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 Ddimensional 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.

Simplifying multiloop integrands and ultraviolet divergences of gauge theory and gravity amplitudes

“…In particular, it has been confirmed that the duality holds for the three-loop four-point amplitude of N = 4 sYM theory [30]. (The one-and two-loop four-point amplitudes [8,52,53] in this theory also manifestly satisfy the duality). Similarly, the duality-satisfying five-point one-, two-and three-loop amplitudes of N = 4 sYM theory have recently been constructed [33].…”

“…At L loops, the four-point amplitude in N = 4 sYM theory is expected to have a representation with at most 2(L − 2) powers of the loop momentum in the numerator of each 1PI cubic graph [8,63]. At three loops, this power counting allows for cancellation of one internal propagator, as in graphs (i), (j) and (k) of fig.…”

“…For example, many types of color correlations are suppressed in the large-N c limit. Furthermore, the information provided by the full-color expression for N = 4 sYM amplitudes, expressed in terms of their loop-momentum integrands, can be used to construct corresponding amplitudes [8][9][10][11] in N = 8 supergravity [12]. From each set of amplitudes one can extract information about ultraviolet divergences in the respective theory.…”

New color decompositions for gauge amplitudes at tree and loop level

“…Finally, we remark that even if a general, multi-loop f abc -based color decomposition is not yet available, the results in this paper can still be employed in the context of specific multi-loop gluonic calculations based on unitarity cuts [21,7,22]. For example, the unitarity cuts of two-loop amplitudes are given in terms of tree-level and one-loop amplitudes.…”

“…We expect that structure-constant-based decompositions will also be useful for general multiloop scattering amplitude calculations, once they have been derived. In a particular case they have already proven useful: Two-loop four-gluon amplitudes in N = 4 supersymmetric Yang-Mills theory were computed via their unitarity cuts and were presented in a trace-based formalism [21], but they can be more compactly expressed in an f abc basis [22]. In the latter basis only two types of color structures appear; these are represented graphically in fig.…”