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 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 in doublet bands M1 and M4.
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