Effects of Bulk Viscosity on p T -Spectra and Elliptic Flow Parameter

1. Effects of Bulk Viscosity on pT-Spectra and Elliptic Flow Parameter Akihiko Monnai Department of Physics, The University of Tokyo, Japan Collaborator: Tetsufumi Hirano Quark Matter 2009 March 30th- April 4th, 2009, Knoxville, TN, U.S.A. arXiv:0903.4436 [nucl-th]

2. Effects of Bulk Viscosity on pT-spectra and Elliptic Flow Parameter Quark Matter 2009, Knoxville, Tennessee, April 2nd 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 Parameter Quark Matter 2009, Knoxville, Tennessee, April 2nd 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 Parameter Quark Matter 2009, Knoxville, Tennessee, April 2nd 2009 Introduction (II) Cooper & Frye (‘74) freezeout hypersurface Σ particles hadron resonance gas QGP variation of the flow 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 Parameter Quark Matter 2009, Knoxville, Tennessee, April 2nd 2009 Relativistic Kinetic Theory • Discussion to express in terms of macroscopic variables for a single-component gas by Israel & Stewart (‘79) • The macroscopic variables: Bulk pressure (1): Energy current (3): Charge current (3): Shear tensor (5): • To ensure thermodynamic stability: Landau matching conditions (2): , where and . : Energy-momentum tensor : Net baryon number current Introduction (II) Relativistic Kinetic Theory In Multi-Component System Grad’s 14-moment method

6. Effects of Bulk Viscosity on pT-spectra and Elliptic Flow Parameter Quark Matter 2009, Knoxville, Tennessee, April 2nd 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) + matching conditions (2) = 14 equations. Relativistic Kinetic Theory In Multi-Component System Decomposition of Moments Grad’s 14-moment method

7. Effects of Bulk Viscosity on pT-spectra and Elliptic Flow Parameter Quark Matter 2009, Knoxville, Tennessee, April 2nd 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 ] In Multi-Component System Decomposition of Moments Comments on Quadratic Ansatz Grad’s 14-moment method

8. Effects of Bulk Viscosity on pT-spectra and Elliptic Flow Parameter Quark Matter 2009, Knoxville, Tennessee, April 2nd 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

9. Effects of Bulk Viscosity on pT-spectra and Elliptic Flow Parameter Quark Matter 2009, Knoxville, Tennessee, April 2nd 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 EoS, Transport Coefficients and Flow

10. Effects of Bulk Viscosity on pT-spectra and Elliptic Flow Parameter Quark Matter 2009, Knoxville, Tennessee, April 2nd 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 EoS, Transport Coefficients and Flow pT-Spectra

11. Effects of Bulk Viscosity on pT-spectra and Elliptic Flow Parameter Quark Matter 2009, Knoxville, Tennessee, April 2nd 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. Prefactors in Viscous Correction EoS, Transport Coefficients and Flow pT-Spectra Elliptic Flow Coefficient v2(pT)

12. Effects of Bulk Viscosity on pT-spectra and Elliptic Flow Parameter Quark Matter 2009, Knoxville, Tennessee, April 2nd 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) Results with Quadratic Ansatz

13. Effects of Bulk Viscosity on pT-spectra and Elliptic Flow Parameter Quark Matter 2009, Knoxville, Tennessee, April 2nd 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) Results with Quadratic Ansatz Summary

14. Effects of Bulk Viscosity on pT-spectra and Elliptic Flow Parameter Quark Matter 2009, Knoxville, Tennessee, April 2nd 2009 Results with 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) Results with Quadratic Ansatz Summary

15. Effects of Bulk Viscosity on pT-spectra and Elliptic Flow Parameter Quark Matter 2009, Knoxville, Tennessee, April 2nd 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. Results with Quadratic Ansatz Summary

16. Effects of Bulk Viscosity on pT-spectra and Elliptic Flow Parameter Quark Matter 2009, Knoxville, Tennessee, April 2nd 2009 Thank You • The numerical code for calculations of ’s, ’s and the prefactors shown in this presentation will become an open source in near future at http://tkynt2.phys.s.u-tokyo.ac.jp/~monnai/distributions.html Thank You

