The 2PI-loop expansion revisited
Anders Tranberga,b ,
in collaboration with
Alejandro Arrizabalagaa,c and Jan Smita ,
a.) University of Amsterdam,
b.) University of Sussex,
c.) NIKHEF, Amsterdam.
hep-ph/0503287→ Phys.Rev.D
Quantum fields out of equilibrium
Out of equilibrium, the density matrix and operator expectation
values evolve in time, (Z = T r[ρ] = 1)
hO(t)iρ = T r[ρ(t)O] = T r[ρ(0) e−iHt O eiHt ].
We can write this as a path-integral, as an evolution along a path
in complex t-space,
Im t
Schwinger−Keldysh contour
C+
C
Re t
0
C−
t
Z
Z[ρ, C, Ji , K] = Dφ+ Dφ− hφ+ |ρ(0)|φ− i
Z
1
,
× Dφ exp i SC [φ] + J φ + φ K φ
2
1
SC [φ] + Jφ + φKφ =
2
Z
C
dt
Z
1
d3 x[L(φ) + J± φ + φKφ].
2
The currents J± and K live on upper (+)/lower (-) branches, and
φ(x, 0+ ) = φ+ , φ(x, 0− ) = φ− . The initial condition in
hφ+ |ρ(0)|φ− i can be included in the currents.
Approximate quantum dynamics
Φ-derivable approximations defined in terms of truncated loop
expansions of the 2PI effective action provide a framework for
approximating the quantum dynamics (in and) out of equilibrium.
Equations of motion are derived in terms of the 1- and 2-point
functions (the mean field φ̄ and the propagator G). The derived
equations of motion have to be solved numerically, by discretizing
the field on a lattice.
[Berges&Cox:2000, Aarts&Berges:2001, Berges:2002, Aarts et al.:2002,
Cassing et al.:2003, Berges et al.:2004], thermalization for scalar fields
in 1+1, 2+1 and 3+1 dimensions, [Berges et al:2003] fermions and
scalars in 3+1 dimensions. [Berges&Serreau:2003, Arrizabalaga et
al.(AT):2004] for studies of preheating. For gauge fields, see
[Arrizabalaga&Smit:2003, Berges:2004]. [van Hees&Knoll:2002, Blaizot
et al.:2003, Berges et al.:2005], renormalization].
2PI effective action
Z[J, K]
φ̄(x) =
=
Z
δW [J,K]
δJ(x) ,
i S[φ]+i J φ+ 2i φ K φ
dφ e
G(x, y)
=2
= ei W [J,K] ,
δW [J, K]
− φ̄(x)φ̄(y).
δK(x, y)
Define:
Γ
2P I
Then:
δΓ
1
[φ̄, G] = W [J] − J φ̄ − K G + φ̄φ̄ .
2
2P I
[φ̄, G]
δ φ̄(x)
δΓ2P I [φ̄, G]
δG(x, y)
= −J(x) −
Z
1
= − K(x, y).
2
d4 y K(x, y)φ̄(y),
Φ-derivable approximations
The 2PI effective action can be written as [Cornwall et al :1974]:
Γ
2P I
i
i
[φ̄, G] = S[φ̄] − Tr ln G + Tr G−1
0 G + Φ[φ̄, G],
2
2
with
iG−1
0 (x, y)
δ 2 S[φ̄]
=
δ(x − y).
δ φ̄(x)δ φ̄(y)
Φ is the sum of all 2PI skeleton diagrams with no external lines,
full propagators and bare vertices. A choice of truncation of this
I
series Φtr → Γ2P
constitutes a Φ-derived approximation.
tr
Equations of motion derived from any truncation
I
δΓ2P
tr [φ̄, G]
= 0,
δG(x, y)
conserve an energy functional.
I
δΓ2P
tr [φ̄, G]
=0
δ φ̄(x)
Equations of motion
The equations of motion for a homogeneous system :
2
(t)F (t, t0 , x)
∂t2 F (t, t0 , x) = ∂i2 F (t, t0 , x) − Meff
Z t
+
dt00 dy Σρ (t, t00 , y)F (t00 , t0 , x − y)
0
−
Z
t0
dt00 dy ΣF (t, t00 , y)ρ(t00 , t0 , x − y),
0
2
∂t2 ρ(t, t0 , x) = ∂i2 ρ(t, t0 , x) − Meff
(t)ρ(t, t0 , x)
Z t
+
dt00 dy ΣF (t, t00 , y)ρ(t00 , t0 , x − y),
t0
2
∂t2 φ̄(t) = − Meff
(t)φ̄(t) −
Z
t
dt00 dy Σφ (t, t00 , y)φ̄(t00 ).
0
with F = h{φ, φ}i, ρ = ih[φ, φ]i. These are the Kadanoff-Baym
equations (if non-relativistic) [Kadanoff&Baym:1962].
