Hypertriton lifetime

Over the last decade, conflicting values of the hypertriton ${}_{\mathrm{\Lambda }}{}^3\mathrm{H}$ lifetime $\tau \left({}_{\mathrm{\Lambda }}{}^3\mathrm{H}\right)$ were extracted from relativistic heavy-ion (RHI) collision experiments, ranging from values compatible with the free-$\mathrm{\Lambda }$ lifetime ${\tau }_{\mathrm{\Lambda }}$—as expected naively for a very weakly bound $\mathrm{\Lambda }$ in ${}_{\mathrm{\Lambda }}{}^3\mathrm{H}$—to lifetimes as short as $\tau \left({}_{\mathrm{\Lambda }}{}^3\mathrm{H}\right)\approx \left(0.4\text{–}0.7\right){\tau }_{\mathrm{\Lambda }}$. In a recent work [Phys. Lett. B 811, 135916 (2020)] we studied this ${}_{\mathrm{\Lambda }}{}^3\mathrm{H}$ lifetime puzzle theoretically using realistic three-body ${}_{\mathrm{\Lambda }}{}^3\mathrm{H}$ and ${}^3\mathrm{He}$ wave functions computed within the ab initio no-core shell model approach with interactions derived from chiral effective field theory to calculate the partial decay rate $\mathrm{\Gamma }({}_{\mathrm{\Lambda }}{}^3\mathrm{H}\to {}^3\mathrm{He}+{\pi }^{-})$. Significant but opposing contributions were found from $\mathrm{\Sigma }NN$ admixtures in ${}_{\mathrm{\Lambda }}{}^3\mathrm{H}$ and from ${\pi }^{-}\text{−}{}^3\mathrm{He}$ final-state interaction. In particular, $\tau \left({}_{\mathrm{\Lambda }}{}^3\mathrm{H}\right)$ was found to be strongly correlated with the $\mathrm{\Lambda }$ separation energy $B_{\mathrm{\Lambda }}$ in ${}_{\mathrm{\Lambda }}{}^3\mathrm{H}$, the value of which is rather poorly known experimentally and, in addition, is known to suffer from sizable theoretical uncertainties inherent in the employed nuclear and hypernuclear interaction models. In the present work we find that these uncertainties propagate into $\tau \left({}_{\mathrm{\Lambda }}{}^3\mathrm{H}\right)$, and thus limit considerably the theoretical precision of its computed value. Although none of the conflicting RHI measured $\tau \left({}_{\mathrm{\Lambda }}{}^3\mathrm{H}\right)$ values can be excluded, but rather can be attributed to a poor knowledge of $B_{\mathrm{\Lambda }}$, we note the good agreement between the lifetime value $\tau \left({}_{\mathrm{\Lambda }}{}^3\mathrm{H}\right)=242\left(28\right)\mathrm{ps}$ computed at the lowest value $B_{\mathrm{\Lambda }}=66\mathrm{keV}$ reached by us and the very recent ALICE measured lifetime value ${\tau }^{\text{ALICE}}\left({}_{\mathrm{\Lambda }}{}^3\mathrm{H}\right)=253\left(11\right)\left(6\right)\mathrm{ps}$ associated with the ALICE measured $B_{\mathrm{\Lambda }}$ value $B_{\mathrm{\Lambda }}^{\text{ALICE}}=102\left(63\right)\left(67\right)\mathrm{keV}$ [S. Acharya et al. (ALICE Collaboration), Phys. Rev. Lett. 131, 102302 (2023)].

Figure 1

${}_{\mathrm{\Lambda }}{}^3\mathrm{H}$ lifetime values $\tau \left({}_{\mathrm{\Lambda }}{}^3\mathrm{H}\right)$ obtained in selected [25] past nuclear BC [14, 15, 17] and recent RHI collision [8, 9, 10, 11, 18, 19, 20] experiments, together with a few (post-1997) microscopic calculations [21, 22, 23, 24]; see text. Error bars and shaded areas indicate the measurement statistical and systematical errors, respectively, as well as the estimated theoretical uncertainties (if available). The vertical dashed line corresponds to the free-$\mathrm{\Lambda }$ lifetime value ${\tau }_{\mathrm{\Lambda }}=263\left(2\right)\mathrm{ps}$ [7].

