Abstract
To any graph with external half-edges and internal masses, we associate canonical integrals which depend non-trivially on particle masses and momenta, and are always finite. They are generalised Feynman integrals which satisfy graphical relations obtained from contracting edges in graphs, and a coproduct involving both ultra-violet and infra-red subgraphs. Their integrands are defined by evaluating bi-invariant forms, which represent stable classes in the cohomology of the general linear group, on a generalised graph Laplacian matrix which depends on the external kinematics of a graph.
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1 Introduction
In the paper [Bro21] we introduced canonical differential forms on moduli spaces of metric graphs, and showed how they provide a connection between the cohomology of the commutative graph complex, the algebraic K-theory of the integers, and Feynman integrals. In this paper, we extend this theory to the case of graphs with external momenta and masses, with an emphasis on physical aspects.
Let us first recall some background on Feynman integrals. Consider a connected graph G, with n external legs (or half-edges), \(h_G\) loops, and \(e_G\) internal edges. Every external leg represents an incoming particle with momentum \(q_i \in {\mathbb {R}}^d\) subject to overall momentum conservation \(\sum _{i=1}^n q_i=0\). The number of spacetime dimensions d will be \(d=2\) or \(d=4\) in this paper. To each edge e one additionally associates one of a finite set of particle masses \(m_e\), which are arbitrary real numbers.
For scalar theories, the parametric Feynman integral is the projective integral
where the edges of G are numbered from 1 to \(e_G\), the form \(\Omega _G\) is defined by
and \(\sigma _G = \{ (\alpha _1: \cdots : \alpha _{e_G} ) \in {\mathbb {P}}^{e_G-1}({\mathbb {R}}): \alpha _i \ge 0\}\) is the coordinate simplex in projective space. The quantity D is typically an even integer, or, in the setting of dimensional regularisation, \(D= 2k - \varepsilon \), for a small positive \(\varepsilon \).
The integrand involves the ‘second Symanzik’ polynomial
which is expressed in terms of the two more basic polynomials \(\Psi _G, \Phi _G(q)\), which are homogeneous in the \(\alpha _e, e\in E_G\) of degrees \(h_G\), \(h_G+1\) respectively. They are defined as sums over spanning forests in the graph G with 1 or 2 connected components (see §6.1). The integral (1.1) diverges in general and can be regularised in a variety of manners including, for example, Laurent expansion in the parameter \(\varepsilon \).
For most quantum field theories of relevance for collider physics, one is led to consider a wider class of Feynman integrals, which in parametric form (see, e.g. [Gol19]) have the following general shape (omitting \(\Gamma \)-factors for simplicity):
where \(a,b\in {\mathbb {Z}}\) (or \(a,b \in {\mathbb {Z}}+ \varepsilon {\mathbb {Z}}\)), and the numerator \(N_G\) is a polynomial in the parameters \(\alpha _i\) with typically complicated coefficients.
Faced with the considerable difficulty in computing integrals (1.1) or (1.2), a common recent theme of research is to seek alternative theoretical frameworks in which the corresponding amplitudes are simpler and more highly structured, with a long term view to unearthing mathematical properties which are valid for general quantum field theories. Notable examples in this direction include the amplituhedron programme [AHT14], which organises certain amplitudes in \(N=4\) SYM according to geometric principles; string perturbation theory, which studies scattering amplitudes defined on punctured Riemann surfaces; or integrable ‘fishnet’ models [Zam80, GK16] which reduce to a small number of Feynman graphs.
In this paper, we introduce a special class of geometrically-defined integrals of the form (1.2), which are always finite irrespective of the graph G. They generalize [Bro21] to incorporate masses and momenta, and satisfy symmetry properties including a family of graphically-encoded relations. The latter involve both contraction of internal edges and also the ‘motic’ coproduct of [Bro17] which has applications to the study of both UV and IR divergences [BBH20, AHHM22].
1.1 Canonical forms and their integrals
Our starting point is the moduli space \({\mathcal {M}}^{\textrm{trop}}_{g}\) of stable tropical curves of genus g, or more precisely the open locus
consisting of graphs whose vertices have weight zero. The quotient \(\left( {\mathcal {M}}^{\textrm{trop}}_{g}\right) _{w=0} /{\mathbb {R}}^{\times }_{>0}\) is isomorphic to the quotient of Culler–Vogtmann’s Outer space \({\mathcal {O}}_g\) [CV86] by the group of outer automorphisms \(\hbox {Out}(F_g)\) of the free group \(F_g\) on g elements. Points in this space are equivalence classes of connected metric graphs (with no external legs), where a metric on a graph is an assignment of a length \(\ell _e>0\) to every internal edge, normalised so that the total length of all edges is 1.
The tropical Torelli map [BMV11] on \({\mathcal {M}}^{\textrm{trop}}_{g}\) restricts to a map
where \({\mathcal {P}}_g\) is the space of positive definite \(g\times g\) symmetric matrices X, upon which \( P \in \textrm{GL}_g({\mathbb {Z}})\) acts via \(X \mapsto P^T X P\). The map (1.3) assigns to a connected metric graph G the \(\textrm{GL}_g({\mathbb {Z}})\)-equivalence class of any choice of graph Laplacian matrix \(\Lambda _G\). The latter is a \(g \times g\) symmetric matrix whose determinant
equals the first Symanzik, or Kirchhoff, graph polynomial which arises in the integrals (1.1). The graph Laplacian may be interpreted as a tropical version of the Riemann polarization form on a compact Riemann surface of genus g. Canonical forms on \(\left( {\mathcal {M}}^{\textrm{trop}}_{g}\right) _{w=0}\) are defined as follows. The forms \(\textrm{tr}( (X^{-1} d X)^{4k+1})\), for \(k\ge 1\), are bi-invariant under left and right multiplication by \(\textrm{GL}_g({\mathbb {R}})\) and were shown by Borel to generate the stable cohomology of the symmetric space \({\mathcal {P}}_g/\textrm{GL}_g({\mathbb {Z}})\). Their pull-backs \(\textrm{tr}( (\Lambda _G^{-1} d \Lambda _G)^{4k+1})\) along the map (1.3) defines a distinguished family of differential forms on the images of cells in \(\left( {\mathcal {M}}^{\textrm{trop}}_{g}\right) _{w=0} /{\mathbb {R}}^{\times }_{>0}\). The integrals of these forms are called canonical since they do not depend on any choices, and were studied in [Bro21]. In that paper it was shown that interesting examples of Feynman residues for vacuum diagrams (see [Sch10] for a survey) of relevance to quantum field theory can arise as canonical integrals. A natural question is whether general Feynman integrals with non-trivial kinematics are also amenable to a ‘canonical’ geometrical interpretation of this kind. This is addressed in this paper.
1.1.1 Canonical forms with kinematic dependence
We consider a generalised moduli space \({\mathcal {M}}^{\textrm{trop}}_{g,n,M}\), first considered in [BM19], of weighted metric graphs which have n external legs labelled from \(1,\ldots , n\), and where each internal edge e is assigned one of M possible particle masses. When \(M=0\), this space coincides with the moduli space \({\mathcal {M}}^{\textrm{trop}}_{g,n}\) of n-marked tropical curves of genus g. Denote the open locus of graphs whose vertex weights are all zero by:
Consider in the first instance external particle momenta \(q_1,\ldots , q_n \in {\mathbb {R}}^2\) in two-dimensional Euclidean space, which we identify with the complex numbers, and any choice of M internal particle masses. For any choice of routing \(\mu _e\) of the external momenta through the internal edges e of G, we consider a generalised graph Laplacian matrix (see (5.4)) whose determinant satisfies
The generalised Laplacian \(\widetilde{\Lambda }_G(\mu ,m)\) is a Hermitian \((g+1) \times (g+1)\) matrix (note that \(g+1\) is the degree of the polynomial \(\Xi _G(q,m)\)). It is the tropical version of a regularised Hermitian polarization form on the cohomology of a compact Riemann surface with n punctures. From this perspective, the external particle momenta are interpreted as tropical versions of the residues of a differential form of the third kind. The case when the external momenta lie in \({\mathbb {R}}^4\) may be treated by replacing complex momenta with momenta in the ring of quaternions \({\mathcal {Q}}\cong {\mathbb {R}}^4\), leading to a quaternionic Hermitian graph Laplacian which satisfies
where \(\hbox {Det}\) of a quaternionic matrix is the determinant of its complex adjoint. One of our key results (§9) is that the generalised graph Laplacian admits an asymptotic decomposition into block matrices as one approaches the boundary of the moduli space \(({\mathcal {M}}^{\textrm{trop}}_{g,n,M})_{w=0}\). This provides a geometric interpretation of asymptotic factorisation formulae for graph polynomials which are important for the study of infra-red and ultra-violet singularities of Feynman integrals [BBH20, AHHM22].
Define canonical formsFootnote 1 of the ‘first’ and ‘second’ kinds for all \(k \ge 1\), by
when the external momenta lie in \({\mathbb {R}}^2 \cong {\mathbb {C}}\). In the case of quaternionic momenta in \({\mathbb {R}}^4 \cong {\mathcal {Q}}\), the definition of canonical forms of the first kind is unchanged, but the canonical forms of the second kind are now defined for all \(k \ge 1\) by:
where the quaternionic trace \(\hbox {Tr}\) is the trace of the complex adjoint. A crucial property of canonical forms (Theorem 9.3) is that they factorise as one approaches infinity along the boundary faces of Feynman polytopes.
When the external momenta lie in \({\mathbb {R}}^2 \cong {\mathbb {C}}\), we define a canonical form \(\omega _G\) to be any homogeneous polynomial in the forms (1.7). When the momenta lie in \({\mathbb {R}}^4 \cong {\mathcal {Q}}\), a (quaternionic) canonical form, denoted \(\omega ^{{\mathcal {Q}}}_G\), is a homogeneous polynomial in \(\omega ^{4k+1}_G\) and \( \varpi ^{{\mathcal {Q}}, 4k+1}_G\). For any connected graph G, and any such form \(\omega _G\) of degree \(e_G+1\), we may consider the integral
Since it is independent of all choices which go into defining generalised graph Laplacian matrices, we call it a canonical integral.
We prove the following facts about the integrals (1.9):
-
(1)
They are always finite.
-
(2)
They are generalised Feynman integrals of the form (1.2).
-
(3)
They satisfy graphical relations involving both the contraction of internal edges and a generalisation of the Connes–Kreimer coproduct which encodes not only ultraviolet, but also infra-red phenomena.
Thus the integrals (1.9), which may be loosely interpreted as certain ‘volumes’ of cells on a moduli space of tropical curves, pick out a distinguished class of generalised Feynman integrals with special properties. The reader may wish to turn to Sects. 11–13 for examples of canonical integrals and their relations.
Remark 1.1
The canonical forms (1.7), (1.8) arise from the graded exterior algebras of invariant differential forms \(I_{H_n}\) on the symmetric spaces associated to the following sequences of classical non-compact Lie groups \(H_n\). The following table is extracted from [Bor74], page 265, and based on results of H. Cartan:
\({H}_{n}\) | \(\underset{\longleftarrow }{\lim }\ \, {I}_{{H}_{n}}\) |
|---|---|
\(\hbox {SL}_n({\mathbb {R}})\) | \({E}[ x_{4k+1}, k\ge 1 ]\) |
\(\hbox {SL}_n({\mathbb {C}})\) | \({E}[ x_{2k+1}, k\ge 1 ]\) |
\(\hbox {SL}_n({\mathcal {Q}})\) | \({E}[x_{4k+1}, k\ge 1 ]\) |
where \(E[\{x_i\}]\) denotes the graded exterior algebra generated by elements \(x_i\) in odd degrees i, and the inverse limit of the spaces of forms \(I_{H_n}\) is taken in the category of graded exterior algebras. The generators are given by \(\beta ^n_X\) in the case of real and complex matrices, and by \(\hbox {Tr} ((X^{-1} d X)^n)\) in the quaternionic case. They always vanish when n is even, and additionally for \(n\equiv 3 \pmod 4\) in the case of \({\mathbb {R}}\) and \({\mathcal {Q}}\).
1.2 Discussion and questions for further research
It would be very interesting to find a momentum or position space formulation for the integrals (1.9), and to relate them more closely to the theory of graphical functions [GPS17, BS21] in the case when all masses vanish. The generalised graph Laplacian is a tropical version of ‘single-valued’ period integrals on a punctured Riemann sphere, which are prevalent in closed string perturbation theory, and suggests a possible connection with string perturbation theory. In a different direction, it was shown in [Bro21] that the canonical integrals for graphs without kinematics are closely connected to the homology of the commutative, even, graph complex. It would be interesting to find such a relation for the integrals (1.9) based on Corollary 10.4, for example. See also [BK20] for related interpretations of Feynman integrals.
We also expect the work in this paper to have applications to the motivic study of Feynman integrals. For example, the existence of the generalised graph Laplacian implies that the graph motives at a fixed loop order have a universal family, and the canonical forms provide universally-defined classes in the relative cohomology of graph hypersurface complements and the de Rham realisation of graph motives.
In particular, since canonical integrals are distinguished elements of period matrices associated to ordinary Feynman integrals, they should be related to the latter via differential equations and also via the motivic coaction. The Stokes’ relations proven here should thus enable one to transfer information about differential equations, periods, and infra-red singularities between graphs with different topologies.
Finally, it would also be interesting to study integrals of the form
in dimensional regularisation, where \(\omega _G\) is canonical. They too will satisfy relations via Stokes’ formula, but of a slightly different form to those considered here.
