Separated structure functions for the exclusive electroproduction of $K^{+}\Lambda$ and $K^{+}{\Sigma }^0$ final states

We report measurements of the exclusive electroproduction of $K^{+}\Lambda$ and $K^{+}{\Sigma }^0$ final states from a proton target using the Continuous Electron Beam Accelerator Facility (CEBAF) large-acceptance spectrometer (CLAS) detector at the Thomas Jefferson National Accelerator Facility. The separated structure functions ${\sigma }_T,{\sigma }_L,{\sigma }_{\mathit{TT}}$, and ${\sigma }_{\mathit{LT}}$ were extracted from the Φ- and ε-dependent differential cross sections taken with electron beam energies of 2.567, 4.056, and 4.247 GeV. This analysis represents the first ${\sigma }_L/{\sigma }_T$ separation with the CLAS detector, and the first measurement of the kaon electroproduction structure functions away from parallel kinematics. The data span a broad range of momentum transfers from $0.5⩽Q^2⩽2.8{\mathrm{GeV}}^2$ and invariant energy from $1.6⩽W⩽2.4$ GeV, while spanning nearly the full center-of-mass angular range of the kaon. The separated structure functions reveal clear differences between the production dynamics for the Λ and ${\Sigma }^0$ hyperons. These results provide an unprecedented data sample with which to constrain current and future models for the associated production of strangeness, which will allow for a better understanding of the underlying resonant and nonresonant contributions to hyperon production.

Figure 1

Kinematics for $K^{+}Y$ (where $Y$ is either a Λ or a ${\Sigma }^0$) electroproduction defining the angles ${\theta }_K^{*}$ and Φ with respect to the c.m. reference frame.

Figure 2

Feynman diagrams of $s$-, $t$-, and $u$-channel exchanges that contribute to the reaction models. Vertex labels $g_{\mathit{MBB}}$ represent the strong coupling constants.

Figure 3

CLAS kinematic coverage in terms of $Q^2$ vs $W$ for $p(e,e^{\prime }{K}^{+})Y$ ($Y=\Lambda ,{\Sigma }^0$) events for two beam energies.

Figure 4

Reconstructed mass for positively charged particles. (a) Mass vs measured momentum. Lines show the mass cuts used to identify kaon candidates. A logarithmic yield density scale is employed. (b) Reconstructed hadron mass. These spectra were made from our kaon-filtered data files.

Figure 5

(a) ${\theta }_K^{*}$ vs $p(e,e^{\prime }{K}^{+})Y$ missing mass, showing $\mathrm{ep}$ elastic events and $e{\pi }^{+}n$ events. Vertical bands correspond to ground state $\Lambda (1116),{\Sigma }^0(1193)$, and $\Lambda (1405)/{\Sigma }^0(1385)$ hyperons. (b) ${\theta }_K$ vs $Q^2$ for $p(e,e^{\prime }{K}^{+})Y$, showing $\mathrm{ep}$ elastic events and the cut used to remove them.

Figure 6

Signal and background fits from the 2.567 GeV data for the $e^{\prime }{K}^{+}$ missing mass spectrum (a) summed over all kinematics and (b) for a typical $\cos {\theta }_K^{*}/\Phi$ bin at $Q^2=1.0{\text{GeV}}^2$ and $W=1.85$ GeV to demonstrate the typical fit quality in our data.

Figure 7

Distribution of the computed $K^{+}\Lambda$ acceptance for CLAS as a function of $\cos {\theta }_K^{*}$ and Φ for the $W=1.85$ GeV and $Q^2=0.65{\text{GeV}}^2$ bin. The depleted area near Φ of $0°$ for forward angles is the forward “hole” in CLAS due to the beam pipe. The number of bins in Φ is different for forward and backward $\cos {\theta }_K^{*}$. The statistical error bars from the Monte Carlo are smaller than the symbol size on this plot. The curves on each plot serve only to guide the eye.

Figure 8

Φ-dependent differential cross sections (nb/sr) and fits for $K^{+}\Lambda$ events from our 2.567 GeV data at $Q^2=0.65{\text{GeV}}^2$ for each of our six $\cos {\theta }_K^{*}$ bins (labeled at the top of each column) for three different $W$ bins (labeled on the left of each row). Curves represent fits to the Φ-dependent differential cross sections.

Figure 9

CLAS acceptance of the scattered electron in terms of ε vs $Q^2$ at 2.567 and 4 GeV for four 100 MeV $W$ bins centered from 1.65 to 1.95 GeV. The vertical line at $Q^2=1.0{\mathrm{GeV}}^2$ marks where we performed the separation of ${\sigma }_T$ and ${\sigma }_L$.

Figure 10

Representative Rosenbluth separation plots of ${\sigma }_U$ (nb/sr) vs ε for our $K^{+}\Lambda$ data at $Q^2=1.0{\text{GeV}}^2$ and $W=1.85$ GeV for our six $\cos {\theta }_K^{*}$ bins. Lines represent fits that determine the slope parameter (${\sigma }_L$) and the intercept parameter (${\sigma }_T$), which are printed on each plot. Error bars on the data points and errors listed for ${\sigma }_L$ and ${\sigma }_T$ on each plot represent the combined statistical and systematic uncertainties.

Figure 11

Comparison of Φ fits to the differential cross sections performed for two different algorithms. In the first approach (dashed curves), cross sections at 2.567 and 4 GeV are fit separately. In the second approach (solid curves), two different beam energy data sets are fit simultaneously. Plots are for kinematics with $Q^2=1.0{\text{GeV}}^2$ and $W=1.85$ GeV for the $K^{+}\Lambda$ (left columns) and $K^{+}{\Sigma }^0$ (right columns) final states at different values of $\cos {\theta }_K^{*}$.

