Holon Pairing Instability based on the Bethe-Salpeter Equation obtained   from the t-J Hamiltonians of both U(1) and SU(2) Slave-boson Symmetries
  We investigate a possibility of holon pairing for bose condensation based on the   Bethe-Salpeter equation obtained from the use of the t-J Hamiltonians of both t he U(1) and SU(2) slave-boson symmetries. It is shown that the vertex function contributed from ladder diagram series involving holon-holon scattering channel in the Bethe-Salpeter equation leads to a singular behavior at a critical temperature at each hole doping concentration, showing the instability of the normal state against holon pairing. We find that the holon pairing instability occurs only in a limited range of hole doping, by showing an "arch" shaped bose condensation line in agreement with observation for high $T_c$ cuprates. It is revealed that this is in agreement with a functional integral approach of the slave-boson theories. 
Sung-Sik Lee and Sung-Ho Suck Salk