Contributions for the kaon pair from $\rho(770)$, $\omega(782)$ and their excited states in the $B\to K\bar{K}h$ decays

Charmless three-body hadronic $B$ meson decays provide us a field to investigate different aspects of weak and strong interactions. The underlying weak decay for $b$-quark is simple which can be described well by the effective Hamiltonian rmp68-1125 , but the strong dynamics in these three-body processes is very complicated, owing to the hadron-hadron interactions, the three-body effects npps199-341 ; prd84-094001 and the rescattering processes 1512-09284 ; prd89-094013 ; epjc78-897 ; prd71-074016 in the final states, and also on account of the resonant contributions which are related to the scalar, vector and tensor resonances and are commonly described by the relativistic Breit-Wigner (BW) formula BW-model as well as the nonresonant contributions which are the rest at the amplitude level for the relevant decay processes. The experimental efforts for the three-body $B$ decays by employing Dalitz plot technique prd94-1046 within the isobar formalism pr135-B551 ; pr166-1731 ; prd11-3165 have revealed valuable information on involved strong and weak dynamics. But a priori model with all reliable and correct strong dynamical components is needed for the Dalitz plot analyses plb665-30 . The expressions of the decay amplitudes for those three-body decays without or have wrong factors for certain intermediate states will have negative impacts on the observables such as the branching fractions and $CP$ violations for the relevant decay processes.

Recently, in the amplitude analysis of the three-body decays $B^{\pm}\to\pi^{\pm}K^{+}K^{-}$, LHCb Collaboration reported an unexpected large fit fraction $(30.7\pm 1.2\pm 0.9)\%$ in Ref. prl123-231802 for the resonance $\rho(1450)^{0}$ decaying into charged kaon pair. This fit fraction implies a branching fraction $(1.60\pm 0.14)\times 10^{-6}$ for the quasi-two-body decay $B^{+}\to\pi^{+}\rho(1450)^{0}\to\pi^{+}K^{+}K^{-}$ PDG-2020 , in view of the branching fractions $(5.38\pm 0.40\pm 0.35)\times 10^{-6}$ from Belle prd96-031101 and $(5.0\pm 0.5\pm 0.5)\times 10^{-6}$ presented by BaBar prl99-221801 for the $B^{+}\to K^{+}K^{-}\pi^{+}$ decays. While in the $\rho$ dominant decay modes $B^{\pm}\to\pi^{\pm}\pi^{+}\pi^{-}$, the contribution for $\pi^{+}\pi^{-}$ pair from the intermediate state $\rho(1450)^{0}$ was found to be small but consistent with the theoretical expectation prd96-036014 by LHCb in their recent works prl124-031801 ; prd101-012006 .

In Ref. prd101-111901 , within flavour $SU(3)$ symmetry, we predicted the branching fraction for $B^{+}\to\pi^{+}\rho(1450)^{0}\to\pi^{+}K^{+}K^{-}$ to be about one tenth of that for the decay $B^{+}\to\pi^{+}\rho(1450)^{0}\to\pi^{+}\pi^{+}\pi^{-}$ and much smaller than the corresponding result in prl123-231802 ; PDG-2020 , and our prediction got the supports from the theoretical analyses in Ref. 2007-02558 . In addition, the virtual contribution plb791-342 ; prd69-112002 ; prd79-112004 ; prd91-092002 ; prd94-072001 for $K^{+}K^{-}$ from the Breit-Wigner (BW) formula BW-model tail of the resonance $\rho(770)^{0}$ which has been ignored by the experimental analysis was found to be the same order but larger than the contribution of $\rho(1450)^{0}\to K^{+}K^{-}$ prd101-111901 . In this work, we shall systematically study the contributions for the kaon pair from the resonances $\rho(770,1450,1700)$ and $\omega(782,1420,1650)$ in the $B\to K\bar{K}h$ decays within the perturbative QCD (PQCD) approach plb504-6 ; prd63-054008 ; prd63-074009 ; ppnp51-85 , where $h$ is the bachelor state pion or kaon. As for the other $J^{PC}=1^{--}$ isovector resonances, like $\rho(1570)$, $\rho(1900)$ and $\rho(2150)$, we will leave their possible contributions for kaon pair to the future studies in view of their ambiguous nature PDG-2020 .

