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Renormalization Group Flow of Scalar Models in Gravity Renormalization group flow of scalar models in gravity DISSERTATION zur Erlangung des akademischen Grades doctor rerum naturalium (Dr. rer. nat.) im Fach Physik eingereicht an der Mathematisch-Naturwissenschaflichen Fakultät I Humboldt-Universität zu Berlin von Herrn Filippo Guarnieri Betreuer der Arbeit Prof. Dr. Hermann Nicolai Präsident der Humboldt-Universität zu Berlin: Prof. Dr. Jan-Hendrik Olbertz Dekan der Mathematisch-Naturwissenschaflichen Fakultät I: Prof. Dr. Stefan Hecht Gutachter: Prof. Dr. Hermann Nicolai Prof. Dr. Martin Reuter Dr. Roberto Percacci Tag der mündlichen Prüfung: 08.04.2014 2 3 Sic unumquicquid paulatim protrahit aetas in medium ratioque in luminis erigit oras De rerum natura, Lucretius 4 Contents Abstract 9 Zusammenfassung 11 Sommario 13 Introduction 15 1 Basics of renormalization group 19 1.1 Gell-Mann Low equation . 20 1.2 Functional renormalization group . 22 1.2.1 Sharp cutoff equation . 26 1.2.2 Exact equation . 29 1.2.3 Proper time equation . 33 2 Renormalization group in quantum gravity 39 2.1 Renormalizability and unitarity . 40 2.2 Asymptotic safety scenario . 42 2.2.1 Gravity in 2 + dimensions . 45 2.2.2 Background field method . 47 2.2.3 Gauge fixing and ghosts . 49 2.2.4 Exact equation for gravity . 51 2.2.5 Polynomial truncations . 53 2.2.6 Non-polynomial truncations . 56 2.3 Hořava-Lifshitz gravity . 58 2.3.1 Spacetime anisotropy . 58 2.3.2 Anisotropic gravitational actions . 59 2.3.3 Detailed balance condition . 61 2.3.4 Anisotropic Weyl invariance . 62 3 Asymptotic safety in conformal gravity 63 3.1 RG flow equation for the Einstein-Hilbert action . 64 3.2 Conformally reduced action . 67 3.3 Polynomial truncations . 70 3.3.1 Sd topology . 71 3.3.2 Rd topology . 72 3.3.3 Fixed points and linearized flow . 74 6 CONTENTS 3.4 Non-polynomial truncations . 77 4 Brans-Dicke theory in the LPA 83 4.1 Scalar-tensor theories . 84 4.2 Quantization procedure . 87 4.2.1 Feynman gauge . 88 4.2.2 Landau gauge . 89 4.3 The flow equation . 90 4.3.1 Feynman gauge . 90 4.3.2 Landau gauge . 92 4.4 Analytical study of the equation . 93 4.4.1 Feynman gauge . 94 4.4.2 Landau gauge . 99 4.5 Numerical results . 102 4.5.1 Feynman Gauge . 102 4.5.2 Landau Gauge . 103 4.6 Quantum equivalence . 109 5 Hořava-Lifshitz gravity in 2+1 dimensions 111 5.1 The action in 2+1 dimensions . 112 5.2 Metric decomposition and gauge fixing . 114 5.3 Setup of the one-loop calculation . 116 5.4 Divergences and β-functions . 117 5.4.1 Heat kernel expansion . 118 5.4.2 β-functions . 120 Conclusions 125 A Functional representation of the effective action 131 B Arnowitt-Deser-Misner splitting 133 C CREH: Critical exponents and β-functions 137 C.1 Critical exponents and universal quantities . 137 C.2 β-functions . 139 C.2.1 The projection on Sd . 139 C.2.2 The projection on Rd .......................140 D Heat kernel techniques 143 D.1 Pseudodifferential operators . 144 D.1.1 Heat kernel expansion . 146 D.1.2 The Laplacian operators . 148 D.1.3 The squared Laplacian operators . 150 D.1.4 Other results . 151 E Other results for the scalar-tensor model 157 E.1 Subleading corrections to singular behavior . 157 E.2 The two-dimensional case . 159 CONTENTS 7 F Anisotropic Weyl invariance 161 8 CONTENTS Abstract In this Ph.D. thesis we will study the issue of renormalizability of gravitation in the context of the renormalization group (RG), employing both perturbative and non-perturbative techniques. In particular, we will focus on different gravitational models and approximations in which a central role is played by a scalar degree of freedom, since their RG flow is easier to analyze. More specifically, we will consider two types of situations: on the one hand, we will study scalar field theories obtained as conformally reduced toy models of gravity, that is, by neglecting the contribution of the spin-2 degrees of freedom of the metric in the quantization procedure. On the other hand, we will study scalar-tensor theories as dynamically equivalent theories of higher derivative models. We restrict our interest in particular to two quantum gravity approaches that have gained a lot of attention recently, namely the asymptotic safety scenario for gravity and the Hořava-Lifshitz quantum gravity. In the so-called asymptotic safety conjecture the high energy regime of gravity is controlled by a non-Gaussian fixed point which ensures non-perturbative renormalizability and finiteness of the corre- lation functions. We will then investigate the existence of such a non trivial fixed point using the functional renormalization group, a continuum version of the non- perturbative Wilson’s renormalization group. In particular we will quantize the sole conformal degree of freedom, which is an approximation that has been shown