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Tales of 1001 Gluons

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Tales of 1001 Gluons MITP/16-109 Tales of 1001 Gluons Stefan Weinzierl PRISMA Cluster of Excellence, Institut für Physik, Johannes Gutenberg-Universität Mainz, D - 55099 Mainz, Germany Abstract This report is centred around tree-level scattering amplitudes in pure Yang-Mills theories, the most prominent example is given by the tree-level gluon amplitudes of QCD. I will discuss several ways of computing these amplitudes, illustrating in this way recent devel- opments in perturbative quantum field theory. Topics covered in this review include colour decomposition, spinor and twistor methods, off- and on-shell recursion, MHV amplitudes and MHV expansion, the Grassmannian and the amplituhedron, the scattering equations and the CHY representation. At the end of this report there will be an outlook on the rela- tion between pure Yang-Mills amplitudes and scattering amplitudes in perturbative quantum gravity. arXiv:1610.05318v2 [hep-th] 20 Jan 2017 Contents 1 Introduction 4 2 The textbook method 7 2.1 TheLagrangianofYang-Millstheory . ...... 7 2.2 Yang-Millstheoryintermsofdifferentialforms . ............ 8 2.3 Gaugefixing..................................... 11 2.4 Feynmanrules.................................... 14 2.5 Amplitudesandcrosssections . ..... 17 2.6 Singularlimitsofscatteringamplitudes . .......... 21 3 Efficiency improvements 24 3.1 Colourdecomposition. ... 24 3.2 Colour-kinematicsduality . ..... 32 3.3 Thespinorhelicitymethod . .... 35 3.3.1 TheDiracequation ............................. 36 3.3.2 MasslessspinorsintheWeylrepresentation . ........ 37 3.3.3 Spinorproducts............................... 41 3.3.4 Polarisationvectors. .. 42 3.3.5 Massivespinors............................... 44 3.3.6 TheMajoranarepresentation . ... 45 3.4 Twistors....................................... 46 3.4.1 Momentumtwistors ............................ 46 3.4.2 Thetwistortransform. 50 3.5 Off-shellrecurrencerelations . ........ 53 4 MHV amplitudes and MHV expansion 58 4.1 TheParke-Taylorformulæ . ... 58 4.2 TheCSWconstruction .............................. 60 4.3 TheLagrangianoftheMHVexpansion . .... 63 5 On-shell recursion 68 5.1 BCFWrecursion .................................. 69 5.2 Momentashiftsandthebehaviouratinfinity . ......... 71 5.3 Theon-shellrecursionalgorithm . ....... 73 5.4 Three-point amplitudes and on-shell constructible theories ............ 76 5.5 Application: Proof of the fundamental BCJ relations . ............ 77 6 Grassmannian geometry 81 6.1 Projective spaces, Grassmannians and flag varieties . ............. 81 6.2 Thelinkrepresentation . .... 84 6.3 Supersymmetricnotation . .... 88 6.4 Theamplituhedron ................................ 91 2 6.5 Infinitytwistors ................................. .. 96 7 The CHY representation 101 7.1 Thescatteringequations . .101 7.2 Polarisation factors and Parke-Taylor factors . .............104 7.3 ThetwoformsoftheCHYrepresentation . .. ..105 7.4 Globalresidues .................................. 109 8 Perturbative quantum gravity 114 8.1 Gaugeinvarianceofgravity. .115 8.2 GravityamplitudesfromFeynmandiagrams . ........116 8.3 GravityamplitudesfromtheCHYrepresentation . ..........120 8.4 Gravity amplitudes from colour-kinematics duality . .............123 8.5 GravityamplitudesfromtheKLTrelations. .........124 9 Outlook 126 A Solutions to the exercises 128 3 1 Introduction Each student of physics has encountered the harmonic oscillator during her or his studies, usually more than once. The harmonic oscillator is a simple system, familiar to the students, and can be used as a pedagogical example to introduce new concepts – just think about the introduction of raising and lowering operators for the quantum mechanical harmonic oscillator. Another system encountered by any physics student is the hydrogen atom. Again, methods to solve this system apply to a wider context. For example, positronium and charmonium are close cousins of the hydrogen atom. Within the context of quantum field theory the computation of tree-level scattering ampli- tudes in pure Yang-Mills theories constitutes another example of physical problems every student should know about: The tree-level scattering amplitudes in pure Yang-Mills theories can be com- puted in several way and each possibility illuminates certain mathematical structures underlying quantum field theories. In this report we will take the tree-level scattering amplitudes in pure Yang-Mills theories as our reference object and explore in a pedagogical way the mathematical structures associated to it. A prominent special case are tree-level gluon amplitudes of QCD, corresponding to the choice G = SU(3) as gauge group. Originally, the study of these amplitudes was motivated by finding efficient methods to compute