Variational Calculations of Positronium Scattering with Hydrogen

Abstract

Positronium-hydrogen (Ps-H) scattering is of interest, as it is a fundamental four-body Coulomb problem. We have investigated low-energy Ps-H scattering below the Ps(n=2) excitation threshold using the Kohn variational method and variants of the method with a trial wavefunction that includes highly correlated Hylleraas-type short-range terms. We give an elegant formalism that combines all Kohn-type variational methods into a single form. Along with this, we have also developed a general formalism for Kohn-type matrix elements that allows us to evaluate arbitrary partial waves with a single codebase. Computational strategies we have developed and use in this work are also discussed. With these methods, we have computed phase shifts for the first six partial waves for both the singlet and triplet states. The $^{1,3}$S and $^{1,3}$P phase shifts are highly accurate results and could potentially be viewed as benchmark results. Resonance positions and widths for the $^1$S-, $^1$P-, $^1$D-, and $^1$F-waves have been calculated. We present elastic integrated, elastic differential, and momentum transfer cross sections using all six partial waves and note interesting features of each. We use multiple effective range theories, including several that explicitly take into account the long-range van der Waals interaction, to investigate scattering lengths and effective ranges.