Quantized meson fields in and out of equilibrium

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Quantized meson fields in and out of equilibrium

• Mamoru Matsuo (KEK), T. Matsui (Univ. Tokyo)

Based on NPA809(2008)211 f4 (N1),
arXiv0812.1853 O(N) linear sigma model
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Chiral Phase Transition in heavy ion collision
Two Lorentz contracted nuclei are approaching
toward each other at the velocity of light.
After two nuclei pass each other,,,
Vacuum between two nuclei are excited and filled
with a quark-gluon plasma. Vacuum chiral
condensate has melted away in this region.
As the system expands, the quark-gluon plasma
will hadronize and the chiral symmetry will be
broken spontaneously again.
We have growing chiral condensate (classical mean
field) and particle excitations (quantum
fluctuations)
Eventually, condensate will be repaired and
particles will fly apart with a frozen momentum
distribution.
We like to formulate a quantum transport theory
to describe the final stages.
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Our Physical Picture Particle excitation by
quantizing the fields
In the past described by Classical fields
(coherent state)
Our formalism put incoherent particle
excitations by quantizing the fields
time-evolution
We develop a quantum kinetic theory of the chiral
condensate and meson excitations using the O(N)
linear sigma model. It is designed to describe
the chiral phase transition both in and out of
equilibrium in a unified way.
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Our Physical picture BEC and Chiral Phase
Transition
Bose-Einstein Condensation in magnetically
trapped dilute gases of alkali-metal atoms
Chiral Phase Transition in relativistic heavy
ion collisions
a gas of elementally excitations Perhaps,
quarks and gluons
Normal gas of excited atoms
Kinetic equation for quantum fluctuations
Kinetic equation for quantum fluctuations
Bose-Einstein Condensate
Chiral condensate
Classical Field Equation for condensate
Classical Field Equation for condensate
Describe space-time evolution of BEC coupled with
Quantum Fluctuations
How quantum fluctuations effect the formation of
chiral condensate?
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Outline of the rest of this talk
I. Derivation of Coupled Equations
II. Uniform Equilibrium
III. Dispersion Relations solutions in
linearized approx. around uniform equilibrium
IV. Open problems How the system time-evolve?
I. derivation of coupled equations to
describe evolution of non-equilibrium systems
Quantum kinetic equation for particle
excitations
Classical field equation for chiral
condensate
Coupled Eqn.
IV. time-evolution of final stages of NN
coll. Numerical solution with Vlasov code gtgtgt
still Working
III. dispersion relations solutions in
linearized approx. around uniform equilibrium
II. Apply to time-independent
equilibrium states
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Outline of Our Formalism
Hamiltonian of O(N) linear sigma model
Separate the fields into Condensate / Non-cond.
part
Statistical average with Gaussian density matrix
odd power gt 0
4th-power decoupled into the product of
2nd-powers
Equation of Motion for the mean field
Classical field equation (Klein-Gordon type)
for condensate
Equation of Motion for fluctuation in terms of
the Wigner functions
Quantum kinetic equation (Vlasov type) for
non-condensate (particle excitations) including
anomalous fluctuations(off-shell effect)
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Classical mean field equation N1
Modelphi4 model for quantized real scalar field
Hamiltonian
Heisenberg eq. of scalar field
Gaussian statistical average
Classical mean field equation
(cf. Gross-Pitaevskii eq for BEC)
?This equation includes the effects of quantum
fluctuation
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Quantum kinetic equationsGeneralized Wigner
functions N1
Define creation/annihilation operator
(µ quasi-particle mass)
Construct Wigner function (quantum version of
number density distribution in phase space)
Equation of Motion for F contains other Wigner
functions
? For a static uniform system, G,Gbar can be
eliminated by the Bogoliubov tr. (corresponds to
redefinition of particle mass) .
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Kinetic Equation for f(p,r,t)ltaagt N1
Equation for f(p,r,t) in long wavelength limit
(quantum Vlasov eq.)
generalized Landau kinetic equation
(collisionless)
Fluctuation of meson self-energy which is not
included in the particle mass
Quasi-particle energy
Mean Field potential
Relativistic drift term including the effect of
local change of particle mass
Vlasov term due to continuous acceleration
generated by the gradient of mean field potential
U
r.h.s sink/source terms due to the local
fluctuation of particle mass can not be
eliminated for a nonuniform system.
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Kinetic Equation for the Wigner function
g(p,r,t)ltaagt N1
Equation for the Wigner function gltaagt
(describing pair creation/annihilation of meson
fields off-shell effect)
no drift/Vlasov term for g purely quantum
mechanical origin looks more like an equation
of a simple ocsillator with frequency 2e
Rapid oscillation between particle and
anti-particle
r.h.s Matter distribution f(p,r,t) disturbs
the oscillation as external perturbation
? If the system is static and uniform, Up0
? f and g are decoupled by Bogoliubov tr.
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Coupled Kinetic Equation N1
Classical mean field equation
Vlasov equation for f(p,r,t)ltaagt
Kinetic equation for g(p,r,t)ltaagt
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Extension to O(N) model
Extend one component model to multi component
model with continuous symmetry (Chiral symmetry
SU(2)L SU(2)R O(4) )
(i1N)
N classical field equations (non-linear
Klein-Gordon eq.)
Define creation/annihilation ops. construct
NN Wigner functions
NN kinetic eqs (Quantum Vlasov equations)
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Equilibrium States
Time-independent solution of Coupled equation for
O(N) (assuming only one component of the meson
field fc0 has non-vanishing expectation value in
equilibrium )
gap equations
Difficulties

• 1st order phase transition

- Always confronted with this problem when
using mean field approximation
II. Goldstone theorem is apparently violated.
(µ1?0 µ2?0)

• We will show later missing Goldstone mode can be
• found in the collective excitations of the
  system.

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Linearized Eqs. and Collective modes
Coupled non-linear eqs for condensates
particle excitations
linearization with respect to small deviations
from equilibrium solutions
(assuming only one component of the condensates
fc0 has non-vanishing expectation value in
equilibrium )
?Coupled linear eqs for these fluctuations
?N decoupled sets of fluctuations
sigma mode
(N-1) pionic modes
...
?Dispersion relations
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Solution near Equilibrium Dispersion relation
O(N) (Chiral limit)
Dispersion relation (s-like mode) in the
direction of condensate
Dispersion relation (p-like mode) in the
direction perpendicular to condensate
Gray area continuum of the sigma-like quasi-parti
cle excitations
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Solutions in and near equilibrium O(N) model
with explicit symm. breaking
Near equilirbrium
s-like mode
p-like mode
MM, TM arXiv0812.1853
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Concluding remarks

• Summary
• We have derived a coupled set of equations for
  quantized self-interacting real scalar field
  (O(N) linear sigma model) containing equations
  for classical mean field and Vlasov equations for
  particle excitations.
• We have solved the equations in static
  equilibrium.
• We have found that anomalous fluctuation gltaagt
  excludes tachyon.
• We have studied dispersion relation of
  excitations and found sigma-like mode and pi-like
  massless modes corresponding to the
  Nambu-Goldstone bosons.
• Open problems
• We have to solve a coupled kinetic equation in
  the case of out of equilibrium numerically.
• Non-hydrodynamic collective flow may be
  generated by acceleration by pionic mean field
  gradient .
• HBT 2 pion particle correlation

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