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0 NONEQUILIBRIUM DYNAMICS OF THE O(N) LINEAR
2
SIGMA MODEL IN THE HARTREE APPROXIMATION
n
a
J
7 S.MICHALSKI
1
Institut fu¨r Physik
Theoretische Physik III
1
Otto-Hahn-Str. 4
v
4 D–44221 Dortmund, Germany
3 E-mail: [email protected]
1
1
Weinvestigatetheout-of-equilibriumevolutionofaclassicalbackgroundfieldand
0
its quantum fluctuations in the scalar O(N) model with spontaneous symmetry
3
breaking 1. We consider the 2-loop 2PI effective action in the Hartree approx-
0
imation, i.e. including bubble resummation but without non-local contributions
/
h tothe Dyson-Schwinger equation. Weconcentrate onthe(nonequilibrium)phase
p structureofthemodelandobserveafirst-ordertransitionbetweenaspontaneously
- broken and a symmetric phase at low and high energy densities, respectively. So
p
typical structures expected inthermal equilibriumareencountered innonequilib-
e
riumdynamicsevenatearlytimesbeforethermalization.
h
:
v
i
X 1. The model
r 1.1. Applications
a
Scalar models have a wide range of applications in quantum field theory.
Normally they are parts of more complex models like e.g. the Standard
Model or Grand Unified Theories but they often serve as toy models for a
simplifieddescriptionofcomplexphenomenasuchasinflationarycosmology
or meson interactions in relativistic heavy ion collisions.
1.2. Nonequilibrium 2PI effective potential
We consider the O(N) model with spontaneous symmetry breaking whose
classical action is
1 λ 2
[Φ~]= d4x L[Φ~]= d4x ∂ Φ~ ∂µΦ~ Φ~2 v2 . (1)
µ
S Z Z (cid:26)2 · − 4 (cid:16) − (cid:17) (cid:27)
Following Refs. 1,2 we can compute the 2PI effective action 3 in the
Hartree approximation. Furthermore, we diagonalize the Green function
1
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by an O(N)-symmetric ansatz and by restricting the background field to
one direction. Since the Greenfunction is local, it canbe describedby two
(time-dependent) mass parameters 2 . For nonequilibrium purposes it
Mσ,π
can be factorized into mode functions
G (x,t ;x′,t )= d3k f (k,t )f¯ (k,t ) eik·(x−x′) , (2)
σ,π > < Z (2π)32ωσ,π σ,π > σ,π <
where ω = k2+ 2 (0). One constructs an expression for the total
σ,π Mσ,π
q
(conserved) energy density of the system in the Hartree approximation
1 1 λ v2
= φ˙2+ 2φ2 φ4 2 +(N 1) 2
E 2 2Mσ − 2 − 2(N +2)(cid:20)Mσ − Mπ(cid:21)
1
(N +1) 4 +3(N 1) 4 2(N 1) 2 2 (3)
−8λ(N +2)(cid:20) Mσ − Mπ− − Mσ Mπ
+2Nλ2v4 + σ(t)+(N 1) π(t) ,
(cid:21) Efl − Efl
where the fluctuation energy densities σ,π are the nonequilibrium analogs
Efl
of the one-loop “log det” terms expressed by mode functions f(k,t)
~ d3k
Efl∗(t)= 2 Z (2π)32ω∗ (cid:20)|f˙∗(k,t)|2+(k2+M2∗) |f∗(k,t)|2(cid:21), ∗=σ,π .(4)
1.3. Equations of motion
The equations of motion follow from the conservation of the energy (3).
The backgroundfield obeys
φ¨+ 2(t) 2λ φ2(t) φ(t)=0 , (5)
Mσ −
(cid:2) (cid:3)
the mass parameters are solutions of the gap equations
2 = λ 3φ2 v2+3~ +(N 1)~ (6)
Mσ (cid:16) 0− Fσ − Fπ(cid:17)
2 = λ φ2 v2+~ +(N +1)~ , (7)
Mπ (cid:16) 0− Fσ Fπ(cid:17)
where ∗ is the fluctuation integral
F
d3k
∗(t)= f∗(k,t)2 with =σ,π
F Z (2π)32ω∗| | ∗
which equals the usual tadpole integral at t = 0 (cf. section 1.4). The
equation for the mode functions is
f¨∗(k,t)+ k2+ 2∗(t) f∗(k,t)=0 . (8)
(cid:20) M (cid:21)
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Thefactthatthemodeequationiscoupledtothegapequations(6)and(7)
hasanimportantinfluenceonthedynamics. Whenatime-dependentmass
square 2(t) acquires a negative value, eq. (8) will imply an exponential
M
growth of the modes which reacts back on the mass squares via the gap
equations by drivingthem back to positive values. Inthe one-loopapprox-
imation (with no gap equations) the system shows unphysical behavior 5
because the modes never stop growing exponentially.