φ4 -model
We study a single real scalar field in a φ4 potential:
Z
Z
2
µ
∂µ φ ∂ φ m 2
λ
+
φ + φ4 .
S = − dt d3 x
2
2
24
In the symmetric phase m2 > 0, hφi ' 0 and in the broken phase
p
2
m < 0, hφi ' 6|m2 |/λ.
The functional Φ
The functional Φ in a loop expansion is
1
Φ[φ̄, G] =
4!
1
+
12
1
+
2
+
1
+
8
1
48
+ ...
Motivation
• Thermalization for scalar fields in 3+1d.
• Most 2PI studies compare O(λ) (Hartree) to O(λ2 )
(Basketball) or LO-1/N to NLO-1/N. hφi = 0.
• Hartree/LO-1/N: Leading contribution to ReΣ.
Basketball/NLO-1/N: Next-to-Leading contribution to ReΣ.
• Basketball/NLO-1/N: Leading contribution to ImΣ. No results
exist at NNLO. Numerically Intractable(?).
• The ImΣ is related to decay/damping rates, equilibration
times.
6 0 induces three point vertex, and an additional diagram
• hφi =
at O(λ). Perturbatively, ImΣ = 0 (on shell). 2PI-resummed?
• Three point vertex present in non-abelian gauge theories.
Initial conditions
We apply two initialization schemes, Top-hat and Thermal,
determined by a choice for the distribution function nk and the
dispersion relation ωk , through:
nk + 1/2
,
hφk φ−k i =
ωk
hπk π−k i = (nk + 1/2)ωk ,
• Top-hat:
nk = 0;
nk = c > 0,
kmin < k < kmax ,
for some c > 0.
• Thermal:
nk = (eωk /T − 1)−1 .
In both cases, ωk2 = k 2 + m2 .
What equilibrium looks like
For small coupling we expect the equilibrium state to have a
quasi-particle dispersion relation and a Bose-Einstein particle
distribution. We will define (also out of equilibrium)
p
ωk = hπk π−k i/hφk φ−k i,
p
nk + 1/2 = hπk π−k ihφk φ−k i.
When plotting ωk2 vs. k 2 the slope is c2 and the intercept
c2 m2eff (T ). In a plot of ln (1 + 1/nk ) vs. ωk , the slope is T −1 and
the intercept −µch /T .
3
25
2
ln(1+1/nk)
2
10
2
c
2
15
ωk /m
20
T
0
5
2
c meff
0
0
1/
1
5
2
- µch/T
-1
10
15
2
2
k /m
20
25
0
1
2
ωk/m
3
4
Φ=
2
φ=0
O( 0 ) O( 0 ) O( λ ) O( 0 ) O( λ )
−1/2
−1
φ=λ
O( λ ) O( 1 ) O( λ ) O( λ ) O( λ2 )
dΦ/dφ
2
Ο(δ ,1) Ο(λ)
d Φ/dG
Ο(0,1)
Ο(λ)
2
Ο(λ )
2
Ο(0,λ) Ο(λ )
Symmetric phase, including basketball
5
4
3
2
1
0
0
5
4
3
2
1
0
0
5
4
3
2
1
0
0
1
1
1
5
mt = 7
4
3
2
1
0
2
3
4
0
5
mt = 28
4
3
2
1
0
2
3
4
0
5
mt = 84
4
3
2
1
0
2
3
4
0
1
1
1
5
mt = 14
4
3
2
1
0
2
3
4
0
5
mt = 42
4
3
2
1
0
2
3
4
0
5
mt = 280
4
3
2
1
0
2
3
4
0
mt = 21
1
2
3
4
mt = 56
1
2
3
4
mt = 476
1
2
3
4
Symmetric phase, including basketball
30
25
20
15
10
5
0
0
30
25
20
15
10
5
0
0
30
25
20
15
10
5
0
0
30
25
mt = 7
20
15
10
5
0
5 10 15 20 25 30 0
30
mt = 28
25
20
15
10
5
0
5 10 15 20 25 30 0
30
25
mt = 84
20
15
10
5
0
5 10 15 20 25 30 0
30
25
mt = 14
20
15
10
5
0
5 10 15 20 25 30 0
30
25
mt = 42
20
15
10
5
0
5 10 15 20 25 30 0
30
25
mt = 280
20
15
10
5
0
5 10 15 20 25 30 0
mt = 21
5 10 15 20 25 30
mt = 56
