∎<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>e4e-mail:Corresponding author:
<EMAIL_ADDRESS>
11institutetext: College of Nuclear Science and Technology, Beijing Normal
University, Beijing, 100875, China 22institutetext: School of Nuclear Science
and Technology, University of South China, Hengyang, 421001, Hunan, China

# Localized direct CP violation in
$B^{\pm}\rightarrow\rho^{0}(\omega)\pi^{\pm}\rightarrow\pi^{+}\pi^{-}\pi^{\pm}$

Chao Wange1,addr1  Zhen-Hua Zhange2,addr2  Zhen-Yang Wange3,addr1  Xin-Heng
Guoe4,addr1

(Received: date / Revised version: date)

###### Abstract

We study the localized direct CP violation in the hadronic decays
$B^{\pm}\rightarrow\rho^{0}(\omega)\pi^{\pm}\rightarrow\pi^{+}\pi^{-}\pi^{\pm}$,
including the effect caused by an interesting mechanism involving the charge
symmetry violating mixing between $\rho^{0}$ and $\omega$. We calculate the
localized integrated direct CP violation when the low invariant mass of
$\pi^{+}\pi^{-}$ [$m(\pi^{+}\pi^{-})_{low}$] is near $\rho^{0}(770)$. For five
models of form factors investigated, we find that the localized integrated
direct CP violation varies from -0.0752 to -0.0290 in the ranges of parameters
in our model when $0.750<m(\pi^{+}\pi^{-})_{low}<0.800$ GeV. This result,
especially the sign, agrees with the experimental data and is independent of
form factor models. The new experimental data shows that the signs of the
localized integrated CP asymmetries in the regions
$0.470<m(\pi^{+}\pi^{-})_{low}<0.770$ GeV and
$0.770<m(\pi^{+}\pi^{-})_{low}<0.920$ GeV are positive and negative,
respectively. We find that $\rho$-$\omega$ mixing makes the localized
integrated CP asymmetry move toward the negative direction, and therefore
contributes to the sign change in those two regions. This behavior is also
independent of form factor models. We also calculate the localized integrated
direct CP violating asymmetries in the regions
$0.470<m(\pi^{+}\pi^{-})_{low}<0.770$ GeV,
$0.770<m(\pi^{+}\pi^{-})_{low}<0.920$ GeV and the whole region
$0.470<m(\pi^{+}\pi^{-})_{low}<0.920$ GeV and find that they agree with the
experimental data in some models of form factors.

###### pacs:

11.30.Er 12.39.-x 13.25.Hw

††journal: Eur. Phys. J. C

##  1 Introduction

CP violation is one of the most fundamental and important properties of weak
interactions. Even though it has been known since 1964 J.C. , we still do not
know the source of CP violation completely. In the Standard Model, CP
violation originates from the weak phase in the Cabibbo-Kobayashi-Maskawa
(CKM) matrix N.C. ; M.K. . Besides the weak phase, a large strong phase is
also needed for direct CP violation to occur in decay processes. Usually, this
large phase is provided by QCD loop corrections and some phenomenological
mechanisms. In the past few years, numerous theoretical studies have been
conducted on CP violation. However, we need a lot of data to test these
approaches because there are many theoretical uncertainties such as CKM matrix
elements, hadronic matrix elements, and nonfactorizable effects. These
uncertainties would be reduced by the increase of experimental data in the
future and the improvement of theoretical methods.

Recently, the LHCb Collaboration focused on three-body final states in the
decays of $B$ and $D$ mesons and a novel strategy to probe CP asymmetry in
their Dalitz plots J.M. ; R.A.1 ; R.A.2 . The local asymmetries in specific
regions of the phase space of charmless three-body decays of bottom mesons,
such as $B^{\pm}\to\pi^{\pm}\pi^{+}\pi^{-}$ and $B^{\pm}\to
K^{\pm}\pi^{+}\pi^{-}$, were measured. It was shown that the local asymmetry
distributions in the Dalitz plots reveal rich structures and are not uniform
J.M. ; R.A.1 ; R.A.2 . These intriguing discoveries offer opportunities to
search for different sources of CP violation, through the study of the
signatures of these sources in certain phase spaces of the Dalitz plots. In
fact, several theoretical studies have been made to explain these
distributions, such as the interference between intermediate states S.K.2 ;
Z.H. ; Z.H.1  and final-state rescatterings R.A.2 ; B.B. ; I.B. ; D.A. . One
can confirm that these complex structures originate from more than one source
R.A.2 .

Charge symmetry is broken at the most fundamental level in strong interaction
physics through the small mass difference between up and down quarks in the
QCD Lagrangian. As a consequence the physical $\rho^{0}$ and $\omega$ mesons
are not eigenstates of isospin but, for example, the physical $\rho^{0}$
contains a small admixture of an $I=0$ state H.B.3 . In previous works, this
phenomenon, known as $\rho$-$\omega$ mixing, was considered to obtain a large
strong phase in $B$ decays R.E. ; S.G. ; X.H.1  and it was found that such a
mixing can lead to a peak of CP violation when the invariant mass of
$\pi^{+}\pi^{-}$ is near $\omega$. The differential CP asymmetry was studied
in the decays $B^{\pm}\to\rho^{0}(\omega)\pi^{\pm}\to\pi^{+}\pi^{-}\pi^{\pm}$
before X.H.2 ; O.L. ; G.L. . $\rho$-$\omega$ mixing provides additional
complex terms to the amplitudes C.W.  and the strong phase passes through
$90\,^{\circ}$ at the $\omega$ resonance S.G. ; X.H.1 ; X.H.2 ; O.L. ; G.L. ;
G.L.2 . In our previous work, we found that it is more useful to investigate
the localized integrated CP asymmetry and studied the localized integrated CP
asymmetry in hadronic $\tau$ decays with this mechanism C.W. . The newest LHCb
experiments showed that the resonances are $\rho^{0}$(770) when
$m^{2}(\pi^{+}\pi^{-})_{low}<1\mathrm{GeV}^{2}/\mathrm{c}^{4}$ for
$B^{\pm}\to\pi^{+}\pi^{-}\pi^{\pm}$ decays [$m(\pi^{+}\pi^{-})_{low}$ is the
low invariance mass of $\pi^{+}\pi^{-}$] R.A.2 . It was claimed that the CP
asymmetry in these decays changes sign around the $\rho^{0}(770)$ peak of
$m^{2}(\pi^{+}\pi^{-})_{low}$ R.A.2 , which contains the significant region of
$\rho$-$\omega$ mixing. In the present paper, we aim at studying the localized
integrated CP asymmetry in $B^{\pm}\to\pi^{+}\pi^{-}\pi^{\pm}$ decays
involving $\rho$-$\omega$ mixing and comparing it with the results obtained
from the LHCb Collaboration.

In this paper, we will investigate the localized integrated CP violation with
five phenomenological models of weak form factors for
$B^{\pm}\rightarrow\rho^{0}(\omega)\pi^{\pm}\rightarrow\pi^{-}\pi^{+}\pi^{\pm}$
with and without $\rho$-$\omega$ mixing. We will determine the allowed range
of $N_{c}$ which is the effective color number describing nonfactorizable
contributions. The model dependence of our results will be discussed in
detail. We will see that for five models of form factors, our result for the
localized integrated CP asymmetry varies from -0.0752 (-0.0626) to -0.0403
(-0.0290) in the ranges of $N_{c}$, $2.07(2.09)<N_{c}<4.54(4.65)$
corresponding to $q^{2}/m_{b}^{2}=0.3(0.5)$ ($q$ is the typical momentum
transfer of the gluon or photon in the penguin diagrams) when
$0.750<m(\pi^{+}\pi^{-})_{low}<0.800$ GeV. From Fig. 4 of Ref. R.A.2 , we can
see the localized integrated CP asymmetries have different signs in
$\cos\theta>0$ and $\cos\theta<0$ regions when
$0.750<m(\pi^{+}\pi^{-})_{low}<0.800\,\text{GeV}$ ($\theta$ is the angle
between the momenta of the unpaired $\pi^{+(-)}$ and the $\rho^{0}$ decay
product with the same-sign charge). If one adds the events in these two
experimental regions together, the total localized integrated CP asymmetry
will be $-0.0294\pm 0.0285$. Our results agree with this experimental data.
The experimental values of the localized integrated CP asymmetries in the
regions $0.470<m(\pi^{+}\pi^{-})_{low}<0.770\,\text{GeV}$ and
$0.770<m(\pi^{+}\pi^{-})_{low}<0.920\,\text{GeV}$ are $0.0508\pm 0.0171$ and
$-0.0256\pm 0.0202$, respectively, with opposite signs R.A.2 . We will find
that $\rho$-$\omega$ mixing can make the localized integrated CP asymmetry
move toward the negative direction, and therefore will contribute to the sign
change from the region $0.470<m(\pi^{+}\pi^{-})_{low}<0.770\,\text{GeV}$ to
$0.770<m(\pi^{+}\pi^{-})_{low}<0.920\,\text{GeV}$. This behavior is
independent of form factor models. Furthermore, we will find that our results
in these two regions and the whole region of them are consistent with the
experimental data for several models of form factors.

The remainder of this paper is organized as follows. In Sect. 2, we present
the form of the effective Hamiltonian and the values of Wilson coefficients
and give the formalism for the CP violating asymmetry in
$B^{\pm}\rightarrow\rho^{0}(\omega)\pi^{\pm}\rightarrow\pi^{+}\pi^{-}\pi^{\pm}$.
Then we show numerical results with several models of form factors in this
section. In Sect. 3, we calculate branching ratios for
$B^{+}\to\rho^{0}\pi^{+}$ and $B^{0}\to\rho^{+}\pi^{-}$ and present numerical
results for the range of $N_{c}$ allowed by the experimental data. In the last
section, we give some discussions and summarize our results.

