New Exciting Approaches to Scattering Amplitudes

Henriette Elvang University of Michigan

Aspen Center for Physics Colloquium July 16, 2015 In the news… #10 Using only pen and paper, he [Arkani-Hamed & Trnka] discovered a new kind of geometric shape called an amplituhedron […] The shape’s dimensions – length width, height and other parameters […] – represent information about the colliding particles […]

Using only pen and paper, he [Arkani-Hamed & Trnka] discovered a new kind of geometric shape called an amplituhedron […] The shape’s dimensions – length width, height and other parameters […] – represent information about the colliding particles […]

“We’ve known for decades that spacetime is doomed,” says Arkani-Hamed.

Amplituhedrons?

Spacetime doomed??

What in the world is this all about??? The story begins with scattering amplitudes in particle physics 1 Introduction

1 Introduction

2 pi =0

In a traditional quantum ﬁeld theory (QFT) course, you learn to extract Feynman rules from a Lagrangian and use them to calculate a scattering amplitude A as a sum of Feynman dia- Scattering Amplitudes grams organized perturbatively in the loop-expansion. From the amplitude you calculate the Scatteringdi↵erential amplitudes cross-section, d A 2, which — if needed — includes a suitable spin-sum average. encode the processes of d⌦ / | | elementaryFinally particle the cross-sectioninteractions can be found by integration of d/d⌦ over angles, with appropriate forsymmetry example factors included for identical ﬁnal-state particles. The quantities and d/d⌦ are electrons, photons, quarks,… the observables of interest for particle physics experiments, but the input for computing them Comptonare thescattering:gauge invariant on-shell scattering amplitudes A. These on-shell amplitudes A are the electronsubject + photon of → this electron review. + photon Annihilation: electronExamples + positron of→ photon processes + photon you have likely encountered in QFT are Mathematical language: quantum field theory Compton scattering e + e + , !

Møller scattering e + e e + e , (1.1) ! + + Bhabha scattering e + e e + e , ! and perhaps also 2 2 gluon scattering ! g + g g + g. (1.2) ! For instance, starting from the Quantum Electrodynamics (QED) Lagrangian you may have calculated the tree-level di↵erential cross-section for Bhabha-scattering. It is typical for such a calculation that the starting point — the Lagrangian in its most compact form — is not too terribly complicated. And the ﬁnal result can be rather compact and simple too. But the intermediate stages of the calculation often explode in an inferno of indices, contracted up-and-down and in all directions — providing little insight of the physics and hardly any hint of simplicity. Thus, while you think back at your QFT course as a class in which (hopefully!) you did a lot of long character-building calculations, you will also note that you were probably never asked to use Feynman diagrams to calculate processes that involved more than four or ﬁve particles, even at tree level: for example, e + e+ e + e+ + or g + g g + g + g. Why not? ! ! Well, one reason is that the number of Feynman diagrams tends to grow fast with the number of particles involved: for gluon scattering at tree level we have

g + g g + g 4 diagrams ! g + g g + g + g 25 diagrams (1.3) ! g + g g + g + g + g 220 diagrams ! 2 Particle Physics Experiments

(gedanken)! How to calculate scattering amplitudes?

Interaction is ! Governed by! perturbation on! Expansion in ! “free theory” ! Small coupling! !

mathematical diagram expressions Feynman rules + + Quantum Electrodynamics (QED) e + e µ + µ ! µ

2 ✓ + 1 e e e ↵ = 4⇡✏0 ~c

µ+ e µ photon tree just one A4 = Feynman diagram e+ µ+

2 d 2 ECM mµ ↵ 2 3 A4 2 (1 + cos ✓)+O(↵ ) d⌦ / ! 4ECM The point: answer depends on energies and scattering angles as well as a dimensionless small coupling, here the ﬁne structure constant ↵ + + Quantum Electrodynamics (QED) e + e µ + µ ! µ

✓ + e e

µ+ e µ photon tree just one A4 = Feynman diagram e+ µ+

2 d 2 ECM mµ ↵ 2 3 A4 2 (1 + cos ✓)+O(↵ ) d⌦ / ! 4ECM The point: answer depends on energies and scattering angles as well as a dimensionless small coupling, here the ﬁne structure constant ↵ Strong force Quantum Chromodynamics (QCD) 9 Introduction Force carrier: gluons!

Unlike photons, gluons interact with each other… so we can of particles involved: for gluonstudy gluon scattering scattering! at tree-level in quantum chromodynamics (QCD) we have !!!!!Relevant for the LHC.! g + g g + g requires four4 diagrams diagrams at tree level ! g + g g + g + g (leading order25 diagramsin perturbation theory) (1.18) ! g + g g + g + g + g 220 diagrams ! and for g + g 8g one needs more than one million diagrams [1]. Another very important point ! is that the mathematical expression for each+ diagram becomes+ + significantly more complicated as the number of external particles grows. So the reason students are not asked to calculate multi-gluon processes from Feynman diagrams is that it would be awful, un-insightful, and in many cases impossible.2 There are tricks for simplifying the calculations; one is to use the spinor helicity formalism as indicated in our example for the high-energy limit of the tree-level process + e + e + . However, for a multi-gluon process even this does not directly provide a way ! to handle the growing number of increasingly complicated Feynman diagram. Other methods are needed and you will learn much more about them in this book. Thus, although visually appealing and seemingly intuitive, Feynman diagrams are not a partic- ularly physical approach to study amplitudes in gauge theories. The underlying reason is that the Feynman rules are non-unique: generally, individual Feynman diagrams are not physical observables. In theories, such as QED or QCD, that have gauge redundancy in their Lagrangian descriptions, we are required to fix the gauge in order to extract the Feynman rules from the Lagrangian. So the Feynman rules depend on the choice of gauge and therefore only gauge invariant sums of diagrams are physically sensible. The on-shell amplitude is at each loop-order such a gauge-invariant sum of Feynman diagrams. Furthermore, field redefinitions in the Lagrangian change the Feynman rules, but not the physics, hence the amplitudes are invariant. So field redefinitions and gauge choices can ‘move’ the physical information between the diagrams in the Feynman expression for the amplitude and this can hugely obscure the appearance of its physical properties. It turns out that despite the complications of the Feynman diagrams, the on-shell amplitudes for processes such as multi-gluon scattering g + g g + g + ...+ g can actually be written as ! remarkably simple expressions. This raises the questions: “why are the on-shell amplitudes so simple?” and “isn’t there a better way to calculate amplitudes?”. These are questions that have been explored in recent years and a lot of progress has been made towards improving calculational techniques and gaining insight into the underlying mathematical structure. What do we mean by ‘mathematical structure’ of amplitudes? At the simplest level, this means 2 the analytic structure. For example, the amplitude (1.10) has a pole at (p1 +p2) = 0 — this pole shows that a physical massless scalar particle is being exchanged in the process. Tree amplitudes are rational functions of the kinematic invariants, and understanding their pole structure is key to the derivation of on-shell recursive methods that provide a very ecient alternative to Feynman diagrams. These recursive methods allow one to construct on-shell n-particle tree amplitudes from input of on-shell amplitudes with fewer particles. The power of this approach is that gauge redundancy is eliminated since all input is manifestly on-shell and gauge invariant.

9 Unlike photons, gluons interact withIntroduction each other… so we can of particles involved: for gluonstudy gluon scattering scattering! at tree-level in quantum chromodynamics (QCD) we have !!!!!Relevant for the LHC.! g + g g + g requires four4 diagrams diagrams at tree level of particles involved: for gluon! scattering at tree-level in quantum chromodynamics (QCD) we g + g g + g + g (leading order25 diagramsin perturbation theory) (1.18) have ! g + g g + g + g + g 220 diagrams How many! gof +youg have usedg + Feynmang diagrams to calculate4 diagrams and for g + g 8g one needs more than one! million diagrams [1]. Another very important point ! g + g g + g + g 25 diagrams (1.18) is that the mathematical expression for each! diagram becomes significantly more complicated or g + g g + g + g + g ???220 diagrams as the number of external particles grows.! So the reason students are not asked to calculate multi-gluonand for processesg + g from8g one Feynman needs morediagrams than is one that million it would diagrams be awful, [1 un-insightful,]. Another very and in important point ! 2 manyis cases that impossible. the mathematicalThere are expression tricks for simplifying for each the diagram calculations; becomes one is significantly to use the spinor more complicated helicityas formalism the number as indicated of external in our particles example grows. for the high-energy So the reason limit students of the tree-level are not process asked to calculate + e + e + . However, for a multi-gluon process even this does not directly provide a way multi-gluon! processes from Feynman diagrams is that it would be awful, un-insightful, and in to handle the growing number of increasingly complicated Feynman diagram. Other methods are many cases impossible.2 There are tricks for simplifying the calculations; one is to use the spinor needed and you will learn much more about them in this book. helicity formalism as indicated in our example for the high-energy limit of the tree-level process Thus, although+ visually appealing and seemingly intuitive, Feynman diagrams are not a partic- e + e + . However, for a multi-gluon process even this does not directly provide a way ! ularlyto physical handle approach the growing to study number amplitudes of increasingly in gauge complicated theories. The Feynman underlying diagram. reason is Other that methods are the Feynman rules are non-unique: generally, individual Feynman diagrams are not physical ob- needed and you will learn much more about them in this book. servables. In theories, such as QED or QCD, that have gauge redundancy in their Lagrangian descriptions,Thus, although we are required visually to appealing fix the gauge and in seemingly order to extract intuitive, the Feynman Feynman rules diagrams from the are not a partic- Lagrangian.ularly physical So the Feynman approach rules to depend study on amplitudes the choice of in gauge gauge and theories. therefore Theonly gauge underlying invari- reason is that ant sums of diagrams are physically sensible. The on-shell amplitude is at each loop-order such the Feynman rules are non-unique: generally, individual Feynman diagrams are not physical ob- a gauge-invariant sum of Feynman diagrams. Furthermore, field redefinitions in the Lagrangian servables. In theories, such as QED or QCD, that have gauge redundancy in their Lagrangian change the Feynman rules, but not the physics, hence the amplitudes are invariant. So field re- definitionsdescriptions, and gauge we choices are required can ‘move’ to the fix physicalthe gauge information in order between to extract the diagramsthe Feynman in the rules from the FeynmanLagrangian. expression So for the the Feynman amplitude rules and this depend can hugely on the obscure choice the of gauge appearance and therefore of its physical only gauge invari- properties.ant sums of diagrams are physically sensible. The on-shell amplitude is at each loop-order such a gauge-invariant sum of Feynman diagrams. Furthermore, field redefinitions in the Lagrangian It turns out that despite the complications of the Feynman diagrams, the on-shell amplitudes for processeschange the such Feynman as multi-gluon rules, scattering but notg the+ g physics,g + g hence+ ...+ theg can amplitudes actually be are written invariant. as So field re- ! remarkablydefinitions simple and expressions. gauge choices This raises can ‘move’ the questions: the physical “why are information the on-shell between amplitudes the so diagrams in the simple?”Feynman and “isn’t expression there a better for the way amplitude to calculate and amplitudes?”. this can hugely These obscure are questions the appearance that have of its physical beenproperties. explored in recent years and a lot of progress has been made towards improving calculational techniques and gaining insight into the underlying mathematical structure. It turns out that despite the complications of the Feynman diagrams, the on-shell amplitudes Whatfor do processes we mean by such ‘mathematical as multi-gluon structure’ scattering of amplitudes?g + g Atg the+ g simplest+ ...+ level,g can this actually means be written as ! 2 the analyticremarkably structure. simple For expressions.example, the amplitude This raises (1.10 the) has questions: a pole at ( “whyp1 +p2) are= the 0 — on-shell this pole amplitudes so shows that a physical massless scalar particle is being exchanged in the process. Tree amplitudes simple?” and “isn’t there a better way to calculate amplitudes?”. These are questions that have are rational functions of the kinematic invariants, and understanding their pole structure is key to the derivationbeen explored of on-shell in recent recursive years methods and a lot that of provide progress a very has e beencient made alternative towards to improving Feynman calculational diagrams.techniques These and recursive gaining methods insight allow into one the to underlying construct on-shell mathematicaln-particle structure. tree amplitudes fromWhat input of do on-shell we mean amplitudes by ‘mathematical with fewer particles. structure’ The of power amplitudes? of this approach At the is simplest that gauge level, this means redundancy is eliminated since all input is manifestly on-shell and gauge invariant. 2 the analytic structure. For example, the amplitude (1.10) has a pole at (p1 +p2) = 0 — this pole 2 Usingshows computers that to a do physical the calculation massless can of scalar course particle be very helpful, is being but not exchanged in all cases. in Sometimes the process. numerical Tree amplitudes evaluation of Feynman diagrams is simply so slow that it is not realistic to do. Moreover, given that there are polesare that rational can cancel functions between diagrams, of the kinematic big numerical invariants, errors can arise and in understanding such evaluations. Therefore their pole compact structure is key to analyticthe derivation expressions for of the on-shell amplitudes recursive are very useful methods in practical that applications. provide a very ecient alternative to Feynman diagrams. These recursive methods allow one to construct on-shell n-particle tree amplitudes from input of on-shell amplitudes with fewer particles. The power of this approach is that gauge redundancy is eliminated since all input is manifestly on-shell and gauge invariant.

9 Unlike photons, gluons interact withIntroduction each other… so we can of particles involved: for gluonstudy gluon scattering scattering! at tree-level in quantum chromodynamics (QCD) we have !!!!!Relevant for the LHC.! g + g g + g requires four4 diagrams diagrams at tree level of particles involved: for gluon! scattering at tree-level in quantum chromodynamics (QCD) we g + g g + g + g (leading order25 diagramsin perturbation theory) (1.18) have ! g + g g + g + g + g 220 diagrams How many! gof +youg have usedg + Feynmang diagrams to calculate4 diagrams and for g + g 8g one needs more than one! million diagrams [1]. Another very important point ! g + g g + g + g 25 diagrams (1.18) is that the mathematical expression for each! diagram becomes significantly more complicated or g + g g + g + g + g ???220 diagrams as the number of external particles grows.! So the reason students are not asked to calculate multi-gluonand for processesg + g from8g one Feynman needs morediagrams than is one thatwith million it would your diagrams bebare awful, hands??? [1 un-insightful,]. Another! very and in important point ! 2 manyis cases that impossible. the mathematicalThere are expression tricks for simplifying for each the diagram calculations; becomes one is significantly to use the spinor more complicated helicityas formalism the number as indicated of external in our particles example grows. for the high-energy So the reason limit students of the tree-level are not process asked to calculate + e + e + . However, for a multi-gluon process even this does not directly provide a way multi-gluon! processes from Feynman diagrams is that it would be awful, un-insightful, and in to handle the growing number of increasingly complicated Feynman diagram. Other methods are many cases impossible.2 There are tricks for simplifying the calculations; one is to use the spinor needed and you will learn much more about them in this book. helicity formalism as indicated in our example for the high-energy limit of the tree-level process Thus, although+ visually appealing and seemingly intuitive, Feynman diagrams are not a partic- e + e + . However, for a multi-gluon process even this does not directly provide a way ! ularlyto physical handle approach the growing to study number amplitudes of increasingly in gauge complicated theories. The Feynman underlying diagram. reason is Other that methods are the Feynman rules are non-unique: generally, individual Feynman diagrams are not physical ob- needed and you will learn much more about them in this book. servables. In theories, such as QED or QCD, that have gauge redundancy in their Lagrangian descriptions,Thus, although we are required visually to appealing fix the gauge and in seemingly order to extract intuitive, the Feynman Feynman rules diagrams from the are not a partic- Lagrangian.ularly physical So the Feynman approach rules to depend study on amplitudes the choice of in gauge gauge and theories. therefore Theonly gauge underlying invari- reason is that ant sums of diagrams are physically sensible. The on-shell amplitude is at each loop-order such the Feynman rules are non-unique: generally, individual Feynman diagrams are not physical ob- a gauge-invariant sum of Feynman diagrams. Furthermore, field redefinitions in the Lagrangian servables. In theories, such as QED or QCD, that have gauge redundancy in their Lagrangian change the Feynman rules, but not the physics, hence the amplitudes are invariant. So field re- definitionsdescriptions, and gauge we choices are required can ‘move’ to the fix physicalthe gauge information in order between to extract the diagramsthe Feynman in the rules from the FeynmanLagrangian. expression So for the the Feynman amplitude rules and this depend can hugely on the obscure choice the of gauge appearance and therefore of its physical only gauge invari- properties.ant sums of diagrams are physically sensible. The on-shell amplitude is at each loop-order such a gauge-invariant sum of Feynman diagrams. Furthermore, field redefinitions in the Lagrangian It turns out that despite the complications of the Feynman diagrams, the on-shell amplitudes for processeschange the such Feynman as multi-gluon rules, scattering but notg the+ g physics,g + g hence+ ...+ theg can amplitudes actually be are written invariant. as So field re- ! remarkablydefinitions simple and expressions. gauge choices This raises can ‘move’ the questions: the physical “why are information the on-shell between amplitudes the so diagrams in the simple?”Feynman and “isn’t expression there a better for the way amplitude to calculate and amplitudes?”. this can hugely These obscure are questions the appearance that have of its physical beenproperties. explored in recent years and a lot of progress has been made towards improving calculational techniques and gaining insight into the underlying mathematical structure. It turns out that despite the complications of the Feynman diagrams, the on-shell amplitudes Whatfor do processes we mean by such ‘mathematical as multi-gluon structure’ scattering of amplitudes?g + g Atg the+ g simplest+ ...+ level,g can this actually means be written as ! 2 the analyticremarkably structure. simple For expressions.example, the amplitude This raises (1.10 the) has questions: a pole at ( “whyp1 +p2) are= the 0 — on-shell this pole amplitudes so shows that a physical massless scalar particle is being exchanged in the process. Tree amplitudes simple?” and “isn’t there a better way to calculate amplitudes?”. These are questions that have are rational functions of the kinematic invariants, and understanding their pole structure is key to the derivationbeen explored of on-shell in recent recursive years methods and a lot that of provide progress a very has e beencient made alternative towards to improving Feynman calculational diagrams.techniques These and recursive gaining methods insight allow into one the to underlying construct on-shell mathematicaln-particle structure. tree amplitudes fromWhat input of do on-shell we mean amplitudes by ‘mathematical with fewer particles. structure’ The of power amplitudes? of this approach At the is simplest that gauge level, this means redundancy is eliminated since all input is manifestly on-shell and gauge invariant. 2 the analytic structure. For example, the amplitude (1.10) has a pole at (p1 +p2) = 0 — this pole 2 Usingshows computers that to a do physical the calculation massless can of scalar course particle be very helpful, is being but not exchanged in all cases. in Sometimes the process. numerical Tree amplitudes evaluation of Feynman diagrams is simply so slow that it is not realistic to do. Moreover, given that there are polesare that rational can cancel functions between diagrams, of the kinematic big numerical invariants, errors can arise and in understanding such evaluations. Therefore their pole compact structure is key to analyticthe derivation expressions for of the on-shell amplitudes recursive are very useful methods in practical that applications. provide a very ecient alternative to Feynman diagrams. These recursive methods allow one to construct on-shell n-particle tree amplitudes from input of on-shell amplitudes with fewer particles. The power of this approach is that gauge redundancy is eliminated since all input is manifestly on-shell and gauge invariant.