1.3 Contents
In §2 we recall some notations and conventions as well as some background on quaternionic matrices. In §3 we describe a moduli space of marked metric graphs with additional edge colourings, which encode particle masses. This space was previously introduced by Berghoff and Mühlbauer in [BM19]. After a brief discussion in §4 of momentum routings, and complex and quaternionic momenta, we define the generalised Laplacians in §5. In the case when momenta are quaternionic and all masses are zero, they coincide with matrices previously considered by Bloch and Kreimer in [BK10]. The way we arrived at the definition is explained in §14; namely by taking the tropical analogue of the single-valued integration pairing on the cohomology of a punctured Riemann surface. In §6 we recall the definitions of Symanzik polynomials, and prove a formula for \(\Phi _G\) in terms of spanning forest polynomials, which is used to deduce (1.5) and (1.6). Starting in §8, we recall the definitions of bi-invariant differential forms. A key result is a formula (§9) for the asymptotic behaviour at infinity of graph Laplacian matrices, which is used to prove the convergence of canonical integrals in §10, along with the Stokes relations. It could also be used to prove (1.5) and (1.6). Sections 11 and 13 provide examples of canonical integrals for some well-studied Feynman diagrams, and §12 illustrates the general Stokes relations by deducing a graphical 5-term functional equation for the massive box diagram in 4 spacetime dimensions.
2 Notations and Conventions
Graphs in this paper are finite, and usually connected. They have a finite number n of external half-edges. The set of internal edges is denoted by E(G), and the set of vertices by V(G). We write \(e_G = |E(G)|\) for the number of internal edges, and \(h_G\) for the number of loops of G. When an internal edge is directed, we denote by s(e) its source, and t(e) its target. An edge with \(s(e)=t(e)\) is called a self-edge or tadpole.
2.1 Reminders on quaternions
Denote the ring of quaternions by
where \({\textsf{1}}\) is the unit, \({\textsf{i}}^2 = {\textsf{j}}^2={\textsf{k}}^2=-{\textsf{1}}\), and \({\textsf{i}}{\textsf{j}}=-{\textsf{j}}{\textsf{i}}= {\textsf{k}}, {\textsf{k}}{\textsf{i}}=-{\textsf{i}}{\textsf{k}}= {\textsf{j}}, {\textsf{j}}{\textsf{k}}=-{\textsf{k}}{\textsf{j}}={\textsf{i}}\). It admits a representation \( \chi : {\mathcal {Q}}\rightarrow M_2({\mathbb {C}})\) via
The ring \({\mathcal {Q}}\) has an anti-involution, denoted \(x \mapsto \overline{x}\), which is induced by Hermitian conjugation \(V \mapsto (\overline{V})^T\) on \(M_2({\mathbb {C}})\). The conjugate of \( x = x_1 {\textsf{1}}+ x_2 {\textsf{i}}+ x_3 {\textsf{j}}+ x_4 {\textsf{k}}\) is \( \overline{x} = x_1 {\textsf{1}}- x_2 {\textsf{i}}- x_3 {\textsf{j}}- x_4 {\textsf{k}}\), and the square of the quaternion norm
coincides with the square of the Euclidean norm.
2.1.1 Quaternionic matrices
Matrices with entries in the ring of quaternions share many properties with rings of matrices over a commutative ring (see [Zha97] for a survey). A convenient method for studying them is via the complex adjoint representation, which is the ring homomorphism
induced by the representation \(\chi \) defined above. One shows that a matrix \(M \in M_n({\mathcal {Q}})\) has a unique left inverse (which is necessarily also a right inverse) if and only if its image \(\chi _M\) is invertible. However, the determinant of a quaternionic matrix is not defined in general, and its trace is not a similarity invariant, meaning that \(\textrm{tr}(P^{-1}MP) \ne \textrm{tr}(M)\) for general P, M. For this reason it is customary to define the determinant and trace of \( M \in M_n({\mathcal {Q}})\) to be \(\hbox {Det} (M) = \det \chi _M\) and \(\hbox {Tr}(M) = \textrm{tr}\,\chi _M\) respectively. They take values in the complex numbers.
Two particular representations of quaternionic matrices are convenient:
-
(i)
If one identifies \(M_n(M_2({\mathbb {C}})) \) with \(M_{2n}({\mathbb {C}})\), then the image of \(\chi \) is the set of matrices \(\{V \in M_{2n}({\mathbb {C}}): \overline{V}=J_n V J_n^{-1}\}\), where \(J_n\) is the block diagonal matrix consisting of n copies of \(\chi _{{\textsf{j}}}\) along the diagonal.
-
(ii)
If one identifies \(M_n(M_2({\mathbb {C}})) \) with \(M_2(M_n({\mathbb {C}}))\), then the image of \(\chi \) consists of block matrices of the form:
$$\begin{aligned} \chi _{A + {\textsf{j}}B} = \begin{pmatrix} A &{} B \\ - \overline{B} &{} \overline{A} \end{pmatrix}, \quad A, B \in M_n({\mathbb {C}}), \end{aligned}$$(2.2)since any quaternionic matrix may be uniquely written \(A + {\textsf{j}}\,B\), with \(A, B \in M_n({\mathbb {C}})\). Here, and later, we identify the complex numbers with \({\mathbb {R}}{\textsf{1}}+ {\mathbb {R}}{\textsf{i}}\subset {\mathcal {Q}}\).
Either representation may be used to compute the quaternionic trace or determinant. We shall use (i) in the examples.
2.1.2 Moore determinant
The first part of the following proposition implies the existence of the reduced Pfaffian norm \(\hbox {Nrp}\) [DR22, Tig99].
Proposition 2.1
Let \(\chi \) be the map (2.1).
-
(i)
Let \(M \in M_n({\mathcal {Q}})\) be a quaternionic Hermitian matrix, i.e., \(M^T = \overline{M}\), where \(\overline{M}\) denotes its quaternionic conjugate. Then \(\chi _M\) is Hermitian and
$$\begin{aligned} \hbox {Det}(M) = \det ( \chi _{M}) = \left( F(\chi _{M})\right) ^2, \end{aligned}$$(2.3)where F is a polynomial in the entries of \(\chi _{M}\) which is invariant under complex conjugation.
-
(ii)
In the case when \(M \in M_n({\mathbb {C}}) \subset M_n({\mathcal {Q}})\) has complex entries
$$\begin{aligned} \hbox {Det}(M)= \det ( \chi _M) = \det (M)^2. \end{aligned}$$(2.4)
Although we shall not use it, the Moore determinant \(F=\hbox {Nrp}(M)\) is a canonical solution to (2.3) which satisfies \(\hbox {Nrp}(M) = \det (M)\) for all \(M \in M_n({\mathbb {C}})\).
3 Moduli of Marked Metric Graphs with Masses and Momenta
We consider a moduli space of metric graphs with external legs and masses. It was previously studied in [BM19].
3.1 Graphs with external legs and massive edges
3.1.1 Combinatorial masses
Let \(M\ge 0\) be a fixed integer. For a connected graph G, the assignment of a mass to each internal edge \(e\in E(G)\) is encoded by
The labelling 0 denotes the zero mass.
3.1.2 Physical masses
We fix a finite set of non-zero masses \(m_i\in {\mathbb {R}}^{\times }\), for \(i=1,\ldots , M\), and set \(m_0=0\) to be the zero mass, i.e., \(m: \{0,1\ldots , M\} \rightarrow {\mathbb {R}}\). Only the mass squares \(m_i^2\) will play a role in the theory. The mass \(m_i\) denotes the mass of a particle of type i, and there is no need for the \(m_i\) to be distinct. When drawing Feynman diagrams, our convention is to depict massless edges with a single line, and massive edges with a doubled line.
Two cases are of particular interest, namely the ‘massless case’ \(M=0\), when all edges have mass zero; and the ‘generic mass case’, when \(M=e_G\), and \(\underline{M}: E(G) \overset{\sim }{\rightarrow } \{1,\ldots , e_G\}\) is a bijection.
3.1.3 Combinatorial momenta
Let \(n\ge 0\) be a fixed integer. If \(n\ge 1\), the data of n external legs or ‘markings’ is given by a map
which can be realised combinatorially by attaching a half-edge, labelled i, to the vertex \(\underline{n}(i)\). These external half-edges will always be oriented towards their endpoint, i.e., all external momenta are incoming. The map \(\underline{n}\) does not need to be injective or surjective. In the case when \(n=0\), or when the image of \(\underline{n}\) reduces to a single vertex, the associated graph is called a vacuum diagram, or momentumless.
3.1.4 Physical momenta
Let \(d \ge 0\). Consider n vectors in Euclidean space
which satisfy momentum conservation
The external half-edge i represents an incoming particle carrying momentum \(q_i\). Given such a collection of momenta, we obtain a vector at every vertex
which is defined to be 0 if vertex v has no external edges attached to it, and otherwise is the sum of all incoming momenta at v.
3.2 A category of graphs
Let us fix integers \(g, n, M\ge 0\).
Definition 3.1
A marked, weighted metric graph with masses and momenta is a tuple \( (G, w, \underline{n}, \underline{M}) \) where G is a connected graph, and
are the data of: a weighting w for every vertex; n external half lines \(\underline{n}\); and an assignment \(\underline{M}\) of a mass label to each internal edge of G. Its genus is
where \(h_G\) is the first Betti (loop) number of G. Such a graph is called stable if, for every vertex \(v\in V(G)\), one has the inequality \( 2 w(v) -2 + d(v) \ge 0 \) where d(v) is the total degree at v (including both internal and external half-edges).
An isomorphism of graphs \((G, w, \underline{n}, \underline{M})\) is an isomorphism of the underlying graphs which respects the additional data (3.3).
For every edge \(e\in E(G)\), the contraction
of e is defined as follows. If e has distinct endpoints, then G/e is the graph with edge e removed and its endpoints \(\{s(e),t(e)\}\) identified. As usual, the weight of the new vertex is the sum of weights \(w(s(e))+ w(t(e))\), and \(\underline{n}'\) is the composite of \(\underline{n}\) with the quotient map \(V(G) \rightarrow V(G')= V(G)/ \{s(e) = t(e)\}\). The map \(\underline{M}' \) is the restriction of \(\underline{M} \) to the subset \(E(G/e) = E(G)\backslash \{e\}\).
When e is a tadpole or self-edge, and its endpoints coincide \(v=s(e)=t(e)\), the contraction G/e is simply the deletion of edge e. The weight of the vertex v in G/e is defined to be one more than its weight in G: \(w'(v)= w(v) +1\). The map \(\underline{n}\) is unchanged, and \(\underline{M}'\) is defined by restriction of \(\underline{M}\) to \(E(G) \backslash \{e\}\) as before.
Edge contraction preserves both the genus and the property of stability.
Definition 3.2
For fixed g, n, M, let \(I_{g,n,M}\) denote the category whose objects \( (G, w, \underline{n}, \underline{M}) \) are connected stable graphs of genus g, and whose morphisms are generated by isomorphisms and edge contractions.
When \(M=0\), it is nothing other than the category of marked stable metric graphs of genus g with n external markings. The category \(I_{g,n,M}\) has a final object given by the graph consisting of a single vertex of weight g, and n external legs.
3.3 Associated moduli space
Based on ideas in [ACP22, BBC+20], one can efficiently define the moduli space of marked metric graphs as follows.
Let \({\mathcal {T}}op\) denote the category whose objects are topological spaces and whose morphisms are continuous maps. Consider a functor \({\textsf{s}}: I \rightarrow {\mathcal {T}}op\), where I is a finite (diagram) category. Its realisation \(X_{{\textsf{s}}}\) is the topological space
A morphism from \({\textsf{s}}: I \rightarrow {\mathcal {T}}op\) to \({\textsf{s}}': J \rightarrow {\mathcal {T}}op\) is the data of a functor \(j: I\rightarrow J\), together with a natural transformation from \({\textsf{s}}\) to \({\textsf{s}}' \circ j\). Such a morphism induces a continuous map \(X_{{\textsf{s}}} \rightarrow X_{{\textsf{s}}'}\) between the associated topological realisations.
Definition 3.3
Define a functor
as follows. To the object \((G,w,\underline{n}, \underline{M})\) it associates the closed cell
given by the space of non-negative edge lengths \(\ell _e\), for \(e\in E(G)\). It only depends on the underlying graph G. An isomorphism \((G,w,\underline{n}, \underline{M}) \cong (G',w',\underline{n}', \underline{M}')\) induces a bijection \(E(G)\cong E(G')\), and a linear isomorphism on the corresponding cells. Contraction of an edge \(e \in E(G)\) induces a natural inclusion of cells
upon identifying \( \overline{C}(G/e)\) with the subset \(\ell _e=0\).
Definition 3.4
Define the tropical moduli space to be its topological realisation
Although the cases when \(g=0\) are topologically interesting, they will play no role in this paper: in the absence of internal edges the canonical differential forms we shall consider are identically zero and only interesting for \(g>0\).
Similarly, one can consider the functor \(L {\textsf{s}}: I^{\textrm{opp}}_{g,n,M} \rightarrow {\mathcal {T}}op\ \) which assigns to every object \((G,w,\underline{n}, \underline{M})\) the closed simplex
where the edge lengths are normalised to sum to 1. If E(G) is empty, then \(\overline{\sigma }_G\) is defined to be the empty set.
It defines a functor for the same reasons as above. The tropical moduli space \({\mathcal {M}}^{\textrm{trop}}_{g,n,M}\) is a cone whose cone point p is the cell associated to the final object in \(I_{g,n,M}\). The link \({\mathcal {M}}^{\textrm{trop}}_{g,n,M} \backslash \{p\}/{\mathbb {R}}^{\times }_{>0} \) is isomorphic to the topological space
Remark 3.5
In the case when \(M=0\), one retrieves the definition of the moduli space \( {\mathcal {M}}^{\textrm{trop}}_{g,n}\) of curves with n marked points. In general, the data of masses \(\underline{M}\) is not completely anodine, since assigning distinct masses to the edges of a graph will in general reduce the size of its automorphism group.
