\documentclass[prd,preprint,showpacs]{revtex4}
\usepackage{epsfig,chay}
\begin{document}
\title{Nonleptonic $B$ decays into two light mesons in soft-collinear
  effective theory} 
\author{Junegone Chay}
\email{chay@korea.ac.kr}
\author{Chul Kim}
\affiliation{Department of Physics, Korea University, Seoul 136-701,
Korea}
\preprint{KUPT--03--02} 
\begin{abstract}
We consider nonleptonic $B$ decays into two light mesons at leading
order in soft-collinear effective theory. The four-quark operators
relevant to $B$ decays in the full theory are matched to the operators
in the effective theory in a gauge invariant way. We include the
analysis of nonfactorizable spectator contributions and spectator
contributions to the heavy-to-light form factor at leading order. The
decay amplitudes at leading order are shown to be factorized to all
orders in $\alpha_s$. As an application, we present the decay
amplitudes for $\overline{B} \rightarrow \pi\pi$ in soft-collinear
effective theory.  
\end{abstract}
\pacs{13.25.Hw, 11.10.Hi, 12.38.Bx, 11.40.-q}

\maketitle

\section{Introduction}
Decay of $B$ mesons plays an important role in particle physics since
it is a testing ground for the Standard Model and a window for
possible new physics. We can obtain good information on
Cabibbo-Kobayashi-Maskawa (CKM) matrix elements and CP
violation from $B$ decays. Since the contribution of QCD in weak
decays changes the whole structure of the theory, the study of $B$
decays is an intertwined field of particle physics. Among these
decays, nonleptonic $B$ decays have been the subject of intense 
interest. Especially the treatment of nonperturbative effects from the
strong interaction is a serious theoretical problem in
nonleptonic decays. Precise experimental observation of these
nonleptonic decays makes it urgent to give a firm theoretical
prediction including the effects of CP violation. There has been a lot
of theoretical progress in nonleptonic $B$ decays and we suggest how
to consider nonleptonic $B$ decays from the viewpoint of the
soft-collinear effective theory (SCET). 


The effective Hamiltonian for nonleptonic decays from the Standard
Model has been derived and the Wilson coefficients of the relevant
operators for nonleptonic $B$ decays have been calculated to
next-to-leading order, and next-to-next-to-leading order for some
operators \cite{buras}.  In order to calculate the hadronic matrix
elements of four-quark operators, naive factorization \cite{naive}
was assumed, in which the matrix elements were reduced to products of
current matrix elements. But there was no justification for this
assumption except the argument of color transparency \cite{trans}.
A big problem in naive factorization is that
decay amplitudes depend on an arbitrary renormalization scale $\mu$
since the Wilson coefficients depend on $\mu$, while the matrix
elements of operators do not.  

Ali et al. \cite{ali} have improved the problem of the scale
dependence by including radiative corrections of the operators before
taking hadronic matrix elements. Then the $\mu$ 
dependence of the Wilson coefficients is cancelled by the $\mu$
dependence of the radiative corrections of the operators. However,
since they choose the off-shell renormalization scheme, the decay
amplitudes depend on calculation schemes \cite{buras2}. 

Politzer and Wise \cite{politzer} suggested to take the heavy quark
mass limit in computing corrections to the decay rate ratio $\Gamma
(\overline{B} \rightarrow D^* \pi)/\Gamma( \overline{B} \rightarrow
D\pi)$. They considered radiative corrections for nonfactorizable
contributions and found that they are finite and the decay amplitudes
are factorized. Beneke et al. \cite{bbns} have extended this approach
to general two-body decays including two light final-state mesons, in
which the decay amplitudes can be expanded in a power series of
$1/m_b$. They show that nonfactorizable contributions including
spectator interactions are factorized as a convolution of the Wilson
coefficients and meson wave functions, and the corrections are
suppressed by powers of $1/m_b$. This is an important step toward
theoretical understanding of nonleptonic $B$ decays. First the
amplitudes can be obtained from first principles and a systematic
$1/m_b$ expansion is possible. Second, since they use on-shell
renormalization, there is no 
scheme dependence. And they improved previous approaches by including
momentum-dependent parts, which had not been included previously.
However, when higher-twist light-cone wave functions are included,
there appear infrared divergences in the amplitudes. These are treated
as theoretical uncertainties, but from the theoretical viewpoint it
is a problem to be solved in this approach.


Bauer et al. \cite{bauer1,bauer2} have proposed an effective field
theory in which massless quarks move with large energy. This
effective theory called the soft-collinear effective theory which is
appropriate for light quarks with large energy. It has been applied to
hard scattering processes and $B$ decays
\cite{bauer5,bauer3,bauer4,chay1,manohar,bauer6}.    
It is an appropriate effective theory for two-body nonleptonic $B$
decays to light mesons. In this paper, we apply SCET to two-body
nonleptonic decays into light mesons. We construct all the relevant
operators in the effective theory at leading order in a
gauge-invariant way by integrating out all the off-shell modes. The
Wilson coefficients are calculated by matching the full theory onto
SCET. We show that the four-quark operators in SCET are factorized to
all orders in $\alpha_s$ and the argument of color transparency is 
explictly shown. And we consider nonfactorizable spectator
contribution in SCET, and find that they are 
also factorized to all orders in $\alpha_s$. 



The basic idea of the factorization properties in $B$ decays
into two light mesons was sketched in Ref.~\cite{chay2}. In this
paper, we extend the 
argument and discuss intricate characteristics of SCET in nonleptonic
decays, the details of the procedure of matching, present all the Wilson
coefficients and technical details in the calculation. We also present
the analysis of $B\rightarrow \pi\pi$ decays as an application, which
is consistent with the calculation in the heavy quark mass limit.

The organization of the paper is as follows:
In Section~\ref{sec2} we construct four-quark operators in SCET, which 
are gauge invariant by integrating out off-shell modes. This is
achieved by attaching gluons to fermion legs and by integrating out
off-shell intermediate states. We discuss the factorization properties
of the operators. In Section~\ref{sec3}, we match the full theory and
SCET and obtain the Wilson coefficients of various four-quark
operators. In Section~\ref{sec4}, we consider nonfactorizable
spectator interaction by including operators at subleading order,
which interacts with an ultrasoft quark inside a $B$ meson and turns it
into a collinear quark. In Section~\ref{sec5}, we include the
subleading operator from the heavy-to-light current, which contributes
to the heavy-to-light form factor. In Section~\ref{sec6} we combine
all the results to apply SCET to nonleptonic decays  $B\rightarrow \pi
\pi$. And finally a conclusion is presented. In Appendix, we show how
the auxiliary field method can be used to derive gauge-invariant
operators. 

\section{Construction of gauge-invariant four-quark
  operators in SCET\label{sec2}} 
The SCET, which describes the
dynamics of  massless, energetic quarks, is described extensively in
Ref.~\cite{bauer1,bauer2,bauer5}. The basic idea of SCET is to
decompose the momentum as 
\begin{equation}
p_n^{\mu} = \frac{\overline{n} \cdot p_n}{2} n^{\mu} + p_{n\perp}^{\mu}
+\frac{n\cdot p_n}{2} \overline{n}^{\mu} = \mathcal{O} (\lambda^0) +
\mathcal{O} (\lambda) + \mathcal{O} (\lambda^2),
\end{equation}
for the collinear particle moving in the $n^{\mu}$ direction with $n^2
= \overline{n}^2=0$, and $n\cdot \overline{n} =2$. And $\lambda \sim
p_{\perp}/\overline{n} \cdot p$ is the small expansion parameter. The
SCET contains three scales $Q$, $Q\lambda$ and $Q\lambda^2$. And we
can obtain effective theories going from the scale $\mu >Q$ down to
the scale $\mu \sim Q\lambda^2$ by integrating out energetic degrees
of freedom step by step and by matching theories at each
boundary. 

In studying nonleptonic decays there are collinear quarks moving in
opposite directions. We choose these two directions as $n^{\mu}$
and $\overline{n}^{\mu}$. For an energetic quark moving in the
$\overline{n}^{\mu}$ direction, the momentum is decomposed as
\begin{equation}
p_{\overline{n}}^{\mu} = \frac{n \cdot p_{\overline{n}}}{2}
    \overline{n}^{\mu} +     p_{\overline{n}\perp}^{\mu} 
+\frac{\overline{n}\cdot p_{\overline{n}}}{2} n^{\mu} = \mathcal{O}
    (\lambda^0) + \mathcal{O} (\lambda) + \mathcal{O} (\lambda^2).
\end{equation}
This situation is simlar to two jets in the opposite direction, which
was considered in Ref.~\cite{bauer4}. But we also have a heavy quark in
the system interacting with collinear gluons either in the
$n^{\mu}$ or in the $\overline{n}^{\mu}$ direction, which makes the
analysis more interesting and complicated.

Let us summarize the notations which will be used throughout the
paper. We denote a collinear field by $\xi$ ($\chi$) which moves in
the $n^{\mu}$ ($\overline{n}^{\mu}$) direction. These fields satisfy
the relations
\begin{equation} 
\FMslash{n} \xi = 0, \ \ \frac{\FMslash{n}\FMslash{\overline{n}}}{4} \xi
  = \xi,  \ \ 
\FMslash{\overline{n}} \chi = 0, \ \  \frac{\FMslash{\overline{n}}
  \FMslash{n}}{4} \chi = \chi, 
\end{equation}
with $n^2 = \overline{n}^2=0$, and $n\cdot \overline{n}=2$. The
collinear gauge field in the $n^{\mu}$ ($\overline{n}^{\mu}$)
direction is written as $A_n^{\mu}$ ($A_{\overline{n}}^{\mu}$), and
$D_n^{\mu}$ ($D_{\overline{n}}^{\mu}$)  for the covariant
derivatives. 

The effective Hamiltonian for $B$ decays to two light mesons
is given in the full theory as
\begin{equation}
H_{\mathrm{eff}} = \frac{G_F}{\sqrt{2}} \sum_{p=u,c} V_{pd}^* V_{pb}
\Bigl( C_1 O_1^p + C_2 O_2^p +\sum_{i=3,\cdots,6,8} C_i O_i\Bigr).
\label{fullop}
\end{equation}
The operators in Eq.~(\ref{fullop}) are local $\Delta B=1$ operators,
which are given as
\begin{eqnarray}
O_1^p &=& (\overline{p}_{\alpha} b_{\alpha})_{V-A}
(\overline{d}_{\alpha} p_{\beta})_{V-A}, \ \ O_2^p =
(\overline{p}_{\beta} b_{\alpha})_{V-A} 
(\overline{d}_{\alpha} p_{\beta})_{V-A}, \nonumber \\
O_3 &=& (\overline{d}_{\alpha} b_{\alpha})_{V-A} \sum_q
(\overline{q}_{\beta} q_{\beta})_{V-A},  \ \ O_4 =
(\overline{d}_{\alpha} b_{\beta})_{V-A} \sum_q 
(\overline{q}_{\beta} q_{\alpha})_{V-A}, \nonumber \\
O_5 &=& (\overline{d}_{\alpha} b_{\alpha})_{V-A} \sum_q
(\overline{q}_{\beta} q_{\beta})_{V+A},  \ \ O_6 =
(\overline{d}_{\alpha} b_{\beta})_{V-A} \sum_q 
(\overline{q}_{\beta} q_{\alpha})_{V+A}, \nonumber \\
O_8 &=& \frac{-g}{8\pi^2} m_b \overline{d}_{\alpha} \sigma^{\mu\nu}
(1+\gamma_5) (T_a)_{\alpha\beta} b_{\beta} G_{\mu\nu}^a,
\end{eqnarray}
where $p=u,c$ and $d$ denotes down-type quarks.

We look for the corresponding operators in SCET. The operators in
SCET should be gauge-invariant under collinear, soft and ultrasoft
(usoft) gauge transformations. The process of obtaining the operators
in SCET requires two-step matching \cite{neuts,form} because the SCET
involves two different scales $\mu\sim m_b$ and $\mu_0\sim 
  \sqrt{m_b  \Lambda_{\mathrm{QCD}}}$. First we match the 
full theory  onto $\mathrm{SCET}_{\mathrm{I}}$ at $\mu = m_b$. But
meson states still have $p_M^2 \sim m_b \Lambda_{\mathrm{QCD}}$ which 
is still too large. So we match successively onto
$\mathrm{SCET}_{\mathrm{II}}$ at $\mu_0$. A concrete example of the
two-step matching was illustrated in Ref.~\cite{pirjol}. 

When we go down to SCET and replace massless quark 
fields by collinear fields, they can be either $\xi$ or $\chi$.  
In order to find the operators, we first have to specify
which quark or antiquark goes to a certain direction to form a light
meson. We set $\overline{n}^{\mu}$ as the direction of a quark and and
antiquark to form a light meson, and $n^{\mu}$ as the direction of the
remaining quark which combines with a spectator quark in a $B$ meson
to form another meson. Therefore the construction of the operators is
process-dependent in the sense that we first specify the direction of
each outgoing quark, and the number of operators in SCET is doubled
because we have two possibilities to assign two quark fields in both
directions. 

A generic four-quark operator for nonleptonic $B$ decays in SCET has
the form $(\overline{\xi} \Gamma_1 h) (\overline{\chi} \Gamma_2
\chi)$, or $(\overline{\chi} \Gamma_1 h)(\overline{\xi} \Gamma_2
\chi)$, where  $\Gamma_1$ and $\Gamma_2$ are arbitrary Dirac matrices,
and $h$ is the heavy quark field in the heavy quark effective
theory. These operators are derived from the operator $\overline{q}_1 
\Gamma_1 b \cdot \overline{q}_2 \Gamma_2 q_3$ in the full theory where
$q_i$ ($i=1,2,3$) are light quarks. The operator $(\overline{\xi}
\Gamma_1 h_v) (\overline{\chi} \Gamma_2 \chi)$ is obtained by
replacing $q_1$ by $\xi$, and $q_2$ and $q_3$ by the quark fields
$\chi$. For the operator 
$(\overline{\chi} \Gamma_1 h) (\overline{\xi} \Gamma_2 \chi)$, we
replace $q_2$ by $\xi$ and $q_1$ and $q_3$ by the $\chi$ fields. 
The second operator produces a light meson, in which one quark comes 
from the heavy-to-light current and another antiquark comes from the
light-to-light current to form a meson in the $\overline{n}^{\mu}$
direction. A remaining quark goes to the $n^{\mu}$ direction. We can
transform this operator to the form $(\overline{\xi} \Gamma_1^{\prime}
h) (\overline{\chi} \Gamma_2^{\prime} \chi)$ by Fierz
transformation. In order to simplify the organization of the
computation, any operator will be written in the form $(\overline{\xi}
\Gamma_1  h)(\overline{\chi}\Gamma_2 \chi)$. If we need Fierz
transformation to obtain this form, we also apply Fierz transformation
in the full theory and perform matching. Therefore we only have to
consider operators of the form  $(\overline{\xi} \Gamma_1
h_v)(\overline{\chi} \Gamma_2  \chi)$, but the two types of the
operators with or without Fierz transformation are regarded as
distinct and their corresponding Wilson coefficients are different as
will be shown in a later section. 




Though we know the form of the operators, the operator
$(\overline{\xi} \Gamma_1 h)(\overline{\chi} \Gamma_2 \chi)$ itself
is not gauge invariant under the collinear, soft and usoft gauge  
transformations. We can construct gauge-invariant operators by
attaching collinear or soft gluons to each fermion leg and integrate
out off-shell modes. Because the ordering of gauge fields is important
due to the nonabelian nature of the gauge fields, we explictly
consider corrections to order $g^2$. Typical Feynman diagrams at
order $g^2$ are shown in Fig.~\ref{fig1}, and other
diagrams in which gluons are attached to the light-to-light current
are omitted. But it is straightforward to attach two gluons 
to other fermion lines making intermediate states off-shell. From this
process, we can obtain gauge-invariant four quark operators in
$\mathrm{SCET}_{\mathrm{II}}$. 


\begin{figure}[t]
\begin{center}
\epsfig{file=nlfig1.eps, width=13.5cm}
\end{center}
\caption{QCD diagrams attaching two gluons to external fermions to
integrate out off-shell modes. All the external gluons $A_n$,
$A_{\overline{n}}$ or $A_s$ are considered to make the intermediate
states off the mass shell.  Diagrams with gluons attached to other
fermions are omitted.}  
\label{fig1}
\end{figure}

For example, for a heavy quark, if the attached gluons are collinear
gluons in the $n^{\mu}$ or $\overline{n}^{\mu}$ direction, or soft gluons
[Fig.~\ref{fig1} (a), (b)], the intermediate heavy quark and gluon states
are off the mass shell and we integrate them out. For the collinear
quark $\xi$, the interaction with a collinear gluon in the
$\overline{n}^{\mu}$ direction or a soft gluon makes the collinear
quark off the mass shell. For the collinear quark $\chi$, when a
collinear gluon in the $n^{\mu}$ direction or a soft gluon interacts,
it becomes off the mass shell. We add all the possible configurations
in which the intermediate states become off the mass shell
and collect the terms at leading order in $\lambda$. 


