Effect of interaction terms on particle
production due to time-varying mass
[work in progress]
Seishi Enomoto (Univ. of Warsaw)
Collaborators : Olga Fuksińska
Zygmunt Lalak
1.
2.
3.
4.
2014/08/23
(Univ. of Warsaw)
(Univ. of Warsaw)
Outline
Introduction
Bogoliubov transformation law with
interaction terms
Application to our model
Summary
Summer Institute 2014 @ Fuji Calm
1/20
1. Introduction
x
■ Particle production from vacuum
■ It is known that a varying background causes production of particles
■ Oscillating Electric field  pair production of electrons
[E. Brezin and C. Itzykson, Phys. Rev. D 2,1191 (1970)]
■ Changing metric  gravitational particle production
[L. Parker, Phys. Rev. 183, 1057 (1969)]
[L. H. Ford, Phys. Rev. D 35, 2955 (1987)]
■ Oscillating inflaton  (p)reheating
[L. Kofman, A. D. Linde, A. A. Starobinsky, Phys. Rev. Lett. 73, 3195 (1994)]
[L. Kofman, A. D. Linde, A. A. Starobinsky, Phys. Rev. D 56, 3258 (1997)]
■ etc…
2014/08/23
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■ Example of scalar particle production
𝑔 : coupling
1
2
■ Let us consider : ℒ𝑖𝑛𝑡 = − 𝑔2 𝜙 2 𝜒 2
𝜙 : complex scalar field
(classical)
𝜒 : real scalar particle
(quantum)
■ If 𝜙 goes near the origin…
 mass of 𝜒 (𝑚𝜒 = 𝑔𝜙) becomes small around 𝜙 = 0
 kinetic energy of 𝜙 converts to 𝜒
 𝜒 particles are produced !!
𝑚𝜒
■ produced occupation number :
𝑛𝜒𝑘 ≡
0in
out† out
𝑎𝜒𝐤
𝑎𝜒𝐤
= 𝑉 ⋅ exp
0in
𝑘 2 +𝑔2 𝜇2
−𝜋
𝑔𝑣
[L. Kofman, A. D. Linde, X. Liu, A. Maloney,
L. McAllister and E. Silverstein,
JHEP 0405, 030 (2004)]
2014/08/23
Summer Institute 2014 @ Fuji Calm
𝝌
𝝌
𝝌
𝝌
𝑡
𝑣
Im 𝜙
𝜇
𝜙
Re 𝜙
3/20
■ Our interests
1. How about supersymmetric model?
2. How do (quantum) interaction terms affect particle production?
■ Usually production rates are calculated in the purely classical
background
 We would like to estimate the contribution of the quantum interaction
term
𝜙
𝜒
quantum
𝜓𝜒
classical
quantum
classical
𝜓𝜙
2014/08/23
Summer Institute 2014 @ Fuji Calm
4/20
■ Model in this talk
Φ = 𝜙 + 2𝜃𝜓𝜙 + 𝜃 2 𝐹𝜙
■ Super potential :
1
Χ = 𝜒 + 2𝜃𝜓𝜒 + 𝜃 2 𝐹𝜒
𝑊 = 𝑔Φ𝑋 2
2
𝑔 : coupling
 Interaction terms in components :
1 2 4
1
2
2
2
ℒ𝑖𝑛𝑡 = −𝑔 𝜙 𝜒 − 𝑔 𝜒 −
𝑔𝜙𝜓𝜒 𝜓𝜒 + 𝑔𝜓𝜙 𝜓𝜒 𝜒 + (ℎ. 𝑐. )
4
2
■ Stationary point :
■ 𝜒 = 𝜓𝜙 = 𝜓𝜒 = 0 , but 𝜙 can have any value
Im 𝜙
■ Masses
■ 𝜙 ≠ 0  𝜒, 𝜓𝜒 : mass = 𝑔𝜙 , 𝜓𝜙 : massless
Production may
be possible
2014/08/23
Impossible…?
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𝜙
Re 𝜙
Our main study!
