# Isospin symmetry breaking effects for the mode ${B}\rightarrow\pi\pi(K)$ in
Perturbative QCD

Gang Lü1111Email<EMAIL_ADDRESS>Liang-Chen Liu 1222Email:
<EMAIL_ADDRESS>Qin-Qin Zhi  1College of Science, Henan University of
Technology, Zhengzhou 450001, China

###### Abstract

We calculate the direct $CP$ violation for the decay process of
${B}^{0}\rightarrow\pi^{0}\pi^{0}$, ${B}^{+}\rightarrow\pi^{0}\pi^{+}$,
${B}^{0}\rightarrow K^{0}\pi^{0}$ and ${B}^{+}\rightarrow K^{+}\pi^{0}$ via
isospin symmetry breaking effects from the $\pi^{0}-\eta-\eta^{\prime}$ mixing
mechanism in perturbative QCD approach. Isospin symmetry breaking originates
from the electroweak interaction and the u-d quark mass difference through the
strong interaction which are known to be tiny. However, we find that isospin
symmetry breaking at the leading order changes the $CP$ violation from the new
strong phases. Our calculation results for the $CP$ violation are within or
including the range of experimental results. We also compare our results with
those from the QCD factorization and the perturbative QCD schemes without
regard to isospin symmetry breaking effects.

##  I Introduction

The theoretical studies and experiment of the two body charmless hadronic B
meson decay play an important part in searching new physics signals beyond the
Standard Model (SM). In the SM, the non-leptonic decays of
${B}\rightarrow\pi\pi$ and ${B}\rightarrow\pi K$ provide valuable information
on the inner angles of the unitarity triangle of Cabibbo-Kobayashi-Maskawa
(CKM)cab ; kob  matrix, and have been widely studied SB . Due to the quark
flavor mixing, the CKM matrix provides us the weak phases. Associated with a
strong stage, the weak phase is the source of CP violation. The strong phase
comes from the dynamics of QCD and the other mechanism.

Since the interference of tree and penguin amplitudes is expected to produce
$CP$ violation, the charmless ${B}\rightarrow\pi\pi$ and ${B}\rightarrow\pi K$
modes are very representative in the two-body B meson decays, and hence have
been studied most extensively. Due to the interaction between the theory and
data of B factories, it is possible to obtain valuable new insights into the
physics of these modes, and also raises a question about the field of
electroweak penguin sector, which can be contributed according to the new
physics to modify. R. . There are some potential contradiction between the
current new B-factory data for ${B}\rightarrow\pi\pi$ and ${B}\rightarrow\pi
K$ decays and the predictions. For example, new experimental data for
${B}^{0}\rightarrow\pi^{0}\pi^{0}$ and ${B}^{0}\rightarrow K^{0}\pi^{0}$ decay
rates are significantly larger than the theoretical predictions Li . In
addition, the prediction of direct CP asymmetry in these modes is also
inconsistent with the data, even if the signs of some processes are reversed
MB ; MB1 . There is conspicuous difference for the $CP$ violation between the
process $B^{\pm}\rightarrow\pi^{0}K^{\pm}$ and
$B^{0}\rightarrow\pi^{\mp}K^{\pm}$. For the scheme of QCD factorization, the
perturbative and non-perturbative contribution are suppressed by
$\mathcal{O}(\alpha_{s})$ and $\mathcal{O}(\Lambda_{QCD})$, respectively.
Therefore, because there are very few violations of $CP$, it is difficult to
explain the difference. The Glauber-gluon effects are discussed for the $CP$
violation of process ${B}\rightarrow\pi K$ from the Glauber phase factors
which lead to the rotation of the tree amplitude by a strong phase in the
framework of perturbative QCD liuxin2016 . New physics contribution is the new
sources of $CP$ violation through electroweak penguins for the decay process
of ${B}\rightarrow\pi K$ Buras2004NB ; Barger2009 ; Beaudy2018 .

Isospin is an approximate symmetry in the Standard model which can be broken
either by electroweak effects or by the strong interaction through the u and d
quark mass difference. Both mechanisms of isospin violation lead to $\triangle
I=\frac{3}{2}$ penguin contributions. The $\pi^{0}-\eta-\eta^{\prime}$ mixing
arises from the latter H.  and permits an $I=1$ amplitude AJ . These latter
contributions convert the triangle relations between the amplitudes to
quadrilaterals. The effect of electroweak penguin diagrams has been studied
earlier in the literature and is estimated to be small NG ; MG ; RF . The
quark flavor basis leads to the mixing of the pseudoscalar mesons. Isospin
symmetry breaking from the QCD dynamics are introduced by the mixing of
$\pi^{0}-\eta-\eta^{\prime}$, which can be applied to the phenomenology by the
quark flavor mixing scheme. The $\pi^{0}-\eta-\eta^{\prime}$ mixing from u and
d quark mass difference can present the strong phase, which may affect the
value of $CP$ violation accordingly. It is interesting to study how it could
affect the $CP$ violation for the decay processes of ${B}\rightarrow\pi\pi$,
${B}\rightarrow\pi K$.

The remainder of this paper is organized as follows. In Sec. II we present the
form of the effective Hamiltonian. In Sec. III we give the calculating
formalism of $CP$ violation from isospin symmetry breaking effects. Input
parameters are presented in Sec.IV. We present the numerical results in Sec.V.
Summary and discussion are included in Sec. VI. The related functions defined
in the text are given in the Appendix.

##  II The effective hamiltonian

From the operator product expansion, the effective weak Hamiltonian can be
expressed as buras

$\displaystyle{\cal H}_{\Delta
B=1}={G_{F}\over\sqrt{2}}[V^{*}_{ub}V_{uq}(C_{1}O^{u}_{1}+C_{2}O^{u}_{2})$
$\displaystyle-V^{*}_{tb}V_{tq}\sum^{10}_{i=3}C_{i}O_{i}]+H.C.,\;\vspace{2mm}$
(1)

where $G_{F}$ is Fermi constant, $C_{i}$ (i=1,…,10) represent the Wilson
coefficients, $V_{ub}$, $V_{uq}$, $V_{tb}$ and $V_{tq}(q=s,d)$ are the CKM
matrix elements. The operators $O_{i}$ have the following forms:

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

where $\alpha$ and $\beta$ are color indices, and $q^{\prime}=u,d$ or $s$
quarks. In Eq.(LABEL:2b) $O_{1}^{u}$ and $O_{2}^{u}$ are tree operators,
$O_{3}$–$O_{6}$ are QCD penguin operators and $O_{7}$–$O_{10}$ are the
operators associated with electroweak penguin diagrams.

We can obtain numerical values of $C_{i}$. When $C_{i}(m_{b})$ LKM ,

$\displaystyle\begin{split}C_{1}&=-0.2703,\;\;\;C_{2}=1.1188,\\\
C_{3}&=0.0126,\;\;\;C_{4}=-0.0270,\\\ C_{5}&=0.0085,\;\;\;C_{6}=-0.0326,\\\
C_{7}&=0.0011,\;\;\;C_{8}=0.0004,\\\
C_{9}&=-0.0090,\;\;\;C_{10}=0.0022.\end{split}$ (3)

One can obtain numerical values of $a_{i}$. The combinations $a_{i}$ of Wilson
coefficients are defined as AG ; YH

$\displaystyle\begin{split}a_{1}&=C_{2}+C_{1}/3,\;\;\;a_{2}=C_{1}+C_{2}/3,\\\
a_{3}&=C_{3}+C_{4}/3,\;\;\;a_{4}=C_{4}+C_{3}/3,\\\
a_{5}&=C_{5}+C_{6}/3,\;\;\;a_{6}=C_{6}+C_{5}/3,\\\
a_{7}&=C_{7}+C_{8}/3,\;\;\;a_{8}=C_{8}+C_{7}/3,\\\
a_{9}&=C_{9}+C_{10}/3,\;\;\;a_{10}=C_{10}+C_{9}/3.\\\ \end{split}$ (4)

##  III $CP$ violation from Isospin symmetry breaking effects

###  III.1 Formalism

By containing or not containing strange quark, we can denote isospin vector
triplet $\pi_{3}$, isospin scalar $\eta_{n}$ and isospin scalar $\eta_{s}$.
One can translate the $SU(3)$ singlet $\eta_{0}$ and octet $\eta_{8}$ into
isospin scalar $\eta_{n}$ and isospin scalar $\eta_{s}$ by the relations
$\eta_{n}=\frac{\sqrt{2}\eta_{0}+\eta_{8}}{\sqrt{3}}$ and
$\eta_{s}=\sqrt{\frac{1}{3}}\eta_{0}-\sqrt{\frac{2}{3}}\eta_{8}$. The states
of $\pi_{3}$, $\eta_{n}$ and $\eta_{s}$ are represented as
$\pi_{3}=\frac{1}{\sqrt{2}}|u\bar{u}-d\bar{d}>$,
$\eta_{n}=\frac{1}{\sqrt{2}}|u\bar{u}+d\bar{d}>$ and $\eta_{s}=|s\bar{s}>$
from the quark model, respectively. The physical meson states $\pi^{0}$,
$\eta$, $\eta^{\prime}$ can be transformed from the $\pi_{3}$, $\eta_{n}$ and
$\eta_{s}$ by unitary matrix $U$ Kroll2005PP :

$\displaystyle\begin{pmatrix}\pi^{0}\\\ \eta\\\
\eta^{\prime}\end{pmatrix}=U(\varepsilon_{1},\varepsilon_{2},\phi)\begin{pmatrix}\pi_{3}\\\
\eta_{n}\\\ \eta_{s}\end{pmatrix},$ (5)

where

$\displaystyle
U(\varepsilon_{1},\varepsilon_{2},\phi)=\begin{pmatrix}1&\varepsilon_{1}+\varepsilon_{2}cos\phi&-\varepsilon_{2}sin\phi\\\
-\varepsilon_{2}-\varepsilon_{1}cos\phi&cos\phi&-sin\phi\\\
-\varepsilon_{1}sin\phi&sin\phi&cos\phi\end{pmatrix},$ (6)

where $\phi$ is the mixing angle AMLi2007 . $\varepsilon_{1}$,
$\varepsilon_{2}$ $\propto\mathcal{O}(\lambda),\lambda\ll 1$ and the higher
order terms are neglected.

