Renormalization Group Evolution of the Standard Model Dimension Six Operators III: Gauge Coupling Dependence and Phenomenology (1312.2014v5)

Abstract: We calculate the gauge terms of the one-loop anomalous dimension matrix for the dimension-six operators of the Standard Model effective field theory (SM EFT). Combining these results with our previous results for the $\lambda$ and Yukawa coupling terms completes the calculation of the one-loop anomalous dimension matrix for the dimension-six operators. There are 1350 $CP$-even and $1149$ $CP$-odd parameters in the dimension-six Lagrangian for 3 generations, and our results give the entire $2499 \times 2499$ anomalous dimension matrix. We discuss how the renormalization of the dimension-six operators, and the additional renormalization of the dimension $d \le 4$ terms of the SM Lagrangian due to dimension-six operators, lays the groundwork for future precision studies of the SM EFT aimed at constraining the effects of new physics through precision measurements at the electroweak scale. As some sample applications, we discuss some aspects of the full RGE improved result for essential processes such as $gg \to h$, $h \to \gamma \gamma$ and $h \to Z \gamma$, for Higgs couplings to fermions, for the precision electroweak parameters $S$ and $T$, and for the operators that modify important processes in precision electroweak phenomenology, such as the three-body Higgs boson decay $h \rightarrow Z \, \ell^+ \, \ell^-$ and triple gauge boson couplings. We discuss how the renormalization group improved results can be used to study the flavor problem in the SM EFT, and to test the minimal flavor violation (MFV) hypothesis. We briefly discuss the renormalization effects on the dipole coefficient $C_{e\gamma}$ which contributes to $\mu \to e \gamma$ and to the muon and electron magnetic and electric dipole moments.

• The paper computes the complete one-loop gauge coupling contributions to the 2499×2499 anomalous dimension matrix for SM EFT dimension-six operators.
• It employs rigorous dimensional analysis of 59 operators, capturing both CP-even and CP-odd effects crucial for new physics signals.
• The results impact precision measurements in Higgs processes and electroweak parameters, advancing constraints on beyond-standard-model physics.

Anomalous Dimensions in the Standard Model EFT

The paper "Renormalization Group Evolution of the Standard Model Dimension Six Operators III: Gauge Coupling Dependence and Phenomenology" by Alonso et al. is a comprehensive paper on the one-loop renormalization group equations (RGEs) of dimension-six operators in the Standard Model Effective Field Theory (SM EFT). This work represents the final piece of a broader effort to compute the complete one-loop anomalous dimension matrix for these operators, providing critical insights for precision studies in particle physics.

Summary of Key Contributions

The central achievement of the paper is the computation of gauge-dependent terms in the one-loop anomalous dimension matrix for the dimension-six operators in the SM EFT. This incorporates previous results addressing the $\lambda$ and Yukawa coupling terms, culminating in the full $2499 \times 2499$ anomalous dimension matrix. By completing this matrix, the authors lay foundational work for setting stringent bounds on new physics effects at the electroweak scale through high-precision measurements.

Highlights:

• Dimensional Analysis: The RGEs are calculated for a set of 59 dimension-six operators, addressing $CP$-even and $CP$-odd parameterizations, both of which contribute significantly to possible new physics signals.
• Numerical Results: The work determines the leading order logarithmic running effects of these operators. Such contributions are pivotal for understanding how potential beyond-the-standard-model (BSM) physics might manifest at accessible energy scales.
• Applications: The authors explore how their results can be applied to key processes, including Higgs production and decay channels ($gg \to h$, $h \to \gamma \gamma$, $h \to Z \gamma$), and modifications to precision electroweak parameters ($S$ and $T$).

Implications and Future Directions

The implications of this work are extensive in both theoretical and phenomenological domains. From a theoretical perspective, the determination of the complete RGE allows for refined modeling of the RG flow in the context of SM EFT, advancing our understanding of how new physics scales down to observable phenomena. Practically, these results provide a framework to better interpret deviations from the Standard Model in current and future experiments, including those at the LHC.

Theoretical Implications:

• Minimal Flavor Violation (MFV): The RGEs support potential tests for the MFV hypothesis, allowing the SM EFT to be a robust tool for exploring flavor symmetries and their violations.
• Operator Basis Reduction: The work exemplifies the importance of choosing a non-redundant operator basis to eliminate unphysical redundancies, simplifying calculations and interpretations.

Experimental Implications:

• By integrating the full one-loop RGE into analysis routines, experiments can achieve higher precision in constraining BSM physics scenarios, significantly reducing the parameter space of viable new theories.
• Precision measurements at the electroweak scale can now be interpreted with a higher degree of confidence about external corrections, improving the trustworthiness of any emergent discrepancies with SM predictions.

Conclusion

This paper significantly advances the EFT approach to particle physics by bringing mathematical precision to its treatment of dimension-six operators. As collider experiments continue to search for hints of new physics, the comprehensive framework established by this work equips researchers with the necessary tools to analyze data through the lens of a finely-tuned Standard Model extension. The future of particle physics, both theoretical exploration and experimental precision, will inevitably benefit from such rigorous foundations.