The contributions for $K\bar{K}$ from the tails of $\rho(770)$ and $\omega(782)$ in the charmless three-body hadronic $B$ meson decays have been ignored in both the theoretical studies and the experimental works. But in the processes of $\pi^{-}p\to K^{-}K^{+}n$ and $\pi^{+}n\to K^{-}K^{+}p$ prd15-3196 ; prd22-2595 , $\bar{p}p\to K^{+}K^{-}\pi^{0}$ plb468-178 ; epjc80-453 , $e^{+}e^{-}\to K^{+}K^{-}$ pl99b-257 ; pl107b-297 ; plb669-217 ; prd76-072012 ; prd88-032013 ; prd94-112006 ; plb779-64 ; prd99-032001 ; zpc39-13 and $e^{+}e^{-}\to K^{0}_{S}K^{0}_{L}$ pl99b-261 ; prd63-072002 ; plb551-27 ; plb760-314 ; prd89-092002 ; jetp103-720 , the resonances $\rho(770)$ and $\omega(782)$ along with their excited states are indispensable for the formation of the kaon pair. In addition, the resonances $\rho(770,1450)^{\pm}$ are the important intermediate states for the $K^{\pm}K^{0}_{S}$ pair in the final state of hadronic $\tau$ decays prd98-032010 ; prd89-072009 ; prd53-6037 ; epjc79-436 . The subprocesses $\rho(1450,1700)\to K\bar{K}$ be concerned for the decay $J/\psi\to K^{+}K^{-}\pi^{0}$ in Refs. prd76-094016 ; prd75-074017 ; prd95-072007 ; prd100-032004 could be mainly attributed to the observation of a resonant broad structure around $1.5$ GeV in the $K^{+}K^{-}$ mass spectrum in prl97-142002 . While for the decays $B\to KKK$ prd65-092005 ; prd71-092003 ; prd74-032003 ; prl99-161802 ; prd82-073011 ; prd85-112010 and $B\to KK\pi$ prd87-091101 ; prl99-221801 , the unsettled $f_{X}(1500)$ which decaying into $K^{+}K^{-}$ channel could probably be related to the resonance $\rho(1450)^{0}$ 2007-13141 .

For the three-body decays $B\to K\bar{K}h$, the subprocesses $\rho\to K\bar{K}$ and $\omega\to K\bar{K}$ can not be calculated in the PQCD approach and will be introduced into the distribution amplitudes of the $K\bar{K}$ system via the kaon vector time-like form factors. The intermediate $\rho(770)$, $\omega(782)$ resonances and their excited states are generated in the hadronization of the light quark-antiquark pair $q\bar{q}^{(\prime)}$ with $q^{(\prime)}=(u,d)$ as demonstrated in the Fig. 1 where the factorizable and nonfactorizable Feynman diagrams have been merged for the sake of simplicity. In the first approximation one can neglect the interaction of the $K\bar{K}$ pair originating from the intermediate states with the bachelor $h$, and study the decay processes $B\to\rho(770,1450,1700)h\to K\bar{K}h$ and $B\to\omega(782,1420,1650)h\to K\bar{K}h$ in the quasi-two-body framework plb763-29 ; 1605-03889 ; prd96-113003 . The $\pi\pi\leftrightarrow KK$ rescattering effects were found have important contributions for $B^{\pm}\to\pi^{\pm}K^{+}K^{-}$ prl123-231802 , which would be investigated in a subsequent work. The final state interaction effect for the $\rho(1450,1700)\to K\bar{K}$ were found to be suppressed in prd75-074017 and will be neglected in the numerical calculation of this work. The quasi-two-body framework based on PQCD approach has been discussed in detail in plb763-29 , which has been followed in Refs. prd101-111901 ; epjc80-815 ; jhep2003-162 ; prd96-036014 ; prd95-056008 ; 2007-13141 ; 2010-12906 ; prd96-093011 ; npb923-54 ; epjc80-394 ; 2102-04691 for the quasi-two-body $B$ meson decays in recent years. Parallel analyses for the related three-body $B$ meson processes within QCD factorization can be found in Refs. 2007-08881 ; jhep2006-073 ; plb622-207 ; plb669-102 ; prd79-094005 ; prd72-094003 ; prd76-094006 ; prd88-114014 ; prd89-074025 ; prd94-094015 ; npb899-247 ; 2007-02558 ; epjc75-536 ; prd89-094007 , and for relevant work within the symmetries one is referred to Refs. plb564-90 ; prd72-075013 ; prd72-094031 ; prd84-056002 ; plb727-136 ; plb726-337 ; prd89-074043 ; plb728-579 ; prd91-014029 .

This paper is organized as follows. In Sec. II, we review the kaon vector time-like form factors, which are the crucial inputs for the quasi-two-body framework within PQCD and decisive for the numerical results of this work. In Sec. III, we give a brief introduction of the theoretical framework for the quasi-two-body $B$ meson decays within PQCD approach. In Sec. IV, we present our numerical results of the branching fractions and direct $CP$ asymmetries for the quasi-two-body decays $B\to\rho(770,1450,1700)h\to K\bar{K}h$ and $B\to\omega(782,1420,1650)h\to K\bar{K}h$, along with some necessary discussions. Summary of this work is given in Sec. V. The wave functions and factorization formulae for the related decay amplitudes are collected in the Appendix.