Figure 2

Dependence of ${}^2\mathrm{H}$ and ${}^3\mathrm{He}$ g.s. energies $E_{\mathrm{g}.\mathrm{s}.}$ on the NCSM model-space truncation $10\le N_{\max }\le 40$ for several values of the HO frequency, $\hslash \omega =10$, 14, and 16 MeV, calculated using the ${\mathrm{NNLO}}_{\mathrm{sim}}$(${\mathrm{\Lambda }}_{NN}$ = 500 MeV, $T_{\mathrm{Lab}}^{\max }$ = 290 MeV) interaction. Indicated by the gray horizontal lines are the corresponding experimental values.

Figure 3

NCSM ${}_{\mathrm{\Lambda }}{}^3\mathrm{H}$ g.s. energies $E^{\mathrm{UV}}\left({}_{\mathrm{\Lambda }}{}^3\mathrm{H}\right)$ (empty and filled symbols) as functions of the IR length $L_{\mathrm{IR}}$ (up to $N_{\max }=68$) calculated using the ${\mathrm{NNLO}}_{\mathrm{sim}}$(${\mathrm{\Lambda }}_{NN}$ = 500 MeV, $T_{\mathrm{Lab}}^{\max }$ = 290 MeV) and LO $YN$ (${\mathrm{\Lambda }}_{YN}$ = 600 MeV) interactions for several fixed values of the UV cutoff ${\mathrm{\Lambda }}_{\mathrm{UV}}$, together with their extrapolations (solid lines). Only the points marked by filled symbols, corresponding to particle-stable ${}_{\mathrm{\Lambda }}{}^3\mathrm{H}$ configurations, $E\left({}_{\mathrm{\Lambda }}{}^3\mathrm{H}\right)<E\left({}^2\mathrm{H}\right)$, are included in the fits. The world-average value of the measured ${}_{\mathrm{\Lambda }}{}^3\mathrm{H}$ energy $E^{\exp .}\left({}_{\mathrm{\Lambda }}{}^3\mathrm{H}\right)=-2.389\left(43\right)\mathrm{MeV}$ [6] is indicated by the gray band.

Figure 4

NCSM ${}_{\mathrm{\Lambda }}{}^3\mathrm{H}$ two-body ${\pi }^{-}$ rates $\mathrm{\Gamma }({}_{\mathrm{\Lambda }}{}^3\mathrm{H}\to {}^3\mathrm{He}+{\pi }^{-})$ (empty and filled symbols) as functions of the IR length $L_{\mathrm{IR}}$ (up to $N_{\max }=68$) calculated using the ${\mathrm{NNLO}}_{\mathrm{sim}}$(${\mathrm{\Lambda }}_{NN}$ = 500 MeV, $T_{\mathrm{Lab}}^{\max }$ = 290 MeV) and LO $YN$ (${\mathrm{\Lambda }}_{YN}$ = 600 MeV) interactions for several fixed values of the UV cutoff ${\mathrm{\Lambda }}_{\mathrm{UV}}$, together with their extrapolations (solid lines). The rates are evaluated using pion DW and considering the $\mathrm{\Sigma }NN$ contributions. Only the points marked by filled symbols, corresponding to particle-stable ${}_{\mathrm{\Lambda }}{}^3\mathrm{H}$ configurations, are included in the fits. The NCSM model-space parameters for ${}^3\mathrm{He}$ are fixed at $N_{\max }=36$ and $\hslash \omega =14\mathrm{MeV}$, corresponding to a well-converged wave function.

