aainstitutetext: Perimeter Institute for Theoretical Physics, Waterloo, ON N2L
2Y5, Canadabbinstitutetext: Department of Physics and Astronomy, Western
University, London, ON N6A 5B7, Canada

# Connecting Scalar Amplitudes using The Positive Tropical Grassmannian

Freddy Cachazo<EMAIL_ADDRESS>a,b and Bruno Giménez Umbert
<EMAIL_ADDRESS>
###### Abstract

The biadjoint scalar partial amplitude, $m_{n}(\mathbb{I},\mathbb{I})$, can be
expressed as a single integral over the positive tropical Grassmannian thus
producing a Global Schwinger Parameterization. The first result in this work
is an extension to all partial amplitudes $m_{n}(\alpha,\beta)$ using a
limiting procedure on kinematic invariants that produces indicator functions
in the integrand. The same limiting procedure leads to an integral
representation of $\phi^{4}$ amplitudes where indicator functions turn into
Dirac delta functions. Their support decomposes into $\textrm{C}_{n/2-1}$
regions, with $\textrm{C}_{q}$ the $q^{\rm th}$-Catalan number. The
contribution from each region is identified with a
$m_{n/2+1}(\alpha,\mathbb{I})$ amplitude. We provide a combinatorial
description of the regions in terms of non-crossing chord diagrams and propose
a general formula for $\phi^{4}$ amplitudes using the Lagrange inversion
construction. We start the exploration of $\phi^{p}$ theories, finding that
their regions are encoded in non-crossing $(p-2)$-chord diagrams. The
structure of the expansion of $\phi^{p}$ amplitudes in terms of $\phi^{3}$
amplitudes is the same as that of Green functions in terms of connected Green
functions in the planar limit of $\Phi^{p-1}$ matrix models. We also discuss
possible connections to recent constructions based on Stokes polytopes and
accordiohedra.

##  1 Introduction

The positive tropical Grassmannian $\textrm{Trop}^{+}G(2,n)$ SWTrop  is the
space of all planar metric trees with $n$ leaves and vertices of any degree.
Hence, in some sense it governs the singularity structure of any planar tree-
level scattering amplitude of massless scalar fields with arbitrary polynomial
interactions.

For example, tree-level partial amplitudes of massless scalars in the
biadjoint representation of $U(N)\times U(\tilde{N})$ with only cubic
interactions, $m_{n}(\alpha,\beta)$, are computed by summing over all
$n$-particle Feynman diagrams which are planar with respect to both, $\alpha$
and $\beta$, orderings. The partial amplitudes with the largest number of
diagrams are the ones where both orderings coincide, e.g.,
$m_{n}(\mathbb{I},\mathbb{I})$. In a recent work by Cachazo and Early
Cachazo:2020wgu , a formula for $m_{n}(\mathbb{I},\mathbb{I})$ as a single
integral over ${\rm Trop}^{+}G(2,n)$ was presented,

$m_{n}(\mathbb{I},\mathbb{I})=\int_{\mathbb{R}^{n-3}}\\!\\!d^{n-3}x\,{\rm
exp}(-F_{n}(x))\,.$ (1.1)

Here the function $F_{n}(x)=\sum_{a,b}t_{[a,b]}f_{[a,b]}(x)$ is a piece-wise
linear function defined on $\textrm{Trop}^{+}G(2,n)$ via the “tropical cross-
ratios” $f_{[a,b]}(x)$, and $t_{[a,b]}:=(p_{a}+p_{a+1}+\cdots+p_{b})^{2}$ are
standard planar kinematic invariants. This construction is reviewed in detail
in section 2.

When restricted to a single cone of ${\rm Trop}^{+}G(2,n)$, the corresponding
integral becomes nothing but the Schwinger parameterization of a single
Feynman diagram. In this sense, ${\rm Trop}^{+}G(2,n)$ provides a global
Schwinger parameterization of the amplitude. The regions in
$\textrm{Trop}^{+}G(2,n)$ corresponding to trees with one or more higher-
degree vertices are of measure zero and therefore do not contribute to the
integral.

In this work we continue the study of global Schwinger parameterizations by
extending the construction to all partial amplitudes $m_{n}(\alpha,\beta)$ and
to amplitudes in scalar theories with $\phi^{p}$-interactions and $p>3$. In
both cases, our construction starts with the global Schwinger formulation of
$m_{n}(\mathbb{I},\mathbb{I})$ and proceeds with a limiting procedure on the
planar kinematic invariants to obtain amplitudes in other theories.

In the case of $m_{n}(\alpha,\mathbb{I})$ partial amplitudes, the limiting
procedure on kinematic invariants produces indicator functions in the
integrand. These indicator functions describe the regions of the original
$\textrm{Trop}^{+}G(2,n)$ that intersect with
$\textrm{Trop}^{+}G_{\alpha}(2,n)$, defined with the ordering $\alpha$. Our
first result is the following,

$m_{n}(\alpha,\mathbb{I})=\int_{\mathbb{R}^{n-3}}\\!\\!d^{n-3}x\,{\rm
exp}\left(-G_{\alpha}(x)\right){\bf 1}_{S(H_{\alpha})}(x)\,,$ (1.2)

where $G_{\alpha}(x)$ is a piece-wise linear function that depends on the
ordering $\alpha$, and ${\bf 1}_{S(H_{\alpha})}(x)$ is an indicator function.
We provide a derivation and examples of this formula in section 3.

The next set of theories we study are a straightforward generalization of
$m_{n}(\mathbb{I},\mathbb{I})$ in which one sums over only planar Feynman
diagrams with $\phi^{p}$-interaction vertices. We denote such amplitudes by
$A^{\phi^{p}}_{n}$. In this work we only consider planarity with respect to
the cannonical ordering $\mathbb{I}$ and therefore there is no need to specify
it. Of course, $A^{\phi^{p}}_{n}$ is defined to be zero if no Feynman diagram
exists for the particular number of external particles.

In order to obtain the global Schwinger formulation of $A^{\phi^{p}}_{n}$, the
limiting procedure on kinematic invariants produces distributions in the
integrand. In particular, the Dirac delta functions localize the integral over
$\mathbb{R}^{n-3}$ to regions of measure zero111Each corresponds to a
polyhedral cone in $\textrm{Trop}^{+}G(2,n)$., those where trees with higher-
degree vertices live. For example, for $\phi^{4}$ amplitudes we find

$A^{\phi^{4}}_{n}=\int_{\mathbb{R}^{n-3}}\\!\\!d^{n-3}x\,{\rm
exp}\left(-\sum_{a<b}^{\rm even}t_{[a,b]}f_{[a,b]}(x)\right)Q(x)\,,$ (1.3)

where $Q(x)$ is the distribution obtained from the limiting procedure and the
sum is over $b-a\equiv 0\\!\mod 2$. This procedure, together with the
derivation of the global Schwinger formula for $A^{\phi^{4}}_{n}$, is
developed in section 4. In section 5 we provide several examples.

Our global Schwinger formula for $\phi^{4}$ amplitudes reveals surprising
connections to cubic amplitudes: we find that each of the $\textrm{C}_{n/2-1}$
regions that define the support of the distributions in the integrand is in
bijection with a $m_{n/2+1}(\alpha,\mathbb{I})$ amplitude. We study this
feature in section 6, where we propose a combinatorial procedure to obtain
such regions from non-crossing chord diagrams. This implies that
$A_{n}^{\phi^{4}}$ can also be expressed as a sum over regions. These results
motivate a formula for the general schematic structure of $A_{n}^{\phi^{4}}$
in terms of cubic amplitudes based on the Lagrange inversion procedure

$A^{\phi^{4}}_{n}=\left(\frac{2}{n\,h_{0}^{n/2-1}}\right)\frac{1}{2\pi
i}\oint_{|z|=\epsilon}dz\left(\frac{h(z)}{z}\right)^{n/2}~{}~{}{\rm
with}~{}~{}h(x)=\sum_{i=0}^{\infty}m_{i+2}(\mathbb{I},\mathbb{I})\,x^{i}\,.$
(1.4)

Here $h_{0}=m_{2}(\mathbb{I},\mathbb{I}):=P^{2}$ and
$m_{3}(\mathbb{I},\mathbb{I}):=1$.

The fact that $A^{\phi^{4}}_{n}$ is computed as a sum over regions is very
reminiscent of the recent constructions based on Stokes polytopes yuliy ;
Banerjee:2018tun ; Kalyanapuram:2019nnf ; Aneesh:2019cvt ; Salvatori:2019phs ;
Srivastava:2020dly , which were motivated by the connection between $\phi^{3}$
amplitudes and the associahedron Mizera:2017cqs ; Arkani-Hamed:2017mur . It is
known that some Stokes polytopes are associahedra and therefore their
contribution could coincide with that of some of the regions we find. However,
we find that only associahedra or intersections of associahedra
Cachazo:2021llu  appear in our construction.

In section 7 we start the exploration of $\phi^{p}$ amplitudes in general. We
propose an analogous limiting procedure and obtain the corresponding global
Schwinger formula

$A_{n}^{\phi^{p}}=\int_{\mathbb{R}^{n-3}}d^{n-3}x\,\textrm{exp}\left(-\sum_{a<b}^{{\cal
K}_{p}}t_{[a,b]}f_{[a,b]}(x)\right)Q(x)\,.$ (1.5)

where ${\cal K}_{p}$ indicates that the sum is over ordered pairs $(a,b)$ such
that $b-a\equiv 0\\!\\!\mod p-2$.

We also propose a diagrammatic construction to find the regions that compute
$A_{n}^{\phi^{p}}$ as non-crossing $(p-2)$-chord diagrams (these are counted
by the Fuss-Catalan numbers222Here $\textrm{FC}_{m}(q,r)$ is the Fuss-Catalan
number given by  $\textrm{FC}_{m}(q,r)\equiv\frac{r}{mq+r}{mq+r\choose m}\,.$
$\textrm{FC}_{(n-2)/(p-2)}(p-2,1)$). The sum over all contributions leads to
the expected number of trees in $A_{n}^{\phi^{p}}$, which is also given by
Fuss-Catalan numbers, $\textrm{FC}_{(n-2)/(p-2)}(p-1,1)$. Moreover, we point
out a connection to $m_{(n+2(p-3))/(p-2)}(\alpha,\mathbb{I})$ amplitudes and
provide some examples. We also propose a formula giving the general structure
of $A_{n}^{\phi^{p}}$ in terms of cubic amplitudes.

Of course, $A_{n}^{\phi^{p}}$ amplitudes have also been recently studied and
found to be related to a class of polytopes known as accordiohedra
2017arXiv170309953M ; Raman:2019utu ; Aneesh:2019ddi ; Kojima:2020tox ;
Kalyanapuram:2020vil ; John:2020jww ; Kalyanapuram:2020axt ; Jagadale:2021iab
(see also Baadsgaard:2015ifa ; Baadsgaard:2016fel  for related work). Some
accordiohedra are associahedra and therefore as in the case of $\phi^{4}$ we
suspect that contributions from such accordiohedra could coincide with that of
some of our regions.

In section 8 we conclude with discussions on possible future research
directions including connections between our schematic formulas for
$A_{n}^{\phi^{p}}$ and those that express general Green functions in terms of
connected Green functions in planar theories, a way to connect to
accordiohedra constructions, and possible extensions to CEGM generalized
amplitudes. We end this work with two appendices. In appendix A we give a
review of the Lagrange inversion formula and provide details on the
derivations of the schematic structures of $A_{n}^{\phi^{p}}$ amplitudes. In
appendix B we provide an example which uses the global Schwinger formula to
compute the contribution to regions without actually carrying out the integral
by identifying the corresponding $\phi^{3}$ amplitude.

##  2 Global Schwinger Formula for $m_{n}(\mathbb{I},\mathbb{I})$

In this section we review the construction of the global Schwinger formula for
$m_{n}(\mathbb{I},\mathbb{I})$ introduced and proved in Cachazo:2020wgu .

Consider a single metric tree ${\cal T}$ with $n$ leaves and all internal
vertices of degree three. We follow the mathematical convention and call these
binary trees333The name stems from the fact that when a leaf is selected as a
root, then walking up along the tree implies that at each internal vertex
there are exactly two possible edges to choose from in order to continue the
walk..

Label the leaves of ${\cal T}$ so that it is planar with respect to the
ordering $\mathbb{I}:=(1,2,\ldots,n)$. Its contribution to an amplitude can be
constructed as follows. First, define the function

$\displaystyle{\cal F}({\cal T}):=-\sum_{1\leq a,b\leq n}d_{ab}s_{ab}\,,$
(2.1)

where $d_{ab}$ represents the matrix of distances, i.e., the distance from
leaf $a$ to leaf $b$. Mandelstam invariants $s_{ab}:=(k_{a}+k_{b})^{2}$
satisfy

$\displaystyle s_{ab}=s_{ba},\quad s_{aa}=0,\quad{\rm
and}\quad\sum_{b=1}^{n}s_{ab}=0\quad\forall a.$ (2.2)

Let $e_{a}$ be the length of the edge containing the $a^{\rm th}$ leaf and
write $d_{ab}=e_{a}+e_{b}+d^{\textrm{int}}_{ab}$ where $d^{\textrm{int}}_{ab}$
is the length of the internal edges along the unique path connecting $a$ and
$b$. Due to momentum conservation (2.2), $e_{a}$ drops out from the function
${\cal F}({\cal T})$ and it can be written as

$\displaystyle{\cal F}({\cal T})=\sum_{i=1}^{n-3}f_{I_{i}}t_{I_{i}}\,,$ (2.3)

where $f_{I}$ denotes the length of an internal edge that partitions the set
leaves of ${\cal T}$ as $I\cup I^{\rm c}=[n]$. The kinematic invariant
multiplying $f_{I}$ is defined as the square of the momentum flowing through
the edge under consideration, i.e.,

$t_{I}:=\left(\sum_{a\in I}k_{a}\right)^{2}=\sum_{\\{a,b\\}\subset
I}s_{ab}\,.$ (2.4)

The conditions (2.2) guarantee that $t_{I}=t_{I^{\rm c}}$.

Finally, the contribution to the amplitude is

$\displaystyle{\cal R}({\cal T})=\int_{O^{+}}\\!\\!\\!\\!d^{n-3}f\,{\rm
exp}(-{\cal F}({\cal
T}))=\prod_{i=1}^{n-3}\int_{0}^{\infty}\\!\\!\\!df_{I_{i}}\,\textrm{exp}(-f_{I_{i}}t_{I_{i}})=\prod_{i=1}^{n-3}\frac{1}{t_{I_{i}}}\,.$
(2.5)

where $O^{+}:=(\mathbb{R}^{+})^{n-3}$ is the positive orthant in
$\mathbb{R}^{n-3}$. Of course, the integral formulas are only defined for
$t_{I_{i}}>0$ but once the answer is in the rational function form, it is
valid for any values $t_{I_{i}}\neq 0$.

Note that the second integral in (2.5) is the standard Schwinger formula and
the edge lengths $f_{I}$ are the Schwinger parameters.

Denoting the set of all binary trees which are planar with respect to an
ordering of the leaves $\alpha:=(\alpha_{1},\alpha_{2},\ldots,\alpha_{n})$ by
${\tt Pl}_{n}(\alpha)$, the amplitude is computed as

$m_{n}(\mathbb{I},\mathbb{I})=\\!\\!\sum_{{\cal T}\in\,{\tt
Pl}_{n}(\mathbb{I})}{\cal R}({\cal T})\,.$ (2.6)

In SSTrop , Speyer and Sturmfels introduced the tropical Grassmannian ${\rm
Trop}\,G(2,n)$ and showed that it agrees with the moduli space of phylogenetic
trees studied by Billera, Holmes and Vogtmann (BHV) BHV .

Motivated by the work of Postnikov Alex  on totally positive Grassmannians,
Speyer and Williams introduced positive tropical Grassmannians SWTrop . In
particular, ${\rm Trop}^{+}\,G(2,n)$ parameterizes the space of planar trees
on $n$ leaves. This means that ${\rm Trop}^{+}\,G(2,n)$ must provide a global
definition of Schwinger parameters which unifies all the individual Schwinger
representations into a single integral.

