  We study the quantum phase transition of $U(1)$ - charged Dirac fermions
Yukawa-coupled to the Kekul\'e valence bond solid order parameter with $Z_3$
symmetry of the honeycomb lattice. The symmetry allows for the presence of the
term in the action which is cubic in the Kekul\'e order parameter, and which is
expected to prevent the quantum phase transition in question from being
continuous. The Gross-Neveu-Yukawa theory for the transition is investigated
using a perturbative renormalization group and within the $\epsilon$ expansion
close to four space-time dimensions. For a vanishing $U(1)$ charge we show that
quantum fluctuations may render the phase transition continuous only
sufficiently far away from 3+1 dimensions, where the validity of the
conclusions based on the leading order $\epsilon$ expansion appear
questionable. In the presence of a fluctuating gauge field, on the other hand,
we find quantum critical behavior even at weak coupling to appear close to 3+1
dimensions, that is, within the domain of validity of the perturbation theory.
We also determine the renormalization group scaling of the cubic coupling at
higher-loop orders and for a large number of Dirac fermions for vanishing
charge.
