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

Abstract

A bound state of a proton, p, and its counterpart antiproton,p, is a protonium atom Pn = (pp). The following three-charge-particle reaction:p + (pμ − ) 1s → (pp) 1s + μ − is considered in this work, where μ − 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/2 p, wherep and p are placed close enough to each other and the effect of thep-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 (FHtype) equations. Unlike the original three-body Faddeev method the FH-type equation approach is 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 the (pμ − ) eigenfunctions. At the same time Ψ 2 is expanded in terms of the output channel two-body wave function, that is in terms of the protonium (pp) 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.

Effect of strong $\bar{\rm p}$-p nuclear forces on the rate of the low-energy three-body protonium formation reaction: $\bar{p} + H_μ(1s) \rightarrow (\bar{p} p)_α + μ^-$

“…Further, the second order partial derivatives on the left side can be discretized by using a three-point rule [49]. This process allows us to obtain a set of linear equations for the unknown coefficients f [34,35]. Then it is possible to ascertain through the symbolic-operator notations that the set of linear equations has the following characteristics [34,35]:…”

“…It then follows that the matrix A should exhibit a well known block-structure. In this case there are four major blocks in the matrix: two of them are related to the differential operators and other two are related to the integral operators [34,35]. Further, each block should contain sub-blocks.…”

“…The solution to the coupled integral-differential equations ( 21) requires one to first compute the angular integrals S [34]. These integrals are independent of the energy, E. To improve efficiency, one can compute them only once and then store them on a computer's hard drive for future reference.…”

“…A modified close coupling approach (MCCA) is applied in this work in order to solve the Faddeev-Hahn-type (FH-type) equations [33][34][35]. In other words, we carry out an expansion of the Faddeev-type components into eigenfunctions of the subsystem Hamiltonians.…”

Nuclear effects in protonium formation low-energy three-body reaction: p̄ + (pμ)1s→ (p̄p)α+μ−: Strong p̄–p interaction in p̄ + (pμ)1s

“…This means to carry out an expansion of the Faddeev-type components into eigenfunctions of the subsystem Hamiltonians. This technique provides an infinite set of one-dimensional integral-differential equations [30,31]. Within this formalism the asymptotic of the full three-body wave function contains two parts corresponding to two open channels [32].…”