text:
\section{Introduction}\label{sec-int}

Charmless three-body hadronic $B$ meson decays provide us a field to investigate different aspects of weak and
strong interactions. The underlying weak decay of $b$-quark is simple and can be described well by the effective
Hamiltonian~\cite{rmp68-1125}, but the strong dynamics in these three-body processes is very complicated, owing
to the hadron-hadron interactions, the three-body effects~\cite{npps199-341,prd84-094001} and the rescattering
processes~\cite{1512-09284,prd89-094013,epjc78-897,prd71-074016} in the final states, and also on account of various
resonance and nonresonance contributions in the relevant decay processes. The experimental efforts for the three-body
$B$ decays by employing Dalitz plot technique~\cite{prd94-1046} within the isobar formalism~\cite{pr135-B551,pr166-1731,
prd11-3165} partly relieve the tension.  Dalitz plot enable us to investigate the properties of the specific scalar, vector and
tensor intermediate states in the related resonance contributions, but the priori model is needed with all reliable and correct
dynamical components~\cite{plb665-30}. The expressions of the decay amplitudes of these three-body $B$ decays for Dalitz
plot analyses 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 $(30.7\pm1.2\pm0.9)\%$ in ref.~\cite{prl123-231802} for the resonance $\rho(1450)^0$
decaying into charged kaon pair. This fit fraction implies a branching fraction $(1.60\pm0.14)\times10^{-6}$ for the
quasi-two-body decay $B^+\to\pi^+\rho(1450)^0 \to\pi^+ K^+K^-$~\cite{PDG-2020}, in view of the branching fractions  $(5.38\pm0.40\pm0.35)\times10^{-6}$ from Belle~\cite{prd96-031101} and $(5.0\pm0.5\pm0.5)\times10^{-6}$ presented
by BaBar~\cite{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 in ref.~\cite{prd96-036014} by LHCb in their recent
works~\cite{prl124-031801,prd101-012006}.

In ref.~\cite{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~\cite{prl123-231802,PDG-2020},
and our prediction got the supports from the theoretical analyses in ref.~\cite{2007-02558}. In addition, the virtual
contribution~\cite{plb791-342,prd69-112002,prd79-112004,prd91-092002,prd94-072001} for $K^+K^-$ from the Breit-Wigner
(BW) formula~\cite{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 $\rho(1450)^0\to K^+K^-$~\cite{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~\cite{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 $K\bar K$ to the future studies in view of their ambiguous
nature~\cite{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 $\pi^+n\to K^-K^+p$~\cite{prd15-3196,prd22-2595}, $\bar p p \to K^+K^-\pi^0$~\cite{plb468-178,
epjc80-453}, $e^+e^- \to K^+K^-$~\cite{pl99b-257,pl107b-297,plb669-217,prd76-072012,prd88-032013,prd94-112006,
plb779-64,prd99-032001,zpc39-13} and $e^+ e^- \to K^0_{S}K^0_{L}$~\cite{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~\cite{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.~\cite{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~\cite{prl97-142002}. While for the decays
$B\to KKK$~\cite{prd65-092005,prd71-092003,prd74-032003,prl99-161802,prd82-073011,prd85-112010} and
$B\to KK\pi$~\cite{prd87-091101,prl99-221801}, the unsettled $f_X(1500)$ which decaying into $K^+K^-$ channel could
probably be related to the resonance $\rho(1450)^0$~\cite{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
figure~\ref{fig-feyndiag} 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~\cite{plb763-29,1605-03889,prd96-113003}.
The $\pi\pi\leftrightarrow KK$ rescattering effects were found have important contributions for
$B^\pm\to \pi^\pm K^+K^-$~\cite{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~\cite{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~\cite{plb763-29}, which has been followed in refs.~\cite{prd101-111901,epjc80-815,jhep2003-162,prd96-036014,
prd95-056008,2007-13141,2010-12906,prd96-093011,npb923-54} 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.~\cite{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}, and for relevant work within the symmetries one
is referred to refs.~\cite{plb564-90,prd72-075013,prd72-094031,prd84-056002,plb727-136,plb726-337,prd89-074043,
plb728-579,prd91-014029}.

\begin{figure}[tbp]
\centerline{\epsfxsize=15cm \epsffile{Feyfig.eps}}
\caption{Typical Feynman diagrams for the processes $B\to R h\to K\bar K h$, 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.
              }
\label{fig-feyndiag}
\end{figure}

This paper is organized as follows. In section~\ref{sec-KaonFFs}, 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 section~\ref{sec-kinot}, we give a brief introduction of the theoretical framework for the quasi-two-body $B$ meson
decays within PQCD approach. In section~\ref{sec-res}, 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 section~\ref{sec-con}. The wave functions and factorization formulas for the related decay amplitudes are
collected in the appendix.

\section{Kaon time-like form factors}  \label{sec-KaonFFs}

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~\cite{prd69-093003,plb779-64,prd97-114025} and are also valuable for the measurements of the
resonance parameters~\cite{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.~\cite{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 $e^+e^- \to K^+K^-$~\cite{zpc39-13,prd76-072012,prd99-032001} and
$e^+e^- \to K^+K^-(\gamma)$~\cite{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~~\cite{prd88-032013}. The
combined analysis of the $e^+e^- \to K^+K^-$ and $e^+e^- \to K_SK_L$ cross sections and the spectral function in the
$\tau^-\to K^-K^0\nu_\tau$ decay allows us to extract the isovector and isoscalar electromagnetic form factors for
kaons~\cite{jetp129-386}.

The vector time-like form factors for charged and neutral kaons are defined by the matrix elements~\cite{zpc29-637,prd72-094003}
\begin{eqnarray}
  \langle K^+(p_1) K^-(p_2) |\bar q\gamma_\mu (1-\gamma_5) q | 0 \rangle &=& (p_1-p_2)_\mu \,F_{K^+K^-}^{q}(s), \\
  \langle K^0(p_1)\bar K^0(p_2) |\bar q\gamma_\mu (1-\gamma_5) q | 0 \rangle &=& (p_1-p_2)_\mu\,F_{K^0\bar K^0}^{q}(s),
\end{eqnarray}
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~\cite{epjc39-41}
\begin{eqnarray}
  \langle K^+(p_1) K^-(p_2) | j^{em}_\mu | 0 \rangle &=& (p_1-p_2)_\mu \,F_{K^+}(s), \\
  \langle K^0(p_1)\bar K^0(p_2) | j^{em}_\mu | 0 \rangle &=& (p_1-p_2)_\mu\,F_{K^0}(s),
\end{eqnarray}
and have the forms~\cite{epjc39-41}
\begin{eqnarray}
  F_{K^+}(s)&=&+\frac12\sum_{\iota=\rho,\rho^\prime,...} c^K_\iota {\rm BW}_\iota(s)
                    +\frac16\sum_{\varsigma=\omega,\omega^\prime,...} c^K_\varsigma {\rm BW}_\varsigma(s)
                   +\frac13\sum_{\kappa=\phi,\phi^\prime,..} c^K_\kappa {\rm BW}_\kappa(s),
  \label{def-F-K+}   \\
  F_{K^0}(s)&=&-\frac12\sum_{\iota=\rho,\rho^\prime,...} c^K_\iota {\rm BW}_\iota(s)
                    +\frac16\sum_{\varsigma=\omega,\omega^\prime,...} c^K_\varsigma {\rm BW}_\varsigma(s)
                   +\frac13\sum_{\kappa=\phi,\phi^\prime,..} c^K_\kappa {\rm BW}_\kappa(s),
  \label{def-F-K0}
\end{eqnarray}
with the electromagnetic current $j^{em}_\mu=\frac23\bar u\gamma_\mu u-\frac13\bar d\gamma_\mu d-\frac13\bar s\gamma_\mu s$
carried by the light quarks $u, d$ and $s$~\cite{npb250-517}. The BW formula in $F_{K^+}(s)$ and $F_{K^0}(s)$ has the
form~\cite{zpc48-445,prd101-012006}
\begin{eqnarray}
   {\rm BW}_R= \frac{m_{R}^2}{m_R^2-s-i m_R \Gamma_{R}(s)}\,,
   \label{eq-BW}
\end{eqnarray}
where the $s$-dependent width is given by
\begin{eqnarray}\label{def-width}
 \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}).
  \label{eq-sdep-Gamma}
\end{eqnarray}
The Blatt-Weisskopf barrier factor~\cite{BW-X} with barrier radius $r^R_{\rm BW}=4.0$ GeV$^{-1}$~\cite{prd101-012006}
is given by
\begin{eqnarray}
     X(z)=\sqrt{\frac{1+z^2_0}{1+z^2}}\,.
\end{eqnarray}
The magnitude of the momentum
\begin{eqnarray}
   \left| \overrightarrow{q} \right|&=&\frac{1}{2\sqrt s}\sqrt{\left[s-(m_K+m_{\bar K})^2\right]\left[s-(m_K-m_{\bar K})^2\right]}\,,
   \label{def-q}
\end{eqnarray}
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~\cite{prd88-032013,prd92-054024,prd92-072008} and the BW formula for the $\rho$ family could be replaced
with the Gounaris-Sakurai (GS) model~\cite{prl21-244} as in refs.~\cite{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 $\langle K^+(p_1) \bar{K}^0(p_2) | \bar u \gamma_\mu d | 0 \rangle =(p_1-p_2)_\mu 2F_{K^+}^{I=1}(s)$~\cite{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~\cite{prd72-094003}
\begin{eqnarray}
  F_{K^+K^-}^{u}(s)&=&F_{K^0\bar K^0}^{d}(s)= +\frac12\sum_{\iota} c^K_\iota {\rm BW}_\iota(s)
                    +\frac12\sum_{\varsigma} c^K_\varsigma {\rm BW}_\varsigma(s),   \label{def-F-u}  \\
  F_{K^+K^-}^{d}(s)&=&F_{K^0\bar K^0}^{u}(s)=-\frac12\sum_{\iota} c^K_\iota {\rm BW}_\iota(s)
                    +\frac12\sum_{\varsigma} c^K_\varsigma {\rm BW}_\varsigma(s).   \label{def-F-d}
\end{eqnarray}
For the pairs $K^+\bar K^0$ and $K^0K^-$ which have no contribution from the neutral resonances $\omega(782,1420,1650)$,
we have~\cite{prd67-034012,epjc39-41,epjc79-436}
\begin{eqnarray}
  F_{K^+\bar K^0}(s)=F_{K^0K^-}(s)=F_{K^+}(s)-F_{K^0}(s)=\sum_{\iota} c^K_\iota {\rm BW}_\iota(s).   \label{def-F-ud}
\end{eqnarray}
One should note that the different constants in eqs.~(\ref{def-F-u})-(\ref{def-F-d}) and eqs.~(\ref{def-F-K+})-(\ref{def-F-K0})
reveal the different definitions of the vector time-like and electromagnetic form factors for kaons in this work.

\begin{table}[]
\begin{center}
\caption{The fitted results of $c^K_R$'s in refs.~\cite{epjc39-41,prd81-094014,jetp129-386}.
               The column Fit-1 (Fit-2) contains the values of the constrained (unconstrained) fits from~\cite{epjc39-41,prd81-094014}.
              }
\label{sum-cKR}
\footnotesize
\begin{tabular}{l l l l l l l} \hline\hline
     ~~\;$c^K_R$~          &   ~~Fit-1~\cite{epjc39-41}               & ~~Fit-2~\cite{epjc39-41}
                                       &   ~~Fit-1~\cite{prd81-094014}        & ~~Fit-2~\cite{prd81-094014}~~
                                       &  ~Model-I~\cite{jetp129-386}          & \,Model-II~\cite{jetp129-386}      \\
\hline
     \,$c^K_{\rho(770)}$   &  ~\,$1.195\pm0.009$                       &  ~\,$1.139\pm0.010$
                                       &  ~\,$1.138\pm0.011$                       &  ~\,$1.120\pm0.007$
                                       &  ~$1.162\pm0.005$                         &  ~$1.067\pm0.041$                     \\
\,$c^K_{\omega(782)}$  &  ~\,$1.195\pm0.009$                       &  ~\,$1.467\pm0.035$
                                       &  ~\,$1.138\pm0.011$                       &  ~\,$1.37\pm0.03$
                                       &  ~$1.26\pm0.06$                             &  ~$1.28\pm0.14$                         \\
     $c^K_{\rho(1450)}$   &  $-0.112\mp0.010$                          &  $-0.124\mp0.012$
                                       &  $-0.043\pm0.014$                          &  $-0.107\pm0.010$
                                       &  $-0.063\pm0.014$                          &  $-0.025\pm0.008$                      \\
$c^K_{\omega(1420)}$  &  $-0.112\mp0.010$                          &  $-0.018\mp0.024$
                                       &  $-0.043\pm0.014$                          &  $-0.173\pm0.003$
                                       &  $-0.13\pm0.03$                              &  $-0.13\pm0.02$                          \\
     $c^K_{\rho(1700)}$   &  $-0.083\mp0.019$                          &  $-0.015\mp0.022$
                                       &  $-0.144\pm0.015$                          &  $-0.028\pm0.012$
                                       &  $-0.160\pm0.014$                          &  $-0.234\pm0.013$                      \\
$c^K_{\omega(1650)}$  &  $-0.083\mp0.019$                          &  $-0.449\mp0.059$
                                       &  $-0.144\pm0.015$                          &  $-0.621\pm0.020$
                                       &  $-0.37\pm0.05$                              &  $-0.234\pm0.013$                       \\
\hline\hline
\end{tabular}
\end{center}
\end{table}

