Vlasov Equation for Chiral Phase Transition

1. T Quark-Gluon Plasma Chiral Phase Transition Hadron CSC μ Vlasov Equation for Chiral Phase Transition M. Matsuo and T. Matsui Univ. of Tokyo ,Komaba [ /14]

2. 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. [ /14]

3. 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 [ /14]

4. 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 Classical field equation for chiral condensate Quantum kinetic equation for particle excitations Coupled Eqn. II. Apply to time-independent equilibrium states III. dispersion relations: solutions in linearized approx. around uniform equilibrium IV. time-evolution >> Sorry! Now Working! [ /14]

5. Our Formalism Heisenberg Equation of Motion for quantum fields *Separate the fields into Condensate / Non-cond. part *Statistical average with Gaussian density matrix *odd power => 0 *4th-power decoupled into the product of 2nd-powers Equation of Motion for the mean field Classical field equation for condensate Equation of Motion for fluctuation in terms of the Wigner functions Quantum kinetic equation for non-condensate (particle excitations) [ /14]

6. A simple model: phi^4 model *Model:phi^4 model for quantized real scalar field *Hamiltonian: *Heisenberg eq. of scalar field: Gaussian statistical average Classical mean field equation (“Non-linear Klein-Gordon eq.) ★This equation includes the effects of quantum fluctuation [ /14]

7. 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]

8. Quantum Kinetic Equation for <a+a> *Equation for f(p,r,t) in long wavelength limit (quantum Vlasov eq.) l.h.s: Landau kinetic equation 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. [ /14]

9. Kinetic Equation for the Wigner function g *Equation for the Wigner function g=<aa> * 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” r.h.s: Matter distribution f(p,r,t) disturbs the oscillation as “external perturbation “ ★ If the system is static and uniform,Up=0  F and G are decoupled by Bogoliubov tr. [ /14]

10. Extension to O(N) model Highly non-linear eas. Extend 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) [ /14]

11. Dispersion relations Equilibrium States Time-independent solution of Coupled equation for O(2) (assuming only one component of the meson field φc0 has non-vanishing expectation value in equilibrium ): gap equations Difficulties • 1st order phase transition - Always confronted with this problem when using mean field approximation 0 T Tc 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. [ /14]

12. 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 φc0 has non-vanishing expectation value in equilibrium ) Coupled linear eqs for these fluctuations: N decoupled sets of fluctuations Dispersion relations: [ /14]

13. w=k • w=k w 1 2 2 1 1 1 2 2 k k 1 1 1 2 2 1 1 1 [N=2] Dispersion relations Dispersion relation (σ-like mode) in the direction of condensate No collective branch => Meson excitation Meson excitation Dispersion relation (π-like mode) in the direction perpendicular to condensate w Massless collective mode => Nambu-Goldstone boson Goldstone theorem is recovered! [ /14]

14. 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 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!! [ /14]

15. Bose-Einstein dist.