  We study the application of the Breitenlohner-Maison-'t Hooft-Veltman (BMHV)
scheme of Dimensional Regularization to the renormalization of chiral gauge
theories, focusing on the specific counterterm structure required by the
non-anticommuting Dirac $\gamma_5$ matrix and the breaking of the BRST
invariance. Calculations are performed at the one-loop level in a massless
chiral Yang-Mills theory with chiral fermions and real scalar fields. We
discuss the setup and properties of the regularized theory in detail. Our
central results are the full counterterm structures needed for the correct
renormalization: the singular UV-divergent counterterms, including evanescent
counterterms that have to be kept for consistency of higher-loop calculations.
  We find that the required singular, evanescent counterterms associated with
vector and scalar fields are uniquely determined but are not gauge invariant.
Furthermore, using the framework of algebraic renormalization, we determine the
symmetry-restoring finite counterterms, that are required to restore the BRST
invariance, central to the consistency of the theory. These are the necessary
building blocks in one-loop and higher-order calculations.
  Finally, renormalization group equations are derived within this framework,
and the derivation is compared with the more customary calculation in the
context of symmetry-invariant regularizations. We explain why, at one-loop
level, the extra BMHV-specific counterterms do not change the results for the
RGE. The results we find complete those that have been obtained previously in
the literature in the absence of scalar fields.
