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

Dynamics of Oscillating Scalar Field in Thermal Environment Kyohei Mukaida (Univ. of Tokyo) Based on 1208.3399 with Kazunori Nakayama 1 Kyohei Mukaida - Univ. of Tokyo

2 Introduction

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 3 Kyohei Mukaida - Univ. of Tokyo

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 e.g., [J. Yokoyama; M. Drewes; A. Berera et al.] ‣ Non-perturbative particle production e.g., [L. Kofman, A. Linde, A. Starobinsky] ‣ (possible) non-topological soliton formation e.g., [E. Copeland et al.; S. Kasuya et al.] ➡ We included these effects simultaneously. 4 Kyohei Mukaida - Univ. of Tokyo

Set Up Let us consider the following simplest set up: Thermal Plasma λ 2 φ 2 ˜ χ 2 φ λφ ¯ χχ , χ χ Scalar Gauge Int. ˜ Condensate χ ✴ w/ “free” quadratic potential χ Sector V ( φ ) = 1 2 φ 2 2 m φ α T ⇠ Γ th � H 5 Kyohei Mukaida - Univ. of Tokyo

Outline Introduction Closed Time Path Formalism Thermal Effects Non-perturbative Particle Production Numerical Results 6 Kyohei Mukaida - Univ. of Tokyo

7 Thermal Effects

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 : δφ + δ ˜ 0 = δ S δφ = δ S δφ � ∂ F Γ ∂φ � Π ret ⇤ δφ + · · · ' δ S δφ � ∂ F Π J ( ω , 0 ) ∂φ � Γ φ ˙ φ ; Γ φ ' lim . 2 ω ω ! 0 e.g., [A. Berera et al., hep-ph/9803394] 8 Kyohei Mukaida - Univ. of Tokyo

Thermal Effects Typically, there are two effects from thermal plasma. ‣ Force from thermal plasma: φ φ = − ∂ F φ + (3 H + Γ φ ) ˙ ¨ φ + m 2 ∂φ Free Energy of thermal plasma w/ the background φ . 9 Kyohei Mukaida - Univ. of Tokyo

Thermal Effects Typically, there are two effects from thermal plasma. ‣ Force from thermal plasma: φ φ = − ∂ F φ + (3 H + Γ φ ) ˙ ¨ φ + m 2 ∂φ Free Energy of thermal plasma w/ the background φ . - Small amplitude ⇒ Thermal mass F / λ 2 T 2 φ 2 ; λφ ⌧ T - Large amplitude ⇒ Thermal log F / α 2 T 4 ln( λ 2 φ 2 / T 2 ); λφ � T [A. Anisimov, M. Dine] 10 Kyohei Mukaida - Univ. of Tokyo

Thermal Effects Typically, there are two effects from thermal plasma. ‣ Dissipation to thermal plasma: φ φ = − ∂ F φ + (3 H + Γ φ ) ˙ ¨ φ + m 2 ∂φ Π J ( ω , 0 ) Friction coefficient from Kubo-formula Γ φ ' lim . 2 ω ω ! 0 11 Kyohei Mukaida - Univ. of Tokyo

Thermal Effects Typically, there are two effects from thermal plasma. ‣ Dissipation to thermal plasma: φ φ = − ∂ F φ + (3 H + Γ φ ) ˙ ¨ φ + m 2 ∂φ Π J ( ω , 0 ) Friction coefficient from Kubo-formula Γ φ ' lim . 2 ω ω ! 0 - Small amplitude: processes including χ . Γ φ ∼ λ 2 α T - Large amplitude: multiple scattering by gauge bosons. α 2 T 3 Γ φ ∼ [D. Bodeker; M. Laine] ln α − 1 φ 2 12 Kyohei Mukaida - Univ. of Tokyo

13 Non-perturb. Particle Production

Non-perturb. Production The adiabaticity of χ particles can be broken down when the scalar passes through the origin. ‣ Adiabaticity is broken if ✏ � 1 . � � ˙ ! � � � q k 2 + m χ � � ✏ : = � ; ω χ = e ff ( T ) 2 + λ 2 φ 2 ( t ) � � ! 2 � � � � ➡ Efficient χ production can occur at Φ ~0. [L. Kofman, A. Linde, A. Starobinsky, hep-ph/9704452] 14 14 Kyohei Mukaida - Univ. of Tokyo

Non-perturb. Production If Φ oscillates w/ the finite T potential (free-energy),  : No efficient prod. λ ⌧ α    : Complicated...  λ � α   If Φ oscillates w/ the zero T potential, the efficient i 1 / 2 h particle production occurs @ m φ ˜ | φ | . φ / λ . k 3 q λ ˜ ∗ n χ ∼ (2 π ) 3 ; k ∗ = φ m φ [ ˜ φ : amplitude] 15 Kyohei Mukaida - Univ. of Tokyo

Non-perturb. Production The produced χ may decay @ t dec Γ χ ( φ ( t dec )) ∼ 1 . [Otherwise the parametric resonance may occur] The Φ ’s energy is partially converted to the radiation, and it is estimated as δρ φ λ 2 4 π 3 α 1 / 2 ; @ every one oscillation. ∼ ρ φ Here we assumed that the decay rate of χ is given by Γ χ ∼ α m χ ( φ ) . [Felder, Kofman, Linde, “Instant preheating”,hep-ph/9812289] 16 Kyohei Mukaida - Univ. of Tokyo

