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

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

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

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  1. 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) ZygmuntLalak (Univ. of Warsaw) Outline Introduction Bogoliubov transformation law with interaction terms Application to our model Summary Summer Institute 2014 @ Fuji Calm

  2. VACUUM 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… Summer Institute 2014 @ Fuji Calm

  3. : coupling • 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 : : complex scalar field (classical) : real scalar particle (quantum) [L. Kofman, A. D. Linde, X. Liu, A. Maloney, L. McAllister and E. Silverstein, JHEP0405, 030 (2004)] Summer Institute 2014 @ Fuji Calm

  4. Our interests • How about supersymmetric model? • 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 Summer Institute 2014 @ Fuji Calm

  5. Model in this talk • Super potential :  Interaction terms in components : • Stationary point : • , but can have any value • Masses •  , : massless : coupling Production may be possible Impossible…? Our main study! Summer Institute 2014 @ Fuji Calm

  6. How do we calculate with interaction term? • Towards the calculation of produced numbers • Solving Equations of Motion for field operators : : : : : • Bringing out creation/annihilation operators from field operators • Estimation of produced numbers Our approach : Using Yang-Feldman equation How do we bring out? ?Relation? ?Relation? ?Relation? ?Relation? Summer Institute 2014 @ Fuji Calm

  7. 2. Bogoliubov transformation law with interaction terms • An example with a real scalar field • Operator field equation : • Commutation relation :  Formal solution (Yang-Feldman equation) : some const. : asymptotic field x x x x x Summer Institute 2014 @ Fuji Calm

  8. Relation between in- & out-field operator • Set “as” = “in” “as” = “out” 1 Summer Institute 2014 @ Fuji Calm

  9. Representation of with field operator • 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. 2 Summer Institute 2014 @ Fuji Calm

  10. Bogoliubov transformation law • Relation between and 2 1 (ordinary) Bogoliubov tf law Interaction effects Summer Institute 2014 @ Fuji Calm

  11. Produced (occupation) number :  Particles can be produced even if ! Summer Institute 2014 @ Fuji Calm

  12. 3. Application to our model • Equation of Motion (again) : • EOM for asymptotic fields (as = in, out): : : : : : : : : : macroscopic , , : microscopic Summer Institute 2014 @ Fuji Calm

  13. Solutions for Asymptotic fields • Assuming for simplicity, then ,  ( valid for ) (analytic continuation)  Summer Institute 2014 @ Fuji Calm

  14. eigen spinor for helicity op. : • Solutions for Asymptotic fields  ( valid for ) (analytic continuation)  Summer Institute 2014 @ Fuji Calm

  15. Produced Particle number  leading term is obtained  need to calculate next to leading order Focus on! Summer Institute 2014 @ Fuji Calm

  16. X • Leading term of  We estimated in special case with steepest decent method steepest decent method Summer Institute 2014 @ Fuji Calm

  17. Analytical Result @ (c.f.) • Produced number of is suppressed by factor comparing with or • This results is consistent with perturbativity Summer Institute 2014 @ Fuji Calm

  18. Leading term of (only final formula) X X Summer Institute 2014 @ Fuji Calm

  19. Numerical Results Preliminary consistent with analytical expected value Summer Institute 2014 @ Fuji Calm

  20. 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 • However, the produced number in case of strong couplings may be comparable to massive particles Summer Institute 2014 @ Fuji Calm

  21. Summer Institute 2014 @ Fuji Calm

  22. Numerical Results 2 Preliminary Summer Institute 2014 @ Fuji Calm

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