Dyson-Schwinger equation approach to Lorentz symmetry breaking with finite temperature and chemical potential

In this work, we investigate the dynamical breakdown of Lorentz symmetry in 4 dimensions by the condensation of a fermionic field described by a Dirac lagrangean with a four-fermion interaction. Using the Keldysh formalism we show that the Lorentz symmetry breaking modifies the Dyson-Schwinger equations of the fermionic propagator. We analyze the nonperturbative solutions for the Dyson-Schwinger equations using the combination of the rainbow and quenched approximations and show that, in equilibrium, the Lorentz symmetry breakdown can occur in the strong coupling regime and new features arise from this approach. Finally, we analyze the contributions of temperature and chemical potential and find the respective phase diagram of the model and analyze the dependence of the critical temperature and chemical potential as functions of the coupling constant.