Missing-mass spectroscopy of the ${}^{12}\mathrm{C}(p,d)$ reaction near the ${\eta }^{\prime }$-meson production threshold

Excitation-energy spectra of ${}^{11}\mathrm{C}$ nuclei near the ${\eta }^{\prime }$-meson production threshold have been measured by missing-mass spectroscopy using the ${}^{12}\mathrm{C}(p,d)$ reaction. A carbon target has been irradiated with a 2.5 GeV proton beam supplied by the synchrotron SIS-18 at GSI to produce ${\eta }^{\prime }$-meson bound states in ${}^{11}\mathrm{C}$ nuclei. Deuterons emitted at $0^{\circ }$ in the reaction have been momentum analyzed by the fragment separator (FRS), used as a high-resolution spectrometer. No distinct structure due to the formation of ${\eta }^{\prime }$-mesic states is observed although a high statistical sensitivity is achieved in the experimental spectra. Upper limits on the formation cross sections of ${\eta }^{\prime }$-mesic states are determined, and thereby a constraint imposed on the ${\eta }^{\prime }$-nucleus interaction is discussed.

Figure 1

Top panel: A schematic view of the experimental setup with the FRS. A 2.5 GeV proton beam impinged on a carbon target. Deuterons emitted in the ${}^{12}\mathrm{C}(p,d)$ reaction were momentum analyzed at F4 and the tracks were reconstructed from measurements by multiwire drift chambers (MWDCs). Sets of 5-mm-thick plastic scintillation counters (SC2H, SC2V, and SC41) and a 20-mm-thick one (SC42) were installed at F2 and F4 for time-of-flight (TOF) measurements. Čerenkov detectors (ACs and TORCH) and time projection chambers (TPCs) were installed but not used in the present analysis. Middle panel: Horizontal beam trajectories with the specially developed optics mode based on calculated third-order transfer matrices. Initial positions and angles are taken from $\{-1,1\}$ mm $\times \{-8,-4,0,4,8\}$ mrad, and a momentum at the central value. Bottom panel: A momentum-dispersion curve of this optics mode.

Figure 2

TOF between SC2H and SC41 and between SC41 and SC42 recorded with a data acquisition system triggered by the SC41 signals (top) and by the TOF-based coincidence between the SC2H and SC41 signals (middle). Typical signal waveforms of SC2H are shown for the deuteron (bottom left) and the multiple-hit proton (bottom right) events based on the TOFs.

Figure 3

Fit results of SC2H waveforms with a single-pulse function. SSR and the height parameter $p_2$ obtained in the fit are plotted for both of the two photomultipliers (left and right) of SC2H. The solid curves are used as boundaries to select single-hit events.

Figure 4

Corrected TOF spectra between F2 and F4 at the FRS scale factor of $f=0.980$. The unshaded histogram shows the total number of events, while the shaded one corresponds to the single-hit events selected by the SC2H waveforms. The dashed lines indicate the $\pm 5\sigma$ region around the deuteron peak used for the TOF selection. The region between the dotted lines is used to estimate the proton contamination fraction.

Figure 5

Calibration data with the elastic proton-deuteron scattering for the FRS scale factors $f=0.980$, 0.990, 1.000, 1.010, 1.020. Left panel: Horizontal position $X$ and angle $X^{\prime }$ at F4 reconstructed by MWDCs. Right panel: Calibrated $\delta$, the momentum deviation relative to the FRS central momentum, is shown against the angle $X^{\prime }$.

Figure 6

Momentum spectrum of deuterons in the ${\mathrm{CD}}_2(p,d)$ calibration measurement with the 1.6 GeV proton beam. The lower axis shows the excitation energy for the production run corresponding to the deuteron momentum. The gray solid line displays a fit with a function consisting of a Gaussian peak (dotted line) and a second-order polynomial (dashed line).

