Narrow Results

Deep inelastic scattering is one of the best place to study hadron structures. In this paper we consider the target fragmentation region deep inelastic scattering process at leading twist. The calculations are carried out by applying the collinear expansion. In the collinear expansion formalism the multiple gluon scattering is taken into account and gauge links are obtained systematically and automatically. Quantum chromodynamics is a non-Abelian gauge theory of strong interactions in which parity symmetry can be violated by the nontrivial $𝜃$-vacuum tunneling effects. As a result, the axial vector current is induced. By defining and decomposing the parity-odd correlator we calculate both the parity-even and parity-odd contributions to the cross-section of the target fragmentation region deep inelastic scattering. We also present the positivity bounds for these fracture functions.

This is intend to provide an overview of the theory and phenomenology parts of the TMD (Transverse Momentum Dependent parton distribution and fragmentation functions) studies. By comparing with the theoretical framework that we have for the inclusive deep inelastic lepton-nucleon scattering and the one-dimensional imaging of the nucleon, I try to outline what we need to do in order to construct a comprehensive theoretical framework for semi-inclusive reactions and the three dimensional imaging of the nucleon. After that, I try to give an overview of what we have already achieved and make an outlook for the future.

We applied collinear expansion to the semi-inclusive deeply inelastic lepton-nucleon (nucleus) scattering process $e+N(A)\to e+q+X$ with both polarized beam and polarized target up to twist-3, and unpolarized process up to twist-4. The differential cross section and azimuthal asymmetries are expressed in terms of gauge invariant twist-3 and twist-4 TMD parton distribution/correlation functions. Measurements of such azimuthal asymmetries provide methods to study different spin and transverse momentum aspects of the partonic structure of nucleon. We further study the nuclear dependence of azimuthal asymmetries and adopt Gaussian ansatz for TMD distribution/correlation functions to estimat the semi-quantitive behaviour of the nuclear dependence. We predict the A-dependence of azimuthal asymmetries which can be tested in the planned EIC’s.

In the two papers published recently,^{1, 2} we apply collinear expansion to both inclusive ($e^{+}{e}^{-}\to h+X$) and semi-inclusive ($e^{+}{e}^{-}\to h+\bar{q}+X$) hadron production in $e^{+}{e}^{-}$ annihilation to derive a formalism suitable for a systematic study of leading as well as higher twist contributions to fragmentation functions at the tree level. We carry out the calculations for hadrons with spin-0, spin-1/2 as well as spin-1. This proceeding is mainly a summary of these two papers.