Parton distribution functions (PDFs) provide important information on
the quark and gluon structure of hadrons which, at leading twist, they
give the probability of finding a specific parton in the hadron
carrying certain momentum and spin, in the infinite momentum frame.
Due to the fact that PDFs are light-cone correlation functions, they
cannot be computed directly on a Euclidean lattice.
Recently, a novel direct approach has been proposed by X. Ji
suggesting that one can compute quasi-distribution functions, which are
accessible in Lattice QCD. This formalism provides a promising means
of studying quark distribution functions in nucleons, as for large
momenta, one can establish connection with the physical PDFs through
a matching procedure.
In this talk we will discuss aspect of this direct approach, with main
focus on the renormalization. In particular, we study gauge invariant
"Wilson line" operators of the type: $\bar\psi(x) \Gamma U(x,y)
\Psi(y)$, where $\bar\Psi$ and $\Psi$ are quark fields, $U(x,y)$ is a
path-ordered exponential of the gluon field and $\Gamma$ is a product
of Dirac gamma matrices.
The extended nature of these "Long-Link" operators results in a
nontrivial renormalization, including a finite factor, as well
as, contributions which diverge linearly and logarithmically with the
lattice spacing. We will discuss possible mixing of such operators
in the Lattice Regularization, as well as, alternative prescriptions
to extract the linear divergence.
