D-instantons and Effective Couplings in Intersecting D-brane Models Nikolas Akerblom M¨unchen 2008 D-instantons and Effective Couplings in Intersecting D-brane Models Nikolas Akerblom Dissertation an der Fakult¨atf¨urPhysik der Ludwig–Maximilians–Universit¨at M¨unchen vorgelegt von Nikolas Akerblom aus J¨onk¨oping/Schweden M¨unchen, April 2008 Erstgutachter: PD Dr. Ralph Blumenhagen Zweitgutachter: Prof. Dr. Dieter L¨ust Tag der m¨undlichen Pr¨ufung:15. Juli 2008 Second Witch When the hurlyburly’s done, When the battle’s lost and won. W. Shakespeare, Macbeth, Act 1, Scene 1. To Heike Contents Zusammenfassung ix Summary xi 1 Introduction and Overview 1 1.1 Why Go Beyond the Standard Model? . 2 1.2 String Theory . 6 1.2.1 Intersecting D6-brane Models . 13 1.2.2 Discussion . 15 1.3 Overview of this Thesis . 17 2 Some Background: Instantons in Field Theory 19 2.1 Introduction . 20 2.2 Instantons in Yang–Mills Theory . 21 2.3 Instantons in the Euclidean Path Integral . 25 2.4 Instanton Resolution of the U(1) Problem . 28 2.5 The CP Problem of QCD . 29 2.6 Baryon and Lepton Number Violations . 31 2.7 ADHM Construction . 32 2.8 The ADS Superpotential . 33 3 The ADS Superpotential from D-instantons 39 3.1 Introduction . 40 3.2 Instanton Generated Superpotential . 40 3.3 SQCD from an Intersecting D6-brane Model . 44 3.3.1 Field Theory Limit . 47 3.3.2 Disk Amplitudes . 49 3.4 Calculating the ADS Superpotential . 53 3.4.1 Fermionic Zero Mode Integration . 53 3.4.2 Bosonic Zero Mode Integration . 54 3.5 Relation to Field Theory . 58 viii Contents 4 One-loop Corrections to Gauge Kinetic Functions 61 4.1 Introduction . 62 4.2 Non-renormalization Theorems . 62 6 4.3 Threshold Corrections for T /Z2 × Z2 Models . 64 4.3.1 Background and Calculations . 65 1-loop 4.4 Extracting fa from Gauge Threshold Corrections . 77 6 4.4.1 Results for T /Z2 × Z2 Models . 78 5 D-instanton Corrections to Low Energy Effective Couplings 83 5.1 Introduction . 84 5.2 Contributions to the Superpotential . 85 5.3 Corrections to Gauge Kinetic Functions . 89 5.4 Consequences for Toroidal Orbifold Models . 90 6 Conclusions 93 A Four Dimensions of Spacetime 97 B Contributions of Instantons 103 C Euclidean Field Theory 105 D Some Relations in Linear Algebra 109 Acknowledgements 113 Bibliography 115 Zusammenfassung Diese Dissertation besch¨aftigt sich mit zwei Aspekten von Intersecting D6-brane Modellen: Wir behandeln einige von E2-Instantonen erzeugte Effekte und betrachten Einschleifenkor- rekturen zu den eichkinetischen Funktionen in der effektiven Niederenergiefeldtheorie. Wir beginnen Kapitel 1 mit einer allgemeinen Einf¨uhrung, in welcher der Frage nachgegan- gen wird, warum man Stringtheorie als einen Kandidat f¨ureine fundamentale Theorie der Naturgesetze in Betracht ziehen sollte. Hiernach behandeln wir in Kapitel 2 einige Themen, die mit Instantonen in Feldtheorie zusammenh¨angen und schließen mit dem von Instan- tonen erzeugten Superpotential der supersymmetrischen QCD, dem sogenannten Affleck– Dine–Seiberg (ADS) Superpotential. In Kapitel 3 gehen wir dann daran, das ADS Superpotential in einem Intersecting D6- brane Modell als den Effekt eines E2-Instantons herzuleiten. Dazu legen wir zun¨achst den f¨urdie Berechnung des Beitrags eines E2-Instantons zum Superpotential notwendigen Formalismus dar, konstruieren daraufhin eine Realisierung von SQCD in einem (lokalen) Intersecting D6-brane Modell und zeigen schließlich, daß der Formalismus in der Tat das Affleck–Dine–Seiberg Superpotential reproduziert. In Kapitel 4 betrachten wir Einschleifenkorrekturen zu eichkinetischen Funktionen. Diese Einschleifenkorrekturen h¨angen in recht komplizierter Art und Weise mit Schwellenkorrek- turen zu Eichkopplungen zusammen und wir berechnen diese f¨ur Z2 × Z2 Torus-Orbifold Modelle. Eine Analyse des Formalismus f¨urdie Berechnung des Superpotentialbeitrages eines E2- Instantons wirft die Frage auf, ob dieser Beitrag eine holomorphe Gr¨oße ist. Wir gehen dieser Frage in Kapitel 5 nach und zeigen, daß der Beitrag in der Tat holomorph ist. Außer- dem diskutieren wir, unter welchen Umst¨andenE2-Instanton Korrekturen der eichkinetis- chen Funktionen zu erwarten sind. Es schließen sich eine Anzahl von Anh¨angen an, von denen einige Themen behandeln, die im Haupttext nur gestreift werden und andere notwendige Formeln bringen. x Zusammenfassung Summary This thesis is concerned with two aspects of intersecting D6-brane models: We treat some of the effects that are induced by E2-instantons and we consider one-loop corrections to the gauge kinetic functions in the low-energy effective field theory. We begin in chapter 1 with a general introduction covering the question why to consider string theory as a candidate theory of nature. After that, in chapter 2, we go over some topics related to field theory instantons and finally discuss the instanton generated su- perpotential in supersymmetric QCD (SQCD), the so-called Affleck–Dine–Seiberg (ADS) superpotential. In