Cancellation of divergences up to three loops in exceptional field theory

TL;DR

This paper computes and cancels divergences in a three-loop exceptional field theory diagram related to M-theory, revealing connections to BPS states and Eisenstein series in the low-energy effective action.

Contribution

It explicitly evaluates three-loop diagrams in exceptional field theory and expresses divergences as Eisenstein series, linking BPS state sums to supersymmetric couplings.

Findings

Divergences up to three loops are canceled in exceptional field theory.

Loop sums over BPS states can be expressed as Eisenstein series.

Results are consistent with expected BPS contributions in M-theory.

Abstract

We consider the tetrahedral three-loop diagram in $E_d$ exceptional field theory evaluated as a scalar diagram for four external gravitons. At lowest order in momenta, this diagram contributes to the $\nabla^6 R^4$ term in the low-energy effective action for M-theory. We evaluate explicitly the sums over the discrete exceptional field theory loop momenta that become sums over 1/2-BPS states in the compact exceptional space. These sums can be rewritten as Eisenstein series that solve the homogeneous differential equations that supersymmetry implies for the $\nabla^6 R^4$ coupling. We also show how our results, even though sums over 1/2-BPS states, are consistent with expected 1/4-BPS contributions to the couplings.