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\begin{document}
%\preprint{NCTU/HEP2001-2}
\title{Subleading power corrections in $B\to PP$ decays}
\author{Tsung-Wen Yeh}
\email{twyeh@cc.nctu.edu.tw}
\affiliation{Institute of Physics, National Chiao-Tung University,
Hsinchu 300, Taiwan}
%\maketitle
\begin{abstract}
The QCD factorization is generalized to include power corrections. The leading and subleading twist contributions are separated by means of collinear expansion. The higher twist corrections from the nonvanishing light quark masses, and the two and three parton distribution amplitudes are considered. The contributions involving three parton distribution amplitudes have the sources: the transverse momenta of the loop partons, the higher Fock state with more quark (antiquark) or gluons, and the wrong spin projector of the meson. The complete tree level twist-3 and twist-4 contributions for the decays of the $B$ meson into two light pseudoscalar mesons are given. The higher twist contributions are shown to be gauge invariant by employing the light-cone and covariant gauges. The new method also exhibits the dynamics of these higher twist contributions. The magnitude of the twist-3 contributions relative to the leading twist contributions is by a factor $1.3\mu_\chi$, with $\mu_\chi=m_\pi^2/m_b(m_u+m_d)$. The twist-4 contributions vanish due to the $G$-parity, or can be ignorable.
\end{abstract}
\pacs{12.38.Bx,12.38.-t,12.40.-y,13.25.Hw,14.40.Nd}
\maketitle
%\narrowtext
\section{introduction}
The interplay between weak and strong interactions makes complicate the calculations of hadronic $B$ decays. In the heavy quark mass limit, some approximations have been proven to be done. The QCD factorization formula proposed by Beneke, Buchalla, Neubert, and Sachrajda (BBNS)\cite{Beneke:1999br,Beneke:2000ry}
demonstrate that the decay amplitude in the heavy quark limit can be expressed into either the product of the transition form factor and the convolution integral of the ejected meson distribution amplitude with hard scattering kernel or the convolution integral of the hard scattering kernel with the relevant meson light-cone distribution amplitudes. The important progress is that the hard scattering kernels are calculable under perturbation theory, and the naive factorization appears as the leading order of the factorization formula. 

As an increasing accumulation of experimental data, for theory, it is indispensable to have more accurate calculations than the leading order approximations. There require the subleading corrections. The corrections could be higher order radiative corrections ordered in $\alpha_s$ or higher order power corrections ordered in $\Lambda_{QCD}/m_b$. For the radiative corrections, there have been many good calculations \cite{Beneke:1999br,Beneke:2000ry,Beneke:2001ev,Neubert:2002ix}. On the other hand, the power corrections seem less known and lack a systematic method. Since the $b$ quark mass $m_b$ is finite, one may expect that power corrections are desirable. There indeed exist power suppressed but chirally enhanced corrections in the studies of $B$ two body nonleptonic charmless decays. There are also other kinds of power corrections: such as the light quark mass, the higher Fock state with more quark (antiquark) or gluon, transverse momenta of quarks in the light meson. In this paper, we would like to propose a systematic method to calculate these power corrections. 

The QCD factorization formula assumes that the external partons to the hard scattering kernel are on-shell and the heavy quark mass is infinite. To include the power corrections, we first need to relax these restrictions: the external partons could be on-shell or off-shell and the heavy quark mass is finite. With these generalizations, for $\bar{B}$ decays into two light mesons $P_1$ and $P_2$, the formula for operator $Q_i$ can be recast as  
\bee\label{QFF1}
\langle P_1P_2|Q_i|\bar{B}\rangle&=&\int \frac{d^4 l}{(2\pi)^4}\Tr[T^{I}_i(l)\Phi_{P_2}(l)]\nn
&&+ \int [\frac{d^4 l_j}{(2\pi)^4}]_{j=1}^2
\Tr[T^{II}_i(l_1,l_2,l_3)\Phi_{P_1}(l_1)
\Phi_{P_2}(l_2)\Phi_B(l_3)]\nn
&&+ \int [\frac{d^4 l_j}{(2\pi)^4}]_{j=1}^{2}\Tr[T^{I}_{i,\alpha}(l_1,l_2)\Phi_{P_2}^{\alpha}(l_1,l_2)]\nn
&&+ \int [\frac{d^4 l_j}{(2\pi)^4}]_{j=1}^4
\Tr[T^{II}_{i,\alpha}(l_1,l_2,l_3,l_4)\Phi_{P_2}^{\alpha}(l_1,l_2)\Phi_{P_1}(l_3)\Phi_B(l_4)]+(P_1\leftrightarrow P_2)\nn
&&+\cdots\ ,
\eee 
where the dots means higher twist correction terms. The notation $[d^4 l_j/(2\pi)^4]_{j=1}^k$ represents the integration measure 
\bee
[\frac{d^4 l_j}{(2\pi)^4}]_{j=1}^k=\frac{d^4 l_1}{(2\pi)^4}\cdots\frac{d^4 l_k}{(2\pi)^4}\ .
\eee
The integration variables $l_j$ represent the loop momenta carried by the loop partons of the external mesons. We also assume that the $P_2$ meson is the ejected meson carrying large momentum and the $P_1$ meson is the one picking up the spectator quark of the $B$ meson. The kernels $T^{I,II}$ correspond to the type-I and type-II color topologies defined in QCD factorization and the kernels $T^{I,II}_{\alpha}$ result from the one gluon insertion of the type-I,II topologies. The $\Phi_{P_i}$ denote the two parton distribution amplitudes and the $\Phi_{P_i}^{\alpha}$ represent three parton distributions. The first two terms generalize the type-I and type-II color topologies in the QCD factorization formula by including noncollinear loop momenta. This generalization is necessary to obtain higher twist contributions. For the third and fourth terms, the situations are a little complicate. The inserted gluon could be soft, collinear, or hard. If the inserted gluon is soft, there are cancellations between diagrams at next-to-leading twist. If the inserted gluon is hard, there needs one additional quark-gluon interaction vertex to make the contributions become radiative corrections. These two situations render the third and fourth terms to the first and second terms, respectively. The remaining configuration for the third and forth terms is that the gluon is collinear to the ejected light meson. The validity of the factorization should be proven order by order in power corrections and in radiative corrections. In order to pick up not only the leading contributions but also subleading power corrections for a specific loop correction, it is necessary to have a systematic method. For this purpose, we will generalize the collinear expansion method \cite{Yeh:2002up,Yeh:2002rd,Yeh:2001ys,Yeh:2001ta,Yeh:2001gu} to include the nonleptonic decay processes of the $B$ meson.
We start at the tree level to investigate how the higher twist contributions can be derived. At tree level, we only need to investigate how the type-I amplitudes can be expanded. The type-II and annihilation diagrams are of at least $O(\alpha_s)$ and will not be considered in this paper. By the power counting method employed in \cite{Beneke:2000ry}, it is not difficult to find out the leading configurations are composed of collinear partons. The scattering kernel in the first term of Eq.~(\ref{QFF1}) can be expanded into a Taylor series with respect to the leading configurations. The expansion series of the scattering kernel are then substituted into the convolution integration with the meson distribution amplitude over the loop momentum. Since each term in the expanded series of the hard scattering kennel is function of the momentum fractions and independent of the loop momenta, the loop momentum integrations can be attributed to the meson distribution amplitudes. Note that the full range of the momentum integrations of the loop partons is taken into account. This is different from the BBNS approach in which only partial range of momentum integrations is considered. 

