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\centerline{\large\bf Two-parton twist-3 factorization in perturbative QCD}
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\centerline{Makiko Nagashima$^1$ and Hsiang-nan Li$^2$}
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\centerline{$^1$Department of Physics, Ochanomizu University,}\par
\centerline{Bunkyo-ku, Tokyo 112-8610, Japan}
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\centerline{$^2$Institute of Physics, Academia Sinica,
Taipei, Taiwan 115, Republic of China}
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\centerline{$^2$Department of Physics, National Cheng-Kung University,}\par
\centerline{Tainan, Taiwan 701, Republic of China}
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\centerline{\bf abstract}
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We prove factorization theorem for the process $\pi\gamma^*\to\pi$ at
the twist-3 level in the covariant gauge by means of the Ward identity
and gauge invariance, concentrating on the two-parton case. It is shown
that soft divergences cancel and collinear divergences are grouped into
the pseudo-scalar and pseudo-tensor two-parton twist-3 pion distribution
amplitudes. The delicate summation of a complete set of diagrams for
achieving factorization in momentum, spin, and color spaces is emphasized.
The proof is then extended to the exclusive semileptonic $B$ meson decay
$B\to\pi l\bar\nu$, assuming the hard scale to be of
$O(\sqrt{\bar\Lambda M_B})$, where $\bar\Lambda\equiv M_B-m_b$ is the $B$
meson and $b$ quark mass difference. We explain the distinction between
the above collinear factorization and the soft factorization for the $B$
meson distribution amplitudes. The gauge invariance and universality of
the two-parton twist-3 pion distribution amplitudes are confirmed. The
proof presented here can also accommodate the leading twist-2 case.


\section{INTRODUCTION}

Recently, we have proposed a simple proof of perturbative QCD (PQCD)
factorization theorem for the exclusive processes
$\pi\gamma^*\to \gamma(\pi)$ and $B\to \gamma(\pi) l\bar\nu$ based on
Ward identity \cite{L1}. According to this theorem, hadronic form factors
are factorized into the convolution of hard amplitudes with hadron
distribution amplitudes in momentum, spin, and color spaces. The former,
being infrared finite, are calculable in perturbation theory. The latter,
absorbing the infrared divergences involved in the processes, are defined
as matrix elements of nonlocal operators. The universality of the
distribution amplitudes and the gauge invariance of the factorization
have been explicitly demonstrated. However, the above proof is restricted
to the leading-twist, {\it i.e.}, twist-2 formalism. As emphasized in
\cite{L0,CKL}, contributions from the two-parton twist-3 pion distribution
amplitudes are of the same power as the leading-twist one in exclusive
$B$ meson decays. Hence, it is necessary to derive corresponding
factorization theorem. This proof can be regarded as an essential step
toward a rigorous construction of factorization theorem for two-body
nonleptonic $B$ meson decays.

Nonperturbative dynamics is reflected by infrared divergences in
radiative corrections. There are two types of infrared divergences, soft
and collinear. Soft divergences come from the region of a loop momentum
$l$, where all its components vanish. Collinear divergences are
associated with a massless quark of momentum $P=(Q,0,0_T)$, $Q$ being a
large scale. In the soft region and in the collinear region with $l$
parallel to $P$, the components of $l$ behave like
\begin{eqnarray}
l^\mu=(l^+,l^-,l_T)\sim (\lambda,\lambda,\lambda)\;,\;\;\;\;
l^\mu\sim (Q,\lambda^2/Q,\lambda)\;,
\label{sog2}
\end{eqnarray}
respectively, where the light-cone variables have been adopted, and
$\lambda$ is a small scale. In both regions the invariant mass of the
radiated gluon diminishes as $\lambda^2$, and the corresponding loop
integrand may diverge as $1/\lambda^4$. As the phase space for loop
integration vanishes like $d^4 l\sim \lambda^4$, logarithmic divergences
are generated.

In this paper we shall derive the factorization formula for the
scattering process $\pi\gamma^*\to\pi$, which involves the pion form
factor, at the next-to-leading twist by means of the Ward identity and
gauge invariance. We concentrate on the two-parton cases, whose
contributions are proportional to the chiral symmetry breaking scale
$m_0$. It will be shown that soft divergences cancel and collinear
divergences, factored out of the whole processes order by order, are
absorbed into the pseudo-scalar (PS) and pseudo-tensor (PT) two-parton
twist-3 pion distribution amplitudes, which are defined by nonlocal
matrix elements. The definition of the hard amplitudes at each order will
be given as a result of the proof. We then prove factorization theorem
for the exclusive semileptonic $B$ meson decay $B\to\pi l\bar\nu$, whose
topology is similar to the scattering process $\pi\gamma^*\to\pi$.
The distinction between the above
collinear factorization and the soft factorization for the $B$ meson
distribution amplitudes will be explained. The universality of the
two-parton twist-3 pion distribution amplitudes is confirmed. Our
derivation is simple, explicitly gauge-invariant, and appropriate for
both the factorizations of the soft and collinear divergences, compared
to those in the literature \cite{BL,BFL,MR,DM}.

In Sec.~II we demonstrate the $O(\alpha_s)$ factorization of the collinear
divergences in the process $\pi\gamma^*\to\pi$. The delicate summation of
a complete set of diagrams for achieving factorization in momentum, spin,
and color spaces is emphasized. The all-order proof based on the Ward
identity and gauge invariance is presented in Sec.~III. The absence of the
soft divergences is also shown. The technique is then generalized to the
decay $B\to\pi l\bar\nu$ in Sec.~IV, assuming the hard scale to be of
$O(\sqrt{\bar\Lambda M_B})$, where $\bar\Lambda\equiv M_B-m_b$ is the
$B$ meson and $b$ quark mass difference. Section V is the conclusion,
where the final goal to the proof of $k_T$ factorization theorem for
two-body nonleptonic $B$ meson decays is proposed. We refer the detailed
calculations of the $O(\alpha_s)$ collinear divergences in the above two
processes to Appendices A, B, and C.



\section{$O(\alpha_s)$ FACTORIZATION of $\pi\gamma^*\to\pi$}

We start with the two-parton twist-3 factorization of the process
$\pi\gamma^*\to\pi$ at the one-loop level, which will serve as the basis
of the all-order proof. The momentum $P_1$ ($P_2$) of the initial-state
(final-state) pion is parametrized as
\begin{eqnarray}
& &P_1=(P_1^+,0,{\bf 0}_T)=\frac{Q}{\sqrt{2}}(1,0,{\bf 0}_T)\;,
\nonumber\\
& &P_2=(0,P_2^-,{\bf 0}_T)=\frac{Q}{\sqrt{2}}(0,1,{\bf 0}_T)\;.
\label{mpp}
\end{eqnarray}
Consider the kinematic region with large $Q^2=-q^2$, $q=P_2-P_1$
being momentum transfer from a virtual photon, where PQCD is applicable.
The lowest-order diagrams are displayed in Fig.~1. The lower valence
quark (an anti-quark ${\bar d}$) in the incoming pion carries the
fractional momentum $x_1P_1$. The lower valence quark in the outgoing pion
carries the fractional momentum $x_2P_2$. Figure 1(a) gives the amplitude,
\begin{eqnarray}
H^{(0)}(x_1,x_2)=\frac{i}{2}eg^2 C_F
\frac{{\bar d}(x_1 P_1)\gamma^\nu {\bar d}(x_2 P_2)u({\bar x}_2 P_2)
\gamma_\nu(\not P_2-x_1\not P_1)\gamma_\mu u({\bar x}_1P_1)}
{(P_2-x_1P_1)^2(x_1P_1-x_2P_2)^2}\;,
\label{h0}
\end{eqnarray}
with ${\bar x}_{1(2)}\equiv 1-x_{1(2)}$ and the color factor $C_F$, where
the averages over spins and colors of the $u$ and ${\bar d}$ quarks have
been done. The $u$ and $\bar d$ quark fields obey the equations of motion,
\begin{eqnarray}
({\bar x}_1\not P_1-m_u)u({\bar x}_1P_1)=0\;,\;\;\;\;
{\bar d}(x_1P_1)(x_1\not P_1-m_d)=0\;,
\label{eqm}
\end{eqnarray}
$m_u$ ($m_d$) being the $u$ ($\bar d$) quark mass.

Insert the Fierz identity,
\begin{eqnarray}
I_{ij}I_{lk}&=&\frac{1}{4}I_{ik}I_{lj}
+\frac{1}{4}(\gamma_\alpha)_{ik}(\gamma^\alpha)_{lj}
+\frac{1}{4}(\gamma_5\not n_-)_{ik}(\not n_+\gamma_5)_{lj}
\nonumber\\
& &+\frac{1}{4}(\gamma_5)_{ik}(\gamma_5)_{lj}
+\frac{1}{4}[\gamma_5(\not n_+\not n_--1)]_{ik}
[(\not n_+\not n_--1)\gamma_5]_{lj}\;,
\label{1f}
\end{eqnarray}
into Fig.~1(a), where $I$ represents the identity matrix, and
$n_+=(1,0,0_T)$ and $n_-=(0,1,0_T)$ are the dimensionless vectors on the
light cone. Different terms in the above identity correspond to
contributions of different twists. The PS structure proportional to
$\gamma_5$ and the PT structure proportional to
$\gamma^5(\not n_+\not n_--1)$ lead to the twist-3 contributions. The
choice of the PT structure in Eq.~(\ref{1f}) and the ordinary one
$(\gamma_5\sigma^{\alpha\beta})_{ik}(\sigma_{\alpha\beta}\gamma_5)_{lj}$
are equivalent in the viewpoint of extracting the collinear divergences
\cite{TLS}. The PS and PT contributions to the process
$\pi\gamma^*\to\pi$ must be included simultaneously in order to form the
gauge interaction vertex of a pseudo-scalar particle, which is
proportional to $(P_1+P_2)_\mu$.

%with $\sigma_{\alpha\beta}\equiv i[\gamma_\alpha,\gamma_\beta]/2$.

The insertion on the initial-state side gives
\begin{eqnarray}
H^{(0)}&\equiv& \phi_S^{(0)}\otimes H^{(0)}_{S}+
\phi_T^{(0)}\otimes H^{(0)}_{T}\;,
\end{eqnarray}
with $\otimes$ representing the convolution in the momentum fraction. The
functions $\phi_{S(T)}^{(0)}$ and $H_{S(T)}^{(0)}$,
\begin{eqnarray}
& &\phi_S^{(0)}=\frac{1}{4m_0}{\bar d}(x_1P_1)\gamma^5
u({\bar x}_1P_1)\;,
\nonumber\\
& &\phi_T^{(0)}=\frac{1}{4m_0}{\bar d}(x_1P_1)
\gamma^5(\not n_+\not n_--1)u({\bar x}_1P_1)\;,
\nonumber\\
& &H_{S}^{(0)}(x_1,x_2)=\frac{i}{2}eg^2 C_Fm_0\frac{tr[\gamma^\nu
{\bar d}(x_2 P_2)u({\bar x}_2 P_2)
\gamma_\nu(\not P_2-x_1\not P_1)
\gamma_\mu\gamma_5]}{(P_2-x_1P_1)^2(x_1P_1-x_2P_2)^2}\;,
\nonumber\\
& &H_{T}^{(0)}(x_1,x_2)=\frac{i}{2}eg^2 C_Fm_0\frac{tr[\gamma^\nu
{\bar d}(x_2 P_2)u({\bar x}_2 P_2)
\gamma_\nu(\not P_2-x_1\not P_1)
\gamma_\mu(\not n_+\not n_--1)\gamma_5]}{(P_2-x_1P_1)^2
(x_1P_1-x_2P_2)^2}\;,
\label{low}
\end{eqnarray}
define the lowest-order perturbative pion PS (PT) distribution amplitude
and the corresponding hard amplitude, respectively, where
$m_0=M_{\pi}^2/(m_u+m_d)$ is the chiral symmetry breaking scale, with
$M_\pi$ being the pion mass.

Consider the $O(\alpha_s)$ radiative corrections to Fig.~1(a),
which are shown in Fig.~2, and identify their infrared divergences.
Self-energy corrections to the internal lines, leading to a
next-to-leading-order hard amplitude, are not included. Here we summarize
only the results of the $O(\alpha_s)$ factorization, and leave the
details to Appendix A. It will be shown that all diagrams in
Fig.~2 can be written as the convolution of the lowest-order hard
amplitudes $H_S^{(0)}$ and $H_T^{(0)}$ in Eq.~(\ref{low}) with the
$O(\alpha_s)$ collinear divergent pion distribution amplitudes
$\phi_S^{(1)}$ and $\phi_T^{(1)}$, respectively, in the region with
the loop momentum $l$ parallel to $P_1$. We shall derive the definitions 
of $\phi_S^{(1)}$ and $\phi_T^{(1)}$ as nonlocal matrix elements,
which then serve as a starting point of the all-order proof
of two-parton twist-3 factorization theorem for the process
$\pi\gamma^*\to\pi$.

Figures 2(a)-2(c) are the two-particle reducible diagrams with the
additional gluon attaching the two valence quarks of the initial pion. It
has been known that soft divergences cancel among these diagrams. The
reason for this cancellation is that soft gluons, being huge in
space-time, do not resolve the color structure of the pion. Collinear
divergences in Figs.~2(a)-2(c) do not cancel, since the loop momentum
flows into the internal lines in Fig.~2(b), but not in Figs.~2(a)
and 2(c). To absorb these collinear divergences, the pion distribution
amplitudes need to be introduced. The factorization of Figs.~2(a)-2(c)
is achieved trivially by inserting the Fierz identity. We obtain
\begin{eqnarray}
I^{(a),(c)}&\approx &\sum_{n=S,T}\phi^{(1)}_{na,nc} H_n^{(0)}(x_1,x_2)\;,
\label{2a2}\\
I^{(b)}&\approx& \sum_{n=S,T}\phi^{(1)}_{nb} H_n^{(0)}(\xi_1,x_2)\;,
\label{2b2}
\end{eqnarray}
with the momentum fraction $\xi_1=x_1-l^+/P_1^+$. Note that the
lowest-order hard amplitude in Eq.~(\ref{2b2}) depends on the loop
momentum $l$, because $l$ flows through the hard gluon.
The $O(\alpha_s)$ PS collinear pieces,
\begin{eqnarray}
\phi^{(1)}_{Sa}&=& \frac{-g^2 C_F}{4m_0}{\bar d}(x_1P_1)
\gamma_5\frac{i}{\bar{x}_1\not P_1}
\gamma_\beta\frac{{\bar x}_1\not P_1 +\not l}
{({\bar x}_1 P_1 +l)^2}\gamma^\beta u({\bar x}_1P_1)\frac{1}{l^2}\;,
\label{p2a}\\
\phi^{(1)}_{Sb}&=& \frac{ig^2 C_F}{4m_0}{\bar d}(x_1P_1)\gamma_\beta
\frac{x_1\not P_1-\not l}{(x_1P_1-l)^2}\gamma_5
\frac{{\bar x}_1\not P_1 +\not l}{({\bar x}_1 P_1 +l)^2}\gamma^\beta
u({\bar x}_1P_1)\frac{1}{l^2}\;,
\label{p2b}\\
\phi^{(1)}_{Sc}&=& \frac{g^2 C_F}{4m_0}{\bar d}(x_1P_1)\gamma_\beta
\frac{x_1\not P_1-\not l}{(x_1P_1-l)^2}\gamma^\beta
\frac{-i}{x_1\not P_1}
\gamma_5u({\bar x}_1P_1)\frac{1}{l^2}\;,
\label{p2c}
\end{eqnarray}
and the PT pieces with $\gamma_5$ in the above expressions replaced by
$\gamma_5(\not n_+\not n_--1)$, contain the collinear divergences in
Figs.~2(a), 2(b), and 2(c), respectively. It is easy to find that the
above integrands diverge as $1/\lambda^4$.

