Three-Loop Five-Point CK-Dual Amplitudes and UV Structure in N=4 SYM and N=8 SUGRA

Abstract

We construct the complete full-color three-loop five-point integrand of N=4 super-Yang--Mills theory in a representation that manifestly satisfies color--kinematics duality. Its double copy gives the corresponding N=8 supergravity integrand. For four-dimensional external states, we evaluate the ultraviolet poles of both amplitudes in the critical dimension, Dc=6. Extending the external-state dependence to D dimensions is subtle. We consider a candidate replacement of the four-dimensional prefactors by expressions built from D-dimensional tree amplitudes. It reproduces the open-string prediction for the SYM pole with generic D-dimensional external states, whereas the corresponding gravity expression differs from the string-inspired one by an evanescent term.