##  2 CP violation in
$B^{\pm}\rightarrow\rho^{0}(\omega)\pi^{\pm}\rightarrow\pi^{+}\pi^{-}\pi^{\pm}$

The amplitude of a decay process described by some amplitudes may have CP-even
and -odd relative phases. Within the Standard Model, the CP-odd relative phase
is always a weak phase difference which is directly determined by the CKM
matrix. On the other hand, CP-even phases, which are called strong phases,
usually originates from nonperturbative effects of strong interactions and are
hard to handle. We consider a $B$ meson weak decay process, $B\to
M_{1}M_{2}M_{3}$, where $M_{i}(i=1,2,3)$ is a light pseudoscalar meson. For a
weak decay process of a heavy meson, a typical form of the decay amplitude $A$
and its CP conjugate one $\bar{A}$ are

$\displaystyle A$ $\displaystyle=$ $\displaystyle
g_{1}r_{1}\mathrm{e}^{\mathrm{i}\phi_{1}}+g_{2}r_{2}\mathrm{e}^{\mathrm{i}\phi_{2}},$
(1) $\displaystyle\bar{A}$ $\displaystyle=$ $\displaystyle
g_{1}^{*}r_{1}\mathrm{e}^{\mathrm{i}\phi_{1}}+g_{2}^{*}r_{2}\mathrm{e}^{\mathrm{i}\phi_{2}},$
(2)

where $g_{1}$ and $g_{2}$ represent CP-odd complex terms which involve CKM
matrix elements, $r_{1}e^{\mathrm{i}\phi_{1}}$ and
$r_{2}e^{\mathrm{i}\phi_{2}}$ terms are even under the CP transformation. Then
one has

$\displaystyle|A|^{2}-|\bar{A}|^{2}=4r_{1}r_{2}\,\mathfrak{I}\mathfrak{m}(g_{1}^{*}g_{2})\,\sin(\phi_{1}-\phi_{2})$
$\displaystyle\quad=4r_{1}r_{2}|g_{1}||g_{2}|\,\sin[\,\mathrm{Arg}(g_{2}/g_{1})]\,\sin(\phi_{1}-\phi_{2}),$
(3)

from which, we can see explicitly that both the CP-odd phase difference
$\mathrm{Arg}(g_{2}/g_{1})$ and the CP-even phase difference
$\phi_{1}-\phi_{2}$ are needed to produce CP violation.

###  2.1 The effective Hamiltonian

In order to calculate the direct CP violating asymmetry in hadronic decays,
one can use the following effective weak Hamiltonian, based on the Operator
Product Expansion G.B. ; M.B.1 ; M.B.2 ; M.B.3 ; M.B.4 :

$\mathcal{H}=\frac{G_{F}}{\sqrt{2}}\big{[}V_{ub}V^{*}_{ud}(c_{1}O^{u}_{1}+c_{2}O^{u}_{2})-V_{tb}V^{*}_{td}\sum^{10}_{i=3}c_{i}O_{i}\big{]}+H.c.,$
(4)

where $V_{ub}$, $V_{ud}$, $V_{tb}$ and $V_{td}$ are CKM matrix elements, and
$c_{i}(i=1,2,..,10)$ are the Wilson coefficients, which are calculable in the
renormalization group improved perturbation theory and are scale dependent. In
the present case, we work with the renormalization scheme independent Wilson
coefficients and use the values of the Wilson coefficients at the
renormalization scale $\mu\approx m_{b}$. The operators $O_{i}$ have the
following forms:

$\displaystyle{}O^{u}_{1}$ $\displaystyle=$
$\displaystyle\overline{d}_{\alpha}\gamma_{\mu}(1-\gamma_{5})u_{\beta}\overline{u}_{\beta}\gamma^{\mu}(1-\gamma_{5})b_{\alpha},$
$\displaystyle O^{u}_{2}$ $\displaystyle=$
$\displaystyle\overline{d}\gamma_{\mu}(1-\gamma_{5})u\overline{u}\gamma^{\mu}(1-\gamma_{5})b,$
$\displaystyle O_{3}$ $\displaystyle=$
$\displaystyle\overline{d}\gamma_{\mu}(1-\gamma_{5})b\sum_{q^{\prime}}\overline{q^{\prime}}\gamma^{\mu}(1-\gamma_{5})q^{\prime},$
$\displaystyle O_{4}$ $\displaystyle=$
$\displaystyle\overline{d}_{\alpha}\gamma_{\mu}(1-\gamma_{5})b_{\beta}\sum_{q^{\prime}}\overline{q^{\prime}}_{\beta}\gamma^{\mu}(1-\gamma_{5})q^{\prime}_{\alpha},$
$\displaystyle O_{5}$ $\displaystyle=$
$\displaystyle\overline{d}\gamma_{\mu}(1-\gamma_{5})b\sum_{q^{\prime}}\overline{q^{\prime}}\gamma^{\mu}(1+\gamma_{5})q^{\prime},$
$\displaystyle O_{6}$ $\displaystyle=$
$\displaystyle\overline{d}_{\alpha}\gamma_{\mu}(1-\gamma_{5})b_{\beta}\sum_{q^{\prime}}\overline{q^{\prime}}_{\beta}\gamma^{\mu}(1+\gamma_{5})q^{\prime}_{\alpha},$
$\displaystyle O_{7}$ $\displaystyle=$
$\displaystyle\frac{3}{2}\overline{d}\gamma_{\mu}(1-\gamma_{5})b\sum_{q^{\prime}}e_{q^{\prime}}\overline{q^{\prime}}\gamma^{\mu}(1+\gamma_{5})q^{\prime},$
$\displaystyle O_{8}$ $\displaystyle=$
$\displaystyle\frac{3}{2}\overline{d}_{\alpha}\gamma_{\mu}(1-\gamma_{5})b_{\beta}\sum_{q^{\prime}}e_{q^{\prime}}\overline{q^{\prime}}_{\beta}\gamma^{\mu}(1+\gamma_{5})q^{\prime}_{\alpha},$
$\displaystyle O_{9}$ $\displaystyle=$
$\displaystyle\frac{3}{2}\overline{d}\gamma_{\mu}(1-\gamma_{5})b\sum_{q^{\prime}}e_{q^{\prime}}\overline{q^{\prime}}\gamma^{\mu}(1-\gamma_{5})q^{\prime},$
$\displaystyle O_{10}$ $\displaystyle=$
$\displaystyle\frac{3}{2}\overline{d}_{\alpha}\gamma_{\mu}(1-\gamma_{5})b_{\beta}\sum_{q^{\prime}}e_{q^{\prime}}\overline{q^{\prime}}_{\beta}\gamma^{\mu}(1-\gamma_{5})q^{\prime}_{\alpha},$
(5)

where $\alpha$ and $\beta$ are color indices, and $q^{\prime}=u,d$ or $s$
quarks. In Eq. (2.1), $O_{1}^{u}$ and $O_{2}^{u}$ are the tree level
operators, $O_{3}-O_{6}$ are QCD penguin operators, and $O_{7}-O_{10}$ arise
from electroweak penguin diagrams.

The Wilson coefficients, $c_{i}$, which are known to the next-to-leading
logarithmic order, take the following values N.G. ; R.F. :

$\displaystyle c_{1}$ $\displaystyle=$ $\displaystyle-0.3125,\qquad
c_{2}=1.1502,$ $\displaystyle c_{3}$ $\displaystyle=$ $\displaystyle
0.0174,\qquad c_{4}=-0.0373,$ $\displaystyle c_{5}$ $\displaystyle=$
$\displaystyle 0.0104,\qquad c_{6}=-0.0459,$ $\displaystyle c_{7}$
$\displaystyle=$ $\displaystyle-1.050\times 10^{-5},\qquad c_{8}=3.839\times
10^{-4},$ $\displaystyle c_{9}$ $\displaystyle=$ $\displaystyle-0.0101,\qquad
c_{10}=1.959\times 10^{-3},$ (6)

at the scale $\mu=m_{b}=5\,\text{GeV}$.

To be consistent, the matrix elements of the operators $O_{i}$ should also be
renormalized to the one-loop order. This results in the effective Wilson
coefficients, $c^{\prime}_{i}$, which satisfy the constraint

${}c_{i}(m_{b})\langle O_{i}(m_{b})\rangle=c^{\prime}_{i}\langle
O_{i}\rangle^{tree},$ (7)

where $\langle O_{i}\rangle^{tree}$ is the matrix element at the tree level,
which will be evaluated in the factorization approach. From Eq. (7), the
relations between $c^{\prime}_{i}$ and $c_{i}$ are N.G. ; R.F.