The $c^K_R$ (with $R=\iota,\varsigma,\kappa$) is proportional to the coupling constant
$g_{R K\bar K}$, and the coefficients have the constraints~\cite{jetp129-386}
\begin{eqnarray}
\sum_{\iota=\rho,\rho^\prime,...} c^K_\iota  =1, \qquad
\frac13\sum_{\varsigma=\omega,\omega^\prime,...} c^K_\varsigma +\frac23\sum_{\kappa=\phi,\phi^\prime,..} c^K_\kappa =1
\end{eqnarray}
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~\cite{epjc39-41}.
In refs.~\cite{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~\ref{sum-cKR}, 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.~\cite{epjc39-41,prd81-094014,jetp129-386}.

With the relations~\cite{epjc39-41}
\begin{eqnarray}
    c^K_{\omega(782)}\approx\sqrt2\cdot\frac{f_{\omega(782)} g_{\omega(782)K^+ K^-}}{m_{\omega(782)}},\quad
    g_{\omega(782)K^+ K^-}=\frac{1}{\sqrt2}g_{\phi(1020)K^+ K^-},  \label{def-cKomega}
\end{eqnarray}
and $\Gamma_{\omega(782)\to ee}=0.60\pm0.02$ keV, $\Gamma_{\phi(1020)}=4.249\pm0.013$ MeV, the branching
fraction $(49.2\pm0.5)\%$ for the decay $\phi(1020)\to K^+K^-$ and the masses for $K^\pm, \omega(782)$ and
$\phi(1020)$ in~\cite{PDG-2020}, it's easy to obtain the result $1.113\pm0.019$ 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
$c^K_{\rho(770)}=1.247\pm0.019$ with $g_{\rho(770)K^+ K^-}=g_{\omega(782)K^+ K^-}$~\cite{epjc39-41}
and the decay constant $f_{\rho(770)}=216\pm3$ MeV~\cite{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~\cite{epjc39-41,prd81-094014,jetp129-386}.
But unlike the results of Fit-2 in refs.~\cite{epjc39-41,prd81-094014} and the values in~\cite{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.~(\ref{def-cKomega}) and then back to the point of the
constrained fit in~\cite{epjc39-41,prd81-094014}. To be sure, the violation of the relation
$g_{\rho(770)K^+ K^-}=g_{\omega(782)K^+ K^-}=\frac{1}{\sqrt2}g_{\phi(1020)K^+ K^-}$ will modify our estimations for
$c^K_{\omega(782)}$ and $c^K_{\rho(770)}$, but the violation was found quite small~\cite{plb779-64}.

In principle, the $c^K_R$ for the couplings can be calculated with the formula~\cite{prd81-094014,plb512-331}
\begin{eqnarray}
    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)},
     \label{fmla-cKR}
\end{eqnarray}
with $\alpha^\prime=1/(2m^2_{R_0})$, and $n=0$ for the ground states $\rho(770), \omega(782)$ and $\phi(1020)$, $n\geq1$
for their radial excitations. The parameters $\beta^K_R$ could be deduced from eq.~(\ref{fmla-cKR}) with the fitted
$c^K_{R_0}$~\cite{prd81-094014}.  With eq.~(\ref{fmla-cKR}) one will deduce the results $c^K_{\rho(1450)}=-0.156\pm0.015$
and $c^K_{\omega(1420)}=-0.066\pm0.014$. The $c^K_{\rho(1450)}$ here is consistent with the result of Fit-2 in~\cite{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.~\cite{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 $g_{\rho(1450)K^+ K^-}\approx \frac12 g_{\rho(1450)\pi^+ \pi^-}$ within flavour $SU(3)$
symmetry~\cite{epjc39-41}, one has
\begin{eqnarray}
    |c^K_{\rho(1450)}|\approx\sqrt2\cdot\frac{|f_{\rho(1450)} g_{\rho(1450)K^+ K^-}|}{m_{\rho(1450)}}
                               \approx\frac{|f_{\rho(1450)} g_{\rho(1450)\pi^+ \pi^-}|}{\sqrt2\,m_{\rho(1450)}}
                               \approx  |c^\pi_{\rho(1450)}|,
        \label{rela-ck1450}
\end{eqnarray}
where the different definitions for the coefficient $c^\pi_{\rho(1450)}$ in~\cite{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~\cite{epjc39-41} and $c^\pi_{\rho(1450)}$ in refs.~\cite{prd86-032013,zpc76-15,prd61-112002,pr421-191,
prd78-072006}, we adopt the $c^K_{\rho(1450)}=-0.156\pm0.015$ deduced from eq.~(\ref{fmla-cKR}) in our numerical calculation.
In ref.~\cite{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 $f_{\rho(1450)}=182\pm5$ MeV
in~\cite{prd77-116009} and the masses for $\omega(1420)$ and $\rho(1450)$ in~\cite{PDG-2020}, one can estimate the ratio
between $c^K_{\omega(1420)}$ and $c^K_{\rho(1450)}$ as $0.748\pm0.040$, then one has
$c^K_{\omega(1420)}=-0.117\pm0.013$, which agree with the constrained result in~\cite{epjc39-41} and the corresponding values
in~\cite{jetp129-386} as shown in table~\ref{sum-cKR}.

The results for $c^K_{\rho(1700)}$ vary dramatically in table~\ref{sum-cKR}, from $-0.015\mp0.022$~\cite{epjc39-41} to
$-0.234\pm0.013$ ~\cite{jetp129-386}. A reliable reference value should come from the measurements of $F_\pi$ rather
than the result deduced from eq.~(\ref{fmla-cKR}) since $\rho(1700)$ is believed to be a $1^3D_1$ state in $\rho$
family~\cite{zpc62-455,prd55-4157,PDG-2020}. With eq.~(\ref{rela-ck1450}) and the replacement $\rho(1450)\to\rho(1700)$
one has $|c^K_{\rho(1700)}|\approx0.081$ with the result $|c_{\rho^{\prime\prime}}|=0.068$ for $F_\pi$ in~\cite{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 $-0.083\mp0.019$ for
$c^K_{\rho(1700)}$~\cite{epjc39-41} in the numerical calculation in this work. As for the coefficient $c^K_{\omega(1650)}$,
we employ the value $-0.083\mp0.019$ of the constrained fits in~\cite{epjc39-41} because of insufficiency of the knowledge
for the properties of $\omega(1650)$.

\section{Kinematics and differential branching fraction}  \label{sec-kinot}

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 $p_B=\frac{m_B}{\sqrt2}(1,1,0_{\rm T})$ in the rest frame of $B$ meson. In the same coordinates, the bachelor
state pion or kaon in the concerned processes has the momentum $p_3=\frac{m_B}{\sqrt2}(1-\zeta, 0, 0_{\rm T})$, and
its spectator quark has the momentum $k_3=(\frac{m_B}{\sqrt2}(1-\zeta)x_3, 0, k_{3{\rm T}})$. 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 as shown in figure~\ref{fig-feyndiag}~(a) has the momenta
$k_B=(\frac{m_B}{\sqrt2}x_B, 0, k_{B{\rm T}})$ 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~\cite{prd101-111901,epjc80-815,plb763-29}
\begin{eqnarray}
  \phi^{P\text{-wave}}_{K\bar K}(x,s)=\frac{-1}{\sqrt{2N_c}}
      \left[\sqrt{s}\,{\epsilon\hspace{-1.5truemm}/}\!_L\phi^0(x,s)
             + {\epsilon\hspace{-1.5truemm}/}\!_L {p\hspace{-1.7truemm}/} \phi^t(x,s)
             +\sqrt s \phi^s(x,s)  \right]\!,
\end{eqnarray}
with
\begin{eqnarray}
   \phi^{0}(x,s)&=&\frac{3C_X F_K(s)}{\sqrt{2N_c}} x(1-x)\left[1+a_R^{0} C^{3/2}_2(1-2x) \right]\!,\label{def-DA-0}\\
   \phi^{t}(x,s)&=&\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]\!,\label{def-DA-t}\\
   \phi^{s}(x,s)&=&\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]\!,\label{def-DA-s}
\end{eqnarray}
where $F_K$ is employed as the abbreviation of the vector time-like form factors in eqs.~(\ref{def-F-u})-(\ref{def-F-ud}) and
gain different component for different resonance contribution in the concerned decay processes from to the expressions
of the eqs.~(\ref{def-F-u})-(\ref{def-F-ud}). 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
\begin{eqnarray}
 C_{\rho^0}=C_{\omega}=\sqrt 2, \qquad C_{\rho^\pm}=1.
\end{eqnarray}
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.~\cite{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 $F^{t,s}_K(s)\approx (f^T_{\omega}/f_{\omega})F_K(s)$~\cite{plb763-29}
with the result $f^T_\rho/f_\rho=0.687$ at the scale $\mu=2$ GeV~\cite{prd78-114509}. The relation
$f^T_\rho/f_\rho\approx f^T_\omega/f_\omega$~\cite{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~\cite{plb561-258,prd89-074031}
\begin{eqnarray}
    {\mathcal A}=\phi_B \otimes {\mathcal H} \otimes  \phi^{P\text{-wave}}_{K\bar K} \otimes \phi_{h}
          \label{def-DA-Q2B}
\end{eqnarray}
according to figure~\ref{fig-feyndiag} 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~\ref{sec-DAs}.

For the $CP$ averaged differential branching fraction ($\mathcal B$), one has the
formula~\cite{prd101-111901,prd79-094005,PDG-2020}
\begin{eqnarray}
 \frac{d{\mathcal B}}{d\zeta}=\tau_B\frac{\left| \overrightarrow{q} \right|^3 \left| \overrightarrow{q_h}\right|^3}
                                                                 {12\pi^3m^5_B}\overline{|{\mathcal A}|^2}\;,
    \label{eqn-diff-bra}
\end{eqnarray}
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
\begin{eqnarray}
   \left| \overrightarrow{q_h}\right|=\frac{1}{2\sqrt s} \sqrt{\left[m^2_{B}-(s+m_{h})^2\right]\left[m^2_{B}-(s-m_{h})^2\right]},
                \label{def-qh}
\end{eqnarray}
with $m_h$ the mass for the bachelor meson pion or kaon. When $m_K=m_{\bar K}$, the eq.~(\ref{def-q}) has a simpler form
\begin{eqnarray}
 \left| \overrightarrow{q} \right|&=&\frac{1}{2}\sqrt{s-4m_K^2}\,.
\end{eqnarray}
Note that the cubic ${\small|\overrightarrow{q}|}$ and
${\small|\overrightarrow{q_h}|}$ in eq.~(\ref{eqn-diff-bra}) 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~\cite{Zemach}. The direct $CP$ asymmetry ${\mathcal A}_{CP}$ is defined as
\begin{eqnarray}
   {\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)}.
\end{eqnarray}
The Lorentz invariant decay amplitudes for the decays $B\to \rho h\to K\bar K h$ and $B\to \omega h\to K\bar K h$ are collected
in the appendix~\ref{sec-decayamp}.

