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

1. Effect of interaction terms on particle production due to time-varying mass Outline Seishi Enomoto (Univ. of Warsaw) Collaborators : Olga Fuksińska (Univ. of Warsaw) ZygmuntLalak (Univ. of Warsaw) Introduction How to calculate particle number Calculation of particle number Summary Planck2014 @ Institut des Cordeliers

2. 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… Planck2014 @ Institut des Cordeliers

3. : coupling • 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 toparticles • produced occupation number : : complex scalar field (classical) : real scalar particle (quantum) Planck2014 @ Institut des Cordeliers

4. Our interests • How about supersymmetric model? • What is the role of the superpartner of the background field? • 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 classical quantum classical classical quantum quantum Planck2014 @ Institut des Cordeliers

5. 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? : coupling Production is possible impossible Planck2014 @ Institut des Cordeliers

6. Equations of Motion for field operators : : : : : How do we calculate produced particle number including interaction term ? Planck2014 @ Institut des Cordeliers

7. 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 Planck2014 @ Institut des Cordeliers

8. Example in a simple case (scalar field) • Operator field equation : • Commutation relation :  Formal solution (Yang-Feldman equations) • If we take or , : some const. : asymptotic field Planck2014 @ Institut des Cordeliers

9. 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 , plane wave (time dependent) wave func. creation/annihilation op. Planck2014 @ Institut des Cordeliers

10. Example in a simple case (scalar field) • Relation between and (usual) Bogoliubov tf low Interaction effects Planck2014 @ Institut des Cordeliers

11. Example in a simple case (scalar field) • Produced (occupation) number :  Particles can be produced even if ! Planck2014 @ Institut des Cordeliers

12. 3. Calculation of particle number • Equation of Motion (again) : • EOM for asymptotic fields (as = in, out): : : : : : : : : : macroscopic , , : microscopic Planck2014 @ Institut des Cordeliers

13. Solutions for Asymptotic fields • Assuming for simplicity, then ,  ( valid for ) (analytic continuation)  Planck2014 @ Institut des Cordeliers

14. eigen spinor for helicity op. : • Solutions for Asymptotic fields  ( valid for ) (analytic continuation)  Planck2014 @ Institut des Cordeliers

15. Produced Particle number  leading term is obtained  need to calculate next to leading order Focus on! Planck2014 @ Institut des Cordeliers

16. Calculation of • This integral is too complicated to perform  We estimated in special case with steepest decent method steepest decent method Planck2014 @ Institut des Cordeliers

17. Result (c.f.) , • Produced number of is suppressed by factor comparing with or • This results is consistent with perturbativity • can be removed by redefinition : ,, Planck2014 @ Institut des Cordeliers

18. 4. Summary • We constructed the Bogoliubov transformation taking into account interaction effects • 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 Planck2014 @ Institut des Cordeliers

19. Planck2014 @ Institut des Cordeliers

20. Introduction • Brief overview of particle production • Our Model • How to calculate particle number • Relation between in-field and out-field • Bogoliubov transformation within interaction term • Calculation of produced particle number • Asymptotic solution, Formal solutions • Produced particle number • Summary Planck2014 @ Institut des Cordeliers

21. : coupling • 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! : real scalar particle (quantum) : complex scalar field (classical) Planck2014 @ Institut des Cordeliers

22. If go through near the origin (ESP), particles are produced • Because  mass of ()becomes small around  kinetic energy of converts toparticles • The produced occupation number : Therefore the mass of produced particle must be varied on time ESP Planck2014 @ Institut des Cordeliers

23. If go through near the origin, particles are produced • Because  mass of ()becomes small around  kinetic energy of converts toparticles • The produced occupation number : Therefore the mass of produced particle must be varied on time Planck2014 @ Institut des Cordeliers

24. : macroscopic , , : microscopic • Minimal Model with Flat Direction • Equations of Motion : : : : : • For simplicity, solution : • , , : free field equations  plane wave expansion ( , ) plane wave (time dependent) wave func. creation/annihilation op. eigen spinor for helicity op. Planck2014 @ Institut des Cordeliers

25. Minimal Model with Flat Direction • Plane wave expansion : • WKB solutions for wave functions Planck2014 @ Institut des Cordeliers

26. : macroscopic , , : microscopic • Minimal Model with Flat Direction • Equations of Motion : : : : : • EOM for asymptotic fields (as = in, out): : : : : How do we calculate particle number ? Planck2014 @ Institut des Cordeliers

27. 4. Summary • We constructed the Bogoliubov transformation taking into account interaction effects • 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 Planck2014 @ Institut des Cordeliers

28. Bogoliubov transformation 2. Estimation of produced particle number • WKB type solution (wave func. of) , : generalized Bogoliubov coefficients ⇒  • Occupation number • Initial conditions ：  In state out state If you want to know the particle number, you only calculate . Planck2014 @ Institut des Cordeliers

29. Bogoliubov transformation • Estimation of produced particle number • WKB type solution (wave func. of) , : generalized Bogoliubov coefficients ⇒ • EOMs : ↓ • Initial conditions ： 　　⇒　 Planck2014 @ Institut des Cordeliers