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License Attribution 4.0 International (CC BY 4.0).

TPDL2022: 26th International Conference on Theory and Practice of Digital
Libraries, 20-23 September 2022, Padua, Italy

[orcid=0000-0002-9232-4840<EMAIL_ADDRESS>]

[1]This doctoral thesis is supervised by Prof. Dr. Adrian Paschke at the Free
University of Berlin.

# A Continual Relation Extraction Approach for Knowledge Graph Completeness

Sefika Efeoglu  Free University of Berlin, Takustrasse 9, 14195 Berlin,
Germany

(2022)

###### Abstract

Representing unstructured data in a structured form is most significant for
information system management to analyze and interpret it. To do this, the
unstructured data might be converted into Knowledge Graphs, by leveraging an
information extraction pipeline whose main tasks are named entity recognition
and relation extraction. This thesis aims to develop a novel continual
relation extraction method to identify relations (interconnections) between
entities in a data stream coming from the real world. Domain-specific data of
this thesis is corona news from German and Austrian newspapers.

###### keywords:

relation extraction continual learning knowledge graph completeness

##  1 Introduction

For various reasons, semantic analysis and interpretation of unstructured
data, such as social media posts, texts on web pages, and chats, are one of
the challenges in information management systems. To analyze and interpret
unstructured text data, it must be represented in a structured form. One way
of representing unstructured data in the structured form is the use of
knowledge graphs (KGs) [1]. An information extraction (IE) pipeline is
generally organized as the product of several analysis components, e.g.: named
entity (NE) tagging; syntactic analysis; coreference resolution within a
document; entity, relation and event extraction (semantic analysis); and
cross-document coreference resolution [2]. Semantic analysis in IE systems
might be carried out on the KG.

Recent KG construction approaches utilize machine learning-based approaches
instead of rule-based techniques for NE and relation extraction (RE). The
machine learning-based approaches obtain impressive results in the NE tagging
[3], while they do not show this performance on RE [4]. Detection of relations
between entity pairs has been addressed with various types of approaches: (i)
supervised techniques including features-based and kernel-based methods, (ii)
a special class of techniques which jointly extract entities and relations (
semi-supervised), (iii) unsupervised, (iv) Open IE and (v) distant supervision
based techniques [5]. Supervised techniques require a large annotated data
set, and its annotation process is time-consuming and expensive [5]. Distant
supervision is amongst one of the popular methods to deal with this annotated
data problem. The distant supervision, based on existing knowledge bases,
brings its own drawback, and it faces the problem of wrongly labeled sentences
troubling the training due to the excessive amount of noise [6]. In addition
to distant supervision, another most popular approach is bootstrapping-based
semantic RE techniques, namely weakly supervised RE [7]. However, the weakly
supervised approach is more error-prone because of semantic drift in a set of
patterns per iteration of a snowball algorithm [7]. On the other hand, in
rule-based RE approaches, finding relations are mostly dependent on predefined
rules [5].

Recently, extracting information from health data published on the web has
become more significant to analyze and take precautions due to the ongoing
pandemic. To analyze and evaluate the processes of the pandemic or its
measurements, its data must be represented in a machine-readable and
understandable format. Moreover, the IE pipeline extracted information from
web documents or pages must run throughout producing new data by web
applications without any interruption.

Concerning the applicability of existing RE methods, information systems have
to cope with different challenges, leveraging KG in real-world applications in
terms of RE: (i) missing extracted relations between the entities throughout
the data stream, since the existing methods run once on a fixed data set, (ii)
heterogeneous representation of data, (iii) requirement of a large annotated
data set and (iv) unexplainable relations extracted by machine learning-based
approaches. The existing RE approaches trained and evaluated on the fixed data
set are mostly dependent on predefined relations [5, 6, 8]; therefore, they
might not discover new relation types in the applications whose data is coming
from the real world. Because of this reason, to discover new relation types,
the learning process must be continuous. Otherwise, using the existing RE
approaches on real-world applications leads to KG incompleteness [8].

The rest of this paper explains the thesis’ problem details in the Problem
Statement section, seeking the solution to research questions in this thesis.
After that, this paper introduces the possible methods for these research
questions in the Research Methodology section, and then the paper gives
possible evaluation approaches in the Evaluation section. Lastly, it details
concluding remarks in the Conclusion section.

##  2 Problem Statement

As stated in the Introduction section, relation extraction is one of the most
significant tasks in knowledge acquisition which is used for managing and
analyzing data on web documents and pages in the health domain. Traditional
machine learning-based relation extraction approaches such as distant
supervision and bootstrapping methods run once on a fixed data set offline.
The main problem with these approaches is that it could not be feasible to
detect relations between entities throughout a data stream coming from the
real world since they run once as offline on the fixed data set [9]. Because
of this reason, they might not keep and transfer knowledge learned from
previous tasks to further tasks.

Furthermore, existing RE approaches for streaming data like meta-continual
learning are applied offline and do not support knowledge retention [10]. To
provide learning over data coming from real-world applications, knowledge
retention is one of the most important requirements for human-like learning.
In addition, this thesis addresses KG incompleteness for RE in terms of non-
stationary text data. This thesis tries to tackle the problem above, seeking
answers to the following research questions (RQ):

  * •

RQ1: How can new relation types between entities be continuously discovered
throughout a data stream coming from the real world? (The ${1}^{st}$)

  * •

RQ2: How can the KG incompleteness in terms of relations between entities be
addressed in the context of continuous streaming data? (The ${2}^{nd}$)

  * •

RQ3: Can semantic drift of a set of patterns in a weakly supervised approach
be tackled by a rule learning method using KG or ontology embeddings? (The
${3}^{rd}$)

  * •

RQ4: How can finally extracted relations be made explainable and
interpretable? since machine learning approaches are black-box. (Last year)

##  3 Research Methodology

The thesis proposal hypothesizes that a weakly supervised continual RE
approach tackles the KG incompleteness problem in terms of relations between
entities, identifying relation types continuously in a data stream coming from
real-world applications like newspapers. In addition, this thesis claims that
continual learning might increase the number of newly discovered different
relation types, feeding the model with new data resources. The proposed
research methods below (See Fig. 1) are developed and evaluated on the corona
news published on Tagesschau 111Tagesschau : https://www.tagesschau.de/ and
Austrian Derstandard 222Derstandard: https://www.derstandard.at/.

Figure 1: This pipeline shows how the thesis is progressing.

###  3.1 Named Entity Tagging for Corona News

Since a corona Named Entity Recognition (NER) model has not been developed
yet, and there is no gold standard benchmark data set in this domain, the NER
model must be developed to identify named entities in a text before developing
and running a RE approach for this data set. However, a weakly supervised
CORD-NER data set was published in [11] and is used in the baseline NER model
of this thesis after reducing the noise in its labels with Wikidata
333Wikidata: https://www.wikidata.org. Then, in order to improve its accuracy
on the corona news text data, fine-tuning is applied to this baseline model
with Corona News corpus (their English versions) described above.

###  3.2 A Continual Relation Extraction Approach for Knowledge Graph
Completeness

Continuously extraction of relations from non-stationary data still has to
address some challenges, such as labeled training data, catastrophic
forgetting (because of continual learning) , and predefined relation types.
This thesis aims to develop a (online) weakly supervised continual RE approach
by using a snowball algorithm which applies an incremental learning method
[7]. However, this algorithm has its own drawbacks like a semantic drift on
pattern extraction for relation types, and seed relation selection in the
first step of the algorithm. Although there are previous attempts to implement
this algorithm for RE, they have fixed constant coefficients in its learning
phase or knowledge retention problems due to their rule-based approach [12],
and also simple transfer learning [13]. Nevertheless, both approaches are
evaluated on stationary data for predefined relation types. With respect to
evaluation of the continual RE algorithms on the streaming data, existing
(offline) continual learning RE algorithms [10, 14] have been tested for the
predefined relations on the benchmarks, like FewRel [15] and SimpleQuestions
[16]. As stated earlier, the previous algorithms have been run offline and not
supported the knowledge retention. An online continual RE algorithm, e.g.,
neurogenesis and memory replay might transfer the learnt knowledge (relation
types) to next tasks [9]. Therefore, this thesis aims to transfer the learnt
knowledge in the previous tasks to the following task with an assist of the
continual learning algorithm to keep the learnt relation types. In addition to
this, the algorithm will consider knowledge graph and category embeddings to
discover new relation types together with dependency parsing of a sentence.

##  4 Evaluation Approaches

This thesis evaluates the continual RE approach’s results with evaluation
metrics in terms of various perspectives. The thesis takes into account the
following metrics: Precision (P), Recall (R), F1 score, P-R curve, area under
the curve [14] to evaluate the relation extraction approach. The existing
relation extraction algorithms using curriculum meta continual learning [10,
14] have been evaluated offline on the partitions of benchmark data sets e.g.,
FewRel [15] and SimpleQuestions [16] for time slots. However, there is no
approach to evaluate online RE algorithms on streaming data. Similarly, this
thesis will evaluate the metrics above for the time slots of the data and
measure its performances on the benchmark data in the first phase of its
implementation. Furthermore, this thesis also considers following metrics to
evaluate the approach’s performance with respect to continual learning:
average accuracy, whole accuracy, forgetting measure, learning curve area, and
error bound [10, 14].

##  5 Conclusion

Consequently, this thesis is planning to develop a continual learning approach
to identify relation types between entities on non-stationary data with a weak
supervision.

