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β-function for the Higgs self-interaction in the Standard Model at three-loop level

K. G. Chetyrkin, M. F. Zoller

TL;DR

The paper computes the three-loop β-functions for the Higgs sector in the unbroken Standard Model, including QCD, electroweak, Higgs self-interaction, and third-generation Yukawa couplings for the Higgs self-coupling λ and the Higgs mass parameter m^2. The authors perform a mass-independent MSbar calculation, decompose the SM Lagrangian into QCD, EW, Yukawa, and Φ sectors, and derive the necessary counterterms up to two-loop order (three-loop for the central results) while regulating infrared subtleties with an auxiliary mass M^2; diagram generation and evaluation are performed with QGRAF, FORM, and MATAD, with extensive cross-checks ensuring gauge-parameter independence and consistency with known lower-loop results. The results include explicit expressions for β_λ^(1-3) and β_{m^2}^(1-3), reproducing established limits in various decoupling scenarios, and numerical studies show that at μ = M_t the three-loop electroweak contributions, though small, are non-negligible due to cancellations among dominant terms. The findings improve the precision of λ running and vacuum-stability analyses, particularly at low scales, and have been independently confirmed in a parallel work.

Abstract

We analytically compute the QCD, electroweak, Higgs and third generation Yukawa contributions to the β-function for the Higgs self-coupling as well as for the Higgs mass parameter in the unbroken phase of the Standard Model at three-loop level.

β-function for the Higgs self-interaction in the Standard Model at three-loop level

TL;DR

The paper computes the three-loop β-functions for the Higgs sector in the unbroken Standard Model, including QCD, electroweak, Higgs self-interaction, and third-generation Yukawa couplings for the Higgs self-coupling λ and the Higgs mass parameter m^2. The authors perform a mass-independent MSbar calculation, decompose the SM Lagrangian into QCD, EW, Yukawa, and Φ sectors, and derive the necessary counterterms up to two-loop order (three-loop for the central results) while regulating infrared subtleties with an auxiliary mass M^2; diagram generation and evaluation are performed with QGRAF, FORM, and MATAD, with extensive cross-checks ensuring gauge-parameter independence and consistency with known lower-loop results. The results include explicit expressions for β_λ^(1-3) and β_{m^2}^(1-3), reproducing established limits in various decoupling scenarios, and numerical studies show that at μ = M_t the three-loop electroweak contributions, though small, are non-negligible due to cancellations among dominant terms. The findings improve the precision of λ running and vacuum-stability analyses, particularly at low scales, and have been independently confirmed in a parallel work.

Abstract

We analytically compute the QCD, electroweak, Higgs and third generation Yukawa contributions to the β-function for the Higgs self-coupling as well as for the Higgs mass parameter in the unbroken phase of the Standard Model at three-loop level.

Paper Structure

This paper contains 4 sections, 26 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Calculation
  3. Results
  4. Conclusions

Figures (4)

  • Figure 1: Some diagrams contributing to the renormalization of the $4$-$\Phi$-vertices
  • Figure 2: Evolution of $\lambda$ around the scale of the top mass the with and without the electroweak three-loop contributions.
  • Figure 3: Evolution of $\lambda$ around the scale of the top mass the with and without the electroweak three-loop contributions.
  • Figure 4: Evolution of $\lambda$ with the scale $\mu$ around $10^{10}$ GeV with and without the electroweak three-loop contributions.