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On the Relationship between Yang-Mills Theory and Gravity and its Implication for Ultraviolet Divergences


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.

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