The 2-loop MSSM finite temperature effective potential with stop condensation

TL;DR

This work analyzes the finite-temperature dynamics of the MSSM with a light right-handed stop by constructing a 3d effective theory and computing the 2-loop finite-temperature potential in a two-field background (Higgs and stop). It demonstrates that a two-stage electroweak phase transition, mediated by stop condensation followed by a transition to the standard EW minimum, is possible only in a narrow region of parameter space (roughly $m_H\lesssim 100$ GeV and $m_{\\tilde{t}_R} \sim 155-160$ GeV). The study highlights substantial 2-loop and gauge/scale uncertainties, analyzes the real-time implications of such a transition, and discusses the need for nonperturbative checks (e.g., lattice simulations) to confirm viability and its potential impact on baryogenesis via sphaleron suppression. Overall, while intriguing, the two-stage scenario remains delicate and highly parameter-dependent, with significant theoretical caveats.

Abstract

We calculate the finite temperature 2-loop effective potential in the MSSM with stop condensation, using a 3-dimensional effective theory. We find that in a part of the parameter space, a two-stage electroweak phase transition appears possible. The first stage would be the formation of a stop condensate, and the second stage is the transition to the standard electroweak minimum. The two-stage transition could significantly relax the baryon erasure bounds, but the parameter space allowing it (m_H \lsim 100 GeV, m_tR \sim 155-160 GeV) is not very large. We estimate the reliability of our results using renormalization scale and gauge dependence. Finally we discuss some real-time aspects relevant for the viability of the two-stage scenario.

Figures (6)

• Figure 1: The 2-loop effective potential at $T=T_c^\phi=T_c^\chi=T_c^{\chi\to\phi}$ for $\tan\!\beta=5$ ($m_H\sim 92$ GeV), $m_{\tilde{t}_R}=158.3$ GeV, $\bar{\mu}=T$, $\xi=\zeta=0$.
• Figure 2: The 1-loop and 2-loop effective potentials at the corresponding critical temperatures, for $\tan\!\beta=5, m_{\tilde{t}_R}=158.3$ GeV. It is seen that there may be a relatively large uncertainty in the prediction for the surface tension due to the large $\bar{\mu}$-dependence, especially for the $\phi$-direction. The figure has been drawn at the special point $m_{\tilde{t}_R}=m_{\tilde{t}_R,c}$ so that the critical temperatures $T_c^{\rm 2-l}$ of the two transitions are equal (see Fig. \ref{['Tc']}). The $\bar{\mu}$-dependence is especially large at this point, since the mass parameters $m_{H3}^2(\bar{\mu})$, $m_{U3}^2(\bar{\mu})$ are close to zero so that the relative change can be significant.
• Figure 3: The critical temperatures of the three transitions as a function of $m_{\tilde{t}_R}$ for $\tan\!\beta=3,12$ (thin lines) and $\tan\!\beta=5$ (thick lines). The two-stage transition would take place to the left of the crossing point of the three critical curves; it is seen that a two-stage transition is possible, but there is not very much parameter space for it. The continuations of $T^\phi_c$ to the left of the crossing point, and $T^\chi_c$ to the right of it, roughly represent the region of metastability of the transition with a higher $T_c$. Error bars indicate the dependences on $\bar{\mu}, \xi, \zeta$ (for $\tan\!\beta=5$), as explained in the text; the actual curves correspond to the Landau gauge and $\bar{\mu}=T$. In terms of the parameter $m_U^2=m_{\tilde{t}_R}^2-m_t^2$, the $x$-axis is from $-(70$ GeV)$^2$ to $-(40$ GeV)$^2$.
• Figure 4: The expectation values $\phi_{\rm min}/T_c$, $\chi_{\rm min}/T_c$ of the three transitions (see Fig. \ref{['Tc']}) in the Landau gauge as a function of $m_{\tilde{t}_R}$ for $\tan\!\beta=3,5,12$. The broken phase transition from $\chi_{\rm min}>0$ to $\phi_{\rm min}>0$ is seen to easily lead to very large values of $\phi_{\rm min}$. For clarity the metastable branches are not shown here.
• Figure 5: The latent heats of the transitions as a function of $m_{\tilde{t}_R}$ for $\tan\!\beta=3,5,12$. Note that at $m_{\tilde{t}_R,c}$, $L^{\chi\to\phi}=L^{\phi}-L^{\chi}$. For smaller $m_{\tilde{t}_R}$, $L^{\chi\to\phi}$ grows rapidly. Metastable branches are not shown.