QCD analysis of nucleon structure functions in deep-inelastic neutrino-nucleon scattering: Laplace transform and Jacobi polynomials approach

We present a detailed QCD analysis of nucleon structure functions $x{F}_3(x,Q^2)$, based on Laplace transforms and the Jacobi polynomials approach. The analysis corresponds to the next-to-leading order and next-to-next-to-leading order approximations of perturbative QCD. The Laplace transform technique, as an exact analytical solution, is used for the solution of nonsinglet Dokshitzer-Gribov-Lipatov-Altarelli-Parisi evolution equations at low- and large-$x$ values. The extracted results are used as input to obtain the $x$ and $Q^2$ evolution of $x{F}_3(x,Q^2)$ structure functions using the Jacobi polynomials approach. In our work, the values of the typical QCD scale ${\mathrm{\Lambda }}_{\overline{\mathrm{MS}}}^{\left(n_f\right)}$ and the strong coupling constant ${\alpha }_s\left(M_Z^2\right)$ are determined for four quark flavors ($n_f=4$) as well. A careful estimation of the uncertainties shall be performed using the Hessian method for the valence-quark distributions, originating from the experimental errors. We compare our valence-quark parton distribution functions sets with those of other collaborations, in particular with the CT14, MMHT14, and NNPDF sets, which are contemporary with the present analysis. The obtained results from the analysis are in good agreement with those from the literature.

Figure 1

$\mathrm{\Delta }{\chi }^2$ as a function of $t$ defined in Refs. [54, 57, 58, 63] in the NLO approximation. The results correspond to some random sample of eigenvectors. The solid line corresponds to the ideal case, $\mathrm{\Delta }{\chi }_{\text{global}}^2=t^2$.

Figure 2

As in Fig. 1, but for the NNLO approximation. For comparison, we also display the Hessian approximation given by the quadratic form $\mathrm{\Delta }{\chi }_{\text{global}}^2=t^2$.

Figure 3

The valence-quark parton densities $x{u}_v$ and $x{d}_v$ at the reference value $Q_0^2=4{\mathrm{GeV}}^2$ obtained from the NLO and NNLO global analyses. The corresponding $\mathrm{\Delta }{\chi }^2=1$ uncertainty bands computed with the standard Hessian error matrix approach are also shown.

Figure 4

The up and down valence parton distributions $x{u}_v$ and $x{d}_v$ for some selected values of $Q^2=10$, 100, 1000, and 10000 ${\mathrm{GeV}}^2$ in NLO order including their error bands. The dashed line is the BBG model [75], dashed dotted is the NNPDF [76] model, short dashed is the MMHT14 [68] model, and short-dashed dotted is the CT14 [66] model.

Figure 5

The up and down valence parton distributions $x{u}_v$ and $x{d}_v$ for some selected values of $Q^2=10$, 100, 1000, and 10000 ${\mathrm{GeV}}^2$ in NNLO order including their error bands. The dashed line is the BBG model [75], dashed dotted is the NNPDF [76] model, short dashed is the MMHT14 [68] model, and short-dashed dotted is the CT14 [66] model.

Figure 6

The quality of the NLO and NNLO fits to the CCFR antineutrino-initiated dimuon production. The dots represent the CCFR neutrino structure function measurements [6]. The solid curve represents our theoretical predictions as a function of $x$ and for some different values of $Q^2$.

Figure 7

The quality of the NLO and NNLO fits to the NUTEV antineutrino-initiated dimuon production. The dots represent the NuTeV neutrino structure function measurements [7]. The solid curve represents our theoretical predictions as a function of $x$ and for some different values of $Q^2$.

Figure 8

The quality of the NLO and NNLO fits to the CHORUS antineutrino-initiated dimuon production. The dots represent the CHORUS neutrino structure function measurements [8]. The solid curve represents our theoretical predictions as a function of $x$ and for some different values of $Q^2$.