# Geometrical approach to causality in multi-loop amplitudes

German F. R. Sborlini a<EMAIL_ADDRESS>a Deutsches Elektronen-
Synchrotron DESY, Platanenallee 6, D–15738 Zeuthen, Germany.

###### Abstract

An impressive effort is being placed in order to develop new strategies that
allow an efficient computation of multi-loop multi-leg Feynman integrals and
scattering amplitudes, with a particular emphasis on removing spurious
singularities and numerical instabilities. In this article, we describe an
innovative geometric approach based on graph theory to unveil the causal
structure of any multi-loop multi-leg amplitude in Quantum Field Theory. Our
purely geometric construction reproduces faithfully the manifestly causal
integrand-level behaviour of the Loop-Tree Duality representation. We found
that the causal structure is fully determined by the _vertex matrix_ , through
a suitable definition of connected partitions of the underlying diagrams.
Causal representations for a given topological family are obtained by summing
over subsets of all the possible causal entangled thresholds that originate
connected and oriented partitions of the underlying topology. These results
are compatible with Cutkosky rules. Moreover, we found that diagrams with the
same number of vertices and multi-edges exhibit similar causal structures,
regardless of the number of loops.

###### pacs:

11.10.Gh, 11.15.Bt, 12.38.Bx

††preprint: DESY 21-017

##  I Introduction

Nowadays, one of the most successful descriptions of Nature is based on
Quantum Field Theories (QFT). Impressive calculations are being performed in
order to extract highly-precise theoretical predictions, which must be
confronted with the highly-accurate data collected from the current and future
experiments Abada  _et al._ (2019a, b, c, d); Blondel  _et al._ (2019);
Bambade  _et al._ (2019); Roloff  _et al._ (2018); Dong  _et al._ (2018). Any
tiny discrepancy in the theory-experiment comparison might open the path to
huge discoveries regarding the fundamental building blocks of the Universe.
Moreover, the challenge to achieve further improvements in the computational
techniques behind QFT is originating several profound discoveries about the
underlying mathematical structures of gauge theories.

Due to the high-complexity of gauge theories and QFT in general, exact
solutions are unknown for most of the experimentally relevant observables. In
the context of high-energy physics, the perturbative approach turns out to be
the most suitable, since it allows to express experimentally accessible
quantities (such as cross-sections) in terms of power series: to increase the
precision of the results, higher-orders must be included. These higher order
contributions involve dealing with vacuum quantum fluctuations which are
encoded through complicated multi-loop multi-leg Feynman diagrams.

In the last twenty years, there was an enormous progress towards more
efficient loop calculations. Several techniques were developed, such as sector
decomposition Binoth and Heinrich (2000); Smirnov and Tentyukov (2009); Carter
and Heinrich (2011); Borowka  _et al._ (2018), Mellin-Barnes transformations
Blumlein (2000); Anastasiou and Daleo (2006); Bierenbaum  _et al._ (2006);
Gluza  _et al._ (2007); Freitas and Huang (2010); Dubovyk  _et al._ (2016),
algebraic reduction of integrands Mastrolia and Ossola (2011); Badger  _et
al._ (2012); Zhang (2012); Mastrolia  _et al._ (2012, 2013); Ita (2015);
Mastrolia  _et al._ (2016); Ossola  _et al._ (2007), integration-by-parts
Chetyrkin and Tkachov (1981); Laporta (2000), semi-numerical integration
Moriello (2020); Bonciani  _et al._ (2020); Czakon (2008), among other highly
creative and powerful ideas Gnendiger  _et al._ (2017); Heinrich (2020);
Torres Bobadilla  _et al._ (2020). In this direction, the Loop-Tree Duality
(LTD) Catani  _et al._ (2008); Bierenbaum  _et al._ (2010, 2013); de Jesús
Aguilera-Verdugo  _et al._ (2021) constitutes a novel strategy to tackle
higher-order calculations by opening loops into trees, thus recasting the
virtual states into configurations that resemble real-radiation processes.

The purpose of LTD is twofold. On one side, expressing the virtual and real-
radiation contributions on similar integration spaces allows to infer a
natural way to combine them at integrand-level. This unified formalism
produces an integrand-level representation of physical observables which is
locally free of infrared singularities Hernandez-Pinto  _et al._ (2016);
Sborlini (2016); Sborlini  _et al._ (2016a, b). On the other hand, there are
tremendous simplifications in the description of the causal and singular
structure of multi-loop Feynman integrals and scattering amplitudes, that
leads to a more compact and numerically stable representation of the loop
integrands in the Euclidean space of the loop three-momenta Buchta  _et al._
(2014); Buchta (2015); Buchta  _et al._ (2017). Several studies that took
advantage of a simplified treatment of singularities within LTD were carried
out Driencourt-Mangin  _et al._ (2018); Plenter (2019); Plenter and Rodrigo
(2020); Driencourt-Mangin (2019); Driencourt-Mangin  _et al._ (2019).
Regarding the causal structure of scattering amplitudes, there are previous
studies using different techniques Tomboulis (2017); Runkel  _et al._ (2019a,
b). Very recently, LTD was applied to remove unphysical threshold
singularities and obtain a manifestly causal integrand-level definition of
multi-loop scattering amplitudes Aguilera-Verdugo  _et al._ (2019); Capatti
_et al._ (2020a, 2019); Aguilera-Verdugo  _et al._ (2020a); Ramírez-Uribe  _et
al._ (2021a); Aguilera-Verdugo  _et al._ (2020b, 2021); Capatti  _et al._
(2020b). All these LTD-based techniques were implemented in an automatized
framework Torres Bobadilla (2021a).

It is known that geometry and graph theory can be used to re-interpret the
physical meaning of scattering amplitudes. Moreover, Cutkosky rules Cutkosky
(1960); Bloch and Kreimer (2015) and Steinmann relations Cahill and Stapp
(1975, 1973); Caron-Huot  _et al._ (2016) establish a deep connection among
geometrical properties of Feynman diagrams (cuts or partitions) and the
structure of discontinuities of the underlying amplitude. Inspired by these
ideas, we investigated similar ideas with the purpose of reconstructing the
whole amplitude at integrand-level using a manifestly causal representation.
Even more, in a recent article, we explored the application of novel quantum
algorithms to efficiently detect causal configurations in multi-loop diagrams,
by identifying acyclic graphs with Grover’s algorithm Ramírez-Uribe  _et al._
(2021b).

The outline of this article is the following. In Sec. II, we briefly recall
the basic ideas behind the Loop-Tree Duality theorem, and we introduce its
connection with the causal structures in multi-loop multi-leg scattering
amplitudes. Then, in Sec. III, we establish the geometrical concepts required
to describe Feynman integrals and multi-loop amplitudes. Introducing the
concepts of multi-edges, vertices and the vertex matrix, we explain how to
generate all the possible causal propagators involved in their causal
representations in Sec. III.1. After that, in Sec. IV, we describe a set of
geometrical rules to unveil the causal structure of any amplitude. These rules
explain how to combine compatible causal propagators, associated to different
thresholds, leading to the concept of _compatible causal entangled
thresholds_. We present a detailed example based on four-vertex topologies in
Sec. V. A discussion about more complicated configurations, including the
causal structure of $N$-vertex topologies at one-loop, is given in Sec. VI.
Finally, the conclusions and outlook are presented in Sec. VII.

##  II Loop-Tree Duality and Causality

In order to provide a proper description of multi-loop multi-leg scattering
amplitudes, it is mandatory to identify and classify the kinematical variables
involved. So, let’s consider a generic $L$-loop $N$-point amplitude. In first
place, we define $L$ primitive loop momenta $\\{\ell_{i}\\}_{i=1,\ldots,L}$
which correspond to the integration variables.

Then, we group the momenta of the internal lines, $I$, associated to a Feynman
diagram (or topology) into $n$ sets, according to their dependence on the
primitive variables. In this way, the set $s$ contains all the internal
momenta of the form

$q_{i_{s}}=\sum_{j}\,\beta_{j}^{s}\ell_{j}\,+\,k_{i_{s}}\,,$ (1)

where $k_{i_{s}}$ represents a linear combination of external momenta
$\\{p_{r}\\}_{r=1,\ldots,N}$ and $\beta_{j}^{s}\in\\{-1,0,1\\}$. The linear
combination of primitive momenta $\beta_{j}^{s}\ell_{j}$ remains fixed for
each $i_{s}\in s$. Here, external momenta are considered outgoing and the
short-hand notation $q_{i}\equiv i$ is used when there is only one element per
set (i.e. $\\#s=1$ for every $s$).

Figure 1: General examples of MLT (a), NMLT (b) and N2MLT (c) topologies. We
indicate with a single line each of the the sets defined by Eq. (1). As
explained in the text, we can attach an arbitrary number of external particles
to each line, which corresponds to enlarging the sets $s$ by adding more
propagators on them.

At this point, we introduce the concept of _Maximal Loop Topology_ (MLT),
which describes those diagrams or families of diagrams with the minimal number
of sets for a given number of loops; i.e. $n-L=1$. This naturally defines a
topological classification of diagrams through the so-called topological
complexity, $\tilde{k}\equiv n-L$, as previously discussed in Refs. Aguilera-
Verdugo  _et al._ (2020a, 2021); Ramírez-Uribe  _et al._ (2021a); Jesús
Aguilera-Verdugo  _et al._ (2021). In this way, MLT corresponds to
$\tilde{k}=1$; the Next-to-Maximal Loop Topology (NMLT) to $\tilde{k}=2$, and
so on. In Fig. 1, we show generic examples of MLT (left), NMLT (center) and
N2MLT (right) diagrams. It is worth appreciating that the lines drawn in the
diagrams represent sets, which might contain several external particles
attached to them.

