As we saw in Chapter 3, the renormalization procedure has considerable arbitrariness: the counterterm for a graph must cancel its divergence but may contain any amount of finite part. A rule for choosing the value of the counterterm we called a renormalization prescription. In one-loop order it was clear from the examples that a change in renormalization prescription can be cancelled by a change in the finite, renormalized couplings corresponding to each divergence. Thus a change in renormalization prescription does not change the theory but only the parametrization by renormalized coupling and mass. What is not so easy is to see that this property is true to all orders. This we will show in Section 7.1. The invariance of the theory under such transformations is called renormalization-group (RG) invariance.

A particularly useful type of change of renormalization prescription is to change the renormalization mass μ. Infinitesimal changes are conveniently described by a differential equation, called the renormalization-group equation, which is derived in Section 7.3. This leads to the concept of the effective momentum-dependent coupling. This concept is very useful in calculations of high-energy behavior, as explained in Section 7.4. The coefficients in the renormalization-group equation are called the renormalization-group coefficients and are important properties of a theory.