Effects of Bulk Viscosity on pT Spectra and Elliptic Flow Coefficients Akihiko Monnai Department of Physics, The University of Tokyo, Japan Collaborator: Tetsufumi Hirano The 12th Heavy Ion Café May 9th 2009, Hongo, The University of Tokyo arXiv:0903.4436 [nucl-th]

Outline Introduction Theories and Methods Numerical Results Summary Effects of Bulk Viscosity on pT Spectra and Elliptic Flow Coefficients Heavy Ion Café, The University of Tokyo, May 9th 2009 Outline Introduction - Hydrodynamic models and the Cooper-Frye formula at freezeout Theories and Methods - Distortion of the distribution from bulk viscosity for a multi-component system in Grad’s 14-moment method Numerical Results - Viscous effects on particle spectra and elliptic flow coefficient v2(pT) Summary Outline Introduction (I) Introduction (II)

Introduction (I) Success of ideal hydrodynamic models Effects of Bulk Viscosity on pT Spectra and Elliptic Flow Coefficients Heavy Ion Café, The University of Tokyo, May 9th 2009 Introduction (I) Success of ideal hydrodynamic models for the QGP created in relativistic heavy ion collisions Development of viscous hydrodynamic models (1) to understand of the hot QCD matter better (2) to constrain the EoS and the transport coefficients from experimental data Importance of bulk viscosity since it would become large near the QCD phase transition. Mizutani et al. (‘88) Paech & Pratt (‘06) Kharzeev & Tuchin (’08) … In this work, we will see the effects of bulk viscosity at freezeout. Outline Introduction (I) Introduction (II) Relativistic Kinetic Theory

Introduction (II) Cooper & Frye (‘74) Effects of Bulk Viscosity on pT Spectra and Elliptic Flow Coefficients Heavy Ion Café, The University of Tokyo, May 9th 2009 Introduction (II) Cooper & Frye (‘74) Hydrodynamic analyses needs the Cooper-Frye formula at freezeout (i) for comparison with experimental data, (ii) as an interface to a cascade model. Viscous corrections come in two ways: particles freezeout hypersurface Σ hadron resonance gas QGP variation of the flow/hypersurface modification of the distribution (3+1)-D viscous hydro required. We estimate this for a multi-component gas. * :normal vector to the freezeout hypersurface element, :distribution of the ith particle, :degeneracy. Introduction (I) Introduction (II) Relativistic Kinetic Theory In Multi-Component System

Relativistic Kinetic Theory Effects of Bulk Viscosity on pT Spectra and Elliptic Flow Coefficients Heavy Ion Café, The University of Tokyo, May 9th 2009 Relativistic Kinetic Theory Discussion to express in terms of macroscopic variables for a single-component gas by Israel & Stewart (‘79) Tensor decomposition and the macroscopic variables: where and . Bulk pressure (1): Energy current (3): Charge current (3): Shear tensor (5): Introduction (II) Relativistic Kinetic Theory In Multi-Component System The Matching Conditions

In Multi-Component System Effects of Bulk Viscosity on pT Spectra and Elliptic Flow Coefficients Heavy Ion Café, The University of Tokyo, May 9th 2009 In Multi-Component System Kinetic definitions for a multi-component gas: where gi is the degeneracy and bi is the baryon number. We express in terms of macroscopic variables for a multi-component system in Grad’s 14-moment method. Note: kinetic definitions = 12 equations Relativistic Kinetic Theory In Multi-Component System The Matching Conditions Grad’s 14-moment method

The Matching Conditions Effects of Bulk Viscosity on pT Spectra and Elliptic Flow Coefficients Heavy Ion Café, The University of Tokyo, May 9th 2009 The Matching Conditions Necessary conditions to ensure thermodynamic stability 　　Landau matching conditions (2): , ! kinetic definitions (12) + matching conditions (2) = 14 equations In Multi-Component System The Matching Conditions Grad’s 14-moment method Decomposition of Moments