## References

  * Sheth et al. [2019] A. Sheth, S. Padhee, A. Gyrard,  Knowledge graphs and knowledge networks: The story in brief,  IEEE Internet Computing 23 (2019) 67–75.
  * Grishman [2015] R. Grishman,  Information extraction,  IEEE Expert 30 (2015) 8–15.
  * Li et al. [2018] J. Li, A. Sun, J. Han, C. Li, A survey on deep learning for named entity recognition, 2018.
  * Al-Aswadi et al. [2019] F. N. Al-Aswadi, H. Y. Chan, K. H. Gan,  Automatic ontology construction from text: a review from shallow to deep learning trend,  Artificial Intelligence Review 53 (2019) 3901 – 3928.
  * Pawar et al. [2017] S. Pawar, G. K. Palshikar, P. Bhattacharyya, Relation extraction : A survey, 2017.
  * Aydar et al. [2020] M. Aydar, O. Bozal, F. Ozbay, Neural relation extraction: a survey, 2020.
  * Agichtein and Gravano [2000] E. Agichtein, L. Gravano,  Snowball: Extracting relations from large plain-text collections,  in: Proceedings of the Fifth ACM Conference on Digital Libraries, DL ’00, Association for Computing Machinery, New York, NY, USA, 2000, p. 85–94.
  * Ji et al. [2022] S. Ji, S. Pan, E. Cambria, P. Marttinen, P. S. Yu,  A survey on knowledge graphs: Representation, acquisition, and applications,  IEEE Transactions on Neural Networks and Learning Systems 33 (2022) 494–514.
  * Parisi et al. [2019] G. I. Parisi, R. Kemker, J. L. Part, C. Kanan, S. Wermter,  Continual lifelong learning with neural networks: A review,  Neural Networks 113 (2019) 54–71.
  * Wu et al. [2021] T. Wu, X. Li, Y.-F. Li, R. Haffari, G. Qi, Y. Zhu, G. Xu, Curriculum-meta learning for order-robust continual relation extraction, 2021. arXiv:2101.01926.
  * Wang et al. [2020] X. Wang, X. Song, B. Li, Y. Guan, J. Han, Comprehensive named entity recognition on cord-19 with distant or weak supervision, 2020.
  * Batista et al. [2015] D. S. Batista, B. Martins, M. J. Silva,  Semi-supervised bootstrapping of relationship extractors with distributional semantics,  in: Proceedings of the 2015 Conference on Empirical Methods in Natural Language Processing, Association for Computational Linguistics, Lisbon, Portugal, 2015.
  * Gao et al. [2019] T. Gao, X. Han, R. Xie, Z. Liu, F. Lin, L. Lin, M. Sun, Neural snowball for few-shot relation learning, 2019.
  * Biesialska et al. [2020] M. Biesialska, K. Biesialska, M. R. Costa-jussà,  Continual lifelong learning in natural language processing: A survey,  in: Proceedings of the 28th International Conference on Computational Linguistics, International Committee on Computational Linguistics, Barcelona, Spain (Online), 2020, pp. 6523–6541. doi:10.18653/v1/2020.coling-main.574.
  * Han et al. [2018] X. Han, H. Zhu, P. Yu, Z. Wang, Y. Yao, Z. Liu, M. Sun,  FewRel: A large-scale supervised few-shot relation classification dataset with state-of-the-art evaluation,  in: Proceedings of the 2018 Conference on Empirical Methods in Natural Language Processing, Association for Computational Linguistics, Brussels, Belgium, 2018, pp. 4803–4809. doi:10.18653/v1/D18-1514.
  * Petrochuk and Zettlemoyer [2018] M. Petrochuk, L. Zettlemoyer,  SimpleQuestions nearly solved: A new upperbound and baseline approach,  in: Proceedings of the 2018 Conference on Empirical Methods in Natural Language Processing, Association for Computational Linguistics, Brussels, Belgium, 2018, pp. 554–558. doi:10.18653/v1/D18-1051.


#  Constraining the weights of Stokes Polytopes using BCFW recursions for
$\phi^{4}$

Ishan Srivastava Email<EMAIL_ADDRESS>Chennai Mathematical Institute,
H1 SIPCOT IT Park, Siruseri, Chennai 603103, India

###### Abstract

The relationship between certain geometric objects called polytopes and
scattering amplitudes has revealed deep structures in QFTs. It has been
developed in great depth at tree and loop-level for
$\mathcal{N}=4~{}\text{SYM}$ theory and has been extended to the scalar
$\phi^{3}$ and $\phi^{4}$ theories at tree-level. In this paper, we use the
generalized BCFW recursion relations for massless planar $\phi^{4}$ theory to
constrain the weights of a class of geometric objects called Stokes polytopes,
which manifest in the geometric formulation of $\phi^{4}$ amplitudes. We
compute the weights of $n=8$ and $n=10$ Stokes polytopes corresponding to
eight- and ten-point amplitudes respectively. We generalize our results to
higher-point amplitudes and show that in general the weights of an
$n$-dimensional Stokes polytopes is fixed precisely in terms of lower-point
weights, by the BCFW recursion relations.

###### Contents

  1. 1 Introduction
  2. 2 Amplitudes for massless planar $\phi^{4}$ theory
    1. 2.1 $\phi^{4}$ amplitudes from Stokes Polytopes
    2. 2.2 $\phi^{4}$ amplitudes from generalized BCFW recursion relations
  3. 3 Determining the weights of Stokes polytopes
    1. 3.1 Six-point amplitudes
    2. 3.2 Eight-point amplitudes
    3. 3.3 Ten-point amplitudes
  4. 4 Generalization to higher-point amplitudes
    1. 4.1 Overview of the proof
    2. 4.2 $n$-point amplitudes
  5. 5 Discussion

##  1 Introduction

In recent years a lot of work has been done in understanding both the analytic
and geometric structure of scattering amplitudes in various classes of quantum
field theories [1, 2, 3, 4]. On-shell methods such as the BCFW recursion
relations [6, 7, 8, 9, 10] have been not only extremely successful in
simplifying the calculations of an infinite class of amplitudes but also
revealed deep connections between physics and broad areas of mathematics such
as Algebraic Geometry and Combinatorics. In addition, generalizations of the
BCFW recursion relations have been successfully devised and are used to
incorporate the boundary contributions, which correspond to pole at infinity,
in a broad class of QFTs [9, 10].

Furthermore, the seminal work of [1] dubbed as the ‘Amplituhedron’ program has
been greatly successful in developing a geometric formulation of the
$\mathcal{N}=4$ SYM theory. An essential feature in this formulation was that
the scattering amplitudes were considered not as functions of particle
momentum but rather as differential forms on certain auxiliary spaces. It was
shown that there is a relation between these differential forms and certain
positive geometries which completely encapsulates the $\mathcal{N}=4$ SYM
amplitudes. A remarkable feature of this formulation is that it is gauge-
invariant and makes no reference to underlying principles of standard
formulation of QFTs such as locality and unitarity, which are emergent in the
formulation.

In [2, 3] the authors extended this program to scalar $\phi^{3}$ and
$\phi^{4}$ theories. In particular, for the massless planar tree-level
$\phi^{3}$ amplitudes, it was shown in [2] that there is a precise relation
between so-called planar scattering forms on kinematic space and a polytope
known as Associahedron. Further, in [3, 4], the authors developed a similar
formulation for the massless planar $\phi^{4}$ amplitudes at tree-level and in
[5] it was extended to the generalized case of $\phi^{p}$ interactions. It was
shown that the planar $\phi^{4}$ amplitudes can be obtained from the geometry
of an object known as Stokes polytopes. However, it was found that the
calculation of $\phi^{4}$ amplitudes from the geometry of Stokes polytopes
presents a peculiarity. The peculiarity lies in the fact that for a given
number of particles $n$ there does not exist a unique Stokes polytope which
completely determines the $\phi^{4}$ amplitudes; in contrast, there exists a
unique Associahedron for a given $n$ which completely encapsulate the
$\phi^{3}$ amplitudes. Each Stokes polytope contains only partial information
about the complete $\phi^{4}$ amplitudes.

In order to determine the complete $\phi^{4}$ amplitudes, a weighted sum over
all the Stokes polytopes is taken, and in general, these weights are not
equal. The problem is made simpler by the fact that a cyclic permutation of
the labels of only a few Stokes polytopes, referred to as the primitive Stokes
polytopes, determine all the Stokes polytopes for a given dimension $n$. As a
result of this fact, the weights can be parametrized only by the primitive
quadrangulation of Stokes polytopes. The weights can be assigned a unique
numerical value, which makes the sum over Stokes polytopes equal to the
$\phi^{4}$ amplitudes.

In this paper, we address the issue of the undetermined weights of Stokes
polytopes using the generalized BCFW recursion relations. We make use of the
fact that the higher-point amplitudes factorize into lower-point amplitudes at
physical poles. The factorization constrains the weights of the higher-point
amplitudes, calculated by summing up over Stokes polytopes, in terms of the
weights of the lower-point amplitudes. The boundary terms in $\phi^{4}$, which
corresponds to the $\mathcal{O}(z^{0})$-behaviour for large $z$, fix the
numerical values of these weights uniquely.

The paper is organized as follows. In section 2.1, we give an overview of the
calculation of amplitudes from the geometry of Stokes polytopes. We briefly
review the important notions of quadrangulation, $Q$-compatibility and convex
realization of Stokes polytopes. The overview is not exhaustive, and we refer
the interested reader to [2, 3, 4] for complete details. In section 2.2, we
review the generalized BCFW recursion relations for $\phi^{4}$ amplitudes as
given in [10], and discuss the calculation of boundary terms in $\phi^{4}$
amplitudes. In section 3, we use the generalized BCFW recursion relations to
constrain the weights of higher-point Stokes polytopes in terms of lower-point
weights. Firstly, we use the boundary terms to show that the lowest-point
weight $\alpha_{6}$ is fixed uniquely. Then, we calculate the weights of
Stokes polytopes of eight- and ten-point amplitudes and show that these are
determined exactly in terms of the six-point weights. By substituting the
value of $\alpha_{6}$, we determine the numerical value of eight- and ten-
point weights and show that these agree with the results in [3]. In section 4,
we generalize our results and prove that the weights of Stokes polytopes for
any number of particles $n$ can be determined exactly in terms of the six-
point weights. We end the paper with a discussion of our results and a brief
commentary on using other methods to determine the weights.

