Dynamical symmetry breaking in the external gravitational and constant magnetic fields

We investigate the effects of the external gravitational and constant magne tic fields to the dynamical symmetry breaking. As simple models of the dyna mical symmetry breaking we consider the Nambu-Jona-Lasinio (NJL) model and the supersymmetric Nambu-Jona-Lasinio (SUSY NJL) model nonminimally interac ting with the external gravitational field and minimally interacting with c onstant magnetic field. The explicit expressions for the scalar and spinor Green functions are found to the first order in the space-time curvature an d exactly for a constant magnetic field. We obtain the effective potential of the above models from the Green functions in the magnetic field in curve d space-time. Calculating the effective potential numerically with the vary ing curvature and/or magnetic fields we show the effects of the external gr avitational and magnetic fields to the phase structure of the theories. In particular, increase of the curvature in the spontaneously broken phase of the chiral symmetry due to the fixed magnetic held makes this phase to be l ess broken. At the same time the strong magnetic field quickly induces chir al symmetry breaking even in the presence of fixed gravitational field with in the nonbroken phase.