Field diffeomorphisms and the algebraic structure of perturbative expansions

TL;DR

This paper explores how non-linear field diffeomorphisms affect the perturbative expansions in scalar field theories, revealing conditions under which the S-matrix remains trivial and analyzing the algebraic structure of the resulting Feynman rules.

Contribution

It demonstrates that tree-level amplitudes satisfy BCFW recursion relations under field diffeomorphisms and identifies conditions for loop computations in massless and massive theories.

Findings

Tree-level amplitudes obey BCFW recursion relations.

Loop relations hold in massless theories.

Massive tadpole integrals must be renormalized to zero.

Abstract

We consider field diffeomorphisms in the context of real scalar field theories. Starting from free field theories we apply non-linear field diffeomorphisms to the fields and study the perturbative expansion for the transformed theories. We find that tree level amplitudes for the transformed fields must satisfy BCFW type recursion relations for the S-matrix to remain trivial. For the massless field theory these relations continue to hold in loop computations. In the massive field theory the situation is more subtle. A necessary condition for the Feynman rules to respect the maximal ideal and co-ideal defined by the core Hopf algebra of the transformed theory is that upon renormalization all massive tadpole integrals (defined as all integrals independent of the kinematics of external momenta) are mapped to zero.