Perturbative Contributions to the Electroweak Interface Tension

TL;DR

The paper analyzes perturbative contributions to the electroweak interface tension at high temperature, focusing on one-loop derivative corrections and the role of the massless infrared mode. It demonstrates that massive-mode corrections are largely encoded by the $Z_H(\varphi)$ factor, while higher-derivative terms are small; the massless mode requires infrared resummation or a nonperturbative cutoff but contributes modestly. The authors compare derivative, multi-local, and heat-kernel expansions, finding broad agreement and proposing a self-consistent framework to include the $Z$-factor in the interface tension calculation. They find perturbation theory remains reliable for moderate Higgs masses in the broken phase, though lattice data at higher masses hint at nonperturbative effects and gauge-dependence that motivate further work, including a two-loop $Z$-factor.

Abstract

The main perturbative contribution to the free energy of an electroweak interface is due to the effective potential and the tree level kinetic term. The derivative corrections are investigated with one-loop perturbation theory. The action is treated in derivative, in heat kernel, and in a multi local expansion. The massive contributions turn out to be well described by the Z-factor. The massless mode, plagued by infrared problems, is numerically less important. Its perturbatively reliable part can by calculated in derivative expansion as well. A self consistent way to include the Z-factor in the formula for the interface tension is presented.

Figures (5)

• Figure 1: $\Gamma_N(m_0)$ vs. $m_0^2$ . Left plot: the Higgs-fluctuation, right plot: the $W$-fluctuation. The limits of extrapolation are expected to be in the hatched range. The parameters are: $~\lambda_T/g^2 = 0.024,\; d = 18.3$ corresponding to $\bar{m}_{\rm H}= 35$ GeV.
• Figure 2: Extrapolated results of the multi local expansion and the $Z$-factor predictions vs. $\bar{m}_{\rm H}$ (eq. \ref{['mH']}). The full lines are the $Z$-factor predictions.
• Figure 3: The integrand of eq. (\ref{['evint']}) and the $Z$-factor prediction vs. $\exp(-z/d)$. The parameters are $\lambda_T/g^2=0.04, d=12.5$
• Figure 4: The one-loop contribution to the surface tension in the derivative expansion. The full line is the $Z$-factor contribution; the dashed line includes the terms up to order $\partial^4$; the dot-dashed line up to $\partial^6$.
• Figure 5: The interface tension vs. $\lambda_T/g^2$. $~\sigma$ of eq. (\ref{['sigma']}) is plotted in the upper two lines; $\tilde{\sigma}$ of eq. (\ref{['sigmalit']}) in the lower two lines. The dashed lines are the Feynman gauge results, the full lines are the Landau gauge values. The two data points are the lattice values of ref. KajantieEA2.