aainstitutetext: School of Physics & Astronomy, University of Southampton,
Southampton, SO17 1BJ, UK
Mathematical Sciences, University of Southampton, Highfield, Southampton SO17
1BJ, UKbbinstitutetext: Perimeter Institute for Theoretical Physics, Waterloo,
ON N2L 2Y5, Canadaccinstitutetext: Department of Combinatorics & Optimization,
University of Waterloo, Waterloo, ON N2L 3G1, Canada

#  $\phi^{p}$ Amplitudes from the Positive Tropical Grassmannian:
Triangulations of Extended Diagrams

Bruno Giménez Umbert<EMAIL_ADDRESS>c,b and Karen Yeats
<EMAIL_ADDRESS>
###### Abstract

The global Schwinger formula, introduced by Cachazo and Early as a single
integral over the positive tropical Grassmannian, provides a way to uncover
properties of scattering amplitudes which are hard to see in their standard
Feynman diagram formulation. In a recent work, Cachazo and one of the authors
extended the global Schwinger formula to general $\phi^{p}$ theories. When
$p=4$, it was conjectured that the integral decomposes as a sum over cones
which are in bijection with non-crossing chord diagrams, and further that
these can be obtained by finding the zeroes of a piece-wise linear function,
$H(x)$. In this note we give a proof of this conjecture. We also present a
purely combinatorial way of computing $\phi^{p}$ amplitudes by triangulating a
trivial extended version of non-crossing $(p-2)$-chord diagrams, called
extended diagrams, and present a proof of the bijection between triangulated
extended diagrams and Feynman diagrams when $p=4$. This is reminiscent of
recent constructions using Stokes polytopes and accordiohedra. However, the
$\phi^{p}$ amplitude is now partitioned by a new collection of objects, each
of which characterizes a polyhedral cone in the positive tropical Grassmannian
in the form of an associahedron or of an intersection of two associahedra.
Moreover, we comment on the bijection between extended diagrams and double-
ordered biadjoint scalar amplitudes. We also conjecture the form of the
general piece-wise linear function, $H^{\phi^{p}}(x)$, whose zeroes generate
the regions in which the $\phi^{p}$ global Schwinger formula decomposes into.

##  1 Introduction

Tropical geometry and tropical Grassmannians play an important role in various
aspects of quantum field theory, as shown by a large amount of recent work on
these topics, see Tourkine:2013rda ; Cachazo:2019ngv ; Cachazo:2022voc ;
Drummond:2019qjk ; Cachazo:2021llu ; Drummond:2019cxm ; Henke:2019hve ;
Arkani-Hamed:2024vna ; Arkani-Hamed:2022cqe ; Lukowski:2020dpn ; Early:2023tjj
for a few examples. The tropical Grassmannian ${\rm Trop}\,G(k,n)$ was first
introduced by Speyer and Sturmfels in SSTrop , where it was shown that it
agreed with the moduli space of phylogenetic trees studied by Billera, Holmes
and Vogtmann BHV . Speyer and Williams then introduced positive tropical
Grassmannians SWTrop , ${\rm Trop}^{+}\,G(k,n)$, where ${\rm
Trop}^{+}\,G(2,n)$ is related to the associahedron and parameterizes the space
of planar trees on $n$ leaves.

For example, figure 1 shows a positive part of the space of metric trees with
$n=5$ leaves, which coincides with $\textrm{Trop}^{+}G(2,5)$ for the canonical
ordering $\mathbb{I}:=(12345)$. The figure requires a bit of explanation. A
point in each quadrant of this space is in correspondence with a metric tree
with only degree $p=3$ vertices (shown in blue), where the internal lengths
$\ell_{i}$ of the edges are interpreted as Schwinger parameters which are
defined by the semi-rays in the quadrant. A point in each semi-ray is,
therefore, in correspondence with a tree with a degree $p=3$ vertex, a degree
$p=4$ vertex and a single edge. In this example, the black tree in the figure
is in correspondence with the planar kinematic invariant
$s_{23}:=(p_{2}+p_{3})^{2}=(p_{4}+p_{5}+p_{1})^{2}$. Notice that this tree in
the figure can be obtained by collapsing the other edge (of length $\ell_{2}$
or $\ell_{4}$) of any of the two trees in the quadrants sharing the semi-ray
$\ell_{3}$. This is due to the fact that any two quadrants sharing a semi-ray
correspond to two trees related by a planar mutation since, after contracting
an edge, there is another possible way of opening up a new edge to generate a
tree planar with respect to the given ordering. The pentagon in blue is the
boundary of the dual associahedron with $n=5$ letters, also known as the link
of the origin, which is given by the union of all hypersurfaces
$\sum_{i}\ell_{i}=1$ defined for each quadrant.

For general $n$ and for a given ordering $\alpha$, $\textrm{Trop}^{+}G(2,n)$
has Cn-2 positive orthants $(\mathbb{R}^{+})^{n-3}$, where Cm is the mth
Catalan number, and this count coincides with the number of metric trees with
$n$ leaves and only degree $p=3$ vertices which are planar with respect to
such ordering $\alpha$. Codimension-$d$ boundaries of the space correspond to
trees with $d$ zero-length edges and at least one degree $p>3$ vertex.

Figure 1: The space of planar metric trees with $n=5$ leaves.

A representative case in which $\textrm{Trop}^{+}G(2,n)$ plays an important
role in physics is the realm of tree-level partial amplitudes of massless
scalars in the biadjoint representation of a $U(N)\times U(\tilde{N})$ flavor
group with cubic interactions, $m_{n}(\alpha,\beta)$ Cachazo:2013iea . These
amplitudes can be computed by summing over all $n$-particle Feynman diagrams
planar with respect to both orderings $\alpha$ and $\beta$. Therefore, when
the two orderings coincide, e.g. $\alpha=\beta=\mathbb{I}$, the partial
amplitude can be computed by summing over all degree $p=3$ metric trees with
$n$ leaves which are planar with respect to $\mathbb{I}:=(12\cdots n)$. For
example, when $n=5$ the partial amplitude $m_{5}(\mathbb{I},\mathbb{I})$ is
given by the sum of the five blue trees in figure 1.

For the general $n$ case, the partial amplitude $m_{n}(\mathbb{I},\mathbb{I})$
is given by summing over all trees which are in correspondence with a positive
orthant $(\mathbb{R}^{+})^{n-3}$ in $\textrm{Trop}^{+}G(2,n)$. The
contribution of a single tree $T$ can be computed as a Schwinger integral of
the form

${\cal
R}(T)=\prod_{a=1}^{n-3}\int_{0}^{\infty}d\ell_{a}\textrm{exp}(-\ell_{a}t_{a})=\prod_{a=1}^{n-3}\frac{1}{t_{a}}\,,$
(1.1)

where $\ell_{a}$ is the Schwinger parameter associated to the internal length
of an edge, and $t_{a}$ is the kinematic invariant associated to that
edge111The integral is only defined for $t_{a}>0$, but after integrating the
resulting rational function is valid for any $t_{a}\neq 0$.. For example, the
tree in figure 1 associated to the quadrant parametrized by $\ell_{1}$ and
$\ell_{2}$ contributes as

${\cal
R}(T)=\int_{0}^{\infty}d\ell_{1}d\ell_{2}\textrm{exp}(-\ell_{1}s_{12}-\ell_{2}s_{45})=\frac{1}{s_{12}s_{45}}\,.$

Having understood the connection between $\textrm{Trop}^{+}G(2,n)$ and partial
biadjoint amplitudes, it is very tempting to try and attempt to unify the sum
of the Cn-2 $(n-3)$-dimensional Schwinger integrals into a single object.
Indeed, in a recent work Cachazo and Early showed that the partial amplitude
$m_{n}(\mathbb{I},\mathbb{I})$ can be computed as a single integral over ${\rm
Trop}^{+}\,G(2,n)$, known as the global Schwinger formula Cachazo:2020wgu
(see Cachazo:2022voc ; Arkani-Hamed:2022cqe ; GimenezUmbert:2023ykk ; Arkani-
Hamed:2023mvg ; Arkani-Hamed:2024vna ; Arkani-Hamed:2023lbd  for more work on
global Schwinger parameterizations)

$m_{n}(\mathbb{I},\mathbb{I})=\int_{\mathbb{R}^{n-3}}\\!\\!d^{n-3}x\,{\rm
exp}(-F_{n}(x))\,,$ (1.2)

which is determined by a piece-wise linear function $F_{n}(x)$ that we call
the tropical potential, thus providing a unification of all the Schwinger
integrals. See figure 2 for a pictorial representation of the formula.

Let us explain how to construct the tropical potential for future convenience.
We start with a positive parameterization of $G^{+}(2,n)$ Alex  and then
tropicalize the Plücker coordinates. A positive parameterization of
$G^{+}(2,n)$ is given by222Where we have suppressed the torus coordinates.

$\displaystyle\begin{pmatrix}1&0&-1&-(1+\tilde{x}_{1})&-(1+\tilde{x}_{1}+\tilde{x}_{2})&\cdots&-(1+\tilde{x}_{1}+\cdots+\tilde{x}_{n-3})\\\
0&1&1&1&1&\cdots&1\end{pmatrix}$ (1.3)

where $\tilde{x}_{a}\in\mathbb{R}^{+}$ and any maximal minor $\Delta_{ab}$
with $a<b$ is positive. We then tropicalize a minor $\Delta_{ab}$ in (1.3) by
replacing addition, ${\tilde{x}_{i}}+{\tilde{x}_{j}}$, with the min-function
${\rm min}(x_{i},x_{j})$, and multiplication,
${\tilde{x}_{i}}{\tilde{x}_{j}}$, with addition $x_{i}+x_{j}$, with now the
tropical variables being unconstrained, i.e. $x_{a}\in\mathbb{R}$. The
tropicalized minors evaluate to

$\Delta^{\rm Trop}_{ab}(x)=\left\\{\begin{array}[]{ccc}{\rm
min}(x_{a-2},x_{a-1},\ldots,x_{b-3})&&2\leq a\leq b-1,\,4\leq b\leq n\\\
0&&{\rm otherwise}\end{array}\right.,$ (1.4)

where $x_{0}:=0$ and whenever there is a single argument we use ${\rm
min}(x):=x$. The tropical potential function is therefore defined as333Here we
are making the connection to the space of planar metric trees by identifying
the matrix of distances $d_{ab}$ from leave $a$ to leave $b$ with the tropical
Plücker coordinates $\Delta_{ab}^{\textrm{Trop}}$. See Cachazo:2022voc  for
more details.

$\displaystyle F_{n}(x):=\sum_{1\leq a<b\leq n}\\!\\!\\!s_{ab}\,\Delta^{\rm
Trop}_{ab}(x)=\sum_{b=4}^{n}\sum_{a=2}^{b-1}\\!s_{ab}\,{\rm
min}(x_{a-2},x_{a-1},\ldots,x_{b-3})\,,$ (1.5)

where $s_{ab}:=(p_{a}+p_{b})^{2}$ are kinematic invariants and $p_{a}$ is the
momentum vector of particle $a$. One can also write the piece-wise linear
function (1.5) in terms of planar kinematic invariants,
$X_{a,b}:=(p_{a}+p_{a+1}+\cdots+p_{b-2}+p_{b-1})^{2}$, as

$F_{n}(x)=\sum_{a<b}X_{a,b+1}f_{[a,b]}(x)\,,$ (1.6)

where $s_{ab}=X_{a,b+1}-X_{a+1,b+1}-X_{a,b}+X_{a+1,b}$ and $X_{c,d+1}=0$
whenever $c\geq d$. The tropical cross-ratios $f_{[a,b]}(x)$ are defined as

$f_{[a,b]}(x):=\Delta^{\rm Trop}_{a,b}(x)-\Delta^{\rm
Trop}_{a,b+1}(x)-\Delta^{\rm Trop}_{a-1,b}(x)+\Delta^{\rm
Trop}_{a-1,b+1}(x)\,.$ (1.7)

Figure 2: The global Schwinger parameterization for $n=5$, viewed as a
projection of $\textrm{Trop}^{+}G(2,5)$, which unifies five 2-dimensional
Schwinger integrals into a single 2-dimensional integral, parametrized by two
tropical variables $x_{1}$ and $x_{2}$. The red lines on the plane define the
domains where the tropical potential $F_{5}(x)$ becomes linear.

