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\hyphenation{ tem-pe-ra-tu-re ap-pa-rent-ly re-pre-sents }

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%
\noindent \hspace{1cm}  \hfill IFUP--TH/2003--10 \hspace{1cm}\\
\mbox{}                 \hfill January 2003 \hspace{1cm}\\

\begin{center}
\vspace*{1.0cm}
{\large\bf Reviewing the problem of the $U(1)$ axial symmetry}\\
{\large\bf and the chiral transition in QCD} \\
\vspace*{1.0cm}
{\large M. Marchi, E. Meggiolaro}\\
\vspace*{0.5cm}{\normalsize
{Dipartimento di Fisica, \\
Universit\`a di Pisa, \\
Via Buonarroti 2, \\
I--56127 Pisa, Italy.}}\\
\vspace*{2cm}{\large \bf Abstract}
\end{center}

\noindent
We discuss the role of the $U(1)$ axial symmetry for the phase structure of
QCD at finite temperature. We expect that, above a certain critical
temperature, also the $U(1)$ axial symmetry will be restored.
We will try to see if this transition has (or has not) anything to do
with the usual chiral transition: various possible scenarios are discussed.
In particular, supported by recent lattice results, we analyse a scenario in
which the $U(1)$ axial symmetry is still broken above the chiral transition.
This scenario can be consistently reproduced using an effective Lagrangian
model. A new order parameter is introduced for the $U(1)$ axial symmetry
and its effects on the slope of the topological susceptibility in the full
theory with quarks are studied: we find that this quantity (in the chiral
limit of zero quark masses) acts as an order parameter for the $U(1)$ axial
symmetry above the chiral transition.
Further information on the new $U(1)$ chiral order parameter is derived
from the study (at zero temperature) of the radiative decays of the ``light''
pseudoscalar mesons in two photons: a comparison of our results with the
experimental data is performed.

\vspace{0.5cm}
\noindent
(PACS codes: 12.38.Aw, 12.39.Fe, 11.15.Pg, 11.30.Rd)
}
%
\vfill\eject

\newsection{Introduction}

\noindent
It is generally believed that a phase transition which occurs in QCD at a
finite temperature is the restoration of the spontaneously broken
$SU(L) \otimes SU(L)$ chiral symmetry in association with $L$ massless quarks.
At zero temperature the chiral symmetry is broken spontaneously by the
condensation of $q\bar{q}$ pairs and the $L^2-1$ $J^P=0^-$ mesons
are just the Nambu--Goldstone (NG) bosons associated with this breaking
\cite{chiral-symmetry}.
At high temperatures the thermal energy breaks up the $q\bar{q}$ condensate,
leading to the restoration of chiral symmetry. We expect that this property
not only holds for massless quarks but also continues for a small mass region.
The order parameter for the chiral symmetry breaking is apparently
$\langle \bar{q}q \rangle \equiv \sum_{i=1}^L \langle \bar{q}_i q_i \rangle$:
the chiral symmetry breaking corresponds to the non--vanishing of
$\langle \bar{q}q \rangle$ in the chiral limit $\sup(m_i) \to 0$.
From lattice determinations of the chiral order parameter
$\langle \bar{q}q \rangle$ one knows that the $SU(L) \otimes SU(L)$ chiral
phase transition temperature $T_{ch}$, defined as the temperature at which
the {\it chiral condensate} $\langle \bar{q}q \rangle$ goes to zero (in the
chiral limit $\sup(m_i) \to 0$), is nearly equal to the deconfining temperature
$T_c$ (see, e.g., Ref. \cite{Blum-et-al.95}).
But this is not the whole story: QCD possesses not only an
approximate $SU(L) \otimes SU(L)$ chiral symmetry, for $L$ light quark
flavours, but also a $U(1)$ axial symmetry (at least at the classical level)
\cite{Weinberg75,tHooft76}.
The role of the $U(1)$ symmetry for the finite temperature phase
structure has been so far not well studied and it is still an open question
of hadronic physics whether the fate of the $U(1)$ chiral symmetry of QCD has
or has not something to do with the fate of the $SU(L) \otimes SU(L)$ chiral
symmetry. In the following sections we will try to answer these questions:
\begin{itemize}
\item{} At which temperature is the $U(1)$ axial symmetry restored?
(if such a critical temperature does exist!)
\item{} Does this temperature coincide with the deconfinement temperature
and with the temperature at which the $SU(L)$ chiral symmetry is restored?
\end{itemize}
In the ``Witten--Veneziano mechanism'' \cite{Witten79a,Veneziano79}
for the resolution of the $U(1)$ problem, a fundamental role is played by
the so--called ``topological susceptibility'' in a QCD without
quarks, i.e., in a pure Yang--Mills (YM) theory, in the large--$N_c$ limit
($N_c$ being the number of colours):
\be
A = \displaystyle\lim_{k \to 0}
\displaystyle\lim_{N_c \to \infty}
\left\{ -i \displaystyle\int d^4 x~ e^{ikx} \langle T Q(x) Q(0) \rangle
\right\} ,
\label{eqn1}
\ee
where $Q(x) = {g^2 \over 64\pi^2}\varepsilon^{\mu\nu\rho\sigma} F^a_{\mu\nu}
F^a_{\rho\sigma}$ is the so--called ``topological charge density''.
This quantity enters into the expression for the squared mass of the $\eta'$:
$m^2_{\eta'} = {2L A \over F^2_\pi}$, where $L$ is the number of light quark
flavours taken into account in the chiral limit.
Therefore, in order to study the role of the $U(1)$ axial symmetry for the
full theory at non--zero temperatures, one should consider the YM topological
susceptibility $A(T)$ at a given temperature $T$, formally defined as
a large--$N_c$ limit of a certain expectation value $\langle \ldots \rangle_T$
in the full theory at the temperature $T$ \cite{EM1998}:
\be
A(T) = \displaystyle\lim_{k \to 0}
\displaystyle\lim_{N_c \to \infty}
\left\{ -i \displaystyle\int d^4 x~ e^{ikx} \langle T Q(x) Q(0) \rangle_T
\right\} .
\label{EQN1.3}
\ee
In other words, $A(T) = \displaystyle\lim_{k \to 0} A_0(k,T) = A_0(0,T)$,
where the quantity
\be
A_0(k,T) =
\displaystyle\lim_{N_c \to \infty}
\left\{ -i \displaystyle\int d^4 x~ e^{ikx} \langle T Q(x) Q(0) \rangle_T
\right\} ,
\label{EQN1.4}
\ee
at four--momentum $k$ and physical temperature $T$, is nothing but the
leading--order term in the $1/N_c$ expansion of the corresponding quantity
\be
\chi(k,T) =
-i \displaystyle\int d^4 x~ e^{ikx} \langle T Q(x) Q(0) \rangle_T ,
\label{EQN1.5}
\ee
of the full theory, at the same four--momentum $k$ and physical temperature
$T$. [From the general theory of $1/N_c$ expansion it is known that
$A_0(k,T)$ is of order ${\cal O}(N_c^0)$, while successive terms coming
from including fermions, are of order ${\cal O}(1/N_c)$.]
It is in this sense that one can talk about the behaviour of $A(T)$
above or below the chiral transition temperature $T_{ch}$.

