Backward ($\theta =180\mathrm{}°$) $\pi N$ Dispersion Relations: Applications to the Interference Model, $P$ and $P^{\prime }$ Trajectories, and to the Mechanical Form Factors of the Nucleon

On the basis of Mandelstam analyticity, crossing, and the observed drop of the backward (180°) ${\pi }^{\pm }p$ differential cross sections with energy, a set of unsubtracted dispersion relations is written for the $\pi N$ amplitudes $A^{\pm }$, $B^{\pm }$ at fixed $\theta =\pi$. A further application of crossing allows the derivation of separate sum rules on $A^{-}$ and $B^{+}$, which are not of the superconvergent variety, and which provide us with information about $\overline{N}N\to \pi \pi$ scattering. In particular, we are able to deduce a value of the spin-flip $f^0\mathrm{}\left(1250\right)-N-N$ coupling constant, which is shown to permit the following observations: (1) The residue function of the $P$ (or $P^{\prime }$) trajectory in $\pi N$ scattering changes sign between $t=0\mathrm{}$ and $t={m_f}^2=1.56\mathrm{}$ ${\mathrm{GeV}}^2$, and (2) universal coupling of the $f^0\mathrm{}\left(1250\right)\mathrm{}$ meson to the gravitational stress-energy density and the knowledge of the aforementioned coupling constant fixes the zero-momentum-transfer values of two of the three mechanical form factors. The values are given in the text. Lastly, we present an extended discussion of the Barger-Cline model within the context of backward dispersion relations.