Determination of the π ± p → π ± π + n cross section near threshold

The total cross section for the ${\ensuremath{\pi}}^{\ensuremath{-}}\stackrel{\ensuremath{\rightarrow}}{p}{\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\pi}}^{+}n$ reaction has been measured at incident pion kinetic energies of 200, 190, 184, 180, and 176 MeV, and cross sections for the ${\ensuremath{\pi}}^{+}\stackrel{\ensuremath{\rightarrow}}{p}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{+}n$ reaction were obtained at 200 and 184 MeV. A fit of these cross sections by heavy baryon chiral perturbation theory yields values of $8.5\ifmmode\pm\else\textpm\fi{}{0.6(m}_{\ensuremath{\pi}}^{\ensuremath{-}3})$ and $2.5\ifmmode\pm\else\textpm\fi{}{0.1(m}_{\ensuremath{\pi}}^{\ensuremath{-}3})$ for the reaction matrix elements ${\mathcal{A}}_{10}$ and ${\mathcal{A}}_{32},$ and, for the s-wave isospin-0 and isospin-2 \ensuremath{\pi}-\ensuremath{\pi} scattering lengths, the values ${a}_{0}=0.23\ifmmode\pm\else\textpm\fi{}{0.08(m}_{\ensuremath{\pi}}^{\ensuremath{-}1})$ and ${a}_{2}=\ensuremath{-}0.031\ifmmode\pm\else\textpm\fi{}{0.008(m}_{\ensuremath{\pi}}^{\ensuremath{-}1}),$ respectively.