Contributions of the subprocesses $ρ(770,1450,1700)\to K \bar{K}$ and $ω(782,1420,1650)\to K \bar{K}$ for the three-body decays $B\to η^{(\prime)} K\bar{K}$

TL;DR

This work addresses how to account for resonance contributions to three-body decays $B\to \eta^{(\prime)} K\bar{K}$ within perturbative QCD by incorporating intermediate $\rho$ and $\omega$ states into the kaon-pair system through the kaon vector timelike form factors $F_K(s)$. The analysis constructs a quasi-two-body framework where the kaon pair dynamics are described by these form factors, embedding $\rho$ and $\omega$ subprocesses via Breit-Wigner components. CP-averaged branching fractions and direct CP asymmetries for the decays through $\rho(770,1450,1700)$ and $\omega(782,1420,1650)$ are calculated, revealing that the BW tails of $\rho(770)$ and $\omega(782)$ are on par with or exceed contributions from higher resonances, making them significant in the $K\bar{K}$ system. The study provides predictions with quantified uncertainties that can be tested by Belle II and LHCb, highlighting the importance of including all relevant resonant contributions and the current lack of measured relative phases among the $K\bar{K}$ form-factor components.

Abstract

As an extension of our prior work, we analyze the resonance contributions for the kaon pair originating from the intermediate $ρ(770)$, $ω(782)$ and their excited states in the three-body decays $B\to η^{(\prime)} K\bar{K}$ within the perturbative QCD approach. The information of subprocesses $ρ(770,1450,1700)\to K\bar K$ and $ω(782,1420,1650)\to K\bar K$ are included in the distribution amplitudes for $K\bar K$ system by using the kaon vector time-like form factors. We calculate the $CP$ averaged branching fractions and the direct $CP$ asymmetries for the relevant quasi-two-body $B$ meson decays. The branching fractions of the virtual contributions for $K\bar K$ from the Breit-Wigner formula tails of $ρ(770)$ and $ω(782)$ for these decays are found comparable to the corresponding contributions from the resonances $ρ(1450,1700)$ and $ω(1420,1650)$. Consequently, they constitute a significant component that should be accounted for in the considered three-body decays. All the predictions in this work are expected to be tested by the LHCb and Belle-II experiments in the future.

Figures (3)

• Figure 1: Schematic view of the cascade decays $B\to\eta^{(\prime)} \rho/\omega \to \eta^{(\prime)}K\bar{K}$, where $\rho/\omega$ stands for the intermediate states $\rho(770,1450,1700)$ or $\omega(782,1420,1650)$ which will decay into kaon pair in this work.
• Figure 2: The differential branching fractions for the decays $B^+\to \eta [\rho(770)^+\to] K^+\bar{K}^0$ and $B^+\to \eta [\rho(770)^+\to] \pi^+\pi^0$. The big diagram is for the comparison for the differential branching fractions of $B^+\to \eta [\rho(770)^+\to] K^+\bar{K}^0$ and $B^+\to \eta [\rho(770)^+\to] \pi^+\pi^0$, in which the solid line for $B^+\to \eta [\rho(770)^+\to] K^+\bar{K}^0$ is magnified by a factor of $10$.
• Figure 3: Typical Feynman diagrams for the processes $B\to\eta_{q(s)} R \to \eta_{q(s)} K\bar{K}$, with $R$ represents the resonances $\rho$, $\omega$ and their excited states. The dots on the quarks connecting the weak vertex $\otimes$ are the switchable vertices for the hard gluons.