$\omega N$ scattering length from $\omega$ photoproduction on the proton near the reaction threshold

Photoproduction of the $\omega$ meson on the proton has been experimentally studied near the threshold. The total cross sections are determined at incident energies ranging from 1.09 to 1.15 GeV. The 1/2 and 3/2 spin-averaged scattering length $a_{\omega p}$ and effective range $r_{\omega p}$ between the $\omega$ meson and proton are estimated from the shape of the total cross section as a function of the incident photon energy: $a_{\omega p}=(-0.{97}_{-0.{16}_{\mathrm{stat}}-0.{00}_{\mathrm{syst}}}^{+0.{16}_{\mathrm{stat}}+0.{03}_{\mathrm{syst}}})+i(0.{07}_{-0.{14}_{\mathrm{stat}}-0.{09}_{\mathrm{syst}}}^{+0.{15}_{\mathrm{stat}}+0.{17}_{\mathrm{syst}}})$ fm and $r_{\omega p}=(+2.{78}_{-0.{54}_{\mathrm{stat}}-0.{12}_{\mathrm{syst}}}^{+0.{67}_{\mathrm{stat}}+0.{11}_{\mathrm{syst}}})+i(-0.{01}_{-0.{50}_{\mathrm{stat}}-0.{00}_{\mathrm{syst}}}^{+0.{46}_{\mathrm{stat}}+0.{06}_{\mathrm{syst}}})$ fm, resulting in a repulsive force. The real and imaginary parts for $a_{\omega p}$ and $r_{\omega p}$ are determined separately for the first time. A small $P$-wave contribution does not affect the obtained values.

Figure 1

Typical $M_{\pi \gamma }$ distributions for the highest incident energy group $(E_{\gamma }=1.144$–1.149 GeV). In each panel, the histogram (blue) shows the experimentally obtained $M_{\pi \gamma }$ distribution, and the solid curve (red) shows the sum of the $M_{\pi \gamma }$ distributions obtained in the simulation for the $\gamma p\to \omega p\to {\pi }^0\gamma p$ and $\gamma p\to {\pi }^0{\pi }^{0}p$ reactions. The dashed (magenta) and dotted (cyan) curves show these contributions. The angular region of $\omega$ emission in the $\gamma p$-c.m. frame is described in each panel. The vertical lines show the lower limit $M_{\pi \gamma }=0.76$ GeV for selecting the $\omega$-produced events.

Figure 2

Typical angular differential cross sections $d\sigma /d\mathrm{\Omega }$ as a function of the $\omega$ emission angle $cos\theta$ in the $\gamma p$-c.m. frame. The range of the incident photon energies is described in each panel. The filled circles (blue) represent the measured $d\sigma /d\mathrm{\Omega }$ in this work. The shaded areas represent the systematic uncertainties of $d\sigma /d\mathrm{\Omega }\mathrm{s}$. The solid curves show the fitted distribution with a $P$-wave contribution of ${\sigma }_{\max }^P/5$ (see text). The $d\sigma /d\mathrm{\Omega }$ results from SAPHIR [10], CLAS [11], and A2 [4] collaborations are depicted by open boxes (green), open triangles (black), and open circles (magenta), respectively. The photon-energy coverages are 25, 18, 15, and $\approx 4.5$ MeV in SAPHIR, CLAS, A2, and our results, respectively. The median of each coverage is described in the same color in the corresponding panel as data points.

Figure 3

Same as Fig. 2 but for different incident energies. The $d\sigma /d\mathrm{\Omega }$ results from other collaborations are plotted by increasing their incident photon energies by 10 MeV.

Figure 4

Total cross section $\sigma$ as a function of $E_{\gamma }$. The filled circles (blue) represent the measured $\sigma$ in this work. Each horizontal bar indicates the incident-energy coverage. Each short vertical bar (blue) represents the statistical error of $\sigma$, and the connected bars (cyan) indicate the upper- and lower-side systematic error of $\sigma$. The vertical line (black) shows $E_{\gamma }$ corresponding to the reaction threshold for production of $\omega$ having the centroid mass. The solid curve (red) shows the calculated excitation function with the parameters: $a_{\omega p}=-0.97+i0.07$ fm and $r_{\omega p}=+2.78-i0.01$ fm. The dashed curves (gray) show $2{\sigma }_{\mathrm{nonFSI}}^S$ where ${\sigma }_{\mathrm{nonFSI}}^S$ denotes the excitation function with $a_{\omega p}=0$ fm and $r_{\omega p}=0$ fm. The dotted curve (purple) shows $5{\sigma }_{\max }^P$ where ${\sigma }_{\max }^P$ stands for the upper limit of the $P$-wave contribution. The $\sigma$ results from SAPHIR [10] and A2 [4] collaborations are depicted by open boxes (green) and open circles (magenta), respectively.

Figure 5

Real and imaginary parts of spin-averaged $a_{\omega N}$ obtained in this work (red). The donut (magenta) represents the experimentally obtained $|a_{\omega N}|$ using a VMD model [4]. Other markers indicate $a_{\omega N}\mathrm{s}$ obtained in the theoretical works on an effective Lagrangian approach [5] (black), a QCD sum-rule analysis [6] (green), coupled-channel analyses [7, 8, 9] (blue).