The factorization of the collinear divergences from the two-particle
irreducible diagrams in Fig.~2(d)-2(k) is written as
\begin{eqnarray}
I^{(d)}&\approx& \sum_{n=S,T}\phi_{nd}^{(1)}\left[H_n^{(0)}(x_1,x_2)
-H_n^{(0)}(\xi_1,x_2)\right]\;,
\label{2d2}\\
I^{(e)}&\approx& \sum_{n=S,T}\phi_{ne}^{(1)}H_n^{(0)}(x_1,x_2)\;,
\label{2e2}\\
I^{(f)}&\approx& -\sum_{n=S,T}\phi_{nf}^{(1)}H_n^{(0)}(\xi_1,x_2)\;,
\label{2f2}
\end{eqnarray}
with the $O(\alpha_s)$ PS collinear pieces,
\begin{eqnarray}
\phi_{Sd}^{(1)}&=&\frac{-ig^2}{2m_0C_F}
{\bar d}(x_1P_1)\gamma_5
\frac{{\bar x}_1\not P_1 +\not l}{({\bar x}_1 P_1 +l)^2}
\gamma^\beta u({\bar x}_1P_1)\frac{1}{l^2}
\frac{n_{-\beta}}{n_-\cdot l}\;,
\label{p2d}\\
\phi_{Se}^{(1)}=\phi_{Sf}^{(1)}
&=&\frac{ig^2}{8m_0N_c}
{\bar d}(x_1P_1)\gamma^5
\frac{{\bar x}_1\not P_1 +\not l}{({\bar x}_1 P_1 +l)^2}\gamma^\beta
u({\bar x}_1P_1)\frac{1}{l^2}
\frac{n_{-\beta}}{n_-\cdot l}\;,
\label{p2e}
\end{eqnarray}
$N_c=3$ being the number of colors. The corresponding PT pieces are
defined similarly with $\gamma_5$ in the above expressions replaced by
$\gamma_5(\not n_+\not n_--1)$. Figure 2(g) does not contain collinear
divergence. Note that Fig.~2(d) is free of soft divergence, because the
additional gluon attaches the hard gluon. The soft divergences cancel
between Figs.~2(e) and 2(f). The absence of the soft divergences is
obvious from the cancellation in Eq.~(\ref{2d2}) and between
Eqs.~(\ref{2e2}) and (\ref{2f2}) as $l\to 0$.

The contribution from Fig.~2(d) has been split into two terms as a
consequence of the Ward identity \cite{L1} (see Appendix A). The first
and second terms correspond to the hard amplitudes without and with the
loop momentum $l$ flowing through the internal lines, respectively. The
Feynman rule $n_{-\beta}/n_-\cdot l$ in the collinear pieces, coming from
the eikonal approximation, is represented by a Wilson line in the
direction of $n_-$. Note that Eqs.~(\ref{p2d}) and (\ref{p2e}) possess
different color factors due to different color flows in Figs.~2(d)-2(f).
Combining Eqs.~(\ref{2d2}), (\ref{2e2}), and
(\ref{2f2}), we arrive at
\begin{eqnarray}
\sum_{i=(d)}^{(g)}I^i&\approx&
\sum_{n=S,T}[\phi_{nd}^{(1)}+\phi_{ne}^{(1)}]H_n^{(0)}(x_1,x_2)
-\sum_{n=S,T}[\phi_{nd}^{(1)}+\phi_{nf}^{(1)}]H_n^{(0)}(\xi_1,x_2)
\nonumber\\
&=&\sum_{n=S,T}\phi_{nu}^{(1)}[H_n^{(0)}(x_1,x_2)-H_n^{(0)}(\xi_1,x_2)]\;,
\label{2dg}
\end{eqnarray}
where the PS piece,
\begin{eqnarray}
\phi_{Su}^{(1)}=\frac{-ig^2C_F}{4m_0}
{\bar d}(x_1P_1)\gamma^5
\frac{{\bar x}_1\not P_1 +\not l}{({\bar x}_1 P_1 +l)^2}\gamma^\beta
u({\bar x}_1P_1)\frac{1}{l^2}
\frac{n_{-\beta}}{n_-\cdot l}\;,
\label{2nu}
\end{eqnarray}
is associated with the collinear gluon emitted from the $u$ quark.
The color factor $C_F$ implies the factorization of the distribution
amplitude from the other part of the process in color space, which can be
achieved only in the summation of diagrams. The first and second terms in
Eq.~(\ref{2dg}) correspond to Figs.~3(a) and 3(b), respectively.

The analysis of Figs.~2(h)-2(k) is similar, and the conclusion is that
Fig.~2(h) is split into two terms as in Eq.~(\ref{2d2}), the collinear
gluons in Figs.~2(i) and 2(j) are eikonalized as in Eqs.~(\ref{2e2}) and
(\ref{2f2}), respectively, and Fig.~2(k), like Fig.~2(g), does not
contribute collinear divergence. The soft divergences cancel among the
above diagrams. The detailed calculation is referred to Appendix A.
Summing the four diagrams, we derive the correct color factor:
\begin{eqnarray}
\sum_{i=(h)}^{(k)}I^i&\approx&
\sum_{n=S,T}[\phi_{nh}^{(1)}+\phi_{ni}^{(1)}]H_n^{(0)}(x_1,x_2)
-\sum_{n=S,T}[\phi_{nh}^{(1)}+\phi_{nj}^{(1)}]H_n^{(0)}(\xi_1,x_2)
\nonumber\\
&=&\sum_{n=S,T}\phi_{n\bar d}^{(1)}[H_n^{(0)}(x_1,x_2)
-H_n^{(0)}(\xi_1,x_2)]\;,
\label{2hk}
\end{eqnarray}
where the PS piece,
\begin{eqnarray}
\phi_{S\bar d}^{(1)}=\frac{ig^2C_F}{4m_0}{\bar d}(x_1P_1)
\gamma^\beta\frac{x_1\not P_1 -\not l}{(x_1 P_1 -l)^2}
\gamma^5 u({\bar x}_1P_1)\frac{1}{l^2}
\frac{n_{-\beta}}{n_-\cdot l}\;,
\label{2nd}
\end{eqnarray}
is associated with the collinear gluon emitted from the $\bar d$ quark.
The first and second terms in Eq.~(\ref{2hk}) correspond to Figs.~3(c)
and 3(d), respectively.

The sum of Eqs.~(\ref{2a2}), (\ref{2b2}), (\ref{2dg}) and (\ref{2hk})
leads to
\begin{eqnarray}
\sum_{i=(a)}^{(k)}I^i\approx
\sum_{n=S,T}\phi_n^{(1)}\left[H_n^{(0)}(x_1,x_2)
-H_n^{(0)}(\xi_1,x_2)\right]\;.
\label{phi1}
\end{eqnarray}
Equations (\ref{p2a})-(\ref{p2c}), (\ref{2nu}) and (\ref{2nd}) imply
the Feynman rules for the perturbative evaluation of the $O(\alpha_s)$
two-parton twist-3 pion distribution amplitudes, from which
$\phi^{(1)}_{S(T)}$ is written as a nonlocal hadronic matrix element with
the structure $\gamma_5$ $[\gamma_5(\not n_+\not n_--1)]$ sandwiched,
\begin{eqnarray}
\phi^{(1)}_{S}(x)&=&i\int \frac{P^+dy^-}{2\pi m_0}e^{ixP^+y^-}
\langle 0|{\bar d}(y^-)\gamma_5(-ig)
\int_0^{y^-}dzn\cdot A(zn)u(0)|\pi^+(P)\rangle\;,
\nonumber\\
\phi^{(1)}_{T}(x)&=&i\int \frac{P^+dy^-}{2\pi m_0}e^{ixP^+y^-}
\langle 0|{\bar d}(y^-)\gamma_5(\not n_+\not n_--1)(-ig)
\int_0^{y^-}dzn\cdot A(zn)u(0)|\pi^+(P)\rangle\;.
\label{ld}
\end{eqnarray}
The integral over $z$ in fact contains two pieces: for the upper eikonal
line in Fig.~3(a), $z$ runs from 0 to $\infty$. For the lower eikonal
line in Fig.~3(b), $z$ runs from $\infty$ back to $y^-$. The light-cone
coordinate $y^-\not =0$ corresponds to the fact that the collinear
divergences in Fig.~2 do not cancel.

The $O(\alpha_s)$ radiative corrections to Fig.~1(b) are displayed in
Fig.~4. The factorization of the collinear divergences from these diagrams
is referred to Appendix B. The result is similar to Eq.~(\ref{phi1}) but
without the PT contributions, because of
$\gamma^\nu (\not n_+\not n_--1)\gamma_\nu=0$, where the gamma
matrices $\gamma^\nu$ and $\gamma_\nu$ come from the gluon vertices
in Fig.~1(b). The definition of the $O(\alpha_s)$ PS pion distribution
amplitude $\phi_S^{(1)}$ is the same as in Eq.~(\ref{ld}). This is
expected due to the universality. The above derivation also applies
straightforwardly to the factorization of the collinear divergences
associated with the final-state pion, which arise from the region with
the loop momentum parallel to $P_2$.

In conclusion, the $O(\alpha_s)$ factorization of the process
$\pi\gamma^*\to\pi$ is written as
\begin{eqnarray}
G^{(1)}=\sum_{n=S,T}\phi^{(1)}_n\otimes H^{(0)}_n
+\sum_{m=S,T}H^{(0)}_m\otimes\phi^{(1)}_m+H^{(1)}\;,
\label{fac1}
\end{eqnarray}
where $G^{(1)}$ denotes the complete set of the $O(\alpha_s)$ corrections
to Fig.~1, $\phi_n^{(1)}$ ($\phi_m^{(1)}$) is the initial-state
(final-state) pion distribution amplitude, and $H_m^{(0)}$ is the hard
amplitude with the Fierz identity inserted into the final-state
side of $H^{(0)}$ for Fig.~1. The first term on the right-hand
side of the above expression does not contain the collinear
divergences from the loop momentum $l$ parallel to $P_2$. In this region
$l^+$ is negligible, $\xi_1$ approaches $x_1$, and the two terms on the
right-hand side of Eq.~(\ref{phi1}) cancel. For the similar reason, the
second term on the right-hand side of Eq.~(\ref{fac1}) does not contain
the collinear divergences from $l$ parallel to $P_1$. That is, the
initial-state and final-state collinear divergences in $G^{(1)}$
have been completely factorized into the first and second terms on the
right-hand side of Eq.~(\ref{fac1}), respectively. The $O(\alpha_s)$ hard
amplitude $H^{(1)}$, defined via Eq.~(\ref{fac1}), is infrared finite.

Adding Eq.~(\ref{fac1}) to the lowest-order diagrams
$G^{(0)}\equiv H^{(0)}$, the factorization formula for the
two-parton twist-3 contributions to the process $\pi\gamma^*\to\pi$ is
given, up to $O(\alpha_s)$, by
\begin{eqnarray}
G^{(0)}+G^{(1)}=\sum_{n,m=S,T}(\phi^{(0)}_n+\phi^{(1)}_n)
\otimes(H_{nm}^{(0)}+H^{(1)}_{nm})\otimes
(\phi^{(0)}_m+\phi^{(1)}_m)\;,
\label{ff1}
\end{eqnarray}
where the trivial factorizations,
\begin{eqnarray}
H_{n}^{(0)}=\sum_{m=S,T}H_{nm}^{(0)}\otimes \phi_{m}^{(0)}\;,
\;\;\;\;
H_{m}^{(0)}=\sum_{n=S,T}\phi_{n}^{(0)}\otimes H_{nm}^{(0)}\;,
\;\;\;\;
H^{(1)}=\sum_{n,m=S,T}\phi_n^{(0)}\otimes H_{nm}^{(1)}
\otimes \phi^{(0)}_m\;,
\label{wi3}
\end{eqnarray}
have been adopted. The last expression defines the $O(\alpha_s)$ hard
amplitude $H^{(1)}_{nm}$. It is obvious that the PS and PT
structures must be included simultaneously for complete two-parton twist-3
collinear factorization.


\section{ALL-ORDER FACTORIZATION of $\pi\gamma^*\to\pi$}

In this section we present the all-order proof of two-parton twist-3
factorization theorem for the process $\pi\gamma^*\to \pi$, and
construct the gauge-invariant pion distribution amplitudes,
\begin{eqnarray}
\phi_S(x)&=&i\int\frac{P^+dy^-}{2\pi m_0}e^{ixP^+y^-}
\langle 0|{\bar d}(y^-)\gamma_5
P\exp\left[-ig\int_0^{y^-}dzn\cdot A(zn)\right]u(0)|\pi^+(P)
\rangle\;,
\nonumber\\
\phi_T(x)&=&i\int\frac{P^+dy^-}{2\pi m_0}e^{ixP^+y^-}
\langle 0|{\bar d}(y^-)\gamma_5(\not n_+\not n_--1)
P\exp\left[-ig\int_0^{y^-}dzn\cdot A(zn)\right]u(0)|\pi^+(P)
\rangle\;,
\label{pwt}
\end{eqnarray}
graphically shown in Fig.~3(e). By expanding the quark field
${\bar d}(y^-)$ and the path-ordered exponential (Wilson line) into
powers of $y^-$, the above matrix elements can be expressed as a series
of covariant derivatives $(D^+)^n{\bar d}(0)$, implying that
Eq.~(\ref{pwt}) is gauge invariant. Models for the above two-parton
twist-3 pion distribution amplitudes have been derived based on
QCD sum rules \cite{PB1}.

It has been mentioned
in the Introduction that factorization of a QCD process in momentum,
spin, and color spaces requires summation of many diagrams, especially
at higher orders. The diagram summation can be handled in an elegant way
by employing the Ward identity,
\begin{eqnarray}
l_\mu G^\mu(l,k_1,k_2,\cdots, k_n)=0\;,
\label{war}
\end{eqnarray}
where $G^\mu$ represents a physical amplitude with an external gluon
carrying the momentum $l$ and with $n$ external quarks carrying the
momenta $k_1$, $k_2$, $\cdots$, $k_n$. All these external particles are
on mass shell. The Ward identity can be easily derived by means of the
Becchi-Rouet-Stora (BRS) transformation \cite{BRS}.

We shall prove two-parton twist-3 factorization theorem for the process
$\pi\gamma^*\to\pi$ to all orders by induction. The factorization of the
$O(\alpha_s)$ collinear divergences has been worked out in Sec.~II.
Assume that factorization theorem holds up to $O(\alpha_s^N)$:
\begin{eqnarray}
G=\sum_{n,m=S,T}\phi_n\otimes H_{nm}\otimes \phi_m\;,
\label{gh1}
\end{eqnarray}
with the definitions,
\begin{eqnarray}
G=\sum_{i=0}^N G^{(i)}\;,\;\;\;
\phi_{n(m)}=\sum_{i=0}^N \phi_{n(m)}^{(i)}\;,\;\;\;
H_{nm}=\sum_{i=0}^N H_{nm}^{(i)}\;.
\label{ghf}
\end{eqnarray}
$G^{(i)}$ denotes the full diagrams of $O(\alpha_s^i)$ with
$G^{(0)}\equiv H^{(0)}$ for Fig.~1, the initial-state pion
distribution amplitudes $\phi_{n}^{(i)}$ are defined by the
$O(\alpha_s^i)$ terms in the perturbative expansion of Eq.~(\ref{pwt}),
the final-state pion distribution amplitudes $\phi_{m}^{(i)}$ defined
similarly by the complex conjugate of Eq.~(\ref{pwt}), and $H_{nm}^{(i)}$
the remaining $O(\alpha_s^i)$ pieces of the process, which do not contain
the collinear divergences. We then have the relations,
\begin{eqnarray}
G^{(k)}=\sum_{n,m=S,T}\sum_{i=0}^{k}\sum_{j=0}^{k-i}
\phi_{n}^{(i)}\otimes
H_{nm}^{(k-i-j)}\otimes \phi_{m}^{(j)}\;,\;\;\;k=0,1,\cdots N\;,
\label{gnf}
\end{eqnarray}
implying that all the initial-state and final-state collinear divergences
in $G^{(k)}$ have been collected into $\phi_{n}^{(i)}$ and
$\phi_{m}^{(j)}$ systematically.
Inserting the Fierz identity, we have also the trivial factorizations of
the pion distribution amplitudes $\phi$ and the diagrams $G$ at arbitrary
orders of $\alpha_s$, similar to Eq.~(\ref{wi3}). Equation (\ref{gnf})
then leads to
\begin{eqnarray}
G_{nm}^{(k)}=\sum_{n',m'=S,T}\sum_{i=0}^{k}\sum_{j=0}^{k-i}
\phi_{nn'}^{(i)}\otimes H_{n'm'}^{(k-i-j)}\otimes
\phi_{m'm}^{(j)}\;.
\label{gnf1}
\end{eqnarray}

Below we prove the collinear factorization of the $O(\alpha_s^{N+1})$
diagrams $G^{(N+1)}$ into the convolution of the $O(\alpha_s^{N})$
diagrams $G^{(N)}$ with the $O(\alpha_s)$ pion distribution amplitudes.
Note that the proof presented here differs slightly from that in
\cite{L1}. Look for the gluon in a complete set of $O(\alpha_s^{N+1})$
diagrams $G^{(N+1)}$, one of whose ends attaches the outer most vertex on
the upper $u$ quark line in the  pion. Such a gluon exists, since
$G^{(N+1)}$ are finite-order diagrams. Let $\alpha$ denote the outer most
vertex, and $\beta$ denote the attachments of the other end of the
radiated gluon inside the rest of the diagrams. There are two types of
collinear configurations associated with this gluon, depending on whether
the vertex $\beta$ is located on an internal line with momentum along
$P_1$. The fermion propagator adjacent to the vertex $\alpha$ is
proportional to $\not P_1$ in the collinear region with the loop momentum
$l$ parallel to $P_1$. If $\beta$ is not located on a collinear line
along $P_1$, the component $\gamma^+$ in $\gamma^\alpha$ and the minus
component of the vertex $\beta$ give the leading contribution. If $\beta$
is located on a collinear line along $P_1$, $\beta$ can not be minus, and
both $\alpha$ and $\beta$ represent the transverse components. This
configuration is the same as of the self-energy correction to an on-shell
particle.

