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Two loops, four tops and two $γ_5$ schemes: a renormalization story

Stefano Di Noi, Ramona Gröber

TL;DR

This work computes the two-loop renormalization effects of four-top SMEFT operators on the running of the strong coupling $g_s$ and the top-quark mass $m_t$, using Naive Dimensional Regularization (NDR) and then translating to the Breitenlohner–Maison–'t Hooft–Veltman (BMHV) scheme. The authors perform an off-shell, diagrammatic renormalization with redundant operators, extract the UV-divergent parts, and derive the corresponding renormalization group equations (RGEs) in NDR, obtaining explicit expressions for $\beta_{m_t}^{\mathrm{NDR}}$ and $\beta_{g_s}^{\mathrm{NDR}}$ in terms of the Warsaw-basis four-top Wilson coefficients. Through the one-loop translation between NDR and BMHV, they obtain $\beta_{g_s}^{\mathrm{BMHV}} = 0$ and a BMHV-running top mass given by $\beta_{m_t}^{\mathrm{BMHV}} = \beta_{m_t}^{\mathrm{NDR}} + \Delta \beta_{m_t}$, with a concrete formula for $\Delta \beta_{m_t}$ and a final compact expression for $\beta_{m_t}^{\mathrm{BMHV}}$. The results provide a concrete two-loop SMEFT contribution to the running of $m_t$ and $g_s$, and illustrate the scheme dependence and translation needed for consistent EFT analyses, aiding UV matching and global SMEFT studies.

Abstract

We calculate the four-top quark operator contributions to the two-loop renormalization constants of the fermion and gluon fields, necessary to obtain the renormalization group equations for the fermion masses and the strong coupling constant. The computation has been carried out in the naïve dimensional regularization scheme for the $γ_5$ matrix. We also discuss a strategy to translate the results into the Breitenlohner--Maison--'t~Hooft--Veltman scheme and present the corresponding beta functions in both schemes.

Two loops, four tops and two $γ_5$ schemes: a renormalization story

TL;DR

This work computes the two-loop renormalization effects of four-top SMEFT operators on the running of the strong coupling and the top-quark mass , using Naive Dimensional Regularization (NDR) and then translating to the Breitenlohner–Maison–'t Hooft–Veltman (BMHV) scheme. The authors perform an off-shell, diagrammatic renormalization with redundant operators, extract the UV-divergent parts, and derive the corresponding renormalization group equations (RGEs) in NDR, obtaining explicit expressions for and in terms of the Warsaw-basis four-top Wilson coefficients. Through the one-loop translation between NDR and BMHV, they obtain and a BMHV-running top mass given by , with a concrete formula for and a final compact expression for . The results provide a concrete two-loop SMEFT contribution to the running of and , and illustrate the scheme dependence and translation needed for consistent EFT analyses, aiding UV matching and global SMEFT studies.

Abstract

We calculate the four-top quark operator contributions to the two-loop renormalization constants of the fermion and gluon fields, necessary to obtain the renormalization group equations for the fermion masses and the strong coupling constant. The computation has been carried out in the naïve dimensional regularization scheme for the matrix. We also discuss a strategy to translate the results into the Breitenlohner--Maison--'t~Hooft--Veltman scheme and present the corresponding beta functions in both schemes.

Paper Structure

This paper contains 7 sections, 34 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Computational set-up
  3. Results in the NDR scheme
  4. Results in the BMHV scheme
  5. Conclusion
  6. Two-point functions
  7. Results in the on-shell basis

Figures (2)

  • Figure 1: Representative diagrams displaying the four-top operators (solid square dot) to the top quark propagator (a-c) and to the gluon propagator (d-e). The internal fermion line can represent either a top or a bottom quark.
  • Figure 2: Representative diagrams displaying the one-loop insertion of the one-loop counterterms to the top quark propagator (a-c) and to the gluon propagator (d-e).