Celestial Amplitudes, Cluster Adjacency, and Symbol Alphabets

Brown University

Doctoral Thesis

Celestial Amplitudes Cluster Adjacency and

Symbol Alphabets

Author

Anders Oslashhrberg Schreiber

Supervisor

Prof Anastasia Volovich

A dissertaton submitted in partial fulfillment of the requirements for

the degree of Doctor of Philosophy

in the

Department of Physics at Brown University

Providence Rhode Island

February 2021

This dissertation by Anders Oslashhrberg Schreiber is accepted in its present form by

the Department of Physics as satisfying the

dissertation requirement for the degree of

Doctor of Philosophy

Anastasia Volovich Advisor

Recommended to the Graduate Council

Antal Jevicki Reader

Chung-I Tan Reader

Approved by the Graduate Council

Andrew G Campbell

Dean of the Graduate School

ldquoAll we have to decide is what to do with the time that is given to usrdquo

mdash JRR Tolkien The Fellowship of the Ring

BROWN UNIVERSITY

Abstract

Anastasia Volovich

Celestial Amplitudes Cluster Adjacency and Symbol Alphabets

by Anders Oslashhrberg Schreiber

In this thesis we present studies of scattering amplitudes on the celestial sphere at null

infinity (celestial amplitudes) the cluster adjacency structure of scattering amplitudes in

planar maximally supersymmetric Yang-Mills theory (N = 4 SYM) and a method to derive

symbol letters for loop amplitudes in N = 4 SYM

First we show that n-particle celestial gluon tree amplitudes take the form of Aomoto-

Gelfand hypergeometric functions and Gelfand A-hypergeometric functions We then study

conformal properties conformal partial wave decomposition and the optical theorem of

four-particle celestial amplitudes in massless scalar φ3 theory and Yang-Mills theory Sub-

sequently we derive single- and multi-soft theorems for celestial amplitudes in Yang-Mills

theory

Second we provide computational evidence that each rational Yangian invariant inN = 4

SYM has poles that are cluster adjacent (belong to the same cluster in the Gr(4 n) cluster

algebra) through the Sklyanin bracket test We also use this bracket test to study cluster

adjacency of the symbol of one-loop NMHV amplitudes in N = 4 SYM

Finally we suggest an algorithm for computing symbol alphabets from plabic graphs

by solving matrix equations of the form C sdot Z = 0 to associate functions on Gr(mn) to

parameterizations of certain cells in Gr(kn) indexed by plabic graphs For m = 4 and n = 8

we show that this association precisely reproduces the 18 algebraic symbol letters of the

two-loop NMHV eight-particle amplitude from four plabic graphs

Curriculum Vitae

Contact and Date of Birth

Date of birth 30 March 1992Country of Citizenship DenmarkAddress Physics Department Barus and Holley Building

Brown University 182 Hope Street Providence RI 02912Phone +1 401 480 3895Email anders_schreiberbrownedu

Research

Dec 2020 - Dec 2021 Postdoctoral Research Associate at University of OxfordPostdoc at the Mathematical Institute under the grant Scattering Ampli-tudes and the Galois Theory of Periods

Jun 2018 - Dec 2020 Research Assistantship at Brown UniversityResearch assistant working under Prof Anastasia Volovich on mathematicalaspects of scattering amplitudes

Education

Feb 2021 PhD in PhysicsBrown University

Aug 2016 Masterrsquos Degree in Physics at the Niels Bohr InstituteUniversity of Copenhagen

Jan 2015 Bachelorrsquos Degree in Physics at the Niels Bohr InstituteUniversity of Copenhagen

May 2014 Exchange Abroad ProgramUniversity of California Berkeley

Teaching

Sep 2016 - May 2018 Teaching assistant at Brown UniversityTaught introductory labs in Physics 0070 Physics 0040 and problem solvingworkshops in Physics 0070

Sep 2014 - Jun 2016 Teaching assistant at The Niels Bohr Institute CopenhagenTaught labs in Electrodynamics 2 and Quantum Mechanics 1 and taught ex-ercise classes in Statistical Physics and Mathematics for Physicists 1 and 2

Jun 2014 - Aug 2014 Physics Teacher at Herning Gymnasium HerningTaught a high school physics B level class in the High School SupplementaryCourse program Teaching involved lectures experimental work correctingproblem sets and experimental reports and examining students an oral final

List of Publications

This thesis is based on the following publications

Jul 2020 ldquoSymbol Alphabets from Plabic Graphswith Jorge Mago Marcus Spradlin and Anastasia VolovichJHEP 10 128 (2020) [arXiv200700646]

May 2020 ldquoA Note on One-loop Cluster Adjacency in N = 4 SYMwith Jorge Mago Marcus Spradlin and Anastasia VolovichAccepted for publication in JHEP [arXiv200507177]

Jun 2019 ldquoYangian Invariants and Cluster Adjacency in N=4 Yang-Millswith Jorge Mago Marcus Spradlin and Anastasia VolovichJHEP 1910 099 (2019) [arXiv190610682]

Apr 2019 ldquoCelestial Amplitudes Conformal Partial Waves and Soft Limitswith Dhritiman Nandan Anastasia Volovich and Michael ZlotnikovJHEP 1910 018 (2019) [arXiv190410940]

Nov 2017 ldquoTree-level gluon amplitudes on the celestial spherewith Anastasia Volovich and Michael ZlotnikovPhys Lett B 781 349 (2018) [arXiv171108435]

Awards Scholarships and Fellowships

May 2020 Physics Merit Fellowship from Brown University Department of Physics

May 2017 Excellence as a Graduate Teaching Assistant from Brown University Depart-ment of Physics

May 2017 Samuel Miller Research Scholarship from the Sigma Alpha Mu Foundation

Schools and Talks

Sep 2020 Conference talk at the DESY Virtual Theory Forum 2020Plabic Graphs and Symbol Alphabets in N=4 super-Yang-Mills Theory

Jan 2020 GGI Lectures on the Theory of Fundamental Interactions

Jan 2020 HET Seminar at NBICluster Adjacency in N=4 Super Yang-Mills Theory

Jul 2019 Poster at Amplitudes 2019Scattering Amplitudes on the Celestial Sphere

Jun 2019 TASI 2019

Jan 2017 Nordic Winter School on Cosmology and Particle Physics 2017

Additional Skills

Languages Danish English German

Computer Literacy MS Windows MS Office LATEX Python Matlab Mathematica

Acknowledgements

The journey of my PhD has been fantastic I have faced many challenges but a lot

of people have been there to help and guide me through these Firstly I would like to

thank my advisor Anastasia Volovich who has been tremendously helpful in making me

grow as a physicist I am grateful for your patience support and guidance throughout my

graduate studies I would also like to thank the other professors in the high energy theory

group including Stephon Alexander Ji Ji Fan Herb Fried Jim Gates Antal Jevicki Savvas

Koushiappas David Lowe Marcus Spradlin and Chung-I Tan You have all stimulated

a rich and exciting research environment on the fifth floor of Barus and Holley and have

made it a pleasure to work in your group I would like to especially thank Antal Jevicki and

Chung-I Tan for being on my thesis committee Thank you also to the postdocs in the high

energy theory group over the years including Cheng Peng Giulio Salvatori David Ramirez

JJ Stankowicz and Akshay Yelleshpur Srikant I have learned a lot from my discussions

with all of you Finally I would like to thank Idalina Alarcon Barbara Cole Mary Ann

Rotondo Mary Ellen Woycik You have all made my life in the physics department infinitely

easier and I have enjoyed the many conversations we have had

I would now like to thank all the other students in the high energy theory group that I

have had the pleasure to work alongside with during my PhD Thank you all for being good

friends and supporting me on my journey Jatan Buch Atreya Chatterjee Tom Harrington

Yangrui Crystal Hu Leah Jenks Michael Toomey Shing Chau John Leung Luke Lippstreu

Sze Ning Hazel Mak Igor Prlina Lecheng Ren Robert Sims Stefan Stanojevic Kenta

Suzuki Jorge Leonardo Mago Trejo and Peter Tsang

I have spent a large chunk of my free time in the Nelson Fitness Center throughout my

PhD where I have enjoyed training for powerlifting I would like to thank all my fellow

lifters in from the Nelson and in the Brown Barbell Club All of you have lifted me up to

be a better powerlifter

I am so thankful for my lovely girlfriend Nicole Ozdowski Thank you for being there for

me and supporting me every day Big thanks to my parents Per Schreiber Tina Schreiber

my brother Jesper Schreiber my grandparents Lizzie Pedersen Bodil Schreiber and Karl-

Johan Schreiber who have been my biggest supporters from day one

Finally I would like to thank all the people I have not listed here I have met so many

people at Brown over the years and you have all had a positive impact on my life and my

journey towards PhD Thank you all

Contents

Abstract v

Acknowledgements xi

1 Introduction 1

11 Celestial Amplitudes and Holography 3

111 Conformal Primary Wavefunctions 3

112 Celestial Amplitudes 4

12 Cluster Algebras in planar N = 4 super Yang-Mills Theory 6

121 Momentum Twistors and Dual Conformal Symmetry 6

122 Cluster Algebras and Cluster Adjacency 8

13 Symbols Alphabet and Plabic Graphs 10

131 Yangian Invariants and Leading Singularities 11

132 Plabic Graphs and Cluster Algebras 11

2 Tree-level Gluon Amplitudes on the Celestial Sphere 15

21 Gluon amplitudes on the celestial sphere 17

22 n-point MHV 19

221 Integrating out one ωi 19

222 Integrating out momentum conservation δ-functions 20

223 Integrating the remaining ωi 22

224 6-point MHV 24

23 n-point NMHV 25

24 n-point NkMHV 28

25 Generalized hypergeometric functions 31

3 Celestial Amplitudes Conformal Partial Waves and Soft Limits 35

31 Scalar Four-Point Amplitude 37

32 Gluon Four-Point Amplitude 42

33 Soft limits 43

34 Conformal Partial Wave Decomposition 47

35 Inner Product Integral 49

4 Yangian Invariants and Cluster Adjacency in N = 4 Yang-Mills 53

41 Cluster Coordinates and the Sklyanin Poisson Bracket 56

42 An Adjacency Test for Yangian Invariants 58

421 NMHV 60

422 N2MHV 62

423 N3MHV and Higher 63

43 Explicit Matrices for k = 2 64

5 A Note on One-loop Cluster Adjacency in N = 4 SYM 69

51 Cluster Adjacency and the Sklyanin Bracket 70

52 One-loop Amplitudes 73

521 BDS- and BDS-like Subtracted Amplitudes 73

522 NMHV Amplitudes 75

53 Cluster Adjacency of One-Loop NMHV Amplitudes 76

531 The Symbol and Steinmann Cluster Adjacency 76

532 Final Entry and Yangian Invariant Cluster Adjacency 76

54 Cluster Adjacency and Weak Separation 79

55 n-point NMHV Transcendental Functions 82

6 Symbol Alphabets from Plabic Graphs 85

61 A Motivational Example 87

62 Six-Particle Cluster Variables 91

63 Towards Non-Cluster Variables 95

64 Algebraic Eight-Particle Symbol Letters 98

65 Discussion 101

66 Some Six-Particle Details 104

67 Notation for Algebraic Eight-Particle Symbol Letters 105

List of Figures

11 A cluster in the Gr(46) cluster algebra Boxed coordinates are frozen and

do not change under mutations while unboxed coordinates are mutable 9

12 An example of a plabic graph of Gr(26) 12

31 Four-Point Exchange Diagrams 37

51 Weak separation graph indicating that if both i and j are within any of the

green regions (or on the green chords) then ⟨iminus1 i jminus1 j⟩ is cluster adjacent

to ⟨k(l l + 1)(mm + 1)(k minus 1k + 1)⟩ 80

to ⟨k(l l + 1)(mm + 1)(pp + 1)⟩ 81

61 The three types of (reduced perfectly orientable bipartite) plabic graphs

corresponding to km-dimensional cells of Gr(kn)ge0 for k = 2 m = 4 and

n = 6 are shown in (a)ndash(c) The associated input and output clusters (see

text) are shown in (d)ndash(f) and (g)ndash(i) respectively Lines connecting two

frozen nodes are usually omitted but we include in (g)ndash(i) the dotted lines

(having ldquoblack on the rightrdquo in the dual plabic graph) that encode (66) (627)

and (629) (up to signs) 93

List of Tables

Dedicated to my family Tina Per Jesper Lizzie Bodil and Karl-Johan

I love you all

Chapter 1

Introduction

The study of elementary particles and their interactions have led to a paradigm shift in our

understanding of the laws of nature in the past 100 years From early discoveries of charged

particles in cloud chambers to deep probing of the structure of hadrons in high powered

particle accelerators we today have an incredible understanding of how the universe works

through the Standard Model of particle physics The enormous success of the Standard

Model of particle physics is hinged on our ability to calculate scattering cross sections which

we measure in particle scattering experiments like the Large Hadron Collider (LHC) The

computation of scattering cross sections in turn depend on our ability to compute scattering

amplitudes

When we are taught quantum field theory in graduate school we learn the method of

Feynman diagrams [1] to compute scattering amplitudes This method originally revolu-

tionized the way one thinks about scattering in quantum field theories as it gives a neat

way to organize computations via simple diagrams However computations of scattering

amplitudes via Feynman diagrams have rapidly scaling complexity with the number of par-

ticles involved in the scattering process For example if we consider 2-to-n gluon scattering

2 Chapter 1 Introduction

at tree level in Yang-Mills theory the following number of Feynman diagrams need to be

calculated

g + g rarr g + g 4 diagrams

g + g rarr g + g + g 25 diagrams

g + g rarr g + g + g + g 220 diagrams

However amplitudes often enjoy dramatic simplifications once all the diagrams are added

up A classic example of this is the Parke-Taylor formula [2] for maximally helicity violating

(MHV) scattering of any number of particles This reduction in complexity hints at hidden

simplicity and potentially more efficient techniques for computing amplitudes

To understand and develop new computational techniques we need to understand the

analytic structure of amplitudes We therefore study amplitudes in various bases and vari-

ables as this can highlight special properties The choice of basis states of external particles

can make various symmetry properties of amplitudes manifest Certain kinematic variables

offer simplifications like in the Parke-Taylor formula but also highlight deeper properties

of the amplitudes like dual superconformal symmetry [3] and when utilizing momentum

twistors [4] cluster algebraic structure [5] in planar maximally supersymmetric Yang-Mills

theory (N = 4 SYM) becomes apparent

In the next three sections we review the three main topics of this thesis scattering

amplitudes on the celestial sphere at null infinity of flat space cluster adjacency in scattering

amplitudes in N = 4 SYM and the determination of symbol alphabets of loop amplitudes

in N = 4 SYM via plabic graphs

11 Celestial Amplitudes and Holography

In the last 23 years theoretical physics has seen a paradigm shift with the introduction of

the anti-de Sitter spaceconformal field theory (AdSCFT) holographic principle [6] Here

observables of string theories in the bulk of the AdS are dual to observables of CFTs that

live on the boundary of AdS This principle has a strongweak coupling duality where for

example observables in the bulk theory at weak coupling are dual to observables of the

boundary CFT at strong coupling This offers a powerful tool as we can use perturbation

theory at weak coupling to do computations and get results in theories at strong coupling

via the duality In flat Minkowski space a similar connection was observed in [7] as it is

possible to slice Minkowski space in four dimensions into slices of AdS3 where one can apply

the tools of AdSCFT This has recently lead to an application in scattering amplitudes in

flat space [8] where it is possible to map plane-waves to the celestial sphere at null infinity

via conformal primary wavefunctions [9]

111 Conformal Primary Wavefunctions

When we compute scattering amplitudes in flat space the initial and final states are chosen

in the basis of plane-waves eplusmniksdotX (for scalars) The plane-wave basis makes translation

symmetry manifest while other features like boosts are obscured A new basis called

conformal primary wavefunctions was introduced in [9] These wavefunctions connect plane-

wave representations of particle wavefunctions at a point in flat space Xmicro to a point on the

celestial sphere at null infinity (z z) (in stereographic coordinates) For a massless scalar

particle the conformal primary wavefunction takes the form of a Mellin transform

φ∆plusmn(X z z) = intinfin

0dω ω∆minus1eplusmniωqsdotX (11)

where ∆ is a free parameter that will take the role of conformal dimension By requiring φ to

form an orthonormal basis with respect to the Klein-Gordon inner product ∆ is restricted to

the principal series ∆ = 1+iλ In the above formula we have parameterized the momentum

associated with the massless scalar as

kmicro = ωqmicro(z z) = ω(1 + zz z + zminusi(z minus z)1 minus zz) (12)

where qmicro is a null vector In four dimensions Lorentz transformations act as two-dimensional

conformal transformations on the celestial sphere [10] and under Lorentz transformations

(11) transforms as

φ∆plusmn (ΛmicroνXν az + bcz + d

az + bcz + d

) = ∣cz + d∣2∆φ∆plusmn(X z z) (13)

which is exactly how scalar conformal primaries transform The formula (11) extends to

massless spinning particles of integer spin given by a Mellin transform of the associated

polarization vector and plane-wave [9]

112 Celestial Amplitudes

Given a scattering amplitudes we can change the basis to conformal primary wavefunctions

by applying a Mellin transform to each external particle involved in the scattering process

This defines the celestial amplitude [9]

AJ1⋯Jn(∆j zj zj) =n

prodj=1int

0dωj ω

∆jminus1j A`1⋯`n (14)

where `j is helicity of particle j and Jj is the spin of the associated conformal primary

wavefunction given by Jj = `j Note that the scattering amplitude A here includes the

overall momentum conservation delta function The celestial amplitude transforms as a

conformal correlator under SL(2C) Lorentz transformations

AJ1⋯Jn (∆j az + bcz + d

az + bcz + d

prodj=1

[(czj + d)∆j+Jj(cz + d)∆jminusJj ] AJ1⋯Jn(∆j zj zj) (15)

Due to the conformal correlator nature of celestial amplitudes it is possible that there exists

a conformal field theory on the celestial sphere that generates scattering amplitudes in the

form of celestial amplitudes In Chapter 2 we will explore how to compute n-point celestial

gluon amplitudes

In Chapter 3 we will explore conformal properties of four-point massless scalar celestial

amplitudes conformal partial wave decomposition and optical theorem For four-point

celestial gluon amplitudes we compute the conformal partial wave decomposition and study

single- and multi-soft theorems

12 Cluster Algebras in planar N = 4 super Yang-Mills Theory

Theories with a large amount of symmetry often see fruitful developments from studying

them in terms of different kinematic variables We will study N = 4 SYM which enjoys su-

perconformal symmetry in spacetime in addition to dual superconformal symmetry in dual

momentum space [3] When kinematics are parameterized in terms of momentum twistors

[4] n-points on P3 dual conformal symmetry enhances the kinematic space to the Grassman-

nian Gr(4 n) [5] This space has a cluster algebraic structure which strongly constrains the

analytic structure of amplitudes in the theory At tree-level amplitudes in N = 4 SYM are

rational functions depending on dual superconformally invariant combinations of momen-

tum twistors called Yangian invariants [11] At loop-level trancendental functions appear

which in the cases of our interest can be described by iterated integrals called generalized

polylogarithms These have a total differential given by a product of d logrsquos which can be

mapped to a tensor product structure called the symbol [12] The structure of both Yangian

invariants and symbols is constrained by cluster adjacency which we will describe below

Cluster adjacency has been used to perform computations of high loop amplitudes in the

cluster bootstrap program [13]

121 Momentum Twistors and Dual Conformal Symmetry

Dual conformal symmetry [3] in N = 4 SYM was discovered by studying scattering ampli-

tudes in dual momentum space We start with scattering amplitudes described by momenta

kmicroi of massless particles We define dual momenta xmicroi as

kmicroi = xmicroi minus x

microi+1 (16)

where the index i labels particles i isin 1 n in an ordered fashion Let us now define a

second set of coordinates called momentum twistors [4] We can define these through inci-

dence relations Since we are considering massless particles the definition of dual momenta

combined with the spinor-helicity formalism (see [14] for a review) allows us to write (16)

⟨i∣axaai = ⟨i∣axaai+1 equiv [microi∣a (17)

We can pair the momentum twistor components [microi∣a with the spinor-helicity angle bracket

to form a joint spinor that we will collectively refer to as a momentum twistor

ZIi = (∣i⟩a [microi∣a) (18)

where I = (a a) is an SU(22) index As the momentum twistor is defined from two points in

dual momentum space this definition maps any two null separated points in dual momentum

space to a point in momentum twistor space With a bit of algebra we can write point in

dual momentum in terms of the momentum twistor variables

xaai = ∣i⟩a[microiminus1∣a minus ∣i minus 1⟩a[microi∣a⟨i minus 1 i⟩ (19)

Due to the construction of the momentum twistor variables via (17) all coordinates in

the momentum twistor ZIi scales uniformly under little group transformations Thus for

n-particle scattering the kinematic space is n-points on P3 also known as twistor space

[15 16] Furthermore dual conformal transformations act as GL(4) transformations on

momentum twistors thus enhancing the momentum twistors from living in P3 to Gr(4 n)

Dual conformal generators act linearly on functions of momentum twistors and we can

construct a dual conformally invariant quantity from the SU(22) Levi-Civita symbol

⟨ijkl⟩ = εIJKLZIi ZJj ZKk ZLl (110)

which will be the central objects that we construct scattering amplitudes from

122 Cluster Algebras and Cluster Adjacency

Cluster algebras [17 18 19 20] can be represented by quivers with cluster coordinates (each

quiver corresponding to a single cluster) equipped with a mutation rule Starting with an

initial cluster we can mutate on individual cluster coordinates and obtain different clusters

As an example consider a cluster in the Gr(46) cluster algebra Figure 11 Here we have

frozen coordinates (in boxes) that we are not allowed to mutate and non-frozen coordinates

(unboxed) that we can mutate on The mutation rule is defined by an adjacency matrix

bij = ( arrows irarr j) minus ( arrows j rarr i) (111)

〈2345〉

〈2346〉〈2356〉〈2456〉〈3456〉

〈1234〉〈1236〉〈1256〉〈1456〉

Figure 11 A cluster in the Gr(46) cluster algebra Boxed coordinates are frozen anddo not change under mutations while unboxed coordinates are mutable

such that when we mutate on a cluster coordinate ak we obtain a new coordinate aprimek given

akaprimek = prod

i∣bikgt0

abiki + prodi∣biklt0

aminusbiki (112)

To complete the mutation we flip all arrows in the quiver connected to aprimek This way we can

generate all clusters in the cluster algebra if it is of finite type We say that a cluster algebra

is of infinite type if it contains an infinite number of clusters Gr(4 n) cluster algebras [21]

are of finite type when n = 67 and of infinite type when n ge 8

The notion of cluster adjacency plays an important role in the analytic structure of

scattering amplitudes Two cluster coordinates are said to be cluster adjacent if and only

they can be found in a common cluster together As an example from Figure 11 we see

that ⟨2346⟩ ⟨2356⟩ ⟨2456⟩ are all cluster adjacent In Chapter 4 we study how cluster

adjacency constrains the pole structure Yangian invariants in N = 4 SYM In Chapter 5 we

explore how cluster adjacency constrains the symbol in one-loop NMHV amplitudes

13 Symbols Alphabet and Plabic Graphs

An outstanding problem in the computation of scattering amplitudes of N = 4 SYM is

the determination of symbol alphabets of amplitudes When amplitudes are computed say

via the cluster bootstrap method the symbol alphabet is an important input but it is only

known in certain cases either via cluster algebras [5] or direct computation [22 23 24] From

cluster algebras we are limited to cases where the cluster algebra is of finite type (n = 67)

Is there an alternative way to predict the symbol alphabet of amplitudes in N = 4 SYM

One approach is using Landau analysis [25 26] but here we will discuss a separate approach

involving plabic graphs that index Grassmannian cells Formulas involving integrals over

Grassmannian spaces are commonplace in N = 4 SYM [27 28] Yangian invariants and

leading singularities are computed as integrals over Grassmannian cells indexed by plabic

graphs [29 30] These integral formulas are localized on solutions to matrix equations of the

form C sdotZ = 0 where C is a ktimesn matrix representation of the auxiliary Grassmannian space

Gr(kn) and Z is the collection of 4 times n momentum twistors As these equations together

with the integral formulas determine the structure of Yangian invariants and leading sin-

gularities it is interesting to ask if we can derive complete symbol alphabets of amplitudes

by collecting coordinates appearing in the solutions to C sdotZ = 0

131 Yangian Invariants and Leading Singularities

We can represent Yangian invariants in N = 4 SYM as integrals over an auxiliary Grass-

mannian space [27 28]

Y (Z ∣η) = int4k

prodi=1

d log fi4

prodI=1

prodα=1

suma=1

Cαa(Z ∣η)aI) (113)

where fi are variables parameterizing the k times n matrix C The integration is localized on

solutions to the matrix equations Cαa(Z ∣η)aI equiv C sdot Z = 0 for a = 1 n I = 1 4 and

α = 1 k Here k corresponds to the level of helicity violation of an NkMHV amplitude

For a n we can consider the finite set of all Gr(kn) cells each with an associated matrix

C such that they exactly localize the integration (113) Thus for each Gr(kn) cell there is

a corresponding Yangian invariant where variables appearing in the Yangian invariant are

dictated by the solutions to C sdotZ = 0

132 Plabic Graphs and Cluster Algebras

Cells of Gr(kn) Grassmannians can be indexed by decorated permutations [29] ie per-

mutations σ of length n with σ(a) if a lt σ(a) and σ(a)+n if σ(a) lt a Furthermore k refers

to the number of entries in a permutation with σ(a) lt a Such decorated permutations can

be represented by plabic graphs - planar bicolored graphs [29]

Example Consider the plabic graph in Figure 12 which has an associated decorated

permutation 345678 To read off the permutation we start at any external point

move through the graph turn to the first left path if we meet a white vertex while we turn

to the first right path if we meet a black vertex

Figure 12 An example of a plabic graph of Gr(26)

We can read off the C-matrix parameterizing the associated cell in Gr(kn) from the

plabic graph We start with a matrix that has the identity in the columns corresponding to

sources in the plabic graph Each entry in the remaining columns is given by the formula

cij = (minus1)s sump∶i↦j

prodαisinp

fα (114)

where s is the number of sources strictly between i and j the sum runs over all allowed

paths p from i to j (allowed paths must traverse each edge only in the direction of its arrow)

and the product runs over all faces α to the right of the path p denoted by p On top of

this the face variables fi over-count the degrees of freedom in a plabic graph by one and

satisfy the relation

fi = 1 (115)

With the construction (114) we will study solutions to the matrix equations C sdotZ = 0

In Chapter 6 we will see how this method can be used to generate all Gr(4 n) cluster

coordinates when n = 67 (which are known to be the n = 67 symbols alphabets) but also

algebraic coordinates that are known to appear in scattering amplitudes but are not cluster

coordinates

Chapter 2

Tree-level Gluon Amplitudes on the

Celestial Sphere

This chapter is based on the publication [31]

The holographic description of bulk physics in terms of a theory living on the boundary

has been concretely realised by the AdSCFT correspondence for spacetimes with global

negative curvature It remains an important outstanding problem to understand suitable

formulations of holography for flat spacetime a goal that has elicited a considerable amount

of work from several complementary approaches [32]

Recently Pasterski Shao and Strominger [8] studied the scattering of particles in four-

dimensional Minkowski space and formulated a prescription that maps these amplitudes to

the celestial sphere at infinity The Lorentz symmetry of four-dimensional Minkowski space

acts as the conformal group SL(2C) on the celestial sphere It has been shown explicitly

that the near-extremal three-point amplitude in massive cubic scalar field theory has the

correct structure to be identified as a three-point correlation function of a conformal field

16 Chapter 2 Tree-level Gluon Amplitudes on the Celestial Sphere

theory living on the celestial sphere [8] The factorization singularities of more general scat-

tering amplitudes in this CFT perspective have been further studied in [33] The map uses

conformal primary wave functions which have been constructed for various fields in arbitrary

dimensions in [9] In [34] it was shown that the change of basis from plane waves to the

conformal primary wave functions is implemented by a Mellin transform which was com-

puted explicitly for three and four-point tree-level gluon amplitudes The optical theorem

in the conformal basis and scattering in three dimensions were studied in [35] One-loop

and two-loop four-point amplitudes have also been considered in [36]

In this note we use the prescription [34] to investigate the structure of CFT correlators

corresponding to arbitrary n-point gluon tree-level scattering amplitudes thus generaliz-

ing their three- and four-point MHV results Gluon amplitudes can be represented in many

different ways that exhibit different complementary aspects of their rich mathematical struc-

ture It is natural to suspect that they may also take a particularly interesting form when

written as correlators on the celestial sphere We find that Mellin transforms of n-point

MHV gluon amplitudes are given by Aomoto-Gelfand generalized hypergeometric functions

on the Grassmannian Gr(4 n) (224) For non-MHV amplitudes the analytic structure of

the resulting functions is more complicated and they are given by Gelfand A-hypergeometric

functions (233) and its generalizations It will be very interesting to explore further the

structure of these functions and possibly make connections to other representations of tree-

level amplitudes [37] which we leave for future work

21 Gluon amplitudes on the celestial sphere

We work with tree-level n-point scattering amplitudes of massless particlesA`1⋯`n(kmicroj ) which

are functions of external momenta kmicroj and helicities `j = plusmn1 where j = 1 n We want

to map these scattering amplitudes to the celestial sphere To that end we can parametrize

the massless external momenta kmicroj as

kmicroj = εjωjqmicroj equiv εjωj(1 + ∣zj ∣2 zj + zj minusi(zj minus zj)1 minus ∣zj ∣2) (21)

where zj zj are the usual complex cordinates on the celestial sphere εj encodes a particle

as incoming (εj = minus1) or outgoing (εj = +1) and ωj is the angular frequency associated with

the energy of the particle [34] Therefore the amplitude A`1⋯`n(ωj zj zj) is a function of

ωj zj and zj under the parametrization (21)

Usually we write any massless scattering amplitude in terms of spinor-helicity angle-

and square-brackets representing Weyl-spinors (see [14] for a review) The spinor-helicity

variables are related to external momenta kmicroj so that in turn we can express them in terms

of variables on the celestial sphere via [34]

[ij] = 2radicωiωj zij ⟨ij⟩ = minus2εiεj

radicωiωjzij (22)

where zij = zi minus zj and zij = zi minus zj

In [9 34] it was proposed that any massless scattering amplitude is mapped to the

celestial sphere via a Mellin transform

AJ1⋯Jn(λj zj zj) =n

prodj=1int

0dωj ω

iλjj A`1⋯`n(ωj zj zj) (23)

The Mellin transform maps a plane wave solution for a helicity `j field in momentum space

to a corresponding conformal primary wave function on the boundary with spin Jj where

helicity `j and spin Jj are mapped onto each other and the operator dimension takes values

in the principal continuous series representation ∆j = 1+iλj [9] Therefore AJ1⋯Jn(λj zj zj)

has the structure of a conformal correlator on the celestial sphere where the symmetry group

of diffeomorphisms is the conformal group SL(2C)

Explicitly under conformal transformations we have the following behavior

ωj rarr ωprimej = ∣czj + d∣2ωj zj rarr zprimej =azj + bczj + d

zj rarr zprimej =azj + bczj + d

where a b c d isin C and ad minus bc = 1 The transformation for zj zj is familiar from the

usual action of SL(2C) on the complex coordinates on a sphere Concerning ωj recall

that qmicroj transforms as qmicroj rarr ∣czj + d∣minus2Λmicroνqνj [9] where Λmicroν is a Lorentz transformation in

Minkowski space corresponding to the celestial sphere conformal transformation Thus ωj

must transform as in (24) to ensure that kmicroj transforms as a Lorentz vector kmicroj rarr Λmicroνkνj

The conformal covariance of AJ1⋯Jn(λj zj zj) on the celestial sphere demands

AJ1⋯Jn (λj azj + bczj + d

azj + bczj + d

prodj=1

[(czj + d)∆j+Jj(czj + d)∆jminusJj ] AJ1⋯Jn(λj zj zj) (25)

22 n-point MHV 19

as expected for a correlator of operators with weights ∆j and spins Jj

22 n-point MHV

The cases of 3- and 4-point gluon amplitudes have been considered in [34] Here we will

map n ge 5-point MHV gluon amplitudes to the celestial sphere

221 Integrating out one ωi

Starting from (23) we can anchor the integration to one of our variables ωi by making a

change of variables for all l ne i

ωl rarrωisiωl (26)

where si is a constant factor that cancels the conformal scaling of ωi in (24) so that the

ratio ωi

siis conformally invariant One choice which is always possible in Minkowski signature

si =∣ziminus1 i+1∣

∣ziminus1 i∣ ∣zi i+1∣ (27)

Since gluon scattering amplitudes scale homogeneously under uniform rescalings col-

lecting all the factors in front we have

AJ1⋯Jn(λj zj zj) = intinfin

dωiωi

(ωisi

j=1 iλj

s1+iλii

⎛⎜⎝

proda=1anei

intinfin

0dωa ω

⎞⎟⎠A`1⋯`n(si ωl zj zj)

where we used that the scaling power of dressed gluon amplitudes is An(Λωi)rarr ΛminusnAn(ωi)

We recognize that the integral over ωi is the Mellin transform of 1 which is given by

intinfin

dωiωi

(ωisi

= 2πδ(z) (29)

With this we simplify the transformation prescription (23) to

AJ1⋯Jn(λj zj zj) = 2πδ⎛⎝n

sumj=1

λj⎞⎠s1+iλii

⎛⎜⎝

proda=1anei

intinfin

0dωa ω

⎞⎟⎠A`1⋯`n(si ωl zj zj) (210)

222 Integrating out momentum conservation δ-functions

For simplicity we choose the anchor variable above to be ω1 and use ωnminus3 ωn to localize

the momentum conservation δ-functions in the amplitude These δ-functions can then be

equivalently rewritten as follows compensating the transformation by a Jacobian

δ4(ε1s1q1 +n

sumi=2

εiωiqi) =4

prodj=nminus3

sjδ (ωj minus ωlowastj )1gt0(ωlowastj ) (211)

where ωlowastj are solutions to the initial set of linear equations

ω⋆j = minussj (U1j

U+nminus4

sumi=2

U) (212)

The Uij and U are minor determinants by Cramerrsquos rule

Uij = det(Mnminus3jrarrin) U = det(Mnminus3n) (213)

22 n-point MHV 21

where j rarr i means that index j is replaced by index i Mabcd denotes the 4 times 4 matrix

Mabcd = (pa pb pc pd) (214)

For the purpose of determinant calculation the column vectors pmicroi = εisiqmicroi can be written

in a manifestly conformally invariant form

pmicro1(z z) = ε1(100minus1) pmicro2(z z) = ε2(1001) pmicro3(z z) = ε3(2200)

pmicroi (z z) = εi1

∣ui∣(1 + ∣ui∣2 ui + uiminusi(ui minus ui)1 minus ∣ui∣2) for i = 45 n

in terms of conformal invariant cross-ratios

ui =z31zi2z32zi1

and ui =z31zi2z32zi1

for i = 45 n (216)

but if and only if we also specify the explicit choice

s1 =∣z32∣

∣z31∣ ∣z12∣ s2 =

∣z31∣∣z32∣ ∣z21∣

and si =∣z12∣

∣z1i∣ ∣zi2∣for i = 3 n (217)

The indicator functions prodni=nminus3 1gt0(ωlowasti ) appear due to the integration range in all ω being

along the positive real line such that the δ-functions can only be localized in this region

Furthermore in order for all the remaining integration variables ωj with j = 2 n minus 4

to be defined on the whole integration range the indicator functions prodni=nminus3 1gt0(ωlowasti ) have

to demand Uij

U lt 0 for all i = 1 n minus 4 and j = n minus 3 n so that we can write them as

prodij 1lt0(Uij

223 Integrating the remaining ωi

In this section we apply (210) to the usual n-point MHV Parke-Taylor amplitude [2] in

spinor-helicity formalism for n ge 5 rewritten via (327)

Aminusminus++(s1 ωj zj zj) =z3

12s1ω2δ4(ε1s1q1 +sumni=2 εiωiqi)

(minus2)nminus4z23z34zn1ω3ω4ωn (218)

Making use of the solutions (211) and performing four of the integrations in (210) we have

Aminusminus++(λi zi zi) = 2πδ(sumnj=1 λj)z3

12 siλ1+21

(minus2)nminus4Uz23z34zn1

nminus4

proda=2int

0dωa ω

ω2prodnb=nminus3 sbωlowastbiλnminus3

ω3ω4ωlowastnprodij

1lt0(Uij

For convenience we transform the remaining integration variables as

ωi = siU1n

uiminus1

1 minussumnminus5j=1 uj

i = 23 n minus 4 (220)

which leads to

Aminusminus++(λi zi zi) simz3

12siλ1+21 siλ2+2

2 siλ33 siλnn

z23z34zn1U1nδ(

sumj=1

λj) ϕ(α x)prodij

1lt0(Uij

U) (221)

Note that the overall factor in (221) accounts for proper transformation weight of the

resulting correlator under conformal transformations (25)

22 n-point MHV 23

Here we recognize a hypergeometric function ϕ(α x) of type (n minus 4 n) as defined in

section 381 of [38] and described in appendix 25 In particular here we have

ϕ(α x) equivintu1ge0unminus5ge01minussuma uage0

prodj=1

Pj(u)αjdϕ dϕ = dP2

P2and and dPnminus4

Pnminus4

Pj(u) =x0j + x1ju1 + + xnminus5 junminus5 1 le j le n

The parameters in (222) corresponding to (221) read1

α1 =1 α2 = 2 + iλ2 α3 = iλ3 αnminus4 = iλnminus4 αnminus3 = iλnminus3 minus 1 αnminus1 = iλnminus1 minus 1

αn =1 + iλ1 x0 i =U1i

U1n xjminus1 i =

Ujnminus U1i

U1n x0n = minus

U1n xjminus1n =

U1n x01 = 1 xjminus1 j = minus

for i = n minus 3 n minus 2 n minus 1 and j = 23 n minus 4 and all other xab = 0

These kinds of functions are also known as Aomoto-Gelfand hypergeometric functions

on the Grassmannian Gr(n minus 4 n)

Making use of eq (324) and (325) from [38] we can write down a dual representation

of the same function which yields a hypergeometric function of type (4 n)

ϕ(α x) equivc2

c1intu1ge0u3ge0

1minussuma uage0

prodj=1

Pj(u)αjdϕ dϕ = dPnminus3

Pnminus3and and dPnminus1

Pnminus1

Pj(u) =x0j + x1ju1 + x2ju2 + x3ju3 1 le j le n

1For n = 5 the normally different cases α2 = 2+iλ2 and αnminus3 = iλnminus3minus1 are reduced to a single α2 = 1+iλ2In this case there also are no integrations so that the result becomes a simple product of factors

In this case the parameters of (224) corresponding to (221) read

α1 =1 α2 = minus2 minus iλ2 α3 = minusiλ3 αnminus4 = minusiλnminus4 αnminus3 = 1 minus iλnminus3 αnminus1 = 1 minus iλnminus1

αn = minus iλn x0j =Ujn

U1n xij =

Ujnminus4+i

U1nminus4+iminus UjnU1n

x0n = minusU

U1n xin =

U1n x01 = 1

x1nminus3 =minusUU1nminus3

Γ(2 + iλ1)Γ(2 + iλ2)prodnminus4j=3 Γ(iλj)

Γ(1 minus iλ1)prod3i=1 Γ(1 minus iλnminusi)

for i = 123 and j = 23 n minus 4 and all other xab = 0

The hypergeometric functions ϕ(α x) form a basis of solutions to a Pfaffian form

equation which defines a Gauss-Manin connection as described in section 38 of [38] This

Pfaffian form equation can be interpreted as a generalized Knizhnik-Zamolodchikov equation

satisfied by our correlators [40 39] Similar generalized hypergeometric functions appeared

in [41] in the context of N = 4 Yang-Mills scattering amplitudes and the deformed Grass-

mannian

224 6-point MHV

In the special case of six gluons there is only one integral in (222) such that the function

reduces to the simpler case of Lauricella function ϕD

ϕD(α x) =( minusUU26

)iλ1+1

( minusUU16

)iλ2+2

iλ3minus1

iλ4minus1

iλ5minus1

times int1

0dt tαminus1(1 minus t)γminusαminus1

prodi=1

(1 minus xit)minusβi (226)

23 n-point NMHV 25

with parameters and arguments given by

α = 2 + iλ2 γ = 4 + iλ1 + iλ2 βi = 1 minus iλi+2 xi = 1 minus U1i+2U26

U16U2i+2for i = 123 (227)

Note that x0j arguments have been factored out of the integrand to achieve this form

23 n-point NMHV

In this section we will map the n-point NMHV split helicity amplitude Aminusminusminus++⋯+ to the

celestial sphere via (210) The spinor-helicity expression for Aminusminusminus++⋯+ can be found eg in

Aminusminusminus++⋯+ =1

nminus1

sumj=4

⟨1∣P2jPj+12∣3⟩3

P 22jP

⟨j + 1 j⟩[2∣P2j ∣j + 1⟩⟨j∣Pj+12∣2]

equivnminus1

sumj=4

Mj (228)

where Fij equiv ⟨i i + 1⟩⟨i + 1 i + 2⟩⋯⟨j minus 1 j⟩ and Pxy equiv sumyk=x ∣k⟩[k∣ where x lt y cyclically

We will work with M4 for the purpose of our calculations Using momentum conser-

vation and writing M4 in terms of spinor-helicity variables we find

⟨34⟩⟨45⟩⋯⟨n minus 1 n⟩⟨n1⟩(⟨12⟩[24]⟨43⟩ + ⟨13⟩[34]⟨43⟩)3

(⟨23⟩[23] + ⟨24⟩[24] + ⟨34⟩[34])⟨34⟩[34]times

times ⟨54⟩([23]⟨35⟩ + [24]⟨45⟩)(⟨43⟩[32]) (229)

Writing this in terms of celestial sphere variables via (327) we find

M4 =ω1ω4(ε2z12z24ω2+ε3z13z34ω3)3

2nminus4z56z67⋯znminus1nzn1z23z34prodnj=2jne4 ωj

(ε3z35z23ω3 + ε4z45z24ω4) (ε2ω2 (ε3∣z23∣2ω3 + ε4∣z24∣2ω4) + ε3ε4∣z34∣2ω3ω4) (230)

The following map of the above formula to the celestial sphere will only be strictly valid for

n ge 8 We will comment on changes at 6- and 7-points in the next section We use the map

(210) anchor the calculation about ω1 make use of solutions (211) and perform a change

of variables

ωi = siuiminus1

i = 2 n minus 4 (231)

to find the resulting term in the n-point NMHV correlator

M4 sim δ⎛⎝n

sumj=1

λj⎞⎠

prodni=1 siλii

z12z23z13z45z56⋯znminus1nz4n

z12z13z45z4ns21s

z34zn1UF(αx)prod

1lt0(Uij

U) (232)

with the function F(αx) being a Gelfand A-hypergeometric function as defined in Appendix

25 In this case it explicitly reads

F(α x) = int u1ge0unminus5ge01minusu1minus⋯minusunminus5ge0

nminus5

proda=1

nminus5

prodj=1

uiλj+1

j u23(u1u2x10 + u1u3x20 + u2u3x30)minus1

times7

prodi=1

(x0i + u1x1i +⋯ + unminus5xnminus5i)αi

where parameters are given by

α1 = 3 α2 = minus1 α3 = iλ1 + 1 α4 = iλnminus3 minus 1 α5 = iλnminus2 minus 1 α6 = iλnminus1 minus 1 α7 = iλn minus 1

23 n-point NMHV 27

and function arguments are given by

x10 = ε2ε3∣z23∣2s2s3 x20 = ε2ε4∣z24∣2s2s4 x30 = ε3ε4∣z34∣2s3s4

x11 = ε2z12z24s2 x21 = ε3z13z34s3 x22 = ε3z35z23s3 x32 = ε4z45z24s4

x03 = 1 xj3 = minus1 j = 1 n minus 5 x04 =U1nminus3

U xj4 =

Ujnminus3 minusU1nminus3

U j = 1 n minus 5

x05 =U1nminus2

U xj5 =

U j = 1 n minus 5 (235)

x06 =U1nminus1

U xj6 =

U j = 1 n minus 5

x07 =U1n

U xj7 =

Ujn minusU1n

U j = 1 n minus 5

Note that the first fraction in (232) accounts for the correct transformaton weight of the

correlator under conformal tranformation (25)

6- and 7-point NMHV

In the cases of 6- and 7-point the results in the previous section change somewhat due

to the presence of ω3 and ω4 in the denominator of (230) These variables are fixed by

momentum conservation δ-functions in the lower point cases such that the parameters and

function arguments of the resulting Gelfand A-hypergeometric functions change

For the 6-point case we find that the resulting correlator part M4 is proportional to

a Gelfand A-hypergeometric function as defined in Appendix 25

F(α x) = int u1ge01minusu1ge0

u1uiλ2

1 (x00 + u1x10 + u21x20)minus1(1 minus u1)iλ1+1

prodi=2

(x0i + u1x1i)αi (236)

α2 = iλ3 minus 1 α3 = iλ4 + 1 α4 = iλ5 minus 1 α5 = iλ6 minus 1 α6 = 3 α7 = minus1 (237)

and function arguments xij depend on εi zi zi and Uij Performing a partial fraction de-

composition on the quadratic denominator in (236) we can reduce the result to a sum of

two Lauricella functions

In the 7-point case we find that the resulting correlator part M4 is proportional to a

Gelfand A-hypergeometric function as defined in Appendix 25

F(α x) = int u1ge0u2ge01minusu1minusu2ge0

u2uiλ2

1 uiλ32 (u1x10 + u2x20 + u1u2x30 + u2

1x40 + u22x50)minus1

times7

prodi=1

(x0i + u1x1i + u2x2i)αi

α1 = iλ1 + 1 α2 = iλ4 + 1 α3 = iλ5 minus 1 α4 = iλ6 minus 1 α5 = iλ7 minus 1 α6 = 3 α7 = minus1 (239)

and function arguments xij again depend on εi zi zi and Uij

24 n-point NkMHV

In this section we discuss the schematic structure of NkMHV amplitudes with higher k under

the Mellin transform (210)

24 n-point NkMHV 29

N2MHV amplitude

In the 8-point N2MHV split helicity case Aminusminusminusminus++++ we consider one of the six terms of

the amplitude found in eg [42] on page 6 as an example

F41F23

⟨1∣P26P72P35P63∣4⟩3

P 226P

⟨76⟩[23]⟨65⟩[2∣P26∣7⟩⟨6∣P72∣2][3∣P35∣6⟩⟨5∣P63∣3]

where Fij is the complex conjugate of Fij Performing the same sequence of steps as in the

previous sections we find a resulting Gelfand A-hypergeometric function of the form

F(α x) = intu1ge0u2ge0u3ge01minusu1minusu2minusu3ge0

u3uα1

1 uα22 uα3

prodi=4

(x0i + u1x1i + u2x2i + u3x3i)αi

times17

prodj=14

(x0j + u1x1j + u2x2j + u3x3j + u1u2x4j + u1u3x5j + u2u3x6j + u21x7j + u2

2x8j + u23x9j)αj

for some parameters αi where P4 is a degree four polynomial in ui and function arguments

xij again depend on εi zi zi and Uij

NkMHV amplitude

More generally a split helicity NkMHV amplitude Aminus⋯minus+⋯+ involves a sum over the terms

described in eq (31) (32) of [42] Terms corresponding in complexity to M4 discussed

in the previous section are always present with constant Laurent polynomial powers at any

k However for higher k the most complicated contributing summands result in hypergeo-

metric integrals schematically given by

F(α x) =int u1unminus4ge01minusu2minus⋯minusunminus4ge0

nminus4

prodl=2

dululuαl

⎛⎝

1 minusnminus4

sumj=2

uj⎞⎠

P32k (prod

(P i1)αi)

⎛⎝prodj

(Pj2)αj

⎞⎠

where αi are parameters and Pd is a degree d polynomial in ua Here we explicitly see an

increase in power of the Laurent polynomials with increasing k in NkMHV The examples

above feature the Gelfand A-hypergeometric function F The increase in Laurent polyno-

mial degree is traced back to the presence of Mandelstam invariants P 2ij for degree two

polynomials as well as the factors ⟨a∣PijPklPrt∣b⟩ for higher degree polynomials The

length of chains of the Pij depends on n and k such that multivariate Laurent polynomials

of any positive degree are present at sufficiently high n k

Similar generalized hypergeometric functions or equivalently generalized Euler integrals

are found in the case of string scattering amplitudes [43 44] It will be interesting to explore

this connection further

25 Generalized hypergeometric functions

The Aomoto-Gelfand hypergeometric functions of type (n + 1m + 1) relevant in this work

can be defined as in section 351 of [38]

ϕ(α x) equivintu1ge0unge01minussuma uage0

prodj=0

Pj(u)αjdϕ (243)

dϕ =dPj1Pj1

and and dPjnPjn

0 le j1 lt lt jn lem (244)

Pj(u) =x0j + x1ju1 + + xnjun 1 le j lem (245)

where here the parameters αi collectively describe all the powers for the factors in the

integrand When all αi are zero the function reduces to the Aomoto polylogarithm

The arguments xij of the hypergeometric function of type (m+ 1 n+ 1) in (245) can be

arranged in a matrix

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

x00 x0m

x10 x1m

⋮ ⋱ ⋮

xn0 xnm

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Each column in this matrix defines a hyperplane in Cn that appears in the hypergeometric

integral as (x0j +sumni=1 xijui)αi Furthermore (n + 1) times (n + 1) minor determinants of the

matrix can be regarded as Pluumlcker coordinates on the Grassmannian Gr(n + 1m + 1) over

the space of arguments xij

Sometimes it is convenient to transform the argument arrangement (246) to the following

gauge fixed form

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

1 0 0 1 1 1

0 1 0 minus1 minusx11 minusx1mminusnminus1

⋮ ⋱ minus1 ⋮ ⋮ ⋮

0 0 1 minus1 minusxn1 minusxnmminusnminus1

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

In this case the hypergeometric function can then be written in the following two equivalent

ways eq (324) of [38]

F ((αi) (βj) γx) =c1intu1ge0unge01minussuma uage0

prodi=1

uαiminus1i sdot (1 minus

suml=1

ul)γminussumi αiminus1mminusnminus1

prodj=1

(1 minusn

sumi=1

xijui)minusβj

c1 =Γ(γ)Γ(γ minusn

sumi=1

αi) sdotn

prodi=1

Γ(αi) (248)

and the dual representation in eq (325) of [38]

F ((αi) (βj) γx) =c2intu1ge0umminusnminus1ge01minussuma uage0

dmminusnminus1umminusnminus1

prodi=1

uβiminus1i sdot (1 minus

mminusnminus1

suml=1

ul)γminussumi βiminus1n

prodj=1

(1 minusmminusnminus1

sumi=1

xjiui)minusαj

c2 =Γ(γ)Γ(γ minusmminusnminus1

sumi=1

βi) sdotmminusnminus1

prodi=1

Γ(βi) (249)

where the parameters are assumed to satisfy the conditions

αi notin Z 1 le i le n βj notin Z 1 le j lem minus n minus 1

γ minusn

sumi=1

αi notin Z γ minusmminusnminus1

sumj=1

βj notin Z(250)

The hypergeometric functions (243) comprise a basis of solutions to the defining set of

differential equations

sumi=0

xijpartϕ

partxij= αjϕ 0 le j lem

sumj=0

xijpartϕ

partxij= minus(1 + αi)ϕ 0 le i le n (251)

(3) part2ϕ

partxijpartxpq= part2ϕ

partxiqpartxpj 0 le i p le n 0 le j q lem

In cases where factors of the integrand are non-linear in the integration variables the

functions can be generalized further to Gelfand A-hypergeometric functions [45 46] defined

F(α x) = intu1ge0ukge01minussuma uage0

Pi(u1 uk)αiuα11 uαk

k du1duk (252)

where αi are complex parameters and Pi now are Laurent polynomials in u1 uk

Chapter 3

Celestial Amplitudes Conformal

Partial Waves and Soft Limits

Pasterski Shao and Strominger (PSS) have proposed a map between S-matrix elements

in four-dimensional Minkowski spacetime and correlation functions in two-dimensional con-

formal field theory (CFT) living on the celestial sphere [8 34] Celestial CFT is interesting

both for understanding the long elusive holographic description of flat spacetime [48] as well

as for exploring the mathematical structures of amplitudes In recent years many remarkable

properties of amplitudes have been uncovered via twistor space momentum twistor space

scattering equations etc(see [49] for review) hence it is quite plausible that exploring prop-

erties of celestial amplitudes may also lead to new insights

A key idea behind the PSS proposal was to transform the plane wave basis to a manifestly

conformally covariant basis called the conformal primary wavefunction basis This basis

was constructed explicitly by Pasterski and Shao [9] for particles of various spins in diverse

dimensions The celestial sphere is the null infinity of four-dimensional Minkowski spacetime

36 Chapter 3 Celestial Amplitudes Conformal Partial Waves and Soft Limits

The double cover of the four-dimensional Lorentz group is identified with the SL(2C)

conformal group of the celestial sphere Two-dimensional correlators on the celestial sphere

will be referred to as celestial amplitudes from here on

The celestial amplitudes of massless particles are given by Mellin transforms of the

corresponding four-dimensional amplitudes

An(zj zj) = intinfin

prodl=1

dωl ω∆lminus1l An(kl) (31)

where ∆l = 1 + iλl with λl isin R [9] are conformal dimensions taking values in the principal

continuous series in order to ensure the orthogonality and completeness of the conformal

primary wavefunction basis Further details are given below

In the spirit of recent developments in understanding scattering amplitudes from the on-

shell perspective by studying symmetries analytic properties and unitarity many recent

studies have delved into similar aspects of celestial amplitudes The structure of factorization

of singularities of celestial amplitudes was investigated in [33] three- and four-point gluon

amplitudes were computed in [34] and arbitrary tree-level ones in [31] Celestial four-point

string amplitudes have been discussed in [50] Unitarity via the manifestation of the optical

theorem on celestial amplitudes has been observed recently [36 35] and the generators of

Poincareacute and conformal groups in the celestial representation were constructed in [51]

This paper is organized as follows In section 31 we compute massless scalar four-point

celestial amplitudes and study its properties such as conformal partial wave decomposition

crossing relations and optical theorem In section 32 we derive conformal partial wave

decomposition for four-point gluon celestial amplitude and in section 33 single and double

Figure 31 Four-Point Exchange Diagrams

soft limits for all gluon celestial amplitudes The conformal partial wave decomposition

formalism is summarized in appendix 34 and details about inner product integrals required

in the main text are evaluated in appendix 35

Note added During this work we became aware of related work by Pate Raclariu and

Strominger [52] which has some overlap with section 4 of our paper

31 Scalar Four-Point Amplitude

In this section we study a tree level four-point amplitude of massless scalars mediated by

exchange of a massive scalar depicted on Figure 311

The corresponding celestial amplitude (31) is

A4(zj zj) = g2intinfin

prodj=1

dωj ω∆jminus1j δ(4) (

sumi=1

ki)( 1

(k1+k2)2+m2+ 1

(k1+k3)2+m2+ 1

(k1+k4)2+m2)

where zj zj are coordinates on the celestial sphere and ωj are the energies Defining εj = minus1

(+1) for incoming (outgoing) particles we can parameterize the momenta kmicroj as

kmicroj = εjωj (1 + ∣zj ∣2 zj + zj izj minus izj 1 minus ∣zj ∣2) (33)

