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# Exposing the threshold structure of loop integrals

Zeno Capatti<EMAIL_ADDRESS>Institute for Theoretical Physics, ETH Zürich,
Wolfgang-Pauli-Str. 27, 8093, Zürich

###### Abstract

The understanding of the physical laws determining the infrared behaviour of
amplitudes is a longstanding and topical problem. In this paper, we show that
energy conservation alone implies strong constraints on the threshold
singularity structure of Feynman diagrams. In particular, we show that it
implies a representation of loop integrals in terms of Fourier transforms of
non-simplicial convex cones. We then engineer a triangulation that has a
direct diagrammatic interpretation in terms of a straightforward edge-
contraction operation. We use it to develop an algorithmic procedure that
performs the Fourier integrations in closed form, yielding the novel Cross-
Free Family three-dimensional representation of loop integrals. Contrary to
the TOPT and LTD representations, its singularity structure is entirely and
elegantly expressed in terms of the graph-theoretic notions of connectedness
and crossing. These results can be used to classify infrared-finite scattering
theories, numerically evaluate loop integrals and to simplify threshold
regularisation procedures.

††preprint: APS/123-QED

##  I Introduction

The study of the threshold singularity structure of Feynman diagrams and
scattering amplitudes is an enduring effort that has, throughout the years,
lead to incredible developments that have shed light on the infrared physics
of quantum scattering phenomena (see Agarwal  _et al._ (2021) for a review and
Cutkosky (1960); Bloch and Nordsieck (1937); Jauch and Rohrlich (1954); Yennie
_et al._ (1961); Kinoshita (1962); Lee and Nauenberg (1964); Sterman (1978a,
b); Frye  _et al._ (2019); Agarwal  _et al._ (2021); Hannesdottir and Mizera
(2022); Hannesdottir and Schwartz (2019); Magnea (2021); Ma (2020); Erdoğan
and Sterman (2015); Akhoury  _et al._ (2011); Binoth and Heinrich (2000);
Libby and Sterman (1978); Collins (2020); Mizera and Telen (2022); Erdoğan
(2014); Heissenberg (2021); Caron-Huot  _et al._ (2018); Agarwal  _et al._
(2020); Catani and Seymour (1997); Catani (1998); Dixon  _et al._ (2008);
Becher and Neubert (2009a, b); Gardi and Magnea (2009); Dixon  _et al._
(2010); Ahrens  _et al._ (2012); Naculich  _et al._ (2013); Becher and Neubert
(2009c); Ferroglia  _et al._ (2009); Gardi  _et al._ (2010); Aguilera-Verdugo
_et al._ (2019); Doria  _et al._ (1980); Henn and Mistlberger (2016); Arkani-
Hamed  _et al._ (2021); Bern  _et al._ (1999); Dennen  _et al._ (2017); Caola
_et al._ (2021); Parisi (1990); Del Duca (1990); Laenen  _et al._ (2009);
Contopanagos and Einhorn (1992); Forde and Signer (2004); Coleman and Norton
(1965) for a historical selection of works on the topic).

Three-dimensional representations of Feynman diagrams, obtained via Time-
Ordered Perturbation Theory (TOPT) (see Sterman (1993, 1995) for a review),
via Flow-Oriented Perturbation Theory (FOPT) Borinsky  _et al._ (2022) or
through the Loop-Tree Duality (LTD) formalism Catani  _et al._ (2008);
Bierenbaum  _et al._ (2010); Runkel  _et al._ (2019, 2020); Capatti  _et al._
(2019); Tomboulis (2017), are famously apt to performing a systematic
singularity analysis of Feynman integrals. Indeed, the interplay of energy
conservation and residue theorem involved in their derivation makes their
singularities directly interpretable in terms of cuts Cutkosky (1960). This in
turn allows to leverage a host of graph-theoretical knowledge to perform a
diagrammatic study of the structure of physical thresholds. Even then, these
three-dimensional representations are plagued by spurious divergences,
corresponding to cuts that divide the graph in more than two connected
components (TOPT) or cuts containing particles that have both positive and
negative on-shell energy (LTD). Even improved LTD formalisms Capatti  _et al._
(2020a); Aguilera-Verdugo  _et al._ (2020); Capatti  _et al._ (2021); Jesús
Aguilera-Verdugo  _et al._ (2021); Sborlini (2021); Kromin  _et al._ (2022);
Bobadilla (2021); Ramírez-Uribe  _et al._ (2021, 2022) that remove spurious
singularities by relying on algebraic manipulations of the integrand or direct
ansatzes are either inadequate for wider generalisations or lack first-
principle justification.

In this paper, we derive the precise relationship between the singularity
structure of a Feynman integral and the graph-theoretic notions of crossing
and connectedness Abreu  _et al._ (2014); Bloch and Kreimer (2015); Arkani-
Hamed  _et al._ (2017); Abreu  _et al._ (2017); Capatti  _et al._ (2021);
Bourjaily  _et al._ (2021); Berghoff and Kreimer (2020); Kreimer (2021);
Kreimer and Yeats (2021); Benincasa  _et al._ (2020); Sborlini (2021);
Hannesdottir  _et al._ (2022), by explicitly constructing a three-dimensional
representation of loop integrals that manifests such properties.

