What is the Geometry of Effective Field Theories?

TL;DR

This paper introduces a geometric framework for scalar effective field theories, using a metric on the functional manifold to make amplitudes covariant and relate geometry to physical predictions.

Contribution

It develops a novel geometric formula for amplitudes that makes on-shell covariance explicit, linking functional geometry with effective field theory calculations.

Findings

Geometric quantities transform covariantly under field redefinitions

A new formula for amplitudes using geometric vertices

Manifest on-shell covariance of amplitudes achieved

Abstract

We elaborate on a recently proposed geometric framework for scalar effective field theories. Starting from the action, a metric can be identified that enables the construction of geometric quantities on the associated functional manifold. These objects transform covariantly under general field redefinitions that relate different operator bases, including those involving derivatives. We present a novel geometric formula for the amplitudes of the theory, where the vertices in Feynman diagrams are replaced by their geometrized counterparts. This makes the on-shell covariance of amplitudes manifest, providing the link between functional geometry and effective field theories.