Towards Positive Geometry of Multi Scalar Field Amplitudes
Accordiohedron and Effective Field Theory

Mrunmay Jagadalea and Alok Laddhab

a California Institute of Technology, Pasadena, CA 91125, USA

bChennai Mathematical Institute, Siruseri, Chennai, India

Siruseri, SIPCOT IT Park

E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
Abstract

The geometric structure of S-matrix encapsulated by the “Amplituhedron
program” has begun to reveal itself even in non-supersymmetric quantum field
theories. Starting with the seminal work of Arkani-Hamed, Bai, He and Yan [1]
it is now understood that for a wide class of scalar quantum field theories,
tree-level amplitudes are canonical forms associated to polytopes known as
accordiohedra. Similarly the higher loop scalar integrands are canonical forms
associated to so called type-D cluster polytopes for cubic interactions or
recently discovered class of polytopes termed pseudo-accordiohedron for higher
order scalar interactions.

In this paper, we continue to probe the universality of these structures for a
wider class of scalar quantum field theories. More in detail, we discover new
realisations of the associahedron in planar kinematic space whose canonical
forms generate (colour-ordered) tree-level S matrix of external massless
particles with $n-4$ massless poles and one massive pole at $m^{2}$. The
resulting amplitudes are associated to
$\lambda_{1}\,\phi_{1}^{3}\,+\,\lambda_{2}\,\phi_{1}^{2}\phi_{2}$ potential
where $\phi_{1}$ and $\phi_{2}$ are massless and massive scalar fields with
bi-adjoint colour indices respectively. We also show how in the “decoupling
limit” (where $m\,\rightarrow\,\infty,\lambda_{2}\,\rightarrow\,\infty$ such
that $g\,:=\,\frac{\lambda_{2}}{m}\,=\,\textrm{finite}$) these associahedra
project onto a specific class of accordiohedron which are known to be positive
geometries of amplitudes generated by
$\lambda\,\phi_{1}^{3}\,+\,g\,\phi_{1}^{4}$.

Dedicated to the memory of Nila;
Teacher, Mentor and Friend.

###### Contents

  1. 1 Introduction
  2. 2 Positive Geometries for polynomial scalar interactions.
    1. 2.1 A Closed Family of Accordiohedra
    2. 2.2 Amplitude as a Sum Over d-$\log$ Forms
  3. 3 Two-Scalar Field Theory with Cubic Interaction
    1. 3.1 From Feynman Graphs to Dual Triangulations
  4. 4 Coloured Causal Diamonds and Convex polytopes
    1. 4.1 A Brief Review of Causal Diamonds
    2. 4.2 From Uncoloured to Coloured Causal Diamonds
    3. 4.3  Coloured Causal diamonds and Convex Blocks in kinematic space
    4. 4.4 Mutation rules from Coloured Causal Diamonds
    5. 4.5 From the Associahedron Blocks to Scattering Amplitudes
    6. 4.6 Factorisation on Massive Channels
  5. 5 The Scattering form for an Effective Field Theory : Accordiohedra in low energy limit
  6. 6 EFT Amplitude from Projected Accordiohedra
  7. 7 Discussion and Open Issues
  8. A Explicit formula for the weights

##  1 Introduction

The analysis of the structure of the S-matrix has witnessed several striking
developments in past two decades. But the scattering amplitudes associated to
scalar field theories display a “dual persona” in many of these developments.
On one hand, S-matrix of bi-adjoint scalar $\phi^{3}$ theory is a prototype
for many of the more sophisticated theories such as Yang-Mills theory or
Gravity111In fact the bi-adjoint $\phi^{3}$ scattering amplitude also plays
the role of a building block in the so-called KLT relations, and on the other
hand the scattering amplitudes associated to higher order scalar interactions
are more intricate and do not appear to share the remarkable simplicity and
elegance of the amplitudes associated to cubic interactions.

BCFW recursion relations provided first hints of such intricacies as it is
only the cubic scalar interactions which are BCFW constructible using a single
BCFW shift [3, 2]. However the “brutality” of generic scalar field amplitudes
manifested itself most clearly in the CHY (Cachazo He, and Yuan) formulation
of the S-matrix. As was shown by CHY in [4] and later in a series of works by
Bourjaily et al [5, 6] the integrand for the bi-adjoint $\phi^{3}$ theory is
the canonical top form in the moduli space but the integrands for other scalar
theories do not appear to have any obvious geometric or cohomological
characterisation. (The amplituhedron program in fact motivates us to look for
lower forms on CHY moduli space or top forms on binary geometries [7], [8] as
integrands for $\phi^{p}$ theories. For some initial attempts in the first
direction we refer the reader to [9]).

The extension of the Amplituhedron program to non super-symmetric quantum
field theories and the discovery of positive geometries (more specifically
convex polytopes) in kinematic space [1] has shed new light on some of these
issues. We now understand how the tree-level (and colour ordered) amplitudes
of a generic massless scalar field theories are directly tied to the existence
of certain very specific Positive geometries (in fact Polytopes) in the
kinematic space.222Even perturbative string amplitudes have intimate
relationship with positive geometries and the associated canonical forms as
discovered in the seminal paper by Mizera [10]. The “Amplituhedron” for a
$\phi^{p}$ interaction for a generic $p$ is a combinatorial polytope known as
the accordiohedron which admits convex realisations in kinematic space . [11,
12, 13, 9, 14, 15]. Each accordiohedron defines a unique canonical form in
kinematic space and a (weighted) sum over all the canonical forms of a given
dimension is the $n$-point amplitude of $\phi^{p}$ theory.333Equivalently,
each accordiohedron is dual to a simple polytope and the canonical form
associated to the accordiohedron induces a volume measure on the kinematic
space such that the sum over volumes of all the dual polytopes of a given
dimension equals the tree-level amplitude.

However, even for tree-level amplitudes, a number of puzzles remain
unresolved. For an $n$ particle colour-ordered amplitude in bi-adjoint scalar
$\phi^{3}$ theory, there is a unique $n-3$ dimensional associahedron and hence
the amplitude is nothing but the canonical top form on the associahedron [1].
For a generic scalar interaction, even for a fixed number of external
particles, there is a whole set of acccordiohedra and they all need to be
accounted for (the forms associated to each of these have to be added with
specific coefficients) when evaluating the tree-level amplitude. Thus the
scattering form for $\phi^{p\,>\,3}$ theory is not a d-$\log$ form but a
weighted sum over $k\,>\,1$ d-$\log$ forms.

The weights are uniquely fixed via combinatorial properties of the
accordiohedron ([17, 18]) but it is not apriori clear why a specific linear
combination of the d $\log$-forms is “special” in the sense that it generates
unitary and local scattering amplitudes. In fact, as was observed in [14], if
we consider the accordiohedra polytopes for polynomial scalar interactions,
then a class of such accordiohedra contribute with vanishing weights and it
has till date remained unclear why certain accordiohedra are redundant as far
as the tree-level S-matrix is concerned.

It has been advocated by Nima Arkani-Hamed, that reason for emergence of
accordiohedron polytopes (as the amplituhedron for higher order scalar
interaction) should be probed via effective field theory ideas. We can think
of generating polynomial scalar interactions from cubic interactions by
integrating out massive fields. In this sense Accordiohedra polytopes should
“naturally” emerge from the kinematic space Associahedra in
$X_{ij}\,<<\,m^{2}$ limit.444We also thank Nemani Suryanarayana for
highlighting this possibility to us in 2018.

In this paper, we attempt to resolve few pieces of this puzzle. We take first
steps towards showing that there is an “amplituhedron” for tree-level S matrix
in which the interaction Hamiltonian includes coupling of two scalars (one of
which is massive and the other massless). More in detail, we show that there
exists a class of polytopes which are topologically equivalent to an
associahedron but whose boundaries come in two possible colours. We refer to
such a polytope as an associahedron block. We show that an associahedron block
admits realisations in kinematic space whose canonical form generates a set of
tree level amplitudes of
$\lambda_{1}\,\phi_{1}^{3}+\lambda_{2}\,\phi_{1}^{2}\phi_{2}$ theories, where
$\phi_{1},\,\phi_{2}$ are two species of scalar fields. The set consists of
$n$-point amplitudes in which all the external states are massless and the
amplitude is expanded upto $\lambda^{2}$. We distinguish the two fields by
taking $\phi_{1}$ to be massless and $\phi_{2}$ to be massive. We then show
that there is a way to “geometrize” the effect of integrating out massive
modes to leading order in $\frac{1}{m}$ ($m$ being mass of $\phi_{2}$) on
positive geometries and show that this leads to the accordiohedron polytope

_Our primary result in the paper is the following_ : Up to order
$\lambda_{2}^{2}$ perturbative expansion of the $n$ point (colour-ordered)
tree-level amplitude in which all the external particles are massless is a sum
over canonical forms of a set of polytopes which are realisations of
associahedra in the positive region of kinematic space. We derive a formula
which computes the (color-ordered) amplitude for the multi-scalar theory in
terms of canonical forms associated to ABHY-type realisations of associahedron
blocks.

In particular, let $A_{n-3}$ be an ABHY associahedron in the positive region
of planar kinematic space, ${\cal K}_{n}^{+}$. As we show below, for each
$(i,j)$ such that $\\{\,|i-j|$ modulo $n\,\in\,1,\,\dots,\,\frac{n}{2}\,\\}$
there exists a polytope that we call associahedron block and denote as
$A_{n-3}^{(i,j)}$. Each associahedron-block has a subset of co-dimension one
facets which are associated with
$(m,n)\,\in\,\\{\,(i,j),\,\dots,\,i+(j-i)-1,j+(j-i)-1)\,\\}$ that are coloured
red while all the other facets being coloured black.

We then show that the causal structure on ${\cal K}_{n}$ introduced in [19]
has enough richness which we use to generalise the ABHY construction. We thus
obtain convex realisation of an associahedron block $A_{n-3}^{(i,j)}$ in
${\cal K}_{n}^{+}$ and we denot this (ABHY) realisation as $A_{n-3}^{{\cal
F}_{ij}}$. Here ${\cal F}_{ij}$ is the set of all the triangulations of an
$n$-gon such that (i) Each triangulation has at most one red diagonal from the
set $\,\\{\,(i,j),\dots,\,(i+|j-i|-1,j+|j-i|\,-1\,\\}$. We then prove the
following theorem.

Let $\Omega_{n}^{{\cal F}_{ij}}$ be the planar scattering form defined by the
associahedron block $A_{n-3}^{(i,j)}$ with $\Omega_{n}^{{\cal
F}_{ij}}(A_{n-3}^{ij})$ being its pull-back on the _unique_ convex realisation
of the same associahedron block in ${\cal K}_{n}^{+}$. Similarly, let
$\Omega_{n}^{\phi^{3}}$ be the planar scattering form defined in [1] with
$\Omega_{n}^{\phi^{3}}(A_{n-3})$ being its pull-back on a ABHY realisation.

The scattering form defined below defines the $n$-point amplitude of our
theory upto sub-leading order in $\lambda_{2}$.

$\displaystyle\Omega_{n}^{Y}\,:=\,\left[\,\sum_{|i-j|=2}^{\frac{n}{2}}\,\frac{1}{|i-j|}\,\sum_{{\cal
F}_{ij}}\,\Omega_{n}^{{\cal
F}_{ij}}(A_{n-3}^{ij})-\,\gamma\,\Omega_{n}^{\phi^{3}}(A_{n-3})\right]$ (1)

where

$\displaystyle\gamma\,=\,\sum_{\textrm{Sum over all diagonals,
(i,j)}}\,\frac{1}{|i-j|}\,-\,(n-3)$ (2)

Hence, the following (sum over) $d-\log$ forms,

$\displaystyle\omega_{n}\,=\,\lambda_{1}^{n-2}\,\Omega_{n}^{\phi^{3}}(A_{n-3})\,+\,\lambda_{1}^{n-4}\,\lambda_{2}^{2}\,\Omega_{n}^{Y}$
(3)

generates the tree-level planar amplitude up to order $\lambda_{2}^{2}$.

We then show that the effect of integrating out the massive $\phi_{2}$ field
is geometrized in the world of positive geometries. It amounts to moving the
associahedra to “infinity” in different directions where they degenerate into
a family of lower dimensional accordiohedra.

The paper is organised as follows. In section 2, we review the analysis of
tree-level amplitudes for color ordered massless scalar theories with
polynomial interactions . Although the main ingredients in this section are a
review, we give an explicit evaluation of generic tree-level amplitudes
generated by polynomial (massless) scalar interaction in terms of canonical
forms in section 2.2. In sections 3 and 3.1, we introduce a two-scalar field
theory with Lagrangian defined in equation (24) and analyse the triangulations
dual to Feynman diagrams. We argue how the naive attempt at using these dual
triangulations to generate positive geometries such as associahedron fail if
the number of particles $n\,\geq\,6$.

In section 4, we use the remarkable causal structure in the planar kinematic
space introduced in [19] to locate convex polytopes whose canonical forms
generate tree-level perturbative amplitude of the two-scalar theory. As we
prove in 4.1, combinatorially all the polytopes are in fact associahedra such
that a unique linear combination over $d\log$ forms associated to their convex
realisation is the scattering amplitude of interest. We show how these
associahedra have boundaries some of which correspond to massless poles and
some to the massive ones. In section 5, we consider a low energy limit of
these associahedra and show that they in fact project onto accordiohedra which
are known to be positive geometries for the effective field theory that arises
once we integrate out the massive field. We end with a discussion of some of
the immediate open questions.

##  2 Positive Geometries for polynomial scalar interactions.

In this section, we review the “Amplituhedron program” in the context of tree-
level (and colour-ordered) scattering amplitudes of massless scalar theories
with generic polynomial interaction.

