Universal behavior of the BCS energy gap

We consider the BCS energy gap \Xi(T) (essentially given by \Xi(T) \approx \Delta(T, \sqrt{\mu}) , the BCS order parameter) at all temperatures 0 \le T \le T_{\textup{c}} up to the critical one, T_{\textup{c}} , and show that, in the limit of weak coupling, the ratio \Xi(T)/T_{\textup{c}} is given by a universal function of the relative temperature T/T_{\textup{c}} . On the one hand, this recovers a recent result by Langmann and Triola [Phys. Rev. B 108 (2023), no. 10, article no. 104503] on three-dimensional s -wave superconductors for temperatures bounded uniformly away from T_{\textup{c}} . On the other hand, our result lifts these restrictions, as we consider arbitrary spatial dimensions d \in \{1,2,3\} , discuss superconductors with non-zero angular momentum (primarily in two dimensions), and treat the perhaps physically most interesting (due to the occurrence of the superconducting phase transition) regime of temperatures close to T_{\textup{c}} .