  We review recent theoretical developments concerning the definition and the
renormalization of equal-time correlators that can be computed on the lattice
and related to Parton Distribution Functions (PDFs) through a factorization
formula. We show how these objects can be studied and analyzed within the
framework of a nongauge theory, gaining insight through a one-loop computation.
We use scalar field theory as a playground to revise, analyze and present the
main features of these ideas, to explore their potential, and to understand
their limitations for extracting PDFs. We then propose a framework that would
allow to include the available lattice QCD data in a global analysis to extract
PDFs.
