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

• Published in: The European physical journal. C, Particles and fields Vol. 81; no. 9; pp. 1 - 28
• Main Authors: Dittmaier, Stefan, Schuhmacher, Sebastian, Stahlhofen, Maximilian
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.09.2021; Springer; Springer Nature B.V; SpringerOpen
• Subjects: Astronomy; Astrophysics and Cosmology; Couplings; Decoupling; Eigenvectors; Elementary Particles; Field theory; Formalism; Hadrons; Heavy Ions; Higgs bosons; Integrals; Measurement Science and Instrumentation; Nuclear Energy; Nuclear Physics; Operators (mathematics); Physics; Physics and Astronomy; Quantum Field Theories; Quantum Field Theory; Regular Article - Theoretical Physics; String Theory
• ISSN: 1434-6044; 1434-6052
• DOI: 10.1140/epjc/s10052-021-09587-7

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.