The integrand of any multi-loop integral is characterised after Feynman parametrisation by two polynomials. In this review we summarise the properties of these polynomials. Topics covered in this article include among others: Spanning trees and spanning forests, the all-minors matrix-tree theorem, recursion relations due to contraction and deletion of edges, Dodgson’s identity and matroids.
Feynman graph polynomials
Christian Bogner and Stefan Weinzierl
Institut für Theoretische Teilchenphysik und Kosmologie, RWTH Aachen,
D - 52056 Aachen, Germany
Institut für Physik, Universität Mainz,
D - 55099 Mainz, Germany
In this review we discuss Feynman graph polynomials. Let us first motivate the interest in these polynomials. The Feynman graph polynomials are of interest from a phenomenological point of view as well as from a more mathematical perspective. We start with the phenomenological aspects: For the practitioner of perturbative loop calculations the integrand of any multi-loop integral is characterised after Feynman parametrisation by two polynomials. These two polynomials, which are called the first and the second Symanzik polynomial, can be read off directly from the Feynman graph and are the subject of this review. They have many special properties and these properties can be used to derive algorithms for the computation of loop integrals.
In recent years the graph polynomials have received additional attention from a more formal point of view [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29]. Feynman integrals are now considered as non-trivial examples of mixed Hodge structures and motives. The zero sets of the graph polynomials play a crucial role in this setting.
Graph polynomials have a long history dating back to Kirchhoff . There are well-established books on this subject [31, 32, 33, 34, 35, 36]. However the field has evolved, new insights have been added and old results have been re-discovered. As a consequence the available information is scattered over the literature. What is missing is a concise summary of the properties of these polynomials. With this review we hope to fill this gap. We have tried to make this article accessible both to the phenomenological oriented physicist interested in loop calculations as well as to the mathematician interested in the properties of Feynman integrals. In addition we include in a few places results which are new and cannot be found in the literature.
This review is organised as follows: In the next section we recall some basic facts about multi-loop Feynman integrals. We introduce the two polynomials associated to a Feynman graph and give a first method for their computation. Sect. 3 is devoted to the singularities of a Feynman integral. There are two aspects to it. If we fix the external momenta and take them to lie in the Euclidean region the singularities of the Feynman integral after integration are of ultraviolet or infrared origin. They arise – apart from a possible overall ultraviolet divergence – from the regions in Feynman parameter space where one of the two graph polynomials vanishes. For the second aspect we give up the restriction on the Euclidean region and view the Feynman integral as a function of the external momenta. As we vary the external momenta, additional threshold singularities may arise. Necessary conditions for them are given by Landau’s equations. In sect. 4 we start to introduce concepts of graph theory and define for a graph its spanning trees and spanning forests. These concepts lead to a second method for the computation of the graph polynomials, such that the two graph polynomials can be directly read off from the topology of the graph. Sect. 5 introduces the Laplacian of a graph and states the matrix-tree theorem. This in turn provides a third method for the computation of the two graph polynomials. This method is well suited for computer algebra, as it involves just the computation of a determinant of a matrix. The matrix is easily constructed from the data defining the graph. In sect. 6 the two operations of deleting and contracting an edge are studied in detail. This leads to a fourth and recursive method for the computation of the two graph polynomials. In addition we discuss in this section the multivariate Tutte polynomial and Dodgson’s identity. Sect. 7 is devoted to the dual of a (planar) graph. The Kirchhoff polynomial and the first Symanzik polynomial exchange their role when going from a graph to its dual. Sect. 8 is of a more formal character. We introduce matroids, which provide a natural generalisation of graphs. Within matroid theory, some things are simpler: For example there is always the dual of a matroid, whereas for graphs we were restricted to planar graphs. Matroid theory provides in addition an answer to the question under which conditions two topologically different graphs have the same Kirchhoff polynomial. Finally, sect. 9 contains our conclusions.
2 Feynman integrals
In this section we recall some basic facts about multi-loop Feynman integrals. We introduce the two polynomials associated to a Feynman graph and give a first method for their computation. We will work in a space-time of dimensions. To set the scene let us consider a scalar Feynman graph with external lines and internal lines. Fig. 1 shows an example. In this example there are four external lines and seven internal lines. The momenta flowing in or out through the external lines are labelled , , and and can be taken as fixed -dimensional vectors. They are constrained by momentum conservation: If all momenta are taken to flow outwards, momentum conservation requires that
At each vertex of a graph we have again momentum conservation: The sum of all momenta flowing into the vertex equals the sum of all momenta flowing out of the vertex.
