The universality class of the electroweak theory

TL;DR

The paper investigates the universality class of the electroweak phase transition endpoint in a three-dimensional SU(2)+Higgs theory, showing it belongs to the 3d Ising universality class. Using lattice Monte Carlo, it analyzes two- and multi-observable probability distributions to locate the critical endpoint and to map M-like and E-like directions, confirming Ising-like critical indices (e.g., γ/ν ≈ 1.96, ν ≈ 0.63) despite sizable asymmetric corrections to scaling. The continuum endpoint occurs at x_c ≈ 0.0983(15), translating to a Higgs mass m_H ≈ 72(2) GeV for sin^2θ_W = 0 and ≈ 77(2) GeV for sin^2θ_W = 0.23, implying no EW phase transition in the SM with the physical Higgs mass; however, in MSSM-like theories a first-order transition could persist for some parameter choices. The work provides a general, nonperturbative framework to determine universality classes from lattice data and has implications for beyond-Standard-Model electroweak dynamics.

Abstract

We study the universality class and critical properties of the electroweak theory at finite temperature. Such critical behaviour is found near the endpoint m_H=m_{H,c} of the line of first order electroweak phase transitions in a wide class of theories, including the Standard Model (SM) and a part of the parameter space of the Minimal Sypersymmetric Standard Model (MSSM). We find that the location of the endpoint corresponds to the Higgs mass m_{H,c} = 72(2) GeV in the SM with sin^2 theta_W = 0, and m_{H,c} < 80 GeV with sin^2 theta_W = 0.23. As experimentally m_H > 88 GeV, there is no electroweak phase transition in the SM. We compute the corresponding critical indices and provide strong evidence that the phase transitions near the endpoint fall into the three dimensional Ising universality class.

Figures (15)

• Figure 1: Left: The phase diagram of the 3d SU(2)+Higgs theory. The datapoints are from nonpert and from this paper. The value of $x\cdot y_c$ at $x\to 0$ is given by the 1-loop effective potential and is hence known analytically. Right: The phase diagram of the 3d scalar $\phi^4$ theory in Eq. (\ref{['isingaction']}). The value of $x_{I}$ at the endpoint has to our knowledge not been determined. The number 0.002 on the vertical axis is symbolic, as the figure is scale invariant in this direction.
• Figure 2: (a) 1000 configurations from the Monte Carlo simulation of the theory in Eq. (\ref{['latticeaction']}), represented by points in the $S_{(\phi^2-1)^2}$ vs. $S_{\hbox{\scriptsize hopping}}$ plane, for $x=0.105253, \beta_G=8,\beta_H=0.349853, V=48^3$. (b) 13822 configurations of the same system, for the same parameter values, after a shift and rotation in the coordinate plane. The angle of rotation is chosen to make the elongated distribution in (a) go approximately horizontally. (c) 20000 configurations of the 3d Ising model on a $58^3$ lattice at the critical point $\beta_c=0.221654,\ h=0$, on "minus the energy" ($0<\sum_{\langle ij\rangle}\delta_{s_is_j}<3\cdot 58^3$) vs. magnetization ($-58^3<\sum_i s_i<58^3$) plane.
• Figure 3: The smoothed and normalized probability distributions, at the critical point, for (a) the Ising model at the volume $58^3$, (b) the O(2) spin model at $64^3$, (c) the O(4) spin model at $64^3$. The $x$-axis is the magnetic direction and the $y$-axis the energy direction.
• Figure 4: The reweighted $x_c$ from simulations at $\beta_G=5,x = 0.112706$, as a function of the volume.
• Figure 5: The probability distributions $P(M,E)$ ( left) and $P(M)$ ( right) at the infinite volume critical point, for the volumes (a) $16^3$, (b) $32^3$, (c) $64^3$. It is seen how the distribution becomes more symmetric for increasing volumes. The $M$ and $E$ directions have been found with a 6-dimensional fluctuation matrix analysis, see Sec. \ref{['sec:extending']}.