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Next-to-Leading Order Running in the SMEFT

Lukas Born, Javier Fuentes-Martín, Anders Eller Thomsen

TL;DR

This work delivers the first complete two-loop renormalization group equations for the baryon-number-conserving sector of the dimension-six SMEFT, a critical step for consistent NLO SMEFT predictions. The authors combine functional methods with an anticommuting γ5, a local R*-operation, and a revised Mainz operator basis to control reading-point ambiguities and evanescent operators; they validate their framework through cross-checks against LEFT results and other SMEFT computations and provide comprehensive supplementary files with the β-functions. The approach enables scheme-independent NLO SMEFT analyses and lays the groundwork for precision Higgs and electroweak studies within SMEFT, while acknowledging residual prescription-dependent ambiguities that are largely under control or numerically subleading. Overall, the paper establishes a robust computational pipeline for high-precision SMEFT running and sets the stage for broader applicability to fler-scale EFT computations.

Abstract

The next-to-leading order (NLO) Standard Model Effective Field Theory (SMEFT) renormalization group equations are needed to account for phenomenologically relevant operator mixing and ensure renormalization scale independence in NLO calculations of observables. For the first time, we present the renormalization group equations of the baryon-number-conserving sector of the dimension-six SMEFT up to two-loop order. Our calculations have been performed using functional methods with an anticommuting $ γ_5 $-scheme. A variety of strategies are employed to mitigate the reading-point ambiguities inherent to this scheme choice. We also describe how a local version of the $ \boldsymbol{R}^\ast $-method is adapted to handle the evanescent operators arising in dimensional regularization. The results are provided in various supplementary files to make them accessible for both human inspection and numerical implementation.

Next-to-Leading Order Running in the SMEFT

TL;DR

This work delivers the first complete two-loop renormalization group equations for the baryon-number-conserving sector of the dimension-six SMEFT, a critical step for consistent NLO SMEFT predictions. The authors combine functional methods with an anticommuting γ5, a local R*-operation, and a revised Mainz operator basis to control reading-point ambiguities and evanescent operators; they validate their framework through cross-checks against LEFT results and other SMEFT computations and provide comprehensive supplementary files with the β-functions. The approach enables scheme-independent NLO SMEFT analyses and lays the groundwork for precision Higgs and electroweak studies within SMEFT, while acknowledging residual prescription-dependent ambiguities that are largely under control or numerically subleading. Overall, the paper establishes a robust computational pipeline for high-precision SMEFT running and sets the stage for broader applicability to fler-scale EFT computations.

Abstract

The next-to-leading order (NLO) Standard Model Effective Field Theory (SMEFT) renormalization group equations are needed to account for phenomenologically relevant operator mixing and ensure renormalization scale independence in NLO calculations of observables. For the first time, we present the renormalization group equations of the baryon-number-conserving sector of the dimension-six SMEFT up to two-loop order. Our calculations have been performed using functional methods with an anticommuting -scheme. A variety of strategies are employed to mitigate the reading-point ambiguities inherent to this scheme choice. We also describe how a local version of the -method is adapted to handle the evanescent operators arising in dimensional regularization. The results are provided in various supplementary files to make them accessible for both human inspection and numerical implementation.
Paper Structure (57 sections, 119 equations, 12 figures, 4 tables)

This paper contains 57 sections, 119 equations, 12 figures, 4 tables.

Table of Contents

  1. Introduction
  2. The SMEFT and a Revised Operator Basis
  3. The SMEFT
  4. The Mainz basis
  5. Dimensional Continuation and Scheme Conventions
  6. Dirac algebra in $d$ dimensions
  7. Reading-point prescription and $\gamma_5$ ambiguities
  8. Identifying ambiguous diagrams
  9. Four-fermion operators ($\psi^4$)
  10. Vector four-fermion operators
  11. Scalar four-fermion operators
  12. Higgs current operators ($\psi^2 H^2 D$)
  13. Dimension-six Yukawas ($\psi^2 H^3$)
  14. Dimension-six dipoles ($\psi^2 H X$)
  15. Bosonic operators
  16. ...and 42 more sections

Figures (12)

  • Figure 1: Diagrams that are unambiguous even though they contain $\gamma_5$-odd traces with six $\gamma$-matrices. The dimension-six operator is indicated with a box.
  • Figure 2: Ambiguous diagrams from a vector four-fermion insertion.
  • Figure 3: Ambiguous diagrams from a scalar four-fermion insertion.
  • Figure 4: Ambiguous diagrams from a Higgs current insertion.
  • Figure 5: Ambiguous diagram from a dimension-six Yukawa insertion.
  • ...and 7 more figures