We refer the reader to Cachazo:2020wgu ; Cachazo:2022voc  for more details and
for comments on the existence and regions of convergence of the integral
(1.2).

At first it might seem puzzling that an integral over all ${\rm
Trop}^{+}\,G(2,n)$ computes the partial amplitude
$m_{n}(\mathbb{I},\mathbb{I})$, which only encodes trees with cubic
interaction vertices, since ${\rm Trop}^{+}\,G(2,n)$ also contains trees with
higher-degree vertices living in higher codimension boundaries of the space.
However, the regions in ${\rm Trop}^{+}\,G(2,n)$ that contain higher-degree
vertices are of measure zero and, therefore, do not contribute to the integral
(1.2).

Recently, Cachazo and one of the authors continued the study of global
Schwinger formulas for amplitudes in scalar theories with general tree-level
$\phi^{p}$-interactions, for any $p>3$ Cachazo:2022voc . Namely, these are
amplitudes obtained by summing over trees with only degree-$p$ interaction
vertices and which are planar with respect to one ordering (in this work we
implicitly consider the canonical ordering $\mathbb{I}$ without loss of
generality).

For the simplest example, i.e. $p=4$, the global Schwinger formula is given by

$A^{\phi^{4}}_{n}=\int_{\mathbb{R}^{n-3}}\\!\\!d^{n-3}x\,{\rm
exp}\left(-\sum_{b-a\equiv 0\,\rm{mod}\,2}X_{a,b+1}f_{[a,b]}(x)\right)Q(x)\,,$
(1.8)

where $Q(x)$ is a sum over distributions that localize the integral to the
regions where trees with only degree-4 vertices live. In Cachazo:2022voc  it
was conjectured that such regions, each of which corresponds to a polyhedral
cone in ${\rm Trop}^{+}\,G(2,n)$ of dimension $n/2-1$ in $\mathbb{R}^{n-3}$,
are given by the solutions to the equation $H(x)=0$, where $H(x)$ is

$\displaystyle
H(x)=\sum_{a=0}^{n-3}x_{a}+2\sum_{a=0}^{n-4}\sum_{b=a+1}^{n-3}(-1)^{b-a}\,\textrm{min}(x_{a},x_{a+1},...,x_{b})\,.$
(1.9)

We give a proof of this conjecture in section 2. This result implies that
$A_{n}^{\phi^{4}}$ amplitudes can be expressed as a sum over regions.
Moreover, a remarkable connection to cubic amplitudes was found. Namely, each
of the Cn/2-1 regions that define the support of the distribution $Q(x)$ is
related to a double-ordered cubic amplitude $m_{n/2-1}(\alpha,\beta)$ with a
smaller number of particles. Therefore, the $\phi^{4}$ amplitude can be
expressed as a sum of products of cubic amplitudes Cachazo:2013iea , something
which is hard to see from the standard Feynman diagram formulation of the
amplitude. In Cachazo:2022voc  it was also shown that each of these regions in
$\textrm{Trop}^{+}G(2,n)$ are classified by non-crossing chord diagrams, which
we review in section 2.

One of the main results of this paper, presented in section 2, is that we show
that one can easily obtain the $\phi^{4}$ amplitude without ever having to
carry out an integral by taking an extended version of the non-crossing chord
diagrams, called extended diagrams, and then further decomposing them in a way
which we call triangulation. In section 2 we also present a formal proof of
the bijection between triangulations of extended diagrams and $\phi^{4}$
trees, here using the fact that the extended diagram triangulations give
quadrangulations of an even n-gon. Therefore, we proof that all triangulated
extended diagrams account for the whole amplitude, showing that the whole
collection of extended diagrams provides a novel way of partitioning
$A_{n}^{\phi^{4}}$ into new objects, that differ e.g. from Stokes polytopes
Banerjee:2018tun ; Salvatori:2019phs ; Aneesh:2019cvt ; Srivastava:2020dly ,
whose introduction was motivated by the connection between $\phi^{3}$
amplitudes and the associahedron Mizera:2017cqs ; Arkani-Hamed:2017mur . The
reason this is different is that these objects are now given by regions in
$\textrm{Trop}^{+}G(2,n)$ in the form of an associahedron or of an
intersection of two associahedra.

For example, figure 3 shows a triangulation of an extended diagram for $n=16$,
together with the tree the triangulation is related to. Here every black chord
separating two colored regions –which are related to cubic amplitudes– and
every triangulating chord –shown in red– corresponds to a propagator in the
tree. For example, the chord going from point 4 to point 9 on the real line
gives rise to $1/X_{4,9}$, while the triangulating chord going from point 9 to
point 14 gives rise to $1/X_{9,14}$. In this correspondence a colored region
with $m-1$ boundary chords will be related to an $m$-point amplitude and the
further decomposition into smaller regions by the triangulating chords will be
related to a triangulation of an $m$-gon for the $m$-point amplitude. This why
we use the term triangulation, even though these smaller regions each have $4$
boundary lines and arcs in the extended diagram.

In section 3 we present an analogous procedure for general $\phi^{p}$
theories, using a more general version of extended diagrams, first introduced
in Cachazo:2022voc , which are also obtained using similar rules. Namely, we
explain how we can obtain any $\phi^{p}$ amplitude by triangulating all
extended diagrams.

Figure 3: Triangulation of an extended diagram for $n=16$ in $\phi^{4}$ and
its associated tree. The triangulating chords are shown in red, and the
colored regions correspond to different cubic subamplitudes.

In section 4 we continue the exploration of the relation between extended
diagrams, like the one in figure 3, with double-ordered cubic amplitudes. We
also comment on the fact that colored regions in an extended diagram must be
related to a cubic subamplitude, by presenting a polygon decomposition of the
extended diagram. We conclude in section 5 with some additional comments and
discussions about future research topics. For example, we conjecture the
analogous form of the function (1.9) for general $p$, and we present some
examples of polygon-decomposed extended diagrams in $\phi^{p}$ that require
further exploration.

##  2 $\phi^{4}$ from Triangulations of Extended Diagrams

A remarkable implication of the global Scwhinger formulation for $\phi^{4}$
theories found in Cachazo:2022voc  is that the amplitude decomposes as a sum
over polyhedral cones in $\textrm{Trop}^{+}G(2,n)$, which are in bijection
with non-crossing chord diagrams. Such diagrams provide a systematic way of
finding the regions which turn into distributions in the integral (1.8),
ultimately computing the amplitude.

In this section we show one of the main results in this paper. Namely, that by
triangulating a trivial extended version of these diagrams, known as extended
non-crossing chord diagrams and originally introduced in Cachazo:2022voc , we
can obtain the $\phi^{4}$ amplitude without having to compute the integral.

Before showing how to obtain $\phi^{4}$ amplitudes from triangulations of
extended diagrams, let us first review what these diagrams are.

###  2.1 Non-Crossing Chord Diagrams and Extended Diagrams

First of all, let us recall the definition of a non-crossing chord diagram,
where we have slightly changed the labelling with respect to Cachazo:2022voc
for convenience.

######  Definition 2.1.

Place $n-2$ points labeled $3,4,\ldots,n$ in increasing order on the real
line. A non-crossing chord diagram is a perfect matching of the points such
that all edges, which we call _chords_ , can be drawn as arcs on the upper
half plane without any crossings. Let us denote the chord connecting points
$a$ and $b$ as $\theta_{ab}$.

This leads to the following conjecture which we will prove in section 2.4.

######  Conjecture 2.2 (Conjecture 6.2 of Cachazo:2022voc ).

The regions contributing to $A^{\phi^{4}}_{n}$ are in bijection with the set
of all $\textrm{C}_{n/2-1}$ possible non-crossing chord diagrams. Moreover,
the region $R$ corresponding to a particular diagram is obtained as follows:

  * •

For each chord $\theta_{ab}$ set $x_{a-3}=x_{b-3}$.

  * •

If a chord $\theta_{ab}$ surrounds another chord $\theta_{cd}$, then
$x_{a-3}=x_{b-3}<x_{c-3}=x_{d-3}$.

In other words, the regions defined by non-crossing chord diagrams are all the
solutions to $H(x)=0$, where $H(x)$ is given by equation (1.9).

For example, in $n=6$ there are two non-crossing chord diagrams as shown in
figure 4.

Figure 4: Non-crossing chord diagrams for $n=6$. On the right, the chord
$\theta_{36}$ surrounds the chord $\theta_{45}$ and therefore the condition
$x_{0}<x_{1}$ is imposed.

The corresponding regions are

$R_{1}=\\{x_{0}=x_{1},\,x_{2}=x_{3}\\}\,,\quad
R_{2}=\\{x_{0}=x_{3},\,x_{1}=x_{2},\,x_{0}<x_{1}\\}\,.$ (2.1)

By evaluating (1.8) in each of these regions one finds

$A_{6}^{\phi^{4}:R_{1}}=\frac{1}{X_{1,4}}+\frac{1}{X_{2,5}}\,,\hskip
42.67912ptA_{6}^{\phi^{4}:R_{2}}=\frac{1}{X_{3,6}}\,,$ (2.2)

and one obtains the amplitude $A_{6}^{\phi^{4}}$ by summing over both regions.

Similarly, for $n=8$ there are five possible non-crossing chord diagrams,
shown in figure 5, which give rise to the five regions

$\displaystyle R_{1}=\,$
$\displaystyle\\{x_{1}=0,\,x_{2}=x_{3},\,x_{4}=x_{5}\\}\,,\hskip
14.22636ptR_{2}=\\{x_{1}=0,\,x_{2}=x_{5},\,x_{3}=x_{4},\,x_{2}<x_{3}\\}\,,$
$\displaystyle R_{3}=\,$
$\displaystyle\\{x_{5}=0,\,x_{1}=x_{2},\,x_{3}=x_{4},\,x_{1}>0,\,x_{3}>0\\}\,,$
$\displaystyle R_{4}=\,$
$\displaystyle\\{x_{3}=0,\,x_{1}=x_{2},\,x_{4}=x_{5},\,x_{1}>0\\}\,,\hskip
2.84526ptR_{5}=\\{x_{5}=0,\,x_{1}=x_{4},\,x_{2}=x_{3},\,x_{2}>x_{1}>0\\}\,,$
(2.3)

each of which contributes to the $\phi^{4}$ amplitude after using the integral
(1.8), as

$\displaystyle A^{\phi^{4}:(1)}_{8}=\,$
$\displaystyle\frac{1}{X_{1,4}X_{4,7}}+\frac{1}{X_{4,7}X_{2,7}}+\frac{1}{X_{2,7}X_{2,5}}+\frac{1}{X_{2,5}X_{1,6}}+\frac{1}{X_{1,6}X_{1,4}}\,,$
$\displaystyle A^{\phi^{4}:(2)}_{8}=\,$
$\displaystyle\frac{1}{X_{5,8}}\left(\frac{1}{X_{1,4}}+\frac{1}{X_{2,5}}\right)\,,\hskip
2.84526ptA^{\phi^{4}:(3)}_{8}=\frac{1}{X_{3,8}}\left(\frac{1}{X_{3,6}}+\frac{1}{X_{5,8}}\right)\,,$
$\displaystyle A^{\phi^{4}:(4)}_{8}=\,$
$\displaystyle\frac{1}{X_{3,6}}\left(\frac{1}{X_{1,6}}+\frac{1}{X_{2,7}}\right)\,,\hskip
2.84526ptA^{\phi^{4}:(5)}_{8}=\frac{1}{X_{4,7}X_{3,8}}\,.$ (2.4)

Figure 5: Non-crossing chord diagrams for $n=8$. In the second diagram
$\theta_{58}$ surrounds $\theta_{67}$ and, therefore, $x_{2}<x_{3}$. In the
third diagram $\theta_{38}$ surrounds both $\theta_{45}$ and $\theta_{67}$ so
we have $x_{0}<x_{1}$ and $x_{0}<x_{3}$. In the fourth diagram $\theta_{36}$
surrounds $\theta_{45}$, so we have $x_{0}<x_{1}$. Finally, in the fifth
diagram $\theta_{38}$ surrounds $\theta_{47}$, which at the same time
surrounds $\theta_{56}$, so $x_{0}<x_{1}<x_{2}$.

The reader familiar with the biadjoint scalar theory will notice from these
examples that the contribution to every region resembles that of
$m_{n/2+1}(\alpha,\beta)$ for some permutations $\alpha$ and $\beta$. This is
an example of how the global Schwinger formula uncovers properties of
scattering amplitudes which are non-obvious from the standard Feynman diagram
formulation. In section 4 we explore this connection.