A graph, where the external momenta determine uniquely all internal momenta is called a tree graph. It can be shown that such a graph does not contain any closed circuit. In contrast, graphs which do contain one or more closed circuits are called loop graphs. If we have to specify besides the external momenta in addition internal momenta in order to determine uniquely all internal momenta we say that the graph contains loops. In this sense, a tree graph is a graph with zero loops and the graph in fig. 1 contains two loops. In more mathematical terms the number is known as the cyclomatic number or the first Betti number of the graph. Let us agree that we label the additional internal momenta by to .
In the example of fig. 1 there are two independent loop momenta. We can choose them to be and . Then all other internal momenta are expressed in terms of , and the external momenta , …, :
In general, each momentum flowing through an internal line is given as a linear combination of the external momenta and the loop momenta with coefficients , or :
We associate to a Feynman graph the Feynman integral
This Feynman integral depends on the Feynman graph and the external momenta . The graph together with the independent loop momenta and the external momenta fixes all internal momenta according to eq. (3). In addition depends on the masses of the particles corresponding to the internal line , as well as integer numbers , specifying the power to which each propagator is raised. The parameter is an arbitrary mass scale. We have multiplied the integral with a factor , this ensures that is dimensionless.
How to perform the integration over the loop momenta? The first step is to convert the products of propagators into a sum. This can be done with the Feynman parameter technique. In its full generality it is also applicable to cases, where each factor in the denominator is raised to some power . The formula reads:
We use this formula with . Applied to eq. (4) we have
Now one can use translational invariance of the -dimensional loop integrals and shift each loop momentum to complete the square, such that the integrand depends only on . Then all -dimensional loop integrals can be performed. As the integrals over the Feynman parameters still remain, this allows us to treat the -dimensional loop integrals for Feynman parameter integrals. One arrives at the following Feynman parameter integral :
The functions and depend on the Feynman parameters . If one expresses
where is a matrix with scalar entries and is a -vector with four-vectors as entries, one obtains
The functions and are called graph polynomials and are the subject of this review. They are polynomials in the Feynman parameters and – as we will show later – can be derived from the topology of the underlying graph. The polynomials and have the following properties:
They are homogeneous in the Feynman parameters, is of degree , is of degree .
is linear in each Feynman parameter. If all internal masses are zero, then also is linear in each Feynman parameter.
In expanded form each monomial of has coefficient .
We call the first Symanzik polynomial and the second Symanzik polynomial. Eqs. (8) and (9) allow us to calculate these polynomials for a given graph. We will learn several alternative ways to determine these polynomials later, but for the moment it is instructive to go through this exercise for the graph of fig. 1. We will consider the case
In comparing with eq. (8) we find
Plugging this into eq. (9) we obtain the graph polynomials as
We see in this example that is of degree and is of degree . Each polynomial is linear in each Feynman parameter. Furthermore, when we write in expanded form
each term has coefficient .
In this section we briefly discuss singularities of Feynman integrals. There are two aspects to it. First we fix the external momenta and take them to lie in the Euclidean region. We may encounter singularities in the Feynman integral. These singularities are of ultraviolet or infrared origin and require regularisation. We briefly discuss how they are related to the vanishing of the two graph polynomials and . For the second aspect we consider the Feynman integrals as a function of the external momenta, which are now allowed to lie in the physical region. Landau’s equations give a necessary condition for a singularity to occur in the Feynman integral as we vary the external momenta.
It often occurs that the Feynman integral as given in eq. (4) or in eq. (7) is an ill-defined and divergent expression when considered in dimensions. These divergences are related to ultraviolet or infrared singularities. Dimensional regularisation is usually employed to regulate these divergences. Within dimensional regularisation one considers the Feynman integral in dimensions. Going away from the integer value regularises the integral. In dimensions the Feynman integral has a Laurent expansion in the parameter . The poles of the Laurent series correspond to the original divergences of the integral in four dimensions.
