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Antihydrogen $(\bar{\rm{H}})$ and muonic antihydrogen $(\bar{\rm{H}}_μ)$ formation in low energy three-charge-particle collisions

A few-body formalism is applied for computation of two different three-charge-particle systems. The first system is a collision of a slow antiproton, $\bar{\rm{p}}$, with a positronium atom: Ps$=(e^+e^-)$ $-$ a bound state of an electron and a positron. The second problem is a collision of $\bar{\rm{p}}$ with a muonic muonium atom, i.e. true muonium $-$ a bound state of two muons one positive and one negative: Ps$_μ=(μ^+μ^-)$. The total cross section of the following two reactions: $\bar{\rm p}+(e^+e^-) \rightarrow \bar{\rm{H}} + e^-$ and $\bar{\rm p}+(μ^+μ^-) \rightarrow \bar{\rm{H}}_μ + μ^-$, where $\bar{\rm{H}}=(\bar{\rm p}e^+)$ is antihydrogen and $\bar{\rm{H}}_μ=(\bar{\rm p}μ^+)$ is a muonic antihydrogen atom, i.e. a bound state of $\bar{\rm{p}}$ and $μ^+$, are computed in the framework of a set of coupled two-component Faddeev-Hahn-type (FH-type) equations. Unlike the original Faddeev approach the FH-type equations are formulated in terms of only two but relevant components: $Ψ_1$ and $Ψ_2$, of the system's three-body wave function $Ψ$, where $Ψ=Ψ_1+Ψ_2$. In order to solve the FH-type equations $Ψ_1$ is expanded in terms of the input channel target eigenfunctions, i.e. in this work in terms of, for example, the $(μ^+μ^-)$ atom eigenfunctions. At the same time $Ψ_2$ is expanded in terms of the output channel two-body wave functions, that is in terms of $\bar{\rm{H}}_μ$ atom eigenfunctions.