STRANGELETS Effects of Finite Size and Exact Color Singletness Dan Mønster Jensen Institute of Physics and Astronomy University of Aarhus YSIK O F G R A O F S T R T O U N T I O T M S I N I IFA arXiv:1904.04534v1 [hep-ph] 9 Apr 2019 July 1996 Preface The present volume has been submitted to the Faculty of Natural Science at the University of Aarhus, in order to fulfill the requirements for the Ph.D. degree in physics. The work presented here has been done in collaboration with my supervisor, Jes Madsen, during my time as a Ph.D student at the Institute of Physics and Astronomy, University of Aarhus, and during a stay at the Physics Department, Brookhaven National Laboratory. I have also had the pleasure of working with another Ph.D. student, Michael B. Christiansen, here at the Institute of Physics and Astronomy. List of Publications The work presented here has been described in part in the following publications: 1. Dan Mønster Jensen and Jes Madsen, Strangelets at Non-Zero Temper- ature, in Strangeness and Quark Matter, edited by G. Vassiliadis, A. D. Panagiotou, S. Kumar, and J. Madsen (World Scientific, Singapore, 1995), pp. 220{229. 2. Jes Madsen, Dan M. Jensen, and Michael B. Christiansen, Color Singlet Suppression of Quark-Gluon Plasma Formation, Phys. Rev. C 53, 1883 (1996). 3. Dan M. Jensen and Jes Madsen, Strangelets with Finite Entropy, Phys. Rev. D 53, R4719 (1996). 4. Dan Mønster Jensen, Jes Madsen, and Michael B. Christiansen, Color Singlet Strangelets, to appear in Heavy Ion Phys. (1996). The contents of Paper 1 has been described in Chapter 3 and Chapter 4, with some additional details not included in the paper. The subject of Paper 2 is described in Chapter 7, but some new material has been added, as discussed in that chapter. Chapter 6 discusses the same problem that is treated in Paper 3, but again some additional aspects are discussed, some of them also presented in Paper 4. iii iv Notation I use the Einstein summation convention, where implicit summation is under- stood whenever an index is repeated, as in X µqNq ≡ µqNq; q but not if one of the indexed quantities appear as an argument X Nq(µq) 6= Nq(µq): q Relativistic convention: µ x = (t; x) xµ = (t; −x) @ µ @ @µ = ( @t ; r) @ = ( @t ; −∇) Minkowski metric: 0 1 0 0 0 1 µν B 0 −1 0 0 C η = ηµν = B C @ 0 0 −1 0 A 0 0 0 −1 Unless otherwise noted I have used so-called natural units, in which the Planck constant, the Boltzmann constant, and the speed of light are unity ~ = kB = c = 1: Acknowledgements First and foremost I would like to thank my supervisor, Jes Madsen. During all of my Ph.D. study he has provided me with constant inspiration, and guidance. Many thanks to Carl Dover for making my visit at the Physics Department, Brookhaven National Laboratory possible. I was sad to learn that Prof. Dover passed away recently. I would also like to thank my office mates Michael B. Christiansen and Georg Bergeton Larsen for many interesting discussions, and for being great company. At Brookhaven I shared office, a single computer terminal, and many good times with Raffaele Mattiello. My Ph.D. study has been financed by a scholarship from the University of Aarhus. I also appreciate the generosity and hospitality of Brookhaven National Laboratory who funded part of my stay. A very special thanks goes to my girlfriend Rikke F¨ursterling,for putting up with me for all these years, especially during the writing of this thesis. Thanks for all your love and support! Last, but not least, I wish to thank my parents, Inge Mønster Jensen and Carl M. Jensen, who have always backed me up in what I did. Thanks! Contents Preface iii List of Publications . iii Notation . iv Acknowledgements . iv 1 Introduction 1 1.1 Basic Ideas . 1 1.2 Phase Diagram of Strongly Interacting Matter . 4 1.3 The Cosmological Quark-Hadron Phase Transition . 6 1.4 Ultrarelativistic Heavy-Ion Collisions . 6 1.5 Strange Stars . 7 1.6 Thesis Outline . 9 2 Quark Matter in the MIT Bag 11 2.1 The MIT Bag Model . 11 2.2 Shell Model . 14 2.3 Liquid Drop Model . 16 2.3.1 Density of States . 16 2.3.2 Partition Function . 20 2.3.3 Mass Formula . 25 2.4 Summary and Discussion . 26 3 Strangelets at Finite Temperature 31 3.1 Equilibrium Conditions . 31 3.2 Mass Formula . 32 3.2.1 Fixed Temperature . 34 3.2.2 Fixed Entropy . 34 3.3 Phase Diagram . 35 3.4 Summary and Discussion . 37 4 Quark Matter { Hadronic Matter Equilibrium 39 4.1 Scenario . 