Three-loop cusp anomalous dimension and a conjecture for $n$ loops

TL;DR

This paper derives explicit three-loop expressions for the cusp anomalous dimension in QCD, explores relations between different loop orders, and conjectures a general formula for n loops, with applications to top-quark production.

Contribution

It provides the first analytical three-loop results for the cusp anomalous dimension and proposes a conjecture for the n-loop generalization based on lower-loop data.

Findings

Explicit three-loop cusp anomalous dimension expressions in QCD.

Relations observed between results at different loop orders.

Numerical results and approximate formulas for top-quark production.

Abstract

I present analytical expressions for the cusp anomalous dimension in QCD through three loops in terms of elementary functions and ordinary polylogarithms. I observe interesting relations between the results at different loops and provide a conjecture for the $n$-loop cusp anomalous dimension in terms of the lower-loop results. I also present numerical results and simple approximate formulas for the cusp anomalous dimension relevant to top-quark production.