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The Topological Suceptibility of QCD at High Temperatures The Topological Susceptibility of QCD at High Temperatures Die topologische Suszeptibilität der QCD bei hohen Temperaturen Zur Erlangung des Grades eines Doktors der Naturwissenschaften (Dr. rer. nat.) genehmigte Dissertation von Peter Thomas Jahn Tag der Einreichung: 15.10.2019, Tag der Prüfung: 27.11.2019 Darmstadt — D 17 1. Gutachten: Prof. Ph.D. Guy D. Moore 2. Gutachten: Prof. Dr. rer. nat. Jens Braun Fachbereich Physik Institut für Kernphysik Theoriezentrum – AG Moore The Topological Susceptibility of QCD at High Temperatures Die topologische Suszeptibilität der QCD bei hohen Temperaturen Genehmigte Dissertation von Peter Thomas Jahn 1. Gutachten: Prof. Ph.D. Guy D. Moore 2. Gutachten: Prof. Dr. rer. nat. Jens Braun Tag der Einreichung: 15.10.2019 Tag der Prüfung: 27.11.2019 Darmstadt — D 17 Bitte zitieren Sie dieses Dokument als: URN: urn:nbn:de:tuda-tuprints-94854 URL: http://tuprints.ulb.tu-darmstadt.de/id/eprint/9485 Dieses Dokument wird bereitgestellt von tuprints, E-Publishing-Service der TU Darmstadt http://tuprints.ulb.tu-darmstadt.de [email protected] Die Veröffentlichung steht unter folgender Creative Commons Lizenz: Namensnennung – Nicht kommerziell – Keine Bearbeitungen 4.0 International https://creativecommons.org/licenses/by-nc-nd/4.0/ To Klary who passed away way too early. I’ve never met a cat as good-hearted and lively as you. Erklärung zur Dissertation Hiermit versichere ich, dass ich die vorliegende Dissertation selbstständig angefertigt und keine anderen als die angegebenen Quellen und Hilfsmittel verwendet habe. Alle wörtlichen und paraphrasierten Zitate wurden angemessen kenntlich gemacht. Die Ar- beit hat bisher noch nicht zu Prüfungszwecken gedient. Darmstadt, den 06. Dezember 2019 (Peter Thomas Jahn) Abstract Two of the most challenging problems in modern physics are the origin of dark matter and the strong CP problem. The latter means the non-observation of the violation of the combined particle-antiparticle and parity (CP) symmetries by the strong interaction which is conceptually allowed. Both problems – although prima facie disparate – could be simultaneously solved by the Peccei-Quinn mechanism. This results in a new particle, the axion. Despite strong experimental efforts, the discovery of the axion is yet to come, making precise theoretical predictions of its properties, especially its mass, highly valu- able. The axion’s properties are closely related to the topological structure of the vacuum of quantum chromodynamics (QCD). The QCD vacuum exhibits topologically non-trivial fluctuations of the gauge fields with the most important fluctuations being instantons. These topological fluctuations are quan- tified by the topological susceptibility χtop that controls the axion mass and therefore is – especially at high temperatures – an important input for axion cosmology. Since topological effects are inherently non-perturbative, lattice QCD is particularly suitable for precisely determining χtop. However, lattice simulations become extremely challenging at high temperatures because χtop is very suppressed. In this work, we develop and establish a novel method based on a combination of gradient flow and reweight- ing that artificially enhances the number of instantons and therefore allows to determine χtop at high temperatures. For computational simplicity, we content ourselves to pure SU(3) Yang-Mills theory for developing the method, but it is explicitly designed to be applicable also in full QCD. In particular, we provide a discretization of the instanton that allows for an analysis of the lattice-spacing effects on a lattice study of χtop. We then present the reweighting method that is eventually used to determine χtop up to 2 GeV in pure SU(3) Yang-Mills theory which constitutes the first direct determination of χtop at such high temperatures. Zusammenfassung Zwei der spannendsten Probleme der modernen Physik sind der Ursprung der dunklen Materie und das starke CP-Problem. Letzteres beschreibt die Nichtbeobachtung einer Verletzung der kombinierten Teilchen-Antiteilchen- und Parität (CP)-Symmetrien durch die starke Wechselwirkung, die grundsätz- lich erlaubt ist. Beide Probleme – obwohl a priori unterschiedlich – könnten gleichzeitig durch den Peccei-Quinn-Mechanismus gelöst werden. Dieser postuliert das Axion als neues Elementarteilchen. Trotz starker experimenteller Bemühungen steht die Entdeckung des Axions noch aus, sodass präzise theoretische Vorhersagen über dessen Eigenschaften, insbesondere dessen Masse, sehr wertvoll sind. Die Axioneigenschaften sind eng mit der topologischen Struktur des Vakuums der Quantenchromody- namik (QCD) verknüpft. Das QCD-Vakuum weist topologisch nichttriviale Fluktuationen der Eichfelder auf, quantifiziert durch die topologische