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Thermalization of scalar and gauge theory with off-shell entropy
Akihiro NishiyamaUniversity of TokyoInstitute of Physics
基研研究会「熱場の量子論とその応用」 2009年9月4日
τ<0 fm/cCGC
√sNN=130, 200 GeV
Glasma
0<τ<0.6~1.0fm/c
Kinetic freezeout
Hadronizationτ~10 fm/c
Figures from P. SORENSEN
QGP τ>1.0fm/c
Background
Observation
1fm/c0 fm/c 10fm/c 15fm/cBlack box
hydro+ hadron gas
HydrodynamicsBefore collision
g
g
g
gg
g
g
g
g
g
g
gg
RHIC experiments
Nonequilibrium dynamics of gluons
Introduction for the kinetic entropy based on Kadanoff-Baym equation
Proof for the H-theorem for scalar O(N) Φ4 theory.
Study for the non-Abelian gauge theory
Introduction for the kinetic entropy based on Kadanoff-Baym equation
Proof for the H-theorem for scalar O(N) Φ4 theory.
Study for the non-Abelian gauge theory
Introduction for the kinetic entropy based on Kadanoff-Baym equation
Proof for the H-theorem for scalar O(N) Φ4 theory.
Study for the non-Abelian gauge theory
Topics in nonequilibrium gluodynamics• Early thermalization for Partons
2-3fm/c (Perturbative Analysis<0.6-1fm/c (Exp.)
gg→gg, gg→gggHydrodynamics
Purpose of this talkPurpose of this talk
Yang Mills equation‘Soft’: field
Vlasov-Boltzmann equation‘Hard’ :parton
However
Dense system (Boltzmann eq. should not be applied)
No consideration of particle number changing process g→gg, g→ggg
(Off-shell effect)
Time Contour
Schwinger-Dyson equationSchwinger-Dyson equation
in terms of statistical (distribution) and spectral functionsKadanoff-Baym equationKadanoff-Baym equation
For a free field
Breit-Wigner typeBoson
in Closed-Time Path formalism2. Kadanoff-Baym eqn
2 Particle-Irreducible Effective Action Γ2 Particle-Irreducible Effective Action Γ
Mean field is omitted.
Merit• Spectral function
Time evolution of spectral function + distribution function
• Off-shell effectWe can trace partons which are unstable by its particle number changing process in addition to collision effects. we can extract gg→g (2 to 1) and ggg→g (3 to 1) and the inverse prohibited kinematically in Boltzmann simulation. This process might contribute the early thermalization.
(Why do we use KB eq, not Boltzmann eq?)
Σ=Self-energies Memory integral
The Kadanoff-Baym equation:Time evolution of statistical (distribution) and spectral function
• O(N) theory with no condensate <φa>=0• Homogeneous in space • 1+1 dimensions • Next Leading Order in 1/N expansion
• O(N) theory with no condensate <φa>=0• Homogeneous in space • 1+1 dimensions • Next Leading Order in 1/N expansion
Scalar theory as a toy model
• φ4 theory with no condensate <φ>=0• Homogeneous in space • 1+1 dimensions • Next Leading Order of coupling
• φ4 theory with no condensate <φ>=0• Homogeneous in space • 1+1 dimensions • Next Leading Order of coupling
O(λ2) LO
NLO
Application for BEC, Cosmology (or reheating) and DCC dynamics?
+・・・
+・・・
Entropy in Rel. Kadanoff-Baym equation• Nonrel. case: Ivanov, Knoll and Voskresenski (2000), Kita (2006)• The first order gradient expansion of the Schwinger-Dyson equation.
For NLO λ2 (Φ4)
In the quasiparticle limit We reproduce the entropy for the boson.
[ ] Entropy flow spectral function
H-theorem needs not to be based on the quasiparticle picture.
For NLO of 1/N (O(N))
Sketch of H-theorem (O(N))Φ4Φ4
~λ2
×(x-y)log(x/y) ≧ 0
O(N)O(N)
+・・・ =
=
+
+
+
+ +
1/N
1/N
1/N 1/N 1/N
1/N
× (x-y)log(x/y) ≧ 0×
Coupling expansion
1/N expansion
=
Self-energy
Evolution of kinetic entropy (Φ4)N
umbe
r den
sity
Entropy density
Ratio of two entropy
Total number density
N/V=∫p np
λ/m2=4 1+1
∫p(1+n)log(1+n)-nlog n
No thermalization for the Boltzmann eq.
Off-shell
1.8
1.85
1.9
1.95
2
2.05
2.1
2.15
2.2
2.25
0 10 20 30 40 50 60 70 80
1.5
2
2.5
3
3.5
4
0 10 20 30 40 50 60 70 80
QPOS
0
1
2
3
4
0 5 10 15 20 25 30 35 40
mX^0=0.075mX^0=7.5
mX^0=15.0mX^0=30.0mX^0=60.0
mX^0=120.0
Evolution of kinetic entropy (O(N))λ/m2=40, N=10 , 1+1 dim
Num
ber d
ensi
ty
px
Entropy density
mX0
Total Number density
N/V=∫p np
mX0
1
1.05
1.1
1.15
1.2
1.25
1.3
1.35
0 10 20 30 40 50 60 70 80
Ratio
Ratio of two entropy
S(OS)/S(QP)
No thermalization for the Boltzmann eq. in 1+1 dimensions.
Non-Abelian Gauge Theory• Controlled gauge dependence of effective
action (Smit and Arrizabaraga (2002), Carrington et al (2005) )
Truncated effective actionTruncated effective action
Gauge invariant Green’s function
ExactExact
Expansion of coupling of self energy
Self-energy
Under gauge transformationUnder gauge transformation
⇔ Schwinger-Dyson equation
at ⇔
Stationary point
Higher order gauge dependence⇒ Energy, Pressure and Entropy derived from δΓ/δT has controlled gauge dependence. Gauge invariance is reliable in the truncated order.
⇒ Gauge invariant Energy, Pressure and Entropy derived from δΓ/δTThis might not the entropy in the previous page.
Nielsen (1975)
4. Summary and Remaining Problems• We have introduced the kinetic entropy based on the
Kadanoff-Baym equation.• The kinetic entropy satisfies H-theorem for NLO of
λ(Φ4) and 1/N (O(N)).• S(OS)/S(QP) is nearly constant, but S(QP) tends to be
affected by the total number density.• Gauge dependence is controlled in deriving
thermodynamic variables (energy, pressure and entropy and so on).
• Longer time simulation in the O(N) case• Thermal solution for the SD eq. for the LO of g2 for the gauge theory (2+1
dimensions)• H-theorem for the gauge theory, gauge invariance of the entropy
Proof of H-theorem (O(N))Φ4Φ4
~λ2
×(x-y)log(x/y) ≧ 0
O(N)O(N)
+・・・
= =
+ +
++
+
+ +
1/N1/N
1/N 1/N
1/N 1/N
× (x-y)log(x/y) ≧ 0×
Coupling expansion
1/N expansion
Time irreversibility
Exact 2PI (no truncation)
Truncation
LO of Gradient expansionH-theorem
λΦ4 O(N) SU(N)
NLO of λ NLO of 1/N LO of g2
Symmetric phase 〈Φ〉=0
× × ×
△ △ ?
?○ ○
Self-energy
Schwinger-Dyson eqn
For perturbative Green’s functions
Imaginary part
contributes to the particle number changing process g⇔gg