Fixation of theoretical ambiguities in the improved fits to $xF_{3}$ CCFR data at the next-to-next-to-leading order and beyond

Using new theoretical information on the NNLO and N$^3$LO perturbative QCD corrections to renormalization-group quantities of odd $xF_3$ Mellin moments, we perform the reanalysis of the CCFR'97 data for $xF_3$ structure function. The fits were done without and with twist-4 power suppressed terms. Theoretical questions of applicability of the renormalon-inspired large-$\beta_0$ approximation for estimating NNLO and N$^3$LO terms in the coefficient functions of odd $xF_3$ moments and even non-singlet moments of $F_2$ are considered. The comparison with [1/1] Pad\'e estimates is presented. The small $x$ behaviour of the phenomenological model for $xF_3$ is compared with available theoretical predictions. The $x$-shape of the twist-4 contributions is determined. Indications of oscillating-type behaviour of $h(x)$ are obtained from more detailed NNLO fits when only statistical uncertainties are taken into account. The scale-dependent uncertainties of $\alpha_s(M_Z)$ are analyzed. The obtained NNLO and approximate N$^3$LO values of $\alpha_s(M_Z)$ turn out to be in agreement with the world average value $\alpha_s(M_Z)\approx 0.118$. The interplay between higher-order perturbative QCD corrections and $1/Q^2$-terms (or high twist duality) is studied in more detail. The problem of performing NNLO scheme-invariant analysis of $xF_3$ data is discussed. The results of our studies are compared with those obtained recently using the NNLO model of the kernel of the DGLAP equation and with the results of the NNLO fits to CCFR'97 $xF_3$ data, performed by the Bernstein polynomial technique.