Download Particle Production and Vacuum Selection with Higher Dimensional

Particle Production and Vacuum Selection
with Higher Dimensional Interaction
Sesihi Enomoto ( Nagoya Univ. )
Collaborators : Nobuhiro Maekawa ( Nagoya Univ. )
Tomohiro Matsuda ( Saitama Institute of Tech. )
Satoshi Iida ( Nagoya Univ. )
Contents
1. Introduction
2. How about Higher Dimensional Interaction?
3. Summary
2012/9/11
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1. Introduction
■ Problem of the Vacuum Selection
■ There exists many vacua in the GUT or the string theory…
 Which vacuum is selected? How?
?
?
?
■ “Beauty is attractive.”
・・・ L.Kofman, A.Linde, X.Liu, A.Maloney, L.McAllister, E.Silverstein (2004)
The vacuum where the symmetry is enhanced tends to be selected.
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■ Moduli Trapping & Particle Creation
[L.Kofman, A.Linde, X.Liu, A.Maloney, L.McAllister, E.Silverstein, JHEP 0405, 030 (2004)]
1
2
𝜙 : complex moduli (classical) ,
1
2
𝜒 : real scalar particle (quantum) ,
■ Lagrangian : ℒ = 𝜕𝜇 𝜙 ∗ 𝜕𝜇 𝜙 + 𝜕𝜇 𝜒𝜕𝜇 𝜒 − 𝑔2 𝜙 2 𝜒 2
Note : 𝜒 becomes massless @ 𝜙 = 0 !
𝑔 : coupling
“Enhanced Symmetry Point” (ESP)
■ EOMs
Wave func.
𝜙=𝜙 𝑡
𝑑3 𝑘
2𝜋 3/2
，𝜒=
†
𝑢𝑘 (𝑡)𝑎𝑘 + 𝑢−𝑘 (𝑡)𝑎−𝑘
𝑒 𝑖𝒌∙𝒙
Creation / Annihilation op.
⇒
1
2
𝜙 + 𝑔2 𝜒 2 𝜙 = 0
𝑢𝑘 +
𝜔𝑘2 𝑢𝑘
=0
( 𝜔𝑘 =
■ The Asymptotic Solutions ( 𝑡 → −∞ , valid in
𝑘 2 + 𝑔2 𝜙
2
𝑑
1
𝑑𝑡 𝜔𝑘
≲1)
: frequency of 𝜒 )
Im 𝜙
𝜙(𝑡) = 𝜙𝑡 + 𝑖𝜇
𝜒
2
∼0
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𝑢𝑘 (𝑡) =
1
2𝜔𝑘 (𝑡)
𝜇
𝑡
′
′
𝑒 −𝑖 −∞ 𝑑𝑡 𝜔𝑘(𝑡 )
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𝜙
ESP
𝜙 ≳
𝜙𝐸𝑆𝑃 /𝑔
Re 𝜙
3/11
■ 𝜒 particles are produced when 𝜙 approaches the ESP ( 𝜙
≲ 𝜙𝐸𝑆𝑃 /𝑔
1/2
).
■ Because 𝜒 becomes massless @ ESP, the kinetic energy of 𝜙 converts to
𝜒 particles.
■ The produced number density : 𝑛𝜒 =
𝑔𝜙𝐸𝑆𝑃
2𝜋
3
3/2
exp −𝜋𝑔𝜇2 /𝜙𝐸𝑆𝑃
■ Once 𝜒 is produced, then the effective potential is established for 𝜙.
𝜌𝜒 =
𝑑3𝑘
𝑛
2𝜋 3 𝑘
𝑘 2 + 𝑔2 𝜙
2
∼ 𝑔 𝜙 𝑛𝜒
𝑤ℎ𝑒𝑛 𝜙 𝑖𝑠 𝑙𝑎𝑟𝑔𝑒
𝑉
Im 𝜙
Potential Energy
＝
𝜒 production!
Initial Kinetic Energy
ESP
𝜒 production!
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Re 𝜙
𝜙 is trapped around
the ESP.
--> vacuum selection
4/11
2. How about Higher Dimensional Interaction?
■ The higher dim. int. becomes important in some case.
■ Lagrangian : ℒ = 𝜕𝜇
𝜙 ∗ 𝜕𝜇 𝜙
+
1
𝜕𝜇 𝜒𝜕𝜇 𝜒
2
−
1 𝑔2
2 Λ2(𝑛−1)
𝜙
2𝑛 𝜒 2
(Λ : cutoff )
■ For example : The established effective potential
The frequency of 𝜒 : 𝜔𝑘 =
⇒ 𝜌𝜒 =
𝑑3 𝑘
𝑛
2𝜋 3 𝑘
𝑘 2 + 𝑔2 𝜙
𝑘 2 + 𝑔2 𝜙
2𝑛 /Λ2(𝑛−1)
2𝑛 /Λ2(𝑛−1)
∼ 𝑔 𝜙 𝑛 𝑛𝜒 /Λ𝑛−1
★ The trapping effect is enhanced steeply @ 𝜙 ∼ Λ.