We do so by exploiting methods that pertain to a recently growing branch of
literature that applies methods of convex geometry, and more precisely the
geometry of polytopes Hodges (2013); Arkani-Hamed  _et al._ (2016); Arkani-
Hamed and Trnka (2014); Arkani-Hamed  _et al._ (2015, 2017, 2022); Benincasa
_et al._ (2020); Benincasa and Bobadilla (2022); Borinsky (2020); Brown
(2015); Binoth and Heinrich (2000); Kaneko and Ueda (2010); Pak and Smirnov
(2011); Tourkine (2017); Franco  _et al._ (2014); Kreimer (2021); Kreimer and
Yeats (2021) and their Fourier transform Borinsky  _et al._ (2022), to
problems in high-energy physics. In particular, we will relate Feynman
diagrams to Fourier transforms of (non-simplicial) convex cones. We will then
perform the Fourier integration analytically using an identity with a
digrammatic interpretation in terms of a straightforward edge-contraction
procedure.

We provide an algorithm based on the recursive application of edge
contraction, resulting in the surprisingly compact and elegant Cross-Free
Family (CFF) representation of loop integrals. Each denominator is identified
with a cross-free family of connected subgraphs of the starting graph. Such
representation can be cast in factorised form and is especially suited for the
numerical evaluation of loop integrals. We provide in the mathematica package
cLTD.m 111 github.com/apelloni/cLTD a generic implementation of the algorithm
presented in this paper.

We finally describe the factorisation properties of the Fourier transform
which, as a result, provide a necessary condition for the non-integrability of
the intersection of any number of threshold singularities for QCD diagrams.
The CFF representation, together with the factorisation argument, de facto
provide a classification of the threshold singularity structure of Feynman
diagrams in terms of crossing and connectedness, a fundamental step in the
long-standing quest for the understanding of the infrared behaviour of Feynman
diagrams and amplitudes.

##  II Energy conservation

Consider a bridgeless digraph $G=(\mathcal{V},\mathcal{E})$ with underlying
undirected graph $G_{u}$, given as a tuple of a set of vertices $\mathcal{V}$
and a set of ordered pairs of vertices $\mathcal{E}$. To each edge
$e\in\mathcal{E}$ of the graph is associated a weight $x_{e}\in\mathbb{R}$,
collected in a vector
$\mathbf{x}=(x_{e})_{e\in\mathcal{E}}\in\mathbb{R}^{|\mathcal{E}|}$. The
standard scalar product of two weight vectors $\mathbf{x}$ and $\mathbf{y}$ is
defined as $\mathbf{x}\cdot\mathbf{y}=\sum_{e\in\mathcal{E}}x_{e}y_{e}$. The
component-wise multiplication of two vectors $\bm{\sigma}$ and $\mathbf{x}$ is
denoted as $\mathbf{x}\odot\bm{\sigma}=(\sigma_{e}x_{e})_{e\in\mathcal{E}}$.
Given any subset of the edges $\mathcal{E}^{\prime}\subseteq\mathcal{E}$, its
characteristic vector $\mathbf{1}^{\mathcal{E}^{\prime}}$ has components
$\mathbf{1}^{\mathcal{E}^{\prime}}_{e}=1$ if $e\in\mathcal{E}^{\prime}$ and
$\mathbf{1}^{\mathcal{E}^{\prime}}_{e}=0$ otherwise. The positive (negative)
boundary of a subset $S\subset\mathcal{V}$ (also called a _cut_ $S$), is
defined as

$\displaystyle\delta^{+}(S)=\\{e=\\{v,v^{\prime}\\}\,|\,v\in
S,v^{\prime}\notin S\\},$ (1)
$\displaystyle\delta^{-}(S)=\\{e=\\{v^{\prime},v\\}\,|\,v\in
S,v^{\prime}\notin S\\},$ (2)

and $\delta(S)=\delta^{+}(S)\cup\delta^{-}(S)$. Consider the space of weight
vectors satisfying energy-conservation constraints,
$\mathcal{Q}^{0}_{G}\subset\mathbb{R}^{|\mathcal{E}|}$. An element
$\mathbf{q}^{0}_{G}\in\mathcal{Q}^{0}_{G}$ satisfies energy conservation at
each vertex:

$\mathbf{q}^{0}_{G}\cdot(\mathbf{1}^{\delta^{+}(v)}-\mathbf{1}^{\delta^{-}(v)})=p_{v}^{0},\quad\forall
v\in\mathcal{V},$ (3)

for fixed vertex weights $\\{p_{v}^{0}\\}_{v\in\mathcal{V}}$ with
$\sum_{v\in\mathcal{V}}p_{v}^{0}=0$. In Feynman diagrams, $p_{v}^{0}$ is the
external momentum that injects into the vertex $v$, which may be chosen to be
vanishing. Elements of $\mathcal{Q}^{0}_{G}$ can be written in terms of
elements of a cycle basis $\mathcal{C}$ of the graph as

$\mathbf{q}^{0}_{G}(\\{k_{c}^{0}\\}_{c\in\mathcal{C}},\\{p_{v}^{0}\\}_{v\in\mathcal{V}})=\sum_{c\in\mathcal{C}}\mathbf{s}_{c}^{G}k_{c}^{0}+\mathbf{p}^{0}_{G}(\\{p_{v}^{0}\\}_{v\in\mathcal{V}}),$
(4)

with $L=|\mathcal{C}|=|\mathcal{E}|-|\mathcal{V}|+1$, and

$\mathbf{p}^{0}_{G}=\sum_{v\in\mathcal{V}}\mathbf{r}_{v}^{G}p_{v}^{0}.$ (5)

