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Quasiparticle Anisotropic Hydrodynamics in Ultra-Relativistic QUASIPARTICLE ANISOTROPIC HYDRODYNAMICS IN ULTRA-RELATIVISTIC HEAVY-ION COLLISIONS A dissertation submitted to Kent State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy by Mubarak Alqahtani December, 2017 c Copyright All rights reserved Except for previously published materials Dissertation written by Mubarak Alqahtani BE, University of Dammam, SA, 2006 MA, Kent State University, 2014 PhD, Kent State University, 2014-2017 Approved by , Chair, Doctoral Dissertation Committee Dr. Michael Strickland , Members, Doctoral Dissertation Committee Dr. Declan Keane Dr. Spyridon Margetis Dr. Robert Twieg Dr. John West Accepted by , Chair, Department of Physics Dr. James T. Gleeson , Dean, College of Arts and Sciences Dr. James L. Blank Table of Contents List of Figures . vii List of Tables . xv List of Publications . xvi Acknowledgments . xvii 1 Introduction ......................................1 1.1 Units and notation . .1 1.2 The standard model . .3 1.3 Quantum Electrodynamics (QED) . .5 1.4 Quantum chromodynamics (QCD) . .6 1.5 The coupling constant in QED and QCD . .7 1.6 Phase diagram of QCD . .9 1.6.1 Quark gluon plasma (QGP) . 12 1.6.2 The heavy-ion collision program . 12 1.6.3 Heavy-ion collisions stages . 13 1.7 Some definitions . 16 1.7.1 Rapidity . 16 1.7.2 Pseudorapidity . 16 1.7.3 Collisions centrality . 17 1.7.4 The Glauber model . 19 1.8 Collective flow . 21 iii 1.8.1 Radial flow . 22 1.8.2 Anisotropic flow . 22 1.9 Fluid dynamics . 26 1.10 Non-relativistic fluid dynamics . 26 1.10.1 Relativistic fluid dynamics . 27 1.10.2 The kinetic theory . 27 1.10.3 Relaxation time approximation . 28 2 Anisotropic hydrodynamics ............................ 30 2.1 Anisotropic hydrodynamics . 31 2.2 Why anisotropic hydrodynamics? . 34 2.2.1 Conformal 0+1d systems . 36 2.2.2 Non-conformal 0+1d systems . 38 2.2.3 Gubser flow in 1+1d . 39 2.3 Quasiparticle anisotropic hydrodynamics . 41 2.3.1 Equation of state . 42 2.3.2 Moments of Boltzmann Equation . 46 2.3.3 Bulk variables . 47 2.3.4 Dynamical equations . 48 2.3.5 The relaxation time τeq .......................... 51 2.4 Anisotropic Freeze-out . 52 2.4.1 Anisotropic Freeze-out between aHydro and vHydro . 56 3 0+1d quasiparticle anisotropic hydrodynamics ................ 58 3.1 Dynamical equations in 0+1d systems . 58 3.1.1 Dynamical equations in aHydroQP . 58 3.1.2 Dynamical equations of Standard aHydro . 61 iv 3.2 Numerical results . 63 4 1+1d quasiparticle anisotropic hydrodynamics ................ 69 4.1 Dynamical equations in 1+1d systems . 69 4.2 Freeze-out in 1+1d . 70 4.3 Results . 71 5 3+1d quasiparticle anisotropic hydrodynamics ................ 81 5.1 Dynamical equations . 81 5.2 Transport coefficients . 82 5.3 3+1d aHydroQP results . 84 5.3.1 Numerical setup . 85 5.3.2 Initial conditions . 85 5.3.3 Numerical results of the 3+1daHydro code . 87 5.3.4 The hadronic freeze-out . 89 5.3.5 Fixing free parameters . 90 5.3.6 The phenomenological results . 92 6 NLO anisotropic hydrodynamics ......................... 100 6.1 next-to-leading-order distribution function . 100 6.2 Bulk variables in NLO aHydro . 102 6.3 First moment . 103 6.4 Second moment . 104 7 Summary and outlook ............................... 108 7.1 Summary . 108 7.2 Outlook . 109 v 8 Appendices ...................................... 111 8.1 Basis Vectors . 111 8.2 Second-order viscous hydrodynamics . 114 8.3 Explicit formulas for derivatives . 119 8.3.1 Simplification for 1+1d . 120 8.3.2 Simplification for 0+1d . 121 8.4 Special functions . 122 8.4.1 In the massive case . 123 8.4.2 0+1d case . 124 8.4.3 In the massless case . 125 8.4.4 A way around doing integrals in ^-functions . 125 H 8.5 NLO special functions . 131 8.5.1 Calculating the conformal NLO -functions . 133 H Bibliography . 