nCTEQ15: Global analysis of nuclear parton distributions with uncertainties in the CTEQ framework

We present the new nCTEQ15 set of nuclear parton distribution functions (PDFs) with uncertainties. This fit extends the CTEQ proton PDFs to include the nuclear dependence using data on nuclei all the way up to ${}^{208}\mathrm{Pb}$. The uncertainties are determined using the Hessian method with an optimal rescaling of the eigenvectors to accurately represent the uncertainties for the chosen tolerance criteria. In addition to the deep inelastic scattering and Drell-Yan processes, we also include inclusive pion production data from the Relativistic Heavy Ion Collider to help constrain the nuclear gluon PDF. Furthermore, we investigate the correlation of the data sets with specific nuclear PDF flavor components and asses the impact of individual experiments. We also provide comparisons of the nCTEQ15 set with recent fits from other groups.

Figure 1

Kinematic reach of DIS and DY data used in the presented nCTEQ fits. The dashed lines represent the kinematic cuts employed in this analysis ($Q>2\mathrm{GeV}$, $W>3.5\mathrm{GeV}$). Only the data points lying above both of these lines are included in the fits.

Figure 2

Approximate $x$-range for the pion data with the Binnewies-Kniehl-Kramer fragmentation function.

Figure 3

$A$ dependence of the fit parameters as given in Eq. (2.6). Specifically, we plot $c_k(A)=c_{k,0}+c_{k,1}(1-A^{-c_{k,2}})$ for each flavor normalized to the corresponding free-proton parameter $c_{k,0}$. The superscripts $\{1,2,\dots \}$ in the legend correspond to the parameters $\{c_1,c_2,\dots \}$ in Eq. (2.5).

Figure 4

Value of ${\chi }^2/\mathrm{dof}$ for (a) individual experiments and (b) per nuclear target used in the nCTEQ15 fit. The numbers on top of the bars represent the number of data points (after kinematic cuts).

Figure 5

${\chi }^2$ function relative to its value at the minimum, $\mathrm{\Delta }{\chi }^2={\chi }^2-{\chi }_0^2$, plotted along the 16 error directions in the eigenvector space, ${\tilde{z}}_i^2$. We display the true ${\chi }^2$ function (solid lines) and the quadratic approximation given by Hessian method $\mathrm{\Delta }{\chi }^2={\tilde{z}}_i^2$ (dashed lines). The eigenvector directions are ordered from the largest to the smallest eigenvalue.

Figure 6

nCTEQ15 bound proton PDFs at the scale $Q=10\mathrm{GeV}$ for a range of nuclei from the free proton ($A=1$) to lead ($A=208$).

Figure 7

Results of the nCTEQ15 fit. On the left, we show nuclear modification factors defined as ratios of proton PDFs bound in lead to the corresponding free-proton PDFs, and on the right we show the actual bound proton PDFs for lead. In both cases, the scale is equal to $Q=1.3\mathrm{GeV}$.

Figure 8

Results of the nCTEQ15 fit. On the left, we show nuclear modification factors defined as ratios of proton PDFs bound in lead to the corresponding free-proton PDFs, and on the right we show the actual bound proton PDFs for lead. In both cases, the scale is equal to $Q=10\mathrm{GeV}$.

Figure 9

We show nuclear modification factors defined as ratios of lead PDFs compared to a lead PDF constructed from free-proton PDFs. The PDFs are constructed using $f^A=\frac{Z}{A}{f}^{p/A}+\frac{A-Z}{A}{f}^{n/A}$ for ${}^{207}\mathrm{Pb}$ and the free proton, with a scale of $Q=10\mathrm{GeV}$.

Figure 10

Comparison of the nCTEQ15 NLO theory predictions for $R=F_2^A(x,Q^2)/F_2^D(x,Q^2)$ as a function of $x$ with nuclear target data. The theory predictions have been calculated at the $Q^2$ values of the corresponding data points. The bands show the uncertainty from the nuclear PDFs.

Figure 11

Same as in Fig. 10 for $R=F_2^A(x,Q^2)/F_2^{A^{\prime }}(x,Q^2)$.

Figure 12

Comparison of the nCTEQ15 NLO theory predictions for $R=F_2^{Sn}/F_2^C$ as a function of $Q^2$ with nuclear target data from the NMC Collaboration. The bands show the uncertainty from the nuclear PDFs.

Figure 13

Ratio of the $F_2$ structure functions for iron and deuteron calculated with the nCTEQ15 fit at (a) $Q^2=5{\mathrm{GeV}}^2$ and (b) $Q^2=20{\mathrm{GeV}}^2$. This is compared with the fitted data from SLAC-E049 [57], SLAC-E139 [51], SLAC-E140 [59], BCDMS-85 [56], and BCDMS-87 [60] experiments and results from EPS09 and HKN07. (The data points shown are within 50% of the nominal $Q^2$ value.)

