  Constructing an effective field theory in terms of doped magnetic impurities
(described by an O(3) vector model with a random mass term), itinerant
electrons of spin-orbit coupled semiconductors (given by a Dirac theory with a
relatively large mass term), and effective interactions between doped magnetic
ions and itinerant electrons (assumed by an effective Zeeman coupling term), we
perform the perturbative renormalization group analysis in the one-loop level
based on the dimensional regularization technique. As a result, we find that
the mass renormalization in dynamics of itinerant electrons acquires negative
feedback effects due to quantum fluctuations involved with the Zeeman coupling
term, in contrast with that of the conventional problem of quantum
electrodynamics, where such interaction effects enhance the fermion mass more
rapidly. Recalling that the applied magnetic field decreases the band gap in
the presence of spin-orbit coupling, this renormalization group analysis shows
that the external magnetic field overcomes the renormalized band gap, allowed
by doped magnetic impurities even without ferromagnetic ordering. In other
words, the Weyl metal physics can be controlled by doping magnetic impurities
into spin-orbit coupled semiconductors, even if the external magnetic field
alone cannot realize the Weyl metal phase due to relatively large band gaps of
semiconductors. Furthermore, we emphasize that quasiparticles do not exist in
this emergent disordered Weyl metal phase due to correlations with strong
magnetic fluctuations. This non-Fermi liquid type Weyl metal state may be
regarded to be a novel metallic phase in the respect that a topologically
nontrivial band structure appears in the vicinity of quantum criticality.