Grad’s 14-moment method NOT equivalent for a multi-component system. Effects of Bulk Viscosity on pT Spectra and Elliptic Flow Coefficients Heavy Ion Café, The University of Tokyo, May 9th 2009 Grad’s 14-moment method Distortion of the distribution expressed with 14 (= 4+10) unknowns: where + for bosons and – for fermions. [tensor term ] vs. [scalar term + traceless tensor term ] The trace part The scalar term particle species dependent (mass dependent) particle species independent (macroscopic quantity) NOT equivalent for a multi-component system. The Matching Conditions Grad’s 14-moment method Decomposition of Moments Comments on Quadratic Ansatz

Decomposition of Moments Effects of Bulk Viscosity on pT Spectra and Elliptic Flow Coefficients Heavy Ion Café, The University of Tokyo, May 9th 2009 Decomposition of Moments Definitions: *Contributions are : [baryons] + [anti-baryons] + [mesons] : [baryons] – [anti-baryons] Grad’s 14-moment method Decomposition of Moments Comments on Quadratic Ansatz Prefactors in Viscous Correction

Comments on Quadratic Ansatz Effects of Bulk Viscosity on pT Spectra and Elliptic Flow Coefficients Heavy Ion Café, The University of Tokyo, May 9th 2009 Comments on Quadratic Ansatz Previous study of the bulk viscosity for a massless gas in QGP with the quadratic ansatz: It violates thermodynamic stability (matching conditions) as , . Note (i) It is not unique; , or ? (ii) Explicit treatment of a multi-component system is necessary. We will derive without this assumption for a multi-component gas. * remains meaningful in the zero net baryon density limit i.e. . Dusling & Teaney (‘08) Decomposition of Moments Comments on Quadratic Ansatz Prefactors in Viscous Correction Summary So Far

Prefactors in Viscous Correction Effects of Bulk Viscosity on pT Spectra and Elliptic Flow Coefficients Heavy Ion Café, The University of Tokyo, May 9th 2009 Prefactors in Viscous Correction Insert the distribution function into the conditions: where , , and . The unique form of the deviation is determined: where, and are functions of ’s and ’s. Scalar terms Vector terms Tensor term Comments on Quadratic Ansatz Prefactors in Viscous Correction Summary So Far EoS, Transport Coefficients and Flow

Effects of Bulk Viscosity on pT Spectra and Elliptic Flow Coefficients Heavy Ion Café, The University of Tokyo, May 9th 2009 Summary So Far Unique for a multi-component gas in Grad’s 14-moment method Thermodynamically stable (Landau matching conditions satisfied) The finite trace tensor in the expansion Dependent on all the components Next: Particle spectra at freezeout with in the C-F formula Models for EoS, transport coefficient, flow and freezeout hypersurface Prefactors in Viscous Correction Summary So Far EoS, Transport Coefficients and Flow pT-Spectra

EoS, Transport Coefficients and Flow Effects of Bulk Viscosity on pT Spectra and Elliptic Flow Coefficients Heavy Ion Café, The University of Tokyo, May 9th 2009 EoS, Transport Coefficients and Flow Equation of state:16-component hadron resonance gas *mesons and baryons with mass up to . is taken. Transport coefficients: , where is the sound velocity and s the entropy density. Freezeout temperature: Tf = 0.16(GeV) and ( ). Kovtun et al.(‘05) Weinberg (‘71) … Profiles of the flow and the freezeout hypersurface : a (3+1)-dimensional ideal hydrodynamic simulation. Hirano et al.(‘06) Summary So Far EoS, Transport Coefficients and Flow pT-Spectra Elliptic Flow Coefficient v2(pT)

pT-Spectra Au+Au, , b = 7.2(fm), pT -spectra of Effects of Bulk Viscosity on pT Spectra and Elliptic Flow Coefficients Heavy Ion Café, The University of Tokyo, May 9th 2009 pT-Spectra Au+Au, , b = 7.2(fm), pT -spectra of Model of the bulk pressure: : free parameter The bulk viscosity lowers <pT> of the particle spectra. EoS, Transport Coefficients and Flow pT-Spectra Elliptic Flow Coefficient v2(pT) Quadratic Ansatz

