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Vlasov Equation for Chiral Phase Transition

M. Matsuo and T. MatsuiUniv. of Tokyo ,Komaba

Hadron CSC

Quark-Gluon Plasma

T

μ

Chiral Phase Transition

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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 and particle excitations

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) (ref. Asakawa, Minakata, Muller)

Our formalism: put incoherent particle excitations by quantizing the fields

time-evolution

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Quantum kinetic equation for particle excitations

Classical field equation for chiral condensate Coupled Eqn.

Outline of the rest of this talkI. 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?

IV. time-evolution >> Sorry! Now Working!

III. dispersion relations: solutions in linearized approx. around uniform equilibrium

II. Apply to time-independent equilibrium states

I. derivation of coupled equations to describe evolution of non-equilibrium systems

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Our FormalismHeisenberg Equation of Motion for quantum fields

Equation of Motion for the mean field

Equation of Motion for fluctuationin terms of the Wigner functions

Quantum kinetic equation for non-condensate (particle excitations)

Classical field equation for condensate

*Statistical average with Gaussian density matrix

*Separate the fields into Condensate / Non-cond. part

*odd power => 0

*4th-power decoupled into the product of 2nd-powers

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A simple model: phi^4 model

*Hamiltonian:

*Model:phi^4 model for quantized real scalar field

Classical mean field equation (“Non-linear Klein-Gordon eq.)

*Heisenberg eq. of scalar field:

★This equation includes the effects of quantum fluctuation

Gaussian statistical average

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Wigner function … ～ Quantum Kinetic Equations～

*Define creation/annihilation operator:

(μ: physical 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) . [ /14]

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Quantum Kinetic Equation for <a+a>

l.h.s: Landau kinetic equation

r.h.s: sink/source terms due to the local fluctuation of “particle mass” can not be eliminated for a nonuniform system.

*Equation for f(p,r,t) in long wavelength limit (quantum Vlasov eq.)

Fluctuation of meson self-energy which is not included in the particle mass

Quasi-particle energy

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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

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Kinetic Equation for the Wigner function g

* no drift/Vlasov term for g* purely quantum mechanical origin* looks more like an equation of a simple ocsillator with frequency 2ε

Rapid oscillation between particle and “anti-particle”

*Equation for the Wigner function g=<aa>

★ If the system is static and uniform, 　 Up=0 F and G are decoupled by Bogoliubov tr.

r.h.s: Matter distribution f(p,r,t) disturbs the oscillation as “external perturbation “

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Extension to O(N) modelExtend one component model to multi component model with continuous symmetry (Chiral symmetry SU(2)L × SU(2)R 　 ～ 　 O(4) )

(i＝ 1～ N)

N classical field equations (non-linear Klein-Gordon eq.)

*Define creation/annihilation ops. & construct N×N Wigner functions:

N×N kinetic eqs (Quantum Vlasov equations)

Highly non-linear eas.

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Equilibrium StatesTime-independent solution of Coupled equation for O(2)(assuming only one component of the meson field φc

0 has

non-vanishing expectation value in equilibrium ):

Difficulties

T0Tc

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II. Goldstone theorem is apparently violated. (μ1≠0 ＆ μ2≠0)

I. 1st order phase transition

- Always confronted with this problem when using mean field approximation

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

gap equations

Dispersion relations

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*linearization with respect to small deviations from equilibrium solutions

Linearized Eqs. and Collective modes

Coupled linear eqs for these fluctuations:

N decoupled sets of fluctuations

Dispersion relations: [ /14]

Coupled non-linear eqs for condensates & particle excitations

(assuming only one component of the condensates φc

0 has non-vanishing expectation value in equilibrium )

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1

1

1

1 2

21

1

[N=2] Dispersion relations

Dispersion relation (π-like mode) in the direction perpendicular to condensate

k

•w=kw

Massless collective mode 　 => Nambu-Goldstone boson

2 2 21

1 1 12

k

w•w=k

Meson excitation

Dispersion relation (σ-like mode) in the direction of condensate

Goldstone theorem is recovered!

No collective branch => Meson excitation

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SUMMARY

* We have derived a coupled set of equations for quantized self-interacting realscalar field (O(N) linear sigma model) containing equations for classical mean field and Vlasov equations for particle excitations.

* We have studied dispersion relation of excitations and found σ-like mode with mass and π-like massless modes corresponding to the Nambu-Goldstone bosons.

Open problems:* We have to solve the equations with more realistic initial condition for final stages of evolution of nucleus-nucleus collision.

* Non-hydrodynamic collective flow may be generated by acceleration by mean field gradient .

* This formalism gives a consistent framework for studying these problems.

Thank you very much indeed for your kind attention!!

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•Bose-Einstein dist.