has been adopted (see mio for details); 5. 5.

for the basically unknown Collins fragmentation function, H_{1}^{\perp q}(z,(z{{\mbox{\boldmath\kappa}}}_{T})^{2}), appearing in Eq. (71), the model calculation of Ref. amr has been used.

Equation (66) has been presented in Ref. mio within PWIA. As already noticed, within GEA the theoretical expression of the nuclear asymmetries does not formally change in presence of FSI. Therefore Eq. (66) can be exploited also in this case, but using the suitable ingredient, i.e. the distorted spin-dependent spectral function, and eventually evaluating the distorted light-cone momentum distributions.

Let us discuss now the crucial issue of the extraction of the neutron information from 3He data. A strategy for extracting the neutron Sivers and Collins asymmetries from 3He data, developed in Ref. mio , is summarized and applied in the following.

If the results of the calculation were able to simulate 3He data, the problem would amount to unfolding the convolution formula. This can be done taking into account that the light-cone momentum distributions $f_{N}(\alpha,Q^{2},{\cal E})$ and $f_{N}^{\perp}(\alpha,Q^{2},{\cal E})$ exhibit sharp maxima at $\alpha\sim 1$, i.e. $f_{N}(\alpha,Q^{2},{\cal E})\sim\delta(\alpha-1)$ even in presence of FSI, as we will show in the next Section. Let us remind that this peak is expected since $\alpha=-(\tilde{p}_{mis}\cdot q)/m_{N}\nu$ plays the role of the Bjorken variable for a bound nucleon. Assuming that the delta-like behavior for the light-cone distributions is a reliable approximation (as shown in what follows), then $\Delta\sigma_{Col(Siv)}^{n}\left(x_{Bj}/\alpha,Q^{2},{\bf S}_{\perp}^{n},z\right)\sim\Delta\sigma_{Col(Siv)}^{n}\left(x_{Bj},Q^{2},{\bf S}_{\perp}^{n},z\right)$, and the calculated asymmetries $A_{3}$ can be written as (notably, the dependence on ${\cal E}$ becomes milder and milder, approaching the Bjorken limit)

[TABLE]

Let us introduce the so-called “dilution” factors as

[TABLE]

where

[TABLE]

Notice that, within PWIA and in the Bjorken limit, when $W_{Y}\rightarrow\infty$, then $\langle N_{p(n)}\rangle$ must strictly be 1, providing an obvious physical meaning. In presence of FSI there is a depletion that spoils the above interpretation in terms of number of nucleons involved in the elementary process.

By using the dilution factors, Eq. (74) can be approximated as follows

[TABLE]

where $A^{Col(Siv)}_{n(p)}$ are the free nucleon asymmetries and $p^{\perp}_{n(p)}$ are the average, or effective, transverse polarizations of the neutron (proton) in a transversely polarized 3He nucleus, given by

[TABLE]

In the Bjorken limit, they are $Q^{2}$-independent and can be obtained directly from the nuclear wave function, without evaluating the complicated final states entering the spectral function. In such a limit, by adopting the nucleon-nucleon AV18 interaction and disregarding relativistic corrections (see Ref. tobe ) one gets that the effective longitudinal and transverse polarizations coincide and are equal to

[TABLE]

It is important to stress that, using another realistic potential, these values change by a few percent at most pskv . We also note that, to obtain Eq. (77), the term $m_{N}/E_{N}$ in the definition of the light cone momentum distibutions $f_{N}^{\perp}$, Eq. (67), and $f_{N}$, Eq. (69), has been neglected in Eq. (74). We checked that this procedure introduces a change in the nuclear asymmetries of the order of a few parts in one thousand, not relevant phenomenologically.

