  We derive an effective field theory for the competition between
superconductivity (SC) and charge density waves (CDWs) by employing the SO(3)
pseudospin representation of the SC and CDW order parameters. One important
feature in the effective nonlinear $\sigma$ model is the emergence of Berry
phase even at half filling, originating from the competition between SC and
CDWs, i.e., the pseudospin symmetry. A well known conflict between the previous
studies of Oshikawa\cite{Oshikawa} and D. H. Lee et al.\cite{DHLee} is resolved
by the appearance of Berry phase. The Berry phase contribution allows a
deconfined quantum critical point of fractionalized charge excitations with $e$
instead of $2e$ in the SC-CDW quantum transition at half filling. Furthermore,
we investigate the stability of the deconfined quantum criticality against
quenched randomness by performing a renormalization group analysis of an
effective vortex action. We argue that although randomness results in a weak
disorder fixed point differing from the original deconfined quantum critical
point, deconfinement of the fractionalized charge excitations still survives at
the disorder fixed point owing to a nonzero fixed point value of a vortex
charge.
