Asymptotic Properties of Virtual Compton Amplitudes in the Ladder Model

Abstract

We study the asymptotic properties of the virtual Compton scattering amplitude for a spacelike spin-1 photon off a scalar target, in the limit $k^2\to \infty$, $\frac{\nu }{k^2}=\rho$ fixed. This is accomplished by means of a perturbative model based on the ladder-diagram series. The absorptive parts of these amplitudes in the forward direction are related, as is well known, to the structure functions $W_1$ and $W_2$ that in turn describe inelastic electron scattering off the same target. In terms of $W_1$ and $W_2$ (using a ${\phi }^3$ model for the hadron dynamics), we find that $W_1\sim \left(\frac{1}{k^2}\right)\ln \left(\frac{k^2}{m^2}\right)F_1\left(\rho \right)$ and $W_2\sim \left(\frac{1}{k^2}\right)F_2\left(\rho \right)$. Notice that in this model $W_1$ does not approach a constant limit. We also find, for large $\rho$, that $F_1\left(\rho \right)\sim {\lambda }_1{\rho }^{\alpha \mathrm{}\left(0\right)\mathrm{}}$ and $F_2\left(\rho \right)\sim {\lambda }_2{\rho }^{\alpha \mathrm{}\left(0\right)-2\mathrm{}}$, where $\alpha \mathrm{}\left(t\right)\mathrm{}$ is the same Regge trajectory that dominates at high energy. To understand these results better, we also study the Bjorken limit for $i{k}_0\to \infty$, both for spin-0 and spin-1 currents. In the first case, we find that the expansion in inverse powers of $i{k}_0$ is valid for the first few terms, while in the second case, one of the invariant amplitudes shows a term ${\left(i{k}_0\right)}^{-2\mathrm{}}\ln \left(i{k}_0\right)$. The similar behavior of $W_1$ for $k^2\to \infty$ is presumably related to that logarithmic term.