Covariant and single-field effective action with the background-field formalism

TL;DR

This paper develops a covariant, single-field dependent effective action in scalar quantum field theory using a nonlinear background-field split, enabling exact solutions to the splitting Ward identity and preserving ultraviolet symmetries.

Contribution

It introduces a novel class of nonlinear quantum-background splits that allow for an exact solution to the splitting Ward identity, resulting in a covariant effective action dependent on a single total field.

Findings

Exact solutions to the splitting Ward identity are achieved.

The constructed effective action maintains covariance and symmetry inheritance.

Applications include the $O(N)$ effective field theory relevant for the Higgs sector.

Abstract

In the context of scalar quantum field theory we introduce a class of generically nonlinear quantum-background splits for which the splitting Ward identity, encoding the single field dependence in the effective action, can be solved exactly. We show that this can be used to construct an effective action which is both covariant and dependent on the background and fluctuation fields only through a single total field in a way independent from the dynamics. Moreover we discuss the criteria under which the ultraviolet symmetries are inherited by the quantum effective action. The approach is demonstrated through some examples, including the $O(N)$ effective field theory, which might be of interest for the Higgs sector of the Standard Model or its extensions.