On the relationship between Yang-Mills theory and gravity and its implication for ultraviolet divergences

Abstract

String theory implies that field theories containing gravity are in a certain sense ‘products’ of gauge theories. We make this product structure explicit up to two loops for the relatively simple case of N = 8 supergravity four-point amplitudes, demonstrating that they are ‘squares’ of N = 4 super-Yang-Mills amplitudes. This is accomplished by obtaining an explicit expression for the D-dimensional two-loop contribution to the four-particle S-matrix for N = 8 supergravity, which we compare to the corresponding N = 4 Yang-Mills result. From these expressions we also obtain the two-loop ultraviolet divergences in dimensions D = 7 through D = 11. The analysis relies on the unitarity cuts of the two theories, many of which can be recycled from a one-loop computation. The two-particle cuts, which may be iterated to all loop orders, suggest that squaring relations between the two theories exist at any loop order. The loop-momentum power-counting implied by our two-particle cut analysis indicates that in four dimensions the first four-point divergence in N = 8 supergravity should appear at five loops, contrary to the earlier expectation, based on superspace arguments, of a three-loop counterterm.