$\displaystyle c_{1}^{\prime}$ $\displaystyle=$ $\displaystyle c_{1},\qquad
c_{2}^{\prime}=c_{2},$ $\displaystyle c_{3}^{\prime}$ $\displaystyle=$
$\displaystyle c_{3}-P_{s}/3,\qquad c_{4}^{\prime}=c_{4}+P_{s},$
$\displaystyle c_{5}^{\prime}$ $\displaystyle=$ $\displaystyle
c_{5}-P_{s}/3,\qquad c_{6}^{\prime}=c_{6}+P_{s},$ $\displaystyle
c_{7}^{\prime}$ $\displaystyle=$ $\displaystyle c_{7}+P_{e},\qquad
c_{8}^{\prime}=c_{8},$ $\displaystyle c_{9}^{\prime}$ $\displaystyle=$
$\displaystyle c_{9}+P_{e},\qquad c_{10}^{\prime}=c_{10},$ (8)

where

$\displaystyle P_{s}$ $\displaystyle=$
$\displaystyle(\alpha_{s}/8\pi)c_{2}[10/9+G(m_{c},\mu,q^{2})],$ $\displaystyle
P_{e}$ $\displaystyle=$
$\displaystyle(\alpha_{em}/9\pi)(3c_{1}+c_{2})[10/9+G(m_{c},\mu,q^{2})],$

with

$\displaystyle
G(m_{c},\mu,q^{2})=4\int^{1}_{0}\text{d}xx(x-1)\text{ln}\frac{m^{2}_{c}-x(1-x)q^{2}}{\mu^{2}}.$

$G(m_{c},\mu,q^{2})$ has the following explicit expression G.K. :

$\displaystyle\mathfrak{R}\mathfrak{e}G$ $\displaystyle=$
$\displaystyle\frac{2}{3}\Bigg{[}\text{ln}\frac{m_{c}^{2}}{\mu^{2}}-\frac{5}{3}-4\frac{m_{c}^{2}}{q^{2}}+\big{(}1+2\frac{m_{c}^{2}}{q^{2}}\big{)}$
$\displaystyle\qquad\times\sqrt{1-4\frac{m_{c}^{2}}{q^{2}}}\text{ln}\frac{1+\sqrt{1-4\frac{m_{c}^{2}}{q^{2}}}}{1-\sqrt{1-4\frac{m_{c}^{2}}{q^{2}}}}\Bigg{]},$
$\displaystyle\mathfrak{I}\mathfrak{m}G$ $\displaystyle=$
$\displaystyle-\frac{2}{3}\Big{(}1+2\frac{m_{c}^{2}}{q^{2}}\Big{)}\sqrt{1-4\frac{m_{c}^{2}}{q^{2}}}.$
(9)

Based on simple arguments at the quark level, the value of $q^{2}$ is chosen
to be in the range $0.3<q^{2}/m^{2}_{b}<0.5$ R.E. . From Eqs. (8) and (9) we
can obtain numerical values for $c^{\prime}_{i}$.

When $q^{2}/m_{b}^{2}=0.3$,

$\displaystyle c^{\prime}_{1}$ $\displaystyle=$ $\displaystyle-0.3125,\qquad
c^{\prime}_{2}=1.1502,$ $\displaystyle c^{\prime}_{3}$ $\displaystyle=$
$\displaystyle 2.433\times 10^{-2}+1.543\times 10^{-3}\text{i},$
$\displaystyle c^{\prime}_{4}$ $\displaystyle=$ $\displaystyle-5.808\times
10^{-2}-4.628\times 10^{-3}\text{i},$ $\displaystyle c^{\prime}_{5}$
$\displaystyle=$ $\displaystyle 1.733\times 10^{-2}+1.543\times^{-3}\text{i},$
$\displaystyle c^{\prime}_{6}$ $\displaystyle=$ $\displaystyle-6.668\times
10^{-2}-4.628\times^{-3}\text{i},$ $\displaystyle c^{\prime}_{7}$
$\displaystyle=$ $\displaystyle-1.435\times 10^{-4}-2.963\times^{-5}\text{i},$
$\displaystyle c^{\prime}_{8}$ $\displaystyle=$ $\displaystyle 3.839\times
10^{-4},$ $\displaystyle c^{\prime}_{9}$ $\displaystyle=$
$\displaystyle-1.023\times 10^{-2}-2.963\times^{-5}\text{i},$ $\displaystyle
c^{\prime}_{10}$ $\displaystyle=$ $\displaystyle 1.959\times 10^{-3},$ (10)

and when $q^{2}/m_{b}^{2}=0.5$, one has

$\displaystyle c^{\prime}_{1}$ $\displaystyle=$ $\displaystyle-0.3125,\qquad
c^{\prime}_{2}=1.1502,$ $\displaystyle c^{\prime}_{3}$ $\displaystyle=$
$\displaystyle 2.120\times 10^{-2}+2.174\times 10^{-3}\text{i},$
$\displaystyle c^{\prime}_{4}$ $\displaystyle=$ $\displaystyle-4.869\times
10^{-2}-1.552\times 10^{-3}\text{i},$ $\displaystyle c^{\prime}_{5}$
$\displaystyle=$ $\displaystyle 1.420\times 10^{-2}+5.174\times^{-3}\text{i},$
$\displaystyle c^{\prime}_{6}$ $\displaystyle=$ $\displaystyle-5.729\times
10^{-2}-1.552\times^{-2}\text{i},$ $\displaystyle c^{\prime}_{7}$
$\displaystyle=$ $\displaystyle-8.340\times 10^{-5}-9.938\times^{-5}\text{i},$
$\displaystyle c^{\prime}_{8}$ $\displaystyle=$ $\displaystyle 3.839\times
10^{-4},$ $\displaystyle c^{\prime}_{9}$ $\displaystyle=$
$\displaystyle-1.017\times 10^{-2}-9.938\times^{-5}\text{i},$ $\displaystyle
c^{\prime}_{10}$ $\displaystyle=$ $\displaystyle 1.959\times 10^{-3},$ (11)

where we have taken $\alpha_{s}(m_{Z})=0.112$, $\alpha_{em}(m_{b})=1/132.2$,
$m_{b}=5$ GeV, and $m_{c}=1.35$ GeV.

###  2.2 Formalism

For the $B^{-}\rightarrow\rho^{0}\pi^{-}$ process, the amplitude can be
written as $M_{B^{-}\rightarrow\rho^{0}\pi^{-}}^{\lambda}=\alpha
p_{B}\cdot\epsilon^{*}(\lambda)$, where $\epsilon$ is the polarization vector
of $\rho^{0}$ and $\lambda$ is its polarization, $p_{B}$ is the momenta of
$B^{-}$ meson, and $\alpha$ is independent of $\lambda$. The amplitude for
$\rho^{0}\rightarrow\pi^{+}\pi^{-}$ is
$M_{\rho^{0}\rightarrow\pi^{+}\pi^{-}}^{\lambda}=g_{\rho}\epsilon(\lambda)(p_{1}-p_{2})$,
where $p_{1}$ and $p_{2}$ are the momenta of $\pi^{+}$ and $\pi^{-}$ produced
by $\rho^{0}$, respectively, and $g_{\rho}$ is the effective coupling constant
for $\rho^{0}\rightarrow\pi^{+}\pi^{-}$. Then, for the sequential decay
$B^{-}\rightarrow\rho^{0}\pi^{-}\rightarrow\pi^{+}\pi^{-}\pi^{-}$, the
amplitude is Z.H. ; Z.H.1 :

$\displaystyle A$ $\displaystyle=$ $\displaystyle\alpha
p_{B}^{\mu}\frac{\sum_{\lambda}\epsilon^{*}_{\mu}(\lambda)\epsilon_{\nu}(\lambda)}{s_{\rho}}g_{\rho}(p_{1}-p_{2})^{\nu}$
(12) $\displaystyle=$ $\displaystyle\frac{g_{\rho}\alpha}{s_{\rho}}\cdot
p_{B}^{\mu}\Big{[}g_{\mu\nu}-\frac{(p_{1}+p_{2})_{\mu}(p_{1}+p_{2})_{\nu}}{s}\Big{]}(p_{1}-p_{2})^{\nu}$
$\displaystyle=$
$\displaystyle\frac{g_{\rho}}{s_{\rho}}\cdot\frac{M_{B^{+}\rightarrow\rho^{0}\pi^{+}}^{\lambda}}{p_{B}\cdot\epsilon^{*}}\cdot(\Sigma-s^{\prime})$
$\displaystyle=$ $\displaystyle(\Sigma-s^{\prime})\cdot\mathcal{M},$

where $\sqrt{s^{\prime}}$ is the high invariance mass of the $\pi^{+}\pi^{-}$
pair, $\Sigma=\frac{1}{2}(s^{\prime}_{max}+s^{\prime}_{min})$ with
$s^{\prime}_{max}$ and $s^{\prime}_{min}$ being the maximum and minimum values
of $s^{\prime}$ for a fixed $s$, respectively, and $\sqrt{s}$ is the low
invariant mass of the $\pi^{+}\pi^{-}$ pair [$m(\pi^{+}\pi^{-})_{low}$] and
$s_{V}$ is from the inverse propagator of the vector meson $V$,
$s_{V}=s-m_{V}^{2}+\mathrm{i}m_{V}\Gamma_{V}$. The factor $\Sigma-s^{\prime}$
is equal to $-2|\vec{p_{2}}||\vec{p_{3}}|\cos{\theta}$ in Ref. I.B.0  , which
accounts for angular momentum conservation for the spin-1 resonance.
$\mathcal{M}$ will be calculated in the following. According to the effective
Hamiltonian, $A$ can be divided into two parts X.H.2 :

$\displaystyle A$ $\displaystyle=$
$\displaystyle\langle\pi^{+}\pi^{-}\pi^{-}|H^{T}|B^{-}\rangle+\langle\pi^{+}\pi^{-}\pi^{-}|H^{P}|B^{-}\rangle,$
(13)

where $H^{T}$ and $H^{P}$ are the Hamiltonians for the tree and penguin
operators, respectively.

