Non-perturbative determination of the sphaleron rate for first-order phase transitions

TL;DR

This work non-perturbatively determines the sphaleron rate in the Higgs phase by simulating a minimal 3D SU(2) + Higgs effective theory, parameterized by $x = \lambda_3/g_3^2$ and $y = m_3^2/g_3^4$. By coupling lattice results for the sphaleron rate with non-perturbative data on bubble nucleation and percolation, the authors identify a critical boundary $x_c \approx 0.025$ below which sphaleron washout is suppressed inside expanding bubbles, translating to $v/T_c \gtrsim 1.33$ for a gauge-invariant Higgs condensate discontinuity. They present a general template for translating high-temperature BSM scenarios with heavy fields into the EFT framework, enabling a rapid viability assessment for electroweak baryogenesis. A comparison with perturbative semi-classical results shows near-agreement in the relevant regime, while highlighting the importance of non-perturbative checks. The study provides a practical, EFT-based pathway to evaluate EW baryogenesis viability across a wide class of BSM theories.

Abstract

In many extensions of the Standard Model electroweak phase transitions at high temperatures can be described in a minimal dimensionally reduced effective theory with SU(2) gauge field and fundamental Higgs scalar. In this effective theory, all thermodynamic information is governed by two dimensionless ratios $x \equiv λ_3/g^2_3$ and $y\equiv m^2_3/g^4_3$, where $λ_3$, $m^2_3$ and $g_3$ are the effective thermal scalar self-interaction coupling, the thermal mass and the effective gauge-coupling, respectively. By using non-perturbative lattice simulations to determine the rate of sphaleron transitions in the entire $(x,y)$-plane corresponding to the Higgs phase, and by applying previous lattice results for the bubble nucleation, we find a condition $x(T_c) \lesssim 0.025$ to guarantee preservation of the baryon asymmetry, which translates to $v/T_c \equiv \sqrt{2 Δ\langle φ^\dagger φ\rangle}/T_c \gtrsim 1.33$ for the (gauge-invariant) discontinuity in Higgs condensate. This indicates that viability of the electroweak baryogenesis requires the phase transition to be slightly stronger than previously anticipated. Finally, we present a general template for analysing such viability in a wide class of beyond the Standard Model theories, in which new fields are heavy enough to be integrated out at high temperature.

Figures (9)

• Figure 1: The sphaleron rate for all the different $x$ and $y$ values we computed. These are for the smallest lattice spacing used $ag_3^2 = 0.25$ with volume $V=32^3a^3$. For each point the rate was computed from \ref{['sph_rate_lattice']} with $\sim 5\times 10 ^5$ samples for the statistical part and $\sim 4000$ trajectories for the dynamical part. The data can be found in zenodo_data.
• Figure 2: The sphaleron rate for large and small $x$ values with different lattice spacings with fixed volume. We show both a linear $1+a$ and quadratic $1+a^2$ fits to the data points extrapolated to $a\to 0$. For large $x$ there is no noticeable lattice spacing dependence for the sizes simulated. For small $x$ there is noticeable dependence on the lattice spacing. For small $x$ the quadratic fits are used to estimate the error from not taking the continuum limit.
• Figure 3: Sphaleron rate at three different $x$ values as a function of $y$. The blue data points are the full rate and the orange points show only the statistical part as defined in \ref{['sph_rate_lattice']}. The solid black lines are the fits to the full rate $\ln\tilde{\Gamma}$. The gray dotted curves are the estimates from a semi-classical computation without a dynamical factor (see Eq. \ref{['rate_semiclassical']}), shifted by an order $O(10)$ number inserted by hand, as explained in the main body. On the right most panel the dot-dashed line is the semi-classical computation for large $x$ values (see Eq. \ref{['eq:large-x-pert']}).
• Figure 4: The sphaleron rate in the $(x,y)$ plane. The contour and black dotted constant rate lines are obtained from linearly interpolating the data obtained from simulations with $a g_3^2 = 0.25$, $V = 32^3 a^3$. The orange almost vertical line corresponds to the SM mapping for temperatures $130-167~$GeV. The gray dots with error bars are the sphaleron freeze-out $y$-values for fixed $x$, i.e. $y_f(x)$. The black solid line is a fit to small $x\lesssim 0.03$ values of $y_f$. The red and blue solid lines are the non-perturbative results for $y_c(x)$ and $y_p(x)$ respectively Gould:2022ran. The star denotes the first-order phase transition end point $x_*=0.0983$Gould:2022ran, i.e. for larger values of $x$ the transition turns into a smooth crossover. On the left panel the dot-dashed box denotes the area that is zoomed in on the right panel which shows the relevant range for strong first-order phase transitions.
• Figure 5: Zoomed in plot of Fig. \ref{['fig:rate_contours']} where the sphaleron freeze-out $y_f$ and the nucleation temperature $y_p$ curves intersect at $0.0251^{+0.0012}_{-0.0002}$. The error bounds for the blue $y_p$ curve are from varying the cosmology condition $\pm 5$. The black solid line is a linear fit to the obtained $y_f(x)$ values for $x < 0.03$ and the gray error bound is a combination of the fit error and the estimated continuum limit error. The linear fit for small $x$ is $y_f = (-5.65\pm 0.14)x + (0.164\pm 0.004)$.