The process of pair creation by a photon in a constant and homogeneous
electromagnetic field of an arbitrary configuration is investigating. At high
energy the correction to the standard quasiclassical approximation (SQA) has
been calculated. In the region of intermediate photon energies where SQA is
inapplicable the new approximation, developed recently by authors, is used. The
influence of weak electric field on the process in a magnetic field is
considered. In particular, in the presence of electric field the root
divergence in the probability of pair creation on the Landau energy levels is
vanished. For smaller photon energies the low energy approximation is used. The
found probability describes the absorption of soft photon by particles created
by field. At low photon energy the electric field action dominates and the
influence of magnetic field on the process is connected with the interaction of
it and the magnetic moment of creating particles.Comment: 11 page

Katkov, V. M.

English

arXiv

The process of pair photoproduction in an electromagnetic field of arbitrary configuration is investigated. At high energy the correction to the standard quasiclassical approximation (SQA) of the process probability has been

calculated. In the region of intermediate photon energies where SQA is inapplicable the new approximation is used. The influence of weak electric field on the process in a magnetic field is considered. In particular, in the presence of this field the root divergence in the probability of pair creation on the Landau energy levels is

vanished. For smaller photon energies the low energy approximations have been derived. At very low photon energy the found probability describes the absorption of soft photon by the particles created by the field. At low photon energy the electric field action dominates and the influence of the magnetic field on the process occurs