In order to obtain a large signal for direct CP violation, we need to appeal
to some phenomenological mechanisms. $\rho$-$\omega$ mixing has the dual
advantages that the strong phase difference is large (passes through
$90\,^{\circ}$ at the $\omega$ resonance) and well known S.G. ; X.H.1 . With
this mechanism, to the first order in isospin violation, the amplitude for
$B^{-}\to\rho^{0}(\omega)\pi^{-}\to\pi^{+}\pi^{-}\pi^{-}$ takes the following
form at a value of $\sqrt{s}$ close to the $\omega$ resonance mass X.H.2 :

$\displaystyle\langle\pi^{+}\pi^{-}\pi^{-}|H^{T}|B^{-}\rangle$
$\displaystyle=$ $\displaystyle(\Sigma-s^{\prime})$ (14)
$\displaystyle\times\left(\frac{g_{\rho}}{s_{\rho}s_{\omega}}\tilde{\Pi}_{\rho\omega}t_{\omega}+\frac{g_{\rho}}{s_{\rho}}t_{\rho}\right),$
$\displaystyle\langle\pi^{+}\pi^{-}\pi^{-}|H^{P}|B^{-}\rangle$
$\displaystyle=$ $\displaystyle(\Sigma-s^{\prime})$ (15)
$\displaystyle\times\left(\frac{g_{\rho}}{s_{\rho}s_{\omega}}\tilde{\Pi}_{\rho\omega}p_{\omega}+\frac{g_{\rho}}{s_{\rho}}p_{\rho}\right),$

where $t_{V}$ ($V=\rho^{0}$ or $\omega$) is the tree amplitude and $p_{V}$ is
the penguin amplitude for producing an intermediate vector meson $V$,
$\tilde{\Pi}_{\rho\omega}$ is the effective $\rho$-$\omega$ mixing amplitude.
From Eqs. (14) and (15), we note that $\rho$-$\omega$ mixing provides an
additional complex term for the tree and penguin amplitudes (the first term in
each equation), respectively. These complex terms will enlarge the CP-even
phase and lead to a peak of CP asymmetry as mentioned before. We will show the
difference between the CP asymmetries with and without $\rho$-$\omega$ mixing
later. Here, we assume that the $B^{\pm}\to\pi^{+}\pi^{-}\pi^{\pm}$ process is
dominated by the resonance $\rho^{0}$ in certain region of its Dalitz plot.

We stress that the direct coupling $\omega\to\pi^{+}\pi^{-}$ is effectively
absorbed into $\tilde{\Pi}_{\rho\omega}$ H.B. , leading to the explicit $s$
dependence of $\tilde{\Pi}_{\rho\omega}$. Making the expansion
$\tilde{\Pi}_{\rho\omega}(s)=\tilde{\Pi}_{\rho\omega}(m_{\omega}^{2})+(s-m_{\omega}^{2})\tilde{\Pi}_{\rho\omega}^{\prime}(m_{\omega}^{2})$,
the $\rho$-$\omega$ mixing parameters were determined in the fit of Gardner
and O’Connell S.G.2 :

$\displaystyle\mathfrak{R}\mathfrak{e}\tilde{\Pi}_{\rho\omega}(m^{2}_{\omega})$
$\displaystyle=$ $\displaystyle{}-3500\pm 300\ \mathrm{MeV^{2}},$
$\displaystyle\mathfrak{I}\mathfrak{m}\tilde{\Pi}_{\rho\omega}(m^{2}_{\omega})$
$\displaystyle=$ $\displaystyle{}-300\pm 300\ \mathrm{MeV^{2}},$
$\displaystyle\tilde{\Pi}_{\rho\omega}^{\prime}(m^{2}_{\omega})$
$\displaystyle=$ $\displaystyle 0.03\pm\ 0.04.$ (16)

In practice, the effect of the derivative term is negligible.

In this work, we only consider $\rho^{0}$ and $\omega$ resonances. Then, for a
fixed $s$, the differential CP asymmetry parameter can be defined as

$\displaystyle
A_{CP}=\frac{|\mathcal{M}|^{2}-|\bar{\mathcal{M}}|^{2}}{|\mathcal{M}|^{2}+|\bar{\mathcal{M}}|^{2}}.$
(17)

By integrating the denominator and numerator of $A_{CP}$, respectively, in the
region $\Omega$ ($s_{1}<s<s_{2}$, $s^{\prime}_{1}<s^{\prime}<s^{\prime}_{2}$),
we obtain the localized integrated CP asymmetry, which can be measured by
experiments and takes the following form:

$\displaystyle
A_{CP}^{\Omega}=\frac{\int_{s_{1}}^{s_{2}}\mathrm{d}s\int_{s^{\prime}_{1}}^{s^{\prime}_{2}}\mathrm{d}s^{\prime}(\Sigma-s^{\prime})^{2}(|\mathcal{M}|^{2}-\bar{|\mathcal{M}|}^{2})}{\int_{s_{1}}^{s_{2}}\mathrm{d}s\int_{s^{\prime}_{1}}^{s^{\prime}_{2}}\mathrm{d}s^{\prime}(\Sigma-s^{\prime})^{2}(|\mathcal{M}|^{2}+\bar{|\mathcal{M}|}^{2})}.$
(18)

According to kinematics of the three body decay,
$\Sigma[=\frac{1}{2}(s^{\prime}_{max}+s^{\prime}_{min})]$ is related to $s$.
In our calculations, $s$ varies in a small region, and therefore $\Sigma$ can
be treated as a constant approximately R.A.2 . Then, the terms
$\int_{s^{\prime}_{1}}^{s^{\prime}_{2}}\mathrm{d}s^{\prime}(\Sigma-s^{\prime})^{2}$
are cancelled, and $A_{CP}^{\Omega}$ becomes independent of the high
invariance mass of $\pi^{+}\pi^{-}$. In practice, to be more precise, we take
into account the $s$-dependence of $s^{\prime}_{max}$ and $s^{\prime}_{min}$
in our calculations. We choose $s^{\prime}_{min}<s^{\prime}<s^{\prime}_{max}$
as the integration interval of the high invariance mass of $\pi^{+}\pi^{-}$
and regard
$\int_{s^{\prime}_{min}}^{s^{\prime}_{max}}\mathrm{d}s^{\prime}(\Sigma-s^{\prime})^{2}$
as a factor which is dependent on $s$.

###  2.3 Calculational details

With the Hamiltonian given in Eq. (4), we are ready to evaluate the matrix
elements for $B^{-}\to\rho^{0}(\omega)\pi^{-}$. According to the theory of QCD
factorization, the naive factorization approach has been shown to be the
leading order result in the framework of QCD factorization when the radiative
QCD corrections of order $\mathcal{O}(\alpha_{s}(m_{b}))$ and the
$\mathcal{O}(1/m_{b})$ corrections in the heavy quark effective theory are
neglected M.B.1 ; M.B.2 ; M.B.3 ; M.B.4 . Since the $b$ quark is very heavy
and $B$ meson decays are very energetic, so the quark-antiquark pair in a
meson in the final state moves very fast away from the weak interaction point.
The hadronization of the quark-antiquark pair occurs far away from the
remaining quarks. Then the meson can be factorized out and the interaction
between the quark pair in the meson and the remaining quark is tiny J.D. ;
M.J. . The deviation of the value of $N_{c}$ from the color number, 3,
measures the nonfactorizable effects in the naive factorization scheme X.H.1 ;
X.H.2 . In the factorization approximation, either $\rho^{0}(\omega)$ or
$\pi^{-}$ is generated by one current which has appropriate quantum numbers in
the Hamiltonian. For this decay process, two kinds of matrix element products
are involved after factorization:
$\langle\rho^{0}(\omega)|J^{\mu}|0\rangle\langle\pi^{-}|J_{\mu}|B^{-}\rangle$
and
$\langle\pi^{-}|J^{\mu}|0\rangle\langle\rho^{0}(\omega)|J_{\mu}|B^{-}\rangle$.
We will calculate them in some phenomenological quark models.

The matrix elements for $B\to P$ and $B\to V$ (where $P$ and $V$ denote
pseudoscalar and vector mesons, respectively) can be decomposed as M.W.

$\displaystyle\langle P|J_{\mu}|B\rangle$ $\displaystyle=$
$\displaystyle\Big{(}p_{B}+p_{P}-\frac{m_{B}^{2}-m_{P}^{2}}{k^{2}}\Big{)}_{\mu}F_{1}^{BP}(k^{2})$
(19)
$\displaystyle+\frac{m_{B}^{2}-m_{P}^{2}}{k^{2}}k_{\mu}F_{0}^{BP}(k^{2}),$
$\displaystyle\langle V|J_{\mu}|B\rangle$ $\displaystyle=$
$\displaystyle\frac{2}{m_{B}+m_{V}}\varepsilon_{\mu\nu\rho\sigma}\epsilon^{*\nu}p_{B}^{\rho}p^{\sigma}_{V}V^{BV}(k^{2})$
(20)
$\displaystyle+\text{i}\Big{\\{}\epsilon^{*}_{\mu}(m_{B}+m_{V})A_{1}^{BV}(k^{2})$
$\displaystyle\qquad-\frac{\epsilon^{*}\cdot
k}{m_{B}+m_{V}}(p_{B}+p_{V})_{\mu}A_{2}^{BV}(k^{2})$
$\displaystyle\qquad-\frac{\epsilon^{*}\cdot k}{k^{2}}2m_{V}\cdot
k_{\mu}A_{3}^{BV}(k^{2})\Big{\\}}$
$\displaystyle+\text{i}\frac{\epsilon^{*}\cdot k}{k^{2}}2m_{V}\cdot
k_{\mu}A^{BV}_{0}(k^{2}),$

where $J_{\mu}$ is the weak current
[$J_{\mu}=\bar{u}\gamma_{\mu}(1-\gamma_{5})b$ or
$\bar{d}\gamma_{\mu}(1-\gamma_{5})b$], $k=p_{B}-p_{P(V)}$, and
$\epsilon_{\mu}$ is the polarization vector of $V$. The form factors included
in our calculations satisfy $F_{1}^{BP}(0)=F_{0}^{BP}(0)$,
$A_{3}^{BV}(0)=A_{0}^{BV}(0)$, and
$A_{3}^{BV}(k^{2})=[(m_{B}+m_{V})/2m_{V}]A_{1}^{BV}(k^{2})-[(m_{B}-m_{V})/2m_{V}]A_{2}^{BV}(k^{2})$.
We define the notation $X$ for matrix elements. For example,
$X^{(B^{-}\rho^{0},\pi^{-})}$ is defined as
$\langle\pi^{-}|\bar{d}\gamma^{\mu}(1-\gamma_{5})u|0\rangle\langle\rho^{0}|\bar{u}\gamma_{\mu}(1-\gamma_{5})b|B^{-}\rangle$.
These matrix elements can be parameterized as the products of decay constants
and form factors. Therefore, the factorized terms $X^{(BM_{1},M_{2})}$ have
the following expressions Y.H. :

