  We investigate the current-current correlation functions or the so-called
response functions of a two-flavor Nambu-Jona-Lasino model at finite
temperature and density. We study the linear response by using the functional
path integral approach and introducing the conjugated gauge fields as external
sources. The response functions can be obtained by expanding the generational
functional in powers of the external sources. We derive the response functions
parallel to two well-established approximations for the equilibrium
thermodynamics: the mean-field theory and a beyond-mean-field theory taking
into account the mesonic contributions. The response functions based on the
mean-field theory recover the so-called quasiparticle random phase
approximation. We calculate the dynamical structure factors for the density
responses in various channels within the random phase approximation. We show
that the dynamical structure factors in the baryon axial vector and isospin
axial vector channels can be used reveal the quark mass gap and the Mott
dissociation of mesons, respectively. Noting that the mesonic contributions are
not taken into account in the random phase approximation, we also derive the
response functions parallel to the beyond-mean-field theory. We show that the
mesonic fluctuations naturally give rise to three kinds of famous diagrammatic
contributions: the Aslamazov-Lakin contribution, the Self-Energy or
Density-of-State contribution, and the Maki-Thompson contribution. Unlike the
equilibrium case, in evaluating the fluctuation contributions, we need to treat
carefully the linear terms in the external sources and the induced
perturbations. These contributions from the mesonic fluctuations are expected
to have significant effects on the transport properties of hot and dense matter
around the chiral phase transition or crossover.
