Groves and Gauge Flips

Having completed the diagrammatical proof of the STI (4.1) for an arbitrary number of loops, we can now extend the notions of groves and gauge flips to the multi-loop case. LetGdenote the expansion of a connected Green’s function into Feynman diagrams atn-loop order. The definition (4.50) of invariant subsets of the forestF(G) is taken over unchanged. Also, the groves ofF(G) are again the elements of the finest possible partitioning ofF(G) into invariant subsets.

A priori, the cancellations in the multi-loop case could require the introduc-tion of further gauge flips, in addiintroduc-tion to the ones constructed for the one-loop case (cf. 4.3). However, the cancellations in B4(G), B5(G), and Bc(G) still in-volve the same reachable four-point or five-point gauge subdiagrams as in the one-loop case. Indeed, the multi-loop case just introduced additional combina-torial factors, but not new kinds of cancellations. Therefore, we conclude that the elementary gauge flips as defined in (4.53), (4.56), and (4.58) constitute the correct set of elementary gauge flips for the construction of invariant subsets of F(G) at an arbitrary order of perturbation theory.

On the other hand, in the multi-loop case we need a more general proce-dure to make sure that diagrams with all possible numbers of ghost loops are produced. Recall that, if overlapping gauge field loops are present, mutually exclusive choices of ghost loops are possible (cf. (4.84)). Therefore, we start from a diagramdwithout ghost loop. (Ifdcontains ghost loops, we just replace them by gauge field loops.)

9Of course, thisd1is not the same as the one in the last paragraph.

Givendwithout ghost loop, we apply all possible elementary gauge flips in reachable subdiagrams as in the one-loop case. This will produce all required diagrams without ghost loop. After that, choose asinglediagram without ghost loop and replace an arbitrary gauge field loop by a ghost loop, if there is any.

All other required diagrams with a ghost loop are then produced by applying gauge flips to this single diagram. This is true, because the ghost-gauge boson subdiagrams have the same gauge flips as pure gauge boson subdiagrams.

Therefore, in order to construct the diagrams with all possible numbers of ghost loops, we need only repeat the procedure just described: Having added all gauge flipped diagrams with one ghost loop, we choose a single diagram with one ghost loop and replace some gauge field loop by a second ghost loop. After that, we apply gauge flips again to obtain all diagrams with two ghost loops.

This process is continued until the maximum possible number of ghost loops is reached.

Having determined a general procedure for the construction of minimal in-variant subsets of a connected Green’s function G in the multi-loop case, we emphasize again that we have defined groves as the equivalence classes ofF(G) under sequences ofallgauge flips, not only gauge flips in reachable subdiagrams.

Therefore, also in the multi-loop case the groves ofF(G) are not necessarily the minimal subsets satisfying (4.50). However, as will be demonstrated in [23], the inclusion of elementary gauge flips in non-reachable subdiagrams is necessary to make the contribution of a single grove to a physical amplitude gauge parameter independent.

Before closing this chapter, we remark that, for practical purposes, a different but equivalent strategy for the construction of groves may be more efficient.

In particular, once we have determined all diagrams without ghost loop, the required diagrams with n ghost loops can be obtained by replacing, in each diagram without ghost loop,nnon-overlapping gauge field loops in all possible ways (which, of course, can be done recursively, increasing the number of ghost loops by one in each step). This procedure will lead to the same set of diagrams as the one described above, because the gauge flips performed in the diagrams with ghost loops occur also in the diagrams without a ghost loop.

—5—

Unflavored Flips

Having described an algorithm for the construction of groves in gauge theories by means of gauge flips, as defined in section4.3, the logical next step would be that we apply this algorithm to investigate the forest of connected Green’s functions in the Standard Model, which is our paradigm for a spontaneously broken gauge theory. However, flips, when interpreted as graphical operations on Feynman diagrams, have a meaning independent of the context of gauge theories. In particular, they can be used for the construction of Feynman diagrams in general perturbative quantum field theories with renormalizable interactions. In this chapter, we shall develop a formalism for flips of unflavored diagrams. We shall derive a number o derive a number of results on the structure of the forest of unflavoredn-loop diagrams with certain properties, e. g. 1PI diagrams or amputated diagrams. Gauge theories are then taken up again in the next chapter.

5.1 Flips Without Flavor: The Basic Tool

In the previous chapter, we have proven that the groves of a connected Green’s function at an arbitrary loop order can be constructed by applying gauge flips to Feynman diagrams, i. e. exchanging a certain four-point gauge subdiagram by another one. Thus, a gauge flip transforms Feynman diagrams into each other, an operation that can be described in purely graphical terms, where no more reference to the meaning of a Feynman diagram as a representative for an analytical expression is made.

We can go even further and detach the notion of flips completely from the context of gauge theories. To this end, we forget about the field types, i. e. the quantum numbers, of the lines in a Feynman diagram and consider transforma-tions in anarbitrary four-point subdiagram. That is, we consider the possible topologies of Feynman diagrams in a general renormalizable gauge theory, and how they can be transformed into each other by exchanging four-point subdia-grams.

The relevant topologies of Feynman diagrams can easily be generated from the Feynman rules corresponding to the theory of a self-interacting real scalar field with cubic and quartic interactions. Such a theory has no quantum numbers connected with internal degrees of freedom, and the interaction degrees cover all renormalizable interactions in non-abelian gauge theories. It can be described

by a Lagrangian of the form:

L=−1

2φ ∂²−m² Φ− g

3!φ³− λ

4!φ⁴ (5.1)

This theory will be referred to in the remainder of this chapter as unflavored φ-theory. We will frequently consider the case λ ≡0 to eliminate the quartic coupling, which is then referred to as unflavored φ³-theory. Of course, such a theory is unsound as a fundamental theory, since the Hamiltonian is not bounded from below. On the other hand, all we need is the Feynman rules of these theories, so that the choice λ= 0 corresponds to omitting any diagram with quartic vertices. In this section, the attribute “unflavored” will usually be omitted, because we do not deal with flavored theories here.