2 Using computers to do the calculation can of course be very helpful, but not in all cases. Sometimes numerical evaluation of Feynman diagrams is simply so slow that it is not realistic to do. Moreover, given that there are poles that can cancel between diagrams, big numerical errors can arise in such evaluations. Therefore compact analytic expressions for the amplitudes are very useful in practical applications. 9 Introduction 9 Strong force Introduction Quantum Chromodynamics (QCD) of particles involved:ofForce for particles gluon carrier: scattering involved: gluons at tree-level for! gluonin scattering quantum chromodynamics at tree-level in (QCD) quantum we chromodynamics (QCD) we have have g + g g + g g + g4 diagramsg + g 4 diagrams ! ! g + g g + g + g g +25g diagramsg + g + g 25 diagrams(1.18) (1.18) ! ! g + g g + g + g + g g +220g diagramsg + g + g + g 220 diagrams ! ! and for g + g 8g one needs more than one million diagrams [1]. Another very important point ! and for g + g 8g one needs more than one million diagrams [1]. Another very important point is that the mathematical expression! for each diagram becomes significantly more complicated is that the mathematical expression for each diagram becomes significantly more complicated as the number of externalHave to particles write down grows. the mathematical So the reason expression students for are each not diagram asked ! to calculate multi-gluon processesasand from the simplify Feynman number the sum…. diagrams of external ! is that particles it would grows. be awful, So un-insightful, the reason studentsand in are not asked to calculate many cases impossible.multi-gluon2 There are tricks processes for simplifying from Feynman the calculations; diagrams one is is that to use it the would spinor be awful, un-insightful, and in helicity formalism asmany indicated cases in our impossible. example2 forThere the high-energy are tricks limit for simplifying of the tree-level the process calculations; one is to use the spinor + e + e + . However, for a multi-gluon process even this does not directly provide a way ! helicity formalism as indicated in our example for the high-energy limit of the tree-level process to handle the growing number+ of increasingly complicated Feynman diagram. Other methods are e + e + . However, for a multi-gluon process even this does not directly provide a way ! needed and you willto learn handle much the more growing about them number in this of book. increasingly complicated Feynman diagram. Other methods are Thus, although visuallyneeded appealing and you and seemingly will learn intuitive, much more Feynman about diagrams them in are this not book.a partic- ularly physical approach to study amplitudes in gauge theories. The underlying reason is that the Feynman rules areThus, non-unique: although generally, visually individual appealing Feynman and seemingly diagrams are intuitive, not physical Feynman ob- diagrams are not a partic- servables. In theories,ularly such as physical QED or approach QCD, that to have study gauge amplitudes redundancy in in gauge their Lagrangian theories. The underlying reason is that descriptions, we arethe required Feynman to fix therules gauge are non-unique:in order to extract generally, the Feynman individual rules Feynman from the diagrams are not physical ob- Lagrangian. So the Feynmanservables. rules In depend theories, on the such choice as of QED gauge or and QCD, therefore that only have gauge gauge invari- redundancy in their Lagrangian ant sums of diagramsdescriptions, are physically we sensible. are required The on-shell to fix amplitude the gauge is at in each order loop-order to extract such the Feynman rules from the a gauge-invariant sum of Feynman diagrams. Furthermore, field redefinitions in the Lagrangian Lagrangian. So the Feynman rules depend on the choice of gauge and therefore only gauge invari- change the Feynman rules, but not the physics, hence the amplitudes are invariant. So field redefinitions and gaugeant choices sums can of ‘move’ diagrams the physical are physically information sensible. between The the on-shell diagrams amplitude in the is at each loop-order such Feynman expression fora gauge-invariant the amplitude and sum this of can Feynman hugely obscure diagrams. the appearance Furthermore, of its physical field redefinitions in the Lagrangian properties. change the Feynman rules, but not the physics, hence the amplitudes are invariant. So field redefinitions and gauge choices can ‘move’ the physical information between the diagrams in the It turns out that despite the complications of the Feynman diagrams, the on-shell amplitudes for processes such asFeynman multi-gluon expression scattering forg + theg amplitudeg + g + ... and+ g can this actually can hugely be written obscure as the appearance of its physical ! remarkably simple expressions.properties. This raises the questions: “why are the on-shell amplitudes so simple?” and “isn’t there a better way to calculate amplitudes?”. These are questions that have It turns out that despite the complications of the Feynman diagrams, the on-shell amplitudes been explored in recent years and a lot of progress has been made towards improving calculational for processes such as multi-gluon scattering g + g g + g + ...+ g can actually be written as techniques and gaining insight into the underlying mathematical structure. ! remarkably simple expressions. This raises the questions: “why are the on-shell amplitudes so What do we mean by ‘mathematical structure’ of amplitudes? At the simplest level, this means simple?” and “isn’t there a better way to calculate2 amplitudes?”. These are questions that have the analytic structure. For example, the amplitude (1.10) has a pole at (p1 +p2) = 0 — this pole shows that a physicalbeen massless explored scalar in particle recent is years being exchanged and a lot of in progressthe process. has Tree been amplitudes made towards improving calculational are rational functionstechniques of the kinematic and invariants, gaining insight and understanding into the underlying their pole structure mathematical is key to structure. the derivation of on-shellWhat recursive do we methods mean by that ‘mathematical provide a very structure’ ecient alternative of amplitudes? to Feynman At the simplest level, this means diagrams. These recursive methods allow one to construct on-shell n-particle tree amplitudes the analytic structure. For example, the amplitude (1.10) has a pole at (p +p )2 = 0 — this pole from input of on-shell amplitudes with fewer particles. The power of this approach is that gauge 1 2 redundancy is eliminatedshows since that all a input physical is manifestly massless on-shell scalar and particle gauge is invariant. being exchanged in the process. Tree amplitudes are rational functions of the kinematic invariants, and understanding their pole structure is key to 2 Using computers to do the calculation can of course be very helpful, but not in all cases. Sometimes numerical evaluation of Feynmanthe diagrams derivation is simply of so on-shell slow that it recursive is not realistic methods to do. Moreover, that provide given that a there very are ecient alternative to Feynman poles that can cancel betweendiagrams. diagrams, These big numerical recursive errors methods can arise in allow such evaluations. one to construct Therefore compact on-shell n-particle tree amplitudes analytic expressions for the amplitudes are very useful in practical applications. from input of on-shell amplitudes with fewer particles. The power of this approach is that gauge redundancy is eliminated since all input is manifestly on-shell and gauge invariant.

2 Using computers to do the calculation can of course be very helpful, but not in all cases. Sometimes numerical evaluation of Feynman diagrams is simply so slow that it is not realistic to do. Moreover, given that there are poles that can cancel between diagrams, big numerical errors can arise in such evaluations. Therefore compact analytic expressions for the amplitudes are very useful in practical applications. 9 Introduction 9 Strong force Introduction Quantum Chromodynamics (QCD) of particles involved:ofForce for particles gluon carrier: scattering involved: gluons at tree-level for! gluonin scattering quantum chromodynamics at tree-level in (QCD) quantum we chromodynamics (QCD) we have have g + g g + g g + g4 diagramsg + g 4 diagrams ! ! g + g g + g + g g +25g diagramsg + g + g 25 diagrams(1.18) (1.18) ! ! g + g g + g + g + g g +220g diagramsg + g + g + g 220 diagrams ! ! and for g + g 8g one needs more than one million diagrams [1]. Another very important point ! and for g + g 8g one needs more than one million diagrams [1]. Another very important point is that the mathematical expression! for each diagram becomes significantly more complicated is that the mathematical expression for each diagram becomes significantly more complicated as the number of externalHave to particles write down grows. the mathematical So the reason expression students for are each not diagram asked ! to calculate multi-gluon processesasand from the simplify Feynman number the sum…. diagrams of external ! is that particles it would grows. be awful, So un-insightful, the reason studentsand in are not asked to calculate many cases impossible.multi-gluon2 There are tricks processes for simplifying from Feynman the calculations; diagrams one is is that to use it the would spinor be awful, un-insightful, and in helicity formalism asmany indicated cases in our impossible.This example would 2 fornotThere thebe popular high-energy are trickshomework… limit for simplifying of the tree-level the process calculations; one is to use the spinor + e + e + . However, for a multi-gluon process even this does not directly provide a way ! helicity formalism as indicated in our example for the high-energy limit of the tree-level process to handle the growing numberBesides+ of the increasingly result of the complicateddiagrams with Feynman be an awful diagram. un-insightful Other mess! methods are e + e + . However, for a multi-gluon process even this does not directly provide a way ! needed and you willto learn handle much the more growing about them number in this of book. increasingly complicated Feynman diagram. Other methods are Thus, although visuallyneeded appealing and you and seemingly will learn intuitive, much more Feynman about diagrams them in are this not book.a partic- ularly physical approach to study amplitudes in gauge theories. The underlying reason is that the Feynman rules areThus, non-unique: although generally, visually individual appealing Feynman and seemingly diagrams are intuitive, not physical Feynman ob- diagrams are not a partic- servables. In theories,ularly such as physical QED or approach QCD, that to have study gauge amplitudes redundancy in in gauge their Lagrangian theories. The underlying reason is that descriptions, we arethe required Feynman to fix therules gauge are non-unique:in order to extract generally, the Feynman individual rules Feynman from the diagrams are not physical ob- Lagrangian. So the Feynmanservables. rules In depend theories, on the such choice as of QED gauge or and QCD, therefore that only have gauge gauge invari- redundancy in their Lagrangian ant sums of diagramsdescriptions, are physically we sensible. are required The on-shell to fix amplitude the gauge is at in each order loop-order to extract such the Feynman rules from the a gauge-invariant sum of Feynman diagrams. Furthermore, field redefinitions in the Lagrangian Lagrangian. So the Feynman rules depend on the choice of gauge and therefore only gauge invari- change the Feynman rules, but not the physics, hence the amplitudes are invariant. So field redefinitions and gaugeant choices sums can of ‘move’ diagrams the physical are physically information sensible. between The the on-shell diagrams amplitude in the is at each loop-order such Feynman expression fora gauge-invariant the amplitude and sum this of can Feynman hugely obscure diagrams. the appearance Furthermore, of its physical field redefinitions in the Lagrangian properties. change the Feynman rules, but not the physics, hence the amplitudes are invariant. So field redefinitions and gauge choices can ‘move’ the physical information between the diagrams in the It turns out that despite the complications of the Feynman diagrams, the on-shell amplitudes for processes such asFeynman multi-gluon expression scattering forg + theg amplitudeg + g + ... and+ g can this actually can hugely be written obscure as the appearance of its physical ! remarkably simple expressions.properties. This raises the questions: “why are the on-shell amplitudes so simple?” and “isn’t there a better way to calculate amplitudes?”. These are questions that have It turns out that despite the complications of the Feynman diagrams, the on-shell amplitudes been explored in recent years and a lot of progress has been made towards improving calculational for processes such as multi-gluon scattering g + g g + g + ...+ g can actually be written as techniques and gaining insight into the underlying mathematical structure. ! remarkably simple expressions. This raises the questions: “why are the on-shell amplitudes so What do we mean by ‘mathematical structure’ of amplitudes? At the simplest level, this means simple?” and “isn’t there a better way to calculate2 amplitudes?”. These are questions that have the analytic structure. For example, the amplitude (1.10) has a pole at (p1 +p2) = 0 — this pole shows that a physicalbeen massless explored scalar in particle recent is years being exchanged and a lot of in progressthe process. has Tree been amplitudes made towards improving calculational are rational functionstechniques of the kinematic and invariants, gaining insight and understanding into the underlying their pole structure mathematical is key to structure. the derivation of on-shellWhat recursive do we methods mean by that ‘mathematical provide a very structure’ ecient alternative of amplitudes? to Feynman At the simplest level, this means diagrams. These recursive methods allow one to construct on-shell n-particle tree amplitudes the analytic structure. For example, the amplitude (1.10) has a pole at (p +p )2 = 0 — this pole from input of on-shell amplitudes with fewer particles. The power of this approach is that gauge 1 2 redundancy is eliminatedshows since that all a input physical is manifestly massless on-shell scalar and particle gauge is invariant. being exchanged in the process. Tree amplitudes are rational functions of the kinematic invariants, and understanding their pole structure is key to 2 Using computers to do the calculation can of course be very helpful, but not in all cases. Sometimes numerical evaluation of Feynmanthe diagrams derivation is simply of so on-shell slow that it recursive is not realistic methods to do. Moreover, that provide given that a there very are ecient alternative to Feynman poles that can cancel betweendiagrams. diagrams, These big numerical recursive errors methods can arise in allow such evaluations. one to construct Therefore compact on-shell n-particle tree amplitudes analytic expressions for the amplitudes are very useful in practical applications. from input of on-shell amplitudes with fewer particles. The power of this approach is that gauge redundancy is eliminated since all input is manifestly on-shell and gauge invariant.

I Exercise 2.22 Use a well-chosen set of reference spinors to show that the entire 4-gluon amplitudes vanish if all four gluons have the same helicity.

I Exercise 2.23 + + Calculate the color-ordered 4-gluon tree amplitude A4[123 4 ] using Feynman rules and a smart choice of reference spinors. Show that the answer can be brought to the form

4 + + 12 A [123 4 ]= h i . (2.79) 4 12 23 34 41 h ih ih ih i Note the cyclic structure of the numerator factor. The result for theNonetheless 4-gluon amplitude the answer is an example for (some) of the famousgluon amplitudesParke-Taylor n-gluon tree is wonderfully simple!!! amplitude: for the case where gluons i and j have helicity 1 and all the n 2 other gluons have helicity +1, the tree amplitude is helicity +/- 1 4 + + ij A [1 ...i ...j ...n ]= h i . (2.80) n 12 23 n1 n=k+2 h ih i ···h i [Parke, Taylor (1986)] We prove this formula in Section 3. The number of Feynman diagrams that generically con- Spinor helicity formalism 9 tribute to an n-gluon tree amplitude1/2 The issum of hundreds (thousands,…, millions…!) ! <12> ~ (2p1.p2) complex version of diagrams simplify to one single simple term!!!! n =34567... #diagrams = 1 3 10 38 154 . . . Why so simple? A fun little trivia point you can impress your friends with in a bar (oh, I mean at the library), is that the number of trivalent graphsBetter that way contribute to calculate? to the n-gluon tree process is counted by the Catalan numbers. It should be clear that even if you have learned now some handy tricks of how to choose the polarization vectors to reduce the complexity of the calculation, it would be no fun trying to calculate these higher-point amplitudes brute force. But despite the complications of the many diagrams and their increased complexity, the answer is just the simple Parke-Taylor expression (2.80) for the ++ + helicity case. And that is the answer no matter which fancy ﬁeld ··· redeﬁnitions we might subject the Lagrangian to and no matter which ugly gauge we could imagine choosing. It is precisely the point of the modern approach to amplitudes to avoid such complications and get to an answer such as (2.80)inasimpleway.

I Exercise 2.24 Rewrite the expression (2.79) to show that the 4-gluon amplitude can also be written

4 + + [34] A [123 4 ]= . (2.81) 4 [12][23][34][41]

9This can be seen by direct counting, but see also analysis in [5].

The surprising simplicity and enticing mathematical structure -- and of course the practical relevance for particle physics -- is what motivates current studies of scattering amplitudes. Three Main Research Directions

1) Try to push as far as possible in a very controlled simple theory: “(planar) N=4 Super Yang Mills Theory” (SYM) (gluons, gluinos, scalars - all massless) Goals: `Solve’ theory at all loop order.! Compact expressions? Understand why!! Three Main Research Directions

2) Adapt lessons from N=4 SYM to phenomenologically relevant theories and invent new methods.

Goals: application to analysis of data from ! LHC and future particle experiments. New physics insights?! Three Main Research Directions

2) Adapt lessons from N=4 SYM to phenomenologically relevant theories and invent new methods.

Goals: application to analysis of data from ! LHC and future particle experiments. New physics insights?!

3) Use new methods to explore perturbative quantum gravity.

Goals: Point-particle quantum gravity perturbatively sensible?! Gravity as (gauge theory)2 ! Structure of string theory amplitudes New on-shell methods for calculating scattering amplitudes

Tree level amplitudes satisfy on-shell recursion relations

n 1 1 An = Ak 2 An k+2 PI kX=3

Has been an immensely powerful approach

- even revealed new symmetries of amplitudes

3 Introduction

3 Introduction Scattering amplitudes have physical relevance3 through their role for the scattering cross-section.Introduction Moreover, it has been realized in recent years that amplitudes have very interesting mathematical 3 Scattering amplitudes have physicalIntroduction relevance through their role for the scattering cross-section. structure. Understanding this structure guides usScattering towards amplitudes more ecient have methods physical for relevance calculation through their role for the scattering cross-section. Moreover, it has been realized in recent years that amplitudes have very interesting mathematical of amplitudes, and also makes it exciting to studyMoreover, scattering it has amplitudes been realized in in their recent own years right that and amplitudes have very interesting mathematical structure. Understanding this structure guides us towards more ecient methods for calculation explore their connections to interesting branchesstructure. of mathematics, Understanding including this combinatorics structure guides and us ge- towards more ecient methods for calculation Scatteringof amplitudes, amplitudes have and physical also makes relevance it exciting through to study their role scattering for the amplitudes scattering cross-section. in their own right and ometry. The purpose of this book is two-fold. First,of amplitudes, we wish to provide and also a makes pedagogical it exciting introduction to study scattering amplitudes in their own right and Moreover,explore it has their been connections realized in recent to interesting years that branches amplitudes of mathematics, have very interesting including mathematical combinatorics and ge- to the ecient modern methods for calculating scatteringexplore their amplitudes. connections Second, to interesting we survey branches several of mathematics, including combinatorics and ge- structure.ometry. Understanding The purpose this of structure this book guides is two-fold. us towards First, more we wish ecient to provide methods a pedagogical for calculation introduction interesting mathematical properties of the amplitudesometry. and The recent purpose research of this book advances. is two-fold. First, we wish to provide a pedagogical introduction of amplitudes,to the e andcient also modern makes it methods exciting for to study calculating scattering scattering amplitudes amplitudes. in their Second, own right we and survey several to the ecient modern methods for calculating scattering amplitudes. Second, we survey several explore theirinteresting connections mathematical to interesting properties branches of ofthe mathematics, amplitudes and including recent combinatorics research advances. and geometry. The purpose of this bookWhat is two-fold. is a First, scattering we wish amplitude? to provide a pedagogical introductioninteresting mathematical properties of the amplitudes and recent research advances. to the ecient modern methods for calculating scattering amplitudes. Second, we survey several + What is a scatteringLet amplitude? us consider a few examples of scattering processes involving electrons e, positrons e , and interesting mathematical properties of the amplitudes and recent research advances.What is a scattering amplitude? photons : + Let us consider a few examples of scattering processes involving electrons e, positrons e , and Let us consider a few examples of scattering processes involving electrons e , positrons e+, and photons : What is a scattering amplitude? Compton scattering e + e + , photons !: + Let us consider a few examples ofCompton scattering scattering processes involvinge + electronse +e,, positrons e , and Møller! scattering e + e e + e ,Compton scattering (1.1)e + e + , photons : ! ! Møller scattering e + e e + e , + (1.1)+ Bhabha! scattering e + e e + e . Møller scattering e + e e + e , (1.1) Compton scattering e + e + , ! ! + + ! Bhabha scattering e + e e + e . + + These processes are described in! the quantum field theory that couplesBhabha Maxwell’s scattering electromag-e + e e + e . Møller scattering e + e e + e , (1.1) ! These processes are describednetism toin the electrons quantum and field! positrons, theory that namely couples quantum Maxwell’s electrodynamics electromag- (QED). The processes are + + These processes are described in the quantum field theory that couples Maxwell’s electromag- netism to electronsBhabha and positrons, scattering namelye + quantume e electrodynamics + e . (QED). The processes are characterized by the types! of particles involvednetism in the to initial electrons and and final positrons, states (herenamely electrons, quantum electrodynamics (QED). The processes are characterized by the types of particles involved in the initial and final states (here electrons, These processes are described inpositrons, the quantum photons) field theory as well that as the couples momentum Maxwell’s andcharacterized electromag- energy of each by the particle. types of This particles input involved is the ‘ex- in the initial and final states (here electrons, positrons, photons) as well as the momentum and energy of each particle. This input is the ‘ex- netism to electrons and positrons,ternal namely data’ quantum for an amplitude: electrodynamics a scattering (QED). The amplitude processespositrons, involving are photons) a total as well of n asinitial the momentum and final and state energy of each particle. This input is the ‘external data’ for an amplitude: a scattering amplitude involving a total of n initial and final state characterized by the types of particlesparticles involved takes the in thelist initialEi, p~i and;type finali statesof external (hereternal electrons, data data’ and for returns an amplitude: a complex a scattering number: amplitude involving a total of n initial and final state particles takes the list E , p~ ;type of external data and returns a complex number: positrons, photons) as well as the momentumi i i and energy of each particle. This input is the ‘ex- particles takes the list Ei, p~i ;typei of external data and returns a complex number: ternal data’ for an amplitude: a scattering amplitude amplitude involvingAn: a totalE ofi,np~initiali ;type andi final stateAn Ei, p~i ;typei C . (1.2) amplitude An: Ei, p~i ;typei An Ei, p~i ;typei !C . (1.2) 2 particles takes the list Ei, p~i ;type of external data and returns! a complex number: 2 ⇣ amplitude An: ⌘ Ei, p~i ;typei An Ei, p~i ;typei C . (1.2) Thei external data has to satisfy certain⇣ constraints. ⌘ The energy and momentum of each particle! 2 The external data has to satisfy certain constraints. The energy and momentum of each particle ⇣ ⌘ must obey the relativistic on-shell condition The external data has to satisfy certain constraints. The energy and momentum of each particle must obeyamplitude the relativisticAn: Eon-shelli, p~i ;typeconditioni An Ei, p~i ;typei C . (1.2) Outline of recursive! method 2 must obey the relativistic on-shell condition ⇣ 2⌘ 2 2 2 4 The external data has to satisfy certain constraints. 2 The2 energy2 2 and4 momentumE i = p~ ofi eachc + particlemi c . (1.3) Ei = p~i c + mi c . | | (1.3) 2 2 2 2 4 Think of the amplitude| | A (p ) as function of momenta p w/ i=1,…,n E = p~i c + m c . (1.3) must obey the relativistic on-shell condition n i i i | | i It is convenient to set cIt= is 1 convenient as it can always to set bec = restored 1 as it by can dimensional always be analysis. restored We by combine dimensional analysis. We combine 2 2 2 2 4 µ It isµ convenient to set c = 1 as it can always be restored by dimensional analysis. We combine the energy and momentumthe energyE intoi = ap~ andi 4-momentumc momentum+ mi c . p into=(E a, p~ 4-momentum), with µ =0p,i 1,=((1.3)2, 3,E andi, p~i), write with µ =0, 1, 2, 3, and write | | i i i the energy and momentum into a 4-momentum pµ =(E , p~ ), with µ =0, 1, 2, 3, and write 2 2 2 p2 = E2 + p~ 2 such that the on-shell2 condition2 becomes p2 = m2. The 4-momenta of thei i i pi = Ei + p~i such thati thei on-shelli condition becomes pi = mi . The2 4-momenta2 i of2 the i 2 2 It is convenient to set| c |= 1 as it can always be| | restored by dimensional analysis.c=1 Wep = combineE + p~i such that the on-shell condition becomes p = m . The 4-momenta of the initial and final state electronsinitial and and final positrons state1µ Introductionin electrons the QED and processes positrons (1.1) in must thei QEDsatisfy i processes the| on-shell| (1.1) must satisfy the on-shell i i the energy and momentum into a 4-momentum p =(Ei, p~i), with µ =0, 1, 2, 3, and write i 2 initial and final2 state electrons and positrons in the QED processes (1.1) must satisfy the on-shell 2 condition2 2 (1.3)withmiconditionequal to the (1.3 electron)withm mass,i equal 5112 to keV/ the2c electron,whilem mass,i is zero 511 for keV/ thec photon,whilemi is zero for the photon 2 pi = Ei + p~i such that the on-shell condition becomes pi = mi . The 4-momentacondition of the (1.3)withmi equal to the electron mass, 511 keV/c ,whilemi is zero for the photon since| it| is massless. since it is massless. initial and final state electrons andAmplitudes positrons inwith the massless QED processes particles: (1.1) must satisfysince the on-shell it is massless. 1 Introduction2 condition (1.3)withmi equal to the electron mass, 511 keV/c ,whilemi is zero for the photon µ µ Conservation of relativisticConservation energy and of momentum relativistic requires energy and the sum momentum of initial requires momenta thepin sumto of initial momenta pin to µ since it is massless. µ µ Conservation of relativistic energy and momentum requires the sum of initial momenta pin to equal the sum of finalequal state the momenta sum ofpout2 final. It stateis convenient momenta top flip. the It signs is convenient of all incoming to flip the signs of allµ incoming p =0 (on-shell) out equal the sum of final state momenta p . It is convenient to flip the signs of all incoming momenta so that the conservation of 4-momentumi simply reads µ out Conservation of relativistic energymomenta and momentum so that the requires conservation the sum of of 4-momentum initial momentamomenta simplypin to so reads that the conservation of 4-momentum simply reads µ n equal the sum of final state momenta pout. It is convenient to flip the signsn of all incoming n µ µ momenta so that the conservation of 4-momentum simplypi =0 reads. (momentum conservation)p =0 . (1.4) µ(1.4) i pi =0. (1.4) i=1 n X i=1 i=1 µ X p =0. (1.4) X Thus, to summarize, theThus, external to summarize, datai for the the amplitude external involves data for a specification the amplitudeThus, ofto 4-momentum summarize, involves a specification the external data of 4-momentum for the amplitude involves a specification of 4-momentum µ 2 i=1 In a traditional quantum field theory (QFT) course, you learn to extract Feynmanµ rules from and particle type for each externalX particle, pi , typei , subject to the on-shellµ and particle constraints typep fori = each external particle, p ,2type , subject to the on-shell constraints p2 = 2 and particle type for each external particle, pi , typei , subject to the on-shell constraints ipi = i i Thus, to summarize,mi and momentum the external conservation data2 for the (1.4 amplitude). a Lagrangian involves a and specification use them of to 4-momentum calculate2 a scattering amplitude A as a sum of Feynman dia- m and momentum conservation (1.4). mi and momentum conservation (1.4). i µ 2 and particle type for each external particle, pi , typegramsi , subject organized to the perturbatively on-shell constraints in the p loop-expansion.i = From the amplitude you calculate the 2 d 2 mi and momentum conservation (1.4). di↵erential cross-section, A , which — if needed — includes a suitable spin-sum average. d⌦ / | | Finally the cross-section can be found by integration of d/d⌦ over angles, with appropriate symmetry factors included for identical final-state particles. The quantities and d/d⌦ are the observables of interest for particle physics experiments, but the input for computing them are the gauge invariant on-shell scattering amplitudes A. These on-shell amplitudes A are the subject of this review. Examples of processes you have likely encountered in QFT are

