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

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