136 vi List of Figures 1.1 This figure shows the forces, particles, forces carriers that the standard model is based on. Figure is taken from Ref. [1] . .4 1.2 The fundamental vertices of the QCD theory: quark-gluon vertex, three-gluon vertex, and four-gluon vertex are shown. Figure is adapted from Ref. [2]. .7 1.3 Panel (a) shows the QED one-loop running coupling constant (α) scaled by (α0) as a function of the scale Q. In panel (b), we show the QCD one-loop running coupling constant (αs) as a function of the scale Q. The plots are adapted from Ref. [2]. .9 1.4 Experimental measurements of the QCD coupling constant (αs) as a function of the energy scale Q. Figure is taken from Ref. [3] where more details and a recent review can be found about all listed measurements in the plot. 10 1.5 This figure shows the QCD phase diagram in the temperature (T) and baryon chemical potential (µB) plane. Figure is taken from Ref. [4]. 11 1.6 Stages of relativistic heavy-ion collisions from QGP formation until hadroniza- tion where particles reach the detector. Figure is made by Chun Shen Ref. [5]. 14 1.7 A sketch showing the psseudorapidity as a function of the angle θ as defined in Eq. (1.22). 17 1.8 The left side of the illustration shows the geometrical definition of the impact parameter while the right side shows the spectators and participants nucleons after the collision. Figure is taken from Ref. [6]. 18 vii 1.9 In panel (a) charged particle distribution is shown as a function of the mul- tiplicity where the centralities classes are shown as well. In panel (b) the number of participating nucleons and binary nucleons, Npart and Nbin respec- tively as a function of the impact parameter for both Pb-Pb and Au-Au at psNN = 2:76 TeV, and 200 GeV respectively. Figure is taken from Ref. [7] where panel (a) is adapted from Ref. [8] . 19 1.10 Schematic sketch of radial flow for most central collisions. Figure is taken from Ref. [9]. 23 1.11 A schematic diagram showing the reaction plane viewed in the transverse plane to the beam (z-direction). It also shows the reaction plane angle (ΨR) and its relation to the impact parameter orientation (b). Figure is taken from Ref. [10]. 24 1.12 A schematic diagram for the directed flow, viewed in the x z plane, showing − the deflection of the spectators along x for spectators moving with y. ± ± Figure is taken from Ref. [11]. 25 1.13 The left sketch shows the shape of the overlap region that generates elliptic flow v2 while the right sketch shows the shape of the overlap region that generates triangular flow v3. Figure is taken from Ref. [12] where left, and right sketches are adapted originally from Ref. [13] and Ref. [14] respectively. 25 2.1 The local rest frame pressure anisotropy, L= T predicted by different studies P P showing QGP is highly anisotropic plasma. Figure is taken from [15]. 32 2.2 Particle production as a function of 4πη=s predicted by Boltzmann equation exact solution compared with 2nd-order viscous hydro, 3rd-order viscous hydro, LO aHydro, and NLO aHydro (vaHydro). Figure taken from Ref. [16] . 34 viii 2.3 Pressure anisotropy as a function of proper time predicted by the exact solu- tion of Boltzmann equation in conformal 0+1d systems versus IS, DNMR, and aHydro methods. In the left panel, ξ0 = 0 where in the right panel ξ0 = 10. In both panels, the top, middle, bottom rows corresponds to 4πη=s = 1, 4πη=s = 3, and 4πη=s = 10 respectively. Figure is taken from Ref. [15] which is adapted from the original Ref. [17]. 37 2.4 Comparisons of temperature evolution (top row) and pressure anisotropy (bot- tom row) as a function of proper time predicted by the exact solution of Boltz- mann equation in conformal 0+1d systems versus aHydro at different τeq or equivalently at different η=s. In the left panel, ξ0 = 0, and in the right panel ξ0 = 100. Figure taken from Ref. [18] . 39 2.5 Comparisons of pressure anisotropy (left column) and bulk pressure (right column) as a function of proper time. In the top, bottom rows ξ0 is taken to be 0 and 100 respectively. The predicted results by exact solution is compared versus vHydro, LO aHydro, and vaHydro for non-conformal 0+1d systems. Figure taken from Ref. [19] . 40 2.6 Comparisons of de Sitter effect temperature (top row) and the scaled shear (bottom row) as.
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