On overlapping Feynman (sub)graphs

Abstract

We discuss, on general grounds, how two subgraphs of a given Feynman graph can overlap with each other. For this, we use the notion of connecting and returning lines that describe how any subgraph is inserted within the original graph. This, in turn, allows us to derive "non-overlap" theorems for one-particle-irreducible subgraphs with 2, 3 and 4 external legs. As an application, we provide a simple justification of the skeleton expansion for vertex functions with more than five legs, in the case of scalar field theories. We also discuss how the skeleton expansion can be extended to other classes of graphs. I. INTRODUCTIONOverlapping divergences make the practical treatment of UV divergences in a quantum field theory cumbersome. In modern approaches, there exist various ways of tackling this issue, all based, in one way or another, on the use of infinitesimal or finite variations of the Feynman graphs with respect to appropriate parameters. The best known among these approaches is certainly the functional renormalization group [1], but there exist other possibilities, such as the one put forward in Ref. [2], see also Ref. [3,4] for even older proposals.The benefit of these approaches is that one does not need to worry about possible overlapping divergences, since they are, if any, automatically disentangled.There might be situations, however, where one needs to assess the absence of overlapping divergences in a given quantity build out of Feynman graphs. One recent example of this situation is reported in Ref. [5] where the overlapping divergences that appear in the twoparticle-irreducible (2PI) formalism for the case of a scalar ϕ 4 theory are disentangled with the help of the functional renormalization group and classified into divergences of the twopoint function, divergences of the four-point function, and divergences of higher derivatives δ n Φ[G]/δG n (with n ≥ 3) of the so-called Luttinger-Ward functional Φ[G], a functional of