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

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 $x\equiv {\lambda }_3\mathrm{/}{g}_3^2$ and $y\equiv m_3^2\mathrm{/}{g}_3^4$, where ${\lambda }_3$, $m_3^2$ 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 $v\mathrm{/}{T}_c\equiv \sqrt{2\mathrm{\Delta }⟨{\phi }^{†}\phi ⟩}\mathrm{/}{T}_c\gtrsim 1.33$ 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.