One-loop four-point amplitudes in pure and matter-coupled $ \mathcal{N}\leq 4 $ supergravity

Abstract

We construct all supergravity theories that can be obtained through factorized orbifold projections of $ \mathcal{N}=8 $ supergravity, exposing their double-copy structure, and calculate their one-loop four-point scattering amplitudes. We observe a unified structure in both matter and gravity amplitudes, and demonstrate that the four-graviton amplitudes are insensitive to the precise nature of the matter couplings. We show that these amplitudes are identical for the two different realizations of $ \mathcal{N}=4 $ supergravity with two vector multiplets, and argue that this feature extends to all multiplicities and loop orders as well as to higher dimensions. We also construct a selected set of supergravities obtained through a non-factorized orbifold action. Furthermore we calculate one-loop four-point amplitudes for all pure super-Yang-Mills theories with less-than-maximal supersymmetry using the duality between color and kinematics, finding here a unified expression that holds for all four gluon amplitudes in these theories. We recover the related amplitudes of factorized $ \mathcal{N}\leq 4 $ supergravities employing the double-copy construction. We observe a requirement that the four-point loop-level amplitudes have non-local integrand representations, exhibiting a mild non-locality in the form of inverse powers of the three external Mandelstam invariants. These are the first loop-level color-kinematic-satisfying representations in reduced supersymmetry theories.