The free nucleon asymmetries $A^{Col(Siv)}_{N}$ can be calculated in terms of the quark distributions and fragmentation functions previously described, using their leading twist definitions barone1

[TABLE]

and

[TABLE]

If Eq. (77) were a good approximation of reality, it would be possible to use it to extract the neutron asymmetry according to the following recipe, suggested in Ref. antico for the polarized DIS case, and in Ref. mio for polarized SIDIS in PWIA and in the Bjorken limit (for $j$=Collins, Sivers):

[TABLE]

A theoretical check of Eq. (81) can be performed if a realistic calculation of the 3He single spin asymmetries, $A^{theo,j}_{3}$, is introduced in Eq. (81) in place of the forthcoming experimental data $A^{exp,j}_{3}$, and models for $A^{exp,j}_{p}$ and $A^{j}_{n}$ are used in the theoretical calculation of $A^{theo,j}_{3}$, and in the r.h.s. of the above equation. If nuclear effects were safely taken care of by Eq. (77), one should be able to extract, according to Eq. (81), the neutron asymmetry used as an input for calculating $A^{theo,j}_{3}$. Namely a self-consistency check can be carried out, in preparation of the future extraction from the experimental $A^{exp,j}_{3}$. It has to be noticed that a more stringent test of Eq. (81) could be attained if SSAs of 3H will become available at some time in the future (let us remind that some steps forward in the actual use of unpolarized 3H target in DIS experiments have been accomplished marathon ).

VI Results and discussion

Now we are ready to present the results of our calculation.

Let us start providing a pictorial view of the main quantity of interest, i.e. the distorted spectral function, evaluated using 3He and $\Psi_{23}$ wave functions computed within the AV18 potential av18 . As an example, the neutron spectral function, in the unpolarized case, is shown in Fig. 2, in PWIA and with FSI between debris and spectator taken into account, within GEA framework. It is clearly seen that, as found in previous studies dedicated to quasi-elastic scattering unfact , the effect of FSI increases with ${\bf p}_{mis}$, as it is easily understood by thinking that, when ${\bf p}_{mis}={\bf p}_{Y}-{\bf q}$ is low, the final debris $Y$ has to be very fast. The low impact of FSI for small values of $|{\bf p}_{mis}|$ is illustrated in more detail in Fig. 3, where it is shown the ratio of the unpolarized distorted spectral function of the neutron, evaluated for $\alpha=1$, to the PWIA one. Also the increase of the relevance of FSI when, at fixed $|{\bf p}_{mis}|$, the removal energy $E=M^{*}_{23}+m_{N}-M_{3}$ increases, is physically expected. As a matter of fact from the energy conservation

[TABLE]

with $M_{Y}\geq m_{\pi}+m_{N}$, one can realize that the momentum $|{\bf p}_{Y}|$ has to decrease (i) for any $|{\bf p}_{mis}|$, when the removal energy increases, and (ii) for any $\epsilon_{23}^{*}$, when $|{\bf p}_{mis}|$ increases. Then, the debris gets slower and FSI sizably affects the distorted spectral function. This is indeed what can be seen in Fig. 3.