The problem of studying the behaviour of $A(T)$ as a function of the
temperature $T$ was first addressed, in lattice QCD,
in Refs. \cite{Teper86,EM1992a,EM1995b}.
Recent lattice results \cite{Alles-et-al.97} (obtained for the $SU(3)$
pure--gauge theory) show that the YM topological susceptibility $A(T)$
is approximately constant up to the critical temperature $T_c \simeq T_{ch}$,
it has a sharp decrease above the transition, but it remains different
from zero up to $\sim 1.2~T_c$.
In the Witten--Veneziano mechanism \cite{Witten79a,Veneziano79},
a (no matter how small!) value different from zero for $A$ is related to the
breaking of the $U(1)$ axial symmetry, since it implies the existence of a
pseudo--Goldstone particle with the same quantum numbers of the $\eta'$
(see also Ref. \cite{Veneziano80}).
Therefore, the available lattice results for the topological susceptibility
show that the $U(1)$ chiral symmetry is restored at a temperature $T_{U(1)}$
greater than $T_{ch}$.

Another way to address the same question is to look at the behaviour at
non--zero temperatures of the susceptibilities related to the
propagators for the following meson channels \cite{Shuryak94}
(we consider for simplicity the case of $L=2$ light flavours):
the isoscalar $I=0$ scalar channel $\sigma$ (also known as $f_0$ in the
modern language of hadron spectroscopy), interpolated by the operator
$O_\sigma = \bar{q} q$;
the isovector $I=1$ scalar channel $\delta$
(also known as $a_0$), interpolated by the operator
$\vec{O}_\delta = \bar{q} {\vec{\tau} \over 2} q$;
the isovector $I=1$ pseudoscalar channel $\pi$, interpolated by the operator
$\vec{O}_\pi = i\bar{q} \gamma_5 {\vec{\tau} \over 2} q$;
the isoscalar $I=0$ pseudoscalar channel $\eta'$, interpolated by the operator
$O_{\eta'} = i\bar{q} \gamma_5 q$.
Under $SU(2)_A$ transformations, $\sigma$ is mixed with $\pi$: thus the
restoration of this symmetry at $T_{ch}$ requires identical correlators
for these two channels. Another $SU(2)$ chiral multiplet is $(\delta,\eta')$.
On the contrary, under the $U(1)_A$ transformations, $\pi$ is mixed
with $\delta$: so, a ``practical restoration'' of the $U(1)$ axial
symmetry should imply that these two channels become degenerate, with
identical correlators. Another $U(1)$ chiral multiplet is $(\sigma,\eta')$.
(Clearly, if both chiral symmetries are restored, then all $\pi$, $\eta'$,
$\sigma$ and $\delta$ correlators should become the same.)
In practice, one can construct, for each meson channel $f$, the
corresponding chiral susceptibility
\be
\chi_f = \displaystyle\int d^4x~ \langle O_f(x) O_f^\dagger(0) \rangle ,
\label{eqn2}
\ee
and then define two order parameters:
$\chi_{SU(2) \otimes SU(2)} \equiv \chi_\sigma - \chi_\pi$, and
$\chi_{U(1)} \equiv \chi_\delta - \chi_\pi$.
If an order parameter is non--zero in the chiral limit, then the
corresponding symmetry is broken.
Present lattice data for these quantities seem to indicate that the $U(1)$
axial symmetry is still broken above $T_{ch}$, up to $\sim 1.2~T_{ch}$,
where the $\delta$--$\pi$ splitting is small but still
different from zero \cite{Bernard-et-al.97,Karsch00,Vranas00}.
In terms of the left--handed and right--handed quark fields
[$q_{L,R} \equiv {1 \over 2} (1 \pm \gamma_5) q$, with $\gamma_5 \equiv
-i\gamma^0\gamma^1\gamma^2\gamma^3$], one has the following
expression for the difference between the correlators for
the $\delta^+$ and $\pi^+$ channels:
\ba
\lefteqn{
{\cal D}_{U(1)}(x) \equiv \langle O_{\delta^+}(x) O_{\delta^+}^\dagger(0)
\rangle - \langle O_{\pi^+}(x) O_{\pi^+}^\dagger(0) \rangle } \nonumber \\
& & = 2 \left[ \langle \bar{u}_R d_L(x) \cdot \bar{d}_R u_L(0) \rangle
+ \langle \bar{u}_L d_R(x) \cdot \bar{d}_L u_R(0) \rangle \right] .
\label{eqn3}
\ea
(The integral of this quantity, $\int d^4x~ {\cal D}_{U(1)}(x)$, is just equal
to the $U(1)$ chiral parameter $\chi_{U(1)} = \chi_\delta - \chi_\pi$.)
What happens below and above $T_{ch}$?
Below $T_{ch}$, in the chiral limit $\sup(m_i) \to 0$, the left--handed
and right--handed components of a given light quark flavour ({\it up} or
{\it down}, in our case with $L=2$) can be connected through the $q\bar{q}$
chiral condensate, giving rise to a non--zero contribution to the
quantity ${\cal D}_{U(1)}(x)$ in Eq. (\ref{eqn3}) (i.e., to the quantity
$\chi_{U(1)}$). But above $T_{ch}$ the $q\bar{q}$ chiral condensate is zero:
so, how can the quantity
${\cal D}_{U(1)}(x)$ (i.e., the quantity $\chi_{U(1)}$) be different from zero
also above $T_{ch}$, as indicated by present lattice data?
The only possibility in order to solve this puzzle seems to be that of
requiring the existence of a genuine four--fermion local condensate,
which is an order parameter for the $U(1)$ axial symmetry and which
remains different from zero also above $T_{ch}$.
This will be discussed in Section 3.

The paper is organized as follows.
In Section 2 we discuss the role of the $U(1)$ axial
symmetry for the phase structure of QCD at finite temperature.
One expects that, above a certain critical temperature $T_{U(1)}$, also the
$U(1)$ axial symmetry will be restored. We will try to see if this transition
has (or has not) anything to do with the usual chiral transition: various
possible scenarios are discussed.
In particular, supported by the above--mentioned lattice results, in Sections
3 and 4 we analyse a scenario in which the $U(1)$ axial symmetry is
still broken above the chiral transition and a new order parameter is
introduced for the $U(1)$ axial symmetry alone \cite{EM1994a,EM1994b,EM1994c}.
A new meson field associated with the new $U(1)$ chiral order parameter
is introduced and then we study the form of a new effective Lagrangian
(originally proposed in Refs. \cite{EM1994a,EM1994b,EM1994c}) including,
in addition to the usual quark--antiquark meson fields $U_{ij}$, also the new
$2L$--fermion field $X$.
(This analysis was originally performed in Refs. \cite{EM1994a,EM1994b,EM1994c}
and is resumed here, in Sections 3 and 4, for the convenience of the reader.
See also Refs. \cite{EM2002a,EM2002b} for a recent review on these problems.)
In Section 5 (which, together with Section 6, contains the main original
results of this paper) we analyse the consequences of our theoretical model
on the slope of the topological susceptibility $\chi'$, in the {\it full}
theory with quarks, showing how this quantity is modified by the presence
of a new $U(1)$ chiral order parameter: we will find that $\chi'$
(in the chiral limit $\sup(m_i) \to 0$) acts as an order parameter
for the $U(1)$ axial symmetry above $T_{ch}$.
Further information on the new $U(1)$ chiral order parameter is derived in
Section 6 from the study (at zero temperature) of the radiative decays of the
``light'' pseudoscalar mesons in two photons: a comparison of our results
with the experimental data is performed.
Finally, the conclusions and an outlook are given in Section 7.

