a double hadronization compared to the single hadronization of Fig. \[Fig:feyn\] (B). The hadronization reverts into a decreased rate for two meson production compared to the single meson of the original $q\bar{q}$, which we can estimate in about one order of magnitude from the experimental rate [@pdg; @LHCb:2012ae] (see Ref. [@weihong] for details), $$\begin{aligned}
\frac{\Gamma (\bar{B}^0_s \to J/\psi f_0(980); f_0(980) \to
\pi^+\pi^-)}{\Gamma (\bar{B}^0_s \to J/\psi \phi)} = 0.14 . \nonumber \\\end{aligned}$$

In the literature there is much discussions about the relevance of the $W$-exchange mechanism. In Ref. [@Cheng:2010vk] an empirical analysis is done based on giving a weight to the different topological mechanisms, and the $W$-exchange mechanism (evaluated under the assumption that the $f_0$ and $a_0$ resonances are $q\bar{q}$ or tetraquark states) appears of the same order of the internal conversion, with opposite sign, that makes the $C-E$ combination in $a_0$ production bigger than the $C+E$ combination in $f_0$ production. [^3] However, in the same paper, a factorization approach is followed (see section V of Ref. [@Cheng:2010vk]) in which the $W$-exchange contribution is claimed to be suppressed and is neglected in that approach. The present work neglects the $W$-exchange mechanism and produces a large $a_0(980)$ production relative to $f_0(980)$ due 