because of its interaction with the magnetic moment of creating particles

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DOI 10.1393/ncc/i2011-10973-yColloquia: Channeling 2010IL NUOVO CIMENTO Vol. 34 C, N. 4 Luglio-Agosto 2011Pair photoproduction in a constant and homogeneouselectromagnetic fieldV. M. Katkov(∗)Budker Institute of Nuclear Physics - 630090 Novosibirsk, Russia(ricevuto il 22 Dicembre 2010; pubblicato online il 21 Settembre 2011)Summary. — The process of pair photoproduction in an electromagnetic fieldof arbitrary configuration is investigated. At high energy the correction to thestandard quasiclassical approximation (SQA) of the process probability has beencalculated. In the region of intermediate photon energies where SQA is inapplicablethe new approximation is used. The influence of weak electric field on the processin a magnetic field is considered. In particular, in the presence of this field theroot divergence in the probability of pair creation on the Landau energy levels isvanished. For smaller photon energies the low energy approximations have beenderived. At very low photon energy the found probability describes the absorptionof soft photon by the particles created by the field. At low photon energy the electricfield action dominates and the influence of the magnetic field on the process occursbecause of its interaction with the magnetic moment of creating particles.PACS 12.20.-m – Quantum electrodynamics.PACS 13.60.Le – Meson production.1. – IntroductionThe pair photoproduction in an electromagnetic field is the basic QED reaction whichcan play a significant role in many processes. This process was considered first in amagnetic field. The investigation was started in 1952 independently by Klepikov andToll [1,2]. In Klepikov’s paper [3], which was based on the solution of the Dirac equation,the probability of photoproduction had been obtained on the mass shell (k2 = 0, k is the4-momentum of photon. We use the system of units with  = c = 1 and the metricab = aμbμ = a0b0 − ab). In 1971 Adler [4] had calculated the photon polarizationoperator in a magnetic field using the proper-time technique developed by Schwinger [5]and Batalin and Shabad [6] had calculated this operator in an electromagnetic field using(∗) E-mail: V.M.Katkov@inp.nsk.suc© Societa` Italiana di Fisica 12 V. M. KATKOVthe Green’s function found by Schwinger [5]. In 1975 the contribution of charged-particlesloop in an electromagnetic field with n external photon lines had been calculated in [7].For n = 2 the explicit expressions for the contribution of scalar and spinor particlesto the polarization operator of photons were given in this work. Making use of theimaginary part of this operator for spinor particles the pair photoproduction probabilitywas analyzed in the pure magnetic [8] and the pure electric [9] field.The probability of pair photoproduction in a constant and homogeneous electric fieldin the quasi-classical approximation had been found by Narozhny [10] using the solution ofthe Dirac equation in the Sauter potential [11]. Nikishov [12] had obtained the differentialdistribution of this process also using the solution of the Dirac equation in the indicatedfield.In the present paper we consider the integral probability of pair creation by an un-polarized photon in a constant and homogeneous electromagnetic field of an arbitraryconfiguration basing on the polarization operator [7]. In sect. 2 the exact expressionfor this probability has been received for the general case k2 = 0. In sect. 3 the stan-dard quasi-classical approximation (SQA) [13,14] is outlined for the high-energy photonω  m (m is the electron mass). The corrections to SQA, determined also the applica-bility region of SQA, have been calculated. The found expressions, given in the Lorentzinvariant form, contain two invariant parameters. In sect. 4 the new approach has beendeveloped for the relatively low energies where SQA is not applicable. This approachis based on the method proposed in [8]. The obtained probability is valid in the wideinterval of photon energy, which is overlapped with SQA. In sect. 5 the case of the “non-relativistic” photon ω  m is analyzed. In particular, in the energy region ω  eE/mwhere the previous approach is inapplicable, the low energy and the very low energyapproximations have been developed basing on the analysis in [9]. In turn the foundresults have an overlapping region of applicability with the previous approach and witheach other. So just as in [9] we have four overlapping approximations which include allphoton energies. At the photon energy ω  eEm/(m2 + eE) the probability has beenfound for arbitrary values of electric E and magnetic B fields.2. – General expressions for the probability of processOur analysis is based on the general expression for the contribution of spinor particlesto the polarization operator obtained in a diagonal form (Baier, Katkov, Strakhovenko,1975). The imaginary part of the eigenvalue