We first begin by quickly reviewing the positive geometries, more specifically
a class of simple polytopes known as accordiohedra (a polytope is a bounded,
convex, higher dimensional generalisation of a polygon) which generate
amplitudes for monomial ($\phi^{p}$.) interactions. A polytope ${\cal P}$ is
called simple if each of the vertex is adjacent to exactly $\textrm{dim}({\cal
P})$ co-dimension one boundaries called facets. Accordiohedron is a simple
polytope whose co-dimension $k$ faces are in bijection with a set of
dissections of polygon. Depending on the nature of dissections, that is, if
the polygon is dissected into triangles or $p\,>\,3$-gons, we get distinct
accordiohedra. If the co-dimension $k$ faces of the simple polytope are in
bijection with $k$-partial triangulation of an $n$-gon then the simple
polytope is the $n-3$ dimensional associahedron $A_{n-3}$.555The precise
definition of positive geometry is not required in this work but can be found
in [20]. For our purpose, we may define positive geometry $X_{\geq\,0}$ as (i)
a closed oriented subset in a projective space $X$ which has boundaries of all
co-dimensions (ii) there is a unique canonical form on $X$ which has simple
poles on (and only on) all the faces of $X_{\geq\,0}$ and (iii) the residue of
the canonical form is itself the canonical form defined intrinsically on the
boundary as a positive geometry. For the purpose of this paper, we will need
only a specific class of positive geometries, namely convex polytopes embedded
in positive region of planar kinematic space ${\cal K}_{n}$.

We then prove that a weighted sum over canonical forms associated to a family
of simple polytopes which are closed under factorisation and gluing generate
scattering amplitudes for a scalar field theories with polynomial
interactions. Although the proof is simply a consequence of results proved in
the literature, we take this opportunity to write down in complete generality
the relationship between tree-level S-matrix for scalar theories and a set of
positive geometries. Reader not interested in this level of generality of
already known results is encouraged to skip the proof in the first
reading.666We thank Ashoke Sen for discussions on this issue and pressing on
us the need to clarify the relationship between positive geometries and
polynomial scalar interactions.

We also caution the reader that the review is not self-contained as we assume
familiarity with the basic notions of positive geometry, especially
associahedron, accordiohedron, their convex realisations and corresponding
canonical forms. Interested reader is encouraged to read the original
references, especially [1, 12, 14, 9, 15] or recent review [16].

As was shown in [11, 13, 14], the tree-level planar (colour-ordered)
amplitudes for massless $\phi^{p}$ interactions are sums over canonical forms
of kinematic space accordiohedra. For quartic interaction, an $n$-particle
amplitude is simply a (weighted) sum over canonical top forms associated to
the accordiohedra of dimension $\frac{n-4}{2}$ 777These accordiohedra were
discovered by Baryshnikov and are known as Stokes polytopes [21].. Although
explicit formulae have been derived for computing tree-level planar amplitudes
associated to monomial-scalar interactions, an explicit classification of
positive geometries whose canonical forms generate the scattering amplitude
for generic polynomial interaction has never been analysed. In this section,
we fill this gap. Although the essential ingredients are simply review of
known results, the final result has never been written down explicitly to the
best of our knowledge.

This result essentially classifies the families of accordiohedra whose
canonical forms generate amplitudes of a local unitary quantum field theory.
We refer to such a set as a “closed family”. A closed family is defined as
family of polytopes whose canonical forms generate amplitude of local unitary
quantum field theories. We will see how different families of accordiohedra
give us the amplitudes for massless scalar field theories. In the first
section we will define accordiohedron and the families of accordiohedra; and
in the second section we will discuss how to obtain a tree level scalar field
theory amplitude from the families of accordiohedra.

###  2.1 A Closed Family of Accordiohedra

Accordiohedra are generalization of a family of polytopes called associahedra.
The $n-3$ dimensional associahedron $\mathcal{A}_{n-3}$ is a simple polytope
(i.e., each vertex is adjacent to $n-3$ co-dimension one faces) whose vertices
are in one-to-one correspondence with triangulations of an $n$-gon and whose
facets (co-dimension one boundaries) are in one-to-one correspondence with
diagonals of $n$-gon. The facets of associahedra satisfy the following
property which we call “factorization”. Any $d$-dimensional facet of
$\mathcal{A}_{n-3}$ is a product of two lower dimensional associahedra
$\mathcal{A}_{r}\times\mathcal{A}_{d-r}$ for some $r$ ($0\leq r\leq d$).

Accordiohedron is a generalization of associahedra where the vertices of the
polytope are associated with various dissections of $n$-gon instead of just
triangulations. For example, we could consider a polytope where the vertices
are associated with quagrangulations of an $n$-gon. Such a polytope is known
as Stokes polytope. It can be easily verified with simple example of a hexagon
that we can not have a simple polytope with vertices in one-to-one
correspondence with quadrangulations of $n$-gon and facets (co-dimension one
boundaries) in one-to-one correspondence with diagonals. There are more than
necessary quandrangulations and not all diagonals can be part of a
quadrangulation.

To deal with this we introduce a notion of “compatiblity” with a reference
quadrangulation. We say a diagonal $\delta$ of an $n$-gon is compatible with
reference quadrangulation $Q$ if the set of diagonals and sides of $Q$ that
intersect the segment $\delta^{\prime}$ obtained by small clock-wise rotation
of $\delta$ is connected. The Stokes polytope associated with reference
quadrangulation $Q$ is a simple polytope whose vertices are in one-to-one
correspondence with quadrangulations formed by diagonals compatible with $Q$
and whose co-dimension one facets are in one-to-one correspondence with
diagonals compatible with $Q$. We should note here that, unlike the family of
associahedra there are more than one Stokes polytopes of a given dimension. We
have one Stokes polytope for every quadrangulation. However, just like
associahedra the facets of Stokes polytopes are products of lower dimensional
Stokes polytope.

Now we define $\mathcal{AC}(D)$ the accordiohedron associated with a reference
dissection $D$. To do that we first have to define the notion of compatibility
of a diagonal with a reference dissection just as defined it for
quadrangulations. We say a diagonal $\delta$ of an $n$-gon is compatible with
reference dissection $D$ if the set of diagonals and sides of $D$ that
intersect the segment $\delta^{\prime}$ obtained by small clock-wise rotation
of $\delta$ is connected.(See figure 1).

Figure 1: The diagonal (38) of the (dual) octagon is compatible with the
reference {13,36,16}

The accordiohedron associated with dissection $D$ is a simple polytope whose
vertices are in one-to-one correspondence with maximal dissections formed by
diagonals compatible with $D$ and whose co-dimension one facets are in one-to-
one correspondence with diagonals compatible with $D$.

We have one accordiohedra for every dissection of a polygon. Thus the space of
accordiohedra is labeled by dissections of polygons. Associahedra and Stokes
polytopes are some examples of accordiohedra where the reference dissections
are triangulations and quadrangulations respectively. Now we will look at
different ways of classifying the space of accordiohedra.

To begin with, we can classify the space of accordiohedra into different
families using the following criteria

  * •

Number of sides (external sides) of the reference dissection

  * •

Dimension of accordiohedra, or equivalently number of diagonals in the
reference dissections.

  * •

Constituents or building blocks of dissection (triangles, quadrilaterals,
pentagons, etc.)

Note, these are the criteria for reference dissection, which is expected as
dissections label the space of accordiohedra. All these criteria are encoded
in the following infinite dimensional vector associated with the dissection,
$\mathbf{v}=(v_{3},v_{4},v_{5},\ldots)$. Where $v_{r}$ is the number of
$r$-gons in the reference dissection. Hence, all but finitely many components
of $\mathbf{v}$ are zero. Number of sides of reference dissection is given by

$n=2+\sum_{i=3}^{\infty}(i-2)v_{i}.$ (4)

The dimension of accordiohedra is given by

$\textrm{Dim}\left[\mathcal{AC}(D)\right]=-1+\sum_{i=3}^{\infty}v_{i}.$ (5)

The family of associahedra consists of accordiohedra with
$\mathbf{v}=(n-2,0,0,\ldots)$ and the family of Stokes polytopes consists of
accordiohedra with $\mathbf{v}=(0,\frac{n-2}{2},0,0,\ldots)$.

There are two further classification of the space of accordiohedra which will
play a role in identifying the family of polytopes whose forms generate tree-
level amplitudes of massless scalars. One of these classifications is coarser
than the three discussed above and the other one is finer.

Coarser classification :
The family of associahedra and Stokes polytopes satisfy the interesting
property that facets of associahedra are products of lower dimensional
associahedra and facets of Stokes polytopes is product of lower dimensional
Stokes polytopes. We would like to classify the set of accordiohedra which are
closed under such factorisation.

We first define pure accordiohedron. Any accordiohedron is called pure if the
reference dissection breaks the polygon into a $p$-cells with $p\,\geq\,3$.
Pure accordiohedra are interesting as each tower of Pure accordiohedra ${\cal
AC}(D)$ (where $D$ is a $p$-gulation of an $n$-gon and tower is defined with
respect to the number of external vertices $n$) is closed under factorization.

Now we introduce the notion of “closed under gluing”. We say a set of
accordiohedra $S$ is closed under gluing if taking the reference dissections
of any two elements of $S$ and gluing them along some side gives you the
reference of some element of $S$ then we say $S$ is closed under gluing. The
sets of associahedra and stokes polytopes are examples of sets closed under
gluing.

If we take two dissections, $D_{1}$ of $n_{1}$-gon and $D_{2}$ of $n_{2}$-gon,
and glue these dissections along some side, we get a dissection $D$ of
$n_{1}+n_{2}-2$-gon. The side along which the dissections $D_{1}$ and $D_{2}$
were glued becomes a diagonal of $D$. Let’s denote that diagonal by $\delta$.
The vectors $\mathbf{v}_{1}$, $\mathbf{v}_{2}$, and $\mathbf{v}$ associated
with $D_{1}$, $D_{2}$, and $D$, respectively, are related by
$\mathbf{v}_{1}+\mathbf{v}_{2}=\mathbf{v}$. Thus, the dimensions of
$\mathcal{AC}(D_{1})$, $\mathcal{AC}(D_{2})$ and $\mathcal{AC}(D)$ are related
by
$\mathrm{Dim}[\mathcal{AC}(D_{1})]+\mathrm{Dim}[\mathcal{AC}(D_{2})]=\mathrm{Dim}[\mathcal{AC}(D)]-1$.
Further, the co-dimension one facet associated with the diagonal $\delta$ is
given by $\mathcal{AC}(D_{1})\times\mathcal{AC}(D_{2})$. That is, the facet
$\delta$ of $\mathcal{AC}(D)$ factorizes into
$\mathcal{AC}(D_{1})\times\mathcal{AC}(D_{2})$. Therefore, the notion of
closed under gluing is closely related to the notion of closed under
factorization. However they are not the same. An example of set closed under
factorization but not closed under gluing is the set of associahedra with
dimension less than 10. An example of a set closed under gluing but not closed
under factorization is the set of associahedra with dimensions greater than 9.

The coarser families of accordiohedra we are interested in are the sets of
accordiohedra which are closed under factorization and closed under gluing. An
example of set closed under factorization and closed under gluing is the set
of accordiohedra whose vector $\mathbf{v}$ is of the type
$(v_{1},v_{2},0,0,\ldots)$. This set contains all associahedra and all Stokes
polytopes along with all accordiohedra where the reference dissection has both
triangles and quadrilaterals.

We claim that all sets which are closed under factorization and closed under
gluing are of the type

$S_{i_{1},i_{2},\ldots,i_{r}}=\\{\mathcal{AC}(D)|\mathbf{v}_{D}=\sum_{j=1}^{r}a_{i_{j}}\hat{e}_{i_{j}}\\}.$
(6)

Where $\hat{e}_{i_{j}}$ is a unit vector with one in $i_{j}-2$ th position and
zero in other positions. So the set of all associahedra is denoted by $S_{3}$,
and the set of all associahedra and all Stokes polytope along with all
accordiohedra where the reference dissection has both triangles and
quadrilaterals is denoted by $S_{3,4}$.

Finer classification:
Now we move on to the second, finer classification. The Dihedral group $D_{n}$
acts on the set of all dissections of $n$-gon. The the action of $D_{n}$
dissection does not change the building blocks of the dissection. That is
$\mathbf{v}_{D}=\mathbf{v}_{g\cdot D}$. Where, $g\cdot D$ denotes the action
of $g\in D_{n}$ on $D$ and $\mathbf{v}_{D}$ and $\mathbf{v}_{g\cdot D}$ are
the infinite dimensional vectors associated with disections $D$ and $g\cdot D$
respectively. Thus, there is an action of $D_{n}$ on all dissections with a
given $\mathbf{v}$. We can classify the set of all dissections with a given
$\mathbf{v}$ by the orbits of this action. We denote the set of orbits by
$\mathcal{P}_{\mathbf{v}}$ and call the representative dissection of a orbit
as the “primitive” of that orbit.

###  2.2 Amplitude as a Sum Over d-$\log$ Forms

A set of accordiohedra which is closed under factorization and closed under
gluing gives us tree level planar scalar field theory amplitudes. In this
section, we will describe how we get tree level planar scalar field theory
amplitudes from $S_{i_{1},i_{2},\ldots,i_{r}}$.

The $n$-point amplitude is obtained from the subset
$S_{i_{1},i_{2},\ldots,i_{r}}^{(n)}$ of $S_{i_{1},i_{2},\ldots,i_{r}}$ where
$\sum_{j=1}^{r}(i_{j}-2)a_{i_{j}}=n-2$ that is number of sides of reference
dissections is n. Suppose

$\mathcal{V}^{(n)}_{i_{1},i_{2},\ldots,i_{r}}=\\{\mathbf{v}|\mathbf{v}=\sum_{j=1}^{r}a_{i_{j}}\hat{e}_{i_{j}},\text{
with }\sum_{j=1}^{r}(i_{j}-2)a_{i_{j}}=n-2\\},$ (7)

for ease of notation we will drop $i_{1},i_{2},\ldots,i_{r}$ in the subscript
and just write $\mathcal{V}_{n}$.

