#  Light-Front Transverse Momentum Distributions for ${\cal J}$=1/2 Hadronic
Systems in Valence Approximation.

Rocco Alessandro  Università di Roma “Tor Vergata”, Via della Ricerca
Scientifica 1, 00133 Rome, Italy  Alessio Del Dotto  Istituto Nazionale di
Fisica Nucleare, Laboratori Nazionali di Frascati, Via Enrico Fermi 40, 00044
Frascati (Roma), Italy
Emanuele Pace  Università di Roma “Tor Vergata”, Via della Ricerca Scientifica
1, 00133 Rome, Italy  Gabriele Perna  Dipartimento di Fisica e Geologia,
Università di Perugia, Via Alessandro Pascoli, 06123 Perugia, Italy  Giovanni
Salmè  Istituto Nazionale di Fisica Nucleare, Sezione di Roma, P.le A. Moro 2,
00185 Rome, Italy  Sergio Scopetta  Dipartimento di Fisica e Geologia,
Università di Perugia, Via Alessandro Pascoli, 06123 Perugia, Italy  Istituto
Nazionale di Fisica Nucleare, Sezione di Perugia, Via Alessandro Pascoli,
06123 Perugia, Italy

###### Abstract

The semi-inclusive correlator for a ${\cal J}$=1/2 bound-system, composed by A
spin-1/2 fermions, is linearly expressed in terms of the light-front Poincaré
covariant spin-dependent spectral function, in valence approximation. The
light-front spin-dependent spectral function is fully determined by six scalar
functions that allow for a complete description of the six T-even transverse-
momentum distributions, suitable for a detailed investigation of the dynamics
inside the bound system. The application of the developed formalism to a case
with a sophisticated dynamical content, like 3He, reaches two goals: (i) to
illustrate a prototype of an investigation path for gathering a rich wealth of
information on the dynamics and also finding valuable constraints to be
exploited from the phenomenological standpoint; (ii) to support for the three-
nucleon system a dedicated experimental effort for obtaining a detailed 3D
picture in momentum space. In particular, the orbital-angular momentum
decomposition of the bound state can be studied through the assessment of
relations among the transverse-momentum distributions, as well as the
relevance of the relativistic effect generated by the implementation of
macrocausality. A fresh evaluation of the longitudinal and transverse
polarizations of the neutron and proton is also provided, confirming
essentially the values used in the standard procedure for extracting the
neutron structure functions from both deep-inelastic scattering and semi-
inclusive reactions, in the same kinematical regime.

##  I Introduction

A fully relativistic treatment is needed to describe hadronic bound systems,
when high energy processes are studied and/or a high degree of accuracy is
required. Among the phenomenological efforts to implement a Poincaré-covariant
description of a bound system, we recall the proposal in Ref. [1], where the
light-front Hamiltonian dynamics (LFHD) [2, 3, 4, 5, 6, 7] was adopted in
order to obtain the light-front (LF) spin-dependent spectral function, whose
diagonal elements yield the distribution probability to find a constituent
with given spin, LF momentum and (off-shell) energy, inside the bound system.
The spectral function is primarily used to study the nucleon momentum
distributions in nuclei, but the formalism can be notably extended to a
hadronic bound system and, as it will be illustrated in detail, to eventually
obtain the six T-even transverse momentum distributions (TMDs) [8]. Through
the latter quantities, one can achieve a detailed description of the system,
much richer than the one given by the usual distribution in terms of the
constituent momentum $|{\bf p}|$ in the lab frame. As a matter of fact, one
can address the correlations between spin and momentum, substantially
deepening our understanding of the inner dynamics. For the nucleon, TMDs (see,
e.g., [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19] are the object of impressive
theoretical and experimental efforts, both in semi-inclusive deep inelastic
scattering (SIDIS) and in Drell-Yan processes (see, e.g., [20, 21, 22, 23, 24,
25, 26, 27, 28] and [29, 30, 31], respectively). In particular light-cone
models and phenomenological approaches for the TMDs have been used, e.g., (i)
to study the three-dimensional nucleon structure [32, 33, 34, 35, 36, 37, 38],
(ii) to address the nucleon-spin puzzle [39] and (iii) hence to disentangle
the contributions of different angular-momentum components to the spin of the
nucleon [40, 41]. Let us notice that the nucleon, namely a spin $1/2$ system,
is composed by three quarks, in valence approximation. Therefore, in this
approximation, the approach we have elaborated in LFHD can be applied both to
the nucleon, as a system of three quarks, and to 3He or 3H, as systems of
three nucleons.

The application of our formalism to the three-nucleon system has a twofold
benefit. On one side, it allows one to illustrate a realistic example, with
specific features of TMDs and constraints among them that can be traced back
to the inner dynamics, e.g. the impact of the orbital-momentum content
generated by the interaction. On the other side, it establishes a first
theoretical basis for supporting future experimental efforts aiming to
investigate the TMDs of 3He, and eventually construct a 3D tomography of the
nucleus in momentum space. In a nucleus with total momentum $P$ in the lab
frame, TMDs describe the nucleon distribution as a function of $x=p^{+}/P^{+}$
111For the definitions of the kinematical variables of the LFHD used in this
paper, the reader can refer to Ref. [1]. Let us only recall here that the
light-front components of a four vector $v$ are $(v^{-},{\bf\tilde{v}})$,
where ${\bf\tilde{v}}=(v^{+},{\bf v}_{\perp})$ with
$v^{\pm}=v^{0}\pm{\bf{\hat{n}}}\cdot{\bf v}$ and ${\bf v}_{\perp}={\bf
v}-{\bf{\hat{n}}}({\bf{\hat{n}}}\cdot{\bf v})$. The vector ${\bf{\hat{n}}}$ is
a generic unit vector. Then the scalar product of two four vectors ${a}$ and
${b}$ is ${a}\cdot{b}={a^{-}b^{+}+a^{+}b^{-}\over 2}-{\bf a}_{\perp}\cdot{\bf
b}_{\perp}$. In this paper we choose ${\bf{\hat{n}}}\equiv{{\hat{z}}}$. and of
the transverse momentum $p_{\perp}$, for any possible orientation of the spin
of the nucleus and of the spin of the nucleon. Hence, it is quite natural to
look for a possible interplay with the spin-dependent spectral function. In
Ref. [1] the LF spectral function, ${\cal P}^{\tau}_{{\cal
M},\sigma^{\prime}\sigma}({\tilde{\mbox{\boldmath{$\kappa$}}}},\epsilon,S)$,
is defined starting from the LF wave function for a three-nucleon system with
polarization vector $\bf S$, spin $1/2$ and third component $\cal M$, using
non symmetric intrinsic variables. The energy $\epsilon$ is the energy of a
fully interacting two-particle (23) subsystem and the variable
$\tilde{\mbox{\boldmath{$\kappa$}}}=(\kappa^{+},\mbox{\boldmath{$\kappa$}}_{\perp})$
is the LF momentum of particle 1 in the intrinsic reference frame of the
cluster [1,(23)]. From the spectator energy $\epsilon$ one can reconstruct the
component $\kappa^{-}$, leading to the off-shell energy of the constituent.
The spectral function is defined through the overlaps between the LF wave
function of the system and tensor products of a plane wave of momentum
$\tilde{\mbox{\boldmath{$\kappa$}}}$ and the intrinsic state of the two-
particle spectator subsystem. The mentioned tensor product allows one to take
care of macrocausality [3] and to introduce a new effect of binding in the
spectral function [1]. With the help of the Bakamjan-Thomas construction of
the Poincaré generators [42], the LF wave function can be obtained from the
usual non relativistic wave function of the system with a realistic
interaction between the nucleons. Then the LF spectral function allows one to
embed the successful phenomenology for few-nucleon systems in a Poincaré-
covariant framework and to satisfy at the same time both normalization of the
three-body system bound state (i.e. the baryon number sum rule) and momentum
sum rule. Interestingly, in Ref. [1] the definition of the LF spin-dependent
spectral function was plainly generalized to a generic system of A spin 1/2
fermions. As a first test of our approach, the calculation of the EMC effect
for 3He is under way [43]. Preliminary results which consider only the
contribution of the two-body bound-state channel show encouraging improvements
[44, 45, 46, 47] with respect to a convolution approach with a momentum
distribution [48].

In this work the most general expressions for the spin-dependent spectral
functions and for the spin-dependent momentum distribution in terms of six
scalar functions, ${\cal B}_{i}$ and $b_{i}$, respectively (i=0,…,5), valid
for any system of spin 1/2 fermions are presented. In valence approximation,
we demonstrate that a linear relation between the LF spectral function and the
semi-inclusive fermion correlator occurs (for preliminary results see Ref.
[49, 50, 51, 52, 53]). In turn, for a spin 1/2 system it is straightforward to
relate the six T-even twist-two TMDs to the LF spectral function and
eventually to the system wave-function. The results for the 3He TMDs
corresponding to a realistic nuclear interaction are also presented. In
particular, the LF longitudinal and transverse effective polarizations,
quantities relevant for the extraction of the neutron information from data
collected with polarized nuclear targets, are evaluated for the proton and the
neutron in 3He and compared with the corresponding non relativistic results
currently used by experimental collaborations. Moreover we have assessed the
approximated relations relations between the TMDs, investigated in Refs. [13,
35] with the aim to offer a guide for the extraction of TMDs from experimental
data.

It is important to stress that the validity of these relations could be tested
using 3He as a playground, since to perform a similar test for the proton
target is much more challenging, at the present stage, due to quark
confinement and hadron fragmentation. Indeed the measurement of 3He TMDs
appear feasible at high luminosity facilities, such as Jefferson Lab and the
future electron-ion collider [54], through ${}^{3}{\overrightarrow{\rm
He}}({\overrightarrow{e}},e^{\prime}p)X$ experiments with proper polarization
set-ups of beams and targets [55].

The paper is organized as follows. In Section II, the most general expressions
of the LF spin-dependent spectral function and of the LF spin-dependent
momentum distribution are presented for any bound-system, composed by A
fermions of spin 1/2 in terms of six suitable scalar functions. Explicit
expressions for the six scalar functions defining the momentum distribution
are given in Appendix B for a three-nucleon system of spin 1/2. In Section
III, the linear relation in valence approximation between the semi-inclusive
correlator and the LF spectral function is derived. In Sections IV and V, the
relation between the LF spectral function and the T-even twist-two TMDs is
illustrated. In Section VI, the numerical results for the 3He nucleus are
presented, ranging from the TMDs to the new calculations of LF longitudinal
and transverse effective polarizations for the proton and the neutron in 3He.
Also the approximate relations between the T-even twist-two TMDs are
discussed. In Section VII, our conclusions are drawn.

##  II The LF spin-dependent spectral function and the LF spin-dependent
momentum distribution

Following Ref. [1], within the LFHD a Poincaré covariant definition of the
spin-dependent spectral function for an $A$-particle bound system polarized
along $\bf{S}$, can be simply obtained by replacing the non relativistic
overlaps
$\langle{\vec{p}_{1},\sigma\tau;\psi}_{f_{(A-1)}}|{\psi}_{\cal{J}\cal{M}};S,TT_{z}\rangle$,
which define the non relativistic spectral function, with their LF
counterparts
${}_{LF}\langle\tau_{S},T_{S};\alpha,\epsilon;J_{z}J;\tau,\sigma,\tilde{\mbox{\boldmath{$\kappa$}}}|{\psi}_{\cal{J}\cal{M}};S,TT_{z}\rangle$.

Hence, the LF spin-dependent spectral function reads

$\displaystyle{\cal P}^{\tau}_{{\cal
M},\sigma^{\prime}\sigma}(\tilde{\mbox{\boldmath{$\kappa$}}},\epsilon,S)=\rho(\epsilon)\sum_{JJ_{z}\alpha}\sum_{T_{S}\tau_{S}}~{}_{LF}\langle\tau_{S},T_{S};\alpha,\epsilon;JJ_{z};\tau\sigma^{\prime},\tilde{\mbox{\boldmath{$\kappa$}}}|{\psi}_{\cal{J}\cal{M}};S,TT_{z}\rangle~{}\langle
S,TT_{z};{\psi}_{\cal{J}\cal{M}}|\tilde{\mbox{\boldmath{$\kappa$}}},\sigma\tau;JJ_{z};\epsilon,\alpha;T_{S},\tau_{S}\rangle_{LF}~{},$
(1)

where $|{\psi}_{\cal{J}\cal{M}};S,TT_{z}\rangle$ is the $A$-fermion ground
state, with total angular momentum ${\cal J}$ (third component ${\cal M}$),
isospin $TT_{z}$ and rest-frame polarization $S\equiv\\{0,{\bf S}\\}$. The
state
$|\tilde{\mbox{\boldmath{$\kappa$}}},\sigma,\tau;T_{S},\tau_{S};\alpha,\epsilon;JJ_{z}\rangle_{LF}$
is the tensor product of (i) a fully interacting intrinsic state of the
$(A-1)$-particle spectator system, with intrinsic energy $\epsilon$ (negative
for bound states and positive for the continuum spectrum ones), spin $J$
(third component $J_{z}$), isospin $T_{S}$ (third component $\tau_{S}$), and
$\alpha$ the set of quantum numbers needed to completely specify the state,
and (ii) a plane wave for the acting particle with LF-momentum
$\tilde{\mbox{\boldmath{$\kappa$}}}$ in the intrinsic reference frame of the
cluster $[1,(A-1)]$. The total LF-momentum of the cluster is ${\tilde{\bf
P}}_{intr}[1,(A-1)]\equiv\\{{\cal M}_{0},{\bf 0}_{\perp}\\}$ with ${\cal
M}_{0}$ its free mass. In Eq. (1), $\rho(\epsilon)$ is the energy density of
the $(A-1)$-particle states. Notice that for $(A-1)$ = 2, one has
$\rho(\epsilon)=1$ for the bound states, e.g. for the deuteron, and
$\rho(\epsilon)=m\sqrt{m\epsilon}/2$ for the continuum, with $m$ the
constituent mass.

Denoting with ${\bf p}_{i}$ (i=1,…,A) and $\bf P$ the momenta of the particles
and the whole system in the lab frame, respectively, one gets the following
expression for the intrinsic momentum $\tilde{\mbox{\boldmath{$\kappa$}}}$

$\displaystyle\kappa^{+}=\xi_{1}~{}{\cal{M}}_{0}[1,(A-1)]~{},\quad{\mbox{\boldmath{$\kappa$}}}_{\perp}={\bf
p}_{1\perp}-\xi_{1}{\bf P}_{\perp}~{},$ (2)

where $\xi_{1}={p^{+}_{1}/P^{+}}$ and ${\cal{M}}_{0}[1,(A-1)]$, the previously
mentioned free mass of the $[1,(A-1)]$ cluster [1], is given by

$\displaystyle{\cal{M}}^{2}_{0}[1,(A-1)]={m^{2}+{\mbox{\boldmath{$\kappa$}}}_{\perp}^{2}\over\xi_{1}}+{M_{S}^{2}+{\mbox{\boldmath{$\kappa$}}}_{\perp}^{2}\over(1-\xi_{1})}~{},$
(3)

with $M_{S}$ the mass of the interacting (A-1) system. Let us assume that the
system is at rest in the laboratory. Then it follows that
${\mbox{\boldmath{$\kappa$}}}_{\perp}={\bf p}_{1\perp}$ (recall that the LF-
momentum $\tilde{\mbox{\boldmath{$\kappa$}}}$ is relative to the cluster
frame). For completeness let us introduce the LF momentum $\tilde{\bf
k}\equiv(k^{+},{\bf k}_{\perp})$ of the acting particle in the intrinsic
$A$-particle system, with transverse component ${\bf k}_{\perp}={\bf
p}_{1\perp}-\xi_{1}{\bf P}_{\perp}={\bf p}_{1\perp}$, and plus component
$k^{+}=\xi_{1}~{}M_{0}\neq p^{+}_{1}$ ($M_{0}$ is the free mass of $A$
particles). In what follows the subscript $1$ will be dropped out and the
notations $\xi_{1}=x$ and ${\bf p}_{1\perp}={\bf p}_{\perp}$ will be adopted.

It is worth reminding that the states
$|\tilde{\mbox{\boldmath{$\kappa$}}},\sigma,\tau;T_{S},\tau_{S};\alpha,\epsilon;JJ_{z}\rangle_{LF}$,
to be used for the definition of the spectral function in the LF overlaps
${}_{LF}\langle\tau_{S},T_{S};\alpha,\epsilon;J_{z}J;\tau\sigma,\tilde{\mbox{\boldmath{$\kappa$}}}|{\psi}_{\cal{J}\cal{M}};S,TT_{z}\rangle$,
fulfill the macrocausality, i.e. the property that observables associated with
different space-time regions must commute for large space-like separations
(see Refs. [1, 3]). Furthermore the use of the momentum
$\tilde{\mbox{\boldmath{$\kappa$}}}$, instead of the momentum ${\bf{p}}$ of
the fermion in the lab frame, introduces a new effect of binding in the
spectral function.

The LF overlaps where the ground state has a generic polarization vector $\bf
S$, i.e. $|{\psi}_{\cal{J}\cal{M}};S,TT_{z}\rangle$, can be obtained from the
overlaps where the ground state is polarized along the $z$ axis, i.e.
$|{{\cal{J}}{m}};\epsilon^{A},\Pi;TT_{z}\rangle_{z}$, by using the Wigner
rotation matrixes, $D^{\cal J}_{m,\cal M}(\alpha,\beta,\gamma)$, viz

$\displaystyle|{\psi}_{\cal{J}\cal{M}};S,TT_{z}\rangle={{\sum}_{m}}|{{\cal{J}}{m}};\epsilon^{A},\Pi;TT_{z}\rangle_{z}D^{\cal
J}_{m,\cal M}(\alpha,\beta,\gamma)~{},$ (4)

where $\alpha,\beta$ and $\gamma$ are the Euler angles describing the proper
rotation from the $z$-axis to the polarization vector $\bf S$ and
$\epsilon^{A}$, $\Pi$ are the energy and the parity of the state,
respectively. Let us recall that the rotations involved act on the bound
system as a whole, and therefore they are interaction-free. Through Eq. (4)
one can relate the spin-dependent spectral function with a given polarization
to the one with polarization $\bf S={\hat{z}}$ (see Appendix A).

As explained in Ref. [1], in the three-nucleon case the overlaps
${}_{LF}\langle\tau_{S},T_{S};\alpha,\epsilon;J_{z}J;\tau\sigma,\tilde{{\mbox{\boldmath{$\kappa$}}}}|{\psi}_{{\cal{J}}{m}};S,TT_{z}\rangle_{z}$
can be evaluated in terms of canonical (or instant-form) two- and three-body
wave functions, replaced by the non relativistic ones, after applying the
Melosh rotation matrixes [56, 57](needed for obtaining the LF spin states from
the canonical ones). This follows, once the Bakamjian-Thomas construction of
the Poincaré generators [42] is adopted. Then, it turns out that the two- and
three-body non relativistic wave functions have all the needed properties with
respect to rotations and translations of the corresponding canonical wave
functions.

In conclusion the LF spin-dependent spectral function is a $2\times 2$ matrix,
${\mbox{\boldmath{$\widehat{\cal{P}}$}}}^{\tau}_{\cal{M}}({\tilde{\mbox{\boldmath{$\kappa$}}}},\epsilon,S)$,
which depends on the direction of the polarization vector $\bf{S}$, and it is
usually normalized for each isospin channel ${{\tau=p(n)}}$, i.e.,

$\displaystyle\int\\!\\!\ \\!\\!\\!\\!\\!\
\\!\\!\\!\\!\\!\sum{d\epsilon}\int{d{\mbox{\boldmath{$\kappa$}}}\over
2E({\bf\kappa})(2\pi)^{3}}~{}Tr\left[{\mbox{\boldmath{$\widehat{\cal
P}$}}}^{\tau}_{\cal
M}(\tilde{{\mbox{\boldmath{$\kappa$}}}},\epsilon,S)\right]~{}=~{}1~{},$ (5)

with $E(\kappa)=\sqrt{m^{2}+|{\mbox{\boldmath{$\kappa$}}}|^{2}}$.

A general expression of
${\mbox{\boldmath{$\widehat{\cal{P}}$}}}^{\tau}_{\cal{M}}({\tilde{\mbox{\boldmath{$\kappa$}}}},\epsilon,S)$
can be obtained in terms of the vectors at our disposal in the rest frame of
the system, i.e. (i) the unit vector ${\hat{\bf n}}$, which defines the LF
components of a four vector, (ii) the polarization vector $\bf S$, and (iii)
the transverse (with respect to the ${\hat{\bf n}}$ axis) momentum component
${\bf k}_{\perp}={\bf p}_{\perp}=\mbox{\boldmath{$\kappa$}}_{\perp}$. Let us
recall that we adopt ${\hat{\bf n}}\equiv\hat{z}$. Then the LF spin-dependent
spectral function reads

$\displaystyle{{\cal P}}^{\tau}_{{\cal
M},\sigma^{\prime}\sigma}({\tilde{\mbox{\boldmath{$\kappa$}}}},\epsilon,S)={1\over
2}\left[{\cal{B}}_{0,{\cal
M}}^{\tau}+{\mbox{\boldmath{$\sigma$}}}\cdot{\mbox{\boldmath{$\cal
F$}}}^{\tau}_{{\cal M}}({\tilde{\mbox{\boldmath{$\kappa$}}}},\epsilon,\bf
S)\right]_{\sigma^{\prime}\sigma},$ (6)

where the function ${\cal{B}}_{0,{\cal M}}^{\tau}$ is the trace of
${{{\widehat{{\mbox{\boldmath{$\cal P$}}}}}^{\tau}_{\cal
M}({\tilde{\mbox{\boldmath{$\kappa$}}}},\epsilon,S)}}$ and yields the
unpolarized spectral function, while

$\displaystyle{\mbox{\boldmath{$\cal F$}}}^{\tau}_{{\cal
M}}({\tilde{\mbox{\boldmath{$\kappa$}}}},\epsilon,{\bf
S})~{}=~{}Tr\left[{\mbox{\boldmath{$\widehat{\cal P}$}}}^{\tau}_{\cal
M}({\tilde{\mbox{\boldmath{$\kappa$}}}},\epsilon,S)~{}\mbox{\boldmath{$\sigma$}}\right]~{}.$
(7)

The quantity ${\mbox{\boldmath{$\cal
F$}}}^{\tau}_{\cal{M}}({\tilde{\mbox{\boldmath{$\kappa$}}}},\epsilon,\bf S)$
is a pseudovector and depends on the direction of the polarization vector
$\bf{S}$. Therefore it can be written as a linear combination of the
independent pseudovectors at our disposal, viz. $\bf{S}$, $~{}\hat{\bf
k}_{\perp}({\bf S}\cdot\hat{\bf k}_{\perp})$, $~{}\hat{\bf k}_{\perp}({\bf
S}\cdot\hat{z})$, $~{}\hat{z}~{}({\bf S}\cdot\hat{\bf k}_{\perp})$, and
$~{}\hat{z}~{}({\bf S}\cdot\hat{z})$. Furthermore ${\mbox{\boldmath{$\cal
F$}}}^{\tau}_{\cal{M}}$ depends on $x$

$\displaystyle x={\kappa^{+}\over{\cal{M}}_{0}[1,(A-1)]}~{},$ (8)

where ${{\cal{M}}_{0}[1,(A-1)]}$ (cf. Eq. (3)), is given in terms of
${\tilde{\mbox{\boldmath{$\kappa$}}}}$ by [1]

$\displaystyle{\cal{M}}_{0}[1,(A-1)]=E(\kappa)+E_{S}={(\kappa^{+})^{2}+(m^{2}+|{\bf
k}_{\perp}|^{2})\over 2~{}\kappa^{+}}$
$\displaystyle+\left\\{\left[(\kappa^{+})^{2}+(m^{2}+|{\bf
k}_{\perp}|^{2})\over
2~{}\kappa^{+}\right]^{2}~{}+~{}M_{S}^{2}~{}-~{}m^{2}\right\\}^{1/2}~{},$ (9)

where $E_{S}=\sqrt{M_{S}^{2}+|{\mbox{\boldmath{$\kappa$}}}|^{2}}$, with
$M_{S}^{2}=4m^{2}+4m\epsilon$ for $(A-1)=2$, and ${\bf
k}_{\perp}={\mbox{\boldmath{$\kappa$}}}_{\perp}={\bf p}_{\perp}$.