5/20
■ Towards the calculation of produced numbers
1. Solving Equations of Motion for field operators :
1
2
𝜙 :
0 = 𝜕 2 + 𝑔2 𝜒
2
𝜙 + 𝑔𝜓𝜒† 𝜓𝜒†
𝜒 :
0 = 𝜕 2 + 𝑔2 𝜙
2
+ 𝑔2 𝜒
𝜓𝜙 :
0 = 𝜎𝜇 𝜕𝜇 𝜓𝜙 + 𝑖𝑔𝜒 ∗ 𝜓𝜒†
1
2
2
† †
𝜒 + 𝑔𝜓𝜙
𝜓𝜒
∗ †
∗ †
𝜓𝜒 :Our
0 =approach
𝜎𝜇 𝜕𝜇 𝜓𝜒 + 𝑖𝑔𝜙
𝜓
+
𝑖𝑔𝜒
𝜓𝜙
: 𝜒
Using Yang-Feldman equation
2. Bringing out creation/annihilation operators from field operators
in
in
in
𝜙 𝑎in
𝜒
𝜓𝜙 𝑎𝜓
𝜓𝜒 𝑎𝜓
𝑎𝜒𝐤
𝐤
𝐤
𝜙𝐤
out
𝑎𝜙𝐤
𝜙
out
𝑎𝜒𝐤
out
𝑎𝜓
𝜙𝐤
𝜒
out
𝑎𝜓
𝜒𝐤
3. Estimation of produced numbers
𝑛𝑘 = 0in 𝑎𝐤out† 𝑎𝐤out 0in
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2. Bogoliubov transformation law
with interaction terms
■ An example with a real scalar field Ψ
■ Operator field equation : 0 = 𝜕 2 + 𝑀2 Ψ + 𝐽
■ Commutation relation : Ψ 𝐱 , Ψ(𝐲) = 𝑖𝛿 3 𝐱 − 𝐲
 Formal solution (Yang-Feldman equation)
𝑥0
Ψ 𝑥 = 𝑍Ψas 𝑥 − 𝑖𝑍
𝑑𝑦 0
𝑑 3 𝑦 Ψ as 𝑥 , Ψ as 𝑦
𝐽(𝑦)
𝑡 as
𝑍 : some const.
Ψ as : asymptotic field
0 = 𝜕 2 + 𝑀2 Ψ as
𝐽(𝑦)
Ψ as (𝑥)
𝐽(𝑦)
𝑥
+
𝐽(𝑦)
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■ Relation between in- & out-field operator
■ Set 𝑥 0 = 𝑡 out ≡ +∞
+∞
Ψ 𝑡 out = 𝑍Ψ as 𝑡 out − 𝑖𝑍
𝑑 4 𝑦 Ψ as 𝑡 out , Ψ as 𝑦
𝐽 𝑦
𝑡 as
+∞
= 𝑍Ψin 𝑡 out − 𝑖𝑍
𝑑 4 𝑦 Ψ in 𝑡 out , Ψ in 𝑦
“as” = “in”
𝐽 𝑦
−∞
“as” = “out”
= 𝑍Ψ out 𝑡 out
1
Ψ out 𝑥 out = Ψ in 𝑥 out − 𝑖 𝑍 𝑑 4 𝑦 Ψ in 𝑥 out , Ψ in 𝑦
out
𝑎𝐤
2014/08/23
𝐽 𝑦
in
𝑎𝐤
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■ Representation of 𝑎𝐤as with field operator
■ Ψ as is free particle, so we can expand with plane waves as
Ψ as 𝑥 =
𝑑 3 𝑘 𝑖𝐤⋅𝐱 as 0 as
as∗ 0 as†
𝑒
Ψ
(𝑥
)𝑎
+
Ψ
𝑥 𝑏−𝐤
𝑘
𝑘
𝐤
3
2𝜋
plane wave
(time dependent) wave func.
creation/annihilation op.