\newpage

\newsection{The phase transitions of QCD}

\noindent
One expects that, above a certain critical temperature, also the $U(1)$ axial
symmetry will be restored. We will try to see if this transition has (or has
not) anything to do with the usual chiral transition.
Let us define the following temperatures:
\begin{itemize}
\item{} $T_{ch}$: the temperature at which the chiral condensate
$\langle \bar{q} q \rangle$ goes to zero. The chiral symmetry
$SU(L) \otimes SU(L)$ is spontaneously broken below $T_{ch}$ and
it is restored above $T_{ch}$.
\item{} $T_{U(1)}$: the temperature at which the $U(1)$ axial symmetry
is (approximately) restored.
If $\langle \bar{q} q \rangle \ne 0$ also the $U(1)$ axial symmetry is
broken, i.e., the chiral condensate is an order parameter also for
the $U(1)$ axial symmetry. Therefore we must have: $T_{U(1)} \ge T_{ch}$.
\item{} $T_\chi$: the temperature at which the pure--gauge topological
susceptibility $A$ (approximately) drops to zero. Present lattice results
indicate that $T_\chi \ge T_{ch}$ \cite{Alles-et-al.97}.
Moreover, the Witten--Veneziano mechanism implies that $T_{U(1)} \ge T_\chi$.
\end{itemize}
The following scenario, that we will call ``\underline{SCENARIO 1}'',
in which $T_\chi < T_{ch}$, is, therefore, immediately ruled out.
In this case, in the range of temperatures between $T_\chi$ and $T_{ch}$
the $U(1)$ axial symmetry is still broken by the chiral condensate, but the
anomaly effects are absent. In other words, in this range of temperatures
the $U(1)$ axial symmetry is spontaneously broken ({\it \`a la} Goldstone)
and the $\eta'$ is the corresponding NG boson, i.e., it is massless in the
chiral limit $\sup(m_i) \to 0$, or, at least, as light as the pion $\pi$,
when including the quark masses.
This scenario was first discussed (and indeed really supported!) in
Ref. \cite{Pisarski-Wilczek84}.
It is known that the $U(1)$ chiral anomaly effects are related to
instantons \cite{tHooft76}. It is also known that at high temperature $T$
the instanton--induced amplitudes are suppressed by the so--called
``Pisarski--Yaffe suppression factor'' \cite{Gross-Pisarski-Yaffe81},
due to the Debye--type screening:
\be
d{\cal N}_{inst}(T) \sim d{\cal N}_{inst}(T=0) \cdot
\exp \left[ -\pi^2 \rho^2 T^2 \left( {2N_c + L \over 3} \right) \right] ,
\label{eqn4}
\ee
$\rho$ being the instanton radius.
The argument of Pisarski and Wilczek in Ref. \cite{Pisarski-Wilczek84} was
the following: {``If instantons themselves are the primary
chiral--symmetry--breaking mechanism, then it is very difficult to
imagine the unsuppressed $U(1)_A$ amplitude at $T_{ch}$.''}
So, what was wrong in their argument?
The problem is that Eq. (\ref{eqn4}) can be applied only in the quark--gluon
plasma phase, since the Debye screening is absent below $T_{ch}$.
Indeed, Eq. (\ref{eqn4}) is applicable only for $T \ge 2T_{ch}$ and
one finds instanton suppression by at least two orders of magnitude
at $T \simeq 2T_{ch}$ (see Ref. \cite{Shuryak94} and references therein).
Moreover, the qualitative picture of instanton--driven chiral symmetry
restoration which is nowadays accepted has significantly changed since
the days of Ref. \cite{Pisarski-Wilczek84}.
It is now believed (see Ref. \cite{Shuryak94} and references therein)
that the suppression of instantons is not the only way to ``kill'' the
$q\bar{q}$ chiral condensate. Not only the number of instantons is important,
but also their relative positions and orientations. Above $T_{ch}$,
instantons and anti-instantons can be rearranged into some finite
clusters with zero topological charge, such as well--formed
``instanton--anti-instanton molecules''.

Therefore, we are left essentially with the two following scenarios.\\
\underline{SCENARIO 2}: $T_{ch} \le T_{U(1)}$, with
$T_{ch} \sim T_\chi \sim T_{U(1)}$.
If $T_{ch} = T_\chi = T_{U(1)}$, then, in the case of $L=2$ light flavours,
the restored symmetry across the transition is $U(1)_A \otimes SU(2)_L \otimes
SU(2)_R \sim O(2) \otimes O(4)$, which may yield a first--order phase
transition (see, for example, Ref. \cite{Kharzeev-et-al.98}).\\
\underline{SCENARIO 3}: $T_{ch} \ll T_{U(1)}$, that is, the complete
$U(L)_L \otimes U(L)_R$ chiral symmetry is restored only well inside the
quark--gluon plasma domain.
In the case of $L=2$ light flavours, we then have at $T=T_{ch}$ the
restoration of $SU(2)_L \otimes SU(2)_R \sim O(4)$.
Therefore, we can have a second--order phase transition with the
$O(4)$ critical exponents. $L=2$ QCD at $T \simeq T_{ch}$ and the $O(4)$
spin system should belong to the same universality class.
An effective Lagrangian describing the softest modes is essentially
the Gell-Mann--Levy linear sigma model, the same as for the $O(4)$
spin systems (see Ref. \cite{Pisarski-Wilczek84}).
If this scenario is true, one should find the $O(4)$ critical indices
for the $q\bar{q}$ chiral condensate and the specific heat:
$\langle \bar{q} q \rangle \sim |(T - T_{ch})/T_{ch}|^{0.38 \pm 0.01}$,
and $C(T) \sim |(T - T_{ch})/T_{ch}|^{0.19 \pm 0.06}$.
Present lattice data partially support these results.