Hence ${\mbox{\boldmath{$\cal F$}}}^{\tau}_{\cal{M}}$ can be expressed as a
sum of the five available independent pseudovectors multiplied by five scalar
quantities, ${\cal B}^{\tau}_{i,{\cal M}}$ ($i=1,...,5)$, viz.

$\displaystyle{\mbox{\boldmath{$\cal F$}}}^{\tau}_{{\cal M}}(x,{\bf
k}_{\perp};\epsilon,{\bf S})={\bf S}{\cal B}^{\tau}_{1,{\cal M}}+\hat{\bf
k}_{\perp}({\bf S}\cdot\hat{\bf k}_{\perp}){\cal B}^{\tau}_{2,{\cal M}}$
$\displaystyle+\hat{\bf k}_{\perp}({\bf S}\cdot\hat{z}){\cal
B}^{\tau}_{3,{\cal M}}+\hat{z}({\bf S}\cdot\hat{\bf k}_{\perp}){\cal
B}^{\tau}_{4,{\cal M}}+\hat{z}({\bf S}\cdot\hat{z}){\cal B}^{\tau}_{5,{\cal
M}}$ (10)

where the dependence upon $\hat{n}\equiv\hat{z}$ is understood for making
light the notation. It should be pointed out that to fully address the issue
of TMDs one has to distinguish between the transverse dof’s and the one
associate to $\hat{n}$. Therefore one cannot anymore use the parametrization
with only three scalar functions considered in Ref. [58].

The six scalar quantities ${\cal B}^{\tau}_{i,{\cal M}}$ in general can depend
on the possible scalars at our disposal, i.e., $x,|{\bf
k}_{\perp}|,\epsilon,({\bf S}\cdot\hat{\bf k}_{\perp})^{2},({\bf
S}\cdot{\hat{z}})^{2},{{(\hat{\bf k}_{\perp}\times{\hat{z}})\cdot{\bf S}}}$.
However, as shown in Appendix A, for a system of total angular momentum ${\cal
J}=1/2$, as 3He or 3H, the quantities ${\cal B}^{\tau}_{i,{\cal M}}$ can
depend only on $x,|{\bf k}_{\perp}|$ and $\epsilon$.

Through the trace of the spectral function ${{{\widehat{{\mbox{\boldmath{$\cal
P$}}}}}^{\tau}_{\cal M}({\tilde{\mbox{\boldmath{$\kappa$}}}},\epsilon,S)}}$
one can define the LF spin-independent nucleon momentum distribution, averaged
on the spin directions, as follows (see Ref. [1])

$\displaystyle n^{\tau}(x,{\bf k}_{\perp})=\int\\!\\!\ \\!\\!\\!\\!\\!\
\\!\\!\\!\\!\\!\sum{d\epsilon~{}}{1\over
2\kappa^{+}~{}(2\pi)^{3}}~{}~{}{\partial\kappa^{+}\over\partial x}~{}Tr{\cal
P}^{\tau}(\tilde{\mbox{\boldmath{$\kappa$}}},\epsilon,S)=\int\\!\\!\
\\!\\!\\!\\!\\!\ \\!\\!\\!\\!\\!\sum{d\epsilon~{}}{1\over
2~{}(2\pi)^{3}}~{}{E_{S}\over(1-{x})~{}\kappa^{+}}~{}\rho(\epsilon)$
$\displaystyle\times~{}\sum_{\sigma}\sum_{JJ_{z}\alpha}\sum_{T_{S}\tau_{S}}~{}_{LF}\langle\tau_{S},T_{S};\alpha,\epsilon;J_{z}J;\tau\sigma,\tilde{\mbox{\boldmath{$\kappa$}}}|{\psi}_{\cal{J}\cal{M}};S,TT_{z}\rangle~{}\langle
TT_{z},S;{\psi}_{\cal{J}\cal{M}}|\tilde{\mbox{\boldmath{$\kappa$}}},\sigma\tau;JJ_{z};\epsilon,\alpha;T_{S},\tau_{S}\rangle_{LF}~{}.$
(11)

The completeness relation of the nonsymmetric basis for three-interacting-
particle systems (see Eq. (51) of Ref. [1]), immediately leads to the
normalization of the nucleon momentum distribution, i.e. the baryon number sum
rule,

$\displaystyle\int dx~{}\int d{\bf k}_{\perp}~{}n^{\tau}(x,{\bf
k}_{\perp})~{}=~{}1~{},$ (12)

and to the momentum sum rule

$\displaystyle\int x~{}dx~{}\int d{\bf k}_{\perp}~{}n^{\tau}(x,{\bf
k}_{\perp})~{}=~{}{1\over 3}~{}.$ (13)

From the LF spin-dependent spectral function, after performing the integration
shown in Eq. (11), one can obtain the LF spin-dependent momentum distribution,
a $2\times 2$ matrix defined by (see also Appendix B)

$\displaystyle\left[{\mbox{\boldmath{$\cal N$}}}_{\cal M}^{\tau}(x,{\bf
k}_{\perp};{\bf S})\right]_{\sigma^{\prime}\sigma}=\int\\!\\!\
\\!\\!\\!\\!\\!\ \\!\\!\\!\\!\\!\sum d\epsilon~{}{1\over
2~{}{(2\pi)^{3}}}~{}{1\over 1-x}~{}{E_{S}\over\kappa^{+}}~{}$
$\displaystyle\times{{\cal P}}^{\tau}_{{\cal
M},\sigma^{\prime}\sigma}({\tilde{\mbox{\boldmath{$\kappa$}}}},\epsilon,S)={\pi\over
4m}~{}\int\frac{d{p^{+}}d{p^{-}}}{(2\pi)^{4}}~{}\delta(p^{+}-\xi
P^{+})~{}P^{+}$ $\displaystyle\times~{}{E_{S}\over\kappa^{+}}~{}{{\cal
P}}^{\tau}_{{\cal
M},\sigma^{\prime}\sigma}({\tilde{\mbox{\boldmath{$\kappa$}}}},\epsilon,S)~{}.$
(14)

As shown in Appendix B, one can obtain the LF spin-dependent momentum
distribution from the three-body wave function, using Eq. (4) and the
expression for the LF spin-dependent spectral function given by Eq. (72) of
Ref. [1].

As it occurs for the spectral function, the momentum distribution can be
expressed through the three independent vectors available in the rest frame of
the system, i.e. $\bf k_{\perp}$, $\bf S$, and $\hat{n}\equiv\hat{z}$, and six
scalar functions $b^{\tau}_{i,{\cal M}}$ ($i=0,1,...,5)$, viz

$\displaystyle{\mbox{\boldmath{$\cal N$}}}_{\cal M}^{\tau}(x,{\bf
k}_{\perp};{\bf S})={1\over 2}\left\\{b_{0,{\cal
M}}+{\mbox{\boldmath{$\sigma$}}}\cdot{\mbox{\boldmath{$f$}}^{\tau}_{{\cal
M}}(x,{\bf k}_{\perp};\bf S)}\right\\}$ (15)

where ${{\mbox{\boldmath{$f$}}}}^{\tau}_{{\cal M}}(x,{\bf k}_{\perp};\bf S)$
is a pseudovector (recall that the dependence upon $\hat{n}\equiv\hat{z}$ has
been dropped out for simplicity) that can be decomposed as follows

$\displaystyle{\mbox{\boldmath{$f$}}}^{\tau}_{{\cal M}}(x,{\bf k}_{\perp};{\bf
S})={\bf S}b^{\tau}_{1,{\cal M}}+\hat{\bf k}_{\perp}({\bf S}\cdot\hat{\bf
k}_{\perp})b^{\tau}_{2,{\cal M}}$ $\displaystyle+\hat{\bf k}_{\perp}({\bf
S}\cdot\hat{z})b^{\tau}_{3,{\cal M}}+\hat{z}({\bf S}\cdot\hat{\bf
k}_{\perp})b^{\tau}_{4,{\cal M}}+\hat{z}({\bf S}\cdot\hat{z})b^{\tau}_{5,{\cal
M}}~{}.$ (16)

The functions $b^{\tau}_{i,{\cal M}}$, that depend upon $x$, $|{\bf
k}_{\perp}|$, $({\bf S}\cdot\hat{\bf k}_{\perp})^{2}$, $({\bf
S}\cdot{\hat{z}})^{2}$ and ${{(\hat{\bf k}_{\perp}\times{\hat{z}})\cdot{\bf
S}}}$ , are integrals over the energy $\epsilon$ of the functions ${\cal
B}^{\tau}_{i,{\cal M}}$ (see Eq. (14)), viz.

$\displaystyle b^{\tau}_{i,{\cal M}}\left[x,|{\bf k}_{\perp}|,({\bf
S}\cdot\hat{\bf k}_{\perp})^{2},({\bf S}\cdot{\hat{z}})^{2},{{(\hat{\bf
k}_{\perp}\times{\hat{z}})\cdot{\bf S}}}\right]$ $\displaystyle={\pi\over
4m}\int{{dp^{+}dp^{-}}\over{(2\pi)^{4}}}~{}\delta[p^{+}-xP^{+}]~{}P^{+}~{}{E_{S}\over\kappa^{+}}$
$\displaystyle\times~{}{\cal B}^{\tau}_{i,{\cal M}}\left[x,|{\bf
k}_{\perp}|,\epsilon,({\bf S}\cdot\hat{\bf k}_{\perp})^{2},({\bf
S}\cdot{\hat{z}})^{2},{{(\hat{\bf k}_{\perp}\times{\hat{z}})\cdot{\bf
S}}}\right]~{}.$ (17)

In Appendix B, for a three-nucleon system of total angular momentum ${\cal
J}=1/2$ explicit expressions for the quantities $b^{\tau}_{i,{\cal M}}$
($i=0,1,...,5$) are obtained in terms of the wave function of the three-
nucleon system, according to the BT procedure. It is also shown that these
functions do not depend on $\bf S$, while they do depend on $|{\bf
k}_{\perp}|$ and $x$. Moreover, the quantity $b_{0}$ is independent of $\cal
M$, while for $i=1,...,5$ the dependence on $\cal M$ is through the factor
$(-1)^{{\cal M}+1/2}$.

From the actual expressions of the quantities $b^{\tau}_{i,{\cal M}}$, the
essential role of the Melosh matrixes to generate the six different quantities
$b^{\tau}_{i,{\cal M}}$ clearly emerges. Their effect is parametrized by the
angle $\varphi$ present in the expression of the Melosh rotations, ${\cal
R}_{M}({\tilde{\bf k}})$, given in Appendix C, viz.

$\displaystyle D^{{1\over 2}}[{\cal R}_{M}({\tilde{\bf
k}})]_{\sigma\sigma^{\prime}}=\left[\cos{\varphi\over
2}~{}+~{}i~{}\sin{\varphi\over 2}~{}\hat{\bf
n}\cdot{\mbox{\boldmath{$\sigma$}}}\right]_{\sigma\sigma^{\prime}}~{},$ (18)

where

$\displaystyle\varphi=2~{}arctg~{}{|{\bf k}_{\perp}|\over k^{+}+m}~{}.$ (19)

It has to be pointed out that $\varphi$ is small if the relevant values of
$|{\bf k}_{\perp}|/m$ do too, making the effect of the Melosh rotations small.

Indeed, in absence of the Melosh matrixes, the quantities $b^{\tau}_{2,{\cal
M}}$, $b^{\tau}_{3,{\cal M}}$, $b^{\tau}_{4,{\cal M}}$, and $b^{\tau}_{5,{\cal
M}}$ are related to each other by factors as $\cos^{2}\theta$,
$\sin^{2}\theta$, and $\cos\theta\sin\theta$, with $\theta$ the angle between
the momentum $\bf k$ and the $z$ axis. Then in this case the spin-dependent
momentum distribution can be expressed in terms of only three independent
scalar quantities, $b^{\tau}_{0}$, $b^{\tau}_{1,{\cal M}}$, and
$b^{\tau}_{2,{\cal M}}/\sin^{2}\theta=b^{\tau}_{3,{\cal
M}}/\cos\theta\sin\theta=b^{\tau}_{4,{\cal
M}}/\cos\theta\sin\theta=b^{\tau}_{5,{\cal M}}/\cos^{2}\theta$, as in the non
relativistic approximation.

Our aim is to obtain an expression of TMDs from the functions
$b^{\tau}_{i,{\cal M}}$, just introduced. In order to accomplish this task,
another ingredient, the fermion correlator for a semi-inclusive process, has
to be added, along with its relation to the LF spectral function This is
detailed in the following section.

##  III  The Semi-Inclusive Correlator for a ${\cal J}=1/2$ bound system and
the LF spectral function

Let $p$ be the momentum in the lab frame of an off-mass-shell spin 1/2
particle, with isospin ${\tau}$, inside a bound system of A spin 1/2 particles
with total momentum $P$ and spin $S$. The semi-inclusive fermion correlator in
terms of the LF coordinates is [8]

$\displaystyle\Phi^{\tau}_{\alpha,\beta}(p,P,S)={1\over
2}\int{d\zeta^{-}d\zeta^{+}d{\mbox{\boldmath{$\zeta$}}}_{\perp}}~{}e^{ip^{-}\zeta^{+}/2}~{}e^{ip^{+}\zeta^{-}/2}~{}e^{-i{\bf
p}_{\perp}\cdot{\mbox{\boldmath{$\zeta$}}}_{\perp}}\langle
P,S,A|\bar{\psi}^{\tau}_{\beta}(0)~{}{{\cal W}(\hat{n}\cdot
A)}~{}\psi^{\tau}_{\alpha}(\zeta)|A,S,P\rangle$ (20)

where $|A,S,P\rangle$ is the A-particle state (e.g. a nucleus or a nucleon),
$\psi^{\tau}_{\alpha}(\zeta)$ the particle field (e.g. a nucleon of isospin
${\tau}$ if the system is a nucleus, or a quark if the system is a nucleon )
and ${\cal W}(\hat{n}\cdot A)$ is a link operator which makes
$\Phi^{\tau}_{\alpha,\beta}(p,P,S)$ gauge invariant. By working in the
$A^{+}=0$ gauge, ${{\cal W}}$ can be reduced to unity. Let us notice that the
$A^{+}$ gauge condition is preserved under Lorentz transformation in the LF
dynamics [59, 60]. Hereafter, we shall assume that the link operator is the
unity operator. Using the translation invariance relation

$\displaystyle\psi^{\tau}_{\alpha}(\zeta)=e^{i\widehat{P}\cdot\zeta}~{}\psi^{\tau}_{\alpha}(0)~{}e^{-i\widehat{P}\cdot\zeta}\,,$
(21)

one can rewrite the correlator as

$\displaystyle\Phi^{\tau}_{\alpha,\beta}(p,P,S)={1\over
2}~{}\int{d\zeta^{-}d\zeta^{+}d{\mbox{\boldmath{$\zeta$}}}_{\perp}}~{}e^{i(p-P)\cdot\zeta}$
$\displaystyle\times~{}\langle
P,S,A|\bar{\psi}^{\tau}_{\beta}(0)~{}e^{i\zeta\cdot\widehat{P}}~{}\psi^{\tau}_{\alpha}(0)|A,S,P\rangle~{},$
(22)

where the explicit expression for the fermion field in $\zeta^{\mu}=0$ is [60]

$\displaystyle\psi^{\tau}_{\alpha}(0)=\int{d\tilde{\bf p}\over
2p^{+}(2\pi)^{3}}$
$\displaystyle\times~{}\sum_{\sigma}\left[b^{\tau}_{\sigma}({\tilde{\bf
p}})~{}u_{\alpha}(\tilde{\bf p},\sigma)+d^{{\tau}\dagger}_{\sigma}(\tilde{\bf
p})~{}v_{\alpha}(\tilde{\bf p},\sigma)\right]\,,$ (23)

with $\tilde{\bf p}=(p^{+},{\bf p}_{\perp})$ the LF particle momentum in the
lab frame and $u(\tilde{\bf p},\sigma)$, $v(\tilde{\bf p},\sigma)$ the
particle and anti-particle LF spinors [60, 61]. The following spinor
normalization is adopted

$\displaystyle\bar{u}({\bf\tilde{p}},\sigma)u({\bf\tilde{p}},\sigma^{\prime})=2~{}m~{}\delta_{\sigma\sigma^{\prime}}~{}.$
(24)

The Fock operators satisfy the canonical anti-commutation relations,

$\displaystyle\left\\{b^{\tau}_{\sigma}(\tilde{{\bf
p}}),b_{\sigma^{\prime}}^{{\tau^{\prime}}{\dagger}}(\tilde{{\bf
p^{\prime}}})\right\\}=\left\\{d^{\tau}_{\sigma}(\tilde{{\bf
p}}),d_{\sigma^{\prime}}^{{\tau}^{\prime}{\dagger}}(\tilde{{\bf
p^{\prime}}})\right\\}$
$\displaystyle=2p^{+}(2\pi)^{3}\delta_{\sigma\sigma^{\prime}}\delta_{\tau\tau^{\prime}}\delta(\tilde{{\bf
p}}-\tilde{{\bf p^{\prime}}}).$ (25)

By inserting in Eq. (22) for the correlator
$\Phi^{\tau}_{\alpha,\beta}(p,P,S)$ the fermionic field given in Eq. (23), the
particle correlator reads

$\displaystyle\Phi^{\tau}_{\alpha,\beta}(p,P,S)=~{}\int{{d\zeta^{+}}\int
d{\bar{\mbox{\boldmath{$\zeta$}}}}~{}e^{{i(p^{-}-P^{-})\zeta^{+}}/2}~{}e^{i({\tilde{\bf
p}}-{\tilde{\bf P}})\cdot{\bar{\mbox{\boldmath{$\zeta$}}}}}~{}\int{d\tilde{\bf
p}^{\prime}\over 2p^{\prime+}(2\pi)^{3}}}$
$\displaystyle\times~{}\int{d\tilde{\bf p}^{\prime\prime}\over
2p^{\prime\prime+}(2\pi)^{3}}\langle
P,S,A|~{}\sum_{\sigma^{\prime\prime}}\left[b^{\tau\dagger}_{\sigma^{\prime\prime}}({\bf\tilde{p}^{\prime\prime}})~{}{\bar{u}}_{\beta}({\bf\tilde{p}^{\prime\prime}},\sigma^{\prime\prime})+d^{\tau}_{\sigma^{\prime\prime}}({\bf\tilde{p}^{\prime\prime}})~{}{\bar{v}}_{\beta}({\bf\tilde{p}^{\prime\prime}},\sigma^{\prime\prime})\right]~{}~{}e^{i\widehat{P}^{-}\zeta^{+}/2}~{}$
$\displaystyle\times\sum_{\sigma^{\prime}}\left[e^{i(\bf{\tilde{P}}-\bf{\tilde{p}^{\prime}})\cdot{\mbox{\boldmath{$\bar{\zeta}$}}}}~{}b^{\tau}_{\sigma^{\prime}}({\bf\tilde{p}^{\prime}})~{}u_{\alpha}({\bf\tilde{p}^{\prime}},\sigma^{\prime})+e^{i(\bf{\tilde{P}}+\bf{\tilde{p}^{\prime}})\cdot{\mbox{\boldmath{$\bar{\zeta}$}}}}~{}d^{\tau\dagger}_{\sigma^{\prime}}({\bf\tilde{p}^{\prime}})~{}v_{\alpha}({\bf\tilde{p}^{\prime}},\sigma^{\prime})\right]|A,S,P\rangle$
(26)

with
${\mbox{\boldmath{$\bar{\zeta}$}}}=(\zeta^{-}/2,{\mbox{\boldmath{$\zeta$}}}_{\perp})$.
Recall that $b^{\tau}_{\sigma^{\prime}}({\bf\tilde{p}^{\prime}})|A,S,P\rangle$
and $d^{\tau\dagger}_{\sigma^{\prime}}({\bf\tilde{p}^{\prime}})|A,S,P\rangle$
are eigenvectors of the LF-momentum operator (it is a kinematical one) with
eigenvalues $\tilde{P}-\tilde{p}^{\prime}$ and $\tilde{P}+\tilde{p}^{\prime}$,
respectively.

By performing the integration on
$\\{\zeta^{+}/2,{\bar{\mbox{\boldmath{$\zeta$}}}}\\}$, the term in (26) with
$v_{\alpha}({\bf\tilde{p}^{\prime}},\sigma^{\prime})$ vanishes, since
$p^{\prime+}$ cannot be negative. Then one has

$\displaystyle\Phi_{\alpha,\beta}^{\tau}(p,P,S)=\sum_{\sigma^{\prime}}u_{\alpha}({\bf\tilde{p}},\sigma^{\prime}){2\pi\over
p^{+}}\int{d\tilde{\bf p}^{\prime\prime}\over
2p^{\prime\prime+}(2\pi)^{3}}~{}\langle
P,S,A|\sum_{\sigma^{\prime\prime}}\left[b^{\tau\dagger}_{\sigma^{\prime\prime}}(\tilde{\bf
p}^{\prime\prime}){\bar{u}}_{\beta}(\tilde{\bf
p}^{\prime\prime},\sigma^{\prime\prime})+d^{\tau}_{\sigma^{\prime\prime}}({\bf\tilde{p}^{\prime\prime}}){\bar{v}}_{\beta}({\bf\tilde{p}^{\prime\prime}},\sigma^{\prime\prime})\right]$
$\displaystyle\times~{}\delta(\hat{P}^{-}+p^{-}-P^{-})b^{\tau}_{\sigma^{\prime}}({\bf\tilde{p}})|A,S,P\rangle$
$\displaystyle={2\pi\over
p^{+}}~{}\sum_{\sigma}\sum_{\sigma^{\prime}}~{}u_{\alpha}({\bf\tilde{p}},\sigma^{\prime})~{}~{}\int{\tilde{\bf
p}^{\prime\prime}\over 2p^{\prime\prime+}(2\pi)^{3}}\langle
P,S,A|~{}b^{\tau\dagger}_{\sigma}({\bf\tilde{p}^{\prime\prime}})~{}\delta(\hat{P}^{-}+p^{-}-P^{-})~{}b^{\tau}_{\sigma^{\prime}}({\bf\tilde{p}})~{}|A,S,P\rangle~{}\bar{u}_{{\beta}}({\bf\tilde{p}^{\prime\prime}},\sigma)~{},$
(27)

where in the last step the anti-particle contribution has been eliminated,
since the LF-momentum has to be conserved and $p^{\prime\prime+}$ cannot be
negative.

In the previous equation, let us introduce the following quantity

$\displaystyle\sum_{JJ_{z}\alpha}\sum_{T_{S}\tau_{S}}\int\\!\\!\
\\!\\!\\!\\!\\!\ \\!\\!\\!\\!\\!\sum{\rho(\epsilon)d\epsilon}\int{d{\tilde{\bf
P}}_{S}\over(2\pi)^{3}2P^{+}_{S}}$ $\displaystyle\times|{\tilde{\bf
P}}_{S};JJ_{z}\epsilon,\alpha;T_{S},\tau_{S}\rangle_{LF}~{}_{LF}\langle\tau_{S},T_{S};\alpha,\epsilon
J_{z}J;{\tilde{\bf P}}_{S}|,$ (28)

that in valence approximation is the completeness for the states of
$(A-1)$-particles (see also Ref. [1]). In Eq. (28) ${\tilde{\bf P}}_{S}$ is
the total LF momentum of the fully interacting $(A-1)$-particle system. The
symbol $~{}{\LARGE{\int}}\\!\\!\ \\!\\!\\!\\!\\!\ \\!\\!\\!\sum$ means a sum
over the bound states of the $(A-1)$ system and an integration over the
continuum.

