Parton distribution functions and QCD coupling constant from LHC and non-LHC data

We present a new QCD analysis of parton distribution functions (PDFs) at next-to-leading order (NLO) and next-to-next-to-leading order (NNLO). In the present paper, the role of special types of experimental measurements for extracting PDFs is investigated, and simultaneously the values of the strong coupling constant are determined. Essential elements of this QCD analysis are the HERA combined data as a base, heavy-quark cross-section measurement together with Tevatron data on jet production and H1 and ZEUS jet cross sections as “non-LHC” data, and especially the “LHC” dataset for top-quark and jet cross sections. These different datasets have allowed detailed information at low $x$, providing worthwhile information on the nucleon’s flavor, and have played an important role for some PDFs at large-$x$ and strong coupling constant. Since the large-$x$ gluon PDF benefits from an accurate determination of quark PDFs, we have enough motivation to focus on the top-quark production and jet cross-section measurements from LHC to find the impact of these data on the gluon PDF and strong coupling constant. These experimental data have an impact on the relative uncertainties of PDF. The gluon PDF at large $x$ and the values of ${\alpha }_s(M_Z^2)$ are affected significantly. Although the main motivation of this paper is to focus on the gluon PDF and its uncertainty at large-$x$, giving notice to heavy PDF behavior in this region is also very important. We study the intrinsic charm (IC) using the BHPS model together with our extracted extrinsic charm PDF at large $x$, which can be very worthwhile for future experiments at the LHC.

Figure 1

Gluon PDF ratios of the NLO (left) and NNLO (right) HERAPDF20 [7], MSHT20 [8], NNPDF [9] and PDF4LHC [10] predictions to the CT18 [6] ones, $g/g_{\mathrm{CT}18}(x,Q^2)$ as a function of $x$ at $Q^2=1.9{\mathrm{GeV}}^2$.

Figure 2

The results of pQCD calculations for the cross sections of DIS processes, jet production, and the differential cross section of top-quark pair production, and their comparison with experimental measurements.

Figure 3

The NLO and NNLO parton distribution of $x{u}_v$, $x{d}_v$, $x\overline{u}$, $x\overline{d}$, $xs$, and $xg$ for Fit A, Fit B, and Fit C extracted as a function of $x$ at $Q^2=1.9{\mathrm{GeV}}^2$. In the left panels, we present our results for NLO, whereas the right panels are for NNLO.

Figure 4

The NLO and NNLO ratios of $xq(x,Q^2)/xq(x,Q^2{)}_{\mathrm{ref}}$ at $Q^2=1.9{\mathrm{GeV}}^2$ for $q=u_v,d_v,\overline{u},\overline{d},s$, and $g$ for Fit A, Fit B, and Fit C, with respect to Fit A. In the left panels, we present our results for NLO, whereas the right panels are for NNLO.

Figure 5

The NLO parton distribution of $x{u}_v$, $x{d}_v$, $x\overline{u}$, $x\overline{d}$, $xs$, and $xg$, as a function of $x$ and for different values of $Q^2=3$, 10, and $100{\mathrm{GeV}}^2$.

Figure 6

The NNLO parton distribution of $x{u}_v$, $x{d}_v$, $x\overline{u}$, $x\overline{d}$, $xs$, and $xg$, as a function of $x$ and for different values of $Q^2=3$, 10, and $100{\mathrm{GeV}}^2$.

Figure 7

The NLO and NNLO results of the relative uncertainties $\delta xq(x,Q^2)/xq(x,Q^2)$ for $q=u_v,d_v,\mathrm{\Sigma }$, and $g$ at the selected scales $Q^2=1.9$, $10{\mathrm{GeV}}^2$ as a function of $x$ and for individual Fits A, B, and C. In the left panels, we present our results for NLO, whereas the right panels are for NNLO.

Figure 8

The impact of the LHC data Fit (C) on the high-$x$ relative uncertainties $\delta xg(x,Q^2)/xg(x,Q^2)$ at NLO and NNLO with comparisons of HERA $\mathrm{I}+\mathrm{II}$ (Fit A), non-LHC data (Fit B), and LHC data (Fit C). The linear plots of the gluon PDF relative ratio as a function of $x$ are presented at 1.9, 3, 5, and $10{\mathrm{GeV}}^2$. In the upper panels, we present our results for NLO, whereas the lower panels are for NNLO.

Figure 9

The comparison of NLO and NNLO gluon relative uncertainties $\delta xg(x,Q^2)/xg(x,Q^2)$ for Fit C as a function of $x$ at $Q^2=10$, 100, 6464, and $8317{\mathrm{GeV}}^2$.

Figure 10

The compatibility of the main extracted PDF (Fit (C) at NNLO with NNPDF [9] and CT18 [6] for $x{u}_v$, $x{d}_v$, $x\mathrm{\Sigma }$, and $xg$ distribution as a function of $x$ at $1.9{\mathrm{GeV}}^2$. The relative uncertainties $\delta xq(x,Q^2)/xq(x,Q^2)$ PDFs with both logarithmic and linear plots are also presented.

Figure 11

The impact of the LHC data Fit (C) on the high-$x$ charm PDF at NNLO with comparisons of HERA $\mathrm{I}+\mathrm{II}$ Fit (A) and non-LHC data (Fit B). The charm PDF with both logarithmic and linear plots and also the relative uncertainties $\delta xc(x,Q^2)/xc(x,Q^2)$ as functions of $x$ at 3, 5, 10 and $100{\mathrm{GeV}}^2$ are presented.

Figure 12

The scan of the QCD strong coupling constant ${\alpha }_s(M_Z^2)$ at NNLO precision. Different curves for $\mathrm{\Delta }{\chi }^2$ as a function of strong coupling ${\alpha }_s(M_Z^2)$ which obtained from the Fit A, B, and C datasets at NNLO and for each type of dataset are presented. The $\mathrm{\Delta }{\chi }_{\mathrm{tot}}^2$ curves for all experiments are also shown.

Figure 13

The intrinsic charm $x{c}_{\mathrm{int}}(x,Q^2)$, extrinsic charm $x{c}_{\mathrm{ext}}(x,Q^2)$, and total charm PDF $xc(x,Q^2)=(x{c}_{\mathrm{int}}+x{c}_{\mathrm{ext}})(x,Q^2)$, extracted from Fit C as a function of $x$ for $Q^2=5$, 10, and $100{\mathrm{GeV}}^2$ with $P_{c\overline{c}/p}=1\%$. The uncertainties for $x{c}_{\mathrm{ext}}(x,Q^2)$ and total charm $xc(x,Q^2)$ are presented.

Figure 14

Comparison between experimental measurements of the cross section of Higgs boson at 8 TeV and 13 TeV center-of-mass energies presented by ATLAS [112, 113] and theoretical calculations by considering modern PDF sets NNPDF [9] and CT18 [6], and also our PDF set from Fits A, B, and C at NNLO.