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Renormalization Group Approach to Dense Baryonic Matter RENORMALIZATIONGROUPAPPROACH TODENSEBARYONICMATTER Matthias Drews October, 2014 Technische Universität München Physik Department Institut für Theoretische Physik T39 Univ.-Prof. Dr. Wolfram Weise RENORMALIZATIONGROUPAPPROACH TODENSEBARYONICMATTER Matthias Johannes Drews, M. Sc. Vollständiger Abdruck der von der Fakultät für Physik der Technischen Universität München zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften (Dr. rer. nat.) genehmigten Dissertation. Vorsitzender: Univ.-Prof. Dr. Stephan Paul Prüfer der Dissertation: 1. Univ.-Prof. Dr. Wolfram Weise (em.) 2. Univ.-Prof. Dr. Nora Brambilla Die Dissertation wurde am 02.10.2014 bei der Technischen Universität München einge- reicht und durch die Fakultät für Physik am 11.11.2014 angenommen. This work has been supported by the BMBF, the Excellence cluster “Origin and Structure of the Universe”, the Fondazione Bruno Kessler, the TUM Graduate School, and the Wilhelm und Else Heraeus-Stiftung. ABSTRACT This work investigates a chiral nucleon-meson model, which is suited to analyze chiral sym- metry breaking and the thermodynamics of nuclear matter at the same time. We go beyond the mean-field approximation by including pionic fluctuations in a non-perturbative way with methods of the functional renormalization group. Within the model, we can exclude a chiral phase transition for temperatures below 100 MeV and densities below about twice the nuclear saturation density. Furthermore, we extend the model to asymmetric nuclear matter and therefore obtain a description for the interior of neutron stars. ZUSAMMENFASSUNG In dieser Arbeit untersuchen wir ein chirales Nukleon-Meson-Modell, welches sowohl chi- rale Symmetriebrechung als auch die Thermodynamik von Kernmaterie beschreiben kann. Wir gehen über die Mean-Field-Näherung hinaus und berücksichtigen Pionfluktuationen auf nichtperturbative Weise im Rahmen der funktionalen Renormierungsgruppe. Innerhalb des Modells kann für Temperaturen unterhalb von 100 MeV und Dichten unter etwa der zweifachen nuklearen Sättigungsdichte ein chiraler Phasenübergang ausgeschlossen werden. Darüber hinaus weiten wir das Modell auf asymmetrische Kernmaterie aus und erhalten damit eine Beschreibung für das Innere von Neutronensternen. vii PUBLICATIONS The text of this thesis contains material which has previously been published in the arti- cles listed below. M. Drews, T. Hell, B. Klein, and W. Weise, “Thermodynamic phases and mesonic fluctua- tions in a chiral nucleon-meson model”, Phys. Rev. D. 88 (2013) 096011, arXiv:1308.5596, M. Drews, T. Hell, B. Klein, and W. Weise, “Dense nucleonic matter and the renormal- ization group”, EPJ web conf. 66 (2014) 04008, arXiv:1307.6973, M. Drews, W. Weise, “Functional renormalization group approach to neutron matter”, Phys. Lett. B 738 (2014) 187–190, arXiv:1404.0882. ix ACKNOWLEDGMENTS In the first place, I would like to express my gratitude to Prof. Dr. Wolfram Weise for giving me the opportunity to write my thesis in his group. He was always open for inspiring discussions and for sharing his profound insights on nuclear physics with a former string theorist. I would further like to thank him for offering me the possibility to participate in numerous conferences all over the world and for initiating my six-months stay at the ECT* in beautiful Trento, Italy. I am thankful to Kenji Fukushima for inviting me to the Yukawa Institute in Kyoto for a very instructive three-weeks visit in 2011. I further thank Kouji Kashiwa for inviting me to a joyful visit at the Brookhaven National Laboratory on Long Island in 2012. I also wish to thank the members of the ECT* for their warm welcome during my stay in Trento. It is a pleasure to thank Prof. Dr. Norbert Kaiser for his helpful advice on all sorts of conceptional and practical issues during my research. I am very grateful to Thomas Hell for his support and his collaboration. Moreover, I am glad to thank my office mates, Robert Lang and Alexander Laschka, for the friendly atmosphere, which made me enjoy my stay at the TUM. I am very grateful to all my current and former colleagues at T39, in particular Michael Altenbuchinger, Nino Bratovic, Maximilian Duell, Salvatore Fior- illa, Lisheng Geng, Jeremy Holt, Bertram Klein, Stefan Petschauer, Paul Springer and Corbinian Wellenhofer for a nice and inspiring environment, many enlightening conversa- tions on physics and for shared hours outside the office. A very special place is reserved for my thanks to Kathrin Helmsauer for her love and support. Last, but not least, I am indebted to express my deepest gratitude to my family for their permanent support during all these years, which made this thesis possible. xi CONTENTS 1 introduction 1 2 from gauge theories to nuclear matter 3 2.1 Gravity . .3 2.2 Quantum Chromodynamics . .5 2.3 Asymptotic freedom and confinement . .6 2.4 Global symmetries . .8 2.5 QCD at finite temperature and density . 