For the bound states $|{\tilde{\bf
P}}_{S};JJ_{z}\epsilon,\alpha;T_{S},\tau_{S}\rangle_{LF}$, the normalization
adopted is

${}_{LF}\langle
T^{\prime},\tau^{\prime};J^{\prime}J^{\prime}_{z}\epsilon^{\prime}\alpha^{\prime},{\tilde{\bf
P}}^{\prime}_{S}|{\tilde{\bf
P}}_{S};JJ_{z}\epsilon\alpha;T,\tau\rangle_{LF}=2~{}P_{S}^{+}~{}(2\pi)^{3}$
$\displaystyle\times~{}\delta^{3}({\tilde{\bf P}}_{S}^{\prime}-{\tilde{\bf
P}}_{S})~{}\delta_{T^{\prime},T}\delta_{\tau^{\prime},\tau}\delta_{\alpha^{\prime},\alpha}\delta_{J^{\prime},J}\delta_{J^{\prime}_{z},J_{z}}\delta_{\epsilon^{\prime},\epsilon}~{},$
(29)

while for the LF continuum states the orthogonality reads (see Appendix A of
Ref. [1])

${}_{LF}\langle
T^{\prime}\tau^{\prime};\alpha^{\prime}\epsilon^{\prime}J^{\prime}_{z}J^{\prime};{\tilde{\bf
P}}^{\prime}_{S}|{\tilde{\bf
P}}_{S};JJ_{z}\epsilon\alpha;T\tau\rangle_{LF}=2P_{S}^{+}(2\pi)^{3}$
$\displaystyle\times~{}\delta^{3}({\tilde{\bf P}}^{\prime}_{S}-{\tilde{\bf
P}}_{S})\delta_{T^{\prime},T}\delta_{\tau^{\prime},\tau}\delta_{\alpha^{\prime},\alpha}\delta_{J^{\prime},J}\delta_{J^{\prime}_{z},J_{z}}{\delta(\epsilon^{\prime}-\epsilon)\over\rho(\epsilon)}~{}.$
(30)

Then, one can define as follows the valence contribution to the particle
correlator

$\displaystyle\left[\Phi^{\tau}_{V}(p,P,S)\right]_{\alpha,\beta}={2\pi\over
p^{+}}\int{d{\bf\tilde{p}}^{\prime\prime}\over(2\pi)^{3}{2{p}}^{\prime\prime+}}~{}\sum_{JJ_{z}\alpha}\sum_{T_{S}\tau_{S}}\sum_{J^{\prime}J^{\prime}_{z}\alpha^{\prime}}\sum_{T^{\prime}_{S}\tau^{\prime}_{S}}\int\\!\\!\
\\!\\!\\!\\!\\!\ \\!\\!\\!\\!\\!\sum{\rho(\epsilon)d\epsilon~{}}\int\\!\\!\
\\!\\!\\!\\!\\!\
\\!\\!\\!\\!\\!\sum{\rho(\epsilon^{\prime})d\epsilon^{\prime}~{}}\int{d{\tilde{\bf
P}}_{S}\over(2\pi)^{3}2P^{+}_{S}}~{}$ $\displaystyle\times~{}\int{d{\tilde{\bf
P}}^{\prime}_{S}\over(2\pi)^{3}2P^{\prime+}_{S}}~{}\sum_{\sigma\sigma^{\prime}}\left[~{}u_{\alpha}({\bf\tilde{p}},\sigma^{\prime})\langle
P,S,A|{\tilde{\bf p}}^{\prime\prime}\sigma\tau;{\tilde{\bf
P}}^{\prime}_{S};J^{\prime}J^{\prime}_{z}\epsilon^{\prime},\alpha^{\prime};T^{\prime}_{S},\tau^{\prime}_{S}\rangle_{LF}~{}_{LF}\langle\tau^{\prime}_{S},T^{\prime}_{S};\alpha^{\prime},\epsilon^{\prime}J^{\prime}_{z}J^{\prime};{\tilde{\bf
P}}^{\prime}_{S}|\right.$
$\displaystyle\left.\times~{}\delta(\hat{P}^{-}+p^{-}-P^{-})~{}|{\tilde{\bf
P}}_{S};JJ_{z}\epsilon,\alpha;T_{S},\tau_{S}\rangle_{LF}~{}_{LF}\langle\tau_{S},T_{S};\alpha,\epsilon
J_{z}J;{\tilde{\bf P}}_{S};{\tilde{\bf
p}}\sigma^{\prime}\tau|A,S,P\rangle~{}{\bar{u}}_{\beta}({\bf\tilde{p}},\sigma)~{}\right]$
$\displaystyle={2\pi\over
p^{+}}\int{d{\bf\tilde{p}}^{\prime\prime}\over(2\pi)^{3}{2{p}}^{\prime\prime+}}\sum_{JJ_{z}\alpha}\sum_{T_{S}\tau_{S}}\int\\!\\!\
\\!\\!\\!\\!\\!\
\\!\\!\\!\\!\\!\sum{\rho(\epsilon)d\epsilon~{}}\int{d{\tilde{\bf
P}}_{S}\over(2\pi)^{3}2P^{+}_{S}}\sum_{\sigma\sigma^{\prime}}\Bigl{[}u_{\alpha}({\bf\tilde{p}},\sigma^{\prime})\langle
P,S,A|{\tilde{\bf p}}^{\prime\prime}\sigma\tau;{\tilde{\bf
P}}_{S};JJ_{z}\epsilon,\alpha;T_{S},\tau_{S}\rangle_{LF}$
$\displaystyle\times~{}\delta(P^{-}_{S}+p^{-}-P^{-})~{}_{LF}\langle\tau_{S},T_{S};\alpha,\epsilon
J_{z}J;{\tilde{\bf P}}_{S};{\tilde{\bf
p}}\sigma^{\prime}\tau|A,S,P\rangle~{}{\bar{u}}_{\beta}({\bf\tilde{p}},\sigma)~{}\Bigr{]}~{},$
(31)

where the equality

$\displaystyle\hat{b}^{\tau\dagger}_{\sigma}({\bf\tilde{p}^{\prime\prime}})~{}|{\tilde{\bf
P}}^{\prime}_{S};J^{\prime}J^{\prime}_{z}\epsilon^{\prime},\alpha^{\prime};T^{\prime}_{S},\tau^{\prime}_{S}\rangle_{LF}$
$\displaystyle=|{\tilde{\bf p}^{\prime\prime}}\sigma\tau;{\tilde{\bf
P}}^{\prime}_{S};J^{\prime}J^{\prime}_{z}\epsilon^{\prime},\alpha^{\prime};T^{\prime}_{S},\tau^{\prime}_{S}~{},\rangle_{LF}$
(32)

has been used, i.e. a free particle $|{\tilde{\bf
p}^{\prime\prime}}\sigma\tau\rangle$, with momentum ${\tilde{\bf
p}^{\prime\prime}}$ in the lab frame, has been created. Moreover, it has been
taken into account that the operator $\hat{P}^{-}$ acts on $|{\tilde{\bf
P}}_{S};JJ_{z}\epsilon,\alpha;T_{S},\tau_{S}\rangle_{LF}$ as follows

$\displaystyle\hat{P}^{-}~{}|{\tilde{\bf
P}}_{S};JJ_{z}\epsilon,\alpha;T_{S},\tau_{S}\rangle_{LF}~{}=P^{-}_{S}~{}|{\tilde{\bf
P}}_{S};JJ_{z}\epsilon,\alpha;T_{S},\tau_{S}\rangle_{LF}~{}$
$\displaystyle=~{}{M^{2}_{S}+|{\bf P}_{S\perp}|^{2}\over
P^{+}_{S}}~{}|{\tilde{\bf
P}}_{S};JJ_{z}\epsilon,\alpha;T_{S},\tau_{S}\rangle_{LF}~{},$ (33)

with $M_{S}$ the mass of the interacting $(A-1)$-particle system (for $A-1=2$,
one has $M_{S}^{2}=4m^{2}+4m\epsilon$).

By considering that the LF-momentum is conserved (the interaction is contained
only in the minus component of the momenta) and the kinematical nature of the
LF-boosts, one has the exact separation of the intrinsic dof’s from the CM one
(see Appendix A of [45]), obtaining

$\displaystyle|{\tilde{\bf p}}\sigma\tau;{\tilde{\bf
P}}_{S};JJ_{z}\epsilon,\alpha;T_{S},\tau_{S}\rangle_{LF}=\sqrt{{E_{S}\over{\cal{M}}_{0}[1,(A-1)]}}$
$\displaystyle\times~{}|{\tilde{\bf p}}+{\tilde{\bf
P}}_{S}\rangle_{LF}~{}|{\mbox{\boldmath{$\tilde{\bf\kappa}$}}}\sigma\tau;JJ_{z}\epsilon,\alpha;T_{S},\tau_{S}\rangle_{LF}~{},$
(34)

where $|{\tilde{\bf p}}+{\tilde{\bf P}}_{S}\rangle_{LF}$ is the total LF
momentum eigenstate of the cluster $[1,(A-1)]$. The intrinsic state $\
|{\mbox{\boldmath{$\tilde{\bf\kappa}$}}}\sigma\tau;JJ_{z}\epsilon,\alpha;T_{S},\tau_{S}\rangle_{LF}$
is composed by a fully-interacting intrinsic state of $(A-1)$ constituents and
a plane wave, describing a constituent that freely moves in the intrinsic
frame of the whole cluster $[1,(A-1)]$ with LF-momentum $\tilde{\bf\kappa}$
(see Eq. (2)).

In Eq. (34) one has
$E_{S}=\sqrt{M_{S}^{2}+|{\mbox{\boldmath{$\kappa$}}}|^{2}}$, and
${\cal{M}}_{0}[1,(A-1)]$ is defined by Eq. (3). The factor
$\sqrt{{E_{S}/{\cal{M}}_{0}[1,(A-1)]}}$ takes care of the proper normalization
of the momentum eigenstates $|{\tilde{\bf p}}\rangle_{LF}$, $|{\tilde{\bf
P}}_{S}\rangle_{LF}$ and $|{\tilde{\bf p}}+{\tilde{\bf P}}_{S}\rangle_{LF}$
(see Ref. [45]).

Summarizing, the following overlap, present in Eq. (31), can be written as
follows

$\displaystyle\langle P,S,A|{\tilde{\bf p}}\sigma\tau;{\tilde{\bf
P}}_{S};JJ_{z}\epsilon,\alpha;T_{S},\tau_{S}\rangle_{LF}$
$\displaystyle=2P^{+}(2\pi)^{3}~{}\sqrt{{E_{S}\over{\cal{M}}_{0}[1,(A-1)]}}~{}~{}\delta^{3}({\tilde{\bf
P}}-{\tilde{\bf P}}_{S}-{\tilde{\bf p}})$ $\displaystyle\times~{}\langle
A,S,{int}|{\mbox{\boldmath{$\tilde{\bf\kappa}$}}}\sigma\tau;JJ_{z}\epsilon,\alpha;T_{S},\tau_{S}\rangle_{LF}~{},$
(35)

where the orthogonality of the plane waves is given by
$\langle{\bf\tilde{P}}|{\bf\tilde{P}^{\prime}}\rangle=2P^{+}(2\pi)^{3}\delta({\bf\tilde{P}}-{\bf\tilde{P}^{\prime}})$
and $|int,S,A\rangle\equiv|{\psi}_{\cal{J}\cal{M}};S,TT_{z}\rangle$ is the
intrinsic eigenstate of the system.

The normalization for the intrinsic overlaps
${}_{LF}\langle\tau_{S},T_{S};\alpha,\epsilon;J_{z}J;{\mbox{\boldmath{$\tilde{\bf\kappa}$}}}\sigma\tau|{int},S,A\rangle$
(see Eq. (5)) reads

$\displaystyle\int{d{\tilde{\mbox{\boldmath{$\kappa$}}}}\over
2\kappa^{+}(2\pi)^{3}}~{}\int\\!\\!\ \\!\\!\\!\\!\\!\
\\!\\!\\!\\!\\!\sum{\rho(\epsilon)d\epsilon}~{}\sum_{\sigma}\sum_{T_{S}\tau_{S}}\sum_{JJ_{z}\alpha}$
$\displaystyle\times_{LF}\langle\tau_{S},T_{S};\alpha,\epsilon
J_{z}J;{\mbox{\boldmath{$\tilde{\bf\kappa}$}}}\sigma\tau|{int},S,A\rangle|^{2}=1~{}.$
(36)

Eventually, with the help of Eq. (35) the general expression for the valence
contribution to the semi-inclusive fermion correlator becomes

$\displaystyle\left[\Phi^{\tau}_{V}(p,P,S)\right]_{\alpha,\beta}=2\pi~{}\left({P^{+}\over
p^{+}}\right)^{2}~{}\sum_{JJ_{z}\alpha}\sum_{T_{S}\tau_{S}}\int\\!\\!\
\\!\\!\\!\\!\\!\
\\!\\!\\!\\!\\!\sum{\rho(\epsilon)d\epsilon~{}{\delta(P^{-}_{S}+p^{-}-P^{-})\over(P^{+}-p^{+})}}{E_{S}\over{\cal{M}}_{0}[1,(A-1)]}~{}\quad\quad\quad\quad\quad$
$\displaystyle\times~{}\sum_{\sigma\sigma^{\prime}}\left[~{}u_{\alpha}(\tilde{\bf
p},\sigma^{\prime})\langle
A,S,{int}|{\mbox{\boldmath{$\tilde{\bf\kappa}$}}}\sigma\tau;JJ_{z}\epsilon,\alpha;T_{S},\tau_{S}\rangle_{LF}~{}_{LF}\langle\tau_{S},T_{S};\alpha,\epsilon
J_{z}J;{\mbox{\boldmath{$\tilde{\bf\kappa}$}}}\sigma^{\prime}\tau|{int},S,A\rangle_{LF}{\bar{u}}_{\beta}(\tilde{\bf
p},\sigma)~{}\right]~{}~{},$ (37)

where $P^{-}_{S}=(M^{2}_{S}+|{\bf P}_{S\perp}|^{2})/P^{+}_{S}$, (see Eq.
(33)). Once the mass $M_{S}$ is expressed in terms of the intrinsic energy,
the integration on $\epsilon$ can be easily performed, obtaining a relation
between the correlator and the spin-dependent LF spectral function defined in
Eq. (1). In the case where $(A-1)=2$ and in the reference frame where ${\bf
P}_{\perp}=0$, the $\delta$ function in Eq. (37) implies the equation

$\displaystyle{M^{2}\over P^{+}}~{}=~{}p^{-}~{}+~{}{4m^{2}+4m\epsilon+|{\bf
P}_{S\perp}|^{2}\over P^{+}-p^{+}}$ (38)

i.e.

$\displaystyle\epsilon={({M^{2}\over
P^{+}}~{}-~{}p^{-})~{}(P^{+}-p^{+})~{}-|{\bf P}_{S\perp}|^{2}\over 4m}-m$ (39)

with ${\bf P}_{S\perp}=-{\bf
p}_{\perp}=-{\mbox{\boldmath{$\kappa$}}}_{\perp}$. Then for $A=3$ one obtains

$\displaystyle\left[\Phi^{\tau}_{V}(p,P,S)\right]_{\alpha,\beta}~{}=~{}{2\pi~{}(P^{+})^{2}\over(p^{+})^{2}~{}4m}~{}{E_{S}\over{\cal{M}}_{0}[1,(23)]}~{}\sum_{\sigma\sigma^{\prime}}$
$\displaystyle\times~{}\left\\{~{}u_{\alpha}({\tilde{\bf
p}},\sigma^{\prime})~{}{\cal P}^{\tau}_{{\cal
M},\sigma^{\prime}\sigma}({\tilde{\mbox{\boldmath{$\kappa$}}}},\epsilon,S)~{}{\bar{u}}_{\beta}({\tilde{\bf
p}},\sigma)~{}\right\\}~{}.$ (40)

Due to Eq. (24), the following equation holds :

$\displaystyle{\bar{u}({\tilde{\bf
p}},\sigma^{\prime})\Phi^{\tau}_{V}(p,P,S)u({\tilde{\bf p}},\sigma)}$
$\displaystyle={2\pi~{}m~{}E_{S}\over{\cal{M}}_{0}[1,(23)]}~{}\left({P^{+}\over
p^{+}}\right)^{2}~{}{\cal
P}^{\tau}_{{\cal{M}},\sigma^{\prime}\sigma}({\tilde{\mbox{\boldmath{$\kappa$}}}},\epsilon,S)~{}.$
(41)

From Eq. (40), using the relation (see Appendix C of Ref. [61])

$\displaystyle\bar{u}({\tilde{\bf
p}}^{\prime},\sigma^{\prime})~{}\gamma^{+}~{}u({\tilde{\bf
p}},\sigma)=\delta_{\sigma^{\prime}\sigma}~{}2~{}\sqrt{{p^{\prime}}^{+}p^{+}}$
(42)

one has

$\displaystyle{\rm
Tr}\Bigl{[}\gamma^{+}\Phi^{\tau}_{V}(p,P,S)\Bigr{]}={\pi~{}{(P^{+})^{2}}\over
m~{}p^{+}}{E_{S}\over{\cal{M}}_{0}[1,(23)]}$
$\displaystyle\times~{}\sum_{\sigma}{\cal
P}^{\tau}_{{\cal{M}},\sigma\sigma}({\tilde{\mbox{\boldmath{$\kappa$}}}},\epsilon,S)$
(43)

Therefore, since (see Ref. [1])

$\displaystyle{\partial x\over\partial\kappa_{z}}={(1-x)~{}\kappa^{+}\over
E_{S}~{}E({\bf\kappa})}$ (44)

and (see Eq. (38))

$\displaystyle{\partial p^{-}\over\partial\epsilon}=-{4m\over(P^{+}-p^{+})}$
(45)

one obtains (recall that $x=p^{+}/P^{+}$ and ${\bf
p}_{\perp}={\mbox{\boldmath{$\kappa$}}}_{\perp}$)

$\displaystyle\frac{1}{2(2\pi)^{4}}\int dp^{-}\int{{dp^{+}\over 2P^{+}}}\int
d{\bf p}_{\perp}~{}Tr\Bigl{[}\gamma^{+}\Phi^{\tau}_{V}(p,P,S)\Bigr{]}$
$\displaystyle=\frac{1}{(2\pi)^{4}}\int d\epsilon\int
d{\mbox{\boldmath{$\kappa$}}}{\pi\over E({\bf\kappa})}\sum_{\sigma}{\cal
P}^{\tau}_{{\cal
M},\sigma\sigma}({\tilde{\mbox{\boldmath{$\kappa$}}}},\epsilon,S)=1,$ (46)

because of the spectral function normalization (see Eq. (5)).

Eventually we have the normalization condition for the particle correlator

$\displaystyle\int\frac{d^{4}p}{(2\pi)^{4}}\frac{1}{2P^{+}}~{}{\rm
Tr}\Bigl{[}\gamma^{+}\Phi^{\tau}_{V}(p,P,S)\Bigr{]}=\frac{1}{2P^{+}}\frac{1}{(2\pi)^{4}}$
$\displaystyle\times\frac{1}{2}\int dp^{-}\int dp^{+}\int d{\bf p}_{\perp}{\rm
Tr}\Bigl{[}\gamma^{+}\Phi^{\tau}_{V}(p,P,S)\Bigr{]}=1.$ (47)

For a generic value of $A$, the above equations can be easily generalized.

##  IV Semi-inclusive fermion correlator, twist-two transverse momentum
distributions and LF spectral function

For the sake of completeness, let us summarize the decomposition of the semi-
inclusive fermion correlator in terms of the twist-two T-even TMDs as
presented in Ref. [8]. Let us recall that ${\bf p}_{\perp}={\bf
k}_{\perp}={\mbox{\boldmath{$\kappa$}}}_{\perp}$, and to simplify the notation
with respect to the previous Section, the isospin index $\tau$ is omitted in
what follows. The semi-inclusive correlator at leading twist is given by

$\displaystyle\Phi(p,P,S)={1\over 2}~{}{/\\!\\!\\!\\!P}~{}A_{1}$
$\displaystyle+{1\over
2}~{}\gamma_{5}~{}{/\\!\\!\\!\\!P}\left[A_{2}\,S_{z}\,+\frac{1}{M}\,\widetilde{A}_{1}\,\bf{p}_{\perp}{\cdot}\bf{S}_{\perp}\right]$
$\displaystyle+{1\over
2}\gamma_{5}~{}{/\\!\\!\\!\\!P}\left[A_{3}\,{/\\!\\!\\!\\!S}_{\perp}+\widetilde{A}_{2}\,\frac{S_{z}}{M}\,{/\\!\\!\\!p}_{\perp}+\widetilde{A}_{3}\,{{\bf{p}_{\perp}{\cdot}\bf{S}_{\perp}}\,{/\\!\\!\\!p}_{\perp}\over
M^{2}}\right]~{},$ (48)

where $M$ is the mass of the system, and the scalar functions, $A_{l}$ and
$\widetilde{A}_{l}$, contain the information on the inner dynamics
($l=1,2,3$). By performing the traces of the correlator with suitable
combinations of Dirac matrixes one has

$\displaystyle\frac{1}{2P^{+}}\,\rm Tr(\gamma^{+}\Phi)=A_{1}\,,$ (49)
$\displaystyle\frac{1}{2P^{+}}\,\rm
Tr(\gamma^{+}\gamma_{5}\Phi)=S_{z}\,A_{2}+\frac{1}{M}\,{\bf{p}_{\perp}{\cdot}\bf{S}_{\perp}}\,\widetilde{A}_{1}\,,$
(50) $\displaystyle\frac{1}{2P^{+}}\,{\rm
Tr}(i\sigma^{j+}\gamma_{5}\Phi)=-\frac{1}{2P^{+}}\,{\rm
Tr}(\gamma^{j}\gamma^{+}\gamma_{5}\Phi)$
$\displaystyle=S_{\perp}^{j}\,A_{3}+\frac{S_{z}}{M}\,p_{\perp}^{j}\,\widetilde{A}_{2}+\frac{1}{M^{2}}\,{\bf{p}_{\perp}{\cdot}\bf{S}_{\perp}}\,p_{\perp}^{j}\,\widetilde{A}_{3}$
(51)

where j=1,2. Finally, by integrating proper combinations of $A_{l}$, and
$\widetilde{A}_{l}$ on $p^{+}$ and $p^{-}$, one gets the TMDs as follows [8]

$\displaystyle f(x,|{\bf
p}_{\perp}|^{2})=\int\Bigl{[}d{p^{+}}d{p^{-}}\Bigr{]}~{}A_{1}\,,$ (52)
$\displaystyle\Delta f(x,|{\bf
p}_{\perp}|^{2})=\int\Bigl{[}d{p^{+}}d{p^{-}}\Bigr{]}~{}A_{2}\,,$ (53)
$\displaystyle g_{1T}(x,|{\bf
p}_{\perp}|^{2})=\int\Bigl{[}d{p^{+}}d{p^{-}}\Bigr{]}~{}\widetilde{A}_{1}\,,$
(54) $\displaystyle\Delta^{\prime}_{T}f(x,|{\bf
p}_{\perp}|^{2})=\int\Bigl{[}d{p^{+}}d{p^{-}}\Bigr{]}\left({A}_{3}+{|{\bf
p}_{\perp}|^{2}\over 2M^{2}}\widetilde{A}_{3}\right),$ (55) $\displaystyle
h^{\perp}_{1L}(x,|{\bf
p}_{\perp}|^{2})=\int\Bigl{[}d{p^{+}}d{p^{-}}\Bigr{]}~{}\widetilde{A}_{2}\,,$
(56) $\displaystyle h^{\perp}_{1T}(x,|{\bf
p}_{\perp}|^{2})=\int\Bigl{[}d{p^{+}}d{p^{-}}\Bigr{]}~{}\widetilde{A}_{3}\,.$
(57)

with

$\displaystyle\int\Bigl{[}d{p^{+}}d{p^{-}}\Bigr{]}=\frac{1}{2}\int\frac{d{p^{+}}d{p^{-}}}{(2\pi)^{4}}\delta[p^{+}-xP^{+}]P^{+}$
(58)