Now we proceed to define the central object of this paper: the extended non-
crossing chord diagram, which from now on we will call extended diagram for
simplicity.

######  Definition 2.3.

An extended diagram associated to $A^{\phi^{4}}_{n}$ is a non-crossing chord
diagram on $n$ points labeled by $\\{2,3,4,5,\ldots,n,1\\}$ in which the chord
$\theta_{21}$ is always included. We also define a meadow of an extended
diagram as any region in the diagram delimited by more than one chord and by
the line on which the points lie. A meadow is an $m$-point meadow if there are
$m-1$ chords delimiting it.

In other words, an extended diagram is a non-crossing chord diagram with an
additional chord, which goes from 2 to 1, surrounding all other chords. In the
combinatorial language of chord diagrams, these are known as indecomposible
non-crossing chord diagrams. Figures 6 and 7 show all the extended diagrams
for $n=6$ and $n=8$, respectively. The reason why we have colored every meadow
will become clear in section 4, but for now it serves us as a visual way to
differentiate between them.

Figure 6: Extended diagrams for $n=6$. Every colored region in the diagram
corresponds to a meadow. Figure 7: Extended diagrams for $n=8$. Every colored
region in the diagram corresponds to a meadow.

###  2.2 $\phi^{4}$ from Triangulations of Extended Diagrams

In Cachazo:2022voc  it was shown that there are Cn/2-1 extended diagrams that
contribute to the $\phi^{4}$ amplitude. In this section we show that
triangulations of extended diagrams, which are counted by Fuss-Catalan
numbers444Here $\textrm{FC}_{m}(q,r)$ is the Fuss-Catalan number given by
$\textrm{FC}_{m}(q,r)\equiv\frac{r}{mq+r}{mq+r\choose m}\,.$
FC${}_{n/2-1}(3,1)$, correspond to $\phi^{4}$ trees participating in the
amplitude. This means that every extended diagram contains at least one
$\phi^{4}$ tree. Therefore, one obtains the amplitude by summing over
triangulations.

A triangulation of an extended diagram is given by adding new chords
$\Omega_{ab}$ in the following way:

  * •

A triangulating chord $\Omega_{ab}$ must start, reading from left to right,
from the label $a$ on the right of a lower chord of the meadow or on the left
of the upper chord of the meadow, and must end on the label $b$ on the left of
a lower chord of the meadow or on the right of the upper chord of the meadow.

  * •

A triangulating chord must surround at least another chord in the diagram.

  * •

The triangulation ends when only regions with 4 delimiting chords and lines
are left.

Once the triangulation of an extended diagram is done, one can easily compute
its contribution to the $\phi^{4}$ amplitude in the following way:

  * •

Every already existing chord $\theta_{ab}$ in the diagram surrounding another
chord, with the exception of $\theta_{21}$, and every triangulating chord
$\Omega_{ab}$, correspond to a propagator of the form $1/X_{a,b}$, where
$X_{a,b}=X_{b,a}$.

  * •

The contribution of a triangulated diagram to the amplitude is given by
multiplying all propagators involved.

In the following subsection we show why every triangulated diagram is in
bijection with a $\phi^{4}$ tree. It is also worth mentioning that the
distinction between two kinds of chords, $\theta_{ab}$ and $\Omega_{ab}$, will
become more clear in section 4.

For example, let us consider the first diagram in figure 6. The two possible
triangulations of this diagram are shown in figure 8, where the triangulating
chords are shown in red.

Figure 8: Triangulations of an extended diagram for $n=6$ and the $\phi^{4}$
tree they correspond to, shown below.

The diagram on the left has one triangulating chord $\Omega_{25}$, thus it
corresponds to a propagator of the form $1/X_{2,5}$, which is the contribution
to the amplitude from the tree below. The diagram on the right has one
triangulating chord $\Omega_{41}$, and it therefore contributes as
$1/X_{1,4}$, the same as the tree below. The sum of the two triangulations
gives $A_{6}^{\phi^{4}:R_{1}}$ in (2.2).

The second diagram in figure 6 is already completely triangulated, i.e. it is
only made of 3-point meadows. This diagram contains a chord of the form
$\theta_{36}$ and it therefore contributes as $1/X_{3,6}$, which is the
contribution of the remaining tree in the $\phi^{4}$ amplitude, i.e.
$A_{6}^{\phi^{4}:R_{2}}$ in (2.2). This is the only valid chord contributing
to the amplitude in the diagram, since the remaining chords are $\theta_{21}$
and one that does not surround any other chord. Therefore, one gets the total
amplitude by summing over triangulated extended diagrams, which gives the same
result as summing over regions like in (2.2).

Now let us move on to a more interesting example, when $n=8$. Consider the
first extended diagram in figure 7, i.e. the one with a single meadow. It has
five possible triangulations, which are represented in figure 9, again with
the triangulating chords in red, together with their associated $\phi^{4}$
trees below.

Figure 9: Triangulations of an extended diagram for $n=8$ and their
corresponding $\phi^{4}$ tree.

For example, the triangulated diagram on the bottom-right of the figure has
two chords of the form $\Omega_{25}$ and $\Omega_{61}$ which correspond to
propagators of the form $1/X_{2,5}$ and $/X_{1,6}$, respectively. Therefore,
this triangulation contributes as

$\frac{1}{X_{2,5}X_{1,6}}\,,$

the same as the tree below. Summing over these five triangulations gives
$A_{8}^{\phi^{4}:(1)}$ in (2.1). The attentive reader might have noticed that
this extended diagram only consists of a 5-point meadow, and the five
triangulations are in bijection with those of a pentagon. This relation will
become clear in section 4.

Figure 10: Triangulations for the rest of the extended diagrams in $n=8$, and
their corresponding $\phi^{4}$ tree shown below, with the exception of the
extended diagram that only has 3-point meadows.

Figure 10 shows the triangulations for other extended diagrams in $n=8$, with
their corresponding trees. For example, the second extended diagram in figure
7 has a 4-point meadow and a 3-point meadow. Its triangulations are the first
two in the first row of figure 10, which are those of a square. In this case,
the first triangulated diagram has two contributing chords $\theta_{58}$ and
$\Omega_{25}$, which give rise to a contribution of the form $1/X_{5,8}\times
1/X_{2,5}$. The second triangulated diagram has two contributing chords
$\Omega_{41}$ and $\theta_{58}$, which give rise to a contribution of the form
$1/X_{1,4}\times 1/X_{5,8}$. Therefore, the contribution from this extended
diagram is given by

$\frac{1}{X_{5,8}}\left(\frac{1}{X_{1,4}}+\frac{1}{X_{2,5}}\right)\,,$

which corresponds to $A_{8}^{\phi^{4}:(2)}$ in (2.1) and is in bijection with
a double-ordered 5-point cubic amplitude. One can perform a similar analysis
for the rest of the diagrams in figure 10.

Notice that the last diagram in figure 7 is already completely triangulated,
this is why we have not included it in figure 10, but it of course corresponds
to the remaining tree, giving rise to $1/X_{3,8}\times 1/X_{4,7}$, which
corresponds to $A_{8}^{\phi^{4}:(5)}$ in (2.1). The sum of all the 12
triangulations, which are in bijection with the 12 $\phi^{4}$ trees, gives the
total amplitude found by summing the terms in (2.1).

As a last example, let us look at the triangulated extended diagram in figure
11, the same presented in section 1, which corresponds to a $\phi^{4}$ tree in
the $n=16$ case.

Figure 11: Triangulation of an extended diagram for $n=16$ and their
corresponding $\phi^{4}$ tree. The triangulating chords are shown in red, and
the colored regions correspond to different meadows of the original extended
diagram.

This triangulation has chords of the form $\theta_{3,16}$, $\theta_{49}$,
$\theta_{10,13}$, $\Omega_{47}$, $\Omega_{9,14}$ and $\Omega_{9,16}$ which
correspond to propagators of the form $1/X_{3,16}$, $1/X_{4,9}$,
$1/X_{10,13}$, $1/X_{4,7}$, $1/X_{9,14}$ and $1/X_{9,16}$, respectively.
Therefore, this triangulation contributes as

$\frac{1}{X_{3,16}X_{4,9}X_{10,13}X_{4,7}X_{9,14}X_{9,16}}\,$

the same as the tree shown in figure 11. In fact, this triangulation comes
from an extended diagram with two 3-point meadows –colored in grey and pink–,
a 5-point meadow –colored in blue– and a 4-point meadow colored in green. The
triangulation is equivalent as that of a square times that of a pentagon,
times three more propagators, giving the 6 propagators in the term. This comes
from the fact that this extended diagram is in bijection with a 9-point double
ordered amplitude, as wee will explain in section 4.

###  2.3 Bijection Betwen Triangulations of Extended Diagrams and $\phi^{4}$
Trees

Let us now write the main result of this construction more formally.

######  Theorem 2.4.

For each $n\geq 2$ the set of triangulations of extended diagrams is in
bijection with the set of 4-valent planar trees with $n$ leaves with leaves
labelled cyclically $1,2,\ldots,n$ according to the planar structure.
Consequently, the extended diagrams give a partition of the $\phi^{4}$
amplitude by collecting tree-level Feynman diagrams according to the extended
diagram whose triangulation gives the tree under this bijection.

Figure 12: Sequence of steps presented in the proof. First row: triangulated
extended diagram for $n=10$ together with the $\phi^{4}$ tree that it
corresponds to. Second row: the quadrangulation of the 10-gon that is
associated to the tree, also represented after cutting edge 1. On the right we
have included a black chord from 2 to 1. Uncolored chords are represented as
dashed lines. Third row: we color the chords using the procedure explained in
the proof. Sequences of black arrows count an odd number of steps, while
sequences of red arrows count an even number of steps, including zero steps
(represented as a dot). Double sided arrows indicate pairs of consecutive
vertices with no black chords as endpoints, thus leading to black chords
joining them.

###### Proof.

We will prove the bijection by exhibiting the function, the purported inverse,
and then checking the purported inverse is a two sided-inverse, which implies
the injectivity and surjectivity of both functions.

The function from triangulations of extended diagrams to leaf-labelled planar
4-valent trees is as described above. It is clear that this function is well-
defined and maps to the correct set with $n$ as described in the statement.

The key step is to describe the inverse function. Dual to a planar 4-valent
tree with $n$ leaves cyclically labelled $1,2,\ldots,n$ is a quadrangulation
of an $n$-gon with the edges cyclically labelled $1,2,\ldots,n$. Shifting each
label clockwise to the neighbouring vertex and then cutting the edge between
vertex $1$ and $2$ (which before the label shift had been edge $1$), we can
draw the remaining edges of the $n$-gon horizontally on a line with vertices
labelled $2,3,\ldots,n,1$ from left to right and the internal edge of the
quadrangulation becoming arcs. This diagram is the triangulated extended
diagram with two exceptions. First, the chords of length 2 are missing and
second the chords which will be internal chords in the extended diagram must
be distinguished from the triangulating chords.

This will be done by a recursive parity rule. For convenience we will call the
chords of the extended diagram (whether internal or external) _black_ and the
triangulating chords _red_ , lining up with the conventions in our figures.

First add the chord from $2$ to $1$ as a black chord. Call $2$ the left root
and $1$ the right root. Count inwards from the left root until reaching a
vertex incident to a chord that is not yet colored. If an even number of steps
(including potentially no steps) were taken to reach this vertex then color
all the uncolored chords at this vertex red and repeat. If an odd number of
steps were taken to reach this vertex then color the outermost chord at this
vertex black, jump to the other end of this black chord, counting this as one
step, and continue. Stop when the right root is met. Likewise count inwards
from the right root until reaching a vertex incident to a chord that is not
yet colored, coloring and proceeding as in the other case but with the
stopping condition being if the right root is met.

We claim that this process is unambiguous in the sense that if a chord is
reached from both directions then it will get the same color from each side.
The claim holds because in a quadrangulation the two vertices at the ends of
any chord must have opposite parity, but our left and right root also had
opposite parity, so counting from each one the parity to the nearest ends of
another chord will be the same. Furthermore, jumping from one end of a chord
to another changes the parity, so counting it as one step when jumping across
a black chord preserves parity considerations.

