$E(5)$-like emerging $\gamma$-softness in $^{82}$Kr

Recently the interacting bosom model with $SU(3)$ higher-order interactions was proposed by one of the authors, wherein unexpected $\gamma$-softness can be emerged in this new model. This stimulates further discussion on the connections of the new $\gamma$-softness and the realistic $\gamma$-soft nuclei. In this paper, $E(5)$-like $\gamma$-softness arises when the $SU(3)$ fourth-order interaction $\hat{C}_{2}^{2}[SU(3)]$ is considered. And the corresponding transitional behaviors are similar to that from the $U(5)$ limit to the $O(6)$ limit in previous IBM-1, which provides a novel perspective for understanding the new model. Low-lying spectra, $B(E2)$ values between the low-lying states and quadrupole moment of the first $2_{1}^{+}$ state are investigated. Experimental data of the $E(5)$-like nucleus $^{82}$Kr are compared with the calculated theoretical results, where the calculated low-lying level energies and the associated $B(E2)$ values fit very well with the experimental data.