Before proceeding we want to add that while this manuscript was being
prepared, we came across [15], which uses ‘BCFW’ like recursion relations as
described in [12, 13] to determine the weights of the Stokes polytopes in
$\phi^{4}$ and in general, $\phi^{p}$ theory.

##  2 Amplitudes for massless planar $\phi^{4}$ theory

In this section, we give an overview of the key results of [3], where the
relationship between planar Feynman graphs in $\phi^{4}$ theory and positive
geometries was established. We focus on the construction of planar massless
$\phi^{4}$ amplitudes at tree-level by summing over the Stokes polytopes.
Further, we also review the key results of [10], where the boundary behaviour
of the $\phi^{4}$ amplitudes was analyzed, and the generalized BCFW recursion
relations was presented. Throughout the paper, we have considered the tree-
level massless planar $\phi^{4}$ amplitudes.

###  2.1 $\phi^{4}$ amplitudes from Stokes Polytopes

In [3, 4], the authors extended the seminal work of [1] to show that the
planar amplitudes for massless $\phi^{4}$ theory can be obtained from the
positive geometries of a polytope of dimension $\left(\frac{n-4}{2}\right)$
known as Stokes polytopes.

A positive geometry, for example polygons and polytopes, is a closed geometry
with boundaries or facets of all co-dimensions. There is a unique meromorphic
differential form $\Omega$ that is canonically associated with a positive
geometry whose form is fixed by the requirement of having logarithmic
singularities at the boundaries and the residue at the boundary is equal to
the canonical form of the boundary [11]. These canonical forms link the
positive geometries to the scattering amplitudes.

For the analysis of planar amplitudes, the planar kinematic variables are
used. These variables are defined as

$X_{i,j}=(P_{ij\ldots j-1})^{2}\equiv(p_{i}+p_{i+1}+\cdots+p_{j-1})^{2},\qquad
1\leq i<j\leq n,$ (2.1)

where $p_{i}$ represents the momentum of the $i$-th particle. The Mandelstam
variables can be expressed in terms of planar variables as

$s_{ij}=2p_{i}\cdot p_{j}=X_{i,j+1}+X_{i+1,j}-X_{i,j}-X_{i+1,j+1}~{}.$ (2.2)

For the $\phi^{4}$ amplitudes, the quadrangulations of an $n$-gon, where n is
always even, are considered (figure 2, 3). The total number of ways to
completely quadrangulate an $n=(2I+2)$-gon is equal to the Fuss-Catalan number
$F_{I}=\frac{1}{2I+1}\left({3I\atop I}\right)$. Each quadrangulation of the
$n$-gon is associated with a planar Feynman graph with propagators
$X_{a_{1}},\ldots,X_{a_{\frac{n-4}{2}}}$. Unlike the planar $\phi^{3}$
amplitudes111The massless planar $\phi^{3}$ amplitudes can be obtained from
the canonical form associated to a polytope known as Associahedron [2]., the
$\phi^{4}$ amplitudes cannot be constructed from a unique canonical form
related to a positive geometry. In $\phi^{4}$, for a given number of particles
$n=2I+2$, there is $F_{I}$ number of Stokes polytopes whose weighted sum gives
the full amplitude. These weights are constrained by the factorization of the
amplitudes at the physical poles.

For the construction of Stokes polytopes, it is important to define the notion
of compatibility of a quadrangulation with the reference quadrangulation. This
follows from the notion of compatibility of a diagonal with the reference
quadrangulation and is given in detail in [3, 4]. The vertices of a Stokes
polytopes with reference quadrangulation $Q$ are the quadrangulations which
are compatible with $Q$ (figure 2). A key result from [3] is that the
construction of Stokes polytopes depends on the reference quadrangulation
chosen and different reference quadrangulations give different Stokes
polytopes denoted by $\mathcal{S}_{n}^{Q}$.

For a given quadrangulation $Q$ of an $n$-gon, the $Q$ dependent planar
scattering form is given as

$\Omega^{Q}_{n}=\sum_{\text{Graphs}}(-1)^{\sigma(\text{flip})}d~{}\text{ln}X_{a_{1}}\wedge
d~{}\text{ln}X_{a_{2}}\ldots\wedge d~{}\text{ln}X_{a_{\frac{n-4}{2}}}~{},$
(2.3)

where $\sigma(\text{flip})=\pm 1$. A single quadrangulation does not capture
the contributions from all the $\phi^{4}$ propagators.

With the Stokes polytopes and its respective canonical differential form
defined, the pullback of the canonical form on the polytopes gives a form
proportional to the partial amplitude corresponding to the quadrangulation
$Q$. To define the pullback, a convex realization of Stokes polytopes is
established by imposing a set of constraints222A positive geometry known as
the ABHY associahedra completely encapsulates the $\phi^{3}$ and $\phi^{4}$
amplitudes. The convex realization of Stokes polytopes never enters the ABHY
formalism [4].. This is done by embedding the Stokes polytope
$\mathcal{S}_{n}^{Q}$ of dimension $\left(\frac{n-4}{2}\right)$ inside an
Associahedra $\mathcal{A}_{n}$ with dimension $(n-3)$. This gives a set of
constraints that locate the Stokes polytope in the kinematic space
$\mathcal{K}_{n}$.
For example, in $n=6$ case the constraints corresponding to reference
quadrangulation $Q=(14)$ and $Q$-compatible quadrangulation $Q_{1}=(36)$ are
given by

$\displaystyle s_{ij}$ $\displaystyle=-c_{ij}\qquad\forall~{}1\leq i<j\leq
n-1=5,~{}~{}|i-j|\geq 2$ (2.4a) $\displaystyle X_{1,3}$
$\displaystyle=d_{13},~{}~{}X_{1,5}=d_{15}~{}~{}~{}\text{with}~{}~{}d_{13},d_{15}\geq
2~{}.$ (2.4b)

The constraints in (2.4a) locate the $3$-D associahedron $\mathcal{A}_{6}$
inside $\mathcal{K}_{6}$. The constraints in (2.4b) locates the $1$-D Stokes
polytopes $\mathcal{S}_{6}^{(14)}$ inside $\mathcal{A}_{6}$.
The pullback of (2.3) on the space of $\mathcal{S}_{n}^{Q}$ gives

$\omega_{n}^{Q}=\left(m_{n}^{Q}\right)~{}dX_{a_{1}}\wedge
dX_{a_{2}}\ldots\wedge dX_{a_{\frac{n-4}{2}}}~{},$ (2.5)

where $m_{n}^{Q}$ is the rational canonical function associated with the
Stokes polytope $\mathcal{S}_{n}^{Q}$ and determines partial amplitude. The
weighted sum of these functions over all Stokes polytopes gives the full
planar scattering amplitude.

The computation of $m_{n}^{Q}$ is greatly aided by a curious fact about
$n$-gon, i.e. all quadrangulations of an $n$-gon can be determined by cyclic
permutation $\sigma$ of a subset of quadrangulations. This subset is referred
to as the primitive quadrangulation. What this implies is that, given $n$
number of particles, once the rational canonical functions for a given set of
primitives $\\{Q_{1},\ldots,Q_{J}\\}$ have been calculated, all the other
$m_{n}^{Q}$ ’s can be computed by a cyclical permutation of the labels of
$m_{n}^{Q_{i}}$, where $Q_{i}\in\\{Q_{1},\ldots,Q_{J}\\}$. The
$m_{n}^{Q_{i}}$’s are referred to as primitives. Therefore the master formula
for evaluating the amplitudes is given as

$\widetilde{\mathcal{M}}_{n}=\sum_{\text{Q}|\text{primitives}}\sum_{\sigma}\alpha_{Q}~{}m_{n}^{(\sigma\cdot
Q)}~{},$ (2.6)

where $\alpha_{Q}$ are the weights. These are parametrized only by the
primitive quadrangulations, i.e.

$\alpha_{Q}=\alpha_{\overline{Q}}~{}~{}\forall~{}~{}\overline{Q}=\sigma\cdot
Q$ (2.7)

There is a unique choice of weights $\alpha_{Q}$, which is constrained by the
factorization of amplitude at the physical poles, such that
$\widetilde{\mathcal{M}}_{n}=\mathcal{M}_{n}$, where $\mathcal{M}_{n}$ is the
$\phi^{4}$ amplitude. The weighted sum over the rational canonical function
(2.6)

###  2.2 $\phi^{4}$ amplitudes from generalized BCFW recursion relations

In [10] it was shown that a generalized BCFW recursion relation, which gives a
prescription to compute the boundary contributions, can be written for
$\phi^{4}$ theory.

Consider the following BCFW shifts for $\phi^{4}$ denoted as $\langle i|j]$

$|i\rangle\rightarrow|\hat{i}\rangle=|i\rangle+z|j\rangle,\qquad|j]\rightarrow|\hat{j}]=|j]-z|i].$
(2.8)

Under the BCFW shifts the amplitude $\mathcal{M}_{n}$ becomes a meromorphic
function of $z$ and in general, develops poles at finite $z$ and at
$z=\infty$. There are two categories of Feynman diagram for $\langle i|j]$
deformation. Category (a) is where the particles $i$, $j$ are attached to the
same vertex and category (b) where $i$, $j$ are attached to different vertices
(figure 1).