The results for the spin-independent and spin-dependent light-cone momentum distributions have already been evaluated and shown in Ref. mio , in PWIA, using the AV18 interaction av18 , but assuming the Bjorken limit ($|\vec{q}|\simeq\nu$). Let us perform a first step forward, by illustrating in Figs. 4 and 5 the effect of JLab kinematics, at finite values of $\nu$ and $Q^{2}$, on the light-cone momentum distributions (67) and (69), using the PWIA spectral function already exploited in Ref. mio . As already mentioned, in the kinematics under scrutiny, the distribution functions $f_{n(p)}(\alpha,Q^{2},{\cal E})$ and $f^{\perp}_{n(p)}(\alpha,Q^{2},{\cal E})$ depend on both the energy $\nu$ and the momentum ${\bf q}$ through the limits of integration $|{\bf p}_{min(max)}|$ and the invariant mass of the debris. Figure 4 shows $|{\bf p}_{min}|$ and $|{\bf p}_{max}|$ as a function of the light-cone variable $\alpha$, for two values of the removal energy $E$, i.e. $E=0$ and $200$ MeV, given the electron beam energy, ${\cal E}$=8.8 GeV, and $Q^{2}=5.73~{}{\rm(GeV/c)^{2}}$. For this kinematical choice, it is seen that one can explore only the region where $\alpha\geq 0.55$ (i.e. when $|{\bf p}_{max}|>|{\bf p}_{min}|$). By changing the kinematics one can investigate a wider interval of $\alpha$. Figure 5, where the PWIA distribution function $f_{N}(\alpha,Q^{2},{\cal E})$ and $f^{\perp}_{N}(\alpha,Q^{2},{\cal E})$ are presented for the above kinematical conditions, shows that, as it happens in the Bjorken limit, the polarization of the 3He nucleus is almost entirely determined by the neutron one, while the contribution of the proton polarization is very small. It is worth mentioning that the existence of a kinematically forbidden region $\alpha\ <0.55$ can lead to slight modifications in the normalization conditions for both the unpolarized and the polarized light-cone momentum distributions.

In Fig. 6, the investigation on the PWIA light-cone distributions becomes more detailed. The functions $f^{\perp}_{n(p)}(\alpha,Q^{2},{\cal E},E)$ and $f_{n(p)}(\alpha,Q^{2},{\cal E},E)$ of Eqs. (68) and (70), respectively, are shown for two different choices of kinematics, corresponding to the planned experiments at JLab, and $\alpha=0.65$. Such a value of $\alpha$ belongs to the region where the neutron light-cone momentum distributions (unpolarized and transversely polarized) have shown the biggest differences in PWIA. In correspondence with the different kinematical choices, the calculated curves are hardly distinguishable and one can conclude that the dependence upon kinematics is rather mild in PWIA.

The extraction procedure shown in Eq. (81) and proposed in Ref. mio for SIDIS adopting PWIA and Bjorken limit, works very well and it has been already applied in the experimental analysis of the JLab data collected at 6 GeV prljlab . In the actual JLab kinematics, a non trivial $Q^{2}$ dependence is introduced in the integration limits of the convolution formula (cf. Fig. 4). This amounts to a deviation of the quantities $\langle N_{n}\rangle,\,\langle N_{p}\rangle,\,p_{n},\,p_{p}$ from their values obtained in the Bjorken limit, namely $1,\,1,\,0.878,\,-0.024$, respectively. In the kinematics of JLab@12 GeV He3exp , this deviation is found to be a few parts in one thousand. In Fig. 7, it is shown that the excellent performance of the extraction procedure of Eq. (81) does not change appreciably when we move from the Bjorken limit to the experimental kinematics of JLab@12 GeV He3exp , corresponding to finite values of $Q^{2}$ and $\nu\neq|\vec{q}|$. Hence, the Sivers (left panel) and the Collins (right panel) asymmetries are well determined when our theoretical check of Eq. (81) is carried out.

Now it comes the basic issue of understanding to what extent FSI effects between debris and remnants can modify the outcomes obtained through Eq. (81) and shown in Fig. 7. This is a crucial step for a reliable extraction of the neutron information. As pointed out in Sections III, IV and V, the formal expressions for the Collins and Sivers asymmetries obtained within PWIA, Eq. (66), still work when FSI are considered within GEA.

