Parton distribution functions and constraints on the intrinsic charm content of the proton using the Brodsky-Hoyer-Peterson-Saka approach

In this work, a new set of parton distribution functions taking into account the intrinsic charm (IC) contribution is presented. We focus on the impact of the European Muon Collaboration (EMC) measurements on the large $x$ charm structure function as the strongest evidence for the intrinsic charm when combined with the HERA, SLAC, and BCDMS data. The main goal of this paper is the simultaneous determination of the intrinsic charm probability $P_{c\overline{c}/p}$ and strong coupling ${\alpha }_s$. This allows us to study the interaction of these two quantities as well as the influence on the PDFs in the presence of IC contributions. By considering ${\alpha }_s$ which can be fix or free parameter from our QCD analysis, we find that although there is not a significant change in the extracted central value of PDFs and their uncertainties, the obtained value of $P_{c\overline{c}/p}$ change by factor 7.8%. The extracted value of $P_{c\overline{c}/p}$ in the present QCD analysis is consistent with the recent reported upper limit of 1.93%, which is obtained for the first time from LHC measurements. We show the intrinsic charm probability is sensitive to the strong coupling constant and also the charm mass. The extracted value of the strong coupling constant ${\alpha }_s(M_Z^2)=0.1191\pm 0.0008$ at NLO is in good agreement with the world average value and available theoretical models.

Figure 1

Our theoretical predictions of the reduced cross section and proton structure function which are obtained from the QCD fits as a function of the momentum fraction $x$ at different values of $Q^2$ for our base, Fits A, B and C. We compare to the experimental data from HERA $\mathrm{I}+\mathrm{II}$ [71], the charm cross section from the combined the H1 and ZEUS [74] [Fig. 1], the proton structure function from SLAC [73] and BCDMS [72], and the charm structure function from EMC data [43] (Fig. 1).

Figure 2

The NLO PDFs of our base, Fits A, B and C results for $x{u}_v$, $x{d}_v$, $x\mathrm{\Sigma }$ and $xg$ at the initial scale of $Q_0^2=1.9{\mathrm{GeV}}^2$ as a function of $x$, with $m_c=1.50\mathrm{GeV}$ (left panels). The relative uncertainties $\delta xq(x,Q^2)/xq(x,Q^2)$ are displayed (right panels) and the Fits B and C have IC contribution.

Figure 3

The NLO extrinsic charm PDF from the base and Fits A, B and C as a function of $x$ for $Q^2=10{\mathrm{GeV}}^2$ and $Q^2=100{\mathrm{GeV}}^2$ (left panels) and the relative uncertainties $\delta xc(x,Q^2)/xc(x,Q^2)$ (right panels) are shown.

Figure 4

The parabolic fit to the QCD ${\chi }^2$ values as a function of charm mass in the RT-opt scheme.

Figure 5

The NLO PDFs from Fit C (this fit) for different charm mass values $m_c=1.43$, 1.5, and 1.55 GeV at the initial scale of $Q_0^2=1.9{\mathrm{GeV}}^2$ as a function of $x$ (left panels). This fit uses the IC contribution. The relative uncertainty ratios $xq(x,Q^2)/xq(x,Q^2{)}_{\mathrm{ref}}$ with respect to $m_c=1.5\mathrm{GeV}$ (right panels) are shown.

Figure 6

The NLO extrinsic charm PDF extracted from Fit C (this fit) with different charm mass values $m_c=1.43$, 1.5 and 1.55 GeV for $Q^2=10{\mathrm{GeV}}^2$ and $Q^2=100{\mathrm{GeV}}^2$ as a function of $x$ (left panels). This fit include the IC contribution. The relative uncertainty ratios $xc(x,Q^2)/xc(x,Q^2{)}_{\mathrm{ref}}$ with respect to $m_c=1.5\mathrm{GeV}$ (right panels) are shown.

Figure 7

The intrinsic charm distribution as a function of $x$ and its uncertainties at $Q^2=4$, 10 and $100{\mathrm{GeV}}^2$ with $P_{c\overline{c}/p}=0.94\pm 0.2\%$.

Figure 8

The comparison of the intrinsic charm $x{c}_{\mathrm{int}}(x,Q^2)$, extrinsic charm $x{c}_{\mathrm{ext}}(x,Q^2)$, and total charm distribution $xc(x,Q^2)=(x{c}_{\mathrm{int}}+x{c}_{\mathrm{ext}})(x,Q^2)$, extracted from Fit C as a function of $x$ for $Q^2=4$ and $100{\mathrm{GeV}}^2$ with $P_{c\overline{c}/p}=0.94\pm 0.2\%$.

Figure 9

The comparison of our NLO results for $x{u}_v$, $x{d}_v$, $x\mathrm{\Sigma }$, and $xg$ PDFs in presence of IC from Fit C (this fit). We compare with HERAPDF2 [71] and NNPDF3IC [38] results, for $Q^2=10{\mathrm{GeV}}^2$ and $Q^2=100{\mathrm{GeV}}^2$ as a function of $x$.

Figure 10

The extrinsic charm PDFs from Fit C (left panels) and the relative uncertainties $\delta xq(x,Q^2)/xq(x,Q^2)$ (right panels) for the selected scales $Q^2=4,10,100$, $8317{\mathrm{GeV}}^2$, as a function of $x$, and compared with HERAPDF2 [71] and NNPDF3IC [38].