# CP asymmetry from the effect of the isospin symmetry breaking during B-meson
decay

Wan-Ying Yao1111Email<EMAIL_ADDRESS>Gang Lü1222Corresponding author
Email<EMAIL_ADDRESS>Hai-Feng Ou1333Email<EMAIL_ADDRESS>1
Institute of Theoretical Physics, College of Science, Henan University of
Technology, Zhengzhou 450001, China

###### Abstract

The direct CP asymmetry in quasi-two-body decays of
$B\rightarrow(V\rightarrow\pi^{+}\pi^{-})P$ is investigated in the
perturbative QCD (PQCD) method, where P represents a pseudoscalar meson and V
refers to $\rho$, $\omega$ and $\phi$ mesons. We present the amplitude of the
quasi-two-body decay process and investigate the effects of mixed resonance
involving $\rho^{0}-\omega$, $\rho^{0}-\phi$ and $\omega-\phi$, while
considering the impact of isospin symmetry breaking. We observe a significant
CP asymmetry when the invariant mass of the $\pi^{+}\pi^{-}$ pair is within
the resonance ranges of $\rho$, $\omega$ and $\phi$. Consequently, we proceed
to quantify the regional CP asymmetry in these resonance regions. A
significant difference is observed when comparing results obtained with and
without interference and isospin conservation. The CP asymmetry results
obtained from the three-body decay process, without interference due to
isospin conservation by the PQCD method, are in agreement with the newly
updated data acquired by the LHCb experiment.

##  I Introduction

CP asymmetry, which plays a crucial role in elucidating the matter-antimatter
asymmetry of the Universe, holds significant theoretical and experimental
significance in the field of Particle physics Shaposhnikov:1991cu ;
Okun:1967cxa . The breaking of CP symmetry can originate from weak decay of
hadron, neutral hadron mixing, or the interference between decay and mixing
Cabibbo:1963yz . In the framework of the Standard Model (SM), CP asymmetry
arises due to the presence of a weak complex phase in the Cabibbo-Kobayashi-
Maskawa (CKM) matrix, which characterizes the transition from mass eigenstates
to weak interaction eigenstates of quarks. CP asymmetry has been
experimentally observed in the K, B, and D meson decay process KTeV:1999aiu ;
BaBar:2001oxa ; Belle:2001zzw . The weak phase difference is determined by the
CKM matrix elements, while the strong phase can arise from the hadronic matrix
elements and interference among intermediate states. Recently, CP asymmetry
has been observed in the course of the B-meson multi-body decay experiment
LHCb:2019jta ; LHCb:2022tuk . The investigation of CP asymmetry in multi-body
decays of B mesons plays an increasingly pivotal role in both scrutinizing the
CKM mechanism of the Standard Model and exploring novel sources of CP
asymmetry.

In contrast to the two-body decay, the three-body decay of B meson encompasses
both resonance and non-resonance contributions, as well as being influenced by
final state re-scattering in $KK\rightarrow\pi\pi$ interactions LHCb:2013ptu .
Therefore, it exhibits a greater range of diverse characteristics pertaining
to CP asymmetry information, rendering it deserving of attention. The
observation of significant CP asymmetry in localized regions of phase space
for charmless three-body B-meson decays has been reported through model-
independent analyses. In the LHCb experiment, researchers extensively
investigated the complete phase space of B meson decay, focusing on a specific
invariant mass region.The data used for the integrated CP asymmetries in full
phase space, $A_{CP}^{inte}$, are obtained from the decay process
$B^{\pm}\rightarrow\pi^{+}\pi^{-}\pi^{\pm}$. Similarly, the CP asymmetry
values are derived from the decay process of
$B^{\pm}\rightarrow\pi^{+}\pi^{-}\pi^{\pm}$, resulting in
$A_{CP}^{low}=58.4\pm 8.2\pm 2.7\pm 0.7$ and in the rescattering regions with
1.0 GeV$<m_{\pi^{+}\pi^{-}}<$ 1.5 GeV, for the low invariant mass region with
$m^{2}_{\pi^{+}\pi^{-}\rm~{}low}<0.4~{}{\rm GeV}^{2}$ and high invariant mass
region with $m^{2}_{\pi^{+}\pi^{-}\rm~{}high}<15~{}{\rm GeV}^{2}$,
respectively, yielding a value of $A_{CP}^{resc}=17.2\pm 2.1\pm 1.2\pm 0.7$
LHCb:2014mir ; LHCb:2013lcl . The data are extracted from the
$B^{\pm}\rightarrow\pi^{+}\pi^{-}K^{\pm}$ decay for the integrated CP
asymmetries in full phase space, where $A_{CP}^{inte}$ is measured to be
$2.5\pm 0.4\pm 0.4\pm 0.7$. Additionally, measurements are performed in
specific regions: the low invariant mass region ($0.08~{}{\rm
GeV}^{2}<m_{\pi^{+}\pi^{-}~{}low}^{2}<0.66~{}{\rm GeV}^{2}$) and the high
invariant mass region ($m_{\pi^{\pm}K^{\pm}~{}high}^{2}<15~{}{\rm GeV}^{2}$),
yielding $A_{CP}^{low}$ of $67.8\pm 7.8\pm 3.2\pm 0.7$ and measurements in
rescattering regions with $1.0~{}{\rm GeV}<m_{\pi^{+}\pi^{-}}<1.5~{}{\rm GeV}$
resulting in $A_{CP}^{resc}$ of $12.1\pm 1.2\pm 1.7\pm 0.7$ LHCb:2013ptu ;
LHCb:2014mir .

From the existing experimental data, we can conclude that there is a large CP
symmetry in the invariant mass region. Therefore, it has more diverse
characteristics in terms of CP symmetry information, which is worthy of
academic attention. The decay of three-body B mesons was analyzed by using the
Dalitz diagram in the LHcb experiment. The characteristics of the resonance
amplitudes can be clearly observed from the diagram, it is found that the
scalar and vector resonances with invariant masses less than about
$1GeV/c^{2}$ dominate. In the decay of charged B mesons, it is clearly
observed from the Dalitz diagram that there are obviously CP symmetry when the
invariant masses of $\pi^{+}\pi^{-}$ are in the $\rho^{0}(770)$ and
$f^{0}(980)$ resonance regions . Until recently, a resonance model that
accurately describes these effects has been lacking LHCb:2019jta ;
LHCb:2022tuk . Although our previous results showed that $\pi^{+}\pi^{-}$ has
a large CP symmetry the $\rho^{0}(770)$ and $\omega(782)$ resonance regions
LHCb:2013ptu ; LHCb:2014mir . Due to the intermediate state resonance between
different isospin mesons and the factor of non-uniformity of phase space
distribution is not considered, they cannot be compared with the experimental
results.

The charmless decay of B mesons to three hadrons is predominantly governed by
a quasi-two-body process involving an intermediate resonance state, and the
effects of resonance can be accurately described using the Breit-Wigner (BW)
formalism. This decay process encompasses two distinct interference amplitudes
between the weak phase and the strong phase. The weak phase is associated with
the Cabibbo-Kobayashi-Maskawa (CKM) matrix, while the strong phase arises from
rescattering or other mechanisms during the two-body decay process. We aim to
investigate the CP asymmetry of vector meson resonance in the three-body decay
of B mesons using the perturbative QCD method (pQCD). With a comprehensive
exploration of CP asymmetry, various approaches have been employed to describe
the multibody decays of B mesons, including QCD factorization (QCDF)
Yuan:2023ujm  and perturbative QCD (pQCD) Shi:2022ggo . The application of
various methodologies can induce significant phase differences, thereby
exerting an impact on the CP asymmetry observed in the weak decay process of B
mesons. The Sudakov factor is incorporated into PQCD to effectively suppress
the non-perturbative contribution, while the non-perturbative component is
included in the hadronic wave function. The determination of the wave function
can be achieved once experimental data are validated. The PQCD method in the
decay of B-meson two-body non-leptonic decay has been demonstrated to be
consistent with experimental observations for the majority of decay channels.
The two-body decay process corresponding to the B meson has indeed been
established and further expanded into various three-body decay processes,
owing to the adoption of the finite width approximation.

According to the Vector Meson Dominated (VMD) model, the polarization of
photons in vacuum leads to the emergence of vector particles such as
$\rho^{0}$, $\omega$, and $\phi$. The annihilation of $e^{+}e^{-}$ into
photons results in the production of a pair of $\pi^{+}\pi^{-}$ through an
intermediate neutral vector meson. The decay of $\rho^{0}$ into
$\pi^{+}\pi^{-}$ is an isospin-conserving process, while the decays of
$\omega$ and $\phi$ into pairs of $\pi^{+}\pi^{-}$ involve isospin asymmetry.
To better describe these decay processes, we utilize a unitary matrix
transformation to convert the ideal field of the intermediate state (non-
physical states) into a physical field. By observing interference between the
resonant mesons ($\rho^{0}$, $\omega$ and $\phi$), valuable insights into
their kinetic mechanisms can be obtained. Furthermore, it should be noted that
the presence of intermediate resonance hadrons contributes to the formation of
a new strong phase, which may have implications for CP asymmetry in hadronic
decays. We investigate the impact of mixed resonance of intermediate particles
on CP asymmetry in the three-body decay process of B mesons. By examining the
decay of vector meson resonances, it will enable more precise measurement of
CP asymmetry in future experimental studies. Additionally, this paper also
discusses regional variations in CP asymmetry values for comparison with
experimental results.

The layout of the present paper is as follows: In Sec. II we discuss resonant
contributions to three-body decays of B meson. The quasi-two-body method $B\to
VP$( $V\to PP$) is employed to investigate the CP asymmetry. In Sec. III is
dedicated to the examination of direct CP asymmetry. The regional CP
asymmetries arising from the resonance effects of $\rho^{0}$, $\omega$, and
$\phi$ are taken into consideration. We illustrate the form of the three-body
decay amplitude of B meson after considering above resonance effect and
present the CP asymmetry. The numerical results and discussion on the local CP
defect will be presented in Sec. IV. Sec. V contains our conclusions.

##  II  EFFECT OF VECTOR MESON RESONANCE ON THREE-BODY DECAY OF B MESONS

###  II.1  mixing mechanism

Based on the Vector Meson Dominated (VMD) model, electrons and positrons
annihilate to generate photons, which are further polarized to form vector
meson $\rho^{0}$, $\omega$, $\phi$. Then, these vector mesons decay into pairs
of $\pi^{+}$$\pi^{-}$ mesons 6Lu2017 . In this mechanism, the mixing
parameters corresponding to two vector particles can be obtained by using the
electromagnetic form factor of the $\pi$ meson. In the VMD model, vector
mesons transfer momentum and the mixed amplitude parameters are correlated
with it.