Also Eqs. (67) - (69) remain formally unchanged if FSI are included: the only difference amounts to use there the distorted spectral function for obtaining the distorted light-cone momentum distributions, instead of adopting the corresponding PWIA expressions. In Figs. 8 and 9, neutron and proton light-cone momentum distributions, obtained within GEA for the unpolarized and the transversely polarized cases, are shown for ${\cal E}$=8.8 GeV, a value of the beam energy typical for the planned JLab@12 experiments, and for $Q^{2}$= 5.73 (GeV/c)2 (i.e. one of the values which will be tested at $\cal E$=8.8 GeV). Moreover, they are compared with the corresponding quantities calculated within PWIA. The differences between the results with and without FSI are quite sizable and therefore the quantities defined in Eqs. (71), (72) and (73), necessary to calculate Collins and Sivers asymmetries, are largely affected by FSI effects, that have to be carefully taken into account. In particular, $\langle N_{n}\rangle,\,\langle N_{p}\rangle,\,p_{n},\,p_{p}$, defined according to Eqs. (76) and (78), respectively, and calculated at the actual JLab kinematics corresponding to $Q^{2}\sim 3\div 7$ (GeV/c)2, are affected by FSI and exhibit deviations from their values in the Bjorken limit (given above) as large as 20 %.

The $Q^{2}$ dependence of the above results is quite important, in view of the possible construction of the EIC (see, e.g. Ref. accardi for the presentation of the physics case), that could open unprecedented possibilities in the studies of the nucleon TMDs. In order to give a first idea of the impact on the future measurements, in Fig. 10, it is shown the ratio of the light-cone spin-independent momentum distribution, evaluated taking into account FSI, to the corresponding quantity obtained in PWIA, for different values of $Q^{2}$, at the peak, i.e. $\alpha=1$. Four different kinematical conditions have been chosen, two of them, namely (i) $\cal E$=8.8 GeV, $Q^{2}\simeq 5.7$ (GeV/c)2, $x_{Bj}\simeq 0.48$ and (ii) $\cal E$=11 GeV, $Q^{2}\simeq 7.7$ (GeV/c)2, $x_{Bj}\simeq 0.48$, are typical for JLab@12 . The third and the fourth ones are kinematics occurring at the planned EIC, namely at ${\cal E}_{collider}=11$ GeV, ${\cal E}_{collider}^{{}^{3}He}$ = 40 GeV, $Q^{2}=$ 10 and 12 (GeV/c)2, $x_{Bj}\simeq 0.48$ (noteworthy, this value of $x_{Bj}$ could be achieved by a beam energy ${\cal E}=293$ GeV for a fixed target experiment). It is important to recall that a single point in Fig. 10 represents the outcome of a one-week run on the ZEFIRO INFN-facility in Pisa, Italy. What is found is that the effects of FSI, evaluated within GEA framework, is almost $Q^{2}$ independent, but rather sizable at JLab and EIC energies. Could one think that the extraction procedure shown in Eq. (81), had to be abandoned in favor of more involved and model dependent techniques? Actually, a crucial observation is now in order. It is clearly seen in Figs. 5, 8 and 9 that the spin-independent and spin-dependent light-cone momentum distributions are strongly peaked around $\alpha=1$, both in PWIA and with FSI effects taken into account. This means that the approximation given in Eq. (74) for the nuclear Sivers and Collins asymmetries (cf Eq. (66)), should basically hold. Moreover, looking at the same figures, it is also rather apparent that FSI produces a decrease of all the distributions in a similar way, both qualitatively and quantitatively. From Eq. (74), it is easy to see that the results for the nuclear asymmetries obtained in PWIA, $A^{PWIA,j}_{3}$, or taking into account FSI, $A^{FSI,j}_{3}$ (recall that $j=$ Sivers or Collins), should not sizably differ from each other, due to a cancellation of effects present in both the numerator and the denominator. The realization of this fact in the actual calculation of Eq. (66) is shown in Fig. 11. In principle, in this figure and in the two following ones, at any $x_{Bj}$ should correspond a slightly different value of $Q^{2}$. Nevertheless, in the $x_{Bj}$ range explored at fixed ${\cal{E}}$, the dependence on $Q^{2}$ of the light-cone momentum distributions $f_{p(n)}(\alpha,Q^{2},{\cal E})$ and $f_{p(n)}^{\perp}(\alpha,Q^{2},{\cal E})$ is rather mild and therefore we will show the results for the nuclear asymmetries, Eq. (66), at a fixed value of $Q^{2}$, namely 5.73 (GeV/c)2.