Notice that in the right hand side of the above equations, a factor of 2 is
missing with respect to the expressions in Ref. [8], because of the different
definitions of the variables $v^{\pm}$. If only the valence contribution to
the correlator is retained, the full $\Phi(p,P,S)$ is approximated by
$\Phi_{V}(p,P,S)$, and in turn $\Phi_{V}(p,P,S)$ is expanded in analogy with
Eq. (48) in terms of $A^{V}_{l}$ and $\widetilde{A}^{V}_{l}$. Hence, instead
of Eqs. (49), (50) and (51) one can write

$\displaystyle\frac{1}{2P^{+}}\,\rm
Tr(\gamma^{+}\Phi)~{}\sim~{}\frac{1}{2P^{+}}\,\rm
Tr(\gamma^{+}\Phi_{V})=A^{V}_{1}\,,$ (59) $\displaystyle\frac{1}{2P^{+}}\,{\rm
Tr}(\gamma^{+}\gamma_{5}\Phi)~{}\sim~{}\frac{1}{2P^{+}}\,{\rm
Tr}(\gamma^{+}\gamma_{5}\Phi_{V})$
$\displaystyle=S_{z}\,A^{V}_{2}+\frac{1}{M}\,{\bf{p}_{\perp}{\cdot}\bf{S}_{\perp}}\,\widetilde{A}^{V}_{1}\,,$
(60) $\displaystyle\frac{1}{2P^{+}}\,{\rm
Tr}(i\sigma^{j+}\gamma_{5}\Phi)~{}\sim-~{}\frac{1}{2P^{+}}\,{\rm
Tr}(\gamma^{j}\gamma^{+}\gamma_{5}\Phi_{V})=$
$\displaystyle=S_{\perp}^{j}\,A^{V}_{3}+\frac{S_{z}}{M}\,p_{\perp}^{j}\,\widetilde{A}^{V}_{2}+\frac{{\bf{p}_{\perp}{\cdot}\bf{S}_{\perp}}}{M^{2}}\,\,p_{\perp}^{j}\widetilde{A}^{V}_{3}\,,$
(61)

where $A^{V}_{l}$ and $\widetilde{A}^{V}_{l}$ are the valence approximations
for $A_{l}$ and $\widetilde{A}_{l}$, respectively. Because of Eqs. (40) and
(41), the above traces of $\Phi_{V}$ can be expressed by means of traces of
the spectral function, as shown in Appendix D (see also Eq. (42)):

$\displaystyle\frac{1}{2P^{+}}\,{\rm Tr}(\gamma^{+}\Phi_{V})={P^{+}\over
p^{+}}{2\pi\over 4m}{E_{S}\over{\cal{M}}_{0}[1,(23)]}$
$\displaystyle\times~{}{\rm Tr}\left[{\mbox{\boldmath{$\widehat{\cal
P}$}}}_{\cal M}(\tilde{\mbox{\boldmath{$\kappa$}}},\epsilon,S)\right]~{},$
(62) $\displaystyle\frac{1}{2P^{+}}\,{\rm
Tr}(\gamma^{+}\gamma_{5}~{}\Phi_{V})={P^{+}\over p^{+}}~{}{2\pi\over
4m}~{}{E_{S}\over{\cal{M}}_{0}[1,(23)]}$ $\displaystyle\times~{}{\rm
Tr}\left[\sigma_{z}~{}{\mbox{\boldmath{$\widehat{\cal P}$}}}_{\cal
M}(\tilde{\mbox{\boldmath{$\kappa$}}},\epsilon,S)\right]\,,$ (63)
$\displaystyle-~{}\frac{1}{2P^{+}}\,{\rm
Tr}(\gamma^{j}~{}\gamma^{+}~{}\gamma_{5}~{}\Phi_{V})={P^{+}\over
p^{+}}~{}{2\pi\over 4m}~{}{E_{S}\over{\cal{M}}_{0}[1,(23)]}$
$\displaystyle\times~{}{\rm
Tr}\left[\sigma^{j}~{}{\mbox{\boldmath{$\widehat{\cal P}$}}}_{\cal
M}(\tilde{\mbox{\boldmath{$\kappa$}}},\epsilon,S)\right]\,.$ (64)

As noted in Section II, the LF spectral function can be written in terms of
six scalar quantities, ${\cal B}_{i,{\cal M}}$.

Therefore the following equations hold :

$\displaystyle{\rm Tr}({\mbox{\boldmath{$\widehat{\cal P}$}}_{\cal
M}}I)=~{}{\cal{B}}_{0,{\cal M}}\,,$ (65) $\displaystyle{\rm
Tr}({\mbox{\boldmath{$\widehat{\cal P}$}}_{\cal M}}\sigma_{x})={S}_{x}~{}{\cal
B}_{1,{\cal M}}~{}+~{}{k_{x}\over k_{\perp}}~{}({\bf S}\cdot\hat{\bf
k}_{\perp})~{}{\cal B}_{2,{\cal M}}$ $\displaystyle+~{}{k_{x}\over
k_{\perp}}~{}({\bf S}\cdot\hat{z})~{}{\cal B}_{3,{\cal M}}\,,$ (66)
$\displaystyle{\rm Tr}({\mbox{\boldmath{$\widehat{\cal P}$}}_{\cal
M}}\sigma_{y})=S_{y}~{}{\cal B}_{1,{\cal M}}~{}+~{}{k_{y}\over
k_{\perp}}~{}({\bf S}\cdot\hat{\bf k}_{\perp})~{}{\cal B}_{2,{\cal M}}$
$\displaystyle+~{}{k_{y}\over k_{\perp}}~{}({\bf S}\cdot\hat{z})~{}{\cal
B}_{3,{\cal M}}\,,$ (67) $\displaystyle{\rm
Tr}({\mbox{\boldmath{$\widehat{\cal P}$}}_{\cal M}}\sigma_{z})=S_{z}~{}{\cal
B}_{1,{\cal M}}~{}+~{}({\bf S}\cdot\hat{\bf k}_{\perp})~{}{\cal B}_{4,{\cal
M}}~{}$ $\displaystyle+~{}({\bf S}\cdot\hat{z})~{}{\cal B}_{5,{\cal M}}~{}.$
(68)

From Eqs. (59) to (68) one obtains

$\displaystyle A^{V}_{1}~{}={\pi\over
2m}{E_{S}\over\kappa^{+}}~{}{\cal{B}}_{0,{\cal M}}$ (69) $\displaystyle
S_{z}\,A^{V}_{2}+\frac{1}{M}\,{\bf{p}_{\perp}{\cdot}\bf{S}_{\perp}}\,\widetilde{A}^{V}_{1}\,={\pi\over
2m}{E_{S}\over\kappa^{+}}$ $\displaystyle\times~{}\Bigl{[}S_{z}~{}{\cal
B}_{1,{\cal M}}~{}+~{}({\bf S}\cdot\hat{\bf k}_{\perp})~{}{\cal B}_{4,{\cal
M}}~{}+~{}({\bf S}\cdot\hat{z})~{}{\cal B}_{5,{\cal M}}\Bigr{]}$ (70)
$\displaystyle
S_{x}A^{V}_{3}+\frac{S_{z}}{M}\,p_{x}\,\widetilde{A}^{V}_{2}+\frac{{\bf{p}_{\perp}{\cdot}\bf{S}_{\perp}}}{M^{2}}\,\,p_{x}\,\widetilde{A}^{V}_{3}={\pi\over
2m}{E_{S}\over\kappa^{+}}\Bigl{[}{S}_{x}~{}{\cal B}_{1,{\cal M}}$
$\displaystyle+~{}{k_{x}\over k_{\perp}}~{}({\bf S}\cdot\hat{\bf
k}_{\perp})~{}{\cal B}_{2,{\cal M}}~{}+~{}{k_{x}\over k_{\perp}}~{}({\bf
S}\cdot\hat{z})~{}{\cal B}_{3,{\cal M}}\Bigr{]}~{},$ (71) $\displaystyle
S_{y}A^{V}_{3}+\frac{S_{z}}{M}\,p_{y}\,\widetilde{A}^{V}_{2}+\frac{{\bf{p}_{\perp}{\cdot}\bf{S}_{\perp}}}{M^{2}}\,\,p_{y}\,\widetilde{A}^{V}_{3}={\pi\over
2m}{E_{S}\over\kappa^{+}}~{}\Bigl{[}S_{y}~{}{\cal B}_{1,{\cal M}}$
$\displaystyle+~{}{k_{y}\over k_{\perp}}~{}({\bf S}\cdot\hat{\bf
k}_{\perp})~{}{\cal B}_{2,{\cal M}}~{}+~{}{k_{y}\over k_{\perp}}~{}({\bf
S}\cdot\hat{z})~{}{\cal B}_{3,{\cal M}}\Bigr{]}~{}.$ (72)

Let us integrate Eqs. (69), (70), (71) and (72) on $p^{+}$ and $p^{-}$ as in
Eqs. (52) - (57). Then in valence approximation one has

$\displaystyle f(x,|{\bf p}_{\perp}|^{2})~{}=~{}{{b}}_{0}\,,$ (73)
$\displaystyle S_{z}\,\Delta
f+\frac{1}{M}\,{\bf{p}_{\perp}{\cdot}\bf{S}_{\perp}}\,g_{1T}\,=~{}\Bigl{[}S_{z}~{}{b}_{1,{\cal
M}}$ $\displaystyle+~{}({\bf S}\cdot\hat{\bf k}_{\perp})~{}{b}_{4,{\cal
M}}~{}+~{}({\bf S}\cdot\hat{z})~{}{b}_{5,{\cal M}}\Bigr{]}\,,$ (74)
$\displaystyle
S_{x}~{}h_{1T}+\frac{S_{z}}{M}\,p_{x}\,h^{\perp}_{1L}+\frac{{\bf{p}_{\perp}{\cdot}\bf{S}_{\perp}}}{M^{2}}\,\,p_{x}\,h^{\perp}_{1T}=\Bigl{[}{S}_{x}~{}{b}_{1,{\cal
M}}$ $\displaystyle+~{}{k_{x}\over k_{\perp}}~{}({\bf S}\cdot\hat{\bf
k}_{\perp})~{}{b}_{2,{\cal M}}~{}+~{}{k_{x}\over k_{\perp}}~{}({\bf
S}\cdot\hat{z})~{}{b}_{3,{\cal M}}\Bigr{]}\,,$ (75) $\displaystyle
S_{y}~{}h_{1T}+\frac{S_{z}}{M}\,p_{y}\,h^{\perp}_{1L}+\frac{{\bf{p}_{\perp}{\cdot}\bf{S}_{\perp}}}{M^{2}}\,\,p_{y}\,h^{\perp}_{1T}=\Bigl{[}S_{y}~{}{b}_{1,{\cal
M}}$ $\displaystyle+~{}{k_{y}\over k_{\perp}}~{}({\bf S}\cdot\hat{\bf
k}_{\perp})~{}{b}_{2,{\cal M}}~{}+~{}{k_{y}\over k_{\perp}}~{}({\bf
S}\cdot\hat{z})~{}{b}_{3,{\cal M}}\Bigr{]}~{},$ (76)

where (see Ref. [62])

$\displaystyle h_{1T}=\int\Bigl{[}d{p^{+}}d{p^{-}}\Bigr{]}~{}A^{V}_{3}\,.$
(77)

Let us recall that for a three-body system with total angular momentum ${\cal
J}=1/2$, the dependence of ${{b}}_{i,{\cal M}}$ ($i=0,...,5$) on ${\bf S}$ is
absent and that the ${{b}}_{i,{\cal M}}$ are invariant for rotations of
${\bf{k}}_{\perp}$ around the $z$ axis (see Appendix B). Then any dependence
on ${\bf S}$ and on the direction of ${\bf{k}}_{\perp}$ in the right-hand side
of Eqs. (74), (75), (76) is explicitly written down.

##  V T-even twist-two Transverse Momentum Distributions

In what follows, we present the relations between the leading-twist TMDs, in
valence approximation, and the scalar functions ${{b}}_{i,{\cal M}}$, that
contain the relevant information on the dynamics inside the bound system. To
obtain these relations from Eqs. (74), (75) and (76) we consider specific
orientations of the target spin. From Eq. (74) one obtains :

  * •

for ${\bf S}=(0,0,1)$

$\displaystyle\Delta f={b}_{1,{\cal M}}+~{}{b}_{5,{\cal M}}$ (78)

  * •

for ${\bf S}=(1,0,0)$ and ${\bf S}=(0,1,0)$

$\displaystyle g_{1T}={{M}\over|{\bf p}_{\perp}|}~{}{b}_{4,{\cal M}}$ (79)

From Eq. (75) one obtains

  * •

for ${\bf S}=(0,0,1)$

$\displaystyle h^{\perp}_{1L}={M\over|{\bf p}_{\perp}|}~{}{b}_{3,{\cal
M}}~{},$ (80)

  * •

for ${\bf S}=(1,0,0)$

$\displaystyle{{h_{1T}(S_{x}=1)}}+{|{\bf p}_{\perp}|^{2}cos^{2}\phi\over
M^{2}}~{}h^{\perp}_{1T}$ $\displaystyle={b}_{1,{\cal M}}+\left({k_{x}\over
k_{\perp}}\right)^{2}{b}_{2,{\cal M}}~{},$ (81)

  * •

for ${\bf S}=(0,1,0)$

$\displaystyle h^{\perp}_{1T}={M^{2}\over|{\bf p}_{\perp}|^{2}}~{}{b}_{2,{\cal
M}}~{}.$ (82)

Eventually, from Eq. (76) one has

  * •

for ${\bf S}=(0,1,0)$

$\displaystyle{{{h_{1T}(S_{y}=1)}}}+{|{\bf p}_{\perp}|^{2}sen^{2}\phi\over
M^{2}}~{}h^{\perp}_{1T}$ $\displaystyle={b}_{1,{\cal M}}+\left({k_{y}\over
k_{\perp}}\right)^{2}{b}_{2,{\cal M}}~{},$ (83)

  * •

for ${\bf S}=(1,0,0)$ an equation identical to (82) is obtained.

  * •

for ${\bf S}=(0,0,1)$ an equation identical to (80) is obtained.

If Eq. (82) is inserted in Eqs. (81) and (83) one obtains

$\displaystyle{{{h_{1T}}}}=~{}{b}_{1,{\cal M}}~{}.$ (84)

The sum of Eqs. (81 ) and (83) gives

$\displaystyle h_{1T}+{|{\bf p}_{\perp}|^{2}\over
2M^{2}}~{}h^{\perp}_{1T}=\Delta^{\prime}_{T}f={1\over
2}~{}\Bigl{(}~{}2~{}{b}_{1,{\cal M}}~{}+~{}{b}_{2,{\cal M}}~{}\Bigr{)}~{}.$
(85)

Summarizing our results, we can write the following expressions for the
leading-twist TMDs in valence approximation (recall ${\bf p}_{\perp}={\bf
k}_{\perp}={\mbox{\boldmath{$\kappa$}}}_{\perp}$ )

$\displaystyle f^{\tau}(x,|{\bf p}_{\perp}|^{2})={{b}}_{0}^{\tau}~{},$ (86)
$\displaystyle\Delta f^{\tau}(x,|{\bf p}_{\perp}|^{2})={b}_{1,{\cal
M}}^{\tau}+~{}{b}_{5,{\cal M}}^{\tau}~{},$ (87) $\displaystyle
g^{\tau}_{1T}(x,|{\bf p}_{\perp}|^{2})={{M}\over|{\bf
p}_{\perp}|}~{}{b}_{4,{\cal M}}^{\tau}~{},$ (88)
$\displaystyle\Delta^{\prime}_{T}f^{\tau}(x,|{\bf p}_{\perp}|^{2})={1\over
2}~{}\Bigl{(}~{}2~{}{b}_{1,{\cal M}}^{\tau}~{}+~{}{b}_{2,{\cal
M}}^{\tau}~{}\Bigr{)}~{},$ (89) $\displaystyle h^{\perp\tau}_{1L}(x,|{\bf
p}_{\perp}|^{2})={M\over|{\bf p}_{\perp}|}~{}{b}_{3,{\cal M}}^{\tau}$ (90)
$\displaystyle h^{\perp\tau}_{1T}(x,|{\bf p}_{\perp}|^{2})={M^{2}\over|{\bf
p}_{\perp}|^{2}}~{}{b}_{2,{\cal M}}^{\tau}~{},$ (91)

where the isospin index $\tau$ has been reintroduced.

Since for a three-body system with total angular momentum ${\cal J}=1/2$, the
dependence of ${{b}}_{i,{\cal M}}$ on ${\bf S}$ is absent and the
${{b}}_{i,{\cal M}}$ are invariant for rotations of ${\bf{k}}_{\perp}$ around
the $z$ axis, the transverse momentum distributions $\Delta f(x,|{\bf
p}_{\perp}|^{2})$, $g_{1T}(x,|{\bf p}_{\perp}|^{2})$,
$\Delta^{\prime}_{T}f^{\tau}(x,|{\bf p}_{\perp}|^{2})$,
$h^{\perp}_{1L}(x,|{\bf p}_{\perp}|^{2})$, $h^{\perp}_{1T}(x,|{\bf
p}_{\perp}|^{2})$ do not depend on the direction of ${\bf k}_{\perp}$, as
expected.

In Appendix B.4, explicit expressions for the functions ${b}_{i,{\cal
M}}^{\tau}$ are obtained in terms of the three-body wave function. From these
expressions, according to Eqs. (86-91), the twist-two T-even transverse
momentum distributions can be evaluated, directly from the wave function,
without a cumbersome analysis of the spectral properties of the system,
described by the spectral function.

##  VI Applications

In this Section, the above formalism developed for the valence contribution to
the leading-twist TMDs for a ${\cal J}=1/2$ system is applied to the proton
and neutron inside 3He. The quantitative analysis allows one to clearly show
the impact of the inner dynamics on the evaluation of the TMDs, confirming the
expectation of reaching a detailed 3D picture of the investigated system in
momentum space. Moreover, the numerical information we have obtained could be
exploited for motivating further experimental efforts for measuring the 3He
TMDs.

###  VI.1  TMDs of the 3He nucleus

The T-even TMDs for 3He are evaluated by using the 3He wave function of Ref.
[63, 64] with the realistic nuclear interaction of Ref.[65], but neglecting
the small effect of the Coulomb repulsion between the protons. The results are
shown in Figs. 1 to 6.

Figure 1: (color online) Nucleon momentum distribution $f^{\tau}(x,|{\bf
p}_{\perp}|^{2})$ in an unpolarized 3He for the proton (left panel) and the
neutron (right panel).

Figure 2: (color online) Absolute value of the nucleon longitudinal-
polarization distribution, $\Delta f^{\tau}(x,|{\bf p}_{\perp}|^{2})$, in a
longitudinally polarized 3He for the proton (left panel) and the neutron
(right panel).

Figure 3: (color online) Absolute value of the nucleon transverse-polarization
distribution, $\Delta^{\prime}_{T}f^{\tau}(x,|{\bf p}_{\perp}|^{2})$, in a 3He
nucleus transversely polarized in the same direction of the nucleon
polarization, for the proton (left panel) and the neutron (right panel).

Figure 4: (color online) Absolute value of the nucleon longitudinal-
polarization distribution, $g^{\tau}_{1T}(x,|{\bf p}_{\perp}|^{2})$, in a
transversely polarized 3He for the proton (left panel) and the neutron (right
panel).

Figure 5: (color online) Absolute value of the nucleon transverse-polarization
distribution, $h^{\perp\tau}_{1L}(x,|{\bf p}_{\perp}|^{2})$ in a
longitudinally polarized 3He for the proton (left panel) and the neutron
(right panel).

Figure 6: (color online) Absolute value of the nucleon transverse-polarization
distribution, $h^{\perp\tau}_{1T}(x,|{\bf p}_{\perp}|^{2})$, in a 3He nucleus
transversely polarized in a direction orthogonal to the direction of the
nucleon polarization, for the proton (left panel) and the neutron (right
panel).

As a first observation one can notice that all of the TMDs are distributed
around $x=1/3$, as expected. The structure of $f^{\tau}(x,|{\bf
p}_{\perp}|^{2})$, presented in Fig. 1, is very smooth, while the other five
distributions have a rich structure, as a function both of $x$ and $|{\bf
p}_{\perp}|$. Another relevant observation arises from the comparison of the
TMDs $\Delta f^{\tau}(x,|{\bf p}_{\perp}|^{2})$, Eq. (87), with
$\Delta^{\prime}_{T}f^{\tau}(x,|{\bf p}_{\perp}|^{2})$, Eq. (89), shown in
Figs. 2 and 3, respectively. As is well known, these distributions should be
equal in a non relativistic framework, where boosts and transverse rotations
commute [66, 67, 68, 69]. The same does not hold in a relativistic treatment,
as the LFHD one adopted here. Indeed, the comparison shows that the two
distributions are actually different, a signature of remarkable relativistic
corrections, even in a system, the 3He nucleus, where the ratio of the average
nucleon momentum to the nucleon mass is rather small.

The TMDs presented in Figs. 4, 5 and 6, namely $g^{\tau}_{1T}(x,|{\bf
p}_{\perp}|^{2})$, $h^{\perp\tau}_{1L}(x,|{\bf p}_{\perp}|^{2})$ and
$h^{\perp\tau}_{1T}(x,|{\bf p}_{\perp}|^{2})$, respectively, have relevant
peaks at low values of $|{\bf p}_{\perp}|$. Interestingly,
$h^{\perp\tau}_{1T}(x,|{\bf p}_{\perp}|^{2})$ shows a sizable secondary bump
at $|{\bf p}_{\perp}|\sim 2.5$ ${fm^{-1}}$, due to the presence of the squared
transverse-momentum in ${b}_{2,{\cal M}}^{\tau}$, differently from the linear
power occurring in the other functions ${b}_{i,{\cal M}}^{\tau}$. It is very
important to notice that in valence approximation $|g^{\tau}_{1T}(x,|{\bf
p}_{\perp}|^{2})|$, shown in Fig. 4, and $|h^{\perp\tau}_{1L}(x,|{\bf
p}_{\perp}|^{2})|$, shown in Fig. 5, are very similar, as expected by
inspecting the two scalar functions, $b^{\tau}_{3,{\cal M}}$ and
$b^{\tau}_{4,{\cal M}}$ given in Eqs. (140), (141), (142) and (143), and
recalling that the effect of the Melosh rotations, parametrized through the
angle $\varphi$, is small (see below Eq. (19)).

The shape of the presented distributions demonstrates that a comparison of
these results with the TMDs extracted from future measurements of appropriate
spin asymmetries in ${}^{3}{\vec{\rm He}}(\vec{e},e^{\prime}p)X$ experiments
[55] could give very detailed information on the 3He wave function, on the
validity of the LF description and consequently on the nuclear interaction,
once the possible final state interaction is properly taken care of.