The resulting expression is now an expanded series of the amplitude. The expanded amplitudes are ordered by the expanded scattering kernel. The higher order terms in the expanded scattering kernel have more loop momentum derivatives than the lower order terms. Therefore, the higher order terms are more power suppressed in $1/m_b$ than the leading term. The scale factor $m_b$ is due to our choice of the largest scale of the process. Since in the $B$ meson decays, the $b$ quark mass $m_b$ is a convenient scale. However, the leading term in the expanded amplitude still contains leading and subleading twist contributions. The transverse and off-shell components of the loop parton momenta can contribute at subleading twist. The second term of the expanded hard scattering kernel has an additional derivative with respect to the loop momenta than the leading term. Because of gauge invariance, the one gluon insertion amplitude is indispensable. The proof of QCD factorization for the one loop radiative corrections and the collinear expansion can be incorporated. Especially, for the collinear radiative corrections to the leading twist diagrams, the loop momentum flowing through the loop partons of the ejected meson might modify the loop parton momenta away from being collinear. There are possibilities that the loop parton is in the off-shell state. The separation of the short distance and long distance contributions for the diagrams with radiative gluons in the collinear region can be accomplished by means of collinear expansion. This is similar to the tree level expansion introduced above. Although the calculation of one loop correction can be systematically performed, but there would involve more complications than the tree level cases. In order to emphasize the meanings of collinear expansion, we shall not consider the radiative corrections and leave this subject to the other works. 

With the one gluon insertion amplitudes, we can obtain a complete set of twist-3 contributions. It is important to retain the gauge invariance of the twist-3 contributions. The traditional approach of dealing with the $(S+P)(S-P)$ matrix element invokes the free equations of motion for the quarks to transform that matrix element to the $(V-A)(V-A)$ matrix element. However, due to gauge invariance, there exists the gauge phase factor in the matrix element. Therefore, the application of free equation of motion is not gauge invariant. Due to the presence of gluon fields, it is better to have a systematic method to employ the full equation of motion. We will show this is possible to be achieved. 

To be specific, we will consider the $\bar{B}_d^0\to\pi^+\pi^-$ decay as an example for discussion. At tree level, the twist-3 contributions can come from the light quark mass corrections, the two parton pseudoscalar distribution amplitude and one gluon insertion of the matrix element of the $(S+P)(S-P)$ operators. The one gluon insertions of the matrix element of the octet $(V-A)(V-A)$ operators result in twist-4 contributions. The organization of this paper is as follows. In the next two sections, the twist-3 and twist-4 contributions are derived by using the light-cone and covariant gauges for the gluon fields. This shows the gauge invariance of these higher twist corrections. The last section discusses the twist-3 and twist-4 three parton distributions amplitudes. The conclusions are given in this section.

\section{twist-3 contributions}
The matrix elements of $(S+P)(S-P)$ operators lead to power suppressed but chirally enhanced corrections. For example, in the process $\bar{B}\to\pi^+\pi^-$, the matrix element $\langle Q_6\rangle$ of operator $Q_6$ is expressed as
\bee
\langle Q_6\rangle=-2\langle \pi^{-}|\bar{d}(0)(1+\gamma_5) u(0)|0\rangle
\langle \pi^{+}|\bar{u}(0)(1-\gamma_5) u(0)|\bar{B}\rangle\ ,
\eee
where we have omitted the gauge phase factor 
\[\hat{P}\int dy_{\alpha} A^{\alpha}(y)\]
in the matrix elements. We will take this convention through the whole text. To relate the $(S+P)(S-P)$ matrix element to the $(V-A)(V-A)$ matrix element, the following approach is always taken. The first step is to employ the free equation of motion for the quark currents
\bee
\partial_{\mu}\bar{d}(x)\gamma^{\mu}\gamma_5u(x)&=&i(m_u+ m_d)\bar{d}(x)\gamma_5 u(x)\ ,\\
\partial_{\mu}\bar{u}(x)\gamma^{\mu}b(x)&=&-i(m_b- m_u)\bar{u}(x)b(x)\ .
\eee
The following step is to assume that the axial vector current is identical to the interpolating field of $\pi^-$
\bee
A_{\mu}^{\pi^-}(x)=\bar{d}(x)\gamma^{\mu}\gamma_5 u(x)\ . 
\eee
With the above results, one can derive
\bee
\langle \pi^{-}|\bar{d}(0)\gamma_5u(0)|0\rangle
=\frac{i\partial^{\mu}}{m_u+m_d}\langle \pi^{+}|\bar{u}(0)\gamma_{\mu}(1-\gamma_5) u(0)|0\rangle
\eee
Similarly, the matrix element $\langle \pi^{+}|\bar{u}(0)(1-\gamma_5) u(0)|\bar{B}\rangle$ can be transformed as
\bee
\langle \pi^{+}|\bar{u}(0)(1-\gamma_5) u(0)|\bar{B}\rangle
=\frac{i\partial_{\mu}}{m_b-m_u}\langle \pi^{+}|\bar{u}(0)\gamma^{\mu}(1-\gamma_5) u(0)|\bar{B}\rangle\ .
\eee
Combining the results, one can obtain
\bee
\langle Q_6\rangle=\frac{2m_\pi^2}{(m_b-m_u)(m_u+m_d)}\langle \pi^{-}|\bar{d}(0)\gamma^{\mu}(1-\gamma_5)u(0)|0\rangle
\langle \pi^{+}|\bar{u}(0)\gamma_{\mu}(1-\gamma_5) u(0)|\bar{B}\rangle\ .
\eee
Although the derivation is simple, but many points require distinguished. First is the application of the free equations of motion. Since the current in the matrix element contains the gauge phase factor, there would have contributions of the same twist order from the gluon field. However, it is difficult to derive such a term using the above method. Second is the assumption of the identification of the free quark current with the interpolating hadronic current for the specific external hadron. This can only be justified for single external meson. It is the problem when one applies the assumption over the transition matrix element, such as the $\bar{B}\to\pi^+$ matrix element. The third question has been noticed that the above method is difficult to extend to the case $\langle \pi\pi|\bar{d}d|0\rangle$. There leads to a divergent result $1/(m_d-m_d)$. In summary, it seems that we need an more universal method to calculate the matrix element involving $(S+P)(S-P)$ operators. In the following, we will demonstrate an approach based on perturbative QCD (pQCD). In this method, we can have a more fundamental basis to perform calculations and can have wider applications. Not only the matrix element of $(S+P)(S-P)$ operators can be dealt with, but also the twist-4 and even higher twist contributions can be derived.

The twist-3 contributions are power suppressed like $m_q/m_b$ with $m_q$ the light quark mass and $m_b$ the $b$ quark mass, or $\Lambda_{QCD}/m_b$. We now evaluate these contributions in the following.

