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\newcommand{\CH}[2]{\chi_{#1,#2}}
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\title{
Factorization, Power Corrections, and the Pion Form Factor}
\author{Ira Z. Rothstein}
\affiliation{Dept. of Physics, Carnegie Mellon University, Pittsburgh, PA 15213\footnote{Permanent address}}\affiliation{ 
Dept. of Physics, University of California at San Diego, La Jolla  , 
    CA 02093 
}


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\begin{abstract}

This letter  is an investigation of the pion form factor utilizing  recently developed effective field theory techniques. The primary results reported  are: Both the transition and electromagnetic form factors are corrected  at order $\Lambda/Q$.  However,  these corrections only arise  due to time ordered products which are sensitive to soft components of the pion. 
The usual higher twist wave function corrections contribute only at order $\Lambda^2/Q^2$, when the quark mass vanishes.
In the case of the electromagnetic form factor the $\Lambda/Q$  power correction is enhanced by a power of $1/\alpha_s(Q)$ relative to the leading order result of Brodsky and Lepage, if the scale $\sqrt{\Lambda Q}$ is non-perturbative. This enhanced correction could explain the discrepancy with the data.\end{abstract}

\maketitle
Making predictions for hadronic observables is extremely difficult given the complexity of the theory of strong interactions (QCD). In the absence of a solution to the theory, we are forced to accept  reduced predictive power. We can make  predictions only after we have extracted some crucial related information from the data. Moreover, the necessary information is usually in the form of non-local matrix elements and not just fixed couplings.
Even with these lowered expectations, it is still highly non-trivial to find observables which we can predict from first principles. The primary tool at our disposal is factorization. Observables which are ``factorizable" can be separated into long and short distance contributions. Asymptotic freedom allows for a calculation of the short distance piece via a perturbative expansion in the coupling, while the long distance piece is in principle calculable, but in practice must be extracted from experiment. The predictive power lies in the fact that factorization implies universality. That is, the same non-perturbative factor can appear in predictions for disparate processes.


Proving that a certain observable is factorizable is a highly non-trivial process.  The pioneering works on the subject \cite{bl} were based upon diagrammatic techniques and can be quite intricate. Using these techniques, factorization was shown to leading in $1/Q$, where $Q$ is the large energy scale in the process.  For example, the photon-pion transition form factor can be written to leading order (in $1/Q$) as
\begin{eqnarray} \label{LOtFF}
  F_{\pi\gamma} 
  =\frac{2f_\pi}{Q^2} \int_0^1\!\! dx\: C_1(x,Q,\mu)
  \: \phi_{\pi}(x,\mu) \,,
\end{eqnarray}
while the pion EM  form factor may be written as
\begin{eqnarray}
F_{\pi}= (Q_u\!-\!Q_d)\, \frac{f_\pi^2}{Q^2}\, 
\int_0^1\!\!\mbox{d}x\!
~\mbox{d}y \: T_1(x,y,\mu) \phi_\pi(x,\mu) \phi_\pi(y,\mu)\,.
\end{eqnarray}
In these expressions $\phi_\pi$ carries the universal,  non-perturbative, information about the structure of the pion and is defined via
\begin{eqnarray} \label{pimom}
&&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!-{i}f_\pi\,\delta^{ab}
   \phi_\pi(x,\mu)=\\ \nn & & \!\!\!\!\!\!\! \int \frac{dy_-}{2 \pi}\, e^{-i x y {\bn\mcdot p} } ~\big\langle ~\pi_{n,p}^a~ \big| 
  ~\bar{q}(y)~ \Gamma_\pi^b W(y,-y)~ q(-y) \big|~ 0~
  \big\rangle,
\end{eqnarray}
where $W$ is a light-like Wilson line needed for manifest gauge invariance.
 $C_1$ and $T_1$ are the perturbatively calculable, process dependent, high energy Wilson coefficients. At asymptotically large values of $Q$, the pion wave function is dominated by its first moment and 
approaches the form $\phi_\pi(x)\propto x(1-x)f_\pi$. Experimentally, the prediction for the transition form factor agrees at the ten percent level, while the EM form factor is off by a factor of order one. Thus, the relevant questions is,  
how large are the subleading corrections in each case? 
This letter undertakes the task of systematically categorizing  the power corrections to these processes, including the contribution from 
the so-called ``Feynman region". The issue of the breakdown of factorization
will also be discussed.


