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Geometric Description of Scattering Amplitudes: Exploring The Geometric Description of Scattering Amplitudes Exploring the Amplituhedron Andrea Orta M¨unchen2017 Geometric Description of Scattering Amplitudes Exploring the Amplituhedron Andrea Orta Dissertation an der Fakult¨atf¨urPhysik der Ludwig{Maximilians{Universit¨at M¨unchen vorgelegt von Andrea Orta aus Torino, Italien M¨unchen, den 30. Oktober 2017 Erstgutachterin: Prof. Dr. Livia Ferro Zweitgutachter: Prof. Dr. Dieter L¨ust Tag der m¨undlichen Pr¨ufung:15. Dezember 2017 Contents Zusammenfassung xi Summary xiii Acknowledgements xv 0 Introduction 1 1 Scattering amplitudes in massless quantum field theories 9 1.1 General framework . .9 1.2 Large-N limit and colour decomposition . 14 1.2.1 Properties of tree-level colour-ordered amplitudes . 16 1.3 N = 4 super Yang{Mills . 19 1.3.1 The superspace formalism . 22 2 On-shell methods for N = 4 SYM theory and its symmetries 25 2.1 Spinor-helicity variables . 25 2.2 Symmetries of tree-level superamplitudes and the Yangian . 31 2.2.1 Superconformal symmetry . 31 2.2.2 Dual superconformal symmetry . 33 2.2.3 Yangian symmetry . 36 2.3 Twistor variables . 38 2.3.1 Spacetime supertwistors . 38 2.3.2 Momentum supertwistors . 39 2.3.3 Yangian generators in supertwistor formulation . 42 2.4 BCFW recursion relations . 43 2.5 Grassmannian formulation of scattering amplitudes . 54 viii CONTENTS 2.5.1 Geometric interpretation of momentum conservation . 56 2.5.2 Grassmannian integrals . 58 2.5.3 On-shell diagrams . 63 3 The Amplituhedron proposal 79 3.1 Positive geometry in arbitrary dimensions . 80 3.2 Tree amplituhedron and scattering amplitudes . 85 3.3 Loop-level amplituhedron . 91 3.4 An i-prescription for volume functions . 94 3.4.1 A mathematical prelude: multivariate residues . 94 3.4.2 NMHV amplitudes . 96 3.4.3 N2MHV amplitudes . 105 4 Volume of the dual tree-level amplituhedron 107 4.1 Symmetries of the volume function and the Capelli differential equations . 108 4.2 Solution for the k = 1 case . 110 4.3 Applying the master formula . 114 4.3.1 Volume in the m = 2 case . 114 4.3.2 Volume in the m = 4 case . 117 4.4 Deformations and higher-k amplituhedron volumes . 120 5 Yangian symmetry for the tree-level amplituhedron 123 5.1 The Heisenberg isotropic spin chain and the algebraic Bethe ansatz . 124 5.1.1 Introduction . 124 5.1.2 Lax operators, R-matrices, monodromy and transfer matrix . 128 5.1.3 The ansatz and the Bethe equations . 130 5.2 Construction of amplituhedron volume functions . 133 5.2.1 From scattering amplitudes to the amplituhedron . 134 5.2.2 On-shell diagrammatics . 136 5.3 Spin chain picture for the amplituhedron volume . 139 5.3.1 Proof of the monodromy relation . 143 5.3.2 Yangian invariance for the amplituhedron volume . 144 6 Summary and outlook 147 Contents ix A More on the i-prescription 151 A.1 Even distribution of poles: a proof . 151 A.2 The five-point N2MHV volume function . 154 B Details on the solution of the Heisenberg spin chain 161 B.1 Permutation and shift operators . 162 B.2 UN and HN from the transfer matrix . 163 B.3 On the ansatz and the Bethe equations . 164 C Yangian for the tree amplituhedron 167 C.1 Construction of volume functions . 167 C.2 Seed mutations . 168 C.3 Intertwining relations . 169 C.4 Action of the monodromy matrix on the seed . 170 x Contents Zusammenfassung Die vorliegende Dissertation befasst sich mit einigen der neuesten Entwicklungen im The- mengebiet der Streuamplituden in schwach gekoppelten Eichtheorien. Der traditionelle st¨orungstheoretische Zugang, der Feynman-Diagramme verwendet, hat sich durchgesetzt und großen Erfolg erzielt, jedoch er ist zunehmend ungeeignet geworden, um mit der Komplexit¨atder Pr¨azisionsberechnungen umzugehen, die heutzutage ben¨otigtsind. Neue Techniken, die zusammenfassend als on-shell Methoden bezeichnet werden, sind entwickelt worden, um die Nachteile und Engp¨assedieses Zugangs zu vermeiden. Sie enth¨ullteneine inh¨arente (und manchmal unerwartete) Einfachheit der Streuamplituden, oft in Verbindung zur Symmetrien; daneben, erlaubten sie die Entwicklung eines grundverschiedenen, ge- ometrischen Verst¨andnisvon Amplituden, bei dem Grassmann-Mannigfaltigkeiten eine zentrale Rolle spielen. Dieses Programm hat sich sehr gut im Rahmen eines speziellen Modells bew¨ahrt,der planaren N = 4 super Yang{Mills Theorie. Im Jahr 2013 ist eine be- merkenswerte Vermutung vorgelegt worden, die besagt, dass jede Amplitude ist insgeheim das Volumen eines verallgemeinerten Polytops, das Amplituhedron heißt. In dieser Doktorarbeit, untersuchen wir, nach einer detaillierten Einf¨uhrungin die on-shell Methoden, den Amplituhedron-Vorschlag ausf¨uhrlich auf Baumniveau und wir berichten ¨uber mehrere Ergebnisse, die zusammen mit anderen Mitarbeitern erreicht wor- den sind. Wir f¨uhreneine Formel ein, die NMHV-Baumniveau Volumenfunktionen als Inte- grale ¨uber eine duale Grassmann-Mannigfaltigkeit darstellt, und damit Triangulationen des Amplituhedrons vermeidet; dann leiten wir eine i-Vorschrift her, die uns erlaubt, sie als Summe von Residuen eines Grassmannschen Integrals aufzubauen, das im Geiste ¨ahnlich