1The same amplitude in three dimensions was studied in [35]

Under conformal transformations by construction [9] the four-point celestial amplitude

behaves as a four-point CFT correlation function of operators with conformal weights

(hj hj) =1

2(∆j + Jj ∆j minus Jj) (34)

where Jj are spins We can split the four-point celestial amplitude into a conformally

invariant function of only the cross-ratios A4(z z) and a universal prefactor

A4(zj zj) =( z24

z14)h12 ( z14

z13)h34

zh1+h212 zh3+h4

z14)h12 ( z14

z13)h34

zh1+h212 zh3+h4

A4(z z) (35)

where we define hij = hi minus hj hij = hi minus hj and cross-ratios

z = z12z34

z13z24 z = z12z34

z13z24with zij = zi minus zj zij = zi minus zj (36)

Letrsquos fix the external points in (32) as z1 = 0 z2 = z z3 = 1 z4 = 1τ with τ rarr 0 and

compute

A4(z) equiv ∣z∣∆1+∆2 limτrarr0

τminus2∆4A4(0 z11τ) (37)

We will consider the case where particles 1 and 2 are incoming while 3 and 4 are outgoing

so ε1 = ε2 = minusε3 = minusε4 = minus1 and denote it as 12harr 34 The s-channel diagram on figure 31 is

A12harr344s (z) sim g2∣z∣∆1+∆2 lim

τrarr0τminus2∆4 int

prodi=1

dωi ω∆iminus1i δ(4)

⎛⎝

sumj=1

kj⎞⎠

m2 minus 4ω1ω2∣z∣2 (38)

The momentum conservation delta functions can be rewritten as

δ(4)⎛⎝

sumj=1

kj⎞⎠= 4τ2

ω1δ(iz minus iz)

prodi=2

δ(ωi minus ωlowasti ) (39)

ωlowast2 = ω1

z minus 1 ωlowast3 = zω1

z minus 1 ωlowast4 = zω1τ

2 (310)

The delta function only has solutions when all the ωlowasti are positive so z gt 1

Then (38) reduces to a single integral

A12harr344s (z) sim g2δ(iz minus iz)z∆1+∆2 lim

τrarr0τ2minus2∆4 int

0dω1ω

∆1minus21

prodi=2

(ωlowasti )∆iminus1 1

m2 minus 4z2

zminus1ω21

= g2 (im2)2αminus2

sin(πα) δ(iz minus iz) z2 (z minus 1)h12minush34 (311)

Adding the s- t- and u-channel contributions we obtain our final result

A12harr344 (z) sim g2 (m2)2αminus2

sin(πα) δ(iz minus iz) z2 (z minus 1)h12minush34 (eπiα + ( z

z minus 1)α

+ zα) (312)

sumi=1

hi minus 2 (313)

Let us discuss some properties of this expression

First it is straightforward to verify that the Poincareacute generators on the celestial sphere

constructed in [51]

L1i = (1 minus z2i )partzi minus 2zihi

P0i = (1 + ∣zi∣2)e(parthi+parthi)2

P2i = minusi(zi minus zi)e(parthi+parthi)2

L2i = (1 + z2i )partzi + 2zihi L3i = 2(zipartzi + hi)

P1i = (zi + zi)e(parthi+parthi)2

P3i = (1 minus ∣zi∣2)e(parthi+parthi)2

annihilate the celestial amplitude on the support of the delta function δ(iz minus iz)

Second we can show that A4 satisfies the crossing relations

A13harr244 (1 minus z) = (1 minus z

2(h2+h3)A13harr24

4 (z) 0 lt z lt 1 (315)

as well as

A13harr244 (z) = z2(h1+h4)A12harr34

4 (1z)

= (1 minus z)2(h12minush34)A14harr234 ( z

z minus 1) 0 lt z lt 1 (316)

The relations (315) and (316) generalize similar relations in [35]

Third the conformal partial wave decomposition of s-channel celestial amplitude

(311)2 is computed in the appendix 34 35 and takes the following form

A12harr344s (z) sim g

2 (im2)2αminus2

2 sin(πα) intC

Γ (1minus∆2 minush12)Γ (∆

2 minush12)Γ (1minus∆2 minush34)Γ (∆

2 minush34)Γ(1 minus∆)Γ(∆ minus 1) Ψ∆

hi(z z)

2The other two channels can be obtained in similar manner

where Ψ∆hi(z z) is given in (345) restricted to the internal scalar case with J = 0 and the

contour C runs from 1 minus iinfin to 1 + iinfin

The gamma functions in (317) unambiguously specify all pole sequences in conformal

dimensions Closing the contour to the right or left of the complex axis in ∆ we find simple

poles at ∆ and their shadows at ∆ given by

2= 1 minus h12 + n

2= 1 minus h34 + n

2= h12 minus n

2= h34 minus n (318)

with n = 0123

Finally letrsquos explicitly check the celestial optical theorem derived by Shao and Lam in

[35] which relates the imaginary part of the four-point celestial amplitude to the product

of two three-point celestial amplitudes with the appropriate integration measure Taking

imaginary part of (317) we obtain

Im [A12harr344s (z)] sim int

Cd∆micro(∆)C(h1 h2 ∆)C(h3 h4 2 minus∆)Ψ∆

hi(z z) (319)

up to some overall constants independent of hi Here C(hi hj ∆) is the coefficient of the

three-point function given by [35]

C(hi hj ∆) = g (m2)hi+hjminus2

4hi+hj

Γ (hij + ∆2)Γ (∆

2 minus hij)Γ(∆) (320)

micro(∆) is the integration measure

micro(∆) = Γ(∆)Γ(2 minus∆)4π3Γ(∆ minus 1)Γ(1 minus∆) (321)

and Ψ∆hi(z z) is

Ψ∆hi(z z) equiv

Γ (1 minus ∆2 minus h12)Γ (∆

2 minus h34)Γ (∆

2 + h12)Γ (1 minus ∆2 + h34)

Ψ∆hi(z z) (322)

32 Gluon Four-Point Amplitude

In this section we study the massless four-point gluon celestial amplitude which has been

computed in [34] and is given by

A12harr34minusminus++ (z) sim δ(iz minus iz)∣z∣3∣1 minus z∣h12minush34minus1 z gt 1 (323)

where the conformal ratios z z are defined in (36)

Evaluating the integral in appendix 35 we find the conformal partial wave expansion is

given by the following simple result3

A12harr34minusminus++ (z) sim 2i

infinsumJ=0

4π2Ψhh

π (1 minus 2h)(2h minus 1 minus 2J)(h34minush12) sin(π(h12minush34))

(Γ(hminush12)Γ(1+Jminush34minush)Γ(h+h12)Γ(1+J+h34minush)

+(h12 harr h34))

where sumprime means that the J = 0 term contributes with weight 12

There is no truncation of the spins J in this case so primary operators of all integer

spins contribute to the OPE expansion of the external gluon operators in contrast with the

previously considered scalar case3When considering J lt 0 take hharr h in the expansion coefficient

33 Soft limits 43

Poles ∆ and shadow poles ∆ are located at

∆ minus J2

= 1 minus h12 + n ∆ minus J

2= 1 minus h34 + n

∆ + J2

= h12 minus n ∆ + J

2= h34 minus n

with n = 0123 These poles are integer spaced as expected

33 Soft limits

Single soft limits

In this section we study the analog of soft limits for celestial amplitudes The universal

soft behavior of color-ordered gluon scattering amplitudes corresponding to ωk rarr 0 is

well-known [53] and takes the form

limωkrarr0

A`k=+1n = ⟨k minus 1k + 1⟩

⟨k minus 1k⟩⟨k k + 1⟩Anminus1

limωkrarr0

A`k=minus1n = [k minus 1k + 1]

[k minus 1k][k k + 1]Anminus1

where `k is the helicity of particle k

The spinor-helicity variables are related to the celestial sphere variables via [34]

radicωiωjzij (327)

Conformal primary wavefunctions become soft (pure gauge) when ∆k rarr 1 (or λk rarr 0) [9 54]

In this limit we can utilize the delta function representation4

δ(x) = 1

2limλrarr0

iλ ∣x∣iλminus1 (328)

such that (31) becomes

limλkrarr0

An(zj zj) =1

prodj=1jnek

intinfin

0dωj ω

iλjj int

0dωk 2 δ(ωk)ωkAn(ωj zj zj) (329)

We see that the λk rarr 0 limit localizes the integral at ωk = 0 and we obtain

limλkrarr0

AJk=+1n = 1

zkminus1k+1

zkminus1kzk k+1Anminus1 (330)

limλkrarr0

AJk=minus1n = 1

zkminus1k+1

An alternative derivation of these relations was given in [55]

Double soft limits

For consecutive soft limits one can apply (330) or (331) multiple times and the con-

secutive soft factors are simply products of single soft factors4See httpmathworldwolframcomDeltaFunctionhtml

33 Soft limits 45

For simultaneous double soft limits energies of particles are simultaneously scaled by δ

so ωk rarr δωk and ωl rarr δωl with δ rarr 0 which for example yields [56 57]

limδrarr0An(δω1 δω2 ωj zk zk) =

⟨n∣1 + 2∣3] ( [13]3⟨n3⟩[12][23]s123

+ ⟨n2⟩3[n3]⟨n1⟩⟨12⟩sn12

)Anminus2(ωj zj zj)

for `1 = +1 `2 = minus1 j = 3 n and k = 1 n Here sijl = (ki + kj + kl)2 More generally

we will write

limδrarr0An(δωk δωl ωj zi zi) = DS(k`k l`l)Anminus2(ωj zj zj) (333)

where DS(k`k l`l) is the simultaneous double soft factor

For celestial amplitudes the analog of the simultaneous double soft limit is to take two

λrsquos scale them by ε λk rarr ελk and λl rarr ελl and take the ε rarr 0 limit To implement this

practically in (31) we change variables for the associated ωrsquos

ωk = r cos(θ) ωl = r sin(θ) 0 le r ltinfin 0 le θ le π2 (334)

The mapping (31) becomes

An(zj zj) =n

prodj=1jnekl

intinfin

0dωj ω

iλjj int

0dr int

0dθ r(iλk+iλl)εminus1

times (cos(θ))iλkε(sin(θ))iλlεr2An(ωj zj zj)

We can use (328) to obtain a delta function in r which enforces the simultaneous double

soft limit for the scattering amplitude as in (332) The result is

limεrarr0An(λkε λlε) = DS(kJk lJl)Anminus2 (336)

where DS(kJk lJl) is the simultaneous double soft factor on the celestial sphere

DS(kJk lJl) = 1

(iλk + iλl)ε[2int

0dθ (cos(θ))iλkε(sin(θ))iλlε [r2DS(k`k l`l)]

r=0]εrarr0

As an example consider the simultaneous double soft factor in (332) We can use (327) to

translate it into celestial sphere coordinates and plug into (337) to obtain

DS(1+12minus1) sim 1

2(iλ1 + iλ2)ε21

zn1z23( 1

zn3z2n

z12z2n+ 1

z3nz31

z12z31) (338)

Explicitly let us check (336) by considering the six-point NMHV split helicity amplitude

A+++minusminusminus = δ(4) (6

sumi=1

4ω1⋯ω6

times⎡⎢⎢⎢⎢⎢⎣

ω21ω

24(ω3z34z13minusω2z24z12)3

(ω3ω4z34z34minusω2ω4z24z24minusω2ω3z23z23)

z23z34z56z61 (ω4z24z54 minus ω3z23z35)+

ω23ω

26(ω4z46z34+ω5z56z35)3

(ω3ω4z34z34+ω3ω5z35z35+ω4ω5z45z45)

z12z16z34z45 (ω3z23z35 + ω4z24z45)

⎤⎥⎥⎥⎥⎥⎦

and map it via (31) Taking the simultaneous double soft limit of particles 3 and 4 as

prescribed in (336) we find

limεrarr0A+++minusminusminus(λ3ε λ4ε) =

2(iλ3 + iλ4)ε21

z23z45( 1

z25z41

z34z42+ 1

z52z53

z34z53) A++minusminus (340)

where the four-point correlator is given by mapping the appropriate MHV amplitude via

A++minusminus = 4iδ(λ1 + λ2 + λ5 + λ6)z3

56 δ(izprime minus izprime)z12z2

25z216z25z61

(z15z61

z25z26)iλ2minus1

(z12z16

z25z56)iλ5+1

(z15z12

z56z26)iλ6+1

where zprime = z12z56

z25z61and zprime = z12z56

z25z61 The conformal soft factor found in (340) matches our

general result by taking the double soft factor [56 57]

⟨2∣3 + 4∣5] ( [35]3⟨25⟩[34][45]s345

+ ⟨24⟩3[25]⟨23⟩⟨34⟩s234

) (342)

and mapping it via (337)

It is straightforward to generalize (336) to m particles taken simultaneously soft by

introducing m-dimensional spherical coordinates as in (334) and scale m λrsquos by ε

34 Conformal Partial Wave Decomposition

In the CFT four-point function defined as (35) we can expand the conformally invariant

part A4(z z) on the basis of conformal partial waves Ψhh

hihi(z z) As can be shown along

the lines of [58 60 59] the expansion takes the following form

A4(z z) = iinfinsumJ=0

intCd∆ Ψhh

hihi(z z)(1 minus 2h)(2h minus 1)

(2π)2⟨A4(z z)Ψhh

hihi(z z)⟩ (343)

where h minus h = J h + h = ∆ = 1 + iλ The contour C runs from 1 minus iinfin to 1 + iinfin The

integration and summation is over all dimensions and spins of exchanged primary operators

in the theory sumprime means that the J = 0 summand contributes with a weight of 12 The

inner product is defined by

⟨G(z z) F (z z)⟩ equiv intdzdz

(zz)2G(z z)F (z z) (344)

The conformal partial waves Ψhh

hihi(z z) have been computed in [61 62 63] and are

given by

hihi(z z) =cprime1F+(z z) + cprime2Fminus(z z) (345)

F+(z z) =1

zh34 zh342F1 (

1 minus h + h34 h + h34

1 + h12 + h341

z) 2F1 (

1 minus h + h34 h + h34

1 + h12 + h341

z) (346)

Fminus(z z) =zh12 zh122F1 (

1 minus h minus h12 h minus h12

1 minus h12 minus h341

z) 2F1 (

cprime1 =(minus1)hminush+h12minush12Γ (minush12 minus h34)

Γ (1 + h12 + h34)Γ (1 minus h + h12)Γ (h + h34)Γ (h + h12)Γ (1 minus h + h34)Γ (1 minus h minus h12)Γ (h minus h34)Γ (h minus h12)Γ (1 minus h minus h34)

cprime2 =(minus1)hminush+h34minush34Γ (h12 + h34)

Γ (1 minus h12 minus h34)

Here we made use of hypergeometric identities discussed in [62] to rewrite the result in a

form which is suited for the region z z gt 1

Conformal partial waves are orthogonal with respect to the inner product (344)

⟨Ψhh

hihi(z z)Ψhprimehprime

hihi(z z)⟩ = (2π)2

(1 minus 2h)(2h minus 1)δJJ primeδ(λ minus λprime) (347)

The basis functions (345) span a complete basis for bosonic fields on each of the ranges

(J isin Z λ isin R+ ∣ J isin Z+ λ isin R ∣ J isin Z λ isin Rminus ∣ J isin Zminus λ isin R) (348)

We can perform the ∆ integration in (343) by collecting residues of poles located to the

left or to the right of the complex axis One can use eg the integral representation of the

conformal partial wave (345) (given by eq (7) in [63]) to make sure that the half-circle

integration at infinity vanishes

35 Inner Product Integral

In this appendix we evaluate the inner product

⟨A4(z z)Ψhh

hihi(z z)⟩ equiv int

(zz)2δ(iz minus iz) ∣z∣2+σ ∣z minus 1∣h12minush34minusσ Ψhh

hihi(z z) (349)

for σ = 0 and σ = 1 where Ψhh

hihi(z z) is given by (345)5

5Note that in both of our examples we have hij = hij and the complex conjugation prescription hrarr 1minus hhrarr 1 minus h hij rarr minushij and zharr z

First we change integration variables to z = x + iy z = x minus iy and localize the delta

function on y = 0 Subsequently we write the hypergeometric functions from (345) in the

following Mellin-Barnes representation

2F1(a b c z) =Γ(c)

Γ(a)Γ(b)Γ(c minus a)Γ(c minus b) intCds

2πi(1 minus z)sΓ(minuss)Γ(c minus a minus b minus s)Γ(a + s)Γ(b + s)

where (1 minus z) isin CRminus and the contour C goes from minus to plus complex infinity while

separating pole sequences in Γ(minuss)Γ(c minus a minus b minus s) from pole sequences in Γ(a + s)Γ(b + s)

The x gt 1 integral then gives a beta function which we express in terms of gamma

functions At this point similarly to section 34 in [64] the gamma function arguments in

the integrand arrange themselves exactly such that one of the Mellin-Barnes integrals (350)

can be evaluated by second Barnes lemma6 The final inverse Mellin transform integral is

then done by closing the integration contour to the left or to the right of the complex axis

Performing the sum over all residues of poles wrapped by the contour in this process we

obtain

⟨A4(z z)Ψhh

hihi(z z)⟩ = π2(minus1)hminush csc (π (h12 minus h34)) csc (π (h12 + h34))Γ(1 minus σ) (351)

⎡⎢⎢⎢⎢⎢⎣

⎛⎜⎝

Γ (1 minus σ + h12 minus h34) 4F3 ( 1minusσ1minush+h12h+h121minusσ+h12minush34

2minushminusσ+h12hminusσ+h12+1h12minush34+1 1)Γ (h12 minus h34 + 1)Γ (1 minus h + h34)Γ (h + h34)Γ (2 minus h minus σ + h12)Γ (h minus σ + h12 + 1)

minus (h12 harr h34)⎞⎟⎠

+( Γ(1minushminush12)Γ(hminush12)Γ(1minusσminush12+h34)

Γ(1minush12+h34)Γ(2minushminusσminush12)Γ(hminusσminush12+1) 4F3 ( 1minusσ1minushminush12hminush121minusσminush12+h34

2minushminusσminush12hminusσminush12+11minush12+h34 1) minus (h12 harr h34))

Γ (1 minus h + h12)Γ (h + h12)Γ (1 minus h + h34)Γ (h + h34)

⎤⎥⎥⎥⎥⎥⎥⎦

6We assume the integrals to be regulated appropriately such that these formal manipulations hold

where we used identities such as sin(x+ πh) sin(y + πh) = sin(x+ πh) sin(y + πh) for integer

J and sin(πx) = π(Γ(x)Γ(1 minus x)) to write (351) in a shorter form

Evaluation for σ = 0

When σ = 0 one upper and one lower parameter in the 4F3 hypergeometric functions

become equal and cancel so that the functions reduce to 3F2 Interestingly an even greater

simplification occurs as

3F2 (1 a minus c + 1 a + ca minus b + 2 a + b + 1

1) =Γ(aminusb+2)Γ(a+b+1)Γ(aminusc+1)Γ(a+c) minus (a minus b + 1)(a + b)

(b minus c)(b + c minus 1) (352)

Then making use of various sine- and gamma function identities as mentioned above it

turns out that the result is proportional to

sin(2πJ)2πJ

= 1 J = 0

0 J ne 0 (353)

Therefore the only non-vanishing inner product in this case comes from the scalar conformal

partial wave Ψ∆hiequiv Ψhh

hihi∣J=0

which simplifies to

⟨A4(z z)Ψ∆hi(z z)⟩ =

2 minus h12)Γ (1 minus ∆2 minus h34)Γ (∆

2 minus h34)Γ(2 minus∆)Γ(∆) (354)

As we take σ rarr 1 the overall factor Γ(1 minus σ) diverges However the rest of the terms

conspire to cancel this pole so that the limit σ rarr 1 is finite The simplification of the result

in all generality is quite tedious here we instead discuss a less rigorous but quick way to

arrive at the end result

The cases for the first few values of J = 01 can be simplified directly eg in Mathe-

matica We recognize that the result is always proportional to csc(π(h12minush34))(h12minush34)

To quickly arrive at the full result start with (351) and divide out the overall factor

csc(π(h12 minus h34))(h12 minus h34) By the previous observation we see that the rest is finite

in h12 minus h34 rarr 0 Sending h34 rarr h12 under a small 1 minus σ deformation the hypergeometric

functions become equal to 1 for σ rarr 1 and the remaining terms simplify To recover the full

h12 h34 dependence it then suffices to match these terms eg to the specific example in the

case J = 1 which then for all J ge 0 leads to

⟨A4(z z)Ψhh

hihi(z z)⟩ = π csc(π(h12 minus h34))

(h34 minus h12)(Γ(h minus h12)Γ(1 minus h34 minus h)

Γ(h + h12)Γ(1 + h34 minus h)+ (h12 harr h34))

To obtain the result for J lt 0 substitute hharr h

Chapter 4

Yangian Invariants and Cluster

Adjacency in N = 4 Yang-Mills

In recent years cluster algebras have shed interesting light on the mathematical properties

of scattering amplitudes in planar N = 4 supersymmetric Yang-Mills (SYM) theory [5]

Cluster algebraic structure manifests itself in several distinct ways notably including the

appearance of certain Gr(4 n) cluster coordinates in the symbol alphabets [5 66 67 68]

cobrackets [5 69 70 71 72] and integrands [30] of n-particle amplitudes

There has been a recent revival of interest in the cluster structure of SYM amplitudes

following the observation [73] that certain amplitudes exhibit a property called cluster adja-

cency Cluster coordinates are grouped into sets called clusters with two coordinates being

called adjacent if there exists a cluster containing both The central problem of the ldquocluster

adjacencyrdquo literature is to identify (and hopefully to explain) correlations between sets of

pairs (or larger groupings) of cluster coordinates and the manner in which those pairs are

observed to appear together in various amplitudes

54 Chapter 4 Yangian Invariants and Cluster Adjacency in N = 4 Yang-Mills

For example for loop amplitudes all evidence available to date [81 22 131 75 76

77 78 80 79 82 89 83] supports the hypothesis that two cluster coordinates appear in

adjacent symbol entries only if they are cluster adjacent In [89] it was shown that this

type of cluster adjacency implies the Steinmann relations [84 85 86] For tree amplitudes a

somewhat analogous version of cluster adjacency was proposed in [81] where it was checked

in several cases and conjectured in general that every Yangian invariant in the BCFW

expansion of tree-level amplitudes in SYM theory has poles given by cluster coordinates

that are all contained in a common cluster

In this paper we provide further evidence for this and the even stronger conjecture that

cluster adjacency holds for every rational Yangian invariant in SYM theory even those that

do not appear in any representation of tree amplitudes

In Sec 2 we review the main tool of our analysis the Sklyanin Poisson bracket [87 88]

which can be used to diagnose whether two cluster coordinates on Gr(4 n) are adjacent

which we will call the bracket test [89] In Sec 3 we review the Yangian invariants of

SYM theory and explain how (in principle) to use the bracket test to provide evidence that

NkMHV Yangian invariants satisfy cluster adjacency We carry out this check for all k le 2

invariants and many k = 3 invariants

Before proceeding we make a few comments clarifying the ways in which our tests are

weaker than the analysis of [81] and the ways in which they are stronger

1 It could have happened that only certain repreresentations of tree-level amplitudes

(depending perhaps on the choice of shifts during intermediate steps of BCFW re-

cursion) satisfy cluster adjacency but as already noted our results suggest that every

Chapter 4 Yangian Invariants and Cluster Adjacency in N = 4 Yang-Mills 55

rational Yangian invariant satisfies cluster adjacency If true this suggests that the

connection between cluster adjacency and Yangian invariants admits a mathematical

explanation independent of the physics of scattering amplitudes

2 For any fixed k there are finitely many functionally independent NkMHV Yangian

invariants If it is known that these all satisfy cluster adjacency it immediately follows

that the n-particle NkMHV amplitude satisfies cluster adjacency for all n Our results

therefore extend the analysis of [81] in both k and n

3 However unlike in [81] we make no attempt to check whether each of the polynomial

factors we encounter is actually a Gr(4 n) cluster coordinate Indeed for n gt 7 there

is no known algorithm for determining in finite time whether or not a given homoge-

neous polynomial in Pluumlcker coordinates is a cluster coordinate The bracket does not

help here it is trivial to write down pairs of polynomials that pass the bracket test

but are not cluster coordinates

4 In the examples checked in [81] it was noted that each term in a BCFW expansion of an

amplitude had the property that there exists a cluster of Gr(4 n) that simultaneously

contains all of the cluster coordinates appearing in the denominator of that term

Our test is much weaker in that it can only establish pairwise cluster adjacency For

example if we encounter a term with three polynomial factors p1 p2 and p3 our test

provides evidence that there is some cluster containing p1 and p2 and also some cluster

containing p2 and p3 and also some cluster containing p1 and p3 but the bracket

cannot provide any evidence for or against the existence of a cluster simultaneously

containing all three

41 Cluster Coordinates and the Sklyanin Poisson Bracket

The objects of study in this paper will be certain rational functions on the kinematic space of

n cyclically ordered massless particles of the type that appear in tree-level gluon scattering

amplitudes A point in this kinematic space is conveniently parameterized by a collection

of n momentum twistors [4] ZI1 ZIn each of which can be regarded as a four-component

(I isin 1 4) homogeneous coordinate on P3

In these variables dual conformal symmetry [3] is realized by SL(4C) transformations

For a given collection of nmomentum twistors the (n4) Pluumlcker coordinates are the SL(4C)-

invariant quantities

⟨i j k l⟩ equiv εIJKLZIi ZJj ZKk ZLl (41)

The Gr(4 n) Grassmannian cluster algebra whose structure has been found to underlie

at least certain amplitudes in SYM theory is a commutative algebra with generators called

cluster coordinates Every cluster coordinate is a polynomial in Pluumlckers that is homogeneous

under a projective rescaling of each momentum twistor separately for example

⟨1 2 6 7⟩⟨2 3 4 5⟩ minus ⟨1 2 4 5⟩⟨2 3 6 7⟩ (42)

Every Pluumlcker coordinate is on its own a cluster coordinate For n lt 8 the number of cluster

coordinates is finite and they can easily be enumerated but for n gt 7 the number of cluster

coordinates is infinite

The cluster coordinates of Gr(4 n) are grouped into non-disjoint sets of cardinality 4nminus15

called clusters Two cluster coordinates are said to be cluster adjacent if there exists a cluster

containing both The n Pluumlcker coordinates ⟨1 2 3 4⟩ ⟨2 3 4 5⟩ ⋯ ⟨n1 2 3⟩ containing four

cyclically adjacent momentum twistors play a special role these are called frozen coordinates

and are elements of every cluster Therefore each frozen coordinate is adjacent to every

cluster coordinate

Two Pluumlcker coordinates are cluster adjacent if and only if they satisfy the so-called weak

separation criterion [90] In order to address the central problem posed in the Introduction

it is desirable to have an efficient algorithm for testing whether two more general cluster

coordinates are cluster adjacent As proposed in [89] the Sklyanin Poisson bracket [87 88]

can serve because of the expectation (not yet completely proven as far as we are aware)

that two cluster coordinates a1 a2 are adjacent if and only if log a1 log a2 isin 12Z

In the next section we use the Sklyanin Poisson bracket to test the cluster adjacency prop-

erties of Yangian invariants To that end let us briefly review following [89] (see also [91])

how it can be computed First any generic 4 times n momentum twistor matrix ZIi can be

brought into the gauge-fixed form

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

1 0 0 0 y15 ⋯ y1

0 1 0 0 y25 ⋯ y2

0 0 1 0 y35 ⋯ y3

0 0 0 1 y45 ⋯ y4

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

by a suitable GL(4C) transformation The Sklyanin Poisson bracket of the yrsquos is defined

yIa yJ b =1

2(sign(J minus I) minus sign(b minus a))yJayI b (44)

Finally the Sklyanin Poisson bracket of two arbitrary functions f g of momentum twistors

can be computed by plugging in the parameterization (43) and then using the chain rule

f(y) g(y) =n

sumab=1

sumIJ=1

partyIa

partyJ byIa yJ b (45)

42 An Adjacency Test for Yangian Invariants

The conformal [92] and dual conformal symmetry of scattering amplitudes in SYM theory

combine to generate a Yangian [11] symmetry Yangian invariants [3 93 94 96 95 28 98

30 97] are the basic building blocks in terms of which amplitudes can be constructed We

say that a Yangian invariant is rational if it is a rational function of momentum twistors

equivalently it has intersection number Γ = 1 in the terminology of [30 99] Any n-particle

tree-level amplitude in SYM theory can be written as the n-particle Parke-Taylor-Nair su-

peramplitude [2 100] times a linear combination of rational Yangian invariants (see for

example [101]) In general the linear combination is not unique since Yangian invariants

satisfy numerous linear relations

Yangian invariants are actually superfunctions an n-particle invariant is a polynomial

of uniform degree 4k in 4kn Grassmann variables χAi where k is the NkMHV degree For a

rational Yangian invariant Y the coefficient of each distinct term in its expansion in χrsquos can

be uniquely factored into a ratio of products of polynomials in Pluumlcker coordinates with

each polynomial having uniform weight in each momentum twistor separately Let pi

denote the union of all such polynomials that appear in the denominator of the expansion

of Y Then we say that Y passes the bracket test if

Ωij equiv log pi log pj isin1

2Z foralli j (46)

As explained in [30] n-particle Yangian invariants can be classified in terms of permuta-

tions on n elements Since the bracket test is invariant1 under the Zn cyclic group that shifts

the momentum twistors Zi rarr Zi+1 modn we only need to consider one member from each

cyclic equivalence class The number of cyclic classes of rational NkMHV Yangian invariants

with nontrivial dependence on n momentum twistors was tabulated for various k and n in

Table 3 of [30] We record these numbers here correcting typos in the (315) and (420)

entries

n5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Total

1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1

2 0 1 2 5 4 1 0 0 0 0 0 0 0 0 0 0 13

3 0 0 1 6 54 177 298 274 134 30 3 0 0 0 0 0 977

4 0 0 0 1 13 263 1988 7862 18532 28204 28377 18925 8034 2047 270 17 114533

When they appear in scattering amplitudes Yangian invariants typically have triv-

ial dependence on several of the particles For example the five-particle NMHV Yan-

gian invariant Y (1)(Z1 Z2 Z3 Z4 Z5) could appear in a nine-particle NMHV amplitude

as Y (1)(Z2 Z4 Z5 Z7 Z8) among other possibilities Fortunately because of the simple1Certainly the value of the Sklyanin Poisson bracket is not in general cyclic invariant since evaluating it

requires making a gauge choice which breaks cyclic symmetry such as in (43) but the binary statement ofwhether some pair does or does not have half-integer valued bracket is cyclic invariant

sign(b minus a) dependence on column number in the definition (44) the bracket test is insen-

sitive to trivial dependence on additional momentum twistors2

Therefore for any fixed k but arbitrary n we can provide evidence for the cluster

adjacency of every rational n-particle NkMHV Yangian invariant by applying the bracket

test described above (46) to each one of the (finitely many) rational Yangian invariants In

the next few subsections we present the results of our analysis beginning with the trivial

but illustrative case of k = 1

421 NMHV

The unique k = 1 Yangian invariant is the well-known five-bracket [93] (originally presented

as an ldquoR-invariantrdquo in [3])

Y (1) = [12345] equiv δ(4)(⟨1 2 3 4⟩χA5 + cyclic)⟨1 2 3 4⟩⟨2 3 4 5⟩⟨3 4 5 1⟩⟨4 5 1 2⟩⟨5 1 2 3⟩ (47)

whose denominator contains the five factors

p1 p5 = ⟨1 2 3 4⟩ ⟨2 3 4 5⟩ ⟨3 4 5 1⟩ ⟨4 5 1 2⟩ ⟨5 1 2 3⟩ (48)

each of which is simply a Pluumlcker coordinate Evaluating these in the gauge (43) gives

p1 p5 = 1minusy15minusy2

5minusy35minusy4

5 (49)

2As in footnote 1 the actual value of the Sklyanin Poisson bracket will in general change if the particlerelabeling affects any of the first four gauge-fixed columns of Z

and evaluating the bracket (46) in this basis using (44) gives

Ω(1)ij = log pi log pj =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0

0 0 12

0 minus12 0 1

0 minus12 minus1

2 0 12

0 minus12 minus1

2 minus12 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Since each entry is half-integer the five-bracket (47) passes the bracket test

We wrote out the steps in detail in order to illustrate the general procedure although

in this trivial case the conclusion was foregone for n = 5 each Pluumlcker coordinate in (47)

is frozen so each is automatically cluster adjacent to each of the others It is however

interesting to note that if we uplift (47) by introducing trivial dependence on additional

particles this simple argument no longer applies For example [13579] still passes the

bracket test even though it does not involve any frozen coordinates The fact that the five-

bracket [i j k lm] passes the bracket test for any choice of indices can be understood in

terms of the weak separation criterion [90] for determining when two Pluumlcker coordinates

are cluster adjacent The connection between the weak separation criterion and all Yangian

invariants with n = 5k will be explored in [102]

422 N2MHV

The 13 rational Yangian invariants with k = 2 are listed in Table 1 of [30] (we disregard the

ninth entry in the table which is algebraic but not rational3) They are given by

1 = [12 (23) cap (456) (234) cap (56)6][23456]

2 = [12 (34) cap (567) (345) cap (67)7][34567]

3 = [123 (345) cap (67)7][34567]

4 = [123 (456) cap (78)8][45678]

5 = [12348][45678]

6 = [123 (45) cap (678)8][45678]

7 = [123 (45) cap (678) (456) cap (78)][45678] (411)

8 = [1234 (456) cap (78)][45678]

9 = [12349][56789]

10 = [1234 (567) cap (89)][56789]

11 = [1234 (56) cap (789)][56789]

12 = ϕ times [123 (45) cap (789) (46) cap (789)][(45) cap (123) (46) cap (123)789]

13 = [12345][678910]

3As mentioned in [81] it would be very interesting if some suitably generalized version of cluster adjacencycould be found which applies to algebraic functions of momentum twistors

(ij) cap (klm) = Zi⟨j k lm⟩ minusZj⟨i k lm⟩ (412)

denotes the point of intersection between the line (ij) and the plane (klm) in momentum

twistor space The Yangian invariant Y (2)12 has the prefactor

ϕ = ⟨4 5 (123) cap (789)⟩⟨4 6 (123) cap (789)⟩⟨1 2 3 4⟩⟨4 7 8 9⟩⟨5 6 (123) cap (789)⟩ (413)

(ijk) cap (lmn) = (ij)⟨k lmn⟩ + (jk)⟨i lmn⟩ + (ki)⟨j lmn⟩ (414)

denotes the line of intersection between the planes (ijk) and (lmn)

Following the same procedure outlined in the previous subsection for each Yangian

invariant Y (2)a listed in (411) we enumerate all polynomial factors its denominator contains

and then compute the associated bracket matrix Ω(2)a Explicit results for these matrices

are given in appendix 43 We find that each matrix is half-integer valued and therefore

conclude that all rational k = 2 Yangian invariants satisfy the bracket test

423 N3MHV and Higher

For k gt 2 it is too cumbersome and not particularly enlightening to write explicit formulas

for each of the 977 rational Yangian invariants We can use [99] to compute a symbolic

formula for each Yangian invariant Y in terms of the parameterization (43) Then we

read off the list of all polynomials in the yIarsquos that appear in the denominator of Y and

compute the bracket matrix (46) We have carried out this test for all 238 rational N3MHV

invariants with n le 10 (and many invariants with n gt 10) and find that each one passes the

bracket test Although it is straightforward in principle to continue checking higher n (and

k) invariants it becomes computationally prohibitive

43 Explicit Matrices for k = 2

Using the notation given in (411) we present here for each rational N2MHV Yangian in-variant the bracket matrix of its polynomial factors

Ω(2)1

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 1 1 0 0 0 12

minus 12

minus1

0 0 0 0 minus 12

0 minus 12

minus 12

minus1

minus1 0 0 minus1 minus 32

0 minus 12

minus 12

minus1

minus1 0 1 0 minus 32

0 minus 12

0 minus1 minus1

minus 12

minus1

0 0 0 0 minus 12

0 minus 12

0 0 0 0

minus 12

0 minus 12

0 0 0 minus 12

minus 12

0 0 12

0 minus 12

1 1 1 1 1 0 0 12

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 1 0 0 0 0 minus1 minus 12

minus 12

0 0 0 minus 12

minus 12

0 minus 12

minus 12

minus1 0 0 minus 32

minus 32

0 minus 12

minus 32

minus 12

0 minus 12

0 minus1 minus 12

minus 12

0 minus1 minus 12

minus 12

0 0 0 minus 12

minus 12

0 minus 12

minus 12

0 0 12

0 minus 12

1 1 12

0 0 0 0 minus 12

0 0 0 12

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)3

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 12

0 0 0 0 minus1 0 minus 12

0 0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus1 minus1 0 minus 12

minus 32

minus 12

1 0 minus 12

0 minus1 0 minus 12

0 0 0 minus 12

minus 12

0 minus 12

minus 12

0 0 12

0 minus 12

1 1 12

0 0 12

0 0 0 0 0 0 minus 12

0 0 0 12

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)4

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 minus1 minus1 0

0 0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus1 minus1 0

0 minus 12

0 minus1 minus1 0

0 0 minus 12

minus 12

0 minus 12

minus 12

0 0 0 12

0 minus 12

minus 12

1 1 1 12

0 minus 12

1 1 1 12

0 0 0 0 0 0 0 0 0 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)5

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 0 0 0

0 0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 0 0 0

0 minus 12

0 0 0 0

0 0 minus 12

minus 12

0 minus 12

minus 12

0 0 0 12

0 minus 12

minus 12

0 0 0 12

0 minus 12

0 0 0 12

0 0 0 0 0 0 0 0 0 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)6

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 0 minus1 0

0 0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 0 minus1 0

0 minus 12

0 0 minus1 0

0 0 minus 12

minus 12

0 minus 12

minus 12

0 0 0 12

0 minus 12

minus 12

0 0 0 12

0 minus 12

1 1 1 12

0 0 0 0 0 0 0 0 0 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)7

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 0 minus1 0

0 0 minus1 minus1 minus1 0 0 minus1 minus1 0

0 1 0 minus 12

minus 12

minus 32

0 1 12

0 minus 12

minus 12

minus 32

0 1 12

minus 12

minus 32

0 0 minus 12

minus 12

0 minus 12

minus 12

0 0 minus 12

minus 12

0 minus 12

minus 12

0 1 12

0 minus 12

1 1 32

0 0 0 0 0 0 0 0 0 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)8

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 0 0 0

0 1 0 minus 12

minus 12

0 1 12

0 minus 12

minus 12

0 1 12

minus 12

0 0 minus 12

minus 12

0 minus 12

minus 12

0 0 minus 12

minus 12

0 minus 12

minus 12

0 1 12

0 minus 12

0 1 12

0 0 0 0 0 0 0 0 0 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)9

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 0 0 0

0 0 minus 12

minus 12

0 0 0 0

0 minus 12

minus 12

0 0 0 0

0 minus 12

0 0 0 0

0 minus 12

minus 12

0 minus 12

minus 12

0 0 0 0 0 12

0 minus 12

minus 12

0 0 0 0 0 12

0 minus 12

minus 12

0 0 0 0 0 12

0 minus 12

0 0 0 0 0 12

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)10

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 0 0 0

0 0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)11

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 0 0 0

0 0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)12

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 1 1 1 32

0 minus1 0 minus 12

minus 12

minus 32

minus 12

0 minus1 12

0 minus 12

minus 12

0 minus 12

0 minus1 12

0 minus 12

minus 12

0 minus 32

0 minus 12

minus 12

2 2 2 12

0 minus 32

0 minus 12

2 2 2 12

0 minus 32

0 2 2 2 12

0 minus 32

minus 12

minus2 minus2 minus2 0 minus 12

minus 12

0 minus 32

minus 12

minus2 minus2 minus2 12

0 minus 12

minus 12

0 minus 32

minus 12

0 minus 12

minus 12

0 minus1 12

0 minus 12

minus 12

0 minus 12

0 minus1 12

0 minus 12

minus 12

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)13

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 0 0 0

0 0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Each matrix Ω(2)i is written in the basis Bi of polynomials shown below

B1 =⟨12 (23) cap (456) (234) cap (56)⟩ ⟨612 (23) cap (456)⟩ ⟨(234) cap (56)612⟩

⟨(23) cap (456) (234) cap (56)61⟩ ⟨2 (23) cap (456) (234) cap (56)6⟩ ⟨2345⟩ ⟨6234⟩ ⟨5623⟩

⟨4562⟩ ⟨3456⟩

B2 =⟨12 (34) cap (567) (345) cap (67)⟩ ⟨712 (34) cap (567)⟩ ⟨(345) cap (67)712⟩ ⟨(34) cap (567)

(345) cap (67)71⟩ ⟨2 (34) cap (567) (345) cap (67)7⟩ ⟨3456⟩ ⟨7345⟩ ⟨6734⟩ ⟨5673⟩

⟨4567⟩

B3 =⟨123 (345) cap (67)⟩ ⟨7123⟩ ⟨(345) cap (67)712⟩ ⟨3 (345) cap (67)71⟩ ⟨23 (345) cap (67)7⟩

⟨3456⟩ ⟨7345⟩ ⟨6734⟩ ⟨5673⟩ ⟨4567⟩

B4 =⟨123 (456) cap (78)⟩ ⟨8123⟩ ⟨(456) cap (78)812⟩ ⟨3 (456) cap (78)81⟩ ⟨23 (456) cap (78)8⟩

⟨4567⟩ ⟨8456⟩ ⟨7845⟩ ⟨6784⟩⟨5678⟩

B5 =⟨1234⟩ ⟨8123⟩ ⟨4812⟩ ⟨3481⟩ ⟨2348⟩ ⟨4567⟩ ⟨8456⟩ ⟨7845⟩ ⟨6784⟩

⟨5678⟩

B6 =⟨123 (45) cap (678)⟩ ⟨8123⟩ ⟨(45) cap (678)812⟩ ⟨3 (45) cap (678)81⟩ ⟨23 (45) cap (678)8⟩

⟨4567⟩ ⟨8456⟩ ⟨7845⟩ ⟨6784⟩⟨5678⟩

B7 =⟨123 (45) cap (678)⟩ ⟨(456) cap (78)123⟩ ⟨(45) cap (678) (456) cap (78)12⟩

⟨3 (45) cap (678) (456) cap (78)1⟩ ⟨23 (45) cap (678) (456) cap (78)⟩ ⟨4567⟩ ⟨8456⟩ ⟨7845⟩

⟨6784⟩⟨5678⟩

B8 =⟨1234⟩ ⟨(456) cap (78)123⟩ ⟨4 (456) cap (78)12⟩ ⟨34 (456) cap (78)1⟩ ⟨234 (456) cap (78)⟩

⟨4567⟩ ⟨8456⟩ ⟨7845⟩ ⟨6784⟩⟨5678⟩

B9 =⟨1234⟩ ⟨9123⟩ ⟨4912⟩ ⟨3491⟩ ⟨2349⟩ ⟨5678⟩ ⟨9567⟩ ⟨8956⟩

⟨7895⟩ ⟨6789⟩

B10 =⟨1234⟩ ⟨(567) cap (89)123⟩ ⟨4 (567) cap (89)12⟩ ⟨34 (567) cap (89)1⟩ ⟨234 (567) cap (89)⟩

⟨5678⟩ ⟨9567⟩ ⟨8956⟩ ⟨7895⟩ ⟨6789⟩

B11 =⟨1234⟩ ⟨(56) cap (789)123⟩ ⟨4 (56) cap (789)12⟩ ⟨34 (56) cap (789)1⟩ ⟨234 (56) cap (789)⟩

⟨5678⟩ ⟨9567⟩ ⟨8956⟩ ⟨7895⟩ ⟨6789⟩

B12 =⟨1234⟩ ⟨4789⟩ ⟨56 (123) cap (789)⟩ ⟨123 (45) cap (789)⟩ ⟨(46) cap (789)123⟩

⟨(45) cap (789) (46) cap (789)12⟩ ⟨3 (45) cap (789) (46) cap (789)1⟩ ⟨23 (45) cap (789) (46) cap (789)⟩

⟨789(45) cap (123)⟩ ⟨(46) cap (123)789⟩

B13 =⟨1234⟩ ⟨5123⟩ ⟨4512⟩ ⟨3451⟩ ⟨2345⟩ ⟨6789⟩ ⟨10678⟩ ⟨91067⟩

⟨89106⟩ ⟨78910⟩

Chapter 5

A Note on One-loop Cluster

Adjacency in N = 4 SYM

Cluster algebras [17 18 19] of Grassmannian type [104 21] have been found to play a

significant role in the mathematical structure of scattering amplitudes in planar maximally

supersymmetric Yang-Mills theory (N = 4 SYM) [5 69] constraining the structure of ampli-

tudes at the level of symbols and cobrackets [67 69 71 72] The recently introduced cluster

adjacency principle [73] has opened a new line of research in this topic shedding light on

even deeper connections between amplitudes and cluster algebras This principle applies

conjecturally to various aspects of the analytic structure of amplitudes in N = 4 SYM The

many guises of cluster adjacency at the level of symbols [89] Yangian invariants [65 105]

and the correlation between them [81] have also been exploited to help compute new am-

plitudes via bootstrap [82] These mathematical properties however are perhaps somewhat

obscure and although it is understood that cluster adjacency of a symbol implies the Stein-

mann relations [73] its other manifestations have less clear physical interpretations (see

70 Chapter 5 A Note on One-loop Cluster Adjacency in N = 4 SYM

however [129] which establishes interesting new connections between cluster adjacency and

Landau singularities) Even finer notions of cluster adjacency that more strictly constrain

pairs of adjacent symbol letters have recently been studied in [108 107]

In this paper we show that that the one-loop NMHV amplitudes in N = 4 SYM theory

satisfy symbol-level cluster adjacency for all n and we check that for n = 9 the amplitude can

be written in a form that exhibits adjacency between final symbol entries and R-invariants

supporting the conjectures of [73 81] The outline of this paper is as follows In Section 2 we

review the kinematics of N = 4 SYM and the bracket test used to assess cluster adjacency

In Section 3 we review formulas for the amplitudes to which we apply the bracket test In

Section 4 we present our analysis and results as well as new cluster adjacency conjectures for

Pluumlcker coordinates and cluster variables that are quadratic in Pluumlckers These conjectures

generalize the notion of weak separation [109 110]

51 Cluster Adjacency and the Sklyanin Bracket

In N = 4 SYM the kinematics of scattering of n massless particles is described by a collection

of n momentum twistors [4] ZI1 ZIn each of which is a four-component (I isin 1 4)

homogeneous coordinate on P3 Thanks to dual conformal symmetry [3] the collection of

momentum twistors have a GL(4) redundancy and thus can be taken to represent points in

Gr(4 n) By an appropriate choice of gauge we can take

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

Z11 ⋯ Z1

Z21 ⋯ Z2

Z31 ⋯ Z3

Z41 ⋯ Z4

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ETHrarrGL(4)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

1 0 0 0 y15 ⋯ y1

0 1 0 0 y25 ⋯ y2

0 0 1 0 y35 ⋯ y3

0 0 0 1 y45 ⋯ y4

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

The degrees of freedom are given by yIa = (minus1)I⟨1234 ∖ I a⟩⟨1234⟩ for a =

56 n with

⟨a b c d⟩ equiv εijklZiaZjbZ

ld (52)

denoting Pluumlcker coordinates on Gr(4 n) Throughout this paper we will make use of the

relation between momentum twistors and dual momenta [3]

x2ij =

⟨iminus1 i jminus1 j⟩⟨iminus1 i⟩⟨jminus1 j⟩ (53)

where ⟨i j⟩ is the usual spinor helicity bracket (that completely drops out of our analysis

due to cancellations guaranteed by dual conformal symmetry)

The fact that (52) are cluster variables of the Gr(4 n) cluster algebra plays a constrain-

ing role in the analytic structure of amplitudes in N = 4 SYM through the notion of cluster

adjacency [73] and it is therefore of interest to test the cluster adjacency properties of ampli-

tudes Two cluster variables are cluster adjacent if they appear together in a common cluster

of the cluster algebra (this notion is also called ldquocluster compatibilityrdquo) To test whether two

given variables are cluster adjacent one can use the Poisson structure of the cluster algebra

[104] which is related to the Sklyanin bracket [87] We call this the bracket test and was

first applied to amplitudes in [89] In terms of the parameters of (51) the Sklyanin bracket

is given by

yIa yJ b =1

which extends to arbitrary functions as

f(y) g(y) =n

sumab=5

sumIJ=1

partyIa

The bracket test then says two cluster variables ai and aj are cluster adjacent iff

Ωij = log ai log aj isin1

2Z (56)

Note that whenever i j k l are cyclically adjacent ⟨i j k l⟩ is a frozen variable and is

therefore automatically adjacent with every cluster variable

The aim of this paper is to provide evidence for two cluster adjacency conjectures for

loop amplitudes of generalized polylogarithm type [73]

Conjecture 1 ldquoSteinmann cluster adjacencyrdquo Every pair of adjacent entries in the symbol of

an amplitude is cluster adjacent

This type of cluster adjacency implies the extended Steinmann relations at all particle

multiplicities [89] In fact it appears to be equivalent to the extended Steinmann conditions

of [111] for all known integrable symbols with physical first entries (that means of the form

⟨i i + 1 j j + 1⟩)

Conjecture 2 ldquoFinal entry cluster adjacencyrdquo There exists a representation of the symbol of

an amplitude in which the final symbol entry in every term is cluster adjacent to all poles

of the Yangian invariant that term multiplies

Support for these conjectures was given for NMHV amplitudes at 6- and 7-points in

[82 81] (to all loop order at which these amplitudes are currently known) and for one- and

two-loop MHV amplitudes (to which only the first conjecture applies) at all multipliticies

in [89]

52 One-loop Amplitudes

To demonstrate the cluster adjacency of NMHV amplitudes with respect to the conjec-

tures in Section 51 we need to work with appropriate finite quantities after IR divergences

have been subtracted To this end we will be working with two types of regulators at one

loop BDS [112] and BDS-like [113] normalized amplitudes In this section we review these

regulators and the one-loop amplitudes relevant for our computations

521 BDS- and BDS-like Subtracted Amplitudes

We start by reviewing the BDS normalized amplitude which was first introduced in [112]

Consider the n-point MHV amplitudeAMHVn in planarN = 4 SYM with gauge group SU(Nc)

coupling constant gYM where the tree-level amplitude has been factored out Evaluating the

amplitude in 4minus2ε dimensions regulates the IR divegences The BDS normalization involves

dividing all amplitudes by the factor

ABDSn = exp [

infinsumL=1

g2L (f(L)(ε)

2A(1)n (Lε) +C(L))] (57)

that encapsulates all IR divergences Here where g2 = g2YMNc

16π2 is the rsquot Hooft coupling the

superscript (L) on any function denotes its O(g2L) term C(L) is a transcendental constant

and f(ε) = 12Γcusp +O(ε) where Γcusp is the cusp anomalous dimension

Γcusp = 4g2 +O(g4) (58)

The BDS-like normalization contrasts with BDS normalization by the inclusion of a

dual conformally invariant function Yn chosen such that the BDS-like normalization only

depends on two-particle Mandelstam invariants

ABDS-liken = ABDS

n exp [Γcusp

4Yn] 4 ∣ n

Yn = minusFn minus 4ABDS-like +nπ2

where Fn is (in our conventions) twice the function in Eq (457) of [112] (one can use an

equivalent representation from [89]) and ABDS-like is given on page 57 of [114] Since ABDS-liken

only depends on two-particle Mandelstam invariants which can be written entirely in terms

of frozen variables of the cluster algebra the BDS-like normalization has the nice feature

of not spoiling any cluster adjacency properties At the same time it means that BDS-like

normalized amplitudes will satisfy Steinmann relations [84 85 86]

Discx2i+1j

[Discx2i+1i+p

(An)] = 0

Discx2i+1i+p

[Discx2i+1j+p+q

(An)] = 0

⎫⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎭

0 lt j minus i le p or q lt i minus j le p + q (510)

522 NMHV Amplitudes

The one-loop n-point NMHV ratio function can be written in the dual conformally invariant

form [115 116]

Pn = VtotRtot + V14nR14n +nminus2

sums=5

sumt=s+2

V1stR1st + cyclic (511)

The transcendental functions Vtot V14n and V1st are given explicitly in Appendix 55 The

function Rtot is given in terms of R-invariants [3]

Rtot =nminus2

sums=3

sumt=s+2

R1st (512)

and Rrst are the five-brackets [93] written in terms of momentum supertwistors as

Rrst = [r s minus 1 s t minus 1 t]

[a b c d e] = δ(4)(χa⟨b c d e⟩ + cyclic)⟨a b c d⟩⟨b c d e⟩⟨c d e a⟩⟨d e a b⟩⟨e a b c⟩

These are special cases of Yangian invariants [3 11] and we will henceforth refer to them as

53 Cluster Adjacency of One-Loop NMHV Amplitudes

In this section we will describe the method we used to test the conjectures in Section 51

and our results

531 The Symbol and Steinmann Cluster Adjacency

To compute the symbol of a transcendental function we follow [12] (see also [117]) Only

weight two polylogarithms appear at one loop so it is sufficient for us to use the symbols

S(log(R1) log(R2)) = R1 otimesR2 +R2 otimesR1 S(Li2(R1)) = minus(1 minusR1)otimesR1 (514)

Once the symbol of an amplitude is computed we expand out any cross ratios using (528)

and (53) and perform the bracket test to adjacent symbol entries It is straightforward

to compute the symbol of the expressions in Appendix 55 using (514) and we find that

the symbol of each of the transcendental functions of (511) V14n V1st and Vtot satisfy

Steinmann cluster adjacency (after dropping spurious terms that cancel when expanded

out) and hence satisfies Conjecture 1

532 Final Entry and Yangian Invariant Cluster Adjacency

To study Conjecture 2 we follow [81] and start with the BDS-like normalized amplitude

expanded as a linear combination of Yangian invariants times transcendental functions

ANMHV BDS-likenL =sum

Yif (2L)i (515)

We seek a representation of this amplitude that satisfies Conjecture 2 Using the bracket

test (56) we determine which final symbol entries are not cluster adjacent to all poles

of the Yangian invariant multiplying that term We then rewrite the non-cluster adjacent

combinations of Yangian invariants and final entries by using the identities [93]

[a b c d e] minus [a b c d f] + [a b c e f] minus [a b d e f] + [a c d e f] minus [b c d e f] = 0

until we are able to reach a form that satisfies final entry cluster adjacency Note that

rewriting in this manner makes the integrability of the symbol no longer manifest The 6-

and 7-point cases were studied in [81] We checked that this conjecture is true in the 9-point

case as well To get a flavor for our 9-point calculation consider the following term that we

encounter which does not manifestly satisfy final entry cluster adjacency

minus 1

2([12345] + [12356] + [12367] minus [12457] minus [12567]

+ [13456] + [13467] + [14567] minus [23457] minus [23567])

times (log(⟨1234⟩⟨1789⟩⟨1278⟩⟨1349⟩)otimes ⟨3478⟩)

To get rid of the non-cluster adjacent combinations of Yangian invariants and final entries

we list all identities (516) and note that there are 14 cyclic classes of Yangian invariants

at 9-points A cyclic class is generated by taking a five-bracket and shifting all indices

cyclically This collection forms a cyclic class Solving the identities (516) for 7 of the

