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Form Factors and the Dilatation Operator in N=4 Super Yang-Mills

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Form Factors and the Dilatation Operator in N=4 Super Yang-Mills Form factors and the dilatation operator in N = 4 super Yang-Mills theory and its deformations Dissertation zur Erlangung des akademischen Grades doctor rerum naturalium (Dr. rer. nat.) im Fach Physik eingereicht an der Mathematisch-Naturwissenschaftlichen Fakult¨at der Humboldt-Universit¨at zu Berlin von M.Sc. B.Sc. Matthias Oliver Wilhelm Pr¨asident der Humboldt-Universit¨at zu Berlin Prof. Dr. Jan-Hendrik Olbertz Dekan der Mathematisch-Naturwissenschaftlichen Fakult¨at Prof. Dr. Elmar Kulke Gutachter/innen: 1. Prof. Dr. Matthias Staudacher 2. Prof. Dr. Jan Plefka 3. Prof. Dr. Tristan McLoughlin Tag der m¨undlichen Pr¨ufung: 12.02.2016 2 Zusammenfassung Seit mehr als einem halben Jahrhundert bietet die Quantenfeldtheorie (QFT) den genausten und erfolgreichsten theoretischen Rahmen zur Beschreibung der fundamentalen Wechsel- wirkungen zwischen Elementarteilchen, wenn auch mit Ausnahme der Gravitation. Den- noch sind QFTs im Allgemeinen weit davon entfernt, vollst¨andig verstanden zu sein. Dies liegt an einem Mangel an theoretischen Methoden zur Berechnung ihrer Observablen sowie an fehlendem Verst¨andnis der auftretenden mathematischen Strukturen. In den letzten anderthalb Jahrzehnten kam es zu bedeutendem Fortschritt im Verst¨andnis von speziellen Aspekten einer bestimmten QFT, der maximal supersymmetrischen Yang-Mills-Theorie in vier Dimensionen, auch = 4 SYM-Theorie genannt. Diese haben die Hoffnung N geweckt, dass die = 4 SYM-Theorie exakt l¨osbar ist. Besonders bemerkenswert war der N Fortschritt auf den Gebiet der Streuamplituden auf Grund der Entwicklung sogenannter Masseschalen-Methoden und auf dem Gebiet der Korrelationsfunktionen zusammengeset- zter Operatoren auf Grund von Integrabilit¨at. In dieser Dissertation gehen wir der Frage nach, ob und in welchem Umfang die in diesem Kontext gefunden Methoden und Struk- turen auch zum Verst¨andnis weitere Gr¨oßen in dieser Theorie sowie zum Verst¨andnis an- derer Theorien beitragen k¨onnen. Formfaktoren beschreiben den quantenfeldtheoretischen Uberlapp¨ eines lokalen, eich- invarianten, zusammengesetzten Operators mit einem asymptotischen Streuzustand. Als solche bilden sie eine Br¨ucke zwischen der Welt der Streuamplituden, deren externe Im- pulse sich auf der Masseschale befinden, auf der einen Seite und der Welt der Korrela- tionsfunktionen von zusammengesetzten Operatoren, welche keine entsprechende Bedin- gung erf¨ullen, auf der anderen Seite. Im ersten Teil dieser Arbeit berechnen wir Form- faktoren von allgemeinen, gesch¨utzten und ungesch¨utzten Operatoren f¨ur verschiedene Schleifenordnungen und Multiplizit¨aten externer Teilchen in der = 4 SYM-Theorie. N Dies gelingt durch Anwendung verschiedener Masseschalen-Methoden, die im Kontext von Streuamplituden entwickelt wurden und sehr erfolgreich angewandt werden konnten, wenn auch erst nach wichtigen Weiterentwicklungen. Insbesondere zeigen wir, wie Formfaktoren und die zuvor genannten Methoden es erm¨oglichen, den Dilatationsoperator zu bestim- men. Dieser Operator liefert das Spektrum der anomalen Skalendimensionen der zusam- mengesetzten Operatoren und wirkt als Hamilton-Operator der integrablen Spin-Kette des Spektralproblems. Auf Einschleifenordnung nutzen wir verallgemeinerte Unitarit¨at, um den aus entsprechenden Schnitten rekonstruierbaren Teil des Formfaktors mit minimaler Multiplizit¨at f¨ur beliebige zusammengesetzte Operatoren zu berechnen, von dem wir den vollst¨andigen Dilatationsoperator auf Einschleifenordnung ablesen k¨onnen. Am Beispiel des Konishi-Operators und Operatoren des SU(2)-Sektors auf Zweischleifenordnung zeigen wir, dass Masseschalen-Methoden und Formfaktoren auch auf h¨oheren Schleifenordnun- gen zur Bestimmung des Dilatationsoperators eingesetzt werden k¨onnen. Die R¨uckstands- funktionen letztgenannter Formfaktoren erf¨ullen interessante universelle Eigenschaften im 3 4 Zusammenfassung Bezug auf ihre Transzendenz. Auf Baumgraphenniveau konstruieren wir Formfaktoren ¨uber erweiterte Masseschalen-Diagramme, Graßmann-Integrale und die integrabilit¨atsin- spirierte Technik der R-Operatoren. Letztere erm¨oglicht es, Formfaktoren als Eigen- zust¨ande der integrablen Transfermatrix zu konstruieren, was die Existenz eines Satzes erhaltener Ladungen impliziert. Deformationen der = 4 SYM-Theorie erlauben es uns, andere Theorien mit den N gleichen speziellen Eigenschaften zu finden und neue Erkenntnisse ¨uber den Ursprung von Integrabilit¨at und der AdS/CFT-Korrespondenz zu gewinnen. Im zweiten Teil dieser Ar- beit untersuchen wir die = 1 supersymmetrische β-Deformation und die nichtsuper- N symmetrische γ -Deformation. Beide teilen viele Eigenschaften mit der = 4 SYM- i N Theorie, speziell im