effects of bulk viscosity on p t -spectra and elliptic flow parameter akihiko monnai department of...

Effects of Bulk Viscosity on pT-Spectra and Elliptic Flow Parameter

Akihiko MonnaiDepartment of Physics, The University of Tokyo, JapanCollaborator: Tetsufumi Hirano

Quark Matter 2009March 30th- April 4th, 2009, Knoxville, TN, U.S.A. arXiv:0903.4436 [nucl-th]

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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

Quark Matter 2009, Knoxville, Tennessee, April 2Quark Matter 2009, Knoxville, Tennessee, April 2ndnd 2009 2009Effects of Bulk Viscosity on Effects of Bulk Viscosity on ppTT-spectra and Elliptic Flow Parameter-spectra and Elliptic Flow Parameter

OutlineOutline Introduction (I)Introduction (I)

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Introduction (I)

Paech & Pratt (‘06)Mizutani et al. (‘88) Kharzeev & Tuchin (’08) …


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.

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

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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)


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

variation of the flow modification of the distribution

* :normal vector to the freezeout hypersurface element,

:distribution of the ith particle, :degeneracy.

particles

hadronresonancegas

QGP

freezeout hypersurface Σ

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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 SystemIn Multi-Component SystemIntroduction (II)Introduction (II)


Relativistic Kinetic Theory Relativistic Kinetic Theory

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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) + matching conditions (2) = 14 equations.

Grad’s 14-moment methodGrad’s 14-moment method


In Multi-Component SystemIn Multi-Component SystemRelativistic Kinetic Theory Relativistic Kinetic Theory

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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 ]

Grad’s 14-moment methodGrad’s 14-moment method

The trace part The scalar term

particle species dependent(mass dependent)

particle species independent(macroscopic quantity)

Decomposition of MomentsDecomposition of Moments


In Multi-Component SystemIn Multi-Component System

NOT equivalent for a multi-component system.

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Decomposition of Moments Definitions:

*Contributions are

: [baryons] + [anti-baryons] + [mesons]

: [baryons] – [anti-baryons]

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


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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 AnsatzComments on Quadratic AnsatzDecomposition of MomentsDecomposition of Moments


Dusling & Teaney (‘08)

Prefactors in Viscous CorrectionPrefactors in Viscous Correction

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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 AnsatzComments on Quadratic Ansatz


Prefactors in Viscous CorrectionPrefactors in Viscous Correction EoS, Transport Coefficients and FlowEoS, Transport Coefficients and Flow

Scalar terms

Vector terms

Tensor term

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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 ( ).


Weinberg (‘71)

Kovtun et al.(‘05)

Hirano et al.(‘06)

…

EoS, Transport Coefficients and FlowEoS, Transport Coefficients and FlowPrefactors in Viscous CorrectionPrefactors in Viscous Correction

Profiles of the flow and the freezeout hypersurface : a (3+1)-dimensional ideal hydrodynamic simulation.

ppTT-Spectra-Spectra

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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.

Elliptic Flow Coefficient Elliptic Flow Coefficient vv22((ppTT))ppTT-Spectra-Spectra


EoS, Transport Coefficients and FlowEoS, Transport Coefficients and Flow

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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

Results with Quadratic AnsatzResults with Quadratic Ansatz


ppTT-Spectra-Spectra Elliptic Flow Coefficient Elliptic Flow Coefficient vv22((ppTT))

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

and the same EoS

Results with Quadratic AnsatzResults with Quadratic Ansatz SummarySummary


Effects of the bulk viscosity is underestimated in quadratic ansatz.

Elliptic Flow Coefficient Elliptic Flow Coefficient vv22((ppTT))

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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.

SummarySummaryResults with Quadratic AnsatzResults with Quadratic Ansatz


pT-spectra : suppressed

v2(pT) : enhancedwhen estimated with an ideal hydrodynamic flow.

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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 YouThank You


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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.

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


pT-spectra : suppressed

v2(pT) : enhancedwhen estimated with an ideal hydrodynamic flow.

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

EoS.

Results for Shear ViscosityResults for Shear ViscositySummarySummary Results for Shear + Bulk ViscosityResults for Shear + Bulk Viscosity


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

EoS.

Results for Shear + Bulk ViscosityResults for Shear + Bulk ViscosityResults for Shear ViscosityResults for Shear Viscosity Shear Viscosity in Blast Wave ModelShear Viscosity in Blast Wave Model


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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.

Shear Viscosity in Blast Wave ModelShear Viscosity in Blast Wave Model

Teaney (‘03)

Viscous Flow vs. Viscous DistributionViscous Flow vs. Viscous DistributionResults for Shear + Bulk ViscosityResults for Shear + Bulk Viscosity


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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 DistributionViscous Flow vs. Viscous Distribution

Song & Heinz (‘08)

Shear Viscosity in Blast Wave ModelShear Viscosity in Blast Wave Model Freezeout HypersurfaceFreezeout Hypersurface


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Freezeout Hypersurface Comparison of freezeout hypersurfaces for the ideal and the (2+1)-D viscous

hydrodynamics:

Freezeout HypersurfaceFreezeout Hypersurface Expansion of DistributionExpansion of DistributionViscous Flow vs. Viscous DistributionViscous Flow vs. Viscous Distribution


Song & Heinz (‘08)

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Expansion of Distribution The distribution function is expanded as follows:

where the momentum expansion is used.

Expansion of DistributionExpansion of Distribution The Matching ConditionsThe Matching Conditions

=

Freezeout HypersurfaceFreezeout Hypersurface


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The Landau matching conditions are necessary to ensure the thermodynamic stability in the 1st order theory:

The Matching Conditions

The Matching ConditionsThe Matching ConditionsExpansion of DistributionExpansion of Distribution


Terms proportional to destabilize the system.

Numerical Results (Prefactors)Numerical Results (Prefactors)

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The prefactors for and in ’s near Tf:

Numerical Results (Prefactors)

Numerical Results (Prefactors)Numerical Results (Prefactors)


Trace Part vs. Scalar TermTrace Part vs. Scalar TermThe Matching ConditionsThe Matching Conditions

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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.

Trace Part vs. Scalar TermTrace Part vs. Scalar Term


Prefactors in Special CasePrefactors in Special CaseNumerical Results (Prefactors)Numerical Results (Prefactors)

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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 CasePrefactors in Special Case


Trace Part vs. Scalar TermTrace Part vs. Scalar Term Explicit Forms of PrefactorsExplicit Forms of Prefactors

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Explicit Forms of Prefactors The prefactors in the viscous correction tensors are:

where,

Explicit Forms of PrefactorsExplicit Forms of Prefactors


Prefactors in Special CasePrefactors in Special Case

effects of bulk viscosity on p t -spectra and elliptic flow parameter akihiko monnai department of...

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