  We study a simple model of three-dimensional fermions close to a quadratic
band touching point, built from the celebrated Luttinger single-particle
Hamiltonian and an attractive contact interaction between the particles. Such a
system displays a novel quantum critical point between the semimetallic and an
s-wave superconducting phase at which the low-energy "Luttinger fermions" are
inextricably coupled to the order parameter fluctuations. The quantum critical
point is perturbatively accessible near four spatial dimensions, where it
features nontrivial scaling with dynamical exponent 1<z<2 and emergent
rotational and particle-hole symmetries. Some features of the criticality, such
as oscillatory corrections to scaling and its enhanced symmetry, are discussed.
