  In light of the new experimental and theoretical important developments in
high-$T_c$ superconductivity, we revisit the fermionic hot-spot model relevant
to the phenomenology of the cuprates. We extend previous results by means of a
complete two-loop order renormalization group (RG) framework. Here, we
explicitly study the effect of the charge-density-wave (CDW) order parameter
with a $d$-wave form factor with the experimentally observed modulation $(\pm
Q_0,0)$ and $(0,\pm Q_0)$ at the infrared-stable nontrivial fixed point
obtained previously for this model. Additionally, we proceed to investigate
also the so-called pair-density-wave (PDW) order that was recently proposed in
the literature as a possible candidate for the "hidden" order to describe the
pseudogap phase observed in underdoped cuprates. We confirm that although the
above two ordering tendencies are also found to be nearly degenerate both at
one-loop and two-loop RG orders and linked by an emergent $SU(2)$ pseudospin
symmetry, they turn out to be subleading for weaker couplings in the present
model to antiferromagnetism, $d$-wave bond-density wave (BDW) order with
modulation along Brillouin zone diagonals $(\pm Q_0,\pm Q_0)$, and $d$-wave
singlet superconductivity (SSC). However, as we increase the strength of the
initial coupling towards moderate values, we do capture a tendency for the
entangled PDW/CDW order to become leading compared to BDW/SSC in the model,
which suggests that the former composite order might be indeed a viable concept
to describe some cuprate superconductors at high temperatures in the underdoped
regime, as has been recently alluded to by many authors in the literature.