13 2.5.1 Heavy-ion collisions . 13 2.5.2 Chiral restoration . 15 2.6 Nuclear matter . 20 3 concepts of the renormalization group 23 3.1 The effective action . 23 3.2 Finite temperature . 26 3.3 Exact renormalization group equations . 28 3.3.1 Wilson’s RG and Polchinski’s RG . 28 3.3.2 The functional RG . 30 3.4 Numerical evaluation . 34 4 functional renormalization group approach to a chiral nucleon-meson model 37 4.1 The chiral nucleon-meson model . 37 4.2 Mean-field approximation . 41 4.2.1 Vacuum constraints . 45 4.2.2 Constraints from the liquid-gas phase transition . 47 4.2.3 Constraints from pure neutron matter . 51 4.3 Wetterich’s flow equation . 54 4.3.1 Taylor-expansion method . 56 4.3.2 Flow equations for the vector fields . 58 4.3.3 Fluctuations around the liquid-gas transition . 61 4.3.4 Renormalization group equations in the mean-field approximation . 62 4.4 Symmetric nuclear matter . 65 4.4.1 Liquid-gas transition . 66 4.4.2 Chiral restoration . 70 4.4.3 Fluctuations . 74 4.5 Asymmetric nuclear matter . 76 4.5.1 Phase coexistence and equation of state . 76 4.5.2 In-medium pion mass . 78 4.5.3 Chiral restoration . 79 4.6 Neutron stars . 81 5 summary and outlook 87 a appendix 89 xiii 1 INTRODUCTION In January 1932, Lev Landau theorized [1] about the interior of heavy stars, a place characterized by “a violation of the law of energy”, where “the laws of ordinary quantum mechanics break down”, and “the density of matter becomes so great that atomic nuclei come in close contact, forming one gigantic nucleus”. Whereas the two former conclusions might sound a bit premature from today’s perspective, the “gigantic nucleus” is often understood as a first description of a neutron star. Remarkably, Landau’s paper was developed independently and even before1 the detection of the neutron by James Chadwick in February 1932 [3]. It took another two years until Walter Baade and Fritz Zwicky constructed the first models of neutron stars built out of actual neutrons, without altering quantum mechanics and the conservation of energy [4–7]. These days we are again in a situation where the smoke starts to clear. In the past, high-precision measurements of neutron-star masses were relatively sparse. Consequently, the interior of a neutron star was up for speculation, and a plethora of theories based on all kinds of exotic matter was on the market. But after the discovery of neutron stars as heavy as twice the mass of the sun [8,9], paired with an extreme accuracy, theoretical considerations are strongly constrained. It turns out that conventional models – working with nucleon and meson rather than quark degrees of freedom – are in agreement with all constraints. More exotic models are either ruled out or require a substantial amount of fine tuning. In the present thesis, we study a generalized linear sigma model of nucleons (protons and neutrons), interacting through exchange of pions and other boson fields. We include fluctuation effects in the framework of the functional renormalization group in a systematic non-perturbative way. The thermodynamic equation of state obtained in this model can be taken to describe the interior of neutron stars. This allows us to compare our results with state-of-the-art approaches such as chiral effective field theory, quantum Monte-Carlo calculations, or phenomenological models. One aspect of the strong force as described by Quantum Chromodynamics (QCD) is chiral symmetry. At small baryon chemical potential chiral symmetry is restored at high temperatures in a crossover and not in a true phase transition. In contrast, the situation at higher baryon chemical potentials remains unclear. Many models predict a first-order transition that ends in a second-order critical endpoint. A lot of theoretical and experi- mental effort is put into the determination of the critical endpoint, whose existence would be a landmark in the phase-diagram of QCD. However, it is far from certain whether such a critical point exists. While we certainly cannot definitely answer this question within our model, we can still exclude the appearance of a chiral first-order phase transition in a broad region. Another aspect of our approach concerns the question of chiral restoration in the context of neutron matter at low temperatures, which is also relevant for the description of neutron 1 See Ref. [2] for the historical backgrounds, in particular how the paper on neutron stars possibly saved Landau’s life. 1 2 introduction stars. The chiral condensate is an order parameter of chiral restoration. In chiral effective field theory it was observed that the chiral condensate in symmetric nuclear matter is stabilized at higher densities [10–13]. An important role is played by two-pion exchange processes and repulsive three-body forces, including the appearance of a virtual ∆-isobar and the action of the Pauli principle in the nucleon Fermi sea.
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