In order to present the formula, one starts with $G^{+}(2,n)$ and then
tropicalizes the Plücker coordinates. Such a positive parameterization of
$G^{+}(2,n)$ is given by444Here we suppress the torus coordinates which under
tropicalization map to the $e_{a}$’s which drop out.

$\displaystyle\begin{pmatrix}1&0&-1&-(1+\tilde{x}_{1})&-(1+\tilde{x}_{1}+\tilde{x}_{2})&\cdots&-(1+\tilde{x}_{1}+\cdots+\tilde{x}_{n-3})\\\
0&1&1&1&1&\cdots&1\end{pmatrix}$ (2.7)

where $\tilde{x}_{a}\in\mathbb{R}^{+}$. Note that any minor $\Delta_{ab}$ with
$a<b$ is positive.

The tropicalization of a minor $\Delta_{ab}$ of (2.7) proceeds 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 $x_{i}+x_{j}$. Note that we drop the tilde to differentiate the two
sets of variables. This is important since while
$\tilde{x}_{a}\in\mathbb{R}^{+}$, the tropical variables are unconstrained,
i.e., $x_{a}\in\mathbb{R}$.

The connection to the space of planar trees is simply the identification of
(minus) the distance matrix $d_{ab}$ with the tropical Plücker coordinates. It
is not difficult to evaluate $\Delta^{\rm Trop}_{ab}(x)$ and find

$-d_{ab}\leftrightarrow\Delta^{\rm
Trop}_{ab}(x)=\left\\{\begin{array}[]{ccc}{\rm
min}(x_{a-2},x_{a-1},\ldots,x_{b-3})&&2\leq a\leq b-1,\,4\leq b\leq n\\\
0&&{\rm otherwise}\end{array}\right.,$ (2.8)

where $x_{0}:=0$ and whenever there is a single argument ${\rm min}(x):=x$.

It might seem strange that some $d_{ab}$ are sent to zero, however, recall
that momentum conservation makes the physics independent of the lengths
$e_{a}$ which can then be used to set to zero some of the entries $d_{ab}$.
The choice corresponds to a choice of frame in (2.7).

Using this in (2.1) one defines the “tropical potential function”

$\displaystyle F_{n}(x):=\sum_{1\leq a<b\leq n}\\!\\!\\!s_{ab}\,\Delta^{\rm
Trop}_{ab}(x)=\sum_{b=4}^{n}\sum_{a=2}^{b-1}\\!s_{ab}\,{\rm
min}(x_{a-2},x_{a-1},\ldots,x_{b-3})\,.$ (2.9)

The scattering amplitude (2.6) now has a single integral representation
Cachazo:2020wgu

$m_{n}(\mathbb{I},\mathbb{I})=\int_{\mathbb{R}^{n-3}}\\!\\!d^{n-3}x\,{\rm
exp}(-F_{n}(x))\,.$ (2.10)

Note that the integral is over all $\mathbb{R}^{n-3}$.

Of course, the integral in (2.10) might not exist for some values of kinematic
invariants. Let us discuss the regions of convergence. The best approach is to
write the tropical potential function $F_{n}(x)$ in terms of planar kinematic
invariants.

This is easily done by introducing the notation $t_{[a,b]}$ to denote $t_{I}$
with $I=\\{a,a+1,\ldots,b-1,b\\}$, a set of consecutive labels, and using

$s_{ab}=t_{[a,b]}-t_{[a+1,b]}-t_{[a,b-1]}+t_{[a+1,b-1]}\,.$ (2.11)

Here one defines $t_{[c,d]}=0$ whenever $c\geq d$. For example,
$s_{14}=t_{[1,4]}-t_{[2,4]}-t_{[1,3]}+t_{[2,3]}$ while $s_{23}=t_{[2,3]}$.

Using (2.11) in $F_{n}(x)$ and arranging by planar kinematic invariants one
finds

$F_{n}(x)=\sum_{a<b}t_{[a,b]}\left(\Delta^{\rm Trop}_{a,b}(x)-\Delta^{\rm
Trop}_{a,b+1}(x)-\Delta^{\rm Trop}_{a-1,b}(x)+\Delta^{\rm
Trop}_{a-1,b+1}(x)\right)\,.$ (2.12)

Figure 1: In a generic tree, the combination of distances
$-d_{a,b}+d_{a,b+1}+d_{a-1,b}-d_{a-1,b+1}$ equals twice the length of the edge
which removal would split the diagram into two parts, one containing $a$ and
$b$ and the other $a-1$ and $b+1$.

The quantity in brackets has a very beautiful interpretation when thought of
as $-d_{a,b}+d_{a,b+1}+d_{a-1,b}-d_{a-1,b+1}$ for a single planar Feynman
diagram. This is nothing but twice the length of the edge partitioning the
labels as $\\{a,a+1,\ldots,b-1,b\\}\cup\\{b+1,b+2,\ldots,a-2,a-1\\}$, i.e.,
what used to be $f_{[a,b]}$, see figure 1. Of course, for this to be the case,
it better be that it is always non-negative. This is easily seen to be the
case by noticing the following general property for any three real numbers
$A,B,C$,

$A-{\rm min}(A,B)-{\rm min}(A,C)+{\rm min}(A,B,C)\geq 0\,.$ (2.13)

The proof is left as an exercise to the reader555Hint: $A-{\rm min}(A,B)=-{\rm
min}(A-A,B-A)=-{\rm min}(0,B-A)\geq 0$. Repeated use leads to $-{\rm
min}(0,B-A)-{\rm min}(0,(B-A)-{\rm min}(0,C-A))$..

It is important to mention that the condition

$\Delta^{\rm Trop}_{a,b}(x)+\Delta^{\rm Trop}_{a-1,b+1}(x)\geq\Delta^{\rm
Trop}_{a,b+1}(x)+\Delta^{\rm Trop}_{a-1,b}(x)$ (2.14)

is part of what is referred to as a positive tropical Plücker relation and it
must be satisfied in order to be in ${\rm Trop}^{+}G(2,n)$.

Finally, the condition for the integral formula (2.10) to exist is simply that
all planar kinematic invariants be positive. In the rest of this work, only
the formula with planar invariants will be used. This why it is convenient to
introduce special notation for the combination of tropical minors in (2.12),

$f_{[a,b]}(x):=\Delta^{\rm Trop}_{a,b}(x)-\Delta^{\rm
Trop}_{a,b+1}(x)-\Delta^{\rm Trop}_{a-1,b}(x)+\Delta^{\rm
Trop}_{a-1,b+1}(x)\,,$ (2.15)

so that $F_{n}(x)=\sum_{a,b}t_{[a,b]}f_{[a,b]}(x)$ and we arrive at the final
form of the global Schwinger formula Cachazo:2020wgu

$m_{n}(\mathbb{I},\mathbb{I})=\int_{\mathbb{R}^{n-3}}\\!\\!d^{n-3}x\,{\rm
exp}\left(-\sum_{a<b}t_{[a,b]}f_{[a,b]}(x)\right)\,.$ (2.16)

##  3 From $m_{n}(\mathbb{I},\mathbb{I})$ to $m_{n}(\alpha,\beta)$

In order to extend the global Schwinger formula (2.16) for
$m_{n}(\mathbb{I},\mathbb{I})$ to all other partial amplitudes,
$m_{n}(\alpha,\beta)$, let us first review their definition. Recall that ${\tt
Pl}_{n}(\alpha)$ denotes the set of all binary trees which are planar with
respect to the ordering of the leaves defined by
$\alpha=(\alpha_{1},\alpha_{2},\ldots,\alpha_{n})$. Given a second ordering
$\beta=(\beta_{1},\beta_{2},\ldots,\beta_{n})$, one can determine the set of
trees which are planar with respect to both orderings by simply finding the
intersection ${\tt Pl}_{n}(\alpha)\cap{\tt Pl}_{n}(\beta)$, and therefore,

$m_{n}(\alpha,\beta)=\\!\\!\sum_{{\cal T}\in\,{\tt Pl}_{n}(\alpha)\cap\,{\tt
Pl}_{n}(\beta)}{\cal R}({\cal T})\,.$ (3.1)

Depending on conventions, there might be an overall sign which depends on the
two orderings chosen. Since our main concern is the kinematic dependence of
the amplitude, we refer the reader to Cachazo:2013iea  for details on the
definition of the sign666For example, one could decide to define partial
amplitudes so that the sign is included in the traces of the flavour groups.
While convenient when individual partial amplitudes are considered, this makes
properties such as the $U(1)$-decoupling identity, which involves several
partial amplitudes, cumbersome..

Without loss of generality we assume that $\beta=\mathbb{I}=(1,2,\ldots,n)$.
Now, recall that $t_{I}$ is planar with respect to an ordering if the set $I$
coincides with the set of labels of an interval in the ordering. For example,
if $\alpha=(1,2,5,4,3,6)$ then $I=\\{1,2,5\\}$ is planar with respect to
$\alpha$ but not with respect to $\mathbb{I}$ while $I=\\{3,4,5\\}$ is planar
with respect to both orderings.

Let ${\tt PK}(\alpha)$ denote the set of all planar kinematic invariants with
respect to $\alpha$.

######  Proposition 3.1.

Consider the set of planar kinematic invariants ${\tt PK}(\mathbb{I})$, set
$t_{I}=1/\epsilon$ whenever $t_{I}\notin{\tt PK}(\alpha)$, and evaluate
$m_{n}(\mathbb{I},\mathbb{I})$ on it to get a function of $\epsilon$ and
kinematic invariants in ${\tt PK}(\alpha)\cap{\tt PK}(\mathbb{I})$,
$m_{n}^{(\epsilon)}(\mathbb{I},\mathbb{I})$. Then

$m_{n}(\alpha,\mathbb{I})=\lim_{\epsilon\to
0}m_{n}^{(\epsilon)}(\mathbb{I},\mathbb{I})\,.$ (3.2)

###### Proof.

Since all trees in $m_{n}^{(\epsilon)}(\mathbb{I},\mathbb{I})$ that do not
contribute to $m_{n}(\alpha,\mathbb{I})$ contain at least one kinematic
invariant that has been set to $1/\epsilon$, in the limit their contribution
to the amplitude vanishes. Since invariants in ${\tt PK}(\alpha)\cap{\tt
PK}(\mathbb{I})$ are $\epsilon$-independent, so are the corresponding Feynman
diagram contributions ${\cal R}({\cal T})$ to (3.1). ∎

The construction of the global Schwinger formula for
$m_{n}(\alpha,\mathbb{I})$ proceeds in exactly the same way. Let us define the
$\epsilon$-dependent tropical potential function, $F_{n}(x,\epsilon)$, by
starting with $F_{n}(x)$ and restricting to the kinematic space of Proposition
3.1. More explicitly,

$F_{n}(x):=\sum_{I\in\,{\tt PK}(\mathbb{I})}t_{I}\,f_{I}(x)\to
F_{n}(x,\epsilon):=\\!\\!\\!\sum_{I\in{\tt PK}(\alpha)\cap\,{\tt
PK}(\mathbb{I})}\\!\\!\\!t_{I}\,f_{I}(x)+\frac{1}{\epsilon}\left(\sum_{I\notin{\tt
PK}(\alpha)\cap\,{\tt PK}(\mathbb{I})}\\!\\!\\!f_{I}(x)\right)\,.$ (3.3)

Let us define the finite and divergent parts to be
$F_{n}(x,\epsilon)=G_{\alpha}(x)+\frac{1}{\epsilon}H_{\alpha}(x)$, i.e.,

$G_{\alpha}(x):=\sum_{I\in{\tt PK}(\alpha)\cap\,{\tt
PK}(\mathbb{I})}\\!\\!\\!t_{I}\,f_{I}(x)\,,\quad
H_{\alpha}(x):=\sum_{I\notin{\tt PK}(\alpha)\cap\,{\tt
PK}(\mathbb{I})}\\!\\!\\!f_{I}(x)\,.$ (3.4)

Note that we have chosen to add the subscript $\alpha$ to indicate that the
form of the functions depends on the $\alpha$-ordering.

Using (3.2) and (2.10) one finds

$m_{n}(\alpha,\mathbb{I})=\lim_{\epsilon\to
0^{+}}\int_{\mathbb{R}^{n-3}}\\!\\!d^{n-3}x\,{\rm exp}(-F_{n}(x,\epsilon))\,.$
(3.5)

Defining the limit as a directional limit from above is necessary for the
convergence of the integral since all planar kinematic invariants must be
positive. Moreover, it also allows the limit to be taken inside the integral.
Using (3.4) one finds

$m_{n}(\alpha,\mathbb{I})=\int_{\mathbb{R}^{n-3}}\\!\\!d^{n-3}x\,{\rm
exp}\left(-G_{\alpha}(x)\right)\lim_{\epsilon\to 0^{+}}{\rm
exp}\left(-\frac{1}{\epsilon}H_{\alpha}(x)\right)\,.$ (3.6)

The function resulting from computing the limit is nothing but an indicator
function. In general, given two sets $S,U$ such that $S\subset U$,

${\bf 1}_{S}:U\to\\{0,1\\},\quad{\bf
1}_{S}(x)=\left\\{\begin{array}[]{cc}1&{\rm if~{}}x\in S,\\\
0&{\rm~{}\,otherwise}.\end{array}\right.$ (3.7)

In the case at hand, we define the set
$S(H_{\alpha}):=\\{x\in\mathbb{R}^{n-3}:H_{\alpha}(x)=0\\}\subset\mathbb{R}^{n-3}$.
This leads to the final formula for the global Schwinger formula,

$m_{n}(\alpha,\mathbb{I})=\int_{\mathbb{R}^{n-3}}\\!\\!d^{n-3}x\,{\rm
exp}\left(-G_{\alpha}(x)\right){\bf 1}_{S(H_{\alpha})}(x)\,.$ (3.8)

######  Example 3.2.

Consider $\alpha=(1324)$. In this case

$G_{\alpha}(x)=-s_{23}\,{\rm min}(0,x_{1})\,,\quad H_{\alpha}(x)=x_{1}-{\rm
min}(0,x_{1})\,.$ (3.9)

The set $S(H_{\alpha})=\\{x_{1}:x_{1}\leq 0\\}=(-\infty,0]$ and therefore,

$m_{4}(1324,\mathbb{I})=\int_{-\infty}^{\infty}\\!\\!dx_{1}\,{\rm
exp}\left(s_{23}\,{\rm min}(0,x_{1})\right){\bf
1}_{(-\infty,0]}(x_{1})=\int_{-\infty}^{0}\\!\\!dx_{1}\,{\rm
exp}\left(s_{23}\,x_{1}\right)=\frac{1}{s_{23}}\,.$ (3.10)

Let us present a more interesting example.

######  Example 3.3.

Consider $\alpha=(123654)$. In this case

$\displaystyle\begin{split}G_{\alpha}(x)=&s_{12}\left(x_{1}-\min\left(0,x_{1}\right)\right)-s_{23}\,\min\left(0,x_{1}\right)+s_{45}\left(x_{2}-\min\left(x_{2},x_{3}\right)\right)+\\\
&s_{56}\left(x_{3}-\min\left(x_{2},x_{3}\right)\right)+t_{123}\left(\min\left(x_{2},x_{3}\right)-x_{1}\right),\\\
H_{\alpha}(x)=&x_{1}-\min(x_{1},x_{2},x_{3})\,.\end{split}$ (3.11)

In the expression for $G_{\alpha}(x)$ we have already used that
$x_{1}\leq\min(x_{2},x_{3})$ is the condition imposed by requiring
$H_{\alpha}(x)=0$ in order to simplify the expression. The set
$S(H_{\alpha})=\\{x_{1}:x_{1}\leq\min(x_{2},x_{3})\\}$. In this case it is
convenient to write the indicator function as a product of two Heaviside step
functions $\theta(x_{2}-x_{1})\theta(x_{3}-x_{1})$ so that

$m_{6}(123654,\mathbb{I})=\int_{\mathbb{R}^{3}}\\!\\!d^{3}x\,{\rm
exp}\left(-G_{\alpha}(x_{1},x_{2},x_{3})\right)\theta(x_{2}-x_{1})\theta(x_{3}-x_{1})\,.$
(3.12)

This integral is easily evaluated to give the expected result

$m_{6}(123654,\mathbb{I})=\left(\frac{1}{s_{12}}+\frac{1}{s_{23}}\right)\left(\frac{1}{s_{45}}+\frac{1}{s_{56}}\right)\frac{1}{t_{123}}\,.$
(3.13)

##  4 From $\phi^{3}$ Amplitudes to $\phi^{4}$ Amplitudes

The positive tropical Grassmannian ${\rm Trop}^{+}\,G(2,n)$ is the space of
all planar metric trees. In other words, trees with vertices of any degree
$3\leq d\leq n$ are part of the space. In the previous section, amplitudes of
theories where only Feynman diagrams corresponding to binary trees were
discussed. At first it might be puzzling that a formula for
$m_{n}(\mathbb{I},\mathbb{I})$ involves an integration over the entire ${\rm
Trop}^{+}\,G(2,n)$. However, this is easily understood by noticing that the
regions in ${\rm Trop}^{+}\,G(2,n)$ which correspond to trees with at least
one vertex of degree $d>3$ are of measure zero and do not contribute to the
integral.