We will soon see that a function
$\omega(\mathcal{AC}(D)):\mathcal{K}_{n}\rightarrow\mathbb{R}$ is associated
with accordiohedra $\mathcal{AC}(D)$. Where $\mathcal{K}_{n}$ is the
$n$-particle kinematics space. The scattering amplitude is a weighted sum of
these functions over the space $S_{i_{1},i_{2},\ldots,i_{r}}^{(n)}$.

$\mathcal{M}_{n}(p_{1},\,p_{2},\,\ldots,p_{n})=\sum_{\mathbf{v}\in\mathcal{V}_{n}}\lambda_{i_{1}}^{a_{i_{1}}}\cdots\lambda_{i_{r}}^{a_{i_{r}}}\sum_{D\in\mathcal{P}_{\mathbf{v}}}\hskip
7.11317pt\sum_{\sigma\in D_{n}/G_{D}}\alpha_{\sigma\cdot D}\hskip
2.84544pt\omega(\sigma\cdot D).$ (8)

Where $G_{D}=\\{g\in D_{n}|g\cdot D=D\\}$ is the stabiliser of $D$ under the
action of dihedral group. It can be shown that we can choose the weights
$\alpha_{D}$ to be equal for $D$s belonging to the same orbit under the action
of dihedral group. For a given $\mathbf{v}$ the term
$\sum_{D\in\mathcal{P}_{\mathbf{v}}}\hskip 7.11317pt\sum_{\sigma\in
D_{n}/G_{D}}\alpha_{\sigma\cdot D}\hskip 2.84544pt\omega(\sigma\cdot D)$ is
independent of the family of accordiohedra
$S_{i_{1},i_{2},\ldots,i_{r}}^{(n)}$.

The combinatorial polytope $\mathcal{AC}(D)$ gives us the canonical form
associated with it on the kinematic space. This form is given by,

$\Omega\left[\mathcal{AC}(D)\right]=\sum_{D_{a}\in\mathcal{AC}(D)}\mathrm{Sgn}(D_{a})\bigwedge_{(ij)\in
D_{a}}d\log(X_{ij}).$ (9)

Where $X_{ij}=(p_{i}+p_{i+1}+\cdots+p_{j-1})^{2}$ are variables on the
kinematic space. Now, to get the function $\omega(\mathcal{AC}(D))$, we have
to realise the combinatorial polytope $\mathcal{AC}(D)$ in the kinematic space
and restrict the canonical form on the kinematic space accordiohedra.

$\Omega\left[\mathcal{AC}(D)\right]|_{\mathcal{AC}(D)_{k}}=\omega(\mathcal{AC}(D))\bigwedge_{(ij)\in
D}dX_{ij}.$ (10)

It can be shown that $\omega(D)=\sum_{D_{i}\in\mathcal{AC}(D)}\psi(D_{i})$
where $\psi(D_{i})=\prod_{(ab)\in D_{i}}\frac{1}{X_{ab}}$.

Now let’s see how we determine the weights $\alpha$. We can write the
amplitude as

$\displaystyle\mathcal{M}_{n}(p_{1},\,p_{2},\,\ldots,\,p_{n})$
$\displaystyle=\sum_{\mathbf{v}\in\mathcal{V}_{n}}\lambda_{3}^{v_{3}}\cdots\lambda_{n}^{v_{n}}\sum_{D\in\mathcal{P}_{\mathbf{v}}}\alpha_{D}\sum_{\sigma\in
D_{n}/G_{D}}\sum_{D_{i}\in\mathcal{AC}(\sigma\cdot D)}\psi(D_{i}).$ (11)

We fix the weights demanding that all poles of $\mathcal{M}(1,2,\ldots,n)$
come with residue one.

To analyse the requirement on the weights it would be useful to define
$\delta(D_{i},D_{j})$, $N([D_{i}],[D_{j}])$, $M([D_{i}],[D_{j}])$, and
$\Psi([D])$ as follows

  * •

$\delta(D_{i},D_{j})$ tells you wether the dissection $D_{j}$ occurs in
$\mathcal{AC}(D_{i})$ or not. That is,

$\delta(D_{i},D_{j})=\begin{cases}1\text{ if $D_{j}\in\mathcal{AC}(D_{i})$
}\\\ 0\text{ if $D_{j}\notin\mathcal{AC}(D_{i})$ }\end{cases}$ (12)

For any $\sigma\in D_{n}$,

$\displaystyle\delta(D_{i},D_{j})=\delta(\sigma\cdot D_{i},\sigma\cdot D_{j})$
(13)

  * •

$N([D_{i}],[D_{j}])$ is the number of times any dissection $D_{k}$ in the
orbit of $D_{j}$ occurs in the set of all accordiohedra of dissections in the
orbit of $D_{i}$. That is,

$N([D_{i}],[D_{j}])=\sum_{D_{\ell}\in[D_{i}]}\delta(D_{\ell},D_{k})\hskip
56.9055pt\text{for any }D_{k}\in[D_{j}].$ (14)

  * •

$M([D_{i}],[D_{j}])$ is the number of elements from the orbit of $D_{i}$ occur
in any accordiohedron of a dissection in the orbit of $D_{j}$. That is,

$M([D_{i}],[D_{j}])=\sum_{D_{k}\in[D_{i}]}\delta(D_{j},D_{k}).$ (15)

They are related as follows,

$N([D_{i}],[D_{j}])=\frac{|G_{D_{j}}|}{|G_{D_{i}}|}M([D_{j}],[D_{i}]).$ (16)

Suppose $[D_{i}]\in\mathcal{P}_{\mathbf{v}_{i}}$ and
$[D_{j}]\in\mathcal{P}_{\mathbf{v}_{j}}$, and if
$\mathbf{v}_{i}\neq\mathbf{v}_{j}$ then $\delta(D_{i},D_{j})=0$ and hence
$N([D_{i}],[D_{j}])=M([D_{i}],[D_{j}])=0$.

  * •

Lastly,

$\Psi([D])=\sum_{\sigma\in D_{n}/G_{D}}\psi(\sigma\cdot D).$ (17)

Now the scattering amplitude can be expressed as

$\mathcal{M}_{n}(p_{1},\,p_{2},\,\ldots,p_{n})=\sum_{\mathbf{v}\in\mathcal{V}_{n}}\lambda_{3}^{v_{3}}\cdots\lambda_{n}^{v_{n}}\sum_{[D_{i}]\in\mathcal{P}_{\mathbf{v}}}\sum_{[D_{j}]\in\mathcal{P}_{\mathbf{v}}}\alpha_{[D_{j}]}N([D_{j}],[D_{i}])\Psi([D_{i}]).$
(18)

The requirement on the weights is equivalent to the following system of linear
equations

$\sum_{[D_{j}]\in\mathcal{P}_{\mathbf{v}}}\alpha_{[D_{j}]}N([D_{j}],[D_{i}])=1\hskip
42.67912pt\forall[D_{i}]\in\mathcal{P}_{\mathbf{v}}.$ (19)

All the vertices (triangulations) of combinatorial associahedra are on equal
footing, whereas in other accordiohedra the reference dissection is special
compared to the rest. This disparity is captured in the weight of the
accordiohera.

We end this section by briefly reviewing the positive geometry for massive bi-
adjoint scalar amplitude. Although our work is concerned with scattering
amplitude of massless particles (but with massive as well as massless
propagators), we review the known extension of the associahedron program to
massive scalar amplitudes.

The positive geometry for tree-level colour ordered amplitude of a massive bi-
adjoint scalar is simply the ABHY associahedron whose facets are located
$X_{ij}\,=\,m^{2}$ [24]. The planar kinematic space of massive particles is
defined as follows. Consider the massive kinematic space ${\cal K}_{n}^{(m)}$
co-ordinatized by the planar variables,

$\displaystyle\tilde{X}_{ij}\,=\,(p_{i}\,+\,\dots,\,+\,p_{j-1})^{2}\,-\,m^{2}$
(20)

Then $\tilde{X}_{i,i+1}\,=\,\tilde{X}_{1n}\,=\,0$.

Just as the massless kinematic space, ${\cal K}_{n}^{(m)}$ is
$\frac{n(n-3)}{2}$ dimensional and we can locate ABHY associahedra in the
positive region ${\cal K}_{n}^{m+}$ of the kinematic space. The corresponding
canonical form equals the tree-level planar amplitude for the massive bi-
adjoint $\phi^{3}$ theory.

##  3 Two-Scalar Field Theory with Cubic Interaction

As stated in the introduction, our aim is to analyse tree-level scattering
amplitudes in a theory with two scalar fields $\phi_{1}^{aA},\,\phi_{2}^{bB}$
that transform in the bi-adjoint representation of
$U(N)\,\times\,U(\tilde{N})$. The Lagrangian is

$\displaystyle\begin{array}[]{lll}L(\,\phi_{1},\,\phi_{2})&=\\\
&\frac{1}{2}\,\sum_{i=1}^{2}\,\partial_{\mu}\,\phi_{iaA}\,\partial^{\mu}\,\phi_{i}^{aA}\,-\,\frac{1}{2}m^{2}\phi_{2}^{aA}\,\phi_{2\,aA}\,-\,\frac{\lambda_{1}}{3!}\,f_{abc}\,\tilde{f}_{ABC}\,\phi_{1}^{aA}\phi_{1}^{bB}\,\phi_{1}^{cC}\\\
&\hskip
209.58255pt-\,\frac{\lambda_{2}}{2!}\,f_{abc}\,\tilde{f}_{ABC}\,\phi_{1}^{aA}\,\phi_{1}^{bB}\,\phi_{2}^{cC}\end{array}$
(24)

Tree-level amplitudes in a multi-scalar field theory with Yukawa-type
interaction between the bi-adjoint scalar field and a “massless” matter field
which transforms in (bi)fundamental representation of
$U(N)\,\times\,U(\tilde{N})$ have been analysed in a beautiful series of works
[15, 22]. The bi-adjoint scalar field self-interaction as well as the Yukawa
interactions were taken to have the same coupling constant $\lambda$ in these
works. It was proved that the positive geometries underlying amplitudes in
such a theory are what are called open associahedra.

While certainly motivated by the analysis in [15, 22], we have a different set
up involving one massive and one massless scalar field with two independent
coupling constants which govern the $\phi_{1}^{3}$ and $\phi_{1}^{2}\phi_{2}$
interactions respectively.

Schematically, we are interested in amplitude generated by interaction
potential,

$\displaystyle
V(\phi_{1},\phi_{2})\,=\,(\lambda_{1}\phi_{1}^{3}\,+\,\lambda_{2}\phi_{1}^{2}\,\phi_{2}\,)$
(25)

where $\phi_{1},\phi_{2}$ are massless and massive scalar fields respectively.

We will be specifically interested in tree-level and colour-ordered (CO)
amplitudes involving _only_ massless external particles. Much of the analysis
in our paper can be generalised for generic configurations of the external
particles, but as we are eventually interested in integrating out the massive
modes, our primary focus is on amplitudes involving massless external states.
We will denote such amplitudes as ${\cal
M}_{n}^{\textrm{CO}}(p_{1},\dots,\,p_{n})$ with $p_{i}^{2}\,=\,0\,\forall\,i$.
Perturbative evaluation of the amplitude to sub-leading order in $\lambda_{2}$
can be written as,

$\displaystyle{\cal
M}^{\textrm{CO}}_{n}(p_{1},\dots,p_{n})\,=\,\lambda_{1}^{n-3}\,{\cal
M}_{n}^{CO\,(1)}(p_{1},\dots,p_{n})\,+\,\lambda_{2}^{2}\,\lambda_{1}^{n-4}\,{\cal
M}_{n}^{\textrm{CO}(2)}(p_{1},\dots,p_{n})$ (26)

${\cal M}_{n}^{\textrm{CO}\,(1)}$ is generated by $\phi_{1}^{3}$ vertices in
which all the propagators are massless and ${\cal M}_{n}^{\textrm{CO}\,(2)}$
is generated by all but two $\phi_{1}^{2}\,\phi_{2}$ vertices with one massive
and $n-4$ massless propagators. Thus the set of all the Feynman graphs that
contribute to the total amplitude ${\cal M}_{n}^{\textrm{CO}}$ has at most one
$\phi_{2}$ propagator.

The effective field theory involving only the massless scalar field is
obtained by taking $m\,\rightarrow\,\infty,\,\lambda_{2}\,\rightarrow\,\infty$
such that $\frac{\lambda_{2}}{m}\,=\,g\,=\,\textrm{fixed}$. This ensures that
in the lower energy limit we obtain amplitudes associated to

$\displaystyle
V_{\textrm{eff}}(\phi_{1})\,=\,\lambda_{1}\,\phi_{1}^{3}\,+\,g\,\phi_{1}^{4}$
(27)

Our goal is to find positive geometries in planar kinematic space ${\cal
K}_{n}$ whose set of co-dimension one facets either correspond to a pole of
the massive propagator or the massless propagator. For the amplitudes of
interest (namely up to order $\lambda_{2}^{2}$) each vertex of such a positive
geometry can be adjacent to at most one “massive facet”.

###  3.1 From Feynman Graphs to Dual Triangulations

In the case of the bi-adjoint cubic scalar coupling with a single massless
field, the positive geometries are convex realisations of the combinatorial
associahedron, each of whose vertices are in bijection with complete “mono-
chromatic triangulations” of an $n$-gon where each diagonal is dual to a
propagator in the Feynman graph.

