A differential equation approach for examining the subtraction schemes

We propose a natural differential equation with respect to mass(es) to analyze the scheme dependence problem. It is shown that the vertex functions subtracted at an arbitrary Euclidean momentum (MOM) do not satisfy such differential equations, as extra unphysical mass dependence is introduced which is shown to lead to the violation of the canonical form of the Slavnov-Taylor identities, a notorious fact with MOM schemes. By the way, the traditional advantage of MOM schemes in decoupling issue is shown to be lost in the context of Callan-Symanzik equations.