Jet Bundle Geometry of Scalar Field Theories

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 $\varphi$ 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 $\varphi$ 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.

arXiv comment84 pages, 3 figures. Included a new proof that higher-jet bundle metrics yield complete EFT bases (up to a fixed number of derivatives); extended and generalised discussion of examples; two figures added; included discussion of topological terms; revised and elaborated on the connection to amplitudes

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