Our full evaluations of the 3He Collins and Sivers asymmetries, presented in Fig. 11, strongly encourage the investigation of the extraction formula, Eq. (81), that relies on the validity of the approximation Eq. (77), where effective polarization and dilution factors are multiplied by each other. In particular, we want to assess if Eq. (81) can be safely (or better with a low degree of uncertainty) applied to the experimental data, where FSI is certainly acting. Noteworthy, the relevant product of effective polarizations and dilution factors is found to have a very little dependence on FSI, as one can straightforwardly realize by inspecting Tables 1 and 2, where the dilution factors, the effective polarizations and their products are presented with or without FSI effects taken into account, by adopting the kinematics of the forthcoming JLab experiments.

Considering that (i) $A^{PWIA,j}_{3}\simeq A^{FSI,j}_{3}$ (see Fig. (11)), and (ii) the products of effective polarizations and dilution factors are almost the same in PWIA and including FSI, one has

[TABLE]

In Fig. 12, the reliability of the above relations in the extraction of $A^{j}_{n}$ is illustrated through our theoretical test, where the experimental $A_{3}^{exp,j}$ is replaced by our full calculation. Indeed, in Fig. 12, the model Collins and Sivers asymmetries for the neutron used in the full calculations of 3He asymmetries are hardly distinguishable from the neutron asymmetries extracted through Eq. (83) by using PWIA effective polarizations and dilution factors, or by considering the corresponding quantities calculated within GEA (a preliminary version of this figure was presented in Ref. dani ). It should be pointed out that these quantities can be evaluated in any kinematical configuration using our model of FSI, which is rather well constrained phenomenologically, and could be improved checking our predictions against the spin-dependent cross sections which will be soon available.

In addition to the above extraction procedure, one could adopt the following one where the experimental inputs are $A_{3}^{exp,j}$ and $A_{p}^{exp,j}$, while the theoretical quantities reduce to the PWIA effective polarization in the Bjorken limit. In this case, one has a nice possibility to extract the neutron information through another extraction scheme, independent of the FSI model. The procedure is based on the following expression

[TABLE]

Indeed, $p_{n(p)}^{PWIA}$ can be obtained from a realistic wave function with very small model dependence (see Ref. pskv for an analysis of the dependence of effective polarizations on different realistic potentials). In Eq. (84), the experimental dilution factors are

[TABLE]

where no dependence on the FSI model is present, differently from Eq. (75). In Fig. 13 one sees that the uncertainty in the extraction procedure based on Eq. (84) is not much bigger than the one occurred by using Eq. (83). In Fig. 13, Eq. (84) has been actually evaluated using $A^{FSI,j}_{3}$ instead of $A^{exp,j}_{3}$, and using, instead of $d_{p(n)}^{exp}$, the dilution factors evaluated with the parameterizations of unpolarized parton distributions gluu and fragmentation functions kret already described in the previous section. Therefore Fig. 13 shows that, for a safe extraction procedure through Eq. (84), the evaluation of distorted effective polarizations and dilution factors, which appear in Eq. (83) and are depending on the adopted FSI model, is actually not required.

Summarizing, the comparisons shown in Figs. 12 and 13 illustrates two methods for the successful extraction of the neutron single spin asymmetries using transversely polarized 3He targets at JLab, and they represent the most relevant outcomes of the present investigation.

One could argue that the very nice results obtained within our FSI model, are actually expected to hold in any description of final state interactions which is (i) factorized and (ii) basically spin-independent, i.e., producing a similar effect in spin-dependent and spin-independent cross sections. This last feature is very likely to be realized for any FSI occurring in processes where the relative energy of the interacting systems is high, as it is the case in the present study.