For each chord colored black in the process above, repeat the process on the
region inside this chord with the leftmost endpoint of the chord as the left
root and the rightmost endpoint of this chord as the left root.

Recursively this colors all the chords either red or black. Furthermore, the
black chords have a nested structure in which the black chords immediately
inside a particular black chord have their left and right endpoints an odd
number of steps from the left and right endpoints of the surrounding black
chord. This implies two things, first no black chords share an endpoint, and
second consecutive sequences of vertices with no endpoint of a black chord
must be of even length. For each such sequence of vertices, add a black chord
to the first and second vertex, the third and fourth vertex, and so on.

This describes the purported inverse function. Now we will make a few remarks
about it. By the observations of the previous paragraph, the black chords
define a chord diagram, and by construction there is a chord from $2$ to $1$
so this is an extended diagram in the sense of this paper. By the parity
conditions on red edges, the red edges are valid triangulation chords and the
result is a quadrangulation so all the meadows are 3-point meadows. Therefore,
the purported inverse function does define a triangulated extended diagram.

It remains to show that the functions are inverses in both directions.
Beginning with a tree, taking the purported inverse as described above and
then taking the original funtion back to trees clearly gives the identity
because the quadrangulation dual to the tree remains intact throughout, with
the chords of length 2 not affecting the tree in the original function. The
other direction is a bit more intricate. Begin with a triangulated extended
diagram. Taking the original function to obtain a tree and then applying the
purported inverse function, we have the same underlying quadrangulation since
it is simply the dual to the tree. What we need to check is that the coloring
of the chords is consistent with the original and that the size 2 chords are
added in a way that is consistent with the original. Since the internal black
chords determine the length two chords, it suffices to check that the internal
chords are correctly colored. This we prove following the recursive structure
of the map (which could be set up as a formal induction should the reader care
to). The other black chords of the meadow touching the chord from $2$ to $1$
must all have an odd left endpoint and an even right endpoint, and so are
correctly colored black in the first iteration of the algorithm defining the
purported inverse, while all the other chords in the meadow are correctly
colored red. Now looking at each meadow adjacent to this meadow notice that
the parity of left and right endpoints has flipped, both of the outer chord
and the inner ones. This leaves the relative parity unchanged and so the
algorithm correctly colors the chords in this meadow and so on.

This proves that the purported inverse really is a two-sided inverse to the
original map and hence completes the proof of the theorem. ∎

In this section we have seen how to compute $\phi^{4}$ amplitudes just by
triangulating the extended diagrams that characterize the regions in
$\textrm{Trop}^{+}G(2,n)$ that turn into distributions in the global Schwinger
formula for $\phi^{4}$. In the next section we show how to analogously compute
$\phi^{p}$ amplitudes for general $p$, by triangulating extended non-crossing
$(p-2)$-chord diagrams. But before finishing with this section, we proceed to
proof Conjecture 2.2.

###  2.4 Proof of Conjecture 2.2

The aim of this section is to prove Conjecture 2.2. We need some preliminary
definitions and lemmas.

######  Definition 2.5.

Let $x_{0},x_{1},\ldots,x_{n-3}$ be totally ordered. Write

$\displaystyle k_{i}$ $\displaystyle=\max\\{j:x_{i-\ell}\geq x_{i}\text{ for
}0\leq\ell\leq i\\},$ $\displaystyle\ell_{i}$
$\displaystyle=\max\\{j:x_{i+\ell}\geq x_{i}\text{ for }0\leq\ell\leq i\\}.$

The idea of $k_{i}$ and $\ell_{i}$ is as follows. Write the string
$x_{0}x_{1}\cdots x_{n-3}$. Then immediately before $x_{i}$ there are $k_{i}$
larger elements (after that either we run out of numbers or find a smaller
one) and immediately after $x_{i}$ there are $\ell_{i}$ larger elements (after
that either we run out of numbers or find a smaller one).

######  Definition 2.6.

Given a non-crossing chord diagram $C$ on $2,3,\ldots,n,1$ let $R_{C}$ be the
region corresponding to $C$ according to Conjecture 2.2, that is, let $R_{C}$
be the region in $\mathbb{R}^{n-2}$ with $x_{a-3}=x_{b-3}$ for every chord
$\theta_{ab}$ of $C$ and with $x_{a-3}=x_{b-3}<x_{c-3}=x_{d-3}$ whenever chord
$\theta_{ab}$ surrounds chord $\theta_{cd}$ in $C$.

The key to the proof is to understand the behaviour of the explicit formula
for $H(x)$ given in (1.9). This formula is an alternating sum of terms of the
form $\min\\{x_{a},x_{a+1},\ldots,x_{b}\\}$. We refer to each such term as a
_window_ of length $b-a+1$ and if $x_{i}$ is the minimum of the $x$s in that
window, then we say that window _contributes_ $x_{i}$. The contribution of
$x_{i}$ to $R(x)$ as a whole is the sum, with signs and factors of $2$ as in
(1.9), of the terms to which it contributes.

######  Lemma 2.1.

Let $x_{0},x_{1},\ldots,x_{n-3}$ be totally ordered. With $k_{i}$ and
$\ell_{i}$ as in Definition 2.5, then the contribution of $x_{i}$ to $R(x)$ is

$\begin{cases}x_{i}&\text{if $k_{i}$, $\ell_{i}$ are both even}\\\
-x_{i}&\text{otherwise}\end{cases}$

###### Proof.

The number of length $j$ windows which contribute $x_{i}$ to $R(x)$ is the
number of length $j$ windows beginning at or after $\max\\{i-j+1,i-k_{i}\\}$
and ending at or before $\min\\{i+j-1,i+\ell_{i}\\}$, which is

$\displaystyle\min\\{i+j-1,i+\ell_{i}\\}-\max\\{i-j+1,i-k_{i}\\}-(j-2)$
$\displaystyle=\min\\{j-1,\ell_{i}\\}-\max\\{-j+1,-k_{i}\\}-(j-2)$
$\displaystyle=\min\\{j-1,\ell_{i}\\}+\min\\{j-1,k_{i}\\}-(j-2).$

Fix $i$ and write $k$ and $\ell$ for $k_{i}$ and $\ell_{i}$ respectively, to
keep the notation lighter. Reversing the order if necessary, we can, without
loss of generality suppose $k\leq\ell$. Then we directly compute the
contribution of $x_{i}$ to $R(x)$ to be

$\displaystyle
x_{i}+2x_{i}\sum_{j=2}^{k+1}(-1)^{j-1}j+2x_{i}\sum_{j=k+2}^{\ell+1}(-1)^{j-1}(k+1)+2x_{i}\sum_{j=\ell+2}^{k+\ell+1}(-1)^{j-1}(k+\ell-j+2)$
$\displaystyle=x_{i}+2x_{i}\sum_{j=2}^{k+1}(-1)^{j-1}j+2x_{i}(k+1)\sum_{j=2}^{\ell-k+1}(-1)^{j+k-1}+2x_{i}\sum_{j=2}^{k+1}(-1)^{j+\ell-1}(k+\ell-j+2)$
$\displaystyle=x_{i}+2x_{i}\sum_{j=2}^{k+1}\left((-1)^{j-1}+(-1)^{j+\ell}\right)+2x_{i}(k+1)\sum_{j=2}^{\ell-k+1}(-1)^{j+k-1}+2x_{i}(k+2)\sum_{j=2}^{k+1}(-1)^{j+\ell-1}$
$\displaystyle=\begin{cases}x_{i}+0+0+0&\text{if $\ell$ even and $k$ even}\\\
x_{i}+0+2x_{i}(k+1)-2x_{i}(k+2)&\text{if $\ell$ even and $k$ odd}\\\
x_{i}+2x_{i}\cdot 2\cdot\frac{k}{2}-2x_{i}(k+1)+0&\text{if $\ell$ odd and $k$
even}\\\ x_{i}-2x_{i}\cdot 2\cdot\frac{k+3}{2}+0+2x_{i}(k+2)&\text{if $\ell$
odd and $k$ odd}\end{cases}$ $\displaystyle=\begin{cases}x_{i}&\text{if $\ell$
even and $k$ even}\\\ -x_{i}&\text{if $\ell$ even and $k$ odd}\\\
-x_{i}&\text{if $\ell$ odd and $k$ even}\\\ -x_{i}&\text{if $\ell$ odd and $k$
odd}\end{cases}$

as desired. ∎

######  Definition 2.7.

Given $x_{0},x_{1},\ldots,x_{n-3}\in\mathbb{R}$ we will call a non-crossing
chord diagram on $\\{3,4,\ldots,n\\}$ a _bounding chord diagram_ for these
$x_{i}$s if there is some total order which refines the order structure of
$x_{0},x_{1},\ldots,x_{n-3}$ as elements of $\mathbb{R}$ (that is, if any
$x_{i}$ are equal, the total order chooses one to be larger and otherwise
agrees with the order of the $x_{i}$ as elements of $\mathbb{R}$) such that,
with respect to this total order, the following two properties hold:

  * •

Suppose $\theta_{ab}$ is a chord and $a<b$. If $x_{a-3}<x_{b-3}$, then
$k_{b-3}$ and $\ell_{b-3}$ are both even while $\ell_{a-3}$ is odd, while if
$x_{a-3}>x_{b-3}$, then $k_{a-3}$ and $\ell_{a-3}$ are both even while
$k_{b-3}$ is odd.

  * •

Whenever a chord $\theta_{ab}$ surrounds another chord $\theta_{cd}$ then
$x_{a-3},x_{b-3}<x_{c-3},x_{d-3}$.

Note that the second condition on a bounding chord diagram is the same as the
second condition in the conjecture. The first condition is designed so that,
using the notation of Lemma 2.1, $k_{b-3}$ and $\ell_{b-3}$ are even and
$\ell_{a-3}$ is odd for $x_{a-3}<x_{b-3}$ and analogously for
$x_{a-3}>x_{b-3}$.

######  Lemma 2.2.

For any $x_{0},x_{1},\ldots,x_{n-3}\in\mathbb{R}$ there is at least one
bounding chord diagram and the bounding chord diagrams are determined only by
the order structure of $x_{0},x_{1},\ldots,x_{n-3}$ as elements of
$\mathbb{R}$.

###### Proof.

The definition of a bounding chord diagram only sees the order structure of
the $x_{i}$ as elements of $\mathbb{R}$ so the second part of the lemma is
immediate.

For the first part the proof will be by induction. The base case with one
chord is apparent. For the inductive case, choose a total order extending the
order structure of $x_{0},\ldots,x_{n-3}$. Using the notation of Definition
2.5, consider $x_{0}$. We have $k_{0}=0$.

Case 1: Suppose first that $\ell_{0}$ is even. In this case put a chord from
$3$ to $\ell_{0}+4$.

Note that $x_{\ell_{0}+1}<x_{0}$ in the total order and
$k_{\ell_{0}+1}=\ell_{0}+1$ which is odd so the first condition holds for this
chord. By construction there are an even number of $x_{i}$ inside the chord
$\theta_{3,\ell_{0}+4}$ and all these $x_{i}$ inside this chord have larger
values than $x_{0}$ so the values of $k_{i}$ and $\ell_{i}$ are the same
whether considered on $x_{0},\ldots,x_{n-3}$ or on
$x_{1},\ldots,x_{\ell_{0}}$, so we can apply the induction hypothesis to
$x_{1},\ldots,x_{\ell_{0}}$ to obtain a bounding chord diagram on
$4,\ldots,\ell_{0}+3$.

Now consider $x_{i}$ with $i>\ell_{0}+1$. The index $\ell_{i}$ is unchanged
whether considered on $0,\ldots,n-3$ or on $\ell_{0}+2,\ldots,n-3$. For the
index $k_{i}$, if $x_{i}>x_{\ell_{0}+1}$ then $k_{i}$ is also unchanged
between $0,\ldots,n-3$ and $\ell_{0}+2,\ldots,n-3$ while if
$x_{i}<x_{\ell_{0}+1}$ then either $k_{i}$ is unchanged or it is increased by
$\ell_{0}+2$. In all cases the parity of $k_{i}$ is unchanged. Apply the
induction hypothesis to $x_{\ell_{0}+2},\ldots,x_{n-3}$ to obtain a bounding
chord diagram on $\ell_{0}+5,\ldots,n$. Since the defining conditions on
bounding chord diagrams only involves the $\ell_{i}$ and $k_{i}$ via their
parity, these chords still satisfy the bounding condition on
$x_{0},\ldots,x_{n-3}$.