Elliptic Flow Coefficient v2(pT) Effects of Bulk Viscosity on pT Spectra and Elliptic Flow Coefficients Heavy Ion Café, The University of Tokyo, May 9th 2009 Elliptic Flow Coefficient v2(pT) Au+Au, , b = 7.2(fm), v2(pT) of The bulk viscosity enhances v2(pT). *Viscous effects might be overestimated for: (1) No relaxation for is from the Navier-Stokes limit. (2) Derivatives of are larger than those of real viscous flow pT-Spectra Elliptic Flow Coefficient v2(pT) Quadratic Ansatz Summary & Outlook

Quadratic Ansatz pT -spectra and v2(pT) of with and the same EoS Effects of Bulk Viscosity on pT Spectra and Elliptic Flow Coefficients Heavy Ion Café, The University of Tokyo, May 9th 2009 Quadratic Ansatz pT -spectra and v2(pT) of with and the same EoS Effects of the bulk viscosity is underestimated in quadratic ansatz. Elliptic Flow Coefficient v2(pT) Quadratic Ansatz Summary & Outlook

Effects of Bulk Viscosity on pT Spectra and Elliptic Flow Coefficients Heavy Ion Café, The University of Tokyo, May 9th 2009 Summary & Outlook Consistent determination of for a multi-particle system A non-zero trace tensor term is needed for the hadron resonance gas up to the mass of Visible effects of on particle spectra Bulk viscosity should be considered to constrain the transport coefficients with better accuracy from experimental data. A (3+1)-dimensional viscous hydrodynamic flow is necessary to see more realistic behavior of the particle spectra. pT-spectra : suppressed v2(pT) : enhanced when estimated with an ideal hydrodynamic flow. Quadratic Ansatz Summary & Outlook

Effects of Bulk Viscosity on pT Spectra and Elliptic Flow Coefficients Heavy Ion Café, The University of Tokyo, May 9th 2009 Thank You The numerical code for calculations of ’s, ’s and the prefactors shown in this presentation will become available in near future at http://tkynt2.phys.s.u-tokyo.ac.jp/~monnai/distributions.html Thank You

Effects of Bulk Viscosity on pT Spectra and Elliptic Flow Coefficients Heavy Ion Café, The University of Tokyo, May 9th 2009 Summary & Outlook Consistent determination of for a multi-particle system A non-zero trace tensor term is needed for the hadron resonance gas up to the mass of Visible effects of on particle spectra Bulk viscosity should be considered to constrain the transport coefficients with better accuracy from experimental data. A (3+1)-dimensional viscous hydrodynamic flow is necessary to see more realistic behavior of the particle spectra. pT-spectra : suppressed v2(pT) : enhanced when estimated with an ideal hydrodynamic flow. Quadratic Ansatz Effects of Shear Viscosity Summary & Outlook Effects of Shear and Bulk Viscosity

Effects of Shear Viscosity Effects of Bulk Viscosity on pT Spectra and Elliptic Flow Coefficients Heavy Ion Café, The University of Tokyo, May 9th 2009 Effects of Shear Viscosity pT -spectra and v2(pT) of with , , and the same EoS. Non-triviality in shear viscous corrections; both pT -spectra and v2(pT) suppressed Summary & Outlook Effects of Shear Viscosity Effects of Shear and Bulk Viscosity Bjorken Model

Effects of Shear and Bulk Viscosity Effects of Bulk Viscosity on pT Spectra and Elliptic Flow Coefficients Heavy Ion Café, The University of Tokyo, May 9th 2009 Effects of Shear and Bulk Viscosity pT -spectra and v2(pT) of with , , and the same EoS. Cancellation in viscous corrections in v2(pT) (rather accidental) Effects of Shear Viscosity Effects of Shear and Bulk Viscosity Bjorken Model Blast Wave Model

Bjorken Model pT -spectra and v2(pT) of with cylindrical geometry, Effects of Bulk Viscosity on pT Spectra and Elliptic Flow Coefficients Heavy Ion Café, The University of Tokyo, May 9th 2009 Bjorken Model pT -spectra and v2(pT) of with cylindrical geometry, Flow: Freezeout hypersurface: Bulk viscosity suppresses pT -spectra. Shear viscosity enhances pT -spectra. Effects of Shear and Bulk Viscosity Bjorken Model Blast Wave Model Shear Viscosity in Blast Wave Model

