Generation of relativistic positrons carrying intrinsic orbital angular momentum

High energy positrons can be efficiently created through high-energy photons splitting into electron-positron pairs under the influence of the Coulomb field. Here we show that a new degree of freedom-the intrinsic orbital angular momentum (OAM) can be introduced into relativistic positrons when the incident photons are twisted. We developed the full-twisted scattering theory to describe the transfer of angular momentum before and after the interaction. It is found that the total angular momentum (TAM) of the photon is equally distributed among the positron and electron. For each photon TAM value, the generated leptons gain higher average OAM number when the photon spin is anti-parallel to its TAM. The impact of photon polarization on the OAM spectrum profile and the scattering probability is more significant at small photon TAM numbers, owing to the various interaction channels influenced by flipping the photon spin. Our work provides the theoretical basis to study OAM physics in particle scattering and to obtain copious relativistic vortex positrons through the Beth-Heitler process.


I. Introduction:
High energy positrons are of great significance in modern particle physics experiments.
Their collision with energetic electrons is essential in generating new particles such as Bmesons/Z-bosons [1,2] and monitoring various reaction processes via Bhabha scattering [3][4][5]. Further, in astrophysics positrons are strongly correlated to black hole physics [6], gamma-ray bursts [7] and pair plasma physics [8,9]. Here interactions mainly concern the energy/momentum and the spin properties of the involved positrons.
In transmission electron microscope (TEM), vortex electrons have been prepared in the 80~300 keV regime to improve the resolution and reveal new information of the subjects [19]. Interactions based on relativistic vortex particles have also been studied in the framework of quantum electrodynamics (QED), showing new features in the Vavilov-Cherenkov radiation [20,21] and the possibility of creating spin polarized particles from spinless ones [22].
Although high energy vortex particles bring novel insights in many interactions, generation of relativistic vortex positrons is extremely challenge. First, positrons, unlike electrons, must be created before introducing any features to the particle states. Second, manipulation of the lepton wave packets in TEM becomes invalid in the relativistic regime since typical wavelengths of high-energy leptons are too small. Recalling that vortex gamma-photons can be readily obtained from Compton backscattering of Laguerre-Gaussian laser beams off high energy electrons [23], we propose a scheme to generate vortex positrons at MeV energies by bombarding the vortex gamma-photons onto high-Z material. Relativistic electron-positron pairs can be created through the Beth-Heitler (BH) process, which facilitates efficient positron sources. By introducing a new degree of freedom-OAM into the interaction, we obtain the law of vorticity transfer from the incident gamma-photons to the created pairs. This is achieved with the first full vortex scattering theory of the BH process developed in this work.

II. Theory of the vortex BH process
The natural unit system 1 c .is applied in all calculations. We consider the process shown in Fig. 1(a). Here a high-energy vortex photon bombards a high-Z target. Under the influence of the Coulomb field, photons may split into electron-positron pairs carrying the initial OAM information. To capture the vortex nature of the interaction, we use the Bessel modes to describe all involved particles [18,23] Here M is the mass of electron/positron and The integer l represents the photon TAM number, 1 m and 2 m are the OAM numbers of electrons and positrons, respectively. Details of the matrix elements 11 S is obtained by exchanging the photon and the Coulomb field in S1, which leads to a different matrix  (see in Appendix). Each matrix element in  and  determines the creation probability of pairs with different spinpolarizations, and the angular momentum (AM) dependent Kronecker delta function gives the corresponding selection rule for the twisted BH process, with Δ=0, ±1. The minus sign before m2 in Eq. (6) with central momenta and energy , is a coefficient associated with wave packet and  is the opening angle of the incident particle in the wave packet tan( ) /    z kk . The pair creation probability is 2 1 2 1 1 2 2 wave packet z z d p p S dp dp dp dp