The electromagnetic form factors for the charged and neutral kaon are important for the precise determination of the hadronic loop contributions to the anomalous magnetic moment of the muon and the running of the QED coupling to the $Z$ boson mass prd69-093003 ; plb779-64 ; prd97-114025 and are also valuable for the measurements of the resonance parameters zpc39-13 ; plb669-217 ; prd88-032013 ; plb779-64 ; prd63-072002 ; prd89-092002 ; plb760-314 . The kaon electromagnetic form factors have been extensively studied in Refs. epjc79-436 ; prd67-034012 ; epjc39-41 ; epjc49-697 ; prd81-094014 on the theoretical side. Up to now the experimental information on these form factors comes from the measurements of the reactions $e^{+}e^{-}\to K^{+}K^{-}$ zpc39-13 ; prd76-072012 ; prd99-032001 and $e^{+}e^{-}\to K^{+}K^{-}(\gamma)$ prd88-032013 . Since $K\bar{K}$ is not an eigenstate of isospin, both isospin $0$ and $1$ resonances need to be considered in components of the form factors of kaon  prd88-032013 . The combined analysis of the $e^{+}e^{-}\to K^{+}K^{-}$ and $e^{+}e^{-}\to K_{S}K_{L}$ cross sections and the spectral function in the $\tau^{-}\to K^{-}K^{0}\nu_{\tau}$ decay allows one to extract the isovector and isoscalar electromagnetic form factors for kaons jetp129-386 .

The vector time-like form factors for charged and neutral kaons are defined by the matrix elements zpc29-637 ; prd72-094003

	$\displaystyle\langle K^{+}(p_{1})K^{-}(p_{2})|\bar{q}\gamma_{\mu}(1-\gamma_{5})q|0\rangle$	$\displaystyle=$	$\displaystyle(p_{1}-p_{2})_{\mu}\,F_{K^{+}K^{-}}^{q}(s),$		(1)
	$\displaystyle\langle K^{0}(p_{1})\bar{K}^{0}(p_{2})|\bar{q}\gamma_{\mu}(1-\gamma_{5})q|0\rangle$	$\displaystyle=$	$\displaystyle(p_{1}-p_{2})_{\mu}\,F_{K^{0}\bar{K}^{0}}^{q}(s),$		(2)

with the invariant mass square $s=p^{2}$ and the $K\bar{K}$ system momentum $p=p_{1}+p_{2}$. These two form factors $F_{K^{+}K^{-}}^{q}$ and $F_{K^{0}\bar{K}^{0}}^{q}$ can be related to kaon electromagnetic form factors $F_{K^{+}}$ and $F_{K^{0}}$, which are defined by epjc39-41

	$\displaystyle\langle K^{+}(p_{1})K^{-}(p_{2})|j^{em}_{\mu}|0\rangle$	$\displaystyle=$	$\displaystyle(p_{1}-p_{2})_{\mu}\,F_{K^{+}}(s),$		(3)
	$\displaystyle\langle K^{0}(p_{1})\bar{K}^{0}(p_{2})|j^{em}_{\mu}|0\rangle$	$\displaystyle=$	$\displaystyle(p_{1}-p_{2})_{\mu}\,F_{K^{0}}(s),$		(4)

and have the forms epjc39-41

	$\displaystyle F_{K^{+}}(s)$	$\displaystyle=$	$\displaystyle+\frac{1}{2}\sum_{\iota=\rho,\rho^{\prime},...}c^{K}_{\iota}{\rm BW}_{\iota}(s)+\frac{1}{6}\sum_{\varsigma=\omega,\omega^{\prime},...}c^{K}_{\varsigma}{\rm BW}_{\varsigma}(s)+\frac{1}{3}\sum_{\kappa=\phi,\phi^{\prime},..}c^{K}_{\kappa}{\rm BW}_{\kappa}(s),\quad$		(5)
	$\displaystyle F_{K^{0}}(s)$	$\displaystyle=$	$\displaystyle-\frac{1}{2}\sum_{\iota=\rho,\rho^{\prime},...}c^{K}_{\iota}{\rm BW}_{\iota}(s)+\frac{1}{6}\sum_{\varsigma=\omega,\omega^{\prime},...}c^{K}_{\varsigma}{\rm BW}_{\varsigma}(s)+\frac{1}{3}\sum_{\kappa=\phi,\phi^{\prime},..}c^{K}_{\kappa}{\rm BW}_{\kappa}(s),\quad$		(6)

with the electromagnetic current $j^{em}_{\mu}=\frac{2}{3}\bar{u}\gamma_{\mu}u-\frac{1}{3}\bar{d}\gamma_{\mu}d-\frac{1}{3}\bar{s}\gamma_{\mu}s$ carried by the light quarks $u,d$ and $s$ npb250-517 . The BW formula in $F_{K^{+}}(s)$ and $F_{K^{0}}(s)$ has the form zpc48-445 ; prd101-012006

