Bulk Viscosity in Nuclear Collisions Other Remarks

1
Bulk Viscosity in Nuclear Collisions Other
Remarks

• Rainer Fries
• Texas AM University RIKEN BNL

RIKEN Workshop Hydrodynamics in HI and QCD
EOS Brookhaven National Laboratory, April 22,
2008
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Outline

• Early time evolution and hydro initial conditions
• Pressure anisotropy
• Radial flow
• Bulk viscosity and equation of state
• Phenomenological connection between bulk
  viscosity and phase transition
• Longitudinal pressure and entropy production in a
  simple hydro model

Thanks to B. Müller and A. Schäfer
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Motivation

• Summary of our task to describe heavy ion
  collisions
• Initial interaction energy deposited between the
  nuclei (gluon field), color glass
• Pre-equilibrium / Glasma decoherence,
  thermalization? particle production,
  instabilities?
• Equilibrium (at least approx.) (viscous)
  hydrodynamics
• Freeze-out / hadronic phase cascade ? free
  streaming

Equilibrium Hydro
CGC
Hadronic Phase
Equilibration?
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Hydrodynamics

• Need the right context (in particular for ideal
  hydro)
• Initial time / degree of thermalization
• Initial conditions
• Break-down at late times, freeze-out
• Correct treatment of viscous corrections.
• Shear bulk
• Connection with phase transition / equation of
  state?

Equilibrium Hydro
CGC
Hadronic Phase
Equilibration?
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Early Time Picture

• Intuitive picture
• Initial longitudinal field
• Subsequent decay into (thermal?) particle
  ensemble at roughly constant proper
  time.
• Energy for the field is taken from
  deceleration of the
  nuclei in the color field.
• RHIC produces tiny color capacitors

Mishustin, Kapusta
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Color Glass Condensate

• Before the collision transverse fields in the
  nuclei
• E and B orthogonal

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Gluon Near Field

• Before the collision transverse fields in the
  nuclei
• E and B orthogonal
• Immediately after overlap in the
  forward light cone
  Strong
  longitudinal electric
  magnetic fields.

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Gluon Near Field

• Before the collision transverse fields in the
  nuclei
• E and B orthogonal
• Immediately after overlap in the
  forward light cone
  Strong
  longitudinal electric
  magnetic fields.
• Transverse E, B fields start to build
  up ?

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Initial Energy Momentum Tensor

• Assuming boost-invariance
• Initial energy density
• Tensor diagonal for ? ? 0
• Precise form to be expected for longitudinal
  vacuum field

Transverse pressure pT ?0
Longitudinal pressure pL ?0
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Initial Pressure

• System starts with maximum pressure anisotropy.

p
T
?
L
Longitudinal Field
(Ideal) plasma
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Initial Longitudinal Pressure

• Negative longitudinal pressure
• Leads to the deceleration of the nuclei.
• BRAHMS baryon number
• dy 2.0 ? 0.4
• Nucleon 100 GeV ? 27 GeV
• With Kapusta, Mishustin model

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Initial Transverse Pressure

• Positive transverse pressure ? transverse
  expansion
• Outward flow of energy for ? ? 0
• Outward flow of energy translates to radial flow
  of particles after equilibration
• Initial radial flow is there,
  no equilibration necessary!
• Important for hydro phase?
• Little time to build up, but pT ?.
• Maybe interesting for v2.

b 8 fm
(Leading behavior for ? ? 0 assuming p ? /3)
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Viscosity

• Recent interesting results for bulk viscosity
• ?/s ltlt ?/s for large temperatures
• Karsch, Kharzeev, Tuchin Meyer ?/s O(1) at
  Tc.
• Seem to be sharply peaked at Tc.
• Possible consequences
• Significant contribution to
  entropy
  production?
• Evolution of the pressure?
• Break-down of (viscous) hydrodynamics?

Kharzeev, Tuchin
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Bulk Viscosity

• Interesting development new focus on the phase
  transition.
• Interplay of bulk viscosity/viscous hydro ?
  equation of state?
• Does cross over vs 1st order phase transition
  matter?
• Try to address some of these questions in a
  simple hydro model.