From the Feynman parameter integral in eq. (7) we see that there are three possibilities how poles in can arise: First of all the Gamma-function of the prefactor can give rise to a (single) pole if the argument of this function is close to zero or to a negative integer value. This divergence is called the overall ultraviolet divergence.
Secondly, we consider the polynomial . Depending on the exponent of the vanishing of the polynomial in some part of the integration region can lead to poles in after integration. As mentioned in the previous section, each term of the expanded form of the polynomial has coefficient , therefore can only vanish if some of the Feynman parameters are equal to zero. In other words, is non-zero (and positive) inside the integration region, but may vanish on the boundary of the integration region. Poles in resulting from the vanishing of are related to ultraviolet sub-divergences.
Thirdly, we consider the polynomial . In an analytic calculation one often considers the Feynman integral in the Euclidean region. The Euclidean region is defined as the region, where all invariants are negative or zero, and all internal masses are positive or zero. The result in the physical region is then obtained by analytic continuation. It can be shown that in the Euclidean region the polynomial is also non-zero (and positive) inside the integration region. Therefore under the assumption that the external kinematics is within the Euclidean region the polynomial can only vanish on the boundary of the integration region, similar to what has been observed for the the polynomial . Depending on the exponent of the vanishing of the polynomial on the boundary of the integration region may lead to poles in after integration. These poles are related to infrared divergences.
The Feynman integral as given in eq. (7) depends through the polynomial on the external momenta . We can also discuss as a function of the ’s without restricting the external kinematics to the Euclidean region. Doing so, the region where the polynomial vanishes is no longer restricted to the boundary of the Feynman parameter integration region and we may encounter zeros of the polynomial inside the integration region. The vanishing of may in turn result in divergences after integration. These singularities are called Landau singularities. Necessary conditions for the occurrence of a Landau singularity are given as follows: A Landau singularity may occur if and if there exists a subset of such that
The case corresponding to is called the leading Landau singularity, and cases corresponding to are called non-leading singularities. It is sufficient to focus on the leading Landau singularity, since a non-leading singularity is the leading Landau singularity of a sub-graph of obtained by contracting the propagators corresponding to the Feynman parameters with .
Let us now consider the leading Landau singularity of a graph with external lines. We view the Feynman integral as a function of the external momenta and a solution of the Landau equations is given by a set of external momenta
satisfying momentum conservation and eq. (3). If the momenta in eq. (17) define a one-dimensional sub-space, we call the Landau singularity a normal threshold. In this case all external momenta are collinear. If on the contrary the momenta in eq. (17) define a higher-dimensional sub-space, we speak of an anomalous threshold.
We give a simple example for a Feynman integral with an ultraviolet divergence and a normal threshold.
The graph in fig. 2 shows a one-loop two-point function. The internal propagators correspond to particles with mass . This graph corresponds in dimensions to the Feynman integral
Introducing Feynman parameters one obtains the form of eq. (7):
This integral is easily evaluated with standard techniques :
Here, denotes Euler’s constant. The -term corresponds to an ultraviolet divergence. As a function of the integral has a normal threshold at . The normal threshold manifests itself as a branch point in the complex -plane.
4 Spanning trees and spanning forests
In this section we start to introduce concepts of graph theory. We define spanning trees and spanning forests. These concepts lead to a second method for the computation of the graph polynomials. We consider a connected graph with external lines and internal lines. Let be the number of vertices of the graph . We denote the set of internal edges of the graph by
and the set of vertices by
As before we denote by the first Betti number of the graph (or in physics jargon: the number of loops). We have the relation
If we would allow for disconnected graphs, the corresponding formula for the first Betti number would be , where is the number of connected components. A spanning tree for the graph is a sub-graph of satisfying the following requirements:
contains all the vertices of ,
the first Betti number of is zero,
If is a spanning tree for , then it can be obtained from by deleting edges. In general a given graph has several spanning trees. We will later obtain a formula which counts the number of spanning trees for a given graph . A spanning forest for the graph is a sub-graph of satisfying just the first two requirements:
contains all the vertices of ,
the first Betti number of is zero.
It is not required that a spanning forest is connected. If has connected components, we say that is a -forest. A spanning tree is a spanning -forest. If is a spanning -forest for , then it can be obtained from by deleting edges.