Once the notation has been established, we can use it to write any $L$-loop
$N$-point diagram in the Feynman representation as

$\mathcal{A}_{N}^{\left(L\right)}=\int_{\ell_{1},\ldots,\ell_{L}}\,\mathcal{N}\left(\left\\{\ell_{i}\right\\}_{L},\left\\{p_{j}\right\\}_{N}\right)\times
G_{F}\left(1,\ldots,n\right)\,,$

where $n=L+\tilde{k}$ is the total number of momentum sets, $\tilde{k}$ is the
corresponding topological complexity of the diagram and ${\cal N}$ represents
a generic numerator that depends on all the kinematical variables, i.e. any
possible scalar product involving $\\{p_{j}\\}$ and $\\{\ell_{i}\\}$. In this
expression,

$\int_{\ell}\,\equiv-\imath\,\mu^{4-d}\int\,\frac{d^{d}\ell}{(2\pi)^{d}}\,,$
(2)

corresponds to the standard $d$-dimensional loop integration measure. The
product of Feynman propagators is given by

$G_{F}\left(1,\ldots,n\right)=\prod_{i\in 1\cup\cdots\cup
n}\,(G_{F}(q_{i}))^{\alpha_{i}}\,,$ (3)

where $\alpha_{i}\in{\mathbb{N}}$. Individually, each Feynman propagator is
expressed according to

$G_{F}(q)=\frac{1}{(q_{0}-q_{0}^{(+)})(q_{0}+q_{0}^{(+)})}\,,$ (4)

where $q=(q_{0},\vec{q})$ is the momentum carried by the particle and
$q_{0}^{(+)}=\sqrt{(\vec{q})^{2}+m^{2}-\imath 0}$ is the associated positive
on-shell energy. This emphasizes that any internal line of a loop amplitude
can be interpreted as the quantum superposition of two off-shell states
flowing forward and backward in time, respectively. We will recall this
interpretation later in Sec. IV.

The LTD representation of Eq. (II) is obtained by integrating out one degree
of freedom per loop through the Cauchy residue theorem. The application of the
nested residues leads to a collection of diagrams with as many on-shell cuts
as loops, in such a way that each loop diagram is open into a sum of non-
disjoint trees. After adding together all the dual terms, the final result
only involves same-sign combinations of on-shell energies in the denominator:
these are the so-called _causal propagators_ Aguilera-Verdugo  _et al._
(2020a, 2021); Ramírez-Uribe  _et al._ (2021a); Jesús Aguilera-Verdugo  _et
al._ (2021). The causal propagators are associated to threshold
discontinuities or singularities, as those predicted by the optical theorem
and reconstructed through Cutkosky’s rules Cutkosky (1960).

As already investigated in previous articles, the LTD offers an excellent
opportunity to disentangle the causal structure of scattering amplitudes. In
particular, starting from Eq. (II) and computing the nested residues, we claim
that Aguilera-Verdugo  _et al._ (2020a, 2021); Ramírez-Uribe  _et al._
(2021a); Torres Bobadilla (2021b)

$\displaystyle{\cal
A}_{N}^{\left(L\right)}=\sum_{\sigma\in\Sigma}\,\int_{\vec{\ell}_{1},\cdots,\vec{\ell}_{L}}\frac{{\cal
N}_{\sigma}(\\{q_{r,0}^{(+)}\\},\\{p_{j,0}\\})}{x_{n}}\,$
$\displaystyle\times\prod_{i=1}^{k}\frac{1}{-\lambda_{\sigma(i)}}\,+\,(\sigma\leftrightarrow\bar{\sigma})\,,$
(5)

fully describes the causal structure of any multi-loop multi-leg Feynman
diagram of order $k$. The order of a diagram is given by

$k=I-L\,,$ (6)

namely, the number of remaining off-shell propagators after opening the loops
into trees through $L$ iterated cuts. In Eq. (5), the causal propagators have
the generic form

$\lambda_{j}^{\pm}\equiv\sum_{i\in o_{j}}q_{i,0}^{(+)}\pm k_{j}\,,$ (7)

where $k_{j}$ denotes a sum of external momenta and $o_{j}$ represents the
internal lines that are on-shell in the associated threshold singularity. The
set $\Sigma$ contains all the subsets of products of $k$ causal propagators,
$\sigma$, which fulfill certain compatibility criteria. Also, $\bar{\sigma}$
is obtained from $\sigma$ through the replacement
$\lambda_{j}^{\pm}\leftrightarrow\lambda_{j}^{\mp}$, namely by reversing
simultaneously the momenta flow of all the internal lines. Additionally, we
introduce the short-hand definitions

$\int_{\vec{\ell}}\equiv\mu^{d-4}\int\frac{d^{d-1}\ell}{(2\pi)^{d-1}}\,,\ \
x_{n}=\prod_{i\in 1\cup...\cup n}2q_{i,0}^{(+)}\,,$ (8)

which encode, respectively, the Euclidean dual integration measure and the
normalization factor coming from the iterated application of Cauchy’s residue
theorem. Regarding the numerator, ${\cal N}_{\sigma}$ is given by the
application of an operator depending on the subset $\sigma$ (whose explicit
form might be obtained from a direct calculation of the residues) and only
involves on-shell energies of the internal lines and the energies for external
particles. It is worth appreciating that, for scalar integrals, we obtain
${\cal N}_{\sigma}\equiv 1$ for all $\sigma\in\Sigma$. Thus, a possible path
to recover Eq. (5) consists in performing a reduction to scalar integrals and
then computing their causal representations Torres Bobadilla (2021a, b).

##  III Geometrical description of multi-loop amplitudes

Multi-loop multi-leg scattering amplitudes are built from Feynman diagrams,
i.e. geometrical structures described by graphs made of _vertices_ and
_lines_. The lines are understood as propagators that carry momenta and
connect the different vertices. The vertices describe interactions among
particles and impose momentum conservation involving internal and/or external
particles. Propagators connect exactly two vertices, and there could be more
than one propagator connecting two vertices. In that case, we substitute the
sum of all the momenta flowing through propagators connecting to two vertices
by a single _multi-edge_ ; thus a _multi-edge_ corresponds to a bunch of
lines, with the same origin and end, which are merged together. The set of all
the multi-edges defines a basis $Q$, which is extended to include external
momenta as well.

Let’s consider a multi-loop Feynman diagram with $N$ external particles, $L$
loops and $V$ interaction vertices. These vertices are connected through $I$
propagators, which can be reduced to $M$ multi-edges by merging those
connecting the same vertices. Thus, the same Feynman diagram can be described
in two equivalent ways:

  * •

Standard Feynman diagram: a graph with $L$ loops, $I$ propagators (lines) and
$V$ vertices satisfying

$V-1=I-L\,,$ (9)

i.e. Euler’s formula.

  * •

Reduced Feynman diagram: a graph with $V$ vertices connected by $M$ multi-
edges, which satisfies an analogous conservation equation,

$V-1=M-\tilde{L}\,,$ (10)

with $\tilde{L}$ the number of _graphical loops_.

Whilst in the standard representation each loop is associated to a loop
integration, the _graphical loops_ only designate a topological characteristic
of the reduced Feynman graph 111The graphical loops correspond to the concept
of _eloop_ introduced in Ref. Ramírez-Uribe  _et al._ (2021b).. In order to
clarify these concepts, we sketch the distinction between them in Fig. 2 for a
four-vertex topology. In the left side, we show the standard Feynman graph
with $L=8$ and $I=11$. By merging lines into multi-edges, we obtain the
reduced graph in the right side, which is composed by $M=5$ multi-edges and
$\tilde{L}=2$ graphical loops. Additionally, if the multi-edge $e_{1}$ is the
result of merging the lines $\\{i_{1},\ldots,i_{r}\\}$, the associated energy
is

$q_{e_{1},0}\equiv\sum_{j=1}^{r}q_{i_{j},0}\,,$ (11)

and the corresponding on-shell energy is given by

$q^{(+)}_{e_{1},0}\equiv\sum_{j=1}^{r}q^{(+)}_{i_{j},0}\,.$ (12)

The last definition is supported by the behaviour of the LTD representation of
MLT-like insertions in multi-loop Feynman diagrams. As rigorously proven in
Ref. Jesús Aguilera-Verdugo  _et al._ (2021), when several lines connect two
vertices, they can be replaced by an equivalent propagator whose equivalent
on-shell energy is the sum of the on-shell energies of each individual line
222For more details, see Sec. 3 of Ref. Jesús Aguilera-Verdugo  _et al._
(2021)..

In general, we notice that combining Eq. (6) with Eqs. (9)-(10), we obtain

$k=V-1\,,$ (13)

which indicates that the order of a diagram is directly related to the number
of vertices. It turns out that the reduced Feynman diagrams are more suitable
to infer the causal structure of Feynman integrals (or amplitudes), as we will
explain in the rest of the article. This assertion is also supported by
similar studies based on algebraic properties of multi-loop Feynman diagrams
constructed from vertices and multi-edges Torres Bobadilla (2021b).

Figure 2: Comparison between the standard Feynman (left) and the reduced graph
(right), for a four-vertex topology. Vertices are labelled with bold numbers.
Whilst the diagram in a has 11 propagators and 8 loops, the reduced one in b
has only 5 multi-edges and 2 graphical loops. Multi-edges 1 and 4 are the
result of collapsing 3 lines; multi-edges 2 and 3 originate from 2
propagators; and multi-edge 5 is composed by a single line in the standard
graph.

