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

Abstract

We calculate structure functions at small $x$ both under the assumption of a hard singularity (a power behaviour $x^{-\lambda },\lambda$ positive, for $x\to 0$) 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 $F_2,F_{\mathrm{Glue}}$ 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 $x\le 0.032,10ge{v}^2\le Q^2\le 1500ge{v}^2$. 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 $Q^2$, $0.31ge{v}^2\le Q^2\le 8.5ge{v}^2$, 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, ${\tilde{\alpha }}_s(Q^2)=4\pi /{\beta }_0\log [(Q^2+{\Lambda }_{eff}^2)/{\Lambda }_{eff}^2]$ The description for low $Q^2$ implies self-consistent values for the parameters in the exponents of $x$. One gets, for the Regge intercepts, ${\alpha }_{\rho }(0)=0.48$ and ${\alpha }_P(0)=1.470$ [$\lambda =0.470$], 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.

Tex file plus .ps figures. This paper includes the results from FTUAM 96-39 [hep-ph/9610380] and FTUAM 96-44 [hep-ph/9612469]

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