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\usepackage{epsfig}
\usepackage{amsmath}

\newcommand{\req}[1]{(\ref{#1})}
\newcommand{\muR}{\mu_{R}^{2}}
\newcommand{\muF}{\mu_{F}^{2}}
\newcommand{\muO}{\mu_{0}^{2}}


\begin{document}

\preprint{IRB-TH-2/01}

\title{
A note on the factorization scale independence of the PQCD
predictions for exclusive processes 
}

\author{B. Meli\'{c}}
\thanks{Alexander von Humboldt Fellow.
On leave of absence from the 
Rudjer Bo\v{s}kovi\'{c} Institute,
Zagreb, Croatia.}
%\email{melic@thphys.irb.hr}
\affiliation{Institut f\"{u}r Physik,
        Universit\"{a}t Mainz,\\
        D-55099 Mainz, Germany} 
\affiliation{Institut f\"{u}r Theoretische Physik,
        Universit\"{a}t W\"{u}rzburg,\\
        D-97074 W\"{u}rzburg, Germany} 
\author{B. Ni\v{z}i\'{c}}
%\email{nizic@thphys.irb.hr}
\author{K. Passek}
%\email{passek@thphys.irb.hr}
\affiliation{Theoretical Physics Division, 
        Rudjer Bo\v{s}kovi\'{c} Institute, \\
        P.O. Box 180, HR-10002 Zagreb, Croatia}

\date{\today}

\begin{abstract}
\vspace*{0.5cm}

We have shown that 
the PQCD prediction for an exclusive process at large momentum transfer 
is free of any residual dependence 
on the factorization scale 
%at every order in the strong coupling constant.
provided the complete results for both the 
hard-scattering and distribution amplitudes
are consistently obtained to the desired order in the strong
coupling constant.
\end{abstract}

\pacs{12.38.Bx,11.15.Me,13.40.Gp,13.60.Le}

\maketitle

The application of perturbative QCD (PQCD) to exclusive processes  
at large momentum transfer $Q^2$ 
is based on the factorization theorems
\cite{LeBr79etc,EfR80etc,DuM80etc,LeBr80}. The main idea 
is the separation of short- from long-distance effects in the sense 
that the high-energy region, being highly off-shell, is factorized 
from the low-energy region, which is characteristic of the 
bound-state formation. The factorization may be carried out order 
by order in perturbation theory. 
%that the large momenta $\sim Q$ in the process are factorized 
%from the small momentum 
%scales (i.e. quark and hadron masses). 
The information concerning  
the long-distance dynamics is accumulated in the distribution amplitude
(DA), one 
for each hadron involved, 
whereas the short-distance dynamics is represented by the hard-scattering 
amplitude. The 
separation of the short- from the long-distance part occurs at 
the factorization scale which is usually chosen by convenience.  

%PQCD calculation to the finite order necessarily asks for the renormalization  
%of the UV divergences 
%and introduces therefore the renormalization scale dependence in the 
%final result.  

As is well known, one of the most critical problems in making reliable 
PQCD predictions for exclusive processes at large momentum transfer is 
how to deal with the dependence of the corresponding truncated perturbation 
series on the choice of the renormalization scale $\mu_R$, the scheme for 
the QCD running coupling constant $\alpha_S(\mu_R^2)$, and the choice 
of the factorization scale $\mu_F$. 
Although the physical quantities depend neither on the
renormalization nor on the factorization scale, the PQCD prediction 
at the given order seems to bear the residual 
dependence on the renormalization and factorization scales, the choice 
of which introduces theoretical uncertainties in the prediction. 
 
%On the one hand, 
A lot of work has been devoted to
the analysis of the renormalization scale and scheme dependence 
\cite{FAC,PMS,BLM,BrJPR98,Max99etc}.
The problem of finding the optimal renormalization scale
in a given scheme has been widely discussed in the literature
and three quite different approaches have been proposed:
the principle of fastest apparent convergence (FAC) \cite{FAC}, 
the principle of minimal sensitivity (PMS) \cite{PMS} 
and the Brodsky-Lepage-Mackenzie (BLM) scale setting \cite{BLM}.
A physically motivated formalism based on which any two
perturbatively calculable observables can be related to each other
without any renormalization scale or scheme ambiguity has been
developed \cite{BrJPR98}. 
%It is based on using scale fixed relations between two physical
%observables derived by applying the BLM scale fixing to their
%respective perturbative predictions. 

In contrast to the renormalization scale, 
less attention 
has been paid to the role played by the  
factorization scale.
As far as this scale is concerned, a 
convenient choice $\mu_F^2 = Q^2$ is mainly used, with the justification 
that for such a choice, $\ln(Q^2/\mu_F^2)$ logarithms, 
giving rise to the growth
of the coefficients in the expansion 
of the hard-scattering amplitude when $Q^2 \gg \muF$, vanish. 
There have been some attempts to justify other choices 
for the factorization scale 
based on the underlying dynamics of the process
%which have been introduced 
%in the calculation of the pion form factor 
 \cite{DiR81,MNP99}. 


