  In this work we introduce the possibility of unparticle mediated
superconductivity. We discuss a theoretical scenario where it can emerge and
show that a superconducting state is allowed by deriving and solving the gap
equation for $s$-wave pairing of electrons interacting through the unparticle
generalization of the Coulomb interaction. The dependence of the gap equation
on the unparticle energy scale $\Lambda_{U}$ and the unparticle scaling
dimension $d_{U}$ enables us to find a richer set of solutions compared to
those of the conventional BCS paradigm. We discuss unconventional features
within this construction, including the resulting insensitivity of pairing to
the density of states at the Fermi energy for $d_{U}=3/2$ of the
superconducting gap and suggest possible experimental scenarios for this
mechanism.
