Polarization-operator approach to electron-positron pair production in combined laser and Coulomb fields

The optical theorem is applied to the process of electron-positron pair creation in the superposition of a nuclear Coulomb and a strong laser field. We derive representations for the total production rate as twofold integrals, both for circular laser polarization and for the general case of elliptic polarization. Our approach allows us to obtain, by analytical means, the asymptotic behavior of the pair creation rate for various limits of interest. In particular, we consider pair production by two-photon absorption and show that, close to the energetic threshold of this process, the rate obeys a power law in the laser frequency with different exponents for linear and circular laser polarization. With the help of the upcoming x-ray laser sources, our results could be tested experimentally.

Figure 1

Pictorial equation in terms of Feynman diagrams describing $e^{+}{e}^{-}$ pair production in combined laser and Coulomb fields. In our approach, the Coulomb field is treated within the first order of perturbation theory, while the effect of the laser wave is nonperturbatively taken into account to all orders. This is expressed by the Feynman graph on the left-hand side, where the double lines represent the exact lepton wave functions in the laser field (Volkov states) and the dashed line stands for their interaction with the Coulomb field. Expanding the Volkov states with respect to the lepton-laser coupling results in a perturbation series, some typical low-order terms of which are shown on the right-hand side. The wavy lines symbolize the laser photons, and the arrows indicate whether the respective laser photon is emitted or absorbed during the process. The first diagram on the right-hand side, e.g., describes the leading order of pair production by the net-absorption of one laser photon (with $\hslash \omega >2m{c}^2$). For later reference, we note that the interference between the first and the fourth diagrams leads to a ${\xi }^2$ correction to the probability for one-photon pair creation [cf. Eq. (34)].

Figure 2

Lowest-order (in $\alpha$) Feynman graph for the polarization operator of a photon in an external laser field. In our case the photon is a virtual one stemming from a Coulomb field (outer dashed lines). The diagram can be viewed as describing the elastic forward scattering of the photon through an intermediate laser-dressed $e^{+}{e}^{-}$ state (double line). By performing, as indicated, a cut along the central dashed line, it is seen that the graph represents the product of the amplitude for the production of a laser-dressed $e^{+}{e}^{-}$ pair by the virtual photon and its complex conjugate, including a sum over all electron states. Application of the optical theorem (i.e., the unitarity of the scattering matrix) therefore relates the imaginary part of the polarization operator with the total probability for pair creation, as expressed by Eqs. (4, 5).

Figure 3

The function $\mathcal{F}(x,\mu )$ in the interval $1⩽x⩽2$ for $\mu =1$ (linear polarization, solid line) and $\mu =0$ (circular polarization, short-dashed line). The long-dashed and dotted lines show the respective threshold laws for $x\approx 1$ according to Eqs. (24, 28).