$\displaystyle X^{(BP,V)}$ $\displaystyle=$ $\displaystyle\langle
V|\bar{q}_{2}\gamma^{\mu}(1-\gamma_{5})q_{3}|0\rangle\langle
P|\bar{q}_{1}\gamma_{\mu}(1-\gamma_{5})b|B\rangle$ (21) $\displaystyle=$
$\displaystyle 2f_{V}m_{V}F_{1}^{BP}(m^{2}_{V})(\epsilon^{*}\cdot p_{B}),$
$\displaystyle X^{(BV,P)}$ $\displaystyle=$ $\displaystyle\langle
P|\bar{q}_{2}\gamma^{\mu}(1-\gamma_{5})q_{3}|0\rangle\langle
V|\bar{q}_{1}\gamma_{\mu}(1-\gamma_{5})b|B\rangle$ (22) $\displaystyle=$
$\displaystyle 2f_{P}m_{V}A_{0}^{BV}(m^{2}_{P})(\epsilon^{*}\cdot p_{B}),$

where $f_{V}$ and $f_{P}$ are the decay constants of vector and pseudoscalar
mesons, respectively. Using the decomposition in Eqs. (19)-(22), one has

$\displaystyle t_{\rho}$ $\displaystyle=$ $\displaystyle
V_{ub}V^{*}_{ud}\Bigg{(}a_{1}\frac{X^{(B^{-}\rho^{0},\pi^{-})}}{\epsilon^{*}\cdot
p_{B}}+a_{2}\frac{X^{(B^{-}\pi^{-},\rho^{0})}}{\epsilon^{*}\cdot
p_{B}}\Bigg{)},$ (23) $\displaystyle t_{\omega}$ $\displaystyle=$
$\displaystyle
V_{ub}V^{*}_{ud}\Bigg{(}a_{1}\frac{X^{(B^{-}\omega,\pi^{-})}}{\epsilon^{*}\cdot
p_{B}}+a_{2}\frac{X^{(B^{-}\pi^{-},\omega)}}{\epsilon^{*}\cdot
p_{B}}\Bigg{)},$ (24)

where all the $a_{i}$ are built up from the effective Wilson coefficients
$c_{i}^{\prime}$’s, and take the form
$a_{i}=c^{\prime}_{i}+c^{\prime}_{i+1}/N_{c}$ for odd $i$ and
$a_{i}=c^{\prime}_{i}+c^{\prime}_{i-1}/N_{c}$ for even $i$. It is noted that
in the factorization approach $N_{c}$ includes nonfactorizable contributions
effectively, and the value of $N_{c}$ should be determined by experiments
since we cannot evaluate nonfactorizable contributions. In the same way, we
obtain the penguin operator contributions:

$\displaystyle p_{\rho}$ $\displaystyle=$ $\displaystyle-
V_{tb}V^{*}_{td}\Bigg{\\{}\Big{[}-a_{4}+\frac{3}{2}a_{7}+\frac{3}{2}a_{9}+\frac{1}{2}a_{10}\Big{]}\frac{X^{(B^{-}\pi^{-},\rho^{0})}}{\epsilon^{*}\cdot
p_{B}}$ (25)
$\displaystyle+\Big{[}a_{4}+a_{10}-2(a_{6}+a_{8})\frac{m^{2}_{\pi}}{(m_{d}+m_{u})(m_{b}+m_{u})}\Big{]}$
$\displaystyle\qquad\times\frac{X^{(B^{-}\rho^{0},\pi^{-})}}{\epsilon^{*}\cdot
p_{B}}\Bigg{\\}},$ $\displaystyle p_{\omega}$ $\displaystyle=$ $\displaystyle-
V_{tb}V^{*}_{td}\Bigg{\\{}\Big{[}2a_{3}+a_{4}+2a_{5}+\frac{1}{2}(a_{7}+a_{9}-a_{10})\Big{]}$
(26)
$\displaystyle\qquad\qquad\times\frac{X^{(B^{-}\pi^{-},\omega)}}{\epsilon^{*}\cdot
p_{B}}$
$\displaystyle+\Big{[}a_{4}+a_{10}-2(a_{6}+a_{8})\frac{m^{2}_{\pi}}{(m_{d}+m_{u})(m_{b}+m_{u})}\Big{]}$
$\displaystyle\qquad\qquad\times\frac{X^{(B^{-}\omega,\pi^{-})}}{\epsilon^{*}\cdot
p_{B}}\Bigg{\\}}.$

We adopt the same decay constants and form factors for the matrix elements
producing $\rho^{0}$ and $\omega$ mesons. Then we have
$X^{(B^{-}\rho^{0},\pi^{-})}=X^{(B^{-}\omega,\pi^{-})}=2f_{\pi}m_{\rho}A_{0}(m_{\pi}^{2})(\epsilon^{*}\cdot
p_{B})$ and
$X^{(B^{-}\pi^{-},\rho^{0})}=X^{(B^{-}\pi^{-},\omega)}=\sqrt{2}f_{\rho}m_{\rho}F_{1}(m_{\rho}^{2})(\epsilon^{*}\cdot
p_{B})$, where
$\langle\rho^{0}(\omega)|J^{\mu}|0\rangle=1/\sqrt{2}f_{\rho}m_{\rho}\epsilon^{*\mu}$
and $\langle\pi^{-}|J_{\mu}|0\rangle=\text{i}f_{\pi}p_{\mu}$.

###  2.4 Numerical results

Table 1: The localized integrated asymmetries $A^{\Omega}_{CP}$ when $q^{2}/m_{b}^{2}=0.3(0.5)$. The first and second lines of each model corresponds to $A^{\Omega}_{CP}$ with and without $\rho$-$\omega$ mixing in the region $0.750<\sqrt{s}<0.800$ GeV, respectively. The third and forth lines of each model correspond to the low-mass region ($0.470<\sqrt{s}<0.770$ GeV) and the high-mass region ($0.770<\sqrt{s}<0.920$ GeV) near the resonance mass, respectively. The fifth and sixth lines of each model corresponds to $A^{\Omega}_{CP}$ with and without $\rho$-$\omega$ mixing in the region $0.470<\sqrt{s}<0.920$ GeV, respectively. The second and third columns correspond to $N_{c}=2.07(2.09)$. The fourth and fifth columns correspond to $N_{c}=4.54(4.65)$. The second and fourth columns correspond to lower limiting values of the CKM matrix elements. The third and fifth columns correspond to upper limiting values of the CKM matrix elements. $N_{c}$ | 2.07(2.09) | 4.54(4.65)
---|---|---
$\rho,\eta$ | min | max | min | max
Model 1 |  |  |  |
$0.750<\sqrt{s}<0.800$ GeV | -0.0647(-0.0520) | -0.0736(-0.0591) | -0.0455(-0.0300) | -0.0517(-0.0341)
| 0.0054(0.0204) | 0.0062(0.0232) | 0.0079(0.0281) | 0.0090(0.0319)
$0.470<\sqrt{s}<0.770$ GeV | -0.0024(0.0061) | -0.0027(0.0070) | 0.0017(0.0278) | 0.0019(0.0290)
$0.770<\sqrt{s}<0.920$ GeV | -0.0352(-0.0128) | -0.0400(-0.0145) | -0.0239(0.0014) | -0.0272(0.0016)
$0.470<\sqrt{s}<0.920$ GeV | -0.0167(-0.0021) | -0.0189(-0.0023) | -0.0093(0.0096) | -0.0106(0.0109)
| 0.0054(0.0204) | 0.0062(0.0232) | 0.0079(0.0281) | 0.0090(0.0319)
Model 2 |  |  |  |
$0.750<\sqrt{s}<0.800$ GeV | -0.0661(-0.0516) | -0.0752(-0.0587) | -0.0469(-0.0290) | -0.0533(-0.0330)
| 0.0062(0.0234) | 0.0071(0.0267) | 0.0092(0.0327) | 0.0105(0.0371)
$0.470<\sqrt{s}<0.770$ GeV | -0.0019(0.0086) | -0.0021(0.0098) | 0.0026(0.0306) | 0.0030(0.0353)
$0.770<\sqrt{s}<0.920$ GeV | -0.0359(-0.0112) | -0.0408(-0.0127) | -0.0247(0.0038) | -0.0280(0.0043)
$0.470<\sqrt{s}<0.920$ GeV | -0.0166(0.00004) | -0.0189(0.00004) | -0.0091(0.0129) | -0.0103(0.0146)
| 0.0062(0.0234) | 0.0071(0.0267) | 0.0092(0.0327) | 0.0105(0.0371)
Model 3 |  |  |  |
$0.750<\sqrt{s}<0.800$ GeV | -0.0647(-0.0520) | -0.0737(-0.0592) | -0.0455(-0.0300) | -0.0515(-0.0341)
| 0.0054(0.0204) | 0.0062(0.0232) | 0.0079(0.0281) | 0.0090(0.0320)
$0.470<\sqrt{s}<0.770$ GeV | -0.0024(0.0061) | -0.0028(0.0069) | 0.0017(0.0278) | 0.0019(0.0292)
$0.770<\sqrt{s}<0.920$ GeV | -0.0350(-0.0125) | -0.0398(-0.0142) | -0.0238(0.0016) | -0.0270(0.0016)
$0.470<\sqrt{s}<0.920$ GeV | -0.0167(-0.0021) | -0.0190(-0.0023) | -0.0094(0.0097) | -0.0106(0.0110)
| 0.0054(0.0204) | 0.0062(0.0232) | 0.0079(0.0281) | 0.0090(0.0320)
Model 4 |  |  |  |
$0.750<\sqrt{s}<0.800$ GeV | -0.0661(-0.0517) | -0.0752(-0.0587) | -0.0469(-0.0290) | -0.0534(-0.0330)
| 0.0062(0.0235) | 0.0071(0.0267) | 0.0092(0.0327) | 0.0105(0.0372)
$0.470<\sqrt{s}<0.770$ GeV | -0.0019(0.0086) | -0.0022(0.0098) | 0.0026(0.0306) | 0.0030(0.0353)
$0.770<\sqrt{s}<0.920$ GeV | -0.0357(-0.0110) | -0.0405(-0.0125) | -0.0245(0.0040) | -0.0278(0.0046)
$0.470<\sqrt{s}<0.920$ GeV | -0.0167(0.00005) | -0.0190(0.00005) | -0.0091(0.0129) | -0.0104(0.0146)
| 0.0062(0.0235) | 0.0071(0.0267) | 0.0092(0.0327) | 0.0105(0.0372)
Model 5 |  |  |  |
$0.750<\sqrt{s}<0.800$ GeV | -0.0577(-0.0550) | -0.0658(-0.0626) | -0.0403(-0.0375) | -0.0459(-0.0427)
| 0.0011(0.0041) | 0.0012(0.0047) | 0.0015(0.0055) | 0.0017(0.0062)
$0.470<\sqrt{s}<0.770$ GeV | -0.0055(-0.0083) | -0.0063(-0.0094) | -0.0034(-0.0046) | -0.0039(-0.0053)
$0.770<\sqrt{s}<0.920$ GeV | -0.0322(-0.0216) | -0.0366(-0.0245) | -0.0222(-0.122) | -0.0252(-0.0139)
$0.470<\sqrt{s}<0.920$ GeV | -0.0170(-0.0140) | -0.0194(-0.0160) | -0.0114(-0.0079) | -0.0130(-0.0090)
| 0.0010(0.0039) | 0.0012(0.0044) | 0.0015(0.0052) | 0.0017(0.0059)