Figure 5

Probability densities ${\mathrm{p}}_{\alpha }\left(p_3\right)\equiv \int \mathrm{d}{p}_{12}{p}_{12}^2{p}_3^2{\left|{\psi }_{\alpha }(p_{12},p_3)\right|}^2$ of the active baryon ($N$ or $\mathrm{\Sigma }$) relative momentum $p_3$ for the $\alpha ={}^{31}S_0,l_3=0{}^3\mathrm{He}$ and the dominant $\mathrm{\Sigma }NN\alpha ={}^{31}S_0,l_3=0,t_3=1{}_{\mathrm{\Lambda }}{}^3\mathrm{H}$ wave-function components (top panel), together with the corresponding distributions ${\mathrm{p}}_{\alpha }\left(p_{12}\right)\equiv \int \mathrm{d}{p}_3{p}_3^2{p}_{12}^2{\left|{\psi }_{\alpha }(p_{12},p_3)\right|}^2$ of the $NN$ relative momentum $p_{12}$ (bottom panel). The densities are calculated using the ${\mathrm{NNLO}}_{\mathrm{sim}}$(${\mathrm{\Lambda }}_{NN}$ = 500 MeV, $T_{\mathrm{Lab}}^{\max }$ = 290 MeV) and LO $YN$ (${\mathrm{\Lambda }}_{YN}$ = 600 MeV) interactions. For ${}_{\mathrm{\Lambda }}{}^3\mathrm{H}$, each band corresponds to a particular value of the ${\mathrm{\Lambda }}_{\mathrm{UV}}$ cutoff, while its width shows the variation with the model-space truncation $N_{\max }$ for $60\le N_{\max }\le 68$. For ${}^3\mathrm{He}$, the NCSM model-space parameters are fixed at $N_{\max }=36$ and $\hslash \omega =14\mathrm{MeV}$, corresponding to a well-converged wave function. The ${\mathrm{p}}_{\alpha }\left(p_{12}\right)$ and ${\mathrm{p}}_{\alpha }\left(p_3\right)$ distributions in ${}^3\mathrm{He}$ are scaled by a factor of $1/250$.

Figure 6

Probability densities ${\mathrm{p}}_{\alpha }\left(p_3\right)$ as in Fig. 5 of the dominant $\alpha ={}^{13}S_1,l_3=0{}^3\mathrm{He}$ and $\alpha ={}^{13}S_1,l_3=t_3=0{}_{\mathrm{\Lambda }}{}^3\mathrm{H}$ wave-function components as functions of the active baryon ($N$ or $\mathrm{\Lambda }$) relative momentum $p_3$. Note that $\int \mathrm{d}{p}_3{\mathrm{p}}_{\alpha }\left(p_3\right)\approx 47\%$ and 96% for ${}^3\mathrm{He}$ and ${}_{\mathrm{\Lambda }}{}^3\mathrm{H}$, respectively.

Figure 7

Variation of the extrapolated $\mathrm{\Lambda }$ separation energies in ${}_{\mathrm{\Lambda }}{}^3\mathrm{H}$ and decay rates $\mathrm{\Gamma }({}_{\mathrm{\Lambda }}{}^3\mathrm{H}\to {}^3\mathrm{He}+{\pi }^{-})$ with the ${\mathrm{\Lambda }}_{NN}$, $T_{\mathrm{Lab}}^{\max }$ and ${\mathrm{\Lambda }}_{YN}$ cutoffs applied in the nuclear ${\mathrm{NNLO}}_{\mathrm{sim}}$ and LO $YN$ hypernuclear interactions. The rates are calculated including contributions from ${\pi }^{-}$ DW and $\mathrm{\Sigma }NN$ contributions for ${\mathrm{\Lambda }}_{\mathrm{UV}}=1200\mathrm{MeV}$. The energies and rates are computed for a fixed value of ${\mathrm{\Lambda }}_{YN}=600\mathrm{MeV}$ and all 42 ${\mathrm{NNLO}}_{\mathrm{sim}}$ Hamiltonians (circles); and all cutoffs ${\mathrm{\Lambda }}_{YN}=550,600,650,700\mathrm{MeV}$ for fixed values of $({\mathrm{\Lambda }}_{NN},T_{\mathrm{Lab}}^{\max })=(600,125)\mathrm{MeV}$ (squares), $(500,290)\mathrm{MeV}$ (triangles), and $(400,290)\mathrm{MeV}$ (diamonds). Results obtained for different values of the HO basis UV scale $800\le {\mathrm{\Lambda }}_{\mathrm{UV}}\le 1200\mathrm{MeV}$ with $({\mathrm{\Lambda }}_{NN},T_{\mathrm{Lab}}^{\max },{\mathrm{\Lambda }}_{YN})=(500,290,600)\mathrm{MeV}$ are marked by black crosses. See text for details.