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\begin{document}
\begin{titlepage}

\begin{center}
\LARGE\bf Fixed-Point Analysis of the Low-Energy Constants
in the Pion-Nucleon Chiral Lagrangian \\  

\end{center}
\vspace{1cm plus 0.5cm minus 0.5cm}
\begin{center}
{\large 
 Youngman Kim$^{(a, b)}$, Fred Myhrer$^{(b)}$ and  
 Kuniharu Kubodera$^{(b)}$}
\end{center}
\vspace{0.5cm plus 0.5cm minus 0.5cm}
\begin{center}
(a)~{\it  School of Physics, Seoul National University,
Seoul 151-742, Korea}\\
(b)~{\it Department of Physics and Astronomy, 
University of South Carolina, 
Columbia, South 
Carolina 29208}


\end{center}
\vspace{1.0cm plus 0.5cm minus 0.5cm}

\begin{abstract}
In the framework of heavy-baryon chiral perturbation theory, 
we investigate the fixed point structure 
of renormalization group equations (RGE) 
for the ratios of the renormalized 
low energy constants (LECs) that feature 
in the pion-nucleon chiral Lagrangian.
The ratios of the LECs deduced from our RGE analysis
are found to be in semi-quantitative agreement with 
those obtained from direct fit to the experimental data.
The naturalness of this agreement
is discussed using a simple dimensional analysis 
combined with Wilsonian RGEs. 

\end{abstract}
\end{titlepage}

\newpage
\renewcommand{\thefootnote}{\#\arabic{footnote}}
\section{Introduction}
As is well known, 
the long-distance or low-energy behaviour of QCD
is not amenable to perturbative approaches.
Chiral perturbation theory (ChPT) provides
a highly useful framework for correlating
hadronic observables 
in the low-energy regime~\cite{chpta}.
%
In ChPT, possible terms 
in the effective Lagrangian, ${\cal L}_{\rm eff}$,
are classified in terms of the chiral index $\nu$;
the contribution of a term with index $\nu$
carries the factor $(Q/\Lambda)^\nu$,
where $Q$ is a typical energy-momentum scale
involved in a given process,
and $\Lambda$ is the scale of short-range processes 
that have been integrated out.
Since ChPT is a non-renormalizable theory, 
one must introduce,
for each given order $\nu$, 
new counter-terms 
containing unknown coupling constants,
called the low energy constants (LECs). 
These LECs reflect short-distance physics
that has been integrated out. 
After renormalization at each chiral order, 
the finite part of the LECs 
are to be determined from empirical data, 
see {\it e.g.},~\cite{fms, fm}. 
Once the LECs are determined, 
${\cal L}_{\rm eff}$ can be used
to make predictions on those observables 
which have not been used as input. 

The present report is concerned with
the renormalization group properties
of the LECs in the pion-nucleon sector.
Our work is motivated by a similar study 
for the pion sector 
by Atance and Schrempp~\cite{as00}.
In Ref.~\cite{as00} it was discussed
that there exist non-trivial fixed points 
in the renormalization group equations 
(RGEs) for the ratios of LECs in the pion sector.  
These fixed points exist 
in the limit $\mu\rightarrow 0$,
where $\mu$ is a renormalization 
scale\footnote{Whether or not 
the fixed points in the limit
$\mu\rightarrow 0$ govern low energy physics 
of the system in question 
is a subtle question, 
reflecting the arbitrariness
of the renormalization scale $\mu$ in ChPT. 
This is in contrast to the case of  
renormalizable theory such as QCD,
where one must choose $\mu$ as a typical energy 
of the process in question
to keep perturbation theory meaningful, 
see {\it e.g.}~\cite{wein} for details.}. 
Atance and Schrempp estimated,
for the $\nu$=$4$ and $\nu=6$ mesonic chiral Lagrangians,
the LEC ratios 
at the relevant fixed points.
The resulting values of the ratios were found to be
in reasonable agreement with those determined 
from data. 
We describe here an application of 
a similar fixed-point analysis 
to the pion-nucleon Lagrangian. 
We show that, by studying the fixed points
in the RGEs for the ratios of LECs,
one can estimate the LEC ratios pertaining 
to the pion-nucleon sector
and that the results are in semi-quantitative agreement
with those obtained from direct fitting 
to the experimental data. 
We argue that this agreement can be understood 
by invoking a simple dimensional analysis, and
we discuss our results in the context 
of the Wilsonian renormalization scheme.