\newsection{The $U(1)$ chiral order parameter}

\noindent
We make the assumption (discussed in the previous sections) that the $U(1)$
chiral symmetry is broken independently from the $SU(L) \otimes SU(L)$
symmetry. The usual chiral order parameter $\langle \bar{q} q \rangle$
is an order parameter both for $SU(L) \otimes SU(L)$ and for $U(1)_A$:
when it is different from zero, $SU(L) \otimes SU(L)$ is broken down to
$SU(L)_V$ and also $U(1)_A$ is broken.
Thus we need another quantity which could be an order parameter only for
the $U(1)$ chiral symmetry \cite{EM1994a,EM1994b,EM1994c,EM1995a}.
The most simple quantity of this kind was found by 'tHooft
in Ref. \cite{tHooft76}.
For a theory with $L$ light quark flavours, it is a $2L$--fermion interaction
that has the chiral transformation properties of:
\be
{\cal L}_{eff} \sim \displaystyle{{\det_{st}}(\bar{q}_{sR}q_{tL})
+ {\det_{st}}(\bar{q}_{sL}q_{tR}) },
\label{eqn5}
\ee
where $s,t = 1, \ldots ,L$ are flavour indices, but the colour indices are
arranged in a more general way  (see Refs. \cite{EM1994c,EM1995a}).
It is easy to verify that ${\cal L}_{eff}$ is invariant under
$SU(L) \otimes SU(L) \otimes U(1)_V$, while it is not invariant under $U(1)_A$.
To obtain an order parameter for the $U(1)$ chiral symmetry, one can
simply take the vacuum expectation value of ${\cal L}_{eff}$:
$C_{U(1)} = \langle {\cal L}_{eff} \rangle$.
The arbitrarity in the arrangement of the colour indices can be removed if we
require that the new $U(1)$ chiral condensate is ``independent'' on the
usual chiral condensate $\langle \bar{q} q \rangle$, as explained in
Refs. \cite{EM1994c,EM1995a}. In other words, the condensate $C_{U(1)}$
is chosen to be a {\it genuine} $2L$--fermion condensate, with a zero
``disconnected part'', the latter being the contribution proportional
to $\langle \bar{q} q \rangle^L$,
corresponding to retaining the vacuum intermediate state in all the channels
and neglecting the contributions of all the other states.
As a remark, we observe that the condensate $C_{U(1)}$ so defined
turns out to be of order ${\cal O}(g^{2L - 2} N_c^L) = {\cal O}(N_c)$
in the large--$N_c$ expansion, exactly as the chiral condensate
$\langle \bar{q} q \rangle$.

The existence of a new $U(1)$ chiral order parameter has of course interesting
physical consequences, which can be revealed by analysing some relevant QCD
Ward Identities (WI's) (see Ref. \cite{EM1994b} and also Ref. \cite{EM2002a}).
In the case of the $SU(L) \otimes SU(L)$ chiral symmetry, one
immediately verifies that:
\be
\langle [Q_A^a(x_0), i\bar{q} \gamma_5 T^b q(y)]_{x_0 = y_0} \rangle
= -i \delta_{ab} {1 \over L} \langle \bar{q} q \rangle ,
\label{comm-su}
\ee
where $Q_A^a(x_0)$ is the charge operator for the $SU(L)$ chiral symmetry
(i.e., $Q_A^a(x_0) \equiv \int d^3 \vec{x}~ A^a_0 (\vec{x},x_0)$,
where $A^a_\mu = \bar{q}\gamma_\mu \gamma_5 T^a q$ is the $SU(L)$ axial
current). From Eq. (\ref{comm-su}) one derives the following WI:
\be
\int d^4 x \langle T\partial^\mu A^a_\mu (x)
i\bar{q} \gamma_5 T^b q(0) \rangle
= i \delta_{ab} {1 \over L} \langle \bar{q} q \rangle .
\label{eqn6}
\ee
If $\langle \bar{q} q \rangle \ne 0$ (in the chiral limit
$\sup(m_i) \to 0$), the anomaly--free WI (\ref{eqn6}) implies the existence
of $L^2-1$ non--singlet NG bosons, interpolated by the hermitian fields
$O_b = i \bar{q} \gamma_5 T^b q$.
Similarly, in the case of the $U(1)$ axial symmetry, one finds that:
\be
\langle [Q_5(x_0), i\bar{q} \gamma_5 q(y)]_{x_0 = y_0} \rangle
= -2i \langle \bar{q} q \rangle ,
\label{comm-u1}
\ee
where $Q_5(x_0)$ is the charge operator for the $U(1)$ chiral symmetry
(i.e., $Q_5(x_0) \equiv \int d^3 \vec{x}$ $J_{5, 0} (\vec{x},x_0)$,
where $J_{5, \mu}= {\bar{q} \gamma_\mu \gamma_5 q}$
is the usual $U(1)$ axial current).
From Eq. (\ref{comm-u1}) one derives the following WI:
\be
\int d^4x~ \langle T\partial^\mu J_{5, \mu}(x) i\bar{q} \gamma_5 q(0)
\rangle = 2i \langle \bar{q} q \rangle .
\label{eqn7}
\ee
But this is not the whole story! One also derives that:
\be
[Q_5(x_0) , O_P(y)]_{x_0 = y_0} =
-2Li \cdot {\cal L}_{eff}(y) ,
\label{EQN10.4}
\ee
where ${\cal L}_{eff} \sim {\det}(\bar{q}_{sR}q_{tL})
+ {\det}(\bar{q}_{sL}q_{tR})$ is the $2L$--fermion operator
discussed in the previous section and the hermitian field $O_P$ is so
defined:
\be
O_P \sim i[ \displaystyle{\det_{st}}(\bar{q}_{sR}q_{tL})
- \displaystyle{\det_{st}}(\bar{q}_{sL}q_{tR}) ].
\label{O_P}
\ee
From Eq. (\ref{EQN10.4}) one derives the following WI:
\be
\int d^4x~ \langle T\partial^\mu J_{5, \mu}(x) O_P(0) \rangle =
2Li \langle {\cal L}_{eff}(0) \rangle .
\label{EQN10.5}
\ee
If the $U(1)$ chiral symmetry is still broken above $T_{ch}$, i.e.,
$\langle {\cal L}_{eff}(0) \rangle \ne 0$ for $T > T_{ch}$
(while $\langle \bar{q} q \rangle = 0$ for $T > T_{ch}$), then this WI
implies the existence of a (pseudo--)Goldstone boson (in the large--$N_c$
limit!) coming from this breaking and interpolated by the hermitian
field $O_P$. Therefore, the $U(1)_A$ (pseudo--)NG boson (i.e.,
the $\eta'$) is an ``exotic'' $2L$--fermion state for $T > T_{ch}$!

\newsection{The new chiral effective Lagrangian}

\noindent
We shall see in this section how the proposed scenario, in which the $U(1)$
axial symmetry is still broken above the chiral transition, can be
consistently reproduced using an effective--Lagrangian model.
This analysis was originally performed in Refs. \cite{EM1994a,EM1994b,EM1994c}
and is resumed here for the convenience of the reader.

