Effective β-functions for Effective Field Theory

We consider the problem of determining the beta-functions for any reduced effective field theory. Even though not all the Green’s functions of a reduced effective field theory are renormalizable, unlike the full effective field theory, certain effective beta-functions for the reduced set of couplings may be calculated without having to introduce vertices in the Feynman rules for redundant operators. These effective beta-functions suffice to apply the renormalization group equation to any transition amplitude (i.e., S-matrix element), thereby rendering reduced effective field theories no more cumbersome than traditionally renormalizable field theories. Thus, this formalism may often be employed to avoid a breakdown in perturbation theory due to the large logarithms encountered in considering phenomena over a wide range of energy scales.