What is the Simplest Quantum Field Theory?

TL;DR

This paper argues that N=8 supergravity has the simplest interacting S-matrix in 4D, with amplitudes exhibiting remarkable properties like vanishing at infinite momentum and being determined by leading singularities, suggesting potential finiteness.

Contribution

It demonstrates that N=8 SUGRA's scattering amplitudes are uniquely simple and can be fully characterized by recursion relations and symmetry considerations, challenging traditional notions of simplicity in field theories.

Findings

All tree amplitudes in N=4 SYM and N=8 SUGRA vanish at infinite complex momentum.

One-loop amplitudes in N=8 SUGRA contain only scalar box integrals, no triangles or bubbles.

Amplitudes are conjectured to be determined by leading singularities, implying perturbative finiteness.

Abstract

Conventional wisdom says that the simpler the Lagrangian of a theory the simpler its perturbation theory, but an increased understanding of the structure of the S-matrix in gauge theories and gravity has been pointing to the opposite conclusion. In this paper we suggest that N=8 SUGRA has the simplest interacting S-matrix in 4D. Using Grassmann coherent states for external particles shows that amplitudes with maximal SUSY are smooth objects, with the action of SUSY manifest. We show that all tree amplitudes in N=4 SYM and N=8 SUGRA vanish at (supersymmetric) infinite complex momentum, and can thus be determined by recursion relations. We also identify the action of the non-linearly realized E_{7(7)} symmetry of N=8 SUGRA on scattering amplitudes. We give a simple discussion of the structure of 1-loop amplitudes in any QFT, in close parallel to recent work of Forde, showing that the coefficients of scalar "triangle" and "bubble" integrals are determined by the "pole at infinite momentum" of tree amplitude products appearing in cuts. The on-shell superspace for maximal SUSY makes it easy to compute the multiplet sums that arise in these cuts, leading to a simple proof of the absence of triangles and bubbles at 1-loop. We also argue that rational terms are absent. This establishes the recent conjecture that 1-loop amplitudes in N=8 SUGRA have only scalar box integrals, just as N=4 SYM. It is natural to conjecture that with maximal SUSY, amplitudes are completely determined by their leading singularities even beyond tree- and 1-loop level; this would directly imply the perturbative finiteness of N=8 SUGRA. The remarkable properties of scattering amplitudes call for an explanation in terms of a "weak-weak" dual formulation of QFT, a holographic dual of flat space.