Investigation of $\pi^+\pi^-, K^+\pi^-$ and $\pi^+K^-$ atoms for pion-pion and pion-kaon scattering length measurements

Theory, using Low Energy QCD, calculated with high precision the $\pi\pi$ and $\pi K$ scattering length. Lifetime of $\pi^+\pi^−$ atoms are related with a difference of pion-pion s-wave scattering lengths with isospin 0 and 2: $|a_0 − a_2|$. Energy splitting between p-states (long-lived atoms) and s-states of $\pi^+\pi^−$ atoms allows to investigate another combination: $2a_0 + a_2$. Lifetime of $\pi K$ atoms gives information about difference of pion-kaon s-wave scattering lengths with isospin 1/2 and 3/2: $|a_{1/2} − a_{3/2}|$. Experimental measurements of pion-kaon lifetimes and observation of long-lived $\pi^+\pi^−$ atoms at experiment DIRAC are presented