\section{RGE fixed points and ratios of LECs}

ChPT is based on a low-energy effective Lagrangian, 
${\cal{L}}_{eff}$,
which is the most general possible Lagrangian
consistent with spontaneously broken chiral symmetry
of QCD.  
${\cal{L}}_{eff}$ is written 
as an expansion in $(Q/\Lambda)^\nu$ 
(see {\it e.g.} \cite{fms, fmms, ecker} for details): 
\ba
{\cal L}_{\rm eff} ={\cal L}_{\pi\pi}^{(2)}+ 
{\cal L}_{\pi\pi}^{(4)}+{\cal L}_{\pi N}^{(1)}+
{\cal L}_{\pi N}^{(2)}+ {\cal L}_{\pi N}^{(3)}
+\cdots\ ,
\ea
where the superscript denotes the chiral order $\nu$,
and ``$\cdots$'' stands for higher order terms 
in the expansion.
In this work we focus on the third-order 
pion-nucleon Lagrangian,
${\cal L}_{\pi N}^{(3)}$.
Using heavy-baryon chiral perturbation theory, 
and adopting the notational conventions of Ref.~\cite{fms},
we write
\ba
{\cal L}_{\pi N}^{(3)}={\cal L}_{\pi N}^{(3), 
{\rm fixed}} +\sum_{i=1}^{23} 
d_i\bar H \tilde O_i H
+\sum_{i=24}^{31}\tilde d_i\bar H \tilde O_i^{\rm div} H \; . 
\label{eq:L3}
\ea 
Here $H$ is the heavy nucleon field,
and $\tilde O_i$ and $\tilde O_i^{\rm div}$ are 
operators constrained by chiral symmetry considerations.
We refer to \cite{fms} for a complete list of $\tilde O_i$ 
and $\tilde O_i^{\rm div}$.
The LECs, $d_i$ and $\tilde d_i$,
cannot be constrained by symmetry,
and their behaviour is our main concern here.
Via dimensional regularization
the ultraviolet (UV) divergences
of loop-diagrams are absorbed in $d_i$, 
which we decompose as 
\ba
d_i=d_i^r(\mu) +\frac{(4\pi )^2}{(4\pi F)^2} 
\; \kappa_i \; L (\mu) \; , 
\label{lecri}
\ea
where $\mu$ is a renormalization scale, and 
\ba
L (\mu ) =\frac{\mu^{d-4}}
{(4\pi)^2}\Big(\frac{1}{d-4} - 
\frac{1}{2} [\log 4\pi +1+
\Gamma^\prime (1)]\Big) \; .
\ea 
The constant $F$ is the leading term 
in the quark-mass expansion 
of the pion decay constant $F_\pi$.   
A complete list of analytic expressions for $\kappa_i$ 
can be found in Refs.~\cite{fms,ecker}.
The values of the 
finite renormalized LECs, $d_i^r(\mu)$, 
are to be fixed by phenomenology or 
estimated using models.

