Constraining the symmetry energy with heavy-ion collisions and Bayesian analyses

Efficiency corrected single ratios of neutron and proton spectra in central $^{112}$Sn+$^{112}$Sn and $^{124}$Sn+$^{124}$Sn collisions at 120 MeV/u are combined with double ratios to provide constraints on the density and momentum dependencies of the isovector mean-field potential. Bayesian analyses of these data reveal that the isoscalar and isovector nucleon effective masses, $m_s^* - m_v^*$ are strongly correlated. The linear correlation observed in $m_s^* - m_v^*$ yields a nearly independent constraint on the effective mass splitting $\Delta m_{np}^*= (m_n^*-m_p^*)/m_N = -0.05_{-0.09}^{+0.09}\delta$. The correlated constraint on the standard symmetry energy, $S_0$ and the slope, $L$ at saturation density yields the values of symmetry energy $S(\rho_s)=16.8_{-1.2}^{+1.2}$ MeV at a sensitive density of $\rho_s/\rho_0 = 0.43_{-0.05}^{+0.05}$.