Moving forward with the formulation of the geometrical framework, we define
the basis of multi-edges and external momenta $Q$. For a given reduced Feynman
diagram, we choose the ordering

$Q=\\{Q_{1},\ldots,Q_{M};p_{1},\ldots,p_{N-1}\\}\,,$ (14)

where we implicitly impose $\sum p_{i}=0$ due to momentum conservation. For
each vertex, we have a unique momentum conservation equation: momentum is
conventionally considered positive (negative) if it is outgoing (incoming). In
this way, a vertex $v\in V$ is spanned over the basis $Q$, as a linear
combination of multi-edge momenta with coefficients $\\{\pm 1,0\\}$. For
instance, given the vertex $v$ with multi-edges $Q_{1}$ and $Q_{2}$ outgoing,
$Q_{4}$ incoming and the external outgoing momentum $p_{2}$ attached to it, we
introduce the representation

$\displaystyle v$ $\displaystyle=$
$\displaystyle(1^{+},2^{+},4^{-};\hat{2}^{+})$ (15) $\displaystyle\equiv$
$\displaystyle(1,1,0,-1,\ldots,0;0,1,\ldots,0)\,,$

with the short-hand notation $\hat{j}=p_{j}$. The vertex $v$ corresponds to
the momentum conservation equation

$v:\longrightarrow Q_{1}+Q_{2}-Q_{4}+p_{2}=0\,.$ (16)

Then, we define the _vertex matrix_ ${\cal V}$ as the $V\times(M+N-1)$
rectangular matrix whose rows corresponds to the coordinates of all the
vertices spanned on the basis $Q$. For example, if the first vertex is the one
given in Eq. (15), the generic structure of ${\cal V}$ would be

$\displaystyle{\cal
V}=\left(\begin{array}[]{@{}c|c@{}}\begin{matrix}1&1&0&-1&\ldots\\\
0&-1&1&0&\ldots\\\ &&\ldots&&\\\ \end{matrix}&\begin{matrix}0&1&0&\ldots\\\
0&0&1&\ldots\\\ &&\ldots&\\\ \end{matrix}\end{array}\right)\,,$ (18)

where the vertical line separates internal (left) and external (right)
momenta.

All the kinematic information encoded in ${\cal V}$ is enough to unveil the
causal structure of the underlying diagram as we will explain in the
following. Global momentum conservation implies that

${\rm Rank}({\cal V})=V-1=k\,,$ (19)

since the information of the momenta entering (exiting) to (from) a given
vertex is constrained by the whole system. We can make two direct
observations. First, the rank of the vertex matrix agrees with the order of
the diagram, by virtue of Eq. (13). Second, we can implement a practical
criterion to identify linear combinations of multi-edge momenta that are
compatible with momentum conservation. Explicitly, given $q=\sum a_{i}Q_{i}$
we have a unique coordinate representation in the $Q$ basis and we can attach
the corresponding row to the vertex matrix: if the extended matrix fulfils
${\rm Rank}({\cal V^{*}})\leq V-1$, then $q\equiv 0$ because of momentum
conservation. This last property is useful to test the compatibility rules
that define all the possible entangled thresholds.

###  III.1 Generation of causal propagators

Here, we explain how the causal propagators are generated using concepts from
graph theory. To do so, we consider reduced Feynman diagrams, i.e. we only
rely on the information concerning vertices and multi-edges. Then, we define a
_binary partition_ of the reduced graph as a non-trivial partition of the set
of vertices $V=\\{v_{1},v_{2},\ldots\\}$ with two components: since one is the
complement of the other, we identify each partition with the smaller subset of
vertices. In general, we have

${\cal P}_{V}=\\{\\{1\\},\\{2\\},\ldots,\\{1,2\\},\\{1,3\\},\ldots\\}\ ,$ (20)

with the notation $j\equiv v_{j}$. ${\cal P}_{V}$ is the quotient set of all
the possible subsets of $V$, constrained by the equivalence relation $r\equiv
r^{c}$ where $r^{c}=V/r$.

Since Feynman diagrams fulfill momentum conservation (and reduced graphs
inherit this property), a physical partition must also fulfil it. This means
that vertices inside each element of the partition must be _connected_ through
oriented multi-edges in a consistent way. We can encode this information by
looking into the vertices contained in each element of ${\cal P}_{V}$ and its
complement. By definition, a single vertex is self-connected. A set of
vertices is connected if there exist multi-edges joining them two-by-two;
since multi-edges converge into a vertex and they have a given orientation,
this implies a consistent momentum flow in each vertex belonging to the
partition. Thus, we define a _connected_ partition ${\cal P}_{V}^{C}$ as the
subset of elements of $p\in{\cal P}_{V}$ such that $p$ and $p^{c}$ are
connected. The connection is defined by the existence of at least one path
that allows to go from one vertex to any other inside the partition.

Let us use the reduced graph in Fig. 2 to illustrate the concept of
connection. The set of all the possible binary partitions is given by

${\cal
P}_{V}=\\{\\{1\\},\\{2\\},\\{3\\},\\{4\\},\\{1,2\\},\\{1,3\\},\\{1,4\\}\\}\ ,$
(21)

but not all of them are connected. This is the case of $p=\\{1,3\\}$, which
includes the vertices 1 and 3 connected through the multi-edge 5. However, its
complement, $p^{c}=\\{2,4\\}$, involves disconnected vertices: there is not a
multi-edge joining 2 and 4. In consequence, $p=\\{1,3\\}\not\in{\cal
P}_{V}^{C}$. Also, with this definition, all the remaining binary partitions
in Eq. (21) are connected.

The connected binary partitions originated from a reduced graph are important
since they codify the threshold structure of the corresponding Feynman
amplitude. Then, we need to establish the relation among partitions and causal
denominators. In order to do this, given $p\in{\cal P}_{V}^{C}$, we define its
_conjugated causal propagator_ as the sum of all the energies of the
associated multi-edge momenta connecting the vertices inside the partition. If
$\beta_{j}\in\\{\pm 1,0\\}$ and $\gamma_{j}\in\\{1,0\\}$, a generic conjugated
causal propagator is given by

$\bar{\lambda}_{p}=\sum_{j}\beta_{j}\,Q_{j,0}+\sum_{i=1}^{N-1}\gamma_{i}\,p_{i,0}\,,$
(22)

where the coefficients $\beta_{j}$ reflects the freedom to choose the
propagator momenta flow in the Feynman representation and
$\bar{\lambda}_{p}=0$ when momentum conservation is fulfilled in all the
vertices associated to the partition $p$. External particles are always
labelled as outgoing, even if they carry negative energy (which is equivalent
to say that they are actually incoming particles). It is worth appreciating
that this definition is consistent, i.e. we recover the same expression for
$\bar{\lambda}_{p^{c}}$ (i.e. by considering $p^{c}$), because of global
momentum conservation. Also, we notice that $\bar{\lambda}_{p}$ has a strong
physical meaning: it is the total momenta flowing from (or to) a given _binary
partition_ of a reduced Feynman diagram.

Once the _conjugated causal propagators_ are defined, we can generate all the
possible _causal propagators_. They originate from the overlap of momentum
conservation and the nested application of Cauchy’s residue theorem, leading
to causal same-sign combinations of on-shell energies. Explicitly, we
introduce the transformation

$\bar{\lambda}_{p}\to\pm\lambda^{\pm}_{p}=\pm\sum_{j}|\beta_{j}|\,Q^{(+)}_{j,0}+\sum_{i=1}^{N-1}\gamma_{i}\,p_{i,0}\,,$
(23)

which is equivalent to fix a partition, evaluate the nested residues (i.e.
replace loop energy components by positive on-shell energies, $Q_{i,0}^{(+)}$)
and consistently align all the momenta involved.

To conclude this Section, we shall recall the discussion given in Eq. (4):
propagators involve the superposition of two off-shell modes, which implies
that they can be aligned in two possible directions once they become on-shell.
Then, we appreciate that $\lambda^{\pm}_{p}$ is determined modulo a global
sign. Besides that, we use the convention in Eq. (23) since it implies
$\lambda^{+}=\lambda^{-}$ in the absence of external momenta attached to some
vertices, thus simplifying the expressions. Finally, we will equivalently
denote the causal propagators by the vertices involved in the associated
connected binary partition, i.e.
$\lambda_{p}\equiv\\{v_{i_{1}},\ldots,v_{i_{r}}\\}$.

##  IV Causality and compatibility conditions

As mentioned before, the connected binary partitions of a reduced Feynman
diagram and all the associated causal propagators are in strict
correspondence. Each causal propagator corresponds to a different physical
threshold, as discussed in Refs. Aguilera-Verdugo  _et al._ (2019, 2020a). On
the other hand, multi-loop multi-leg Feynman diagrams involve a superposition
of several thresholds, that might occur when different combinations of
internal states become on-shell. Thus, from a geometrical point of view, the
threshold or causal structure of a given Feynman diagram is determined by a
set of specific combinations of connected binary partitions. Each possible
term in a representation like Eq. (5) corresponds to _entangled causal
thresholds_ , using the concepts defined in Ref. Aguilera-Verdugo  _et al._
(2021). So, in this Section, we explain how to determine all the allowed
entangled causal thresholds imposing geometrical selection rules to combine
the different causal propagators.

In first place, we notice that:

  1. 1.

_Each possible entangled combination of causal denominators involves the on-
shell energies of all the propagators_.

This condition is related to the fact that we are cutting the diagram into
trees, and classifying these trees according to their threshold structure. At
one-loop, this agrees with the original Cutkosky’s rules Cutkosky (1960).
Beyond one-loop, there are more possibilities to decompose the topologies into
trees, but the idea remains the same: identify all the _causal compatible_
tree-level structures inside a diagram. Here, two tree-level blocks are
_causal compatible_ if they fulfill momenta conservation and if they can be
combined in such a way that the momenta exiting one block consistently enter
into the other.

