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 Σ. We 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 Σ from the 1-jet bundle that is associated with maps from Σ to M. More precisely, our starting point is to trade the field space M for a fibre bundle π:E →Σ, with fibre M, of which the scalar field Φ 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 J1E which, as a manifold, is the space of sections Φ that agree in their zeroth and first derivatives above each spacetime point. Equipped with a notion of (spacetime and internal) symmetry on J1E, the idea is that one can write down the most general metric on J1E consistent with symmetries, in the spirit of the effective field theorist, and pull it back to spacetime to build an invariant Lagrangian; because J1E 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.