All together this builds a bounding chord diagram on $x_{0},\ldots,x_{n-3}$.

Case 2: Now suppose $\ell_{0}$ is odd. Let $i_{1},i_{2},\ldots,i_{m}$ be the
indices where minima within prefixes of $x_{1},\ldots,x_{\ell_{0}}$ are
achieved, that is, $x_{i_{j}}<x_{1},\ldots,x_{i_{j}-1}$. Note that $i_{1}=1$
since the prefix of size 1’s minimum is its one element. Then we have
$k_{i_{j}}=i_{j}-1$ and $\ell_{i_{j}}=i_{j+1}-i_{j}-1$ with the convention
that $i_{m+1}=\ell_{0}+1$. If all the $\ell_{i_{j}}$ were odd then all the
differences $i_{j+1}-i_{j}$ would be even and so summing these differences
would give $i_{m+1}-i_{1}=\ell_{0}+1-1=\ell_{0}$. But $\ell_{0}$ is odd, so
cannot be the sum of even numbers. Therefore at least one $\ell_{i_{j}}$ is
even. Let $j$ be minimal with $\ell_{i_{j}}$ even. Then
$k_{i_{j}}=i_{j}-1=(i_{2}-i_{1})+(i_{3}-i_{2})+\cdots+(i_{j}-i_{j-1})$ which
is a sum of even numbers by the minimality of $j$ and hence is itself even.
Therefore, we can put in the chord $\theta_{3,i_{j}+3}$ and this chord will
satisfy the conditions of Definition 2.7.

Apply the induction hypothesis to obtain a bounding chord diagram on
$x_{1},\ldots,x_{i_{j}-1}$. All the elements strictly between $x_{0}$ and
$x_{i_{j}}$ are larger than $x_{0}$ and $x_{i_{j}}$ so the $k$s and $\ell$s
for these elements are unchanged when viewed on $x_{0},\ldots x_{n-3}$ and
hence these chords continue to satisfy the condition of a bounding chord
diagram when viewed on $x_{0},\ldots,x_{n-3}$.

Also apply the induction hypothesis to obtain a bounding chord diagram on
$x_{i_{j}+1},\ldots,$ $x_{n-3}$. The $\ell$s for these elements are unchanged
when viewed on $x_{0},\ldots,x_{n-3}$. For the $k$s, for elements beyond
$x_{\ell_{0}}$ the parity is unchanged when viewed on $x_{0},\ldots,x_{n-3}$
for the same reason as in the case when $\ell_{0}$ was even. For elements
between $x_{i_{j}+1}$ and $x_{\ell_{0}}$ (inclusive) either $k$ is unchanged
or is increased by $i_{j}+1$ which is even as discussed above. In all cases
the parity of the $k$s and $\ell$s is unchanged and so these chords continue
to satisfy the bounding condition on $x_{0},\ldots,x_{n-3}$.

This constructs a bounding chord diagram on $x_{0},\ldots,x_{n-3}$. ∎

######  Lemma 2.3.

For $x_{0},x_{1},\ldots,x_{n-3}\in\mathbb{R}$, $R(x)>0$ unless the $x_{i}$
have a bounding chord diagram $C$ with the additional property that for every
chord $\theta_{ab}$, $x_{a-3}=x_{b-3}$, and in this case the point
$(x_{0},x_{1},\ldots,x_{n-3})$ is within the region $R_{C}$.

###### Proof.

Take a bounding chord diagram for these $x_{i}$, which must exist by Lemma
2.2. If there are any equalities among the $x_{i}$ perturb their values
slightly to agree with the total order corresponding to the bounding chord
diagram. For each chord $\theta_{ab}$ with $x_{a-3}<x_{b-3}$ then by Lemma 2.1
the contribution of $x_{a-3}$ and $x_{b-3}$ to $R(x)$ is $x_{b-3}-x_{a-3}>0$
(and analogously if $x_{a-3}>x_{b-3}$ then the contribution is
$x_{a-3}-x_{b-3}>0$. All $x_{i}$ belong to some chord so $R(x)$ is a sum of
contribitions of each chord and these contribitions are all positive. Now take
limits to return to the original values of the $x_{i}$, which may have
equalities. Since the total order extended the original order structure on the
$x_{i}$, in taking these limits the contribution of each chord remains non-
negative and is zero precisely if $x_{a-3}=x_{b-3}$. Therefore $R(x)>0$ unless
$x_{a-3}=x_{b-3}$ for all chords $\theta_{ab}$, in which case $R(x)=0$ and
both conditions defining the chord diagram of the conjecture hold, hence
$(x_{0},x_{1},\ldots,x_{n-3})$ is within the region defined by this chord
diagram. ∎

In a noncrossing chord diagram, say two chords are _siblings_ if they are both
nested directly under the same chord.

######  Lemma 2.4.

Every noncrossing chord diagram defines a region where $R(x)=0$.

###### Proof.

Fix a noncrossing diagram $C$. Let $x_{0},\ldots,x_{n-3}$ be in the region
$R_{C}$. The claim is that the noncrossing diagram is a bounding diagram for
$x_{0},\ldots,x_{n-3}$. The second condition is the same between the
conjecture and the definition of a bounding diagram, so we only need to check
the first condition

Choose a total order extending the order structure on $x_{0},\ldots,x_{n-3}$
as follows. Let $\epsilon$ be smaller than the difference between any two
nonequal $x_{i}$. If multiple chords of the chord diagram have their $x_{i}$
equal add $k\epsilon/n$ to the $x_{i}$s corresponding to both ends of the
$k$th chord whose variables had this value from left to right. Then for each
chord of the chord diagram add $\epsilon/(2n)$ to the variable corresponding
to its right end point. By this construction we get a total order such that
for each chord $\theta_{ab}$ of the chord diagram with $a-3<b-3$ we have
$x_{a-3}<x_{b-3}$ and $x_{b-3}$ is the least element of the total order which
is larger than $x_{a-3}$.

For any chord $\theta_{ab}$ with $a-3<b-3$, consider
$k_{a-3},k_{b-3},\ell_{a-3},\ell_{b-3}$. Note that all chords in a noncrossing
chord diagram are of even length. By the choice of the total order
$k_{b-3}=b-a-2$ which is even. $\ell_{b-3}$ must also be even, since moving
rightwards, the first element we encounter which is less than $x_{b-3}$ is
either the left endpoint of a sibling of $\theta_{ab}$ (anything nested within
a sibling is larger than the left endpoint of the sibling) or is the left
endpoint of the chord immediately above $\theta_{ab}$, so the number of
elements passed on the way is the entirety of 0 or more siblings and the
chords they contain, which is even. Finally, $\ell_{a-3}=\ell_{b-3}+k_{b-3}+1$
which is odd since we chose our total order so that no elements are between
$x_{a-3}$ and $x_{b-3}$. Therefore $\theta_{ab}$ satisfies the definition of a
bounding diagram for $x_{0},\ldots,x_{n-3}$, but this holds for all chords,
and so this chord diagram is a bounding chord diagram.

Hence, by the second part of Lemma 2.3 $R(x)=0$ for this point
$x_{0},\ldots,x_{n-3}$, but this was an arbitrary point in the region so
$R(x)=0$ on the region. ∎

###### Proof of Conjecture 2.2.

By Lemma 2.3 and Lemma 2.4, $R(x)=0$ if and only if $x$ is in one of the
regions defined by a noncrossing chord diagram as in the statement of the
conjecture. ∎

Note, we can give a direct combinatorial proof that $H(x)$ is invariant under
an overall shift of all the variables (a fact that we we also know by physical
arguments, but that we did not use above) as follows. Suppose we shift all
variables by adding $y$. This shifts all minima by $y$ and so, using that $n$
is even,

$\displaystyle H(x+y)$
$\displaystyle=H(x)+\sum_{a=0}^{n-3}y+2\sum_{a=0}^{n-4}\sum_{b=a+1}^{n-3}(-1)^{b-a}y$
$\displaystyle=H(x)+y(n-3)+2y\sum_{a=0}^{n-4}\sum_{c=1}^{n-a-3}(-1)^{c}$
$\displaystyle=H(x)+y(n-3)+2y\sum_{a=0}^{n-4}\begin{cases}-1&\text{if $n-a-3$
odd}\\\ 0&\text{if $n-a-3$ even}\end{cases}$
$\displaystyle=H(x)+y(n-3)+2y\sum_{\begin{subarray}{c}0\leq a\leq n-4\\\
a\text{ even}\end{subarray}}-1$
$\displaystyle=H(x)+y(n-3)-2y\left(\frac{n-4}{2}+1\right)$
$\displaystyle=H(x)$

##  3 $\phi^{p}$ from Triangulations of Extended Diagrams

For general $\phi^{p}$ theories, the global Schwinger formulation presented in
Cachazo:2022voc  is analogous to that of $\phi^{4}$ but now the regions that
compute $A_{n}^{\phi^{p}}$ are given by non-crossing $(p-2)$-chord diagrams,
see definition below, which are counted by $\textrm{FC}_{(n-2)/(p-2)}(p-2,1)$.
Such regions are of dimension $n/(p-2)-1$ in $\mathbb{R}^{n-3}$. The sum over
all regions leads to the expected amplitude $A_{n}^{\phi^{p}}$, where now each
region is connected to a double-ordered cubic amplitude
$m_{(n+2(p-3))/(p-2)}(\alpha,\beta)$, as we will comment in section 5. We
refer the reader to Cachazo:2022voc  for more details on the global Schwinger
formula for $\phi^{p}$ theories and on the role non-crossing $(p-2)$-chord
diagrams play in this formulation.

In this section, by following the same philosophy as in section 2, we perform
the study of $\phi^{p}$ theories. In particular, we will show how we can
compute the $\phi^{p}$ amplitude by triangulating an extended version of non-
crossing $(p-2)$-chord diagrams, without ever having to compute the integral
of the corresponding global Schwinger formula. The number of triangulated
extended diagrams coincides with the number of $\phi^{p}$ trees, which are
counted by $\textrm{FC}_{(n-2)/(p-2)}(p-1,1)$.

To start with, let us recall the definitions of $(p-2)$-chord diagrams and
their extension, which in this paper we will call again extended diagrams for
simplicity. Notice that we will slightly change again the convention on the
labelling with respect to Cachazo:2022voc  for convenience.

######  Definition 3.1.

Place $n-2$ points labeled $3,4,\ldots,n$ on the real line in increasing
order. A non-crossing $(p-2)$-chord diagram is a partition of the points into
sets of size $p-2$ where each part we call a $(p-2)$-chord, and furthermore,
where the $(p-2)$-chords can be drawn on the upper half plane without any
crossings. Let us denote the $(p-2)$-chord connecting points
$a_{1},a_{2},\ldots,a_{p-2}$ as $\theta_{a_{1},a_{2},...,a_{p-2}}$. We also
call $\Theta_{ab}$ to the unique path in a $(p-2)$-chord joining two points
$a$ and $b$.

Note, this notion of non-crossing agrees with the usual combinatorial notion
of non-crossing set partition, which can be defined rigorously by the
condition that there do not exist $a<b<c<d$ with $a,c$ in one part and $b,d$
in another.

######  Definition 3.2.

An extended diagram in this context is a non-crossing $(p-2)$-chord diagram on
$n$ points labelled by $\\{2,3,4,5,\ldots,n,1\\}$ in which $\Theta_{21}$ is
always included555In the definition presented in Cachazo:2022voc  this chord
was a $(p-2)$-chord. In this paper we use a 2-chord to surround all others for
simplicity. We will therefore abuse notation since, strictly speaking, now the
extended diagram is not a $(p-2)$-chord diagram anymore. One can imagine that
we are still talking about a $(p-2)$-chord diagram with an additional
$(p-2)$-chord surrounding all others but only the lower part of it, i.e. the
path $\Theta_{21}$, is what matters for the argument.. We also define a meadow
of an extended diagram as any region in the diagram delimited by more than one
chord and by the line where the points lie. A meadow is an $m$-point meadow if
there are $m-1$ $(p-2)$-chords delimiting it.

In section 5.2 we will comment on the fact that a meadow which is delimited by
$m-1$ such paths and the real line corresponds to a biadjoint $m$-subamplitude
participating in $m_{(n+2(p-3))/(p-2)}(\alpha,\beta)$. Moreover, the upper
boundary of a meadow, $\Theta_{ab}$, actually corresponds to a propagator of
the form $1/X_{a,b}$, with the exception of $\Theta_{21}$.