With the previous ideas, we can re-interpret the generalization of Cutkosky’s
rules in terms of vertices and causal propagators. We claim that the
fundamental objects to build the causal representation are the elements of
${\cal P}_{V}^{C}$, since they contain information about the momenta flow and
the propagators that can be simultaneously set on-shell in each contribution.
Each $\lambda_{p}$ is associated to a connected binary partition of the
original reduced Feynman diagram in which internal lines are all entering or
exiting the partition. Thus, we motivate the following two compatibility
criteria:

  1. 2.

_Absence of crossing_ : Two compatible causal propagators $\lambda_{p}$ and
$\lambda_{q}$ must fulfill that the associated connected sets of vertices are
disjoint or one is totally included in the other. Explicitly, if $p$ is
smaller than $q$, then $p\subset q$ or $p\subset q^{c}$.

  2. 3.

_Consistent multi-edge momenta orientation_ : A set of causal propagators is
compatible if the associated $\bar{\lambda}_{p}$ can be consistently oriented,
i.e. if all the multi-edges contribute to the entangled cuts with the same
orientation.

Regarding the last criterion, let’s remember that a causal propagator
$\lambda^{\pm}_{p}$ corresponds to a binary partition which splits the reduced
diagram into two connected pieces. At this point, the causal denominators can
be thought as aligned contributions obtained from $\bar{\lambda}_{p}$ by
selecting either negative or positive energy modes. Then, this criterion
implies that all the multi-edges crossing the cut must have the same
direction, i.e. they are all entering or exiting the partition. In order to
achieve this, we might have to reverse some multi-edges. If a diagram involves
several cuts or causal thresholds, we have to look for a consistent
orientation of all the multi-edges of the reduced diagram. In consequence, it
could happen that when combining several $\lambda_{p}$, some multi-edges can
not be consistently aligned (for instance, because $q_{j}\equiv-q_{j}$ is
required).

It turns out that imposing criterion 3 leads to a reduced Feynman diagram
without cycles, i.e. an acyclic oriented graph. We call _causal orientation_
of the multi-edges to a configuration of momenta flow which converts the
reduced graph into an acyclic oriented graph. Thus, given a reduced graph, we
can first find all the causal orientations and then look for those
combinations of $\lambda_{p}$ which fulfil criteria 1-3. This problem can be
tackled by studying the eigenvalues of the adjacency matrix, which is build
from the vertex matrix introduced in Eq. (18) 333Recent developments on
quantum algorithms to speed-up the identification of causal orientations were
presented in Ref. Ramírez-Uribe  _et al._ (2021b)..

Figure 3: (Left) Example of two incompatible causal propagators with non-
disjoint sets of vertices. (Right) Example of a forbidden combination of
causal propagators due to incompatible momenta orientation.

To illustrate the application of criteria 1-3, let’s consider a typical 3-loop
Mercedes-Benz diagram as shown in Fig. 3. We assume that there is only one
line per multi-edge, in such a way that the reduced and the standard Feynman
diagram are the same. First, we define

$\lambda_{1}\equiv\\{1,3\\}\ ,\ \lambda_{2}\equiv\\{1\\}\ ,\
\lambda_{3}\equiv\\{3\\}\ ,\ \lambda_{6}\equiv\\{1,4\\}\,,$ (24)

which corresponds to the identification among partitions of vertices and
conjugated causal propagators. On the left graph,
$\\{1,3\\}\not\subset\\{1,4\\}$ and
$\\{1,3\\}\not\subset\\{2,3\\}=\\{1,4\\}^{c}$ since these are non-disjoint
sets of vertices: criterion 2 is not fulfilled here. On the right graph, the
entangled threshold corresponding to $(\lambda_{1}\,\lambda_{2}\,\lambda_{3})$
is considered. If we fix the orientation of the lines exiting the partition
$\\{1,3\\}\equiv\lambda_{1}$, the multi-edge connecting the vertices 1 and 3
can not be consistently oriented. Thus, this combination of causal thresholds
is forbidden from the causal representation by virtue of criterion 3.

After applying criterion 3 and ordering the multi-edges to represent acyclic
directed graphs, we need to keep the information about those multi-edges that
were reversed, i.e. the causal orientations. This is important to apply the
transformation given in Eq. (23) and properly determine whether it corresponds
to select $\lambda_{p}^{+}$ or $\lambda_{p}^{-}$. In fact, we have the
following rule:

  1. 4.

_Causal propagator orientation_ : Given a connected binary partition in a
causally oriented reduced graph, if the external momenta and the oriented
multi-edges are both outgoing, then $\bar{\lambda}_{p}\to\lambda_{p}^{+}$.
Otherwise, $\bar{\lambda}_{p}\to\lambda_{p}^{-}$.

Summarizing the procedure described up to now and applying the criteria 1-4 to
any reduced Feynman graph, we can obtain the set of all the allowed entangled
causal thresholds describing that diagram. We will denote this set as
$\bar{\Sigma}$.

However, some of the entangled thresholds defined by criteria 1-4 might be
degenerated due to global momentum conservation. This is true when the number
of multi-edges is not enough to fully constrain the flow of the different cuts
involved in an entangled threshold. Equivalently, the degeneration takes place
if the number of multi-edges is not maximal for the associated topology. This
observation motivates the following definition: a reduced Feynman diagram is a
_maximally connected graph_ (MCG) if all the vertices are connected to each
other. In other words, the associated adjacency matrix can be transformed into
an upper-triangular $\\#V\times\\#V$ matrix with all the entries equal to 1.
After exploring several topologies, we find the last rule:

  1. 5.

_Removing the threshold degeneration_ : Given a non-maximally connected graph,
we select a pair of disconnected vertices, $i$ and $j$, and force the
condition $p_{i}=-p_{j}\equiv q_{M+1}$. We repeat the procedure for all the
disconnected vertices, till we generate a maximally connected graph. Then, we
force the validity of criterion 3 including the fictitious multi-edges
$\\{q_{M+1},\ldots,q_{M+R-1}\\}$.

This criterion determines the set $\Sigma\subset\bar{\Sigma}$ in Eq. (5) and
fixes a causal representation of the diagram. Of course, this also shows that
there might be several equivalent causal representations for a given Feynman
graph, being all of them related due to momentum conservation. Only when the
reduced Feynman graph is maximally connected, we have $\Sigma=\bar{\Sigma}$.
This condition is also true when the diagram is _next-to-maximally connected_
, in the sense that only two vertices are disconnected. For more general
cases, criterion 5 might take a more complicated explicit form; we defer for
future studies these configurations and the exploration of their corresponding
geometrical properties.

##  V Application to four-vertex topologies

After presenting the abstract geometrical formalism in the previous Sections,
we will provide a concrete application. But first, we want to recall that all
the information about the causal structure of any Feynman diagram is encoded
within the vertex matrix. Independently of the number of loops and external
legs, the computational complexity of a Feynman diagram is given by the number
of vertices and how they are connected. Rephrasing a bit the last two
sentences, all the selection criteria described in Sec. IV can be implemented
through operations performed on ${\cal V}$.

Figure 4: Four-vertex topologies with 4, 5 and 6 multi-edges, respectively.
These topologies involve four external particles.

So, let us illustrate our framework for the case of the scalar four-vertex
topologies. We start with the case with 4 multi-edges depicted in Fig. 4
(left), i.e. a one-loop four-point function. For the sake of simplicity, we
consider that each multi-edge is composed by a single propagator (i.e. the
reduced and standard Feynman graphs are equal), whose associated momenta are

$\displaystyle q_{1}=\ell_{1}\,,\ q_{2}=\ell_{1}-p_{1}\,,\
q_{3}=\ell_{1}-p_{1}-p_{2}\,,$ $\displaystyle
q_{4}=\ell_{1}-p_{1}-p_{2}-p_{3}\,,$ (25)

and $\\{p_{i}\\}_{i=1,\ldots,4}$ are the external momenta fulfilling $\Sigma
p_{i}=0$. The basic set of momenta is

$Q=\\{1,2,3,4;\hat{1},\hat{2},\hat{3}\\}\,,$ (26)

and $\hat{4}=-\hat{1}-\hat{2}-\hat{3}$ because of global momentum
conservation. The vertices are

$\displaystyle v_{1}=(1^{-},2^{+};\hat{1}^{+})\,,\quad
v_{2}=(2^{-},3^{+};\hat{2}^{+})\,,$ $\displaystyle
v_{3}=(3^{-},4^{+};\hat{3}^{+})\,,\quad v_{4}=(4^{-},1^{+};\hat{4}^{+})\,,$
(27)

which leads to the vertex matrix

$\displaystyle{\cal
V}=\left(\begin{array}[]{@{}c|c@{}}\begin{matrix}-1&1&0&0\\\ 0&-1&1&0\\\
0&0&-1&1\\\ 1&0&0&-1\end{matrix}&\begin{matrix}1&0&0\\\ 0&1&0\\\ 0&0&1\\\
-1&-1&-1\end{matrix}\end{array}\right)\,.$ (29)

By looking into the connected partitions of vertices, we find

${\cal
P}_{V}^{C}=\\{\\{1\\},\\{2\\},\\{3\\},\\{4\\},\\{1,2\\},\\{1,4\\}\\}\,,$ (30)

where we only keep the equivalence classes determined by $p\equiv p^{c}$. We
notice that $\\{1,3\\}\equiv\\{2,4\\}\not\in{\cal P}_{V}^{C}$ because both of
them involve non-adjacent vertices which can not be connected. Associated to
each element of ${\cal P}_{V}^{C}$, we have the corresponding conjugated
causal propagator, i.e.

