Exploiting Dirac equations solution for exact integral calculations in processes of muonic atoms

Precise muon wave functions are calculated by solving the Dirac equations within a method based on modern neural networks and stochastic optimization techniques. The reduced radial wave functions (up and bottom) of a bound muon in muonic atoms are then given as linear combinations of the well behaved sigmoid functions. As concrete applications, the known muonnucleus overlap integrals, which determine the structure and evolution of muonic atoms (twobody quantum systems bound together due to the fundamental electromagnetic interactions), may accurately be calculated. In this work, the muon-nucleus integrals for a set of nuclei including the 28Si and 64Zn entering the ordinary muon capture rates are exactly obtained through numerical integrations using Gauss integration algorithms.