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

TL;DR

The paper develops a systematic UT-integral framework to attack the four-loop Sudakov form factor in ${\mathcal N}=4$ SYM, including the nonplanar sector. It identifies UT candidates via residue tests and $d$Log criteria, then expresses the full form factor as a rational UT basis in both planar and nonplanar sectors. Numerically evaluating the nonplanar UT integrals with Mellin-Barnes representations and sector decomposition yields the nonplanar cusp and collinear anomalous dimensions, revealing a nonzero nonplanar CAD and a sign pattern for the nonplanar collinear AD that challenges planar intuition and quadratic Casimir scaling. The work also provides a detailed error analysis and shows that UT integrals simplify the numerics, offering a broadly applicable methodology for high-loop computations in gauge theories.

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.

Figures (6)

• Figure 1: Integral topologies that contribute only to the planar form factor at four loops.
• Figure 2: Sample integral topologies that contribute to the nonplanar form factor at four loops.
• Figure 3: Integral topologies that do not have dLog numerators.
• Figure 4: Example of rational IBP relations.
• Figure 5: Topologies for which it is straightforward to construct a $d$Log form.