What is the simplest quantum field theory?

Abstract

Conventional wisdom says that the simpler the Lagrangian of a theory the simpler its perturbation theory. An ever-increasing understanding of the structure of scattering amplitudes has however been pointing to the opposite conclusion. At tree level, the BCFW recursion relations that completely determine the S-matrix are valid not for scalar theories but for gauge theories and gravity, with gravitational amplitudes exhibiting the best UV behavior at infinite complex momentum. At 1-loop, amplitudes in $ \mathcal{N} = 4 $ SYM only have scalar box integrals, and it was recently conjectured that the same property holds for $ \mathcal{N} = 8 $ SUGRA, which plays an important role in the suspicion that this theory may be finite. In this paper we explore and extend the S-matrix paradigm, and suggest that $ \mathcal{N} = 8 $ SUGRA has the simplest scattering amplitudes in four dimensions. Labeling external states by supercharge eigenstates-Grassmann coherent states-allows the amplitudes to be exposed as completely smooth objects, with the action of SUSY manifest. We show that under the natural supersymmetric extension of the BCFW deformation of momenta, all tree amplitudes in $ \mathcal{N} = 4 $ SYM and $ \mathcal{N} = 8 $ SUGRA vanish at infinite complex momentum, and can therefore be determined by recursion relations. An important difference between $ \mathcal{N} = 8 $ SUGRA and $ \mathcal{N} = 4 $ SYM is that the massless S-matrix is defined everywhere on moduli space, and is acted on by a non-linearly realized E7(7) symmetry. We elucidate how non-linearly realized symmetries are reflected in the more familiar setting of pion scattering amplitudes, and go on to identify the action of E7(7) on amplitudes in $ \mathcal{N} = 8 $ SUGRA. Moving beyond tree level, we give a simple general 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 products of tree amplitudes appearing in cuts. In $ \mathcal{N} = 4 $ SYM and $ \mathcal{N} = 8 $ SUGRA, the on-shell superspace makes it easy to compute the multiplet sums that arise in these cuts by relating them to the best behaved tree amplitudes of highest spin, leading to a straightforward proof of the absence of triangles and bubbles at 1-loop. We also argue that rational terms are absent. This establishes that 1-loop amplitudes in $ \mathcal{N} = 8 $ SUGRA only have scalar box integrals. We give an explicit expression for 1-loop amplitudes for both $ \mathcal{N} = 4 $ SYM and $ \mathcal{N} = 8 $ SUGRA in terms of tree amplitudes that can be determined recursively. These amplitudes satisfy further relations in $ \mathcal{N} = 8 $ SUGRA that are absent in $ \mathcal{N} = 4 $ SYM. Since both tree and 1-loop amplitudes for maximally supersymmetric theories can be completely determined by their leading singularities, it is natural to conjecture that this property holds to all orders of perturbation theory. This is the nicest analytic structure amplitudes could possibly have, and if true, would directly imply the perturbative finiteness of $ \mathcal{N} = 8 $ SUGRA. All these 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.