In this paper we  reexamine the factorization scale dependence 
of the PQCD predictions for exclusive processes
(obtained in the, so called, standard hard-scattering approach
\cite{LeBr79etc,EfR80etc,DuM80etc,LeBr80}). 
%We are going to show that the 
%factorization scale is not an artifact of QCD in a sense in which the 
%renormalization scale is, and 
We show that the residual factorization scale 
dependence in the finite order of PQCD calculation reflects the 
failure of the proper resummation of all $\ln(\mu_F^2)$ logarithms. 
Thus, 
taking into account the 
factorization scale dependence of the hard-scattering amplitude 
and of the distribution amplitude by consistently
including all terms 
that are effectively of the same 
order in $\alpha_S$,  
the PQCD prediction for 
an exclusive process is free of any residual dependence 
on the factorization scale at every order of the PQCD calculation. 
The unavoidable theoretical uncertainty 
of a particular order of the PQCD calculation remains to be only due to the 
renormalization procedure. 

For definiteness, notational simplicity, and clarity of
presentation, we 
consider the high-energy behavior of 
the simplest (from the theoretical point of view) exclusive 
quantity, 
the pion transition form factor 
$F_{\gamma \pi}(Q^2)$, 
defined in terms
of the 
$\gamma^*(q,\mu) + \gamma(k,\nu) \rightarrow \pi(P)$
amplitude.
%In the standard hard scattering approach
For large  momentum transfer $Q^2(=-q^2)$, 
the general factorization formula 
\cite{LeBr79etc,EfR80etc,DuM80etc,LeBr80} 
 for $F_{\gamma \pi}(Q^2)$ reads 
%\footnote{Here $\otimes$ stands for the usual convolution
%symbol defined by 
%$A(x) \otimes B(x) = \int_0^1 dx A(x) B(x)$.
%To simplify the expressions 
%the convolution ($\otimes$) is where appropriate
%replaced by the matrix multiplication in x-y space 
%(the unit matrix is defined as $\openone = \delta(x-y)$):
%$A(x,y) \otimes B(y,z) \equiv A \, B$. 
%}
\begin{equation}
    F_{\gamma \pi}(Q^{2})= 
       \Phi^{*}(x,\muF) \, \otimes \, T_{H}(x,Q^{2},\muF) 
          %\,, \quad \otimes = \int_0^1 dx 
           \, .  
\label{eq:tffcf}
\end{equation}
Here, 
$\Phi(x,\muF)$ is the pion distribution amplitude; 
$T_{H}(x,Q^{2},\muF)$ is the hard-scattering amplitude; 
$\muF$ is the factorization scale, and 
$x$ denotes the pion constituent's momentum
fraction, while 
$\otimes \equiv \int_0^1 dx$. 
%stands for the usual convolution symbol.
%The factorization scale $\muF$ represents
%the boundary between the short- and long-distance part.

The hard-scattering amplitude (HSA) $T_{H}$ can be calculated in 
perturbation theory and represented as a
series in the QCD running coupling constant $\alpha_S(\muR)$
by
\begin{eqnarray}
  T_{H}(x,Q^2,\muF) &=&
         T_{H}^{(0)}(x,Q^2)
         + \frac{\alpha_{S}(\muR)}{4 \pi} \, 
             T_{H}^{(1)}(x, Q^2,\muF) 
             \nonumber \\ & &
         + \frac{\alpha_{S}^2(\muR)}{(4 \pi)^2} \, 
             T_{H}^{(2)}(x, Q^2,\muF,\muR) 
                + \cdots  \, , 
             \nonumber \\
\label{eq:TH}
\end{eqnarray}
where $\muR$ is the renormalization scale. 
The dependence of the 
coefficients of the expansion \req{eq:TH}
on the scales $\muR$ and $\muF$ is of the 
form 
$\ln^n(\muR/Q^2)$ and $\ln^n(\muF/Q^2)$, respectively. 

