Nonmesonic weak decay of $\mathrm{\Lambda }$ hypernuclei: The $\mathrm{\Lambda }N-\mathrm{\Sigma }N$ coupling

The nonmesonic weak decay of $\mathrm{\Lambda }$ hypernuclei is studied within a microscopic diagrammatic approach which is extended to include the $\mathrm{\Lambda }N-\mathrm{\Sigma }N$ strong interaction coupling. We adopt a nuclear matter formalism which, through the local density approximation, allows us to model finite hypernuclei. One-meson-exchange potentials are used for the weak and strong four-baryon interactions. The rates for the neutron- and proton-induced weak decays, $\mathrm{\Lambda }N\to nN$ and $\mathrm{\Lambda }N\to \mathrm{\Sigma }N\to nN$ ($N=n$ or $p$; $\mathrm{\Sigma }={\mathrm{\Sigma }}^{-}, {\mathrm{\Sigma }}^0$, and ${\mathrm{\Sigma }}^{+}$), are predicted for ${}_{\mathrm{\Lambda }}{}^{12}\mathrm{C}, {}_{\mathrm{\Lambda }}{}^{28}\mathrm{Si}, {}_{\mathrm{\Lambda }}{}^{56}\mathrm{Fe}$, and ${}_{\mathrm{\Lambda }}{}^{208}\mathrm{Pb}$. The $\mathrm{\Lambda }N-\mathrm{\Sigma }N$ coupling turns out to provide a contribution of about 23% (5%) to the total neutron-induced rate (proton-induced rate). Again as an effect of the hyperon coupling, the total one-nucleon-induced rate increases by about 9%, while the ${\mathrm{\Gamma }}_n/{\mathrm{\Gamma }}_p$ ratio increases by about 17%. The saturation property of the nonmesonic rates with increasing hypernuclear mass number is clearly obtained and explained. The above percentage modifications remain almost unaltered through the hypernuclear mass number. All the non-negligible changes introduced by the $\mathrm{\Lambda }N-\mathrm{\Sigma }N$ coupling lead to a definite improvement of our predictions once compared with the available KEK and FINUDA data.

Figure 1

The $\mathrm{\Lambda }N\to \mathrm{\Sigma }N\to nN$ transition amplitudes originated by the $\mathrm{\Lambda }N-\mathrm{\Sigma }N$ coupling. Diagrams (a) and (b) [(c) to (f)] contribute to the neutron-induced (proton-induced) rate ${\mathrm{\Gamma }}_n$ (${\mathrm{\Gamma }}_p$). In Appendix pp1 we give the isospin wave functions for each contribution.

Figure 2

Goldstone diagrams for the evaluation of the one-nucleon-stimulated decay rates in infinite nuclear matter once the $\mathrm{\Lambda }N-\mathrm{\Sigma }N$ strong coupling is taken into account.

Figure 3

Goldstone diagram contributing to ${\mathrm{\Gamma }}_1^{\mathrm{\Sigma }\mathrm{\Sigma }}$. Concerning the values of the flowing energy-momenta, it is easy to see that $p_2=h+q-p, p_2^{\prime }=h+q, p_2^{\prime \prime }=h+q-p+p^{\prime }$, and $Q=q-p=q^{\prime }-p^{\prime }$. In this diagram, the dotted line shows the final state.

Figure 4

Goldstone diagram corresponding to the interference rate $\mathrm{\Delta }{\mathrm{\Gamma }}_1^{0\mathrm{\Sigma }}$. The energy-momentum transfer is shown along each line (note that $p_2=h+q-p$ and $p_2^{\prime }=h+q$).