\subsection{light quark mass contributions}
We now derive the twist-3 contributions due to nonvanishing light quark masses. The matrix element of $Q_6$ is written as 
\bee
\langle Q_6\rangle=-2\langle\pi^-|\bar{d}(0)(1+\gamma_5)u(0)|0\rangle
\langle\pi^+|\bar{u}(0)(1-\gamma_5)b(0)|\bar{B}_d^0\rangle\ .
\eee
By inserting the quark mass terms for $\langle\pi^-|\bar{d}(0)(1+\gamma_5)u(0)|0\rangle$, we can have 
\bee
\int d^4z \langle\pi^-|\bar{d}(z)(-im_d)d(z)\bar{d}(0)(1+\gamma_5)u(0)|0\rangle\ .
\eee
With Wick contraction for $d(z)\bar{d}(0)$, it is easy to derive
\bee
\int d^4z \int\frac{d^4 l_d}{(2\pi)^4}e^{il_d\cdot z}\langle\pi^-|\bar{d}(z)(-im_d)\frac{i\s{l}_d}{l_d^2+i\epsilon}(1+\gamma_5)u(0)|0\rangle\ .
\eee
The $d$ quark momentum $l_d$ can be parameterized as
\bee
l_d^{\mu}=\hat{l}_d^{\mu}+\frac{l^2_d}{2x}n^{\mu}\ ,
\eee
where we have defined the momentum $\hat{l}_d$ as on-shell $\hat{l}_d^2=0$ and a light-like vector $n$ with property $n\cdot q=1$. This is convenient for extraction of the higher twist contributions. We have taken the convention that the internal quark propagator is independent of the light quark mass. This is consistent with that convention with nonvanishing light quark mass in the internal propagator. The separation of $d$ quark propagator into the long distance part $i\hat{\s{l}}_d/(l_d^2+i\epsilon)$ and the short distance part $i\s{n}/2x$ can separate the matrix into the twist-3 term from the higher twist ones. We also take the $q$ to mean the momentum of the ejected meson. The $x$ denotes the momentum fraction for the $d$ quark. The term containing the short distance part is of twist-3 and the term with the long distance part is of higher twist than three. Taking into account of the contributions from the $\bar{u}$ side, gives contributions of the form
\bee
\int^1_0 dx \phi_{\pi}(x)\frac{(m_d(1-x)+m_ux)}{2x(1-x)}\ ,
\eee
where we have employed the definition for the twist-2 pion distribution amplitude
\bee
\langle\pi^-|\bar{d}(0)\s{n}\gamma_5 u(0)|0\rangle=-if_{\pi}\int dx\phi_{\pi}(x)\ .
\eee
Integration out of the momentum fraction $x$ by taking the symmetric pion distribution amplitude $\phi_{\pi}(x)=6x(1-x)$ gives the result
\bee
-\frac{3}{2}(m_u+m_d)if_{\pi}.
\eee
Similar consideration can be applied for the matrix element $\langle\pi^+|\bar{u}(0)(1-\gamma_5)b(0)|\bar{B}_d^0\rangle$. The insertion of the $b$ quark and $u$ quark mass terms can lead to contributions as
\bee
&&\langle\pi^+|\bar{u}(0)(1-\gamma_5)b(0)|\bar{B}_d^0\rangle\nn
&&=\frac{1}{2m_b}\langle\pi^+|\bar{u}(0)\s{q}
(1+\gamma_5)b(0)|\bar{B}_d^0\rangle
+\frac{m_u}{2m_b^2}\langle\pi^+|\bar{u}(0)\s{q}(1-\gamma_5)b(0)|\bar{B}_d^0\rangle\ .
\eee
Combining the result of the $\pi^-$ matrix element and ignoring the terms with $m_u/m_b^2$, we can derive 
\bee
\langle Q_6\rangle_{m_q} =\frac{3(m_u+m_d)}{2m_b}\langle\pi^-|\bar{d}(z)\s{n}(1-\gamma_5)u(0)|0\rangle
\langle\pi^+|\bar{u}(0)\s{q}(1-\gamma_5)b(0)|\bar{B}_d^0\rangle
\eee
where we have modified the expression to match with that matrix element of operator $Q_1$ . The current quark mass $m_u+m_d$ at scale $\mu=m_b$ is about few MeVs. It implies that the light quark mass effect is about $0.1\%$ of the leading twist contributions. The above approach can also be applied for $(V-A)(V\pm A)$ operators. For example, the light quark mass effects from the operator $Q_1$ can be written as
\bee
\langle Q_1\rangle_{m_q} &=& if_{\pi}\frac{m_u m_d}{2m_b^2}\int dx\frac{\phi_{\pi}(x)}{x(1-x)}
\langle\pi^+|\bar{u}(0)\s{p}^{\prime}(1-\gamma_5)b(0)|\bar{B}_d^0\rangle\ ,
\eee
where $p^\prime=P_b-q$ with $P_b$ the $b$ quark momentum. The equation of motion $\bar{u}(0)\s{p}^{\prime}=0$ leads to the vanishing result $\langle Q_1\rangle_{m_q}=0$.   

\subsection{two parton contributions}
The incorrect spin projection for the ejected light meson can give twist-3 contributions. There are two parton twist-3 contributions from the pseudoscalar and pseudotensor distribution amplitudes \footnote{The author thanks H-n. Li for a discussion on this point}. The contributions, for operator $Q_6$, can be expressed as
\bee
\langle Q_6\rangle=2if_\pi\frac{m_\pi^2}{m_u+m_d}\int dx \phi_P(x)
\langle\pi^+|\bar{u}(0)(1-\gamma_5)b(0)|\bar{B}_d^0\rangle\ ,
\eee 
where we have employed the pseudoscalar distribution amplitude
\bee
\langle\pi^-|\bar{d}(0)\gamma_5 u(0)|0\rangle=-if_{\pi}\frac{m_\pi^2}{m_u+m_d}\int dx\phi_{P}(x)\ .
\eee
After inserting the $b$ quark mass term, the result then reads as
\bee
\langle Q_6\rangle=if_\pi\frac{m_\pi^2}{m_b(m_u+m_d)}\int dx \phi_P(x)
\langle\pi^+|\bar{u}(0)\s{q}(1-\gamma_5)b(0)|\bar{B}_d^0\rangle\ ,
\eee
and can be recast as
\bee
\langle Q_6\rangle=\frac{m_\pi^2}{m_b(m_u+m_d)}\langle Q_1\rangle_f\ ,
\eee
where we have used the notation
\bee
\langle Q_1\rangle_f=\langle \pi^-|\bar{d}(0)\gamma^{\mu}(1-\gamma_5)u(0)|0\rangle
\langle\pi^+|\bar{u}(0)\gamma_{\mu}(1-\gamma_5)b(0)|\bar{B}_d^0\rangle\ .
\eee

The two parton twist-3 contributions have only one half of the result derived by using free equations of motion. This difference is because, in our approach, only the off-shell components of the $b$ quark propagator can contribute at this order and on-shell parts are of higher twist order. The free equation method takes into account the full free $b$ quark propagator.
 