####  VI.1.1 Effective polarizations

As a first application of our results, let us evaluate the LF longitudinal,
$p^{\tau}_{||}$, and transverse, $p^{\tau}_{\perp}$, effective polarizations
for the proton and for the neutron, viz.

$\displaystyle p^{\tau}_{||}=\int_{0}^{1}dx~{}\int d{\bf p}_{\perp}~{}\Delta
f^{\tau}(x,|{\bf p}_{\perp}|^{2})$ (92) $\displaystyle
p^{\tau}_{\perp}=\int_{0}^{1}dx~{}\int d{\bf
p}_{\perp}~{}\Delta^{\prime}_{T}f^{\tau}(x,|{\bf p}_{\perp}|^{2})~{}.$ (93)

They are used in the extraction of neutron polarized structure functions and
of neutron Collins and Sivers single spin asymmetries, respectively, from the
corresponding quantities measured for 3He (see, e.g., Refs. [70, 71], and
Refs. [72, 73, 74] respectively). This kind of extraction is based on the
validity of the Impulse Approximation (IA), i.e. no final state interaction
between the struck particle and the interacting spectator system, in the
kinematics of the corresponding experiments. In the SIDIS case and in the
kinematics of both Jefferson Lab and the future EIC, it has been shown [73]
that effective polarizations, can be used for the single spin asymmetries,
even beyond the IA, including final state interaction effects.

From the the discussion in the previous section, and from the expressions of
Eqs. (147) and (148) it is clear that the longitudinal and the transverse
effective polarizations are not anymore equal, as it occurs in the non
relativistic approximation. The difference between the two effective
polarizations is due to the effect of the Melosh rotations for the spin.
Indeed, without this effect one has (see Appendix (B.5) )

$\displaystyle p^{\tau}_{||}=p^{\tau}_{\perp}={(-1)^{{\cal
M}+1/2}}~{}{\sqrt{{3}}}\int d{k}_{23}~{}k^{2}_{23}$
$\displaystyle\times~{}\int_{0}^{\infty}{k}^{2}~{}d{k}~{}2~{}{\cal
H}^{\tau}(0,1,k_{23},k)\,,$ (94)

where the function ${\cal H}^{\tau}(0,1,k_{23},k)$ can be obtained from Eq.
(124). The LF results obtained for the effective polarizations of proton and
neutron in 3He with the nuclear interaction AV18 of Ref. [65], without the
Coulomb repulsion, are shown in Table 1, together with the corresponding
normalizations, and compared with the non relativistic result. In the first
and in the second line the normalizations obtained from the proton and neutron
spectral functions through Eq. (5) and directly from the wave function are
shown, respectively. In the following three lines the LF calculations for the
longitudinal and the transverse polarizations and for the polarizations
without the Melosh rotations are presented. Eventually, in the last line, the
results for the non relativistic polarizations with the same wave function are
shown.

Normalization and effective polarizations | proton | neutron
---|---|---
normalization from the spectral function |  0.99915 |  0.99885
normalization from the wave function | 0.99929 | 0.99897
LF longitudinal polarization | -0.02299 | 0.87261
LF transverse polarization | -0.02446 | 0.87314
LF polarization without Melosh rotations | -0.02407 | 0.87698
non relativistic polarization | -0.02118 | 0.89337
Table 1: Normalization and effective longitudinal and transverse polarizations
for the proton and the neutron in ${}^{3}{{He}}$

The comparison between the two normalizations allows one to assess the
numerical accuracy, that can be estimated of the order of a few parts in
$10^{4}$. Therefore the difference of a few parts in $10^{3}$ between
longitudinal and transverse polarizations is a meaningful result. However,
these small difference indicates that the effects of the Melosh rotations are
tiny (see the result in the last but one line and the comment below Eq. (19)),
although for the proton it becomes sizable in percentage. Interestingly, the
difference of the LF polarizations with respect to the non relativistic
results are larger, up to $2\%$ in the neutron case and should be ascribed to
the transformations performed between the symmetric and non symmetric
intrinsic coordinate systems, not to the Melosh rotations involving the spins.
In any case, the important point is that the relativistic results do not
differ too much from the non relativistic ones, currently used by the
experimental collaborations. Therefore, the values of the effective
polarizations used to extract the neutron polarized structure functions and
the Collins and Sivers asymmetries from 3He data can be considered reliable
also from a Poincaré-covariant point of view, although for a more precise
determination the new values for the effective polarizations should be
adopted. In closing this subsection, we observe that the longitudinal and
transverse polarizations of the nucleons in 3He are analogous to the axial and
tensor charges of the nucleon in terms of its constituent quarks, in valence
approximation. Since the beginning of the transversity studies, their
difference has been always considered a signature of the relativistic content
of the system [66, 67, 68, 69, 33].

####  VI.1.2 Approximate relations

In Refs. [13, 33] approximate relations between the TMDs were discussed, i.e.,

$\displaystyle\Delta f(x,|{\bf p}_{\perp}|^{2})=\Delta^{\prime}_{T}f(x,|{\bf
p}_{\perp}|^{2})+{|{\bf p}_{\perp}|^{2}\over 2M^{2}}~{}h^{\perp}_{1T}(x,|{\bf
p}_{\perp}|^{2})\,,$ (95)

and

$\displaystyle g_{1T}(x,|{\bf p}_{\perp}|^{2})=-h^{\perp}_{1L}(x,|{\bf
p}_{\perp}|^{2})\,.$ (96)

In principle, these relations put clear-cut phenomenological constraints on
the number of independent T-even twist-two TMDs. Therefore, it is interesting
to raise the question to what extent the above relations are valid. To attempt
a realistic answer, we tested Eqs. (95) and (96) in the case with a refined
dynamical content as the 3He nucleus, assuming the nucleons as constituents.
The first relation, Eq. (95), is not exactly satisfied, since the equality
should hold if $b^{\tau}_{2,{\cal M}}=b^{\tau}_{5,{\cal M}}$, as it follows
from Eqs. (87), (89) and (91). By inspecting Eqs. (138), (139), (144) and
(145), one gets $b^{\tau(0)}_{2,{\cal M}}=b^{\tau(0)}_{5,{\cal M}}$, while
$b^{\tau(2)}_{2,{\cal M}}$ and $b^{\tau(2)}_{5,{\cal M}}$ have not the same
expressions. The quantitative difference between the lhs and rhs of Eq. (95)
is quite small for the neutron, while is not negligible for the proton as
shown in Fig. 7.

From the evaluation of the effective polarizations we learned that the effects
of the Melosh rotations are tiny, and if these effects are neglected in
${{b}}_{i,{\cal M}}$, i.e. $sin(\varphi/2)\to 0$ in the expressions presented
at the end of Appendix B.4, one finds that the second relation, Eq. (96),
holds, but with the opposite sign. It is very interesting to analyze such a
different sign, that could have far reaching consequences. The functions
$b^{\tau}_{4,{\cal M}}$ and $b^{\tau}_{3,{\cal M}}$ determine $g_{1T}$ and
$h^{\perp}_{1L}$, respectively, as shown in Eqs. (88) and (90). From Eqs.
(140) and (142), where the contribution with $L=0$ is considered, one has
$b^{\tau(0)}_{3,{\cal M}}=-b^{\tau(0)}_{4,{\cal M}}$, but the two functions
are of the order $sin(\varphi)$ and therefore very small. The contribution
with $L=2$ leads to $b^{\tau(2)}_{3,{\cal M}}\sim b^{\tau(2)}_{4,{\cal M}}$
(see Eqs. (141) and (143)) and generates the leading term, proportional to
$cos^{2}(\varphi)$, of the two TMDs. Therefore, the sign in Eq. (96), has to
be considered as a signature of the bound-state orbital content that prevails
in the actual value of $g_{1T}$ and $h^{\perp}_{1L}$. As clear from Appendix B
(where it is shown that only the values $L=0$ and $L=2$ are possible), if only
the contribution from $L=0$ is present, this implies a vanishing value of the
active fermion orbital angular-momentum $L_{\rho}$ in the bound-system wave
function (see Eqs. (112) - (122)). On the other hand, a non-vanishing value of
$L_{\rho}$ implies a non-vanishing value of the contribution from $L=2$ to the
momentum distribution.

For 3He, Eq. (96) is fulfilled in modulus, as pictorially shown in Figs. 4 and
5.

Figure 7:  Absolute value of the difference $\Delta^{\prime}_{T}f(x,|{\bf
p}_{\perp}|^{2})~{}+~{}{|{\bf p}_{\perp}|^{2}\over
2M^{2}}~{}h^{\perp}_{1T}(x,|{\bf p}_{\perp}|^{2})-\Delta f(x,|{\bf
p}_{\perp}|^{2})$ in 3He for the proton (left panel) and the neutron (right
panel).

For a system with a high average value of the ratio $|{\bf p}_{\perp}|/m$, as
occurs when we shift from the 3He case to the nucleon one, where the
constituents are the quarks with $m_{q}<m_{N}$, (see Ref. [75, 76]), Melosh
rotations become relevant. If this is the case, from the previous discussion,
one expects that the relations (95) and (96) hold exactly in valence
approximation when the contribution to the transverse momentum distributions
from $L=2$ is very small or totally absent.

Therefore, the validity of the equalities (95) and (96) in a bound-system with
a high average value of $|{\bf p}_{\perp}|/m$ represent a well-defined
constraint on the allowed dynamics inside the three-body bound system.

Another relation proposed in Refs. [13, 33] i.e.,

$\displaystyle\left[g_{1T}(x,|{\bf
p}_{\perp}|^{2})\right]^{2}=-2~{}\Delta^{\prime}_{T}f(x,|{\bf
p}_{\perp}|^{2})~{}h^{\perp}_{1T}(x,|{\bf p}_{\perp}|^{2})\,,$ (97)

does not hold in our approach, even if the contribution from the angular
momentum $L=2$ is absent, because of the presence of the integration on
$k_{23}$ in the expressions of the transverse-momentum distributions (see Eq.
(133)). Namely, a genuine relativistic approach, related to the implementation
of the macrocausality, spoils the above relation. Therefore, a signature of
such a relativistic effect is given by the violation of Eq. (97).

##  VII Conclusions

A Poincaré-covariant description of bound systems, with $A$ constituents of
spin $1/2$, has been developed within the light-front Hamiltonian dynamics in
Ref. [1] and applied in the present work to the T-even twist-two transverse-
momentum distributions of ${\cal{J}}=1/2$ systems. The explicit expressions
for the TMDs are obtained in terms of LF overlaps, between the bound state and
states described by tensor products of a constituent plane wave and an $(A-1)$
fully interacting system. These LF overlaps are the basic ingredients of the
spin-dependent spectral function [1], that is a $2\times 2$ matrix with the
main-diagonal terms yielding the distribution probability to find a
constituent with given spin and LF momentum, once the $(A-1)$-spectator system
has an assigned mass. Indeed, the leading-twist TMDs, in valence
approximation, is linked to proper traces of the valence contribution to the
semi-inclusive correlation function, that is linearly related to the spin-
dependent spectral function. The formalism has been applied to the 3He
nucleus, keeping a twofold aim in mind: (i) illustrating a realistic case,
where a theoretical description of the bound state takes into account a highly
non trivial dynamics, and consequently assessing the impact of dynamics on the
actual TMDs; (ii) cumulating quantitative analyses for supporting future
experimental efforts dedicated to achieve a detailed 3D picture of the 3He. As
a matter of fact, for the 3He nucleus there exist a reliable LF spin-dependent
spectral function, obtained within the Bakamjian-Thomas framework [42],
suitable for embedding the wide knowledge on the nuclear interaction and the
successful phenomenology developed for few-nucleon systems in a Poincaré-
covariant approach (preliminary calculations of the EMC effect in 3He were
presented in Refs. [46, 47, 44, 45] and a full calculation will be soon
available [43]).

Besides the Melosh rotations, the peculiar feature of our LF approach is the
macrocausality, implemented through the use of non symmetric intrinsic
variables, i.e. intrinsic internal variables for the $A-1$ system and the LF
momentum of the active fermion in the intrinsic reference frame of the
$[1,(A-1)]$ cluster [1]. Their impact on the 3He TMDs has been discussed, with
a particular attention to the relevance from the experimental point of view.
To this end, we have evaluated, as a first application, the LF longitudinal
and transverse effective polarizations for the proton and the neutron in 3He.
These two quantities are widely used to extract the neutron polarized
structure functions and the neutron Collins and Sivers asymmetries from the
corresponding quantities measured in DIS and SIDIS off 3He, respectively.
Although this procedure is generally used assuming the validity of the impulse
approximation, in Ref. [73] it was shown how it can be applied even including
the final state interaction.

We found that, for a system with a small average value of $p_{\perp}/m$, as
the three-nucleon system, the effect of the spin Melosh rotation is tiny and
the longitudinal and the transverse effective polarizations differ very little
from each other, both for the proton and the neutron. On the contrary, the
difference with respect to the non relativistic result is not negligible and
this effect, ascribed to the use of LF intrinsic variables, has to be
considered for a more accurate extraction of neutron properties from 3He data.

A second important result we have discussed is the validation of the linear
relations proposed in [13, 35] between the T-even twist-two TMDs. Although
those relations were introduced in the context of the nucleon studies in
valence approximation, we have investigated their validity for the 3He case,
in order to show how a rich dynamics actually impacts the extraction of
important information on the orbital-momentum decomposition of the three-body
bound-system. Namely, one recovers the above mentioned relations if the state
has a simple S-wave structure, while in presence of a D-wave and a small
effect of the Melosh rotations, i.e. a small average value of $p_{\perp}/m$,
the second relation even changes sign. The other relation proposed in Refs.
[13, 35], between the TMDs, a quadratic one, does not hold in any case, since
the TMDs are given by integrals over the relative momentum of the spectator
interacting pair, which represent an unavoidable feature of a genuinely
Poincaré-covariant framework. Hence, an experimental investigation, performed
at high luminosity facilities, could open a window directly on the orbital-
momentum content of the bound state and the relativistic regime of the inner
dynamics.

The twist-three TMDs can be also evaluated in our scheme and will be the
object of another study.

##  Appendix A Dependence of the spectral function on the polarization vector
$\bf S$

The LF spin-dependent spectral function for any direction of the polarization
vector $\bf S$ can be written in terms of the Wigner rotation matrixes
$D^{\cal J}_{m,\cal M}(\alpha,\beta,\gamma)$ (see Eqs. (1) and (4))

$\displaystyle{\cal P}_{{\cal
M},\sigma^{\prime}\sigma}(\tilde{\mbox{\boldmath{$\kappa$}}},\epsilon,S)~{}=~{}\sum_{m,m^{\prime}}~{}D^{{\cal
J}*}_{m,\cal M}(\alpha,\beta,\gamma)~{}D^{\cal J}_{m^{\prime},\cal
M}(\alpha,\beta,\gamma)$ $\displaystyle\times~{}{\cal
P}^{m^{\prime},m}_{\sigma^{\prime}\sigma}(\tilde{\mbox{\boldmath{$\kappa$}}},\epsilon,S_{z})~{},$
(98)

where $S_{z}=(0,0,0,1)$ and

$\displaystyle{\cal
P}^{m^{\prime},m}_{\sigma^{\prime}\sigma}(\tilde{\mbox{\boldmath{$\kappa$}}},\epsilon,S_{z})=\rho(\epsilon)$
$\displaystyle\times~{}\sum_{JJ_{z}\alpha}\sum_{T_{S}\tau_{S}}~{}_{LF}\langle\tau_{S}T_{S};\alpha,\epsilon;JJ_{z};\tau\sigma^{\prime},\tilde{\mbox{\boldmath{$\kappa$}}}|{\cal
J}m^{\prime};\epsilon^{A},\Pi;TT_{z}\rangle_{z}$
$\displaystyle\times~{}_{z}\langle TT_{z};\Pi,\epsilon^{A};{\cal
J}m|\tilde{\mbox{\boldmath{$\kappa$}}},\sigma\tau;JJ_{z};\epsilon,\alpha;T_{S}\tau_{S}\rangle_{LF}\,.$
(99)

with $|{\cal J}m;\epsilon^{A},\Pi;TT_{z}\rangle_{z}$ the system ground state
with energy $\epsilon^{A}$, parity $\Pi$ and isospin $T,T_{z}$, polarized
along $\hat{z}$.

For a system with ${\cal J}=1/2$ Eq. (98) can be written as follows

$\displaystyle{\cal P}_{{\cal
M},\sigma^{\prime}\sigma}(\tilde{\mbox{\boldmath{$\kappa$}}},\epsilon,S)~{}=~{}\sum_{m,m^{\prime}}~{}D^{1/2}_{-m,-\cal
M}(\alpha,\beta,\gamma)~{}(-1)^{m-{\cal M}}$
$\displaystyle\times~{}D^{1/2}_{m^{\prime},\cal
M}(\alpha,\beta,\gamma)~{}{\cal
P}^{m^{\prime},m}_{\sigma^{\prime}\sigma}(\tilde{\mbox{\boldmath{$\kappa$}}},\epsilon,S_{z})~{},$
(100)

Let us take advantage of the following property for the product of two Wigner
rotation matrixes with the same arguments (see Eq. (1) at p. 84 of [77])

$\displaystyle
D^{j}_{m,k}(\alpha,\beta,\gamma)~{}D^{j^{\prime}}_{m^{\prime},k^{\prime}}(\alpha,\beta,\gamma)=\sum_{J=|j-j^{\prime}|}^{j+j^{\prime}}\langle
jm,j^{\prime}m^{\prime}|J(m+m^{\prime})\rangle$ $\displaystyle\times~{}\langle
jk,j^{\prime}k^{\prime}|J(k+k^{\prime})\rangle~{}D^{J}_{(m+m^{\prime}),(k+k^{\prime})}(\alpha,\beta,\gamma)~{}.$
(101)

In our case it reads

$\displaystyle D^{1/2}_{-m,-{\cal
M}}(\alpha,\beta,\gamma)~{}D^{1/2}_{m^{\prime},{\cal M}}(\alpha,\beta,\gamma)$
$\displaystyle=\sum_{J=0}^{1}\langle\frac{1}{2}-m,\frac{1}{2}m^{\prime}|J(m^{\prime}-m)\rangle~{}\langle\frac{1}{2}-{\cal
M},\frac{1}{2}{\cal M}|J0\rangle$ $\displaystyle\times
D^{J}_{(m^{\prime}-m),0}(\alpha,\beta,\gamma)$
$\displaystyle=\sum_{J=0}^{1}\hskip
2.84544pt\langle\frac{1}{2}-m,\frac{1}{2}m^{\prime}|J(m^{\prime}-m)\rangle~{}\langle\frac{1}{2}-{\cal
M},\frac{1}{2}{\cal M}|J0\rangle$
$\displaystyle\times~{}(-1)^{m^{\prime}-m}\sqrt{{4\pi\over
2J+1}}~{}Y_{J(m-m^{\prime})}(\beta,\alpha)~{}.$ (102)

Then the product of the two $D^{j}_{m,k}(\alpha,\beta,\gamma)$ matrixes is a
sum of two terms. The first one, with $J=0$, is independent of $\beta,\alpha$
(i.e. of $\bf S$). In the second one, with $J=1$, the spherical harmonics
$Y_{1(m-m^{\prime})}(\beta,\alpha)$ can be raplaced by the spherical
components $S_{i}$ of the polarization vector [77]

$\displaystyle Y_{J(m-m^{\prime})}(\beta,\alpha)=\sqrt{{3\over
4~{}\pi}}~{}S_{(m-m^{\prime})}~{}.$ (103)

Hence the spectral function depends linearly on $\bf S$. Since in the
expansion (10) of the pseudovector ${\cal F}^{\tau}_{\cal M}$ a linear
dependence on $\bf S$ already explicitly appears in the five pseudovectors
$\bf{S}$, $~{}\hat{\bf k}_{\perp}({\bf S}\cdot\hat{\bf k}_{\perp})$,
$~{}\hat{\bf k}_{\perp}({\bf S}\cdot\hat{z})$, $~{}\hat{z}~{}({\bf
S}\cdot\hat{\bf k}_{\perp})$, and $~{}\hat{z}~{}({\bf S}\cdot\hat{z})$, as a
consequence for a system with total angular momentum ${\cal J}=1/2$ the
quantities ${\cal B}_{i}$ for $i=1,...,5$ can depend only on $|{\bf
k}_{\perp}|,x$, and $\epsilon$.

##  Appendix B The spin-dependent momentum distribution and the three-body
wave function

One can obtain the LF momentum distribution dependent upon the spin
directions, $\left[{\mbox{\boldmath{$\cal N$}}}_{\cal M}^{\tau}(x,{\bf
k}_{\perp};\bf S)\right]_{\sigma\sigma^{\prime}}$, Eq. (14), for any direction
of the polarization vector $\bf S$ of the system, using Eq. (98) and Eq. (99)

$\displaystyle\left[{\mbox{\boldmath{$\cal N$}}}_{\cal M}^{\tau}(x,{\bf
k}_{\perp};\bf S)\right]_{\sigma\sigma^{\prime}}~{}=~{}\int\\!\\!\
\\!\\!\\!\\!\\!\ \\!\\!\\!\\!\\!\sum{d\epsilon~{}}{1\over
2~{}(2\pi)^{3}}~{}{E_{S}\over(1-x)~{}\kappa^{+}}~{}\rho(\epsilon)\sum_{JJ_{z}\alpha}\sum_{T_{S}\tau_{S}}\sum_{m}D^{\cal
J}_{m,{\cal M}}(\alpha,\beta,\gamma)_{LF}$
$\displaystyle\times\langle\tau_{S}T_{S};\alpha,\epsilon;JJ_{z};\tau\sigma,\tilde{\mbox{\boldmath{$\kappa$}}}|{\cal
J}j_{z}=m;\epsilon^{3},\Pi;TT_{z}\rangle\sum_{m^{\prime}}[D^{\cal
J}_{m^{\prime},{\cal M}}(\alpha,\beta,\gamma)]^{*}\langle
TT_{z};\Pi,\epsilon^{3};{\cal
J}j_{z}=m^{\prime};|\tilde{\mbox{\boldmath{$\kappa$}}},\sigma^{\prime}\tau;JJ_{z};\epsilon,\alpha;T_{S}\tau_{S}\rangle_{LF}~{},$
(104)

where $|{\cal J}j_{z}=m;\epsilon^{3},\Pi;TT_{z}\rangle$ is the three-body
ground state, polarized along $\hat{z}$. Using the explicit expression for the
overlaps given by Eq. (62) of Ref. [1], the two-body completeness

$\displaystyle\sum_{J,J_{z}\alpha}\sum_{TT_{z}}\int\\!\\!\ \\!\\!\\!\\!\\!\
\\!\\!\\!\\!\\!\sum d\epsilon~{}\rho(\epsilon)\langle{\bf
k}^{\prime}|JJ_{z};\epsilon,\alpha;T_{S}\tau_{S}\rangle$
$\displaystyle\times~{}\langle\tau_{S}T_{S};\alpha,\epsilon;J_{z}J|{\bf
k}\rangle=\delta^{3}({\bf k}^{\prime}-{\bf k})~{},$ (105)

and the unitarity of the $D$ matrixes, we have

$\displaystyle\left[{\mbox{\boldmath{$\cal N$}}}_{\cal M}^{\tau}(x,{\bf
k}_{\perp};\bf S)\right]_{\sigma\sigma^{\prime}}~{}=~{}\sum_{m}~{}D^{{\cal
J}}_{m,{\cal M}}(\alpha,\beta,\gamma)$
$\displaystyle\times~{}\sum_{m^{\prime}}~{}[D^{{\cal J}}_{m^{\prime},{\cal
M}}(\alpha,\beta,\gamma)]^{*}~{}{\cal F}^{\tau
mm^{\prime}}_{\sigma\sigma^{\prime}}(x,{\bf k}_{\perp})~{},$ (106)

where

$\displaystyle{\cal F}^{\tau mm^{\prime}}_{\sigma\sigma^{\prime}}(x,{\bf
k}_{\perp})={1\over(1-x)}~{}\sum_{\tau_{2}\tau_{3}}~{}\int d{\bf
k}_{23}~{}E({\bf k})~{}{E_{23}\over
k^{+}}~{}\sum_{\sigma_{1}{\sigma^{\prime}_{1}}}~{}D^{{1\over 2}}[{\cal
R}_{M}({\tilde{\bf k}})]_{\sigma\sigma_{1}}~{}D^{{1\over 2}*}[{\cal
R}_{M}({\tilde{\bf k}})]_{\sigma^{\prime}{\sigma}^{\prime}_{1}}~{}$
$\displaystyle\times\sum_{\sigma^{\prime}_{2},\sigma^{\prime}_{3}}~{}\langle\sigma^{\prime}_{3},\sigma^{\prime}_{2},\sigma_{1};\tau_{3},\tau_{2},\tau;{\bf
k}_{23},{\bf
k}|j,j_{z}=m;\epsilon^{3}_{int},\Pi;TT_{z}\rangle~{}\langle{\sigma}^{\prime}_{3},{\sigma}^{\prime}_{2},{\sigma}^{\prime}_{1};{\tau}_{3},{\tau}_{2},{\tau};{\bf
k}_{23},{\bf
k}|j,j_{z}=m^{\prime};\epsilon^{3}_{int},\Pi;TT_{z}\rangle^{*}~{},$ (107)

with ${\cal R}_{M}$ the Melosh rotation (see Appendix C) and
$\langle\sigma^{\prime}_{3},\sigma^{\prime}_{2},\sigma^{\prime}_{1};\tau_{3},\tau_{2},\tau;{\bf
k}_{23},{\bf k}|j,j_{z}=m;\epsilon^{3}_{int},\Pi;TT_{z}\rangle$ the momentum-
space instant-form wave function.

