Near-threshold neutral pion electroproduction at high momentum transfers and generalized form factors

We report the measurement of near-threshold neutral pion electroproduction cross sections and the extraction of the associated structure functions on the proton in the kinematic range $Q^2$ from 2 to $4.5$ GeV${}^2$ and $W$ from $1.08$ to $1.16$ GeV. These measurements allow us to access the dominant pion-nucleon $s$-wave multipoles $E_{0+}$ and $S_{0+}$ in the near-threshold region. In the light-cone sum-rule framework (LCSR), these multipoles are related to the generalized form factors $G_1^{{\pi }^{0}p}\left(Q^2\right)$ and $G_2^{{\pi }^{0}p}\left(Q^2\right)$. The data are compared to these generalized form factors and the results for $G_1^{{\pi }^{0}p}\left(Q^2\right)$ are found to be in good agreement with the LCSR predictions, but the level of agreement with $G_2^{{\pi }^{0}p}\left(Q^2\right)$ is poor.

Figure 1

Neutral pion electroproduction in the c.m. frame.

Figure 2

(a) A three-dimensional view of CLAS showing the superconducting coils of the torus, the three regions of drift chambers (R1–R3), the Čerenkov counters, the time-of-flight system, and the electromagnetic calorimeters. The positive $Z^{ˆ}$ axis is out of the page along the symmetry axis. (b) A schematic view of a typical near-threshold event showing the reconstructed electron and proton tracks with the corresponding detector hits in two opposite CLAS sectors. The ${\pi }^0$ is reconstructed using the missing mass technique as discussed in the text.

Figure 3

EC sampling fraction as a function of electron momentum for one of the CLAS sectors for (a) data and (b) Monte Carlo (MC) simulation. The dashed lines show the parametrized mean and the solid line indicates the $3\sigma$ cut.

Figure 4

$\Delta t$ as a function of $p$. The curves show the $\pm 3.5\sigma$ cut (solid lines) from the mean fit (dashed line) for one of the CLAS sectors for (a) experimental and (b) Monte Carlo simulated events.

Figure 5

Electron ${\varphi }_e$ distribution for CLAS Sector 4 for $p_e=4.1\pm 0.1$ GeV shown for different ${\theta }_e$ slices. The unshaded curves show ${\varphi }_e$ distribution after electron selection and the shaded curves show the ${\varphi }_e$ distribution after applying electron drift chambers fiducial cuts.

Figure 6

Proton ${\varphi }_p$ vs ${\theta }_p$ distribution for CLAS Sector 4 for $p_p=2.85\pm 0.15$ GeV. Rejected tracks are shown in black.

Figure 7

The Bethe-Heitler process $ep\to ep\gamma$ diagrams for (a) a photon emitted from an incident electron (preradiation) and for (b) a photon emitted from a scattered electron (postradiation).

Figure 8

(a) $M_X^2$ vs $\Delta {\theta }_1^p$ for $W=1.09\pm 0.01$ GeV. The red dashed line indicates the expected pion peak position. The left red spot centered around zero degrees corresponds to the elastic scattering events in which the incident electrons have undergone Bethe-Heitler radiation (preradiative) and the one on the right to the elastic postradiative events. The events below the linear polynomial and outside the ellipse are selected as pions. (b) $M_X^2$ for events with $W=1.09\pm 0.01$ GeV. The black solid curve shows events prior to any Bethe-Heitler subtraction cuts, the blue dash-dotted curve shows events rejected from the cuts, and the red dashed curve shows those events that survive the Bethe-Heitler subtraction cuts.

Figure 9

An example of the $M_X^2\left(ep\right)$ distribution with a double Gaussian fit after applying the elliptical cuts (black circles) of Fig. 8 and after residual Bethe-Heitler and other contamination subtractions (green triangles) for $Q^2=2.75\pm 0.25$ GeV${}^2$ and $W=1.09\pm 0.01$ GeV (top) and $W=1.11\pm 0.01$ GeV (bottom) integrated over all ${\varphi }_{\pi }^{*}$ and $\cos {\theta }_{\pi }^{*}$. The black dashed lines indicate the $\pm 3\sigma$ cuts applied to select the pions. The ${\chi }^2$ is the goodness of fit per degree of freedom. See Sec. 4d for details.

Figure 10

Acceptance corrections for $W=1.09$ GeV and $Q^2=2.75$ GeV${}^2$ as a function of ${\varphi }_{\pi }^{*}$. Each subplot shows the correction for a different $\cos {\theta }_{\pi }^{*}$ bin.