Let us introduce the factors
\begin{equation}
A=\frac{\overline{n} \cdot A_n}{\overline{n}\cdot q_n}, \ B=
\frac{n\cdot A_{\overline{n}}}{n\cdot q_{\overline{n}}}, \ C=
\frac{\overline{n} \cdot A_s}{\overline{n} \cdot q_s}, \ D=
\frac{n\cdot A_s}{n\cdot q_s},
\end{equation}
where $q_n^{\mu}$ ($q_{\overline{n}}^{\mu}$) is the momentum of the
collinear gluon $A_n^{\mu}$ ($A_{\overline{n}}^{\mu}$), and
$q_s^{\mu}$ is the soft momentum of the soft gluon $A_s^{\mu}$. 
The Wilson lines corresponding to each type of gluons are obtained by
exponentiating the above factors as
\begin{eqnarray}
W&=& \sum_{\mathrm{perm}} \exp \Bigl[ -g \frac{1}{\overline{n} \cdot
    \mathcal{P}} \overline{n}\cdot A_n \Bigr], \ \ \overline{W}=
    \sum_{\mathrm{perm}}  \exp \Bigl[ -g \frac{1}{n \cdot
    \mathcal{Q}} n\cdot A_{\overline{n}} \Bigr], \nonumber \\
\overline{S}&=&
    \sum_{\mathrm{perm}}  \exp \Bigl[ -g \frac{1}{\overline{n} \cdot
    \mathcal{R}} \overline{n}\cdot A_s \Bigr], \ \ 
S= \sum_{\mathrm{perm}} \exp \Bigl[ -g \frac{1}{n \cdot
    \mathcal{R}}n \cdot A_s \Bigr]. 
\end{eqnarray}
Here $\mathcal{P}^{\mu} = \overline{n}\cdot \mathcal{P} n^{\mu}/2 +
\mathcal{P}_{\perp}^{\mu}$ is the label momentum operator for
collinear fields in the $n^{\mu}$ direction, and $\mathcal{Q}^{\mu} = 
n\cdot \mathcal{Q} \overline{n}^{\mu}/2 + \mathcal{Q}_{\perp}^{\mu}$ is
the label momentum operator for collinear fields in the
$\overline{n}^{\mu}$ direction. And the operator $\mathcal{R}$ is the
operator extracting the soft momentum from soft fields.

When we sum over all these diagrams, the singlet operators of the form
$(\overline{\xi}_{\alpha} \Gamma_1 h_{\alpha})(\overline{\chi}_{\beta}
\Gamma_2 \chi_{\beta})$ and the nonsinglet operators of the form
$(\overline{\xi}_{\beta} \Gamma_1 h_{\alpha})(\overline{\chi}_{\alpha} 
\Gamma_2 \chi_{\beta})$ are affected by the gauge fields differently and
the  final form of the four-quark opearators can be written as
\begin{eqnarray}
O_S&=& (\overline{\xi}_{\alpha} \Gamma_1
h_{\alpha})(\overline{\chi}_{\beta} \Gamma_2 \chi_{\beta}) \rightarrow
H_{\alpha\beta}^S L_{\gamma\delta}^S  (\overline{\xi}_{\alpha} \Gamma_1
h_{\beta})(\overline{\chi}_{\gamma} \Gamma_2 \chi_{\delta}), \nonumber
\\
O_N&=& (\overline{\xi}_{\beta} \Gamma_1
h_{\alpha})(\overline{\chi}_{\alpha} \Gamma_2 \chi_{\beta}) \rightarrow
H_{\gamma\beta}^N L_{\alpha\delta}^N  (\overline{\xi}_{\alpha} \Gamma_1
h_{\beta})(\overline{\chi}_{\gamma} \Gamma_2 \chi_{\delta}),
\end{eqnarray}
where $H_{\alpha \beta}^O$, $L_{\gamma\delta}^O$ ($O=S,N$) are color
factors which contain $A$, $B$, $C$ and $D$. 

One interesting case arises when we attach $A_n^{\mu}$ and
$A_{\overline{n}}^{\mu}$ to a heavy quark. At leading order in $\lambda$,
the sum of the  amplitude in Fig.~\ref{fig1} (a) with $A_n^{\mu}$
($A_{\overline{n}}^{\mu}$) carrying the incoming momentum $q_n^{\mu}$
($q_{\overline{n}}^{\mu}$), and the amplitude with the gluons
switched, is given by  
\begin{eqnarray}
M_a&=& \frac{g^2}{m_b (\overline{n} \cdot q_n +n \cdot q_{\overline{n}}
  ) +\overline{n}\cdot q_n n\cdot q_{\overline{n}}}  \nonumber \\
&\times&\overline{q}
  \Gamma_1 \Bigl[ \Bigl( m_b n\cdot A_{\overline{n}} + \overline{n}
  \cdot q_n n\cdot A_{\overline{n}} \frac{\FMslash{n}
  \FMslash{\overline{n}}}{4} \Bigr) \frac{\overline{n} \cdot
  A_n}{\overline{n} \cdot q_n} +\Bigl(m_b \overline{n} \cdot A_n +
  n\cdot q_{\overline{n}} 
  \overline{n} \cdot A_n \frac{\FMslash{\overline{n}} \FMslash{n}}{4}
  \Bigr) \frac{n\cdot A_{\overline{n}}}{n\cdot q_{\overline{n}}}
  \Bigr] b,
\end{eqnarray}
and the amplitude for Fig.~\ref{fig1} (b) with $A_n^{\mu}$ and
$A_{\overline{n}}^{\mu}$ is written as
\begin{eqnarray}
M_b &=& \frac{ig^2}{2} f_{abc} \overline{q} \Gamma_1 T_a \frac{n\cdot
  A_{\overline{n}}^b \overline{n} \cdot A_n^c}{n\cdot q_{\overline{n}}
  \overline{n} \cdot q_n}b \nonumber \\
&-&\frac{ig^2 f_{abc}}{m_b (\overline{n} \cdot q_n +n \cdot
  q_{\overline{n}} ) +\overline{n}\cdot q_n n\cdot q_{\overline{n}}}
  \overline{q} 
  \Gamma_1 T^a n\cdot A_{\overline{n}}^b 
  \frac{\overline{n} \cdot A_n^c}{\overline{n} \cdot q_n} \Bigl( m_b
  +\overline{n} \cdot q_n \frac{\FMslash{n} \FMslash{\overline{n}}}{4}
  \Bigr) b.
\end{eqnarray}
At first sight, these amplitudes include a complicated denominator
which cannot be expressed in terms of $A$, $B$, $C$ or $D$. However,
if we add $M_a$ and $M_b$, we obtain
\begin{equation}
M_a + M_b =\frac{ig^2}{2} f_{abc} \overline{q} \Gamma_1 T_a \frac{n\cdot
  A_{\overline{n}}^b \overline{n} \cdot A_n^c}{n\cdot q_{\overline{n}}
  \overline{n} \cdot q_n}b+ g^2 \overline{q} \Gamma_1
  \frac{\overline{n} \cdot A_n}{\overline{n} 
  \cdot q_n} \frac{n\cdot A_{\overline{n}}}{n\cdot q_{\overline{n}}}
  b,
\label{heavy}
\end{equation}
in which the denominator ${m_b (\overline{n} \cdot q_n +n \cdot
q_{\overline{n}}) +\overline{n}\cdot q_n n\cdot
q_{\overline{n}}}$ disappears and there appear only $A$ and
$B$. Therefore the role of the triple gluon vertex is critical not
only in  determining the order of the Wilson lines but also in making
the final expression simple. We can derive gauge-invariant operators
using the auxiliary fields, and this cancellation is important in
constructing the Lagrangian for the auxiliary fields. The auxiliary
field method is explained in Appendix. 



For singlet operators, when we sum over all the Feynman diagrams, we
have  
\begin{equation}
H^S_{\alpha\beta} = \Bigl[ 
 g(-A+D) -g^2 AD \Bigr]_{\alpha \beta}, \ L^S_{\gamma\delta}
 =\delta_{\gamma\delta}, 
\label{dfsing}
\end{equation}
to order $g^2$. Here we show only the products of two different
gauge fields, since they indicate the ordering
of the Wilson lines. From Eq.~(\ref{dfsing}), $H^S_{\alpha\beta}$
suggests the Wilson line to be $(WS^{\dagger})_{\alpha\beta}$. 
For color nonsiglet operators, we have
\begin{eqnarray}
H^N_{\gamma\beta} &=& \Bigl[g(-A+B-C+D) +g^2
  (AC+DB-CB-DC-AB-AD)\Bigr]_{\gamma\beta}, \nonumber \\ 
L^N_{\alpha\delta} &=&\Bigl[ g(-B+C) -g^2 BC\Bigr]_{\alpha\delta}.
\label{dfnon}
\end{eqnarray}
From Eq.~(\ref{dfnon}), $H^N_{\gamma\beta}$ suggests the Wilson line in
the order $(\overline{W} \overline{S}^{\dagger})_{\gamma\beta}$, and
$L^N_{\alpha\delta}$ suggests the form $(WS^{\dagger}
\overline{S}\overline{W}^{\dagger})_{\alpha\delta}$. Therefore the
gauge-invariant singlet and nonsinglet operators are given by
\begin{eqnarray}
O_S &=& \Bigl((\overline{\xi} W)_{\alpha} \Gamma_1
(S^{\dagger}h)_{\alpha} \Bigr)
\Bigl((\overline{\chi}\overline{W})_{\beta} \Gamma_2
(\overline{W}^{\dagger} \chi)_{\beta}\Bigr), \nonumber \\
O_N &=& \Bigl(
(\overline{\xi} WS^{\dagger} \overline{S})_{\beta} \Gamma_1
(\overline{S}^{\dagger} h )_{\alpha} \Bigr)
\Bigl( (\overline{\chi} \overline{W})_{\alpha} \Gamma_2
(\overline{W}^{\dagger} \chi)_{\beta} \Bigr).
\label{gio2}
\end{eqnarray}
The operator $O_N$ can be written as $\Bigl(
(\overline{\xi} WS^{\dagger})_{\beta} \Gamma_1
(\overline{S}^{\dagger} h )_{\alpha} \Bigr)
\Bigl( (\overline{\chi} \overline{W})_{\alpha} \Gamma_2
(\overline{S}\overline{W}^{\dagger} \chi)_{\beta} \Bigr)$, but this
is identical to $O_N$ due to the identity $(WS^{\dagger}
\overline{S})_{\alpha\gamma} \otimes
(\overline{W}^{\dagger})_{\gamma\beta} =
(WS^{\dagger})_{\alpha\gamma}  \otimes
(\overline{S}\overline{W}^{\dagger})_{\gamma\beta}$. 


All the four-quark operators for nonleptonic $B$ decays are of the
form $O_S$ or $O_N$ with different Dirac structure. And the form of
the operators $O_S$ and $O_N$ in Eq.~(\ref{gio2}) manifestly shows the
factorization of 
four-quark operators at leading order in SCET and to all orders in
$\alpha_s$. In the operators $O_S$ and $O_N$, the interactions of
collinear gluons $A_n^{\mu}$ and soft gluons $A_s^{\mu}$ occur
only in the heavy-to-light current sector, while the interactions of
collinear gluons $A_{\overline{n}}^{\mu}$ occur only in the
light-to-light current sector. Due to this property, the form of the
operators is preserved even though there are any possible exchange of
gluons to all orders. This is an explicit proof of color
transparency in SCET, and we can safely calculate the matrix elements
of the operators in terms of a product of the matrix elements of the
two currents.  

Note that the terminology ``factorization'' has been used in two
ways. First, it means that the matrix elements of the
four-quark operators are reduced to products of the matrix elements of
the currents. That is, we can write the matrix element as
\begin{equation}
\langle M_1 M_2 | j_1 \otimes j_2 |B\rangle = \langle M_1 |j_1 |B\rangle
\langle M_2|j_2 |0\rangle.
\end{equation}
This was first assumed in naive factorization. It is possible only
when there is no gluon exchange which connects the two currents and it
is explicitly shown in SCET to all orders in $\alpha_s$.  

Another meaning of factorization appears in high-energy processes in
which a physical amplitude can be separated into a short-distance
part and a long-distance part. For example, exclusive hadronic form
factors at momentum transfer $Q^2 \gg \Lambda_{\mathrm{QCD}}^2$ factor
into nonperturbative light-cone wave functions $\phi$ for mesons,
convoluted with a hard scattering kernel $T$ as \cite{brodsky}
\begin{equation}
F(Q^2) =\frac{f_a f_b}{Q^2} \int dx dy T(x,y,\mu) \phi_a (x,\mu)
\phi_b (y,\mu) +\cdots.
\end{equation}
We discuss both types of factorization in this paper. So far, we have
considered the factorization of matrix elements. The second meaning of
factorization will be considered after we include spectator
contributions. 



By matching the four-quark operators in the full theory and SCET, the
effective Hamiltonian for $B$ decays in SCET can be written as 
\begin{equation}
H_{\mathrm{eff}} = \frac{G_F}{\sqrt{2}} \sum_{T=R,C}
\Bigl[V_{ub} V_{ud}^* \Bigl(
  C_{1T} O_{1T} +   C_{2T} O_{2T} \Bigr)
+ \sum_{p=u,c} V_{pb} V_{pd}^*  \sum_{i=3,\cdots,6} C_{iT}^p 
  O_{iT} \Bigr) \Bigr], 
\label{scetham}
\end{equation}
where
\begin{eqnarray}
O_{1R} &=& \Bigl( (\overline{\xi}^u W)_{\alpha} (S^{\dagger}
h)_{\alpha} \Bigr)_{V-A} \Bigl( (\overline{\chi}^d \overline{W})_{\beta}
(\overline{W}^{\dagger}\chi^u)_{\beta} \Bigr)_{V-A}, \nonumber \\
O_{2R} &=&  \Bigl( (\overline{\xi}^u
W S^{\dagger} \overline{S})_{\beta} (\overline{S}^{\dagger}
h)_{\alpha} \Bigr)_{V-A} \Bigl( (\overline{\chi}^d \overline{W} )_{\alpha}
(\overline{W}^{\dagger}\chi^u)_{\beta} \Bigr)_{V-A}, \nonumber \\
O_{3R} &=& \Bigl( (\overline{\xi}^d W)_{\alpha} (S^{\dagger}
h)_{\alpha} \Bigr)_{V-A} \sum_q \Bigl( (\overline{\chi}^q
\overline{W})_{\beta} (\overline{W}^{\dagger}\chi^q)_{\beta}
\Bigr)_{V-A}, \nonumber \\ 
O_{4R} &=&  \Bigl( (\overline{\xi}^d
W S^{\dagger} \overline{S})_{\beta} (\overline{S}^{\dagger}
h)_{\alpha} \Bigr)_{V-A} \sum_q \Bigl( (\overline{\chi}^q \overline{W}
)_{\alpha} 
(\overline{W}^{\dagger}\chi^q)_{\beta} \Bigr)_{V-A}, \nonumber \\
O_{5R} &=& \Bigl( (\overline{\xi}^d W)_{\alpha} (S^{\dagger}
h)_{\alpha} \Bigr)_{V-A} \sum_q \Bigl( (\overline{\chi}^q
\overline{W})_{\beta} (\overline{W}^{\dagger} \chi^q)_{\beta}
\Bigr)_{V+A}, \nonumber \\ 
O_{6R} &=&  \Bigl( (\overline{\xi}^d
W S^{\dagger} \overline{S})_{\beta} (\overline{S}^{\dagger}
h)_{\alpha} \Bigr)_{V-A} \sum_q \Bigl( (\overline{\chi}^q \overline{W}
)_{\alpha} 
(\overline{W}^{\dagger}\chi^q)_{\beta} \Bigr)_{V+A}, \nonumber \\
O_{1C} &=&  \Bigl( (\overline{\xi}^d
W S^{\dagger} \overline{S})_{\beta} (\overline{S}^{\dagger}
h)_{\alpha} \Bigr)_{V-A} \Bigl( (\overline{\chi}^u \overline{W} )_{\alpha}
(\overline{W}^{\dagger}\chi^u)_{\beta} \Bigr)_{V-A}, \nonumber \\
O_{2C} &=& \Bigl( (\overline{\xi}^d W)_{\alpha} (S^{\dagger}
h)_{\alpha} \Bigr)_{V-A} \Bigl( (\overline{\chi}^u \overline{W})_{\beta}
(\overline{W}^{\dagger}\chi^u)_{\beta} \Bigr)_{V-A}, \nonumber \\
O_{3C} &=&  \sum_q \Bigl( (\overline{\xi}^q
W S^{\dagger} \overline{S})_{\beta} (\overline{S}^{\dagger}
h)_{\alpha} \Bigr)_{V-A}  \Bigl( (\overline{\chi}^d \overline{W}
)_{\alpha} 
(\overline{W}^{\dagger}\chi^q)_{\beta} \Bigr)_{V-A}, \nonumber \\
O_{4C} &=& \sum_q \Bigl( (\overline{\xi}^q W)_{\alpha} (S^{\dagger}
h)_{\alpha} \Bigr)_{V-A}  \Bigl( (\overline{\chi}^d \overline{W})_{\beta}
(\overline{W}^{\dagger} \chi^q)_{\beta} \Bigr)_{V-A}.
\end{eqnarray}
Here the summation over $q$ goes over to light massless quarks, say,
$u$, $d$, and $s$ quarks. Since we specify the direction of each
quark, we have the original operators $O_{iR}$, obtained from
Eq.~(\ref{fullop}), in which the light-to-light current forms a meson
in the $\overline{n}^{\mu}$ direction. When a light quark in the
heavy-to-light current moves in the $\overline{n}^{\mu}$ direction and
forms a meson with an antiquark in the light-to-light current, we use
the Fierz transformation first in the full theory and match the
operators. 

