Jet bundle geometry of scalar field theories

Abstract 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>π</mml:mi> <mml:mo>:</mml:mo> <mml:mi>E</mml:mi> <mml:mo accent="false" stretchy="false">→</mml:mo> <mml:mi mathvariant="normal">Σ</mml:mi> </mml:mrow> </mml:math> , 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msup> <mml:mi>J</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:mi>E</mml:mi> </mml:mrow> </mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msup> <mml:mi>J</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:mi>E</mml:mi> </mml:mrow> </mml:math> , the idea is that one can write down the most general metric on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msup> <mml:mi>J</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:mi>E</mml:mi> </mml:mrow> </mml:math> consistent with symmetries, in the spirit of the effective field theorist, and pull it back to spacetime to build an invariant Lagrangian; because <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msup> <mml:mi>J</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:mi>E</mml:mi> </mml:mrow> </mml:math> 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.