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

Outline 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 v 2 (p T ) Summary Quark Matter 2009, Knoxville, Tennessee, April 2 nd 2009 Effects of Bulk Viscosity on p T -spectra and Elliptic Flow Parameter Outline Introduction (I) Introduction (II)

Outline Introduction (I) Paech & Pratt (‘06)Mizutani et al. (‘88)Kharzeev & Tuchin (’08) … Quark Matter 2009, Knoxville, Tennessee, April 2 nd 2009 Effects of Bulk Viscosity on p T -spectra and Elliptic Flow Parameter Success of ideal hydrodynamic models Development of viscous hydrodynamic models for the QGP created in relativistic heavy ion collisions (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. In this work, we will see the effects of bulk viscosity at freezeout. Introduction (I) Introduction (II) Outline Relativistic Kinetic Theory

Outline Introduction (II) 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: (3+1)-D viscous hydro required. We estimate this for a multi-component gas. Cooper & Frye (‘74) Quark Matter 2009, Knoxville, Tennessee, April 2 nd 2009 Effects of Bulk Viscosity on p T -spectra and Elliptic Flow Parameter Introduction (II) Introduction (I) Relativistic Kinetic Theory variation of the flowmodification of the distribution * :normal vector to the freezeout hypersurface element, :distribution of the i th particle, :degeneracy. particles hadron resonance gas QGP freezeout hypersurface Σ In Multi-Component System

Outline 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 In Multi-Component System Grad’s 14-moment method Introduction (II) Quark Matter 2009, Knoxville, Tennessee, April 2 nd 2009 Effects of Bulk Viscosity on p T -spectra and Elliptic Flow Parameter Relativistic Kinetic Theory

Outline In Multi-Component System Kinetic definitions for a multi-component gas: where g i is the degeneracy and b i 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) + matching conditions (2) = 14 equations. Decomposition of Moments Grad’s 14-moment method Quark Matter 2009, Knoxville, Tennessee, April 2 nd 2009 Effects of Bulk Viscosity on p T -spectra and Elliptic Flow Parameter In Multi-Component System Relativistic Kinetic Theory

Outline 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 ] Comments on Quadratic Ansatz Grad’s 14-moment method The trace partThe scalar term particle species dependent (mass dependent) particle species independent (macroscopic quantity) Decomposition of Moments Quark Matter 2009, Knoxville, Tennessee, April 2 nd 2009 Effects of Bulk Viscosity on p T -spectra and Elliptic Flow Parameter In Multi-Component System NOT equivalent for a multi-component system.

Outline Decomposition of Moments Definitions: *Contributions are : [baryons] + [anti-baryons] + [mesons] : [baryons] – [anti-baryons] Comments on Quadratic Ansatz Grad’s 14-moment method Decomposition of Moments Quark Matter 2009, Knoxville, Tennessee, April 2 nd 2009 Effects of Bulk Viscosity on p T -spectra and Elliptic Flow Parameter Prefactors in Viscous Correction

Outline 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.. Comments on Quadratic Ansatz Decomposition of Moments Quark Matter 2009, Knoxville, Tennessee, April 2 nd 2009 Effects of Bulk Viscosity on p T -spectra and Elliptic Flow Parameter Dusling & Teaney (‘08) Prefactors in Viscous Correction EoS, Transport Coefficients and Flow

Outline 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. Comments on Quadratic Ansatz Quark Matter 2009, Knoxville, Tennessee, April 2 nd 2009 Effects of Bulk Viscosity on p T -spectra and Elliptic Flow Parameter Prefactors in Viscous Correction EoS, Transport Coefficients and Flow Scalar terms Vector terms Tensor term p T -Spectra

Outline 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: T f = 0.16(GeV) and ( ). Quark Matter 2009, Knoxville, Tennessee, April 2 nd 2009 Effects of Bulk Viscosity on p T -spectra and Elliptic Flow Parameter Weinberg (‘71) Kovtun et al.(‘05) Hirano et al.(‘06) … EoS, Transport Coefficients and Flow Prefactors in Viscous Correction Profiles of the flow and the freezeout hypersurface : a (3+1)-dimensional ideal hydrodynamic simulation. p T -Spectra Elliptic Flow Coefficient v 2 (p T )

Outline p T -Spectra Au+Au,, b = 7.2(fm), p T -spectra of Model of the bulk pressure: : free parameter The bulk viscosity lowers of the particle spectra. Elliptic Flow Coefficient v 2 (p T ) p T -Spectra Quark Matter 2009, Knoxville, Tennessee, April 2 nd 2009 Effects of Bulk Viscosity on p T -spectra and Elliptic Flow Parameter Results with Quadratic Ansatz EoS, Transport Coefficients and Flow

Outline Elliptic Flow Coefficient v 2 (p T ) Au+Au,, b = 7.2(fm), v 2 (p T ) of The bulk viscosity enhances v 2 (p T ). *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 Summary Results with Quadratic Ansatz Quark Matter 2009, Knoxville, Tennessee, April 2 nd 2009 Effects of Bulk Viscosity on p T -spectra and Elliptic Flow Parameter p T -Spectra Elliptic Flow Coefficient v 2 (p T )

Outline Results with Quadratic Ansatz p T -spectra and v 2 (p T ) of with and the same EoS Results with Quadratic Ansatz Summary Quark Matter 2009, Knoxville, Tennessee, April 2 nd 2009 Effects of Bulk Viscosity on p T -spectra and Elliptic Flow Parameter Effects of the bulk viscosity is underestimated in quadratic ansatz. Elliptic Flow Coefficient v 2 (p T )

Outline 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. Summary Results with Quadratic Ansatz Quark Matter 2009, Knoxville, Tennessee, April 2 nd 2009 Effects of Bulk Viscosity on p T -spectra and Elliptic Flow Parameter p T -spectra : suppressed v 2 (p T ) : enhanced when estimated with an ideal hydrodynamic flow.

Outline 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 Thank You Quark Matter 2009, Knoxville, Tennessee, April 2 nd 2009 Effects of Bulk Viscosity on p T -spectra and Elliptic Flow Parameter

Outline 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. Summary Results for Shear Viscosity Results for Shear + Bulk Viscosity Results with Quadratic Ansatz Quark Matter 2009, Knoxville, Tennessee, April 2 nd 2009 Effects of Bulk Viscosity on p T -spectra and Elliptic Flow Parameter p T -spectra : suppressed v 2 (p T ) : enhanced when estimated with an ideal hydrodynamic flow.

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

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

Outline Shear Viscosity in Blast Wave Model Effects of the shear viscosity on v 2 (p T ) 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 1 st order theory because in Boltzmann approximation. Shear Viscosity in Blast Wave Model Teaney (‘03) Viscous Flow vs. Viscous Distribution Results for Shear + Bulk Viscosity Freezeout Hypersurface Quark Matter 2009, Knoxville, Tennessee, April 2 nd 2009 Effects of Bulk Viscosity on p T -spectra and Elliptic Flow Parameter

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

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

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

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

Outline The prefactors for and in ’s near T f : Numerical Results (Prefactors) Quark Matter 2009, Knoxville, Tennessee, April 2 nd 2009 Effects of Bulk Viscosity on p T -spectra and Elliptic Flow Parameter Trace Part vs. Scalar Term The Matching Conditions Prefactors in Special Case

Outline 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 T c. Trace Part vs. Scalar Term Quark Matter 2009, Knoxville, Tennessee, April 2 nd 2009 Effects of Bulk Viscosity on p T -spectra and Elliptic Flow Parameter Prefactors in Special Case Explicit Forms of Prefactors Numerical Results (Prefactors)

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

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