High-Energy Limit ofπ+π0Total Cross Section

The high-energy limit of the ${\pi }^{+}{\pi }^0$ total cross section $\sigma \left(\infty \right)$ is estimated upon the assumption that the ${\pi }^{+}{\pi }^0$ forward scattering amplitude is dominated by the $S$ wave below the $\rho$ resonance and becomes purely imaginary fairly rapidly above the $\rho$ resonance. The underlying assumption is that the ${\pi }^{+}{\pi }^0$ forward amplitude satisfies the usual analyticity assumption. Using the experimental values of the mass and the width of the $\rho$ resonance and also assuming an effective-range expansion for the $S$ wave, we estimate $\sigma \left(\infty \right)$ as a function of the scattering length and the effective range in the channel of total isospin 2. It is found that, for negative scattering length (corresponding to a repulsive interaction), $\sigma \left(\infty \right)$ rises fairly steeply as the scattering length increases its magnitude for all the effective ranges. In the case of positive scattering lengeth (corresponding to an attractive interaction), however, $\sigma \left(\infty \right)$ is much less dependent on the scattering length up to $0.8{\mu }^{-1\mathrm{}}$ especially when the effective range lies between ${\mu }^{-1\mathrm{}}$ and $2{\mu }^{-1\mathrm{}}$ (${\mu }^{-1\mathrm{}}$ being the pion Compton wave-length). In particular, $\sigma \left(\infty \right)$ hardly decreases below 20 mb as long as the effective range is not too much greater than $2{\mu }^{-1\mathrm{}}$. It is argued that various corrections to be considered can quite likely increase the above estimate of $\sigma \left(\infty \right)$ 10% to 20% in the case of a weakly attractive $S$-wave interaction, but hardly can decrease it. It is therefore felt that the present analysis excludes the possibility of a repulsive $S$-wave interaction and also implies that the high-energy limit of the ${\pi }^{+}{\pi }^0$ total cross section is about the same as that for the pion-nucleon total cross section.