Dynamical Chiral Symmetry Breaking, Color Superconductivity, and   Bose-Einstein Condensation in an $SU(N_{c})\times U(N_{f})_{L}\times   U(N_{f})_{R}$-invariant Supersymmetric Nambu$-$Jona-Lasinio Model
  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. 
Tadafumi Ohsaku