$\mathbf{s}^{G}_{c}$ is constructed in the following way: first,
$(\mathbf{s}^{G}_{c})_{e}=s^{G}_{ce}=0$ if and only if $e$ does not belong to
$c$. Second, $s^{G}_{ce}=1$ if $e$ belongs to the cycle $c$ and the cycle $c$
is oriented in $G$. Finally, $s^{G^{\prime}}_{ce}=-s^{G}_{ce}$, if the
orientation for the arc $e$ in $G$ is swapped, yielding a graph $G^{\prime}$.
The cycle vector so constructed is ambiguous up to a sign, which can be chosen
arbitrarily. Analogously, we have $r^{G^{\prime}}_{ve}=-r^{G}_{ve}$ (for the
purposes of this paper, this is all we need to know about
$\mathbf{r}_{v}^{G}$).

##  III Feynman diagrams as Fourier transforms of convex cones

A bridgeless Feynman diagram
$I_{G_{\text{u}}}(\mathbf{E},\\{p_{v}^{0}\\}_{v\in\mathcal{V}})$, stripped of
its spatial loop integrations, reads

$I_{G_{\text{u}}}=\int\left[\prod_{c\in\mathcal{C}}\frac{\mathrm{d}k_{c}^{0}}{2\pi}\right]\frac{\mathcal{N}_{G}(\mathbf{q}_{G}^{0})}{\prod_{e\in\mathcal{E}}((q_{e}^{0})^{2}-E_{e}^{2}+i\varepsilon)},$
(6)

where $q_{e}^{0}=(\mathbf{q}^{0}_{G})_{e}$ is the linear function of the loop
momenta given in eq. (4), for an arbitrarily chosen digraph $G$ with
underlying graph $G_{u}$, and $\mathcal{N}_{G}$ is a polynomial numerator.
$I_{G_{\text{u}}}$ is parametric in $\mathbf{E}$, which for applications in
QFT is the vector of on-shell energies assigned to the edges of the graph,
$E_{e}=\sqrt{|\vec{q}_{e}|^{2}+m_{e}^{2}}$. We introduce an auxiliary
integration $1=\int\mathrm{d}x_{e}\delta(x_{e}-q_{e}^{0})$ for each edge and
use the Fourier representation of the Dirac delta function, which yields

$I_{G_{u}}=\int\prod_{c\in\mathcal{C}}\frac{\mathrm{d}k_{c}^{0}}{2\pi}\int\mathrm{d}\mathbf{x}\mathrm{d}\bm{\tau}\,\frac{\mathcal{N}_{G}(\mathbf{x})e^{i\bm{\tau}\cdot(\mathbf{x}-\mathbf{q}^{0}_{G})}}{\prod_{e\in\mathcal{E}}2\pi(x_{e}^{2}-E_{e}^{2}+i\varepsilon)}.$
(7)

This step requires one auxiliary variable for each edge (as in the FOPT
derivation Borinsky  _et al._ (2022)) and not for each vertex (as in TOPT).
Residue integration in the variables $x_{e}$ appearing in the propagators of
eq. (7) is trivial, as they are not subject to energy conservation. Each
propagator contributes with two terms, associated to the sign of $\tau_{e}$
and corresponding to the poles $x_{e}=\pm E_{e}$. Thus, we express $I_{G_{u}}$
as a sum over $2^{|\mathcal{E}|}$ contributions

$I_{G_{u}}=\sum_{\bm{\sigma}\in\\{\pm
1\\}^{|\mathcal{E}|}}\frac{\mathcal{N}_{G}(\bm{\sigma}\odot\mathbf{E})}{\prod_{e\in\mathcal{E}}2iE_{e}}\hat{\mathds{1}}_{\bm{\sigma}}(\mathbf{E},\\{p_{v}^{0}\\}_{v\in\mathcal{V}}),$
(8)

where we introduced, with some fore-shadowing, the function
$\hat{\mathds{1}}_{\bm{\sigma}}(\mathbf{E},\\{p_{v}^{0}\\}_{v\in\mathcal{V}})$
defined as

$\displaystyle\hat{\mathds{1}}_{\bm{\sigma}}=\int$
$\displaystyle\prod_{c\in\mathcal{C}}\frac{\mathrm{d}k_{c}^{0}}{2\pi}\int\prod_{e\in\mathcal{E}}\mathrm{d}\tau_{e}e^{i\tau_{e}(\sigma_{e}E_{e}-q_{e}^{0})}\Theta(-\sigma_{e}\tau_{e}).$
(9)

Note that, in order for the residue at infinity to vanish, $\mathcal{N}$ must
be a polynomial in which the energy arguments appear with power at most one
222The same condition holds for the derivation of TOPT presented in Sterman
(1995, 1993). We observe that any diagram can itself be algebraically
decomposed into diagrams satisfying this numerator property. Changing
integration variables from $\tau_{e}$ to $-\sigma_{e}\tau_{e}$ gives

$\hat{\mathds{1}}_{\bm{\sigma}}=\int\prod_{c\in\mathcal{C}}\frac{\mathrm{d}k_{c}^{0}}{2\pi}\int_{\mathbb{R}_{+}^{|\mathcal{E}|}}\mathrm{d}\bm{\tau}e^{-i\bm{\tau}\cdot(\mathbf{E}-\bm{\sigma}\odot\mathbf{q}^{0}_{G})}.$
(10)