The functor \({\textsf{s}}\) defined above factors through a category of rational polyhedral cones as in [ACP22, BBC+20], which can in turn be upgraded to a category built out of affine spaces equipped with certain linear embeddings. There is a variant for the links \(L{\textsf{s}}\) involving projective spaces. These notions will be pursued elsewhere.
3.4 Homology and graph complexes
The homology of the spaces \( {\mathcal {M}}^{\textrm{trop}}_{g,n,M}\) can be expressed in terms of complexes of graphs in the category \(I_{g,n,M}\) via a variant of the classical result which relates cellular and singular homology (see [BBC+20, Proposition 2.1], [ACP22, Theorem 4.2]). We refer the reader to [BM19] for specific results about the homology of \( {\mathcal {M}}^{\textrm{trop}}_{g,n,M}\) when \(M>0\).
4 Interpretations of Momenta
4.1 Genericity
Let G be a connected graph with external momenta \(q_i \in {\mathbb {R}}^d\), and internal masses \(m_e\). We shall say that momenta are generic if
for all strict subsets I of the set of external half edges [Bro17, (1.18)]. It is automatic from our definitions that all non-trivial edge masses \(m_e\) are non-zero. Note that the genericity conditions for complex-valued momenta, which will not be considered in this paper, are more general than the Euclidean condition above.
4.2 Momentum routing
Choose an orientation on every edge of G, such that external edges are oriented inwards.
Definition 4.1
A momentum routing relative to the edge orientation is the data, for every internal edge \(e\in E(G)\), of a momentum vector
such that momentum conservation holds at every vertex \(v\in V(G)\), i.e.,
where the first sum is over all edges emanating from v, and the second is over all edges terminating at v. In the second sum we include external half edges, for which we write \(\mu _e= q_e\) for the incoming external momentum along e.
To see that momentum routings exist, we can reformulate (4.2) as follows. Let us write \(R = {\mathbb {R}}^d\), and consider the exact sequence:
where \(\partial (e) = t(e) - s(e)\). The data of external momenta defines a vector \( \underline{q} \in R^{V_G}\) where \( \underline{q}= (q_v)_{v\in V_G}\) via (3.2). The condition of momentum conservation (3.1) is precisely the statement that its image under the map \(R^{V_G} \rightarrow R\) is zero. Therefore \(\underline{q} \in \hbox {Im} (\partial )\). A choice of momentum routing is any element
Any two choices of momentum routing differ by an element of \(H_1(G;R)\).
Note that one may always assume that \(\mu _e=0\) for every self-edge or tadpole e.
Remark 4.2
One can define a canonical momentum routing by demanding that \(\underline{\mu } \in R^{E_G}\) be orthogonal to the subspace \(H_1(G;R)\) with respect to the inner product on \(R^{E_G}\) for which the edges form an orthonormal basis. However, the canonical routing is not respected by contraction of edges.
4.3 Complex momenta
Suppose that the number of spacetime dimensions is \(d=2\), and the external momentum associated with the \(\textrm{k}\)th external half edge is
Having fixed a choice of square root i of \(-1\), there is an identification
which is compatible with the Euclidean norm. In other words, if \(q_k = (x_k, y_k)\) and \({\textsf{q}}_k = x_k+iy_k\) for \(k=1,2\), then the Euclidean inner product is
and \(|\!|q_1|\!|^2 = {\textsf{q}}_1 \overline{{\textsf{q}}}_1\). We can always assume (4.3) holds when the total number of external momenta is at most 3, since by momentum conservation, the Feynman integral only depends on two external momenta, which lie in a Euclidean plane.
4.4 Quaternionic momenta
Suppose that G is a connected graph as above, but the number of Euclidean dimensions is \(d=4\). The external momenta satisfy
Let us denote the corresponding quaternions §2.1 by \({\textsf{q}}_k \in {\mathcal {Q}}\), where
which we may identify with their image in \(M_2({\mathbb {C}})\) under \(\chi \) (2.1). The inner product will be written interchangably in either notation via the identity:
5 Generalised Laplacians for Graphs with External Legs
We define a generalised graph Laplacian of a connected graph G with external momenta subject to momentum conservation, and internal particle masses. In the following, let \({\mathcal {C}}\) be a ring, which is not necessarily commutative, equipped with an anti-commutative involution \(\iota : {\mathcal {C}}\rightarrow {\mathcal {C}}\) such that its invariant subspace \({\mathcal {R}} = \{ c\in {\mathcal {C}}: \iota (c) = c\}\) is commutative.
Examples include:
-
(1)
\({\mathcal {C}}= {\mathbb {C}}\) the complex numbers, equipped with complex conjugation.
-
(2)
\({\mathcal {C}}= {\mathcal {Q}}\) the ring of quaternions, with quaternionic conjugation.
-
(3)
\({\mathcal {C}}= {\mathbb {Q}}[ \mu _e, \overline{\mu }_e, m_i] \) an abstract ring of kinematic variables, where \(\iota \) is the \({\mathbb {Q}}\) linear map acting trivially on \(m_i\), and such that \(\iota (\mu _e) = \overline{\mu }_e\), \(\iota (\overline{\mu }_e) = \mu _e\).
-
(4)
\({\mathcal {C}}= {\mathbb {Q}}[ \lambda _e, \overline{\lambda }_e, \nu _e, \overline{\nu }_e, m_i] \) as in (3), where \(\iota ( \lambda _e) = \overline{\lambda }_e\), and \(\iota ( \nu _e) = -\nu _e\), \(\iota ( \overline{\nu }_e) = -\overline{\nu }_e\). The multiplication is given by viewing \(\mu _e = \lambda _e + {\textsf{j}}\nu _e\) as an abstract quaternion, or equivalently as a \(2 \times 2\) matrix \(\left( {\begin{matrix} \lambda _e &{} \nu _e \\ -\overline{\nu }_e &{} \overline{\lambda }_e \end{matrix}} \right) \).
In the first instance, the reader may wish to consider only the case (1) where \({\mathcal {C}}={\mathbb {C}}\). The generalised graph Laplacian requires the data of:
-
an external momentum vector \(\underline{q} \in {\mathcal {C}}^{V_G}\) subject to momentum conservation
-
a vector of internal masses \(\underline{m}\in {\mathcal {C}}^{E_G}\) which is invariant under \(\iota \).
It is defined in two stages: first we define the graph Laplacian for zero masses, and then modify it very slightly to take into account the internal masses.
5.1 Definition of the generalised graph Laplacian
Choose an orientation on the internal edges of G. Consider the exact sequence
Since the momentum vector \(\underline{q} \in {\mathcal {C}}^{V_G}\) maps to zero, we obtain an extension
where \({\mathcal {E}}_G =\{ \underline{\mu } \in {\mathcal {C}}^{E_G}: \partial ( \underline{\mu }) \in {\mathcal {C}}\underline{q} \}\) and the element \(1\in {\mathcal {C}}\) in the right-hand term is identified with \(\underline{q} \in {\mathcal {C}}^{V_G}\). We shall call this the extension by momenta.
Consider the pairing satisfying
It restricts to a Hermitian form (with respect to the involution \(\iota \)) on \({\mathcal {E}}_G\), which we also denote by \(\langle \, \rangle _0\), and which may be interpreted as a \({\mathcal {C}}\)-linear map \( y\mapsto \left( x\mapsto \langle y, x \rangle _0\right) \) which we call the ‘graph Laplacian for zero masses’:
Here, \(\hbox {Hom}_{({\mathcal {C}},\iota )}(A,B)\) denotes anti-linear maps \(\phi :A \rightarrow B\) satisfying \(\phi (\lambda x) = \iota (\lambda ) \phi (x)\) for all \(\lambda \in {\mathcal {C}}\). Equivalently, (5.2) may be viewed as a \({\mathcal {C}}\)-linear map:
using which one has the identity \( \widetilde{\Lambda }_G(\mu ,0)= \widetilde{{\mathcal {H}}}_G^* D_G \widetilde{{\mathcal {H}}}_G \) where \(\widetilde{{\mathcal {H}}}_G: {\mathcal {E}}_G \rightarrow {\mathcal {C}}^{E_G}\) is inclusion and \(X\mapsto X^*\) denotes Hermitian conjugation, i.e., \(X^* = \iota (X)^T\).
The subspace \(H_1(G;{\mathcal {C}}) \subset {\mathcal {E}}_G\) contains the subspace \(H_1(G;{\mathbb {Z}})\), which is invariant under the involution \(\iota \). Restricting the Hermitian form \(\langle \, \ \rangle _0\) to the space \(H_1(G;{\mathbb {Z}})\) therefore defines a symmetric bilinear form which is nothing other than the usual graph Laplacian \(\Lambda _G\), which depends neither on masses nor momenta.
The masses of G are encoded by a single element
where \(m_e = \iota (m_e)\). The norm of this element with respect to (5.2) is
We define the mass-correction Hermitian inner product \(\langle \, \ \rangle _m\) on \( {\mathcal {E}}_G\) to be the unique inner product vanishing identically on \(H_1(G;{\mathcal {C}})\):
but for which \( \langle f, f \rangle _m = |\!| \underline{m} |\!|^2\) for any \(f\in {\mathcal {E}}_G\) whose image is \(1\in {\mathcal {C}}\) under the natural map in (5.1). Consider the sum of the two Hermitian forms
The associated \({\mathcal {C}}\)-linear map defines the generalised graph Laplacian
5.2 Generalised graph Laplacian matrix
The generalised graph Laplacian may be computed by splitting (5.1). Number the edges of G from \(1,\ldots , N\). Choose a basis \(c_1,\ldots , c_h\) of \(H_1(G;{\mathbb {Z}})\) and a routing of edge momenta \(\underline{\mu } \in {\mathcal {C}}^{E_G}\).
Define a matrix \(\widetilde{{\mathcal {H}}}_G\) with N rows, \(h+1\) columns and entries in \({\mathcal {C}}\) as follows. If \(1 \le k \le h\) then the entries \((\widetilde{{\mathcal {H}}}_G)_{e,k}\) count the number of times (with multiplicity) that the oriented edge e appears in the cycle \(c_k\). Thus the first h columns consists of the usual edge-cycle incidence matrix \({\mathcal {H}}_{G}\). The final column is defined by
Now consider the matrix \({\widetilde{M}}_G\) with N rows and \(h+1\) columns
Thus \({\widetilde{M}}_G\) is zero except for the last column, which is the vector of masses \(m_e\).
The generalised graph Laplacian matrix is defined to be
where \(X\mapsto X^*\) denotes Hermitian conjugation, i.e., \(X^* = \iota (X)^T\). Since \({\widetilde{M}}_G\) has entries which are invariant under \(\iota \), we have \({\widetilde{M}}_G^{*} = {\widetilde{M}}_G^T\). Note that the matrix \( {\widetilde{M}}_G^* D_G {\widetilde{M}}_G\) only has a single non-zero entry in the bottom right-hand corner:
In general, the matrix \(\widetilde{\Lambda }_G\) has the following block-matrix form:
where \(\Lambda _G = {\mathcal {H}}_G^T D {\mathcal {H}}_G \) is the usual graph Laplacian matrix,
and where for any \({\mathbb {Z}}\)-linear combination of edges \(c = \sum _{e\in E(G)} p_e e\), we write
and \(c(\alpha , \iota (\mu ))\) for its image under \(\iota \).
5.2.1 Change of edge orientations
Changing the orientation of an edge does not modify the matrix \(\widetilde{\Lambda }_G\). To see this, observe that switching the orientation of any number of edges amounts to multiplying the matrix \(\widetilde{{\mathcal {H}}}_G\) on the left by a diagonal matrix E with entries in \(\{1, -1\}\): reversing the orientation of edge e results in a change of sign for the associated momentum \(\mu _e\). The claim follows since \(E^* D_GE=D_G\) and since \({\widetilde{M}}_G\) does not depend on the edge orientations.
5.2.2 Change of basis
The matrix \(\widetilde{\Lambda }_G\) depends on the choice of basis for the homology \(H_1(G;{\mathbb {Z}})\). Changing basis modifies the matrix \(\widetilde{\Lambda }_G\) by
where \({\widetilde{P}}\) is an invertible block matrix of the form
where \(P \in \textrm{GL}(H_1(G;{\mathbb {Z}})) \) is a change of basis matrix with integer entries. In particular, one has \(\det (P)^2 =\det ({\widetilde{P}})^2=1\).
5.2.3 Change of momentum routing
Two momentum routings \(\overline{\mu }\), \(\overline{\mu }'\) differ by an element of \(H_1(G;{\mathcal {C}})\). Therefore changing momentum routing (or changing a choice of splitting of (5.1)) is equivalent to modifying the matrix \(\widetilde{\Lambda }_G\) by
where \({\widetilde{S}}\) is an invertible block matrix of the form
where I denotes the \(h_G \times h_G\) identity matrix and \(s_1,\ldots , s_h\in {\mathcal {C}}\).
5.2.4 Group of indeterminacy
The graph Laplacian matrix \(\widetilde{\Lambda }_G\) is therefore ambiguous up to the action of the semi-direct product
where \(P \in \textrm{GL}_g({\mathbb {Z}})\) acts on \(S\in {\mathcal {C}}^g\) via \(S\mapsto PSP^{-1} \). Note that it only involves the additive structure on \({\mathcal {C}}\), which is commutative, and not the multiplicative structure, which may not be. The group (5.10) can be viewed as the group of automorphisms of the extension \({\mathcal {E}}_G\) (5.1) which respects the integral structure \(H_1(G;{\mathbb {Z}}) \subset H_1(G;{\mathcal {C}})\). Recall that \( {\mathcal {C}}^{g} \rtimes \textrm{GL}_g({\mathbb {Z}}) \) is the set \((S, P) \in {\mathcal {C}}^g \times \textrm{GL}_g({\mathbb {Z}}),\) which can be identified with the set of matrices
equipped with the group law \((S_1,P_1) (S_2,P_2) = (S_1 P_1 S_2 P_1^{-1}, P_1P_2)\).