VII Conclusions

Measurements of the Sivers and Collins asymmetries for both proton and deuteron have shown a strong flavor dependence, motivating independent further investigations using different targets to safely access the same quantities for the neutron. As for any polarized neutron observable, 3He is the natural target, due to its specific spin structure. Two experiments, aimed at measuring azimuthal asymmetries in the production of $\pi^{\pm}$ from transversely polarized 3He, were performed at JLab. From the gathered 3He data prljlab , the Collins and Sivers neutron asymmetries were extracted using a procedure proposed in Ref. mio . However, such an extraction procedure was not considering some relevant nuclear effects, properly evaluated in the present paper, which strengthens a posteriori the method used in Ref. prljlab to obtain the neutron information. In particular, the extraction procedure proposed in Ref. mio and used in Ref. prljlab was able to take care of (i) the spin structure of 3He and (ii) the momentum and energy distributions of bound nucleons, through a realistic spin-dependent spectral function evaluated by using nuclear wave functions obtained from the AV18 interaction, in plane wave impulse approximation. The results of Ref. mio were obtained in the Bjorken limit, namely without considering possible effects of the kinematics of JLab, dominated by finite values of the energy and momentum transfers, and, more important, without FSI effects. The problem whether or not the extraction procedure based on PWIA calculations can be extended to a scenario where final state interactions between the debris, originated from the struck nucleon, and the interacting spectator system are allowed to play a role, as it likely happens in the actual JLab kinematics, has been thoroughly analyzed in the present paper. We were able to quantitatively show that the extraction procedure is basically independent of FSI, evaluated within the generalized eikonal approximation. In particular, in order to perform the needed full evaluation of the FSI effects, we have extended the calculation of a realistic distorted spin-dependent spectral function, introduced in a previous paper of ours nostro , where it was taken into account the two-body break up channel only. Actually, we have performed a highly non trivial (from the numerical point of view) computation of the contribution to the distorted spin-dependent spectral function from the three-body break-up channel, essential to obtain reliable cross sections and in turn to robustly extract valuable neutron information. Once, such a refined spectral function became available, we have exploited our results for calculating both Sivers and Collins single spin asymmetries. FSI effects have been found to produce sizable effects in both the unpolarized and polarized cross sections. Differently, the SSAs have resulted slightly affected by FSI, since they are ratio of cross sections, and therefore the FSI effects cancel to a large extent. As a result, the very same extraction procedure proven to be successful in PWIA can be used also in a scenario where FSI effects are relevant. This means that all the complexities related to Fermi motion, binding and FSI effects can be summarized in the nucleon effective polarizations, quantities known from accurate few-body calculations in a rather model independent way. This scheme is valid in a wide range of FSI models, every time that FSI are basically spin-independent, as expected to happen at high energies (i.e. in the case of JLab or the planned Electron Ion Collider) and lead to convolution formulas for the nuclear cross sections, namely a folding of cross sections off bound nucleons and distorted spin-dependent spectral functions.

The importance of these results for both the planning and the analysis of experiments with transversely polarized 3He target is clear. Further studies of the same issue will involve the implementation of GEA in the relativistic nuclear overlaps, defined in tobe , so that a light-front, distorted, spin dependent spectral function can be evaluated and relativistic effects can be taken into account in a consistent framework.

Acknowledgments

Calculations were in part performed on the ZEFIRO facility of INFN, Commissione 4, Pisa, Italy. L.P.K. thanks INFN, Perugia, for partial financial support during his stay in Perugia in 2015 and 2016, and the Dipartimento di Fisica e Geologia of Perugia University for warm hospitality.

Appendix A Overlaps for the distorted spectral function

The overlaps

[TABLE]

corresponding to Eq. (48) with the index $f_{23}$ removed for simplicity, are built in terms of two- and three-body wave functions.