5 10 15 20 25 30
mt = 476
5 10 15 20 25 30
Kinetic equilibration
4
T/m = 1.0
ln(1+1/nk)
3
T/m = 1.36
T1
T2
T3
T/m = 1.93
2
T/m = 2.86
1
0
0
1
2
3
ωk/m
4
5
6
7
Kinetic equilibration
3.5
2.5
2
ω /m
2
3
2
T1
T2
T3
Thermal, T/m = 1.36
Thermal, T/m = 1.93
Thermal, T/m = 2.86
1.5
1
0
0.5
1
2
k /m
2
1.5
2
Chemical equilibration
2
MH (T)/m
2
1.4
2
M (T)/m
2
T/m
1.2
1
µch/m
0.8
0
2000
4000
mt
6000
8000
Φ=
2
φ=0
O( 0 ) O( 0 ) O( λ ) O( 0 ) O( λ )
−1/2
−1
φ=λ
O( λ ) O( 1 ) O( λ ) O( λ ) O( λ2 )
dΦ/dφ
2
Ο(δ ,1) Ο(λ)
d Φ/dG
Ο(0,1)
Ο(λ)
2
Ο(λ )
2
Ο(0,λ) Ο(λ )
Mean field damping
0.1
0.05
φ(t) 0
T/m = 0.0
T/m = 1.43
T/m = 2.86
-0.05
-0.1
0
100
mt
200
Mean field damping
0.02
Continuum, perturbative
Mean field damping, Basketball
Mean field damping, Two-loop
Zero mode damping, Basketball
Lattice, perturbative
Γ /m
0.01
0
0
0.5
1
T/m
1.5
2
Mean field damping
1.25
M(T)/m
1.2
Continuum gap equation
Lattice gap equation
Zero mode mass, Basketball
Mean field mass, Two-loop
Mean field mass, Basketball
1.15
1.1
1.05
1
0
0.5
1
T/m
1.5
2
Φ=
2
φ=0
O( 0 ) O( 0 ) O( λ ) O( 0 ) O( λ )
−1/2
−1
φ=λ
O( λ ) O( 1 ) O( λ ) O( λ ) O( λ2 )
dΦ/dφ
2
Ο(δ ,1) Ο(λ)
d Φ/dG
Ο(0,1)
Ο(λ)
2
Ο(λ )
2
Ο(0,λ) Ο(λ )
Broken phase
5
4
3
2
1
0
0
5
4
3
2
1
0
0
5
4
3
2
1
0
0
1
1
1
5
mt = 28
4
3
2
1
0
2
3
4
0
5
mt = 196
4
3
2
1
0
2
3
4
0
5
mt = 364
4
3
2
1
0
2
3
4
0
1
1
1
5
mt = 84
4
3
2
1
0
2
3
4
0
5
mt = 252
4
3
2
1
0
2
3
4
0
5
mt = 420
4
3
2
1
0
2
3
4
0
mt = 140
1
2
3
4
mt = 308
1
2
3
4
mt = 476
1
2
3
4
Broken phase
30
30
30
25 mt = 28
25 mt = 84
25 mt = 140
20
20
20
15
15
15
10
10
10
5
5
5
0
0
0
0 5 10 15 20 25 30 0 5 10 15 20 25 30 0 5 10 15 20 25 30
30
30
30
25 mt = 196
25 mt = 252
25 mt = 308
20
20
20
15
15
15
10
10
10
5
5
5
0
0
0
0 5 10 15 20 25 30 0 5 10 15 20 25 30 0 5 10 15 20 25 30
30
30
30
25 mt = 364
25 mt = 420
25 mt = 476
20
20
20
15
15
15
10
10
10
5
5
5
0
0
0
0 5 10 15 20 25 30 0 5 10 15 20 25 30 0 5 10 15 20 25 30
Equilibration
6
ln(1+1/nk)
5
T1: Basketball
T1: Two-loop
T/m = 1.0
4
T/m = 1.36
3
T/m = 1.93
2
T/m = 2.86
1
0
0
1
2
3
ωk/m
4
5
6
7
( φ (t)-v)/( φ (0)-v)
Mean field damping
1.7
1.6
T/m = 2.14, Basketball
T/m = 2.14, Two-loop
1.5
1.4
1.3
0
50
mt
100
150
Mean field damping
3
2.5
2
1.5
1
Meff(T)/m
200 Γ (T)/m
v(T)/m
0.5
0
0
1
2
T/m
3
4
Conclusion
• In the symmetric phase, ReΣ(λ2 )/ReΣ(λ) ' 0.3 at λ = 6.
Expansion will break down for much larger λ. Renormalization.
• ImΣ(λ2 ) is slightly larger than non-resummed analytical
calculations.
• In the broken phase, resummation introduces ImΣ 6= 0 at O(λ).
Qualitatively different from perturbation theory.
• Equilibration time: O(λ2 )/O(λ) ∼ O(1).
• O(λ) does not dominate O(λ2 ). Probably because first level of
resummation is O(λ4 v 4 ) = O(λ2 ).