From the quark model, the physical states $\eta$ and $\eta^{\prime}$ are
related to quark flavor basis $\eta_{n}$ and $\eta_{s}$. The matrix
$U(\varepsilon_{1},\varepsilon_{2},\phi)$ reduces to the $U^{\prime}(\phi)$ as
$\varepsilon_{1},\varepsilon_{2}\rightarrow 0$ when one considers the isospin
symmetry. Hence, the physical state $\pi^{0}$ is equivalent to the $\pi_{3}$
state and the formula of Eq.(5)(6) translates into the $\eta-\eta^{\prime}$
mixing in Eq.(7):

$\left(\begin{matrix}\eta\\\
\eta^{\prime}\end{matrix}\right)=U^{\prime}(\phi)\left(\begin{matrix}\eta_{n}\\\
\eta_{s}\end{matrix}\right)=\left(\begin{array}[]{ccc}\cos\phi&-sin\phi\\\
\sin\phi&cos\phi\end{array}\right)\left(\begin{matrix}\eta_{n}\\\
\eta_{s}\end{matrix}\right).$ (7)

The mechanism of isospin symmetry breaking depends on the electroweak
interaction and the strong interaction by the $u-d$ quark mass difference in
Standard Model. For the isospin symmetry breaking, the effect of the
electroweak interaction is tinyNG ; MG ; RF . The $u-d$ quark mass difference
is responsible for $\pi^{0}-\eta-\eta^{\prime}$ mixing and make major
contribution for the isospin symmetry breaking. One can calculate the effects
from isospin symmetry breaking by the chiral perturbative theory. To the
leading order of isospin symmetry breaking via $\pi^{0}-\eta-\eta^{\prime}$
mixing, the physical eigenstates of mesons $\pi^{0}$, $\eta$ and
$\eta^{\prime}$ from Eq.(5)(6) can be written as

$\displaystyle\begin{split}|\pi^{0}\rangle&=|\pi_{3}\rangle+(\varepsilon_{1}+\varepsilon_{2}\cos\phi)|\eta_{n}\rangle-\varepsilon_{2}\sin\phi|\eta_{s}\rangle,\\\
|\eta\rangle&=(-\varepsilon_{2}-\varepsilon_{1}\cos\phi)|\pi_{3}\rangle+\cos\phi|\eta_{n}\rangle-\sin\phi|\eta_{s}\rangle,\\\
|\eta^{\prime}\rangle&=-\varepsilon_{1}\sin\phi|\pi_{3}\rangle+\sin\phi|\eta_{n}\rangle+\cos\phi|\eta_{s}\rangle,\end{split}$
(8)

where $\pi_{3}$ refer to the isospin $I=1$ component in the triplet. One can
define $\varepsilon=\varepsilon_{2}+\varepsilon_{1}cos\phi$,
$\varepsilon^{\prime}=\varepsilon_{1}sin\phi$, and the values are
$\varepsilon=0.017\pm 0.002$ , $\varepsilon^{\prime}=0.004\pm 0.001$,
$\phi=39.0^{\circ}$ Kroll2005PP .

We need to deal with the hadronic matrix element to calculate the $CP$
violation. Due to the high-energy of heavy meson, the factorization method is
widely used to estimate the hadronic matrix element from the heavy quarks
effective theory and the large momenta of quarks. Because of the QCD
correction, we can calculate the decay amplitudes of the heavy $B$ meson since
soft gluons is not enough time to exchange between the final states mesons
with large momenta. The contribution of hard gluons can be estimated by
perturbative theory. In the framework of perturbative QCD,the transverse
momenta of quark is considered to avoid the divergence. One need pay close
attention to three scales: $m_{W}$ scale ($m_{W}$ refers to the W-boson mass),
the hard scale $t$, and factorization scale $1/b$ ($b$ is the conjugate
variable of the parton transverse momenta $k_{T}$). The Wilson coefficients
$C(t)$ is related to contribution of short distance by the leading logarithm
order. One can resume the results from the order of $m_{W}$ down to hadronic
scale $t$ through the renormalization group equation. The non-perturbative
contribution can be absorbed into hadronic wave function $\Phi(x)$ below
factorization scale $k_{T}$. The overlap of soft divergence and collinear
divergence will result in double logarithms $ln^{2}(Pb)$ ($P$ refers to the
light-cone component of meson momentum). The resummation of the double
logarithms generates a Sudakov factor $exp[-s(P,b)]$, which can suppress the
long distance contribution from the non-perturbative effects in the large $b$
region, and vanishes as $b>1/\Lambda_{QCD}$ fortunately. We can calculate the
remaining finite subamplitude $H(x,t)$ by pertubative theory. Hence, we
present the $CP$ violation via $\pi^{0}-\eta-\eta^{\prime}$ mixing in the
pertubative QCD scheme.

Since the wave function is independent on the decay mode, we use the model for
the $B$ meson functionAMLi2007 ; wangz2014

$\displaystyle\phi_{B}(x,b)=N_{B}{x^{2}}(1-x)^{2}exp[-\frac{{M_{B}^{2}}x^{2}}{2{\omega_{b}}^{2}}-\frac{1}{2}(\omega_{b}{b})^{2}],$
(9)

where the normalization factor $N_{B}$ is dependent of the free parameter
$\omega_{b}$. $b$ is the conjugate variable of the parton transverse momenta
$k_{T}$. $M_{B}$ refers to the mass of the $B$ meson. For the $B$ meson, we
can obtain the value of $\omega_{b}=0.50\pm 0.05$ from the light cone sum rule
hnl2002 .

The wave function of the final state pseudoscalar mesons
$M=(\pi,K,\eta_{n},\eta_{s})$ are the same in form and can be defined in
Ref.VM2004 ; ZH2014 ; VM1990 ; YY2013

$\displaystyle\Phi_{M_{i}}(P_{i},x_{i})\equiv\frac{1}{\sqrt{6}}\gamma_{5}[\not{P_{i}}\phi_{M_{i}}^{A}(x_{i})+m_{0i}\phi_{M_{i}}^{P}(x_{i})+\zeta(\not{n}\not{\upsilon}-1)m_{0i}\phi^{T}(x_{i})],$
(10)

where $m_{0i}$ is the chiral mass of the meson $M_{i}$. $P_{i}$ and $x_{i}$
are the momentum and the fraction of the momentum of $M_{i}$. Depending on the
assignment of the momentum fraction $x$, the parameter $\zeta$ should be
chosen as +1 or -1.

In this paper, we will use those distribution amplitudes of the $\pi$ meson
and $\phi_{\eta_{d\bar{d}}}^{A,P,T}$ Lu2006 :

$\displaystyle\begin{split}\phi_{\pi}^{A}(x)&=\frac{3f_{\pi}}{\sqrt{6}}x(1-x)[1+0.44C_{2}^{3/2}(2x-1)+0.25C_{4}^{3/2}(2x-1)],\\\
\phi_{\pi}^{P}(x)&=\frac{f_{\pi}}{2\sqrt{6}}[1+0.43C_{2}^{1/2}(2x-1)+0.09C_{4}^{1/2}(2x-1)],\\\
\phi_{\pi}^{T}(x)&=\frac{f_{\pi}}{2\sqrt{6}}(1-2x)[1+0.55(10x^{2}-10x+1)],\\\
\phi_{\eta_{d\bar{d}}}^{A}(x)&=\frac{3}{{2\sqrt{2N_{c}}}}f_{x}x(1-x)[1+a_{2}^{\eta_{d\bar{d}}}\frac{3}{2}(5(1-2x)^{2}-1)\\\
&+a_{4}^{\eta_{d\bar{d}}}\frac{15}{8}(21(1-2x)^{4}-14(1-2x)^{2}-1)],\\\
\phi_{\eta_{d\bar{d}}}^{P}(x)&=\frac{1}{2\sqrt{2N_{c}}}f_{x}[1+\frac{1}{2}(30\eta_{3}-\frac{5}{2}\rho_{\eta_{d\bar{d}}}^{2})(3(1-2x)^{2}-1)\\\
&+\frac{1}{8}(-3\eta_{3}\omega_{3}-\frac{27}{20}\rho_{\eta_{d\bar{d}}}^{2}-\frac{81}{10}\rho_{\eta_{d\bar{d}}}^{2}a_{2}^{\eta_{d\bar{d}}})(35(1-2x)^{4}-30(1-2x)^{2}+3)],\\\
\phi_{\eta_{d\bar{d}}}^{T}(x)&=\frac{3}{2\sqrt{2N_{c}}}f_{x}(1-2x)[\frac{1}{6}(5\eta_{3}-\frac{1}{2}\eta_{3}\omega_{3}-\frac{7}{20}\rho_{\eta_{d\bar{d}}}^{2}a_{2}^{\eta_{d\bar{d}}}-\frac{3}{5}\rho_{\eta_{d\bar{d}}}^{2}a_{2}^{\eta_{d\bar{d}}})(10x^{2}-10x+1)],\\\
\end{split}$

where $\rho_{\eta_{d\bar{d}}}={m_{\pi}}/{m_{0}^{\eta_{d\bar{d}}}}$. The
distribution amplitudes $\phi_{\eta_{d\bar{d}}}^{A,P,T}$ represent the axial
vector, pseudoscalar and tensor components of the wave function, respectively.
In Ref.TF1998 , isospin symmetry is assumed for $f_{x}$ and $f_{x}=f_{\pi}$.
The Gegenbauer polynomials are given as:

$\begin{array}[]{ll}C^{1/2}_{2}(t)=\frac{1}{2}(3t^{2}-1),&C^{1/2}_{4}(t)=\frac{1}{8}(3-30t^{2}+35t^{4})\\\
C^{3/2}_{1}(t)=3t,&C^{3/2}_{2}(t)=\frac{3}{2}(5t^{2}-1),\\\
C^{3/2}_{4}(t)=\frac{15}{8}(1-14t^{2}+21t^{4}).\end{array}$ (11)

where $t=2x-1$.

The expressions of the distribution amplitudes of the $K$ meson and
$\phi_{\eta_{n}}^{A,P,T}$ are given as X2008 :

$\displaystyle\begin{split}\phi_{K}^{A}(x)&=\frac{f_{K}}{{2\sqrt{2N_{c}}}}6x(1-x)[1+a_{1}^{K}C_{1}^{3/2}(t)++a_{2}^{K}C_{2}^{3/2}(t)++a_{4}^{K}C_{4}^{3/2}(t)],\\\
\phi_{K}^{P}(x)&=\frac{f_{K}}{2\sqrt{2N_{c}}}[1+(30\eta_{3}-\frac{5}{2}\rho_{K}^{2})C_{2}^{1/2}(t)-3(\eta_{3}\omega_{3}+\frac{9}{20}\rho_{K}^{2}(1+6a_{2}^{K}))C_{4}^{1/2}],\\\
\phi_{K}^{T}(x)&=-\frac{f_{K}}{2\sqrt{2N_{c}}}t[1+6(5\eta_{3}-\frac{1}{2}\eta_{3}\omega_{3}-\frac{7}{20}\rho_{K}^{2}-\frac{3}{5}\rho_{K}^{2}a_{2}^{K})(1+10x+10x^{2})],\\\
\phi_{\eta_{n}}^{A}(x)&=\frac{f_{n}}{{2\sqrt{2N_{c}}}}6x(1-x)[1+a_{1}^{\eta_{n}}C_{1}^{3/2}(2x-1)+a_{2}^{\eta_{n}}C_{2}^{3/2}(2x-1)+a_{4}^{\eta_{n}}C_{4}^{3/2}(2x-1)],\\\
\phi_{\eta_{n}}^{P}(x)&=\frac{f_{n}}{2\sqrt{2N_{c}}}[1+(30\eta_{3}-\frac{5}{2}\rho_{\eta_{n}}^{2})C_{2}^{1/2}(2x-1)-3(\eta_{3}\omega_{3}+\frac{9}{20}\rho_{\eta_{n}}^{2}(1+6a_{2}^{\eta_{n}}))C_{4}^{1/2}(2x-1)],\\\
\phi_{\eta_{n}}^{T}(x)&=\frac{f_{n}}{2\sqrt{2N_{c}}}(1-2x)[1+6(5\eta_{3}-\frac{1}{2}\eta_{3}\omega_{3}-\frac{7}{20}\rho_{\eta_{n}}^{2}-\frac{3}{5}\rho_{\eta_{n}}^{2}a_{2}^{\eta_{n}})(1+10x+10x^{2})],\\\
\end{split}$ (12)

where $\rho_{K}=\frac{m_{K}}{m_{0K}}$ and
$\rho_{\eta_{n}}=\frac{2m_{q}}{m_{qq}}$ .The wave function of the $u\bar{u}$
is same as the wave function $d\bar{d}$. The Gegenbauer moments and the other
parameters are given as X2008 :

