# What are the low-$Q$ and large-$x$ boundaries of collinear QCD factorization theorems?

## TL;DR

This paper investigates the limitations of collinear QCD factorization at low momentum transfer and high Bjorken-x, emphasizing the importance of parton virtuality and proposing the need for refined factorization theorems.

## Contribution

The study uses an idealized model to analyze the boundaries of factorization assumptions at low Q and large x, highlighting the role of parton virtuality in power corrections.

## Key findings

- Target mass and intrinsic transverse momentum are significant at low Q and large x.
- Parton virtuality is crucial for accurate power correction treatments.
- Existing factorization theorems may need modifications for moderate Q and high x regimes.

## Abstract

Familiar factorized descriptions of classic QCD processes such as deeply-inelastic scattering (DIS) apply in the limit of very large hard scales, much larger than nonperturbative mass scales and other nonperturbative physical properties like intrinsic transverse momentum. Since many interesting DIS studies occur at kinematic regions where the hard scale, $Q \sim$ 1-2 GeV, is not very much greater than the hadron masses involved, and the Bjorken scaling variable $x_{bj}$ is large, $x_{bj} \gtrsim 0.5$, it is important to examine the boundaries of the most basic factorization assumptions and assess whether improved starting points are needed. Using an idealized field-theoretic model that contains most of the essential elements that a factorization derivation must confront, we retrace the steps of factorization approximations and compare with calculations that keep all kinematics exact. We examine the relative importance of such quantities as the target mass, light quark masses, and intrinsic parton transverse momentum, and argue that a careful accounting of parton virtuality is essential for treating power corrections to collinear factorization. We use our observations to motivate searches for new or enhanced factorization theorems specifically designed to deal with moderately low-$Q$ and large-$x_{bj}$ physics.