The quantities ${{\bf k}}$, $E({\bf k})$, and $E_{23}$ in Eq. (107) are easily
determined from the variables $x$, $\bf{k_{\perp}}$, ${{\bf k}_{23}}$ [1]

$\displaystyle k^{+}=x~{}M_{0}(1,2,3)$ (108)

with

$\displaystyle M^{2}_{0}(1,2,3)={{m^{2}+{\bf k}^{2}_{\perp}}\over
x}+{{M^{2}_{23}+{\bf k}^{2}_{\perp}}\over(1-x)}`,$ $\displaystyle
M_{23}=2~{}\sqrt{{m^{2}+|\bf k}_{23}|^{2}}$ (109)

and eventually

$\displaystyle k_{z}={1\over 2}(k^{+}-k^{-})={1\over
2}\left(k^{+}-{{m^{2}+{\bf k}^{2}_{\perp}}\over k^{+}}\right)~{},$ (110)

while

$\displaystyle E({\bf k})=\sqrt{{m^{2}+|{\bf k}|^{2}}}~{}~{},\quad
E_{23}=\sqrt{M^{2}_{23}+{{\bf k}}^{2}}~{}.$ (111)

Let us recall that in the system rest frame one has
$\mbox{\boldmath{$P$}}_{\perp}$ = 0 .

###  B.1 Evaluation of ${\cal F}^{\tau mm^{\prime}}_{\sigma\sigma^{\prime}}$

Let us take the instant-form three-body wave function as the 3He non-
relativistic wave function [1]. The 3He wave function in momentum space can be
written as follows from the wave function in coordinate space [63, 64]

$\displaystyle\langle\sigma_{1},\sigma_{2},\sigma_{3};\tau_{1},\tau_{2},\tau_{3};{\bf
k}_{23},{\bf k}|^{3}{\rm He};{1\over 2}m;{1\over
2}T_{z}\rangle=\sum_{l_{23}\mu_{23}}\sum_{L_{\rho}M_{\rho}}{Y}_{l_{23}\mu_{23}}(\hat{\bf
k}_{23})~{}{Y}_{L_{\rho}M_{\rho}}(\hat{\bf
k})\sum_{T_{23},\tau_{23}}\langle{1\over 2}\tau_{2}{1\over
2}\tau_{3}|T_{23}\tau_{23}\rangle\langle T_{23}\tau_{23}{1\over
2}\tau_{1}|{1\over 2}T_{z}\rangle$
$\displaystyle\times\sum_{XM_{X}}\sum_{j_{23}m_{23}}\langle
XM_{X}L_{\rho}M_{\rho}|{1\over 2}m\rangle\langle j_{23}\ m_{23}{1\over
2}\sigma_{1}|XM_{X}\rangle\sum_{s_{23}\sigma_{{23}}}\langle{1\over
2}\sigma_{2}{1\over 2}\sigma_{3}|s_{23}\sigma_{{23}}\rangle\langle
l_{23}\mu_{23}s_{23}\sigma_{{23}}|j_{23}\ m_{23}\rangle{\mbox{\boldmath{$\cal
G$}}}_{{L_{\rho}}X}^{j_{23}l_{23}s_{23}}(k_{23},k)$ (112)

with

$\displaystyle{\mbox{\boldmath{$\cal
G$}}}_{{L_{\rho}}X}^{j_{23}l_{23}s_{23}}(k_{23},k)={2(-1)^{{l_{23}+L_{\rho}}\over
2}\over\pi}\int r^{2}d{r}~{}j_{l_{23}}(k_{23}r)$
$\displaystyle\times~{}\int{\rho}^{2}~{}d{\rho}~{}j_{L_{\rho}}(k\rho)~{}\phi^{j_{23}l_{23}s_{23}}_{L_{\rho}X}(|{\bf
r}|,|{\mbox{\boldmath{$\rho$}}}|)~{}.$ (113)

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.

Then, inserting Eq. (112) in Eq. (107), the function ${\cal F}^{\tau
mm^{\prime}}_{\sigma\sigma^{\prime}}(x,{\bf k}_{\perp})$ becomes

$\displaystyle{\cal F}^{\tau mm^{\prime}}_{\sigma\sigma^{\prime}}(x,{\bf
k}_{\perp})={1\over(1-x)}~{}\sum_{\tau_{2}\tau_{3}}~{}\sum_{\sigma^{\prime}_{2}\sigma^{\prime}_{3}}\int
d{\bf k}_{23}~{}{E_{23}~{}E({\bf k})\over
k^{+}_{1}}~{}\sum_{\sigma_{1}}D^{{1\over 2}}[{\cal R}_{M}({\tilde{\bf
k}})]_{\sigma\sigma_{1}}\sum_{l_{23}\mu_{23}}\sum_{L_{\rho}M_{\rho}}{Y}_{l_{23}\mu_{23}}(\hat{\bf
k}_{23})~{}{Y}_{L_{\rho}M_{\rho}}(\hat{\bf k})$
$\displaystyle\times~{}\sum_{T_{23},\tau_{23}}\langle{1\over 2}\tau_{2}{1\over
2}\tau_{3}|T_{23}\tau_{23}\rangle~{}\langle T_{23}\tau_{23}{1\over
2}\tau|{1\over 2}T_{z}\rangle\sum_{XM_{X}}\sum_{j_{23}m_{23}}\langle
XM_{X}L_{\rho}M_{\rho}|{1\over 2}m\rangle~{}\langle j_{23}\ m_{23}{1\over
2}\sigma_{1}|XM_{X}\rangle\sum_{s_{23}\sigma_{{23}}}~{}\langle{1\over
2}\sigma^{\prime}_{2}{1\over 2}\sigma^{\prime}_{3}|s_{23}\sigma_{{23}}\rangle$
$\displaystyle\times~{}\langle l_{23}\mu_{23}s_{23}\sigma_{{23}}|j_{23}\
m_{23}\rangle~{}{\mbox{\boldmath{$\cal
G$}}}_{{L_{\rho}}X}^{j_{23}l_{23}s_{23}}(k_{23},k)\sum_{{\sigma}^{\prime}_{1}}D^{{1\over
2}*}[{\cal R}_{M}({\tilde{\bf
k}}]_{\sigma^{\prime}{\sigma}^{\prime}_{1}}\sum_{l^{\prime}_{23}\mu^{\prime}_{23}}\sum_{L^{\prime}_{\rho}M^{\prime}_{\rho}}{Y}^{*}_{l^{\prime}_{23}\mu^{\prime}_{23}}(\hat{\bf
k}_{23})~{}{Y}^{*}_{L^{\prime}_{\rho}M^{\prime}_{\rho}}(\hat{\bf k})$
$\displaystyle\times\sum_{T^{\prime}_{23},\tau^{\prime}_{23}}\langle{1\over
2}\tau_{2}{1\over
2}\tau_{3}|T^{\prime}_{23}\tau^{\prime}_{23}\rangle~{}\langle
T^{\prime}_{23}\tau^{\prime}_{23}{1\over 2}\tau|{1\over
2}T_{z}\rangle\sum_{X^{\prime}M^{\prime}_{X}}\sum_{j^{\prime}_{23}m^{\prime}_{23}}\langle
X^{\prime}M^{\prime}_{X}L^{\prime}_{\rho}M^{\prime}_{\rho}|{1\over
2}m^{\prime}\rangle~{}\langle j^{\prime}_{23}\ m^{\prime}_{23}{1\over
2}\sigma^{\prime}_{1}|X^{\prime}M^{\prime}_{X}\rangle$
$\displaystyle\times\sum_{s^{\prime}_{23}\sigma^{\prime}_{{23}}}~{}\langle{1\over
2}\sigma^{\prime}_{2}{1\over
2}\sigma^{\prime}_{3}|s^{\prime}_{23}\sigma^{\prime}_{{23}}\rangle~{}\langle
l^{\prime}_{23}\mu^{\prime}_{23}s^{\prime}_{23}\sigma^{\prime}_{{23}}|j^{\prime}_{23}\
m^{\prime}_{23}\rangle~{}{\mbox{\boldmath{$\cal
G^{*}$}}}_{{L^{\prime}_{\rho}}X^{\prime}}^{j^{\prime}_{23}l^{\prime}_{23}s^{\prime}_{23}}(k_{23},k)~{}.$
(114)

Since ${{\bf k}}$ is only function of $|{\bf k}_{23}|$, then we are allowed to
integrate the spherical harmonics in Eq. (114) over $d{\Omega_{\hat{\bf
k}_{23}}}$. Therefore, taking care of the orthogonality of the Clebsch-Gordan
coefficients, we can write:

$\displaystyle{\cal F}^{\tau
mm^{\prime}}_{\sigma\sigma^{\prime}}={1\over(1-x)}~{}\int_{0}^{\infty}d{k}_{23}~{}k^{2}_{23}~{}{E_{23}~{}E({\bf
k})\over k^{+}}~{}\sum_{\sigma_{1}}D^{{1\over 2}}[{\cal R}_{M}({\tilde{\bf
k}})]_{\sigma\sigma_{1}}\sum_{{\sigma}^{\prime}_{1}}D^{{1\over 2}*}[{\cal
R}_{M}({\tilde{\bf
k}}]_{\sigma^{\prime}{\sigma}^{\prime}_{1}}~{}\sum_{s_{23}}\sum_{j_{23}}\sum_{l_{23}}\sum_{T_{23},\tau_{23}}$
$\displaystyle\times\langle T_{23}\tau_{23}{1\over 2}\tau|{1\over
2}T_{z}\rangle\langle T_{23}\tau_{23}{1\over 2}\tau|{1\over
2}T_{z}\rangle\sum_{L_{\rho}}\sum_{L^{\prime}_{\rho}}\sum_{X}\sum_{X^{\prime}}{\cal
R}^{j_{23},mm^{\prime},\sigma_{1}{{\sigma}^{\prime}_{1}}}_{L_{\rho}L^{\prime}_{\rho}XX^{\prime}}(k_{23},{\bf
k}){\mbox{\boldmath{$\cal
G$}}}_{{L_{\rho}}X}^{j_{23}l_{23}s_{23}}(k_{23},k){\mbox{\boldmath{$\cal
G^{*}$}}}_{{L^{\prime}_{\rho}}X^{\prime}}^{j_{23}l_{23}s_{23}}(k_{23},k)~{},$
(115)

where

$\displaystyle{\cal
R}^{j_{23},mm^{\prime},\sigma_{1}{{\sigma}^{\prime}_{1}}}_{L_{\rho}L^{\prime}_{\rho}XX^{\prime}}(k_{23},{\bf
k})=\sum_{M_{\rho}}~{}{Y}_{L_{\rho}M_{\rho}}(\hat{\bf
k})\sum_{M_{X}}\sum_{m_{23}}~{}\langle XM_{X}L_{\rho}M_{\rho}|{1\over
2}m\rangle~{}\langle j_{23}\ m_{23}{1\over 2}\sigma_{1}|XM_{X}\rangle$
$\displaystyle\times\sum_{M^{\prime}_{\rho}}~{}(-1)^{-M^{\prime}_{\rho}}{Y}_{L^{\prime}_{\rho}-M^{\prime}_{\rho}}(\hat{\bf
k})\sum_{M^{\prime}_{X}}~{}\langle
X^{\prime}M^{\prime}_{X}L^{\prime}_{\rho}M^{\prime}_{\rho}|{1\over
2}m^{\prime}\rangle~{}\langle j_{23}\ m_{23}{1\over
2}\sigma^{\prime}_{1}|X^{\prime}M^{\prime}_{X}\rangle~{}.$ (116)

The quantity ${\cal
R}^{j_{23},mm^{\prime},\sigma_{1}{{\sigma}^{\prime}_{1}}}_{L_{\rho}L^{\prime}_{\rho}XX^{\prime}}(k_{23},{\bf
k})$ is invariant for parity, since $L_{\rho}$ and $L^{\prime}_{\rho}$ have
the same parity.

###  B.2 Sum of products of five 3j symbols

By using the properties of the product of two spherical harmonics with the
same argument [77] and 3j symbols, Eq. (116) becomes

$\displaystyle{\cal
R}^{j_{23},mm^{\prime},\sigma_{1}{{\sigma}^{\prime}_{1}}}_{L_{\rho}L^{\prime}_{\rho}XX^{\prime}}(k_{23},{\bf
k})=\sum_{LM}\sqrt{{(2L_{\rho}+1)(2L^{\prime}_{\rho}+1)}\over{4\pi(2L+1)}}~{}\langle
L_{\rho}~{}0~{}L^{\prime}_{\rho}~{}0|L~{}0\rangle~{}Y_{LM}(\theta,\phi)~{}(-1)^{(L_{\rho}-L^{\prime}_{\rho}+M)}$
$\displaystyle\times\sum_{M_{\rho}M^{\prime}_{\rho}}~{}(-1)^{-M^{\prime}_{\rho}}~{}\sqrt{2L+1}\begin{pmatrix}L_{\rho}&L^{\prime}_{\rho}&L\\\
M_{\rho}&-M^{\prime}_{\rho}&-M\end{pmatrix}\sum_{M_{X}M^{\prime}_{X}}~{}\sum_{m_{23}}~{}(-1)^{(X-L_{\rho}+m)}~{}(-1)^{(j_{23}-{1\over
2}+M_{X})}$
$\displaystyle\times(-1)^{(X^{\prime}-L^{\prime}_{\rho}+m^{\prime})}~{}(-1)^{(j_{23}-{1\over
2}+M^{\prime}_{X})}~{}\sqrt{2}\sqrt{2}~{}\sqrt{2X+1}~{}\sqrt{2X^{\prime}+1}~{}$
$\displaystyle\times\begin{pmatrix}X&L_{\rho}&{1\over 2}\\\
M_{X}&M_{\rho}&-m\end{pmatrix}\begin{pmatrix}j_{23}&{1\over 2}&X\\\
m_{23}&\sigma_{1}&-M_{X}\end{pmatrix}\begin{pmatrix}X^{\prime}&L^{\prime}_{\rho}&{1\over
2}\\\
M^{\prime}_{X}&M^{\prime}_{\rho}&-m^{\prime}\end{pmatrix}\begin{pmatrix}j_{23}&{1\over
2}&X^{\prime}\\\
m_{23}&{\sigma^{\prime}_{1}}&-M^{\prime}_{X}\end{pmatrix}~{},$ (117)

where the angles $\theta$ and $\phi$ define the direction of $\hat{\bf k}$.
Only even values of $L$ are allowed to satisfy the parity invariance of ${\cal
R}^{j_{23},mm^{\prime},\sigma_{1}{{\sigma}^{\prime}_{1}}}_{L_{\rho}L^{\prime}_{\rho}XX^{\prime}}(k_{23},{\bf
k})$.

Through permutations of the columns in the 3j symbols, to have the indices $m$
and $m^{\prime}$ in the middle, and changing the sign of the third momentum
components in the 3j symbols where $X$ appears, we obtain (see Eq. (16) at p.
457 of Ref. [77] )

$\displaystyle{\cal
R}^{j_{23},mm^{\prime},\sigma_{1}{{\sigma}^{\prime}_{1}}}_{L_{\rho}L^{\prime}_{\rho}XX^{\prime}}(k_{23},{\bf
k})=-2~{}(-1)^{\sigma_{1}^{\prime}}~{}(-1)^{m}~{}(-1)^{(X+m)}~{}(-1)^{(X^{\prime}+m^{\prime})}~{}(-1)^{j_{23}}~{}\sqrt{{(2L_{\rho}+1)(2L^{\prime}_{\rho}+1)}\over{4\pi}}~{}$
$\displaystyle\times~{}\langle
L_{\rho}~{}0~{}L^{\prime}_{\rho}~{}0|L~{}0\rangle~{}\sqrt{2X+1}~{}\sqrt{2X^{\prime}+1}~{}\sum_{LM}(-1)^{(L+M)}~{}Y_{LM}(\theta,\phi)~{}(-1)^{(X^{\prime}-L^{\prime}_{\rho}-1/2-1/2-L-1/2)}$
$\displaystyle\times~{}\sum_{a\alpha
y\eta}(-1)^{(a-\alpha+y-\eta)}\Pi^{2}_{ay}\begin{pmatrix}{1\over 2}&{1\over
2}&a\\\ \\\
{\sigma^{\prime}_{1}}&-\sigma_{1}&\alpha\end{pmatrix}\begin{pmatrix}a&L&y\\\
\\\ -\alpha&-M&-\eta\end{pmatrix}\begin{pmatrix}y&{1\over 2}&{1\over 2}\\\ \\\
\eta&m&-m^{\prime}\end{pmatrix}\begin{Bmatrix}{1\over 2}&{1\over 2}&a\\\ \\\
X&X^{\prime}&j_{23}\end{Bmatrix}\begin{Bmatrix}{1\over 2}&{1\over 2}&y\\\ \\\
X&X^{\prime}&a\\\ \\\ L_{\rho}&L^{\prime}_{\rho}&L\end{Bmatrix}~{}.$ (118)

###  B.3 Sums involving the $D$ matrixes for the system polarization and the
Melosh factors

Let us consider the following combination of Wigner functions and Melosh
rotations

$\displaystyle D^{j{\cal M},a\,y}_{\sigma\sigma^{\prime}}({\bf S},{\tilde{\bf
k}})~{}=-~{}2~{}~{}\sum_{LM}(-1)^{M}~{}~{}Y_{LM}(\theta,\phi)\sum_{m}~{}D^{j}_{m,{\cal
M}}(\alpha,\beta,\gamma)~{}\sum_{m^{\prime}}~{}[D^{j}_{m^{\prime},{\cal
M}}(\alpha,\beta,\gamma)]^{*}~{}~{}\sum_{\sigma_{1}}~{}D^{{1\over 2}}[{\cal
R}_{M}({\tilde{\bf k}})]_{\sigma\sigma_{1}}$
$\displaystyle\times\sum_{{\sigma^{\prime}_{1}}}D^{{1\over 2}*}[{\cal
R}_{M}({\tilde{\bf
k}})]_{\sigma^{\prime}{\sigma}^{\prime}_{1}}~{}\sum_{\mu,\eta}~{}(-1)^{-\mu-\eta}\begin{pmatrix}1\over
2&{1\over 2}&a\\\
{\sigma^{\prime}_{1}}&-\sigma_{1}&\mu\end{pmatrix}\begin{pmatrix}a&L&y\\\
-\mu&-M&-\eta\end{pmatrix}\begin{pmatrix}y&{1\over 2}&{1\over 2}\\\
\eta&m&-m^{\prime}\end{pmatrix}~{}(-1)^{\sigma_{1}}~{}(-1)^{2m+m^{\prime}}~{}.$
(119)

If one applies Eq. (101) to the product $D^{j}_{m,{\cal
M}}(\alpha,\beta,\gamma)~{}D^{j}_{-m^{\prime},-{\cal M}}(\alpha,\beta,\gamma)$
, then Eq. (119) becomes

$\displaystyle D^{j{\cal M},a\,y}_{\sigma\sigma^{\prime}}({\bf S},{\tilde{\bf
k}})~{}=~{}\sqrt{2}~{}{(-1)^{a+1}\over\sqrt{2a+1}}~{}\sum_{LM}~{}(-1)^{M}~{}Y_{LM}(\theta,\phi)~{}\sum_{J=0}^{2j}~{}\langle
j{\cal M},j-{\cal
M}|J0\rangle~{}\sum_{\mu,\eta}~{}(-1)^{-\eta}~{}\sum_{mm^{\prime}}(-1)^{m^{\prime}-{\cal
M}}~{}$ $\displaystyle\times~{}\langle
jm,j-m^{\prime}|J(m-m^{\prime})\rangle~{}~{}D^{J}_{(m-m^{\prime}),0}(\alpha,\beta,\gamma)~{}~{}\sum_{\sigma_{1}}~{}D^{{1\over
2}}[{\cal R}_{M}({\tilde{\bf
k}})]_{\sigma\sigma_{1}}~{}\sum_{{\sigma^{\prime}_{1}}}D^{{1\over 2}*}[{\cal
R}_{M}({\tilde{\bf
k}})]_{\sigma^{\prime}{\sigma}^{\prime}_{1}}~{}(-1)^{{\sigma_{1}}}~{}(-1)^{({1}-a)}$
$\displaystyle\times~{}\langle{1\over 2}\sigma_{1},{1\over
2}-{\sigma^{\prime}_{1}}~{}|~{}a\mu\rangle~{}\begin{pmatrix}a&L&y\\\
-\mu&-M&-\eta\end{pmatrix}~{}(-1)^{1/2+2y}~{}(-1)^{1/2-m}\sqrt{{2\over
2y+1}}~{}\langle{1\over 2}m,{1\over 2}~{}-m^{\prime}~{}|~{}y-\eta\rangle~{}.$
(120)

One has to recall that $j=1/2$ and that $(m-m^{\prime})$ has to be equal to
$-\eta$. Then we obtain

$\displaystyle D^{1/2{\cal M},a\,y}_{\sigma\sigma^{\prime}}({\bf
S},{\tilde{\bf
k}})~{}=-~{}{2}~{}{1\over\sqrt{2a+1}}~{}\sum_{LM}~{}(-1)^{M}~{}~{}Y_{LM}(\theta,\phi)~{}~{}\langle{1\over
2}{\cal M},{1\over 2}-{\cal M}|y0\rangle~{}\sum_{\mu,\eta}~{}(-1)^{-{\cal
M}}~{}Y^{*}_{y-\eta}(\beta,\alpha)~{}\sqrt{{4\pi\over 2y+1}}$
$\displaystyle\times~{}~{}\sum_{\sigma_{1}}~{}D^{{1\over 2}}[{\cal
R}_{M}({\tilde{\bf
k}})]_{\sigma\sigma_{1}}~{}\sum_{{\sigma^{\prime}_{1}}}D^{{1\over 2}*}[{\cal
R}_{M}({\tilde{\bf
k}})]_{\sigma^{\prime}{\sigma}^{\prime}_{1}}~{}\langle{1\over
2}\sigma_{1},{1\over
2}-{\sigma^{\prime}_{1}}~{}|~{}a\mu\rangle~{}(-1)^{{\sigma_{1}}}~{}\begin{pmatrix}a&L&y\\\
-\mu&-M&-\eta\end{pmatrix}~{}~{}{1\over\sqrt{2y+1}}~{}.$ (121)