39 4.2 Description of the Model . 40 4.2.1 The Quark Phase . 40 v vi Contents 4.2.2 The Hadronic Phase . 41 4.2.3 Equilibrium Criteria . 42 4.3 Results . 42 4.3.1 The Phase Diagram at T = 0 . 43 4.3.2 The Phase Diagram for T 6= 0 . 44 4.4 Conclusion and Discussion . 46 5 The Color Singlet Projection 49 5.1 Some Results from Group Theory . 49 5.1.1 Lie Groups . 50 5.1.2 Representations . 50 5.2 The Group Theoretical Projection Method . 52 5.3 A Simple Example: U(1) . 54 5.4 The Color Singlet Partition Function . 55 5.4.1 Background . 56 5.4.2 The Partition Function . 58 5.5 Evaluation of the Partition Function . 62 5.5.1 Massless Quarks and Gluons . 62 5.5.2 Saddle-Point Approximation . 64 5.5.3 Numerical Evaluation . 66 5.6 Summary and Discussion . 67 6 Color Singlet Strangelets 69 6.1 Massless Quarks . 69 6.1.1 Saddle-Point Approximation . 69 6.1.2 Numerical Calculation . 72 6.2 Massive Quarks . 74 6.3 Conclusion and Discussion . 76 7 Color Singlet Suppressed Nucleation 77 7.1 Nucleation Rate . 77 7.1.1 Free Energy Barrier . 77 7.2 Rate Suppression . 79 7.3 Conclusion and Discussion . 81 8 Finale 85 8.1 Thesis Conclusion and Summary . 85 8.2 Outlook . 87 A The Multiple Reflection Expansion 89 A.1 Trace Formula for the Density of States . 89 A.2 Density of States for a Free Field . 92 A.3 Expansion of the Green's function . 94 A.4 Expansion of the Density of States . 98 Contents vii B Fermi-Dirac and Bose-Einstein Integrals 99 B.1 Fermions . 99 B.2 Bosons . 100 B.3 Complex Chemical Potential . 100 C The SHELL Program 103 C.1 The Main Loop . 103 C.2 Minimizing the Energy . 103 C.2.1 MinimumEnergy . 104 C.2.2 Energy ............................104 C.2.3 Find chempot . 105 C.2.4 Number ............................105 C.3 The Optimal Composition . 106 C.3.1 Find composition . 106 C.3.2 Sort levels . 108 C.4 The Eigenvalues . 109 C.4.1 Find levels . 109 C.4.2 Get eigenvalues . 110 C.4.3 Dirac .............................111 C.4.4 fkx ..............................112 Bibliography 119 Index 121 Chapter I Introduction This thesis deals with quark matter, in particular, strange quark matter with the focus being on very small systems, the so-called strangelets. Quarks are fundamental particles, just like the electron, but they are a sort of ghosts, who never show themselves. A free quark has thus never been seen, and it is believed to be a law of nature that quarks may not be observed directly. They always come either as three quarks, making up what we call baryons, of which the familiar proton and neutron are the most well-known, or as a pair of quark and antiquark, called a meson, of which the π-meson is an example. The complicated rules stating what combinations of quarks are allowed (the so-called color singlet states) do not limit the spectrum to the baryons and the mesons, but also allow for the existence of conglomerates of an integer multiple of three quarks. The lightest of these are the baryons, whereas a six quark state is sometimes called a dibaryon. All of these heavier states are collectively called quark matter, and unlike nuclei which cease to exist at a baryon number of ∼240, strange quark matter, if stable, could span a range in baryon number all the way from light nuclei to stars, as heavy as or heavier than our own sun. The purpose of this chapter is to give an introduction to strange quark matter for the benefit of the non-specialist reader, as well as to give an overview of the properties of this exotic phase of matter, and where it might play a role. 1.1 Basic Ideas Quark matter is best explained by contrasting it with ordinary nuclear matter. Whereas the quarks in a nucleus are confined within the nucleons (protons, neutrons) the quarks in quark matter are free to roam the entire phase (see Fig. 1.1). Thus one can imagine quark matter as a nucleus made up of quarks that are not correlated in such a way as to form nucleons. From nuclear physics we know that in fact quarks do lump together to form nucleons which in turn make up the nucleus. So what makes contemplations on quark matter more than a purely academic exercise? The answer is strangeness. Nature has supplied us with three so-called 1 2 Chapter 1. Introduction Figure 1.1: A nucleus (left) consisting of 7 nucleons (e.g., 7Li), with each nucleon made up of three quarks; and a strangelet (right) composed of 3×7 = 21 quarks.