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 University of Timi¸ soara Faculty of Physics Strings and Fields 2020 ) Kyoto, Japan, November 18 1 / 14 Scalar pair production in a magnetic field in de Sitter universe

Outline Introduction The transition amplitude Probability of transition Graphical results Conclusions M. B˘ aloi, C.Crucean, D.Popescu (West University of Timi¸ soara Faculty of Physics Strings and Fields 2020 ) Kyoto, Japan, November 18 2 / 14 Scalar pair production in a magnetic field in de Sitter universe

Introduction The line element of de Sitter expanding universe is: 1 ds 2 = dt 2 − e 2 ω t d � x 2 = dt 2 x 2 � � c − d � , (1) ( ω t c ) 2 where the conformal time is given in terms of the proper time by t c = − e − ω t and ω ω > 0 is the expansion factor. The exact solution of the Klein-Gordon equation with a defined momentum has the following expression: � p � π e − 3 ω t / 2 p ( x ) = 1 ω e − ω t � x , (2 π ) 3 / 2 e − πµ/ 2 H (1) e i � p · � f � (2) i µ 2 ω where H 1 µ ( z ) is the Hankel function of first kind, p = | � p | is the momentum modulus. We also use the notations: k = m � k 2 − 9 µ = 4 , ω , (3) with m > 3 ω/ 2. The fundamental solutions of negative frequencies are obtained by complex conjugation f ∗ p ( x ). � M. B˘ aloi, C.Crucean, D.Popescu (West University of Timi¸ soara Faculty of Physics Strings and Fields 2020 ) Kyoto, Japan, November 18 3 / 14 Scalar pair production in a magnetic field in de Sitter universe

Introduction The vector potential that produces the dipole magnetic field on Minkowski spacetime reads: � M × � x � A M = , (4) x | 3 | � 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: x ( � x ) − � A M = 3 � M · � M ( � x · � x ) � B M = � ∇ × � . (5) x | 5 | � The expression of � A in de Sitter geometry is established by using the conformal invariance of Maxwell equations. Knowing that � A M is the vector potential in Minkowski space, then the vector potential in de Sitter geometry is: A µ = Ω − 1 A µ M , (6) where Ω = ( ω t c ) − 2 is the conformal factor transformation. Taking A 0 ( x ) = 0, we obtain for � A the following expression: � M × � x � e − 2 ω t . A ( x ) = (7) | � x | 3 M. B˘ aloi, C.Crucean, D.Popescu (West University of Timi¸ soara Faculty of Physics Strings and Fields 2020 ) Kyoto, Japan, November 18 4 / 14 Scalar pair production in a magnetic field in de Sitter universe

The transition amplitude The transition amplitude of scalar pair production in external field, defined in the first order of perturbation theory reads: � � � � ↔ A i ( x ) d 4 x . A i → f = − e − g ( x ) f ∗ p ′ ( x ) ∂ i f ∗ p ( x ) (8) � � By taking the bilateral derivative we obtain the following expression of the spatial integral: � d 3 x � x x = − 4 π i ( � p + � p ′ ) x | 3 e − i ( � p + � p ′ ) � p ′ | 2 . (9) | � | � p + � For solving the temporal integral we use the following relation that connects the Hankel functions and Bessel K functions: � 2 i � H (1 , 2) e ∓ i πν/ 2 K ν ( ∓ iz ) . ( z ) = ∓ (10) ν π M. B˘ aloi, C.Crucean, D.Popescu (West University of Timi¸ soara Faculty of Physics Strings and Fields 2020 ) Kyoto, Japan, November 18 5 / 14 Scalar pair production in a magnetic field in de Sitter universe

The transition amplitude The transition amplitude can be expressed as follows: e p ′ | 2 ( � = − M × ( � p + � p ′ )) · ( � p − � p ′ ) A i → f 2 π 3 | � p + � � ∞ dzzK − i µ ( ipz ) K − i µ ( ip ′ z ) , × (11) 0 where we pass in the temporal integral to the new variable of integration z = e − ω t /ω . The general form of the temporal integral is given below: � ∞ dzz − λ K µ ( az ) K ν ( bz ) = 2 − 2 − λ a − ν + λ − 1 b ν � 1 − λ + µ + ν � Γ Γ(1 − λ ) 2 0 � 1 − λ − µ + ν � � 1 − λ + µ − ν � � 1 − λ − µ − ν � × Γ Γ Γ 2 2 2 ; 1 − λ ; 1 − b 2 � 1 − λ + µ + ν , 1 − λ − µ + ν � × 2 F 1 , 2 2 a 2 Re ( a + b ) > 0 , Re ( λ ) < 1 − | Re ( µ ) | − | Re ( ν ) | . (12) M. B˘ aloi, C.Crucean, D.Popescu (West University of Timi¸ soara Faculty of Physics Strings and Fields 2020 ) Kyoto, Japan, November 18 6 / 14 Scalar pair production in a magnetic field in de Sitter universe

