Exclusive $\eta$ electroproduction at $W>2$ GeV with CLAS and transversity generalized parton distributions

The cross section of the exclusive $\eta$ electroproduction reaction $ep\to e^{\prime }{p}^{\prime }\eta$ was measured at Jefferson Laboratory with a 5.75 GeV electron beam and the CLAS detector. Differential cross sections $d^4\sigma /dtd{Q}^{2}d{x}_{B}d{\varphi }_{\eta }$ and structure functions ${\sigma }_U={\sigma }_T+\epsilon {\sigma }_L,{\sigma }_{TT}$, and ${\sigma }_{LT}$, as functions of $t$, were obtained over a wide range of $Q^2$ and $x_B$. The $\eta$ structure functions are compared with those previously measured for ${\pi }^0$ at the same kinematics. At low $t$, both ${\pi }^0$ and $\eta$ are described reasonably well by generalized parton distributions (GPDs) in which chiral-odd transversity GPDs are dominant. The ${\pi }^0$ and $\eta$ data, when taken together, can facilitate the flavor decomposition of the transversity GPDs.

Figure 1

The handbag diagram for deeply virtual $\eta$ and ${\pi }^0$ production. The helicities of the initial and final nucleons are denoted by $\nu$ and ${\nu }^{\prime }$, of the incident photon and produced meson by $\mu$ and ${\mu }^{\prime }$ and of the active initial and final quark by $\lambda$ and ${\lambda }^{\prime }$. The arrows in the figure represent schematically the corresponding positive and negative helicities, respectively. For final-state pseudoscalar mesons ${\mu }^{\prime }=0$.

Figure 2

Schematic view of the CLAS detector in the plane of the beamline constructed by the Monte Carlo simulation program gsim. The notation is as follows: inner calorimeter (IC), electromagnetic calorimeter (EC), large-angle electromagnetic calorimeter (LAC), Cherenkov counter (CC), scintillation hodoscope (SC), drift chambers (DCs). The LAC was not used in this analysis. The tracks correspond, from top to bottom, to a photon (blue online), an electron (red online) curving toward the beam line, and a proton (purple online) curving away from the beam line.

Figure 3

A blowup of Fig. 2 showing the CLAS target region in detail. IC is the inner calorimeter and DC Region 1 represents the drift chambers closest to the target.

Figure 4

The kinematic coverage and binning as a function of $Q^2$ and $x_B$. The accepted region (yellow online) is determined by the following cuts: $W>2$ GeV, $E^{\prime }>0.8$ GeV, ${21}^{\circ }<\theta <{45}^{\circ }$. $W$ is the ${\gamma }^{*}p$ center-of-mass energy, $E^{\prime }$ is the scattered electron energy and $\theta$ is the electron's polar angle in the laboratory frame. The accepted yellow region within each grid boundary represents the kinematic regions for which the cross sections are calculated and presented.

Figure 5

Yield distributions for kinematic variables $Q^2, x_B, -t$ and ${\varphi }_{\eta }$ in arbitrary units. The data are in black (solid) and the results of Monte Carlo simulations (see Sec. 6) are in red (dotted). The areas under the curves are normalized to each other. The curves for both the data and Monte Carlo simulations are the final distributions obtained after tracking and include acceptances and efficiencies.

Figure 6

The number of CC photoelectrons for events that pass all cuts.

Figure 7

The $z$ coordinate of the electron vertex. The vertical lines are the positions of the applied cuts. Note that, in panel (a), the small peak to the right of the target that is due to a foil placed at $z=-62.5$ cm downstream of the target window. In panel (b) the peak due to the foil disappears after the selection of the exclusive reaction.

Figure 8

The two-photon invariant-mass distribution $M_{\gamma \gamma }$ after all exclusivity cuts have been applied, for the case where the two photons are detected by the IC. The large peak at lower $M_{\gamma \gamma }$ is due to ${\pi }^0$ electroproduction and the smaller peak at higher $M_{\gamma \gamma }$ is due to $\eta$ electroproduction. The inset magnifies the region around the $\eta$ peak. The filled regions above and below the peak (red online) are the sidebands that are used for background subtraction, as discussed in the text.

Figure 9

The differential cross section $d^2\sigma /dtd{\varphi }_{\eta }$ for the reaction ${\gamma }^{*}p\to p^{\prime }\eta$ for the kinematic interval at $Q^2=1.75{\mathrm{GeV}}^2, x_B=0.23$, and $t=-0.8{\mathrm{GeV}}^2$. The error bars indicate statistical uncertainties. Systematic uncertainties are indicated by the cyan bars. The red curve is a fit in terms of the structure functions in Eq. ((7)).

Figure 10

Feynman diagrams contributing to the $\eta$ electroproduction cross section. Left to right: Born process, Bremsstrahlung (by the initial and the final electron), vertex correction, and vacuum polarization.

Figure 11

Radiative corrections ${\delta }_{\mathrm{RC}}$ for $\eta$ electroproduction as a function of ${\varphi }_{\eta }$ for the kinematic interval at $Q^2=1.15{\mathrm{GeV}}^2, x_B=0.13$, and $t=-0.12{\mathrm{GeV}}^2$.

Figure 12

The relative systematic uncertainties, $\delta {\sigma }_{\mathrm{sys}}/\sigma$ of the fourfold differential cross section [see Eq. (1)] for all kinematic points. These do not include the overall normalization uncertainty.

Figure 13

The structure functions vs $t$ for the different $(Q^2,x_B)$ bins, extracted from the present experiment. Black circles: $d{\sigma }_U/dt$. Red squares: $d{\sigma }_{LT}/dt$. Blue triangles: $d{\sigma }_{TT}/dt$. The black, red, and blue curves are the corresponding results of the handbag-based calculation of Ref. [8]. The inset is an enlarged view of the bin with $x_B=0.17$ and $Q^2=1.87{\mathrm{GeV}}^2$. The error bars are statistical only.

Figure 14

The ratio $R$ of the unpolarized structure functions for $\eta$ and ${\pi }^0$ extracted from the present experiment and Ref. [5], as functions of $t$ for $(Q^2,x_B)$ bins. The leading-order handbag calculations [27] predict asymptotically $R\sim 1$. The curves are the result of a handbag-based calculation of Ref. [8]. The inset is an enlarged view of the bin with $x_B=0.28$ and $Q^2=2.2{\mathrm{GeV}}^2$. The error bars are statistical only.

Figure 15

Slope parameters $B$ for different $x_B$ and $Q^2$ bins. The error bars are statistical only.

Figure 16

The extracted structure functions vs $t$ for the ${\pi }^0$ (left column) [20] and $\eta$ (right column). The top row presents data for the kinematic point ($Q^2=1.38{\mathrm{GeV}}^2, x_B=0.17$) and bottom row for the kinematic point ($Q^2=2.21{\mathrm{GeV}}^2, x_B=0.28$). The data for the $\eta$ is identical to that shown in Fig. 13, with the vertical axis rescaled to highlight the difference in the magnitude of the cross sections for ${\pi }^0$ and $\eta$ electroproduction. The data and curves are as follows: black circles, $d{\sigma }_U/dt=d{\sigma }_T/dt+\epsilon d{\sigma }_L/dt$; blue triangles, $d{\sigma }_{TT}/dt$; red squares, $d{\sigma }_{LT}/dt$. The error bars are statistical only. The gray bands are our estimates of the absolute normalization systematic uncertainties on $d{\sigma }_U/dt$. The curves are theoretical predictions produced with the models of Ref. [8].