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 2nd-4th, 2009, Nagoya, Japan

2. Outline Introduction Theories and Methods Numerical Results Summary
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)
Introduction (II)

3. Introduction (I) Success of ideal hydrodynamic models for
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.
In this work, we see the effects of bulk viscosity at freezeout.
Mizutani et al. (‘88)
Paech & Pratt (‘06)
Kharzeev & Tuchin (’08) …
Outline
Introduction (I)
Introduction (II)
Kinetic Theory (I)

4. Introduction (II) Cooper & Frye (‘74)
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.
where,
:normal vector to the freezeout
hypersurface element
:distribution function of the ith particle
:degeneracy
Viscous effects are taken into account via
(1) variation of the flow
(2) modification of the distribution function
QGP
hadron resonance gas
freezeout hypersurface Σ
Particles
dσμ
This needs (3+1)-D viscous hydro.
We focus on the contributions of
the bulk viscosity to this phenomenon.
Introduction (I)
Introduction (II)
Kinetic Theory (I)
Kinetic Theory (II)

5. 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.
Tensor decompositions of the energy-momentum tensor and the net baryon number current:
where , and
Bulk pressure:
Energy current: Charge current:
Shear stress tensor:
Israel & Stewart (‘79)
Introduction (II)
Kinetic Theory (I)
Kinetic Theory (II)
Grad’s 14-moment method

6. Kinetic Theory (II) Kinetic definitions for a multi-particle system:
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
Decomposition of Moments

7. 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
(mass dependent)
particle species independent
(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
Comments on Quadratic Anzatz

8. 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
Prefactors (I)

9. 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:
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.
Dusling & Teaney (‘08)
Decomposition of Moments
Comments on Quadratic Ansatz
Prefactors (I)
Prefactors (II)

10. 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)
Prefactors in Special Case

11. Prefactors (II) The solutions are
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
Models (I)

12. 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)
Models (II)

13. Models (I) Equation of State - 16-component hadron resonance gas
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:
where (sound velocity) and
s is the entropy density.
The freezeout temperature: Tf = 0.16(GeV)
where and ( ).
Kovtun et al.(‘05)
Arnold et al.(‘06)
Prefactors in Special Case
Models (I)
Models (II)
Numerical Results (Prefactors)

14. 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.
For numerical calculations we take the Landau frame ( ) and the zero net baryon density limit ( ).
Hirano et al.(‘06)
Models (I)
Models (II)
Numerical Results (Prefactors)
Numerical Results (Particle Spectra)

15. 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)
Numerical Results (v2(pT) )

16. 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) )
Summary

17. 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
Summary

18. 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

19. 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
Results for Shear Viscosity
Summary
Results for Shear + Bulk Viscosity

20. Results for Shear 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

21. 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

22. 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
Thank You