5 loops in 24/5 dimensions

TL;DR

This paper investigates the ultraviolet divergences of maximally supersymmetric field theories using a pure spinor particle approach, revealing a potential five-loop divergence in 24/5 dimensions and implications for supergravity's divergence protection.

Contribution

It introduces a first quantised pure spinor particle method to systematically analyze loop amplitudes and divergences in supersymmetric theories, uncovering new divergence patterns at five loops.

Findings

Identifies momentum factors multiplying R^4 in supergravity and F^4 in Yang-Mills.

Shows leading divergences at 2-4 loops align with previous field theory results.

Predicts a five-loop divergence at 24/5 dimensions and suggests all loops contribute to certain divergences.

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 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 d*2L 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 d*8 R**4. We further argue that d*8 R**4 should receive contributions from all loops, which would imply that N=8 supergravity (with D=4) is not protected by supersymmetry from a seven-loop divergence.