Relativistic Three-Body Recombination with the QED Vacuum

Electron-positron pair annihilation into a single photon is studied when a second free electron is present. Focussing on the relativistic regime, we show that the photon emitted in the three-lepton interaction may exhibit distinct angular distributions and polarization properties. Moreover, the process can dominate over two-photon annihilation in relativistic electron-positron plasmas of few-MeV temperature. An analogy with three-body recombination of electrons with ions is drawn.

Figure 1

(Left) Furry-Feynman diagram of $e^{+}{e}^{-}$ pair annihilation upon scattering off another electron in the presence of an external plane-wave photon field. The zigzag lines represent field-dressed (Volkov) states; they are labeled by the particle momenta outside the field. $n$ and $n^{\prime }$ are the numbers of real photons absorbed at each vertex, respectively. (Right) In the limit of a weak external field, the Furry-Feynman diagram can be expanded to leading order into four ordinary Feynman graphs (a)–(d). The corresponding exchange Furry-Feynman diagram (with $p^{-}\leftrightarrow p$) accounts in the same manner for the remaining four field-free Feynman graphs.

Figure 2

Total rate for annihilation of an $e^{+}{e}^{-}$ pair at rest into a single photon, as a function of the energy of the second electron [see Eq. (1)]. The particle densities are chosen as ${\rho }_p={\rho }_{p^{-}}={10}^{24}{\mathrm{cm}}^{-3}$. The solid line is a logarithmic fit function to the numerical results shown by the open circles. Also shown are the separate contributions to the rate stemming from photons with the polarization vector perpendicular (${ϵ}_{\perp }$) or parallel (${ϵ}_{\parallel }$) to the plane spanned by $\mathbf{k}$ and $\mathbf{p}$.

Figure 3

Angular distribution of the photon emitted in reaction (1) for $E^{+}=E^{-}\approx m$ and various $E_p$ values: $E_p=40\mathrm{MeV}$ (solid line), $E_p=50\mathrm{MeV}$ (long-dashed line), $E_p=60\mathrm{MeV}$ (short-dashed line). The angle ${\theta }_k$ is measured with respect to the momentum vector $\mathbf{p}$. The inset shows the $E_p$ dependence of the angle where the emission maximum lies.

Figure 4

Temperature dependence of the total rates per volume for single-photon annihilation (blue dots) and two-photon annihilation (purple squares) in an $e^{+}{e}^{-}$ plasma. The inset displays the ratio of both annihilation channels.