Compton scattering e + e + , !

g + g g + g 4 diagrams ! g + g g + g + g 25 diagrams (1.3) ! g + g g + g + g + g 220 diagrams ! 2 3 On-shell recursion relations at tree-level

3 On-shell recursion3 relations On-shell recursion at tree-level relations at tree-level

Recursion relations provide a method3 On-shell for building recursion higher-point relations amplitudes at from tree-level lower-point information. In 1988, Berends-Giele developed o↵-shell recursion relations [11] to construct n- point parton amplitudes from buildingRecursion blocks relations with one provide leg o↵ a-shell method (see for the building reviews [ higher-point1, 3]). This amplitudes from lower-point o↵-shell method remains usefulinformation. as an algorithm In 1988, for e Berends-Gielecient numerical developed evaluationo↵-shell of scatteringrecursion relations [11] to construct n- amplitudes. In this review, wepoint focus parton on the amplitudes newer (2005) from recursive building blocks methods with whose one leg building o↵-shell (see the reviews [1, 3]). This blocks are themselves on-shell amplitudes.o↵-shell method These remainson-shell useful recursion as an algorithm relations forare e elegantcient numerical in evaluation of scattering that they use input only from gauge-invariantamplitudes. In objects this review, and they we focus have on proven the newer very powerful (2005) recursive for methods whose building elucidating the mathematical structureblocks are of on-shell themselves scatteringon-shell amplitudes.amplitudes. These on-shell recursion relations are elegant in In the modern approaches, a keythat idea they is to use use input the only power from of complex gauge-invariant analysis objects and exploit and they the have proven very powerful for analytic properties of on-shell scatteringelucidating amplitudes. the mathematical The derivation structure of on-shell on-shell scattering recursion amplitudes. relations is a great example of this,In the as we modern shall approaches, see soon. The a key most idea famous is to use on-shell the power recursion of complex analysis and exploit the relations are the “BCFW recursionanalytic relations” properties by Britto, of on-shell Cachazo, scattering Feng, amplitudes. and Witten The [12, derivation13], of on-shell recursion re- but there are other versions basedlations on the is same a great idea example as BCFW, of this, namely as we the shall use of see complex soon. The defor- most famous on-shell recursion mations of the external momenta.relations We describe are the this “BCFW idea here, recursion ﬁrst in relations” a very general by Britto, formulation Cachazo, Feng, and Witten [12, 13], (Section 3.1), then specialize thebut results there to are derive other the versions BCFW based recursion on the same relations idea (Section as BCFW,3.2 namely). the use of complex defor- We illustrate the BCFW methodsmations with of a the selection external of momenta. examples, We including describe an this inductive idea here, proof ﬁrst in a very general formulation of the Parke-Taylor formula (2.80(Section). Section3.1),3.3 thencontains specialize a discussion the results of to when derive to the expect BCFW exis- recursion relations (Section 3.2). tence of recursion relations in generalWe illustrate local QFTs. the BCFW Finally, methods in Section with3.4 a selectionwe outline of the examples, CSW including an inductive proof construction (Cachazo-Svrcek-Wittenof the [ Parke-Taylor14]), also called formula the MHV (2.80). vertex Section expansion.3.3 contains a discussion of when to expect exis- tence of recursion relations in general local QFTs. Finally, in Section 3.4 we outline the CSW construction (Cachazo-Svrcek-Witten [14]), also called the MHV vertex expansion. 3.1 Complex shifts & Cauchy’s theorem

An on-shell amplitude An is characterized3.1 Complex by the shifts momenta & Cauchy’s of the external theorem particles and their type (for example a helicity label hi for massless particles). We focus here on massless particles 2 n µ so pi = 0 for all i =1, 2,...,nAn. on-shell Of course, amplitude momentumAn is characterized conservation byi the=1 p momentai = 0 is ofalso the external particles and their imposed. type (for example a helicity label hi for massless particles). We focus here on massless particles P so p2 = 0 for all i =1, 2,...,n. Of course, momentum conservation n pµ = 0 is also Let us now introduce n complex-valuedi vectors rµ (some of which may be zero) such that i=1 i imposed. i n P µ Let us now introduce n complex-valued vectors rµ (some of which may be zero) such that (i) ri =0, i i=1 n X µ (ii) r r = 0 for all i, j =1, 2(i),...,n. Inri particular=0, r2 = 0 , and i · j i Xi=1 (iii) pi ri = 0 for each i (no sum). 2 · (ii) ri rj = 0 for all i, j =1, 2,...,n. In particular ri = 0 , and · Outline of recursive method These are used to define n shifted momenta (iii) pi ri = 0 for each i (no sum). · Think of the amplitude An(pi) as function of momenta pi. pˆµ pµ + zrµ with z . (3.1) iThesei are usedi to defineShift na shiftedsubsetC of momenta the momenta: 3 On-shell⌘ recursion relations2 at tree-level 3.1 Complex shifts & Cauchy’s theorem µ µ µ Note that pî pi + zri with z C . (3.1) ⌘ n 2 µ such that 2 and (A) By property (i), momentum(B)Note conservationBy that (ii) and holds (iii), for we the have shiftedp î = momenta: 0, so the shiftedpî = momenta 0. are on-shell. n (on-shell) i=1 13 X (momentum conservation)µ µ µ ˆ2 (C)(A)ForBy a property non-trivial (i), momentumsubset of conservation generic momenta holds for thepi i shiftedI ,define momenta:PI = i pÎip=i .Then 0. PI is { } 2 2 34 Now An is a function of z: An(z) i=1 linear in z: X 2 2 2 P Pˆ = pî = P + z 2 PI RI with RI = ri , (3.2) Tree-level:I I 34 · A (z) only simplei I poles (by choice of shift vectors) i I n X2 X2 because the z2 term vanishes by property (ii). We can write P 2 P 2 Pˆ2 = I (z z )withz = I . (3.3) I z I I 2P R I I · I

As a result of (A) and (B), we can consider our amplitude An in terms of the shifted momenta µ µ pˆi instead of the original momenta pi . In particular, it is useful to study the shifted amplitude as a function of z; by construction it is holomorphic, Aˆn(z). The amplitude with unshifted µ ˆ momenta pi is obtained by setting z = 0, An = An(z = 0).

We specialize to the case where An is a tree-level amplitude. In that case, the analytic structure of Aˆn(z) is very simple. For example, it does not have any branch cuts — there are no log’s, square-roots, etc, at tree-level. Its analytic structure is captured by its poles, and it can have only simple poles. To see this, consider the Feynman diagrams: the only places we ˆ2 ˆ can get poles is from the shifted propagators 1/PI ,wherePI is a sum of a nontrivial subset of ˆ2 the shifted momenta. By (C) above, 1/PI gives a simple pole at zI , and for generic momenta zI = 0. For generic momenta, no Feynman tree diagram can have more than one power of a 6 ˆ2 given propagator 1/PI ; and poles of di↵erent propagators are located at di↵erent positions in the z-plane. Hence, for generic momenta, Aˆn(z) only has simple poles and they are all located away from the origin. Note the implicit assumption of locality, i.e. that the amplitudes can be derived from some local Lagrangian: the propagators determine the poles.

Aˆn(z) Let us then look at z in the complex z-plane. Pick a contour that surrounds the simple pole at the origin. The residue at this pole is nothing but the unshifted amplitude, An = Aˆn(z = 0). Deforming the contour to surround all the other poles, Cauchy’s theorem tells us that

Aˆ (z) A = Res n + B , (3.4) n z=zI z n z XI where B is the residue of the pole at z = . By taking z 1/w it is easily seen that B is n 1 ! n the O(z0)terminthez expansion of A . !1 n ˆ2 Now, then, so what? Well, at a zI -pole the propagator 1/PI goes on-shell. In that limit, the shifted amplitude factorizes into two on-shell parts, AˆL and AˆR. Using (3.3), we ﬁnd

^ ^ ^ P Aˆ (z) 1 ^ I ^ Res n = Aˆ (z ) Aˆ (z )= L R . (3.5) z=zI z L I P 2 R I I ^ ^ Note that — as opposed to Feynman diagrams — the momentum of the internal line in (3.5)is ˆ2 on-shell, PI = 0, and the vertex-blobs represent shifted on-shell amplitudes evaluated at z = zI ; 13Non-trivial means at least two and no more than n 2momentasuchthatP 2 =0. I 6

35 3 On-shell recursion relations at tree-level

3 On-shell recursion3 relations On-shell recursion at tree-level relations at tree-level

An on-shell amplitude An is characterized3.1 Complex by the shifts momenta & Cauchy’s of the external theorem particles and their type (for example a helicity label hi for massless particles). We focus here on massless particles 2 n µ so pi = 0 for all i =1, 2,...,nAn. on-shell Of course, amplitude momentumAn is characterized conservation byi the=1 p momentai = 0 is ofalso the external particles and their imposed. type (for example a helicity label hi for massless particles). We focus here on massless particles P so p2 = 0 for all i =1, 2,...,n. Of course, momentum conservation n pµ = 0 is also Let us now introduce n complex-valuedi vectors rµ (some of which may be zero) such that i=1 i imposed. i n P µ Let us now introduce n complex-valued vectors rµ (some of which may be zero) such that (i) ri =0, i i=1 n X µ (ii) r r = 0 for all i, j =1, 2(i),...,n. Inri particular=0, r2 = 0 , and i · j i Xi=1 (iii) pi ri = 0 for each i (no sum). 2 · (ii) ri rj = 0 for all i, j =1, 2,...,n. In particular ri = 0 , and · Outline of recursive method These are used to define n shifted momenta (iii) pi ri = 0 for each i (no sum). · Think of the amplitude An(pi) as function of momenta pi. pˆµ pµ + zrµ with z . (3.1) iThesei are usedi to defineShift na shiftedsubsetC of momenta the momenta: 3 On-shell⌘ recursion relations2 at tree-level 3.1 Complex shifts & Cauchy’s theorem µ µ µ Note that pî pi + zri with z C . (3.1) ⌘ n 2 µ such that 2 and (A) By property (i), momentum(B)Note conservationBy that (ii) and holds (iii), for we the have shiftedp î = momenta: 0, so the shiftedpî = momenta 0. are on-shell. n (on-shell) i=1 13 X (momentum conservation)µ µ µ ˆ2 (C)(A)ForBy a property non-trivial (i), momentumsubset of conservation generic momenta holds for thepi i shiftedI ,define momenta:PI = i pÎip=i .Then 0. PI is { } 2 2 34 Now An is a function of z: An(z) i=1 linear in z: X 2 2 2 P Pˆ = pî = P + z 2 PI RI with RI = ri , (3.2) Tree-level:I I 34 · A (z) only simplei I poles i I n X2 X2 2 because the z term vanishes by propertyOh, (ii). that makes We can us want write to use complex analysis! P 2 P 2 Pˆ2 = I (z z )withz = I . (3.3) I z I I 2P R I I · I

35 3 On-shell recursion relations at tree-level 3.2 BCFW recursion relations we call them subamplitudes. The rule for the internal line in the diagrammatic representation 2 (3.5) is to write the scalar propagator 1/PI of the unshifted momenta. Each subamplitude necessarily involves fewer than n external particles, hence all the residues at ﬁnite z can be determined in terms of on-shell amplitudes with less then n particles. This is the basis of the recursion relations.

The contribution Bn from theOutline pole at inﬁnityof recursive has in method general no similar expression in terms of lower-point amplitudes; there has recently been various approaches to try to compute the form of Bn systematically (see for exampleTree-level: [15, A16n(z)]), butonly theresimple is poles currently not a general constructive method. Thus, in most applications, one assumes — or, much preferably, proves — that Bn = 0. Apply Cauchy’s theorem: This is most often justiﬁed by demonstrating that If Aˆ (z) 0 for z . (3.6) n ! !1 Then use that the amplitude factorizes on the simple poles to get If (3.6) holds, we say that the shift (3.1)isvalid (or good), and in that case the n-point on-shell amplitude is completely determined in terms of lower-point on-shell amplitudes as

^ ^ ^ ^ P ˆ 1 ˆ L I R ^ An = AL(zI ) 2 AR(zI )= . (3.7) “BCFWPI shift” [Britto, Cachazo, Feng, Witten (2004)] diagramsX I diagramsX I ^ ^ “all-line shifts” [Freedman, Kiermaier, HE (2008)] The sum is over all possibleWhen factorization does it work? channels [Cohen,Kiermaier,HEI. There is (2010)]+[Cheung,Shen,Trnka also implicitly a sum (2015)] over all possible on-shell particle states that can be exchanged on the internal line: for example, for a gluon we have to sum the possible helicity assignments. The recursive formula (3.7) gives a manifestly gauge invariant construction of scattering amplitudes. This is the general form of the “on-shell recursion relations” for tree-level amplitudes with the property (3.6). We did not use any special properties of d = 4 spacetime, so the recursion relations are valid in d spacetime dimensions. In the following, we specialize to d = 4 again.

3.2 BCFW recursion relations

Above we shifted all external momenta democratically, but with a parenthetical remark that µ µ some of the lightlike shift-vectors ri might be trivial, ri = 0. The BCFW shift is one in which exactly two lines, say i and j, are selected as the only ones with non-vanishing shift-vectors. In d = 4 spacetime dimension, the shift is implemented on angle and square spinors of the two chosen momenta:

î]= i]+z j] , ˆj]= j] , î = i , ˆj = j z i . (3.8) | | | | | | i | i | i | i | i No other spinors are shifted. We call this a [i, j -shift. Note that [îk] and ˆjk are linear in z i h i for k = i, j while îˆj = ij ,[îˆj]=[ij], îk = ik , and [ˆjk]=[jk] remain unshifted. 6 h i h i h i h i I Exercise 3.1 µ µ Use (2.15) to calculate the shift vectors ri and rj corresponding to the shift (3.8). Then show that your shift vectors satisfy the properties (i)-(iii) of the Section 3.1.

36 2 Spinor helicity formalism 2.5 Yang-Mills theory, QCD, and color-ordering

I Exercise 2.22 Use a well-chosen set of reference spinors to show that the entire 4-gluon amplitudes vanish if all four gluons have the same helicity.

I Exercise 2.23 + + Calculate the color-ordered 4-gluon tree amplitude A4[123 4 ] using Feynman rules and a smart choice of reference spinors. Show that the answer can be brought to the form

4 + + 12 A [123 4 ]= h i . (2.79) 4 12 23 34 41 h ih ih ih i Note the cyclic structure of the numerator factor. The result for the 4-gluon amplitude is anExample example of the famous Parke-Taylor n-gluon tree amplitude: for the case where gluons i and j have helicity 1 and all the n 2 other gluons BCFW recursion relations lets you derive the have helicity +1, the tree amplitude is n-gluon tree amplitudes 4 + + ij A [1 ...i ...j ...n ]= h i . (2.80) n 12 23 n1 h ih i ···h i We prove this formula in Sectionfrom3. Theone diagram number!!! of! Feynman diagrams that generically con- 9 tribute to an n-gluon tree amplitude is vs. hundreds (or worse) of Feynman diagrams n =34567...This class of amplitudes with 2 negative helicity gluons and #diagramsn-2 = positive 1 3 helicity 10 gluons 38 154are called . . . “Maximally helicity violating”: MHV A fun little trivia point you can impress your friends with in a bar (oh, I mean at the library), is that the numberThey of trivalent are the simplest graphs amplitudes that contribute in Yang-Mills to thetheoryn-gluon. tree process is counted by the Catalan numbers. It should be clear that even if you have learned now some handy tricks of how to choose the polarization vectors to reduce the complexity of the calculation, it would be no fun trying to calculate these higher-point amplitudes brute force. But despite the complications of the many diagrams and their increased complexity, the answer is just the simple Parke-Taylor expression (2.80) for the ++ + helicity case. And that is the answer no matter which fancy ﬁeld ··· redeﬁnitions we might subject the Lagrangian to and no matter which ugly gauge we could imagine choosing. It is precisely the point of the modern approach to amplitudes to avoid such complications and get to an answer such as (2.80)inasimpleway.

I Exercise 2.24 Rewrite the expression (2.79) to show that the 4-gluon amplitude can also be written

4 + + [34] A [123 4 ]= . (2.81) 4 [12][23][34][41]

9This can be seen by direct counting, but see also analysis in [5].

24 Examples

Amplitudes with k+2 negative helicity gluons and n-k-2 positive helicity gluons are called “(Next-to)k-Maximally helicity violating”: NkMHV

They are the (Next-to)k-simplest amplitudes in Yang-Mills theory. 3 On-shell recursion relations at tree-level 3.2 BCFW recursion relations 9 Grassmannia 9.3 Yangian invariants as residues in the Grassmannian

