What exactly is a parton density?

TL;DR

This paper clarifies the precise quantum mechanical definitions of parton densities, both integrated and unintegrated, addressing divergences and proposing improved definitions to connect with non-perturbative QCD.

Contribution

It provides a rigorous, divergence-free definition of parton densities as quantum operators, enhancing the theoretical foundation of parton distribution functions.

Findings

Identifies divergences in naive operator definitions

Proposes corrected definitions for unintegrated densities

Establishes a clear link to non-perturbative QCD

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.