Neutron structure function via a maximum entropy analysis

TL;DR

This work addresses the limited direct information on the neutron’s parton structure by applying a Maximum Entropy Method to infer a nonperturbative three-valence-quark input at $Q_0^2=0.067$ GeV$^2$, incorporating quark-model constraints, quark-hadron duality, and confinement via Heisenberg uncertainty. The inferred neutron PDFs are evolved to higher $Q^2$ using DGLAP with recombination corrections, yielding predictions for $F_2^{ m n}$ and the ratio $F_2^{ m n}/F_2^{ m p}$, which are then compared to world DIS data, as well as JLab MARATHON and BONuS results with appropriate duality and resonance considerations. The results show good overall agreement with experimental data, especially in the large-$x$ region when duality is invoked, and reveal that the large-$x$ behavior of $F_2^{ m n}/F_2^{ m p}$ and $u^{ m n}/d^{ m n}$ is consistent with perturbative QCD and quark-counting rules but deviates from SU(6) symmetry predictions; isospin symmetry breaking is found to be small and mainly confined to intermediate $x$. Together, these findings provide a viable, data-guided nonperturbative baseline for neutron structure and offer insights into the $d/u$ dynamics and isospin-breaking effects in nucleon structure.

Abstract

We employ the maximum entropy method to extract the valence quark distributions of the neutron at a low scale, \( Q_0^2 \). At this initial scale, the neutron is defined to contain only three valence quarks, with no contributions from sea quarks or gluons. The distributions of these initial valence quarks are constrained by principles from quark models, quark-hadron duality, and quark confinement. Employing the DGLAP equations supplemented by parton-parton recombination corrections, we derive the neutron structure function \( F_2^{\rm n} \) at higher scales \( Q^2 \). The resulting ratio of the neutron to proton structure functions, $F_2^{\rm n}$/$F_{2}^{\rm p}$, aligns well with the world deep inelastic scattering data at Bjoken variable $x<0.7$, particularly when accounting for uncertainties from model-dependent corrections. Notably, this ratio is in agreement with the JLab MARATHON data after considering the quark-hadron duality assumption, especially in the region of $x \gtrsim 0.7$. Additionally, our findings for $F_2^{\rm n}$/$F_{2}^{\rm p}$ correspond well with the JLab BONuS experimental results after considering the impact of nucleon resonance contamination in the region $x \gtrsim 0.4, 0.5, 0.6$. We further compare our predictions for $F_2^{\rm n}$/$F_{2}^{\rm p}$ and the \( u/d \) ratios in the limit as \( x \rightarrow 1 \) with existing theoretical calculations. Finally, we observe a minor violation of isospin symmetry between the proton and neutron, evidenced by the differences in valence quark distributions and the first-order moments of these distributions.

Figures (8)

• Figure 1: Entropy $S$ of valence quark distributions of neutron at $Q^{2}_{0}$ is plotted as a function of the free parameter $B_{u}$ and $B_{d}$, where subfigures (b) and (c) are projections of subfigure (a).
• Figure 2: The predicted valence quark, sea quark and gluon distributions of neutron at $Q^{2} = 1.5$ GeV$^{2}$ by performing DGLAP evolution equations with the parton-parton recombination corrections to the pure valence quark nonperturbative input from MEM.
• Figure 3: The predicted structure function of neutron $F_{2}^{n}$ as a function of Bjorken variable $x$ compared with the BONuS experiment CLAS:2011qvjCLAS:2014jvt, where Analysis 1 (triangles) of BONuS experiment performs the Monte Carlo method and the Analysis 2 (squares) performs the ratio method. The solid curve includes the contribution of the sea quarks and the valence quarks, while the dashed curve includes only the valence quark contribution. The systematic uncertainties of the Monte Carlo method in $F_2^{n}$ extraction are shown as the green shaded band.
• Figure 4: Comparisons of the structure functions of proton (solid curve) and neutron (dashed curve). The structure function calculations are performed with and without the sea quark distribution.
• Figure 5: The obtained structure function ratio of neutron to proton $F_{2}^{n}/F_{2}^{p}$ from MEM, compared with the previous experimental extractions. NMC data (squares) are extracted from the simultaneous measurements on hydrogen and deuterium with the incident muon beam NewMuon:1991exl. The analysis by J. Arrington et al. (circles) are corrected with the nucleon motion in deuterium Arrington:2008zh. The triangles show $F^{n}_{2}/F^{p}_{2}$ data extracted from the deuteron in-medium correction Weinstein:2010rt. The errors plotted display the total experimental uncertainties.