Thus this is the first time we see a lower-point NMHV amplitude shows up in the recursion relations.3 On-shell This recursion is quite relations generic: at the tree-level BCFW relations are recursive3.2 both BCFW in particle recursion number relationsn K and in N MHV level K. M M 2 2 M M The two BCFW representations (3.31) and1 (3.35) look quite di↵erent.1 In order for both to Thus this is the first time3 On-shell we see recursionM a lower-point relations at tree-level= NMHV amplitudeM 3.2 shows BCFW recursion up in relations the recursion describe the same amplitude, there6 has to be a certain identity6 that ensures that diagrams M M relations. This is quite generic:4 the BCFW relations are4 recursive both in particle number n A+B = AK0 +B0+C0.Thus To thisshow is the that first time this we identity see a lower-point holds NMHV requires amplitude a shows nauseating up in the recursion trip through and in N MHV level K. M M relations. This is quite3 generic:M the BCFW relations are recursive3 bothM in particle number n Schouten identities and momentum conservation5 relations in order to manipulate5 the angle and and in NK MHV level K. squareThe two brackets BCFW into representations the right form: (3.31 numerical) and ( checks3.35) look can save quite you di↵ aerent. lot of energy In order when for dealing both to The two BCFW representations (3.31) and (3.35) look quite di↵erent. In order for both to withdescribe amplitudes the same with amplitude, moredescribe than the there 5 same external amplitude, has to lines. there be has a It certain turns to be a certainout identity that identity the that that identities ensures ensures that that diagrams that guarantee diagrams Figure 11: The “tree-contour” in the Grassmannian. It circles the residues of the poles A+B = A0 +B0+C0. To show that this identity holds requires a nauseating trip through theA+B equivalence =M A,0 +BM of,0+C BCFWM 0.. To Through expressions show contour that such this deformation, as identity A+B and holds the A result0 requires+B is0+C equivalent a0 nauseatingactually to minus originate trip the through sumfrom of { 2} { 4} { 6} Schouten identities and momentumExamples conservation relations in order to manipulate the angle and powerfulSchoutenM residue identities, M theorems, M and. square momentum [21 brackets] related into conservation the to right quite form: di numerical↵ relationserent checks formulations in can order save you to a lot manipulate of of the energy amplitudes. when dealingthe angle This and { 1} { 3} { 5} hassquare to do brackets with the into description thewithAmplitudes right amplitudes of form: amplitudes with with numerical morek+2 thannegative in 5 external checks the helicity Grassmannian lines. can It gluons turns save out youand that — an-k-2 the lot we identities ofpositive get energy to that that guarantee when in Section dealing the equivalence of BCFW expressions such as A+B and A +B+C actually originate from helicity gluons are called 0 0 0 10with, but amplitudesassociated wanted to with with give the more youpowerful minor a than hint residueM 5 ofin external theorems this the curious cyclically [21] lines. related point toIt invariant quite turns here. di↵erent out Grassmannian formulations that the of identities the integral. amplitudes. that So This guarantee for =4 “(Next-to)i k-Maximally helicity violating”: NkMHV N the equivalenceSYM, individual of BCFW BCFWhas to expressions do diagrams with the description such are in of as amplitudes one-to-one A+B andin the correspondence Grassmannian A0 +B0+C — we0 actually get with to that the in originate residuesSection of from the I Exercise 3.8 10 , but wanted to give you a hint of this curious point here. powerfulGrassmannian residue theorems integral.They [21 Sinceare] relatedthe the (Next-to) Grassmannian to quitek-simplest di↵erent amplitudes integral formulations was in Yang-Mills constructed of the theory. to amplitudes. produce Yangian This I Exercise 3.8 has toShowinvariants, do with that the the we descriptionnow BCFW understand recursion of amplitudes that relations each super-BCFW in based the Grassmannian on the diagram [2, 3 -shift is — a we Yangian give get the to invariant. thatfollowing in Section rep- Show that the BCFW recursion relations based on the [2, 3 -shift give the following rep- BCFW recursion for a 6-gluon NMHV amplitude igivesi 10, butresentation wanted to of give the 6-point youresentation a hint ‘alternating of of the this 6-point curious helicity’ ‘alternating point gluon helicity’ here. amplitude: gluon amplitude: Consider the contour that encircles the minors M2 , M4 , M6 . It is this contour that + + + { } { } { } gives a Yangian invariant rational+ + function+A6[1 2 that3 45 is6 local]= M2 and+ M free4 + ofM6 spurious, singularities.(3.36) The I Exercise 3.8 A6[1 23 45 6]= M2 +{ M} 4 { +} M{ 6 } , (3.36) statement of locality haswhere become a choice of contour.{ } { } { } Show that the BCFW recursion relations basedi, i +2 on4 the[i +3,i [2, 31]4 -shift give the following rep- where with Mi = h i . { } P˜2 i P˜ i + 3] i +2P˜ i 1] i, i +1 i +1,i+2i [i +3,i 2][i 2,i 1] Throughresentation contour of the deformation, 6-point ‘alternating illustratedi h | i| h helicity’ schematically| i| gluonh ih in amplitude: Figurei 11,wehave i, i +2 4[i +3,i 1]4 (3.37) Mi = and P˜ = P . [Hint:h M isi the value of the 12-channel diagram.] . { } ˜2 ˜ i + i,i+1+˜,i+2 + { 4} Pi i POri i +InAM 3] Section62[1i++2210M3weP4i discover4i+5 1]M6 that6i,]= each i=+1MMi M2ican+11+ be,i understoodM+2M4 3 +[i as+3M theM6,i residue5 ,. 2][ associatedi 2,i with 1](3.36)(9.50) h | | { h} { | }| { h } {{{ih} } } { { }}i {{ }} a very interesting contour integral (di↵erent from the one used in the BCFW argument). (3.37) Thiswhere means that the tree amplitude A [1+2 3+4 5+6 ] can also be represented by (minus) the and P˜i = Pi,i+1,i+2.dependingThe [Hint:fourth thingM onworth4 howis discussing the 6you value do further BCFW… of are the the poles 12-channel of scattering diagram.] amplitudes. Color-ordered { } 4 4 Insum Section of M101 ,weM discover3tree, and amplitudes thatM5 can each. haveIndeed,M physicali, thiscan i +2 poles be is only the[ understoodi +3 when representation,i the momenta1] as the of that adjacent residue one external associated obtains lines from with the M{ =} { go} collinear.{ We} touched thishi point alreadyi when we discussed the 3-particle poles. In fact, . BCFW-shifti [3˜,22 , the˜ ‘parity conjugate’˜ { } of the shift [2, 3 that produced the M , M , M a very{ interesting} P i contourPyoui i can+ see 3] integral fromi +2 theP Parke-Taylori (dii ↵erent1] i, formula i from+1 that thei MHV+1 one,i amplitudes+2 used[i in do+3 not the,i have BCFW2][ multi-particlei 2 2 argument).,i 4 1] 6 i hi | | h | | h ih i i { } { } { } representation. Actually,poles, only 2-particle this is poles. a little And you too have quick, seen that because the 6-gluon for NMHV the amplitude component-amplitude has both (3.37) + + + 2- and 3-particle poles. But as you stare intensely at (3.34), you will also note that there is The fourthAand6[1 P thing˜2=3 P4worth5 6 discussing],. the [Hint: shift furtherM [3, 2 iswould are the the value be poles an of illegal the of scattering 12-channel [+, -shift; amplitudes. diagram.] the precise Color-ordered statement is i i,i+1,i+2a strange denominator-factor4 i 5 1+62] in the result fromi each BCFW diagram. This does tree amplitudesthat the M can, haveM , physicalM {representation poles} onlyh | when is| the result the momenta of a [3, 2 ofBCFW adjacentsuper external-shift recursion lines In Section{ 101}we{ discovernot3} correspond{ 5 that} to each a physicalM polei can of the be scattering understood amplitude: as iti the is a spurious residue pole associated.The with go collinear.relation We with touched a projectionresidue this of point tothis the unphysical already (+, { pole, + whenbetter,} , + be, we zero) gluon discussed — and helicityit is: the the spurious states. 3-particle pole cancels poles. in the In fact, a very interestingsum contour of the two integral BCFW diagrams (di↵erent in (3.34 from). It is the typical one that used BCFW in packs the the BCFW information argument). you canThe see insight from the gained Parke-Taylorof here the amplitudes is that formula into the compact mysterious that expressions, MHV six-term but amplitudes the cost identity is the appearance do (9.50 not) of have that spurious multi-particle arises poles; from the The fourth thing worththis discussing means that infurther the BCFW are representation the poles the of locality scattering of the underlying amplitudes. field theory Color-ordered is not poles,equivalence only 2-particle of the poles. two conjugate And you haveBCFW seen super-shifts that the [2 6-gluon, 3 and NMHV [3, 2 is amplitude simply a consequence has both tree amplitudes can havemanifest. physical Elimination poles of spurious only poles when in the representations the momentai of amplitudes ofi adjacent leads to interesting external lines 2- andof 3-particle the residue poles. theoremresults But of [ as22] the that you Grassmannian we stare discuss in intensely a later section. integral at (3.34),! you Actually, will also the identity note that (9.50 there)isthe is go collinear. We touched this point already when we discussedLn,k the 3-particle poles. In fact, + + + a strange5-bracket denominator-factor six-termFinally identity, let5 us (1+65.58 for completeness)projectedtothe(+2] in the note that result the color-ordered from, , each+ amplitudes, , BCFW+, A)6[1 gluon diagram.23 4 helicity5 6 ] and This states. does In you can see from the Parke-Taylor+ h |+ + formula| that MHV amplitudes do not have multi-particle not correspondsection 10 towe a expose physicalA6[1 an2 underlying3 pole4 56 of] with the geometric other scattering arrangements interpretation amplitude: of helicities are of itinequivalent such is a identities.spurious to the split-helicity pole.The poles, only 2-particle poles. And you have seen that the 6-gluon NMHV amplitude has both residue of this unphysical pole better be zero — and43 it is: the spurious pole cancels in the 2- and 3-particle poles. But as you stare intensely at (3.34), you will also note that there is sum ofThe the Grassmannian, two BCFW diagrams the tree in ( contour,3.34). It and is typical the twistor that BCFW string packs the information a strange denominator-factor 5 1+62] in the result from each BCFW diagram.K This does of theIn amplitudes Witten’s twistor into compact string [h55 expressions,|], mentioned| but briefly the at cost the is end the of appearance Section 5.2,theN of spuriousMHV poles; super- not correspond to a physical pole of the scattering amplitude: it is a spurious pole.The this meansamplitudes that in in the= BCFW 4 SYM representation are calculated the as open locality string of current the underlying algebra correlators field theory integrated is not residue of this unphysicalN pole better be zero — and it is: the spurious pole cancels in the manifest.over Elimination the moduli space of spurious of degree poles (K+1) in the curves representations in supertwistor of amplitudes space. It turns leads out to that interesting the RSV sum of the two BCFW diagrams in (3.34). It is typical that BCFW packs the information resultsconnected [22] that prescriptionwe discuss in[115 a later] for section. the twistor string has a direct relation to the BCFW recur- of thesion amplitudes relations. into Moreover, compact di↵ expressions,erent BCFW but representations the cost is the are appearance related via (higher-dimensional of spurious poles; + + + Finallythis means, let us that for incompleteness the BCFW note representation that the color-ordered the locality amplitudes of the underlyingA6[12 field3 4 theory5 6 ] is and not + + + Amanifest.6[12 3 Elimination4 56 ] with of other spurious arrangements poles in the of representations helicities147 are inequivalent of amplitudes to leads the split-helicity to interesting results [22] that we discuss in a later section.

43 + + + Finally, let us for completeness note that the color-ordered amplitudes A6[123 45 6 ] and + + + A6[12 34 56 ] with other arrangements of helicities are inequivalent to the split-helicity

43 3 On-shell recursion relations at tree-level 3.2 BCFW recursion relations 9 Grassmannia 9.3 Yangian invariants as residues in the Grassmannian

Thus this is the first time we see a lower-point NMHV amplitude shows up in the recursion relations.3 On-shell This recursion is quite relations generic: at the tree-level BCFW relations are recursive3.2 both BCFW in particle recursion number relationsn K and in N MHV level K. M M 2 2 M M The two BCFW representations (3.31) and1 (3.35) look quite di↵erent.1 In order for both to Thus this is the first time3 On-shell we see recursionM a lower-point relations at tree-level= NMHV amplitudeM 3.2 shows BCFW recursion up in relations the recursion describe the same amplitude, there6 has to be a certain identity6 that ensures that diagrams M M relations. This is quite generic:4 the BCFW relations are4 recursive both in particle number n A+B = AK0 +B0+C0.Thus To thisshow is the that first time this we identity see a lower-point holds NMHV requires amplitude a shows nauseating up in the recursion trip through and in N MHV level K. M M relations. This is quite3 generic:M the BCFW relations are recursive3 bothM in particle number n Schouten identities and momentum conservation5 relations in order to manipulate5 the angle and and in NK MHV level K. squareThe two brackets BCFW into representations the right form: (3.31 numerical) and ( checks3.35) look can save quite you di↵ aerent. lot of energy In order when for dealing both to The two BCFW representations (3.31) and (3.35) look quite di↵erent. In order for both to withdescribe amplitudes the same with amplitude, moredescribe than the there 5 same external amplitude, has to lines. there be has a It certain turns to be a certainout identity that identity the that that identities ensures ensures that that diagrams that guarantee diagrams Figure 11: The “tree-contour” in the Grassmannian. It circles the residues of the poles A+B = A0 +B0+C0. To show that this identity holds requires a nauseating trip through theA+B equivalence =M A,0 +BM of,0+C BCFWM 0.. To Through expressions show contour that such this deformation, as identity A+B and holds the A result0 requires+B is0+C equivalent a0 nauseatingactually to minus originate trip the through sumfrom of { 2} { 4} { 6} Schouten identities and momentumExamples conservation relations in order to manipulate the angle and powerfulSchoutenM residue identities, M theorems, M and. square momentum [21 brackets] related into conservation the to right quite form: di numerical↵ relationserent checks formulations in can order save you to a lot manipulate of of the energy amplitudes. when dealingthe angle This and { 1} { 3} { 5} hassquare to do brackets with the into description thewithAmplitudes right amplitudes of form: amplitudes with with numerical morek+2 thannegative in 5 external checks the helicity Grassmannian lines. can It gluons turns save out youand that — an-k-2 the lot we identities ofpositive get energy to that that guarantee when in Section dealing the equivalence of BCFW expressions such as A+B and A +B+C actually originate from helicity gluons are called 0 0 0 10with, but amplitudesassociated wanted to with with give the more youpowerful minor a than hint residueM 5 ofin external theorems this the curious cyclically [21] lines. related point toIt invariant quite turns here. di↵erent out Grassmannian formulations that the of identities the integral. amplitudes. that So This guarantee for =4 “(Next-to)i k-Maximally helicity violating”: NkMHV N the equivalenceSYM, individual of BCFW BCFWhas to expressions do diagrams with the description such are in of as amplitudes one-to-one A+B andin the correspondence Grassmannian A0 +B0+C — we0 actually get with to that the in originate residuesSection of from the I Exercise 3.8 10 , but wanted to give you a hint of this curious point here. powerfulGrassmannian residue theorems integral.TheyCan [21 Sinceare] relatedbethe thewritten (Next-to) Grassmannian to in quitek -simplest di↵erent amplitudes integral formulations was in Yang-Mills constructed of the theory. to amplitudes. produce Yangian This I Exercise 3.8 has toShowinvariants, do with that the the we descriptionnow BCFW understanddifferent recursion of amplitudes thatways. relations each super-BCFW in based the Grassmannian onmore the diagram [2 complicated, 3 -shift is — a we Yangian give getthan the to MHV invariant. thatfollowing in Section rep- Show that the BCFW recursion relations based on the [2, 3 -shift give the following rep- BCFW recursion for a 6-gluon NMHV amplitude igivesi 10, butresentation wanted to of give the 6-point youresentation a hint ‘alternating of of the this 6-point curious helicity’ ‘alternating point gluon helicity’ here. amplitude: gluon amplitude: Consider the contour that encircles the minors M2 , M4 , M6 . It is this contour that + + + { } { } { } gives a Yangian invariant rational+ + function+A6[1 2 that3 45 is6 local]= M2 and+ M free4 + ofM6 spurious, singularities.(3.36) The I Exercise 3.8 A6[1 23 45 6]= M2 +{ M} 4 { +} M{ 6 } , (3.36) statement of locality haswhere become a choice of contour.{ } { } { } Show that the BCFW recursion relations basedi, i +2 on4 the[i +3,i [2, 31]4 -shift give the following rep- where with Mi = h i . { } P˜2 i P˜ i + 3] i +2P˜ i 1] i, i +1 i +1,i+2i [i +3,i 2][i 2,i 1] Throughresentation contour of the deformation, 6-point ‘alternating illustratedi h | i| h helicity’ schematically| i| gluonh ih in amplitude: Figurei 11,wehave i, i +2 4[i +3,i 1]4 (3.37) Mi = and P˜ = P . [Hint:h M isi the value of the 12-channel diagram.] . { } ˜2 ˜ i + i,i+1+˜,i+2 + { 4} Pi i POri i +InAM 3] Section62[1i++2210M3weP4i discover4i+5 1]M6 that6i,]= each i=+1MMi M2ican+11+ be,i understoodM+2M4 3 +[i as+3M theM6,i residue5 ,. 2][ associatedi 2,i with 1](3.36)(9.50) h | | { h} { | }| { h } {{{ih} } } { { }}i {{ }} a very interesting contour integral (di↵erent from the one used in the BCFW argument). (3.37) Thiswhere means that the tree amplitude A [1+2 3+4 5+6 ] can also be represented by (minus) the and P˜i = Pi,i+1,i+2.dependingThe [Hint:fourth thingM onworth4 howis discussing the 6you value do further BCFW… of are the the poles 12-channel of scattering diagram.] amplitudes. Color-ordered { } 4 4 Insum Section of M101 ,weM discover3tree, and amplitudes thatM5 can each. haveIndeed,M physicali, thiscan i +2 poles be is only the[ understoodi +3 when representation,i the momenta1] as the of that adjacent residue one external associated obtains lines from with the M{ =} { go} collinear.{ We} touched thishi point alreadyi when we discussed the 3-particle poles. In fact, . BCFW-shifti [3˜,22 , the˜ ‘parity conjugate’˜ { } of the shift [2, 3 that produced the M , M , M a very{ interesting} P i contourPyoui i can+ see 3] integral fromi +2 theP Parke-Taylori (dii ↵erent1] i, formula i from+1 that thei MHV+1 one,i amplitudes+2 used[i in do+3 not the,i have BCFW2][ multi-particlei 2 2 argument).,i 4 1] 6 i hi | | h | | h ih i i { } { } { } representation. Actually,poles, only 2-particle this is poles. a little And you too have quick, seen that because the 6-gluon for NMHV the amplitude component-amplitude has both (3.37) + + + 2- and 3-particle poles. But as you stare intensely at (3.34), you will also note that there is The fourthAand6[1 P thing˜2=3 P4worth5 6 discussing],. the [Hint: shift furtherM [3, 2 iswould are the the value be poles an of illegal the of scattering 12-channel [+, -shift; amplitudes. diagram.] the precise Color-ordered statement is i i,i+1,i+2a strange denominator-factor4 i 5 1+62] in the result fromi each BCFW diagram. This does tree amplitudesthat the M can, haveM , physicalM {representation poles} onlyh | when is| the result the momenta of a [3, 2 ofBCFW adjacentsuper external-shift recursion lines In Section{ 101}we{ discovernot3} correspond{ 5 that} to each a physicalM polei can of the be scattering understood amplitude: as iti the is a spurious residue pole associated.The with go collinear.relation We with touched a projectionresidue this of point tothis the unphysical already (+, { pole, + whenbetter,} , + be, we zero) gluon discussed — and helicityit is: the the spurious states. 3-particle pole cancels poles. in the In fact, a very interestingsum contour of the two integral BCFW diagrams (di↵erent in (3.34 from). It is the typical one that used BCFW in packs the the BCFW information argument). you canThe see insight from the gained Parke-Taylorof here the amplitudes is that formula into the compact mysterious that expressions, MHV six-term but amplitudes the cost identity is the appearance do (9.50 not) of have that spurious multi-particle arises poles; from the The fourth thing worththis discussing means that infurther the BCFW are representation the poles the of locality scattering of the underlying amplitudes. field theory Color-ordered is not poles,equivalence only 2-particle of the poles. two conjugate And you haveBCFW seen super-shifts that the [2 6-gluon, 3 and NMHV [3, 2 is amplitude simply a consequence has both tree amplitudes can havemanifest. physical Elimination poles of spurious only poles when in the representations the momentai of amplitudes ofi adjacent leads to interesting external lines 2- andof 3-particle the residue poles. theoremresults But of [ as22] the that you Grassmannian we stare discuss in intensely a later section. integral at (3.34),! you Actually, will also the identity note that (9.50 there)isthe is go collinear. We touched this point already when we discussedLn,k the 3-particle poles. In fact, + + + a strange5-bracket denominator-factor six-termFinally identity, let5 us (1+65.58 for completeness)projectedtothe(+2] in the note that result the color-ordered from, , each+ amplitudes, , BCFW+, A)6[1 gluon diagram.23 4 helicity5 6 ] and This states. does In you can see from the Parke-Taylor+ h |+ + formula| that MHV amplitudes do not have multi-particle not correspondsection 10 towe a expose physicalA6[1 an2 underlying3 pole4 56 of] with the geometric other scattering arrangements interpretation amplitude: of helicities are of itinequivalent such is a identities.spurious to the split-helicity pole.The poles, only 2-particle poles. And you have seen that the 6-gluon NMHV amplitude has both residue of this unphysical pole better be zero — and43 it is: the spurious pole cancels in the 2- and 3-particle poles. But as you stare intensely at (3.34), you will also note that there is sum ofThe the Grassmannian, two BCFW diagrams the tree in ( contour,3.34). It and is typical the twistor that BCFW string packs the information a strange denominator-factor 5 1+62] in the result from each BCFW diagram.K This does of theIn amplitudes Witten’s twistor into compact string [h55 expressions,|], mentioned| but briefly the at cost the is end the of appearance Section 5.2,theN of spuriousMHV poles; super- not correspond to a physical pole of the scattering amplitude: it is a spurious pole.The this meansamplitudes that in in the= BCFW 4 SYM representation are calculated the as open locality string of current the underlying algebra correlators field theory integrated is not residue of this unphysicalN pole better be zero — and it is: the spurious pole cancels in the manifest.over Elimination the moduli space of spurious of degree poles (K+1) in the curves representations in supertwistor of amplitudes space. It turns leads out to that interesting the RSV sum of the two BCFW diagrams in (3.34). It is typical that BCFW packs the information resultsconnected [22] that prescriptionwe discuss in[115 a later] for section. the twistor string has a direct relation to the BCFW recur- of thesion amplitudes relations. into Moreover, compact di↵ expressions,erent BCFW but representations the cost is the are appearance related via (higher-dimensional of spurious poles; + + + Finallythis means, let us that for incompleteness the BCFW note representation that the color-ordered the locality amplitudes of the underlyingA6[12 field3 4 theory5 6 ] is and not + + + Amanifest.6[12 3 Elimination4 56 ] with of other spurious arrangements poles in the of representations helicities147 are inequivalent of amplitudes to leads the split-helicity to interesting results [22] that we discuss in a later section.

