The D^6 R^4 term from three loop maximal supergravity

TL;DR

This paper computes the moduli-dependent coefficient of the D^6 R^4 term in three-loop maximal supergravity, revealing its structure through integrals over Schwinger parameters and moduli space, and discusses its renormalization.

Contribution

It provides a compact expression for the D^6 R^4 term's coefficient in 11D supergravity, involving SL(3,Z) invariant integrals and addresses divergence renormalization.

Findings

The D^6 R^4 term is BPS and has a simple renormalized amplitude.

Only Mercedes skeleton diagrams contribute to the amplitude.

The coefficient involves an SL(3,Z) invariant integral over T^3 moduli.

Abstract

We consider the D^6 R^4 term which is the leading contribution in the low momentum expansion of the three loop, four graviton amplitude in maximal supergravity. We calculate the moduli dependent coefficient of this term in 11 dimensional supergravity compactified on T^2. Only diagrams that involve the Mercedes skeleton contribute resulting in a compact expression involving an integral over 6 Schwinger parameters. We express this integral as an integral over the moduli of an auxiliary T^3. This includes an SL(3,Z) invariant integral over the shape moduli of the T^3. We discuss the renormalization of the ultraviolet divergences of this amplitude that arise from the boundaries of moduli space. The renormalized amplitude is simple which is a consequence of the fact that the D^6 R^4 term is BPS.