Extraction of the neutron $F_2$ structure function from inclusive proton and deuteron deep-inelastic scattering data

The available world deep-inelastic scattering (DIS) data on proton and deuteron structure functions $F_2^p$, $F_2^d$, and their ratios are leveraged to extract the free neutron $F_2^n$ structure function, the $F_2^n/F_2^p$ ratio, and associated uncertainties using the latest nuclear effect calculations in the deuteron. Special attention is devoted to the normalization of the proton and deuteron experimental datasets and to the treatment of correlated systematic errors, as well as the quantification of procedural and theoretical uncertainties. The extracted $F_2^n$ dataset is utilized to evaluate the $Q^2$ dependence of the Gottfried sum rule and the nonsinglet $F_2^p-F_2^n$ moments. To facilitate replication of our study, as well as for general applications, we provide a comprehensive DIS database including all recent Jefferson Lab 6 GeV measurements, the extracted $F_n^2$, a modified CTEQ-JLab global parton distribution function fit named CJ15nlo_mod, and grids with calculated proton, neutron, and deuteron DIS structure functions.

Figure 1

$d/u$ PDF ratio from the CJ15 global QCD analysis [7]. The lower three bands show the size of the PDF and off-shell parameter variations compatible with the experimental uncertainties of the fitted experimental data (mustard colored band), as well as the impact of theoretical uncertainties from an extended $d$-quark parametrization (gray band) and the choice of different deuteron wave functions in the calculation of the deuteron structure function (maroon band). The upper green band, with only statistical uncertainties displayed, shows the result of a fit where nuclear corrections are neglected entirely; this fit is strongly disfavored over the nuclear corrected CJ15 fit, with a ${\chi }^2$ increase of 262 units, largely driven by tensions between deuteron DIS and DØ $W$ asymmetry data.

Figure 2

Deuteron-to-nucleon $F_2$ structure function ratio from the CJ15 [7] and AKP17 [12] global QCD analyses compared to the BONuS experimental extraction [80]. The bands display the fit uncertainty at the 90% CL (the published 68% CL AKP band was rescaled by a factor 1.646). Data at $x\gtrsim 0.5$ have an average invariant mass squared $2<⟨W^2⟩<3{\mathrm{GeV}}^2$, and the theoretical calculations here should be understood in the spirit of quark-hadron duality [81].

Figure 3

“Matched” kinematic range (upper) and $F_2$ value binned in $x$ (lower) of the neutron structure functions extracted from world data (see also Table 1). The $F_2$ values are cut at $W^2=3.5{\mathrm{GeV}}^2$.

Figure 4

Kinematic range (upper) and $F_2$ value binned in $x$ (lower) of the $n/p$ ratio extracted from world data (see also Table 2). The $F_2$ ratios are cut at $W^2=3.5{\mathrm{GeV}}^2$.

Figure 5

Extracted $F_2$ neutron structure function (upper) and $n/p$ ratio (lower) as a function of $x$ in selected $Q^2$ bins, compared with the CJ15 fit (dashed black line). The data are bin centered in $Q^2$ and $x$ for clarity, and the experimental (${\delta }^u$) and procedural (${\delta }^{\mathrm{CJ}}$) uncertainties are added in quadrature.

Figure 6

Neutron-to-deuteron ratios $F_2^n/F_2^d$ from this extraction (blue circles) compared with the JLab BONuS data (red diamonds) and the ratio calculated from CJ15 (dashed black lines). The $F_2^n/F_2^d$ values extracted in this paper are selected within a $0.1{\mathrm{GeV}}^2$ slice of each quoted $Q^2$ value.

Figure 7

Isovector nucleon structure function $F_2^p-F_2^n$ versus $x$ from the combined reanalysis at $Q^2=4{\mathrm{GeV}}^2$ (filled circles), compared with the CJ15 global fit (solid black line) and the CJ15 $2F_2^p-F_2^d$ structure function combination (dashed green line), which would coincide with the isovector combination in the absence of nuclear effects. The pseudodata points in the extrapolated region at $x>0.603$ (open circles) are generated from the CJ15 calculation with 5% nominal model uncertainties on $F_2^p$ and $F_2^n$.