43 + + + Finally, let us for completeness note that the color-ordered amplitudes A6[123 45 6 ] and + + + A6[12 34 56 ] with other arrangements of helicities are inequivalent to the split-helicity

43 It turns out…. that amplitudes of planar N=4 super Yang-Mills theory (which includes the gluon amplitudes) have a geometric interpretation That’s what the amplituhedron story is about It turns out….

that amplitudes of planar N=4 super Yang-Mills theory (which includes the gluon amplitudes) have a geometric interpretation That’s what the amplituhedron story is about

Amplituhedron From Wikipedia, the free encyclopedia

An amplituhedron is a geometric structure that enables simplified calculation of particle interactions in some quantum field theories. In planar N = 4 supersymmetric Yang–Mills theory, an amplituhedron is defined as a mathematical space known as the Positive Grassmannian.[1]

Amplituhedron theory challenges the notion that space-time locality and unitarity are necessary components of a model of particle interactions. Instead, they are treated as properties that emerge from an underlying phenomenon.[2][3]

The connection between the amplituhedron and scattering amplitudes is at present a conjecture that has passed many non-trivial checks, including an understanding of how locality and unitarity arise as consequences of positivity.[1]

Research has been led by Nima Arkani-Hamed. Edward Witten described the work as “very unexpected" and said that "it is difficult to guess what will happen or what the lessons will turn out to be."[4]

Contents

1 Description 2 Implications 3 See also 4 References 4.1 Notes 4.2 Bibliography

Description

In the approach, the on-shell scattering process "tree" is described by a positive Grassmannian, a structure in algebraic geometry analogous to a convex polytope, that generalizes the idea of a simplex in projective space.[2] A polytope is a kind of higher dimensional polyhedron, and the values being calculated are scattering amplitudes, and so the object is called an amplituhedron.[5][1]

Using Twistor theory, BCFW recursion relations involved in the scattering process may be represented as a small number of Twistor diagrams. These diagrams effectively provide the recipe for constructing the positive Grassmannian, i.e. the amplituhedron, which may be captured in a single equation.[2] The scattering amplitude can thus be thought of as the volume of a certain polytope, the positive Grassmannian, in momentum twistor space.[1]

When the volume of the amplituhedron is calculated in the planar limit of N = 4 D = 4 supersymmetric Yang– Mills theory, it describes the scattering amplitudes of subatomic particles.[5] The amplituhedron thus provides a more intuitive geometric model for calculations whose underlying principles were until then highly abstract.[6] 5 Symmetries of = 4 SYM3 On-shell recursion relations at tree-level 3.25.4 BCFW Momentumrecursion relations twistors 10 Polytopes 10.23 On-shell NMHV recursion tree relations superamplitude at tree-level as the3.2 volume BCFW of recursion a polytope relations 3 On-shell recursionN relations at tree-level 3.2 BCFW recursion relations

Thus this is the first time we see a lower-point NMHV amplitude shows up in the recursion Thus this is the first time werelations. see a lower-point This is quite NMHV generic: amplitudeThus the this BCFW shows is the up relations in first the time recursion are we recursive see a lower-point both in particle NMHV number amplituden shows up in the recursion The 5-bracket in (5.55) correspond to the termsrelations. in This the is super-BCFW quite generic: the BCFW expansion relations are of recursive the both super- in particle number n relations.10.2 This is quite NMHV generic:and the in N BCFW treeK MHV relations superamplitude level K are. recursive both in particle as the number volumen of a polytope K K and in N MHV level K. amplitude;and in specifically N MHV level K we. have seen in Section 4.4.2 how each Rn2k arises from an MHV MHV The two BCFW representations (3.31) and (3.35) look quite di↵erent. In order for both to The two BCFW representations (3.31) and (3.35) look quite di↵erent.⇥ In order for both to BCFW diagramThe twoThe BCFW while simplest representations thedescribe remainingNMHV (3.31) the and samecase (3.35R amplitude,) isnjk look the’s quite with 5-point there di↵erent. hasj> to (anti-MHV)superamplitude In2 be order appear a certain for both identity via to recursion that ensures that from diagrams the BCFW describe the same amplitude, there has to be a certain identitydescribe that the ensures same that amplitude, diagrams there has to be a certain identity that ensures that diagrams A+B = A0 +B0+C0. To show that this identity holds requires a nauseating trip through diagram with NMHV anti-MHV subamplitudes.A+B = As A0 +B we0+C have0. To show discussed, that this identity this holds means requires that a nauseating the trip through A+B = A0 +B0+C0. To showSchouten that thisidentities identity and holds momentum requires conservation a nauseating relations trip through in order to manipulate the angle and Schouten identities and⇥ momentum conservation relationsNMHV in orderSchouten to manipulate identities theand angle momentumMHV and conservation relations in order to manipulate the angle and representation (5.55) is notsquare be brackets unique, intoA the since5 right thereform:[1, 2 numerical, 3 are, 4, 5] many checks = canA equivalent5 save you a lot1, of BCFW2 energy, 3, 4, when5 expansions dealing for (10.20) square brackets into the right form: numerical checks can savesquare you a lotbrackets of energy into when the right dealing form: numerical checks can save you a lot of energy when dealing with amplitudesTowards with more than 5amplituhedrons external lines. It turns out that⇥ the identities that guarantee a given amplitude,with amplitudes with depending more than 5 external on the lines. choice It turns out ofwith that lines the amplitudes identities in the with that BCFWmore guarantee than 5 externalshift. lines. This It turns implies out that that the identities the that guarantee the equivalence of BCFW expressionsNMHV such as A+B and A0 +B0+C⇥ 0 actually originate⇤ from4 the equivalenceThus, of BCFW up to expressions the MHV such as factor, A+B andA A0 +Bthe0 equivalence+Cis0 actually the of volume originate BCFW expressions from of a 4-simplex such as A+B andin A0 +B. 0+C0 actually originate from powerful residue theorems [215] related to quite di↵erent formulations of the amplitudes.CP This dual conformalpowerful residue invariants theorems [21 (]5.54 related) to are quite linearly di↵erent formulations dependent.powerful of residue the amplitudes. theorems For example, [21 This] related to compare quite di↵erent for formulationsn =6the of the amplitudes. This To gethas an to doidea with theof descriptionthe geometric of amplitudes picture, in the Grassmannian we will — we write get to that in Section result ofhas the to do recursionsNext, with the fordescription relationsthe of NMHV amplitudes based 6-point in the on Grassmannian the superamplitude,has BCFW to — do we with get supershifts theto that description consider in Section of [6 amplitudes, the1 and [2 in, 3 the [1, Grassmanniansuper-BCFW2 :theyhaveto — we get representation to that in Section 10, but wanted to givethe you 6-gluon a10 hint, but of this wanted amplitude curious to give point you here. a as hint10, butof this wanted curious to give point you here. a hint of thisi curiousi pointi here. give the sameon result, the LHS so of (10.1): I Exercise 3.8 I Exercise 3.8 I Exercise 3.8 Show that the BCFW recursionShow relations that the based BCFW on the recursion [2, 3 -shiftShow relations give that the the based following BCFW on the recursionrep- [2, 3 -shift relations give the based following on the rep-[2, 3 -shift give the following rep- NMHV i i i [6, 1, 2, 3,resentation4] + [6 of, 1A the, 2 6-point, 4, 5][1 ‘alternating +,resentation2 [6, 3,,24, helicity’,35, of,46] the, 5] gluon 6-point = amplitude: [1 ‘alternating,22,,33, ,44, 6, helicity’5], 1 ++ [1 gluon, 2,2 amplitude:3, 3, 5, 4, ,6]5, +6 [1,+3, 42,,54,,6]5,.6, 1(5.56). (10.21) 6 / resentation of the 6-point ‘alternating helicity’ gluon amplitude: + + + + + + + + + A6[1 23 45 6]= M2 +AM6[144,26+,13M,246, 5, 26,3,]=4,6 M2 +6(3.36)A,26M,[134,42+,3 4M4,565,6,,62]= M2,24,5+,6(3.36)M, 4 6+,1,2M,46 , (3.36) { } { h ⇥} { i} h ⇤ {i }h ⇥{ }i h{ } ⇤ i {h ⇥ } {i h} { ⇤ }i Using the cyclicwhere property of thewhere 5-bracket, we canwhere write this 4 4 Apart from the overalli, i +2 MHV[i +3,i factor,1]| {z thei, i 6-point+2} 4[i +3|,i NMHV1]4{z superamplitude}i, i +2 4[|i +3,i{z1]4 is} the sum of the Mi = Mh = i Mhi = i . h i . . { } P˜2 i P˜ i + 3] i +2P˜ i i 1] i, i 2+1 i +1,i+2 [i +3,i 2][i ˜22,i˜ 1] ˜ i“5-bracketsi {i ”: } [P˜i,i j,P˜ k,i + l, 3] i m4+2]=det(P˜{i }1] i,P i Z+1i PZiii+1+Z 3],i+2iZ+2[ZiP+3i i ,i)1] i,2][ ii+12,ii +11],i+2 [i +3,i 2][i 2,i 1] [2, 3, 4, 6, 1] + [2, 3h, |4,| 5, 6]h +| [2|,4, 5h , 6i , 1]ihi = [3, 4i , 5i, 6, 1] +i h [3i| ,| 5j, 6kh, 1,l2]| |m + [3h , 4, 5ih, 1, 2] . i (5.57) volumes of three 4-simplicesh | in| CPh ; we| | expect h thisih (3.37) to bei the volume of (3.37) a polytope obtained(3.37) by and P˜ = P . [Hint: M is the value of the 12-channel diagram.]˜ i i,i+1,i+2 and4 P˜ = P . [Hint: MandisP thei = valuePi,i+1,i of+2 the. [Hint: 12-channelM4 is diagram.] the value of the 12-channel diagram.] somehow gluing{ theI} threei i,i+1 simplices,i+2 together.4 But how{ exactly} does this work? To address this In Section 10 weVectors discover that Z each liveMi incan a be 5-dimensional understood{ as the} residue associatedspace. with Now you see that the LHS looksIn Section like{ } 10 thewe discover result that of eachIn a Section [2M, 3i 10cansupershift,we be discover understood that as each while theM residuei thecan associated be RHS understood comes with as the from residue associated with a very interesting contour integral (di↵erent from the one used in the{ BCFW} argument). { } question,(5? it Think is usefula verythe to interesting 4-momenta examine contour the integral (E,p)a poles very (di interesting↵ erentand ini the from “particle contour 5-brackets. the one integral used type”.) in (di the↵erent BCFW from argument). the one used in the BCFW argument). a[3, 4 supershift.The fourth thing worth In fact, discussing you’ll further note are the poles that of scattering the LHS amplitudes. and independentlyColor-ordered the RHS are invariant i The fourth thing worth discussingThe furtherfourth thing are theworth poles discussing of scattering further amplitudes. are the poles Color-ordered of scattering amplitudes. Color-ordered tree amplitudesRecall can from have physical (5.45 poles) and only ( when5.47 the) that momenta momentum of adjacent external twistor lines 4-brackets i 1,i,j 1,j in the de- under i i + 2. WeTo can help alsotree us amplitudes reverse make can the contact have labels physicaltree with in amplitudes poles the the only 5-brackets can whenamplituhedron, have the physical momenta at poles no of cost adjacent only we when to externalh get the momenta lines of adjacenti external lines !go collinear.nominator We touched gives this point local already poles, when we whereas discussedgo collinear. the 4-brackets 3-particle We touched poles. like In this fact,k, point i already1,i,j whengive we discussed spurious the 3-particle ‘non-local’ poles. Inpoles. fact, you can see from the Parke-Taylorgo collinear. formula We that touched MHVi amplitudes this point do already not have when multi-particle we discussed the 3-particle poles. In fact, can restrict the Z ’s to ayou lower can see dimensional from the Parke-Taylorh subspace formula thati MHV amplitudes do not have multi-particle poles, only 2-particle poles. Andyou can you seehave from seen the that Parke-Taylor the 6-gluon NMHV formula amplitude that MHV has amplitudes both do not have multi-particle [2, 3, 4, 6, 1]Examining + [2, 3, 4, 5 the, 6] denominator + [2, 4, 5, 6, 1] terms = [3poles,, of1, only the6, 5 2-particle, 5-brackets4] + [3 poles., 2, And1 in, 6 (you,10.215] have +), [3seen, we2 that, 1 find, the5, 6-gluon4] that. NMHV each(5.58) amplitude of them has has both 2 2- and 3-particle poles. Butpoles, as you only stare 2-particle intensely poles.at (3.34 And), you you will have also seen note that that the there 6-gluon is NMHV amplitude has both 2- and 3-particle poles. But as you stare intensely at (3.34), you will also note that there is a strangespurious denominator-factor poles;2-5 and1+6 they 3-particle2] are in the listed poles. result fromBut under as each you BCFW each stare intensely diagram. 5-bracket. at This (3.34 does The), you spurious will also note poles that there come is in pairs — for ex- h | | a strange denominator-factor 5 1+62] in the result from each BCFW diagram. This does This statesnot correspond that the to a ‘parity physicala pole conjugate’strange of the denominator-factor scattering supershifts amplitude:5 1+6 it is [22] a, spurious3 in theand result pole [3 from,.The2 h eachgive| BCFW| identical diagram. results. This does In fact, ample 4, 6, 1, 2 and 6, 1, 2, 4 inhnot the| correspond first| and to a third physical 5-brackets pole of the scattering — and amplitude: cancel it in is a thespurious sum pole (10.21.The), residue of this unphysical polenot better correspond be zero to — a and physical it is: pole the spurious of the scatteringi pole cancels amplitude: ini the it is a spurious pole.The h i h i residue of this unphysical pole better be zero — and it is: the spurious pole cancels in the we can nowsum of conclude theas two required BCFW that diagrams byresidue any locality in (of adjacent3.34 this). of unphysical It is the typical supershifts physical pole that better BCFW amplitude. be packs are zero the equivalent.— and information it In is: the the spuriousgeometric The identity pole cancels description ( in5.58 the )shows of a 5-bracket as sum of the two BCFW diagrams in (3.34). It is typical that BCFW packs the information of the amplitudes into compactsum expressions, of4 the two but BCFW the cost diagrams is the appearance in (3.34). of It spurious is typical poles; that BCFW packs the information up again in Sectionsa 4-simplex9 and in10 where, each we of the will five understandof the factors amplitudes in its intothe origins compact denominator expressions, better. ofbut the the cost volume-expression is the appearance of spurious (i.e. poles; the this means that in the BCFWof representationCP the amplitudes the into locality compact of the expressions, underlying fieldbut the theory cost is is not the appearance of spurious poles; this means that in the BCFW representation the locality of the underlying field theory is not manifest.5-bracket) Elimination of spurious isthis determined poles means in that the representations in by the BCFW a vertex of representation amplitudes of the leads the associated to locality interesting of the 4-simplex. underlying field In theory particular, is not spurious poles Exercise 5.15 manifest. Elimination of spurious poles in the representations of amplitudes leads to interesting I results [22] that we discuss in a later section. 4 must be associatedmanifest. Elimination with vertices of spuriousresults in poles [22 in] that thethat representations we discuss somehow in a of later amplitudes section. ‘disappear’ leads to interesting from the polytope whose CP + + + Finally, let us for completenessresults note that [22] the that color-ordered we discuss in amplitudes a later section.A6[123 45 6 ] and + + + Use cyclicity+ + of+ the 5-brackets to show thatFinally the, let us tree-level for completeness 6-point note that NMHV the color-ordered superamplitude amplitudes A6[123 45 6 ] and A6[12 3volume4 56 ] with equals other arrangements the sum of of helicities the simplex-volumes are inequivalent to the split-helicity in (10.21). We now discuss how the ‘spurious’ Finally, let us for completeness note that+ the+ color-ordered+ amplitudes A [1 2 3+4 5+6+] and A6[12 34 56 ] with other arrangements6 2 of helicities are inequivalent to the split-helicity can be writtenvertices in disappear theA form[1 2+3 in4+ the5 6+] sum with other of simplices. arrangements of Let helicities us start are inequivalent in towhere the split-helicity the polytopes are easier 6 43 CP to draw. 1 43 NMHV = MHV R43 +cyclic , (5.59) A6 A6 ⇥ 2 146 2 Polytopes in CP ⇣ ⌘ where “cyclic” means the sum over advancing the labels cyclically, i.e. R146 + R251+4 In 2 dimensions, consider the 4-edge polytope more terms. (2,3) 3 (3,4) The presence of these equivalence-relations between the2 dual conformal invariants may strike you as rather peculiar and you may wonder if it has(1,2) a deeper meaning. Furthermore, while the expressions in (5.53) and (5.55) are extremely simple, they lack one key aspect when compared to 4 the Parke-Taylor superamplitude: cyclic invariance. The1 presence of dual conformal symmetry relies heavily on the cyclic ordering of the amplitude, and hence it is somewhat surprising that the manifestly dual conformal invariant form of the superamplitude(4,1) (5.55) breaks manifest (10.22) cyclic invariance. One might say that we are asking too much of the amplitude, but considering 2 the payo↵ weAll have vertices reaped for from this CP the“amplitude” innocent chase are defined of manifest by adjacent momentum edges andconservation, is in this sense we local. We will boldly pushwould ahead like with to compute our pursuit the area of of “having the 2-polytope cakes and (10.22 eating) using them” 2-simplex in Sections volumes9 and [a, b, c]. There 10. are several di↵erent ways to do this, corresponding to di↵erent triangulations of the polytope.

159 84 10 Polytopes 10.2 NMHV tree superamplitude as the volume of a polytope

As an example, introduce a ‘non-local’ point (1, 3) as the intersection of lines 1 and 3. The resulting triangulation is

(1,3) (1,3) (2,3) 3 (3,4) 3 (3,4) 2 (1,3) 3 (2,3) (1,2) = 1 1 4 4 2 1 (1,2)

(4,1) (4,1) = 4, 1, 3 2, 1, 3 = ⇥4, 1, 3⇤ + ⇥1, 2, 3⇤ . (10.23)

The area of the 4-edge polytope is given⇥ by the⇤ di↵⇥ erence⇤ of two triangular areas. The non-local vertex (1, 3) appears in both triangles. Comparing the last two lines, the sign of the 3-bracket 10 Polytopes indicates10.2 the orientation NMHV tree of the superamplitude triangle with respect as the volume to a particular of a polytope predetermined ordering of all edges (or, in higher dimensions, boundaries). As an example, introduceIt is useful a ‘non-local’ to also considerpoint (1, another3) as the triangulation, intersection so of introduce lines 1 and the 3. point The (2, 4): resulting triangulation is (2,4) (2,4)

(4,1)Then the analogue(4,1) of the(4,1) 6-gluon NMHV amplitude(4,1) takes the form = 4, 1, 3 2, 1, 3 = 1, 2, 4 2, 4, 3 Amplitude =~ ⇥4, 1, 3⇤ + ⇥1, 2, 3⇤ .== ⇥1, 2, 4⇤ + ⇥2, 3, 4⇤ . (10.23) (10.24) ⇥ ⇤ ⇥ ⇤ The area of the 4-edgeThe polytope two triangulations is given⇥ by the⇤ (10.23 di↵⇥ erence) and⇤ of (10.24 two) triangular compute theareas. same The area non-local (“amplitude”), so we have 2 vertex (1, 3) appears ina bothCP version triangles. of the Comparing identity ( the10.1), last namely two lines, [4, 1, the3] + sign [1, 2, of3] the = [1 3-bracket, 2, 4] + [2, 3, 4] which can also indicates the orientationbe of written the triangle with respect to a particular predetermined ordering of all edges (or, in higher dimensions, boundaries).2, 3, 4 1, 3, 4 + 1, 2, 4 1, 2, 3 =0. (10.25) It is useful to also considerI anotherExercise triangulation, 10.3 ⇥ so⇤ introduce⇥ ⇤ the⇥ point⇤ (2,⇥4): ⇤

(2,4)Suppose the 4-vertex polytope in(2,4) the example above was not convex as drawn in (10.24): show that the volume of a non-convex 4-vertex polytope can also be written [1, 2, 4] + (2,3) 2 3 [2, 3, 4]. 2 (3,4) (2,4) 2 (1,2) (1,2)4 Polytopes in=CP 4 4 (1,2) 4 2 4 (4,1) Extending the simple CP example to CP , one ﬁnds3 that the BCFW representation of a 6- 1 1 point NMHV tree superamplitude corresponds to a triangulation of the associated polytope by

(4,1) introduction of three new(4,1) auxiliary vertices. This allows one to use the given external data, the = 1, 2, 4 2, 4, 3 160 = ⇥1, 2, 4⇤ + ⇥2, 3, 4⇤ . (10.24)

The two triangulations (10.23) and (10.24⇥ ) compute⇤ ⇥ the⇤ same area (“amplitude”), so we have 2 a CP version of the identity (10.1), namely [4, 1, 3] + [1, 2, 3] = [1, 2, 4] + [2, 3, 4] which can also be written 2, 3, 4 1, 3, 4 + 1, 2, 4 1, 2, 3 =0. (10.25) I Exercise 10.3 ⇥ ⇤ ⇥ ⇤ ⇥ ⇤ ⇥ ⇤ Suppose the 4-vertex polytope in the example above was not convex as drawn in (10.24): show that the volume of a non-convex 4-vertex polytope can also be written [1, 2, 4] + [2, 3, 4].

4 Polytopes in CP 2 4 Extending the simple CP example to CP , one ﬁnds that the BCFW representation of a 6- point NMHV tree superamplitude corresponds to a triangulation of the associated polytope by introduction of three new auxiliary vertices. This allows one to use the given external data, the

160 10 Polytopes 10.2 NMHV tree superamplitude as the volume of a polytope

As an example, introduce a ‘non-local’ point (1, 3) as the intersection of lines 1 and 3. The resulting triangulation is

(1,3) (1,3) (2,3) 3 (3,4) 3 (3,4) 2 (1,3) 3 (2,3) (1,2) = 1 1 4 4 2 1 (1,2)

(4,1) (4,1) = 4, 1, 3 2, 1, 3 = ⇥4, 1, 3⇤ + ⇥1, 2, 3⇤ . (10.23)

(1,3) (2,3) (1,3) 3 3 (3,4) (2,3) 3 (3,4) 2 2 2 (3,4) (2,4) (1,3) 3 (2,3) 2 (1,2) (1,2) Towards amplituhedrons= (1,2) 4 (1,2) 4 = 1 1 4 4 4 2 3 (4,1) 1 To help us make1 contact with the amplituhedron,1 (1,2) we can restrict the Zi’s to a 3-dimensional subspace. 10 Polytopes 10.2 NMHV tree superamplitude as the volume of a polytope (4,1)Then the analogue(4,1) of the(4,1) 6-gluon NMHV amplitude(4,1) takes the form = 4, 1, 3 2, 1, 3 = 1, 2, 4 2, 4, 3 = As an example, introduce a ‘non-local’ point (1Amplitude, 3) as the =~ intersection ⇥4, 1, 3⇤ + ⇥1 of, 2, lines3⇤ .= 1⇥1, and2, 4⇤ + 3.⇥2 The, 3, 4⇤ . (10.23) (10.24) ⇥ ⇤ ⇥ ⇤ ⇥ ⇤ ⇥ ⇤ resulting triangulation is The area of the 4-edgeIt The polytopeis very two interesting triangulations is given by thethat (10.23 di the↵erence) mathematical and of (10.24 two) triangular compute expression theareas. same Thethat area non-local (“amplitude”), so we have 2 vertex (1, 3) appearshides ina bothCP behindversion triangles. the of the3-bracket Comparing identity ([1,2,3] the10.1), last namely is two actually lines, [4, 1, the3] exactly + sign [1, 2, of3] the = [1 3-bracket, 2, 4] + [2, 3, 4] which can also (1,3) (1,3) (2,3) 3 indicates(3,4) the orientationthebe3 area of written the of triangle(3,4) a triangle with those respect edges to a particularare given predetermined by the vectors ordering of all edges (or, in higherZ1, dimensions, Z2, and Z3 boundaries).. 2, 3, 4 1, 3, 4 + 1, 2, 4 1, 2, 3 =0. (10.25) 2 (1,3) 3 (2,3) (1,2) It is useful to also considerI anotherExercise triangulation, 10.3 ⇥ so⇤ introduce⇥ ⇤ the⇥ point⇤ (2,⇥4): ⇤ = 1 1 4 (2,4)Suppose4 the 4-vertex polytope in(2,4) the example above was not convex as drawn in (10.24): 1 2 show that the volume(1,2) of a non-convex 4-vertex polytope can also be written [1, 2, 4] + (2,3) 2 3 [2, 3, 4]. 2 (3,4) (2,4) (4,1) (4,1) 2 (1,2) (1,2)4 Polytopes in=CP 4 = 4, 1, 3 2, 1, 3 4 (1,2) 4 2 4 (4,1) Extending the simple CP example to CP , one ﬁnds3 that the BCFW representation of a 6- = 41, 1, 3 + 1, 2, 3 . 1 (10.23) ⇥ ⇤ point⇥ NMHV⇤ tree superamplitude corresponds to a triangulation of the associated polytope by introduction of three new auxiliary vertices. This allows one to use the given external data, the The area of the 4-edge polytope is given⇥ by(4,1) the⇤ di↵⇥ erence⇤ of two triangular(4,1) areas. The non-local = 1, 2, 4 2, 4, 3 vertex (1, 3) appears in both triangles. Comparing the last two lines, the sign of the 3-bracket160 indicates the orientation of the triangle with respect to a particular= ⇥1, 2, 4⇤ predetermined+ ⇥2, 3, 4⇤ . ordering of (10.24) all edges (or, in higher dimensions,The two boundaries). triangulations (10.23) and (10.24⇥ ) compute⇤ ⇥ the⇤ same area (“amplitude”), so we have a 2 version of the identity (10.1), namely [4, 1, 3] + [1, 2, 3] = [1, 2, 4] + [2, 3, 4] which can also It is useful to also consider anotherCP triangulation, so introduce the point (2, 4): be written

(2,4) 2, 3, 4 (2,4)1, 3, 4 + 1, 2, 4 1, 2, 3 =0. (10.25) I Exercise 10.3 ⇥ ⇤ ⇥ ⇤ ⇥ ⇤ ⇥ ⇤ (2,3) 3 2 2 (3,4) Suppose the 4-vertex polytope in the example above was(2,4) not convex as drawn in (10.24): 2 (1,2) show that the volume of a non-convex 4-vertex polytope can also be written [1, 2, 4] + = (1,2) 4 [2, 3, 4]. 4 (1,2) 4 3 (4,1) 1 1 4 Polytopes in CP 2 4 Extending the simple CP example to CP , one ﬁnds that the BCFW representation of a 6- (4,1) (4,1) point NMHV tree superamplitude corresponds to a triangulation of the associated polytope by = 1, 2, 4 2, 4, 3 introduction of three new auxiliary vertices. This allows one to use the given external data, the = ⇥1, 2, 4⇤ + ⇥2, 3, 4⇤ . (10.24) 160 The two triangulations (10.23) and (10.24⇥ ) compute⇤ ⇥ the⇤ same area (“amplitude”), so we have 2 a CP version of the identity (10.1), namely [4, 1, 3] + [1, 2, 3] = [1, 2, 4] + [2, 3, 4] which can also be written 2, 3, 4 1, 3, 4 + 1, 2, 4 1, 2, 3 =0. (10.25) I Exercise 10.3 ⇥ ⇤ ⇥ ⇤ ⇥ ⇤ ⇥ ⇤ Suppose the 4-vertex polytope in the example above was not convex as drawn in (10.24): show that the volume of a non-convex 4-vertex polytope can also be written [1, 2, 4] + [2, 3, 4].