to lead to a qualitatively correct picture. The use of such a scalar toy model, moreover, will help us to investigate in a non-perturbative way the role that non-local operators have in the emergence of a symmetry-broken phase in the infrared. The question of the existence of a non-Gaussian fixed point in an infinite- dimensional parameter space, that is for a generic f(R) theory, cannot however be studied using such a conformally reduced model. Hence we will study it by quantizing a dynamically equivalent scalar-tensor theory, i.e. a generic Brans-Dicke theory with ! = 0 in the local potential approximation. We will then debate the breaking of the equivalence at a quantum level. Finally, we will investigate, using a perturbative RG scheme, the asymptotic freedom of the Hořava-Lifshitz gravity, that is an approach based on the emergence of an anisotropy between space and time which lifts the Newton’s constant to a marginal coupling and explicitly preserves unitarity. In particular we will evaluate the one-loop correction in 2+1 dimensions quantizing only the conformal degree of freedom. For this dimensionality it is in fact the only physical one. We obtain in this way the first results on the RG flow of the model in the ultraviolet. 10 ABSTRACT Zusammenfassung In dieser Doktorarbeit werden wir das Renormierungsproblem von Gravitationsthe- orien im Kontext der Renormierungsgruppe (RG) unter Anwendung von perturba- tiven und nicht-perturbativen Methoden untersuchen. Insbesondere werden wir uns auf verschiedene Gravitationsmodelle und Näherungen konzentrieren, in welchen die zentrale Rolle von einem skalaren Freiheitsgrad eingenommen wird, da sein RG Fluss einfacher zu analysieren ist. Wir werden zwei Fälle genauer betrachten: einerseits werden wir skalare Feldtheorien aus konform reduzierten vereinfachten Gravitation- smodellen untersuchen, indem der Beitrag der Gravitonen bei der Quantisierung vernachlässigt wird. Andererseits untersuchen wir Skalar-Tensortheorien als dy- namische Äquivalentstheorie zu höheren Ableitungstheorien. Wir konzentrieren uns besonders auf zwei Ansätze für Quantengravitation, die in letzter Zeit viel Aufmerksamkeit erhalten haben, nämlich den asymptotisch sicheren Fall der Gravitation und die Hořava-Lifshitz Quantengravitation. Das Prinzip der Asymptotischen Sicherheit beruht auf der Annahme, dass das hochenergetische Gravitationsregime von einem nicht-Gaußschen Fixpunkt bestimmt wird, der nicht- perturbative Renormierung und Endlichkeit der Korrelationsfunktionen sicherstellt. Wir werden die Existenz eines solchen nicht-trivialen Fixpunktes mit Hilfe der funk- tionalen Renormierungsgruppe, einer kontinuierlichen Version der nicht-perturbativen Wilson Renormierungsgruppe, untersuchen. Insbesondere werden wir den einzigen konformen Freiheitsgrad quantisieren. In diesem Fall konnte gezeigt werden, dass dies eine qualitativ gute Näherung darstellt. Diese skalare Feldtheorie hilft uns auf nicht-perturbative Weise die Rolle der nicht-lokalen Operatoren zu verstehen, die diese bei der Entstehung einer Symmetrie-gebrochenen Phase im Infraroten haben. Die Frage nach der Existenz eines nicht-Gaußschen Fixpunktes in einem unendlich- dimensionalen Parameterraum, das heißt für eine generische f(R)-Theorie, kann je- doch nicht mit einem solchen konform reduzierten Model analysiert werden. Deshalb werden wir es untersuchen, indem wir eine skalare dynamische Äquivalentstheorie, das heißt eine generische Brans-Dicke Theorie in der lokal-Potential Näherung mit ! = 0, quantisieren. Wir werden dann die Äquivalenzbrechung auf dem Quanten- level debattieren. Schließlich werden wir mittels einer perturbativen RG Methode die asympto- tische Freiheit der Hořava-Lifshitz Gravitationstheorie analysieren. Diese Gravita- tionstheorie beruht auf der Entstehung einer Anisotropie zwischen Raum und Zeit, die Newtons Konstante zu einer marginalen Koppelung werden lässt und explizit die Unitarität bewahrt. Insbesondere werden wir die Einschleifenkorrektur in 2+1 Dimensionen berechnen, indem wir nur den konformen Freiheitsgrad, den einzigen physikalischen für diese Dimensionalität, quantisieren. Wir erhalten somit die ersten Resultate über den RG Fluss dieses Models im ultravioletten Regime. 12 ZUSAMMENFASSUNG Sommario In questa tesi di dottorato studieremo la rinormalizzabilità di teorie gravitazionali nel contesto del gruppo di rinormalizzazione (RG), impiegando sia tecniche per- turbative che non perturbative. In particolare, ci concentreremo su diversi modelli gravitazionali ed approssimazioni in cui un ruolo centrale viene svolto da una grado di libertà scalare; il flusso di rinormalizzazione di questi modelli risulta infatti più semplice da analizzare. Più nello specifico prenderemo in considerazione due situ- azioni: in una studieremo teorie di campo
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