The two-gap BCS model in the large-$N$ approximation within a field-theory approach

We study the continuum version of the two-gap BCS model in (3+1)D within the large-N approximation. We calculate the effective potential of the model which depends on two independent energy gaps $\sigma$ and $\Delta$, where $\sigma$ describes the Cooper pair made of electrons that belong to the same internal symmetry whereas $\Delta$ describes the Cooper pair of electrons that have a different internal symmetry. The effective potential is calculated by considering that the Debye frequency is an ultraviolet cutoff $\Lambda$, which is meant to describe the physical lattice of 3D superconductors. Our main result shows that the extra gap provides a possible inter-band phase transition that may be either a stable or metastable phase, depending on the competion between the coupling constants of the model. We also derive the critical temperature below which the phases may be observed.