$\begin{array}[]{ll}a_{1}^{K}=0.06,&a_{2}^{K}=0.25,\\\
a_{4}^{K}=0,&a_{1}^{\eta_{n}}=0,\\\
a_{2}^{\eta_{n},\eta_{d\bar{d}}}=0.44,&a_{4}^{\eta_{n},\eta_{d\bar{d}}}=0.25\\\
a_{2}^{\pi}=0.25,&a_{4}^{\pi}=-0.015,\\\
\eta_{3}=0.015,&\omega_{3}=-3.0.\end{array}$ (13)

The wave function of the meson is non-perturbative and process-independent. We
can describe the formation of hadrons from the positive and negative quarks by
the wave function, which provides the distribution of the momentum carried by
the parton. In the framework of PQCD, we calculate the QCD correction by
introducing the transverse momentum. Hence, the meson wave function should be
dependent on transverse momentum. The relevant results show that transverse
momentum has large effects on the heavy meson function. However, there is
little effect on the wave function of light meson by transverse momentum. At
present, only the Light-cone QCD sum rule and Lattice QCD are credible and
used to calculate non-perturbative contributions. Generally, one obtains the
wave function of optical pseudoscalar meson by Light-cone QCD sum rule.Within
the framework of PQCD, The branching ratios and CP violations have been
calculated. The theoretical prediction is inconsistent with the experimental
results for the most decay processes of $B/B_{s}\rightarrow M_{2}M_{3}$ LKM ;
X2008 . Hence, the wave function is credible ganggang2019 .

The relevant decay constants can be written as tpbm :

$\displaystyle\begin{split}\langle
0|\bar{n}\gamma^{\mu}\gamma_{5}n|\eta_{n}(P)\rangle&=\frac{i}{\sqrt{2}}\,f_{n}\,P^{\mu}\;,\\\
\langle
0|\bar{s}\gamma^{\mu}\gamma_{5}s|\eta_{s}(P)\rangle&=if_{s}\,P^{\mu}\;,\end{split}$
(14)

where $P$ refers to the momenta of $\eta_{n}$ or $\eta_{s}$. The relation
between the decay constants $f_{n}$ and $f_{s}$ can be found in Ref.X2008 .

###  III.2 Calculation details

In the framework of PQCD, we can calculate the $CP$ violation for the decay
modes ${B}\rightarrow\pi\pi$ and ${B}\rightarrow\pi K$ from the effects of
isospin symmetry breaking mechanism via $\pi^{0}-\eta-\eta^{\prime}$ mixing.
The tree level amplitude $T$ and penguin level amplitude $P$ are obtained by
perturbative theory. For the sake of simplification, we take the decay process
${B}^{0}\rightarrow\pi^{0}\pi^{0}$ as example for showing the
$\pi^{0}-\eta-\eta^{\prime}$ mixing mechanism.

The decay amplitudes $\mathcal{A}$ for ${B}^{0}\rightarrow\pi^{0}\pi^{0}$ via
$\pi^{0}-\eta-\eta^{\prime}$ mixing can be written as :

$\displaystyle\mathcal{A}$
$\displaystyle=\langle\pi^{0}\pi^{0}|{\mathcal{H}}_{eff}|B^{0}\rangle={\mathcal{A}}_{1}+{\mathcal{A}}_{2}+{\mathcal{A}}_{3}$
(15)

where

$\displaystyle\begin{split}{\mathcal{A}}_{1}&=\langle\pi_{3}\pi^{0}|{\mathcal{H}}_{eff}|B^{0}\rangle\\\
&=\langle\pi_{3}\pi_{3}|{\mathcal{H}}_{eff}|B^{0}\rangle+(\varepsilon_{1}+\varepsilon_{2}\cos\phi)\langle\pi_{3}\eta_{n}|{\mathcal{H}}_{eff}|B^{0}\rangle-\varepsilon_{2}\sin\phi\langle\pi_{3}\eta_{s}|{\mathcal{H}}_{eff}|B^{0}\rangle,\end{split}$
(16)
$\displaystyle\begin{split}{\mathcal{A}}_{2}&=(\varepsilon_{1}+\varepsilon_{2}\cos\phi)\langle\pi^{0}\eta_{n}|{\mathcal{H}}_{eff}|B^{0}\rangle\\\
&=(\varepsilon_{1}+\varepsilon_{2}\cos\phi)\langle\pi_{3}\eta_{n}|{\mathcal{H}}_{eff}|B^{0}\rangle+\mathcal{O}(\varepsilon),\end{split}$
(17)

and

$\displaystyle\begin{split}{\mathcal{A}}_{3}&=-\varepsilon_{2}\sin\phi\langle\pi^{0}\eta_{s}|{\mathcal{H}}_{eff}|B^{0}\rangle\\\
&=-\varepsilon_{2}\sin\phi\langle\pi_{3}\eta_{s}|{\mathcal{H}}_{eff}|B^{0}\rangle+\mathcal{O}(\varepsilon),\end{split}$
(18)

and we have ignored the higher order term of $\varepsilon$ and
$\mathcal{O}(\varepsilon)=\mathcal{O}(\varepsilon_{1})+\mathcal{O}(\varepsilon_{2})$.

We obtained the amplitude forms required in Eq.(16)(17)(18) by calculation and
they can be written as

$\displaystyle\langle\pi_{3}\pi_{3}|{\mathcal{H}}_{eff}|B^{0}\rangle$
$\displaystyle=$
$\displaystyle-\frac{1}{\varepsilon_{1}}\Big{\\{}F_{e\pi}\left[\xi_{u}a_{2}-\xi_{t}(2a_{3}+a_{4}-2a_{5}-\frac{1}{2}a_{7}+\frac{1}{2}a_{9}-\frac{1}{2}a_{10})f_{\eta}^{d}-\xi_{t}(a_{3}-a_{5}+\frac{1}{2}a_{7}-\frac{1}{2}a_{9})f_{\eta}^{s}\right]$
(19)
$\displaystyle+F_{e\pi}^{P2}\xi_{t}(a_{6}-\frac{1}{2}a_{8})f_{\eta}^{d}\Big{\\}},$

$\displaystyle\langle\pi_{3}\eta_{n}|{\mathcal{H}}_{eff}|B^{0}\rangle$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\Big{\\{}F_{e\eta}\left[\xi_{u}a_{2}-\xi_{t}(-a_{4}-\frac{3}{2}a_{7}+\frac{3}{2}a_{9}+\frac{1}{2}a_{10})\right]f_{\pi}+(M_{a\eta}^{P1}+M_{a\pi}^{P1})(C_{5}-\frac{1}{2}C_{7})$
(20)
$\displaystyle+M_{e\pi}^{P2}\xi_{t}\left[(2C_{6}+\frac{1}{2}C_{8})\right]+(M_{a\pi}+M_{e\eta}+M_{a\eta})\left[\xi_{u}C_{2}-\xi_{t}(-C_{3}+\frac{1}{2}C_{9}+\frac{3}{2}C_{10})\right]$
$\displaystyle-\frac{3}{2}(M_{a\pi}^{P2}+M_{e\eta}^{P2}+M_{a\eta}^{P2})\xi_{t}C_{8}+F_{e\pi}^{P2}\xi_{t}(a_{6}-\frac{1}{2}a_{8})\Big{\\}}$
$\displaystyle+\frac{\varepsilon_{1}}{\varepsilon_{2}}\Big{\\{}F_{e\pi}\left[\xi_{u}a_{2}-\xi_{t}(2a_{3}+a_{4}-2a_{5}-\frac{1}{2}a_{7}+\frac{1}{2}a_{9}-\frac{1}{2}a_{10})f_{\eta}^{d}-\xi_{t}(a_{3}-a_{5}+\frac{1}{2}a_{7}-\frac{1}{2}a_{9})f_{\eta}^{s}\right]$
$\displaystyle+F_{e\pi}^{P2}\xi_{t}(a_{6}-\frac{1}{2}a_{8})f_{\eta}^{d}\Big{\\}}+\frac{\varepsilon_{1}}{2\varepsilon_{2}}M_{e\pi}\left[\xi_{u}C_{2}-\xi_{t}(C_{3}+C_{4}-\frac{1}{2}C_{9}+\frac{1}{2}C_{10})\right],$

and

$\displaystyle\langle\pi_{3}\eta_{s}|{\mathcal{H}}_{eff}|B^{0}\rangle$
$\displaystyle=$ $\displaystyle
M_{e\pi}^{P2}\xi_{t}(2C_{6}+\frac{1}{2}C_{8})+\frac{\varepsilon_{1}}{2\sqrt{2}\varepsilon_{2}}M_{e\pi}\left[-\xi_{t}(C_{4}-\frac{1}{2}C_{10})\right].$
(21)

Depending on the CKM matrix elements, we can express the decay amplitudes
$\mathcal{A}$ for ${B}^{0}\rightarrow\pi^{0}\pi^{0}$ as following :

$\displaystyle\begin{split}-\sqrt{2}\mathcal{A}({B}^{0}\rightarrow\pi^{0}\pi^{0})=\xi_{u}T-\xi_{t}P\end{split}$
(22)

where $\xi_{u}={V_{ub}^{*}}{V_{ud}}$, $\xi_{t}={V_{tb}^{*}}{V_{td}}$. $T$ and
$P$ refer to the tree and penguin contributions from $\mathcal{A}$ in Eq.(15),
respectively. The amplitudes T and P from the decay process of
${B}^{0}\rightarrow\pi^{0}\pi^{0}$ with $\pi^{0}-\eta-\eta^{\prime}$ mixing
can be written as:

