Intrinsic finite-energy Cooper pairing in $j=3/2$ superconductors

We show that Cooper pairing can occur intrinsically away from the Fermi surface in $j=3/2$ superconductors with strong spin-orbit coupling and equally curved bands in the normal state. In contrast to conventional pairing between spin-$1/2$ electrons, we derive that pairing can happen between inter-band electrons having different total angular momenta, i.e., $j=1/2$ with $j=3/2$ electrons. Such superconducting correlations manifest themselves by a pair of indirect gap-like structures at finite excitation energies. An observable signature of this exotic pairing is the emergence of a pair of symmetric superconducting coherence peaks in the density of states at finite energies. We argue that finite-energy pairing is a generic feature of high-spin superconductors, both in presence and absence of inversion symmetry.