  We analyze the $\beta$-functions of Yukawa and electromagnetic theories with
Lorentz violation (LV) and propose an alternative method to find the scale
dependence of the different fields that parametrize such violations. The method
of solution consists of decomposing a family of parameters into their
irreducible representations and thus generating a group of subfamilies that
obey the same symmetries and transformation rules. This method allows us to
decouple the differential equations describing the $\beta$-functions and find
out if whether they are positive or not. For a set of parameters describing a
Lorentz-violating theory, we expect their associated $\beta$-functions to be
nonnegative or, otherwise, their scale dependence to be weak enough. These
conditions rely on the fact that asymptotically-free parameters can leave high
imprints of LV at low energies, which are ruled out by observations. Besides
imposing some constrains on the coefficients that describe LV, this method can
be used to extract irrelevant coefficients with no scale dependence.