Figure 8

Variation with $Q^2$ of the contributions to the Gottfried integral $I_G(x_{\min },x_{\max };Q^2)$ from different $x$ intervals, including from the measured (blue circles), $x\to 0$ extrapolated (green triangles), and $x\to 1$ extrapolated (black diamonds) regions, along with the total integral (red squares). The blue shaded bands show the integral in the measured region calculated from the CJ15 PDFs.

Figure 9

Dependence of the Gottfried integral $I_G(x_{\min },1;Q^2)$ on the lower limit of the $x$ integration $x_{\min }$ at $Q^2=4{\mathrm{GeV}}^2$ in our analysis (red diamonds), compared with the NMC extraction [87, 95] (open blue circles) and the analysis by Abbate and Forte [42] (open green squares).

Figure 10

Gottfried integral $I_G(0,1;Q^2=4{\mathrm{GeV}}^2)$ from data extractions (bottom) and from theoretical calculations using PDF obtained in recent global QCD analyses (top). The smaller CJ15 uncertainty compared to the other QCD calculations is due to the collaboration’s choice of a nominal $t=1.646$ tolerance factor, which is smaller than in the dynamical tolerance criteria adopted in the other analyses. The large CT18 uncertainty is due to bigger PDF uncertainties at $x\lesssim 0.001$ compared to NNPDF4.0 and MSHT20.

Figure 11

Nonsinglet Nachtmann moment $M_2^{\mathrm{NS}}$ extracted from world DIS data at various values of $Q^2$. The total moments (red diamonds) are broken down into contributions from different $x$ intervals: the measured region $[x_{\mathrm{low}},x_{\mathrm{high}}]$ (blue triangles), the $x>x_{\mathrm{high}}$ extrapolation region (black circles), and the $x<x_{\mathrm{low}}$ extrapolation region (green squares). The blue shaded bands represent the integral in the measured region calculated using CJ15 structure functions instead of data points.

Figure 12

A comparison of the nonsinglet PDF moment $⟨x{⟩}_{u^{+}-d^{+}}$ extracted from DIS structure functions data (solid symbols) and from various lattice QCD (LQCD) calculations (open circles) at $Q^2=4{\mathrm{GeV}}^2$. The green vertical band shows the unweighted combination of moments from recent global PDF analyses discussed in the text.

Figure 13

Kinematic range (upper) and proton $F_2^p$ data included in this database, binned in $x$ (lower). The proton $F_2^p$ values are cut on $W^2=3.5{\mathrm{GeV}}^2$. Note that the available ${\sigma }_{\mathrm{red}}$ data will have a different (larger) kinematic coverage.

Figure 14

As in Fig. 13, but for the deuteron $F_2^d$ data.

Figure 15

Neutron-to-proton $F_2$ structure function ratio (left) and isoscalar corrections for ${}^{56}\mathrm{Fe}$ (right) at $Q^2=10{\mathrm{GeV}}^2$ from this work and several empirical fits including the F2ALLM model with and without $Q^2$ dependence [160] and more recent fits by Arrington et al. [3] and Segarra et al. [161]. Models with $Q^2$ dependence are labeled by $†$.

Figure 16

Left: neutron-to-proton $F_2$ ratio calculated at $Q^2=20{\mathrm{GeV}}^2$ using CJ15 PDFs. The positively charged CC $W^{+}$ ratio, that at LO is equal to the reciprocal of the negatively charged current (CC) $W^{-}$ ratio and grows as $x\to 1$, is compared to the neutral current (NC) ratio and the CC $W^{-}$ ratio on a linear scale. Right: the isoscalar correction factor for ${}^{56}\mathrm{Fe}$ calculated from corresponding neutron-to-proton ratios.