Δ ( 1232 ) in π N → π π N reaction

The contribution of $\ensuremath{\Delta}(1232)$ to the $\ensuremath{\pi}N\ensuremath{\rightarrow}\ensuremath{\pi}\ensuremath{\pi}N$ reaction is examined by making use of the chiral reduction formula developed by Yamagishi and Zahed. The influence of the $\ensuremath{\pi}\ensuremath{\Delta}\ensuremath{\Delta}$ and $\ensuremath{\rho}N\ensuremath{\Delta}$ interactions on this reaction, which has not been regarded as important so far, is considered for all channels with the initial ${\ensuremath{\pi}}^{\ifmmode\pm\else\textpm\fi{}}p$ states. The total cross sections are calculated to tree level for the energy up to ${T}_{\ensuremath{\pi}}=400\phantom{\rule{0.3em}{0ex}}\text{MeV}$. Although the $\ensuremath{\pi}\ensuremath{\Delta}\ensuremath{\Delta}$ and $\ensuremath{\rho}N\ensuremath{\Delta}$ interactions give a small effect on the ${\ensuremath{\pi}}^{+}p\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{+}n$, ${\ensuremath{\pi}}^{\ensuremath{-}}p\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}n$, and ${\ensuremath{\pi}}^{\ensuremath{-}}p\ensuremath{\rightarrow}{\ensuremath{\pi}}^{0}{\ensuremath{\pi}}^{0}n$ channels, the ${\ensuremath{\pi}}^{\ifmmode\pm\else\textpm\fi{}}p\ensuremath{\rightarrow}{\ensuremath{\pi}}^{\ifmmode\pm\else\textpm\fi{}}{\ensuremath{\pi}}^{0}p$ channels are found to be sensitive to these interactions.