Now let us proceed to explain how to triangulate an extended diagram. The
rules are analogous to those seen in section 2 for $\phi^{4}$, where the
triangulating chords are all $2$-chords:

  * •

A triangulating chord $\Omega_{ab}$ must start, reading from left to right,
from the label $a$ on the right of a lower $(p-2)$-chord of the meadow or on
the left of the upper path $\Theta_{ad}$ in the meadow, and must end on the
label $b$ on the left of a lower chord of the meadow or on the right of
$\Theta_{cb}$.

  * •

A triangulating chord must surround at least another $(p-2)$-chord in the
diagram.

  * •

The triangulation ends when only regions with 4 delimiting chords and lines
are left.

Again these are called triangulations because they will correspond to
triangulations in the context of section 5.2. Once the triangulation of an
extended diagram is done, one can easily compute its contribution to the
$\phi^{p}$ amplitude in the following way:

  * •

Every already existing path $\Theta_{ab}$ which defines the upper path of a
meadow, with the exception of $\Theta_{21}$, and every triangulating chord
$\Omega_{ab}$ correspond to a propagator of the form $1/X_{a,b}$.

  * •

The contribution of a triangulated diagram to the amplitude is given by
multiplying all propagators involved.

Now we proceed to show some examples to see how the triangulation works. We
start with $\phi^{6}$ and $n=10$, which involves four extended diagrams. In
figure 13 the $4$-chords are shown in black, the different meadows are shaded
in colors and the triangulating chords are the red ones.

Figure 13: Triangulated extended diagrams for $n=10$ and their corresponding
$\phi^{6}$ tree. The triangulating chords are shown in red, and the colored
regions correspond to different meadows. There are four extended diagrams in
total, which give rise to five possible triangulations.

The first row of extended diagrams in figure 13 are those which only consist
of 3-point meadows and are therefore already triangulated. For example, the
first extended diagram has two 3-point meadows –shown in gray and blue– and
only one propagator associated to the upper boundary of the blue meadow, i.e.
$\Theta_{49}$, which corresponds to a contribution of the form $1/X_{4,9}$,
the same as the $\phi^{6}$ tree below. One can do a similar analysis for the
other two diagrams in the first row.

The second row of figure 13 shows the two possible triangulations of the
remaining extended diagram, which contains a single 4-point meadow shown in
gray. The first diagram in the second row has a triangulating chord of the
form $\Omega_{27}$, which gives rise to a propagator of the form $1/X_{2,7}$,
the same given by the $\phi^{6}$ tree below. Similarly, the second diagram has
a triangulating chord $\Omega_{61}$, which corresponds to a propagator of the
form $1/X_{1,6}$ which is also given by the tree below.

Summing over the five triangulated extended diagrams is equivalent to summing
over the five existing $\phi^{6}$ trees for $n=10$, and one obtains the
amplitude $A_{10}^{\phi^{6}}$.

Now let us see how by triangulating the extended diagrams for $\phi^{5}$ and
$n=11$ leads to the full amplitude. In this case, there are 12 extended
diagrams in total. First of all, we show in figure 14 all the 5 extended
diagrams which only consist of 3-point meadows and are therefore already fully
triangulated.

Figure 14: The 5 extended diagrams with only 3-point meadows, colored in
different ways, for $n=11$ and the corresponding $\phi^{5}$ tree shown below.

For example, the first diagram in figure 14 has three 3-point meadows with
upper boundaries of the form $\Theta_{3,10}$ and $\Theta_{48}$, which
correspond to the propagators $1/X_{3,10}$ and $1/X_{4,8}$, the same appearing
in the $\phi^{5}$ tree below. Hence, this diagram contributes

$\frac{1}{X_{3,10}X_{4,8}}$

to the amplitude. As another example, the last diagram contains three 3-point
meadows with upper boundaries $\Theta_{37}$ and $\Theta_{7,11}$ corresponding
to the propagators $1/X_{3,7}$ and $1/X_{7,11}$, which one can also read from
the tree below, and it contributes

$\frac{1}{X_{3,7}X_{7,11}}$

to the amplitude. For the $\phi^{5}$ and $n=11$ case, we also have 6 extended
diagrams with one 3-point meadow and one 4-point meadow, and therefore
$6\times 2=12$ possible triangulations, as shown in figure 15.

Figure 15: The 6 extended diagrams in $n=11$ with one 3-point meadow and one
4-point meadow. Every row corresponds to an extended diagram, and the two
columns correspond to the two possible triangulations, with the corresponding
$\phi^{5}$ tree shown on the right.

In this case, every extended diagram has two possible triangulations, due to
the fact that each extended diagram contains one 4-point meadow. The extended
diagram on the first row in figure 15 has an upper boundary of the form
$\Theta_{6,10}$ in the blue meadow, associated to a propagator $1/X_{6,10}$.
Each of its triangulations, corresponding to every column in the first row,
has a red triangulating chord of the form $\Omega_{26}$ and $\Omega_{51}$,
respectively, giving rise to propagators of the form $1/X_{2,6}$ and
$1/X_{1,5}$.

Therefore, the contribution of the first extended diagram in figure 15 to the
$\phi^{5}$ amplitude is given by

$\frac{1}{X_{6,10}}\left(\frac{1}{X_{2,6}}+\frac{1}{X_{1,5}}\right)\,,$

which is the same given by summing over the two trees on the first row of
figure 15. Again, this expression resembles that of a double-ordered 5-point
cubic amplitude $m_{5}(\alpha,\beta)$, for some orderings $\alpha$ and
$\beta$. Finally, one can perform a similar analysis to the rest of the
extended diagram and their triangulations.

There is one more extended diagram in $\phi^{5}$ and $n=11$ that we have not
talked about yet, which corresponds to the one containing a single 5-point
meadow. We show its 5 possible triangulations in figure 16.

Figure 16: The extended diagram in $n=11$ with one 5-point meadow has 5
possible triangulations, shown together with the corresponding $\phi^{5}$ tree
below.

The triangulation corresponding to the left diagram in the first row of figure
16, has two triangulating chords $\Omega_{26}$ and $\Omega_{81}$. Therefore,
its contribution to the $\phi^{5}$ amplitude is given by

$\frac{1}{X_{2,6}X_{1,8}}\,,$

which is the same as that of the $\phi^{5}$ tree below the triangulated
diagram. By a similar analysis done for the rest of the triangulations of this
extended diagram one ends up with its contribution to $A_{11}^{\phi^{5}}$,
given by

$\frac{1}{X_{2,6}X_{1,8}}+\frac{1}{X_{2,9}X_{2,6}}+\frac{1}{X_{2,9}X_{5,9}}+\frac{1}{X_{5,9}X_{1,5}}+\frac{1}{X_{1,5}X_{1,8}}\,,$

which resembles that of a 5-point diagonal amplitude $m_{5}(\alpha,\alpha)$.
This is not a coincidence, as we will discuss in section 5.2.

To end with this example, one can sum over the contributions of all the
FC${}_{3}(3,1)=12$ extended diagrams, which are given by $5\times 1+6\times
2+1\times 5=$FC${}_{3}(4,1)=22$ triangulations –which are in correspondence
with the 22 $\phi^{5}$ trees in $n=11$–, to end up with the expected
$A_{11}^{\phi^{5}}$ amplitude.

##  4 Comments on Relation with Cubic Amplitudes

One of the powers of the global Schwinger formula is that it provides a
framework to discover some properties of scattering amplitudes which are hard
to see from the standard Feynman diagram formulation. For example, in section
2 we saw how extended diagrams contribute to the $\phi^{4}$ amplitude in a
form which resembles that of a double-ordered cubic amplitude
$m_{n/2+1}(\alpha,\beta)$, as first noticed in Cachazo:2022voc . In fact, in
this work it was proven that, for a given number of particles $n$, extended
diagrams consisting of a single $(n/2+1)$-point meadow666This corresponds to
the region in $\textrm{Trop}^{+}G(2,n)$ coming from the non-crossing chord
diagram in which all the chords join two consecutive points, i.e.
$\\{x_{0}=x_{1},x_{2}=x_{3},...,x_{n-4}=x_{n-3}\\}$. are in bijection with a
diagonal amplitude $m_{n/2+1}(\alpha,\alpha)$ for some bijection of the
kinematic invariants.

In Cachazo:2013iea  it was shown that double-ordered cubic amplitudes can be
expressed as a product of cubic subamplitudes, using a polygon decomposition
that we review in the next subsection. Therefore, in Cachazo:2022voc  it was
proposed that $\phi^{4}$ amplitudes could be expressed as a sum of products of
cubic amplitudes, and a formula for the general schematic structure of
$A_{n}^{\phi^{4}}$ was given based on the Lagrange inversion procedure. For
example, using this formula one could write

$A_{8}^{\phi^{4}}=m_{5}+3m_{3}m_{4}\frac{1}{P^{2}}+m_{3}^{3}\left(\frac{1}{P^{2}}\right)^{2}\,,$

where $m_{i}$ is a cubic subamplitude with $i$ points and $1/P^{2}$ is a
generic propagator.

Going back to the $n=8$ example in section 2, we know that in this case we
have an extended diagram consisting of a single 5-point meadow, three extended
diagrams consisting of a 3-point meadow together with a 4-point meadow, and
one extended diagram consisting of three 3-point meadows, as shown in figure
7. This is precisely the schematic structure of $A_{8}^{\phi^{4}}$ shown
above, if one relates the $m$-point meadows with the subamplitudes in the
expression. Moreover, the number of propagators coincides with the number of
chords separating different meadows in the extended diagrams.

In Cachazo:2022voc  it was also pointed out that extended diagrams for general
$\phi^{p}$ theories are also connected to double-ordered amplitudes
$m_{(n+2(p-3))/(p-2)}(\alpha,\beta)$, and a formula for the general schematic
structure of $A_{n}^{\phi^{p}}$ in terms of products of cubic subamplitudes
was also given.

In this section we delve more into the connection between extended diagrams in
$\phi^{4}$ and double-ordered amplitudes. In fact, we propose a combinatorial
way to relate any extended diagram with such a schematic structure in lines
with the standard polygon decomposition used in double-ordered amplitudes,
hence extending the results of Cachazo:2022voc . However, we must mention that
our new proposed polygon-decomposition does not generate the two permutations
$\alpha$ and $\beta$ of the double-ordered amplitude. Instead, it provides an
alternative way to understand an extended diagram as a polygon-decomposed
object, with different triangulating rules to produce the desired amplitude,
which are of course equivalent to the triangulating rules of extended
diagrams, hence justifying the term triangulation in the extended diagram
case.

###  4.1 Review of the Polygon Decomposition of $m_{n}(\alpha,\beta)$

In section 1 we saw the definition of what double-ordered cubic
amplitudes777In the literature these are also known as off-diagonal
amplitudes. $m_{n}(\alpha,\beta)$ are. In this subsection we review a very
efficient diagrammatic procedure to compute them, first introduced in
Cachazo:2013iea , which will turn out to be very convenient to explain the
main point of this section.

The procedure is the following. Given $m_{n}(\alpha,\beta)$, which is defined
by any two permutations $\alpha$ and $\beta$, we first draw points on a circle
following the permutation $\alpha$. We next join the points according to the
ordering defined by $\beta$ inside the circle888Or viceversa. Notice that
$m_{n}(\alpha,\beta)=m_{n}(\beta,\alpha)$.. The resulting object is a set of
polygons which correspond to cubic subamplitudes. Each vertex in the polygon
is dual to a leg of a tree that would appear after triangulating the polygon
(note this is a slightly different notion of duality to that giving the dual
quadrangulation used in section 2.3). Also, the internal vertices joining two
subamplitudes define propagators.

There is also a global sign that multiplies the double-ordered amplitude. We
can obtain the sign e.g. by following the convention of Mizera:2016jhj , which
consists of computing the relative winding number $w_{\alpha,\beta}$ between
the two permutations. The overall sign is given by
$(-1)^{1+w_{\alpha,\beta}}$.

Figure 17: The permutations $\alpha=(12678345)$ in blue and
$\beta=\mathbb{I}=(12345678)$ in black.

