Production of a neutron-rich ${}_{\mathrm{\Lambda }}{}^6\mathrm{H}$ hypernucleus in the ${}^6\mathrm{Li}({\pi }^{-},K^{+})$ reaction

We study phenomenologically the production of the neutron-rich hypernucleus ${}_{\mathrm{\Lambda }}{}^6\mathrm{H}$ in the ${}^6\mathrm{Li}({\pi }^{-},K^{+})$ reaction at 1.2 GeV/$c$, using a distorted-wave impulse approximation in a one-step mechanism, ${\pi }^{-}p\to K^{+}{\mathrm{\Sigma }}^{-}$ via ${\mathrm{\Sigma }}^{-}$ doorways caused by ${\mathrm{\Sigma }}^{-}p\leftrightarrow \mathrm{\Lambda }n$ coupling. The production cross section of ${}_{\mathrm{\Lambda }}{}^6\mathrm{H}\left(1_{\mathrm{exc}.}^{+}\right)$ is evaluated by a coupled $({}^5\mathrm{H}-\mathrm{\Lambda })+({}^5\mathrm{He}-{\mathrm{\Sigma }}^{-})$ model with a spreading potential, in comparison with the data of the missing mass spectrum at the J-PARC E10 experiment. The result indicates that the ${\mathrm{\Sigma }}^{-}$ mixing probabilities in ${}_{\mathrm{\Lambda }}{}^6\mathrm{H}\left(1_{\mathrm{exc}.}^{+}\right)$ are $P_{{\mathrm{\Sigma }}^{-}}<\sim 0.2\%$ both for $s_{\mathrm{\Sigma }}$ state and for $p_{\mathrm{\Sigma }}$ state in order to reproduce no significant peak in the $\mathrm{\Lambda }$ production data, so that the cross section of ${}_{\mathrm{\Lambda }}{}^6\mathrm{H}$ is less than on the order of 0.4 nb/sr. The sensitivity of the $\mathrm{\Sigma }\mathrm{\Lambda }$ coupling and $\mathrm{\Lambda }$ potentials to the near-$\mathrm{\Lambda }$-threshold spectrum is discussed. The shape and magnitude of the spectrum provide valuable information on the $\mathrm{\Sigma }\mathrm{\Lambda }$ coupling in the production mechanism and also the nuclear structure of ${}_{\mathrm{\Lambda }}{}^6\mathrm{H}\left(1_{\mathrm{exc}.}^{+}\right)$.

Figure 1

Diagrams of a one-step mechanism, ${\pi }^{-}p\to K^{+}{\mathrm{\Sigma }}^{-}$ via ${\mathrm{\Sigma }}^{-}$ doorways caused by the ${\mathrm{\Sigma }}^{-}p\leftrightarrow \mathrm{\Lambda }n$ coupling, for production of $\mathrm{\Lambda }$ hypernuclear states by the DCX nuclear $({\pi }^{-},K^{+})$ reactions.

Figure 2

Energy spectrum and decay threshold of the ${}_{\mathrm{\Lambda }}{}^6\mathrm{H}$ hypernucleus. Binding energies of $B_{\mathrm{\Lambda }}\left({}_{\mathrm{\Lambda }}{}^6\mathrm{H}\right)$ and $B_{2n}\left({}_{\mathrm{\Lambda }}{}^6\mathrm{H}\right)$ are defined with respect to ${}^5\mathrm{H}+\mathrm{\Lambda }$ and ${}_{\mathrm{\Lambda }}{}^4\mathrm{H}\left(1^{+}\right)+2n$ thresholds, respectively. The threshold-energy difference between ${}^5\mathrm{H}(1/2_{\mathrm{g}.\mathrm{s}.}^{+})$ and the ${}^3\mathrm{H}+2n$ threshold is assumed to be 4.0 MeV.

Figure 3

Schematic illustration of shell-model configurations for (a) $p_p{s}_{{\mathrm{\Sigma }}^{-}}\to p_n{s}_{\mathrm{\Lambda }}$ transitions from $s_p^{-1}{s}_{\mathrm{\Sigma }}$ components, and (b) $s_p{p}_{{\mathrm{\Sigma }}^{-}}\to p_n{s}_{\mathrm{\Lambda }}$ transitions from $p_p^{-1}{p}_{\mathrm{\Sigma }}$ components in ${}_{\mathrm{\Lambda }}{}^6\mathrm{H}\left(1_{\mathrm{exc}.}^{+}\right)$.

