Scalar pair production in a magnetic field in de Sitter universe M. - - PowerPoint PPT Presentation
Scalar pair production in a magnetic field in de Sitter universe M. B aloi, C.Crucean, D.Popescu West University of Timi soara Faculty of Physics Strings and Fields 2020 Kyoto, Japan, November 18 M. B aloi, C.Crucean, D.Popescu (West
aloi, C.Crucean, D.Popescu
West University of Timi¸ soara Faculty of Physics Strings and Fields 2020
Kyoto, Japan, November 18
aloi, C.Crucean, D.Popescu (West University of Timi¸ soara Faculty of Physics Strings and Fields 2020) Scalar pair production in a magnetic field in de Sitter universe Kyoto, Japan, November 18 1 / 14
Introduction The transition amplitude Probability of transition Graphical results Conclusions
aloi, C.Crucean, D.Popescu (West University of Timi¸ soara Faculty of Physics Strings and Fields 2020) Scalar pair production in a magnetic field in de Sitter universe Kyoto, Japan, November 18 2 / 14
The line element of de Sitter expanding universe is: ds2 = dt2 − e2ωtd x2 = 1 (ωtc)2
c − d
x2 , (1) where the conformal time is given in terms of the proper time by tc = − e−ωt
ω
and ω > 0 is the expansion factor. The exact solution of the Klein-Gordon equation with a defined momentum has the following expression: f
p (x) = 1
2 π ω e−3ωt/2 (2π)3/2 e−πµ/2H(1)
iµ
p ω e−ωt ei
p· x ,
(2) where H1
µ(z) is the Hankel function of first kind, p = |
p| is the momentum
µ =
4 ,
k = m ω , (3) with m > 3ω/2. The fundamental solutions of negative frequencies are obtained by complex conjugation f ∗
aloi, C.Crucean, D.Popescu (West University of Timi¸ soara Faculty of Physics Strings and Fields 2020) Scalar pair production in a magnetic field in de Sitter universe Kyoto, Japan, November 18 3 / 14
The vector potential that produces the dipole magnetic field on Minkowski spacetime reads:
x | x|3 , (4) where M is the magnetic dipole moment. The expression of the field of a magnetic dipole can be obtained as the curl of the vector potential:
∇ × AM = 3 x ( M · x ) − M( x · x ) | x|5 . (5) The expression of A in de Sitter geometry is established by using the conformal invariance of Maxwell equations. Knowing that AM is the vector potential in Minkowski space, then the vector potential in de Sitter geometry is: Aµ = Ω−1Aµ
M,
(6) where Ω = (ωtc)−2 is the conformal factor transformation. Taking A0(x) = 0, we
A the following expression:
x | x|3 e−2ωt. (7)
aloi, C.Crucean, D.Popescu (West University of Timi¸ soara Faculty of Physics Strings and Fields 2020) Scalar pair production in a magnetic field in de Sitter universe Kyoto, Japan, November 18 4 / 14
The transition amplitude of scalar pair production in external field, defined in the first order of perturbation theory reads: Ai→f = −e −g(x)
↔
∂i f ∗
(8) By taking the bilateral derivative we obtain the following expression of the spatial integral:
x | x|3 e−i(
p+ p ′) x = −4πi(
p + p ′) | p + p ′|2 . (9) For solving the temporal integral we use the following relation that connects the Hankel functions and Bessel K functions: H(1,2)
ν
(z) = ∓ 2i π
(10)
aloi, C.Crucean, D.Popescu (West University of Timi¸ soara Faculty of Physics Strings and Fields 2020) Scalar pair production in a magnetic field in de Sitter universe Kyoto, Japan, November 18 5 / 14
The transition amplitude can be expressed as follows: Ai→f = − e 2π3| p + p ′|2 ( M × ( p + p ′)) · ( p − p ′) × ∞ dzzK−iµ(ipz)K−iµ(ip ′z), (11) where we pass in the temporal integral to the new variable of integration z = e−ωt/ω. The general form of the temporal integral is given below: ∞ dzz−λKµ(az)Kν(bz) = 2−2−λa−ν+λ−1b ν Γ(1 − λ) Γ 1 − λ + µ + ν 2
1 − λ − µ + ν 2
1 − λ + µ − ν 2
1 − λ − µ − ν 2
1 − λ + µ + ν 2 , 1 − λ − µ + ν 2 ; 1 − λ; 1 − b2 a2
Re(a + b) > 0 , Re(λ) < 1 − |Re(µ)| − |Re(ν)|. (12)
aloi, C.Crucean, D.Popescu (West University of Timi¸ soara Faculty of Physics Strings and Fields 2020) Scalar pair production in a magnetic field in de Sitter universe Kyoto, Japan, November 18 6 / 14
The final result of the transition amplitude reads: Ai→f = − e 4π3| p + p ′|2 ( M × ( p + p ′)) · ( p − p ′) × θ(p − p ′) p2 fk p ′ p
p ′2 fk p p ′
(13) The functions fk
p
fk p ′ p
p ′ p −iµ Γ(1 − iµ)Γ(1 + iµ) 2F1
p ′ p 2 . (14) We mention that the functions fk
p ′
between them.
