Separated kaon electroproduction cross section and the kaon form factor from 6 GeV JLab data

The ${}^1\mathrm{H}\left(e,e^{\prime }{K}^{+}\right)\mathrm{\Lambda }$ reaction was studied as a function of the Mandelstam variable $-t$ using data from the E01-004 (FPI-2) and E93-018 experiments that were carried out in Hall C at the 6 GeV Jefferson Laboratory. The cross section was fully separated into longitudinal and transverse components, and two interference terms at four-momentum transfers $Q^2$ of 1.00, 1.36, and 2.$07{\mathrm{GeV}}^2$. The kaon form factor was extracted from the longitudinal cross section using the Regge model by Vanderhaeghen et al. [Phys. Rev. C 57, 1454 (1998)]. The results establish the method, previously used successfully for pion analyses, for extracting the kaon form factor. Data from 12 GeV Jefferson Laboratory experiments are expected to have sufficient precision to distinguish between theoretical predictions, for example, recent perturbative QCD calculations with modern parton distribution amplitudes. The leading-twist behavior for light mesons is predicted to set in for values of $Q^2$ between 5 and $10{\mathrm{GeV}}^2$, which makes data in the few-GeV regime particularly interesting. The $Q^2$ dependence at fixed $x$ and $-t$ of the longitudinal cross section that we extracted seems consistent with the QCD factorization prediction within the experimental uncertainty.

Figure 1

Missing mass distribution of events within the real (solid line) and random coincidence (green, hatched-shaded area) time window for the FPI-2 setting $Q^2=2.07{\mathrm{GeV}}^2$ and $-t=0.39$ (GeV/${\mathrm{c})}^2$, integrated over the azimuthal angle. Also shown is the dummy target distribution within the real (blue, cross-shaded area) and random (gray, solid-shaded area) coincidence time window.

Figure 2

Unseparated differential cross section extracted in different bins in $\varphi$ for the two $\epsilon$ settings of the kinematics with $Q^2=2.07$ ${\mathrm{GeV}}^2(W=2.31$ GeV, $-t=0.39$ (GeV/${\mathrm{c})}^2)$ of the FPI-2 experiment. The solid lines in panels (a) and (b) are fits to the data using the functional form of Eq. (2). The dashed lines in both panels (a) and (b) indicate the cross section averaged over all $\varphi$ bins.

Figure 3

Kaon form factor from E93-018 and FPI-2 (filled squares) compared to data from Refs. [20, 34]. The error bars include the statistical and systematic uncertainty. Also shown are two kaon form factor points (filled circles) we extracted from the cross section data of Ref. [35] using our form factor extraction method. The blue open squares indicate the projected reach and accuracy of data that are anticipated from JLab 12 GeV experiment E12-09-011 [36]. The two error estimates are based on different assumptions about the $-t$ and model dependence of the form factor extractions, with the larger uncertainty being more conservative. The dashed curve shows the monopole using a kaon charge radius of 0.56 fm obtained at low $Q^2$ in Ref. [20]. The remaining curves are calculations with different kaon parton distribution amplitudes (PDAs). The solid blue line shows the kaon form factor obtained in Ref. [1] using the Dyson-Schwinger equation with rainbow ladder truncation. The curve is consistent within 15% with that obtained using the leading-order (LO), leading-twist (LT) QCD prediction with a pion valence-quark PDA evaluated at a scale appropriate to the experiment. The dotted pink curve corresponds to a LO/LT calculation obtained with the conformal-limit PDA.

Figure 4

The $Q^2$ dependence of the separated cross sections at fixed values of $-t=0.4$ (GeV/${\mathrm{c})}^2$ and $x_B=0.3$. The filled symbols are the scaled cross sections from E93-018 and FPI-2 and the open symbols from E98-108 data. The error bars denote the statistical and systematic uncertainties combined in quadrature. The solid curves show a fit of the form $Q^{-n}$, for ${\sigma }_L$ and $Q^{-m}$ for ${\sigma }_T$, where $n=5.69\pm 0.52$ and $m=8.47\pm 0.46$. Calculations using the VR (dashed) and the VGL (dotted) models are also shown.