Study of four-body decays $B_{(s)}\to (\pi \pi )(\pi \pi )$ in the perturbative QCD approach

Abstract

In this work, we make a systematical study on the four-body $B_{(s)}\to (\pi \pi )(\pi \pi )$ decays in the perturbative QCD approach, where the $\pi \pi$ invariant mass spectra are dominated by the vector resonance $\rho (770)$ and the scalar resonance $f_0(980)$. We improve the Gengenbauer moments for the longitudinal $P$-wave two-pion distribution amplitudes (DAs) by fitting the PQCD factorization formulas to measured branching ratios of three-body and four-body $B$ decays. With the fitted Gegenbauer moments, we make predictions for the branching ratios and direct $CP$ asymmetries of four-body $B_{(s)}\to (\pi \pi )(\pi \pi )$ decays. We extract the branching ratios of two-body $B_{(s)}\to \rho \rho$ from the corresponding four-body decay modes and calculate the relevant polarization fractions. We find that the $\mathcal{B}(B^0\to {\rho }^{+}{\rho }^{-})$ is consistent with the previous theoretical predictions and data. The leading-order PQCD calculations of the $\mathcal{B}(B^{+}\to {\rho }^{+}{\rho }^0)$, $\mathcal{B}(B^0\to {\rho }^0{\rho }^0)$ and the $f_0(B^0\to {\rho }^0{\rho }^0)$ are a bit lower than the experimental measurements, which should be further examined. In addition, the "true" and "fake" triple-product asymmetries (TPAs) in the $B_{(s)}\to (\pi \pi )(\pi \pi )$ decays are also analyzed. The sizable averaged TPA ${\mathcal{A}}_{\text{T-true}}^{1,\text{ave}}=25.26\%$ of the color-suppressed decay $B^0\to {\rho }^0{\rho }^0\to ({\pi }^{+}{\pi }^{-})({\pi }^{+}{\pi }^{-})$ is predicted for the first time, which deviates a lot from the so-called "true" TPA ${\mathcal{A}}_{\text{T-true}}^1=7.92\%$ due to the large direct $CP$ violation. A large "fake" TPA ${\mathcal{A}}_{\text{T-fake}}^1=24.96\%$ of the decay $B^0\to {\rho }^0{\rho }^0\to ({\pi }^{+}{\pi }^{-})({\pi }^{+}{\pi }^{-})$ is also found, which indicates the significance of the final-state interactions. The predictions in this work can be tested by LHCb and Belle-II experiments in the near future.

Comments: 24 pages, 4 figures. Several new references are added. arXiv admin note: text overlap with arXiv:2204.01092, arXiv:2107.10684