\section{Numerical results and discussions}  \label{sec-res}

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~\cite{prd98-074512}, respectively, and the mean lifetimes
$\tau_{B^0}=(1.519\pm0.004)\times 10^{-12}$~s, $\tau_{B^\pm}=(1.638\pm0.004)\times 10^{-12}$~s and
$\tau_{B^0_s}=(1.515\pm0.004)\times 10^{-12}$~s~\cite{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~\ref{params}.

\begin{table}[thb]
\begin{center}
\caption{Masses for the relevant particles, the full widths for $\rho(770,1450,1700)$ and $\omega(782,1420,1650)$ (in units of GeV)
               and the Wolfenstein parameters~\cite{PDG-2020}.}
\label{params}
\begin{tabular}{l}
 \hline\hline
  $m_{B^{0}}=5.280  \quad m_{B^{\pm}}=5.279 \qquad m_{B^0_s}\,=5.367 \quad\;\,m_{K^0}\;\,=0.498 \quad\;
    m_{K^\pm}\;\;=0.494$ \\
  $m_{\pi^0}\,=0.135 \quad m_{\pi^\pm}\,=0.140 \quad m_{\rho(770)}=0.775 \quad \Gamma_{\rho(770)}=0.149 \quad\!
    m_{\omega(782)}=0.783$ \\
  $\Gamma_{\omega(782)}\;\,=0.00849  \qquad\qquad\;   m_{\omega(1420)}=1.410\pm0.060 \quad\;\;
      \Gamma_{\omega(1420)}=\,0.290\pm0.190$\\
  $m_{\rho(1450)}=1.465\pm0.025 \qquad\!  \Gamma_{\rho(1450)}\;=0.400\pm0.060  \quad\;\;
    m_{\omega(1650)}\!=1.670\pm0.030$\\
  $\Gamma_{\omega(1650)}\,=0.315\pm0.035 \qquad\! m_{\rho(1700)}=1.720\pm0.020 \quad\;\;\,
    \Gamma_{\rho(1700)}=0.250\pm0.100$\\
  $\lambda=0.22650\pm 0.00048  \quad\;\;  A=0.790^{+0.017}_{-0.012} \quad\;\; \bar{\rho} = 0.141^{+0.016}_{-0.017}
      \quad\;\;\; \bar{\eta}= 0.357\pm0.01$ \\
 \hline\hline
\end{tabular}
\end{center}
\end{table}

\begin{table}[]
\begin{center}
\caption{PQCD predictions of the $CP$ averaged branching fractions and the direct $CP$ asymmetries for the
               quasi-two-body $B\to \rho(770)h \to K\bar K h$ and $B\to \omega(782)h \to K\bar K h$ decays.
               The decays with the subprocess $\rho(770)^0\to K^0\bar K^0$ or $\omega(782)\to K^0\bar K^0$
               have the same results as their corresponding decay modes with $\rho(770)^0\to K^+K^-$ or
               $\omega(782)\to K^+K^-$.  }
\label{Res-770}
\begin{tabular}{l l l} \hline\hline
\quad Decay modes       & \quad\quad${\mathcal B}$    &  \quad\quad\;${\mathcal A}_{CP}$ \\
\hline  %
 $B^+\to \pi^0 [\rho(770)^+\to] K^+\bar K^0$\;
       &$2.01^{+0.38+0.29+0.24+0.10+0.06}_{-0.35-0.26-0.20-0.07-0.06}\times10^{-8}$
       &$-0.16^{+0.18+0.20+0.10+0.00}_{-0.20-0.18-0.10-0.00}$    \\
 $B^+\to \pi^+ [\rho(770)^0\to] K^+K^-$\;
       &$1.43^{+0.26+0.19+0.11+0.06+0.04}_{-0.25-0.17-0.10-0.05-0.04}\times10^{-7}$
       &$-0.22^{+0.04+0.01+0.01+0.01}_{-0.04-0.01-0.01-0.01}$    \\
 $B^+\to\pi^+ [\omega(782)\;\to] K^+K^-$\;
       &$4.21^{+1.67+1.03+0.08+0.21+0.14}_{-1.34-0.96-0.08-0.17-0.14}\times10^{-8}$
       &\;\;\,$0.02^{+0.01+0.01+0.02+0.00}_{-0.01-0.01-0.01-0.00}$    \\
 $B^+\to K^0 [ \rho(770)^+ \to] K^+\bar K^0$\;
       &$2.21^{+0.51+0.51+0.34+0.10+0.07}_{-0.45-0.46-0.29-0.08-0.07}\times10^{-7}$
       &\;\;\,$0.17^{+0.04+0.04+0.01+0.00}_{-0.05-0.03-0.02-0.00}$    \\
 $B^+\to K^+ [\rho(770)^0 \to] K^+K^-$\;
       &$5.15^{+0.91+0.99+0.69+0.25+0.16}_{-0.85-0.98-0.66-0.21-0.16}\times10^{-8}$
       &\;\;\,$0.39^{+0.03+0.04+0.04+0.00}_{-0.04-0.04-0.05-0.01}$    \\
 $B^+\to K^+ [\omega(782)\;\to] K^+K^-$\;
       &$8.92^{+1.67+2.33+1.19+0.43+0.30}_{-1.47-2.18-1.07-0.34-0.30}\times10^{-8}$ \;
       &\;\;\,$0.22^{+0.04+0.05+0.04+0.00}_{-0.04-0.04-0.04-0.00}$    \\
\hline
 $B^0\to \pi^- [\rho(770)^+ \to] K^+\bar K^0$\;
       &$1.02^{+0.21+0.28+0.14+0.06+0.03}_{-0.17-0.25-0.13-0.05-0.03}\times10^{-7}$
       &\;\;\,$0.15^{+0.04+0.04+0.00+0.00}_{-0.03-0.03-0.00-0.00}$    \\
 $B^0\to \pi^+ [\rho(770)^-\to] K^- K^0$\;
       &$9.59^{+3.25+1.96+0.22+0.46+0.29}_{-2.90-1.88-0.19-0.33-0.29}\times10^{-8}$
       &$-0.27^{+0.11+0.02+0.02+0.00}_{-0.08-0.01-0.02-0.00}$    \\
 $B^0\to \pi^0\; [\rho(770)^0\,\to] K^+K^-$\;
       &$1.47^{+0.96+0.53+0.19+0.13+0.04}_{-0.78-0.49-0.14-0.07-0.04}\times10^{-9}$
       &\;\;\,$0.19^{+0.17+0.07+0.06+0.05}_{-0.15-0.06-0.04-0.05}$    \\
 $B^0\to \pi^0\; [\omega(782)\;\,\to] K^+K^-$\;
       &$4.96^{+0.73+1.25+0.63+0.24+0.17}_{-0.87-1.36-0.65-0.22-0.17}\times10^{-9}$
       &\;\;\,$0.58^{+0.19+0.11+0.14+0.04}_{-0.18-0.11-0.14-0.04}$    \\
 $B^0\to K^+ [\rho(770)^-\!\to] K^- K^0$\;
       &$1.77^{+0.30+0.41+0.27+0.08+0.05}_{-0.25-0.39-0.25-0.06-0.05}\times10^{-7}$
       &\;\;\,$0.20^{+0.07+0.03+0.03+0.00}_{-0.08-0.02-0.03-0.00}$    \\
 $B^0\to K^0\; [\rho(770)^0\to] K^+K^-$\;
       &$5.44^{+0.88+1.26+0.82+0.24+0.17}_{-0.81-1.19-0.76-0.18-0.17}\times10^{-8}$
       &$-0.01^{+0.01+0.01+0.00+0.00}_{-0.01-0.01-0.01-0.00}$    \\
 $B^0\to K^0\; [\omega(782)\;\to] K^+K^-$\;
       &$5.99^{+1.15+1.60+0.88+0.22+0.20}_{-0.96-1.39-0.75-0.19-0.20}\times10^{-8}$
       &\;\;\,$0.01^{+0.02+0.00+0.01+0.00}_{-0.02-0.00-0.01-0.00}$    \\
\hline  %
 $B_s^0\to\pi^- [\rho(770)^+\to] K^+\bar K^0$\;
       &$2.31^{+0.75+0.51+0.30+0.11+0.07}_{-0.62-0.39-0.26-0.08-0.07}\times10^{-9}$
       &$-0.66^{+0.17+0.04+0.03+0.01}_{-0.16-0.06-0.03-0.01}$    \\
 $B_s^0\to \pi^+ [\rho(770)^-\to] K^- K^0$\;
       &$5.43^{+1.47+0.57+0.79+0.24+0.17}_{-1.45-0.48-0.77-0.20-0.17}\times10^{-9}$
       &\;\;\,$0.04^{+0.03+0.01+0.01+0.00}_{-0.04-0.01-0.01-0.00}$    \\
 $B_s^0\to\pi^0\; [\rho(770)^0\to] K^+K^-$\;
       &$1.63^{+0.98+0.46+0.18+0.07+0.05}_{-0.81-0.41-0.16-0.06-0.05}\times10^{-9}$
       &$-0.35^{+0.13+0.05+0.12+0.03}_{-0.14-0.06-0.14-0.03}$    \\
 $B_s^0\to\pi^0\; [\omega(782)\;\to] K^+K^-$\;
       &$8.17^{+3.83+2.37+1.22+0.51+0.28}_{-3.28-2.14-1.21-0.45-0.28}\times10^{-11}$
       &\;\;\,$0.11^{+0.03+0.00+0.02+0.00}_{-0.04-0.00-0.02-0.01}$    \\
 $B_s^0\to K^- [\rho(770)^+\to] K^+ \bar K^0$\;
       &$2.04^{+0.03+0.43+0.22+0.11+0.06}_{-0.02-0.41-0.21-0.09-0.06}\times10^{-7}$
       &\;\;\,$0.25^{+0.04+0.03+0.00+0.01}_{-0.04-0.03-0.00-0.01}$    \\
 $B_s^0\to\bar K^0\; [\rho(770)^0\to] K^+K^-$\;
       &$1.03^{+0.63+0.19+0.18+0.08+0.03}_{-0.45-0.17-0.16-0.05-0.03}\times10^{-9}$
       &\;\;\,$0.60^{+0.24+0.03+0.16+0.02}_{-0.22-0.04-0.14-0.04}$    \\
 $B_s^0\to \bar K^0\; [\omega(782) \;\to] K^+K^-$\;
       &$1.39^{+0.68+0.17+0.12+0.07+0.05}_{-0.57-0.14-0.14-0.07-0.05}\times10^{-9}$
       &$-0.34^{+0.29+0.06+0.01+0.03}_{-0.21-0.06-0.03-0.03}$    \\
\hline\hline
\end{tabular}
\end{center}
\end{table}

Utilizing the differential branching fractions the eq.~(\ref{eqn-diff-bra}) and the decay amplitudes collected in the
appendix~\ref{sec-decayamp}, we obtain the $CP$ averaged branching fractions and the direct $CP$ asymmetries in
tables~\ref{Res-770}, \ref{Res-1450}, \ref{Res-1700} for the concerned quasi-two-body decay processes
$B\to \rho(770,1450,1700) h\to K\bar K h$ and $B\to\omega(782,1420,1650) h \to K\bar K h$. For these PQCD predictions,
the uncertainties of the Gegenbauer moments $a_R^{0}=0.25\pm0.10$, $a_R^t=-0.50\pm0.20$ and $a_R^s=0.75\pm0.25$
along with the decay widths of the intermediate states contribute the first error. The second error for each result in
tables~\ref{Res-770}, \ref{Res-1450}, \ref{Res-1700} comes from the shape parameter $\omega_B=0.40\pm0.04$ or
$\omega_{B_s}=0.50 \pm 0.05$  in eq.~(\ref{phib}) for the $B^{+,0}$ or $B^0_s$ meson. The third one is induced by the
chiral scale parameters $m^h_0=\frac{m^2_{h}}{m_q+m_{q^\prime}}$ with $m^\pi_0=1.4\pm0.1$ GeV and
$m^K_0=1.9\pm0.1$ GeV~\cite{prd76-074018} and the Gegenbauer moment $a^h_2=0.25\pm0.15$ for the bachelor
final state pion or kaon. The fourth one comes from the Wolfenstein parameters $A$ and $\bar \rho$ listed in
table~\ref{params}. The uncertainties of $c^K_{\rho(770)}=1.247\pm0.019$, $c^K_{\omega(782)}=1.113\pm0.019$,
$c^K_{\rho(1450)}=-0.156\pm0.015$, $c^K_{\omega(1420)}=-0.117\pm0.013$ and
$c^K_{\omega(1650),\rho(1700)}=-0.083\pm0.019$ result in the fifth error for the predicted branching fractions in this work,
while these coefficients $c^K_R$ which exist only in the kaon time-like form factors will not change the direct $CP$ asymmetries
for the relevant decay processes. There are other errors for the PQCD predictions in this work, which come from the masses and
the decay constants of the initial and final states, from the parameters in the distribution amplitudes for bachelor pion or kaon,
from the uncertainties of the Wolfenstein parameters $\lambda$ and $\bar{\eta}$, etc., are small and have been neglected.