14 cyclic classes in Mathematica (yielding (147) = 3432 different solutions) we find that at

least one solution for each final entry brings the symbol to a final entry cluster adjacent

form For the example (517) one of the combinations from these solutions that is cluster

adjacent takes the form

minus 1

2([12348] minus [12378] + [12478] minus [13478]

+ [23478] + [34567]) (log(⟨1234⟩⟨1789⟩⟨1278⟩⟨1349⟩)otimes ⟨3478⟩)

One can check that the complete set of Yangian invariants that are cluster adjacent to

⟨3478⟩ is given by

[12347] [12348] [12349] [12378] [12379] [12389]

[12478] [12479] [12489] [12789] [13478] [13479]

[13489] [13789] [14789] [23478] [23479] [23489]

[23789] [24789] [34567] [34568] [34578] [34678]

[34789] [35678] [45678]

At 10-points this method becomes much more computationally intensive as we have 26

cyclic classes If we follow the same procedure as for 9-points we would have to check

cluster adjacency of (2613) = 10400600 solutions per final entry with non cluster adjacent

Yangian invariants

54 Cluster Adjacency and Weak Separation

In our study of one-loop NMHV amplitudes we observed some general cluster adjacency

properties of symbol entries and Yangian invariants involved in the one-loop NMHV ampli-

tude Let us denote the various types of symbol letters by

a1ij = ⟨i minus 1 i j minus 1 j⟩ (520)

a2ijk = ⟨i(j j + 1)(k k + 1)(i minus 1 i + 1)⟩

= ⟨i j j + 1 i minus 1⟩⟨i k k + 1 i + 1⟩ minus ⟨i j j + 1 i + 1⟩⟨i k k + 1 i minus 1⟩ (521)

a3ijkl = ⟨i(j j + 1)(k k + 1)(l l + 1)⟩

= ⟨i j k k + 1⟩⟨i j + 1 l l + 1⟩ minus ⟨i j + 1 k k + 1⟩⟨i j l l + 1⟩ (522)

In this section we summarize their cluster adjacency properties as determined by the bracket

First consider a1ij and a2klm We observe that these variables are adjacent if they

satisfy a generalized notion of weak separation [109 110] In particular we find that

⟨i minus 1 i j minus 1 j⟩ and ⟨k(l l + 1)(mm + 1)(k minus 1k + 1)⟩ are cluster adjacent iff

i j isin k + 1 l + 1 or l + 1 m + 1 or m + 1 k or

i = k j = l + 1 or i = k j =m + 1 or i = k + 1 j = l + 1 or i = k + 1 j =m + 1

This adjacency statement can be represented by drawing a circle with labeled points 1 n

appearing in cyclic order as in Figure 51 For the variables a1ij and a3klmp we observe

Figure 51 Weak separation graph indicating that if both i and j are within any ofthe green regions (or on the green chords) then ⟨i minus 1 i j minus 1 j⟩ is cluster adjacent to⟨k(l l + 1)(mm + 1)(k minus 1k + 1)⟩

⟨i minus 1 i j minus 1 j⟩ and ⟨k(l l + 1)(mm + 1)(pp + 1)⟩ are cluster adjacent iff

i j isin k + 1 l + 1 or l + 1 m + 1 or m + 1 p + 1 or p + 1 k + 1

or i = k + 1 j = l + 1 or i = l + 1 j =m + 1 or i =m + 1 j = p + 1

or i = p + 1 j = k + 1 or i = k + 1 j =m + 1 or i = l + 1 j = p + 1

This statement is represented in Figure 52

For Pluumlcker coordinate of type (520) and Yangian invariants (513) we observe

⟨i minus 1 i j minus 1 j⟩ and [a b c d e] are cluster adjacent iff

a b c d e sub (i minus 1 i j minus 1 j5

) cup (j minus 1 j i minus 1 i5

)(525)

Figure 52 Weak separation graph indicating that if both i and j are within any ofthe green regions (or on the green chords) then ⟨i minus 1 i j minus 1 j⟩ is cluster adjacent to⟨k(l l + 1)(mm + 1)(pp + 1)⟩

Next up the variables (521) and Yangian invariants (513) are observed to have the adjacency

condition

⟨i(j j + 1)(k k + 1)(i minus 1 i + 1)⟩ and [a b c d e] are cluster adjacent iff

a b c d e sub i j j + 1 k k + 1 cup (i i + 1 j j + 15

cup (j j + 1 k k + 15

) cup (k k + 1 i minus 1 i5

Finally for variables (522) and Yangian invariants (513) we observe adjacency when

⟨i(j j + 1)(k k + 1)(l l + 1)⟩ and [a b c d e] are cluster adjacent iff

a b c d e sub (i j j + 15

) cup (i j j + 1 k k + 15

cup (i k k + 1 l l + 15

) cup (l l + 1 i5

The statements about cluster adjacency in this section hint at a generalization of the notion

of weak separation for Pluumlcker coordinates [109 110] We are only able to verify these

statements ldquoexperimentallyrdquo via the bracket test To prove such statements we look to

Theorem 16 of [110] which states that given a subset C of (1n4

) the set of Pluumlcker

coordinates pIIisinC forms a cluster in the Gr(4 n) cluster algebra iff C is a maximally

weakly separated collection Maximally weakly separated means that if C sube (1n4

) is a

collection of pairwise weakly separated sets and C is not contained in any larger set of of

pairwise weakly separated sets then the collection C is maximally weakly separated To

prove the cluster adjacency statements made in this section we would have to prove that

there exists a maximally weakly separated collection containing all the weakly separated

sets proposed in for each pair of coordinatesYangian invariants considered in this section

We leave this to future work

55 n-point NMHV Transcendental Functions

In this Appendix we present the transcendental functions contributing to the NMHV ratio

function (511) from [116] All functions are written in a dual conformally invariant form

in terms of cross ratios

uijkl =x2ikx

of dual momenta (53) The functions V1st are given by

V1st = Li2(1 minus u12t4) minus Li2(1 minus u12ts) +s

sumi=5

[Li2(1 minus u2i+2iminus1i) minus Li2(1 minus u12iiminus1)

minus Li2(1 minus u1i+2iminus1i) minus1

2ln(u21ii+2) ln(u1i+2iminus1i) minus

2ln(u12ti) ln(u1timinus1i)

minus 1

2ln(u2iminus1ti+2) ln(u12iiminus1)] for 5 le s t le n minus 1

where 5 le s le n minus 2 and s + 2 le t le n and

V1sn = Li2(1 minus u2snnminus1) + Li2(1 minus u214nminus1) + ln(u2snnminus1) ln(u21snminus1)

sumi=5

[Li2(1 minus u2i+2iminus1i) minus Li2(1 minus u12iiminus1) minus Li2(1 minus u1i+2iminus1i)

minus 1

2ln(u12nminus1i) ln(u1nminus1iminus1i)

minus 1

2ln(u2iminus1nminus1i+2) ln(u12iiminus1)] minus

6 for 4 le s le n minus 3

where the sum empty sum is understood to vanish for s = 4 The function V1nminus2n is given

V1nminus2n = Li2(1 minus u2nnminus3nminus2) minus Li2(1 minus u12nminus2nminus3) + Li2(1 minus u2nminus3nnminus1)

+ Li2(1 minus u214nminus1) minus ln(un1nminus3nminus2) ln( u12nminus2nminus1

u2nminus3nminus1n)

+ ln(u2nminus3nnminus1) ln(u21nminus3nminus1) +nminus3

sumi=5

[Li2(1 minus u2i+2iminus1i)

minus Li2(1 minus u12iiminus1) minus Li2(1 minus u1i+2iminus1i) minus1

2ln(u21ii+2) ln(u1i+2iminus1i)

minus 1

2ln(u12nminus1i) ln(u1nminus1iminus1i) minus

Finally Vtot is given by two different formulas one for n = 8 and one for n gt 8 For n = 8 we

8Vn=8tot = minusLi2(1 minus uminus1

1247) +1

sumi=4

Li2(1 minus uminus112ii+1) +

4ln(u8145) ln(u1256u3478

u2367) + cyclic (532)

while for n gt 8 we have

nVtot = minusLi2(1 minus uminus1124nminus1) +

nminus2

sumi=4

Li2(1 minus uminus112ii+1)

2ln(un134) ln(u136nminus2) minus

2ln(un145) ln(u236nminus2u2367) + vn + cyclic

n odd ∶ vn =nminus1

sumi=4

ln(un1ii+1)iminus1

sumj=1

ln(ujj+1i+jnminusi+j) (534)

n even ∶ vn =nminus1

sumi=4

ln(un1ii+1)iminus1

sumj=1

ln(ujj+1i+jnminusi+j) +1

4ln(un1n

2n2+1)

nminus22

sumi=1

ln(uii+1i+n2i+n

Chapter 6

Symbol Alphabets from Plabic

Graphs

A central problem in studying the scattering amplitudes of planar N = 4 super-Yang-

Mills (SYM) theory is to understand their analytic structure Certain amplitudes are known

or expected to be expressible in terms of generalized polylogarithm functions The branch

points of any such amplitude are encoded in its symbol alphabetmdasha finite collection of multi-

plicatively independent functions on kinematic space called symbol letters [12] In [5] it was

observed that for n = 67 the symbol alphabet of all (then-known) n-particle amplitudes is

the set of cluster variables [17 119] of the Gr(4 n) Grassmannian cluster algebra [21] The

hypothesis that this remains true to arbitrary loop order provides the bedrock underlying

a bootstrap program that has enabled the computation of these amplitudes to impressively

high loop order and remains supported by all available evidence (see [13] for a recent review)

For n gt 7 the Gr(4 n) cluster algebra has infinitely many cluster variables [119 21]

While it has long been known that the symbol alphabets of some n gt 7 amplitudes (such

86 Chapter 6 Symbol Alphabets from Plabic Graphs

as the two-loop MHV amplitudes [22]) are given by finite subsets of cluster variables there

was no candidate guess for a ldquotheoryrdquo to explain why amplitudes would select the sub-

sets that they do At the same time it was expected [25 26] that the symbol alphabets

of even MHV amplitudes for n gt 7 would generically require letters that are not cluster

variablesmdashspecifically that are algebraic functions of the Pluumlcker coordinates on Gr(4 n)

of the type that appear in the one-loop four-mass box function [120 121] (see Appendix 67)

(Throughout this paper we use the adjective ldquoalgebraicrdquo to specifically denote something that

is algebraic but not rational)

As often the case for amplitudes guesses and expectations are valuable but explicit

computations are king Recently the two-loop eight-particle NMHV amplitude in SYM

theory was computed [23] and it was found to have a 198-letter symbol alphabet that can

be taken to consist of 180 cluster variables on Gr(48) and an additional 18 algebraic letters

that involve square roots of four-mass box type (Evidence for the former was presented

in [26] based on an analysis of the Landau equations the latter are consistent with the

Landau analysis but less constrained by it) The result of [23] provided the first concrete

new data on symbol alphabets in SYM theory in over eight years We will refer to this as

ldquothe eight-particle alphabetrdquo in this paper since (turning again to hopeful speculation) it

may turn out to be the complete symbol alphabet for all eight-particle amplitudes in SYM

theory at all loop order

A few recent papers have sought to explain or postdict the eight-particle symbol alphabet

and to clarify its connection to the Gr(48) cluster algebra In [122] polytopal realizations

of certain compactifications of (the positive part of) the configuration space Conf8(P3)

of eight particles in SYM theory were constructed These naturally select certain finite

subsets of cluster variables including those in the eight-particle alphabet and the square

roots of four-mass box type make a natural appearance as well At the same time an

equivalent but dual description involving certain fans associated to the tropical totally

positive Grassmannian [123] appeared simultaneously in [124 108] Moreover [124] proposed

a construction that precisely computes the 18 algebraic letters of the eight-particle symbol

alphabet by (roughly speaking) analyzing how the simplest candidate fan is embedded within

the (infinite) Gr(48) cluster fan

In this paper we show that the algebraic letters of the eight-particle symbol alphabet are

precisely reproduced by an alternate construction that only requires solving a set of simple

polynomial equations associated to certain plabic graphs This raises the possibility that

symbol alphabets of SYM theory could be encoded more generally in certain plabic graphs

In Sec 61 we introduce our construction with a simple example and then complete the

analysis for all graphs relevant to Gr(46) in Sec 62 In Sec 63 we consider an example

where the construction yields non-cluster variables of Gr(36) and in Sec 64 we apply it

to graphs that precisely reproduce the algebraic functions on Gr(48) that appear in the

symbol of [23]

61 A Motivational Example

Motivated by [125] in this paper we consider solutions to sets of equations of the form

C sdotZ = 0 (61)

which are familiar from the study of several closely connected or essentially equivalent

amplitude-related objects (leading singularities Yangian invariants on-shell forms see for

example [27 93 94 28 30])

For the application to SYM theory that will be the focus of this paper Z is the n times 4

matrix of momentum twistors describing the kinematics of an n-particle scattering event

but it is often instructive to allow Z to be n timesm for general m

The k timesn matrix C(f0 fd) in (61) parameterizes a d-dimensional cell of the totally

non-negative Grassmannian Gr(kn)ge0 Specifically we always take it to be the boundary

measurement of a (reduced perfectly oriented) plabic graph expressed in terms of the face

weights fα of the graph (see [29 30]) One could equally well use edge weights but using

face weights allows us to further restrict our attention to bipartite graphs and to eliminate

some redundancy the only residual redundancy of face weights is that they satisfy proda fα = 1

for each graph

For an illustrative example consider

which affords us the opportunity to review the construction of the associated C-matrix

from [29] The graph is perfectly oriented because each black (white) vertex has all incident

arrows but one pointing in (out) The graph has two sources 12 and four sinks 3456

and we begin by forming a 2 times (2 + 4) matrix with the 2 times 2 identity matrix occupying the

source columns

C =⎛⎜⎜⎜⎝

1 0 c13 c14 c15 c16

0 1 c23 c24 c25 c26

⎞⎟⎟⎟⎠ (63)

The remaining entries are given by

prodαisinp

fα (64)

and the product runs over all faces α to the right of p denoted by p In this manner we find

c13 = minusf0f1f2f3f4f5f6 c23 = f0f1f2f3f4f5f6f8

c14 = minusf0f1f2f3f4(1 + f6) c24 = f0f1f2f3f4f6f8

c15 = minusf0f1f2(1 + f4 + f4f6) c25 = f0f1f2f4f6f8

c16 = minusf0(1 + f2 + f2f4 + f2f4f6) c26 = f0f2f4f6f8

Then form = 4 (61) is a system of 2times4 = 8 equations for the eight independent face weights

which has the solution

f0 = minus⟨1234⟩⟨2346⟩ f1 = minus

⟨2346⟩⟨2345⟩ f2 =

⟨2345⟩⟨1236⟩⟨1234⟩⟨2356⟩

f3 = minus⟨2356⟩⟨2346⟩ f4 =

⟨2346⟩⟨1256⟩⟨2456⟩⟨1236⟩ f5 = minus

⟨2456⟩⟨2356⟩

f6 =⟨2356⟩⟨1456⟩⟨3456⟩⟨1256⟩ f7 = minus

⟨3456⟩⟨2456⟩ f8 = minus

⟨2456⟩⟨1456⟩

where ⟨ijkl⟩ = det(ZiZjZkZl) are Pluumlcker coordinates on Gr(46)

We pause here to point out two features evident from (66) First we see that on

the solution of (61) each face weight evaluates (up to sign) to a product of powers of

Gr(46) cluster variables ie to a symbol letter of six-particle amplitudes in SYM theory [12]

Moreover the cluster variables that appear (⟨2346⟩ ⟨2356⟩ ⟨2456⟩ and the six frozen

variables) constitute a single cluster of the Gr(46) algebra

The fact that cluster variables of Gr(mn) seem to arise at least in this example raises

the possibility that the symbol alphabets of amplitudes in SYM theory might be given more

generally by the face weights of certain plabic graphs evaluated on solutions of C sdotZ = 0 A

necessary condition for this to have a chance of working is that the number of independent

face weights should equal the number of equations (both eight in the above example) oth-

erwise the equations would have no solutions or continuous families of solutions For this

reason we focus exclusively on graphs for which (61) admits isolated solutions for the face

weights as functions of generic ntimesm Z-matrices in particular this requires that d = km In

such cases the number of isolated solutions to (61) is called the intersection number of the

The possible connection between plabic graphs and symbol alphabets is especially tanta-

lizing because it manifestly has a chance to account for both issues raised in the introduction

(1) while the number of cluster variables of Gr(4 n) is infinite for n gt 7 the number of (re-

duced) plabic graphs is certainly finite for any fixed n and (2) graphs with intersection

number greater than 1 naturally provide candidate algebraic symbol letters Our showcase

example of (2) is presented in Sec 64

62 Six-Particle Cluster Variables

The problem formulated in the previous section can be considered for any k m and n In

this section we thoroughly investigate the first case of direct relevance to the amplitudes of

SYM theory m = 4 and n = 6 Although this case is special for several reasons it allows us

to illustrate some concepts and terminology that will be used in later sections

Modulo dihedral transformations on the six external points there are a total of four

different types of plabic graph to consider We begin with the three graphs shown in Fig 61

(a)ndash(c) which have k = 2 These all correspond to the top cell of Gr(26)ge0 and are related

to each other by square moves Specifically performing a square move on f2 of graph (a)

yields graph (b) while performing a square move on f4 of graph (a) yields graph (c) This

contrasts with more general cases for example those considered in the next two sections

where we are in general interested in lower-dimensional cells

The solution for the face weights of graph (a) (the same as (62)) were already given

in (66) and those of graphs (b) and (c) are derived in (627) and (629) of Appendix 66 The

latter two can alternatively be derived from the former via the square move rule (see [29 30])

In particular for graph (b) we have

f(b)0 = f (a)0 (1 + f (a)2 )

f(b)1 = f

1 + 1f (a)2

f(b)2 = 1

f(b)3 = f (a)3 (1 + f (a)2 )

f(b)4 = f

1 + 1f (a)2

with f5 f6 f7 and f8 unchanged while for graph (c) we have

f(c)2 = f (a)2 (1 + f (a)4 )

f(c)3 = f

1 + 1f (a)4

f(c)4 = 1

f(c)5 = f (a)5 (1 + f (a)4 )

f(c)6 = f

1 + 1f (a)4

with f0 f1 f7 and f8 unchanged

To every plabic graph one can naturally associate a quiver with nodes labeled by Pluumlcker

coordinates of Gr(kn) In Fig 61 (d)ndash(f) we display these quivers for the graphs under

consideration following the source-labeling convention of [126 127] (see also [128]) Because

in this case each graph corresponds to the top cell of Gr(26)ge0 each labeled quiver is a

seed of the Gr(26) cluster algebra More generally we will have graphs corresponding to

lower-dimensional cells whose labeled quivers are seeds of subalgebras of Gr(kn)

Henceforth we refer to a labeled quiver associated to a plabic graph in this manner as

an input cluster taking the point of view that solving the equations C sdot Z = 0 associates a

collection of functions on Gr(mn) to every such input At the same time there is a natural

way to graphically organize the structure of each of (66) (627) and (629) in terms of an

output cluster as we now explain

First of all we note from (627) and (629) that like what happened for graph (a) consid-

ered in the previous section each face weight evaluates (up to sign) to a product of powers

(a) (b) (c)

⟨16⟩ ⟨12⟩

⟨23⟩

⟨34⟩⟨45⟩

⟨56⟩

⟨26⟩

⟨36⟩

⟨46⟩

ampamppp

⟨16⟩ ⟨12⟩

⟨23⟩

⟨34⟩⟨45⟩

⟨56⟩

⟨26⟩

⟨36⟩

⟨35⟩

⟨16⟩ ⟨12⟩

⟨23⟩

⟨34⟩⟨45⟩

⟨56⟩

⟨26⟩

⟨24⟩⟨46⟩ oo

(d) (e) (f)

⟨3456⟩ ⟨1456⟩

⟨1256⟩

⟨1236⟩⟨1234⟩

⟨2345⟩

⟨2456⟩

⟨2356⟩

⟨2346⟩

ampamppp

⟨3456⟩ ⟨1456⟩

⟨1256⟩

⟨1236⟩⟨1234⟩

⟨2345⟩

⟨2456⟩

⟨2356⟩

⟨1235⟩

⟨3456⟩ ⟨1456⟩

⟨1256⟩

⟨1236⟩⟨1234⟩

⟨2345⟩

⟨2456⟩

⟨1246⟩⟨2346⟩ oo

(g) (h) (i)

Figure 61 The three types of (reduced perfectly orientable bipartite)plabic graphs corresponding to km-dimensional cells of Gr(kn)ge0 for k = 2m = 4 and n = 6 are shown in (a)ndash(c) The associated input and output clus-ters (see text) are shown in (d)ndash(f) and (g)ndash(i) respectively Lines connectingtwo frozen nodes are usually omitted but we include in (g)ndash(i) the dottedlines (having ldquoblack on the rightrdquo in the dual plabic graph) that encode (66)

(627) and (629) (up to signs)

of Gr(46) cluster variables Second again we see that for each graph the collection of

variables that appear precisely constitutes a single cluster of Gr(46) suppressing in each

case the six frozen variables we find ⟨2346⟩ ⟨2356⟩ and ⟨2456⟩ for graph (a) ⟨1235⟩ ⟨2356⟩

and ⟨2456⟩ for graph (b) and ⟨1456⟩ ⟨2346⟩ and ⟨2456⟩ for graph (c) Finally in each case

there is a unique way to label the nodes of the quiver not with cluster variables of the ldquoinputrdquo

cluster algebra Gr(26) as we have done in Fig 61 (d)ndash(f) but with cluster variables of the

ldquooutputrdquo cluster algebra Gr(46) We show these output clusters in Fig 61 (g)ndash(i) using

the convention that the face weight (also known as the cluster X -variable) attached to node

i is prodj abjij where bji is the (signed) number of arrows from j to i

For the sake of completeness we note that there is also (modulo Z6 cyclic transforma-

tions) a single relevant graph with k = 1

for which the boundary measurement is

C = (0 1 f0f1f2f3 f0f1f2 f0f1 f0) (69)

and the solution to C sdotZ = 0 is given by

f0 =⟨2345⟩⟨3456⟩ f1 = minus

⟨2346⟩⟨2345⟩ f2 = minus

⟨2356⟩⟨2346⟩ f3 = minus

⟨2456⟩⟨2356⟩ f4 = minus

⟨3456⟩⟨2456⟩

Again the face weights evaluate (up to signs) to simple ratios of Gr(46) cluster variables

though in this case both the input and output quivers are trivial This graph is an example

of the general feature that one can always uplift an n-point plabic graph relevant to our

analysis to any value of nprime gt n by inserting any number of black lollipops (Graphs with

white lollipops never admit solutions to C sdotZ = 0 for generic Z) In the language of symbology

this is in accord with the expectation that the symbol alphabet of an nprime-particle amplitude

always contains the Znprime cyclic closure of the symbol alphabet of the corresponding n-particle

amplitude

In this section we have seen that solving C sdotZ = 0 induces a map from clusters of Gr(26)

(or subalgebras thereof) to clusters of Gr(46) (or subalgebras thereof) Of course these two

algebras are in any case naturally isomorphic Although we leave a more detailed exposition

for future work we have also checked for m = 2 and n le 10 that every appropriate plabic

graph of Gr(kn) maps to a cluster of Gr(2 n) (or a subalgebra thereof) and moreover that

this map is onto (every cluster of Gr(2 n) is obtainable from some plabic graph of Gr(kn))

However for m gt 2 the situation is more complicated as we see in the next section

63 Towards Non-Cluster Variables

Here we discuss some features of graphs for which the solution to C sdotZ = 0 involves quantities

that are not cluster variables of Gr(mn) A simple example for k = 2 m = 3 n = 6 is the

whose boundary measurement has the form (63) with

c13 = minus0 c15 = minusf0f1(1 + f3) c23 = f0f1f2f3f4f5 c25 = f0f1f3f5

c14 = minusf0f1f2f3 c16 = minusf0(1 + f3) c24 = f0f1f2f3f5 c26 = f0f3f5

The solution to C sdotZ = 0 is given by

f0 =⟨123⟩⟨145⟩

⟨1 times 42 times 35 times 6⟩ f1 = minus⟨146⟩⟨145⟩

f2 =⟨1 times 42 times 35 times 6⟩

⟨234⟩⟨146⟩ f3 = minus⟨234⟩⟨156⟩⟨123⟩⟨456⟩

f4 = minus⟨124⟩⟨456⟩

⟨1 times 42 times 35 times 6⟩ f5 =⟨1 times 42 times 35 times 6⟩

⟨134⟩⟨156⟩

f6 = minus⟨134⟩⟨124⟩

which involves four mutable cluster variables ⟨124⟩ ⟨134⟩ ⟨145⟩ and ⟨146⟩ which comprise

a cluster of the Gr(36) algebra together with the quantity

⟨1 times 42 times 35 times 6⟩ = ⟨123⟩⟨456⟩ minus ⟨234⟩⟨156⟩ (614)

which is not a cluster variable of Gr(36)

We can gain some insight into the origin of (614) by considering what happens under a

square move on f3 This transforms the face weights to

f0 =⟨145⟩⟨456⟩ f1 = minus

⟨146⟩⟨145⟩ f2 = minus

⟨156⟩⟨146⟩ f3 = minus

⟨123⟩⟨456⟩⟨234⟩⟨156⟩

f4 = minus⟨124⟩⟨123⟩ f5 = minus

⟨234⟩⟨134⟩ f6 = minus

⟨134⟩⟨124⟩

leaving four mutable cluster variables ⟨124⟩ ⟨134⟩ ⟨145⟩ and ⟨146⟩ which comprise a cluster

of the Gr(36) algebra However it is not possible to associate a labeled ldquooutputrdquo quiver

to (615) in the usual way as we did for the examples in the previous section

To turn this story around had we started not with (611) but with its square-moved

partner we would have encountered (615) and then noted that performing a square move

back to (611) would necessarily introduce the multiplicative factor

1 + f3 = minus⟨1 times 42 times 35 times 6⟩

⟨234⟩⟨156⟩ (616)

into four of the face weights

The example considered in this section provides an opportunity to comment on the

connection of our work to the study of cluster adjacency for Yangian invariants In [81 65]

it was noted in several examples and conjectured to be true in general that the set of

factors appearing in the denominator of any Yangian invariant with intersection number 1

are cluster variables of Gr(4 n) that appear together in a cluster This was proven to be true

for all Yangian invariants in the m = 2 toy model of SYM theory in [105] and for all m = 4

N2MHV Yangian invariants in [129] We recall from [30 130] that the Yangian invariant

associated to a plabic graph (or to use essentially equivalent language the form associated

to an on-shell diagram) is given by d log f1and⋯andd log fd One of our motivations for studying

the conjecture that the face weights associated to any plabic graph always evaluate on the

solution of C sdotZ = 0 to products of powers of cluster variables was that it would immediately

imply cluster adjacency for Yangian invariants Although the graph (611) violates this

stronger conjecture it does not violate cluster adjacency because on-shell forms are invariant

under square moves [30] Therefore even though ⟨1 times 42 times 35 times 6⟩ appears in individual

face weights of (613) it must drop out of the associated on-shell form because it is absent

from (615)

64 Algebraic Eight-Particle Symbol Letters

One reason it is obvious that the solutions of C sdotZ = 0 cannot always be written in terms of

cluster variables of Gr(mn) is that for graphs with intersection number greater than 1 the

solutions will necessarily involve algebraic functions of Pluumlcker coordinates whereas cluster

variables are always rational

The simplest example of this phenomenon occurs for k = 2 m = 4 and n = 8 for which

there are four relevant plabic graphs in two cyclic classes Let us start with

which has boundary measurement

C =⎛⎜⎜⎜⎝

1 c12 0 c14 c15 c16 c17 c18

0 c32 1 c34 c35 c36 c37 c38

⎞⎟⎟⎟⎠

c12 = f0f1f2f3f4f5f6f7 c14 = minus0 c15 = minusf0f1f2f3f4 (619)

c16 = minusf0f1f2f3 c17 = minusf0f1(1 + f3) c18 = minusf0(1 + f3) (620)

c32 = 0 c34 = f0f1f2f3f4f5f6f8 c35 = f0f1f2f3f4f6f8 (621)

c36 = f0f1f2f3f6f8 c37 = f0f1f3f6f8 c38 = f0f3f6f8 (622)

The solution to C sdotZ = 0 for generic Z isin Gr(48) can be written as

f0 =iquestAacuteAacuteAgrave ⟨7(12)(34)(56)⟩ ⟨1234⟩

a5 ⟨2(34)(56)(78)⟩ ⟨3478⟩ f5 =iquestAacuteAacuteAgravea1a6a9 ⟨3(12)(56)(78)⟩ ⟨5678⟩

a4a7 ⟨6(12)(34)(78)⟩ ⟨3478⟩

f1 = minusiquestAacuteAacuteAgravea7 ⟨8(12)(34)(56)⟩

⟨7(12)(34)(56)⟩ f6 = minusiquestAacuteAacuteAgravea3 ⟨1(34)(56)(78)⟩ ⟨3478⟩

a2 ⟨4(12)(56)(78)⟩ ⟨1278⟩

f2 = minusiquestAacuteAacuteAgravea4 ⟨5(12)(34)(78)⟩ ⟨3478⟩

a8 ⟨8(12)(34)(56)⟩ ⟨3456⟩ f7 = minusiquestAacuteAacuteAgravea2 ⟨4(12)(56)(78)⟩

a1⟨3(12)(56)(78)⟩

f3 =iquestAacuteAacuteAgravea8 ⟨1278⟩ ⟨3456⟩

a9 ⟨1234⟩ ⟨5678⟩ f8 = minusiquestAacuteAacuteAgravea5 ⟨2(34)(56)(78)⟩

a3 ⟨1(34)(56)(78)⟩

f4 = minusiquestAacuteAacuteAgrave ⟨6(12)(34)(78)⟩

a6 ⟨5(12)(34)(78)⟩

⟨a(bc)(de)(fg)⟩ equiv ⟨abde⟩⟨acfg⟩ minus ⟨abfg⟩⟨acde⟩ (624)

and the nine ai provide a (multiplicative) basis for the algebraic letters of the eight-particle

symbol alphabet that contain the four-mass box square rootradic

∆1357 as reviewed in Ap-

pendix 67

The nine face weights shown in (623) satisfy prod fα = 1 so only eight are multiplicatively

independent It is easy to check that they remain multiplicatively independent if one sets

all of the Pluumlcker coordinates and brackets of the form (624) to one Therefore the fα

(multiplicatively) only span an eight-dimensional subspace of the full nine-dimensional space

spanned by the nine algebraic letters We could try building an eight-particle alphabet by

taking any subset of eight of the face weights as basis elements (ie letters) but we would

always be one letter short

Fortunately there is a second plabic graph relevant toradic

∆1357 the one obtained by

performing a square move on f3 of (617) As is by now familiar performing the square

move introduces one new multiplicative factor into the face weights

1 + f3 =iquestAacuteAacuteAgrave ⟨1256⟩⟨3478⟩

a9⟨1234⟩⟨5678⟩ (625)

which precisely supplies the ninth missing letter To summarize the union of the nine face

weights associated to the graph (617) and the nine associated to its square-move partner

multiplicatively span the nine-dimensional space ofradic

∆1357-containing symbol letters in the

eight-particle alphabet of [23]

The same story applies to the graphs obtained by cycling the external indices on (617)

by onemdashtheir face weights provide all nine algebraic letters involvingradic

∆2468

Of course it would be very interesting to thoroughly study the numerous plabic graphs

65 Discussion 101

relevant tom = 4 n = 8 that have intersection number 1 In particular it would be interesting

to see if they encode all 180 of the rational (ie Gr(48) cluster variable) symbol letters

of [23] and whether they generate additional cluster variables such as those obtained from

the constructions of [124 122 108]

Before concluding this section let us comment briefly on ldquokrdquo since one may be confused

why the plabic graph (617) which has k = 2 and is therefore associated to an N2MHV

leading singularity could be relevant for symbol alphabets of NMHV amplitudes The

symbol letters of an NkMHV amplitude reveal all of its singularities including multiple

discontinuities that can be accessed only after a suitable analytic continuation Physically

these are computed by cuts involving lower-loop amplitudes that can have kprime gt k Indeed

the expectation that symbol letters of lower-loop higher-k amplitudes influence those of

higher-loop lower-k amplitudes is manifest in the Q-bar equation technology [22 131 132]

underlying the computation of [23] Moreover there is indirect evidence [133] that the symbol

alphabet of the L-loop n-particle NkMHV amplitude in SYM theory is independent of both k

and L (beyond certain accidental shortenings that may occur for small k or L) This suggests

that for the purpose of applying our construction to ldquothe n-particle symbol alphabetrdquo one

should consider all relevant n-point plabic graphs regardless of k

65 Discussion

The problem of ldquoexplainingrdquo the symbol alphabets of n-particle amplitudes in SYM theory

apparently requires for n gt 7 a mechanism for identifying finite sets of functions on Gr(4 n)

that include some subset of the cluster variables of the associated cluster algebra together

with certain non-cluster variables that are algebraic functions of the Pluumlcker coordinates

In this paper we have initiated the study of one candidate mechanism that manifestly

satisfies both criteria and may be of independent mathematical interest Specifically to

every (reduced perfectly oriented) plabic graph of Gr(kn)ge0 that parameterizes a cell of

dimensionmk one can naturally associate a collection ofmk functions of Pluumlcker coordinates

on Gr(mn)

We have seen that for some graphs the output of this procedure is naturally associated

to a seed of the Gr(mn) cluster algebra for some graphs the output is a clusterrsquos worth of

cluster variables that do not correspond to a seed but rather behave ldquobadlyrdquo under mutations

(this means they transform into things which are not cluster variables under square moves

on the input plabic graph) and finally for some graphs the output involves non-cluster

variables including when the intersection number is greater than 1 algebraic functions

We leave a more thorough investigation of this problem for future work The ldquosmoking

gunrdquo that this procedure may be relevant to symbol alphabets in SYM theory is provided

by the example discussed in Sec 64 which successfully postdicts precisely the 18 multi-

plicatively independent algebraic letters that were recently found to appear in the two-loop

eight-particle NMHV amplitude [23] Our construction provides an alternative to the similar

postdiction made recently in [124]

It is interesting to note that since form = 4 n = 8 there are no other relevant plabic graphs

having intersection number gt 1 beyond those already considered Sec 64 our construction

has no room for any additional algebraic letters for eight-particle amplitudes Therefore if

it is true that the face weights of plabic graphs evaluated on the locus C sdot Z = 0 provide

symbol alphabets for general amplitudes then it necessarily follows that no eight-particle

65 Discussion 103

amplitude at any loop order can have any algebraic symbol letters beyond the 18 discovered

in [23]

At first glance this rigidity seems to stand in contrast to the constructions of [122 124

108] which each involve some amount of choicemdashhaving to do with how coarse or fine one

chooses onersquos tropical fan or equivalently how many factors to include in the Minkowski

sum when building the dual polytope But in fact our construction has a choice with a

similar smell When we say that we start with the C-matrix associated to a plabic graph

that automatically restricts us to very special clusters of Gr(kn)mdashthose that contain only

Pluumlcker coordinates Clusters containing more complicated non-Pluumlcker cluster variables

are not associated to plabic graphs One certainly could contemplate solving the C sdot Z = 0

equations for C given by a ldquonon-plabicrdquo cluster parameterization of some cell of Gr(kn)ge0

and it would be interesting to map out the landscape of possibilities

It has been a long-standing problem to understand the precise connection between the

Gr(kn) cluster structure exhibited [30] at the level of integrands in SYM theory and the

Gr(4 n) cluster structure exhibited [5] by integrated amplitudes It was pointed out in [125]

that the C sdot Z = 0 equations provide a concrete link between the two and our results shed

some initial light on this intriguing but still very mysterious problem In some sense we can

think of the ldquoinputrdquo and ldquooutputrdquo clusters defined in Sec 62 as ldquointegrandrdquo and ldquointegratedrdquo

clusters with respect to the auxiliary Grassmannian space (See the last paragraph of Sec 64

for some comments on why k ldquodisappearsrdquo upon integration) Although we have seen that

the latter are not in general clusters at all the example of Sec 64 suggests that they may

be even better exactly what is needed for the symbol alphabets of SYM theory

Note Added The preprint [134] appeared on arXiv shortly after and has significant overlap

with the result presented in this note

66 Some Six-Particle Details

Here we assemble some details of the calculation for graphs (b) and (c) of Fig 61 The

boundary measurement for graph (b) has the form (63) with

c15 = minusf0f1(1 + f4 + f2f4 + f4f6 + f2f4f6) c25 = f0f1f4f6f8(1 + f2)

c16 = minusf0(1 + f4 + f4f6) c26 = f0f4f6f8

f(b)0 = minus⟨1235⟩

⟨2356⟩ f(b)1 = minus⟨1236⟩

⟨1235⟩ f(b)2 = ⟨1234⟩⟨2356⟩

⟨2345⟩⟨1236⟩

f(b)3 = minus⟨1235⟩

⟨1234⟩ f(b)4 = ⟨2345⟩⟨1256⟩

⟨1235⟩⟨2456⟩ f(b)5 = minus⟨2456⟩

⟨2356⟩

f(b)6 = ⟨2356⟩⟨1456⟩

⟨3456⟩⟨1256⟩ f(b)7 = minus⟨3456⟩

⟨2456⟩ f(b)8 = minus⟨2456⟩

⟨1456⟩

The boundary measurement for graph (c) has

c14 = minusf0f1f2f3(1 + f6 + f4f6) c24 = f0f1f2f3f6f8(1 + f4)

c15 = minusf0f1f2(1 + f6) c25 = f0f1f2f6f8

and the solution to C sdotZ = 0 is

f(c)0 = minus⟨1234⟩

⟨2346⟩ f(c)1 = minus⟨2346⟩

⟨2345⟩ f(c)2 = ⟨2345⟩⟨1246⟩

⟨1234⟩⟨2456⟩

f(c)3 = minus⟨1256⟩

⟨1246⟩ f(c)4 = ⟨2456⟩⟨1236⟩

⟨2346⟩⟨1256⟩ f(c)5 = minus⟨1246⟩

⟨1236⟩

f(c)6 = ⟨1456⟩⟨2346⟩

⟨3456⟩⟨1246⟩ f(c)7 = minus⟨3456⟩

⟨2456⟩ f(c)8 = minus⟨2456⟩

⟨1456⟩

67 Notation for Algebraic Eight-Particle Symbol Letters

Here we review some details from [23] to set the notation used in Sec 64 There are two

basic square roots of four-mass box type that appear in symbol letters of eight-particle

amplitudes These areradic

∆1357 andradic

∆2468 with

∆1357 = (⟨1256⟩⟨3478⟩ minus ⟨1278⟩⟨3456⟩ minus ⟨1234⟩⟨5678⟩)2 minus 4⟨1234⟩⟨3456⟩⟨5678⟩⟨1278⟩ (630)

and ∆2468 given by cycling every index by 1 (mod 8)

The eight-particle symbol alphabet can be written in terms of 180 Gr(48) cluster vari-

ables plus 9 letters that are rational functions of Pluumlcker coordinates andradic

∆1357 and

another 9 that are rational functions of Pluumlcker coordinates andradic

∆2468 We focus on the

first 9 as the latter is a cyclic copy of the same story

There are many different ways to write a basis for the eight-particle symbol alphabet

as the various letters one can form satisfy numerous multiplicative identities among each

other For the sake of definiteness we use the basis provided in the ancillary Mathematica

file attached to [23] The choice of basis made there starts by defining

2(1 + u minus v +

radic(1 minus u minus v)2 minus 4uv)

2(1 + u minus v minus

in terms of the familiar eight-particle cross ratios

u = ⟨1278⟩⟨3456⟩⟨1256⟩⟨3478⟩ v = ⟨1234⟩⟨5678⟩

⟨1256⟩⟨3478⟩ (632)

Note that the square root appearing in (631) is

radic(1 minus u minus v)2 minus 4uv =

radic∆1357

⟨1256⟩⟨3478⟩ (633)

Then a basis for the algebraic letters of the symbol alphabet is given by

a1 =xa minus zxa minus z

∣irarri+6

a2 =xb minus zxb minus z

∣irarri+6

a3 = minusxc minus zxc minus z

∣irarri+6

a4 = minusxd minus zxd minus z

∣irarri+4

∣irarri+6

a6 =xe minus zxe minus z

∣irarri+4

∣irarri+6

z a9 =

1 minus z1 minus z

where the xrsquos are defined in (13) of [23] While the overall sign of a symbol letter is irrelevant

we have taken the liberty of putting a minus sign in front of a3 a4 and a5 to ensure that

each of the nine ai indeed each individual factor appearing in (623) is positive-valued for

Z isin Gr(48)gt0

Bibliography

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[2] S J Parke and T R Taylor ldquoAn Amplitude for n Gluon Scatteringrdquo Phys Rev Lett

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[8] S Pasterski S H Shao and A Strominger ldquoFlat Space Amplitudes and Conformal

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[9] S Pasterski and S H Shao ldquoA Conformal Basis for Flat Space Amplitudesrdquo

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[10] R Penrose ldquoThe Apparent shape of a relativistically moving sphererdquo Proc Cambridge

Phil Soc 55 137-139 (1959) doi101017S0305004100033776

[11] J M Drummond J M Henn and J Plefka ldquoYangian symmetry of scattering am-

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[12] A B Goncharov M Spradlin C Vergu and A Volovich ldquoClassical Polyloga-

rithms for Amplitudes and Wilson Loopsrdquo Phys Rev Lett 105 151605 (2010)

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[13] S Caron-Huot L J Dixon J M Drummond F Dulat J Foster Ouml Guumlrdoğan

M von Hippel A J McLeod and G Papathanasiou ldquoThe Steinmann Cluster Boot-

strap for N = 4 Super Yang-Mills Amplitudesrdquo PoS CORFU2019 003 (2020)

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[14] M Srednicki ldquoQuantum field theoryrdquo

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[23] S He Z Li and C Zhang ldquoTwo-loop Octagons Algebraic Letters and Q Equa-

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[26] I Prlina M Spradlin J Stankowicz and S Stanojevic ldquoBoundaries of Amplituhedra

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[27] N Arkani-Hamed F Cachazo C Cheung and J Kaplan ldquoA Duality For The S Matrixrdquo

JHEP 03 020 (2010) doi101007JHEP03(2010)020 [arXiv09075418 [hep-th]]

[28] J M Drummond and L Ferro ldquoThe Yangian origin of the Grassmannian integralrdquo

[29] A Postnikov ldquoTotal Positivity Grassmannians and Networksrdquo httpmathmit

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[30] N Arkani-Hamed J L Bourjaily F Cachazo A B Goncharov A Post-

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[hep-th] A Ashtekar ldquoAsymptotic Quantization Based On 1984 Naples Lec-

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drum ldquo4D scattering amplitudes and asymptotic symmetries from 2D CFTrdquo JHEP

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th0303006] G Barnich and C Troessaert ldquoSymmetries of asymptotically flat 4 di-

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scattering amplitudes and asymptotic symmetries from 2D CFTrdquo JHEP 1701 112

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its of Gluons and Gravitonsrdquo JHEP 1507 135 (2015) doi101007JHEP07(2015)135

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Lorentzian OPE inversion formulardquo arXiv171103816 [hep-th]

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Analogs of the SYK Modelrdquo JHEP 1708 146 (2017) doi101007JHEP08(2017)146

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[65] J Mago A Schreiber M Spradlin and A Volovich ldquoYangian invariants and cluster

adjacency in N = 4 Yang-Millsrdquo JHEP 10 099 (2019) doi101007JHEP10(2019)099

[66] J Golden and M Spradlin ldquoThe differential of all two-loop MHV amplitudes in

N = 4 Yang-Mills theoryrdquo JHEP 1309 111 (2013) doi101007JHEP09(2013)111

[67] J Golden and M Spradlin ldquoA Cluster Bootstrap for Two-Loop MHV Amplitudesrdquo

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athanasiou and B Verbeek ldquoMulti-Regge kinematics and the moduli space of Riemann

spheres with marked pointsrdquo JHEP 1608 152 (2016) doi101007JHEP08(2016)152

[69] J Golden M F Paulos M Spradlin and A Volovich ldquoCluster Polylogarithms for

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[71] T Harrington and M Spradlin ldquoCluster Functions and Scattering Amplitudes

for Six and Seven Pointsrdquo JHEP 1707 016 (2017) doi101007JHEP07(2017)016

[72] J Golden and A J Mcleod ldquoCluster Algebras and the Subalgebra Con-

structibility of the Seven-Particle Remainder Functionrdquo JHEP 1901 017 (2019)

[73] J Drummond J Foster and Ouml Guumlrdoğan ldquoCluster Adjacency Properties of Scattering

Amplitudes in N = 4 Supersymmetric Yang-Mills Theoryrdquo Phys Rev Lett 120 no

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[74] S Caron-Huot and S He ldquoJumpstarting the All-Loop S-Matrix of Planar N = 4 Super

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[75] L J Dixon and M von Hippel ldquoBootstrapping an NMHV amplitude through three

loopsrdquo JHEP 1410 065 (2014) doi101007JHEP10(2014)065 [arXiv14081505 [hep-

[76] J M Drummond G Papathanasiou and M Spradlin ldquoA Symbol of Uniqueness

The Cluster Bootstrap for the 3-Loop MHV Heptagonrdquo JHEP 1503 072 (2015)

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[77] L J Dixon M von Hippel and A J McLeod ldquoThe four-loop six-gluon NMHV ratio

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[hep-th]]

[78] S Caron-Huot L J Dixon A McLeod and M von Hippel ldquoBootstrapping a Five-Loop

Amplitude Using Steinmann Relationsrdquo Phys Rev Lett 117 no 24 241601 (2016)

[79] L J Dixon M von Hippel A J McLeod and J Trnka ldquoMulti-loop positiv-

ity of the planar N = 4 SYM six-point amplituderdquo JHEP 1702 112 (2017)

[80] L J Dixon J Drummond T Harrington A J McLeod G Papathanasiou and

M Spradlin ldquoHeptagons from the Steinmann Cluster Bootstraprdquo JHEP 1702 137

[81] J Drummond J Foster and Ouml Guumlrdoğan ldquoCluster adjacency beyond MHVrdquo JHEP

1903 086 (2019) doi101007JHEP03(2019)086 [arXiv181008149 [hep-th]]

[82] J Drummond J Foster Ouml Guumlrdoğan and G Papathanasiou ldquoCluster

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[83] S Caron-Huot L J Dixon F Dulat M von Hippel A J McLeod and G Papathana-

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Mills Theoryrdquo Phys Lett 100B 245 (1981) doi1010160370-2693(81)90326-9

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[94] N Arkani-Hamed F Cachazo and C Cheung ldquoThe Grassmannian Origin Of Dual

Superconformal Invariancerdquo JHEP 1003 036 (2010) doi101007JHEP03(2010)036

[95] N Arkani-Hamed J Bourjaily F Cachazo and J Trnka ldquoLocal Spacetime Physics

from the Grassmannianrdquo JHEP 1101 108 (2011) doi101007JHEP01(2011)108

[96] N Arkani-Hamed J Bourjaily F Cachazo and J Trnka ldquoUnification of Residues

and Grassmannian Dualitiesrdquo JHEP 1101 049 (2011) doi101007JHEP01(2011)049

[97] J M Drummond and L Ferro ldquoYangians Grassmannians and T-dualityrdquo JHEP 1007

[98] S K Ashok and E DellrsquoAquila ldquoOn the Classification of Residues of the Grassman-

nianrdquo JHEP 1110 097 (2011) doi101007JHEP10(2011)097 [arXiv10125094 [hep-

[99] J L Bourjaily ldquoPositroids Plabic Graphs and Scattering Amplitudes in Mathematicardquo

[100] V P Nair ldquoA Current Algebra for Some Gauge Theory Amplitudesrdquo Phys Lett B

214 215 (1988) doi1010160370-2693(88)91471-2

BIBLIOGRAPHY 123

[101] J M Drummond and J M Henn ldquoAll tree-level amplitudes in N = 4 SYMrdquo JHEP

0904 018 (2009) doi1010881126-6708200904018 [arXiv08082475 [hep-th]]

[102] L Lippstreu J Mago M Spradlin and A Volovich ldquoWeak Separation Positivity and

Extremal Yangian Invariantsrdquo JHEP 09 093 (2019) doi101007JHEP09(2019)093

[103] J Mago A Schreiber M Spradlin and A Volovich ldquoA Note on One-loop Cluster

Adjacency in N = 4 SYMrdquo [arXiv200507177 [hep-th]]

[104] M Gekhtman M Z Shapiro and A D Vainshtein Mosc Math J 3 no3 899 (2003)

[arXivmath0208033 [mathQA]]

[105] T Łukowski M Parisi M Spradlin and A Volovich ldquoCluster Adjacency for

m = 2 Yangian Invariantsrdquo JHEP 10 158 (2019) doi101007JHEP10(2019)158

[106] Ouml Guumlrdoğan and M Parisi ldquoCluster patterns in Landau and Leading Singularities

via the Amplituhedronrdquo [arXiv200507154 [hep-th]]

[107] J Drummond J Foster Ouml Guumlrdoğan and C Kalousios ldquoTropical fans scattering

equations and amplitudesrdquo [arXiv200204624 [hep-th]]

[108] N Henke and G Papathanasiou ldquoHow tropical are seven- and eight-particle ampli-

tudesrdquo [arXiv191208254 [hep-th]]

[109] B Leclerc and A Zelevinsky ldquoQuasicommuting families of quantum Pluumlcker coordi-

natesrdquo Adv Math Sci (Kirillovrsquos seminar) AMS Translations 181 85 (1998)

124 BIBLIOGRAPHY

[111] S Caron-Huot L J Dixon F Dulat M Von Hippel A J McLeod and G Pap-

athanasiou ldquoThe Cosmic Galois Group and Extended Steinmann Relations for Pla-

nar N = 4 SYM Amplitudesrdquo JHEP 09 061 (2019) doi101007JHEP09(2019)061

[112] Z Bern L J Dixon and V A Smirnov ldquoIteration of planar amplitudes in maximally

supersymmetric Yang-Mills theory at three loops and beyondrdquo Phys Rev D 72 085001

(2005) doi101103PhysRevD72085001 [arXivhep-th0505205 [hep-th]]

[113] L F Alday D Gaiotto and J Maldacena ldquoThermodynamic Bubble Ansatzrdquo JHEP

[114] L F Alday J Maldacena A Sever and P Vieira ldquoY-system for Scattering

Amplitudesrdquo J Phys A 43 485401 (2010) doi1010881751-81134348485401

[115] J Drummond J Henn G Korchemsky and E Sokatchev ldquoGeneralized

unitarity for N=4 super-amplitudesrdquo Nucl Phys B 869 452-492 (2013)

[116] H Elvang D Z Freedman and M Kiermaier ldquoDual conformal symmetry

of 1-loop NMHV amplitudes in N = 4 SYM theoryrdquo JHEP 03 075 (2010)

BIBLIOGRAPHY 125

[117] A B Goncharov ldquoGalois symmetries of fundamental groupoids and noncommutative

geometryrdquo Duke Math J 128 no2 209 (2005) [arXivmath0208144 [mathAG]]

[118] J Mago A Schreiber M Spradlin and A Volovich ldquoSymbol Alphabets from Plabic

Graphsrdquo [arXiv200700646 [hep-th]]

[119] S Fomin and A Zelevinsky ldquoCluster algebras II Finite type classificationrdquo Invent

Math 154 no 1 63 (2003) [arXivmath0208229]

[120] A Hodges Twistor Newsletter 5 1977 reprinted in Advances in twistor theory

eds LP Hugston and R S Ward (Pitman 1979)

[121] G rsquot Hooft and M J G Veltman ldquoScalar One Loop Integralsrdquo Nucl Phys B 153

365 (1979)

[122] N Arkani-Hamed T Lam and M Spradlin ldquoNon-perturbative geometries for planar

N = 4 SYM amplitudesrdquo [arXiv191208222 [hep-th]]

[123] D Speyer and L Williams ldquoThe tropical totally positive Grassmannianrdquo J Algebr

Comb 22 no 2 189 (2005) [arXivmath0312297]

[124] J Drummond J Foster Ouml Guumlrdoğan and C Kalousios ldquoAlgebraic singularities of

scattering amplitudes from tropical geometryrdquo [arXiv191208217 [hep-th]]

[125] N Arkani-Hamed ldquoPositive Geometry in Kinematic Space (I) The Amplituhedronrdquo

Spacetime and Quantum Mechanics Master Class Workshop Harvard CMSA October

30 2019 httpswwwyoutubecomwatchv=6TYKM4a9ZAUampt=3836

126 BIBLIOGRAPHY

[126] G Muller and D Speyer ldquoCluster algebras of Grassmannians are locally acyclicrdquo

Proc Am Math Soc 144 no 8 3267 (2016) [arXiv14015137 [mathCO]]

[127] K Serhiyenko M Sherman-Bennett and L Williams ldquoCombinatorics of cluster struc-

tures in Schubert varietiesrdquo arXiv181102724 [mathCO]

[128] M F Paulos and B U W Schwab ldquoCluster Algebras and the Positive Grassmannianrdquo

JHEP 10 031 (2014) [arXiv14067273 [hep-th]]

[129] Ouml Guumlrdoğan and M Parisi [arXiv200507154 [hep-th]]

[130] N Arkani-Hamed H Thomas and J Trnka ldquoUnwinding the Amplituhedron in Bi-

naryrdquo JHEP 01 016 (2018) [arXiv170405069 [hep-th]]

Yang-Millsrdquo JHEP 07 174 (2012) [arXiv11121060 [hep-th]]

[132] M Bullimore and D Skinner ldquoDescent Equations for Superamplitudesrdquo

[133] I Prlina M Spradlin and S Stanojevic ldquoAll-loop singularities of scattering am-

plitudes in massless planar theoriesrdquo Phys Rev Lett 121 no8 081601 (2018)

[134] S He and Z Li ldquoA Note on Letters of Yangian Invariantsrdquo [arXiv200701574 [hep-th]]

