Lepton effects on the proto-neutron stars with the hadron-quark mixed phase in the Nambu-Jona-Lasinio model

We study the structures of hybrid stars with leptons at finite temperature under beta equilibrium. For the quark phase, we use the three flavor Nambu-Jona-Lasinio (NJL) model. For the hadron phase, we adopt nuclear equation of state (EOS) by Shen et al.. This EOS is in the framework of the relativistic mean field theory including the tree body effects. For the hadron-quark phase transition, we impose the bulk Gibbs construction or the Maxwell construction to take into account uncertainties by {\it finite size effects}. We find that the pure quark phase does not appear in stable star cores in all cases. With the phase transition, the maximum masses increase $\sim 10 %$ for high lepton fraction. On the contrary, without the transition, they decrease $\sim 10 %$. We also find that, in the NJL model, the lepton fraction is more important for structures of unstable stars than the temperature. This result is important for many astrophysical phenomena such as the core collapse of massive stars.