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Effect of interaction terms on particle production due to time-varying mass. [work in progress] Seishi Enomoto (Univ. of Warsaw) Collaborators : Olga Fuksińska (Univ . of Warsaw) Zygmunt Lalak (Univ. of Warsaw). Outline. Introduction - PowerPoint PPT Presentation
Citation preview
1/20
Effect of interaction terms on particle production due to time-varying mass
[work in progress]
Seishi Enomoto (Univ. of Warsaw)
Collaborators : Olga Fuksińska (Univ. of Warsaw)Zygmunt Lalak (Univ. of Warsaw)
2014/08/23 Summer Institute 2014 @ Fuji Calm
1. Introduction2. Bogoliubov transformation law with
interaction terms3. Application to our model4. Summary
Outline
2/20
1. Introduction■ 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 Summer Institute 2014 @ Fuji Calm
VACUUM
x
)) ))
3/20
𝑚𝜒
𝑡
■ Example of scalar particle production■ Let us consider :
■ If goes near the origin… mass of () becomes small around kinetic energy of converts to particles are produced !!
■ produced occupation number :
2014/08/23 Summer Institute 2014 @ Fuji Calm
𝑣
ℜ𝜙
ℑ𝜙𝜇𝜙
: complex scalar field (classical)
: real scalar particle (quantum)
: coupling
𝝌 𝝌𝝌 𝝌
[L. Kofman, A. D. Linde, X. Liu, A. Maloney, L. McAllister and E. Silverstein, JHEP 0405, 030 (2004)]
4/20
■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
2014/08/23 Summer Institute 2014 @ Fuji Calm
𝜙
𝜓𝜙
𝜒
𝜓 𝜒
classical
classicalquantum
quantum
quantumclassical
5/20
■Model in this talk
■ Super potential :
Interaction terms in components :
■ Stationary point :
■ , but can have any value
■ Masses
■ , : massless
Φ=𝜙+√2𝜃𝜓𝜙+𝜃2𝐹𝜙
Χ=𝜒+√2𝜃𝜓𝜒+𝜃2𝐹 𝜒
: coupling
2014/08/23 Summer Institute 2014 @ Fuji Calm
ℑ𝜙
ℜ𝜙
𝜙
Production may be possible Impossible…? Our main study!
6/20
■ Towards the calculation of produced numbers
1. Solving Equations of Motion for field operators : : : : :
2. Bringing out creation/annihilation operators from field operators
3. Estimation of produced numbers
2014/08/23 Summer Institute 2014 @ Fuji Calm
How do we calculate
with interaction term?
𝑎𝜓 𝜒𝐤out𝑎𝜓 𝜒𝐤
¿
𝑎𝜓𝜙 𝐤out𝑎𝜓𝜙 𝐤
¿
𝑎𝜒 𝐤out𝑎𝜒 𝐤
¿
𝑎𝜙𝐤out𝑎𝜙𝐤
¿
?Relation?
?Relation?
?Relation?
?Relation?How do we
bring out?
Our approach : Using Yang-Feldman equation
7/20
2. Bogoliubov transformation law with interaction terms
■ An example with a real scalar field
■ Operator field equation : ■ Commutation relation :
Formal solution (Yang-Feldman equation)
2014/08/23 Summer Institute 2014 @ Fuji Calm
: some const. : asymptotic field
𝐽 (𝑦 )
𝐽 (𝑦 )
𝐽 (𝑦 )Ψ as(𝑥 ) 𝑥
xxx
xx
+¿
8/20
■ Relation between in- & out-field operator■ Set
2014/08/23 Summer Institute 2014 @ Fuji Calm
𝑎𝐤out 𝑎𝐤
¿
“as” = “in”
“as” = “out”
1
9/20
■ Representation of with field operator
■ is free particle, so we can expand with plane waves as
■ inner product relation :
which comes from conditions ,
2014/08/23 Summer Institute 2014 @ Fuji Calm
(time dependent) wave func. creation/annihilation op.plane wave
2
10/20
■ Bogoliubov transformation law■ Relation between and
2014/08/23 Summer Institute 2014 @ Fuji Calm
Ψ out (𝑥out )=Ψ ¿ (𝑥out )−𝑖√𝑍∫𝑑4 𝑦 [Ψ ¿ (𝑥out ) ,Ψ ¿ ,∗ (𝑦 ) ] 𝐽 ( 𝑦 )
Interaction effects(ordinary) Bogoliubov tf law
2
1
11/20
■ Produced (occupation) number :
Particles can be produced even if !