0 = Ψ𝑘as + 𝐤 2 + 𝑀2 Ψ𝑘as
■ inner product relation : Ψ𝑘as∗ Ψ𝑘as − Ψ𝑘as∗ Ψ𝑘as = 𝑖/𝑍
which comes from conditions
𝑎𝐤as , 𝑎𝐤as†
= 2𝜋 3 𝛿 3 𝐤 − 𝐤 ′ , 𝑍 Ψ as 𝐱 , Ψ as 𝐲
′
2
2014/08/23
𝑎𝐤as = −𝑖𝑍
𝑡→𝑡 as
= 𝑖ℏ𝛿 3 𝐱 − 𝐲
𝑑 3 𝑥 𝑒 −𝑖𝐤⋅𝐱 Ψ𝑘as∗ Ψas − Ψ𝑘as∗ Ψas
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■ Bogoliubov transformation law
■ Relation between 𝑎𝐤in and 𝑎𝐤out
out
2 𝑎𝐤 = −𝑖𝑍
1
𝑑 3 𝑥 𝑒 −𝑖𝐤⋅𝐱 Ψ𝑘out∗ Ψout − Ψ𝑘out∗ Ψout
Ψ out 𝑥 out = Ψ in 𝑥 out − 𝑖 𝑍 𝑑 4 𝑦 Ψ in 𝑥 out , Ψ in,∗ 𝑦
𝐽 𝑦
in†
𝑎𝐤out = 𝛼𝑘 𝑎𝐤in + 𝛽𝑘 𝑎−𝐤
− 𝑖 𝑍 𝑑 4 𝑥 𝑒 −𝑖𝐤⋅𝐱 𝛼𝑘 Ψ𝑘in∗ − 𝛽𝑘 Ψ𝑘in 𝐽(𝑦)
(ordinary) Bogoliubov tf law
𝛼𝑘 ≡ −𝑖𝑍
Ψ𝑘out∗ Ψ𝑘in
−
Interaction effects
Ψ𝑘out∗ Ψ𝑘in
𝛽𝑘 ≡ −𝑖𝑍 Ψ𝑘out∗ Ψ𝑘in∗ − Ψ𝑘out∗ Ψ𝑘in∗
2014/08/23
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Ψ𝑘in = 𝛼𝑘 Ψ𝑘out + 𝛽𝑘 Ψ𝑘out∗
Ψ𝑘out = 𝛼𝑘∗ Ψ𝑘in − 𝛽𝑘∗ Ψ𝑘in∗
𝛼𝑘
2
− 𝛽𝑘
2
=1
10/20
■ Produced (occupation) number :
𝑛𝑘 = 0in 𝑎𝐤out† 𝑎𝐤out 0in
=
=
in†
𝛽𝑘 𝑎−𝐤
−𝑖 𝑍
𝑉 ⋅ 𝛽𝑘
0
2
𝑑4 𝑥
𝑒 −𝑖𝐤⋅𝐱
𝛼𝑘 Ψ𝑘in∗
−
𝛽𝑘 Ψ𝑘in
𝐽
2
𝛽𝑘 ≠ 0
+⋯
+𝑍
|0in
𝑑 4 𝑥 𝑒 −𝑖𝐤⋅𝐱 Ψ𝑘in∗ 𝐽 |0in
2
+⋯
𝛽𝑘 = 0
 Particles can be produced even if 𝛽𝑘 = 0 !