It is well known that the low--energy dynamics of the pseudoscalar mesons,
including the effects due to the anomaly and the $q\bar{q}$ chiral condensate,
and expanding to the first order in the light quark masses, can be described,
in the large--$N_c$ limit, by an effective Lagrangian
\cite{DiVecchia-Veneziano80,Witten80,Rosenzweig-et-al.80,Nath-Arnowitt81,Ohta80}
written in terms of the mesonic field $U_{ij} \sim \bar{q}_{jR} q_{iL}$
(up to a multiplicative constant) and the topological charge density $Q$.
If we make the assumption that the $U(1)$ chiral symmetry is restored at
a temperature $T_{U(1)}$ greater than $T_{ch}$, we need another order parameter
for the $U(1)$ chiral symmetry, the form of which has been
discussed in the previous section. We must now define a field variable $X$,
associated with this new condensate, to be inserted in the chiral Lagrangian.
The translation from the fundamental quark fields to the
effective--Lagrangian meson fields is done as follows. The operators
$i \bar{q} \gamma_5 q$ and $\bar{q} q$ entering in the WI (\ref{eqn7})
are essentially equal to (up to a multiplicative constant)
$i(\Tr U - \Tr U^\dagger)$ and $\Tr U + \Tr U^\dagger$ respectively.
Similarly, the operators
${\cal L}_{eff} \sim {\det}(\bar{q}_{sR}q_{tL})
+ {\det}(\bar{q}_{sL}q_{tR})$ and
$O_P \sim i[ {\det}(\bar{q}_{sR}q_{tL})
- {\det}(\bar{q}_{sL}q_{tR}) ]$
entering in the WI (\ref{EQN10.5}) can be put equal to (up to a multiplicative
constant) $X + X^\dagger$ and $i(X - X^\dagger)$ respectively, where
$X \sim {\det} \left( \bar{q}_{sR} q_{tL} \right)$
is the new field variable (up to a multiplicative constant),
related to the new $U(1)$ chiral condensate, which must be inserted
in the chiral effective Lagrangian.
It was shown in Refs. \cite{EM1994a,EM1994b,EM1994c} that the most simple
effective Lagrangian, constructed with the fields $U$, $X$ and $Q$, is:
\ba
\lefteqn{
{\cal L}(U,U^\dagger ,X,X^\dagger ,Q) =
{1 \over 2}\Tr(\partial_\mu U\partial^\mu U^\dagger )
+ {1 \over 2}\partial_\mu X\partial^\mu X^\dagger } \nonumber \\
& & -V(U,U^\dagger ,X,X^\dagger ) +{1 \over 2}iQ(x)\omega_1 \Tr(\ln U -
\ln U^\dagger) \nonumber \\
& & +{1 \over 2}iQ(x)(1-\omega_1)(\ln X-\ln X^\dagger)+{1 \over 2A}Q^2(x),
\label{eqn9}
\ea
where the potential term $V(U,U^{\dagger},X,X^{\dagger})$ has the form:
\ba
\lefteqn{
V(U,U^\dagger ,X,X^\dagger )={1 \over 4}\lambda_{\pi}^2 \Tr[(U^\dagger U
-\rho_\pi \cdot {\bf I})^2] +
{1 \over 4}\lambda_X^2 (X^\dagger X-\rho_X )^2 } \nonumber \\
& & -{B_m \over 2\sqrt{2}}\Tr(MU+M^\dagger U^\dagger)
-{c_1 \over 2\sqrt{2}}[\det(U)X^\dagger + \det(U^\dagger )X].
\label{eqn10}
\ea
{\bf I} is the identity matrix.
$M$ represents the quark mass matrix, $M={\rm diag}(m_1,\ldots,m_L)$,
which enters in the QCD Lagrangian as $\delta {\cal L}^{(mass)}_{QCD} =
-\bar{q}_R M q_L -\bar{q}_L M^\dagger q_R$,
and $A$ is the topological susceptibility in the pure--YM theory.
All the parameters appearing in the Lagrangian must be considered as
functions of the physical temperature $T$. In particular, the parameters
$\rho_{\pi}$ and $\rho_X$ are responsible for the behaviour of the theory
respectively across the $SU(L) \otimes SU(L)$
and the $U(1)$ chiral phase transitions, according to the following table:
$$
\vbox{\tabskip=0pt \offinterlineskip
\halign to320pt{\strut#
& \vrule#\tabskip=1em plus2em& \hfil#&\vrule#
& \hfil#&\vrule#
& \hfil#&\vrule#
& \hfil#&\vrule#
\tabskip=0pt\cr\noalign{\hrule}
&& \omit\hidewidth { } \hidewidth
&& \omit\hidewidth $T<T_{ch}$ \hidewidth
&& \omit\hidewidth $T_{ch}<T<T_{U(1)}$ \hidewidth
&& \omit\hidewidth $T>T_{U(1)}$ \hidewidth& \cr
\noalign{\hrule}
&& { } & & { } & & { } & & { } &\cr
&& $\rho_\pi$ & & ${1\over 2}F_\pi^2>0$ & & $-{1\over 2}B_\pi^2<0$ &
& $-{1\over 2}B_\pi^2<0$ &\cr
&& { } & & { } & & { } & & { } &
\cr\noalign{\hrule}
&& { } & & { } & & { } & & { } &\cr
&& $\rho_X$ & & ${1\over 2}F_X^2>0$ & & ${1\over 2}F_X^2>0$ &
& $-{1\over 2}B_X^2<0$ &\cr
&& { } & & { } & & { } & & { } &
\cr\noalign{\hrule}\noalign{\smallskip}& \multispan7 [Tab.1]
\hfil\cr}}
$$
[That is: $\rho_{\pi}(T_{ch})=0$ and $\rho_X(T_{U(1)})=0$.]
For $T<T_{ch}$, $\rho_\pi > 0$ and therefore, by virtue of the form
(\ref{eqn10}) of the potential,\footnote{As it has been stressed in Ref.
\cite{EM1994a} (see also Ref. \cite{DiVecchia-Veneziano80}), the {\it linear
$\sigma$--type} model (\ref{eqn9})--(\ref{eqn10}) contains {\it redundant}
scalar fields (we are only interested in the pseudoscalar NG, or
{\it pseudo}--NG, bosons!), which can be eliminated by taking the limit
$\lambda_\pi^2 \to +\infty$ and $\lambda_X^2 \to +\infty$ in Eq. (\ref{eqn10}):
so an expansion is performed not only in powers of the light quark masses
$m_i$, but also in powers of $1/\lambda_\pi^2$ and $1/\lambda_X^2$.}
one finds that $\langle U \rangle \ne 0$,
or, in other words $\langle \bar{q} q \rangle \ne 0$ (being
$U_{ij} \sim \bar{q}_{jR} q_{iL}$, up to a multiplicative constant):
i.e., the $SU(L) \otimes SU(L)$ chiral symmetry is broken.
Instead, for $T>T_{ch}$, $\rho_\pi < 0$ and then, always from
Eq. (\ref{eqn10}), one has that $\langle U \rangle = 0$, i.e.,
$\langle \bar{q} q \rangle = 0$.
The $U(1)$ chiral symmetry remains broken also in the region of
temperatures $T_{ch} < T < T_{U(1)}$, where, on the contrary, the
$SU(L) \otimes SU(L)$ chiral symmetry is restored.
In fact, for $T<T_{U(1)}$, $\rho_X > 0$ and therefore, from Eq. (\ref{eqn10}),
one finds that $\langle X \rangle \ne 0$, or, in other words, $C_{U(1)} =
\langle {\cal L}_{eff} \rangle \ne 0$ (being
$X \sim {\det} \left( \bar{q}_{sR} q_{tL} \right)$, up to a multiplicative
constant). The $U(1)$ chiral symmetry is restored above $T_{U(1)}$,
where $\rho_X < 0$ and then, from Eq. (\ref{eqn10}), $\langle X \rangle = 0$,
i.e., $C_{U(1)} = \langle {\cal L}_{eff} \rangle = 0$.
According to what we have said in the Introduction and in Section 2,
we also assume that the topological susceptibility $A(T)$ of the pure--YM
theory drops to zero at a temperature $T_{\chi}$ greater than $T_{ch}$
(but smaller than, or equal to, $T_{U(1)}$).

