Propagators for Scalar Bound States at Finite Temperature in an NJL Model

We re-examine physical causal propagators for scalar and pseudoscalar bound states at finite temperaturein a chiral Ut(1) x UR(1) NJL model, defined by four-point amputated fimctions subtracted through the gap equation,and prove that they are completely equivalent in the imaginary-time and real-time formalisms by separating carefiullythe imaginary part of the zero-temperature loop integral. It is shown that the same thermal transformation matrix ofthe matrix propagators for these bound states in the real-time formalism is precisely the one of the matrix propagatorfor an elementary scalar particle and this fact shows the similarity of thermodynamic property between a composite andelementary scalar particle. The retarded and advanced propagators for these bound states are also given explicitly fromthe imaginary-time formalism.