Oscillating Scalar Field in Thermal Environment Kyohei Mukaida - - PowerPoint PPT Presentation

Kyohei Mukaida - Univ. of Tokyo

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Dynamics of

Oscillating Scalar Field

in Thermal Environment

Kyohei Mukaida (Univ. of Tokyo)

Based on 1208.3399 with Kazunori Nakayama

Introduction

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Introduction

At the early universe, a scalar field may condensate homogeneously w/ a far from equilibrium value. It begins to oscillate coherently at H ~ m and behaves as “Matter”.

• Examples: Inflaton, Moduli, Curvaton, Affleck-Dine field...

To avoid the “overclosure”, the scalar condensate should decay to the “Radiation” sector at an appropriate epoch.

➡ Interaction btw the scalar and radiation

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Introduction

Such interactions btw scalar and radiation induce various effects that make the scalar dynamics complicated:

• Thermal Effects
• Thermally modified effective potential of the scalar field
• Dissipation of scalar condensate into the radiation sector
• Non-perturbative particle production
• (possible) non-topological soliton formation

➡We included these effects simultaneously.

e.g., [J. Yokoyama; M. Drewes; A. Berera et al.] e.g., [L. Kofman, A. Linde, A. Starobinsky] e.g., [E. Copeland et al.; S. Kasuya et al.]

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Set Up

Let us consider the following simplest set up:

λ2φ2 ˜ χ2

λφ ¯ χχ,

χ ˜ χ χ

Thermal Plasma

φ

Scalar Condensate

Gauge Int.

χ Sector

✴ w/ “free” quadratic potential

αT ⇠ Γth H

V(φ) = 1 2mφ

2φ2

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Outline

Introduction Closed Time Path Formalism Thermal Effects Non-perturbative Particle Production Numerical Results

Thermal Effects

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Thermal Effects

Typically, there are two effects from thermal plasma.

• Force from thermal plasma
• Dissipation to thermal plasma

✴If the Bkg plasma remains in thermal equilibrium.

➡Reduced to Coarse-Grained EOM of Φ:

0 = δS δφ + δ˜ Γ δφ = δS δφ ∂F ∂φ Πret ⇤ δφ + · · · ' δS δφ ∂F ∂φ Γφ ˙ φ;

Γφ ' lim

ω!0

ΠJ(ω, 0) 2ω .

e.g., [A. Berera et al., hep-ph/9803394]

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Thermal Effects

Typically, there are two effects from thermal plasma.

• Force from thermal plasma:

Free Energy of thermal plasma w/ the background

¨ φ + (3H + Γφ) ˙ φ + m2

φφ = −∂F

∂φ φ.

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Typically, there are two effects from thermal plasma.

• Force from thermal plasma:

Free Energy of thermal plasma w/ the background

¨ φ + (3H + Γφ) ˙ φ + m2

φφ = −∂F

∂φ

• Small amplitude ⇒ Thermal mass
• Large amplitude ⇒ Thermal log

[A. Anisimov, M. Dine]

φ.

Thermal Effects

F / α2T4 ln(λ2φ2/T2); λφ T F / λ2T2φ2; λφ ⌧ T

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Typically, there are two effects from thermal plasma.

• Dissipation to thermal plasma:

¨ φ + (3H + Γφ) ˙ φ + m2

φφ = −∂F

∂φ

Friction coefficient from Kubo-formula Γφ ' lim

ω!0

ΠJ(ω, 0) 2ω .

Thermal Effects

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Typically, there are two effects from thermal plasma.

• Dissipation to thermal plasma:

¨ φ + (3H + Γφ) ˙ φ + m2

φφ = −∂F

∂φ

Friction coefficient from Kubo-formula

• Small amplitude: processes including χ.
• Large amplitude: multiple scattering by gauge bosons.

[D. Bodeker; M. Laine]

Γφ ∼ λ2αT Γφ ∼ α2 ln α−1 T3 φ2

Γφ ' lim

ω!0

ΠJ(ω, 0) 2ω .

Thermal Effects

Non-perturb. Particle Production

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Non-perturb. Production

The adiabaticity of χ particles can be broken down when the scalar passes through the origin.

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• Adiabaticity is broken if ✏ 1.