$\displaystyle\bar{\lambda}_{1}=(-1,1,0,0;1,0,0)\equiv\\{1\\}\,,$
$\displaystyle\bar{\lambda}_{2}=(0,-1,1,0;0,1,0)\equiv\\{2\\}\,,$
$\displaystyle\bar{\lambda}_{3}=(0,0,-1,1;0,0,1)\equiv\\{3\\}\,,$
$\displaystyle\bar{\lambda}_{4}=(1,0,0,-1;-1,-1,-1)\equiv\\{4\\}\,,$
$\displaystyle\bar{\lambda}_{5}=(-1,0,1,0;1,1,0)\equiv\\{1,2\\}\,,$
$\displaystyle\bar{\lambda}_{6}=(0,1,0,-1;0,-1,-1)\equiv\\{1,4\\}\,,$ (31)

being the internal (external) coordinates located to the left (right) of the
semicolon. Applying the transformation described in Eq. (23) to Eq. (31), we
obtain all the causal propagators for this topology. Explicitly, we have

$\displaystyle\lambda_{1}^{\pm}$ $\displaystyle=$ $\displaystyle
q_{1,0}^{(+)}+q_{2,0}^{(+)}\pm p_{1,0}\,,$ $\displaystyle\lambda_{2}^{\pm}$
$\displaystyle=$ $\displaystyle q_{2,0}^{(+)}+q_{3,0}^{(+)}\pm p_{2,0}\,,$
$\displaystyle\lambda_{3}^{\pm}$ $\displaystyle=$ $\displaystyle
q_{3,0}^{(+)}+q_{4,0}^{(+)}\pm p_{3,0}\,,$ $\displaystyle\lambda_{4}^{\pm}$
$\displaystyle=$ $\displaystyle
q_{1,0}^{(+)}+q_{4,0}^{(+)}\pm(p_{1,0}+p_{2,0}+p_{3,0})\,,$
$\displaystyle\lambda_{5}^{\pm}$ $\displaystyle=$ $\displaystyle
q_{1,0}^{(+)}+q_{3,0}^{(+)}\pm(p_{1,0}+p_{2,0})\,,$
$\displaystyle\lambda_{6}^{\pm}$ $\displaystyle=$ $\displaystyle
q_{2,0}^{(+)}+q_{4,0}^{(+)}\pm(p_{2,0}+p_{3,0})\,,$ (32)

by replacing the energy component of each multi-edge with the associated
aligned positive on-shell energies, i.e. $q_{i,0}^{(+)}$.

Once we generated all the connected binary partitions, we need to identify the
allowed entangled thresholds. First, we notice that the order of this diagram
is $k=3$, because it is a four vertex topology and we apply Eq. (13). Since
there are 6 causal propagators, the number of potential combinations of
thresholds is 20. The application of criteria 1 and 2 reduces the
possibilities to only 16. More combinations are discarded after considering
criteria 3-4, which involve a compatible ordering of the multi-edges. Thus,
given an entangled threshold, we test all the possible orderings of multi-
edges that leads to an acyclic graph; we retain only those configurations
where multi-edges are aligned when entering/exiting all the binary partitions.
This information allows to define an _ordering matrix_ that is used to
distinguish between $\lambda_{i}^{+}$ and $\lambda_{i}^{-}$. Thus, applying
the transformation rules in Eq. (23), we obtain

$\displaystyle\bar{\Sigma}=\\{(1^{+},2^{-},3^{+}),(1^{+},2^{-},4^{+}),(1^{+},2^{-},6^{-}),$
$\displaystyle(1^{+},3^{+},4^{+}),(1^{+},3^{-},5^{+}),(1^{+},3^{-},6^{-}),(1^{+},4^{+},5^{+}),$
$\displaystyle(2^{+},3^{-},4^{-}),(2^{+},3^{-},5^{+}),(2^{+},4^{+},5^{+}),(2^{+},4^{+},6^{+}),$
$\displaystyle(3^{+},4^{+},6^{+})\\}\,,$ (33)

which is the set of all the compatible entangled thresholds. Here, we use the
short-hand notation $i^{\pm}\equiv\lambda_{i}^{\pm}$. Notice that
$\\#(\bar{\Sigma})=12$.

At this point, we appreciate that setting $\Sigma=\bar{\Sigma}$ and ${\cal
N}_{\sigma}\equiv 1$ for all $\sigma\in\Sigma$ in Eq. (5) does not agree with
the result of the explicit nested residue calculation. This is because there
are degenerated entangled thresholds: in fact,

$\\{(1,2,3),(1,3,4)\\}\equiv\\{(1,2,4),(2,3,4)\\}\ ,$ (34)

because of momentum conservation. By including these configurations we over-
count the effect of some entangled thresholds 444An alternative approach
consists in calculating symmetry factors for all the possible entangled
thresholds. In this example, if we set ${\cal N}_{\sigma}=1/2!$ for the
configurations in Eq. (34) and ${\cal N}_{\sigma}=1$ for the others, we obtain
the proper result. We defer for a future publication the exploration of this
alternative representation and its properties.. Thus, we need to break the
degeneration by applying criterion 5. So, we can choose to close the loop by
joining vertices 1 and 3, or 2 and 4. In the first case, we obtain

$\displaystyle\Sigma_{1}=\\{(1^{+},2^{-},4^{+}),(1^{+},2^{-},6^{-}),(1^{+},3^{-},5^{+}),$
$\displaystyle(1^{+},3^{-},6^{-}),(1^{+},4^{+},5^{+}),(2^{+},3^{-},4^{-}),(2^{+},3^{-},5^{+}),$
$\displaystyle(2^{+},4^{+},5^{+}),(2^{+},4^{+},6^{+}),(3^{+},4^{+},6^{+})\\}\,,$
(35)

whilst in the second

$\displaystyle\Sigma_{2}=\\{(1^{+},2^{-},3^{+}),(1^{+},2^{-},6^{-}),(1^{+},3^{+},4^{+}),$
$\displaystyle(1^{+},3^{-},5^{+}),(1^{+},3^{-},6^{-}),(1^{+},4^{+},5^{+}),(2^{+},3^{-},5^{+}),$
$\displaystyle(2^{+},4^{+},5^{+}),(2^{+},4^{+},6^{+}),(3^{+},4^{+},6^{+})\\}\,,$
(36)

where both sets contain 10 elements. Following the functional form presented
in Eq. (5), causal representations of the scalar one-loop four-vertex topology
are given by

$\displaystyle{\cal
A}_{\text{4-vertex}}^{\left(\text{1-loop}\right)}=\int_{\vec{\ell}_{1}}\frac{1}{x_{4}}\,\sum_{\sigma\in\Sigma_{r}}\,\prod_{i=1}^{3}\frac{-1}{\lambda_{\sigma(i)}}\,\
+(\lambda_{i}^{+}\leftrightarrow\lambda_{i}^{-})\,,$ (37)

with

$x_{4}^{-1}=16q_{1,0}^{(+)}q_{2,0}^{(+)}q_{3,0}^{(+)}q_{4,0}^{(+)}\,,$ (38)

using either $r=1$ or $r=2$, from Eqs. (35) and (36) respectively.

###  V.1 Maximally and next-to-maximally connected four-vertex topologies

Then, let’s consider the remaining four-vertex topologies with 5 and 6 multi-
edges. Starting from the two-loop four-vertex topology in Fig. 4 (center), we
generate all the possible connected binary partitions. We immediately realize
that they are the same as for the one-loop four-vertex case; i.e. the set
${\cal P}_{V}^{C}$ is also given by Eq. (30). However, the functional form of
the corresponding causal propagators changes because of the additional multi-
edge. Explicitly, we have

$\displaystyle\lambda_{1}^{\pm}$ $\displaystyle=$ $\displaystyle
q_{1,0}^{(+)}+q_{2,0}^{(+)}+q_{5,0}^{(+)}\pm p_{1,0}\,,$
$\displaystyle\lambda_{2}^{\pm}$ $\displaystyle=$ $\displaystyle
q_{2,0}^{(+)}+q_{3,0}^{(+)}\pm p_{2,0}\,,$ $\displaystyle\lambda_{3}^{\pm}$
$\displaystyle=$ $\displaystyle q_{3,0}^{(+)}+q_{4,0}^{(+)}+q_{5,0}^{(+)}\pm
p_{3,0}\,,$ $\displaystyle\lambda_{4}^{\pm}$ $\displaystyle=$ $\displaystyle
q_{1,0}^{(+)}+q_{4,0}^{(+)}\pm(p_{1,0}+p_{2,0}+p_{3,0})\,,$
$\displaystyle\lambda_{5}^{\pm}$ $\displaystyle=$ $\displaystyle
q_{1,0}^{(+)}+q_{3,0}^{(+)}+q_{5,0}^{(+)}\pm(p_{1,0}+p_{2,0})\,,$
$\displaystyle\lambda_{6}^{\pm}$ $\displaystyle=$ $\displaystyle
q_{2,0}^{(+)}+q_{4,0}^{(+)}+q_{5,0}^{(+)}\pm(p_{2,0}+p_{3,0})\,,$ (39)

Again, there are 20 potential candidates to be entangled thresholds (6 causal
propagators to be grouped in sets of $k=3$ elements). Applying criteria 1-2,
we discard 8 configurations. When imposing criteria 3-4, we realize that the
presence of an additional multi-edge (w.r.t. the one-loop case) leads to
stricter constraints and 2 other configurations are eliminated. Thus, we
obtain

$\displaystyle\bar{\Sigma}=\\{(1^{+},2^{-},4^{+}),(1^{+},2^{-},6^{-}),(1^{+},3^{-},5^{+}),$
$\displaystyle(1^{+},3^{-},6^{-}),(1^{+},4^{+},5^{+}),(2^{+},3^{-},4^{-}),(2^{+},3^{-},5^{+}),$
$\displaystyle(2^{+},4^{+},5^{+}),(2^{+},4^{+},6^{+}),(3^{+},4^{+},6^{+})\\}\,,$
(40)

which is the same set $\Sigma_{1}$ presented in Eq. (35). This is not a
coincidence: this two-loop topology is a next-to-maximally connected graph
(NMCG), and removing the multi-edge $q_{5}$ leads to the one-loop box
described in the previous discussion. Thus, using Eq. (5), we get

$\displaystyle{\cal
A}_{\text{4-vertex}}^{\left(\text{2-loop}\right)}=\int_{\vec{\ell}_{1}\vec{\ell}_{2}}\frac{1}{x_{5}}\,\sum_{\sigma\in\Sigma}\,\prod_{i=1}^{3}\frac{-1}{\lambda_{\sigma(i)}}\,\
+(\lambda_{i}^{+}\leftrightarrow\lambda_{i}^{-})\,,$ (41)

as a causal representation for the scalar two-loop four-vertex topology, where

$x_{5}^{-1}=32q_{1,0}^{(+)}q_{2,0}^{(+)}q_{3,0}^{(+)}q_{4,0}^{(+)}q_{5,0}^{(+)}\,,$
(42)

and $\Sigma\equiv\bar{\Sigma}$ given by Eq. (40).