Note that there are no operators $O_{5C}$ and $O_{6C}$ in SCET, which
are obtained by Fierzing $O_{5R}$ and $O_{6R}$. Neglecting color
structure and the Wilson lines, $O_{5C}$ and $O_{6C}$ have the form $-2
\overline{\xi} (1+\gamma_5) h \cdot \overline{\chi} (1-\gamma_5) \chi$,
which is identically zero at leading order in $\lambda$ in SCET
because 
\begin{equation}
\overline{\chi} (1+\gamma_5)\chi = \overline{\chi} \frac{\FMslash{n}
  \FMslash{\overline{n}}}{4} (1+\gamma_5)\chi \nonumber \\
=\overline{\chi} (1+\gamma_5) \frac{\FMslash{n}
  \FMslash{\overline{n}}}{4} \chi =0,
\end{equation}
since $\FMslash{\overline{n}} \chi =0$. In the literature, the effects
of the operators $O_{5C}$ and $O_{6C}$ are sometimes known as
chirally-enhanced contributions. Even though the effect is formally
suppressed, the numerical values are not negligible. The enhancement
is obtained by applying the equation of motion for the currents and
the result is such that there is an enhancement factor $m_M^2/m_q m_b$,
where $m_M$ is a meson mass and $m_q$ is the constituent quark
mass. But in SCET there is simply no such operator as $O_{5C}$ and
$O_{6C}$ at leading order in $\lambda$. This contribution is formally
suppressed in powers of $\lambda$ in SCET, but the coefficient of
these operators can be large. In phenomenological applications, we
have to know how to treat the chirally-enhanced contributions, but
we will not consider them here.




\section{Matching and the Wilson coefficients in SCET\label{sec3}}
We can determine the Wilson coefficients of the four-quark operators
in SCET given by Eq.~(\ref{scetham}) by matching the full theory onto
SCET. We require that the matrix elements of an operator in the
full theory be equal to the matrix elements of the corresponding
operator in SCET. 
\begin{figure}[t]
\begin{center}
\epsfig{file=nlfig2.eps, width=12.0cm}
\end{center}
\caption{Radiative corrections at one loop in the full theory, which
contribute to the Wilson coefficients of the effective operators. $p_1$, 
  $p_2$, $p$ are outgoing momenta with $p_b = p
  +p_1+p_2$. Infrared divergences exist in diagrams (a) to
  (f). Diagrams (g) and (h) are infrared finite. In (h), the square is
  the chromomagnetic operator $O_8$.} 
\label{fig2}
\end{figure}
We use the $\overline{\mathrm{MS}}$ scheme with naive dimensional
regularization scheme with anticommuting $\gamma_5$. All the external
particles are on the mass shell. The radiative corrections for the
four-quark operators in the full theory are shown in Fig.~\ref{fig2}. 
As a specific example, the amplitudes for Fig.~\ref{fig2} (a) to (d)
for the operator $O_1 = 
(\overline{d} u)_{V-A} (\overline{u} b)_{V-A}$ are given as
\begin{eqnarray}
iM_a^{(1)} &=& \frac{\alpha_s}{4\pi} (T_a)_{jk} (T_a)_{il}
(\overline{d}_j u_k)_{V-A} (\overline{u}_i b_l)_{V-A} \nonumber \\
&\times& \Bigl[\frac{1}{\epsilon_{\mathrm{UV}}} -\frac{1}{\epsilon^2}
  +\frac{2}{\epsilon} \bigl( \ln \frac{x_1 m_b}{\mu}-1\Bigr)
 -4 +2\ln \frac{m_b}{\mu} 
-2\ln^2 \frac{x_1 m_b}{\mu} \nonumber \\
&+&\frac{2-3x_1}{1-x_1}\ln x_1 - 2\mathrm{Li}_2 (1-x_1)
-\frac{\pi^2}{12}\Bigr], \nonumber \\
iM_b^{(1)} &=&  \frac{\alpha_s}{4\pi} (T_a)_{jk} (T_a)_{il}
(\overline{d}_j u_k)_{V-A} (\overline{u}_i b_l)_{V-A} \nonumber \\
&\times& \Bigl[ -\frac{4}{\epsilon_{\mathrm{UV}}}
  +\frac{1}{\epsilon^2} -\frac{2}{\epsilon} \Bigl( \ln \frac{x_2
    m_b}{\mu} -1 \Bigr) -5 + 4\ln \frac{m_b}{\mu} +2\ln^2 \frac{x_2
    m_b}{\mu} \nonumber \\
&-&\frac{2}{1-x_2} \ln x_2 +2\mathrm{Li}_2 (1-x_2)+\frac{\pi^2}{12}
  \Bigr], \nonumber \\
iM_c^{(1)} &=&  \frac{\alpha_s}{4\pi} (T_a)_{jk} (T_a)_{il}
(\overline{d}_j u_k)_{V-A} (\overline{u}_i b_l)_{V-A} \nonumber \\
&\times& \Bigl[  -\frac{4}{\epsilon_{\mathrm{UV}}}
  +\frac{2}{\epsilon^2} +\frac{2}{\epsilon} \Bigl(2-\ln
  \frac{-xx_1 m_b^2}{\mu^2}  \Bigr) -1 +\ln^2 \Bigl(
  \frac{-xx_1 m_b^2}{\mu^2} \Bigr) -\frac{\pi^2}{6} \Bigr], \nonumber
\\
iM_d^{(1)} &=&  \frac{\alpha_s}{4\pi} (T_a)_{jk} (T_a)_{il}
(\overline{d}_j u_k)_{V-A} (\overline{u}_i b_l)_{V-A} \nonumber \\
&\times& \Bigl[ \frac{1}{\epsilon_{\mathrm{UV}}} -\frac{2}{\epsilon^2}
  -\frac{2}{\epsilon} \Bigl(2 -\ln \frac{-xx_2 m_b^2}{\mu^2}  \Bigr)
  \nonumber \\
&&-8 +3 \ln \frac{-xx_2m_b^2}{\mu^2}  -\ln^2 \frac{-xx_2
   m_b^2}{\mu^2} +\frac{\pi^2}{6} \Bigr], 
\label{mat1}
 \end{eqnarray}
where $\mathrm{Li}_2 (x)$ is the dilogarithmic function. And the
prescription for $\ln (-x_i)$ is given by
$\ln (-x_i-i\epsilon) = \ln x_i -i\pi$ for $i=1,2$. 
If we add all these ``nonfactorizable'' contributions, the infrared
divergence cancels and the only infrared divergence comes from the
vertex corrections of the currents [Fig.~\ref{fig2} (e) and
  (f)]. Since the vertex correction of the light-to-light current is
both the same in the full theory and in SCET, we will neglect it. 
And Fig.~\ref{fig2} (f) is given by
\begin{eqnarray}
iM_f^{(1)} &=&  \frac{\alpha_s C_F}{4\pi} (\overline{d} u)_{V-A}
(\overline{u} b)_{V-A}  \nonumber \\
&\times& \Bigl[ \frac{1}{\epsilon_{\mathrm{UV}}} -\frac{1}{\epsilon^2}
  +\frac{2}{\epsilon} \Bigl( \ln \frac{xm_b}{\mu} -1 \Bigr) -3  +2
  \ln \frac{m_b}{\mu}-2\ln^2 \frac{xm_b}{\mu} \nonumber \\
&+&\frac{2-x}{1-x} \ln x  - 2\mathrm{Li}_2 (1-x)-\frac{\pi^2}{12}
  \Bigr]. 
\end{eqnarray}

For the operator $O_5 = (\overline{d} b)_{V-A} \sum_q (\overline{q}
q)_{V+A}$, the corresponding amplitudes are given as
\begin{eqnarray}
iM_a^{(5)} &=& \frac{\alpha_s}{4\pi} (T_a)_{jk} (T_a)_{il}
(\overline{d}_i b_l )_{V-A} (\overline{q}_j q_k)_{V+A} \nonumber \\
&\times& \Bigl[\frac{4}{\epsilon_{\mathrm{UV}}} -\frac{1}{\epsilon^2}
    +\frac{2}{\epsilon} \Bigl( \ln \frac{x_1 m_b}{\mu} -1 \Bigr) +2-4\ln
    \frac{m_b}{\mu} -2\ln^2 \frac{x_1 m_b}{\mu} \nonumber \\
&&+ \frac{2}{1-x_1} \ln x_1 -2\mathrm{Li}_2 (1-x_1) -\frac{\pi^2}{12}
    \Bigr], \nonumber \\
iM_b^{(5)} &=& \frac{\alpha_s}{4\pi} (T_a)_{jk} (T_a)_{il}
(\overline{d}_i b_l )_{V-A} (\overline{q}_j q_k)_{V+A} \nonumber \\
&\times& \Bigl[-\frac{1}{\epsilon_{\mathrm{UV}}} +\frac{1}{\epsilon^2}
  -\frac{2}{\epsilon} \Bigl( \ln \frac{x_2 m_b}{\mu} -1 \Bigr) +1-2\ln
  \frac{m_b}{\mu} +2\ln^2 \frac{x_2 m_b}{\mu} \nonumber \\
&&-\frac{2-3x_2}{1-x_2} \ln x_2 +2\mathrm{Li}_2 (1-x_2)
  +\frac{\pi^2}{12} \Bigr], \nonumber \\
iM_c^{(5)} &=& \frac{\alpha_s}{4\pi} (T_a)_{jk} (T_a)_{il}
(\overline{d}_i b_l )_{V-A} (\overline{q}_j q_k)_{V+A} \nonumber \\
&\times& \Bigl[-\frac{1}{\epsilon_{\mathrm{UV}}} +\frac{2}{\epsilon^2}
  +\frac{2}{\epsilon} \Bigl( 2-\ln \frac{-xx_1 m_b^2}{\mu^2} \Bigr) +5
  -3 \ln  \frac{-xx_1 m_b^2}{\mu^2} \nonumber \\
&&+ \ln^2  \frac{-xx_1 m_b^2}{\mu^2} -\frac{\pi^2}{6} \Bigr],
\nonumber \\
iM_d^{(5)} &=& \frac{\alpha_s}{4\pi} (T_a)_{jk} (T_a)_{il}
(\overline{d}_i b_l )_{V-A} (\overline{q}_j q_k)_{V+A} \nonumber \\
&\times& \Bigl[\frac{4}{\epsilon_{\mathrm{UV}}} -\frac{2}{\epsilon^2}
  -\frac{2}{\epsilon} \Bigl( 2-\ln \frac{-xx_2 m_b^2}{\mu^2} \Bigr) -2
  -\ln^2  \frac{-xx_2 m_b^2}{\mu^2} +\frac{\pi^2}{6}\Bigr], 
\label{mat5}
\end{eqnarray}
and $iM_f^{(5)}$ is the same as $iM_f^{(1)}$ except the operator
structure. Here also the infrared divergence of the nonfactorizabie
contribution in Eq.~(\ref{mat5}) cancels, and the only infrared
divergence comes from the vertex corrections of the currents.

Using the color identity
\begin{equation}
(T_a)_{jk} (T_a)_{il} =\frac{1}{2} \delta_{jl} \delta_{ik}
  -\frac{1}{2N} \delta _{jk} \delta_{il},
\end{equation}
the operators in Eq.~(\ref{mat1}) becomes $O_1/2 -O_2/(2N)$, similary
the operators in Eq.~(\ref{mat5}) becomes $O_5/2 -O_6/(2N)$. In the
radiative corrections of $O_2$, the color factors in Fig.~\ref{fig2}
(a) and (d) make the operator as $C_F O_2$, while (b) and (c) become
$O_1/2-O_2/(2N)$. Therefore the radiative corrections for $O_2$ or
nonsinglet operators for that matter have infrared divergences in all
the diagrams. 


\begin{figure}[b]
\begin{center}
\epsfig{file=nlfig3.eps, height=7.0cm}
\end{center}
\caption{Radiative corrections at one loop in the SCET. Curly lines
  with a line represent collinear gluons, and curly lines represent
 soft gluons. }
\label{fig3}
\end{figure}

The corresponding radiative corrections for the four-quark operators
in SCET are shown in Fig.~\ref{fig3}. For singlet operators
the radiative corrections exist only in the heavy-to-light current
sector [Fig.~\ref{fig3} (a), (b)], in which the infrared divergence
appearing in the calculation is exactly the same as the infrared
divergence in the full theory.  The diagram (c) should be included,
but since the result is the same in the full theory, it is cancelled
in matching. 
For nonsinglet operators we need all the diagrams in Fig.~\ref{fig3}
with additional collinear gluon interactions in the light-to-light
current sector [Fig.~\ref{fig3} (e) and (f)] and usoft gluon
interactions [Fig.~\ref{fig3} (b) and (d)]. When we calculate these
diagrams, the infrared divergence is exactly the same as the infrared
divergence in the full theory. Therefore we can safely match the full
theory onto SCET since the infrared divergence cancels, and the
Wilson coefficients can be calculated.



Note that any loop diagram in SCET is zero when we treat both
ultraviolet and infrared divergences using dimensional regularization
because there is no explicit scale in the theory. If there is an
infrared divergence, it is 
cancelled by an ultraviolet divergence. Ultraviolet divergences are
removed by adding counterterms and infrared divergences cancel in
matching with the full theory. Therefore the Wilson coefficients of
various operators in SCET can be obtained by calculating radiative
corrections in the full theory. In the full theory we also have to
consider the Feynman diagrams with fermion loops and the effect of the
chromomagnetic operator. These contributions are included in
Fig.~\ref{fig1} (g) and (h).  