According to the above classification, we decompose the tensor
$g_{\alpha\beta}$ appearing in the propagator of the radiated gluon into
\begin{eqnarray}
g_{\alpha\beta}=\delta_{\alpha +}\delta_{\beta -}
-\delta_{\alpha \perp}\delta_{\beta \perp}
+\delta_{\alpha -}\delta_{\beta +}\;.
\label{dec}
\end{eqnarray}
The first (second) term on the right-hand side of Eq.~(\ref{dec})
extracts the first (second) type of initial-state collinear divergences
mentioned above. The third term does not contribute due to the equations of
motion in Eq.~(\ref{eqm}). We shall concentrate on the collinear
factorization corresponding to the first term, and argue that the
factorization corresponding to the second term can be achieved simply by
the requirement of gauge invariance. The replacement for the first term
\cite{L1},
\begin{eqnarray}
\delta_{\alpha +}\delta_{\beta -}\to
\frac{n_{-\alpha} l_\beta}{n_-\cdot l}\;.
\label{rep}
\end{eqnarray}
extracts the first type of collinear divergences, since the light-like
vector $n_{-\alpha}$ selects the plus component of $\gamma^\alpha$, and
$l_\beta$ selects the minus component of the vertex $\beta$ in the
collinear region. Therefore, the left-hand and right-hand sides of
Eq.~(\ref{rep}) lead to the identical collinear configuration.

As stated before, the radiated collinear gluon with $\alpha=+$ and
$\beta=-$ does not attach the upper or lower quark line directly, which
carries momentum along $P_1$. That is, those diagrams with Figs.~2(a) and
2(b) as the $O(\alpha_s)$ subdiagrams are excluded from the set of
$G^{(N+1)}$ as discussing the first type of collinear configurations. The
contraction of $l_\beta$ hints the application of the Ward identity in
Eq.~(\ref{war}) to the case, in which the on-shell external $u$ quark,
$\bar d$ quark and gluon carry the momenta ${\bar \xi}_1 P_1$, $x_1P_1$
and $l$, respectively. Figure 5(a), describing the Ward identity,
contains a complete set of contractions of $l_\beta$, since the second
and third diagrams, excluded here, have been added back. The contractions
of $l$ are represented by arrows in Fig.~5(a). The second and third
diagrams in Fig.~5(a) then give
\begin{eqnarray}
& &l_\beta \frac{1}{{\bar \xi}_1\not P_1-\not l}\gamma^\beta
u({\bar \xi}_1 P_1)
=\frac{1}{{\bar \xi}_1\not P_1-\not l}[\not l-{\bar \xi}_1\not P_1 +
{\bar \xi}_1\not P_1]u({\bar \xi}_1 P_1)
=-u({\bar \xi}_1 P_1)\;,
\label{ide}\\
& &l_\beta{\bar d}(x_1P_1)\gamma^\beta\frac{1}{x_1\not P_1-\not l}
=-{\bar d}(x_1P_1)\;,
\label{ide2}
\end{eqnarray}
respectively, where the equations of motion in Eq.~(\ref{eqm}) have been
employed. The terms $u({\bar \xi}_1 P_1)$ and ${\bar d}(x_1P_1)$ at the
ends of the above expressions correspond to the $O(\alpha_s^N)$ diagrams.

We then insert the Fierz identity into Fig.~5(a), and factor the
lowest-order expressions ${\bar d}(x_1P_1)\Gamma u({\bar \xi}_1 P_1)$
with $\Gamma$ being the PS or PT structure considered in this
work. The result is a relation shown in Fig.~5(b), where the cuts on the
quark lines denote the insertion of the Fierz identity. Figure 5(b)
implies that the diagrams $G^{(N+1)}$ with the replacement in
Eq.~(\ref{rep}) for the tensor $g_{\alpha\beta}$ are factorized into the
convolution of the full diagrams $G_n^{(N)}$ with $\phi_{nu}^{(1)}$,
$n=S,T$. The factor $n_{-\alpha}/n_-\cdot l$ is exactly the Feynman rule
associated with the Wilson line in the direction of $n_-$, which appears
in Eq.~(\ref{pwt}). The same discussion applies to the factorization of
the diagrams $G^{(N+1)}$ with the collinear gluon emitted from the outer
most vertex on the ${\bar d}$ quark line. Similarly, the collinear
divergences in $G^{(N+1)}$ from the loop momentum parallel to $P_2$ are
factorized into the final-state pion distribution amplitudes. Note that
it is the third term in Eq.~(\ref{dec}) that corresponds to the first
type of collinear configurations in this case. We conclude that
$G^{(N+1)}_{\parallel\cdot}$ with the replacement in Eq.~(\ref{rep})
on the initial-state side and $G^{(N+1)}_{\cdot \parallel}$ with the
corresponding replacement on the final-state side are written as the
convolutions,
\begin{eqnarray}
G^{(N+1)}_{\parallel\cdot}&\approx& \sum_{n=S,T}(\phi_{nu}^{(1)}
+\phi_{n\bar d}^{(1)})\otimes G_n^{(N)}\;,
\label{wic}\\
G^{(N+1)}_{\cdot\parallel}&\approx& \sum_{m=S,T} G_m^{(N)}\otimes
(\phi_{mu}^{(1)}+\phi_{m\bar d}^{(1)})\;,
\label{wicr}
\end{eqnarray}
respectively. The first formula is displayed in Fig.~6.

The above procedures are also applicable to the $O(\alpha_s^{j+1})$
initial-state and final-state pion distribution amplitudes
$\phi_n^{(j+1)}$ and $\phi_m^{(j+1)}$. We identify the gluon in a complete
set of the $O(\alpha_s^{j+1})$ diagrams $\phi_n^{(j+1)}$, one of whose
ends attaches the outer most vertex $\alpha$ on the $u$ quark line. The
other end attaches the vertex $\beta$ inside the rest of the diagrams. For
the first term on the right-hand side of Eq.~(\ref{dec}), we have a Ward
identity similar to Fig.~5(a). Figure 5(b) then leads to the
factorizations of the initial-state and final-state distribution
amplitudes,
\begin{eqnarray}
\phi_{n\parallel}^{(j+1)}&\approx& \sum_{n'=S,T}
(\phi_{n'u}^{(1)}+\phi_{n'\bar d}^{(1)})\otimes \phi_{n'n}^{(j)}\;,
\nonumber\\
\phi_{m\parallel}^{(j+1)}&\approx&\sum_{m'=S,T}\phi_{mm'}^{(j)}\otimes
(\phi_{m'u}^{(1)}+\phi_{m'\bar d}^{(1)})\;,
\label{wi2}
\end{eqnarray}
where the PS and PT structures in Eq.~(\ref{1f}) have been inserted.

We sum Eqs.~(\ref{wic}) and (\ref{wicr}), and subtract the
double-counted diagrams $G^{(N+1)}_{\parallel\;\parallel}$ with
the replacements on both the initial-state and final-state sides.
The result is
\begin{eqnarray}
G^{(N+1)}_{\parallel\cdot}+G^{(N+1)}_{\cdot \parallel}
-G^{(N+1)}_{\parallel\;\parallel}
&\approx&\sum_{n,m=S,T}\left[\phi_{nu\bar d}^{(1)}\otimes
G_{nm}^{(N)}\otimes\phi_{m}^{(0)}
+\phi_n^{(0)}\otimes G_{nm}^{(N)}\otimes \phi_{mu\bar d}^{(1)}\right.
\nonumber\\
& &\left.-\phi_{nu\bar d}^{(1)}\otimes G_{nm}^{(N-1)}
\otimes \phi_{mu\bar d}^{(1)}\right]\;,
\label{gf}
\end{eqnarray}
where $\phi_{n(m)u\bar d}^{(1)}$ is a brief notation of
\begin{eqnarray}
\phi_{n(m)u\bar d}^{(1)}=\phi_{n(m)u}^{(1)}+\phi_{n(m)\bar d}^{(1)}\;.
\end{eqnarray}
The trivial factorizations, similar to
Eq.~(\ref{wi3}), have been inserted. Substituting Eqs.~(\ref{gnf1}) and
(\ref{wi2}) into Eq.~(\ref{gf}), a simple algebra gives
\begin{eqnarray}
G^{(N+1)}_{\parallel\cdot}+G^{(N+1)}_{\cdot \parallel}
-G^{(N+1)}_{\parallel\;\parallel}
&\approx&\sum_{n,m=S,T}\sum_{i=0}^{N}\sum_{j=0}^{N-i}
\left[\phi_{n\parallel}^{(i+1)}\otimes H_{nm}^{(N-i-j)}\otimes
\phi_{m}^{(j)} +\phi_{n}^{(i)}\otimes H_{nm}^{(N-i-j)}\otimes
\phi_{m\parallel}^{(j+1)}\right.
\nonumber\\
& &\left.-\phi_{n\parallel}^{(i+1)}\otimes H_{nm}^{(N-i-j-1)}\otimes
\phi_{m\parallel}^{(j+1)}\right]\;.
\label{gnf2}
\end{eqnarray}

Next we consider the collinear factorization of the $O(\alpha_s^{N+1})$
diagrams $G^{(N+1)}_{\perp\perp}$, which involve the tensors
$-\delta_{\alpha \perp}\delta_{\beta \perp}$ in Eq.~(\ref{dec}) on both
the initial-state and final-state sides. We argue that the
additional collinear divergences from these diagrams can be factorized
into the initial-state and final-state pion distribution amplitudes
simply by substituting $\phi_{n(m)}$ for $\phi_{n(m)\parallel}$ in
Eq.~(\ref{gnf2}). The complete collinear factorization of the diagrams
$G^{(N+1)}$ is then given by
\begin{eqnarray}
G^{(N+1)}&=&G^{(N+1)}_{\parallel\cdot}+G^{(N+1)}_{\cdot \parallel}
-G^{(N+1)}_{\parallel\;\parallel}+G^{(N+1)}_{\perp\perp}\;,
\nonumber\\
&\approx&\sum_{n,m=S,T}\sum_{i=0}^{N}\sum_{j=0}^{N-i}
\left[\phi_{n}^{(i+1)}\otimes H_{nm}^{(N-i-j)}\otimes \phi_{m}^{(j)}
+\phi_{n}^{(i)}\otimes H_{nm}^{(N-i-j)}\otimes \phi_{m}^{(j+1)}\right.
\nonumber\\
& &\left.-\phi_{n}^{(i+1)}\otimes H_{nm}^{(N-i-j-1)}\otimes
\phi_{m}^{(j+1)}\right]+F^{(N+1)}\;,
\label{gnf3}
\end{eqnarray}
with the $O(\alpha_s^{N+1})$ function,
\begin{eqnarray}
F^{(N+1)}&=&G^{(N+1)}_{\perp\perp}-\left[\sum_{n=S,T}
\sum_{i=1}^{N+1}\phi_{n\perp}^{(i)}\otimes H_{n}^{(N+1-i)}
+\sum_{m=S,T}\sum_{j=1}^{N+1} H_{m}^{(N+1-j)}\otimes \phi_{m\perp}^{(j)}
\right.
\nonumber\\
& &\left.+\sum_{n,m=S,T}\sum_{i=1}^{N+1}\sum_{j=1}^{N+1-i}
\phi_{n\perp}^{(i)}\otimes H_{nm}^{(N+1-i-j)}\otimes
\phi_{m\perp}^{(j)}\right]\;,
\label{fn1}
\end{eqnarray}
and $\phi_{n(m)\perp}^{(i)}=\phi_{n(m)}^{(i)}-\phi_{n(m)\parallel}^{(i)}$.

We show that $F^{(N+1)}$ is free of collinear divergences by means of
gauge invariance. The diagrams $G^{(N+1)}$ and the terms on
the right-hand side of Eq.~(\ref{gnf3}) other than $F^{(N+1)}$ are gauge
invariant, because both the pion distribution amplitudes $\phi_{n(m)}$ and
the hard amplitudes $H_{nm}$ are gauge invariant. 
Hence, $F^{(N+1)}$ must be gauge
invariant. It implies that a Ward identity similar to Fig.~5(a) holds for
the diagrams associated with $F^{(N+1)}$, where the external quarks and
the radiated gluon identified above are all on-shell in the collinear
regions. We contract the loop momentum $l$ to the vertices $\beta$ from
the initial-state or final-state side in the above diagrams.
$G^{(N+1)}_{\perp\perp}$ and $H$ represent complete sets of diagrams,
and their contractions diminish. However, $\phi_{n\perp}$ and
$\phi_{m\perp}$ do not form complete sets of diagrams, since the radiated
gluon does not attach the eikonal lines. Note that the eikonal vertices
$n_{-\alpha}$ and $n_{+\alpha}$ contain only the longitudinal
components. Therefore, the contractions to the terms in the brackets do
not vanish. The Ward identity can not hold, unless $F^{(N+1)}=0$ in the
collinear regions with $l$ parallel to $P_1$ and to $P_2$. Moreover, it
has been stated that the second type of collinear configurations arises
from the attachment of the identified gluon to a collinear internal line.
That is, all collinear divergences in $F^{(N+1)}$ disappear together with
the vanishing of the collinear divergence from the identified gluon. 

Taking into account the infrared finite piece corresponding to the
difference between $\delta_{\alpha +}\delta_{\beta -}$ and
$n_{-\alpha} l_\beta/n_-\cdot l$, Eq.~(\ref{gnf3}) becomes
\begin{eqnarray}
G^{(N+1)}=\sum_{n,m=S,T}\sum_{i=0}^{N+1}\sum_{j=0}^{N+1-i}
\phi_n^{(i)}\otimes H_{nm}^{(N+1-i-j)}\otimes \phi_m^{(j)}\;,
\label{gf1}
\end{eqnarray}
where the $O(\alpha_s^{N+1})$ hard amplitude $H_{nm}^{(N+1)}$, defined via
\begin{eqnarray}
F^{(N+1)}=\sum_{n,m=S,T}\phi_n^{(0)}\otimes H_{nm}^{(N+1)}
\otimes \phi_m^{(0)}\;,
\label{hn}
\end{eqnarray}
is infrared finite, because $F^{(N+1)}$ is from the above argument.
Equation (\ref{gf1}) states that all the two-parton twist-3 collinear
divergences in the process $\pi\gamma^*\to\pi$ can be factorized into the
pion distribution amplitudes in Eq.~(\ref{pwt}) order by order.

At last, we prove by induction that soft divergences do not exist in the
process $\pi\gamma^*\to\gamma$ at the two-parton twist-3 level. The
$O(\alpha_s)$ soft cancellation has been shown in Sec.~II. Assume
that the $O(\alpha_s^N)$ diagrams $G^{(N)}$ do not contain any soft
divergence, though they contain collinear divergences. They are then
dominated either by hard, by collinear dynamics associated with the
initial-state pion, or by collinear dynamics associated with the
final-state pion. Consider the $O(\alpha_s^{N+1})$ full diagrams
$G^{(N+1)}$. Similarly, we look for the gluon radiated from the outer
most vertex on the $u$ quark line in the initial-state pion, and adopt
the decomposition of the tensor $g_{\alpha\beta}$ in Eq.~(\ref{dec}). The
attachment of a soft gluon to an off-shell internal line does not
introduce infrared divergence, since an off-shell propagator behaves at
least like $1/Q$. Employing the similar argument, the first (second) term
on the right-hand side of Eq.~(\ref{dec}) leads to soft divergence, which
arises from the attachment of the soft gluon to a collinear line along
$P_2$ ($P_1$). Again, the third term on the right-hand side of
Eq.~(\ref{dec}) does not contribute due to the equations of motion.

