The Unpolarized Gluon Anomalous Dimension at Small x

TL;DR

The paper analyzes how resummation of small-x contributions to twist-2 anomalous dimensions, extended beyond NLO to NLx, affects gluon evolution within a renormalization-group framework. Using a Bethe-Salpeter equation with an infrared-finite kernel and Mellin-space diagonalization, it expresses the NLx gluon anomalous dimension $\gamma_{gg}(N,\alpha_s)$ and decomposes the NLx contribution into conformal and conformal-breaking parts with explicit forms. Numerical results show large negative NLx corrections to $P_{gg}$, driving it negative at small x, and the analysis stresses that less singular terms are equally important. The study also analyzes the exponent $\omega$ governing $s^{\omega}$ growth, revealing that perturbative estimates are highly sensitive to which NLx pieces are included and to running-coupling and nonperturbative effects. Overall, the work highlights substantial uncertainties in small-x gluon dynamics and the need to account for subleading terms when predicting high-energy QCD behavior relevant to HERA and related phenomena.

Abstract

We discuss the quantitative consequences of the resummation of the small-x contributions to the anomalous dimensions beyond next-to-leading order in alpha_s and up to next order in ln(1/x) (NLx) in a framework based on the renormalization group equations. We find large and negative effects leading to negative values for the {\sf total} splitting function P_{gg}(x,alpha_s) already for x \lsim 0.01 at Q^2 \simeq 20 GeV^2. Terms less singular than those under consideration turn out to be quantitatively as important and need to be included. We derive the effects of the conformal part of the NLx contributions to the anomalous dimensions and discuss the exponent omega describing the s^omega behavior of inclusive cross sections.