\begin{table}[]
\begin{center}
\caption{PQCD predictions of the $CP$ averaged branching ratios and the direct $CP$ asymmetries for the
               quasi-two-body $B\to \rho(1450)h \to K\bar K h$ and $B\to \omega(1420)h \to K\bar K h$ decays.
               The decays with the subprocess $\rho(1450)^0\to K^0\bar K^0$ or $\omega(1420)\to K^0\bar K^0$
               have the same results as their corresponding decay modes with $\rho(1450)^0\to K^+K^-$ or
               $\omega(1420)\to K^+K^-$.  }
\label{Res-1450}
\begin{tabular}{l l l} \hline\hline
\quad Decay modes       & \quad\quad${\mathcal B}$    &  \quad\quad\;${\mathcal A}_{CP}$ \\
\hline
 $B^+\to \pi^0 [\rho(1450)^+\to] K^+\bar K^0$\;
       &$1.27^{+0.26+0.22+0.10+0.06+0.24}_{-0.22-0.18-0.12-0.04-0.24}\times10^{-8}$
       &$-0.14^{+0.24+0.21+0.11+0.00}_{-0.22-0.17-0.09-0.00}$    \\
 $B^+\to \pi^+ [\rho(1450)^0\to] K^+K^-$\;
       &$9.46^{+1.79+1.16+0.72+0.49+1.82}_{-1.65-1.14-0.69-0.38-1.82}\times10^{-8}$
       &$-0.22^{+0.04+0.01+0.01+0.01}_{-0.04-0.01-0.01-0.01}$    \\
 $B^+\to\pi^+ [\omega(1420)\;\to] K^+K^-$\;
       &$1.62^{+0.61+0.45+0.03+0.08+0.36}_{-0.52-0.39-0.02-0.07-0.36}\times10^{-8}$
       &\;\;\,$0.01^{+0.01+0.02+0.01+0.01}_{-0.02-0.02-0.02-0.01}$    \\
 $B^+\to K^0 [ \rho(1450)^+ \to] K^+\bar K^0$\;
       &$1.20^{+0.29+0.24+0.17+0.05+0.23}_{-0.25-0.23-0.16-0.04-0.23}\times10^{-7}$
       &\;\;\,$0.20^{+0.04+0.03+0.02+0.00}_{-0.05-0.02-0.02-0.00}$    \\
 $B^+\to K^+ [\rho(1450)^0 \to] K^+K^-$\;
       &$3.36^{+0.62+0.67+0.47+0.16+0.65}_{-0.56-0.64-0.43-0.13-0.65}\times10^{-8}$
       &\;\;\,$0.42^{+0.03+0.04+0.05+0.01}_{-0.03-0.03-0.05-0.01}$    \\
 $B^+\to K^+ [\omega(1420)\;\to] K^+K^-$\;
       &$3.09^{+0.64+0.80+0.42+0.15+0.69}_{-0.57-0.73-0.37-0.12-0.69}\times10^{-8}$
       &\;\;\,$0.32^{+0.05+0.05+0.03+0.01}_{-0.05-0.05-0.03-0.01}$    \\
\hline
 $B^0\to \pi^- [\rho(1450)^+ \to] K^+\bar K^0$\;
       &$7.39^{+1.58+2.20+1.01+0.41+1.42}_{-1.31-1.86-0.96-0.33-1.42}\times10^{-8}$
       &\;\;\,$0.16^{+0.02+0.05+0.01+0.01}_{-0.03-0.03-0.01-0.01}$    \\
 $B^0\to \pi^+ [\rho(1450)^-\to] K^- K^0$\;
       &$6.94^{+2.04+1.40+0.14+0.33+1.33}_{-1.94-1.38-0.14-0.25-1.33}\times10^{-8}$
       &$-0.27^{+0.12+0.02+0.02+0.00}_{-0.08-0.01-0.02-0.00}$    \\
 $B^0\to \pi^0\; [\rho(1450)^0\,\to] K^+K^-$\;
       &$8.48^{+5.96+3.07+0.81+0.68+1.63}_{-5.14-3.01-0.78-0.49-1.63}\times10^{-10}$
       &\;\;\,$0.20^{+0.21+0.10+0.09+0.06}_{-0.17-0.08-0.07-0.05}$    \\
 $B^0\to \pi^0\; [\omega(1420)\;\,\to] K^+K^-$\;
       &$2.08^{+0.32+0.58+0.28+0.10+0.46}_{-0.37-0.66-0.32-0.08-0.46}\times10^{-9}$
       &\;\;\,$0.58^{+0.17+0.10+0.11+0.02}_{-0.16-0.11-0.09-0.02}$    \\
 $B^0\to K^+ [\rho(1450)^-\!\to] K^- K^0$\;
       &$1.18^{+0.20+0.27+0.18+0.05+0.23}_{-0.17-0.25-0.17-0.04-0.23}\times10^{-7}$
       &\;\;\,$0.22^{+0.08+0.03+0.04+0.00}_{-0.08-0.02-0.04-0.00}$    \\
 $B^0\to K^0\; [\rho(1450)^0\to] K^+K^-$\;
       &$3.69^{+0.67+0.84+0.55+0.16+0.71}_{-0.60-0.82-0.51-0.12-0.71}\times10^{-8}$
       &$-0.01^{+0.01+0.01+0.00+0.00}_{-0.02-0.01-0.01-0.00}$    \\
$B^0\to K^0\; [\omega(1420)\;\to] K^+K^-$\;
       &$2.07^{+0.48+0.48+0.29+0.08+0.46}_{-0.46-0.45-0.26-0.06-0.46}\times10^{-8}$
       &$-0.02^{+0.04+0.03+0.01+0.00}_{-0.02-0.03-0.01-0.00}$    \\
\hline  %
 $B_s^0\to\pi^- [\rho(1450)^+\to] K^+\bar K^0$\;
       &$1.55^{+0.39+0.30+0.16+0.07+0.30}_{-0.33-0.28-0.14-0.05-0.30}\times10^{-9}$
       &$-0.66^{+0.15+0.04+0.05+0.02}_{-0.16-0.08-0.04-0.01}$    \\
 $B_s^0\to \pi^+ [\rho(1450)^-\to] K^- K^0$\;
       &$4.54^{+1.30+0.37+0.69+0.20+0.87}_{-1.27-0.40-0.67-0.16-0.87}\times10^{-9}$
       &\;\;\,$0.04^{+0.03+0.01+0.02+0.00}_{-0.05-0.01-0.01-0.00}$    \\
 $B_s^0\to\pi^0\; [\rho(1450)^0\to] K^+K^-$\;
       &$1.15^{+0.72+0.35+0.09+0.05+0.22}_{-0.59-0.30-0.12-0.04-0.22}\times10^{-9}$
       &$-0.36^{+0.12+0.05+0.10+0.02}_{-0.16-0.04-0.14-0.03}$    \\
 $B_s^0\to\pi^0\; [\omega(1420)\;\to] K^+K^-$\;
       &$3.67^{+1.59+1.17+0.65+0.21+0.82}_{-1.38-0.97-0.58-0.19-0.82}\times10^{-11}$
       &\;\;\,$0.14^{+0.03+0.00+0.01+0.00}_{-0.02-0.01-0.01-0.01}$    \\
 $B_s^0\to K^- [\rho(1450)^+\to] K^+ \bar K^0$\;
       &$1.49^{+0.07+0.31+0.16+0.08+0.29}_{-0.06-0.30-0.15-0.06-0.29}\times10^{-7}$
       &\;\;\,$0.25^{+0.04+0.03+0.00+0.01}_{-0.04-0.03-0.00-0.01}$    \\
 $B_s^0\to\bar K^0\; [\rho(1450)^0\to] K^+K^-$\;
       &$6.86^{+4.09+0.81+1.03+0.44+1.32}_{-3.56-0.75-0.94-0.39-1.32}\times10^{-10}$
       &\;\;\,$0.64^{+0.29+0.02+0.08+0.05}_{-0.27-0.01-0.12-0.07}$    \\
 $B_s^0\to \bar K^0\; [\omega(1420) \;\to] K^+K^-$\;
       &$5.79^{+3.28+0.53+0.63+0.34+1.29}_{-2.39-0.42-0.57-0.31-1.29}\times10^{-10}$
       &$-0.54^{+0.29+0.13+0.05+0.01}_{-0.33-0.12-0.05-0.03}$    \\
\hline\hline
\end{tabular}
\end{center}
\end{table}