The results in this paper are derived  utilizing recent developments in the so-called Soft-Collinear Effective Theory (SCET)\cite{SCET}. In this approach, factorization proofs simplify because modes with varying virtualities are separated at the level of the Lagrangian\cite{bfprs}. Proving factorization becomes tantamount to determining if the theory properly accounts for the IR physics  of  the process under consideration. 
Perhaps the true power of the effective field theory approach, as applied to  exclusive processes,  is that it enables one to account for  power corrections in a systematic fashion including the so-called ``soft" or ``end-point" contributions.  

To build the proper effective field theory  one isolates the degrees of freedom responsible for the non-analytic behavior of the low energy theory. Usually this corresponds to simply integrating out massive fields, but there are cases \cite{LMR}where one wishes to explicitly separate  certain subset of fluctuations of the fields. This separation allows for  manifest power counting, which
 in turn makes the classification of power corrections relatively simple. In SCET the relevant modes in the IR are:  
collinear $p_c = (p_c^+, p_c^-, p_c^\perp)=(n\mcdot p_c, \bn \mcdot p_c,
p_c^\perp) \sim Q(\lambda^2, 1,
\lambda)$, soft $p_s \sim Q(\lambda, \lambda, \lambda)$ and usoft $p_{us} \sim
Q(\lambda^2, \lambda^2, \lambda^2)$, where $n^2=\bn^2=0$, $n\mcdot\bn=2$, and
$\lambda \ll 1$ is the expansion parameter. The relevant modes are fixed by the external momenta and use of the Coleman-Norton theorem. In \cite{bfprs}, the authors chose  $\lambda\propto \Lambda/Q$, as this fixes the transverse momenta of the external lines to be of order $\Lambda$, as it should be physically. This also means that the usoft modes have virtuality less than $\Lambda$.
 However, at leading order in $\lambda$, this point is irrelevant, as all such modes cancel explicitly in all matrix elements.
When trying to understand the scaling of end-point contributions, it is necessary to include subleading operators, and the issue of the proper choice of $\lambda$ is crucial, as was recently pointed out in \cite{bps3}, where it was shown that in general one needs to split the effective theory into two stages. At the high scale $Q$, ones matches onto SCETI where   $\lambda\propto \sqrt{\Lambda/Q}$. One then matches onto SCETII at the scale $\sqrt{\Lambda Q}$, at which point the scaling becomes $\lambda \propto \Lambda/Q$. 



In the effective theory
each mode is interpolated by a distinct field  and  scales homogeneously in $\lambda$. For instance, collinear modes with large light-cone momentum in the $n$ direction are interpolated by $\xi_n$ and $A_n^\mu$ for fermions and gauge bosons respectively.
These fields have support only over momenta of order $\lambda^2$, as their large light cone and transverse momenta have been scaled out.
 The leading order Lagrangian for these collinear fields with momenta 
in the $n$ direction, is given by
\begin{eqnarray} \label{L0}
 && {\cal L}^{(0)}_c = \bar \xi_n \left[ i n \mcdot D 
  + i \Slash{D}^c_\perp \frac{1}{i \bn \mcdot D_c}  i \Slash{D}^c_\perp\right]  
  \frac{\bnslash}{2} \xi_n + {\cal L}^{(0)}_{cg} \,,\hspace{0.9cm}
\end{eqnarray}with $i\bn\mcdot D_c=\bnP\!+\!g\bn\mcdot A_n$, $iD_c^\perp=\cP^\perp\! +
\!gA^\perp_c$, $in\mcdot D=in\mcdot\partial+gn\mcdot A_{us}+gn\mcdot A_n$.
The operators $\bnP$ and $\cP^\perp$ are derivative like operators whose eigenvalues are the large light cone and transverse momenta respectively. 
 The 
gluon action, ${\cal L}^{(0)}_{cg}$, can be found in Ref. [2d].  

