Charge-Dependent Corrections to Pion-Nucleon Scattering

If account is taken of the mass difference between neutral and charged pions and of the possibility that the three coupling constants (${\pi }^0-n$, ${\pi }^0-p$, ${\pi }^{\pm }$ nucleon) may differ, then the pion-nucleon system no longer conserves isotopic spin. This effect has been investigated using Chew-Low theory with a $p$-state interaction. For each $J$ value there are ten scattering amplitudes, replacing the two of the charge-independent case. Only eight of these amplitudes are independent due to time reversal invariance, and the mass difference effect can be related to a change in the energy scale. The amplitudes are determined as solutions to a set of linear integral equations which may be solved approximately in the one-meson approximation. Corrections to the differential cross sections are then calculated. These corrections go through a maximum at about 125 Mev and can affect the magnitude of the ${\pi }^{-}$ cross sections by as much as 35% in this region, as well as the slope of the ${\pi }^{-}$ cross section in the region 125-175 Mev. The effect on the ${\pi }^{+}$ cross section is small. Attempts are made to correlate the calculation with available data.