\begin{table}[]
\begin{center}
\caption{PQCD predictions of the $CP$ averaged branching ratios and the direct $CP$ asymmetries for the
               quasi-two-body $B\to \rho(1700)h \to K\bar K h$ and $B\to \omega(1650)h \to K\bar K h$ decays.
               The decays with the subprocess $\rho(1700)^0\to K^0\bar K^0$ or $\omega(1650)\to K^0\bar K^0$
               have the same results as their corresponding decay modes with $\rho(1700)^0\to K^+K^-$ or
               $\omega(1650)\to K^+K^-$.  }
\label{Res-1700}
\begin{tabular}{l l l} \hline\hline
\quad Decay modes       & \quad\quad${\mathcal B}$    &  \quad\quad\;${\mathcal A}_{CP}$ \\
\hline  %
 $B^+\to \pi^0 [\rho(1700)^+\to] K^+\bar K^0$\;
       &$1.03^{+0.21+0.20+0.09+0.05+0.47}_{-0.18-0.17-0.10-0.04-0.47}\times10^{-8}$
       &$-0.15^{+0.22+0.23+0.13+0.01}_{-0.23-0.21-0.12-0.00}$    \\
 $B^+\to \pi^+ [\rho(1700)^0\to] K^+K^-$\;
       &$8.71^{+1.47+1.20+0.61+0.46+3.99}_{-1.34-1.17-0.59-0.36-3.99}\times10^{-8}$
       &$-0.25^{+0.03+0.02+0.01+0.01}_{-0.03-0.01-0.01-0.01}$    \\
 $B^+\to\pi^+ [\omega(1650)\;\to] K^+K^-$\;
       &$1.48^{+0.42+0.32+0.02+0.01+0.68}_{-0.34-0.28-0.02-0.01-0.68}\times10^{-9}$
       &\;\;\,$0.02^{+0.01+0.00+0.00+0.00}_{-0.01-0.00-0.00-0.00}$    \\
 $B^+\to K^0 [ \rho(1700)^+ \to] K^+\bar K^0$\;
       &$1.08^{+0.27+0.21+0.18+0.05+0.49}_{-0.25-0.19-0.15-0.03-0.49}\times10^{-7}$
       &\;\;\,$0.21^{+0.05+0.04+0.03+0.00}_{-0.06-0.03-0.02-0.00}$    \\
 $B^+\to K^+ [\rho(1700)^0 \to] K^+K^-$\;
       &$2.85^{+0.50+0.49+0.35+0.14+1.30}_{-0.49-0.48-0.32-0.11-1.30}\times10^{-8}$
       &\;\;\,$0.47^{+0.02+0.04+0.05+0.01}_{-0.02-0.03-0.05-0.01}$    \\
 $B^+\to K^+ [\omega(1650)\;\to] K^+K^-$\;
       &$2.81^{+0.53+0.66+0.36+0.13+1.29}_{-0.47-0.59-0.32-0.10-1.29}\times10^{-8}$
       &\;\;\,$0.36^{+0.03+0.05+0.05+0.01}_{-0.04-0.05-0.05-0.01}$    \\
\hline
 $B^0\to \pi^- [\rho(1700)^+ \to] K^+\bar K^0$\;
       &$4.38^{+0.80+1.17+0.50+0.23+2.01}_{-0.73-1.06-0.48-0.19-2.01}\times10^{-8}$
       &\;\;\,$0.18^{+0.03+0.03+0.01+0.01}_{-0.02-0.03-0.01-0.01}$    \\
 $B^0\to \pi^+ [\rho(1700)^-\to] K^- K^0$\;
       &$6.66^{+1.78+1.41+0.13+0.32+3.05}_{-1.69-1.40-0.12-0.24-3.05}\times10^{-8}$
       &$-0.29^{+0.12+0.02+0.02+0.01}_{-0.08-0.02-0.02-0.01}$    \\
 $B^0\to \pi^0\; [\rho(1700)^0\,\to] K^+K^-$\;
       &$8.11^{+5.46+3.02+0.82+0.68+3.71}_{-4.98-2.97-0.80-0.54-3.71}\times10^{-10}$
       &\;\;\,$0.18^{+0.20+0.08+0.07+0.04}_{-0.18-0.06-0.07-0.04}$    \\
 $B^0\to \pi^0\; [\omega(1650)\;\,\to] K^+K^-$\;
       &$1.48^{+0.31+0.44+0.15+0.06+0.68}_{-0.34-0.39-0.16-0.06-0.68}\times10^{-9}$
       &\;\;\,$0.57^{+0.21+0.07+0.09+0.01}_{-0.17-0.09-0.07-0.01}$    \\
 $B^0\to K^+ [\rho(1700)^-\!\to] K^- K^0$\;
       &$9.95^{+1.87+1.83+1.31+0.44+4.56}_{-1.60-1.61-1.15-0.32-4.56}\times10^{-8}$
       &\;\;\,$0.28^{+0.07+0.01+0.05+0.00}_{-0.09-0.01-0.04-0.00}$    \\
 $B^0\to K^0\; [\rho(1700)^0\to] K^+K^-$\;
       &$2.94^{+0.54+0.57+0.38+0.13+1.35}_{-0.53-0.56-0.36-0.09-1.35}\times10^{-8}$
       &$-0.01^{+0.01+0.00+0.01+0.00}_{-0.01-0.00-0.01-0.00}$    \\
 $B^0\to K^0\; [\omega(1650)\;\to] K^+K^-$\;
       &$1.89^{+0.43+0.39+0.22+0.08+0.87}_{-0.38-0.36-0.19-0.07-0.87}\times10^{-8}$
       &$-0.01^{+0.04+0.00+0.01+0.00}_{-0.03-0.00-0.00-0.00}$    \\
\hline  %
 $B_s^0\to\pi^- [\rho(1700)^+\to] K^+\bar K^0$\;
       &$1.37^{+0.34+0.29+0.14+0.06+0.63}_{-0.31-0.27-0.14-0.05-0.63}\times10^{-9}$
       &$-0.70^{+0.16+0.04+0.01+0.01}_{-0.15-0.07-0.04-0.01}$    \\
 $B_s^0\to \pi^+ [\rho(1700)^-\to] K^- K^0$\;
       &$3.57^{+0.94+0.30+0.54+0.16+1.63}_{-0.86-0.32-0.52-0.13-1.63}\times10^{-9}$
       &\;\;\,$0.07^{+0.04+0.01+0.02+0.00}_{-0.05-0.02-0.02-0.00}$    \\
 $B_s^0\to\pi^0\; [\rho(1700)^0\to] K^+K^-$\;
       &$1.01^{+0.59+0.35+0.09+0.04+0.46}_{-0.51-0.30-0.11-0.03-0.46}\times10^{-9}$
       &$-0.29^{+0.11+0.06+0.12+0.01}_{-0.18-0.08-0.15-0.01}$    \\
 $B_s^0\to\pi^0\; [\omega(1650)\;\to] K^+K^-$\;
       &$3.14^{+1.35+1.10+0.53+0.19+1.44}_{-1.29-0.98-0.49-0.16-1.44}\times10^{-11}$
       &\;\;\,$0.15^{+0.06+0.02+0.02+0.01}_{-0.05-0.01-0.03-0.01}$    \\
 $B_s^0\to K^- [\rho(1700)^+\to] K^+ \bar K^0$\;
       &$1.14^{+0.07+0.25+0.12+0.06+0.52}_{-0.07-0.24-0.12-0.05-0.52}\times10^{-7}$
       &\;\;\,$0.29^{+0.04+0.04+0.01+0.01}_{-0.04-0.03-0.01-0.01}$    \\
 $B_s^0\to\bar K^0\; [\rho(1700)^0\to] K^+K^-$\;
       &$4.21^{+1.90+0.47+0.55+0.29+1.93}_{-1.70-0.42-0.50-0.26-1.93}\times10^{-10}$
       &\;\;\,$0.67^{+0.25+0.03+0.12+0.04}_{-0.26-0.02-0.16-0.03}$    \\
 $B_s^0\to \bar K^0\; [\omega(1650) \;\to] K^+K^-$\;
       &$4.18^{+1.44+0.42+0.50+0.27+1.91}_{-1.17-0.38-0.43-0.23-1.91}\times10^{-10}$
       &$-0.64^{+0.26+0.08+0.09+0.03}_{-0.19-0.08-0.12-0.05}$    \\
\hline\hline
\end{tabular}
\end{center}
\end{table}

The PQCD predictions for those quasi-two-body decays with the subprocesses $\rho(770, 1450, 1700)^0\to K^0\bar K^0$ and
$\omega(782, 1420, 1650)\to K^0\bar K^0$ are omitted in tables~\ref{Res-770}, \ref{Res-1450}, \ref{Res-1700}. The variations
caused by the small mass difference between $K^\pm$ and $K^0$ for the branching fraction and direct $CP$ asymmetry of a
decay mode with one of these intermediate states decaying into $K^0\bar K^0$ or $K^+K^-$ are tiny.
As the examples, we calculate the the branching fractions for the decays $B^+\to \pi^+ \rho(770)^0$, $B^+\to K^+ \rho(770)^0$,
$B^+\to \pi^+ \omega(782)$ and $B^+\to K^+ \omega(782)$ with the resonances $\rho(770)^0$ and $\omega(782)$ decay into
the final state $K^0\bar K^0$. Their four branching fractions with the same sources for the errors as these results in
table~\ref{Res-770} are predicted to be
\begin{eqnarray}
{\mathcal B}(B^+\to \pi^+ \rho(770)^0\to \pi^+ K^0\bar K^0)
                    &=& 1.40^{+0.26+0.17+0.10+0.06+0.04}_{-0.24-0.17-0.10-0.06-0.04} \times 10^{-7},   \\
{\mathcal B}(B^+\to K^+ \rho(770)^0 \to K^+  K^0\bar K^0)
                    &=& 5.08^{+0.92+1.00+0.70+0.25+0.15}_{-0.83-0.97-0.65-0.20-0.15} \times 10^{-8},     \\
{\mathcal B}(B^+\to \pi^+ \omega(782)\to \pi^+ K^0\bar K^0)
                    &=& 4.14^{+1.64+1.02+0.07+0.20+0.14}_{-1.32-0.94-0.08-0.16-0.14} \times 10^{-8},    \\
{\mathcal B}(B^+\to K^+ \omega(782) \to K^+ K^0\bar K^0)
                    &=& 8.79^{+1.65+2.30+1.17+0.42+0.30}_{-1.44-2.15-1.03-0.33-0.30} \times 10^{-8}.
\end{eqnarray}
It's easy to check that these branching fractions are very close to the results in table~\ref{Res-770} for the corresponding
decay modes with $\rho(770)^0$ and $\omega(782)$ decaying into $K^+K^-$. The impacts from the mass difference
of $K^\pm$ and $K^0$ for the direct $CP$ asymmetries for the relevant processes are even smaller, which could be inferred
from the comparison of the results in table~\ref{Res-770} with
\begin{eqnarray}
{\mathcal A}_{CP}(B^+\to \pi^+ \rho(770)^0\to \pi^+ K^0\bar K^0)
                   &=& -0.22^{+0.04+0.01+0.01+0.01}_{-0.04-0.01-0.01-0.01},    \\
{\mathcal A}_{CP}(B^+\to K^+ \rho(770)^0 \to K^+  K^0\bar K^0)
                   &=& \;\;\;0.39^{+0.03+0.04+0.04+0.01}_{-0.04-0.04-0.04-0.01}.
\end{eqnarray}
For the decay modes $B^+\to \pi^+ \rho(1450)^0$ and $B^+\to K^+ \rho(1450)^0$ with $\rho(1450)^0\to K^0\bar K^0$,
we have the central values $9.32\times10^{-8}$ and $-0.22$, $3.30\times10^{-8}$ and $0.42$ as their branching fractions and
direct $CP$ asymmetries, respectively, which are also very close to the results in table~\ref{Res-1450} for the corresponding decay
processes with $\rho(1450)^0\to K^+K^-$. In view of the large errors for the predictions in tables~\ref{Res-770}, \ref{Res-1450},
\ref{Res-1700}, we set the concerned decays with the subprocess $\rho(770, 1450, 1700)^0\to K^0\bar K^0$ or
$\omega(782, 1420, 1650)\to K^0\bar K^0$ have the same results as their corresponding decay modes with the resonances
decaying into $K^+K^-$. It should be stressed that the $K^0\bar K^0$ with the $P$-wave resonant origin in the final state of
$B\to K\bar K h$ decays can not generate the $K^0_SK^0_S$ system because of the Bose-Einstein statistics.

From the branching fractions in tables~\ref{Res-770}, \ref{Res-1450}, one can find that the virtual contributions for $K\bar K$
from the BW tails of the intermediate states $\rho(770)$ and $\omega(782)$ in those quasi-two-body decays which have
been ignored in experimental and theoretical studies are all larger than the corresponding results from $\rho(1450)$ and
$\omega(1420)$. Specifically, the branching fractions in tables~\ref{Res-770} with the resonances $\rho(770)^0$ and
$\rho(770)^\pm$ are about $1.2$-$1.8$ times of the corresponding results in tables~\ref{Res-1450} for the decays with
$\rho(1450)^0$ and $\rho(1450)^\pm$, while the six predictions for the branching fractions in tables~\ref{Res-770} with
$\omega(782)$ in the quasi-two-body decay processes are about $2.2$-$2.9$ times of the corresponding values for the
decays with the resonance $\omega(1420)$ in table~\ref{Res-1450}. The difference of the multiples between the results
of the branching fractions with the resonances $\rho$ and $\omega$ in table~\ref{Res-770} and table~\ref{Res-1450}
should mainly be attributed to the relatively small value for the $c^K_{\omega(1420)}$ adopted in this work comparing with
$c^K_{\rho(1450)}$.

\begin{figure}[tbp]
\centerline{\epsfxsize=9.0cm \epsffile{dep770.eps}   \hspace{-0.6cm}
                  \epsfxsize=9.0cm \epsffile{dep782.eps}}
\caption{The differential branching fractions for the decays $B^+\to \pi^+ [\rho(770)^0\to] K^+K^-$ (left) and
               $B^+\to \pi^+ [\omega(782)\to] K^+K^-$ (right). The big diagram in the left is for the comparison
               for the differential branching fractions of $B^+\to \pi^+ [\rho(770)^0\to] K^+K^-$ and
               $B^+\to \pi^+ [\rho(770)^0\to] \pi^+\pi^-$, in which the solid line for $B^+\to \pi^+ [\rho(770)^0\to] K^+K^-$
               is magnified by a factor of $10$.
              }
\label{fig-dB-770}
\end{figure}

It is remarkable about these virtual contributions in table~\ref{Res-770} that their differential branching fractions are nearly
unaffected by the full widths of $\rho(770)$ and $\omega(782)$, which could be concluded from the figure~\ref{fig-dB-770}.
In this figure, the lines in the left diagram for $B^+\to \pi^+ [\rho(770)^0\to] K^+K^-$ and in the right diagram for
$B^+\to \pi^+ [\omega(782)\to] K^+K^-$ have very similar shape although there is a big difference between the values for
the widths of $\rho(770)$ and $\omega(782)$ as listed in table~\ref{params}. The best explanation for figure~\ref{fig-dB-770} is
that the imaginary part of the denominator in the BW formula the eq.~(\ref{eq-BW}) which hold the energy dependent width
for the resonances $\rho(770)$ or $\omega(782)$ becomes unimportant when the invariant mass square $s$ is large enough
even if one employs the effective mass defined by the {\it ad hoc} formula~\cite{prd91-092002,prd90-072003} to replace
the $m^2_R$ in $\left| \overrightarrow{q_0}\right|$ in eq.~(\ref{eq-sdep-Gamma}) or calculates the energy dependent width
with the partial widths and the branching ratios for the intermediate state as in refs.~\cite{plb779-64,prd88-032013,
plb760-314,prd76-072012}. At this point, the BW expression for $\rho(770)$ or $\omega(782)$ is charged only by
the coefficient $c^K_R$ in the time-like form factors for kaons and the gap between the invariant mass square $s$
for kaon pair and the squared mass of the resonance. Although the threshold of kaon pair is not far from the pole masses
of $\rho(770)$ and $\omega(782)$, thanks to the strong suppression from the factor $\left| \overrightarrow{q}\right|^3$ in
eq.~(\ref{eqn-diff-bra}), the differential branching fractions for those processes with $\rho(770)$ or $\omega(782)$ decaying
into kaon pair will reach their peak at about $1.35$ GeV as shown in figure~\ref{fig-dB-770}.

