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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)
Nagoya Mini Workshop, Nagoya University, March 3rd 2009
Effects of Bulk Viscosity at Freezeout
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
QGP
dσμ
Particles
where,
hadron resonance gas
Viscous effects are taken into account via
(1) variation of the flow
(2) modification of the distribution function
:normal vector to the freezeout
hypersurface element
:distribution function of the ith particle
:degeneracy

Introduction (I)
Introduction (II)
This needs (3+1)-D viscous hydro.
We focus on the contributions of
the bulk viscosity to this phenomenon.
Kinetic Theory (I)
Nagoya Mini Workshop, Nagoya University, March 3rd 2009
Effects of Bulk Viscosity at Freezeout
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)
Nagoya Mini Workshop, Nagoya University, March 3rd 2009
Effects of Bulk Viscosity at Freezeout
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
Nagoya Mini Workshop, Nagoya University, March 3rd 2009
Effects of Bulk Viscosity at Freezeout
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
particle species dependent
(mass dependent)
]
The scalar term
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
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
,
(3) Hydrodynamic simulations need discussion for a resonance gas.

or
.
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)
Nagoya Mini Workshop, Nagoya University, March 3rd 2009
Effects of Bulk Viscosity at Freezeout
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)
Nagoya Mini Workshop, Nagoya University, March 3rd 2009
Effects of Bulk Viscosity at Freezeout
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)
Nagoya Mini Workshop, Nagoya University, March 3rd 2009
Effects of Bulk Viscosity at Freezeout
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 (
baryon density limit (
).
Models (I)
Models (II)
) and the zero net
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
Nagoya Mini Workshop, Nagoya University, March 3rd 2009
Effects of Bulk Viscosity at Freezeout
Results with Quadratic Ansatz

pT -spectra and v2(pT) of π - with
the same EoS.
Numerical Results (v2(pT) )
Results with Quadratic Ansatz
,
and
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
Results for
Summary
Shear Viscosity
Results for Shear + Bulk Viscosity
Nagoya Mini Workshop, Nagoya University, March 3rd 2009
Effects of Bulk Viscosity at Freezeout
Results for Shear Viscosity

pT -spectra and v2(pT) of π - with
EoS.
Summary
Results for Shear Viscosity
,
, and the same
Results for Shear + Bulk Viscosity
Nagoya Mini Workshop, Nagoya University, March 3rd 2009
Effects of Bulk Viscosity at Freezeout
Results for Shear + Bulk Viscosity

pT -spectra and v2(pT) of π -, with
EoS.
Results for Shear Viscosity
Results for Shear + Bulk Viscosity
,
, and the same
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