Suszeptibilität χtop, welche die Axionenmasse kontrolliert und daher – insbesondere bei hohen Temperaturen – ein wichtiger Parameter für die Axionkosmologie ist. Die wichtigsten Fluktuationen sind dabei Instantone. Da topologische Effekte intrinsisch nichtperturba- tiv sind, eignen sich Gittersimulationen besonders gut zur präzisen Bestimmung von χtop. Diese werden jedoch bei hohen Temperaturen sehr anspruchsvoll, da χtop stark unterdrückt ist. In dieser Arbeit stellen wir eine neue Methode vor, basierend auf einer Kombination aus Gradientenfluss und Regewichtung, die die Anzahl der Instantone künstlich erhöht und es daher ermöglicht, χtop bei hohen Temperaturen zu bestimmen. Zur Entwicklung der Methode beschränken wir uns auf SU(3) Yang-Mills-Theorie, was numerisch deutlich günstiger ist. Die Methode ist jedoch so konzipiert, dass sie auch auf volle QCD angewandt werden kann. Insbesondere entwickeln wir eine Diskretisierung des Instantons, was eine Analyse der Diskretisierungseffekte bei eine Gitterberechnung von χtop ermöglicht. Weitherin stellen wir die Regewichtungs-Methode vor, die schließlich verwendet wird, um χtop bis zu Temperaturen von 2 GeV in reiner SU(3) Yang-Mills-Theorie zu bestimmen, was die erste direkte Bestimmung von χtop bei solch hohen Temperaturen darstellt. Contents 1. Introduction 1 2. Quantum Chromodynamics 11 2.1. The QCD Lagrangian........................................... 11 2.2. Path Integral Formalism and Thermal Field Theory........................ 14 2.3. Chiral Symmetry and the Chiral Anomaly.............................. 17 2.4. Topology and the QCD Vacuum..................................... 19 2.4.1. Instantons and the Vacuum Structure of QCD....................... 20 2.4.2. The Theta Vacua......................................... 30 2.4.3. The Topological Susceptibility................................ 31 2.4.4. The Dilute Instanton Gas................................... 33 3. Lattice Gauge Theory 37 3.1. Basic Concepts of Lattice Gauge Theory............................... 37 3.2. Pure Gauge Theory on the Lattice................................... 38 3.3. Symanzik Improvement Program................................... 40 3.4. The Lattice Field-Strength Tensor................................... 41 3.4.1. Clover Field-Strength Tensor.................................. 42 3.4.2. Improved Field-Strength Tensor................................ 42 3.4.3. The Topological Charge..................................... 43 3.5. Gradient Flow............................................... 44 3.5.1. Wilson Flow............................................ 45 3.5.2. Zeuthen Flow........................................... 46 3.6. Scale Setting................................................ 48 3.7. Monte Carlo Simulations........................................ 49 3.7.1. Hybrid Monte Carlo Algorithm................................ 51 3.8. Autocorrelation and Error Analysis.................................. 53 3.8.1. Binning and Jackknife...................................... 54 3.8.2. Example of a Statistical Analysis............................... 54 4. Topological Configurations on the Lattice 57 4.1. Instanton and Caloron Discretization................................. 58 4.2. Lattice Instanton and Caloron Properties.............................. 60 4.2.1. Caloron vs. Instanton...................................... 60 4.2.2. Actions............................................... 61 4.2.3. Topological Charges....................................... 62 4.2.4. Critical Radius.......................................... 63 4.3. Estimated a2 Errors in the Topological Susceptibility....................... 65 4.4. Conclusions................................................. 68 5. The Topological Susceptibility via Reweighting 69 5.1. The Reweighting Method........................................ 70 5.1.1. Definition of Reweighting................................... 70 5.1.2. Choice of Reweighting Variable................................ 71 5.1.3. Updating with an Arbitrary Weight Function....................... 71 5.1.4. Building the Reweighting Function............................. 72 5.1.5. Parameters to Tune....................................... 75 5.2. Results.................................................... 76 5.3. Discussion.................................................. 81 6. Improvement of the Reweighting Method: χtop up to 2 GeV 85 6.1. The Method................................................ 85 6.1.1. The Low Barrier......................................... 85 6.1.2. The High Barrier......................................... 87 6.1.3. The Middle Region....................................... 88 6.1.4. Reweighting with Multiple Regions.............................. 88 6.1.5. Parameters to Tune......................................
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