Investigations of the $\pi N$ total cross sections at high energies using a new finite-energy sum rule: $\log \nu$ or $(\log \nu {)}^2$

We propose to use rich information on $\pi p$ total cross sections below $N\left(\sim 10\hspace{0.5em}\mathrm{GeV}\right)$ in addition to high-energy data in order to discern whether these cross sections increase like $\log \nu$ or $(\log \nu {)}^2$ at high energies, since it is difficult to discriminate between asymptotic $\log \nu$ and $(\log \nu {)}^2$ fits from high-energy data alone. A finite-energy sum rule (FESR) which is derived in the spirit of the $P^{\prime }$ sum rule as well as the $n=1\mathrm{}$ moment FESR is required to constrain the high-energy parameters. We then search for the best fit of ${\sigma }_{\mathrm{tot}}^{\mathrm{}(+)\mathrm{}}$ above 70 GeV in terms of high-energy parameters constrained by these two FESR’s. We can show from this analysis that the $(\log \nu {)}^2$ behavior is preferred to the $\log \nu$ behavior.