One can study the mass spectrum of the theory for $T < T_{ch}$ and
$T_{ch} < T < T_{U(1)}$. First of all, let us see what happens for
$T<T_{ch}$, where both the $SU(L) \otimes SU(L)$ and the $U(1)$ chiral
symmetry are broken. Integrating out the field variable $Q$ and taking only
the quadratic part of the Lagrangian, one finds that, in the chiral limit
$\sup(m_i) \to 0$, there are $L^2-1$ zero--mass states, which represent the
$L^2-1$ Goldstone bosons coming from the breaking of the $SU(L) \otimes SU(L)$
chiral symmetry down to $SU(L)_V$. Then there are two singlet eigenstates
with non--zero masses:
\ba
\eta' &=& {1 \over \sqrt{F_\pi^2 + LF_X^2}}(\sqrt{L}F_X S_X + F_\pi S_\pi),
\nonumber \\
\eta_X &=& {1 \over \sqrt{F_\pi^2 + LF_X^2}}(-F_\pi S_X + \sqrt{L}F_X S_\pi),
\label{eqn11}
\ea
where $S_\pi$ is the usual $SU(L)$--singlet meson field associated with $U$,
while $S_X$ is the meson field associated with $X$ (see Refs.
\cite{EM1994a,EM1994b,EM1994c} and Eqs. (\ref{u,x})--(\ref{Phi}) below).
The field $\eta'$ has a ``light'' mass, in the sense of the
$N_c \to \infty$ limit, being
\be
m^2_{\eta'} = {2LA \over F_\pi^2 + LF_X^2} = {\cal O}({1 \over N_c}).
\label{eqn12}
\ee
This mass is intimately related to the anomaly and they both vanish in the
$N_c \to \infty$ limit. On the contrary, the field $\eta_X$ has a sort of
``heavy hadronic'' mass of order ${\cal O}(1)$ in the large--$N_c$ limit.
We immediately see that, if we put $F_X=0$ in the above--written
formulae (i.e., if we neglect the new $U(1)$ chiral condensate), then
$\eta' = S_\pi$ and $m^2_{\eta'}$ reduces to ${2LA \over F^2_\pi}$, which is
the ``usual'' $\eta'$ mass in the chiral limit \cite{Witten79a,Veneziano79}.
Yet, in the general case $F_X \ne 0$, the two states which diagonalize the
squared mass matrix are linear combinations of the ``quark--antiquark''
singlet field $S_\pi$ and of the ``exotic'' field $S_X$.
Both the $\eta'$ and the $\eta_X$ have the same quantum numbers (spin,
parity and so on), but they have a different quark content: one is mostly
$\sim i(\bar{q}_{L}q_{R}-\bar{q}_{R}q_{L})$, while the other is mostly
$\sim i[ {\det}(\bar{q}_{sL}q_{tR}) - {\det}(\bar{q}_{sR}q_{tL}) ]$.
What happens when approaching the chiral transition temperature $T_{ch}$?
We know that $F_\pi(T) \to 0$ when $T \to T_{ch}$. From Eq. (\ref{eqn12})
we see that $m^2_{\eta'}(T_{ch}) = {2A \over F_X^2}$
and, from the first Eq. (\ref{eqn11}), $\eta'(T_{ch}) = S_X$.
We have continuity in the mass spectrum of the theory through the chiral
phase transition at $T=T_{ch}$.
In fact, if we study the mass spectrum of the theory in the region of
temperatures $T_{ch} < T < T_{U(1)}$ (where the $SU(L) \otimes SU(L)$ chiral
symmetry is restored, while the $U(1)$ chiral symmetry is still broken),
one finds that there is a singlet meson field $S_X$ (associated with the
field $X$ in the chiral Lagrangian) with a squared mass given by (in the
chiral limit): $m^2_{S_X} = {2A \over F_X^2}$.
This is nothing but the {\it would--be} Goldstone particle
coming from the breaking of the $U(1)$ chiral symmetry, i.e., the $\eta'$,
which, for $T>T_{ch}$, is a sort of ``exotic'' matter field of the form
$\sim i[ {\det}(\bar{q}_{sL}q_{tR})
- {\det}(\bar{q}_{sR}q_{tL}) ]$.
Its existence could be proved perhaps in the near
future by heavy--ions experiments.

\newsection{A relation between $\chi'$ and the new $U(1)$ chiral condensate}

\noindent
In this section and in the following one we want to describe some methods which
provide us with some information about the parameter $F_X$. This quantity
controls the $U(1)$ axial symmetry restoration: indeed (as we can see from
the table written in the previous section), for $T<T_{U(1)}$,
$\rho_X = {1 \over 2} F_X^2 > 0$ and therefore, from Eq. (\ref{eqn10}),
$\langle X \rangle = F_X/\sqrt{2} \ne 0$. Remembering that
$X \sim {\det} \left( \bar{q}_{sR} q_{tL} \right)$, up to a multiplicative
constant, we find that $F_X$ is proportional to the new $2L$--fermion
condensate $C_{U(1)} = \langle {\cal L}_{eff} \rangle$ introduced above.\\
In the same way, the pion decay constant $F_{\pi}$, which controls the breaking
of the $\sgru$ symmetry, is related to the $q\bar{q}$ chiral condensate
by a simple and well--known proportionality relation that we are going to
re--derive for the benefit of the reader.
Due to the fact that the derivative of the QCD Hamiltonian with respect to
$m_i$ is the operator $\bar{q}_i q_i$ (being $\delta {\cal L}_{QCD}^{(mass)} =
-\sum_{i=1}^L{m_i \bar{q}_i q_i}$), the corresponding derivative of the vacuum
energy represents the vacuum expectation value of $\bar{q}_i q_i$.
So, in our case, it must be:
\be
\langle \bar{q}_i q_i \rangle = {\partial \over \partial m_i}
\langle V(U,U^\dagger ,X,X^\dagger ) \rangle ,
\label{EQN8.8}
\ee
where $V(U,U^{\dagger},X,X^{\dagger})$ is the potential term written in
Eq. (\ref{eqn10}). It is immediate to calculate the expectation value of the
potential $V$ in the two regions $T<T_{ch}$ and $T_{ch} < T < T_{U(1)}$ (this
was originally done in Ref. \cite{EM1994a}).
In particular, in  the region $T<T_{ch}$ we can find that (at the leading
order in $m_i$, $1/\lambda_{\pi}^2$, $1/\lambda_X^2$):
\be
\langle \bar{q}_i q_i \rangle_{T<T_{ch}} \simeq -{1 \over 2}B_m F_\pi .
\label{EQN8.9}
\ee
In the case of $L$ light quarks, taken, for simplicity, with the same mass $m$,
one immediately derives from this equation the so--called
\emph{Gell-Mann--Oakes--Renner relation} \cite{GOR68}:
\be
\label{GOR}
m_{NS}^{2}F_{\pi}^{2}\simeq-\frac{2m}{L}\langle\bar{q}q\rangle_{T<T_{ch}},
\ee
where, as usual,
$\langle \bar{q}q \rangle \equiv \sum_{i=1}^L \langle \bar{q}_i q_i \rangle$,
and, moreover, $m^2_{NS} = m B_m/F_\pi$, $m_{NS}$ being the mass of the
non--singlet pseudoscalar mesons.
Eq. (\ref{GOR}) relates, on the left--hand side, the pion decay constant
$F_{\pi}$ and the mass $m_{NS}$ of the non--singlet mesons with, on the
right--hand side, the chiral condensate $\langle\bar{q}q\rangle$ and the
quark mass $m$.\\
Obviously it is not possible to find, in an analogous way, the relation between
$F_X$ and the new condensate $C_{U(1)} = \langle {\cal L}_{eff} \rangle$,
since the QCD Lagrangian does not contain any term proportional to the
$2L$--fermion operator ${\cal L}_{eff}$.