The pion distribution amplitude $\Phi(x,\muF)$, although intrinsically 
nonperturbative, satisfies the Brodsky-Lepage (BL) evolution equation 
\begin{equation}
  \muF \frac{\partial}{\partial \muF} \Phi(x,\muF)   =
   V(x,u,\muF) \, \otimes \, \Phi(u,\muF)
         \, ,
\label{eq:eveq}
\end{equation}
where $V(x,u,\muF)$ is the 
perturbatively calculable evolution kernel
\begin{eqnarray}
  \lefteqn{V(x,u,\muF)}
           \nonumber \\ &= & 
        \frac{\alpha_S(\muF)}{4 \pi} \, V_1(x,u) +
                 \frac{\alpha_S^2(\muF)}{(4 \pi)^2}  V_2(x,u) +
                 \cdots \, . \quad
\label{eq:kernel}
\end{eqnarray}
The solution of Eq. \req{eq:eveq} can be represented
as 
\begin{equation}
   \Phi(x, \muF) =
   \Phi^{LO}(x, \muF)
 + \frac{\alpha_S(\muF)}{4 \pi} \;
   \, \Phi^{NLO}(x, \muF) + \cdots
          \, ,
\label{eq:PhiLONLO}
\end{equation}
where $\Phi^{LO}$ and $\Phi^{NLO}$ denote
the leading order (LO) and next-to-leading order (NLO) 
parts, respectively.
When convoluting  the finite-order results 
\req{eq:TH} and \req{eq:PhiLONLO}
according to \req{eq:tffcf},
one is left with the residual dependence on
both $\muR$ and $\muF$.

In order to be able to examine the origin of this
residual dependence on $\muF$,
we first reexamine the calculational
procedure and the ingredients
of the standard hard-scattering picture
for $F_{\gamma \pi}(Q^2)$. 

The HSA $T_H$ 
is obtained by evaluating 
the $\gamma^* + \gamma \rightarrow q \overline{q}$
amplitude, which we denote by $T$. 
Owing to the fact that final-state quarks are taken to be
massless and on-shell, the amplitude 
contains collinear singularities.
Since $T_H$ is a finite quantity by definition,
collinear singularities have to be subtracted.
Therefore, $T$ factorizes as 
\begin{equation}
    T(u,Q^2) = T_H(x, Q^2, \muF) \, \otimes \, Z_{T,col}(x, u; \muF)
        \, ,
\label{eq:TTHZ}
\end{equation}
with collinear singularities being subtracted at the scale 
$\muF$ and absorbed into the constant $Z_{T,col}$. 
The UV singularities are removed by
the renormalization of the fields
and by the coupling-constant renormalization at the 
(renormalization) scale $\muR$. 
It should be pointed out that calculation to any finite order 
in $\alpha_S(\muR)$
introduces the $\muR$ dependence of $T$ (and $T_H$), 
but not of $Z_{T,col}$ 
\cite{MNP01}.
%(this can be easily shown in explicit calculation,
%taking the proper care  
%when  changing the scale of the coupling constant).

The process-independent pion DA 
in a frame where
$P^+=P^0+P^3=1$, $P^-=P^0-P^3=0$, and $P_{\perp}=0$
is defined \cite{LeBr80,Ka85etc,BrD86} 
as
\begin{eqnarray}
 \Phi (u)
 & = &  \int \frac{dz^-}{2 \pi} e^{i(u-(1-u))z^- /2}
      \nonumber \\ & & \times
    \left< 0 \left| 
  \bar{\Psi}(-z) \, \frac{\gamma^+ \gamma_5}{2\sqrt{2}}
           \, \Omega \, \Psi(z) 
    \right| \pi \right> _{(z^+=z_{\perp}=0)}
                , 
\label{eq:PhiOPi}
\end{eqnarray}
where 
%\begin{eqnarray}
% \Omega & = & 
%%P \mbox{exp} \left\{ i g \int_{-1}^{1} ds A(z s)z \right\}
%%               \nonumber \\
%%        & = &  
%  \mbox{exp} \left\{ i g \int_{-1}^{1} ds A^+(z s)z^-/2 \right\}
%\label{eq:Omega}
%\end{eqnarray}
$ \Omega  =  
  \mbox{exp} \left\{ i g \int_{-1}^{1} ds A^+(z s)z^-/2 \right\} $
is a path-ordered factor
making $\Phi$ gauge invariant. 
Owing to the light-cone singularity at $z^2=0$ \cite{LeBr80,BrD86} 
the matrix element in \req{eq:PhiOPi} is UV divergent. 
After regularization and renormalization at the scale $\tilde{\mu}_R^2$, 
$z^2$ is effectively smeared over a region of order
$z^2=-z_{\perp}^2\sim 1/\tilde{\mu}_R^2$.
As a result, a finite quantity, namely, 
the pion DA $\Phi(v,\tilde{\mu}_R^2)$, 
is obtained and corresponds to 
the pion wave function integrated
over the pion intrinsic transverse momentum up to the scale 
$\tilde{\mu}_R^2$.

