$C_5^A$ axial form factor from bubble chamber experiments

A careful reanalysis of both Argonne National Laboratory and Brookhaven National Laboratory data for weak single pion production is done. We consider deuteron nuclear effects and normalization (flux) uncertainties in both experiments. We demonstrate that these two sets of data are in good agreement. For the dipole parametrization of $C_5^A(Q^2)$, we obtain $C_5^A(0)=1.19\pm 0.08$, $M_A=0.94\pm 0.03\mathrm{GeV}$. As an application we present the discussion of the uncertainty of the neutral current $1{\pi }^0$ production cross section, important for the T2K neutrino oscillation experiment.

Figure 1

Differential cross sections for $\nu +d\to {\mu }^{-}+n+p+{\pi }^{+}$ scattering computed for the ANL beam. The solid/dashed lines denote the best fit obtained with $C_5^A(0)$ fitted/fixed (with a value of 1.15). In the left panel the cross sections computed with the dipole functional form of $C_5^A(Q^2)$ (32) are shown, while in the right panel results obtained with Adler parametrizations (30) are presented. The black squares denote the experimental data [5], while the black triangles denote the experimental data multiplied by factors of 1.08 (left panel) and 1.11 (right panel)—the renormalization factors were obtained from the global fits (see Tables ). The theoretical cross sections are modified to include deuteron nuclear effects. The cut $W<1.4\mathrm{GeV}$ on the hadronic invariant mass was imposed.

Figure 2

Distribution of the number of events of the BNL experiment for $\nu +d\to {\mu }^{-}+n+p+{\pi }^{+}$ scattering. The solid/dashed lines denote the best fit obtained with $C_5^A(0)$ fitted/fixed (with a value of 1.15). In the left panel the cross sections computed with the dipole functional form of $C_5^A(Q^2)$ (32) are shown, while in the right panel results obtained with Adler parametrizations (30) are presented. The black dots denote the experimental data [6]. The theoretical cross sections are modified to include deuteron nuclear effects. The cut $W<1.4\mathrm{GeV}$ on the hadronic invariant mass was imposed. For higher $Q^2$ (above 1.15 GeV) the data bins were joined, to get better statistics. The new $Q^2$ bins have the following widths: 0.1 for $Q^2\in (1.15,1.55)$, $0.2{\mathrm{GeV}}^2$ for $Q^2\in (1.55,1.95){\mathrm{GeV}}^2$, $0.4{\mathrm{GeV}}^2$ for $Q^2\in (1.95,2.35){\mathrm{GeV}}^2$, and $0.65{\mathrm{GeV}}^2$ for $Q^2\in (2.35,3){\mathrm{GeV}}^2$.

Figure 3

$C_5^A(0)$ fits and $1\sigma$ error bounds obtained by assuming dipole [Eq. (32)] and Adler [Eq. (30)] parametrizations of axial form factors. The solid perpendicular line denotes the PCAC value quoted in 28 together with $1\sigma$ error bounds.

Figure 4

Uncertainties of the fitted parameters. On the left the $1\sigma$ error contour in the $M_A$ and $C_5^A(0)$ plane is plotted. The black squares and triangles denote the points on the contour with maximal and minimal errors for a total cross section for $\nu +p\to {\mu }^{-}+p+{\pi }^{+}$ at different values of neutrino energy $E$. The obtained points are then mapped in the form of $(C_5^A(0){)}_{\min ,\max }$ and $(M_A{)}_{\min ,\max }$ dependence on $E$ (middle and right figures).

Figure 5

Total cross section for $\nu +p\to {\mu }^{-}+p+{\pi }^{+}$. In the left panel the ANL data [5] with the cut $W=1.4$ are shown (black squares), while the right panel presents the BNL data [42] (without cuts in $W$)—black triangles. The overall normalization error is not plotted. The best fit curves were obtained with a corresponding cut in $W$. The theoretical curves were obtained with dipole parametrization Eq. (32) with $M_A=0.94\mathrm{GeV}$ and $C_5^A(0)=1.19$. The shaded areas denote the $1\sigma$ uncertainties of the best fit. The theoretical curves are not modified by the deuteron correction effect.

Figure 6

The $1\sigma$ (solid line), $2\sigma$ (dotted line), and $3\sigma$ (dashed line) error contours for the $C_5^A(0)$ and $M_A$ fit parameters [see Eq. (32)]. The best fit is denoted by the open circle. The best fit obtained for only ANL/BNL data is represented by the black square/triangle. The open triangle denotes the best fit obtained with $p_{\mathrm{ANL}}=p_{\mathrm{BNL}}=15\%$ (the $1\sigma$ contour is plotted with a dashed line). The open diamond denotes the best fit obtained with $p_{\mathrm{ANL}}=p_{\mathrm{BNL}}=20\%$ (the $1\sigma$ contour is plotted with the dashed-dotted line).

Figure 7

Cross sections of ${\nu }_{\mu }$ SPP production on water. From top to bottom—CC ${\pi }^{+}$ production absolute cross section, NC ${\pi }^0$ production absolute cross section, CC ${\pi }^{+}$ production cross section normalized to Nuance results without NEO, and NC ${\pi }^0$ production cross section normalized to Nuance results without NEO. NuWro points show $1\sigma$ error contours. Nuance points are shown here for reference.

Figure 8

The ratio $\mathrm{CC}1{\pi }^{+}/\mathrm{CCQE}$ on the ${\mathrm{CH}}_2$ target. Data points are taken from 16. The dipole fit of this paper was used in the NuWro Monte Carlo generator of events with QE axial mass $M_A^{\mathrm{QE}}=1.03\mathrm{GeV}$.