Tree Amplitudes and Two-loop Counterterms in D=11 Supergravity

TL;DR

Addressing whether $D=11$ SUGRA admits a local two-loop counterterm, the paper constructs the bosonic part of a linearized SUSY invariant by computing tree-level 4-point amplitudes and localizing the nonlocal pieces with a $stu$ factor. It computes explicit amplitudes in sectors $R^4$, $F^4$, $F^3R$, and $R^2F^2$, expressing the localized results in BR currents and $t_8$-type structures, yielding a 12-derivative $R^4$-type invariant. The two-loop pure gravity divergence coefficient matches the result of Bern et al., and the formion sector agrees with independent calculations, confirming the predicted nonrenormalizability of $D=11$ SUGRA and informing expectations for M-theory corrections. Collectively, the work demonstrates that no hidden symmetry obstructs such divergences and provides a concrete local invariant consistent with M-theory's slope expansion.

Abstract

We compute the tree level 4-particle bosonic scattering amplitudes in D=11 supergravity. By construction, they are part of a linearized supersymmetry-, coordinate- and 3-form gauge-invariant. While this on-shell invariant is nonlocal, suitable SUSY-preserving differentiations turn it into a local one with correct dimension to provide a natural lowest (two-loop) order counterterm candidate. Its existence shows explicitly that no symmetries protect this ultimate supergravity from the nonrenormalizability of its lower-dimensional counterparts.