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Viscosity Equation of State

• Realistic Scenario lattice results
• Convenient parametrizations of lattice results
• p(T) , (?-3p)(T) from Cheng et al.,
  arXiv0710.0354
• ?/s( (?-3p)/(?p) )
  from
  Meyer, arXiv0710.3717
• Pure glue large error bars.

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Viscosity Equation of State

• Extract ?/s(T)
• Keep ?/s 1/(4?) fixed
  always
• Caution
• Two very different lattice calculations.
• Indeed Meyer ?/s(T) 0.5 2 just above Tc.
• Strategy here
• Take shape of temperature-dependence as suggested
  above (also on the hadron side).
• Perform calculations scaling ? with factors c?
  between 0.5 and 4.

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Hydro Implementation

• Test with 01 D boost-invariant hydro

Two scenarios I (lattice) lattice EOS
lattice-inspired bulk viscosity II (toy model
1st order) 1st order phase transition, use same
?/s(T) as in I
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Hydro Implementation

• Fixed relaxation times
• Initial time ?0 0.6 fm/c
• Fix initial entropy density at ?0 by setting s(?)
  seq(?) for asymptotically long times ? and
  extrapolating back.
• For most cases set initial shear and bulk
  pressure at ?0 to zero.

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Results Longitudinal Pressure
I Lattice
II 1st order

• Pressure, bulk pressure and shear
• Relative long. pressure, bulk pressure and shear

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Results Entropy
I Lattice
II 1st order

• Entropy ?s total, shear, bulk contributions
• Shear, bulk contributions relative to final
  entropy

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Results Entropy

• So far c? 1 now vary bulk viscosity using
  model I.
• Entropy ?s shear bulk contributions relative
  to final entropy
• In this hydro model ?/smax 0.4 produces roughly
  as much entropy as ?/s 1/(4?) over the lifetime
  of the fireball.
• Caution half of S? comes from T lt Tc need
  realistic ?had.

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Comparison

• Strong sensitivity to equation of state/phase
  transition.
• Both scenarios use the same ?/s.
• Entropy 50 change in S? depening on EOS.
• Longitudinal pressure large change in pz.
• Sharp phase transition and/or large ?(Tc) might
  lead to complete breakdown of pz.
• A. Dumitru and others negative longitudinal
  pressure at TTc?
• Applicability of (2nd order) hydrodynamics around
  the phase transition might need investigation.
• Effects on transverse pressure?

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Longitudinal Pressure Revisited

• Longitudinal pressure with reasonable initial
  conditions.
• I.e. ?p(?) and ? (?) are smooth functions around
  ? ?0
• Even with c? 1 longitudinal pressure pz ½ p
  during entire QGP phase.
• Observable consequences?

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Summary

• Importance of Initial Conditions
• Evolution of longitudinal pressure.
• Transverse pressure early flow, also v2.
• No equilibration necessary for initial flow.
• Interesting interplay bulk viscosity ? phase
  transition/equation of state.
• Focus on Tc is it useful or annoying?
• Our model I (Lattice) ? important for p and s,
  but doesnt overwhelm entropy production from ?.

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Backup
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The Color Capacitor

• Gauge potential (light cone gauge)
• In sectors 1 and 2 single nucleus solutions Ai1,
  Ai2.
• In sector 3 (forward light cone)
• YM in forward direction
• Set of non-linear differential
  equations
• Boundary conditions at ? 0 given by the
  fields of the single nuclei

McLerran, Venugopalan Kovner, McLerran,
Weigert Jalilian-Marian, Kovner, McLerran,
Weigert
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Estimating Energy Density

• Initial energy density in the MV model with
  cutoff Q0.
• Similar to previous estimate using screened
  abelian boundary fields (modulo logarithmic
  term).

T. Lappi (2006)
RJF, Kapusta, Li (2006)
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Coupling to the Plasma Phase

• How to get an equilibrated (?) plasma?
• Difficult problem. We just assume it happens.
• Use energy-momentum conservation to constrain the
  plasma phase
• Total energy momentum tensor of the system
• r(?) interpolating function
• Enforce