In this section we extend the idea used to obtain $m_{n}(\alpha,\mathbb{I})$
from $m_{n}(\mathbb{I},\mathbb{I})$ by a limiting procedure in order to obtain
a global Schwinger formula for $A^{\phi^{4}}_{n}$. The main difference is that
while the limiting procedure produced indicator functions leading to
$m_{n}(\alpha,\mathbb{I})$, here it produces Dirac delta functions that
localize the integral to the regions of measure zero where $\phi^{4}$ planar
trees are located. The process unearths a surprising connection to
$m_{n/2+1}(\alpha,\mathbb{I})$ amplitudes.

######  Proposition 4.1.

Consider the space of kinematic invariant of $n=2m$ massless particles with
$t_{[a,b]}=1/\epsilon$ whenever $b-a\equiv 1\\!\\!\mod 2$ and let
$m_{n}^{(\epsilon)}(\mathbb{I},\mathbb{I})$ denote
$m_{n}(\mathbb{I},\mathbb{I})$ evaluated on it. Then

$A^{\phi^{4}}_{n}=\lim_{\epsilon\to
0}\frac{1}{(2\epsilon)^{n/2-1}}m_{n}^{(\epsilon)}(\mathbb{I},\mathbb{I})\,.$
(4.1)

###### Proof.

Consider any Feynman diagram ${\cal T}^{(4)}$ contributing to
$A^{\phi^{4}}_{n}$, that is, any completely ternary planar tree on $n$-leaves.
Such a diagram has $n/2-1$ vertices of degree $4$. The strategy is to find out
how many planar binary trees give rise to ${\cal T}^{(4)}$ by collapsing
edges, i.e., taking their length to zero. This is easily done by realizing
that for each degree-four vertex of ${\cal T}^{(4)}$ there are exactly two
ways, compatible with planarity, of growing an edge to produce two degree-
three vertices. This means that there are $2^{n/2-1}$ binary trees that give
rise to ${\cal T}^{(4)}$. Of course, not all binary trees descend to a ternary
diagram. Note that under the kinematics in the proposition, $\phi^{3}$ Feynman
diagrams that collapse to $\phi^{4}$ diagrams have exactly $n/2-2$
$\epsilon$-independent propagators and $(n-3)-(n/2-2)=n/2-1$ which become
$1/t=\epsilon$. Diagrams that do not produce a $\phi^{4}$ diagram have at
least one extra propagator of the form $1/t=\epsilon$. Therefore, in the limit
$\epsilon\to 0$ the $\phi^{4}$ amplitude is recovered. ∎

###  4.1 Global Schwinger Formula for $A^{\phi^{4}}_{n}$

Following the same steps as in section 3 we start the derivation of the global
Schwinger formulation of $A^{\phi^{4}}_{n}$ by using Proposition 4.1 and the
representation for $m_{n}(\mathbb{I},\mathbb{I})$ given in (2.16), i.e.

$A^{\phi^{4}}_{n}=\lim_{\epsilon\to
0^{+}}\frac{1}{(2\epsilon)^{n/2-1}}\int_{\mathbb{R}^{n-3}}\\!\\!d^{n-3}x\,{\rm
exp}\left(-\sum_{a<b}t_{[a,b]}(\epsilon)f_{[a,b]}(x)\right)\,.$ (4.2)

Rewrite

$F_{n}(x,\epsilon)=\sum_{a<b}t_{[a,b]}f_{[a,b]}(x)=G(x)+\frac{1}{\epsilon}H(x)$
(4.3)

with

$G(x)=\sum_{a<b}^{\rm even}t_{[a,b]}f_{[a,b]}(x)\,,\quad H(x)=\sum_{a<b}^{\rm
odd}f_{[a,b]}(x)\,,$ (4.4)

where the sums are over ordered pairs $(a,b)$ such that $b-a\equiv 0\\!\mod 2$
(even) or $b-a\equiv 1\\!\mod 2$ (odd).

Unlike the cases considered in the previous section, the limit $\epsilon\to 0$
of $m_{n}^{(\epsilon)}(\mathbb{I},\mathbb{I})$ does not lead to a finite
answer and therefore commuting the limit and the integration in (4.2) must be
carefully defined.

We take the approach in which

$Q(x):=\lim_{\epsilon\to
0^{+}}\frac{1}{(2\epsilon)^{n/2-1}}\exp\left(-\frac{1}{\epsilon}H(x)\right)$
(4.5)

is to be treated as a distribution. Since $H(x)\geq 0$, it is clear that
$Q(x)$ only has support in regions where $H(x)=0$.

In the next section we show that solutions to $H(x)=0$ are regions of
dimension $n/2-1$ in $\mathbb{R}^{n-3}$ which are classified by non-crossing
chord diagrams. Here we show the explicit form of $H(x)$,

$\displaystyle
H(x)=\sum_{a=0}^{n-3}x_{a}+2\sum_{a=0}^{n-4}\sum_{b=a+1}^{n-3}(-1)^{b-a}\,\textrm{min}(x_{a},x_{a+1},...,x_{b})\,.$
(4.6)

Thus, the distribution $Q(x)$ becomes a sum over distributions that localize
the integral to the regions. This gives the first form of the global Schwinger
formula for the $\phi^{4}$ theory,

$A^{\phi^{4}}_{n}=\int_{\mathbb{R}^{n-3}}\\!\\!d^{n-3}x\,{\rm
exp}\left(-\sum_{a<b}^{\rm even}t_{[a,b]}f_{[a,b]}(x)\right)Q(x)\,.$ (4.7)

In the next section we present some examples that motivate a second version of
the formula as a sum over regions labelled by non-crossing chord diagrams.

##  5 Computing $\phi^{4}$ Amplitudes Using the Global Schwinger Formula

In this section we illustrate the use of the global Schwinger formula (4.7) by
considering several examples.

###  5.1 Four-Point Amplitude

The four-particle kinematic space is only two dimensional, $s_{12},s_{23}$.
Therefore $G(x)=0$ and

$H(x)=x_{1}-2\min(0,x_{1})=|x_{1}|.$ (5.1)

The distribution in the integral is

$Q(x)=\lim_{\epsilon\to
0^{+}}\frac{1}{2\epsilon}\exp\left(-\frac{1}{\epsilon}|x_{1}|\right)=\delta(x_{1})\,.$
(5.2)

This implies that (4.7) becomes

$A^{\phi^{4}}_{4}=\int_{\mathbb{R}}\\!\\!dx_{1}\delta(x_{1})=1\,.$ (5.3)

###  5.2 Six-Point Amplitude

The six-particle kinematic space is nine dimensional,
$s_{12},s_{23},\ldots,s_{61},t_{[1,3]},t_{[2,4]},t_{[3,5]}$. Evaluating $H(x)$
one finds

$\displaystyle H(x)=$
$\displaystyle\,x_{1}+x_{2}+x_{3}-2\min(0,x_{1})-2\min(x_{1},x_{2})-2\min(x_{2},x_{3})$
$\displaystyle+2\min(0,x_{1},x_{2})+2\min(x_{1},x_{2},x_{3})-2\min(0,x_{1},x_{2},x_{3})\,.$
(5.4)

Setting $H(x)$ to zero gives rise to two regions,

$R_{1}=\\{x_{1}=0,\,x_{2}=x_{3}\\}\,,\quad
R_{2}=\\{x_{1}=x_{2}>0,\,x_{3}=0\\}\,.$ (5.5)

The distribution $Q(x)$ then becomes

$Q(x)=Q_{1}(x)+Q_{2}(x)\,,\,\,{\rm
with}\,\,Q_{1}(x):=\delta(x_{1})\delta(x_{2}-x_{3})\,,\,\,Q_{2}(x):=\theta(x_{1})\delta(x_{1}-x_{2})\delta(x_{3})\,.$
(5.6)

Instead of computing (4.7) as a single object, let us split it by regions

$A^{\phi^{4}:(1)}_{6}=\int_{\mathbb{R}^{3}}\\!\\!d^{3}x\exp(-G(x))Q_{1}(x)=\frac{1}{t_{123}}+\frac{1}{t_{234}}\,.$
(5.7)
$A^{\phi^{4}:(2)}_{6}=\int_{\mathbb{R}^{3}}\\!\\!d^{3}x\exp(-G(x))Q_{2}(x)=\frac{1}{t_{345}}\,.$
(5.8)

Adding up the two contributions leads to the amplitude

$A^{\phi^{4}}_{4}=\frac{1}{t_{123}}+\frac{1}{t_{234}}+\frac{1}{t_{345}}\,.$
(5.9)

###  5.3 Eight-Point Amplitude

The eight-particle kinematic space is twenty dimensional,
$s_{12},s_{23},\ldots,s_{81}$, $t_{123},t_{234},\ldots,t_{812}$, and
$t_{[1,4]},t_{[2,5]},t_{[3,6]},t_{[4,7]}$. Evaluating $H(x)$ using (4.6) one
finds five regions:

$\displaystyle R_{1}=\,$
$\displaystyle\\{x_{1}=0,\,x_{2}=x_{3},\,x_{4}=x_{5}\\}\,,$ $\displaystyle
R_{2}=\,$
$\displaystyle\\{x_{1}=0,\,x_{2}=x_{5},\,x_{3}=x_{4},\,x_{2}<x_{3}\\}\,,$
$\displaystyle R_{3}=\,$
$\displaystyle\\{x_{5}=0,\,x_{1}=x_{2},\,x_{3}=x_{4},\,x_{1}>0,\,x_{3}>0\\}\,,$
$\displaystyle R_{4}=\,$
$\displaystyle\\{x_{3}=0,\,x_{1}=x_{2},\,x_{4}=x_{5},\,x_{1}>0\\}\,,$
$\displaystyle R_{5}=\,$
$\displaystyle\\{x_{5}=0,\,x_{1}=x_{4},\,x_{2}=x_{3},\,x_{2}>x_{1}>0\\}\,.$
(5.10)

The distribution $Q(x)$ then becomes

$Q(x)=Q_{1}(x)+Q_{2}(x)+Q_{3}(x)+Q_{4}(x)+Q_{5}(x)$ (5.11)

with

$\displaystyle Q_{1}(x):=\,$
$\displaystyle\delta(x_{1})\delta(x_{2}-x_{3})\delta(x_{4}-x_{5})\,,$
$\displaystyle Q_{2}(x):=\,$
$\displaystyle\theta(x_{3}-x_{2})\delta(x_{1})\delta(x_{2}-x_{5})\delta(x_{3}-x_{4})\,,$
$\displaystyle Q_{3}(x):=\,$
$\displaystyle\theta(x_{1})\theta(x_{3})\delta(x_{5})\delta(x_{1}-x_{2})\delta(x_{3}-x_{4})\,,$
$\displaystyle Q_{4}(x):=\,$
$\displaystyle\theta(x_{1})\delta(x_{3})\delta(x_{1}-x_{2})\delta(x_{4}-x_{5})\,,$
$\displaystyle Q_{5}(x):=\,$
$\displaystyle\theta(x_{1})\theta(x_{2}-x_{1})\delta(x_{5})\delta(x_{1}-x_{4})\delta(x_{2}-x_{3})\,.$

The contributions from each region are:

$\displaystyle A^{\phi^{4}:(1)}_{8}=\,$
$\displaystyle\frac{1}{t_{123}t_{456}}+\frac{1}{t_{456}t_{781}}+\frac{1}{t_{781}t_{234}}+\frac{1}{t_{234}t_{678}}+\frac{1}{t_{678}t_{123}}\,,$
$\displaystyle A^{\phi^{4}:(2)}_{8}=\,$
$\displaystyle\frac{1}{t_{567}}\left(\frac{1}{t_{123}}+\frac{1}{t_{234}}\right)\,,$
$\displaystyle A^{\phi^{4}:(3)}_{8}=\,$
$\displaystyle\frac{1}{t_{812}}\left(\frac{1}{t_{345}}+\frac{1}{t_{567}}\right)\,,$
$\displaystyle A^{\phi^{4}:(4)}_{8}=\,$
$\displaystyle\frac{1}{t_{345}}\left(\frac{1}{t_{678}}+\frac{1}{t_{781}}\right)\,,$
$\displaystyle A^{\phi^{4}:(5)}_{8}=\,$
$\displaystyle\frac{1}{t_{456}t_{812}}\,.$ (5.12)

The amplitude $A^{\phi^{4}}_{8}$ is the sum over all five contributions and
gives rise to the familiar expression in terms of $12$ Feynman diagrams.

###  5.4 One Region for All $n$

In the next section we provide a diagrammatic technique for finding all
regions contributing to $A^{\phi^{4}}_{n}$. In this last example, we study the
contribution from the analog to $R_{1}$ for all $n$. The region is defined in
the following proposition.

######  Proposition 5.1.

The function

$H(x)=\sum_{a<b}^{\rm odd}f_{[a,b]}(x)\,,$ (5.13)

defined in (4.4), vanishes in the region

$R_{1}=\\{x_{0}=x_{1},\,x_{2}=x_{3},\,x_{4}=x_{5},\ldots,\,x_{n-4}=x_{n-3}\\}\,.$
(5.14)

###### Proof.

Since $H(x)$ is the sum of non-negative functions, $f_{[a,b]}(x)$, we have to
show that each such function vanishes on $R_{1}$. Using the definitions
(2.15), (2.8), one has

$\displaystyle f_{[a,b]}(x)$
$\displaystyle=\min(x_{a-2},x_{a-1},\ldots,x_{b-3})-\min(x_{a-2},x_{a-1},\ldots,x_{b-3},x_{b-2})$
(5.15)
$\displaystyle-\min(x_{a-3},x_{a-2},\ldots,x_{b-3})+\min(x_{a-3},x_{a-2},\ldots,x_{b-3},x_{b-2})\,.$

By definition, $H(x)$ only contains $f_{[a,b]}(x)$ with $b-a\equiv 1\\!\mod
2$. This means that on $R_{1}$, either $x_{a-3}=x_{a-2}$ or $x_{b-3}=x_{b-2}$.
This is easily seen by considering two cases: If $a\in 2\mathbb{Z}$ then $b\in
2\mathbb{Z}+1$ and therefore $a-3\in 2\mathbb{Z}+1$, $b-3\in 2\mathbb{Z}$, and
$x_{b-3}=x_{b-2}$ on $R_{1}$. The same can be repeated when $a\in
2\mathbb{Z}+1$ to conclude that $x_{a-3}=x_{a-2}$.

Finally, note that if $x_{a-3}=x_{a-2}$ then the first and third terms in
(5.15) cancel each other while the second and fourth do too. If
$x_{b-3}=x_{b-2}$ then the first and second cancel while the third and fourth
do too. ∎

In order to evaluate the contribution from $R_{1}$ it is convenient to define
the following combination of kinematic invariants,

$r_{ab}:=s_{ab}+s_{a,b+1}+s_{a+1,b}+s_{a+1,b+1}\,.$ (5.16)

It is also useful to write $r_{ab}$ in terms of planar invariants using
(2.11),

$r_{ab}=-t_{[a,b-1]}+t_{[a,b+1]}+t_{[a+2,b-1]}-t_{[a+2,b+1]}\,.$ (5.17)

Note that if $a\in 2\mathbb{Z}$ and $b\in 2\mathbb{Z}+1$, then all four
invariants in (5.17) belong to the set of $\phi^{4}$ invariants.

