4.4 STI at n-loop
5.1.5 An Explicit Example
To illustrate the ideas developed in the foregoing subsections, we discuss the connectedness in the forestF(3,2) of connected two-loop three-point diagrams.
F(3,2) itself contains 217 diagrams, too many to provide a practical example.
Therefore, we restrict ourselves to the subset F∗(3,2) of amputated diagrams without one-point insertions. Doing so will render this example manageable and at the same time confirm the results derived above.
F∗(3,2) contains 40 diagrams, still a lot of diagrams to draw. Fortunately, in unflavoredφ-theory many diagrams will differ only by permutations of exter-nal lines. If we consider exterexter-nal lines as indistinguishable, we can choose one diagram to represent all permutations. This will give us the possible topologies of diagrams in F∗(3,2). It turns out that there are thirteen topologies. Fur-thermore, since we know thatF∗(3,2) is connected, we need only demonstrate the connectedness of the topologies without a quartic vertex (i. e. the topologies of F3∗(3,2)), of which there are just three. The diagrams with quartic vertices are necessarily given by performing all possible contractions inF3∗(3,2).
We first illustrate in detail how the three topologies ofF3∗(3,2) are connected
by flips:
1
2 3
4 5 =
5
4
2 1 3 =
5
1 3
2 4
l 1
2 3
4 5 = 3
4
1 5 2
= 5
1 3 2
4
l
1
2 3
5
4 = 3
4 5 1
2
(5.28)
The diagrams are drawn in the left column in what could be called a conven-tional style. The middle and right column show how the flipped four-point subdiagrams—displayed in dashed line style—are embedded in these diagrams, leading to rather unconventional drawings. We denote the three topologies of F3∗(3,2) from top down ascrossed ladder, ladder andself energy topology, re-spectively.
Note that the vertices are labeled in the above diagrams only only to ease recognition of the topologies. Whencomparing topologies, these labels have to be forgotten. On the other hand, when we discuss the diagrams inF3∗(3,2), the labels 1, 2, and 3 must be kept to distinguish external lines.
Let us first count the number of diagrams inF3∗(3,2): The permutations of external lines (i. e. of the labels 1, 2, and 3) produce three different diagrams for the ladder and self energy topology. On the other hand, every permutation of external lines leaves the crossed ladder invariant. Thus, F3∗(3,2) consists of seven diagrams, one crossed ladder, three ladders and three self energies.
Next we need the flips in F3∗(3,2). We begin with the most symmetric diagram, the crossed ladder, which we abbreviate asC. Due to the symmetry ofC, every flip must lead to a ladder:
C =
1
2 3
→
1
2 3
,
2 3
1 ,
3 1 2
≡ {L1, L2, L3}
(5.29)
We have introduced the notationL1,L2, andL3for the ladder diagrams, where the index refers to the label of the external line at the “top” of the ladder.
Next, consider the flips of a ladder diagram. We shall ignore flips which leave the diagram invariant. Of course, we can flip back to the crossed ladder C. Furthermore, from (5.28) we know that a ladder can be flipped to a self energy diagram, where the self energy can appear on both “sides” of the ladder.
But a ladder can also be flipped into another ladder, e. g.
1
2 3
→
2 3
1
. (5.30)
Clearly, a similar flip can be performed at the opposite side of the ladder. Thus, every ladder diagram can be flipped into every other ladder diagram, the crossed ladder, and two self energy diagrams:
1
2 3
→
2 3
1 ,
3 1 2
,
1
2 3
,
2 3
1 ,
3 1 2
(5.31)
Denoting bySj the self energy diagram with external linej opposite to the self energy insertion, this can be written symbolically as
L1→ {L2, L2, C, S2, S3} . (5.32) Finally, we have to determine the flips of a self energy diagram. From the symmetries of the diagrams and the above discussion we see that there must be flips to two different ladders. In addition, every self energy diagram is connected with every other self energy diagram, as demonstrated by the flip below:
3 1 2
→
2 3
1
(5.33)
Altogether, the flips of the self energy diagramS1 are given by
S1→ {L2, L3, S2, S3} . (5.34) We have now determined the structure of the subset F3∗(3,2). In figure 5.2 it is represented as a graph, nodes corresponding to the diagrams and edges corresponding to a single flip. Figure 5.3 then displays how the topologies
S2
S3
S1
L1 L2
L3 C
L1
L2
L3
S1 S2
S3
C
Figure 5.2: Two views on the graph of the subset F3∗(3,2). On the left hand side, the triangles in the planes indicated by the dotted circles correspond to the three ladder diagrams (top layer) and the three self energy diagrams (bottom layer), respectively. The top node represents the crossed ladder diagram, which is completely symmetric under permutation of external lines. The right hand side then represents a view “from above”. The edges of the upper tetrahedron have been drawn in bold style to ease recognition.
