### Speaker

Dr
Giovanni Salme'
(Istituto Nazionale di Fisica Nucleare -)

### Description

The formalism based on the Poincare' covariant spin-dependent spectral function, recently introduced [1] within the light-front (LF) Hamiltonian dynamics, allows one to develop a
phenomenological tool for investigating three-fermion bound systems, in particular their dynamical observables. As it
is well-known, several and quite
relevant efforts are underway at JLAB with the aim of getting information
on both the nucleon transverse-momentum
distributions [2] and
$^{\rm 3}$He and $^{\rm 3}$H structure functions [3].
In this contribution, it will be presented
some new relations among the six T-even leading-twist transverse-momentum distributions
(TMDs) of a three-fermion system, e.g. the nucleon in valence approximation. Such relations can be formally obtained
once the valence contribution to the correlator for a J=1/2 bound system
is expressed in terms of the spin-dependent
spectral function, whose diagonal terms yield the probability distributions to
find a constituent with given spin and LF momentum. Moreover, within the same
framework,
some preliminary results of the EMC effect
for the three-nucleon system will be also provided.
The spin-dependent spectral function is defined through the overlaps between the ground state wave function of the three-body system and the states given by a Cartesian product of a plane wave of LF momentum $\tilde{ \kappa}$ (describing the struck particle in the intrinsic reference frame of the cluster [1,(23)]) and a state describing the intrinsic motion of
a fully interacting two-particle spectator subsystem. As an example of the
rich wealth of information one can extract and check against the experimental
outcomes, one could mention
the decomposition of the spin-dependent momentum distributions, obtained
after properly integrating the spectral
function on the relative intrinsic momentum $k_{23}$ of the interacting spectator pair, viz
$$
n^\tau_{\sigma \sigma '}(x,{\bf k}_{\perp};{\cal M},{\bf S}) ~
=
~ {2 (-1)^{{\cal M} + 1/2} \over (1- x)}~
\int d { k}_{23}~
%k^2_{23} ~{E_{23}~E({\bf k_1})\over k^+_1}~
\left \{ ~{\cal Z}_{\sigma \sigma'}(k_{23}, L=0) ~ + ~
{\cal Z}_{\sigma \sigma'}(k_{23}, L=2) ~ \right \}
~
$$
where $L$ is the orbital angular momentum of the one-body off-diagonal density matrix. By exploiting a similar result,
but for the T-even leading twist TMDS, it will be shown that
the linear relations proposed in Ref. [4] between the transverse parton distributions, i.e.
$$ \Delta f(x, |{\bf p}_{\perp}|^2 )= \Delta'_T f(x, |{\bf p}_{\perp}|^2 ) ~ + ~ {|{\bf p}_{\perp}|^2 \over 2 M^2}~ h^{\perp}_{1T}(x, |{\bf p}_{\perp}|^2 ) \quad \quad
g_{1T}(x, |{\bf p}_{\perp}|^2 ) = - h^{\perp}_{1L}(x, |{\bf p}_{\perp}|^2 )
$$
exactly hold in valence approximation when the contribution to the transverse momentum distributions from the angular momentum $L=2$ is absent.
**References**
[1] A. Del Dotto, E. Pace, G. Salme', S. Scopetta, Phys. Rev. C **95**, 014001 (2017).
[2] Hall A,
H. Gao et al, PR12-09-014: Target Single Spin
Asymmetry in Semi-Inclusive Deep-Inelastic $(e, e' \pi^\pm)$
Reaction on a Transversely Polarized $^{3}$He Target;
J.P. Chen et al, PR12-11-007:
``Asymmetries in Semi-Inclusive Deep-Inelastic $(e, e' \pi^\pm)$ Reactions
on a Longitudinally Polarized $^{3}$He Target''.
[3] MARATHON Coll., P. Gerassimov et al E12-10-103:
``MeAsurement of the $F_{2n}/F_{2p}$, d/u Ratios and A=3 EMC Effect in Deep
Inelastic Electron Scattering Off the Tritium and Helium MirrOr Nuclei''.
[4] R. Jacob, P.J. Mulders, and J. Rodrigues, Nucl. Phys.
A **626,** 937 (1997); B. Pasquini, S. Cazzaniga, S. Boffi,
Phys. Rev. D **78**, 034025 (2008); C. Lorce', B. Pasquini, Phys. Rev. D **84**, 034039 (2011).

### Primary author

Dr
Giovanni Salme'
(Istituto Nazionale di Fisica Nucleare -)

### Co-authors

Prof.
Emanuele Pace
(Rome Univ. "Tor Vergata" and INFN)
Prof.
Sergio Scopetta
(Perugia University)