Journal of Physics: Conference Series PAPER OPEN ACCESS Influence of an Electric Field on the Propagation of a Photon in a Magnetic field To cite this article: V M Katkov 06 J. Phys.: Conf. Ser. 73 0003 View the article online for updates and enhancements. Related content - Threshold Electric Field for Domain Formation in Semiconductive CdS Tomonobu Hata and Seijiro Furukawa - Influence of the electric field on the silver and iron whiskers growth T Hoffmann, J Mazur, J Nikliborc et al. - Electric Field and Polarity Dependence of Photocurrent in Polyethylene Teruyoshi Mizutani, Yoshiaki Takai and Masayuki Ieda This content was downloaded from IP address 8.5.3.83 on 7/07/08 at 09:6
RREPS05 Journal of Physics: Conference Series 73 06 0003 doi:0.088/7-6596/73//0003 Influence of an Electric Field on the Propagation of a Photon in a Magnetic field V M Katkov Budker Institute of Nuclear Physics, Akademika Lavrentieva Ave., Novosibirsk, 630090, Russian Federation E-mail: katkov@inp.nsk.su Abstract. In this work, a constant and uniform magnetic field is less than the Schwinger critical value. In turn, an additional constant and uniform electric field is taken much smaller than the magnetic field value. The propagation of a photon in this electromagnetic field is investigating. In particular, in the presence of a weak electric field, the root divergence is absent in the photon effective mass near the thresholds of pair creation. The effective mass of a real photon with a preset polarization is considered in the quantum energy region as well as in the quasiclassical one.. Introduction The photon propagation in electromagnetic fields and the dispersive properties of the space region with magnetic fields is of very much interest. This propagation is accompanied by the photon conversion into a pair of charged particles when the transverse photon momentum is larger than the process threshold value k > m the system of units = c = is used. In 97 Adler [] had calculated the photon polarization operator in a magnetic field using the proper-time technique developed by Schwinger [] and Batalin and Shabad [3] had calculated this operator in an electromagnetic field using the Green function found by Schwinger []. In 97 Batalin and Shabad [3] had calculated polarization operator in an electromagnetic field using the Green function found by Schwinger []. In 975 the contribution of charged-particles loop in an electromagnetic field with n external photon lines had been calculated in []. For n = the explicit expressions for the contribution of scalar and spinor particles to the polarization operator of photon were given in []. For the contribution of spinor particles obtained expressions coincide with the result of [3], but another form is used. In this work, a constant and uniform magnetic field is less than the Schwinger critical value H 0 = m /e =. 0 3 G. In turn, an additional constant and uniform electric field is taken much smaller than the magnetic field value. In these fields, we consider the polarization operator on mass shell k = 0, the metric ab = a 0 b 0 ab is used at arbitrary value of the photon energy ω.. General expressions Our analysis is based on the general expression for the contribution of spinor particles to the polarization operator obtained in a diagonal form in [] see equations 3.9 and 3.33. The Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the authors and the title of the work, journal citation and DOI. Published under licence by Ltd
RREPS05 Journal of Physics: Conference Series 73 06 0003 doi:0.088/7-6596/73//0003 eigenvalue κ i of this operator on the mass shell k = 0 determines the effective mass of the real photon with the polarization e i directed along the corresponding eigenvector: Π µν = κ i β µ i βν i, β i β j = δ ij, β i k = 0; e µ i = i=,3 bµ i, b µ b = Bkµ + Ω Ω Ckµ, i κ = r Ω Ω Ω b µ 3 = Ckµ Ω Ω Bkµ ;, κ 3 = r Ω 3 + Ω Ω Ω = Ω 3 Ω + Ω 3 Ω + Ω, r = ω k3 m., 3 The consideration realizes in the frame where electric E and magnetic H fields are parallel and directed along the axis 3. In this frame the tensor of electromagnetic field F µν and tensors F µν, B µν and C µν have a form Here F µν = C µν E + B µν H, F µν = C µν H B µν E, C µν = g 0 µg 3 ν g 3 µg 0 ν, B µν = g µg ν g µg ν; ee/m = E/E 0 ν, eh/m = H/H 0 µ; Ω i = αm π µ dv 0 f i v, x exp iψv, x dx. 5 νx f cosµxv coshνxv cosµx coshνx sinµxv sinhνxv = sinµx sinhνx sin µx sinh, νx νx