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Effect of interaction terms on particle production due to time-varying mass

Seishi Enomoto (Univ. of Warsaw)

Collaborators : Olga Fuksińska (Univ. of Warsaw)Zygmunt Lalak (Univ. of Warsaw)

2014/05/29 Planck2014 @ Institut des Cordeliers

1. Introduction2. How to calculate particle number3. Calculation of particle number4. 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…

2014/05/29 Planck2014 @ Institut des Cordeliers

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𝑚𝜒

𝑡

■ Example of scalar particle production■ Let us consider :

■ If goes near the origin, particles are produced

■ Because mass of () becomes small around kinetic energy of converts to particles

■ produced occupation number :

2014/05/29 Planck2014 @ Institut des Cordeliers

𝑣

ℜ𝜙

ℑ𝜙𝜇𝜙

: complex scalar field (classical)

: real scalar particle (quantum)

: coupling

𝝌𝝌𝝌

𝝌𝝌𝝌

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■ Our interests

1. How about supersymmetric model?■ What is the role of the superpartner of the background field?

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

2014/05/29 Planck2014 @ Institut des Cordeliers

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■ However, ’s mass may be influenced by through loop effects…?■ Is quantum part of also influenced?

Φ=𝜙+√2𝜃𝜓𝜙+𝜃2𝐹𝜙

Χ=𝜒+√2𝜃𝜓𝜒+𝜃2𝐹 𝜒

: coupling

2014/05/29 Planck2014 @ Institut des Cordeliers

ℑ𝜙

ℜ𝜙

𝜙Production is possible impossible

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■ Equations of Motion for field operators : : : : :

2014/05/29 Planck2014 @ Institut des Cordeliers

How do we calculate produced particle number including interaction term ?

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2. How to calculate particle number■ Definition of (occupation) number :

Information about in-state (@ far past) and out-state (@ far future) of field needs for the calculation

■ How are they related to each other?

(in-state) (out-state)

Asymptotic field expansion

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■ Example in a simple case (scalar field)

■ Operator field equation : ■ Commutation relation :

Formal solution (Yang-Feldman equations)

■ If we take or ,

2014/05/29 Planck2014 @ Institut des Cordeliers

: some const. : asymptotic field𝑎𝐤out𝑎𝐤

¿

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■ Example in a simple case (scalar field)

■ is free particle, so we can expand with plane waves as

■ inner product relation :

which comes from conditions ,

2014/05/29 Planck2014 @ Institut des Cordeliers

(time dependent) wave func. creation/annihilation op.plane wave

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■ Example in a simple case (scalar field)

■ Relation between and

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Interaction effects(usual) Bogoliubov tf low

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■ Example in a simple case (scalar field)

■ Produced (occupation) number :

Particles can be produced even if !

2014/05/29 Planck2014 @ Institut des Cordeliers

𝛽𝑘=0𝛽𝑘≠0

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■ Equation of Motion (again) :

■ EOM for asymptotic fields (as = in, out): : : : :

3. Calculation of particle number

2014/05/29 Planck2014 @ Institut des Cordeliers

: : : :

: macroscopic , , : microscopic

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■ Solutions for Asymptotic fields

■ Assuming for simplicity, then

,

■ ( valid for )

(analytic continuation)

2014/05/29 Planck2014 @ Institut des Cordeliers

ℑ𝜙

ℜ𝜙

𝜙𝑣𝜇

𝜙𝑘out=𝛼𝑘

∗𝜙𝑘¿− 𝛽𝑘

∗𝜙𝑘¿∗

ℑ𝑡ℜ𝑡

𝜒 ¿ 𝜒out

0=𝜕2𝜙as

¿ (𝜕2+𝑔2|⟨𝜙as ⟩|2 ) 𝜒 as

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■ Solutions for Asymptotic fields

( valid for )

(analytic continuation)

2014/05/29 Planck2014 @ Institut des Cordeliers

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

2014/05/29 Planck2014 @ Institut des Cordeliers

Focus on!