Figure 1: The differential asymmetry, $A_{CP}$, in Model 1. (a) For
$q^{2}/m_{b}^{2}=0.3$: Solid and dot lines correspond $N_{c}=2.78$ and
$N_{c}=9.68$ with maximum CKM matrix elements, respectively. Dot-dashed and
dashed line correspond $N_{c}=2.78$ and $N_{c}=9.68$ with minimum CKM matrix
elements, respectively; (b) Same as (a) but for $q^{2}/m_{b}^{2}=0.5$ and
$N_{c}=2.85$ and 9.02.

In our numerical calculations we have several parameters: $q^{2}$, $N_{c}$,
and the CKM matrix elements in the Wolfenstein parametrization. As mentioned
in Sect. 2.1, the value of $q^{2}$ is conventionally chosen to be in the range
$0.3<q^{2}/m_{b}^{2}<0.5$. The CKM matrix, which should be determined from the
experimental data, has the following form in terms of the Wolfenstein
parameters, $A$, $\lambda$, $\rho$, $\eta$ K.A. :

$\left(\begin{array}[]{ccc}1-\tfrac{1}{2}\lambda^{2}&\lambda&A\lambda^{3}(\rho-\mathrm{i}\eta)\\\
-\lambda&1-\tfrac{1}{2}\lambda^{2}&A\lambda^{2}\\\
A\lambda^{3}(1-\rho-\mathrm{i}\eta)&-A\lambda^{2}&1\\\ \end{array}\right),$
(27)

where $O(\lambda^{4})$ corrections are neglected. Since $\lambda$ and $A$ are
well determined and the uncertainties due to the CKM matrix elements are
mostly from $\rho$ and $\eta$, we take the central values of $\lambda(=0.225)$
and $A(=0.814)$ in the following. The ranges for $\rho$ and $\eta$ are

$\bar{\rho}=0.117\pm 0.021,\qquad\bar{\eta}=0.353\pm 0.013,$ (28)

with

$\bar{\rho}=\rho\Bigg{(}1-\frac{\lambda^{2}}{2}\Bigg{)},\qquad\bar{\eta}=\eta\Bigg{(}1-\frac{\lambda^{2}}{2}\Bigg{)}.$

The form factors $F_{1}(m^{2}_{\rho})$ and $A_{0}(m^{2}_{\pi})$ depend on the
inner structure of the hadrons. Under the nearest pole dominance assumption,
we take the following $k^{2}$ dependence of the form factors: for Model 1(2)
M.B. ; Y.H.2 ; D.C. ; P.B.2 ; A.A. ; H.Y.4 :

$F_{1}(k^{2})=\frac{h_{1}}{1-\frac{k^{2}}{m_{1}^{2}}},\qquad
A_{0}(k^{2})=\frac{h_{A_{0}}}{1-\frac{k^{2}}{m_{A_{0}}^{2}}},$ (29)

where $h_{1}=0.25(0.292)$, $h_{A_{0}}=0.30(0.366)$,
$m_{1}=5.32\,\mathrm{GeV}$, and $m_{A_{0}}=5.27\,\mathrm{GeV}$; for Model 3(4)
Y.H.2 ; D.C. ; P.B.2 ; A.A. ; H.Y.4 ; Y.K.2 :

$F_{1}(k^{2})=\frac{h_{1}}{\big{(}1-\frac{k^{2}}{m_{1}^{2}}\big{)}^{2}},\qquad
A_{0}(k^{2})=\frac{h_{A_{0}}}{\big{(}1-\frac{k^{2}}{m_{A_{0}}^{2}}\big{)}^{2}},$
(30)

where $h_{1}=0.25(0.292)$, $h_{A_{0}}=0.30(0.366)$,
$m_{1}=5.32\,\mathrm{GeV}$, and $m_{A_{0}}=5.27\,\mathrm{GeV}$; for Model 5
P.B.1 ; H.B.F. :

$\displaystyle F_{1}(k^{2})$ $\displaystyle=$
$\displaystyle\frac{h_{1}}{1-x_{1}\frac{k^{2}}{m_{1}^{2}}+y_{1}\big{(}\frac{k^{2}}{m_{1}^{2}}\big{)}^{2}},$
$\displaystyle A_{0}(k^{2})$ $\displaystyle=$
$\displaystyle\frac{h_{A_{0}}}{1-x_{0}\frac{k^{2}}{m_{A_{0}}^{2}}+y_{0}\big{(}\frac{k^{2}}{m_{A_{0}}^{2}}\big{)}^{2}},$
(31)

where $h_{1}=0.261$, $h_{A_{0}}=0.302$, $x_{1}=2.03$, $y_{1}=1.29$,
$x_{0}=-1.49$, $y_{0}=6.61$, $m_{1}=5.32\,\mathrm{GeV}$, and
$m_{A_{0}}=5.27\,\mathrm{GeV}$. The decay constants used in our calculations
are $f_{\rho}=216\,\mathrm{MeV}$ and $f_{\pi}=132\,\mathrm{MeV}$ H.Y.4 .

In the numerical calculations, it is found that for a fixed $N_{c}$, there is
a maximum value for the differential CP violating parameter, when the low
invariant mass of the $\pi^{+}\pi^{-}$ pair is in the vicinity of the $\omega$
resonance, $0.780-0.785\,\mathrm{GeV}$. Five models with different form
factors were investigated to study the model dependence of $A_{CP}$ in Ref.
X.H.2 . To be more specific, in Figs. 1a, b we display the results for the
form factors in Model 1. These results show explicitly the dependence of the
CP violating asymmetry on $q^{2}/m_{b}^{2}$, CKM matrix elements and the
effective parameter $N_{c}$. The dependence on $N_{c}$ comes form the fact
that $N_{c}$ is related to the hadronization effects, and consequently, we
cannot exactly determine $N_{c}$ in our calculations. Therefore, we treat
$N_{c}$ as a free effective parameter and take it in the range
$2.07(2.09)<N_{c}<4.54(4.65)$ when $q^{2}/m_{b}^{2}=0.3(0.5)$ for reasons
which will be explained later (Sect. 3).

Then we calculate the localized integrated CP asymmetries. According to Eq.
(18), we integrate $A_{CP}$ over the low invariant mass of $\pi^{+}\pi^{-}$
($\sqrt{s}$) and obtain the localized integrated asymmetries
$A_{CP}^{\Omega}$. Considering the significant region of $\rho$-$\omega$
mixing, we choose the integration interval of $\sqrt{s}$ to be from 0.750 to
0.800 GeV. In order to compare with the newest result of the LHCb experiments,
we also calculate $A_{CP}^{\Omega}$ when $\sqrt{s}$ is in the low-mass region
($0.470<\sqrt{s}<0.770$ GeV), the high-mass region ($0.770<\sqrt{s}<0.920$
GeV) and the whole region ($0.470<\sqrt{s}<0.920$ GeV) near the $\rho^{0}$
resonance R.A.2 . The numerical results are displayed in Table 1. We also
display $A^{\Omega}_{CP}$ with and without $\rho$-$\omega$ mixing when
$0.750<\sqrt{s}<0.800$ GeV and $0.470<\sqrt{s}<0.920$ GeV in Table 1.

