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

Transcription

1 Effect of interaction terms on particle production due to time-varying mass Seishi Enomoto (Univ. of Warsaw) [work in progress] Collaborators : Olga Fuksińska Zygmunt Lalak (Univ. of Warsaw) (Univ. of Warsaw) Outline 1. Introduction 2. How to calculate particle number 3. Calculation of particle number 4. Summary 1/22

2 1. Introduction Particle production from vacuum x 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 Oscillating inflaton (p)reheating etc [L. Parker, Phys. Rev. 183, 1057 (1969)] [L. H. Ford, Phys. Rev. D 35, 2955 (1987)] [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)] 2/22

3 Example of scalar particle production Let us consider : L int = 1 2 g2 φ 2 χ 2 φ : complex scalar field (classical) g : coupling χ : real scalar particle (quantum) If φ goes near the origin, χ particles are produced Because mass of χ (m χ = gφ) becomes small around φ = 0 kinetic energy of φ converts to χ particles m χ produced occupation number : n χk = V exp π k2 +g 2 μ 2 gv χ χ χ χ χ χ Im φ v t [L. Kofman, A. D. Linde, X. Liu, A. Maloney, L. McAllister and E. Silverstein, JHEP 0405, 030 (2004)]. μ Re φ φ 3/22

4 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 φ χ ψ φ ψ χ 4/22

5 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 φ classical χ classical ψ φ ψ χ 5/22

6 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 φ classical χ quantum classical ψ φ quantum ψ χ 6/22

7 Model in this talk Super potential : Φ = φ + 2θψ φ + θ 2 F φ W = 1 2 gφx2 Χ = χ + 2θψ χ + θ 2 F χ Interaction terms in components : g : coupling L int = g 2 φ 2 χ g2 χ 4 g 1 2 φψ χψ χ + ψ φ ψ χ χ + (h. c. ) Stationary point : χ = ψ φ = ψ χ = 0, but φ can have any value Masses Production is possible φ 0 χ, ψ χ : mass = gφ, ψ φ : massless However, ψ φ s mass may be influenced by φ through loop effects? Is quantum part of φ also influenced? impossible Im φ φ Re φ 7/22

8 Equations of Motion for field operators : φ : 0 = 2 + g 2 χ 2 φ + 1 gψ 2 χ ψ χ χ : 0 = 2 + g 2 φ g2 χ 2 χ + gψ φ ψ χ ψ φ : 0 = σ μ μ ψ φ + igχ ψ χ ψ χ : 0 = σ μ μ ψ χ + igφ ψ χ + igχ ψ φ How do we calculate produced particle number including interaction term? 8/22

9 2. How to calculate particle number Definition of (occupation) number : n k = 0 in a out k a out k 0 in 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 9/22

10 An example with a scalar field (mass) (source term) Operator field equation : 0 = 2 + M 2 Ψ + J Commutation relation : Ψ x, Ψ (y) = iδ 3 x y Formal solution (Yang-Feldman equations) Ψ x = ZΨ as x x iz 0 dy 0 t as d 3 y Ψ as x, Ψ as, y J y Z : some const. Ψ as : asymptotic field 0 = 2 + M 2 Ψ as J(y) Ψ as (x) J(y) J(y) J(y) If we take t as = t in = or t as = t out = +, Ψ out x out = Ψ in x out i Z d 4 y Ψ in x out, Ψ in y J y 10/22

11 An example with a scalar field (mass) (source term) Operator field equation : 0 = 2 + M 2 Ψ + J Commutation relation : Ψ x, Ψ (y) = iδ 3 x y Formal solution (Yang-Feldman equations) Ψ x = ZΨ as x x iz 0 dy 0 t as d 3 y Ψ as x, Ψ as, y J y Z : some const. a k out Ψ as : asymptotic field 0 = 2 + M 2 Ψ as a k in J(y) Ψ as (x) J(y) J(y) J(y) If we take t as = t in = or t as = t out = +, Ψ out x out = Ψ in x out i Z d 4 y Ψ in x out, Ψ in y J y 11/22

12 An example with a scalar field Ψ as is free particle, so we can expand with plane waves as Ψ as d 3 k x = 2π 3 eik x Ψ as k (x 0 )a as k + Ψ as k x 0 as b k plane wave (time dependent) wave func. 0 = Ψ k as + k 2 + M 2 Ψ k as creation/annihilation op. inner product relation : Ψ k as Ψ k as Ψ k as Ψ k as = i/z which comes from conditions a as k, a as k = 2π 3 δ 3 k k, Z Ψ as x, Ψ as y t t as = iħδ 3 x y a k as = iz d 3 x e ik x Ψ k as Ψ as Ψ k as Ψ as 12/22

13 An example with a scalar field Relation between a k in and a k out a k out = iz d 3 x e ik x Ψ k out Ψ out Ψ k out Ψ out Ψ out x out = Ψ in x out i Z d 4 y Ψ in x out, Ψ in, y J y a out k = α k a in k + β k a in k i Z d 4 x e ik x α k Ψ in in k β k Ψ k J(y) (usual) Bogoliubov tf low α k iz Ψ k out Ψ k in Ψ k out Ψ k in β k iz Ψ k out Ψ k in Ψ k out Ψ k in Interaction effects Ψ in k = α k Ψ out out k + β k Ψ k Ψ out k = α k Ψ in k β in k Ψ k α 2 k β 2 k = 1 13/22

