  Background: Non-yrast states in neutron-rich nuclei are being investigated
experimentally. These states reveal various aspects and details of the nuclear
structure, such as the fluctuation around the axially symmetric shape. Purpose:
The beyond-mean-field effects in neutron-rich nuclei with $N \simeq 28$ are
investigated. We focus on the role of collective mass in triaxial-shape
dynamics. Method: We employ the five-dimensional quadrupole collective
Hamiltonian method with the potential obtained in a constrained
Hartree--Fock--Bogoliubov approach with a Skyrme energy-density functional and
the collective-mass functions obtained by the cranking approximation. The
method includes triaxial deformations. Results: We find that $^{42}$Mg,
$^{40}$Si, $^{44}$S, and $^{46}$S show $\gamma$-soft: A flat behavior in the
potential energy surface along the triaxial deformation. Their low-lying
spectra show a strong nucleus dependence, while those obtained with a
collective mass assumed as constant are similar to each other. The energy ratio
$E(0_2^+)/E(2_1^+)$ and the $B(E2)$ ratio $B(E2;0_2^+\to 2_1^+)/B(E2;2_1^+\to
0_1^+)$ 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 not
depend on nucleus so much. Conclusions: Low-lying spectra are determined by not
only the potential energy but also the collective mass. We clarify the
important role of the collective mass in low-energy dynamics in the
neutron-rich $N\sim28$ nuclei.