Abstract

Acknowledgements

Introduction

Celestial Amplitudes and Holography

Conformal Primary Wavefunctions

Celestial Amplitudes

Cluster Algebras in planar N=4 super Yang-Mills Theory

Momentum Twistors and Dual Conformal Symmetry

Cluster Algebras and Cluster Adjacency

Symbols Alphabet and Plabic Graphs

Yangian Invariants and Leading Singularities

Plabic Graphs and Cluster Algebras

Tree-level Gluon Amplitudes on the Celestial Sphere

Gluon amplitudes on the celestial sphere

n-point MHV

Integrating out one i

Integrating out momentum conservation -functions

Integrating the remaining i

6-point MHV

n-point NMHV

n-point NkMHV

Generalized hypergeometric functions

Celestial Amplitudes Conformal Partial Waves and Soft Limits

Scalar Four-Point Amplitude

Gluon Four-Point Amplitude

Soft limits

Conformal Partial Wave Decomposition

Inner Product Integral

Yangian Invariants and Cluster Adjacency in N=4 Yang-Mills

Cluster Coordinates and the Sklyanin Poisson Bracket

An Adjacency Test for Yangian Invariants

NMHV

NNMHV

NNNMHV and Higher

Explicit Matrices for k=2

A Note on One-loop Cluster Adjacency in N=4 SYM

Cluster Adjacency and the Sklyanin Bracket

One-loop Amplitudes

BDS- and BDS-like Subtracted Amplitudes

NMHV Amplitudes

Cluster Adjacency of One-Loop NMHV Amplitudes

The Symbol and Steinmann Cluster Adjacency

Final Entry and Yangian Invariant Cluster Adjacency

Cluster Adjacency and Weak Separation

n-point NMHV Transcendental Functions

Symbol Alphabets from Plabic Graphs

A Motivational Example

Six-Particle Cluster Variables

Towards Non-Cluster Variables

Algebraic Eight-Particle Symbol Letters

Discussion

Some Six-Particle Details

Notation for Algebraic Eight-Particle Symbol Letters

Chung-I Tan Reader

Andrew G Campbell

BROWN UNIVERSITY

Abstract

Anastasia Volovich

theory

Curriculum Vitae

Research

Education

Teaching

Schools and Talks

Jun 2019 TASI 2019

Additional Skills

Acknowledgements

Contents

Abstract v

Acknowledgements xi

1 Introduction 1

22 n-point MHV 19

224 6-point MHV 24

23 n-point NMHV 25

24 n-point NkMHV 28

33 Soft limits 43

421 NMHV 60

422 N2MHV 62

65 Discussion 101

List of Figures

to ⟨k(l l + 1)(mm + 1)(k minus 1k + 1)⟩ 80

to ⟨k(l l + 1)(mm + 1)(pp + 1)⟩ 81

List of Tables

I love you all

Chapter 1

Introduction

amplitudes

calculated

(11) transforms as

az + bcz + d

) = ∣cz + d∣2∆φ∆plusmn(X z z) (13)

prodj=1int

0dωj ω

az + bcz + d

prodj=1

gluon amplitudes

microi+1 (16)

〈2345〉

〈2346〉〈2356〉〈2456〉〈3456〉

〈1234〉〈1236〉〈1256〉〈1456〉

akaprimek = prod

i∣bikgt0

aminusbiki (112)

Y (Z ∣η) = int4k

prodi=1

d log fi4

prodI=1

prodα=1

suma=1

Cαa(Z ∣η)aI) (113)

prodαisinp

fα (114)

fi = 1 (115)

coordinates

Chapter 2

Celestial Sphere

radicωiωjzij (22)

prodj=1int

0dωj ω

azj + bczj + d

prodj=1

22 n-point MHV 19

22 n-point MHV

ratio ωi

dωiωi

(ωisi

j=1 iλj

s1+iλii

⎛⎜⎝

proda=1anei

intinfin

0dωa ω

intinfin

dωiωi

(ωisi

= 2πδ(z) (29)

sumj=1

λj⎞⎠s1+iλii

⎛⎜⎝

proda=1anei

intinfin

0dωa ω

⎞⎟⎠A`1⋯`n(si ωl zj zj) (210)

δ4(ε1s1q1 +n

sumi=2

εiωiqi) =4

prodj=nminus3

U+nminus4

sumi=2

U) (212)

22 n-point MHV 21

ui =z31zi2z32zi1

for i = 45 n (216)

s1 =∣z32∣

∣z31∣ ∣z12∣ s2 =

∣z31∣∣z32∣ ∣z21∣

and si =∣z12∣

∣z1i∣ ∣zi2∣for i = 3 n (217)

to demand Uij

prodij 1lt0(Uij

12 siλ1+21

nminus4

proda=2int

0dωa ω

1lt0(Uij

ωi = siU1n

uiminus1

i = 23 n minus 4 (220)

which leads to

12siλ1+21 siλ2+2

2 siλ33 siλnn

z23z34zn1U1nδ(

sumj=1

λj) ϕ(α x)prodij

1lt0(Uij

U) (221)

22 n-point MHV 23

prodj=1

P2and and dPnminus4

Pnminus4

αn =1 + iλ1 x0 i =U1i

U1n xjminus1 i =

Ujnminus U1i

U1n x0n = minus

U1n xjminus1n =

ϕ(α x) equivc2

c1intu1ge0u3ge0

1minussuma uage0

prodj=1

Pnminus1

U1n xij =

Ujnminus4+i

x0n = minusU

U1n xin =

U1n x01 = 1

mannian

224 6-point MHV

)iλ1+1

( minusUU16

)iλ2+2

iλ3minus1

iλ4minus1

iλ5minus1

times int1

prodi=1

23 n-point NMHV 25

U16U2i+2for i = 123 (227)

23 n-point NMHV

nminus1

sumj=4

⟨1∣P2jPj+12∣3⟩3

P 22jP

⟨j + 1 j⟩[2∣P2j ∣j + 1⟩⟨j∣Pj+12∣2]

equivnminus1

sumj=4

Mj (228)

⟨34⟩⟨45⟩⋯⟨n minus 1 n⟩⟨n1⟩(⟨12⟩[24]⟨43⟩ + ⟨13⟩[34]⟨43⟩)3

(⟨23⟩[23] + ⟨24⟩[24] + ⟨34⟩[34])⟨34⟩[34]times

times ⟨54⟩([23]⟨35⟩ + [24]⟨45⟩)(⟨43⟩[32]) (229)

M4 =ω1ω4(ε2z12z24ω2+ε3z13z34ω3)3

of variables

ωi = siuiminus1

i = 2 n minus 4 (231)

M4 sim δ⎛⎝n

sumj=1

λj⎞⎠

prodni=1 siλii

z12z13z45z4ns21s

z34zn1UF(αx)prod

1lt0(Uij

U) (232)

nminus5

proda=1

nminus5

prodj=1

uiλj+1

times7

prodi=1

23 n-point NMHV 27

U xj4 =

U j = 1 n minus 5

x05 =U1nminus2

U xj5 =

U j = 1 n minus 5 (235)

x06 =U1nminus1

U xj6 =

U j = 1 n minus 5

x07 =U1n

U xj7 =

Ujn minusU1n

U j = 1 n minus 5

6- and 7-point NMHV

u1uiλ2

prodi=2

(x0i + u1x1i)αi (236)

u2uiλ2

1 uiλ32 (u1x10 + u2x20 + u1u2x30 + u2

times7

prodi=1

24 n-point NkMHV

24 n-point NkMHV 29

N2MHV amplitude

F41F23

⟨1∣P26P72P35P63∣4⟩3

P 226P

⟨76⟩[23]⟨65⟩[2∣P26∣7⟩⟨6∣P72∣2][3∣P35∣6⟩⟨5∣P63∣3]

u3uα1

1 uα22 uα3

prodi=4

times17

prodj=14

2x8j + u23x9j)αj

NkMHV amplitude

nminus4

prodl=2

dululuαl

⎛⎝

1 minusnminus4

sumj=2

uj⎞⎠

P32k (prod

(P i1)αi)

⎛⎝prodj

(Pj2)αj

⎞⎠

prodj=0

Pj(u)αjdϕ (243)

dϕ =dPj1Pj1

and and dPjnPjn

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

x00 x0m

x10 x1m

⋮ ⋱ ⋮

xn0 xnm

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

gauge fixed form

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

1 0 0 1 1 1

⋮ ⋱ minus1 ⋮ ⋮ ⋮

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ways eq (324) of [38]

prodi=1

suml=1

prodj=1

(1 minusn

sumi=1

xijui)minusβj

sumi=1

αi) sdotn

prodi=1

Γ(αi) (248)

prodi=1

mminusnminus1

suml=1

prodj=1

sumi=1

xjiui)minusαj

sumi=1

prodi=1

Γ(βi) (249)

γ minusn

sumi=1

sumj=1

βj notin Z(250)

sumi=0

xijpartϕ

sumj=0

xijpartϕ

(3) part2ϕ

k du1duk (252)

Chapter 3

prodl=1

prodj=1

sumi=1

ki)( 1

(k1+k2)2+m2+ 1

(k1+k3)2+m2+ 1

(k1+k4)2+m2)

(hj hj) =1

A4(zj zj) =( z24

z14)h12 ( z14

z13)h34

zh1+h212 zh3+h4

z14)h12 ( z14

z13)h34

zh1+h212 zh3+h4

A4(z z) (35)

z = z12z34

z13z24 z = z12z34

compute

τminus2∆4A4(0 z11τ) (37)

prodi=1

⎛⎝

sumj=1

kj⎞⎠

m2 minus 4ω1ω2∣z∣2 (38)

δ(4)⎛⎝

sumj=1

kj⎞⎠= 4τ2

ω1δ(iz minus iz)

prodi=2

ωlowast2 = ω1

2 (310)

0dω1ω

∆1minus21

prodi=2

m2 minus 4z2

zminus1ω21

= g2 (im2)2αminus2

z minus 1)α

+ zα) (312)

sumi=1

hi minus 2 (313)

constructed in [51]

2(h2+h3)A13harr24

4 (z) 0 lt z lt 1 (315)

as well as

4 (1z)

z minus 1) 0 lt z lt 1 (316)

2 (im2)2αminus2

2 sin(πα) intC

hi(z z)

2= 1 minus h12 + n

2= 1 minus h34 + n

2= h12 minus n

2= h34 minus n (318)

with n = 0123

hi(z z) (319)

4hi+hj

Γ (hij + ∆2)Γ (∆

and Ψ∆hi(z z) is

Ψ∆hi(z z) equiv

2 minus h34)Γ (∆

2 + h12)Γ (1 minus ∆2 + h34)

Ψ∆hi(z z) (322)

infinsumJ=0

4π2Ψhh

+(h12 harr h34))

33 Soft limits 43

∆ minus J2

2= 1 minus h34 + n

∆ + J2

2= h34 minus n

33 Soft limits

Single soft limits

limωkrarr0

A`k=+1n = ⟨k minus 1k + 1⟩

limωkrarr0

δ(x) = 1

2limλrarr0

limλkrarr0

An(zj zj) =1

prodj=1jnek

intinfin

0dωj ω

iλjj int

limλkrarr0

AJk=+1n = 1

zkminus1k+1

limλkrarr0

AJk=minus1n = 1

zkminus1k+1

Double soft limits

33 Soft limits 45

⟨n∣1 + 2∣3] ( [13]3⟨n3⟩[12][23]s123

+ ⟨n2⟩3[n3]⟨n1⟩⟨12⟩sn12

we will write

An(zj zj) =n

prodj=1jnekl

intinfin

0dωj ω

iλjj int

0dr int

DS(kJk lJl) = 1

r=0]εrarr0

2(iλ1 + iλ2)ε21

zn1z23( 1

zn3z2n

z12z2n+ 1

z3nz31

z12z31) (338)

sumi=1

4ω1⋯ω6

ω21ω

ω23ω

26(ω4z46z34+ω5z56z35)3

(ω3ω4z34z34+ω3ω5z35z35+ω4ω5z45z45)

z12z16z34z45 (ω3z23z35 + ω4z24z45)

⎤⎥⎥⎥⎥⎥⎦

2(iλ3 + iλ4)ε21

z23z45( 1

z25z41

z34z42+ 1

z52z53

25z216z25z61

(z15z61

z25z26)iλ2minus1

(z12z16

z25z56)iλ5+1

(z15z12

z56z26)iλ6+1

⟨2∣3 + 4∣5] ( [35]3⟨25⟩[34][45]s345

+ ⟨24⟩3[25]⟨23⟩⟨34⟩s234

) (342)

intCd∆ Ψhh

hihi(z z)⟩ (343)

(zz)2G(z z)F (z z) (344)

given by

F+(z z) =1

zh34 zh342F1 (

1 minus h + h34 h + h34

1 + h12 + h341

z) 2F1 (

1 minus h + h34 h + h34

1 + h12 + h341

z) (346)

z) 2F1 (

⟨Ψhh

hihi(z z)⟩ = (2π)2

⟨A4(z z)Ψhh

hihi(z z) (349)

2F1(a b c z) =Γ(c)

obtain

⟨A4(z z)Ψhh

⎡⎢⎢⎢⎢⎢⎣

⎛⎜⎝

⎤⎥⎥⎥⎥⎥⎥⎦

sin(2πJ)2πJ

= 1 J = 0

0 J ne 0 (353)

hihi∣J=0

which simplifies to

2 minus h34)Γ(2 minus∆)Γ(∆) (354)

⟨A4(z z)Ψhh

Chapter 4

⟨1 2 6 7⟩⟨2 3 4 5⟩ minus ⟨1 2 4 5⟩⟨2 3 6 7⟩ (42)

cluster coordinate

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

1 0 0 0 y15 ⋯ y1

0 1 0 0 y25 ⋯ y2

0 0 1 0 y35 ⋯ y3

0 0 0 1 y45 ⋯ y4

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

yIa yJ b =1

f(y) g(y) =n

sumab=1

sumIJ=1

partyIa

2Z foralli j (46)

entries

n5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Total

1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1

2 0 1 2 5 4 1 0 0 0 0 0 0 0 0 0 0 13

3 0 0 1 6 54 177 298 274 134 30 3 0 0 0 0 0 977

4 0 0 0 1 13 263 1988 7862 18532 28204 28377 18925 8034 2047 270 17 114533

421 NMHV

p1 p5 = ⟨1 2 3 4⟩ ⟨2 3 4 5⟩ ⟨3 4 5 1⟩ ⟨4 5 1 2⟩ ⟨5 1 2 3⟩ (48)

5minusy35minusy4

5 (49)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0

0 0 12

0 minus12 0 1

0 minus12 minus1

2 0 12

0 minus12 minus1

2 minus12 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

422 N2MHV

1 = [12 (23) cap (456) (234) cap (56)6][23456]

2 = [12 (34) cap (567) (345) cap (67)7][34567]

3 = [123 (345) cap (67)7][34567]

4 = [123 (456) cap (78)8][45678]

5 = [12348][45678]

6 = [123 (45) cap (678)8][45678]

7 = [123 (45) cap (678) (456) cap (78)][45678] (411)

8 = [1234 (456) cap (78)][45678]

9 = [12349][56789]

10 = [1234 (567) cap (89)][56789]

11 = [1234 (56) cap (789)][56789]

13 = [12345][678910]

ϕ = ⟨4 5 (123) cap (789)⟩⟨4 6 (123) cap (789)⟩⟨1 2 3 4⟩⟨4 7 8 9⟩⟨5 6 (123) cap (789)⟩ (413)

Ω(2)1

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 1 1 0 0 0 12

minus 12

minus1

0 0 0 0 minus 12

0 minus 12

minus 12

minus1

0 minus 12

minus 12

minus1

0 minus 12

0 minus1 minus1

minus 12

minus1

0 0 0 0 minus 12

0 minus 12

0 0 0 0

minus 12

0 minus 12

0 0 0 minus 12

minus 12

0 0 12

0 minus 12

1 1 1 1 1 0 0 12

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 1 0 0 0 0 minus1 minus 12

minus 12

0 0 0 minus 12

minus 12

0 minus 12

minus 12

minus1 0 0 minus 32

minus 32

0 minus 12

minus 32

minus 12

0 minus 12

0 minus1 minus 12

minus 12

0 minus1 minus 12

minus 12

0 0 0 minus 12

minus 12

0 minus 12

minus 12

0 0 12

0 minus 12

1 1 12

0 0 0 0 minus 12

0 0 0 12

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)3

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 12

0 0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

minus 32

minus 12

1 0 minus 12

0 minus1 0 minus 12

0 0 0 minus 12

minus 12

0 minus 12

minus 12

0 0 12

0 minus 12

1 1 12

0 0 12

0 0 0 0 0 0 minus 12

0 0 0 12

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)4

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 minus1 minus1 0

0 0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus1 minus1 0

0 minus 12

0 minus1 minus1 0

0 0 minus 12

minus 12

0 minus 12

minus 12

0 0 0 12

0 minus 12

minus 12

1 1 1 12

0 minus 12

1 1 1 12

0 0 0 0 0 0 0 0 0 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)5

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 0 0 0

0 0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 0 0 0

0 minus 12

0 0 0 0

0 0 minus 12

minus 12

0 minus 12

minus 12

0 0 0 12

0 minus 12

minus 12

0 0 0 12

0 minus 12

0 0 0 12

0 0 0 0 0 0 0 0 0 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)6

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 0 minus1 0

0 0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 0 minus1 0

0 minus 12

0 0 minus1 0

0 0 minus 12

minus 12

0 minus 12

minus 12

0 0 0 12

0 minus 12

minus 12

0 0 0 12

0 minus 12

1 1 1 12

0 0 0 0 0 0 0 0 0 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)7

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 0 minus1 0

0 1 0 minus 12

minus 12

minus 32

0 1 12

0 minus 12

minus 12

minus 32

0 1 12

minus 12

minus 32

0 0 minus 12

minus 12

0 minus 12

minus 12

0 0 minus 12

minus 12

0 minus 12

minus 12

0 1 12

0 minus 12

1 1 32

0 0 0 0 0 0 0 0 0 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)8

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 0 0 0

0 1 0 minus 12

minus 12

0 1 12

0 minus 12

minus 12

0 1 12

minus 12

0 0 minus 12

minus 12

0 minus 12

minus 12

0 0 minus 12

minus 12

0 minus 12

minus 12

0 1 12

0 minus 12

0 1 12

0 0 0 0 0 0 0 0 0 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)9

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 0 0 0

0 0 minus 12

minus 12

0 0 0 0

0 minus 12

minus 12

0 0 0 0

0 minus 12

0 0 0 0

0 minus 12

minus 12

0 minus 12

minus 12

0 0 0 0 0 12

0 minus 12

minus 12

0 0 0 0 0 12

0 minus 12

minus 12

0 0 0 0 0 12

0 minus 12

0 0 0 0 0 12

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)10

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 0 0 0

0 0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)11

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 0 0 0

0 0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)12

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 1 1 1 32

0 minus1 0 minus 12

minus 12

minus 32

minus 12

0 minus1 12

0 minus 12

minus 12

0 minus 12

0 minus1 12

0 minus 12

minus 12

0 minus 32

0 minus 12

minus 12

2 2 2 12

0 minus 32

0 minus 12

2 2 2 12

0 minus 32

0 2 2 2 12

0 minus 32

minus 12

0 minus 32

minus 12

0 minus 12

minus 12

0 minus 32

minus 12

0 minus 12

minus 12

0 minus1 12

0 minus 12

minus 12

0 minus 12

0 minus1 12

0 minus 12

minus 12

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)13

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 0 0 0

0 0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

B1 =⟨12 (23) cap (456) (234) cap (56)⟩ ⟨612 (23) cap (456)⟩ ⟨(234) cap (56)612⟩

⟨(23) cap (456) (234) cap (56)61⟩ ⟨2 (23) cap (456) (234) cap (56)6⟩ ⟨2345⟩ ⟨6234⟩ ⟨5623⟩

⟨4562⟩ ⟨3456⟩

(345) cap (67)71⟩ ⟨2 (34) cap (567) (345) cap (67)7⟩ ⟨3456⟩ ⟨7345⟩ ⟨6734⟩ ⟨5673⟩

⟨4567⟩

B3 =⟨123 (345) cap (67)⟩ ⟨7123⟩ ⟨(345) cap (67)712⟩ ⟨3 (345) cap (67)71⟩ ⟨23 (345) cap (67)7⟩

⟨3456⟩ ⟨7345⟩ ⟨6734⟩ ⟨5673⟩ ⟨4567⟩

B4 =⟨123 (456) cap (78)⟩ ⟨8123⟩ ⟨(456) cap (78)812⟩ ⟨3 (456) cap (78)81⟩ ⟨23 (456) cap (78)8⟩

⟨4567⟩ ⟨8456⟩ ⟨7845⟩ ⟨6784⟩⟨5678⟩

B5 =⟨1234⟩ ⟨8123⟩ ⟨4812⟩ ⟨3481⟩ ⟨2348⟩ ⟨4567⟩ ⟨8456⟩ ⟨7845⟩ ⟨6784⟩

⟨5678⟩

B6 =⟨123 (45) cap (678)⟩ ⟨8123⟩ ⟨(45) cap (678)812⟩ ⟨3 (45) cap (678)81⟩ ⟨23 (45) cap (678)8⟩

⟨4567⟩ ⟨8456⟩ ⟨7845⟩ ⟨6784⟩⟨5678⟩

B7 =⟨123 (45) cap (678)⟩ ⟨(456) cap (78)123⟩ ⟨(45) cap (678) (456) cap (78)12⟩

⟨3 (45) cap (678) (456) cap (78)1⟩ ⟨23 (45) cap (678) (456) cap (78)⟩ ⟨4567⟩ ⟨8456⟩ ⟨7845⟩

⟨6784⟩⟨5678⟩

B8 =⟨1234⟩ ⟨(456) cap (78)123⟩ ⟨4 (456) cap (78)12⟩ ⟨34 (456) cap (78)1⟩ ⟨234 (456) cap (78)⟩

⟨4567⟩ ⟨8456⟩ ⟨7845⟩ ⟨6784⟩⟨5678⟩

B9 =⟨1234⟩ ⟨9123⟩ ⟨4912⟩ ⟨3491⟩ ⟨2349⟩ ⟨5678⟩ ⟨9567⟩ ⟨8956⟩

⟨7895⟩ ⟨6789⟩

B10 =⟨1234⟩ ⟨(567) cap (89)123⟩ ⟨4 (567) cap (89)12⟩ ⟨34 (567) cap (89)1⟩ ⟨234 (567) cap (89)⟩

⟨5678⟩ ⟨9567⟩ ⟨8956⟩ ⟨7895⟩ ⟨6789⟩

B11 =⟨1234⟩ ⟨(56) cap (789)123⟩ ⟨4 (56) cap (789)12⟩ ⟨34 (56) cap (789)1⟩ ⟨234 (56) cap (789)⟩

⟨5678⟩ ⟨9567⟩ ⟨8956⟩ ⟨7895⟩ ⟨6789⟩

B12 =⟨1234⟩ ⟨4789⟩ ⟨56 (123) cap (789)⟩ ⟨123 (45) cap (789)⟩ ⟨(46) cap (789)123⟩

⟨789(45) cap (123)⟩ ⟨(46) cap (123)789⟩

B13 =⟨1234⟩ ⟨5123⟩ ⟨4512⟩ ⟨3451⟩ ⟨2345⟩ ⟨6789⟩ ⟨10678⟩ ⟨91067⟩

⟨89106⟩ ⟨78910⟩

Chapter 5

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

Z11 ⋯ Z1

Z21 ⋯ Z2

Z31 ⋯ Z3

Z41 ⋯ Z4

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ETHrarrGL(4)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

1 0 0 0 y15 ⋯ y1

0 1 0 0 y25 ⋯ y2

0 0 1 0 y35 ⋯ y3

0 0 0 1 y45 ⋯ y4

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

56 n with

ld (52)

x2ij =

is given by

yIa yJ b =1

f(y) g(y) =n

sumab=5

sumIJ=1

partyIa

2Z (56)

⟨i i + 1 j j + 1⟩)

in [89]

ABDSn = exp [

infinsumL=1

g2L (f(L)(ε)

2A(1)n (Lε) +C(L))] (57)

Γcusp = 4g2 +O(g4) (58)

ABDS-liken = ABDS

n exp [Γcusp

4Yn] 4 ∣ n

Discx2i+1j

[Discx2i+1i+p

(An)] = 0

Discx2i+1i+p

[Discx2i+1j+p+q

(An)] = 0

⎫⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎭

522 NMHV Amplitudes

form [115 116]

sums=5

sumt=s+2

Rtot =nminus2

sums=3

sumt=s+2

R1st (512)

and our results

Yif (2L)i (515)

minus 1

2([12345] + [12356] + [12367] minus [12457] minus [12567]

+ [13456] + [13467] + [14567] minus [23457] minus [23567])

times (log(⟨1234⟩⟨1789⟩⟨1278⟩⟨1349⟩)otimes ⟨3478⟩)

minus 1

2([12348] minus [12378] + [12478] minus [13478]

+ [23478] + [34567]) (log(⟨1234⟩⟨1789⟩⟨1278⟩⟨1349⟩)otimes ⟨3478⟩)

[12347] [12348] [12349] [12378] [12379] [12389]

[12478] [12479] [12489] [12789] [13478] [13479]

[13489] [13789] [14789] [23478] [23479] [23489]

[23789] [24789] [34567] [34568] [34578] [34678]

[34789] [35678] [45678]

Yangian invariants

a3ijkl = ⟨i(j j + 1)(k k + 1)(l l + 1)⟩

)(525)

condition

cup (j j + 1 k k + 15

) cup (i j j + 1 k k + 15

cup (i k k + 1 l l + 15

) cup (l l + 1 i5

) is a

uijkl =x2ikx

sumi=5

minus 1

sumi=5

minus 1

u2nminus3nminus1n)

sumi=5

minus 1

1247) +1

sumi=4

4ln(u8145) ln(u1256u3478

u2367) + cyclic (532)

nminus2

sumi=4

ln(un1ii+1)iminus1

sumj=1

sumi=4

ln(un1ii+1)iminus1

sumj=1

4ln(un1n

2n2+1)

nminus22

sumi=1

ln(uii+1i+n2i+n

Chapter 6

Graphs

symbol of [23]

C sdotZ = 0 (61)

example [27 93 94 28 30])

for each graph

source columns

C =⎛⎜⎜⎜⎝

1 0 c13 c14 c15 c16

0 1 c23 c24 c25 c26

⎞⎟⎟⎟⎠ (63)

prodαisinp

fα (64)

f0 = minus⟨1234⟩⟨2346⟩ f1 = minus

⟨2346⟩⟨2345⟩ f2 =

⟨2345⟩⟨1236⟩⟨1234⟩⟨2356⟩

f3 = minus⟨2356⟩⟨2346⟩ f4 =

⟨2346⟩⟨1256⟩⟨2456⟩⟨1236⟩ f5 = minus

⟨2456⟩⟨2356⟩

f6 =⟨2356⟩⟨1456⟩⟨3456⟩⟨1256⟩ f7 = minus

⟨3456⟩⟨2456⟩ f8 = minus

⟨2456⟩⟨1456⟩

f(b)0 = f (a)0 (1 + f (a)2 )

f(b)1 = f

1 + 1f (a)2

f(b)2 = 1

f(b)3 = f (a)3 (1 + f (a)2 )

f(b)4 = f

1 + 1f (a)2

f(c)2 = f (a)2 (1 + f (a)4 )

f(c)3 = f

1 + 1f (a)4

f(c)4 = 1

f(c)5 = f (a)5 (1 + f (a)4 )

f(c)6 = f

1 + 1f (a)4

(a) (b) (c)

⟨16⟩ ⟨12⟩

⟨23⟩

⟨34⟩⟨45⟩

⟨56⟩

⟨26⟩

⟨36⟩

⟨46⟩

ampamppp

⟨16⟩ ⟨12⟩

⟨23⟩

⟨34⟩⟨45⟩

⟨56⟩

⟨26⟩

⟨36⟩

⟨35⟩

⟨16⟩ ⟨12⟩

⟨23⟩

⟨34⟩⟨45⟩

⟨56⟩

⟨26⟩

⟨24⟩⟨46⟩ oo

(d) (e) (f)

⟨3456⟩ ⟨1456⟩

⟨1256⟩

⟨1236⟩⟨1234⟩

⟨2345⟩

⟨2456⟩

⟨2356⟩

⟨2346⟩

ampamppp

⟨3456⟩ ⟨1456⟩

⟨1256⟩

⟨1236⟩⟨1234⟩

⟨2345⟩

⟨2456⟩

⟨2356⟩

⟨1235⟩

⟨3456⟩ ⟨1456⟩

⟨1256⟩

⟨1236⟩⟨1234⟩

⟨2345⟩

⟨2456⟩

⟨1246⟩⟨2346⟩ oo

(g) (h) (i)

C = (0 1 f0f1f2f3 f0f1f2 f0f1 f0) (69)

f0 =⟨2345⟩⟨3456⟩ f1 = minus

⟨2346⟩⟨2345⟩ f2 = minus

⟨2356⟩⟨2346⟩ f3 = minus

⟨2456⟩⟨2356⟩ f4 = minus

⟨3456⟩⟨2456⟩

amplitude

f0 =⟨123⟩⟨145⟩

⟨234⟩⟨146⟩ f3 = minus⟨234⟩⟨156⟩⟨123⟩⟨456⟩

f4 = minus⟨124⟩⟨456⟩

⟨134⟩⟨156⟩

f6 = minus⟨134⟩⟨124⟩

f0 =⟨145⟩⟨456⟩ f1 = minus

⟨146⟩⟨145⟩ f2 = minus

⟨156⟩⟨146⟩ f3 = minus

⟨123⟩⟨456⟩⟨234⟩⟨156⟩

f4 = minus⟨124⟩⟨123⟩ f5 = minus

⟨234⟩⟨134⟩ f6 = minus

⟨134⟩⟨124⟩

⟨234⟩⟨156⟩ (616)

from (615)

C =⎛⎜⎜⎜⎝

1 c12 0 c14 c15 c16 c17 c18

0 c32 1 c34 c35 c36 c37 c38

⎞⎟⎟⎟⎠

a4a7 ⟨6(12)(34)(78)⟩ ⟨3478⟩

a2 ⟨4(12)(56)(78)⟩ ⟨1278⟩

a1⟨3(12)(56)(78)⟩

a3 ⟨1(34)(56)(78)⟩

a6 ⟨5(12)(34)(78)⟩

pendix 67

a9⟨1234⟩⟨5678⟩ (625)

∆2468

65 Discussion 101

65 Discussion

on Gr(mn)

65 Discussion 103

in [23]

f(b)0 = minus⟨1235⟩

⟨2356⟩ f(b)1 = minus⟨1236⟩

⟨1235⟩ f(b)2 = ⟨1234⟩⟨2356⟩

⟨2345⟩⟨1236⟩

f(b)3 = minus⟨1235⟩

⟨1234⟩ f(b)4 = ⟨2345⟩⟨1256⟩

⟨1235⟩⟨2456⟩ f(b)5 = minus⟨2456⟩

⟨2356⟩

f(b)6 = ⟨2356⟩⟨1456⟩

⟨3456⟩⟨1256⟩ f(b)7 = minus⟨3456⟩

⟨2456⟩ f(b)8 = minus⟨2456⟩

⟨1456⟩

f(c)0 = minus⟨1234⟩

⟨2346⟩ f(c)1 = minus⟨2346⟩

⟨2345⟩ f(c)2 = ⟨2345⟩⟨1246⟩

⟨1234⟩⟨2456⟩

f(c)3 = minus⟨1256⟩

⟨1246⟩ f(c)4 = ⟨2456⟩⟨1236⟩

⟨2346⟩⟨1256⟩ f(c)5 = minus⟨1246⟩

⟨1236⟩

f(c)6 = ⟨1456⟩⟨2346⟩

⟨3456⟩⟨1246⟩ f(c)7 = minus⟨3456⟩

⟨2456⟩ f(c)8 = minus⟨2456⟩

⟨1456⟩

∆1357 andradic

∆2468 with

∆1357 and

2(1 + u minus v +

u = ⟨1278⟩⟨3456⟩⟨1256⟩⟨3478⟩ v = ⟨1234⟩⟨5678⟩

⟨1256⟩⟨3478⟩ (632)

radic∆1357

⟨1256⟩⟨3478⟩ (633)

∣irarri+6

∣irarri+4

∣irarri+6

∣irarri+4

∣irarri+6

z a9 =

1 minus z1 minus z

Z isin Gr(48)gt0

Bibliography

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th9711200 [hep-th]]

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[hep-th0503198]

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doi101007BF01077848

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Abstract

Acknowledgements

Introduction












n-point MHV




6-point MHV

n-point NMHV

n-point NkMHV





Soft limits






NMHV

NNMHV

NNNMHV and Higher




One-loop Amplitudes


NMHV Amplitudes











Discussion



Chung-I Tan Reader

Andrew G Campbell

BROWN UNIVERSITY

Abstract

Anastasia Volovich

theory

Curriculum Vitae

Research

Education

Teaching

Schools and Talks

Jun 2019 TASI 2019

Additional Skills

Acknowledgements

Contents

Abstract v

Acknowledgements xi

1 Introduction 1

22 n-point MHV 19

224 6-point MHV 24

23 n-point NMHV 25

24 n-point NkMHV 28

33 Soft limits 43

421 NMHV 60

422 N2MHV 62

65 Discussion 101

List of Figures

to ⟨k(l l + 1)(mm + 1)(k minus 1k + 1)⟩ 80

to ⟨k(l l + 1)(mm + 1)(pp + 1)⟩ 81

List of Tables

I love you all

Chapter 1

Introduction

amplitudes

calculated

(11) transforms as

az + bcz + d

) = ∣cz + d∣2∆φ∆plusmn(X z z) (13)

prodj=1int

0dωj ω

az + bcz + d

prodj=1

gluon amplitudes

microi+1 (16)

〈2345〉

〈2346〉〈2356〉〈2456〉〈3456〉

〈1234〉〈1236〉〈1256〉〈1456〉

akaprimek = prod

i∣bikgt0

aminusbiki (112)

Y (Z ∣η) = int4k

prodi=1

d log fi4

prodI=1

prodα=1

suma=1

Cαa(Z ∣η)aI) (113)

prodαisinp

fα (114)

fi = 1 (115)

coordinates

Chapter 2

Celestial Sphere

radicωiωjzij (22)

prodj=1int

0dωj ω

azj + bczj + d

prodj=1

22 n-point MHV 19

22 n-point MHV

ratio ωi

dωiωi

(ωisi

j=1 iλj

s1+iλii

⎛⎜⎝

proda=1anei

intinfin

0dωa ω

intinfin

dωiωi

(ωisi

= 2πδ(z) (29)

sumj=1

λj⎞⎠s1+iλii

⎛⎜⎝

proda=1anei

intinfin

0dωa ω

⎞⎟⎠A`1⋯`n(si ωl zj zj) (210)

δ4(ε1s1q1 +n

sumi=2

εiωiqi) =4

prodj=nminus3

U+nminus4

sumi=2

U) (212)

22 n-point MHV 21

ui =z31zi2z32zi1

for i = 45 n (216)

s1 =∣z32∣

∣z31∣ ∣z12∣ s2 =

∣z31∣∣z32∣ ∣z21∣

and si =∣z12∣

∣z1i∣ ∣zi2∣for i = 3 n (217)

to demand Uij

prodij 1lt0(Uij

12 siλ1+21

nminus4

proda=2int

0dωa ω

1lt0(Uij

ωi = siU1n

uiminus1

i = 23 n minus 4 (220)

which leads to

12siλ1+21 siλ2+2

2 siλ33 siλnn

z23z34zn1U1nδ(

sumj=1

λj) ϕ(α x)prodij

1lt0(Uij

U) (221)

22 n-point MHV 23

prodj=1

P2and and dPnminus4

Pnminus4

αn =1 + iλ1 x0 i =U1i

U1n xjminus1 i =

Ujnminus U1i

U1n x0n = minus

U1n xjminus1n =

ϕ(α x) equivc2

c1intu1ge0u3ge0

1minussuma uage0

prodj=1

Pnminus1

U1n xij =

Ujnminus4+i

x0n = minusU

U1n xin =

U1n x01 = 1

mannian

224 6-point MHV

)iλ1+1

( minusUU16

)iλ2+2

iλ3minus1

iλ4minus1

iλ5minus1

times int1

prodi=1

23 n-point NMHV 25

U16U2i+2for i = 123 (227)

23 n-point NMHV

nminus1

sumj=4

⟨1∣P2jPj+12∣3⟩3

P 22jP

⟨j + 1 j⟩[2∣P2j ∣j + 1⟩⟨j∣Pj+12∣2]

equivnminus1

sumj=4

Mj (228)

⟨34⟩⟨45⟩⋯⟨n minus 1 n⟩⟨n1⟩(⟨12⟩[24]⟨43⟩ + ⟨13⟩[34]⟨43⟩)3

(⟨23⟩[23] + ⟨24⟩[24] + ⟨34⟩[34])⟨34⟩[34]times

times ⟨54⟩([23]⟨35⟩ + [24]⟨45⟩)(⟨43⟩[32]) (229)

M4 =ω1ω4(ε2z12z24ω2+ε3z13z34ω3)3

of variables

ωi = siuiminus1

i = 2 n minus 4 (231)

M4 sim δ⎛⎝n

sumj=1

λj⎞⎠

prodni=1 siλii

z12z13z45z4ns21s

z34zn1UF(αx)prod

1lt0(Uij

U) (232)

nminus5

proda=1

nminus5

prodj=1

uiλj+1

times7

prodi=1

23 n-point NMHV 27

U xj4 =

U j = 1 n minus 5

x05 =U1nminus2

U xj5 =

U j = 1 n minus 5 (235)

x06 =U1nminus1

U xj6 =

U j = 1 n minus 5

x07 =U1n

U xj7 =

Ujn minusU1n

U j = 1 n minus 5

6- and 7-point NMHV

u1uiλ2

prodi=2

(x0i + u1x1i)αi (236)

u2uiλ2

1 uiλ32 (u1x10 + u2x20 + u1u2x30 + u2

times7

prodi=1

24 n-point NkMHV

24 n-point NkMHV 29

N2MHV amplitude

F41F23

⟨1∣P26P72P35P63∣4⟩3

P 226P

⟨76⟩[23]⟨65⟩[2∣P26∣7⟩⟨6∣P72∣2][3∣P35∣6⟩⟨5∣P63∣3]

u3uα1

1 uα22 uα3

prodi=4

times17

prodj=14

2x8j + u23x9j)αj

NkMHV amplitude

nminus4

prodl=2

dululuαl

⎛⎝

1 minusnminus4

sumj=2

uj⎞⎠

P32k (prod

(P i1)αi)

⎛⎝prodj

(Pj2)αj

⎞⎠

prodj=0

Pj(u)αjdϕ (243)

dϕ =dPj1Pj1

and and dPjnPjn

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

x00 x0m

x10 x1m

⋮ ⋱ ⋮

xn0 xnm

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

gauge fixed form

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

1 0 0 1 1 1

⋮ ⋱ minus1 ⋮ ⋮ ⋮

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ways eq (324) of [38]

prodi=1

suml=1

prodj=1

(1 minusn

sumi=1

xijui)minusβj

sumi=1

αi) sdotn

prodi=1

Γ(αi) (248)

prodi=1

mminusnminus1

suml=1

prodj=1

sumi=1

xjiui)minusαj

sumi=1

prodi=1

Γ(βi) (249)

γ minusn

sumi=1

sumj=1

βj notin Z(250)

sumi=0

xijpartϕ

sumj=0

xijpartϕ

(3) part2ϕ

k du1duk (252)

Chapter 3

prodl=1

prodj=1

sumi=1

ki)( 1

(k1+k2)2+m2+ 1

(k1+k3)2+m2+ 1

(k1+k4)2+m2)

(hj hj) =1

A4(zj zj) =( z24

z14)h12 ( z14

z13)h34

zh1+h212 zh3+h4

z14)h12 ( z14

z13)h34

zh1+h212 zh3+h4

A4(z z) (35)

z = z12z34

z13z24 z = z12z34

compute

τminus2∆4A4(0 z11τ) (37)

prodi=1

⎛⎝

sumj=1

kj⎞⎠

m2 minus 4ω1ω2∣z∣2 (38)

δ(4)⎛⎝

sumj=1

kj⎞⎠= 4τ2

ω1δ(iz minus iz)

prodi=2

ωlowast2 = ω1

2 (310)

0dω1ω

∆1minus21

prodi=2

m2 minus 4z2

zminus1ω21

= g2 (im2)2αminus2

z minus 1)α

+ zα) (312)

sumi=1

hi minus 2 (313)

constructed in [51]

2(h2+h3)A13harr24

4 (z) 0 lt z lt 1 (315)

as well as

4 (1z)

z minus 1) 0 lt z lt 1 (316)

2 (im2)2αminus2

2 sin(πα) intC

hi(z z)

2= 1 minus h12 + n

2= 1 minus h34 + n

2= h12 minus n

2= h34 minus n (318)

with n = 0123

hi(z z) (319)

4hi+hj

Γ (hij + ∆2)Γ (∆

and Ψ∆hi(z z) is

Ψ∆hi(z z) equiv

2 minus h34)Γ (∆

2 + h12)Γ (1 minus ∆2 + h34)

Ψ∆hi(z z) (322)

infinsumJ=0

4π2Ψhh

+(h12 harr h34))

33 Soft limits 43

∆ minus J2

2= 1 minus h34 + n

∆ + J2

2= h34 minus n

33 Soft limits

Single soft limits

limωkrarr0

A`k=+1n = ⟨k minus 1k + 1⟩

limωkrarr0

δ(x) = 1

2limλrarr0

limλkrarr0

An(zj zj) =1

prodj=1jnek

intinfin

0dωj ω

iλjj int

limλkrarr0

AJk=+1n = 1

zkminus1k+1

limλkrarr0

AJk=minus1n = 1

zkminus1k+1

Double soft limits

33 Soft limits 45

⟨n∣1 + 2∣3] ( [13]3⟨n3⟩[12][23]s123

+ ⟨n2⟩3[n3]⟨n1⟩⟨12⟩sn12

we will write

An(zj zj) =n

prodj=1jnekl

intinfin

0dωj ω

iλjj int

0dr int

DS(kJk lJl) = 1

r=0]εrarr0

2(iλ1 + iλ2)ε21

zn1z23( 1

zn3z2n

z12z2n+ 1

z3nz31

z12z31) (338)

sumi=1

4ω1⋯ω6

ω21ω

ω23ω

26(ω4z46z34+ω5z56z35)3

(ω3ω4z34z34+ω3ω5z35z35+ω4ω5z45z45)

z12z16z34z45 (ω3z23z35 + ω4z24z45)

⎤⎥⎥⎥⎥⎥⎦

2(iλ3 + iλ4)ε21

z23z45( 1

z25z41

z34z42+ 1

z52z53

25z216z25z61

(z15z61

z25z26)iλ2minus1

(z12z16

z25z56)iλ5+1

(z15z12

z56z26)iλ6+1

⟨2∣3 + 4∣5] ( [35]3⟨25⟩[34][45]s345

+ ⟨24⟩3[25]⟨23⟩⟨34⟩s234

) (342)

intCd∆ Ψhh

hihi(z z)⟩ (343)

(zz)2G(z z)F (z z) (344)

given by

F+(z z) =1

zh34 zh342F1 (

1 minus h + h34 h + h34

1 + h12 + h341

z) 2F1 (

1 minus h + h34 h + h34

1 + h12 + h341

z) (346)

z) 2F1 (

⟨Ψhh

hihi(z z)⟩ = (2π)2

⟨A4(z z)Ψhh

hihi(z z) (349)

2F1(a b c z) =Γ(c)

obtain

⟨A4(z z)Ψhh

⎡⎢⎢⎢⎢⎢⎣

⎛⎜⎝

⎤⎥⎥⎥⎥⎥⎥⎦

sin(2πJ)2πJ

= 1 J = 0

0 J ne 0 (353)

hihi∣J=0

which simplifies to

2 minus h34)Γ(2 minus∆)Γ(∆) (354)

⟨A4(z z)Ψhh

Chapter 4

⟨1 2 6 7⟩⟨2 3 4 5⟩ minus ⟨1 2 4 5⟩⟨2 3 6 7⟩ (42)

cluster coordinate

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

1 0 0 0 y15 ⋯ y1

0 1 0 0 y25 ⋯ y2

0 0 1 0 y35 ⋯ y3

0 0 0 1 y45 ⋯ y4

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

yIa yJ b =1

f(y) g(y) =n

sumab=1

sumIJ=1

partyIa

2Z foralli j (46)

entries

n5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Total

1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1

2 0 1 2 5 4 1 0 0 0 0 0 0 0 0 0 0 13

3 0 0 1 6 54 177 298 274 134 30 3 0 0 0 0 0 977

4 0 0 0 1 13 263 1988 7862 18532 28204 28377 18925 8034 2047 270 17 114533

421 NMHV

p1 p5 = ⟨1 2 3 4⟩ ⟨2 3 4 5⟩ ⟨3 4 5 1⟩ ⟨4 5 1 2⟩ ⟨5 1 2 3⟩ (48)

5minusy35minusy4

5 (49)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0

0 0 12

0 minus12 0 1

0 minus12 minus1

2 0 12

0 minus12 minus1

2 minus12 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

422 N2MHV

1 = [12 (23) cap (456) (234) cap (56)6][23456]

2 = [12 (34) cap (567) (345) cap (67)7][34567]

3 = [123 (345) cap (67)7][34567]

4 = [123 (456) cap (78)8][45678]

5 = [12348][45678]

6 = [123 (45) cap (678)8][45678]

7 = [123 (45) cap (678) (456) cap (78)][45678] (411)

8 = [1234 (456) cap (78)][45678]

9 = [12349][56789]

10 = [1234 (567) cap (89)][56789]

11 = [1234 (56) cap (789)][56789]

13 = [12345][678910]

ϕ = ⟨4 5 (123) cap (789)⟩⟨4 6 (123) cap (789)⟩⟨1 2 3 4⟩⟨4 7 8 9⟩⟨5 6 (123) cap (789)⟩ (413)

Ω(2)1

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 1 1 0 0 0 12

minus 12

minus1

0 0 0 0 minus 12

0 minus 12

minus 12

minus1

0 minus 12

minus 12

minus1

0 minus 12

0 minus1 minus1

minus 12

minus1

0 0 0 0 minus 12

0 minus 12

0 0 0 0

minus 12

0 minus 12

0 0 0 minus 12

minus 12

0 0 12

0 minus 12

1 1 1 1 1 0 0 12

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 1 0 0 0 0 minus1 minus 12

minus 12

0 0 0 minus 12

minus 12

0 minus 12

minus 12

minus1 0 0 minus 32

minus 32

0 minus 12

minus 32

minus 12

0 minus 12

0 minus1 minus 12

minus 12

0 minus1 minus 12

minus 12

0 0 0 minus 12

minus 12

0 minus 12

minus 12

0 0 12

0 minus 12

1 1 12

0 0 0 0 minus 12

0 0 0 12

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)3

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 12

0 0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

minus 32

minus 12

1 0 minus 12

0 minus1 0 minus 12

0 0 0 minus 12

minus 12

0 minus 12

minus 12

0 0 12

0 minus 12

1 1 12

0 0 12

0 0 0 0 0 0 minus 12

0 0 0 12

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)4

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 minus1 minus1 0

0 0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus1 minus1 0

0 minus 12

0 minus1 minus1 0

0 0 minus 12

minus 12

0 minus 12

minus 12

0 0 0 12

0 minus 12

minus 12

1 1 1 12

0 minus 12

1 1 1 12

0 0 0 0 0 0 0 0 0 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)5

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 0 0 0

0 0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 0 0 0

0 minus 12

0 0 0 0

0 0 minus 12

minus 12

0 minus 12

minus 12

0 0 0 12

0 minus 12

minus 12

0 0 0 12

0 minus 12

0 0 0 12

0 0 0 0 0 0 0 0 0 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)6

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 0 minus1 0

0 0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 0 minus1 0

0 minus 12

0 0 minus1 0

0 0 minus 12

minus 12

0 minus 12

minus 12

0 0 0 12

0 minus 12

minus 12

0 0 0 12

0 minus 12

1 1 1 12

0 0 0 0 0 0 0 0 0 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)7

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 0 minus1 0

0 1 0 minus 12

minus 12

minus 32

0 1 12

0 minus 12

minus 12

minus 32

0 1 12

minus 12

minus 32

0 0 minus 12

minus 12

0 minus 12

minus 12

0 0 minus 12

minus 12

0 minus 12

minus 12

0 1 12

0 minus 12

1 1 32

0 0 0 0 0 0 0 0 0 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)8

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 0 0 0

0 1 0 minus 12

minus 12

0 1 12

0 minus 12

minus 12

0 1 12

minus 12

0 0 minus 12

minus 12

0 minus 12

minus 12

0 0 minus 12

minus 12

0 minus 12

minus 12

0 1 12

0 minus 12

0 1 12

0 0 0 0 0 0 0 0 0 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)9

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 0 0 0

0 0 minus 12

minus 12

0 0 0 0

0 minus 12

minus 12

0 0 0 0

0 minus 12

0 0 0 0

0 minus 12

minus 12

0 minus 12

minus 12

0 0 0 0 0 12

0 minus 12

minus 12

0 0 0 0 0 12

0 minus 12

minus 12

0 0 0 0 0 12

0 minus 12

0 0 0 0 0 12

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)10

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 0 0 0

0 0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)11

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 0 0 0

0 0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)12

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 1 1 1 32

0 minus1 0 minus 12

minus 12

minus 32

minus 12

0 minus1 12

0 minus 12

minus 12

0 minus 12

0 minus1 12

0 minus 12

minus 12

0 minus 32

0 minus 12

minus 12

2 2 2 12

0 minus 32

0 minus 12

2 2 2 12

0 minus 32

0 2 2 2 12

0 minus 32

minus 12

0 minus 32

minus 12

0 minus 12

minus 12

0 minus 32

minus 12

0 minus 12

minus 12

0 minus1 12

0 minus 12

minus 12

0 minus 12

0 minus1 12

0 minus 12

minus 12

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)13

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 0 0 0

0 0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

B1 =⟨12 (23) cap (456) (234) cap (56)⟩ ⟨612 (23) cap (456)⟩ ⟨(234) cap (56)612⟩

⟨(23) cap (456) (234) cap (56)61⟩ ⟨2 (23) cap (456) (234) cap (56)6⟩ ⟨2345⟩ ⟨6234⟩ ⟨5623⟩

⟨4562⟩ ⟨3456⟩

(345) cap (67)71⟩ ⟨2 (34) cap (567) (345) cap (67)7⟩ ⟨3456⟩ ⟨7345⟩ ⟨6734⟩ ⟨5673⟩

⟨4567⟩

B3 =⟨123 (345) cap (67)⟩ ⟨7123⟩ ⟨(345) cap (67)712⟩ ⟨3 (345) cap (67)71⟩ ⟨23 (345) cap (67)7⟩

⟨3456⟩ ⟨7345⟩ ⟨6734⟩ ⟨5673⟩ ⟨4567⟩

B4 =⟨123 (456) cap (78)⟩ ⟨8123⟩ ⟨(456) cap (78)812⟩ ⟨3 (456) cap (78)81⟩ ⟨23 (456) cap (78)8⟩

⟨4567⟩ ⟨8456⟩ ⟨7845⟩ ⟨6784⟩⟨5678⟩

B5 =⟨1234⟩ ⟨8123⟩ ⟨4812⟩ ⟨3481⟩ ⟨2348⟩ ⟨4567⟩ ⟨8456⟩ ⟨7845⟩ ⟨6784⟩

⟨5678⟩

B6 =⟨123 (45) cap (678)⟩ ⟨8123⟩ ⟨(45) cap (678)812⟩ ⟨3 (45) cap (678)81⟩ ⟨23 (45) cap (678)8⟩

⟨4567⟩ ⟨8456⟩ ⟨7845⟩ ⟨6784⟩⟨5678⟩

B7 =⟨123 (45) cap (678)⟩ ⟨(456) cap (78)123⟩ ⟨(45) cap (678) (456) cap (78)12⟩

⟨3 (45) cap (678) (456) cap (78)1⟩ ⟨23 (45) cap (678) (456) cap (78)⟩ ⟨4567⟩ ⟨8456⟩ ⟨7845⟩

⟨6784⟩⟨5678⟩

B8 =⟨1234⟩ ⟨(456) cap (78)123⟩ ⟨4 (456) cap (78)12⟩ ⟨34 (456) cap (78)1⟩ ⟨234 (456) cap (78)⟩

⟨4567⟩ ⟨8456⟩ ⟨7845⟩ ⟨6784⟩⟨5678⟩

B9 =⟨1234⟩ ⟨9123⟩ ⟨4912⟩ ⟨3491⟩ ⟨2349⟩ ⟨5678⟩ ⟨9567⟩ ⟨8956⟩

⟨7895⟩ ⟨6789⟩

B10 =⟨1234⟩ ⟨(567) cap (89)123⟩ ⟨4 (567) cap (89)12⟩ ⟨34 (567) cap (89)1⟩ ⟨234 (567) cap (89)⟩

⟨5678⟩ ⟨9567⟩ ⟨8956⟩ ⟨7895⟩ ⟨6789⟩

B11 =⟨1234⟩ ⟨(56) cap (789)123⟩ ⟨4 (56) cap (789)12⟩ ⟨34 (56) cap (789)1⟩ ⟨234 (56) cap (789)⟩

⟨5678⟩ ⟨9567⟩ ⟨8956⟩ ⟨7895⟩ ⟨6789⟩

B12 =⟨1234⟩ ⟨4789⟩ ⟨56 (123) cap (789)⟩ ⟨123 (45) cap (789)⟩ ⟨(46) cap (789)123⟩

⟨789(45) cap (123)⟩ ⟨(46) cap (123)789⟩

B13 =⟨1234⟩ ⟨5123⟩ ⟨4512⟩ ⟨3451⟩ ⟨2345⟩ ⟨6789⟩ ⟨10678⟩ ⟨91067⟩

⟨89106⟩ ⟨78910⟩

Chapter 5

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

Z11 ⋯ Z1

Z21 ⋯ Z2

Z31 ⋯ Z3

Z41 ⋯ Z4

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ETHrarrGL(4)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

1 0 0 0 y15 ⋯ y1

0 1 0 0 y25 ⋯ y2

0 0 1 0 y35 ⋯ y3

0 0 0 1 y45 ⋯ y4

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

56 n with

ld (52)

x2ij =

is given by

yIa yJ b =1

f(y) g(y) =n

sumab=5

sumIJ=1

partyIa

2Z (56)

⟨i i + 1 j j + 1⟩)

in [89]

ABDSn = exp [

infinsumL=1

g2L (f(L)(ε)

2A(1)n (Lε) +C(L))] (57)

Γcusp = 4g2 +O(g4) (58)

ABDS-liken = ABDS

n exp [Γcusp

4Yn] 4 ∣ n

Discx2i+1j

[Discx2i+1i+p

(An)] = 0

Discx2i+1i+p

[Discx2i+1j+p+q

(An)] = 0

⎫⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎭

522 NMHV Amplitudes

form [115 116]

sums=5

sumt=s+2

Rtot =nminus2

sums=3

sumt=s+2

R1st (512)

and our results

Yif (2L)i (515)

minus 1

2([12345] + [12356] + [12367] minus [12457] minus [12567]

+ [13456] + [13467] + [14567] minus [23457] minus [23567])

times (log(⟨1234⟩⟨1789⟩⟨1278⟩⟨1349⟩)otimes ⟨3478⟩)

minus 1

2([12348] minus [12378] + [12478] minus [13478]

+ [23478] + [34567]) (log(⟨1234⟩⟨1789⟩⟨1278⟩⟨1349⟩)otimes ⟨3478⟩)

[12347] [12348] [12349] [12378] [12379] [12389]

[12478] [12479] [12489] [12789] [13478] [13479]

[13489] [13789] [14789] [23478] [23479] [23489]

[23789] [24789] [34567] [34568] [34578] [34678]

[34789] [35678] [45678]

Yangian invariants

a3ijkl = ⟨i(j j + 1)(k k + 1)(l l + 1)⟩

)(525)

condition

cup (j j + 1 k k + 15

) cup (i j j + 1 k k + 15

cup (i k k + 1 l l + 15

) cup (l l + 1 i5

) is a

uijkl =x2ikx

sumi=5

minus 1

sumi=5

minus 1

u2nminus3nminus1n)

sumi=5

minus 1

1247) +1

sumi=4

4ln(u8145) ln(u1256u3478

u2367) + cyclic (532)

nminus2

sumi=4

ln(un1ii+1)iminus1

sumj=1

sumi=4

ln(un1ii+1)iminus1

sumj=1

4ln(un1n

2n2+1)

nminus22

sumi=1

ln(uii+1i+n2i+n

Chapter 6

Graphs

symbol of [23]

