Running Couplings in High-Temperature Effective Field Theory

Abstract

In this work, we study the renormalization-group evolution of parameters in the three-dimensional effective field theory (3D EFT) that describes the thermally driven electroweak phase transition of the Higgs field. We focus on the first-order case, triggered by beyond the Standard Model physics. We compute the two-loop running of the 3D EFT couplings, including the effect of the leading non-renormalizable terms. We then analyze how the new contributions to the beta functions compare with those in the super-renormalizable case, highlighting their impact on perturbative computations of the scalar potential, which describes the vacuum structure of the theory. By incorporating higher-order corrections in the mass parameter evolution, as well as the running of other effective operators, we set the stage for testing their impact on phase transition dynamics in lattice simulations.

Figures (6)

• Figure 1: Example diagrams for the renormalization of $c_{\phi^8}$. Gray blobs represent effective operators of the second line of \ref{['eq:lagrangian3d']}, while black dots stand for renormalizable gauge, quartic and sextic scalar couplings.
• Figure 2: Running of the effective couplings $m_3^2$, $\lambda_3$, $c_{\phi^6}$ and $c_{\phi^8}$ from the hard scale $2\pi T$ down to $\mu_{3D}$. We take $T = 100\,\text{GeV}\xspace$, and fix the effective couplings $m^2_{3}=100\,\text{GeV}\xspace^2$, $\lambda_{3}=-4\,\text{GeV}\xspace$, $c_{\phi^6}=0.04$, $c_{\phi^4D^2}^{(1)} = c_{\phi^4D^2}^{(2)}= 10^{-4} \text{GeV}\xspace^{-1}$, $c_{\phi^2W^2}=c_{\phi^8}= 10^{-6}\,\text{GeV}\xspace^{-1}$, $g_3^2 = 2\,\text{GeV}\xspace$ at the hard scale.
• Figure 3: Relative correction to the $m_3^2$ beta function from the effective operators. The black dashed line denotes the parameter values for which two minima of the scalar potential are degenerate.
• Figure 4: Scalar potential $V (\phi)$ without running, with LO running, and with NLO running of the effective couplings. Parameters run from $2\pi T$ down to $g^2 T$, with $g=0.1$ and $T=100$ GeV.
• Figure 5: Relative change of the position of the broken minimum of the scalar potential ($v$) with the inclusion of the LO mass parameter running only (left) and NLO running of the effective parameters (right). Parameters run from $2\pi T$ down to $g^2 T$, with $g=0.1$ and $T=100$ GeV. Grey regions represent the parameter space where the broken minimum is absent in case no running is included. The black-dashed line denotes the parameter values for which two minima of the potential are degenerate in the absence of RG running.