κi of this operator on the mass shell (k2 = 0)determines the probability per unit length Wi of the e−e+ pair creation by the real photonwith the polarization ei directed along the corresponding eigenvector. The considerationrealizes in the frame where the electric E and magnetic B fields are parallel and directedalong the axis 3. The probability of pair creation by the unpolarized photon has theformW =αm2r2πiωμν∫ 1−1dv∫ ∞−i0−∞−i0f(v, x) exp[iψ(v, x)]xdx;(1)r =ω2 − k234m2, ν =eEm2=EE0, μ =eBm2=BB0,(2)E0 = 1.32 · 1016 V/cm, B0 = 4.41 · 1013 G.PAIR PHOTOPRODUCTION IN A CONSTANT AND HOMOGENEOUS ETC. 3Heref(v, x) =cosh(νx)(cos(μx)− cos(μxv))sinh(νx) sin3(μx)+cos(μx)(cosh(νx)− cosh(νxv))sin(μx) sinh3(νx),(3)ψ(v, x) = 2r(cosh(νx)− cosh(νxv)ν sinh(νx)+cos(μx)− cos(μxv)μ sin(μx))− x.(4)After all calculations have been fulfilled we can return to a covariant form of theprocess description using the following expressions:(5) ν2 − μ2 = 2F = E2E20− B2B20, νμ = G = EBE0B0.The SQA is valid for ultrarelativistic created particles (r  1) and can be derivedfrom eqs. (1)-(4) by expanding the functions f(v, x), ψ(v, x) over x powers. Retainingthe leading powers of x one obtainsW (SQA) =αm23√3πω∫ 109− v21− v2K2/3(z)dv, z =83(1− v2)κ ,(6)κ2 = 4r(μ2 + ν2) = − e2m6(Fμνkν)2.To get the correction to the probability in SQA we shall keep the next powers of x. Wehave(7) W (1) = − αm2F30√3πωκ∫ 10dv1− v2G(v, z),where(8) G(v, z) = 2(1 + v2 − 27z2)K1/3(z) + 3(7− v2)zK2/3(z).It is seen that in this order of decomposition the correction does not depend on theinvariant parameter G, because G is the pseudoscalar. The asimptotics of the integralsincoming in the correction terms have been given in Appendix C [8]. The asymptotic atκ 1 will become necessary further(9) W (1) =6αm2F5ωκ2√23exp[− 83κ],W (1)W (SQA)=64F15κ3.3. – Region of intermediate photon energiesIn the field, which is weak in comparison with the critical field E/E0 = ν  1,B/B0 = μ 1 and at the relatively low photon energies r  ν−2/3, the standard quasi-classical approximation is non-applicable. At these energies, if the condition r  ν2 isfulfilled, the saddle-point method can be applied to integration over x. In this case the4 V. M. KATKOVsmall values of v contribute to the integral over v. So one can expand the phase ψ(v, x)over v and extend the integration limit to the infinity. We get(10) W =αm2r2πiωμν∫ ∞−∞dv∫ ∞−∞f(0, x) exp[−i [ϕ(x) + v2χ(x)]]xdx,whereϕ(x) = 2r(1μtanμx2− 1νtanhνx2)+ x,(11)χ(x) = rx2(νsinh(νx)− μsin(μx)).(12)From the equation ϕ′(x0) = 0 we find the saddle point x0(13) tan2νs2+ tanh2μs2=1r, x0 = −is.At r  1 we have(14) W =3αm2κ16ω√32exp[− 83κ+64F15κ3].This expression is valid at κ  1 and coincides with eq. (9) for F  κ3. So theoverlapping region of both approximations exists.It is interesting to consider the photon energy region |r − 1|  1 in the presence of aweak electric field (ν  μ) where in the absence of an electric field the approach underconsideration is valid if the condition r − 1  μ is fulfilled. In this case the followingapproximate equations are valid:ξ2y2016 exp[−y0] + 1− r4 , y0 = μs, ξ =νμ;(15)y0  2 ln 2ξ ln 4ξ(1− r − 12ξ2 ln 2ξ ln3 4ξ), |r − 1|  ξ2;(16)y0  ln 4r − 1(1− ξ24(r − 1) ln4r − 1), r − 1 ξ2;(17)ξy0 = νs  2√1− r, 1− r  ξ2.(18)The applicability of the using saddle-point method is connected with the large value ofthe coefficient to the second power (y − y0)2 of the decomposition in the phase ϕ(x). Inthe energy region under consideration we have(19) iϕ′′(x0)(x− x0)2/2  ξ24μ[y0 +y202+2(r − 1)ξ2](y − y0)2.So, in the case ν/μ = ξ  1, |r − 1|  ξ2 this approximation is valid if the conditionξ2/μ  1 is fulfilled. In the case 1  r − 1  ξ2 the condition r − 1  μ has toPAIR PHOTOPRODUCTION IN A CONSTANT AND HOMOGENEOUS ETC. 5be available for that. And in the case 1  1 − r  ξ2 the condition √1− rξ/μ =√(ξ2/μ)(1− r)/μ 1 is necessary.At energy r  1 (ν2  r  ν2/3) we have νs  π − 2√r,(20) W =αm2μ4ω√rcoth(πη) exp[−1ν(π − 4√r + 2rηtanhπη2)].At η  1 the probability W has been increased by the factor ηπ exp[πr/ν] in comparisonwith the case of the absence of magnetic field.4. – Approximations at low photon energyAt r ∼ ν2 the above approximation becomes non-applicable and another approachhas to be applied. We close the integration over x contour in the lower half-plane andrepresent eq. (1) in the following form:(21) W =αm2r2πiωμν∫ 1−1dv∞∑n=1∮f(v, x) exp[iψ(v, x)]xdx,where the path of integration is any simple closed contour around the point −iπnupslopeν. Letus choose the contour near this point in the following way νx = −iπn+ξn, |ξn| ∼√r ∼ νand expand the function entering in over the variables ξn. In the case ν  1, because ofthe appearance of the factor exp[−iπnupslopeν], the main contribution to the sum gives theterm n = 1. Near the point −iπupslopeν the main terms of expansion (ξ ≡ ξ1) are(22) f =2iξ3coth(πη) cos2πv2, ψ =4rξνcos2πv2− ξν+iπν.Using eq. (7.3.1) and eq. (7.7.1 (11)) in [15] we find after integration over ξ and v(23) W =αm2ωηπ coth(πη) exp[−πν]I21 (z), z =2√rν,where In(z) is the Bessel function of the imaginary argument. The found probability isapplicable for r  ν.For r  ν2 the asymptotic representation In(z)  exp[z]upslope√2πz can be used. As aresult one obtains eq. (20) if in the exponent of the last one leaves out the term ∝ r/ν.At very low photon energy r  ν2, using the expansion of the Bessel functions for thesmall value of argument, we have(24) W =αm2rων2ηπ coth(πη) exp[−πν].The probability under consideration draws the interest of theoreticians for arbitraryvalues μ and ν. For r  ν2upslope(1+ ν2) one can conserve in the phase ψ(v, x) the term −x6 V. M. KATKOVonly. After integrating over v we get the following equation for the probability of photonabsorption:W =αm2riπω∞∑n=1∮F (yn) exp[−iynν]dyn, yn = −inπ + y,(25)F (y) =cosh(y) (ηy cos(ηy)− sin(ηy))sinh y sin3 ηy+η cos(ηy)(y cosh y − sinh y)sinh3 y sin(ηy).(26)Summing the residues in the points yn = −inπ one obtainsW =αm2rω∞∑n=1exp[−πnν]Φ(zn), zn = ηπn,(27)Φ(zn) =znν2coth zn +2sinh2 zn[ηznν+(1 + η2)zn coth zn − 1].(28)At η → 0, zn → 0 we have (compare with eq. (28) in [9])Φ =1ν2+2νπn+2π2n2+23,(29)W =αm2rω[(1ν2+23)1eπ/ν − 1 −2πνln(1− e−π/ν)+2π2Li2(e−π/ν)],(30)where Li2(z) is the Euler dilogarithm. In the opposite case η →∞, zn →∞ one obtains(31) Φ =πηnν2, W =αm2rων2πη4sinh−2π2ν.5. – ConclusionThe probability of the process has been calculated using four different overlappingapproximations. In the region of SQA applicability the particles created by a photonhave ultrarelativistic energies. The role of fields in this case is to transfer the requiredtransverse momentum and the electric and magnetic field actions are equivalent. Buteven in this case it is necessary to note the special significance of the weak electric fieldE = ξB (ξ  1) in the removal of the root divergence of the probability when theparticles of the pair are created on the Landau levels with the electron and positronmomentum p3 = 0. In the frame used, k3 = 0.Generally speaking, at ξ  1 the formation time tc of the process under considerationis 1/μ. Here we use units  = c = m = 1. At this time the particle of the pair gets themomentum δp3 ∼ ξ because of the electric field. If the value ξ2 becomes more larger thanthe distance apart Landau levels 2μ (ν2  μ3) all levels have been overlapped. Underthis condition the divergence of the probability is vanished and the new quasi-classicalapproach is valid even in the energy region r−1  μ where it has been inapplicable in theabsence of the electric field. In the opposite case ν2  μ3 for the small value of p3  √μin the region where the influence of the electric field can be neglected, the formation timeof the process tf is 1/p23 and δp3 ∼ ν/p23  p3. It follows from the above that in thiscase the condition ν1/3  p3  √μ has to be satisfied. At this condition the value ofPAIR PHOTOPRODUCTION IN A CONSTANT AND HOMOGENEOUS ETC. 7discontinuity is√tf/tc ∼ √μ/p3. For ν1/3  p3 the time tf is determined by the self-consistent equation δε2 ∼ 1/tf ∼ ν2t2f , tf ∼ ν−2/3 and the value of discontinuity becomes√μtf ∼ (μ3/ν2)1/6 instead of √μ/p3. In the region ω  2m (r  1) the energy transferfrom the electric field to the created particles becomes appreciable and for ω  m itdetermines mainly the probability of the process. At ω  eE/m the photon assistancein the pair creation comes to an end and the probability under consideration definesthe probability of photon absorption by the particles created by electromagnetic fields.The influence of the magnetic field on the process is connected with the interaction ofthe magnetic moment of the created particles and magnetic field. This interaction, inparticular, has appeared in the distinction of the pair creation probability by field forscalar and spinor particles [5].∗ ∗ ∗This work was partly supported by Grant 14.740.11.0082 of Federal Program “Per-sonnel of Innovational Russia”.REFERENCES[1] Klepikov N. P., Ph.D. dissertation, unpublished (Moscow State University) 1952.[2] Toll J. S., Ph.D. dissertation, unpublished (Prinston University) 1952.[3] Klepikov N. P., Zh. Eksp. Teor. Fiz., 26 (1954) 19.[4] Adler S. L., Ann. Phys. (N.Y.), 67 (1971) 599.[5] Schwinger J., Phys. Rev., 82 (1951) 664.[6] Batalin I. A. and Shabad A. E., Sov. Phys. JETP, 33 (1971) 483.[7] Baier V. N., Katkov V. M. and Strakhovenko V. M., Sov. Phys. JETP, 41 (1975)198.[8] Baier V. N. and Katkov V. M., Phys. Rev. D, 75 (2007) 073009.[9] Baier V. N. and Katkov V. M., Phys. Lett. A, 374 (2010) 2201.[10] Narozhny N. B., Zh. Eksp. Teor. Fiz., 54 (1968) 676.[11] Sauter F., Z. Phys., 69 (1931) 742.[12] Nikishov A. I., Zh. Eksp. Teor. Fiz., 59 (1970) 1262.[13] Baier V. N. and Katkov V. M., Sov. Phys. JETP, 26 (1968) 854.[14] Tsai W. and Erber T., Phys. Rev. D, 10 (1974) 492.[15] Bateman H. and Erde´lyi A., Higher Transcendental Functions, Vol. II (McGraw-HillBook Co, New York) 1953.

Pair Photoproduction in Constant and Homogeneous Electromagnetic Fields

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