Let us start by considering the pion transition form factor. This process involves the scattering of a highly virtual
photon and a quark-anti-quark constituent pair off of an on-shell photon, which we treat as point-like to leading order in $\alpha_{em}$, 
%\footnote{If we were to treat the photon like a hadron, then the analysis is al%most identical to the case of the EM form factor discussed in later paragraphs.}
 into a pion. By integrating out the hard off-shell intermediate state in the $\gamma^\star +\gamma\rightarrow q+\bar{q}$ process, we generate a two quark operator in the SCETI.
The
most general spin structures for currents with two collinear particles
moving in the same or opposite directions are \cite{bfprs}
%\begin{eqnarray} \label{Jspin}
%  && \bar\xi_{n}\,\Gamma_1\, \, \xi_{n}\qquad\quad 
%  \Gamma_1 = \big\{ \bnslash\,, \bnslash\gamma_5\,,
%                  \bnslash\gamma^\mu_\perp \big\}\,, \nn\\
%  && \bar\xi_{\bn} \,\Gamma_2\, \, \xi_{n}\qquad\quad 
%  \Gamma_2 = \big\{ 1 \,, \gamma_5 \,, \gamma^\mu_\perp \big\}\,.
%\end{eqnarray}

\begin{eqnarray} \label{Jspin}
   \bar\xi_{n}\,\big\{ \bnslash\,, \bnslash\gamma_5\,,
                 \bnslash\gamma^\mu_\perp \big\} \xi_{n}~~~;~~~
     \bar\xi_{\bn} \big\{ 1 \,, \gamma_5 \,, \gamma^\mu_\perp \big\}  \, \xi_{n}\end{eqnarray}
From this result we can see that for the case at hand there is only one relevant operator structure which interpolates for the pion namely $\bnslash \gamma_5$, thus the leading order matching result is of the form
\begin{eqnarray} \label{Opig}
  O_0(\omega_1,\omega_2) &=&  (\bar{\xi}_{n}W)_{\omega_1}\, \bar n \!\!\! \slash \gamma_5
    (W^\dagger \xi_{n})_{\omega_2}~, 
\end{eqnarray}
where the isospin structure has been suppressed.
Here the $W$'s are the Fourier transforms of light-like Wilson lines and $\omega_{1,2}$ the total collinear momenta of the jet like structure $W^\dagger \xi$ and $\bar{\xi} W$ respectively.
The operators have numbers $(\omega_1,\omega_2)$ associated with them, the sum of which is fixed by the momentum of the pion.
We may then decouple the usoft modes from the collinear modes in the Lagrangian via field redefinitions [2d]. \begin{eqnarray}\label{redef}
 \xi_n^{(0)} = Y^\dagger \xi_n\,, \quad 
 A_n^{(0)} = Y^\dagger A_n Y~, 
\end{eqnarray}
where $Y$ is an usoft Wilson line. This has the effect of decoupling usoft lines from collinear lines in the action, at the cost of introducing $Y$ factors into the operator $O_0$. However, we can see that since the $Y$ are usoft and carry no large light-cone momenta they will cancel in $O_0$ as a consequence of unitarity. The matching onto SCETII is trivial at this order. The off-shellness of the external collinear lines is reduced to being less than $\Lambda Q$.
Taking the matrix element of this operator between the vacuum and one pion state yields the usual leading order result in terms of the pion wave function~(\ref{LOtFF}), except one makes the replacement $f_\pi\rightarrow f_\pi^{SCET}$, with
\begin{equation}
f_\pi=f_\pi^{SCET}+O(\lambda).
\end{equation}