multi-parton amplitudes, where the multiplicity ranges from 4 (corresponding to a 2 2 scattering process) to roughly 10 (corresponding to a 2 8 scattering process). This is relevant→ for hadron collider experiments. In particular at the LHC→ gluon-initiated processes constitute the bulk of scattering events. Although we focus on tree- level amplitudes in this review, many of the techniques discussed in this report have an extension towards higher orders in perturbation theory. This is useful for precision calculations. Finally, let me mention that the study of scattering amplitudes has developed in recent years in a field of its own, revealing hidden structures of quantum field theory which are not manifest in the conventional Lagrangian formulation. There are other excellent review articles and lecture notes, covering some of the topics pre- sented here, for example [1–4]. Furthermore the content of this report is not meant to cover the field completely. More on the contrary, by focusing solely on the tree-level amplitudes in Yang-Mills theory we try to explain the basic principles and leave advanced topics to the current research literature. Not included in this report are for example the extension of pure Yang-Mills theory to QCD, i.e. the inclusion of fermions in the fundamental representation of the gauge group, massless or massive. Furthermore, not included is the important field of loop calcula- tions, this field alone would fill another report. Neither covered in detail are supersymmetric extensions of Yang-Mills theory, the most popular example of the latter would be N= 4 super- symmetric Yang-Mills theory. We will start with a review of the textbook method based on Feynman diagrams in section (2). Feynman diagrams allows us in principle to compute a tree-level amplitude for any number n of external particles. However, in practice this method is highly inefficient. In section (3) we will discuss efficient methods for the computation of tree-level scattering amplitudes in Yang-Mills theory. We will learn about colour decomposition, the spinor helicity method and off-shell recurrence relations. The combination of these three tools constitutes one 4 of the best (probably the best) methods to compute tree-level scattering amplitudes numerically. In section (3) we treat in addition the topics of colour-kinematics duality and twistors. These two topics are not directly targeted at efficiency. However, colour-kinematics duality is closely related to colour decomposition and twistors are closely related to the spinor helicity method. Therefore these topics are best discussed together. Section (4) is centred around maximally helicity violating amplitudes. We present the Parke- Taylor formulæ and show that an arbitrary tree-level Yang-Mills amplitude may be computed from an effective scalar theory with an infinite tower of vertices, where the new interaction vertices are given by the Parke-Taylor formulæ. In section (5) we present the on-shell recursion relations. On-shell recursion relations are one of the best (probably the best) methods to compute tree-level scattering amplitudes analytically. In addition, they are a powerful tool to prove the correctness of new representations of tree-level scattering amplitudes in Yang-Mills theory. Section (6) is devoted to a geometric interpretation of a scattering amplitude as the volume of a generalised polytope (the amplituhedron) in a certain auxiliary space. We discuss Grassmanni- ans and the representation of scattering amplitude in this formalism. In section (7) we will learn about the Cachazo-He-Yuan (CHY) representation. Again, we 1 look at some auxiliary space and associate to each external particle a variable z j CP . Of par- ∈ ticular interest are the values of the z j’s, which are solutions of the so-called scattering equations. We will present the scattering equations and show that the scattering amplitude can be expressed as a global residue at the zeros of the scattering equations. In section (8) we explore tree-level amplitudes in perturbative quantum gravity. This may at first sight seem to be disconnected from the rest of this review, but we will see that there is a close relation between scattering amplitudes in Yang-Mills theory and gravity. The link is either provided by colour-kinematics duality, the CHY representation or the Kawai-Lewellen- Tye (KLT) relations. The latter are introduced in this section. Finally, section (9) gives an outlook. Let me add a word on the notation used in this review: Although it is desirable to have a unique notation throughout these notes, with the variety of topics treated in this course we face the challenge that one topic is best presented in one notation, while another topic is more clearly exposed in another notation. An example is given by a central quantity of this report,
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