Baryon and lepton number violation rates across the electroweak crossover

TL;DR

Problem: Baryon and lepton number violation rates across the electroweak crossover strongly influence low-temperature baryogenesis scenarios. Approach: derives rate equations for $B$ and $L_i$, introduces the temperature-dependent diffusion rate $\Gamma_{\text{diff}}(T)$ and related functions, and develops a practical method to estimate $\Gamma_{\text{diff}}(T)$ for $m_H = 100$–$300$ GeV using a two-loop effective potential, polynomial fits, and lattice-inspired corrections. Key contributions: provides a framework for evolving $B(t)$ and $L_i(t)$ with a parametric fit for $\ln[\Gamma_{\text{diff}}(T)/T^4]$ and a decoupling temperature $T_*$, enabling accurate baryogenesis computations across the crossover. Significance: improves reliability of SM-like baryogenesis/leptogenesis predictions and highlights the need for dedicated lattice studies at the physical Higgs mass to reduce remaining uncertainties.

Abstract

We point out that the results of many baryogenesis scenarios operating at or below the TeV scale are rather sensitive to the rate of anomalous fermion number violation across the electroweak crossover. Assuming the validity of the Standard Model of electroweak interactions, and making use of previous theoretical work at small Higgs masses, we estimate this rate for experimentally allowed values of the Higgs mass (m_H = 100 ... 300 GeV). We also elaborate on how the rate makes its appearance in (leptogenesis based) baryogenesis computations.

Figures (4)

• Figure 1: The temperatures for which specific values of $v_{\hbox{\scriptsize min}}/T$ (in Landau gauge) are reached, as a function of the Higgs mass $m_H$. For $v_{\hbox{\scriptsize min}}/T = 1.0$ we also show the effects of the variations $m_{\hbox{\scriptsize top}} = 174.3 \pm 5.1$ GeV (dashed lines) and $\Delta = 0.25 ... 4.0$ (dotted lines).
• Figure 2: $\ln[\Gamma_{\hbox{\scriptsize diff}}(T)/T^4]$ as a function of the Higgs mass and temperature. The overall error is estimated in Eq. (\ref{['Gammaerror']}). The dotted horizontal line indicates the value which all curves approach at large $T$. The values in the range 100 GeV $\le m_H \le$ 200 GeV can be roughly approximated by Eq. (\ref{['fit']}). Note that the rate falls off more slowly at large Higgs masses.
• Figure 3: The solid line indicates the decoupling temperature $T_*$ as defined in the text (assuming a constant $g_* \simeq 106.75$), with an error band following from changing $\Gamma_{\hbox{\scriptsize diff}}(T_*)/T_*^4$ within the range of Eq. (\ref{['Gammaerror']}). The dashed lines show the corresponding anomalous rate.
• Figure 4: The function $1-\omega(t';t)$ appearing in Eq. (\ref{['soln']}), as a function of the temperature $T'$ corresponding to the time $t'$ (the final moment $t$ is fixed to the point where $T=100$ GeV). We indicate temperatures instead of times, because this significantly reduces the dependence on the constant $g_* \simeq 106.75$, which has non-negligible radiative corrections gv. This figure can be used to gauge the accuracy of the sudden decoupling approximation shown in Fig. \ref{['fig:Tstar']}.