Charmed nuclei within a microscopic many-body approach

Single-particle energies of the ${\mathrm{\Lambda }}_c$ charmed baryon are obtained in several nuclei from the relevant self-energy constructed within the framework of a perturbative many-body approach. Results are presented for a charmed baryon-nucleon ($Y_{c}N$) potential based on a SU(4) extension of the meson-exchange hyperon-nucleon potential $\tilde{A}$ of the Jülich group. Three different models (A, B, and C) of this interaction, which differ only on the values of the couplings of the scalar $\sigma$ meson with the charmed baryons, are considered. Phase shifts, scattering lengths, and effective ranges are computed for the three models and compared with those predicted by the $Y_{c}N$ interaction derived by Haidenbauer and Krein [Eur. Phys. A 54, 199 (2018)] from the extrapolation to the physical pion mass of recent results of the HAL QCD Collaboration. Qualitative agreement is found for two of the models (B and C) considered. Our results for ${\mathrm{\Lambda }}_c$ nuclei are compatible with those obtained by other authors based on different models and methods. We find a small spin-orbit splitting of the $p$-, $d$-, and $f$-wave states as in the case of single $\mathrm{\Lambda }$ hypernuclei. The level spacing of ${\mathrm{\Lambda }}_c$ single-particle energies is found to be smaller than that of the corresponding one for hypernuclei. The role of the Coulomb potential and the effect of the coupling of the ${\mathrm{\Lambda }}_{c}N$ and ${\mathrm{\Sigma }}_{c}N$ channels on the single-particle properties of ${\mathrm{\Lambda }}_c$ nuclei are also analyzed. Our results show that, despite the Coulomb repulsion between the ${\mathrm{\Lambda }}_c$ and the protons, even the less attractive one of our $Y_{c}N$ models (model C) is able to bind the ${\mathrm{\Lambda }}_c$ in all the nuclei considered. The effect of the ${\mathrm{\Lambda }}_{c}N\text{−}{\mathrm{\Sigma }}_{c}N$ coupling is found to be almost negligible due to the large mass difference of the ${\mathrm{\Lambda }}_c$ and ${\mathrm{\Sigma }}_c$ baryons.

Figure 1

Single-meson exchange contributions included in our model for the $Y_{c}N$ interaction.

Figure 2

${}^1{S}_0$ [panel (a)] and ${}^3{S}_1$ [panel (b)] ${\mathrm{\Lambda }}_{c}N$ phase shifts as a function of the center-of-mass kinetic energy. Results are shown for models A, B, and C. The thick band shows the extrapolation to the physical pion mass of the recent results of the HAL QCD Collaboration [37] made by Haidenbauer and Krein in Ref. [38].

Figure 3

${}^1{S}_0$ [panel (a)] and ${}^3{S}_1$ [panel (b)] ${\mathrm{\Lambda }}_{c}N\to {\mathrm{\Lambda }}_{c}N$ diagonal matrix elements as a function of the relative momentum $q$. Results are shown for models A, B, and C.

Figure 4

Brueckner-Hartree-Fock approximation to the finite nucleus ${\mathrm{\Lambda }}_c$ self-energy (diagram a), split into the sum of a first-order contribution (diagram (b)) and a second-order 2p1h correction (diagram c).

Figure 5

Contributions of the kinetic energy, the $Y_{c}N$ interaction, and the Coulomb potential to the energy of the ${\mathrm{\Lambda }}_c$ single-particle bound state $1s_{1/2}$ as a function of the mass number of the ${\mathrm{\Lambda }}_c$ nuclei considered.

Figure 6

${\mathrm{\Lambda }}_c$ probability density distribution for the $1s_{1/2}$ state in the six ${\mathrm{\Lambda }}_c$ nuclei considered. Results are shown for the three models A, B, and C of the $Y_{c}N$ interaction. Thin solid, dashed, and dotted lines show the results when the Coulomb interaction is artificially switched off.