For category (b) of Feynman diagrams, there is at least one propagator on the
line connecting $i$ and $j$ that depends linearly on $z$. This gives a factor
of $\frac{1}{P^{2}-z\langle j|P|i]}$ in the expression. In the limit,
$z\to\infty$, such factors have a zero contribution, and therefore category
(b) diagrams do not contribute to the boundary terms.

For category (a) diagrams, there is a cancellation of the $z$ terms in the
summation of momenta, and therefore it has no $z$ dependence in the
expression. Consequently, the boundary terms are equal to Feynman diagrams
where particles $i$ and $j$ have a common vertex.

From the above analysis, the generalized on-shell BCFW recursion relations is
given as

$\mathcal{M}_{n}=\mathcal{P}_{n}+\mathcal{B}_{n}~{},$ (2.9)

where $\mathcal{P}_{n}$ denotes the pole part corresponding to category (b)
diagrams and is given as

$\mathcal{P}_{n}=\sum_{\mathcal{I}}{\mathcal{M}_{L}(z_{\mathcal{I}})}~{}\frac{1}{P^{2}_{\mathcal{I}}}~{}{\mathcal{M}_{R}(z_{\mathcal{I}})}~{},$
(2.10)

and $\mathcal{B}_{n}$ denotes the boundary contributions corresponding to
category (a) diagrams and is given as

$\mathcal{B}_{n}=\sum_{\mathcal{I}^{\prime}\cup\mathcal{J}^{\prime}=\\{n\\}\setminus\\{i,j\\}}{\mathcal{M}_{\mathcal{I}^{\prime}}}\frac{1}{P^{2}_{\mathcal{I}^{\prime}}}\frac{1}{P^{2}_{\mathcal{J}^{\prime}}}{\mathcal{M}_{\mathcal{J}^{\prime}}}~{}.$
(2.11)

Figure 1:  (a) Diagrams contributing to boundary term $\mathcal{B}_{n}$. (b)
Diagrams contributing to the pole part $\mathcal{P}_{n}$.

When the particles $i$, $j$ are color ordered and separated by a distance of
more than two, there are no diagrams where they will be attached to the same
vertex, and the boundary terms will vanish. For example, in the $\langle 1|4]$
shift the boundary terms in $\mathcal{M}_{n}$ are absent.

##  3 Determining the weights of Stokes polytopes

In this section, we use the generalized BCFW recursion relations, as discussed
in section 2.2, to constrain the weights of the Stokes polytopes. Firstly we
give the results for six-point amplitudes, calculated in full detail in [3,
4]. Then, we use the six-point amplitudes as input for the calculation of
eight- and ten-point amplitudes. We read the coefficients of the individual
terms appearing in the amplitudes calculated from generalized BCFW and match
them with the coefficients of respective terms in the amplitudes calculated
from the summation over Stokes polytopes. We show that this along with the
six-point weight uniquely determines the weights. We use the notation
$\alpha_{n}^{Q}$ to denote the $n$-point weights corresponding to the
primitive quadrangulation $Q$. Also, we have referred to the summation over
rational canonical function $m_{n}^{Q}$ as in (2.6), as a summation over the
Stokes polytopes $S_{n}^{Q}$ throughout the text.

###  3.1 Six-point amplitudes

There are three possible quadrangulations of the $6$-gon as shown in figure 2.
The weighted sum over the Stokes polytopes $\mathcal{S}_{6}^{Q}$ where
$Q\in\\{(14),(25),(36)\\}$ is given as

$\widetilde{\mathcal{M}}_{6}=\alpha_{6}^{Q(14)}\left(\frac{1}{X_{1,4}}+\frac{1}{X_{3,6}}\right)+\alpha_{6}^{Q(25)}\left(\frac{1}{X_{2,5}}+\frac{1}{X_{1,4}}\right)+\alpha_{6}^{Q(36)}\left(\frac{1}{X_{3,6}}+\frac{1}{X_{2,5}}\right).$
(3.1)

The $n=6$ case has only one primitive333The choice of a primitive is
arbitrary. The only condition is that two members of the set of primitive
quadrangulations must be related unrelated by a cyclic permutation.,
$\left(\frac{1}{X_{1,4}}+\frac{1}{X_{3,6}}\right)$, and its cyclic permutation
gives the canonical rational functions corresponding to other quadrangulations
as can be seen from (3.1). This implies that the three weights are equal and
the six-point amplitude is given as

$\widetilde{\mathcal{M}}_{6}=2\alpha_{6}\left(\frac{1}{X_{1,4}}+\frac{1}{X_{2,5}}+\frac{1}{X_{3,6}}\right)=2\alpha_{6}\left(\frac{1}{P^{2}_{123}}+\frac{1}{P^{2}_{234}}+\frac{1}{P^{2}_{345}}\right),$
(3.2)

where we used the equation (2.2) to express the planar variables $X$ in terms
of $P^{2}$. We also dropped the label for quadrangulation in the weight
$\alpha_{6}$. Note that (3.2) gives the correct $\phi^{4}$ amplitude upon
substituting the correct numerical value for $\alpha_{6}$. This value is
determined in the next section.

Figure 2:  All possible quadrangulations of a hexagon. The solid-line
diagonals represents the reference quadrangulation $Q$ and the dashed-line
represents the $Q$-compatible quadrangulation.

###  3.2 Eight-point amplitudes

To determine the eight-point amplitudes, we use the $\langle 1|2]$-shift and
then apply BCFW recursion relations. The boundary term has three contributions
from the diagrams $(3|45678)$, $(345|678)$ and $(34567|8)$ and is given as

$\begin{split}\widetilde{\mathcal{B}}^{\langle
1|2]}_{8}(1,2,...,8)=&~{}\widetilde{\mathcal{M}}_{4}(1,2,3,-P)\frac{1}{P^{2}_{123}}\widetilde{\mathcal{M}}_{6}(P,4,5,6,7,8)\\\
&+~{}\widetilde{\mathcal{M}}_{4}(8,1,2,-P)\frac{1}{P^{2}_{128}}\widetilde{\mathcal{M}}_{6}(P,3,4,5,6,7)\\\
&+~{}\widetilde{\mathcal{M}}_{4}(3,4,5,-P)\frac{1}{P^{2}_{345}P^{2}_{678}}\widetilde{\mathcal{M}}_{4}(P,6,7,8)~{},\end{split}$
(3.3)

which on substituting (3.2) simplifies as

$\widetilde{\mathcal{B}}^{\langle
1|2]}_{8}=\frac{2\alpha_{6}}{P^{2}_{123}}\left(\frac{1}{P^{2}_{456}}+\frac{1}{P^{2}_{567}}+\frac{1}{P^{2}_{678}}\right)+\frac{2\alpha_{6}}{P^{2}_{128}}\left(\frac{1}{P^{2}_{345}}+\frac{1}{P^{2}_{456}}+\frac{1}{P^{2}_{567}}\right)+\frac{1}{P^{2}_{345}P^{2}_{678}}.$
(3.4)

The pole contribution is from two diagrams and is given as

$\begin{split}\widetilde{\mathcal{P}}^{\langle
1|2]}_{8}(1,2,...,8)=&~{}\widetilde{\mathcal{M}}_{4}(7,8,\widehat{1},-\widehat{P})\frac{1}{P^{2}_{178}}\widetilde{\mathcal{M}}_{6}(\widehat{P},\widehat{2},3,4,5,6)\\\
&+~{}\widetilde{\mathcal{M}}_{6}(5,6,7,8,\widehat{1},-\widehat{P})\frac{1}{P^{2}_{234}}\widetilde{\mathcal{M}}_{4}(\widehat{P},\widehat{2},3,4)\\\
=&~{}\frac{2\alpha_{6}}{P^{2}_{178}}\left(\frac{1}{\widehat{P}^{2}_{\widehat{2}34}}+\frac{1}{P^{2}_{345}}+\frac{1}{P^{2}_{456}}\right)+\frac{2\alpha_{6}}{P^{2}_{234}}\left(\frac{1}{P^{2}_{567}}+\frac{1}{P^{2}_{678}}+\frac{1}{\widehat{P}^{2}_{\widehat{1}78}}\right)~{}.\end{split}$
(3.5)

The above equation can be further simplified by using the following relations

$\displaystyle\frac{1}{P^{2}_{178}\widehat{P}^{2}_{\widehat{2}34}}+\frac{1}{\widehat{P}^{2}_{\widehat{1}78}P^{2}_{234}}=\frac{1}{P^{2}_{178}P^{2}_{234}}\left(\frac{1}{1-\frac{z_{1}}{z_{2}}}+\frac{1}{1-\frac{z_{2}}{z_{1}}}\right)=\frac{1}{P^{2}_{178}P^{2}_{234}}~{},$
(3.6)

where we used the identity

$\displaystyle\sum^{n}_{i=1}\prod^{n}_{j=1,j\neq
i}\frac{1}{1-\frac{z_{i}}{z_{j}}}=1~{},$ (3.7)

and $z_{i}$ and $z_{j}$ are the locations of poles. We get

$\displaystyle\widetilde{\mathcal{P}}^{\langle
1|2]}_{8}=2\alpha_{6}\left(\frac{1}{P^{2}_{178}P^{2}_{234}}+\frac{1}{P^{2}_{178}P^{2}_{345}}+\frac{1}{P^{2}_{178}P^{2}_{456}}+\frac{1}{P^{2}_{234}P^{2}_{567}}+\frac{1}{P^{2}_{234}P^{2}_{678}}\right)~{}.$
(3.8)

Figure 3:  The two primitives of $n=8$ Stokes polytopes corresponding to
quadrangulations $Q=(14,58)$ and $\widetilde{Q}=(14,16)$ respectively.

