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Renormalization-group analysis of the SM: Loops, uncertainties, and vacuum stability

A. V. Bednyakov1,*, A. S. Fedoruk2,†, and D. I. Kazakov1,‡

  • *Contact author: bednya@jinr.ru
  • †Contact author: fedoruk.as@phystech.edu
  • ‡Contact author: kazakovd@theor.jinr.ru
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Phys. Rev. D 113, 036018 – Published 19 February, 2026

DOI: https://doi.org/10.1103/lgnl-2nxv

Abstract

Renormalization-group equations (RGE) is one of the key tools in studying high-energy behavior of the Standard Model (SM). We begin by reviewing one-loop RGE for the dimensionless couplings of the SM and proceed to the state-of-the-art results. Our study focuses on the RGE solutions at different loop orders. We compare not only the standard (“diagonal”) loop counting when one considers gauge, Yukawa, and scalar self-coupling beta functions at the same order but also “nondiagonal” ones, inspired by the so-called Weyl consistency conditions. We discuss the initial conditions for RGE (“matching”) for different loop configurations and study the uncertainties of running couplings both related to the limited precision of the experimental input (“parametric”) and the missing high-order corrections (“theoretical”). As an application of our analysis we also estimate the electroweak vacuum decay probability and study how the uncertainties in the running parameters affect the latter. We argue that nondiagonal beta functions, if coupled with a more consistent nondiagonal matching, lead to larger theoretical uncertainty than diagonal ones.

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Phys. Rev. D 113, 036018– Published 19 February, 2026
  • Received 19 September 2025
  • Accepted 29 January 2026
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