Exclusive neutral pion electroproduction in the deeply virtual regime

We present measurements of the $\mathit{ep}\to \mathit{ep}{\pi }^0$ cross section extracted at two values of four-momentum transfer $Q^2=1.9\hspace{0.5em}{\mathrm{GeV}}^2$ and $Q^2=2.3\hspace{0.5em}{\mathrm{GeV}}^2$ at Jefferson Lab Hall A. The kinematic range allows one to study the evolution of the extracted cross section as a function of $Q^2$ and $W$. Results are confronted with Regge-inspired calculations and GPD predictions. An intepretation of our data within the framework of semi-inclusive deep inelastic scattering is also discussed.

Figure 1

Diagram of the forward ${\pi }^0$ electroproduction reaction (top), and of the dominant ${\pi }^0$ decay mode (bottom). The kinematic invariants of this reaction are defined as $Q^2=-(k-k^{\text{'}}){}^2$, $x_{\text{Bj}}=Q^2/(2\mathit{pq})$, $t=(q-q^{\text{'}}){}^2$, $W^2=s=M_p+Q^2(1/x_{\text{Bj}}-1)$, and $t_{\text{min}}=\frac{(Q^2-m_{\pi }^2){}^2}{4s}-(|q^{\text{c.m.}}|-|q^{\text{'}\text{c.m.}}|){}^2$, with $\left|q^{\text{c.m.}}\right|$ and $\left|q^{\text{'}\text{c.m.}}\right|$ the norms of $\mathrm{q⃗}$, $\overrightarrow{q^{\text{'}}}$ in the $p{\pi }^0$ final state center-of-mass frame.

Figure 2

Distribution of $H(e,e^{\text{'}}{\pi }^0)X$ events in the [$x_{\text{Bj}}$, $Q^2$] plane, for Kin2 ($x_{\text{Bj}}=0.36$, $Q^2=1.9\hspace{0.5em}{\mathrm{GeV}}^2$) and Kin3 ($x_{\text{Bj}}=0.36$, $Q^2=2.3\hspace{0.5em}{\mathrm{GeV}}^2$). Events for Kin$X$2 ($x_{\text{Bj}}=0.40$, $Q^2=2.1\hspace{0.5em}{\mathrm{GeV}}^2$) and Kin$X$3 ($x_{\text{Bj}}=0.33$, $Q^2=2.1\hspace{0.5em}{\mathrm{GeV}}^2$) are bounded by the two horizontal lines.

Figure 3

[(a),(b)] Distributions of $H(e,e^{\text{'}}\gamma \gamma )X$ events within cuts in the [$M_X^2$,$m_{\gamma \gamma }$] plane for Kin3. (a) Raw distribution showing a clear correlation between these two variables. (b) The same distribution after a rotation around ($M_p^2$,$m_{{\pi }^0}$) to improve the $M_X^2$ resolution. [(c),(d)] Projections on the $M_X^2$ axis of the [$M_X^2$,$m_{\gamma \gamma }$] distributions shown, respectively, in (a) and (b). The lower right panel shows that the resolution is indeed improved by the rotation.

Figure 4

Projection on the calorimeter of the virtual photons ${\gamma }^{*}$ within cuts for Kin3. Also shown is the block relabeling used for the calorimeter calibration described in the text. The calorimeter is viewed from the rear, with the downstream beam passing to the right.

Figure 5

Different iterations of the calibration for Kin2. The differences between simulation and data of the missing-mass peak position (a) and resolution (b) are shown before calibration (crosses), after calibration (open circles), and at a random iteration during calibration (asterisks).

Figure 6

(a) Raw $H(e,e^{\text{'}}{\pi }^0)X$ missing-mass distribution for Kin3 (solid histogram) compared to the simulation (dashed histogram), and the difference between the two (dotted histogram). (b) $H(e,e^{\text{'}}{\pi }^0)X$ missing-mass distribution at different values for the calorimeter threshold, corrected with a factor $1/[1-2(E_{\text{thr}}/\left|\overrightarrow{p_{\pi }}\right|)]$. This correction adds to the distribution all ${\pi }^0$ events missed because of the threshold value.

Figure 7

Raw $H(e,e^{\text{'}}\gamma \gamma )X$ distribution in the [$t_{\text{min}}-t$, $Q^2$] plane with cuts for Kin3. The vertical lines delimit the bins we chose in $t_{\text{min}}-t$ for our analysis. Superimposed is the ($t_{\text{min}}-t$) resolution for each alternate bin, showing that each bin is larger than the resolution.

Figure 8

Total cross section integrated over $t_{\text{min}}-t$ and ${\varphi }_{\pi }$, for Kin3, as a function of the $M_X^2$ cut. The vertical lines indicate, from left to right, the minimal, optimal, and maximal $M_X^2$ cut values of the stability domain.