Hence we first need to classify triangulations which are dual to the Feynman
graphs that produce amplitude defined in equation (26). The set of Feynman
graphs that produce the amplitude proportional to $\lambda_{1}^{n-3}$ is dual
to triangulations of an $n$-gon where the diagonals have no additional labels.
As ABHY has taught us, it is the immensely deep structure hidden in the
combinatorics of these traingulations which generate positive geometry for a
single scalar field amplitude.

However, for the amplitude contribution proportional to $\lambda_{2}^{2}$,
such mono-chromatic triangulations are not appropriate as the contributing
Feynman graphs have precisely one $\phi_{2}$ propagator. Hence the dual
triangulations are such that precisely one of the diagonals in a _complete_
triangulation is distinguished from the other diagonals. We colour such
diagonals red to differentiate them from the black diagonals. We refer to
triangulations in which all but one diagonals are black as “mostly black
triangulations” ( See figure 2).

Figure 2: Mostly Black triangulations with (13) red and (14),(15) black on the
left, (35) red and (13),(15) black in the center and (14) red and (24), (15)
black on the right.

It is immediately obvious that mostly black triangulations are in one to one
correspondence with cubic Feynman graphs where all but one propagators are
massless and one of the propagator is massive.

The simplest mutation rule one can define on the mostly black triangulations
is the one where mutation does not change colour of the diagonal. We can then
immediately see that mostly black triangulations produce combinatorial
polytopes in one and two dimensions . We denote a black diagonal between
vertices $i$ and $j$ as $(i,j)$ and the red diagonal as
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}(k,l)}$.888When
we write a set of diagonals, we will indicate each diagonal as $(ij)$ instead
of explicitly writing it out as $(i,j)$.

  * •

Triangulation of a $4$-gon with precisely one red diagonal and with the
mutation rule which does not change colour produces one dimensional
associahedron with the vertices being labelled by two red-diagonals
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}(1,3)}$ and
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}(2,4)}$.

  * •

Mostly black triangulation of a $5$-gon produces a two dimensional polygon
with ten edges, whose vertex set is in bijection with
$((i,j),\,{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}(k,l)})$

Let us first start with $n\,=\,4$ case. In this case the requirement of mostly
black triangulations which has at least one red diagonal has precisely one red
(and no black diagonal) and the resulting polytope is simply the one
dimensional assocaihedron $A_{1}$.

In the case of $n\,=\,5$ the resulting polytope is a $10$-gon and is known in
the literature as the two dimensional colorful associahedron $A_{2}^{c}$.
[25]. An $n$-dimensional colorful associahedron $A_{n-3}^{c}$ is combinatorial
polytope associated with colored triangulations of $n+3$-gons in which the
diagonals are assigned a color from a set of $n$ colors. $A_{2}^{c}$ is shown
in the figure 3 below. However these simple mutation rules do not work for two
reasons.

Figure 3: $A_{2}^{c}$

  * •

If we consider complete mostly black triangulations of an $n$-gon where
$n\,\geq\,6$ then there are no closed combinatorial polytopes. This can be
seen by looking at the example of $n\,=\,6$. Let us assume on the contrary
that there exists an polytope $A_{3}^{r}$ whose vertices are in bijection with
complete triangulation of the hexagon in which precisely one diagonal is red.
If ${\cal A}_{3}^{r}$ exists, then each of it’s co-dimension one facets are in
bijection with one partial triangulation of the hexagon. Hence we can have
facets corresponding to partial triangulation where a black diagonal $(i,j)$
or a red diagonal
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}(i,j)}$ is
deleted. Clearly, each such facet is either a square (red diagonal with
$j-i\,=\,3$ (modulo 6) ), pentagon (red diagonal with $j-i\,=\,2$ (modulo 6))
or a decagon (black diagonal with $j-i\,=\,2$ (modulo 6)). However there is no
unique facet associated to black diagonal $(i,j)$ with $j-i\,=\,2$ (modulo 6).
This can be seen as follows. Consider e.g. partial triangulation obtained by
deleting the diagonal
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}(1,4)}$.
The corresponding facet is adjacent to Four Facets obtained by deleting the
diagonals

$\displaystyle\\{\,(1,3),\,(2,4),\,(4,6),\,(1,5)\,\\}$

respectively. However there is no unique facet associated to $(1,4)$ (and in
general any $(i,j)$), as adjacent to such a facet, there are two possible sets
of vertices,

$\displaystyle\begin{array}[]{lll}S_{1}\,=\,\\{\,({\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}13},14,15)\,({\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}24},14,15),\,({\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}24},14,46)\,({\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}13},14,46)\,\\}\end{array}$
(29)

and

$\displaystyle\begin{array}[]{lll}S_{2}\,=\,\\{\,({\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}15},14,13)\,({\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}15},14,24),\,({\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}46},14,24)\,({\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}46},14,13)\,\\}\end{array}$
(31)

Clearly if we choose one of the $S_{1}$ or $S_{2}$ as being the set of
vertices belonging to the $(1,3)$-facet then we immediately see that $(1,3)$
and $(1,5)$ which are adjacent in the neighbourhood of
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}(1,4)}$ are
not adjacent in the neighbourhood of the facet associated to $(1,4)$. A moment
of meditation will convince the reader that this example clearly indicates why
the set of all mostly-black triangulations can not form a closed
polytope.999We thank Vincent Pilaud for discussion on this issue. This is the
first obstruction towards realising positive geometry for scattering
amplitudes in a theory with $V(\phi_{1},\phi_{2})\,=\,\phi_{1}^{2}\phi_{2}$
where all the external particles are massless.

  * •

The second obstruction to simple minded mutation rules comes from the
impossibility of convex realisation of colorful associahedron in ${\cal
K}_{n}^{+}$.

######  Lemma 3.1.

The two dimensional colorful associahedron $A_{2}^{c}$ which is the
combinatorial polytope for $n,=\,5$ particles can not be realised in ${\cal
K}_{n}^{+}$.

Proof : We denote the massive facet $X_{ij}\,=\,m^{2}$ as
$\tilde{X}_{ij}\,=\,0$. $A_{2}^{c}$ is a 10-gon obtained by unfolding $A_{2}$
by painting alternating faces red. (See figure 3). There is no planar
polytopal realisation of $A_{2}^{c}$ such that $X_{13}\,=\,0$ is parallel to
$\tilde{X}_{13}\,=\,0$, $\tilde{X}_{25}\,=\,0$ and $X_{25}\,=\,0$.101010Any
other realisation will not have the canonical form all whose residues vertices
are $\pm 1$ and hence will not define any tree-level amplitude. It can be
immediately checked that canonical form associated to any other realisation
will not produce the tree-level amplitude for the bi-scalar theory.

However these obstructions in fact guides us in our pursuit of positive
geometries whose boundaries correspond to all the singularities of the desired
amplitude. In the next section, we show that although $A_{2}^{c}$ can not be
realised as a convex polytope in the (positive region of) kinematic space,
union of all the facets of $A_{2}^{c}$ and $A_{2}$ arrange themselves into a
family combinatorial polytopes.

##  4 Coloured Causal Diamonds and Convex polytopes

###  4.1 A Brief Review of Causal Diamonds

The convex realisation of associahedron discovered by ABHY in [1] was
formalised in terms of polytopal fans generated by type-A clusters in [23].
The “naturalness” of these specific realisations arises from the fact that
associahedra are naturally associated to type-A cluster algebra and the
cluster algebra has enough structure to in turn produce convex realisations of
$A_{n-3}$ in ${\cal K}_{n}^{+}$.

However, the connection and dependence on cluster algebra, though striking can
be unnerving for a physicist and it would certainly be pleasing to know if the
naturalness underlying ABHY realisations could be understood in terms of some
simple “physics” principles. A remarkable progress in answering this question
was made by Arkani-Hamed et. al. in [19] who discovered a “causal” structure
on the planar kinematic space ${\cal K}_{n}$. The causal structure is realised
via interpreting $X_{ij}$ variables as discretization of a two dimensional
massless scalar field $X(u,v)$ with $u$ and $v$ being (abstract) retarded and
advanced co-ordinates on the two dimensional flat space-time.

More in detail, in [19], the authors introduced a causal diamond in ${\cal
K}_{n}$ which is a null lattice with vertices labelled by the planar kinematic
variables. $X_{ij}$s with fixed $i$ are placed along the
$u\,=\,\textrm{const}$ “null lines and cyclic symmetry of the $X$ variables,
$X_{i,j+n}\,=\,X_{ij}$ can be broken by choosing a strip in the causal
kinematic space such that $X_{i,i\pm 1}\,=\,0$. Kinematic space causal
diamonds are best understood by simply staring at the figure 4 below.

Figure 4: Causal diamond for $n=5$

Causality in kinematic space beautifully captures compatibility of the
diagonals111111Here by compatible diagonals we mean non-intersecting, or
equivalently diagonals which together can be a part some triangulation.. [19]
: Given any diagonal $(i,j)$ of an $n$-gon which labels a planar variable
$X_{ij}$, all the “compatible” diagonals are outside or on the light cone of
the $X_{ij}$ and all of the incompatible diagonals are in the union of
(strictly) future and (strictly) past light cone of $X_{ij}$. We show this
beautiful map from a combinatorial structure to a causal structure in the
figure 5 below.

Figure 5: Compatible diagonals

There are several features of the causal diamond which will be central to our
analysis below.

  * •

A temporal evolution in causal diamond corresponds to mutation such that
$X_{i,j}$ and $X_{i\,\pm\,1,j\,\pm\,1}$ are incompatible diagonals.

  * •

All the variables on a constant $u$ or $v$ slice triangulate the $n$-gon.

  * •

Even after the redundancy encoded in cyclic symmetry is broken by working in a
given strip bounded by $X_{i,i\pm 1}\,=\,0$, there is a further redundancy in
the causal diamond description. This is due to the fact that temporal
evolution of $X_{ij}$ eventually leads us to $X_{ij}$. Thus the smallest
domain in the causal diamond in which all the $X_{ij}$ variables occur
precisely once is called non-redundant domain. Non-redundant domains are not
unique as the initial slice (or equivalently, an initial choice of
triangulation) is not unique. It is precisely the multiplicity of non-
redundant domains which lead to distinct ABHY realisations discovered and
analysed in [23, 11].

  * •

The discretized massless scalar field equation with a source precisely
generate all the linear equations which leads to convex realisation of
$A_{n-3}$ in ${\cal K}_{n}^{+}$.

###  4.2 From Uncoloured to Coloured Causal Diamonds

The causal diamonds (reviewed in section 4.1) give a novel perspective on ABHY
realisations of the associahedron. There are many non-redundant domains inside
the causal diamond. The non-redundant domains are the collection of vertices
inside the causal diamond which is isomorphic to the set of all the diagonals.
“Initial slice” in the non-redundant domain corresponds to a complete
triangulation $T$ that generates the ABHY realisations via the constraints
$s_{ij}\,=\,-\,c_{ij}\,\forall\,(i,j)\,\notin\,T^{c}$.

For the case of mostly black triangulations the simple minded mutation rules
face obstructions. Hence we can ask if there are a different set of mutation
rules which generate closed combinatorial poytopes. Rather remarkably, if we
let the abstract structure of a causal diamond guide us, then it leads us to a
set of mutation rules and a rather natural definition of a (set of) positive
geometries with multi-coloured facet structure. In order to adapt the causal
diamond to mostly black triangulations, we let the vertices of causal diamond
have one of two possible colours, namely black and red. As any mostly black
triangulation has all but one diagonals black, we start with any slice in the
causal diamond which corresponds to a complete triangulation and colour one of
the vertices in it as red (see figure 6).

Figure 6: Colored Causal Diamond with initial slice $(13),(14),(15)$ where
$(14)$ is red

We then keep the mutation rules as they are for uncoloured causal diamonds so
that a vertex evolves into another vertex with the same colour. _We refer to
the resulting diamonds as coloured causal diamonds_. Certain properties of
coloured causal diamonds can immediately be noted.

  * •

If any slice on the causal diamond which corresponds to a triangulation of the
$n$-gon has a red vertex $X_{ij}$ then any vertex in the causal diamond will
be red if $|k-l|\,=\,|i-j|\,\textrm{modulo}\,n$ and if it is in the same
vertical line as $X_{ij}$.

  * •

There are $\lceil\frac{n-3}{2}\rceil$ disconnected copies of coloured causal
diamonds that contain all possible mostly black triangulations.

We now consider specific non-redundant domains inside the coloured causal
diamond (described in section 4.3). We refer to these domains as fundamental
domains. The initial “null slice” of the fundamental domain has one red and
all other black vertices. The final slice obtained within the fundamental
domain ensures that combinatorially we have an associahedron polytope but
whose facets have additional (red or black) labelling.

Remarkably enough, distinct fundamental domains (differentiated by the choice
of colouring of vertices on the initial slice) produce $\frac{n(n-3)}{2}$
number of combinatorial associahedra. Union of vertices belonging to all such
associahedra contain (1) (multiple copies) of all the mono-chromatic and
mostly black triangulations and (2) do not contain any other species of
triangulations (such as a triangulation with two diagonals being red). _We
refer to this (coloured) associahedron as associahedron block_.

As each associahedron block is identified with a fundamental domain in the
coloured causal diamond, we have a realisation of each block in the (positive
region) of kinematic space. One of our primary results in this paper is to
show that the associahedron blocks are the positive geometries for ${\cal
M}_{n}^{Y}$.

###  4.3  Coloured Causal diamonds and Convex Blocks in kinematic space

We will now give a detailed analysis of the ideas outlined in section 4.2 and
colour the causal diamonds in the kinematic space. We will then show that the
coloured causal structure can be used to locate a class of polytopes in ${\cal
K}_{n}^{+}$ which constitute the “amplituhedron” for the amplitudes of two-
scalar field theory.

