Hyperspherical elliptic coordinates treatment of muon transfer from muonic hydrogen to atomic oxygen

Quantum-mechanical calculations of muon transfer between muonic hydrogen and an oxygen nuclei for $s$ waves and collision energies in the range $10^{-3} - 10^3$ eV, are presented. Close-coupling time-independent Schr\"odinger equations, written in terms of hyperspherical elliptic coordinates were integrated along the hyper-radius to obtain the partial and total muon-transfer probabilities. The results show the expected Wigner-Bethe threshold behavior up to collision energies of the order of $10^{-2}$ eV and pronounced maxima at $10^2$ eV which can be interpreted in terms of crossings between potential energy curves corresponding to the entrance channel state $(\mu p)_{1s} + \mO$ and two product channels which asymptotically correlate to $p + (\mO\mu)_{n=5,6}$. The population of the final states with different orbital angular momenta is found to be essentially independent of energy in the range considered in this work. This can be attributed to a strong selection rule for the conservation of the quantum number associated to one of the elliptic hyperangles.