$B^0$ and $B^0_s$ decays into $J/\psi$ $f_0(980)$ and $J/\psi$ $f_0(500)$ and the nature of the scalar resonances

We describe the $B^0$ and $B^0_s$ decays into $J/\psi$ $f_0(500)$ and $J/\psi$ $f_0(980)$ by taking into account the dominant process for the weak decay of $B^0$ and $B^0_s$ into $J/\psi$ and a $q \bar q$ component. After hadronization of this $q \bar q$ component into pairs of pseudoscalar mesons we obtain certain weights for the meson-meson components and allow them to interact among themselves. The final state interaction of the meson-meson components, described in terms of chiral unitary theory, gives rise to the $f_0(980)$ and $f_0(500)$ resonances and we can obtain the $\pi^+ \pi^- $ invariant mass distributions after the decay of the resonances, which allows us to compare directly to the experiments. We obtain ratios of $J/\psi$ $f_0(980)$ and $J/\psi$ $f_0(500)$ for each of the $B$ decays in quantitative agreement with experiment, with the $f_0(980)$ clearly dominant in the $B^0_s$ decay and the $f_0(500)$ in the $B^0$ decay.