$\displaystyle{\rm BW}_{R}=\frac{m_{R}^{2}}{m_{R}^{2}-s-im_{R}\Gamma_{R}(s)}\,,$

(7)

where the $s$-dependent width is given by

$\displaystyle\Gamma_{R}(s)=\Gamma_{R}\frac{m_{R}}{\sqrt{s}}\frac{\left|\overrightarrow{q}\right|^{3}}{\left|\overrightarrow{q_{0}}\right|^{3}}X^{2}(\left|\overrightarrow{q}\right|r^{R}_{\rm BW}).$

(8)

The Blatt-Weisskopf barrier factor BW-X with barrier radius $r^{R}_{\rm BW}=4.0$ GeV^-1 prd101-012006 is given by

$\displaystyle X(z)=\sqrt{\frac{1+z^{2}_{0}}{1+z^{2}}}\,.$

(9)

The magnitude of the momentum

$\displaystyle\left|\overrightarrow{q}\right|$

$\displaystyle=$

$\displaystyle\frac{1}{2\sqrt{s}}\sqrt{\left[s-(m_{K}+m_{\bar{K}})^{2}\right]\left[s-(m_{K}-m_{\bar{K}})^{2}\right]}\,,$

(10)

and the $\left|\overrightarrow{q_{0}}\right|$ is $\left|\overrightarrow{q}\right|$ at $s=m^{2}_{R}$. One should note that $\bar{c}\gamma_{\mu}c$ can also contribute to $F_{K^{+}}$ and $F_{K^{0}}$ in the high-mass region prd88-032013 ; prd92-054024 ; prd92-072008 and the BW formula for the $\rho$ family could be replaced with the Gounaris-Sakurai (GS) model prl21-244 as in Refs. epjc39-41 ; prd81-094014 ; prd86-032013 . The $F_{K^{+}}$ and $F_{K^{0}}$ can be separated into the isospin $I=0$ and $I=1$ components as $F_{K^{+(0)}}=F_{K^{+(0)}}^{I=1}+F_{K^{+(0)}}^{I=0}$, with the $F_{K^{+}}^{I=0}=F_{K^{0}}^{I=0}$ and $F_{K^{+}}^{I=1}=-F_{K^{0}}^{I=1}$, and one has $\langle K^{+}(p_{1})\bar{K}^{0}(p_{2})|\bar{u}\gamma_{\mu}d|0\rangle=(p_{1}-p_{2})_{\mu}2F_{K^{+}}^{I=1}(s)$ epjc39-41 ; prd96-113003 .

When concern only the contributions for ${K^{+}K^{-}}$ and ${K^{0}\bar{K}^{0}}$ from the resonant states $\iota=\rho(770,1450,1700)$ and $\varsigma=\omega(782,1420,1650)$, we have prd72-094003

	$\displaystyle F_{K^{+}K^{-}}^{u}(s)$	$\displaystyle=$	$\displaystyle F_{K^{0}\bar{K}^{0}}^{d}(s)=+\frac{1}{2}\sum_{\iota}c^{K}_{\iota}{\rm BW}_{\iota}(s)+\frac{1}{2}\sum_{\varsigma}c^{K}_{\varsigma}{\rm BW}_{\varsigma}(s),$		(11)
	$\displaystyle F_{K^{+}K^{-}}^{d}(s)$	$\displaystyle=$	$\displaystyle F_{K^{0}\bar{K}^{0}}^{u}(s)=-\frac{1}{2}\sum_{\iota}c^{K}_{\iota}{\rm BW}_{\iota}(s)+\frac{1}{2}\sum_{\varsigma}c^{K}_{\varsigma}{\rm BW}_{\varsigma}(s).$		(12)

For the $K^{+}\bar{K}^{0}$ and $K^{0}K^{-}$ pairs which have no contribution from the neutral resonances $\omega(782,1420,1650)$, we have prd67-034012 ; epjc39-41 ; epjc79-436

$\displaystyle F_{K^{+}\bar{K}^{0}}(s)=F_{K^{0}K^{-}}(s)=F_{K^{+}}(s)-F_{K^{0}}(s)=\sum_{\iota}c^{K}_{\iota}{\rm BW}_{\iota}(s).$

(13)

One should note that the different constants in Eqs. (11)-(12) and Eqs. (5)-(6) reveal the different definitions of the vector time-like and electromagnetic form factors for kaons in this work.

