Propagators for scalar bound states at finite temperature in a NJL model

We reexamine physical causal propagators for scalar and pseudoscalar bound states at finite temperature in a chiral $U_L(1)\times U_R(1)$ NJL model, defined by four-point amputated functions subtracted through the gap equation, and prove that they are completely equivalent in the imaginary-time and real-time formalism by separating carefully the imaginary part of the zero-temperature loop integral. It is shown that the thermal transformation matrix of the matrix propagators for these bound states in the real-time formalism is precisely the one of the matrix propagator for an elementary scalar particle and this fact shows similarity of thermodynamic property between a composite and an elementary scalar particle. The retarded and advanced propagators for these bound states are also given explicitly from the imaginary-time formalism.