The complete eight-point amplitude
$\widetilde{\mathcal{M}}_{8}=\widetilde{\mathcal{P}}_{8}+\widetilde{\mathcal{B}}_{8}$
is given as

$\displaystyle\widetilde{\mathcal{M}}_{8}=2\alpha_{6}$
$\displaystyle\left(\frac{1}{P^{2}_{178}P^{2}_{234}}+\frac{1}{P^{2}_{178}P^{2}_{345}}+\frac{1}{P^{2}_{178}P^{2}_{456}}+\frac{1}{P^{2}_{234}P^{2}_{567}}+\frac{1}{P^{2}_{234}P^{2}_{678}}+\frac{1}{P^{2}_{123}P^{2}_{456}}\right.$
(3.9)
$\displaystyle\left.+\frac{1}{P^{2}_{123}P^{2}_{567}}+\frac{1}{P^{2}_{123}P^{2}_{678}}+\frac{1}{P^{2}_{128}P^{2}_{345}}+\frac{1}{P^{2}_{128}P^{2}_{456}}+\frac{1}{P^{2}_{128}P^{2}_{567}}\right)+\frac{1}{P^{2}_{345}P^{2}_{678}}~{}.$

Now, we calculate the eight-point amplitude by summing up over all the Stokes
polytopes $\mathcal{S}^{Q}_{8}$. The $n=8$ case has two primitives and in
total twelve quadrangulations444The complete set of quadrangulations for $n=8$
Stokes polytopes are given as
$\sigma\cdot
Q(14,58)\Rightarrow\mathcal{C}_{1}=\\{(14,58),(25,16),(36,27),(47,38)\\}$ and
$\sigma\cdot\widetilde{Q}(14,16)\Rightarrow\mathcal{C}_{2}=\\{(14,16),(25,27),(36,38),(47,14),(58,25),(16,36),(27,47),(38,58)\\}$
(figure 3), which are the cyclic permutation of labels of the either of the
two primitives. The canonical function corresponding to these primitives are
given as

$\begin{split}m_{8}^{Q}&=\left(\frac{1}{X_{1,4}X_{5,8}}+\frac{1}{X_{3,8}X_{4,7}}+\frac{1}{X_{1,4}X_{4,7}}+\frac{1}{X_{3,8}X_{5,8}}\right)\\\
&=\left(\frac{1}{P^{2}_{123}P^{2}_{567}}+\frac{1}{P^{2}_{128}P^{2}_{456}}+\frac{1}{P^{2}_{123}P^{2}_{456}}+\frac{1}{P^{2}_{128}P^{2}_{567}}\right)\\\\[5.0pt]
m_{8}^{\widetilde{Q}}&=\left(\frac{1}{X_{1,4}X_{1,6}}+\frac{1}{X_{1,4}X_{5,8}}+\frac{1}{X_{3,6}X_{1,6}}+\frac{1}{X_{3,6}X_{3,8}}+\frac{1}{X_{5,8}X_{3,8}}\right)\\\
&=\left(\frac{1}{P^{2}_{123}P^{2}_{678}}+\frac{1}{P^{2}_{123}P^{2}_{567}}+\frac{1}{P^{2}_{345}P^{2}_{678}}+\frac{1}{P^{2}_{345}P^{2}_{128}}+\frac{1}{P^{2}_{567}P^{2}_{128}}\right)~{}.\end{split}$
(3.10)

Taking the weighted sum over all the $\mathcal{S}_{8}^{Q}$ using (2.6) and
comparing each term to respective term in equation (3.9), we get the following
relations

$\begin{split}2\alpha_{8}^{Q}+2\alpha_{8}^{\widetilde{Q}}=2\alpha_{6},\qquad\alpha_{8}^{Q}+4\alpha_{8}^{\widetilde{Q}},=2\alpha_{6}~{}.\end{split}$
(3.11)

Also, the last term in (3.9) gives the relation

$\displaystyle\alpha_{8}^{Q}+4\alpha_{8}^{\widetilde{Q}}=1~{}.$ (3.12)

Using (3.11) and (3.12) we determined the weight of six-point Stokes polytopes
as $\alpha_{6}=\frac{1}{2}$ and the weights for eight-point Stokes polytopes
as

$\alpha_{8}^{Q}=\frac{2\alpha_{6}}{3}=\frac{1}{3},\qquad\alpha_{8}^{\widetilde{Q}}=\frac{\alpha_{6}}{3}=\frac{1}{6}~{},$
(3.13)

where the $Q$ and $\widetilde{Q}$ denote the primitive quadrangulations. Note
that the boundary term in (3.9) uniquely fixes $\alpha_{6}$.

###  3.3 Ten-point amplitudes

To determine the relation between the weights of $n=10$ Stokes polytopes and
$n=6$ weights, firstly we express the $n=8$ amplitude in terms of
$\alpha_{8}^{Q}$ and $\displaystyle{\alpha_{8}^{\widetilde{Q}}}$ i.e. as
weighted sum over $\mathcal{S}_{8}^{Q}$ as 555Note that we are using the
weighted sum of Stokes polytopes as our input in BCFW to determine higher-
point weights in terms of lower-points weights.

$\displaystyle\widetilde{\mathcal{M}}_{8}(1,2,...,8)=$
$\displaystyle~{}(2\alpha_{8}^{Q}+2\alpha_{8}^{\widetilde{Q}})\sum_{\sigma\in\mathbb{Z}_{8}}\left(\frac{1}{P^{2}_{\sigma(1)\sigma(2)\sigma(3)}P^{2}_{\sigma(5)\sigma(6)\sigma(7)}}\right)$
(3.14)
$\displaystyle+(\alpha_{8}^{Q}+4\alpha_{8}^{\widetilde{Q}})\sum_{\sigma\in\mathbb{Z}_{8}}\left(\frac{1}{P^{2}_{\sigma(1)\sigma(2)\sigma(3)}P^{2}_{\sigma(4)\sigma(5)\sigma(6)}}\right).$

Again, we use the $\langle 1|2]$-shift and apply generalized BCFW recursion
relations. The boundary terms has four contributions form the diagrams
$(23|456789(10))$, $(345|6789(10))$,
$(34567|89(10))$ and $(3456789|10)$ and is given as

$\displaystyle\widetilde{\mathcal{B}}^{\langle 1|2]}_{10}(1,2,...,10)=$
$\displaystyle~{}\widetilde{\mathcal{M}}_{4}(1,2,3,-P)\frac{1}{P^{2}_{123}}\widetilde{\mathcal{M}}_{8}(P,4,...10)$
(3.15)
$\displaystyle+\widetilde{\mathcal{M}}_{4}(1,2,-P_{1},-P_{2})\left(\frac{1}{P^{2}_{345}}\widetilde{\mathcal{M}}_{4}(P_{1},3,4,5)\right)\left(\frac{1}{P^{2}_{12345}}\widetilde{\mathcal{M}}_{6}(P_{2},6,...,10)\right)$
$\displaystyle+\widetilde{\mathcal{M}}_{4}(1,2,-P_{1},-P_{2})\left(\frac{1}{P^{2}_{345}}\widetilde{\mathcal{M}}_{6}(P_{1},3,..,7)\right)\left(\frac{1}{P^{2}_{12345}}\widetilde{\mathcal{M}}_{4}(P_{2},8,9,10)\right)$
$\displaystyle+\widetilde{\mathcal{M}}_{4}(10,1,2,-P)\frac{1}{P^{2}_{12(10)}}\widetilde{\mathcal{M}}_{8}(P,3,..,9)~{}.$

Using (3.2) and (3.14) the boundary term is simplified to

$\displaystyle\widetilde{\mathcal{B}}^{\langle 1|2]}_{10}=$
$\displaystyle~{}\frac{1}{P_{123}^{2}}\left[\left(2\alpha_{8}^{Q}+2\alpha_{8}^{\widetilde{Q}}\right)\left(\frac{1}{P_{567}^{2}P_{45678}^{2}}+\frac{1}{P_{678}^{2}P_{56789}^{2}}+\frac{1}{P_{789}^{2}P_{6789(10)}^{2}}+\frac{1}{P_{456}^{2}P_{89(10)}^{2}}\right)\right.$
(3.16)
$\displaystyle\left.+\left(\alpha_{8}^{Q}+4\alpha_{8}^{\widetilde{Q}}\right)\left(\frac{1}{P_{567}^{2}P_{89(10)}^{2}}+\frac{1}{P_{456}^{2}P_{45678}^{2}}+\frac{1}{P_{678}^{2}P_{45678}^{2}}+\frac{1}{P_{567}^{2}P_{56789}^{2}}+\frac{1}{P_{678}^{2}P_{6789(10)}^{2}}\right.\right.$
$\displaystyle\left.\left.+\frac{1}{P_{89(10)}^{2}P_{6789(10)}^{2}}+\frac{1}{P_{456}^{2}P_{789}^{2}}+\frac{1}{P_{56789}^{2}P_{789}^{2}}\right)\right]+\frac{2\alpha_{6}}{P_{345}^{2}P_{12345}^{2}}\left(\frac{1}{P_{789}^{2}}+\frac{1}{P_{89(10)}^{2}}+\frac{1}{P_{678}^{2}}\right)$
$\displaystyle+\frac{2\alpha_{6}}{P_{89(10)}^{2}P_{34567}^{2}}\left(\frac{1}{P_{456}^{2}}+\frac{1}{P_{567}^{2}}+\frac{1}{P_{345}^{2}}\right)+\frac{1}{P_{12(10)}^{2}}\left[\left(2\alpha_{8}^{Q}+2\alpha_{8}^{\widetilde{Q}}\right)\left(\frac{1}{P_{456}^{2}P_{34567}^{2}}+\frac{1}{P_{567}^{2}P_{45678}^{2}}\right.\right.$
$\displaystyle\left.\left.+\frac{1}{P_{678}^{2}P_{56789}^{2}}+\frac{1}{P_{345}^{2}P_{789}^{2}}\right)+\left(\alpha_{8}^{Q}+4\alpha_{8}^{\widetilde{Q}}\right)\left(\frac{1}{P_{456}^{2}P_{789}^{2}}+\frac{1}{P_{345}^{2}P_{34567}^{2}}+\frac{1}{P_{567}^{2}P_{34567}^{2}}\right.\right.$
$\displaystyle\left.\left.+\frac{1}{P_{456}^{2}P_{45678}^{2}}+\frac{1}{P_{567}^{2}P_{56789}^{2}}+\frac{1}{P_{789}^{2}P_{56789}^{2}}+\frac{1}{P_{345}^{2}P_{678}^{2}}+\frac{1}{P_{45678}^{2}P_{678}^{2}}\right)\right].$

