  We investigate the phenomena of the dynamical chiral symmetry breaking
(DCSB), color superconductivity (CSC), and Bose-Einstein condensation (BEC) in
an ${\cal N}=1$ four-dimensional generalized SUSY Nambu$-$Jona-Lasinio model at
finite temperature and density. Both the ${\cal N}=1$ four-dimensional and
${\cal N}=2$ three-dimensional cases are considered. The ${\cal N}=2$
three-dimensional theory is obtained by a simple dimensional reduction scheme
of the four-dimensional counterpart. In order to realize the DCSB and BCS-type
CSC in this model, we introduce a SUSY soft mass term. After adopting the
method of SUSY auxiliary fields with the Fierz transformation in color and
flavor spaces, we discuss several possible breaking schemes of the global
symmetries of the model. The integrations of the auxiliary fields of composites
in the effective potential are performed by using the steepest descent
approximation. The roles of both the boson and fermion sectors in the BEC, DCSB
and CSC are examined by the quasiparticle excitation spectra and the gap
equations. The physical properties of the DCSB and the CSC are studied in
detail, while that of the BEC is beyond scope of this paper. It is found that
the BEC, DCSB and CSC can coexist under a condition of model parameters.

High Energy Physics - Phenomenology;High Energy Physics - Theory