$\displaystyle\begin{split}T&=T_{1}+(\varepsilon_{1}+\varepsilon_{2}\cos\phi)T_{n}-\varepsilon_{2}\sin\phi
T_{s},\\\
P&=P_{1}+(\varepsilon_{1}+\varepsilon_{2}\cos\phi)P_{n}-\varepsilon_{2}\sin\phi
P_{s}.\end{split}$ (23)

where $T_{1}$, $P_{1}$ are from the decay process of
${B}^{0}\rightarrow\pi^{0}\pi^{0}$ without $\pi^{0}-\eta-\eta^{\prime}$
mixing. $T_{n}$, $P_{n}$, $T_{s}$ and $P_{s}$ come from the decay amplitudes
${B}^{0}\rightarrow\pi^{0}\eta_{n}$ and ${B}^{0}\rightarrow\pi^{0}\eta_{s}$,
respectively. The tree level amplitude and penguin level amplitude can be
given as

$\displaystyle
T_{1}=f_{\pi}F_{e}\left[c_{1}+\frac{1}{3}c_{1}\right]+M_{e}[C_{2}]-M_{a}[C_{2}],$
(24)

$\displaystyle P_{1}$ $\displaystyle=$ $\displaystyle
f_{\pi}F_{e}\left[\frac{1}{3}C_{3}+C_{4}+\frac{3}{2}C_{7}+\frac{1}{2}C_{8}-\frac{5}{3}C_{9}-C_{10}\right]+f_{\pi}F_{e}^{P}\left[C_{6}+\frac{1}{3}C_{5}-\frac{1}{6}C_{7}-\frac{1}{2}C_{8}\right]$
(25) $\displaystyle-
M_{e}\left[-C_{3}+\frac{3}{2}C_{8}+\frac{1}{2}C_{9}+\frac{3}{2}C_{10}\right]+M_{e}\left[C_{3}+2C_{4}++2C_{6}+\frac{1}{2}C_{8}-\frac{1}{2}C_{9}+\frac{1}{2}C_{10}\right]$
$\displaystyle+f_{B}F_{a}\left[\frac{1}{3}C_{5}+C_{6}-\frac{1}{6}C_{7}-\frac{1}{2}C_{8}\right],$

$\displaystyle T_{n}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}[F_{e\eta}f_{\pi}a_{2}+(M_{a\pi}+M_{e\eta}+M_{a\eta})C_{2}]+\frac{\varepsilon_{1}}{\varepsilon_{2}}F_{e\pi}a_{2}+\frac{\varepsilon_{1}}{2\varepsilon_{2}}M_{e\pi}C_{2},$
(26)

$\displaystyle P_{n}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\Big{\\{}F_{e\eta}\left[-(-a_{4}-\frac{3}{2}a_{7}+\frac{3}{2}a_{9}+\frac{1}{2}a_{10})\right]f_{\pi}+M_{e\pi}^{P2}\left[(2C_{6}+\frac{1}{2}C_{8})\right]$
(27)
$\displaystyle+(M_{a\pi}+M_{e\eta}+M_{a\eta})\left[-(-C_{3}+\frac{1}{2}C_{9}+\frac{3}{2}C_{10})\right]-\frac{3}{2}(M_{a\pi}^{P2}+M_{e\eta}^{P2}+M_{a\eta}^{P2})C_{8}+F_{e\pi}^{P2}\xi_{t}(a_{6}-\frac{1}{2}a_{8})\Big{\\}}$
$\displaystyle+\frac{\varepsilon_{1}}{\varepsilon_{2}}\Big{\\{}F_{e\pi}\left[-(2a_{3}+a_{4}-2a_{5}-\frac{1}{2}a_{7}+\frac{1}{2}a_{9}-\frac{1}{2}a_{10})f_{\eta}^{d}-(a_{3}-a_{5}+\frac{1}{2}a_{7}-\frac{1}{2}a_{9})f_{\eta}^{s}\right]$
$\displaystyle+F_{e\pi}^{P2}(a_{6}-\frac{1}{2}a_{8})f_{\eta}^{d}\Big{\\}}+\frac{\varepsilon_{1}}{2\varepsilon_{2}}M_{e\pi}\left[-(C_{3}+C_{4}-\frac{1}{2}C_{9}+\frac{1}{2}C_{10})\right],$

$\displaystyle\begin{split}T_{s}=0,\end{split}$ (28)

and

$\displaystyle\begin{split}P_{s}&=M_{e\pi}^{P2}(2C_{6}+\frac{1}{2}C_{8})+\frac{\varepsilon_{1}}{2\sqrt{2}\varepsilon_{2}}M_{e\pi}\left[-(C_{4}-\frac{1}{2}C_{10})\right],\end{split}$
(29)

with the $f_{\pi}$ and $f_{B}$ refer to the decay constant. The individual
decay amplitudes in the above equations, such as $F_{e}$, $F_{e}^{P}$,
$F_{a}$, $M_{a}$ and $M_{e}$ arise from the $(V-A)(V-A)$, $(V-A)(V+A)$ and
$(S-P)(S+P)$ operators, respectively, and will be given in Appendix.

One can see that the Eq.(23) without $\pi^{0}-\eta-\eta^{\prime}$ mixing is
reduced to

$\displaystyle T=T_{1},\;\;\;\;\;\;\;P=P_{1},$ (30)

which are expressed in Eq.(24) and Eq.(25).

The relevant weak phase $\theta$ and strong phase $\delta$ are obtained as
following

$\displaystyle re^{i\delta}e^{i\theta}$ $\displaystyle=$
$\displaystyle\frac{P}{T}\times\frac{V_{td}V_{tb}^{*}}{V_{ud}V_{ub}^{*}},$
(31)

where the parameter $r$ represents the absolute value of the ratio of penguin
and tree amplitudes:

$\displaystyle
r\equiv\Bigg{|}\frac{\langle\pi^{0}\pi^{0}|H^{P}|{B}^{0}\rangle}{\langle\pi^{0}\pi^{0}|H^{T}|\bar{B}_{s}^{0}\rangle}\Bigg{|}.$
(32)

The strong phase associated with $r$ can be given

$\displaystyle re^{i\delta}$ $\displaystyle=$
$\displaystyle\frac{P}{T}\times\bigg{|}\frac{V_{td}V_{tb}^{*}}{V_{ud}V_{ub}^{*}}\bigg{|}=r\cos\delta+\mathrm{i}r\sin\delta,$
(33)

where

$\left|\frac{V_{td}V^{*}_{tb}}{V_{ud}V^{*}_{ub}}\right|=\frac{\sqrt{[\rho(1-\rho)-\eta^{2}]^{2}+\eta^{2}}}{(1-\lambda^{2}/2)(\rho^{2}+\eta^{2})}.\vspace{2mm}$
(34)

where $\rho$, $\eta$, $\lambda$ is the Wolfenstein parameters.

The $CP$ violation, $A_{CP}$, can be written as

$\displaystyle
A_{CP}\equiv\frac{|A|^{2}-|\bar{A}|^{2}}{|A|^{2}+|\bar{A}|^{2}}=\frac{-2r{\rm{sin}}\delta{\rm{sin}}\theta}{1+2r{\rm{cos}}\delta{\rm{cos}}\theta+r^{2}}.$
(35)

##  IV INPUT PARAMETERS

The CKM matrix, which elements are determined from experiments, can be
expressed in terms of the Wolfenstein parameters $A$, $\rho$, $\lambda$ and
$\eta$ wol :

$\left(\begin{array}[]{ccc}1-\tfrac{1}{2}\lambda^{2}&\lambda&A\lambda^{3}(\rho-\mathrm{i}\eta)\\\
-\lambda&1-\tfrac{1}{2}\lambda^{2}&A\lambda^{2}\\\
A\lambda^{3}(1-\rho-\mathrm{i}\eta)&-A\lambda^{2}&1\\\ \end{array}\right),$
(36)

where $\mathcal{O}(\lambda^{4})$ corrections are neglected. The latest values
for the parameters in the CKM matrix are PDG2018 :

$\displaystyle\lambda=0.22506\pm 0.00050,\quad A=0.811\pm 0.026,$
$\displaystyle\bar{\rho}=0.124_{-0.018}^{+0.019},\quad\bar{\eta}=0.356\pm{0.011}.$
(37)

where

$\displaystyle\bar{\rho}=\rho(1-\frac{\lambda^{2}}{2}),\quad\bar{\eta}=\eta(1-\frac{\lambda^{2}}{2}).$
(38)

From Eqs. (37) ( 38) we have

$\displaystyle 0.109<\rho<0.147,\quad 0.354<\eta<0.377.$ (39)

The other parameters are given as following Lu2006 ; xiao2007 ; PDG2018 :

$\displaystyle f_{\pi}$ $\displaystyle=$ $\displaystyle 0.13\text{GeV},\hskip
83.93553ptf_{B}=0.19\text{GeV},$ $\displaystyle m_{B}$ $\displaystyle=$
$\displaystyle 5.2792\text{GeV},\hskip 75.39963ptf_{K}=0.16\text{GeV},$
$\displaystyle f_{s}$ $\displaystyle=$ $\displaystyle 0.17\text{GeV},\hskip
83.93553ptf_{n}=0.14\text{GeV},$ $\displaystyle m_{\pi}$ $\displaystyle=$
$\displaystyle 0.14\text{GeV},\hskip 83.93553ptm_{W}=80.41\text{GeV},$
$\displaystyle m_{0}^{\eta_{s\bar{s}}}$ $\displaystyle=$ $\displaystyle
2.4\text{GeV},\hskip 83.93553ptm_{0}^{\eta_{d\bar{d}}}=1.4\text{MeV}.$ (40)

##  V Numerical results

The $CP$ violation depends on the weak phase differences from the CKM matrix
elements and the strong phase differences which comes from the dynamics of QCD
and the other mechanism. The CKM matrix elements are related to $A$, $\rho$,
$\eta$ and $\lambda$, but the results for the $CP$ violation are less reliant
on $A$ and $\lambda$ in the course of calculations. Hence, we present the $CP$
violation from the weak phases associated with the $\rho$ and $\eta$ in the
CKM matrix while the $A$ and $\lambda$ are assigned for the central values.
The major uncertainties for the $CP$ violations is from the parameters, which
arises from the uncertainties of parameter $\omega_{b}=0.50\pm 0.05Gev$ the
variations of $m_{0}^{K}=1.6\pm 0.1Gev$, $m_{0}^{\pi}=1.4\pm 0.1Gev$ and the
Gegenbauer Coefficients $a_{2}^{K}=0.25\pm 0.15Gev$, $a_{2}^{\eta}=0.44\pm
0.22Gev$. From Table.I, we can find that the isospin breaking via
$\pi^{0}-\eta-\eta^{\prime}$ mixing changes the sign of the $CP$ violation for
the decay channel of ${B}^{0}\rightarrow\pi^{0}\pi^{0}$ and $B^{0}\rightarrow
K^{0}\pi^{0}$, which compared data from three other methods for the central
value. Compared with the PQCD, one can find that the central value of the $CP$
violation via $\pi^{0}-\eta-\eta^{\prime}$ mixing has changed $0.25$ for the
decay channel of ${B}^{0}\rightarrow\pi^{0}\pi^{0}$. This is due to the
breaking of isospin symmetry, the interference between the $\pi^{0}$ and
$\eta(^{\prime})$ mesons is stronger than other decay modes. And there is
twice contribution from $\pi^{0}-\eta-\eta^{\prime}$ mixing. We can find the
$CP$ violation of the decay mode ${B}^{+}\rightarrow\pi^{+}\pi^{0}$ agree well
with the QCDF and PQCD predictions in Table.I. For the decay mode
${B}^{+}\rightarrow K^{+}\pi^{0}$, our result presents the large region for
the $CP$ violation, and include the experiment data. In comparison with the
results of PQCD without $\pi^{0}-\eta-\eta^{\prime}$ mixing, the isospin
symmetry breaking generates large effect.