The Wilson coefficients are, in general, functions of the operators
$\overline{n}\cdot \mathcal{P}/m_b$, $n\cdot \mathcal{Q}/m_b$ and
$n\cdot \mathcal{Q}^{\dagger}/m_b$. The matrix elements of the
four-quark operators can be written in terms of a convolution as
\begin{eqnarray}
M&=&\int_0^1 dx dx_1 dx_2 C(x,x_1,x_2) \langle \overline{\xi} W \delta
  \Bigl(x-\frac{\overline{n} \cdot \mathcal{P}^{\dagger}}{m_b} \Bigr)
  \Gamma_1 S^{\dagger} h   \nonumber \\
&&\times \overline{\chi} \overline{W} \delta \Bigl( x_1 -\frac{n\cdot
  \mathcal{Q}^{\dagger}}{m_b} \Bigr) \Gamma_2 \delta \Bigl( x_2
  +\frac{n\cdot \mathcal{Q}}{m_b} \Bigr) \overline{W}^{\dagger} \chi
  \rangle   
\label{mael}
\end{eqnarray}
Due to the momentum conservation $x$, $x_1$ and $x_2$ satisfiy $x+x_1
+x_2 =2$. For nonleptonic decays into two light mesons at leading
order in SCET, we can set $x=1$ and $x_1 =u$, and $x_2 = \overline{u}
\equiv 1-x_1$. In this case, the matrix element can be written as
\begin{equation}
M\rightarrow \int d\eta \ C(\eta) \langle \overline{\xi} W\Gamma_1
  S^{\dagger} 
  h \cdot \overline{\chi}\overline{W} \delta (\eta -
  \mathcal{Q}_+ ) \Gamma_2 \overline{W}^{\dagger}
  \chi\rangle.
\label{fqop}
\end{equation}
where $u=x_1=1-x_2=\eta/(4E) +1/2$ with $\mathcal{Q}_+ = n\cdot
\mathcal{Q}^{\dagger} + n\cdot \mathcal{Q}$. However, we list all the
Wilson coefficients for general $x$, $x_1$ and $x_2$, which will be
useful for other decay modes, or nonleptonic $B$ decays at subleading
order in SCET.



By adding the wave function renormalization of the heavy quark, the
Wilson coefficients at next-to-leading order are given as
\begin{eqnarray}
C_{1R}  &=& \Bigl[1-\frac{A_1}{2N} + C_F A_2 \Bigr] C_1
+\frac{A_3}{2} C_2 , \  C_{2R} = \frac{A_1}{2} C_1 +\Bigl[
  1-\frac{A_3}{2N} +C_F A_4 \Bigr] C_2, \nonumber \\
C_{3R}^p &=& \Bigl[1-\frac{A_1}{2N}  + C_F A_2 \Bigr] C_3 +\frac{A_3}{2}
C_4, \ C_{4R}^p = \frac{A_1}{2} C_3 +\Bigl[ 1-\frac{A_3}{2N} +C_F A_4
  \Bigr] C_4, \nonumber \\
C_{5R}^p &=& \Bigl[ 1-\frac{A_5}{2N} +C_F A_2 \Bigr] C_5 +\frac{A_6}{2}
C_6, \ C_{6R}^p = \frac{A_5}{2} C_5 +\Bigl[1-\frac{A_6}{2N} +C_F A_7
  \Bigr] C_6, \nonumber \\
C_{1C}&=& \Bigl[ 1-\frac{A_3}{2N} +C_F A_4 \Bigr] C_1 + \frac{A_1}{2}
C_2, \ C_{2C} = \frac{A_3}{2} C_1 + \Bigl[1-\frac{A_1}{2N} + C_F A_2
  \Bigr] C_2\nonumber \\ 
C_{3C}^p&=&\Bigl[ 1-\frac{A_3}{2N} +C_F A_4 \Bigr] C_3 +  \frac{A_1}{2}
C_4-\frac{1}{2N} C_l^p, \nonumber \\
C_{4C}^p &=&  \frac{A_3}{2} C_3 +   \Bigl[1-\frac{A_1}{2N}  + C_F A_2
  \Bigr] C_4 +\frac{1}{2}  C_l^p,
\label{wilson}
\end{eqnarray}
where $C_i$'s are the Wilson coefficients from the full theory. And
the coefficients $A_i$ to order $\alpha_s$ are given as 
\begin{eqnarray}
A_1&=& \frac{\alpha_s}{4\pi} \Bigl[ -18 +12 \ln
  \frac{m_b}{\mu} + 3\ln (-x)  +\ln \frac{x^2}{x_1 x_2} \ln \frac{x_1}{x_2}
+\frac{2-3x_1}{1-x_1} \ln x_1 \nonumber \\
&&+\frac{1-3x_2}{1-x_2} \ln x_2 -2
  \mathrm{Li}_2 (1-x_1 ) +2 \mathrm{Li}_2 (1-x_2)  \Bigr]
  \nonumber \\ 
A_2&=& \frac{\alpha_s}{4\pi} \Bigl[-5 +5\ln \frac{m_b}{\mu}
  -2 \ln^2 \frac{xm_b}{\mu} +\frac{2-x}{1-x} \ln x -2 \mathrm{Li}_2
  (1-x) -\frac{\pi^2}{12} \Bigr]    \nonumber \\
A_3 &=& \frac{\alpha_s}{4\pi} \Bigl[ -9 +6 \ln \frac{m_b}{\mu} 
 +\frac{2-x}{1-x} \ln x  -2 \mathrm{Li}_2 ( 1-x)
+2\ln^2 \frac{x_2 m_b}{\mu}\nonumber \\
&& +2\ln^2 \frac{-x_1 m_b}{\mu} -\ln^2 \frac{-x_1}{x} -\frac{2}{1-x_2}
  \ln x_2 +2 \mathrm{Li}_2(1-x_2) -\frac{\pi^2}{6}\Bigr],  \nonumber \\
A_4 &=& \frac{\alpha_s}{4\pi} \Bigl[-14 + 11 \ln \frac{m_b}{\mu}
  -2\ln^2 \frac{x_1 m_b}{\mu} -\ln^2 \frac{-xx_2 m_b^2}{\mu^2}
  \nonumber \\
&& +3\ln (-xx_2) +\frac{2-3x_1}{1-x_1} \ln x_1 -2\mathrm{Li}_2 (1-x_1)
  +\frac{\pi^2}{12} \Bigr],  \nonumber \\
A_5  &=&  \frac{\alpha_s}{4\pi} \Bigl[6 -12 \ln \frac{m_b}{\mu} -3 \ln
 (- x)  +\ln \frac{x^2}{x_1 x_2} \ln \frac{x_1}{x_2}
  -\frac{1-3x_1}{1-x_1} \ln x_1 
  \nonumber \\
&&-\frac{2-3x_2}{1-x_2} \ln x_2 -2 \mathrm{Li}_2 (1-x_1) +2
  \mathrm{Li}_2  ( 1-x_2) \Bigr],
  \nonumber \\
A_6 &=& \frac{\alpha_s}{4\pi} \Bigl[3 -6 \ln  \frac{m_b}{\mu} +2\ln^2
  \frac{x_2 m_b}{\mu} +2\ln^2 \frac{-x_1 m_b}{\mu} -\ln^2
  \frac{-x_1}{x} \nonumber \\
&&-\frac{1-2x}{1-x} \ln x -3\ln (-x_1)
  -\frac{2-3x_2}{1-x_2} \ln x_2 \nonumber \\
&&-2 \mathrm{Li}_2 (1-x) +2 \mathrm{Li}_2 ( 1-x_2)
  -\frac{\pi^2}{6} \Bigr], \nonumber \\
A_7 &=&\frac{\alpha_s}{4\pi} \Bigl[ -2 - \ln \frac{m_b}{\mu} -2\ln^2
  \frac{x_1 m_b}{\mu} -\ln^2 \frac{-xx_2 m_b^2}{\mu^2}
+\frac{2}{1-x_1} \ln x_1 \nonumber \\
&&-2\mathrm{Li}_2 (1-x_1) +\frac{\pi^2}{12} \Bigr]. 
\end{eqnarray}
And the contribution $C_l^p$ from fermion loops and the
chromomagnetic operator [Fig.~\ref{fig2} (g) and (h)]  are given by
\begin{eqnarray}
C_l^p&=& \frac{\alpha_s}{4\pi} \Bigl[ C_1 \Bigl( \frac{2}{3}
  +\frac{4}{3} \ln \frac{m_b}{\mu} -G(s_p) \Bigr) +C_3 \Bigl(
  \frac{4}{3} +\frac{8}{3} \ln \frac{m_b}{\mu} -G(0) -G(1) \Bigr)
  \nonumber \\
&&+C_4 \Bigl( \frac{2n_f}{3} +\frac{4n_f}{3} \ln \frac{m_b}{\mu}
  -3G(0) -G(s_c) -G(1) \Bigr) \nonumber \\
&&+C_6 \Bigl( \frac{4n_f}{3} \ln \frac{m_b}{\mu} -3G(0) -G(s_c) -G(1)
  \Bigr) -(C_5+C_8)\frac{2}{1-x_1} \Bigr], 
\label{clp}
\end{eqnarray}
where $s_p=m_p^2/m_b^2$ ($s_c = m_c^2/m_b^2$), and $G(s)$ is given by
\begin{equation}
G(s) =-4\int_0^1 dz \, z(1-z) \Bigl(s-z(1-z) (1-x_1) -i\epsilon \Bigr).
\end{equation}
We can add $C_l^p/(2N)$ in $C_3^R$ and $C_5^R$,
and $C_l^p/2$ in $C_4^R$ and $C_6^R$ with $x_1 \rightarrow x$ in
Eq.~(\ref{clp}). But in physical processes in which the 
final-state consists of color singlet mesons, the contribution of
$C_l^p$ cancels, hence omitted in Eq.~(\ref{wilson}).  

\section{Nonfactorizable spectator contributions\label{sec4}}
We also have to consider nonfactorizable spectator contributions.
Nonfactorizable spectator contributions arise from the interactions of
the collinear gluons in the $n^{\mu}$ direction in the light-to-light
current with a spectator quark, which becomes a collinear quark as a
result. However the operators $O_S$ and $O_N$ at leading order
in SCET do not involve the interaction of $A_n^{\mu}$. Therefore we take
into account a subleading operator which involves collinear gluons
$A_n^{\mu}$ in the light-to-light current sector. And the Lagrangian
describing the interaction of collinear and usoft quarks begins with
$\mathcal{O} (\lambda)$ compared to the leading-order collinear
Lagrangian. But the propagator of the exchanged gluon is of order
$\lambda^2$. Therefore the nonfactorizable spectator
contribution is of the same order as the leading
contributions from the four-quark operators. In order to evaluate
decay amplitudes at leading order in SCET, we need to include the
nonfactorizable spectator contributions. The
nonfactorizable spectator contribution is shown in
Fig.~\ref{fig4}. Here we consider a collinear gluon $A_n^{\mu}$ from the
light-to-light current sector interacting with a spectator quark
inside a $B$ meson to produce a collinear quark $\xi$.


\begin{figure}[b]
\begin{center}
\epsfig{file=nlfig4.eps, width=4.0cm}
\end{center}
\caption{A Feynman diagram for nonfactorizable spectator
contributions from the subleading operator in the light-to-light
current. The soft momentum $k$ is incoming, and $p_i$ ($i=1,2,3,4$)
are outgoing.} 
\label{fig4}
\end{figure}

Considering operators interacting with $A_n^{\mu}$,  there can be also
a collinear gluon $A_n^{\mu}$ from the heavy-to-light current starting
at leading order. These operators can interact with a spectator quark,
but they contribute to the heavy-to-light form factor. This was
considered in Ref.~\cite{form}, which will be discussed in the next
section.  


The Lagrangian for the  collinear and usoft quarks at order $\lambda$
is given by \cite{pirjol,beneke1}
\begin{equation}
\mathcal{L}_{\xi q} = \overline{q}_{us} W^{\dagger}
i\FMslash{D}_n^{\perp} \xi + \mathrm{h.c.}.
\end{equation}
There is a single four-quark operator $O^{(1)}$ at order $\lambda$,
which contributes to nonfactorizable spectator interactions. In
$\mathrm{SCET}_{\mathrm{I}}$ it is given by
\begin{eqnarray}
O^{(1)} &=& \Bigl( (\overline{\xi}W)_{\beta} \Gamma_1 h_{\alpha}
 \Bigr) \Biggl\{ \Bigl(\overline{\chi} \overline{W} \frac{1}{n\cdot
    \mathcal{Q}^{\dagger}} \frac{\FMslash{n}}{2} \Bigl[ W^{\dagger}
  i \overleftarrow{\FMSlash{D}}_{n\perp} W\Bigr]\Bigl)_{\alpha} \Gamma_2
(\overline{W}^{\dagger} \chi)_{\beta} \nonumber \\
&&+(\overline{\chi}
\overline{W})_{\alpha} \Gamma_2 \Bigl( \Bigl[ W^{\dagger}
  i\overrightarrow{\FMSlash{D}}_{n\perp} W\Bigr] \frac{\FMslash{n}}{2}
  \frac{1}{n \cdot \mathcal{Q}} \overline{W}^{\dagger} \chi
 \Bigr)_{\beta}  \Biggr\}.
\label{osub}
\end{eqnarray}
Here we do not include subleading operators which do not involve the
gauge field $A_n^{\mu}$.
The form of the operator $O^{(1)}$ can be obtained in a
straightforward manner, but the ordering of the Wilson lines is 
nontrivial. It will be explained using the auxiliary field method in
Appendix. 

In $\mathrm{SCET}_{\mathrm{I}}$, the nonfactorizable spectator
contribution is given by the matrix elements of the time-ordered
product
\begin{equation}
T^{(1)} = i\int d^4 x T \Bigl[ O^{(1)} (0), \mathcal{L}_{\xi q}^{(1)}
  (x)   \Bigr].
\end{equation}
In order to go down to $\mathrm{SCET}_{\mathrm{II}}$, we decouple the
collinear-usoft interaction using the field redefinitions \cite{bauer6} 
\begin{equation}
\xi^{(0)} = Y^{\dagger} \xi, \ \ A_n^{(0)} = Y^{\dagger}A_n Y,
\ \ 
Y(x) = \mathrm{P} \exp \Bigl( ig \int_{-\infty}^x ds n\cdot
  A_{us} (ns) \Bigr).
\end{equation}
Matching at $\mu_0 \sim \sqrt{m_b \Lambda}$, the usoft fields become
soft ($Y\rightarrow S$) and the operators are matched onto the
operators in $\mathrm{SCET}_{\mathrm{II}}$. 
The nonfactorizable spectator contribution comes from the matrix
elements of six-quark operators. In calculating the matrix elements of
$T^{(1)}$, we first project out color indices in such a way that the
quark bilinears forming a meson are color singlets. Then the matrix
element is given by
\begin{eqnarray}
\langle T^{(1)} \rangle &=& -4\pi \alpha_s \frac{C_F}{N^2} \int
d\overline{n}\cdot x \int \frac{dn\cdot k}{4\pi} e^{i n\cdot k
    \overline{n} \cdot x/2}
\nonumber \\
&\times& \Biggl\{ 
\langle \Bigl[(\overline{\xi} W)_{\beta} \Gamma_1 (S^{\dagger}
  h)_{\alpha} \Bigr] 
\Bigl[(\overline{\chi} \overline{W} )_{\gamma} \frac{1}{n\cdot
    \mathcal{Q}^{\dagger}} 
\frac{\FMslash{n}}{2} \gamma_{\perp}^{\mu} \Gamma_2
(\overline{W}^{\dagger} \chi)_{\gamma} \Bigr] \nonumber \\
&&\times\Bigl[
  (\overline{q}_s
S)_{\alpha} (\overline{n}\cdot x) \frac{1}{n\cdot \mathcal{R}^{\dagger}}
\gamma_{\mu}^{\perp}\frac{1}{\overline{n}\cdot   \mathcal{P}}
  (W^{\dagger} \xi)_{\beta} (0) \Bigr] 
\rangle \nonumber \\
&+& \langle \Bigl[(\overline{\xi} W)_{\beta}
  \Gamma_1 (S^{\dagger}   h)_{\alpha} \Bigr] 
\Bigl[(\overline{\chi} \overline{W} )_{\gamma}
\Gamma_2  \gamma_{\perp}^{\mu} \frac{\FMslash{n}}{2} \frac{1}{n\cdot
  \mathcal{Q}} 
(\overline{W}^{\dagger} \chi)_{\gamma} \Bigr] \nonumber \\
&&\times \Bigl[  (\overline{q}_s S)_{\alpha}(\overline{n}\cdot x)
  \frac{1}{n\cdot 
  \mathcal{R}^{\dagger}}\gamma_{\mu}^{\perp}
\frac{1}{\overline{n}\cdot 
  \mathcal{P}} (W^{\dagger} \xi)_{\beta} (0)\Bigr]
\rangle
\Biggr\}. 
\label{spec}
\end{eqnarray}

