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

TL;DR

The paper formalizes a double-copy paradigm in which gravity amplitudes are squares of corresponding gauge-theory amplitudes, verified explicitly at two loops for N=8 supergravity by expressing the four-point amplitude in terms of squared N=4 YM coefficients and scalar integrals. Using unitarity cuts and KLT relations, the authors show that two-particle cuts can be iterated to all loops for entirely two-particle constructible terms, and they extend the analysis with three-particle cuts to fully reconstruct the two-loop N=8 amplitude. They extract UV divergences in various dimensions, finding no divergences below certain dimensions and predicting a five-loop finite-term counterterm in D=4, consistent with some string-theory expectations but challenging previous superspace finiteness assumptions. The work provides strong evidence for a gravity–gauge double-copy structure at the level of the S-matrix, with significant implications for understanding ultraviolet behavior in maximal supergravity and for the broader connection between gravity and gauge theories.

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.

Figures (14)

• Figure 1: Tree-level gauge theory four-gluon amplitudes and gravity four-graviton amplitudes expressed in terms of scalar $\phi^3$ diagrams. The coefficients of the graviton amplitudes are squares of the coefficients for four-gluon amplitudes.
• Figure 2: One-loop gauge theory and gravity four-point contributions expressed in terms of scalar diagrams. The coefficient of a given scalar diagram in the $N=8$ four-graviton amplitude is the square of the corresponding coefficient in the $N=4$ four-gluon amplitude.
• Figure 3: The planar and non-planar scalar integrals.
• Figure 4: This three-loop non-planar graph is not 'entirely two-particle constructible'. In fact, it has no two-particle cuts at all.
• Figure 5: Starting from an $L$-loop planar integral function appearing in an $N=4$ Yang-Mills amplitude we may add an extra line, using this rule. The two-lines on the left represent two lines buried in some $L$-loop integral.