We denote by the set of spanning forests of and by the set of spanning -forests of . Obviously, we can write as the disjoint union
is the set of spanning trees. For an element of we write
The are the connected components of the -forest. They are necessarily trees. We denote by the set of external momenta attached to . For the example of the -forest in the right picture of fig. 3 we have
The spanning trees and the spanning -forests of a graph are closely related to the graph polynomials and of the graph. We have
The sum is over all spanning trees for , and over all spanning -forests in the first term of the formula for . Eq. (4) provides a second method for the computation of the graph polynomials and . Let us first look at the formula for . For each spanning tree we take the edges , which have been removed from the graph to obtain . The product of the corresponding Feynman parameters gives a monomial. The first formula says, that is the sum of all the monomials obtained from all spanning trees. The formula for has two parts: One part is related to the external momenta and the other part involves the masses. The latter is rather simple and we write
We focus on the polynomial . Here the -forests are relevant. For each -forest we consider again the edges , which have been removed from the graph to obtain . The product of the corresponding Feynman parameters defines again a monomial, which in addition is multiplied by a quantity which depends on the external momenta. We define the square of the sum of momenta through the cut lines of by
Here we assumed for simplicity that the orientation of the momenta of the cut internal lines are chosen such that all cut momenta flow from to (or alternatively that all cut momenta flow from to , but not mixed). From momentum conservation it follows that the sum of the momenta flowing through the cut lines out of is equal to the negative of the sum of the external momenta of . With the same reasoning the sum of the momenta flowing through the cut lines into is equal to the sum of the external momenta of . Therefore we can equally write
and is given by
Since we have to remove edges from to obtain a spanning tree and edges to obtain a spanning -forest, it follows that and are homogeneous in the Feynman parameters of degree and , respectively. From the fact, that an internal edge can be removed at most once, it follows that and are linear in each Feynman parameter. Finally it is obvious from eq. (4) that each monomial in the expanded form of has coefficient .
Let us look at an example. Fig. 4 shows the graph of a two-loop two-point integral. We take again all internal masses to be zero.
The set of all spanning trees for this graph is shown in fig. 5. There are eight spanning trees.
Fig. 6 shows the set of spanning -forests for this graph. There are ten spanning -forests. The last example in each row of fig. 6 does not contribute to the graph polynomial , since the momentum sum flowing through all cut lines is zero. Therefore we have in this case . In all other cases we have . We arrive therefore at the graph polynomials
5 The matrix-tree theorem
In this section we introduce the Laplacian of a graph. The Laplacian is a matrix constructed from the topology of the graph. The determinant of a minor of this matrix where the -th row and column have been deleted gives us the Kirchhoff polynomial of the graph, which in turn upon a simple substitution leads to the first Symanzik polynomial. We then show how this construction generalises for the second Symanzik polynomial. This provides a third method for the computation of the two graph polynomials. This method is very well suited for computer algebra systems, as it involves just the computation of a determinant of a matrix. The matrix is easily constructed from the data defining the graph.
We begin with the Kirchhoff polynomial of a graph. This polynomial is defined by
The definition is very similar to the expression for the first Symanzik polynomial in eq. (4). Again we have a sum over all spanning trees, but this time we take for each spanning tree the monomial of the Feynman parameters corresponding to the edges which have not been removed. The Kirchhoff polynomial is therefore homogeneous of degree in the Feynman parameters. There is a simple relation between the Kirchhoff polynomial and the first Symanzik polynomial :
These equations are immediately evident from the fact that and are homogeneous polynomials which are linear in each variable together with the fact that a monomial corresponding to a specific spanning tree in one polynomial contains exactly those Feynman parameters which are not in the corresponding monomial in the other polynomial.
The graph may contain multiple edges and self-loops.
We speak of a multiple edge, if two vertices are connected by more than one edge. We speak of a self-loop if an edge starts and ends at the same vertex. In the physics literature a self-loop is known as a tadpole. Fig. 7 shows a simple example for a double edge and a self-loop. If the vertices and are connected by two edges and , then the Laplacian depends only on the sum . If an edge is a self-loop attached to a vertex , then it does not contribute to the Laplacian.
Let us consider an example: The Laplacian of the two-loop two-point graph of fig. 4 is given by
In the sequel we will need minors of the matrix and it is convenient to introduce the following notation: For a matrix we denote by the matrix, which is obtained from by deleting the rows , …, and the columns , …, . For we will simply write .
