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.

• Received 3 March 2022
• Revised 22 November 2022
• Accepted 4 January 2023

DOI:https://doi.org/10.1103/PhysRevLett.130.041603

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Published by the American Physical Society