Nucleon axial form factor from a Bayesian neural-network analysis of neutrino-scattering data

The Bayesian approach for feedforward neural networks has been applied to the extraction of the nucleon axial form factor from the neutrino-deuteron-scattering data measured by the Argonne National Laboratory bubble-chamber experiment. This framework allows to perform a model-independent determination of the axial form factor from data. When the low $0.05<Q^2<0.10-{\mathrm{GeV}}^2$ data are included in the analysis, the resulting axial radius disagrees with available determinations. Furthermore, a large sensitivity to the corrections from the deuteron structure is obtained. In turn, when the low-$Q^2$ region is not taken into account with or without deuteron corrections, no significant deviations from previous determinations have been observed. A more accurate determination of the nucleon axial form factor requires new precise measurements of neutrino-induced quasielastic scattering on hydrogen and deuterium.

Figure 1

Feedforward neural network in an MLP configuration $\mathcal{N}\text{:}\mathbb{R}\mapsto \mathbb{R}$. It consists of: (i) an input layer with one input unit $Q^2$ (open square); (ii) one hidden layer with three hidden $M=3$ units (filled blue circles); (iii) an output layer consisting of one output unit (black square). Each line denotes a weight parameter $w_j$. The bias weights are denoted by dashed lines, whereas the bias units are represented by open blue circles.

Figure 2

The $i\mathrm{th}$ unit in the $k\mathrm{th}$ layer [Eq. (5)].

Figure 3

The error Eq. (21) as a function of the logarithm of the evidence Eqs. (17) and (18). Each point denotes the result obtained for one MLP fit to the BIN1 data, including deuteron corrections (see Sec. 3 for details). The fits with a logarithm of the evidence smaller than $-100$ are not shown in the figure.

Figure 4

The dependence of $r_A^2$, defined in Eq. (3), on the logarithm of the evidence (17) and (18). Results for the MLP fits to BIN1 data (without the deuteron correction). The MLPs consist of $M=1–4$ hidden units.

Figure 5

Distribution of the ANL number of events and the best fits obtained for MLPs with $M=1–4$ hidden units. The figure shows the results of the analysis of BIN0 data (deuteron corrections included). For each fit, the value of the logarithm of evidence (ev) is given.

Figure 6

Best fits of the axial form factor obtained from the analysis of the BIN0, BIN1, and BIN2 data sets. The top (bottom) panel presents the results obtained without (with) deuteron corrections. The shaded areas denote $1\sigma$ uncertainties. Additionally, the relative uncertainty $\mathrm{\Delta }{F}_A/F_A$ is plotted.

Figure 7

Impact of the deuteron corrections on the axial form factor fits. Results of the best fits to the BIN0, BIN1, and BIN2 data sets with and without the deuteron correction together with relative uncertainties. All curves for the BIN1 and BIN2 cases nearly overlap.

Figure 8

Dependence of $r_A^2$ on the logarithm of evidence. Open and full triangles denote (the best) fits to the BIN0 data without and with the deuteron corrections, respectively. Analogously, the fits to the BIN1 and BIN2 data are denoted by circles and squares, respectively.

Figure 9

Impact of the $\mathrm{\Delta }{g}_A$ uncertainty on the extraction of the axial form factor. The best fits to BIN0, BIN1, and BIN2 data (with deuteron correction). The thin lines denote results obtained assuming a 10% uncertainty for $g_A$, whereas the thick lines denote fits with the PDG uncertainty.

Figure 10

Comparison of the dipole with the neural network fits to the BIN0 and BIN1 data, the deuteron corrections included. The shaded areas denote $1\sigma$ uncertainties of $F_A$.