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

× × ×

△ △ ?

?○ ○

Time evolution of number density

For quasiparticle approximation

0⇔4

1⇔3

2⇔2

3⇔1

Microscopic processes

Spectral function

Coupling dependence

Self-energy

Schwinger-Dyson eqn

For perturbative Green’s functions

Imaginary part

contributes to the particle number changing process g⇔gg

O(N): 1/N expansion

LO O(N)

NLO O(1)

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

0 1 2 3 4 5 6 7 8

mX^0=7.5mX^0=15.0mX^0=22.5mX^0=30.0mX^0=37.5

O(N): spectral function