Quantum Corrections to the Weizsäcker--Williams Distribution Function as Small x

TL;DR

The authors compute quantum corrections to the small-$x$ gluon distribution in a non-Abelian Weizsäcker–Williams background field generated by valence quarks in a large nucleus, deriving results valid to all orders in $α_s μ$. They extract the leading $ abla(1/x)$-type logarithms and show that loop corrections to the background field merely renormalize the coupling $g$ (and the background field) rather than introduce new singular terms. The analysis reveals that the effective perturbative expansion is governed by $α_s \ln(1/x)$, indicating the necessity of resumming leading logs at very small $x$. The central results are encapsulated in a general expression for quantum corrections to the distribution, with explicit perturbative corrections and a clear separation between fluctuation-induced effects and background renormalization.

Abstract

We compute the quantum corrections to the gluon distribution function in the background of a non--Abelian Weizsäcker--Williams field. These corrections are valid to all orders in the effective coupling $α_s μ$, where $μ^2$ denotes the average valence quark color charge squared per unit area. We find $\ln(1/x)$ logarithmic corrections to the classical gluon distribution function. The one loop corrections to the classical Weizsäcker-Williams field do not contribute to these singular terms in the distribution function. Their effect is to cause the running of $α_s$.