Since the $\rho^{0}_{I}$ ($\omega_{I}$, $\phi_{I}$) resonance state is a non-
physical state, we use the unitary matrix R to transform the isospin field
into the physical field: $\rho_{I}^{0}$ ($\omega_{I}$,$\phi_{I}$ )
$\rightarrow\ \rho^{0}$ ($\omega$, $\phi$). The diagonal elements of the
unitary matrix R are 1, and the non-diagonal elements represent the
information of the $\rho^{0}-\omega-\phi$ mixture. The contribution of high
order terms is ignored in the process of this transformation. The resonance
effects for $A_{\rho\omega},B_{\omega\phi}$, and $C_{\rho\phi}$ are defined by
a set of mixed amplitude parameters, where it holds that
$A_{\rho\omega}=-A_{\omega\rho}$, $B_{\omega\phi}=-B_{\phi\omega}$, and
$C_{\rho\phi}=-C_{\phi\rho}$. The amplitude represents a first-order
approximation that is dependent on the variable s, which in turn is related to
the square of momentum. The expression for R is

$R=\left(\begin{array}[]{lll}~{}~{}~{}~{}1&\hskip 14.22636pt-
A_{\rho\omega}(s)&\hskip 14.22636pt-C_{\rho\phi}(s)\\\\[14.22636pt]
\displaystyle A_{\rho\omega}(s)&\hskip 14.22636pt~{}~{}~{}1&\hskip 14.22636pt-
B_{\omega\phi}(s)\\\\[14.22636pt] \displaystyle C_{\rho\phi}(s)&\hskip
14.22636pt\displaystyle B_{\omega\phi}(s)&\hskip
28.45274pt1\end{array}\right).$ (1)

According to the transformation relation of the R matrix, the expression form
of the resonance state in the physical state is given:
$\phi=C_{\rho\phi}(s)\rho_{I}^{0}+B_{\omega\phi}(s)\omega_{I}+\phi_{I}$,
$\omega=A_{\rho\omega}(s)\rho_{I}^{0}+\omega_{I}-B_{\omega\phi}(s)\phi_{I}$,
$\rho^{0}=\rho_{I}^{0}-A_{\rho\omega}(s)\omega_{I}-C_{\rho\phi}(s)\phi_{I}$.
The relationship between the mixing parameters $\Pi_{V_{i}V_{j}}$ and
$A_{V_{i}V_{j}}$ ($V_{i}$ and $V_{j}$ represent distinct vector mesons) can be
deduced from the subsequent equation:

$\begin{split}A_{\rho\omega}=-A_{\omega\rho}=\frac{\Pi_{\rho\omega}}{({s-m_{\rho}^{2}+im_{\rho}\varGamma_{\rho}})-({s-m_{\omega}^{2}+im_{\omega}\varGamma_{\omega}})}\end{split}.$
(2)
$\begin{split}B_{\omega\phi}=-B_{\phi\omega}=\frac{\Pi_{\omega\phi}}{({s-m_{\omega}^{2}+im_{\omega}\varGamma_{\omega}})-({s-m_{\phi}^{2}+im_{\phi}\varGamma_{\phi}})}\end{split}$
(3)
$\begin{split}C_{\rho\phi}=-C_{\phi\rho}=\frac{\Pi_{\rho\phi}}{({s-m_{\rho}^{2}+im_{\rho}\varGamma_{\rho}})-({s-m_{\phi}^{2}+im_{\phi}\varGamma_{\phi}})}\end{split}$
(4)

where $\sqrt{s}$ represents the invariant mass of the $\pi^{+}\pi^{-}$ meson
pair, $m_{V}$ denotes the mass of the vector meson, and $\varGamma_{V}$
signifies the decay width of the vector meson. Considering the mixing
mechanism, we define the corresponding parameters as $\Pi_{V_{1}V_{2}}(s)$,
which are associated with momentum. Our objective is to investigate the impact
of this mixing mechanism on CP asymmetry in the ${\omega}$ and ${\phi}$
resonance regions.

The parameters $A_{\rho\omega},B_{\omega\phi},C_{\rho\phi}$, and
${\Pi_{ViVj}}$ are considered as first-order approximations. Any product of
two parameters can be disregarded due to its higher-order nature. We hereby
define:

$\widetilde{\Pi}_{\rho\omega}=\frac{({s-m_{\rho}^{2}+im_{\rho}\varGamma_{\rho}})\Pi_{\rho\omega}}{({s-m_{\rho}^{2}+im_{\rho}\varGamma_{\rho}})-({s-m_{\omega}^{2}+im_{\omega}\varGamma_{\omega}})}$
(5)
$~{}~{}\widetilde{\Pi}_{\rho\phi}=\frac{({s-m_{\rho}^{2}+im_{\rho}\varGamma_{\rho}})\Pi_{\rho\phi}}{({s-m_{\rho}^{2}+im_{\rho}\varGamma_{\rho}})-({s-m_{\phi}^{2}+im_{\phi}\varGamma_{\phi}})}$
(6)
$~{}~{}\widetilde{\Pi}_{\omega\phi}=\frac{({s-m_{\phi}^{2}+im_{\phi}\varGamma_{\phi}})\Pi_{\omega\phi}}{({s-m_{\omega}^{2}+im_{\omega}\varGamma_{\omega}})-({s-m_{\phi}^{2}+im_{\phi}\varGamma_{\phi}})}$
(7)

Through the derivation process of the mixed parameters, it can be inferred
that a correlation exists between the mixed parameters and $s$. These mixed
parameters amalgamate the contributions from both resonant and non-resonant
components arising due to isospin symmetry breaking effects in the direct
decay process, while also exhibiting momentum dependence. Based on the
available experimental results, we are able to accurately determine the mixing
parameters $\Pi_{\rho\omega}=-4470\pm 250\pm 160-i(5800\pm 2000\pm
1100)\mathrm{MeV}^{2}$ in the vicinity of the $\rho$ meson. The mixing
parameters $\Pi_{\phi\rho}=720\pm 180-i(870\pm 320)\mathrm{MeV}^{2}$ were
obtained near the $\phi$ meson CE2009 ; MN2000 . Furthermore, a mixing
parameter of $\Pi_{\omega\phi}=19000+i(2500\pm 300)\mathrm{MeV}^{2}$ was
determined in close proximity to the $\phi$ meson CE2011 .

###  II.2  CP asymmetry induced by interference

The decay $B\rightarrow(V\rightarrow\pi^{+}\pi^{-})\pi(K)$ is modeled as a
quasi-two-body process, where V represents an intermediate vector meson. The
presence of vector meson resonances in the decay process significantly impacts
the strong phase, tree, and penguin amplitudes. In Fig.1, for the decay of
$B\rightarrow[\rho^{0}(\omega,\phi)\rightarrow\pi^{+}\pi^{-}]\pi(K)$, we
investigate the impact of various vector meson resonances.

Figure 1: The diagrams of ${B}\rightarrow\pi^{+}\pi^{-}\pi(K)$ decay process.

According to the three-vector mixing mechanism and neglecting the contribution
of higher order terms, we can derive the decay diagram for the
$B\rightarrow[\rho^{0}(\omega,\phi)\rightarrow\pi^{+}\pi^{-}]\pi(K)$ decay
process, as illustrated in Figs $\left(a\right)\thicksim\left(i\right)$. In
Fig.1(a), it can be observed that the $\bar{B}_{d}^{0}$ undergoes direct decay
to produce a pair of $\pi^{+}\pi^{-}$ mesons through the intermediate state of
$\rho^{0}$: $\rho\rightarrow\pi\pi$. Simultaneously, it is evident that both
$\omega$ ($\phi$) also contribute to the production of pairs of
$\pi^{+}\pi^{-}$ mesons via direct decays: $\phi\rightarrow\pi\pi$ (d), and
$\omega\rightarrow\pi\pi$ (g).

In addition to the direct decay of vector mesons ($\rho^{0}$, $\omega$,
$\phi$) into pairs of $\pi^{+}\pi^{-}$ mesons, the production of such meson
pairs can also occur through vector meson resonances (as depicted in figure
(b)), including mixed resonances involving $\rho^{0}-\phi$. During this decay
process, isospin symmetry is violated. The black dots in the figure represent
these mixing parameters expressed as ${\Pi}_{\mathrm{\rho^{0}\phi}}$.
Subsequent figures illustrate various effects arising from mixed resonances
and introduce their corresponding mixing parameters.

The decay rate of $\rho\rightarrow\pi^{+}\pi^{-}$ is observed to be $100\%$,
suggesting that the contribution of $\omega$ and $\phi$ to the decay of
$\pi^{+}\pi^{-}$ can be considered negligible in a rough approximation.
However, it is still necessary to take into account the contribution from this
mixing mechanism which produces the strong phases. It should be noted that the
mixed parameters ${\Pi}_{\mathrm{\rho^{0}\phi}}$,
${\Pi}_{\mathrm{\rho^{0}\omega}}$, and ${\Pi}_{\mathrm{\omega\phi}}$ are
approximations at first order. Multiplying any two of these parameters yields
a higher-order contribution that can be neglected; hence, cases involving
multiple mixed parameters are disregarded.

In conjunction with the significant contributions from the aforementioned
diagrams, we present the amplitude expression for the three-body decay process
$B\rightarrow\pi^{+}\pi^{-}\pi(K)$:

$\displaystyle
A=A_{T}+A_{P}=<\pi^{+}\pi^{-}\pi(K)|\mathcal{H}_{T}|B>+<\pi^{+}\pi^{-}\pi(K)|\mathcal{H}_{P}|B>.$
(8)

In the above expression, the matrix element
$<\pi^{+}\pi^{-}\pi(K)|\mathcal{H}_{T}|B>$
($<\pi^{+}\pi^{-}\pi(K)|\mathcal{H}_{P}|B>$) contributes to the tree (penguin)
graph within an effective Hamiltonian framework. An amplitude analysis of this
decay channel reveals that $\rho^{0}$ is the dominant contribution. Based on
the physical information depicted in the decay diagram, we derive a detailed
formulation for both the tree and penguin amplitudes.

$\displaystyle<\pi^{+}\pi^{-}\pi(K)|H_{T}|B>$
$\displaystyle=\frac{g^{\rho^{0}\rightarrow\pi^{+}\pi^{-}}}{(s-m_{\rho^{0}}^{2}+im_{\rho^{0}}\varGamma_{\rho^{0}})({s-m_{\omega}^{2}+im_{\omega}\varGamma_{\omega}})}\overset{\sim}{\Pi}_{\mathrm{\rho\omega}}t_{\omega}$
(9)
$\displaystyle+\frac{g^{\rho^{0}\rightarrow\pi^{+}\pi^{-}}}{(s-m_{\rho^{0}}^{2}+im_{\rho^{0}}\varGamma_{\rho^{0}})({s-m_{\phi}^{2}+im_{\phi}\varGamma_{\phi}})}\overset{\sim}{\Pi}_{\mathrm{\rho^{0}\phi}}t_{\phi}$
$\displaystyle+\frac{g^{\rho^{0}\rightarrow\pi^{+}\pi^{-}}}{s-m_{\rho^{0}}^{2}+im_{\rho}\varGamma_{\rho^{0}}}t_{\rho^{0}},$

$\displaystyle<\pi^{+}\pi^{-}\pi(K)|H_{P}|B>$
$\displaystyle=\frac{g^{\rho^{0}\rightarrow\pi^{+}\pi^{-}}}{(s-m_{\rho^{0}}^{2}+im_{\rho^{0}}\varGamma_{\rho^{0}})({s-m_{\omega}^{2}+im_{\omega}\varGamma_{\omega}})}\overset{\sim}{\Pi}_{\mathrm{\rho^{0}\omega}}p_{\omega}$
(10)
$\displaystyle+\frac{g^{\rho^{0}\rightarrow\pi^{+}\pi^{-}}}{(s-m_{\rho^{0}}^{2}+im_{\rho^{0}}\varGamma_{\rho^{0}})({s-m_{\phi}^{2}+im_{\phi}\varGamma_{\phi}})}\overset{\sim}{\Pi}_{\mathrm{\rho^{0}\phi}}p_{\phi}$
$\displaystyle+\frac{g^{\rho^{0}\rightarrow\pi^{+}\pi^{-}}}{s-m_{\rho^{0}}^{2}+im_{\rho^{0}}\varGamma_{\rho^{0}}}p_{\rho^{0}},$

where $t_{V}$ and $p_{V}$ denote tree contribution and penguin contribution
during the decay process $B\rightarrow V(V=\rho^{0},\omega,\phi)\pi(K)$,
respectively. ${g^{\rho^{0}\rightarrow\pi^{+}\pi^{-}}}$ represents the
coupling constant of $\rho\rightarrow\pi\pi$ LW1964 . We define the
differential parameter of CP asymmetry as:

$A_{CP}=\frac{\left|A\right|^{2}-\left|\overline{A}\right|^{2}}{\left|A\right|^{2}+\left|\overline{A}\right|^{2}}.$
(11)

###  II.3  Regional CP asymmetry

Recently, the LHCb team has made a significant discovery indicating the
presence of substantial values in a specific region of phase space through
direct measurements of CP asymmetry in charmless decays of B mesons involving
three-body final states. For instance, the decay process
${B}_{u}^{-}\rightarrow\pi^{+}\pi^{-}\pi^{-}$ exhibits a significant CP
asymmetry within the invariant mass region
($m^{2}_{\pi^{+}\pi^{-}\rm~{}low}<0.4~{}{\rm GeV}^{2}$) LHCb:2014mir . The
$A_{CP}$ integrals are computed in this study to facilitate the comparison
with experimental observations.