Restricting the tropical potential function (2.9) to $R_{1}$ one finds

$F_{n}(x)=G_{n}(x)=\sum_{a=2}^{\rm even}\sum_{b=5}^{\rm
odd}r_{ab}\min(x_{a-2},x_{a},\ldots,x_{b-5},x_{b-3})+\sum_{a=2}^{{\rm
even}}t_{a,a+1,a+2}x_{a-2}\,.$ (5.18)

In sums labeled “even” (“odd”) the index only takes even (odd) values. The
first equality is due to the fact that on $R_{1}$ the function $H_{n}(x)=0$.

The function $G_{n}(x)$ has exactly the structure of a tropical potential for
$m_{n/2+1}(\mathbb{I},\mathbb{I})$ if the labels are identified as $x_{a}\to
x_{a/2}$. This is well-defined since $a$ only takes even values in (5.18).
Instead of using the mapping, we keep the original labels and write the
tropical potential for $m_{n/2+1}(\mathbb{I},\mathbb{I})$ as

$F_{n/2+1}^{\phi^{3}}(x_{0},x_{2},\ldots,x_{n}):=\sum_{a=2}^{\rm
even}\sum_{b=5}^{\rm
odd}s_{a,b+1}\min(x_{a-2},x_{a},\ldots,x_{b-5},x_{b-3})+\sum_{a=2}^{{\rm
even}}s_{a,a+2}x_{a-2}\,.$ (5.19)

Matching the coefficients gives the map of kinematic invariants,

$r_{ab}=s_{a,b+1}\,,\quad t_{a,a+1,a+2}=s_{a,a+2}\,,\quad
a\in\\{2,4,6\ldots\\}\,,\,\,b\in\\{5,7,9,\ldots\\}\,.$ (5.20)

We conclude that the contribution of region $R_{1}$ to $A^{\phi^{4}}_{n}$ is
nothing but $m_{n/2+1}(\mathbb{I},\mathbb{I})$ with kinematic invariants given
by (5.20).

This result prompts the following proposition.

######  Proposition 5.2.

Consider $A^{\phi^{4}}_{n}$ evaluated on the following subspace of kinematic
invariants,

$t_{[a,b]}=\frac{1}{\epsilon}\,,\quad
a\in\\{3,5,\ldots,n-3\\}\,,\,\,b\in\\{a+2,a+4,\ldots,n-1\\}\,.$ (5.21)

to produce a function $A^{\phi^{4}(\epsilon)}_{n}$. Then,

$\lim_{\epsilon\to
0}A^{\phi^{4}(\epsilon)}_{n}=m_{n/2+1}(\mathbb{I},\mathbb{I}),$ (5.22)

for some bijection of the set of planar kinematic invariants.

###### Proof.

Let us start by proving the spaces of kinematic invariants possess the same
cardinality. The set of planar kinematic invariants of $A^{\phi^{4}}_{n}$ has
cardinality of $n(n-4)/4$. For example, it is $0,3,8,15$ for $n=4,6,8,10$. In
the statement of the proposition, $(n-2)(n-4)/8$ of the kinematic invariants
are set to $1/\epsilon$ so there are only $(n-4)(n+2)/8$ left. Let us
introduce $n_{3}:=n/2+1$, then $(n-4)(n+2)/8=n_{3}(n_{3}-3)/2$ which is the
cardinality of the set of planar kinematic invariants for
$m_{n_{3}=n/2+1}(\mathbb{I},\mathbb{I})$.

The rest of the proof is based on the fact that in the kinematic space of
interest and the corresponding limit, only region $R_{1}$ contributes to the
amplitude. Since the result from region $R_{1}$ was shown to be
$m_{n_{3}=n/2+1}(\mathbb{I},\mathbb{I})$ for some bijetion of the kinematic
invariants in this subsection, then the statement of the proposition follows.
In order to actually complete the proof, we first need to classify all regions
and this is done in the next section. We therefore postpone the completion of
the proof to the end of the next section in subsection 6.3.

∎

##  6 Combinatorial Description of Regions

In the previous section we provided some examples of how the global Schwinger
formula for $A^{\phi^{4}}_{n}$ is evaluated. The result decomposes as a sum
over regions (cones) which are in bijection with noncrossing chord diagrams.
In this section we provide a systematic study of the structure of the regions.

The unexpected appearance of $m_{n/2+1}(\mathbb{I},\mathbb{I})$ in the
contribution from one of the regions, explained in section 5.4, motivates a
similar interpretation for the other regions. Indeed, in all examples we have
studied we find that all contributions are related to
$m_{n/2+1}(\alpha,\mathbb{I})$ for some choice of ordering $\alpha$.

In order to make the study systematic, we propose a diagrammatic procedure for
finding all the regions that contribute to $A^{\phi^{4}}_{n}$ and show how
each such region is in bijection with a cubic $m_{n/2+1}(\alpha,\mathbb{I})$
amplitude.

###  6.1 Regions for $A_{n}^{\phi^{4}}$: Non-Crossing Chord Diagrams

Let us start by defining non-crossing chord diagrams in our context.

######  Definition 6.1.

Place $n-2$ points labeled $0,1,\ldots,n-3$ in increasing order on the real
line. A non-crossing chord diagram is a perfect matching of the points such
that all edges can be drawn as chords on the upper half plane without any
crossings. Let us denote the chord connecting points $a$ and $b$ as
$\theta_{ab}$.

######  Conjecture 6.2.

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 defined in
6.1. Moreover, the region $R$ corresponding to a particular diagram is
obtained as follows:

  * •

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

  * •

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

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

Let us note that the case in which no chord surrounds any other chord
corresponds to

$R=\\{x_{0}=x_{1},\,x_{2}=x_{3},\,\ldots,\,x_{n-4}=x_{n-3}\\}\,.$ (6.1)

This is nothing but the region $R_{1}$ which was proven to set $H(x)=0$ in
Proposition (5.1).

######  Example 6.3.

Consider two of the examples presented in section 5. For $n=4$ there is a
single chord diagram. It has a single chord $\theta_{01}$ and therefore the
region is given by $x_{0}=x_{1}$. Recall that $x_{0}=0$ and so $x_{1}=0$.

For $n=6$ there are two non-crossing chord diagrams as shown in figure 2.

Figure 2: Non-crossing chord diagrams for $n=6$. On the right, the chord
$\theta_{03}$ surrounds the chord $\theta_{12}$ and therefore the condition
$x_{0}<x_{1}$ is imposed.

The corresponding regions can be seen to match $R_{1}$ and $R_{2}$ in (5.5),
i.e.,

$R_{1}=\\{x_{0}=x_{1},\,x_{2}=x_{3}\\}\,,\quad
R_{2}=\\{x_{0}=x_{3},\,x_{1}=x_{2},\,x_{0}<x_{1}\\}\,.$ (6.2)

Finally, we leave as an exercise to the reader to check that the five regions
for $n=8$ presented in (5.3) correspond to the diagrams in figure 3.

Figure 3: Non-crossing chord diagrams for $n=8$. In the second diagram
$\theta_{25}$ surrounds $\theta_{34}$ and therefore $x_{2}<x_{3}$. In the
third diagram $\theta_{05}$ surrounds both $\theta_{12}$ and $\theta_{34}$ and
therefore $x_{0}<x_{1}$ and $x_{0}<x_{3}$. In the fourth diagram $\theta_{03}$
surrounds $\theta_{12}$ so $x_{0}<x_{1}$. Finally, in the fifth diagram
$\theta_{05}$ surrounds $\theta_{14}$ which surrounds $\theta_{23}$ so
$x_{0}<x_{1}<x_{2}$.

In the last example of section 5 we found that the contribution to region
$R_{1}$ was computed by a biadjoint $\phi^{3}$ amplitude with $n/2+1$
particles, i.e., $m_{n/2+1}(\mathbb{I},\mathbb{I})$. The attentive reader
might have noticed that in all examples provided so far, the structure of the
answer resembles that of $m_{n/2+1}(\alpha,\mathbb{I})$ for some permutation
$\alpha$. We leave the precise connection between $\alpha$ and a region for
future work and here we concentrate on the schematic structure of
$A^{\phi^{4}}_{n}$ for which we have an all $n$ proposal.

###  6.2 Connecting Regions to Products of $\phi^{3}$ Amplitudes: Towards
$m_{n/2+1}(\alpha,\mathbb{I})$

In order to understand the structure of each region, it is useful to introduce
an additional chord to the non-crossing chord diagrams described above. More
precisely, we introduce two new points, which could be denoted $-1$ and $n-2$,
and we always draw a chord between them. The point $-1$ is located to the left
of $0$ and $n-2$ is to the right of $n-3$ so that the chord $\theta_{-1,n-2}$
surrounds the whole diagram. This can be understood as a way of introducing
into the figure the fixed particles $1$ and $2$ in the parameterization (2.7).

######  Definition 6.4.

An extended non-crossing chord diagram associated to $A^{\phi^{4}}_{n}$ is a
non-crossing chord diagram on $n$ points labeled by
$\\{-1,0,1,2,\ldots,n-3,n-2\\}$ in which $\theta_{-1,n-2}$ is always included.
We also define a meadow of an extended non-crossing chord diagram as any
region in the diagram delimited by more than one chord and by the line where
the points lie.

The claim is that a meadow delimited by $m$ chords and the real line
corresponds to a biadjoint ($m+1$)-subamplitude participating in
$m_{n/2+1}(\alpha,\mathbb{I})$. Moreover, we also claim that any chord
$\theta_{ab}$ shared by two meadows corresponds to a propagator in
$m_{n/2+1}(\alpha,\mathbb{I})$ of the form $1/t_{[a+3,b+2]}$. This also fixes
the topology of the cubic double-ordered amplitude.

Before describing the consequences of this proposal, let us give some examples
to illustrate it.

######  Example 6.5.

Consider the region described by the diagram in figure 4 for $n=10$.

Using the diagram it is easy to recognize the region as

$R=\\{x_{0}=x_{1},\,x_{2}=x_{5},\,x_{3}=x_{4},\,x_{6}=x_{7},\,x_{2}<x_{3}\\}\,.$
(6.3)

Since the green meadow is delimited by 4 chords and the real line then it
corresponds to a 5-particle subamplitude of $m_{6}(\alpha,\mathbb{I})$, while
the blue meadow is delimited by 2 chords and the real line and thus
corresponds to a 3-particle subamplitude.

The disk diagram on the right is intended to represent the topology of
$m_{6}(\alpha,\mathbb{I})$ given the diagram on the left. The reader familiar
with the CHY description of biadjoint partial amplitudes would recognize the
disk diagram as encoding the two orderings $\alpha$ and $\mathbb{I}$.

Figure 4: Left: An extended non-crossing chord diagram for $n=10$ where the
meadows have been coloured. The additional points $-1$ and $8$ together with
the chord $\theta_{-1,8}$ joining them are coloured in magenta. Right: Disc
diagram of an $m_{6}(\alpha,\mathbb{I})$ amplitude corresponding to the
contribution of the region on the left.

Finally, notice that the chord $\theta_{25}$ is shared by two meadows, hence
it generates the propagator $1/t_{567}$ in $m_{6}(\alpha,\mathbb{I})$. The
conclusion is that the contribution of this region to $A_{10}^{\rm\phi^{4}}$
is schematically given by

$m_{3}\times m_{5}\times\frac{1}{t_{567}}\,.$ (6.4)

######  Example 6.6.

Consider another region contributing to $A_{10}^{\rm\phi^{4}}$, described by
the diagram in figure 5.

Figure 5: Left: An extended non-crossing chord diagram for $n=10$ where the
meadows have been coloured. Right: Disc diagram of an
$m_{6}(\alpha,\mathbb{I})$ amplitude corresponding to the contribution of the
region on the left.

In this case we have two 4-particle subamplitudes and one propagator of the
form $1/t_{[3,7]}$. The contribution of this region to $A_{10}^{\rm\phi^{4}}$
is schematically given by

$(m_{4})^{2}\times\frac{1}{t_{[3,7]}}\,.$ (6.5)

Note that $(m_{4})^{2}$ stands for the product of two distinct four-point
$\phi^{3}$ amplitudes. Since we are only interested in the schematic
structure, i.e. in the number of amplitudes of a given type, we keep track of
that using exponents.

######  Example 6.7.

Consider now a region contributing to $A_{8}^{\rm\phi^{4}}$. The region is
defined by the diagram in figure 6. We leave as an exercise for the reader to
show that this corresponds to region $R_{5}$ in the example given in section
5.3.

Figure 6: Left: An extended non-crossing chord diagram for $n=8$ where the
meadows have been coloured. Right: Disc diagram of an
$m_{5}(\alpha,\mathbb{I})$ amplitude corresponding to the contribution of the
region on the left.

The extended non-crossing chord diagram contains three meadows delimited by
two chords (i.e. three 3-particle subamplitudes) and two propagators
corresponding to the chords $\theta_{05}$ and $\theta_{14}$. Accoding to our
proposal, these propagators are $1/t_{[3,7]}=1/t_{812}$ and $1/t_{456}$,
respectively. Once again, the schematic form of the contribution is

$(m_{3})^{3}\times\frac{1}{t_{812}}\times\frac{1}{t_{456}}\,.$ (6.6)

If we define $m_{3}:=1$ this is exactly the contribution
$A_{8}^{\phi^{4}:(5)}$ presented in (5.3).

Based on these and many other examples, we have found a formula that
reproduces the schematic structure of $A_{n}^{\rm\phi^{4}}$ in every case. The
formula is based on the Lagrange inversion procedure relating the series
expansion of a function $f(x)$ with that of its compositional inverse. We
review some related material in appendix A. Here we simply present the final
form of the proposal. Let

$h(x)=\sum_{i=0}^{\infty}h_{i}x^{i}:=\sum_{i=0}^{\infty}m_{i+2}x^{i}\,,$ (6.7)

where $m_{i+2}$ represents a generic $(i+2)$-particle amplitude in the
biadjoint $\phi^{3}$ scalar theory of the form
$m_{i+2}(\mathbb{I},\mathbb{I})$. Since the mass dimension of
$m_{i+2}(\mathbb{I},\mathbb{I})$ is $-2(i-1)$ we are motivated to define
$m_{2}:=P^{2}$ and $m_{3}:=1$. Here $1/P^{2}$ represents a generic propagator.

The claim is that the schematic form of the amplitude $A^{\phi^{4}}_{n}$ is
given by

$A^{\phi^{4}}_{n}=\left(\frac{2}{n\,h_{0}^{n/2-1}}\right)\frac{1}{2\pi
i}\oint_{|z|=\epsilon}dz\left(\frac{h(z)}{z}\right)^{n/2}\,.$ (6.8)

Let us compute the first few cases of (6.8),

$\displaystyle A^{\phi^{4}}_{4}=$ $\displaystyle\,h_{1}=m_{3}\,,$
$\displaystyle A^{\phi^{4}}_{6}=$
$\displaystyle\,\frac{h_{0}^{2}h_{2}+h_{0}h_{1}^{2}}{h_{0}^{2}}=m_{4}+m_{3}^{2}\frac{1}{P^{2}}\,,$
$\displaystyle A^{\phi^{4}}_{8}=$
$\displaystyle\,\frac{h_{3}h_{0}^{3}+3h_{1}h_{2}h_{0}^{2}+h_{1}^{3}h_{0}}{h_{0}^{3}}=m_{5}+3m_{3}m_{4}\frac{1}{P^{2}}+m_{3}^{3}\left(\frac{1}{P^{2}}\right)^{2}\,,$
$\displaystyle A^{\phi^{4}}_{10}=$
$\displaystyle\,m_{6}+4m_{5}m_{3}\frac{1}{P^{2}}+2m_{4}^{2}\frac{1}{P^{2}}+6m_{4}m_{3}^{2}\left(\frac{1}{P^{2}}\right)^{2}+m_{3}^{4}\left(\frac{1}{P^{2}}\right)^{3}\,.$

There are several consistency checks that can be done on (6.8).

