High-Energy Behavior of Total Cross Sections

A proof of Pomeranchuk's theorem regarding the high-energy limits of the total cross sections is presented. The proof consists of assuming the usual analyticity for the forward elastic amplitudes and the assumption that these amplitudes become pure imaginary in the high-energy limit. This proof does not require that the total cross sections have finite limits. It is also shown that the total cross-section $\sigma \mathrm{}\left(s\right)\mathrm{}$ as a function of $s$, the total c.m. energy squared, behaves asymptotically as $\sigma \mathrm{}\left(s\right)=\sigma \left(\infty \right)+\frac{\delta }{s^{\frac{1}{2}}}+\mathrm{O}\left(\frac{1}{s}\right)$ when $\delta \left(\infty \right)$ is nonzero, where $\delta$ is some constant. This asymptotic form is based upon a more specific assumption that high-energy elastic scattering is described by an effective complex and energy-dependent potential which satisfies a dispersion relation in the energy variable. However, the above asymptotic form is valid independently of the dependence of this effective potential on the spacial coordinate. It is argued that the term $\frac{\delta }{s^{\frac{1}{2}}}$ in the above asymptotic form should be regarded as genuinely asymptotic, while the term of the order of $\frac{1}{s}$ is not. According to this criterion, the available high-energy $p-p$ data are not so close to the asymptotic region as the ${\pi }^{\pm }-p$ data in the same laboratory momentum range.