\subsection{three parton contributions}
The short distance quark gluon interactions can also give twist-3 contributions. For operator $Q_6$, the contributions  can be written as
\bee\label{twist3-1}
&&\int d^4y \int d^4z \int\frac{d^4 l}{(2\pi)^4}\frac{d^4 k}{(2\pi)^4}e^{il\cdot z}e^{ik\cdot(y- z)}
\langle\pi^-|\bar{d}(z)[(-ig\s{A}(y))
\frac{i(\s{l}+\s{k})}{(l+k)^2+i\epsilon}
(1+\gamma_5)\nn
&&+\frac{-i(\s{q}-\s{l}+\s{k})}{(q-l+k)^2+i\epsilon}
(-ig\s{A}(y))(1+\gamma_5)]u(0)|0\rangle\ ,
\eee
where we have omitted the following term in the expression
\bee
\langle\pi^+|\bar{u}(0)(1-\gamma_5)b(0)|\bar{B}\rangle\ .
\eee
The $l$ and $k$ denote the momenta carried by the $d$ quark and gluon in Fig.~1(a) and (b). Since the derivation will depend on the gauge for the gluon field, we shall investigate the cases of the light-cone gauge $n\cdot A(y)=0$ and the covariant gauge $\partial\cdot A(y)=0$. The gluon field $A^{\alpha}(y)$ represents for $A^{\alpha,a}(y)T^a$. For the light-cone gauge, we may recast Eq.~(\ref{twist3-1}) as 
\bee\label{twist3-2}
\int\frac{d^4 l}{(2\pi)^4}\frac{d^4 k}{(2\pi)^4}\Tr[H_{\alpha}(k,l)w^{\alpha}_{\alpha^\prime}\Phi^{\alpha^\prime}(k,l)]
\eee
where the kernel $H_{\alpha}(k,l)$ is defined as
\bee
H_{\alpha}(k,l)=[i\gamma_{\alpha}\frac{i(\s{l}+\s{k})}{(l+k)^2+i\epsilon}
+\frac{-i(\s{q}-\s{l}+\s{k})}{(q-l+k)^2+i\epsilon}i\gamma_{\alpha}](1+\gamma_5)
\eee
and the soft function $\Phi^{\alpha^\prime}(k,l)$
\bee
\Phi^{\alpha^\prime}(k,l)=\int d^4y \int d^4z 
e^{il\cdot z}e^{ik\cdot(y- z)}
\langle\pi^-|\bar{d}(z)(-gA^{\alpha^\prime}(y))u(0)|0\rangle\ . 
\eee
The tensor $w^{\alpha}_{\alpha\prime}=g^{\alpha}_{\alpha\prime}-q^{\alpha}n_{\alpha\prime}$ has been introduced for convenience. Under the assumption that the ejected $\pi^-$ meson is highly energetic such that the naive factorization can be applied, the leading contributions can come from the configuration where the momentum $l$ is collinear $l\approx x q$. This allows us to expand the kernel $H_{\alpha}(k,l)$ with respect to $\hat{l}=x q$
\bee
H_{\alpha}(k,l)=H_{\alpha}(k,\hat{l})
+\frac{\partial H_{\alpha}(k,l)}{\partial l^\rho}|_{l=\hat{l}}(l-\hat{l})+\cdots \ .
\eee
By substituting the first term in the above expansion into the convolution integral, we can have
\bee
\int\frac{d^4 l}{(2\pi)^4}\frac{d^4 k}{(2\pi)^4}\Tr[H_{\alpha}(k,\hat{l})w^{\alpha}_{\alpha^\prime}\Phi^{\alpha^\prime}(k,l)]+\cdots
\eee
where the dots mean the higher twist terms. Since $H_{\alpha}(k,\hat{l})$ is independent of $l$, we can use the identity
\bee
\int_0^1 dx\delta(x-l\cdot n)=1
\eee
to simplify the above equation into
\bee\label{twist3-3}
\int dx\frac{d^4 k}{(2\pi)^4}\Tr[H_{\alpha}(k,x)w^{\alpha}_{\alpha^\prime}\Phi^{\alpha^\prime}(k,x)]
\eee
with
\bee
\Phi^{\alpha^\prime}(k,x)=
\int\frac{d^4 l}{(2\pi)^4}\delta(x-l\cdot n)\int d^4y \int d^4z 
e^{il\cdot z}e^{ik\cdot(y- z)}
\langle\pi^-|\bar{d}(z)(-gA^{\alpha^\prime}(y))u(0)|0\rangle\ .
\eee 
Note that the full momentum range of $l$ is integrated. This is different from the treatment of the loop momentum integration in the BBNS approach, in which only the longitudinal and transverse parts of $l$ are integrated. The gluon could be soft, collinear or hard. If the gluon is soft $k\propto (\lambda,\lambda,\lambda)$ with $\lambda\sim\Lambda_{QCD}$, the eikonal approximation can be applied such that the kernel $H_{\alpha}(k,x)$ is vanishing. If the gluon is hard $k^{\alpha}=(k^+,k^-,k_\perp)\approx(m_b,m_b,m_b)$, the gluon is short lived and one additional quark-gluon interaction vertex should be put in. There are reducible and irreducible diagrams. The reducible diagrams involve the radiative gluons inside the ejected meson, and the irreducible diagrams have the radiative gluons connected between the loop partons of ejected meson  and the other partons from the other external mesons. The detailed analysis of these radiative gluons has been made in \cite{Beneke:2000ry}. We will not and don't need to repeat the same things. The radiative corrections being soft or collinear would require extending the analysis made below. We leave this to the other work. The remaining situation is that the gluon is collinear. In this case, the gluon is long lived with the other partons to form one possible Fock state of the ejected meson. This possibility can be described by a three parton pion distribution amplitude. Since the gluon is collinear, we can assign its momentum $k$ as $\hat{k}=(x^\prime-x)q$ with momentum fraction variables $x$ and $x^{\prime}$. This means that we can expand the kernel $H_{\alpha}(k,x)$ with respect to $\hat{k}=(x^\prime-x)q$
as
\bee
H_{\alpha}(k,x)=H_{\alpha}(\hat{k},x)
+\frac{\partial H_{\alpha}(k,x)}{\partial k^\rho}|_{k=\hat{k}}(k-\hat{k})+\cdots\ .
\eee
Repeating the previous considerations, we can introduce the integration over $x^\prime$ to obtain
\bee
\int dx \int dx^\prime
\Tr[H_{\alpha}(x^\prime,x)w^{\alpha}_{\alpha^\prime}
\Phi^{\alpha^\prime}(x^\prime,x)]
\eee
with
\bee
\Phi^{\alpha^\prime}(x^\prime,x)=
&&\int\frac{d^4 l}{(2\pi)^4}\delta(x-l\cdot n)
\int\frac{d^4 k}{(2\pi)^4}\delta(x^\prime-x-k\cdot n)
\int d^4y \int d^4z \nn
&&\times e^{il\cdot z}e^{ik\cdot(y- z)}
\langle\pi^-|\bar{d}(z)(-gA^{\alpha^\prime}(y))u(0)|0\rangle\ .
\eee
However, there are infrared divergences as $x^\prime\to 0$ in the denominator of internal quark propagators of $H_{\alpha}(\hat{k},x)$ as e.g.
\bee
\frac{ix^\prime \s{q}}{(x^\prime q)^2+i\epsilon}\ .
\eee
We need to regularize this divergence by replacing the quark propagators with corresponding special propagators as
\bee
\frac{ix^\prime \s{q}}{(x^\prime q)^2+i\epsilon}\to
\frac{i\s{n}}{2x+i\epsilon}\frac{x^\prime-x}{x^\prime-x+i\epsilon}\ .
\eee  
For the light-cone gauge $n\cdot A(y)=0$, gauge invariance requires transforming the gluon fields $A^{\alpha}(y)$ into the field strength $G^{+\alpha}(y)$ by the following replacement
\bee
A^{\alpha}(y)\to\frac{-i n_{\beta}G^{\beta\alpha}(y)}{(x-x^\prime)}\ .
\eee
Combining every thing, we can obtain
\bee
\int dx \int dx^\prime \frac{G_{\mu}^{\beta}(x^\prime,x)n^{\mu}n_{\beta}}{2(x^\prime-x)x(1-x^\prime)}.
\eee
The distribution amplitude $G_{\mu}^{\beta}(x^\prime,x)$ is defined as
\bee
G_{\mu}^{\beta}(x^\prime,x)&=&\int\frac{d^4 l}{(2\pi)^4}\delta(x-l\cdot n)
\int\frac{d^4 k}{(2\pi)^4}\delta(x^\prime-x-k\cdot n)
\int d^4y \int d^4z \nn
&&\times e^{il\cdot z}e^{ik\cdot(y- z)}
\langle\pi^-|\bar{d}(z)\sigma_{\mu\alpha}w^{\alpha}_{\alpha^\prime}gG^{\beta\alpha^\prime}(y)u(0)|0\rangle\ .
\eee
Refer to the definition for the twist-3 matrix element \cite{Ball:1998je}
\bee
&&\langle\pi^-|\bar{d}(z)\sigma_{\mu\nu}\gamma_5 gG_{\alpha\beta}(y)u(0)|0\rangle\nn
&=&-i\frac{f_\pi m_\pi^2}{m_u+m_d}
(q_{\alpha}q_\mu d_{\nu\beta}-q_{\alpha}q_{\nu}d_{\mu\beta}-q_{\beta}q_{\mu}d_{\nu\alpha}+q_{\beta}q_{\nu}d_{\alpha\mu})
T(z,y)+\cdots\ ,
\eee
where we have defined
\bee
T(z,y)=\int_0^1 dx \int_0^x dx^\prime e^{-ixq\cdot z}e^{-i(x-x^\prime)q\cdot(y-z)}T(x^\prime,x)\ .
\eee
It is not difficult to derive
\bee
&&\int dx \int dx^\prime \frac{G_{\mu}^{\beta}(x^\prime,x)n^{\mu}n_{\beta}}
{(x^\prime-x)x(1-x^\prime)}\nn
&=&-\frac{if_\pi m_\pi^2}{m_u+m_d}\int dx \int dx^\prime \frac{T(x^\prime,x)}{(x-x^\prime)x(1-x^\prime)}\ .
\eee
The insertion of the $b$ quark mass term for $\langle\pi^+|\bar{u}(0)(1-\gamma_5)b(0)|\bar{B}\rangle$ leads to 
\bee
\langle\pi^+|\bar{u}(0)\frac{\s{q}}{2m_b}(1-\gamma_5)b(0)|\bar{B}\rangle\ .
\eee
The one gluon part of $\langle Q_6\rangle$ is read as
\bee
\langle Q_6\rangle_{1-gluon}=\frac{A m_\pi^2}{m_b(m_u+m_d)}\langle \pi^-|\bar{d}(0)\gamma^{\mu}(1-\gamma_5)u(0)|0\rangle
\langle\pi^+|\bar{u}(0)\gamma_{\mu}(1-\gamma_5)b(0)|\bar{B}_d^0\rangle
\eee
with
\bee
A=\int dx \int dx^\prime \frac{T(x^\prime,x)}{(x-x^\prime)x(1-x)}\ .
\eee
The total twist-3 contribution from operator $Q_6$ is then equal to
\bee
\langle Q_6\rangle^{t=3}=(\frac{3(m_u+m_d)}{2m_b}+\frac{(1+A) m_\pi^2}{m_b(m_u+m_d)})\langle Q_1\rangle_f\ .
\eee