Alternatively, the quantity $F_{X}$ can be written in terms of a certain
two--point Green function of the topological charge--density operator
$Q(x)$ in the {\it full} theory with quarks.\\
If we want to derive the two--point function of $Q(x)$, we need to consider the
effective Lagrangian in the form (\ref{eqn9}), where the field variable $Q(x)$
has not yet been integrated. Therefore:
\be
\chi(k)\equiv-i\int d^{4}x\;e^{ikx}\langle{TQ(x)Q(0)}\rangle=
({\cal K}^{-1}(k))_{11},
\label{chik}
\ee
where ${\cal K}^{-1}(k)$ is the inverse of the matrix ${\cal K}(k)$ associated
with the quadratic part of the Lagrangian (\ref{eqn9}) in the momentum space,
for the ensemble of fields $(Q(x),\ldots )$.\\
In particular, for $T<T_{ch}$, one has to consider the following quadratic
Lagrangian, in the chiral limit $\sup(m_i) \to 0$:
\ba
\label{l2ch}
\lefteqn{\La_{2}=
\unme\sum_{a=1}^{L^2-1}\partial_{\mu}\pi_{a}\partial^{\mu}\pi_{a}+
\unme \partial_{\mu}S_{\pi}\partial^{\mu}S_{\pi}+
\unme \partial_{\mu}S_{X}\partial^{\mu}S_{X} }&&\nonumber\\
&&-\unme c\Bigg(\frac{\sqrt{2L}}{F_{\pi}}S_{\pi}
-\frac{\sqrt{2}}{F_{X}}S_{X}\Bigg)^{2} +\frac{1}{2A}Q^{2} \nonumber \\
&& -\omega_{1}\frac{\sqrt{2L}}{F_{\pi}}S_{\pi}Q
-(1-\omega_{1})\frac{\sqrt2}{F_{X}}S_{X}Q.
\ea
We have used for $U$ and $X$ the following exponential form, valid for
$T<T_{ch}$ \cite{EM1994a,EM1994b,EM1994c}:
\be
U= \frac{F_\pi}{\sqrt2}\exp\left( {i\sqrt{2}\over F_\pi}\Phi\right),~~~
X= \frac{F_X}{\sqrt2}\exp\left({i\sqrt{2}\over F_X} S_X\right),
\label{u,x}
\ee
with:
\be
\label{Phi}
\Phi = \displaystyle\sum_{a=1}^{L^2-1}
\pi_{a}\tau_{a}+\frac{S_{\pi}}{\sqrt L}\cdot\I ,
\ee
where the matrices $\tau_a$ ($a=1,\ldots,L^2-1$) are the
generators of the algebra of $SU(L)$ in the fundamental representation,
with normalization: $$\Tr(\tau_a) = 0\,,\;\;\Tr(\tau_a \tau_b) = \delta_{ab}.$$
(In other words, $\tau_a=\lambda_a/\sqrt2$, where the $\lambda_a$ are the
usual Gell-Mann matrices, with normalization
$\Tr(\lambda_a\lambda_b)=2\delta_{ab}$.)\\
As already pointed out in Section 4, $S_\pi$ is the usual ``quark--antiquark''
$SU(L)$--singlet meson field associated with $U$, while $S_X$ is the ``exotic''
$2L$--fermion meson field associated with $X$.
The $\pi_a$ ($a=1,\ldots,L^2-1$) are the self--hermitian fields describing
the $L^2-1$ pions: they are massless in the chiral limit $\sup(m_i) \to 0$.\\
From the quadratic part of the Lagrangian (\ref{l2ch}) in the momentum space,
we derive the following matrix ${\cal K}(k)$ for the ensemble of fields
$(Q,S_X,S_{\pi})$ [the contribution of the pion fields $\pi_a$ is simply
diagonal, diag($k^2,\ldots,k^2$), and therefore can be trivially factorized
out]:
\be
\label{matrice1}
{\cal K}(k)=
\left( \begin{array}{ccc}
\frac{1}{A} & -\frac{\sqrt{2}(1-\omega_{1})}{F_{X}} &
-\frac{\omega_1 \sqrt{2L}}{F_{\pi}} \\
 & & \\
-\frac{\sqrt{2}(1-\omega_{1})}{F_{X}} & k^{2}-\frac{2c}{F^{2}_{X}} &
\frac{2c\sqrt{L}}{F_{\pi}F_{X}} \\
 & & \\
-\frac{\omega_1 \sqrt{2L}}{F_{\pi}} & \frac{2c\sqrt{L}}{F_{\pi}F_{X}} &
k^{2}-\frac{2Lc}{F^{2}_{\pi}}
\end{array} \right).
\ee
The calculation of the right--hand side of Eq. (\ref{chik}) can then be
performed explicitly, using Eq. (\ref{matrice1}), obtaining:
\ba
\label{chisotto}
\chi(k)=\frac{A\Big[k^{4}-\frac{2c(\csti)}{\cstib}k^{2}\Big]}
{\Big[k^{4}-\frac{2c(\csti)}{\cstib}k^{2}\Big]
-2A\Big[\frac{(1-\omega_{1})^{2}F_{\pi}^{2}+L\omega_{1}^{2}F_{X}^{2}}
{\cstib}\Big]k^{2}
+\frac{4LAc}{\cstib}}.
\ea
One immediately sees that $\chi \equiv \chi(0)=0$, i.e., the topological
susceptibility in the full theory with quarks vanishes in the chiral limit
$\sup(m_i) \to 0$, as expected.\\
But the most interesting result is found when considering the so--called
``slope'' of the topological susceptibility, defined as:
\ba
\label{chiprimo}
\chi'\equiv\frac{1}{8}{\partial \over \partial k_\mu}
{\partial \over \partial k^\mu} \chi(k)\Bigg|_{k=0}=
\frac{i}{8}\int d^{4}x \;x^{2} \langle TQ(x)Q(0)\rangle.
\ea
Moreover, whenever $\chi(k)$ is a function of $k^2$ (such as in the theory
at $T=0$, by virtue of the Lorentz invariance), one can also write:
\ba
\label{chiprimo-bis}
\chi'=\frac{d}{dk^{2}}\chi(k)\Bigg|_{k=0}.
\ea
In the theory at finite temperature $T \ne 0$ the Lorentz invariance is
broken down to the $O(3)$ invariance under spatial rotations only,
so that $\chi(k)$ is, in general, a function of $\vec{k}^2$ and $k^0$.
However, the propagator $\chi(k)$ obtained in Eq. (\ref{chisotto}) is a
function of $k^2 = (k^0)^2 - \vec{k}^2$ also in the case of non--zero
temperature. Therefore, we can explicitly calculate the quantity $\chi'$
in our effective model in the chiral limit $\sup(m_i) \to 0$ (and we shall
call ``$\chi'_{ch}$'' its value), using Eqs. (\ref{chisotto}) and
(\ref{chiprimo-bis}), obtaining:
\ba
\label{chi'sotto}
\chi'_{ch}=-\frac{1}{2L}(\csti)\equiv-\frac{1}{2L}F_{\eta'}^2,
\ea
where $F_{\eta'}\equiv\sqrt{\csti}$ is the decay constant of the $\eta'$ (at
the leading order in the $1/N_c$ expansion), modified by the presence of the
new $U(1)$ chiral order parameter \cite{EM1994c}.