Table 1 shows that the values of $A^{\Omega}_{CP}$ in our calculations vary
from -0.0752 (-0.0626) to -0.0403 (-0.0290) corresponding to
$q^{2}/m_{b}^{2}=0.3(0.5)$, in the regions of $N_{c}$, the CKM matrix
elements, and the form factors in five models when $0.750<\sqrt{s}<0.800$ GeV.
From Fig. 4 of Ref. R.A.2 , we can see the localized integrated CP asymmetries
have different signs in $\cos\theta>0$ and $\cos\theta<0$ regions when
$0.750<m(\pi^{+}\pi^{-})_{low}<0.800\,\text{GeV}$. If one adds the events in
these two experimental regions together, the total localized integrated CP
asymmetry obtained from experiment becomes $-0.0294\pm 0.0285$ when
$0.750<m(\pi^{+}\pi^{-})_{low}<0.800\,\text{GeV}$. The values in our
calculations agree with this experimental data. We stress that
$A_{CP}^{\Omega}$ in our calculations is always negative in this integration
region and its sign is independent of form factor models. We note that the
signs of $A^{\Omega}_{CP}$ are positive when $\rho$-$\omega$ mixing is not
considered in this region. This indicates that $\rho$-$\omega$ mixing is vital
for $A_{CP}^{\Omega}$ to be negative in this region.

From Table 1, we can see $\rho$-$\omega$ mixing changes the sign of
$A^{\Omega}_{CP}$ from positive to negative. Figs. 1(a) and 1(b) show that the
peak of the differential asymmetry $A_{CP}$ involving $\rho$-$\omega$ mixing
is on the right of $0.770$ GeV. Therefore, comparing with $A_{CP}^{\Omega}$ in
the range $0.470<\sqrt{s}<0.770$ GeV, the localized integrated CP asymmetries
move towards the negative direction when $0.770<\sqrt{s}<0.920$ GeV due to
$\rho$-$\omega$ mixing. This behavior is also independent of form factor
models. In fact, in our calculations the difference between the localized
integrated CP asymmetries in the regions $0.470<\sqrt{s}<0.770$ GeV and
$0.770<\sqrt{s}<0.920$ GeV varies from 0.0076 to 0.0387. We add the
experimental events for positive and negative $\cos\theta$ in the regions
$0.470<m(\pi^{+}\pi^{-})_{low}<0.770$ GeV and $0.770<\sqrt{s}<0.920$ GeV and
obtain $A^{\Omega}_{CP}$ as $0.0508\pm 0.0171$ and $-0.0256\pm 0.0202$,
respectively, from the data in TABLE IV of Ref. R.A.2 . We can see they have
opposite signs. After we combine these two regions together, $A^{\Omega}_{CP}$
in the whole region $0.470<m(\pi^{+}\pi^{-})_{low}<0.920$ GeV is $0.0173\pm
0.0130$ R.A.2 . The values of $A^{\Omega}_{CP}$ with $\rho$-$\omega$ mixing
shown in Table 1 differ a lot between Model 1 (or 2,3,4) and Model 5.
$A^{\Omega}_{CP}$ is also sensitive to the choice of $q^{2}/m_{b}^{2}$. When
$0.470<\sqrt{s}<0.770$ GeV, it appears that $A^{\Omega}_{CP}$ varies from
0.0278 to 0.0353 in Models 1,2,3 and 4 for $q^{2}/m_{b}^{2}=0.5$ and
$N_{c}=4.65$. This result is consistent with the experimental data. When
$0.770<\sqrt{s}<0.920$ GeV, except for $q^{2}/m_{b}^{2}=0.5$ and $N_{c}=4.65$
in Models 1,2,3 and 4, $A^{\Omega}_{CP}$ varies from -0.0408 to -0.0110. This
result is consistent with the experimental data. In the whole region of
$0.470<\sqrt{s}<0.920$ GeV, $A^{\Omega}_{CP}$ varies from 0.0096 to 0.0146 in
Models 1,2,3 and 4 for $q^{2}/m_{b}^{2}=0.5$ and $N_{c}=4.65$. This result is
also consistent with the experimental data.

From above discussions, we can see $A^{\Omega}_{CP}$ with $\rho$-$\omega$
mixing, especially the signs, agree with the experimental result when
$0.750<\sqrt{s}<0.800$ GeV. We also find that the localized integrated CP
asymmetries move toward the negative direction due to $\rho$-$\omega$ mixing.
$A^{\Omega}_{CP}$ in the region $0.770<\sqrt{s}<0.920$ GeV contains the
contribution of $\rho$-$\omega$ mixing while that in the region
$0.470<\sqrt{s}<0.770$ GeV does not. Therefore, $\rho$-$\omega$ mixing
contributes to the sign change of CP asymmetry around the $\rho^{0}(770)$ peak
of $m(\pi^{+}\pi^{-})_{low}$. Our results in these two regions and the whole
region are consistent with the experimental data for several choices of
$q^{2}/m^{2}$ and models of form factors. One should take the effect of
$\rho$-$\omega$ mixing into account in order to answer the question why the
sign of CP asymmetry changes around the $\rho^{0}(770)$ peak of
$m(\pi^{+}\pi^{-})_{low}$.

##  3 Extraction of $N_{c}$ from data of branching ratios

###  3.1 Formalism

Figure 2: The branching ratio for $B^{+}\to\rho^{0}\pi^{+}$ in Model 2. Solid
(dashed) line stands for $q^{2}/m_{b}^{2}=0.3$ and maximum (miximum) CKM
matrix elements. Dot (dot-dashed) line stands for $k^{2}/m_{b}^{2}=0.5$ and
maximum (miximum) CKM matrix elements.

As mentioned before, $N_{c}$ includes nonfactorizable effects which are
difficult to deal with at present. Therefore, we treat $N_{c}$ as an effective
parameter to be determined by the experimental data. With the factorized decay
amplitudes, we can calculate the decay rates by using the following expression
Y.H. :

$\displaystyle\Gamma(B\to VP)=\frac{|\vec{p}_{\rho}|^{3}}{8\pi
m^{2}_{V}}\Big{|}\frac{A(B\to VP)}{\epsilon^{*}\cdot p_{B}}\Big{|}^{2},$ (32)

where

$\displaystyle|\vec{p}_{\rho}|=\frac{\sqrt{[m_{B}^{2}-(m_{V}+m_{P})^{2}][m_{B}^{2}-(m_{V}-m_{P})^{2}]}}{2m_{B}}$
(33)

is the c.m. momentum of the decay particles, and $A(B\to VP)$ is the decay
amplitude. In our case, to be consistent, we should also take into account the
$\rho$-$\omega$ mixing contribution when we calculate the branching ratio
since we are working to the first order of isospin violation. Explicitly, for
$B^{+}\to\rho^{0}\pi^{+}$, we obtain

$\displaystyle BR(B^{+}\to\rho^{0}\pi^{+})$ (34) $\displaystyle=$
$\displaystyle\frac{G_{F}^{2}|\vec{p}_{\rho}|^{3}}{16\pi
m_{\rho}^{2}\Gamma_{B^{+}}}\Big{|}[V_{ud}V^{*}_{ub}A^{T}_{\rho^{0}}-V_{td}V^{*}_{tb}A^{P}_{\rho^{0}}]/(\epsilon^{*}\cdot
p_{B})$
$\displaystyle\qquad+[V_{ud}V^{*}_{ub}A^{T}_{\omega}-V_{td}V^{*}_{tb}A^{P}_{\omega}]/(\epsilon^{*}\cdotp_{B})$
$\displaystyle\qquad\qquad\quad\times\frac{\tilde{\Pi}_{\rho\omega}}{(s_{\rho}-m_{\omega}^{2})+\text{i}m_{\omega}\Gamma_{\omega}}\Big{|}^{2},$

where the tree and penguin amplitudes are

$\displaystyle A^{T}_{\rho^{0}}$ $\displaystyle=$ $\displaystyle
a_{1}X^{(B^{+}\rho^{0},\pi^{+})}+a_{2}X^{(B^{+}\pi^{+},\rho^{0})},$
$\displaystyle A^{P}_{\rho^{0}}$ $\displaystyle=$
$\displaystyle\Big{(}-a_{4}+\frac{3}{2}a_{7}+\frac{3}{2}a_{9}+\frac{1}{2}a_{10}\Big{)}X^{(B^{+}\pi^{+},\rho^{0})}$
$\displaystyle+\Big{[}a_{4}+a_{10}-2(a_{6}+a_{8})\frac{m^{2}_{\pi}}{(m_{u}+m_{d})(m_{b}+m_{u})}\Big{]}$
$\displaystyle\qquad\qquad\times X^{(B^{+}\rho^{0},\pi^{+})},$ $\displaystyle
A^{T}_{\omega}$ $\displaystyle=$ $\displaystyle
a_{1}X^{(B^{+}\omega,\pi^{+})}+a_{2}X^{(B^{+}\pi^{+},\omega)},$ $\displaystyle
A^{P}_{\omega}$ $\displaystyle=$
$\displaystyle\Big{[}2a_{3}+a_{4}+2a_{5}+\frac{1}{2}(a_{7}+a_{9}-a_{10})\Big{]}X^{(B^{+}\pi^{+},\omega)}$
(35)
$\displaystyle+\Big{[}a_{4}+a_{10}-2(a_{6}+a_{8})\frac{m^{2}_{\pi}}{(m_{u}+m_{d})(m_{b}+m_{u})}\Big{]}$
$\displaystyle\qquad\qquad\times X^{(B^{+}\omega,\pi^{+})}.$