The pion DA as given in
\req{eq:PhiOPi}, with $\left| \pi \right>$ being the physical
pion state, cannot be determined by perturbation theory. 
If the meson state $\left| \pi \right>$ is replaced
by a $\left| q \overline{q}; t \right>$ state composed of a 
free (collinear, massless, and on-shell) quark and antiquark 
(carrying momenta $t P$ and $(1-t) P$),
then the amplitude \req{eq:PhiOPi} becomes
\begin{eqnarray}
  \tilde{\phi} (u, t)
& = &  \int \frac{dz^-}{2 \pi} e^{i(u-(1-u))z^- /2}
   \nonumber \\ & & \times
    \left< 0 \left| 
  \overline{\Psi}(-z) \, \frac{\gamma^+ \gamma_5}{2\sqrt{2}}
           \, \Omega \, \Psi(z) \right| q \overline{q}; t \right>
          \, \frac{1}{\sqrt{N_c}}
              \, . \quad
\label{eq:phiOqq}
\end{eqnarray}
Taking \req{eq:phiOqq} into account,
we can express Eq. \req{eq:PhiOPi} as 
\begin{equation}
   \Phi(u)  = 
       \tilde{\phi}(u,t) \otimes
            \, \left< q\bar{q}; t | \pi \right> \, \sqrt{N_c}
               \, .
\label{eq:Phitphirest}
\end{equation}
The distribution $\tilde{\phi}(u,t)$ can be treated perturbatively,
which enables us to investigate
the high-energy tail of
the pion DA and its evolution.
The $\tilde{\phi}(u,t)$ distribution is
multiplicatively renormalizable 
(owing to the multiplicative renormalizability
of the composite operator
  $\overline{\Psi}(-z) \, \gamma^+ \gamma_5
           \, \Omega \, \Psi(z)$). 
This means that the UV singularities 
that are not removed by the renormalization of the
fields and by  
the coupling-constant renormalization 
factorize in the renormalization constant 
$Z_{\phi,ren}$
at the (renormalization) scale $\tilde{\mu}_R^2$. 
Apart from UV singularities, the matrix element in 
\req{eq:phiOqq} contains also collinear singularities. 
Subtracting these singularities at the scale $\mu_0^2$ and 
absorbing in $Z_{\phi,col}$, 
we can write Eq. \req{eq:phiOqq}
as 
\begin{eqnarray}
   \lefteqn{\tilde{\phi}(u,t)} \nonumber \\
& = & Z_{\phi,ren}(u,v; \tilde{\mu}_R^2) \otimes 
              \phi_V(v,s; \tilde{\mu}_R^2, \mu_0^2) \otimes
               Z_{\phi,col}(s,t; \mu_0^2)
                 \, .
           \nonumber \\ 
\label{eq:ZfVZ}
\end{eqnarray}

By combining \req{eq:Phitphirest} and \req{eq:ZfVZ},
we obtain the distribution $\Phi(u)$ in the form
\begin{eqnarray}
   \Phi(u) & = &
            Z_{\phi,ren}(u,v; \tilde{\mu}_R^2) \otimes
              \Phi(v, \tilde{\mu}_R^2)
             \, ,
\label{eq:PhiZPhi}
\end{eqnarray}
where 
%the distribution amplitude $\Phi(v,\tilde{\mu}_R^2)$ 
%given by
\begin{equation}
   \Phi(v,\tilde{\mu}_R^2)=
         \phi_V(v,s; \tilde{\mu}_R^2, \mu_0^2) \otimes
         \Phi(s, \mu_0^2)
           \, .
\label{eq:PhifVPhi}
\end{equation}
Here, 
\begin{equation}
\Phi(s, \mu_0^2) = Z_{\phi,col}(s,t; \mu_0^2) \otimes 
 \, \left< q\bar{q}; t | \pi \right> \, \sqrt{N_c}
               \, 
\label{eq:PhiNP}
\end{equation}
represents the nonperturbative input (containing all effects 
of collinear singularities, confinement, and pion bound-state 
dynamics) determined at the scale 
$\mu_0^2$, while 
$\phi_V(v,s; \tilde{\mu}_R^2, \mu_0^2)$
governs the evolution of $\Phi(v,\mu_0^2)$ 
to the scale $\tilde{\mu}_R^2$.
By differentiating \req{eq:PhiZPhi}
with respect to $\tilde{\mu}_R^2$, one obtains
the evolution equation \req{eq:eveq},
with the evolution potential $V$ given by% 
%\footnote{To simplify the expressions 
%the convolution ($\otimes$) is here and where appropriate
%replaced by the matrix multiplication in x-y space 
%(the unit matrix is defined as $\openone = \delta(x-y)$).} 
\begin{equation}
   V(\tilde{\mu}_R^2) = -Z_{\phi, ren}^{-1}(\tilde{\mu}_R^2) 
          \, \left( \tilde{\mu}_R^2
       \frac{\partial}{\partial \tilde{\mu}_R^2} Z_{\phi, ren}(\tilde{\mu}_R^2)
              \right)
          \, .
\label{eq:VZ}
\end{equation}
To simplify the expressions, 
the convolution ($\otimes$) is here and where appropriate
replaced by the matrix multiplication in $x$-$y$ space 
(the unit matrix is defined as $\openone = \delta(x-y)$),
while the $x$, $y$ variables are suppressed. 

