  We calculate spectral functions associated with hadronic current correlation
functions for vector and pseudoscalar currents at finite temperature. We make
use of the Nambu--Jona--Lasinio (NJL) model with temperature-dependent coupling
constants and temperature-dependent momentum cutoff parameters. At low
energies, good fits are obtained for the spectral functions that were extracted
from lattice data by means of the maximum entropy method (MEM). Our model has
two parameters which are used to fix the magnitude and position of the large
peak seen in the spectral functions. With those two parameters fixed, we obtain
a satisfactory fit to the width of the peak. The model then also reproduces the
energy of a second peak seen in the spectral functions. In the case of the
pseudoscalar spectral function, the calculated peak is about 20 percent higher
than that found for the spectral function obtained from the lattice data.
However, it appears that the second peak is a lattice artifact [ P. Petreczky,
private communication ] and our fit to the second peak may not be meaningful.
We conclude that the NJL model may have a broader range of application than
previously considered to be the case, if one allows for significant temperature
dependence of the parameters of the model, as well as rather large values of
the momentum cutoff parameter. Our treatment of temperature-dependent coupling
constants and cutoff parameters is analogous to the procedure introduced by R.
Casalbuoni, R. Gatto, G. Nardulli, and M. Ruggieri, [ Phys. Rev. D \textbf{68},
034024 (2003) ], who make use of the NJL model at finite density and find that
they need to use the density-dependent coupling constants and density
-dependent cutoff parameters to study matter at high density.
