Two-Loop Renormalization in the Standard Model Part I: Prolegomena

TL;DR

This work lays a comprehensive groundwork for two‑loop renormalization in the Standard Model by detailing Higgs tadpole treatments (β_h and β_t schemes), the order‑by‑order diagonalization of the neutral sector, and the derivation of Dyson resummed gauge boson propagators that satisfy Ward–Slavnov–Taylor identities. It introduces a consistent set of special vertices and a Gamma/Γ–tadpole framework, and develops the LQ basis to cleanly factor out θ dependence in two‑loop self‑energies. Together, these elements establish a robust formalism for ultraviolet counterterms and finite renormalization, paving the way for Part II (counterterms) and Part III (renormalization scheme with unstable particles). The results have direct implications for precise electroweak calculations and the interpretation of pseudo‑observables in a gauge‑invariant, gauge‑parameter‑independent manner.

Abstract

In this paper the building blocks for the two-loop renormalization of the Standard Model are introduced with a comprehensive discussion of the special vertices induced in the Lagrangian by a particular diagonalization of the neutral sector and by two alternative treatments of the Higgs tadpoles. Dyson resummed propagators for the gauge bosons are derived, and two-loop Ward-Slavnov-Taylor identities are discussed. In part II, the complete set of counterterms needed for the two-loop renormalization will be derived. In part III, a renormalization scheme will be introduced, connecting the renormalized quantities to an input parameter set of (pseudo-)experimental data, critically discussing renormalization of a gauge theory with unstable particles.