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Encouraged by the global agreement between theoretical predictions andexperimental measurements for $B \to J/\psi V$ decays, we extend thatperturbative QCD formalism to $B_{d,s}^0 \to J/\psi f_0(500) [f_0(980)]$ decaysat the presently known next-to-leading order in the quark-antiquark descriptionof $f_0(500)$ and $f_0(980)$. With the angle $\phi_f \approx 25^\circ$ of the$f_0(500)-f_0(980)$ mixing in the quark-flavor basis, we find that thebranching ratios of the $B_d^0 \to J/\psi f_0(500) (\to \pi^+ \pi^-)$ and$B_{d,s}^0 \to J/\psi f_0(980) (\to \pi^+ \pi^-)$ modes generally agree withthe current data or the upper limits within uncertainties, except for theseemingly challenging $B_s^0 \to J/\psi f_0(500) (\to \pi^+\pi^-)$ one. Then,we further explore the relevant observables of the $B_{d,s}^0 \to J/\psif_0(500) [f_0(980)]$ decays, which could provide further constraints on themixing angle $\phi_f$ and/or SU(3) flavor symmetry breaking effects. As abyproduct, we predict ${\rm BR}(B_{d}^0 \to J/\psi f_0(980)(\to K^+K^-))=5.8^{+3.1}_{-2.9} \times 10^{-7}$ and ${\rm BR}(B_{s}^0 \to J/\psif_0(980)(\to K^+ K^-)) =4.6^{+2.6}_{-2.3} \times 10^{-5}$. All theoreticalpredictions await the future examinations with high precision.

Phenomenological studies on the $B_{d,s}^0 \to J/\psi f_0(500) [f_0(980)]$ decays