Algebra of Current Components and the Hypothesis of Partially Conserved Axial-Vector Current Applied at High Energies

Using the hypothesis of partially conserved axial-vector currents, the algebra of current components, and the assumption that the pion-hadron total cross section $\sigma \mathrm{}\left(s\right)\mathrm{}$ approaches its asymptotic value rapidly, a method is developed which allows a calculation of the elastic amplitude at high energies and small momentum transfers. This method uses the fact that asymptotically the dynamics is given by the commutator on the light cone. The results are ${\sigma }_{\pi p}\left(\infty \right)=25.7\mathrm{}\pm 4.2$ mb and $\frac{d\sigma }{\mathrm{dt}}={\left(\frac{d\sigma }{\mathrm{dt}}\right)}_{t=0\mathrm{}}[G_E^{}{}_{}{}^2-(\frac{t}{4M^2}\left)G_M^{}{}_{}{}^2\right]{\left(\frac{1-t}{4M^2}\right)}^{-1\mathrm{}}$ (for small values of the momentum transfer $t$), where $G_E\mathrm{}\left(t\right)\mathrm{}$ and $G_M\mathrm{}\left(t\right)\mathrm{}$ are the electric and magnetic form factors of the proton. It is shown that possible Schwinger terms in the equal-time commutators are without importance for our results. An important feature of our calculation is that the energy and the momentum are allowed to go to infinity simultaneously; our method therefore deviates essentially from the Bjorken limit, which in general involves a continuation of the amplitude infinitely off the mass shell.