Photon splitting in a laser field

Photon splitting due to vacuum polarization in a laser field is considered. Using an operator technique, we derive the amplitudes for arbitrary strength, spectral content, and polarization of the laser field. The case of a monochromatic circularly polarized laser field is studied in detail, and the amplitudes are obtained as threefold integrals. The asymptotic behavior of the amplitudes for various limits of interest is investigated also in the case of a linearly polarized laser field. Using the results obtained, the possibility of experimental observation of the process is discussed.

Figure 1

Feynman diagram for the photon splitting amplitude given by Eq. (2). This diagram corresponds to the Furry representation and the thick line denotes the electron propagator (Green’s function) in the laser field. The wavy lines symbolize the external photons. The diagram with the permutation $k_2\leftrightarrow k_3$ has to be added.

Figure 2

The photon splitting rate $\dot{W}$ in a circularly polarized laser field in units of $\left({\pi }^2{m}^2{\alpha }^3∕{\omega }_1\right)\times {10}^{-4}$ as a function of $\chi$ at $\eta \ll 1$. The solid line is obtained by using the amplitudes (39), the dashed line is based on the small-$\chi$ asymptotics (38). The rate is averaged over the polarization of the initial photon, summed up over the polarizations of the final ones and integrated over ${\nu }_2$ and ${\varphi }_q$.

Figure 3

The ratio $\mathcal{R}$ defined in Eq. (42) calculated at $\chi =1$ with the help of Eq. (39) as a function of the angle ${\vartheta }_2$, measured in units of $\sqrt{{\omega }_0∕{\omega }_1}$.

Figure 4

The photon splitting rate $\dot{W}$ in a linearly polarized laser field in units of $\left({\pi }^2{m}^2{\alpha }^3∕{\omega }_1\right)\times {10}^{-4}$ as a function of the number $n$ of laser photons absorbed. The rate is calculated from the amplitudes in Eqs. (43) with $\chi =1$, is averaged over the polarization of the initial photon, summed up over the polarizations of the final ones, and integrated over ${\nu }_2$ and ${\varphi }_q$.