C sdotZ = 0 (61)

example [27 93 94 28 30])

for each graph

source columns

C =⎛⎜⎜⎜⎝

1 0 c13 c14 c15 c16

0 1 c23 c24 c25 c26

⎞⎟⎟⎟⎠ (63)

prodαisinp

fα (64)

f0 = minus⟨1234⟩⟨2346⟩ f1 = minus

⟨2346⟩⟨2345⟩ f2 =

⟨2345⟩⟨1236⟩⟨1234⟩⟨2356⟩

f3 = minus⟨2356⟩⟨2346⟩ f4 =

⟨2346⟩⟨1256⟩⟨2456⟩⟨1236⟩ f5 = minus

⟨2456⟩⟨2356⟩

f6 =⟨2356⟩⟨1456⟩⟨3456⟩⟨1256⟩ f7 = minus

⟨3456⟩⟨2456⟩ f8 = minus

⟨2456⟩⟨1456⟩

f(b)0 = f (a)0 (1 + f (a)2 )

f(b)1 = f

1 + 1f (a)2

f(b)2 = 1

f(b)3 = f (a)3 (1 + f (a)2 )

f(b)4 = f

1 + 1f (a)2

f(c)2 = f (a)2 (1 + f (a)4 )

f(c)3 = f

1 + 1f (a)4

f(c)4 = 1

f(c)5 = f (a)5 (1 + f (a)4 )

f(c)6 = f

1 + 1f (a)4

(a) (b) (c)

⟨16⟩ ⟨12⟩

⟨23⟩

⟨34⟩⟨45⟩

⟨56⟩

⟨26⟩

⟨36⟩

⟨46⟩

ampamppp

⟨16⟩ ⟨12⟩

⟨23⟩

⟨34⟩⟨45⟩

⟨56⟩

⟨26⟩

⟨36⟩

⟨35⟩

⟨16⟩ ⟨12⟩

⟨23⟩

⟨34⟩⟨45⟩

⟨56⟩

⟨26⟩

⟨24⟩⟨46⟩ oo

(d) (e) (f)

⟨3456⟩ ⟨1456⟩

⟨1256⟩

⟨1236⟩⟨1234⟩

⟨2345⟩

⟨2456⟩

⟨2356⟩

⟨2346⟩

ampamppp

⟨3456⟩ ⟨1456⟩

⟨1256⟩

⟨1236⟩⟨1234⟩

⟨2345⟩

⟨2456⟩

⟨2356⟩

⟨1235⟩

⟨3456⟩ ⟨1456⟩

⟨1256⟩

⟨1236⟩⟨1234⟩

⟨2345⟩

⟨2456⟩

⟨1246⟩⟨2346⟩ oo

(g) (h) (i)

C = (0 1 f0f1f2f3 f0f1f2 f0f1 f0) (69)

f0 =⟨2345⟩⟨3456⟩ f1 = minus

⟨2346⟩⟨2345⟩ f2 = minus

⟨2356⟩⟨2346⟩ f3 = minus

⟨2456⟩⟨2356⟩ f4 = minus

⟨3456⟩⟨2456⟩

amplitude

f0 =⟨123⟩⟨145⟩

⟨234⟩⟨146⟩ f3 = minus⟨234⟩⟨156⟩⟨123⟩⟨456⟩

f4 = minus⟨124⟩⟨456⟩

⟨134⟩⟨156⟩

f6 = minus⟨134⟩⟨124⟩

f0 =⟨145⟩⟨456⟩ f1 = minus

⟨146⟩⟨145⟩ f2 = minus

⟨156⟩⟨146⟩ f3 = minus

⟨123⟩⟨456⟩⟨234⟩⟨156⟩

f4 = minus⟨124⟩⟨123⟩ f5 = minus

⟨234⟩⟨134⟩ f6 = minus

⟨134⟩⟨124⟩

⟨234⟩⟨156⟩ (616)

from (615)

C =⎛⎜⎜⎜⎝

1 c12 0 c14 c15 c16 c17 c18

0 c32 1 c34 c35 c36 c37 c38

⎞⎟⎟⎟⎠

a4a7 ⟨6(12)(34)(78)⟩ ⟨3478⟩

a2 ⟨4(12)(56)(78)⟩ ⟨1278⟩

a1⟨3(12)(56)(78)⟩

a3 ⟨1(34)(56)(78)⟩

a6 ⟨5(12)(34)(78)⟩

pendix 67

a9⟨1234⟩⟨5678⟩ (625)

∆2468

65 Discussion 101

65 Discussion

on Gr(mn)

65 Discussion 103

in [23]

f(b)0 = minus⟨1235⟩

⟨2356⟩ f(b)1 = minus⟨1236⟩

⟨1235⟩ f(b)2 = ⟨1234⟩⟨2356⟩

⟨2345⟩⟨1236⟩

f(b)3 = minus⟨1235⟩

⟨1234⟩ f(b)4 = ⟨2345⟩⟨1256⟩

⟨1235⟩⟨2456⟩ f(b)5 = minus⟨2456⟩

⟨2356⟩

f(b)6 = ⟨2356⟩⟨1456⟩

⟨3456⟩⟨1256⟩ f(b)7 = minus⟨3456⟩

⟨2456⟩ f(b)8 = minus⟨2456⟩

⟨1456⟩

f(c)0 = minus⟨1234⟩

⟨2346⟩ f(c)1 = minus⟨2346⟩

⟨2345⟩ f(c)2 = ⟨2345⟩⟨1246⟩

⟨1234⟩⟨2456⟩

f(c)3 = minus⟨1256⟩

⟨1246⟩ f(c)4 = ⟨2456⟩⟨1236⟩

⟨2346⟩⟨1256⟩ f(c)5 = minus⟨1246⟩

⟨1236⟩

f(c)6 = ⟨1456⟩⟨2346⟩

⟨3456⟩⟨1246⟩ f(c)7 = minus⟨3456⟩

⟨2456⟩ f(c)8 = minus⟨2456⟩

⟨1456⟩

∆1357 andradic

∆2468 with

∆1357 and

2(1 + u minus v +

u = ⟨1278⟩⟨3456⟩⟨1256⟩⟨3478⟩ v = ⟨1234⟩⟨5678⟩

⟨1256⟩⟨3478⟩ (632)

radic∆1357

⟨1256⟩⟨3478⟩ (633)

∣irarri+6

∣irarri+4

∣irarri+6

∣irarri+4

∣irarri+6

z a9 =

1 minus z1 minus z

Z isin Gr(48)gt0

Bibliography

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56 2459 (1986) doi101103PhysRevLett562459

th9711200 [hep-th]]

110 BIBLIOGRAPHY

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[hep-th0503198]

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Phys 90 438 (1975) doi1010160003-4916(75)90006-8

doi101007BF01077848

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365 (1979)

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JHEP 10 031 (2014) [arXiv14067273 [hep-th]]

Abstract

Acknowledgements

Introduction












n-point MHV




6-point MHV

n-point NMHV

n-point NkMHV





Soft limits






NMHV

NNMHV

NNNMHV and Higher




One-loop Amplitudes


NMHV Amplitudes











Discussion



BROWN UNIVERSITY

Abstract

Anastasia Volovich

theory

Curriculum Vitae

Research

Education

Teaching

Schools and Talks

Jun 2019 TASI 2019

Additional Skills

Acknowledgements

Contents

Abstract v

Acknowledgements xi

1 Introduction 1

22 n-point MHV 19

224 6-point MHV 24

23 n-point NMHV 25

24 n-point NkMHV 28

33 Soft limits 43

421 NMHV 60

422 N2MHV 62

65 Discussion 101

List of Figures

to ⟨k(l l + 1)(mm + 1)(k minus 1k + 1)⟩ 80

to ⟨k(l l + 1)(mm + 1)(pp + 1)⟩ 81

List of Tables

I love you all

Chapter 1

Introduction

amplitudes

calculated

(11) transforms as

az + bcz + d

) = ∣cz + d∣2∆φ∆plusmn(X z z) (13)

prodj=1int

0dωj ω

az + bcz + d

prodj=1

gluon amplitudes

microi+1 (16)

〈2345〉

〈2346〉〈2356〉〈2456〉〈3456〉

〈1234〉〈1236〉〈1256〉〈1456〉

akaprimek = prod

i∣bikgt0

aminusbiki (112)

Y (Z ∣η) = int4k

prodi=1

d log fi4

prodI=1

prodα=1

suma=1

Cαa(Z ∣η)aI) (113)

prodαisinp

fα (114)

fi = 1 (115)

coordinates

Chapter 2

Celestial Sphere

radicωiωjzij (22)

prodj=1int

0dωj ω

azj + bczj + d

prodj=1

22 n-point MHV 19

22 n-point MHV

ratio ωi

dωiωi

(ωisi

j=1 iλj

s1+iλii

⎛⎜⎝

proda=1anei

intinfin

0dωa ω

intinfin

dωiωi

(ωisi

= 2πδ(z) (29)

sumj=1

λj⎞⎠s1+iλii

⎛⎜⎝

proda=1anei

intinfin

0dωa ω

⎞⎟⎠A`1⋯`n(si ωl zj zj) (210)

δ4(ε1s1q1 +n

sumi=2

εiωiqi) =4

prodj=nminus3

U+nminus4

sumi=2

U) (212)

22 n-point MHV 21

ui =z31zi2z32zi1

for i = 45 n (216)

s1 =∣z32∣

∣z31∣ ∣z12∣ s2 =

∣z31∣∣z32∣ ∣z21∣

and si =∣z12∣

∣z1i∣ ∣zi2∣for i = 3 n (217)

to demand Uij

prodij 1lt0(Uij

12 siλ1+21

nminus4

proda=2int

0dωa ω

1lt0(Uij

ωi = siU1n

uiminus1

i = 23 n minus 4 (220)

which leads to

12siλ1+21 siλ2+2

2 siλ33 siλnn

z23z34zn1U1nδ(

sumj=1

λj) ϕ(α x)prodij

1lt0(Uij

U) (221)

22 n-point MHV 23

prodj=1

P2and and dPnminus4

Pnminus4

αn =1 + iλ1 x0 i =U1i

U1n xjminus1 i =

Ujnminus U1i

U1n x0n = minus

U1n xjminus1n =

ϕ(α x) equivc2

c1intu1ge0u3ge0

1minussuma uage0

prodj=1

Pnminus1

U1n xij =

Ujnminus4+i

x0n = minusU

U1n xin =

U1n x01 = 1

mannian

224 6-point MHV

)iλ1+1

( minusUU16

)iλ2+2

iλ3minus1

iλ4minus1

iλ5minus1

times int1

prodi=1

23 n-point NMHV 25

U16U2i+2for i = 123 (227)

23 n-point NMHV

nminus1

sumj=4

⟨1∣P2jPj+12∣3⟩3

P 22jP

⟨j + 1 j⟩[2∣P2j ∣j + 1⟩⟨j∣Pj+12∣2]

equivnminus1

sumj=4

Mj (228)

⟨34⟩⟨45⟩⋯⟨n minus 1 n⟩⟨n1⟩(⟨12⟩[24]⟨43⟩ + ⟨13⟩[34]⟨43⟩)3

(⟨23⟩[23] + ⟨24⟩[24] + ⟨34⟩[34])⟨34⟩[34]times

times ⟨54⟩([23]⟨35⟩ + [24]⟨45⟩)(⟨43⟩[32]) (229)

M4 =ω1ω4(ε2z12z24ω2+ε3z13z34ω3)3

of variables

ωi = siuiminus1

i = 2 n minus 4 (231)

M4 sim δ⎛⎝n

sumj=1

λj⎞⎠

prodni=1 siλii

z12z13z45z4ns21s

z34zn1UF(αx)prod

1lt0(Uij

U) (232)

nminus5

proda=1

nminus5

prodj=1

uiλj+1

times7

prodi=1

23 n-point NMHV 27

U xj4 =

U j = 1 n minus 5

x05 =U1nminus2

U xj5 =

U j = 1 n minus 5 (235)

x06 =U1nminus1

U xj6 =

U j = 1 n minus 5

x07 =U1n

U xj7 =

Ujn minusU1n

U j = 1 n minus 5

6- and 7-point NMHV

u1uiλ2

prodi=2

(x0i + u1x1i)αi (236)

u2uiλ2

1 uiλ32 (u1x10 + u2x20 + u1u2x30 + u2

times7

prodi=1

24 n-point NkMHV

24 n-point NkMHV 29

N2MHV amplitude

F41F23

⟨1∣P26P72P35P63∣4⟩3

P 226P

⟨76⟩[23]⟨65⟩[2∣P26∣7⟩⟨6∣P72∣2][3∣P35∣6⟩⟨5∣P63∣3]

u3uα1

1 uα22 uα3

prodi=4

times17

prodj=14

2x8j + u23x9j)αj

NkMHV amplitude

nminus4

prodl=2

dululuαl

⎛⎝

1 minusnminus4

sumj=2

uj⎞⎠

P32k (prod

(P i1)αi)

⎛⎝prodj

(Pj2)αj

⎞⎠

prodj=0

Pj(u)αjdϕ (243)

dϕ =dPj1Pj1

and and dPjnPjn

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

x00 x0m

x10 x1m

⋮ ⋱ ⋮

xn0 xnm

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

gauge fixed form

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

1 0 0 1 1 1

⋮ ⋱ minus1 ⋮ ⋮ ⋮

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ways eq (324) of [38]

prodi=1

suml=1

prodj=1

(1 minusn

sumi=1

xijui)minusβj

sumi=1

αi) sdotn

prodi=1

Γ(αi) (248)

prodi=1

mminusnminus1

suml=1

prodj=1

sumi=1

xjiui)minusαj

sumi=1

prodi=1

Γ(βi) (249)

γ minusn

sumi=1

sumj=1

βj notin Z(250)

sumi=0

xijpartϕ

sumj=0

xijpartϕ

(3) part2ϕ

k du1duk (252)

Chapter 3

prodl=1

prodj=1

sumi=1

ki)( 1

(k1+k2)2+m2+ 1

(k1+k3)2+m2+ 1

(k1+k4)2+m2)

(hj hj) =1

A4(zj zj) =( z24

z14)h12 ( z14

z13)h34

zh1+h212 zh3+h4

z14)h12 ( z14

z13)h34

zh1+h212 zh3+h4

A4(z z) (35)

z = z12z34

z13z24 z = z12z34

compute

τminus2∆4A4(0 z11τ) (37)

prodi=1

⎛⎝

sumj=1

kj⎞⎠

m2 minus 4ω1ω2∣z∣2 (38)

δ(4)⎛⎝

sumj=1

kj⎞⎠= 4τ2

ω1δ(iz minus iz)

prodi=2

ωlowast2 = ω1

2 (310)

0dω1ω

∆1minus21

prodi=2

m2 minus 4z2

zminus1ω21

= g2 (im2)2αminus2

z minus 1)α

+ zα) (312)

sumi=1

hi minus 2 (313)

constructed in [51]

2(h2+h3)A13harr24

4 (z) 0 lt z lt 1 (315)

as well as

4 (1z)

z minus 1) 0 lt z lt 1 (316)

2 (im2)2αminus2

2 sin(πα) intC

hi(z z)

2= 1 minus h12 + n

2= 1 minus h34 + n

2= h12 minus n

2= h34 minus n (318)

with n = 0123

hi(z z) (319)

4hi+hj

Γ (hij + ∆2)Γ (∆

and Ψ∆hi(z z) is

Ψ∆hi(z z) equiv

2 minus h34)Γ (∆

2 + h12)Γ (1 minus ∆2 + h34)

Ψ∆hi(z z) (322)

infinsumJ=0

4π2Ψhh

+(h12 harr h34))

33 Soft limits 43

∆ minus J2

2= 1 minus h34 + n

∆ + J2

2= h34 minus n

33 Soft limits

Single soft limits

limωkrarr0

A`k=+1n = ⟨k minus 1k + 1⟩

limωkrarr0

δ(x) = 1

2limλrarr0

limλkrarr0

An(zj zj) =1

prodj=1jnek

intinfin

0dωj ω

iλjj int

limλkrarr0

AJk=+1n = 1

zkminus1k+1

limλkrarr0

AJk=minus1n = 1

zkminus1k+1

Double soft limits

33 Soft limits 45

⟨n∣1 + 2∣3] ( [13]3⟨n3⟩[12][23]s123

+ ⟨n2⟩3[n3]⟨n1⟩⟨12⟩sn12

we will write

An(zj zj) =n

prodj=1jnekl

intinfin

0dωj ω

iλjj int

0dr int

DS(kJk lJl) = 1

r=0]εrarr0

2(iλ1 + iλ2)ε21

zn1z23( 1

zn3z2n

z12z2n+ 1

z3nz31

z12z31) (338)

sumi=1

4ω1⋯ω6

ω21ω

ω23ω

26(ω4z46z34+ω5z56z35)3

(ω3ω4z34z34+ω3ω5z35z35+ω4ω5z45z45)

z12z16z34z45 (ω3z23z35 + ω4z24z45)

⎤⎥⎥⎥⎥⎥⎦

2(iλ3 + iλ4)ε21

z23z45( 1

z25z41

z34z42+ 1

z52z53

25z216z25z61

(z15z61

z25z26)iλ2minus1

(z12z16

z25z56)iλ5+1

(z15z12

z56z26)iλ6+1

⟨2∣3 + 4∣5] ( [35]3⟨25⟩[34][45]s345

+ ⟨24⟩3[25]⟨23⟩⟨34⟩s234

) (342)

intCd∆ Ψhh

hihi(z z)⟩ (343)

(zz)2G(z z)F (z z) (344)

given by

F+(z z) =1

zh34 zh342F1 (

1 minus h + h34 h + h34

1 + h12 + h341

z) 2F1 (

1 minus h + h34 h + h34

1 + h12 + h341

z) (346)

z) 2F1 (

⟨Ψhh

hihi(z z)⟩ = (2π)2

⟨A4(z z)Ψhh

hihi(z z) (349)

2F1(a b c z) =Γ(c)

obtain

⟨A4(z z)Ψhh

⎡⎢⎢⎢⎢⎢⎣

⎛⎜⎝

⎤⎥⎥⎥⎥⎥⎥⎦

sin(2πJ)2πJ

= 1 J = 0

0 J ne 0 (353)

hihi∣J=0

which simplifies to

2 minus h34)Γ(2 minus∆)Γ(∆) (354)

⟨A4(z z)Ψhh

Chapter 4

⟨1 2 6 7⟩⟨2 3 4 5⟩ minus ⟨1 2 4 5⟩⟨2 3 6 7⟩ (42)

cluster coordinate

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

1 0 0 0 y15 ⋯ y1

0 1 0 0 y25 ⋯ y2

0 0 1 0 y35 ⋯ y3

0 0 0 1 y45 ⋯ y4

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

yIa yJ b =1

f(y) g(y) =n

sumab=1

sumIJ=1

partyIa

2Z foralli j (46)

entries

n5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Total

1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1

2 0 1 2 5 4 1 0 0 0 0 0 0 0 0 0 0 13

3 0 0 1 6 54 177 298 274 134 30 3 0 0 0 0 0 977

4 0 0 0 1 13 263 1988 7862 18532 28204 28377 18925 8034 2047 270 17 114533

421 NMHV

p1 p5 = ⟨1 2 3 4⟩ ⟨2 3 4 5⟩ ⟨3 4 5 1⟩ ⟨4 5 1 2⟩ ⟨5 1 2 3⟩ (48)

5minusy35minusy4

5 (49)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0

0 0 12

0 minus12 0 1

0 minus12 minus1

2 0 12

0 minus12 minus1

2 minus12 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

422 N2MHV

1 = [12 (23) cap (456) (234) cap (56)6][23456]

2 = [12 (34) cap (567) (345) cap (67)7][34567]

3 = [123 (345) cap (67)7][34567]

4 = [123 (456) cap (78)8][45678]

5 = [12348][45678]

6 = [123 (45) cap (678)8][45678]

7 = [123 (45) cap (678) (456) cap (78)][45678] (411)

8 = [1234 (456) cap (78)][45678]

9 = [12349][56789]

10 = [1234 (567) cap (89)][56789]

11 = [1234 (56) cap (789)][56789]

13 = [12345][678910]

ϕ = ⟨4 5 (123) cap (789)⟩⟨4 6 (123) cap (789)⟩⟨1 2 3 4⟩⟨4 7 8 9⟩⟨5 6 (123) cap (789)⟩ (413)

Ω(2)1

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 1 1 0 0 0 12

minus 12

minus1

0 0 0 0 minus 12

0 minus 12

minus 12

minus1

0 minus 12

minus 12

minus1

0 minus 12

0 minus1 minus1

minus 12

minus1

0 0 0 0 minus 12

0 minus 12

0 0 0 0

minus 12

0 minus 12

0 0 0 minus 12

minus 12

0 0 12

0 minus 12

1 1 1 1 1 0 0 12

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 1 0 0 0 0 minus1 minus 12

minus 12

0 0 0 minus 12

minus 12

0 minus 12

minus 12

minus1 0 0 minus 32

minus 32

0 minus 12

minus 32

minus 12

0 minus 12

0 minus1 minus 12

minus 12

0 minus1 minus 12

minus 12

0 0 0 minus 12

minus 12

0 minus 12

minus 12

0 0 12

0 minus 12

1 1 12

0 0 0 0 minus 12

0 0 0 12

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)3

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 12

0 0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

minus 32

minus 12

1 0 minus 12

0 minus1 0 minus 12

0 0 0 minus 12

minus 12

0 minus 12

minus 12

0 0 12

0 minus 12

1 1 12

0 0 12

0 0 0 0 0 0 minus 12

0 0 0 12

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)4

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 minus1 minus1 0

0 0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus1 minus1 0

0 minus 12

0 minus1 minus1 0

0 0 minus 12

minus 12

0 minus 12

minus 12

0 0 0 12

0 minus 12

minus 12

1 1 1 12

0 minus 12

1 1 1 12

0 0 0 0 0 0 0 0 0 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)5

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 0 0 0

0 0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 0 0 0

0 minus 12

0 0 0 0

0 0 minus 12

minus 12

0 minus 12

minus 12

0 0 0 12

0 minus 12

minus 12

0 0 0 12

0 minus 12

0 0 0 12

0 0 0 0 0 0 0 0 0 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)6

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 0 minus1 0

0 0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 0 minus1 0

0 minus 12

0 0 minus1 0

0 0 minus 12

minus 12

0 minus 12

minus 12

0 0 0 12

0 minus 12

minus 12

0 0 0 12

0 minus 12

1 1 1 12

0 0 0 0 0 0 0 0 0 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)7

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 0 minus1 0

0 1 0 minus 12

minus 12

minus 32

0 1 12

0 minus 12

minus 12

minus 32

0 1 12

minus 12

minus 32

0 0 minus 12

minus 12

0 minus 12

minus 12

0 0 minus 12

minus 12

0 minus 12

minus 12

0 1 12

0 minus 12

1 1 32

0 0 0 0 0 0 0 0 0 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)8

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 0 0 0

0 1 0 minus 12

minus 12

0 1 12

0 minus 12

minus 12

0 1 12

minus 12

0 0 minus 12

minus 12

0 minus 12

minus 12

0 0 minus 12

minus 12

0 minus 12

minus 12

0 1 12

0 minus 12

0 1 12

0 0 0 0 0 0 0 0 0 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)9

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 0 0 0

0 0 minus 12

minus 12

0 0 0 0

0 minus 12

minus 12

0 0 0 0

0 minus 12

0 0 0 0

0 minus 12

minus 12

0 minus 12

minus 12

0 0 0 0 0 12

0 minus 12

minus 12

0 0 0 0 0 12

0 minus 12

minus 12

0 0 0 0 0 12

0 minus 12

0 0 0 0 0 12

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)10

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 0 0 0

0 0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)11

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 0 0 0

0 0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)12

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 1 1 1 32

0 minus1 0 minus 12

minus 12

minus 32

minus 12

0 minus1 12

0 minus 12

minus 12

0 minus 12

0 minus1 12

0 minus 12

minus 12

0 minus 32

0 minus 12

minus 12

2 2 2 12

0 minus 32

0 minus 12

2 2 2 12

0 minus 32

0 2 2 2 12

0 minus 32

minus 12

0 minus 32

minus 12

0 minus 12

minus 12

0 minus 32

minus 12

0 minus 12

minus 12

0 minus1 12

0 minus 12

minus 12

0 minus 12

0 minus1 12

0 minus 12

minus 12

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)13

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 0 0 0

0 0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

B1 =⟨12 (23) cap (456) (234) cap (56)⟩ ⟨612 (23) cap (456)⟩ ⟨(234) cap (56)612⟩

⟨(23) cap (456) (234) cap (56)61⟩ ⟨2 (23) cap (456) (234) cap (56)6⟩ ⟨2345⟩ ⟨6234⟩ ⟨5623⟩

⟨4562⟩ ⟨3456⟩

(345) cap (67)71⟩ ⟨2 (34) cap (567) (345) cap (67)7⟩ ⟨3456⟩ ⟨7345⟩ ⟨6734⟩ ⟨5673⟩

⟨4567⟩

B3 =⟨123 (345) cap (67)⟩ ⟨7123⟩ ⟨(345) cap (67)712⟩ ⟨3 (345) cap (67)71⟩ ⟨23 (345) cap (67)7⟩

⟨3456⟩ ⟨7345⟩ ⟨6734⟩ ⟨5673⟩ ⟨4567⟩

B4 =⟨123 (456) cap (78)⟩ ⟨8123⟩ ⟨(456) cap (78)812⟩ ⟨3 (456) cap (78)81⟩ ⟨23 (456) cap (78)8⟩

⟨4567⟩ ⟨8456⟩ ⟨7845⟩ ⟨6784⟩⟨5678⟩

B5 =⟨1234⟩ ⟨8123⟩ ⟨4812⟩ ⟨3481⟩ ⟨2348⟩ ⟨4567⟩ ⟨8456⟩ ⟨7845⟩ ⟨6784⟩

⟨5678⟩

B6 =⟨123 (45) cap (678)⟩ ⟨8123⟩ ⟨(45) cap (678)812⟩ ⟨3 (45) cap (678)81⟩ ⟨23 (45) cap (678)8⟩

⟨4567⟩ ⟨8456⟩ ⟨7845⟩ ⟨6784⟩⟨5678⟩

B7 =⟨123 (45) cap (678)⟩ ⟨(456) cap (78)123⟩ ⟨(45) cap (678) (456) cap (78)12⟩

⟨3 (45) cap (678) (456) cap (78)1⟩ ⟨23 (45) cap (678) (456) cap (78)⟩ ⟨4567⟩ ⟨8456⟩ ⟨7845⟩

⟨6784⟩⟨5678⟩

B8 =⟨1234⟩ ⟨(456) cap (78)123⟩ ⟨4 (456) cap (78)12⟩ ⟨34 (456) cap (78)1⟩ ⟨234 (456) cap (78)⟩

⟨4567⟩ ⟨8456⟩ ⟨7845⟩ ⟨6784⟩⟨5678⟩

B9 =⟨1234⟩ ⟨9123⟩ ⟨4912⟩ ⟨3491⟩ ⟨2349⟩ ⟨5678⟩ ⟨9567⟩ ⟨8956⟩

⟨7895⟩ ⟨6789⟩

B10 =⟨1234⟩ ⟨(567) cap (89)123⟩ ⟨4 (567) cap (89)12⟩ ⟨34 (567) cap (89)1⟩ ⟨234 (567) cap (89)⟩

⟨5678⟩ ⟨9567⟩ ⟨8956⟩ ⟨7895⟩ ⟨6789⟩

B11 =⟨1234⟩ ⟨(56) cap (789)123⟩ ⟨4 (56) cap (789)12⟩ ⟨34 (56) cap (789)1⟩ ⟨234 (56) cap (789)⟩

⟨5678⟩ ⟨9567⟩ ⟨8956⟩ ⟨7895⟩ ⟨6789⟩

B12 =⟨1234⟩ ⟨4789⟩ ⟨56 (123) cap (789)⟩ ⟨123 (45) cap (789)⟩ ⟨(46) cap (789)123⟩

⟨789(45) cap (123)⟩ ⟨(46) cap (123)789⟩

B13 =⟨1234⟩ ⟨5123⟩ ⟨4512⟩ ⟨3451⟩ ⟨2345⟩ ⟨6789⟩ ⟨10678⟩ ⟨91067⟩

⟨89106⟩ ⟨78910⟩

Chapter 5

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

Z11 ⋯ Z1

Z21 ⋯ Z2

Z31 ⋯ Z3

Z41 ⋯ Z4

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ETHrarrGL(4)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

1 0 0 0 y15 ⋯ y1

0 1 0 0 y25 ⋯ y2

0 0 1 0 y35 ⋯ y3

0 0 0 1 y45 ⋯ y4

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

56 n with

ld (52)

x2ij =

is given by

yIa yJ b =1

f(y) g(y) =n

sumab=5

sumIJ=1

partyIa

2Z (56)

⟨i i + 1 j j + 1⟩)

in [89]

ABDSn = exp [

infinsumL=1

g2L (f(L)(ε)

2A(1)n (Lε) +C(L))] (57)

Γcusp = 4g2 +O(g4) (58)

ABDS-liken = ABDS

n exp [Γcusp

4Yn] 4 ∣ n

Discx2i+1j

[Discx2i+1i+p

(An)] = 0

Discx2i+1i+p

[Discx2i+1j+p+q

(An)] = 0

⎫⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎭

522 NMHV Amplitudes

form [115 116]

sums=5

sumt=s+2

Rtot =nminus2

sums=3

sumt=s+2

R1st (512)

and our results

Yif (2L)i (515)

minus 1

2([12345] + [12356] + [12367] minus [12457] minus [12567]

+ [13456] + [13467] + [14567] minus [23457] minus [23567])

times (log(⟨1234⟩⟨1789⟩⟨1278⟩⟨1349⟩)otimes ⟨3478⟩)

minus 1

2([12348] minus [12378] + [12478] minus [13478]

+ [23478] + [34567]) (log(⟨1234⟩⟨1789⟩⟨1278⟩⟨1349⟩)otimes ⟨3478⟩)

[12347] [12348] [12349] [12378] [12379] [12389]

[12478] [12479] [12489] [12789] [13478] [13479]

[13489] [13789] [14789] [23478] [23479] [23489]

[23789] [24789] [34567] [34568] [34578] [34678]

[34789] [35678] [45678]

Yangian invariants

a3ijkl = ⟨i(j j + 1)(k k + 1)(l l + 1)⟩

)(525)

condition

cup (j j + 1 k k + 15

) cup (i j j + 1 k k + 15

cup (i k k + 1 l l + 15

) cup (l l + 1 i5

) is a

uijkl =x2ikx

sumi=5

minus 1

sumi=5

minus 1

u2nminus3nminus1n)

sumi=5

minus 1

1247) +1

sumi=4

4ln(u8145) ln(u1256u3478

u2367) + cyclic (532)

nminus2

sumi=4

ln(un1ii+1)iminus1

sumj=1

sumi=4

ln(un1ii+1)iminus1

sumj=1

4ln(un1n

2n2+1)

nminus22

sumi=1

ln(uii+1i+n2i+n

Chapter 6

Graphs

symbol of [23]

C sdotZ = 0 (61)

example [27 93 94 28 30])

for each graph

source columns

C =⎛⎜⎜⎜⎝

1 0 c13 c14 c15 c16

0 1 c23 c24 c25 c26

⎞⎟⎟⎟⎠ (63)

prodαisinp

fα (64)

f0 = minus⟨1234⟩⟨2346⟩ f1 = minus

⟨2346⟩⟨2345⟩ f2 =

⟨2345⟩⟨1236⟩⟨1234⟩⟨2356⟩

f3 = minus⟨2356⟩⟨2346⟩ f4 =

⟨2346⟩⟨1256⟩⟨2456⟩⟨1236⟩ f5 = minus

⟨2456⟩⟨2356⟩

f6 =⟨2356⟩⟨1456⟩⟨3456⟩⟨1256⟩ f7 = minus

⟨3456⟩⟨2456⟩ f8 = minus

⟨2456⟩⟨1456⟩

f(b)0 = f (a)0 (1 + f (a)2 )

f(b)1 = f

1 + 1f (a)2

f(b)2 = 1

f(b)3 = f (a)3 (1 + f (a)2 )

f(b)4 = f

1 + 1f (a)2

f(c)2 = f (a)2 (1 + f (a)4 )

f(c)3 = f

1 + 1f (a)4

f(c)4 = 1

f(c)5 = f (a)5 (1 + f (a)4 )

f(c)6 = f

1 + 1f (a)4

(a) (b) (c)

⟨16⟩ ⟨12⟩

⟨23⟩

⟨34⟩⟨45⟩

⟨56⟩

⟨26⟩

⟨36⟩

⟨46⟩

ampamppp

⟨16⟩ ⟨12⟩

⟨23⟩

⟨34⟩⟨45⟩

⟨56⟩

⟨26⟩

⟨36⟩

⟨35⟩

⟨16⟩ ⟨12⟩

⟨23⟩

⟨34⟩⟨45⟩

⟨56⟩

⟨26⟩

⟨24⟩⟨46⟩ oo

(d) (e) (f)

⟨3456⟩ ⟨1456⟩

⟨1256⟩

⟨1236⟩⟨1234⟩

⟨2345⟩

⟨2456⟩

⟨2356⟩

⟨2346⟩

ampamppp

⟨3456⟩ ⟨1456⟩

⟨1256⟩

⟨1236⟩⟨1234⟩

⟨2345⟩

⟨2456⟩

⟨2356⟩

⟨1235⟩

⟨3456⟩ ⟨1456⟩

⟨1256⟩

⟨1236⟩⟨1234⟩

⟨2345⟩

⟨2456⟩

⟨1246⟩⟨2346⟩ oo

(g) (h) (i)

C = (0 1 f0f1f2f3 f0f1f2 f0f1 f0) (69)

f0 =⟨2345⟩⟨3456⟩ f1 = minus

⟨2346⟩⟨2345⟩ f2 = minus

⟨2356⟩⟨2346⟩ f3 = minus

⟨2456⟩⟨2356⟩ f4 = minus

⟨3456⟩⟨2456⟩

amplitude

f0 =⟨123⟩⟨145⟩

⟨234⟩⟨146⟩ f3 = minus⟨234⟩⟨156⟩⟨123⟩⟨456⟩

f4 = minus⟨124⟩⟨456⟩

⟨134⟩⟨156⟩

f6 = minus⟨134⟩⟨124⟩

f0 =⟨145⟩⟨456⟩ f1 = minus

⟨146⟩⟨145⟩ f2 = minus

⟨156⟩⟨146⟩ f3 = minus

⟨123⟩⟨456⟩⟨234⟩⟨156⟩

f4 = minus⟨124⟩⟨123⟩ f5 = minus

⟨234⟩⟨134⟩ f6 = minus

⟨134⟩⟨124⟩

⟨234⟩⟨156⟩ (616)

from (615)

C =⎛⎜⎜⎜⎝

1 c12 0 c14 c15 c16 c17 c18

0 c32 1 c34 c35 c36 c37 c38

⎞⎟⎟⎟⎠

a4a7 ⟨6(12)(34)(78)⟩ ⟨3478⟩

a2 ⟨4(12)(56)(78)⟩ ⟨1278⟩

a1⟨3(12)(56)(78)⟩

a3 ⟨1(34)(56)(78)⟩

a6 ⟨5(12)(34)(78)⟩

pendix 67

a9⟨1234⟩⟨5678⟩ (625)

∆2468

65 Discussion 101

65 Discussion

on Gr(mn)

65 Discussion 103

in [23]

f(b)0 = minus⟨1235⟩

⟨2356⟩ f(b)1 = minus⟨1236⟩

⟨1235⟩ f(b)2 = ⟨1234⟩⟨2356⟩

⟨2345⟩⟨1236⟩

f(b)3 = minus⟨1235⟩

⟨1234⟩ f(b)4 = ⟨2345⟩⟨1256⟩

⟨1235⟩⟨2456⟩ f(b)5 = minus⟨2456⟩

⟨2356⟩

f(b)6 = ⟨2356⟩⟨1456⟩

⟨3456⟩⟨1256⟩ f(b)7 = minus⟨3456⟩

⟨2456⟩ f(b)8 = minus⟨2456⟩

⟨1456⟩

f(c)0 = minus⟨1234⟩

⟨2346⟩ f(c)1 = minus⟨2346⟩

⟨2345⟩ f(c)2 = ⟨2345⟩⟨1246⟩

⟨1234⟩⟨2456⟩

f(c)3 = minus⟨1256⟩

⟨1246⟩ f(c)4 = ⟨2456⟩⟨1236⟩

⟨2346⟩⟨1256⟩ f(c)5 = minus⟨1246⟩

⟨1236⟩

f(c)6 = ⟨1456⟩⟨2346⟩

⟨3456⟩⟨1246⟩ f(c)7 = minus⟨3456⟩

⟨2456⟩ f(c)8 = minus⟨2456⟩

⟨1456⟩

∆1357 andradic

∆2468 with

∆1357 and

2(1 + u minus v +

u = ⟨1278⟩⟨3456⟩⟨1256⟩⟨3478⟩ v = ⟨1234⟩⟨5678⟩

⟨1256⟩⟨3478⟩ (632)

radic∆1357

⟨1256⟩⟨3478⟩ (633)

∣irarri+6

∣irarri+4

∣irarri+6

∣irarri+4

∣irarri+6

z a9 =

1 minus z1 minus z

Z isin Gr(48)gt0

Bibliography

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56 2459 (1986) doi101103PhysRevLett562459

th9711200 [hep-th]]

110 BIBLIOGRAPHY

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[hep-th0503198]

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Phys 90 438 (1975) doi1010160003-4916(75)90006-8

doi101007BF01077848

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365 (1979)

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JHEP 10 031 (2014) [arXiv14067273 [hep-th]]

Abstract

Acknowledgements

Introduction












n-point MHV




6-point MHV

n-point NMHV

n-point NkMHV





Soft limits






NMHV

NNMHV

NNNMHV and Higher




One-loop Amplitudes


NMHV Amplitudes











Discussion



BROWN UNIVERSITY

Abstract

Anastasia Volovich

theory

Curriculum Vitae

Research

Education

Teaching

Schools and Talks

Jun 2019 TASI 2019

Additional Skills

Acknowledgements

Contents

Abstract v

Acknowledgements xi

1 Introduction 1

22 n-point MHV 19

224 6-point MHV 24

23 n-point NMHV 25

24 n-point NkMHV 28

33 Soft limits 43

421 NMHV 60

422 N2MHV 62

65 Discussion 101

List of Figures

to ⟨k(l l + 1)(mm + 1)(k minus 1k + 1)⟩ 80

to ⟨k(l l + 1)(mm + 1)(pp + 1)⟩ 81

List of Tables

I love you all

Chapter 1

Introduction

amplitudes

calculated

(11) transforms as

az + bcz + d

) = ∣cz + d∣2∆φ∆plusmn(X z z) (13)

prodj=1int

0dωj ω

az + bcz + d

prodj=1

gluon amplitudes

microi+1 (16)

〈2345〉

〈2346〉〈2356〉〈2456〉〈3456〉

〈1234〉〈1236〉〈1256〉〈1456〉

akaprimek = prod

i∣bikgt0

aminusbiki (112)

Y (Z ∣η) = int4k

prodi=1

d log fi4

prodI=1

prodα=1

suma=1

Cαa(Z ∣η)aI) (113)

prodαisinp

fα (114)

fi = 1 (115)

coordinates

Chapter 2

Celestial Sphere

radicωiωjzij (22)

prodj=1int

0dωj ω

azj + bczj + d

prodj=1

22 n-point MHV 19

22 n-point MHV

ratio ωi

dωiωi

(ωisi

j=1 iλj

s1+iλii

⎛⎜⎝

proda=1anei

intinfin

0dωa ω

intinfin

dωiωi

(ωisi

= 2πδ(z) (29)

sumj=1

λj⎞⎠s1+iλii

⎛⎜⎝

proda=1anei

intinfin

0dωa ω

⎞⎟⎠A`1⋯`n(si ωl zj zj) (210)

δ4(ε1s1q1 +n

sumi=2

εiωiqi) =4

prodj=nminus3

U+nminus4

sumi=2

U) (212)

22 n-point MHV 21

ui =z31zi2z32zi1

for i = 45 n (216)

s1 =∣z32∣

∣z31∣ ∣z12∣ s2 =

∣z31∣∣z32∣ ∣z21∣

and si =∣z12∣

∣z1i∣ ∣zi2∣for i = 3 n (217)

to demand Uij

prodij 1lt0(Uij

12 siλ1+21

nminus4

proda=2int

0dωa ω

1lt0(Uij

ωi = siU1n

uiminus1

i = 23 n minus 4 (220)

which leads to

12siλ1+21 siλ2+2

2 siλ33 siλnn

z23z34zn1U1nδ(

sumj=1

λj) ϕ(α x)prodij

1lt0(Uij

U) (221)

22 n-point MHV 23

prodj=1

P2and and dPnminus4

Pnminus4

αn =1 + iλ1 x0 i =U1i

U1n xjminus1 i =

Ujnminus U1i

U1n x0n = minus

U1n xjminus1n =

ϕ(α x) equivc2

c1intu1ge0u3ge0

1minussuma uage0

prodj=1

Pnminus1

U1n xij =

Ujnminus4+i

x0n = minusU

U1n xin =

U1n x01 = 1

mannian

224 6-point MHV

)iλ1+1

( minusUU16

)iλ2+2

iλ3minus1

iλ4minus1

iλ5minus1

times int1

prodi=1

23 n-point NMHV 25

U16U2i+2for i = 123 (227)

23 n-point NMHV

nminus1

sumj=4

⟨1∣P2jPj+12∣3⟩3

P 22jP

⟨j + 1 j⟩[2∣P2j ∣j + 1⟩⟨j∣Pj+12∣2]

equivnminus1

sumj=4

Mj (228)

⟨34⟩⟨45⟩⋯⟨n minus 1 n⟩⟨n1⟩(⟨12⟩[24]⟨43⟩ + ⟨13⟩[34]⟨43⟩)3

(⟨23⟩[23] + ⟨24⟩[24] + ⟨34⟩[34])⟨34⟩[34]times

times ⟨54⟩([23]⟨35⟩ + [24]⟨45⟩)(⟨43⟩[32]) (229)

M4 =ω1ω4(ε2z12z24ω2+ε3z13z34ω3)3

of variables

ωi = siuiminus1

i = 2 n minus 4 (231)

M4 sim δ⎛⎝n

sumj=1

λj⎞⎠

prodni=1 siλii

z12z13z45z4ns21s

z34zn1UF(αx)prod

1lt0(Uij

U) (232)

nminus5

proda=1

nminus5

prodj=1

uiλj+1

times7

prodi=1

23 n-point NMHV 27

U xj4 =

U j = 1 n minus 5

x05 =U1nminus2

U xj5 =

U j = 1 n minus 5 (235)

x06 =U1nminus1

U xj6 =

U j = 1 n minus 5

x07 =U1n

U xj7 =

Ujn minusU1n

U j = 1 n minus 5

6- and 7-point NMHV

u1uiλ2

prodi=2

(x0i + u1x1i)αi (236)

u2uiλ2

1 uiλ32 (u1x10 + u2x20 + u1u2x30 + u2

times7

prodi=1

24 n-point NkMHV

24 n-point NkMHV 29

N2MHV amplitude

F41F23

⟨1∣P26P72P35P63∣4⟩3

P 226P

⟨76⟩[23]⟨65⟩[2∣P26∣7⟩⟨6∣P72∣2][3∣P35∣6⟩⟨5∣P63∣3]

u3uα1

1 uα22 uα3

prodi=4

times17

prodj=14

2x8j + u23x9j)αj

NkMHV amplitude

nminus4

prodl=2

dululuαl

⎛⎝

1 minusnminus4

sumj=2

uj⎞⎠

P32k (prod

(P i1)αi)

⎛⎝prodj

(Pj2)αj

⎞⎠

prodj=0

Pj(u)αjdϕ (243)

dϕ =dPj1Pj1

and and dPjnPjn

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

x00 x0m

x10 x1m

⋮ ⋱ ⋮

xn0 xnm

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

gauge fixed form

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

1 0 0 1 1 1

⋮ ⋱ minus1 ⋮ ⋮ ⋮

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ways eq (324) of [38]

prodi=1

suml=1

prodj=1

(1 minusn

sumi=1

xijui)minusβj

sumi=1

αi) sdotn

prodi=1

Γ(αi) (248)

prodi=1

mminusnminus1

suml=1

prodj=1

sumi=1

xjiui)minusαj

sumi=1

prodi=1

Γ(βi) (249)

γ minusn

sumi=1

sumj=1

βj notin Z(250)

sumi=0

xijpartϕ

sumj=0

xijpartϕ

(3) part2ϕ

k du1duk (252)

Chapter 3

prodl=1

prodj=1

sumi=1

ki)( 1

(k1+k2)2+m2+ 1

(k1+k3)2+m2+ 1

(k1+k4)2+m2)

(hj hj) =1

A4(zj zj) =( z24

z14)h12 ( z14

z13)h34

zh1+h212 zh3+h4

z14)h12 ( z14

z13)h34

zh1+h212 zh3+h4

A4(z z) (35)

z = z12z34

z13z24 z = z12z34

compute

τminus2∆4A4(0 z11τ) (37)

prodi=1

⎛⎝

sumj=1

kj⎞⎠

m2 minus 4ω1ω2∣z∣2 (38)

δ(4)⎛⎝

sumj=1

kj⎞⎠= 4τ2

ω1δ(iz minus iz)

prodi=2

ωlowast2 = ω1

2 (310)

0dω1ω

∆1minus21

prodi=2

m2 minus 4z2

zminus1ω21

= g2 (im2)2αminus2

z minus 1)α

+ zα) (312)

sumi=1

hi minus 2 (313)

constructed in [51]

2(h2+h3)A13harr24

4 (z) 0 lt z lt 1 (315)

as well as

4 (1z)

z minus 1) 0 lt z lt 1 (316)

2 (im2)2αminus2

2 sin(πα) intC

hi(z z)

2= 1 minus h12 + n

2= 1 minus h34 + n

2= h12 minus n

2= h34 minus n (318)

with n = 0123

hi(z z) (319)

4hi+hj

Γ (hij + ∆2)Γ (∆

and Ψ∆hi(z z) is

Ψ∆hi(z z) equiv

2 minus h34)Γ (∆

2 + h12)Γ (1 minus ∆2 + h34)

Ψ∆hi(z z) (322)

infinsumJ=0

4π2Ψhh

+(h12 harr h34))

33 Soft limits 43

∆ minus J2

2= 1 minus h34 + n

∆ + J2

2= h34 minus n

33 Soft limits

Single soft limits

limωkrarr0

A`k=+1n = ⟨k minus 1k + 1⟩

limωkrarr0

δ(x) = 1

2limλrarr0

limλkrarr0

An(zj zj) =1

prodj=1jnek

intinfin

0dωj ω

iλjj int

limλkrarr0

AJk=+1n = 1

zkminus1k+1

limλkrarr0

AJk=minus1n = 1

zkminus1k+1

Double soft limits

33 Soft limits 45

⟨n∣1 + 2∣3] ( [13]3⟨n3⟩[12][23]s123

+ ⟨n2⟩3[n3]⟨n1⟩⟨12⟩sn12

we will write

An(zj zj) =n

prodj=1jnekl

intinfin

0dωj ω

iλjj int

0dr int

DS(kJk lJl) = 1

r=0]εrarr0

2(iλ1 + iλ2)ε21

zn1z23( 1

zn3z2n

z12z2n+ 1

z3nz31

z12z31) (338)

sumi=1

4ω1⋯ω6

ω21ω

ω23ω

26(ω4z46z34+ω5z56z35)3

(ω3ω4z34z34+ω3ω5z35z35+ω4ω5z45z45)

z12z16z34z45 (ω3z23z35 + ω4z24z45)

⎤⎥⎥⎥⎥⎥⎦

2(iλ3 + iλ4)ε21

z23z45( 1

z25z41

z34z42+ 1

z52z53

25z216z25z61

(z15z61

z25z26)iλ2minus1

(z12z16

z25z56)iλ5+1

(z15z12

z56z26)iλ6+1

⟨2∣3 + 4∣5] ( [35]3⟨25⟩[34][45]s345

+ ⟨24⟩3[25]⟨23⟩⟨34⟩s234

) (342)

intCd∆ Ψhh

hihi(z z)⟩ (343)

(zz)2G(z z)F (z z) (344)

given by

F+(z z) =1

zh34 zh342F1 (

1 minus h + h34 h + h34

1 + h12 + h341

z) 2F1 (

1 minus h + h34 h + h34

1 + h12 + h341

z) (346)

z) 2F1 (

⟨Ψhh

hihi(z z)⟩ = (2π)2

⟨A4(z z)Ψhh

hihi(z z) (349)

2F1(a b c z) =Γ(c)

obtain

⟨A4(z z)Ψhh

⎡⎢⎢⎢⎢⎢⎣

⎛⎜⎝

⎤⎥⎥⎥⎥⎥⎥⎦

sin(2πJ)2πJ

= 1 J = 0

0 J ne 0 (353)

hihi∣J=0

which simplifies to

2 minus h34)Γ(2 minus∆)Γ(∆) (354)

⟨A4(z z)Ψhh

Chapter 4

⟨1 2 6 7⟩⟨2 3 4 5⟩ minus ⟨1 2 4 5⟩⟨2 3 6 7⟩ (42)

cluster coordinate

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

1 0 0 0 y15 ⋯ y1

0 1 0 0 y25 ⋯ y2

0 0 1 0 y35 ⋯ y3

0 0 0 1 y45 ⋯ y4

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

yIa yJ b =1

f(y) g(y) =n

sumab=1

sumIJ=1

partyIa

2Z foralli j (46)

entries

n5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Total

1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1

2 0 1 2 5 4 1 0 0 0 0 0 0 0 0 0 0 13

3 0 0 1 6 54 177 298 274 134 30 3 0 0 0 0 0 977

4 0 0 0 1 13 263 1988 7862 18532 28204 28377 18925 8034 2047 270 17 114533

421 NMHV

p1 p5 = ⟨1 2 3 4⟩ ⟨2 3 4 5⟩ ⟨3 4 5 1⟩ ⟨4 5 1 2⟩ ⟨5 1 2 3⟩ (48)

5minusy35minusy4

5 (49)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0

0 0 12

0 minus12 0 1

0 minus12 minus1

2 0 12

0 minus12 minus1

2 minus12 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

422 N2MHV

1 = [12 (23) cap (456) (234) cap (56)6][23456]

2 = [12 (34) cap (567) (345) cap (67)7][34567]

3 = [123 (345) cap (67)7][34567]

4 = [123 (456) cap (78)8][45678]

5 = [12348][45678]

6 = [123 (45) cap (678)8][45678]

7 = [123 (45) cap (678) (456) cap (78)][45678] (411)

8 = [1234 (456) cap (78)][45678]

9 = [12349][56789]

10 = [1234 (567) cap (89)][56789]

11 = [1234 (56) cap (789)][56789]

13 = [12345][678910]

ϕ = ⟨4 5 (123) cap (789)⟩⟨4 6 (123) cap (789)⟩⟨1 2 3 4⟩⟨4 7 8 9⟩⟨5 6 (123) cap (789)⟩ (413)

Ω(2)1

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 1 1 0 0 0 12

minus 12

minus1

0 0 0 0 minus 12

0 minus 12

minus 12

minus1

0 minus 12

minus 12

minus1

0 minus 12

0 minus1 minus1

minus 12

minus1

0 0 0 0 minus 12

0 minus 12

0 0 0 0

minus 12

0 minus 12

0 0 0 minus 12

minus 12

0 0 12

0 minus 12

1 1 1 1 1 0 0 12

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 1 0 0 0 0 minus1 minus 12

minus 12

0 0 0 minus 12

minus 12

0 minus 12

minus 12

minus1 0 0 minus 32

minus 32

0 minus 12

minus 32

minus 12

0 minus 12

0 minus1 minus 12

minus 12

0 minus1 minus 12

minus 12

0 0 0 minus 12

minus 12

0 minus 12

minus 12

0 0 12

0 minus 12

1 1 12

0 0 0 0 minus 12

0 0 0 12

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)3

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 12

0 0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

minus 32

minus 12

1 0 minus 12

0 minus1 0 minus 12

0 0 0 minus 12

minus 12

0 minus 12

minus 12

0 0 12

0 minus 12

1 1 12

0 0 12

0 0 0 0 0 0 minus 12

0 0 0 12

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)4

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 minus1 minus1 0

0 0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus1 minus1 0

0 minus 12

0 minus1 minus1 0

0 0 minus 12

minus 12

0 minus 12

minus 12

0 0 0 12

0 minus 12

minus 12

1 1 1 12

0 minus 12

1 1 1 12

0 0 0 0 0 0 0 0 0 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)5

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 0 0 0

0 0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 0 0 0

0 minus 12

0 0 0 0

0 0 minus 12

minus 12

0 minus 12

minus 12

0 0 0 12

0 minus 12

minus 12

0 0 0 12

0 minus 12

0 0 0 12

0 0 0 0 0 0 0 0 0 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)6

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 0 minus1 0

0 0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 0 minus1 0

0 minus 12

0 0 minus1 0

0 0 minus 12

minus 12

0 minus 12

minus 12

0 0 0 12

0 minus 12

minus 12

0 0 0 12

0 minus 12

1 1 1 12

0 0 0 0 0 0 0 0 0 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)7

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 0 minus1 0

0 1 0 minus 12

minus 12

minus 32

0 1 12

0 minus 12

minus 12

minus 32

0 1 12

minus 12

minus 32

0 0 minus 12

minus 12

0 minus 12

minus 12

0 0 minus 12

minus 12

0 minus 12

minus 12

0 1 12

0 minus 12

1 1 32

0 0 0 0 0 0 0 0 0 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)8

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 0 0 0

0 1 0 minus 12

minus 12

0 1 12

0 minus 12

minus 12

0 1 12

minus 12

0 0 minus 12

minus 12

0 minus 12

minus 12

0 0 minus 12

minus 12

0 minus 12

minus 12

0 1 12

0 minus 12

0 1 12

0 0 0 0 0 0 0 0 0 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)9

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 0 0 0

0 0 minus 12

minus 12

0 0 0 0

0 minus 12

minus 12

0 0 0 0

0 minus 12

0 0 0 0

0 minus 12

minus 12

0 minus 12

minus 12

0 0 0 0 0 12

0 minus 12

minus 12

0 0 0 0 0 12

0 minus 12

minus 12

0 0 0 0 0 12

0 minus 12

0 0 0 0 0 12

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)10

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 0 0 0

0 0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)11

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 0 0 0

0 0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)12

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 1 1 1 32

0 minus1 0 minus 12

minus 12

minus 32

minus 12

0 minus1 12

0 minus 12

minus 12

0 minus 12

0 minus1 12

0 minus 12

minus 12

0 minus 32

0 minus 12

minus 12

2 2 2 12

0 minus 32

0 minus 12

2 2 2 12

0 minus 32

0 2 2 2 12

0 minus 32

minus 12

0 minus 32

minus 12

0 minus 12

minus 12

0 minus 32

minus 12

0 minus 12

minus 12

0 minus1 12

0 minus 12

minus 12

0 minus 12

0 minus1 12

0 minus 12

minus 12

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)13

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 0 0 0

0 0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

B1 =⟨12 (23) cap (456) (234) cap (56)⟩ ⟨612 (23) cap (456)⟩ ⟨(234) cap (56)612⟩

⟨(23) cap (456) (234) cap (56)61⟩ ⟨2 (23) cap (456) (234) cap (56)6⟩ ⟨2345⟩ ⟨6234⟩ ⟨5623⟩