2014/08/23
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3. Application to our model
■ Equation of Motion (again) :
1
𝜙 :
0 = 𝜕 2 + 𝑔2 𝜒
2
𝜙 + 2 𝑔𝜓𝜒† 𝜓𝜒†
𝜒 :
0 = 𝜕 2 + 𝑔2 𝜙
2
+ 2 𝑔2 𝜒
𝜓𝜙 :
0 = 𝜎𝜇 𝜕𝜇 𝜓𝜙 + 𝑖𝑔𝜒 ∗ 𝜓𝜒†
𝜓𝜒 :
†
0 = 𝜎𝜇 𝜕𝜇 𝜓𝜒 + 𝑖𝑔𝜙 ∗ 𝜓𝜒† + 𝑖𝑔𝜒 ∗ 𝜓𝜙
1
2
† †
𝜒 + 𝑔𝜓𝜙
𝜓𝜒
■ EOM for asymptotic fields (as = in, out):
𝜙 as : 0 = 𝜕 2 𝜙 as
𝜒 as : 0 = 𝜕 2 + 𝑔2 𝜙 2 𝜒 as
as
as
𝜓𝜙
: 0 = 𝜎𝜇 𝜕𝜇 𝜓𝜙
𝜓𝜒as :
2014/08/23
𝜙 : macroscopic
𝜒 , 𝜓𝜙 , 𝜓𝜒 : microscopic
0 = 𝜎𝜇 𝜕𝜇 𝜓𝜒as + 𝑖𝑔 𝜙 ∗ 𝜓𝜒as†
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■ Solutions for Asymptotic fields
Im 𝜙
■ Assuming 𝜙 = 𝜙 (𝑡) for simplicity, then
■
𝜙 as
=
𝜙 as
+
𝑑 3 𝑘 𝑖𝐤⋅𝐱
𝑒
2𝜋 3
as
𝜙𝑘as 𝑎𝜙𝐤
■ 𝜒
as
𝑍𝜒 𝜒𝑘in ∼
= 𝜕 2 + 𝑔2 𝜙
2
𝜒 as
1
2𝐤
𝑒 −𝑖 𝐤 𝑡
as†
as
𝜒𝑘as 𝑎𝜒𝐤
+ 𝜒𝑘as∗ 𝑏𝜒−𝐤
1
2𝜔𝑘
exp −𝑖
𝜔𝑘 ≡
𝑡
𝑑𝑡 ′ 𝜔𝑘 (𝑡 ′ )
𝐤 2 + 𝑍𝜙 𝑔2 𝜙
(analytic continuation)  𝛽𝜒𝑘 ∼ −𝑖 exp
2014/08/23
𝑍𝜙 𝜙𝑘out =
𝜙𝑘out = 𝛼𝑘∗ 𝜙𝑘in − 𝛽𝑘∗ 𝜙𝑘in∗
𝑑 3 𝑘 𝑖𝐤⋅𝐱
𝑒
2𝜋 3
=
Re 𝜙
as†
𝜙𝑘as∗ 𝑏𝜙−𝐤
 𝛽𝜙𝑘 = 0
𝜙
𝜇
𝑍𝜙 𝜙𝑘in =
𝑍𝜙 𝜙 in = 𝑣𝑡 + 𝑖𝜇 ,
0 = 𝜕 2 𝜙 as
+
𝑣
( valid for 𝑡 ≳ 1/ 𝑔𝑣 )
2
𝑘 2 +𝑔2 𝜇2
−𝜋
2𝑔𝑣
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Im 𝑡
Re 𝑡
𝜒 out
𝜒 in
13/20
■ Solutions for Asymptotic fields
■
as
𝜓𝜙
=
𝑑 3 𝑘 𝑖𝐤⋅𝐱
𝑒
2𝜋 3
±,in
𝑍𝜙 𝜓𝜙𝑘
=
eigen spinor for helicity op. :
𝑘 𝑖 𝜎 𝑖 𝑒𝐤± = ± 𝐤 𝜎 0 𝑒𝐤±
+,as +as
−,as∗ −as†
𝑒𝐤+ 𝜓𝜙𝑘
𝑎𝜓𝜙𝐤 + 𝑒𝐤− 𝜓𝜙𝑘
𝑎𝜓𝜙 −𝐤
±,out
𝑍𝜙 𝜓𝜙𝑘
= 𝑒 −𝑖 𝐤 𝑡
as
0 = 𝜎𝜇 𝜕𝜇 𝜓𝜙
 𝛽𝜓±𝜙 𝑘 = 0
■
𝜓𝜒as
=
𝑑 3 𝑘 𝑖𝐤⋅𝐱
𝑒
2𝜋 3
± 𝑠,in
𝑍𝜒 𝜓𝜒𝑘
∼
1
2
+ 𝑠,as 𝑠
𝑎𝜓𝜒𝐤
