Sphalerons in the MSSM

TL;DR

This work computes sphaleron solutions in the MSSM at zero and finite temperature, incorporating leading one-loop corrections to the effective potential in the sphaleron background. The MSSM sphaleron energy $E_{MSSM}$ largely tracks the effective Higgs mass $m_h^{eff}$ and agrees with the SM value $m_h^{SM}=m_h^{eff}$ within a few percent. At finite temperature, daisy-resummed one-loop corrections reveal that $E_{sph}(T)$ decreases with $T$, and the scaling law $E_{sph}^{scal}(T)=E_{sph}(0)rac{v(T)}{v}$ is accurate to about 3% for weak transitions and up to ~10% for strong transitions. The authors identify a baryogenesis window in the light-stop MSSM region characterized by $m_h\lesssim 80$ GeV, $m_A\gtrsim 110$ GeV, $A_t\lesssim 0.4 m_Q$, and $ an\beta\lesssim 3$, consistent with LEP constraints, thereby providing concrete parameter-space guidance for MSSM electroweak baryogenesis. These results quantify how MSSM parameters influence sphaleron dynamics and the viability of baryogenesis, informing both theory and experimental searches.

Abstract

We construct the sphaleron solution, at zero and finite temperature, in the Minimal Supersymmetric Standard Model as a function of the supersymmetric parameters, including the leading one-loop corrections to the effective potential in the presence of the sphaleron. At zero temperature we have included the one-loop radiative corrections, dominated by the top/stop sector. The sphaleron energy $E_{\rm MSSM}$ mainly depends on an effective Higgs mass ${\displaystyle m_h^{\rm eff}=\lim_{m_A\gg m_W} m_h}$, where $m_h$ is the lightest CP-even Higgs mass and $m_A$ the pseudoscalar mass. We have compared it with the Standard Model result, with $m_h^{\rm SM}=m_h^{\rm eff}$, and found small differences (1-2\%) in all cases. At finite temperature we have included the one-loop effective potential improved by daisy diagram resummation. The sphaleron energy at the critical temperature can be encoded in the temperature dependence of the vacuum expectation value of the Higgs field with an error $\simlt 10$\%. The light stop scenario has been re-examined and the existence of a window where baryon asymmetry is not erased after the phase transition, confirmed. Although large (low) values of $m_h$ ($m_A$) are disfavoured by the strength of the phase transition, that window (along with LEP results) allows for $m_h\simlt 80$ GeV, $m_A\simgt 110$ GeV and $A_t\simlt 0.4\; m_Q$.

Figures (15)

• Figure 1: Plots of $h_1(r)/v$, $h_2(r)/v$ and $f(r)$ at zero (solid lines) and the critical temperature $T=T_c$ (dashed lines) for $m_t=175$ GeV and the values of supersymmetric parameters: $\tan\beta=1.5$, $m_A=100$ GeV, $m_Q=500$ GeV, $m_U=0$ and $A_t=\mu=0$.
• Figure 2: Plot of $E_{\rm sph}$ as a function of $\tan\beta$ for $m_Q=500$ GeV, $A_t=\mu=0$ and: a) $m_A=100$ GeV, $m_U=0$; b) $m_A=500$ GeV, $m_U=0$; c) $m_A=100$ GeV, $m_U=400$ GeV; and, d) $m_A=500$ GeV, $m_U=400$ GeV.
• Figure 3: Plot of $E_{\rm sph}$ as a function of $m_A$ for $m_Q=500$ GeV, $A_t=\mu=0$ and: a) $\tan\beta=2$, $m_U=0$; b) $\tan\beta=15$, $m_U=0$; c) $\tan\beta=2$, $m_U=400$ GeV; and, d) $\tan\beta=15$, $m_U=400$ GeV.
• Figure 4: The same as in Fig. 2 but as a function of the effective Higgs boson mass, $m_h^{\rm eff}$. The dashed line is $E_{\rm sph}$ for the Standard Model with a Higgs boson with mass $m_h^{\rm eff}$.
• Figure 5: Plot of $E_{\rm sph}$ as a function of $A_t$ for $m_A=m_Q=500$ GeV, $\mu=m_U=0$, and: a) $\tan\beta=2$; b) $\tan\beta=15$.