Finally, let’s consider the four-vertex topology with 6 multi-edges, shown in
Fig. 4 (right). This topology is straightforwardly a maximally connected
graph; all the vertices are connected. Using the identification between
conjugated causal propagators and the elements of the connected binary
partitions ${\cal P}_{V}^{C}$, we have

$\displaystyle\bar{\lambda}_{1}\equiv\\{1\\}\,,\
\bar{\lambda}_{2}\equiv\\{2\\}\,,\ \bar{\lambda}_{3}\equiv\\{3\\}\,,\
\bar{\lambda}_{4}\equiv\\{4\\}\,,$
$\displaystyle\bar{\lambda}_{5}\equiv\\{1,2\\}\,,\
\bar{\lambda}_{6}\equiv\\{1,4\\}\,,\ \bar{\lambda}_{7}\equiv\\{1,3\\}\,.$ (43)

Notice that $\\{1,3\\}\equiv\\{2,4\\}$ was not a connected binary partition
for the two previous topologies, but it contributes to this one. The
introduction of additional edges allows to define a path connecting the
vertices 1 and 3, as well as 2 and 4. Explicitly, the causal propagators are
given by

$\displaystyle\lambda_{1}^{\pm}$ $\displaystyle=$ $\displaystyle
q_{1,0}^{(+)}+q_{2,0}^{(+)}+q_{5,0}^{(+)}\pm p_{1,0}\,,$
$\displaystyle\lambda_{2}^{\pm}$ $\displaystyle=$ $\displaystyle
q_{2,0}^{(+)}+q_{3,0}^{(+)}+q_{6,0}^{(+)}\pm p_{2,0}\,,$
$\displaystyle\lambda_{3}^{\pm}$ $\displaystyle=$ $\displaystyle
q_{3,0}^{(+)}+q_{4,0}^{(+)}+q_{5,0}^{(+)}\pm p_{3,0}\,,$
$\displaystyle\lambda_{4}^{\pm}$ $\displaystyle=$ $\displaystyle
q_{1,0}^{(+)}+q_{4,0}^{(+)}+q_{6,0}^{(+)}\pm(p_{1,0}+p_{2,0}+p_{3,0})\,,$
$\displaystyle\lambda_{5}^{\pm}$ $\displaystyle=$ $\displaystyle
q_{1,0}^{(+)}+q_{3,0}^{(+)}+q_{5,0}^{(+)}+q_{6,0}^{(+)}\pm(p_{1,0}+p_{2,0})\,,$
$\displaystyle\lambda_{6}^{\pm}$ $\displaystyle=$ $\displaystyle
q_{2,0}^{(+)}+q_{4,0}^{(+)}+q_{5,0}^{(+)}+q_{6,0}^{(+)}\pm(p_{2,0}+p_{3,0})\,,$
$\displaystyle\lambda_{7}^{\pm}$ $\displaystyle=$ $\displaystyle
q_{1,0}^{(+)}+q_{2,0}^{(+)}+q_{3,0}^{(+)}+q_{4,0}^{(+)}\pm(p_{1,0}+p_{3,0})\,$
(44)

Regarding possible causal representations for this topology, we have 35
entangled thresholds (i.e. all the possible subsets of 3 causal propagators
taken from the 7 available ones). Imposing criteria 1-2 eliminates 19
combinations, and we further reduce this set by requiring criteria 3-4 to be
fulfilled. The remaining 12 allowed causal entangled thresholds are given by

$\displaystyle\bar{\Sigma}$
$\displaystyle=\\{(1^{+},2^{-},6^{-}),(1^{+},2^{-},7^{+}),(1^{+},3^{-},5^{+}),$
(45)
$\displaystyle(1^{+},3^{-},6^{-}),(1^{+},4^{+},5^{+}),(1^{+},4^{+},7^{+}),$
$\displaystyle(2^{+},3^{-},5^{+}),(2^{+},3^{-},7^{-}),(2^{+},4^{+},5^{+}),$
$\displaystyle(2^{+},4^{+},6^{+}),(3^{+},4^{+},6^{+}),(3^{+},4^{+},7^{+})\\}\,.$

Again, we notice that _the_ causal structure of this scalar four-vertex
diagram is

$\displaystyle{\cal
A}_{\text{4-vertex}}^{\left(\text{3-loop}\right)}=\int_{\vec{\ell}_{1}\vec{\ell}_{2}\vec{\ell}_{3}}\frac{1}{x_{6}}\,\sum_{\sigma\in\Sigma}\,\prod_{i=1}^{3}\frac{-1}{\lambda_{\sigma(i)}}\,\
+(\lambda_{i}^{+}\leftrightarrow\lambda_{i}^{-})\,,$ (46)

with

$x_{6}^{-1}=64q_{1,0}^{(+)}q_{2,0}^{(+)}q_{3,0}^{(+)}q_{4,0}^{(+)}q_{5,0}^{(+)}q_{6,0}^{(+)}\,,$
(47)

and $\Sigma\equiv\bar{\Sigma}$ given by Eq. (45). As in the case of the two-
loop box, the set of causal entangled thresholds is not degenerated (i.e.
criterion 5 is immediately fulfilled). For this reason, we emphasize that it
is the only possible causal representation compatible with Eq. (5).

To conclude this Section, we highlight that the procedure followed here is
only based on geometrical concepts. In the three examples reported, we
compared the reconstructed causal representations with the integrand-level
result of the LTD representation (which was obtained through the explicit
computation of the nested residues). A perfect agreement was found in all the
cases.

##  VI Causal structure of multi-vertex diagrams

Taking a step forward in complexity, we consider the causal structure of
scalar $N$-vertex reduced diagrams. The possible number of multi-edges, $M$,
fulfills

$N\leq M\leq\frac{(N-1)N}{2}\,,$ (48)

where the upper bound corresponds to the maximally connected topologies. By
direct computation, we explored several examples and computed the possible
causal entangled thresholds.

In first place, we studied the generation of all the possible causal
propagators $\lambda_{p}^{\pm}$ starting from the sets of binary connected
partitions for different topologies. In particular, we compared the results
obtained with the geometric algorithm with the expressions reported in Refs.
Aguilera-Verdugo  _et al._ (2021); Ramírez-Uribe  _et al._ (2021a), finding
complete agreement.

Then, we center into the generation of the allowed causal entangled thresholds
for different topologies. In particular, we studied the causal structure of
scalar maximally and next-to-maximally connected graphs. Based on explicit
computational exploration, we conjecture that their causal structures are
given by

$\displaystyle{\cal
A}_{\text{N-vertex}}^{\text{(N)MCG}}=\int_{\vec{\ell}_{1}\ldots\vec{\ell}_{L}}\frac{1}{x_{M}}\,\sum_{\sigma\in\Sigma}\,\prod_{i=1}^{N-1}\frac{-1}{\lambda_{\sigma(i)}}\,\
+(\lambda_{i}^{+}\leftrightarrow\lambda_{i}^{-})\,,$ (49)

where $\Sigma\equiv\bar{\Sigma}$, i.e. the set of allowed causal entangled
thresholds after applying criteria 1-4. It is worth appreciating that the
number of loops, $L$, only enters in this formula through the integration
measure and the explicit dependence of each $q_{i,0}^{(+)}$ (with
$i\in\\{1,\ldots,M\\}$). Thus, this supports the initial claim that the causal
structure is independent of the number of loops and propagators (only depends
of the vertices and multi-edges, as also reported in Ref. Torres Bobadilla
(2021b)).

Finally, we explored the opposite limit, i.e. the possible causal
representations of $N$-vertex diagrams at one-loop. It might sound counter-
intuitive that one-loop topologies are more complicated to describe than
multi-loop ones. The point is that the geometrical reconstruction algorithm
exploits the restrictions imposed by the momentum flow among vertices. Thus,
maximally connected graphs are very constrained and the criteria 1-4 lead to a
set of causal entangled thresholds that is not degenerated. One-loop $N$-point
amplitudes are described by _minimally connected graphs_ , and there is an
over-counting of configurations. To remove the degeneration, we need to apply
criterion 5 and force global momentum conservation. However, we found an
alternative way to select a non-degenerated set of causal thresholds.
Explicitly, for one-loop diagrams, we realized that a particular choice of the
selections rules dictated by criterion 5 is:

  1. 5.