Blast Wave Model pT -spectra and v2(pT) of Effects of Bulk Viscosity on pT Spectra and Elliptic Flow Coefficients Heavy Ion Café, The University of Tokyo, May 9th 2009 Blast Wave Model pT -spectra and v2(pT) of ( ) Shear viscosity enhances pT -spectra and suppresses v2(pT). Bjorken Model Blast Wave Model Shear Viscosity in Blast Wave Model (2+1)-Viscous Flow/Hypersurface

Shear Viscosity in Blast Wave Model Effects of Bulk Viscosity on pT Spectra and Elliptic Flow Coefficients Heavy Ion Café, The University of Tokyo, May 9th 2009 Shear Viscosity in Blast Wave Model Effects of the shear viscosity on v2(pT) at freezeout was previously estimated for a pion gas. Different when in a single component gas and in a multi-component gas - Shear viscous correction is not so different when and is taken from the 1st order theory because in Boltzmann approximation. Teaney (‘03) Blast Wave Model Shear Viscosity in Blast Wave Model (2+1)-Viscous Flow/Hypersurface Expansion of Distribution

(2+1)-Viscous Flow/Hypersurface Effects of Bulk Viscosity on pT Spectra and Elliptic Flow Coefficients Heavy Ion Café, The University of Tokyo, May 9th 2009 (2+1)-Viscous Flow/Hypersurface For shear viscosity, (effects of ) (effects of ) Freezeout hypersurfaces are roughly the same for the ideal and for the (2+1)-D viscous. Song & Heinz (‘08) Shear Viscosity in Blast Wave Model (2+1)-Viscous Flow/Hypersurface Expansion of Distribution Numerical Results (Prefactors)

Expansion of Distribution Effects of Bulk Viscosity on pT Spectra and Elliptic Flow Coefficients Heavy Ion Café, The University of Tokyo, May 9th 2009 Expansion of Distribution The distribution function is expanded as follows: where the momentum expansion is used. = (2+1)-Viscous Flow/Hypersurface Expansion of Distribution Numerical Results (Prefactors) Trace Part vs. Scalar Term

Numerical Results (Prefactors) Effects of Bulk Viscosity on pT Spectra and Elliptic Flow Coefficients Heavy Ion Café, The University of Tokyo, May 9th 2009 Numerical Results (Prefactors) The prefactors for and in ’s near Tf: Highly non-trivial, but effects of bulk viscosity can be large. Expansion of Distribution Numerical Results (Prefactors) Trace Part vs. Scalar Term Dependence on Components

Trace Part vs. Scalar Term Effects of Bulk Viscosity on pT Spectra and Elliptic Flow Coefficients Heavy Ion Café, The University of Tokyo, May 9th 2009 Trace Part vs. Scalar Term Separation of the trace part: works as a free parameter; a single mass dependence can be absorbed. If we have the scalar term in , , , and diverge at a temperature below Tc. Numerical Results (Prefactors) Trace Part vs. Scalar Term Dependence on Components Prefactors in Special Case

Dependence on Components Effects of Bulk Viscosity on pT Spectra and Elliptic Flow Coefficients Heavy Ion Café, The University of Tokyo, May 9th 2009 Dependence on Components Distortion depends on all the components of the gas. : ith component Consistent discussion needed for a resonance gas. and , and thus and , are “macroscopic” quantities. Trace Part vs. Scalar Term Dependence on Components Prefactors in Special Case Explicit Forms of Prefactors

Prefactors in Special Case Effects of Bulk Viscosity on pT Spectra and Elliptic Flow Coefficients Heavy Ion Café, The University of Tokyo, May 9th 2009 Prefactors in Special Case We consider : the Landau frame i.e. the zero net baryon density limit i.e. for analyses of heavy ion collisions. - Apparently vanishes, BUT it does yield a finite relation: because the expansion of around is and the chemical potential ’s cancel out. The number of equations does not change in the process. Dependence on Components Prefactors in Special Case Explicit Forms of Prefactors

Explicit Forms of Prefactors Effects of Bulk Viscosity on pT Spectra and Elliptic Flow Coefficients Heavy Ion Café, The University of Tokyo, May 9th 2009 Explicit Forms of Prefactors The prefactors in the viscous correction tensors are: where, Prefactors in Special Case Explicit Forms of Prefactors