New Approach to the Renormalization Group

A new set of renormalization-group equations is presented. These equations are based on a renormalization procedure in which counterterms are calculated for zero unrenormalized mass. Unlike the Gell-Mann-Low and Callan-Symanzik equations, they can be solved for arbitrary momenta. The solutions involve a momentum-dependent effective mass as well as a momentum-dependent effective coupling constant. By studying these solutions at large momenta, it can be shown that the nonleading terms discarded by previous authors do, in fact, remain negligible when the perturbation series is summed to all orders if, and only if, the effective mass vanishes at large momentum, which will be the case if a certain anomalous dimension is less than unity, as it is in asymptotically free theories. In this case, the new renormalization-group equations can be used at large momentum to derive not only the leading term, but the first three terms in an asymptotic expansion of any Green's function. These results are also applied to Wilson coefficient functions, and an important cancellation of anomalous dimensions is noted.