The pole part is given as

$\displaystyle\widetilde{\mathcal{P}}^{\langle 1|2]}_{10}(1,2,...,10)=$
$\displaystyle~{}\widetilde{\mathcal{M}}_{4}(9,10,\widehat{1},-\widehat{P})\frac{1}{P^{2}_{19(10)}}\widetilde{\mathcal{M}}_{8}(\widehat{P},\widehat{2},3,...,8)$
(3.17)
$\displaystyle+\widetilde{\mathcal{M}}_{6}(7,8,9,10,\widehat{1},-\widehat{P})\frac{1}{P^{2}_{23456}}\widetilde{\mathcal{M}}_{6}(\widehat{P},\widehat{2},3,4,5,6)$
$\displaystyle+\widetilde{\mathcal{M}}_{8}(5,6,7,8,9,10,\widehat{1},-\widehat{P})\frac{1}{P^{2}_{234}}\widetilde{\mathcal{M}}_{4}(\widehat{P},\widehat{2},3,4)~{}.$

which is expanded as

$\displaystyle\widetilde{\mathcal{P}}^{\langle 1|2]}_{10}=$
$\displaystyle~{}\frac{1}{P_{19(10)}^{2}}\left[\left(2\alpha_{8}^{Q}+2\alpha_{8}^{\widetilde{Q}}\right)\left(\frac{1}{P_{456}^{2}P_{34567}^{2}}+\frac{1}{P_{567}^{2}P_{45678}^{2}}+\frac{1}{P_{345}^{2}P_{\widehat{2}3456}^{2}}+\frac{1}{P_{678}^{2}P_{\widehat{2}34}^{2}}\right)\right.$
(3.18)
$\displaystyle\left.+\left(\alpha_{8}^{Q}+4\alpha_{8}^{\widetilde{Q}}\right)\left(\frac{1}{P_{567}^{2}P_{\widehat{2}34}^{2}}+\frac{1}{P_{345}^{2}P_{34567}^{2}}+\frac{1}{P_{567}^{2}P_{34567}^{2}}+\frac{1}{P_{456}^{2}P_{45678}^{2}}+\frac{1}{P_{456}^{2}P_{\widehat{2}3456}^{2}}\right.\right.$
$\displaystyle\left.\left.+\frac{1}{P_{\widehat{2}34}^{2}P_{\widehat{2}3456}^{2}}+\frac{1}{P_{345}^{2}P_{678}^{2}}+\frac{1}{P_{45678}^{2}P_{678}^{2}}\right)\right]+\frac{4\alpha_{6}^{2}}{P_{23456}^{2}}\left(\frac{1}{P_{89(10)}^{2}}+\frac{1}{P_{9(10)\widehat{1}}^{2}}+\frac{1}{P_{789}^{2}}\right)\times$
$\displaystyle\left(\frac{1}{P_{456}^{2}}+\frac{1}{P_{\widehat{2}34}^{2}}+\frac{1}{P_{345}^{2}}\right)+\frac{1}{P_{234}^{2}}\left[\left(2\alpha_{8}^{Q}+2\alpha_{8}^{\widetilde{Q}}\right)\left(\frac{1}{P_{678}^{2}P_{56789}^{2}}+\frac{1}{P_{789}^{2}P_{6789(10)}^{2}}+\frac{1}{P_{89(10)}^{2}P_{\widehat{1}789(10)}^{2}}\right.\right.$
$\displaystyle\left.\left.+\frac{1}{P_{567}^{2}P_{9(10)\widehat{1}}^{2}}\right)+\left(\alpha_{8}^{Q}+4\alpha_{8}^{\widetilde{Q}}\right)\left(\frac{1}{P_{678}^{2}P_{9(10)\widehat{1}}^{2}}+\frac{1}{P_{567}^{2}P_{56789}^{2}}+\frac{1}{P_{789}^{2}P_{56789}^{2}}+\frac{1}{P_{678}^{2}P_{6789(10)}^{2}}\right.\right.$
$\displaystyle\left.\left.+\frac{1}{P_{789}^{2}P_{\widehat{1}789(10)}^{2}}+\frac{1}{P_{9(10)\widehat{1}}^{2}P_{\widehat{1}789(10)}^{2}}+\frac{1}{P_{567}^{2}P_{89(10)}^{2}}+\frac{1}{P_{6789(10)}^{2}P_{89(10)}^{2}}\right)\right].$

Now, making repeated use of the identity (3.7), the above equation can be
further simplified to

$\displaystyle\widetilde{\mathcal{P}}^{\langle 1|2]}_{10}=$
$\displaystyle~{}\frac{1}{P_{19(10)}^{2}}\left[\left(2\alpha_{8}^{Q}+2\alpha_{8}^{\widetilde{Q}}\right)\left(\frac{1}{P_{456}^{2}P_{34567}^{2}}+\frac{1}{P_{567}^{2}P_{45678}^{2}}+\frac{1}{P_{345}^{2}P_{23456}^{2}}+\frac{1}{P_{678}^{2}P_{234}^{2}}\right)\right.$
(3.19)
$\displaystyle\left.+~{}\left(\alpha_{8}^{Q}+4\alpha_{8}^{\widetilde{Q}}\right)\left(\frac{1}{P_{567}^{2}P_{234}^{2}}+\frac{1}{P_{345}^{2}P_{34567}^{2}}+\frac{1}{P_{567}^{2}P_{34567}^{2}}+\frac{1}{P_{456}^{2}P_{45678}^{2}}+\frac{1}{P_{456}^{2}P_{23456}^{2}}\right.\right.$
$\displaystyle\left.\left.+\frac{1}{P_{234}^{2}P_{23456}^{2}}+\frac{1}{P_{345}^{2}P_{678}^{2}}+\frac{1}{P_{45678}^{2}P_{678}^{2}}\right)\right]+\frac{4\alpha_{6}^{2}}{P_{23456}^{2}}\left(\frac{1}{P_{89(10)}^{2}}+\frac{1}{P_{789}^{2}}\right)\left(\frac{1}{P_{456}^{2}}+\frac{1}{P_{345}^{2}}\right)$
$\displaystyle+\frac{1}{P_{234}^{2}}\left[\left(2\alpha_{8}^{Q}+2\alpha_{8}^{\widetilde{Q}}\right)\left(\frac{1}{P_{678}^{2}P_{56789}^{2}}+\frac{1}{P_{789}^{2}P_{6789(10)}^{2}}+\frac{1}{P_{89(10)}^{2}P_{23456}^{2}}\right)\right.+\left(\alpha_{8}^{Q}+4\alpha_{8}^{\widetilde{Q}}\right)\times$
$\displaystyle\left.\left(\frac{1}{P_{567}^{2}P_{56789}^{2}}+\frac{1}{P_{789}^{2}P_{56789}^{2}}+\frac{1}{P_{678}^{2}P_{6789(10)}^{2}}+\frac{1}{P_{567}^{2}P_{89(10)}^{2}}+\frac{1}{P_{789}^{2}P_{23456}^{2}}+\frac{1}{P_{6789(10)}^{2}P_{89(10)}^{2}}\right)\right],$

where we have used the relations in (3.11), (3.12), and the fact that
$2\alpha_{6}=4\alpha^{2}_{6}$ since $\alpha_{6}=\frac{1}{2}$, in making the
simplifications.

