Energy Dependence of the Phase Shifts in Pion-Proton Scattering

It is shown that all pion-proton scattering data up to 300 Mev can be fit within limits of experimental error by using the $s$-waves linearly extrapolated from the low-energy slopes of ${\alpha }_1=0.16\mathrm{}\eta$ and ${\alpha }_3=-0.11\mathrm{}\eta$. All other phase shifts are assumed zero except ${\alpha }_{33}$ which is given the energy dependence proposed by Chew and Low, $\left(\frac{{\eta }^3}{{\omega }^{*}}\right){cot\alpha }_{33}=8.05-3.8\mathrm{}{\omega }^{*}$. The two constants in the ${\alpha }_{33}$ energy dependence fit the initial slope of ${\alpha }_{33}=0.235\mathrm{}{\eta }^3$ as determined from low-energy experiments and make ${\alpha }_{33}$ go through 90° at 192 Mev. According to the cutoff theory this determines a $k_{max}\approx \frac{6\mu c}{\hslash }$, corresponding to a pion-nucleon interaction of much shorter range than $\frac{\hslash }{\mu c}$. A theoretical implication of the fit proposed here is that also the $s$-wave interaction range is less than $\frac{\hslash }{\mu c}$