Let us consider in detail the matrix elements in Eq.~(\ref{spec}) for
different Dirac structure $\Gamma_1 \otimes \Gamma_2$. For simplicity
we omit all the Wilson lines and the momentum operators in the
following calculation. For
$\gamma_{\nu} (1-\gamma_5) \otimes \gamma^{\nu} (1-\gamma_5)$, we can
evaluate the first term in the parenthesis in Eq.~(\ref{spec}) as 
\begin{eqnarray}
&&\langle \overline{\xi}_{\beta} \gamma_{\nu} (1-\gamma_5) h_{\alpha}
  \cdot \overline{\chi} \frac{\FMslash{n}}{2} \gamma_{\perp}^{\mu}
  \gamma^{\nu} (1-\gamma_5) \chi \cdot \overline{q}_{\alpha}
  \gamma^{\perp}_{\mu} \xi_{\beta} \rangle \nonumber \\
&&=\langle \overline{\xi}_{\beta} \gamma_{\nu}^{\perp} (1-\gamma_5)
  h_{\alpha} \cdot \overline{\chi} \frac{\FMslash{n}}{2}
  (2g_{\perp}^{\mu\nu} -\gamma_{\perp}^{\nu} \gamma_{\perp}^{\mu})
  (1-\gamma_5) \chi \cdot \overline{q}_{\alpha} \gamma^{\perp}_{\mu}
  \xi_{\beta} \rangle \nonumber \\
&&= 2\langle \overline{\xi}_{\beta} \gamma_{\perp}^{\mu} (1-\gamma_5)
  h_{\alpha} \cdot \overline{q}_{\alpha} \gamma_{\mu}^{\perp}
  \xi_{\beta} \cdot \overline{\chi} \frac{\FMslash{n}}{2} (1-\gamma_5)
  \chi \rangle \nonumber \\
&&-\langle \overline{\xi}_{\beta} \gamma_{\nu}^{\perp} (1-\gamma_5)
  h_{\alpha} \cdot \overline{\chi} \frac{\FMslash{n}}{2}
  \gamma_{\perp}^{\nu} \gamma_{\perp}^{\mu} (1-\gamma_5) \chi \cdot
  \overline{q}_{\alpha} \gamma_{\mu}^{\perp} \xi_{\beta}\rangle.
\label{matcal}
\end{eqnarray}
We suppress the color indices for $\chi$ fields since they already
form a color singlet. In the first line we can replace $\gamma_{\nu}$
by $\gamma_{\nu}^{\perp}$ and we use the Fierz
transformation to arrive at the last relation. The first term in
Eq.~(\ref{matcal}) can be further simplified as 
\begin{eqnarray}
&&2\overline{\xi}_{\beta} \gamma_{\perp}^{\mu} (1-\gamma_5)
  h_{\alpha} \cdot \overline{q}_{\alpha} \gamma_{\mu}^{\perp}
  \xi_{\beta} \cdot \overline{\chi} \frac{\FMslash{n}}{2} (1-\gamma_5)
  \chi  \nonumber \\
&&= \frac{1}{2} \overline{\xi}_{\beta} \gamma^{\mu} (1-\gamma_5)
  h_{\alpha} \cdot \overline{q}_{\alpha} \Bigl[ \gamma_{\mu}
  (1-\gamma_5) +\gamma_{\mu} (1+\gamma_5) \Bigr] \xi_{\beta} \cdot
  \overline{\chi} \FMslash{n} (1-\gamma_5)  \chi  \nonumber \\
&&=\frac{1}{2}  \overline{\xi} \gamma^{\mu} (1-\gamma_5) \xi \cdot
  \overline{q} \gamma_{\mu} (1-\gamma_5) h  \cdot \overline{\chi}
  \FMslash{n} (1-\gamma_5)  \chi  \nonumber \\
&&= \frac{1}{4}  
  \overline{\xi} \FMslash{\overline{n}} (1-\gamma_5) \xi \cdot
  \overline{q} \FMslash{n} (1-\gamma_5) h \cdot  \overline{\chi}
  \FMslash{n} (1-\gamma_5)  \chi. 
\label{firterm}
\end{eqnarray}
In the second line the part proportional to $\gamma^{\mu}
(1+\gamma_5)$ vanishes when we apply the Fierz transformation. In the
third line we use the Fierz transformation for the product of the
first two currents. Similarly, the second term in Eq.~(\ref{matcal})
is simplied as
\begin{eqnarray}
&&-\overline{\xi}_{\beta} \gamma_{\nu} (1-\gamma_5) h_{\alpha}
  \cdot \overline{\chi} \frac{\FMslash{n}}{2} \gamma_{\perp}^{\nu}
  \gamma_{\perp}^{\mu} (1-\gamma_5) \chi \cdot \overline{q}_{\alpha}
  \gamma_{\mu}^{\perp} \xi_{\beta}  \nonumber \\
&& = -\overline{\xi}_{\beta} \gamma_{\nu} (1-\gamma_5) h_{\alpha}
  \cdot \overline{\chi} \frac{\FMslash{n}}{2} \gamma^{\nu}
 (1+\gamma_5) \gamma_{\perp}^{\mu}\chi \cdot \overline{q}_{\alpha}
  \gamma_{\mu}^{\perp} \xi_{\beta}  \nonumber \\
&&= 2 \overline{\xi}_{\beta} \gamma_{\perp}^{\mu} (1-\gamma_5)
  \chi_{\gamma} \cdot \overline{\chi}_{\gamma} \frac{\FMslash{n}}{2}
  (1-\gamma_5) h_{\alpha} \cdot \overline{q}_{\alpha}
  \gamma_{\mu}^{\perp} \xi_{\beta} \nonumber \\
&&=   \overline{\xi}_{\beta} \gamma^{\mu} (1-\gamma_5)
  \chi_{\gamma} \cdot \overline{q}_{\alpha}
  \gamma_{\mu} (1-\gamma_5) \xi_{\beta}\cdot \overline{\chi}_{\gamma}
  \frac{\FMslash{n}}{2}   (1-\gamma_5) h_{\alpha} \nonumber \\
&&=\frac{1}{2}  \overline{\xi} \FMslash{\overline{n}} (1-\gamma_5)
  \xi \cdot \overline{q}_{\alpha} \FMslash{n} (1-\gamma_5)
  \chi_{\gamma} \cdot \overline{\chi}_{\gamma} (1+\gamma_5)
  \frac{\FMslash{n}}{2} h_{\alpha} \nonumber \\
&&=- \frac{1}{4} \overline{\xi} \FMslash{\overline{n}} (1-\gamma_5)\xi
  \cdot \overline{q} \FMslash{n} (1-\gamma_5) h \cdot \overline{\chi}
  \FMslash{n} (1-\gamma_5) \chi.
\label{secterm}
\end{eqnarray}
As can be seen in each step, we use Fierz transformations successively to
obtain the final form. When we add Eqs.~(\ref{firterm}) and
(\ref{secterm}), they cancel. The second term in Eq.~(\ref{spec}) is
given by Eq.~(\ref{secterm}) with an opposite sign. As a result, 
for $\Gamma_1 \otimes \Gamma_2= \gamma_{\nu} (1-\gamma_5)
\otimes \gamma^{\nu} (1-\gamma_5)$, only the second term in
Eq.~(\ref{spec}) contributes and the matrix element is given as
\begin{equation}
\frac{1}{4} \langle \overline{\xi} W \frac{1}{\overline{n} \cdot
  \mathcal{P}}  W^{\dagger}\FMslash{\overline{n}}
  (1-\gamma_5)  \xi \rangle \langle \overline{q}_s S
  \frac{1}{n\cdot   \mathcal{R}^{\dagger}} 
S^{\dagger} \FMslash{n} (1-\gamma_5) h\rangle 
\langle \overline{\chi}
\overline{W} \frac{1}{n\cdot \mathcal{Q}} \overline{W}^{\dagger}
\FMslash{n} (1-\gamma_5) \chi\rangle.
\label{me1}
\end{equation}
And for $\gamma_{\nu} (1-\gamma_5) \otimes
\gamma^{\nu} (1+\gamma_5)$, only the first term contributes and the
matrix element is given by
\begin{equation}
\frac{1}{4} \langle \overline{\xi} W \frac{1}{\overline{n} \cdot
  \mathcal{P}}  W^{\dagger} \FMslash{\overline{n}}  (1-\gamma_5) 
 \xi \rangle \langle \overline{q}_s S
  \frac{1}{n\cdot \mathcal{R}^{\dagger}} 
S^{\dagger} \FMslash{n} (1-\gamma_5) h \rangle 
\langle \overline{\chi} \overline{W} 
\frac{1}{n\cdot \mathcal{Q}^{\dagger}} \overline{W}^{\dagger} \FMslash{n}
(1+\gamma_5)\chi\rangle.
\label{me2}
\end{equation}
And for $(1-\gamma_5)\otimes (1+\gamma_5)$, the matrix element is
given as
\begin{eqnarray}
&&\frac{1}{16} \langle \overline{\xi} W \frac{1}{\overline{n} \cdot
    \mathcal{P}} W^{\dagger}\FMslash{\overline{n}} (1+\gamma_5) \xi
    \rangle \langle \overline{q}_s S 
\frac{1}{n\cdot   \mathcal{R}^{\dagger}} 
S^{\dagger} \FMslash{n} \gamma_{\perp}^{\mu} (1-\gamma_5) h \rangle
    \nonumber \\
&&\times \langle \overline{\chi}
\overline{W} 
\Bigl(\frac{1}{n\cdot \mathcal{Q}^{\dagger}} -\frac{1}{n\cdot
    \mathcal{Q}}\Bigr)  \overline{W}^{\dagger}
\FMslash{n} \gamma_{\mu}^{\perp} (1+\gamma_5)\chi\rangle.
\label{me3}
\end{eqnarray}
As can be clearly seen in the final expressions in
Eqs.~(\ref{me1})--(\ref{me3}), the gluons $A_n^{\mu}$,
$A_{\overline{n}}^{\mu}$, and $A_s^{\mu}$ can be exchanged only inside
each meson. Therefore the nonfactorizable spectator contribution is
factorized to all orders. 

Let us calculate the matrix element $T^{(1)}$ explicitly for $\Gamma_1
\otimes \Gamma_2 =\gamma_{\nu} (1-\gamma_5) \otimes \gamma^{\nu}
(1-\gamma_5)$. It is given by
\begin{eqnarray}
\langle T^{(1)} \rangle &=& -4\pi \alpha_s \frac{C_F}{N^2} \int
d\overline{n} \cdot x \int \frac{dn\cdot k}{4\pi} \frac{e^{in\cdot k
\overline{n}\cdot x/2}}{n\cdot k n\cdot p_2\overline{n} \cdot p_3}
\nonumber \\
&\times& \frac{1}{4} \langle M_1| \overline{\xi} W
\FMslash{\overline{n}} (1-\gamma_5) W^{\dagger} \xi |0\rangle \langle
M_2| \overline{\chi} \overline{W} \FMslash{n} (1-\gamma_5)
\overline{W}^{\dagger} \chi|0\rangle \nonumber \\
&\times& \langle 0| \overline{q}_S S (\overline{n} \cdot x) \FMslash{n}
(1-\gamma_5) S^{\dagger} h (0)|B\rangle,
\end{eqnarray}
where we integrate out $n\cdot x$ and $x_{\perp}$ explicitly. The
matrix element involving the $B$ meson can be calculated as
\begin{eqnarray}
&&\langle 0| \overline{q}_s S (\overline{n} \cdot x) \FMslash{n}
(1-\gamma_5) S^{\dagger} h |B\rangle  \nonumber \\
&&=\int dr_+ e^{-ir_+ \overline{n} \cdot x/2} \mathrm{Tr} \ \Bigl[
    \Psi_B (r_+) \FMslash{n} (1-\gamma_5) \Bigr] \nonumber \\
&&=-\frac{if_B m_B}{4} \int dr_+  e^{-ir_+ \overline{n} \cdot x/2}
  \mathrm{Tr}\ \Bigl[ \frac{1+\FMslash{v}}{2} \FMslash{\overline{n}}
    \gamma_5 \FMslash{n} (1-\gamma_5) \Bigr] \phi_B^+ (r_+) \nonumber
  \\
&&= -if_B m_B \int dr_+  e^{-ir_+ \overline{n} \cdot x/2}
  \phi_B^+ (r_+),
\end{eqnarray}
where the leading-twist $B$ meson light-cone wave function is defined
through the projection of the $B$ meson as \cite{grozin,feldmann}
\begin{equation}
\Psi_B (r_+) =-\frac{if_B m_B}{4} \Bigl[ \frac{1+\FMslash{v}}{2}
    \Bigl( \FMslash{\overline{n}} \phi_B^+ (r_+) + \FMslash{n}
    \phi_B^- (r_+) \Bigr) \gamma_5 \Bigr].
\end{equation}
And the light-cone wave function for the light mesons is defined as
\begin{equation}
\langle M_2| \overline{\chi}\overline{W} \FMslash{n} \gamma_5
  \delta (\eta -   \mathcal{Q}_+ ) 
  \overline{W}^{\dagger}   \chi\rangle |0\rangle 
 =-if_{M2} 2E \int_0^1
  du \delta [\eta -(4u-2)E] \phi_{M2} (u).
\end{equation}



For $\gamma_{\nu} (1-\gamma_5) \otimes \gamma^{\mu} (1\mp \gamma_5)$,
we have the same matrix element which is given by
\begin{equation}
\langle T^{(1)}\rangle = i\frac{C_F \pi \alpha_s}{N^2}f_B f_{M1}
f_{M2} m_B 
\int dr_+ \frac{\phi_B^+ (r_+)}{r_+} \int du \frac{\phi_{M_1}(u)}{u}
\int dv \frac{\phi_{M2} (v)}{v}.
\label{specc}
\end{equation}
For $(1-\gamma_5) \otimes (1+\gamma_5)$, the matrix
elements are zero if we use the leading-twist $B$ meson wave function
because
\begin{eqnarray}
\langle 0| \overline{q}_s \FMslash{n} \gamma_{\perp}^{\mu}
(1+\gamma_5) h| B\rangle &=& -\frac{if_B m_B}{4} \mathrm{tr} \ \Bigl[
  \FMslash{n} \gamma_{\perp}^{\mu} (1+\gamma_5)
  \frac{1+\FMslash{v}}{2} \Bigl( \FMslash{\overline{n}} \phi_B^+
  +\FMslash{n} \phi_B^- \Bigr) \gamma_5 \Bigr] \nonumber \\
&=& -\frac{if_B m_B}{8} \phi_B^+ \mathrm{tr} \ \FMslash{v}
  \FMslash{\overline{n}} \FMslash{n} \gamma_{\perp}^{\mu} =0.
\end{eqnarray}
If we use higher-twist wave function for the $B$ meson, there may be
nonzero contributions, but this is expected to be suppressed. 

\section{Spectator contribution to the form factor\label{sec5}}
In Section~\ref{sec4}, we have considered the nonfactorizable spectator
contributions arising from the subleading four-quark operators in which
we include only the subleading part from the light-to-light
current. However, there is also a subleading operator coming from the
heavy-to-light current, but this contributes to the heavy-to-light
form factor. It is considered first in Ref.~\cite{form}, and we
discuss in detail here in the context of nonleptonic $B$ decays.


The four-quark operators in $\mathrm{SCET}_{\mathrm{I}}$
which contribute to the form factor are given by
\begin{eqnarray}
J^{(0)} &=& (\overline{\xi} W \Gamma_1 h) (\overline{\chi}
\overline{W} \Gamma_2 \overline{W}^{\dagger}\chi), \nonumber \\ 
J^{(1a)} &=& \Bigl(\overline{\xi} W
(W^{\dagger} i\overleftarrow{\FMSlash{D}}_{n\perp} W)
  \frac{\Gamma_1}{\overline{n}\cdot \mathcal{P}^{\dagger}}
  h\Bigr)(\overline{\chi} \overline{W} \Gamma_2 \overline{W}^{\dagger}
  \chi), \nonumber \\
 J^{(1b)} &=&\Bigl( \overline{\xi} W
(W^{\dagger} i\overrightarrow{\FMSlash{D}}_{n\perp} W)
  \frac{\Gamma_1}{m_b} h\Bigr) (\overline{\chi} \overline{W} \Gamma_2
\overline{W}^{\dagger} \chi).
\label{fsubop}
\end{eqnarray}
Here we list only singlet operators. There are nonsinglet operators
with different color structure, but when we take the color projections
in taking the matrix elements, the matrix elements of the nonsinglet
operators are the same as those of the singlet operators except that
they are suppressed by $1/N$. We will consider only singlet
operators from now on. The explicit proof of the dependence on the
Wilson lines is given in Appendix using the auxiliary field method.