Now, for the $n=10$ case there are seven primitive Stokes polytopes and in
total fifty-five quadrangulations. These correspond to Cube type, Snake type,
Lucas type and Mixed type Stokes polytopes [3]. Taking the weighted sum over
all the Stokes polytope $\mathcal{S}_{10}^{Q}$ and comparing to the amplitude
$\displaystyle{\widetilde{\mathcal{M}}_{10}=\widetilde{\mathcal{P}}^{\langle
1|2]}_{10}+\widetilde{\mathcal{B}}^{\langle 1|2]}_{10}}$ computed by BCFW
recursions, we get the following set of equations that constrain the ten-point
weights as

$\begin{split}4\alpha_{10}^{Q_{b}}+2\alpha_{10}^{Q_{c}}+2\alpha_{10}^{Q_{d}}+2\alpha_{10}^{Q_{e}}+2\alpha_{10}^{Q_{f}}+2\alpha_{10}^{Q_{g}}&=2\alpha_{6}~{},\\\\[4.0pt]
2\alpha_{10}^{Q_{a}}+\alpha_{10}^{Q_{c}}+\alpha_{10}^{Q_{d}}+2\alpha_{10}^{Q_{f}}+2\alpha_{10}^{Q_{g}}&=2\alpha_{8}^{Q}+2\alpha_{8}^{\widetilde{Q}}~{},\\\\[4.0pt]
\alpha_{10}^{Q_{a}}+2\alpha_{10}^{Q_{b}}+\alpha_{10}^{Q_{c}}+2\alpha_{10}^{Q_{e}}+2\alpha_{10}^{Q_{f}}+2\alpha_{10}^{Q_{g}}&=2\alpha_{6}~{},\\\\[4.0pt]
2\alpha_{10}^{Q_{b}}+\alpha_{10}^{Q_{c}}+\alpha_{10}^{Q_{d}}+4\alpha_{10}^{Q_{e}}+2\alpha_{10}^{Q_{f}}+2\alpha_{10}^{Q_{g}}&=2\alpha_{8}^{Q}+2\alpha_{8}^{\widetilde{Q}}~{},\\\\[4.0pt]
\alpha_{10}^{Q_{a}}+4\alpha_{10}^{Q_{b}}+3\alpha_{10}^{Q_{c}}+2\alpha_{10}^{Q_{d}}+2\alpha_{10}^{Q_{e}}+2\alpha_{10}^{Q_{f}}&=2\alpha_{6}~{},\\\\[4.0pt]
\alpha_{10}^{Q_{a}}+2\alpha_{10}^{Q_{b}}+\alpha_{10}^{Q_{d}}+2\alpha_{10}^{Q_{e}}+2\alpha_{10}^{Q_{f}}+2\alpha_{10}^{Q_{g}}&=4\alpha_{6}^{2}~{},\\\\[4.0pt]
\alpha_{10}^{Q_{a}}+4\alpha_{10}^{Q_{b}}+2\alpha_{10}^{Q_{c}}+3\alpha_{10}^{Q_{d}}+2\alpha_{10}^{Q_{e}}+2\alpha_{10}^{Q_{g}}&=\alpha_{8}^{Q}+4\alpha_{8}^{\widetilde{Q}}~{}\end{split}$
(3.20)

where $\\{Q_{a},Q_{b},Q_{c},Q_{d},Q_{e},Q_{f},Q_{g}\\}$ correspond to set of
quadrangulations that form the primitives of $n=10$ Stokes polytopes.

Substituting (3.11) in the above equation and solving for the seven
undetermined $\alpha_{10}$’s in terms of $\alpha_{6}$ we get

$\displaystyle\alpha_{10}^{Q_{a}}$
$\displaystyle=\frac{1}{12}(12\alpha_{6}^{2}-\alpha_{6})=\frac{5}{24},\qquad\alpha_{10}^{Q_{b}}=\frac{1}{12}(12\alpha_{6}^{2}-5\alpha_{6})=\frac{1}{24},$
(3.21) $\displaystyle\alpha_{10}^{Q_{c}}$
$\displaystyle=\frac{1}{12}(19\alpha_{6}-36\alpha_{6}^{2})=\frac{1}{24},\qquad\alpha_{10}^{Q_{d}}=\frac{1}{12}(12\alpha_{6}^{2}-5\alpha_{6})=\frac{1}{24}~{},$
$\displaystyle\alpha_{10}^{Q_{e}}$
$\displaystyle=\frac{\alpha_{6}}{6}=\frac{2}{24}~{},\qquad\qquad\qquad~{}~{}~{}~{}\alpha_{10}^{Q_{f}}=\frac{1}{4}(4\alpha_{6}^{2}-\alpha_{6})=\frac{3}{24}~{},$
$\displaystyle\alpha_{10}^{Q_{g}}$
$\displaystyle=\frac{1}{4}(3\alpha_{6}-4\alpha^{2}_{6})=\frac{3}{24}~{},$

where we substituted $\alpha_{6}=\frac{1}{2}$ in the end. The weights
determined in (3.13) and (3.21) are in perfect agreement with the results in
[3].

##  4 Generalization to higher-point amplitudes

###  4.1 Overview of the proof

It is useful to introduce the following notations. Let
$N=\left(\frac{n-4}{2}\right)$ denote the dimensions of the Stokes polytopes.
For example, six and eight particles correspond to $N=1$ and $N=2$ dimensional
Stokes polytope respectively. Let $\mathcal{F}_{n}$ denote the complete set of
primitive quadrangulations of Stokes polytopes of dimension $N$. Let
$\alpha_{n}^{Q_{n}}$ be the set of weights corresponding to the primitive
quadrangulations $Q_{n}\in\mathcal{F}_{n}$ of the $N$-dimensional Stokes
polytopes $\mathcal{S}_{n}^{Q}$ and let $\widetilde{\alpha}_{6}=2\alpha_{6}$.
Then we would like to prove that if the $\langle i|j]$-shift uniquely fixes
the weights of $\widetilde{\mathcal{M}}_{n-2}$ in terms of
$\widetilde{\alpha}_{6}$, then the weights of the higher-point amplitudes
$\widetilde{\mathcal{M}}_{n}$ is uniquely fixed in terms of
$\widetilde{\alpha}_{6}$ as well. Without any loss of generality, we chose the
$\langle 1|4]$-shift to prove our statement. This is because for the $\langle
1|4]$-shift the boundary terms are absent in color-ordered $\phi^{4}$, and the
amplitude $\widetilde{\mathcal{M}}_{n}$ follows a simple factorization scheme.

For the purpose of the proof, it is important to determine the dependence of
amplitudes on the six-point weight $\widetilde{\alpha}_{6}$. Consider the six-
point amplitude $\widetilde{\mathcal{M}}_{6}$, which is
$\propto\widetilde{\alpha}_{6}$ as can be seen in (3.2). Since we take the
six-point amplitude as the input to recursively construct the higher-point
amplitudes, each product of lower-point amplitudes appearing in the
factorization of the $n$-point amplitude must be proportional to some power of
$\widetilde{\alpha}_{6}$ 666It does not matter which power of
$\widetilde{\alpha}_{6}$ because $\widetilde{\alpha}_{6}$ raised to any power
is equal to one, since $\widetilde{\alpha}_{6}=1$.. For example, the eight-
point amplitude obey a factorization which is schematically given as
$\widetilde{\mathcal{M}}_{8}\sim\widetilde{\mathcal{M}}_{4}\times\widetilde{\mathcal{M}}_{6}$
and is therefore $\propto\widetilde{\alpha}_{6}$. The ten-point amplitude has
the following factorization, schematically given as

$\widetilde{\mathcal{M}}_{10}\sim\widetilde{\mathcal{M}}_{4}\times\widetilde{\mathcal{M}}_{8}+\widetilde{\mathcal{M}}_{6}\times\widetilde{\mathcal{M}}_{6},$
(4.1)

where the first term is $\propto\widetilde{\alpha}_{6}$ and the second terms
is $\propto(\widetilde{\alpha}_{6})^{2}$. Consequently, we have that the
linear combinations of weights $\alpha_{8}^{Q}$ and $\alpha_{10}^{Q}$ that
appear in the sum over all $\mathcal{S}_{8}^{Q}$ and $\mathcal{S}_{10}^{Q}$
respectively, are equal to some power of $\widetilde{\alpha}_{6}$ as in (3.11)
and (3.20). Let us assume that the above statement is true for $(n-2)$-point
amplitude, i.e. each product of lower-point amplitudes appearing in the
factorization of $\widetilde{\mathcal{M}}_{n-2}$ is proportional to some power
of $\widetilde{\alpha}_{6}$. And, the linear combinations of weights that
appear in the summation over all $\mathcal{S}_{n-2}^{Q}$ are equal to some
power of $\widetilde{\alpha}_{6}$.

We prove the above statement for a general $n$-point amplitude in the next
section. The proof follows from the following steps

  * •

We use the $\langle 1|4]$-shift to obtain the correct factorization of the
$n$-point amplitude $\widetilde{\mathcal{M}}_{n}$.

  * •

We show by induction that each term appearing in the factorization of
$\widetilde{\mathcal{M}}_{n}$ can only be proportional to some power of
$\widetilde{\alpha}_{6}$.

  * •

Using the fact that the weights of the primitives are parametrized only by
$\mathcal{F}_{n}$, we show that there are $J$ unique linear combinations of
$J$ number of weights $\alpha_{n}^{Q}$ that appear in the weighted sum over
all Stokes polytopes $\mathcal{S}_{n}^{Q}$, where $J$ is the number of
primitives for an $n$-point amplitude.

  * •

Using the factorization property of $\widetilde{\mathcal{M}}_{n}$, proved by
induction as in the second bullet point above, we show that the $J$ linear
combinations of $\alpha_{n}^{Q}$ are equal to powers of
$\widetilde{\alpha}_{6}$.

  * •

Finally, we use the above to prove our claim that the $\alpha_{n}^{Q}$ can be
uniquely determined in terms of the six-point weight $\widetilde{\alpha}_{6}$
using BCFW factorization.

###  4.2 $n$-point amplitudes

Consider the above statement for dimension $N=2$ and $N=3$ 777The case $N=1$
is the trivial case of six-point amplitude itself.. The statement holds true
for these as can be seen from equations (3.11), (3.13), (3.20), and (3.21).
Let us assume that the statement holds true for dimension $N=(k-1)$, i.e. the
weights of Stokes polytopes $\mathcal{S}^{Q}_{n-2}$ are fixed uniquely in
terms of $\widetilde{\alpha}_{6}$ by the $\langle 1|4]$-shift.