In contrast to the traditional derivation of the twist-3 contribution,
our derivation exhibits the QCD dynamics more transparent. In fact, the complete twist-3 contributions can be classified into two types: one is in terms of the gluon field strength and the other with covariant derivative. Which type of twist-3 corrections can contribute is dependent on the hard scattering kernel. If the hard scattering kernel involves the momenta of the loop partons composed of the ejected meson, the covariant derivative type will contribute. On the other hand, the remaining situations then require the gluon field strength type corrections. There are twist-3 corrections from the two parton twist-3 distribution amplitudes (the pseudoscalar and pseudotensor distribution amplitudes). The two parton twist-3 corrections and the covariant derivative type corrections are not independent and can be combined to form more compact expressions. Therefore, we identify the two parton twist-3 corrections to the covariant type corrections. 

We now derive the twist-3 one gluon contribution by employing the covariant gauge $\partial\cdot A=0$. For this gauge, the dominant contribution is from the longitudinal component of gluon field. We can write $A^{\alpha}(y)$ into the decomposition $A^{\alpha}(y)=w^{\alpha}_{\alpha^\prime}A^{\alpha^\prime}(y)
+n\cdot A(y)q^{\alpha}$. The transverse part $w^{\alpha}_{\alpha^\prime}A^{\alpha^\prime}(y)$ will contribute to higher twist order. The longitudinal part $n\cdot A(y)q^{\alpha}(y)$ can anticipate at tree level. To transform the gluon field into the field strength, we need one transversal momentum $k_\perp$ from the kernel function $H_{\alpha}(k,x)$ in Eq.~(\ref{twist3-3}). To achieve this purpose, we can expand $H_{\alpha}(k,x)$ with respect to $\hat{k}=(x^\prime-x)q$
\bee
H_{\alpha}(k,x)=H_{\alpha}(\hat{k},x)+\frac{\partial H_{\alpha}(k,x)}{\partial k^{\rho}}|_{k=\hat{k}}(k-\hat{k})^{\rho}+O(k^2)
\eee 
where the first term will vanish as it contracts with $q^{\alpha}$, the second term will be discussed below and the higher order terms are of higher twist. Since $\hat{k}$ is proportional to $q$, $k-\hat{k}$ can be proportional to $n$ or transverse. That is we can write it as $(k-\hat{k})^{\rho}=d^{\rho}_{\perp\rho^\prime}(k-\hat{k})^{\rho^\prime}+q\cdot k n^{\rho}$. We now argue that only the transverse part $k^{\rho}_{\perp}=d^{\rho}_{\perp\rho^\prime}(k-\hat{k})^{\rho^\prime}$ can contribute at this twist. The tensor $d^{\alpha\mu}_{\perp}=g^{\alpha\mu}-q^{\alpha}n^{\mu}-q^{\mu}n^{\alpha}$ is defined for convenience. First note that the kernel $\partial H_{\alpha}/\partial k^{\rho}$ can have terms proportional to $g_{\alpha\rho}$ and $\sigma_{\alpha\rho}$. For the $n$ component $k^{-}=q\cdot k n^{\rho}$, the $g_{\alpha\rho}$ terms will be absorbed into the gauge phase factor and the $\sigma_{\alpha\rho}$ terms will lead to vanishing result, since $\sigma_{\alpha\rho} q^{\alpha}n^{\rho}=0$. 

