Analytic derivation of the next-to-leading order proton structure function $F_2^p(x,Q^2)$ based on the Laplace transformation

An analytical solution based on the Laplace transformation technique for the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) evolution equations is presented at next-to-leading order accuracy in perturbative QCD. This technique is also applied to extract the analytical solution for the proton structure function, $F_2^p(x,Q^2)$, in the Laplace $s$ space. We present the results for the separate parton distributions of all parton species, including valence quark densities, the antiquark and strange sea parton distribution functions (PDFs), and the gluon distribution. We successfully compare the obtained parton distribution functions and the proton structure function with the results from GJR08 [Gluck, Jimenez-Delgado, and Reya, Eur. Phys. J. C 53, 355 (2008)] and KKT12 [Khanpour, Khorramian, and Tehrani, J. Phys. G 40, 045002 (2013)] parametrization models as well as the $x$-space results using QCDnum code. Our calculations show a very good agreement with the available theoretical models as well as the deep inelastic scattering (DIS) experimental data throughout the small and large values of $x$. The use of our analytical solution to extract the parton densities and the proton structure function is discussed in detail to justify the analysis method, considering the accuracy and speed of calculations. Overall, the accuracy we obtain from the analytical solution using the inverse Laplace transform technique is found to be better than 1 part in ${10}^4$ to ${10}^5$. We also present a detailed QCD analysis of nonsinglet structure functions using all available DIS data to perform global QCD fits. In this regard we employ the Jacobi polynomial approach to convert the results from Laplace $s$ space to Bjorken $x$ space. The extracted valence quark densities are also presented and compared to the JR14, MMHT14, NNPDF, and CJ15 PDFs sets. We evaluate the numerical effects of target mass corrections (TMCs) and higher twist (HT) terms on various structure functions, and compare fits to data with and without these corrections.

Figure 1

Our results for the nonsinglet distribution, $x{u}_v(x,Q^2)$ and $x{d}_v(x,Q^2)$, using Eq. (31), and comparison with the global QCD analysis of GJR08. The $x$-space results from the QCD evolution package, QCDnum, are also presented (dashed line).

Figure 2

Sea quarks and singlet distributions in comparison with the next-to-leading order results of GJR08 model. The gluon distribution are also shown. The solid line corresponds to $Q^2=10{\mathrm{GeV}}^2$ and the dashed line corresponds to $Q^2=20{\mathrm{GeV}}^2$. The results from the QCD evolution package, QCDnum, are also presented (dash-dotted and dash–double-dotted lines).

Figure 3

The strange sea distribution $xs=x\overline{s}$ in comparison with the next-to-leading order results of GJR08 model. The solid line corresponds to $Q^2=10{\mathrm{GeV}}^2$ and the dashed line corresponds to $Q^2=20{\mathrm{GeV}}^2$. The results from the QCD evolution package, QCDnum, are also presented (dash-dotted and dash–double-dotted lines).

Figure 4

Sea quarks, gluon, and nonsinglet distributions and comparison with the results from next-to-leading order KKT12 global QCD analysis at $Q^2=100{\mathrm{GeV}}^2$.

Figure 5

The next-to-leading order approximation of the total proton structure function, $F_2^{p,\mathrm{total}}(x,Q^2)$, as a function of $x$ at $Q^2=9.795{\mathrm{GeV}}^2$. The input distributions are obtained from the GJR08 model [38]. Here the straight line represents our result, using the Laplace transform technique, and the red circles represent the proton structure function arising from the GJR08 global QCD analysis. A comparison with the E665 experimental data [52] is also shown.

Figure 6

The next-to-leading order approximation of the total proton structure function, $F_2^{p,\mathrm{total}}(x,Q^2)$, as a function of $x$ at $Q^2=25{\mathrm{GeV}}^2$. The input distributions are obtained from the GJR08 model [38]. Here the straight line represents our result, using the Laplace transform technique, and the red circles represent the proton structure function arising from the GJR08 global QCD analysis. The square and up-triangle signs represent the the E665 experimental data [52] and H1 inclusive deep inelastic neutral current data [8], respectively.

Figure 7

The parton densities $x{u}_v$ and $x{d}_v$ at the input scale $Q_0^2=2{\mathrm{GeV}}^2$. The uncertainties of our PDFs (yellow band) correspond to a 68% confidence level (C.L.) with $\mathrm{\Delta }{\chi }_{\text{global}}^2=5.86$. The dashed line is the BBG PDF [90], the dashed-dotted line is the CJ15 PDF [63], and the dashed–double-dotted line is the result from GJR08 [46].

Figure 8

The parton densities $x{u}_v(x,Q^2)$ and $x{d}_v(x,Q^2)$ at the scale of $Q^2=10{\mathrm{GeV}}^2$. The dashed line is the JR14 PDF [57], the dashed-dotted line is the NNPDF2.3 model [91], and the dashed–double-dotted line is the result from the MMHT14 group [19].

Figure 9

Comparison of proton structure function $F_2^p$ data from BCDMS, SLAC, NMC, H1, and ZEUS with our theory predictions, as a function of $Q^2$ for fixed values of $x$. The pure QCD fit in next-to-leading order is shown as a solid line, the contributions from target mass corrections (TMCs) are shown as a dashed line, and the higher twist (HT) correction is shown as a dashed-dotted line.

Figure 10

Comparison of the structure function $F_2^d$ data from BCDMS, SLAC, and NMC with our theory predictions, as a function of $Q^2$ for the fixed values of $x$. The pure QCD fit in next-to-leading order is shown by a solid line, the contributions from target mass corrections (TMCs) shown as a dashed line, and the higher twist (HT) correction is shown as a dashed-dotted line.

Figure 11

Comparison of BCDMS and NMC data for the nonsinglet structure function $F_2^{\mathrm{NS}}$ with our QCD predictions at next-to-leading order.