The $c^{K}_{R}$ (with $R=\iota,\varsigma,\kappa$) is proportional to the coupling constant $g_{RK\bar{K}}$, and the coefficients have the constraints jetp129-386

$\displaystyle\sum_{\iota=\rho,\rho^{\prime},...}c^{K}_{\iota}=1,\qquad\frac{1}{3}\sum_{\varsigma=\omega,\omega^{\prime},...}c^{K}_{\varsigma}+\frac{2}{3}\sum_{\kappa=\phi,\phi^{\prime},..}c^{K}_{\kappa}=1$

(14)

to provide the proper normalizations $F_{K^{+}}(0)=1$ and $F_{K^{0}}(0)=0$, but the possibility of $SU(3)$ violations are allowed which will become manifest in differences between the fitted normalization coefficients epjc39-41 . In Refs. epjc39-41 ; prd81-094014 ; jetp129-386 , the coefficients $c^{K}_{R}$’s for the resonances $\rho(770),\omega(782),\phi(1020)$ and their excited states have been fitted to the data, the results for $\rho(770,1450,1700)$ and $\omega(782,1420,1650)$ are summarised in Table 1, from which one can find that the fitted values for the $c^{K}_{\rho(1450)}$, $c^{K}_{\rho(1700)}$, $c^{K}_{\omega(1420)}$ or $c^{K}_{\omega(1650)}$ are quite different in Refs. epjc39-41 ; prd81-094014 ; jetp129-386 .

With the relations epjc39-41

$\displaystyle c^{K}_{\omega(782)}\approx\sqrt{2}\cdot\frac{f_{\omega(782)}g_{\omega(782)K^{+}K^{-}}}{m_{\omega(782)}},\quad\;g_{\omega(782)K^{+}K^{-}}=\frac{1}{\sqrt{2}}g_{\phi(1020)K^{+}K^{-}},$

(15)

and $\Gamma_{\omega(782)\to ee}=0.60\pm 0.02$ keV, $\Gamma_{\phi(1020)}=4.249\pm 0.013$ MeV, the branching fraction $(49.2\pm 0.5)\%$ for the decay $\phi(1020)\to K^{+}K^{-}$ and the masses for $K^{\pm},\omega(782)$ and $\phi(1020)$ in PDG-2020 , it’s easy to obtain the result $1.113\pm 0.019$ for the coefficient $c^{K}_{\omega(782)}$, where the error comes from the uncertainties of $\Gamma_{\omega(782)\to ee}$ and $\Gamma_{\phi(1020)}$, while the errors come from the uncertainties of the relevant masses are very small and have been neglected. Similarly, we have $c^{K}_{\rho(770)}=1.247\pm 0.019$ with $g_{\rho(770)K^{+}K^{-}}=g_{\omega(782)K^{+}K^{-}}$ epjc39-41 and the decay constant $f_{\rho(770)}=216\pm 3$ MeV jhep1608-098 , where the error comes from the uncertainties of $f_{\rho(770)}$ and $\Gamma_{\phi(1020)}$. Our estimations for $c^{K}_{\omega(782)}$ and $c^{K}_{\rho(770)}$ are consistent with the results in epjc39-41 ; prd81-094014 ; jetp129-386 . But unlike the results of Fit-2 in Refs. epjc39-41 ; prd81-094014 and the values in jetp129-386 , we have $c^{K}_{\omega(782)}$ slightly less than $c^{K}_{\rho(770)}$, because the decay constant (mass) for ${\omega(782)}$ is slightly smaller (larger) than that for ${\rho(770)}$. Supposing $f_{\rho(770)}=f_{\omega(782)}$ and $m_{\rho(770)}=m_{\omega(782)}$, one will have $c^{K}_{\omega(782)}=c^{K}_{\rho(770)}$ with Eq. (15) and then back to the point of the constrained fit in epjc39-41 ; prd81-094014 . To be sure, the violation of the relation $g_{\rho(770)K^{+}K^{-}}=g_{\omega(782)K^{+}K^{-}}=\frac{1}{\sqrt{2}}g_{\phi(1020)K^{+}K^{-}}$ will modify our estimations for $c^{K}_{\omega(782)}$ and $c^{K}_{\rho(770)}$, but the violation was found quite small plb779-64 .