In the former case the vertex $\beta$ must be dominated by the minus
component. For example, as the loop momentum $l$, flowing through a
quark line with momentum parallel to $P_2$, is small, we have
\begin{eqnarray}
\frac{\not P_2+\not l}{(P_2+l)^2}\gamma^\beta\not P_2
\approx \frac{P_2^\beta}{P_2\cdot l}\not P_2\;.
\label{eik}
\end{eqnarray}
The gamma matrix $\gamma^\alpha$ is dominated by the component $\gamma^+$
stated above. Therefore, the replacement in Eq.~(\ref{rep}) still
applies. Following the same procedures, we arrive at Eq.~(\ref{wic})
for the $O(\alpha_s^{N+1})$ diagrams $G^{(N+1)}_{\parallel\cdot}$. The
soft factorization of the diagrams $G^{(N+1)}_{\perp\cdot}$ in the latter
case can be achieved simply by substituting $\phi_{n}^{(1)}$ for
$\phi_{nu\bar d}^{(1)}$ in Eq.~(\ref{wic}):
\begin{eqnarray}
G^{(N+1)}&\approx& \sum_{n=S,T}\phi_n^{(1)}\otimes G_n^{(N)}
+{\bar F}^{(N+1)}\;,
\label{gnf4}
\end{eqnarray}
with the $O(\alpha_s^{N+1})$ function,
\begin{eqnarray}
{\bar F}^{(N+1)}&=&G^{(N+1)}_{\perp\cdot}-\sum_{n=S,T}
\phi_{n\perp}^{(1)}\otimes G_n^{(N)}\;,
\label{fn2}
\end{eqnarray}
where $\phi_{n\perp}^{(1)}=\phi_{n}^{(1)}-\phi_{n\parallel}^{(1)}$
for $n=S$ represents the sum of Eqs.~(\ref{p2a})-(\ref{p2c}).

The first term on the right-hand side of Eq.~(\ref{gnf4}) does not
contain soft divergences, since $G_n^{(N)}$ is free of soft divergences
by assumption, and the soft contributions in Eqs.~(\ref{p2a}), (\ref{p2b})
and (\ref{p2c}), those from Figs.~3(a) and 3(b) in $\phi_{nu}^{(1)}$ and
those from Figs.~3(c) and 3(d) in $\phi_{n\bar d}^{(1)}$ cancel exactly.
The function ${\bar F}^{(N+1)}$ should be gauge invariant. Hence, a Ward
identity similar to Fig.~5(a) holds for the diagrams, where the external
quarks and the soft gluon are all on-shell. We contract the loop momentum
$l$ to the vertex $\beta$ from the initial-state side in the diagrams
associated with ${\bar F}^{(N+1)}$. $G^{(N+1)}_{\perp\cdot}$ represent a
complete set of diagrams, and their contraction diminishes. However,
$\phi_{n\perp}$ do not form a complete set of diagrams, because the
soft gluon does not attach the eikonal lines. Therefore, the Ward
identity can not hold, unless ${\bar F}^{(N+1)}=0$ in the soft region.
The above argument also applies to the soft gluon emitted from the
valence quark in the final-state pion. We conclude that the diagrams
$G^{(N+1)}$ do not contain soft divergences either. Extending $N$ to
infinity, the absence of soft divergences is proved. Here we complete the
all-order proof of two-parton twist-3 factorization theorem for the
process $\pi\gamma^*\to\pi$.


\section{FACTORIZATION OF $B \to \pi \ell {\bar \nu}$ }

As emphasized in the Introduction, the contributions to the exclusive
semileptonic decay $B\to\pi l\bar\nu$ from the two-parton twist-3 pion
distribution amplitudes are in fact of the same power as those from the
twist-2 one. This is our motivation to prove corresponding factorization
theorem in the heavy quark limit. It has been shown \cite{KPY} that the
transverse degrees of freedom of partons involved in exclusive $B$ meson
decays are not negligible. However, we shall not consider these degrees
of freedom in the following derivation of factorization formulas. This
simplification will be acceptable, if the hard amplitudes are
evaluated to lowest order, as applying factorization theorem. In this
case the transverse parton momenta do not appear explicitly, which can
then be integrated out of the $B$ meson and pion wave functions.
Moreover, we do not concern the end-point singularities of the hard
amplitudes from the vanishing momentum fractions associated with the
pion \cite{ASY,SHB}. These singularities will be smeared by Sudakov
resummation eventually \cite{L3}.
 
The momentum $P_1$ of the $B$ meson and the momentum $P_2$ of the
outgoing pion are parametrized as
\begin{eqnarray}
P_1=\frac{M_B}{\sqrt{2}}(1,1,{\bf 0}_T)\;,\;\;\;
P_2=\frac{M_B}{\sqrt{2}}(0,\eta,{\bf 0}_T)\;,
\label{bmpp}
\end{eqnarray}
where $\eta$ denotes the energy fraction carried by the pion. Consider
the kinematic region with small $q^2$, $q=P_1-P_2$ being the lepton
pair momentum, {\it i.e.}, with large $\eta$, where PQCD is applicable.
In the heavy quark limit the mass difference between the $B$ meson and
the $b$ quark, $\bar\Lambda=M_B-m_b$, represents a small scale. Let the
light spectator $\bar d$ quark in the $B$ meson (pion) carry
the momentum $k_1$ ($k_2$), and the $b$ quark carry the momentum
$P_1-k_1$. We have the equations of motion,
\begin{eqnarray}
(\not P_1-\not k_1-m_b)b(P_1-k_1)=0\;,\;\;\;\;
{\bar d}(k_1)(\not k_1-m_d)=0\;,
\label{eqb}
\end{eqnarray}
and those for the valence quarks in the pion. Without the end-point
singularity, the pion distribution amplitudes are dominated by collinear
dynamics as in the pion form factor. It is then appropriate to
parameterize $k_2$ as $k_2=x_2P_2$.


\subsection{$O(\alpha_s)$ Factorization}

We start with the $O(\alpha_s)$ collinear factorization of the
final-state pion distribution amplitudes. The lowest-order diagrams for
the $B\to\pi l\bar\nu$ decay are the same as in Fig.~1, but with the
upper quark line in the initial state representing a $b$ quark and with
the symbol $\times$ representing a weak decay vertex. Figure~1(a) gives
the amplitude,
\begin{eqnarray}
H^{(0)}(x_1,x_2)&=&\frac{-g^2C_F}{2}
\frac{u({\bar x}_2P_2)\gamma^\nu(\not P_2-\not k_1)\gamma_\mu
b(P_1-k_1){\bar d}(k_1)\gamma_\nu {\bar d}(x_2P_2)}
{(P_2-k_1)^2(k_1-x_2P_2)^2}
\nonumber\\
&\approx&\frac{-g^2C_F}{2\eta^2 M_B^4}
\frac{u({\bar x}_2P_2)\gamma^\nu\not P_2\gamma_\mu
b(P_1-k_1){\bar d}(k_1)\gamma_\nu {\bar d}(x_2P_2)}{x_1^2x_2}\;,
\label{b1a}
\end{eqnarray}
where we have dropped the higher-power term $\not k_1$ in the numerator.
It is then observed that at leading power and leading order,
$H^{(0)}$ depends only on the single plus component of $k_1$ through
$k_1\cdot P_2$, which defines the momentum fraction $x_1=k_1^+/P_1^+$
carried the $\bar d$ quark in the $B$ meson.
Inserting the Fierz identity in Eq.~(\ref{1f}) into the above expression,
we obtain the lowest-order pion distribution amplitude and hard
amplitude,
\begin{eqnarray}
& &\phi_S^{(0)}=\frac{1}{4m_0}u({\bar x}_2P_2)\gamma^5
{\bar d}(x_2P_2)\;,
\nonumber\\
& &H_{S}^{(0)}(x_1,x_2)=\frac{-g^2C_F}{2\eta^2M_B^4}m_0
\frac{tr[\gamma^\nu\not P_2\gamma_\mu
b(P_1-k_1){\bar d}(k_1)\gamma_\nu\gamma^5]}{x_1^2x_2}\;.
\label{b1aS}
\end{eqnarray}
The PT structure does not contribute because of
$\gamma^\nu(\not n_+\not n_--1)\gamma^5\gamma_\nu=0$.

Consider $O(\alpha_s)$ corrections to Fig.~1(a), which are displayed
in Fig.~4 with the initial and final states flipped. Here we summarize
only the results of their factorization, and refer the details to
Appendix C. The factorization of the two-particle reducible diagrams in
Figs.~4(a)-4(c) is straightforward. After inserting the Fierz identity,
the loop integrands associated with Figs.~4(a), 4(b), and 4(c) are
written as
\begin{eqnarray}
I^{(a),(c)}&\approx& H_S^{(0)}(x_1,x_2)\phi^{(1)}_{Sa,Sc}\;,
\label{4a2}\\
I^{(b)}&\approx& H_{S}^{(0)}(x_1,\xi_2)\phi_{Sb}^{(1)}\;,
\label{b4b}
\end{eqnarray}
where the PS collinear pieces are given by
\begin{eqnarray}
\phi_{Sa}^{(1)}&=&\frac{-ig^2C_F}{4m_0}u({\bar x}_2P_2)\gamma_\beta
\frac{{\bar x}_2\not P_2+\not l}{({\bar x}_2P_2+l)^2}
\gamma^\beta \frac{1}{\bar{x}_2\not P_2}
\gamma^5{\bar d}(x_2P_2)\frac{1}{l^2}\;,
\label{p4a}\\
\phi_{Sb}^{(1)}&=&\frac{ig^2C_F}{4m_0}
u({\bar x}_2P_2)\gamma_\beta
\frac{{\bar x}_2\not P_2+\not l}{({\bar x}_2P_2+l)^2}\gamma^5
\frac{x_2\not P_2-\not l}{(x_2P_2-l)^2}\gamma^\beta
{\bar d}(x_2P_2)\frac{1}{l^2}\;,
\label{p4b} \\
\phi_{Sc}^{(1)}&=&\frac{-ig^2C_F}{4m_0}u({\bar x}_2P_2)
\gamma^5\frac{1}{x_2\not P_2}
\gamma_\beta\frac{x_2\not P_2-\not l}{(x_2P_2-l)^2}\gamma^\beta
{\bar d}(x_2P_2)\frac{1}{l^2}\;,
\label{p4c}
\end{eqnarray}
with the convolution variable $\xi_2=x_2-l^-/P_2^-$. Note that the
lowest-order hard amplitude from Fig.~4(b) depends on the loop
momentum $l$.

The collinear factorization of Figs.~4(d)-4(g) is summarized as
\begin{eqnarray}
I^{(d)} &\approx & 
\left[H_{S}^{(0)}(x_1,x_2)-H_{S}^{(0)}(x_1,\xi_2)\right]\phi_{Sd}^{(1)}\;,
\label{4bd1}\\
I^{(e)}+I^{(g)}&\approx&
H_{S}^{(0)}(x_1,x_2)\phi_{Se}^{(1)}\;,
\label{4beg}\\
I^{(f)}&\approx& -H_{S}^{(0)}(x_1,\xi_2)\phi_{Sf}^{(1)}\;,
\label{4f2}
\end{eqnarray}
with the PS collinear pieces,
\begin{eqnarray}
\phi_{Sd}^{(1)}&=&\frac{-ig^2}{2m_0C_F}u({\bar x}_2P_2)\gamma^\beta
\frac{{\bar x}_2\not P_2 +\not l}{({\bar x}_2P_2+l)^2}
\gamma^5 {\bar d}(x_2P_2)\frac{1}{l^2}
\frac{n_{+\beta}}{n_+\cdot l}\;,
\label{pb4d}\\
\phi_{Se}^{(1)}=\phi_{Sf}^{(1)}
&=&\frac{ig^2}{8m_0N_c}u({\bar x}_2P_2)\gamma^\beta
\frac{{\bar x}_2\not P_2+\not l}{({\bar x}_2P_2+l)^2}\gamma^5
{\bar d}(x_2P_2)\frac{1}{l^2}
\frac{n_{+\beta}}{n_+\cdot l}\;.
\label{pb4e}
\end{eqnarray}

The contribution from Fig.~4(d) has been split into two terms as a
consequence of the Ward identity \cite{L1} (see Appendix C). The first
and second terms correspond to the hard amplitudes without and with the
loop momentum $l$ flowing through the hard gluon, respectively. The
Feynman rule $n_{+\beta}/n_+\cdot l$ in the collinear divergent pieces,
coming from the eikonal approximation, can be represented by a Wilson
line in the direction of $n_+$. Note that each of the hard amplitudes
from Figs.~4(e) and 4(g) contains a residual dependence on $l$. It is
their sum that does not depend on $l$, and possesses the desired
factorization form as shown in Eq.~(\ref{4beg}).
Equations (\ref{pb4d}) and (\ref{pb4e})
carry different color factors due to different color flows in
Figs.~4(d)-4(g). The sum of Eqs.~(\ref{4bd1}), (\ref{4beg}) and
(\ref{4f2}) then leads to the factorization in the color space:
\begin{eqnarray}
\sum_{i=(d)}^{(g)}I^i&\approx&
H_S^{(0)}(x_1,x_2)[\phi_{Sd}^{(1)}+\phi_{Se}^{(1)}]
-H_S^{(0)}(x_1,\xi_2)[\phi_{Sd}^{(1)}+\phi_{Sf}^{(1)}]
\nonumber\\
&=&[H_S^{(0)}(x_1,x_2)-H_S^{(0)}(x_1,\xi_2)]\phi_{Su}^{(1)}\;,
\label{4bdg}
\end{eqnarray}
where the PS collinear piece,
\begin{eqnarray}
\phi_{Su}^{(1)}=\frac{-ig^2C_F}{4m_0}u({\bar x}_2P_2)\gamma^\beta
\frac{{\bar x}_2\not P_2 +\not l}{({\bar x}_2P_2 +l)^2}\gamma^5
{\bar d}(x_2P_2) \frac{1}{l^2}
\frac{n_{+\beta}}{n_+\cdot l}\;,
\label{4nu}
\end{eqnarray}
is associated with the collinear gluon emitted from the $u$ quark.

The collinear factorization of Figs.~4(h)-4(k), derived in a similar
way, is written as
\begin{eqnarray}
\sum_{i=(h)}^{(k)}I^i&\approx&
H_S^{(0)}(x_1,x_2)[\phi_{Sh}^{(1)}+\phi_{Si}^{(1)}]
-H_S^{(0)}(x_1,\xi_2)[\phi_{Sh}^{(1)}+\phi_{Sj}^{(1)}]
\nonumber\\
&=&[H_S^{(0)}(x_1,x_2)
-H_S^{(0)}(x_1,\xi_2)]\phi_{S\bar d}^{(1)}\;,
\label{4bhk}
\end{eqnarray}
where the PS collinear piece,
\begin{eqnarray}
\phi_{S\bar d}^{(1)}=\frac{ig^2C_F}{4m_0}u({\bar x}_2P_2)
\gamma^5\frac{x_2\not P_2 -\not l}{(x_2P_2 -l)^2}
\gamma^\beta {\bar d}(x_2P_2)\frac{1}{l^2}
\frac{n_{+\beta}}{n_+\cdot l}\;,
\label{4nd}
\end{eqnarray}
is associated with the collinear gluon emitted from the $\bar d$ quark.
The explicit collinear divergent functions $\phi_{Sh}^{(1)}$,
$\phi_{Si}^{(1)}$, and $\phi_{Sj}^{(1)}$ are referred to Appendix C.
Note that Fig.~4(k) also contributes to collinear divergences, and we
have to combine the results from Figs.~4(j) and 4(k) in order to arrive at
the desired factorization form. At last, the sum of
Eqs.~(\ref{4a2}), (\ref{b4b}), (\ref{4bdg}) and (\ref{4bhk}) gives
\begin{eqnarray}
\sum_{i=(a)}^{(k)}I^i\approx 
\left[H_S^{(0)}(x_1,x_2)-H_S^{(0)}(x_1,\xi_2)\right]\phi_S^{(1)}\;,
\end{eqnarray}
where the PS collinear piece $\phi_S^{(1)}$ is defined by the complex
conjugate of Eq.~(\ref{ld}), consistent with the universality.

