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									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 2nd-4th, 2009, Nagoya, Japan
Effects of Bulk Viscosity at Freezeout                Nagoya Mini Workshop, Nagoya University, March 3rd 2009



                                            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 v2(pT)

       Summary




                                                    Outline                             Introduction (I)
Effects of Bulk Viscosity at Freezeout             Nagoya Mini Workshop, Nagoya University, March 3rd 2009



                                    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

       The bulk viscosity is expected to become large near the QCD phase transition.
                     Mizutani et al. („88) Paech & Pratt („06) Kharzeev & Tuchin (‟08) …

        In this work, we see the effects of bulk viscosity at freezeout.




              Outline                       Introduction (I)                         Introduction (II)
Effects of Bulk Viscosity at Freezeout                       Nagoya Mini Workshop, Nagoya University, March 3rd 2009



                                   Introduction (II)                   Cooper & Frye („74)
       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.
                        freezeout hypersurface Σ

                                                            where,
                                               Particles
                                         dσμ                              :normal vector to the freezeout
                             QGP
                                                                           hypersurface element
                                                                          :distribution function of the ith particle
          hadron resonance gas                                            :degeneracy

       Viscous effects are taken into account via
     (1) variation of the flow                                             This needs (3+1)-D viscous hydro.
     (2) modification of the distribution function                        We focus on the contributions of
                                                                          the bulk viscosity to this phenomenon.

          Introduction (I)                            Introduction (II)                           Kinetic Theory (I)
Effects of Bulk Viscosity at Freezeout               Nagoya Mini Workshop, Nagoya University, March 3rd 2009



                                Kinetic Theory (I)
       We express the phase space distribution in terms of macroscopic variables for a
        multi-component system.                           Israel & Stewart („79)
       Tensor decompositions of the energy-momentum tensor and the net baryon number
        current:




    where                                ,                          and


    Bulk pressure:
    Energy current:                                      Charge current:
    Shear stress tensor:

          Introduction (II)                  Kinetic Theory (I)                       Kinetic Theory (II)
Effects of Bulk Viscosity at Freezeout             Nagoya Mini Workshop, Nagoya University, March 3rd 2009



                              Kinetic Theory (II)
       Kinetic definitions for a multi-particle system:




        where gi is the degeneracy and bi is the baryon number.

       We need      to see viscous corrections at freezeout. We introduce Landau matching
        conditions to ensure the thermodynamic stability in the 1st order theory.
                 Landau matching conditions:                         ,
        Together with the kinetic definitions we have 14 equations.




         Kinetic Theory (I)                Kinetic Theory (II)                   Grad’s 14-moment method
Effects of Bulk Viscosity at Freezeout                     Nagoya Mini Workshop, Nagoya University, March 3rd 2009



                   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       ]
          The trace part                                                         The scalar term



               particle species dependent                       particle species independent
                    (mass dependent)                             (thermodynamic quantity)

                         - Equivalent for a single particle system (e.g. pions).
                         - NOT equivalent for a multi-particle system.




        Kinetic Theory (II)                      Grad’s 14-moment method                 Decomposition of Moments
Effects of Bulk Viscosity at Freezeout              Nagoya Mini Workshop, Nagoya University, March 3rd 2009



                Decomposition of Moments
        Definitions:




        *The former has contributions from both baryons and mesons, while the latter only
         from baryons.




     Grad’s 14-moment method             Decomposition of Moments               Comments on Quadratic Ansatz
Effects of Bulk Viscosity at Freezeout                Nagoya Mini Workshop, Nagoya University, March 3rd 2009



          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:       Dusling & Teaney („08)


        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.




     Decomposition of Moments            Comments on Quadratic Ansatz                    Prefactors (I)
Effects of Bulk Viscosity at Freezeout              Nagoya Mini Workshop, Nagoya University, March 3rd 2009



                                         Prefactors (I)
        Insert the distribution function into the kinetic definitions and the Landau matching
         conditions:




        where                     ,           ,                  and                                     .

        They are three independent sets of equations.




   Comments on Quadratic Ansatz               Prefactors (I)                           Prefactors (II)
Effects of Bulk Viscosity at Freezeout                  Nagoya Mini Workshop, Nagoya University, March 3rd 2009



                                         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,                              .



