  Quantum critical behaviors induced by a putative quantum phase transition are
vigilantly investigated, which separates a $d$-wave superconducting state and
$d$-wave superconducting+$X$ state below the superconducting dome of the
$d-$wave superconductors with tuning the non-thermal doping variable. Within
the framework of renormalization group approach, we start with a
phenomenological effective theory originated from the Landau-Ginzburg-Wilson
theory and practice one-loop calculations to construct a set of coupled flows
of all interaction parameters. After extracting related physical information
from these coupled evolutions, we address that both fermion velocities and
critical temperatures exhibit critical behaviors, which are robust enough
against the initial conditions due to strong quantum fluctuations. At first,
the evolution of Yukawa coupling between $X$-state order parameter and nodal
fermions in tandem with quantum fluctuations heavily renormalize fermion
velocities and generally drive them into certain finite anisotropic fixed point
at the lowest-energy limit, whose concrete value relies upon the very quantum
phase transition. In addition, these unique properties of fermion velocities
largely reshape the fate of superfluid density, giving rise to either an
enhancement or a dip of critical temperature. Moreover, we find that
fermion-fermion interactions bring non-ignorable quantitative corrections to
quantum critical behaviors despite they are subordinate to quantum fluctuations
of order parameters.