We start with an “intial configuration” where precisely one of the vertex is
red. We then “evolve the vertices” following our mutation rule which does not
change the colour. After finite number of walks (or evolution in discrete time
in the causal diamond) we reach the initial configuration again. If the
initial configuration has a red vertex $X_{ij}$, then all the vertices
$X_{kl}$ where $|\,l-k\,|\,=\,|\,j-i\,|$ modulo $n$ and where $X_{kl}$ is in
the same vertical line as $X_{ij}$ will eventually be coloured red in this
causal diamond. However, all $X_{kl}$ where
$|\,l-k\,|\,\neq\,|j-i|\,\textrm{modulo}\,n$ will remain black. Hence there
exists disconnected copies of coloured causal diamonds, on each of which the
reference triangulation (or a lattice points on a null slice) has precisely
one red vertex $X_{ij}$. Clearly, the number of disconnected copies of
coloured causal diamonds is $\lceil\frac{n-3}{2}\rceil$.

We then consider specific “non-redundant domains” ([19]) inside the coloured
causal diamond each of which contain all the vertices precisely once and the
smallest possible number of red vertices and each vertex $(k,l)$ (irrespective
of it’s colour) occur exactly once. Hence each of these domains is classified
by the set of red vertices
$\tilde{X}_{ij},\tilde{X}_{i+1,j+1},\,\dots,\,\tilde{X}_{i+k,j+k}$ (where
$k\,=\,|j-i|$ modulo $n$ and where we identify $n+1$ with $1$.) We refer to
this class of non-redundant domains as _fundamental domains_ and denote them
by ${\cal F}_{ij}$. Before giving some examples of the fundamental domains, we
make a couple of observations :
(1) Strictly speaking, the fundamental domain even with same set of red
vertices are not unique and depend on initial choice of triangulation. However
we will suppress the explicit dependence on the reference (initial)
triangulation as fundamental domains with same set of red vertices but
different initial triangulations are simply different ABHY realisations of the
same associahedron block.
(2) We will always choose the initial triangulation containing red diagonal
$(i,j)$ to be such that all the diagonals emanate in the same vertex $j$.
Following examples of ${\cal F}_{ij}$ illustrate these ideas explicitly.

One choice of fundamental domain in the case of $n\,=\,5$ and $n\,=\,6$ are
given below and shown in figures 7 and 8.

Figure 7: Fundamental domain $\mathcal{F}_{13}$ in $n=5$ Figure 8: Fundamental
domains $\mathcal{F}_{26}$ and $\mathcal{F}_{36}$ in $n=6$

$\displaystyle\begin{array}[]{lll}\textrm{n = 5 fundamental
domains}\,=\,\\{\,{\cal F}_{13},{\cal F}_{24},{\cal F}_{35},{\cal
F}_{14},{\cal F}_{25}\,\\}\\\ \textrm{n=6 fundamental domains}\,=\,\\{\,{\cal
F}_{13},{\cal F}_{24},{\cal F}_{35},{\cal F}_{46},{\cal F}_{15},{\cal
F}_{26}\,\\}\,\cup\,\\{\,{\cal F}_{14}{\cal F}_{25},\,{\cal
F}_{36}\,\\}\end{array}$ (34)

Several comments are in order.

  * •

We note that ${\cal F}_{ij}$ is a domain which contains all the diagonals of
the $n$-gon such that the minimal number of them (in the case of $n\,=\,5$, 2
vertices) are coloured red.

  * •

In the $n\,=\,6$ case there are two disconnected copies of the coloured causal
diamonds as evolution of a red vertex $(k,l)$ can never produce $(i,j)$ if
$|j-i|\,\neq\,|k-l|\,\textrm{mod}\,6$.

  * •

The fundamental domains ${\cal F}_{25},\,{\cal F}_{36}$ are redundant, as the
configuration of red and black diagonals in these domains is same as that in
${\cal F}_{14}$. We do not have to include them, but as we will see this
redundant inclusion of ${\cal F}_{ij}$ for
$|j-i|\,=\,\left\lfloor\frac{n}{2}\right\rfloor$ is useful for writing the
final formula for (weighted) sum over canonical forms that produce the
scattering amplitude.

  * •

As can be seen from the figure 7 and figure 8, the examples of fundamental
domains above are such that the initial and final triangulations emanate from
a given vertex. But we can start with different initial triangulation. In the
$n\,=\,6$ case for example, we can consider the fundamental domain ${\cal
F}_{36}$ in which the intial quiver(or the reference triangulation) is
$T\,=\,\\{\,13,{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}36},35\,\\}$.
The corresponding Fundamental domain has precisely three red vertices
$\\{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}36},\,{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}14},\,{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}25}\,\\}$
again and simply gives us a different realisation of the associahedron block
$A_{2}^{(14)}$.

Let us look at the fundamental domains in $n=7$ case. We start with
$\\{\,(i,j),\,\dots,\,(j-1,i-1)\,\\}$ where $|j-i|\,\geq\,2$. Hence in $n=7$
case we have the following _set_ of _disjoint_ fundamental domains each of
which defines a coloured causal diamond.

$\displaystyle\begin{array}[]{lll}\textrm{A complete set of fundamental
domains with ($|j-i|$ = 2 modulo 7})=\\\ \\{\,{\cal F}_{13},\,{\cal
F}_{24},{\cal F}_{35},\,{\cal F}_{46},{\cal F}_{57},\,{\cal F}_{16},{\cal
F}_{27}\,\\}\\\ \textrm{A complete set of fundamental domains with ($|j-i|$ =
3 modulo 7)}=\\\ \\{\,{\cal F}_{14},{\cal F}_{36},{\cal F}_{15},{\cal
F}_{37},{\cal F}_{25},{\cal F}_{47},{\cal F}_{26}\,\\}\end{array}$ (38)

We thus get, seven associahedron blocks ${\cal
A}_{4}^{(i,j)}|i-j|\,=\,2\,\textrm{modulo}\,7$ with two non-adjacent red
facets and seven more associahedron blocks ${\cal
A}_{4}^{(m,n)}|m-n|\,=\,3\,\textrm{modulo}\,7$ each of which has three non-
adjacent red facets.

###  4.4 Mutation rules from Coloured Causal Diamonds

As we have defined the associahedron block $A_{n-3}^{(i,j)}$ directly through
coloured causal diamonds, their convex realisation in ${\cal K}_{n}^{+}$ is
precisely given by the ABHY equations (with certain $X_{ij}$ replaced by
$\tilde{X}_{ij}$ such that $\tilde{X}_{ij}\,\geq\,0$).However, the mutation
rules and the corresponding abstract definition of the combinatorial polytopes
remained implicit in our construction. In this section we give the
combinatorial definition of the associahedron blocks and then describe their
realisations in kinematic space.

An associahedron block is an associahedron defined by a reference
triangulation $T_{i,j}$ which has precisely one red diagonal $(i,j)$. We now
consider all possible triangulations obtained from $T_{i,j}$ via mutation such
that “walks along the causal diamond” (as reviewed in section 4) always
produce triangulations which are mostly black . That is, they correspond to
mutations which do not change the colour of the dianonal being mutated.
However any mutation which maps a diagonal $(k,l)\,\rightarrow\,(i,j)$ is
colour changing from

$\displaystyle\begin{array}[]{lll}{\bf(i)}\,\textrm{Red to black if $(k,l)$
labels the red vertex and $(i,j)$ does not and}\\\ {\bf(ii)}\,\textrm{Black to
red if $(k,l)$ labels the black vertex and $(i,j)$ is red.}\end{array}$

We thus note that the (combinatorial) associahedron block depends on the
choice of reference $(i,j)$.

With these observations we define the associahedron block as follows :
An associahedron block associated to a reference diagonal $(ij)$ is
combinatorially an associahedron in which the diagonals from the set
$\\{\,(i,j),\,\dots,\,(i+|j-i|-1,j+|j-i|-1)\,\\}$ are red while others are
black. One example of the associahedron block in two dimensions is shown in
the figure 9.

Figure 9: Associahedron block $A^{14}_{2}$

By inspection, any associahedron-block is a simple polytope that has the
following properties.

  * •

Combinatorially, $A_{n-3}^{{\cal F}_{ij}}$ is an associahedron.

  * •

Vertices of $A_{n-3}^{{\cal F}_{ij}}$ are in bijection with either mostly
black or uncoloured triangulation of an $n$-gon.

  * •

Co-dimension k faces of $A_{n-3}^{{\cal F}_{ij}}$ are in one to one
correspondence with k-partial triangulation which is either mostly black or
uncoloured.

  * •

Each of its face is a product of a lower associahedron block and an
associahedra.

We also note that the dynamical equations (ABHY constraints) governing the
evolution inside the fundamental domain generates convex realisations of each
of the blocks. In particular for $n=5$, the evolution equations inside the
fundamental domain ${\cal F}_{13}$ generates the following realisation of an
associahedron in kinematic space.

$\displaystyle\begin{array}[]{lll}X_{13}\,+\,X_{24}\,-\,\tilde{X}_{14}\,=\,c_{13}\\\
\tilde{X}_{14}\,+\,X_{35}\,=\,c_{14}+c_{24}\\\
X_{13}\,+\,\tilde{X}_{25}\,=\,c_{13}+c_{14}\end{array}$ (43)

These equations admit solutions in the positive region ${\cal K}_{n}^{+}$ as
long as $c_{ij}\,>\,0$.

In general, given a fundamental domain ${\cal F}_{ij}$ in which only
$\\{\,(i,j),\,\dots,\,(i+|j-i|-1,j+|j-i|-1)\,\\}$ are red, the reference
triangulation generated by initial configuration consists of
$T\,=\,\\{\,(j-2,j),\,\dots,\,(i,j),\,\dots,\,(j+2,j)\,\\}$ (where without
loss of generality we have assumed that if $j-i\,=\,k$ mod $n$ then
$k\,<\,n-k$.) Hence the ABHY equations which produce the convex realisation of
the corresponding associahedron block can be written as follows. We first
define abstract variables ${\bf s}_{ij}$ as,

$\displaystyle{\bf s}_{ij}\,=\,{\bf X}_{ij}\,+\,{\bf X}_{i+1,j+1}\,-\,{\bf
X}_{i,j+1}\,-\,{\bf X}_{i+1,j}$ (44)

where ${\bf X}_{mn}\,=\,\tilde{X}_{mn},\,\textrm{if}(m,n)\,\in\,S_{(i,j)}$ and
${\bf X}_{mn}\,=\,X_{mn}$ otherwise. The convex realisation of the
associahedron block is generated by

$\displaystyle{\bf s}_{ij}\,=\,-\,c_{ij}\,\forall\,(i,j)\,\notin\,T^{c}$ (45)

where $c_{ij}$ are positive constants such that $X_{mn}\,\geq\,m^{2}$ for
$(m,n)\,\in\,S_{i,j}$ and $X_{mn}\,\geq\,0$ otherwise.

###  4.5 From the Associahedron Blocks to Scattering Amplitudes

In this section, we prove one of our primary results. We show that the
$n$-point tree-level (and colour-ordered) scattering amplitudes in our theory
when expanded to sub-leading order in $\lambda_{2}$ are a _unique_ weighted
sum over canonical forms of all the associahedron blocks in a given dimension
As each block is an (ABHY) associahedron, there is a canonical form associated
to it and a weighted sum of all the canonical forms when added to the form
associated to ABHY associahedron for $\phi_{1}^{3}$ theory produce the desired
amplitude. We start with the example of two dimensional associahedron-blocks
(i.e. $n\,=\,5$ case) before proving the master formula in equations (1, 3).

In the $n\,=\,5$ example, we have five associahedron blocks

$\displaystyle\\{\,A_{2}^{(i,j)}|\,(i,j)\,\in\,\\{(1,3),(2,4),(3,5),(1,4),(2.5)\,\\}$

each of which has two non-adjacent red facets that correspond to
$\tilde{X}_{ij}\,=\,\tilde{X}_{i+1,j+1}\,=\,0$. There is a unique canonical
form on ${\cal K}_{2}$ defined by each of these associahedron blocks, which
are simple polytopes. [20]. As an example, we consider the planar scattering
form defined by $A_{2}^{(1,3)}$. We recall that this is an ABHY associahedron
two of whose facets correspond to the massive poles
$X_{13}\,=\,X_{24}\,=\,m^{2}$.

$\displaystyle\begin{array}[]{lll}\Omega_{n=5}^{A_{2}^{{\cal F}_{13}}}\,=\\\
d\log\tilde{X}_{13}\,\wedge\,d\log
X_{14}\,-\,d\log\tilde{X}_{13}\,\wedge\,d\log X_{35}\,+\,d\log
X_{25}\,\wedge\,d\log X_{35}\,-\\\ \hskip 166.2212pt\,d\log
X_{25}\,\wedge\,d\log\tilde{X}_{24}\,+\,d\log
X_{14}\,\wedge\,d\log\tilde{X}_{24}\end{array}$ (49)

The forms associated to other blocks can be written down similarly. It is now
easy to verify that the following sum of canonical forms restricted to the
five associahedron blocks,

$\displaystyle\Omega_{n=5}^{Y}\,:=\,\frac{1}{2}\,\sum_{(i,j)}^{5}\,\Omega_{n=5}^{(i,j)}|_{A_{2}^{(i,j)}}-\,\frac{1}{2}\,\Omega_{n=5}|_{A_{2}}$
(50)

generate the contribution to the scattering amplitude at
$\lambda_{1}\lambda_{2}^{2}$ order. That is, it produces all the terms which
has one massless and one massive pole. And hence the following form generates
the complete amplitude upto $O(\lambda_{2}^{2})$.

