  We investigate the effects of long-ranged Coulomb interactions in a tilted
Dirac semimetal in two dimensions by using the perturbative
renormalization-group method. Depending on the magnitude of the tilting
parameter, the undoped system can have either Fermi points (type-I) or Fermi
lines (type-II). Previous studies usually performed the renormalization-group
transformations by integrating out the modes with large momenta. This is
problematic when the Fermi surface is open, like type-II Dirac fermions. In
this work, we study the effects of Coulomb interactions, following the spirit
of Shankar\cite{Shankar}, by introducing a cutoff in the energy scale around
the Fermi surface and integrating out the high-energy modes. For type-I Dirac
fermions, our result is consistent with that of the previous work. On the other
hand, we find that for type-II Dirac fermions, the magnitude of the tilting
parameter increases monotonically with lowering energies. This implies the
stability of type-II Dirac fermions in the presence of Coulomb interactions, in
contrast with previous results. Furthermore, for type-II Dirac fermions, the
velocities in different directions acquire different renormalization even if
they have the same bare values. By taking into account the renormalization of
the tilting parameter and the velocities due to the Coulomb interactions, we
show that while the presence of a charged impurity leads only to charge
redistribution around the impurity for type-I Dirac fermions, for type-II Dirac
fermions, the impurity charge is completely screened, albeit with a very long
screening length. The latter indicates that the temperature dependence of
physical observables are essentially determined by the RG equations we derived.
We illustrate this by calculating the temperature dependence of the
compressibility and specific heat of the interacting tilted Dirac fermions.
