Renormalization Group Evolution of the Standard Model Dimension Six Operators I: Formalism and lambda Dependence (1308.2627v4)

Abstract: We calculate the order \lambda, \lambda^2 and \lambda y^2 terms of the 59 x 59 one-loop anomalous dimension matrix of dimension-six operators, where \lambda and y are the Standard Model Higgs self-coupling and a generic Yukawa coupling, respectively. The dimension-six operators modify the running of the Standard Model parameters themselves, and we compute the complete one-loop result for this. We discuss how there is mixing between operators for which no direct one-particle-irreducible diagram exists, due to operator replacements by the equations of motion.

• The paper presents a detailed one-loop RGE analysis of dimension-six operators in the SM, focusing on λ, λ², and λy² contributions.
• It employs naive dimensional analysis to normalize 59 independent operators, revealing the systematic structure of the anomalous dimension matrix.
• The study uncovers significant effects on key SM parameters like the Higgs mass, providing insights crucial for probing BSM physics.

An Analysis of Renormalization Group Evolution in the Standard Model Dimension Six Operators

This paper presents a detailed examination of the renormalization group evolution (RGE) for dimension-six operators within the Standard Model (SM) framework. The focus of this paper is on the $\lambda$, $\lambda^2$, and $\lambda y^2$ contributions to the $59 \times 59$ anomalous dimension matrix, where $\lambda$ represents the Higgs self-coupling and $y$ denotes a generic Yukawa coupling. The paper employs a rescaling mechanism using naive dimensional analysis to normalize these operators effectively.

Overview of Dimension-Six Operators

Dimension-six operators provide a natural means to explore beyond the Standard Model (BSM) physics by parameterizing the effects of new physics occurring at a higher energy scale $\Lambda$. The paper identifies 59 independent dimension-six operators that form the basis for this analysis. These operators are classified into categories based on their field content: scalar sector ($H^6$), gauge boson interactions ($X^3$, $X^2 H^2$), Yukawa interactions ($\psi^2 H^3$, $\psi^2 X H$), and four-fermion interactions ($\psi^4$).

Key Contributions

1. Anomalous Dimension Matrix Structure: The structure of the anomalous dimension matrix is rigorously detailed, revealing a systematic pattern that links operator classes based on the leading order in various coupling constants: gauge couplings ($g$), Yukawa couplings ($y$), and the Higgs self-coupling ($\lambda$). Importantly, the matrix classification highlights instances where operator mixing, typically absent in one-particle irreducible diagrams, emerges due to the redefinition of operators using equations of motion (EOM).
2. Explicit RGE Calculations: The paper meticulously calculates the one-loop anomalous dimensions, specifically focusing on terms involving powers of $\lambda$, $\lambda^2$, and $\lambda y^2$. These calculations underscore the significant contributions to the RGE of dimension-six operators from these terms, which predominantly affect deviations that are not captured by loop corrections alone.
3. Implications for Standard Model Parameters: Beyond detailing the evolution of dimension-six operators, the paper provides a complete analysis of how these operators influence the running of SM parameters, particularly the Higgs mass and self-coupling constant. This extension is critical for understanding the stability of the electroweak vacuum and the effective potential of the Higgs field.
4. Mixing Between “Tree” and “Loop” Operators: The paper also explores mixing phenomena between operators that manifest through higher-order corrections in EFT, indicating that even nominally unrelated operator classes (e.g., $\psi^2 X H$ and $\psi^4$) can exhibit intertwined evolution paths due to the standard model dynamics.

Numerical and Theoretical Implications

The numerical results indicate that the $\lambda$-dependent terms, particularly those with large combinatorial factors, have substantial effects on the evolution of operator coefficients. This insight is essential for devising strategies to incorporate higher-dimension operator effects into current SM analyses, especially in collider experiments like those conducted at the LHC where precision measurements of Higgs boson properties are paramount.

Future Directions

This research sets the groundwork for further exploration into the complete RGE behavior of dimension-six operators, suggesting that subsequent studies could address full gauge, Yukawa, and Higgs coupling dependencies. Additionally, the insights from this paper can guide future theoretical advancements in understanding the cosmological implications of BSM models that affect early universe physics via higher-dimension operator contributions.

Conclusion

This paper provides foundational insights into the renormalization dynamics of dimension-six operators within the Standard Model. By elucidating the complex interplay between these operators and standard model parameters, the research enhances our understanding of potential BSM physics and guides ongoing efforts in high-energy physics to probe new phenomena beyond the Standard Model.