The first is that the number of non-crossing chord diagrams with $n/2-1$
chords is $\textrm{C}_{n/2-1}$. Therefore if we set all $m_{r}:=1$ so that the
contribution from each region is unity, one must find that
$A^{\phi^{4}}_{n}=\textrm{C}_{n/2-1}$.

This means that we must set

$h(x)=\sum_{i=1}^{\infty}x^{i}=\frac{1}{1-x}\,,$ (6.9)

and evaluate

$A^{\phi^{4}}_{n}=\left(\frac{2}{n}\right)\frac{1}{2\pi
i}\oint_{|z|=\epsilon}dz\left(\frac{1}{z(1-z)}\right)^{n/2}=\textrm{C}_{n/2-1}\,.$
(6.10)

The last equality follows from the Lagrange inversion formula with
$f(x)=x(1-x)$ and $g(x)=xB_{2}(x)$, where $B_{2}(x)$ is the generating
function of Catalan numbers.

The second check is that if $A^{\phi^{4}}_{n}$ is evaluated on “planar
kinematics” Cachazo:2013iea ; Early:2019eun ; Cachazo:2020wgu , i.e. on the
kinematic point where all planar Mandelstam invariants that participate in
$A^{\phi^{4}}_{n}$ are unity, $t_{[a,b]}=1$, then $A^{\phi^{4}}_{n}$ simply
counts the number of planar ternary trees (with all internal vertices of
degree four). The numbers are known to be given by the Fuss-Catalan sequence,
${\rm FC}_{n/2-1}(3,1)$. For $n=4,6,8,10$ one has ${\rm
FC}_{n/2-1}(3,1)=1,3,12,55$. This check can be done by realizing that on
planar kinematics $m_{n}=\textrm{C}_{n-2}$ and therefore

$h(x)=\sum_{i=1}^{\infty}\textrm{C}_{i}\,x^{i}=B_{2}(x)=\frac{1-\sqrt{1-4x}}{2x}\,.$
(6.11)

As shown in appendix A in (A.8), it is indeed the case that

${\rm FC}_{r}(3,1)=\frac{1}{2\pi
i}\oint_{|z|=\epsilon}\frac{dz}{r+1}\left(\frac{1-\sqrt{1-4z}}{2z^{2}}\right)^{r+1}\,,$
(6.12)

which gives the required relation when $r=n/2-1$.

###  6.3 Completing the Proof of Proposition 5.2

In order to complete the proof of Proposition 5.2 we have to show that all
regions that contribute to $A^{\phi^{4}}_{n}$, except for $R_{1}$, are ${\cal
O}(\epsilon)$ when

$t_{[a,b]}=\frac{1}{\epsilon}\,,\quad
a\in\\{3,5,\ldots,n-3\\}\,,\,\,b\in\\{a+2,a+4,\ldots,n-1\\}\,.$ (6.13)

Recall that $R_{1}$ is the region corresponding to $n/2-1$ non-crossing chords
so that none is surrounded by any other. According to the rules explained in
this section, this means that no propagator is generated. One the other hand,
every single other region has at least one chord surrounded by another, say
$\theta_{ef}$ and therefore there is at least one propagator in the region’s
contribution to the amplitude. The propagator is $1/t_{[e+3,f+2]}$. Clearly
$f-e\geq 3$ so that the chord can contain at least another one. This means
that the chords of interest can only have $e\in\\{0,1,\ldots,n-6\\}$ and
$f\in\\{e+3,e+4,\ldots,n-3\\}$. Therefore each region different from $R_{1}$
contains at last one propagator of the form $1/t_{[a,b]}$ with
$a\in\\{3,4,\ldots,n-3\\}$ and $b\in\\{a+2,a+3,\ldots,n-1\\}$. But this is
exactly the range of propagators set to $\epsilon$ and this concludes the
proof.

##  7 From $\phi^{3}$ Amplitudes to $\phi^{p}$ Amplitudes

In this section we extend the limiting procedure used to obtain
$A_{n}^{\phi^{4}}$ from $m_{n}(\mathbb{I},\mathbb{I})$ to make a general
conjecture for any $A_{n}^{\phi^{p}}$ amplitude and its global Schwinger
formulation. We also propose a diagrammatic procedure for finding all the
regions that contribute to $A_{n}^{\phi^{p}}$ and point out a connection with
$m_{(n+2(p-3))/(p-2)}(\alpha,\mathbb{I})$ amplitudes.

To start with, the limiting procedure that generates $A_{n}^{\phi^{p}}$ from
$m_{n}(\mathbb{I},\mathbb{I})$ is the following.

######  Proposition 7.1.

Consider the region of the kinematic space of $n$ massless particles where
$t_{[a,b]}=1/\epsilon$ whenever $b-a\not\equiv 0\\!\\!\mod p-2$ and let
$m_{n}^{(\epsilon)}(\mathbb{I},\mathbb{I})$ denote
$m_{n}(\mathbb{I},\mathbb{I})$ evaluated on it. Then

$A^{\phi^{p}}_{n}=\lim_{\epsilon\to
0}\frac{1}{(\textrm{C}_{p-2}\,\epsilon^{p-3})^{\frac{n-2}{p-2}}}m_{n}^{(\epsilon)}(\mathbb{I},\mathbb{I}),$
(7.1)

where $\textrm{C}_{m}$ is the $m^{th}$ Catalan number.

###### Proof.

The proof is analogous to that of Proposition 4.1. First, consider any Feynman
diagram ${\cal T}^{(p)}$ of $A^{\phi^{p}}_{n}$, that is, any $(p-1)$-ary
planar tree on $n$-leaves. Such a diagram has $(n-2)/(p-2)$ vertices of degree
$p$. The strategy is again to find out how many planar binary trees give rise
to ${\cal T}^{(p)}$ by collapsing edges, i.e., taking their length to zero.
This is easily done by realizing that for each degree-$p$ vertex of ${\cal
T}^{(p)}$ there are exactly $\textrm{C}_{p-2}$ ways, compatible with
planarity, of growing a tree to produce $\textrm{C}_{p-2}$ degree-three
vertices. This means that there are $\textrm{C}_{p-2}^{(n-2)/(p-2)}$ binary
trees that give rise to ${\cal T}^{(p)}$. Of course, not all binary trees
descend to a $(p-1)$-ary diagram. Note that under the kinematics in the
proposition, $\phi^{3}$ Feynman diagrams that collapse to $\phi^{p}$ diagrams
have exactly $(n-p)/(p-2)$ $\epsilon$-independent propagators and therefore
$(n-3)-(n-p)/(p-2)=(p-3)(n-2)/(p-2)$ which become $1/t=\epsilon$. Diagrams
that do not produce a $\phi^{p}$ diagram have at least one extra propagator of
the form $1/t=\epsilon$. Therefore, in the limit $\epsilon\to 0$ the
$\phi^{p}$ amplitude is recovered. ∎

As in section 4, one can write the global Schwinger formula for
$A_{n}^{\phi^{p}}$ as a single integral

$A_{n}^{\phi^{p}}=\int_{\mathbb{R}^{n-3}}d^{n-3}x\,\textrm{exp}\left(-\sum_{a<b}^{{\cal
K}_{p}}t_{[a,b]}f_{[a,b]}(x)\right)Q(x)\,,$ (7.2)

where ${\cal K}_{p}$ means that the sum is over ordered pairs $(a,b)$ such
that $b-a\equiv 0\\!\\!\mod p-2$. Here $Q(x)$ is defined as

$Q(x):=\lim_{\epsilon\to
0^{+}}\frac{1}{(\textrm{C}_{p-2}\,\epsilon^{p-3})^{\frac{n-2}{p-2}}}\,\textrm{exp}\left(-\frac{1}{\epsilon}H(x)\right)$
(7.3)

with

$H(x)=\sum_{a<b:(a,b)\notin{\cal K}_{p}}f_{[a,b]}(x)\,.$ (7.4)

Note that due to the non-negativity of $H(x)$, $Q(x)$ only has support in
regions where $H(x)=0$. Again, the distribution $Q(x)$ becomes a sum over
distributions that localize the integral to these regions. This means that
equation (7.2) can also be understood as a sum over regions, where these as
associated to diagrams as explained in the next subsection.

###  7.1 Combinatorial Description of Regions

In this subsection we conjecture that the solutions of $H(x)=0$ are regions of
dimension $n/(p-2)-1$ in $\mathbb{R}^{n-3}$ which are classified by non-
crossing $(p-2)$-chord diagrams. The definition of non-crossing $(p-2)$-chord
diagrams, in our context, is the following.

######  Definition 7.2.

Place $n-2$ points labeled $0,1,\ldots,n-3$ on the real line in increasing
order. A non-crossing $(p-2)$-chord diagram is a perfect matching of the
points such that each matching involves $(p-2)$ points joined by a
$(p-2)$-chord and 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
$\theta_{a_{1},a_{2},...,a_{p-2}}$ (for general k-chord diagrams see e.g.
kChords .)

######  Conjecture 7.3.

The regions contributing to $A^{\phi^{p}}_{n}$ are in bijection with the set
of all $\textrm{FC}_{(n-2)/(p-2)}(p-2,1)$777Recall that $\textrm{FC}_{m}(q,r)$
is the Fuss-Catalan number given by
$\textrm{FC}_{m}(q,r)\equiv\frac{r}{mq+r}{mq+r\choose m}\,.$ Note that for
$q=2$ and $r=1$ the Fuss-Catalan numbers coincide with the Catalan numbers,
i.e. $\textrm{FC}_{m}(2,1)=\textrm{C}_{m}$. possible non-crossing
$(p-2)$-chord diagrams. Moreover, the region $R$ corresponding to a particular
diagram is obtained as follows:

  * •

For each $(p-2)$-chord $\theta_{a_{1},a_{2},...,a_{p-2}}$ set
$x_{a_{1}}=x_{a_{2}}=\cdots=x_{a_{p-2}}$.

  * •

If a $(p-2)$-chord $\theta_{a_{1},a_{2},...,a_{p-2}}$ surrounds another
$(p-2)$-chord $\theta_{b_{1},b_{2},...,b_{p-2}}$, then
$x_{a_{1}}=x_{a_{2}}=\cdots=x_{a_{p-2}}<x_{b_{1}}=x_{b_{2}}=\cdots=x_{b_{p-2}}$.

In other words, the regions defined by the non-crossing $(p-2)$-chord diagrams
are all the solutions to $H(x)=0$, where $H(x)$ is given by (7.4), and the sum
of their contributions produces all the $\textrm{FC}_{(n-2)/(p-2)}(p-1,1)$
trees of $\phi^{p}$.

######  Example 7.4.

Consider the $n=10$ amplitude for $\phi^{6}$. There are four non-crossing
$4$-chord diagrams and are shown in figure 7.

Figure 7: All possible non-crossing 4-chord diagrams for $n=10$ and $p=6$.
Each diagram contains exactly two 4-chords. In the top diagram one 4-chord
joins points $0,1,2,3$ while the second 4-chord joins $4,5,6,7$. In the second
diagram the 4-chord $\theta_{0127}$ surrounds $\theta_{3456}$. In the third,
$\theta_{0167}$ surrounds $\theta_{2345}$. In the last diagram,
$\theta_{0567}$ surrounds $\theta_{1234}$.

Reading from top to bottom and recalling that $x_{0}=0$, the four regions
generated by these diagrams correspond, respectively, to

$\displaystyle R_{1}=$
$\displaystyle\,\\{x_{1}=x_{2}=x_{3}=0,\,x_{4}=x_{5}=x_{6}=x_{7}\\}\,,$
$\displaystyle R_{2}=$
$\displaystyle\,\\{x_{1}=x_{2}=x_{7}=0,\,x_{3}=x_{4}=x_{5}=x_{6},\,x_{3}>0\\}\,,$
$\displaystyle R_{3}=$
$\displaystyle\,\\{x_{1}=x_{6}=x_{7}=0,\,x_{2}=x_{3}=x_{4}=x_{5},\,x_{2}>0\\}\,,$
$\displaystyle R_{4}=$
$\displaystyle\,\\{x_{5}=x_{6}=x_{7}=0,\,x_{1}=x_{2}=x_{3}=x_{4},\,x_{1}>0\\}\,.$

Therefore, the distribution $Q(x)$ is given by

$Q(x)=Q_{1}(x)+Q_{2}(x)+Q_{3}(x)+Q_{4}(x)$

with

$\displaystyle Q_{1}:=$
$\displaystyle\,\delta(x_{1})\delta(x_{2})\delta(x_{3})\delta(x_{4}-x_{5})\delta(x_{5}-x_{6})\delta(x_{6}-x_{7})\,,$
$\displaystyle Q_{2}:=$
$\displaystyle\,\theta(x_{3})\delta(x_{1})\delta(x_{2})\delta(x_{7})\delta(x_{3}-x_{4})\delta(x_{4}-x_{5})\delta(x_{5}-x_{6})\,,$
$\displaystyle Q_{3}:=$
$\displaystyle\,\theta(x_{2})\delta(x_{1})\delta(x_{6})\delta(x_{7})\delta(x_{2}-x_{3})\delta(x_{3}-x_{4})\delta(x_{4}-x_{5})\,,$
$\displaystyle Q_{4}:=$
$\displaystyle\,\theta(x_{1})\delta(x_{5})\delta(x_{6})\delta(x_{7})\delta(x_{1}-x_{2})\delta(x_{2}-x_{3})\delta(x_{3}-x_{4})\,.$

The contributions from each region are

$\displaystyle A_{10}^{\phi^{6}:(1)}=$
$\displaystyle\,\frac{1}{t_{[1,5]}}+\frac{1}{t_{[2,6]}}\,,\hskip
14.22636ptA_{10}^{\phi^{6}:(2)}=\,\frac{1}{t_{[5,9]}}\,,$ $\displaystyle
A_{10}^{\phi^{6}:(3)}=$ $\displaystyle\,\frac{1}{t_{[4,8]}}\,,\hskip
51.21495ptA_{10}^{\phi^{6}:(4)}=\,\frac{1}{t_{[3,7]}}\,.$ (7.5)

The amplitude $A_{10}^{\phi^{6}}$ is the sum over all $\textrm{FC}_{2}(4,1)=4$
contributions and gives rise to the familiar expression with
$\textrm{FC}_{2}(5,1)=5$ Feynman diagrams.

######  Example 7.5.

Consider now the $n=11$ case for $\phi^{5}$. There are 12 non-crossing
$3$-chord diagrams and are represented in figure 8.

Figure 8: All possible non-crossing 3-chord diagrams for $n=11$ and $p=5$.