𝑠=±
𝑒𝐤𝑠 𝜓𝜒𝑘
1−
𝑔𝑣𝑡
𝜔𝑘
± 1+
𝑔𝑣𝑡 𝑖𝜃 𝑠
𝑒 𝑘
𝜔𝑘
0 = 𝜎𝜇 𝜕𝜇 𝜓𝜒as + 𝑖𝑔 𝜙 ∗ 𝜓𝜒as†
𝜔𝑘 ≡
exp −𝑖
𝑡
𝑑𝑡 ′ 𝜔𝑘 𝑡 ′
( valid for 𝑡 ≳ 1/ 𝑔𝑣 )
𝐤2
+ 𝑍𝜙
(analytic continuation) 
2014/08/23
− 𝑠,as∗ 𝑠†
𝑎𝜓𝜒−𝐤
𝑠†
− 𝜎 0 𝑒−𝐤
𝜓𝜒𝑘
𝑔2
𝜙
𝛽𝜓𝑠 𝜒𝑘
2
,
𝑠
𝑖𝜃
𝑘
𝑒
≡
∼ −𝑠 ⋅ exp
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𝑠 𝐤 −𝑖𝑔𝜇
𝐤 2 +𝑔2 𝜇2
𝑘 2 +𝑔2 𝜇2
−𝜋
2𝑔𝑣
14/20
■ Produced Particle number
■ 𝑛𝜒𝑘 = 2 × 𝛽𝜒𝑘
■ 𝑛𝜓𝜒 𝑘 =
2
+ ⋯ = 2 × exp
𝑠
𝛽
𝑠=± 𝜓𝜒 𝑘
2
𝑘 2 +𝑔2 𝜇 2
−𝜋
𝑔𝑣
+ ⋯ = 2 × exp
+⋯
𝑘 2 +𝑔2 𝜇 2
−𝜋
𝑔𝑣
+⋯
 leading term is obtained
■ 𝑛𝜙𝑘 = 2 × 𝛽𝜙𝑘
■ 𝑛𝜓𝜙 𝑘 =
2
+⋯=0+⋯
𝑠
𝑠=± 𝛽𝜓𝜙 𝑘
2
+⋯=0+⋯
Focus on!
 need to calculate next to leading order
2014/08/23
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15/20
■ Leading term of 𝑛𝜓𝜙𝑘
𝑛𝜓𝜙 𝑘 =
=
in
𝑍𝜙 𝜓𝜙𝑘
𝑍𝜒 𝜒𝑘in
s,out† s,out in
0in 𝑎𝜓
𝑎𝜓𝜙 𝐤 0
𝜙𝐤
𝑠
𝑔2 𝑍𝜙 𝑍𝜒2
∼𝑉⋅𝑔
=
∼
𝑠 𝑟,in
𝑍𝜒 𝜓𝜒𝑘
𝐤+𝐩
2
𝑠
𝑍𝜙 𝑍𝜒2
𝑑4 𝑥
in∗
𝑒 −𝑖𝐤⋅𝐱 𝜓𝜙𝑘
𝑑3𝑝
2𝜋 3
1
𝑠,𝑟 2
⋅
1−
𝜒 in∗
⋅
𝑠† in† in 2
𝑒𝐤 𝜓𝜒 |0
𝐩⋅𝐤
𝑠𝑟
𝑝𝑘
𝑑𝑡
𝑡
1
−𝑖 𝑑𝑡 ′ 𝜔𝑘 𝑡 ′
𝑒
2𝜔𝑘
∼
𝜒
𝜓𝜙
𝐩
X
𝜓𝜒
+⋯
𝜒 in
𝐤+𝐩
𝑠 𝑟,in
𝜓𝜒𝑝
2
Im 𝑡
𝑒 −𝑖 𝐤 𝑡
1
2
in∗
𝜓𝜙𝑘
𝐤
1
𝑔𝑣𝑡
−𝜔
𝑘
Re 𝑡
+𝑠 1+
𝑔𝑣𝑡 𝑖𝜃 𝑟
𝑒 𝑘
𝜔𝑘
𝑒
−𝑖
𝑡
𝑑𝑡 ′ 𝜔𝑘 𝑡 ′
steepest decent
method
 We estimated in special case 𝐤 = 0 with steepest decent method
2014/08/23
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16/20
■ Analytical Result @ 𝑘 = 0
𝑔2
𝑔2 𝜇 2
𝑛𝜓𝜙 𝑘=0 /𝑉 = 𝐶 ⋅
⋅ exp −𝜋
4𝜋
𝑔𝑣
Γ 4/3 2 6
𝐶=
𝜋𝑒 1/3 𝜋
5/6
(c.f.) 𝑛𝜒𝑘=0 /𝑉 ∼ 𝑛𝜓𝜒 𝑘=0 /𝑉 ∼ 2 × exp
∼ 0.32
𝑔2 𝜇 2
−𝜋
𝑔𝑣
■ Produced number of 𝜓𝜙 is suppressed by factor 𝑔2 comparing with 𝜒