Figure 14

Comparison of the nCTEQ15 NLO theory predictions for $R={\sigma }_{\mathrm{DY}}^A/{\sigma }_{\mathrm{DY}}^{A^{\prime }}$ with data for several nuclear targets from the Fermilab experiments E772 (left) and E866 (right). The error bands show the uncertainty from the nuclear PDFs.

Figure 15

We display the comparison of the nCTEQ15 and EPS09 fits with the PHENIX [67] and STAR [68] data for the ratio $R_{\mathrm{dAu}}^{\pi }$. The plotted PHENIX and STAR data are shifted by our fitted normalization.

Figure 16

We compare the impact of different fragmentation functions on the observable $R_{\mathrm{dAu}}^{\pi }$. The nCTEQ15 error bands are computed using asymmetric uncertainties to match EPS09.

Figure 17

Comparison of the nCTEQ15 fit (blue) with the nCTEQ15-np fit without pion data (gray). On the left, we show nuclear modification factors defined as ratios of proton PDFs bound in lead to the corresponding free-proton PDFs, and on the right we show the actual bound proton PDFs for lead. In both cases, the scale is equal to $Q=1.3\mathrm{GeV}$.

Figure 18

Comparison of the predictions of the nCTEQ15 (solid blue) and nCTEQ15-np (dashed gray) fits to inclusive pion production data from PHENIX and STAR demonstrating the effect of including these data sets. Note that the dark blue area is the overlap between the blue and gray bands.

Figure 19

Contribution of different experiments to the total $\mathrm{\Delta }{\chi }^2={\chi }^2-{\chi }_0^2$ function (solid black line) for a selection of gluon parameters (a) $c_{1,1}^g$, (b) $c_{4,1}^g$, (c) $c_{0,2}^g$. On the $x$ axis, we show the shift from the best fit value (indicated in the parenthesis, cf. Table 5).

Figure 20

Correlation measures for lead and carbon at $Q=10\mathrm{GeV}$ for the gluon of the nCTEQ15 fit. The left panels display the effective ${\chi }^2$, and the right panels display the correlation cosine as a function of $x$.

Figure 21

Correlation measures for lead at $Q=10\mathrm{GeV}$ for the $u$-quark and $d$-quark distributions of the nCTEQ15 fit. The left panels display the effective ${\chi }^2$, and the right panels display the correlation cosine as a function of $x$.

Figure 22

Comparison of the nCTEQ15 fit (blue) with results from other groups: EPS09 [12] (green), DSSZ [11] (orange), and HKN07 [14] (red). The left panel shows nuclear modification factors for lead, and the right panel shows the actual PDFs of a proton bound in lead. The scale is $Q=2\mathrm{GeV}$. The wide spread of the ratios at large $x$ is an unphysical artifact due to the vanishing of the PDFs in this region.

Figure 23

Same as Fig. 22, with $Q=10\mathrm{GeV}$.

Figure 24

(upper panel) Comparison of the full nuclear lead distributions, $f^{Pb}=\frac{82}{207}{f}^{p/Pb}+\frac{207-82}{207}{f}^{n/Pb}$, for nCTEQ15 (blue), EPS09 (green), and HKN07 (red) at $Q=10\mathrm{GeV}$. The lower panel shows the same distributions compared to the lead PDF, $f^{Pb}$, constructed of free-proton distributions. The wide spread of the ratios at large $x$ is an unphysical artifact due to the vanishing of the PDFs in this region.

Figure 25

The 90% confidence level limits from different data sets in the eigenvector direction ${\tilde{z}}_1$. The ${\chi }^2$-minimum for each experiment is denoted by a black square, and the green band demonstrates the interval of the eigenvector parameter corresponding to the final $\mathrm{\Delta }{\chi }^2$.

Figure 26

These plots display the Hessian before the “rescaling” procedure. The ${\chi }^2$ function relative to its value at the minimum, $\mathrm{\Delta }{\chi }^2={\chi }^2-{\chi }_0^2$, plotted along the 16 fitting parameters of the original space (left) and along the $z_i$ directions in the eigenvector space (right). The actual ${\chi }^2$ function is plotted with solid lines, and the Hessian approximation $\mathrm{\Delta }{\chi }^2={\tilde{z}}_i^2$ is shown with dashed lines.

Figure 27

These plots display the Hessian after the rescaling procedure. The ${\chi }^2$ function relative to its value at the minimum, $\mathrm{\Delta }{\chi }^2={\chi }^2-{\chi }_0^2$, plotted along the 16 fitting parameters of the original space (left) and along the $z_i$ directions in the eigenvector space (right). The actual ${\chi }^2$ function is plotted with solid lines, and the Hessian approximation $\mathrm{\Delta }{\chi }^2={\tilde{z}}_i^2$ is shown with dashed lines.