Inversion doublets of 3N+N cluster structure in excited states of $^4$He

Excited states of $^4$He are studied in four-body calculations with explicitly correlated Gaussian bases. All the levels below $E_{\rm x}$=26 MeV are reproduced reasonably well using realistic potentials. An analysis is made to show how the $0^+_2$ state becomes a resonance but those having almost the same structure as this state in different spin-isospin channels are not observed as resonances. The role of tensor force is stressed with a particular attention to the level spacing between the two $0^-$ states. The calculation of spectroscopic amplitudes, nucleon decay widths, and spin-dipole transition strengths demonstrates that the $0^+_2$ state and the three lowest-lying negative-parity states with $0^-$ and $2^-$ have $^3$H+$p$ and $^3$He+$n$ cluster configurations, leading to the interpretation that these negative-parity states are the inversion doublet partners of the $0^+_2$ state.