Virtual Compton scattering and neutral pion electroproduction in the resonance region up to the deep inelastic region at backward angles

We have made the first measurements of the virtual Compton scattering (VCS) process via the $\mathrm{H}$($e$, $e^{\text{'}}p$)$\gamma$ exclusive reaction in the nucleon resonance region, at backward angles. Results are presented for the $W$-dependence at fixed $Q^2=1\hspace{0.5em}{\mathrm{GeV}}^2$ and for the $Q^2$ dependence at fixed $W$ near 1.5 GeV. The VCS data show resonant structures in the first and second resonance regions. The observed $Q^2$ dependence is smooth. The measured ratio of $\mathrm{H}$($e$, $e^{\text{'}}p$)$\gamma$ to $\mathrm{H}$($e$, $e^{\text{'}}p$)${\pi }^0$ cross sections emphasizes the different sensitivity of these two reactions to the various nucleon resonances. Finally, when compared to real Compton scattering (RCS) at high energy and large angles, our VCS data at the highest $W$ ($1.8\text{−}1.9$ GeV) show a striking $Q^2$ independence, which may suggest a transition to a perturbative scattering mechanism at the quark level.

Figure 1

Kinematics for photon electroproduction on the proton (a) and lowest-order amplitudes for Bethe-Heitler ($\mathrm{b}$, $\mathrm{c}$), VCS Born ($\mathrm{d}$, $\mathrm{e}$), VCS non-Born (f), and $t$-channel ${\pi }^0$-exchange (g) processes. The particle four-momenta are indicated in parenthesis in (a).

Figure 2

Squared missing mass $M_X^2$ for an experimental setting at $W=1.2$ GeV (a) and zoom around the $\gamma$ peak (b). The shaded window [–0.005, 0.005] ${\mathrm{GeV}}^2$ is used to select the $\gamma$ events. The FWHM of the peak increases from 0.0022 to 0.0050 ${\mathrm{GeV}}^2$ when $W$ goes from 1.1 to 1.9 GeV.

Figure 3

Excitation curves for the $\mathrm{H}$($e$, $e^{\text{'}}p$)$\gamma$ reaction at $Q^2=1\hspace{0.5em}{\mathrm{GeV}}^2$, $\cos {\theta }_{\gamma \gamma }^{*}=-0.975$, $k_{\mathrm{lab}}=4.032$ GeV and six values of ϕ, as marked. The thick solid curve up through the Δ resonance is our DR fit of the generalized polarizabilities [1]. The thin solid curve is the BH$+$Born cross section, and the dashed curve is the BH$+$Born$+$${\pi }^0$-exchange cross section [18]. The dotted curve is the pure Bethe-Heitler cross section.

Figure 4

The $Q^2$ dependence of the reduced cross section $\langle d^2{\sigma }_{\gamma }\rangle$ in photon electroproduction (see text), at fixed $\cos {\theta }_{\gamma \gamma }^{*}=-0.975,k_{\mathrm{lab}}=4.032$ GeV and $W=1.53$ GeV (statistical error only). The three different data sets are labeled by circles, squares, and triangles. (Left panel) Comparison with theoretical calculations at $W=1.53$ GeV. (Right panel) The same experimental points with two different types of fits, having ($a_1$, ${\Lambda }^2$) or ($a_2$, $b$) as free parameters. The dot-dashed curve is the standard nucleon dipole squared, $G_D^2=[1+Q^2({\text{GeV}}^2)/0.71]{}^{-4}$, normalized arbitrarily to the data point at $Q^2=1\hspace{0.5em}{\mathrm{GeV}}^2$.

Figure 5

The $Q^2$ dependence of the reduced cross section $\langle d^2{\sigma }_{{\pi }^0}\rangle$ in ${\pi }^0$ electroproduction, at fixed $\cos {\theta }_{\gamma \gamma }^{*}=-0.975$, $k_{\mathrm{lab}}=4.032$ GeV and $W=1.53$ GeV (statistical error only). (Left panel) Comparison with theoretical calculations at $W=1.53$ GeV. (Right panel) The same experimental points with two different types of fits. Same comments as in the previous figure.

Figure 6

The slope parameters ${\Lambda }^2$ and $b$ introduced in the two previous figures, determined here in each elementary $W$ bin of width 20 MeV (statistical error only). Left and right panels are for photon and ${\pi }^0$ electroproduction, respectively. The solid curve (${\pi }^0$ channel) gives the result of the MAID2003 calculation, limited to $W⩾1.49$ GeV. This is the only region in $W$ for which the MAID calculation is well described by a simple dipole or exponential fit in the $Q^2$ range [0.5, 2.0] ${\mathrm{GeV}}^2$.

Figure 7

Ratio of the reduced cross sections $\langle d^2\sigma \rangle$ of the processes $\mathrm{H}$($e$, $e^{\text{'}}p$)$\gamma$ and $\mathrm{H}$($e$, $e^{\text{'}}p$)${\pi }^0$ at $Q^2=1\hspace{0.5em}{\mathrm{GeV}}^2,\cos {\theta }_{\mathrm{c}.\mathrm{m}.}=-0.975$ and $k_{\mathrm{lab}}=4.032$ GeV (full circles; statistical error only). The full curve and the dotted curve for $W<1.3$ GeV are this ratio, at the same kinematics, calculated using different theoretical models (see Sec. 3c). The stars represent the ratio $r_{N^{*}}$ (see Sec. 3c) for the listed individual resonances, as obtained from Ref. [29].

Figure 8

Comparison of VCS data from this experiment (•) at ($Q^2=1\hspace{0.5em}{\mathrm{GeV}}^2$, ${\theta }_{\gamma \gamma }^{*}={167.2}^{°}$) and RCS data for $W<2$ GeV at ${\theta }_{\mathrm{c}.\mathrm{m}.}={159}^{°}\text{−}{162}^{°}$ ($\star$) [34], ${\theta }_{\mathrm{c}.\mathrm{m}.}={128}^{°}\text{−}{132}^{°}$ (◊) [32], ${\theta }_{\mathrm{c}.\mathrm{m}.}={141}^{°}$ (▵) [33], ${\theta }_{\mathrm{c}.\mathrm{m}.}={130}^{°}\text{−}{133}^{°}$ (∘) [39] and ${\theta }_{\mathrm{c}.\mathrm{m}.}={131}^{°}(\square )$ [30], and for $W>2$ GeV at ${\theta }_{\mathrm{c}.\mathrm{m}.}={105}^{°}\text{−}{128}^{°}$ (*) [36], ${\theta }_{\mathrm{c}.\mathrm{m}.}={113}^{°}$ ($\star$) and ${128}^{°}$ (◊) [38]. The dotted curves labeled (1) and (2) are the BH$+$Born$+$${\pi }^0$-exchange cross section (see text) and the BH one, respectively. For $W>2$ GeV, the solid curve is an $s^{-6}$ power law normalized to the $W=2.55$ GeV Cornell point of Ref. [36], the dot-dashed and dashed curves are $s^{-8.1}$ and $s^{-5.3}$ power laws fitted to the JLab data [38] at ${\theta }_{\mathrm{c}.\mathrm{m}.}={113}^{°}$ and ${128}^{°}$, respectively.