The amplitude corresponding to Fig.~1(b) is written as
\begin{eqnarray}
H^{(0)}(x_1,x_2)&=&\frac{-g^2 C_F}{2}
\frac{u({\bar x}_2P_2)\gamma_\mu(\not P_1-x_2\not P_2+m_b)
\gamma^\nu b(P_1-k_1){\bar d}(k_1)\gamma_\nu {\bar d}(x_2P_2)}
{[(P_1-x_2P_2)^2-m_b^2](k_1-x_2P_2)^2}\;,
\label{b1b}
\end{eqnarray}
which, after inserting the Fierz identity, gives the lowest-order
pion distribution amplitudes and hard amplitudes of the PS and PT
structures,
\begin{eqnarray}
& &\phi_S^{(0)}=\frac{1}{4m_0}u({\bar x}_2P_2)\gamma^5
{\bar d}(x_2P_2)\;,
\nonumber\\
& &\phi_T^{(0)}=\frac{1}{4m_0}u({\bar x}_2P_2)
\gamma^5(\not n_+\not n_--1){\bar d}(x_2P_2)\;,
\nonumber\\
& &H_{S}^{(0)}(x_1,x_2)=\frac{-g^2 C_F}{2}m_0\frac{tr[\gamma_\mu
(\not P_1-x_2\not P_2+m_b)\gamma^\nu
b(P_1-k_1){\bar d}(k_1)
\gamma_\nu\gamma_5]}{[(P_1-x_2P_2)^2-m_b^2](k_1-x_2P_2)^2}\;,
\nonumber\\
& &H_{T}^{(0)}(x_1,x_2)=\frac{-g^2 C_F}{2}m_0\frac{tr[\gamma_\mu
(\not P_1-x_2\not P_2+m_b)\gamma^\nu
b(P_1-k_1){\bar d}(k_1)
\gamma_\nu(\not n_+\not n_--1)\gamma_5]}
{[(P_1-x_2P_2)^2-m_b^2](k_1-x_2P_2)^2}\;.
\end{eqnarray}

Below we discuss the collinear divergences in the $O(\alpha_s)$
corrections to Fig.~1(b), which are displayed in Fig.~2 with the initial
states and final states flipped. The details are referred to Appendix C.
Figures 2(a)-2(c) are factorized straightforwardly, leading to
\begin{eqnarray}
I^{(a),(c)}&\approx& \sum_{m=S,T}H_m^{(0)}(x_1,x_2)\phi^{(1)}_{ma,mc}\;,
\label{2ba2}\\
I^{(b)}&\approx&\sum_{m=S,T}H_{m}^{(0)}(x_1,\xi_2)\phi_{mb}^{(1)}\;.
\label{b2b}
\end{eqnarray}
The PS part $H_{S}^{(0)}(x_1,\xi_2)$ is written as
\begin{eqnarray}
H_{S}^{(0)}(x_1,\xi_2)&=& \frac{-g^2C_F}{2}m_0
\frac{tr[\gamma_\mu(\not P_1-x_2\not P_2+\not l+m_b)\gamma^\nu
b(P_1-k_1){\bar d}(k_1)\gamma_\nu\gamma^5]}
{[(P_1-x_2P_2+l)^2-m_b^2](k_1-x_2P_2+l)^2}\;,
\nonumber\\
&\approx& \frac{-g^2C_F}{2}m_0
\frac{tr[\gamma_\mu(\not P_1-\xi_2\not P_2+m_b)\gamma^\nu
b(P_1-k_1){\bar d}(k_1)\gamma_\nu\gamma^5]}
{[(P_1-\xi_2P_2)^2-m_b^2](k_1-\xi_2P_2)^2}\;,
\label{pb2b}
\end{eqnarray}
with the convolution variable $\xi_2=x_2-l^-/P_2^-$. The PS collinear
divergent functions $\phi^{(1)}_{Sa,Sb,Sc}$ are the same as those
shown in Eqs.~(\ref{p4a})-(\ref{p4c}), respectively. The PT pieces have
the similar expressions with $\gamma^5$ replaced by
$\gamma^5(\not n_+\not n_--1)$.

For the irreducible diagrams Figs.~2(d)-2(g), a summation of their
contributions is necessary for obtaining the desired collinear
factorization. To simplify the discussion, we show only the PS parts below.
The results are
\begin{eqnarray}
I^{(d)}&\approx &
\Bigg\{\frac{-g^2 C_F}{2}m_0\frac{tr[\gamma_\mu
(\not P_1-x_2\not P_2+\not l+m_b)\gamma^\nu b(P_1-k_1)
{\bar d}(k_1)\gamma_\nu\gamma^5]}
{[(P_1-x_2P_2+l)^2-m_b^2](k_1-x_2P_2)^2}-H^{(0)}_{S}(x_1,\xi_2)\Bigg\}
\phi^{(1)}_{Sd}\;,
\label{2bd}\\
I^{(e)}&\approx &
\Bigg\{\frac{-g^2 C_F}{2}m_0\frac{tr[\gamma_\mu
(\not P_1-x_2\not P_2+\not l+m_b)\gamma^\nu b(P_1-k_1)
{\bar d}(k_1)\gamma_\nu\gamma^5]}
{[(P_1-x_2P_2+l)^2-m_b^2](k_1-x_2P_2)^2}\Bigg\}
\phi^{(1)}_{Se}\;,
\label{2be}\\
I^{(f)}&\approx& -H^{(0)}_{S}(x_1,\xi_2)\phi^{(1)}_{Sf}\;,
\label{2bf}
\end{eqnarray}
with the collinear divergent functions $\phi_{Sd,Se,Sf}^{(1)}$ shown in
Eqs.~(\ref{pb4d}) and (\ref{pb4e}). Combining the above expressions,
we have
\begin{eqnarray}
I^{(d)}+I^{(e)}+I^{(f)}&\approx &
\Bigg\{\frac{-g^2 C_F}{2}m_0\frac{tr[\gamma_\mu
(\not P_1-x_2\not P_2+\not l+m_b)\gamma^\nu b(P_1-k_1)
{\bar d}(k_1)\gamma_\nu\gamma^5]}
{[(P_1-x_2P_2+l)^2-m_b^2](k_1-x_2P_2)^2}-H^{(0)}_{S}(x_1,\xi_2)\Bigg\}
\phi^{(1)}_{Su}\;,
\label{2bdf}
\end{eqnarray}
with the same PS collinear function $\phi_{Su}^{(1)}$ as in
Eq.~(\ref{4nu}).

Consider the loop integrand associated with Fig.~2(g) under the
approximation,
\begin{eqnarray}
\frac{\not P_1-x_2\not P_2+\not l +m_b}{(P_1-x_2P_2+l)^2-m_b^2}
\gamma_\beta\frac{\not P_1-x_2\not P_2+m_b}{(P_1-x_2P_2)^2-m_b^2}
&\approx&\frac{2P_{1\beta}}{(P_1-x_2P_2+l)^2-m_b^2}
\frac{\not P_1+m_b}{(P_1-x_2P_2)^2-m_b^2}
\nonumber \\
&\approx &\frac{2P_1\cdot l}{(P_1-x_2P_2+l)^2-m_b^2}
\frac{n_{+\beta}}{n_+\cdot l}
\frac{\not P_1+m_b}{(P_1-x_2P_2)^2-m_b^2}\;.
\nonumber
\end{eqnarray}
The neglect of $\xi_2\not P_2$ and $x_2\not P_2$ in the first and second
propagators, respectively, is due to $\gamma_\beta=\gamma^+$ in the
collinear region. The integrand $I^{(g)}$ reduces to
\begin{eqnarray}
I^{(g)}\approx \frac{2P_1\cdot l}{(P_1-x_2P_2+l)^2-m_b^2}
\left(\frac{-g^2}{2}C_Fm_0\right)\frac{tr[\gamma_\mu
(\not P_1+m_b)\gamma^\alpha b(P_1-k_1)
{\bar d}(k_1)\gamma_\alpha\gamma^5]}
{[(P_1-x_2P_2)^2-m_b^2](k_1-x_2P_2)^2}\phi^{(1)}_{Su}\;.
\label{2bg}
\end{eqnarray}
The combination of the first term in Eq.~(\ref{2bdf}) and Eq.~(\ref{2bg})
then leads to the the correct factorization form with the hard amplitude
$H_{S}^{(0)}(x_1,x_2)$ independent of $l$.

Summing Figs.~2(d)-2(g) with the PT parts included, we arrive at
\begin{eqnarray}
\sum_{i=(d)}^{(g)}I^i&\approx& \sum_{m=S,T}
[H_m^{(0)}(x_1,x_2)-H_m^{(0)}(x_1,\xi_2)]\phi_{mu}^{(1)}\;.
\label{2bdg}
\end{eqnarray}
The collinear factorization of Fig.~2(h)-2(k), derived in 
a similar way, is written as
\begin{eqnarray}
\sum_{i=(h)}^{(k)}I^i&\approx& \sum_{m=S,T}
[H_m^{(0)}(x_1,x_2)-H_m^{(0)}(x_1,\xi_2)]\phi_{m{\bar d}}^{(1)}\;,
\label{2bhk}
\end{eqnarray}
with the PS collinear piece $\phi_{S\bar d}^{(1)}$
shown in Eq.~(\ref{4nd}). At last, the sum of
Eqs.~(\ref{2ba2}), (\ref{b2b}), (\ref{2bdg}) and (\ref{2bhk}) gives
\begin{eqnarray}
\sum_{i=(a)}^{(k)}I^i\approx
\sum_{m=S,T}\left[H_m^{(0)}(x_1,x_2)
-H_m^{(0)}(x_1,\xi_2)\right]\phi_m^{(1)}\;,
\label{phi2}
\end{eqnarray}
where the collinear divergent functions $\phi_m^{(1)}$ are defined by
the complex conjugate of Eq.~(\ref{ld}).

The factorization of the soft divergences associated with the
initial-state $B$ meson has been discussed in \cite{L1}, which results
in two light-cone $B$ meson distribution amplitudes $\phi_{+}$ and
$\phi_-$ \cite{GN,BF}. The gauge invariance of the two distribution
amplitudes can be achieved, only if the hard scale for the exclusive $B$
decays is of $O(\bar\Lambda M_B)$. For their explicit definitions as
nonlocal matrix elements, refer
to \cite{L1}. Following the similar procedures in Sec.~II, we derive the
$O(\alpha_s)$ factorization of the process $B\to \pi l\bar \nu$, 
\begin{eqnarray}
G^{(1)}=\sum_{n=+,-}\phi^{(1)}_n\otimes H^{(0)}_n
+\sum_{m=S,T}H^{(0)}_m\otimes\phi^{(1)}_m+H^{(1)}\;.
\end{eqnarray}
Consequently, the factorization formula for the two-parton twist-3
contributions is written, up to $O(\alpha_s)$, as
\begin{eqnarray}
G^{(0)}+G^{(1)}=\sum_{n=+,- \atopwithdelims . .m=S,T}
(\phi^{(0)}_n+\phi^{(1)}_n)
\otimes(H_{nm}^{(0)}+H^{(1)}_{nm})\otimes
(\phi^{(0)}_m+\phi^{(1)}_m)\;,
\end{eqnarray}
with the lowest-order diagrams $G^{(0)}\equiv H^{(0)}$. The definitions
for the hard amplitudes $H_{nm}^{(1)}$ and $H_{nm}^{(0)}$ are similar
to those in Eq.~(\ref{wi3}).



\subsection{All-order Factorization}

The all-order proof of two-parton twist-3 factorization theorem for the
process $\pi\gamma^*\to\pi$ in Sec.~II can be generalized to the
$B\to\pi l\bar\nu$ decay straightforwardly. Here we simply summarize the
different points of the proof, and neglect the details. In the case of
$B$ meson decays there is no collinear divergence associated with the
initial state, since the $b$ quark is massive, and the light spectator
$\bar d$ quark is soft \cite{L1}. The important infrared divergences are
then classified into the soft type with small loop momentum $l$ and the
collinear type with $l$ parallel to $P_2$. Identify the
gluon emitted from the outer most vertex $\alpha$ on the $b$ quark line
in the $O(\alpha_s^{N+1})$ diagrams $G^{(N+1)}$. Let $\beta$ denote the
attachments of the other end of the identified gluon inside the diagrams.
Note that the attachment of a soft gluon to a hard line off-shell by
$O(\bar\Lambda M_B)$ also leads to soft divergences \cite{L1}. Hence, we
decompose the tensor $g_{\alpha\beta}$ into
\begin{eqnarray}
g_{\alpha\beta}=\frac{n_{-\alpha} l_\beta}{n_-\cdot l}
+\left(g_{\alpha\beta}-\frac{n_{-\alpha} l_\beta}{n_-\cdot l}\right)\;.
\label{dec1}
\end{eqnarray}
The first term on the right-hand side of Eq.~(\ref{dec1}) corresponds to
the configurations with the identified gluon attaching a collinear line
along $P_2$, attaching a hard line off-shell by $O(\bar\Lambda M_B)$,
and attaching a soft line. The second term corresponds only to the
configuration with the identified gluon attaching a soft line. It will
be shown that it plays the role similar to the second term on the
right-hand side of Eq.~(\ref{dec}).

The decomposition in Eq.~(\ref{dec1}) also applies to the soft gluon
emitted from the light spectator $\bar d$ quark in the $B$ meson.
Following the similar reasoning, we derive
\begin{eqnarray}
G^{(N+1)}_{\parallel\cdot}&\approx&
\sum_{n=+,-}(\phi_{nb}^{(1)}+\phi_{n\bar d}^{(1)})
\otimes G_n^{(N)}\;.
\label{wic1}
\end{eqnarray}
To derive the above expression, the diagrams containing the $O(\alpha_s)$
reducible subdiagrams, {\it i.e.}, those in Figs.~2(a)-2(c) and in
Fig.~4(a)-4(c), have been
excluded from the set of $G^{(N+1)}_{\parallel\cdot}$. Equation
(\ref{wicr}) holds for the final-state collinear factorization, which
comes from the replacement,
\begin{eqnarray}
g_{\alpha\beta}\to \frac{n_{+\alpha} l_\beta}{n_+\cdot l}\;,
\end{eqnarray}
$l$ being mainly in the minus direction. The above procedures, applied
to the $O(\alpha_s^{j+1})$ $B$ meson distribution amplitudes
$\phi_n^{(j+1)}$, give
\begin{eqnarray}
\phi_{n\parallel}^{(j+1)}&\approx& \sum_{n'=+,-}
(\phi_{n'b}^{(1)}+\phi_{n'\bar d}^{(1)})\otimes \phi_{n'n}^{(j)}\;.
\label{wi2b}
\end{eqnarray}
Again, the second formula in Eq.~(\ref{wi2}) holds.

We then obtain the expressions for the decay $B\to\pi l\bar\nu$ similar
to Eqs.~(\ref{gf}) and (\ref{gnf2}). After including the second term on
the right-hand side of Eq.~(\ref{dec1}) and those diagrams containing
the $O(\alpha_s)$ reducible subdiagrams, which have been excluded in the
derivation of Eqs.~(\ref{wic1}) and (\ref{wi2b}), we arrive at
\begin{eqnarray}
G^{(N+1)}&\approx&
\sum_{n=+,- \atopwithdelims . .m=S,T}
\sum_{i=0}^{N}\sum_{j=0}^{N-i}
\left[\phi_{n}^{(i+1)}\otimes H_{nm}^{(N-i-j)}\otimes \phi_{m}^{(j)}
+\phi_{n}^{(i)}\otimes H_{nm}^{(N-i-j)}\otimes \phi_{m}^{(j+1)}\right.
\nonumber\\
& &\left.-\phi_{n}^{(i+1)}\otimes H_{nm}^{(N-i-j-1)}\otimes
\phi_{m}^{(j+1)}\right]+F^{(N+1)}\;,
\label{gnfb}
\end{eqnarray}
with the $O(\alpha_s^{N+1})$ function,
\begin{eqnarray}
F^{(N+1)}&=&G^{(N+1)}_{\perp\perp}-\left[\sum_{n=+,-}
\sum_{i=1}^{N+1}\phi_{n\perp}^{(i)}\otimes H_{n}^{(N+1-i)}
+\sum_{m=S,T}\sum_{j=1}^{N+1} H_{m}^{(N+1-j)}\otimes \phi_{m\perp}^{(j)}
\right.
\nonumber\\
& &\left.+\sum_{n=+,- \atopwithdelims . .m=S,T}
\sum_{i=1}^{N+1}\sum_{j=1}^{N+1-i}
\phi_{n\perp}^{(i)}\otimes H_{nm}^{(N+1-i-j)}\otimes
\phi_{m\perp}^{(j)}\right]\;.
\label{fnb}
\end{eqnarray}
Here the subscript $\perp$ associated with
the $B$ meson side corresponds to the second term on the right-hand
side of Eq.~(\ref{dec1}), and those diagrams containing
the $O(\alpha_s)$ reducible subdiagrams.

We argue that $F^{(N+1)}$ is free of infrared divergences by means of
gauge invariance. A Ward identity similar to Fig.~5(a) holds for the
diagrams associated with $F^{(N+1)}$, where the external quarks and the
identified gluon are all on-shell in the soft and collinear regions. We
contract the loop momentum $l$ to the vertices $\beta$ from the
initial-state or final-state side of the diagrams for $F^{(N+1)}$.
$G^{(N+1)}_{\perp\perp}$ and $H$ represent complete sets of diagrams,
and their contractions diminish. However, $\phi_{n\perp}$ and
$\phi_{m\perp}$ do not form complete sets of diagrams, since the
identified gluon does not attach the eikonal lines, which come from the
eikonal approximation of the collinear lines. Therefore, the Ward
identity can not hold, unless $F^{(N+1)}=0$ in the soft region for the
initial state and in the collinear region for the final state. Moreover,
the soft configuration arises from the attachment of the identified gluon
to a soft internal line. It indicates that all soft divergences in
$F^{(N+1)}$ disappear together with the vanishing of the soft divergences
from the identified gluon.