2014/08/23 Summer Institute 2014 @ Fuji Calm
𝛽𝑘=0𝛽𝑘≠0
12/20
■ Equation of Motion (again) :
■ EOM for asymptotic fields (as = in, out): : : : :
3. Application to our model
2014/08/23 Summer Institute 2014 @ Fuji Calm
: : : :
: macroscopic , , : microscopic
13/20
■ Solutions for Asymptotic fields
■ Assuming for simplicity, then
,
■ ( valid for )
(analytic continuation)
2014/08/23 Summer Institute 2014 @ Fuji Calm
ℑ𝜙
ℜ𝜙
𝜙𝑣𝜇
𝜙𝑘out=𝛼𝑘
∗𝜙𝑘¿− 𝛽𝑘
∗𝜙𝑘¿∗
ℑ𝑡ℜ𝑡
𝜒 ¿ 𝜒out
0=𝜕2𝜙as
¿ (𝜕2+𝑔2|⟨𝜙 ⟩|2 ) 𝜒 as
14/20
■ Solutions for Asymptotic fields
( valid for )
(analytic continuation)
2014/08/23 Summer Institute 2014 @ Fuji Calm
0=𝜎𝜇𝜕𝜇𝜓𝜙as
0=𝜎𝜇𝜕𝜇𝜓 𝜒as+𝑖𝑔 ⟨𝜙∗ ⟩𝜓𝜒
as†
eigen spinor for helicity op. :
15/20
■ Produced Particle number
leading term is obtained
need to calculate next to leading order
2014/08/23 Summer Institute 2014 @ Fuji Calm
Focus on!
16/20
■ Leading term of
We estimated in special case with steepest decent method
2014/08/23 Summer Institute 2014 @ Fuji Calm
steepest decent
method
ℑ𝑡
ℜ𝑡
X𝜓𝜙 𝜓 𝜒
𝜒𝐤𝐤+𝐩
𝐩
17/20
■ Analytical Result @
(c.f.)
■ Produced number of is suppressed by factor comparing with or ■ This results is consistent with perturbativity
2014/08/23 Summer Institute 2014 @ Fuji Calm
𝐶=Γ (4 /3 )2
𝜋𝑒1/3 ( 6𝜋 )5/6
∼0.32
18/20
■ Leading term of (only final formula)
2014/08/23 Summer Institute 2014 @ Fuji Calm
X𝜒𝐤
𝐤+𝐩
𝐩𝜙⟨𝜙 ⟩
𝜒
X𝜓 𝜒𝐤
𝐤+𝐩
𝐩𝜙𝜓 𝜒
Summer Institute 2014 @ Fuji Calm 19/202014/08/23
■Numerical Results Preliminary
consistent with analytical expected value
0.15×10−3
2 .7×10−3
Summer Institute 2014 @ Fuji Calm 20/20
4. Summary1. We constructed the Bogoliubov transformation taking into account
interaction effects
2. We calculated produced particle’s (occupation) number
■ ,
■ Massless particle can be produced, however the production is suppressed by the coupling
■ However, the produced number in case of strong couplings may be comparable to massive particles
2014/08/23
Summer Institute 2014 @ Fuji Calm 21/202014/08/23
Summer Institute 2014 @ Fuji Calm 22/202014/08/23
■Numerical Results 2 Preliminary