The Lagrangian for the interactions of collinear and usoft gluons is
given by \cite{form}
\begin{eqnarray} 
\mathcal{L}_{\xi q}^{(1)} &=& ig \overline{\xi}
\frac{1}{i\overline{n}\cdot D_n} \FMSlash{B}_{\perp}^n W
  q_{us} +\mathrm{h. c.}, \ \mathcal{L}_{\xi q}^{(2a)} =ig
  \overline{\xi} \frac{1}{i\overline{n} \cdot D_n} \FMSlash{M}
  Wq_{us} +\mathrm{h.c.}, \nonumber \\
\mathcal{L}_{\xi q}^{(2b)} &=& ig \overline{\xi} \frac{\FMslash{n}}{2}
i\FMSlash{D}_{\perp}^n \frac{1}{(i\overline{n} \cdot D_n)^2}
\FMSlash{B}_{\perp}^n W q_{us} +\mathrm{h.c.},
\end{eqnarray}
where
\begin{equation}
ig \FMSlash{B}_{\perp}^n = [i\overline{n}\cdot D_n,
  i\FMSlash{D}_{\perp}^n], \ ig \FMSlash{M} =[i\overline{n} \cdot D^n,
  i\FMSlash{D}^{us} +\frac{\FMslash{\overline{n}}}{2} gn\cdot
  A_n].
\end{equation}
We couple the current to the interaction which turns an usoft quark
into a collinear quark. At leading order in SCET, the relevant
time-ordered products are given as
\begin{eqnarray}
T_0^F &=&\int d^4 x T[J^{(0)} (0) i\mathcal{L}_{\xi q}^{(1)} (x) ], \
T_1^F = \int d^4 x T[J^{(1a)} (0) i\mathcal{L}_{\xi q}^{(1)} (x)],
\nonumber \\ 
T_2^F &=&
\int d^4 x T[J^{(1b)} (0) i\mathcal{L}_{\xi q}^{(1)} (x) ], \ T_3^F =
\int d^4 x T[J^{(0)} (0) i\mathcal{L}_{\xi q}^{(2b)} (x) ], \nonumber
\\ 
T_4^{NF} &=&
\int d^4 x T[J^{(0)} (0) i\mathcal{L}_{\xi q}^{(2a)} (x) ], \nonumber
\\  T_5^{NF}
&=& \int d^4 x d^4 y T[J^{(0)} (0) i\mathcal{L}_{\xi \xi}^{(1)} (x)
  i\mathcal{L}_{\xi q}^{(1)} (y) ], \nonumber \\
T_6^{NF}
&=& \int d^4 x d^4 y T[J^{(0)} (0) i\mathcal{L}_{cg}^{(1)} (x)
  i\mathcal{L}_{\xi q}^{(1)} (y) ], 
\label{formtime}
\end{eqnarray}
and these are shown in Fig.~\ref{fig5} schematically.
\begin{figure}[t]
\begin{center}
\epsfig{file=nlfig5.eps, width=10.0cm}
\end{center}
\caption{Tree-level graphs in $\mathrm{SCET}_{\mathrm{I}}$ for the
  spectator contribution to the heavy-to-light form factor. The first
  diagram contributes to $T_{1,2,4}$, and the second diagram
  contributes to $T_{0,1,3,4,5,6}$.}
\label{fig5}
\end{figure}

Note that this is different from the case of the nonfactorizable
spectator interaction. The
leading-order heavy-to-light current $J^{(0)}$ contains a collinear
gluon $A_n^{\mu}$, therefore it contributes to the heavy-to-light form
factor starting from the leading order. However in the
nonfactorizable spectator contribution, there is no leading current
which involves a collinear gluon $A_n$ which turns an usoft quark into
a collinear quark. That is why there is only a single contribution in
the nonfactorizable spectator contribution, which
corresponds to $T_1^F$ and $T_2^F$ in Eq.~(\ref{formtime}). Other
terms especailly nonfactorizable terms $T_i^{NF}$, which includes
$J^{(0)}$ do not exist in the nonfactorizable spectator contribution. 

As suggested in Ref.~\cite{form}, we absorb the nonfactorizable part
$T_i^{NF}$ into the definition of the soft nonperturbative effects for
the form factors at this order. Among the factorizable contributions
$T_i^F$, only $T_2^F$ is nonzero. In evaluating the matrix elements,
we go through the same procedure of matching from
$\mathrm{SCET}_{\mathrm{I}}$ to $\mathrm{SCET}_{\mathrm{II}}$ as in
the nonfactorizable spectation contributions. For $\Gamma_1 \otimes
\Gamma_2 = \gamma_{\nu} (1-\gamma_5) \otimes \gamma^{\nu} (1-\gamma_5)$,
the matrix element of $T_2^F$ is given by
\begin{eqnarray}
&&\langle M_1 | i\int d^4 x T [J^{(1b)} (0), \mathcal{L}^{(1)}_{\xi q}
  (x)]|B \rangle \nonumber \\
&&=\frac{g^2}{4\pi} \frac{1}{m_b \overline{n} \cdot p_1} \int
  d\overline{n} \cdot x \int \frac{dn\cdot k}{n\cdot k} e^{in\cdot k
  \overline{n} \cdot x/2} \nonumber \\
 &&\times \frac{C_F}{4N} \langle \overline{\xi} W \FMslash{\overline{n}}
  (1-\gamma_5) W^{\dagger} \xi \cdot \overline{\chi} \overline{W}
  \FMslash{n}   (1-\gamma_5) \overline{W}^{\dagger} \chi 
  \cdot \overline{q}_s S (\overline{n} \cdot x) \FMslash{n}
  (1-\gamma_5) S^{\dagger} h \rangle \nonumber \\
&&= \frac{\alpha_s}{4\pi} \frac{4\pi^2 C_F}{N}  if_{M_1} f_{M2} f_B
  \frac{2E}{m_b} m_B \int du \frac{\phi_{M_1} (u)}{u} \int
  \frac{dr_+}{r_+} \phi_B^+ (r_+), 
\label{t1mat}
\end{eqnarray}
where we use Fierz transformations successively. For $\Gamma_1 \otimes
\Gamma_2 =\gamma_{\nu} (1-\gamma_5) \otimes \gamma^{\nu}
(1+\gamma_5)$, we obtain the same result as in Eq.~(\ref{t1mat}). For
$(1+\gamma_5) \otimes (1-\gamma_5)$, it vanishes.


The form factors for $B$ decays into light pseudoscalar mesons are
defined as
\begin{equation}
\langle P(p)| \overline{q} \gamma^{\mu} b | \overline{B}
(p_B)\rangle = f_+ (q^2) \Bigl[ p_B^{\mu} + p^{\mu}
  -\frac{m_B^2-m_P^2}{q^2} q^{\mu} \Bigr] 
 +f_0 (q^2) \frac{m_B^2-m_P^2}{q^2} q^{\mu},
\end{equation}
where $q^{\mu}= p_B^{\mu} -p^{\mu}$. The form factor at order
$\alpha_s$ is given by  
\begin{eqnarray}
f_+ (0) &=& \pi \alpha_s\frac{C_F}{N} \frac{f_{M1}f_B m_B}{4E^2}
\frac{2E}{m_b}  \int dx dr_+ \frac{\alpha_s (\mu_0)}{xr_+} \phi_{M1} (x)
  \phi_B^+ (r_+) \nonumber \\
&&+ (1+K) \zeta (E,\mu_0),
\label{fplus}
\end{eqnarray}
where $\zeta (E,\mu)$ is a nonperturbative function introduced in
Ref.~\cite{charles} in large-energy effective theory
\cite{grinstein}. A similar nonperturbative function is
introduced in Refs.~\cite{bauer2,chay1}, based on SCET. The procedure in
obtaining independent nonperturabtive functions in the form factor is
different in the large-energy effective theory and in SCET, but the
number of independent nonperturbative functions is the same.
And $K$ is given by
\begin{equation}
K=\frac{\alpha_s C_F}{4\pi} \Bigl(-6+5\ln
\frac{m_b}{\mu} -2\ln^2 \frac{m_b}{\mu} -\frac{\pi^2}{12}\Bigr). 
\end{equation}
The expression for $f_+$ coincides with the result in Ref.~\cite{form}
with $m_B=2E$, and $K$ is calculated in Refs.~\cite{bauer2,chay1}.


\section{Application to $\overline{B} \rightarrow \pi\pi$
  decays\label{sec6}} 
We can apply SCET to $B$ decays to two light mesons. As an example, we
analyze the decay amplitudes for $\overline{B} \rightarrow \pi \pi$. 
The decay amplitudes can be written as
\begin{equation}
\langle \pi \pi | H_{\mathrm{eff}} |\overline{B}\rangle
=\frac{G_F}{\sqrt{2}} \sum_{p=u,c} V_{pb} V_{pd}^* \langle \pi \pi |
\mathcal{A}_p |\overline{B}\rangle,
\end{equation}
where
\begin{eqnarray}
\mathcal{A}_p &=& a_1^p  \Bigl[(\overline{\xi}^u h)_{V-A} 
(\overline{\chi}^d \chi^u )_{V-A} \Bigr] + a_2^p \Bigl[ (\overline{\xi}^d
h)_{V-A} (\overline{\chi}^u \chi^u )_{V-A}\Bigr] 
+a_3^p \Bigl[ (\overline{\xi}^d h)_{V-A}
(\overline{\chi}^q \chi^q )_{V-A} \Bigr] \nonumber \\
&&+ a_4^p \Bigl[ (\overline{\xi}^q
h)_{V-A} (\overline{\chi}^d \chi^q )_{V-A} \Bigr] 
+a_5^p \Bigl[(\overline{\xi}^d h)_{V-A}
(\overline{\chi}^q \chi^q )_{V+A} \Bigr].
\end{eqnarray}
The notation $a_i^p [\mathcal{O}]$ means that $a_i^p$ are the sum of
the amplitudes initiated by the operator $\mathcal{O}$. They include the
contribution from the operator itself, and the spectator
contributions. That is,  $a_i^p [\mathcal{O}]$ can be written as
\begin{equation}
a_i^p [\mathcal{O}] = T_i^p + N_i^p + F_i^p,
\end{equation}
where $T_i^p$ is the contribution from the four-quark operators,
$N_i^p$ is the nonfactorizable spectator contribution, and $F_i^p$ is
the spectator contribution for the heavy-to-light form
factor with $a_1^c=a_2^c=0$. In calculating $T_i^p$, we use the relation in
Eq.~(\ref{fqop}) 
\begin{eqnarray}
M&=& \int d\eta \ C(\eta) \langle \overline{\xi} W\gamma_{\mu}
  (1-\gamma_5)   S^{\dagger}   h \cdot \overline{\chi}\overline{W}
  \delta (\eta -   \mathcal{Q}_+ ) \gamma^{\mu} (1\mp \gamma_5)
  \overline{W}^{\dagger}   \chi\rangle \nonumber \\
&=& \pm if_{M2} 2E \int_0^1 du C(u) \phi_{M1} (u) \langle M_1 |
  \overline{\xi} W \frac{\FMslash{\overline{n}}}{2} (1-\gamma_5)
  S^{\dagger} h |\overline{B} \rangle \nonumber \\
&=& \pm im_B^2 \zeta f_{M2} \int_0^1 du C(u) \phi_{M2} (u),
\end{eqnarray}
and  
\begin{equation}
\langle M_1 |\overline{\xi} W\gamma_{\mu} (1-\gamma_5)   S^{\dagger}
h |\overline{B} \rangle = 2m_B \zeta.
\label{bpi}
\end{equation}
We need to comment on Eq.~(\ref{bpi}). Since we are working to order
$\alpha_s$, we may regard the radiative correction on this
relation. But as we can see later, the $\alpha_s$ correction to the
heavy-to-light form factor is taken
into account either in the Wilson coefficients or in the spectator
contribution. Therefore there should be no $\alpha_s$ correction
involved in Eq.~(\ref{bpi}) to be consistent. 


The contributions $T_i^p$ are obtained by the convolutions of the
following combinations of the effective Wilson coefficients 
\begin{eqnarray}
T_1^p &\rightarrow& C_{1R} +\frac{C_{2R}}{N}, \ T_2^p \rightarrow C_{2C}
+\frac{C_{1C}}{N}, \ T_3^p\rightarrow C_{3R} +\frac{C_{4R}}{N},
\nonumber \\ T_4^p &\rightarrow& C_{4C}^p
+\frac{C_{3C}^p}{N}, \ T_5^p  \rightarrow C_{5R}
+\frac{C_{6R}}{N},
\end{eqnarray}
with the hadronic matrix elements of the four-quark operators.
If we are interested in the decay amplitudes at leading order, we can
put $x=1$ and let $u=x_1$ and $\overline{u} = x_2 = 1-u$. Then the
amplitudes $T_i^p$'s are given as
\begin{eqnarray}
T_1^p &=& im_B^2 \zeta f_{\pi} \Bigl[\Bigl(C_1 +\frac{C_2}{N}\Bigr)
  (1+K) +\frac{\alpha_s}{4\pi} \frac{C_F}{N} C_2 F \Bigr], \nonumber
  \\ 
T_2^p &=&im_B^2 \zeta f_{\pi} \Bigl[
\Bigl(C_2 +\frac{C_1}{N}\Bigr)  (1+K) 
+\frac{\alpha_s}{4\pi} \frac{C_F}{N} C_1 F \Bigr],\nonumber \\
T_3^p &=&im_B^2 \zeta f_{\pi} \Bigl[ \Bigl(C_3 +\frac{C_4}{N}\Bigr)
  (1+K) +\frac{\alpha_s}{4\pi} \frac{C_F}{N} C_4 F \Bigr], \nonumber
  \\ 
T_4^p  &=& im_B^2 \zeta f_{\pi} \Biggl\{ \Bigl[\Bigl(C_4
  +\frac{C_3}{N}\Bigr)  (1+K) 
+\frac{\alpha_s}{4\pi} \frac{C_F}{N} \Bigl[ C_3 F  \nonumber \\
&+& C_1 \Bigl( \frac{2}{3}
  +\frac{4}{3} \ln \frac{m_b}{\mu} -G(s_p) \Bigr) 
 +C_3\Bigl( \frac{4}{3}  +\frac{8}{3} \ln \frac{m_b}{\mu} -G(0)
  -G(1) \Bigr) \nonumber \\
&+& C_4 \Bigl( \frac{2n_f}{3}  +\frac{4n_f}{3} \ln \frac{m_b}{\mu} -3G(0)
  -G(s_c) -G(1) \Bigr) \nonumber \\
&+& C_6 \Bigl( \frac{4n_f}{3} \ln \frac{m_b}{\mu} -3G(0)
  -G(s_c) -G(1) \Bigr) +G_{\pi,8} (C_5+C_8) \Bigr] \Biggr\}, \nonumber \\
T_5^p &=&-im_B^2 \zeta f_{\pi} \Bigl[
\Bigl(C_5 +\frac{C_6}{N}\Bigr)  (1+K) 
+\frac{\alpha_s}{4\pi} \frac{C_F}{N} C_6 (-F-12) \Bigr],
\end{eqnarray}
where $C_F =(N^2-1)/(2N)$, $N=3$, $n_f=5$ And $F$ is given as
\begin{equation}
F= -18 +12 \ln \frac{m_b}{\mu} +f_{\pi}^{\mathrm{I}},
\end{equation}
where 
\begin{equation}
f_{\pi}^{\mathrm{I}} = \int_0^1 du g(u) \phi_{\pi} (u), \ G_{\pi,8}  =
\int_0^1 du \frac{-2}{1-u} \phi_{\pi} (u) 
\end{equation}
with 
\begin{equation}
g(u) = 3 \frac{1-2u}{1-u} \ln u -3i\pi 
-\Bigl[ 2 \mathrm{Li}_2 (1-u)
+\frac{1-3u}{1-u} \ln u +\ln^2 u -(u\leftrightarrow \overline{u})
  \Bigr].
\end{equation}
Here we use the notation used
by Beneke et al. \cite{bbns} to facilitate the comparison. 