■ How about the produced particle number?
■ How about the size of particle production area?
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■ The particle production area from the ESP can be evaluated roughly,
𝑑
1
𝑑𝑡 𝜔𝑘
1
≳1
⇒
𝜙 ≲
Λ𝑛−1 𝜙/𝑔 𝑛+1 .
★ If 𝜙 ≲ 𝑔Λ2 , then the production area is larger than the case of 𝑛 = 1.
■ Estimation of produced particle number
1
■ WKB type solution (wave func. of 𝜒)
𝑢𝑘 (𝑡) =
𝛼𝑘 (𝑡)
2𝜔𝑘 (𝑡)
𝑒
−𝑖
𝑡
𝑑𝑡 ′ 𝜔𝑘 (𝑡 ′ )
+
𝛽𝑘 (𝑡)
2𝜔𝑘 (𝑡)
𝑒
+𝑖
𝑡
𝑑𝑡 ′ 𝜔𝑘 (𝑡 ′ )
𝑒 −𝑖
𝑡
𝑑𝑡 ′ 𝜔𝑘 (𝑡 ′ )
Bogoliubov
transformation
2𝜔𝑘 (𝑡)
( 𝛼𝑘 , 𝛽𝑘 : Bogoliubov coefficients )
■ EOMs
𝜔
𝛼𝑘 = 2𝜔𝑘 exp +2𝑖
𝑡
𝑑𝑡 ′ 𝜔𝑘 (𝑡 ′ ) 𝛽𝑘
𝑡
𝑑𝑡 ′ 𝜔𝑘 (𝑡 ′ ) 𝛼𝑘
𝑘
𝜔
𝛽𝑘 = 2𝜔𝑘 exp −2𝑖
𝑘
■ Initial conditions ： 𝛼𝑘 𝑡 = −∞ = 1, 𝛽𝑘 (𝑡 = −∞) = 0
⇒ 𝑛𝑘 = 𝛽𝑘 2
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■ Solution by the leading order,
𝛽𝑘 ∼
𝑡
′ 𝜔𝑘
𝑑𝑡
exp
−∞
2𝜔𝑘
−2𝑖
・ 𝜙 = 𝜙𝑡 + 𝑖𝜇 ⇒ 𝜔𝑘 =
𝑡′
𝑑𝑡 ′′ 𝜔𝑘 𝑡 ′′
𝑘 2 + 𝑔2 𝜙 2 𝑡 2 + 𝜇2
𝑛
(cosnt.)
Im 𝜙
Im 𝑡
𝜙
𝜔𝑘
𝜇
ESP
Re 𝑡
Re 𝜙
・ Using the steepest descent method , we obtain as
[D. Chung, Phys. Rev. D 67, 083514 (2004)]
★ 𝑛𝜒 =
𝑑3 𝑘
2𝜋 3
𝛽𝑘
2
∼
𝜙
𝑔Λ2
3 𝑛−1
2 𝑛+1
𝑔𝜙
3/2
𝑛−1
𝑛≥2, 𝜇 ≲ Λ
𝜙/𝑔
1
𝑛+1
∼ 𝑛𝜒 (𝑛 = 1, 𝜇 = 0)
This result does not contain the impact parameter.
So, it is expected that particle production area is broad in this region.
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■ Comparing with the numerical calculation
■ EOMs
𝜙+
𝑢𝑘 +
𝑑3𝑘
2𝜋 3
𝜔𝑘2 𝑢𝑘
𝑢𝑘
2
−
1
2𝜔𝑘
=0
∙
𝑛𝑔2
𝜙
Λ2 𝑛−1
( 𝜔𝑘 =
2 𝑛−1
𝑘 2 + 𝑔2 𝜙
=0
2𝑛 /Λ2(𝑛−1)
)
■ We calculate the above EOMs numerically, and obtain the number
density ( 𝑛𝜒 ) from 𝑢𝑘 .
■ The number density can be calculated from the wave function by
𝑛𝜒 =
𝑑3 𝑘
2𝜋 3
𝑢𝑘
2
+ 𝜔𝑘2 𝑢𝑘
2𝜔𝑘
2
1
−
2
[B. Garbrecht, T. Prokopec, M. Schmidt, Eur. Phys. J. C 38, 135 (2004)]
■ We evaluate the number density 𝑛𝜒 by changing 𝑛 and the impact
parameter 𝜇.
