Multi-Leg One-Loop Gravity Amplitudes from Gauge Theory

TL;DR

The paper leverages KLT relations and unitarity to derive two infinite families of one-loop gravity amplitudes from gauge theory: all-plus amplitudes for generic gravity theories and N=8 supergravity MHV amplitudes. By performing D-dimensional unitarity cuts for up to six external gravitons and applying gravity–gauge dimension-shifting, the authors obtain explicit results and then extrapolate to arbitrary n using soft and collinear constraints. The work reveals gravity amplitudes as squares of gauge-theory structures and introduces half-soft functions to encode the necessary kinematic dependence, yielding compact, symmetry-respecting ansatze for all-plus and N=8 MHV amplitudes. The findings illuminate the UV behavior and factorization properties of gravity at one loop and illustrate a robust, gauge-theory–driven path to understanding quantum gravity amplitudes.

Abstract

By exploiting relations between gravity and gauge theories, we present two infinite sequences of one-loop n-graviton scattering amplitudes: the `maximally helicity-violating' amplitudes in N=8 supergravity, and the `all-plus' helicity amplitudes in gravity with any minimally coupled massless matter content. The all-plus amplitudes correspond to self-dual field configurations and vanish in supersymmetric theories. We make use of the tree-level Kawai-Lewellen-Tye (KLT) relations between open and closed string theory amplitudes, which in the low-energy limit imply relations between gravity and gauge theory tree amplitudes. For n < 7, we determine the all-plus amplitudes explicitly from their unitarity cuts. The KLT relations, applied to the cuts, allow us to extend to gravity a previously found `dimension-shifting' relation between (the cuts of) the all-plus amplitudes in gauge theory and the maximally helicity-violating amplitudes in N=4 super-Yang-Mills theory. The gravitational version of the relation lets us determine the n < 7 N=8 supergravity amplitudes from the all-plus gravity amplitudes. We infer the two series of amplitudes for all n from their soft and collinear properties, which can also be derived from gauge theory using the KLT relations.

Figures (9)

• Figure 1: Kinematics of the two-mass box integrals ${\cal I}_4^{a K_1 b K_2}$ that enter $n$-point MHV amplitudes in both $N=4$ super-Yang-Mills theory and $N=8$ supergravity. Here $a$ and $b$ label the external massless legs for the integral, which coincide with two external momenta for the amplitude, $k_a$ and $k_b$. The massive legs carry momenta $K_1$ and $K_2$, which are sums of the remaining external momenta. The four Lorentz invariants are the masses $K_1^2$ and $K_2^2$, and the Mandelstam invariants $S_{1a} = (K_1+k_a)^2$ and $S_{1b} = (K_1+k_b)^2$.
• Figure 2: Relations between infinite sequences of one-loop amplitudes in four different theories. The vertical arrows correspond to the 'dimension-shifting' relations of ref. DimShift, within (super-) Yang-Mills theory and (super-) gravity. (These remain a conjecture for $n\ge 7$ legs.) The horizontal arrows correspond to the gauge-gravity relations which follow from the KLT equations.
• Figure 3: As two momenta become collinear, the gravity $S$-matrix element develops a phase singularity which can be detected by rotating the two momenta about the axis formed by their sum.
• Figure 4: The class of tree diagrams in a gravity theory that can have a phase singularity factorizes in the collinear limit $a\parallel b$. The appearance of the same three-vertex for any number of external legs implies the universality of the tree-level splitting amplitudes. The splitting amplitudes are given by evaluating the three-vertex in the collinear limit in a helicity basis.
• Figure 5: The class of tree diagrams in a gravity theory that contribute in the soft limit, where leg $s$ is soft. The soft functions are found by summing over all three-vertices containing a soft leg.