Three-body Coulomb description of pionic helium

Abstract

The three-body Schr\"odinger equation of the system composed of an $\ensuremath{\alpha}$ particle ${\mathrm{He}}^{2+}$, a negatively charged pion ${\ensuremath{\pi}}^{\ensuremath{-}}$, and an electron interacting through Coulomb forces is solved in perimetric coordinates with the Lagrange-mesh method. The ground state, quasibound states, and resonances are obtained for total orbital momenta $L=0\ensuremath{-}20$. Mean distances between the particles allow for identifying the structure of these states. The widths of resonances broader than ${10}^{\ensuremath{-}5}$ atomic units are derived with the complex scaling method. A transition from atomic to molecular structure occurs between $L=10$ and $L=13$. Excited levels obtained for $L=0\ensuremath{-}5$ display hydrogenlike properties for both the $\ensuremath{\alpha}$-pion system and the electron excitations. Excited levels with $L\ensuremath{\ge}14$ correspond to a vibrational excitation of the relative motion of the heavy particles. The validity of the Born-Oppenheimer approximation is found to be fair over the whole range of total orbital momenta.