On-Shell Covariance of Quantum Field Theory Amplitudes

Understand

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.