Asymptotic Sum Rules at Infinite Momentum

Abstract

By combining the $q_0\to i\infty$ method for asymptotic sum rules with the $P\to \infty$ 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 $\nu W_2(\nu , q^2)\to f_2\left(\frac{\nu }{q^2}\right)$ and $W_1(\nu , q^2)\to f_1\left(\frac{\nu }{q^2}\right)$ as $\nu$ and $q^2$ tend to $\infty$. 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 $\nu p$ and $\tilde{\nu }p$ cross sections is determined by the equal-time commutator of the Cabibbo current with its time derivative, taken between proton states at infinite momentum.