✏ :=

• ˙

! !2

• ; ωχ =

q k2 + mχ

eff(T)2 + λ2φ2(t)

[L. Kofman, A. Linde, A. Starobinsky, hep-ph/9704452]

➡Efficient χ production can occur at Φ~0.

Kyohei Mukaida - Univ. of Tokyo

If Φ oscillates w/ the finite T potential (free-energy), If Φ oscillates w/ the zero T potential, the efficient particle production occurs @

15        λ ⌧ α λ α : No efficient prod. : Complicated... |φ| . h mφ ˜ φ/λ i1/2 .

Non-perturb. Production

nχ ∼ k3

∗

(2π)3 ; k∗ = q λ ˜ φmφ

[ ˜ φ : amplitude]

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The produced χ may decay @ The Φ’s energy is partially converted to the radiation, and it is estimated as

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Non-perturb. Production

tdec Γχ(φ(tdec)) ∼ 1. δρφ ρφ ∼ λ2 4π3α1/2 ; @ every one oscillation.

[Otherwise the parametric resonance may occur]

Here we assumed that the decay rate of χ is given by

Γχ ∼ α mχ(φ).

[Felder, Kofman, Linde, “Instant preheating”,hep-ph/9812289]

Numerical Results

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Numerical Results

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Contour plot of

m2

φ ˜

φ2/2 ρrad + ρinf

• H=Γφ

.

1010 1011 1012 1013 1014 1015 1016 1017 1018 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 1e-30 1e-25 1e-20 1e-15 1e-10 1e-05 1 100000 i [GeV] 1010 1011 1012 1013 1014 1015 1016 1017 1018 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 1e-18 1e-16 1e-14 1e-12 1e-10 1e-08 1e-06 0.0001 0.01 1 100 i [GeV]

φi [GeV] λ

1

10−5 10−10 10−15

φi [GeV]

1

10−5 10−10 10−15 Initial amplitude: Initial amplitude: Coupling btw Φ & radiation:

α = 0.05; TR = 109 GeV; mφ = 1 TeV.

[ ˜ φ : amplitude]

Kyohei Mukaida - Univ. of Tokyo

Summary

Interactions btw scalar and radiation are often introduced to avoid overclosure. Such interactions induce various effects that make the dynamics of scalar condensate complicated. We take into account all these effects properly, and show that such effects can change the abundance of scalar condensate by orders in a broad range of parameters.

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Back Up

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CTP formalism

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Closed Time Path (CTP) formalism gives us useful tools to study an evolution of expectation value.

• Fig. from M. Garny et al., 0904.3600
• An expectation value

can be obtained from CTP integral:

e.g., [J. Berges, hep-ph/0409233]

h ˆ OH(t)i = tr " ˆ ρ TC exp i Z

C

dt0HI(t0) ! ˆ OI(t) #

ρ : density matrix ˆ OH(t) : Heisenberg Picture ˆ OI(t) : Interaction Picture TC : contour C ordering

CTP Formalism

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It is useful to consider n-point functions on CTP.

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φ = tr h ˆ ρ ˆ φ i G = tr h ˆ ρ TC ˆ φ ˆ φ icon Vn = tr h ˆ ρ TC ˆ φ · · · ˆ φ icon

TC : contour C ordering

[con. stands for connected part]

• Kadanoff-Baym: Follow the evolution of Φ and G self-consistently.

➡Schwinger Dyson equations on CTP [w/ skeleton diagram expansion]

G−1 = G−1

0 + Π[φ, G];

Π = −2iδ˜ Γ[φ, G] δG

0 = δS δφ + δ˜ Γ[φ, G] δφ

[Cornwall, Jackiw, Tomboulis,1974]

CTP Formalism

Kyohei Mukaida - Univ. of Tokyo

CTP Formalism

Top-down approach:

24 Kadanoff-Baym eqs. Coarse-grained eq. for Φ

• Some fields may be kept in thermal

equilibrium during the course of Φ’s dynamics.

χ ˜ χ χ

Thermal Plasma

φ

Reduce

• Propagators of these fields

➡ Thermal Propagators

• Thermal mass
• Thermal width

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Coherent Oscillation

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Scalar condensate is obtained from one point func. If the scalar oscillates adiabatically w.r.t. thermal plasma...

