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Background: Non-yrast states in neutron-rich nuclei are being investigatedexperimentally. These states reveal various aspects and details of the nuclearstructure, such as the fluctuation around the axially symmetric shape. Purpose:The beyond-mean-field effects in neutron-rich nuclei with $N \simeq 28$ areinvestigated. We focus on the role of collective mass in triaxial-shapedynamics. Method: We employ the five-dimensional quadrupole collectiveHamiltonian method with the potential obtained in a constrainedHartree--Fock--Bogoliubov approach with a Skyrme energy-density functional andthe collective-mass functions obtained by the cranking approximation. Themethod includes triaxial deformations. Results: We find that $^{42}$Mg,$^{40}$Si, $^{44}$S, and $^{46}$S show $\gamma$-soft: A flat behavior in thepotential energy surface along the triaxial deformation. Their low-lyingspectra show a strong nucleus dependence, while those obtained with acollective mass assumed as constant are similar to each other. The energy ratio$E(0_2^+)/E(2_1^+)$ and the $B(E2)$ ratio $B(E2;0_2^+\to 2_1^+)/B(E2;2_1^+\to0_1^+)$ show a unique property of the $0_2^+$ state, while the energy and$B(E2)$ ratios in neutron-deficient $\gamma$-soft nuclei with $N=78$ do notdepend on nucleus so much. Conclusions: Low-lying spectra are determined by notonly the potential energy but also the collective mass. We clarify theimportant role of the collective mass in low-energy dynamics in theneutron-rich $N\sim28$ nuclei.

Triaxial-shape dynamics in the low-lying excited $0^+$ state: Role of the collective mass