The nonfactorizable spectator contributions $N_i^p$ ($i=1, \cdots, 5$)
are given by Eq.~(\ref{specc}) as
\begin{equation}
N_i^p =i \frac{C_F}{N^2} \pi \alpha_s f_B f_{\pi}^2 m_B B_i 
\int_0^1 dr_+ \frac{\phi_B^+ (r_+)}{r_+} \Bigl( \int_0^1
  \frac{\phi_{\pi} (u)}{u} \Bigr)^2,
\end{equation}
where 
\begin{equation}
B_1=C_2, \ B_2 = C_1, \ B_3 = C_4, \ B_4 = C_3, \ B_5 = C_6, \ B_6 =
C_5.
\end{equation}
And the spectator contributions to the heavy-to-light form factor $F_i^p$
are given as
\begin{equation}
F_i^p = i \frac{C_F}{N} \pi \alpha_s f_{\pi}^2 f_B m_B D_i \int
du\frac{\phi_{\pi} (u)}{u} \int dr_+ \frac{\phi_B^+ (r_+)}{r_+},
\end{equation}
where
\begin{equation}
D_1 = C_1 +\frac{C2}{N}, \ D_2 = C_2 +\frac{C_1}{N}, \ D_3 =
C_3+\frac{C_4}{N}, \
D_4 = C_4 +\frac{C_5}{N}, \ D_5 = C_5 +\frac{C_6}{N}.
\end{equation}

The final expression can be simplied when we use the definition of
$f_+$ given in Eq.~(\ref{fplus}). For example, $T_1^p+ F_1^p$ is
written as
\begin{eqnarray}
T_1^p+ F_1^p &=&im_B^2 \zeta f_{\pi} \Bigl[\Bigl(C_1 +\frac{C_2}{N}\Bigr)
  (1+K) +\frac{\alpha_s}{4\pi} \frac{C_F}{N} C_2 F \Bigr], \nonumber
  \\ 
&&+i \frac{C_F}{N} \pi \alpha_s f_{\pi}^2 f_B m_B \Bigl(C_1
  +\frac{C_2}{N}\Bigr) \int 
du\frac{\phi_{\pi} (u)}{u} \int dr_+ \frac{\phi_B^+ (r_+)}{r_+}
  \nonumber \\
&=& im_B^2 \zeta f_{\pi} \frac{\alpha_s}{4\pi} \frac{C_F}{N} C_2 F  +
  im_B^2 f_{\pi} \Bigl( C_1 +\frac{C_2}{N} \Bigr) \Bigl[ \zeta (1+K) 
  \nonumber \\
&&  +\pi  \alpha_s \frac{C_F}{N} \frac{f_{\pi} f_B}{m_B}  \int 
du\frac{\phi_{\pi} (u)}{u} \int dr_+ \frac{\phi_B^+ (r_+)}{r_+} \Bigr]
  \nonumber \\
&&\approx im_B^2 f_+ f_{\pi}  \Bigl[\Bigl(C_1 +\frac{C_2}{N}\Bigr)
 +\frac{\alpha_s}{4\pi} \frac{C_F}{N} C_2 F \Bigr],
\end{eqnarray}
where the definition of $f_+$ in Eq.~(\ref{fplus}) is used in the last
line with $m_b \approx m_B =2E$. And we replace $\zeta$ by $f_+$ in the term
proportional to $F$. This induces terms of $\mathcal{O}
(\alpha_s^2)$, which is neglected. This relation also
holds for the combinations  $T_i^p+ F_i^p$ for all $i$.

In summary, the decay amplitudes $a_i^p$ are given by
\begin{eqnarray}
a_1^p &=& im_B^2 f_+ f_{\pi} \Bigl[\Bigl(C_1 +\frac{C_2}{N}\Bigr)
  +\frac{\alpha_s}{4\pi} \frac{C_F}{N} C_2 F^{\prime} \Bigr], \nonumber
  \\ 
a_2^p &=&im_B^2 f_+ f_{\pi} \Bigl[
\Bigl(C_2 +\frac{C_1}{N}\Bigr)   
+\frac{\alpha_s}{4\pi} \frac{C_F}{N} C_1 F^{\prime} \Bigr],\nonumber \\
a_3^p &=&im_B^2 f_+ f_{\pi} \Bigl[ \Bigl(C_3 +\frac{C_4}{N}\Bigr)
   +\frac{\alpha_s}{4\pi} \frac{C_F}{N} C_4 F^{\prime} \Bigr], \nonumber
  \\ 
a_4^p  &=& im_B^2 f_+ f_{\pi} \Biggl\{ \Bigl[\Bigl(C_4
  +\frac{C_3}{N}\Bigr)   
+\frac{\alpha_s}{4\pi} \frac{C_F}{N} \Bigl[ C_3 F^{\prime}  \nonumber \\
&+& C_1 \Bigl( \frac{2}{3}
  +\frac{4}{3} \ln \frac{m_b}{\mu} -G(s_p) \Bigr) 
 +C_3\Bigl( \frac{4}{3}  +\frac{8}{3} \ln \frac{m_b}{\mu} -G(0)
  -G(1) \Bigr) \nonumber \\
&+& C_4 \Bigl( \frac{2n_f}{3}  +\frac{4n_f}{3} \ln \frac{m_b}{\mu} -3G(0)
  -G(s_c) -G(1) \Bigr) \nonumber \\
&+& C_6 \Bigl( \frac{4n_f}{3} \ln \frac{m_b}{\mu} -3G(0)
  -G(s_c) -G(1) \Bigr) +G_{\pi,8} (C_5+C_8) \Bigr] \Biggr\}, \nonumber \\
a_5^p &=&-im_B^2 f_+ f_{\pi} \Bigl[
\Bigl(C_5 +\frac{C_6}{N}\Bigr)   
+\frac{\alpha_s}{4\pi} \frac{C_F}{N} C_6 (-F^{\prime}-12) \Bigr],
\label{final}
\end{eqnarray}
where
\begin{equation}
F^{\prime} = F +  \frac{4\pi^2}{N} \frac{f_{\pi} f_B}{f_+ m_B^2} m_B 
\int_0^1 dr_+ \frac{\phi_B^+ (r_+)}{r_+} \Bigl( \int_0^1
  \frac{\phi_{\pi} (u)}{u} \Bigr)^2.
\end{equation}
The decay amplitudes in Eq.~(\ref{final}) are the same as the result
obtained by Beneke et al. \cite{bbns}. Therefore we have verified
that the leading-order result in SCET is consistent the leading-order
result in the heavy quark mass limit. 

\section{Conclusion\label{sec7}}
We have considered the four-quark operators relevant to nonleptonic $B$
decays into two light mesons in SCET at leading order in
$\lambda$ and to next-to-leading order in $\alpha_s$. The construction
of the four-quark operators in SCET is process-dependent since we
first have to specify the directions of the outgoing quarks and
construct the operators accordingly. In matching onto
the final $\mathrm{SCET}_{\mathrm{II}}$, we integrate out off-shell
modes by attaching collinear and soft gluons to fermion lines. The
result is given as gauge-invariant four-quark operators. The form of
the gauge-invariant operators is obtained to all orders in $\alpha_s$
by using the auxiliary field method. The Wilson
coefficients of these operators can be computed by matching the
amplitudes between the full theory and SCET since the
infrared divergence of the full theory is reproduced in SCET. 


When the effects of collinear and 
soft gluons are included, we can obtain gauge-invariant operators, and
the explicit form of these operators guarantees the color transparency
at leading order in $\lambda$ but to all orders in $\alpha_s$. Now the
idea of the naive factorization in which the matrix elements of
four-quark operators are reduced to products of the matrix elements of
two currents has a theoretical basis.  Furthermore when we include
spectator interactions which contribute to the nonfactorizable
contribution and to the heavy-to-light form factor, the amplitudes are
factorized to all orders in $\alpha_s$. That is, the amplitudes can
be written as a convolution of short-distance effects represented by
the effective Wilson coefficients and long-distance effects
represented by the light-cone wave functions of mesons.

Note that we do not include renormalization group running of the
effective Wilson coefficients in matching from
$\mathrm{SCET}_{\mathrm{I}}$ to $\mathrm{SCET}_{\mathrm{II}}$. This
can be achieved by calculating the anomalous dimensions of the
operators, say, from Eqs.~(\ref{mat1}) and (\ref{mat5}). But the scale
changes from $\mu=m_b$ to $\mu=\sqrt{m_b \Lambda_{\mathrm{QCD}}}$, and
in phenomenological considerations, the running of the Wilson
coefficients may not be appreciable at this order.


The decay amplitudes  for $\overline{B} \rightarrow \pi \pi$ at
leading order in SCET is consistent with the 
approach in the heavy quark mass limit, which is shown explicitly in
the paper. This is not a coincidence because the leading-order decay
amplitudes in the heavy quark mass limit employing leading-twist meson
wave functions correspond to the calculation in SCET at leading
order. Furthermore SCET extends the analysis to all orders in
$\alpha_s$. The two types of the factorization properties, which
correspond to the color transparency and the separation of long- and
short-distance effects are satisfied to all orders in $\alpha_s$. 


We can go beyond the leading-order calculation and consider subleading
 corrections  in order to check the validity of the approach
using SCET. For example, we can ask questions on how chirally-enhaced
 contributions can be treated in SCET, or how to include higher-twist
 wave functions of mesons, and how we can organize higher-order
 corrections in SCET.  However  we stress that this is a first step
 toward understanding nonleptonic $B$ decays, and those questions are
 under investigation. 


\begin{acknowledgments}
This work was supported by Korea Research Foundation Grant
(KRF-2002-041-C00052). 
\end{acknowledgments}

\begin{appendix}
\section{Derivation of the gauge-invariant four-quark operators using
  the auxiliary field method} 
In Section~\ref{sec2} we derived gauge-invariant four-quark operators
through a matching calculation at order $g^2$ in which all the
off-shell modes were integrated out. We can derive the gauge-invariant
four-quark operators of the form $(\overline{\xi} \Gamma_1 h)\cdot
(\overline{\chi} \Gamma_2 \chi)$  using the auxiliary field method. 
In order to integrate out the off-shell modes that arise in QCD to all
orders, we introduce auxiliary fields for these fluctuation as an
intermediate step. First we match onto a Lagrangian with couplings
between the on-shell and off-shell fields, and then the off-shell
fields are explicitly integrated out. This process for four-quark
operators is nontrivial since we have to consider the interaction of
the heavy quark with two types of collinear gluons $A_n^{\mu}$ and
$A_{\overline{n}}^{\mu}$. In this case, only after adding all the
Feynman diagrams including the triple gluon vertex, we have a simple
form for the Wilson lines without the heavy quark mass $m_b$. 

Here we will consider in detail the construction of gauge-invariant
operators at leading order in $\mathrm{SCET}_{\mathrm{I}}$ by
considering off-shell modes of order $\lambda^0$. We are especially
interested in the gauge-invariant form for the heavy-to-light current
in the presence of two types of collinear gluons. It is
straightforward to extend this process to
$\mathrm{SCET}_{\mathrm{II}}$ by including off-shell modes of order
$\lambda$. The auxiliary field method in $\mathrm{SCET}_{\mathrm{II}}$
was treated in detail in Refs.~\cite{bauer4,bauer6}, and here we 
describe the prescription going from  $\mathrm{SCET}_{\mathrm{I}}$ to
$\mathrm{SCET}_{\mathrm{II}}$. 

We have different types of off-shell modes. They can have momenta
$q_n^{\mu}+q_{\overline{n}}^{\mu}$ at order $\lambda^0$, and
$q_n^{\mu}+q_{\overline{n}}^{\mu} +q_s^{\mu}$, $q_n^{\mu}+ q_s^{\mu}$,
$q_{\overline{n}}^{\mu} +q_s^{\mu}$ at order $\lambda$. We list all
the on-shell fields and the off-shell auxiliary fields in
Table~1. The fields and the Wilson lines with the index $X$ denote
that they are auxiliary fields in $\mathrm{SCET}_{\mathrm{II}}$. We
will consider the auxiliary fields in $\mathrm{SCET}_{\mathrm{I}}$
(without the index $X$), and prescribe how they change in
$\mathrm{SCET}_{\mathrm{II}}$. 

\begin{table}[t]
\caption{\label{intmod} List of the on-shell modes and the auxiliary
  off-shell modes.} 
\vspace{0.3cm}
\begin{tabular}{ccccc} \hline
&Type& Momentum scaling& Fields& Wilson lines \\  \hline
on-shell&  $n^{\mu}$& $q_n^{\mu}$ & $\xi$, $A_n^{\mu}$& $W$ \\
& $\overline{n}^{\mu}$& $q_{\overline{n}}^{\mu}$& $\chi$,
$A_{\overline{n}}^{\mu}$ & $\overline{W}$ \\
&soft& $q_s^{\mu}$& $q_s$,  $A_s^{\mu}$& $S$, $\overline{S}$ \\
&usoft& $q_{us}^{\mu}$& $q_{us}$, $A_{us}^{\mu}$&
$Y$, $\overline{Y}$ \\
off-shell& & $q_n^{\mu}$, $q_{\overline{n}}^{\mu}$& $\psi_n$,
$\psi_{\overline{n}}$ \\
&&$q_n^{\mu} +q_{\overline{n}}^{\mu}$& $\psi_H$, $A_Q^{\mu}$&
$W_Q$, $\overline{W}_Q$, $W_Q^X$, $\overline{W}_Q^X$ \\
&&$q_n^{\mu} +q_s^{\mu}$ & $\psi_n^X$, $\xi_X$, $A_{nX}^{\mu}$ &
$W_X$, $S_X$ \\ 
&&$q_{\overline{n}}^{\mu} +q_s^{\mu}$ & $\psi_{\overline{n}}^X$,
$\chi_X$,  $A_{\overline{n} X}^{\mu}$ & $\overline{W}_X$,
$\overline{S}_X$ \\ 
\hline 
\end{tabular}
\end{table}


In order to integrate out off-shell modes,
consider first matching onto an action with the auxiliary fields
$\psi_n$, $\psi_{\overline{n}}$  
and $\psi_H$. $\psi_n$ ($\psi_{\overline{n}}$) denotes the off-shell
heavy quark which becomes off-shell by a collinear momentum
$q_n^{\mu}$ ($q_{\overline{n}}^{\mu}$). $\psi_H$
is the off-shell heavy quark which has collinear momentum $q_n
^{\mu}+q_{\overline{n}}^{\mu}$.  The auxiliary gauge field $A_Q^{\mu}$
has the momentum $q_n^{\mu} +q_{\overline{n}}^{\mu}$. Then we
integrate out off-shell fields $\psi_n$, $\psi_{\overline{n}}$ and
$\psi_H$. 