or 𝜓𝜒
■ This results is consistent with perturbativity
2014/08/23
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17/20
■ Leading term of 𝑛𝜙𝑘 (only final formula)
out† out in
out† out in
■ 𝑛𝜙𝑘 ≡ 0in 𝑎𝜙𝐤
𝑎𝜙𝐤 0 + 0in 𝑏𝜙𝐤
𝑏𝜙𝐤 0
𝑎 ≡ 𝑎 − 𝑎 ,𝑏 ≡ 𝑏 − 𝑏
⋮
=𝑉⋅𝑔
𝐤+𝐩
𝜙
+
X
𝑑𝑡
1
4
2
+ 𝑍𝜙2 𝑍𝜓
𝜒
𝜒
𝜙
×
𝐤+𝐩
𝐤
𝜙
in
in
𝑑𝑡 𝜙𝑘out 𝜒 in
𝜒
⋅
𝑔
𝜙
𝐤+𝐩 𝑝
𝑍𝜙2 𝑍𝜒2
𝜒
𝐤
𝐩
𝑑3 𝑝
2𝜋 3
2
𝐩
𝑑𝑡
𝜙𝑘out
𝑠,𝑟
X
+
1 + 𝑠𝑟
𝜙𝑘out
𝜓𝜒
𝑑𝑡
𝜒 in
𝐤+𝐩
⋅𝑔
2
in∗
𝜙
𝐩⋅ 𝐤+𝐩
𝑝 𝐤+𝐩
+ 𝑠,in
𝜓𝜒 𝐤+𝐩
𝜙𝑘out
𝜒𝑝in
2
+ 𝑟,in
𝜓𝜒𝑝
− 𝑠,in
𝜓𝜒 𝐤+𝐩
2
− 𝑟,in
𝜓𝜒𝑝
2
+⋯
𝜓𝜒
2014/08/23
Summer Institute 2014 @ Fuji Calm
18/20
■ Numerical Results ( 𝑔 = 1, 𝑣 = 0.5, 𝜇 = 0.05 )
2.7 × 10−3
0.15 × 10−3
0.32 ×
𝑔2
×
4𝜋
2014/08/23
consistent with analytical expected value
𝑘 2 + 𝑔2 𝜇2
∼ 2 × exp −𝜋
𝑔𝑣
Summer Institute 2014 @ Fuji Calm
19/20
4. Summary
1. We constructed the Bogoliubov transformation taking into account
interaction effects
2. We calculated produced particle’s (occupation) number
■ 𝑛𝜒𝑘 = 𝑛𝜓𝜒 𝑘 ∼ 𝑉 ⋅ 2 exp
𝑘 2 +𝑔2 𝜇 2
−𝜋
𝑔𝑣
𝑠
■ 𝑛𝜓
, 𝑛𝜙𝑘 ≠ 0
𝜙𝑘
■ Massless particle can be produced, however the production is
suppressed by the coupling ( ∼ 𝑔2 /4𝜋 )
■ However, the produced number in case of strong couplings may be
comparable to massive particles
2014/08/23
Summer Institute 2014 @ Fuji Calm
20/20
2014/08/23
Summer Institute 2014 @ Fuji Calm
21/20
■ Numerical Results 2 ( 𝑔 = 4, 𝑣 = 0.5, 𝜇 = 0.05 )
2014/08/23
Summer Institute 2014 @ Fuji Calm
22/20