The amplitude of the ${B}_{u}^{-}\rightarrow\rho^{0}\pi^{-}$ decay can be
expressed as $M_{{B}_{u}^{-}\rightarrow\rho^{0}\pi^{-}}^{\lambda}=\alpha
P_{{B}_{u}^{-}}\cdot\epsilon^{*}\left(\lambda\right),$ where $\alpha$
represents the effective coupling constant, $P_{{B}_{u}^{-}}$ denotes the
momentum of ${B}_{u}^{-}$, $\epsilon$ is the polarization vector of
$\rho^{0}$, and $\lambda$ corresponds to the helicity of the vector meson. And
the amplitude for $\rho^{0}\rightarrow\pi^{+}\pi^{-}$ can be written as
follows:
$M_{\rho^{0}\rightarrow\pi^{+}\pi^{-}}^{\lambda}={g^{\rho^{0}\rightarrow\pi^{+}\pi^{-}}}\epsilon\left(\lambda\right)\cdot\left(p_{\pi}^{+}-p_{\pi}^{-}\right)$,
where ${g^{\rho^{0}\rightarrow\pi^{+}\pi^{-}}}$ is the effective coupling
constant and $\epsilon$ refers to the plolarization vector of $\rho^{0}$.
Hence, the amplitude of
${B}_{u}^{-}\rightarrow\rho^{0}\pi^{-}\rightarrow\pi^{+}\pi^{-}\pi^{-}$ is:

$\displaystyle A$
$\displaystyle=\frac{g^{\rho^{0}\rightarrow\pi^{+}\pi^{-}}}{\alpha}{(s-m_{\rho^{0}}^{2}+im_{\rho^{0}}\varGamma_{\rho^{0}})}P_{{B}_{u}^{-}}^{\mu}\sum_{\lambda=\pm
1,0}{\epsilon_{\mu}^{*}\left(\lambda\right)\epsilon_{r}\left(\lambda\right)\cdot\left(p_{\pi}^{+}-p_{\pi}^{-}\right)^{r}}$
(12)
$\displaystyle=-\frac{{g^{\rho^{0}\rightarrow\pi^{+}\pi^{-}}}\alpha}{(s-m_{\rho^{0}}^{2}+im_{\rho^{0}}\varGamma_{\rho^{0}})}P_{{B}_{u}^{-}}^{\mu}\left[g_{\mu
r}-\frac{\left(p_{\pi}^{+}+p_{\pi}^{-}\right)_{\mu}\left(p_{\pi}^{+}+p_{\pi}^{-}\right)_{r}}{m_{\rho^{0}}^{2}}\right]\left(p_{\pi}^{+}-p_{\pi}^{-}\right)^{r}$
$\displaystyle=\frac{{g^{\rho^{0}\rightarrow\pi^{+}\pi^{-}}}}{(s-m_{\rho^{0}}^{2}+im_{\rho^{0}}\varGamma_{\rho^{0}})}\cdot\frac{M_{{B}_{u}^{-}\rightarrow\rho^{0}\pi^{-}}^{\lambda}}{p_{{B}_{u}^{-}}\cdot\epsilon^{*}}\cdot\left(\Sigma-s^{\prime}\right).$

In the three-body decay process, we employ the principles of momentum and
energy conservation to transform the equation:
$P_{\bar{B}_{d}^{0}}=p_{\pi}^{+}+p_{\pi}^{-}+p_{\pi}^{0}$ and
$m_{ij}^{2}=p_{ij}^{2}$, aiming to facilitate computational procedures. To
streamline calculations, we introduce the following definitions:

$\displaystyle\mathcal{M}=$
$\displaystyle\frac{{g^{\rho^{0}\rightarrow\pi^{+}\pi^{-}}}}{(s-m_{\rho^{0}}^{2}+im_{\rho^{0}}\varGamma_{\rho^{0}})}\cdot\frac{M_{{B}_{u}^{-}\rightarrow\rho^{0}\pi^{-}}^{\lambda}}{p_{{B}_{u}^{-}}\cdot\epsilon^{*}}.$
(13)

Thus, the amplitude can be written as:

$\displaystyle A=\left(\xi-s^{{}^{\prime}}\right)\cdot\mathcal{M}$ (14)

where $\sqrt{s}$ and $\sqrt{s^{\prime}}$ represent the low and the high
invariant mass of the $\pi^{+}\pi^{-}$ pair, respectively. Based on the
dependency between the values of $s$, $s_{\max}^{{}^{\prime}}$ and
$s_{\min}^{{}^{\prime}}$ by setting a fixed value for the variable $s$, we can
determine an appropriate number $s^{{}^{\prime}}$ that satisfies the formula
$\Sigma=\frac{1}{2}\left(s_{\max}^{\prime}+s_{\min}^{\prime}\right)$, where
$s_{\max}^{{}^{\prime}}$ and $s_{\min}^{{}^{\prime}}$ represent the maximum
and minimum values, respectively Qi:2018lxy .

According to the kinetic principles of the three-body decay process, we can
derive the regional CP violation in the
${B}_{u}^{-}\rightarrow\pi^{+}\pi^{-}\pi^{-}$ decay within a certain invariant
mass ranges:

$\displaystyle
A_{CP}^{\varOmega}=\frac{\int_{s_{1}}^{s_{2}}{ds\int_{s_{1}^{{}^{\prime}}}^{s_{2}^{{}^{\prime}}}{ds^{{}^{\prime}}}}\left(\Sigma-s^{{}^{\prime}}\right)^{2}\left(\left|A\right|^{2}-\left|\overline{A}\right|^{2}\right)}{\int_{s_{1}}^{s_{2}}{ds\int_{s_{1}^{{}^{\prime}}}^{s_{2}^{{}^{\prime}}}{ds^{{}^{\prime}}}}\left(\Sigma-s^{{}^{\prime}}\right)^{2}\left(\left|A\right|^{2}+\left|\overline{A}\right|^{2}\right)}.$
(15)

The numerator and denominator of $A_{CP}$ can be integrated over the range
$\Omega\left(s_{1}<s<s_{2},s_{1}^{\prime}<s^{\prime}<s_{2}^{\prime}\right)$.
The integration interval for high-invariant mass of $\pi^{+}\pi^{-}$ meson
pairs is $s_{1}^{\prime}<s^{\prime}<s_{2}^{\prime}$, where
$\int_{s_{1}^{{}^{\prime}}}^{s_{2}^{{}^{\prime}}}{ds^{\prime}}\left(\Sigma-s^{\prime}\right)^{2}$
represents a factor dependent on $s^{\prime}$. Through kinematic analysis, the
correlation between $\Sigma$ and $s^{\prime}$ can be readily determined.
Assuming a finite range, we can consider $\Sigma$ to be constant.
Consequently, the term
$\int_{s_{1}^{{}^{\prime}}}^{s_{2}^{{}^{\prime}}}{ds^{{}^{\prime}}}\left(\Sigma-s^{{}^{\prime}}\right)^{2}$
becomes negligible in the calculation, rendering $A_{CP}^{\varOmega}$
independent of the high invariant mass of positive and negative particles.

##  III Calculation process and Analysis of CP asymmetry

The narrow width approximation (NWA) is employed in our calculation to
decompose the tree body decay process into two successive quasi-two-body
decays: ${B}\rightarrow(R\rightarrow\pi^{+}\pi^{-})P$, where R denotes the
resonance state and P represents either a $\pi$ or K meson. Considering the
intermediate resonance state R, we introduce the correction factor $\eta_{R}$
to define the expression:

$\displaystyle\eta_{R}=\frac{\mathcal{B}\left(B\rightarrow RP_{3}\rightarrow
P_{1}P_{2}P_{3}\right)}{\mathcal{B}\left(B\rightarrow
RP_{3}\right)\mathcal{B}\left(B\rightarrow P_{1}P2\right)},$ (16)

where shows the relationship between the branch ratio measured around the
resonance region and the branch ratio of the three-body decay of the B meson
Chua:2022owm . According to the QCD factorization method, the correction
factor $\eta_{R}$ is about $7\%$ during the decay of
${B}_{u}^{-}\rightarrow\pi^{+}\pi^{-}\pi^{-}$ Cheng:2022ysn ; Cheng:2020mna .
When calculating CP asymmetry, the numerator and denominator terms containing
the correction factor $\eta_{R}$. Therefore, we does not consider the width
approximation in subsequent calculations.

In calculating CP asymmetry, we consider the contribution of the mixed
resonance mechanism to the three-body decay amplitude. The PQCD method is
employed in this study to investigate the CP asymmetry effects of vector meson
mixing resonance mechanism on B meson decay, encompassing both resonant and
non-resonant contributions. The resonant contribution arising from
intermediate states is expressed in terms of mixing parameters, while the
coupling constants are derived from the decay process of intermediate states
into final-state particles. The parameters used in the calculation are derived
from Table 1 .

Table 1: The decay constants are taken from Refs. Other parameters are from PDG 2022 ParticleDataGroup:2022pth  Mass(GeV) | $m_{B_{u}}=5.27934$ | $m_{B_{d}}=5.27965$ | $m_{\pi^{\pm}}=0.140$ | $m_{\pi^{0}}=0.135$
---|---|---|---|---
Wolfenstein parameters | $\lambda=0.22650$ | $A=0.790$ | $\bar{\rho}=0.141$ | $\bar{\eta}=0.357$
Decay constants (GeV) | $f_{B_{s}}=0.23$ | $f_{B}=0.21$ | $f_{\rho(770)}=0.216$ | $f_{\rho(1020)}^{T}=0.184$
Decay width (GeV) |  $\Gamma_{\rho}$ = $0.15$ |  $\Gamma_{\omega}$ = $8.49\times 10^{-3}$ |  $\Gamma_{\phi}$ = $4.23\times 10^{-3}$ |

###  III.1  CP asymmetry analysis of the
${B}_{u}^{-}\rightarrow(\rho^{0},\omega,\phi\rightarrow\pi^{+}\pi^{-})\pi^{-}$
decay process

The Dalitz diagram analysis of the decay amplitude
${B}^{\pm}\rightarrow\pi^{+}\pi^{-}\pi^{\pm}$ in two-dimensional phase space
is performed, as indicated by relevant research literature. Phenomenological
investigations have primarily focused on exploring regional CP asymmetries in
the decay process ${B}^{\pm}\rightarrow\pi^{+}\pi^{-}\pi^{\pm}$, with certain
studies emphasizing the significance of $\rho^{0}-\omega$ mixing effects
between $\rho^{0}$ and $\omega$, and highlighting potential interference
between the resonance of $\rho^{0}$ and the broad S-wave contribution
LHCb:2014mir ; LHCb:2013lcl . The resonance contributions of $\rho-\omega$,
$\omega-\phi$, and $\rho-\phi$ can give rise to a novel strong phase. To
facilitate a clearer observation, we select the energy range of 0.7 GeV to 1.1
GeV, which corresponds to the primary region where the decay processes
involving $\rho^{0}$, $\omega$, and $\phi$ resonances exhibit significant
effects.

In experiments, the amplitude analysis of B meson decay in Dalitz plots is
commonly used to reconstruct the three-body decay process of B mesons and
investigate the momentum relationship in the decay process. However, due to
the fact that the B meson phase space allows for multiple types of resonances
to exist, there may be a large number of intermediate states. Investigating
these resonance structures is a complex task and it should be noted that it is
not possible to distinguish between the contributions made by $\rho^{0}$ meson
and $\omega$ meson in experiments. Recently, in the decay process of HCLb to
$B_{u}^{-}\rightarrow(\rho^{0},\omega,\phi\rightarrow\pi^{+}\pi^{-})\pi^{-}$,
clear observations have been made regarding the CP asymmetry effects related
to interference between $\rho^{0}$ resonance and s-wave. In this work, we
initially present the amplitude formulation of the
$B_{u}^{-}\rightarrow(\rho^{0},\omega,\phi\rightarrow\pi^{+}\pi^{-})\pi^{-}$
decay process within the framework of PQCD. By calculating the energy-momentum
relationship of the initial meson $B_{u}^{-}$ and final meson $\pi^{+}\pi^{-}$
decay process using a low-energy effective Hamiltonian, selecting well-
established quasi-two-body decay amplitudes, conducting comprehensive analysis
on the momentum analysis of the decay process under the coupling constant $\
g^{v\rightarrow\pi^{+}\pi^{-}}$, as well as summing up the polarization vector
under the momentum relationship, a new form of amplitude is obtained.