###  B.4 Spin-dependent momentum distribution

Making use of Eq. (121) to express the quantity $D^{j{\cal
M},a\,y}_{\sigma\sigma^{\prime}}({\bf S},{\tilde{\bf k}})$, we can now
summarize our results for the spin-dependent momentum distribution, Eq. (14),
as follows

$\displaystyle{\left[{\mbox{\boldmath{$\cal N$}}}_{\cal M}^{\tau}(x,{\bf
k}_{\perp};\bf S)\right]_{\sigma\sigma^{\prime}}}=~{}{(-1)^{{\cal
M}}~{}2\over(1-x)}~{}\int_{0}^{\infty}d{k}_{23}~{}k^{2}_{23}~{}{E_{23}~{}E({\bf
k})\over
k^{+}}~{}\sum_{s_{23}}\sum_{j_{23}}\sum_{l_{23}}\sum_{T_{23},\tau_{23}}\langle
T_{23}\tau_{23}{1\over 2}\tau|{1\over 2}T_{z}\rangle\langle
T_{23}\tau_{23}{1\over 2}\tau|{1\over 2}T_{z}\rangle$
$\displaystyle\times~{}(-1)^{j_{23}}~{}\sum_{L_{\rho}}\sum_{L^{\prime}_{\rho}}\sum_{X}\sum_{X^{\prime}}~{}\sqrt{{(2L_{\rho}+1)(2L^{\prime}_{\rho}+1)}}~{}\langle
L_{\rho}~{}0~{}L^{\prime}_{\rho}~{}0|L~{}0\rangle~{}\sqrt{2X+1}~{}\sqrt{2X^{\prime}+1}~{}(-1)^{(-L^{\prime}_{\rho}-1/2)}$
$\displaystyle\times~{}{\mbox{\boldmath{$\cal
G$}}}_{{L_{\rho}}X}^{j_{23}l_{23}s_{23}}(k_{23},k)~{}{\mbox{\boldmath{$\cal
G^{*}$}}}_{{L^{\prime}_{\rho}}X^{\prime}}^{j_{23}l_{23}s_{23}}(k_{23},k)~{}(-1)^{X}~{}\sum_{a~{}y}(-1)^{a}~{}\Pi^{2}_{ay}~{}\sum_{LM}~{}Y_{LM}(\theta,\phi)~{}{1\over\sqrt{2a+1}}~{}~{}\langle{1\over
2}{\cal M},{1\over 2}-{\cal M}|y0\rangle$
$\displaystyle\times~{}\sum_{\mu,\eta}~{}Y^{*}_{y-\eta}(\beta,\alpha)~{}{1\over\sqrt{2y+1}}~{}~{}\sum_{\sigma_{1}}~{}D^{{1\over
2}}[{\cal R}_{M}({\tilde{\bf
k}})]_{\sigma\sigma_{1}}~{}\sum_{{\sigma^{\prime}_{1}}}D^{{1\over 2}*}[{\cal
R}_{M}({\tilde{\bf
k}})]_{\sigma^{\prime}{\sigma}^{\prime}_{1}}~{}\langle{1\over
2}\sigma_{1},{1\over
2}-{\sigma^{\prime}_{1}}~{}|~{}a\mu\rangle~{}(-1)^{{\sigma_{1}}}$
$\displaystyle\times~{}\langle a\mu
LM|~{}y-\eta\rangle~{}(-1)^{\mu}~{}{1\over{2y+1}}\begin{Bmatrix}{1\over
2}&{1\over 2}&a\\\ \\\ X&X^{\prime}&j_{23}\end{Bmatrix}\begin{Bmatrix}{1\over
2}&{1\over 2}&y\\\ \\\ X&X^{\prime}&a\\\ \\\
L_{\rho}&L^{\prime}_{\rho}&L\end{Bmatrix}~{}.$ (122)

It has to be noticed that $(a+y)$ is an even number, as can be easily shown
exchanging $X$ with $X^{\prime}$ and $L_{\rho}$ with $L^{\prime}_{\rho}$.
Furthermore both $a$ and $y$ can be only $0$ or $1$.

Then, if $L=0$ one has $a=y=0$, or $a=y=1$. If $L=2$, only $a=y=1$ is
possible.

Let us define the quantity

$\displaystyle{\cal Z}^{\tau}_{\sigma\sigma^{\prime}}(k_{23},{\bf k},{\bf
S},a,y,L)=\Pi^{2}_{ay}~{}{1\over\sqrt{2a+1}}~{}{1\over{(2y+1)}^{3/2}}~{}\langle{1\over
2}{\cal M},{1\over 2}-{\cal M}|y0\rangle~{}{\cal
H}^{\tau}(L,a,k_{23},k)\sum_{M}~{}Y_{LM}(\theta,\phi)$
$\displaystyle\times~{}\sum_{\sigma_{1}}~{}D^{{1\over 2}}[{\cal
R}_{M}({\tilde{\bf
k}})]_{\sigma\sigma_{1}}~{}\sum_{{\sigma^{\prime}_{1}}}D^{{1\over 2}*}[{\cal
R}_{M}({\tilde{\bf
k}})]_{\sigma^{\prime}{\sigma}^{\prime}_{1}}~{}\sum_{\mu\eta}~{}Y^{*}_{y-\eta}(\beta,\alpha)~{}\langle
a\mu,LM|y-\eta\rangle~{}(-1)^{\mu}$ $\displaystyle\times~{}\langle{1\over
2}\sigma_{1},{1\over
2}-{\sigma^{\prime}_{1}}~{}|~{}a\mu\rangle~{}(-1)^{{\sigma_{1}}-1/2}~{}$ (123)

where

$\displaystyle{\cal
H}^{\tau}(L,a,k_{23},k)=(-1)^{a}~{}\sum_{j_{23}}\sum_{l_{23}}\sum_{s_{23}}\sum_{T_{23},\tau_{23}}\langle
T_{23}\tau_{23}{1\over 2}\tau|{1\over 2}T_{z}\rangle\langle
T_{23}\tau_{23}{1\over 2}\tau|{1\over 2}T_{z}\rangle~{}(-1)^{j_{23}}~{}$
$\displaystyle\times~{}\sum_{L_{\rho}}\sum_{L^{\prime}_{\rho}}~{}\sum_{X}\sum_{X^{\prime}}~{}(-1)^{(X+1/2)}~{}\sqrt{{(2L_{\rho}+1)(2L^{\prime}_{\rho}+1)}}~{}~{}\langle
L_{\rho}~{}0~{}L^{\prime}_{\rho}~{}0|L~{}0\rangle~{}\sqrt{2X+1}~{}\sqrt{2X^{\prime}+1}$
$\displaystyle\times~{}(-1)^{X+X^{\prime}}~{}\begin{Bmatrix}{1\over 2}&{1\over
2}&a\\\ \\\ X&X^{\prime}&j_{23}\end{Bmatrix}\begin{Bmatrix}L_{\rho}&X&{1\over
2}\\\ \\\ L^{\prime}_{\rho}&X^{\prime}&{1\over 2}\\\ \\\
L&a&y\end{Bmatrix}~{}(-1)^{l_{23}}~{}{\mbox{\boldmath{$\cal
G$}}}_{{L_{\rho}}X}^{j_{23}l_{23}s_{23}}(k_{23},k)~{}{\mbox{\boldmath{$\cal
G^{*}$}}}_{{L^{\prime}_{\rho}}X^{\prime}}^{j_{23}l_{23}s_{23}}(k_{23},k)~{}.$
(124)

Then the momentum distribution can be written as a function of the three
independent quantities ${\cal Z}^{\tau}_{\sigma\sigma^{\prime}}(k_{23},{\bf
k},{\bf S},0,0,0)$, ${\cal Z}^{\tau}_{\sigma\sigma^{\prime}}(k_{23},{\bf
k},{\bf S},1,1,0)$, and ${\cal Z}^{\tau}_{\sigma\sigma^{\prime}}(k_{23},{\bf
k},{\bf S},1,1,2)$ that notably depend on ${\cal H}^{\tau}(0,0,k_{23},k)$,
${\cal H}^{\tau}(0,1,k_{23},k)$, and ${\cal H}^{\tau}(2,1,k_{23},k)$, viz.

$\displaystyle{\left[{\mbox{\boldmath{$\cal N$}}}_{\cal M}^{\tau}(x,{\bf
k}_{\perp};\bf S)\right]_{\sigma\sigma^{\prime}}}~{}=(-1)^{{\cal
M}+1/2}~{}{2\over(1-x)}~{}\int_{0}^{\infty}d{k}_{23}~{}k^{2}_{23}~{}{E_{23}~{}E({\bf
k})\over k^{+}}~{}\left\\{~{}{\cal
Z}^{\tau}_{\sigma\sigma^{\prime}}(k_{23},{\bf k},{\bf S},0,0,0)~{}\right.$
$\displaystyle\left.+~{}{\cal Z}^{\tau}_{\sigma\sigma^{\prime}}(k_{23},{\bf
k},{\bf S},1,1,0)~{}+~{}{\cal Z}^{\tau}_{\sigma\sigma^{\prime}}(k_{23},{\bf
k},{\bf S},1,1,2)~{}\right\\}~{}.$ (125)

We evaluate separately ${\cal Z}^{\tau}_{\sigma\sigma^{\prime}}(k_{23},{\bf
k},{\bf S},a,y,L)$ for the three possible cases of the variables $L,a,y$. The
first two quantities to evaluate are

$\displaystyle{\cal Z}^{\tau}_{\sigma\sigma^{\prime}}(k_{23},{\bf k},{\bf
S},0,0,0)=\Pi^{2}_{00}~{}~{}\langle{1\over 2}{\cal M},{1\over 2}-{\cal
M}|00\rangle~{}{\cal H}^{\tau}(0,0,k_{23},k)~{}\sum_{\sigma_{1}}~{}D^{{1\over
2}}[{\cal R}_{M}({\tilde{\bf k}})]_{\sigma\sigma_{1}}$
$\displaystyle\times~{}~{}\sum_{{\sigma^{\prime}_{1}}}D^{{1\over 2}*}[{\cal
R}_{M}({\tilde{\bf
k}})]_{\sigma^{\prime}{\sigma}^{\prime}_{1}}~{}Y_{00}(\theta,\phi)~{}Y^{*}_{00}(\beta,\alpha)~{}\langle
00,00|00\rangle~{}\langle{1\over 2}\sigma_{1},{1\over
2}-{\sigma_{1}^{\prime}}~{}|~{}00\rangle~{}(-1)^{{\sigma_{1}}-1/2}$
$\displaystyle=~{}\delta_{\sigma\sigma^{\prime}}~{}(-1)^{1/2-{\cal M}}~{}{\cal
H}^{\tau}(0,0,k_{23},k)~{}{1\over 8\pi}~{}$ (126)

and

$\displaystyle{\cal Z}^{\tau}_{\sigma\sigma^{\prime}}(k_{23},{\bf k},{\bf
S},1,1,0)=\Pi^{2}_{1,1}~{}{1\over\sqrt{3}}~{}~{}{1\over{(3)}^{3/2}}~{}\langle{1\over
2}{\cal M},{1\over 2}-{\cal M}|10\rangle~{}{\cal
H}^{\tau}(0,1,k_{23},k)~{}\sum_{\sigma_{1}}~{}D^{{1\over 2}}[{\cal
R}_{M}({\tilde{\bf k}})]_{\sigma\sigma_{1}}$
$\displaystyle\times~{}\sum_{{\sigma^{\prime}_{1}}}D^{{1\over 2}*}[{\cal
R}_{M}({\tilde{\bf
k}})]_{\sigma^{\prime}{\sigma}^{\prime}_{1}}~{}\sum_{\mu}~{}~{}Y_{00}(\theta,\phi)\sum_{\eta}~{}~{}Y^{*}_{1-\eta}(\beta,\alpha)~{}~{}\langle
1\mu,00|1-\eta\rangle~{}(-1)^{\mu}~{}\langle{1\over 2}\sigma_{1},{1\over
2}-{\sigma^{\prime}_{1}}~{}|~{}1\mu\rangle~{}(-1)^{{\sigma_{1}}-1/2}$
$\displaystyle=~{}~{}{1\over\sqrt{2}}~{}{1\over\sqrt{4\pi}}~{}{\cal
H}^{\tau}(0,1,k_{23},k)~{}\sum_{\sigma_{1}}~{}D^{{1\over 2}}[{\cal
R}_{M}({\tilde{\bf
k}})]_{\sigma\sigma_{1}}~{}\sum_{{\sigma^{\prime}_{1}}}D^{{1\over 2}*}[{\cal
R}_{M}({\tilde{\bf k}})]_{\sigma^{\prime}{\sigma}^{\prime}_{1}}~{}{1\over
2}~{}\sqrt{{3\over 2\pi}}~{}\left[{\mbox{\boldmath{$\sigma$}}}\cdot{\hat{\bf
S}}\right]_{\sigma_{1}{\sigma}^{\prime}_{1}}~{},$ (127)

where the identity (see Ref. [77])

$\displaystyle\sum_{\mu}~{}Y_{1-\mu}(\beta,\alpha)~{}~{}(-1)^{\mu}~{}\langle{1\over
2}\sigma_{1},{1\over
2}-{\sigma^{\prime}_{1}}~{}|~{}1\mu\rangle~{}(-1)^{{{\sigma}_{1}^{\prime}}}=~{}{1\over
2}~{}(-1)^{1/2}\sqrt{{3\over
2\pi}}~{}\left[{\mbox{\boldmath{$\sigma$}}}\cdot{\hat{\bf
S}}\right]_{\sigma_{1}{\sigma}^{\prime}_{1}}~{},$ (128)

has been used.

Through the actual expressions of the Melosh rotations (see Appendix C and in
particular Eqs. (156) and (158)), Eq. (127) becomes

$\displaystyle{\cal Z}^{\tau}_{\sigma\sigma^{\prime}}(k_{23},{\bf k},{\bf
S},1,1,0)={1\over 2}~{}{1\over\sqrt{8\pi}}~{}\sqrt{{3\over 2\pi}}~{}{\cal
H}^{\tau}(0,1,k_{23},k)\left[\cos{\varphi\over 2}~{}+~{}i~{}\sin{\varphi\over
2}~{}\hat{\bf
n}\cdot{\mbox{\boldmath{$\sigma$}}}\right]_{\sigma\sigma_{1}}\left[{\mbox{\boldmath{$\sigma$}}}\cdot{\hat{\bf
S}}\right]_{\sigma_{1}{\sigma}^{\prime}_{1}}$
$\displaystyle\times~{}\left[\cos{\varphi\over 2}~{}-~{}i~{}\sin{\varphi\over
2}~{}\hat{\bf
n}\cdot{\mbox{\boldmath{$\sigma$}}}\right]_{{\sigma}^{\prime}_{1}\sigma^{\prime}}$
$\displaystyle={\sqrt{{3}}\over{8\pi}}~{}{\cal
H}^{\tau}(0,1,k_{23},k)~{}\left\\{\left[{\mbox{\boldmath{$\sigma$}}}\cdot{{\bf
S}}\right]_{{\sigma}\sigma^{\prime}}~{}-~{}2~{}\sin{\varphi\over
2}~{}\cos{\varphi\over 2}~{}\left\\{\left[\left({\bf S}\cdot\hat{\bf
k}_{\perp}\right){\hat{z}}-({\bf S}\cdot\hat{z})~{}{\hat{\bf
k}_{\perp}}\right]\cdot{\mbox{\boldmath{$\sigma$}}}\right\\}_{{\sigma}\sigma^{\prime}}\right.$
$\displaystyle\left.-~{}2~{}\sin^{2}{\varphi\over
2}~{}\left\\{\left[\left({\bf S}\cdot\hat{z}\right)\hat{z}-\left({\bf
S}\cdot\hat{\bf k}_{\perp}\right)\hat{\bf
k}_{\perp}~{}\right]\cdot\sigma\right\\}_{{\sigma}\sigma^{\prime}}\right\\}~{},$
(129)

where $\sin{\varphi/2}$ and $\cos{\varphi/2}$ are defined in Eq. (154).

The last quantity, ${\cal Z}^{\tau}_{\sigma\sigma^{\prime}}(k_{23},{\bf
k},{\bf S},1,1,2)$, is

$\displaystyle{\cal Z}^{\tau}_{\sigma\sigma^{\prime}}(k_{23},{\bf k},{\bf
S},1,1,2)=~{}\Pi^{2}_{1,1}~{}{1\over\sqrt{3}}~{}~{}{1\over{(3)}^{3/2}}~{}\langle{1\over
2}{\cal M},{1\over 2}-{\cal M}|10\rangle~{}{\cal
H}^{\tau}(2,1,k_{23},k)\sum_{\sigma_{1}}~{}D^{{1\over 2}}[{\cal
R}_{M}({\tilde{\bf k}})]_{\sigma\sigma_{1}}~{}$
$\displaystyle\times~{}\sum_{{\sigma^{\prime}_{1}}}D^{{1\over 2}*}[{\cal
R}_{M}({\tilde{\bf
k}})]_{\sigma^{\prime}{\sigma}^{\prime}_{1}}~{}\sum_{M}~{}Y_{2M}(\theta,\phi)\sum_{\mu\eta}~{}Y^{*}_{1-\eta}(\beta,\alpha)~{}~{}\langle
1\mu,2M|1-\eta\rangle~{}(-1)^{\mu}~{}\langle{1\over 2}\sigma_{1},{1\over
2}-{\sigma^{\prime}_{1}}~{}|~{}1\mu\rangle~{}(-1)^{{\sigma_{1}}-1/2}$
$\displaystyle=~{}-{1\over{2}}~{}{\cal
H}^{\tau}(2,1,k_{23},k)~{}\sum_{\sigma_{1}}~{}D^{{1\over 2}}[{\cal
R}_{M}({\tilde{\bf
k}})]_{\sigma\sigma_{1}}~{}\sum_{{\sigma^{\prime}_{1}}}D^{{1\over 2}*}[{\cal
R}_{M}({\tilde{\bf
k}})]_{\sigma^{\prime}{\sigma}^{\prime}_{1}}~{}\sum_{\mu}~{}(-1)^{\mu}~{}\langle{1\over
2}\sigma_{1},{1\over
2}-{\sigma^{\prime}_{1}}~{}|~{}1\mu\rangle~{}(-1)^{{\sigma_{1}}-1/2}$
$\displaystyle\times~{}\left[Y^{*}_{1\mu}(\theta,\phi)~{}{\sqrt{3}}~{}Y_{1,0}({\hat{\bf
k}}\cdot\hat{\bf S})-Y^{*}_{1\mu}(\hat{\bf
S})~{}{1\over\sqrt{4\pi}}\right]~{}.$ (130)

In the last step the first of the bipolar harmonics in Eq. (A5) of Ref. [58]
has been used. Therefore, using again Eq. (128) and the results of Appendix C
(see Eq. (156)) we have

$\displaystyle{\cal Z}^{\tau}_{\sigma\sigma^{\prime}}(k_{23},{\bf k},{\bf
S},1,1,2)=~{}-{1\over{4}}~{}\sqrt{{3\over 8}}~{}{1\over\pi}~{}{\cal
H}^{\tau}(2,1,k_{23},k)\left[~{}{3}~{}{\hat{\bf k}}\cdot\hat{\bf
S}~{}\sum_{\sigma_{1}{\sigma^{\prime}_{1}}}~{}D^{{1\over 2}}[{\cal
R}_{M}({k^{+},{\bf
k}_{\perp}})]_{\sigma\sigma_{1}}~{}\left[{\mbox{\boldmath{$\sigma$}}}\cdot{\hat{\bf
k}}\right]_{\sigma_{1}{\sigma}^{\prime}_{1}}\right.$
$\displaystyle\left.\times~{}D^{{1\over 2}}[{\cal R}_{M}({{k^{+},-{\bf
k}_{\perp}}})]_{{\sigma}^{\prime}_{1}\sigma^{\prime}}~{}-~{}\sum_{\sigma_{1}{\sigma^{\prime}_{1}}}~{}D^{{1\over
2}}[{\cal R}_{M}({k^{+},{\bf
k}_{\perp}})]_{\sigma\sigma_{1}}~{}\left[{\mbox{\boldmath{$\sigma$}}}\cdot{\hat{\bf
S}}\right]_{\sigma_{1}{\sigma}^{\prime}_{1}}~{}D^{{1\over 2}}[{\cal
R}_{M}({k^{+}_{1},-{\bf
k}_{\perp}})]_{{\sigma}^{\prime}_{1}\sigma^{\prime}}\right]~{}.$ (131)

Eventually from Eqs. (158, 159) we have

$\displaystyle{\cal Z}^{\tau}_{\sigma\sigma^{\prime}}(k_{23},{\bf k},{\bf
S},1,1,2)=~{}-{1\over{4}}~{}\sqrt{{3\over 8}}~{}{1\over\pi}~{}{\cal
H}^{\tau}(2,1,k_{23},k)~{}\left\\{~{}{{3}\over k}~{}{\hat{\bf k}}\cdot\hat{\bf
S}~{}\left[~{}\left(\cos^{2}{\varphi\over 2}-\sin^{2}{\varphi\over
2}\right)~{}\left[{\mbox{\boldmath{$\sigma$}}}\cdot{{\bf
k}}\right]_{{\sigma}\sigma^{\prime}}\right.\right.$
$\displaystyle\left.\left.-~{}2~{}\sin{\varphi\over 2}~{}\cos{\varphi\over
2}~{}\left[\left({k}_{\perp}~{}{\hat{z}}-{k}_{z}~{}{\hat{\bf
k}}_{\perp}\right)\cdot{\mbox{\boldmath{$\sigma$}}}\right]_{{\sigma}\sigma^{\prime}}\right]~{}-~{}\left\\{~{}\left[{\mbox{\boldmath{$\sigma$}}}\cdot{{\bf
S}}\right]_{{\sigma}\sigma^{\prime}}-~{}2~{}\sin{\varphi\over
2}~{}\cos{\varphi\over 2}~{}\left[\left(({\bf S}\cdot\hat{\bf
k}_{\perp})~{}{\hat{z}}-({\bf S}\cdot\hat{z})~{}{\hat{\bf
k}_{\perp}}\right)\cdot{\mbox{\boldmath{$\sigma$}}}\right]_{{\sigma}\sigma^{\prime}}\right.\right.$
$\displaystyle\left.\left.-~{}2~{}\sin^{2}{\varphi\over 2}~{}\left[\left(({\bf
S}\cdot\hat{z})~{}{\hat{z}}+({\bf S}\cdot\hat{\bf k}_{\perp})~{}{\hat{\bf
k}_{\perp}}\right)\cdot{\mbox{\boldmath{$\sigma$}}}~{}\right]_{{\sigma}\sigma^{\prime}}~{}\right\\}\right\\}~{}.$
(132)

Summarizing our results, the spin-dependent momentum distribution in terms of
the pseudovectors $\bf{S}$, $~{}\hat{\bf k}_{\perp}({\bf S}\cdot\hat{\bf
k}_{\perp})$, $~{}\hat{\bf k}_{\perp}({\bf S}\cdot\hat{z})$,
$~{}\hat{z}~{}({\bf S}\cdot\hat{\bf k}_{\perp})$, and $~{}\hat{z}~{}({\bf
S}\cdot\hat{z})$ is