Reading from left to right and top to bottom, the 12 regions generated by
these diagrams correspond, respectively, to

$\displaystyle R_{1}=$
$\displaystyle\,\\{x_{1}=x_{2}=0,\,x_{3}=x_{4}=x_{5},\,x_{6}=x_{7}=x_{8}\\}\,,$
$\displaystyle R_{2}=$
$\displaystyle\,\\{x_{1}=x_{2}=0,\,x_{3}=x_{7}=x_{8},\,x_{4}=x_{5}=x_{6},\,x_{3}<x_{4}\\}\,,$
$\displaystyle R_{3}=$
$\displaystyle\,\\{x_{1}=x_{2}=0,\,x_{5}=x_{6}=x_{7},\,x_{3}=x_{4}=x_{8},\,x_{3}<x_{5}\\}\,,$
$\displaystyle R_{4}=$
$\displaystyle\,\\{x_{4}=x_{5}=0,\,x_{1}=x_{2}=x_{3},\,x_{6}=x_{7}=x_{8},\,x_{1}>0\\}\,,$
$\displaystyle R_{5}=$
$\displaystyle\,\\{x_{1}=x_{5}=0,\,x_{2}=x_{3}=x_{4},\,x_{6}=x_{7}=x_{8},\,x_{2}>0\\}\,,$
$\displaystyle R_{6}=$
$\displaystyle\,\\{x_{7}=x_{8}=0,\,x_{1}=x_{2}=x_{3},\,x_{4}=x_{5}=x_{6},\,x_{1}>0,\,x_{4}>0\\}\,,$
$\displaystyle R_{7}=$
$\displaystyle\,\\{x_{4}=x_{8}=0,\,x_{1}=x_{2}=x_{3},\,x_{5}=x_{6}=x_{7},\,x_{1}>0,\,x_{5}>0\\}\,,$
$\displaystyle R_{8}=$
$\displaystyle\,\\{x_{1}=x_{8}=0,\,x_{2}=x_{3}=x_{4},\,x_{5}=x_{6}=x_{7},\,x_{2}>0,\,x_{5}>0\\}\,,$
$\displaystyle R_{9}=$
$\displaystyle\,\\{x_{7}=x_{8}=0,\,x_{1}=x_{5}=x_{6},\,x_{2}=x_{3}=x_{4},\,x_{2}>x_{1}>0\\}\,,$
$\displaystyle R_{10}=$
$\displaystyle\,\\{x_{7}=x_{8}=0,\,x_{1}=x_{2}=x_{6},\,x_{3}=x_{4}=x_{5},\,x_{3}>x_{1}>0\\}\,,$
$\displaystyle R_{11}=$
$\displaystyle\,\\{x_{1}=x_{8}=0,\,x_{2}=x_{6}=x_{7},\,x_{3}=x_{4}=x_{5},\,x_{3}>x_{2}>0\\}\,,$
$\displaystyle R_{12}=$
$\displaystyle\,\\{x_{1}=x_{8}=0,\,x_{2}=x_{3}=x_{7},\,x_{4}=x_{5}=x_{6},\,x_{4}>x_{2}>0\\}\,.$

We leave as an exercise to the reader to find the distributions associated to
these regions and to show that the contributions from each region are:

$\displaystyle A_{11}^{\phi^{5}:(1)}=$
$\displaystyle\,\frac{1}{t_{[2,5]}t_{[8,11]}}+\frac{1}{t_{[2,5]}t_{[2,8]}}+\frac{1}{t_{[5,8]}t_{[2,8]}}+\frac{1}{t_{[5,8]}t_{[5,11]}}+\frac{1}{t_{[8,11]}t_{[5,11]}}\,,$
$\displaystyle A_{11}^{\phi^{5}:(2)}=$
$\displaystyle\,\frac{1}{t_{[6,9]}}\left(\frac{1}{t_{[2,5]}}+\frac{1}{t_{[5,11]}}\right)\,,\hskip
14.22636ptA_{11}^{\phi^{5}:(3)}=\,\frac{1}{t_{[7,10]}}\left(\frac{1}{t_{[2,5]}}+\frac{1}{t_{[5,11]}}\right)\,,$
$\displaystyle A_{11}^{\phi^{5}:(4)}=$
$\displaystyle\,\frac{1}{t_{[3,6]}}\left(\frac{1}{t_{[8,11]}}+\frac{1}{t_{[2,8]}}\right)\,,\hskip
14.22636ptA_{11}^{\phi^{5}:(5)}=\,\frac{1}{t_{[4,7]}}\left(\frac{1}{t_{[8,11]}}+\frac{1}{t_{[2,8]}}\right)\,,$
$\displaystyle A_{11}^{\phi^{5}:(6)}=$
$\displaystyle\,\frac{1}{t_{[3,9]}}\left(\frac{1}{t_{[3,6]}}+\frac{1}{t_{[6,9]}}\right)\,,\hskip
17.07164ptA_{11}^{\phi^{5}:(7)}=\,\frac{1}{t_{[7,10]}t_{[3,6]}}\,,$
$\displaystyle A_{11}^{\phi^{5}:(8)}=$
$\displaystyle\,\frac{1}{t_{[4,10]}}\left(\frac{1}{t_{[4,7]}}+\frac{1}{t_{[7,10]}}\right)\,,\hskip
11.38109ptA_{11}^{\phi^{5}:(9)}=\,\frac{1}{t_{[4,7]}t_{[3,9]}}\,,$
$\displaystyle A_{11}^{\phi^{5}:(10)}=$
$\displaystyle\,\frac{1}{t_{[5,8]}t_{[3,9]}}\,,\hskip
14.22636ptA_{11}^{\phi^{5}:(11)}=\,\frac{1}{t_{[5,8]}t_{[4,10]}}\,,\hskip
14.22636ptA_{11}^{\phi^{5}:(12)}=\,\frac{1}{t_{[6,9]}t_{[4,10]}}\,.$ (7.6)

The amplitude $A_{11}^{\phi^{5}}$ is the sum over all
$\textrm{FC}_{3}(3,1)=12$ contributions and gives rise to the familiar
expression with $\textrm{FC}_{3}(4,1)=22$ Feynman diagrams.

From these examples note that even for $p>4$ the structure of the contribution
of each region also resembles that of a cubic amplitude. In particular, it has
the structure of $m_{(n+2(p-3))/(p-2)}(\alpha,\mathbb{I})$ for some
permutation $\alpha$. Here we will only concentrate on the schematic structure
of $A_{n}^{\phi^{p}}$ for all $n$, leaving again the precise connection
between $\alpha$ and the region to future work.

###  7.2 From Regions to Products of $\phi^{3}$ Amplitudes: Towards
$m_{(n+2(p-3))/(p-2)}(\alpha,\mathbb{I})$

As in the $p=4$ case, in order to understand the structure of each region it
is useful to introduce an additional $(p-2)$-chord to the non-crossing
$(p-2)$-chord diagrams from Definition 7.2. This is done by adding $p-2$ new
points labelled $-p+3,-p+4,\dots,-1$ and $n-2$ so that the new set of points
is $\\{-p+3,-p+4,\ldots-1,0,1,2,\ldots,n-3,n-2\\}$ and points are located in
increasing order on the real line888In fact, any choice where at least one
point is on the left of 0 and one point is on the right on $n-3$ is valid..

######  Definition 7.6.

An extended non-crossing $(p-2)$-chord diagram is a non-crossing $(p-2)$-chord
diagram on $n$ points labelled by
$\\{-p+3,-p+4,\ldots-1,0,1,2,\ldots,n-3,n-2\\}$ in which
$\theta_{-p+3,-p+4,...,-1,n-2}$ is always included. We also define a meadow of
an extended non-crossing $(p-2)$-chord diagram as any region in the diagram
delimited by more than one $(p-2)$-chord and by the line where the points lie.

From now on we will abuse notation and use $\theta_{ab}$ to refer to the
unique path in a $(p-2)$-chord joining two points $a$ and $b$. Therefore, the
general claim is that a meadow delimited by $m$ such paths and the real line
corresponds to a biadjoint ($m+1$)-subamplitude participating in
$m_{(n+2(p-3))/(p-2)}(\alpha,\mathbb{I})$. We also claim that the upper
boundary of a meadow, $\theta_{ab}$, corresponds to a propagator in
$m_{(n+2(p-3))/(p-2)}(\alpha,\mathbb{I})$ of the form $1/t_{[a+3,b+2]}$, with
the exception of the pair $\\{a,b\\}=\\{-1,n-2\\}$. This also fixes the
topology of the cubic double-ordered amplitude.

Let us again give some examples to illustrate the proposal.

######  Example 7.7.

Consider the extended non-crossing 4-chord diagram of $\phi^{6}$ for $n=14$
shown in figure 9.

Figure 9: Left: An extended non-crossing chord diagram of $\phi^{6}$ for
$n=14$ where the meadows have been coloured. Right: Disc diagram of an
$m_{5}(\alpha,\mathbb{I})$ amplitude corresponding to the contribution of the
region on the left.

In this extended diagram the points $-3$, $-2$, $-1$ and $12$ together with
the 4-chord $\theta_{-3,-2,-1,12}$ that joins them are coloured in magenta.
One can see that there are two meadows coloured in green and blue. The green
meadow is delimited by the real line and by 2 paths $\theta_{-1,12}$ and
$\theta_{0,11}$, thus it corresponds to a 3-point subamplitude appearing in
$m_{5}(\alpha,\mathbb{I})$. Similarly, the blue meadow is delimited by 3 paths
$\theta_{2,11}$, $\theta_{36}$ and $\theta_{7,10}$ and the real line, thus it
corresponds to a 4-point subamplitude of $m_{5}(\alpha,\mathbb{I})$. The upper
boundary of the blue meadow is $\theta_{2,11}$ and this means that there is a
propagator of the form $1/t_{[5,13]}$. The upper boundary of the green meadow
is of the form $\theta_{-1,n-2}$ and it does not generate a propagator.
Therefore, the schematic form of the contribution is

$m_{3}\times m_{4}\times\frac{1}{t_{[5,13]}}\,.$

######  Example 7.8.

Now consider another extended non-crossing 4-chord diagram of $\phi^{6}$ for
$n=14$ shown in figure 10.

Figure 10: Left: An extended non-crossing chord diagram of $\phi^{6}$ for
$n=14$ where the meadows have been coloured. Right: Disc diagram of an
$m_{5}(\alpha,\mathbb{I})$ amplitude corresponding to the contribution of the
region on the left.

As in the previous example, one can see that there are three meadows coloured
in green, blue and red. The green meadow is delimited by the real line and by
2 paths $\theta_{-1,12}$ and $\theta_{0,11}$ and gives rise to a 3-point
subamplitude appearing in $m_{5}(\alpha,\mathbb{I})$. Similarly, the blue
meadow is delimited by the real line and 2 paths $\theta_{05}$ and
$\theta_{14}$, thus it corresponds to a 3-point subamplitude of
$m_{5}(\alpha,\mathbb{I})$. Likewise, the red meadow is delimited by the real
line and 2 paths $\theta_{5,10}$ and $\theta_{69}$ and gives rise to a 3-point
subamplitude of $m_{5}(\alpha,\mathbb{I})$. We also have two propagators of
the form $1/t_{[3,7]}$ and $1/t_{[8,12]}$. Therefore, the schematic form of
the contribution is

$(m_{3})^{3}\times\frac{1}{t_{[3,7]}}\times\frac{1}{t_{[8,12]}}\,.$

###  7.3 Schematic Structure of $A_{n}^{\phi^{p}}$

Before proposing a Lagrange inversion-like formula to reproduce the schematic
structure of $A_{n}^{\phi^{p}}$, we present more examples:

$\displaystyle A_{8}^{\phi^{5}}=$
$\displaystyle\,m_{4}+2m_{3}^{2}\frac{1}{P^{2}}\,,\quad
A_{11}^{\phi^{5}}=\,m_{5}+6m_{3}m_{4}\frac{1}{P^{2}}+5m_{3}^{3}\left(\frac{1}{P^{2}}\right)^{2},$
$\displaystyle A_{14}^{\phi^{5}}=$
$\displaystyle\,m_{6}+4m_{4}^{2}\frac{1}{P^{2}}+8m_{5}m_{3}\frac{1}{P^{2}}+28m_{4}m_{3}^{2}\left(\frac{1}{P^{2}}\right)^{2}+14m_{3}^{4}\left(\frac{1}{P^{2}}\right)^{3},$
$\displaystyle A_{10}^{\phi^{6}}=$
$\displaystyle\,m_{4}+3m_{3}^{2}\frac{1}{P^{2}}\,,\quad
A_{14}^{\phi^{6}}=\,m_{5}+9m_{3}m_{4}\frac{1}{P^{2}}+12m_{3}^{3}\left(\frac{1}{P^{2}}\right)^{2},$
$\displaystyle A_{12}^{\phi^{7}}=$
$\displaystyle\,m_{4}+4m_{3}^{2}\frac{1}{P^{2}}\,,$ $\displaystyle
A_{14}^{\phi^{8}}=$ $\displaystyle\,m_{4}+5m_{3}^{2}\frac{1}{P^{2}}\,.$ (7.7)

Let us make a proposal for the all $n$ structure of $A^{\phi^{p}}_{n}$
amplitudes in terms of biadjoint cubic amplitudes and then perform the same
consistency check as done for $\phi^{4}$.

The proposal is motivated by the fact, proven in appendix A, that
$f(x)=x/B_{k-1}(x)$ and $g(x)=xB_{k}(x)$ are compositional inverses of each
other if $B_{r}(x)$ is the generating function of the Fuss-Catalan numbers
$\textrm{FC}_{m}(r,1)$. This led us to propose a recursive structure in which
we define

$h_{3}(x):=\sum_{i=0}^{\infty}m_{i+2}x^{i}\,$ (7.8)

and

$h_{k}(x)=\sum_{j=0}^{\infty}h_{k,j}x^{j}:=\sum_{j=0}^{\infty}\frac{1}{2\pi
i}\oint_{|z|=\epsilon}\frac{dz}{j+1}\left(\frac{h_{k-1}(z)}{z}\right)^{j+1}x^{j}\,.$
(7.9)

The structure of the $A^{\phi^{p}}_{n}$ amplitude is then given by

$A^{\phi^{p}}_{n}=\frac{h_{p,(n-2)/(p-2)}}{h_{0}^{(p-3)(n-2)/(p-2)}}\,.$
(7.10)

The first consistency check is that the number of non-crossing $(p-2)$-chord
diagrams with $(n-2)/(p-2)$ chords is $\textrm{FC}_{(n-2)/(p-2)}(p-2,1)$.
Therefore if one sets all $m_{i+2}:=1$ so that the contribution from each
region is unity, one must find that
$A^{\phi^{p}}_{n}=\textrm{FC}_{(n-2)/(p-2)}(p-2,1)$.

In section 6.2 we showed that setting

$h_{3}(x)=\sum_{i=1}^{\infty}x^{i}=\frac{1}{1-x}=B_{1}(x)\,,$ (7.11)

turns $h_{4}(x)$ into the generating function of the numbers
$\textrm{FC}_{m}(3,1)$. Iterating the procedure one finds that $h_{k}(x)$
turns into the generating function of the numbers $\textrm{FC}_{m}(k-1,1)$.

The second check is evaluating $A^{\phi^{p}}_{n}$ on planar kinematics so that
$A^{\phi^{p}}_{n}$ counts the number of unrooted planar $(p-1)$-ary trees
(with all internal vertices of degree $p$). The numbers are known to be given
by the Fuss-Catalan sequence, ${\rm FC}_{(n-2)/(p-2)}(p-1,1)$. This check can
again be done by realizing that on planar kinematics $m_{n}=\textrm{C}_{n-2}$
and therefore

$h_{3}(x)=\sum_{i=1}^{\infty}\textrm{C}_{i}\,x^{i}=B_{2}(x)=\frac{1-\sqrt{1-4x}}{2x}\,.$
(7.12)

Iterating one finds that $h_{k}(x)=B_{k-1}(x)$, the generating function of the
Fuss-Catalan numbers $\textrm{FC}_{m}(k-1,1)$ as required.

In appendix A we provide several examples that illustrate the iteration
procedure and the resulting formulas for $A_{n}^{\phi^{p}}$.

##  8 Discussions

In this work we have extended the global Schwinger formulation to all partial
amplitudes $m_{n}(\alpha,\beta)$, and also to amplitudes in $\phi^{p}$
theories. $A_{n}^{\phi^{p}}$ is given as a sum over regions, each of which is
proposed to be in bijection with a $\phi^{3}$ biadjoint partial amplitude.
This leads to the statement that $A_{n}^{\phi^{p}}$ amplitudes can be
understood as a sum of products of cubic amplitudes.

A very simple diagrammatic procedure for listing all regions contributing to
an amplitude was found in terms of non-crossing k-chord diagrams. Given one
such diagram, we have provided an algorithm for determining the structure of
the contribution in terms of $\phi^{3}$ amplitudes (meadows) and propagators
(“frontiers” separating meadows). Every meadow can be seen to be related to a
cubic amplitude participating in $m_{(n+2(p-3))/(p-2)}(\alpha,\mathbb{I})$.
Our identification so far is lacking a direct way of determining the
permutation $\alpha$ from the non-chord diagram. It would also be very
important to find a purely combinatorial method to determine the precise
bijection between the set of planar kinematic invariants in each object.

Our main focus has been on a combinatorial prescription for $\phi^{p}$
amplitudes. However, it would be interesting to find a diagrammatic procedure,
in the lines of that for $\phi^{p}$ amplitudes, to determine the regions that
compute $m_{n}(\alpha,\beta)$ so that $H_{\alpha}(x)=0$ in (3.4).

We end this work with three topics for future research.

