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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)

2014/08/23 Summer Institute 2014 @ Fuji Calm

1. Introduction2. Bogoliubov transformation law with

interaction terms3. Application to our model4. Summary

Outline

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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

)) ))

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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)]

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■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

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■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!

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■ 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

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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

+¿

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■ Relation between in- & out-field operator■ Set

2014/08/23 Summer Institute 2014 @ Fuji Calm

𝑎𝐤out 𝑎𝐤

¿

“as” = “in”

“as” = “out”

1

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■ 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

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■ 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

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■ Produced (occupation) number :

Particles can be produced even if !

2014/08/23 Summer Institute 2014 @ Fuji Calm

𝛽𝑘=0𝛽𝑘≠0

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■ 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

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■ 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

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■ 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. :

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■ Produced Particle number

leading term is obtained

need to calculate next to leading order

2014/08/23 Summer Institute 2014 @ Fuji Calm

Focus on!

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■ 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

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■ 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

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■ 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