Hyperspherical close-coupling study of hyperfine transitions in low-energyp+pμande±+Psscattering

Abstract

We apply the hyperspherical close-coupling method to the elastic and spin-flip processes in $p+p\ensuremath{\mu}$ and ${e}^{\ensuremath{-}}+\mathrm{Ps}$ (or equivalently, ${e}^{+}+\mathrm{Ps})$ scattering at low collision energies. The spin-spin interactions ${V}_{S}$ are included explicitly in the Hamiltonian, producing the correct hyperfine-splitting energy $\ensuremath{\Delta}\ensuremath{\epsilon}$ in the separated-atom limit. A comparison is made with a simpler method in which the effect of ${V}_{S}$ is represented only by the symmetry of the wave functions with respect to the interchange of the two identical particles. Explicit inclusion of ${V}_{S}$ is found to be of vital importance for the calculation of the spin-flip cross sections for collision energies lower than $\ensuremath{\Delta}\ensuremath{\epsilon}$ for both $p+p\ensuremath{\mu}$ and ${e}^{\ensuremath{-}}{(e}^{+})+\mathrm{Ps}$ collisions. The elastic cross sections are less sensitive to ${V}_{S},$ but those for $p+p\ensuremath{\mu}$ are still affected by ${V}_{S}$ for collision energies lower than $\ensuremath{\Delta}\ensuremath{\epsilon}$.