N=4 SUSY Yang--Mills: three loops made simple(r)

TL;DR

This work leverages ${{{\rm N}=4}}$ SYM to illuminate the structure of multi-loop QCD anomalous dimensions by formulating a reciprocity-respecting evolution equation (RRE) with a universal kernel ${\cal P}$. Through harmonic-sum representations and analytic construction, the authors derive ${\cal P}_1, {\cal P}_2, {\cal P}_3$ and show that ${\cal P}_3$ is compact and obeys Gribov--Lipatov reciprocity, while the space- and time-like anomalous dimensions ${\gamma}^{(T/S)}$ are generated perturbatively via ${\gamma} = {\cal P} + \sigma {\cal P}\dot{ } + \tfrac{1}{2}{\cal P}^2 \ddot{ } + \cdots$. They demonstrate how higher-order contributions are inherited from lower orders, with double-logarithmic structures and negative-index harmonic sums tied to crossing and non-planar effects, and they discuss the implications for BFKL-like limits and small- and large-$x$ behavior. The results provide a compact, GL-consistent framework that could simplify the dominant part of multi-loop QCD anomalous dimensions by exploiting universal soft-gluon radiation physics, potentially connecting to LBK principles and integrability in ${{{ m N}=4}}$ SYM. A contemporaneous note links the RRE to conformal symmetry results, reinforcing the fundamental origin of reciprocity in this context.

Abstract

We construct universal parton evolution equation that produces space- and time-like anomalous dimensions for the maximally super-symmetric N=4 Yang--Mills field theory model, and find that its kernel satisfies the Gribov--Lipatov reciprocity relation in three loops. Given a simple structure of the evolution kernel, this should help to generate the major part of multi-loop contributions to QCD anomalous dimensions, due to classical soft gluon radiation effects.

Figures (1)

• Figure 1: On the origin of negative index harmonic sums