5.3 Complex adjoint of the generalised graph Laplacian
Let G be a connected graph as before, and suppose that the external momenta lie in \({\mathcal {C}}= {\mathcal {Q}}\), the ring of quaternions. The generalised graph Laplacian \( \widetilde{\Lambda }_G(\mu , m) \) is a quaternionic Hermitian form of rank \(g+1\), and may be represented by a \((g+1) \times (g+1)\) Hermitian matrix with entries in \({\mathcal {Q}}\). Its image under the complex adjoint map (2.1) is a Hermitian complex matrix of rank \(2g+2\).
In block matrix form §2.1.1(i), one may represent a choice of complex adjoint graph Laplacian in the form \(M_{2n}({\mathbb {C}}) \cong M_n(M_2({\mathbb {C}}))\) as follows:
where the quaternionic momenta are \(\mu _e = \lambda _e + {\textsf{j}}\, \nu _e \), for \(\lambda _e, \nu _e \in {\mathbb {C}}\), where
is the complex adjoint of the ordinary graph Laplacian \(\Lambda _G= (\Lambda _{ij})_{ij}\), and
In the massless case \(m_e=0\), these matrices were considered in [BK10].
the reader may prefer to use the equivalent matrix representation §2.1.1(ii).
6 Symanzik and Spanning Forest Polynomials
We recall the definition of Symanzik polynomials and relate the polynomial \(\phi _G(q)\) to a sum of spanning forest polynomials relative to a choice of momentum routing. For further background on Symanzik polynomials and some of their basic properties, we refer the reader to [Bro17, §1], for example.
6.1 Definition of Symanzik polynomials
We recall the definition of the graph polynomials which arise in Feynman integrals in parametric form, since conventions can vary slightly in the literature. Let \(d\ge 0\) be any non-negative integer. In this section, G is any connected graph with external edges and external momenta \( q_i \in {\mathbb {R}}^d.\) Every internal edge e is assigned a mass \(m_e \in {\mathbb {R}}\).
6.1.1 1st Symanzik polynomial
Recall that the first Symanzik polynomial (also known as the ‘Kirchhoff’ or graph polynomial), is defined by
where the sum is over all spanning trees of G. It does not depend on the external half-edges of G. It is equal to the determinant of the ordinary graph Laplacian:
It is homogeneous of degree \(h_G\) and is not identically zero.
6.1.2 2nd Symanzik polynomial
The second Symanzik polynomial depends on external momenta. It is the homogeneous polynomial of degree \(h_G+1\) defined by
where the sum is over all spanning 2-trees of G (or spanning forests with exactly two connected components), and \(q_{T_i}\) denotes the total momentum entering \(T_i\). By momentum conservation \(q^{T_1} + q^{T_2}=0\).
6.1.3 Dependence on masses
The ‘2nd Symanzik’ polynomial which occurs in the (Euclidean) parametric representation of Feynman integrals is
which is also homogenous of degree \(h_G+1\).
6.2 Vanishing of Symanzik polynomials
Definition 6.1
[Bro17, §1.4] A subgraph \(\gamma \subset E_G\) is momentum-spanning if all non-zero external momenta \(q_v\) of G meet vertices v which lie in a single connected component of \(\gamma \). It is called mass-spanning if it contains all massive edges of G.
It is called mass-momentum spanning (or m.m. for short) if both hold.
It is shown in [Bro17, §1.6] that if \(\gamma \subset E_G\) and genericity (4.1) holds, then
is identically zero if and only if \(\gamma \) is an m.m. subgraph. In this paper we work in the Euclidean region (loc. cit. §1.7) which implies that the coefficients of every monomial in the variables \(\alpha _i\) in \(\Xi _G\) are non-negative (this would not be true in Minkowski space). If \(\gamma \) is not m.m. then \(G/\gamma \) has a scale and one shows that
in the region \(\alpha _e>0\), for all \(e\in E_{G/\gamma }=E_G \backslash E_{\gamma }\).
6.3 Spanning forest and Dodgson polynomials
The second Symanzik can be expressed using spanning forest polynomials, which were defined in [BY11].
6.3.1 Spanning forest polynomials
Definition 6.2
Let \(P= P_1 \cup \cdots \cup P_r\) be any partition of a subset of vertices of G into disjoint sets. The associated spanning forest polynomial is defined by
where the sum is over spanning forests \({\mathcal {F}} = T_1 \cup \cdots \cup T_r\) of G with exactly r connected components \(T_i\) such that
In other words, each tree \(T_i\) contains the vertices in \(P_i\) and no other vertices of P. Trees consisting of a single vertex are permitted. We set \(\phi _G^P=0\) if the sets \(P_i\) are not all disjoint.
6.3.2 An identity for the 2nd Symanzik polynomial
Proposition 6.3
Let G be as in the previous paragraph. For any choice of momentum routing, the second Symanzik polynomial satisfies:
where we recall that \(\phi _G^{P_1, P_2}\) is zero if \(P_1 \cap P_2 \) is non-empty. Note that the second summand is symmetric in e and f and therefore each term is counted twice.
Proof
Let \(T= T_1 \cup T_2\) be a spanning forest in G with two connected components \(T_1,T_2\). By momentum conservation (4.2), the momentum flowing into \(T_1\) equals
where each sum is over all internal edges \(e\in E(G)\) such that \(e\cup T\) has no loops, and is hence a spanning tree of G. In particular, e has one endpoint in \(V(T_1)\) and the other in \(V(T_2)\).
Suppose that e is an edge with distinct endpoints. It follows from (6.7) that the coefficient of \(\mu _e. \mu _e\) in \(-q^{T_1}. q^{T_2}\) is 1 if \(e \cup T_1 \cup T_2\) is a tree, and 0 otherwise. By (6.3), the coefficient of \(\mu _e. \mu _e\) in \(\Phi _G(q)\) is therefore equal to
where the sum is over the set of spanning forests T with two connected components such that \(T \cup e\) is a tree. The map \(T \mapsto T \cup e\) is a bijection between the latter and the set of spanning trees which contain e, which in turn are in one-to-one correspondence with the set of spanning trees in G/e. We conclude that the coefficient of \(\mu _e. \mu _e\) in the formula (6.3) is precisely \(\alpha _e \Psi _{G/e}\) (which also equals \( \alpha _e \Psi _{G/\!/e}\)).
In the case when e is a loop, \(\Psi _{G/\!/e}\) vanishes. By (6.7), \(\mu _e\) does not contribute to \(q^{T_1}\) and therefore does not appear in \(\Phi _G(q)\). This establishes the first line of (6.6).
Now consider two distinct edges e, f and assume first of all that they have in total four distinct endpoints. Formula (6.7) implies that \(\mu _e. \mu _f\) does not arise in \(-q^{T_1}.q^{T_2}\) unless both \(T \cup e \) and \(T\cup f\) are trees. In this situation the coefficient is 1 if s(e), s(f) lie in the same connected component of T, and \(-1\) if they lie in different components. In the former case, this means precisely that \(T=T_1 \cup T_2\) is a spanning forest with 2 components such that the vertices of one component \(V(T_1)\) contains \(\{s(e), s(f)\}\) and the other, \(V(T_2)\), contains \(\{t(e), t(f)\}\). In the latter case, \(V(T_1)\) contains \(\{s(e), t(f)\}\) and \(V(T_2)\) contains \( \{s(f), t(e)\}\), or vice-versa. From (6.3), it follows that the coefficient of \(\mu _e. \mu _f\) in \(\Phi _G(q)\) is exactly
It remains to check that (6.8) remains true in the cases when \(\{s(e),s(f), t(e), t(f)\}\) has 3 or fewer elements. The reader may verify this in every situation (e.g., \(s(e) = s(f)\), \(s(e)= t(f)\), etc), using a similar argument and the fact that \(\phi _G^{P_1,P_2}\) vanishes if \(P_1\) and \(P_2\) are not disjoint. Note that if e or f is a loop (i.e., \(s(e)=t(e)\)) then \(\mu _e\) does not occur in (6.7), and indeed, both of the spanning forest polynomials in (6.8) are defined to be zero in this case. \(\square \)
6.3.3 Spanning forests and minors of graph Laplacians
The difference of spanning forest polynomials occurring in Proposition 6.3 can be interpreted as a ‘Dodgson polynomial’ \(\Psi _G^{e,f}\). We will make frequent use of the following well-known lemma.
Lemma 6.4
Let G be a connected graph with directed edges. Suppose that \(e_1,\ldots , e_n \) are distinct edges such that \(G\backslash \{e_1,\ldots , e_n\}\) is connected. Then \(h=h_G \ge n\) and we may choose cycles \(c_1,\ldots , c_h \in {\mathbb {Z}}^{E_G}\) representing a basis for \(H_1(G;{\mathbb {Z}})\) such that:
In other words, for each \(1\le i\le n\), the edge \(e_i\) appears in cycle \(c_i\) with coefficient 1 and occurs in no other cycle.
For a matrix M, let \(M^{i,j}\) denote the matrix with row i and column j removed.
Lemma 6.5
Let G be a connected graph with directed edges.
-
(i)
Let \(e\in E(G)\) be an edge such that \(G\backslash \{e\}\) is connected (i.e., e is not a bridge). Choose cycles \(c_1,\ldots , c_h\) as in the previous lemma and let \(\Lambda _G\) be the graph Laplacian matrix with respect to this basis. Then we have
$$\begin{aligned} \det \Lambda _G^{1,1} = \Psi _{G\backslash e}. \end{aligned}$$ -
(ii)
Let \(e_1,e_2\in E(G)\) be two edges such that \( G\backslash \{e_1,e_2\}\) is connected, and choose cycles \(c_1,\ldots , c_h\) as in the previous lemma. Then
$$\begin{aligned} \det \Lambda _G^{1,2} = \pm \left( \phi _G^{\{s_1,s_2\}, \{t_1,t_2\} } - \phi _G^{\{s_1,t_2\}, \{s_2,t_1\}} \right) . \end{aligned}$$where \(s_i\) is the source of the directed edge \(e_i\), and \(t_i\) its target, for \(i=1,2.\)
Proof
-
(i)
The variable \(\alpha _e\) only occurs in the matrix \(\Lambda _G\) in a single place, namely the top left entry \((\Lambda _G)_{1,1}\). A row expansion shows that the coefficient of \(\alpha _1\) in \(\det (\Lambda _G)\) equals \(\det (\Lambda _G^{1,1})\). On the other hand, the coefficient of \(\alpha _1\) in \(\Psi _G\) is precisely \(\Psi _{G\backslash e_1}\) by the contraction-deletion identity \(\Psi _G = \alpha _1 \Psi _{G\backslash e} + \Psi _{G/\!/e}\).
-
(ii)
By the same argument as in (i), we have \(\det (\Lambda _G^{i,i}) = \Psi _{G \backslash e_i} \) for \(i=1,2\) and \(\det (\Lambda _G^{12,12}) = \Psi _{G\backslash \{e_1,e_2\}}\). Now the ‘Dodgson identity’
$$\begin{aligned} \left( \det (\Lambda _{G}^{1,2})\right) ^2 = \Psi _{G \backslash e_1} \Psi _{G \backslash e_2} - \Psi _{G \backslash \{e_1,e_2\}} \Psi _G \end{aligned}$$implies that \(\det (\Lambda _{G}^{1,2}) \) is equal, up to a possible sign, to the ‘Dodgson polynomial’ \(\Psi ^{1,2}_G\) which satisfies the identical identity (but is defined using the graph matrix, and not the Laplacian). An expression for the latter in terms of the requisite forest polynomials follows from [BY11, §2]. \(\square \)
7 Determinants of Generalised Graph Laplacians
Let G be a connected graph as in §5 with internal particle masses and external momenta lying in 2-dimensional Euclidean space \({\mathbb {R}}^2 \cong {\mathbb {C}}\).
Theorem 7.1
Let \(\widetilde{\Lambda }_G(\mu ,m)\) be a generalised graph Laplacian for any choice of momentum routing \(\mu _e \in {\mathbb {C}}^{E_G}\). It satisfies
Now let G be as above with external momenta in \({\mathbb {R}}^4\cong {\mathcal {Q}}\).
Theorem 7.2
Let \(\widetilde{\Lambda }_G(\mu , m)\) be a generalised quaternionic graph Laplacian with respect to a choice of momentum routing \(\mu \in {\mathcal {Q}}^{E_G}\). It satisfies
Remark 7.3
The right-hand side of (7.1) or (7.2) only depends on external kinematics. It follows that the determinant of the generalised graph Laplacian with generic momentum variables \(\mu _e\) only depends on the quantities
where the sum is over all internal edges incident to a vertex \(v \in V(G)\).
We first prove Theorem 7.1 and use it to deduce Theorem 7.2. It is instructive to prove Theorem 7.1 by direct computation of the determinant of the left-hand side, making use of Proposition 6.3. An alternative approach could exploit the asymptotic factorisation identities proven in §9, which were shown in [Bro17, §4] to determine \(\Xi _G(q,m)\) essentially uniquely.
7.1 Preliminary reductions
It follows from the presentation (5.4) of \(\widetilde{\Lambda }_G(\mu ,m)\) in block matrix form that
Since \(\det (\Lambda _G) = \Psi _G\), it is enough by definition of \(\Xi _G(q,m)\) to prove (7.1) in the case when all masses are zero. Thus, Theorem 7.1 is equivalent to:
Now recall that \(\widetilde{\Lambda }_G (\mu ,0)\) takes the form:
By §5.2.2, its determinant does not depend on the choices of cycles \(c_i\). To verify (7.3), we use Proposition 6.3 and compare the coefficients of \(\mu _{e_1} \overline{\mu }_{e_2}\) on both sides of (7.3). First we treat the case \(e_1=e_2 =e\).