f coshνxcosµx cosµxv = sinhνx sin 3 + f, µx νx f cosµxcoshνx coshνxv 3 = sinµx sinh 3 f, νx νx f cosµx cosµxv coshνx coshνxv sinµxv sinhνxv = sin µx sinh + νx sinµx sinhνx ; 6 coshνx coshνxv cosµx cosµxv ψv, x = r + x. 7 ν sinhνx µ sinµx Let us note that the integration contour in the equation 5 is passing slightly below the real axis. After all calculations have been fulfilled we can return to a covariant form of the process description using the following expressions E, H = F + G / ± F, F = E H /, G = EH, C µν = Fµν + H g µν / E + H, C µν B µν = g µν. 8 The real part of κ i determines the refractive index n i of photon with the polarization e µ i = βµ i, i =, : κ i = m ef i, n i = Re κ i ω. 9
RREPS05 Journal of Physics: Conference Series 73 06 0003 doi:0.088/7-6596/73//0003 At r >, the proper value of polarization operator κ i includes the imaginary part, which determines the probability per unit length of pair production: W i = ω Im κ i. 0 Let us not that in considered case, we have µ, ν µ. The high energy region, r /µ, is contained in the region of the standard quasiclassical approximation SQA [5]. In SQA, the effective mass of a photon depends on the parameter κ only: κ = rµ +ν = F k /m H 0. For κ the influence of weak electric field on the polarization operator is small. Because of this we consider now the case of energies r /µ. 3. Region of intermediate photon energies Choose a point x 0 in the following way: ϕ x 0 = 0, were µx ϕ x = ψ0, x = r tan µ ν tanh νx Then, we have the following equation for x 0 : tan νs We represent the integral for Ω i as where a i = µ b i = µ + x. + tanh µs = r, x 0 = is. Ω i = αm π a i + b i, 3 dv dv x 0 0 f i v, x exp iψv, x dx, x 0 f i v, x exp iψv, x dx. 5 In the integral a i in the equation 3, the small values x contribute. We calculate this integral expanding the entering functions over x, take into account that in the region under consideration, the condition rµ is fulfilled. Then in exponent, we keep the term x only and extend the integration over x to infinity. In the result of not complicated integration over v, we have: a = 6 µ + ν, a 3 = 8 µ + ν, 5 5 κ a = αm κ 5π, κa 3 = 7αm κ 5π, F κ k =. 6 m These asymptotics are well known. In the integral b i equation 5, small values v contribute. Expanding entering functions over v and extending the integration over v to infinity, we get b i = µ dv x 0 [ f i 0, x exp i ϕ x + v χ x ] dx, 7 χ x = rx ν sinhνx µ. 8 sinµx H 0 3
RREPS05 Journal of Physics: Conference Series 73 06 0003 doi:0.088/7-6596/73//0003 After the integration over v, one has b i = µ π exp i π x 0 f i 0, x exp iϕx dx. 9 χx The case µ r µ /5 was considered in [6] see an equation 3. The result has the following invariant form: κ = αm κ 3 8 exp 8 6 F + 3κ 5κ 3, W 3 = W, 0 κ = r µ + nu ν, F µ =. Leaving the main terms of the expansion in the parameter ξ = ν/µ, we have [ κ ξ /κ 0 = exp 3 + ]. r r. The quantum energy region near the lower thresholds We consider now the energy region where r when the moving of created particles is nonrelativistic. In this case, the equation and its solutions are given by the following approximate equations ξ l 6 r exp l +, l = µs, ξ = ν µ ; 3 l ln ξ ln /ξ, r ξ l ; l ln r, r ξ l ; 5 Leaving the main terms of the expansion, we obtain iϕx βr γe iz izq + i x 3 /, 6 x = z il /µ, q = r /µ, = ξ /µ, 7 βr = r/µ qlr, γ = q + l /, 8 χx x, f,, 0, x 0, f 3 0, x i. 9 In the region of lower thresholds, where the particles occupy not very high energy levels, we present the equation 9 for b 3 in the form b 3 = i πµ exp i π exp βr 0 dz z il k=0 { [ γ k k! exp i q k z ]} z il3. 30 If δ, δ = q n,, γ n, the large z contributes and we have after the change of variables b 3 i exp i π µπ n n dz exp βr exp [ i δz z 3 / ]. 3 n! z 0
RREPS05 Journal of Physics: Conference Series 73 06 0003 doi:0.088/7-6596/73//0003 After integration over z, we have the following approximate expressions were b 3 = µπ nn exp βr d δ,, 3 n! d δ, = exp i π ϑδ π d δ, = 6 exp i π Γ 6 6 δ, δ 3 ; 33 /6, δ /3, 3 where ϑz is Heaviside function: ϑz = for z 0, ϑz = 0 for z < 0. The expression for κ 3 with the accepted accuracy can be rewritten in the following form: µ κ b 3 αm δ exp i π ϑδ ζ ζn e, δ 3, ζ = r/µ. 35 n! 3 + i µ /6 κ b 3 αm π Γ ζ ζn e, δ /3, 36 6 n! In considered quantum case n the real part of κ b 3 is exponentially small compared to κa 3 see the equation 6, but the imaginary part of the effective mass is given by the equation 35. At, we have γ, and the small z contributes to the integral in the equation 9. In this case, it is possible to use the method of stationary phase κ b 3 iαm µ q + l / l exp βr + γ + l 3 /. 