As we have stated in ref.~\cite{prd101-111901}, the bumps in figure~\ref{fig-dB-770} for $B^+\to \pi^+ [\rho(770)^0\to] K^+K^-$
and $B^+\to \pi^+ [\omega(782)^0\to] K^+K^-$ are generated by the tails of the BW formula for the resonances $\rho(770)$ and
$\omega(782)$ along with the phase space factors in eq.~(\ref{eqn-diff-bra}) and should not be taken as the evidence for a new
resonant state at about $1.35$ GeV. When we compare the curves for the differential branching fractions for
$B^+\to \pi^+ [\rho(770)^0\to] K^+K^-$ and $B^+\to \pi^+ [\rho(770)^0\to] \pi^+\pi^-$, we can understand this point well.
In order to make a better contrast, the differential branching fraction for $B^+\to \pi^+ [\rho(770)^0\to] K^+K^-$ is magnified $10$
times in the big one of the left diagram of figure~\ref{fig-dB-770}. The dash-dot line for $B^+\to \pi^+ [\rho(770)^0\to] \pi^+\pi^-$
shall climb to its peak at about the pole mass of $\rho(770)^0$ and then descend as exhibited in figure~\ref{fig-dB-770}.
While this pattern is inapplicable to the decay process for $B^+\to \pi^+ [\rho(770)^0\to] K^+K^-$, which can only show its
existence from the threshold of kaon pair which has already crossed the peak of BW for $\rho(770)^0$. As $s$ becoming larger,
the effect of the full width for $\rho(770)$ fade from the stage, the ratio between the differential branching fractions for the
quasi-two-body decays $B^+\to \pi^+ [\rho(770)^0\to] K^+K^-$ and $B^+\to \pi^+ [\rho(770)^0\to] \pi^+\pi^-$ will tend to
be a constant which is proportional to the value of $|g_{\rho(770)K^+ K^-}/g_{\rho(770)\pi^+\pi^-}|^2$ if the phase space for
the decay process is large enough. This conclusion can also be demonstrated well from the curve of the ratio
\begin{eqnarray}
   R_{\rho(1450)}(\sqrt s)=\frac{d{\mathcal B}(B^+\to \pi^+ [\rho(1450)^0\to] K^+K^-)/d\sqrt{s}}
                                   {d{\mathcal B}(B^+\to \pi^+ [\rho(1450)^0\to] \pi^+\pi^-)/d\sqrt{s}}
   \label{def-R-1450}
\end{eqnarray}
for the decays $B^+\to \pi^+ [\rho(1450)^0\to] K^+K^-$ and $B^+\to \pi^+ [\rho(1450)^0\to] \pi^+\pi^-$ in figure~\ref{fig-dep1450}.
The solid line which stands for the $B^+\to \pi^+ [\rho(1450)^0\to] K^+K^-$ decay and has been magnified $10$ times will arise at
the threshold of kaon pair in in figure~\ref{fig-dep1450} and contribute the zero for $R_{\rho(1450)}$ because of the factor
$\left| \overrightarrow{q}\right|^3$ in eq.~(\ref{eqn-diff-bra}), and the following for $R_{\rho(1450)}$ is a rapid rise to the value
about $0.1$ in the region where the main portion of the branching fractions for $B^+\to \pi^+ [\rho(1450)^0\to] K^+K^-$ and
$B^+\to \pi^+ [\rho(1450)^0\to] \pi^+\pi^-$ concentrated, then $R_{\rho(1450)}$ is going to the value
$|g_{\rho(1450)K^+ K^-}/g_{\rho(1450)\pi^+\pi^-}|^2$ as the rise of $s$.

\begin{figure}[tbp]
\centerline{\epsfxsize=10.0cm \epsffile{dep1450.eps}}
\caption{The differential branching fractions for the decays $B^+\to \pi^+ [\rho(1450)^0\to] K^+K^-$ (solid line) which
                is magnified by a factor of $10$ and $B^+\to \pi^+ [\rho(1450)^0\to] \pi^+\pi^-$(dash-dot line) in the large diagram
                and curve for the $\sqrt s$ dependent ratio $R_{\rho(1450)}$ in the small one.
              }
\label{fig-dep1450}
\end{figure}

With the factorization relation
$\Gamma(B^+\to \rho(1450)^0\pi^+\to h^+h^-\pi^+)\approx\Gamma(B^+\to \rho(1450)^0\pi^+){\mathcal B}(\rho^0\to h^+h^-)$
\cite{2011-03201,2011-07468}, the ratio $R_{\rho(1450)}$ can be related to the the coupling constants $g_{\rho(1450)^0\pi^+\pi^-}$
and $g_{\rho(1450)^0K^+K^-}$ with the expression
\begin{eqnarray}
  g_{\rho(1450)^0h^+h^-}=\sqrt{\frac{6\pi m^2_{\rho(1450)}\Gamma_{\rho(1450)}{\mathcal B}_{\rho(1450)^0\to h^+h^-}}{q^3}}\,,
  \label{def-cocont}
\end{eqnarray}
here $q=\frac{1}{2}\sqrt{m^2_{\rho(1450)}-4m_h^2}$ and $h$ is pion or kaon. Utilizing the relation
$g_{\rho(1450)^0K^+K^-}\approx \frac12 g_{\rho(1450)^0\pi^+\pi^-}$~\cite{epjc39-41} one has~\cite{prd101-111901}
\begin{eqnarray}
  R_{\rho(1450)} &=& \frac{{\mathcal B}(\rho(1450)^0\to K^+ K^-)}{{\mathcal B}(\rho(1450)^0 \to\pi^+\pi^-)}
                           \approx \frac{g^2_{\rho(1450)^0K^+K^-} (m^2_{\rho(1450)}-4m^2_K)^{3/2}}
                                                {g^2_{\rho(1450)^0\pi^+\pi^-}(m^2_{\rho(1450)}-4m^2_\pi)^{3/2}} \nonumber\\
                         &=&  0.107.
   \label{th-Rho-1450}
\end{eqnarray}
For the quasi-two-body decay $B^+\to \pi^+ [\rho(1450)^0\to] \pi^+\pi^-$, we have its branching fraction as
$8.73^{+2.73}_{-2.54}\times 10^{-7}$ with the BW formula for $\rho(1450)^0$ and the relation
$|c^K_{\rho(1450)}|  \approx  |c^\pi_{\rho(1450)}|$ in eq.~(\ref{rela-ck1450}), where the error has the same sources as the branching
fractions in table~\ref{Res-1450} but have been added in quadrature.
This result are consistent with the measurements ${\mathcal B}=1.4^{+0.6}_{-0.9}\times10^{-6}$~\cite{prd79-072006,PDG-2020}
from BaBar and ${\mathcal B}=(7.9\pm3.0)\times 10^{-7}$~\cite{prl124-031801,prd101-012006} by LHCb and agree with the
prediction $(9.97\pm2.25)\times 10^{-7}$ in~\cite{prd101-111901} with the GS model for the resonance $\rho(1450)^0$. Then we
have the ratio $R_{\rho(1450)}= 0.108^{+0.000}_{-0.001}$ which is very close to the $0.107$ in eq.~(\ref{th-Rho-1450}) and
the result in figure~\ref{fig-dep1450} for the ratio $R_{\rho(1450)}(\sqrt s)$ in the region around the mass of $\rho(1450)$ where the
main portion of the branching fractions for $B^+\to \pi^+ [\rho(1450)^0\to] K^+K^-$ and $B^+\to \pi^+ [\rho(1450)^0\to] \pi^+\pi^-$
concentrated. The small error for $R_{\rho(1450)}$ from the PQCD predictions is caused by the cancellation, which means that the
increase or decrease for the relevant numerical results from the uncertainties of those parameters will result in nearly identical change
of the weight for these two decays. When the ${\rho(1450)^0}$ in eq.~(\ref{th-Rho-1450}) is replaced by ${\rho(1700)^0}$, one will
have the ratio $R_{\rho(1700)}\approx 0.143$~\cite{prd101-111901}. With the results ${\mathcal B}(\rho(1450)^0\to\pi^+\pi^-)=15\%$
and ${\mathcal B}(\rho(1700)^0\to\pi^+\pi^-)=14\%$ in ref.~\cite{jhep2001-112} from CMD-3 collaboration, one can estimate the
branching fractions ${\mathcal B}(\rho(1450)^0\to K^+K^-)\approx1.6\%$ and ${\mathcal B}(\rho(1700)^0\to K^+K^-)\approx2.0\%$.

It is important to notice that the definition of the coupling constant the eq.~(\ref{def-cocont}) for the resonant states $\rho(770)$ and
$\omega(782)$ decaying to the final state $K\bar K$ are invalid, or rather, one could not define the partial decay width such as
$\Gamma_{\rho(770)\to K^+K^-}=\Gamma_{\rho(770)}{\mathcal B}_{\rho(770)^0\to K^+ K^-}$ or
$\Gamma_{\omega(782)\to K^+K^-}=\Gamma_{\omega(782)}{\mathcal B}_{\omega(782)\to K^+ K^-}$ for the virtual contribution.
This conclusion can be extended to other strong decay processes with the virtual contributions which come from the tails of the
resonances.

In ref.~\cite{prl123-231802}, the fit fraction of $\rho(1450)^0\to  K^+K^-$ for the three-body decays $B^\pm\to \pi^\pm K^+K^-$
was measured to be $(30.7\pm1.2\pm0.9)\%$ by LHCb collaboration, implying ${\mathcal B}=(1.60\pm0.14)\times10^{-6}$ for the
quasi-two-body decay $B^+\to\pi^+\rho(1450)^0 \to\pi^+ K^+K^-$~\cite{PDG-2020}. This branching fraction is close to the
measurement ${\mathcal B}=1.4^{+0.6}_{-0.9}\times10^{-6}$ in~\cite{prd79-072006,PDG-2020} and larger than the result
${\mathcal B}=(7.9\pm3.0)\times 10^{-7}$ from LHCb~\cite{prl124-031801,prd101-012006} for the
$B^+\to \pi^+ [\rho(1450)^0\to] \pi^+\pi^-$ process. In view of the mass difference between kaon and pion, the factor
$\left| \overrightarrow{q}\right|^3$ in eq.~(\ref{eqn-diff-bra}) will be about $4.76$ times larger for the subprocess
$\rho(1450)^0\to \pi^+\pi^-$ when comparing with the $\rho(1450)^0\to K^+K^-$ for the decay $B^+\to\pi^+\rho(1450)^0$ at
$s=m^2_{\rho(1450)}$. It means that the coupling constant for $\rho(1450)^0\to K^+K^-$ should roughly be $\sqrt{4.76}$ times
larger than that for $\rho(1450)^0\to \pi^+\pi^-$ in order to achieve the comparable branching fractions for the quasi-two-body
decays $B^+\to\pi^+[\rho(1450)^0 \to] K^+K^-$ and $B^+\to \pi^+ [\rho(1450)^0\to] \pi^+\pi^-$. Clearly, a larger coupling constant
for $\rho(1450)^0\to K^+K^-$ is contrary to the naive expectation~\cite{2007-02558} and the discussions in literature~\cite{epjc39-41,
plb779-64}.

