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

TL;DR

This work provides a comprehensive renormalization-group analysis of all 59 dimension-six SMEFT operators, focusing on the one-loop contributions that depend on the Higgs self-coupling $\lambda$ and Yukawa couplings $y$. It develops a formalism for the full anomalous-dimension matrix, clarifies how equations of motion induce operator mixing, and presents explicit $\lambda$-dependent RGEs both for the dimension-six operators and for the Standard Model parameters themselves via the $\mathcal{L}^{(6)}$ terms. A key result is the explicit presentation of the $\lambda$, $\lambda^2$, and $\lambda y^2$ pieces of the one-loop matrix, including large combinatorial factors and the nuanced interplay between tree-level and loop-induced operators. These findings enable more precise SMEFT predictions for Higgs processes and potential implications for Higgs potential stability and precision fits at high scales.

Abstract

We calculate the order λ, λ^2 and λy^2 terms of the 59 x 59 one-loop anomalous dimension matrix of dimension-six operators, where λand 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.

Figures (4)

• Figure 1: Penguin diagram contributing to $s \to d$ transitions.
• Figure 2: (a) A $H^6 - \psi^2 H^2 D$ anomalous dimension graph which vanishes. (b) A $g^3 X^3-g^2X^2 H^2$ anomalous dimension graph of order $1$. (c) A $g^2X^2 H^2- g^3 X^3$ anomalous dimension graph of order $g^4$. The solid square is a vertex from $\mathcal{L}^{(6)}$ and the dots are vertices from $\mathcal{L}_{\rm SM}$.
• Figure 3: Diagram contributing to the $\psi^2 X H- \psi^4$ anomalous dimension $\gamma_{68}$ given in Eq. (\ref{['68']}). The solid square is a $\psi^4$ vertex from $\mathcal{L}^{(6)}$ and the dots are gauge and Yukawa vertices from $\mathcal{L}_{\rm SM}$.
• Figure 4: Graph contributing to the $H^4 D^2-H^4D^2$ anomalous dimension and to EOM operators. The solid square is a $H^4 D^2$ vertex from $\mathcal{L}^{(6)}$ and the dot is the $\lambda (H^\dagger H)^2$ vertex from $\mathcal{L}_{\rm SM}$.