1.3 Overview
2.1.1 Preliminaries on graph theory
Parametric Feynman integrals
世上无难事，只怕有心人
.
Nothing in this world is difficult, but thinking makes it seem so.
吴承恩 (Wu Cheng’en), 西游记(Journey to the West), 1592
2.1 Feynman graphs, rules and integrals
2.1.1 Preliminaries on graph theory
Graphs are the central combinatorial object of study for us, so we introduce all the necessary basics that will be needed. For more extensive reviews of graph theory that also keep the connection to physics in mind, the reader is referred to [16]
and [105].
Graphs
A graph G is an ordered pair (VG, EG) of the set of vertices VG = {v1, . . . , v|VG|} and the set ofedges EG={e1, . . . , e|EG|}, together with a map ∂ :EG→VG×VG. In this thesis we will always assume that G is connected, unless it is explicitly defined as a disjoint union G=tiGi with connected components Gi.
Often we will need directed graphs, in which a direction is assigned to each edge e∈EG by specifying an ordered pair ∂(e) = (∂−(e), ∂+(e)), where the vertex
∂−(e) ∈ VG is called start or initial vertex while ∂+(e) ∈ VG is called target or final vertex. The particular choice of direction for each edge will usually have no influence on the end results that we are interested in but in some cases we will fix a particular physically motivated choice to simplify notation.
2. Parametric Feynman integrals
When considering general graphs we make no restrictions on the edges. In particular, multiple edges between the same vertices (∂(ei) = ∂(ej) for ei 6= ej) as well as edges with identical start and target (∂−(e) = ∂+(e), “self-cycles” or
“tadpoles”) are allowed. Only when we explicitly work with Feynman graphs we have the usual physical constraints that may exclude such types of graphs.
Subgraphs g ⊆ G are usually identified with their edge set Eg ⊆ EG and it is implicitly assumed that g does not contain isolated vertices, i.e. Vg = ∂+(Eg)∪
∂−(Eg). The notable exception to this are forests, which are one of a number of types of graph that we will be interested in:
• A tree T is a graph that is connected and simply connected.
• A disjoint union of trees F =tki=1Ti is called a k-forest, such that a tree is a 1-forest.
• A subgraph g ⊂ G that contains all vertices of G, i.e. Vg = VG, is called spanning.
• A bond B ⊂ G is a minimal subgraph G such that G\B has exactly two connected components.
• A simple cycle C ⊂G is a subgraph of G that is 2-regular, i.e. all vertices have exactly two edges incident to it, and has only one connected component.
For a given graph G we denote the sets of all spanning k-forests, bonds, and simple cycles with TG[k], BG, and CG[1] respectively. The number of independent cycles (loops, in physics nomenclature) is denoted h1(G), the first Betti number of the graph. If the graph is unambiguously clear from context we often just write h1 ≡h1(G).
Example 2.1.1. Let G be the banana graph with three edges depicted in fig. 2.1.
It has three spanning trees
T1 ={e1} T2 ={e2} T3 ={e3},
each consisting of a single edge. The only spanning 2-forest has no edges but only the two isolated vertices. There is also only one bond, B1 = {e1, e2, e3}, since all edges have to be removed to separate the graph into two components.
Remark 2.1.2. Consider the vector space of edge subsets of a graph G over Z2, where addition is given by the symmetric difference
E14E2 ..= (E1\E2)∪(E2\E1) = (E1∪E2)\(E1∩E2) (2.1)
2.1. Feynman graphs, rules and integrals
Figure 2.1: The banana graph with 3 edges and examples for a spanning tree, a bond and a cycle subgraph of it.
and the (degenerate) inner product is
hE1, E2i..=
are each others orthogonal complement in this vector space and thus span it. While we do not really explicitly use this anywhere, this duality between bonds and cycles in a graph underlies many of the combinatoric results of this thesis.
Two operations on graphs that we make extensive use of are the deletion and contraction of an edge. Deletion is rather self-explanatory – the edge is simply removed from the edge set. The resulting graph is denotedG\e..= (VG, EG\ {e}).
If the removal of an edge disconnects the graph then e is called a bridge, and if none of its edges are bridges then G is called bridgeless, bridgefree, 2-edge-connected1 or, in physics literature, one-particle irreducible (1PI). Contraction additionally identifies the two end points of a deleted edge, i.e. the resulting graph is G//e..= (VG|∂+(e)=∂−(e), EG\ {e}).
For edge subsetsE ⊂EGwith more than one element the operations also apply and the order of contraction or deletion does not matter. As long as E ∩E0 =∅
1Note that these notions vary slightly for disconnected graphs. A disconnected graph is bridgeless if each connected component is 2-edge-connected.
11
2. Parametric Feynman integrals
they can also be combined to yield (G\E)//E0 = (G//E0)\E.
Note that we use the double slash in the contraction to differentiate between two slightly different notions. Since we usually identify graphs with their edge sets we often have notation like G//g =G//Eg for the (edge) contraction of some subgraph g ⊂ G. On the other hand, we use G/g to denote the quotient graph in the algebraic sense (see section 3.1.1). These notions often coincide, but differ slightly for propagator Feynman graphs, as we will discuss in chapter 3.
G
Figure 2.2: A graph G, one of its spanning trees, and the graph that results from contraction of one of its edges.