$\displaystyle\omega_{n=5}\,:=\,\lambda_{1}\lambda_{2}^{2}\,\Omega_{n=5}^{Y}\,+\,\lambda_{1}^{3}\,\Omega_{n=5}|_{{A}_{2}}$
(51)

It is a happy surprise to see that there exists a linear combination of the
canonical forms on the associahedron blocks $\Omega_{n=5}^{Y}$ that produces
the amplitude contribution _at_ $\lambda_{1}\lambda_{2}^{2}$ order. That is,
it contains only those channels which have one massive and one massless pole.
The existence of $\Omega_{n=5}^{Y}$ is thus necessary and sufficient to obtain
the tree-level amplitude for independent couplings
$\lambda_{1},\,\lambda_{2}$. However, as the number of particles increase, so
do the number of assocaihedron blocks and then it is not at all obvious if
there exists any formula for $\Omega_{n\,>\,5}^{Y}$. We now derive a general
formula for any $n$ and verify it using a few non-trivial examples.

######  Lemma 4.1.

The following weighted sum of canonical forms restricted to their respective
associahedron blocks generate the contribution to the amplitude at
$\lambda_{1}^{n-4}\,\lambda_{2}^{2}$.

$\displaystyle\Omega_{n}^{Y}\,:=\,\left[\,\sum_{|i-j|=2}^{\left\lfloor\frac{n}{2}\right\rfloor}\,\frac{1}{j-i}\,\sum_{{\cal
F}_{ij}}\,\Omega_{n}^{(i,j)}(A_{n-3}^{{\cal
F}_{ij}})-\,\gamma\,\Omega_{n}^{\phi^{3}}(A_{n-3})\right]$ (52)

where

$\displaystyle\gamma\,=\,\sum_{\textrm{sum over all diagonals,
(i,j)}}\,\frac{1}{|j-i|}\,-\,(n-3)$ (53)

###### Proof.

When we sum over fundamental domains ${\cal F}_{ij}$ and span over all the
associahedron-blocks, each vertex (complete triangulation) is covered at least
once. These includes monochromatic triangulations where none of the diagonals
are red. Hence finding an $\Omega_{n}^{Y}$ amounts to showing that there is a
_unique_ choice of co-efficients which removes the contribution of all the
mono chromatic triangulations while simultaneously ensuring that each mostly
black triangulation contributes precisely once. We determine these co-
efficients as follows.
In the fundamental domain, ${\cal F}_{ij}$, all the vertices co-ordinatized by
$I\,=\,\\{\,(i,j),\,(i+1,j+1)\,\dots,\,(j-1,2j\,-\,i\,-1)\,\\}$ are red and
the remaining vertices are black. Hence given ${\cal F}_{ij}$, any
triangulation which has one of the diagonals in $I$ is a mostly black
triangulation and all the other triangulations (which contain no elements from
$I$) are monochromatic triangulations. Now as we scan over all of the
fundamental domains, the vertex $(i,j)$ is labelled red in $|\,j\,-\,i\,|$
number of fundamental domains. Hence any mostly black triangulation ( i.e.
vertex of the associahedron block) containing $\tilde{X}_{ij}$ occurs
$|\,j-i\,|$ times. We thus see that the _unique_ weighted sum over the
canonical forms $\Omega_{n}^{{\cal F}_{ij}}|_{{\cal A}_{n-3}^{ij}}$ which
generates all the terms in the amplitude that have one massive and $n-4$
massless poles is given by,

$\displaystyle\sum_{|i-j|\,=\,2}^{\left\lfloor\frac{n}{2}\right\rfloor}\,\frac{1}{|\,i-j\,|}\sum_{{\cal
F}_{ij}\,||i-j|\,\textrm{modulo}\,n}\,\Omega_{n}^{{\cal F}_{ij}}$ (54)

The uniqueness of the weights simply follows by inspection. As all the $(i,j)$
for which $|j-i|$ modulo $n$ is the same occur with frequency $|j-i|$, there
is no other choice of weights for which we get unit residue over all channels.

However the above sum will contain contribution over mono-chromatic
triangulations which are in bijection with purely massless channels with
varying weights. We now argue that all the mono-chromatic triangulations in
fact occur with equal frequency $\gamma$ and hence can be removed by simply
subtracting $\gamma\,\Omega_{n}({A_{n}})$ from the above form. This then
defines $\Omega_{n}^{Y}$ and completes our proof.

Let us for a moment assume that there exists such a $\Omega_{n}^{Y}$ . Then

$\displaystyle\sum_{|,i-j\,|=2}^{\frac{n}{2}}\,\frac{1}{|,i-j\,|}\sum_{{\cal
F}_{ij}|(j-i)\,\textrm{modulo}\,n}\,\Omega_{n}^{{\cal F}_{ij}}({\cal
A}_{n-3}^{ij})\,-\,\Omega_{n}^{Y}$ (55)

contains is the form of ABHY associahedron (in which all the vertices
correspond to massless poles with residue one). Let us now remove all the red-
colouring from the mostly black triangulations turning each such mostly black
triangulation into a vertex of the (uncoloured) associahedron. This would
imply that

$\displaystyle\Omega_{n}^{{\cal
F}_{ij}}\,\rightarrow\,\Omega_{n}\,\forall\,(i,j)$ (56)
$\displaystyle\Omega_{n}^{Y}\,\rightarrow\,(n-3)\,\Omega_{n}$ (57)

where the second equation follows from the fact that there are $n-3$ mostly
black triangulations that map to monochromatic triangulation when we strip off
the colour. As the map of mostly black to monochromatic triangulation does not
effect the massless channels, we have the following result.

$\displaystyle\sum_{|i-j|\,=1}^{\left\lfloor\frac{n}{2}\right\rfloor}\,\frac{1}{|i-j\,|}\,\sum_{{\cal
F}_{ij}}\,\Omega_{n}(A_{n})\,-\,(n-3)\,\Omega_{n}(A_{n})\,=\,\gamma\,\Omega_{n}$
(58)

and hence $\gamma$ is fixed to

$\displaystyle\gamma\,=\,\sum_{|\,i-j\,|=1}^{\left\lfloor\frac{n}{2}\right\rfloor}\,\frac{1}{\,|i-j\,|}\,\sum_{{\cal
F}_{ij}}\,1-(n-3)$ (59)

∎

We consider the equation (55) to be a striking result. It is apriori not
obvious that the scattering amplitude ${\cal M}_{n}^{\textrm{CO}(2)}$ (defined
in equation (26)) should be a sum over canonical forms of associahedron
blocks. Hence we view the result of the above lemma as a non-trivial evidence
in support of the universality of the Amplituhedron program.

One can verify the validity of the master formula in certain lower dimensional
cases. Here we give two explicit examples of $n=6$ and $n=7$.

$\displaystyle\Omega_{n=6}^{Y}\,=\,\frac{1}{2}\sum_{{\cal
F}_{ij}|j-i|\textrm{even mod 6}}\,\Omega_{6}^{{\cal
F}_{ij}}\,+\,\frac{1}{3}\sum_{{\cal F}_{ij}|j-i|\textrm{odd mod
6}}\,\Omega_{6}^{{\cal F}_{ij}}\,-\,\Omega_{\phi^{3}}$ (60)
$\displaystyle\Omega_{n=7}^{Y}\,=\,\frac{1}{2}\sum_{{\cal
F}_{ij}|j-i|\textrm{2 mod 7}}\,\Omega_{6}^{{\cal
F}_{ij}}\,+\,\frac{1}{3}\sum_{{\cal F}_{ij}|j-i|\textrm{3 mod
7}}\,\Omega_{6}^{{\cal F}_{ij}}\,-\,\frac{11}{6}\,\Omega_{\phi^{3}}$ (61)

It can be in fact verified that these formulae indeed produce the correct
scattering amplitudes.

We thus see that the remarkable richness of such an “doubled” causal diamond
structure (where each vertex can be either black or red) lies in the
generalisation of the above construction to higher dimensional polytopes.

###  4.6 Factorisation on Massive Channels

It was shown in [1] that combinatorial factorisation of an associahedron
implies the geometric factorisation of it’s convex realisation in ${\cal
K}_{n}^{+}$. The geometric factorisation can be quantified as,

$\displaystyle
A_{n-3}\,|_{X_{ij}\,=\,0}\,=\,A_{L}(i,i+1,\,\dots\,\overline{I})\,\times\,A_{R}(1,\,\dots,\,i-1\,I,\,\dots,\,n)$
(62)

$A_{L},\,A_{R}$ are ABHY associahedra of dimensions $|j-i-2|$ and
$n-(j-i)\,-\,1$ respectively. ABHY proved that the geometric factorisation of
the associahedron leads to the factorisation of amplitudes through the residue
formula for the canonical forms.

$\displaystyle\textrm{Res}_{X_{ij}\,=\,0}\,\Omega_{n-3}(A_{n-3})\,=\,\Omega(A_{L})\,\wedge\,\Omega(A_{R})$
(63)

where $\Omega(A_{L}),\,\Omega(A_{R})$ are canonical forms associated to the
lower dimensional associahedra $A_{L},\,A_{R}$ respectively. These results
generalise rather trivially to the associahedron blocks reaffirming one of the
striking results of the amplituhedron program : Locality and Unitarity are
consequences of the underlying positive geometry.

As we saw in an earlier section, a co-dimension one boundary of an
associahedron block is either a product of two lower dimensional associahedron
blocks, or an associahedron block and an associahedron or product of two
associahedra.121212A vertex is zero dimensional associahedron and hence we
identify an associahedron with cartesian product of the associahedron with a
zero dimensional associahedron. It can be verified that exactly as in the case
of associahedron [1] this combinatorial factorisation of the associahedron-
block implies it’s geometric factorisation as well. The geometric
factorisation can be explicitly written as

$\displaystyle\begin{array}[]{lll}A_{n-3}^{{\cal
F}_{ij}}\,|_{X_{mn}\,=\,0}&=\,A_{L}^{{\cal
F}^{L}_{ij}}(m,m+1,\dots,\,\overline{I})\,\times\,A_{R}^{{\cal
F}_{ij}^{R}}(1,\,\dots,\,m-1,\,I\,\dots,\,n)\,\textrm{or}\\\ &=\,A_{L}^{{\cal
F}_{ij}}(m,m+1,\dots,\,\overline{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}I}})\,\times\,A_{R}^{{\cal
F}_{ij}}(1,\,\dots,\,m-1,\,{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}I}\,\dots,\,n)\end{array}$
(66)

In the first case, we are on a boundary associated to massless pole,
$X_{ij}\,=\,0$ and $A_{L/R}^{{\cal F}_{ij}^{L/R}}$ corresponds to the lower
dimensional associahedron blocks. ${\cal F}_{ij}^{L/R}$ indicates the set of
red diagonals which are to the left (right) of the diagonal $(m,n)$. If ${\cal
F}_{ij}^{L/R}$ is empty then the corresponding associahedron block is simply a
convex ABHY associahedron. In the second case, we consider the boundary facet
associated to a massive pole. In this case, both the factorized positive
geometries are associahedra whose canonical forms are the scattering
amplitudes involving one massive and remaining massless particles. The fact
that we are on $X_{mn}\,=\,m^{2}$ hyper-plane automatically implies that the
intermediate particle $I$ is massive.

From the geometric factorisation of the associahedron block, we get the
factorisation property of the amplitudes.

$\displaystyle\begin{array}[]{lll}\textrm{Res}_{X_{mn}\,=\,0}\,\Omega_{n-3}(A_{n-3}^{{\cal
F}_{ij}})=\,\Omega_{L}(A_{L}^{{\cal
F}_{ij}^{L}}\,\wedge\,\Omega_{R}(A_{R}^{{\cal F}_{ij}^{R}})\\\
\textrm{Res}_{X_{mn}\,=\,m^{2}}\,\Omega_{n-3}(A_{n-3}^{{\cal
F}_{ij}})=\,\Omega_{L}(A_{L})\,\wedge\,\Omega_{R}(A_{R})\end{array}$ (68)

The residue of the $n-3$ canonical form on black or red facets equals product
of lower forms associated to associahedron blocks or associahedra
respectively.
We thus see that just as in the case of massless scattering amplitudes the
locality and unitarity of the S-matrix (even in the presence of intermediate
massive states), follows from the combinatorial factorisation property of the
associahedron!

##  5 The Scattering form for an Effective Field Theory : Accordiohedra in low
energy limit

In this section we study the $m\,\rightarrow\,\infty$ limit of the
associahedron blocks. We claim that every block projects onto a _set of_
simple polytopes in this limit as the block “moves towards infinity in various
possible directions”. This gives us a natural perspective on emergence of
positive geometries from associahedra in the effective field theory when
massive field is integrated out! The ABHY realisations basically guarantees
that a specific direction dependent $m\,\rightarrow\,\infty$ limit is a
bijection from the associahedron block onto an the polytope that we will refer
to as “projected accordiohedron”.

Let us first consider the fundamental domain ${\cal F}_{13}$ whose initial
quiver corresponds to the mostly black triangulation
$\\{\,{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}13},35\,\\}$.
The ABHY realisation of the corresponding associahedron block is given by,

$\displaystyle\tilde{X}_{13}\,+\,\tilde{X}_{24}\,-\,X_{14}\,=\,c_{13}$ (69)
$\displaystyle X_{14}\,+\,X_{25}\,-\,\tilde{X}_{24}\,=\,c_{14}$ (70)
$\displaystyle X_{35}\,+\,X_{14}\,-\,\tilde{X}_{13}\,=\,c_{35}$ (71)

where all $c_{ij}\,>\,0$.

For the purpose of this section, we will
(i) Write the ABHY constraints in terms $X_{ij}$ instead of
$\tilde{X}_{ij}\,=\,X_{ij}\,-,m^{2}$ and
(ii) Assume that all the $c_{ij}\,>\,m^{2}$. This non-trivial bound on
$c_{ij}$ ensures that we can scan the hyper-planes satisfying ABHY constraints
where one of the $m^{2}\,>X_{ij}\,>\,0$.

