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Outer Space as a Combinatorial Backbone for Cutkosky Rules and Coactions

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Anti-Differentiation and the Calculation of Feynman Amplitudes

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We consider a coaction which exists for any bridge-free graph. It is based on the cubical chain complex associated to any such graph by considering two boundary operations: shrinking edges or removing them. Only if the number of spanning trees of a graph G equals its number of internal edges we find that the graphical coaction Δ G constructed here agrees with the coaction Δ Inc proposed by Britto and collaborators. The graphs for which this is the case are one-loop graphs or their duals, multi-edge banana graphs. They provide the only examples discussed by Britto and collaborators so far. We call such graphs simple graphs. The Dunce’s cap graph is the first non-simple graph. The number of its spanning trees (five) exceeds the number of its edges (four). We compare the two coactions which indeed do not agree and discuss this result. We also point out that for kinematic renormalization schemes the coaction Δ G simplifies.

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  1. 1.

    For any 4-vector r we have \(r^2=r_0^2-\mathbf {r}\cdot \mathbf {r}\). Let q be a time-like 4-vector, p an arbitrary 4-vector. Then, (q ⋅ p 2 − q 2 p 2)∕q 2 = λ(q 2, p 2,  (q + p)2)∕4q 2 and in the rest frame of q, (q ⋅ p 2 − q 2 p 2)∕q 2 = p ⋅p where λ(a, b, c) = a 2 + b 2 + c 2 − 2(ab + bc + ca), as always.

  2. 2.

    In the parametric representation \(\mathfrak {o}\) orders them by length.


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It is a pleasure to thank Marko Berghoff, Spencer Bloch, Karen Vogtmann and Karen Yeats for helpful advice. I thank Spencer in particular for numerous discussions on the matrices investigated here. I am grateful to Johannes Blümlein and all the organizers for their efforts.

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Correspondence to Dirk Kreimer .

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Appendix 1: The Cubical Chain Complex

We assume the reader is familiar with the notion of a graph and of spanning trees and forests. See [15] where these notions are reviewed. We follow the notation there. In particular |G| is the number of independent cycles of G, e G the number of internal edges and v G the number of vertices of G. For a pair of a graph and a spanning forest we write (G, F) or G F. If a spanning forest has k connected components we call it a k-forest. A spanning tree T is a 1-forest. Its number of edges hence e T. \(\mathcal {T}(G)\) is the set of spanning trees of G.

Pairs (G, F) are elements of a Hopf algebra H GF based on the core Hopf algebra H core of bridgefree graphs [15].

As an algebra H GF is the free commutative Q-algebra generated by such pairs. Product is disjoint union and the empty graph and empty tree provide the unit.

A k-cube is a k-dimensional cube \([0,1]^k\subsetneq \mathbb {R}^k\).

Consider G T. We define a cube complex for e T-cubes \(\mathsf {Cub}_G^T\) assigned to G. There are e T! orderings \(\mathfrak {o}=\mathfrak {o}(T)\) which we can assign to the internal edges of T.

We define a boundary for any elements G F ≡ (G, F) of H GF. For this consider such an ordering

$$\displaystyle \begin{aligned} \mathfrak{o}:E_F\to [1,\ldots,e_F] \end{aligned}$$

of the e F edges of F. There might be other labels assigned to the edges of G and we assume that removing an edge or shrinking an edge will not alter the labels of the remaining edges. In fact the whole Hopf algebra structure of H core and H GF is preserved for arbitrarily labeled graphs [21].

The (cubical) boundary map d is defined by d := d 0 + d 1 where

$$\displaystyle \begin{aligned} d_0(G_F^{\mathfrak{o}(F)}):= \sum_{j=1}^{e_F} (-1)^j(G_{F \! \setminus \! e_j}^{\mathfrak{o}(F\! \setminus \! e_j)}), \quad d_1 (G_F^{\mathfrak{o}(F)}) := \sum_{j=1}^{e_F} ({G/e_j}^{\mathfrak{o}(F/e_j)}_{F/e_j}). \end{aligned} $$

We understand that all edges \(e_k,k\gneq j\) on the right are relabeled by e k → e k−1 which defines the corresponding \(\mathfrak {o}(T/e_j)\) or \(\mathfrak {o}(T \! \setminus \! e_j)\). Similar if T is replaced by F.