 Power corrections 
 %\footnote{In the rest of this paper the term ``higher order" will always refer to the $\lambda$ %expansion and not the $\alpha_s$ expansion for the Wilson coefficients.},
 arise from either matching onto higher order operators or from including corrections from the sub-leading Lagrangian into time ordered products (TOP's).
The order $\lambda$ action introduces couplings between usoft and collinear fields and is given by
~\cite{chay,beneke,rpi}
\begin{eqnarray}
&& {\cal L}_{\xi\xi}^{(1)} = \bar \xi_n  i \Slash{D}^{us}_\perp 
  \frac{1}{i \bn \mcdot D_c}  i \Slash{D}^c_\perp \frac{\bnslash}{2} \xi_n
  \mbox{ + h.c.}\,,  \\
&& {\cal L}_{cg}^{(1)} = \frac{2}{g^2} {\rm tr} 
  \Big\{ \big[i {\cal D}^\mu , iD_c^{\perp\nu} \big] 
     \big[i {\cal D}_\mu , iD_{us\,\nu}^\perp \big] \Big\} + {\rm g.f.}\,, \nn \\
    && {\cal L}^{(1)}_{\xi q} = ig\: \bar\xi_n \: \frac{1}{i\bn\mcdot D_c}\: 
 \Bslash_\perp^c W  q_{us} \mbox{ + h.c.}\,. \nn
\end{eqnarray}
with ${\cal D}^\mu = n^\mu \bn\mcdot D_c/2 + D_c^{\perp\mu} +\bn^\mu n\mcdot
D/2$ and g.f.~denotes gauge fixing terms.
We will also need the order $\lambda^2$ Lagrangian
\begin{eqnarray} \label{L12}
{\cal L}^{(2)}_{\xi \xi}\!\!\! &=& 
\!\!\!\bar \xi_n \!\!\left(\!
\Dslash_{us}^\perp \frac{i}{\bn\cdot D_c} \Dslash_{us}^\perp
 \!-\!
\Dslash^\perp_c \frac{i}{\bn\cdot D_c}
\bn\cdot D_u \frac{1}{\bn\cdot D_c} \Dslash^\perp_c
\! \!\right)\!
\frac{\bnslash}{2}\xi_n.\nn \\
\end{eqnarray} 
 It is important to understand that  time ordered products corrections correspond to perturbations of  the states in the effective theory. That is, in the effective theory the pion state is not the physical pion state, it contains only collinear modes in SCETII. The true pion is an eigenstate of the full Hamiltonian, so including perturbations into the time ordered product accounts for this difference in a systematic fashion. 

 Matching onto SCETI we may generate order $\lambda$ operators by inserting  $B ^\perp_\mu$ into $O_0$,
\begin{equation}
O^{(1)}=(\bar \xi _n  W)_{\omega_1}\bar{n}\!\!\! \slash  \gamma_5
 (W^\dagger B ^{\mu \perp}_nW)_{\omega_2} (W^\dagger \xi_n)_{\omega_3}~,
\end{equation}
 where
   $B^{\mu_\perp}_n\equiv\bn_\nu ({
G}_{n})^{\nu\mu_\perp}$ and the collinear gauge invariant field strength is
\begin{eqnarray} \label{covG}
  ({G}_{n})^{\mu\nu} = -\frac{i}{g}\Big[
   [i{\cal D}_n^\mu + gA_{n,q}^\mu, 
i{\cal D}_n^\nu+gA_{n,q'}^\nu ] 
   \Big] \,.
\end{eqnarray}
But this operator does not interpolate for the pion. Operators with insertions of $D^c_\perp$, can be absorbed into $O^{(1)}$.
 We may also consider quark mass effects, 
which may arise from either the expansion of the fields via
\begin{equation}
q=\left(1+\frac{1}{\bn \cdot iD_c}(iD^c_\perp \!\!\!\!\!\!\!\slash ~-m_q)\frac{\bnslash}{2}\right) \xi_n, 
\end{equation} 
 or from including  mass effects in perturbative matching.
Using a simple spurion analysis and the fact that there is only one possible non-vanishing Dirac structure in the effective theory which violates chirality, it is simple to show that matching cannot generate any $O(\lambda^0)$ terms which are linear in the quark mass to all orders in perturbation theory. 
However, the introduction of a quark mass  generates new $O(\lambda)$ operator
\begin{equation}
O^{1}_m=\!\!\ (\bar \xi_n W)_{\omega_1}\epsilon^\perp_{\mu \nu}(W [ i\overleftarrow{D}^\mu_\perp \gamma^\nu_\perp -i\overrightarrow{D}^\mu_\perp 
\gamma^\nu_\perp] W^\dagger)_{\omega_2}
(W^\dagger\xi_n )_{\omega_3}.
\end{equation}