Plugging eq. (4) in $\bm{\tau}\cdot(\bm{\sigma}\odot\mathbf{q}^{0})$ we
factorise integration on $k_{c}^{0}$, which is then performed by using the
Fourier representation of the Dirac delta function, but in reverse,

$\hat{\mathds{1}}_{\bm{\sigma}}=\int_{\mathbb{R}_{+}^{|\mathcal{E}|}}\mathrm{d}\bm{\tau}e^{-i\bm{\tau}\cdot(\mathbf{E}-\bm{\sigma}\odot\mathbf{p}^{0}_{G})}\prod_{c\in\mathcal{C}}\delta\left(\bm{\tau}\cdot\left(\bm{\sigma}\odot\mathbf{s}_{c}^{G}\,\right)\right).$
(11)

The Dirac delta functions enforce the integration domain to be the
intersection of $|\mathcal{C}|$ hyperplanes with the positive orthant
$\mathbb{R}_{+}^{|\mathcal{E}|}$. Observe that, by definition,
$\bm{\sigma}\odot\mathbf{s}_{c}^{G}=\mathbf{s}_{c}^{G^{\prime}}$,
$\bm{\sigma}\odot\mathbf{p}^{0}_{G}=\mathbf{p}^{0}_{G^{\prime}}$ and
$\mathcal{N}_{G^{\prime}}(\mathbf{E})=\mathcal{N}_{G}(\bm{\sigma}\odot\mathbf{E})$
where $G^{\prime}$ is the graph obtained from $G$ by swapping the orientation
of the edge $e$ if $\sigma_{e}=-1$. We can thus substitute the sum over
$\bm{\sigma}$ with a sum over digraphs. Let us also observe that, if the graph
has an oriented cycle $c$, then we must have
$\bm{\tau}\cdot\mathbf{s}_{c}^{G^{\prime}}=\sum_{e\in c}\tau_{e}=0$ in virtue
of the Dirac deltas of eq. (11). Since $\tau_{e}>0$ for any $e\in\mathcal{E}$,
this implies that the integration domain for the Fourier transform is empty.
Consequently, the orientations $G$ that contribute with non-vanishing Fourier
transform correspond to directed acyclic graphs. In summary, let
$\text{dag}(G_{u})$ be the set of all acyclic digraphs with underlying graph
$G_{u}$, so that

$I_{G_{u}}=\sum_{G\in\text{dag}(G_{u})}\frac{\mathcal{N}_{G}(\mathbf{E})}{\prod_{e\in\mathcal{E}}2iE_{e}}\hat{\mathds{1}}_{\mathcal{K}_{G}},$
(12)

with

$\hat{\mathds{1}}_{\mathcal{K}_{G}}=\int_{\mathcal{K}_{G}}\mathrm{d}\bm{\tau}e^{-i\bm{\tau}\cdot(\mathbf{E}-\mathbf{p}_{G}^{0})},$
(13)

where $\hat{\mathds{1}}_{\mathcal{K}_{G}}$ is the Fourier transform of the
characteristic function $\mathds{1}_{\mathcal{K}_{G}}$ of the cone
$\mathcal{K}_{G}$, defined as

$\displaystyle\mathcal{K}_{G}=\left\\{\bm{\tau}\in\mathbb{R}_{+}^{|\mathcal{E}|}\
\Big{|}\ \bm{\tau}\cdot\mathbf{s}_{c}^{G}=0,\ \forall
c\in\mathcal{C}\right\\}.$ (14)

Only acyclic digraphs contribute to the sum. This is analogous to TOPT,
expressed as a sum over vertex orderings, or _topological orderings_ in
technical jargon. To each topological ordering corresponds a unique acyclic
orientation (but not viceversa). The relevance of acyclic graphs has also been
recognised in the context of causal representations Sborlini (2021). In FOPT
Borinsky  _et al._ (2022), a dual phenomenon happens: in that case, the three-
dimensional representation is expressed as a sum over strongly-connected
digraphs.

##  IV  Triangulations and the edge-contraction operation

Before engineering a triangulation of the convex cones $\mathcal{K}_{G}$, let
us discuss the case in which $G$ is a multigraph, i.e. it has multiple edges
connecting the same two vertices. Let ${e_{1},...,e_{n}}$ be a set of edges
that connect the same two vertices, $v$ and $v^{\prime}$. Let $G^{\prime}$ be
the graph in which all such edges have been substituted by a unique edge $e$.
Then

$\hat{\mathds{1}}_{\mathcal{K}_{G}}(\\{E_{e}\\}_{e\in\mathcal{E}})=\hat{\mathds{1}}_{\mathcal{K}_{G^{\prime}}}(\\{E_{e}\\}_{e\in\mathcal{E}^{\prime}})\big{|}_{E_{e}=\sum_{j=1}^{n}E_{j}}.$
(15)

In plain words, parallel edges are substituted with a unique edge whose energy
equals the sum of their energies. This replacement will result in a
significant simplification of the combinatorial factors involved in the
computation of $\hat{\mathds{1}}_{\mathcal{K}_{G}}$. This property of the
Fourier transform of the cone $\mathcal{K}_{G}$ mirrors an analogous treatment
of multigraphs presented in other works Bloch and Kreimer (2015); Kreimer
(2021); Sborlini (2021).