\vspace*{5mm} 

{}From Eq. (\ref{lecri}) one can derive~\cite{ecker} 
a renormalization group equation (RGE)
for the renormalized LEC, $d_i^r (\mu)$,
\ba
\mu\frac{d}{d\mu} d_i^r(\mu)
=-\frac{\kappa_i}
{(4\pi F)^2} \; , 
\label{eq:rge}
\ea
which can be integrated to yield  
\ba
d_i^r(\mu )=d_i^r (\mu_0)-\frac{\kappa_i}
{(4\pi F)^2}\log 
\frac{\mu}{\mu_0} \; .  
\label{solr} 
\ea
Eq.(\ref{eq:rge}) leads to the
 RGE for the ratio $d_i^r/d_j^r$ \cite{as00} ,  
\ba
\mu\frac{d}{d\mu}\left (\frac{ d_i^r(\mu)}
{ d_j^r(\mu)}\right )=
\frac{1}{(4\pi F)^2}\frac{\kappa_j}{d_j^r (\mu)}
\left ( \frac{d_i^r(\mu)}{d_j^r (\mu)}
-\frac{\kappa_i}{\kappa_j}\right ) \; . 
\label{eq:ratio}
\ea 
We note that, 
as can be easily seen from Eq.(\ref{solr}), 
$\kappa_j/d_j^r (\mu)$ in Eq.(\ref{eq:ratio})
is positive 
in the limit $\mu\rightarrow 0$. 
Therefore Eq.(\ref{eq:ratio}) gives 
a non-trivial stable fixed point (f.p.) 
in the limit 
$\mu\rightarrow 0$\footnote{Eq.(\ref{eq:ratio}) 
has a non-trivial stable f.p. 
in the limit $\mu\rightarrow \infty$ as well. 
However, we will focus on the f.p.'s 
in the limit $\mu\rightarrow 0$  
for a reason to be discussed in Section 3. }, 
\ba
\frac{d_i^r}{d_j^r}\mid_{\rm{f.p.}}=
\frac{\kappa_i}{\kappa_j}~~{\rm for}~ \kappa_i,~\kappa_j\neq 0 \; . 
\label{fp}
\ea 
We note that the derivation of this equation 
involves only few assumptions,
and we shall study the consequences 
of Eq.(\ref{fp}) in what follows.
Since $\kappa_i$'s in the pion-nucleon sector  
are known, we can predict the l.h.s. of 
Eq.(\ref{fp}).
Meanwhile, the value of the l.h.s.
can also be obtained from the LECs 
that have been deduced directly
from the experimental data~\cite{fms,fm}.
The comparison of these two determinations
will check the validity of Eq.(\ref{fp}).

For this comparison,
it is useful to consider
the scale-independent LEC,
$\bar d_i$, defined by 
\ba
d_i=\bar d_i + \frac{\kappa_i}{(4\pi F)^2}
\left( (4\pi)^2 L(\mu ) 
-\log \frac{\mu}{M}\right) , 
\label{lecsi}
\ea
where $M$ is the leading term 
in the quark mass expansion 
of the pion mass.
The $\bar d_i$'s have been determined 
phenomenologically in Ref.~\cite{fms}.
Eqs.(\ref{lecri}) and (\ref{lecsi})  
lead to a relation between $\bar d_i$ and 
$d_i^r(\mu)$:
\ba
\bar d_i =d_i^r(\mu) + 
\frac{\kappa_i}{(4\pi F)^2} \log 
\frac{\mu}{M} \;, 
\label{dbr}
\ea 
which implies
\ba
d_i^r(M)=\bar d_i.
\ea
Since $\bar d_i$'s, and hence the ratios 
${\bar d}_i /{\bar d}_j$ as well, 
are scale-independent, 
it is reasonable to replace Eq.(\ref{fp}) with
\ba
\kappa_i /\kappa_j= \bar d_i /\bar d_j.
\label{comparison}
\ea
More discussion about this replacement
will be given later in the text.  
In the following, we compare 
the left- and right-hand sides 
of Eq. (\ref{comparison}).