17. Effects of Bulk Viscosity on pT-spectra and Elliptic Flow Parameter Quark Matter 2009, Knoxville, Tennessee, April 2nd 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. Results with Quadratic Ansatz Results for Shear Viscosity Summary Results for Shear + Bulk Viscosity

18. Effects of Bulk Viscosity on pT-spectra and Elliptic Flow Parameter Quark Matter 2009, Knoxville, Tennessee, April 2nd 2009 Results for Shear Viscosity • pT -spectra and v2(pT) of with , , and the same EoS. Summary Results for Shear Viscosity Results for Shear + Bulk Viscosity Shear Viscosity in Blast Wave Model

19. Effects of Bulk Viscosity on pT-spectra and Elliptic Flow Parameter Quark Matter 2009, Knoxville, Tennessee, April 2nd 2009 Results for Shear + Bulk Viscosity • pT -spectra and v2(pT) of with , , and the same EoS. Results for Shear Viscosity Results for Shear + Bulk Viscosity Viscous Flow vs. Viscous Distribution Shear Viscosity in Blast Wave Model

20. Effects of Bulk Viscosity on pT-spectra and Elliptic Flow Parameter Quark Matter 2009, Knoxville, Tennessee, April 2nd 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. Results for Shear + Bulk Viscosity Shear Viscosity in Blast Wave Model Viscous Flow vs. Viscous Distribution Freezeout Hypersurface

21. Effects of Bulk Viscosity on pT-spectra and Elliptic Flow Parameter Quark Matter 2009, Knoxville, Tennessee, April 2nd 2009 Viscous Flow vs. Viscous Distribution Song & Heinz (‘08) • Corrections via the distortion of distribution function are of the same order as that of the viscous flow: Shear Viscosity in Blast Wave Model Viscous Flow vs. Viscous Distribution Freezeout Hypersurface Expansion of Distribution

22. Effects of Bulk Viscosity on pT-spectra and Elliptic Flow Parameter Quark Matter 2009, Knoxville, Tennessee, April 2nd 2009 Freezeout Hypersurface Song & Heinz (‘08) • Comparison of freezeout hypersurfaces for the ideal and the (2+1)-D viscous hydrodynamics: Viscous Flow vs. Viscous Distribution Freezeout Hypersurface Expansion of Distribution The Matching Conditions

23. Effects of Bulk Viscosity on pT-spectra and Elliptic Flow Parameter Quark Matter 2009, Knoxville, Tennessee, April 2nd 2009 Expansion of Distribution = • The distribution function is expanded as follows: where the momentum expansion is used. Freezeout Hypersurface Expansion of Distribution The Matching Conditions Numerical Results (Prefactors)

24. Effects of Bulk Viscosity on pT-spectra and Elliptic Flow Parameter Quark Matter 2009, Knoxville, Tennessee, April 2nd 2009 The Matching Conditions Terms proportional to destabilize the system. • The Landau matching conditions are necessary to ensure the thermodynamic stability in the 1st order theory: Expansion of Distribution The Matching Conditions Numerical Results (Prefactors) Trace Part vs. Scalar Term

25. Effects of Bulk Viscosity on pT-spectra and Elliptic Flow Parameter Quark Matter 2009, Knoxville, Tennessee, April 2nd 2009 Numerical Results (Prefactors) • The prefactors for and in ’s near Tf: The Matching Conditions Numerical Results (Prefactors) Trace Part vs. Scalar Term Prefactors in Special Case

26. Effects of Bulk Viscosity on pT-spectra and Elliptic Flow Parameter Quark Matter 2009, Knoxville, Tennessee, April 2nd 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 Prefactors in Special Case Explicit Forms of Prefactors

27. Effects of Bulk Viscosity on pT-spectra and Elliptic Flow Parameter Quark Matter 2009, Knoxville, Tennessee, April 2nd 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: Here, ratios of two ’s remain finite assince and the chemical potential ’s cancel out. The number of equations does not change in the process. Trace Part vs. Scalar Term Prefactors in Special Case Explicit Forms of Prefactors

28. Effects of Bulk Viscosity on pT-spectra and Elliptic Flow Parameter Quark Matter 2009, Knoxville, Tennessee, April 2nd 2009 Explicit Forms of Prefactors • The prefactors in the viscous correction tensors are: where, Prefactors in Special Case Explicit Forms of Prefactors