In principle, the $c^{K}_{R}$ for the couplings can be calculated with the formula prd81-094014 ; plb512-331

$\displaystyle c^{K}_{R_{n}}=\frac{(-1)^{n}\Gamma(\beta^{K}_{R}-1/2)}{\alpha^{\prime}\sqrt{\pi}m^{2}_{R_{n}}\Gamma(n+1)\Gamma(\beta^{K}_{R}-1-n)},$

(16)

with $\alpha^{\prime}=1/(2m^{2}_{R_{0}})$, and $n=0$ for the ground states $\rho(770),\omega(782)$ and $\phi(1020)$, $n\geq 1$ for their radial excitations. The parameters $\beta^{K}_{R}$ could be deduced from Eq. (16) with the fitted $c^{K}_{R_{0}}$ prd81-094014 . With Eq. (16) one will deduce the results $c^{K}_{\rho(1450)}=-0.156\pm 0.015$ and $c^{K}_{\omega(1420)}=-0.066\pm 0.014$. The $c^{K}_{\rho(1450)}$ here is consistent with the result of Fit-2 in epjc39-41 but some larger than the latter for the magnitude. If we take into account the relation $g_{\omega(1420)K^{+}K^{-}}\approx g_{\rho(1450)K^{+}K^{-}}$, the big difference between $c^{K}_{\omega(1420)}$ and $c^{K}_{\rho(1450)}$ seems not reasonable. In view of the consistency for the coefficient $c_{\rho(1450)}$ of the pion electromagnetic form factor $F_{\pi}$ in Refs. prd86-032013 ; zpc76-15 ; prd61-112002 ; pr421-191 ; prd78-072006 by different collaborations, we here propose a constraint for $c^{K}_{\rho(1450)}$ from the coefficient $c^{\pi}_{\rho(1450)}$ of $F_{\pi}$. With the relation $g_{\rho(1450)K^{+}K^{-}}\approx\frac{1}{2}g_{\rho(1450)\pi^{+}\pi^{-}}$ within flavour $SU(3)$ symmetry epjc39-41 , one has

$\displaystyle|c^{K}_{\rho(1450)}|\approx\sqrt{2}\cdot\frac{|f_{\rho(1450)}g_{\rho(1450)K^{+}K^{-}}|}{m_{\rho(1450)}}\approx\frac{|f_{\rho(1450)}g_{\rho(1450)\pi^{+}\pi^{-}}|}{\sqrt{2}\,m_{\rho(1450)}}\approx|c^{\pi}_{\rho(1450)}|,$

(17)

where the different definitions for the coefficient $c^{\pi}_{\rho(1450)}$ in prd86-032013 ; zpc76-15 ; prd61-112002 ; pr421-191 ; prd78-072006 and the differences for the BW and GS models should be taken into account. In view of the results for $c^{K}_{\rho(1450)}$ in epjc39-41 and $c^{\pi}_{\rho(1450)}$ in Refs. prd86-032013 ; zpc76-15 ; prd61-112002 ; pr421-191 ; prd78-072006 , we adopt the $c^{K}_{\rho(1450)}=-0.156\pm 0.015$ deduced from Eq. (16) in our numerical calculation. In Ref. zpc62-455 , with the analyses of the $e^{+}e^{-}$ annihilation data, $\Gamma_{\omega^{\prime}\to ee}$ was estimated to be $0.15$ keV, implies the decay constant $f_{\omega(1420)}=131$ MeV. With the $f_{\rho(1450)}=182\pm 5$ MeV in prd77-116009 and the masses for $\omega(1420)$ and $\rho(1450)$ in PDG-2020 , one can estimate the ratio between $c^{K}_{\omega(1420)}$ and $c^{K}_{\rho(1450)}$ as $0.748\pm 0.040$, then one has $c^{K}_{\omega(1420)}=-0.117\pm 0.013$, which agree with the constrained result in epjc39-41 and the corresponding values in jetp129-386 as shown in Table 1.

The results for $c^{K}_{\rho(1700)}$ vary dramatically in Table 1, from $-0.015\mp 0.022$ epjc39-41 to $-0.234\pm 0.013$  jetp129-386 . A reliable reference value should come from the measurements of $F_{\pi}$ rather than the result deduced from Eq. (16) since $\rho(1700)$ is believed to be a $1^{3}D_{1}$ state in $\rho$ family zpc62-455 ; prd55-4157 ; PDG-2020 . With Eq. (17) and the replacement $\rho(1450)\to\rho(1700)$ one has $|c^{K}_{\rho(1700)}|\approx 0.081$ with the result $|c_{\rho^{\prime\prime}}|=0.068$ for $F_{\pi}$ in prd86-032013 . The difference between the $|c^{K}_{\rho(1700)}|$ and $|c_{\rho^{\prime\prime}}|$ is induced by the differences of the BW and GS models and the different definitions for them. Then we adopt the fitted result $-0.083\mp 0.019$ for $c^{K}_{\rho(1700)}$ epjc39-41 in the numerical calculation in this work. As for the coefficient $c^{K}_{\omega(1650)}$, we employ the value $-0.083\mp 0.019$ of the constrained fits in epjc39-41 because of insufficiency of the knowledge for the properties of $\omega(1650)$.

