  We investigate the role of four-fermion interactions, rotational and
particle-hole asymmetries, and their interplay in three-dimensional systems
with a quadratic band touching point by virtue of the renormalization group
approach, which allows to treat all these facets unbiasedly. The coupled flow
evolutions of interaction parameters are derived by taking into account
one-loop corrections in order to explore the behaviors of low-energy states. We
find four-fermion interaction can drive Gaussian fixed points to be unstable in
the low-energy regime. In addition, the rotational and particle-hole
asymmetries, together with the fermion-fermion interactions conspire to split
the trajectories of distinct types of fermionic couplings and induce
superconductivity instability with appropriate starting conditions.
Furthermore, we present the schematic phase diagrams in the parameter space,
showing the overall behaviors of states in the low-energy regime caused by both
fermionic interactions and asymmetries.