We now present the fundamental relation required to triangulate the cone
$\mathcal{K}_{G}$. Let $G$ be a simple digraph and let us consider a cut
$S\subset V$. Furthermore, let us impose that $\delta(S)=\delta^{+}(S)$. The
existence of such a cut is guaranteed by the acyclic property. Then

$\hat{\mathds{1}}_{\mathcal{K}_{G}}=\frac{1}{i(\mathbf{E}-\mathbf{p}_{G}^{0})\cdot\bm{1}^{\delta(S)}}\sum_{a\in\delta(S)}\hat{\mathds{1}}_{\mathcal{K}_{G_{a}}}.$
(16)

where $G_{a}=(\mathcal{V}_{a},\mathcal{E}_{a})$ is the graph obtained from $G$
by contracting the edge $a$. Note that
$\mathbf{p}_{G}^{0}\cdot\mathbf{1}^{\delta(S)}=\sum_{v\in S}p_{v}^{0}$ if
$\delta^{+}(S)=\delta(S)$. Comparison with general formulas for the Fourier
transform of cones Robins  establishes that $\bm{1}^{\delta(S)}$ is one of the
edge vectors emanating from the vertex of the cone. Eq. (16) is readily
derived given that

$\prod_{e\in\delta(S)}\Theta(\tau_{e})=\sum_{a\in\delta(S)}\Theta(\tau_{a})\prod_{e\in\delta(S)\setminus\\{a\\}}\Theta(\tau_{e}-\tau_{a}),$
(17)

which expresses the triangulation of the positive orthant
$\mathbb{R}_{+}^{|\delta(S)|}$ in $|\delta(S)|$ cones. Once substituted in
$\hat{\mathds{1}}_{\mathcal{K}_{G}}$, we obtain a sum of $|\delta(S)|$
integrals, labelled by an index $a\in\delta(S)$. We then perform the change of
variables $\tau_{e}^{\prime}=\tau_{e}-\tau_{a}$,
$e\in\delta(S)\setminus\\{a\\}$ for the integral labelled by $a$. This
factorises integration over $\tau_{a}$ completely, which is then trivially
performed, yielding eq. (16).

##  V Applications

###  V.1 Singularities, connectedness and crossing

$v_{1}$

$v_{4}$

$v_{5}$

$v_{2}$

$v_{3}$

$\displaystyle{=\frac{-i}{E_{1}+E_{2}+E_{3}+p_{1}^{0}}}\Bigg{[}$

$v_{12}$

$v_{4}$

$v_{5}$

$v_{3}$

$+$

$v_{15}$

$v_{4}$

$v_{2}$

$v_{3}$

$+$

$v_{13}$

$v_{4}$

$v_{5}$

$v_{2}$

$\Bigg{]}$ $\displaystyle{=\frac{-i}{E_{1}+E_{2}+E_{3}+p_{1}^{0}}}$

$v_{15}$

$v_{4}$

$v_{2}$

$v_{3}$

$\displaystyle{=\frac{-i}{E_{1}+E_{2}+E_{3}+p_{1}^{0}}}\frac{-i}{E_{3}+E_{5}+E_{7}-p_{3}^{0}}\Bigg{[}$

$v_{135}$

$v_{4}$

$v_{2}$

$\displaystyle{+}$

$v_{15}$

$v_{23}$

$v_{4}$

$\displaystyle{+}$

$v_{15}$

$v_{34}$

$v_{2}$

$\Bigg{]}$

$\displaystyle{=\frac{-i}{E_{1}+E_{2}+E_{3}+p_{1}^{0}}}\frac{-i}{E_{3}+E_{5}+E_{7}-p_{3}^{0}}\frac{-i}{E_{2}+E_{3}+E_{4}+E_{6}+p_{1}^{0}+p_{5}^{0}}\Bigg{[}$

$v_{154}$

$v_{23}$

$+$

$v_{1235}$

$v_{4}$

$+$

$v_{125}$

$v_{34}$

$+$

$v_{1534}$

$v_{2}$

$\Bigg{]}$

$\displaystyle{=\frac{-i}{E_{1}+E_{2}+E_{3}+p_{1}^{0}}}\frac{-i}{E_{3}+E_{5}+E_{7}-p_{3}^{0}}\frac{-i}{E_{2}+E_{3}+E_{4}+E_{6}+p_{1}^{0}+p_{5}^{0}}\Bigg{[}\frac{-i}{E_{2}+E_{3}+E_{4}+E_{7}-p_{2}^{0}-p_{3}^{0}}+\frac{-i}{E_{3}+E_{5}+E_{6}-p_{3}^{0}-p_{4}^{0}}\Bigg{]}$

