Phase of direct CP violation from threshold pion production.

Abstract

A recent series of experiments on pion production near threshold provides accurate values of the $s$-wave $\ensuremath{\pi}\ensuremath{-}\ensuremath{\pi}$ isospin scattering lengths. Combined with dispersion relations we find the $s$-wave phase shift difference at the $K$ mass is ${\ensuremath{\delta}}_{0}\ensuremath{-}{\ensuremath{\delta}}_{2}=42\ifmmode\pm\else\textpm\fi{}4\ifmmode^\circ\else\textdegree\fi{}$ and the phase of direct $\mathrm{CP}$ violation is $\mathrm{arg}{\ensuremath{\epsilon}}^{\ensuremath{'}}=\frac{\ensuremath{\pi}}{2}\ensuremath{-}({\ensuremath{\delta}}_{0}\ensuremath{-}{\ensuremath{\delta}}_{2})=48\ifmmode\pm\else\textpm\fi{}4\ifmmode^\circ\else\textdegree\fi{}$. We also evaluate the $p$-wave scattering length to be ${a}_{1}=0.035\ifmmode\pm\else\textpm\fi{}0.001{m}_{\ensuremath{\pi}}^{\ensuremath{-}3}$.