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Effects of Bulk Viscosity on p T Spectra and Elliptic Flow Coefficients

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

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Effects of Bulk Viscosity on p T Spectra and Elliptic Flow Coefficients

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  1. 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 May9th2009, Hongo, The University of Tokyo arXiv:0903.4436 [nucl-th]

  2. 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)

  3. 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

  4. 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) freezeout hypersurface Σ particles hadron resonance gas QGP variation of the flow/hypersurface modification of the distribution We estimate this for a multi-component gas. (3+1)-D viscous hydro required. * :normal vector to the freezeout hypersurface element, :distribution of the ith particle, :degeneracy. • 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: Introduction (I) Introduction (II) Relativistic Kinetic Theory In Multi-Component System

  5. 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

  6. 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 giis the degeneracy and biis 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

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

  8. 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 The trace part The scalar term particle species independent (macroscopic quantity) particle species dependent (mass dependent) NOT equivalent for a multi-component system. • 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 Matching Conditions Decomposition of Moments Comments on Quadratic Ansatz Grad’s 14-moment method

  9. 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] Decomposition of Moments Comments on Quadratic Ansatz Prefactors in Viscous Correction Grad’s 14-moment method

  10. 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 Dusling & Teaney (‘08) • 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. . Decomposition of Moments Comments on Quadratic Ansatz Prefactors in Viscous Correction Summary So Far

  11. 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 Scalar terms Vector terms Tensor term • 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. Comments on Quadratic Ansatz Prefactors in Viscous Correction Summary So Far EoS, Transport Coefficients and Flow

  12. 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

  13. 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 Kovtun et al.(‘05) … Weinberg (‘71) Hirano et al.(‘06) • 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 ( ). • Profiles of the flow and the freezeout hypersurface : a (3+1)-dimensional ideal hydrodynamic simulation. Summary So Far EoS, Transport Coefficients and Flow pT-Spectra Elliptic Flow Coefficient v2(pT)

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

  15. 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) 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 • Au+Au, , b = 7.2(fm), v2(pT) of pT-Spectra Elliptic Flow Coefficient v2(pT) Quadratic Ansatz Summary & Outlook

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

  17. Effects of Bulk Viscosity on pT Spectra and Elliptic Flow Coefficients Heavy Ion Café, The University of Tokyo, May 9th 2009 Summary & Outlook when estimated with an ideal hydrodynamic flow. • 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. Quadratic Ansatz Summary& Outlook

  18. 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

  19. Effects of Bulk Viscosity on pT Spectra and Elliptic Flow Coefficients Heavy Ion Café, The University of Tokyo, May 9th 2009 Summary & Outlook when estimated with an ideal hydrodynamic flow. • 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. Effects of Shear Viscosity Quadratic Ansatz Summary & Outlook Effects of Shear and Bulk Viscosity

  20. 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 Non-triviality in shear viscous corrections; both pT -spectra and v2(pT) suppressed • pT -spectra and v2(pT) of with , , and the same EoS. Summary & Outlook Effects of Shear Viscosity Effects of Shear and Bulk Viscosity Bjorken Model

  21. 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 Cancellation in viscous corrections in v2(pT) (rather accidental) • pT -spectra and v2(pT) of with , , and the same EoS. Effects of Shear Viscosity Effects of Shear and Bulk Viscosity Bjorken Model Blast Wave Model

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

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

  24. 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 Teaney (‘03) • 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. Blast Wave Model Shear Viscosity in Blast Wave Model (2+1)-Viscous Flow/Hypersurface Expansion of Distribution

  25. 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 Song & Heinz (‘08) • For shear viscosity, (effects of ) (effects of ) • Freezeout hypersurfaces are roughly the same for the ideal and for the (2+1)-D viscous. Shear Viscosity in Blast Wave Model (2+1)-Viscous Flow/Hypersurface Expansion of Distribution Numerical Results (Prefactors)

  26. 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

  27. 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

  28. 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

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

  30. 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, BUTit 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

  31. 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

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