  An important non-perturbative effect in quantum physics is the energy gap of
superconductors, which is exponentially small in the coupling constant. A
natural question is whether this effect can be incorporated in the theory of
resurgence. In this paper we take some steps in this direction. We conjecture
that the perturbative series for the ground state energy of a superconductor is
factorially divergent, and that its leading Borel singularity is governed by
the superconducting energy gap. We test this conjecture in detail in the
attractive Gaudin-Yang model, an exactly solvable model in one dimension with a
BCS-like ground state. In order to do this, we develop techniques to calculate
the exact perturbative series of its ground state energy up to high order. We
also argue that the Borel singularity is of the renormalon type, and we
identify a class of diagrams leading to factorial growth. We give additional
evidence for the conjecture in other models.
