Simplifying Multiloop Integrands and Ultraviolet Divergences of Gauge Theory and Gravity Amplitudes

TL;DR

This paper leverages the color-kinematics duality to simplify four-loop amplitudes in N=4 super-Yang-Mills and N=8 supergravity, revealing ultraviolet divergences and supporting the duality's validity at loop level.

Contribution

It provides a simplified construction of four-loop amplitudes using duality and double-copy, and uncovers the ultraviolet divergence structure in supergravity.

Findings

Amplitude integrand determined by two planar graphs

Ultraviolet divergence in supergravity matches specific vacuum integrals

Ultraviolet power counting aligns between supergravity and gauge theory

Abstract

We use the duality between color and kinematics to simplify the construction of the complete four-loop four-point amplitude of N=4 super-Yang-Mills theory, including the nonplanar contributions. The duality completely determines the amplitude's integrand in terms of just two planar graphs. The existence of a manifestly dual gauge-theory amplitude trivializes the construction of the corresponding N=8 supergravity integrand, whose graph numerators are double copies (squares) of the N=4 super-Yang-Mills numerators. The success of this procedure provides further nontrivial evidence that the duality and double-copy properties hold at loop level. The new form of the four-loop four-point supergravity amplitude makes manifest the same ultraviolet power counting as the corresponding N=4 super-Yang-Mills amplitude. We determine the amplitude's ultraviolet pole in the critical dimension of D=11/2, the same dimension as for N=4 super-Yang-Mills theory. Strikingly, exactly the same combination of vacuum integrals (after simplification) describes the ultraviolet divergence of N=8 supergravity as the subleading-in-1/N_c^2 single-trace divergence in N=4 super-Yang-Mills theory.