Figure 1:  Graphical depiction of the recursion on the graph, with edges
labelled $e_{1}=\\{v_{1},v_{5}\\},\ e_{2}=\\{v_{1},v_{2}\\},\
e_{3}=\\{v_{1},v_{3}\\},\ e_{4}=\\{v_{2},v_{5}\\},\ e_{5}=\\{v_{2},v_{3}\\},\
e_{6}=\\{v_{5},v_{4}\\},\ e_{7}=\\{v_{3},v_{4}\\}$. The cross-free families
generated for this acyclic graph are $\mathcal{F}_{G}=\\{F_{1},F_{2}\\}$ with
$F_{1}=\\{\\{v_{1}\\},\\{v_{3}\\},\\{v_{1},v_{5}\\},\\{v_{1},v_{5},v_{4}\\}\\}$
and
$F_{2}=\\{\\{v_{1}\\},\\{v_{3}\\},\\{v_{1},v_{5}\\},\\{v_{1},v_{2},v_{5}\\}\\}$.
At the first step, we choose the source $v_{1}$, and obtain three graphs
obtained by contracting the edges adjacent to $v_{1}$. The second equality is
obtained by observing that two of the graphs have directed cycles (highlighted
in blue), so the Fourier transform of their associated cone evaluates to zero.
The third equality is obtained by now choosing the sink $v_{3}$, which yields
three graphs, of which two are acyclic. Observe that we contract all parallel
edges that connect two vertices. The fourth equality is obtained by choosing
the source $v_{15}$ for the two acyclic graphs. Each yields two contracted
graphs, for a total of two acyclic graphs. The final equality is obtained by
choosing the sources $v_{154}$ and $v_{125}$ for the two acyclic graphs
obtained at the previous step.

The contraction operation allows to recursively perform all the Fourier
integrations defining $\hat{\mathds{1}}_{\mathcal{K}_{G}}$, a procedure that
is equivalent to finding a special triangulation of the convex cone
$\mathcal{K}_{G}$. Such a triangulation is different, for example, than the
one needed to obtain the TOPT representation. The specific triangulation we
choose makes the relationship between crossing, connectedness and the
threshold singularity structure of Feynman diagrams manifest. A large body of
recent works Abreu  _et al._ (2014); Bloch and Kreimer (2015); Arkani-Hamed
_et al._ (2017); Abreu  _et al._ (2017); Capatti  _et al._ (2021); Bourjaily
_et al._ (2021); Berghoff and Kreimer (2020); Kreimer (2021); Kreimer and
Yeats (2021); Benincasa  _et al._ (2020); Sborlini (2021); Hannesdottir  _et
al._ (2022) refers to this correspondence with varying degree of rigour and in
separate contexts, which motivates looking for an effective and concise
description of it. In particular, we will provide a combinatorial and generic
construction procedure for the _Cross-Free Family representation_ of Feynman
integrals that was conjectured in eq. (3.44) of Capatti  _et al._ (2021).

First, given a simple digraph $G$ with vertices labelled as
$\mathcal{V}=\\{v_{I_{1}},...,v_{I_{n}}\\}$ for given subsets
$I_{1},...,I_{n}\subseteq\mathcal{X}$ of a ground set $\mathcal{X}$ of
discrete elements, and given an edge $a=\\{v_{I_{i}},v_{I_{j}}\\}$ to be
contracted, we denote with

$G_{a}=(\mathcal{V}\setminus\\{v_{I_{i}},v_{I_{j}}\\}\cup\\{v_{I_{i}\cup
I_{j}}\\},\mathcal{E}\setminus\\{a\\}).$ (18)

the contracted graph. This notation allows to map quantities in the contracted
graph to quantities in the original graph. Then given, as an input, the
simple, acyclic graph $G_{i}$ and the family
$F_{i}\subset\mathcal{P}(\mathcal{X})$ ($\mathcal{P}(\mathcal{X})$ being the
powerset of $\mathcal{X}$):

  * $\bullet$

Find a source or a sink $v_{I}$ of the graph $G_{i}$ such that
$(\mathcal{V}_{i}\setminus\\{v_{I}\\},\mathcal{E}_{i}\setminus\delta(v_{I}))$
is a connected graph. The existence of such a vertex is guaranteed by the
acyclic property of the graph $G_{i}$. Let $F_{i+1}=F_{i}\cup\\{I\\}$.

  * $\bullet$

Let $\\{(G_{i})_{a}\\}_{a\in\delta(v)}$ be the collection of graphs contracted
according to the rule of eq. (18).

  * $\bullet$

For each of the graphs $(G_{i})_{a}$, if the graph is not acyclic, terminate
and output nothing. If the graph $(G_{i})_{a}$ consists of a single vertex,
terminate and output $F_{i+1}$. If none of the two applies, fuse all parallel
edges, as described in eq. (15), so that the resulting graph is also simple,
and iterate.

This algorithmic procedure is guaranteed to end with a graph that consists of
a single vertex. The starting graph $G_{\text{init}}$ is defined to have the
set of vertices $\mathcal{V}_{\text{init}}=\\{v_{\\{1\\}},...,v_{\\{n\\}}\\}$,
and $F_{\text{init}}=\emptyset$. It follows that, at each iteration, the set
$I$, as well as any set in $F_{i}$, can be mapped to a subset of
$\mathcal{V}_{\text{init}}$. The output of the algorithm is a collection of
families $\mathcal{F}_{G}$, dependent on the choice of source or sink at each
iteration. Each family $F_{\text{out}}\in\mathcal{F}_{G}$ is a collection of
cuts $S\in F_{\text{out}}$, themselves collections of vertices
$S\subset\mathcal{V}_{\text{init}}$.

The families of cuts $F_{\text{out}}=\\{S_{1},...,S_{|V|-1}\\}$ obtained in
such a way satisfy the following properties:

  1. a1)

$S_{i}$ is connected

  2. a2)

$V\setminus S_{i}$ is connected

  3. b)

$F_{\text{out}}$ is a laminar family, that is for any two sets
$S,S^{\prime}\in F_{\text{out}}$ of the family, either $S$ and $S^{\prime}$
are contained one in the other, or $S\cap S^{\prime}=\emptyset$.