###  8.1 Relation to Green Functions in Planar Theories

The standard way of computing Green functions, $G_{n}$, from connected Green
functions, $G_{n}^{c}$, is via an exponentiation procedure. However, it is
well-known that in planar theories this does not work Brezin:1977sv . This is
because planarity forces points of the Green function
$G_{n}(x_{1},x_{2},\ldots,x_{n})$, to be on the boundary of a disk and a
connected Green function for points in a subset
$J\subset\\{x_{1},x_{2},\ldots,x_{n}\\}$ can be thought of as cutting the disk
into regions so that one of them only contains the points in $J$. Having done
this, another connected Green function can only be constructed from the pieces
left, and so on. In Brezin:1977sv , a theory that only admits Green functions
with an even number $n=2q$ of points was considered and gave rise to the
following combinatorial problem: in how many ways can $2q$ points on a disk be
clustered in non-overlapping sets so that there are $r_{1}$ pairs, $r_{2}$
quadruplets, etc. As shown in Brezin:1977sv  this is solved by the
coefficients of the formula relating Green functions

$G_{2q}=\sum_{r_{i}\geq
0}\delta_{q,\left(\sum_{i}i\,r_{i}\right)}\frac{(2q)!}{(2q+1-\sum_{i}r_{i})!}\frac{\left(G_{2}^{c}\right)^{r_{1}}}{r_{1}!}\frac{\left(G_{4}^{c}\right)^{r_{2}}}{r_{2}!}\ldots\frac{\left(G_{2q}^{c}\right)^{r_{q}}}{r_{q}!}\,.$
(8.1)

Here the Kronecker delta guarantees that each of the $2q$ points participates
in each term.

Up to this point, this discussion seems to be completely independent of the
formulas found in this work. In order to see the connection, let us list the
first few cases as done in eq. (31) of Brezin:1977sv ,

$\displaystyle G_{4}=$ $\displaystyle\,G_{4}^{c}+2\left(G_{2}^{c}\right)^{2},$
$\displaystyle G_{6}=$
$\displaystyle\,G_{6}^{c}+6\,G_{4}^{c}G_{2}^{c}+5\left(G_{2}^{c}\right)^{3},$
$\displaystyle G_{8}=$
$\displaystyle\,G_{8}^{c}+4\left(G_{4}^{c}\right)^{2}+8\,G_{6}^{c}G_{2}^{c}+28\,G_{4}^{c}\left(G_{2}^{c}\right)^{2}+14\left(G_{2}^{c}\right)^{4}\,.$

Comparing to the expressions for $\phi^{5}$ amplitudes in (7.3), i.e.

$\displaystyle A_{8}^{\phi^{5}}=$
$\displaystyle\,m_{4}+2m_{3}^{2}\frac{1}{P^{2}}\,,$ $\displaystyle
A_{11}^{\phi^{5}}=$
$\displaystyle\,m_{5}+6m_{3}m_{4}\frac{1}{P^{2}}+5m_{3}^{3}\left(\frac{1}{P^{2}}\right)^{2}\,,$
$\displaystyle A_{14}^{\phi^{5}}=$
$\displaystyle\,m_{6}+4m_{4}^{2}\frac{1}{P^{2}}+8m_{5}m_{3}\frac{1}{P^{2}}+28m_{4}m_{3}^{2}\left(\frac{1}{P^{2}}\right)^{2}+14m_{3}^{4}\left(\frac{1}{P^{2}}\right)^{3}\,,$

it is clear that there must be a relation. The fact that the coincidence of
the structure continues to all multiplicities is shown using the Lagrange
inversion formula in appendix A.

It is natural to expect that the relation extends to all $\phi^{p}$ amplitudes
as follows. Let $m=p-3$, and place $mq$ points on a disk. Now count all
possible ways of clustering the points in non-overlapping sets so that there
are $r_{1}$ groups of $m$ points each, $r_{2}$ groups of $2m$ points, etc.
Then the formula that relates $A^{\phi^{p}}_{n}$ amplitudes and Green
functions is given by the natural generalization of (8.1),

$G_{mq}=\sum_{r_{i}\geq
0}\delta_{q,\left(\sum_{i}i\,r_{i}\right)}\frac{(mq)!}{(mq+1-\sum_{i}r_{i})!}\frac{\left(G_{m}^{c}\right)^{r_{1}}}{r_{1}!}\frac{\left(G_{2m}^{c}\right)^{r_{2}}}{r_{2}!}\ldots\frac{\left(G_{mq}^{c}\right)^{r_{q}}}{r_{q}!}\,.$
(8.2)

We leave it as an exercise for the reader to check that the coefficients we
have presented in the text and the ones in appendix A are indeed the correct
values of the combinatorial problem and the coefficients in (8.2).

It would be very interesting to explore this connection further, in particular
to matrix models with $\Phi^{p-1}$ interactions as the one studied in
Brezin:1977sv .

###  8.2 Possible Connection with Stokes Polytopes

Recent work on the computation of $\phi^{p}$ amplitudes as a sum over
contributions obtained from various polytopes known as accordiohedra is very
reminiscent of the structures we have uncovered using the global Schwinger
formulation. Developing a connection between the two approaches is certainly
an important problem. Here we restrict to $\phi^{4}$ amplitudes and therefore
to Stokes polytopes in order to point out some possible directions. Most of
the formulations using Stokes polytopes construct the amplitudes as (see e.g
equation (5) of Raman:2019utu )

$A^{\phi^{4}}_{n}=\sum_{{\rm Symmetry:}\,\sigma}\,\,\sum_{{\rm
Primitive:}\,P}\alpha_{P}\,m_{P,n}^{(\sigma.P)}$ (8.3)

where the sum is over all primitive Stokes polytopes and the symmetry classes
into which they fall. The $m_{P,n}^{(\sigma.P)}$ are the contributions
obtained from the corresponding polytope. Here the $\alpha_{P}$ are the so-
called weights, which are in general rational numbers.

Consider for example,

$A^{\phi^{4}}_{6}=\alpha_{1}\left(\frac{1}{X_{1,4}}+\frac{1}{X_{3,6}}\right)+\alpha_{2}\left(\frac{1}{X_{2,5}}+\frac{1}{X_{1,4}}\right)+\alpha_{3}\left(\frac{1}{X_{3,6}}+\frac{1}{X_{2,5}}\right)\,.$
(8.4)

Here there are three polytopes and the weights have to be chosen to be
$\alpha_{a}=1/2$.

In Salvatori:2019phs , Salvatori and Stanojevic propose a way to simplify
(8.3) by reducing the redundancy by taking certain limits of kinematic
invariants in each term. Let us rewrite Eq. 4.8 of Salvatori:2019phs  for
$n=6$,

$A^{\phi^{4}}_{6}=\left(\frac{1}{X_{1,4}}+\frac{1}{X_{3,6}}\right)+\lim_{X_{1,4}\to\infty}\left(\frac{1}{X_{2,5}}+\frac{1}{X_{1,4}}\right)\,.$
(8.5)

In this formula, the first bracket comes from the Stokes polytope with
reference $1,4$ while the second bracket comes from the reference $2,5$. Here
$X_{i,j}$ can be identified with the planar invariants $t_{[a,b]}$ in a simple
way. Note that (8.5) groups the three terms in the same way as that found in
our construction (5.7) and (5.8) coming from the two possible non-crossing
chord diagrams. In Salvatori:2019phs , the $n=8$ amplitude is also computed.
The amplitude is given as a sum over five Stokes polytopes. Our formula (5.3)
also has five regions. However, while our regions all contribute with a factor
of one, Eq. 4.10 of Salvatori:2019phs  has four terms with coefficient $+1$
and one with $-1$. In fact, only the first region can be matched directly; it
coincides with the first polytope, i.e. the one with no limits and which gives
rise, in our language, to $m_{5}(\mathbb{I},\mathbb{I})$. We suspect that
there exist other combinations with different limits which could match our
formula term by term. One hint is that every one of our terms is isomorphic to
either an associahedron or to intersections of two associahedra.

###  8.3 Towards Generalized $\phi^{p}$ Amplitudes

Another intriguing feature of our procedure for constructing the regions from
the extended non-crossing chord diagrams for $\phi^{4}$ is the introduction of
two additional points ($-1$ and $n-2$) and a chord joining them. The relevance
of this additional chord lies in the way each meadow is associated to a cubic
amplitude. For now we have conceived these diagrams simply as combinatorial
objects, but if one attempts to relate each of the labels $0,1,\ldots,n-3$ in
the diagram to the particles $3,4,\ldots,n$ respectively, then the new chord
$\theta_{-1,n-2}$ has the interpretation of identifying particles 1 and 2,
which in their tropicalized variables are set to $-\infty$ and $+\infty$.
However, one has to be careful in that the variable $x_{a-3}$ coming from the
parameterization (2.7) does not exactly correspond to a single particle $a$ as
it appears in all rows $r\geq a$. It might seem puzzling that for a general
value of $p$, we introduce $p-2$ additional points to the non-crossing
$(p-2)$-chord diagrams and join them with another $(p-2)$-chord. Strikingly,
$p-2$ is precisely the number of particles with tropicalized variables set at
infinity that appear in the higher-$k$ version of the global Schwinger
parameterization using $\textrm{Trop}^{+}G(k,n)$ for $k=p-2$ (see
Cachazo:2020wgu  for its construction). It would be interesting to explore if
there is a connection with these generalized objects and CEGM generalized
amplitudes Cachazo:2019ngv .

In fact, one direction to tackle is to try and find an analog of $\phi^{p}$
amplitudes for higher-$k$ theories, using a similar limiting procedure. One
possible direction is the following. In Borges:2019csl ; Cachazo:2019xjx  it
was shown that planar arrays of Feynman diagrams computed higher-$k$
amplitudes, also known as CEGM amplitudes Cachazo:2019ngv . An element in the
planar array ${\cal A}_{i_{1},i_{2}\ldots,i_{k-2}}$ is an $(n-k+2)$-particle
cubic Feynman diagram with particle labels
$\\{1,2,\ldots,n\\}\setminus\\{i_{1},i_{2}\ldots,i_{k-2}\\}$. By introducing
the notation ${\cal I}:=\\{i_{1},i_{2}\ldots,i_{k-2}\\}$ one can parameterize
the higher-$k$ invariants as Cachazo:2019xjx ; Early:2020hap

$\mathsf{s}_{i_{1}i_{2}\ldots i_{k}}:=\sum_{{\cal
I}\cup\\{j_{1},j_{2}\\}=\\{i_{1},i_{2},\ldots,i_{k}\\}}s_{j_{1},j_{2}}^{({\cal
I})}\,,$ (8.6)

where the sum is over all possible ways of decomposing
$\\{i_{1},i_{2},\ldots,i_{k}\\}$ into two sets of $k-2$ and $2$ elements
respectively. For example, for $k=3$ we have

$\mathsf{s}_{ijk}:=s^{(i)}_{jk}+s^{(j)}_{ki}+s^{(k)}_{ij}\,.$ (8.7)

This means that we can write each kinematic invariant appearing in an element
in the array in terms of the planar invariants of that particular Feynman
diagram, and then apply the familiar limiting procedure to it. After this we
obtain the rank-$k$ kinematic invariants which are sent to $1/\epsilon$.

As an example, consider $m^{(3)}_{7}(\mathbb{I},\mathbb{I})$. Each element in
an array, or when $k=3$ also known as a collection, is a 6-particle binary
tree. The proposed limiting procedure to get a $p=4$ object imposes that
kinematic invariants of the form $s^{(a)}_{bc}$ are sent to $1/\epsilon$ if
$\\{b,c\\}$ are consecutive in the ordering $(1,2,\ldots,\hat{a},\ldots,n)$.

In this case we have that, e.g.,
$\\{s^{(1)}_{23},s^{(1)}_{34},\ldots,s^{(1)}_{67},s^{(1)}_{72},s^{(2)}_{13},s^{(2)}_{34},\ldots,s^{(2)}_{67},s^{(2)}_{71}\\}$
are sent to $1/\epsilon$. Notice that after the limiting procedure each of the
693 collections will either end up being of order ${\cal O}(\epsilon^{4})$ or
higher. Therefore one can define $A^{\phi^{4}}_{3,7}$ as the coefficient of
$\epsilon^{4}$ in a series expansion around $\epsilon\to 0$.

Rather than starting with the CEGM generalized amplitude, a better understood
object is the set of PK amplitudes introduced by Early in Early:2021tce . We
leave the study of this construction to future research.

## Acknowledgements

The authors thank N. Early for helpful discussions and comments on the draft,
and for identifying the regions that define the support of the distribution
$Q(x)$ with polyhedral cones. We also thank N. Kalyanapuram for bringing some
references about Stokes polytopes and accordiohedra to our attention. FC
thanks S. Mizera, G. Zhang, and K. Yeats for discussions related to
constructions of CHY formulations of $\phi^{p}$ amplitudes. This research was
supported in part by a grant from the Gluskin Sheff/Onex Freeman Dyson Chair
in Theoretical Physics and by Perimeter Institute. 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.

##  Appendix A Lagrange Inversion Formula and Fuss-Catalan Numbers

Given a function $f(x)$ that admits a series expansion around $x=0$ and
$f(0)=0$ while $f^{\prime}(0)\neq 0$, the Lagrange inversion formula gives a
series expansion for the compositional inverse of $f(x)$, i.e. for a function
$g(x)$ such that $g(f(x))=x$, in terms of the series coefficients of $f(x)$.

Let us review one particular formulation which is relevant for this work.
Start by defining an auxiliary function $h(x)$ such that $h(x)=x/f(x)$. Let

$h(x)=\sum_{i=0}^{\infty}h_{i}x^{i}$ (A.1)

be the series expansion of $h(x)$ around $x=0$. Now define

$g_{r}:=\frac{1}{2\pi
i}\oint_{|z|=\epsilon}\frac{dz}{r+1}\left(\frac{h(z)}{z}\right)^{r+1}.$ (A.2)

The Lagrange inversion formula states that the series expansion of $g(x)$ is
of the form

$g(x)=x\sum_{r=0}^{\infty}g_{r}x^{r}\,.$ (A.3)

The proof is fairly simple. Consider the RHS of (A.2) and write it in terms of
$f(z)$ and then write $z=g(u)$,

$\frac{1}{2\pi
i}\oint_{|z|=\epsilon}\frac{dz}{r+1}\left(\frac{1}{f(z)}\right)^{r+1}\\!\\!\\!=\frac{1}{2\pi
i}\oint_{|u|=\epsilon}\frac{du}{r+1}g^{\prime}(u)\left(\frac{1}{f(g(u))}\right)^{r+1}\\!\\!\\!=\frac{1}{2\pi
i}\oint_{|u|=\epsilon}\frac{du}{(r+1)}\frac{g^{\prime}(u)}{u^{r+1}}.$ (A.4)

Now, taking the derivative of (A.3),

$g^{\prime}(x)=\sum_{r=0}^{\infty}(r+1)g_{r}x^{r}$ (A.5)

and plugging in it into the last expression on the right in (A.4) one finds
(A.2).

Before seeing explicitly how this works in the context of interest, let us
review some well-known facts about generating functions of Fuss-Catalan
numbers, in particular, how they are interconnected via the Lagrange inversion
formula.

Let $B_{k}(x)$ be the generating of the Fuss-Catalan numbers ${\rm
FC}_{n}(k,1)$. The function $B_{k}(x)$ satisfies the equation
$B_{k}(x)=1+xB_{k}(x)^{k}$. Now let us prove that

$f(x):=\frac{x}{B_{k-1}(x)},\quad g(x):=xB_{k}(x)$ (A.6)

are compositional inverses of each other. Start with
$B_{k}(x)=1+xB_{k}(x)^{k}$ and multiply by $x^{k-1}$ so that we get an
equation for $g(x)$ of the form $x^{k-2}g(x)=x^{k-1}+g(x)^{k}$. Now let $p(x)$
be the compositional inverse of $g(x)$, i.e., $g(p(x))=x$. Letting $x=p(u)$ in
$x^{k-2}g(x)=x^{k-1}+g(x)$ leads to $u\,p(u)^{k-2}=p(u)^{k-1}+u^{k}$. Let us
prove that $f(x)$ satisfies the same equation as $p(x)$. Starting with
$B_{k-1}(x)=1+xB_{k-1}(x)^{k-1}$ and substituting $B_{k-1}(x)=x/f(x)$ gives
$x/f(x)=1+x^{k}/f(x)^{k-1}$. Multiplying by $f(x)^{k-1}$ we obtain the same
equation satisfied by $p(x)$.

