On-Shell Covariance of Quantum Field Theory Amplitudes

Scattering amplitudes in quantum field theory are independent of the field parametrization, which has a natural geometric interpretation as a form of coordinate invariance. Amplitudes can be expressed in terms of Riemannian curvature tensors, which makes the covariance of amplitudes under nonderivative field redefinitions manifest. We present a generalized geometric framework that extends this manifest covariance to all allowed field redefinitions. Amplitudes satisfy a recursion relation to all orders in perturbation theory that closely resembles the application of covariant derivatives to increase the rank of a tensor. This allows us to argue that tree-level amplitudes possess a notion of on-shell covariance, in that they transform as a tensor under any allowed field redefinition up to a set of terms that vanish when the equations of motion and on-shell momentum constraints are imposed. We highlight a variety of immediate applications to effective field theories.

Automated summary

Scattering amplitudes in quantum field theory are geometrically invariant under field redefinitions and can be expressed in terms of Riemannian curvature tensors. A generalized geometric framework is presented to extend the manifest covariance of amplitudes to all allowed field redefinitions. The amplitudes exhibit a recursive relation that resembles the application of covariant derivatives, indicating a notion of on-shell covariance at tree-level.