Figure 11

The radiative corrections for $W=1.11$ GeV and $Q^2=3.25$ GeV${}^2$ as a function of $\cos {\theta }_{\pi }^{*}$ and ${\varphi }_{\pi }^{*}$ obtained from exclurad using the MAID2007 model.

Figure 12

The differential cross section $ep\to ep{\pi }^0$ for the $\Delta \left(1232\right)$ resonance region, $W=1.23\pm 0.01$ GeV, for typical kinematic bins. The squares are the measured cross sections after applying the normalization correction factor (see text for details). The dashed curves are from Ref. [32] and the dash-dotted curves are from the MAID2007 model. The corrected values agree with the two curves to within $5\%$ on average.

Figure 13

The differential cross sections in $\mu$b/sr for a few kinematic bins near threshold as a function of ${\varphi }_{\pi }^{*}$. Experimental points (squares) are shown with statistical uncertainties only. The sizes of the estimated systematic uncertainties are shown in gray boxes below. The predictions from LCSR, MAID2007, and SAID are shown as dashed, dash-dotted, and dash-double-dotted curves, respectively. The horizontal line at zero serves as a visual aid. The fit to the distributions is shown as a solid curve. See Sec. 8 for details.

Figure 14

The structure function ${\sigma }_{\mathrm{T}}+\varepsilon {\sigma }_{\mathrm{L}}$ as a function of $\cos {\theta }_{\pi }^{*}$ in $\mu$b/sr for $W=1.08–1.16$ GeV and $Q^2=2.0–4.5$ GeV${}^2$. Predictions from LCSR that include only $s$-wave contribution (dashed line), MAID2007 (dash-dotted line), and SAID (dash-double-dotted line) are shown. The error bars represent statistical uncertainties only and the estimated systematic uncertainties are shown as gray boxes. The solid curve corresponds to the results obtained from the fit to the cross sections (see Sec. 9 for details). The values of $Q^2$ (on top of the panels) and $W$ (on the right side of the panels) are the central values of the bins.

Figure 15

The structure function ${\sigma }_{\mathrm{TT}}$ as a function of $\cos {\theta }_{\pi }^{*}$ in $\mu$b/sr for $W=1.08-1.16$ GeV and $Q^2=2.0-4.5$ GeV${}^2$. Predictions from MAID2007 (dash-dotted line) and SAID (dash-double-dotted line) are shown. The LCSR predictions do not include any ${\sigma }_{\mathrm{TT}}$ contributions, so they are not shown. The error bars represent statistical uncertainties only and the estimated systematic uncertainties are shown as gray boxes. The solid curve corresponds to the results obtained from the fit to the cross sections (see Sec. 9 for details). The values of $Q^2$ (on top of the panels) and $W$ (on the right side of the panels) are the central values of the bins. The horizontal line at zero serves as a visual aid.

Figure 16

The structure function ${\sigma }_{\mathrm{LT}}$ as a function of $\cos {\theta }_{\pi }^{*}$ in $\mu$b/sr for $W=1.08–1.16$ GeV and $Q^2=2.0–4.5$ GeV${}^2$. Predictions from LCSR that include only $s$-wave contribution (dashed line), MAID2007 (dash-dotted line), and SAID (dash-double-dotted line) are shown. The error bars represent statistical uncertainties only and the estimated systematic uncertainties are shown as gray boxes. The solid curve corresponds to the results obtained from the fit to the cross sections (see Sec. 9 for details). The values of $Q^2$ (on top of the panels) and $W$ (on the right side of the panels) are the central values of the bins. The horizontal line at zero serves as a visual aid.

Figure 17

The $s$-wave multipoles (a) $E_{0+}$ and (b) $S_{0+}$ normalized to the dipole formula $G_D$ are plotted as a function of $Q^2$. The error bars include statistical and systematic uncertainties added in quadrature. The size of the estimated systematic uncertainties are shown at the bottom. The LCSR-based model predictions and the LET predictions are also shown as curves. The horizontal line at zero serves as a visual aid.

Figure 18

The generalized form factors (a) $G_1^{{\pi }^{0}p}$ and (b) $G_2^{{\pi }^{0}p}$ normalized to the dipole formula $G_D$ are plotted as a function of $Q^2$. The error bars include statistical and systematic uncertainties added in quadrature. The size of the estimated systematic uncertainties are shown in the bottom. The LCSR-based model predictions and the LET predictions are also shown as curves. The horizontal line at zero serves as a visual aid.