_Removing the threshold degeneration (one-loop case)_ : Given an entangled
threshold, we keep the ordering matrix obtained after criteria 3-4 and apply
it to the corresponding conjugated causal propagators $\bar{\lambda}_{p}$.
Then, we adjust the direction of the multi-edges in such a way that they are
all outgoing from the associated partition. If they were entering the
partition, we reverse both of their fluxes ($q_{i}\to-q_{i}$) and the
direction of the external momenta attached to that partition. If the external
momenta are consistently aligned, we include the configuration in $\Sigma$;
otherwise, we exclude it.

Criterion 5 can be implemented by constructing a matrix whose rows are the
coordinates of $\bar{\lambda}_{p}$ in the basis $Q$, and applying
transformations on the rows and columns. Also, we notice that this modified
version of criterion 5 implicitly uses global momentum conservation, since the
direction of the external momenta is fixed by the condition $p_{N}=-\sum
p_{i}$.

In this way, causal representations for scalar $N$-point one-loop functions
can be also described by Eq. (49) with $M=N$, $L=1$ and the set $\Sigma$
determined with criteria 1-5. We checked the validity of our claim with
several scalar one-loop $N$-point functions ($N\leq 9$), finding a complete
agreement with the results computed through nested residues.

##  VII Conclusions

In this article, we presented a geometrical study of the entangled causal
structure of multi-loop multi-leg Feynman integrals and amplitudes. We showed
that all the information concerning the causal decomposition of a topology is
encoded in the _vertex matrix_. Diagrams with different numbers of legs and
loops, but sharing the same number of vertices and multi-edges, exhibit a
similar causal structure. We exemplified this situation studying the four-
vertex topologies with 4, 5 and 6 multi-edges, respectively. Moreover, we
introduced a classification of the different topologies based on graph theory:
the order of the diagram is given by $k=V-1$ ($V$ number of vertices) and it
indicates the number of causal thresholds that must be entangled.

In order to unveil the causal structure, we implemented an algorithm to
generate all the possible causal propagators associated to a given (reduced)
Feynman diagram by inquiring into a more fundamental object: the connected
binary partitions of vertices. Then, we developed 4 criteria to select all the
allowed causal entangled thresholds. We relied only on combinatorics, geometry
and graph theory. Specifically, we imposed restrictions that allow to split
the diagram into disconnected and non-overlapping tree-level graphs with a
consistent momentum flow through their multi-edges. To consistently align the
momentum flow of the multi-edges, we implemented transformations in the vertex
matrix to identify all the possible acyclic directed graphs associated to the
reduced Feynman diagram.

For the case of maximally and next-to-maximally connected graphs (i.e. those
where all the vertices are connected or only one multi-edge is missing), it
turns out that criteria 1-4 lead to a set of entangled thresholds that seems
to reconstruct the causal structure conjectured in Eq. (5). However, for
topologies with $M<(N-1)N/2$, some causal entangled thresholds are degenerated
due to global momentum conservation. In fact, criteria 1-4 only use
information regarding the vertices and how they are _internally_ connected
through multi-edges. The momentum conservation associated to external
particles can be understood as _adding an additional vertex in which all
external momenta converge_.

Thus, we introduced an additional selection criterion and explored its
consequences. In particular, we studied the case of $N$-vertex one-loop
diagrams (which we called _minimally connected graphs_) and obtained a
simplified recipe to eliminate the degeneration due to momentum conservation.
Our results were in complete agreement with the ones obtained through the
explicit calculation of nested residues.

The findings regarding our geometrical approach suggest an strong connection
with the algebraic framework proposed in Ref. Torres Bobadilla (2021b). In
that work, causal representations were obtained from maximally connected
graphs and applying algebraic reduction relations. Thus, it would be highly
interesting to understand the interplay between these two frameworks, since
they tackle the same problem with two very different approaches.

Finally, the discoveries reported in this article establish an interesting
connection between geometry, algebra and causality in Quantum Field Theories.
Moreover, the effects of imposing global momentum conservation on scattering
amplitudes deserve to be better understood, since they lead to several
equivalent causal representations. Inquiring more on these findings might open
new and more powerful paths to explore and compute higher-orders, breaking the
precision frontier and unveiling the hidden mathematical structures in QFT.

## Acknowledgments

We gratefully acknowledge the inspiring discussions with G. Rodrigo, R.
Hernández Pinto and W. J. Torres Bobadilla. We thank the author of Ref. Torres
Bobadilla (2021b) for making his results available for comparison, prior to
the publication. We also thank N. S. Ramírez-Uribe and J. J. Aguilera-Verdugo
for additional cross-checks. This article is based upon work from COST Action
PARTICLEFACE CA16201, supported by COST (European Cooperation in Science and
Technology). www.cost.eu.

