Electroweak phase transition, critical bubbles and sphaleron decoupling condition in the MSSM

TL;DR

This work reevaluates electroweak baryogenesis in the MSSM by computing the finite-temperature effective potential and the sphaleron decoupling condition, including zero-mode fluctuations, at both the critical and nucleation temperatures. Focusing on the Light Higgs boson scenario and the decoupling limit, the authors quantify $v/T$ enhancements at the nucleation temperature $T_N$ relative to the critical temperature $T_C$, and determine whether sphaleron processes are sufficiently suppressed after the transition. They find a typical ~10% increase in $v/T$ at $T_N$, but the resulting values ($v_N/T_N\sim1.01$–$1.05$) still fail to meet the MSSM sphaleron decoupling threshold of about $1.38$, even before considering higher-order corrections. Their results, constrained by LEP and $B$-physics data, suggest electroweak baryogenesis remains unlikely in the MSSM under current bounds, though they outline plausible loopholes—such as two-loop effects or metastable vacua—that could alter the outcome.

Abstract

The electroweak phase transition and the sphaleron decoupling condition in the MSSM are revisited taking the latest experimental data into account. The light Higgs boson scenario and the ordinary decoupling limit which are classified by the relative size between the CP-odd Higgs boson mass and Z boson mass are considered within the context of electroweak baryogenesis. We investigate v/T at not only the critical temperature at which the effective potential has two degenerate minima but also the nucleation temperature of the critical bubbles, where v is a vacuum expectation value of the Higgs boson and T denotes a temperature. It is found that v/T at the nucleation temperature can be enhanced by about 10% compared to that at the critical temperature. We also evaluate the sphaleron decoupling condition including the zero mode factors of the fluctuations around sphaleron. It is observed that the sphaleron decoupling condition at the nucleation temperature is given by v/T>1.38 for the typical parameter sets. In any phenomenologically allowed region, v/T at both the critical and nucleation temperatures cannot be large enough to satisfy such a sphaleron decoupling condition.

Figures (5)

• Figure 1: The temperature dependences of the sphaleron energy $\mathcal{E}=E_{\rm sph}/(4\pi v/g_2)$ and the zero mode factors $\mathcal{N}_{\rm tr, rot}$.
• Figure 2: The allowed region of $\overline{B}\to X_s\gamma$ in the $|\mu|$-$|A|$ plane. $\tan\beta=12$ and $m_{H^\pm}=130$ GeV are taken.
• Figure 3: The allowed region in the LHS. We take $A_t=A_b=-300$ GeV, $\mu=100$ GeV, $m_{\tilde{q}}=1200$ GeV, $m_{\tilde{t}_R}=10^{-4}$ GeV, and other input parameters are presented in the text.
• Figure 4: $E_{\rm cb}(T)/T$ as a function of $T$ in the LHS. The blue dotted line shows $E_{\rm cb}(T_N)/T_N=150.386$, where $T_N=115.59$ GeV. We take $\tan\beta=10.1$ and $m_{H^\pm}=127.4$ GeV and other parameters are the same as in the Fig. \ref{['fig:EWPT_LHS']}
• Figure 5: Left: $h_{1}(\xi)$ (red solid curve) and $h_2(\xi)$ (blue dotted curve) are plotted for $T=116.00, 115.59~(=T_N), 114.00$ GeV. Right: $\Delta\beta(r)$ as function of $r=\xi/v_N$. The input parameters are the same as in Fig. \ref{['fig:Ecb_LHS']}