Lemma 7.4
Let e be an edge of G, then the coefficients of \(\mu _e \overline{\mu }_e\) on the left and right hand sides of (7.3) agree.
Proof
Consider the coefficient of \(\mu _e \overline{\mu }_e\) in \(\det \widetilde{\Lambda }_G (\mu ,0)\). If e is a bridge, then we can find cycles \(c_1,\ldots , c_h\in {\mathbb {Z}}^{E_G}\) representing a basis of \(H_1(G;{\mathbb {Z}})\) which do not involve e at all. In this case, \(\mu _e\) only appears in \(\widetilde{\Lambda }_G (\mu ,0)\) in the bottom right hand corner, and the coefficient of \(\mu _e \overline{\mu }_e\) is
If e is not a bridge, then choose cycles \(c_1,\ldots , c_h\) as in Lemma 6.4. There are two possible types of terms in an expansion of the determinant which can involve \(\mu _e \overline{\mu }_e\), namely those involving the top right and bottom left corners \(c_1(\alpha , \mu )\) and \(c_1(\alpha , \overline{\mu })\), and those involving the bottom right hand corner as before. In total they contribute
The minus sign on the left comes from \((-1)^h (-1)^{h-1}=-1\); the first equality is a consequence of Lemma 6.5(i); and the second equality is the contraction-deletion identity: \(\Psi _G = \alpha _e \Psi _{G\backslash e}+ \Psi _{G /\!/e}\). In all cases, we conclude that the coefficient of \(\mu _e \overline{\mu }_e\) in \(\det \widetilde{\Lambda }_G (\mu ,0)\) is \(\alpha _e \Psi _{G/\!/e}\), in agreement with (6.6). \(\square \)
Before proceeding to the case \(e_1, e_2\) distinct, we consider some boundary cases.
Lemma 7.5
Let e be a tadpole in G. Then
Proof
By Lemma 6.4, choose the first cycle in a basis \(c_1,\ldots , c_h\) of \(H_1(G)\) to be \(c_1= e\), and all other \(c_i\) to be independent of e. The matrix \(\widetilde{\Lambda }_G(\mu ,0)\) has the form
The result follows by subtracting \(\overline{\mu }_{e}\) times the first row from the final row, and performing an expansion of the determinant along the first column. \(\square \)
Next we verify that the theorem is true in the massless case for ‘banana’ graphs, i.e., graphs with exactly two vertices joined by \(\ge 3\) internal edges.
Example 7.6
Let G have two vertices \(v_1,v_2\), and \(n \ge 3\) edges. Orient them so that they all have source \(v_1\) and target \(v_2\), except for \(e_n\), which has source \(v_2\) and target \(v_1\). Consider the basis of cycles:
where \(h= n-1.\) With these conventions, the graph Laplacian is
It follows from the explicit description of \(\widetilde{\Lambda }_G(\mu ,m)\) in the form (5.4) that the coefficient of \(\mu _1 \overline{\mu }_2\) in \(\det \widetilde{\Lambda }_G(\mu ,0)\) is \(\alpha _1 \alpha _2 \det \Lambda _G^{1,2}\), where
By subtracting the first row from all other rows, we see that \(\det \Lambda _G^{1,2} = \alpha _3\cdots \alpha _n\), and therefore the coefficient of \(\mu _1\overline{\mu }_2\) in \(\det (\Lambda _G(\mu ,0))\) is \(\alpha _1\cdots \alpha _n\). By symmetry, the same holds for any \(\mu _i \overline{\mu }_j\), where \(i\ne j\). This agrees with the coefficient of \(\mu _i \overline{\mu }_j\) in \(\Phi _G(q)\), by its definition (there is only one spanning two-tree). The case when \(i=j\) follows from Lemma 7.4, and it follows that \(\det \widetilde{\Lambda }_G(\mu ,0) = \Phi _G(q) \) as claimed.
7.2 Proof of Theorem 7.1
Now let \(e_1,e_2\in E(G)\) be two distinct edges and consider the coefficient of \(\mu _{e_1} \overline{\mu }_{e_2}\) in \( \det ( \widetilde{\Lambda }_G(\mu ,0)) \). Suppose first of all that one of the edges, say \(e_1\), is a bridge. By the argument given above, no terms in the matrix \( \widetilde{\Lambda }_G(\mu ,0) \) depend on \(\mu _{e_1}\) except for the bottom right-hand corner, and so the coefficient of \(\mu _{e_1} \overline{\mu }_{e_2}\) in \( \det ( \widetilde{\Lambda }_G(\mu ,0)) \) is zero. Now since \(e_1\) is a bridge, it follows that \(e_2\) lies in just one of the two connected components of \(G \backslash e_1\), and therefore:
Now suppose that neither \(e_1\) nor \(e_2\) is a bridge, yet \(G\backslash \{e_1,e_2\}\) has two connected components. In particular, \(e_2\) is a bridge in \(G\backslash \{e_1\}\). In this case we may choose the cycles \(c_1, \ldots , c_h\) such that \(c_2, \ldots , c_h\) are cycles in \(G\backslash \{e_1\}\) which do not depend on \(e_2\). Thus we can assume that only cycle \(c_1\) involves either \(e_1\) or \(e_2\), and furthermore we can direct the edges \(e_1,e_2\) in such a way that their coefficient in \(c_1\) are both 1. In this case, the coefficient of \(\mu _{e_1} \overline{\mu }_{e_2}\) in \( \det ( \widetilde{\Lambda }_G(\mu ,0)) \) is
Since the sources of \(e_1,e_2\) lie in different components of \(G\backslash \{e_1,e_2\}\), we have
There is a one-to-one correspondence between the set of spanning forests with two components \(T_1 \cup T_2\) such that \( \{s(e_1), t(e_2)\} \subset V(T_1)\) and \(\{t(e_1), s(e_2)\} \subset V(T_2)\), and the set spanning trees \(T= T_1 \cup T_2 \cup e_2\) of \(G\backslash \{e_1\}\). This implies that
and again the coefficient of \(\mu _{e_1} \overline{\mu }_{e_2}\) in \(\det ( \widetilde{\Lambda }_G(\mu ,0)) \) and \(\Phi _G(q)\) are in agreement.
Finally, consider the case when \(G \backslash \{e_1, e_2\}\) is connected. Choose cycles \(c_1,\ldots , c_h\) as in Lemma 6.4. By expanding \(\det \,\widetilde{\Lambda }_G(\mu , 0)\), its coefficient of \(\mu _{e_1} \overline{\mu }_{e_2}\) is
by Lemma 6.5. By Proposition 6.3, the coefficient of \(\mu _{e_1} \overline{\mu }_{e_2}\) in \( \det \widetilde{\Lambda }_G(\mu ,0)\) is equal to that of \(\mu _{e_1}. \mu _{e_2}\) in \(\Phi _{G}(q)\) up to a possible sign. It remains to determine the sign.
If the coefficient of \(\mu _{e_1} \overline{\mu }_{e_2} \) vanishes in \( \det \widetilde{\Lambda }_G(q,0)\) then there is nothing to prove. Otherwise, it is enough to select a single non-zero monomial m in the coefficient of \(\mu _{e_1} \overline{\mu }_{e_2} \) in \( \det \widetilde{\Lambda }_G(\mu ,0)\), and compare with the corresponding coefficient in \(\Phi _G(q)\). Such a monomial corresponds to a unique spanning forest \({\mathcal {F}}=T_1 \cup T_2\) via the formula \(m = \prod _{e\notin {\mathcal {F}}} \alpha _e\). The sign is determined by setting all \(\alpha _e\) to 0 for \(e \in {\mathcal {F}}\), which corresponds to contracting all edges in \({\mathcal {F}}\). Since
it therefore suffices to prove the identity for the graph \(G'=G / {\mathcal {F}}\). Because \({\mathcal {F}}\) has two components, \(G'\) has exactly two vertices. We can assume that \(G' \backslash \{e_1,e_2\}\) is connected since we have already considered the disconnected case. By Lemma 7.5, we can remove all tadpoles from \(G'\), and thus assume that \(G'\) is a banana graph with two vertices. The result then follows from Example 7.6.
7.3 Quaternionic case (proof of Theorem 7.2)
It follows from Proposition 2.1 that there exists a well-defined polynomial F in the \(\alpha _i, m_i\) and the components \(\lambda _e, \nu _e\) of the momenta \(\mu _e = \lambda _e + {\textsf{j}}\nu _e\), and their conjugates, such that
By inspection of (5.12), the middle term is at most quadratic in the mass squares \(m_e^2\), and therefore F may be written in the form
More precisely, by expanding \(\det (\chi _{\widetilde{\Lambda }_G(\mu ,m)} ) \) using the representation (5.12), one sees that the quadratic terms in the mass squares all arise from the term
using Proposition 2.1(ii). By comparing terms in \(m_e^4\), it follows from this that
and therefore
We are thus reduced to proving the case when all masses are zero.
By inspection of the matrix (5.12), the determinant is of degree at most four in the momentum components \(\lambda _e, \nu _e, \overline{\lambda }_e, \overline{\nu }_e\), and therefore F is of degree at most two in them. It suffices to check that for any two edges e, f, the coefficient of \(\lambda _e \overline{\lambda }_f\) (or \(\overline{\nu }_e \nu _f\), etc) in F coincides with that of \(\Phi _G(q)\). Since F does not depend on the choice of momentum routing, we can assume that the momenta through edges e, f are purely complex, since they lie in a copy of \({\mathbb {R}}^2\cong {\mathbb {C}}\) inside \({\mathbb {R}}^4 \cong {\mathcal {Q}}\). In other words, we may assume that \(\nu _e = \nu _f=0\) is zero. Since the coefficient of \(\lambda _e \overline{\lambda }_f\) in F does not depend on the other momentum variables, we can assume that the latter are complex as well. Now if \(\nu _e=0\) for all edges e, the complex adjoint Laplacian \( \chi _{\widetilde{\Lambda }_G(\mu ,m)} \) is simply the image under \(\chi \) of the generalised Laplacian with complex momenta. Thus Theorem 7.2 follows from Theorem 7.1 and Proposition 2.1(ii).
8 Invariant Forms Associated to Graphs
We define invariant differential forms associated to generalised graph Laplacians.
8.1 Invariant forms
Let us define a graded exterior algebra
in which each element \(\beta ^{2k+1}\) has degree \(2k+1\). It is a graded-commutative Hopf algebra with respect to the coproduct
for which the generators \(\beta ^{2k+1}\) are primitive:
We shall call a primitive element odd or even depending on whether k is odd or even. The Hopf subalgebra generated by the even forms \(\beta ^{4k+1}\) was denoted by \(\Omega _{\textrm{can}} \subset B_{\textrm{can}}\) in [Bro21]. Let \(R= \bigoplus _{n\ge 0} R^0\) be a graded-commutative, unitary, differential graded algebra with differential \(d: R^n\rightarrow R^{n+1}\) of degree 1.
Definition 8.1
For any invertible matrix \(X \in \hbox {GL}_h(R_0)\), let us denote by
By taking exterior products, we can define \(\beta _X\) for any \(\beta \in B_{\textrm{can}}\).
We briefly recall properties of these forms. Self-contained proofs of these facts, which are classical, may be found in [Bro21].
-
(1)
The forms \(\beta ^{2k+1}_X\) are closed: \(d \beta ^{2k+1}_X =0 \)
-
(2)
One has \( \beta ^{2k+1}_{X^T} = (-1)^k \beta ^{2k+1}_X. \) In particular, the odd primitive forms vanish on the space of symmetric matrices.
-
(3)
The forms \(\beta ^{2k+1}_X\) are bi-invariant:
$$\begin{aligned} \beta ^{2k+1}_{AXB} = \beta ^{2k+1}_X \end{aligned}$$for any matrices \(A, B \in \textrm{GL}_h(R_0)\) which are constant, i.e., \(dA= dB=0\).
-
(4)
The primitive forms are additive: \(\beta ^{2k+1}_{X_1 \oplus X_2} = \beta ^{2k+1}_{X_1} + \beta ^{2k+1}_{X_2}\) with respect to direct sums. It follows for general \(\beta \in B_{\textrm{can}}\) that
$$\begin{aligned} \beta _{X_1 \oplus X_2 } = \sum \beta '_{X_1} \wedge \beta ''_{X_2} \end{aligned}$$where we write \(\Delta _{\textrm{can}} \beta = \sum \beta ' \otimes \beta ''\).
-
(5)
Let \(\lambda \in \left( R^0\right) ^{\times }\) be invertible. Then
$$\begin{aligned} \beta _{\lambda X} = \beta _X \quad \hbox { for any } \ \beta \in B_{\textrm{can}} \end{aligned}$$ -
(6)
If X has rank h, then
$$\begin{aligned} \beta ^{2k+1}_{X} = 0 \qquad \hbox { if } \quad k\ge h. \end{aligned}$$
8.2 Canonical forms associated to graphs
The canonical forms introduced in [Bro21], which do not depend on kinematics, will henceforth be called ‘of the first kind’ since their denominators involve the first Symanzik polynomial.
Definition 8.2
Let \(\omega \in B_{\textrm{can}}\). For any connected graph G with internal particle masses and external momenta in \({\mathbb {R}}^2\cong {\mathbb {C}}\), define:
-
(i)
The first canonical form to be
$$\begin{aligned} \omega _G = \omega _{\Lambda _G}, \end{aligned}$$(8.2)where \(\Lambda _G\) is any choice of graph Laplacian matrix.
-
(ii)
Define the second canonical form to be
$$\begin{aligned} \varpi _G (\mu ,m)= \omega _{\widetilde{\Lambda }_G(\mu ,m)} \end{aligned}$$(8.3)where \(\widetilde{\Lambda }_G(\mu ,m)\) is any choice of generalised graph Laplacian matrix. It depends a priori on the external momenta \(\mu _e\) and particle masses \(m_e\).