The transition amplitude The final result of the transition amplitude reads: e p ′ | 2 ( � p ′ )) · ( � p ′ ) A i → f = − M × ( � p + � p − � 4 π 3 | � p + � � p + θ ( p ′ − p ) � θ ( p − p ′ ) � p ′ � �� × f k f k . (13) p 2 p ′ 2 p p ′ � � p ′ The functions f k that enter in the definition of the transition amplitude are: p � � 2 � � − i µ � p ′ � � p ′ � p ′ = Γ(1 − i µ )Γ(1 + i µ ) 2 F 1 1 , 1 − i µ ; 2; 1 − . (14) f k p p p � � p are obtained when we interchange p ′ ⇄ p We mention that the functions f k p ′ between them. M. B˘ aloi, C.Crucean, D.Popescu (West University of Timi¸ soara Faculty of Physics Strings and Fields 2020 ) Kyoto, Japan, November 18 7 / 14 Scalar pair production in a magnetic field in de Sitter universe

Probability of transition The probability of pair production is obtained by taking the square modulus of the transition amplitude: e 2 P = |A i → f | 2 = p ′ | 4 | ( � p ′ ) | 2 M × ( � p + � p ′ )) · ( � p − � 16 π 6 | � p + � � p � 2 + θ ( p ′ − p ) 2 � θ ( p − p ′ ) � � p ′ �� � �� � � � � × . (15) � f k � f k � � � � p 4 p ′ 4 p p ′ � � e 3 direction such that � We fix the magnetic dipole moment on the � M = M � e 3 . Then, taking the polar coordinates for the momenta vectors � p ,� p ′ : p 1 = p cos β ; p 2 = p sin β ′ = p ′ cos ϕ ; p 2 ′ = p ′ sin ϕ, p 1 (16) p ′ is just β − ϕ . we obtain that the angle between momenta vectors � p and � M. B˘ aloi, C.Crucean, D.Popescu (West University of Timi¸ soara Faculty of Physics Strings and Fields 2020 ) Kyoto, Japan, November 18 8 / 14 Scalar pair production in a magnetic field in de Sitter universe

Probability of transition The final expression for the probability of scalar pair production is: ( p 2 + p ′ 2 − 2 pp ′ cos( β − ϕ )) e 2 M 2 P = ( p 2 + p ′ 2 + 2 pp ′ cos( β − ϕ )) 16 π 6 � p � 2 + θ ( p ′ − p ) 2 � θ ( p − p ′ ) � � p ′ �� � �� � � � � × . (17) � f k � f k � � � � p 4 p ′ 4 p p ′ � � 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. M. B˘ aloi, C.Crucean, D.Popescu (West University of Timi¸ soara Faculty of Physics Strings and Fields 2020 ) Kyoto, Japan, November 18 9 / 14 Scalar pair production in a magnetic field in de Sitter universe

Graphical results Figure: P as a function of k for p / p ′ = 0 . 9 solid line and p / p ′ = 0 . 8 the point line. Angle β − ϕ = 0 in the left figure and β − ϕ = π in the right figure. M. B˘ aloi, C.Crucean, D.Popescu (West University of Timi¸ soara Faculty of Physics Strings and Fields 2020 ) Kyoto, Japan, November 18 10 / 14 Scalar pair production in a magnetic field in de Sitter universe

Graphical results Figure: P as a function of k for p / p ′ = 0 . 01 solid line and p / p ′ = 0 . 03 the point line. Angle β − ϕ = 0 in the left figure and β − ϕ = π in the right figure. M. B˘ aloi, C.Crucean, D.Popescu (West University of Timi¸ soara Faculty of Physics Strings and Fields 2020 ) Kyoto, Japan, November 18 11 / 14 Scalar pair production in a magnetic field in de Sitter universe

Graphical results 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. M. B˘ aloi, C.Crucean, D.Popescu (West University of Timi¸ soara Faculty of Physics Strings and Fields 2020 ) Kyoto, Japan, November 18 12 / 14 Scalar pair production in a magnetic field in de Sitter universe

Conclusions The first order transition amplitude and probability are nonvanishing only in the strong gravitational fields of the early universe; 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. M. B˘ aloi, C.Crucean, D.Popescu (West University of Timi¸ soara Faculty of Physics Strings and Fields 2020 ) Kyoto, Japan, November 18 13 / 14 Scalar pair production in a magnetic field in de Sitter universe

THANK YOU FOR LISTENING! M. B˘ aloi, C.Crucean, D.Popescu (West University of Timi¸ soara Faculty of Physics Strings and Fields 2020 ) Kyoto, Japan, November 18 14 / 14 Scalar pair production in a magnetic field in de Sitter universe