QCD equation of state and the structure of up-down quark stars in NJL model

The chiral and pion superfluidity phase transitions are studied within NJL model. The model parameters are fitted by pion mass and decay constant together with recent isospin density from lattice QCD. The dense and cold QCD matter suffered the first order chiral phase transition during the quark chemical potential increase. Based on the studies of phase structures, the equation of states~(EoS) of dense QCD matter are calculated with the conditions of $\beta$ equilibrium and electric charge neutrality. Take the EoS as input, the mass-radius relation of neutron stars are predicted, which is consistent with recent observations of heaviest mass and radius of $1.4M_\odot$.