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Solution to the $β$-functions in Lorentz-violating theories as a decomposition into irreducible representations

We analyze the $β$-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 $β$-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 $β$-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.