  We study the renormalization group flow of the velocities in the field theory
describing the coupling of the massless quasi-relativistic fermions to the
bosons through the Yukawa coupling, as well as with both bosons and fermions
coupled to a fluctuating $U(1)$ gauge field in two and three spatial
dimensions. Different versions of this theory describe quantum critical
behavior of interacting Dirac fermions in various condensed-matter systems. We
perform an analysis using one-loop $\epsilon-$expansion about three spatial
dimensions, which is the upper critical dimension in the problem. In two
dimensions, we find that velocities of both charged fermions and bosons
ultimately flow to the velocity of light, independently of the initial
conditions, the number of fermionic and bosonic flavors, and the value of the
couplings at the critical point. In three dimensions, due to the analyticity of
the gauge field propagator, both the $U(1)$ charge and the velocity of light
flow, which leads to a richer behavior than in two dimensions. We show that all
three velocities ultimately flow to a common terminal velocity, which is
non-universal and different from the original velocity of light. Therefore,
emergence of the Lorentz symmetry in the ultimate infrared regime seems to be a
rather universal feature of this class of theories in both two and three
dimensions.
