A Lattice Monte Carlo Study of the Hot Electroweak Phase Transition

TL;DR

This study combines dimensional reduction with 3D lattice Monte Carlo simulations to investigate the hot electroweak phase transition. By reducing the 4D SU(2) + fundamental Higgs system to a 3D SU(2) + adjoint Higgs + fundamental Higgs theory with temperature-dependent coefficients, the authors separate perturbative Debye screening from nonperturbative 3D dynamics and test predictions via lattice computations. They find that, while the 1-loop continuum analysis suggests a second-order transition at small Higgs masses, the lattice simulations show a clear first-order transition for $m_H=35$ GeV with a transition temperature $T_c$ close to but below the perturbative value, highlighting the importance of nonperturbative effects beyond the $\phi^3$ term. For larger Higgs masses, the results are inconclusive due to long correlation lengths and finite-volume effects, underscoring the need for larger lattices and possibly fermionic contributions to fully resolve the phase structure and its implications for electroweak baryogenesis.

Abstract

We study the finite temperature electroweak phase transition with lattice perturbation theory and Monte Carlo techniques. Dimensional reduction is used to approximate the full four-dimensional SU(2) + a fundamental doublet Higgs theory by an effective three-dimensional SU(2) + adjoint Higgs + fundamental Higgs theory with coefficients depending on temperature via screening masses and mass counterterms. Fermions contribute to the effective theory only via the $N_F$ and $m_{\rm top}$ dependence of the coefficients. For sufficiently small lattices ($N^3 < 30^3$ for $m_H$ = 35 GeV) the study of the one-loop lattice effective potential shows the existence of the {\em second} order phase transition even for the small Higgs masses. At the same time, a clear signal of a {\em first order} phase transition is seen on the lattice simulations with a transition temperature close to but less than the value determined from the perturbative calculations. This indicates that the dynamics of the first order electroweak phase transition depends strongly on non-perturbative effects and is not exclusively related to the so-called $φ^3$ term in the effective potential.

Figures (6)

• Figure 1: The relation between $T$ and $\beta_H$ as given by eq. (\ref{['betahT']}) for $m_H=$ 35 and 80 GeV, $\beta_G=20$ and $N_F=0$. The arrows denote the perturbative values of $T_c$ as given in Table 1.
• Figure 2: The distribution of $L = \hbox{Tr,} V^\dagger(\hbox{\bf x})U_i(\hbox{\bf x})V(\hbox{\bf x}+i)$ for $m_H=35$ GeV for $8^3$ and $20^3$ lattices for $\beta_G = 20$ and different values of $\beta_H$.
• Figure 3: The Monte Carlo time history of the lattice simulations for $m_H$ = 35 GeV on a $20^3$ lattice near critical $\beta_H$.
• Figure 4: The dependence of the second moment of the order parameter $L$ on $\beta_H$ for different lattice sizes and $m_H = 35$ GeV.
• Figure 5: The distribution of $L = \hbox{Tr,} V^\dagger(\hbox{\bf x})U_i(\hbox{\bf x})V(\hbox{\bf x}+i)$ for $m_H=80$ GeV and a $16^3$ lattice for different values of $\beta_H$.