21 - The Renormalization Group Equations

Summary

The scaling of finite parameters with energy — providing an efficient approach to computing predictions at high energy and a link between high-energy theories and low energy phenomenology.

Introduction

In this final chapter, we present the renormalization group equations. These equations arise from the freedom to reparameterize a Lagrangian density while preserving the S matrix. We have seen in counterterm renormalization that coupling parameters like the electric charge are defined by experimental values with specific initial and final states. Clearly, we can change the experimental definition of the coupling parameter without changing the theory. To carry out such a program in detail is complicated. It is more efficient to define scaling in terms of the generic energy parameter introduced by regularization. Commonly, the renormalization group equations are differential equations which express the change of all finite parameters in the Lagrangian as a function of such a generic energy parameter.

The general framework of the renormalization group is developed in the first three sections. Section 21.1 gives a more detailed introduction to the concept of the renormalization group. Section 20.2 links dependence on the generic energy parameter to energy-dependence in Green functions, thereby setting the stage for interpretation of the renormalization group.