## References

  * Abada  _et al._ (2019a) A. Abada _et al._ (FCC), Eur. Phys. J. C 79, 474 (2019a).
  * Abada  _et al._ (2019b) A. Abada _et al._ (FCC), Eur. Phys. J. ST 228, 261 (2019b).
  * Abada  _et al._ (2019c) A. Abada _et al._ (FCC), Eur. Phys. J. ST 228, 755 (2019c).
  * Abada  _et al._ (2019d) A. Abada _et al._ (FCC), Eur. Phys. J. ST 228, 1109 (2019d).
  * Blondel  _et al._ (2019) A. Blondel, J. Gluza, S. Jadach, P. Janot, and T. Riemann, eds.,  _Theory for the FCC-ee: Report on the 11th FCC-ee Workshop Theory and Experiments_, CERN Yellow Reports: Monographs, Vol. 3/2020 (CERN, Geneva, 2019) arXiv:1905.05078 [hep-ph] .
  * Bambade  _et al._ (2019) P. Bambade _et al._ , (2019), arXiv:1903.01629 [hep-ex] .
  * Roloff  _et al._ (2018) P. Roloff, R. Franceschini, U. Schnoor, A. Wulzer, _et al._ (CLIC, CLICdp), (2018), arXiv:1812.07986 [hep-ex] .
  * Dong  _et al._ (2018) M. Dong _et al._ (CEPC Study Group), (2018), arXiv:1811.10545 [hep-ex] .
  * Binoth and Heinrich (2000) T. Binoth and G. Heinrich, Nucl. Phys. B 585, 741 (2000), arXiv:hep-ph/0004013 .
  * Smirnov and Tentyukov (2009) A. Smirnov and M. Tentyukov, Comput. Phys. Commun. 180, 735 (2009), arXiv:0807.4129 [hep-ph] .
  * Carter and Heinrich (2011) J. Carter and G. Heinrich, Comput. Phys. Commun. 182, 1566 (2011), arXiv:1011.5493 [hep-ph] .
  * Borowka  _et al._ (2018) S. Borowka, G. Heinrich, S. Jahn, S. Jones, M. Kerner, J. Schlenk, and T. Zirke, Comput. Phys. Commun. 222, 313 (2018), arXiv:1703.09692 [hep-ph] .
  * Blumlein (2000) J. Blumlein, Comput. Phys. Commun. 133, 76 (2000), arXiv:hep-ph/0003100 .
  * Anastasiou and Daleo (2006) C. Anastasiou and A. Daleo, JHEP 10, 031 (2006), arXiv:hep-ph/0511176 .
  * Bierenbaum  _et al._ (2006) I. Bierenbaum, J. Blumlein, and S. Klein, Nucl. Phys. B Proc. Suppl. 160, 85 (2006), arXiv:hep-ph/0607300 .
  * Gluza  _et al._ (2007) J. Gluza, K. Kajda, and T. Riemann, Comput. Phys. Commun. 177, 879 (2007), arXiv:0704.2423 [hep-ph] .
  * Freitas and Huang (2010) A. Freitas and Y.-C. Huang, JHEP 04, 074 (2010), arXiv:1001.3243 [hep-ph] .
  * Dubovyk  _et al._ (2016) I. Dubovyk, J. Gluza, T. Riemann, and J. Usovitsch, PoS LL2016, 034 (2016), arXiv:1607.07538 [hep-ph] .
  * Mastrolia and Ossola (2011) P. Mastrolia and G. Ossola, JHEP 11, 014 (2011), arXiv:1107.6041 [hep-ph] .
  * Badger  _et al._ (2012) S. Badger, H. Frellesvig, and Y. Zhang, JHEP 1204, 055 (2012), arXiv:1202.2019 [hep-ph] .
  * Zhang (2012) Y. Zhang, JHEP 09, 042 (2012), arXiv:1205.5707 [hep-ph] .
  * Mastrolia  _et al._ (2012) P. Mastrolia, E. Mirabella, G. Ossola, and T. Peraro, Phys. Lett. B718, 173 (2012), arXiv:1205.7087 [hep-ph] .
  * Mastrolia  _et al._ (2013) P. Mastrolia, E. Mirabella, G. Ossola, and T. Peraro, Phys. Rev. D87, 085026 (2013), arXiv:1209.4319 [hep-ph] .
  * Ita (2015) H. Ita, (2015), arXiv:1510.05626 [hep-th] .
  * Mastrolia  _et al._ (2016) P. Mastrolia, T. Peraro, and A. Primo, JHEP 08, 164 (2016), arXiv:1605.03157 [hep-ph] .
  * Ossola  _et al._ (2007) G. Ossola, C. G. Papadopoulos, and R. Pittau, Nucl. Phys. B763, 147 (2007), arXiv:hep-ph/0609007 [hep-ph] .
  * Chetyrkin and Tkachov (1981) K. G. Chetyrkin and F. V. Tkachov, Nucl. Phys. B192, 159 (1981).
  * Laporta (2000) S. Laporta, Int. J. Mod. Phys. A15, 5087 (2000), arXiv:hep-ph/0102033 [hep-ph] .
  * Moriello (2020) F. Moriello, JHEP 01, 150 (2020), arXiv:1907.13234 [hep-ph] .
  * Bonciani  _et al._ (2020) R. Bonciani, V. Del Duca, H. Frellesvig, J. Henn, M. Hidding, L. Maestri, F. Moriello, G. Salvatori, and V. Smirnov, JHEP 01, 132 (2020), arXiv:1907.13156 [hep-ph] .
  * Czakon (2008) M. Czakon, Phys. Lett. B 664, 307 (2008), arXiv:0803.1400 [hep-ph] .
  * Gnendiger  _et al._ (2017) C. Gnendiger _et al._ , Eur. Phys. J. C77, 471 (2017), arXiv:1705.01827 [hep-ph] .
  * Heinrich (2020) G. Heinrich, (2020), arXiv:2009.00516 [hep-ph] .
  * Torres Bobadilla  _et al._ (2020) W. Torres Bobadilla _et al._ , (2020), arXiv:2012.02567 [hep-ph] .
  * Catani  _et al._ (2008) S. Catani, T. Gleisberg, F. Krauss, G. Rodrigo, and J.-C. Winter, JHEP 09, 065 (2008), arXiv:0804.3170 [hep-ph] .
  * Bierenbaum  _et al._ (2010) I. Bierenbaum, S. Catani, P. Draggiotis, and G. Rodrigo, JHEP 10, 073 (2010), arXiv:1007.0194 [hep-ph] .
  * Bierenbaum  _et al._ (2013) I. Bierenbaum, S. Buchta, P. Draggiotis, I. Malamos, and G. Rodrigo, JHEP 03, 025 (2013), arXiv:1211.5048 [hep-ph] .
  * de Jesús Aguilera-Verdugo  _et al._ (2021) J. de Jesús Aguilera-Verdugo _et al._ , Symmetry 13, 1029 (2021), arXiv:2104.14621 [hep-ph] .
  * Hernandez-Pinto  _et al._ (2016) R. J. Hernandez-Pinto, G. F. R. Sborlini, and G. Rodrigo, JHEP 02, 044 (2016), arXiv:1506.04617 [hep-ph] .
  * Sborlini (2016) G. F. Sborlini, PoS RADCOR2015, 082 (2016), arXiv:1601.04634 [hep-ph] .
  * Sborlini  _et al._ (2016a) G. F. R. Sborlini, F. Driencourt-Mangin, R. Hernandez-Pinto, and G. Rodrigo, JHEP 08, 160 (2016a), arXiv:1604.06699 [hep-ph] .
  * Sborlini  _et al._ (2016b) G. F. R. Sborlini, F. Driencourt-Mangin, and G. Rodrigo, JHEP 10, 162 (2016b), arXiv:1608.01584 [hep-ph] .
  * Buchta  _et al._ (2014) S. Buchta, G. Chachamis, P. Draggiotis, I. Malamos, and G. Rodrigo, JHEP 11, 014 (2014), arXiv:1405.7850 [hep-ph] .
  * Buchta (2015) S. Buchta,  _Theoretical foundations and applications of the Loop-Tree Duality in Quantum Field Theories_ , Ph.D. thesis, Valencia U. (2015), arXiv:1509.07167 [hep-ph] .
  * Buchta  _et al._ (2017) S. Buchta, G. Chachamis, P. Draggiotis, and G. Rodrigo, Eur. Phys. J. C77, 274 (2017), arXiv:1510.00187 [hep-ph] .
  * Driencourt-Mangin  _et al._ (2018) F. Driencourt-Mangin, G. Rodrigo, and G. F. Sborlini, Eur. Phys. J. C 78, 231 (2018), arXiv:1702.07581 [hep-ph] .
  * Plenter (2019) J. Plenter, Acta Phys. Polon. B 50, 1983 (2019).
  * Plenter and Rodrigo (2020) J. Plenter and G. Rodrigo, (2020), arXiv:2005.02119 [hep-ph] .
  * Driencourt-Mangin (2019) F. Driencourt-Mangin,  _Four-dimensional representation of scattering amplitudes and physical observables through the application of the Loop-Tree Duality theorem_ , Ph.D. thesis, U. Valencia (main) (2019), arXiv:1907.12450 [hep-ph] .
  * Driencourt-Mangin  _et al._ (2019) F. Driencourt-Mangin, G. Rodrigo, G. F. Sborlini, and W. J. Torres Bobadilla, (2019), arXiv:1911.11125 [hep-ph] .
  * Tomboulis (2017) E. Tomboulis, JHEP 05, 148 (2017), arXiv:1701.07052 [hep-th] .
  * Runkel  _et al._ (2019a) R. Runkel, Z. Szőr, J. P. Vesga, and S. Weinzierl, Phys. Rev. Lett. 122, 111603 (2019a), [Erratum: Phys. Rev. Lett.123,no.5,059902(2019)], arXiv:1902.02135 [hep-ph] .
  * Runkel  _et al._ (2019b) R. Runkel, Z. Szőr, J. P. Vesga, and S. Weinzierl, (2019b), arXiv:1906.02218 [hep-ph] .
  * Aguilera-Verdugo  _et al._ (2019) J. J. Aguilera-Verdugo, F. Driencourt-Mangin, J. Plenter, S. Ramírez-Uribe, G. Rodrigo, G. F. Sborlini, W. J. Torres Bobadilla, and S. Tracz, JHEP 12, 163 (2019), arXiv:1904.08389 [hep-ph] .
  * Capatti  _et al._ (2020a) Z. Capatti, V. Hirschi, D. Kermanschah, A. Pelloni, and B. Ruijl, JHEP 04, 096 (2020a), arXiv:1912.09291 [hep-ph] .
  * Capatti  _et al._ (2019) Z. Capatti, V. Hirschi, D. Kermanschah, and B. Ruijl, Phys. Rev. Lett. 123, 151602 (2019), arXiv:1906.06138 [hep-ph] .
  * Aguilera-Verdugo  _et al._ (2020a) J. J. Aguilera-Verdugo, F. Driencourt-Mangin, R. J. Hernandez Pinto, J. Plenter, S. Ramirez-Uribe, A. E. Renteria Olivo, G. Rodrigo, G. F. Sborlini, W. J. Torres Bobadilla, and S. Tracz, Phys. Rev. Lett. 124, 211602 (2020a), arXiv:2001.03564 [hep-ph] .
  * Ramírez-Uribe  _et al._ (2021a) S. Ramírez-Uribe, R. J. Hernández-Pinto, G. Rodrigo, G. F. R. Sborlini, and W. J. Torres Bobadilla, JHEP 04, 129 (2021a), arXiv:2006.13818 [hep-ph] .
  * Aguilera-Verdugo  _et al._ (2020b) J. Aguilera-Verdugo, R. Hernandez-Pinto, S. Ramirez-Uribe, G. Rodrigo, G. Sborlini, and W. Torres Bobadilla, in  _SnowMass 2021_ (2020).
  * Aguilera-Verdugo  _et al._ (2021) J. J. Aguilera-Verdugo, R. J. Hernandez-Pinto, G. Rodrigo, G. F. R. Sborlini, and W. J. Torres Bobadilla, JHEP 01, 069 (2021), arXiv:2006.11217 [hep-ph] .
  * Capatti  _et al._ (2020b) Z. Capatti, V. Hirschi, D. Kermanschah, A. Pelloni, and B. Ruijl, (2020b), arXiv:2009.05509 [hep-ph] .
  * Torres Bobadilla (2021a) W. J. Torres Bobadilla, Eur. Phys. J. C 81, 514 (2021a), arXiv:2103.09237 [hep-ph] .
  * Cutkosky (1960) R. E. Cutkosky, J. Math. Phys. 1, 429 (1960).
  * Bloch and Kreimer (2015) S. Bloch and D. Kreimer, (2015), arXiv:1512.01705 [hep-th] .
  * Cahill and Stapp (1975) K. E. Cahill and H. P. Stapp, Annals Phys. 90, 438 (1975).
  * Cahill and Stapp (1973) K. E. Cahill and H. P. Stapp, Phys. Rev. D 8, 2714 (1973).
  * Caron-Huot  _et al._ (2016) S. Caron-Huot, L. J. Dixon, A. McLeod, and M. von Hippel, Phys. Rev. Lett. 117, 241601 (2016), arXiv:1609.00669 [hep-th] .
  * Ramírez-Uribe  _et al._ (2021b) S. Ramírez-Uribe, A. E. Rentería-Olivo, G. Rodrigo, G. F. R. Sborlini, and L. Vale Silva, (2021b), arXiv:2105.08703 [hep-ph] .
  * Jesús Aguilera-Verdugo  _et al._ (2021) J. Jesús Aguilera-Verdugo, R. J. Hernández-Pinto, G. Rodrigo, G. F. R. Sborlini, and W. J. Torres Bobadilla, JHEP 02, 112 (2021), arXiv:2010.12971 [hep-ph] .
  * Torres Bobadilla (2021b) W. J. Torres Bobadilla, JHEP 04, 183 (2021b), arXiv:2102.05048 [hep-ph] .
  * Note (1) The graphical loops correspond to the concept of _eloop_ introduced in Ref. Ramírez-Uribe  _et al._ (2021b).
  * Note (2) For more details, see Sec. 3 of Ref. Jesús Aguilera-Verdugo  _et al._ (2021).
  * Note (3) Recent developments on quantum algorithms to speed-up the identification of causal orientations were presented in Ref. Ramírez-Uribe  _et al._ (2021b).
  * Note (4) An alternative approach consists in calculating symmetry factors for all the possible entangled thresholds. In this example, if we set ${\cal N}_{\sigma}=1/2!$ for the configurations in Eq. (34) and ${\cal N}_{\sigma}=1$ for the others, we obtain the proper result.