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■ The relation of impact parameter and produced particle number
3 𝑛−1
𝑛𝜒 ∼ 𝐶 𝑛
𝜙 2 𝑛+1
𝑔Λ2
𝐶 𝑛 =
1
𝑛
18 𝑛+1
Γ
𝑔𝜙
3𝑛
𝑛+1
𝑛𝜒 Λ3
3/2
𝑛−1
𝑚=0
𝑛−1
𝑛≥2, 𝜇 ≲ Λ
4 1 + cos 2𝑛
2𝑚+1
2𝑛
sin
𝜙/𝑔
2𝑚+1
1
𝑛+1
Im 𝜙
3𝑛
−
𝑛+1
𝜇
2𝑛
( 𝜙/Λ2 = 0.01 , 𝑔 = 1 )
ESP
Re 𝜙
5.0E-06
n=1 (numerical)
n=2 (numerical)
n=3 (numerical)
n=1 (analytical)
n=2 (analytical)
n=3 (analytical)
4.0E-06
3.0E-06
2.0E-06
𝑔𝜙
2𝜋
3/2
3
exp −𝜋𝑔𝜇 2 /𝜙𝐸𝑆𝑃
By the first introduced
article.
1.0E-06
0.0E+00
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
𝜇/Λ
■ The analytic solution for 𝑛 ≥ 2 is roughly good.
■ The produced particle number at small 𝜇 is reduced when 𝑛 is larger.
■ However, the production area is larger.
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■ Parametric resonance and trapping effect
𝑛=3, 𝑔 =1,
𝜙0 /Λ = 5 + 0.4𝑖 ,
𝜙0 /Λ2 = −0.1
𝜇/Λ = 0.1
𝜙/Λ2 = 0.1
𝑔=1
𝑛=1
𝑛𝜒 Λ3 =
9.3 × 10−5
𝜙 =5
■ In case of approaching the ESP
many time, 𝜒 particles are
exponentially produced by the
parametric resonance.
■ The trapping effect is enhanced in
case of larger 𝑛.
■ Therefore, particle production
effect is enhanced because of
approaching to the ESP many
times.
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𝑛𝜒 Λ3
= 0.07
𝑛=2
𝜙 =5
𝑛𝜒 Λ3
= 0.07
𝜙 =5
𝑛=3
10/11
Same time scale each other
3. Summary
■ We studied about the particle production and the trapping effect
with higher dimensional interaction.
The region of the
particle production
The effective potential
due to the particle
production
The produced particle
number density
𝑛−1
𝜙 ≲ Λ
𝜙/𝑔
1
𝑛+1
𝑉
𝑉 ∼ 𝑔 𝜙 𝑛 𝑛𝜒 /Λ𝑛−1
𝑛𝜒 ∼
3 𝑛−1
𝜙/Λ2 2 𝑛+1
𝑔𝜙
𝑛≥2
𝑛=1
Im 𝜙
3/2
∼Λ
Re 𝜙
■ The trapping effect is enhanced at cutoff scale.
■ The particle production area is larger for larger 𝑛.
■ And also, the particle production effect is enhanced due to
approaching to the ESP many times.
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Back up
4.5E-06
4.0E-06
n=1 (numerical)
3.5E-06
n=2 (numerical)
3.0E-06
n=3 (numerical)
2.5E-06
n=1 (analytical)
n=2 (analytical)
2.0E-06
n=3 (analytical)
1.5E-06
1.0E-06
5.0E-07
0.0E+00
0
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0.05
0.1
0.15
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0.2
0.25
0.3
0.35
12/11
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
1.0E-05
1.0E-06
1.0E-07
n=1 (numerical)
1.0E-08
n=2 (numerical)
n=3 (numerical)
n=1 (analytical)
1.0E-09
n=2 (analytical)
n=3 (analytical)
1.0E-10
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3. Summary
■ We studied about the particle production and the trapping effect
with higher dimensional interaction.
The region of the
particle production
The effective potential
due to the particle
production
The produced particle
number density
𝜙 ≲
1
𝑛−1
𝑛+1
Λ 𝜙/𝑔
𝑉
𝑉 ∼ 𝑔 𝜙 𝑛 𝑛𝜒 /Λ𝑛−1
𝑛𝜒 ∼
3 𝑛−1
2
𝜙/Λ 2 𝑛+1
𝑔𝜙
Im 𝜙
3/2
∼Λ
Re 𝜙
■ The produced particle number due to approaching to the ESP 1 time
is reduced for larger.
■ However, the particle production area is larger for larger 𝑛.
■ The trapping effect is enhanced at cutoff scale, and also the particle
production effect is enhanced due to approaching to the ESP many
times.
14/11
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