By convoluting 
%in the sense of collinear singularities, the ``unrenormalized'' 
%hard-scattering amplitude $T(u,Q^2)$ \req{eq:TTHZ}  
%with the unrenormalized pion distribution amplitude
%$\Phi(u)$ \req{eq:PhiZPhi},
the amplitudes $T(u,Q^2)$ and $\Phi(u)$
(Eqs. \req{eq:TTHZ} and \req{eq:PhiZPhi})
we obtain
(in analogy with \cite{CuF80,LeBr80}) 
the pion transition form factor $F_{\gamma \pi}(Q^2)$
\begin{equation}
  F_{\gamma \pi}(Q^2) =
       \Phi^{\dagger}(u) \, \otimes \,  T(u, Q^2)
          \, .
\label{eq:Fpiur1}
\end{equation}
Now, choosing $\tilde{\mu}_R^2$ to coincide with $\muF$ 
and making use of the fact that%
\footnote{We note here that the same
factorization (and renormalization) scheme
is employed in the hard-scattering and DA part, i.e.,
in Eqs. \req{eq:TTHZ} and \req{eq:PhiZPhi}.}
\begin{equation}
      Z_{T,col}(x,u; \muF) \otimes Z_{\phi, ren}(u,v; \muF)
           = \delta(x-v)
              \, ,
\label{eq:ZTZf}
\end{equation}
the divergences of $T(u,Q^2)$ and $\Phi(u)$ in \req{eq:Fpiur1} cancel 
%for $\tilde{\mu}_R^2=\muF$ due to 
and we are left with
the finite perturbative expression  for the pion
transition form factor %\req{eq:tffcf}
\begin{equation}
    F_{\gamma \pi}(Q^{2})=
      T_{H}(x,Q^{2},\muF)  \, \otimes \,  \Phi^{*}(x,\muF)
                    \,.
\label{eq:tffcf1}
\end{equation}
%Note that the $\muF$ scale at which the hard and soft physics
%factorize corresponds 
%to the factorization scale of collinear singularities in $T(u,Q^2)$ 
%and the renormalization scale of $\Phi(u)$.
It is worth pointing out that
the scale $\muF$ representing the boundary between the low- and 
high-energy parts in
\req{eq:tffcf1} plays the role of the separation scale for
collinear singularities
in $T(u,Q^2)$, on the one hand,
and of the renormalization scale for
UV singularities appearing in the perturbatively calculable part
of the distribution amplitude $\Phi(u)$, on the other hand.
%{\bf
%Because the $\mu_F^2$ scale represents the boundary between the low- and
%high-energy parts in \req{eq:tffcf1}, it appears to be             
%regularization scale of UV-divergences in the low-scale dominated 
%distribution $\Phi (u)$ and the scale at which 
%collinear divergences of on-shell massless valence partons separate
%from the 
%highly off-shell part of the scattering amplitude $T_H$.}
The calculational procedure explained above is 
illustrated in Fig. \ref{f:FpiDA}.
\begin{figure}
  \centerline{\epsfig{file=fancy.ps,height=150pt,width=240pt,silent=}}
 \caption{Pictorial representation of the 
    calculational ingredients
    of the pion transition form factor. 
    $T$ represents the perturbatively calculable 
    $\gamma^* + \gamma \rightarrow q \overline{q}$
    %hard-scattering
    amplitude , while $\Phi$ denotes the (unrenormalized)
    pion distribution
    amplitude given by \protect\req{eq:PhiOPi}, which can be
    expressed, as in \protect\req{eq:Phitphirest},
    in terms of the perturbatively calculable part
    $\tilde{\phi}$
    \protect\req{eq:phiOqq}
    and the perturbatively uncalculable part.}  
 \label{f:FpiDA}
\end{figure}
%

We next turn to the discussion of the $\mu_F^2$ dependence of the 
pion transition form factor defined as in \req{eq:tffcf1}. 

