1. Vlasov Equation for Chiral Phase Transition
M. Matsuo and T. Matsui
Univ. of Tokyo ,Komaba
Hadron
CSC
Quark-Gluon Plasma T μ
Chiral Phase Transition
講演30分　 —09　理研WS　17:15-17:45
Thank you.(=>chairperson)
In this presentation, I would like to discuss our recent work on Vlasov equation for chiral phase transition.
This work is done in collaboration with Professor Matsui at University of Tokyo on Komaba Campus.
This work is aimed at describing chiral phase transition as shown in this canonical phase diagram
In the course of relativistic heavy ion collision.
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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
単位rapidityあたり１０００こ
I like to first discuss our physical picture how the chiral phase transition will take place in relativistic heavy ion collision.
First figure shows two highly Lorentz contracted nuclei approaching toward each other
at the velocity of light.
After two nuclei pass each other, the region behind two nuclei is a highly excited and contains quark-gluon plasma.
And in this region, chiral symmetry is restored and vacuum condensate will melt away.
As system expands, the quark gluon plasma will hadronize and the chiral symmetry will be broken spontaneously again.
In the intermediate state, we expect to have growing chiral condensate shown by light blue region filled with particle excitations.
And eventually, vacuum condensate will be repaired and we have particle moving apart at a frozen momentum distribution
We like to formulate a transport theory to describe final stages of this transition in terms of effective quantum theory.
In the past, most of the works which were aimed to describe these final stages in terms of classical field equations,
But problem of these works is there are no particle excitations because the fields are not quantized.
We like to put particle excitations by properly quantizing the fields.
Now let me discuss more details of our formalism.
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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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
In the past, most of the works described this stage of evolution in terms of classical field equations,
But problem is there is no particle excitations.
We like to put additional incoherent particle excitations by quantizing the fields.
time-evolution
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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.
I first present the derivation of coupled equations to describe evolution of non-equilibrium systems.
I will then apply this formalism to describe time-independent equilibrium states.
Next, we calculate dispersion relations of collective excitations.
The main purpose of this work is to describe time-evolution of final stages of nucleus-nucleus collision with realistic initial conditions.
We are still working on this part of the program.
Unfortunately, we are not able to give you results of such calculation today.
II. Apply to
time-independent
equilibrium states
III. dispersion relations:
solutions in linearized
approx. around uniform
equilibrium
IV. time-evolution
>> Sorry! Now Working!
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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
We start with
Separate into two
Expectation value are taken by statistical average with Gaussian density matrix,
Satisfying these relations
Then, we get eq of motion for the mean field phi_c corresponding to classical field equation for condensate,
And equation of motion for fluctuation …
Equation of Motion
for fluctuation
in terms of the Wigner functions
Quantum kinetic equation
for non-condensate
(particle excitations)
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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.)
We would like to present our formalism,
We start with phi 4 model for quantized real scalar field.
Hamiltonian is like this, and these field operators are canonically quantized.
Then, we separate the meson field into two parts:
“classical mean field” part
and
“quantum fluctuation” part.
Classical mean field is defined by the expectation value of the field operator
And
Quantum fluctuation is defined by this fomula.
After this separation,
we derive coupled equations containing
classical field equation for condensate
kinetic equations for quantum fluctuation
★This equation includes the effects of quantum fluctuation
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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, we can eliminate G and G bar by the Bogoliubov tr.
And this transformation is equivalent to mass renormalization.
Generally, however, we can not remove G and Gbar .
This B. tr corresponds to redefinition of particle mass including effect of interaction.
The system we aim to describe is not static and uniform.
★ For a static uniform system,
G,Gbar can be eliminated by the Bogoliubov tr. (corresponds to redefinition of particle mass) .
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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
Quasi-particle energy
Mean Field potential
Fluctuation of meson self-energy
which is not included in the particle mass
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
l.h.s is equivalent to the Landau kinetic equation if you choose quasi-particle energy in this way.
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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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”
Equation of motion of g looks very different from Vlasov equation.
It has no drift term and no Vlasov term.
It looks more like an equation of a simple oscillator with frequency twice the particle energy.
Matter distribution appears on the right hand side and this will disturb the oscillation as external perturbation.
This equation may be interpreted as describing particle anti-particle oscillation.
This is purely relativistic quantum mechanical effect.
local fluctuation
If the system is static and uniform, delta Pi can be eliminated by Bogoliubov transformation and
F and G are decoupled.
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.
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10. 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) )
(i＝1～N)
Highly non-linear eas.
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)
It is strait forward to extend what we have shown in the case of one component scalar field model
to more realistic multi component scalar field model with continuous symmetry to discuss chiral symmetry
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11. 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
Violation of Goldstone’s theorem will be remedied later by the calculation of excitation of the system.
We will show later missing Goldstone mode can be found in the collective excitations of the system. T
II. Goldstone theorem is apparently violated.
(μ1≠0 ＆ μ2≠0) Tc
We will show later missing Goldstone mode can be
found in the collective excitations of the system.
Dispersion relations
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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
We have derived coupled equations of motion for classical fields and the Wigner functions.
These are highly non-linear equations,
And they can be solved for specific initial conditions by numerical integration
Here we like to first study solutions of these equations by linearizing equations with respect to small deviations from a uniform equilibrium solution.
We assume only one component of the meson field has non-vanishing expectation value in equilibrium.
Also, equilibrium particle distribution gives non-vanishing value only for diagonal parts of the Wigner functions.
And we add fluctuations for each components of the classical fields and Wigner functions.
★We get very large coupled linear differential eqs which can be decomposed into N-decoupled sets of fluctuations.
Each sets of fluctuations has characteristic mode of oscillation and corresponding dispersion relation.
Delta phi1 is the fluctuation in the direction of the condensate which coupled with the fluctuation of diagonal components of Wigner functions.
On the other hand, the fluctuation in the direction perpendicular to the condensate couples with fluctuations in the off-diagonal components of the Wigner functions.
Now, I like to show you the dispersion relation I calculated for each mode in the case of O(2) model.
We can obtain dispersion relation of collective modes
Dispersion relations:
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13. [N=2] Dispersion relations
Dispersion relation (σ-like mode)
in the direction of condensate
No collective branch
=> Meson excitation k w
w=k 1 2
Meson excitation
Dispersion relation (π-like mode)
in the direction perpendicular to condensate k
w=k w
Massless collective mode
　=> Nambu-Goldstone boson
Goldstone theorem recovered!!
Goldstone theorem
is recovered! 2 1
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14. Thank you very much indeed for your kind attention!!
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.
Now, let me summarize my talk.
We have derived a coupled set of equations for self-interacting real/complex scalar model
containing
>>equation for classical mean field
>>Vlasov equations for the Wigner functions (one-body distribution functions) which originate from quantum fluctuation of meson fields.
And then
We have
We found instead an uninvited tachyonic mode in the…
And,we have many open problems :
We have to improve model with continuous symmetry (related to NG boson),
And
Include collision term to analyze thermalization problem,
Finally we wish to apply this theory to simulate heavy ion collision
and look for some signals of phase transition.
Thank you.
Thank you very much indeed for your kind attention.
Thank you very much indeed for your kind attention!!
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15. Bose-Einstein dist.