Experimental information available 
for this comparison may be summarized as follows.
In Refs.~\cite{fms,fm}, 
$\bar d_i$'s have been determined 
by analyzing pion-nucleon scattering.
Table 1 gives the values 
of $\bar d_i=d_i^r(\mu\!=\!M)$ determined 
by Fettes {\it et al.}~\cite{fms,fm}. 
%at the energy scale $\mu=M$;
In Ref.~\cite{fms}, 
the authors investigated pion-nucleon scattering
in a $\nu$=$3$ ChPT calculation
assuming exact isospin symmetry. 
A more general ChPT $\nu$=$3$ 
analysis of pion-nucleon scattering
was implemented in Ref.~\cite{fm}, 
wherein electromagnetic corrections 
as well as $\nu=2$ isospin violating terms 
of ${\cal L}_{\pi N}^{(2)}$ were taken into account.
One naturally expects 
that the determination of 
the counter-terms in ${\cal L}_{\pi N}^{(3)}$
is influenced by the improvement
in the treatment of ${\cal L}_{\pi N}^{(2)}$,
and Table 1 indeed  shows 
drastic differences between the values of the LECs  
obtained in Ref.~\cite{fm} and Ref.~\cite{fms}.
In the following we shall be primarily concerned
with the $\nu=3$ LECs obtained in 
the more general analysis in Ref.~\cite{fm}.  
%
%%%%%%%%%%%%%%% Table 1 %%%%%%%%%%%%%%%%%%
\vskip 0.3cm
  \begin{center}
  Table 1 : \parbox[t]{5.3in}{Phenomenologically
  determined values of the LECs (in ${\rm GeV}^{-2}$).
%at the energy scale $\mu=M$.  
The values in the rows labelled 
EXP(Fit 1), EXP(Fit 2) and EXP(Fit 3) are 
taken from Ref.~\cite{fms}, 
while those in the row EXP(2001) are taken from
Ref.~\cite{fm}.}
  \end{center}
  $$
  \begin{array}{|r||r|r|r|r|r|}
  \hline 
 &{\bar d_1} + {\bar d_2} & \bar d_3 & \bar d_5 &\bar d_{14}  
  -\bar d_{15}& \bar d_{18} \\
  \hline
  \hline
 {\rm EXP( Fit}~ 1) & 3.06\pm 0.21 &-3.27\pm 0.73 & 
 0.45\pm 0.42 &-5.65\pm 0.41 
& -1.40 \pm 0.24    \\ \hline
{\rm EXP( Fit}~ 2) & 3.31\pm 0.14&-2.75\pm 0.18
& -0.48\pm 0.06&-5.69\pm 0.28& -
0.78\pm 0.27    \\ \hline
{\rm EXP( Fit}~ 3) & 2.68\pm 0.15&-3.11\pm 0.79
& 0.43\pm 0.49&-5.74\pm 0.29& -
0.83\pm 0.06    \\ \hline
{\rm EXP( 2001 )} & -2.24\pm 0.16 &0.81\pm 0.16  
&0.67\pm 0.11 & -0.63\pm 0.75 & 
-10.14\pm 0.45   \\ \hline
  \end{array}
  $$
  \vskip 0.3cm
%%%%%%%%%%%%%% End of Table 1 %%%%%%%%%%%% 
In addition to the LECs presented in Table 1, 
Eq.(\ref{eq:L3}) contains one more $\nu$=$3$ LEC 
determined from experiment.
In Ref.~\cite{bkm}, $d_6^r(\mu)$ was deduced 
from the isovector charge radius at 
$\mu=\stackrel{\circ}{m}$,
where $\stackrel{\circ}{m}$ is the value of the 
nucleon mass in the chiral limit;
the result is
$d_6^r(\stackrel{\circ}{m})
=-0.13~ {\rm GeV}^{-2}$.   
For our present purposes,
we need to scale $d_6^r(\mu)$ 
down to $\mu\!=\!M$. 
Carrying out  this rescaling using Eq.(\ref{solr}), 
we obtain $d_6^r(M)=\bar d_6 =-2.96~ {\rm GeV}^{-2}$. 
This value of $d_6^r(M)$ 
will also be considered below. 

