Recursive Subtraction of UV - Divergences by R - Operation

The recursive counterterm procedure developed by Bogoliubov, Parasiuk, and Hepp [1], which was briefly described at the beginning of Chapter 9, yields a finite result for each Feynman diagram, order by order in perturbation theory. It is useful to interprete this procedure as an application of a R-operation to the Feynman diagram. All the finite results on higher loops in this book will be derived via the R-operation. We shall not follow a possible alternative approach based on a general solution of the re-cursion relation developed by Zimmermann [2], who specified the R-operation by a complete diagrammatic expansion in the form of a so-called forest formula. With this formula, the renor-malization procedure is known as the BPHZ formalism. It was applied by Zimmermann directly to the integrands of the Feynman integrals, making them manifestly convergent without regu-larization. Subsequently, several authors showed that the R-operation leads to the same finite results, if applied individually to the Feynman integrals regularized by minimal subtraction [3]. The chapter will end with a discussion of the general structure of the counterterms. This will show how the minimal subtraction scheme considerably reduces the number of relevant diagrams to be evaluated explicitly. In the following manipulations, the negative coupling constant −λ associated with each vertex will be omitted, unless otherwise stated, since they will be of no relevance to the analytic expressions represented by the Feynman diagrams. For recursive diagrammatic subtractions of divergences, some graph-theoretic concepts are useful. The first important concept of a subdiagram was introduced before, after Eq. (9.3). It will also be useful to admit to the set of all subdiagrams the original diagram itself. If this is excluded , we speak otherwise of a proper subdiagram. The notation will be γ ⊂ G for subdiagrams and ¯ γ ⊂ G for proper subdiagrams of the diagram G. Then we define • a shrunk diagram G/γ, obtained from G by shrinking all lines in γ to zero length, so that their endpoints coincide. Each connected part of γ then collapses to a single vertex.