$\mathrm{\Xi }$ hyperons in the nuclear medium described by chiral NLO interactions

Properties of the baryon-baryon interactions in the strangeness $S=-2$ sector of chiral effective field theory at the next-to-leading order (NLO) level are explored by calculating $\mathrm{\Xi }$ single-particle potentials in symmetric nuclear matter. The results are transformed to the $\mathrm{\Xi }$ potential in finite nuclei by a local-density approximation with convolution by a Gaussian form factor to simulate finite-range effects. The $\mathrm{\Xi }$ potential is repulsive in a central region, and attractive in a surface area when the $\mathrm{\Xi }$ energy is low. The attractive pocket can lower the ${\mathrm{\Xi }}^{-} s$ and $p$ atomic states. The obtained binding energies in ${}^{12}\mathrm{C}$ and ${}^{14}\mathrm{N}$ are found to be conformable with those found in emulsion experiments at Japan's National Laboratory for High Energy Physics (KEK). $K^{+}$ spectra of $(K^{-},K^{+}) \mathrm{\Xi }$ production inclusive processes on ${}^9\mathrm{Be}$ and ${}^{12}\mathrm{C}$ are also evaluated, using a semiclassical distorted wave method. The absolute values of the cross section are properly reproduced for ${}^9\mathrm{Be}$, but the peak locates at a lower energy position than that of the experimental data. The calculated spectrum of ${}^{12}\mathrm{C}$ should be compared with the forthcoming result from the new experiments recently carried out at KEK with better resolution than before. The comparison would be valuable to improve the understanding of the $\mathrm{\Xi }N$ interaction, the parametrization of which has still large uncertainties.

Figure 1

Momentum dependence of the real and imaginary parts of $\mathrm{\Xi }$ single-particle potentials in SNM at various Fermi momenta $k_F$, using the ChEFT NLO interactions in the strangeness $S=-2$ sector [4]. $\rho /{\rho }_0$ is the ratio of the density $\rho =2k_F^3/\left(3{\pi }^2\right)$ to the normal density ${\rho }_0$ at $k_F=1.35{\mathrm{fm}}^{-1}$. The thick and thin curves in the real part represent the results with and without baryon-channel coupling effects.

Figure 2

Partial wave contributions to the real part of $\mathrm{\Xi }$ single-particle potentials in SNM at $k_F=1.07{\mathrm{fm}}^{-1}$ and $k_F=1.35{\mathrm{fm}}^{-1}$. The thick curves denote the results of the full calculations. The thin curves in the lower panel represent the results with switched off baryon-channel coupling. Note that the $\underset{1}{{}^{3}S} T=0$ and $\underset{1}{{}^{1}P} T=0$ states have no coupling to other baryon channels. The thick and thin curves are indistinguishable in some partial waves.

Figure 3

Momentum dependence of the contributions to the $\mathrm{\Xi }$ single-particle potential from the $s, p$, and other partial waves after summing over the total $J$ in the isospin $T=0$ or $T=1$ channel in SNM with $k_F=1.07{\mathrm{fm}}^{-1}$ and $k_F=1.35{\mathrm{fm}}^{-1}$.

Figure 4

Radial dependence of the $\mathrm{\Xi }$ potentials in finite nuclei ${}^9\mathrm{Be}, {}^{12}\mathrm{C}$, and ${}^{14}\mathrm{N}$, calculated from those in SNM by a local-density approximation for various $\mathrm{\Xi }$ energies $E$. Results before and after folding the Gaussian form factor of Eq. (2) with $\beta =1$ fm are represented by dashed and solid curves, respectively.

Figure 5

$\mathrm{\Xi }$ wave functions of the $0s, 0p$, and $0d$ bound states given in Table 3 for the ${\mathrm{\Xi }}^{-}-{}^{14}\mathrm{N}$ system. Solid curves represent results of the potential given in Eq. (3) with $E=0$. For comparison, results with a single Woods-Saxon potential with the depth of $-14$ MeV and the geometry of $r_0=1.1A^{1/3}$ and $a_0=0.65$ are shown by dotted curves. The dotted curve for the $0d$ state is almost indistinguishable from the solid curve.

Figure 6

$\mathrm{\Xi }$ potential values $U_{\mathrm{\Xi }}(E,k_F)$ calculated in SNM at various Fermi momenta from $k_F=0.8$ to $1.5{\mathrm{fm}}^{-1}$ are shown by the dots for the $\mathrm{\Xi }$ energies of $E=0$, 50, and 100 MeV. The solid curves represent the cubic spline fit as a function of the density $\rho =\frac{2k_F^3}{3{\pi }^2}$. The dashed curves show the results of the cubic spline fit as a function of the Fermi momentum $k_F$. The interpolation as a function of $k_F$ tends to yield a more attractive potential at low densities than that with respect to the density $\rho$.

Figure 7

$K^{+}$ spectrum of $(K^{-},K^{+}) \mathrm{\Xi }$ inclusive production reactions on ${}^9\mathrm{Be}$. The incident $K^{-}$ momentum is $p_{K^{-}}=1.8$ GeV/$c$. The reaction angle in the laboratory frame is set to be ${\theta }_{K^{+}}=5^{\circ }$. The width of $\mathrm{\Gamma }/2=5$ MeV is applied for Lorentz-type smearing. The experimental resolution of $\mathrm{\Delta }E=6.1$ MeV is also taken into account, as explained in the text. Experimental data, which were an average over $1.5^{\circ }<{\theta }_{K^{+}}<8.5^{\circ }$, are taken from Ref. [26].

Figure 8

$K^{+}$ spectrum of $(K^{-},K^{+}) \mathrm{\Xi }$ inclusive production reactions on ${}^{12}\mathrm{C}$. The incident $K^{-}$ momentum is $p_{K^{-}}=1.8$ GeV/$c$ and the reaction angle in the laboratory frame is ${\theta }_{K^{+}}=5^{\circ }$. The width of $\mathrm{\Gamma }/2=5$ MeV is applied for Lorentz-type smearing. The experimental resolution of $\mathrm{\Delta }E-2$ MeV is assumed.

Figure 9

$K^{+}$ spectrum of $(K^{-},K^{+}) \mathrm{\Xi }$ inclusive production reactions on ${}^{12}\mathrm{C}$ around the threshold region with the incident $K^{-}$ momentum of $p_{K^{-}}=1.8$ GeV/$c$ and the reaction angle in the laboratory frame of ${\theta }_{K^{+}}=5^{\circ }$. The width of $\mathrm{\Gamma }/2=2$ MeV is applied, instead of 5 MeV in Fig. 8. The experimental resolution of $\mathrm{\Delta }E-2$ MeV is assumed.