A simple approach to counterterms in N=8 supergravity

TL;DR

This paper introduces a straightforward method to analyze potential counterterms in N=8 supergravity by examining on-shell matrix elements and SUSY Ward identities, effectively excluding many candidates at various loop levels.

Contribution

It provides a systematic approach to determine the supersymmetrization of counterterms using on-shell matrix elements, avoiding complex operator details, and constructs explicit superamplitudes for allowed operators.

Findings

Excluded R^n, D^2 R^n, D^4 R^n, D^6 R^n as counterterms for n>4 at certain levels

Constructed explicit superamplitude for 7-loop D^4 R^6 operator

Identified conditions under which counterterms are allowed or excluded

Abstract

We present a simple systematic method to study candidate counterterms in N=8 supergravity. Complicated details of the counterterm operators are avoided because we work with the on-shell matrix elements they produce. All n-point matrix elements of an independent SUSY invariant operator of the form D^{2k} R^n +... must be local and satisfy SUSY Ward identities. These are strong constraints, and we test directly whether or not matrix elements with these properties can be constructed. If not, then the operator does not have a supersymmetrization, and it is excluded as a potential counterterm. For n>4, we find that R^n, D^2 R^n, D^4 R^n, and D^6 R^n are excluded as counterterms of MHV amplitudes, while only R^n and D^2 R^n are excluded at the NMHV level. As a consequence, for loop order L<7, there are no independent D^{2k}R^n counterterms with n>4. If an operator is not ruled out, our method constructs an explicit superamplitude for its matrix elements. This is done for the 7-loop D^4 R^6 operator at the NMHV level and in other cases. We also initiate the study of counterterms without leading pure-graviton matrix elements, which can occur beyond the MHV level. The landscape of excluded/allowed candidate counterterms is summarized in a colorful chart.