In the case at hand, we are interested in $k=3$ so that

$h(x)=B_{2}(x)=\frac{1-\sqrt{1-4z}}{2z}$ (A.7)

is the generating function of Catalan numbers, and $g_{r}$ becomes the Fuss-
Catalan number ${\rm FC}_{r}(3,1)$. In this case (A.2) reads

${\rm FC}_{r}(3,1)=\frac{1}{2\pi
i}\oint_{|z|=\epsilon}\frac{dz}{r+1}\left(\frac{1-\sqrt{1-4z}}{2z^{2}}\right)^{r+1}$
(A.8)

with

$g(x)=xB_{3}(x),\quad{\rm with}\quad B_{3}(x):=\sum_{r=0}^{\infty}{\rm
FC}_{r}(3,1)x^{r}\,.$ (A.9)

Let us see how this applies to our construction in section 6. Let us consider
the following choice for the function $h(x)$

$h(x)=\sum_{i=0}^{\infty}m_{i+2}x^{i}$ (A.10)

where $m_{i+2}$ represents a generic $(i+2)$-particle amplitude in the
biadjoint $\phi^{3}$ scalar theory of the form
$m_{i+2}(\mathbb{I},\mathbb{I})$.

Since the mass dimension of $m_{i+2}(\mathbb{I},\mathbb{I})$ is $-2(i-1)$ we
are motivated to define $m_{2}:=P^{2}$ and $m_{3}:=1$. Here $1/P^{2}$
represents a generic propagator. We will soon see why this somewhat strange
definition of $m_{2}$ is useful. Let us start by noticing that the number of
Feynman diagrams contributing to $m_{i+2}(\mathbb{I},\mathbb{I})$ is the
Catalan number $\textrm{C}_{i}$.

The claim is that the form of the amplitude $A^{\phi^{4}}_{n}$ is determined
by the coefficient $g_{n/2-1}$ divided by $h_{0}^{n/2-1}$.

Let us compute the first few cases of (A.2) in order to illustrate the use of
the formula,

$\displaystyle A^{\phi^{4}}_{4}=\frac{g_{1}}{h_{0}}=$ $\displaystyle
h_{1}=m_{3}$ $\displaystyle A^{\phi^{4}}_{6}=\frac{g_{2}}{h_{0}^{2}}=$
$\displaystyle\frac{h_{0}^{2}h_{2}+h_{0}h_{1}^{2}}{h_{0}^{2}}=m_{4}+m_{3}^{2}\frac{1}{P^{2}}$
$\displaystyle A^{\phi^{4}}_{8}=\frac{g_{3}}{h_{0}^{3}}=$
$\displaystyle\frac{h_{3}h_{0}^{3}+3h_{1}h_{2}h_{0}^{2}+h_{1}^{3}h_{0}}{h_{0}^{3}}=m_{5}+3m_{3}m_{4}\frac{1}{P^{2}}+m_{3}^{3}\left(\frac{1}{P^{2}}\right)^{2}.$

Finally, specializing to what is called planar kinematics, in which all planar
invariants are set to unity, one finds that $A^{\phi^{4}}_{n}$ counts the
number of Feynman diagrams contributing to the amplitude. This is the number
of ternary planar unrooted trees with $n$ leaves which is known to be the
given by the Fuss-Catalan numbers. Applying the same kinematics to the
$\phi^{3}$ amplitudes one can replace each by the corresponding Catalan
numbers and therefore we reproduce the relation (A.2).

###  A.1 Extension to $\phi^{p}$: Iterated Structure

Let us explicitly construct the iteration used in section 7.2 to propose the
schematic structure of $\phi^{p}$ amplitudes.

Let us start by defining generating functions

$h_{k}(x)=\sum_{j=0}^{\infty}h_{k,j}x^{j}\,.$ (A.11)

The goal is to construct a recursive procedure that determines all
coefficients $h_{k,j}$ as functions of the base case defined to be

$h_{3}(x)=\sum_{j=0}^{\infty}h_{j}x^{j}\,.$ (A.12)

Note that for the base function we have denoted the coefficients by $h_{j}$
instead of $h_{3,j}$. This was done in order not clutter the formulas. Using
the expression in (7.9)

$h_{k}(x)=\sum_{j=0}^{\infty}h_{k,j}x^{j}:=\sum_{j=0}^{\infty}\frac{1}{2\pi
i}\oint_{|z|=\epsilon}\frac{dz}{j+1}\left(\frac{h_{k-1}(z)}{z}\right)^{j+1}x^{j}$
(A.13)

let us present some results for the expansions.

For $\phi^{4}$ amplitudes we have $h_{4}(x)$ with coefficients

$\begin{array}[]{l}h_{0}\\\ h_{0}h_{1}\\\ h_{0}h_{1}^{2}+h_{0}^{2}h_{2}\\\
h_{0}h_{1}^{3}+3h_{0}^{2}h_{1}h_{2}+h_{0}^{3}h_{3}\\\
h_{0}h_{1}^{4}+6h_{0}^{2}h_{1}^{2}h_{2}+2h_{0}^{3}h_{2}^{2}+4h_{0}^{3}h_{1}h_{3}+h_{0}^{4}h_{4}\\\
h_{0}h_{1}^{5}+10h_{0}^{2}h_{1}^{3}h_{2}+10h_{0}^{3}h_{1}h_{2}^{2}+10h_{0}^{3}h_{1}^{2}h_{3}+5h_{0}^{4}h_{2}h_{3}+5h_{0}^{4}h_{1}h_{4}+h_{0}^{5}h_{5}.\\\
\end{array}$ (A.14)

These coefficients are a refinement of the Narayana numbers. Let us see this
more explicitly. Consider first the table of coefficients (see OEIS entry
A134264, oeis ),

$\begin{array}[]{l}1\\\ 1\\\ 1,1\\\ 1,3,1\\\ 1,6,2,4,1\\\ 1,10,10,10,5,5,1\\\
1,15,30,5,20,30,3,15,6,6,1\\\ 1,21,70,35,35,105,21,21,35,42,7,21,7,7,1.\\\
\end{array}$ (A.15)

If we now set $h_{0}=x$ and all other $h_{i}=1$, then terms with the same
power of $h_{0}$ are combined. For example,
$2h_{2}^{2}h_{0}^{3}+4h_{1}h_{3}h_{0}^{3}$ in the fifth row of (A.14) becomes
$2x^{3}+4x^{3}=6x^{3}$. Carrying this out one gets

$\begin{array}[]{l}x\\\ x\\\ x^{2}+x\\\ x^{3}+3x^{2}+x\\\
x^{4}+6x^{3}+6x^{2}+x\\\ x^{5}+10x^{4}+20x^{3}+10x^{2}+x\\\
x^{6}+15x^{5}+50x^{4}+50x^{3}+15x^{2}+x\\\
x^{7}+21x^{6}+105x^{5}+175x^{4}+105x^{3}+21x^{2}+x.\\\ \end{array}$ (A.16)

These coefficients are the Narayana numbers (OEIS entry A001263, oeis ).

Let us consider $\phi^{5}$ amplitudes, so we have $h_{5}(x)$ with coefficients

$\begin{array}[]{l}h_{0}\\\ h_{0}^{2}h_{1}\\\
2h_{0}^{3}h_{1}^{2}+h_{0}^{4}h_{2}\\\
5h_{0}^{4}h_{1}^{3}+6h_{0}^{5}h_{1}h_{2}+h_{0}^{6}h_{3}\\\
14h_{0}^{5}h_{1}^{4}+28h_{0}^{6}h_{1}^{2}h_{2}+4h_{0}^{7}h_{2}^{2}+8h_{0}^{7}h_{1}h_{3}+h_{0}^{8}h_{4}\\\
42h_{0}^{6}h_{1}^{5}+120h_{0}^{7}h_{1}^{3}h_{2}+45h_{0}^{8}h_{1}h_{2}^{2}+45h_{0}^{8}h_{1}^{2}h_{3}+10h_{0}^{9}h_{2}h_{3}+10h_{0}^{9}h_{1}h_{4}+h_{0}^{10}h_{5}.\\\
\end{array}$ (A.17)

Listing only the coefficient allows us to present one more row (OEIS entry
A338135, oeis ),

$\begin{array}[]{l}1\\\ 1\\\ 2,1\\\ 5,6,1\\\ 14,28,4,8,1\\\
42,120,45,45,10,10,1\\\ 132,495,330,22,220,132,6,66,12,12,1.\\\ \end{array}$
(A.18)

Once again, if we set $h_{0}=x$ and all other $h_{i}=1$, then (A.17) becomes
the generating functions for the $2$-Narayana numbers. In general one finds
the triangle of $m$-Narayana numbers, where the standard ones correspond to
$m=1$. The $2$-Narayana numbers are then given by the coefficients in (see
e.g. section 6.8 of nara )

$\begin{array}[]{l}x\\\ x^{2}\\\ x^{4}+2x^{3}\\\ x^{6}+6x^{5}+5x^{4}\\\
x^{8}+12x^{7}+28x^{6}+14x^{5}\\\ x^{10}+20x^{9}+90x^{8}+120x^{7}+42x^{6}\\\
x^{12}+30x^{11}+220x^{10}+550x^{9}+495x^{8}+132x^{7}\\\ \end{array}$ (A.19)

###  A.2 One Function to Compute Them All

There is one more interesting property of these representation of $\phi^{p}$
amplitudes which interconnects them. Consider the coefficients of the function
$h_{4}(x)$. Some of them are explicitly shown in (A.14).

The claim is that the coefficients of the function $h_{k}(x)$ can be obtained
from those of $h_{4}(x)$ by simply setting to zero all $h_{a}$ with
$a\notin(k-3)\mathbb{Z}$ (see text in OEIS entry A338135 for $k=5$ case, oeis
). For example, $h_{5}(x)$ is obtained by setting all $h_{a}$ with $a$ odd to
zero. Of course, every other coefficient of $h_{4}(x)$ vanishes completely but
the ones that do not reproduce $h_{5}(x)$.

One direct way to understand the relation among the different generating
functions $h_{k}(x)$ is by recalling the combinatorial problem they solve. As
explained in the discussions, set $m=k-3$ and place $mq$ points on a disk. Now
count all possible ways of clustering the points in non-overlapping sets so
that there are $r_{1}$ groups of $m$ points each, $r_{2}$ groups of $2m$
points, etc. Clearly, $h_{4}(x)$, for which $m=1$, contains all other problems
counted by $h_{k}(x)$ with $k>4$ as special cases.

##  Appendix B Computing a Region for $n=12$ that Leads to
$m_{7}(1234576,\mathbb{I})$

Directly computing amplitudes, $A^{\phi^{4}}_{n}$, using the global Schwinger
formula becomes harder as $n$ grows. In this appendix, we show how to use the
global Schwinger formula to find an explicit map from a region to
$m_{n/2+1}(\alpha,\mathbb{I})$. Having the precise bijection of kinematic
invariants, one gets the contribution of the region without ever carrying out
an integral. The region under consideration is

$R=\\{x_{0}=x_{3}<x_{1}=x_{2},\,x_{4}=x_{5},\,x_{6}=x_{7},\,x_{8}=x_{9}\\}\,.$
(B.1)

Let us consider the behavior of the $G_{12}(x)$ part of the tropical potential
on a region $R_{\rm ext}$ where the condition $x_{3}<x_{1}$ is relaxed an
therefore contains $R$, i.e.,

$R\subset R_{\rm
ext}=\\{x_{0}=x_{3}\,x_{1}=x_{2},\,x_{4}=x_{5},\,x_{6}=x_{7},\,x_{8}=x_{9}\\}\,.$
(B.2)

The function $G_{12}(x)$ can be seen to be a linear combination of the
following $14$ piecewise linear functions

$\displaystyle\min\left(x_{1},x_{3}\right),\,\min\left(x_{3},x_{5}\right),\,\min\left(x_{5},x_{7}\right),\,\min\left(x_{7},x_{9}\right),$
$\displaystyle\min\left(x_{1},x_{3},x_{5}\right),\,\min\left(x_{3},x_{5},x_{7}\right),\,\min\left(x_{5},x_{7},x_{9}\right),\,$
$\displaystyle\min\left(x_{1},x_{3},x_{5},x_{7}\right),\min\left(x_{3},x_{5},x_{7},x_{9}\right),\min\left(x_{1},x_{3},x_{5},x_{7},x_{9}\right),$
$\displaystyle x_{1},x_{5},x_{7},x_{9}.$

Note that $x_{1}$ is always accompanied by $x_{3}$ when it is an argument in a
$\min$ function. This means that when restricting to $R$, i.e. imposing
$x_{3}<x_{1}$ on the functions, $x_{1}$ drops out and we are left with the
following $11$ functions,

$\displaystyle\min\left(x_{3},x_{5}\right),\,\min\left(x_{5},x_{7}\right),\,\min\left(x_{7},x_{9}\right),$
$\displaystyle\min\left(x_{3},x_{5},x_{7}\right),\,\min\left(x_{5},x_{7},x_{9}\right),\,$
$\displaystyle\min\left(x_{3},x_{5},x_{7},x_{9}\right),\,x_{1},\,x_{3},\,x_{5},\,x_{7},\,x_{9}.$

It is easy to compute the coefficients of each of the $11$ functions to be

$\displaystyle
t_{[2,8]}-t_{[2,6]}-t_{[6,8]},\,-t_{[6,8]}+t_{[6,10]}-t_{[8,10]},\,-t_{[8,10]}+t_{[8,12]}-t_{[10,12]},$
$\displaystyle
t_{[2,10]}-t_{[2,8]}+t_{[6,8]}-t_{[6,10]},\,-t_{[6,10]}+t_{[6,12]}+t_{[8,10]}-t_{[8,12]},$
$\displaystyle
t_{[6,10]}-t_{[2,10]}-t_{[6,12]},\,t_{[3,5]},\,t_{[2,6]}-t_{[3,5]},\,t_{[6,8]},\,t_{[8,10]},\,t_{[10,12]}.$

Note that the coefficient of $x_{1}$ is $t_{[3,5]}$, which is precisely the
invariant in the propagator that must appear according to the rules for the
non-crossing diagram corresponding to the region $R$. The only other place
where $t_{[3,5]}$ appears is in the coefficient of $x_{3}$. This means that we
can write the integral over $x_{1}$ as

$\int_{-\infty}^{\infty}dx_{1}\theta(x_{1}-x_{3})\exp\left(-t_{[3,5]}(x_{1}-x_{3})\right)=\frac{1}{t_{[3,5]}}\,.$
(B.3)

Combining the left over terms and relabeling variables so that $x_{a}\to
x_{(a-3)/2}$ one finds the “effective” potential

$\displaystyle F_{6}$
$\displaystyle=(t_{[2,8]}-t_{[2,6]}-t_{[6,8]})\min\left(x_{0},x_{1}\right)+(-t_{[6,8]}+t_{[6,10]}-t_{[8,10]})\min\left(x_{1},x_{2}\right)$
$\displaystyle+(-t_{[8,10]}+t_{[8,12]}-t_{[10,12]})\min\left(x_{2},x_{3}\right)+(t_{[2,10]}-t_{[2,8]}+t_{[6,8]}-t_{[6,10]})\min\left(x_{0},x_{1},x_{2}\right)$
$\displaystyle+(-t_{[6,10]}+t_{[6,12]}+t_{[8,10]}-t_{[8,12]})\min\left(x_{1},x_{2},x_{3}\right)$
$\displaystyle+(t_{[6,10]}-t_{[2,10]}-t_{[6,12]})\min\left(x_{0},x_{1},x_{2},x_{3}\right)+t_{[2,6]}x_{0}+t_{[6,8]}x_{1}+t_{[8,10]}x_{2}+t_{[10,12]}x_{3}\,.$

It is a simple exercise to match the coefficients with that of the tropical
potential function for $m_{6}(\mathbb{I},\mathbb{I})$. The non-trivial fact is
that the result is not only a map but a bijection between the corresponding
sets of planar invariants. This is left as an exercise for the reader.

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