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

TL;DR

This paper analyzes scalar and pseudoscalar bound state propagators at finite temperature within a NJL model, demonstrating their equivalence in different formalisms and comparing their thermodynamic properties to elementary particles.

Contribution

It establishes the equivalence of propagators in imaginary-time and real-time formalisms and links the thermal transformation matrices of bound states to those of elementary scalars.

Findings

Propagators are equivalent in both formalisms.

Thermal transformation matrix matches that of elementary scalars.

Explicit forms of retarded and advanced propagators are provided.

Abstract

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.