It is instructive to see an example of how the whole procedure works.
Consider, e.g., the double-ordered amplitude $m_{8}(12678345,\mathbb{I})$. The
two permutations are pictorially represented in figure 17. Notice from this
figure that we can see four different polygons whose interior has been shaded,
i.e. two triangles and two squares, and three internal vertices. This means
that this double-ordered amplitude has a schematic form

$m_{8}(12678345,\mathbb{I})=m_{3}^{2}m_{4}^{2}\left(\frac{1}{P^{2}}\right)^{3}\,.$

The square formed by vertices 3, 4, 5 and an internal one corresponds to a
4-point subamplitude given by

$\frac{1}{s_{34}}+\frac{1}{s_{45}}\,.$

Similarly, the square formed by vertices 6, 7, 8 and another internal one also
corresponds to a 4-point subamplitude, in this case

$\frac{1}{s_{67}}+\frac{1}{s_{78}}\,.$

The two triangles correspond to 3-point subamplitudes, which are constants
that we take as 1 by convention. There are also 3 internal vertices, which are
propagators of the form $1/s_{12}$, $1/X_{3,6}$ and $1/X_{1,6}$. Moreover, the
relative winding number in this case is given by $w_{\alpha,\beta}=2$ and the
amplitude is, therefore,

$m_{8}(12678345,\mathbb{I})=-\frac{1}{s_{12}X_{3,6}X_{1,6}}\left(\frac{1}{s_{34}}+\frac{1}{s_{45}}\right)\left(\frac{1}{s_{67}}+\frac{1}{s_{78}}\right)\,.$

###  4.2 Polygon Decomposition of a $\phi^{4}$ Extended Diagram

In this subsection we start by proposing a polygon-decomposition like
procedure for an extended diagram participating in $\phi^{4}$ theories,
motivated by the nested structure that extended diagrams have in terms of
meadows. The rules to express a generic extended diagram into a polygon-
decomposed object are the following:

  * •

Given an extended diagram, we draw an $m$-gon for every $m$-point meadow in
the diagram.

  * •

For every $m$-gon, two of its vertices are given by the two labels of the
upper chord defining the meadow where it comes from. The rest of the vertices
are labelled by the pair of points joined by lower chords $\theta_{ab}$
defining the meadow, in the form $[a,b]:=a,a+1,\cdots,b-1,b$.

  * •

Every propagator $\theta_{ab}$ separating two meadows in the extended diagram
corresponds to an internal vertex, which is labelled $b$ and lies inside the
polygon corresponding to the meadow surrounding the other one from above –with
$b>a$– and which joins the two corresponding $m$-gons. These kind of vertices
lie strictly inside a circle. The rest of the vertices lie on the boundary of
the circle.

Let us see some examples to help us clarify the procedure. Recall that for
$n=6$ we have two different extended diagrams, which are shown in figure 6.
The proposed polygon decomposition is shown in figure 18.

Figure 18: Polygon decomposition of the extended diagrams in $n=6$.

The extended diagram on the left consists of a single $4$-point meadow, so its
polygon decomposition corresponds to a square, as shown below. The extended
diagram on the right consists of two 3-point meadows, shaded in gray and blue,
which give rise to the polygon decomposition shown below and which corresponds
to two triangles. Note that the triangles are joined by an internal vertex,
labelled by 6, which is the greater label in the propagator $\theta_{36}$ in
the extended diagram.

Figure 19: Triangulations of a polygon-decomposed extended diagram in $n=6$.

One might be tempted to say that the extended diagram on the left corresponds
to a diagonal amplitude $m_{{4}}(\alpha,\alpha)$ with
$\alpha=({1},{2},{34},{56})$. Similarly, the diagram on the right might
correspond to an off-diagonal amplitude of the form $m_{{4}}(\alpha,\beta)$,
with the permutations defined by $\alpha=({1},{2},{3},{45})$ and
$\beta=({1},{2},{45},{3})$. However, the $m_{{4}}(\alpha,\beta)$ amplitudes in
general cannot be read in a standard way from the permutations presented in
this picture, since there is a caveat when it comes to triangulating the
polygons.

Notice that the two possible triangulations of the extended diagram on the
left, which are those in figure 8, are equivalent to the triangulations of the
square, shown in figure 19. The triangulation on the left would give us the
expected contribution of $1/s_{{2},{34}}=1/s_{{56},{1}}$. But the
triangulation on the right, if not interpreted carefully, would give us
$1/s_{{34},{56}}=1/s_{{1},{2}}$. This is obviously not what we want, and the
reason is that the corresponding triangulating chord in the extended diagram
started from label 4 and did not include label 3.

Figure 20: Extended diagrams in $n=8$ and their polygon decomposition shown
below.

This motivates us to introduce a couple more new rules in order to read a
triangulation of a polygon-decomposed extended diagram:

  * •

We say that a triangulating chord $\Omega_{a,b}$ inside a polygon starts at a
vertex $a$ and ends at a vertex $b$ when $a<b$ in the ordering
$(2,3,4,...,n,1)$. If a triangulating chord inside a polygon starts at a
vertex labelled by a pair of points $ij$, it partitions the pair and label $j$
goes along with the following vertex in the polygon.

  * •

If an $m$-gon with vertices $\\{a_{1},a_{2},...,a_{i},...a_{m}\\}$ has an
internal vertex $a_{i}$, the propagator corresponding to that vertex is
$1/X_{{a}_{1},{a_{m}+1}}$. It is also read as $1/X_{{b}_{1},{b_{l}+1}}$, where
$\\{b_{1},b_{2},...,b_{l}\\}$ are the remaining vertices of the polygon that
shares the same internal vertex as the previous one, with
$a_{i}\notin\\{b_{1},b_{2},...,b_{l}\\}$, and which corresponds to a meadow
below the chord in the extended diagram containing $a_{i}$.

Therefore, now the triangulation on the left of figure 19 reads as
$1/s_{{2},{34}}=1/s_{{56},{1}}$ and the one on the rights reads as
$1/s_{{4},{56}}$, as we expected. Moreover, the already triangulated diagram
in figure 18 contributes as $1/s_{{3},{45}}$.

Now let us consider the more interesting case of $n=8$. In figure 20 we show
the polygon decomposition of the five extended diagrams previously shown in
figure 7, where we have colored the meadows and their corresponding n-gons in
the same way to make the correspondence more clear. The extended diagram on
the bottom-right only consists of 3-point meadows, which are given by
triangles in its polygon decomposition. Therefore, it is already triangulated.
Its contribution to the amplitude $A_{8}^{\phi^{4}}$ is then given by the
product of its two internal vertices or propagators, which are of the form
$1/X_{3,8}$ and $1/X_{4,7}$.

In figure 21 we show the five triangulations of the pentagon, which correspond
to the five triangulations of the extended diagram shown in figure 9, and
presented in the same order.

Figure 21: Triangulations of the pentagon corresponding to the extended
diagram with a single 5-point meadow in $n=8$. They are the same as those in
figure 9.

Going from left to right, top to bottom, the first triangulation gives a
contribution of the form $1/X_{4,7}\times 1/X_{1,4}$. This is because the
triangulating chords $\Omega_{34,78}$ and $\Omega_{34,1}$ start from the pair
$34$, thus splitting it into 3 and 4. Similarly, the rest of the
triangulations in figure 21 give contributions of the form

$\frac{1}{X_{2,7}X_{4,7}}\,,\hskip 5.69054pt\frac{1}{X_{2,7}X_{2,5}}\,,\hskip
5.69054pt\frac{1}{X_{1,4}X_{1,6}}\,,\hskip
5.69054pt\frac{1}{X_{2,5}X_{1,6}}\,,$

respectively. Thus summing over the five triangulations of the pentagon we
obtain the contribution $A_{8}^{\phi^{4}:(1)}$ seen in (2.1).

Figure 22: Rest of the triangulations of the polygon-decomposed extended
diagrams in $n=8$. They are the same as those in figure 10.

Figure 22 shows the triangulations for the rest of the polygon-decomposed
extended diagrams that appear in $n=8$. The first two triangulations in the
first row are those of the same polygon decomposition, and their sum gives a
contribution of

$\frac{1}{X_{5,8}}\left(\frac{1}{X_{2,5}}+\frac{1}{X_{1,4}}\right)\,,$

which agrees with that of $A_{8}^{\phi^{4}:(2)}$ seen in (2.1).

Similarly, by summing over the first two triangulations in the second row of
figure 22, which are those of the same polygon decomposition, we obtain a
contribution of

$\frac{1}{X_{3,6}}\left(\frac{1}{X_{1,6}}+\frac{1}{X_{2,7}}\right)\,,$

which agrees with that of $A_{8}^{\phi^{4}:(4)}$ seen in (2.1).

Finally, the two triangulations in the third column give a contribution of

$\frac{1}{X_{3,8}}\left(\frac{1}{X_{3,6}}+\frac{1}{X_{5,8}}\right)\,,$

which agrees with that of $A_{8}^{\phi^{4}:(3)}$ in (2.1).

Therefore, the sum of all the triangulations of the polygon-decomposed
diagrams in $n=8$ compute the full amplitude $A_{8}^{\phi^{4}}$.

To finish with the section, we will have a look at two triangulations of a
more complicated polygon decomposition. In fact, we will study that of the
extended diagram appearing in figure 11, now represented in figure 23 on the
left with the polygons in same color as their corresponding meadows.

Figure 23: Triangulations of a polygon decomposition in $n=16$. The
triangulation on the left is equivalent to that in figure 11.

In the case on the left of figure 23, we have three internal vertices which
correspond to propagators. The vertex joining the triangle $\\{16,1,2\\}$ and
the pentagon gives a propagator of the form $1/X_{3,16}$. Likewise, the vertex
joining the pentagon and the triangle with external vertices $10$ and $11,12$
corresponds to a propagator of the form $1/X_{10,13}$. Finally, the vertex
joining the pentagon and the square gives rise to $1/X_{4,9}$.

In this diagram, we have three triangulating chords, shown in red. The
triangulating chord $\Omega_{4,78}$ gives rise to $1/X_{4,7}$. The chord
$\Omega_{9,1415}$ generates $1/X_{9,14}$ and the chord $\Omega_{9,16}$
generates $1/X_{9,16}$. By multiplying all the propagators one ends up with
the same result as that seen from figure 11.

In the case on the right of figure 23, the only difference is that now the
triangulation of the square, i.e. $\Omega_{56,9}$, gives a contribution of
$1/X_{6,9}$, according to our triangulating rules.

Hence, in this subsection we have provided a polygon-decomposition pictorial
procedure to connect extended diagrams with products of cubic subamplitudes,
which slightly differs from the standard polygon decomposition for off-
diagonal amplitudes. Of course, this procedure is exactly equivalent to the
triangulations of extended diagrams. The main problem is that it is non-
obvious how $\phi^{4}$ trees coming from a triangulation of an extended
diagram can be read as $\phi^{3}$ trees contributing to the off-diagonal
amplitude. One of the main obstacles we encounter in this regard is that one
has to be careful in that the variable $x_{a-3}$ coming from the
parameterization (1.3), which we relate with the label $a$ on the real line in
extended diagrams, does not exactly correspond to a single particle $a$, as
this variable appears in all rows $r\geq a$. However, due to the similarities
with the polygonal decomposition of double-oredered amplitudes, we hope this
approach might provide a step further into finding the permutations $\alpha$
and $\beta$ of the expected amplitude from an extended diagram.

##  5 Discussions

In this note we have shown that we can obtain tree-level $\phi^{4}$ scattering
amplitudes by triangulating extended diagrams. We have proved that these
diagrams are in bijection with the regions in $\textrm{Trop}^{+}G(2,n)$
contributing to the global Schwinger formula for $A_{n}^{\phi^{4}}$. This
means that every extended diagram contains at least one $\phi^{4}$ tree. The
reason is that, in general, an extended diagram by itself does not account for
all the propagators of a tree, but needs to be triangulated. In fact, we have
also proven that every triangulated extended diagram is in bijection with a
$\phi^{4}$ Feynman diagram, and that all the triangulated extended diagrams
add up to $A_{n}^{\phi^{4}}$. This implies that every triangulating chord in
an extended diagram is a quadrangulating chord in an even n-gon. However, to
account for all the quadrangulating chords of the n-gon, i.e. the propagators
of the tree, one must also consider the chords appearing in the original non-
triangulated extended diagram. So in a sense, the collection of extended
diagrams, each of which characterizes a polyhedral cone in
$\textrm{Trop}^{+}G(2,n)$ which is related to a double-ordered cubic
amplitude, partitions the $\phi^{4}$ amplitude in a new way.