III. Results
In order to calculate the scattering probabilities, we set the central energy and momenta of the photon wave packet as 5MeV , the photon polarization is 1   and TAM number is 6 l  . We consider photons interacting with copper atoms   Z 29  . By integrating the momentum of the electron in Eq. (9) we obtain the creation probability of the positron. As known in plane-wave scattering, the interaction follows energy and momentum conservation, leading to a thin resonance line in the momentum space, as represented by the dashed circle in Fig. 1(b). However, in vortex scattering the conservation is for energy. The resonance condition is significantly relaxed such that probability is distributed in a broad region in Fig. 1(b). Furthermore, it is seen that the probability peaks along a certain angle. In Fig. 1(c)   We also compare the distributions of final positron with spin-up and down states in Fig.   1(c). For the case we considered, the spin-down channel contributes to the large part of the interaction probability. Thus, the generated particle is obviously polarized. This is consistent with the propagation of polarization from polarized photons to positrons [24][25][26]. The polarization is however not 100%, indicating spin-orbital coupling during the scattering process.
A central question about the vortex scattering is how OAM is distributed among the final particles. In the following, we show OAM spectra for the generated pair and their relationship with the TAM of the initial photon. We integrate the momentum and sum over the spins of created pairs to get the total probabilities at different OAM number, as shown in Fig. 2(a) with photon TAM 5,10,15, 20  l . The electron primarily carries OAM with the same sign as the photon while that of the positron is opposite. This is because the defined direction of positron OAM is opposite to that of electron. As the photon TAM increases, the central OAM shifts towards large values accordingly. Furthermore, the total creation probability declines and the OAM spectrum width increase at larger photon TAM values, as illustrated in Fig. 2(b).  Flip of the photon polarization also changes the orientation of lepton spins. It is seen in Fig. 2(d) that the averaged spin number turns over when switching from =1 to -1 which leads to higher average OAM in Fig.2(c) (red line). In fact, the photon spin effect is also imprinted in the asymmetric OAM distributions in Fig. 2(a) at relatively small photon TAM numbers, e.g., l=5 and 10. This effect becomes less significant when l is large. We show the total scattering probability as a function of photon TAM number in Fig. 2(c) with different photon spins. It is seen that at each TAM value the probability is notably higher for =1. The difference is much more suppressed when increasing the TAM.
To reveal how photon's polarization affects the distribution of positron and electron OAM, we divide the interaction into four channels determined by the final spin states of the leptons and compare their OAM spectra in Fig. 3 with photon TAM l=10 and 20 respectively. First of all, for 1   we find that the (-1/2, -1/2) channel in Fig. 3 is significantly suppressed as compared to the (1/2, 1/2) one in Fig. 3  For l=20 the spectra of Fig. 3(f) and (g) much more smooth and monopole-like. After summing up the four channels, the OAM distribution shows good symmetry as seen in Fig. 2(a). This is because the ratio between photon spin and TAM is much smaller here.
The obervation suggests that the photon spin has little effect on the total probability when keeping l-λ (rather than l) constant.

IV. Discussion
Spin-orbit coupling naturally exists in the quantum vortex state in the relativistic regime. The average OAM and spin angular momentum (SAM) of the vortex positron state after considering the SOI are [18]: Here   2 1 / sin    ME denotes the SOI induced change. We take photon TAM 5  l with different polarization to calculate the possibility as a function of the average OAM and SAM in Fig. 4. As the opening angle increases, both the average OAM m and SAM s deviate from the quantized integrals. In general, peak position of average SAM is slightly lower than 1/ 2  s in all cases and the one for m is also quite close to the OAM quantum number. These results validates our approximations in the analysis.

V. Conclusions
We employed the twisted states of photons and electrons/positrons to calculate the scattering probabilities of twisted photons into twisted electron-positron pairs under the influence of the Coulomb field. We found that the photon TAM is efficiently transferred to the final leptons. We discussed the effect of spin-orbit coupling on average OAM and SAM and noticed that angular momentum is also transferred from spin to orbit. This work lays the foundation of generating relativistic vortex positron beams for new physics in a number of twisted scattering processes.