Electron-positron pair production by linearly polarized photon in the nuclear field

Process of lepton pair production by polarized photon on nuclei can be used to measure the degree of linear polarization of high energy photon. The differential cross section and the analyzing power are calculated with taking into account higher powers of expansion on $Z\alpha$. Pure Coulomb and screened potential are considered.


II. INTRODUCTION
Studying of process of pair creation starts from celebrated papers of 1953-1969 [1,2,3,4,5,6] and continue to attract the attention up to now. Main interests nowadays is to use this process as a polarimeter. Really it have rather large cross section and the polarization effects can reach 14 percents, [12]. Two different mechanisms of pair creation must be taken into account: the Bethe-Heitler one, when the pair is produced in collision of two photonone real and other virtual and the bremsstrahlung mechanism,when pair is created by single virtual photon. It was shown in fundamental papers of E. Haug [13] that at photon energies exceeding 50MeV in the target Laboratory frame the contribution of bremsstrahlung mechanism as well as the interference of corresponding amplitude with two-photon ones do not exceed 5 percents and decreased with further photon energy growth. Taking into account the lowest order radiative corrections (RC) do not change the situation. For the case of target such as proton or light nuclei the main contribution to (RC) are connected with final state interaction between pair components. Two virtual photon exchange between pair and nuclei amplitude do not interfered with Born amplitude as well as they have different signatures.
Pure two photon exchange amplitude contribution do not contain any enhancement factors such as "large logarithms" of ratio of photon energy ω to lepton mass m,and have of order α 2 . It can be neglected compared with contribution of order α/π coming from interference of Born amplitude with 1-loop ones connected with lepton pair interaction.
The situation changed when one consider the pair creation on heavy nuclei with charge parameter ν = Zα not too small. Main contribution arises from many photon exchanges mechanism between pair component with nuclei.
Total cross section of pair creation process by photon on a nuclei of charge Z for the case of unpolarized photon is [7] σ = 28 9 .
This result was recently reproduced in informative paper of Ivanov an Melnikov [9], where the differential cross section was as well considered.
Direction of e + and e − emitting correlates with degree and direction of photon linear polarization and so this process can serve as polarimetric reaction for the measuring linear This approximation is valid for high energy of produced particles, m/ǫ ± ≪ 1, and for small emitting angles θ ± ∼ m/ǫ ± where ǫ ± ≤ 10 −3 , is energy of positron or electron, m is electron mass.
It is well known that main contribution to cross section of considered process gives just the region of small emitting angles. But it should be noted that for that for the purposes of highenergy photon polarimetry, where restriction on the value of observable in the experiment angles is θ * ± In this paper we use formalism of [9] paper to consider the case of linearly polarized photon.
First we briefly sketch the relevant results of paper [9]. Sudakov's parameterization of 4-momenta is used below: with a-euclidean two dimensional vector a = (0, 0, a x , a y ), orthogonal to photon 4-momentum The conservation low and on mass shell conditions leads to Matrix element corresponding to N photon exchange is where J N γ ll -are impact factor which rewritten in the simple form [8,9,10] with The quantities S, T obey the recurrent relations and the similar expression for T N . The initial values, which corresponds to one photon exchange are Introducing the values and their Fourier transform: the recurrent relations can be written in form: In Moliere approximation of atomic form-factor in Tomas-Fermi model (we use it below) the expression for form-factor is [11]: with this case the analytic expressions con be obtained: For pure Coulomb potential F (q 2 ) = 1, we have Φ c ( r 1 , r 2 ) = ln Boundary of recurrent relations are I (1) with K 0,1 (z)-modified Bessel functions. The summation on the number of exchanged photons can be performed: Differential cross section have a form: with polarization degree of photon, described by means of ξ 1,3 -it's Stokes parameters, ϕ is the angle between the vector J T and the direction of maximal polarization of photon (if we choice the x axes along the direction of maximal polarization of photon, we put ξ 1 = 0; P = ξ 3 ), and For the case of screened potential (ignoring the experimental conditions of pairs component detection) we use the expression for phase given above. Performing the integration on pair momenta we obtain: with the azimuthal angle ϕ 1 is the angle between the direction of maximal photon polarization and the plane containing the direction of initial photon and electron (positron) from the pair.
For the case of pure Coulomb potential integration in (19) diverge and must be regularized. We leave here this academic problem. For the case of screened potential we obtain: The ν dependence of coefficients a(ν), b(ν) is shown in Fig. 1.
Further we consider the realistic case of nonzero momentum, transferred to nuclei | q| 2 ≫ m 2 e = m 2 . For the case of pure Coulomb potential we have with , the value of transverse component of pair are For the case of small transferred to nuclei momentum m 2 ≪ q 2 ≪ p 2 1 ≈ p 2 2 we can put in (22)z = 1 and using F 1 = F 2 = |Γ(1 − iν)| −2 [14], we reproduce the cross section in Born And the similar expression for the case of screened potential with the replacement We note that the quantity in square brackets in the rhs of (23) equation is proportional to In Appendix we give the explicit expression of two virtual photons exchange to matrix element. The limit of large transversal momenta is also considered.

III. DISCUSSION
In a famous papers of Bethe, Maximon and Olsen [2][3][4] the general theory of pair production and bremsstrahlung was build basing on on the knowledge of electron wave function in Coulomb field. Part of these results were reproduced in perturbation theory in [9]. Unfortunately the expression for the differential cross sections which can be used in current experiments with some cuts were rather poorly presented. It is the motivation of this paper.
Part of distributions, needed for experiment we derive above. It is the differential distri- Asymmetry calculated by formulae (20) (see Fig.2) is as well large compared with the results obtained in [3]. The reason: is in [3] an restriction on emission angles was θ ± > 10 −3 put on. The main contribution which is relevant in (20), arises from the emission angles much more smaller corresponding to p 1,2 << m.
Results for cross sections given above do not depend on photon energy which in accordance with the results obtained in paper [13]. Namely the other mechanisms of pair creation give negligible contribution (on the level of several percents compared with the one considered above) starting from ω > 50MeV .
To estimate the order of magnitude of cross section we calculate it for unpolarized case for Z = 79 to be σ = 170mb.
The accuracy of our calculations is determined by the omitted terms The quantity of errors is of order of several percents. The last term corresponds to the final state interaction of the pair component, which was not considered here.

IV. ACKNOWLEDGEMENTS
We are grateful to V. Bytev, A. Korchin and E. Vinokurov for discussions and help. Two

V. APPENDIX
For the case of screened potential the matrix element, corresponding to two photon exchange have a form: with N P = 1 sū (P ′ )ku(P ), where c l was defined above, c lk = ( k 1 − p l ) 2 + m 2 .