Calculation of quantum partition function of the BCS model in the disordered phase

Functional integration is used to make an exact calculation of the quantum partition function of the reduced BCS model in the thermodynamic limit. The expression for the quantum partition function and the estimate of the temperature $T_c$ of the phase transition in the ordered phase agree with the well-known results [1]. In the disordered phase, a singularity with respect to the temperature appears in the quantum partition function, which diverges at temperatures below a certain critical temperature $T_c^*$. This satisfies $T_c^*>T_c$ and the difference $T_c^*-T_c$ is not small. An interpretation of the temperature $T_c^*$ is given.