160 10 Polytopes 10.2 NMHV tree superamplitude as the volume of a polytope

As an example, introduce a ‘non-local’ point (1, 3) as the intersection of lines 1 and 3. The resulting triangulation is

(1,3) (1,3) (2,3) 3 (3,4) 3 (3,4) 2 (1,3) 3 (2,3) 10 Polytopes (1,2) 10.2 NMHV tree superamplitude as the volume of a polytope = 1 4 4 1 1 2 10.2 NMHV tree superamplitude as the volume of a polytope (1,2)

The simplest NMHV case is the(4,1) 5-point (anti-MHV)superamplitude(4,1)

NMHV = MHV4, 1, 3 2, 1, 3 A5 [1, 2, 3, 4, 5] = A5 1, 2, 3, 4, 5 (10.20) = 4, 1⇥, 3 + 1, 2, 3 . (10.23) NMHV ⇥ ⇥ ⇤ ⇥ ⇤ ⇤ 4 Thus, up to the MHV factor, A5 is the volume of a 4-simplex in CP . ⇥ ⇤ ⇥ ⇤ Next,The for the area NMHV of the 6-point 4-edge superamplitude, polytope is given consider by the [2 di, ↵3 erencesuper-BCFW of two representation triangular areas. The non-local i on thevertex LHS of (1 (10.1, 3)): appears in both triangles. Comparing the last two lines, the sign of the 3-bracket 10 Polytopes indicates10.2 the orientation NMHV tree of the superamplitude triangle with as respect the volume to a particular of a polytope predetermined ordering of ANMHV[1, 2, 3, 4, 5, 6] 2, 3, 4, 6, 1 + 2, 3, 4, 5, 6 + 2, 4, 5, 6, 1 . (10.21) all6 edges (or, in higher/ dimensions, boundaries). 4,6,1,2 , 2,3,4,6 6,2,3,4 , 4,5,6,2 2,4,5,6 , 6,1,2,4 h ⇥ i h ⇤ i h ⇥ i h ⇤ i h ⇥ i h ⇤ i As an example, introduceApartIt from is a useful the ‘non-local’ overall to also MHV point consider factor,| (1, another3){z the as 6-point} the triangulation,| intersection NMHV{z superamplitude} so of| introduce lines{z 1 is} and the the sum 3. point The of the (2, 4): 4 resulting triangulationvolumes is of three 4-simplices in CP ; we(2,4) expect this to be the volume of a polytope(2,4) obtained by somehow gluing the three simplices together. But how exactly does this work? To address this (1,3) (1,3) question,(2,3) it3 is(3,4) useful to examine(2,3) the3 poles3 in the(3,4) 5-brackets. 2 2 (3,4) (2,4) 2Recall from (5.45) and (5.47) that momentum twistor 4-brackets i 1,i,j 1,j in the de- (1,3) h 3 (2,3) i 2 (1,2) nominator gives(1,2) local poles, whereas 4-brackets= like (1,2)k, i 1,i,j give spurious ‘non-local’ poles. h i 4 (1,2) 4 Examining the denominator= terms1 of the 5-brackets in (10.211), we find that each of them has 2 4 4 4 2 3 (4,1) 1 spurious poles; they1 are listed under each 5-bracket. The spurious1 (1,2) poles come in pairs — for example 4, 6, 1, 2 and 6, 1, 2, 4 Towardsin the first and amplituhedrons third 5-brackets — and cancel in the sum (10.21), h i h i as(4,1) required by locality of(4,1) the physical(4,1) amplitude. In the geometric(4,1) description of a 5-bracket as 4 a 4-simplex inThatCP , helps each of theus fiveunderstand factors in the the denominator two expressions of the volume-expression (i.e. the = 4, 1, 3 2, 1, 3 = 1, 2, 4 2, 4, 3 5-bracket) is determined by a vertex of the associated 4-simplex. In particular, spurious poles 4 must be associated= with⇥ vertices4, 1, 3⇤ in+ CP⇥1, 2that, 3⇤ . = somehow= ⇥1, 2, ‘disappear’4⇤ + ⇥2, 3, from4⇤ . the polytope(10.23) whose (10.24) volume equals the sum of the simplex-volumes in (10.21). We now discuss how the ‘spurious’ ⇥ ⇤ 2 ⇥ ⇤ The area of the 4-edgevertices polytopeThe disappear twoand triangulations is why in given the ⇥they sum by the of have⇤ simplices. (10.23 di↵⇥ erenceto) be and Let⇤ the of us (10.24 two start same. triangular) in computeCP where areas. the the same polytopes The area non-local are (“amplitude”), easier so we have to draw. 2 vertex (1, 3) appears ina bothCP version triangles. of the Comparing identity ( the10.1 last), namely two lines, [4, 1, the3] + sign [1, 2, of3] the = [1 3-bracket, 2, 4] + [2, 3, 4] which can also We have four lines involved, labeled 1,2,3,4. indicates the orientationbe of written the triangle2 with respect to a particular predetermined ordering of Polytopes inFourCP lines in a plane make a polygon: all edges (or, in higher dimensions, boundaries).2, 3, 4 1, 3, 4 + 1, 2, 4 1, 2, 3 =0. (10.25) In 2 dimensions, consider the 4-edge polytope It is useful to also consider anotherExercise triangulation, 10.3 ⇥ so introduce⇤ ⇥(2,3) 3 ⇤(3,4) the⇥ point⇤ (2, 4):⇥ ⇤ I 2

(2,4)Suppose the 4-vertex(1,2) polytope in(2,4) the example above was not convex as drawn in (10.24): show that the volume of a non-convex4 4-vertex polytope can also be written [1, 2, 4] + (2,3) 3 2 1 2 (3,4)[2, 3, 4]. (2,4) (4,1) 2 (10.22) (1,2) = (1,2)4 Polytopes in CP 4 (1,2) 4 2 All vertices4 for this CP “amplitude”2 are deﬁned by adjacent4 edges and3 is in this(4,1) sense local. We 1 Extending the simple example to , one ﬁnds that the BCFW representation of a 6- would like to compute the area1 ofCP the 2-polytope (10.22CP) using 2-simplex volumes [a, b, c]. There are severalpoint di NMHV↵erent ways tree to superamplitude do this, corresponding corresponds to di↵erent to triangulations a triangulation of the of polytope. the associated polytope by (4,1) introduction of three new(4,1) auxiliary vertices. This allows one to use the given external data, the = 1, 2, 4 2, 4, 3 159 160 = ⇥1, 2, 4⇤ + ⇥2, 3, 4⇤ . (10.24)

160 Towards amplituhedrons

If we want to calculate the area of a polygon, one strategy 10 Polytopesis to triangulate it10.2 and NMHV use that tree superamplitudethe 3-bracket as theis the volume area of a polytope of a triangle:

As an example,One triangulation: introduce a ‘non-local’ point (1, 3) as the intersection of lines 1 and 3. The resulting triangulation is

(1,3) (1,3) (2,3) 3 (3,4) 3 (3,4) 2 (1,3) 3 (2,3) (1,2) = 1 1 4 4 2 1 (1,2)

(4,1) (4,1) = 4, 1, 3 2, 1, 3 = ⇥4, 1, 3⇤ + ⇥1, 2, 3⇤ . (10.23)

The area of the 4-edge polytope is given⇥ by the⇤ di↵⇥ erence⇤ of two triangular areas. The non-local vertex (1, 3) appears in both triangles. Comparing the last two lines, the sign of the 3-bracket indicates the orientation of the triangle with respect to a particular predetermined ordering of all edges (or, in higher dimensions, boundaries). It is useful to also consider another triangulation, so introduce the point (2, 4):

(2,4) (2,4)

(2,3) 3 2 2 (3,4) (2,4) 2 (1,2) = (1,2) 4 4 (1,2) 4 3 (4,1) 1 1

(4,1) (4,1) = 1, 2, 4 2, 4, 3 = ⇥1, 2, 4⇤ + ⇥2, 3, 4⇤ . (10.24)

160 10 Polytopes 10.2 NMHV tree superamplitude as the volume of a polytope

As an example, introduce a ‘non-local’ point (1, 3) as the intersection of lines 1 and 3. The resulting triangulation is

(1,3) (1,3) (2,3) 3 (3,4) 3 (3,4) 2 (1,3) 3 (2,3) (1,2) = 1 1 4 4 2 1 (1,2)

(4,1) (4,1) = 4, 1, 3 2, 1, 3 Towards= 4 , amplituhedrons1, 3 + 1, 2, 3 . (10.23) ⇥ ⇤ ⇥ ⇤ ⇥ ⇤ ⇥ ⇤ The areaIf of we the 4-edgewant polytopeto calculate is given bythe the area di↵erence of a of polygon, two triangular one areas. strategy The non-local vertex (1is, 3) to appears triangulate in both triangles. it and use Comparing that the last 3-bracket two lines, theis the sign ofarea the 3-bracket indicatesof the a orientationtriangle: of the triangle with respect to a particular predetermined ordering of all edges (or, in higher dimensions, boundaries). It is usefulAnother to also consider triangulation: another triangulation, so introduce the point (2, 4):

(2,4) (2,4)

(2,3) 3 2 2 (3,4) (2,4) 2 (1,2) = (1,2) 4 4 (1,2) 4 3 (4,1) 1 1

(4,1) (4,1) = 1, 2, 4 2, 4, 3 = ⇥1, 2, 4⇤ + ⇥2, 3, 4⇤ . (10.24)

160 10 Polytopes 10.2 NMHV tree superamplitude as the volume of a polytope

As an example, introduce a ‘non-local’ point (1, 3) as the intersection of lines 1 and 3. The resulting triangulation is

(1,3) (1,3) (2,3) 3 (3,4) 3 (3,4) 2 (1,3) 3 (2,3) (1,2) = 1 1 4 4 2 1 (1,2)

(4,1) (4,1) 10 Polytopes = 410.2, 1, 3 NMHV2, 1, 3 tree superamplitude as the volume of a polytope = ⇥4, 1, 3⇤ + ⇥1, 2, 3⇤ . (10.23) As an example, introduce a ‘non-local’ point (1, 3) as the intersection of lines 1 and 3. The The area of the 4-edge polytope is given⇥ by the⇤ di↵⇥ erence⇤ of two triangular areas. The non-local resulting triangulation10 is Polytopes 10.2 NMHV tree superamplitude as the volume of a polytope vertex (1, 3) appears in both triangles. Comparing the last two lines, the sign of the 3-bracket 10 Polytopes 10.2 NMHV(1,3) tree superamplitude as the(1,3) volume of a polytope indicates the orientation of the triangle(2,3) 3 (3,4) with respect to a particular3 (3,4) predetermined ordering of As an example, introduce a ‘non-local’ point (1, 3) as the intersection of lines 1 and 3. The all edges (or, in higher dimensions,2 boundaries). resulting triangulation is (1,3) 3 (2,3) (1,2) As an example, introduceIt is a useful ‘non-local’ to also point consider (1, another3) as the triangulation, intersection so of introduce lines 1 and the 3. point The (2, 4): (1,3)= (1,3) (2,3)1 3 (3,4) 3 1 (3,4) resulting triangulation is 4 4 2 1(2,4) 2 (2,4) (1,2) (1,3) 3 (2,3) (1,2) (1,3) (1,3) (2,3) (3,4) (2,3) 3 (3,4) 2 = 1 3 3 (4,1) 4 (4,1) 4 1 (3,4) 2 2 2 1 (2,4) (1,2) = 4(1,3), 1, 33 2(2,3), 1, 3 2 (1,2) (1,2) = (1,2) (4,1) 4 (1,2)(4,1) 4 = 1 = ⇥4, 1, 3⇤ + ⇥1, 2, 3⇤ . (10.23) 4 4 4 1 3 (4,1) 1 1 2 = 4, 1, 3 2, 1, 3 1 (1,2)⇥ ⇤ ⇥ ⇤ The area of theTowards 4-edge polytope amplituhedrons is given by the di ↵erence= ⇥4, of1, 3 two⇤ + ⇥ triangular1, 2, 3⇤ . areas. The non-local (10.23)

(4,1) vertex(4,1) (1, 3) appears(4,1) inThe both area triangles. of the 4-edge(4,1) Comparing polytope is given the⇥ lastby the⇤ two di↵⇥ erence lines,⇤ of the two sign triangular of the areas. 3-bracket The non-local 10 Polytopes So now we10.2 understand NMHV the tree identity superamplitude as the volume of a polytope 10 Polytopesindicates= 4 the, 1, 3 orientation2, 1vertex, 310.2 of= (1 the, NMHV3)1 triangle appears, 2, 4 tree in with superamplitude2 both, 4, respect3 triangles. to Comparing aas particular the volume the predetermined last of a two polytope lines, the ordering sign of the of 3-bracket all edges (or, in higherindicates dimensions, the orientation boundaries). of the triangle with respect to a particular predetermined ordering of = 4, 1, 3 + 1, 2, 3 . == 1, 2, 4 + 2, 3, 4 . (10.23) (10.24) ⇥ ⇤ ⇥ all edges⇤ (or,⇥ in higher⇤ ⇥ dimensions,⇤ boundaries). As an example, introduceAs anIt example, is useful a ‘non-local’ introduce to also considerpoint a ‘non-local’ (1, another3) as point the triangulation, (1 intersection, 3) as the intersection so of introduce lines 1 of and the lines 3. point 1 The and (2 3., 4): The ⇥ ⇤ ⇥ ⇤ ⇥ ⇤ ⇥ ⇤ The arearesulting of the 4-edge triangulation polytopeTheresulting twoas triangulationstwo is given triangulation different by the (triangulations10.23 di is↵erence)It and is useful of (10.24 two to of triangular also) the compute consider area areas. the anotherof a same polygon: The triangulation, area non-local (“amplitude”), so introduce so the we point have (2, 4): 2 (2,4) (2,4) vertex (1, 3) appears ina bothCP version triangles. of the Comparing identity ( the10.1 last), namely two lines, [4, 1, the3] + sign(2,4) [1, 2, of3] the = [1 3-bracket, 2, 4] + [2, 3, 4] which(2,4) can also (1,3) (2,3) (1,3) 3 (1,3) (2,3) (1,3)3 (3,4) 3 (3,4) indicates the orientationbe of written the triangle3 (3,4) with respect(2,3) to a particular(3,4)(2,3) predetermined2 ordering of 2 3 3 2 2 (3,4) 2 (3,4) (2,4) (2,4) all edges (or, in higher dimensions,2 boundaries).2, 3, 4 1, 3, 4 + 1, 2, 4 1, 2, 3 =0(1,3). 3 (2,3) (10.25) (1,2) (1,3) 3 2 (1,2) (2,3) 2 (1,2) (1,2) = 1 (1,2) = (1,2) 4 ⇥ ⇤ ⇥ ⇤ =⇥ ⇤ ⇥ ⇤ 1 4 (1,2)4 It is useful to also considerI anotherExercise triangulation, 10.3 = 4 so introduce the point (24, 4): 4 2 (1,2) 1 1 4 1 (1,2) 3 (4,1) 4 4 1 4 2 1 3 (4,1) 1(2,4)Suppose the 4-vertex1 polytope in(2,4) the example1 above(1,2) was not convex as drawn in (10.24): (4,1) (4,1) show that the volume of a non-convex(4,1) 4-vertex polytope can also(4,1) be written [1, 2, 4] + (2,3) 2 3 [2That(4,1), 3, 4]. means that(4,1) we can interpret(4,1)= 4, 1, 3 the “amplitude”2, 1, 3 (4,1) = as1, 2 , 4 2, 4, 3 2 (3,4) (2,4) the area of the= polygon:4, 1, 3 =2, 1⇥4,,31,=3⇤ + ⇥11,,22,,43⇤ .2 2,=4, 3⇥1, 2, 4⇤ + ⇥2, 3, 4⇤ . (10.23) (10.24) (1,2) (1,2) = 4 4 Polytopes in CP 4 ⇥ ⇤ ⇥(1,2) ⇤ The areaAmplitude of the 4-edge =~ polytope⇥4, 1,The3⇤ is+ two given⇥1 triangulations, 2, by3⇤ the .== di⇥↵1erence, (210.23, 4⇤ +) of and⇥ two2, (310.24 triangular, 4⇥⇤ .) compute⇤ areas.⇥ the(10.23)⇤ The same non-local area (“amplitude”),(10.24) so we have 4 2 4 3 (4,1) 1 Extending the simple CP example2 to CP , one finds that the BCFW representation of a 6- vertex (1, 3) appears1 in⇥ botha triangles.CP⇤ version⇥ Comparing of⇤ the⇥ identity the⇤ last (10.1⇥ two), namely lines,⇤ the[4, 1, sign3] + of[1, the2, 3] 3-bracket = [1, 2, 4] + [2, 3, 4] which can also The area of thepoint 4-edgeindicates NMHVThe polytope the two tree orientation triangulations superamplitude is given by of the thebe ( writtentriangle10.23 di corresponds↵erence) and with of ( respect10.24 two to a) triangular triangulation to compute a particular theareas. of same predetermined the The area associated non-local (“amplitude”), ordering polytope of by so we have 2 vertex (1, 3) appears ina bothCP version triangles. of the Comparing identity ( the10.1), last namely two lines,2 [4, 3,,14, the3] +1, sign [13,,42, of+3] the1 =, 2 [1, 4 3-bracket, 2, 4]1, +2, [23 , 3=0, 4]. which can also (10.25) (4,1) introductionall edges (or, of three in higher new(4,1) dimensions, auxiliary vertices. boundaries). This allows one to use the given external data, the indicates the orientationbe of written the triangle with respect to a particular predetermined ordering of It is useful= to also1, 2 consider, 4 2 another, 4,I3 Exercise triangulation, 10.3 so⇥ introduce⇤ ⇥ the point⇤ ⇥ (2, 4):⇤ ⇥ ⇤ all edges (or, in higher dimensions, boundaries). 2, 3, 4 1, 3, 4 + 1, 2, 4 1, 2, 3 =0. (10.25) Suppose the160 4-vertex polytope in the example above was not convex as drawn in (10.24): = ⇥1, 2, 4⇤ + ⇥2(2,4), 3, 4⇤ . (2,4) (10.24) It is useful to also considerI anotherExercise triangulation, 10.3 ⇥ show so⇤ that introduce⇥ the volume⇤ the⇥ of point a non-convex⇤ (2,⇥4): 4-vertex⇤ polytope can also be written [1, 2, 4] + The two triangulations (10.23) and (10.24⇥ )(2,3) compute⇤ 3 ⇥ the⇤[2 same, 3, 4]. area (“amplitude”),2 so we have 2 (3,4) (2,4) 2 (2,4)Suppose the 4-vertex polytope in(2,4) the example above was not convex as drawn in (10.24): a version of the identity (10.1), namely [4, 1, 3] + [1, 2, 3] = [1, 24, 4] + [2, 3, 4] which can also2 CP (1,2) Polytopes(1,2) in show that the volume= of a non-convexCP 4-vertex polytope can also4 be written [1, 2, 4] + be written (2,3) 2 2 4 4 (1,2) 3 4 Extending the simple CP example to CP , one finds that(4,1) the BCFW representation of a 6- (3,4)[2, 3, 4]. 3 2 2, 3, 4 11, 3, 4 + 1, 2, 4 point1 NMHV, 2, 3 tree=01 superamplitude. corresponds(10.25)(2,4) to a triangulation of the associated polytope by 2 (1,2) (1,2)4 introduction of three new auxiliary vertices. This allows one to use the given external data, the Exercise 10.3 ⇥ Polytopes⇤ ⇥ ⇤ in=⇥CP ⇤ ⇥ ⇤ 4 I (4,1) 4 (4,1) (1,2) 4 2 4 3 (4,1) Extending the simple CP =example1, 2, 4 to CP2, 4,, 3 one finds that160 the BCFW representation of a 6- Suppose the 4-vertex1 polytope in the example above1 was not convex as drawn in (10.24): point NMHV tree superamplitude corresponds to a triangulation of the associated polytope by show that the volume of a non-convex 4-vertex polytope= ⇥1, 2 can, 4⇤ + also⇥2, 3 be, 4⇤ . written [1, 2, 4] + (10.24) (4,1) introduction of three new(4,1) auxiliary vertices. This allows one to use the given external data, the [2, 3, 4]. The two triangulations (10.23) and (10.24⇥ ) compute⇤ ⇥ the⇤ same area (“amplitude”), so we have a 2 version of the= identity1, 2, (410.1), namely2, 4, 3 [4, 1, 3] + [1, 2, 3] = [1, 2, 4] + [2, 3, 4] which can also 4 CP Polytopes in CP 160 be written = ⇥1, 2, 4⇤ + ⇥2, 3, 4⇤ . (10.24) 2 4 Extending the simple CP example to CP , one2 finds, 3, 4 that1, 3 the, 4 BCFW+ 1, 2, 4 representation1, 2, 3 =0. of a 6- (10.25) ⇥ ⇤ ⇥ ⇤ point NMHVThe two tree triangulations superamplitudeExercise (10.23 corresponds) and 10.3 (10.24 to⇥ a) triangulation compute⇤ ⇥ the⇤ of same⇥ the associatedarea⇤ ⇥ (“amplitude”),⇤ polytope by so we have 2 I introductiona CP ofversion three new of the auxiliary identity vertices. (10.1), This namely allows [4, 1 one, 3] + to [1 use, 2, the3] = given [1, 2, external4] + [2, 3, data,4] which the can also Suppose the 4-vertex polytope in the example above was not convex as drawn in (10.24): be written show that the volume of a non-convex 4-vertex polytope can also be written [1, 2, 4] + 2, 3, 4 1160, 3, 4 + 1, 2, 4 1, 2, 3 =0. (10.25) [2, 3, 4]. ⇥ ⇤ ⇥ ⇤ ⇥ ⇤ ⇥ ⇤ I Exercise 10.3 4 Polytopes in CP 2 4 Suppose theExtending 4-vertex polytope the simple inCP theexample example to aboveCP , one was finds not that convex the as BCFW drawn representation in (10.24): of a 6- show that thepoint volume NMHV of tree a non-convex superamplitude 4-vertex corresponds polytope to a triangulation can also be of written the associated [1, 2, 4] polytope + by [2, 3, 4]. introduction of three new auxiliary vertices. This allows one to use the given external data, the