In particular, when the energy of the pair is $\epsilon_{23}^{*}=t^{2}/m$, the two-body wave function reads:

[TABLE]

with the tensor spherical harmonics defined as

[TABLE]

When the pair is in the deuteron state, with binding energy $E_{D}$, the two-body wave function reads:

[TABLE]

The three body wave function in pisa is defined according to the following scheme

[TABLE]

The antisymmetrization of the wave function requires $l_{23}+s_{23}+T_{23}$, where $T_{23}$ is the isospin of the pair $23$, to be odd. In addition, $l_{23}+L_{\rho}$ has to be even, due to the parity of 3He.

Using these wave functions, one has, in the 3bbu channel:

[TABLE]

where $\{\alpha\}=\{L_{\rho},M_{\rho},X,j_{23},l_{f},m_{f},l_{23},\mu_{23},l,J_{f}\}$ and

[TABLE]

When the active nucleon $N$ is a proton $p$, besides the $3bbu$ channel, one can have also the 2bbu channel, for which the overlap becomes

[TABLE]

where

[TABLE]

and $\{\beta\}=\{L_{\rho},M_{\rho},X,j_{23},l_{23},\mu_{23},L_{D}=0,2,m_{L}\}$.

Appendix B Properties of the Glauber distorted Spectral Function

Let us consider a reference frame with the $z$-axis along the momentum transfer ${\bf q}$. If in such a reference frame a nucleus with $J_{A}=1/2$ has a polarization ${\bf S}_{A}$, one can expand the nucleus state by using pure states polarized with respect to the quantization axis $\hat{q}\equiv\hat{e}_{z}$, i.e. $\left|{1\over 2},\pm{1\over 2}\right\rangle_{\hat{q}}$. In this case, a generic state with $J_{A}=1/2$ and polarization directed along some direction is written as follows

[TABLE]

where $\cos\beta={\hat{S}_{A}}\cdot{\hat{q}}$ and $\left|{1\over 2},{1\over 2}\right\rangle_{\hat{S}_{A}}$ is a pure state polarized with respect to the quantization axis $\hat{S}_{A}$ (see Eq. (43)). In Eq. (19) of cda1 one can find a general expression of the PWIA spectral function,

[TABLE]

where ${\bf p}=-{\bf p}_{mis}$ is the nucleon three-momentum inside the target, the index ${\cal M}$ refers to the third component with respect to the quantization axis $\hat{S}_{A}$ and ${\bm{\cal F}}_{{\cal M}}({\bf p},E)$ is a pseudovector depending upon the vector $\hat{\bf p}$ and the peudovector ${\bf S}_{A}$

[TABLE]

In the case were the FSI is considered through a Glauber operator at high momentum transfer, there is a further dependence of the spectral function upon the vector ${\bf q}$ and Eqs. (96) and (97) are to be replaced by

[TABLE]

[TABLE]

The above expressions for the spectral function, put in evidence the dependence upon ${\bf S}_{A}$, as well as the dependence of the scalar functions $B_{i}$ ($i=1,...,6$) by the possible scalars $|{\bf p}|,E,({\bf S}_{A}\cdot\hat{\bf p})^{2},|{\bf q}|,({\bf S}_{A}\cdot\hat{\bf q})^{2},\hat{\bf p}\cdot\hat{\bf q}$. If ${\bf S}_{A}$ is orthogonal to the $z$ axis, ${\bm{\cal F}}_{{\cal M}}({\bf p},E,{\bf q})$ reduces to

[TABLE]

From Eq. (98) one has

[TABLE]

Let us now express the distorted spectral function with a polarization axis along ${\bf S}_{A}$ (cf Eqs. (40) and (41)) in terms of the components given in Eq. (47), that correspond to a polarization axis along $\hat{q}$ by using Eq. (95). Since we are interested in a transversely-polarized target, i.e. ${\bf S}_{A}\equiv\{1,0,0\}$, one has to consider $\beta=90^{o}$, and the components of the spectral functions are

[TABLE]

If the nucleus is polarized along $-{\bf S}_{A}$, the state of the nucleus can be written as follows

[TABLE]

and for the spectral function becomes

[TABLE]