The possible order $\lambda^2$ operators which are bi-linear in the collinear quarks are \begin{eqnarray} \label{loc}
 O^{(2)}_a \!\!\!&=&\!\!\!(\bar \xi_n W)_{\omega_1} (W^\dagger_n n \mcdot D W^\dagger )_{\omega_2}  \bnslash\gamma_5(W^\dagger \xi_n)_{\omega_3} \\
 O^{(2)}_b\!\!\!&=& \!\!\! (\bar \xi_n W)_{\omega_1} (W \Slash{D}^c_\perp W^\dagger)_{\omega_2}  (W\Slash{D}^c_\perp W^\dagger)_{\omega_3}
 \bnslash\gamma_5(W^\dagger\xi_n )_{\omega_4} \nn
\end{eqnarray}
where the appearance of the momentum subscript implies the existence of non-trivial Wilson coefficients which account for possible insertions of the operator 
$\bn\cdot D_c$.  $ O^{(2)}_b$, is a representative of a class of operators with two transverse covariant derivatives acting in all possible ways.
Other operators involving collinear field strength operators can be expressed in terms of linear combinations of these operators. 
In addition, it is possible to generate operators with one collinear and one usoft quark,
i.e. $\bar \xi_n W \bar n \!\!\!\slash ~\gamma_5 q_{us}$, but the contribution from this operator will be order $\lambda^3$, since an additional insertion of a subleading operator is necessary to get a non-vanishing matrix element between pion and vacuum. 

 We now match onto the lower theory SCETII, where external virtualities are restricted to be less than $\Lambda Q$. In doing so, the external states will pick out a subset of the  collinear modes, whose transverse label momentum is of order $\Lambda$. In addition, the usoft modes get relabeled to be soft modes. For Wilson lines this transformation is denoted by $Y\rightarrow S$.
The operators in the class of $O_b^{(2)}$ will have vanishing Wilson coefficients when matching onto SCETII since we set the external transverse momenta to zero. Formally this occurs because
\begin{equation}
\cP^\perp \xi^{II}=0.
\end{equation}
 However, there are a class of identical operators with usoft derivatives as well which will give $\Lambda^2/Q^2$ corrections in SCETII. This scaling arises because usoft in SCETI  match onto soft fluctuation, which scale as $\Lambda/Q$ in SCETII.
In addition, we will generate new operators by considering the time ordered products in SCETI
\begin{eqnarray}
T_1&=&\int d^4 x d^4y T\left( O^{(0)} {\cal L}_{i}^{(1)}(x) {\cal L}_{i}^{(1)}(y)\right) \nn \\
T_2&=&\int d^4 x   T\left( O^{(0)} {\cal L}_{j}^{(2)}(x) \right), 
\end{eqnarray}
where ${\cal L}_{i}^{(1)}$ and ${\cal L}_{i}^{(2)}$ are any of the first and second order Lagrangian corrections respectively. Since the external states have vanishing transverse momenta as far as the label operators in SCETI are concerned, any operator with an odd number of transverse derivatives vanishes.
The TOPs $T_1$ and $T_2$ however,  will have non-vanishing matching coefficients and will contribute at order $\Lambda/Q$.
To see how this scaling comes about, note that integrating out modes which are off-shell by $\Lambda Q$, generates powers of $\lambda^{-1}$. For instance, in the case  $T_2$ the Wilson coefficient will contain a factor of
$1/(\bn \cdot p n \cdot s)$  which scales as $Q/\Lambda$ since the soft momentum $s$ is of 
order $\Lambda$. Together with the the suppression of two soft fields in the external states which scale as $\Lambda^2 /Q^2$, we find that $T_2$ scales as $\Lambda/Q$. Notice that  the ``local'' power corrections, such as  (\ref{loc}), 
will, in general, contribute at  the same order in $\alpha_s(Q)$ as
 the leading order contribution, while the TOP contributions 
 will be proportional to  higher powers of $\alpha_s(\sqrt{\Lambda Q})$.



