Symmetries of the Renormalization Group Equations

The determination of the renormalization group equations in quantum field theory is a very laborious task. For example in the Standard Model the full set of these equations is known only up to two loops, while only some partial results are obtained at higher orders. We argue that one can simplify the calculation of the renormalization group equations by using the symmetry of a system. The origin of the simplification lies in the use of the invariant polynomials of the symmetry group of the theory. We consider a quantum field theory with three scalar fields that is invariant under the action of the permutation group S₃. We show, using the theorem of Molien, that for such a model there is a significant reduction of the amount of work needed for the derivation of the renormalization group equations.