  We attentively investigate the effects of short-range fermion-fermion
interactions on the low-energy properties of both two-dimensional type-I and
type-II tilted Dirac semimetals by means of the renormalization group
framework. Practicing the standard renormalization group procedures via taking
into account all one-loop corrections gives rise to the coupled
energy-dependent evolutions of all interaction parameters, which are adopted to
carefully examine whether and how the fermion-fermion interactions influence
the low-energy physical behaviors of tilted Dirac fermions. After carrying out
the detailed analysis of coupled flows, we figure out the tilting parameter
dictates the low-energy states of tilted Dirac fermions in conjunction with
starting values of fermion-fermion couplings. With proper variations of these
two kinds of parameters, the tilted Dirac fermions can either flow towards the
Gaussian fixed point or undergo certain instability that is conventionally
accompanied by a phase transition in the low-energy regime. In addition, all
potential instabilities can be clustered into five distinct classes owing to
the competitions between the tilting parameter and initial fermionic
interactions. Moreover, the dominant phases accompanied by the instabilities
are determined via computing and comparing the susceptibilities of eight
potential phases.