-
(iii)
Define a mixed canonical form, or simply canonical form to be any polynomial in the canonical forms \(\omega _G\), \(\varpi _G\) of the first or second kinds.
The case when momenta lie in \({\mathbb {R}}^4\) is discussed below.
Since the graph Laplacian \(\Lambda _G\) is symmetric, odd canonical forms of the first kind are identically zero. This is not the case for the forms of the second kind. A mixed canonical form is abstractly represented, therefore, by an element of
The following theorem summarizes some basic properties of these forms.
Theorem 8.3
The primitive canonical graph forms are well-defined, projectively invariant closed differential forms of the shape:
where \(N_1, N_2\) are polynomials with rational coefficients. Let
denote the generalised graph hypersurface. For any canonical form \(\omega \in \mho _{\textrm{can}}\) of degree n, \(\omega _G\) is a closed, projectively invariant form of degree n:
which, under the assumption of generic momenta (§4.1), is smooth on the open coordinate simplex \(\sigma _G \subset {\mathbb {P}}^{E_G}({\mathbb {R}})\) defined by \(\alpha _1, \ldots , \alpha _{e_G}>0\).
Proof
The fact that the forms are closed and well-defined follows from bi-invariance §8.1, as does projective invariance. It is clear from the definition that their denominators only involve a power of the determinant: the order of this power follows from [Bro21, Theorem 5.2], and the expression for the determinant as a graph polynomial follows from Theorem 7.1. The numerator \(N_2\) is a priori only a polynomial in the momentum routing \(\mu _e\), but again by bi-invariance, it is independent of the choice of momentum routing, and hence only depends on external momenta \({\textsf{q}}_v, \overline{{\textsf{q}}}_v\). The fact that \(\omega \) is smooth on the open simplex follows from the assumption that kinematics are generic, and hence \(\Xi _{G}(q,m)\) and \(\Psi _G\) are strictly positive on \(\sigma _G\). \(\square \)
Remark 8.4
For any Hermitian matrix X, one has
As a consequence, \(\varpi _G^{2k+1}\) is real if k is even, and imaginary if k is odd. In the case when k is even, \(\varpi _G\) will be a rational function of the inner products \(q_i. q_j\) of external momenta. In the case when k is odd, \(\varpi ^{2k+1}_G\) may also depend on the quantities
Remark 8.5
If G has only two external momenta \(q_1=-q_2\), \(\widetilde{\Lambda }_G(\mu ,m)\) is equivalent to a symmetric matrix because we can find a momentum routing so that the \(\mu _e\) lie on the real line spanned by \(q_1\). In this case, \(\varpi ^{2k+1}_G\) vanishes when k is odd.
8.3 Further properties of canonical forms
The following properties of canonical forms are almost immediate from the definitions.
Lemma 8.6
-
(i)
Let \(\pi \) be an automorphism of G. It acts on \(R^0\) by permutation of the edge variables \(x_e\). For any canonical form of mixed type \(\omega \in \mho _{\textrm{can}}\),
$$\begin{aligned} \pi ^* \omega _G = \omega _G \end{aligned}$$ -
(ii)
Let \(\gamma \subset G\) be any subset of edges such that \(\gamma \) neither contains a loop, nor is mass-momentum spanning. Then
$$\begin{aligned} \omega _G\Big |_{L_{\gamma }} = \omega _{G/\gamma }\ \end{aligned}$$where \(L_{\gamma } \subset {\mathbb {P}}^{E_G}\) denotes the linear subspace defined by \(\alpha _e=0\) for \(e\in \gamma \).
Proof
The proof for a general canonical form are essentially the same as for the first canonical form [Bro21, §6.2]. \(\square \)
8.4 Additional form in degree 1
The bi-invariant form
is excluded because it is poorly behaved. In particular, the forms
are not projectively invariant since \(\partial \omega ^1_G=h_G\) and \(\partial \varpi ^1_G =h_{G}+1\), where \(\partial \) denotes the Euler vector field. Nonetheless, the combination
is projectively invariant since \(\partial ^2=0\), which provides an additional form in degree one. However, it has poles at infinity and so products of canonical forms involving \(\o ^1_G\) often give rise to divergent integrals, and will be excluded on this account. See however, §11.1, for an example of a convergent integral associated to \(\o ^1_G\).
Remark 8.7
The significance of the form \(\o ^1_G\) is that it appears in the connection which defines the twisted cohomology underlying dimensional regularisation. For example, one may consider the Mellin transform
where the form \(\omega _G\) in the integrand is to be viewed as a cohomology class with respect to the twisted connection \(\nabla (\omega ) = - s \o ^1_G \wedge \omega + d \omega \).
8.5 Quaternionic forms
Let G be a connected graph with external momenta in \({\mathbb {R}}^4\), which we identify with quaternions via §4.4.
Definition 8.8
For any \(\omega \in \Omega ^{\textrm{can}}\), define the quaternionic canonical form of the second kind associated to G to be
where \(\chi _{\widetilde{\Lambda }_G(\mu ,m)}\) is the image under (2.1) of any choice of quaternionic Laplacian.
Remark 8.9
Since the quaternionic graph Laplacian is Hermitian, its complex adjoint \(X= \chi _{\widetilde{\Lambda }_G(\mu ,m)}\) satisfies the following identity, where \(J_n\) was defined in §2.2,
and is therefore similar to its transpose. For any such matrix \(\beta ^{2k+1}_X=0\) if k is odd. For this reason only the even canonical forms are relevant here.
The quaternionic forms of the second kind have very similar properties to the complex case. We briefly summarise these properties in the following theorem.
Theorem 8.10
Every quaternionic canonical form of degree n of the second kind is a well-defined, closed, projectively-invariant differential form
which is smooth on the open simplex \(\sigma _G\) when momenta are generic (4.1). It is invariant with respect to automorphisms of the graph G and compatible with contraction of subgraphs which have no loops and are not mass-momentum spanning.
Remark 8.11
The primitive quaternionic forms may be defined directly using the quaternionic trace instead of passing via complex adjoints, using the identity:
This uses the fact that \(M\in M_n({\mathcal {Q}})\) is invertible if and only if \(\chi _M \in M_{2n}({\mathbb {C}})\) is invertible §2.2.
9 Canonical Forms at Infinity
This section consists of technical results concerning the asymptotic behaviour of canonical forms in the limit when the generalised graph Laplacians become singular.
9.1 Asymptotic behaviour of generalised Laplacian matrices
Let \(I \subset E_G\) denote any subset of the set of edges of G. Consider the map
Suppose that G has internal masses and external momenta in \({\mathbb {R}}^2 \cong {\mathbb {C}}\), for the time being. Recall that \({\mathcal {E}}_G\) was the extension defined in (5.1).
Lemma 9.1
i). Let \(\gamma \subset E_G\) be a subgraph. Then, writing matrices in block matrix form with respect to a choice of splitting
the rescaled generalised graph Laplacian satisfies:
where B, C, D are matrices with entries in \({\mathbb {C}}[z, \alpha _e, e\in E_G]\) and
ii). Now let \(\gamma \subset E_G\) be a mass-momentum spanning subgraph. If we write matrices in block matrix form with respect to a splitting
then the rescaled generalised graph Laplacian satisfies:
for some matrices A, B, C with entries in \({\mathbb {C}}[z, \alpha _e, e\in E_G]\).
Proof
i). The short exact sequence
defines a filtration \(0 \subset H_1(\gamma ; {\mathbb {Z}}) \subset H_1(G; {\mathbb {Z}})\). We may choose representatives \(c_1,\ldots , c_g \in {\mathbb {Z}}^{E_G}\) for a basis of \(H_1(G;{\mathbb {Z}})\) with the property that \(c_1,\ldots , c_h \subset {\mathbb {Z}}^{E_{\gamma }}\) represent a basis for \(H_1(\gamma ; Z)\). A choice of momentum routing \(\mu \) defines a splitting \({\mathcal {E}}_G= H_1(G;{\mathbb {C}}) \oplus {\mathbb {C}}\) of (5.1), and hence we may write
The statement (9.2) follows from the general form (5.4) of the matrix \(\widetilde{\Lambda }_G(\mu ,m)\) using the fact that \(s_{\gamma }^* c_i(\alpha , \mu ) = z c_i(\alpha ,\mu )\) for \(1 \le i \le h\), and the fact that
ii). Now let \(\gamma \) be mass-momentum spanning. As above, a choice of momentum routing \(\mu \) defines a splitting \({\mathcal {E}}_G \cong H_1(G;{\mathbb {C}}) \oplus {\mathbb {C}}\), and we may write
Let \(c_1, \ldots , c_g\) be a system of representative cycles for a basis of \(H_1(G;{\mathbb {Z}})\), adapted to the filtration \(H_1(\gamma ; {\mathbb {Z}}) \subset H_1(G;{\mathbb {Z}})\). Consider the form (5.4) of the matrix \(\widetilde{\Lambda }_G(\mu ,m)\) with respect to this basis. Since \(\gamma \) is momentum spanning, one has by definition \(s_{\gamma }^* c_i(\alpha , \mu ) = z c_i(\alpha ,\mu )\) for all \(1 \le i \le g\). Since \(\gamma \) is mass and momentum spanning, one has \(s_{\gamma }^* X_G = z X_{\gamma }\), where \(X_{\bullet }\) is defined by formula (5.5). The statement therefore follows from \(s_{\gamma }^* \widetilde{\Lambda }_{G}(\mu ,m) \big |_{z=0} = \Lambda _{G/\gamma }\). \(\square \)
Remark 9.2
By taking the determinant and using Theorem 7.1, the previous lemma gives an interpretation of the ‘asymptotic’ factorisation formulae
stated in [Bro10, Theorem 2.7] for generic kinematics. This observation leads to another proof of Theorem 7.1 since the above factorisation formulae determine \(\Xi _G(q,m)\) essentially uniquely [Bro10, Proposition 4.8].
Note that in the case when \(\gamma \) is mass-momentum spanning, both Eqs. (9.2) and (9.3) are simultaneously valid. Since in this case \(G/\gamma \) is scaleless, \(\widetilde{\Lambda }_{G/\gamma }(\mu ,m)\) has vanishing determinant and (9.2) is not relevant.
9.2 Reminders on motives for graphs with generic kinematics
We assume generic kinematics (4.1) throughout this section. Let us denote by
the union of graph hypersurfaces. By blowing up linear subspaces as in [BEK06] corresponding to motic subgraphs (the notion of a motic subgraph is defined in [Bro17, Definition 3.1]), one obtains a space [Bro10, Definition 6.3]
which contains a distinguished simple normal crossing divisor D whose image under \(\pi _G\) is the union of coordinate hyperplanes. The irreducible components of D are of two types: exceptional divisors \(D_{\gamma }\) corresponding to \(\gamma \) a motic subgraph of G, and divisors \(D_{e}\) where e is an edge of \(E_G\), which are strict transforms of the coordinate hyperplanes \(\alpha _e=0\). We denote the strict transform of \(X_G\) by \(Y_G \subset P^G\).
Every exceptional irreducible component \(D_{\gamma }\) of D admits a canonical isomorphism \(D_{\gamma } \cong P^{\gamma } \times P^{G/\gamma }\) which induces an isomorphism
9.3 Canonical forms along exceptional divisors
For any canonical form \(\eta \), denote its pull-back to \(P^G \backslash (Y_G \cup {\mathcal {E}})\) along (9.4) by
where \({\mathcal {E}}\) denotes the exceptional divisor of \(\pi _G\). It could a priori have poles along \({\mathcal {E}}\), but the following theorem shows that in fact it does not.
Theorem 9.3
Let \(\eta _G \in \mho ^n_{\textrm{can}}\) be any canonical form. Then \(\widetilde{\eta }_G\) has no poles along D and extends to a smooth form on \(P^G \backslash Y_G\), i.e.,
Its restriction to an irreducible boundary component \(D_e\) of D satisfies:
if \(D_e\) is the strict transform of the hyperplane \(L_e = V(\alpha _e)\) corresponding to a single edge e of G. The restriction of \(\eta _G\) to exceptional divisors depend on its kind. For simplicity, assume that \(\eta _G \in \{\omega _G, \varpi _G\}\) is primitive (the general case follows by multiplying primitive forms together), and that \(\gamma \subset G\) is motic and \(e_{\gamma }>1\).
-
(i)
Suppose that \(\eta _G=\omega _G\) is of the first kind. Then
$$\begin{aligned} \widetilde{\omega }_G\big |_{D_{\gamma }} = \widetilde{\omega }_{\gamma } \wedge 1 + 1 \wedge \widetilde{\omega }_{G/\gamma } \end{aligned}$$(9.6)if \(\gamma \) is a core (bridgeless but not necessarily connected) subgraph, and \(\widetilde{\omega }_G\big |_{D_{\gamma }}\) vanishes if \(D_{\gamma }\) is the exceptional divisor corresponding to a m.m. subgraph which is not core.
-
(ii)
Suppose that \(\eta _G = \varpi _G\) is of the second kind. Then
$$\begin{aligned} \widetilde{\varpi }_G\big |_{D_{\gamma }}= & {} \widetilde{\omega }_{\gamma } \wedge 1 + 1 \wedge \widetilde{\varpi }_{G/\gamma } \quad \hbox {if}\quad \gamma \hbox { core, not m.m.} \nonumber \\ \widetilde{\varpi }_G\big |_{D_{\gamma }}= & {} \widetilde{\varpi }_{\gamma } \wedge 1 + 1 \wedge \widetilde{\omega }_{G/\gamma }\quad \hbox {if}\quad \gamma \hbox { is m.m.}. \end{aligned}$$(9.7)
Proof
The argument in [Bro21, Theorem 7.4] proves the case when \(\eta \) is of the first kind and \(\gamma \) is a core subgraph. The general result follows in an identical manner from a local description of the blow-ups in affine coordinates, combined with the asymptotic formulae of Lemma 9.1. The essential part of the argument is the calculation (loc. cit.) of the behaviour of the bi-invariant form \(\beta ^n_X\) where X is a matrix which is asymptotically block-diagonal as in (9.2) or (9.3). \(\square \)
Remark 9.4
An interesting feature of Eq. (9.7) is that the right-hand side involves canonical forms of both the first and second kinds. In particular, if \(\omega \) is primitive of degree \(n\equiv 3 \pmod 4\) then the corresponding form of the first kind \(\omega _G\) vanishes, and the right-hand sides of (9.7) reduce to a single term.
