Structure of ${}_{\mathrm{\Lambda }}^{10}$Be and ${}_{\mathrm{\Lambda }}^{10}$B hypernuclei studied with the four-body cluster model

The structure of the isodoublet hypernuclei, ${}_{\mathrm{\Lambda }}^{10}$B and ${}_{\mathrm{\Lambda }}^{10}$Be within the framework of an $\alpha +\alpha +\mathrm{\Lambda }+N$ four-body cluster model is studied. Interactions between the constituent subunits are determined so as to reproduce reasonably well the observed low-energy properties of the $\alpha \alpha$, $\alpha N$, $\alpha \mathrm{\Lambda }$, $\alpha \alpha \mathrm{\Lambda }$ and $\alpha \alpha N$ subsystems. Furthermore, the two-body $\mathrm{\Lambda }N$ interaction is adjusted so as to reproduce the $0^{+}$-$1^{+}$ splitting of ${}_{\mathrm{\Lambda }}^4$H. The $\mathrm{\Lambda }$ binding energies of ${}_{\mathrm{\Lambda }}^{10}$B and ${}_{\mathrm{\Lambda }}^{10}$Be are 8.76 MeV and 8.94 MeV, respectively. The energy splitting of the $1^{-}$-$2^{-}$ levels in ${}_{\mathrm{\Lambda }}^{10}$B is 0.08 MeV, which does not contradict the experimental report in BNL-E930. An even-state $\mathrm{\Lambda }N$ charge symmetry breaking (CSB) interaction determined from the A=4 systems works repulsively by +0.1 MeV (attractively by -0.1 MeV) in ${}_{\mathrm{\Lambda }}^{10}$Be (${}_{\mathrm{\Lambda }}^{10}$B). We discuss a possibility that an odd-state CSB interaction improves the fitting to the experimental data of A=10 double $\mathrm{\Lambda }$ hypernuclei.