\vspace*{3mm}

We now examine the two sides of Eq.(\ref{comparison}).
Using the analytic expressions of $\kappa_i$  
given in Ref.~\cite{fms,ecker} and 
$g_A=1.26$, we obtain 
$\kappa_1 +\kappa_2 =-0.42$, $\kappa_3=  0.18$,
$\kappa_5=0.37$ and $\kappa_6=-1.49$. 
The use of these values of $\kappa_i$
in Eq.(\ref{comparison}) leads to RGE-based predictions
of the LEC ratios at the scale $\mu\!=\!M$.
The results are given in the last column (labelled ``RGE")
in Table 2.
This table also shows the LEC ratios determined from  
the experimental information summarized in Table 1.
(For simplicity and because of 
the semi-quantitative nature
of the present study, 
we do not quote errors in the experimental data.)  
Since $\bar d_{15}$ and $\bar d_{18}$ 
are finite LECs and independent of $\mu$~\cite{fms},  
the ratios involving these LECs cannot be determined
from our analysis based on fixed points. 
%%%%%%%%%%%%%%% Table 2 %%%%%%%%%%%%%%%%%%
\vskip 0.3cm
  \begin{center}
  Table 2 : \parbox[t]{5.3in}{The ratios of the
  LECs.
  The data in the columns 
  labeled EXP(Fit 1), EXP(Fit 2) and  EXP(Fit 3)
  are taken from Ref.~\cite{fms},
  while those in the column labeled EXP(2001)
  are taken from Ref.~\cite{fm}.}
  \end{center}
  $$
  \begin{array}{|r||r|r|r|r|r|}
  \hline
 {\rm Ratio~ of ~LEC}s &{\rm EXP}~({\rm Fit}~1) 
 &{\rm EXP}~({\rm Fit}~2)& {\rm EXP}~({\rm Fit}~3)&
{\rm EXP}~ (2001) &{\rm  RGE} \\
  \hline
  \hline
  (\bar d_1+\bar d_2)/\bar d_3 &-0.94& -1.23&-0.86 &-2.77 & -2.33   \\ 
\hline
(\bar d_1+\bar d_2)/\bar d_5 &6.8& -6.9&6.23 &-3.34&  -1.14  \\ \hline
{\bar d_3/\bar d_5} &-7.27& 5.79& -7.23& 1.21& 0.49  \\ \hline
(\bar d_1+\bar d_2)/\bar d_6 &-1.03& -1.12&-0.91 &0.76& 0.28  \\ \hline
\bar d_3/\bar d_6 &1.1& 0.93&1.1 &-0.27& -0.12   \\ \hline
\bar d_5/\bar d_6 &-0.15& 0.16&-0.15 &-0.23& -0.25   \\ \hline
  \end{array}
  $$
  \vskip 0.3cm
%%%%%%%%%%%%%% End of Table 2 %%%%%%%%%%%%
In Table 2 we note 
that the ratios given in the column
labeled EXP (2001) and those obtained
in our RGE analysis
show semi-quantitative agreement; 
the signs are all in agreement
and the magnitudes exhibit 
a similar general tendency.
We note that the results of our RGE analysis
show definite disagreement with
those of the earlier (presumably less reliable)
empirical determinations of the LECs.   
This illustrates the possibility of
using a fixed-point analysis like the one described here  
as a constraint in determining
the LECs from data. 