planaren Limes. Sie zeigen jedoch auch neue Merkmale, insbesondere das Auftreten von Doppelspurtermen in ihrem Wirkungsfunktional. Zwar scheinen diese Terme im planaren Limes zu verschwinden, doch k¨onnen sie durch einen neuen Effekt der endlichen Systemgr¨oße, welchen wir Vorwickeln nennen, in f¨uhrender Ordnung beitra- gen. In der β-Deformation werden diese Terme f¨ur die konforme Invarianz ben¨otigt und wir berechnen die durch sie entstehenden Korrekturen zum vollst¨andigen planaren Dilatations- operator auf Einschleifenordnung und dessen Spektrum. In der γi-Deformation zeigen wir, dass Quantenkorrekturen rennende Doppelspurkopplungen ohne Fixpunkte induzieren, was die konforme Invarianz bricht. Dann berechnen wir die planaren anomalen Skalendimensio- nen von Einspuroperatoren, die aus L identischen Skalarfeldern bestehen, bei der kritischen Wickelordnung ℓ = L f¨ur alle L 2. F¨ur L 3 stimmen die Ergebnisse unser feldtheo- ≥ ≥ retischen Rechnung exakt mit den durch Integrabilit¨at gewonnenen Vorhersagen ¨uberein. F¨ur L = 2, wo die Vorhersage durch Integrabilit¨at divergiert, finden wir ein endliches, ra- tionales Ergebnis. Dieses h¨angt jedoch von der rennenden Doppelspurkopplung und durch sie vom Renormierungsschema ab. Abstract For more than half a century, quantum field theory (QFT) has been the most accurate and successful framework to describe the fundamental interactions among elementary particles, albeit with the notable exception of gravity. Nevertheless, QFTs are in general far from being completely understood. This is due to a lack of calculational techniques and tools as well as our limited understanding of the mathematical structures that emerge in them. In the last one and a half decades, tremendous progress has been made in understanding certain aspects of a particular QFT, namely the maximally supersymmetric Yang-Mills theory in four dimensions, termed = 4 SYM theory, which has risen the hope that N this theory could be exactly solvable. In particular, this progress occurred for scattering amplitudes due to the development of on-shell methods and for correlation functions of gauge-invariant local composite operators due to integrability. In this thesis, we address the question to which extend the methods and structures found there can be generalised to other quantities in the same theory and to other theories. Form factors describe the overlap between a gauge-invariant local composite operator on the one hand and an asymptotic on-shell scattering state on the other hand. Thus, they form a bridge between the purely off-shell correlation functions and the purely on- shell scattering amplitudes. In the first part of this thesis, we calculate form factors of general, protected as well as non-protected, operators at various loop orders and numbers of external points in = 4 SYM theory. This is achieved using many of the successful on- N shell methods that were developed in the context of scattering amplitudes, albeit after some important extensions. In particular, we show how form factors and on-shell methods allow us to obtain the dilatation operator, which yields the spectrum of anomalous dimensions of composite operators and acts as Hamiltonian of the integrable spin chain of the spectral problem. At one-loop level, we calculate the cut-constructible part of the form factor with minimal particle multiplicity for any operator using generalised unitarity and obtain the complete one-loop dilatation operator from it. We demonstrate that on-shell methods and form factors can be used to calculate the dilatation operator also at higher loop orders, using the Konishi operator and the SU(2) sector at two loops as examples. Remarkably, the finite remainder functions of the latter form factors possess universal properties with respect to their transcendentality. Moreover, form factors of non-protected operators share many features of scattering amplitudes in QCD, such as UV divergences and rational terms. At tree level, we show how to construct form factors via extended on-shell diagrams, a Graßmannian integral as well as the integrability-based technique of R operators. Using the latter technique, form factors can be constructed as eigenstates of an integrable transfer matrix, which implies the existence of a tower of conserved charges. Deformations of = 4 SYM theory allow us to find further theories with its special N properties and to shed light on the origins of integrability and of the AdS/CFT correspon- dence. In the second part of this thesis, we study the = 1 supersymmetric β-deformation N 5 6 Abstract and the non-supersymmetric γi-deformation. While they share many properties of their undeformed parent theory, in particular in the planar limit, also new features arise. These new features are related to the occurrence of double-trace terms in the action. Although apparently suppressed, double-trace
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