Concerning the pion distribution amplitude $\Phi(x,\muF)$, its 
dependence on $\mu_F^2$ is completely specified 
%The dependence of the pion distribution
%amplitude $\Phi(x,\muF)$ on the
%factorization scale $\muF$ is described
by the evolution equation \req{eq:eveq}.
As can be seen from Eq. \req{eq:PhifVPhi},
this dependence is completely contained in the evolutional
part $\phi_V$.  
By calculating the perturbatively obtainable amplitude 
$\tilde{\phi}$ \cite{MNP01} and making use of the relation \req{eq:ZfVZ},
the result obtained for $\phi_V$ can be organized as 
\begin{eqnarray}
\phi_V(\muF,\muO) & = &
      \phi_{V}^{LO}(\muF,\muO)
 + \frac{\alpha_S(\muF)}{4 \pi}
      \phi_{V}^{NLO}(\muF,\muO) 
         \nonumber \\ & & + \cdots
             \, ,
\label{eq:phiVevLONLO}
\end{eqnarray}
where
\begin{subequations}
\begin{eqnarray}
 \phi_{V}^{LO}(\muF,\muO) &=&
     \openone
 + \frac{\alpha_S(\muF)}{4 \pi}
    \ln \frac{\muF}{\mu_0^2} \, V_1
       \nonumber  \\ & &
 + \frac{\alpha_S^2(\muF)}{(4 \pi)^2}
    \ln^2 \frac{\muF}{\mu_0^2}\,
   \frac{1}{2} \, (V_1^2 + \beta_0 \, V_1)
   \nonumber \\ & & + \cdots \, ,
    \label{eq:phiVevLO} \\
\phi_{V}^{NLO}(\muF,\muO) &=&
  \frac{\alpha_S(\muF)}{4 \pi}
    \ln \frac{\muF}{\mu_0^2}
           \, V_2 \,
      + \cdots
           \, ,
    \label{eq:phiVevNLO}
\end{eqnarray}
\label{eq:phiVev}
\end{subequations}
and functions $V_n$ represent the $n$-loop evolutional kernels
from Eq. \req{eq:kernel}.
The terms explicitly given in \req{eq:phiVev}
correspond to the results of the two-loop calculation \cite{MNP01}. 
In writing \req{eq:phiVev}, 
the use has been made of
%we have used
\begin{displaymath}
%\begin{equation}
\frac{\alpha_S(\muF)}{4 \pi} 
  \ln \frac{\muF}{\mu_0^2} 
  \approx \frac{1}{\beta_0} \left( 1 - 
\frac{\alpha_S(\muF)}{\alpha_S(\mu_0^2)} \right)
 = O(\alpha_S^0)
     \, .
%\end{equation}
\end{displaymath}

On the other hand,
the complete LO and NLO behavior of $\phi_V(v,s; \muF, \muO)$
and, consequently, of $\Phi(v,\muF)$
can be determined by solving the
evolution equation \req{eq:eveq} or equivalently
\begin{eqnarray}
       \lefteqn{\muF 
       \frac{\partial}{\partial \muF} \phi_V (v,s,\muF,\muO)}
           \nonumber \\  
        \qquad & \qquad =& V(v,s',\muF) \, \otimes \, \phi_V(s',s,\muF,\muO)
             \, . \qquad 
\label{eq:VfV}
\end{eqnarray}
The LO result is of the form 
\begin{eqnarray}
 \phi_V^{LO}(v,s; \muF, \muO) &=&
      \sum_{n=0}^{\infty} {}' 
         \,
      \frac{v (1-v)}{N_n}
         \,
        C_n^{3/2}(2 v-1)  
    \nonumber \\ &  & \times \, 
        C_n^{3/2}(2 s -1) \;
        \left(
    \frac{\alpha_S(\muF)}{\alpha_S(\mu_0^2)}
        \right)^{-\gamma_n^{(0)}/\beta_0}
          \hspace{-1.0cm} , \hspace{1.0cm}
\label{eq:phiVLOcom}
\end{eqnarray}
where $N_n=(n+1)(n+2)/(4 (2 n+3) )$,
while $C_n^{3/2}(2 x -1)$ are the Gegenbauer polynomials (
the eigenfunctions of the LO kernel $V_1$
with the corresponding eigenvalues $\gamma_n^{(0)}$ \cite{MNP99}).
The above given complete LO prediction 
represents the summation of all
$(\alpha_S \ln \muF/\mu_0^2)^n$ terms from \req{eq:phiVevLO}.
The complete formal solution of the NLO evolution equation
was obtained in \cite{Mu94etc} by using conformal constraints
and the form of $\phi_V^{NLO}$ (corresponding
to the resummation of \req{eq:phiVevNLO}) can
be extracted from the results listed in \cite{MNP99}.



The method employed above to study the $\mu_F^2$ behavior of 
$\phi_V$ can be used to examine 
the dependence of the hard-scattering amplitude $T_{H}(x,Q^{2},\muF)$
on the scale $\muF$.  

