The Four Loop QCD Rapidity Anomalous Dimension

TL;DR

This paper calculates the four-loop rapidity anomalous dimension in QCD by relating it to the soft anomalous dimension, using three-loop soft function calculations and recent perturbative data.

Contribution

It provides the first derivation of the four-loop rapidity anomalous dimension in QCD, extending previous three-loop results and clarifying its relation to the soft anomalous dimension.

Findings

Derived the four-loop rapidity anomalous dimension in QCD.

Calculated the three-loop fully differential soft function to  order.

Established the relation between rapidity and soft anomalous dimensions in QCD.

Abstract

The rapidity anomalous dimension controls the scaling of transverse momentum dependent observables in the Sudakov region. In a conformal theory it is equivalent to the soft anomalous dimension, but in QCD this relation is broken by anomalous terms proportional to the $\beta$-function. In this paper we first give a simple proof of this relation using two different representations of the energy-energy correlator observable. We then calculate the anomalous terms to three loops by computing the three-loop fully differential soft function to $\mathcal{O}(\epsilon)$. Combined with recent perturbative data from the study of on-shell form factors and splitting functions, this allows us to derive the four loop rapidity anomalous dimension in QCD.