On-Shell Covariance of Quantum Field Theory Amplitudes

TL;DR

This paper introduces a geometric framework demonstrating that quantum field theory amplitudes are covariant under all allowed field redefinitions, revealing an on-shell tensorial transformation property with applications to effective field theories.

Contribution

It extends the geometric interpretation of amplitude covariance to all field redefinitions, establishing on-shell covariance and recursion relations for amplitudes.

Findings

Amplitudes are invariant under non-derivative field redefinitions.

Amplitudes transform as tensors on-shell, up to terms vanishing on equations of motion.

The framework applies to effective field theories with immediate practical implications.

Abstract

Scattering amplitudes in quantum field theory are independent of the field parameterization, 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 non-derivative field redefinitions manifest. We present a generalized geometric framework that extends this manifest covariance to $all$ allowed field redefinitions. Amplitudes satisfy a recursion relation 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.