We prove that this statement is true for $N=k$. The amplitude
$\widetilde{\mathcal{M}}_{n}$ obeys the following factorization schematically
given as

$\begin{split}\widetilde{\mathcal{M}}_{n}^{\langle
1|4]}\sim\widetilde{\mathcal{M}}_{4}\times\widetilde{\mathcal{M}}_{n-2}+\widetilde{\mathcal{M}}_{6}\times\widetilde{\mathcal{M}}_{n-4}+\ldots+\widetilde{\mathcal{M}}_{q}\times\widetilde{\mathcal{M}}_{n-q+2}~{},\end{split}$
(4.2)

where

$q=\Bigg{\\{}{~{}\frac{n}{2}\qquad~{}~{}~{}\text{if}~{}\frac{n}{2}~{}\text{is
even}\atop\left(\frac{n}{2}+1\right)~{}~{}\text{if}~{}\frac{n}{2}~{}\text{is
odd .}}~{}$ (4.3)

Every term in (4.2) has an additional $\left(\frac{1}{P^{2}}\right)$ factor
which we have omitted safely for the purpose of the proof.

Substituting (2.6) for each $\widetilde{\mathcal{M}}$ in the factorization in
(4.2) we get

$\begin{split}\widetilde{\mathcal{M}}_{n}^{\langle
1|4]}\sim\sum^{q}_{l=4,6,\ldots}\left(\sum_{Q_{l},\sigma}\alpha_{l}^{Q_{l}}m_{l}^{(\sigma\cdot
Q_{l})}\right)\times\left(\sum_{\widetilde{Q}_{l},\sigma^{\prime}}\alpha_{n-l+2}^{\widetilde{Q}_{l}}m_{n-l+2}^{(\sigma^{\prime}\cdot\widetilde{Q}_{l})}\right)~{},\end{split}$
(4.4)

where $Q_{l}\in\mathcal{F}_{l}$ and $\widetilde{Q}_{l}\in\mathcal{F}_{n-l+2}$
. The $l=4$ case in (4.4) corresponds to $\widetilde{\mathcal{M}}_{4}$ and is
equal to one. The $\alpha_{l}^{Q}$’s are polynomial functions in
$\widetilde{\alpha}_{6}$ by our assumption, and therefore the product
$\alpha^{Q_{l}}_{l}\alpha^{\widetilde{Q}_{l}}_{n-l+2}$ will be polynomial
functions in $\widetilde{\alpha}_{6}$ as well.

The left-hand side of (4.2) is equal to the sum over $\mathcal{S}_{n}^{Q}$ and
is given as

$\begin{split}\widetilde{\mathcal{M}}_{n}=\sum_{Q_{n}}\sum_{\sigma}\alpha_{n}^{Q_{n}}m_{n}^{(\sigma\cdot
Q_{n})}~{},\end{split}$ (4.5)

where $Q_{n}\in\mathcal{F}_{n}$.

Under the $\langle 1|4]$-shift, the amplitude
$\widetilde{\mathcal{M}}^{\langle 1|4]}_{n}$ has two types of terms. Terms of
type (A) do not depend on the shifted variables $|\hat{1}\rangle$ and
$|\hat{4}]$. Terms of type (B) are functions of these shifted variables. For
terms of type (B), repeated use of the identity (3.7) removes the dependence
on $|\hat{1}\rangle$ and $|\hat{4}]$. After explicitly performing the sums in
(4.4) and (4.5), we can choose a term with a particular momentum dependence in
the denominator and match its coefficients in both the equations888This is
done after putting the correct factors of $\frac{1}{P^{2}}$ in (4.4)..
This gives us the relations

$\sum_{Q_{n}}\mathcal{C}_{Q_{n}}\alpha_{n}^{Q_{n}}=\left(\sum_{Q_{l}}\mathcal{C}_{Q_{l}}~{}\alpha_{l}^{Q_{l}}\right)\left(\sum_{\widetilde{Q}_{l}}\mathcal{C}_{\widetilde{Q}_{l}}\alpha_{n-l+2}^{\widetilde{Q}_{l}}\right)~{},$
(4.6)

for some $l\in\\{4,6,...,q\\}$, where $q$ is defined in (4.3) and,
$\mathcal{C}_{Q_{n}}$ are positive constants. The summation is over a subset
of primitive quadrangulations. This is because a given term with a particular
momentum dependence in the denominator appears only in a subset of the
primitives. The coefficients $\mathcal{C}_{Q_{n}}$ count the number of times a
particular term with weight $\alpha_{n}^{Q_{n}}$ appears in the summation over
Stokes polytopes of a given dimension $N$.

Further, for $n$ particles there are $F_{I}$ quadrangulations of the Stokes
polytopes $\displaystyle{\mathcal{S}_{n}^{Q}}$, where $n=(2I+2)$. A subset of
these quadrangulations forms the primitive quadrangulations999A method to
count the number of primitives in a given dimension $n$ $(n=1,2,3)$ is given
in [5].. Since the weights of the Stokes polytopes are parametrized only by
the primitive quadrangulations, there are precisely $J$ number of weights,
where $J$ is the number of primitives and is strictly less than the total
number of quadrangulations $F_{I}$. Thus, equation (4.6) encodes $J$ linear
equations. Each linear equation is a unique linear combination of weights
$\alpha_{n}^{Q}$, where $Q\in\mathcal{F}_{n}$, and is equal to some power of
$\widetilde{\alpha}_{6}$. This follows from our assumption about
$\widetilde{\mathcal{M}}_{n-2}$ that the products of the lower-point
amplitudes appearing in its factorization are equal to some power of
$\widetilde{\alpha}_{6}$, implying that the linear combinations of weights
that appear in the sum over $\mathcal{S}_{n-2}^{Q}$ are equal to some power of
$\widetilde{\alpha}_{6}$. Consequently, each of the sums in the right side of
equation (4.6) is equal to some power of $\widetilde{\alpha}_{6}$ implying
that the sum on the left is equal to a power of $\widetilde{\alpha}_{6}$

The $J$ linear equations encoded in (4.6) can be solved for $J$ weights
$\alpha_{n}^{Q}$ in terms of $\widetilde{\alpha}_{6}$. This determines the
$\alpha_{n}^{Q}$’s exactly, and upon substituting $\widetilde{\alpha}_{6}=1$
fixes their numerical values for which
$\widetilde{\mathcal{M}}_{n}=\mathcal{M}_{n}$, completing our proof. Q.E.D.

##  5 Discussion

The geometric formulation of scattering amplitudes is opening up new ways of
thinking about QFTs. The results have been striking in the supersymmetric
theories such as the $\mathcal{N}=4$ SYM [1], where the geometry of the
polytope referred as ‘Amplituhedron’, completely encapsulates the amplitudes
at all orders. A great understanding of amplitude at tree-level in non-
supersymmetric theories such as the scalar massless planar $\phi^{3}$,
$\phi^{4}$ and in general, $\phi^{p}$ theories has been propelled by the work
in [2, 3, 4, 5]. However, it was shown in [3] that there is no single polytope
structure for a given dimension that completely encapsulates the $\phi^{4}$
amplitudes at tree-level. Rather there is a family of Stokes polytopes, whose
weighted sum gives the complete $\phi^{4}$ amplitude.

In this paper, we addressed the issue of computing the weights. We showed that
the factorization of the $\phi^{4}$ amplitudes at the physical poles put
strong constraints on the weights. We showed that the boundary terms of $n=8$
amplitudes uniquely fixed the value of the lowest-point weight as
$\alpha_{6}=\frac{1}{2}$. Further, we explicitly calculated the weights for
$n=8$ and $n=10$ cases in section 3.2 and 3.3 and showed that the weights can
be solved in terms of the six-point weight $\alpha_{6}$. In section 4, we
generalized our result to higher-point amplitudes. Using mathematical
induction, we proved that the correct factorization of an $n$-point amplitude
fixes the weights $\alpha_{n}^{Q}$ exactly in terms of $\alpha_{6}$.

A key feature of our analysis of the weights relied on the boundary terms of
$\phi^{4}$ amplitudes, which correspond to the $\mathcal{O}(z^{0})$-behaviour
of the amplitudes at large $z$. As we saw that the higher-point weights are
fixed uniquely in terms of the lowest-point weight $\alpha_{6}$, it is crucial
to determine the value of $\alpha_{6}$. Further, the value of the weight
$\alpha_{6}=\frac{1}{2}$ is highly non-trivial, as it is only for this
particular value that the dependence of the shifted variables in the
factorization of a general $n$-point amplitude as in (4.4), can be removed.

A shortcoming of our analysis was that we could not derive an explicit formula
for the higher-point weights. One of the limiting factors was that there does
not exist a general method to count the number of primitive Stokes polytopes
for dimensions greater than three. However, despite this limitation, we could
make a general statement about the weights in our proof in section 4, which is
a powerful result. Also, the computation of $\alpha_{n}^{Q}$ relied on the
correct factorization of the amplitudes at the poles $\widehat{P}=0$. This is
a step back from the Amplituhedron program, where the geometry of the
polytopes is sufficient to determine the amplitudes fully. Further, we believe
that the extension of the ‘BCFW’-type recursion relations for $\phi^{3}$
amplitudes, as presented in [12, 13, 14], to $\phi^{4}$ amplitudes can help to
fix the weights by purely geometrical inputs. A key feature of these recursion
relations is the fact that the $\phi^{3}$ amplitudes factorize at the poles in
correspondence with the geometric factorization of the associahedron
$\mathcal{A}_{n}$. However, the extension to $\phi^{4}$ is not immediately
evident.

Acknowledgements
We are extremely thankful to Alok Laddha for suggesting the problem and for
enormously useful discussions on the topic. We are thankful to Dileep Jatkar
and PB Aneesh for going through the draft and for providing valuable comments.
We would like to thank the Chennai Mathematical Institute, Chennai, for the
hospitality provided during the completion of this project.

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