aloi, C.Crucean, D.Popescu (West University of Timi¸ soara Faculty of Physics Strings and Fields 2020) Scalar pair production in a magnetic field in de Sitter universe Kyoto, Japan, November 18 7 / 14
The probability of pair production is obtained by taking the square modulus of the transition amplitude: P = |Ai→f |2 = e2 16π6| p + p ′|4 |( M × ( p + p ′)) · ( p − p ′)|2 ×
p4
p ′ p
+ θ(p ′ − p) p ′4
p p ′
. (15) We fix the magnetic dipole moment on the e3 direction such that M = M e3. Then, taking the polar coordinates for the momenta vectors p , p ′: p1 = p cos β; p2 = p sin β p1
′ = p ′ cos ϕ; p2 ′ = p ′ sin ϕ,
(16) we obtain that the angle between momenta vectors p and p ′ is just β − ϕ.
aloi, C.Crucean, D.Popescu (West University of Timi¸ soara Faculty of Physics Strings and Fields 2020) Scalar pair production in a magnetic field in de Sitter universe Kyoto, Japan, November 18 8 / 14
The final expression for the probability of scalar pair production is: P = e2M2 16π6 (p2 + p ′2 − 2pp ′ cos(β − ϕ)) (p2 + p ′2 + 2pp ′ cos(β − ϕ)) ×
p4
p ′ p
+ θ(p ′ − p) p ′4
p p ′
. (17) From the above equation it is observed that: The probability is minimum when β − ϕ = 0; it is very probable for the scalar pair to annihilate. The probability is maximum when β − ϕ = π; it is very probable for the scalar pair to separate.
aloi, C.Crucean, D.Popescu (West University of Timi¸ soara Faculty of Physics Strings and Fields 2020) Scalar pair production in a magnetic field in de Sitter universe Kyoto, Japan, November 18 9 / 14
Figure: P as a function of k for p/p ′ = 0.9 solid line and p/p ′ = 0.8 the point
aloi, C.Crucean, D.Popescu (West University of Timi¸ soara Faculty of Physics Strings and Fields 2020) Scalar pair production in a magnetic field in de Sitter universe Kyoto, Japan, November 18 10 / 14
Figure: P as a function of k for p/p ′ = 0.01 solid line and p/p ′ = 0.03 the point
aloi, C.Crucean, D.Popescu (West University of Timi¸ soara Faculty of Physics Strings and Fields 2020) Scalar pair production in a magnetic field in de Sitter universe Kyoto, Japan, November 18 11 / 14
Figure: P as a function of β − ϕ. Left figure: p/p ′ = 0.9 solid line and p/p ′ = 0.8 the point line; Right figure: p/p ′ = 0.3 solid line and p/p ′ = 0.1 the point line. Parameter k = 1.52 in both figures.
aloi, C.Crucean, D.Popescu (West University of Timi¸ soara Faculty of Physics Strings and Fields 2020) Scalar pair production in a magnetic field in de Sitter universe Kyoto, Japan, November 18 12 / 14
The first order transition amplitude and probability are nonvanishing
The scalar particles will most probable be emitted perpendicular to the magnetic field direction; In the Minkowski limit the amplitude and probability are vanishing, since in the Minkowski scalar QED this process is forbidden by the energy-momentum conservation.
aloi, C.Crucean, D.Popescu (West University of Timi¸ soara Faculty of Physics Strings and Fields 2020) Scalar pair production in a magnetic field in de Sitter universe Kyoto, Japan, November 18 13 / 14
aloi, C.Crucean, D.Popescu (West University of Timi¸ soara Faculty of Physics Strings and Fields 2020) Scalar pair production in a magnetic field in de Sitter universe Kyoto, Japan, November 18 14 / 14