Integrating out heavy fields in the path integral using the background-field method: general formalism

TL;DR

This paper presents a systematic, transparent method using the background-field formalism to integrate out heavy fields in the path integral, deriving effective Lagrangians with one-loop effects, demonstrated through a heavy Higgs in the Standard Model extension.

Contribution

It introduces a general, algorithmic procedure for integrating out heavy fields in the path integral, including one-loop effects, with a focus on non-decoupling phenomena and proper renormalization schemes.

Findings

Effective Lagrangian including one-loop contributions derived

Method applied to non-decoupling effects of a heavy Higgs

Proper renormalization scheme is essential for consistency

Abstract

Building on an older method used to derive non-decoupling effects of a heavy Higgs boson in the Standard Model, we describe a general procedure to integrate out heavy fields in the path integral. The derivation of the corresponding effective Lagrangian including the one-loop contributions of the heavy particle(s) is particularly transparent, flexible, and algorithmic. The background-field formalism allows for a clear separation of tree-level and one-loop effects involving the heavy fields. Using expansion by regions the one-loop effects are further split into contributions from large and small momentum modes. The former are contained in Wilson coefficients of effective operators, the latter are reproduced by one-loop diagrams involving effective tree-level couplings. The method is illustrated by calculating potential non-decoupling effects of a heavy Higgs boson in a singlet Higgs extension of the Standard Model. In particular, we work in a field basis corresponding to mass eigenstates and properly take into account non-vanishing mixing between the two Higgs fields of the model. We also show that a proper choice of renormalization scheme for the non-standard sector of the underlying full theory is crucial for the construction of a consistent effective field theory.