zu dem f¨urAmplituden relevanten ist. Anschließend, angeregt von den Schwierigkeiten bei der Verallgemeinerung unserer Ergebnisse, suchen wir nach einer Realisation der Yangschen Symmetrie im Rahmen des Amplituhedrons. Wir zeigen, dass mit den Volumenfunktio- nen eng verwandte Objekte wirklich invariant unter den Transformationen des Yangians Y gl(m + k) sind und denken ¨uber die Implikationen davon nach. xii 0. Zusammenfassung Summary This dissertation is concerned with some of the most recent developments in the under- standing of gauge theory scattering amplitudes at weak coupling. The traditional pertur- bative approach in terms of Feynman diagrams is well established and has achieved great triumphs, however it has become increasingly unsuited to handle the complexity of the precision calculations needed nowadays. In looking for new ways to avoid its drawbacks and bottlenecks, new techniques have been developed, collectively going under the name of on-shell methods. Beyond unveiling an inherent (and at times unexpected) simplicity of scattering amplitudes { often in connection to symmetries { they helped the rise of a rad- ically different, geometric understanding of them, in which Grassmannian manifolds play a central role. This program has proven very successful in the context of a special model, planar N = 4 super Yang{Mills theory. In 2013 a striking conjecture was put forward, stating that every amplitude is secretly the volume of a generalised polytope, called the amplituhedron. In this thesis, after providing a broad introduction about on-shell methods, we investi- gate this proposal at tree-level in detail and report on several new results that were obtained in collaboration with other authors. In particular, we present a closed formula for express- ing NMHV tree-level volume functions as integrals over a dual Grassmannian, avoiding triangulations of the amplituhedron; we subsequently derive an i-prescription that allows to alternatively construct them as a sum of residues of a Grassmannian integral, similar in spirit to the one relevant for scattering amplitudes. Motivated by the difficulties in mov- ing to more general tree-level volume functions, we then look for a realisation of Yangian symmetry in the framework of the amplituhedron. We prove that objects closely related to the volume functions are in fact invariant under the Yangian Y gl(m + k) and ponder on the implications of this result. xiv 0. Summary Acknowledgements First and foremost, I am very grateful to my supervisor, Livia Ferro, who introduced me to a thriving field to investigate a fascinating problem. Beyond always being available to answer my questions and sharing her deep knowledge, she was supportive and encouraging throughout the difficulties of my first research experiences. Moreover, she gave me the opportunity to travel to attend schools and conferences, where I had the opportunity to learn from numerous outstanding scientists: in relation to this, I would like to thank Nima Arkani-Hamed for a kind invitation to the IAS in Princeton. It has been a pleasure to share my research projects with TomekLukowski, whose expertise and insight have been of inspiration and who contributed to create a pleasant atmosphere throughout our collaborations. I also want to thank Matteo Parisi for the many fruitful discussions that improved my understanding. This dissertation has been heavily influenced by many people: I am indebted to Reinke Isermann for helping me gain familiarity with the on-shell methods; to Jos`eFrancisco Morales and Congkao Wen for collaborating on my first project; to Nils Kanning for sharing his knowledge on integrability and proofreading the Zusammenfassung of this thesis (any remaining mistake is of course my own!). I also wish to thank Jake Bourjaily, Hugh Thomas and Jaroslav Trnka for useful discussions about details of this work; finally, I owe Matteo Rosso and especially Michela Amicone for helping me code a complicated matrix in LATEX. I thoroughly enjoyed my time in Munich, also thanks to the lively atmosphere created by all members of the Mathematical physics and String theory research group. I want to personally thank for the time spent together my fellow PhD students Enrico Brehm, Fabrizio Cordonier-Tello, Nicol´asGonzalez, Daniel Jaud, Benedikt Richter and the Post- Docs Reinke Isermann, Daniel Junghans, Nils Kanning, Emanuel Malek, Stefano Massai, Christoph Mayrhofer, Erik Plauschinn, Felix Rudolph, Cornelius Schmidt-Colinet, Angnis Schmidt-May, as well as Michael Haack for the friendly help and advice on many techni- cal issues. Finally, I express my gratitude to Dieter L¨ust,for agreeing to be my second xvi 0. Acknowledgements Gutachter. I would like to thank Ruth Britto for giving me the chance to continue my career in Dublin as a PostDoc. In this regard, I am grateful to Livia Ferro, Lorenzo Magnea, Dieter L¨ustand TomekLukowski for supporting my applications with their recommenda- tion letters, as well as to Reinke Isermann, Matteo Rosso, Cornelius Schmidt-Colinet and Johannes Broedel for sharing their experience and offering me their friendly advice on the matter.
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