14 An example with a scalar field Produced (occupation) number : n k = 0 in a k out a k out 0 in = β k a in k i Z d 4 x e ik x α k Ψ in in k β k Ψ k J 0 in 2 = V β k Z d 4 x e ik x Ψ k in J 0 in 2 + β k 0 β k = 0 Particles can be produced even if β k = 0! 14/22

15 3. Calculation of particle number Equation of Motion (again) : φ : 0 = 2 + g 2 χ 2 φ + 1 gψ 2 χ ψ χ χ : 0 = 2 + g 2 φ g2 χ 2 χ + gψ φ ψ χ ψ φ : 0 = σ μ μ ψ φ + igχ ψ χ ψ χ : 0 = σ μ μ ψ χ + igφ ψ χ + igχ ψ φ EOM for asymptotic fields (as = in, out): φ as : 0 = 2 φ as χ as : 0 = 2 + g 2 φ 2 χ as ψ φ as : ψ χ as : 0 = σ μ μ ψ φ as 0 = σ μ μ ψ χ as + ig φ ψ χ as φ : macroscopic χ, ψ φ, ψ χ : microscopic 15/22

16 Solutions for Asymptotic fields Assuming φ = φ (t) for simplicity, then 0 = 2 φ as φ as = φ as + d3 k 2π 3 eik x φ as k a as φk + φ as as k b φ k Im φ Z φ φ in = vt + iμ, Z φ φ k in = Z φ φ k out = 1 2 k μ v e i k t Re φ φ β φk = 0 φ k out = α k φ k in β k φ k in χ as = d3 k 2π 3 eik x χ as k a as χk + χ as as k b χ k Z χ χ in k 1 t exp i dt ω 2ω k (t ) k ( valid for t 1/ gv ) = 2 + g 2 φ 2 χ as ω k k 2 + Z φ g 2 φ 2 (analytic continuation) β χk i exp π k2 +g 2 μ 2 2gv χ in Im t Re t χ out 16/22

17 Solutions for Asymptotic fields eigen spinor for helicity op. : k i σ i e k ± = ± k σ 0 e k ± ψ φ as = d3 k 2π 3 eik x ± β ψφ k Z φ ψ ±,in φk = = 0 e + k ψ +,as φk a +as ψφ k Z φ ψ ±,out φk = e i k t + e k ψ,as as φk a ψφ k 0 = σ μ μ ψ φ as ψ χ as = d3 k 2π 3 eik x s=± e s + k ψ s,as s χk a ψχ k σ 0 e s k ψ s,as s χk a ψχ k Z χ ψ χk ± s,in gvt ω k ± 1 + gvt e iθ k s t exp i dt ω ω k t k 0 = σ μ μ ψ χ as + ig φ ψ χ as ( valid for t 1/ gv ) ω k k 2 + Z φ g 2 φ 2, e iθ k s s k igμ k 2 +g 2 μ 2 s (analytic continuation) β ψχ k s exp π k2 +g 2 μ 2 2gv 17/22

18 Produced Particle number n χk = β χk 2 + = exp π k 2 +g 2 μ 2 s n ψχ k s = β ψχ k gv 2 + = exp π k 2 +g 2 μ 2 leading term is obtained gv + + n φk = β φk 2 + = 0 + s n ψφ k s = β ψφ k 2 + = 0 + Focus on! need to calculate next to leading order 18/22

19 Calculation of n ψφ k n ψφ k = 0 in a s,out ψφ k a s,out ψφ k 0 in = g 2 Z φ Z2 χ d 4 x e ik x ψ in φk χ in e s k ψ in χ 0 in 2 + V g 2 Z φ Z χ 2 d 3 p 1 2π 3 r=± 2 1 sr p k pk dt ψ in in φk χ k+p ψ χp + r,in 2 Z φ ψ in φk = e i k t Z χ χ k in 1 2ω k e i dt ω k t Z χ ψ χk + s,in 1 2 t 1 gvt ω k ± 1 + gvt e iθ k s e i t dt ω k t ω k This integral is too complicated to perform Im t steepest decent method Re t We estimated in special case k = 0 with steepest decent method 19/22

20 Analytical Result s n ψφ k=0/v = C g2 4π exp π g2 μ 2 gv C = /6 Γ 4/3 2 π 2 e 2/ s (c.f.) n χk /V n ψχ k/v exp π k2 +g 2 μ 2 gv Produced number of ψ φ is suppressed by factor g 2 comparing with χ or ψ χ This results is consistent with perturbativity 20/22

21 Numerical Results 0.28 g 2 4π consistent with analytical expected value 2 exp π k2 + g 2 μ 2 gv 21/22

22 4. Summary 1. We constructed the Bogoliubov transformation taking into account interaction effects 2. We calculated produced particle s (occupation) number s n χk /V n ψχ k/v exp π k2 +g 2 μ 2 gv s n ψφ k=0/v 0.28 g2 exp π g2 μ 2 0 4π gv Massless particle can be produced, however the production is suppressed by the coupling n φk=0 /V now in progress The numerical result shows massless bosonic statistics. It would be interesting to see modification induced by supersymmetry breaking terms - this is issue under investigation 22/22