The case of a non-primitive form is easily deduced from Theorem 9.3 by taking products. For example, for a general form \(\omega \), Eqs. (9.6), (9.7) take the form
where \(\Delta _{\textrm{can}} \omega = \sum \omega ' \otimes \omega ''\) is the coproduct of \(\omega \).
Remark 9.5
The differential form \(\o ^1_G\) in general has poles along boundary divisors. Let G be as above, and consider a core subgraph \(\gamma \subset E_G\). Then by the partial factorisations for the first Symanzik polynomial (or by Lemma 9.1(i)), one has:
It follows that
and so \(\o ^1_G\) has a simple pole along the exceptional boundary divisor \(D_{\gamma }\) (which is given by \(z=0\) in local coordinates) and its residue depends only on the loop number of \(\gamma \). A similar argument shows that \(\o ^1_G\) also has poles along exceptional divisors \(D_{\gamma }\), when \(\gamma \) is mass-momentum spanning. Differential forms of the type \(\o ^1_G \wedge \eta \), where \(\eta \) is canonical, will therefore in general have poles along the divisor D, and will not be considered here. They naturally arise, however, when considering Stokes’ formula for canonical forms in dimensional-regularisation.
9.4 Quaternionic case
The above arguments similarly apply to the case of a canonical form of the second kind \(\varpi ^{{\mathcal {Q}}}_{G}\) associated to a quaternionic generalised Laplacian, by replacing \(\widetilde{\Lambda }_G(\mu ,m)\) with the complex adjoint matrix \(\chi _{\widetilde{\Lambda }_G(\mu ,m)}\). Theorem 9.3 holds verbatim (with the only difference that odd canonical forms of the second kind vanish in the quaternionic setting). In particular, \(\varpi ^{{\mathcal {Q}}}_{G}\) has no poles along D and extends to a smooth differential form on \(P^G \backslash Y^G\). Its restriction to irreducible components of D are given by (9.8).
10 Canonical Integrals and Relations from Stokes Theorem
Throughout this section we assume generic (Euclidean) kinematics (4.1).
10.1 Canonical integrals
Let G be a connected graph with an orientation \(e_1\wedge \cdots \wedge e_{N} \in \left( \bigwedge {\mathbb {Z}}^{E(G)}\right) ^{\times } \). It induces an orientation on the simplex \(\sigma _G\).
10.1.1 Complex momenta
Suppose that external momenta lie in \({\mathbb {R}}^2 \cong {\mathbb {C}}\).
Theorem 10.1
Let \(\omega \in \mho _{\textrm{can}}\) be a canonical form of degree \(N+1\) and let G be a connected graph with N edges. Then under assumption (4.1), the integral
is finite and defines an analytic function of the external kinematics.
Proof
This is a corollary of Theorem 9.3. Let \(\widetilde{\sigma }_G = \overline{\pi ^{-1}_G (\sigma _G)}\) denote the closure in the analytic topology of the pull-back of the open simplex \(\sigma _G\) to \(P^G\). One has
The right-most integral converges because of the fact [Bro17] that \(\widetilde{\sigma }_G\), which is compact, does not meet \(Y_G\), which is where the poles of \(\widetilde{\omega }_G\) are located. \(\square \)
Reversing the orientation on G reverses the sign of the canonical integral (10.1). Furthermore, if \(\tau : G \overset{\sim }{\rightarrow }\ G'\) is an isomorphism of oriented graphs (with external legs and masses, etc), then since \(\tau ^* \omega _G = \omega _{G'}\), we have:
Thus the integrals (10.1) are invariants of isomorphism classes of oriented graphs.
10.1.2 Quaternionic case
The previous theorem, and comments which follow, hold verbatim for graphs with momenta in 4-dimensional Euclidean space with
The only difference is that in this case we must assume \(\omega \in \Omega _{\textrm{can}}\) is even, since odd quaternionic canonical forms vanish by Remark 8.9.
10.2 Stokes’ formula
In [Bro17], it was shown that one can extend the usual Connes–Kreimer coproduct to an enlarged coproduct
where the sum is over all motic subgraphs of G. It has two types of terms: if \(\gamma \subset G\) is a core subgraph which is not m.m., then \(\gamma \) is considered to be scaleless, and encodes ultraviolet divergences of the graph. If \(\gamma \subset G\) is m.m. then \(G/\gamma \) is scaleless, and \(\gamma \) encodes certain kinds of infrared divergences of G.
10.2.1 Complex momenta
Theorem 10.2
Let \(\omega \in \mho _{\textrm{can}}\) be a canonical form of degree k. Write its coproduct in the form \(\Delta _{\textrm{can}} \omega = \sum _i \omega '_i \otimes \omega ''_i\). Suppose that G is a connected oriented graph with external kinematics as above with \(k+2\) edges. Then
where the rightmost sum is over motic subgraphs \(\gamma \subsetneq G\) not including G itself (which are not necessarily connected) such that \(e_{\gamma } = \deg \omega _i'+1>1\), and the orientations on \(\sigma _{G / e}, \sigma _{\gamma } \times \sigma _{G/\gamma }\) are induced from those on \(\sigma _G\).
Proof
Identical to the proof of [Bro21, Theorem 8.5], using Theorem 9.3. \(\square \)
Note that in the case when \(\omega \in B^{\textrm{can}}\) is purely of the second kind, one has
where we have decomposed the set of motic subgraphs \(\gamma \) of G, such that \(1< e_{\gamma }<e_G\) into two types according to whether \(\gamma \) is m.m. or not.
Note that the Stokes’ formulae above remain valid by analytic continuation to a larger region of kinematic space wherever the integrals are finite.
10.2.2 Quaternionic case
The statement is identical on replacing \(\omega \) with \(\omega ^{{\mathcal {Q}}}\).
Remark 10.3
In [Bro21, Theorem 8.5], the Stokes’ formula is not quite correct as stated in the case when a graph G has a tadpole \(\gamma \). In that case, the corresponding term was counted twice: once in the term G/e, where \(e= \gamma \), and again in the form \(\gamma \times G/\gamma \). The condition imposed above, that \(e_{\gamma }>1\), fixes this problem.
Similarly, if \(\gamma \) is an m.m. subgraph which has only a single edge, then it only contributes a single term in the formula (10.2), which is counted in the left-most summation over all edges, and not in the right-hand double sum.
10.3 Canonical forms of compact type
Let \(h\ge 1\). Let us say that a canonical form \(\omega \in B^{\textrm{can}}\) is of compact type relative to h if it is divisible by the primitive form \(\omega ^{2h-1}\) of degree \(2h-1\), i.e., \(\omega = \omega ^{2h-1} \wedge \eta \) for some \(\eta \in B^{\textrm{can}}\). It follows from the property (6) stated in §8.1, that \(\omega \) vanishes on all matrices of rank \(<h\).
Corollary 10.4
Let G be as in Theorem 10.2 and let \(\varpi \in B^{\textrm{can}}\) be of the second kind of degree \(e_G -2\), and of compact type relative to \(h_G+1\). Thus \(\varpi = \varpi ^{2 h_{G}+1} \wedge \eta \) for some \(\eta \in B^{\textrm{can}}\) of the second kind. If G is core, then formula (10.3) reduces to
Proof
Let \(\gamma \subset G\) be a strict motic subgraph with at least two edges. Suppose that \(\gamma \) is not m.m. and therefore core. We need to check that a term of the form
vanishes in the right-hand side of (10.3). Either \(\omega '\) or \(\omega ''\) is of compact type relative to \(h_G+1\). If the former, then \(\omega '_{\gamma }= \omega '_{\Lambda _{\gamma }}\) vanishes since \(h_{\gamma } < h_G+1\). If \(\omega ''\) is of compact type, then \(\varpi ''_{G/\gamma }= \omega ''_{\widetilde{\Lambda }_{G/\gamma }}\) vanishes if \(h_{G/\gamma } +1 < h_G+1\). This inequality holds since \(h_{\gamma } + h_{G/\gamma }+1 = h_G+1\), and \(h_{\gamma }>0\) because \(\gamma \) is core.
Now suppose that \(\gamma \) is m.m.. We need to check that any term of the form
vanishes. If \(\omega '\) is of compact type, then \(\varpi '_{\gamma }= \omega '_{\widetilde{\Lambda }_{\gamma }}\) vanishes if \(h_{\gamma } + 1 < h_{G}+1\). Since \(\gamma \) is a strict subgraph, this follows from \(h_{\gamma } < h_{G}\) since G is core. If \(\omega ''\) is of compact type, then \(\omega ''_{G/\gamma }= \omega ''_{\Lambda _{G/\gamma }}\) vanishes since \(h_{G/\gamma } < h_G+1\). \(\square \)
11 Examples of Canonical Amplitudes (Complex Momenta)
For a graph G with edges numbered \(1,\ldots , N\), we write
We consider canonical differential forms in low degrees and their integrals. For now, we consider only the case when external momenta lie in \({\mathbb {R}}^2 \cong {\mathbb {C}}\).
11.1 Digression: the divergent form \(\o ^1\)
The integrals involving the differential form \(\o ^1\) diverge, and for that reason are not considered to be ‘canonical’ forms according to our definitions. However, this form provides some interesting first examples, which may on occasion lead to convergent integrals.
Graphs which pair with \(\o ^1\) have two edges. Consider the moduli space \({\mathcal {M}}^{\textrm{trop}}_{1,2,2}\) of metric graphs with 1 loop, 2 external legs, and 2 distinct non-zero masses. It has two metric graphs of maximal dimension (see Fig. 1). The one on the right has a non-trivial subdivergence and so the integral of \(\o ^1\) is logarithmically divergent. Therefore, consider the graph G on the left. It is important that its edge masses \(m_1, m_2\) be distinct, for otherwise, this graph has a symmetry which induces an odd permutation of its set of edges, implying that its canonical integral vanishes.
The two metric graphs of maximal dimension in \({\mathcal {M}}^{\textrm{trop}}_{1,2,2}\)
Orient and number the edges as shown in the figure. A representative for a generator of the homology \(H_1(G;{\mathbb {Z}})\) is \(c_1 = e_1 +e_2\). Momentum conservation implies that \(q_2=- q_1 \in {\mathbb {R}}^2\cong {\mathbb {C}}\). Any momentum routing is given by a solution to:
The generalised graph Laplacian matrix with respect to these choices is:
The relevant graph polynomials are given by
where we write \(q_1^2\) for the Euclidean norm, which equals \(\overline{{\textsf{q}}}_1 {\textsf{q}}_1\), where \({\textsf{q}}_1\) is \(q_1\) viewed in \({\mathbb {C}}\). Using (8.7), the exceptional form of degree 1 is
Thus the associated integral is
where the last equality follows by computing the projective integral on the affine chart \(\alpha _2=1\). The integral reduces to
which is an agreement with the renormalised amplitude for a massive bubble. As expected, it vanishes when \(m_1=m_2\) for the symmetry reasons mentioned above.
11.2 Canonical amplitudes associated to \(\varpi _3\)
We now turn to bona fide canonical amplitudes, whose integrals are always finite by Theorem 10.1. The canonical form of smallest degree is \(\varpi ^3 \in B_{\textrm{can}}\). It pairs with connected graphs which have 4 edges, and vanishes for graphs with fewer than 1 loop. Thus there are 3 cases:
-
\(h_G=1, v_G=4\)
-
\(h_G=2, v_G=3\)
-
\(h_G=3, v_G=2\)
which we consider in turn for the maximal number of masses.
11.2.1 \({\mathcal {M}}^{\textrm{trop}}_{1,4,4}\)
The massive box diagram (Fig. 2) is the unique graph which gives a cell of maximal dimension in \({\mathcal {M}}^{\textrm{trop}}_{1,4,4}\).
Massive box graph
Choose any complex numbers \(\mu _e\) for each edge e such that momentum conservation (4.2) holds: \( \mu _{i+1} - \mu _{i} = {\textsf{q}}_i, \) where \(q_i \in {\mathbb {R}}^2\), corresponding to \({\textsf{q}}_i \in {\mathbb {C}}\), is the external momentum at vertex i subject to momentum conservation \({\textsf{q}}_1+\cdots +{\textsf{q}}_4=0\), and where we write \(\mu _{5} =\mu _{1}\). The generalised graph Laplacian with respect to the cycle \(c_1=e_1+\cdots +e_4\) is
The associated graph polynomials are:
The canonical form \(\varpi ^3_G\) is proportional to the Feynman differential form:
where the numerator has the symmetric form:
It can be re-expressed as a function of the external momenta:
In conclusion, we find that
which is proportional to the Feynman integral for the massive box graph (viewed in \(D=4\) spacetime dimensions).
11.2.2 Examples from \({\mathcal {M}}^{\textrm{trop}}_{2,3,4}\)
Figure 3 depicts cells of dimension 4 on \({\mathcal {M}}^{\textrm{trop}}_{2,3,4}\).
Left: The dunce’s cap has a non-trivial divergent subgraph given by the subgraph spanned by edges 3, 4. Right: A double bubble graph, upon which the canonical form vanishes
The graph on the right is an example of a 1-vertex join. There is a similar graph in which the central vertex has no incoming momentum, and the left-most vertex has two incoming vertices. In both cases, the graph Laplacian has the form
and one verifies that \(\varpi ^3\) vanishes for this matrix. Therefore the only non-zero contribution is from the dunce’s cap, depicted on the left of Fig. 3. It is one of the first examples of a graph with a non-trivial subdivergence (given by the subgraph spanned by edges 3, 4), so the associated Feynman integral diverges. By contrast, the canonical integral is necessarily finite and it is instructive to see why.
