Generation of Diagrams

In Section 5.4, we have developed a functional formalism to find all connected vacuum diagrams and their multiplicities. These serve as a basis for deriving all diagrams contributing to two-and four-point functions. This is done by removing from the connected vacuum diagrams one line and two lines, respectively. Instead of the two lines we may also remove a vertex, as pointed out in Section 5.5, where we also indicated a way to implement the recursive formalism for the generation of all graphs on a computer [1], and the reader is referred to the detailed original paper for a detailed explanation of the techniques. In this chapter, we shall describe a somewhat shortened way of generating all diagrams which was used in the initial work on the five-loop determinations of the critical exponents [2]. Because of the special features of the regularization by minimal subtraction discussed in Section 11.6, we can restrict the generation to • all 1PI four-point diagrams without tadpole parts and their weight factors for Z g , • all 1PI two-point diagrams without tadpole parts and their weight factors for Z φ , • all 1PI four-point diagrams without tadpole parts which result from differentiation with respect to m 2 , and the weight factors of the corresponding two-point diagrams for Z m 2. The generation starts with all connected vacuum diagrams. For the generation of tadpole-free two-and four-point diagrams, only vacuum diagrams without cutvertices are needed. From these, the connected two-and four-point diagrams are constructed by cutting lines and vertices, respectively. These diagrams may now contain cutvertices. In the last step, we select the 1PI diagrams. Since we shall eventually be interested only in the renormalization group functions of the theory, we can reduce the number of diagrams further by selecting only those two-and four-point 1PI diagrams without cutvertices. Recall that the pole part K ¯ RG of a diagram G with a cutvertex contains only ε −n-poles with n ≥ 2, as explained on page 191. It contributes to the renormalization constants, but not to the renormalization group functions due to the cancellation of all higher ε-poles (see Section 10.3). Nevertheless, we find it useful to include diagrams with cutvertices at each stage of the calculations, since they permit us to check the correctness of the ε-poles via the necessary cancellations. They appear among the other diagrams in Appendix A, for example, the first two-, …