We now consider the contribution from the remaining transverse part $k_\perp$. Only the $\sigma_{\alpha\rho}$ terms can contribute, since $\rho=\perp$ but $\alpha=+$. Substituting the $k^{\rho}_{\perp}$ into the function $\Phi^{\alpha}(k,l)$ and employing the replacement $k^{\rho}_{\perp}A^{\alpha}(y)\to -i G^{\rho\alpha}(y)$ can result in the following expression
\bee\label{twist-3Cov}
\langle Q_6\rangle^{t=3}_{1-gluon}&=&-2\int dx \int dx^\prime 
\Tr[\frac{\partial H_{\alpha}(x,x^\prime)}{\partial k^{\rho}}G^{\rho\alpha}(x,x^\prime)]\nn
&&\times\langle\pi^+|\bar{u}(0)(1-\gamma_5)b(0)|\bar{B}_d^0\rangle\ ,
\eee
where we have defined the hard kernel
\bee
\frac{\partial H_{\alpha}(x^\prime,x)}{\partial k^{\rho}}=\frac{i\sigma_{\alpha\rho}}{2(x-x^\prime)x^\prime(1-x^\prime)q^2}
\eee
and the soft function
\bee
G^{\rho\alpha}(x^\prime,x)&=&\int\frac{d^4 l}{(2\pi)^4}\delta(x-l\cdot n)
\int\frac{d^4 k}{(2\pi)^4}\delta(x^\prime-x-k\cdot n)
\int d^4y \int d^4z \nn
&&\times e^{il\cdot z}e^{ik\cdot(y- z)}
\langle\pi^-|\bar{d}(z)igG^{\rho\alpha}(y)(1+\gamma_5)u(0)|0\rangle\ .
\eee
The contraction of $\sigma_{\alpha\rho}$ with $G^{\rho\alpha}(x,x^\prime)$ gives
\bee
\Tr[i\sigma_{\alpha\rho}G^{\rho\alpha}(x^\prime,x)]=\frac{-2if_{\pi}m_\pi^2 q^2}{(m_u+m_d)}T(x^\prime,x)\ .
\eee
Note that the pole $q^2$ in the hard kernel ${\partial H_{\alpha}(x^\prime,x)}/{\partial k^{\rho}}$ is cancelled by the $q^2$ factor in the contraction $\Tr[i\sigma_{\alpha\rho}G^{\rho\alpha}(x^\prime,x)]$.
It is easy to see that Eq.~(\ref{twist-3Cov}) is equal to the result derived from the light-cone gauge. This explicitly shows the gauge invariance of $\langle Q_6\rangle^{t=3}$ from the one gluon contribution.
The situations of Fig.~1(c) and (d) result in higher twist contributions for octet operator $Q_6^{(8)}$, since the loop partons can not form a correct spin for ejected meson at twist-3. This can be shown by calculations similar to the above.

\section{twist-4 contributions}
The tree level amplitude may contain twist-4 contributions from the one gluon insertion for the matrix element of $(V-A)(V-A)$ octet operator. For example, the matrix element of octet operator $Q_1^{(8)}$ vanishes under naive  factorization
\bee
\langle Q_1^{(8)}\rangle_f=0\ .
\eee
But this statement may not be valid at higher twist. Considering the one gluon insertion topologies as Fig.~1(c) and (d), we can obtain
\bee
\langle Q_1^{(8)}\rangle_{1-gluon}&=&2\int d^4y \langle\pi^-|\bar{d}(0)\gamma^{\mu}(1-\gamma_5)T^{a}u(0)|0\rangle\nn
&&\times[\langle\pi^+|\bar{u}(0)\gamma_{\mu}(1-\gamma_5)T^{a}b(0)\bar{b}(y)
(-ig\s{A}^b(y)T^b(y))b(y)|\bar{B}\rangle\nn
&&+\langle\pi^+|\bar{u}(y)(-ig\s{A}^b(y)T^b)u(y)\bar{u}(0)\gamma_{\mu}(1-\gamma_5)T^{a}b(0)|\bar{B}\rangle\ ,
\eee
where we have shown the color matrix explicitly. After a little algebra, it is not difficult to derive
\bee
\langle Q_2^{(8)}\rangle_{1-gluon}&=&2\times\frac{1}{2}\int \frac{d^4 k}{(2\pi)^4} T_{\alpha\mu}(k)G^{\alpha\mu}(k)\ ,
\eee
where the factor $1/2$ is from the color matrix. The kernel function $T_{\alpha\mu}(k)$ is expressed as
\bee
T_{\alpha\mu}(k)&=&\langle\pi^+|\bar{u}(0)[\gamma_{\mu}(1-\gamma_5)\frac{i(\s{P}_b-\s{k}+m_b)}{(P_b-k)^2-m_b^2+i\epsilon}(i\gamma_{\alpha})\nn
&&+(i\gamma_{\alpha})\frac{i(\s{P}_b-\s{q}+\s{k})}{(P_b-q+k)^2+i\epsilon}\gamma_{\mu}(1-\gamma_5)b(0)
|\bar{B}_d^0\rangle\ ,
\eee 
and the soft function is defined
\bee
G^{\alpha\mu}(k)=\int\frac{d^4 l}{(2\pi)^4}\int d^4 y
\int d^4 z e^{il\cdot z}e^{ik\cdot( y-z)}
\langle\pi^-|\bar{d}(z)\frac{in_{\beta}gG^{\beta\alpha}(y)}
{k\cdot n}\gamma^{\mu}(1-\gamma_5)u(0)|0\rangle\ .
\eee
We have chosen the light-cone gauge $n\cdot A=0$. The gluon field $A^{\alpha}(y)$ has been transformed into the field tensor $G^{\beta\alpha}(y)$. The gluon could be soft, collinear, or hard. For soft gluon, the kernel $T_{\alpha\mu}$ vanishes by using the eikonal approximation for the internal propagators. If the gluon is hard, there need additional quark gluon interaction vertices. This is just similar to the previous twist-3 situations. The gluon now can only be collinear. Since $\alpha$ is equal to $\perp$, the kernel $T_{\alpha\mu}$ can be simplified as
\bee
T_{\alpha\mu}(k)&=&\frac{1}{m_b^2}\langle\pi^+|\bar{u}(0)[\gamma_{\mu}\s{q}\gamma_{\alpha}+\gamma_{\alpha}\s{q}\gamma_{\mu}](1-\gamma_5)b(0)|\bar{B}_d^0\rangle\ .
\eee
It is easy to observe that the kernel $T_{\alpha\mu}(k)$ is independent of $k$ and can be factorized from the loop momentum integrals. The integration of $G^{\alpha\mu}(k)$ over $k$ gives the result
\bee
\int \frac{d^4k}{(2\pi)^4} \frac{G^{\alpha\mu}(k)}{k\cdot n}=\frac{1}{2}if_{\pi}m_\pi^2 d^{\alpha\mu}_{\perp}\int dx \int dx^\prime \frac{A_{\perp}(x,x^\prime)}{(x-x^\prime)}\ ,
\eee
where the tensor $d^{\alpha\mu}_{\perp}$ has been defined perviously. The twist-4 distribution amplitude $A_{\perp}(x^\prime,x)$ has been introduced.