In the light-cone coordinates, the momentum $p_{B}$ for the initial state $B^{+},B^{0}$ or $B^{0}_{s}$ with the mass $m_{B}$ is written as $p_{B}=\frac{m_{B}}{\sqrt{2}}(1,1,0_{\rm T})$ in the rest frame of $B$ meson. In the same coordinates, the bachelor state pion or kaon in the concerned processes has the momentum $p_{3}=\frac{m_{B}}{\sqrt{2}}(1-\zeta,0,0_{\rm T})$, and its spectator quark has the momentum $k_{3}=(\frac{m_{B}}{\sqrt{2}}(1-\zeta)x_{3},0,k_{3{\rm T}})$. For the resonances $\rho$, $\omega$ and their excited states, and the $K\bar{K}$ system generated from them by the strong interaction, we have the momentum $p=\frac{m_{B}}{\sqrt{2}}(\zeta,1,0_{\rm T})$ and the longitudinal polarization vector $\epsilon_{L}=\frac{1}{\sqrt{2}}(-\sqrt{\zeta},1/\sqrt{\zeta},0_{\rm T})$. It’s easy to check the variable $\zeta=s/m^{2}_{B}$ with the invariant mass square $s=m^{2}_{K\bar{K}}\equiv p^{2}$. The spectator quark comes out from $B$ meson and goes into the intermediate states in hadronization shown in Fig. 1 (a) has the momenta $k_{B}=(\frac{m_{B}}{\sqrt{2}}x_{B},0,k_{B{\rm T}})$ and $k=(0,\frac{m_{B}}{\sqrt{2}}x,k_{\rm T})$ before and after it pass through the hard gluon vertex. The $x_{B}$, $x$ and $x_{3}$, which run from zero to one in the numerical calculation, are the momentum fractions for the $B$ meson, the resonances and the bachelor final state, respectively.

For the $P$-wave $K\bar{K}$ system along with the subprocesses $\rho\to K\bar{K}$ and $\omega\to K\bar{K}$, the distribution amplitudes are organized into prd101-111901 ; epjc80-815 ; plb763-29

$\displaystyle\phi^{P\text{-wave}}_{K\bar{K}}(x,s)=\frac{-1}{\sqrt{2N_{c}}}\left[\sqrt{s}\,{\epsilon/}\!_{L}\phi^{0}(x,s)+{\epsilon/}\!_{L}{p/}\phi^{t}(x,s)+\sqrt{s}\phi^{s}(x,s)\right]\!,$

(18)

with

	$\displaystyle\phi^{0}(x,s)$	$\displaystyle=$	$\displaystyle\frac{3C_{X}F_{K}(s)}{\sqrt{2N_{c}}}x(1-x)\left[1+a_{R}^{0}C^{3/2}_{2}(1-2x)\right]\!,$		(19)
	$\displaystyle\phi^{t}(x,s)$	$\displaystyle=$	$\displaystyle\frac{3C_{X}F^{t}_{K}(s)}{2\sqrt{2N_{c}}}(1-2x)^{2}\left[1+a_{R}^{t}C^{3/2}_{2}(1-2x)\right]\!,$		(20)
	$\displaystyle\phi^{s}(x,s)$	$\displaystyle=$	$\displaystyle\frac{3C_{X}F^{s}_{K}(s)}{2\sqrt{2N_{c}}}(1-2x)\left[1+a_{R}^{s}\left(1-10x+10x^{2}\right)\right]\!,$		(21)

where $F_{K}$ is employed as the abbreviation of the vector time-like form factors in Eqs. (11)-(13) and gain different component for different resonance contribution from to the expressions of the Eqs. (11)-(13) in the concerned decay processes. Moreover, we have factored out the normalisation constant $C_{X}$ to make sure the the proper normalizations for the time-like form factors for kaon, and $C_{X}$ are given by

$\displaystyle C_{\rho^{0}}=C_{\omega}=\sqrt{2},\qquad C_{\rho^{\pm}}=1.$

(22)

The Gegenbauer polynomial $C^{3/2}_{2}(\chi)=3\left(5\chi^{2}-1\right)/2$ for the distribution amplitudes $\phi^{0}$ and $\phi^{t}$, and the Gegenbauer moments have been catered to the data in Ref. plb763-29 for the quasi-two-body decays $B\to K\rho\to K\pi\pi$. Within flavour $SU(2)$ symmetry, we adopt the same Gegenbauer moments for the $P$-wave $K\bar{K}$ system originating from the intermediate states $\omega$ and $\rho$ in this work. The vector time-like form factors $F^{t}_{K}$ and $F^{s}_{K}$ for the twist-$3$ distribution amplitudes are deduced from the relations $F^{t,s}_{K}(s)\approx(f^{T}_{\rho}/f_{\rho})F_{K}(s)$ and $F^{t,s}_{K}(s)\approx(f^{T}_{\omega}/f_{\omega})F_{K}(s)$ plb763-29 with the result $f^{T}_{\rho}/f_{\rho}=0.687$ at the scale $\mu=2$ GeV prd78-114509 . The relation $f^{T}_{\rho}/f_{\rho}\approx f^{T}_{\omega}/f_{\omega}$ jhep1608-098 is employed because of the lack of a lattice QCD determination for $f^{T}_{\omega}$.

