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Perturbative QCD analysis of neutral $B$-meson decays into $σσ, σf_0$ and $f_0 f_0$

The decays of $B_{d,s}^0 \to σσ, σf_0, f_0 f_0$, with $σ$ and $f_0$ denoting the light scalar mesons $f_0(500)$ and $f_0(980)$ in the two-quark picture, are studied in the perturbative QCD approach based on $k_T$ factorization. With the referenced value of the mixing angle $|φ| \sim 25^\circ$ for the $σ-f_0$ mixing in the quark-flavor basis, it is of great interest to obtain that: (a) these neutral $B$-meson decays into $σσ, σf_0$, and $f_0 f_0$ have large branching ratios in the order of $10^{-6} \sim 10^{-4}$, which mean the possibly constructive interferences existed in the decays with different flavor states, and then are expected to be tested at the Large Hadron Collider beauty and/or Belle-II experiments in the (near) future; (b) the large direct CP violations could be easily found in the $B_d^0 \to σσ, f_0 f_0$ and $B_{d,s}^0 \to σf_0$ decays, which indicate the considerable interferences between the tree and the penguin decay amplitudes involved in these four modes, and would be confronted with the future measurements; (c) these neutral $B$-meson decays could be examined through the secondary decay chain $σ/f_0 \to π^+ π^-$, namely, the four-body decays of $B_{d,s}^0 \to (π^+ π^-)_{σ(f_0)} (π^+ π^-)_{σ(f_0)}$. On the other side, it seems that other 4 four-body decays of $B_d^0 \to (π^+ π^-)_σ (K^+ K^-)_{f_0}$, $B_s^0 \to (π^+ π^-)_{σ(f_0)} (K^+ K^-)_{f_0}$, and $B_s^0 \to (K^+ K^-)_{f_0} (K^+ K^-)_{f_0}$ could also be detected at the relevant experiments, if the $f_0 \to K^+ K^-$ could be identified from the $ϕ\to K^+ K^-$ clearly.