By contracting the tensor $d^{\alpha\mu}_{\perp}$ with the kernel $T_{\alpha\mu}$, we can obtain
\bee
d^{\alpha\mu}_{\perp}T_{\alpha\mu}(q)=-\frac{4}{m_b^2}\langle\pi^+|\bar{u}(0)\s{q}(1-\gamma_5)b(0)|\bar{B}_d^0\rangle\ .
\eee
This implies that the leading contribution from $Q_1^{(8)}$ is equal to
\bee
\langle Q_1^{(8)}\rangle^{t=4}=-\frac{4m_\pi^2}{m_b^2}
\langle Q_1\rangle_f\int dx \int dx^\prime\frac{A_{\perp}(x,x^\prime)}{(x-x^\prime)}\  ,
\eee
which is indeed a twist-4 effect. 

We now calculate the twist-4 contribution for $\langle Q_1^{(8)}\rangle^{t=4}$ by employing the covariant gauge. The first step is to derive the relevant hard kernel. The expansion of $T_{\alpha\mu}(k)$ with respect to $k=\hat{k}$ leads to
\bee
T_{\alpha\mu}(k)=T_{\alpha\mu}(\hat{k})+\frac{\partial T_{\alpha\mu}(k)}{\partial k^{\rho}}(k-\hat{k})^{\rho}+\cdots\ .
\eee
The first term vanishes as it contracts with $q^{\alpha}$. In covariant gauge, the gauge field can be expanded as
$A^{\alpha}(y)=w^{\alpha}_{\alpha^\prime}A^{\alpha^\prime}(y)
+n\cdot A(y)q^{\alpha}$. The dominant contributions are from the longitudinal part $A^{+}(y)=n\cdot A(y)q^{\alpha}$. By contracting $q^{\alpha}$ with the second term, we can obtain
\bee
q^{\alpha}\frac{\partial T_{\alpha\mu}(k)}{\partial k^{\rho}}=\frac{-2d_{\perp\mu\rho}}{m_b^2(x-x^\prime)}\langle \pi^+|\bar{u}(0)\s{q}(1-\gamma_5)b(0)|\bar{B}\rangle\ .
\eee
The transversal momentum $k_\perp^{\rho}=(k-\hat{k})^\rho$ is absorbed into the soft function 
\bee
G^{\alpha\mu}(k)=\int d^4 y\int d^4 z e^{il\cdot z}e^{ik\cdot( y-z)}\langle\pi^-|\bar{d}(z)(-gA^{\alpha}(y))\gamma^{\mu}(1-\gamma_5)u(0)|0\rangle\ .
\eee
By the transformation $k_{\perp}^\rho A^{\alpha}(y)\to-iG^{\rho\alpha}(y)$, we can derive the result
\bee
\langle Q_1^{(8)}\rangle^{t=4}=\frac{2id_{\perp\mu\rho}}{m_b^2}\int dx\int dx^\prime\frac{G^{\mu\rho\alpha}(x,x^\prime)n_{\alpha}}{(x-x^\prime)}
\langle \pi^+|\bar{u}(0)\s{q}(1-\gamma_5)b(0)|\bar{B}\rangle
\eee
with 
\bee
G^{\mu\rho\alpha}(x,x^\prime)&=&
\int\frac{d^4 l}{(2\pi)^4}\delta(x-l\cdot n)
\int\frac{d^4 k}{(2\pi)^4}\delta(x-x^\prime-k\cdot n)
\int d^4 y\int d^4 z 
e^{il\cdot z}e^{ik\cdot( y-z)}\nn
&&\times\langle\pi^-|\bar{d}(z)gG^{\rho\alpha}(y)\gamma^{\mu}(1-\gamma_5)u(0)|0\rangle\ .
\eee
It is an easy task to show the equivalence between the results of the light-cone gauge and the covariant gauge. 

In the chiral limit, the G-parity requires that the distribution amplitude $A_{\perp}(x,x^\prime)$ is antisymmetric under the exchange of $x^\prime\leftrightarrow 1-x$. This leads to vanishing contributions. Thus we can neglect the twist-4 contributions from the one gluon insertion of tree level diagrams. This clearly shows the similar conclusion obtained in \cite{Beneke:2000ry}. The remaining twist-4 contributions from the tree level amplitudes are the light quark mass effects. They are power suppressed as $(m_u+m_d)m_u/m_b^2$, which is very small and can be ignored confidently. 

\section{discussions and conclusions}
Since the tree level twist-3 and twist-4 contributions only involve the integration over the distribution amplitude of the ejected meson, the integration can be performed completely. For the twist-3 contributions, there involves the integration
\bee
A=\int dx \int dx^\prime \frac{T(x^\prime,x)}{(x-x^\prime)x(1-x^\prime)}\ ,
\eee
where the three parton twist-3 distribution amplitude $T(x^\prime,x)$ is defined as \cite{Ball:1998je}
\bee
T(x^\prime,x)=360\eta_3(1-x)x^\prime(x-x^\prime)^2[1+\omega_3\frac{1}{2}(7(x-x^\prime)-3)]\ .
\eee 
The parameters $\eta_3$ and $\omega_3$ depend on the external meson.
%For the pion, they are given as
%\bee
%\eta_3=\frac{f_{3\pi}}{f_\pi}\frac{m_u+m_d}{m_\pi^2}\ .
%\eee
%and $w_3=w_{10}$.
The parameters $\eta_3$ and $\omega_3$ are scale dependent \cite{Ball:1998je}
\bee
\eta_3(Q^2)&=&(\frac{\alpha_s(Q^2)}{\alpha_s(\mu^2)})^{\gamma_3^\eta/b}
\eta_3(\mu^2)\ ,\;\; \gamma_3^\eta=\frac{16}{3}C_F+C_A\ ,\nn
\omega_3(Q^2)&=&(\frac{\alpha_s(Q^2)}{\alpha_s(\mu^2)})^{\gamma_3^\omega/b}
\omega_3(\mu^2)\ ,\;\; \gamma_3^\omega=-\frac{25}{6}C_F+\frac{7}{3}C_A\ .
\eee

