First-order Born and first-order distorted-wave Born cross sections forμ++(μ−p)→(μ+μ−)+p

As one of the possible ways to produce a (${\mu }^{+}$ ${\mu }^{\mathrm{-}}$) atom, we have theoretically investigated the cross section of the process ${\mu }^{+}$+(${\mu }^{\mathrm{-}}$p)→(${\mu }^{+}$ ${\mu }^{\mathrm{-}}$)+p. The general behaviors of the cross sections obtained by the first-order Born approximation (FBA) and the first-order distorted-wave Born approximation (DWBA) are discussed. From the comparison of the present results with those of classical-trajectory Monte Carlo (CTMC) calculation and from the reliability of the present method which was discussed in the case of $e^{+}$+H→($e^{+}$ $e^{\mathrm{-}}$)+p, we can conclude that the maximum cross section is of the order of ${10}^{\mathrm{-}20}$ ${\mathrm{cm}}^2$ at about 2.8 keV impact energy, which is of the order of the geometrical cross section. The cross sections by FBA are of the same order with those by DWBA, agreeing to within 10–20 %. The DWBA cross section is larger than that by CTMC by 30–50 % in the velocity range of 5–10 keV. Scaling relationships between the cross section and impact velocity are discussed in the cases of ${\mu }^{+}$+(${\mu }^{\mathrm{-}}$p) and $e^{+}$+H.