By differentiating \req{eq:tffcf} with respect to $\muF$ and
by taking into account \req{eq:eveq},
one finds that the hard-scattering amplitude satisfies
the evolution equation
\begin{equation}
  \muF \frac{\partial}{\partial \muF} 
        T_H(x, Q^2, \muF) = -
        T_H(y, Q^2, \muF)  \, \otimes \,
            V(y,x;\muF) 
         \, . 
\label{eq:EvEqT}
\end{equation}
This equation% 
\footnote{The Eq. \req{eq:EvEqT} can be also obtained by combining 
Eq. \req{eq:TTHZ} with Eqs. \req{eq:VZ} and 
\req{eq:ZTZf}.}
is analogous to the DA evolution equation 
\req{eq:eveq}.
Similarly to the discussed
solution of the DA evolution equation, 
the finite-order solution of \req{eq:EvEqT}
contains the complete dependence on $\muF$,
%to that order,
in contrast to
the expansion \req{eq:TH} 
truncated at the same order.
%The explicit expressions for the hard-scattering amplitude
%$T_{H}(x,Q^{2},\muF)$, evaluated
%up to $n_f$-proportional NNLO terms, are given in \cite{MNP01}.

The $\muF$ dependence of $T_H(x, Q^2, \muF)$
can be, similarly to \req{eq:PhifVPhi}, 
factorized in the function $\phi_V(y,x,Q^2,\muF)$ as
\begin{equation}
  T_H(x,Q^2,\muF) =  T_H(y,Q^2,\muF=Q^2) \otimes
                   \phi_V(y,x,Q^2,\muF)
            \, .
\label{eq:tTHfV}
\end{equation}
Using \req{eq:VfV} 
one can show by partial integration that
\req{eq:tTHfV} 
indeed represents the solution of
the evolution equation \req{eq:EvEqT}%
\footnote{In Ref. \cite{MNP01} 
the factorization \req{eq:tTHfV}
is  illustrated by
an explicit calculation of
\req{eq:TH} and \req{eq:phiVev} up to 
$n_f$-proportional NNLO terms.}. 

When calculating to finite order in $\alpha_S$,
it is inappropriate that the
$\Phi(x,\muF)$ distribution obtained by solving 
the evolution equation \req{eq:eveq} 
is convoluted
with $T_H(x,Q^2,\muF)$ obtained by the
truncation of the expansion \req{eq:TH}. 
In the latter case, only the partial dependence
on $\muF$ is included,
in contrast to the former case.
The proper procedure is to convolute the solution
of the DA evolutional equation \req{eq:eveq} 
with the solution of the $T_H$ evolutional equation 
\req{eq:EvEqT}
and both solutions are expressed in terms
of the same (up to NLO order known) function
$\phi_V$ (see \req{eq:PhifVPhi} and \req{eq:tTHfV}).


Furthermore, substituting \req{eq:PhifVPhi} and \req{eq:tTHfV} 
in \req{eq:tffcf1}, we obtain
\begin{eqnarray}
  \lefteqn{F_{\gamma \pi}(Q^2)} \nonumber \\
    & = & 
   T_H(y, Q^2, Q^2) 
          \, \otimes \phi_V(y,s, Q^2, \mu_0^2) 
          \, \otimes \,\Phi^*(s, \mu_0^2)
                 \, , \qquad 
\label{eq:Fpi2q2}
\end{eqnarray}
where
\begin{equation}
   \phi_V(y,x, Q^2, \muF)\, \otimes \phi_V(x,s,\muF,\mu_0^2) 
          = \phi_V(y,s, Q^2, \mu_0^2) 
               \, ,
\label{eq:fvfvfv}
\end{equation}
has been taken into account. 

