Apparent Violation of Unitarity in Elastic ${\pi }^{-}p$ Scattering

We deduce that if $f\left(cos\theta \right)$ has nonnegative partial waves, then ${\left[\frac{f\left(cos\theta \right)}{f\left(1\right)\mathrm{}}\right]}^2<~\frac{1}{3}+\frac{2}{3}\frac{f\left(\frac{1}{2}\right(3\mathrm{}{cos}^2\theta -1\left)\right)}{f\left(1\right)\mathrm{}}$ for $-1\mathrm{}<~cos\theta <~1$. Applying this, we show that recent determinations of the ${\pi }^{-}p$ slope parameter for $\mathrm{}\left|t\right|\mathrm{}<~0.04$ (${\mathrm{G}\mathrm{e}\mathrm{V}/\mathit{c})}^2$ at 100 GeV/c at CERN, together with those at larger $\mathrm{}\left|t\right|\mathrm{}$ at Fermilab, imply a substantial violation of unitarity, though each determination by itself is consistent with unitarity.