$\displaystyle{\cal
A}(B_{u}^{-}{\to}(\rho^{0}\rightarrow\pi^{+}\pi^{-}){\pi}^{-})=$
$\displaystyle\frac{<\rho^{0}K^{-}|H_{eff}|{B}_{u}^{-}><\pi^{+}\pi^{-}|H_{\rho^{0}\rightarrow\pi^{+}\pi^{-}}|\rho^{0}>}{{s-m_{\rho^{0}}^{2}+im_{\rho^{0}}\varGamma_{\rho^{0}}}}$
(17) $\displaystyle=\sum_{\lambda=0,\pm
1}\frac{G_{F}P_{(B_{u}^{-})}\cdot\epsilon^{*}\left(\lambda\right)\
g^{\rho^{0}\rightarrow\pi^{+}\pi^{-}}\epsilon\left(\lambda\right)\cdot\left(p_{\pi^{+}}-p_{\pi^{-}}\right)}{\sqrt{2}({s-m_{\rho^{0}}^{2}+im_{\rho^{0}}\varGamma_{\rho^{0}}})}$
$\displaystyle\times\bigg{\\{}V_{ub}\,V_{ud}^{\ast}\,\big{\\{}a_{1}\,\big{[}{\cal
A}_{ab}^{LL}({\pi},{\rho})+{\cal A}_{ef}^{LL}({\pi},{\rho})-{\cal
A}_{ef}^{LL}({\rho},{\pi})\big{]}+a_{2}\,{\cal A}_{ab}^{LL}({\rho},{\pi})$
$\displaystyle+C_{2}\,\big{[}{\cal A}_{cd}^{LL}({\pi},{\rho})+{\cal
A}_{gh}^{LL}({\pi},{\rho})-{\cal
A}_{gh}^{LL}({\rho},{\pi})\big{]}+C_{1}\,{\cal
A}_{cd}^{LL}({\rho},{\pi})\big{\\}}-V_{tb}\,V_{td}^{\ast}\,\big{\\{}(a_{4}$
$\displaystyle+a_{10})\,\big{[}{\cal A}_{ab}^{LL}({\pi},{\rho})+{\cal
A}_{ef}^{LL}({\pi},{\rho})-{\cal
A}_{ef}^{LL}({\rho},{\pi})\big{]}+(a_{6}+a_{8})\,\big{[}{\cal
A}_{ab}^{SP}({\pi},{\rho})$ $\displaystyle+{\cal
A}_{ef}^{SP}({\pi},{\rho})-{\cal
A}_{ef}^{SP}({\rho},{\pi})\big{]}-(a_{4}-\frac{3}{2}\,a_{7}-\frac{3}{2}\,a_{9}-\frac{1}{2}\,a_{10})\,{\cal
A}_{ab}^{LL}({\rho},{\pi})+(C_{3}$ $\displaystyle+C_{9})\,\big{[}{\cal
A}_{cd}^{LL}({\pi},{\rho})+{\cal A}_{gh}^{LL}({\pi},{\rho})-{\cal
A}_{gh}^{LL}({\rho},{\pi})\big{]}+(C_{5}+C_{7})\,\big{[}{\cal
A}_{cd}^{SP}({\pi},{\rho})$ $\displaystyle+{\cal
A}_{gh}^{SP}({\pi},{\rho})-{\cal
A}_{gh}^{SP}({\rho},{\pi})\big{]}-(C_{3}-\frac{3}{2}\,C_{10}-\frac{1}{2}\,C_{9})\,{\cal
A}_{cd}^{LL}({\rho},{\pi})+\frac{3}{2}\,C_{8}\,{\cal
A}_{cd}^{LR}({\rho},{\pi})$ $\displaystyle-(C_{5}-\frac{1}{2}\,C_{7})\,{\cal
A}_{cd}^{SP}({\rho},{\pi})\big{\\}}\big{\\}}.$

The momentum parameter $P_{i}$ ($P_{i}=P_{{B}_{u}^{-}}$, $p_{\pi^{+}}$,
$p_{\pi^{-}}$) is defined, where $\epsilon$ represents the polarization vector
of the vector meson. The Fermi coupling constant is denoted as $G_{F}$. Three
flow structures are labeled as $LL,LR,$ and $SP$. ${\cal A}_{ab}$ refers to
factorizable emission diagrams, while ${\cal A}_{cd}$ represents
nonfactorizable emission diagrams. Similarly, ${\cal A}_{ef}$ denotes
factorizable annihilation diagrams and ${\cal A}_{gh}$ corresponds to
nonfactorizable annihilation diagrams.

$\displaystyle{\cal
A}(B_{u}^{-}{\to}(\phi\rightarrow\pi^{+}\pi^{-}){\pi}^{-})=$
$\displaystyle\frac{<{\phi}\pi^{-}|H_{eff}|{B}_{u}^{-}><\pi^{+}\pi^{-}|H_{\phi\rightarrow\pi^{+}\pi^{-}}|\phi>}{{s-m_{\phi}^{2}+im_{\phi}\varGamma_{\phi}}}$
(18) $\displaystyle=\sum_{\lambda=0,\pm
1}\frac{G_{F}P_{{B}_{u}^{-}}\cdot\epsilon^{*}\left(\lambda\right)\
g^{\phi\rightarrow\pi^{+}\pi^{-}}\epsilon\left(\lambda\right)\cdot\left(p_{\pi^{+}}-p_{\pi^{-}}\right)}{\sqrt{2}({s-m_{\phi}^{2}+im_{\phi}\varGamma_{\phi}})}$
$\displaystyle\times\bigg{\\{}V_{tb}\,V_{td}^{\ast}\,\big{\\{}(a_{3}+a_{5}-\frac{1}{2}\,a_{7}-\frac{1}{2}\,a_{9})\,{\cal
A}_{ab}^{LL}({\phi},{\pi})+(C_{4}-\frac{1}{2}\,C_{10})\,{\cal
A}_{cd}^{LL}({\phi},{\pi})$ $\displaystyle+(C_{6}-\frac{1}{2}\,C_{8})\,{\cal
A}_{cd}^{LR}({\phi},{\pi})\big{\\}}.$

The additional representations of the three-body decay amplitudes that must be
taken into account for calculating CP asymmetry through the mixing mechanism
are as follows: $V_{tb}V_{ts}^{*}=\lambda$,
$V_{ub}V_{us}^{*}=A\lambda^{4}(\rho-i\eta)$,
$V_{ub}V_{ud}^{*}=A\lambda^{3}(\rho-i\eta)(1-\frac{\lambda^{2}}{2})$, and
$V_{tb}V_{td}^{*}=A\lambda^{3}(1-\rho+i\eta)$.

$\displaystyle{\cal
A}(B_{u}^{-}{\to}(\omega\rightarrow\pi^{+}\pi^{-}){\pi}^{-})=$
$\displaystyle\frac{<{\omega}\pi^{-}|H_{eff}|{B}_{u}^{-}><\pi^{+}\pi^{-}|H_{\omega\rightarrow\pi^{+}\pi^{-}}|\omega>}{{s-m_{\omega}^{2}+im_{\omega}\varGamma_{\omega}}}$
(19) $\displaystyle=\sum_{\lambda=0,\pm
1}\frac{G_{F}P_{{B}_{u}^{-}}\cdot\epsilon^{*}\left(\lambda\right)\
g^{\omega\rightarrow\pi^{+}\pi^{-}}\epsilon\left(\lambda\right)\cdot\left(p_{\pi^{+}}-p_{\pi^{-}}\right)}{\sqrt{2}({{s-m_{\omega}^{2}+im_{\omega}\varGamma_{\omega}}})}$
$\displaystyle\times\bigg{\\{}V_{ub}\,V_{ud}^{\ast}\,\big{\\{}a_{1}\,\big{[}{\cal
A}_{ab}^{LL}({\pi},{\omega})+{\cal A}_{ef}^{LL}({\pi},{\omega})+{\cal
A}_{ef}^{LL}({\omega},{\pi})\big{]}+a_{2}\,{\cal A}_{ab}^{LL}({\omega},{\pi})$
$\displaystyle+C_{2}\,\big{[}{\cal A}_{cd}^{LL}({\pi},{\omega})+{\cal
A}_{gh}^{LL}({\pi},{\omega})+{\cal
A}_{gh}^{LL}({\omega},{\pi})\big{]}+C_{1}\,{\cal
A}_{cd}^{LL}({\omega},{\pi})\big{\\}}$ $\displaystyle-
V_{tb}\,V_{td}^{\ast}\,\big{\\{}(a_{4}+a_{10})\,\big{[}{\cal
A}_{ab}^{LL}({\pi},{\omega})+{\cal A}_{ef}^{LL}({\pi},{\omega})+{\cal
A}_{ef}^{LL}({\omega},{\pi})\big{]}$
$\displaystyle+(a_{6}+a_{8})\,\big{[}{\cal A}_{ab}^{SP}({\pi},{\omega})+{\cal
A}_{ef}^{SP}({\pi},{\omega})+{\cal A}_{ef}^{SP}({\omega},{\pi})\big{]}$
$\displaystyle+(2\,a_{3}+a_{4}+2\,a_{5}+\frac{1}{2}\,a_{7}+\frac{1}{2}\,a_{9}-\frac{1}{2}\,a_{10})\,{\cal
A}_{ab}^{LL}({\omega},{\pi})$ $\displaystyle\ +(C_{3}+C_{9})\,\big{[}{\cal
A}_{cd}^{LL}({\pi},{\omega})+{\cal A}_{gh}^{LL}({\pi},{\omega})+{\cal
A}_{gh}^{LL}({\omega},{\pi})\big{]}$
$\displaystyle+(C_{5}+C_{7})\,\big{[}{\cal A}_{cd}^{SP}({\pi},{\omega})+{\cal
A}_{gh}^{SP}({\pi},{\omega})+{\cal A}_{gh}^{SP}({\omega},{\pi})\big{]}$
$\displaystyle+(C_{3}+2\,C_{4}-\frac{1}{2}\,C_{9}+\frac{1}{2}\,C_{10})\,{\cal
A}_{cd}^{LL}({\omega},{\pi})$
$\displaystyle+(2\,C_{6}+\frac{1}{2}\,C_{8})\,{\cal
A}_{cd}^{LR}({\omega},{\pi})+(C_{5}-\frac{1}{2}\,C_{7})\,{\cal
A}_{cd}^{SP}({\omega},{\pi})\big{\\}}\big{\\}}.$

We present a diagram illustrating the relationship between CP asymmetry and
the invariant mass $\sqrt{s}$ (where $\sqrt{s}$ represents the invariant mass
of $\pi^{+}\pi^{-}$) in the quasi-two-body decay
$B_{u}^{-}\rightarrow(\rho^{0},\omega,\phi\rightarrow\pi^{+}\pi^{-})\pi^{-}$,
which serves as a theoretical reference for investigating CP asymmetry within
the interference range.

Figure 2: Plot of $A_{CP}$ as a function of $\sqrt{s}$ corresponding to
central parameter values of CKM matrix elements for the decay channel of
${B}^{-}\rightarrow\pi^{+}\pi^{-}\pi^{-}$.

Figure 3: CP asymmetry plot near the invariant mass of $\phi$ during the decay
channel of ${B}^{-}\rightarrow\pi^{+}\pi^{-}\pi^{-}$ from the central
parameter values of CKM matrix elements.

The results for the $B_{u}^{-}\rightarrow\pi^{+}\pi^{-}\pi^{-}$ decay process
are presented in Fig.2 and Fig.3. As depicted in Fig.2 and Fig.3, it is
evident that the CP violation of the
$B_{u}^{-}\rightarrow\pi^{+}\pi^{-}\pi^{-}$ channel undergoes a change when
the invariant masses of the $\pi^{+}\pi^{-}$ pair encompass the resonance
range of $\omega$ and $\phi$, with a maximum achievable CP asymmetry of
$94\%$. The decay process $B_{u}^{-}\rightarrow\pi^{+}\pi^{-}\pi^{-}$ exhibits
a significant variation in CP asymmetry when the invariant masses of the
$\pi^{+}\pi^{-}$ pair approaches 0.76 GeV, reaching a peak value of $94\%$.
This behavior can be attributed to the effect arising from the mixing
mechanism between $\rho^{0}$ and $\omega$. Consequently, interference effects
are expected within the range of 0.7 GeV-0.8 GeV as indicated by Fig.2, along
with small peaks observed in the invariant mass range corresponding to $\phi$
according to Fig.3 analysis results. The resulting CP asymmetry is measured at
$1.6\%$, noting that this contribution arises solely from penguin diagram
contributions associated with
$B_{u}^{-}\rightarrow(\phi\rightarrow\pi^{+}\pi^{-})\pi^{-}$.

