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 everincreasing 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 Smatrix 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 1loop, 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 Smatrix paradigm, and suggest that mathcal{N} = 8 SUGRA has the simplest scattering amplitudes in four dimensions. Labeling external states by supercharge eigenstatesGrassmann coherent statesallows 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 Smatrix is defined everywhere on moduli space, and is acted on by a nonlinearly realized E _{7(7)} symmetry. We elucidate how nonlinearly realized symmetries are reflected in the more familiar setting of pion scattering amplitudes, and go on to identify the action of E _{7(7)} on amplitudes in mathcal{N} = 8 SUGRA. Moving beyond tree level, we give a simple general discussion of the structure of 1loop 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 onshell 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 1loop. We also argue that rational terms are absent. This establishes that 1loop amplitudes in mathcal{N} = 8 SUGRA only have scalar box integrals. We give an explicit expression for 1loop 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 1loop 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 "weakweak" dual formulation of QFT, a holographic dual of flat space.
 Publication:

Journal of High Energy Physics
 Pub Date:
 September 2010
 DOI:
 10.1007/JHEP09(2010)016
 arXiv:
 arXiv:0808.1446
 Bibcode:
 2010JHEP...09..016A
 Keywords:

 Extended Supersymmetry;
 Supergravit Models;
 Global Symmetries;
 High Energy Physics  Theory
 EPrint:
 V1: 71+23 pages, 14 figures, less telegraphic abstract in body of paper V2: 74 + 24 pages. Typos fixed. Added discussion of parity invariance. Corrected the relationship between the N=4 IR equations and recursion relations. Streamlined discussion of the absence of rational terms V3: 75+24 pages. Typos fixed. Corrected beta function computation and N=4 soft scalar discussion