\section{Discussion and summary}

We observed that the RGE for the ratio 
$d_i^r/d_j^r$ of the renormalized
LEC has a non-trivial fixed point given 
by  Eq.(\ref{fp}). We now
discuss the implication 
of Eq.(\ref{fp}) in the context of 
a simple dimensional analysis~\cite{georgi,manohar}.
To this end, we first briefly review the naive dimensional
analysis given in Ref.~\cite{georgi}.
Consider the $\pi$-$\pi$ scattering amplitude 
at order $Q^4$.
The amplitude of a one-loop diagram involving 
the lowest order Lagrangian 
${\cal L}^{(2)}_{\pi \pi}$ is given by
\ba
 \frac{Q^4}{F^4}\frac{1}{(4\pi)^2}\log\mu 
 + \cdots \; . 
\label{eq:pipiloop}
\ea
{}The necessary counter-terms come from 
the $\nu$=$4$ Lagrangian ${\cal L}^{(4)}_{\pi \pi}$. 
A term like
\ba
\frac{F^2}{\mu_{\chi SB}^2} 
tr (\partial^\mu\Sigma\partial^\nu\Sigma
\partial_\mu\Sigma^\dagger\partial_\nu\Sigma^\dagger)
\ea
gives a scattering amplitude of order $cQ^4/F^4$, 
where $c$ is given by 
$c=F^2/\mu_{\chi SB}^2$.
Since the total $\pi$-$\pi$ scattering amplitude 
should not depend on the renormalization 
scale $\mu$, 
a shift in $\mu$ should be 
compensated by a shift in $c$. 
Therefore a change in $\mu$ of order one 
produces a change in $c$ of 
order $\delta c\sim 1/(4\pi )^2$. 
Then, barring the accidental fine tuning 
of the parameters, $c$ must be of the 
order of $\delta c$,
\ba
c\sim \delta c\sim \frac{1}{(4\pi)^2}.
\ea
This implies that $\mu_{\chi SB}\sim 4\pi F $~\cite{georgi}. 
Now, an advantage of using the RGE 
is that one can calculate $\delta c$ explicitly. 
To show this point in our case, 
we return to Eq.(\ref{solr}).  
As stated, observables should be independent 
of $\mu$.
{}From Eq.(\ref{solr}) we can see 
that a change in $\mu$ 
produces a change in the LEC $d_i^r (\mu)$: 
\ba
\delta d_i^r&\equiv& d_i^r 
(\mu )-d_i^r (\mu_0)\no
&=& - \frac{ \kappa_i}{(4\pi F)^2} 
\log\frac{\mu}{\mu_0} \; . 
\label{eq:deltaLEC}
\ea
As discussed above, from
$\delta d_i^r(\mu ) \sim d_i^r(\mu )$, 
we may infer
\ba
\frac{d_i^r (\mu)}{d_j^r 
(\mu)}\sim \frac{\kappa_i}{\kappa_j}.
\ea 
%
We now discuss, 
in the context of a Wilsonian renormalization
group, whether the fixed points 
featuring in Eq.(\ref{fp}) are relevant to
low energy physics.\footnote{We thank M. Rho for 
suggesting this viewpoint.}
To do this, we follow Ref.\cite{hy00}
and derive a Wilsonian RGE
that includes quadratic divergences as well as 
logarithmic divergences. 
To preserve chiral symmetry, 
dimensional regularization is adopted,
and the quadratic divergences
are identified by the following replacement 
\cite{hy00, velt} :
\ba
\int \frac{d^nk}{i(2\pi)^n}\frac{1}{-k^2}
\rightarrow\frac{\Lambda^2}{(4\pi 
)^2}~,~~~
\int \frac{d^nk}{i(2\pi)^n}\frac{k_\mu k\nu}
{[-k^2]^2}\rightarrow -\frac{\Lambda^2}{2(4\pi )^2}g_{\mu\nu}\; ,
\label{c1hy00}
\ea
where the scale $\Lambda$ has the meaning of a naive cutoff.
These replacements are known to preserve 
chiral symmetry at the 
one-loop order \cite{harada}. 
For illustration,
we again use the one-loop diagram 
of the $\pi$-$\pi$ scattering. 
Since the logarithmic divergences in 
the  $\pi$-$\pi$ scattering are
well known, we focus on the quadratic divergences.
Let us take a one-loop diagram
 with one vertex from ${\cal L}_{\pi\pi}^{(2)}$
and the other one from  ${\cal L}_{\pi\pi}^{(4)}$.
The vertex from  ${\cal L}_{\pi\pi}^{(2)}$ is of the form 
 $p^2/F^2$ and the one from  ${\cal L}_{\pi\pi}^{(4)}$ 
 has the form $L_i~ p^4/F^4$, 
where $L_i$ is a LEC 
 in ${\cal L}_{\pi\pi}^{(4)}$. 
We consider the contribution of a particular term in which  
 only two of the six derivatives act on 
the internal lines. 
{}If we apply Eq.(\ref{c1hy00}) to this term, 
its quadratic divergence is given schematically by
\ba
L_i~ \frac{p^4}{F^6}\int \frac{d^nk}{(2\pi)^n}\frac{1}{k^2}
\rightarrow L_i~\frac{p^4}{F^4}\frac{\Lambda^2}{(4\pi F)^2}\; .
\label{qdex}
\ea
%where we used Eq.(\ref{c1hy00}).
We may assume that there are similar quadratic divergences 
in the $\pi$-$N$ sector.  We can then generalize the RGE
in Eq. (\ref{eq:rge}) and write: 
\ba
\mu\frac{d}{d\mu} d_i^r(\mu)=-\frac{\kappa_i}{\Lambda_{\chi SB}^2} 
+ g_i\frac{\mu^2}{\Lambda_{\chi SB}^2}d_i^r (\mu)
+h_i\frac{\mu^2}
{\Lambda_{\chi SB}^2}~,\label{grg}
\ea
where $\Lambda_{\chi SB}=4\pi F$ 
and $g_i$ and $h_i$ are calculable
constants.
The term with $g_i$ in Eq. (\ref{grg}) is similar to the quadratic
divergence in Eq. (\ref{qdex}),
and the term with $h_i$ is a possible
quadratic divergence with all vertices 
coming from ${\cal L}_{\pi N}^{(1)}$
and ${\cal L}_{\pi N}^{(2)}$.
{}It is easy to obtain from Eq. (\ref{grg})
a modified RGE for the ratio of the LECs
\ba
\mu\frac{d}{d\mu}\left (\frac{ d_i^r(\mu)}
{ d_j^r (\mu)}\right )=\frac{\kappa_j}{d_j^r (\mu)}
\left (\frac{1}{\Lambda_{\chi SB}^2} 
(\frac{d_i^r(\mu)}{d_j^r (\mu)}
-\frac{\kappa_i}{\kappa_j}) +\frac{1}{\kappa_j}
\frac{\mu^2}{\Lambda_{\chi SB}^2}
\Big[ g_id_i^r-g_jd_i^r +h_i-h_j\frac{ d_i^r(\mu)}
{ d_j^r (\mu)}\Big] \right ) .
\label{WRGE}
\ea
In this Wilsonian RGE, 
the terms with $\mu^2/\Lambda_{\chi SB}^2$
becomes negligible in a low-energy regime (${\mu^2}\ll 
{\Lambda_{\chi SB}^2}$), and, as a result, we recover the
RGE in Eq.(\ref{eq:ratio}).
This feature suggests that the fixed points in 
Eq.(\ref{fp}) governs low
energy physics qualitatively up to
$\mu^2/\Lambda_{\chi SB}^2$ corrections.