Moreover, we have also seen how triangulations of the extended diagrams that
participate in the global Schwinger formulation of $A_{n}^{\phi^{p}}$ also
lead to $\phi^{p}$ amplitudes, although we lack of a formal proof for the
general case.

An interesting property of this procedure is that extended diagrams are very
easily obtained by using a simple combinatorial operation, and so are their
triangulations. The procedure to compute $\phi^{p}$ amplitudes presented in
this work actually differs from the Stokes polytopes and accordiohedra
literature, as well as other approaches Banerjee:2018tun ; Salvatori:2019phs ;
Aneesh:2019cvt ; Srivastava:2020dly ; Raman:2019utu ; Aneesh:2019ddi ;
Kojima:2020tox ; Kalyanapuram:2020vil ; John:2020jww ; Kalyanapuram:2020axt ;
Jagadale:2021iab ; Baadsgaard:2015ifa , as pointed out in Cachazo:2022voc ,
since each of our regions/extended diagrams is isomorphic to an associahedron
or to an intersection of two associahedra with a numerical weight of 1
associated to them. It would be very interesting to make a connection between
the two approaches in the future.

It is important to mention that a global Schwinger formula for general
positive tropical Grassmannians $\textrm{Trop}^{+}G(k,n)$ also exists
Cachazo:2020wgu , which is naturally related to generalized Grassmannian
amplitudes, also known as CEGM amplitudes Cachazo:2019ngv . This framework
provides an intriguing generalization of the notion of quantum field theory,
with many exciting results so far, see e.g. Early:2021tce ; Arkani-
Hamed:2019mrd ; Cachazo:2022vuo ; GarciaSepulveda:2019jxn ; Early:2023dlt .
For example, the analogous objects to Feynman diagrams that compute CEGM
amplitudes consist of arrays of lower-point Feynman diagrams Borges:2019csl ;
Cachazo:2019xjx ; Cachazo:2022pnx ; Cachazo:2023ltw . It would be interesting
to explore the possibility of finding what the $k>2$ analogous to $\phi^{p}$
amplitudes are as sums of $\phi^{p}$-like arrays of Feynman diagrams, that
could be obtained by triangulating a more general version of extended
diagrams.

Moreover, the global Schwinger formula was recently generalized to higher loop
order in Arkani-Hamed:2023lbd . All these frameworks leave plenty of room for
exciting future research directions.

To finish with the paper, we comment on a few interesting research lines, more
related to the work presented here.

###  5.1 Prospects for Extending Proofs in $\phi^{p}$

It would be interesting to have a proof of how triangulations of the extended
diagrams that participate in the global Schwinger formulation of
$A_{n}^{\phi^{p}}$ also lead to $\phi^{p}$ amplitudes, analagously to the
proof of section 2.3 in the $\phi^{4}$ case. Hopefully similar ideas would
succeed, but there are subtleties on account of the interplay between the
$(p-2)$-chords and the $2$-chords.

Furthermore, in section 2.4 we gave a proof of Conjecture 2.2, based on the
expression (1.9) for $H(x)$ in $\phi^{4}$. Namely, we proved that the
solutions to $H(x)=0$ are in bijection with the regions defined by non-
crossing chord diagrams that contribute to the amplitude $A_{n}^{\phi^{4}}$.
An analogous conjecture was proposed in Cachazo:2022voc  for general
$\phi^{p}$ theories. In particular, Conjecture 7.3 of Cachazo:2022voc
proposes a similar bijection between regions contributing to
$A^{\phi^{p}}_{n}$ or, equivalently, regions where $H^{\phi^{p}}(x)=0$ and
non-crossing $(p-2)$-chord diagrams. However, no such function
$H^{\phi^{p}}(x)$ was given.

Here we conjecture that, in fact, the function whose zeros give the desired
regions is

$H^{\phi^{p}}(x)=\sum_{a=0}^{n-3}x_{a}-\sum_{{\cal
X}}\textrm{min}(x_{a},x_{a+1},...,x_{b})+2\sum_{{\cal
Y}}\textrm{min}(x_{a},x_{a+1},...,x_{b})\,,$

where ${\cal X}$ means summing for $a<b$ over ordered pairs $(a,b)$ such that
$b-a\in\\{1,p-3,p-1,2p-5,2p-3,3p-7,3p-5,...,n-3\\}$. The domain ${\cal Y}$
means summing for $a<b$ over ordered pairs $(a,b)$ such that $b-a\equiv 0$ mod
$p-2$, up to $n-3$.

Because of the very similar form of the conjectured form of $H^{\phi^{p}}$,
one would hope that similar methods, like those in section 2.4, would work to
prove Conjecture 7.3 of Cachazo:2022voc .

###  5.2 Polygon Decomposition of an Extended Diagram in $\phi^{p}$

Of course, one of the most pressing problems is to be able to figure out the
precise bijection between an extended diagram and a double-ordered cubic
amplitude. We have made a small step towards this goal in section 4, by
proposing a way to read the polygonal decomposition of an extended diagram in
$\phi^{4}$. However, we still lack of a precise combinatorial method that
would give the desired permutations from it.

In this subsection we start the exploration of the polygon decomposition of
extended diagrams for general $\phi^{p}$ theories.

For example, let us start with $p=6$ and $n=10$. Figure 24 shows the already
triangulated extended diagrams, together with their corresponding proposed
polygon decomposition. Note from the three top diagrams in this example that
only one of the labels in the 4-chord separating the two meadows becomes an
external vertex. Namely, the label on the left of the upper chord of the lower
meadow. The rest of the labels become internal vertices inside the upper
polygon. One difference from the $\phi^{4}$ case is that now we must consider
internal labels on the left and on the right, see e.g. the first polygon
decomposition. Moreover, by looking at the two bottom diagrams in the figure,
we see again how a triangulating chord only picks the higher label between
those where it starts from.

Figure 24: Triangulated extended for $p=6$ and $n=10$ and their corresponding
polygon decomposition shown below.

Now let us have a look at figure 25, which belongs to the $p=5$, $n=11$ case.
This is the first example in which we encounter with a case where we have a
single 3-chord separating three meadows (last diagram in the figure). In this
case, we have a triangle with two labelled internal vertices, one of them
being label 2, which is a label from an upper chord in a meadow. This is
reminiscent of the $\phi^{4}$ example for $n=16$ (see figure 11), in which a
5-point meadow was separating three different meadows, and which corresponded
to a pentagon with two labelled internal vertices.

Figures 26 and 27 show the rest of the polygon decompositions for the
remaining triangulated extended diagrams in $p=5$ and $n=11$.

Figure 25: Polygon decomposition of the extended diagrams for $p=5$ and $n=11$
appearing in figure 14, presented in the same order. Figure 26: Triangulations
of the polygon decompositions for the extended diagrams in $p=5$ and $n=11$
appearing in figure 15. The first two columns correspond to the first three
rows in figure 15, while the last two columns correspond to the last three
rows in figure 15, ordered in the same way. Figure 27: Polygon decomposition
of the extended diagram for $p=5$ and $n=11$ appearing in figure 16, presented
in the same order.

Figures 28 and 29 correspond to two similar decompositions for $p=5$ and
$n=14$.

Figure 28: One of the extended diagrams for $p=5$ and $n=14$, together with
its polygon decomposition. Note that this is the same extended diagram as the
last one in figure 14, which corresponds to $p=5$ and $n=11$, but with an
additional 3-chord joining the new points 12, 13 and 14. In the polygon
decomposition representation this is equivalent to adding a new vertex
$\overline{12,13,14}$ to the upper triangle in figure 25. Figure 29: Another
similar example of an extended diagram for $p=5$ and $n=14$, together with its
polygon decomposition.

Finally, in figure 30 we show a more exotic polygon decomposition of an
extended diagram in $p=6$ and $n=14$. Interestingly, now one of the triangles
has three internal labels, but one of them is not naturally associated to a
vertex.

Figure 30: An example of an extended diagram for $p=6$ and $n=14$, together
with its polygon decomposition. Adding another 4-chord joining points 15, 16,
17 and 18 in the $n=18$ case adds an extra vertex to the upper triangle,
forming a square. The triangulation of this square is not affected by the
internal label 8 in the middle.

From these examples we conclude that a deeper understanding of the relation
between extended diagrams and off-diagonal amplitudes is needed. We leave the
study of this important and intriguing connection to future research.

###  5.3 Towards Smoothly Splitting $\phi^{p}$ Amplitudes

In Cachazo:2021wsz  a new intriguing property of tree-level scattering
amplitudes was found. This property was called a 3-split, which comes into
play in certain subspaces of the kinematic space where the scattering
amplitude smoothly splits into three amputated currents Dixon:1996wi ;
Mafra:2016ltu  without becoming singular. This and related phenomena,
including factorization near zeros in “stringy” amplitudes Arkani-
Hamed:2023swr , or extensions of the splitting behavior Cao:2024gln , have
recently been studied.

In this subsection we comment on the possibility of finding subspaces in the
kinematic space that would lead to similar smooth splits in $\phi^{p}$
amplitudes.

For example, it is reasonable to start with $\phi^{4}$ and $n=10$, since
extended diagrams are related to 6-point cubic amplitudes, which is the
minimum number of particles required for the smooth split to happen in
$\phi^{3}$ amplitudes. In this case, the extended diagram corresponding to a
single 6-point meadow is a diagonal amplitude $m_{{6}}(\mathbb{I},\mathbb{I})$
under the bijection

$\\{X_{1,4}\to s_{{12}},X_{2,5}\to s_{{23}},X_{4,7}\to s_{{34}},X_{6,9}\to
s_{{45}},X_{1,8}\to s_{{56}},X_{2,9}\to s_{{61}},X_{1,6}\to X_{{1},{4}},$
$X_{2,7}\to X_{{2},{5}},X_{4,9}\to X_{{3},{6}}\\}\,.$

The conditions $X_{1,6}=X_{1,4}+X_{2,5}$, $X_{2,7}=X_{1,8}+X_{2,9}$ and
$X_{4,9}=X_{4,7}+X_{6,9}$ are equivalent, under the bijection, to the split
kinematics subspace $({2},{4},{6})$ on 6 particles, given by the conditions
$X_{{1},{4}}=s_{{12}}+s_{{23}}$, $X_{{3},{6}}=s_{{34}}+s_{{45}}$ and
$X_{{2},{5}}=s_{{56}}+s_{{61}}$. The contribution of the extended diagram in
this subspace smoothly splits into

$\left(\frac{1}{X_{2,5}}+\frac{1}{X_{4,7}}\right)\left(\frac{1}{X_{6,9}}+\frac{1}{X_{1,8}}\right)\left(\frac{1}{X_{1,4}}+\frac{1}{X_{2,9}}\right)\,.$

It would be interesting to study if the whole $\phi^{4}$ amplitude can
therefore smoothly split. At first it might seem impossible since other
extended diagrams are given by products of 4-point or even 3-point and 5-point
subamplitudes, but the whole combination might allow for smooth splits, maybe
using a different kinematic subspace.

One interesting possibility would be to try and understand how the smooth
splitting works at the level of extended diagrams. Of course, by directly
looking at Feynman diagrams it is non-obvious what is the combinatorial
picture that underlies smooth splits. But, similarly to what is seen in other
contexts like the CHY formula Cachazo:2021wsz  or from the kinematic mesh
Arkani-Hamed:2023swr , extended diagrams in some sense resum Feynman diagrams,
so it might be possible to see this behavior in a more obvious way. We leave
this fascinating question for future research.

## Acknowledgements

We would like to thank F. Cachazo for reading a previous version of this
draft, as well as for helpful comments and discussions. We also thank J.
Drummond and Ö. Gürdoğan for discussions about the topic of this work, and C.
Chowdhury for explaining how to use TikZ, which helped to generate the figures
of the paper. BGU is supported by the STFC consolidated grant ST/X000583/1. KY
is supported by an NSERC Discovery grant, the Canada Research Chairs program,
and was supported by the Emmy Noether Fellows program at Perimeter Institute
which is supported in part by the Simons Foundation. Research at Perimeter
Institute is supported in part by the Government of Canada through the
Department of Innovation, Science and Economic Development Canada and by the
Province of Ontario through the Ministry of Colleges and Universities.

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