In PQCD approach, the factorization formula for the decay amplitude ${\mathcal{A}}$ of the quasi-two-body decays $B\to\rho h\to K\bar{K}h$ and $B\to\omega h\to K\bar{K}h$ is written as plb561-258 ; prd89-074031

$\displaystyle{\mathcal{A}}=\phi_{B}\otimes{\mathcal{H}}\otimes\phi^{P\text{-wave}}_{K\bar{K}}\otimes\phi_{h}$

(23)

according to Fig. 1 at leading order in the strong coupling $\alpha_{s}$. The hard kernel ${\mathcal{H}}$ here contains only one hard gluon exchange, and the symbol $\otimes$ means convolutions in parton momenta. For the $B$ meson and bachelor final state $h$ in this work, their distribution amplitudes $\phi_{B}$ and $\phi_{h}$ are the same as those widely adopted in the PQCD approach, we attach their expressions and parameters in the Appendix A.

For the $CP$ averaged differential branching fraction ($\mathcal{B}$), one has the formula prd101-111901 ; prd79-094005 ; PDG-2020

$\displaystyle\frac{d{\mathcal{B}}}{d\zeta}=\tau_{B}\frac{\left|\overrightarrow{q}\right|^{3}\left|\overrightarrow{q_{h}}\right|^{3}}{12\pi^{3}m^{5}_{B}}\overline{|{\mathcal{A}}|^{2}}\;,$

(24)

where $\tau_{B}$ is the mean lifetime for $B$ meson. The magnitude of the momentum ${\small|\overrightarrow{q_{h}}|}$ for the state $h$ in the rest frame of the intermediate states is written as

$\displaystyle\left|\overrightarrow{q_{h}}\right|=\frac{1}{2\sqrt{s}}\sqrt{\left[m^{2}_{B}-(\sqrt{s}+m_{h})^{2}\right]\left[m^{2}_{B}-(\sqrt{s}-m_{h})^{2}\right]},$

(25)

with $m_{h}$ the mass for the bachelor meson pion or kaon. When $m_{K}=m_{\bar{K}}$, the Eq. (10) has a simpler form

$\displaystyle\left|\overrightarrow{q}\right|$

$\displaystyle=$

$\displaystyle\frac{1}{2}\sqrt{s-4m_{K}^{2}}\,.$

(26)

Note that the cubic ${\small|\overrightarrow{q}|}$ and ${\small|\overrightarrow{q_{h}}|}$ in Eq. (24) are caused by the introduction of the Zemach tensor ${\small-2\overrightarrow{q}}\cdot{\small\overrightarrow{q_{h}}}$ which is employed to describe the angular distribution for the decay of spin $1$ resonances Zemach . The direct $CP$ asymmetry ${\mathcal{A}}_{CP}$ is defined as

$\displaystyle{\mathcal{A}}_{CP}=\frac{{\mathcal{B}}(\bar{B}\to\bar{f})-{\mathcal{B}}(B\to f)}{{\mathcal{B}}(\bar{B}\to\bar{f})+{\mathcal{B}}(B\to f)}.$

(27)

The Lorentz invariant decay amplitudes according to Fig. 1 for the decays $B\to\rho h\to K\bar{K}h$ and $B\to\omega h\to K\bar{K}h$ are given in the Appendix B.

In the numerical calculation, we employ the decay constants $f_{B}=0.189$ GeV and $f_{B_{s}}=0.231$ GeV for the $B^{0,\pm}$ and $B^{0}_{s}$ mesons prd98-074512 , respectively, and the mean lifetimes $\tau_{B^{0}}=(1.519\pm 0.004)\times 10^{-12}$ s, $\tau_{B^{\pm}}=(1.638\pm 0.004)\times 10^{-12}$ s and $\tau_{B^{0}_{s}}=(1.515\pm 0.004)\times 10^{-12}$ s PDG-2020 . The masses for the relevant particles in the numerical calculation of this work, the full widths for the resonances $\rho(770,1450,1700)$ and $\omega(782,1420,1650)$, and the Wolfenstein parameters of the CKM matrix are presented in Table 2.