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On-Shell Renormalization of Dim-8 SMEFT from Complete Amplitude Basis: I. Four-Fermion Operators

Chao Wu, Ming-Lei Xiao, Jiang-Hao Yu, Yu-Hui Zheng

TL;DR

This work delivers the first complete one-loop renormalization group equations for all dimension-8 four-fermion operators in the SMEFT, computed entirely within an on-shell, unitarity framework. By employing the Young Tensor amplitude basis (the y-basis) and a full flavor-aware reduction (p-basis to f-basis, then to y-basis), the authors automatically generate nonredundant, flavor-general RGEs from two-particle unitarity cuts, with ultraviolet information encoded in bubble integrals. They explicitly include both linear and quadratic-in-Coeficient contributions, validate against known dimension-6 results, and cross-check against an independent dim-8 calculation, providing a Mathematica file RGEresultfor4fermions.m to enable phenomenologists to evolve Wilson coefficients across scales. The approach cleanly separates UV and IR structures, using standard collinear anomalous dimensions for SM fields to complete the IR sector, and the results have broad implications for precision Higgs, electroweak, and flavor phenomenology where dimension-8 effects and their scale dependence matter. This framework therefore offers a systematic, flavor-complete method to study higher-dimensional SMEFT RG evolution and paves the way for completing the remaining dim-8 sectors and extending to HEFT or higher-loop analyses.

Abstract

We compute the complete one-loop renormalization group equations (RGEs) for all the four-fermion operators at dimension-8 Standard Model Effective Field Theory (SMEFT). We adopt the on-shell framework, where the RGEs are obtained from the unitarity cuts of the bubble integrals. To construct a consistent set of RGEs without redundancy, we utilize the Young Tensor amplitude/operator basis as the building blocks of the tree-level amplitudes that constitute the unitarity cuts, which incorporates the full flavor structures of the effective operators. Due to the large number of effective operators for a dimension-8 type, it is crucial to reduce the integrated cuts to the same RGE amplitude basis, which is also made possible by the algorithm in the Young Tensor method. With the Mathematica package ABC4EFT that implements the method, we obtain the full result of the dimension-8 four-fermion RGEs with the output file attached as the supplementary material.

On-Shell Renormalization of Dim-8 SMEFT from Complete Amplitude Basis: I. Four-Fermion Operators

TL;DR

This work delivers the first complete one-loop renormalization group equations for all dimension-8 four-fermion operators in the SMEFT, computed entirely within an on-shell, unitarity framework. By employing the Young Tensor amplitude basis (the y-basis) and a full flavor-aware reduction (p-basis to f-basis, then to y-basis), the authors automatically generate nonredundant, flavor-general RGEs from two-particle unitarity cuts, with ultraviolet information encoded in bubble integrals. They explicitly include both linear and quadratic-in-Coeficient contributions, validate against known dimension-6 results, and cross-check against an independent dim-8 calculation, providing a Mathematica file RGEresultfor4fermions.m to enable phenomenologists to evolve Wilson coefficients across scales. The approach cleanly separates UV and IR structures, using standard collinear anomalous dimensions for SM fields to complete the IR sector, and the results have broad implications for precision Higgs, electroweak, and flavor phenomenology where dimension-8 effects and their scale dependence matter. This framework therefore offers a systematic, flavor-complete method to study higher-dimensional SMEFT RG evolution and paves the way for completing the remaining dim-8 sectors and extending to HEFT or higher-loop analyses.

Abstract

We compute the complete one-loop renormalization group equations (RGEs) for all the four-fermion operators at dimension-8 Standard Model Effective Field Theory (SMEFT). We adopt the on-shell framework, where the RGEs are obtained from the unitarity cuts of the bubble integrals. To construct a consistent set of RGEs without redundancy, we utilize the Young Tensor amplitude/operator basis as the building blocks of the tree-level amplitudes that constitute the unitarity cuts, which incorporates the full flavor structures of the effective operators. Due to the large number of effective operators for a dimension-8 type, it is crucial to reduce the integrated cuts to the same RGE amplitude basis, which is also made possible by the algorithm in the Young Tensor method. With the Mathematica package ABC4EFT that implements the method, we obtain the full result of the dimension-8 four-fermion RGEs with the output file attached as the supplementary material.
Paper Structure (31 sections, 139 equations, 4 figures, 27 tables)

This paper contains 31 sections, 139 equations, 4 figures, 27 tables.

Table of Contents

  1. Introduction
  2. Spinor Helicity Formalism and On-shell Amplitude Basis
  3. Operator Y-basis and Basis Reduction Rules
  4. Flavor Basis
  5. One-loop Renormalization
  6. Calculating the One-Loop Renormalization with Helicity Spinor
  7. Infrared Collinear Anomalous Dimensions in the Standard Model
  8. The RGEs for the Four-Fermion Operators
  9. $Q^2Q^{\dagger 2}$
  10. $QQ^{\dagger}u_{\mathbb{C}}u_{\mathbb{C}}^{\dagger}$
  11. $QQ^{\dagger}d_{\mathbb{C}}d_{\mathbb{C}}^{\dagger}$
  12. $u_{\mathbb{C}}^2u_{\mathbb{C}}^{\dagger 2}$
  13. $d_{\mathbb{C}}^2d_{\mathbb{C}}^{\dagger 2}$
  14. $u_{\mathbb{C}}u_{\mathbb{C}}^{\dagger}d_{\mathbb{C}}d_{\mathbb{C}}^{\dagger}$
  15. $Q^2 u_{\mathbb{C}}d_{\mathbb{C}}$
  16. ...and 16 more sections

Figures (4)

  • Figure 1: Procedure for calculating the renormalization group equation using on-shell amplitudes.
  • Figure 2: One-loop unitarity cut for effective operators renormalization. While the renormalization contribution is from the same dimension.
  • Figure 3: One-loop unitarity cut for effective operators renormalization, while the renormalization contribution are from the loeer-dimension.
  • Figure 4: The one-loop diagram when calculating the renormalization group contribution from the effective operator $\mathcal{O}_{L^2L^{\dagger 2}}$ to $\mathcal{O}_{L^2L^{\dagger 2}}$. The black arrows indicate the direction of the fermion lines, while the blue arrows represent the direction of momentum flow.