Figure 7

Excitation-energy spectra of ${}^{11}\mathrm{C}$ measured in the ${}^{12}\mathrm{C}(p,d)$ reaction with seven momentum settings ($f=0.980–1.020$) of the FRS. The lower abscissa is the excitation energy $E_{\mathrm{ex}}$ relative to the ${\eta }^{\prime }$ production threshold $E_0=957.78$ MeV, and the upper axis shows the corresponding deuteron momentum. The gray solid curves display a third-order polynomial simultaneously fitted to the seven spectra.

Figure 8

Top panel: Combined spectrum of the ${}^{11}\mathrm{C}$ excitation energy $E_{\mathrm{ex}}$ near the ${\eta }^{\prime }$ production threshold $E_0=957.78$ MeV. A third-order polynomial fit is given by the solid gray curve. The horizontal bars indicate excitation-energy regions covered by each FRS momentum setting. Bottom panel: Residues of the polynomial fit. The dashed lines display envelopes of two standard deviations.

Figure 9

Fitted values (solid dots) and resulting 95% C.L. upper limits (solid curves) of the Lorentzian-shaped formation cross section of ${\eta }^{\prime }$-mesic nuclei plotted as a function of the assumed peak position $E_{\mathrm{ex}}$ for the widths of $\mathrm{\Gamma }=5,10,15$ MeV. Upper limits evaluated by simultaneous fitting of the seven spectra (Fig. 7) are shown by the dashed curves. The shaded areas indicate the systematic errors on the upper limits.

Figure 10

Theoretically calculated spectra of the ${}^{12}\mathrm{C}(p,d){}^{11}\mathrm{C}\otimes {\eta }^{\prime }$ reaction at 2.5 GeV [29]. The ${\eta }^{\prime }$-nucleus potential parameters are taken as $(V_0,W_0)=(-150,-10)$ MeV (top) and $(-50,-10)$ MeV (bottom). Total formation cross sections are shown by the thick lines, and major configurations of the ${\eta }^{\prime }$-mesic states ${\ell }_{{\eta }^{\prime }}$ coupling with the neutron hole states ${\left(n{\ell }_j\right)}_n^{-1}$ are shown by the thin lines.

Figure 11

A contour plot of ${\mu }_{95}$ (solid curves), the 95% C.L. upper limit of the scale parameter $\mu$ for the theoretical formation spectra, on the real and imaginary potential plane $(V_0,W_0)$. The limits have been analyzed at the potential sets listed in Table 2 and linearly interpolated in-between. The systematic errors on the ${\mu }_{95}=1$ contour are shown by the dashed curves. The region with ${\mu }_{95}\le 1$ is excluded by the present analysis. See the text for further explanation.

Figure 12

Obtained constraint and currently known information on the ${\eta }^{\prime }$-nucleus potential $(V_0+i{W}_0)$ at normal nuclear density. The shaded region (${\mu }_{95}\le 1$) represents the region excluded within the present analysis. The rectangular box shows real and imaginary potentials evaluated in ${\eta }^{\prime }$ photoproduction experiments by the CBELSA/TAPS Collaboration [20, 21, 22, 23]. Theoretical predictions based on the NJL model [9, 10], the linear sigma model [11], the QMC model [12], and the chiral unitary approach [41] are indicated by the dashed lines.

Figure 13

Simulated kinetic energy distributions of protons emitted in the decay of ${\eta }^{\prime }$-mesic nuclei. Three decay modes are considered: ${\eta }^{\prime }N\to \eta N$ (thin gray), ${\eta }^{\prime }N\to \pi N$ (thin black), and ${\eta }^{\prime }NN\to NN$ (thick). The integral of each distribution is normalized to unity.

Figure 14

A schematic view of an experimental setup for the semiexclusive measurement. A 2.5 GeV proton beam impinges on a carbon target. The forward deuteron is momentum analyzed by the downstream FRS section. The proton emitted backward in the decay of ${\eta }^{\prime }$-mesic nuclei is identified by the WASA central detector [47, 48].