Jet Bundle Geometry of Scalar Field Theories

For scalar field theories, such as those effective field theories (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 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 Σ from the 1-jet bundle that is (roughly) 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 J 1E 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 J 1E, the idea is that one can write down the most general metric on J 1E consistent with symmetries, in the spirit of the effective field theorist, and pull it back to spacetime to build an invariant Lagrangian; because J 1E 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.