chapter 3 we then proceed to reproduce the ADS superpotential in an intersecting D6- brane realization of SQCD as the effect of a single E2-instanton. In order to do so, we first explicate the necessary formalism for calculating the E2-instanton contribution to the superpotential, then construct a (local) intersecting D6-brane realization of SQCD, and finally show that the formalism indeed leads to the Affleck–Dine–Seiberg superpotential. In chapter 4 we consider one-loop corrections to the gauge kinetic functions. These one-loop corrections are related quite intricately to gauge threshold corrections which we compute for intersecting D6-brane models on the Z2 × Z2 toroidal orbifold. An analysis of the formulas for calculating the E2-instanton contribution to the superpo- tential raises the question whether this contribution is a holomorphic quantity. In chapter 5 we take up this question showing that indeed it is holomorphic. In addition, we discuss under which circumstances one can expect E2-instanton corrections to the gauge kinetic functions. There follow a number of appendices, some elaborating on points touched upon in the main text, some collecting necessary formulas. xii Summary Chapter 1 Introduction and Overview 2 1. Introduction and Overview 1.1 Why Go Beyond the Standard Model? Physics is not a finished logical system. Rather, at any moment it spans a great confusion of ideas, some that survive like folk epics from the heroic periods of the past, and others that arise like utopian novels from our dim premonitions of a future grand synthesis. Steven Weinberg, Gravitation and Cosmology. As this is being written, the SU(3) × SU(2) × U(1) model of particle physics (patched suitably to take into account neutrino masses), the standard model, accounts for every experimental fact of the subatomic world and it does so with marvelous precision For instance, its QED sector alone accounts for the anmalous magnetic moment of the electron to nine significant figures. The experimental value preferred by the particle data group [153] is g − 2 = 1159.6521810(7) × 10−6 , (1.1) 2 exp which is to be contrasted with the current best theoretical value [14] (to eigth order in the fine structure constant α) g − 2 = 1159.65218279(7.71) × 10−6 . (1.2) 2 th The reason why the QED sector alone can account for the electron’s anomalous magnetic moment to such high accuracy is because its mass is comparatively small, so that loop corrections from massive particles are more or less negligible. The situation is different for the muon (mµ ' 200 me), making measurements of its magnetic moment a potential laboratory for detecting hitherto unknown particles. As another success of the standard model (and there are a quasi infinity of them) let us only mention that QCD lattice calculations determine the mass of e.g. the proton to an accuracy of a few percent, given the constituent quark masses [58]. In comparison to the theoretical value of the electron’s anomalous magnetic moment, a few percent is, of course, not very precise but given the fact that in the scheme of things the proton mass looks quite random it is nevertheless a remarkable achievement. However, the bain of fundamental physics has always been that in some places things don’t seem to fit. This was so around 1900 where applications of classical statistical mechanics gave nonsensical answers for black-body radiation, culminating in the end in quantum mechanics, and this certainly is so today. The standard model has shortcomings and downright defects. Some Open Points in the Standard Model. One particular shortcoming of the standard model which comes to mind is, in fact, a gap in our analytic understanding of 1.1 Why Go Beyond the Standard Model? 3 the QCD sector of the standard model. It is currently not possible to understand the confinement of quarks observed in nature in an analytic way, although numerical lattice calculations gives confidence that QCD indeed shows confining behavior (see, for instance, [106]). Another shortcoming is the arbitrariness of the standard model. The gauge group and particle content is put in by hand, as are the masses of the particles, their representantions, couplings etc. All of these could in principle be just about anything, the only restriction being cancellation of gauge anomalies. In the 19th century such arbitrariness of a physical theory (cf. materials constants) was not considered a problem but nowadays physicists feel confident that large chunks of the world can not only be described but explained. If explanation, then, is the goal, it is a desideratum to further and further reduce the number of parameters in fundamental physical theory. The list of shortcomings of the standard model clearly could be extended almost indefinitely but let us just leave it at that and mention a few of its defects. The most conspicuous defect clearly is that the standard model, by design, does not en- compass the gravitational interaction.