with quartic vertices can be obtained fromF3∗(3,2) by performing contractions.
Arrows indicate a single flip, and the pair of contracted cubic vertices is indicated by white dots, reading flips from left to right. Of course, the displayed topologies are connected by more flips than the ones explicitely shown. The graph of the complete subsetF∗(3,2) can be viewed in figure5.4.
→
→ →
→
→ →
→ →
→ →
Figure 5.3: Contractions yielding the complete subset F∗(3,2). White dots indicate the contracted pair of cubic vertices.
Figure 5.4: The graph of the complete subsetF∗(3,2). Solid lines correspond to rotations, dotted lines to contractions or expansions. In the center of the forest, we recognize the subsetF3∗(3,2). The view corresponds to the one displayed in the right hand side of figure5.2.
—6—
Flips and Groves in Gauge Theories
In this chapter, we study the groves of connected Green’s functions in gauge theories, employing the gauge flips, as defined in section 4.3. We shall argue that the formalism of gauge flips and groves is mainly useful in spontaneously broken gauge theories. To this end, we briefly discuss the situation in QCD, as an example of an unbroken gauge theory. After that, we turn to our main goal, the classification of groves of connected Green’s functions in the (minimal) Standard Model, the external lines of which correspond to physical particles.
It will then prove extremely useful to express the possible actions of gauge flips in terms of more intuitive, higher level graphical transformations of Feyn-man diagrams, such as shifting a certain line along other lines of a diagram, breaking up a loop or joining two loops into a single one. These operations, calledgauge motionswill enable us to determine the structure of SM forests for very general SM forests. We then turn to a specific example, which we discuss in much detail. Finally, we compare the theoretical results thus obtained with the results of an investigation by means of a computer program implementing the decomposition of the forest by gauge flips as we have described it.
6.1 Flips in Gauge Theories
In the previous chapter we have investigated the forest F(E, L) of connected Feynman diagrams in unflavoredφ-theory. Flips were interpreted as higher level graphical operations on Feynman diagrams, through which the connectedness of the complete forestF(E, L) as well as various subsets could be proven. While the connectedness of these subsets is of considerable practical interest, it has no physical significance. In particular, amplitudes in unflavored φ-theory are not constrained by any symmetries. Consequently, there are no relations among Feynman diagrams induced by a gauge symmetry.
The situation is different in gauge theories. As we have demonstrated at length in the first part of this work, the STIs in gauge theories induce relations among diagrams. In particular, in chapter 4 we proved that the groves of a connected Green’s function G, i. e. the minimal gauge invariant subsets of G, can be constructed by applying gauge flips to diagrams of G.
Now, if we forget about the flavors pertaining to the lines of a diagramdinG, we can treatdas a diagram in unflavoredφ-theory, because the renormalizable interactions of a gauge theory are covered by the interaction degrees ofφ-theory.
The gauge flips indare then just a subset of all possible flips ofd. Consequently,
under gauge flips the forestF(G) ofGwill fall into disjoint subsets, which are just the groves ofG.
Since gauge flips have already been defined in4.3—more specifically, in (4.53), (4.56), and (4.58)—we are now in principle in a position to analyze the struc-ture of the groves in a general gauge theory. However, it turns out that, in order to obtain useful results, we have to be sufficiently explicit about the field content of the gauge theory. Therefore, we will discuss the QCD and the SM, as paradigms for an unbroken and broken gauge theory, respectively, in detail.
The generalization of our results to other gauge theories should then be obvious.