It is interesting to note that in the SCET formalism {\it all} the power corrections are factorizable, in the sense that they can be written as products of matrix elements of various types of fields \footnote{I thank Mark Wise and Iain 
Stewart for emphasizing this point to me.}.
That is, factorization is manifest in all the power corrections, since after the field redefinition, none of the various types of fields communicate\footnote{Though individual contributions may not be manifestly gauge invariant.}. As a consequence of this, all the soft effects will be controlled by vacuum matrix elements which are independent of the hadron. This implies that, under our working assumption that SCET as formulated in \cite{SCET,bps3} is the appropriate effective field theory for the above processes, the soft pieces of all hadrons are universal. 
Thus, there is hope that we can extract the soft structure functions and use them to make predictions in disparate processes.






Let us now consider the EM form factor. As we will see, the existence of two jets will have important ramifications. 
In this case,  Lagrangian splits into two pieces, one for each type of collinear mode, which do not communicate,
\begin{equation}
{\cal L}_{tot}={\cal L}_{\bn}+{\cal L}_{n}.
\end{equation}
At subleading order in the $\alpha_s(Q)$, matching onto SCETI will generate operators which connect the two different collinear sectors, but they will not play role here.
The usual leading order Brodsky-Lepage (BL) result
 was regained in SCET by matching the full QCD current onto four quark operators which are generated at order $\alpha_s(Q)$ \cite{bfprs}, 
\begin{eqnarray}O^{(0)}_{1}\!\!\!&=&\!\!\!(\bar \xi _n  W_n)_{\omega_1} \bar{n}\!\!\! \slash  \gamma_5 (W^\dagger_n \xi_n)_{\omega_2}(\bar \xi _\bn  W_{\bn})_{\omega_3} n\!\!\! \slash  \gamma_5 (W^\dagger_\bn \xi_\bn)_{\omega_4} \\
O^{(0)}_{8}\!\!&=&\!\!(\bar \xi _n  W_n)_{\omega_1} \bar{n}\!\!\! \slash  \gamma_5 
T^a(W^\dagger_n \xi_n)_{\omega_2}(\bar \xi _\bn  W_\bn)_{\omega_3} n\!\!\! \slash  \gamma_5 T^a(W^\dagger_\bn \xi_\bn)_{\omega_4}.\nn
\end{eqnarray}
Scaling the fields by the usoft Wilson lines has no effect.  The fields of opposing directions do not couple and therefore, the matrix element of these operators between pion states of opposing light like directions factorizes into the result (\ref{pimom}), with the octet contribution vanishing. 
\begin{figure}[t]
 \centerline{
  \mbox{\epsfysize=2.2truecm \hbox{\epsfbox{newpion.ps}} }
  }
\vskip-0.6cm
\caption[1]{Integrating out modes with virtuality $\sqrt{\Lambda Q}$ lead to operators which represent the ``soft" piece of the wave function. The solid (dashed) circle represents modes off-shell by $Q$ ($\sqrt{\Lambda Q}$).}