17 Numerical Results

Numerical Results φ ˜ m 2 φ 2 / 2 � � α = 0 . 05; Contour plot of � . [ ˜ � φ : amplitude] T R = 10 9 GeV; � ρ rad + ρ inf � Coupling � H = Γ φ m φ = 1 TeV . btw Φ & radiation: λ 10 -2 10 -2 10 -3 10 -3 10 − 15 10 -4 10 -4 10 − 15 10 -5 10 -5 10 -6 10 -6 10 -7 10 -7 10 − 10 10 − 10 10 − 5 1 10 − 5 1 10 -8 10 -8 10 -9 10 -9 100000 1e-30 1e-20 1e-15 1e-10 1e-25 1e-05 1 0.0001 1e-14 1e-16 1e-06 1e-08 1e-10 1e-12 1e-18 0.01 100 1 10 10 10 11 10 12 10 13 10 14 10 15 10 16 10 17 10 18 10 10 10 11 10 12 10 13 10 14 10 15 10 16 10 17 10 18 � i [GeV] � i [GeV] φ i [GeV] φ i [GeV] Initial amplitude: Initial amplitude: 18 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. 19 Kyohei Mukaida - Univ. of Tokyo

20 Back Up

21 CTP formalism

CTP Formalism Closed Time Path (CTP) formalism gives us useful tools to study an evolution of expectation value . ‣ An expectation value can be obtained from CTP integral: " ! # Z h ˆ ˆ dt 0 H I ( t 0 ) O H ( t ) i = tr ρ T C exp � i O I ( t ) ˆ C T C : contour C ordering ρ : density matrix ˆ O H ( t ) : Heisenberg Picture ˆ O I ( t ) : Interaction Picture Fig. from M. Garny et al., 0904.3600 e.g., [J. Berges, hep-ph/0409233] 22 Kyohei Mukaida - Univ. of Tokyo

CTP Formalism It is useful to consider n-point functions on CTP. h i ρ ˆ φ = tr ˆ φ i con h ρ T C ˆ φ ˆ G = tr φ ˆ T C : contour C ordering i con h ρ T C ˆ φ · · · ˆ V n = tr ˆ φ [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] Π = − 2 i δ ˜ Γ [ φ , G ] G − 1 = G − 1 0 + Π [ φ , G ]; δ G δφ + δ ˜ [Cornwall, Jackiw, Tomboulis,1974] Γ [ φ , G ] 0 = δ S δφ 23 Kyohei Mukaida - Univ. of Tokyo

CTP Formalism Top-down approach: Kadanoff-Baym eqs. ‣ Some fields may be kept in thermal Thermal Plasma equilibrium during the course of Φ ’s dynamics. φ ‣ Propagators of these fields χ χ ➡ Thermal Propagators - Thermal mass ˜ χ - Thermal width Reduce Coarse-grained eq. for Φ 24 Kyohei Mukaida - Univ. of Tokyo

Coherent Oscillation 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 : δφ + δ ˜ 0 = δ S δφ = δ S δφ � ∂ F Γ ∂φ � Π ret ⇤ δφ + · · · ' δ S δφ � ∂ F Π J ( ω , 0 ) ∂φ � Γ φ ˙ φ ; Γ φ ' lim . 2 ω ω ! 0 e.g., [A. Berera et al., hep-ph/9803394] 25 Kyohei Mukaida - Univ. of Tokyo

26 Coleman-Weinberg potential

CW potential In non-SUSY, there exists Coleman Weinberg (CW) potential: m 4 m 2 2 3 � ( � ) � ( � ) − 3 X 6 7 4 ln V CW = ✏ F 6 7 6 7 64 ⇡ 2 µ 2 6 2 7 5 F  + 1 for real scalar   ✏ F : =  .  − 2 for Weyl fermion   In SUSY, such radiative corrections to flat directions are suppressed and remain only small logs due to the SUSY breaking effects. 27 Kyohei Mukaida - Univ. of Tokyo

28 Numerical Results

Numerical Results The beginning of oscillation: T R = 10 9 GeV Coupling T R = 10 9 GeV m φ = 10 6 GeV m φ = 10 3 GeV btw Φ & radiation: λ φ i [GeV] φ i [GeV] Initial amplitude: Initial amplitude: (a): thermal log, (b): thermal mass, (c): zero T mass 29 Kyohei Mukaida - Univ. of Tokyo

Numerical Results Oscillation w/ thermal log: 1e+06 m φ = 1 TeV T R = 10 9 GeV 10000 Γ φ ∼ α 2 T 3 λ = 2 × 10 − 3 100 φ 2 φ i = 10 15 GeV 1 0.01 0.0001 1e-06 m eff [GeV] 1e-08 H [GeV] � � [GeV] 1e-10 � � / � tot � rad / � tot 1e-12 10000 1000 100 H [GeV] 30 30 Kyohei Mukaida - Univ. of Tokyo

Numerical Results Oscillation w/ thermal mass: m φ = 1 TeV 100000 T R = 10 9 GeV λ = 10 − 5 1 φ i = 10 14 GeV Γ φ ∼ λ 2 α T 1e-05 1e-10 m eff [GeV] H [GeV] � � [GeV] � � / � tot 1e-15 � rad / � tot 1e+06 10000 100 1 0.01 0.0001 1e-06 H [GeV] 31 Kyohei Mukaida - Univ. of Tokyo