Effect of the variation of mass on fermion localization on thick branes

In a recent paper published in this journal, Zhao and collaborators [Phys. Rev. D \textbf{82}, 084030 (2010)] analyze a toy model of thick branes generated by a real scalar field with the potential $V(\phi)=a\phi^{2}-b\phi^{4}+c\phi^{6}$, and investigate the variation of the mass parameter $a$ on the branes as well as the localization and resonances of fermions. In that research the simplest Yukawa coupling $\eta\bar{\Psi}\phi\Psi$ was considered. In that work does not analyze the zero mode in details and also contains some misconceptions. In this paper, the effect of the variation of the mass parameter $a$ on the brane is reinvestigated and it is associated to the phenomenon of brane splitting. Furthermore, it is shown that the zero mode for the left-handed fermions can be localized on the brane depending on the values for the coupling constant $\eta$ and the mass parameter $a$.