Table 1: The CP violation of $B$ decay mode via isospin symmetry breaking via $\pi^{0}-\eta-\eta^{{}^{\prime}}$ mixing in PQCD. decay mode |  QCDFMB1  |  PQCDlhn2005  |  $\pi^{0}-\eta-\eta^{{}^{\prime}}$ mixing |  DataHFAG ; MP2014
---|---|---|---|---
$B^{0}\rightarrow\pi^{0}\pi^{0}$ |  $0.451^{+0.184}_{-0.028}$ |  $0.07\pm 0.03$ |  $-0.18^{+0.19}_{-0.01}$ |  $0.03\pm 0.17$
$B^{0}\rightarrow\pi^{+}\pi^{0}$ |  $0.000^{+0.000}_{-0.000}$ |  $0.00\pm 0.00$ |  $0.00^{+0.00}_{-0.00}$ |  $0.026\pm 0.039$
$B^{0}\rightarrow K^{0}\pi^{0}$ |  $-0.033^{+0.010}_{-0.008}$ |  $-0.09^{+0.06}_{-0.08}$ |  $0.03^{+0.00}_{-0.04}$ |  $-0.01\pm 0.10$
$B^{0}\rightarrow K^{+}\pi^{0}$ |  $0.071^{+0.017}_{-0.018}$ |  $-0.01^{+0.03}_{-0.05}$ |  $-0.45^{+0.63}_{-0.03}$ |  $0.040\pm 0.021$

##  VI SUMMARY AND DISCUSSION

In this paper, we study the $CP$ violation for the
${B}^{0}\rightarrow\pi^{0}\pi^{0}$, ${B}^{+}\rightarrow\pi^{+}\pi^{0}$,
${B}^{0}\rightarrow K^{0}\pi^{0}$ and ${B}^{+}\rightarrow K^{+}\pi^{0}$ decays
from isospin symmetry breaking effects via $\pi^{0}-\eta-\eta^{\prime}$ in
perturbative QCD. It is found that the $CP$ violation can be changed via
$\pi^{0}-\eta-\eta^{\prime}$ mixing from the isospin symmetry breaking from
the strong phase difference.

The traditional factorization method is the leading order of QCD factorization
scheme with regard to the decay processes of bottom mesons and $1/m_{b}$
corrections is ignored. The QCD factorization scheme is affected by the
singularities of endpoint, which are not well controlled. Recently, next-to-
leading order corrections for the decay of $B\rightarrow\pi\rho(\omega)$ have
been implemented in the PQCD approach lu-next . Some questions of the NLO
accurate calculation in PQCD approach are discussed by the author in article
beneke-proc . The complete NLO calculation in the PQCD method needs to compute
all one-loop spectator-scattering diagrams. Nevertheless, the paper lu-next
only takes into account the 1-loop BBNS kernel (similar to the kernel in QCD
factorization) for the decay of $B\rightarrow\pi\rho(\omega)$. The vertex
diagram can be far away from the shell as a subdiagram of a large diagram with
hard collinear commutation. Besides, the Wilson coefficients are evaluated at
a very low scale where the perturbation theory is destroyed. The non-physical
strengthen of the Wilson coefficients at small scales is also the source of
the large penguin and annihilation. Based on above discussion, we investigate
the $CP$ violation of $B\rightarrow\pi\pi(K)$ via $\pi^{0}-\eta-\eta^{\prime}$
mixing from the leading order contribution in PQCD.

Theoretical errors lead to the uncertainty of results. In general,the power
corrections in the heavy quark limit will give the main theoretical
uncertainty. It means that introduce $1/m_{b}$ power corrections is essential.
Unluckily, there are many possibilities of $1/m_{b}$ power inhibiting effects
and they are nonperturbative essentially, so they unable to calculate by the
perturbative method. This scheme has more uncertainty. The first error relates
to the change in the CKM parameters. The second error comes from hadron
parameters, that is, the shape parameter, shape factor, decay constant, and
the wave function of $B$ meson. The third error corresponds to the selection
of the hard scales, which ranges from 0.75t to 1.25t, indicating the size of
next-to-leading order in QCD contributions.

##  VII Acknowledgments

This work was supported by National Natural Science Foundation of China
(Project Numbers 11605041), Excellent Thesis Cultivation Project of Henan
University of Technology and Science and Education Integration Project of
Henan University of Technology.

##  VIII APPENDIX: Related functions defined in the text

The functions related with the tree and penguin contributions are presented
with PQCD approach LKM ; AMLi2007 ; YK2001 .

The hard scales $t$ are chosen as

$\displaystyle t_{a}$ $\displaystyle=$
$\displaystyle\mbox{max}\\{{\sqrt{x_{2}}m_{B},1/b_{2},1/b_{3}}\\},$ (41)
$\displaystyle t_{e}^{1}$ $\displaystyle=$
$\displaystyle\mbox{max}\\{{\sqrt{x_{3}}m_{B},1/b_{1},1/b_{3}}\\},$ (42)
$\displaystyle t_{e}^{2}$ $\displaystyle=$
$\displaystyle\mbox{max}\\{{\sqrt{x_{1}}m_{B},1/b_{1},1/b_{3}}\\},$ (43)
$\displaystyle t_{e}^{3}$ $\displaystyle=$
$\displaystyle\mbox{max}\\{\sqrt{x_{3}}m_{B},1/b_{2},1/b_{3}\\},$ (44)
$\displaystyle t_{e}^{4}$ $\displaystyle=$
$\displaystyle\mbox{max}\\{\sqrt{x_{2}}m_{B},1/b_{2},1/b_{3}\\},$ (45)
$\displaystyle t_{f}$ $\displaystyle=$
$\displaystyle\mbox{max}\\{\sqrt{x_{1}x_{3}}m_{B},\sqrt{(x_{1}-x_{2})x_{3}}m_{B},1/b_{1},1/b_{2}\\},$
(46) $\displaystyle t_{f}^{1}$ $\displaystyle=$
$\displaystyle\mbox{max}\\{{\sqrt{x_{2}x_{3}}m_{B},1/b_{1},1/b_{2}}\\},$ (47)
$\displaystyle t_{f}^{2}$ $\displaystyle=$
$\displaystyle\mbox{max}\\{{\sqrt{x_{2}x_{3}}m_{B},\sqrt{x_{2}+x_{3}-x_{2}x_{3}}m_{B},1/b_{1},1/b_{2}}\\},$
(48) $\displaystyle t_{f}^{3}$ $\displaystyle=$
$\displaystyle\mbox{max}\\{\sqrt{x_{1}+x_{2}+x_{3}-x_{1}x_{3}-x_{2}x_{3}}m_{B},\sqrt{x_{2}x_{3}}m_{B},1/b_{1},1/b_{2}\\},$
(49) $\displaystyle t_{f}^{4}$ $\displaystyle=$
$\displaystyle\mbox{max}\\{\sqrt{x_{2}x_{3}}m_{B},\sqrt{(x_{1}-x_{2})x_{3}}m_{B},1/b_{1},1/b_{2}\\}.$
(50)

The function $h$ coming from the Fourier transformations of the function
$H^{(0)}$ xiao2007 . They are defined by

