The estimation of production rates of ${\pi }^{+}{K}^{-}$, ${\pi }^{-}{K}^{+}$ and ${\pi }^{+}{\pi }^{-}$ atoms in proton–nucleus interactions at 450 GeV c$^{-1}$

Short-lived ($\tau \sim 3\times {10}^{-15}\,s$) ${\pi }^{+}{K}^{-}$, ${K}^{+}{\pi }^{-}$ and ${\pi }^{+}{\pi }^{-}$ atoms as well as long-lived ($\tau \geqslant 1\times {10}^{-11}$ s) ${\pi }^{+}{\pi }^{-}$ atoms produced in proton–nucleus interactions at 24 GeV c(–)(1) are observed and studied in the DIRAC experiment at the CERN Proton Synchroton. The purpose of this paper is to show that the yields of the short-lived ${\pi }^{+}{K}^{-}$, ${K}^{+}{\pi }^{-}$ and ${\pi }^{+}{\pi }^{-}$ atoms in proton–nucleus interactions at 450 GeV c(–)(1) and ${\theta }_{{\rm{lab}}}=4^\circ $ are estimated to be, respectively 67 ± 13, 31 ± 6 and 15 ± 2 times higher. This may allow a significant improvement of the precision of their lifetime measurement and $\pi \pi $ and $\pi K$ scattering length combinations $| {a}_{0}-{a}_{2}| $ and $| {a}_{1/2}-{a}_{3/2}| $. The yields of the long-lived ${\pi }^{+}{K}^{-}$, ${K}^{+}{\pi }^{-}$ and ${\pi }^{+}{\pi }^{-}$ atoms at 450 GeV c(–)(1) are estimated to be 265 ± 53, 120 ± 24 and 60 ± 9 times higher per time unit than at 24 GeV c(–)(1). This may allow the resonance method to be used for measuring the Lamb shift in the $\pi \pi $ atom and a new $\pi \pi $ scattering length combination $2{a}_{0}+{a}_{2}$ to be obtained.