Scaling and asymptotic behavior on the mass shell deduced from the Dyson–Jost–Lehmann representation $\psi(\mathbf u,\lambda^2)$

The Dyson–Jost–Lehmann representation is used to investigate the relationship between sealing behavior in the deep inelastic region and the asymptotic behavior on the mass shell. Examples are construced of weight functions that lead to scaling for deep inelastic ep scattering and to a constant total cross section for the real Comptoa effect. The corresponding weight functions $\psi(\mathbf u,\lambda^2)$ in the general case have singularities at $\mathbf u=0$ (in the sense of generalized functions), which determine the high-energy behavior on the mass shell, whereas the scaling behavior is determined by the asymptotic behavior of $\psi(\mathbf u,\lambda^2)$ with respect to $\lambda^2$. Examples are given of causal structure functions that establish the importance of the space-like region $q^2>0$ for the determination of the leading singularity of the current commutator on the light cone.