37 5. Low energy region r < 0 The case r ν /3 was discussed above see the equation 36 where we have to put n = 0. For the case r ν /3, the following expression l ξ r, 38 is valid and one can use the method of stationary phase. Then we have [ ] κ b 3 i αm µ exp r r3/. 39 r µ 3ν This method is valid for lower photon energy when ν /r ν /3. Then the approximate solution of the equation has form s ν arctan r, χ x 0 ϕ x 0 νs, νs f 3 x 0 i r r r, f,,0, x 0, 0 κ b αm µ 3 i exp r r r µ ν arctan r + r ν r. 5
RREPS05 Journal of Physics: Conference Series 73 06 0003 doi:0.088/7-6596/73//0003 For r, expanding the incoming in the equation functions we have the equation 39. When the value r small enough ν 3 r 3/ ν, leaving the leading term of decomposition over r, we get κ b 3 i αm µ r exp r µ π ν + r ν. The imaginary part of this expression coincides with the equation 0 [6]. The ratio of this expression to the corresponding formula in a pure electric field has the form κ b 3/κ b 3 µ = 0 = π π [r ξ exp ν ]. 3 µ The equations and 3 are valid for the photon energy r ν. For the region of the photon energy r ν, we can use the results of [6]. 6. Conclusion We have considered the polarization operator of a photon in a constant magnetic fields in the presence of a weak electric field. The effective mass of a photon was calculated using three different overlapping approximation. In the region of SQA applicability, the created by a photon particles have ultrarelativistic energies. The role of fields in this case is to transfer the required transverse momentum and the electric field actions less than that of the magnetic field. At lower energies, the role of the electric field increases. It is necessary to note a special significance of a weak electric field E = ξh ξ in the removal of the root divergence of the probability when the particles of pair are created on the Landau levels with the electron and positron momentum p 3 = 0 [5]. The frame is used where k 3 = 0. Generally speaking, at ξ the formation time t c of the process under consideration is /µ. Here we use units = c = m =. At this time the particles of creating pair gets the momentum δp 3 ξ. If the value ξ becomes larger than the distance apart Landau levels µ = ξ /µ all levels overlapped. Under this condition the divergence of the probability is vanished and the method of stationary phase is valid even in the energy region r µ, whereas that is inapplicable in the absence of electric field [5]. In the opposite case ν µ 3 for the small value of p 3 µ, in the region where the influence of electric field is negligible, the formation time of the process t f is /p 3 and δp 3 ν/p 3 p 3. It is follows from above that ν /3 p 3 µ. At this condition the value of discontinuity is t f /t c µ/p 3. For ν /3 p 3 the time t f is determined by the self-consistent equation δε /t f ν t f, t f ν /3 and the value of discontinuity becomes µt f µ 3 /ν /6 instead of µ/p 3. In the region ω m r the energy transfer from electric field to the created particles becomes appreciable and for ω m it determines the probability of the process mainly. At ω ee/m r ν, the photon assistance in the pair creation comes to the end and the probability under consideration defines the probability of photon absorption by the particles created by electromagnetic fields. The influence of a magnetic field on the process is connected with the interaction of the magnetic moment of the created particles and magnetic field. This interaction, in particular, has appeared in the distinction of the pair creation probability by field for scalar and spinor particles []. Acknowledgments The work was supported by the Ministry of Education and Science of the Russian Federation. The author is grateful to the Russian Foundation for Basic Research grant No. 5-0-067 for partial support of the research. 6
RREPS05 Journal of Physics: Conference Series 73 06 0003 doi:0.088/7-6596/73//0003 References [] Adler S L 97 Ann. Phys. 67 599 [] Schwinger J 95 Phys. Rev. 8 66 [3] Batalin I A and Shabad A E 97 Sov. Phys. JETP 33 83 [] Baier V N, Katkov V M and Strakhovenko V M 975 Sov. Phys. JETP 98 [5] Baier V N and Katkov V M 007 Phys. Rev. D 75 073009 [6] Katkov V M 0 Sov. Phys. JETP 6 7