5 loops in 24/5 dimensions

TL;DR

Using a pure spinor quantum-mechanical framework, the authors count fermionic zero modes to determine the momentum factors multiplying R^4 and F^4 in four-particle amplitudes of maximal SUSY theories. They reproduce known results for L≤4 and identify a qualitative change at L=5 where a ∂^8 R^4 contribution in D=24/5 emerges, suggesting ∂^8 R^4 receives contributions from all loops. They argue that this interaction is unprotected by supersymmetry, which would allow a seven-loop divergence in D=4 N=8 supergravity, challenging perturbative finiteness. The work aligns with string theory expectations and provides a unifying view of UV structures across loops.

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.

Figures (7)

• Figure 1: The unique two-loop skeleton diagram. The amplitude is obtained by attaching vertex operators to points on the lines, which are integrated around the diagram. The circular arrows denote the different $b_I$-cycles. The propagators in the skeleton are numbered from 1 to 3 and the arrows on each line indicate the direction of increasing proper time along the line.
• Figure 2: (a) The three-loop ladder skeleton. (b) The "Mercedes" skeleton.
• Figure 3: A three-loop diagram with one contact term that arises in maximal supergravity and in maximal Yang--Mills. While its contribution makes no qualitative change to the leading behaviour of the supergravity amplitude, in the Yang--Mills case its presence is responsible for the leading behaviour, $\partial^2\, {\mathrm{Tr}} F^4$.
• Figure 4: The five four-loop skeleton diagrams.
• Figure 5: A four-loop diagram with two contact terms that gives the leading behaviour of $\partial^2\, {\mathrm{Tr}} F^4$.