\vspace*{5mm}
In summary,  we have estimated the ratios
of the scale invariant LECs, $\bar d_i/\bar d_j$,
by studying the fixed point structure of the RGE 
for the ratios of the renormalized LECs.
We have found that the ratios of the LECs 
determined from experiments~\cite{fm,bkm}
and the ones estimated by our RGE analysis 
agree semi-quantitatively.  
We have given a plausible explanation 
for this agreement,
invoking a simple dimensional analysis 
and the Wilsonian RGE. 
We have argued that the fixed points in Eq.(\ref{fp})
govern low-energy physics qualitatively up to
$\mu^2/\Lambda_{\chi SB}^2$ corrections. 
This argument combined with the fact that 
the pion mass $M$ is a small scale in ChPT leads us to presume that 
the ratio of the LECs at $\mu = M$, 
$d_i^r(M)/d_j^r (M)$, is close to its fixed-point value. It is hoped that
an analysis like the one described here
may be useful in placing constraints
on the values of other LECs in ChPT as well.

\section*{Acknowledgements}
We thank S. Ando for discussion. Our thanks 
are also due to M. Harada, 
U.-G. Meissner, and
M. Rho for their helpful comments.
This work is supported in part by 
the U.S. National Science Foundation, 
Grant Nos.  and .
The work of YK is supported in part 
by the Brain Korea 21 Project 
of the Ministry of Education,
by the KOSEF R01-1999-000-00017-0
and by the KRF 2001-015-DP0085.

\begin{thebibliography}{99}
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for reviews, see e.g.,
 G. Colangelo and G. Isidori, 
 ``An introduction to CHPT,''  and 
 U.-G. Meissner, Rept. Prog. Phys. {\bf 56}, 903 (1993).


\bibitem{fms} 
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