From [18] we know that d is a boundary:

Theorem 4.1

[ 18 ]

$$\displaystyle \begin{aligned} d\circ d=0,\,d_0\circ d_0=0,\, d_1\circ d_1=0. \end{aligned}$$

Starting from G T for any chosen \(T\in \mathcal {T}(G)\) each chosen order \(\mathfrak {o}\) defines one of e T! simplices of a e T-cube \(\mathsf {Cub}_G^T\). We write \(T_{\mathfrak {o}}\) for a spanning tree T with a chosen order \(\mathfrak {o}\) of its edges. It identifies one such simplex.

Such simplices will each provide one of the lower triangular matrices defining our coactions. If spt(G) is the number of spanning trees of a graph G, we get spt(G) × e T! such matrices where we use that e T = e G −|G| is the same for all spanning trees T of G, as there are e T! different matrices for each of the spt(G) different e T-cubes \(\mathsf {Cub}_G^T\).

Appendix 2: The Lower Triangular Matrices M(G, T o)

Consider a pair \((G,T_{\mathfrak {o}})\) where G is a bridgeless Feynman graph and \(T_{\mathfrak {o}}\) a spanning tree T of G with an ordering \(\mathfrak {o}\) of its edges e ∈ E T. There are e T! such orderings where e T is the number of edges of T.Footnote 2

To such a pair we associate a (e T + 1) × (e T + 1) lower triangular square matrix

$$\displaystyle \begin{aligned} M=M(G,T_{\mathfrak{o}}) \end{aligned}$$

with M ij ∈ H GF.

More precisely, \(M_{ij}\in \mathsf {Gal}(M)\equiv \mathsf {Gal}(G,T_{\mathfrak {o}})\), where \(\mathsf {Gal}(G,T_{\mathfrak {o}})\subsetneq H_{GF}\) is the set of Galois correspondents of \((G,T_{\mathfrak {o}})\), i.e. the graphs which can be obtained from G by removing or shrinking edges of T in accordance with \(\mathfrak {o}\).

As stated above for a pair (G, T) there are e T! such matrices \(M(G,T_{\mathfrak {o}})\) generated by the corresponding e T-cube of the cubical chain complex associated to any pair (G;T) ∈ G F [15].

M is defined through its entries \(M_{ij}\in \mathsf {Gal}(G,T_{\mathfrak {o}})\), j ≤ i,

$$\displaystyle \begin{aligned} M_{ij}:=(G/E_j,(T/E_j\! \setminus \! E^i). \end{aligned}$$

Here E j is the set given by the first (e T − j + 1)-entries of the set

$$\displaystyle \begin{aligned} \{\emptyset, e_{e_T},e_{j-1},\ldots, e_1\} \end{aligned}$$

and E i by the first i entries of \(\{\emptyset , e_1,\ldots ,e_{e_T}\}\). We shrink edges in reverse order and remove them in order.

Define the map \(\varDelta ^M\equiv \varDelta ^{(G,T_{\mathfrak {o}})}:\mathsf {Gal}(G,T_{\mathfrak {o}})\to \mathsf {Gal}(G,T_{\mathfrak {o}})\otimes \mathsf {Gal}(G,T_{\mathfrak {o}})\),

$$\displaystyle \begin{aligned} \varDelta^{(G,T_{\mathfrak{o}})}\left(M\right)_{jk}=\sum_{i=1}^{e_T+1} \left(M\right)_{ik}\otimes \left(M\right)_{ji}, \end{aligned} $$

as before. We often omit the superscript \(\{\}^{(G,T_{\mathfrak {o}})}\) when not necessary.

Let us now define

$$\displaystyle \begin{aligned} V_1=: \mathsf{Gal}(G,T_{\mathfrak{o}})_/\subsetneq \mathsf{Gal}(G,T_{\mathfrak{o}}), \end{aligned}$$

as the \(\mathbb {Q}\)-span of elements M j,1, j ≥ 2.

Then we can regard the coproduct Δ M as a coaction

$$\displaystyle \begin{aligned} \rho_{\varDelta^M}:\,\mathsf{Gal}(G,T_{\mathfrak{o}})_/\to \mathsf{Gal}(G,T_{\mathfrak{o}})_/\otimes \mathsf{Gal}(G,T_{\mathfrak{o}}). \end{aligned}$$

Soon we will evaluate entries in \(M_{\mathbb {I}}\) by Feynman rules.

$$\displaystyle \begin{aligned} (M_{\mathbb{I}}){ij}\to \varPhi_R(M_{ij})/\varPhi_R(M_{jj}). \end{aligned}$$

This normalization \(M\to M_{\mathsf {D}}\times M_{\mathbb {I}}\) to the leading singularities is common [1].

