Ab initio calculations of scattering cross sections of the three-body system ($\overline{p},e^{+},e^{-}$) between the $e^{-}+\overline{\mathrm{H}}(n=2)$ and $e^{-}+\overline{\mathrm{H}}(n=3)$ thresholds

The ab initio method based on the Faddeev-Merkuriev equations is used to calculate cross sections involving the $(\overline{p},e^{+},e^{-})$ three-body system, with an emphasis on antihydrogen formation $\left(\overline{\mathrm{H}}\right)$ via antiproton $\left(\overline{p}\right)$ scattering on positronium. This system is studied in the energy range between the $e^{-}+\overline{\mathrm{H}}(n=2)$ and the $e^{-}+\overline{\mathrm{H}}(n=3)$ thresholds, where precisely calculated cross sections can be useful for future experiments (GBAR, AEGIS, etc.) aiming to produce antihydrogen atoms. A special treatment is developed to take into account the long-range charge-dipole interaction effect on the wave function. Emphasis is placed on the impact of Feshbach resonances and Gailitis-Damburg oscillations appearing in the vicinity of the $\overline{p}+\mathrm{Ps}(n=2)$ threshold.

Figure 1

The three possible configurations, where $\{{\mathbf{x}}_i,{\mathbf{y}}_i\}$ are the three possible sets of Jacobi coordinates.

Figure 2

$S$-wave partial cross section of the reaction $\overline{p}+\mathrm{Ps}(n=1)\to e^{-}+\overline{\mathrm{H}}(n=2)$. Vertical lines labeled $\overline{\mathrm{H}}\left(2\right)$ and $\mathrm{Ps}\left(2\right)$ indicate, respectively, the positions of the $\overline{\mathrm{H}}(n=2)$ and $\mathrm{Ps}(n=2)$ thresholds.

Figure 3

$S$-wave partial cross section of the reaction $\overline{p}+\mathrm{Ps}(n=1)\to e^{-}+\overline{\mathrm{H}}(n=1)$.

Figure 4

$S$-wave partial cross section of the reaction $\overline{p}+\mathrm{Ps}(n=1)\to \overline{p}+\mathrm{Ps}(n=1)$.

Figure 5

$P$-wave partial cross sections of the reaction $\overline{p}+\mathrm{Ps}(n=1)\to e^{-}+\overline{\mathrm{H}}(n=2)$, separated to show the production of the different degenerate channels.

Figure 6

$S$-wave partial cross section of the reaction $\overline{p}+\mathrm{Ps}(n=2)\to e^{-}+\overline{\mathrm{H}}(n\le 2)$.

Figure 7

$S$-wave partial cross section of the reaction $\overline{p}+\mathrm{Ps}(2s,2p)\to \overline{p}+Ps(2s,2p)$.

Figure 8

$S$-wave partial cross sections of the reactions $\overline{p}+\mathrm{Ps}\left(2s\right)\to \overline{p}+\mathrm{Ps}\left(2s\right)$ (triangles) and $\overline{p}+\mathrm{Ps}\left(2p\right)\to \overline{p}+\mathrm{Ps}\left(2p\right)$ (circles).

Figure 9

$P$-wave partial cross section of the reaction $e^{-}+\overline{\mathrm{H}}(n=1)\to e^{-}+\overline{\mathrm{H}}(n=2)$.

Figure 10

$D$-wave partial cross section of the reaction $\overline{p}+\mathrm{Ps}(n=2)\to e^{-}+\overline{\mathrm{H}}(n\le 2)$.

Figure 11

Total cross section of the reaction $\overline{p}+\mathrm{Ps}(n=1)\to e^{-}+\overline{\mathrm{H}}(n=1,2)$. Crosses represent results from Ref. [16].

Figure 12

Total cross section of the reaction $e^{-}+\overline{\mathrm{H}}(n=1)\to e^{-}+\overline{\mathrm{H}}(n=2)$.

Figure 13

Total antihydrogen production cross section from the $\overline{p}+\mathrm{Ps}(n=2)\to e^{-}+\overline{\mathrm{H}}(n\le 2)$ reaction. The blue line summarizes our calculations. Crosses and triangles represent results from Refs. [16] and [15], respectively.

Figure 14

Total cross section of the reaction $\overline{p}+\mathrm{Ps}\left(2p\right)\to e^{-}+\overline{\mathrm{H}}(n\le 2)$. The blue line represents the results in this work; crosses, results from Ref. [16].

Figure 15

Total cross section of the reaction $e^{-}+\overline{\mathrm{H}}(n=2)\to \overline{p}+\mathrm{Ps}(n\le 2)$.

Figure 16

Differential cross sections of the reaction $\overline{p}+\mathrm{Ps}(n=1)\to e^{-}+\overline{\mathrm{H}}(n=1)$ at $E_{3b}=-0.08$ a.u. (squares) and $E_{3b}=-0.115$ a.u. (triangles).

Figure 17

Differential cross sections of the reaction $\overline{p}+\mathrm{Ps}(n=1)\to e^{-}+\overline{\mathrm{H}}(n=2)$ at $E_{3b}=-0.10\text{a.u.}$, $e^{-}({\ell }_{y_1}=L)+\overline{\mathrm{H}}\left(2s\right)$ (open squares), $e^{-}({\ell }_{y_1}=L-1)+\overline{\mathrm{H}}\left(2p\right)$ (triangles), and $e^{-}({\ell }_{y_1}=L+1)+\overline{\mathrm{H}}\left(2p\right)$ (filled squares).

Figure 18

Differential cross sections of the reaction $\overline{p}+\mathrm{Ps}(n=1)\to e^{-}+\overline{\mathrm{H}}(n=2)$ at $E_{3b}=-0.08$ a.u. (open squares), $E_{3b}=-0.10$ a.u. (filled squares), and $E_{3b}=-0.115$ a.u. (triangles).