Three-loop color-kinematics duality: A 24-dimensional solution space induced by new generalized gauge transformations

TL;DR

This paper computes three-loop form factors in N=4 SYM using color-kinematics duality, revealing a large solution space with 24 free parameters due to new gauge transformations, and confirms their infrared properties.

Contribution

It introduces a new class of generalized gauge transformations that generate a 24-dimensional solution space for three-loop form factors in N=4 SYM.

Findings

Full-color three-loop form factors verified by unitarity cuts

Presence of 24 free parameters due to generalized gauge transformations

Numerical results match known infrared divergences and planar remainders

Abstract

We obtain full-color three-loop three-point form factors of the stress-tensor supermultiplet and also of a length-3 half-BPS operator in N=4 SYM based on the color-kinematics duality and on-shell unitarity. The integrand results are verified by all planar and non-planar unitarity cuts, and they satisfy the minimal power-counting of loop momenta and diagrammatic symmetries. Interestingly, these three-loop solutions, while manifesting all dual Jacobi relations, contain a large number of free parameters; in particular, there are 24 free parameters for the form factor of stress-tensor supermultiplet. Such degrees of freedom are due to a new type of generalized gauge transformation associated with the operator insertion for form factors. We also perform numerical integration and obtain consistent full-color infrared divergences and the known planar remainder. The form factors we obtain can be understood as the N=4 SYM counterparts of three-loop Higgs plus three-gluon amplitudes in QCD and are expected to provide the maximally transcendental parts of the latter.