  In this paper, we study the local well-posedness of the $(1+3)$-dimensional
Yang-Mills-Higgs (YMH) and the Yang-Mills-Dirac (YMD) system in the Lorenz
gauge. Since there is some bilinear term in (YMH), which is a lack of null
structure, one may obtain the well-posedness at most the energy space. However,
we attain the scaling critical regularity of (YMH) by imposing the extra
weighted regularity in the angular variables. In (YMD), for the coupled system
to persist in time, it is required to impose the angular regularity on the
Dirac spinor as much as the Yang-Mills gauge potential and curvature. We can
then prove the scaling critical regularity of (YMD) using angular regularity
instead of the null structure of the spinor field. In this manner, we present
an approach to attack the scaling critical regularity of (YMH) and (YMD)
simultaneously. This result is an application of our recent study on the
Yang-Mills system in the Lorenz gauge \cite{hong}.