To obtain the real and the imaginary parts of the quantity $\left[{{\cal P}}^{N\,\frac{1}{2}-\frac{1}{2}}_{\frac{1}{2}-\frac{1}{2}}(E,{\bf p}_{mis})\ +~{}{{\cal P}}^{N\,-\frac{1}{2}\frac{1}{2}}_{\frac{1}{2}-\frac{1}{2}}(E,{\bf p}_{mis})\ \right]$, needed to evaluate the single spin asymmetries (see Eq. (59)), let us first consider the $x$ and the $y$ components of ${\bm{\cal F}}_{\frac{1}{2}}({\bf p},E,{\bf q})$ with ${\bf S}_{A}={\bf S}_{3}$ along the $x$ axis.

From Eq. (100) one has

[TABLE]

where the angles $\theta$ and $\phi$ define the direction of the nucleon momentum $\bm{p}$. From Eq. (102) and Eq. (103) one obtains

[TABLE]

Then let us consider the $x$ and the $y$ components of ${\bm{\cal F}}_{\frac{1}{2}}({\bf p},E,{\bf q})$ with ${\bf S}_{3}$ opposite to the $x$ axis. From Eq. (100) one has

[TABLE]

while from Eq. (102) and Eq. (103) one obtains

[TABLE]

The difference of Eqs. (106) and (110) is equal to the difference of Eqs. (108) and (112)

[TABLE]

and the difference of Eqs. (107) and (111) is equal to the difference of Eqs. (109) and (113)

[TABLE]

Let us stress that the scalar functions $B_{1}$ and $B_{2}$ do depend on the variable $\phi$ only through $({\bf S}_{A}\cdot\hat{\bf p})^{2}=(\sin\theta\cos\phi)^{2}$, since $\hat{\bf p}\cdot\hat{\bf q}=\cos\theta$.

In the nucleon tensor operators ${\hat{w}}_{\mu\nu}^{sN}$ that give rise to the Collins and the Sivers effect, the nucleon momentum can appear directly or through the nucleon spin operator. Therefore terms of zero order in ${\bm{p}}_{\perp}/m_{N}$ can appear, as well as terms of the first, second and third order ($\perp$ means orthogonal to the $\hat{q}=\hat{z}$ axis) barone . Once multiplied by the spectral function and integrated over the nucleon momentum, the terms of the second and third order can be discarded, since the spectral function decreases rapidly as a function of the nucleon momentum (see, e.g., Fig. 2).

In the imaginary part, \Im m\Bigl{[}w_{\mu\nu}^{sN\frac{1}{2}-\frac{1}{2}}\Bigr{]}, the terms of zero order and of the first order in ${\bm{p}}_{\perp}/m_{N}$, once multiplied by the left hand side of Eq. (115) and integrated over $\phi$ , do not give contribution to the hadronic tensor, since one has to integrate quantities like $(\cos\phi~{}\sin\phi)$, $(\cos^{2}\phi~{}\sin\phi)$ or $(\cos\phi~{}\sin^{2}\phi)$ times a function of $\cos^{2}\phi$. Then the product of the imaginary quantities in Eq. (58) does not give contribution to the cross section.

An analogous analysis can be performed on the real part \Re e\Bigl{[}w_{\mu\nu}^{sN\frac{1}{2}-\frac{1}{2}}\Bigr{]} of the nucleon tensor. In this case the terms of zero order in ${\bm{p}}_{\perp}$ give a non-zero contribution, while the first order terms yield zero, once the integration over $\phi$ is performed.

Let us finally notice that since the transverse components of ${\bf p}$ can be disregarded, as discussed above, the expressions for $\Delta\sigma_{Col}^{N}$ and for $\Delta\sigma_{Siv}^{N}$ of Eqs. (66) and (67) of our paper, that were obtained in a reference frame where $\bm{p}_{\perp}=0$ (see, e.g., Eqs. (6.5.18) and (6.5.17) of Ref. barone ), can be safely used.