Dynamics of the Chiral Transition:

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Dynamics of the Chiral Transition Dissipation
Inhomogeneities
Eduardo S. Fraga
Instituto de Física Universidade Federal do Rio
de Janeiro
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Outline

• Introduction and motivation
• Dissipation in spinodal decomposition dynamics
• for the chiral transition
• Inhomogeneity corrections
• Some results for pure gauge (deconfinement)
• Final remarks

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Introduction and motivation

• Depending on the nature of the chiral
  transition, phase
• conversion can occur in a number of ways, e.g.
  
• nucleation vs. spinodal decomposition
• Short time scales in HIC -gt explosive
  decomposition
• Most theoretical attempts rapid change in
  effective
• potential (Polyakov loop model, effective
  chiral models,
• etc) -gt very fast spinodal decomposition
• (explosive phase conversion)
• Signatures e.g., Scavenius, Dumitru Jackson
  (2001) Dumitru Pisarski (2002)
• Randrup (2004/2005) Rafelski
  (2005) Koch, Majumder, Randrup (2005)
• Greiner Xu (2005/2006).

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• Questions
• What are the roles of dissipation and noise
• in the evolution of the order parameter?
• Are memory effects important?
• How is the evolution modified by inhomogeneities
  and
• by the finite size of the system?
• Can one describe the two transitions
  simultaneously?
• Method Langevin description (from nonequilibrium
  QFT)

Phase transitions real-time dynamics!
eff. QFT models
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Langevin equation from QFT e.g. Gleiser Ramos
(1994) Rischke (1998)

• Two-loop effective action (e.g., symmetric lf4)

• Closed-time path contour (Schwinger-Keldysh)

()
(-)

• Decomposition into classical and response
  fields

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Contributions from the 2-loop diagrams can be put
into the form
and the imaginary part can be represented by
functional integrals over gaussian fluctuation
fields (noise)
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• Equations of motion approximations
• approximately homogeneous fields
• adiabatic approximation
• high temperature T gtgt mT

yield the following effective Langevin equation
In general, there is also additive noise (besides
multiplicative)
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Effective theory for the chiral transition

• Assumptions 1st order c phase trans. ( exp.
  system finite size)
• Framework coarse-grained Landau-Ginzburg
  effective potential
• from linear s model quarks , Nf 2
• Csernai Mishustin (1995) Scavenius
  Dumitru (1999)

Parameters such that SU(2) ? SU(2) broken in the
vacuum, ltsgtfp , ltpgt0, H from PCAC, masses, etc
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Integrate over fermions (heat bath for the chiral
fields) ? effective potential for f(s,p)
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Explosive scenario
Time scales (for nucleation)
Phase conversion is likely to proceed via
explosive spinodal decomposition!
similar results for instabilities
from Rafelski Letessier (2000/2005) Dumitru
Pisarski (2001) Scavenius, Dumitru Jackson
(2001) Paech, Stoecker Dumitru
(2003/2005) Randrup (2004/2005) ...
Scavenius, Dumitru, ESF, Lenaghan Jackson
(2001)
System would go directly to spinodal and
explode, mostly bypassing bubble nucleation
-gt Very short time scales
However, shouldnt dissipation effects be
relevant?
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Dissipation in spinodal decomposition for the c
transition
ESF
G. Krein (2005) Additive-noise Langevin
dynamics
GG(T) response coefficient (intensity of
dissipation) f(x,t) non-conserved order
parameter (s field for cPT) Stochastic (noise)
force assumed gaussian and white ()
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Results
For t lt 5 fm/c the solution for the linearized
eqn is very close to the complete one, then is
delayed -gt O(2) potential much shallower than
complete Veff
Even for conservative dissipation G2T, the
retardation effect for exponential growth is
100 !!
For expansion times 5 fm/c ( RHIC times
scales) there might be not enough time for the
onset of the spinodal explosion !
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(Additive Multiplicative)-noise Langevin
dynamics
ESF (2005) ESF, G. Krein R.O. Ramos, to
appear

• A more complete QFT description of
  nonequilibrium dissipative dynamics
• predicts the existence of additional terms in
  the Langevin equation.
• In fact, dissipation effects should depend on
  the local density and, accordingly,
• the noise should contain a multiplicative term
  (fluctuation-dissipation theorem).
• Effects of multiplicative noise seem to be
  rather significant (e.g., in the
• Kibble-Zurek scenario of defect formation in
  one spatial dimension)
• Antunes, Gandra Rivers (2005)

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• Preliminary results
• For degenerate double-well potential at given T
• Still in arbitrary units (dimensional quantities
  in units of m)
• Lattice-size independence assured by
  counterterms
• Farakos, Kajantie, Rummukainen
  Shaposhnikov (1994)
• Ga still T-independent

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Lattice-size independence (T1 m1) e.g.
Borrill Gleiser (1997) Bettencourt, Habib
Lythe (1999) Gagne Gleiser (2000)
Bettencourt, Rajagopal Steele (2001)
additive multiplicative noises
(G1 G2 1)
additive noise only (G1 1)
ESF (2005) ESF, G. Krein R.O. Ramos, to
appear
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Effect of multiplicative noise
ESF (2005) ESF, G. Krein R.O. Ramos, to
appear
Thermalization is clearly delayed!
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Symmetry restoration for high T T-independent
Ga still incomplete, work in progress
additive multiplicative noises
(G1 G2 1)
additive noise only (G1 1)
ESF, G. Krein R.O. Ramos, to appear
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Inhomogeneity corrections
B.G. Taketani ESF, hep-ph/0604036
Thermodynamic potential
Equation of motion and fermionic source term
Homogeneous background field result
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Inhomogeneity corrections
Fermionic source term
Gradient expansion
Inhomogeneity corrections
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1st contribution
2nd contribution
After the k0 integration
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where H is
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Modification of the effective potential
TTsp
TTc

• Smoothening of the potential
• Higher Tc
• Easier to nucleate bubbles
• Significant effect !

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• In the thin-wall approximation ( Landau-Ginzburg
  approx. for Veff)

Nucleation seems to be stimulated - influence on
explosive scenario?
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• Still missing in the dynamical picture
• Finite-size effects important right from the
• start in heavy-ion collisions
• ESF and R. Venugopalan (2004)
• Expansion of the fluid (more dissipation)
• Scavenius, Dumitru Jackson (2001)
• Effects from Polyakov loop sector
• A.J. Mizher, ESF G. Krein (2006)
• Non-Markovian (memory) effects
• L.F. Palhares, ESF, T. Kodama G. Krein
  (2006)

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Langevin dynamics of the deconfinement
transition for pure gauge theory - a few
results A.J. Mizher, ESF G.
Krein (2006) G. Ananos, ESF, G. Krein A.J.
Mizher, to appear
SU(2)
SU(3)
See also poster by O. Bazavov
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Final remarks

• Effects of dissipation seem to be important in
  the process
• of hadronization of the QGP. In particular, it
  could
• dramatically modify the explosive behavior
  scenario.
• Multiplicative noise (and density-dependent
  dissipation) do
• play a role, and retard even more
  thermalization.
• Inhomogeneity effects modify appreciably the
  effective
• potential making nucleation easier.
• Non-Markovian corrections can, in principle, be
  incorporated
• systematically in the Langevin evolution
  (simpler structure).
• Numerical methods for efficient treatment of
  colored
• noise and complete memory kernel being
  developed.