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
May 9th 2009, Hongo, The University of Tokyo
arXiv: [nucl-th]

2. Outline Introduction Theories and Methods Numerical Results Summary
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. Introduction (I) Success of ideal hydrodynamic models
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. Introduction (II) Cooper & Frye (‘74)
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)
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:
particles
freezeout hypersurface Σ
hadron
resonance
gas
QGP
variation of the flow/hypersurface
modification of the distribution
(3+1)-D viscous hydro required.
We estimate this for a multi-component gas.
* :normal vector to the freezeout hypersurface element,
:distribution of the ith particle, :degeneracy.
Introduction (I)
Introduction (II)
Relativistic Kinetic Theory
In Multi-Component System

5. Relativistic Kinetic Theory
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. In Multi-Component System
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 gi is the degeneracy and bi 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 equations
Relativistic Kinetic Theory
In Multi-Component System
The Matching Conditions
Grad’s 14-moment method

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

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

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

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

11. Prefactors in Viscous Correction
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
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.
Scalar terms
Vector terms
Tensor term
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. EoS, Transport Coefficients and Flow
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
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 ( ).
Kovtun et al.(‘05)
Weinberg (‘71) …
Profiles of the flow and the freezeout hypersurface : a (3+1)-dimensional ideal hydrodynamic simulation.
Hirano et al.(‘06)
Summary So Far
EoS, Transport Coefficients and Flow
pT-Spectra
Elliptic Flow Coefficient v2(pT)

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

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

16. Quadratic Ansatz pT -spectra and v2(pT) of with and the same EoS
Effects of Bulk Viscosity on pT Spectra and Elliptic Flow Coefficients
Heavy Ion Café, The University of Tokyo, May 9th 2009
Quadratic Ansatz
pT -spectra and v2(pT) of with
and the same EoS
Effects of the bulk viscosity is underestimated in quadratic ansatz.
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
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.
pT-spectra : suppressed
v2(pT) : enhanced
when estimated with an ideal hydrodynamic flow.
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
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
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.
pT-spectra : suppressed
v2(pT) : enhanced
when estimated with an ideal hydrodynamic flow.
Quadratic Ansatz
Effects of Shear Viscosity
Summary & Outlook
Effects of Shear and Bulk Viscosity

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

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

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

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

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

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

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

30. Prefactors in Special Case
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, BUT it 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. Explicit Forms of Prefactors
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