Mellin-Transform Analysis of Light-Cone Structure and Scaling in Inelastic Electron Scattering

Abstract

Light-cone singularities in the current-commutator matrix element of the nucleon and scaling behavior of the structure function are characterized in terms of the singularities of the Mellin transform $\hat{\Phi }(u, s)$ of the relevant Jost-Lehmann weight. Absence of singularities in $\hat{\Phi }(u, s)$ for $\mathrm{Re}s\le 1$ is shown under some assumptions, to characterize the existence of the scaling limit. For $\mathrm{Re}s<\mathrm{}2\mathrm{}$, either the left-most singularity of $\hat{\Phi }(u, s)$ in the $s$ plane or the canonical singularity at $s=1\mathrm{}$ determines both the leading light-cone singularity of the matrix element and the leading behavior, for large $\nu$ and $Q^2$ with fixed scaling variable, of the structure function. Thus the leading light-cone singularity dominates the leading behavior of the structure function regardless of the existence of the scaling limit. In the Appendix, the contributions of $x$-space singularities of type $\delta (x^2-a^2)$, $a^2>0\mathrm{}$, are proved to decrease in the scaling limit as ${\nu }^{-1\mathrm{}}$ relative to contributions of $\delta \left(x^2\right)$ singularities.