Figure 9

(a) Total cross section integrated over $t_{\text{min}}-t$ and ${\varphi }_{\pi }$, for Kin2, as a function of $E_{\text{thr}}$. The vertical lines indicate, from left to right, the minimal, optimal, and maximal $E_{\text{thr}}$ values of the stability domain (see Table 8 for Kin3 values). (b) The number of events as a function of $E_{\text{thr}}$. The stability domain for $E_{\text{thr}}$ shows the statistics linearly decreasing with $E_{\text{thr}}$.

Figure 10

(a) Separated $H(e,e^{\text{'}}{\pi }^0)p$ cross section ${\sigma }_T+{\epsilon }_L{\sigma }_L$ as a function of $t_{\text{min}}-t$ for $x_{\text{Bj}}=0.36$. Error bars represent statistical errors only. (b) Ratio of ${\sigma }_T+{\epsilon }_L{\sigma }_L$ for the two kinematics as a function of $t_{\text{min}}-t$. The fit of this ratio (dashed line) indicates the $Q^2$ dependence of the cross section.

Figure 11

${\sigma }_{\mathit{TL}}$ (a), ${\sigma }_{\mathit{TT}}$ (b), and ${\sigma }_{{\mathit{TL}}^{\text{'}}}$ (c) $H(e,e^{\text{'}}{\pi }^0)p$ cross-section components as a function of $t_{\text{min}}-t$ for the two $Q^2$ values. Kin2 is represented by the open circles and Kin3 by the solid circles. Error bars represent statistical errors only. The bands (light for Kin2 and dark for Kin3) show fits proportional to $\sin {\theta }_{\pi }^{\text{c.m.}}$, $\sin {}^2{\theta }_{\pi }^{\text{c.m.}}$, and $\sin {\theta }_{\pi }^{\text{c.m.}}$, respectively. Refer to Table 6 for more detailed cross-section values, with statistical and systematic errors.

Figure 12

(a) Separated $H(e,e^{\text{'}}{\pi }^0)p$ cross section ${\sigma }_T+{\epsilon }_L{\sigma }_L$ as a function of $t_{\text{min}}-t$ for $Q^2=2.1\hspace{0.5em}{\mathrm{GeV}}^2$. Error bars represent statistical errors only. (b) Ratio of ${\sigma }_T+{\epsilon }_L{\sigma }_L$ for the two kinematics as a function of $t_{\text{min}}-t$. The fit of this ratio (dashed line) indicates the $W$ dependence of the cross section.

Figure 13

${\sigma }_{\mathit{TL}}$ (a), ${\sigma }_{\mathit{TT}}$ (b), and ${\sigma }_{{\mathit{TL}}^{\text{'}}}$ (c) $H(e,e^{\text{'}}{\pi }^0)p$ cross-section components as a function of $t_{\text{min}}-t$ for the two $x_{\text{Bj}}$ values. Kin$X$2 is represented by the open circles and Kin$X$3 by the solid circles. Error bars represent statistical errors only. The bands (light for Kin$X$2 and dark for Kin$X$3) show fits proportional to $\sin \hspace{0.5em}{\theta }_{\pi }^{\text{c.m.}}$, $\sin {}^2\hspace{0.5em}{\theta }_{\pi }^{\text{c.m.}}$, and $\sin \hspace{0.5em}{\theta }_{\pi }^{\text{c.m.}}$, respectively. Refer to Table 7 for more detailed cross-section values, with statistical and systematic errors.

Figure 14

New calculations at Kin2 (left panels) and Kin3 (right panels) of the $t$-channel meson-exchange model, including charge-pion rescattering with $\pi N$ and $\pi \Delta$ intermediate states [6]. Dashed lines: pole contributions and Pomeron cut alone. Dash-dotted lines: without $\rho \Delta$ cuts. Full lines: $\rho \Delta$ cuts included.

Figure 15

Ratio $\frac{\sqrt{2{\epsilon }_L(1+\epsilon )}{\sigma }_{\mathit{TL}}}{{\sigma }_T+{\epsilon }_L{\sigma }_L}$ for Kin2 (open circles) and Kin3 (solid circles) plotted as a function of $p_{\pi \perp }$. Error bars represent statistical errors only. We fitted to each kinematics a model by Anselmino et al. in [29] using $\langle p_{\perp }^2\rangle /\langle k_{\perp }^2\rangle$ as a free parameter, where $\langle k_{\perp }^2\rangle$ is the intrinsic transverse momentum of quarks and $\langle p_{\perp }^2\rangle$ is the transverse momentum transferred during the hadronization process. The reduced ${\chi }^2$ of the fits are 2.12 for Kin3 and 2.65 for Kin2.