  4. c)

$F_{\text{out}}$ is obstruction-free, that is no element of the family can be
written as a union of sets contained within it.

Given the set $\mathcal{F}_{G}$, we are able to evaluate the Fourier transform
of the cone $\mathcal{K}_{G}$. The algorithm presented above effectively
performs diagrammatically the triangulation of the cone $\mathcal{K}_{G}$ in
$|\mathcal{F}_{G}|$ simplicial cones, with each cross-free family
$F\in\mathcal{F}_{G}$ representing a simplicial cone. Using it, one can
evaluate all Fourier integrations and obtain the CFF representation

$\definecolor{cb@saved}{rgb}{0,0,0}\color[rgb]{0,0.8,0.2}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.8,0.2}\boxed{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\,I_{G_{u}}=\sum_{G\in\text{dag}(G_{u})}\sum_{F\in\mathcal{F}_{G}}\frac{i^{L}(\prod_{e\in\mathcal{E}}2E_{e})^{-1}\mathcal{N}_{G}(\mathbf{E})}{\prod_{S\in
F}(\mathbf{p}_{G}^{0}-\mathbf{E})\cdot\bm{1}^{\delta(S)}},\,}$ (19)

which provides a proof by construction of the representation conjectured in
Capatti  _et al._ (2021). We provide in the mathematica package cLTD.m 333
github.com/apelloni/cLTD a generic implementation of this algorithmic
procedure, resulting in a ready-to-evaluate integrand.

###  V.2 Diagram-level factorisation and iterated connectedness

Factorisation formulas for Fourier transforms of cones and polytopes can be
used to effectively study the leading behaviour of the integrand in singular
limits Borinsky  _et al._ (2022). In particular, the Fourier transform
$\hat{\mathds{1}}_{\mathcal{K}_{G}}$ satisfies a factorisation formula. Let us
consider a _connected_ cut $S$ with _connected_ complement and such that
$\delta(S)=\delta^{+}(S)$, let
$(\mathbf{E}-\mathbf{p}^{0}_{G})\cdot\bm{1}^{\delta(S)}=\epsilon$ and consider
the leading contribution to
$\hat{\mathds{1}}_{\mathcal{K}_{G}}(\\{p_{v}^{0}\\}_{v\in\mathcal{V}})$ in the
expansion in $\epsilon$, which is

$\hat{\mathds{1}}_{\mathcal{K}_{G}}=\frac{\hat{\mathds{1}}_{\mathcal{K}_{G_{1}}}(\\{p_{v}^{G_{1}}\\}_{v\in\mathcal{V}_{1}})\hat{\mathds{1}}_{\mathcal{K}_{G_{2}}}(\\{p_{v}^{G_{2}}\\}_{v\in\mathcal{V}_{2}})}{\epsilon}+o(1).$
(20)

$p_{v}^{G_{i}}$, $v\in\mathcal{V}_{i}$, $i=1,2$, are the capacities for the
graphs $G_{1}=(\mathcal{V}_{1}=S,\mathcal{E}_{1})$ and
$G_{2}=(\mathcal{V}_{2}=\mathcal{V}\setminus S,\mathcal{E}_{2})$, obtained
from $G$ by deleting the edges in $\delta(S)$. The external energies for the
two graphs are defined as follows: if, for a given edge $e$, we let
$v_{e}^{(1)}$ be its departing vertex and $v_{e}^{(2)}$ its arriving one, then

$p^{G_{i}}_{v}=\begin{cases}p_{v}^{0}+(-1)^{i}(E_{e}-p_{e}^{0})\quad\text{if
}v=v_{e}^{(i)},\ e\in\delta(S)\\\ p_{v}^{0}\quad\text{otherwise}\end{cases}.$
(21)

Eq. (20) is readily obtained by direct evaluation of the integrals in the
variables $\tau_{e}$, $e\in\delta(S)$ at leading order in $\epsilon$.
Importantly, if the cut $S$ is disconnected or has disconnected complement,
then $\mathds{1}_{\mathcal{K}_{G}}=o(1)$.

Given eq. (20), we now iterate the argument. The singularities of the two
graphs $G_{1}$ and $G_{2}$ correspond to connected cuts such that the
complement is also connected. Furthermore, these singularities must correspond
to singularities of the original graph $G$, evaluated at
$(\mathbf{E}-\mathbf{p}^{0}_{G})\cdot\mathbf{1}^{\delta(S)}=0$. Thus, let us
consider a cross-free family $F$ of cuts, and let
$(\mathbf{E}-\mathbf{p}^{0}_{G})\cdot\mathbf{1}^{\delta(S)}=o(\epsilon)$, for
all $S\in F$, and
$(\mathbf{E}-\mathbf{p}^{0}_{G})\cdot\mathbf{1}^{\delta(S)}=o(1)$ for $S\notin
F$. By iterating the factorisation argument, we obtain that a necessary and
sufficient condition for

$\hat{\mathds{1}}_{\mathcal{K}_{G}}(\mathbf{E},\\{p_{v}^{0}\\}_{v\in\mathcal{V}})=\frac{w(\mathbf{E},\\{p_{v}^{0}\\}_{v\in\mathcal{V}})}{\epsilon^{|F|}}+o(\epsilon^{-|F|+1})$
(22)

to hold with a non-vanishing function $w$ is that the cuts in $F$ divide the
graph in $|F|+1$ connected components, which is the lowest possible value. We
then say that $F$ satisfies the _iterated connectedness_ property.

