The three-loop form factor in N=4 super Yang-Mills

TL;DR

This work computes the Sudakov form factor in ${\cal N}=4$ SYM up to three loops, revealing a representation in which all loop integrals have uniform transcendentality and confirming the expected infrared exponentiation via cusp and collinear anomalous dimensions. By employing unitarity cuts in a manifestly supersymmetric framework, the authors derive the three-loop result in terms of UT integrals and show that the non-planar sector contributes at leading order in $N$, enriching the connection between form factors and amplitudes. They also demonstrate a leading-transcendentality correspondence with QCD form factors and uncover cancellations that improve ultraviolet behaviour, rendering the three-loop result finite in $D=16/3$ and unveiling a UV structure that is manifest in a special integral basis. The study highlights structural simplifications, such as UT bases and reparametrisation identities, suggesting deeper organizing principles for loop computations beyond planar ${\cal N}=4$ SYM and offering potential avenues for recursion and generalized form factors. Overall, the paper strengthens the link between form factors and amplitudes, clarifies IR/UV properties at three loops, and reinforces the special role of transcendentality in highly symmetric gauge theories.

Abstract

In this paper we study the Sudakov form factor in N=4 super Yang-Mills theory to the three-loop order. The latter is expressed in terms of planar and non-planar loop integrals. We show that it is possible to choose a representation in which each loop integral has uniform transcendentality. We verify analytically the expected exponentiation of the infrared divergences with the correct values of the three-loop cusp and collinear anomalous dimensions in dimensional regularisation. We find that the form factor in N=4 super Yang-Mills can be related to the leading transcendentality part of the quark and gluon form factors in QCD. We also study the ultraviolet properties of the form factor in D>4 dimensions, and find unexpected cancellations, resulting in an improved ultraviolet behaviour.