The twist-4 contributions involve the distribution amplitude $A_{\perp}(x^\prime,x)$ \cite{Ball:1998je}
\bee
A_{\perp}(x^\prime,x)=30(x-x^\prime)^2(1-x-x^\prime)[h_{00}+h_{01}(x-x^\prime)+\frac{1}{2}h_{10}(5(x-x^\prime)-1)]\ ,
\eee
where the parameters are defined as
\bee
h_{00}&=&-\frac{1}{3}\eta_4\ ,\;\; h_{01}=\frac{7}{4}\eta_4 w_{4}-\frac{3}{20}a_2\ ,\;\;
h_{10}=\frac{7}{4}\eta_4 w_{4}+\frac{3}{20}a_2\ .
\eee
The quantities $\eta_4, w_4$ and $a_2$ are scale dependent \cite{Ball:1998je}
\bee
a_2(Q^2)&=&(\frac{\alpha_s(Q^2)}{\alpha_s(\mu^2)})^{\gamma_2/b}
a_2(\mu^2)\ ,\;\; a_2=4C_F(\psi(4)+\gamma_E-\frac{5}{12})\ ,\nn
\eta_4(Q^2)&=&(\frac{\alpha_s(Q^2)}{\alpha_s(\mu^2)})^{\gamma_3^\eta/b}
\eta_4(\mu^2)\ ,\;\; \gamma_4^\eta=\frac{8}{3}C_F\ ,\nn
\omega_4(Q^2)&=&(\frac{\alpha_s(Q^2)}{\alpha_s(\mu^2)})^{\gamma_3^\omega/b}
\omega_4(\mu^2)\ ,\;\; \gamma_4^\omega=-\frac{8}{3}C_F+\frac{10}{3}C_A\ .
\eee
The $b$ is defined as $b=(11N_c-2N_f)/3$ and $C_A=N_c$.
Since $A_{\perp}(x,x^\prime)$ is antisymmetric under the exchange of $x\leftrightarrow (1-x^\prime)$, we have vanishing result for the integration
\bee
\int dx \int dx^\prime \frac{A_{\perp}(x^\prime,x)}{(x-x^\prime)}\ .
\eee

It is obvious that the $A$, the integration of $T(x^\prime,x)$, is scale dependent, since $T(x^\prime,x)$ has an intrinsic scale dependence in the variables $\eta_3$ and $\omega_3$. The scale dependence of the combination
\bee
R_{\chi}(\mu)=\frac{A(\mu)m_\pi^2}{\bar{m}_b(\mu)[\bar{m}_u(\mu)+\bar{m}_d(\mu)]}
\eee
would be cancelled by that from the Wilson coefficient $a_{6,8}$. That means the following RG equations 
\bee
\mu\frac{d}{d\mu}(a_{6,8}R_{\chi})=0\ ,
\eee 
or, the identity of the anomalous dimensions
\bee
\gamma_{a_{6,8}}+\gamma_{A}=-2\gamma_{\bar{m}}
\eee
should be valid. The $\bar{m}_i, i=b,u,d$ are the current masses of the valence quark in the $\bar{B}$ and $\pi^-$ mesons. However, the presence of the one gluon insertion contributions requires that the Wilson coefficient $a_{6,8}$ must be calculated by including the diagrams with topologies containing triple partons of the ejected meson. This is beyond the scope of this paper. Thus, the present result for the twist-3 contributions is not invariant under the renormalization group. There would remain the scale and scheme dependence. 

After integration, the $A$ is equal to $A=\eta_3(C_1-C_2\omega_3)$ with $C_1=(60-6\pi^2)$ and $C_2=(2965-300\pi^2)$. At present, we assume that the $A$ is independent of the scale. The values for the relevant parameters obtained from QCD sum rule \cite{Ball:1998je} are collected in Table I.
\begin{table}[h]
\begin{tabular}[b]{cccc}
\\\hline
         & $\pi$ & $K$ & $\eta$\\\hline
$\eta_3$ & 0.015 & 0.015 & 0.013 \\
$\omega_3$ & -3 & -3 & -3 \\\hline\\
%$\mu^2$[GeV]$^2$ & 0.0077 & 0.096 & 0.12\\\hline\\
\end{tabular}
\caption{}
\end{table}
Using these parameter, the values of $A$ for $\pi, K$ and $\eta$ are calculated as $A_{\pi}=A_K=0.31$ and $A_{\eta}=0.26$. 
 
There are proposals \cite{Ciuchini:1997hb,Ciuchini:1997rj,Colangelo:1989gi,Buras:1995pz} 
that there might exist the so-called charming and GIM penguin contributions. These kinds of corrections involving the penguin loops have the order of twist-3. Including these corrections into the predictions of QCD factorization, it is possible to explain the experimental data. Due to the smaller effects of twist-3 contributions derived here, it implies that these charming and GIM penguin corrections become more important than the usual analysis. How to calculate these contributions from QCD is then an important problem.

For diagrams of one loop radiative corrections, there are other types of twist-3 and twist-4 contributions. For example, there are contributions from the matrix element containing the covariant derivative. This is different from the cases discussed above. Due to loop corrections, the Dirac spin structures of the amplitudes are also different from the tree level cases. This also lead to new higher twist contributions. We leave these interesting issues for the future work.

There is also a PQCD factorization formula for $B$ decays \cite{Li:1994ck,Li:jr,Li:1994iu,Chang:1996dw,Yeh:1997rq,Keum:2000ph,Keum:2000wi}. The basis of the PQCD factorization formula is different from that of the QCD factorization formula. The QCD factorization formula is based on the collinear factorization \cite{sterman}, wherein the internal parton propagator containes only the collinear momenta of the loop partons. The PQCD factorization is based on the $k_\perp$ factorization \cite{Catani:1990xk,Catani:1990eg,Collins:1991ty,Levin:1991ry}, in which the internal parton propagator is a function of both collinear and transverse momenta of the loop partons. In the $k_\perp$ factorization, the hard scattering kernel has the $k_\perp$ as a natural infrared regulator for the end point divergences and its external partons are off-shell. There also exist higher twist contributions in the PQCD factorization \cite{Nagashima:2002ia}
. However, there is no available power expansion method for PQCD factorization. Straightforward generalization of the collinear expansion may not be valid, since the basic physical ideas are different. It is thus an intriguent issue.

In summary, we have derived the twist-3 and twist-4 contributions from the tree level amplitudes in the hadronic decays of $B$ meson into two pseudoscalar mesons. The method can be easily generalized to other cases, such as the final states are composed of one pseudoscalar and one vector mesons, or, two vector mesons. The gauge invariance of the twist-3 and twist-4 contributions are shown by employing the light-cone and covariant gauges. The tree level twist-3 contributions come from the two parton and three parton distribution amplitudes. This explicitly exhibits the nature of the twist-3 contributions. The tree level twist-4 corrections have vanishing contributions due to the $G$-parity. The total twist-3 contribution is about $1.3\mu_{\chi}$ with $\mu_{\chi}=m_\pi^2/(m_b(m_u+m_d))$ times of the leading twist (twist-2) contributions. This is smaller than the usual result with a factor of $2\mu_{\chi}$. 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vspace{0.5cm}
\noindent
\begin{center}
{\bf Acknowledgments}
\end{center}

This work was supported in part by the National
Science Council of R.O.C. under Grant No. NSC89-2811-M-009-0024.
\noindent

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\end{thebibliography}
\newpage
\begin{figure*}
\includegraphics{twist3QFFig1}% Here is how to import EPS art
\caption{\label{fig:fig1}The one gluon insertion Feynman diagrams for $\bar{B}\to P_1 P_2$. The square symbol represents the vertex of weak interactions. The propagator with one bar denotes the special propagator.}
\end{figure*}
\end{document}