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■ Calculation of

■ This integral is too complicated to perform We estimated in special case with steepest decent method

2014/05/29 Planck2014 @ Institut des Cordeliers

steepest decent

method

ℑ𝑡

ℜ𝑡

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■ Result

(c.f.) ,

■ Produced number of is suppressed by factor comparing with or ■ This results is consistent with perturbativity

■ can be removed by redefinition : ,,

2014/05/29 Planck2014 @ Institut des Cordeliers

𝐶=( 59𝜋 𝑒)

5/6 Γ (1 /3 )2

31/3⋅ 2𝜋∼0.06765

Planck2014 @ Institut des Cordeliers 18/18

4. Summary1. We constructed the Bogoliubov transformation taking into account

interaction effects2. We calculated produced particle’s (occupation) number

■ , ■

■ Massless particle can be produced, however the production is suppressed by the coupling

■ … now in progress■ In the case of exact SUSY one may find the same result■ It would be interesting to see modification induced by supersymmetry

breaking terms- this is issue under investigation

2014/05/29

Planck2014 @ Institut des Cordeliers 19/182014/05/29

Planck2014 @ Institut des Cordeliers 20/18

1. Introduction1. Brief overview of particle production2. Our Model

2. How to calculate particle number1. Relation between in-field and out-field2. Bogoliubov transformation within interaction term

3. Calculation of produced particle number1. Asymptotic solution, Formal solutions2. Produced particle number

4. Summary

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■ Essense[L.Kofman, A.Linde, X.Liu, A.Maloney, L.McAllister, E.Silverstein, JHEP 0405, 030 (2004)]

■ Lagrangian :

■ For simplicity, we take EOM :

■ In case : solution :

★ can be any value and can move!

: complex scalar field (classical)

: real scalar particle (quantum)

: coupling

2014/05/29 Planck2014 @ Institut des Cordeliers

ℑ𝜙

ℜ𝜙

𝜙

𝜙 𝜙

𝑣

𝑣

𝑣

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■ If go through near the origin (ESP), particles are produced

■ Because mass of () becomes small around kinetic energy of converts to particles

■ The produced occupation number :

Therefore the mass of produced particle must be varied on time2014/05/29 Planck2014 @ Institut des Cordeliers

𝑣

ℜ𝜙

ℑ𝜙

ESP𝜇

𝜒 𝜒

𝜙

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■ If go through near the origin, particles are produced

■ Because mass of () becomes small around kinetic energy of converts to particles

■ The produced occupation number :

Therefore the mass of produced particle must be varied on time

2014/05/29 Planck2014 @ Institut des Cordeliers

ℑ𝜙

ℜ𝜙

𝜙𝑣𝜇

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(time dependent) wave func.

creation/annihilation op.eigen spinor for helicity op.

plane wave

■ Minimal Model with Flat Direction■ Equations of Motion :

: : : :

■ For simplicity, solution :

■ , , : free field equations plane wave expansion

( , )

2014/05/29 Planck2014 @ Institut des Cordeliers

: macroscopic , , : microscopic

ℑ𝜙

ℜ𝜙

𝜙𝑣𝜇

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■ Minimal Model with Flat Direction■ Plane wave expansion :

■ WKB solutions for wave functions

2014/05/29 Planck2014 @ Institut des Cordeliers

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■ Minimal Model with Flat Direction

■ Equations of Motion : : : : :

■ EOM for asymptotic fields (as = in, out): : : : :

2014/05/29 Planck2014 @ Institut des Cordeliers

: macroscopic , , : microscopic

How do we calculate particle number ?

Planck2014 @ Institut des Cordeliers 27/18

4. Summary1. We constructed the Bogoliubov transformation taking into account

interaction effects2. We calculated produced particle’s (occupation) number

■ ,

■ ■ Massless particle can be produced, however the production is

suppressed by the coupling

■ … now in progress■ It may become same quantity because of SUSY■ However, it is interesting to find out whether it is same or not

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2. Estimation of produced particle number

■ WKB type solution (wave func. of )

, : generalized Bogoliubov coefficients ⇒

■ Occupation number　　

■ Initial conditions ：

2014/05/29

Bogoliubov transformation

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In state

out state

If you want to know the particle number, you only calculate .

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■ Estimation of produced particle number

■ WKB type solution (wave func. of )

, : generalized Bogoliubov coefficients ⇒

■ EOMs : ↓

■ Initial conditions ： 　　⇒　

2014/05/29

Bogoliubov transformation

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