Asymptotic Sum Rules at Infinite Momentum

By combining the ${q}_{0}\ensuremath{\rightarrow}i\ensuremath{\infty}$ method for asymptotic sum rules with the $P\ensuremath{\rightarrow}\ensuremath{\infty}$ method of Fubini and Furlan, we relate the structure functions ${W}_{2}$ and ${W}_{1}$ in inelastic lepton-nucleon scattering to matrix elements of commutators of currents at almost equal times at infinite momentum. We argue that the infinite-momentum limit for these commutators does not diverge, but may vanish. If the limit is nonvanishing, we predict $\ensuremath{\nu}{W}_{2}(\ensuremath{\nu}, {q}^{2})\ensuremath{\rightarrow}{f}_{2}(\frac{\ensuremath{\nu}}{{q}^{2}})$ and ${W}_{1}(\ensuremath{\nu}, {q}^{2})\ensuremath{\rightarrow}{f}_{1}(\frac{\ensuremath{\nu}}{{q}^{2}})$ as $\ensuremath{\nu}$ and ${q}^{2}$ tend to $\ensuremath{\infty}$. From a similar analysis for neutrino processes, we conclude that at high energies the total neutrino-nucleon cross sections rise linearly with neutrino laboratory energy until nonlocality of the weak current-current coupling sets in. The sum of $\ensuremath{\nu}p$ and $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\nu}}p$ cross sections is determined by the equal-time commutator of the Cabibbo current with its time derivative, taken between proton states at infinite momentum.