Equation (69) can be written as,

$\displaystyle X_{13}\,+\,X_{24}\,-\,X_{14}\,=\,c_{13}+2m^{2}$ (72)
$\displaystyle X_{14}\,+\,X_{25}\,-\,X_{24}\,=\,c_{14}-m^{2}$ (73)
$\displaystyle X_{35}\,+\,X_{14}\,-\,X_{13}\,=\,c_{35}-m^{2}$ (74)

We now consider the domain in positive region of kinematic space (but outside
$A_{2}^{F_{13}}$) parametrized by $X_{13}\,<<\,m^{2}$. In this case, we
immediately see that the above equations reduce to,

$\displaystyle X_{14}\,+\,X_{35}\,=\,c_{35}\,-\,m^{2}$ (75) $\displaystyle
X_{25}\,=\,c_{13}+c_{14}+m^{2}$ (76) $\displaystyle
X_{24}\,=\,X_{14}\,+\,c_{13}\,+\,2m^{2}$ (77)

We thus see that for $\overline{c}_{35}\,=\,c_{35}\,-\,m^{2}$ we have a one
dimensional accordiohedra given by

$\displaystyle X_{14}\,+\,X_{35}\,=\,\overline{c}_{35}$ (78)

$X_{25}$ is a positive constant and the pole $X_{24}\,=\,0$ is ruled out as
$X_{14}\,\geq\,0$. We thus see that viewed from “infinite distance” in
kinematic space when $)\,\leq\,X_{13}\,<<\,m^{2}$, the associahedron block
$A_{2}^{{\cal F}_{13}}$ “projects onto” a one dimensional accordiohedron with
boundaries as $X_{14},\,X_{35}\,=\,0$ respectively.

We now give another example : Consider $A_{2}^{F_{35}}$ associated to the
fundamental domain ${\cal F}_{35}$. When this block is realised in
$X_{13},X_{14}$ positive quadrant, with $\tilde{X}_{14}=0,\,\tilde{X}_{35}=0$
moving towards infinity in $X_{14}$ direction such that for
$X_{14}\,<<\,m^{2}$ we get $X_{13},X_{24}$ accordiohedron. This can be seen as
follows. The ABHY equations are,

$\displaystyle X_{14}+X_{35}$ $\displaystyle=\,c_{14}+c_{24}$ $\displaystyle
X_{13}\,+\,X_{25}\,$ $\displaystyle=\,c_{13}+c_{14}$ (79) $\displaystyle
X_{13}\,+\,X_{24}\,-\,X_{14}$ $\displaystyle=c_{13}.$ (80)

Where $c_{14}+c_{24}>2m^{2}$, as we want the two dimensional space cut out by
these equations to intersect $X_{14}=m^{2}$ with other $X_{ij}>0$ and
$X_{35}=m^{2}$ with other $X_{ij}>0$. If $X_{14}\,<<\,m^{2}$ we see from the
first equation that $X_{35}\,\approx\,c_{14}+c_{24}>2m^{2}$, while the second
and the third equations imply,

$\displaystyle X_{13}+X_{24}\,$ $\displaystyle=\,c_{13},$ (81)

and $X_{25}=X_{24}+c_{14}$. Therefore, region with $X_{14}\,<<\,m^{2}$ where
$X_{35}>m^{2}$ and all other $X_{ij}>0$ is the one dimensional accordiohedron
$(24,13)$ given by,

$\displaystyle X_{13}+X_{24}\,$ $\displaystyle=\,c.$ (82)

Figure 10: Acoordiohedra from Associahedra. The Orange strip is the region
with $X_{14}<<m^{2}$ where we get the accordiohedron $(24,13)$, and the green
strip is the region with $X_{35}<<m^{2}$ where we get the accordiohedron
$(13,25)$

On the other hand, if $X_{35}\,<<\,m^{2}$ then using similar logic we see
that,

$\displaystyle X_{13}\,+\,X_{25}\,=\,\textrm{const}$ (83)

is the accordiohedron that one projects onto. This is because in this
kinematic regime, we obtain

$\displaystyle X_{13}\,+\,X_{25}\,=\,c_{13}\,+\,c_{14}$ (84)

and $X_{24}=X_{25}+c_{24}$. Therefore, region with $X_{35}\,<<\,m^{2}$ where
$X_{14}>m^{2}$ and all other $X_{ij}>0$ is the one dimensional accordiohedron
$(13,25)$ given by,

$\displaystyle X_{13}+X_{25}\,$ $\displaystyle=\,c.$ (85)

As there are Five blocks in two dimensions, in the
$m^{2}\,\rightarrow\,\infty$ limit, we get ten one dimensional accordiohedra.
Each of the following accordiohedra occurs twice :

$\displaystyle\\{(13),\,(24)\,\\},\,(13,25),\,(14,25),\,(14,35),\,(24,35)\,\\}$

Now let’s look at an example at $n=6$. Consider the three dimensional block
associated to ${\cal F}_{14}$. We consider the geometric realisation of the
three dimensional block in the kinematic space given by,

$\displaystyle X_{13}+X_{26}$ $\displaystyle=c_{13}+c_{14}+c_{15}$
$\displaystyle X_{35}+X_{14}-X_{15}$ $\displaystyle=c_{14}+c_{24}$ (86)
$\displaystyle X_{14}+X_{36}$ $\displaystyle=c_{14}+c_{15}+c_{24}+c_{25}$
$\displaystyle X_{25}+X_{13}-X_{15}$ $\displaystyle=c_{13}+c_{14}$ (87)
$\displaystyle X_{15}+X_{46}$ $\displaystyle=c_{15}+c_{25}+c_{35}$
$\displaystyle X_{24}+X_{13}-X_{14}$ $\displaystyle=c_{13}.$ (88)
$\displaystyle X_{13}$ $\displaystyle>0$ $\displaystyle X_{24}$
$\displaystyle>0$ $\displaystyle X_{35}$ $\displaystyle>0$ (89) $\displaystyle
X_{46}$ $\displaystyle>0$ $\displaystyle X_{15}$ $\displaystyle>0$
$\displaystyle X_{26}$ $\displaystyle>0$ (90) $\displaystyle X_{14}$
$\displaystyle>m^{2}$ $\displaystyle X_{25}$ $\displaystyle>m^{2}$
$\displaystyle X_{36}$ $\displaystyle>m^{2},$ (91)

such that $c_{14}+c_{15}+c_{24}+c_{25}>2m^{2},c_{13}+c_{14}>m^{2}.$ If we take
$X_{14}\,<<\,m^{2}$ we get the following equations,

$\displaystyle X_{26}$ $\displaystyle\approx X_{24}+c_{14}+c_{15}$
$\displaystyle X_{35}$ $\displaystyle\approx X_{15}+c_{14}+c_{24}$ (92)
$\displaystyle X_{36}$ $\displaystyle\approx
c_{14}+c_{15}+c_{24}+c_{25}>2m^{2}$ $\displaystyle X_{25}$
$\displaystyle\approx X_{15}+X_{24}+c_{14}>0$ (93) $\displaystyle
X_{15}+X_{46}$ $\displaystyle=c_{15}+c_{25}+c_{35}$ $\displaystyle
X_{24}+X_{13}$ $\displaystyle\approx c_{13}.$ (94)

Therefore the region where $X_{14}<<m^{2}$ and all other $X_{ij}>0$ is the two
dimensional accordiohedra $(24,13)\times(15,46)$, given by

$\displaystyle X_{15}+X_{46}$ $\displaystyle=c_{15}+c_{25}+c_{35}$
$\displaystyle X_{24}+X_{13}$ $\displaystyle=c_{13}.$ (95)

In general, if in a given $n-3$ dimensional associahedron block
$\tilde{X}_{ij},\tilde{X}_{i+1,j+1},\,\tilde{X}_{i+(j-i),j+(j-i)}$ are the red
vertices then taking $X_{ij}\,<<\,m^{2}$, we get a $n-4$ dimensional
accordiohedra parametrized by facets which are generated by (colorless)
diagonals that don’t intersect the diagonal $(ij)$.

##  6 EFT Amplitude from Projected Accordiohedra

Tree-level amplitudes with mixed interactions of massless scalars have been
analysed previously in the literature [14]. At each order in perturbative
expansion, the positive geometry associated to the scattering amplitude
generated by $\lambda\phi_{1}^{3}\,+\,g\,\phi_{1}^{4}$ interaction is an
accordiohedron [26]. In particular at order $g$ in the quartic coupling, each
accordiahedron ${\cal AC}_{n-4}({\cal D})$ is a simple polytope which is
defined using a reference dissection ${\cal D}$ of an $n$-gon into $n-4$
triangles and one quadrilateral. It is thus a natural question to ask if there
is a bijection between “projected accordiohedra” ${\cal PAC}_{n-4}[(i,j)]$
that emerge in the “low energy” regions of kinematic space and the
accordiohedra ${\cal AC}_{n-4}({\cal D})$.

Naive expectation for such a bijection is the following : Consider the
reference triangulation $T_{ij}$ defined by the reference triangulation in
${\cal F}_{ij}$. Consider e.g. one of the projected accordiohedra, say ${\cal
PAC}_{n-4}[(i,j)]$ that emerges in $X_{ij}\,<<\,m^{2}$ region. We can
interpret the low energy limit as projecting $T_{ij}$ on to a reference
dissection which consists of $n-4$ triangles and one quadrilateral obtained by
removing the $(i,j)$ diagonal. The dissection $T_{ij}/(i,j)$ can be used to
define the ${\cal AC}_{n-4}(T_{ij}/(i,j))$ as in [26]. We may expect the
resulting polytope to be isomorphic to the projected accordiohedron. However
this expectation turns out to be wrong. This can immediately by checked by
specific examples.

  * •

Consider the $n\,=\,5$ case where we obtained 5 associahedron blocks and 10
accordiohedra in the low energy limit. Consider one of the blocks say ${\cal
A}_{2}^{{\cal F}_{13}}$ in which $\tilde{X}_{13}$ and $\tilde{X}_{24}$ are red
. The initial quiver in ${\cal F}_{13}$ is $\\{(1,3)_{R},(3,5)\,\\}$. Taking
$X_{13}\,<<\,m^{2}$ amounts to deleting the $(1,3)$ diagonal and there by
obtaining the reference dissection $(3,5)$. However, as can be readily
verified using the definition of ${\cal AC}_{1}$, the compatible set of
diagonals is $\\{(3,5),(2,4)\,\\}$ instead of $\\{\,(1,4),(3,5)\,\\}$. That
is, the ${\cal AC}_{1}$ defined using $(3,5)$ as a reference is a one
dimensional polytope with vertices at $X_{35}\,=\,X_{24}\,=\,0$ as opposed to
the ${\cal PAC}_{1}$ we obtained with vertices at $X_{35}\,=\,0$ and
$X_{14}\,=\,0$.

  * •

In the $n\,=\,6$ case, the difference between ${\cal AC}_{n-4}(T_{ij}/(i,j))$
and ${\cal PAC}_{n-4}$ which are located in $X_{ij}\,<<\,m^{2}$ is even more
pronounced. If we consider the associahedron block $A_{3}^{{\cal F}_{14}}$ and
anticipate that the projection onto the $X_{14}\,<<\,m^{2}$ region is
equivalent to considering the reference dissection $\\{(1,3),(1,5)\,\\}$ then
we see an immediate contradiction. Starting with ${\cal
D}\,=\,\\{\,(1,3),\,(1,5)\,\\}$ We obtain a two dimensional ${\cal
AC}_{2}({\cal D})$ which is a pentagon with five facets labelled by

$\displaystyle\\{(1,3),(2,6),(1,5),(2,6),(4,5)\,\\}$

However as there is no diagonal which does not intersect all of these five
facets, this particular accordiohedron is not a (co-dimension one) face of the
three dimensional associahedron. On the other hand, ${\cal PAC}_{2}[(1,4)]$
sitting in the $X_{14}\,<<\,m^{2}$ region, is by construction (parallel) to
the $X_{14}\,=\,m^{2}$ facet of the associahedron block $A_{3}^{{\cal
F}_{14}}$.

In summary, we see that,
(i) For all the accordiohedra which are in fact projections on to co-dimension
one facets of an associahedra, we need to identify the reference dissection so
that they are projected accordiohedra emerging in the low energy limit.
(ii) And in fact, there is a class of accordiohedra that are defined using
specific reference dissections consisting of $(n-4)$ triangles and one
quadrilateral which can not be realised as projected accordiohedra.

The second example cited above pertains to a larger structure of an
accordiohedron. Namely, not all accordiohedra are facets of higher dimensional
associahedra. It is hence puzzling to see that both the descriptions, one via
accordiohedron as positive geometry and one via projected facets ${\cal
PAC}_{n-4}$ of the associahedron blocks in the low energy limit can produce
the same tree-level amplitude.

Resolution to this contradiction was already hinted at in [14] where it was
noticed in certain examples that any combinatorial accordiohedron ${\cal
AC}_{n-4}({\cal D})$ which can not be realised as a co-dimension one facet of
a higher dimensional associahedron does not contribute to the scattering
amplitude! We prove this statement in appendix A.

We will now show that, given an ${\cal F}_{ij}$ with red vertices co-
ordinatized by ${\cal S}_{ij}\,=\,\\{\,(i,j),\,\dots$
$\dots\,(i+|j-i|,j\,+\,|j-i|)\,\\}$, any one of the projected accordiohedron
${\cal PAC}_{n-4}[(p.q)]$ that emerge in
$\\{\,X_{p,q}\,<<\,m^{2}\,|\,(p.q)\,\in\,{\cal S}_{ij}\,\\}$ region can indeed
be realised as ${\cal AC}_{n-4}({\cal D}_{p,q})$. Where ${\cal D}_{p,q}$ is
defined as follows.