###  III.2 CP asymmetry analysis of the
${B}_{u}^{-}\rightarrow(\rho^{0},\omega,\phi\rightarrow\pi^{+}\pi^{-})K^{-}$
decay process

The CP asymmetry in the three-body decay of B meson is related to the
interference between two-body resonant and three-body non-resonant decays, as
well as tree diagram and penguin diagram interference. In experiments, it has
been observed that certain local CP asymmetry in phase space are larger than
the overall CP asymmetry in phase space. Ever since the perturbative QCD
framework for the three-body decay of B meson was proposed, it has been widely
applied to various decay processes. perturbative QCD involves calculating the
hard part of QCD using perturbation theory, while non-perturbative parts such
as interactions between final-state particles are absorbed into hadron wave
functions. These hadron wave functions can be determined from experiments and
are process-independent, possessing universality. In perturbative QCD, the
transverse momentum $(k_{t})$ is preserved. To address the issue of endpoint
divergence, Sudakov factors are introduced to suppress long-range interactions
in a small range of transverse momenta, ensuring that the entire process can
be calculated perturbativelyLi:2004ep ; Chen:2002th . This paper adopts the
PQCD framework and combines experimental phenomena to study the three-body
final state resulting from ${B}\rightarrow VP$ meson decay induced by V
resonance decay.Considering the resonance effect of vector mesons, namely
$\rho^{0}$, $\omega$, and $\phi$, we present the decay process
${B}_{u}^{-}\rightarrow(\rho^{0},\omega,\phi\rightarrow\pi^{+}\pi^{-})K^{-}$
following a similar mechanism.

$\displaystyle{\cal A}(B_{u}^{-}{\to}(\omega\rightarrow\pi^{+}\pi^{-})K^{-})=$
$\displaystyle\frac{<{\omega}K^{-}|H_{eff}|{B}_{u}^{-}><\pi^{+}\pi^{-}|H_{\omega\rightarrow\pi^{+}\pi^{-}}|\omega>}{{s-m_{\omega}^{2}+im_{\omega}\varGamma_{\omega}}}$
(20) $\displaystyle=\sum_{\lambda=0,\pm
1}\frac{G_{F}P_{(B_{u}^{-}}\cdot\epsilon^{*}\left(\lambda\right)\
g^{\omega\rightarrow\pi^{+}\pi^{-}}\epsilon\left(\lambda\right)\cdot\left(p_{\pi^{+}}-p_{\pi^{-}}\right)}{\sqrt{2}({{s-m_{\omega}^{2}+im_{\omega}\varGamma_{\omega}}})}$
$\displaystyle\times\bigg{\\{}V_{ub}\,V_{us}^{\ast}\,\big{\\{}a_{1}\,\big{[}{\cal
A}_{ab}^{LL}({K},{\omega})+{\cal A}_{ef}^{LL}{K},{\omega})\big{]}+a_{2}\,{\cal
A}_{ab}^{LL}({\omega},{K})$ $\displaystyle+C_{2}\big{[}{\cal
A}_{cd}^{LL}({K},{\omega})+{\cal
A}_{gh}^{LL}({K},{\omega})\big{]}+C_{1}\,{\cal
A}_{cd}^{LL}({\omega},{K})\big{\\}}-V_{tb}\,V_{ts}^{\ast}\,\big{\\{}(a_{4}$
$\displaystyle+a_{10})\,\big{[}{\cal A}_{ab}^{LL}({K},{\omega})+{\cal
A}_{ef}^{LL}({K},{\omega})\big{]}+(a_{6}+a_{8})\,\big{[}{\cal
A}_{ab}^{SP}({K},{\omega})$ $\displaystyle+{\cal
A}_{ef}^{SP}(\overline{K},{\omega})\big{]}+(2\,a_{3}+2\,a_{5}+\frac{1}{2}\,a_{7}+\frac{1}{2}\,a_{9})\,{\cal
A}_{ab}^{LL}({\omega},{K})+(C_{3}$ $\displaystyle+C_{9})\,\big{[}{\cal
A}_{cd}^{LL}({K},{\omega})+{\cal
A}_{gh}^{LL}({K},{\omega})\big{]}+(C_{5}+C_{7})\,\big{[}{\cal
A}_{cd}^{SP}({K},{\omega})+{\cal A}_{gh}^{SP}({K},{\omega})\big{]}$
$\displaystyle+(2\,C_{6}+\frac{1}{2}\,C_{8})\,{\cal
A}_{cd}^{LR}({\omega},{K})+(2\,C_{4}+\frac{1}{2}\,C_{10})\,{\cal
A}_{cd}^{LL}({\omega},{K})\big{\\}}\big{\\}},$ $\displaystyle{\cal
A}\left(B_{u}^{-}{\to}\left(\rho^{0}\rightarrow\pi^{+}\pi^{-}\right)K^{-}\right)=$
$\displaystyle\frac{<\rho^{0}K^{-}|H_{eff}|{B}_{u}^{-}><\pi^{+}\pi^{-}|H_{\rho^{0}\rightarrow\pi^{+}\pi^{-}}|\rho^{0}>}{{s-m_{\rho^{0}}^{2}+im_{\rho^{0}}\varGamma_{\rho^{0}}}}$
(21) $\displaystyle=\sum_{\lambda=0,\pm
1}\frac{G_{F}P_{(B_{u}^{-}}\cdot\epsilon^{*}\left(\lambda\right)\
g^{\rho^{0}\rightarrow\pi^{+}\pi^{-}}\epsilon\left(\lambda\right)\cdot\left(p_{\pi^{+}}-p_{\pi^{-}}\right)}{\sqrt{2}({{s-m_{\rho^{0}}^{2}+im_{\rho^{0}}\varGamma_{\rho^{0}}}})}$
$\displaystyle\times\bigg{\\{}V_{ub}\,V_{us}^{\ast}\big{\\{}a_{1}\,\big{[}{\cal
A}_{ab}^{LL}({K},{\rho})+{\cal A}_{ef}^{LL}({K},{\rho})\big{]}+a_{2}\,{\cal
A}_{ab}^{LL}({\rho},{K})+C_{2}\big{[}{\cal A}_{cd}^{LL}({K},{\rho})$
$\displaystyle+{\cal A}_{gh}^{LL}({K},{\rho})\big{]}+C_{1}\,{\cal
A}_{cd}^{LL}({\rho},{K})\big{\\}}-V_{tb}\,V_{ts}^{\ast}\,\big{\\{}(a_{4}+a_{10})\,\big{[}{\cal
A}_{ab}^{LL}({K},{\rho})+{\cal A}_{ef}^{LL}({K},{\rho})\big{]}$
$\displaystyle+(a_{6}+a_{8})\,\big{[}{\cal A}_{ab}^{SP}({K},{\rho})+{\cal
A}_{ef}^{SP}({K},{\rho})\big{]}+\frac{3}{2}\,(a_{7}+a_{9})\,{\cal
A}_{ab}^{LL}({\rho},{K})+(C_{3}$ $\displaystyle+C_{9})\,\big{[}{\cal
A}_{cd}^{LL}({K},{\rho})+{\cal
A}_{gh}^{LL}({K},{\rho})\big{]}+(C_{5}+C_{7})\,\big{[}{\cal
A}_{cd}^{SP}({K},{\rho})+{\cal A}_{gh}^{SP}({K},{\rho})\big{]}$
$\displaystyle+\frac{3}{2}\,C_{8}\,{\cal
A}_{cd}^{LR}({\rho},{K})+\frac{3}{2}\,C_{10}\,{\cal
A}_{cd}^{LL}({\rho},{K})\big{\\}}\big{\\}},$

$\displaystyle{\cal A}(B_{u}^{-}{\to}(\phi\rightarrow\pi^{+}\pi^{-})K^{-})=$
$\displaystyle\frac{<{\phi}K^{-}|H_{eff}|{B}_{u}^{-}><\pi^{+}\pi^{-}|H_{\phi\rightarrow\pi^{+}\pi^{-}}|\phi>}{{s-m_{\phi}^{2}+im_{\phi}\varGamma_{\phi}}}$
(22) $\displaystyle=\sum_{\lambda=0,\pm
1}\frac{G_{F}P_{(B_{u}^{-}}\cdot\epsilon^{*}\left(\lambda\right)\
g^{\phi\rightarrow\pi^{+}\pi^{-}}\epsilon\left(\lambda\right)\cdot\left(p_{\pi^{+}}-p_{\pi^{-}}\right)}{\sqrt{2}({{s-m_{\phi}^{2}+im_{\phi}\varGamma_{\phi}}})}$
$\displaystyle\times\bigg{\\{}V_{tb}\,V_{ts}^{\ast}\,\big{\\{}(a_{3}+a_{4}+a_{5}-\frac{1}{2}\,a_{7}-\frac{1}{2}\,a_{9}-\frac{1}{2}\,a_{10})\,{\cal
A}_{ab}^{LL}({\phi},{K})+(a_{4}$ $\displaystyle+a_{10})\,{\cal
A}_{ef}^{LL}({\phi},{K})+(a_{6}+a_{8})\,{\cal
A}_{ef}^{SP}({\phi},{K})(C_{3}+C_{4}-\frac{1}{2}\,C_{9}$
$\displaystyle-\frac{1}{2}\,C_{10})\,{\cal
A}_{cd}^{LL}({\phi},{K})+(C_{6}-\frac{1}{2}\,C_{8})\,{\cal
A}_{cd}^{LR}({\phi},{K})+(C_{5}-\frac{1}{2}\,C_{7})\,$ $\displaystyle{\cal
A}_{cd}^{SP}({\phi},{K})+(C_{3}+C_{9})\,{\cal
A}_{gh}^{LL}({\phi},{K})+(C_{5}+C_{7})\,{\cal
A}_{gh}^{SP}({\phi},{K})\big{\\}}\big{\\}}.$

Figure 4: Plot of $A_{CP}$ as a function of $\sqrt{s}$ corresponding to
central parameter values of CKM matrix elements for the decay channel of
${B}_{u}^{-}\rightarrow\pi^{+}\pi^{-}K^{-}$.

In Fig. 4, the observed behavior of the CP asymmetry in the
$B_{u}^{-}\rightarrow\pi^{+}\pi^{-}K^{-}$ decay process provides valuable
insights into the dynamics of this decay channel. As we approach the resonance
range of $\omega$, a significant change in the CP asymmetry is evident,
reaching a peak value of $70\%$. This suggests that there is a strong
influence from the $\omega$ resonance on the decay process.

On the other hand, when we consider the resonance range of $\phi$, only slight
variations in CP asymmetry are observed. This can be attributed to several
factors. It should be noted that for the specific decay process
$B_{u}^{-}\rightarrow(\phi\rightarrow\pi^{+}\pi^{-})K^{-}$, which involves
penguin diagram contribution exclusively, its overall amplitude is further
diminished due to interference from $\phi-\rho$ mixing resonance. This
interference effect leads to a reduction in observable changes in CP asymmetry
near the resonance range of $\phi$.

###  III.3 CP asymmetry analysis of the
$\bar{B}_{d}^{0}\rightarrow(\rho^{0},\omega,\phi\rightarrow\pi^{+}\pi^{-})\pi^{0}$
decay process

Considering the vector meson $\rho^{0}$ \- $\omega$ \- $\phi$ resonance
effect, the process of $\bar{B}^{0}_{d}\rightarrow\pi^{+}\pi^{-}\pi^{0}$ can
be further elaborated. The presence of these vector mesons in the decay
channel introduces additional dynamics and interactions that contribute to the
overall behavior of this process. They play a crucial role in understanding
strong interactions. In particular, their resonant behavior is associated with
a peak in the CP asymmetry for certain energy ranges.

