Dimensional Regularization

The problem of perturbative renormalization theory is to give a meaning to certain divergent integrals arising from Feynman diagrams. The analytical difficulty is to regularize the occurring divergent integrals, associating to each Feynman diagram some finite value. The physical difficulty is to do this in such a way that the outcome has a physical meaning. The contributions of the various Feynman diagrams are not independent of each other: If a graph contains divergent subgraphs, then the way they have been regularized must be taken into account. In this talk we will concentrate on the analytical part of the problem. We will explain how to associate a finite value to a Feynman diagram. This value is physically correct only if the diagram contains no divergent subgraphs. The combinatorics needed for general diagrams will be explained in other lectures. The procedure we use to regularize an integral is called dimensional regularization. It is used in the BPHZ renormalization scheme. The basic idea behind dimensional regularization is that renormalization is much easier in low dimensions. Therefore, we try to write down the divergent integrals that we have to regularize in such a way that the dimension of the physical spacetime becomes an external parameter that can be varied. Then we boldly allow the dimension D to be an arbitrary complex number. As expected, our integral becomes convergent for ReD 0 and defines an analytic function there. This function can be continued meromorphically to all of . However, it may have a pole at the physical dimension d of space-time. Finally, the regularized value is obtained by minimal subtraction: We subtract the pole part in the Laurent expansion around d and evaluate the remaining function at d. The pole part is called the counterterm and has to be recorded because it is needed when dealing with divergent subgraphs. Although the meaning of physics in a D-dimensional space-time with D is rather opaque, dimensional regularization is used by physicists because of its simplicity and good invariance properties. This scheme is always gauge and Lorentz invariant. These properties will not become apparent in the discussion below, however, because we will limit ourselves to scalar theories, thus excluding gauge theories. The reason for this restriction is that a scalar field theory can automatically be formulated in all non-negative integer dimensions. Hence we have sufficiently