Three-Body Protonium Formation in a Collision Between a Slow Antiproton ( $${\bar{\rm p}}$$ p ¯ ) and Muonic Hydrogen: $${{\rm H}_{\mu}}$$ H μ —Low Energy $${\bar{\rm p} + ({\rm p} \mu^-)_{1s} \rightarrow (\bar{\rm p} {\rm p})_{1s} + \mu^-}$$ p ¯ + ( p μ - ) 1 s → ( p ¯ p ) 1 s + μ - Reaction

A bound state of a proton, p, and its counterpart antiproton, $${\bar{\rm p}}$$ , is a protonium atom $${Pn = (\bar{\rm p} {\rm p})}$$ . The following three-charge-particle reaction: $${\bar{\rm p} +({\rm p} \mu^-)_{1s} \rightarrow (\bar{\rm p} \rm{p})_{1s} + \mu^-}$$ is considered in this work, where $${\mu^-}$$ is a muon. At low-energies muonic reaction $${Pn}$$ can be formed in the short range state with α = 1s or in the first excited state: α = 2s/2p, where $${\bar{\rm p}}$$ and p are placed close enough to each other and the effect of the $${\bar{\rm p}}$$ –p nuclear interaction becomes significantly stronger. The cross sections and rates of the Pn formation reaction are computed in the framework of a few-body approach based on the two-coupled Faddeev-Hahn-type (FH-type) equations. Unlike the original three-body Faddeev method the FH-type equation approach is formulated in terms of only two but relevant components: $${{\it \Psi}_1}$$ and $${\it \Psi_2}$$ , of the system’s three-body wave function $${\it \Psi}$$ , where $${{\it \Psi}={\it \Psi}_1+{\it \Psi}_2}$$ . In order to solve the FH-type equations $${\it \Psi_1}$$ is expanded in terms of the input channel target eigenfunctions, i.e. in this work in terms of the $${(\rm{p} \mu^-)}$$ eigenfunctions. At the same time $${\it \Psi_2}$$ is expanded in terms of the output channel two-body wave function, that is in terms of the protonium $${(\bar{\rm{p}} \rm{p})}$$ eigenfunctions. A total angular momentum projection procedure is performed, which leads to an infinite set of one-dimensional coupled integral–differential equations for unknown expansion coefficients.