Renormalization group equation (RGE)

Derivation of the RGE

The spirit of the renormalization group approach lies in the observation discussed in Chapter 4 that in a specific theory the renormalized constants such as the couplings or the masses are mathematical parameters which can be varied by arbitrarily changing the renormalization prescription. Once the infinities of the theory have been subtracted out by a renormalization prescription R one is still free to perform further finite renormalizations resulting in each case in a different effective renormalization R′. Each renormalization prescription can be interpreted as a particular reordering of the perturbative expansion and expressing it in terms of the new renormalized constants. The latter are, of course, in each case related differently to physical constants (for example, the mass defined as the pole of the propagator is a physical constant) which are directly measurable and therefore renormalization-invariant. A change in renormalization prescription is compensated by simultaneous changes of the parameters of the theory so as to leave, by construction, all exact physical results renormalization-invariant. (In practice, there is a residual renormalization scheme dependence in each order of perturbation theory.)

Most often, and also in this section, only subsets of arbitrary renormalization prescription transformations which can be parametrized by a single mass scale parameter μ are discussed. The parameter can be, for example, the value of the subtraction point in the μ-subtraction schemes or the dimensionful parameter μ in the minimal subtraction prescription. For a single-mass-scale-dependent subset of renormalization transformations we derive in the following a differential equation which controls the changes in the renormalized parameters induced by changing μ: the RGE.