One-proton emission from the ${}_{\mathrm{\Lambda }}{}^6\mathrm{Li}$ hypernucleus

One-proton ($1p$) radioactive emission under the influence of the ${\mathrm{\Lambda }}^0$-hyperon inclusion is discussed. I investigate the hyper-$1p$ emitter, ${}_{\mathrm{\Lambda }}{}^6\mathrm{Li}$, with a time-dependent three-body model. Two-body interactions for $\alpha$-proton and $\alpha -{\mathrm{\Lambda }}^0$ subsystems are determined consistently to their resonant and bound energies, respectively. For a proton-${\mathrm{\Lambda }}^0$ subsystem, a contact interaction, which can be linked to the vacuum-scattering length of the proton-${\mathrm{\Lambda }}^0$ scattering, is employed. A noticeable sensitivity of the $1p$-emission observables to the scattering length of the proton-${\mathrm{\Lambda }}^0$ interaction is shown. The ${\mathrm{\Lambda }}^0$-hyperon inclusion leads to a remarkable fall of the $1p$-resonance energy and width from the hyperonless $\alpha$-proton resonance. For some empirical values of the proton-${\mathrm{\Lambda }}^0$ scattering length, the $1p$-resonance width is suggested to be of the order of $0.1–0.01$ MeV. Thus, the $1p$ emission from ${}_{\mathrm{\Lambda }}{}^6\mathrm{Li}$ may occur in the time scale of ${10}^{-20}–{10}^{-21}$ seconds, which is sufficiently shorter than the self-decay lifetime of ${\mathrm{\Lambda }}^0,{10}^{-10}$ seconds. By taking the spin-dependence of the proton-${\mathrm{\Lambda }}^0$ interaction into account, a remarkable split of the $J^{\pi }=1^{-}$ and $2^{-} 1p$-resonance states is predicted. It is also suggested that, if the spin-singlet proton-${\mathrm{\Lambda }}^0$ interaction is sufficiently attractive, the $1p$ emission from the $1^{-}$ ground state is forbidden. From these results, I conclude that the $1p$ emission can be a suitable phenomenon to investigate the basic properties of the hypernuclear interaction, for which a direct measurement is still difficult.

Figure 1

Three-body model of ${}_{\mathrm{\Lambda }}{}^6\mathrm{Li}$.

Figure 2

Probability-density distribution at $t=0$ obtained with the confining potential procedure.

Figure 3

Time-dependent probability-density distribution of the decay state, ${\rho }_d\left(t\right)$, for $J^{\pi }=1^{-}$.

Figure 4

Survival probability and resonance width of the $1p$ emission from the ground state of ${}_{\mathrm{\Lambda }}{}^6\mathrm{Li}$ ($J^{\pi }=1^{-}$). The survival probability is plotted in the logarithmic scale. In the default case, $R_{\mathrm{box}}=120$ fm and $E_{\mathrm{cut}}=15$ MeV. In the case A, $R_{\mathrm{box}}=140$ fm and $E_{\mathrm{cut}}=15$ MeV. In the case B, $R_{\mathrm{box}}=120$ fm and $E_{\mathrm{cut}}=20$ MeV. The mean three-body energy is obtained as $E_{3b}=-2.521,-2.518$, and $-2.522$ MeV in the default case, case A, and case B, respectively.

Figure 5

Survival probability and resonance width of the $1p$ emission from the ${}_{\mathrm{\Lambda }}{}^6\mathrm{Li}$ nucleus ($J^{\pi }=1^{-}$). Results with $a_v=-1.6,-1.8,-2.0$, and $-2.2$ fm, in correspondence to $E_{3\mathrm{b}}=-2.52,-2.65,-2.75$, and $-2.83$ MeV, respectively, are presented. The survival probability is plotted in logarithmic scale.

Figure 6

Same to Fig. 5 but for the case of $J^{\pi }=2^{-}$. Results with $a_v=-1.6,-1.8,-2.0$, and $-2.2$ fm, in correspondence to $E_{3\mathrm{b}}=-2.49,-2.60,-2.71$, and $-2.79$ MeV, respectively, are presented.

Figure 7

(i) Top: three-body energies obtained with several values of the input parameter, $a_v$. The corresponding $-B_{\mathrm{\Lambda }}\left({}_{\mathrm{\Lambda }}{}^6\mathrm{Li}\right)$ values are presented in the right vertical axis according to Eq. (16). To convert from $E_{3b}$ to the $Q_{1p}$ value, see Eq. (15). (ii) Bottom: $1p$-emission width of ${}_{\mathrm{\Lambda }}{}^6\mathrm{Li}$, obtained as the mean value during $ct=900–1200$ fm in Figs. 5 and 6.

Figure 8

One-proton resonance width as a function of the emitted $Q_{1p}$ value. From $E_{3b}$ values in Fig. 7, Eq. (15) is utilized to evaluate $Q_{1p}$.

Figure 9

The $1p$-emission width of ${}_{\mathrm{\Lambda }}{}^6\mathrm{Li}$. These results are obtained with the sets of parameters, $(a_v^{\left(s\right)},a_v^{\left(t\right)})$, tabulated in Table 2. In the case III, $1p$ emission is forbidden from the $1^{-}$ ground state.