Cross section for Rydberg antihydrogen production via charge exchange between Rydberg positroniums and antiprotons in a magnetic field

Abstract

The antihydrogen formation by charge exchange between cold antiprotons and Rydberg positronium ${\text{P}}_{s}^{*}$ is studied by using the classical trajectory Monte Carlo method. In the absence of external magnetic field the cross section scaled by the fourth power of the ${\text{P}}_{s}^{*}$ principal quantum number ${n}_{{\text{P}}_{s}}$ shows a universal behavior as a function of the ratio ${k}_{v}$ between the velocity of the ${\text{P}}_{s}$ center of mass and that of the positron in the classical circular orbit. At low velocity, below about ${k}_{v}\ensuremath{\simeq}0.2--0.3$, we show for Rydberg positronium that the cross section increases as $1/{k}_{v}^{2}$ or, in an equivalent way, as $1/{E}_{{\text{P}}_{s}}^{\text{cm}}$ with ${E}_{{\text{P}}_{s}}^{\text{cm}}$ being the ${\text{P}}_{s}^{*}$ center-of-mass energy. In this regime the distribution of the principal quantum number of the antihydrogen state is narrow and it shows a peak at about $\sqrt{2}{n}_{{\text{P}}_{s}}$ while at higher ${k}_{v}$ values a broad distribution of antihydrogen states is produced. The study of the collision process in the presence of moderate magnetic field (0.5--2 T) shows that there is an experimentally interesting region of ${k}_{v}$ with the cross section slightly higher than that in the absence of field. However the presence of a magnetic field changes significantly the cross section behavior as a function of ${k}_{v}$, especially at low velocities, where reductions of the cross sections and deviations from the $1/{k}_{v}^{2}$ ($1/{E}_{{\text{P}}_{s}}^{\text{cm}}$) are observed. Our calculations show a dependance of the cross section upon the angle between the magnetic field and the flight direction of the incoming ${\text{P}}_{s}^{*}$.