$v_{1}$

$v_{2}$

$v_{5}$

$v_{4}$

$v_{3}$

$=$ $+$ $+$

Figure 2:  A diagrammatic way to represent the result of performing by
contraction the integrals of $\mathds{1}_{\mathcal{K}_{G}}$ is to draw the
resulting cross-free families. The three diagrams that are obtained as a
result of the triangulation correspond to the cross-free families
$F_{1}=\\{\\{v_{5}\\},\\{v_{1},v_{5}\\},\\{v_{1},v_{4},v_{5}\\},\\{v_{2}\\}\\}$,
$F_{2}=\\{\\{v_{5}\\},\\{v_{4},v_{5}\\},\\{v_{1},v_{2}\\},\\{v_{2}\\}\\}$,
$F_{3}=\\{\\{v_{5}\\},\\{v_{2}\\},\\{v_{2},v_{3}\\},$
$\\{v_{1},v_{2},v_{3}\\}\\}$, respectively. Each of the three terms scales
like $\epsilon^{-2}$ in the limit in which the two thresholds $\\{v_{5}\\}$
and $\\{v_{2}\\}$ vanish. However, their sum scales like $\epsilon^{-1}$ since
they divide the graph in four connected components instead of three, in
agreement with the bound of sect. V.2.

##  VI Outlook

Let us summarise our findings and their implications. In sect. V.1 we found a
new three-dimensional representation of Feynman diagrams, called the _cross-
free family representation_ , that is free of spurious singularities and in
which the relation between thresholds and connected subgraphs as well as the
relation between intersections of thresholds and crossing of subgraphs are
manifest. In sect. V.2, we derived an upper bound on the scaling of three-
dimensional representations close to the intersection of a number of
thresholds. When combined with a simple scaling argument, the bound determines
that if $n$ thresholds of a QCD diagram intersect at a point, then the
singularity at that point is non-integrable only if the corresponding cuts
divide the graph in $n+1$ connected components. This principle may be used to
drastically simplify regularisation procedures of pinched and non-pinched
thresholds Anastasiou and Sterman (2019); Anastasiou  _et al._ (2021); Ma
(2020); Capatti  _et al._ (2020b); Kermanschah (2022); Becker  _et al._ (2010,
2012); Becker and Weinzierl (2012); Gong  _et al._ (2009); Buchta  _et al._
(2017); Soper (2000).

We conclude by the discussing the deep implications of the factorisation
formula of sect. V.2 on the KLN-based cancellation pattern Kinoshita (1962);
Lee and Nauenberg (1964); Frye  _et al._ (2019); Hannesdottir and Schwartz
(2019); Capatti  _et al._ (2021); Hannesdottir and Mizera (2022); Khalil and
Horowitz (2017a); Capatti  _et al._ (2022); Khalil and Horowitz (2017b);
Blažek and Maták (2021); Lavelle and McMullan (2006); Akhoury  _et al._
(1997); Sborlini  _et al._ (2016); Gómez  _et al._ (2018); Carney  _et al._
(2017); Melnikov and Serbo (1997); Di’Lieto  _et al._ (1981); Frazer and van
Hove (1958); Carneiro  _et al._ (1981); Kinoshita (1950); Muta and Nelson
(1982); Axelrod and Nelson (1985) of infrared singularities. One important
consequence is the following: consider the quantity

$P=\sum_{n,m=0}^{\infty}a_{nm}\mathrm{Tr}[\,\hat{\bm{\mathrm{\rho}}}\,\hat{\mathbf{S}}_{\mathrm{c}}^{n}\,\hat{\bm{\mathrm{P}}}\,(\hat{\mathbf{S}}_{\mathrm{c}}^{\dagger})^{m}],$
(23)

where $\hat{\bm{\mathrm{\rho}}}$ and $\hat{\bm{\mathrm{P}}}$ are matrices in
the Fock space of the free theory that describe the density of initial states
and measurement respectively. They both are diagonal in the momenta and
quantum numbers of the single particle states. $\hat{\mathbf{S}}_{\mathrm{c}}$
is the connected scattering matrix which schematically takes the form

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with an implicit sum over all possible ways to partition $\bm{\alpha}$ and
$\bm{\beta}$ into spectator and scattering states. $P$ is IR-finite for any
value of the coefficients $a_{nm}$, given IR-safety contraints on
$\hat{\bm{\mathrm{\rho}}}$ and $\hat{\bm{\mathrm{P}}}$. This agrees with the
principle implied by eq. (22), that KLN cancellations happen between
interference diagrams that, after the deletion of the cut edges, have the same
amount of connected components.

##  VII Acknowledgments

I would like to thank Michael Borinsky, Eric Laenen and Alex Salas-Bernádez as
my collaboration with them inspired me with some of the proof techniques used
in this paper. I am also grateful to Valentin Hirschi and Ben Ruijl for
countless discussions on the topic, and for their help with the testing of the
cross-free family representation as well as with its implementation. I would
also like to thank Babis Anastasiou, Hofie Hannesdottir, Dario Kermanschah and
Andrea Pelloni for interesting exchanges on the topic, and Michael Borinsky,
Valentin Hirschi, Lucien Huber and Ben Ruijl for their comments on a draft of
this paper. This project has received funding from ETH Zürich under grant
agreement ETH-53 19-2.

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