Figure 12

Cross section models for the $K^{+}\Lambda$ (left) and $K^{+}{\Sigma }^0$ (right) structure functions vs $W$ compared with our CLAS data (square points) at $\cos {\theta }_K^{*}=0.05$ and 0.35. The ad hoc models (dashed and dot-dashed curves, discussed in Sec. 5) were based on the Mart-Bennhold model [44] (solid curve) as a starting point. Plots are for the 2.567 GeV data set at $Q^2=0.65{\text{GeV}}^2$.

Figure 13

Total kinematic-independent systematic uncertainties from Eq. (8) on the structure functions ${\sigma }_U,{\sigma }_{\mathit{TT}}$, and ${\sigma }_{\mathit{LT}}$ normalized to ${\sigma }_U$ for the $K^{+}\Lambda$ data as a function of bin number for the 2.567 GeV (top) and 4 GeV (bottom) data sets. The wide vertical boundaries indicate the $Q^2$ bins, the narrow vertical boundaries indicate the $W$ bins within each $Q^2$ range, and the six points in each $W$ bin represent the angle bins from $\cos {\theta }_K^{*}=-0.60$ to 0.90.

Figure 14

Structure functions ${\sigma }_U,{\sigma }_{\mathit{TT}}$, and ${\sigma }_{\mathit{LT}}$ (in nb/sr) for $K^{+}\Lambda$ production vs $\cos {\theta }_K^{*}$ at 2.567 GeV for $Q^2=0.65{\text{GeV}}^2$ and $W$ from 1.650 to 2.050 GeV. Error bars represent statistical uncertainties only. Relative systematic uncertainties to ${\sigma }_U$ are given in Table 4. Models are described in the text.

Figure 15

Same as Fig. 14, but for $K^{+}{\Sigma }^0$ production.

Figure 16

Structure functions ${\sigma }_U,{\sigma }_{\mathit{TT}}$, and ${\sigma }_{\mathit{LT}}$ (in nb/sr) for $K^{+}\Lambda$ production vs $W$ at 2.567 GeV for $Q^2=0.65{\text{GeV}}^2$ for our six $\cos {\theta }_K^{*}$ bins. Error bars represent the statistical uncertainties only. Relative systematic uncertainties to ${\sigma }_U$ are given in Table 4. Data (and calculations) for ${\sigma }_U$ for the two back-angle points have been scaled by a factor of 3 for clarity.

Figure 17

Same as Fig. 16, but for$K^{+}{\Sigma }^0$ production.

Figure 18

$W$ distributions of ${\sigma }_U$ for the $K^{+}\Lambda$ final state from our 4 GeV data set (evolved to 4.056 GeV) for each of our four points in $Q^2$ for $\cos {\theta }_K^{*}=-0.25$ (left) and 0.35 (right). Relative systematic uncertainties to ${\sigma }_U$ are given in Table 4. Curves are from calculations of the MB [44] (dot-dashed), JB [45] (solid), and GLV (dashed) models.

Figure 19

Same as Fig. 18, but for the $K^{+}{\Sigma }^0$ final state.

Figure 20

$Q^2$ distributions of ${\sigma }_U$ for the $K^{+}\Lambda$ and $K^{+}{\Sigma }^0$ final states from our 4 GeV data set (dark filled circles, evolved to 4.056 GeV) at $\cos {\theta }_K^{*}=-0.25$ (a) and 0.90 (b) for three $W$ points. Solid curves are from a dipole mass fit to the 4 GeV data of the form $C(Q^2+M^2){}^{-2}$. The $Q^2=0$ points (solid squares) come from Bradford et al. [21]; two data points from our 2.567 GeV data (light crosses) are not included in the fits. Dashed lines represent error bands from the fits.

Figure 21

Results for the ratio $R={\sigma }_L/{\sigma }_T$ for the $K^{+}\Lambda$ reaction for the Rosenbluth technique (diamonds) and the simultaneous $\epsilon \text{−}\Phi$ fit (filled circles). The Rosenbluth results have been offset in angle for clarity. The data are plotted vs $\cos {\theta }_K^{*}$ for our four $W$ points at $Q^2=1.0{\text{GeV}}^2$. Inner error bars are statistical only; outer error bars are combined statistical and systematic. Curves are from calculations of MB [44] (dot-dashed), JB [45] (solid), and GLV (dashed) models. Parallel kinematics data points come from Mohring et al. [14] (open square) and Raue-Carman [17] (open triangles).

Figure 22

Same as Fig. 21, but for the $K^{+}{\Sigma }^0$ reaction. The parallel kinematics data point comes from Mohring et al. [14] (open square).

Figure 23

Structure function ${\sigma }_T$ vs $\cos {\theta }_K^{*}$ for the $K^{+}\Lambda$ final state for our different $W$ points at $Q^2=1.0{\text{GeV}}^2$ from the $\epsilon \text{−}\Phi$ fit. Inner error bars are statistical only; outer error bars are combined statistical and systematic. Curves are from calculations of MB [44] (dot-dashed), JB [45] (solid), and GLV (dashed) models. The parallel kinematics data point comes from Mohring et al. [14] (open square).

Figure 24

Same as Fig. 23, but for ${\sigma }_L$.

Figure 25

Structure function ${\sigma }_T$ vs $\cos {\theta }_K^{*}$ for the $K^{+}{\Sigma }^0$ final state for our different $W$ points at $Q^2=1.0{\text{GeV}}^2$ from the $\epsilon \text{−}\Phi$ fit. Details are the same as in Fig. 23.

Figure 26

Same as Fig. 25, but for ${\sigma }_L$.