  In this study, we investigate the mass spectrum of $\pi$ and $\sigma$ mesons at finite chemical potential using the self-consistent NJL model and the Fierz-transformed interaction Lagrangian. The model introduces an arbitrary parameter $\alpha$ to reflect the weights of the Fierz-transformed interaction channels. We show that when $\alpha$ exceeds a certain threshold value, the chiral phase transition transforms from a first-order one to a smooth crossover, which is evident from the behaviors of the chiral condensates and meson masses. Additionally, at high chemical potential, the smaller the value of $\alpha$, the higher the masses of the $\pi$ and $\sigma$ mesons become. Moreover, the Mott and dissociation chemical potentials both increase with the increase in $\alpha$. Thus, the meson mass emerges as a valuable experimental observable for determining the value of $\alpha$ and investigating the properties of the chiral phase transition in dense QCD matter. 
(pseudo)Scalar mesons in a self-consistent NJL model