⟨4562⟩ ⟨3456⟩

(345) cap (67)71⟩ ⟨2 (34) cap (567) (345) cap (67)7⟩ ⟨3456⟩ ⟨7345⟩ ⟨6734⟩ ⟨5673⟩

⟨4567⟩

B3 =⟨123 (345) cap (67)⟩ ⟨7123⟩ ⟨(345) cap (67)712⟩ ⟨3 (345) cap (67)71⟩ ⟨23 (345) cap (67)7⟩

⟨3456⟩ ⟨7345⟩ ⟨6734⟩ ⟨5673⟩ ⟨4567⟩

B4 =⟨123 (456) cap (78)⟩ ⟨8123⟩ ⟨(456) cap (78)812⟩ ⟨3 (456) cap (78)81⟩ ⟨23 (456) cap (78)8⟩

⟨4567⟩ ⟨8456⟩ ⟨7845⟩ ⟨6784⟩⟨5678⟩

B5 =⟨1234⟩ ⟨8123⟩ ⟨4812⟩ ⟨3481⟩ ⟨2348⟩ ⟨4567⟩ ⟨8456⟩ ⟨7845⟩ ⟨6784⟩

⟨5678⟩

B6 =⟨123 (45) cap (678)⟩ ⟨8123⟩ ⟨(45) cap (678)812⟩ ⟨3 (45) cap (678)81⟩ ⟨23 (45) cap (678)8⟩

⟨4567⟩ ⟨8456⟩ ⟨7845⟩ ⟨6784⟩⟨5678⟩

B7 =⟨123 (45) cap (678)⟩ ⟨(456) cap (78)123⟩ ⟨(45) cap (678) (456) cap (78)12⟩

⟨3 (45) cap (678) (456) cap (78)1⟩ ⟨23 (45) cap (678) (456) cap (78)⟩ ⟨4567⟩ ⟨8456⟩ ⟨7845⟩

⟨6784⟩⟨5678⟩

B8 =⟨1234⟩ ⟨(456) cap (78)123⟩ ⟨4 (456) cap (78)12⟩ ⟨34 (456) cap (78)1⟩ ⟨234 (456) cap (78)⟩

⟨4567⟩ ⟨8456⟩ ⟨7845⟩ ⟨6784⟩⟨5678⟩

B9 =⟨1234⟩ ⟨9123⟩ ⟨4912⟩ ⟨3491⟩ ⟨2349⟩ ⟨5678⟩ ⟨9567⟩ ⟨8956⟩

⟨7895⟩ ⟨6789⟩

B10 =⟨1234⟩ ⟨(567) cap (89)123⟩ ⟨4 (567) cap (89)12⟩ ⟨34 (567) cap (89)1⟩ ⟨234 (567) cap (89)⟩

⟨5678⟩ ⟨9567⟩ ⟨8956⟩ ⟨7895⟩ ⟨6789⟩

B11 =⟨1234⟩ ⟨(56) cap (789)123⟩ ⟨4 (56) cap (789)12⟩ ⟨34 (56) cap (789)1⟩ ⟨234 (56) cap (789)⟩

⟨5678⟩ ⟨9567⟩ ⟨8956⟩ ⟨7895⟩ ⟨6789⟩

B12 =⟨1234⟩ ⟨4789⟩ ⟨56 (123) cap (789)⟩ ⟨123 (45) cap (789)⟩ ⟨(46) cap (789)123⟩

⟨789(45) cap (123)⟩ ⟨(46) cap (123)789⟩

B13 =⟨1234⟩ ⟨5123⟩ ⟨4512⟩ ⟨3451⟩ ⟨2345⟩ ⟨6789⟩ ⟨10678⟩ ⟨91067⟩

⟨89106⟩ ⟨78910⟩

Chapter 5

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

Z11 ⋯ Z1

Z21 ⋯ Z2

Z31 ⋯ Z3

Z41 ⋯ Z4

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ETHrarrGL(4)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

1 0 0 0 y15 ⋯ y1

0 1 0 0 y25 ⋯ y2

0 0 1 0 y35 ⋯ y3

0 0 0 1 y45 ⋯ y4

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

56 n with

ld (52)

x2ij =

is given by

yIa yJ b =1

f(y) g(y) =n

sumab=5

sumIJ=1

partyIa

2Z (56)

⟨i i + 1 j j + 1⟩)

in [89]

ABDSn = exp [

infinsumL=1

g2L (f(L)(ε)

2A(1)n (Lε) +C(L))] (57)

Γcusp = 4g2 +O(g4) (58)

ABDS-liken = ABDS

n exp [Γcusp

4Yn] 4 ∣ n

Discx2i+1j

[Discx2i+1i+p

(An)] = 0

Discx2i+1i+p

[Discx2i+1j+p+q

(An)] = 0

⎫⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎭

522 NMHV Amplitudes

form [115 116]

sums=5

sumt=s+2

Rtot =nminus2

sums=3

sumt=s+2

R1st (512)

and our results

Yif (2L)i (515)

minus 1

2([12345] + [12356] + [12367] minus [12457] minus [12567]

+ [13456] + [13467] + [14567] minus [23457] minus [23567])

times (log(⟨1234⟩⟨1789⟩⟨1278⟩⟨1349⟩)otimes ⟨3478⟩)

minus 1

2([12348] minus [12378] + [12478] minus [13478]

+ [23478] + [34567]) (log(⟨1234⟩⟨1789⟩⟨1278⟩⟨1349⟩)otimes ⟨3478⟩)

[12347] [12348] [12349] [12378] [12379] [12389]

[12478] [12479] [12489] [12789] [13478] [13479]

[13489] [13789] [14789] [23478] [23479] [23489]

[23789] [24789] [34567] [34568] [34578] [34678]

[34789] [35678] [45678]

Yangian invariants

a3ijkl = ⟨i(j j + 1)(k k + 1)(l l + 1)⟩

)(525)

condition

cup (j j + 1 k k + 15

) cup (i j j + 1 k k + 15

cup (i k k + 1 l l + 15

) cup (l l + 1 i5

) is a

uijkl =x2ikx

sumi=5

minus 1

sumi=5

minus 1

u2nminus3nminus1n)

sumi=5

minus 1

1247) +1

sumi=4

4ln(u8145) ln(u1256u3478

u2367) + cyclic (532)

nminus2

sumi=4

ln(un1ii+1)iminus1

sumj=1

sumi=4

ln(un1ii+1)iminus1

sumj=1

4ln(un1n

2n2+1)

nminus22

sumi=1

ln(uii+1i+n2i+n

Chapter 6

Graphs

symbol of [23]

C sdotZ = 0 (61)

example [27 93 94 28 30])

for each graph

source columns

C =⎛⎜⎜⎜⎝

1 0 c13 c14 c15 c16

0 1 c23 c24 c25 c26

⎞⎟⎟⎟⎠ (63)

prodαisinp

fα (64)

f0 = minus⟨1234⟩⟨2346⟩ f1 = minus

⟨2346⟩⟨2345⟩ f2 =

⟨2345⟩⟨1236⟩⟨1234⟩⟨2356⟩

f3 = minus⟨2356⟩⟨2346⟩ f4 =

⟨2346⟩⟨1256⟩⟨2456⟩⟨1236⟩ f5 = minus

⟨2456⟩⟨2356⟩

f6 =⟨2356⟩⟨1456⟩⟨3456⟩⟨1256⟩ f7 = minus

⟨3456⟩⟨2456⟩ f8 = minus

⟨2456⟩⟨1456⟩

f(b)0 = f (a)0 (1 + f (a)2 )

f(b)1 = f

1 + 1f (a)2

f(b)2 = 1

f(b)3 = f (a)3 (1 + f (a)2 )

f(b)4 = f

1 + 1f (a)2

f(c)2 = f (a)2 (1 + f (a)4 )

f(c)3 = f

1 + 1f (a)4

f(c)4 = 1

f(c)5 = f (a)5 (1 + f (a)4 )

f(c)6 = f

1 + 1f (a)4

(a) (b) (c)

⟨16⟩ ⟨12⟩

⟨23⟩

⟨34⟩⟨45⟩

⟨56⟩

⟨26⟩

⟨36⟩

⟨46⟩

ampamppp

⟨16⟩ ⟨12⟩

⟨23⟩

⟨34⟩⟨45⟩

⟨56⟩

⟨26⟩

⟨36⟩

⟨35⟩

⟨16⟩ ⟨12⟩

⟨23⟩

⟨34⟩⟨45⟩

⟨56⟩

⟨26⟩

⟨24⟩⟨46⟩ oo

(d) (e) (f)

⟨3456⟩ ⟨1456⟩

⟨1256⟩

⟨1236⟩⟨1234⟩

⟨2345⟩

⟨2456⟩

⟨2356⟩

⟨2346⟩

ampamppp

⟨3456⟩ ⟨1456⟩

⟨1256⟩

⟨1236⟩⟨1234⟩

⟨2345⟩

⟨2456⟩

⟨2356⟩

⟨1235⟩

⟨3456⟩ ⟨1456⟩

⟨1256⟩

⟨1236⟩⟨1234⟩

⟨2345⟩

⟨2456⟩

⟨1246⟩⟨2346⟩ oo

(g) (h) (i)

C = (0 1 f0f1f2f3 f0f1f2 f0f1 f0) (69)

f0 =⟨2345⟩⟨3456⟩ f1 = minus

⟨2346⟩⟨2345⟩ f2 = minus

⟨2356⟩⟨2346⟩ f3 = minus

⟨2456⟩⟨2356⟩ f4 = minus

⟨3456⟩⟨2456⟩

amplitude

f0 =⟨123⟩⟨145⟩

⟨234⟩⟨146⟩ f3 = minus⟨234⟩⟨156⟩⟨123⟩⟨456⟩

f4 = minus⟨124⟩⟨456⟩

⟨134⟩⟨156⟩

f6 = minus⟨134⟩⟨124⟩

f0 =⟨145⟩⟨456⟩ f1 = minus

⟨146⟩⟨145⟩ f2 = minus

⟨156⟩⟨146⟩ f3 = minus

⟨123⟩⟨456⟩⟨234⟩⟨156⟩

f4 = minus⟨124⟩⟨123⟩ f5 = minus

⟨234⟩⟨134⟩ f6 = minus

⟨134⟩⟨124⟩

⟨234⟩⟨156⟩ (616)

from (615)

C =⎛⎜⎜⎜⎝

1 c12 0 c14 c15 c16 c17 c18

0 c32 1 c34 c35 c36 c37 c38

⎞⎟⎟⎟⎠

a4a7 ⟨6(12)(34)(78)⟩ ⟨3478⟩

a2 ⟨4(12)(56)(78)⟩ ⟨1278⟩

a1⟨3(12)(56)(78)⟩

a3 ⟨1(34)(56)(78)⟩

a6 ⟨5(12)(34)(78)⟩

pendix 67

a9⟨1234⟩⟨5678⟩ (625)

∆2468

65 Discussion 101

65 Discussion

on Gr(mn)

65 Discussion 103

in [23]

f(b)0 = minus⟨1235⟩

⟨2356⟩ f(b)1 = minus⟨1236⟩

⟨1235⟩ f(b)2 = ⟨1234⟩⟨2356⟩

⟨2345⟩⟨1236⟩

f(b)3 = minus⟨1235⟩

⟨1234⟩ f(b)4 = ⟨2345⟩⟨1256⟩

⟨1235⟩⟨2456⟩ f(b)5 = minus⟨2456⟩

⟨2356⟩

f(b)6 = ⟨2356⟩⟨1456⟩

⟨3456⟩⟨1256⟩ f(b)7 = minus⟨3456⟩

⟨2456⟩ f(b)8 = minus⟨2456⟩

⟨1456⟩

f(c)0 = minus⟨1234⟩

⟨2346⟩ f(c)1 = minus⟨2346⟩

⟨2345⟩ f(c)2 = ⟨2345⟩⟨1246⟩

⟨1234⟩⟨2456⟩

f(c)3 = minus⟨1256⟩

⟨1246⟩ f(c)4 = ⟨2456⟩⟨1236⟩

⟨2346⟩⟨1256⟩ f(c)5 = minus⟨1246⟩

⟨1236⟩

f(c)6 = ⟨1456⟩⟨2346⟩

⟨3456⟩⟨1246⟩ f(c)7 = minus⟨3456⟩

⟨2456⟩ f(c)8 = minus⟨2456⟩

⟨1456⟩

∆1357 andradic

∆2468 with

∆1357 and

2(1 + u minus v +

u = ⟨1278⟩⟨3456⟩⟨1256⟩⟨3478⟩ v = ⟨1234⟩⟨5678⟩

⟨1256⟩⟨3478⟩ (632)

radic∆1357

⟨1256⟩⟨3478⟩ (633)

∣irarri+6

∣irarri+4

∣irarri+6

∣irarri+4

∣irarri+6

z a9 =

1 minus z1 minus z

Z isin Gr(48)gt0

Bibliography

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th9711200 [hep-th]]

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JHEP 10 031 (2014) [arXiv14067273 [hep-th]]

Abstract

Acknowledgements

Introduction












n-point MHV




6-point MHV

n-point NMHV

n-point NkMHV





Soft limits






NMHV

NNMHV

NNNMHV and Higher




One-loop Amplitudes


NMHV Amplitudes











Discussion



Curriculum Vitae

Research

Education

Teaching

Schools and Talks

Jun 2019 TASI 2019

Additional Skills

Acknowledgements

Contents

Abstract v

Acknowledgements xi

1 Introduction 1

22 n-point MHV 19

224 6-point MHV 24

23 n-point NMHV 25

24 n-point NkMHV 28

33 Soft limits 43

421 NMHV 60

422 N2MHV 62

65 Discussion 101

List of Figures

to ⟨k(l l + 1)(mm + 1)(k minus 1k + 1)⟩ 80

to ⟨k(l l + 1)(mm + 1)(pp + 1)⟩ 81

List of Tables

I love you all

Chapter 1

Introduction

amplitudes

calculated

(11) transforms as

az + bcz + d

) = ∣cz + d∣2∆φ∆plusmn(X z z) (13)

prodj=1int

0dωj ω

az + bcz + d

prodj=1

gluon amplitudes

microi+1 (16)

〈2345〉

〈2346〉〈2356〉〈2456〉〈3456〉

〈1234〉〈1236〉〈1256〉〈1456〉

akaprimek = prod

i∣bikgt0

aminusbiki (112)

Y (Z ∣η) = int4k

prodi=1

d log fi4

prodI=1

prodα=1

suma=1

Cαa(Z ∣η)aI) (113)

prodαisinp

fα (114)

fi = 1 (115)

coordinates

Chapter 2

Celestial Sphere

radicωiωjzij (22)

prodj=1int

0dωj ω

azj + bczj + d

prodj=1

22 n-point MHV 19

22 n-point MHV

ratio ωi

dωiωi

(ωisi

j=1 iλj

s1+iλii

⎛⎜⎝

proda=1anei

intinfin

0dωa ω

intinfin

dωiωi

(ωisi

= 2πδ(z) (29)

sumj=1

λj⎞⎠s1+iλii

⎛⎜⎝

proda=1anei

intinfin

0dωa ω

⎞⎟⎠A`1⋯`n(si ωl zj zj) (210)

δ4(ε1s1q1 +n

sumi=2

εiωiqi) =4

prodj=nminus3

U+nminus4

sumi=2

U) (212)

22 n-point MHV 21

ui =z31zi2z32zi1

for i = 45 n (216)

s1 =∣z32∣

∣z31∣ ∣z12∣ s2 =

∣z31∣∣z32∣ ∣z21∣

and si =∣z12∣

∣z1i∣ ∣zi2∣for i = 3 n (217)

to demand Uij

prodij 1lt0(Uij

12 siλ1+21

nminus4

proda=2int

0dωa ω

1lt0(Uij

ωi = siU1n

uiminus1

i = 23 n minus 4 (220)

which leads to

12siλ1+21 siλ2+2

2 siλ33 siλnn

z23z34zn1U1nδ(

sumj=1

λj) ϕ(α x)prodij

1lt0(Uij

U) (221)

22 n-point MHV 23

prodj=1

P2and and dPnminus4

Pnminus4

αn =1 + iλ1 x0 i =U1i

U1n xjminus1 i =

Ujnminus U1i

U1n x0n = minus

U1n xjminus1n =

ϕ(α x) equivc2

c1intu1ge0u3ge0

1minussuma uage0

prodj=1

Pnminus1

U1n xij =

Ujnminus4+i

x0n = minusU

U1n xin =

U1n x01 = 1

mannian

224 6-point MHV

)iλ1+1

( minusUU16

)iλ2+2

iλ3minus1

iλ4minus1

iλ5minus1

times int1

prodi=1

23 n-point NMHV 25

U16U2i+2for i = 123 (227)

23 n-point NMHV

nminus1

sumj=4

⟨1∣P2jPj+12∣3⟩3

P 22jP

⟨j + 1 j⟩[2∣P2j ∣j + 1⟩⟨j∣Pj+12∣2]

equivnminus1

sumj=4

Mj (228)

⟨34⟩⟨45⟩⋯⟨n minus 1 n⟩⟨n1⟩(⟨12⟩[24]⟨43⟩ + ⟨13⟩[34]⟨43⟩)3

(⟨23⟩[23] + ⟨24⟩[24] + ⟨34⟩[34])⟨34⟩[34]times

times ⟨54⟩([23]⟨35⟩ + [24]⟨45⟩)(⟨43⟩[32]) (229)

M4 =ω1ω4(ε2z12z24ω2+ε3z13z34ω3)3

of variables

ωi = siuiminus1

i = 2 n minus 4 (231)

M4 sim δ⎛⎝n

sumj=1

λj⎞⎠

prodni=1 siλii

z12z13z45z4ns21s

z34zn1UF(αx)prod

1lt0(Uij

U) (232)

nminus5

proda=1

nminus5

prodj=1

uiλj+1

times7

prodi=1

23 n-point NMHV 27

U xj4 =

U j = 1 n minus 5

x05 =U1nminus2

U xj5 =

U j = 1 n minus 5 (235)

x06 =U1nminus1

U xj6 =

U j = 1 n minus 5

x07 =U1n

U xj7 =

Ujn minusU1n

U j = 1 n minus 5

6- and 7-point NMHV

u1uiλ2

prodi=2

(x0i + u1x1i)αi (236)

u2uiλ2

1 uiλ32 (u1x10 + u2x20 + u1u2x30 + u2

times7

prodi=1

24 n-point NkMHV

24 n-point NkMHV 29

N2MHV amplitude

F41F23

⟨1∣P26P72P35P63∣4⟩3

P 226P

⟨76⟩[23]⟨65⟩[2∣P26∣7⟩⟨6∣P72∣2][3∣P35∣6⟩⟨5∣P63∣3]

u3uα1

1 uα22 uα3

prodi=4

times17

prodj=14

2x8j + u23x9j)αj

NkMHV amplitude

nminus4

prodl=2

dululuαl

⎛⎝

1 minusnminus4

sumj=2

uj⎞⎠

P32k (prod

(P i1)αi)

⎛⎝prodj

(Pj2)αj

⎞⎠

prodj=0

Pj(u)αjdϕ (243)

dϕ =dPj1Pj1

and and dPjnPjn

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

x00 x0m

x10 x1m

⋮ ⋱ ⋮

xn0 xnm

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

gauge fixed form

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

1 0 0 1 1 1

⋮ ⋱ minus1 ⋮ ⋮ ⋮

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ways eq (324) of [38]

prodi=1

suml=1

prodj=1

(1 minusn

sumi=1

xijui)minusβj

sumi=1

αi) sdotn

prodi=1

Γ(αi) (248)

prodi=1

mminusnminus1

suml=1

prodj=1

sumi=1

xjiui)minusαj

sumi=1

prodi=1

Γ(βi) (249)

γ minusn

sumi=1

sumj=1

βj notin Z(250)

sumi=0

xijpartϕ

sumj=0

xijpartϕ

(3) part2ϕ

k du1duk (252)

Chapter 3

prodl=1

prodj=1

sumi=1

ki)( 1

(k1+k2)2+m2+ 1

(k1+k3)2+m2+ 1

(k1+k4)2+m2)

(hj hj) =1

A4(zj zj) =( z24

z14)h12 ( z14

z13)h34

zh1+h212 zh3+h4

z14)h12 ( z14

z13)h34

zh1+h212 zh3+h4

A4(z z) (35)

z = z12z34

z13z24 z = z12z34

compute

τminus2∆4A4(0 z11τ) (37)

prodi=1

⎛⎝

sumj=1

kj⎞⎠

m2 minus 4ω1ω2∣z∣2 (38)

δ(4)⎛⎝

sumj=1

kj⎞⎠= 4τ2

ω1δ(iz minus iz)

prodi=2

ωlowast2 = ω1

2 (310)

0dω1ω

∆1minus21

prodi=2

m2 minus 4z2

zminus1ω21

= g2 (im2)2αminus2

z minus 1)α

+ zα) (312)

sumi=1

hi minus 2 (313)

constructed in [51]

2(h2+h3)A13harr24

4 (z) 0 lt z lt 1 (315)

as well as

4 (1z)

z minus 1) 0 lt z lt 1 (316)

2 (im2)2αminus2

2 sin(πα) intC

hi(z z)

2= 1 minus h12 + n

2= 1 minus h34 + n

2= h12 minus n

2= h34 minus n (318)

with n = 0123

hi(z z) (319)

4hi+hj

Γ (hij + ∆2)Γ (∆

and Ψ∆hi(z z) is

Ψ∆hi(z z) equiv

2 minus h34)Γ (∆

2 + h12)Γ (1 minus ∆2 + h34)

Ψ∆hi(z z) (322)

infinsumJ=0

4π2Ψhh

+(h12 harr h34))

33 Soft limits 43

∆ minus J2

2= 1 minus h34 + n

∆ + J2

2= h34 minus n

33 Soft limits

Single soft limits

limωkrarr0

A`k=+1n = ⟨k minus 1k + 1⟩

limωkrarr0

δ(x) = 1

2limλrarr0

limλkrarr0

An(zj zj) =1

prodj=1jnek

intinfin

0dωj ω

iλjj int

limλkrarr0

AJk=+1n = 1

zkminus1k+1

limλkrarr0

AJk=minus1n = 1

zkminus1k+1

Double soft limits

33 Soft limits 45

⟨n∣1 + 2∣3] ( [13]3⟨n3⟩[12][23]s123

+ ⟨n2⟩3[n3]⟨n1⟩⟨12⟩sn12

we will write

An(zj zj) =n

prodj=1jnekl

intinfin

0dωj ω

iλjj int

0dr int

DS(kJk lJl) = 1

r=0]εrarr0

2(iλ1 + iλ2)ε21

zn1z23( 1

zn3z2n

z12z2n+ 1

z3nz31

z12z31) (338)

sumi=1

4ω1⋯ω6

ω21ω

ω23ω

26(ω4z46z34+ω5z56z35)3

(ω3ω4z34z34+ω3ω5z35z35+ω4ω5z45z45)

z12z16z34z45 (ω3z23z35 + ω4z24z45)

⎤⎥⎥⎥⎥⎥⎦

2(iλ3 + iλ4)ε21

z23z45( 1

z25z41

z34z42+ 1

z52z53

25z216z25z61

(z15z61

z25z26)iλ2minus1

(z12z16

z25z56)iλ5+1

(z15z12

z56z26)iλ6+1

⟨2∣3 + 4∣5] ( [35]3⟨25⟩[34][45]s345

+ ⟨24⟩3[25]⟨23⟩⟨34⟩s234

) (342)

intCd∆ Ψhh

hihi(z z)⟩ (343)

(zz)2G(z z)F (z z) (344)

given by

F+(z z) =1

zh34 zh342F1 (

1 minus h + h34 h + h34

1 + h12 + h341

z) 2F1 (

1 minus h + h34 h + h34

1 + h12 + h341

z) (346)

z) 2F1 (

⟨Ψhh

hihi(z z)⟩ = (2π)2

⟨A4(z z)Ψhh

hihi(z z) (349)

2F1(a b c z) =Γ(c)

obtain

⟨A4(z z)Ψhh

⎡⎢⎢⎢⎢⎢⎣

⎛⎜⎝

⎤⎥⎥⎥⎥⎥⎥⎦

sin(2πJ)2πJ

= 1 J = 0

0 J ne 0 (353)

hihi∣J=0

which simplifies to

2 minus h34)Γ(2 minus∆)Γ(∆) (354)

⟨A4(z z)Ψhh

Chapter 4

⟨1 2 6 7⟩⟨2 3 4 5⟩ minus ⟨1 2 4 5⟩⟨2 3 6 7⟩ (42)

cluster coordinate

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

1 0 0 0 y15 ⋯ y1

0 1 0 0 y25 ⋯ y2

0 0 1 0 y35 ⋯ y3

0 0 0 1 y45 ⋯ y4

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

yIa yJ b =1

f(y) g(y) =n

sumab=1

sumIJ=1

partyIa

2Z foralli j (46)

entries

n5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Total

1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1

2 0 1 2 5 4 1 0 0 0 0 0 0 0 0 0 0 13

3 0 0 1 6 54 177 298 274 134 30 3 0 0 0 0 0 977

4 0 0 0 1 13 263 1988 7862 18532 28204 28377 18925 8034 2047 270 17 114533

421 NMHV

p1 p5 = ⟨1 2 3 4⟩ ⟨2 3 4 5⟩ ⟨3 4 5 1⟩ ⟨4 5 1 2⟩ ⟨5 1 2 3⟩ (48)

5minusy35minusy4

5 (49)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0

0 0 12

0 minus12 0 1

0 minus12 minus1

2 0 12

0 minus12 minus1

2 minus12 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

422 N2MHV

1 = [12 (23) cap (456) (234) cap (56)6][23456]

2 = [12 (34) cap (567) (345) cap (67)7][34567]

3 = [123 (345) cap (67)7][34567]

4 = [123 (456) cap (78)8][45678]

5 = [12348][45678]

6 = [123 (45) cap (678)8][45678]

7 = [123 (45) cap (678) (456) cap (78)][45678] (411)

8 = [1234 (456) cap (78)][45678]

9 = [12349][56789]

10 = [1234 (567) cap (89)][56789]

11 = [1234 (56) cap (789)][56789]

13 = [12345][678910]

ϕ = ⟨4 5 (123) cap (789)⟩⟨4 6 (123) cap (789)⟩⟨1 2 3 4⟩⟨4 7 8 9⟩⟨5 6 (123) cap (789)⟩ (413)

Ω(2)1

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 1 1 0 0 0 12

minus 12

minus1

0 0 0 0 minus 12

0 minus 12

minus 12

minus1

0 minus 12

minus 12

minus1

0 minus 12

0 minus1 minus1

minus 12

minus1

0 0 0 0 minus 12

0 minus 12

0 0 0 0

minus 12

0 minus 12

0 0 0 minus 12

minus 12

0 0 12

0 minus 12

1 1 1 1 1 0 0 12

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 1 0 0 0 0 minus1 minus 12

minus 12

0 0 0 minus 12

minus 12

0 minus 12

minus 12

minus1 0 0 minus 32

minus 32

0 minus 12

minus 32

minus 12

0 minus 12

0 minus1 minus 12

minus 12

0 minus1 minus 12

minus 12

0 0 0 minus 12

minus 12

0 minus 12

minus 12

0 0 12

0 minus 12

1 1 12

0 0 0 0 minus 12

0 0 0 12

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)3

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 12

0 0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

minus 32

minus 12

1 0 minus 12

0 minus1 0 minus 12

0 0 0 minus 12

minus 12

0 minus 12

minus 12

0 0 12

0 minus 12

1 1 12

0 0 12

0 0 0 0 0 0 minus 12

0 0 0 12

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)4

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 minus1 minus1 0

0 0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus1 minus1 0

0 minus 12

0 minus1 minus1 0

0 0 minus 12

minus 12

0 minus 12

minus 12

0 0 0 12

0 minus 12

minus 12

1 1 1 12

0 minus 12

1 1 1 12

0 0 0 0 0 0 0 0 0 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)5

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 0 0 0

0 0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 0 0 0

0 minus 12

0 0 0 0

0 0 minus 12

minus 12

0 minus 12

minus 12

0 0 0 12

0 minus 12

minus 12

0 0 0 12

0 minus 12

0 0 0 12

0 0 0 0 0 0 0 0 0 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)6

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 0 minus1 0

0 0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 0 minus1 0

0 minus 12

0 0 minus1 0

0 0 minus 12

minus 12

0 minus 12

minus 12

0 0 0 12

0 minus 12

minus 12

0 0 0 12

0 minus 12

1 1 1 12

0 0 0 0 0 0 0 0 0 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)7

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 0 minus1 0

0 1 0 minus 12

minus 12

minus 32

0 1 12

0 minus 12

minus 12

minus 32

0 1 12

minus 12

minus 32

0 0 minus 12

minus 12

0 minus 12

minus 12

0 0 minus 12

minus 12

0 minus 12

minus 12

0 1 12

0 minus 12

1 1 32

0 0 0 0 0 0 0 0 0 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)8

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 0 0 0

0 1 0 minus 12

minus 12

0 1 12

0 minus 12

minus 12

0 1 12

minus 12

0 0 minus 12

minus 12

0 minus 12

minus 12

0 0 minus 12

minus 12

0 minus 12

minus 12

0 1 12

0 minus 12

0 1 12

0 0 0 0 0 0 0 0 0 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)9

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 0 0 0

0 0 minus 12

minus 12

0 0 0 0

0 minus 12

minus 12

0 0 0 0

0 minus 12

0 0 0 0

0 minus 12

minus 12

0 minus 12

minus 12

0 0 0 0 0 12

0 minus 12

minus 12

0 0 0 0 0 12

0 minus 12

minus 12

0 0 0 0 0 12

0 minus 12

0 0 0 0 0 12

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)10

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 0 0 0

0 0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)11

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 0 0 0

0 0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)12

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 1 1 1 32

0 minus1 0 minus 12

minus 12

minus 32

minus 12

0 minus1 12

0 minus 12

minus 12

0 minus 12

0 minus1 12

0 minus 12

minus 12

0 minus 32

0 minus 12

minus 12

2 2 2 12

0 minus 32

0 minus 12

2 2 2 12

0 minus 32

0 2 2 2 12

0 minus 32

minus 12

0 minus 32

minus 12

0 minus 12

minus 12

0 minus 32

minus 12

0 minus 12

minus 12

0 minus1 12

0 minus 12

minus 12

0 minus 12

0 minus1 12

0 minus 12

minus 12

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)13

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 0 0 0

0 0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

B1 =⟨12 (23) cap (456) (234) cap (56)⟩ ⟨612 (23) cap (456)⟩ ⟨(234) cap (56)612⟩

⟨(23) cap (456) (234) cap (56)61⟩ ⟨2 (23) cap (456) (234) cap (56)6⟩ ⟨2345⟩ ⟨6234⟩ ⟨5623⟩

⟨4562⟩ ⟨3456⟩

(345) cap (67)71⟩ ⟨2 (34) cap (567) (345) cap (67)7⟩ ⟨3456⟩ ⟨7345⟩ ⟨6734⟩ ⟨5673⟩

⟨4567⟩

B3 =⟨123 (345) cap (67)⟩ ⟨7123⟩ ⟨(345) cap (67)712⟩ ⟨3 (345) cap (67)71⟩ ⟨23 (345) cap (67)7⟩

⟨3456⟩ ⟨7345⟩ ⟨6734⟩ ⟨5673⟩ ⟨4567⟩

B4 =⟨123 (456) cap (78)⟩ ⟨8123⟩ ⟨(456) cap (78)812⟩ ⟨3 (456) cap (78)81⟩ ⟨23 (456) cap (78)8⟩

⟨4567⟩ ⟨8456⟩ ⟨7845⟩ ⟨6784⟩⟨5678⟩

B5 =⟨1234⟩ ⟨8123⟩ ⟨4812⟩ ⟨3481⟩ ⟨2348⟩ ⟨4567⟩ ⟨8456⟩ ⟨7845⟩ ⟨6784⟩

⟨5678⟩

B6 =⟨123 (45) cap (678)⟩ ⟨8123⟩ ⟨(45) cap (678)812⟩ ⟨3 (45) cap (678)81⟩ ⟨23 (45) cap (678)8⟩

⟨4567⟩ ⟨8456⟩ ⟨7845⟩ ⟨6784⟩⟨5678⟩

B7 =⟨123 (45) cap (678)⟩ ⟨(456) cap (78)123⟩ ⟨(45) cap (678) (456) cap (78)12⟩

⟨3 (45) cap (678) (456) cap (78)1⟩ ⟨23 (45) cap (678) (456) cap (78)⟩ ⟨4567⟩ ⟨8456⟩ ⟨7845⟩

⟨6784⟩⟨5678⟩

B8 =⟨1234⟩ ⟨(456) cap (78)123⟩ ⟨4 (456) cap (78)12⟩ ⟨34 (456) cap (78)1⟩ ⟨234 (456) cap (78)⟩

⟨4567⟩ ⟨8456⟩ ⟨7845⟩ ⟨6784⟩⟨5678⟩

B9 =⟨1234⟩ ⟨9123⟩ ⟨4912⟩ ⟨3491⟩ ⟨2349⟩ ⟨5678⟩ ⟨9567⟩ ⟨8956⟩

⟨7895⟩ ⟨6789⟩

B10 =⟨1234⟩ ⟨(567) cap (89)123⟩ ⟨4 (567) cap (89)12⟩ ⟨34 (567) cap (89)1⟩ ⟨234 (567) cap (89)⟩

⟨5678⟩ ⟨9567⟩ ⟨8956⟩ ⟨7895⟩ ⟨6789⟩

B11 =⟨1234⟩ ⟨(56) cap (789)123⟩ ⟨4 (56) cap (789)12⟩ ⟨34 (56) cap (789)1⟩ ⟨234 (56) cap (789)⟩

⟨5678⟩ ⟨9567⟩ ⟨8956⟩ ⟨7895⟩ ⟨6789⟩

B12 =⟨1234⟩ ⟨4789⟩ ⟨56 (123) cap (789)⟩ ⟨123 (45) cap (789)⟩ ⟨(46) cap (789)123⟩

⟨789(45) cap (123)⟩ ⟨(46) cap (123)789⟩

B13 =⟨1234⟩ ⟨5123⟩ ⟨4512⟩ ⟨3451⟩ ⟨2345⟩ ⟨6789⟩ ⟨10678⟩ ⟨91067⟩

⟨89106⟩ ⟨78910⟩

Chapter 5

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

Z11 ⋯ Z1

Z21 ⋯ Z2

Z31 ⋯ Z3

Z41 ⋯ Z4

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ETHrarrGL(4)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

1 0 0 0 y15 ⋯ y1

0 1 0 0 y25 ⋯ y2

0 0 1 0 y35 ⋯ y3

0 0 0 1 y45 ⋯ y4

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

56 n with

ld (52)

x2ij =

is given by

yIa yJ b =1

f(y) g(y) =n

sumab=5

sumIJ=1

partyIa

2Z (56)

⟨i i + 1 j j + 1⟩)

in [89]

ABDSn = exp [

infinsumL=1

g2L (f(L)(ε)

2A(1)n (Lε) +C(L))] (57)

Γcusp = 4g2 +O(g4) (58)

ABDS-liken = ABDS

n exp [Γcusp

4Yn] 4 ∣ n

Discx2i+1j

[Discx2i+1i+p

(An)] = 0

Discx2i+1i+p

[Discx2i+1j+p+q

(An)] = 0

⎫⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎭

522 NMHV Amplitudes

form [115 116]

sums=5

sumt=s+2

Rtot =nminus2

sums=3

sumt=s+2

R1st (512)

and our results

Yif (2L)i (515)

minus 1

2([12345] + [12356] + [12367] minus [12457] minus [12567]

+ [13456] + [13467] + [14567] minus [23457] minus [23567])

times (log(⟨1234⟩⟨1789⟩⟨1278⟩⟨1349⟩)otimes ⟨3478⟩)

minus 1

2([12348] minus [12378] + [12478] minus [13478]

+ [23478] + [34567]) (log(⟨1234⟩⟨1789⟩⟨1278⟩⟨1349⟩)otimes ⟨3478⟩)

[12347] [12348] [12349] [12378] [12379] [12389]

[12478] [12479] [12489] [12789] [13478] [13479]

[13489] [13789] [14789] [23478] [23479] [23489]

[23789] [24789] [34567] [34568] [34578] [34678]

[34789] [35678] [45678]

Yangian invariants

a3ijkl = ⟨i(j j + 1)(k k + 1)(l l + 1)⟩

)(525)

condition

cup (j j + 1 k k + 15

) cup (i j j + 1 k k + 15

cup (i k k + 1 l l + 15

) cup (l l + 1 i5

) is a

uijkl =x2ikx

sumi=5

minus 1

sumi=5

minus 1

u2nminus3nminus1n)

sumi=5

minus 1

1247) +1

sumi=4

4ln(u8145) ln(u1256u3478

u2367) + cyclic (532)

nminus2

sumi=4

ln(un1ii+1)iminus1

sumj=1

sumi=4

ln(un1ii+1)iminus1

sumj=1

4ln(un1n

2n2+1)

nminus22

sumi=1

ln(uii+1i+n2i+n

Chapter 6

Graphs

symbol of [23]

C sdotZ = 0 (61)

example [27 93 94 28 30])

for each graph

source columns

C =⎛⎜⎜⎜⎝

1 0 c13 c14 c15 c16

0 1 c23 c24 c25 c26

⎞⎟⎟⎟⎠ (63)

prodαisinp

fα (64)

f0 = minus⟨1234⟩⟨2346⟩ f1 = minus

⟨2346⟩⟨2345⟩ f2 =

⟨2345⟩⟨1236⟩⟨1234⟩⟨2356⟩

f3 = minus⟨2356⟩⟨2346⟩ f4 =

⟨2346⟩⟨1256⟩⟨2456⟩⟨1236⟩ f5 = minus

⟨2456⟩⟨2356⟩

f6 =⟨2356⟩⟨1456⟩⟨3456⟩⟨1256⟩ f7 = minus

⟨3456⟩⟨2456⟩ f8 = minus

⟨2456⟩⟨1456⟩

f(b)0 = f (a)0 (1 + f (a)2 )

f(b)1 = f

1 + 1f (a)2

f(b)2 = 1

f(b)3 = f (a)3 (1 + f (a)2 )

f(b)4 = f

1 + 1f (a)2

f(c)2 = f (a)2 (1 + f (a)4 )

f(c)3 = f

1 + 1f (a)4

f(c)4 = 1

f(c)5 = f (a)5 (1 + f (a)4 )

f(c)6 = f

1 + 1f (a)4

(a) (b) (c)

⟨16⟩ ⟨12⟩

⟨23⟩

⟨34⟩⟨45⟩

⟨56⟩

⟨26⟩

⟨36⟩

⟨46⟩

ampamppp

⟨16⟩ ⟨12⟩

⟨23⟩

⟨34⟩⟨45⟩

⟨56⟩

⟨26⟩

⟨36⟩

⟨35⟩

⟨16⟩ ⟨12⟩

⟨23⟩

⟨34⟩⟨45⟩

⟨56⟩

⟨26⟩

⟨24⟩⟨46⟩ oo

(d) (e) (f)

⟨3456⟩ ⟨1456⟩

⟨1256⟩

⟨1236⟩⟨1234⟩

⟨2345⟩

⟨2456⟩

⟨2356⟩

⟨2346⟩

ampamppp

⟨3456⟩ ⟨1456⟩

⟨1256⟩

⟨1236⟩⟨1234⟩

⟨2345⟩

⟨2456⟩

⟨2356⟩

⟨1235⟩

⟨3456⟩ ⟨1456⟩

⟨1256⟩

⟨1236⟩⟨1234⟩

⟨2345⟩

⟨2456⟩

⟨1246⟩⟨2346⟩ oo

(g) (h) (i)

C = (0 1 f0f1f2f3 f0f1f2 f0f1 f0) (69)

f0 =⟨2345⟩⟨3456⟩ f1 = minus

⟨2346⟩⟨2345⟩ f2 = minus

⟨2356⟩⟨2346⟩ f3 = minus

⟨2456⟩⟨2356⟩ f4 = minus

⟨3456⟩⟨2456⟩

amplitude

f0 =⟨123⟩⟨145⟩

⟨234⟩⟨146⟩ f3 = minus⟨234⟩⟨156⟩⟨123⟩⟨456⟩

f4 = minus⟨124⟩⟨456⟩

⟨134⟩⟨156⟩

f6 = minus⟨134⟩⟨124⟩

f0 =⟨145⟩⟨456⟩ f1 = minus

⟨146⟩⟨145⟩ f2 = minus

⟨156⟩⟨146⟩ f3 = minus

⟨123⟩⟨456⟩⟨234⟩⟨156⟩

f4 = minus⟨124⟩⟨123⟩ f5 = minus

⟨234⟩⟨134⟩ f6 = minus

⟨134⟩⟨124⟩

⟨234⟩⟨156⟩ (616)

from (615)

C =⎛⎜⎜⎜⎝

1 c12 0 c14 c15 c16 c17 c18

0 c32 1 c34 c35 c36 c37 c38

⎞⎟⎟⎟⎠

a4a7 ⟨6(12)(34)(78)⟩ ⟨3478⟩

a2 ⟨4(12)(56)(78)⟩ ⟨1278⟩

a1⟨3(12)(56)(78)⟩

a3 ⟨1(34)(56)(78)⟩

a6 ⟨5(12)(34)(78)⟩

pendix 67

a9⟨1234⟩⟨5678⟩ (625)

∆2468

65 Discussion 101

65 Discussion

on Gr(mn)

65 Discussion 103

in [23]

f(b)0 = minus⟨1235⟩

⟨2356⟩ f(b)1 = minus⟨1236⟩

⟨1235⟩ f(b)2 = ⟨1234⟩⟨2356⟩

⟨2345⟩⟨1236⟩

f(b)3 = minus⟨1235⟩

⟨1234⟩ f(b)4 = ⟨2345⟩⟨1256⟩

⟨1235⟩⟨2456⟩ f(b)5 = minus⟨2456⟩

⟨2356⟩

f(b)6 = ⟨2356⟩⟨1456⟩

⟨3456⟩⟨1256⟩ f(b)7 = minus⟨3456⟩

⟨2456⟩ f(b)8 = minus⟨2456⟩

⟨1456⟩

f(c)0 = minus⟨1234⟩

⟨2346⟩ f(c)1 = minus⟨2346⟩

⟨2345⟩ f(c)2 = ⟨2345⟩⟨1246⟩

⟨1234⟩⟨2456⟩

f(c)3 = minus⟨1256⟩

⟨1246⟩ f(c)4 = ⟨2456⟩⟨1236⟩

⟨2346⟩⟨1256⟩ f(c)5 = minus⟨1246⟩

⟨1236⟩

f(c)6 = ⟨1456⟩⟨2346⟩

⟨3456⟩⟨1246⟩ f(c)7 = minus⟨3456⟩

⟨2456⟩ f(c)8 = minus⟨2456⟩

⟨1456⟩

∆1357 andradic

∆2468 with

∆1357 and

2(1 + u minus v +

u = ⟨1278⟩⟨3456⟩⟨1256⟩⟨3478⟩ v = ⟨1234⟩⟨5678⟩

⟨1256⟩⟨3478⟩ (632)

radic∆1357

⟨1256⟩⟨3478⟩ (633)

∣irarri+6

∣irarri+4

∣irarri+6

∣irarri+4

∣irarri+6

z a9 =

1 minus z1 minus z

Z isin Gr(48)gt0

Bibliography

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th9711200 [hep-th]]

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Abstract

Acknowledgements

Introduction












n-point MHV




6-point MHV

n-point NMHV

n-point NkMHV





Soft limits






NMHV

NNMHV

NNNMHV and Higher




One-loop Amplitudes


NMHV Amplitudes











Discussion



Curriculum Vitae

Research

Education

Teaching

Schools and Talks

Jun 2019 TASI 2019

Additional Skills

Acknowledgements

Contents

Abstract v

Acknowledgements xi

1 Introduction 1

22 n-point MHV 19

224 6-point MHV 24

23 n-point NMHV 25

24 n-point NkMHV 28

33 Soft limits 43

421 NMHV 60

422 N2MHV 62

65 Discussion 101

List of Figures

to ⟨k(l l + 1)(mm + 1)(k minus 1k + 1)⟩ 80

to ⟨k(l l + 1)(mm + 1)(pp + 1)⟩ 81

List of Tables

I love you all

Chapter 1

Introduction

amplitudes

calculated

(11) transforms as

az + bcz + d

) = ∣cz + d∣2∆φ∆plusmn(X z z) (13)

prodj=1int

0dωj ω

az + bcz + d

prodj=1

gluon amplitudes

microi+1 (16)

〈2345〉

〈2346〉〈2356〉〈2456〉〈3456〉

〈1234〉〈1236〉〈1256〉〈1456〉

akaprimek = prod

i∣bikgt0

aminusbiki (112)

Y (Z ∣η) = int4k

prodi=1

d log fi4

prodI=1

prodα=1

suma=1

Cαa(Z ∣η)aI) (113)

prodαisinp

fα (114)

fi = 1 (115)

coordinates

Chapter 2

Celestial Sphere

radicωiωjzij (22)

prodj=1int

0dωj ω

azj + bczj + d

prodj=1

22 n-point MHV 19

22 n-point MHV

ratio ωi

dωiωi

(ωisi

j=1 iλj

s1+iλii

⎛⎜⎝

proda=1anei

intinfin

0dωa ω

intinfin

dωiωi

(ωisi

= 2πδ(z) (29)

sumj=1

λj⎞⎠s1+iλii

⎛⎜⎝

proda=1anei

intinfin

0dωa ω

⎞⎟⎠A`1⋯`n(si ωl zj zj) (210)

δ4(ε1s1q1 +n

sumi=2

εiωiqi) =4

prodj=nminus3

U+nminus4

sumi=2

U) (212)

22 n-point MHV 21

ui =z31zi2z32zi1

for i = 45 n (216)

s1 =∣z32∣

∣z31∣ ∣z12∣ s2 =

∣z31∣∣z32∣ ∣z21∣

and si =∣z12∣

∣z1i∣ ∣zi2∣for i = 3 n (217)

to demand Uij

prodij 1lt0(Uij

12 siλ1+21

nminus4

proda=2int

0dωa ω

1lt0(Uij

ωi = siU1n

uiminus1

i = 23 n minus 4 (220)

which leads to

12siλ1+21 siλ2+2

2 siλ33 siλnn

z23z34zn1U1nδ(

sumj=1

λj) ϕ(α x)prodij

1lt0(Uij

U) (221)

22 n-point MHV 23

prodj=1

P2and and dPnminus4

Pnminus4

αn =1 + iλ1 x0 i =U1i

U1n xjminus1 i =

Ujnminus U1i

U1n x0n = minus

U1n xjminus1n =

ϕ(α x) equivc2

c1intu1ge0u3ge0

1minussuma uage0

prodj=1

Pnminus1

U1n xij =

Ujnminus4+i

x0n = minusU

U1n xin =

U1n x01 = 1

mannian

224 6-point MHV

)iλ1+1

( minusUU16

)iλ2+2

iλ3minus1

iλ4minus1

iλ5minus1

times int1

prodi=1

23 n-point NMHV 25

U16U2i+2for i = 123 (227)

23 n-point NMHV

nminus1

sumj=4

⟨1∣P2jPj+12∣3⟩3

P 22jP

⟨j + 1 j⟩[2∣P2j ∣j + 1⟩⟨j∣Pj+12∣2]

equivnminus1

sumj=4

Mj (228)

⟨34⟩⟨45⟩⋯⟨n minus 1 n⟩⟨n1⟩(⟨12⟩[24]⟨43⟩ + ⟨13⟩[34]⟨43⟩)3

(⟨23⟩[23] + ⟨24⟩[24] + ⟨34⟩[34])⟨34⟩[34]times

times ⟨54⟩([23]⟨35⟩ + [24]⟨45⟩)(⟨43⟩[32]) (229)

M4 =ω1ω4(ε2z12z24ω2+ε3z13z34ω3)3

of variables

ωi = siuiminus1

i = 2 n minus 4 (231)

M4 sim δ⎛⎝n

sumj=1

λj⎞⎠

prodni=1 siλii

z12z13z45z4ns21s

z34zn1UF(αx)prod

1lt0(Uij

U) (232)

nminus5

proda=1

nminus5

prodj=1

uiλj+1

times7

prodi=1

23 n-point NMHV 27

U xj4 =

U j = 1 n minus 5

x05 =U1nminus2

U xj5 =

U j = 1 n minus 5 (235)

x06 =U1nminus1

U xj6 =

U j = 1 n minus 5

x07 =U1n

U xj7 =

Ujn minusU1n

U j = 1 n minus 5

6- and 7-point NMHV

u1uiλ2

prodi=2

(x0i + u1x1i)αi (236)

u2uiλ2

1 uiλ32 (u1x10 + u2x20 + u1u2x30 + u2

times7

prodi=1

24 n-point NkMHV

24 n-point NkMHV 29

N2MHV amplitude

F41F23

⟨1∣P26P72P35P63∣4⟩3

P 226P

⟨76⟩[23]⟨65⟩[2∣P26∣7⟩⟨6∣P72∣2][3∣P35∣6⟩⟨5∣P63∣3]

u3uα1

1 uα22 uα3

prodi=4

times17

prodj=14

2x8j + u23x9j)αj

NkMHV amplitude

nminus4

prodl=2

dululuαl

⎛⎝

1 minusnminus4

sumj=2

uj⎞⎠

P32k (prod

(P i1)αi)

⎛⎝prodj

(Pj2)αj

⎞⎠

prodj=0

Pj(u)αjdϕ (243)

dϕ =dPj1Pj1

and and dPjnPjn

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

x00 x0m

x10 x1m

⋮ ⋱ ⋮

xn0 xnm

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

gauge fixed form

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

1 0 0 1 1 1

⋮ ⋱ minus1 ⋮ ⋮ ⋮

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ways eq (324) of [38]

prodi=1

suml=1

prodj=1

(1 minusn

sumi=1

xijui)minusβj

sumi=1

αi) sdotn

prodi=1

Γ(αi) (248)

prodi=1

mminusnminus1

suml=1

prodj=1

sumi=1

xjiui)minusαj

sumi=1

prodi=1

Γ(βi) (249)

γ minusn

sumi=1

sumj=1

βj notin Z(250)

sumi=0

xijpartϕ

sumj=0

xijpartϕ

(3) part2ϕ

k du1duk (252)

Chapter 3

prodl=1

prodj=1

sumi=1

ki)( 1

(k1+k2)2+m2+ 1

(k1+k3)2+m2+ 1

(k1+k4)2+m2)

(hj hj) =1

A4(zj zj) =( z24

z14)h12 ( z14

z13)h34

zh1+h212 zh3+h4

z14)h12 ( z14

z13)h34

zh1+h212 zh3+h4

A4(z z) (35)

z = z12z34

z13z24 z = z12z34

compute

τminus2∆4A4(0 z11τ) (37)

prodi=1

⎛⎝

sumj=1

kj⎞⎠

m2 minus 4ω1ω2∣z∣2 (38)

δ(4)⎛⎝

sumj=1

kj⎞⎠= 4τ2

ω1δ(iz minus iz)

prodi=2

ωlowast2 = ω1

2 (310)

0dω1ω

∆1minus21

prodi=2

m2 minus 4z2

zminus1ω21

= g2 (im2)2αminus2

z minus 1)α

+ zα) (312)

sumi=1

hi minus 2 (313)

constructed in [51]

2(h2+h3)A13harr24

4 (z) 0 lt z lt 1 (315)

as well as

4 (1z)

z minus 1) 0 lt z lt 1 (316)

2 (im2)2αminus2

2 sin(πα) intC

hi(z z)

2= 1 minus h12 + n

2= 1 minus h34 + n

2= h12 minus n

2= h34 minus n (318)

with n = 0123

hi(z z) (319)

4hi+hj

Γ (hij + ∆2)Γ (∆

and Ψ∆hi(z z) is

Ψ∆hi(z z) equiv

2 minus h34)Γ (∆

2 + h12)Γ (1 minus ∆2 + h34)

Ψ∆hi(z z) (322)

infinsumJ=0

4π2Ψhh

+(h12 harr h34))

33 Soft limits 43

∆ minus J2

2= 1 minus h34 + n

∆ + J2

2= h34 minus n

33 Soft limits

Single soft limits

limωkrarr0

A`k=+1n = ⟨k minus 1k + 1⟩

limωkrarr0

δ(x) = 1

2limλrarr0

limλkrarr0

An(zj zj) =1

prodj=1jnek

intinfin

0dωj ω

iλjj int

limλkrarr0

AJk=+1n = 1

zkminus1k+1

limλkrarr0

AJk=minus1n = 1

zkminus1k+1

Double soft limits

33 Soft limits 45

⟨n∣1 + 2∣3] ( [13]3⟨n3⟩[12][23]s123

+ ⟨n2⟩3[n3]⟨n1⟩⟨12⟩sn12

we will write

An(zj zj) =n

prodj=1jnekl

intinfin

0dωj ω

iλjj int

0dr int

DS(kJk lJl) = 1

r=0]εrarr0

2(iλ1 + iλ2)ε21

zn1z23( 1

zn3z2n

z12z2n+ 1

z3nz31

z12z31) (338)

sumi=1

4ω1⋯ω6

ω21ω

ω23ω

26(ω4z46z34+ω5z56z35)3

(ω3ω4z34z34+ω3ω5z35z35+ω4ω5z45z45)

z12z16z34z45 (ω3z23z35 + ω4z24z45)

⎤⎥⎥⎥⎥⎥⎦

2(iλ3 + iλ4)ε21

z23z45( 1

z25z41

z34z42+ 1

z52z53

25z216z25z61

(z15z61

z25z26)iλ2minus1

(z12z16

z25z56)iλ5+1

(z15z12

z56z26)iλ6+1

⟨2∣3 + 4∣5] ( [35]3⟨25⟩[34][45]s345

+ ⟨24⟩3[25]⟨23⟩⟨34⟩s234

) (342)

intCd∆ Ψhh

hihi(z z)⟩ (343)

(zz)2G(z z)F (z z) (344)

given by

F+(z z) =1

zh34 zh342F1 (

1 minus h + h34 h + h34

1 + h12 + h341

z) 2F1 (

1 minus h + h34 h + h34

1 + h12 + h341

z) (346)

z) 2F1 (

⟨Ψhh

hihi(z z)⟩ = (2π)2

⟨A4(z z)Ψhh

hihi(z z) (349)

2F1(a b c z) =Γ(c)

obtain

⟨A4(z z)Ψhh

⎡⎢⎢⎢⎢⎢⎣

⎛⎜⎝

⎤⎥⎥⎥⎥⎥⎥⎦

sin(2πJ)2πJ

= 1 J = 0

0 J ne 0 (353)

hihi∣J=0

which simplifies to

2 minus h34)Γ(2 minus∆)Γ(∆) (354)

