What is the Simplest Quantum Field Theory?

TL;DR

The paper argues that in four dimensions, theories with maximal supersymmetry—${\cal N}=4$ SYM and ${\cal N}=8$ SUGRA—exhibit the simplest S-matrices, with tree amplitudes fully determined by on-shell recursion and 1-loop amplitudes reducible to scalar box integrals. It introduces an on-shell superspace using Grassmann variables to make SUSY manifest, proving that all tree amplitudes vanish at infinity under a supersymmetric BCFW shift and can be built from lower-point amplitudes. A central result is the no-triangle/no-bubble/no-rational-term structure at 1-loop in these theories, with box functions entirely capturing the quantum corrections and their coefficients computable from tree amplitudes via quadruple cuts. The authors extend these ideas to soft emission and the action of the E7(7) symmetry, and they speculate that leading singularities could determine higher-loop amplitudes, suggesting a holographic, non-local reformulation of QFT for flat space. Collectively, the work points toward a deep, amplitude-centered understanding of quantum field theory where maximal SUSY yields a highly constrained and potentially finite perturbative structure, hinting at new dual descriptions of flat-space physics.

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.

Figures (13)

• Figure 1: The BCFW recursion relation computes an $n$-point amplitude by sewing together lower-point amplitudes with complex on-shell momenta.
• Figure 2: The BCFW recursion relations for ${\cal N} = 4$ SYM and ${\cal N} = 8$ Supergravity. Note that we must analytically continue $\eta_1 \to \eta_1(z_P) = \eta_1 + z_P \eta_2$, but $\eta_2$ is not continued.
• Figure 3: There is an Adler zero when a single pion becomes soft.
• Figure 4: When we take the double soft limit of two pions, the result is a momentum-dependent $H$ rotation on each of the remaining hard states.
• Figure 5: Here we consider the amplitude for emitting a single soft graviton at tree level. The figure shows the only terms that contribute in the soft limit when we compute the amplitude using the BCFW Recursion Relations, analytically continuing the soft graviton momentum and some other hard leg. Terms with more than one hard leg associated with the soft graviton do not contribute directly because they do not have a soft emission pole. They do not contribute through an additional pole in the soft graviton sub-amplitude because in such a sub-amplitude, $p_s$ is analytically continued to a value $p_s(z_P)$ that does not vanish as $p_s \to 0$.