At last, Eq.~(\ref{gnfb}) becomes
\begin{eqnarray}
G^{(N+1)}=\sum_{n=+,- \atopwithdelims . .m=S,T}
\sum_{i=0}^{N+1}\sum_{j=0}^{N+1-i}
\phi_n^{(i)}\otimes H_{nm}^{(N+1-i-j)}\otimes \phi_m^{(j)}\;,
\label{gf2}
\end{eqnarray}
where the $O(\alpha_s^{N+1})$ hard amplitude $H_{nm}^{(N+1)}$,
defined via
\begin{eqnarray}
F^{(N+1)}=\sum_{n=+,- \atopwithdelims . .m=S,T}
\phi_n^{(0)}\otimes H_{nm}^{(N+1)}
\otimes \phi_m^{(0)}\;,
\label{hn1}
\end{eqnarray}
is infrared finite. Equation (\ref{gf2}) indicates that all the soft and
collinear divergences in the semileptonic decay $B\to\pi l\bar\nu$ can be
factorized into the $B$ meson and pion distribution amplitudes,
respectively, order by order. We then complete the proof of corresponding
two-parton twist-3 factorization theorem.


\section{CONCLUSION}

In this paper we have investigated the infrared divergences in the
process $\pi\gamma^*\to\pi$ at the two-parton twist-3 level. We
summarize our observations below. There are no soft divergences
associated with the pion, since they are either absent or cancel among
sets of diagrams. The absence of the soft divergences is related to the
fact that a soft gluon, being huge in space-time, does not resolve the
color structure of the color-singlet pion. In the collinear region
with the loop momentum parallel to the pion momentum, we have shown that
the delicate summation of different diagrams leads to the $O(\alpha_s)$
factorization in the momentum, spin and color spaces. We have presented
an all-order proof of two-parton twist-3 factorization theorem for the
process $\pi\gamma^*\to\pi$ by means of the Ward identity and gauge
invariance. This proof can also accommodate leading-twist factorization
theorem presented in \cite{L1}. The idea is to decompose the tensor
$g_{\alpha\beta}$ for the identified collinear gluon
into the longitudinal and transverse pieces shown in
Eq.~(\ref{dec}). The longitudinal (transverse) piece corresponds to the
configuration without (with) the attachment of the collinear gluon to
a line along the external momentum. The former configuration can be
factorized using the Ward identity as hinted by the replacement in
Eq.~(\ref{rep}). The latter configuration is then merged into the former
one by requiring gauge invariance of the factorization. We have
constructed the definitions of the two-parton twist-3 pion distribution
amplitudes, where the path-ordered integral appears as a consequence of
the Ward identity.

We have generalized the proof to the more complicated semileptonic decay
$B\to\pi l\bar\nu$. The collinear factorization for the final-state pion
is the same as in the process $\pi\gamma^*\to\pi$. The identical collinear
structures in both processes are consistent with the concept of
universality of hadron distribution amplitudes in PQCD. The soft
factorization for the initial-state $B$ meson has been discussed in
\cite{L1}. The conceptual differences are summarized as follows. The
decomposition of the tensor $g_{\alpha\beta}$ for the soft gluon in
Eq.~(\ref{dec1}) works. In this case the longitudinal piece corresponds
to the configuration with the attachment of the soft gluon to all internal
lines, including the hard lines off-shell by $O(\bar\Lambda M_B)$. The
remaining piece corresponds only to the attachment to a soft line, not to
a collinear line along the final-state pion momentum. This configuration
is similar to the one associated with the transverse piece in the process
$\pi\gamma^*\to\pi$. The procedures of the proof then follow those for
the pion form factor. Note that our technique is simple compared to that
based on the ''$\Delta$-forest" prescription in \cite{DM}, and explicitly
gauge invariant, compared to that perform in the light-cone gauge
\cite{BL}.

For a practical application to the $B\to\pi l\bar\nu$ decay, the parton
transverse momenta $k_T$ must be taken into account in order to smear the
end-point singularities in the hard amplitudes \cite{TLS,LY1}. On the
other hand, it has been shown that the $k_T$ dependence does not decouple
from the hard amplitudes in the exclusive $B$ meson decays \cite{KPY}.
The above observations imply the necessity of proving $k_T$ factorization
theorem \cite{BS,LS}. The proof will be eventually generalized to
nonleptonic $B$ meson decays, such as $B\to K\pi$ and $\pi\pi$
\cite{KLS,LUY}. The corresponding factorization theorem is more
complicated, since nonleptonic decays involve three characteristic
scales: the $W$ boson mass $M_W$, $M_B$, and the factorization scale of
$O(\bar\Lambda)$, such as $k_T$ \cite{CL,YL}.

\vskip 0.5cm

We thank S. Brodsky, A.I. Sanda, and G. Sterman for useful discussions.
This work was supported in part by the National Science Council of R.O.C.
under the Grant No. NSC-90-2112-M-001-077, by National Center for
Theoretical Sciences of R.O.C., and by Grant-in Aid for Special Project
Research (Physics of CP Violation) from the Ministry of Education,
Science and Culture, Japan.



\appendix

\section{$O(\alpha_s)$ CORRECTIONS TO FIG. 1(A)}

In this Appendix we present the details of the collinear factorization
of Fig.~2. The loop integrand of Fig.~2(b) is given by
\begin{eqnarray}
I^{(b)}&=& -\frac{1}{2}eg^4 C_F^2\frac{1}{(x_2P_2-x_1P_1+l)^2}
{\bar d}(x_1P_1)\gamma_\beta
\frac{x_1\not P_1-\not l}{(x_1P_1-l)^2}\gamma_\alpha
\nonumber\\
& &\times
{\bar d}(x_2 P_2)u({\bar x}_2 P_2)
\gamma^\alpha\frac{\not P_2-x_1\not P_1+\not l}{(P_2-x_1P_1+l)^2}
\gamma_\mu \frac{{\bar x}_1\not P_1 +\not l}{({\bar x}_1 P_1 +l)^2}
\gamma^\beta u({\bar x}_1P_1)\frac{1}{l^2}\;,
\label{2b}
\end{eqnarray}
Inserting the Fierz identity, we obtain
\begin{eqnarray}
I^{(b)}&=&\phi^{(1)}_{Sb}\left(\frac{1}{2}ieg^2C_Fm_0\right)
tr\Bigg\{\frac{\gamma_\alpha
{\bar d}(x_2 P_2)u({\bar x}_2 P_2)
\gamma^\alpha(\not P_2-x_1\not P_1+\not l)\gamma_\mu
\gamma_5}{(P_2-x_1P_1+l)^2(x_2P_2-x_1P_1+l)^2}\Bigg\}
\nonumber\\
& & +\phi^{(1)}_{Tb}\left(\frac{1}{2}ieg^2C_Fm_0\right)
tr\Bigg\{\frac{\gamma_\alpha
{\bar d}(x_2 P_2)u({\bar x}_2 P_2)
\gamma^\alpha(\not P_2-x_1\not P_1+\not l)\gamma_\mu
(\not n_+\not n_--1)\gamma_5}
{(P_2-x_1P_1+l)^2(x_2P_2-x_1P_1+l)^2}\Bigg\}\;.
\label{2bc}
\end{eqnarray}
To derive the above expression, the twist-2 structure
$(\gamma_5\not n_-)_{ik}(\not n_+\gamma_5)_{jl}$ has been dropped.
Equation (\ref{2bc}), as integrated over $l$, is rewritten as the
convolution of the lowest-order hard amplitude $H_S^{(0)}(\xi_1,x_2)$
$[H_T^{(0)}(\xi_1,x_2)]$ with $\phi^{(1)}_{Sb}$ ($\phi^{(1)}_{Tb}$) in
the momentum fraction $\xi_1=x_1-l^+/P_1^+$.
The dependences on $l^-$ and on $l_T$ in $H_S^{(0)}$ and
in $H_T^{(0)}$, being subleading according to Eq.~(\ref{sog2}), have
been neglected. The collinear factorization of Figs.~2(a) and 2(c)
can be performed in a similar way, leading to Eq.~(\ref{2a2}).

The loop integrand from Fig.~2(d) is written as
\begin{eqnarray}
I^{(d)}&=&\frac{ieg^4}{2N_c}{\bar d}(x_1P_1)\gamma^\lambda
{\bar d}(x_2 P_2)u({\bar x}_2 P_2)
\gamma^\beta\frac{\not P_2-x_1\not P_1+\not l}{(P_2-x_1P_1+l)^2}
\gamma_\mu\frac{{\bar x}_1\not P_1 +\not l}{({\bar x}_1 P_1 +l)^2}
\nonumber\\
& &\times \gamma^\alpha u({\bar x}_1P_1)
\frac{tr(T^cT^bT^a)\Gamma^{cba}_{\lambda\beta\alpha}}
{l^2(x_1P_1-x_2P_2-l)^2(x_1P_1-x_2P_2)^2}\;,
\label{2d0}
\end{eqnarray}
with the color matrices $T^{a,b,c}$ and the triple-gluon vertex,
\begin{eqnarray}
\Gamma^{cba}_{\lambda\beta\alpha}&=&f^{cba}
[g_{\alpha\beta}(2l-x_1P_1+x_2P_2)_\lambda
+g_{\beta\lambda}(2x_1P_1-2x_2P_2-l)_\alpha
\nonumber\\
& &+g_{\lambda\alpha}(x_2P_2-x_1P_1-l)_\beta]\;,
\label{tg}
\end{eqnarray}
$f^{abc}$ being an antisymmetric tensor. The above color structure can be
simplified by employing the identities,
\begin{eqnarray}
tr(T^aT^bT^c)=\frac{1}{4}(d^{abc}+if^{abc})\;,\;\;\;
d^{abc}f^{abc}=0\;,\;\;\;f^{abc}f^{abc}=24\;,
\label{tco}
\end{eqnarray}
$d^{abc}$ being a symmetric tensor.

In the collinear region with $l$ parallel to $P_1$, the terms proportional
to $g_{\alpha\beta}$ and $g_{\lambda\alpha}$ do not contribute. Since
the gamma matrices must be $\gamma^\alpha=\gamma^+$,
$\gamma^\beta=\gamma^-$, and $\gamma_\mu=\gamma_-=\gamma^+$, the quark
propagator between $\gamma_\mu$ and $\gamma^\beta$ is proportional
to $l_T$, which is subleading in the collinear region. The factor
$g_{\lambda\alpha}$ indicates that $\gamma^\lambda$ must be $\gamma^-$,
because of $\gamma^\alpha=\gamma^+$. According to Eq.~(\ref{eqm}), the
contribution from ${\bar d}(x_1P_1)\gamma^\lambda={\bar d}(x_1P_1)\gamma^-$
is subleading and negligible. The second term associated with
$g_{\beta\lambda}$ in Eq.~(\ref{tg}) contains collinear divergence.
Due to $\gamma^\alpha=\gamma^+$, only the term $-2x_2P_{2\alpha}$
contributes. In the collinear region we have the approximation,
\begin{eqnarray}
\frac{-2x_2P_{2\alpha}}{(x_1P_1-x_2P_2)^2(x_1P_1-x_2P_2-l)^2}
\approx -\frac{n_{-\alpha}}{n_-\cdot l}\biggl[\frac{1}{(x_1P_1-x_2P_2)^2}
-\frac{1}{(x_1P_1-x_2P_2-l)^2}\biggr]\;.
\label{pi}
\end{eqnarray}
The expression in Eq.~(\ref{2d0}) is then split, after the insertion of
the Fierz identity, into Eq.~(\ref{2d2}). The hard amplitude
$H_{S(T)}^{(0)}(x_1,x_2)$ $[H_{S(T)}^{(0)}(\xi_1,x_2)]$ in Eq.~(\ref{2d2})
comes from the first (second) term in Eq.~(\ref{pi}).

The loop integrand associated with Fig.~2(e) is given by
\begin{eqnarray}
I^{(e)}&=&-\frac{eg^4 C_F}{4N_c}{\bar d}(x_1P_1)\gamma_\alpha
{\bar d}(x_2 P_2)u({\bar x}_2 P_2)
\gamma_\beta\frac{{\bar x}_2\not P_2+\not l}{({\bar x}_2P_2+l)^2}
\gamma^\alpha\frac{\not P_2-x_1\not P_1 +\not l}{(P_2-x_1 P_1 +l)^2}
\gamma_\mu
\nonumber\\
& &\times \frac{{\bar x}_1\not P_1 +\not l}{({\bar x}_1P_1+l)^2}
\gamma^\beta u({\bar x}_1P_1)\frac{1}{l^2(x_1P_1-x_2P_2)^2}\;.
\label{2e}
\end{eqnarray}
In the collinear region $\gamma^\beta$, $\gamma_\beta$,
$\gamma^\alpha$ and $\gamma_\alpha$ must be $\gamma^+$, $\gamma^-$,
$\gamma^T$ and $\gamma^T$, respectively. Using the eikonal approximation,
\begin{eqnarray}
u({\bar x}_2P_2)\gamma_\beta
\frac{{\bar x}_2\not P_2+\not l}{({\bar x}_2P_2+l)^2}
\approx u({\bar x}_2P_2)\frac{n_{-\beta}}{n_-\cdot l}\;,
\label{eap}
\end{eqnarray}
and inserting Eq.~(\ref{1f}), Eq.~(\ref{2e}) leads to Eq.~(\ref{2e2}).

Following the similar treatment, the loop integrand associated with
Fig.~2(f) reduces to
\begin{eqnarray}
I^{(f)}&=&\frac{eg^4 C_F}{4N_c}{\bar d}(x_1P_1)\gamma^\alpha
\frac{x_2\not P_2+\not l}{(x_2P_2+l)^2}\gamma_\beta
{\bar d}(x_2 P_2)u({\bar x}_2 P_2)
\gamma_\alpha\frac{\not P_2-x_1\not P_1 +\not l}{(P_2-x_1 P_1 +l)^2}
\gamma_\mu
\nonumber\\
& &\times \frac{\bar{x}_1\not P_1 +\not l}{(\bar{x}_1P_1+l)^2}
\gamma^\beta u({\bar x}_1P_1)\frac{1}{l^2(x_1P_1-x_2P_2-l)^2}\;,
\end{eqnarray}
which then leads to Eq.~(\ref{2f2}).

The integrand of Fig.~2(g) is written as,
\begin{eqnarray}
I^{(g)}&=&\frac{eg^4 C_F^2}{2} 
{\bar d}(x_1P_1)\gamma_\alpha
{\bar d}(x_2 P_2)u({\bar x}_2 P_2)
\gamma^\alpha\frac{\not P_2-x_1\not P_1}{(P_2-x_1P_1)^2}\gamma_\beta
\frac{\not P_2-x_1\not P_1+\not l}{(P_2-x_1P_1+l)^2}\gamma_\mu
\nonumber\\
& &\times \frac{\bar{x}_1\not P_1 +\not l}{(\bar{x}_1P_1 +l)^2}
\gamma^\beta u({\bar x}_1P_1)\frac{1}{l^2(x_1P_1-x_2P_2)^2}\;.
\label{2g}
\end{eqnarray}
In collinear region with $l$ parallel to $P_1$, we have the sequence of
the gamma matrices $\gamma^\beta=\gamma^+$, $\gamma_\mu=\gamma^+$, and
$\gamma_\beta=\gamma^-$. The quark propagator between $\gamma_\mu$ and
$\gamma_\beta$ is proportional to $l_T$, which gives subleading
contribution.

Figure 2(h) gives the loop integrand,
\begin{eqnarray}
I^{(h)}&=&\frac{-ieg^4}{2N_c}{\bar d}(x_1P_1)\gamma^\lambda
\frac{x_1\not P_1 -\not l}{(x_1 P_1 -l)^2}
\gamma^\beta{\bar d}(x_2P_2)u({\bar x}_2P_2)\gamma^\alpha
\frac{\not P_2-x_1\not P_1}{(P_2-x_1P_1)^2}
\nonumber\\
& &\times \gamma_\mu u({\bar x}_1P_1)
\frac{tr(T^cT^bT^a)\Gamma^{cba}_{\lambda\beta\alpha}}
{l^2(x_1P_1-x_2P_2-l)^2(x_1P_1-x_2P_2)^2}\;,
\label{2h0}
\end{eqnarray}
with the triple-gluon vertex,
\begin{eqnarray}
\Gamma^{cba}_{\lambda\beta\alpha}&=&f^{cba}
[g_{\beta\lambda}(2l-x_1P_1+x_2P_2)_\alpha
+g_{\alpha\beta}(2x_1P_1-2x_2P_2-l)_\lambda
\nonumber\\
& &+g_{\lambda\alpha}(x_2P_2-x_1P_1-l)_\beta]\;.
\label{tge}
\end{eqnarray}
In the collinear region with $l$ parallel to $P_1$, 
the terms proportional to $g_{\beta\lambda}$ 
and $g_{\lambda\alpha}$ do not contribute. Because $\gamma^\lambda$
must be $\gamma^+$, the factor $g_{\beta\lambda}$ implies that
$\gamma^\beta$ must be $\gamma^-$ and the quark propagator between
$\gamma^\lambda$ and $\gamma^\beta$ is proportional to $l_T$, which is
subleading. The factor $g_{\lambda\alpha}$ implies
$\gamma^\alpha =\gamma^-$. Since $\gamma_\mu$ is adjacent to the spinor
$u(\bar{x}_1P_1)$, it must be $\gamma^+$. The quark propagator between
$\gamma_\mu$ and $\gamma^\alpha$ gives the subleading contribution
proportional to $l_T$. Only the second term in Eq.~(\ref{tge}) contains
the collinear divergence.