$\displaystyle h_{e}(x_{1},x_{3},b_{1},b_{3})$ $\displaystyle=$ $\displaystyle
K_{0}(\sqrt{x_{1}x_{3}}m_{B}b_{1}\left[\theta(b_{1}-b_{3})K_{0}(\sqrt{x}_{3}m_{B}b_{1})I_{0}(\sqrt{x}_{3}m_{B}b_{3})\right.$
$\displaystyle\left.+\theta(b_{3}-b_{1})K_{0}(\sqrt{x}_{3}m_{B}b_{3})I_{0}(\sqrt{x}_{3}m_{B}b_{1})\right]S_{t}(x_{3}),$
$\displaystyle h_{e}^{1}(x_{1},x_{2},b_{1},b_{2})$ $\displaystyle=$
$\displaystyle
K_{0}(\sqrt{x_{1}x_{2}}m_{B}b_{1}\left[\theta(b_{1}-b_{2})K_{0}(\sqrt{x}_{2}m_{B}b_{1})I_{0}(\sqrt{x}_{2}m_{B}b_{2})\right.$
$\displaystyle\left.+\theta(b_{2}-b_{1})K_{0}(\sqrt{x}_{2}m_{B}b_{2})I_{0}(\sqrt{x}_{2}m_{B}b_{1})\right],$
$\displaystyle h_{a}(x_{2},x_{3},b_{2},b_{3})$ $\displaystyle=$ $\displaystyle
K_{0}(i\sqrt{x_{2}x_{3}}m_{B}b_{2}\left[\theta(b_{3}-b_{2})K_{0}(i\sqrt{x}_{3}m_{B}b_{3})I_{0}(i\sqrt{x}_{3}m_{B}b_{2})\right.$
$\displaystyle\left.+\theta(b_{2}-b_{3})K_{0}(i\sqrt{x}_{3}m_{B}b_{2})I_{0}(i\sqrt{x}_{3}m_{B}b_{3})\right]S_{t}(x_{3}),$
$\displaystyle h_{d}(x_{1},x_{2},x_{3},b_{1},b_{2})$ $\displaystyle=$
$\displaystyle
K_{0}(-i\sqrt{x_{2}x_{3}}m_{B}b_{2}\left[\theta(b_{1}-b_{2})K_{0}(\sqrt{x}_{1}x_{2}m_{B}b_{1})I_{0}(\sqrt{x}_{1}x_{2}m_{B}b_{2})\right.$
$\displaystyle\left.+\theta(b_{2}-b_{1})K_{0}(\sqrt{x}_{1}x_{2}m_{B}b_{2})I_{0}(\sqrt{x}_{1}x_{2}m_{B}b_{1})\right],$
$\displaystyle h_{f}(x_{1},x_{2},x_{3},b_{1},b_{2})$ $\displaystyle=$
$\displaystyle[\theta(b_{2}-b_{1})I_{0}(M_{B}\sqrt{x_{1}x_{3}}b_{1})K_{0}(M_{B}\sqrt{x_{1}x_{3}}b_{2})$
(58) $\displaystyle+(b_{1}\longleftrightarrow
b_{2})]\cdot\left\\{\begin{array}[]{ll}K_{0}(M_{B}F_{(1)}b_{2}),&for\;\;F_{(1)}^{2}>0\\\
\frac{i\pi}{2}H_{0}^{(1)}(M_{B}\sqrt{|F_{(1)^{2}}|}b_{2}),&for\;\;F_{(1)}^{2}<0\end{array}\right.,$
$\displaystyle h_{f}^{1}(x_{1},x_{2},x_{3},b_{1},b_{2})$ $\displaystyle=$
$\displaystyle
K_{0}(-i\sqrt{x_{2}x_{3}}m_{B}b_{1}\left[\theta(b_{1}-b_{2})K_{0}(-i\sqrt{x}_{2}x_{3}m_{B}b_{1})J_{0}(\sqrt{x}_{2}x_{3}m_{B}b_{2})\right.$
$\displaystyle\left.+\theta(b_{2}-b_{1})K_{0}(-i\sqrt{x}_{2}x_{3}m_{B}b_{2})J_{0}(\sqrt{x}_{2}x_{3}m_{B}b_{3})\right],$
$\displaystyle h_{f}^{2}(x_{1},x_{2},x_{3},b_{1},b_{2})$ $\displaystyle=$
$\displaystyle
K_{0}(i\sqrt{x_{2}+x_{3}-x_{2}x_{3}}m_{B}b_{1}\left[\theta(b_{1}-b_{2})K_{0}(-i\sqrt{x}_{2}x_{3}m_{B}b_{1})J_{0}(\sqrt{x}_{2}x_{3}m_{B}b_{2})\right.$
$\displaystyle\left.+\theta(b_{2}-b_{1})K_{0}(-i\sqrt{x}_{2}x_{3}m_{B}b_{2})J_{0}(\sqrt{x}_{2}x_{3}m_{B}b_{1})\right],$
$\displaystyle h_{f}^{3}(x_{1},x_{2},x_{3},b_{1},b_{2})$ $\displaystyle=$
$\displaystyle[\theta(b_{1}-b_{2})K_{0}(i\sqrt{x_{2}x_{3}}b_{1}M_{B})I_{0}(i\sqrt{x_{2}x_{3}}b_{2}M_{B})$
$\displaystyle+(b_{1}\longleftrightarrow b_{2})]\cdot
K_{0}(\sqrt{x_{1}+x_{2}+x_{3}-x_{1}x_{3}-x_{2}x_{3}}b_{1}M_{B})$
$\displaystyle h_{f}^{4}(x_{1},x_{2},x_{3},b_{1},b_{2})$ $\displaystyle=$
$\displaystyle[\theta(b_{1}-b_{2})K_{0}(i\sqrt{x_{2}x_{3}}b_{1}M_{B})I_{0}(i\sqrt{x_{2}x_{3}}b_{2}M_{B})$
(65) $\displaystyle+(b_{1}\longleftrightarrow
b_{2})]\cdot\left\\{\begin{array}[]{ll}K_{0}(M_{B}F_{(2)}b_{2}),&for\;\;F_{(2)}^{2}>0\\\
\frac{i\pi}{2}H_{0}^{(1)}(M_{B}\sqrt{|F_{(2)^{2}}|}b_{2}),&for\;\;F_{(2)}^{2}<0\end{array}\right.,$

where $J_{0}$ is the Bessel function and $K_{0}$, $I_{0}$ are the modified
Bessel functions
$K_{0}(-ix)=-\frac{\pi}{2}\mathrm{y}_{0}(x)+i\,\frac{\pi}{2}\mathrm{J}_{0}(x)$,
and $F_{(j)}$’s are defined by

$\displaystyle
F_{(1)}^{2}=(x_{1}-x_{2})x_{3},\;\;\;\;F_{(2)}^{2}=(x_{1}-x_{2})x_{3}.$ (66)

The $S_{t}$ re-sums the threshold logarithms $\ln^{2}x$ appearing in the hard
kernels to all orders and it has been parameterized as

$\displaystyle
S_{t}(x)=\frac{2^{1+2c}\Gamma(3/2+c)}{\sqrt{\pi}\Gamma(1+c)}[x(1-x)]^{c},$
(67)

with $c=0.3$. In the nonfactorizable contributions, $S_{t}(x)$ gives a very
small numerical effect on the amplitude L4 .

The Sudakov exponents are defined as

$\displaystyle S_{B}(t)$ $\displaystyle=$ $\displaystyle
s\left(x_{1}\frac{m_{B}}{\sqrt{2}},b_{1}\right)-\frac{1}{\beta_{1}}\ln\frac{\ln(t/\Lambda)}{-\ln(b_{1}\Lambda)},$
(68) $\displaystyle S_{\pi}^{1}(t)$ $\displaystyle=$ $\displaystyle
s\left(x_{2}\frac{m_{B}}{\sqrt{2}},b_{2}\right)+s\left((1-x_{2})\frac{m_{B}}{\sqrt{2}},b_{2}\right)-\frac{1}{\beta_{1}}\ln\frac{\ln(t/\Lambda)}{-\ln(b_{2}\Lambda)},$
(69) $\displaystyle S_{\pi}^{2}(t)$ $\displaystyle=$ $\displaystyle
s\left(x_{3}\frac{m_{B}}{\sqrt{2}},b_{3}\right)+s\left((1-x_{3})\frac{m_{B}}{\sqrt{2}},b_{3}\right)-\frac{1}{\beta_{1}}\ln\frac{\ln(t/\Lambda)}{-\ln(b_{3}\Lambda)},$
(70) $\displaystyle S_{ab}(t)$ $\displaystyle=$ $\displaystyle
s\left(x_{1}\frac{m_{B}}{\sqrt{2}},b_{1}\right)+s\left(x_{3}\frac{m_{B}}{\sqrt{2}},b_{3}\right)+s\left((1-x_{3})\frac{m_{B}}{\sqrt{2}},b_{3}\right)-\frac{1}{\beta_{1}}\left[\ln\frac{\ln(t/\Lambda)}{-\ln(b_{1}\Lambda)}+\ln\frac{\ln(t/\Lambda)}{-\ln(b_{3}\Lambda)}\right],$
(71) $\displaystyle S_{cd}(t)$ $\displaystyle=$ $\displaystyle
s\left(x_{1}\frac{m_{B}}{\sqrt{2}},b_{1}\right)+s\left(x_{2}\frac{m_{B}}{\sqrt{2}},b_{2}\right)+s\left((1-x_{2})\frac{m_{B}}{\sqrt{2}},b_{2}\right)+s\left(x_{3}\frac{m_{B}}{\sqrt{2}},b_{1}\right)+s\left((1-x_{3})\frac{m_{B}}{\sqrt{2}},b_{1}\right)$
(73)
$\displaystyle-\frac{1}{\beta_{1}}\left[\ln\frac{\ln(t/\Lambda)}{-\ln(b_{1}\Lambda)}+\ln\frac{\ln(t/\Lambda)}{-\ln(b_{2}\Lambda)}\right],$
$\displaystyle S_{ef}(t)$ $\displaystyle=$ $\displaystyle
s\left(x_{1}\frac{m_{B}}{\sqrt{2}},b_{1}\right)+s\left(x_{2}\frac{m_{B}}{\sqrt{2}},b_{2}\right)+s\left((1-x_{2})\frac{m_{B}}{\sqrt{2}},b_{2}\right)+s\left(x_{3}\frac{m_{B}}{\sqrt{2}},b_{2}\right)+s\left((1-x_{3})\frac{m_{B}}{\sqrt{2}},b_{2}\right)$
(75)
$\displaystyle-\frac{1}{\beta_{1}}\left[\ln\frac{\ln(t/\Lambda)}{-\ln(b_{1}\Lambda)}+2\ln\frac{\ln(t/\Lambda)}{-\ln(b_{2}\Lambda)}\right],$
$\displaystyle S_{gh}(t)$ $\displaystyle=$ $\displaystyle
s\left(x_{2}\frac{m_{B}}{\sqrt{2}},b_{2}\right)+s\left(x_{3}\frac{m_{B}}{\sqrt{2}},b_{3}\right)+s\left((1-x_{2})\frac{m_{B}}{\sqrt{2}},b_{2}\right)+s\left((1-x_{3})\frac{m_{B}}{\sqrt{2}},b_{3}\right)$
(77)
$\displaystyle-\frac{1}{\beta_{1}}\left[\ln\frac{\ln(t/\Lambda)}{-\ln(b_{2}\Lambda)}+\ln\frac{\ln(t/\Lambda)}{-\ln(b_{3}\Lambda)}\right].$

The explicit form for the function $s(k,b)$ is LKM :

$\displaystyle s(k,b)$ $\displaystyle=$
$\displaystyle\frac{2}{3\beta_{1}}\left[\hat{q}\ln\left(\frac{\hat{q}}{\hat{b}}-\hat{q}+\hat{b}\right)\right]+\frac{A^{(2)}}{4\beta_{1}^{2}}\left(\frac{\hat{q}}{\hat{b}}-1\right)$
(80)
$\displaystyle-\left[\frac{A^{(2)}}{4\beta_{1}^{2}}-\frac{1}{3\beta_{1}}(2\gamma_{E}-1-\ln
2)\right]\ln\left(\frac{\hat{q}}{\hat{b}}\right),$

where the variables are defined by

$\displaystyle\hat{q}\equiv\mbox{ln}[k/(\sqrt{\Lambda})],~{}~{}~{}\hat{b}\equiv\mbox{ln}[1/(b\Lambda)],$
(81)

and the coefficients $A^{(i)}$ and $\beta_{i}$ are

$\displaystyle\beta_{1}$ $\displaystyle=$ $\displaystyle\frac{33-2n_{f}}{12},$
(82) $\displaystyle A^{(2)}$ $\displaystyle=$
$\displaystyle\frac{67}{9}-\frac{\pi^{2}}{3}-\frac{10}{27}n_{f}+\frac{8}{3}\beta_{1}\mbox{ln}(\frac{1}{2}e^{\gamma_{E}}),$
(83)

$n_{f}$ is the number of the quark flavors and $\gamma_{E}$ is the Euler
constant.