Appendix 3: Summing Orders and Trees

Let us first consider the sectors we are integrating over. A graph G provides e G! sectors. We partition them as follows. We have e G = e T + |G|. Then

$$\displaystyle \begin{aligned} \frac{e_G!}{e_T!\times|G|!}\geq spt(G), \end{aligned}$$

with equality only for |G| = 1 and the dual of one-loop graphs (‘bananas’) and spt(G) is the number of spanning trees of G (see also [15]). We note that e T! ×|G|! is the number of sectors

$$\displaystyle \begin{aligned} a_{e_i}\geq a_{e_f} \Leftrightarrow e_i\in E_G\setminus E_T\wedge e_f\in E_T, \end{aligned}$$

where each edge not in the spanning tree is larger than each edge in the spanning tree. This allows to shrink all e T edges in the spanning tree in any order in accordance with the spine being a deformation retract in the Culler–Vogtmann Outer Space [9].

The difference

$$\displaystyle \begin{aligned} e_G!-spt(G)\times e_T!\times |G|! \end{aligned}$$

are the sectors where at least one loop shrinks. Any spanning tree T defines a basis of |G| loops l i, 1 ≤ i ≤|G|, provided by a path p i in T connecting the two ends of an edge e i ∈ E G ∖ E T. We say that e i generates l i.

For any given T the sectors where a loop shrinks fulfill two conditions

  1. (i)

    for any l i, \(a_{e_i}\geq a_e,\,\forall e\in E_{p_i}\),

  2. (ii)

    it is not a sector for which \(a_{e_i}\geq a_{e_f} \Leftrightarrow e_i\in E_G\setminus E_T\wedge e_f\in E_T, \) holds.

The latter condition (ii) ensures that when shrinking e T edges at least one edge in E G ∖ E T and hence a loop shrinks. The former condition (i) ensures that each loop l i retracts to its generator e i.

Example 4.2

As an example we consider the Dunce’s cap and the wheel with three spokes graph.

The Dunce’s cap: Each spanning tree T gives rise to 2! × 2! sectors e T! ×|G|!. There are five spanning trees, so this covers 20 sectors where no loop shrinks. There are four edges in the Dunce’s cap so we get 4! sectors. For the four missing sectors four spanning trees provide one each.

The wheel with three spoke graph:

e T! ×|G|! = 3! × 3! = 36 and there are 16 spanning trees giving us 576 sectors. The 16 spanning trees correspond to 16 choices of three edges while there are such choices altogether. There are 6! = 720 sectors. The missing 144 = (20 − 16) × 3! × 3! sectors come from the four triangle subgraphs providing 4 × 3! × 3! sectors.

This ends our example.

As a result if we let \(n(T_{\mathfrak {o}})\) be the number of sectors provided by an ordered spanning tree we have

Lemma 4.3

$$\displaystyle \begin{aligned} e_G!=\sum_{T\in\mathcal{T}}\sum_{\mathfrak{o}}n(T_{\mathfrak{o}}). \end{aligned}$$

It thus makes sense to assign a union of sectors to each ordered spanning tree \(T_{\mathfrak {o}}\). Here \(\mathsf {sec}_j\in \mathcal {S}\mathcal {E}\mathcal {C}_T^{\mathfrak {o}}\), the set of sectors compatible with T and its order of edges \(\mathfrak {o}\).

We have a coaction \(\rho _{\varDelta ^{T^{\mathfrak {o}}}}\) and coproduct \(\varDelta ^{T^{\mathfrak {o}}}\) for each ordered spanning tree \(T^{\mathfrak {o}}\) with a corresponding set \(\mathsf {Gal}(G,T_{\mathfrak {o}})\) for each.

We define

This gives rise to a corresponding matrix M G formed from \(M(G,T_{\mathfrak {o}})\) and corresponding coproduct and coaction Δ G.

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Kreimer, D. (2021). Outer Space as a Combinatorial Backbone for Cutkosky Rules and Coactions. In: Blümlein, J., Schneider, C. (eds) Anti-Differentiation and the Calculation of Feynman Amplitudes. Texts & Monographs in Symbolic Computation. Springer, Cham.

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