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\begin{document}
\title{Chiral symmetry restoration with $N_f$ and the scalar mesons}

\author{Bachir Moussallam}

\address{ I.P.N., Groupe de Physique Th\'eorique\\
Universit\'e Paris-Sud, F-91406 Orsay C\'edex, France}

\maketitle

\abstracts{ 
Motivated by the possible vicinity of a chiral phase transition, 
the evolution of the order parameters $<\bar u u>$ and $F_\pi$ 
from chiral $SU(2)$ to chiral $SU(3)$ is considered. Combining
CHPT with  sum rules we relate this evolution to the
physics of the scalar resonances. A significant chiral restoration
is found, in particular for $<\bar u u>$.
}

QCD is expected to undergo phase transitions if one
varies its fermion content. This can be seen already in the perturbative 
regime: with $N_c=3$, if $N_f>16.5$ 
the theory is no longer asymptotically free. Decreasing
$N_f$ below 16.5 one first encounters the  ``conformal phase''
then, for $N_f\le N_f^{crit}$ the phase which displays
confinement and, assuming the  $N_f$ fermions to be massless, the spontaneous
breaking of the $SU(N_f)\times SU(N_f)$
chiral symmetry. The exact value of $N_f^{crit}$ is not yet
known, but recent lattice simulations\cite{iwasaki}
suggest that $N_f^{crit}$ could be not
much larger than $N_f=3$, which is the number of light flavours in actual QCD.
If this is true, one should observe a significant decrease of the chiral
order parameters in going from chiral $SU(2)$ to chiral $SU(3)$. Here,
I will summarize recent work\cite{moi1}
which shows how to relate this evolution to experiment. 
Consider the two basic order parameters 
\be
<\bar u u>_{N_f}\ {\rm and}\ (F_\pi)_{N_f}, 
\ee
where the subscript  ${N_f}$ means that the masses of the  ${N_f}$ lightest
quarks are extrapolated to zero, while the masses of the remaining quarks
are kept at their physical values. Chiral perturbation theory enables one 
to express the evolution from $N_f=2$ to $N_f=3$ as an 
expansion in powers of $m_K/\Lambda$, in terms of low-energy coupling
constants (LEC's). At chiral order $p^4$ two LEC's are involved $L_4$ and
$L_6$ and at $O(p^6)$ there appears  $C_{16}$ and the combination 
$C_{20}+3C_{21}$ in the nomenclature of refs.\cite{gl85} and \cite{bce2}
respectively.

Unfortunately, little is 
known about these LEC's which were usually simply
assumed to be suppressed, according to large $N_c$ arguments\cite{gl85}. These
arguments might fail here, because $N_c\to \infty$  implies 
$N^{crit}_f\to \infty$. More information can be obtained by considering
scalar form-factors, i.e. matrix elements between the vacuum and 
$\pi\pi$ or $K \bar K$ states of the scalar currents $S_u=\bar u u+\bar d d$ 
and $S_s=\bar s s$. Contrary to the vector (or axial-vector) form-factors,
the scalar ones are not directly measurable, but a method for constructing 
them, in an energy range 
$E\le 1$ GeV, using input from $\pi\pi$ scattering data in the S-wave
(which brings in the physics of the scalar mesons), was proposed some
time ago\cite{dgl}.
From these form--factors, one can then
construct (in the same energy range)
the spectral function associated with the correlator 
$\Pi_6(p^2)$ of $S_u(x)$ and $S_s(0)$ , 
it can  be shown that this spectral function satisfies
a super-convergence relation. At chiral order $p^4$, these considerations
are sufficient to determine the LEC's $L_4$ and $L_6$\cite{moi1}. However, it
is important to consider $O(p^6)$ corrections. At this order, it is
necessary to make an assumption concerning the rate of convergence of
the chiral expansion.  Results are shown in table 1 in which we have
assumed that the ratio of successive terms in the expansion of
$\Pi_6(0)$ lies between 10\% and 100\%. We find a strong decrease of the 
chiral condensate with $N_f$
by approximately a factor of two. A decrease of $F_\pi$ is also found,
of the order of 20\%. These results reflect
the tendency of the light scalar mesons to violate the OZI rule. 
This discussion shows that this puzzling feature of the scalars
is linked to chiral symmetry (and its restoration). If the chiral condensate
is too small when $N_f=3$, the conventional chiral expansion will
break down. This situation is discussed in ref.\cite{descotes}.
%---------------table 1-----------------------------------------
\begin{table}[t]
\caption{For a given ratio 
of the successive terms in the  expansion of $\Pi_6(0)$,
the values of the
LEC's $L^r_6(m_\eta)$ and 
the  $C^r_{20}(\mu)+3C^r_{21}(\mu)$ are computed
for $\mu=m_\eta$ (in brackets, $\mu=m_\rho$).
The last column shows the successive terms in the expansion of 
$\langle\bar u u\rangle_{SU(2)}/\langle\bar u u\rangle_{SU(3)}-1$.}
\begin{center}
\footnotesize
\begin{tabular}{|c|c|c|c|}\hline
$O(p^6)/O(p^4)$ & $10^3\,L_6$ & $10^5\,(C_{20}+3C_{21})$ &
$\langle\bar u u\rangle_{SU(2)}/\langle\bar u u\rangle_{SU(3)}-1$\\ \hline
0.10 & 0.73\ [0.56]& 0.29\ [-0.02]& 0.85  + 0.14 \\
0.30 & 0.64\ [0.47]& 0.37\ [0.06] & 0.76  + 0.18 \\
0.50 & 0.57\ [0.41]& 0.43\ [0.12] & 0.69  + 0.21 \\
1    & 0.46\ [0.30]& 0.52\ [0.21] & 0.58  + 0.26 \\ \hline
\end{tabular}
\end{center}
\end{table}
%----------------------------------------------------------------------
 
\begin{thebibliography}{99}
\bibitem{iwasaki}Y. Iwasaki et al., {\it Prog. Theor. Phys. Suppl.} 
{\bf 131} (1998) 415, R.D. Mawhinney, {\it Nucl. Phys. (Proc. Sup.)}
{\bf 60A} (1998) {306}, C. Sui, {\it Nucl. Phys. (Proc. Sup.)} {\bf 73}
(1999) 228.
\bibitem{moi1}B. Moussallam, \zp{C14}(2000){111},{JHEP} {\bf 0008} (2000)005.
\bibitem{gl85}J. Gasser and H. Leutwyler, \np{B250}(1985) {465}.
\bibitem{bce2}J. Bijnens, G. Colangelo and G. Ecker, JHEP 
{\bf 9902}(1999) {020}.
\bibitem{dgl}J.F. Donoghue, J. Gasser and H. Leutwyler, \np{B343} (1990){341}.
\bibitem{descotes}S. Descotes and J. Stern, \pl{B488} (2000) 274.

\end{thebibliography}

\end{document}
