QCD Factorization for Deep-Inelastic Scattering At Large Bjorken $x_B \sim 1-{\cal O}(Λ_{\rm QCD}/Q$

TL;DR

This paper analyzes deep-inelastic scattering in the end-point regime where $(1-x)Q\sim\Lambda_{\rm QCD}$ and demonstrates that standard factorization remains valid to leading order in $1-x$. It develops an effective-field-theory (SCET) refactorization that separates physics at $Q^2$ from the lower scale $(1-x)Q^2$, enabling systematic resummation of large logarithms via RG evolution and a clean interpretation of jet, soft, and parton-distribution components. It then critiques Sterman-style factorization for introducing a spurious scale $(1-x)Q$ in end-point kinematics and presents a regulator-based approach with a rapidity cutoff that reproduces the EFT factorization and clarifies soft-subtraction requirements. The results provide a coherent framework that unifies SCET-based refactorization with generalized Sterman-type factorization and offers practical avenues for controlling end-point logarithms in DIS analyses.

Abstract

We study deep-inelastic scattering factorization on a nucleon in the end-point regime $x_B \sim 1-{\cal O}(Λ_{\rm QCD}/Q)$ where the traditional operator product expansion is supposed to fail. We argue, nevertheless, that the standard result holds to leading order in $1-x_B$ due to the absence of the scale dependence on $(1-x_B)Q$. Refactorization of the scale $(1-x_B)Q^2$ in the coefficient function can be made in the soft-collinear effective theory and remains valid in the end-point regime. On the other hand, the traditional refactorization approach introduces the spurious scale $(1-x_B)Q$ in various factors, which drives them nonperturbative in the region of our interest. We show how to improve the situation by introducing a rapidity cut-off scheme, and how to recover the effective theory refactorization by choosing appropriately the cut-off parameter. Through a one-loop calculation, we demonstrate explicitly that the proper soft subtractions must be made in the collinear matrix elements to avoid double counting.

Figures (1)

• Figure 1: The leading reduced diagram contributing to the deep-inelastic structure function in $x\rightarrow 1$ regime.