⟨A4(z z)Ψhh

Chapter 4

⟨1 2 6 7⟩⟨2 3 4 5⟩ minus ⟨1 2 4 5⟩⟨2 3 6 7⟩ (42)

cluster coordinate

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

1 0 0 0 y15 ⋯ y1

0 1 0 0 y25 ⋯ y2

0 0 1 0 y35 ⋯ y3

0 0 0 1 y45 ⋯ y4

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

yIa yJ b =1

f(y) g(y) =n

sumab=1

sumIJ=1

partyIa

2Z foralli j (46)

entries

n5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Total

1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1

2 0 1 2 5 4 1 0 0 0 0 0 0 0 0 0 0 13

3 0 0 1 6 54 177 298 274 134 30 3 0 0 0 0 0 977

4 0 0 0 1 13 263 1988 7862 18532 28204 28377 18925 8034 2047 270 17 114533

421 NMHV

p1 p5 = ⟨1 2 3 4⟩ ⟨2 3 4 5⟩ ⟨3 4 5 1⟩ ⟨4 5 1 2⟩ ⟨5 1 2 3⟩ (48)

5minusy35minusy4

5 (49)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0

0 0 12

0 minus12 0 1

0 minus12 minus1

2 0 12

0 minus12 minus1

2 minus12 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

422 N2MHV

1 = [12 (23) cap (456) (234) cap (56)6][23456]

2 = [12 (34) cap (567) (345) cap (67)7][34567]

3 = [123 (345) cap (67)7][34567]

4 = [123 (456) cap (78)8][45678]

5 = [12348][45678]

6 = [123 (45) cap (678)8][45678]

7 = [123 (45) cap (678) (456) cap (78)][45678] (411)

8 = [1234 (456) cap (78)][45678]

9 = [12349][56789]

10 = [1234 (567) cap (89)][56789]

11 = [1234 (56) cap (789)][56789]

13 = [12345][678910]

ϕ = ⟨4 5 (123) cap (789)⟩⟨4 6 (123) cap (789)⟩⟨1 2 3 4⟩⟨4 7 8 9⟩⟨5 6 (123) cap (789)⟩ (413)

Ω(2)1

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 1 1 0 0 0 12

minus 12

minus1

0 0 0 0 minus 12

0 minus 12

minus 12

minus1

0 minus 12

minus 12

minus1

0 minus 12

0 minus1 minus1

minus 12

minus1

0 0 0 0 minus 12

0 minus 12

0 0 0 0

minus 12

0 minus 12

0 0 0 minus 12

minus 12

0 0 12

0 minus 12

1 1 1 1 1 0 0 12

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 1 0 0 0 0 minus1 minus 12

minus 12

0 0 0 minus 12

minus 12

0 minus 12

minus 12

minus1 0 0 minus 32

minus 32

0 minus 12

minus 32

minus 12

0 minus 12

0 minus1 minus 12

minus 12

0 minus1 minus 12

minus 12

0 0 0 minus 12

minus 12

0 minus 12

minus 12

0 0 12

0 minus 12

1 1 12

0 0 0 0 minus 12

0 0 0 12

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)3

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 12

0 0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

minus 32

minus 12

1 0 minus 12

0 minus1 0 minus 12

0 0 0 minus 12

minus 12

0 minus 12

minus 12

0 0 12

0 minus 12

1 1 12

0 0 12

0 0 0 0 0 0 minus 12

0 0 0 12

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)4

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 minus1 minus1 0

0 0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus1 minus1 0

0 minus 12

0 minus1 minus1 0

0 0 minus 12

minus 12

0 minus 12

minus 12

0 0 0 12

0 minus 12

minus 12

1 1 1 12

0 minus 12

1 1 1 12

0 0 0 0 0 0 0 0 0 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)5

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 0 0 0

0 0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 0 0 0

0 minus 12

0 0 0 0

0 0 minus 12

minus 12

0 minus 12

minus 12

0 0 0 12

0 minus 12

minus 12

0 0 0 12

0 minus 12

0 0 0 12

0 0 0 0 0 0 0 0 0 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)6

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 0 minus1 0

0 0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 0 minus1 0

0 minus 12

0 0 minus1 0

0 0 minus 12

minus 12

0 minus 12

minus 12

0 0 0 12

0 minus 12

minus 12

0 0 0 12

0 minus 12

1 1 1 12

0 0 0 0 0 0 0 0 0 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)7

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 0 minus1 0

0 1 0 minus 12

minus 12

minus 32

0 1 12

0 minus 12

minus 12

minus 32

0 1 12

minus 12

minus 32

0 0 minus 12

minus 12

0 minus 12

minus 12

0 0 minus 12

minus 12

0 minus 12

minus 12

0 1 12

0 minus 12

1 1 32

0 0 0 0 0 0 0 0 0 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)8

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 0 0 0

0 1 0 minus 12

minus 12

0 1 12

0 minus 12

minus 12

0 1 12

minus 12

0 0 minus 12

minus 12

0 minus 12

minus 12

0 0 minus 12

minus 12

0 minus 12

minus 12

0 1 12

0 minus 12

0 1 12

0 0 0 0 0 0 0 0 0 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)9

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 0 0 0

0 0 minus 12

minus 12

0 0 0 0

0 minus 12

minus 12

0 0 0 0

0 minus 12

0 0 0 0

0 minus 12

minus 12

0 minus 12

minus 12

0 0 0 0 0 12

0 minus 12

minus 12

0 0 0 0 0 12

0 minus 12

minus 12

0 0 0 0 0 12

0 minus 12

0 0 0 0 0 12

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)10

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 0 0 0

0 0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)11

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 0 0 0

0 0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)12

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 1 1 1 32

0 minus1 0 minus 12

minus 12

minus 32

minus 12

0 minus1 12

0 minus 12

minus 12

0 minus 12

0 minus1 12

0 minus 12

minus 12

0 minus 32

0 minus 12

minus 12

2 2 2 12

0 minus 32

0 minus 12

2 2 2 12

0 minus 32

0 2 2 2 12

0 minus 32

minus 12

0 minus 32

minus 12

0 minus 12

minus 12

0 minus 32

minus 12

0 minus 12

minus 12

0 minus1 12

0 minus 12

minus 12

0 minus 12

0 minus1 12

0 minus 12

minus 12

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)13

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 0 0 0

0 0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

B1 =⟨12 (23) cap (456) (234) cap (56)⟩ ⟨612 (23) cap (456)⟩ ⟨(234) cap (56)612⟩

⟨(23) cap (456) (234) cap (56)61⟩ ⟨2 (23) cap (456) (234) cap (56)6⟩ ⟨2345⟩ ⟨6234⟩ ⟨5623⟩

⟨4562⟩ ⟨3456⟩

(345) cap (67)71⟩ ⟨2 (34) cap (567) (345) cap (67)7⟩ ⟨3456⟩ ⟨7345⟩ ⟨6734⟩ ⟨5673⟩

⟨4567⟩

B3 =⟨123 (345) cap (67)⟩ ⟨7123⟩ ⟨(345) cap (67)712⟩ ⟨3 (345) cap (67)71⟩ ⟨23 (345) cap (67)7⟩

⟨3456⟩ ⟨7345⟩ ⟨6734⟩ ⟨5673⟩ ⟨4567⟩

B4 =⟨123 (456) cap (78)⟩ ⟨8123⟩ ⟨(456) cap (78)812⟩ ⟨3 (456) cap (78)81⟩ ⟨23 (456) cap (78)8⟩

⟨4567⟩ ⟨8456⟩ ⟨7845⟩ ⟨6784⟩⟨5678⟩

B5 =⟨1234⟩ ⟨8123⟩ ⟨4812⟩ ⟨3481⟩ ⟨2348⟩ ⟨4567⟩ ⟨8456⟩ ⟨7845⟩ ⟨6784⟩

⟨5678⟩

B6 =⟨123 (45) cap (678)⟩ ⟨8123⟩ ⟨(45) cap (678)812⟩ ⟨3 (45) cap (678)81⟩ ⟨23 (45) cap (678)8⟩

⟨4567⟩ ⟨8456⟩ ⟨7845⟩ ⟨6784⟩⟨5678⟩

B7 =⟨123 (45) cap (678)⟩ ⟨(456) cap (78)123⟩ ⟨(45) cap (678) (456) cap (78)12⟩

⟨3 (45) cap (678) (456) cap (78)1⟩ ⟨23 (45) cap (678) (456) cap (78)⟩ ⟨4567⟩ ⟨8456⟩ ⟨7845⟩

⟨6784⟩⟨5678⟩

B8 =⟨1234⟩ ⟨(456) cap (78)123⟩ ⟨4 (456) cap (78)12⟩ ⟨34 (456) cap (78)1⟩ ⟨234 (456) cap (78)⟩

⟨4567⟩ ⟨8456⟩ ⟨7845⟩ ⟨6784⟩⟨5678⟩

B9 =⟨1234⟩ ⟨9123⟩ ⟨4912⟩ ⟨3491⟩ ⟨2349⟩ ⟨5678⟩ ⟨9567⟩ ⟨8956⟩

⟨7895⟩ ⟨6789⟩

B10 =⟨1234⟩ ⟨(567) cap (89)123⟩ ⟨4 (567) cap (89)12⟩ ⟨34 (567) cap (89)1⟩ ⟨234 (567) cap (89)⟩

⟨5678⟩ ⟨9567⟩ ⟨8956⟩ ⟨7895⟩ ⟨6789⟩

B11 =⟨1234⟩ ⟨(56) cap (789)123⟩ ⟨4 (56) cap (789)12⟩ ⟨34 (56) cap (789)1⟩ ⟨234 (56) cap (789)⟩

⟨5678⟩ ⟨9567⟩ ⟨8956⟩ ⟨7895⟩ ⟨6789⟩

B12 =⟨1234⟩ ⟨4789⟩ ⟨56 (123) cap (789)⟩ ⟨123 (45) cap (789)⟩ ⟨(46) cap (789)123⟩

⟨789(45) cap (123)⟩ ⟨(46) cap (123)789⟩

B13 =⟨1234⟩ ⟨5123⟩ ⟨4512⟩ ⟨3451⟩ ⟨2345⟩ ⟨6789⟩ ⟨10678⟩ ⟨91067⟩

⟨89106⟩ ⟨78910⟩

Chapter 5

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

Z11 ⋯ Z1

Z21 ⋯ Z2

Z31 ⋯ Z3

Z41 ⋯ Z4

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ETHrarrGL(4)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

1 0 0 0 y15 ⋯ y1

0 1 0 0 y25 ⋯ y2

0 0 1 0 y35 ⋯ y3

0 0 0 1 y45 ⋯ y4

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

56 n with

ld (52)

x2ij =

is given by

yIa yJ b =1

f(y) g(y) =n

sumab=5

sumIJ=1

partyIa

2Z (56)

⟨i i + 1 j j + 1⟩)

in [89]

ABDSn = exp [

infinsumL=1

g2L (f(L)(ε)

2A(1)n (Lε) +C(L))] (57)

Γcusp = 4g2 +O(g4) (58)

ABDS-liken = ABDS

n exp [Γcusp

4Yn] 4 ∣ n

Discx2i+1j

[Discx2i+1i+p

(An)] = 0

Discx2i+1i+p

[Discx2i+1j+p+q

(An)] = 0

⎫⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎭

522 NMHV Amplitudes

form [115 116]

sums=5

sumt=s+2

Rtot =nminus2

sums=3

sumt=s+2

R1st (512)

and our results

Yif (2L)i (515)

minus 1

2([12345] + [12356] + [12367] minus [12457] minus [12567]

+ [13456] + [13467] + [14567] minus [23457] minus [23567])

times (log(⟨1234⟩⟨1789⟩⟨1278⟩⟨1349⟩)otimes ⟨3478⟩)

minus 1

2([12348] minus [12378] + [12478] minus [13478]

+ [23478] + [34567]) (log(⟨1234⟩⟨1789⟩⟨1278⟩⟨1349⟩)otimes ⟨3478⟩)

[12347] [12348] [12349] [12378] [12379] [12389]

[12478] [12479] [12489] [12789] [13478] [13479]

[13489] [13789] [14789] [23478] [23479] [23489]

[23789] [24789] [34567] [34568] [34578] [34678]

[34789] [35678] [45678]

Yangian invariants

a3ijkl = ⟨i(j j + 1)(k k + 1)(l l + 1)⟩

)(525)

condition

cup (j j + 1 k k + 15

) cup (i j j + 1 k k + 15

cup (i k k + 1 l l + 15

) cup (l l + 1 i5

) is a

uijkl =x2ikx

sumi=5

minus 1

sumi=5

minus 1

u2nminus3nminus1n)

sumi=5

minus 1

1247) +1

sumi=4

4ln(u8145) ln(u1256u3478

u2367) + cyclic (532)

nminus2

sumi=4

ln(un1ii+1)iminus1

sumj=1

sumi=4

ln(un1ii+1)iminus1

sumj=1

4ln(un1n

2n2+1)

nminus22

sumi=1

ln(uii+1i+n2i+n

Chapter 6

Graphs

symbol of [23]

C sdotZ = 0 (61)

example [27 93 94 28 30])

for each graph

source columns

C =⎛⎜⎜⎜⎝

1 0 c13 c14 c15 c16

0 1 c23 c24 c25 c26

⎞⎟⎟⎟⎠ (63)

prodαisinp

fα (64)

f0 = minus⟨1234⟩⟨2346⟩ f1 = minus

⟨2346⟩⟨2345⟩ f2 =

⟨2345⟩⟨1236⟩⟨1234⟩⟨2356⟩

f3 = minus⟨2356⟩⟨2346⟩ f4 =

⟨2346⟩⟨1256⟩⟨2456⟩⟨1236⟩ f5 = minus

⟨2456⟩⟨2356⟩

f6 =⟨2356⟩⟨1456⟩⟨3456⟩⟨1256⟩ f7 = minus

⟨3456⟩⟨2456⟩ f8 = minus

⟨2456⟩⟨1456⟩

f(b)0 = f (a)0 (1 + f (a)2 )

f(b)1 = f

1 + 1f (a)2

f(b)2 = 1

f(b)3 = f (a)3 (1 + f (a)2 )

f(b)4 = f

1 + 1f (a)2

f(c)2 = f (a)2 (1 + f (a)4 )

f(c)3 = f

1 + 1f (a)4

f(c)4 = 1

f(c)5 = f (a)5 (1 + f (a)4 )

f(c)6 = f

1 + 1f (a)4

(a) (b) (c)

⟨16⟩ ⟨12⟩

⟨23⟩

⟨34⟩⟨45⟩

⟨56⟩

⟨26⟩

⟨36⟩

⟨46⟩

ampamppp

⟨16⟩ ⟨12⟩

⟨23⟩

⟨34⟩⟨45⟩

⟨56⟩

⟨26⟩

⟨36⟩

⟨35⟩

⟨16⟩ ⟨12⟩

⟨23⟩

⟨34⟩⟨45⟩

⟨56⟩

⟨26⟩

⟨24⟩⟨46⟩ oo

(d) (e) (f)

⟨3456⟩ ⟨1456⟩

⟨1256⟩

⟨1236⟩⟨1234⟩

⟨2345⟩

⟨2456⟩

⟨2356⟩

⟨2346⟩

ampamppp

⟨3456⟩ ⟨1456⟩

⟨1256⟩

⟨1236⟩⟨1234⟩

⟨2345⟩

⟨2456⟩

⟨2356⟩

⟨1235⟩

⟨3456⟩ ⟨1456⟩

⟨1256⟩

⟨1236⟩⟨1234⟩

⟨2345⟩

⟨2456⟩

⟨1246⟩⟨2346⟩ oo

(g) (h) (i)

C = (0 1 f0f1f2f3 f0f1f2 f0f1 f0) (69)

f0 =⟨2345⟩⟨3456⟩ f1 = minus

⟨2346⟩⟨2345⟩ f2 = minus

⟨2356⟩⟨2346⟩ f3 = minus

⟨2456⟩⟨2356⟩ f4 = minus

⟨3456⟩⟨2456⟩

amplitude

f0 =⟨123⟩⟨145⟩

⟨234⟩⟨146⟩ f3 = minus⟨234⟩⟨156⟩⟨123⟩⟨456⟩

f4 = minus⟨124⟩⟨456⟩

⟨134⟩⟨156⟩

f6 = minus⟨134⟩⟨124⟩

f0 =⟨145⟩⟨456⟩ f1 = minus

⟨146⟩⟨145⟩ f2 = minus

⟨156⟩⟨146⟩ f3 = minus

⟨123⟩⟨456⟩⟨234⟩⟨156⟩

f4 = minus⟨124⟩⟨123⟩ f5 = minus

⟨234⟩⟨134⟩ f6 = minus

⟨134⟩⟨124⟩

⟨234⟩⟨156⟩ (616)

from (615)

C =⎛⎜⎜⎜⎝

1 c12 0 c14 c15 c16 c17 c18

0 c32 1 c34 c35 c36 c37 c38

⎞⎟⎟⎟⎠

a4a7 ⟨6(12)(34)(78)⟩ ⟨3478⟩

a2 ⟨4(12)(56)(78)⟩ ⟨1278⟩

a1⟨3(12)(56)(78)⟩

a3 ⟨1(34)(56)(78)⟩

a6 ⟨5(12)(34)(78)⟩

pendix 67

a9⟨1234⟩⟨5678⟩ (625)

∆2468

65 Discussion 101

65 Discussion

on Gr(mn)

65 Discussion 103

in [23]

f(b)0 = minus⟨1235⟩

⟨2356⟩ f(b)1 = minus⟨1236⟩

⟨1235⟩ f(b)2 = ⟨1234⟩⟨2356⟩

⟨2345⟩⟨1236⟩

f(b)3 = minus⟨1235⟩

⟨1234⟩ f(b)4 = ⟨2345⟩⟨1256⟩

⟨1235⟩⟨2456⟩ f(b)5 = minus⟨2456⟩

⟨2356⟩

f(b)6 = ⟨2356⟩⟨1456⟩

⟨3456⟩⟨1256⟩ f(b)7 = minus⟨3456⟩

⟨2456⟩ f(b)8 = minus⟨2456⟩

⟨1456⟩

f(c)0 = minus⟨1234⟩

⟨2346⟩ f(c)1 = minus⟨2346⟩

⟨2345⟩ f(c)2 = ⟨2345⟩⟨1246⟩

⟨1234⟩⟨2456⟩

f(c)3 = minus⟨1256⟩

⟨1246⟩ f(c)4 = ⟨2456⟩⟨1236⟩

⟨2346⟩⟨1256⟩ f(c)5 = minus⟨1246⟩

⟨1236⟩

f(c)6 = ⟨1456⟩⟨2346⟩

⟨3456⟩⟨1246⟩ f(c)7 = minus⟨3456⟩

⟨2456⟩ f(c)8 = minus⟨2456⟩

⟨1456⟩

∆1357 andradic

∆2468 with

∆1357 and

2(1 + u minus v +

u = ⟨1278⟩⟨3456⟩⟨1256⟩⟨3478⟩ v = ⟨1234⟩⟨5678⟩

⟨1256⟩⟨3478⟩ (632)

radic∆1357

⟨1256⟩⟨3478⟩ (633)

∣irarri+6

∣irarri+4

∣irarri+6

∣irarri+4

∣irarri+6

z a9 =

1 minus z1 minus z

Z isin Gr(48)gt0

Bibliography

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th9711200 [hep-th]]

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Abstract

Acknowledgements

Introduction












n-point MHV




6-point MHV

n-point NMHV

n-point NkMHV





Soft limits






NMHV

NNMHV

NNNMHV and Higher




One-loop Amplitudes


NMHV Amplitudes











Discussion



Teaching

Schools and Talks

Jun 2019 TASI 2019

Additional Skills

Acknowledgements

Contents

Abstract v

Acknowledgements xi

1 Introduction 1

22 n-point MHV 19

224 6-point MHV 24

23 n-point NMHV 25

24 n-point NkMHV 28

33 Soft limits 43

421 NMHV 60

422 N2MHV 62

65 Discussion 101

List of Figures

to ⟨k(l l + 1)(mm + 1)(k minus 1k + 1)⟩ 80

to ⟨k(l l + 1)(mm + 1)(pp + 1)⟩ 81

List of Tables

I love you all

Chapter 1

Introduction

amplitudes

calculated

(11) transforms as

az + bcz + d

) = ∣cz + d∣2∆φ∆plusmn(X z z) (13)

prodj=1int

0dωj ω

az + bcz + d

prodj=1

gluon amplitudes

microi+1 (16)

〈2345〉

〈2346〉〈2356〉〈2456〉〈3456〉

〈1234〉〈1236〉〈1256〉〈1456〉

akaprimek = prod

i∣bikgt0

aminusbiki (112)

Y (Z ∣η) = int4k

prodi=1

d log fi4

prodI=1

prodα=1

suma=1

Cαa(Z ∣η)aI) (113)

prodαisinp

fα (114)

fi = 1 (115)

coordinates

Chapter 2

Celestial Sphere

radicωiωjzij (22)

prodj=1int

0dωj ω

azj + bczj + d

prodj=1

22 n-point MHV 19

22 n-point MHV

ratio ωi

dωiωi

(ωisi

j=1 iλj

s1+iλii

⎛⎜⎝

proda=1anei

intinfin

0dωa ω

intinfin

dωiωi

(ωisi

= 2πδ(z) (29)

sumj=1

λj⎞⎠s1+iλii

⎛⎜⎝

proda=1anei

intinfin

0dωa ω

⎞⎟⎠A`1⋯`n(si ωl zj zj) (210)

δ4(ε1s1q1 +n

sumi=2

εiωiqi) =4

prodj=nminus3

U+nminus4

sumi=2

U) (212)

22 n-point MHV 21

ui =z31zi2z32zi1

for i = 45 n (216)

s1 =∣z32∣

∣z31∣ ∣z12∣ s2 =

∣z31∣∣z32∣ ∣z21∣

and si =∣z12∣

∣z1i∣ ∣zi2∣for i = 3 n (217)

to demand Uij

prodij 1lt0(Uij

12 siλ1+21

nminus4

proda=2int

0dωa ω

1lt0(Uij

ωi = siU1n

uiminus1

i = 23 n minus 4 (220)

which leads to

12siλ1+21 siλ2+2

2 siλ33 siλnn

z23z34zn1U1nδ(

sumj=1

λj) ϕ(α x)prodij

1lt0(Uij

U) (221)

22 n-point MHV 23

prodj=1

P2and and dPnminus4

Pnminus4

αn =1 + iλ1 x0 i =U1i

U1n xjminus1 i =

Ujnminus U1i

U1n x0n = minus

U1n xjminus1n =

ϕ(α x) equivc2

c1intu1ge0u3ge0

1minussuma uage0

prodj=1

Pnminus1

U1n xij =

Ujnminus4+i

x0n = minusU

U1n xin =

U1n x01 = 1

mannian

224 6-point MHV

)iλ1+1

( minusUU16

)iλ2+2

iλ3minus1

iλ4minus1

iλ5minus1

times int1

prodi=1

23 n-point NMHV 25

U16U2i+2for i = 123 (227)

23 n-point NMHV

nminus1

sumj=4

⟨1∣P2jPj+12∣3⟩3

P 22jP

⟨j + 1 j⟩[2∣P2j ∣j + 1⟩⟨j∣Pj+12∣2]

equivnminus1

sumj=4

Mj (228)

⟨34⟩⟨45⟩⋯⟨n minus 1 n⟩⟨n1⟩(⟨12⟩[24]⟨43⟩ + ⟨13⟩[34]⟨43⟩)3

(⟨23⟩[23] + ⟨24⟩[24] + ⟨34⟩[34])⟨34⟩[34]times

times ⟨54⟩([23]⟨35⟩ + [24]⟨45⟩)(⟨43⟩[32]) (229)

M4 =ω1ω4(ε2z12z24ω2+ε3z13z34ω3)3

of variables

ωi = siuiminus1

i = 2 n minus 4 (231)

M4 sim δ⎛⎝n

sumj=1

λj⎞⎠

prodni=1 siλii

z12z13z45z4ns21s

z34zn1UF(αx)prod

1lt0(Uij

U) (232)

nminus5

proda=1

nminus5

prodj=1

uiλj+1

times7

prodi=1

23 n-point NMHV 27

U xj4 =

U j = 1 n minus 5

x05 =U1nminus2

U xj5 =

U j = 1 n minus 5 (235)

x06 =U1nminus1

U xj6 =

U j = 1 n minus 5

x07 =U1n

U xj7 =

Ujn minusU1n

U j = 1 n minus 5

6- and 7-point NMHV

u1uiλ2

prodi=2

(x0i + u1x1i)αi (236)

u2uiλ2

1 uiλ32 (u1x10 + u2x20 + u1u2x30 + u2

times7

prodi=1

24 n-point NkMHV

24 n-point NkMHV 29

N2MHV amplitude

F41F23

⟨1∣P26P72P35P63∣4⟩3

P 226P

⟨76⟩[23]⟨65⟩[2∣P26∣7⟩⟨6∣P72∣2][3∣P35∣6⟩⟨5∣P63∣3]

u3uα1

1 uα22 uα3

prodi=4

times17

prodj=14

2x8j + u23x9j)αj

NkMHV amplitude

nminus4

prodl=2

dululuαl

⎛⎝

1 minusnminus4

sumj=2

uj⎞⎠

P32k (prod

(P i1)αi)

⎛⎝prodj

(Pj2)αj

⎞⎠

prodj=0

Pj(u)αjdϕ (243)

dϕ =dPj1Pj1

and and dPjnPjn

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

x00 x0m

x10 x1m

⋮ ⋱ ⋮

xn0 xnm

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

gauge fixed form

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

1 0 0 1 1 1

⋮ ⋱ minus1 ⋮ ⋮ ⋮

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ways eq (324) of [38]

prodi=1

suml=1

prodj=1

(1 minusn

sumi=1

xijui)minusβj

sumi=1

αi) sdotn

prodi=1

Γ(αi) (248)

prodi=1

mminusnminus1

suml=1

prodj=1

sumi=1

xjiui)minusαj

sumi=1

prodi=1

Γ(βi) (249)

γ minusn

sumi=1

sumj=1

βj notin Z(250)

sumi=0

xijpartϕ

sumj=0

xijpartϕ

(3) part2ϕ

k du1duk (252)

Chapter 3

prodl=1

prodj=1

sumi=1

ki)( 1

(k1+k2)2+m2+ 1

(k1+k3)2+m2+ 1

(k1+k4)2+m2)

(hj hj) =1

A4(zj zj) =( z24

z14)h12 ( z14

z13)h34

zh1+h212 zh3+h4

z14)h12 ( z14

z13)h34

zh1+h212 zh3+h4

A4(z z) (35)

z = z12z34

z13z24 z = z12z34

compute

τminus2∆4A4(0 z11τ) (37)

prodi=1

⎛⎝

sumj=1

kj⎞⎠

m2 minus 4ω1ω2∣z∣2 (38)

δ(4)⎛⎝

sumj=1

kj⎞⎠= 4τ2

ω1δ(iz minus iz)

prodi=2

ωlowast2 = ω1

2 (310)

0dω1ω

∆1minus21

prodi=2

m2 minus 4z2

zminus1ω21

= g2 (im2)2αminus2

z minus 1)α

+ zα) (312)

sumi=1

hi minus 2 (313)

constructed in [51]

2(h2+h3)A13harr24

4 (z) 0 lt z lt 1 (315)

as well as

4 (1z)

z minus 1) 0 lt z lt 1 (316)

2 (im2)2αminus2

2 sin(πα) intC

hi(z z)

2= 1 minus h12 + n

2= 1 minus h34 + n

2= h12 minus n

2= h34 minus n (318)

with n = 0123

hi(z z) (319)

4hi+hj

Γ (hij + ∆2)Γ (∆

and Ψ∆hi(z z) is

Ψ∆hi(z z) equiv

2 minus h34)Γ (∆

2 + h12)Γ (1 minus ∆2 + h34)

Ψ∆hi(z z) (322)

infinsumJ=0

4π2Ψhh

+(h12 harr h34))

33 Soft limits 43

∆ minus J2

2= 1 minus h34 + n

∆ + J2

2= h34 minus n

33 Soft limits

Single soft limits

limωkrarr0

A`k=+1n = ⟨k minus 1k + 1⟩

limωkrarr0

δ(x) = 1

2limλrarr0

limλkrarr0

An(zj zj) =1

prodj=1jnek

intinfin

0dωj ω

iλjj int

limλkrarr0

AJk=+1n = 1

zkminus1k+1

limλkrarr0

AJk=minus1n = 1

zkminus1k+1

Double soft limits

33 Soft limits 45

⟨n∣1 + 2∣3] ( [13]3⟨n3⟩[12][23]s123

+ ⟨n2⟩3[n3]⟨n1⟩⟨12⟩sn12

we will write

An(zj zj) =n

prodj=1jnekl

intinfin

0dωj ω

iλjj int

0dr int

DS(kJk lJl) = 1

r=0]εrarr0

2(iλ1 + iλ2)ε21

zn1z23( 1

zn3z2n

z12z2n+ 1

z3nz31

z12z31) (338)

sumi=1

4ω1⋯ω6

ω21ω

ω23ω

26(ω4z46z34+ω5z56z35)3

(ω3ω4z34z34+ω3ω5z35z35+ω4ω5z45z45)

z12z16z34z45 (ω3z23z35 + ω4z24z45)

⎤⎥⎥⎥⎥⎥⎦

2(iλ3 + iλ4)ε21

z23z45( 1

z25z41

z34z42+ 1

z52z53

25z216z25z61

(z15z61

z25z26)iλ2minus1

(z12z16

z25z56)iλ5+1

(z15z12

z56z26)iλ6+1

⟨2∣3 + 4∣5] ( [35]3⟨25⟩[34][45]s345

+ ⟨24⟩3[25]⟨23⟩⟨34⟩s234

) (342)

intCd∆ Ψhh

hihi(z z)⟩ (343)

(zz)2G(z z)F (z z) (344)

given by

F+(z z) =1

zh34 zh342F1 (

1 minus h + h34 h + h34

1 + h12 + h341

z) 2F1 (

1 minus h + h34 h + h34

1 + h12 + h341

z) (346)

z) 2F1 (

⟨Ψhh

hihi(z z)⟩ = (2π)2

⟨A4(z z)Ψhh

hihi(z z) (349)

2F1(a b c z) =Γ(c)

obtain

⟨A4(z z)Ψhh

⎡⎢⎢⎢⎢⎢⎣

⎛⎜⎝

⎤⎥⎥⎥⎥⎥⎥⎦

sin(2πJ)2πJ

= 1 J = 0

0 J ne 0 (353)

hihi∣J=0

which simplifies to

2 minus h34)Γ(2 minus∆)Γ(∆) (354)

⟨A4(z z)Ψhh

Chapter 4

⟨1 2 6 7⟩⟨2 3 4 5⟩ minus ⟨1 2 4 5⟩⟨2 3 6 7⟩ (42)

cluster coordinate

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

1 0 0 0 y15 ⋯ y1

0 1 0 0 y25 ⋯ y2

0 0 1 0 y35 ⋯ y3

0 0 0 1 y45 ⋯ y4

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

yIa yJ b =1

f(y) g(y) =n

sumab=1

sumIJ=1

partyIa

2Z foralli j (46)

entries

n5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Total

1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1

2 0 1 2 5 4 1 0 0 0 0 0 0 0 0 0 0 13

3 0 0 1 6 54 177 298 274 134 30 3 0 0 0 0 0 977

4 0 0 0 1 13 263 1988 7862 18532 28204 28377 18925 8034 2047 270 17 114533

421 NMHV

p1 p5 = ⟨1 2 3 4⟩ ⟨2 3 4 5⟩ ⟨3 4 5 1⟩ ⟨4 5 1 2⟩ ⟨5 1 2 3⟩ (48)

5minusy35minusy4

5 (49)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0

0 0 12

0 minus12 0 1

0 minus12 minus1

2 0 12

0 minus12 minus1

2 minus12 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

422 N2MHV

1 = [12 (23) cap (456) (234) cap (56)6][23456]

2 = [12 (34) cap (567) (345) cap (67)7][34567]

3 = [123 (345) cap (67)7][34567]

4 = [123 (456) cap (78)8][45678]

5 = [12348][45678]

6 = [123 (45) cap (678)8][45678]

7 = [123 (45) cap (678) (456) cap (78)][45678] (411)

8 = [1234 (456) cap (78)][45678]

9 = [12349][56789]

10 = [1234 (567) cap (89)][56789]

11 = [1234 (56) cap (789)][56789]

13 = [12345][678910]

ϕ = ⟨4 5 (123) cap (789)⟩⟨4 6 (123) cap (789)⟩⟨1 2 3 4⟩⟨4 7 8 9⟩⟨5 6 (123) cap (789)⟩ (413)

Ω(2)1

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 1 1 0 0 0 12

minus 12

minus1

0 0 0 0 minus 12

0 minus 12

minus 12

minus1

0 minus 12

minus 12

minus1

0 minus 12

0 minus1 minus1

minus 12

minus1

0 0 0 0 minus 12

0 minus 12

0 0 0 0

minus 12

0 minus 12

0 0 0 minus 12

minus 12

0 0 12

0 minus 12

1 1 1 1 1 0 0 12

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 1 0 0 0 0 minus1 minus 12

minus 12

0 0 0 minus 12

minus 12

0 minus 12

minus 12

minus1 0 0 minus 32

minus 32

0 minus 12

minus 32

minus 12

0 minus 12

0 minus1 minus 12

minus 12

0 minus1 minus 12

minus 12

0 0 0 minus 12

minus 12

0 minus 12

minus 12

0 0 12

0 minus 12

1 1 12

0 0 0 0 minus 12

0 0 0 12

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)3

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 12

0 0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

minus 32

minus 12

1 0 minus 12

0 minus1 0 minus 12

0 0 0 minus 12

minus 12

0 minus 12

minus 12

0 0 12

0 minus 12

1 1 12

0 0 12

0 0 0 0 0 0 minus 12

0 0 0 12

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)4

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 minus1 minus1 0

0 0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus1 minus1 0

0 minus 12

0 minus1 minus1 0

0 0 minus 12

minus 12

0 minus 12

minus 12

0 0 0 12

0 minus 12

minus 12

1 1 1 12

0 minus 12

1 1 1 12

0 0 0 0 0 0 0 0 0 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)5

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 0 0 0

0 0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 0 0 0

0 minus 12

0 0 0 0

0 0 minus 12

minus 12

0 minus 12

minus 12

0 0 0 12

0 minus 12

minus 12

0 0 0 12

0 minus 12

0 0 0 12

0 0 0 0 0 0 0 0 0 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)6

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 0 minus1 0

0 0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 0 minus1 0

0 minus 12

0 0 minus1 0

0 0 minus 12

minus 12

0 minus 12

minus 12

0 0 0 12

0 minus 12

minus 12

0 0 0 12

0 minus 12

1 1 1 12

0 0 0 0 0 0 0 0 0 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)7

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 0 minus1 0

0 1 0 minus 12

minus 12

minus 32

0 1 12

0 minus 12

minus 12

minus 32

0 1 12

minus 12

minus 32

0 0 minus 12

minus 12

0 minus 12

minus 12

0 0 minus 12

minus 12

0 minus 12

minus 12

0 1 12

0 minus 12

1 1 32

0 0 0 0 0 0 0 0 0 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)8

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 0 0 0

0 1 0 minus 12

minus 12

0 1 12

0 minus 12

minus 12

0 1 12

minus 12

0 0 minus 12

minus 12

0 minus 12

minus 12

0 0 minus 12

minus 12

0 minus 12

minus 12

0 1 12

0 minus 12

0 1 12

0 0 0 0 0 0 0 0 0 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)9

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 0 0 0

0 0 minus 12

minus 12

0 0 0 0

0 minus 12

minus 12

0 0 0 0

0 minus 12

0 0 0 0

0 minus 12

minus 12

0 minus 12

minus 12

0 0 0 0 0 12

0 minus 12

minus 12

0 0 0 0 0 12

0 minus 12

minus 12

0 0 0 0 0 12

0 minus 12

0 0 0 0 0 12

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)10

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 0 0 0

0 0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)11

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 0 0 0

0 0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)12

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 1 1 1 32

0 minus1 0 minus 12

minus 12

minus 32

minus 12

0 minus1 12

0 minus 12

minus 12

0 minus 12

0 minus1 12

0 minus 12

minus 12

0 minus 32

0 minus 12

minus 12

2 2 2 12

0 minus 32

0 minus 12

2 2 2 12

0 minus 32

0 2 2 2 12

0 minus 32

minus 12

0 minus 32

minus 12

0 minus 12

minus 12

0 minus 32

minus 12

0 minus 12

minus 12

0 minus1 12

0 minus 12

minus 12

0 minus 12

0 minus1 12

0 minus 12

minus 12

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)13

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 0 0 0

0 0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

B1 =⟨12 (23) cap (456) (234) cap (56)⟩ ⟨612 (23) cap (456)⟩ ⟨(234) cap (56)612⟩

⟨(23) cap (456) (234) cap (56)61⟩ ⟨2 (23) cap (456) (234) cap (56)6⟩ ⟨2345⟩ ⟨6234⟩ ⟨5623⟩

⟨4562⟩ ⟨3456⟩

(345) cap (67)71⟩ ⟨2 (34) cap (567) (345) cap (67)7⟩ ⟨3456⟩ ⟨7345⟩ ⟨6734⟩ ⟨5673⟩

⟨4567⟩

B3 =⟨123 (345) cap (67)⟩ ⟨7123⟩ ⟨(345) cap (67)712⟩ ⟨3 (345) cap (67)71⟩ ⟨23 (345) cap (67)7⟩

⟨3456⟩ ⟨7345⟩ ⟨6734⟩ ⟨5673⟩ ⟨4567⟩

B4 =⟨123 (456) cap (78)⟩ ⟨8123⟩ ⟨(456) cap (78)812⟩ ⟨3 (456) cap (78)81⟩ ⟨23 (456) cap (78)8⟩

⟨4567⟩ ⟨8456⟩ ⟨7845⟩ ⟨6784⟩⟨5678⟩

B5 =⟨1234⟩ ⟨8123⟩ ⟨4812⟩ ⟨3481⟩ ⟨2348⟩ ⟨4567⟩ ⟨8456⟩ ⟨7845⟩ ⟨6784⟩

⟨5678⟩

B6 =⟨123 (45) cap (678)⟩ ⟨8123⟩ ⟨(45) cap (678)812⟩ ⟨3 (45) cap (678)81⟩ ⟨23 (45) cap (678)8⟩

⟨4567⟩ ⟨8456⟩ ⟨7845⟩ ⟨6784⟩⟨5678⟩

B7 =⟨123 (45) cap (678)⟩ ⟨(456) cap (78)123⟩ ⟨(45) cap (678) (456) cap (78)12⟩

⟨3 (45) cap (678) (456) cap (78)1⟩ ⟨23 (45) cap (678) (456) cap (78)⟩ ⟨4567⟩ ⟨8456⟩ ⟨7845⟩

⟨6784⟩⟨5678⟩

B8 =⟨1234⟩ ⟨(456) cap (78)123⟩ ⟨4 (456) cap (78)12⟩ ⟨34 (456) cap (78)1⟩ ⟨234 (456) cap (78)⟩

⟨4567⟩ ⟨8456⟩ ⟨7845⟩ ⟨6784⟩⟨5678⟩

B9 =⟨1234⟩ ⟨9123⟩ ⟨4912⟩ ⟨3491⟩ ⟨2349⟩ ⟨5678⟩ ⟨9567⟩ ⟨8956⟩

⟨7895⟩ ⟨6789⟩

B10 =⟨1234⟩ ⟨(567) cap (89)123⟩ ⟨4 (567) cap (89)12⟩ ⟨34 (567) cap (89)1⟩ ⟨234 (567) cap (89)⟩

⟨5678⟩ ⟨9567⟩ ⟨8956⟩ ⟨7895⟩ ⟨6789⟩

B11 =⟨1234⟩ ⟨(56) cap (789)123⟩ ⟨4 (56) cap (789)12⟩ ⟨34 (56) cap (789)1⟩ ⟨234 (56) cap (789)⟩

⟨5678⟩ ⟨9567⟩ ⟨8956⟩ ⟨7895⟩ ⟨6789⟩

B12 =⟨1234⟩ ⟨4789⟩ ⟨56 (123) cap (789)⟩ ⟨123 (45) cap (789)⟩ ⟨(46) cap (789)123⟩

⟨789(45) cap (123)⟩ ⟨(46) cap (123)789⟩

B13 =⟨1234⟩ ⟨5123⟩ ⟨4512⟩ ⟨3451⟩ ⟨2345⟩ ⟨6789⟩ ⟨10678⟩ ⟨91067⟩

⟨89106⟩ ⟨78910⟩

Chapter 5

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

Z11 ⋯ Z1

Z21 ⋯ Z2

Z31 ⋯ Z3

Z41 ⋯ Z4

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ETHrarrGL(4)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

1 0 0 0 y15 ⋯ y1

0 1 0 0 y25 ⋯ y2

0 0 1 0 y35 ⋯ y3

0 0 0 1 y45 ⋯ y4

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

56 n with

ld (52)

x2ij =

is given by

yIa yJ b =1

f(y) g(y) =n

sumab=5

sumIJ=1

partyIa

2Z (56)

⟨i i + 1 j j + 1⟩)

in [89]

ABDSn = exp [

infinsumL=1

g2L (f(L)(ε)

2A(1)n (Lε) +C(L))] (57)

Γcusp = 4g2 +O(g4) (58)

ABDS-liken = ABDS

n exp [Γcusp

4Yn] 4 ∣ n

Discx2i+1j

[Discx2i+1i+p

(An)] = 0

Discx2i+1i+p

[Discx2i+1j+p+q

(An)] = 0

⎫⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎭

522 NMHV Amplitudes

form [115 116]

sums=5

sumt=s+2

Rtot =nminus2

sums=3

sumt=s+2

R1st (512)

and our results

Yif (2L)i (515)

minus 1

2([12345] + [12356] + [12367] minus [12457] minus [12567]

+ [13456] + [13467] + [14567] minus [23457] minus [23567])

times (log(⟨1234⟩⟨1789⟩⟨1278⟩⟨1349⟩)otimes ⟨3478⟩)

minus 1

2([12348] minus [12378] + [12478] minus [13478]

+ [23478] + [34567]) (log(⟨1234⟩⟨1789⟩⟨1278⟩⟨1349⟩)otimes ⟨3478⟩)

[12347] [12348] [12349] [12378] [12379] [12389]

[12478] [12479] [12489] [12789] [13478] [13479]

[13489] [13789] [14789] [23478] [23479] [23489]

[23789] [24789] [34567] [34568] [34578] [34678]

[34789] [35678] [45678]

Yangian invariants

a3ijkl = ⟨i(j j + 1)(k k + 1)(l l + 1)⟩

)(525)

condition

cup (j j + 1 k k + 15

) cup (i j j + 1 k k + 15

cup (i k k + 1 l l + 15

) cup (l l + 1 i5

) is a

uijkl =x2ikx

sumi=5

minus 1

sumi=5

minus 1

u2nminus3nminus1n)

sumi=5

minus 1

1247) +1

sumi=4

4ln(u8145) ln(u1256u3478

u2367) + cyclic (532)

nminus2

sumi=4

ln(un1ii+1)iminus1

sumj=1

sumi=4

ln(un1ii+1)iminus1

sumj=1

4ln(un1n

2n2+1)

nminus22

sumi=1

ln(uii+1i+n2i+n

Chapter 6

Graphs

symbol of [23]

C sdotZ = 0 (61)

example [27 93 94 28 30])

for each graph

source columns

C =⎛⎜⎜⎜⎝

1 0 c13 c14 c15 c16

0 1 c23 c24 c25 c26

⎞⎟⎟⎟⎠ (63)

prodαisinp

fα (64)

f0 = minus⟨1234⟩⟨2346⟩ f1 = minus

⟨2346⟩⟨2345⟩ f2 =

⟨2345⟩⟨1236⟩⟨1234⟩⟨2356⟩

f3 = minus⟨2356⟩⟨2346⟩ f4 =

⟨2346⟩⟨1256⟩⟨2456⟩⟨1236⟩ f5 = minus

⟨2456⟩⟨2356⟩

f6 =⟨2356⟩⟨1456⟩⟨3456⟩⟨1256⟩ f7 = minus

⟨3456⟩⟨2456⟩ f8 = minus

⟨2456⟩⟨1456⟩

f(b)0 = f (a)0 (1 + f (a)2 )

f(b)1 = f

1 + 1f (a)2

f(b)2 = 1

f(b)3 = f (a)3 (1 + f (a)2 )

f(b)4 = f

1 + 1f (a)2

f(c)2 = f (a)2 (1 + f (a)4 )

f(c)3 = f

1 + 1f (a)4

f(c)4 = 1

f(c)5 = f (a)5 (1 + f (a)4 )

f(c)6 = f

1 + 1f (a)4

(a) (b) (c)

⟨16⟩ ⟨12⟩

⟨23⟩

⟨34⟩⟨45⟩

⟨56⟩

⟨26⟩

⟨36⟩

⟨46⟩

ampamppp

⟨16⟩ ⟨12⟩

⟨23⟩

⟨34⟩⟨45⟩

⟨56⟩

⟨26⟩

⟨36⟩

⟨35⟩

⟨16⟩ ⟨12⟩

⟨23⟩

⟨34⟩⟨45⟩

⟨56⟩

⟨26⟩

⟨24⟩⟨46⟩ oo

(d) (e) (f)

⟨3456⟩ ⟨1456⟩

⟨1256⟩

⟨1236⟩⟨1234⟩

⟨2345⟩

⟨2456⟩

⟨2356⟩

⟨2346⟩

ampamppp

⟨3456⟩ ⟨1456⟩

⟨1256⟩

⟨1236⟩⟨1234⟩

⟨2345⟩

⟨2456⟩

⟨2356⟩

⟨1235⟩

⟨3456⟩ ⟨1456⟩

⟨1256⟩

⟨1236⟩⟨1234⟩

⟨2345⟩

⟨2456⟩

⟨1246⟩⟨2346⟩ oo

(g) (h) (i)

C = (0 1 f0f1f2f3 f0f1f2 f0f1 f0) (69)

f0 =⟨2345⟩⟨3456⟩ f1 = minus

⟨2346⟩⟨2345⟩ f2 = minus

⟨2356⟩⟨2346⟩ f3 = minus

⟨2456⟩⟨2356⟩ f4 = minus

⟨3456⟩⟨2456⟩

amplitude

f0 =⟨123⟩⟨145⟩

⟨234⟩⟨146⟩ f3 = minus⟨234⟩⟨156⟩⟨123⟩⟨456⟩

f4 = minus⟨124⟩⟨456⟩

⟨134⟩⟨156⟩

f6 = minus⟨134⟩⟨124⟩

f0 =⟨145⟩⟨456⟩ f1 = minus

⟨146⟩⟨145⟩ f2 = minus

⟨156⟩⟨146⟩ f3 = minus

⟨123⟩⟨456⟩⟨234⟩⟨156⟩

f4 = minus⟨124⟩⟨123⟩ f5 = minus

⟨234⟩⟨134⟩ f6 = minus

⟨134⟩⟨124⟩

⟨234⟩⟨156⟩ (616)

from (615)

C =⎛⎜⎜⎜⎝

1 c12 0 c14 c15 c16 c17 c18

0 c32 1 c34 c35 c36 c37 c38

⎞⎟⎟⎟⎠

a4a7 ⟨6(12)(34)(78)⟩ ⟨3478⟩

a2 ⟨4(12)(56)(78)⟩ ⟨1278⟩

a1⟨3(12)(56)(78)⟩

a3 ⟨1(34)(56)(78)⟩

a6 ⟨5(12)(34)(78)⟩

pendix 67

a9⟨1234⟩⟨5678⟩ (625)

∆2468

65 Discussion 101

65 Discussion

on Gr(mn)

65 Discussion 103

in [23]

f(b)0 = minus⟨1235⟩

⟨2356⟩ f(b)1 = minus⟨1236⟩

⟨1235⟩ f(b)2 = ⟨1234⟩⟨2356⟩

⟨2345⟩⟨1236⟩

f(b)3 = minus⟨1235⟩

⟨1234⟩ f(b)4 = ⟨2345⟩⟨1256⟩

⟨1235⟩⟨2456⟩ f(b)5 = minus⟨2456⟩

⟨2356⟩

f(b)6 = ⟨2356⟩⟨1456⟩

⟨3456⟩⟨1256⟩ f(b)7 = minus⟨3456⟩

⟨2456⟩ f(b)8 = minus⟨2456⟩

⟨1456⟩

f(c)0 = minus⟨1234⟩

⟨2346⟩ f(c)1 = minus⟨2346⟩

⟨2345⟩ f(c)2 = ⟨2345⟩⟨1246⟩

⟨1234⟩⟨2456⟩

f(c)3 = minus⟨1256⟩

⟨1246⟩ f(c)4 = ⟨2456⟩⟨1236⟩

⟨2346⟩⟨1256⟩ f(c)5 = minus⟨1246⟩

⟨1236⟩

f(c)6 = ⟨1456⟩⟨2346⟩

⟨3456⟩⟨1246⟩ f(c)7 = minus⟨3456⟩

⟨2456⟩ f(c)8 = minus⟨2456⟩

⟨1456⟩

∆1357 andradic

∆2468 with

∆1357 and

2(1 + u minus v +

u = ⟨1278⟩⟨3456⟩⟨1256⟩⟨3478⟩ v = ⟨1234⟩⟨5678⟩

⟨1256⟩⟨3478⟩ (632)

radic∆1357

⟨1256⟩⟨3478⟩ (633)

∣irarri+6

∣irarri+4

∣irarri+6

∣irarri+4

∣irarri+6

z a9 =

1 minus z1 minus z

Z isin Gr(48)gt0

Bibliography

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th9711200 [hep-th]]

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Abstract

Acknowledgements

Introduction












n-point MHV




6-point MHV

n-point NMHV

n-point NkMHV





Soft limits






NMHV

NNMHV

NNNMHV and Higher




One-loop Amplitudes


NMHV Amplitudes











Discussion



Schools and Talks

Jun 2019 TASI 2019

Additional Skills

Acknowledgements

Contents

Abstract v

Acknowledgements xi

1 Introduction 1

22 n-point MHV 19

224 6-point MHV 24

23 n-point NMHV 25

24 n-point NkMHV 28

33 Soft limits 43

421 NMHV 60

422 N2MHV 62

65 Discussion 101

List of Figures

to ⟨k(l l + 1)(mm + 1)(k minus 1k + 1)⟩ 80

to ⟨k(l l + 1)(mm + 1)(pp + 1)⟩ 81

List of Tables

I love you all

Chapter 1

Introduction

amplitudes

calculated

(11) transforms as

az + bcz + d

) = ∣cz + d∣2∆φ∆plusmn(X z z) (13)

prodj=1int

0dωj ω

az + bcz + d

prodj=1

gluon amplitudes

microi+1 (16)

〈2345〉

〈2346〉〈2356〉〈2456〉〈3456〉

〈1234〉〈1236〉〈1256〉〈1456〉

akaprimek = prod

i∣bikgt0

aminusbiki (112)

Y (Z ∣η) = int4k

prodi=1

d log fi4

prodI=1

prodα=1

suma=1

Cαa(Z ∣η)aI) (113)

prodαisinp

fα (114)

fi = 1 (115)

coordinates

Chapter 2

Celestial Sphere

radicωiωjzij (22)

prodj=1int

0dωj ω

azj + bczj + d

prodj=1

22 n-point MHV 19

22 n-point MHV

ratio ωi

dωiωi

(ωisi

j=1 iλj

s1+iλii

⎛⎜⎝

proda=1anei

intinfin

0dωa ω

intinfin

dωiωi

(ωisi

= 2πδ(z) (29)

sumj=1

λj⎞⎠s1+iλii

⎛⎜⎝

proda=1anei

intinfin

0dωa ω

⎞⎟⎠A`1⋯`n(si ωl zj zj) (210)

δ4(ε1s1q1 +n

sumi=2

εiωiqi) =4

prodj=nminus3

U+nminus4

sumi=2

U) (212)

22 n-point MHV 21

ui =z31zi2z32zi1

for i = 45 n (216)

s1 =∣z32∣

∣z31∣ ∣z12∣ s2 =

∣z31∣∣z32∣ ∣z21∣

and si =∣z12∣

∣z1i∣ ∣zi2∣for i = 3 n (217)

to demand Uij

prodij 1lt0(Uij

12 siλ1+21

nminus4

proda=2int

0dωa ω

1lt0(Uij

ωi = siU1n

uiminus1

i = 23 n minus 4 (220)

which leads to

12siλ1+21 siλ2+2

2 siλ33 siλnn

z23z34zn1U1nδ(

sumj=1

λj) ϕ(α x)prodij

1lt0(Uij

U) (221)

22 n-point MHV 23

prodj=1

P2and and dPnminus4

Pnminus4

αn =1 + iλ1 x0 i =U1i

U1n xjminus1 i =

Ujnminus U1i

U1n x0n = minus

U1n xjminus1n =

ϕ(α x) equivc2

c1intu1ge0u3ge0

1minussuma uage0

prodj=1

Pnminus1

U1n xij =

Ujnminus4+i

x0n = minusU

U1n xin =

U1n x01 = 1

mannian

224 6-point MHV

)iλ1+1

( minusUU16

)iλ2+2

iλ3minus1

iλ4minus1

iλ5minus1

times int1

prodi=1

23 n-point NMHV 25

U16U2i+2for i = 123 (227)

23 n-point NMHV

nminus1

sumj=4

⟨1∣P2jPj+12∣3⟩3

P 22jP

⟨j + 1 j⟩[2∣P2j ∣j + 1⟩⟨j∣Pj+12∣2]

equivnminus1

sumj=4

Mj (228)

⟨34⟩⟨45⟩⋯⟨n minus 1 n⟩⟨n1⟩(⟨12⟩[24]⟨43⟩ + ⟨13⟩[34]⟨43⟩)3

(⟨23⟩[23] + ⟨24⟩[24] + ⟨34⟩[34])⟨34⟩[34]times

times ⟨54⟩([23]⟨35⟩ + [24]⟨45⟩)(⟨43⟩[32]) (229)

M4 =ω1ω4(ε2z12z24ω2+ε3z13z34ω3)3

of variables

ωi = siuiminus1

i = 2 n minus 4 (231)

M4 sim δ⎛⎝n

sumj=1

λj⎞⎠

prodni=1 siλii

z12z13z45z4ns21s

z34zn1UF(αx)prod

1lt0(Uij

U) (232)

nminus5

proda=1

nminus5

prodj=1

uiλj+1

times7

prodi=1

23 n-point NMHV 27

U xj4 =

U j = 1 n minus 5

x05 =U1nminus2

U xj5 =

U j = 1 n minus 5 (235)

x06 =U1nminus1

U xj6 =

U j = 1 n minus 5

x07 =U1n

U xj7 =

Ujn minusU1n

U j = 1 n minus 5

6- and 7-point NMHV

u1uiλ2

prodi=2

(x0i + u1x1i)αi (236)

u2uiλ2

1 uiλ32 (u1x10 + u2x20 + u1u2x30 + u2

times7

prodi=1

24 n-point NkMHV

24 n-point NkMHV 29

N2MHV amplitude

F41F23

⟨1∣P26P72P35P63∣4⟩3

P 226P

⟨76⟩[23]⟨65⟩[2∣P26∣7⟩⟨6∣P72∣2][3∣P35∣6⟩⟨5∣P63∣3]

u3uα1

1 uα22 uα3

prodi=4

times17

prodj=14

2x8j + u23x9j)αj

NkMHV amplitude

nminus4

prodl=2

dululuαl

⎛⎝

1 minusnminus4

sumj=2

uj⎞⎠

P32k (prod

(P i1)αi)