Applying the approximation in Eq.~(\ref{pi}), and inserting the Fierz
identity in Eq.~(\ref{1f}), Eq.~(\ref{2h0}) becomes
\begin{eqnarray}
I^{(h)}\approx \sum_{n=S,T}\phi_{nh}^{(1)}\left[H_n^{(0)}(x_1,x_2)
-H_n^{(0)}(\xi_1,x_2)\right]\;,
\label{2h2}
\end{eqnarray}
with the $O(\alpha_s)$ collinear divergent PS piece,
\begin{eqnarray}
\phi_{Sh}^{(1)}=\frac{ig^2}{2m_0C_F}
{\bar d}(x_1P_1)\gamma^\beta
\frac{x_1\not P_1 -\not l}{(x_1 P_1 -l)^2}
\gamma^5u({\bar x}_1P_1)\frac{1}{l^2}
\frac{n_{-\beta}}{n_-\cdot l}\;.
\label{p2h}
\end{eqnarray}

The loop integrand associated with Fig.~2(i) is written as
\begin{eqnarray}
I^{(i)}&=&\frac{-eg^4 C_F}{4N_c}{\bar d}(x_1P_1)\gamma^\alpha
\frac{x_1\not P_1-\not l}{(x_1P_1-l)^2}\gamma^\beta
\frac{x_2\not P_2-\not l}{(x_2P_2-l)^2}\gamma_\alpha{\bar d}(x_2P_2)
\nonumber \\
& &\times u({\bar x}_2P_2)\gamma_\beta
\frac{\not P_2-x_1\not P_1}{(P_2-x_1 P_1)^2}
\gamma_\mu u({\bar x}_1P_1)\frac{1}{l^2(x_1P_1-x_2P_2)^2}\;.
\label{2i}
\end{eqnarray}
After applying the eikonal approximation similar to Eq.~(\ref{eap}), and
inserting the Fierz identity in Eq.~(\ref{1f}), the above expression is
simplified to
\begin{eqnarray}
I^{(i)}\approx  \sum_{n=S,T}\phi_{ni}^{(1)}H_n^{(0)}(x_1,x_2)\;,
\label{2i2}
\end{eqnarray}
with the collinear divergent PS piece,
\begin{eqnarray}
\phi_{Si}^{(1)}=\frac{-ig^2}{8m_0N_c}{\bar d}(x_1P_1)\gamma^\beta
\frac{x_1\not P_1 -\not l}{(x_1 P_1 -l)^2}
\gamma^5u({\bar x}_1P_1)\frac{1}{l^2}
\frac{n_{-\beta}}{n_-\cdot l}\;.
\label{p2i}
\end{eqnarray}

Figure 2(j) gives the loop integrand,
\begin{eqnarray}
I^{(j)}&=&\frac{eg^4 C_F}{4N_c}{\bar d}(x_1P_1)\gamma^\alpha
\frac{x_1\not P_1-\not l}{(x_1P_1-l)^2}\gamma^\beta{\bar d}(x_2P_2)
u({\bar x}_2P_2)\gamma_\alpha
\frac{\bar{x}_2\not P_2-\not l}{(\bar{x}_2P_2-l)^2}
\nonumber \\
& &\times \gamma_\beta\frac{\not P_2-x_1\not P_1}{(P_2-x_1 P_1)^2}
\gamma_\mu u({\bar x}_1P_1)\frac{1}{l^2(x_1P_1-x_2P_2-l)^2}\;.
\label{2j}
\end{eqnarray}
Following the same procedure as for Eq.~(\ref{2i}), we obtain
\begin{eqnarray}
I^{(j)}\approx -\sum_{n=S,T}\phi_{nj}^{(1)}H_n^{(0)}(\xi_1,x_2)\;,
\label{2j2}
\end{eqnarray}
with the collinear divergent PS piece $\phi_{Sj}^{(1)}$
the same as $\phi_{Si}^{(1)}$ in Eq.~(\ref{p2i}).

At last, the integrand associated with Fig.~2(k) is given by
\begin{eqnarray}
I^{(k)}&=&\frac{-eg^4 C_F^2}{2}{\bar d}(x_1P_1)\gamma^\alpha
\frac{x_1\not P_1 -\not l}{(x_1 P_1 -l)^2}
\gamma^\beta{\bar d}(x_2P_2)u({\bar x}_2P_2)\gamma_\beta
\frac{\not P_2-x_1\not P_1+\not l}{(P_2-x_1P_1+l)^2}\gamma_\alpha
\nonumber \\
& &\times \frac{\not P_2-x_1\not P_1}{(P_2-x_1P_1)^2}
\gamma_\mu u({\bar x}_1P_1)\frac{1}{l^2(x_1P_1-x_2P_2-l)^2}\;.
\label{2k}
\end{eqnarray}
It is easy to find that the collinear divergence does not exist for
the same reason as for Fig.~2(g): in the collinear region we have
the gamma matrices $\gamma^\alpha=\gamma^+$ and $\gamma^\beta=\gamma^+$
being adjacent to the the spinor ${\bar d}(x_2P_2)$. 
The contribution is then subleading because of the equations of 
motion for the final-state quarks.


\section{$O(\alpha_s)$ CORRECTIONS TO FIG.~1(B)}

In this Appendix we discuss the factorization of collinear divergences
in the $O(\alpha_s)$ radiative corrections to Fig.~1(b), which are shown
in Fig.~4. Figure 1(b) gives only the lowest-order PS distribution
amplitude $\phi_S^{(0)}$ and the hard amplitude $H_S^{(0)}$,
\begin{eqnarray}
& &\phi_S^{(0)}=\frac{1}{4m_0}{\bar d}(x_1P_1)\gamma^5
u({\bar x}_1P_1)\;,
\nonumber\\
& &H_{S}^{(0)}(x_1,x_2)=\frac{i}{2}eg^2 C_Fm_0
\frac{tr[\gamma_\nu
{\bar d}(x_2P_2)u({\bar x}_2P_2)\gamma_\mu
(\not P_1-x_2\not P_2)\gamma^\nu\gamma^5]}
{(P_1-x_2P_2)^2(x_1P_1-x_2P_2)^2}\;,
\label{1bS}
\end{eqnarray}
because of $\gamma^\nu (\not n_+\not n_--1)\gamma_\nu=0$, 
where the gamma matrices $\gamma^\nu$ and $\gamma_\nu$ 
come from the gluon vertices in Fig.~1(b). 

The integrand from Fig.~4(b) is written as
\begin{eqnarray}
I^{(b)}&=& -\frac{1}{2}eg^4 C_F^2\frac{1}{(x_2P_2-x_1P_1+l)^2}
{\bar d}(x_1P_1)\gamma_\beta
\frac{x_1\not P_1-\not l}{(x_1P_1-l)^2}\gamma_\alpha
\nonumber\\
& &\times
{\bar d}(x_2P_2)u({\bar x}_2P_2)
\gamma_\mu\frac{\not P_1-x_2\not P_2}{(P_1-x_2P_2)^2}
\gamma^\alpha \frac{{\bar x}_1\not P_1 +\not l}{({\bar x}_1 P_1 +l)^2}
\gamma^\beta u({\bar x}_1P_1)\frac{1}{l^2}\;.
\label{4b}
\end{eqnarray}
Following the same procedure as for Eq.~(\ref{2b}), we obtain
\begin{eqnarray}
I^{(b)}&\approx& \phi^{(1)}_{Sb} H_S^{(0)}(\xi_1,x_2)\;.
\label{4b1}
\end{eqnarray}
with the $O(\alpha_s)$ hard amplitude,
\begin{eqnarray}
H_S^{(0)}(\xi_1,x_2)=\frac{i}{2}eg^2 C_Fm_0
\frac{tr[\gamma_\alpha
{\bar d}(x_2P_2)u({\bar x}_2P_2)\gamma_\mu
(\not P_1-x_2\not P_2)\gamma^\alpha\gamma^5]}
{(P_1-x_2P_2)^2(\xi_1P_1-x_2P_2)^2}\;.
\label{4bc}
\end{eqnarray}
The expression of $\phi_{Sb}^{(1)}$ has been given in Eq.~(\ref{p2b}). 
The factorization of Figs.~4(a) and 4(c) is performed in a similar
way, leading to $\phi_{Sa,Sc}^{(1)}$ in Eqs.~(\ref{p2a}) and (\ref{p2c}).

The loop integrand associated with Fig.~4(d) is given by
\begin{eqnarray}
I^{(d)}&=&\frac{ieg^4}{2N_c}
{\bar d}(x_1P_1)\gamma^\lambda{\bar d}(x_2P_2)u({\bar x}_2P_2)
\gamma_\mu\frac{\not P_1-x_2\not P_2}{(P_1-x_2P_2)^2}\gamma^\beta
\frac{{\bar x}_1\not P_1 +\not l}{({\bar x}_1 P_1 +l)^2}
\nonumber\\
& &\times \gamma^\alpha u({\bar x}_1P_1)
\frac{tr(T^cT^bT^a)\Gamma^{cba}_{\lambda\beta\alpha}}
{l^2(x_1P_1-x_2P_2-l)^2(x_1P_1-x_2P_2)^2}\;,
\label{4d}
\end{eqnarray}
with the triple-gluon vertex in Eq.~(\ref{tg}). The same procedure as for
Eq.~(\ref{2d0}) leads to
\begin{eqnarray}
I^{(d)} &\approx &\phi_{Sd}^{(1)}
\left[H_{S}^{(0)}(x_1,x_2)-H_{S}^{(0)}(\xi_1,x_2)\right]\;,
\label{4d1}
\end{eqnarray}
where $\phi_{Sd}^{(1)}$ has been shown in Eq.~(\ref{p2d}).

The loop integrand associated with Fig.~4(e) is written as
\begin{eqnarray}
I^{(e)}&=&\frac{-eg^4 C_F}{4N_c}
{\bar d}(x_1P_1)\gamma^\alpha{\bar d}(x_2P_2)u({\bar x}_2P_2)\gamma_\beta
\frac{\bar{x}_2\not P_2+\not l}{(\bar{x}_2P_2+l)^2}
\gamma_\mu\frac{\not P_1-x_2\not P_2}{(P_1-x_2 P_2)^2}
\gamma_\alpha
\nonumber \\
& &\times
\frac{\bar{x}_1\not P_1 +\not l}{(\bar{x}_1P_1+l)^2}
\gamma^\beta u({\bar x}_1P_1)\frac{1}{l^2(x_1P_1-x_2P_2)^2}\;,
\label{4e}
\end{eqnarray}
which is simplified to
\begin{eqnarray}
I^{(e)}&\approx&\frac{(P_1-x_2 P_2)^2}{(P_1-x_2 P_2+l)^2}
\phi_{Se}^{(1)} H_{S}^{(0)}(x_1,x_2) \;,
\label{4e1}
\end{eqnarray}
with the function $\phi^{(1)}_{Se}$ shown in  Eq.~(\ref{p2e}). 

The loop integrand associated with Fig.~4(g) is written as
\begin{eqnarray}
I^{(g)}&=&\frac{-eg^4 C_F}{4N_c}
{\bar d}(x_1P_1)\gamma^\alpha{\bar d}(x_2P_2)u({\bar x}_2P_2)\gamma_\mu
\frac{\not P_1-x_2\not P_2}{(P_1-x_2P_2)^2}\gamma_\beta
\frac{\not P_1-x_2\not P_2+\not l}{(P_1-x_2P_2+l)^2}
\gamma_\alpha
\nonumber \\
& &\times
\frac{\bar{x}_1\not P_1+l}{(\bar{x}_1P_1+l)^2}
\gamma^\beta u({\bar x}_1P_1)\frac{1}{l^2(x_1P_1-x_2P_2)^2}\;,
\label{4g}
\end{eqnarray}
which reduces to
\begin{eqnarray}
I^{(g)}&\approx&
\frac{-2x_2P_2\cdot l}{(P_1-x_2 P_2+l)^2}
\phi_{Se}^{(1)}H_{S}^{(0)}(x_1,x_2)\;.
\label{4g1}
\end{eqnarray}
This result differs from that of Fig.~2(g), which does not contain
collinear divergence. Neither Eq.~(\ref{4e1}) nor (\ref{4g1}) is in
the desired form. However, their combination leads to the correct
factorization,
\begin{eqnarray}
I^{(e)}+I^{(g)}&\approx&
\phi_{Se}^{(1)}H_{S}^{(0)}(x_1,x_2)\;.
\label{4eg}
\end{eqnarray}

The loop integrand associated with Fig.~4(f),
\begin{eqnarray}
I^{(f)}&=&\frac{eg^4 C_F}{4N_c}
{\bar d}(x_1P_1)\gamma^\alpha
\frac{x_2\not P_2+\not l}{(x_2P_2+l)^2}
\gamma_\beta{\bar d}(x_2P_2)u({\bar x}_2P_2)
\gamma_\mu\frac{\not P_1-x_2\not P_2}{(P_1-x_2 P_2)^2}
\gamma_\alpha
\nonumber \\
& &\times
\frac{\bar{x}_1\not P_1 +\not l}{(\bar{x}_1P_1+l)^2}
\gamma^\beta u({\bar x}_1P_1)\frac{1}{l^2(x_1P_1-x_2P_2-l)^2}\;,
\label{4f}
\end{eqnarray}
leads to
\begin{eqnarray}
I^{(f)}&\approx&-\phi_{Sf}^{(1)}H_{S}^{(0)}(\xi_1,x_2)\;,
\label{4f1}
\end{eqnarray}
where the function $\phi^{(1)}_{Sf}$ has been given in Eq.~(\ref{p2e}).

The integrand associated with Fig.~4(h) is given by
\begin{eqnarray}
I^{(h)}&=&\frac{-ieg^4}{2N_c}
{\bar d}(x_1P_1)\gamma^\lambda
\frac{x_1\not P_1 -\not l}{(x_1 P_1 -l)^2}
\gamma^\beta{\bar d}(x_2P_2)u({\bar x}_2P_2)\gamma_\mu
\frac{\not P_1-x_2\not P_2}{(P_1-x_2P_2)^2}
\nonumber\\
& &\times \gamma^\alpha u({\bar x}_1P_1)
\frac{tr(T^cT^bT^a)\Gamma^{cba}_{\lambda\beta\alpha}}
{l^2(x_1P_1-x_2P_2-l)^2(x_1P_1-x_2P_2)^2}\;,
\label{4h}
\end{eqnarray}
with the triple-gluon vertex in Eq.~(\ref{tge}). Following the same
procedure as for Fig.~4(d), the integrand is split into
\begin{eqnarray}
I^{(h)}&\approx&\phi_{Sh}^{(1)}
\left[H_{S}^{(0)}(x_1,x_2)-H_{S}^{(0)}(\xi_1,x_2)\right]\;.
\label{4h1}
\end{eqnarray}
Similarly, we obtain 
\begin{eqnarray}
I^{(i)}\approx 
& & \phi_{Si}^{(1)}H_{S}^{(0)}(x_1,x_2)\;,
\label{4i1}
\\
I^{(j)}+I^{(k)}\approx
& & -\phi_{Sj}^{(1)}H_{S}^{(0)}(\xi_1,x_2)\;.
\label{4jk}
\end{eqnarray}
The functions $\phi^{(1)}_{Sh}$ and $\phi^{(1)}_{Si}=\phi^{(1)}_{Sj}$
have been presented in Eqs.~(\ref{p2h}) and (\ref{p2i}),
respectively.

Combining all the above contributions from Figs.~4(a)-4(k), we derive
the factorization similar to Eq.~(\ref{phi1}).