Since there are only 3 external legs, we can always assume that the momenta are complex numbers \({\textsf{q}}_1, {\textsf{q}}_2, {\textsf{q}}_3\) subject to momentum conservation \( {\textsf{q}}_1 + {\textsf{q}}_2 + {\textsf{q}}_3 =0. \) For the basis of cycles given by \(c_1 = e_1+ e_3+ e_2\), \(c_2= e_4-e_3\), and the orientations of edges as shown in Fig. 3, the generalised graph Laplacian takes the form:
where (4.2) takes the form:
One finds that
A computation shows that the canonical form of the second kind in degree 3 is
where we note that the factor \( {\textsf{q}}_1 \overline{{\textsf{q}}}_2 - {\textsf{q}}_2 \overline{{\textsf{q}}}_1 = {\textsf{q}}_3 \overline{{\textsf{q}}}_1 - {\textsf{q}}_1\overline{{\textsf{q}}}_3\) is indeed symmetric, by momentum conservation. The numerator vanishes at \(\alpha _3 =\alpha _4=0\), to compensate the pole arising from the vanishing of \(\Xi _G(q,m)\) along this locus. Note also that if \(m_3=m_4\) then the integral of \(\varpi ^3_G\) will vanish, due to the fact that interchanging edges 3 and 4 induces an odd permutation on the edges of G. In conclusion, for a suitable choice of orientation of G one has
It is interesting to compare this to the renormalised amplitude associated to the graph G, which was discussed in [BK13].
11.2.3 The space \({\mathcal {M}}^{\textrm{trop}}_{3,2,4}\)
The associated graphs have at most two external momenta. By momentum conservation, they lie on a real line and so we may choose a real momentum routing for which the generalised Laplacian \(\widetilde{\Lambda }_G(q,m)\) will be symmetric. Therefore \(\varpi ^3_{G}\) vanishes, and there are no non-zero canonical integrals.
11.3 Canonical amplitudes in degree 5
In degree 5 there exist canonical forms \(\omega ^5\) and \(\varpi ^5\) of both kinds. There is a unique vacuum diagram, namely the wheel with 3 spokes and no external legs, which pairs non-trivially with the former. Its amplitude is \(6 \zeta (3)\), and is discussed in detail in [Bro21].
Henceforth we shall focus on the form \(\varpi ^5\). Since \(\beta ^5\) vanishes on matrices of rank \(\le 2\), the form \(\varpi ^5_G\) is zero unless \(h_G\ge 2\). There are several graphs G with 6 edges and two or more loops which could potentially pair non-trivially with \(\varpi ^G\), but we shall only focus on a couple of examples for reasons of space.
11.3.1 Examples on \({\mathcal {M}}^{\textrm{trop}}_{2,5,6}\)
One shows by direct calculation that the two graphs depicted in Fig. 4 satisfy \(\varpi ^5_G =0\).
Some graphs with 6 edges for which \(\varpi ^5_G\) vanishes
There is a single potentially non-zero canonical integral on \({\mathcal {M}}^{\textrm{trop}}_{2,5,6}\), given by the box-triangle graph depicted in Fig. 5. A computer calculation shows that
where \(N_2\) is a certain polynomial in the \(m_i^2\) and \(q_i, \overline{q}_i\), and the polynomial \(N_1\) factorises into a product of two terms
where f is a homogeneous linear form in \(\alpha _1,\alpha _2,\alpha _3\) which does not depend on the masses and g is of degree 2 in the \(\alpha _i\).
The box-triangle graph
The integral corresponding to the term \(N_2\) is the usual Feynman integral in 4 space time dimensions. It would be interesting to know if the canonical integral admits a natural interpretation in terms of Feynman integrals via integration by parts identities, or if it may be obtained from the usual Feynman integral by application of a natural differential operator in the external kinematics.
12 Example of a Stokes Relation for the Massive Box Diagram
Consider the massive pentagon diagram with arbitrary masses \(m_1,\ldots , m_5 \in {\mathbb {R}}\) and external particle momenta in \(q_1,\ldots , q_5\in {\mathbb {R}}^2 \cong {\mathbb {C}}\) subject to momentum conservation (Fig. 6). Contracting each of its edges leads to five massive box diagrams (Fig. 7).
Massive pentagon diagram
Since \(\varpi ^3\) is of compact type for 1-loop graphs, we may apply Corollary 10.4 to deduce a five-term relation for the massive box diagram. Indeed (10.4) implies that
where \(G_1,\ldots , G_5\) are the five diagrams pictured in Fig. 7. This relation can easily be checked in the massless case using Panzer’s Hyperint [Pan15].
Five-term relation from pentagon diagram
Since the canonical integrals for massive boxes are proportional to their Feynman integrals, we deduce a 5-term relation for massive box Feynman integrals. It is known [DD98] that the latter are expressible as a linear combination of a number of dilogarithms. The graphical relation above is presumably equivalent to Abel’s 5-term equation for the dilogarithm. This functional equation should, in addition, be compatible with the motivic coaction which was computed in [Tap21].
13 Example of a Canonical Amplitude in the Quaternionic Case
13.1 A canonical amplitude for \(\varpi ^5\)
Let G be the one-loop hexagon diagram with 6 external momenta in \({\mathbb {R}}^4\), and 6 masses. The generalised graph Laplacian is
where \(\mu _i \in {\mathcal {Q}}\) is a quaternionic momentum routing. Its complex adjoint may be represented by
where we write \(\mu _i = \lambda _i + {\textsf{j}}\nu _i\). We find by computer calculation that
where \(N_G\) is the determinant of the following matrix:
where we write \( ||\mu _i||^2 = \lambda _i \overline{\lambda }_i + \nu _i \overline{\nu }_i\). Thus the canonical integral associated to the massive hexagon is proportional to the usual Feynman integral (viewed in 6 spacetime dimensions). Stokes’ formula applied to a massive quaternionic heptagon diagram leads to a 7-term functional equation for these integrals.
14 Tropical Single-Valued Integration on Curves
In this section, which is not required for the paper, we provide some motivation for the definition of the generalised graph Laplacian by analogy with smooth projective algebraic curves with a finite number of punctures.
14.1 Periods on smooth algebraic curves
Let X be a smooth compact curve over a field \(k \subset {\mathbb {C}}\), and let \(\Sigma \subset X({\mathbb {C}})\) denote a finite set of points. Consider the Gysin (residue) sequence in Rham cohomology
and suppose that we are given an element (‘external momentum’):
which may be interpreted as a divisor of degree 0 on X with coefficients in \({\mathbb {C}}\) supported on \(\Sigma \). By pulling back the above sequence along the copy of \( {\mathbb {C}}(-1) \subset H_{dR}^0(\Sigma )(-1)\otimes _k {\mathbb {C}}\) spanned by \(\alpha _q\), we obtain a simple extension over \({\mathbb {C}}\):
Since \(F^1 {\mathbb {C}}(-1) = {\mathbb {C}}(-1)\), it splits if and only if the sequence
splits. Such a splitting is given by an element (‘momentum routing’)
whose image in \({\mathbb {C}}(-1)\) is \(\alpha _q\). In other words, \(\hbox {Res}\, \mu = \alpha _q\). Since \(H^0(X, \Omega _X^1 (\log \Sigma )) \overset{\sim }{\rightarrow } \hbox {gr}_F^1(H^1_{dR}(X\backslash \Sigma ))\), we can assume that \(\mu \) is represented by a differential of the third kind with only logarithmic poles along \(\Sigma \). It is uniquely defined only up to adding a linear combination of holomorphic forms:
where \(\omega _1,\ldots , \omega _g \in H^0(X, \Omega ^1_{X}) \cong F^1H^1_{dR}(X)\) and \(\lambda _i \in {\mathbb {C}}\).
We wish to consider a Hermitian form Q on \(F^1 {\mathcal {E}}_X\) defined formally by:
Alas, the integrals on the right are not always convergent. There are three cases:
-
(1)
\(\eta _1=\omega _i\), and \(\eta _2=\omega _j\) are holomorphic. Then the restriction of Q to \(H^1_{dR}(X)\) is given by the convergent integrals
$$\begin{aligned} \omega _i \otimes \omega _j \mapsto \frac{1}{2\pi i} \int \omega _i \wedge \overline{\omega }_j \end{aligned}$$from which one retrieves the Riemann polarisation form. If k is real, then this actually defines a symmetric quadratic form on \(H^1_{dR}(X)\).
-
(2)
\(\eta _1 =\omega _i\) is holomorphic, and \(\eta _2= \mu \) is of the third kind (or vice-versa). Then the singular integral
$$\begin{aligned} c_i(\mu )= \frac{1}{2\pi i}\int _X \omega _i \wedge \overline{\mu } \end{aligned}$$is convergent since it defines a single-valued period of \(H^1_{dR}(X\backslash \Sigma )\). To see this, note that \(F^1H^1(X, \Sigma ) \overset{\sim }{\rightarrow }\ F^1H^1(X)\) is an isomorphism and since \(\omega _i\) vanishes along \(\Sigma \) it may canonically be viewed as an element of \(H^1(X,\Sigma )\). The integral above is an instance of the single-valued pairing [BD21, §6.4]:
$$\begin{aligned} {\textsf{s}}: H^1_{dR}(X, \Sigma ) \otimes H^1_{dR}(X \backslash \Sigma ) \longrightarrow {\mathbb {C}}\end{aligned}$$ -
(3)
Only in the case \(\eta _1 =\eta _2= \mu \) does one have an ill-defined integral:
$$\begin{aligned} \frac{1}{2\pi i}\int \mu \wedge \overline{\mu } \end{aligned}$$It is singular in a neighbourhood of a point \(\sigma \in \Sigma \), since the integrand may be expressed in local polar coordinates \(z = \rho e^{i \theta }\) based at \(\sigma \) in the form
$$\begin{aligned} \mu \wedge \overline{\mu } = \lambda \, \frac{d\rho }{\rho } d \theta + \hbox {regular terms}, \end{aligned}$$for some \(\lambda \in {\mathbb {C}}\), which is not ‘polar-smooth’ in the terminology of [BD21]. One can regularise these integrals around each point of \(\Sigma \) by a number of methods (see also [BD21, Remark 3.22]). The method used is unimportant for this discussion but the value of the integral will in general depend upon it. In any case we may simply put
$$\begin{aligned} \frac{1}{2\pi i}\int ^{\textrm{reg}} \mu \wedge \overline{\mu }: =x_m \end{aligned}$$where \(x_m\) is any real and positive number of our choosing. The possibility of choosing \(x_m\) relates to the presence of masses in Feynman graphs.
Thus, with respect to the basis \(\omega _1, \ldots , \omega _g, \mu \) of \({\mathcal {E}}_X\), the form Q is represented by a Hermitian matrix
This is precisely the form of the generalised Laplacian matrix (compare (5.4)).
14.2 Tropical analogy
Let G be a connected graph (with no external half edges in the first instance). Choose an orientation on its edges. According to [MZ08], a differential form on G is a linear combination
where \(\lambda _e \in {\mathbb {Z}}\) such that for all vertices v one has \(\sum _{s(e)= v} \lambda _e = \sum _{t(e)= v} \lambda _e\). Thus the space of differential forms \(\Omega _G^1\) on G lies in an exact sequence
and may be identified with \(H_1(G;{\mathbb {Z}})\). The inner product on \({\mathbb {Z}}^{E_G}\) is given by
and restricts to a positive definite quadratic form on \(\Omega _G^1\).
Now suppose that G has external half-edges, and that we are given an element \({\textsf{q}}\in \hbox {ker} ({\mathbb {Z}}^{V_G} \rightarrow {\mathbb {Z}}) \otimes {\mathbb {C}}\). As in §5.1, we deduce an extension \(\widetilde{\Omega }^1_G\) satisfying
and the above inner product (extended \({\mathbb {C}}\)-linearly) restricts to a Hermitian form
which is a tropical analogue of (14.3). The generalised graph Laplacian matrix \(\widetilde{\Lambda }_G(q,0)\) for zero masses is precisely the matrix of this Hermitian form with respect to a splitting of \(\widetilde{\Omega }^1_G\), which is given by any differential form
which satisfies the ‘residue’ equation \(\sum _{s(e)= v} \lambda _e - \sum _{t(e)= v} \lambda _e = {\textsf{q}}_v\) for all vertices v. The presence of masses corresponds, as above, to modifying only the value of \(\langle \nu , \overline{\nu } \rangle \) in the bottom-right hand corner, and from this perspective, we see that the generalised graph Laplacian (5.4) is a tropical analogue of the regularised single-valued integration pairing discussed in §14.1.
Notes
The reader familiar with Chern–Simons theory may recognise canonical forms as a special case: for example, the Chern–Simons 3-form \(\textrm{tr}( dA \wedge A +\frac{2}{3} A\wedge A \wedge A)\) reduces to \(-\frac{1}{3} \textrm{tr}((\Lambda ^{-1} d\Lambda )^3)\) upon substituting \(A=\Lambda ^{-1} d\Lambda \). The higher degree forms are similar.
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Acknowledgements
Many thanks to Marko Berghoff, Paul Fendley, Lionel Mason, Erik Panzer, and all the participants of the seminar at Oxford in 2022 on graph complexes. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 724638).
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Brown, F. Generalised Graph Laplacians and Canonical Feynman Integrals with Kinematics. Commun. Math. Phys. 405, 41 (2024). https://doi.org/10.1007/s00220-023-04879-3
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DOI: https://doi.org/10.1007/s00220-023-04879-3