\section{Summary}  \label{sec-con}

In this work, we studied the contributions for kaon pair originating from the resonances $\rho(770)$, $\omega(782)$ and their
excited states $\rho(1450,1700)$ and $\omega(1420,1650)$ in the three-body decays $B\to K\bar K h$ in the PQCD approach.
The subprocesses $\rho(770,1450,1700)\to K\bar K$ and $\omega(782,1420,1650)\to K\bar K$, which can not be calculated in
the PQCD, were introduced into the distribution amplitudes for $K\bar K$ system via the kaon vector time-like form factors.
With the coefficients $c^K_{\rho(770)}=1.247\pm0.019$, $c^K_{\omega(782)}=1.113\pm0.019$, $c^K_{\rho(1450)}=-0.156\pm0.015$,
$c^K_{\omega(1420)}=-0.117\pm0.013$ and $c^K_{\omega(1650),\rho(1700)}=-0.083\pm0.019$ in the time-like form factors
for kaons, we predicted the $CP$ averaged branching fractions and the direct $CP$ asymmetries for the quasi-two-body
processes $B\to \rho(770,1450,1700) h\to K\bar K h$ and $B\to\omega(782,1420,1650) h \to K\bar K h$.

The branching fractions of the virtual contributions for $K\bar K$ from the BW tails of the intermediate states $\rho(770)$ and
$\omega(782)$ in the concerned decays in this work which have been ignored in experimental and theoretical studies were
found larger than the corresponding results from $\rho(1450,1700)$ and $\omega(1420,1650)$. A remarkable phenomenon
for the virtual contributions discussed in this work is that the differential branching fractions for $B\to \rho(770) h\to K\bar K h$
and $B\to\omega(782) h \to K\bar K h$ are nearly unaffected by the quite different values of the full widths for $\rho(770)$ and
$\omega(782)$. The definition of the partial decay width such as
$\Gamma_{\rho(770)\to K^+K^-}=\Gamma_{\rho(770)}{\mathcal B}_{\rho(770)^0\to K^+ K^-}$ for the virtual contribution are
invalid. This conclusion can be extended to other strong decay processes with the virtual contributions come from the tails
of the resonances. The bumps of the lines for the differential branching fractions for those virtual contributions, which are
generated by the phase space factors and the tails of the BW formula of $\rho(770)$ or $\omega(782)$, should not be taken
as the evidence for a new resonant state at about $1.35$ GeV.

The PQCD predicted results for the branching fractions of the quasi-two-body decays $B^+\to \pi^+ \rho(1450)^0\to \pi^+K^+K^-$
and $B^+\to \pi^+ \rho(1450)^0\to \pi^+\pi^+\pi^-$ meet the requirement of the $SU(3)$ symmetry relation
$g_{\rho(1450)^0K^+K^-}\approx \frac12 g_{\rho(1450)^0\pi^+\pi^-}$. The larger coupling constant for $\rho(1450)^0\to K^+K^-$
deduced from the fit fraction $(30.7\pm1.2\pm0.9)\%$ for $\rho(1450)^0\to  K^+K^-$ in the $B^\pm\to \pi^\pm K^+K^-$ decays
by LHCb collaboration is contrary to the discussions in literature. We estimated the branching fractions to be about $1.6\%$ and
$2.0\%$ for the decays $\rho(1450)^0\to K^+K^-$ and $\rho(1700)^0\to K^+K^-$, respectively, according to the measurement
results from CMD-3 collaboration for $\rho(1450,1700)^0\to\pi^+\pi^-$.

\begin{acknowledgments}
This work was supported in part by the National Natural Science Foundation of China
under Grants No.~11547038 and No.~11575110.
\end{acknowledgments}

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text:
City of Mission Viejo, CA
Provide input during The Gardens property owner's environmental scoping[...]
Idaho Farm Bureau Federation
Sensor helps farmers make real-time decisions to raise crops
Duke Energy Corporation
Duke Energy crews restore power to more than 250,000 customers after[...]
Ministry of Road Transport and Highways of the Republic of India
Shri Nitin Gadkarisays National automobile Scrappage policy is win-win policy for all the stakeholders
Ministry of Road Transport & Highways
Posted On: 23 NOV 2021 2:37PM by PIB Delhi
Union Minister for Road Transport and Highways Shri Nitin Gadkari has said that National automobile Scrappage policy is mutual win-win policy for all the stakeholders. Inaugurating the the Vehicle Scrapping and Recycling Facility set upby Maruti Suzuki Toyotsu India Private Limited (MSTI)at Noida today he said the policy aims at creating an ecosystem for phasing out unfit and polluting vehicles from the Indian roads and to meet this objective state-of-the-art scrapping and recycling units are needed.
The Minister said the policy will help in increasing Automobile sales,provide employment,reduce import cost,generate incremental GST revenue and help to solve the global shortage of semi conductor chip. Shri Gadkari said policy provides an important link in creating circular economy (waste to wealth) for the country.He said three to four scrapping centres should be set up in every district.
The Ministry of Road Transport and Highways has recently launched thevisionary Voluntary Vehicle-Fleet ModernisationProgram (Vehicle Scrapping Policy). This Policy is aimed at creating an ecosystem for phasing out old unsafe, polluting vehicles and replacing them with newer safe and fuel-efficient vehicles. The Ministry has also notified the Registered Vehicle Scrapping Facility (RVSF) Rules to enable setting up of modern vehicle scrapping facilities in India which can recycle the vehicles in an environment friendly manner.
The existing vehicle dismantling and recycling industry in Indiais unorganized and recycling of older vehicles is not done in an environment-friendly manner. This not only leads to inefficient resource management, but also non-realisation of the full value of high-strength steel alloys and recovery of valuable rare-earth metals. The existing ELV scrapping cycle has a low recovery percentage and many materials get wasted or are not properly recycled. As per unofficial industry estimates, the recovery percentage in India is about 70-75%, while global benchmarks for recovery from scrapped vehicle are in the range of 85-95%.
Since the prevailing processes in the unorganized sector for recycling/ scrapping are not carried out in an environment-friendly manner, it is the endeavour of this Ministry to promote setting up of RVSF, to enable use of environmentally-friendly processes in this regard.The Vehicle Scrapping Policy thereby createssuch formal business opportunity & promote scientific recycling within the scrapping industry.
The RVSF inaugurated by Minister has been set up in an area of 11,000 sqm with a capacity of handling 24,000 vehicle per year. The plant has been set up by Maruti Suzuki India Ltd in collaboration with Toyota Tshusho Group.Shri Gadkari congratulated Maruti Suzuki and Toyota Tshoshu Group and also requested other stakeholders to come forward to create similar state-of-the-art scrapping and recycling facilities to help achieve the objectives of the Policy.
H.E.Mr Satoshi Suzuki, Ambassador Extraordinary and Plenipotentiary, Embassy of Japan, India was also present at the inauguration function.
MJPS
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text:
Dr. Brinker has performed more than 1,000 Ilizarov surgeries
Dr. Brinker has treated more than 1,200 fracture nonunions
Dr. Brinker has corrected more than 400 skeletal deformities
Dr. Brinker has treated more than 250 cases with bone infection
Mark R. Brinker, MD
About Dr. Brinker
Invited Faculty/Lecturer
Publications - Journals
Publications - Textbooks
Publications - Book Chapters
Publications - Abstracts
Publications - Proceedings
Posters & Exhibits
Submitted For Publication
Courses/Meetings Hosted
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Fracture Nonunions
Infected Nonunions
Complex Fractures
Bone Infections
Periprosthetic Fractures/Non unions
Limb Lengthening
Bone/Joint
Nonunion Fracture Case Studies
Infected Nonunion Case Studies
Deformity Case Studies
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Bone Infection Case Studies
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Bone Transport
Edwin G. Bovill Award Winning Highlight Paper, Presented at the American Academy of Orthopaedic Surgeons 75th Annual Meeting Specialty Day, San Francisco, California, March, 2008.
Most Outstanding Lecturer, Colorado Orthopaedic Review Course, Colorado Springs, Colorado, May, 2002.
First Runner-up, Most Outstanding Lecturer, Colorado Orthopaedic Review Course, Colorado Springs, Colorado, May, 2001.
American Orthopaedic Association, Elected June 4, 1999.
Faculty Advisor and Co-Author, Runner-up for the Outstanding Paper at the Haddad Memorial Lectureship, Tulane University, New Orleans, Louisiana, April, 1999.
Faculty Advisor and Co-Author, First Place Winner of the Duncan Lectureship Resident Research Award, St. Lukes Medical Center, Department of Orthopaedics, Cleveland, Ohio, May, 1997.
Faculty Advisor, Association for Academic Surgery Surgical Student Research Award. Award presented by the Association for Academic Surgery, 1995.
Faculty Advisor and Co-Author, First Place Winner of the Meyerding Essay Contest, 1995. Award presented at the 57th Annual Meeting and Scientific Program of the American Fracture Association, San Juan, Puerto Rico, May, 1995.
American Orthopaedic Association John J. Fahey, MD, Memorial North American Traveling Fellowship, 1994. Award presented at the 108th Annual Meeting of the American Orthopaedic Association, White Sulfur Springs, West Virginia, June 1995.
Faculty Advisor and Co-Author, American Orthopaedic Association Residents Travel Award for Excellence in Orthopaedic Research, 1994. Award presented at the 107th Annual American Orthopaedic Association, Sun Valley, Idaho, June, 1994.
Rancho Los Amigos Award for Excellence in Orthopaedic Research, 1991. Award presented at the Rancho Los Amigos Annual Winter Orthopaedic Research Meeting, Downey, California, December, 1991.
Southern Orthopaedic Association First Place Award for the Most Outstanding Research Paper Presented at the 7th Annual Residents and Fellows' Conference, Nashville, Tennessee, October, 1990. Award presented at the 8th Annual Southern Orthopaedic Association, Colorado Springs, Colorado, August, 1991.
Harry D. Morris Award for the Most Outstanding Research Paper Presented at the 37th Annual
Louisiana Orthopaedic Association, New Orleans, Louisiana, October, 1990.
Mid-America Orthopaedic Association Annual Traveling Resident Award, 1990-1991. Award presented at the 10th Annual Mid-America Orthopaedic Association, Orlando, Florida, April, 1992.
Smith and Nephew Richards Residents Award for Orthopaedic Research (First Runner-up). Award presented at the 8th Annual Smith and Nephew Richards Invitational Orthopaedic Resident Conference, Memphis, Tennessee, August, 1990.
American Orthopaedic Association Residents Travel Award for Excellence in Orthopaedic Research, 1990. Award presented at the 103rd Annual American Orthopaedic Association, Boston, Massachusetts, June, 1990.
Veterans Administration Outstanding Rating Certificate, 1989.
Tulane University Department of Orthopaedic Surgery Annual Award for Excellence in
the Field of Spine Research, 1989.
Choose Your Affected Bone/Joint
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The Hubble Space Telescope (HST) is celebrating its 25th year in space. Launched in 1990, the HST fit snugly within the confines of the Space Shuttle's Payload Bay, which had a length of 18 meters, or almost 60 feet. The HST overcame its initial flaws and went on to become a historic telescope, thanks to the intrepid high–flying Shuttle repair crews and the folks on the ground. Indeed, after the fixes, the effective Focal Length of the HST lived up to its design of a whopping 57.6 meters (189 feet)!
Wait. The Space Shuttle Payload Bay was 18 m long, and the HST has a focal length of almost 58 m long. That's greater than the entire length of the Shuttle itself at 37 m (122 ft)! So how did the HST fit inside the Space Shuttle?
Enter the Cassegrain Telescope, a design that allows for a magnification factor! The HST is a Ritchey–Chrétien variant, which uses hyperbolic mirrors. A hole in the center of the Primary Mirror (M1)allows the reflected light from the smaller Secondary Mirror (M2)to pass through to the Focal Surface (F) located behind the Primary Mirror.
Extended diagram of a Cassegrain telescope.
So let's check those Effective Focal Length figures, shall we?
Looking up the specifications of the HST, we find that the Focal Length of the Primary Mirror (Fp) is 5.52 m, the mirrors are separated by a distance (Md) of 4.91 m, and the Focal Surface (F) is 1.5 m behind the Primary Mirror.
Our calculation of 58.0 m is very close to the published figure of 57.6 m. This means that we can indeed fit a telescope with a large Focal Length inside a smaller cylindrical length. Sometimes great things really do come in small packages!
This also means that the HST magnifies the Primary Mirror Focal Length by a factor of ten. Holy Hubble Constant, Batman! No wonder it can see so far.
Despite its rough start, the HST has certainly earned its place in history, not only through the efforts of teams of people acting as one to give us mind–altering images of the Universe, but also by the use of clever mirror reflection techniques that allowed us to reach back even further in time and space.
Happy 25th Anniversary, Hubble Space Telescope!