The auxiliary gluon Lagrangian is given by
\begin{equation}
\mathcal{L}^g_{\mathrm{aux}} [A_Q]= \frac{1}{2g^2} \mathrm{tr} \,
\Bigl( [iD_Q^{\mu} +gA_Q^{\mu}, iD_Q^{\nu} +gA_Q^{\nu}]\Bigr)^2
+\frac{1}{\alpha_L} \mathrm{tr}\, \Bigl(\Bigl[iD_{Q\mu}, A_Q^{\mu}]
\Bigr)^2,
\label{gaux}
\end{equation}
where
\begin{eqnarray}
iD_Q^{\mu} &=&\frac{n^{\mu}}{2}(\overline{n} \cdot \mathcal{P} +
g \overline{n} \cdot A_n) +\frac{\overline{n}^{\mu}}{2}
(n\cdot \mathcal{Q} +gn\cdot A_{\overline{n}}) \nonumber \\
&&+\mathcal{P}_{\perp}^{\mu} +gA_{n\perp}^{\mu}
+\mathcal{Q}_{\perp}^{\mu} +g A_{\overline{n}\perp}^{\mu}.
\end{eqnarray}

We can separately obtain the solution of the first term and the second
term in Eq.~(\ref{gaux}). The equation of motion of
the first term is given by
\begin{equation}
[iD_{Q\mu} +gA_{Q\mu} ,[iD_Q^{\mu} + gA_Q^{\mu}, iD_Q^{\nu}
    +gA_Q^{\nu}]] =0.
\label{eom}
\end{equation}
The leading-order solution is obtained by making an ansatz 
\begin{equation}
\overline{W}_Q^{\dagger} W_Q = W \overline{W}^{\dagger}.
\label{lan}
\end{equation}
Here $W_Q$ and $\overline{W}_Q$ are essentially the Fourier transform
of the Wilson lines
\begin{eqnarray}
W_Q (y) &=& \mathrm{P} \exp \Biggl\{ ig \int_{-\infty}^y ds \Bigl[
  \overline{n} \cdot A_Q (s\overline{n}) +\overline{n} \cdot A_n
  (s\overline{n}) \Bigr] \Biggr\}, \nonumber \\
\overline{W}_Q (y) &=& \mathrm{P} \exp \Biggl\{ ig \int_{-\infty}^y ds
  \Bigl[ n \cdot A_Q (sn) +n \cdot A_{\overline{n}} (sn) \Bigr]
  \Biggr\}, 
\label{gluong}
\end{eqnarray}
which satisfy
\begin{equation}
\overline{n} \cdot \Bigl(\mathcal{P} +gA_Q+g A_n \Bigr) W_Q =0, \
n \cdot \Bigl(\mathcal{Q} +g A_Q +g A_{\overline{n}}
\Bigr) \overline{W}_Q =0.
\end{equation}



The construction of the
Lagrangian with auxiliary heavy fields is complicated because only the
sum of three graphs shown in Eq.~(\ref{heavy}) is simple, but the
individual diagrams show complex behavior. We construct the Lagrangian
such that the first term in Eq.~(\ref{heavy}) is reproduced by the
auxiliary field attached with the triple gluon vertex with $A_n^{\mu}$
and $A_{\overline{n}}^{\mu}$, and the second term is produced by
attaching $A_n^{\mu}$ and $A_{\overline{n}}^{\mu}$ starting from the
vertex in this order. Then the Lagrangian for the heavy quark with the
auxiliary fields is given as 
\begin{eqnarray}
\mathcal{L}_{\mathrm{aux}}^h  &=& \overline{\psi}_n g\overline{n}\cdot
  A_n h   +\overline{\psi}_{\overline{n}} gn\cdot A_{\overline{n}} h
  +\overline{\psi}_H g\overline{n}   \cdot A_Q  h \nonumber \\
&&+\overline{\psi}_H
  (g\overline{n} \cdot A_n +g\overline{n} 
  \cdot A_Q)  \psi_{\overline{n}} +\overline{\psi}_H g\overline{n}
  \cdot A_Q \psi_n 
  \nonumber \\
&&+\overline{\psi}_n (\overline{n}\cdot \mathcal{P} +g\overline{n}
  \cdot A_n)   \psi_n + \overline{\psi}_{\overline{n}} (n\cdot
  \mathcal{Q} +gn \cdot A_{\overline{n}} )\psi_{\overline{n}}
  \nonumber   \\
&&+\overline{\psi}_H (\overline{n} \cdot \mathcal{P} +g\overline{n}
  \cdot A_Q +g\overline{n} \cdot A_n ) \psi_H. 
\label{hqaux}
\end{eqnarray}
The inclusion of the auxiliary field $A_Q^{\mu}$ in the second line of
Eq.~(\ref{hqaux}) should be verified at order $g^3$, but these terms
are included since they are kinematically allowed. 


Solving for $\psi_{\overline{n}}$, $\psi_n$ and $\psi_H$ from
Eq.~(\ref{hqaux}),  we obtain 
\begin{eqnarray}
&&gn\cdot A_{\overline{n}} h +n\cdot (\mathcal{Q} + A_{\overline{n}} )
\psi_{\overline{n}} =0, \
g\overline{n}\cdot A_n h +\overline{n} \cdot (\mathcal{P}
+g A_n )\psi_n =0, \label{eqmo} \\
&& g\overline{n} \cdot A_Q (h +\psi_n) +g\overline{n}\cdot (A_n +A_Q)
\psi_{\overline{n}}  + \Bigl(\overline{n} \cdot \mathcal{P} +
g\overline{n} \cdot (A_n +A_Q) \Bigr) \psi_H =0. \nonumber
\end{eqnarray}
The first equation in Eq.~(\ref{eqmo}) can be solved for
$\psi_{\overline{n}}$ as 
\begin{equation}
\psi_{\overline{n}} = (\overline{W} -1)h,
\end{equation}
and adding the second and the third equations in Eq.~(\ref{eqmo})
yields
\begin{equation}
\psi_n +\psi_H = (W_Q-1) (h+\psi_{\overline{n}}).
\end{equation}
Therefore the heavy quark field can be written as
\begin{equation}
h+\psi_{\overline{n}}+\psi_n +\psi_H = h+\psi_{\overline{n}} + (W_Q-1)
(h+\psi_{\overline{n}}) = W_Q \overline{W} h.
\end{equation}

We can arrive at an equivalent conclusion by writing Eq.~(\ref{heavy})
as
\begin{equation}
M_a + M_b =\frac{ig^2}{2} f_{abc} \overline{q} \Gamma_1 T_a \frac{n\cdot
  A_{\overline{n}}^c \overline{n} \cdot A_n^b}{n\cdot q_{\overline{n}}
  \overline{n} \cdot q_n}b+ g^2 \overline{q} \Gamma_1
  \frac{n\cdot A_{\overline{n}}}{n\cdot q_{\overline{n}}}
  \frac{\overline{n} \cdot A_n}{\overline{n}  
  \cdot q_n},
\label{heavy2}
\end{equation}
and define a Lagrangian for the auxiliary fields accordingly. In that
case, we obtain 
\begin{equation}
h+\psi_n+\psi_{\overline{n}}  +\psi_H = \overline{W}_Q W h = W_Q
\overline{W} h,
\end{equation}
where the last equality comes from the ansatz
$\overline{W}_Q^{\dagger} W_Q = W\overline{W}^{\dagger}$.  

If we consider the off-shell modes of order $\lambda$ in
$\mathrm{SCET}_{\mathrm{II}}$, we have to integrate out all the
off-shell modes in Table~1 by constructing the Lagrangian with
additional auxiliary fields. At the end of the calculation, the
prescription from $\mathrm{SCET}_{\mathrm{I}}$ to
$\mathrm{SCET}_{\mathrm{II}}$ is to put the index $X$ to all the
Wilson lines and the 
fermions transform as  
\begin{equation}
h \rightarrow W_Q^X \overline{W}_X h, \ \xi \rightarrow \overline{W}_Q^X
S_X \xi, \ \chi \rightarrow W_Q^X \overline{S}_X \chi,
\label{trans}
\end{equation}
where $W_Q^X$, and $S_X$ are the Fourier transforms of the Wilson lines
\begin{eqnarray}
W_Q^X (y) &=& \mathrm{P} \exp \Biggl\{ ig \int_{-\infty}^y ds \Bigl[
  \overline{n} \cdot A_Q (s\overline{n}) + \overline{n} \cdot A_{nX}
  (s\overline{n}) +
\overline{n} \cdot A_n
  (s\overline{n}) \Bigr] \Biggr\}, \nonumber \\
S_X (y) &=& \mathrm{P} \exp \Biggl\{ ig \int_{-\infty}^y ds
  \Bigl[ n \cdot A_{nX} (sn) +n \cdot A_s (sn) \Bigr]
  \Biggr\},
\label{gluong2}
\end{eqnarray}
and $\overline{W}_Q^X$ and $\overline{S}_X$ are obtained by replacing
$n^{\mu}$ by $\overline{n}^{\mu}$ in Eq.~(\ref{gluong2}).
The Wilson lines satisfy
\begin{equation}
\overline{W}_Q^{X\dagger} W_Q^X = W_X \overline{W}_X^{\dagger}, \
S_X^{\dagger} W_X = WS^{\dagger}, \ \overline{S}_X \overline{W}_X =
\overline{W} \overline{S}^{\dagger}.
\end{equation}
The last two relations in Eq.~(\ref{trans}) were obtained in
Ref.~\cite{bauer4} and the first relation is new.


The singlet four-quark operator in $\mathrm{SCET}_{\mathrm{II}}$
is given by
\begin{eqnarray}
&& \overline{\xi} S_X^{\dagger} \overline{W}_Q^{X\dagger} \Gamma_1 W_Q^X
  \overline{W}_X h \cdot \overline{\chi} \overline{S}_X^{\dagger}
  W_Q^{X\dagger} \Gamma_2 W_Q^X \overline{S}_X \chi \nonumber \\
&&= \overline{\xi} S_X^{\dagger} W_X \Gamma_1 \overline{W}_X^{\dagger}
  \overline{W}_X h \cdot \chi \Gamma_2 \chi =\overline{\xi} W \Gamma_1
  S^{\dagger} h \cdot \overline{\chi}   \Gamma_2 \chi,
\end{eqnarray}
and the nonsinglet four-quark operator is given as
\begin{eqnarray}
&& \Bigl(\overline{\xi} S_X^{\dagger} \overline{W}_Q^{X\dagger}
  \Bigr)_{\beta} \Gamma_1 \Bigl( W_Q^X \overline{W}_X h\Bigr)_{\alpha}
  \cdot \Bigl( \overline{\chi} \overline{S}_X^{\dagger} W_Q^{X\dagger}
  \Bigr)_{\alpha}   \Gamma_2 \Bigl(W_Q^X \overline{S}_X
  \chi\Bigr)_{\beta} \nonumber \\
&& = \Bigl(\overline{\xi} S_X^{\dagger} W_X \Bigr)_{\beta} \Gamma_1
  \Bigl( \overline{S}_X^{\dagger} \overline{W}_X h\Bigr)_{\alpha}
  \cdot \overline{\chi}_{\alpha} \Gamma_2 \Bigl(
  \overline{W}_X^{\dagger} \overline{S}_X \chi\Bigr)_{\beta} \nonumber
  \\
&&= \Bigl( \overline{\xi} WS^{\dagger} \Bigr)_{\beta} \Gamma_1 \Bigl(
  \overline{S}^{\dagger} h\Bigr)_{\alpha} \cdot \Bigl(\overline{\chi}
  \overline{W} \Bigr)_{\alpha} \Gamma_2 \Bigl(\overline{S}
  \overline{W}^{\dagger} \chi\Bigr)_{\beta} \nonumber \\
&&= \Bigl(\overline{\xi} WS^{\dagger} \overline{S} \Bigr)_{\beta}
  \Gamma_1 \Bigl( \overline{S}^{\dagger} h \Bigr)_{\alpha} \cdot
  \Bigl( \overline{\chi} \overline{W} \Bigr)_{\alpha} \Gamma_2 \Bigl(
  \overline{W}^{\dagger} \chi\Bigr)_{\beta}.
\end{eqnarray}
This result is the same as the explicit calculation obtained in
Section 2, and it is a proof to all orders in $\alpha_s$ using the
auxiliary field method. 



\section{Derivation of the subleading operators using the auxiliary
  field method}
We can also derive the subleading operators $O^{(1)}$ in
Eq.~(\ref{osub}) and $J^{(1a)}$ and $J^{(1b)}$ in Eq.~(\ref{fsubop})
using the auxiliary field method. Here we have to consider off-shell
modes from the collinear gluons both in the $n^{\mu}$ and
$\overline{n}^{\mu}$ directions. This was first considered in
Ref.~\cite{bauer4} including the off-shellness from soft modes. In
our approach, we consider the operators in
$\mathrm{SCET}_{\mathrm{I}}$ disregarding the off-shell modes by soft
gluons. It can be obtained in a straightforward way as in
Ref.~\cite{bauer4}. Here we obtain a new result which derives
gauge-invariant operators at subleading order.  The solutions of
Eq.~(\ref{eom}) at subleading order are given as 
\begin{eqnarray}
[W_Q \overline{n} \cdot \mathcal{P} W_Q^{\dagger}, [\overline{W}_Q
    n\cdot \mathcal{Q} \overline{W}_Q^{\dagger},
    i\mathcal{D}_{Q\perp}^{\nu} ]]     &=&0, \nonumber \\ 
{}[\overline{W}_Q n\cdot \mathcal{Q}
    \overline{W}_Q^{\dagger}, [W_Q 
\overline{n} \cdot \mathcal{P} W_Q^{\dagger},
    i\mathcal{D}_{Q\perp}^{\nu} ]]     &=&0,
\label{subeom}
\end{eqnarray}
where $i\mathcal{D}_{Q\perp}^{\mu} = iD_{Q\perp}^{\mu}
+gA_{Q\perp}^{\mu}$.  
Here we make an ansatz 
\begin{equation}
\overline{W}_Q^{\dagger} i\mathcal{D}_{n\perp Q}^{\nu} =
iD_{n\perp}^{\nu} 
\overline{W}_Q^{\dagger} \ \mbox{or} \ W_Q^{\dagger}
i\mathcal{D}_{\overline{n}\perp Q}^{\nu} = iD_{\overline{n}\perp}^{\nu}
W_Q^{\dagger}, 
\label{san}
\end{equation}
and it satisfies Eq.~(\ref{subeom}). 


First the intermediate form of the subleading four-quark operators
relevant to the nonfactorizable spectator contribution and the
heavy-to-light form factor can be obtained by neglecting the off-shell
modes in the $\overline{n}^{\mu}$ direction since what we are
interested in is the operators proportional to $gA_n^{\mu}$ at leading
order in $g$. As a definite example, let us concentrate on the second
part of the operator $O^{(1)}$ and obtain the gauge-invariant form. 
The derivation for the rest of the operators is straightforward. In
$\mathrm{SCET}_{\mathrm{I}}$, without the off-shell modes in the
$\overline{n}^{\mu}$ direction, it is given by
\begin{equation}
\overline{\xi} \Gamma_1 W h \cdot \overline{\chi} W^{\dagger} \Gamma_2
\overline{W} \frac{1}{n\cdot \mathcal{Q}} \overline{W}^{\dagger} [
  i\FMSlash{D}_{n\perp} W ] \frac{\FMslash{n}}{2} \chi.
\label{sample}
\end{equation}
When we include off-shell modes, each field changes to
\begin{equation}
h \rightarrow W_Q \overline{W} h, \ \xi \rightarrow \overline{W}_Q
\xi, \ \chi \rightarrow W_Q \chi,
\end{equation}
and $W$ changes to $W_Q$. Then the operator in Eq.~(\ref{sample}) is
written as
\begin{equation}
\overline{\xi} \overline{W}_Q^{\dagger} \Gamma_1 W_Q \overline{W} h
\cdot \overline{\chi} W_Q^{\dagger} \Gamma_2 \overline{W}_Q
\frac{1}{n\cdot \mathcal{Q}} \overline{W}_Q^{\dagger}
[i\FMSlash{\mathcal{D}}_{n\perp Q} W_Q] \frac{\FMslash{n}}{2} \chi. 
\end{equation}
If we use the ansatz in Eqs.~(\ref{lan}) and (\ref{san}), it becomes
\begin{eqnarray}
\mathrm{singlet}: &&\overline{\xi} W \Gamma_1 h \cdot \overline{\chi}
\overline{W} \Gamma_2 \frac{1}{n\cdot \mathcal{Q}} [W^{\dagger}
  i\FMSlash{D}_{n\perp} W] \frac{\FMslash{n}}{2}
\overline{W}^{\dagger} \chi, \nonumber \\
\mathrm{nonsinglet}: && \Bigl( \overline{\xi} W\Bigr)_{\beta} \Gamma_1
h_{\alpha} \cdot \Bigl(\overline{\chi} \overline{W} \Bigr)_{\alpha}
\Gamma_2 \Bigl( [W^{\dagger} i\FMSlash{D}_{n\perp} W] \frac{1}{n \cdot
  \mathcal{Q}} \frac{\FMslash{n}}{2}
\overline{W}^{\dagger} \chi\Bigr)_{\beta},
\end{eqnarray}
as presented in $O^{(1)}$. Other operators can be shown to have the
forms presented in the paper in a similar way. These operators are
used in $\mathrm{SCET}_{\mathrm{I}}$ and we match them in
$\mathrm{SCET}_{\mathrm{II}}$ to evaluate the matrix elements. 
\end{appendix}



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\end{document}