In the decay process of $\bar{B}^{0}_{d}\rightarrow\pi^{+}\pi^{-}\pi^{0}$, the
vector meson resonance effect refers to how these particles can influence or
modify the decay process. By considering this resonance effect, we gain
insights into various aspects of
$\bar{B}^{0}_{d}\rightarrow\pi^{+}\pi^{-}\pi^{0}$. For instance, it helps us
understand how different intermediate states involving vector mesons
contribute to the final state particles (pions) observed experimentally. It
also provides information about possible interference patterns between
different amplitudes contributing to this decay process. The amplitudes of the
decay process $\bar{B}^{0}_{d}\rightarrow\pi^{+}\pi^{-}\pi^{0}$ from the
different intermediate vector meson can be expressed as

$\displaystyle{\cal
A}(\bar{B}^{0}_{d}{\to}(\rho^{0}\rightarrow\pi^{+}\pi^{-})\pi^{0})=$
$\displaystyle\frac{<\rho^{0}\pi^{0}|H_{eff}|\bar{B}^{0}><\pi^{+}\pi^{-}|H_{\rho^{0}\rightarrow\pi^{+}\pi^{-}}|\rho^{0}>}{{s-m_{\rho^{0}}^{2}+im_{\rho^{0}}\varGamma_{\rho^{0}}}}$
(23) $\displaystyle=\sum_{\lambda=0,\pm
1}\frac{G_{F}P_{\bar{B}^{0}}\cdot\epsilon^{*}\left(\lambda\right)g^{\rho^{0}\rightarrow\pi^{+}\pi^{-}}\epsilon\left(\lambda\right)\cdot\left(p_{\pi^{+}}-p_{\pi^{-}}\right)}{\sqrt{2}({{s-m_{\rho^{0}}^{2}+im_{\rho^{0}}\varGamma_{\rho^{0}}}})}$
$\displaystyle\times\big{\\{}V_{ub}\,V_{ud}^{\ast}\,\big{\\{}a_{2}\,\big{[}-{\cal
A}_{ab}^{LL}({\pi},{\rho})-{\cal A}_{ab}^{LL}({\rho},{\pi})+{\cal
A}_{ef}^{LL}({\pi},{\rho})+{\cal A}_{ef}^{LL}({\rho},{\pi})\big{]}$
$\displaystyle+C_{1}\,\big{[}-{\cal A}_{cd}^{LL}({\pi},{\rho})-{\cal
A}_{cd}^{LL}({\rho},{\pi})+{\cal A}_{gh}^{LL}({\pi},{\rho})+{\cal
A}_{gh}^{LL}({\rho},{\pi})\big{]}\big{\\}}$ $\displaystyle-
V_{tb}\,V_{td}^{\ast}\,\big{\\{}(a_{4}-\frac{3}{2}\,a_{9}-\frac{1}{2}\,a_{10})\,\big{[}{\cal
A}_{ab}^{LL}({\pi},{\rho})+{\cal
A}_{ab}^{LL}({\rho},{\pi})\big{]}+\frac{3}{2}\,a_{7}\,\big{[}{\cal
A}_{ab}^{LL}({\pi},{\rho})$ $\displaystyle-{\cal
A}_{ab}^{LL}({\rho},{\pi})\big{]}+(C_{3}-\frac{1}{2}\,C_{9}-\frac{3}{2}\,C_{10})\,\big{[}{\cal
A}_{cd}^{LL}({\pi},{\rho})+{\cal A}_{cd}^{LL}({\rho},{\pi})\big{]}$
$\displaystyle-\frac{3}{2}\,C_{8}\,\big{[}{\cal
A}_{cd}^{LR}({\pi},{\rho})+{\cal
A}_{cd}^{LR}({\rho},{\pi})\big{]}+(a_{6}-\frac{1}{2}\,a_{8})\,\big{[}{\cal
A}_{ab}^{SP}({\pi},{\rho})+{\cal A}_{ef}^{SP}({\pi},{\rho})$
$\displaystyle+{\cal
A}_{ef}^{SP}({\rho},{\pi})\big{]}+(C_{5}-\frac{1}{2}\,C_{7})\,\big{[}{\cal
A}_{cd}^{SP}({\pi},{\rho})+{\cal A}_{cd}^{SP}({\rho},{\pi})+{\cal
A}_{gh}^{SP}({\pi},{\rho})$
$\displaystyle+(2\,a_{3}+a_{4}-2\,a_{5}-\frac{1}{2}\,a_{7}+\frac{1}{2}\,a_{9}-\frac{1}{2}\,a_{10})\,\big{[}{\cal
A}_{ef}^{LL}({\pi},{\rho})+{\cal A}_{ef}^{LL}({\rho},{\pi})\big{]}$
$\displaystyle+(C_{3}+2\,C_{4}-\frac{1}{2}\,C_{9}+\frac{1}{2}\,C_{10})\,\big{[}{\cal
A}_{gh}^{LL}({\pi},{\rho})+{\cal A}_{gh}^{LL}({\rho},{\pi})\big{]}$
$\displaystyle+(2\,C_{6}+\frac{1}{2}\,C_{8})\,\big{[}{\cal
A}_{gh}^{LR}({\pi},{\rho})+{\cal
A}_{gh}^{LR}({\rho},{\pi})\big{]}\big{\\}}\big{\\}},$

$\displaystyle{\cal
A}(\bar{B}^{0}_{d}{\to}(\omega\rightarrow\pi^{+}\pi^{-})\pi^{0})=$
$\displaystyle\frac{<{\omega}\pi^{0}|H_{eff}|\bar{B}^{0}><\pi^{+}\pi^{-}|H_{\omega\rightarrow\pi^{+}\pi^{-}}|\omega>}{{s-m_{\omega}^{2}+im_{\omega}\varGamma_{\omega}}}$
(24) $\displaystyle=$ $\displaystyle\sum_{\lambda=0,\pm
1}\frac{G_{F}P_{(\bar{B}^{0}}\cdot\epsilon^{*}\left(\lambda\right)\
g^{\omega\rightarrow\pi^{+}\pi^{-}}\epsilon\left(\lambda\right)\cdot\left(p_{\pi^{+}}-p_{\pi^{-}}\right)}{\sqrt{2}({{s-m_{\omega}^{2}+im_{\omega}\varGamma_{\omega}})}}$
$\displaystyle\times\big{\\{}V_{ub}\,V_{ud}^{\ast}\,\big{\\{}a_{2}\,\big{[}{\cal
A}_{ab}^{LL}({\pi},{\omega})-{\cal A}_{ab}^{LL}({\omega},{\pi})+{\cal
A}_{ef}^{LL}({\pi},{\omega})+{\cal A}_{ef}^{LL}({\omega},{\pi})\big{]}$
$\displaystyle+$ $\displaystyle C_{1}\,\big{[}{\cal
A}_{cd}^{LL}({\pi},{\omega})-{\cal A}_{cd}^{LL}({\omega},{\pi})+{\cal
A}_{gh}^{LL}({\pi},{\omega})+{\cal
A}_{gh}^{LL}({\omega},{\pi})\big{]}\big{\\}}$ $\displaystyle-$ $\displaystyle
V_{tb}\,V_{td}^{\ast}\,\big{\\{}-(2\,a_{3}+a_{4}+2\,a_{5}+\frac{1}{2}\,a_{7}+\frac{1}{2}\,a_{9}-\frac{1}{2}\,a_{10})\,{\cal
A}_{ab}^{LL}({\omega},{\pi})$ $\displaystyle-$
$\displaystyle(C_{3}+2\,C_{4}-\frac{1}{2}\,C_{9}+\frac{1}{2}\,C_{10})\,{\cal
A}_{cd}^{LL}({\omega},{\pi})-(2\,C_{6}+\frac{1}{2}\,C_{8})\,{\cal
A}_{cd}^{LR}({\omega},{\pi})$ $\displaystyle-$
$\displaystyle(a_{4}+\frac{3}{2}\,a_{7}-\frac{3}{2}\,a_{9}-\frac{1}{2}\,a_{10})\,\big{[}{\cal
A}_{ab}^{LL}({\pi},{\omega})+{\cal A}_{ef}^{LL}({\pi},{\omega})+{\cal
A}_{ef}^{LL}({\omega},{\pi})\big{]}$ $\displaystyle-$
$\displaystyle(C_{3}-\frac{1}{2}\,C_{9}-\frac{3}{2}\,C_{10})\,\big{[}{\cal
A}_{cd}^{LL}({\pi},{\omega})+{\cal A}_{gh}^{LL}({\pi},{\omega})+{\cal
A}_{gh}^{LL}({\omega},{\pi})\big{]}$ $\displaystyle+$
$\displaystyle\frac{3}{2}\,C_{8}\,\big{[}{\cal
A}_{cd}^{LR}({\pi},{\omega})+{\cal A}_{gh}^{LR}({\pi},{\omega})+{\cal
A}_{gh}^{LR}({\omega},{\pi})\big{]}$ $\displaystyle-$
$\displaystyle(a_{6}-\frac{1}{2}\,a_{8})\,\big{[}{\cal
A}_{ab}^{SP}({\pi},{\omega})+{\cal A}_{ef}^{SP}({\pi},{\omega})+{\cal
A}_{ef}^{SP}({\omega},{\pi})\big{]}$ $\displaystyle-$
$\displaystyle(C_{5}-\frac{1}{2}\,C_{7})\,\big{[}{\cal
A}_{cd}^{SP}({\pi},{\omega})+{\cal A}_{cd}^{SP}({\omega},{\pi})+{\cal
A}_{gh}^{SP}({\pi},{\omega})+{\cal
A}_{gh}^{SP}({\omega},{\pi})\big{]}\big{\\}}\big{\\}},$

$\displaystyle{\cal
A}(\bar{B}^{0}_{d}{\to}(\phi\rightarrow\pi^{+}\pi^{-})\pi^{0})$
$\displaystyle=\frac{<{\phi}\pi^{0}|H_{eff}|\bar{B}^{0}><\pi^{+}\pi^{-}|H_{\phi\rightarrow\pi^{+}\pi^{-}}|\phi>}{{s-m_{\phi}^{2}+im_{\phi}\varGamma_{\phi}}}$
(25) $\displaystyle=\sum_{\lambda=0,\pm
1}\frac{G_{F}P_{(\bar{B}^{0}}\cdot\epsilon^{*}\left(\lambda\right)\
g^{\phi\rightarrow\pi^{+}\pi^{-}}\epsilon\left(\lambda\right)\cdot\left(p_{\pi^{+}}-p_{\pi^{-}}\right)}{\sqrt{2}({{s-m_{\phi}^{2}+im_{\phi}\varGamma_{\phi}}})}$
$\displaystyle\times\big{\\{}V_{tb}\,V_{td}^{\ast}\,\big{\\{}(a_{3}+a_{5}-\frac{1}{2}\,a_{7}-\frac{1}{2}\,a_{9})\,{\cal
A}_{ab}^{LL}({\phi},{\pi})+(C_{4}-\frac{1}{2}\,C_{10})\,{\cal
A}_{cd}^{LL}({\phi},{\pi})$ $\displaystyle+(C_{6}-\frac{1}{2}\,C_{8})\,{\cal
A}_{cd}^{LR}({\phi},{\pi})\big{\\}}\big{\\}}.$

Figure 5: Plot of $A_{CP}$ as a function of $\sqrt{s}$ corresponding to
central parameter values of CKM matrix elements for the decay channel of
$\bar{B}_{d}^{0}\rightarrow\pi^{+}\pi^{-}\pi^{0}$.

Figure 6: CP asymmetry plot near the invariant mass of $\phi$ during the decay
channel of ${B}_{d}^{0}\rightarrow\pi^{+}\pi^{-}\pi^{0}$ from the central
parameter values of CKM matrix elements.

In Fig.5, we present the relationship between the invariant mass of the
$\pi^{+}\pi^{-}$ meson pair and the CP asymmetry in the decay process
$\bar{B}_{d}^{0}\rightarrow\pi^{+}\pi^{-}\pi^{0}$. We aim to investigate how
changes in the meson pair invariant mass affect CP asymmetry.