The Kirchhoff polynomial
Graphs have many invariants that happen to be polynomials. Most famously there is the Tutte polynomial [128,129] and its various specialisations like the chromatic polynomial [11, 138], the Jones polynomial in knot theory [85] or the partition function of the Potts model in statistical physics [112]. The one that we are interested in differs from these in that it is a polynomial in variables α= (αe)e∈EG
assigned to the edges of a graph, whereas the others are usually univariate or bivariate2.
TheKirchhoff polynomial, which is especially in the physics literature also often
2However, the Kirchhoff polynomial can in fact be seen as a limiting case of the multivariate generalisation of the Tutte polynomial, as described in [16, section 6], [93].
12
2.1. Feynman graphs, rules and integrals called the first Symanzik polynomial, is defined as
ΨG(α)..= X
T∈TG[1]
Y
e /∈T
αe. (2.3)
It has been known for a very long time and was first introduced by Kirchhoff in his study of electrical circuits [88]. We will often make use of the abbreviation
αS ..= Y
e∈S
αe (2.4)
for any edge subset S ⊂EG, such that ΨG(α) = PT∈T[1]
G
αEG\T.
Two important properties are obvious directly from the definition: ΨG is
• homogeneous of degree h1(G) in α, and
• linear in each αe.
Moreover, it also satisfies the famous contraction-deletion relation,
ΨG= ΨG//e+αeΨG\e (2.5)
which means that the polynomials belonging to graphs that are related via contrac-tion or delecontrac-tion of edges can be recovered easily from the original graph polynomial:
ΨG//e= ΨG|αe=0 (2.6)
ΨG\e= ∂
∂αe
ΨG (2.7)
Note that two cases have to be excluded: One is the contraction of a tadpole edge, which is the same as just deleting the edge since its endpoints are already identified. The other is deletion of a bridge, which would disconnect the graph.
Since a bridge is necessarily contained in all spanning trees ΨG is independent of the corresponding edge variable such that the derivative vanishes. While this is not in itself inconsistent it would conflict with the common and useful definition
ΨG..=Y
i
ΨGi, (2.8)
for disjoint unions G = FiGi. This product is the Kirchhoff polynomial of a vertex-1-connected graph that consists of the components Gi arranged in a chain, each component overlapping with the next in only one vertex3. It is sensible to define the polynomials for disconnected graphs like this since there is clearly a one-to-one correspondence between spanning trees of such a vertex-1-connected graph and tuples of spanning trees, containing one tree from each component.
3See also [29] and the “circular joins” used therein.
2. Parametric Feynman integrals
Example 2.1.3. We have already seen one example in the introductory chapter.
The polynomial α1α2 +α1α3 +α2α3 in the integral in eq. (1.2) is the Kirchhoff polynomial of the graph in fig. 2.1 and also the Feynman graph in fig. 1.3. For a more elaborate example let G be the graph from fig. 2.2. The spanning tree depicted in that figure corresponds to the monomial α2α4α5. The full polynomial is a sum over 12 spanning trees:
ΨG = α1α3α4+α1α3α5+α1α3α6+α1α4α5 +α1α4α6+α2α3α4 +α2α3α5+α2α3α6+α2α4α5+α2α4α6+α3α4α5+α3α4α6
=α3α4(α1+α2+α5+α6) + (α3+α4)(α1α5+α1α6+α2α5+α2α6) (2.9) Matrices
Many properties of graphs can be captured by matrices4, and we discuss here some of the well known relations between graphs, matrices and the Kirchhoff polynomial.
The incidence matrix I is an |EG| × |VG| matrix Iev ..=
±1 if v =∂±(e)
0 if e is not incident to v. (2.10) The second matrix we need is the Laplacian matrix L. It is defined as the difference of the degree and adjacency matrices of the graph. Since we will not need either of those two going forward we instead use a well known identity to define the Laplacian as the product of incidence matrix and its transpose
L..=ITI. (2.11)
This is a standard result discussed in many graph theory courses [18]. A detailed proof can also be found in the author’s master thesis [72, Lemma 1.2.14].
Instead of the full matrices we will actually always need the smaller matrices in which one column (of I) or one column and one row (of L) corresponding to an arbitrarily chosen vertex of G are deleted. From now on we use I0 and L0 for these |EG| × |VG| −1 and |VG| −1× |VG| −1 matrices, called reduced incidence and reduced Laplacian matrix.
Finally, let A be the diagonal |EG| × |EG| matrix with entries Aij ..= δijαei. With this setup the well known Matrix-Tree-Theorem [35] tells us that
ΨG =αEGdet(I0TA−1I0). (2.12)
4Or, more generally, by matroids [107], [97], [16, section 8]. While we do not use them in this thesis, matroids seem like a useful tool that is currently woefully underused in physics and should be kept in mind for future work.
2.1. Feynman graphs, rules and integrals The original theorem states that the determinant ofL0 =I0TI0counts the spanning trees of the corresponding graph. ReplacingL0 by the weighted LaplacianI0TA−1I0 yields the sum over spanning trees with monomials αT−1 for each spanning tree, so multiplying with all edge parameters turns it into the Kirchhoff polynomial.
Remark 2.1.4. The polynomialΨ∗G = det(I0TAI0)is sometimes called dual Kirch-hoff polynomial. If G is planar then it is the Kirchhoff polynomial of its planar dual graph G∗.
Often I0 and A are arranged in a block matrix M ..=
A I0
−I0T 0
. (2.13)
This is called thegraph matrix ofG[13,28], and with the block matrix determinant identity
det S T U V
!
= det(S) det(V −U S−1T) (2.14) one sees that indeed
det(M) = det(A) det(I0TA−1I0) = ΨG. (2.15)