What exactly is a parton density?
John C. Collins
TL;DR
The paper investigates the precise operator definitions of parton densities in QCD, highlighting divergent pitfalls in naive formulations and the need for gauge-invariant, renormalized constructions. It analyzes integrated and unintegrated (transverse-momentum dependent) PDFs, detailing UV and light-cone gauge divergences, and surveys strategies to regulate them, including non-light-like Wilson lines and generalized renormalization for light-like Wilson lines. The work emphasizes the Collins-Soper-Sterman framework, the relation between unintegrated and integrated PDFs, and the ongoing quest for a universal, rigorous foundation that connects perturbative factorization to non-perturbative hadron structure. These developments are essential for reliable QCD predictions, accurate PDF evolution, and robust interfaces to non-perturbative models and Monte Carlo event generators.
Abstract
I give an account of the definitions of parton densities, both the conventional ones, integrated over parton transverse momentum, and unintegrated transverse-momentum-dependent densities. The aim is to get a precise and correct definition of a parton density as the target expectation value of a suitable quantum mechanical operator, so that a clear connection to non-perturbative QCD is provided. Starting from the intuitive ideas in the parton model that predate QCD, we will see how the simplest operator definitions suffer from divergences. Corrections to the definition are needed to eliminate the divergences. An improved definition of unintegrated parton densities is proposed.