—B—
Tree Level STIs
In this appendix we collect all tree level STIs used in the proof of the STI for connected Green’s functions in chapter 4.
B.1 Propagator and Inverse Propagator STIs
= − (B.1)
= − . (B.2)
Note in particular the minus sign.
− = . (B.6)
B.2.2 Quartic Vertices
0 = + + + . (B.7)
0 = + + . (B.8)
0 = + − . (B.9)
B.2.3 Five-Point Vertices
0 = + + + . (B.10)
—C—
Automated Grove Construction
In this appendix we sketch our implementation of a computer program for the automated construction of groves in gauge theories, specifically the Standard Model.
C.1 Implementation
C.1.1 Representing Feynman Diagrams
In order to manipulate Feynman diagrams with a computer, we must first choose a representation of Feynman diagrams as a data structure in computer memory.
This can be done by indicating which pairs of vertices are connected by propa-gators. To this end, we must be able to reference the vertices of the Feynman diagram. The simplest way to do this is to attach an internal index to each vertex (an integer, say) and then refer to a vertex by its index.
The particular representation we have chosen can be described as a map, specifying, for each vertex, to which neighbors it is connected, and the corre-sponding flavors. Consider the following simple example:
→ 91 1 92 2
→
−1 → 1 → {Z0}
−2 → 2 → {Z0}
1 →
−1 → {Z0} 2 → {W+, W−}
2 →
−1 → {Z0} 2 → {W+, W−}
(C.1)
At the left, the Feynman diagram is displayed in the usual way, without any indices at the vertices. By introducing indices to reference vertices, we get to the next drawing. Observe that the ends of external lines have also been given an index. In fact, in our representation the introduction of external vertices is necessary for the representation of external lines. Of course, these external vertices are mere artifacts of the representation and have no physical meaning.
We have chosen positive and negative indices for internal and external vertices, respectively, because this allows for an easy distinction between both in our computer program.
The indexed Feynman diagram is easily translated into the displayed map.
For instance, one should read the first entry as stating that “vertex91 is con-nected to vertex 1 by aZ0-line”. Observe that the notation for the map is still an abstract representation. There may be many different ways to implement such a map as a data structure. However, it turns out that, in order to deal with comparisons efficiently (cf. the next section), the chosen data structure should satisfy the constraint that identical maps correspond to identical data structures.
C.1.2 Comparing Feynman Diagrams
The algorithm for the construction of groves instructs us to repeatedly add all gauge flipped diagrams to a given set of diagrams, starting from a single diagram, until no more new diagrams are created. In order to implement this algorithm, we have to be able to test the representations of Feynman diagrams in computer memory for equality, or else we will not be able to assert whether or not a diagram has already been produced.
We have already argued that our representation of Feynman diagrams with loops introduces indices to identify the vertices of the diagram. However, these indices are unrelated to the physical meaning of Feynman diagrams. Therefore, two representations of Feynman diagrams that differ only by a permutation of the vertex indices actually denote the same Feynman diagram. On the other hand, different permutations of indices will usually lead to different data struc-tures (unless the permutation corresponds to an automorphism of the Feynman diagram). This means that we cannotnaivelycompare representations, i. e. data structures in computer memory, in order to test two Feynman diagrams for equality.
Wecan, however, compare representations if we find a prescription to pro-duce auniqueindexation for every possible Feynman diagram. In graph theory, this strategy is formalized by considering permutations of indices as an equiva-lence relation on the set of representations of Feynman diagrams. A Feynman diagram corresponds then actually to an equivalence class of representations. A unique prescription to pick a representative from an equivalence class is said to be a canonical label. Accordingly, the corresponding representative is called a canonical representative. Given a canonical label, two Feynman diagrams can be compared by comparing the data structures of their canonical representatives.
We have based our canonical label algorithm on the algorithm nauty,[28]
by adapting it to the peculiarities of Feynman diagrams with loops. Due to the nature of the nauty algorithm, we are able to produce not only canonical representatives of Feynman diagrams, where external lines are assumed to carry distinct momenta, but also topologies of Feynman diagrams, by ignoring the momenta of some or all external lines. Recall that, for forests with external fermions, the groves can be characterized by permutations of identical fermions.
Our canonical label algorithm allows for a simple way to produce just one of all possible permutations.
C.1.3 Constructing Groves
Given an implementation of flips for canonically labeled Feynman diagrams, the construction of groves is performed precisely as described in the main part of
this work, by partitioning the forest.
At present, we use an external program,qgraf,[29] for the production of the forest. The groves can then be constructed by a simple depth-first-search.
On the other hand, since the unflavored forest is connected according to the results of chapter 5, it is also possible to construct the groves incrementally, starting from a single diagram. In particular, once the grove corresponding to the first diagram has been completed, we try to obtain a new diagram by performing unflavored non-gauge flips. If we find one, it can be used to construct a new grove, and we can repeat the procedure. If no more new diagram is found, the forest must be complete.
This strategy is, however, not very well suited for theories with many differ-ent fermion flavors, like the SM. The reason is that, in order for the forest to be connected, we must ignore the fermion flavors completely. But this will generally lead to the production of redundant diagrams, because the unflavored diagrams generated may not be compatible with the flavors of the actual external state.