For $B^{0}\to\rho^{+}\pi^{-}$, we obtain

$\displaystyle
BR(B^{0}\to\rho^{+}\pi^{-})=\frac{G_{F}^{2}|\vec{p}_{\rho}|^{3}}{16\pi
m_{\rho}^{2}\Gamma_{B^{0}}}$
$\displaystyle\qquad\times|(V_{ub}V^{*}_{ud}A^{T}_{\rho^{+}}-V_{tb}V^{*}_{td}A^{P}_{\rho^{+}})/(\epsilon^{*}\cdot
p_{B})|^{2},$ (36)

where

$\displaystyle A^{T}_{\rho^{+}}$ $\displaystyle=$ $\displaystyle
a_{1}X^{(B^{0}\pi^{+},\rho^{-})},$ $\displaystyle A^{P}_{\rho^{+}}$
$\displaystyle=$ $\displaystyle(a_{4}+a_{10})X^{(B^{0}\rho^{-},\pi^{+})},$
(37)

where
$X^{(B^{0}\rho^{+},\pi^{-})}=2f_{\pi}m_{\rho}A_{0}(m_{\pi}^{2})(\epsilon^{*}\cdot
p_{B})$ and
$X^{(B^{0}\pi^{-},\rho^{+})}=\sqrt{2}f_{\rho}m_{\rho}F_{1}(m_{\rho}^{2})(\epsilon^{*}\cdot
p_{B})$.

###  3.2 Numerical results

Table 2: Summary of the ranges of $N_{c}$ which are determined from the experimental data for various models and input parameters [numbers outside (inside) brackets are for $q^{2}/m_{b}^{2}=0.3(0.5)$]. The notation (number, number) means the lower and upper limits for $N_{c}$. (-, -) means that there is no range of $N_{c}$ which is consistent with the experimental data. | $B^{+}$ | $B^{0}$
---|---|---
Model 1 |  |
$\rho_{max},\eta_{max}$ | (2.70, 4.09) [(2.76, 4.22)] | (-, -) [(-, -)]
$\rho_{min},\eta_{min}$ | (2.14, 3.05) [(2.18, 3.13)] | (-, -) [(-, -)]
Model 2 |  |
$\rho_{max},\eta_{max}$ | (-,-) [(-,-)] | (-,-) [(-,-)]
$\rho_{min},\eta_{min}$ | (-,-) [(-,-)] | (-,-) [(-,-)]
Model 3 |  |
$\rho_{max},\eta_{max}$ | (2.24, 3.23) [(2.28, 3.32)] | (0.05, 0.07) [(0.10, 0.14)]
$\rho_{min},\eta_{min}$ | (2.85, 4.40) [(2.92, 4.55)] | (0.06, 0.13) [(0.12, 0.15)]
Model 4 |  |
$\rho_{max},\eta_{max}$ | (-,-) [(-,-)] | (-,-) [(-,-)]
$\rho_{min},\eta_{min}$ | (-,-) [(-,-)] | (-,-) [(-,-)]
Model 5 |  |
$\rho_{max},\eta_{max}$ | (2.71,4.54) [(2.76,4.65)] | (0.11,0.12) [(0.11,0.12)]
$\rho_{min},\eta_{min}$ | (2.07,3.13) [(2.09,3.18)] | (0.10,0.11) [(0.10,0.11)]

The latest experimental data of branching ratios of $B^{+}\to\rho^{0}\pi^{+}$
and $B^{0}\to\rho^{+}\pi^{-}$ from Particle Data Group (PDG) are K.A.

$\displaystyle BR(B^{+}\to\rho^{0}\pi^{+})$ $\displaystyle=$
$\displaystyle(8.3\pm 1.2)\times 10^{-6},$ $\displaystyle
BR(B^{0}\to\rho^{+}\pi^{-})$ $\displaystyle=$ $\displaystyle(2.3\pm
0.23)\times 10^{-5}.$

We can determine the range of $N_{c}$ by comparing the theoretical values of
the branching ratios with those of two-body decay channels. We calculate the
branching ratios with the formula given in Eqs. (32), (34) and (36) in five
models for the weak form factors which are mentioned in the previous
subsection. In Fig. 2, we show the results for $B^{+}\to\rho^{0}\pi^{+}$ in
Model 1 as an example. The numerical results are sensitive to uncertainties
coming from the experimental data. In addition, the branching ratio also
depends on the CKM matrix elements which are parameterized by $\lambda$, $A$,
$\rho$, and $\eta$. In the allowed ranges for the parameters $\rho$ and
$\eta$, we obtain the range of $N_{c}$. We summarize the allowed range of
$N_{c}$ in Table 2. It is found that the experimental data constrain the value
of $N_{c}$ into two regions. We note that if $N_{c}$ approaches to zero,
nonperturbative effects would be very large. Considering this, we drop the
range $N_{c}<1$. Therefore, $N_{c}$ could be in the range
$2.07(2.09)<N_{c}<4.54(4.65)$ for $q^{2}/m_{b}^{2}=0.3(0.5)$. These values
have been used in Sect. 2.

##  4 Conclusion and discussion

The first aim of the present work is to study the localized integrated CP
asymmetry for the decays
$B^{\pm}\to\rho^{0}(\omega)\pi^{\pm}\to\pi^{+}\pi^{-}\pi^{\pm}$ with the
inclusion of $\rho$-$\omega$ mixing. The second aim is to study the sign
change caused by $\rho$-$\omega$ mixing.

In the calculation of CP violating asymmetry parameters, we need the Wilson
coefficients for the tree and penguin operators at the scale $m_{b}$. We
worked with the renormalization scheme independent Wilson coefficients. One of
the major uncertainties in our calculations is due to the fact that hadronic
matrix elements of both tree and penguin operators involve nonperturbative QCD
effects. We worked in the factorization approximation, with $N_{c}$ being
treated as an effective parameter to include nonfactorizable contributions. We
compared our theoretical results with the latest experimental data from PDG to
determine the range of $N_{c}$ as $2.07(2.09)<N_{c}<4.54(4.65)$ for
$q^{2}/m_{b}^{2}=0.3(0.5)$. It has been pointed out that the factorization
approach is quite reliable in energetic weak decays H.Y.1 .

We explicitly showed that the CP violating asymmetry is very sensitive to
$N_{c}$, the CKM matrix elements and the form factors. There is a maximum
value for the differential CP violating parameter when the low invariant mass
of the $\pi^{+}\pi^{-}$ pair is near the vicinity of the $\omega$ resonance,
$0.780-785\,\mathrm{GeV}$. We determined the range of the localized integrated
CP asymmetry with and without $\rho$-$\omega$ mixing in the ranges of $N_{c}$,
the CKM matrix elements, and $q^{2}/m_{b}^{2}$. For all the models
investigated, we found that the localized integrated CP violating asymmetry
with $\rho$-$\omega$ mixing varies from -0.0403 (-0.0290) to -0.0752 (-0.0626)
corresponding to $q^{2}/m_{b}^{2}=0.3(0.5)$ when
$0.750<m(\pi^{+}\pi^{-})_{low}<0.800\,\text{GeV}$. If one adds the events in
the $\cos\theta<0$ and $\cos\theta>0$ experimental regions together when
$0.750<m(\pi^{+}\pi^{-})_{low}<0.800\,\text{GeV}$, the total localized
integrated CP asymmetry is $-0.0294\pm 0.0285$. Our results, especially the
signs, agree with the experimental data when
$0.750<m(\pi^{+}\pi^{-})_{low}<0.800\,\text{GeV}$. We note that the signs are
positive in this region when $\rho$-$\omega$ mixing is not considered. This
indicates that $\rho$-$\omega$ mixing is vital for $A_{CP}^{\Omega}$ to be
negative in this region. It was shown that the maximum localized integrated
asymmetry in the range $0.750<\sqrt{s}<0.800$ GeV can reach -0.0752. We also
found that $\rho$-$\omega$ mixing can make the localized integrated CP
asymmetries move toward the negative direction, and therefore contributes to
the sign change around the $\rho^{0}(770)$ peak of $m(\pi^{+}\pi^{-})_{low}$.
This behavior is independent of the form factor models. In our calculations,
the difference between the localized integrated CP asymmetries in the regions
$0.470<\sqrt{s}<0.770$ GeV and $0.770<\sqrt{s}<0.920$ GeV varies from 0.0076
to 0.0387. Our results by including the $\rho$-$\omega$ mixing mechanism in
these two regions and the whole region around the $\rho^{0}(770)$ peak are
consistent with the experimental results for some models of the form factors.

At this stage, we cannot explain the LHCb experimental data in the regions of
positive and negative $\cos\theta$ individually. This is because three-body
decays of heavy mesons are more complicated than two-body decays as they
receive more contributions from different mechanisms, for example,
nonresonants H.Y. , the interference between intermediate resonances and
final-state $KK\longleftrightarrow\pi\pi$ rescattering. We will investigate
the angle distribution of $A_{CP}^{\Omega}$ when considering both the
$\rho$-$\omega$ mixing mechanism and the interference between different spin
intermediate resonances Z.H. . We will also apply more accurate data in the
future to further decrease the uncertainties in the calculations. With
parameters with smaller uncertainties, we expect to be able to obtain the
effects of $\rho$-$\omega$ mixing more precisely. This is important to
interpret the angle distribution and the sign change of the CP asymmetry
around the $\rho^{0}(770)$ peak of $m^{2}(\pi^{+}\pi^{-})_{low}$ more
accurately.

##  5 Acknowledge

This work was supported in part by National Natural Science Foundation of
China (Project Nos. 11175020, 11275025, 11447021, and 11575023).

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