           Prefactors (I)                         Prefactors (II)                      Prefactors in Special Case
Effects of Bulk Viscosity at Freezeout                      Nagoya Mini Workshop, Nagoya University, March 3rd 2009



                    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.




            Prefactors                           Prefactors in Special Case                     Models (I)
Effects of Bulk Viscosity at Freezeout                     Nagoya Mini Workshop, Nagoya University, March 3rd 2009



                                             Models (I)
       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:
                             Kovtun et al.(„05)

                                         Arnold et al.(„06)

          where               (sound velocity) and
          s is the entropy density.

          The freezeout temperature: Tf = 0.16(GeV)
           where                           and                                       (          ).



        Prefactors in Special Case                     Models (I)                               Models (II)
Effects of Bulk Viscosity at Freezeout          Nagoya Mini Workshop, Nagoya University, March 3rd 2009



                                         Models (II)
       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.
                                                                   Hirano et al.(„06)




       For numerical calculations we take the Landau frame (               ) and the zero net
        baryon density limit (         ).




            Models (I)                       Models (II)                     Numerical Results (Prefactors)
Effects of Bulk Viscosity at Freezeout                Nagoya Mini Workshop, Nagoya University, March 3rd 2009



            Numerical Results (Prefactors)
       The prefactors for         and   near the freezeout temperature Tf:



                                                                   The prefactors of bulk viscosity are
                                                                   generally larger than that of shear
                                                                   viscosity.




                                                                   Contribution of the bulk viscosity
                                                                   to     is expected to be large
                                                                   compared with that of the shear
                                                                   viscosity.




            Models (II)                  Numerical Results (Prefactors)          Numerical Results (Particle Spectra)
Effects of Bulk Viscosity at Freezeout                    Nagoya Mini Workshop, Nagoya University, March 3rd 2009



         Numerical Results (Particle Spectra)
        Au+Au,                          , b = 7.2(fm), pT -spectra of π -



                                                                                Model of the bulk pressure:




                                                                                Parameter α is set to
                                                                                and         for the results.

                                                                                The bulk viscosity lowers
                                                                                <pT> of the particle spectra.




    Numerical Results (Prefactors)       Numerical Results (Particle Spectra)               Numerical Results (v2(pT) )
Effects of Bulk Viscosity at Freezeout                  Nagoya Mini Workshop, Nagoya University, March 3rd 2009



                    Numerical Results (v2(pT) )
        Au+Au,                          , b = 7.2(fm), v2(pT) of π -




                                                                          The bulk viscosity enhances
                                                                          v2(pT) in the high pT region.

                                                                          *Viscous effects may have
                                                                          been overestimated:
                                                                          (1) No relaxation time for
                                                                                           is from the 1st
                                                                          order theory.
                                                                          (2) Derivatives of
                                                                          are larger than those of real
                                                                          viscous flow.


 Numerical Results (Particle Spectra)       Numerical Results (v2(pT) )              Results with Quadratic Ansatz
Effects of Bulk Viscosity at Freezeout                Nagoya Mini Workshop, Nagoya University, March 3rd 2009



              Results with Quadratic Ansatz
       pT -spectra and v2(pT) of π - with                                      ,                   and
        the same EoS.




     Numerical Results (v2(pT) )         Results with Quadratic Ansatz                    Summary
Effects of Bulk Viscosity at Freezeout                 Nagoya Mini Workshop, Nagoya University, March 3rd 2009



                              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 v2(pT) in the high pT 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 pT-spectra and v2(pT).

        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.


    Results with Quadratic Ansatz                  Summary
                                           Results for Shear Viscosity            Results for Shear + Bulk Viscosity
Effects of Bulk Viscosity at Freezeout               Nagoya Mini Workshop, Nagoya University, March 3rd 2009



                  Results for Shear Viscosity
       pT -spectra and v2(pT) of π - with                             ,               , and the same
        EoS.




            Summary                      Results for Shear Viscosity            Results for Shear + Bulk Viscosity
Effects of Bulk Viscosity at Freezeout                   Nagoya Mini Workshop, Nagoya University, March 3rd 2009



        Results for Shear + Bulk Viscosity
        pT -spectra and v2(pT) of π -, with                                  ,            , and the same
         EoS.




     Results for Shear Viscosity         Results for Shear + Bulk Viscosity
Effects of Bulk Viscosity at Freezeout           Nagoya Mini Workshop, Nagoya University, March 3rd 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

								
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