The individual decay amplitudes $F_{e}$, $F_{e}^{P}$, $F_{a}$, $M_{a}$ and
$M_{e}$ arise from the $(V-A)(V-A)$ and $(V-A)(V+A)$ operators, respectively.
The sum of their amplitudes are

  * •

$(V-A)(V-A)$ operators:

$\displaystyle F_{e}$ $\displaystyle=$ $\displaystyle-16\pi
C_{F}m_{B}^{4}\int^{1}_{0}dx_{1}dx_{2}\int^{\infty}_{0}b_{1}db_{1}b_{2}db_{2}\phi_{B}(x_{1},b_{1})\alpha_{s}(t_{e}^{1})h_{e}^{1}(x_{1},x_{2},b_{1},b_{2})$
(84)
$\displaystyle\times\Big{\\{}\Big{[}(1+x_{2})\phi_{\pi}(x_{2},b_{2})+(1-2x_{2})r_{\pi}\phi^{\prime}_{\pi}(x_{2},b_{2})\Big{]}\times\exp[-S_{B}(t_{e}^{1})-S_{\pi}(t_{e}^{1})]$
$\displaystyle\;\;+2r_{\pi}\phi^{\prime}_{\pi}(x_{2},b_{2})\alpha_{s}(t_{e}^{2})h_{e}^{1}(x_{2},x_{1},b_{2},b_{1})\times\exp[-S_{B}(t_{e}^{2})-S_{\pi}(t_{e}^{2})]\Big{\\}},$

  * •

$(V-A)(V+A)$ operators:

$\displaystyle F_{e}^{P}$ $\displaystyle=$ $\displaystyle-32\pi
C_{F}m_{B}^{4}r_{\pi}\int^{1}_{0}dx_{1}dx_{2}\int^{\infty}_{0}b_{1}db_{1}b_{2}db_{2}\phi_{B}(x_{1},b_{1})\cdot\alpha_{s}(t_{e}^{1})h_{e}^{1}(x_{1},x_{2},b_{1},b_{2})$
(85)
$\displaystyle\times\Big{\\{}\Big{[}\phi_{\pi}(x_{2},b_{2})+(2+x_{2})\phi_{\pi}(x_{2},b_{2})+(1-2x_{2})r_{\pi}\phi^{\prime}_{\pi}(x_{2},b_{2})\Big{]}\times\exp[-S_{B}(t_{e}^{1})-S_{\pi}(t_{e}^{1})]$
$\displaystyle\;\;+x_{1}\phi_{\pi}(x_{2},b_{2})+2(1-x_{1})r_{\pi}\phi^{\prime}_{\pi}(x_{2},b_{2})\alpha_{s}(t_{e}^{2})h_{e}^{1}(x_{2},x_{1},b_{2},b_{1})\times\exp[-S_{B}(t_{e}^{2})-S_{\pi}^{1}(t_{e}^{2})]\Big{\\}},$

$\displaystyle F_{a}^{P}$ $\displaystyle=$ $\displaystyle-64\pi
C_{F}m_{B}^{4}r_{\pi}\int^{1}_{0}dx_{1}dx_{2}\int^{\infty}_{0}b_{1}db_{1}b_{2}db_{2}\phi_{B}(x_{1},b_{1})\alpha_{s}(t_{a})h_{a}(x_{2},x_{3},b_{2},b_{3})$
(86)
$\displaystyle\times\Big{[}2\phi_{\pi}(x_{2},b_{2})\phi^{\prime}_{\pi}(x_{3},b_{3})+x_{2}\phi_{\pi}(x_{3},b_{3})\phi^{\prime}_{\pi}(x_{2},b_{2})\Big{]}\exp[-S_{\pi}^{1}(t_{a})-S_{\pi}^{2}(t_{a})],$

where the color factor ${C_{F}}=4/3$ and $a_{i}$ represents the corresponding
Wilson coefficients from differen decay channels.
$r_{\pi}=\frac{m_{0}}{m_{B}}=\frac{m_{\pi}^{2}}{m_{u+m_{d}}}$. Note that
$F_{a}^{P}$ vanishes in the limit of $m_{0}=0$. So the $m_{0}$ term in the
pion wave function has a significant impact on the $CP$ violation.

The function are related to the annihilation type process, whose contribution
are:

$\displaystyle M_{e}$ $\displaystyle=$ $\displaystyle-\frac{32}{3}\pi
C_{F}\sqrt{2N_{C}}m_{B}^{2}\int^{1}_{0}dx_{1}dx_{2}dx_{3}\int^{\infty}_{0}b_{1}db_{1}b_{2}db_{2}\phi_{B}(x_{1},b_{1})\phi_{\pi}(x_{2},b_{2})$
(87)
$\displaystyle\times\phi_{\pi}(x_{3},b_{1})\alpha_{s}(t_{d})h_{d}(x_{1},x_{2},x_{3},b_{1},b_{2})\exp[-S_{B}(t_{d})-S_{\pi}^{1}(t_{d})-S_{\pi}^{2}(t_{d})],$

$\displaystyle M_{a}$ $\displaystyle=$ $\displaystyle-\frac{32}{3}\pi
C_{F}\sqrt{2N_{C}}m_{B}^{2}\int^{1}_{0}dx_{1}dx_{2}dx_{3}\int^{\infty}_{0}b_{1}db_{1}b_{2}db_{2}\phi_{B}(x_{1},b_{1})$
(88)
$\displaystyle\times\Big{\\{}-\Big{\\{}x_{2}\phi_{\pi}(x_{2},b_{2})\phi_{\pi}(x_{3},b_{2})+(x_{2}+x_{3}+2)r_{\eta}^{2}\phi^{\prime}_{\pi}(x_{2},b_{2})\phi^{\prime}_{\pi}(x_{3},b_{2})\Big{\\}}$
$\displaystyle\alpha_{s}(t_{f^{1}})h_{f^{1}}(x_{1},x_{2},x_{3},b_{1},b_{2})\exp[-S_{B}(t_{f^{1}})-S_{\pi}^{1}(t_{f^{1}})-S_{\pi}^{2}(t_{f^{1}})]$
$\displaystyle+\Big{\\{}x_{2}\phi_{\pi}(x_{2},b_{2})\phi_{\pi}(x_{3},b_{2})+(2+x_{2}+x_{3})r_{\eta}^{2}\phi^{\prime}_{\pi}(x_{2},b_{2})\phi^{\prime}_{\pi}(x_{3},b_{2})\Big{\\}}$
$\displaystyle\times\alpha_{s}(t_{f^{2}})h_{f^{2}}(x_{1},x_{2},x_{3},b_{1},b_{2})\exp[-S_{B}(t_{f^{2}})-S_{\pi}^{1}(t_{f^{2}})-S_{\pi}^{2}(t_{f^{2}})]\Big{\\}}.$

The explicit expressions of individual decay amplitude $F_{e\eta}$ and
$F_{e\eta}^{P1,P2}$, $M_{e\eta}$ and $M_{e\eta}^{P1,P2}$, $F_{a\eta}$ and
$F_{a\eta}^{P1,P2}$, $M_{a\eta}$ and $M_{a\eta}^{P1,P2}$ can be found

  * •

$(V-A)(V-A)$ operators:

$\displaystyle F_{e\eta}$ $\displaystyle=$ $\displaystyle 8\pi
C_{F}m_{B}^{4}\int^{1}_{0}dx_{1}dx_{3}\int^{\infty}_{0}b_{1}db_{1}b_{3}db_{3}\phi_{B}(x_{1},b_{1})$
(89)
$\displaystyle\times\Big{\\{}\Big{[}(1+x_{3})\phi_{\eta}(x_{3},b_{3})+(1-2x_{3})r_{\eta}(\phi_{\eta}^{p}(x_{3})+\phi_{\eta}^{t}(x_{3}))\Big{]}\alpha_{s}(t_{e}^{1})h_{e}(x_{1},x_{3},b_{1},b_{3})\exp[-S_{ab}(t_{e}^{1})]$
$\displaystyle\;\;+2r_{\eta}\phi_{\eta}^{p}(x_{3},b_{3})\alpha_{s}(t_{e}^{2})h_{e}(x_{3},x_{1},b_{3},b_{1})\exp[-S_{ab}(t_{e}^{2})]\Big{\\}},$

  * •

$(V-A)(V+A)$ operators:

$\displaystyle F^{P1}_{e\eta}$ $\displaystyle=$ $\displaystyle-F_{e\eta},$
(90)

  * •

$(S-P)(S+P)$ operators:

$\displaystyle F^{P2}_{e\eta}$ $\displaystyle=$ $\displaystyle 16\pi
C_{F}m_{B}^{4}r_{\eta(^{\prime})}\int^{1}_{0}dx_{1}dx_{3}\int^{\infty}_{0}b_{1}db_{1}b_{3}db_{3}\phi_{B}(x_{1},b_{1})$
(91)
$\displaystyle\times\Big{\\{}\Big{[}\phi_{\eta}(x_{3},b_{3})+r_{\eta}((2+x_{3})\phi_{\eta}^{p}(x_{3},b_{3})-x_{3}\phi_{\eta}^{t}(x_{3},b_{3}))\Big{]}\alpha_{s}(t_{e}^{1})h_{e}(x_{1},x_{3},b_{1},b_{3})\exp[-S_{ab}(t_{e}^{1})]$
$\displaystyle\;\;\;+x_{1}\phi_{\eta}(x_{3},b_{3})-2(x_{1}-1)r_{\eta}\phi_{\eta}^{p}(x_{3},b_{3})\alpha_{s}(t_{e}^{2})h_{e}(x_{3},x_{1},b_{3},b_{1})\exp[-S_{ab}(t_{e}^{2})]\Big{\\}},$

where $r_{i}=\frac{m_{0i}}{m_{B}}$ and $m_{0i}$ refers to the chiral scale
parameter.

  * •

$(V-A)(V-A)$ operators:

$\displaystyle M_{e\eta}$ $\displaystyle=$
$\displaystyle\frac{16\sqrt{6}}{3}\pi
C_{F}m_{B}^{4}/\int^{1}_{0}dx_{1}dx_{2}dx_{3}\int^{\infty}_{0}b_{1}db_{1}b_{2}db_{2}\phi_{B}(x_{1},b_{1})\phi_{\eta(^{\prime})}(x_{2},b_{2})$
(92)
$\displaystyle\times\Big{\\{}\Big{[}2x_{3}r_{\eta}\phi_{\eta}^{t}(x_{3},b_{1})-x_{3})\phi_{\eta}(x_{3},b_{1})\Big{]}\alpha_{s}(t_{f})h_{f}(x_{1},x_{2},x_{3},b_{1},b_{2})\exp[-S_{cd}(t_{f})]\Big{\\}},$

  * •

$(V-A)(V+A)$ and $(S-P)(S+P)$ operators:

$\displaystyle M^{P1}_{e\eta}=0,\;\;\;M^{P2}_{e\eta}=-M_{e\eta}.$ (93)

The functions are related with the annihilation type process, whose
contributions are:

  * •

$(V-A)(V-A)$ operators:

$\displaystyle F_{a\eta}$ $\displaystyle=$ $\displaystyle-8\pi
C_{F}m_{B}^{4}\int^{1}_{0}dx_{2}dx_{3}\int^{\infty}_{0}b_{2}db_{2}b_{3}db_{3}\phi_{B}(x_{1},b_{1})$
(94)
$\displaystyle\times\Big{\\{}\Big{[}x_{3}\phi_{\eta}(x_{3},b_{3})\phi_{\eta(^{\prime})}(x_{2},b_{2})+2r_{\eta}r_{\eta(^{\prime})}((x_{3}+1)\phi_{\eta}^{p}(x_{3},b_{3})$
$\displaystyle+(x_{3}-1)\phi_{\eta}^{t}(x_{3},b_{3}))\phi_{\eta(^{\prime})}^{p}(x_{2},b_{2})\Big{]}\alpha_{s}(t_{e}^{3})h_{a}(x_{2},x_{3},b_{2},b_{3})\exp[-S_{gh}(t_{e}^{3})]$
$\displaystyle\;\;-\Big{[}x_{2}\phi_{\eta}(x_{3},b_{3})\phi_{\eta(^{\prime})}(x_{2},b_{2})+2r_{\eta}r_{\eta(^{\prime})}((x_{2}+1)\phi_{\eta(^{\prime})}^{p}(x_{2},b_{2})$
$\displaystyle+(x_{2}-1)\phi_{\eta(^{\prime})}^{t}(x_{2},b_{2}))\phi_{\eta}^{p}(x_{3},b_{3})\Big{]}\alpha_{s}(t_{e}^{4})h_{a}(x_{3},x_{2},b_{3},b_{2})\exp[-S_{gh}(t_{e}^{4})]\Big{\\}},$

  * •

$(V-A)(V+A)$ operators:

$\displaystyle F_{a\eta}^{P1}=F_{a\eta},$ (95)

  * •

$(S-P)(S+P)$ operators:

$\displaystyle F_{a\eta}^{P2}$ $\displaystyle=$ $\displaystyle-16\pi
C_{F}m_{B}^{4}\int^{1}_{0}dx_{2}dx_{3}\int^{\infty}_{0}b_{2}db_{2}b_{3}db_{3}\phi_{B}(x_{1},b_{1})$
(96)
$\displaystyle\times\Big{\\{}\Big{[}x_{3}r_{\eta}(\phi_{\eta}(x_{3},b_{3})-\phi_{\eta}^{t}(x_{3},b_{3}))\phi_{\eta(^{\prime})}(x_{2},b_{2})+2r_{\eta(^{\prime})}\phi_{\eta}^{p}(x_{3},b_{3})+\phi_{\eta(^{\prime})}^{p}(x_{2},b_{2})\Big{]}$
$\displaystyle\times\alpha_{s}(t_{e}^{3})h_{a}(x_{2},x_{3},b_{2},b_{3})\exp[-S_{gh}(t_{e}^{3})]+\Big{[}x_{2}r_{\eta(^{\prime})}(\phi_{\eta(^{\prime})}(x_{2},b_{2})\\-\phi_{\eta(^{\prime})}^{t}(x_{2},b_{2}))\phi_{\eta}(x_{3},b_{3})$
$\displaystyle+2r_{\eta}\phi_{\eta(^{\prime})}(x_{2},b_{2})\phi_{\eta}^{p}(x_{3},b_{3})\Big{]}\alpha_{s}(t_{e}^{4})h_{a}(x_{3},x_{2},b_{3},b_{2})\exp[-S_{gh}(t_{e}^{4})]\Big{\\}}.$

  * •

$(V-A)(V-A)$ operators:

$\displaystyle M_{a\eta}$ $\displaystyle=$
$\displaystyle\frac{16\sqrt{6}}{3}\pi
C_{F}m_{B}^{4}/\int^{1}_{0}dx_{1}dx_{2}dx_{3}\int^{\infty}_{0}b_{1}db_{1}b_{2}db_{2}\phi_{B}(x_{1},b_{1})\times\Big{\\{}-\Big{\\{}x_{2}\phi_{\eta}(x_{3},b_{2})\phi_{\eta(^{\prime})}(x_{2},b_{2})$
(97)
$\displaystyle+r_{\eta}r_{\eta(^{\prime})}[(x_{2}+x_{3}+2)\phi_{\eta(^{\prime})}^{p}(x_{2},b_{2})+(x_{2}-x_{3})\phi_{\eta(^{\prime})}^{t}(x_{2},b_{2})]\phi_{\eta}^{p}(x_{3},b_{2})+r_{\eta}r_{\eta(^{\prime})}[(x_{2}-x_{3})\phi_{\eta(^{\prime})}^{p}(x_{3},b_{2})$
$\displaystyle+(x_{2}+x_{3}-2)\phi_{\eta(^{\prime})}^{t}(x_{2},b_{2})]\phi_{\eta}^{t}(x_{3},b_{2})\Big{\\}}\alpha_{s}(t_{f}^{3})h_{f}^{3}(x_{1},x_{2},x_{3},b_{1},b_{2})\exp[-S_{ef}(t_{f}^{3})]$
$\displaystyle+\Big{\\{}x_{3}\phi_{\eta}(x_{3},b_{2})\phi_{\eta(^{\prime})}(x_{2},b_{2})+r_{\eta}r_{\eta(^{\prime})}[(x_{2}+x_{3})\phi_{\eta(^{\prime})}^{p}(x_{2},b_{2})+(x_{3}-x_{2})\phi_{\eta(^{\prime})}^{t}(x_{2},b_{2})]\phi_{\eta}^{p}(x_{3},b_{2})$
$\displaystyle+r_{\eta}r_{\eta(^{\prime})}[(x_{3}-x_{2})\phi_{\eta(^{\prime})}^{p}(x_{2},b_{2})+(x_{2}+x_{3})\phi_{\eta(^{\prime})}^{t}(x_{2},b_{2})]\phi_{\eta}^{t}(x_{3},b_{2})\Big{\\}}$
$\displaystyle\times\alpha_{s}(t_{f}^{4})h_{f}^{3}(x_{1},x_{2},x_{3},b_{1},b_{2})\exp[-S_{ef}(t_{f}^{4})]\Big{\\}},$

  * •

$(V-A)(V+A)$ operators:

$\displaystyle M_{a\eta}^{P1}$ $\displaystyle=$
$\displaystyle\frac{16\sqrt{6}}{3}\pi
C_{F}m_{B}^{4}/\int^{1}_{0}dx_{1}dx_{2}dx_{3}\int^{\infty}_{0}b_{1}db_{1}b_{2}db_{2}\phi_{B}(x_{1},b_{1})$
(98)
$\displaystyle\times\Big{\\{}\Big{[}(x_{3}-2)r_{\eta}\phi_{\eta(^{\prime})}(x_{2},b_{2})(\phi_{\eta}^{p}(x_{3},b_{2})+\phi_{\eta}^{t}(x_{3},b_{2}))-(x_{2}-2)r_{\eta(^{\prime})}\phi_{\eta}(x_{3},b_{2})(\phi_{\eta(^{\prime})}^{p}(x_{2},b_{2})$
$\displaystyle+\phi_{\eta(^{\prime})}^{t}(x_{2},b_{2}))\Big{]}\alpha_{s}(t_{f}^{3})h_{f}^{3}(x_{1},x_{2},x_{3},b_{1},b_{2})\exp[-S_{ef}(t_{f}^{3})]-\Big{[}x_{3}r_{\eta}\phi_{\eta(^{\prime})}(x_{2},b_{2})(\phi_{\eta}^{p}(x_{3},b_{2})+\phi_{\eta}^{t}(x_{3},b_{2}))$
$\displaystyle-
x_{2}r_{\eta(^{\prime})}\phi_{\eta}(x_{3},b_{2})(\phi_{\eta(^{\prime})}^{p}(x_{2},b_{2})+\phi_{\eta(^{\prime})}^{t}(x_{2},b_{2}))\Big{]}\alpha_{s}(t_{f}^{4})h_{f}^{3}(x_{1},x_{2},x_{3},b_{1},b_{2})\exp[-S_{ef}(t_{f}^{4})]\Big{\\}},$

  * •

$(S-P)(S+P)$ operators:

$\displaystyle M_{a\eta}^{P2}$ $\displaystyle=$
$\displaystyle\frac{16\sqrt{6}}{3}\pi
C_{F}m_{B}^{4}/\int^{1}_{0}dx_{1}dx_{2}dx_{3}\int^{\infty}_{0}b_{1}db_{1}b_{2}db_{2}\phi_{B}(x_{1},b_{1})\times\Big{\\{}\Big{\\{}x_{3}\phi_{\eta}(x_{3},b_{2})\phi_{\eta(^{\prime})}(x_{2},b_{2})$
(99)
$\displaystyle+r_{\eta}r_{\eta(^{\prime})}[(x_{2}+x_{3}+2)\phi_{\eta(^{\prime})}^{p}(x_{2},b_{2})+(x_{3}-x_{2})\phi_{\eta(^{\prime})}^{t}(x_{2},b_{2})]\phi_{\eta}^{p}(x_{3},b_{2})+r_{\eta}r_{\eta(^{\prime})}[(x_{3}-x_{2})\phi_{\eta(^{\prime})}^{p}(x_{3},b_{2})$
$\displaystyle+(x_{2}+x_{3}-2)\phi_{\eta(^{\prime})}^{t}(x_{2},b_{2})]\phi_{\eta}^{t}(x_{3},b_{2})\Big{\\}}\alpha_{s}(t_{f}^{3})h_{f}^{3}(x_{1},x_{2},x_{3},b_{1},b_{2})\exp[-S_{ef}(t_{f}^{3})]$
$\displaystyle-\Big{\\{}x_{2}\phi_{\eta}(x_{3},b_{2})\phi_{\eta(^{\prime})}(x_{2},b_{2})+r_{\eta}r_{\eta(^{\prime})}[(x_{2}+x_{3})\phi_{\eta(^{\prime})}^{p}(x_{2},b_{2})+(x_{2}-x_{3})\phi_{\eta(^{\prime})}^{t}(x_{2},b_{2})]\phi_{\eta}^{p}(x_{3},b_{2})$
$\displaystyle+r_{\eta}r_{\eta(^{\prime})}[(x_{2}-x_{3})\phi_{\eta(^{\prime})}^{p}(x_{2},b_{2})+(x_{2}+x_{3})\phi_{\eta(^{\prime})}^{t}(x_{2},b_{2})]\phi_{\eta}^{t}(x_{3},b_{2})\Big{\\}}$
$\displaystyle\times\alpha_{s}(t_{f}^{4})h_{f}^{3}(x_{1},x_{2},x_{3},b_{1},b_{2})\exp[-S_{ef}(t_{f}^{4})]\Big{\\}}.$

If we change the $\pi$ and $\eta(^{\prime})$ meson, the corresponding
expressions of amplitudes for new diagrams will be similar with those as given
in Eqs.(89)-(99), since the $\pi$ and $\eta(^{\prime})$ are all pseudoscalar
mesons and have the similar wave functions Lu2006 . For example, we can find
that

$\displaystyle
F_{a\eta(^{\prime})}=-F_{a\pi},\;\;\;\;F_{a\eta(^{\prime})}^{P1}=F_{a\pi}^{P1},\;\;\;\;F_{a\eta(^{\prime})}^{P2}=F_{a\pi}^{P2}.$
(100)

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