  The open problem on whether or not the chirality exists in doublet bands M1
and M4 in light-mass even-even nucleus $^{60}$Ni is studied by adopting the
recently developed fully quantal four-$j$ shells triaxial particle rotor model.
The corresponding experimental energy spectra, energy differences between
doublet bands, and the available $B(M1)/B(E2)$ values are successfully
reproduced. The analyses on the basis of the angular momentum components, the
azimuthal plots, and the $K$-plots suggest that the chiral modes exist at
$I\geq 12\hbar$ in doublet bands M1 and M4.
