Unitarity Lower Bound on the Longitudinal Contribution to Inelastic Electron-Proton Scattering

The diffractive electroproduction of longitudinal $\rho$ mesons sets, via unitarity and the Schwarz inequality, a lower bound to the longitudinal cross section in inelastic electron-proton scattering. Recent $\rho$ electroproduction data may be used to infer that the longitudinal-to-transverse ratio $R$ in total electroproduction at $\omega \simeq 16$, $q^2\simeq 1$ Ge${\mathrm{V}}^2$ is an order of magnitude greater than is predicted by models yielding $R=\frac{q^2}{{\nu }^2}=\left(\frac{4{m_N}^2}{q^2}\right){\omega }^{-2\mathrm{}}$.