\label{fig_FACT} 
\vskip-0.5cm
\end{figure}
In the EM  case we may also match onto a quark bilinear 
\begin{equation}
   O^{(-2)}(\omega_1,\omega_2)=(\bar\xi_{\bn}W_{\bn})_{\omega_1} \Gamma 
(W^\dagger_n \xi_n)_{\omega_2} \,,
\end{equation}
where the possible Dirac structures, $\Gamma$ are given in (\ref{Jspin}). 
This operator is enhanced by a factor $\lambda^{-2}$ relative to the four quark operator since each collinear fields scales as $\lambda$.
However, charge conjugation implies that the only structure with non-vanishing Wilson coefficient is $\gamma_\mu^\perp$, which will not contribute for the case of the pion but will for the case of the rho. At next order in the matching the operators 
\begin{eqnarray}
O^{(-1)}_a(\omega_i)\!=\! (\bar\xi_{\bn}W_{\bn})_{\omega_1} (W i\overrightarrow{D_c}^\perp\!\!\!\!\!\!\!\!\!\!\slash~~~ W^\dagger)_{\omega_2} 
(W^\dagger_n \xi_n)_{\omega_3} \nn \\
O^{(-1)}_b(\omega_i)\!= \!(\bar\xi_{\bn}W_{\bn})_{\omega_1} (W i\overleftarrow{D_c}^\perp\!\!\!\!\!\!\!\!\!\!\slash~~~ W^\dagger)_{\omega_2} 
(W^\dagger_n \xi_n)_{\omega_3} 
\end{eqnarray}
are generated at order $\alpha_s^0$. The factor of $(W D_c^\perp\!\!\!\!\!\!\!\!\slash~~~ W^\dagger)_{\omega_2}$ can be composed of fields which are collinear in either the $n$ or $\bn$ direction. 
The leading order matrix elements between back to back pions of this operator vanishes. To generate a non-zero overlap, all that is needed is the proper insertion of sub-leading operators which
will inject one collinear quark into each jet.
Furthermore, to get a non-vanishing Wilson coefficient the TOP should contain an even number of 
insertions of $D\!\!\!\slash_c^\perp$.
This can be accomplished via a usoft (which become soft in SCETII) partonic fluctuation. The TOP
\begin{equation}
T_{us}=\int d^4 x ~d^4y ~d^4z ~T\left( O^{(-1)} {\cal L}_{\xi \xi}^{\bn (1)}(x) {\cal L}_{\xi q}^{\bn (1)}(y){\cal L}_{\xi q}^{n (1)}(z)\right)
\end{equation}
gives an order $\Lambda/Q$ contributions to the EM pion form factor.
%Each intermediate line gives one power of $Q/\Lambda$ while the insertion of 
%the usoft derivative is of order $\Lambda/Q$ in SCETII. The external quarks, which are soft, scal%e as $\Lambda^3/Q^3$. This contribution corresponds to the so-called Feynman region and arises from usoft quark corrections to the pion state as depicted in figure 1. There is an additional operator which gives an identical contribution but 
with $n\leftrightarrow \bn$. TOPs at this order involving insertions of the quark mass have zero matching coefficients as they involve powers of the external transverse momentum. However, 
there will quark  mass depedence in the non-perturbative matrix elements in SCETII which will be relevant for $SU(3)$\cite{Wise}.

Notice that the usual leading order BL result is proportional to $\alpha_s(Q)$ while this soft contribution scales as $\alpha_s(\sqrt{Q\Lambda})^2$. The data only reaches $Q^2=10~GeV^2$, with error bars growing to be order the 
signal itself at larger $Q$\cite{emdata}. Given that $\Lambda$ is of order $1$ GeV, the scale $\sqrt{\Lambda Q}$ is likely  non-perturbative over most of the range of the 
data, giving the TOP an enhancement of $1/\alpha_s(Q)$ relative to the leading order BL result.
Thus, we see that the lack of concordance between theory and data may be due to the enhanced power correction in the EM form factor. Of course, there are other possible reasons for the discrepancy. It could be that using the asymptotic wave function is a poor approximation for these values of $Q$.
However, the fact that the transition form factor seems to agree with the 
data\cite{data}, within theory errors, lends credence to the possibility that the enhanced power correction discussed here could be the real culprit. 
Finally, the discrepancy could also be due to the extrapolation of the $\gamma^\star p\rightarrow \pi n$ data to the pion pole\cite{Carlson}.
Furthermore, there are other logarithms due to the anomalous dimensions of the SCETI operators which will
also effect the final result. A complete operator analysis with RG improvement will follow in a subsequent publication.

This work was supported in part by the DOE under grants DOE-ER-03-40682-143 and
DE-AC02-76CH03000. 


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\end{thebibliography}

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