  We consider the emergence of a non-Fermi liquid fixed point in a
two-dimensional metal, at the onset of a quantum phase transition from a Fermi
liquid state to an incommensurate charge density wave (CDW) ordered phase. The
momentum of the CDW boson is centred at the wavevector $ \boldsymbol{ \mathcal
Q} $, which connects a single pair of antipodal points on the Fermi surface
with antiparallel tangent vectors. We employ the dimensional regularization
technique in which the co-dimension of the Fermi surface is extended to a
generic value, while keeping the dimension of the Fermi surface itself fixed at
one. Although the system is strongly coupled at dimension $d=2$, the
interactions become marginal at the upper critical dimension $d=d_c$, whose
value is found to be $ 5/2$. Using a controlled perturbative expansion in the
parameter $\epsilon = d_c - d $, we compute the critical exponents of the
stable infrared fixed point characterizing the quantum critical point. The
scalings of the original theory are determined by setting $\epsilon =1/2$,
where the fermion self-energy is seen to scale with the frequency with a
fractional power law of $2/3$, which is the telltale signature of a typical
non-Fermi liquid phase.
