Determination of the real part of the ${\eta }^{\prime }$-Nb optical potential

The excitation function and momentum distribution of ${\eta }^{\prime }$ mesons have been measured in photoproduction off ${}^{93}\mathrm{Nb}$ in the energy range of 1.2–2.9 GeV. The experiment has been performed with the combined Crystal Barrel and MiniTAPS detector system, using tagged photon beams from the ELSA electron accelerator. Information on the sign and magnitude of the real part of the ${\eta }^{\prime }$-Nb potential has been extracted from a comparison of the data with model calculations. An attractive potential of $-(41\pm 10(\mathrm{stat})\pm 15(\mathrm{syst}\left)\right)$ MeV depth at normal nuclear matter density is deduced within model uncertainties. This value is consistent with the potential depth of $-(37\pm 10(\mathrm{stat})\pm 10(\mathrm{syst}\left)\right)$ MeV obtained in an earlier measurement for a light nucleus (carbon). This relatively shallow ${\eta }^{\prime }$-nucleus potential will make the search for ${\eta }^{\prime }$-nucleus bound states more difficult.

Figure 1

The ${\pi }^0{\pi }^0\eta$ invariant mass distribution measured in photoproduction off Nb in the incident photon energy range of 1.2–2.9 GeV. The solid curve represents a fit to the data using a Gaussian function combined with a polynomial function for the background. The fit parameters are $\sigma =11.9\pm 0.3$ MeV/$c^2$ (corresponding to the instrumental resolution), $m=957.1\pm 0.4$ MeV/$c^2$.

Figure 2

Two-dimensional reconstruction efficiency for ${\eta }^{\prime }$ photoproduction off Nb as a function of the ${\eta }^{\prime }$ momentum and laboratory angle for the incident photon energy range of 1.2–2.9 GeV.

Figure 3

Differential cross sections for photoproduction of ${\eta }^{\prime }$ mesons off Nb for different bins in the incident photon energy range 1.2–2.9 GeV determined in five $cos{\theta }_{{\eta }^{\prime }}^{\text{c.m.}}$ bins of width 0.4.

Figure 4

Left: Total cross section for ${\eta }^{\prime }$ photoproduction off Nb obtained by integrating the differential cross sections (full circles) and by direct measurement of the ${\eta }^{\prime }$ yield in different incident photon energy bins (open circles). The data are compared to calculations for an inelastic ${\eta }^{\prime }$-nucleon cross section ${\sigma }_{\text{inel}}^{{\eta }^{\prime }}=13$ mb [21, 46] and for potential depths $V=0$ MeV (black line), $-25$ MeV (green), $-50$ MeV (blue),$-75$ MeV (black dashed), $-100$ MeV (red), and $-150$ MeV (magenta) at normal nuclear density, respectively, and using the full nucleon spectral function. All calculated cross sections have been multiplied by a factor 0.91 to match the experimentally observed cross section above $E_{\gamma }>2.2$ GeV (see text). This normalization is within the systematic error quoted for the experimental cross section. Middle: The experimental data and the predicted curves for $V=-25, -50, -75, -100$, and $-150$ MeV divided by the calculation for the scenario $V=0$ MeV and presented on a linear scale. Right: ${\chi }^2$ fit of the data with the excitation functions calculated for the different scenarios over the full incident photon energy range.

Figure 5

Left: Momentum distribution for ${\eta }^{\prime }$ photoproduction off Nb for the incident photon energy range 1.3–2.6 GeV. The calculations are for ${\sigma }_{\text{inel}}^{{\eta }^{\prime }}=13$ mb and for potential depths $V=0, -25, -50, -75, -100$, and $-150$ MeV at normal nuclear density, respectively. All calculated cross sections have been multiplied by a factor 0.83 (see text). The color code is identical to the one in Fig. 4. Middle: The experimental data and the predicted momentum distributions for $V=-25, -50, -75, -100$, and $-150$ MeV divided by the calculation for the scenario $V=0$ MeV and presented on a linear scale. Right: ${\chi }^2$ fit of the data with the momentum distributions calculated for the different scenarios.

Figure 6

Depths of the real part of the ${\eta }^{\prime }$-nucleus potential determined by analyzing the excitation function and the momentum distributions for C [22] (full black circles) and for Nb (this work, red triangles). The weighted overall average is indicated by a blue square and the shaded area. The vertical hatched lines mark the range of systematic uncertainties.