Valence quark distributions of the proton from maximum entropy approach

We present an attempt using the maximum entropy principle to determine valence quark distributions in the proton at a very low resolution scale $Q_0^2$. The initial three valence quark distributions are obtained with limited dynamical information from quark model and QCD theory. Valence quark distributions from this method are compared to the lepton deep inelastic scattering data, and the widely used CT10 and MSTW08 data sets. The obtained valence quark distributions are consistent with experimental observations and the latest global fits of parton distribution functions. The maximum entropy method is expected to be particularly useful in cases where relatively little information from QCD theory is given.

Figure 1

Information entropy $S$ is plotted as a function of the parameter $B_d$.

Figure 2

Comparisons of our predicted structure function $x{F}_3$ (solid and dot-dashed lines) with experimental data from NuTeV (squares) [46] and CCFR (open circles) [47]. Only statistical errors of the experimental data are plotted. Results of CT10 (dashed line) [2] and MSTW08(LO) (dotted line) [3] from a global fit are also shown here.

Figure 3

Solid lines and dot-dashed lines are LO and NNLO predictions, respectively. The combined HERA data [48] are shown in circles. Errors shown in the figure are the total experimental uncertainties. Our predicted $F_2$’s are from valence contribution only, assuming sea quarks are negligible at large $x$.

Figure 4

The predicted $F_2$ ratios of neutron to proton (solid and dot-dashed lines) are shown with the experimental data. Our predicted $F_2$ ratios are calculated without the contributions of sea quarks. NMC data (open squares) is taken from [49]. Detailed analysis data (circles) [50] is from Arrington et al. The plotted errors of experimental data are the total uncertainties.

Figure 5

Comparisons of our predicted $d_v/u_v$ ratios (solid and dot-dashed lines) with experimental results from CDHS (open triangle) [51], WA21 (open circle) [52], and HERMES (squares) [53]. The plotted errors are the total errors. Our predicted ratios are at $Q^2=4{\mathrm{GeV}}^2$. HERMES data is at mean $Q^2=2.4{\mathrm{GeV}}^2$. $Q^2$ of the CDHS data varies from 3.3 to $42.9{\mathrm{GeV}}^2$; $Q^2$ of the WA21 data varies from 3.4 to $36.5{\mathrm{GeV}}^2$.

Figure 6

Comparisons of our predicted valence quark momentum distributions (solid and dot-dashed lines) with global QCD fit CT10 (dashed lines) [2] and MSTW08(LO) (dotted lines) [3].