5 loops in 24/5 dimensions

Abstract

A first quantised approach to loop amplitudes based on the pure spinor particle is applied to the systematics of four-particle amplitudes in maximally supersymmetric field theories. Counting of fermionic zero modes allows the identification of momentum factors multiplying \( {\mathcal{R}^4} \) in the case of supergravity (and F 4 in the Yang-Mills case) thereby making manifest their ultraviolet properties as a function of dimension, D. For L = 2, 3, 4 loops the leading supergravity divergence is in D = 4 + 6/L dimensions and proportional to \( {\partial^{2L}}\ {\mathcal{R}^4} \), in line with earlier field theory calculations. However, at five loops there is a radical change in the systematics, suggesting the presence of a contribution with an explicit L = 5 logarithmic ultraviolet divergence when D = 24/5 that is proportional to \( {\partial^8}{\mathcal{R}^4} \). We further argue that \( {\partial^8}{\mathcal{R}^4} \) should receive contributions from all loops, which would imply that \( \mathcal{N} = 8 \) supergravity (with D = 4) is not protected by supersymmetry from a seven-loop divergence.