Within a specific range of invariant masses, we observe a significant
variation in CP asymmetry within the $\rho^{0}$\- $\omega-\phi$ mass region.
The maximum value of CP asymmetry reaches an impressive $91\%$. Notably,
slight peaks are observed at positions corresponding to the mass of the $\phi$
meson.

To gain further insights into these observations, we investigate the impact of
both three-particle and two-particle mixing effects on CP asymmetry in this
decay process. By conducting regional integration analysis of CP and examining
resonance effects on CP asymmetry, we enhance our understanding of the
contributions from different factors to variations in CP violation. Detailed
discussions regarding this topic are presented in the subsequent section.

Additionally, it is worth mentioning that another related decay process
involving penguins diagrams is investigated:
$\bar{B}_{d}^{0}{\to}(\phi\rightarrow\pi^{+}\pi^{-})\pi^{0}$. Unlike our
previous discussion on tree contributions, this particular decay process
solely relies on penguins diagrams without any involvement from tree
contributions.

###  III.4  CP asymmetry analysis of the
$\bar{B}_{d}^{0}\rightarrow(\rho^{0},\omega,\phi\rightarrow\pi^{+}\pi^{-})\bar{K}^{0}$
decay process

In the above amplitude analysis, we investigated the impact of vector meson
resonance effects on decay processes. During the decay process, vector meson
resonances can induce the generation of strong phases. In CP asymmetry
diagrams, different vector meson resonances can be observed to have varying
effects on the CP asymmetry of decay processes. Currently, there is no
experimental data available on the
$\bar{B}_{d}^{0}\rightarrow(\rho^{0},\omega,\phi\rightarrow\pi^{+}\pi^{-})\bar{K}^{0}$
decay process. Studying the vector meson resonance effect not only provides
rich physical information to explore the energy-momentum relationship in
multi-body decay processes, but also offers a new approach for future
experimental investigations of CP asymmetry values and intermediate mesons in
$\bar{B}_{d}^{0}\rightarrow(\rho^{0},\omega,\phi\rightarrow\pi^{+}\pi^{-})\bar{K}^{0}$
decay processes. We present the amplitude form of the
$\bar{B}_{d}^{0}\rightarrow(\rho^{0},\omega,\phi\rightarrow\pi^{+}\pi^{-})\bar{K}^{0}$
decay process under the PQCD framework.

$\displaystyle{\cal
A}(\overline{B}_{d}^{0}{\to}(\omega\rightarrow\pi^{+}\pi^{-})\overline{K}^{0})=$
$\displaystyle\frac{<{\omega}\overline{K}^{0}|H_{eff}|\bar{B}^{0}><\pi^{+}\pi^{-}|H_{\omega\rightarrow\pi^{+}\pi^{-}}|\omega>}{{s-m_{\omega}^{2}+im_{\omega}\varGamma_{\omega}}}$
(26) $\displaystyle=$ $\displaystyle\sum_{\lambda=0,\pm
1}\frac{G_{F}P_{(\bar{B}^{0}}\cdot\epsilon^{*}\left(\lambda\right)\
g^{\omega\rightarrow\pi^{+}\pi^{-}}\epsilon\left(\lambda\right)\cdot\left(p_{\pi^{+}}-p_{\pi^{-}}\right)}{2({{s-m_{\omega}^{2}+im_{\omega}\varGamma_{\omega}}})}$
$\displaystyle\times\big{\\{}V_{ub}\,V_{us}^{\ast}\,\big{\\{}a_{2}\,\big{[}{\cal
A}_{ab}^{LL}({\omega},\overline{K})\big{]}+C_{1}\,\big{[}{\cal
A}_{cd}^{LL}({\omega},\overline{K})\big{]}\big{\\}}$
$\displaystyle\times\big{\\{}V_{tb}\,V_{ts}^{\ast}\,\big{\\{}(a_{4}-\frac{1}{2}\,a_{10})\,\big{[}{\cal
A}_{ab}^{LL}(\overline{K},{\omega})+{\cal
A}_{ef}^{LL}(\overline{K},{\omega})\big{]}+(C_{3}$
$\displaystyle-\frac{1}{2}\,C_{9})\,\big{[}{\cal
A}_{cd}^{LL}(\overline{K},{\omega})+{\cal
A}_{gh}^{LL}(\overline{K},{\omega})\big{]}+(a_{6}-\frac{1}{2}\,a_{8})\,\big{[}{\cal
A}_{ab}^{SP}(\overline{K},{\omega})$ $\displaystyle+{\cal
A}_{ef}^{SP}(\overline{K},{\omega})\big{]}+(C_{5}-\frac{1}{2}\,C_{7})\,\big{[}{\cal
A}_{cd}^{SP}(\overline{K},{\omega})+{\cal
A}_{gh}^{SP}(\overline{K},{\omega})\big{]}$
$\displaystyle+(2\,a_{3}+2\,a_{5}+\frac{1}{2}\,a_{7}+\frac{1}{2}\,a_{9})\,{\cal
A}_{ab}^{LL}({\omega},\overline{K})+(2\,C_{4}$
$\displaystyle+\frac{1}{2}\,C_{10})\,{\cal
A}_{cd}^{LL}({\omega},\overline{K})+(2\,C_{6}+\frac{1}{2}\,C_{8})\,{\cal
A}_{cd}^{LR}({\omega},\overline{K})\big{\\}}\big{\\}},$

$\displaystyle{\cal
A}(\overline{B}_{d}^{0}{\to}(\rho^{0}\rightarrow\pi^{+}\pi^{-})\overline{K}^{0})=$
$\displaystyle\frac{<\rho^{0}\overline{K}^{0}|H_{eff}|\bar{B}^{0}><\pi^{+}\pi^{-}|H_{\rho^{0}\rightarrow\pi^{+}\pi^{-}}|\rho^{0}>}{{s-m_{\rho^{0}}^{2}+im_{\rho^{0}}\varGamma_{\rho^{0}}}}$
(27) $\displaystyle=$ $\displaystyle\sum_{\lambda=0,\pm
1}\frac{G_{F}P_{\bar{B}^{0}}\cdot\epsilon^{*}\left(\lambda\right)g^{\rho^{0}\rightarrow\pi^{+}\pi^{-}}\epsilon\left(\lambda\right)\cdot\left(p_{\pi^{+}}-p_{\pi^{-}}\right)}{2({{s-m_{\rho^{0}}^{2}+im_{\rho^{0}}\varGamma_{\rho^{0}}}})}$
$\displaystyle\times\big{\\{}V_{ub}\,V_{us}^{\ast}\,\big{\\{}a_{2}\,\big{[}{\cal
A}_{ab}^{LL}({\rho},\overline{K})\big{]}+C_{1}\,\big{[}{\cal
A}_{cd}^{LL}({\rho},\overline{K})\big{]}\big{\\}}\times\big{\\{}V_{tb}\,V_{ts}^{\ast}\,$
$\displaystyle\big{\\{}(a_{4}-\frac{1}{2}\,a_{10})\,\big{[}{\cal
A}_{ab}^{LL}(\overline{K},{\rho})+{\cal
A}_{ef}^{LL}(\overline{K},{\rho})\big{]}+(C_{3}-\frac{1}{2}\,C_{9})\,\big{[}{\cal
A}_{cd}^{LL}(\overline{K},{\rho})$ $\displaystyle+{\cal
A}_{gh}^{LL}(\overline{K},{\rho})\big{]}+(a_{6}-\frac{1}{2}\,a_{8})\,\big{[}{\cal
A}_{ab}^{SP}(\overline{K},{\rho})+{\cal
A}_{ef}^{SP}(\overline{K},{\rho})\big{]}+(C_{5}-$
$\displaystyle\frac{1}{2}\,C_{7})\,\big{[}{\cal
A}_{cd}^{SP}(\overline{K},{\rho})+{\cal
A}_{gh}^{SP}(\overline{K},{\rho})\big{]}-\frac{3}{2}\,(a_{7}+a_{9})\,{\cal
A}_{ab}^{LL}({\rho},\overline{K})$ $\displaystyle-\frac{3}{2}\,C_{8}\,{\cal
A}_{cd}^{LR}({\rho},\overline{K})-\frac{3}{2}\,C_{10}\,{\cal
A}_{cd}^{LL}({\rho},\overline{K})\big{\\}}\big{\\}},$

$\displaystyle{\cal
A}(\overline{B}_{d}^{0}{\to}(\phi\rightarrow\pi^{+}\pi^{-})\overline{K}^{0})$
$\displaystyle=\frac{<{\phi}\overline{K}^{0}|H_{eff}|\bar{B}^{0}><\pi^{+}\pi^{-}|H_{\phi\rightarrow\pi^{+}\pi^{-}}|\phi>}{{s-m_{\phi}^{2}+im_{\phi}\varGamma_{\phi}}}\
$ (28) $\displaystyle=\sum_{\lambda=0,\pm
1}\frac{G_{F}P_{(\bar{B}^{0}}\cdot\epsilon^{*}\left(\lambda\right)\
g^{\phi\rightarrow\pi^{+}\pi^{-}}\epsilon\left(\lambda\right)\cdot\left(p_{\pi^{+}}-p_{\pi^{-}}\right)}{\sqrt{2}({{s-m_{\phi}^{2}+im_{\phi}\varGamma_{\phi}})}}$
$\displaystyle\times\big{\\{}V_{tb}\,V_{ts}^{\ast}\,\big{\\{}(a_{3}+a_{4}+a_{5}-\frac{1}{2}\,a_{7}-\frac{1}{2}\,a_{9}-\frac{1}{2}\,a_{10})\,{\cal
A}_{ab}^{LL}({\phi},\overline{K})$
$\displaystyle+(a_{4}-\frac{1}{2}\,a_{10})\,{\cal
A}_{ef}^{LL}({\phi},\overline{K})+(a_{6}-\frac{1}{2}\,a_{8})\,{\cal
A}_{ef}^{SP}({\phi},\overline{K})$
$\displaystyle+(C_{3}+C_{4}-\frac{1}{2}\,C_{9}-\frac{1}{2}\,C_{10})\,{\cal
A}_{cd}^{LL}({\phi},\overline{K})+(C_{6}-\frac{1}{2}\,C_{8})\,{\cal
A}_{cd}^{LR}({\phi},\overline{K})$
$\displaystyle+(C_{5}-\frac{1}{2}\,C_{7})\,\big{[}{\cal
A}_{cd}^{SP}({\phi},\overline{K})+{\cal
A}_{gh}^{SP}({\phi},\overline{K})\big{]}+(C_{3}-\frac{1}{2}\,C_{9})\,{\cal
A}_{gh}^{LL}({\phi},\overline{K})\big{\\}}\big{\\}},$

Figure 7: Plot of $A_{CP}$ as a function of $\sqrt{s}$ corresponding to
central parameter values of CKM matrix elements for the decay channel of
$\bar{B}_{d}^{0}\rightarrow\pi^{+}\pi^{-}\bar{K}^{0}$.

The results for the $\bar{B}_{d}^{0}\rightarrow\pi^{+}\pi^{-}\bar{K}^{0}$
decay process are presented in Fig. 6. As depicted in Fig. 6, it is evident
that the CP asymmetry of the
$\bar{B}_{d}^{0}\rightarrow\pi^{+}\pi^{-}\bar{K}^{0}$ channel undergoes
variations when the invariant masses of the $\pi^{+}\pi^{-}$ pair encompass
the ranges corresponding to resonances such as $\omega$ and $\phi$. Notably, a
maximum CP asymmetry of $18\%$ can be achieved.

The CP asymmetry in the decay process
$\bar{B}_{d}^{0}\rightarrow\pi^{+}\pi^{-}\bar{K}^{0}$ exhibits a significant
change when the invariant mass of the $\pi^{+}\pi^{-}$ pair approaches 0.75
GeV, reaching a peak value of $18\%$. This behavior can be attributed to the
interference effect arising from the $\rho-\omega$ mixing mechanism, which is
expected to occur within the range of 0.7 GeV - 0.8 GeV. Additionally, small
peaks are observed in the invariant mass range corresponding to $\phi$. The CP
asymmetry in this case amounts to $9\%$. In
${B}_{d}^{0}{\to}(\phi\rightarrow\pi^{+}\pi^{-})\bar{K}^{0}$ decay, penguin
graph contributions play a role. Notably, the resonance effect due to
$\rho-\phi$ mixing significantly influences CP violation with an associated CP
asymmetry value of $9\%$.