\section{$O(\alpha_s)$ CORRECTIONS TO $B\to \pi \ell\bar \nu$}

In this Appendix we discuss the factorization of the collinear divergences
in Fig.4 for the semileptonic decay $B\to\pi l\bar\nu$ with the
initial state and the final state flipped. Figures 4(a)-4(c) can be
factorized straightforwardly, leading to
Eqs.~(\ref{4a2}) and (\ref{b4b}). The hard amplitude in Eq.~(\ref{b4b}),
which depends on the loop momentum $l$, is written as
\begin{eqnarray}
H_{S}^{(0)}(x_1,\xi_2)&= & \frac{-g^2C_F}{2}m_0 
\frac{tr[\gamma_\alpha(\not P_2-\not k_1)\gamma_\mu 
b(P_1-k_1){\bar d}(k_1)\gamma^\alpha\gamma^5]}
{(P_2-k_1)^2(k_1-x_2P_2+l)^2}
\nonumber \\
&\approx & \frac{-g^2C_F}{2\eta^2M_B^4}m_0 
\frac{tr[\gamma_\alpha\not P_2\gamma_\mu 
b(P_1-k_1){\bar d}(k_1)\gamma^\alpha\gamma^5]}{x_1^2\xi_2}\;.
\label{pb4b}
\end{eqnarray}

The loop integrand from Fig.~4(d) is given by
\begin{eqnarray}
I^{(d)}&=&\frac{-g^4}{2N_c}u({\bar x}_2P_2)\gamma^\beta
\frac{{\bar x}_2\not P_2+\not l}{({\bar x}_2P_2+l)^2}\gamma^\alpha
\frac{\not P_2-\not k_1}{(P_2-k_1)^2}\gamma_\mu
b(P_1-k_1){\bar d}(k_1)
\nonumber\\
& &\times\gamma^\lambda {\bar d}(x_2P_2)
\frac{tr(T^cT^bT^a)\Gamma^{cba}_{\lambda\beta\alpha}}
{l^2(k_1-x_2P_2+l)^2(k_1-x_2P_2)^2}\;,
\label{4bd0}
\end{eqnarray}
with the triple-gluon vertex,
\begin{eqnarray}
\Gamma^{cba}_{\lambda\beta\alpha}&=&f^{cba}
[g_{\alpha\beta}(2l+k_1-x_2P_2)_\lambda
+g_{\beta\lambda}(k_1-x_2P_2-l)_\alpha
\nonumber\\
& &+g_{\lambda\alpha}(2x_2P_2-2k_1-l)_\beta]\;.
\end{eqnarray}
The color factor is simplified according to Eq.~(\ref{tco}).

In the collinear region with $l$ parallel to $P_2$, only the term
proportional to $g_{\alpha\lambda}$ contributes by employing the argument
the same as for Fig.~2(d) in Sec.~II. Since $\gamma^\beta$ must be
$\gamma^-$, only the plus component of $k_1$ survives. Applying the
approximation similar to Eq.~(\ref{pi}),
\begin{eqnarray}
\frac{-2k_{1\beta}}{(k_1-x_2P_2)^2(k_1-x_2P_2+l)^2}
\approx -\frac{n_{+\beta}}{n_+\cdot l}\biggl[\frac{1}{(k_1-x_2P_2)^2}
-\frac{1}{(k_1-x_2P_2+l)^2}\biggr]\;,
\end{eqnarray}
and inserting the Fierz identity in Eq.~(\ref{1f}), we obtain
Eq.~(\ref{4bd1}).

The collinear factorization of Fig.~4(e) can be achieved by applying the
eikonal approximation to the $b$ quark propagator,
\begin{eqnarray}
\frac{\not P_1-\not k_1+\not l+m_b}{(P_1-k_1+l)^2-m_b^2}
\gamma_\beta b(P_1-k_1)
&\approx&\frac{2(P_1-k_1)_\beta
-\gamma_\beta(\not P_1-\not k_1-m_b)}
{2(P_1-k_1)\cdot l}b(P_1-k_1)
\nonumber \\
&\approx &\frac{n_{+\beta}}{n_+\cdot l}b(P_1-k_1)\;.
\end{eqnarray}
The neglect of $\not l$ is due to $\gamma_\beta=\gamma^+$ in the collinear
region. The second term on the right-hand side of the first line vanishes
because of Eq.~(\ref{eqb}). To derive the final expression, we further
dropped the power-suppressed $k_1$ term. The integrand associated with
Fig.~4(e) then becomes
\begin{eqnarray}
I^{(e)}&\approx&
\frac{ig^2}{8m_0N_c} u({\bar x}_2P_2)\gamma^\beta
\frac{{\bar x}_2\not P_2+\not l}{({\bar x}_2P_2+l)^2}\gamma^5
{\bar d}(x_2P_2)\frac{1}{l^2}\frac{n_{+\beta}}{n_+\cdot l}
\nonumber \\
& &\times \frac{(P_2-k_1)^2}{(P_2-k_1+l)^2}
\left(\frac{-g^2}{2}C_Fm_0\right)\frac{tr[\gamma^\alpha(\not P_2-\not k_1)
\gamma_\mu b(P_1-k_1){\bar d}(k_1)\gamma_\alpha\gamma^5]}
{(P_2-k_1)^2(k_1-x_2P_2)^2}\;.
\label{4be}
\end{eqnarray}
Note that the prefactor $(P_2-k_1)^2/(P_2-k_1+l)^2$ in the second line
indicates that Eq.~(\ref{4be}) has not yet reached the expected
factorization form.

The integrand for Fig.~4(g) reduces, in a similar way, to
\begin{eqnarray}
I^{(g)}&\approx&
\frac{ig^2}{8m_0N_c}u({\bar x}_2P_2)\gamma^\beta
\frac{{\bar x}_2\not P_2+\not l}{({\bar x}_2P_2+l)^2}\gamma^5
{\bar d}(x_2P_2)\frac{1}{l^2}\frac{n_{+\beta}}{n_+\cdot l}
\nonumber \\
& &\times \frac{-2k_1\cdot l}{(P_2-k_1+l)^2}
\left(\frac{-g^2}{2}C_Fm_0\right)\frac{tr[\gamma^\alpha(\not P_2-\not k_1)
\gamma_\mu b(P_1-k_1){\bar d}(k_1)\gamma_\alpha\gamma^5]}
{(P_2-k_1)^2(k_1-x_2P_2)^2}\;.
\label{4bg}
\end{eqnarray}
Combining Eqs.~(\ref{4be}) and (\ref{4bg}), we arrive at the desired
factorization form in Eq.~(\ref{4beg}). It is observed that the hard part
is independent of the loop momentum $l$.

The integrand associated with Fig.~4(f) is factorized into
Eq.~(\ref{4f2}) straightforwardly. In this case the eikonal propagator
arises from the approximation,
\begin{eqnarray}
{\bar d}(k_1)\gamma_\beta
\frac{\not k_1+\not l}{(k_1+l)^2}
\approx {\bar d}(k_1)\frac{2k_{1\beta}-\not k_1\gamma_\beta}
{2k_1\cdot l}
={\bar d}(k_1)\frac{n_{+\beta}}{n_+\cdot l}\;.
\label{bfap}
\end{eqnarray}
Again, the neglect of $\not l$ is due to $\gamma_\beta=\gamma^+$ in the
collinear region, and Eq.~(\ref{eqb}) has been employed to derive the
final expression.

The loop integrand from Fig.~4(h) is written as
\begin{eqnarray}
I^{(h)}&=&\frac{g^4}{2N_c}u({\bar x}_2P_2)\gamma^\alpha
\frac{\not P_2-\not k_1}{(P_2-k_1)^2}\gamma_\mu
b(P_1-k_1){\bar d}(k_1)
\gamma^\lambda
\frac{x_2\not P_2-\not l}{(x_2P_2-l)^2}
\nonumber\\
& &\times\gamma^\beta{\bar d}(x_2P_2)
\frac{tr(T^cT^bT^a)\Gamma^{cba}_{\lambda\beta\alpha}}
{l^2(k_1-x_2P_2+l)^2(k_1-x_2P_2)^2}\;,
\label{4bh0}
\end{eqnarray}
with the triple-gluon vertex,
\begin{eqnarray}
\Gamma^{cba}_{\lambda\beta\alpha}&=&f^{cba}
[g_{\beta\lambda}(2l+k_1-x_2P_2)_\alpha
+g_{\alpha\beta}(k_1-x_2P_2-l)_\lambda
\nonumber\\
& &+g_{\lambda\alpha}(2x_2P_2-2k_1-l)_\beta]\;.
\end{eqnarray}
Following the same procedure of Fig.~4(d), Eq.~(\ref{4bh0}) reduces to
\begin{eqnarray}
I^{(h)}\approx \left[H_S^{(0)}(x_1,x_2)
-H_S^{(0)}(x_1,\xi_2)\right] \phi_{Sh}^{(1)}\;,
\label{4bh2}
\end{eqnarray}
with the $O(\alpha_s)$ collinear divergent function,
\begin{eqnarray}
\phi_{Sh}^{(1)}=\frac{ig^2}{2m_0C_F}
u({\bar x}_2P_2)\gamma^5
\frac{x_2\not P_2 -\not l}{(x_2 P_2 -l)^2}
\gamma^\beta{\bar d}(x_2P_2)\frac{1}{l^2}
\frac{n_{+\beta}}{n_+\cdot l}\;.
\label{p4bh}
\end{eqnarray}

The loop integrand associated with Fig.~4(i) is given by
\begin{eqnarray}
I^{(i)}&=&\frac{-ig^4 C_F}{4N_c}u({\bar x}_2P_2)\gamma^\alpha
\frac{\not P_2-\not k_1}{(P_2-k_1)^2}\gamma_\mu
b(P_1-k_1){\bar d}(k_1)\gamma_\beta
\frac{\not k_1-\not l}{(k_1-l)^2}\gamma_\alpha
\nonumber \\
& & \times \frac{x_2\not P_2-\not l}{(x_2P_2-\l)^2}
\gamma^\beta{\bar d}(x_2P_2)
\frac{1}{l^2(k_1-x_2P_2)^2}\;.
\label{4bi}
\end{eqnarray}
Using the approximation similar to Eq.~(\ref{bfap}),
the above expression is factorized as
\begin{eqnarray}
I^{(i)}\approx  H_S^{(0)}(x_1,x_2)\phi_{Si}^{(1)}\;,
\label{4bi2}
\end{eqnarray}
with the collinear divergent function,
\begin{eqnarray}
\phi_{Si}^{(1)}=\frac{-ig^2}{8m_0N_c}u({\bar x}_2P_2)\gamma^5 
\frac{x_2\not P_2 -\not l}{(x_2 P_2 -l)^2}
\gamma^\beta{\bar d}(x_2P_2)\frac{1}{l^2}
\frac{n_{+\beta}}{n_+\cdot l}\;.
\label{p4bi}
\end{eqnarray}

Following the same procedure as for Eqs.~(\ref{4be}) and (\ref{4bg}),
Figs.~4(j) and 4(k) lead to
\begin{eqnarray}
I^{(j)}&\approx&
\frac{(P_2-k_1)^2}{(P_2-k_1-l)^2}H_{S}^{(0)}(x_1,\xi_2)\phi_{Sj}^{(1)}\;,
\nonumber\\
I^{(k)}&\approx&
\frac{2k_1\cdot l}{(P_2-k_1-l)^2}H_{S}^{(0)}(x_1,\xi_2)\phi_{Sj}^{(1)}\;,
\label{4bjk0}
\end{eqnarray}
with the collinear divergent piece,
\begin{eqnarray}
\phi_{Sj}^{(1)}=\frac{-ig^2}{8m_0N_c}u({\bar x}_2P_2)\gamma^5
\frac{x_2\not P_2-\not l}{(x_2P_2-l)^2}\gamma^\nu
{\bar d}(x_2P_2)\frac{1}{l^2}
\frac{n_{+\nu}}{n_+\cdot l}\;.
\label{pb4j}
\end{eqnarray}
Combining the expressions in Eq.~(\ref{4bjk0}), we arrive at the desired
factorization form,
\begin{eqnarray}
I^{(j)}+I^{(k)}&\approx&
-H_{S}^{(0)}(x_1,\xi_2)\phi_{Sj}^{(1)}\;.
\label{4bjk}
\end{eqnarray}

We then discuss the factorization of the collinear divergences
from Fig.~2 with the initial and final states flipped, which represents
the $O(\alpha_s)$ corrections to Fig.~1(b).  To simplify the discussion,
we neglect the PT pieces below.
Since the factorization of Figs.~2(a)-2(g) has been discussed in Sec.~IV,
we start with the loop integrand from Fig.~2(h), which is
simplified to
\begin{eqnarray}
I^{(h)}& \approx &
\Bigg\{H^{(0)}_{S}(x_1,x_2)-
\frac{-g^2}{2}C_Fm_0\frac{tr[\gamma_\mu
(\not P_1-x_2\not P_2+m_b)\gamma^\alpha b(P_1-k_1)
{\bar d}(k_1)\gamma_\alpha\gamma^5]}
{[(P_1-x_2P_2)^2-m_b^2](k_1-x_2P_2+l)^2}\Bigg\}\phi^{(1)}_{Sh}\;,
\label{2bh}
\end{eqnarray}
with the $O(\alpha_s)$ collinear divergent piece $\phi_{Sh}^{(1)}$
given in Eq.~(\ref{p4bh}).

The loop integrand associated with Fig.~2(i) reduces to
\begin{eqnarray}
I^{(i)}&\approx & H_{S}^{(0)}(x_1,x_2)\phi_{Si}^{(1)}\;,
\label{2bi}
\end{eqnarray}
with the collinear divergent piece $\phi_{Si}^{(1)}$ shown in
Eq.~(\ref{p4bi}). Figure 2(j) leads to
\begin{eqnarray}
I^{(j)}& \approx &-
\Bigg\{\frac{-g^2}{2}C_Fm_0\frac{tr[\gamma_\mu
(\not P_1-x_2\not P_2+m_b)\gamma^\alpha b(P_1-k_1)
{\bar d}(k_1)\gamma_\alpha\gamma^5]}
{[(P_1-x_2P_2)^2-m_b^2](k_1-x_2P_2+l)^2}\Bigg\}\phi^{(1)}_{Sj}\;,
\label{2bj}
\end{eqnarray}
where $\phi_{Sj}^{(1)}$ has the
same expression as in Eq.~(\ref{pb4j}).
Combining Eqs.~(\ref{2bh}), (\ref{2bi}) and (\ref{2bj}),
we arrive at
\begin{eqnarray}
I^{(h)}+I^{(i)}+I^{(j)}
&\approx &
\Bigg\{H^{(0)}_{S}(x_1,x_2)-
\frac{-g^2}{2}C_Fm_0\frac{tr[\gamma_\mu
(\not P_1-x_2\not P_2+m_b)\gamma^\alpha b(P_1-k_1)
{\bar d}(k_1)\gamma_\alpha\gamma^5]}
{[(P_1-x_2P_2)^2-m_b^2](k_1-x_2P_2+l)^2}\Bigg\}\phi^{(1)}_{S{\bar d}}\;,
\label{2bhj}
\end{eqnarray}
with $\phi_{S{\bar d}}^{(1)}$ given in
Eq.~(\ref{4nd}).

The integrand associated with Fig.~2(k) becomes
\begin{eqnarray}
I^{(k)}\approx \frac{2P_1\cdot l}{(P_1-x_2P_2)^2-m_b^2}
\left(\frac{-g^2}{2}C_Fm_0\right)\frac{tr[\gamma_\mu
(\not P_1+m_b)\gamma^\alpha b(P_1-k_1)
{\bar d}(k_1)\gamma_\alpha\gamma^5]}
{[(P_1-x_2P_2+l)^2-m_b^2](k_1-x_2P_2+l)^2}\phi^{(1)}_{S{\bar d}}\;.
\label{2bk}
\end{eqnarray}
Combining the second term in Eq.~(\ref{2bhj}) and Eq.~(\ref{2bk}),
we obtain the correct factorization with the hard amplitude
$H_S^{(0)}(x_1,\xi_2)$ in Eq.~(\ref{pb2b}).



\vskip 0.5cm

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\newpage
%\vskip 0.5cm

{\bf \Large Figure Captions}
\vspace{10mm}

\begin{enumerate}

\item Fig. 1: Lowest-order diagrams for $\pi\gamma^*\to\pi$
($B\to\pi l\bar\nu$), where the symbol $\times$ represents the
virtual photon (weak decay) vertex.

\item Fig. 2: $O(\alpha_s)$ radiative corrections to Fig.~1(a).

\item Fig. 3: (a)-(d) Infrared divergent diagrams factored out of
Fig.~2(d)-2(k). (e) The graphic definition of the two-parton twist-3 pion
distribution amplitudes.

\item Fig. 4: $O(\alpha_s)$ radiative corrections to Fig.~1(b).

\item Fig. 5: (a) The Ward identity. (b) Factorization of
$O(\alpha_s^{N+1})$ diagrams as a result of (a).

\item Fig. 6: Factorization of $O(\alpha_s^{N+1})$ diagrams corresponding
to Eq.~(\ref{rep}).

\end{enumerate}





\end{document}

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