$\displaystyle
D_{p.q}\,=\,\\{\,X_{p,p-2},\,\dots,\,X_{p,q+1},\,X_{p+1,q},\,X_{p+2,q},\,\dots,\,X_{p+2,q+3},\,\dots,\,X_{p,p-2}\,\\}$
(96)

Before proving this claim we first verify it in a couple of examples.

  * •

In the case of $n\,=\,5$ and the fundamental domain ${\cal F}_{13}$ if we
consider a reference dissection as $\\{\,(1,3)_{\textrm{R}},(1,4)\,\\}$ then
the resulting one dimensional projected accordiohedron in $X_{13}\,<<\,m^{2}$
domain is precisely a polytope with vertices labelled by
$\\{\,(1,4),\,(3,5)\,\\}$.

  * •

In the same way in the case of $n\,=\,6$ and fundamental domain ${\cal
F}_{14}$ choosing the reference as $\\{\,(2,4),(1,4),(1,5)\,\\}$ and deleting
the red diagonal $(1,4)$ in $X_{14}\,<<\,m^{2}$ limit results in a two
dimensional accordiohedron with four vertices.

######  Lemma 6.1.

Given a fundamental domain ${\cal F}_{ij}$ with $(p,q)$ co-ordinatizing one of
the red vertices, consider the projected accordiohedron ${\cal
PAC}_{n-4}[(p,q)]$ obtained by analysing the associahdron block
$A_{n-3}^{{\cal F}_{ij}}$ from $X_{pq}\,<<\,m^{2}$ region. We claim that

$\displaystyle{\cal PAC}_{n-4}[(p,q)]\,=\,{\cal AC}_{n-4}(D_{p,q})$ (97)

where ${\cal D}_{p,q}$ is defined in equation (96).

Proof : The proof is simply by inspection. We first analyze $(p,q)\,=\,(i,j)$
case. Consider the initial quiver in ${\cal F}_{ij}$ which is the reference
triangulation $T_{ij}$ (generating the ABHY equations).

$\displaystyle T_{ij}\,=\,\\{X_{i,i-1},\,\dots,\,X_{i,i+1}\,\\}$ (98)

Where we span the vertex label $i-1$ to $i+1$ counter-clockwise from $i$ and
traverse through $X_{ij}$. ${\cal D}_{ij}$ is simply obtained from $T_{ij}$
by,

$\displaystyle
D_{ij}\,=\,\cup_{m=j}^{i+1}\,T_{ij}\backslash(i,m)\,\cup\,(i+1,m)$ (99)

A moment of meditation reveals that the all the diagonals which are not
transversely intersect $(i,j)$ diagonal are necessarily not compatible with
the dissection ${\cal D}_{ij}$, (see figure 11). This completes the proof.

Figure 11: A diagonal intersecting $(ij)$ is not compatible with the
dissection ${\cal D}_{ij}$

Similarly ${\cal PAC}_{n-4}(p,q)$ (where $(p,q)\,=\,(i+k,j+k)$) obtained by
projecting onto the face parallel to $\tilde{X}_{i+k,j+k}\,=\,0$ is isomorphic
to the accoriohedron ${\cal AC}_{n-4}(D_{i+k,j+k})$ where $D_{i+k,j+k}$ is
obtained by rigidly translating the quiver associated to $D_{ij}$ or
equivalently rotating the dissection $D_{ij}$. Finally, the complete colour-
ordered tree-level amplitude can be computed by using weighted sum over
accordiohedra as in [14]. This result is simply based on the claim proved in
the previous section. Thanks to the identification established through
equation (96), the planar scattering form $\Omega_{n-4}$ determined by ${\cal
PAC}_{n-4}[(p,q)]$ equals the planar scattering form defined by ${\cal
AC}_{n-4}({\cal D}_{p,q})$. Hence we have the following.

$\displaystyle\Omega_{\textrm{EFT}}^{g^{2}}\,=\,\sum_{{\cal
D}_{p,q}}\,\alpha_{[{\cal D}_{p,q}]}\,\Omega^{{\cal D}_{p,q}}_{n-4}|_{{\cal
AC}_{n-4}({\cal D}_{p,q})}$ (100)

where $[{\cal D}]$ was introduced in section 2.1. It is the equivalence class
of (reference) dissection defined as the orbit of ${\cal D}$ under the
dihedral group $D_{n}$.
$\alpha_{[{\cal D}_{p,q}]}$ are determined in Appendix A.

##  7 Discussion and Open Issues

Perturbative expansion at higher orders
A rather natural question that arises out of our analysis is the folliwng :
Can similar ideas be used to compute amplitude for
$\lambda_{1}\,\phi_{1}^{3}\,+\,\lambda_{2}\,\phi_{1}^{2}\phi_{2}$ theory to
arbitrary orders in $\lambda_{2}$, that is including amplitude contributions
that have (at least) two massive and remaining massless poles. We can see that
there is an immediate limitation to generalising the structure of coloured
causal diamonds to define polytopes which includes such contributions. Given
any two diagonals $(ij),\,(ik)$ in an $n$-gon, there is always a complete
triangulation that
(a) includes $(ij),\,(ik)$ and,
(b) the triangulation is such that these two diagonals are adjacent to each
other.

Thus, irrespective of what mutation rules we define, (so as to define a
fundamental domain which has an “initial dissection” that is associated to a
diagram with two massive propagators in our theory) this domain will always
include a triangulation which is dual to the a Feynman diagram in scalar
theory with $\phi_{1}\,\phi_{2}^{2}$ vertex. Hence positive geometries for
higher order perturbative expansion of scattering amplitude in
$\lambda_{1}\,\phi_{1}^{3}\,+\,\lambda_{2}\,\phi_{1}^{2}\,\phi_{2}$ theory
remain unknown. Based on simple examples (such as $n\,=\,6$ case) we do expect
the positive geometries to be open polytopes such as those discovered in [15].
However a detailed analysis of such “open associahedron blocks” is outside the
scope of this work.

Amplitude for most general cubic interactions
Yet another related question which we do not answer in this paper is the
following: Can the perturbative expansion of (planar and tree-level)
scattering amplitude generated by the most general cubic interaction involving
a massless and a massive scalar field be obtained from positive geometries
located in positive regime of ${\cal K}_{n}$. We suspect that the answer is
yes: By expanding the definition of fundamental domain to include “un-
coloured” triangulations more than once, we do end up including vertices of
the associahedra that are dual to poles with the generic interactions.

Relationship with CHY Formulation :
One of the most striking consequences of the discovery of the kinematic space
associahedron in [1] was a derivation of the CHY formula for bi-adjoint
$\phi^{3}$ theory. In [1] it was shown that the compactification of the (real
section) of the CHY moduli space, namely ${\bf M}_{0,n}({\bf R})$ was
diffeomorphic to the kinematic space associahedron where the diffeomorphisms
were defined by the scattering equations. It is thus a natural question to ask
if the convex realisation of associahedron blocks are diffeomorphic to the CHY
moduli space with diffeomorphism defined by the scattering equations
corresponding to mixed scalar interactions. A proposal for this class of
scattering equations is given in [27].

In [27] a proposal was given for generalisation of CHY scattering equations to
theories with several species of scalar fields. The numerator in the CHY
scattering equations which depend on Mandelstam invariants are deformed based
on whether the corresponding channel has a massive pole or a massless pole.
Lam has given the CHY formula for $\phi_{1}^{2}\,\phi_{2}$ interaction A
detailed analysis of the relationship between CHY integrand for bi-scalar
theory and weighted sum over d log forms is outside the scope of the paper and
will be attempted elsewhere but we are tempted to offer some speculation.

If the (generalised) scattering equations proposed in [27] do turn out to be
the diffeomorphisms which map the (compactified) CHY moduli space to the
associahedron block then (1) this will imply that the weighted sum of
canonical forms on the associahedron blocks is a pushforward of a CHY
integrand and (2) the inverse of diffeomorphism between CHY moduli space and
ABHY associahedron can be composed with the diffeomorphism from CHY moduli
space to associahedron block to have a diffeomorphism between massless
$\phi^{3}$ amplitude and perturbative amplitude (up to $\lambda_{2}^{\,2}$) in
two-scalar field theory. At this stage this is a pure speculation but if true
it may give rise to interesting inter-relationships between tree-level
amplitudes of physically inequivalent theories.

Perturbative Amplitude at All Orders:
In this paper, we focused solely on tree level colour-ordered amplitude at
order $\lambda_{2}^{\,2}$. But our eventual goal is to find the positive
geometry in kinematic space whose canonical form generate amplitude at any
order in the coupling. However there is a clear structural difference between
amplitude at order $\lambda_{2}^{\,2}$ from higher order contributions. As we
argued earlier, the polytope with two or more adjacent red facets are bound to
be open if we fix the interaction between the two fields to be
$\phi_{1}^{2}\phi_{2}$. Whether the causal structure in kinematic space
encodes combinatorics and realisations of such polytopes remains to be seen.

This question is intimately tied to seeing how accordiohedra for perturbative
amplitudes in EFT at higher order in $g$ emerge in the decoupling limit from
positive geometry of the bi-scalar field theory.

Coloured Causal diamonds and Positive geometries for several fields:
Finally it would be interesting to see if the ideas proposed in this paper can
be generalised suitably to find positive geometry for tree-level amplitudes of
several fields.131313We are grateful to Nima Arkani-Hamed for posing this
question. As “friends” of associahedra with various possible colorings of the
facets remain to be found, we hope that a colored causal structure in the
kinematic space can be used to define such polytopes.

Local QFTS and Positive Geometries
Our analysis centered around a specific mixed cubic interaction between a
massless and a massive field and we computed tree-level amplitudes with only
massless external particles. It is an important question to analyse the
correspondence between positive geometries in kinematic space and local
unitary scalar scattering amplitudes. That is, how do we classify all the
local multi-scalar interactions for which _any_ scattering amplitude equals
sum over $d\log$ forms associated to positive geometries. We should note that
computation of generic bi-adjoint amplitude with a single massless scalar or
the amplitudes in the Yukawa type interaction analysed in [15] require us to
consider so-called open polytopes in which some of the facets lie at infinity
in the kinematic space. Hence our organisation principle should allow for open
associahedra (or more generally open polytopes) as possible positive
geometries for scattering amplitudes.

We hope to return to at least some of these questions in the future.

##  Appendix A Explicit formula for the weights

In this appendix, we derive an explicit formula for the weights
$\alpha_{[{\cal D}_{p,q}]}$ that appeared in equation (100). We then prove
that if a combinatorial accordiohedron ${\cal AC}({\cal D})$ of dimension
$n-4$ is not (combinatorially) equivalent to one of the co-dimension one facet
of $A_{n-3}$ then the weight $\alpha_{[{\cal D}]}$ is zero.

The explicit formula for the weights when the accordiohedron polytope is
generated a reference dissection consists of $n\,-\,4$ triangles and one
quadrilateral is derived as follows.

######  Lemma A.1.

Consider an accordiohedron whose reference dissection consists of $n-4$
triangles and one quadrilateral. If the accordiohedron is a facet of
associahedron associated to one partial triangulation obtained by removing
$(i,j)$ then,

$\displaystyle\begin{array}[]{lll}\alpha_{[{\cal
D}_{i,j}]}\,=\,\frac{1}{2C_{k-2}\,C_{n-k-2}}\,\textrm{if}\
|\,j-i|\,=\,k\,\textrm{modulo}\,n\end{array}$ (102)

where $C_{l}$ is the Catalan Number defined as

$\displaystyle C_{l}\,=\,\frac{(2l)\,!}{l\,!\,(l+1)\,!}$ (103)

Note that if $|i-j|\,=\,2\,\textrm{modulo}\,n$ then the above formula reduces
to,

$\displaystyle\alpha_{[{\cal D}_{i,j}]}\,=\,\frac{1}{2\,C_{n-4}}\,$ (104)

###### Proof.

Let $(\partial A)_{n-4}(i,j)$ be one of the facets of $A_{n-3}$ that
corresponds to one-partial triangulation obtained by removing the diagonal
$(i,j)$. $(\partial A)_{n-4}(i,j)$ corresponds to a set of $\\{\,{\cal
AC}(B^{\prime}_{ij})\,\\}$ where $B^{\prime}_{ij}$ is the set of all
dissections which lead to the same combinatorial accordiohedron.
Now let $|j-i|\,=\,2\,\textrm{modulo}\,n$. In this case the reader can
convince themselves easily that all the triangulations of the $n-1$ gon
obtained by chopping off the triangle $i,\,i+1\,j$ lead to the same
accordiohedron. Hence the number of such accordiohedra is $C_{n-4}$. Similarly
if $(\partial A)_{n-4}(i,j)\,=\,A_{r}\,\times\,A_{n-r-4}$ then the total
number of dissections which lead to the this facet as an accordiohedron is
$C_{r}\,\times\,C_{n-r-4}$.
We also note that each reference dissection $B^{\prime}_{ij}$ leads to an
accordiohedron which is one of two possible facets : $(\partial A)_{n-4}(i,j)$
and either $(\partial A)_{n-4}(i+1,i\,+1)$ or $(\partial A)_{n-4}(i-1,i\,-1)$
and hence weight that we assign to each boundary accordiohedron for a given
$(i,j)$ is $\frac{1}{2\,C_{r}\,C_{n-r-4}}$. This completes the proof. ∎

## Acknowledgement

We are indebted to Nima Arkani-Hamed for a number of insightful discussions,
many clarifications and encouragement.
We would like to thank Ashoke Sen and Nemani Suryanarayana for valuable inputs
and Sujay Ashok, Pinaki Banerjee, Miguel Campiglia, Dileep Jatkar, Nikhil
Kalyanapuram, Madhusudan Raman, Prashanth Raman and Arnab Priya Saha for many
discussions over the years on related issues. We also thank Pinaki Banerjee
for comments on the manuscript.
We would especially like to thank Vincent Pilaud for his guidance and crucial
insights in the early stages of this work.

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