##  IV Analysis of regional CP asymmetry in the decay process

Table 2: The peak regional $(0.70GeV\leq\sqrt{s}\leq 1.10GeV)$ integral of $\mathrm{A}^{\Omega}_{\mathrm{CP}}$ from different resonance rangs for $\bar{B}_{d}\rightarrow\pi^{+}\pi^{-}K^{0}$, $\bar{B}_{d}\rightarrow\pi^{+}\pi^{-}\pi^{0}$ and $\ B_{u}^{-}\rightarrow\pi^{+}\pi^{-}K^{-}$, $\ B_{u}^{-}\rightarrow\pi^{+}\pi^{-}\pi^{-}$ decay processes. Decay channel | This work | Previous measurements(no mixing)
---|---|---
$\bar{B}_{d}\rightarrow\pi^{+}\pi^{-}\bar{K}^{0}$ |  $\mathrm{-0.1683\pm 0.0013\pm 0.0000}$ ($\rho$) |
|  $\mathrm{-0.0847\pm 0.0000\pm 0.0084}$ ($\rho-\omega$ mixing ) | —————————–
|  $\mathrm{-0.0813\pm 0.0064\pm 0.0052}$ ($\rho-\phi$ mixing) |
|  $\mathrm{-0.0847\pm 0.0056\pm 0.0089}$ ($\phi-\rho-\omega$ mixing ) |
$\bar{B}_{d}\rightarrow\pi^{+}\pi^{-}\pi^{0}$ |  $\mathrm{-0.0054\pm 0.0004\pm 0.0011}$ ($\rho$) |
|  $\mathrm{0.0173\pm 0.0109\pm 0.0015}$ ($\rho-\omega$ mixing ) | —————————–
|  $\mathrm{-0.0055\pm 0.0006\pm 0.0011}$ ($\rho-\phi$ mixing ) |
|  $\mathrm{0.0147\pm 0.0014\pm 0.0086}$ ($\phi-\rho-\omega$ mixing) |
$\ B_{u}^{-}\rightarrow\pi^{+}\pi^{-}K^{-}$ |  $\mathrm{0.2093\pm 0.0206\pm 0.0044}$ ($\rho$) | 0.150$\pm$0.019$\pm$0.011 LHCbLHCb:2022tuk
|  $\mathrm{0.1771\pm 0.0084\pm 0.0061}$ ($\rho-\omega$ mixing ) | 0.44$\pm$0.10$\pm$0.04 BaBarBaBar:2008lpx
|  $\mathrm{0.2109\pm 0.0207\pm 0.0023}$ ($\rho-\phi$ mixing) | 0.30$\pm$0.11$\pm$0.02 BelleBelle:2005rpz
|  $\mathrm{0.3470\pm 0.0310\pm 0.0709}$ ($\phi-\rho-\omega$ mixing) |
$\ B_{u}^{-}\rightarrow\pi^{+}\pi^{-}\pi^{-}$ |  $\mathrm{0.0065\pm 0.0014\pm 0.0031}$($\rho$) | -0.004$\pm$0.017$\pm$0.009 LHCbLHCb:2022tuk
|  $\mathrm{0.0256\pm 0.0013\pm 0.0016}$ ($\rho-\omega$ mixing) | 0.30$\pm$0.11$\pm$0.02 BelleBelle:2005rpz
|  $\mathrm{0.0076\pm 0.0006\pm 0.0023}$ ($\rho-\phi$ mixing) |
|  $\mathrm{0.0260\pm 0.0034\pm 0.0047}$ ($\phi-\rho-\omega$ mixing) |

In Table 2, we present a comparison between our calculated results and the
existing experimental data. Notably, intriguing observations of significant
regional CP asymmetry have been reported in experiments conducted by LHCb,
BaBar, and Belle collaborations. Subsequently, we investigate the influence of
different resonance effects on regional CP asymmetry. Due to the close masses
of $\rho$ and $\omega$ mesons in these experiments, distinguishing between
them becomes challenging. To address this issue, we incorporate this scenario
into our calculations using the PQCD method within the resonance framework for
decay channels such as $\bar{B}_{d}\rightarrow\pi^{+}\pi^{-}\bar{K}^{0}$,
$\bar{B}_{d}\rightarrow\pi^{+}\pi^{-}\pi^{0}$,
$B_{u}^{-}\rightarrow\pi^{+}\pi^{-}K^{-}$ and
$B_{u}^{-}\rightarrow\pi^{+}\pi^{-}\pi^{-}$ . Our results align with
experimental error ranges, validating the accuracy of our calculation method.
Specifically in the decay channel of
$\bar{B}_{d}\rightarrow\pi^{+}\pi^{-}\bar{K}^{0}$ , we observe that resonance
contributions from three-particle mixtures significantly impact CP asymmetry.

For vector particle resonances, the decay processes
$\bar{B}_{d}\rightarrow\pi^{+}\pi^{-}\pi^{0}$ and
$B_{u}^{-}\rightarrow\pi^{+}\pi^{-}\pi^{-}$ exhibit significant contributions
to CP asymmetry compared to non-resonant decays. It is noteworthy that clear
manifestations of CP asymmetry are observed within the resonance range. In the
decay channel of $B_{u}^{-}\rightarrow\pi^{+}\pi^{-}K^{-}$, resonance effects
contribute more prominently to CP asymmetry, resulting in increased regional
values similar to those observed in the decay channel
$\bar{B}_{d}\rightarrow\pi^{+}\pi^{-}K^{0}$ within their respective invariant
mass ranges.

The calculation of the $1/m_{b}$ power correction by perturbation theory is
uncertain. Data errors can introduce uncertainties in the results, with part
of these errors arising from variations in CKM parameters and the remaining
part originating from parameters such as form factors, decay constants, and
B-meson wave functions that involve distributed amplitudes in vector meson
wave functions. By employing the central values of these parameters, we
initially compute numerical results for CP asymmetry and incorporate standard
deviations into the error range presented in Table 2. The influence of mixed
parameters on regional CP asymmetry is negligible; hence it can be disregarded
without further discussion.

##  V SUMMARY AND DISCUSSION

We conducted a comprehensive analysis on the CP asymmetry in the three-body
decay of B meson, focusing specifically regions on the invariant mass of
$\pi^{+}\pi^{-}$ meson pairs. The results revealed an intriguing observation -
a distinct change in CP asymmetry across different resonance regions,
specifically the $\rho$, $\omega$, and $\phi$ meson resonances. This finding
suggests that these resonances play a significant role in influencing the
decay dynamics and subsequent CP asymmetry. The presence of such distinct
changes in CP asymmetry across different resonance regions provides valuable
insights into our understanding of fundamental particle interactions. It
highlights how specific energy regimes can impact particle decays and their
associated symmetries.

We quantify regional CP asymmetry by integrating over the phase space. In
decays such as $\bar{B}_{d}\rightarrow\pi^{+}\pi^{-}\bar{K}^{0}$,
$\bar{B}_{d}\rightarrow\pi^{+}\pi^{-}\pi^{0}$, and
$B_{u}^{-}\rightarrow\pi^{+}\pi^{-}K^{-}$,
$B_{u}^{-}\rightarrow\pi^{+}\pi^{-}\pi^{-}$, we observe CP asymmetry arising
from contributions of two-meson mixing and three-meson mixing processes.
Notably, significant regional CP asymmetry is observed when involving
$\rho-\phi-\omega$ mixing. Experimental detection of regional CP asymmetry can
be achieved by reconstructing resonant states of $\rho$ , $\omega$ and $\phi$
mesons within their respective resonance regions.

Recently, the LHCb experimental group has made significant progress in
investigating the three-body decay of B meson and has obtained noteworthy
findings. By analyzing previous experimental data, they have measured direct
CP asymmetry in various decay modes such as $B^{\pm}\rightarrow
K^{+}K^{-}K^{\pm}$, $B^{\pm}\rightarrow\pi^{+}\pi^{-}K^{\pm}$,
$B^{\pm}\rightarrow\pi^{+}\pi^{-}\pi^{\pm}$, and $B^{\pm}\rightarrow
K^{+}K^{-}\pi^{\pm}$. Building upon the achievements of LHCb experiments,
future investigations are expected to primarily focus on exploring regional CP
asymmetry of the three-body decays of $B$ meson at the resonance regions of
$\rho$ , $\omega$ and $\phi$ mesons.

## Acknowledgements

This work was supported by Natural Science Foundation of Henan (Project No.
232300420115) .

## References

  * (1) L. B. Okun,Sov. Phys. Usp. 9 (1967), 574-601.
  * (2) M. E. Shaposhnikov,Phys. Lett. B 277 (1992), 324-330.
  * (3) N. Cabibbo,Phys. Rev. Lett. 10 (1963), 531-533.
  * (4) A. Alavi-Harati et al. [KTeV],Phys. Rev. Lett. 84 (2000), 408-411.
  * (5) K. Abe et al. [Belle],Phys. Rev. Lett. 87 (2001), 091802.
  * (6) B. Aubert et al. [BaBar],Phys. Rev. D 65 (2002), 051101.
  * (7) R. Aaij et al. [LHCb],Phys. Rev. Lett. 124 (2020) no.3, 031801.
  * (8) R. Aaij et al. [LHCb],Phys. Rev. D 108 (2023) no.1, 012013.
  * (9) R. Aaij et al. [LHCb],Phys. Rev. Lett. 111 (2013), 101801.
  * (10) R. Aaij et al. [LHCb],Phys. Rev. Lett. 112 (2014) no.1, 011801.
  * (11) R. Aaij et al. [LHCb],Phys. Rev. D 90 (2014) no.11, 112004.
  * (12) X. L. Yuan, G. Lü, N. Wang, L. Y. Zhang and X. H. Guo,Chin. Phys. C 47 (2023) no.11, 113101.
  * (13) D. S. Shi, G. Lü, Y. L. Zhao, Na-Wang and X. H. Guo,Eur. Phys. J. C 83 (2023) no.4, 345.
  * (14) J. H. Christenson, J. W. Cronin, Phys. Rev. Lett. 13, 138-140 (1964).
  * (15) Gang. L, Y. Lu, S.-T. Li, Y.-T. Wang, Eur. Phys. J. C77, 518 (2017).
  * (16) C. E. Wolfe, K. Maltman, Phys. Rev. D80, 114024 (2009).
  * (17) M. N. Achasov, et.al., Nucl. Phys. B569, 158 (2000).
  * (18) C. E. Wolfe, K. Maltman, Phys. Rev. D83, 077301 (2011).
  * (19) L. Wolfenstein, Phys. Rev. Lett. 51,1945 (1983); Phys. Rev. Lett. 13, 562 (1964).
  * (20) J. J. Qi, Z. Y. Wang, X. H. Guo, X. W. Kang and Z. H. Zhang,Nucl. Phys. B 948 (2019), 114788.
  * (21) C. K. Chua, H. Y. Cheng and C. W. Chiang,PoS EPS-HEP2021 (2022), 506.
  * (22) H. Y. Cheng,Phys. Rev. D 106 (2022) no.11, 113004
  * (23) H. Y. Cheng, C. W. Chiang and C. K. Chua, Phys. Lett. B 813 (2021), 136058
  * (24) R. L. Workman et al. (Particle Data Group). Review of Particle Physics, PTEP 2022, 083(2022).
  * (25) Y. Li, C. D. Lu, Z. J. Xiao and X. Q. Yu,Phys. Rev. D 70 (2004), 034009
  * (26) C. H. Chen and H. n. Li,Phys. Lett. B 561 (2003), 258-265
  * (27) B. Aubert et al. [BaBar],Phys. Rev. D 79 (2009), 072006.
  * (28) B. Aubert et al. [BaBar],Phys. Rev. D 78 (2008), 012004.
  * (29) A. Garmash et al. [Belle],Phys. Rev. Lett. 96 (2006), 251803.
  * (30) LHCb collaboration, R. Aaij et al. Phys. Rev. Lett. 124 (2020) 031801.