How to perform a QCD analysis of DIS in analytic perturbation theory

We apply (fractional) analytic perturbation theory (FAPT) to the QCD analysis of the nonsinglet nucleon structure function $F_2(x,Q^2)$ in deep inelastic scattering up to the next leading order and compare the results with ones obtained within the standard perturbation QCD. Based on a popular parametrization of the corresponding parton distribution we perform the analysis within the Jacobi polynomial formalism and under the control of the numerical inverse Mellin transform. To reveal the main features of the FAPT two-loop approach, we consider a wide range of momentum transfer from high $Q^2\sim 100{\mathrm{GeV}}^2$ to low $Q^2\sim 0.3{\mathrm{GeV}}^2$ where the approach still works.

Figure 1

Nonsinglet SF $F_2(x,Q^2)$ vs $Q^2$ at (a) LO and (b) NLO. The Bjorken $x=0.25$ for (a.1) and (b.1), and $x=0.7$ for (a.2) or (b.2). The solid line represents the FAPT results and the dashed line—the pQCD ones.

Figure 2

Nonsinglet SF $F_2(x,Q^2)$ vs $x$ at (a) LO and (b) NLO. The energy scale is $Q^2=0.3{\mathrm{GeV}}^2$ in (a.1), (b.1), $Q^2=1{\mathrm{GeV}}^2$ in (a.2), (b.2) and $Q^2=100{\mathrm{GeV}}^2$ in (a.3), (b.3). The solid line represents the FAPT results and the dashed line—the pQCD ones.

Figure 3

Nonsinglet SF $F_2(x)$ vs $x$ in LO (a), at the energy scale $Q^2=0.3{\mathrm{GeV}}^2$ and (b) $Q^2=100{\mathrm{GeV}}^2$. The solid (red) line represents the FAPT result and the dashed (blue) line the pQCD one in the JP method. The thick squares (red) and spheres (blue) represent the result of the “exact” numerical inverse Mellin transform.

Figure 4

Nonsinglet SF $F_2(x)$ vs $x$ at LO, at energy scale (a) $Q^2=0.3{\mathrm{GeV}}^2$, and (b) $Q^2=100{\mathrm{GeV}}^2$. The solid (red) line represents the FAPT outcome and the dashed (blue) line the pQCD one in the JP method. The thick squares (red) and spheres (blue) represent the result of the “exact” numerical Inverse Mellin transform.