Single-particle spectral function of the $\Lambda$ hyperon in finite nuclei

The spectral function of the $\Lambda$ hyperon in finite nuclei is calculated from the corresponding $\Lambda$ self-energy, which is constructed within a perturbative many-body approach using some of the hyperon-nucleon interactions of the J\"{u}lich and Nijmegen groups. Binding energies, wave functions and disoccupation numbers of different single-particle states are obtained for various hypernuclei from $^5_{\Lambda}$He to $^{209}_{\,\,\,\,\,\Lambda}$Pb. The agreement between the calculated binding energies and experimental data is qualitatively good. The small spin-orbit splitting of the $p-, d-, f-$ and $g-$wave states is confirmed. The discrete and the continuum contributions of the single-$\Lambda$ spectral function are computed. Their appearance is qualitatively similar to that of the nucleons. The $Z$-factor, that measures the importance of correlations, is also calculated. Our results show that its value is relatively large, indicating that the $\Lambda$ hyperon is less correlated than nucleons. This is in agreement with the results obtained by other authors for the correlations of the $\Lambda$ in infinite nuclear matter. The disoccupation numbers are obtained by integrating the spectral function over the energy. Our results show that the discrete contribution to the disoccupation number decreases when increasing the momentum of the $\Lambda$. This indicates that, in the production reactions of hypernuclei, the $\Lambda$ hyperon is mostly formed in a quasi-free state.