Probability density of the Yukawa particle at the origin: Numerical calculation and analytical solution

Abstract

The probability density of the Yukawa particle at the origin plays significant roles in fundamental physical quantities such as the darkonium production cross sections and matter-anti-matter annihilation rates. Inspired by the recent work of Napsuciale and Rodríguez (2021) [12], [13] where the authors proposed a complete analytical solution of the Yukawa problem at small screening parameters to analyze the bound state eigenenergies and probability densities at the origin, we perform in this work high-precision numerical calculation of these quantities by employing the generalized pseudospectral method. The variational method based on one-, two-, and three-parameter trial wave functions is also developed to provide an alternative analytical solution to the Yukawa problem. It is shown that these trial functions manifest simple implementation and high accuracy in the entire range of screening parameters.