Effects of Bulk Viscosityat Freezeout

Akihiko MonnaiDepartment of Physics, The University of TokyoCollaborator: Tetsufumi Hirano

Nagoya Mini-Workshop “Photons and Leptons in Hot/Dense QCD”March 2nd-4th, 2009, Nagoya, Japan

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

OutlineOutline Introduction (I)Introduction (I)

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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

IntroductionIntroduction (I)(I) Introduction (II)Introduction (II)

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OutlineOutline

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)

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 ith 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 3Nagoya Mini Workshop, Nagoya University, March 3rdrd 2009 2009Effects of Bulk Viscosity at FreezeoutEffects of Bulk Viscosity at Freezeout

This needs (3+1)-D viscous hydro.

We focus on the contributions ofthe bulk viscosity to this phenomenon.

Introduction (II)Introduction (II)Introduction (I)Introduction (I)

QGPQGP

hadron resonance gas

freezeout hypersurface Σ

Particles

Kinetic Theory (I)Kinetic Theory (I)

dσμ

Cooper & Frye (‘74)

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)Kinetic Theory (II)

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Introduction (II)Introduction (II) Kinetic Theory (I)Kinetic Theory (I)

Israel & Stewart (‘79)

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.

Nagoya Mini Workshop, Nagoya University, March 3Nagoya Mini Workshop, Nagoya University, March 3rdrd 2009 2009Effects of Bulk Viscosity at FreezeoutEffects of Bulk Viscosity at Freezeout

Grad’s 14-moment methodGrad’s 14-moment methodKinetic Theory (I)Kinetic Theory (I) Kinetic Theory (II)Kinetic Theory (II)

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 ]

Nagoya Mini Workshop, Nagoya University, March 3Nagoya Mini Workshop, Nagoya University, March 3rdrd 2009 2009Effects of Bulk Viscosity at FreezeoutEffects of Bulk Viscosity at Freezeout

Kinetic Theory (II)Kinetic Theory (II) Grad’s 14-moment methodGrad’s 14-moment method

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.

Decomposition of MomentsDecomposition of Moments

Decomposition of Moments Definitions:

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

Nagoya Mini Workshop, Nagoya University, March 3Nagoya Mini Workshop, Nagoya University, March 3rdrd 2009 2009Effects of Bulk Viscosity at FreezeoutEffects of Bulk Viscosity at Freezeout

Comments on Quadratic AnsatzComments on Quadratic AnsatzGrad’s 14-moment methodGrad’s 14-moment method Decomposition of MomentsDecomposition of Moments

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 3Nagoya Mini Workshop, Nagoya University, March 3rdrd 2009 2009Effects of Bulk Viscosity at FreezeoutEffects of Bulk Viscosity at Freezeout

Comments on Quadratic AnsatzComments on Quadratic Ansatz

Dusling & Teaney (‘08)

Decomposition of MomentsDecomposition of Moments Prefactors (I)Prefactors (I)

Prefactors (I) 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 3Nagoya Mini Workshop, Nagoya University, March 3rdrd 2009 2009Effects of Bulk Viscosity at FreezeoutEffects of Bulk Viscosity at Freezeout

Prefactors (I)Prefactors (I) Prefactors (II)Prefactors (II)Comments on Quadratic AnsatzComments on Quadratic Ansatz

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 3Nagoya Mini Workshop, Nagoya University, March 3rdrd 2009 2009Effects of Bulk Viscosity at FreezeoutEffects of Bulk Viscosity at Freezeout

Prefactors (I)Prefactors (I) Prefactors (II)Prefactors (II) Prefactors in Special CasePrefactors in Special Case

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 3Nagoya Mini Workshop, Nagoya University, March 3rdrd 2009 2009Effects of Bulk Viscosity at FreezeoutEffects of Bulk Viscosity at Freezeout

Prefactors Prefactors Prefactors in Special CasePrefactors in Special Case Models (I)Models (I)

Models (I)

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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 ( ).

Models (I)Models (I) Models (II)Models (II)

Arnold et al.(‘06)

Kovtun et al.(‘05)

Prefactors in Special CasePrefactors in Special Case

Models (II)

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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)Models (II) Numerical Results (Prefactors)Numerical Results (Prefactors)Models (I)Models (I)

The prefactors for and near the freezeout temperature Tf:

Numerical Results (Prefactors)

Nagoya Mini Workshop, Nagoya University, March 3Nagoya Mini Workshop, Nagoya University, March 3rdrd 2009 2009Effects of Bulk Viscosity at FreezeoutEffects of Bulk Viscosity at Freezeout

Numerical Results (Prefactors)Numerical Results (Prefactors) Numerical Results (Particle Spectra)Numerical Results (Particle Spectra)

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

Models (II)Models (II)

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

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

Nagoya Mini Workshop, Nagoya University, March 3Nagoya Mini Workshop, Nagoya University, March 3rdrd 2009 2009Effects of Bulk Viscosity at FreezeoutEffects of Bulk Viscosity at Freezeout

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 (Numerical Results (vv22((ppTT) )) )Numerical Results (Particle Spectra)Numerical Results (Particle Spectra)Numerical Results (Prefactors)Numerical Results (Prefactors)

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

Nagoya Mini Workshop, Nagoya University, March 3Nagoya Mini Workshop, Nagoya University, March 3rdrd 2009 2009Effects of Bulk Viscosity at FreezeoutEffects of Bulk Viscosity at Freezeout

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 (Particle Spectra) Numerical Results (Numerical Results (vv22((ppTT) )) ) Results with Quadratic AnsatzResults with Quadratic Ansatz

Results with Quadratic Ansatz pT -spectra and v2(pT) of π - with , and

the same EoS.

Results with Quadratic AnsatzResults with Quadratic Ansatz

Nagoya Mini Workshop, Nagoya University, March 3Nagoya Mini Workshop, Nagoya University, March 3rdrd 2009 2009Effects of Bulk Viscosity at FreezeoutEffects of Bulk Viscosity at Freezeout

Numerical Results (Numerical Results (vv22((ppTT) )) ) SummarySummary

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.

SummarySummary

Nagoya Mini Workshop, Nagoya University, March 3Nagoya Mini Workshop, Nagoya University, March 3rdrd 2009 2009Effects of Bulk Viscosity at FreezeoutEffects of Bulk Viscosity at Freezeout

Results for Shear ViscosityResults for Shear Viscosity Results for Shear + Bulk ViscosityResults for Shear + Bulk ViscosityResults with Quadratic AnsatzResults with Quadratic Ansatz

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

EoS.

Results for Shear ViscosityResults for Shear Viscosity

Nagoya Mini Workshop, Nagoya University, March 3Nagoya Mini Workshop, Nagoya University, March 3rdrd 2009 2009Effects of Bulk Viscosity at FreezeoutEffects of Bulk Viscosity at Freezeout

SummarySummary Results for Shear + Bulk ViscosityResults for Shear + Bulk Viscosity

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

EoS.

Results for Shear + Bulk ViscosityResults for Shear + Bulk Viscosity

Nagoya Mini Workshop, Nagoya University, March 3Nagoya Mini Workshop, Nagoya University, March 3rdrd 2009 2009Effects of Bulk Viscosity at FreezeoutEffects of Bulk Viscosity at Freezeout

Results for Shear ViscosityResults for Shear Viscosity

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 3Nagoya Mini Workshop, Nagoya University, March 3rdrd 2009 2009Effects of Bulk Viscosity at FreezeoutEffects of Bulk Viscosity at Freezeout

Thank YouThank You