4 Polytopes in CP 160 2 4 Extending the simple CP example to CP , one ﬁnds that the BCFW representation of a 6- point NMHV tree superamplitude corresponds to a triangulation of the associated polytope by introduction of three new auxiliary vertices. This allows one to use the given external data, the

160 5 Symmetries of = 4 SYM 5.4 Momentum twistors 10 PolytopesN 10.2 NMHV tree superamplitude as the volume of a polytope 5 Symmetries of = 4 SYM 5.4 Momentum twistors N The 5-bracket10.2 in (5.55 NMHV) correspond tree to superamplitude the terms in the as super-BCFW the volume expansion of a polytope of the superamplitude; specifically we have seen in Section 4.4.2 how each R arises from an MHV MHV The 5-bracket in (5.55n2)k correspond to the terms⇥ in the super-BCFW expansion of the super- BCFW diagramThe while simplest the remaining NMHV caseRnjk is the’s with 5-pointj> (anti-MHV)superamplitude2 appear via recursion from the BCFW amplitude; specifically we have seen in Section 4.4.2 how each R arises from an MHV MHV diagram with NMHV anti-MHV subamplitudes. As we have discussed, this means that the n2k ⇥ ⇥ NMHVBCFW diagram whileMHV the remaining Rnjk’s with j>2 appear via recursion from the BCFW representation (5.55) is not be unique,A since5 there[1, 2, 3 are, 4, 5] many = A equivalent5 1, BCFW2, 3, 4, 5 expansions for (10.20) 10 Polytopes 10.2 NMHVdiagram tree superamplitude with NMHV asanti-MHV⇥ the volume subamplitudes. of a polytope As we have discussed, this means that the a given amplitude, depending on the choice of lines in the BCFW⇥ shift. This implies that the Thus, up to the MHV factor,representationANMHV is the volume (5.55) is of not a 4-simplex⇥ be unique, in⇤ since4. there are many equivalent BCFW expansions for dual conformal invariants (5.54) areTowards linearly5 dependent. amplituhedrons For example, compare forCPn =6the a given amplitude, depending on the choice of lines in the BCFW shift. This implies that the result of the recursionsNext, for relations the NMHV based 6-point on the superamplitude, BCFW supershifts consider [6, 1 theand [2, 3 [1,super-BCFW2 :theyhaveto representation 10.2 NMHV tree superamplitude asdual the conformal volume invariants of a polytope (5.54i ) arei linearlyi dependent. For example, compare for n =6the give the sameon result, theThe LHS so “real” of (10.1 version): of this is result of the recursions relations based on the BCFW supershifts [6, 1 and [1, 2 :theyhaveto i i The simplest[6, 1, NMHV2, 3, 4] + case [6, 1A, is2NMHV, the4, 5][1 5-point +, 2 [6, 3,,24,,35 (anti-MHV)superamplitude,,46], 5]give = [1 the,22,,33 same,,44,,65], 1 result, + [1+, 2,2 so3, ,35, 4, 6], 5, +6 [1,+3, 42, 5, 4, ,6]5,.6, 1(5.56). (10.21) 6 / 4,6,1,2 , 2,3,4,6 6,2,3,4 , 4,5,6,2 2,4,5,6 , 6,1,2,4 NMHV h ⇥MHVi h ⇤ i h ⇥ i h ⇤ i h ⇥ i h ⇤ i Using the cyclic propertyA5 of the[1, 2 5-bracket,, 3, 4, 5] = we[6A can,51, 2 write, 3, 4]1 this +, 2 [6, 3, 1, ,42,,54, 5] + [6, 2, 3, 4, 5] = [1(10.20), 2, 3, 4, 5] + [1, 2, 3, 5, 6] + [1, 3, 4, 5, 6] . (5.56) Apart from the overall MHV factor,| {z the⇥ 6-point} | NMHV{z superamplitude} | {z is} the sum of the [2, 3, 4, 6, 1] + [2, 3Recall,, 4, 5, 6] NMHVthese + [2, 4two, 5, 6 different, 1]Using = [34, 4 theexpressions, 5, cyclic6, 1]⇥ + property [3, 5came, 6, 1⇤, of 2]from the+4 [3 5-bracket,two, 4, 5 , 1, 2] . we(5.57) can write this Thus, up to the MHVvolumes factor, of threeA5 4-simplicesis the volume in CP ; we of expect a 4-simplex this to in beCP the. volume of a polytope obtained by 196 differentPolytopes applications of the BCFW recursion relations. somehow gluing the three simplices together. But how exactly does this work? To address this Next,Now for the you NMHV see that 6-point the LHS superamplitude, looks like the result consider[2 of, 3, a4 [2, 6,,3 the1]supershift, + [2 [2,,33, 4super-BCFW, 5, while6] + [2 the, 4, RHS5, representation6, 1] comes = [3, from4, 5, 6, 1] + [3, 5, 6, 1, 2] + [3, 4, 5, 1, 2] . (5.57) question, it is useful to examine the poles ini the 5-brackets.i a[3, 4 supershift. InThe fact, 5-bracket you’ll note encodes that the the LHS volume and independently of a 4-simplex the RHS are invariant Anonn-simplex the LHSisi of the ( convex10.1): hull of a set of n + 1 points. Examples: under i i +Recall 2. We from can also (5.45 reverse) and ( the5.47 labelsNow) that you in momentum the see 5-brackets that the twistor LHS at 4-brackets no looks cost like to thegeti result1,i,j of1 a,j [2,in3 thesupershift, de- while the RHS comes from 0-simplex = a point h i i NMHV! nominator gives local poles, whereasa[3, 4 4-bracketsThesupershift. higher-d like In version fact,k, i you’ll1,i,j of a notegive polygon spurious that the ‘non-local’ LHS and independently poles. the RHS are invariant A6 [1, 2, 3, 4, 5, 6]1-simplex2 =, line3, 4, segment6, 1 + i2, 3, 4, 5, 6 +h 2,4, 5, 6,i1 . (10.21) / under iis calledi + 2. a We polytope can also(10.11) reverse the labels in the 5-brackets at no cost to get [2, 3, 4, 6, 1]Examining + [2, 3,2-simplex4, 5 the, 6] denominator+ = [2, triangle4, 5, 6, 1] terms = [3, of1, 6 the, 5, 5-brackets4] + [3, 2, 1 in, 6 (,10.215] +), [3, we2, 1 find, 5, 4] that. each(5.58) of them has 2 4,6,1,2 , 2,3,4,6 6,2,!3,4 , 4,5,6,2 2,4,5,6 , 6,1,2,4 spurious3-simplex poles;h they⇥ = tetrahedron arei h listed⇤ i under. h ⇥ eachi 5-bracket.h ⇤ i h The⇥ spuriousi h ⇤ polesi come in pairs — for ex- This states that the ‘parity conjugate’ supershifts[2, 3, [24,,63, 1]and + [2 [3,,32, 4give, 5, 6] identical + [2, 4, 5 results., 6, 1] = In [3, fact,1, 6, 5, 4] + [3, 2, 1, 6, 5] + [3, 2, 1, 5, 4] . (5.58) ample 4, 6, 1, 2 |and {z6, 1, 2, 4} in the| first{zi and third} n 5-bracketsi+1| {z —} and cancel in the sum (10.21), AnApartn-simplex from is bounded the overall by n+1 MHVh (n 1)-simplices factor,i h the who 6-point intersecti NMHV each other superamplitude in 2 (n 2)- is the sum of the we can now concludeas required that byn any locality adjacent of the supershifts physical amplitude. are equivalent. In the geometric The n identity description (5.58)shows of a 5-bracket as simplices. For n+1 generic points in R , an 4n-simplex has an n-dimensional volume. (For CP volumes of three 4-simplices in CP 4; we expectThis this states to be that the the volume ‘parity of conjugate’ a polytope supershifts obtained [2 by, 3 and [3, 2 give identical results. In fact, it will be upn-complex again in dimensional.) Sectionsa 4-simplex9 Theand in volume10CPwhere, of each a polytope we of thewill five understand can factors be calculated in its the origins by denominator ‘tessellating’ better. of the volume-expressioni (i.e. the i somehow gluing the three simplices together.we But can how now exactly conclude does that this any work? adjacent To address supershifts this are equivalent. The identity (5.58)shows it into simplices,Exercise whose volumes5-bracket) 5.15 are easier is determined to calculate. by a vertex of the associated 4-simplex. In particular, spurious poles question,I it is useful to examine the poles inup the again 5-brackets. in4 Sections 9 and 10 where we will understand its origins better. Now that we know what simplicesmust be and associated polytopes with are, let vertices us progress in CP towardsthat understanding somehow ‘disappear’ how from the polytope whose Use cyclicity of the 5-brackets to show that4 the tree-level 6-point NMHV superamplitude the integrand in (10.6) representsvolume the equals volume the of sum a 4-simplex of the simplex-volumes inICPExercise, as claimed. 5.15 in As (10.21 a warm-up,). We now discuss how the ‘spurious’ Recall from (5.45) and (5.47) that momentum twistor 4-brackets i 1,i,j2 1,j in the de- we begin in 2 dimensionscan be written withvertices a 2-simplex in disappear the form (a triangle). in the sum of simplices. Let us starth in CP where thei polytopes are easier nominator gives local poles, whereas 4-brackets likeUsek, cyclicity i 1,i,j ofgive the 5-brackets spurious ‘non-local’ to show that poles. the tree-level 6-point NMHV superamplitude to draw.2 NMHV MHV h 1 i Area of a 2-simplex in CP = can beR written+cyclic in the, form (5.59) Examining the denominator termsA of6 the 5-bracketsA6 ⇥ in2 (10.21146 ), we find that each of them has 2 The area of a triangle in a 2-dimensional plane2 can be computed as spurious poles; theyPolytopes are listed inunderCP each 5-bracket. The⇣ spurious poles⌘ comeNMHV in pairs —MHV for ex-1 where “cyclic” means thex,y sum over advancing the labels cyclically, i.e.6 R146= +6R251+4 R146 +cyclic , (5.59) ample 4, 6, 1, 2 andIn 26, dimensions,1, 2, 4 in the consider first and the third4-edge 5-brackets polytope — and cancelA in the sumA (10.21⇥ 2), h morei terms.h 2 i x1 x2 x3 ⇣ ⌘ 2 3 1 as required byArea locality of the physical amplitude.= y1 Inwherey2 they3 geometric “cyclic”, (2,3) means description(3,4)(10.12) the sum of over a 5-bracket advancing as the labels cyclically, i.e. R + R +4 1 2 3 146 251 The presence4 of these equivalence-relations3 between111 the dual conformal invariants may strike a 4-simplex in 6, each of the five factors7 in the denominatormore terms.2 of the volume-expression (i.e. the CP6 7 you as rather6 peculiar and you may wonder7 if it has a deeper meaning. Furthermore, while the 5-bracket) is determined4 by a vertex of5 the associated (1,2) 4-simplex. In particular, spurious poles where theexpressions (xi,yi) are the in coordinates (5.53) and of (5.55 the) three are extremely vertices.The simple, presence they of lack these one equivalence-relations key aspect when compared between to the dual conformal invariants may strike must be associated with vertices in 4 that somehow ‘disappear’4 from the polytope whose the Parke-Taylor superamplitude:CP cyclic invariance.you as rather The1 peculiar presence and of dualyou may conformal wonder symmetry if it has a deeper meaning. Furthermore, while the I Exercise 10.1 volumerelies equals heavily the sum on the of cyclic the simplex-volumes ordering of theexpressions in amplitude, (10.21 in). (5.53 and We) hence now and ( discuss5.55 it is) are somewhat how extremely the surprising ‘spurious’ simple, they lack one key aspect when compared to If the area formula (10.12) is not familiar, you should derive it by showing that2 it is equivalent verticesthat disappear the manifestly in the dual sum conformal of simplices. invariant Letthe form us Parke-Taylor start of the in superamplitudeCP(4,1) superamplitude:where the (5.55 polytopes cyclic) breaks invariance. are manifest easier The(10.22) presence of dual conformal symmetry to the “ 1 base height”-formula that was imprinted on your brain in elementary school. to draw.cyclic2 ⇥ invariance.⇥ One might say that we arerelies asking heavily too much on the of thecyclic amplitude, ordering but of the considering amplitude, and hence it is somewhat surprising All vertices for this 2 “amplitude” are defined by adjacent edges and is in this sense local. We The 1’s inthe the payo last row↵ we of (have10.12) reaped are redundant fromCP the as we innocent canthat write the chase the manifestly same of manifest formula dual as momentum a conformal sum of invariant conservation, form we of the superamplitude (5.55) breaks manifest 2 thePolytopes 2 2will minors. boldly in In physics, pushwould ahead when like faced with to compute with our a pursuit redundancy the area ofcyclic “havingof we the invariance. can 2-polytope choosecakes and to One ( eliminate10.22 eating might) using itthem” say or 2-simplex that in weSections are volumes asking9 and [a, too b, c]. much There of the amplitude, but considering ⇥ CP promote it to a feature. Choosingare several the di latter,↵erent we ways define to three do this, 3-vectors corresponding along with to a reference di↵erent triangulations of the polytope. In 2 dimensions,10. consider the 4-edge polytopethe payo↵ we have reaped from the innocent chase of manifest momentum conservation, we vector: will boldly push ahead with our pursuit of “having cakes and eating them” in Sections 9 and xi 0 (2,3) (3,4) I 10. 3 WiI = yi , Z = 0 ,I=1, 2, 3 . 159 (10.13) 0 1 0 0 1 2 1 1 84 @ A (1,2)@ A The area can now be written x,y 4 84 2 1 1 1, 2, 3 Area 2 3 = h i , (10.14) 1 3 2 (Z0 W1)(Z0 W2)(Z0 W3) 6 7 · · · 6 7 (4,1) 6 7 (10.22) 4 5 where the 3-bracket is the contraction of a 3-index Levi-Civita tensor with the three Wi vectors: IJK 2 1All, 2, 3 vertices= ✏ W for1I W this2J W3K . Using“amplitude” (10.13), the are1, defined2, 3 -numerator by adjacent exactly edges equals andthe 3 is in3- this sense local. We h i CP h i ⇥ determinant in (10.12), so you might consider the trivial dot-products Z W = ZI W =1in would like to compute the area of the 2-polytope (10.22) using0 · i 2-simplex0 iI volumes [a, b, c]. There the denominator a provocation of your sense of humor. However, written in the form (10.14), the are several di↵erent ways to do this, corresponding to di↵erent triangulations of the polytope.

