A Non-Perturbative Analysis of the Finite T Phase Transition in SU(2)xU(1) Electroweak Theory

TL;DR

Kajantie, Laine, Rummukainen, and Shaposhnikov study the finite-temperature phase structure of the 3d SU(2)×U(1)+Higgs theory using lattice Monte Carlo, focusing on how the U(1) sector affects non-perturbative dynamics. They map 4d high-temperature theories to the 3d effective theory and examine two representative Higgs masses, comparing lattice results with perturbative predictions. The inclusion of U(1) shifts observables in line with perturbative expectations, but does not alter the qualitative phase diagram: a first-order line ends at moderate x and a massless photon persists in both phases, with no spontaneous parity breaking observed. At higher Higgs masses, the transition becomes a smooth cross-over, implying that magnetic screening remains absent in the hot electroweak plasma non-perturbatively.

Abstract

The continuum 3d SU(2)$\times$U(1)+Higgs theory is an effective theory for a large class of 4d high-temperature gauge theories, including the minimal standard model and some of its supersymmetric extensions. We study the effects of the U(1) subgroup using lattice Monte Carlo techniques. When $g'^2/g^2$ is increased from the zero corresponding to pure SU(2)+Higgs theory, the phase transition gets stronger. However, the increase in the strength is close to what is expected perturbatively, and the qualitative features of the phase diagram remain the same as for $g'^2=0$. In particular, the first order transition still disappears for $m_H>m_{H,c}$. We measure the photon mass and mixing angle, and find that the mass vanishes in both phases within the statistical errors.

Figures (15)

• Figure 1: The expected schematic phase diagram of the SU(2)$\times$U(1)+Higgs theory in the space of the parameters of eq.(\ref{['3dvariables']}). There is a 1st order transition which terminates in a line of 2nd order transitions.
• Figure 2: The value of $z=g_3'^2/g_3^2=z_c$ at $T=T_c$ as a function of the physical Higgs mass $m_H$ and the top quark mass $m_{\rm top}$ for the minimal standard model. The dependence shown arises mainly from the dependence of $m_W$ on $m_H$, $m_{\rm top}$ (see generic for the formulas used), and gives an estimate of the uncertainty in the physical value of $z_c$. The value of $T_c$ is computed from $m_3^2(g_3^2)=0$; the dependence of $z_c$ on $T_c$ is only logarithmic.
• Figure 3: Examples of parameter values corresponding to $x=0.06444$ in the MSSM. Here $m_{\tilde{t}_R}$ is the right-handed stop mass, $m_H$ is the lightest CP-even Higgs mass and $m_A$ is the CP-odd Higgs mass. The squark mixing parameters have been put to zero.
• Figure 4: The curves show the values of $y_c(x,z)$ computed from the 3d 2-loop effective potential of the SU(2)$\times$U(1)+Higgs model. Lattice Monte Carlo results are also shown. The perturbative curves continue to large values of $x$, but on the lattice the first order transition ends at $x\sim 1/8$. Beyond that in the cross-over region, the values obtained for $z=0$ with the max($\chi_{R^2}$)-method (see Sec. 6.1) are $y_c(0.274,0)=-0.066,\,y_c(0.624,0)=-0.16$.
• Figure 5: The same as Fig. \ref{['xyz']} but for the jump of the order parameter and the interface tension. At $x\approx 0.096$, the interface tension was estimated in leip to be $\sigma_3\sim 1\times 10^{-3}$. For $x\mathop{\hbox{$>$$\sim$}} 1/8$, there is no transition and $\Delta\ell_3$, $\sigma_3$ vanish.