1.4. Initial conditions
At the beginning of the nonequilibrium evolution we fix the classicalback-
ground field to a certain value φ(0)=φ . The mode functions are those of
0
free fields: f (k,0)=1, f˙(k,0)= iω , and the mass parameters and
i i i σ
− M
are solutions of the gap equations (6) and (7) at t=0.
π
M
2. Results and discussion
2.1. Time evolutions
HerewewillpresenttheresultsforN =4,λ=1andfortherenormalization
scale set equal to the tree level sigma mass µ2 = 2λv2. We have only
consideredinitialvaluesφ >v forthebackgroundfieldbecauseforsmaller
0
values the initial value of the parameter (0) is imaginary. This means
π
M
that the region v <φ<v can only be explored dynamically. We display
−
time evolutions of the background field for two different initial conditions
in Fig. 1
φ(t) φ(t)
1.2 1
1 0
0.8 −1
t t
0.6
0 50 100 0 50 100 150 200
Figure1. Timeevolutionsofthebackgroundfieldforφ0=1.3v,φ0=1.6v. Amplitudes
areinunitsofv andtimeismeasuredinunitsof(√λv)−1.
It can be seen that there are two phases depending on the value of
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φ : a symmetric phase when the field φ(t) is oscillating about zero, and
0
a phase of broken symmetry when φ(t) is oscillating about a finite value.
The“critical”valueofφ seemstobeclosetotheclassicallyexpectedvalue
0
√2λv where the total energy is equal to the height of the barrier.
2.2. Phase structure
InordertoanalyzethephasestructureofthemodelintheHartreeapprox-
imation we define φ∞ as a time averaged amplitude at late times, i.e. the
valueaboutwhichthefieldoscillates. φ∞ playstheroleofanorderparam-
eter whose dependence on the total energy of the system is investigated.
The total energy of the system is given here by the initial value φ which
0
is analogous to the temperature in equilibrium field theory.
φ∞
1
0,5
φ0
0
1 1,2 1,4 1,6 1,8
Figure2. Thelate-timeamplitudeasfunctionoftheinitialamplitudeforN =4.
Fig. 2 clearly shows a discontinuous jump of φ∞ at φ0 √2v — a
≃
typical sign of a first order phase transition as found in equilibrium (see
e.g. Refs. 4,2).
2.3. A dynamical effective potential
This nonequilibrium system shows a phase structure which is comparable
toasysteminthermalequilibrium,soitwouldbeniceiftherewasanother
correspondence. We define a dynamical, i.e. time-dependent, potential
which can be compared to the finite temperature potential in equilibrium
1
V (t)= φ˙2(t) . (9)
pot
E − 2
This potential can only be measured within the oscillation range of the
background field φ(t). For two different initial conditions it is shown in
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Fig. 3. In the broken symmetry phase the minimum at φ = v moves
8 −0.8
Vpot7 Vpot
6 −0.85
5
4 −0.9
3
2 −0.95
1
0 φ −1 φ
−1
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2
Figure 3. Evolution of the potential energy (9) for φ0 = 2v (unbroken phase) and
φ0=1.2v (brokenphase).
to smaller values but eventually remains different from zero, so the system
settlesatafiniteexpectationvalue. Inthesymmetricphasethetwominima
entirelydisappearafter afewoscillationsandanew (symmetric)minimum
at φ=0 appears.
3. Conclusions and summary
The analysis of the nonequilibrium dynamics of the O(N) model in the
Hartree approximation allowed us to study new features of the system
which are not accessible in the one-loop or infinite component (N )
→ ∞
approximations. Though thermalization is only expected at approxima-
tions beyond the Hartree level, i.e., when including nonlocal corrections,
thenonequilibriumsystematlatetimesshowsstrikingsimilaritiestoasys-
tem in thermal equilibrium. One can define an order parameter which is
dependentonthetotalenergyofthesystem,givenbytheinitialconditions.
Analyzing the dependence of the order parameter on the initial conditions
one finds a first order phase transition.
References
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