$\displaystyle{\left[{\mbox{\boldmath{$\cal N$}}}_{\cal M}^{\tau}(x,{\bf
k}_{\perp};\bf S)\right]_{\sigma\sigma^{\prime}}}~{}=~{}~{}{(-1)^{{\cal
M}+1/2}\over
4\pi~{}(1-x)}~{}\int_{0}^{\infty}d{k}_{23}~{}k^{2}_{23}~{}{E_{23}~{}E({\bf
k})\over k^{+}}~{}\left\\{~{}\delta_{\sigma\sigma^{\prime}}~{}(-1)^{1/2-{\cal
M}}~{}{\cal H}^{\tau}(0,0,k_{23},k)~{}\right.$
$\displaystyle+~{}{\sqrt{{3}}}~{}\left[~{}{\cal
H}^{\tau}(0,1,k_{23},k)+~{}\sqrt{{1\over 2}}~{}{\cal
H}^{\tau}(2,1,k_{23},k)~{}\right]\left[~{}\left[{\mbox{\boldmath{$\sigma$}}}\cdot{{\bf
S}}\right]_{{\sigma}\sigma^{\prime}}-~{}2~{}\sin{\varphi\over
2}~{}\cos{\varphi\over 2}~{}\left[\left(({\bf S}\cdot{\hat{\bf
k}}_{\perp})~{}{\hat{z}}-\left({\bf S}\cdot{\hat{z}}\right)~{}{\hat{\bf
k}}_{\perp}\right)\cdot{\mbox{\boldmath{$\sigma$}}}\right]_{{\sigma}\sigma^{\prime}}\right.$
$\displaystyle\left.-~{}2~{}\sin^{2}{\varphi\over 2}~{}\left[\left(\left({\bf
S}\cdot{\hat{z}}\right)~{}{\hat{z}}+({\bf S}\cdot{\hat{\bf
k}}_{\perp})~{}{\hat{\bf
k}_{\perp}}\right)\cdot{\mbox{\boldmath{$\sigma$}}}~{}\right]_{{\sigma}\sigma^{\prime}}\right]-\sqrt{{3\over
2}}~{}{\cal H}^{\tau}(2,1,k_{23},k)~{}{3\over k^{2}}\left[{{\bf
k}}_{\perp}\cdot{\bf S}\left[\left(\cos^{2}{\varphi\over
2}-\sin^{2}{\varphi\over 2}\right)\left({\bf
k}_{\perp}\cdot{\mbox{\boldmath{$\sigma$}}}\right)_{{\sigma}\sigma^{\prime}}\right.\right.$
$\displaystyle\left.\left.+\left(\cos^{2}{\varphi\over
2}-\sin^{2}{\varphi\over
2}\right){k}_{z}({\hat{z}}\cdot{\mbox{\boldmath{$\sigma$}}})_{{\sigma}\sigma^{\prime}}-~{}2\sin{\varphi\over
2}\cos{\varphi\over
2}\left[\left(k_{\perp}~{}{\hat{z}}~{}-~{}{k}_{z}~{}{\hat{\bf
k}}_{\perp}\right)\cdot{\mbox{\boldmath{$\sigma$}}}\right]_{{\sigma}\sigma^{\prime}}+~{}{k}_{z}~{}\left({\hat{z}}\cdot{\bf
S}\right)\left[\left(\cos^{2}{\varphi\over 2}-\sin^{2}{\varphi\over
2}\right)\right.\right.\right.$
$\displaystyle\left.\left.\left.\times~{}\left({\bf
k}_{\perp}\cdot{\mbox{\boldmath{$\sigma$}}}\right)_{{\sigma}\sigma^{\prime}}+\left(\cos^{2}{\varphi\over
2}-\sin^{2}{\varphi\over
2}\right){k}_{z}({\hat{z}}\cdot{\mbox{\boldmath{$\sigma$}}})_{{\sigma}\sigma^{\prime}}-2\sin{\varphi\over
2}\cos{\varphi\over 2}\left[\left(k_{\perp}~{}{\hat{z}}-{k}_{z}~{}{\hat{\bf
k}}_{\perp}\right)\cdot{\mbox{\boldmath{$\sigma$}}}\right]_{{\sigma}\sigma^{\prime}}\right]\right]\right\\}~{}.$
(133)

From a comparison of Eqs. (15), (16) and (133) one can immediately obtain
explicit expressions for the functions $b^{\tau}_{i,{\cal M}}\left[|{\bf
k}_{\perp}|,x,({\bf S}\cdot\hat{\bf k}_{\perp})^{2},({\bf
S}\cdot{\hat{z}})^{2},{{(\hat{\bf k}_{\perp}\times{\hat{z}})\cdot{\bf
S}}}\right]$ ($i=0,1,...,5)$ and verify that actually they do not depend on
${\bf S}$ and then they do not depend on the direction of ${\bf k}_{\perp}$.
For $i=0$ one has

$\displaystyle b^{\tau}_{0,{\cal M}}\left[x,|{\bf
k}_{\perp}|\right]=~{}{(-1)\over
2\pi~{}(1-x)}~{}\int_{0}^{\infty}d{k}_{23}~{}k^{2}_{23}~{}{E_{23}~{}E({\bf
k})\over k^{+}}~{}{\cal H}^{\tau}(0,0,k_{23},k)~{}.$ (134)

It can be useful to decompose the functions $b^{\tau}_{i,{\cal M}}$
($i=1,...,5$) according to the values $0$ or $2$ of the momentum $L$

$\displaystyle b^{\tau}_{i,{\cal M}}=b^{\tau(L=0)}_{i,{\cal
M}}+b^{\tau(L=2)}_{i,{\cal M}}~{},$ (135)

and one obtains

$\displaystyle b^{\tau(0)}_{1,{\cal M}}\left(x,|{\bf
k}_{\perp}|\right)=~{}{(-1)^{{\cal M}+1/2}\over
2\pi~{}(1-x)}~{}{\sqrt{{3}}}\int_{0}^{\infty}d{k}_{23}~{}k^{2}_{23}~{}{E_{23}~{}E({\bf
k})\over k^{+}}~{}{\cal H}^{\tau}(0,1,k_{23},k)~{},$ (136) $\displaystyle
b^{\tau(2)}_{1,{\cal M}}\left(x,|{\bf k}_{\perp}|\right)=~{}{(-1)^{{\cal
M}+1/2}\over
2\pi~{}(1-x)}~{}{\sqrt{{3}}}\int_{0}^{\infty}d{k}_{23}~{}k^{2}_{23}~{}{E_{23}~{}E({\bf
k})\over k^{+}}~{}~{}\sqrt{{1\over 2}}~{}{\cal H}^{\tau}(2,1,k_{23},k)~{},$
(137) $\displaystyle b^{\tau(0)}_{2,{\cal M}}\left(x,|{\bf
k}_{\perp}|\right)=~{}-~{}{(-1)^{{\cal M}+1/2}\over
2\pi~{}(1-x)}~{}\sqrt{{3}}\int_{0}^{\infty}d{k}_{23}~{}k^{2}_{23}~{}{E_{23}~{}E({\bf
k})\over k^{+}}~{}{\cal H}^{\tau}(0,1,k_{23},k)~{}~{}2~{}\sin^{2}{\varphi\over
2}~{},$ (138) $\displaystyle b^{\tau(2)}_{2,{\cal M}}\left(x,|{\bf
k}_{\perp}|\right)=~{}-~{}{(-1)^{{\cal M}+1/2}\over
2\pi~{}(1-x)}~{}\sqrt{{3}}\int_{0}^{\infty}d{k}_{23}~{}k^{2}_{23}~{}{E_{23}~{}E({\bf
k})\over k^{+}}~{}\sqrt{{1\over 2}}~{}{\cal H}^{\tau}(2,1,k_{23},k)$
$\displaystyle\left\\{~{}2~{}\sin^{2}{\varphi\over 2}~{}+~{}3~{}{1\over
k^{2}}\left[\left(\cos^{2}{\varphi\over 2}-\sin^{2}{\varphi\over
2}\right)~{}k_{\perp}^{2}+2~{}\sin{\varphi\over 2}~{}\cos{\varphi\over
2}~{}k_{\perp}~{}k_{z}\right]\right\\}~{},$ (139) $\displaystyle
b^{\tau(0)}_{3,{\cal M}}\left(x,|{\bf k}_{\perp}|\right)=~{}{(-1)^{{\cal
M}+1/2}\over
2\pi~{}(1-x)}~{}{\sqrt{{3}}}~{}\int_{0}^{\infty}d{k}_{23}~{}k^{2}_{23}~{}{E_{23}~{}E({\bf
k})\over k^{+}}~{}2~{}\sin{\varphi\over 2}~{}\cos{\varphi\over 2}~{}{\cal
H}^{\tau}(0,1,k_{23},k)~{},$ (140) $\displaystyle b^{\tau(2)}_{3,{\cal
M}}\left(x,|{\bf k}_{\perp}|\right)=~{}{(-1)^{{\cal M}+1/2}\over
2\pi~{}(1-x)}~{}{\sqrt{{3}}}~{}\int_{0}^{\infty}d{k}_{23}~{}k^{2}_{23}~{}{E_{23}~{}E({\bf
k})\over k^{+}}~{}\sqrt{{1\over 2}}~{}{\cal H}^{\tau}(2,1,k_{23},k)$
$\displaystyle\times~{}\left\\{~{}2~{}\sin{\varphi\over 2}~{}\cos{\varphi\over
2}~{}-~{}{3}~{}{1\over
k^{2}}~{}\left[k_{\perp}~{}k_{z}\left(\cos^{2}{\varphi\over
2}-\sin^{2}{\varphi\over 2}\right)~{}+~{}k_{z}^{2}~{}2~{}\sin{\varphi\over
2}~{}\cos{\varphi\over 2}\right]\right\\}~{},$ (141) $\displaystyle
b^{\tau(0)}_{4,{\cal M}}\left(x,|{\bf k}_{\perp}|\right)=-b^{\tau(0)}_{3,{\cal
M}}\left(x,|{\bf k}_{\perp}|\right)~{},$ (142) $\displaystyle
b^{\tau(2)}_{4,{\cal M}}\left(x,|{\bf k}_{\perp}|\right)=~{}-~{}{(-1)^{{\cal
M}+1/2}\over
2\pi~{}(1-x)}~{}{\sqrt{{3}}}\int_{0}^{\infty}d{k}_{23}~{}k^{2}_{23}~{}{E_{23}~{}E({\bf
k})\over k^{+}}~{}\sqrt{{1\over 2}}~{}{\cal H}^{\tau}(2,1,k_{23},k)$
$\displaystyle\times~{}\left\\{~{}2~{}\sin{\varphi\over 2}~{}\cos{\varphi\over
2}~{}+~{}{3}~{}{1\over
k^{2}}~{}\left[k_{\perp}~{}k_{z}\left(\cos^{2}{\varphi\over
2}-\sin^{2}{\varphi\over 2}\right)-k^{2}_{\perp}~{}2~{}\sin{\varphi\over
2}~{}\cos{\varphi\over 2}\right]\right\\}~{},$ (143) $\displaystyle
b^{\tau(0)}_{5,{\cal M}}\left(x,|{\bf k}_{\perp}|\right)=b^{\tau(0)}_{2,{\cal
M}}\left(x,|{\bf k}_{\perp}|\right)~{},$ (144) $\displaystyle
b^{\tau(2)}_{5,{\cal M}}\left(x,|{\bf k}_{\perp}|\right)=~{}-~{}{(-1)^{{\cal
M}+1/2}\over
2\pi~{}(1-x)}~{}\sqrt{{3}}\int_{0}^{\infty}d{k}_{23}~{}k^{2}_{23}~{}{E_{23}~{}E({\bf
k})\over k^{+}}~{}\sqrt{{1\over 2}}~{}{\cal H}^{\tau}(2,1,k_{23},k)$
$\displaystyle\times\left\\{~{}2~{}\sin^{2}{\varphi\over 2}~{}+~{}3~{}{1\over
k^{2}}\left[\left(\cos^{2}{\varphi\over 2}-\sin^{2}{\varphi\over
2}\right)~{}k_{z}^{2}-2~{}\sin{\varphi\over 2}~{}\cos{\varphi\over
2}~{}k_{\perp}~{}k_{z}\right]\right\\}~{},$ (145)

where the dependence upon $\varphi$ is generated by the Melosh rotations (see
Appendix C). It has to be pointed out that, in the case of a three-nucleon
bound system, $\varphi$ is small, as one can deduce from its definition in Eq.
(154), and therefore $sin(\varphi/2)/cos(\varphi/2)<<1$.

One can immediately recognize that the quantities $b^{\tau}_{i,{\cal M}}$
actually are invariant for rotations around the $z$ axis, while they do depend
on $|{\bf k}_{\perp}|$ and $x$. The quantity $b^{\tau}_{0}$ is independent of
$\cal M$. For $i=1,...,5$, the dependence on $\cal M$ is through the factor
$(-1)^{{\cal M}+1/2}$.

###  B.5 Effective polarizations

Using the equations (see Appendix B of [1])

$\displaystyle{dx\over dk^{+}}={1-x\over E_{23}}~{},$
$\displaystyle{dk^{+}\over dk_{z}}={k^{+}\over E({\bf k})}~{},$ (146)

the longitudinal and transverse effective polarizations of Eqs. (92) and (93),
respectively, are given by

$\displaystyle p^{\tau}_{||}={(-1)^{{\cal
M}+1/2}}~{}{\sqrt{{3}}}\int_{0}^{\infty}d{k}_{23}~{}k^{2}_{23}~{}\int_{0}^{\infty}{k}^{2}~{}d{k}$
$\displaystyle\times~{}\int_{-1}^{1}d\cos\theta~{}f_{||}({\bf k},k_{23})~{},$
(147)

and

$\displaystyle p^{\tau}_{\perp}={(-1)^{{\cal
M}+1/2}}~{}{\sqrt{{3}}}\int_{0}^{\infty}d{k}_{23}~{}k^{2}_{23}~{}\int_{0}^{\infty}{k}^{2}~{}d{k}$
$\displaystyle\times~{}\int_{-1}^{1}d\cos\theta~{}f_{\perp}({\bf k},k_{23})$
(148)

where

$\displaystyle f_{||}({\bf k},k_{23})={\cal
H}^{\tau}(0,1,k_{23},k)\left(1-2~{}\sin^{2}{\varphi\over
2}\right)+\sqrt{{1\over 2}}$ $\displaystyle\times~{}{\cal
H}^{\tau}(2,1,k_{23},k)\left\\{1-~{}2~{}\sin^{2}{\varphi\over
2}~{}-~{}3~{}\left[\left(\cos^{2}{\varphi\over 2}\right.\right.\right.$
$\displaystyle\left.\left.\left.-\sin^{2}{\varphi\over
2}\right){\cos^{2}\theta}~{}-2\sin{\varphi\over 2}\cos{\varphi\over
2}{\sin\theta}{\cos\theta}\right]\right\\}~{}$ (149)

and

$\displaystyle f_{\perp}({\bf k},k_{23})={\cal
H}^{\tau}(0,1,k_{23},k)\left(1-\sin^{2}{\varphi\over
2}\right)+~{}\sqrt{{1\over 2}}$ $\displaystyle\times~{}{\cal
H}^{\tau}(2,1,k_{23},k)\left\\{1-\sin^{2}{\varphi\over
2}~{}-~{}\frac{3}{2}~{}\left[\left(\cos^{2}{\varphi\over
2}\right.\right.\right.$
$\displaystyle\left.\left.\left.-\sin^{2}{\varphi\over
2}\right)\sin^{2}\theta+2\sin{\varphi\over 2}\cos{\varphi\over
2}\sin\theta\cos\theta\right]\right\\}~{}.$ (150)

In Eqs. (149) and (150) we have defined ${\cos\theta}=k_{z}/k$ and
$\sin\theta=k_{\perp}/k$.

Let us emphasize that without the effect of the Melosh rotations one has
$\sin{\varphi\over 2}=0$ and $\cos{\varphi\over 2}=1$. Then the two
polarizations become equal, viz

$\displaystyle p^{\tau}_{||}=p^{\tau}_{\perp}={(-1)^{{\cal
M}+1/2}}~{}2~{}{\sqrt{{3}}}$
$\displaystyle\times~{}\int_{0}^{\infty}d{k}_{23}~{}k^{2}_{23}~{}\int_{0}^{\infty}{k}^{2}~{}d{k}~{}{\cal
H}^{\tau}(0,1,k_{23},k)$ (151)

##  Appendix C Products of Melosh and Pauli matrixes

The Melosh matrix

$\displaystyle D^{{1\over 2}}[{\cal R}_{M}({\tilde{\bf
k}})]_{\sigma\sigma^{\prime}}=\chi^{\dagger}_{\sigma}{m+k^{+}-\imath{\mbox{\boldmath{$\sigma$}}}\cdot(\hat{z}\times{\bf
k}_{\perp})\over\sqrt{\left(m+k^{+}\right)^{2}+|{\bf
k}_{\perp}|^{2}}}\chi_{\sigma^{\prime}}$ (152)

can be rewritten as follows

$\displaystyle D^{{1\over 2}}[{\cal R}_{M}({\tilde{\bf
k}})]_{\sigma\sigma^{\prime}}=\left[\cos{\varphi\over
2}~{}+~{}i~{}\sin{\varphi\over 2}~{}\hat{\bf
n}\cdot{\mbox{\boldmath{$\sigma$}}}\right]_{\sigma\sigma^{\prime}}~{},$ (153)

where

$\displaystyle\varphi=2~{}arctg~{}{|{\bf k}_{\perp}|\over k^{+}+m}~{},$
$\displaystyle\cos{\varphi\over 2}={k^{+}+m\over\sqrt{(k^{+}+m)^{2}+{\bf
k}^{2}_{\perp}}}~{},$ $\displaystyle\sin{\varphi\over 2}={|{\bf
k}_{\perp}|\over\sqrt{(k^{+}+m)^{2}+{\bf k}^{2}_{\perp}}}~{},$ (154)

and

$\displaystyle\hat{\bf n}=-{\hat{z}\times{\bf k}_{\perp}\over{|{\bf
k}_{\perp}|}}~{}.$ (155)

Then the equality

$\displaystyle D^{{1\over 2}*}[{\cal R}_{M}({k^{+},{\bf
k}_{\perp}})]_{\sigma\sigma^{\prime}}=D^{{1\over 2}}[{\cal R}_{M}({k^{+},-{\bf
k}_{\perp}})]_{\sigma^{\prime}\sigma}$ $\displaystyle=\left[\cos{\varphi\over
2}~{}-~{}i~{}\sin{\varphi\over 2}~{}\hat{\bf
n}\cdot{\mbox{\boldmath{$\sigma$}}}\right]_{\sigma^{\prime}\sigma}$ (156)

holds.

Let us now evaluate the sandwich of
$\left[{\mbox{\boldmath{$\sigma$}}}\cdot\hat{\bf e}\right]$ between two Melosh
matrixes (see Eqs. (127),(129), (131)), with $\hat{\bf e}$ a unit vector. One
gets

$\displaystyle{\cal D}_{\sigma\sigma^{\prime}}({k^{+},{\bf
k}_{\perp}},\hat{\bf
e})=\sum_{\sigma^{\prime}_{1}{\tilde{\sigma}^{\prime}_{1}}}~{}D^{{1\over
2}}[{\cal R}_{M}({k^{+},{\bf
k}_{\perp}})]_{\sigma\sigma^{\prime}_{1}}~{}~{}\left[{\mbox{\boldmath{$\sigma$}}}\cdot{\hat{\bf
e}}\right]_{\sigma^{\prime}_{1}{\tilde{\sigma}}^{\prime}_{1}}$
$\displaystyle\times~{}D^{{1\over 2}}[{\cal R}_{M}({k^{+},-{\bf
k}_{\perp}})]_{{\tilde{\sigma}}^{\prime}_{1}\sigma^{\prime}}=\left[\cos^{2}{\varphi\over
2}-\sin^{2}{\varphi\over
2}\right]~{}\left[{\mbox{\boldmath{$\sigma$}}}\cdot{\hat{\bf
e}}\right]_{{\sigma}\sigma^{\prime}}$ $\displaystyle-~{}2~{}\sin{\varphi\over
2}~{}\cos{\varphi\over 2}~{}\left\\{[(\hat{\bf e}\cdot\hat{\bf
k}_{\perp})~{}{\hat{z}}-(\hat{\bf e}\cdot{\hat{z}})~{}\hat{\bf
k}_{\perp}]\cdot{\mbox{\boldmath{$\sigma$}}}\right\\}_{{\sigma}\sigma^{\prime}}$
$\displaystyle+~{}2~{}\sin^{2}{\varphi\over 2}~{}\left\\{\left(\hat{\bf
n}\cdot\hat{\bf e}\right)~{}(\hat{\bf
n}\cdot{\mbox{\boldmath{$\sigma$}}})~{}\right\\}_{{\sigma}\sigma^{\prime}}~{}.$
(157)

For $\hat{\bf e}={\bf S}$ one has

$\displaystyle{\cal D}_{\sigma\sigma^{\prime}}({k^{+},{\bf k}_{\perp}},{\bf
S})~{}=~{}\left[{\mbox{\boldmath{$\sigma$}}}\cdot{{\bf
S}}\right]_{{\sigma}\sigma^{\prime}}~{}$
$\displaystyle-~{}2~{}\sin{\varphi\over 2}~{}\cos{\varphi\over
2}~{}\left\\{\left[\left({\bf S}\cdot\hat{\bf
k}_{\perp}\right)~{}{\hat{z}}-\left({\bf S}\cdot{\hat{z}}\right)~{}{\hat{\bf
k}_{\perp}}\right]\cdot{\mbox{\boldmath{$\sigma$}}}\right\\}_{{\sigma}\sigma^{\prime}}$
$\displaystyle-~{}2~{}\sin^{2}{\varphi\over 2}~{}\left\\{\left[\left({\bf
S}\cdot{\hat{z}}\right)~{}{\hat{z}}+\left({\bf S}\cdot\hat{\bf
k}_{\perp}\right)~{}{\hat{\bf
k}_{\perp}}\right]\cdot{\mbox{\boldmath{$\sigma$}}}~{}\right\\}_{{\sigma}\sigma^{\prime}}~{},$
(158)

For ${\hat{\bf e}}=\hat{\bf k}$, one obtains

$\displaystyle{\cal D}_{\sigma\sigma^{\prime}}({k^{+},{\bf
k}_{\perp}},\hat{\bf k})=~{}\left[\cos^{2}{\varphi\over
2}-\sin^{2}{\varphi\over
2}\right]~{}\left[{\mbox{\boldmath{$\sigma$}}}\cdot{\hat{\bf
k}}\right]_{{\sigma}\sigma^{\prime}}~{}$
$\displaystyle-~{}2~{}\sin{\varphi\over 2}~{}\cos{\varphi\over 2}~{}{1\over
k}\left[[{k}_{\perp}~{}{\hat{z}}-{k}_{z}~{}{\hat{\bf
k}}_{\perp}]\cdot{\mbox{\boldmath{$\sigma$}}}\right]_{{\sigma}\sigma^{\prime}}~{},$
(159)

where $~{}{{k}}_{\perp}=\sqrt{k_{x}^{2}+k_{y}^{2}}~{}$ and $~{}k_{z}={\bf
k}\cdot{\hat{z}}~{}={1\over 2}(k^{+}-k^{-})={1\over 2}(k^{+}-{{m^{2}+{\bf
k}^{2}_{\perp}}\over k^{+}})$.

##  Appendix D Traces of the Valence Correlator and of the light-front
Spectral Function

Let us denote by $\Gamma$ a generic $4\times 4$ matrix. The traces of
$\left[\Gamma~{}\Phi_{V}\right]$ can be expressed through traces of the
spectral function or of the spectral function times $\sigma$ matrixes. Indeed
with the help of Eqs. (40) and (41) one has