It is important to realize that the expression \req{eq:fvfvfv} is valid at 
every order of a PQCD calculation%
\footnote{The Eq. \req{eq:fvfvfv} can be easily checked to the NLO order 
\cite{MNP01} 
by using  the LO result \req{eq:phiVLOcom} and the NLO results of 
Ref. \cite{Mu94etc}.}, and hence even the finite order prediction
for $F_{\gamma \pi}(Q^2)$ does not depend on the choice of the 
$\muF$ scale. 
Let us, following Eqs. \req{eq:kernel} and \req{eq:phiVevLONLO},
define the finite  order quantities
\begin{eqnarray}
\lefteqn{\phi_V^{(n)}(\muF,\muO)} \nonumber \\
       & = &
      \phi_{V}^{LO}(\muF,\muO)
 + \cdots + \frac{\alpha_S^{n}(\muF)}{(4 \pi)^{n}}
      \phi_{V}^{N \cdots NLO}(\muF,\muO) 
             \, , \quad
       \nonumber \\ & &
\label{eq:phiVevLONLOfin}
\end{eqnarray}
and
\begin{equation}
  V^{(n)}(\muF) = 
        \frac{\alpha_S(\muF)}{4 \pi} \, V_1 +
                    \cdots
               +  \frac{\alpha_S^{n+1}(\muF)}{(4 \pi)^{n+1}}  V_n 
                  \,  \quad
\label{eq:kernelfin}
\end{equation}
(here $n=0, \ldots $).
The functions $\phi_V^{(n)}(Q^2, \muF)$ and $\phi_V^{(n)}(\muF,\mu_0^2)$
represent the solutions of the evolutional
equations
\begin{subequations}
\begin{eqnarray}
 \muF \frac{\partial}{\partial \muF} \phi_V^{(n)} (\muF,\muO)
        & = & V^{(n)}(\muF) \, \otimes \, \phi_V^{(n)}(\muF,\muO)
             \, , \nonumber \\
             & & \\
 \muF \frac{\partial}{\partial \muF} \phi_V^{(n)} (Q^2,\muF)
        & = & - \phi_V^{(n)}(Q^2,\muF)\, \otimes \, V^{(n)}(\muF) 
             \, . \qquad 
             \nonumber \\
\end{eqnarray}
\label{eq:VfVfVV}
\end{subequations}
By differentiating 
\begin{equation}
   \phi_V^{(n)}(Q^2, \muF)\, \otimes \phi_V^{(n)}(\muF,\mu_0^2) 
        \, 
\label{eq:fvfv}
\end{equation}
with respect to $\muF$,
and taking into account Eqs. \req{eq:VfVfVV}, 
one trivially sees that the convolution \req{eq:fvfv} 
indeed does not depend on $\muF$.

The  expression \req{eq:fvfvfv} represents the resummation of the 
$\ln(Q^2/\mu_0^2)$ logarithms over the intermediate $\muF$ scale, 
performed in such a way that, first, 
the logarithms $\ln(\muF/\mu_0^2)$ originating from the 
perturbative part of the DA are resummed, and then the summation of 
$\ln(Q^2/\muF)$ logarithms from the hard-scattering part is performed. 
Therefore, the summations of the $\muF$ logarithms can be accomplished with 
any choice of $\muF$, because the effect in the final prediction,
at every order, 
is the same as if we had performed 
the complete renormalization-group resummation of $\ln(Q^2/\mu_0^2)$ 
logarithms. 
We note here that
by adopting the common choice $\mu_F^2 = Q^2$,
we avoid the need
for the resummation of the $\ln(Q^2/\muF)$ logarithms in 
the hard-scattering part, making the calculation simpler 
and hence, for practical purposes, the preferable form of
$F_{\gamma \pi}(Q^{2})$ is given by
\begin{equation}
    F_{\gamma \pi}(Q^{2})= 
   T_H(x, Q^2, Q^2) 
        \, \otimes \,  \Phi^{*}(x,Q^2)
                    \,. 
\label{eq:tffcfN}
\end{equation}

The $F_{\gamma \pi}$
prediction \req{eq:tffcf}, as well as the prediction for any
other exclusive quantity obtained in
the standard hard-scattering picture,
is independent 
of the factorization scale $\muF$  
at every order in $\alpha_S$,
when  both $T_H$ and $\Phi$ are 
consistently treated regarding the $\muF$ dependence. 
The choice of the factorization scale is nonessential 
and the predictions obtained using any choice of $\muF$ 
are equivalent to the results obtained using, 
for practical purposes, the simplest choice $\muF=Q^2$,
where $Q^2$ represents the characteristic scale of the process. 
The true expansion parameter left is 
$\alpha_S(\muR)$, 
with $\muR$ representing
the renormalization scale of the complete
perturbatively calculable part of the pion
transition form factor \req{eq:Fpi2q2}, i.e., 
of
\begin{equation} 
T_H(s, Q^2,\muO)=
 T_H(y, Q^2, Q^2) \otimes \phi_V(y,s, Q^2, \muO)
    \, .
\label{eq:pertpart}
\end{equation} 
  
Therefore, although 
$F_{\gamma \pi}(Q^2)$
depends exclusively on the scale of the process,
we are left with
the residual dependence on the $\muR$ scale, 
when calculating to finite order.
The intermediate 
scale at which the short- and long-distance dynamics separate, 
the factorization scale  
disappears from the final prediction at every order in $\alpha_S$
and therefore does not introduce any theoretical uncertainty into the 
PQCD calculation for exclusive processes. 

\begin{acknowledgments}
  One of us (B.M.) acknowledges the support 
  by the Alexander von Humboldt Foundation and the hospitality
  of the theory groups at the Institut f\"{u}r Physik,
  Universit\"{a}t Mainz \& Institut f\"{u}r Theoretische Physik,
  Universit\"{a}t W\"{u}rzburg.
  This work was supported by the Ministry of Science and Technology
  of the Republic of Croatia under Contract No. 00980102.
\end{acknowledgments}

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\end{document}