• Propagators in thermal bath → thermal ones

➡Reduced to Coarse-Grained EOM of Φ:

0 = δS δφ + δ˜ Γ δφ = δS δφ ∂F ∂φ Πret ⇤ δφ + · · · ' δS δφ ∂F ∂φ Γφ ˙ φ;

Γφ ' lim

ω!0

ΠJ(ω, 0) 2ω .

e.g., [A. Berera et al., hep-ph/9803394]

Coleman-Weinberg potential

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CW potential

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In non-SUSY, there exists Coleman Weinberg (CW) potential: In SUSY, such radiative corrections to flat directions are suppressed and remain only small logs due to the SUSY breaking effects.

VCW = X

F

✏F m4

()

64⇡2 2 6 6 6 6 4ln m2

()

µ2 − 3 2 3 7 7 7 7 5

✏F :=        +1 for real scalar −2 for Weyl fermion .

Numerical Results

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Numerical Results

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The beginning of oscillation: φi [GeV] φi [GeV]

Initial amplitude: Initial amplitude:

λ

Coupling btw Φ & radiation:

(a): thermal log, (b): thermal mass, (c): zero T mass

TR =109 GeV mφ =103 GeV TR =109 GeV mφ =106 GeV

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Numerical Results

Oscillation w/ thermal log:

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1e-12 1e-10 1e-08 1e-06 0.0001 0.01 1 100 10000 1e+06 100 1000 10000 meff [GeV] H [GeV] [GeV] /tot rad/tot

H [GeV]

mφ = 1 TeV TR = 109 GeV λ = 2 × 10−3 φi = 1015 GeV

Γφ ∼ α2 T3 φ2

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Numerical Results

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Oscillation w/ thermal mass:

H [GeV]

1e-15 1e-10 1e-05 1 100000 1e-06 0.0001 0.01 1 100 10000 1e+06 meff [GeV] H [GeV] [GeV] /tot rad/tot

mφ = 1 TeV TR = 109 GeV λ = 10−5 φi = 1014 GeV

Γφ ∼ λ2αT

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Numerical Results

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Oscillation w/ zero T mass:

H [GeV]

1e-05 0.0001 0.001 0.01 0.1 1 10 100 1000 10000 0.01 0.1 1 10 100 1000 meff [GeV] H [GeV] [GeV] /tot rad/tot

mφ = 1 TeV TR = 109 GeV λ = 10−2 φi = 1018 GeV

Γφ ∼ α2 T3 φ2

Oscillon

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Oscillon (I-ball)

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A coherently oscillating scalar with a potential flatter than the quadratic one has an instability and may fragment into classical lumps. Even if there is no conserved charge, their stability is guaranteed by the adiabatic invariant. Such a non-topological soliton is dubbed as oscillon

• r I-ball.

e.g., [Copeland, Gleiser, Muller, hep-ph/9503217] [Kasuya, Kawasaki, Takahashi, hep-ph/0209358] [Kasuya, Kawasaki, Takahashi, hep-ph/0209358]

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Oscillon (I-ball)

The region where the I-ball may be formed.

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Coupling btw Φ & radiation: λ

φi [GeV]

Initial amplitude:

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Oscillon (I-ball)

The region where the I-ball may be formed.

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Coupling btw Φ & radiation: λ

φi [GeV]

Initial amplitude:

• It is also possible that the scalar

condensation evaporates due to the dissipation before the formation of oscillon.

• Even if this is the case, the delayed type
• scillon may be formed.

➡ Further study is needed to

say something conclusively.

Bulk Viscosity

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Bulk Viscosity

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The dissipation rate at large amplitude regime is directly related to the bulk viscosity of Yang-Mills plasma.

Γφ = lim

ω!0

ΠJ(ω, 0) 2ω = lim

ω!0

1 2ω Z d4x eiωth[ ˆ O(t, x), ˆ O(0)]i; ˆ O(x) = A 8π2φFaµν(x)Fa

µν(x)

Bulk Viscosity: ζ = 1 9 Z d4x eiωth[Tµ

µ(t, x), Tν ν(0, 0)]i

[D. Bodeker; M. Laine]

ζ ∼ α2T3 ln[1/α]; @ weak coupling

[Arnold, Dogan. Moore, hep-ph/0608012]