⎛⎝prodj

(Pj2)αj

⎞⎠

prodj=0

Pj(u)αjdϕ (243)

dϕ =dPj1Pj1

and and dPjnPjn

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

x00 x0m

x10 x1m

⋮ ⋱ ⋮

xn0 xnm

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

gauge fixed form

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

1 0 0 1 1 1

⋮ ⋱ minus1 ⋮ ⋮ ⋮

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ways eq (324) of [38]

prodi=1

suml=1

prodj=1

(1 minusn

sumi=1

xijui)minusβj

sumi=1

αi) sdotn

prodi=1

Γ(αi) (248)

prodi=1

mminusnminus1

suml=1

prodj=1

sumi=1

xjiui)minusαj

sumi=1

prodi=1

Γ(βi) (249)

γ minusn

sumi=1

sumj=1

βj notin Z(250)

sumi=0

xijpartϕ

sumj=0

xijpartϕ

(3) part2ϕ

k du1duk (252)

Chapter 3

prodl=1

prodj=1

sumi=1

ki)( 1

(k1+k2)2+m2+ 1

(k1+k3)2+m2+ 1

(k1+k4)2+m2)

(hj hj) =1

A4(zj zj) =( z24

z14)h12 ( z14

z13)h34

zh1+h212 zh3+h4

z14)h12 ( z14

z13)h34

zh1+h212 zh3+h4

A4(z z) (35)

z = z12z34

z13z24 z = z12z34

compute

τminus2∆4A4(0 z11τ) (37)

prodi=1

⎛⎝

sumj=1

kj⎞⎠

m2 minus 4ω1ω2∣z∣2 (38)

δ(4)⎛⎝

sumj=1

kj⎞⎠= 4τ2

ω1δ(iz minus iz)

prodi=2

ωlowast2 = ω1

2 (310)

0dω1ω

∆1minus21

prodi=2

m2 minus 4z2

zminus1ω21

= g2 (im2)2αminus2

z minus 1)α

+ zα) (312)

sumi=1

hi minus 2 (313)

constructed in [51]

2(h2+h3)A13harr24

4 (z) 0 lt z lt 1 (315)

as well as

4 (1z)

z minus 1) 0 lt z lt 1 (316)

2 (im2)2αminus2

2 sin(πα) intC

hi(z z)

2= 1 minus h12 + n

2= 1 minus h34 + n

2= h12 minus n

2= h34 minus n (318)

with n = 0123

hi(z z) (319)

4hi+hj

Γ (hij + ∆2)Γ (∆

and Ψ∆hi(z z) is

Ψ∆hi(z z) equiv

2 minus h34)Γ (∆

2 + h12)Γ (1 minus ∆2 + h34)

Ψ∆hi(z z) (322)

infinsumJ=0

4π2Ψhh

+(h12 harr h34))

33 Soft limits 43

∆ minus J2

2= 1 minus h34 + n

∆ + J2

2= h34 minus n

33 Soft limits

Single soft limits

limωkrarr0

A`k=+1n = ⟨k minus 1k + 1⟩

limωkrarr0

δ(x) = 1

2limλrarr0

limλkrarr0

An(zj zj) =1

prodj=1jnek

intinfin

0dωj ω

iλjj int

limλkrarr0

AJk=+1n = 1

zkminus1k+1

limλkrarr0

AJk=minus1n = 1

zkminus1k+1

Double soft limits

33 Soft limits 45

⟨n∣1 + 2∣3] ( [13]3⟨n3⟩[12][23]s123

+ ⟨n2⟩3[n3]⟨n1⟩⟨12⟩sn12

we will write

An(zj zj) =n

prodj=1jnekl

intinfin

0dωj ω

iλjj int

0dr int

DS(kJk lJl) = 1

r=0]εrarr0

2(iλ1 + iλ2)ε21

zn1z23( 1

zn3z2n

z12z2n+ 1

z3nz31

z12z31) (338)

sumi=1

4ω1⋯ω6

ω21ω

ω23ω

26(ω4z46z34+ω5z56z35)3

(ω3ω4z34z34+ω3ω5z35z35+ω4ω5z45z45)

z12z16z34z45 (ω3z23z35 + ω4z24z45)

⎤⎥⎥⎥⎥⎥⎦

2(iλ3 + iλ4)ε21

z23z45( 1

z25z41

z34z42+ 1

z52z53

25z216z25z61

(z15z61

z25z26)iλ2minus1

(z12z16

z25z56)iλ5+1

(z15z12

z56z26)iλ6+1

⟨2∣3 + 4∣5] ( [35]3⟨25⟩[34][45]s345

+ ⟨24⟩3[25]⟨23⟩⟨34⟩s234

) (342)

intCd∆ Ψhh

hihi(z z)⟩ (343)

(zz)2G(z z)F (z z) (344)

given by

F+(z z) =1

zh34 zh342F1 (

1 minus h + h34 h + h34

1 + h12 + h341

z) 2F1 (

1 minus h + h34 h + h34

1 + h12 + h341

z) (346)

z) 2F1 (

⟨Ψhh

hihi(z z)⟩ = (2π)2

⟨A4(z z)Ψhh

hihi(z z) (349)

2F1(a b c z) =Γ(c)

obtain

⟨A4(z z)Ψhh

⎡⎢⎢⎢⎢⎢⎣

⎛⎜⎝

⎤⎥⎥⎥⎥⎥⎥⎦

sin(2πJ)2πJ

= 1 J = 0

0 J ne 0 (353)

hihi∣J=0

which simplifies to

2 minus h34)Γ(2 minus∆)Γ(∆) (354)

⟨A4(z z)Ψhh

Chapter 4

⟨1 2 6 7⟩⟨2 3 4 5⟩ minus ⟨1 2 4 5⟩⟨2 3 6 7⟩ (42)

cluster coordinate

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

1 0 0 0 y15 ⋯ y1

0 1 0 0 y25 ⋯ y2

0 0 1 0 y35 ⋯ y3

0 0 0 1 y45 ⋯ y4

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

yIa yJ b =1

f(y) g(y) =n

sumab=1

sumIJ=1

partyIa

2Z foralli j (46)

entries

n5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Total

1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1

2 0 1 2 5 4 1 0 0 0 0 0 0 0 0 0 0 13

3 0 0 1 6 54 177 298 274 134 30 3 0 0 0 0 0 977

4 0 0 0 1 13 263 1988 7862 18532 28204 28377 18925 8034 2047 270 17 114533

421 NMHV

p1 p5 = ⟨1 2 3 4⟩ ⟨2 3 4 5⟩ ⟨3 4 5 1⟩ ⟨4 5 1 2⟩ ⟨5 1 2 3⟩ (48)

5minusy35minusy4

5 (49)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0

0 0 12

0 minus12 0 1

0 minus12 minus1

2 0 12

0 minus12 minus1

2 minus12 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

422 N2MHV

1 = [12 (23) cap (456) (234) cap (56)6][23456]

2 = [12 (34) cap (567) (345) cap (67)7][34567]

3 = [123 (345) cap (67)7][34567]

4 = [123 (456) cap (78)8][45678]

5 = [12348][45678]

6 = [123 (45) cap (678)8][45678]

7 = [123 (45) cap (678) (456) cap (78)][45678] (411)

8 = [1234 (456) cap (78)][45678]

9 = [12349][56789]

10 = [1234 (567) cap (89)][56789]

11 = [1234 (56) cap (789)][56789]

13 = [12345][678910]

ϕ = ⟨4 5 (123) cap (789)⟩⟨4 6 (123) cap (789)⟩⟨1 2 3 4⟩⟨4 7 8 9⟩⟨5 6 (123) cap (789)⟩ (413)

Ω(2)1

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 1 1 0 0 0 12

minus 12

minus1

0 0 0 0 minus 12

0 minus 12

minus 12

minus1

0 minus 12

minus 12

minus1

0 minus 12

0 minus1 minus1

minus 12

minus1

0 0 0 0 minus 12

0 minus 12

0 0 0 0

minus 12

0 minus 12

0 0 0 minus 12

minus 12

0 0 12

0 minus 12

1 1 1 1 1 0 0 12

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 1 0 0 0 0 minus1 minus 12

minus 12

0 0 0 minus 12

minus 12

0 minus 12

minus 12

minus1 0 0 minus 32

minus 32

0 minus 12

minus 32

minus 12

0 minus 12

0 minus1 minus 12

minus 12

0 minus1 minus 12

minus 12

0 0 0 minus 12

minus 12

0 minus 12

minus 12

0 0 12

0 minus 12

1 1 12

0 0 0 0 minus 12

0 0 0 12

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)3

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 12

0 0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

minus 32

minus 12

1 0 minus 12

0 minus1 0 minus 12

0 0 0 minus 12

minus 12

0 minus 12

minus 12

0 0 12

0 minus 12

1 1 12

0 0 12

0 0 0 0 0 0 minus 12

0 0 0 12

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)4

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 minus1 minus1 0

0 0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus1 minus1 0

0 minus 12

0 minus1 minus1 0

0 0 minus 12

minus 12

0 minus 12

minus 12

0 0 0 12

0 minus 12

minus 12

1 1 1 12

0 minus 12

1 1 1 12

0 0 0 0 0 0 0 0 0 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)5

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 0 0 0

0 0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 0 0 0

0 minus 12

0 0 0 0

0 0 minus 12

minus 12

0 minus 12

minus 12

0 0 0 12

0 minus 12

minus 12

0 0 0 12

0 minus 12

0 0 0 12

0 0 0 0 0 0 0 0 0 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)6

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 0 minus1 0

0 0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 0 minus1 0

0 minus 12

0 0 minus1 0

0 0 minus 12

minus 12

0 minus 12

minus 12

0 0 0 12

0 minus 12

minus 12

0 0 0 12

0 minus 12

1 1 1 12

0 0 0 0 0 0 0 0 0 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)7

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 0 minus1 0

0 1 0 minus 12

minus 12

minus 32

0 1 12

0 minus 12

minus 12

minus 32

0 1 12

minus 12

minus 32

0 0 minus 12

minus 12

0 minus 12

minus 12

0 0 minus 12

minus 12

0 minus 12

minus 12

0 1 12

0 minus 12

1 1 32

0 0 0 0 0 0 0 0 0 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)8

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 0 0 0

0 1 0 minus 12

minus 12

0 1 12

0 minus 12

minus 12

0 1 12

minus 12

0 0 minus 12

minus 12

0 minus 12

minus 12

0 0 minus 12

minus 12

0 minus 12

minus 12

0 1 12

0 minus 12

0 1 12

0 0 0 0 0 0 0 0 0 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)9

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 0 0 0

0 0 minus 12

minus 12

0 0 0 0

0 minus 12

minus 12

0 0 0 0

0 minus 12

0 0 0 0

0 minus 12

minus 12

0 minus 12

minus 12

0 0 0 0 0 12

0 minus 12

minus 12

0 0 0 0 0 12

0 minus 12

minus 12

0 0 0 0 0 12

0 minus 12

0 0 0 0 0 12

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)10

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 0 0 0

0 0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)11

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 0 0 0

0 0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)12

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 1 1 1 32

0 minus1 0 minus 12

minus 12

minus 32

minus 12

0 minus1 12

0 minus 12

minus 12

0 minus 12

0 minus1 12

0 minus 12

minus 12

0 minus 32

0 minus 12

minus 12

2 2 2 12

0 minus 32

0 minus 12

2 2 2 12

0 minus 32

0 2 2 2 12

0 minus 32

minus 12

0 minus 32

minus 12

0 minus 12

minus 12

0 minus 32

minus 12

0 minus 12

minus 12

0 minus1 12

0 minus 12

minus 12

0 minus 12

0 minus1 12

0 minus 12

minus 12

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)13

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 0 0 0

0 0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

B1 =⟨12 (23) cap (456) (234) cap (56)⟩ ⟨612 (23) cap (456)⟩ ⟨(234) cap (56)612⟩

⟨(23) cap (456) (234) cap (56)61⟩ ⟨2 (23) cap (456) (234) cap (56)6⟩ ⟨2345⟩ ⟨6234⟩ ⟨5623⟩

⟨4562⟩ ⟨3456⟩

(345) cap (67)71⟩ ⟨2 (34) cap (567) (345) cap (67)7⟩ ⟨3456⟩ ⟨7345⟩ ⟨6734⟩ ⟨5673⟩

⟨4567⟩

B3 =⟨123 (345) cap (67)⟩ ⟨7123⟩ ⟨(345) cap (67)712⟩ ⟨3 (345) cap (67)71⟩ ⟨23 (345) cap (67)7⟩

⟨3456⟩ ⟨7345⟩ ⟨6734⟩ ⟨5673⟩ ⟨4567⟩

B4 =⟨123 (456) cap (78)⟩ ⟨8123⟩ ⟨(456) cap (78)812⟩ ⟨3 (456) cap (78)81⟩ ⟨23 (456) cap (78)8⟩

⟨4567⟩ ⟨8456⟩ ⟨7845⟩ ⟨6784⟩⟨5678⟩

B5 =⟨1234⟩ ⟨8123⟩ ⟨4812⟩ ⟨3481⟩ ⟨2348⟩ ⟨4567⟩ ⟨8456⟩ ⟨7845⟩ ⟨6784⟩

⟨5678⟩

B6 =⟨123 (45) cap (678)⟩ ⟨8123⟩ ⟨(45) cap (678)812⟩ ⟨3 (45) cap (678)81⟩ ⟨23 (45) cap (678)8⟩

⟨4567⟩ ⟨8456⟩ ⟨7845⟩ ⟨6784⟩⟨5678⟩

B7 =⟨123 (45) cap (678)⟩ ⟨(456) cap (78)123⟩ ⟨(45) cap (678) (456) cap (78)12⟩

⟨3 (45) cap (678) (456) cap (78)1⟩ ⟨23 (45) cap (678) (456) cap (78)⟩ ⟨4567⟩ ⟨8456⟩ ⟨7845⟩

⟨6784⟩⟨5678⟩

B8 =⟨1234⟩ ⟨(456) cap (78)123⟩ ⟨4 (456) cap (78)12⟩ ⟨34 (456) cap (78)1⟩ ⟨234 (456) cap (78)⟩

⟨4567⟩ ⟨8456⟩ ⟨7845⟩ ⟨6784⟩⟨5678⟩

B9 =⟨1234⟩ ⟨9123⟩ ⟨4912⟩ ⟨3491⟩ ⟨2349⟩ ⟨5678⟩ ⟨9567⟩ ⟨8956⟩

⟨7895⟩ ⟨6789⟩

B10 =⟨1234⟩ ⟨(567) cap (89)123⟩ ⟨4 (567) cap (89)12⟩ ⟨34 (567) cap (89)1⟩ ⟨234 (567) cap (89)⟩

⟨5678⟩ ⟨9567⟩ ⟨8956⟩ ⟨7895⟩ ⟨6789⟩

B11 =⟨1234⟩ ⟨(56) cap (789)123⟩ ⟨4 (56) cap (789)12⟩ ⟨34 (56) cap (789)1⟩ ⟨234 (56) cap (789)⟩

⟨5678⟩ ⟨9567⟩ ⟨8956⟩ ⟨7895⟩ ⟨6789⟩

B12 =⟨1234⟩ ⟨4789⟩ ⟨56 (123) cap (789)⟩ ⟨123 (45) cap (789)⟩ ⟨(46) cap (789)123⟩

⟨789(45) cap (123)⟩ ⟨(46) cap (123)789⟩

B13 =⟨1234⟩ ⟨5123⟩ ⟨4512⟩ ⟨3451⟩ ⟨2345⟩ ⟨6789⟩ ⟨10678⟩ ⟨91067⟩

⟨89106⟩ ⟨78910⟩

Chapter 5

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

Z11 ⋯ Z1

Z21 ⋯ Z2

Z31 ⋯ Z3

Z41 ⋯ Z4

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ETHrarrGL(4)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

1 0 0 0 y15 ⋯ y1

0 1 0 0 y25 ⋯ y2

0 0 1 0 y35 ⋯ y3

0 0 0 1 y45 ⋯ y4

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

56 n with

ld (52)

x2ij =

is given by

yIa yJ b =1

f(y) g(y) =n

sumab=5

sumIJ=1

partyIa

2Z (56)

⟨i i + 1 j j + 1⟩)

in [89]

ABDSn = exp [

infinsumL=1

g2L (f(L)(ε)

2A(1)n (Lε) +C(L))] (57)

Γcusp = 4g2 +O(g4) (58)

ABDS-liken = ABDS

n exp [Γcusp

4Yn] 4 ∣ n

Discx2i+1j

[Discx2i+1i+p

(An)] = 0

Discx2i+1i+p

[Discx2i+1j+p+q

(An)] = 0

⎫⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎭

522 NMHV Amplitudes

form [115 116]

sums=5

sumt=s+2

Rtot =nminus2

sums=3

sumt=s+2

R1st (512)

and our results

Yif (2L)i (515)

minus 1

2([12345] + [12356] + [12367] minus [12457] minus [12567]

+ [13456] + [13467] + [14567] minus [23457] minus [23567])

times (log(⟨1234⟩⟨1789⟩⟨1278⟩⟨1349⟩)otimes ⟨3478⟩)

minus 1

2([12348] minus [12378] + [12478] minus [13478]

+ [23478] + [34567]) (log(⟨1234⟩⟨1789⟩⟨1278⟩⟨1349⟩)otimes ⟨3478⟩)

[12347] [12348] [12349] [12378] [12379] [12389]

[12478] [12479] [12489] [12789] [13478] [13479]

[13489] [13789] [14789] [23478] [23479] [23489]

[23789] [24789] [34567] [34568] [34578] [34678]

[34789] [35678] [45678]

Yangian invariants

a3ijkl = ⟨i(j j + 1)(k k + 1)(l l + 1)⟩

)(525)

condition

cup (j j + 1 k k + 15

) cup (i j j + 1 k k + 15

cup (i k k + 1 l l + 15

) cup (l l + 1 i5

) is a

uijkl =x2ikx

sumi=5

minus 1

sumi=5

minus 1

u2nminus3nminus1n)

sumi=5

minus 1

1247) +1

sumi=4

4ln(u8145) ln(u1256u3478

u2367) + cyclic (532)

nminus2

sumi=4

ln(un1ii+1)iminus1

sumj=1

sumi=4

ln(un1ii+1)iminus1

sumj=1

4ln(un1n

2n2+1)

nminus22

sumi=1

ln(uii+1i+n2i+n

Chapter 6

Graphs

symbol of [23]

C sdotZ = 0 (61)

example [27 93 94 28 30])

for each graph

source columns

C =⎛⎜⎜⎜⎝

1 0 c13 c14 c15 c16

0 1 c23 c24 c25 c26

⎞⎟⎟⎟⎠ (63)

prodαisinp

fα (64)

f0 = minus⟨1234⟩⟨2346⟩ f1 = minus

⟨2346⟩⟨2345⟩ f2 =

⟨2345⟩⟨1236⟩⟨1234⟩⟨2356⟩

f3 = minus⟨2356⟩⟨2346⟩ f4 =

⟨2346⟩⟨1256⟩⟨2456⟩⟨1236⟩ f5 = minus

⟨2456⟩⟨2356⟩

f6 =⟨2356⟩⟨1456⟩⟨3456⟩⟨1256⟩ f7 = minus

⟨3456⟩⟨2456⟩ f8 = minus

⟨2456⟩⟨1456⟩

f(b)0 = f (a)0 (1 + f (a)2 )

f(b)1 = f

1 + 1f (a)2

f(b)2 = 1

f(b)3 = f (a)3 (1 + f (a)2 )

f(b)4 = f

1 + 1f (a)2

f(c)2 = f (a)2 (1 + f (a)4 )

f(c)3 = f

1 + 1f (a)4

f(c)4 = 1

f(c)5 = f (a)5 (1 + f (a)4 )

f(c)6 = f

1 + 1f (a)4

(a) (b) (c)

⟨16⟩ ⟨12⟩

⟨23⟩

⟨34⟩⟨45⟩

⟨56⟩

⟨26⟩

⟨36⟩

⟨46⟩

ampamppp

⟨16⟩ ⟨12⟩

⟨23⟩

⟨34⟩⟨45⟩

⟨56⟩

⟨26⟩

⟨36⟩

⟨35⟩

⟨16⟩ ⟨12⟩

⟨23⟩

⟨34⟩⟨45⟩

⟨56⟩

⟨26⟩

⟨24⟩⟨46⟩ oo

(d) (e) (f)

⟨3456⟩ ⟨1456⟩

⟨1256⟩

⟨1236⟩⟨1234⟩

⟨2345⟩

⟨2456⟩

⟨2356⟩

⟨2346⟩

ampamppp

⟨3456⟩ ⟨1456⟩

⟨1256⟩

⟨1236⟩⟨1234⟩

⟨2345⟩

⟨2456⟩

⟨2356⟩

⟨1235⟩

⟨3456⟩ ⟨1456⟩

⟨1256⟩

⟨1236⟩⟨1234⟩

⟨2345⟩

⟨2456⟩

⟨1246⟩⟨2346⟩ oo

(g) (h) (i)

C = (0 1 f0f1f2f3 f0f1f2 f0f1 f0) (69)

f0 =⟨2345⟩⟨3456⟩ f1 = minus

⟨2346⟩⟨2345⟩ f2 = minus

⟨2356⟩⟨2346⟩ f3 = minus

⟨2456⟩⟨2356⟩ f4 = minus

⟨3456⟩⟨2456⟩

amplitude

f0 =⟨123⟩⟨145⟩

⟨234⟩⟨146⟩ f3 = minus⟨234⟩⟨156⟩⟨123⟩⟨456⟩

f4 = minus⟨124⟩⟨456⟩

⟨134⟩⟨156⟩

f6 = minus⟨134⟩⟨124⟩

f0 =⟨145⟩⟨456⟩ f1 = minus

⟨146⟩⟨145⟩ f2 = minus

⟨156⟩⟨146⟩ f3 = minus

⟨123⟩⟨456⟩⟨234⟩⟨156⟩

f4 = minus⟨124⟩⟨123⟩ f5 = minus

⟨234⟩⟨134⟩ f6 = minus

⟨134⟩⟨124⟩

⟨234⟩⟨156⟩ (616)

from (615)

C =⎛⎜⎜⎜⎝

1 c12 0 c14 c15 c16 c17 c18

0 c32 1 c34 c35 c36 c37 c38

⎞⎟⎟⎟⎠

a4a7 ⟨6(12)(34)(78)⟩ ⟨3478⟩

a2 ⟨4(12)(56)(78)⟩ ⟨1278⟩

a1⟨3(12)(56)(78)⟩

a3 ⟨1(34)(56)(78)⟩

a6 ⟨5(12)(34)(78)⟩

pendix 67

a9⟨1234⟩⟨5678⟩ (625)

∆2468

65 Discussion 101

65 Discussion

on Gr(mn)

65 Discussion 103

in [23]

f(b)0 = minus⟨1235⟩

⟨2356⟩ f(b)1 = minus⟨1236⟩

⟨1235⟩ f(b)2 = ⟨1234⟩⟨2356⟩

⟨2345⟩⟨1236⟩

f(b)3 = minus⟨1235⟩

⟨1234⟩ f(b)4 = ⟨2345⟩⟨1256⟩

⟨1235⟩⟨2456⟩ f(b)5 = minus⟨2456⟩

⟨2356⟩

f(b)6 = ⟨2356⟩⟨1456⟩

⟨3456⟩⟨1256⟩ f(b)7 = minus⟨3456⟩

⟨2456⟩ f(b)8 = minus⟨2456⟩

⟨1456⟩

f(c)0 = minus⟨1234⟩

⟨2346⟩ f(c)1 = minus⟨2346⟩

⟨2345⟩ f(c)2 = ⟨2345⟩⟨1246⟩

⟨1234⟩⟨2456⟩

f(c)3 = minus⟨1256⟩

⟨1246⟩ f(c)4 = ⟨2456⟩⟨1236⟩

⟨2346⟩⟨1256⟩ f(c)5 = minus⟨1246⟩

⟨1236⟩

f(c)6 = ⟨1456⟩⟨2346⟩

⟨3456⟩⟨1246⟩ f(c)7 = minus⟨3456⟩

⟨2456⟩ f(c)8 = minus⟨2456⟩

⟨1456⟩

∆1357 andradic

∆2468 with

∆1357 and

2(1 + u minus v +

u = ⟨1278⟩⟨3456⟩⟨1256⟩⟨3478⟩ v = ⟨1234⟩⟨5678⟩

⟨1256⟩⟨3478⟩ (632)

radic∆1357

⟨1256⟩⟨3478⟩ (633)

∣irarri+6

∣irarri+4

∣irarri+6

∣irarri+4

∣irarri+6

z a9 =

1 minus z1 minus z

Z isin Gr(48)gt0

Bibliography

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Abstract

Acknowledgements

Introduction












n-point MHV




6-point MHV

n-point NMHV

n-point NkMHV





Soft limits






NMHV

NNMHV

NNNMHV and Higher




One-loop Amplitudes


NMHV Amplitudes











Discussion



Acknowledgements

Contents

Abstract v

Acknowledgements xi

1 Introduction 1

22 n-point MHV 19

224 6-point MHV 24

23 n-point NMHV 25

24 n-point NkMHV 28

33 Soft limits 43

421 NMHV 60

422 N2MHV 62

65 Discussion 101

List of Figures

to ⟨k(l l + 1)(mm + 1)(k minus 1k + 1)⟩ 80

to ⟨k(l l + 1)(mm + 1)(pp + 1)⟩ 81

List of Tables

I love you all

Chapter 1

Introduction

amplitudes

calculated

(11) transforms as

az + bcz + d

) = ∣cz + d∣2∆φ∆plusmn(X z z) (13)

prodj=1int

0dωj ω

az + bcz + d

prodj=1

gluon amplitudes

microi+1 (16)

〈2345〉

〈2346〉〈2356〉〈2456〉〈3456〉

〈1234〉〈1236〉〈1256〉〈1456〉

akaprimek = prod

i∣bikgt0

aminusbiki (112)

Y (Z ∣η) = int4k

prodi=1

d log fi4

prodI=1

prodα=1

suma=1

Cαa(Z ∣η)aI) (113)

prodαisinp

fα (114)

fi = 1 (115)

coordinates

Chapter 2

Celestial Sphere

radicωiωjzij (22)

prodj=1int

0dωj ω

azj + bczj + d

prodj=1

22 n-point MHV 19

22 n-point MHV

ratio ωi

dωiωi

(ωisi

j=1 iλj

s1+iλii

⎛⎜⎝

proda=1anei

intinfin

0dωa ω

intinfin

dωiωi

(ωisi

= 2πδ(z) (29)

sumj=1

λj⎞⎠s1+iλii

⎛⎜⎝

proda=1anei

intinfin

0dωa ω

⎞⎟⎠A`1⋯`n(si ωl zj zj) (210)

δ4(ε1s1q1 +n

sumi=2

εiωiqi) =4

prodj=nminus3

U+nminus4

sumi=2

U) (212)

22 n-point MHV 21

ui =z31zi2z32zi1

for i = 45 n (216)

s1 =∣z32∣

∣z31∣ ∣z12∣ s2 =

∣z31∣∣z32∣ ∣z21∣

and si =∣z12∣

∣z1i∣ ∣zi2∣for i = 3 n (217)

to demand Uij

prodij 1lt0(Uij

12 siλ1+21

nminus4

proda=2int

0dωa ω

1lt0(Uij

ωi = siU1n

uiminus1

i = 23 n minus 4 (220)

which leads to

12siλ1+21 siλ2+2

2 siλ33 siλnn

z23z34zn1U1nδ(

sumj=1

λj) ϕ(α x)prodij

1lt0(Uij

U) (221)

22 n-point MHV 23

prodj=1

P2and and dPnminus4

Pnminus4

αn =1 + iλ1 x0 i =U1i

U1n xjminus1 i =

Ujnminus U1i

U1n x0n = minus

U1n xjminus1n =

ϕ(α x) equivc2

c1intu1ge0u3ge0

1minussuma uage0

prodj=1

Pnminus1

U1n xij =

Ujnminus4+i

x0n = minusU

U1n xin =

U1n x01 = 1

mannian

224 6-point MHV

)iλ1+1

( minusUU16

)iλ2+2

iλ3minus1

iλ4minus1

iλ5minus1

times int1

prodi=1

23 n-point NMHV 25

U16U2i+2for i = 123 (227)

23 n-point NMHV

nminus1

sumj=4

⟨1∣P2jPj+12∣3⟩3

P 22jP

⟨j + 1 j⟩[2∣P2j ∣j + 1⟩⟨j∣Pj+12∣2]

equivnminus1

sumj=4

Mj (228)

⟨34⟩⟨45⟩⋯⟨n minus 1 n⟩⟨n1⟩(⟨12⟩[24]⟨43⟩ + ⟨13⟩[34]⟨43⟩)3

(⟨23⟩[23] + ⟨24⟩[24] + ⟨34⟩[34])⟨34⟩[34]times

times ⟨54⟩([23]⟨35⟩ + [24]⟨45⟩)(⟨43⟩[32]) (229)

M4 =ω1ω4(ε2z12z24ω2+ε3z13z34ω3)3

of variables

ωi = siuiminus1

i = 2 n minus 4 (231)

M4 sim δ⎛⎝n

sumj=1

λj⎞⎠

prodni=1 siλii

z12z13z45z4ns21s

z34zn1UF(αx)prod

1lt0(Uij

U) (232)

nminus5

proda=1

nminus5

prodj=1

uiλj+1

times7

prodi=1

23 n-point NMHV 27

U xj4 =

U j = 1 n minus 5

x05 =U1nminus2

U xj5 =

U j = 1 n minus 5 (235)

x06 =U1nminus1

U xj6 =

U j = 1 n minus 5

x07 =U1n

U xj7 =

Ujn minusU1n

U j = 1 n minus 5

6- and 7-point NMHV

u1uiλ2

prodi=2

(x0i + u1x1i)αi (236)

u2uiλ2

1 uiλ32 (u1x10 + u2x20 + u1u2x30 + u2

times7

prodi=1

24 n-point NkMHV

24 n-point NkMHV 29

N2MHV amplitude

F41F23

⟨1∣P26P72P35P63∣4⟩3

P 226P

⟨76⟩[23]⟨65⟩[2∣P26∣7⟩⟨6∣P72∣2][3∣P35∣6⟩⟨5∣P63∣3]

u3uα1

1 uα22 uα3

prodi=4

times17

prodj=14

2x8j + u23x9j)αj

NkMHV amplitude

nminus4

prodl=2

dululuαl

⎛⎝

1 minusnminus4

sumj=2

uj⎞⎠

P32k (prod

(P i1)αi)

⎛⎝prodj

(Pj2)αj

⎞⎠

prodj=0

Pj(u)αjdϕ (243)

dϕ =dPj1Pj1

and and dPjnPjn

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

x00 x0m

x10 x1m

⋮ ⋱ ⋮

xn0 xnm

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

gauge fixed form

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

1 0 0 1 1 1

⋮ ⋱ minus1 ⋮ ⋮ ⋮

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ways eq (324) of [38]

prodi=1

suml=1

prodj=1

(1 minusn

sumi=1

xijui)minusβj

sumi=1

αi) sdotn

prodi=1

Γ(αi) (248)

prodi=1

mminusnminus1

suml=1

prodj=1

sumi=1

xjiui)minusαj

sumi=1

prodi=1

Γ(βi) (249)

γ minusn

sumi=1

sumj=1

βj notin Z(250)

sumi=0

xijpartϕ

sumj=0

xijpartϕ

(3) part2ϕ

k du1duk (252)

Chapter 3

prodl=1

prodj=1

sumi=1

ki)( 1

(k1+k2)2+m2+ 1

(k1+k3)2+m2+ 1

(k1+k4)2+m2)

(hj hj) =1

A4(zj zj) =( z24

z14)h12 ( z14

z13)h34

zh1+h212 zh3+h4

z14)h12 ( z14

z13)h34

zh1+h212 zh3+h4

A4(z z) (35)

z = z12z34

z13z24 z = z12z34

compute

τminus2∆4A4(0 z11τ) (37)

prodi=1

⎛⎝

sumj=1

kj⎞⎠

m2 minus 4ω1ω2∣z∣2 (38)

δ(4)⎛⎝

sumj=1

kj⎞⎠= 4τ2

ω1δ(iz minus iz)

prodi=2

ωlowast2 = ω1

2 (310)

0dω1ω

∆1minus21

prodi=2

m2 minus 4z2

zminus1ω21

= g2 (im2)2αminus2

z minus 1)α

+ zα) (312)

sumi=1

hi minus 2 (313)

constructed in [51]

2(h2+h3)A13harr24

4 (z) 0 lt z lt 1 (315)

as well as

4 (1z)

z minus 1) 0 lt z lt 1 (316)

2 (im2)2αminus2

2 sin(πα) intC

hi(z z)

2= 1 minus h12 + n

2= 1 minus h34 + n

2= h12 minus n

2= h34 minus n (318)

with n = 0123

hi(z z) (319)

4hi+hj

Γ (hij + ∆2)Γ (∆

and Ψ∆hi(z z) is

Ψ∆hi(z z) equiv

2 minus h34)Γ (∆

2 + h12)Γ (1 minus ∆2 + h34)

Ψ∆hi(z z) (322)

infinsumJ=0

4π2Ψhh

+(h12 harr h34))

33 Soft limits 43

∆ minus J2

2= 1 minus h34 + n

∆ + J2

2= h34 minus n

33 Soft limits

Single soft limits

limωkrarr0

A`k=+1n = ⟨k minus 1k + 1⟩

limωkrarr0

δ(x) = 1

2limλrarr0

limλkrarr0

An(zj zj) =1

prodj=1jnek

intinfin

0dωj ω

iλjj int

limλkrarr0

AJk=+1n = 1

zkminus1k+1

limλkrarr0

AJk=minus1n = 1

zkminus1k+1

Double soft limits

33 Soft limits 45

⟨n∣1 + 2∣3] ( [13]3⟨n3⟩[12][23]s123

+ ⟨n2⟩3[n3]⟨n1⟩⟨12⟩sn12

we will write

An(zj zj) =n

prodj=1jnekl

intinfin

0dωj ω

iλjj int

0dr int

DS(kJk lJl) = 1

r=0]εrarr0

2(iλ1 + iλ2)ε21

zn1z23( 1

zn3z2n

z12z2n+ 1

z3nz31

z12z31) (338)

sumi=1

4ω1⋯ω6

ω21ω

ω23ω

26(ω4z46z34+ω5z56z35)3

(ω3ω4z34z34+ω3ω5z35z35+ω4ω5z45z45)

z12z16z34z45 (ω3z23z35 + ω4z24z45)

⎤⎥⎥⎥⎥⎥⎦

2(iλ3 + iλ4)ε21

z23z45( 1

z25z41

z34z42+ 1

z52z53

25z216z25z61

(z15z61

z25z26)iλ2minus1

(z12z16

z25z56)iλ5+1

(z15z12

z56z26)iλ6+1

⟨2∣3 + 4∣5] ( [35]3⟨25⟩[34][45]s345

+ ⟨24⟩3[25]⟨23⟩⟨34⟩s234

) (342)

intCd∆ Ψhh

hihi(z z)⟩ (343)

(zz)2G(z z)F (z z) (344)

given by

F+(z z) =1

zh34 zh342F1 (

1 minus h + h34 h + h34

1 + h12 + h341

z) 2F1 (

1 minus h + h34 h + h34

1 + h12 + h341

z) (346)

z) 2F1 (

⟨Ψhh

hihi(z z)⟩ = (2π)2

⟨A4(z z)Ψhh

hihi(z z) (349)

2F1(a b c z) =Γ(c)

obtain

⟨A4(z z)Ψhh

⎡⎢⎢⎢⎢⎢⎣

⎛⎜⎝

⎤⎥⎥⎥⎥⎥⎥⎦

sin(2πJ)2πJ

= 1 J = 0

0 J ne 0 (353)

hihi∣J=0

which simplifies to

2 minus h34)Γ(2 minus∆)Γ(∆) (354)

⟨A4(z z)Ψhh

Chapter 4

⟨1 2 6 7⟩⟨2 3 4 5⟩ minus ⟨1 2 4 5⟩⟨2 3 6 7⟩ (42)

cluster coordinate

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

1 0 0 0 y15 ⋯ y1

0 1 0 0 y25 ⋯ y2

0 0 1 0 y35 ⋯ y3

0 0 0 1 y45 ⋯ y4

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

yIa yJ b =1

f(y) g(y) =n

sumab=1

sumIJ=1

partyIa

2Z foralli j (46)

entries

n5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Total

1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1

2 0 1 2 5 4 1 0 0 0 0 0 0 0 0 0 0 13

3 0 0 1 6 54 177 298 274 134 30 3 0 0 0 0 0 977

4 0 0 0 1 13 263 1988 7862 18532 28204 28377 18925 8034 2047 270 17 114533

421 NMHV

p1 p5 = ⟨1 2 3 4⟩ ⟨2 3 4 5⟩ ⟨3 4 5 1⟩ ⟨4 5 1 2⟩ ⟨5 1 2 3⟩ (48)

5minusy35minusy4

5 (49)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0

0 0 12

0 minus12 0 1

0 minus12 minus1

2 0 12

0 minus12 minus1

2 minus12 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

422 N2MHV

1 = [12 (23) cap (456) (234) cap (56)6][23456]

2 = [12 (34) cap (567) (345) cap (67)7][34567]

3 = [123 (345) cap (67)7][34567]

4 = [123 (456) cap (78)8][45678]

5 = [12348][45678]

6 = [123 (45) cap (678)8][45678]

7 = [123 (45) cap (678) (456) cap (78)][45678] (411)

8 = [1234 (456) cap (78)][45678]

9 = [12349][56789]

10 = [1234 (567) cap (89)][56789]

11 = [1234 (56) cap (789)][56789]

13 = [12345][678910]

ϕ = ⟨4 5 (123) cap (789)⟩⟨4 6 (123) cap (789)⟩⟨1 2 3 4⟩⟨4 7 8 9⟩⟨5 6 (123) cap (789)⟩ (413)

Ω(2)1

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 1 1 0 0 0 12

minus 12

minus1

0 0 0 0 minus 12

0 minus 12

minus 12

minus1

0 minus 12

minus 12

minus1

0 minus 12

0 minus1 minus1

minus 12

minus1

0 0 0 0 minus 12

0 minus 12

0 0 0 0

minus 12

0 minus 12

0 0 0 minus 12

minus 12

0 0 12

0 minus 12

1 1 1 1 1 0 0 12

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 1 0 0 0 0 minus1 minus 12

minus 12

0 0 0 minus 12

minus 12

0 minus 12

minus 12

minus1 0 0 minus 32

minus 32

0 minus 12

minus 32

minus 12

0 minus 12

0 minus1 minus 12

minus 12

0 minus1 minus 12

minus 12

0 0 0 minus 12

minus 12

0 minus 12

minus 12

0 0 12

0 minus 12

1 1 12

0 0 0 0 minus 12

0 0 0 12

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)3

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 12

0 0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

minus 32

minus 12

1 0 minus 12

0 minus1 0 minus 12

0 0 0 minus 12

minus 12

0 minus 12

minus 12

0 0 12

0 minus 12

1 1 12

0 0 12

0 0 0 0 0 0 minus 12

0 0 0 12

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)4

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 minus1 minus1 0

0 0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus1 minus1 0

0 minus 12

0 minus1 minus1 0

0 0 minus 12

minus 12

0 minus 12

minus 12

0 0 0 12

0 minus 12

minus 12

1 1 1 12

0 minus 12

1 1 1 12

0 0 0 0 0 0 0 0 0 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)5

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 0 0 0

0 0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 0 0 0

0 minus 12

0 0 0 0

0 0 minus 12

minus 12

0 minus 12

minus 12

0 0 0 12

0 minus 12

minus 12

0 0 0 12

0 minus 12

0 0 0 12

0 0 0 0 0 0 0 0 0 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)6

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 0 minus1 0

0 0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 0 minus1 0

0 minus 12

0 0 minus1 0

0 0 minus 12

minus 12

0 minus 12

minus 12

0 0 0 12

0 minus 12

minus 12

0 0 0 12

0 minus 12

1 1 1 12

0 0 0 0 0 0 0 0 0 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)7

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 0 minus1 0

0 1 0 minus 12

minus 12

minus 32

0 1 12

0 minus 12

minus 12

minus 32

0 1 12

minus 12

minus 32

0 0 minus 12

minus 12

0 minus 12

minus 12

0 0 minus 12

minus 12

0 minus 12

minus 12

0 1 12

0 minus 12

1 1 32

0 0 0 0 0 0 0 0 0 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)8

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 0 0 0

0 1 0 minus 12

minus 12

0 1 12

0 minus 12

minus 12

0 1 12

minus 12

0 0 minus 12

minus 12

0 minus 12

minus 12

0 0 minus 12

minus 12

0 minus 12

minus 12

0 1 12

0 minus 12

0 1 12

0 0 0 0 0 0 0 0 0 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)9

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 0 0 0

0 0 minus 12

minus 12

0 0 0 0

0 minus 12

minus 12

0 0 0 0

0 minus 12

0 0 0 0

0 minus 12

minus 12

0 minus 12

minus 12

0 0 0 0 0 12

0 minus 12

minus 12

0 0 0 0 0 12

0 minus 12

minus 12

0 0 0 0 0 12

0 minus 12

0 0 0 0 0 12

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)10

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 0 0 0

0 0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)11

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 0 0 0

0 0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)12

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 1 1 1 32

0 minus1 0 minus 12

minus 12

minus 32

minus 12

0 minus1 12

0 minus 12

minus 12

0 minus 12

0 minus1 12

0 minus 12

minus 12

0 minus 32

0 minus 12

minus 12

2 2 2 12

0 minus 32

0 minus 12

2 2 2 12

0 minus 32

0 2 2 2 12

0 minus 32

minus 12

0 minus 32

minus 12

0 minus 12

minus 12

0 minus 32

minus 12

0 minus 12

minus 12

0 minus1 12

0 minus 12

minus 12

0 minus 12

0 minus1 12

0 minus 12

minus 12

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Ω(2)13

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 0 0 0 0 0

0 0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

minus 12

0 minus 12

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

B1 =⟨12 (23) cap (456) (234) cap (56)⟩ ⟨612 (23) cap (456)⟩ ⟨(234) cap (56)612⟩

⟨(23) cap (456) (234) cap (56)61⟩ ⟨2 (23) cap (456) (234) cap (56)6⟩ ⟨2345⟩ ⟨6234⟩ ⟨5623⟩

⟨4562⟩ ⟨3456⟩

(345) cap (67)71⟩ ⟨2 (34) cap (567) (345) cap (67)7⟩ ⟨3456⟩ ⟨7345⟩ ⟨6734⟩ ⟨5673⟩

⟨4567⟩

B3 =⟨123 (345) cap (67)⟩ ⟨7123⟩ ⟨(345) cap (67)712⟩ ⟨3 (345) cap (67)71⟩ ⟨23 (345) cap (67)7⟩

⟨3456⟩ ⟨7345⟩ ⟨6734⟩ ⟨5673⟩ ⟨4567⟩

B4 =⟨123 (456) cap (78)⟩ ⟨8123⟩ ⟨(456) cap (78)812⟩ ⟨3 (456) cap (78)81⟩ ⟨23 (456) cap (78)8⟩

⟨4567⟩ ⟨8456⟩ ⟨7845⟩ ⟨6784⟩⟨5678⟩

B5 =⟨1234⟩ ⟨8123⟩ ⟨4812⟩ ⟨3481⟩ ⟨2348⟩ ⟨4567⟩ ⟨8456⟩ ⟨7845⟩ ⟨6784⟩

⟨5678⟩

B6 =⟨123 (45) cap (678)⟩ ⟨8123⟩ ⟨(45) cap (678)812⟩ ⟨3 (45) cap (678)81⟩ ⟨23 (45) cap (678)8⟩

⟨4567⟩ ⟨8456⟩ ⟨7845⟩ ⟨6784⟩⟨5678⟩

B7 =⟨123 (45) cap (678)⟩ ⟨(456) cap (78)123⟩ ⟨(45) cap (678) (456) cap (78)12⟩

⟨3 (45) cap (678) (456) cap (78)1⟩ ⟨23 (45) cap (678) (456) cap (78)⟩ ⟨4567⟩ ⟨8456⟩ ⟨7845⟩

⟨6784⟩⟨5678⟩

B8 =⟨1234⟩ ⟨(456) cap (78)123⟩ ⟨4 (456) cap (78)12⟩ ⟨34 (456) cap (78)1⟩ ⟨234 (456) cap (78)⟩

⟨4567⟩ ⟨8456⟩ ⟨7845⟩ ⟨6784⟩⟨5678⟩

B9 =⟨1234⟩ ⟨9123⟩ ⟨4912⟩ ⟨3491⟩ ⟨2349⟩ ⟨5678⟩ ⟨9567⟩ ⟨8956⟩

⟨7895⟩ ⟨6789⟩

B10 =⟨1234⟩ ⟨(567) cap (89)123⟩ ⟨4 (567) cap (89)12⟩ ⟨34 (567) cap (89)1⟩ ⟨234 (567) cap (89)⟩

⟨5678⟩ ⟨9567⟩ ⟨8956⟩ ⟨7895⟩ ⟨6789⟩

B11 =⟨1234⟩ ⟨(56) cap (789)123⟩ ⟨4 (56) cap (789)12⟩ ⟨34 (56) cap (789)1⟩ ⟨234 (56) cap (789)⟩

⟨5678⟩ ⟨9567⟩ ⟨8956⟩ ⟨7895⟩ ⟨6789⟩

B12 =⟨1234⟩ ⟨4789⟩ ⟨56 (123) cap (789)⟩ ⟨123 (45) cap (789)⟩ ⟨(46) cap (789)123⟩

⟨789(45) cap (123)⟩ ⟨(46) cap (123)789⟩

B13 =⟨1234⟩ ⟨5123⟩ ⟨4512⟩ ⟨3451⟩ ⟨2345⟩ ⟨6789⟩ ⟨10678⟩ ⟨91067⟩

⟨89106⟩ ⟨78910⟩

Chapter 5

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

Z11 ⋯ Z1

Z21 ⋯ Z2

Z31 ⋯ Z3

Z41 ⋯ Z4

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ETHrarrGL(4)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

1 0 0 0 y15 ⋯ y1

0 1 0 0 y25 ⋯ y2

0 0 1 0 y35 ⋯ y3

0 0 0 1 y45 ⋯ y4

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

56 n with

ld (52)

x2ij =

is given by

yIa yJ b =1

f(y) g(y) =n

sumab=5

sumIJ=1

partyIa

2Z (56)

⟨i i + 1 j j + 1⟩)

in [89]

ABDSn = exp [

infinsumL=1

g2L (f(L)(ε)

2A(1)n (Lε) +C(L))] (57)

Γcusp = 4g2 +O(g4) (58)

ABDS-liken = ABDS

n exp [Γcusp

4Yn] 4 ∣ n

Discx2i+1j

[Discx2i+1i+p

(An)] = 0

Discx2i+1i+p

[Discx2i+1j+p+q

(An)] = 0

⎫⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎭

522 NMHV Amplitudes

form [115 116]

sums=5

sumt=s+2

Rtot =nminus2

sums=3

sumt=s+2

R1st (512)

and our results

Yif (2L)i (515)

minus 1

2([12345] + [12356] + [12367] minus [12457] minus [12567]

+ [13456] + [13467] + [14567] minus [23457] minus [23567])

times (log(⟨1234⟩⟨1789⟩⟨1278⟩⟨1349⟩)otimes ⟨3478⟩)

minus 1

2([12348] minus [12378] + [12478] minus [13478]

+ [23478] + [34567]) (log(⟨1234⟩⟨1789⟩⟨1278⟩⟨1349⟩)otimes ⟨3478⟩)

[12347] [12348] [12349] [12378] [12379] [12389]

[12478] [12479] [12489] [12789] [13478] [13479]

[13489] [13789] [14789] [23478] [23479] [23489]

[23789] [24789] [34567] [34568] [34578] [34678]

[34789] [35678] [45678]

Yangian invariants

a3ijkl = ⟨i(j j + 1)(k k + 1)(l l + 1)⟩

)(525)

condition

cup (j j + 1 k k + 15

) cup (i j j + 1 k k + 15

cup (i k k + 1 l l + 15

) cup (l l + 1 i5

) is a

uijkl =x2ikx

sumi=5

minus 1

sumi=5

minus 1

u2nminus3nminus1n)

sumi=5

minus 1

1247) +1

sumi=4

4ln(u8145) ln(u1256u3478

u2367) + cyclic (532)

nminus2

sumi=4

ln(un1ii+1)iminus1

sumj=1

sumi=4

ln(un1ii+1)iminus1

sumj=1

4ln(un1n

2n2+1)

nminus22

sumi=1

ln(uii+1i+n2i+n

Chapter 6

Graphs

symbol of [23]

C sdotZ = 0 (61)

example [27 93 94 28 30])

for each graph

source columns

C =⎛⎜⎜⎜⎝

1 0 c13 c14 c15 c16

0 1 c23 c24 c25 c26

⎞⎟⎟⎟⎠ (63)

prodαisinp

fα (64)

f0 = minus⟨1234⟩⟨2346⟩ f1 = minus

⟨2346⟩⟨2345⟩ f2 =

⟨2345⟩⟨1236⟩⟨1234⟩⟨2356⟩

f3 = minus⟨2356⟩⟨2346⟩ f4 =

⟨2346⟩⟨1256⟩⟨2456⟩⟨1236⟩ f5 = minus

⟨2456⟩⟨2356⟩

f6 =⟨2356⟩⟨1456⟩⟨3456⟩⟨1256⟩ f7 = minus

⟨3456⟩⟨2456⟩ f8 = minus

⟨2456⟩⟨1456⟩

f(b)0 = f (a)0 (1 + f (a)2 )

f(b)1 = f

1 + 1f (a)2

f(b)2 = 1

f(b)3 = f (a)3 (1 + f (a)2 )

f(b)4 = f

1 + 1f (a)2

f(c)2 = f (a)2 (1 + f (a)4 )

f(c)3 = f

1 + 1f (a)4

f(c)4 = 1

f(c)5 = f (a)5 (1 + f (a)4 )

f(c)6 = f

1 + 1f (a)4

(a) (b) (c)

⟨16⟩ ⟨12⟩

⟨23⟩

⟨34⟩⟨45⟩

⟨56⟩

⟨26⟩

⟨36⟩

⟨46⟩

ampamppp

⟨16⟩ ⟨12⟩

⟨23⟩

⟨34⟩⟨45⟩

⟨56⟩

⟨26⟩

⟨36⟩

⟨35⟩

⟨16⟩ ⟨12⟩

⟨23⟩

⟨34⟩⟨45⟩

⟨56⟩

⟨26⟩

⟨24⟩⟨46⟩ oo

(d) (e) (f)

⟨3456⟩ ⟨1456⟩

⟨1256⟩

⟨1236⟩⟨1234⟩

⟨2345⟩

⟨2456⟩

⟨2356⟩

⟨2346⟩

ampamppp

⟨3456⟩ ⟨1456⟩

⟨1256⟩

⟨1236⟩⟨1234⟩

⟨2345⟩

⟨2456⟩

⟨2356⟩

⟨1235⟩

⟨3456⟩ ⟨1456⟩

⟨1256⟩

⟨1236⟩⟨1234⟩

⟨2345⟩

⟨2456⟩

⟨1246⟩⟨2346⟩ oo

(g) (h) (i)

C = (0 1 f0f1f2f3 f0f1f2 f0f1 f0) (69)

f0 =⟨2345⟩⟨3456⟩ f1 = minus

⟨2346⟩⟨2345⟩ f2 = minus

⟨2356⟩⟨2346⟩ f3 = minus

⟨2456⟩⟨2356⟩ f4 = minus

⟨3456⟩⟨2456⟩

amplitude

f0 =⟨123⟩⟨145⟩

⟨234⟩⟨146⟩ f3 = minus⟨234⟩⟨156⟩⟨123⟩⟨456⟩

f4 = minus⟨124⟩⟨456⟩

⟨134⟩⟨156⟩

f6 = minus⟨134⟩⟨124⟩

f0 =⟨145⟩⟨456⟩ f1 = minus

⟨146⟩⟨145⟩ f2 = minus

⟨156⟩⟨146⟩ f3 = minus

⟨123⟩⟨456⟩⟨234⟩⟨156⟩

f4 = minus⟨124⟩⟨123⟩ f5 = minus

⟨234⟩⟨134⟩ f6 = minus

⟨134⟩⟨124⟩

⟨234⟩⟨156⟩ (616)

from (615)

C =⎛⎜⎜⎜⎝

1 c12 0 c14 c15 c16 c17 c18

0 c32 1 c34 c35 c36 c37 c38

⎞⎟⎟⎟⎠

a4a7 ⟨6(12)(34)(78)⟩ ⟨3478⟩

a2 ⟨4(12)(56)(78)⟩ ⟨1278⟩

a1⟨3(12)(56)(78)⟩

a3 ⟨1(34)(56)(78)⟩

a6 ⟨5(12)(34)(78)⟩

pendix 67

a9⟨1234⟩⟨5678⟩ (625)

∆2468

65 Discussion 101

65 Discussion

on Gr(mn)

65 Discussion 103

in [23]

f(b)0 = minus⟨1235⟩

⟨2356⟩ f(b)1 = minus⟨1236⟩

⟨1235⟩ f(b)2 = ⟨1234⟩⟨2356⟩

⟨2345⟩⟨1236⟩

f(b)3 = minus⟨1235⟩

⟨1234⟩ f(b)4 = ⟨2345⟩⟨1256⟩

⟨1235⟩⟨2456⟩ f(b)5 = minus⟨2456⟩

⟨2356⟩

f(b)6 = ⟨2356⟩⟨1456⟩

⟨3456⟩⟨1256⟩ f(b)7 = minus⟨3456⟩

⟨2456⟩ f(b)8 = minus⟨2456⟩

⟨1456⟩

f(c)0 = minus⟨1234⟩

⟨2346⟩ f(c)1 = minus⟨2346⟩

⟨2345⟩ f(c)2 = ⟨2345⟩⟨1246⟩

⟨1234⟩⟨2456⟩

f(c)3 = minus⟨1256⟩

⟨1246⟩ f(c)4 = ⟨2456⟩⟨1236⟩

⟨2346⟩⟨1256⟩ f(c)5 = minus⟨1246⟩

⟨1236⟩

f(c)6 = ⟨1456⟩⟨2346⟩

⟨3456⟩⟨1246⟩ f(c)7 = minus⟨3456⟩

⟨2456⟩ f(c)8 = minus⟨2456⟩

⟨1456⟩

∆1357 andradic

∆2468 with

∆1357 and

2(1 + u minus v +

u = ⟨1278⟩⟨3456⟩⟨1256⟩⟨3478⟩ v = ⟨1234⟩⟨5678⟩

⟨1256⟩⟨3478⟩ (632)

radic∆1357

⟨1256⟩⟨3478⟩ (633)

∣irarri+6

∣irarri+4

∣irarri+6

∣irarri+4

∣irarri+6

z a9 =

1 minus z1 minus z

Z isin Gr(48)gt0

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Abstract

Acknowledgements

Introduction












n-point MHV




6-point MHV

n-point NMHV

n-point NkMHV





Soft limits






NMHV

NNMHV

NNNMHV and Higher




One-loop Amplitudes


NMHV Amplitudes











Discussion



Documents

Celestial Amplitudes, Cluster Adjacency, and Symbol Alphabets