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The scalars $f_0$ closest to 1.5 GeV contain the mesons $f_0(1370)$,$f_0(1500)$ and $f_0(1710)$, and the latter two ones are usually viewed as thepotential candidates for the scalar glueballs. In this work, by including theimportant contributions from the vertex corrections, we study the decays $B^0\to J/\psi f_0$ within the improved perturbative QCD approach and analyze thepossible scalar glueball hunting. Together with the two mixing models, namely,$f_0(1500) (f_0(1710))$ being the primary scalar glueball in model I (II), andtwo classification scenarios, namely, $f_0$ being the $q\bar q$ excited(ground) states in scenario 1 (2), the branching fractions associated withtheir ratios for $B^0 \to J/\psi f_0$ are evaluated comprehensively. Thepredictions with still large uncertainties in the considered two mixing modelsare roughly consistent with currently limited data, which indicates that bothmore rich data and more precise predictions are urgently demanded to figure outthe scalar glueball clearly in the future. Moreover, several interesting ratiosbetween the branching fractions of $B^0 \to J/\psi f_0(\to \pi^+ \pi^-/K^+K^-)$ and $B^0 \to J/\psi \rho^0/\phi (\to \pi^+ \pi^-/ K^+ K^-)$ that couldhelp us to understand the nature of scalar $f_0$ are defined and predictedtheoretically. These ratios should be examined in future experiments.

The $B^0 \to J/\psi f_0(1370,1500,1710)$ decays: an opportunity for scalar glueball hunting