Jet Bundle Geometry of Scalar Field Theories

TL;DR

This paper extends the geometric framework for scalar field theories to include higher-derivative operators by utilizing jet bundle geometry, providing a systematic way to construct EFT Lagrangians with up to four derivatives.

Contribution

It generalizes the geometric approach to scalar EFTs from 2-derivatives to include 4-derivatives using jet bundle formalism, incorporating symmetries and field redefinitions.

Findings

Derived a geometric method to obtain 4-derivative operators from jet bundle metrics.

Constructed a non-redundant basis of operators for scalar EFTs using this geometry.

Connected geometric invariants to scattering amplitudes.

Abstract

For scalar field theories, such as those EFTs describing the Higgs, it is well-known that the 2-derivative Lagrangian is captured by geometry. That is, the set of operators with exactly 2 derivatives can be obtained by pulling back a metric from a field space manifold $M$ to spacetime $\Sigma$. We here generalise this geometric understanding of scalar field theories to higher- (and lower-) derivative Lagrangians. We show how the entire EFT Lagrangian with up to 4-derivatives can be obtained from geometry by pulling back a metric to $\Sigma$ from the 1-jet bundle that is (roughly) associated with maps from $\Sigma$ to $M$. More precisely, our starting point is to trade the field space $M$ for a fibre bundle $\pi:E \to \Sigma$, with fibre $M$, of which the scalar field $\phi$ is a local section. We discuss symmetries and field redefinitions in this bundle formalism, before showing how everything can be `prolongated' to the 1-jet bundle $J^1 E$ which, as a manifold, is the space of sections $\phi$ that agree in their zeroth and first derivatives above each spacetime point. Equipped with a notion of (spacetime and internal) symmetry on $J^1 E$, the idea is that one can write down the most general metric on $J^1 E$ consistent with symmetries, in the spirit of the effective field theorist, and pull it back to spacetime to build an invariant Lagrangian; because $J^1 E$ has `derivative coordinates', one naturally obtains operators with more than 2-derivatives from this geometry. We apply this formalism to various examples, including a single real scalar in 4d and a quartet of real scalars with $O(4)$ symmetry that describes the Higgs EFTs. We show how an entire non-redundant basis of 0-, 2-, and 4-derivative operators is obtained from jet bundle geometry in this way. Finally, we study the connection to amplitudes and the role of geometric invariants.