159 5 Symmetries of = 4 SYM 5.4 Momentum twistors 10 PolytopesN 10.2 NMHV tree superamplitude as the volume of a polytope 5 Symmetries of = 4 SYM 5.4 Momentum twistors N The 5-bracket10.2 in (5.55 NMHV) correspond tree to superamplitude the terms in the as super-BCFW the volume expansion of a polytope of the superamplitude; specifically we have seen in Section 4.4.2 how each R arises from an MHV MHV The 5-bracket in (5.55n2)k correspond to the terms⇥ in the super-BCFW expansion of the super- BCFW diagramThe while simplest the remaining NMHV caseRnjk is the’s with 5-pointj> (anti-MHV)superamplitude2 appear via recursion from the BCFW amplitude; specifically we have seen in Section 4.4.2 how each R arises from an MHV MHV diagram with NMHV anti-MHV subamplitudes. As we have discussed, this means that the n2k ⇥ ⇥ NMHVBCFW diagram whileMHV the remaining Rnjk’s with j>2 appear via recursion from the BCFW representation (5.55) is not be unique,A since5 there[1, 2, 3 are, 4, 5] many = A equivalent5 1, BCFW2, 3, 4, 5 expansions for (10.20) 10 Polytopes 10.2 NMHVdiagram tree superamplitude with NMHV asanti-MHV⇥ the volume subamplitudes. of a polytope As we have discussed, this means that the a given amplitude, depending on the choice of lines in the BCFW⇥ shift. This implies that the Thus, up to the MHV factor,representationANMHV is the volume (5.55) is of not a 4-simplex⇥ be unique, in⇤ since4. there are many equivalent BCFW expansions for dual conformal invariants (5.54) areTowards linearly5 dependent. amplituhedrons For example, compare forCPn =6the a given amplitude, depending on the choice of lines in the BCFW shift. This implies that the result of the recursionsNext, for relations the NMHV based 6-point on the superamplitude, BCFW supershifts consider [6, 1 theand [2, 3 [1,super-BCFW2 :theyhaveto representation 10.2 NMHV tree superamplitude asdual the conformal volume invariants of a polytope (5.54i ) arei linearlyi dependent. For example, compare for n =6the give the sameon result, theSo LHS so NMHV of (10.1 amplitudes): in N=4 SYM have interpretations as volumes of polytopes!result Formulas of the recursions such as relations based on the BCFW supershifts [6, 1 and [1, 2 :theyhaveto i i The simplest[6, 1, NMHV2, 3, 4] + case [6, 1A, is2NMHV, the4, 5][1 5-point +, 2 [6, 3,,24,,35 (anti-MHV)superamplitude,,46], 5]give = [1 the,22,,33 same,,44,,65], 1 result, + [1+, 2,2 so3, ,35, 4, 6], 5, +6 [1,+3, 42, 5, 4, ,6]5,.6, 1(5.56). (10.21) 6 / 4,6,1,2 , 2,3,4,6 6,2,3,4 , 4,5,6,2 2,4,5,6 , 6,1,2,4 NMHV h ⇥MHVi h ⇤ i h ⇥ i h ⇤ i h ⇥ i h ⇤ i Using the cyclic propertyA5 of the[1, 2 5-bracket,, 3, 4, 5] = we[6A can,51, 2 write, 3, 4]1 this +, 2 [6, 3, 1, ,42,,54, 5] + [6, 2, 3, 4, 5] = [1(10.20), 2, 3, 4, 5] + [1, 2, 3, 5, 6] + [1, 3, 4, 5, 6] . (5.56) Apart from the overall MHV factor,| {z the⇥ 6-point} | NMHV{z superamplitude} | {z is} the sum of the encodeNMHV different waysUsing of 4triangulating the cyclic⇥ property the polytope⇤ of the4 5-bracket, volume we can write this Thus, up[2 to, 3 the, 4, 6 MHV, 1]volumes + [2 factor,, 3, 4 of, 5 three,A6] + 4-simplices [2, 4is, 5 the, 6, 1] volume in =CP [3, ;4, we of5, 6 expect a, 1] 4-simplex + [3 this, 5, to6, in1 be, 2]CP the +. [3volume, 4, 5, 1 of, 2] a. polytope(5.57) obtained by into 4-simplices.5 somehow gluing the three simplices together. But how exactly does this work? To address this Next,Now for the you NMHV see that 6-point the LHS superamplitude, looks like the result consider[2 of, 3, a4 [2, 6,,3 the1]supershift, + [2 [2,,33, 4super-BCFW, 5, while6] + [2 the, 4, RHS5, representation6, 1] comes = [3, from4, 5, 6, 1] + [3, 5, 6, 1, 2] + [3, 4, 5, 1, 2] . (5.57) question,This it generalizes is useful to examine to n-particle the poles NMHV ini the amplitudes. 5-brackets.i a[3, 4 supershift. In fact, you’ll note that the LHS and independently the RHS are invariant on the LHS of (10.1): i Recall from (5.45) and (5.47Now) that you momentum see that the twistor LHS 4-brackets looks like thei result1,i,j of1 a,j [2,in3 thesupershift, de- while the RHS comes from under i i + 2. We can also reverse the labels in the 5-brackets at no cost toh get i i NMHV! nominator gives local poles, whereasa[3, 4 4-bracketssupershift. like In fact,k, i you’ll1,i,j notegive spurious that the ‘non-local’ LHS and independently poles. the RHS are invariant A6 [1, 2, 3, 4, 5, 6] 2, 3, 4, 6, 1 + i2, 3, 4, 5, 6 +h 2,4, 5, 6,i1 . (10.21) [2, 3, 4, 6, 1]Examining + [2, 3, 4, 5/ the, 6] denominator+ [2, 4, 5, 6, 1] termsunder = [3, of1i, 6 the, 5i, 5-brackets+4] 2. + [3We, 2, can1 in, 6 ( also,10.215] + reverse), [3, we2, 1 find, the5, 4] that labels. each(5.58) in of the them 5-brackets has 2 at no cost to get 4,6,1,2 , 2,3,4,6 6,2,!3,4 , 4,5,6,2 2,4,5,6 , 6,1,2,4 spurious poles;h they⇥ arei h listed⇤ i underh ⇥ eachi 5-bracket.h ⇤ i h The⇥ spuriousi h ⇤ polesi come in pairs — for ex- This states that the ‘parity conjugate’ supershifts[2, 3, [24,,63, 1]and + [2 [3,,32, 4give, 5, 6] identical + [2, 4, 5 results., 6, 1] = In [3, fact,1, 6, 5, 4] + [3, 2, 1, 6, 5] + [3, 2, 1, 5, 4] . (5.58) ample 4, 6, 1, 2 |and {z6, 1, 2, 4} in the| first{zi and third} 5-bracketsi | {z —} and cancel in the sum (10.21), Apartwe from can the now overall conclude MHVh that anyfactor,i adjacenth the 6-point supershiftsi NMHV are equivalent. superamplitude The identity is the (5.58 sum)shows of the as required by locality4 of the physical amplitude. In the geometric description of a 5-bracket as volumes of three 4-simplices in CP 4; we expectThis this states to be that the the volume ‘parity of conjugate’ a polytope supershifts obtained [2 by, 3 and [3, 2 give identical results. In fact, up again in Sectionsa 4-simplex9 and in10CPwhere, each we of thewill five understand factors in its the origins denominator better. of the volume-expressioni (i.e. the i somehow gluing the three simplices together.we But can how now exactly conclude does that this any work? adjacent To address supershifts this are equivalent. The identity (5.58)shows Exercise5-bracket) 5.15 is determined by a vertex of the associated 4-simplex. In particular, spurious poles I up again in4 Sections 9 and 10 where we will understand its origins better. question, it is usefulmust to examine be associated the poles with vertices in the 5-brackets. in CP that somehow ‘disappear’ from the polytope whose Use cyclicity of the 5-brackets to show that the tree-level 6-point NMHV superamplitude volume equals the sum of the simplex-volumesI Exercise 5.15 in (10.21). We now discuss how the ‘spurious’ Recall from (5.45) and (5.47) that momentum twistor 4-brackets i 1,i,j2 1,j in the de- can be writtenvertices in disappear the form in the sum of simplices. Let us starth in CP where thei polytopes are easier nominator gives local poles, whereas 4-brackets likeUsek, cyclicity i 1,i,j ofgive the 5-brackets spurious ‘non-local’ to show that poles. the tree-level 6-point NMHV superamplitude to draw. NMHV MHV h 1 i = can beR written+cyclic in the, form (5.59) Examining the denominator termsA of6 the 5-bracketsA6 ⇥ in2 (10.21146 ), we find that each of them has 2 2 spurious poles; theyPolytopes are listed inunderCP each 5-bracket. The⇣ spurious poles⌘ comeNMHV in pairs —MHV for ex-1 where “cyclic” means the sum over advancing the labels cyclically, i.e.6 R146= +6R251+4 R146 +cyclic , (5.59) ample 4, 6, 1, 2 andIn 26, dimensions,1, 2, 4 in the consider first and the third4-edge 5-brackets polytope — and cancelA in the sumA (10.21⇥ 2), h morei terms.h i ⇣ ⌘ as required by locality of the physical amplitude. Inwhere the geometric “cyclic”(2,3) means3 description(3,4) the sum of over a 5-bracket advancing as the labels cyclically, i.e. R146 + R251+4 The presence4 of these equivalence-relations between the dual conformal invariants may strike a 4-simplex in CP , each of the five factors in the denominatormore terms.2 of the volume-expression (i.e. the you as rather peculiar and you may wonder if it has a deeper meaning. Furthermore, while the 5-bracket) is determined by a vertex of the associated(1,2) 4-simplex. In particular, spurious poles expressions in (5.53) and (5.55) are extremelyThe simple, presence they of lack these one equivalence-relations key aspect when compared between to the dual conformal invariants may strike must be associated with vertices in 4 that somehow ‘disappear’4 from the polytope whose the Parke-Taylor superamplitude:CP cyclic invariance.you as rather The1 peculiar presence and of dualyou may conformal wonder symmetry if it has a deeper meaning. Furthermore, while the volumerelies equals heavily the sum on the of cyclic the simplex-volumes ordering of theexpressions in amplitude, (10.21 in). (5.53 and We) hence now and ( discuss5.55 it is) are somewhat how extremely the surprising ‘spurious’ simple, they lack one key aspect when compared to 2 verticesthat disappear the manifestly in the dual sum conformal of simplices. invariant Letthe form us Parke-Taylor start of the in superamplitudeCP(4,1) superamplitude:where the (5.55 polytopes cyclic) breaks invariance. are manifest easier The(10.22) presence of dual conformal symmetry to draw.cyclic invariance. One might say that we arerelies asking heavily too much on the of thecyclic amplitude, ordering but of the considering amplitude, and hence it is somewhat surprising 2 the payo↵ weAll have vertices reaped for from this CP the“amplitude” innocentthat the chase are manifestly defined of manifest by dual adjacent momentum conformal edges invariant conservation, and is in this form sense we of the local. superamplitude We (5.55) breaks manifest 2 Polytopeswill boldly in CP pushwould ahead like with to compute our pursuit the area ofcyclic “havingof the invariance. 2-polytope cakes and One (10.22 eating might) using them” say 2-simplex that in weSections are volumes asking9 and [a, too b, c]. much There of the amplitude, but considering are several di↵erent ways to do this, corresponding to di↵erent triangulations of the polytope. In 2 dimensions,10. consider the 4-edge polytopethe payo↵ we have reaped from the innocent chase of manifest momentum conservation, we will boldly push ahead with our pursuit of “having cakes and eating them” in Sections 9 and 10. (2,3) 3 (3,4) 2 159 84 (1,2) 4 84 1

(4,1) (10.22)

2 All vertices for this CP “amplitude” are deﬁned by adjacent edges and is in this sense local. We would like to compute the area of the 2-polytope (10.22) using 2-simplex volumes [a, b, c]. There are several di↵erent ways to do this, corresponding to di↵erent triangulations of the polytope.

159 5 Symmetries of = 4 SYM 5.4 Momentum twistors 10 PolytopesN 10.2 NMHV tree superamplitude as the volume of a polytope 5 Symmetries of = 4 SYM 5.4 Momentum twistors N The 5-bracket10.2 in (5.55 NMHV) correspond tree to superamplitude the terms in the as super-BCFW the volume expansion of a polytope of the superamplitude; specifically we have seen in Section 4.4.2 how each R arises from an MHV MHV The 5-bracket in (5.55n2)k correspond to the terms⇥ in the super-BCFW expansion of the super- BCFW diagramThe while simplest the remaining NMHV caseRnjk is the’s with 5-pointj> (anti-MHV)superamplitude2 appear via recursion from the BCFW amplitude; specifically we have seen in Section 4.4.2 how each R arises from an MHV MHV diagram with NMHV anti-MHV subamplitudes. As we have discussed, this means that the n2k ⇥ ⇥ NMHVBCFW diagram whileMHV the remaining Rnjk’s with j>2 appear via recursion from the BCFW representation (5.55) is not be unique,A since5 there[1, 2, 3 are, 4, 5] many = A equivalent5 1, BCFW2, 3, 4, 5 expansions for (10.20) 10 Polytopes 10.2 NMHVdiagram tree superamplitude with NMHV asanti-MHV⇥ the volume subamplitudes. of a polytope As we have discussed, this means that the a given amplitude, depending on the choice of lines in the BCFW⇥ shift. This implies that the Thus, up to the MHV factor,representationANMHV is the volume (5.55) is of not a 4-simplex⇥ be unique, in⇤ since4. there are many equivalent BCFW expansions for dual conformal invariants (5.54) areTowards linearly5 dependent. amplituhedrons For example, compare forCPn =6the a given amplitude, depending on the choice of lines in the BCFW shift. This implies that the result of the recursionsNext, for relations the NMHV based 6-point on the superamplitude, BCFW supershifts consider [6, 1 theand [2, 3 [1,super-BCFW2 :theyhaveto representation 10.2 NMHV tree superamplitude asdual the conformal volume invariants of a polytope (5.54i ) arei linearlyi dependent. For example, compare for n =6the give the sameon result, theSo LHS so NMHV of (10.1 amplitudes): in N=4 SYM have interpretations as volumes of polytopes!result Formulas of the recursions such as relations based on the BCFW supershifts [6, 1 and [1, 2 :theyhaveto i i The simplest[6, 1, NMHV2, 3, 4] + case [6, 1A, is2NMHV, the4, 5][1 5-point +, 2 [6, 3,,24,,35 (anti-MHV)superamplitude,,46], 5]give = [1 the,22,,33 same,,44,,65], 1 result, + [1+, 2,2 so3, ,35, 4, 6], 5, +6 [1,+3, 42, 5, 4, ,6]5,.6, 1(5.56). (10.21) 6 / 4,6,1,2 , 2,3,4,6 6,2,3,4 , 4,5,6,2 2,4,5,6 , 6,1,2,4 NMHV h ⇥MHVi h ⇤ i h ⇥ i h ⇤ i h ⇥ i h ⇤ i Using the cyclic propertyA5 of the[1, 2 5-bracket,, 3, 4, 5] = we[6A can,51, 2 write, 3, 4]1 this +, 2 [6, 3, 1, ,42,,54, 5] + [6, 2, 3, 4, 5] = [1(10.20), 2, 3, 4, 5] + [1, 2, 3, 5, 6] + [1, 3, 4, 5, 6] . (5.56) Apart from the overall MHV factor,| {z the⇥ 6-point} | NMHV{z superamplitude} | {z is} the sum of the encodeNMHV different waysUsing of 4triangulating the cyclic⇥ property the polytope⇤ of the4 5-bracket, volume we can write this Thus, up[2 to, 3 the, 4, 6 MHV, 1]volumes + [2 factor,, 3, 4 of, 5 three,A6] + 4-simplices [2, 4is, 5 the, 6, 1] volume in =CP [3, ;4, we of5, 6 expect a, 1] 4-simplex + [3 this, 5, to6, in1 be, 2]CP the +. [3volume, 4, 5, 1 of, 2] a. polytope(5.57) obtained by into 4-simplices.5 somehow gluing the three simplices together. But how exactly does this work? To address this Next,Now for the you NMHV see that 6-point the LHS superamplitude, looks like the result consider[2 of, 3, a4 [2, 6,,3 the1]supershift, + [2 [2,,33, 4super-BCFW, 5, while6] + [2 the, 4, RHS5, representation6, 1] comes = [3, from4, 5, 6, 1] + [3, 5, 6, 1, 2] + [3, 4, 5, 1, 2] . (5.57) question,This it generalizes is useful to examine to n-particle the poles NMHV ini the amplitudes. 5-brackets.i a[3, 4 supershift. In fact, you’ll note that the LHS and independently the RHS are invariant on the LHS of (10.1): i Recall from (5.45) and (5.47Now) that you momentum see that the twistor LHS 4-brackets looks like thei result1,i,j of1 a,j [2,in3 thesupershift, de- while the RHS comes from under i i + 2. WeThe can leap also reverseto all amplitudes the labels in of the N=4 5-brackets SYM is what at no is cost called toh get i i NMHV! nominator gives local poles, whereasa[3, 4 4-bracketssupershift. like In fact,k, i you’ll1,i,j notegive spurious that the ‘non-local’ LHS and independently poles. the RHS are invariant A6 [1, 2, 3, 4, 5, 6] 2, 3 , 4, 6, 1 + i2, 3, 4, 5, 6 +h 2,4, 5, 6,i1 . (10.21) [2, 3, 4, 6, 1]Examining + [2, 3, 4, 5/ the, 6] denominator+ [2, 4, 5, 6, 1] termsunder = [3, of1i, 6 the, 5i, 5-brackets+4] 2. + [3We, 2, can1 in, 6 ( also,10.215] + reverse), [3, we2, 1 find, the5, 4] that labels. each(5.58) in of the them 5-brackets has 2 at no cost to get 4,6,1,2 , 2,3,4,6 6,2,!3,4 , 4,5,6,2 2,4,5,6 , 6,1,2,4 spurious poles;h they⇥ arei h listed⇤ i underh ⇥ eachi 5-bracket.h ⇤ i h The⇥ spuriousi h ⇤ polesi come in pairs — for ex- This states that the ‘parity conjugate’ supershifts[2, 3, [24,,63, 1]and + [2 [3,,32, 4give, 5, 6] identical + [2, 4, 5 results., 6, 1] = In [3, fact,1, 6, 5, 4] + [3, 2, 1, 6, 5] + [3, 2, 1, 5, 4] . (5.58) ample 4, 6, 1, 2 |and {z6, 1, 2, 4} in the| first{zi and third} 5-bracketsi | {z —} and cancel in the sum (10.21), Apartwe from can the now overall conclude MHVh that anyfactor,i adjacenth the 6-point supershiftsi NMHV are equivalent. superamplitude The identity is the (5.58 sum)shows of the as required by locality4 of the physical amplitude. In the geometric description of a 5-bracket as volumes of three 4-simplices in CP 4; we expectThis this states to be that the the volume ‘parity of conjugate’ a polytope supershifts obtained [2 by, 3 and [3, 2 give identical results. In fact, up again in Sectionsa 4-simplex9 and in10CPwhere, each we of thewill five understand factors in its the origins denominator better. of the volume-expressioni (i.e. the i somehow gluing the three simplices together.we But can how now exactly conclude does that this any work? adjacent To address supershifts this are equivalent. The identity (5.58)shows Exercise5-bracket) 5.15 is determined by a vertex of the associated 4-simplex. In particular, spurious poles I up again in4 Sections 9 and 10 where we will understand its origins better. question, it is usefulmust to examine be associated the poles with vertices in the 5-brackets. in CP that somehow ‘disappear’ from the polytope whose Use cyclicity of the 5-brackets to show that the tree-level 6-point NMHV superamplitude volume equals the sum of the simplex-volumesI Exercise 5.15 in (10.21). We now discuss how the ‘spurious’ Recall from (5.45) and (5.47) that momentum twistor 4-brackets i 1,i,j2 1,j in the de- can be writtenvertices in disappear the form in the sum of simplices. Let us starth in CP where thei polytopes are easier nominator gives local poles, whereas 4-brackets likeUsek, cyclicity i 1,i,j ofgive the 5-brackets spurious ‘non-local’ to show that poles. the tree-level 6-point NMHV superamplitude to draw. NMHV MHV h 1 i = can beR written+cyclic in the, form (5.59) Examining the denominator termsA of6 the 5-bracketsA6 ⇥ in2 (10.21146 ), we find that each of them has 2 2 spurious poles; theyPolytopes are listed inunderCP each 5-bracket. The⇣ spurious poles⌘ comeNMHV in pairs —MHV for ex-1 where “cyclic” means the sum over advancing the labels cyclically, i.e.6 R146= +6R251+4 R146 +cyclic , (5.59) ample 4, 6, 1, 2 andIn 26, dimensions,1, 2, 4 in the consider first and the third4-edge 5-brackets polytope — and cancelA in the sumA (10.21⇥ 2), h morei terms.h i ⇣ ⌘ as required by locality of the physical amplitude. Inwhere the geometric “cyclic”(2,3) means3 description(3,4) the sum of over a 5-bracket advancing as the labels cyclically, i.e. R146 + R251+4 The presence4 of these equivalence-relations between the dual conformal invariants may strike a 4-simplex in CP , each of the five factors in the denominatormore terms.2 of the volume-expression (i.e. the you as rather peculiar and you may wonder if it has a deeper meaning. Furthermore, while the 5-bracket) is determined by a vertex of the associated(1,2) 4-simplex. In particular, spurious poles expressions in (5.53) and (5.55) are extremelyThe simple, presence they of lack these one equivalence-relations key aspect when compared between to the dual conformal invariants may strike must be associated with vertices in 4 that somehow ‘disappear’4 from the polytope whose the Parke-Taylor superamplitude:CP cyclic invariance.you as rather The1 peculiar presence and of dualyou may conformal wonder symmetry if it has a deeper meaning. Furthermore, while the volumerelies equals heavily the sum on the of cyclic the simplex-volumes ordering of theexpressions in amplitude, (10.21 in). (5.53 and We) hence now and ( discuss5.55 it is) are somewhat how extremely the surprising ‘spurious’ simple, they lack one key aspect when compared to 2 verticesthat disappear the manifestly in the dual sum conformal of simplices. invariant Letthe form us Parke-Taylor start of the in superamplitudeCP(4,1) superamplitude:where the (5.55 polytopes cyclic) breaks invariance. are manifest easier The(10.22) presence of dual conformal symmetry to draw.cyclic invariance. One might say that we arerelies asking heavily too much on the of thecyclic amplitude, ordering but of the considering amplitude, and hence it is somewhat surprising 2 the payo↵ weAll have vertices reaped for from this CP the“amplitude” innocentthat the chase are manifestly defined of manifest by dual adjacent momentum conformal edges invariant conservation, and is in this form sense we of the local. superamplitude We (5.55) breaks manifest 2 Polytopeswill boldly in CP pushwould ahead like with to compute our pursuit the area ofcyclic “havingof the invariance. 2-polytope cakes and One (10.22 eating might) using them” say 2-simplex that in weSections are volumes asking9 and [a, too b, c]. much There of the amplitude, but considering are several di↵erent ways to do this, corresponding to di↵erent triangulations of the polytope. In 2 dimensions,10. consider the 4-edge polytopethe payo↵ we have reaped from the innocent chase of manifest momentum conservation, we will boldly push ahead with our pursuit of “having cakes and eating them” in Sections 9 and 10. (2,3) 3 (3,4) 2 159 84 (1,2) 4 84 1

(4,1) (10.22)

159 But what about spacetime? Traditional formulation of quantum field theory is based on a LOCAL Lagrangian: it encodes the causal structure, local interactions – i.e. no action at a distance.

For amplitudes, this translates into their analytic structure: the physical poles of an amplitude correspond exactly to where propagating modes go on-shell.

Traditional formulation of quantum field theory is based on a LOCAL Lagrangian: it encodes the causal structure, local interactions – i.e. no action at a distance.

For amplitudes, this translates into their analytic structure: the physical poles of an amplitude correspond exactly to where propagating modes go on-shell.

New calculational methods, such as BCFW recursion, exploits this aspects of locality heavily, but produces results for the amplitudes that are not manifestly “local”.

The 5-brackets have “unphysical” poles. But they cancel in the sum of 5-brackets that equals the amplitude.

Traditional formulation of quantum field theory is based on a LOCAL Lagrangian: it encodes the causal structure, local interactions – i.e. no action at a distance.

For amplitudes, this translates into their analytic structure: the physical poles of an amplitude correspond exactly to where propagating modes go on-shell.

New calculational methods, such as BCFW recursion, exploits this aspects of locality heavily, but produces results for the amplitudes that are not manifestly “local”.

The 5-brackets have “unphysical” poles. But they cancel in the sum of 5-brackets that equals the amplitude.

In the Amplituhedron formulation the cancellation of unphysical poles is geometrized. Locality is statement of defining the appropriate geometric object.

Many different directions are being studied:

• Generalized unitarity cuts [Bern,Dixon,Kosower] • Loop-integrand recursion relations [Arkani-Hamed,Bourjaily,Cachazo,Trnka] • Integrability techniques [Basso,Sever,Vieira;Plefka,Ferro,…] • Exploit physical limits to fix the integrated amplitude [Dixon,Drummond,Duhr,von Hippel,Pennington] • String theory amplitudes [Green,Stieberger,Mafra,Schlotterer,…] What are we doing in Aspen?! What are we doing in Aspen?!

From Scattering Amplitudes to the Conformal Bootstrap Organizers: David Poland, Henriette Elvang, Leonardo Rastelli, Jaroslav Trnka What are we doing in Aspen?!

From Scattering Amplitudes to the Conformal Bootstrap Organizers: David Poland, Henriette Elvang, Leonardo Rastelli, Jaroslav Trnka

Scattering Amplitudes: we talked about those

Conformal Bootstrap: understand the space of possible conformal field theories using symmetry, unitarity, and physical constraints. Specifically, exploit crossing relations

Figure borrowed from Simmons-Duffin et al Bootstrap & Amplitudes

Shared approach: Exploit symmetries, physical constraints, analytic properties (instead of standard brute force calc) to learn about the physical observables as well as the properties of quantum field theories. Bootstrap & Amplitudes

Shared philosophy: Exploit symmetries, physical constraints, analytic properties (instead of standard brute force calc) to learn about the physical observables as well as the properties of quantum field theories.

Conformal bootstrap Amplitudes 3-pt correlation fct fixed by conf sym. 3-pt ampl of massless particles fixed by little group scaling (in 4d).

Crossing relation for 4-pt correlators recursion relations => bootstrap constraint conf field theories. of the S-matrix, constrain QFTs

Ward identities constrain correlators. Ward identities constrain S-matrix.

Q: which CFTs exist? Q: how to efficiently calc amplitudes? Q: characterize CFTs. Q: understand mathematical structure of amplitudes & QFTs. Bootstrap & Amplitudes

Overall shared interests: Wish to understand properties of quantum field theories.

Workshop Brings together the communities => new insights & collaborations. People working together to build a tree (and loops) Thank you!

Niklas Beisert (Zurich ETH) Filipe Rudrigues Yu-tin Huang (Michigan ugrad) (Taiwan) Cindy Keeler (Niels Bohr Inst) Tim Olson Sam Roland (Michigan grad) (Michigan grad)

Dan Freedman Thomas Lam (MIT/Stanford) (Michigan Math) Alejandro H Morales Stephan Stieberger (UCLA Math) (Munich) Tim Cohen (Oregon)

Cheng Peng David E Speyer (Zurich ETH) Michael Kiermaier (Michigan Math)