Figure 4

Calculated $\mathrm{\Sigma }\mathrm{\Lambda }$ coupling potentials $U_X\left(r\right)$ between $[{}^5\mathrm{He}\left(J_C^{\pi }\right)\otimes {\left(\ell j\right)}_{{\mathrm{\Sigma }}^{-}}]$ and $[{}^5\mathrm{H}(1/2_{\mathrm{g}.\mathrm{s}.}^{+})\otimes {\left(s_{1/2}\right)}_{\mathrm{\Lambda }}]$ with $\mathrm{\Delta }\ell =|{\ell }_{{\mathrm{\Sigma }}^{-}}-{\ell }_{\mathrm{\Lambda }}|=0$, 1, 2 in ${}_{\mathrm{\Lambda }}{}^6\mathrm{H}\left(1_{\mathrm{exc}.}^{+}\right)$ at $E_{\mathrm{\Lambda }}=0$ MeV in Eq. (11), as a function of the relative distance between ${}^5\mathrm{H}\left({}^5\mathrm{He}\right)$ and $\mathrm{\Lambda }\left({\mathrm{\Sigma }}^{-}\right)$. ${\tilde{v}}_{\mathrm{\Sigma }N,\mathrm{\Lambda }N}^1=-900$ MeV and ${\tilde{v}}_{\mathrm{\Sigma }N,\mathrm{\Lambda }N}^0=500$ MeV are used. The dot-dashed curve denote the ${}^5\mathrm{H}-\mathrm{\Lambda }$ potential as a guide.

Figure 5

Single-particle density distributions of $\mathrm{\Lambda }$ and ${\mathrm{\Sigma }}^{-}$ in ${}_{\mathrm{\Lambda }}{}^6\mathrm{H}\left(1_{\mathrm{exc}.}^{+}\right)$, ${\rho }_{{\alpha }_Y}\left(r\right)$ with $\alpha =\left\{n\ell j\right\}$ and $Y=\mathrm{\Lambda },{\mathrm{\Sigma }}^{-}$, as a function of the relative distance between ${}^5\mathrm{H}\left({}^5\mathrm{He}\right)$ and $\mathrm{\Lambda }\left({\mathrm{\Sigma }}^{-}\right)$. The $\mathrm{\Sigma }\mathrm{\Lambda }$ coupling potentials given in Fig. 4 are used. Solid, dashed, and dotted curves denote the components of the hyperon densities in ${\left(s_{1/2}\right)}_Y$, ${\left(p_{3/2,1/2}\right)}_Y$, and ${\left(d_{5/2}\right)}_Y$ orbits, respectively.

Figure 6

Calculated missing mass spectra of the ${}^6\mathrm{Li}({\pi }^{-},K^{+})$ reactions near the $\mathrm{\Lambda }$ threshold at 1.2 GeV/$c$ and ${\theta }_{\mathrm{lab}}=7^{\circ }$, with a detector resolution of 3.2 MeV FWHM. The $\mathrm{\Sigma }\mathrm{\Lambda }$ coupling strengths of ${\tilde{v}}_{\mathrm{\Sigma }N,\mathrm{\Lambda }N}^1=$ (a) $-1800$, (b) $-1350$, (c) $-900$, and (d) $-450$ MeV [${\tilde{v}}_{\mathrm{\Sigma }N,\mathrm{\Lambda }N}^0=$ (a) 1000, (b) 750, (c) 500, and (d) 250 MeV] are used, together with $V_{\mathrm{\Lambda }}=-19$ MeV for the $\mathrm{\Lambda }-{}^5\mathrm{H}$ potential. Solid, dashed and dot-dashed curves denote contribution of total, $p$-hole, and $s$-hole spectra, respectively. The data are taken from Ref. [1]. The bins with a finite width of 1 MeV denote the cross sections for ${}_{\mathrm{\Lambda }}{}^6\mathrm{H}\left(1_{\mathrm{exc}.}^{+}\right)$ which is located in particle decay channels.

Figure 7

Dependence of the calculated inclusive $\mathrm{\Lambda }$ spectrum in the ${}^6\mathrm{Li}({\pi }^{-},K^{+})$ reaction at $p_{{\pi }^{-}}=1.2\text{GeV}/c\left(7^{\circ }\right)$ on various strengths of $V_{\mathrm{\Lambda }}$, together with the experimental data [1]. Solid curves denote the spectra by $V_{\mathrm{\Lambda }}=-28$, $-24$, $-19$, and $-11$ MeV when ${\tilde{v}}_{\mathrm{\Sigma }N,\mathrm{\Lambda }N}^1=-900$ MeV and ${\tilde{v}}_{\mathrm{\Sigma }N,\mathrm{\Lambda }N}^0=500$ MeV with a detector resolution of 3.2 MeV FWHM.

Figure 8

Comparison between the calculated inclusive $\mathrm{\Lambda }$ spectrum of $E_r=4.0$ MeV and that of $E_r=1.7$ MeV for the energy of ${}^5\mathrm{H}(1/2_{\mathrm{g}.\mathrm{s}.}^{+})$ in the ${}^6\mathrm{Li}({\pi }^{-},K^{+})$ reaction at $p_{{\pi }^{-}}=1.2\text{GeV}/c\left(7^{\circ }\right)$. $V_{\mathrm{\Lambda }}=-19$ MeV is used. See also the caption in Fig. 7.