The Sudakov form factor at four loops in maximal super Yang-Mills theory

TL;DR

This paper computes the four-loop Sudakov form factor in maximally supersymmetric Yang-Mills theory, revealing new insights into its mathematical structure and anomalous dimensions, and demonstrating the breakdown of quadratic Casimir scaling.

Contribution

It introduces a basis of integrals with uniform transcendentality for four-loop calculations and analyzes nonplanar contributions using numerical methods, advancing understanding of Feynman integral properties.

Findings

Explicit four-loop form factor expression in a uniform transcendental basis

Numerical evaluation of nonplanar integrals using sector-decomposition and Mellin-Barnes methods

Demonstration of quadratic Casimir scaling violation at four loops

Abstract

The four-loop Sudakov form factor in maximal super Yang-Mills theory is analysed in detail. It is shown explicitly how to construct a basis of integrals that have a uniformly transcendental expansion in the dimensional regularisation parameter, further elucidating the number-theoretic properties of Feynman integrals. The physical form factor is expressed in this basis for arbitrary colour factor. In the nonplanar sector the required integrals are integrated numerically using a mix of sector-decomposition and Mellin-Barnes representation methods. Both the cusp as well as the collinear anomalous dimension are computed. The results show explicitly the violation of quadratic Casimir scaling at the four-loop order. A thorough analysis concerning the reliability of reported numerical uncertainties is carried out.