Let be an arbitrary vertex of . The matrix-tree theorem states 
i.e. the Kirchhoff polynomial is given by the determinant of the minor of the Laplacian, where the -th row and column have been removed. One can choose for any number between and .
The matrix-tree theorem allows to determine the number of spanning trees of a given graph . Setting , each monomial in and reduces to . There is exactly one monomial for each spanning tree, therefore one obtains
The matrix-tree theorem as in eq. (37) relates the determinant of the minor of the Laplacian, where the -th row and the -th column have been deleted to a sum over the spanning trees of the graph. There are two generalisations we can think of:
We delete more than one row and column.
We delete different rows and columns, i.e. we delete row and column with .
The all-minors matrix-tree theorem relates the determinant of the corresponding minor to a specific sum over spanning forests [41, 42, 43]. To state this theorem we need some notation: We consider a graph with vertices. Let with denote the rows, which we delete from the Laplacian, and let with denote the columns to be deleted from the Laplacian. We set and . We denote by the spanning -forests, such that each tree of an element of contains exactly one vertex and exactly one vertex . The set is a sub-set of all spanning -forests. We now consider an element of . Since the element is a -forest, it consists therefore of trees and we can write it as
We can label the trees such that , …, . By assumption, each tree contains also exactly one vertex from the set , although not necessarily in the order . In general it will be in a different order, which we can specify by a permutation :
The all-minors matrix-tree theorem reads then
In the special case this reduces to
If we specialise further to , the sum equals the sum over all spanning trees (since each spanning -forest of necessarily contains the vertex ). We recover the classical matrix-tree theorem:
Let us illustrate the all-minors matrix-tree theorem with an example. We consider again the two-loop two-point graph with the labelling of the vertices as shown in fig. 8. Taking as an example
we find for the determinant of :
On the other hand there are exactly two -forests, such that in each -forest the vertices and are contained in one tree, while the vertex is contained in the other tree. These two -forests are shown in fig. 8. The monomials corresponding to these two -trees are and , respectively. The permutation is in both cases the identity and with , we have an overall minus sign
Eq. (37) together with eq. (34) allows to determine the first Symanzik polynomial from the Laplacian of the graph. We may ask if also the polynomial can be obtained in a similar way. We consider again a graph with internal edges , internal vertices and external legs. We attach additional vertices , …, to the ends of the external legs and view the former external legs as additional edges , …, . This defines a new graph . As before we associate the parameters to the edges () and new parameters to the edges (). The Laplacian of is a matrix. Now we consider the polynomial
is a polynomial of degree in the variables and . We can expand in polynomials homogeneous in the variables :
where is homogeneous of degree in the variables . We further write
The sum is over all indices with . The are homogeneous polynomials of degree in the variables . For and one finds
for any . is related to :
with and if is an internal edge or if is an external edge. The sum is over all -forests of , such that each tree in an -forest contains exactly one of the added vertices , …, . Each -forest has connected components. The polynomial by definition does not contain any variable . would therefore correspond to forests where all edges connecting the external vertices , …, have been cut. The external vertices appear therefore as isolated vertices in the forest. Such a forest must necessarily have more than connected components. This is a contradiction with the requirement of having exactly connected components and therefore . Next, we consider . Each term is linear in the variables . Therefore vertices of the added vertices , …, appear as isolated vertices in the -forest. The remaining added vertex is connected to a spanning tree of . Summing over all possibilities one sees that is given by the product of with the Kirchhoff polynomial of . Finally we consider . Here, of the added vertices appear as isolated vertices. The remaining two are connected to a spanning -forest of the graph , one to each tree of the -forest. Summing over all possibilities one obtains eq. (53).
Eq. (52) and eq. (53) together with eq. (28) allow the computation of the first and second Symanzik polynomial from the Laplacian of the graph . This provides a third method for the computation of the graph polynomials and .
As an example we consider the double-box graph of fig. 1. We attach an additional vertex to every external line.
Fig. 9 shows the labelling of the vertices and the Feynman parameters for the graph . The Laplacian of is a -matrix. We are interested in the minor, where – with the labelling of fig. 9 – we delete the rows and columns , , and . The determinant of this minor reads
For the polynomials and one finds