1. Effects of Bulk Viscosity at Freezeout Akihiko Monnai Department of Physics, The University of Tokyo Collaborator: Tetsufumi Hirano Nagoya Mini-Workshop “Photons and Leptons in Hot/Dense QCD” March 2 nd -4 th, 2009, Nagoya, Japan

2. Outline Introduction - Ideal and viscous hydrodynamics, the Cooper-Frye formula at freezeout Theories and Methods - An overview of the kinetic theory to express the distribution with macroscopic variables Numerical Results - Particle spectra and elliptic flow parameter v 2 (p T ) Summary Outline Introduction (I) Nagoya Mini Workshop, Nagoya University, March 3 rd 2009 Effects of Bulk Viscosity at Freezeout Introduction (II)

3. Introduction (I) Success of ideal hydrodynamic models for the quark-gluon plasma created in relativistic heavy ion collisions Importance of viscous hydrodynamic models for (1) better understanding of the hot QCD matter (2) constraining the equation of state and the transport coefficients from experimental data Introduction (I) Introduction (II) Nagoya Mini Workshop, Nagoya University, March 3 rd 2009 Effects of Bulk Viscosity at Freezeout Outline Kinetic Theory (I) The bulk viscosity is expected to become large near the QCD phase transition. In this work, we see the effects of bulk viscosity at freezeout. Paech & Pratt (‘06)Kharzeev & Tuchin (’08) …Mizutani et al. (‘88)

4. Introduction (II) In hydrodynamic analyses, the Cooper-Frye formula is necessary at freezeout: (1) to convert into particles for comparison with experimental data, (2) as an interface from a hydrodynamic model to a cascade model. where, :normal vector to the freezeout hypersurface element :distribution function of the i th particle :degeneracy Viscous effects are taken into account via (1) variation of the flow (2) modification of the distribution function Nagoya Mini Workshop, Nagoya University, March 3 rd 2009 Effects of Bulk Viscosity at Freezeout This needs (3+1)-D viscous hydro. We focus on the contributions of the bulk viscosity to this phenomenon. Introduction (II) Introduction (I) QGP hadron resonance gas freezeout hypersurface Σ Particles Kinetic Theory (I) Kinetic Theory (II) dσ μ Cooper & Frye (‘74)

5. Kinetic Theory (I) We express the phase space distribution in terms of macroscopic variables for a multi-component system. Tensor decompositions of the energy-momentum tensor and the net baryon number current: where, and Bulk pressure: Energy current: Charge current: Shear stress tensor: Kinetic Theory (II) Grad’s 14-moment method Nagoya Mini Workshop, Nagoya University, March 3 rd 2009 Effects of Bulk Viscosity at Freezeout Introduction (II) Kinetic Theory (I) Israel & Stewart (‘79)

6. Kinetic Theory (II) Kinetic definitions for a multi-particle system: where g i is the degeneracy and b i is the baryon number. We need to see viscous corrections at freezeout. We introduce Landau matching conditions to ensure the thermodynamic stability in the 1 st order theory. Landau matching conditions:, Together with the kinetic definitions we have 14 equations. Decomposition of Moments Nagoya Mini Workshop, Nagoya University, March 3 rd 2009 Effects of Bulk Viscosity at Freezeout Grad’s 14-moment method Kinetic Theory (I) Kinetic Theory (II)

7. Grad’s 14-moment method Distortion of the distribution function is expressed with 14 (= 4+10) unknowns: where the sign is + for bosons and – for fermions. [tensor term ] vs. [scalar term + traceless tensor term ] Comments on Quadratic Anzatz Nagoya Mini Workshop, Nagoya University, March 3 rd 2009 Effects of Bulk Viscosity at Freezeout Kinetic Theory (II) Grad’s 14-moment method The trace partThe scalar term particle species dependent (mass dependent) particle species independent (thermodynamic quantity) - Equivalent for a single particle system (e.g. pions). - NOT equivalent for a multi-particle system. Decomposition of Moments

8. Definitions: *The former has contributions from both baryons and mesons, while the latter only from baryons. Prefactors (I) Nagoya Mini Workshop, Nagoya University, March 3 rd 2009 Effects of Bulk Viscosity at Freezeout Comments on Quadratic Ansatz Grad’s 14-moment method Decomposition of Moments

9. Comments on Quadratic Ansatz Effects of the bulk viscosity on the distribution function was previously considered for a massless gas in QGP with the quadratic ansatz: Note (1) This does not satisfy the Landau matching conditions: (2) It is not unique; the bulk viscous term could have been, or. (3) Hydrodynamic simulations need discussion for a resonance gas. We are going to derive the form of the viscous correction without this assumption, for a multi-component gas. Nagoya Mini Workshop, Nagoya University, March 3 rd 2009 Effects of Bulk Viscosity at Freezeout Comments on Quadratic Ansatz Prefactors (II) Dusling & Teaney (‘08) Decomposition of Moments Prefactors (I)

10. Insert the distribution function into the kinetic definitions and the Landau matching conditions: where,, and. They are three independent sets of equations. Nagoya Mini Workshop, Nagoya University, March 3 rd 2009 Effects of Bulk Viscosity at Freezeout Prefactors (I) Prefactors (II) Comments on Quadratic Ansatz Prefactors in Special Case

