Theory of Small $x$ Deep Inelastic Scattering NLO Evaluations, and low $Q^2$ Analysis

We calculate structure functions at small $x$ both under the assumption of a hard singularity (a power behaviour $x^{-λ}, λ$ positive, for $x\rightarrow 0$) or that of a soft-Pomeron dominated behaviour, also called double scaling limit, for the singlet component. A full next to leading order (NLO) analysis is carried for the functions $F_2, F_{\rm Glue}$ and the longitudinal one $F_L$ in $ep$ scattering, and for $x F_3$ in neutrino scattering. The results of the calculations are compared with data (HERA) in the range $x\leq 0.032, 10 gev^2\leq Q^2\leq 1 500 gev^2$. We get reasonable fits, with a chi-squared/d.o.f.$\sim 2$, for both assumptions, but none of them gives a fully satisfactory description. The results improve substantially if combining a soft and a hard component; in this case it is even possible to extend the analysis, phenomenologically, to small values of $Q^2$, $0.31 gev^2\leq Q^2\leq 8.5 gev^2$, and in the $x$ range $6\times10^{-6}\lsim x \lsim 0.04$, with the same hard plus soft Pomeron hypothesis by assuming a saturating expression for the strong coupling, $\tildeα_s(Q^2)=4π/β_0\log[(Q^2+Λ_{eff}^2)/Λ_{eff}^2]$ The description for low $Q^2$ implies self-consistent values for the parameters in the exponents of $x$. One gets, for the Regge intercepts, $α_ρ(0)=0.48$ and $α_P(0)=1.470$ [$λ=0.470$], in uncanny agreement with other determinations of these parameters, in particular the results of the large $Q^2$ fits. The fit to is so good that we may look (at large $Q^2$) for signals of a "triple Pomeron" vertex; some evidence is found.

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