  We construct a low-energy effective theory of topological $s$-wave pairing
superconductors, focusing on the mean-field model of superconductor
$\mbox{Cu}_{x}\mbox{Bi}_{2}\mbox{Se}_{3}$. Our approach is second-order
perturbation with respect to the inverse of the mass (i.e., large-mass
expansion) in the Dirac-type electron dispersion from topological insulator
$\mbox{Bi}_{2}\mbox{Se}_{3}$. Since the Dirac-type dispersion with a large mass
describes non-relativistic electrons, the large-mass expansion corresponds to a
low-energy theory with respect to the original setup. We show that the
effective gap function has not only a $p$-wave-like component as the primary
contribution, but also an $s$-wave-like one as higher-order corrections. The
mixture of $p$- and $s$-wave explains the numerical results [Phys. Rev. B 89
(2014) 214506] of the non-magnetic impurity effects.