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Taking care by Mr. David Kaipeng NGO, this center is 50 Km away from main City. People in this area find difficult in basic needs in medical and first aids .Thus Rahi A Social and Cultural Upliftment Society lunch Free Medical camp every year. This Globe Education and learning with NGO RAHI to make sure that their initiatives have actual effect. Ngo RAhi likewise help all various other NGO in India enhance their systems as well as train regional team to be much more reliable. Globe Education and learning likewise links our programs to NGO in India that give solutions outside the range of our job, such as healthcare. This collaboration makes it possible for individuals we offer to take advantage of a bigger variety of aid.


text:
Boston, Massachusetts is home to many listings on the National Register of Historic Places. This list encompasses those locations that are located north of the Massachusetts Turnpike. See National Register of Historic Places listings in southern Boston for listings south of the Turnpike. Properties and districts located elsewhere in Suffolk County's other three municipalities are also listed separately.

There are 341 properties and districts listed on the National Register in Suffolk County, including 58 National Historic Landmarks. The northern part of the city of Boston is the location of 148 of these properties and districts, including 44 National Historic Landmarks.

Current listings

|}

Former listing

|}

See also
 List of National Historic Landmarks in Massachusetts
 National Register of Historic Places listings in Suffolk County, Massachusetts

References

Northern

redpajama_set_name: RedPajamaWikipedia


text:
Mobile: mobile.captivasanibel.com
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James Hermes
June 12, 2015 James Hermes, 74, of Naples, FL, formerly of Sanibel Island, FL and Youngstown, Ohio passed away in his home on June 5, 2015. Jim was a beloved husband, father, grandfather, and brother. more »»
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June 5, 2015 Jean G. Loubier, 80, of Fort Myers, FL died on June 3, 2015. Arrangements by Affordable Cremations and Horizon Funeral Home, Fort Myers, FL. www.horizonfunerals.com. more »»
Kenneth R. Karasin
May 1, 2015 Kenneth R. Karasin, a loving brother, father, grandfather, uncle, and friend died peacefully in his sleep Saturday April 25th, 2015 in his home in Sanibel, Florida. more »»
Patricia Ann Clemons
April 29, 2015 Patricia Ann Clemons, 86, of Sanibel passed away on April 16, 2015. Cremation arrangements entrusted to Harvey-Engelhardt Funeral and Cremation Services. www.harvey-engelhardt. more »»
John B. Biscello
April 28, 2015 John B. Biscello 89 of Fort Myers passed away on April 23, 2015. Please visit www.harvey-engelhardt.com to leave a condolence. more »»
Loren Richard Yeakley
April 28, 2015 Loren Richard Yeakley 83 of Newark, OH passed away on April 21, 2015. Cremation arrangements entrusted to Harvey-Engelhardt Funeral and Cremation Services. Please visit www.harvey-engelhard. more »»
Elsie F. Grieger
April 28, 2015 Elsie F. Grieger 102 of Fort Myers, FL passed away on April 23, 2015. Please visit www.harvey-engelhardt.com to leave a condolence. more »»
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April 22, 2015 Judy Michie was born December 12, 1938 in Kansas City to Stanley and Arleta Hamilton. She attended Kansas State, majoring in Journalism. more »»
Melissa Rae Godfrey
April 22, 2015 Melissa Rae Godfrey was born on March 12, 1980 in Coupeville, WA. She went peacefully to be with the angels on April 19, 2015 in Fort Myers, FL. She was 35 years old. more »»
Michael Alteri
April 17, 2015 Michael Alteri of Wolcott, New York died peacefully in his sleep on Sunday, March 8, 2015. more »»
Mary Cudahy Keogh-Stringer
April 15, 2015 Mary Cudahy Keogh-Stringer was born on November 11, 1920 in Milwaukee, Wisconsin. Her parents were the Honorable John Cudahy and Katherine Reed Cudahy of Milwauke. more »»
William Francis Olinger, Esq.
April 15, 2015 William Francis Olinger, Esq., 76, of Sanibel, FL died Feb. 25, 2015 at Health Park Memorial Center, Fort Myers, FL after a short battle with lung cancer. more »»
William Paul Wollschlager
April 15, 2015 William Paul Wollschlager 86 died peacefully on March 6, 2015 at Hope Hospice at Shell Point. He was born December 9, 1928 in Glendive, MT. more »»
William Albert Snyder
April 15, 2015 William Albert Snyder, 67, of Sanibel, FL died on March 9, 2015. Arrangements by Affordable Cremations and Horizon Funeral Home, Fort Myers. www.horizonfunerals.com. more »»
Howard L. "Howie" Cosyns,
April 15, 2015 Howard L. "Howie" Cosyns, age 93, of Tucson, AZ, Sanibel, FL (1985-2003), and Naperville, IL (1947-1985) passed away peacefully on March 17, 2015. more »»
Roderick M. Heitman
April 15, 2015 Roderick M. Heitman, 73 of Fort Myers passed away on March 10, 2015. Cremation arrangements entrusted to Harvey-Engelhardt Funeral Home. Please visit www.harvey-engelhardt.com to leave a condolence. more »»
George Downing
April 15, 2015 George Downing passed away on February 26, 2015 in Sanibel, Florida. Beloved husband of Molly Bruce Downing. Dear father of Matthew H. Downing (Debby) and Gwynne Downing Gretter (Craig). more »»
Howard L. "Howie" Cosyns
March 25, 2015 Howard L. "Howie" Cosyns, age 93, of Tucson, AZ, Sanibel, FL (1985-2003), and Naperville, IL (1947-1985) passed away peacefully on March 17, 2015. more »»
Martha "Cookie" Cruickshank Mlakar
March 3, 2015 Martha "Cookie" Cruickshank Mlakar Much loved daughter, sister, wife, mother, grandmother, great-grandmother and friend to many. Au Revoir Please- be not sad If I should Leave you all. more »»
Theresa Elaine Louwers
March 3, 2015 Theresa Elaine Louwers, 65, a longtime resident of Sanibel, passed away on February 17, 2015. She was a beloved wife, mother, and grandmother and a friend to all that knew her. more »»
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text:
Browne: Americas summit productive for Trinidad and Tobago
Clint Chan Tack
Prime Minister Dr Keith Rowley with US President Joe Biden and First Lady Dr Jill Biden at the inaugural event of the IX Summit of the Americas in Los Angeles, California on June 8 - COURTESY OFFICE OF THE PRIME MINISTER
THE Ninth Summit of the Americas in Los Angeles, California, was very productive for Trinidad and Tobago, Foreign and Caricom Affairs Minister Dr Amery Browne said on Saturday.
Browne and Energy Minister Stuart Young were part of a delegation to the summit that was led by the Prime Minister. They left TT for Los Angeles on June 6. The summit ended on June 10.
"The summit was extremely productive for TT and for the Caribbean Community (Caricom)," he said.
Browne said the week and the build-up to the summit "marked one of our most intensive periods of diplomatic engagement." He thanked TT Ambassador to Washington DC Anthony Phillips-Spencer, the staff at the TT Embassy in Washington DC and ministry officials for their contributions towards a significant volume of preparatory work.
All of it provided the platform for Dr Rowley to "make fruitful and effective interventions, and for the Trinidad and Tobago delegation to fulfill its mandate within a coordinated Caricom presence."
Browne said he was proud to witness Rowley's "display of majestic statesmanship during his contributions to an intense Caricom caucus meeting on Wednesday (June 8)."
Rowley, he continued, performed at the same level on June 9 during his presentation of key perspectives and recommendations at a highly significant subsequent meeting on June 8 between Caribbean leaders, US President Joe Biden and Vice-President Kamala Harris
Browne also held discussions with US Secretary of State Antony Blinken; Organisation of American States (OAS) Secretary General Luis Almagro; his Costa Rican, Argentine and Caricom counterparts and other top foreign diplomats at the summit last week.
Those talks dealt with national priorities as well as Caricom agenda items such as food security, energy security and regional financing.
On June 9, Young held talks with US Special Presidential Envoy for Climate, John Kerry.
Those discussions dealt with TT's agreement to sign the Global Methane Pledge as well as its plans to identify methane emissions and reduction.
Young said he also discussed with Kerry, "plans to utilise renewable energy in our electricity grid, starting with Project Lara and the 112.2 MW to be produced via solar and the intention to pursue more solar and studies being done to ascertain the feasibility of wind generated power."
In an address at the summit's second plenary session on June 10, Rowley said the summit "provides a platform of opportunity and promise that should not be missed."
He referred to the challenges posed to the world by the covid19 pandemic over the last two years and by the Russian invasion of Ukraine on February.
"It is my considered view that the compounded and reinforced challenges presented to global supply chains, energy and food security, among others should sharpen our focus on the risks that we all face now and the damage that can be done to future generations if we don't get it right."
Finance Minister Colm Imbert is acting prime minister until Rowley returns.
"Browne: Americas summit productive for Trinidad and Tobago"


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In addition to specific DG listed on the DG Goods Announcement page, Wan Hai Lines also complies with all international laws and regulations; cargo booking of whale products nor shark products will be strictly refused.


text:
Edward William Burns Jr.
Wednesday, May 22nd, 1946 - Saturday, August 20th, 2022
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New Milford – Edward W. Burns Jr., age 76, of New Milford passed away at the Candlewood Nursing Home on August 20, 2022. Burns was born on May 22, 1946 in Torrington, Connecticut, son of the late Edward and Mary Burns.
He graduated from the ... Read More
He graduated from the Litchfield High in 1964. He served in the Army a Veteran of the Vietnam War. He was employed at Kimberly-Clark for 30 years, a part time worker at Northville Store and Home Depot.
Edward is survived by his wife, Lorraine Burns of 51 years, two children; Stacy and her husband Chris White and EJ and his wife, Megan, five grandchildren; Jack, Nathan, Erin, Colin and Abby and several nieces and nephews. He is also survived by his brother, Jim Burns from Torrington and his four sisters, Pat Switzer of Torrington, CT, Laurene Ford of Ellsworth, Maine, Mary Samok of Sarasota, Florida and Kathleen Hanecak of Westerly, Rhode Island.
Burnsey was predeceased by his parents, Edward and Mary Burns from Bantam, Connecticut and a brother, William and sister, Eileen.
The Burn's family would like to thank the staff at the Candlewood Nursing Home for their love, support and kindness that they gave to Ed and his entire family.
In Lieu of flowers memorial contributions may be made to Tunnell to Towers, The Fisher House, connect.fisher.org or a charity of your choice.
A Mass of Christian Burial will be celebrated on Friday, August 26, 2022 at Our Lady of Grace Church, 715 Bantam Road, Bantam. Burial will follow in Bantam Cemetery. Calling hours will be held on Thursday, August 25, 2022 from 4:00-7:00pm at Thurston Rowe Funeral Home, 283 Torrington Road, Litchfield.
To send flowers to the family of Edward William Burns Jr., please visit our Heartfelt Sympathies Store.
Thursday, August 25th, 2022 | 4:00pm - 7:00pm
Thursday, August 25th, 2022 4:00pm - 7:00pm
Friday, August 26th, 2022 | 11:00am
Friday, August 26th, 2022 11:00am
Our Lady of Grace Church
715 Bantam Road
Bantam, Connecticut 06750
The Rev. Robert F. Tucker
Bantam Cemetery
715 Bantam Road (Route 202)
Edward Burns
We encourage you to share your most beloved memories of Edward here, so that the family and other loved ones can always see it. You can upload cherished photographs, or share your favorite stories, and can even comment on those shared by others.
Provide comfort for the family of Edward William Burns Jr by sending flowers.
Jennie Burns
I am so sorry to all of Eddie family. May he Rest In Peace.
Gary Hults
So sorry for your loss. I worked with Ed at KC. May God's peace and love be with you at this time.
Toni Partrick Kowalec
Sorry to hear. R. I.P. Condolences for family and friend Dickie.
A Memorial Tree was planted for Edward Burns


The Jews were singled out for special hatred by the Nazis, who viewed them as symbols of capitalism. "The party . . . combats the Jewish-materialist spirit within and without us," they wrote in their "25-Point Platform of the Nazi Party." And as Nazi apologist Paul Lensch wrote in his book, Three Years of World Revolution (p. 176), the ideas of "freedom and civic right, of constitutionalism and parliamentarianism . . . derived from that individualistic conception of the world," must be gotten "rid of to assist in the growth of a new conception of State and Society. In this sphere also Socialism must present a conscious and determined opposition to individualism" (emphasis added).