  The interplay among Coulomb interaction, electron-phonon coupling, and
phonon-phonon coupling has a significant impact on the low-energy behavior of
three-dimensional type-I tilted Dirac semimetals. To investigate this
phenomenon, we construct an effective theory, calculate one-loop corrections
arising from all these interactions, and establish the coupled energy-dependent
flows of all associated interaction parameters by adopting the renormalization
group approach. Deciphering such coupled evolutions allows us to determine a
series of low-energy critical properties for these materials. At first, we
present the low-energy tendencies of all interaction parameters. The tilting
parameter exhibits distinct tendencies that depend heavily upon the initial
anisotropy of fermion velocities. In comparison, the latter is mainly dominated
by its initial value but is less sensitive to the former. Variations in these
two quantities drive certain interaction parameters toward the strong
anisotropy in the low energy regime, indicating the screened interaction in
specific directions, and others toward an approximate isotropy. Additionally,
we observe that the tendencies of interaction parameters can be qualitatively
clustered into three distinct types of fixed points, accompanied by the
potential instabilities that induce certain interaction-driven phase
transition. Furthermore, approaching these fixed points leads to the critical
behavior of physical quantities, such as the density of states,
compressibility, and specific heat, which exhibit quite different from their
non-interacting counterparts and even deviate slightly from Fermi-liquid
behavior. Our investigation sheds light on the intricate relationship between
different types of interactions in these semimetals and provides useful
insights into their fundamental properties.