11. Prefactors (II) The solutions are where, and are functions of ’s and ’s. The explicit form of the deviation can be uniquely determined: with Here,. Nagoya Mini Workshop, Nagoya University, March 3 rd 2009 Effects of Bulk Viscosity at Freezeout Prefactors (I) Prefactors (II) Prefactors in Special Case Models (I)

12. Prefactors in Special Case We consider the Landau frame i.e. and the zero net baryon density limit i.e., which are often employed for analyses of heavy ion collisions. - Apparently, the matching condition for the baryon number current vanishes. BUT it should be kept because it yields a finite relation even in this limit: Here, ratios of two ’s remain finite as μ → 0 for and the chemical potential μ’s cancel out. The number of equations does not change in the process. Nagoya Mini Workshop, Nagoya University, March 3 rd 2009 Effects of Bulk Viscosity at Freezeout Prefactors Prefactors in Special Case Models (I) Models (II)

13. Models (I) Nagoya Mini Workshop, Nagoya University, March 3 rd 2009 Effects of Bulk Viscosity at Freezeout Equation of State - 16-component hadron resonance gas [mesons and baryons with mass up to Δ(1232)]. μ → 0 is implied. - The models for transport coefficients: where (sound velocity) and s is the entropy density. The freezeout temperature: T f = 0.16(GeV) where and ( ). Models (I) Models (II) Arnold et al.(‘06) Kovtun et al.(‘05) Numerical Results (Prefactors) Prefactors in Special Case

14. Models (II) Nagoya Mini Workshop, Nagoya University, March 3 rd 2009 Effects of Bulk Viscosity at Freezeout Profiles of the flow and the freezeout hypersurface for the calculations of the Cooper-Frye formula were taken from a (3+1)-dimensional ideal hydrodynamic simulation. For numerical calculations we take the Landau frame ( ) and the zero net baryon density limit ( ). Hirano et al.(‘06) Models (II) Numerical Results (Prefactors) Numerical Results (Particle Spectra) Models (I)

15. The prefactors for and near the freezeout temperature T f : Numerical Results (Prefactors) Nagoya Mini Workshop, Nagoya University, March 3 rd 2009 Effects of Bulk Viscosity at Freezeout Numerical Results (Prefactors) Numerical Results (Particle Spectra) Numerical Results (v 2 (p T ) ) The prefactors of bulk viscosity are generally larger than that of shear viscosity. Models (II) Contribution of the bulk viscosity to is expected to be large compared with that of the shear viscosity.

16. Numerical Results (Particle Spectra) Au+Au,, b = 7.2(fm), p T -spectra of π - Nagoya Mini Workshop, Nagoya University, March 3 rd 2009 Effects of Bulk Viscosity at Freezeout Model of the bulk pressure: Parameter α is set to and for the results. The bulk viscosity lowers of the particle spectra. Numerical Results (v 2 (p T ) ) Numerical Results (Particle Spectra) Numerical Results (Prefactors) Summary

17. Numerical Results (v 2 (p T ) ) Au+Au,, b = 7.2(fm), v 2 (p T ) of π - Nagoya Mini Workshop, Nagoya University, March 3 rd 2009 Effects of Bulk Viscosity at Freezeout The bulk viscosity enhances v 2 (p T ) in the high p T region. *Viscous effects may have been overestimated: (1) No relaxation time for is from the 1 st order theory. (2) Derivatives of are larger than those of real viscous flow. Numerical Results (Particle Spectra) Numerical Results (v 2 (p T ) ) Summary Results with Quadratic Ansatz

18. p T -spectra and v 2 (p T ) of π - with, and the same EoS. Results with Quadratic Ansatz Nagoya Mini Workshop, Nagoya University, March 3 rd 2009 Effects of Bulk Viscosity at Freezeout Numerical Results (v 2 (p T ) ) Summary

19. Summary & Outlook We determined δ f i uniquely and consistently for a multi-particle system. - For the 16-component hadron resonance gas, a non-zero trace tensor term is needed. - The matching conditions remain meaningful in zero net baryon density limit. Modification of f due to the bulk viscosity suppresses particle spectra and enhances the elliptic flow parameter v 2 (p T ) in the high p T region. The viscous effects may have been overestimated because (1) we considered the ideal hydrodynamic flow, and (2) the bulk pressure is estimated with the first order theory. A full (3+1)-dimensional viscous hydrodynamic flow is necessary to see more realistic behavior of p T -spectra and v 2 (p T ). The bulk viscosity may have a visible effect on particle spectra, and should be treated with care to constrain the transport coefficients with better accuracy from experimental data. Summary Nagoya Mini Workshop, Nagoya University, March 3 rd 2009 Effects of Bulk Viscosity at Freezeout Results for Shear Viscosity Results for Shear + Bulk Viscosity Results with Quadratic Ansatz

20. Results for Shear Viscosity p T -spectra and v 2 (p T ) of π - with,, and the same EoS. Results for Shear Viscosity Nagoya Mini Workshop, Nagoya University, March 3 rd 2009 Effects of Bulk Viscosity at Freezeout Summary Results for Shear + Bulk Viscosity

21. p T -spectra and v 2 (p T ) of π -, with,, and the same EoS. Results for Shear + Bulk Viscosity Nagoya Mini Workshop, Nagoya University, March 3 rd 2009 Effects of Bulk Viscosity at Freezeout Results for Shear Viscosity

22. 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 Nagoya Mini Workshop, Nagoya University, March 3 rd 2009 Effects of Bulk Viscosity at Freezeout Thank You