For the benefit of the reader, we here briefly repeat the arguments developed
in Ref. \cite{EM1994c}, leading to the result $F_{\eta'}\equiv\sqrt{\csti}$:
this will also provide us with an alternative derivation of
Eq. (\ref{chi'sotto}). We shall consider the $T=0$ case for simplicity.\\
It turns out that $\eta'$ is just the meson state, with a squared mass of order
$1/N_c$, whose contribution to the full topological susceptibility $\chi$
exactly cancels out (in the chiral limit of massless quarks!) the pure--gauge
part $A$ of $\chi$, so making $\chi = 0$: this is the so--called
Witten's mechanism. To see how this picture comes out in our theory, one first
determines the $U(1)$ axial current, starting from our effective Lagrangian.
This is easily done remembering how the fields $U$ and $X$ transform under a
$U(1)$ chiral transformation and one ends up with the following expression
\cite{EM1994c,EM2002a}:
\be
J_{5, \mu} = i[\Tr(U^\dagger \partial_\mu U - U \partial_\mu U^\dagger)
+ L(X^\dagger \partial_\mu X - X \partial_\mu X^\dagger)] .
\label{EQN5.15}
\ee
After having inserted here the expressions (\ref{u,x})--(\ref{Phi}) in place
of $U$ and $X$, the current $J_{5, \mu}$ takes the following form:
\be
J_{5, \mu} = -\sqrt{2L}\,\partial_\mu (F_\pi S_\pi + \sqrt{L} F_X S_X) .
\label{EQN5.16}
\ee
In the case in which $S_X = 0$ (i.e., if we only have the usual chiral
condensate $\langle \bar{q} q \rangle$), Eq. (\ref{EQN5.16}) reduces to
$J_{5, \mu} = -\sqrt{2L}\,F_S\,\partial_\mu S_\pi$,
where $F_S = F_\pi$ is the well--known expression \cite{Witten79a,Veneziano79}
for the ``quark--antiquark'' $SU(L)$--singlet ($S_\pi$) decay constant,
at the leading order in the $1/N_c$ expansion. Instead, in the general case
that we are considering, the first Eq. (\ref{eqn11}) allows us to write
Eq. (\ref{EQN5.16}) as:
\be
J_{5, \mu} = -\sqrt{2L} F_{\eta'} \partial_\mu \eta' ,
\label{EQN5.17}
\ee
with  $F_{\eta'}=\sqrt{\csti}$ [see Eq. (\ref{EQN5.18}) below].
This means that the $U(1)$ axial current $J_{5, \mu}$ only couples to the
``{\it light}'' field $\eta'$, and not to the ``{\it heavy}'' field $\eta_X$
(as we have already said ``{\it light}'' and ``{\it heavy}'' are in the sense
of $N_c$--order).
The relative coupling between $J_{5, \mu}$ and $\eta'$, i.e., the
$SU(L)$--singlet ($\eta'$) decay constant defined as
$\langle 0|J_{5, \mu}(0)|\eta'(p)\rangle = i\sqrt{2L}\,p_\mu\,F_{\eta'}$,
is now given by:
\be
F_{\eta'} = \sqrt{\csti} .
\label{EQN5.18}
\ee
Let us now recall the Witten's argument and write the two--point function (at
four--momentum $k$) of the topological charge density $Q(x)$ as a sum over
one--hadron poles, i.e., one--hadron intermediate states:
\be
\chi(k) = -i\int d^4x\, e^{ikx} \langle TQ(x)Q(0) \rangle = A_0(k)
+ \displaystyle\sum_{mesons}{|\langle 0|Q|n \rangle |^2 \over k^2 - m^2_n} ,
\label{EQN5.19}
\ee
where $A_0(k)$ is the pure Yang--Mills contribution from the glueball
intermediate states and it is the leading--order term in $1/N_c$ [being of
order $\ord(N_c^0)$].
In the chiral limit in which we have $L$ massless quarks, the full topological
susceptibility $\chi\equiv\chi(k=0)$ must vanish: so there must be a meson
state, with squared mass $m^2_n = {\cal O}(1/N_c)$ [since $A_0(0)=\ord(N_c^0)$,
while $|\langle 0|Q|n \rangle |^2 = {\cal O}(1/N_c)$], which exactly cancels
out $A_0(0)$. This is the meson that we usually call $\eta'$: the other meson
states in (\ref{EQN5.19}) have squared masses of order $\ord(N_c^0)$, so that
their contributions to the summation in (\ref{EQN5.19}) are suppressed by a
factor of $1/N_c$. Therefore we obtain that:
\be
{|\langle 0|Q|\eta' \rangle |^2 \over m^2_{\eta'}} = A ,
\label{EQN5.20}
\ee
where $A \equiv A_0(0)$ is the pure Yang--Mills topological susceptibility
in the large--$N_c$ limit.
It is well known that, in the chiral limit of $L$ massless quarks, the
topological charge density $Q(x)$ is directly related to the
four--divergence of the axial current $J_{5, \mu}$, {\it via} the anomaly
equation:
\be
\partial^\mu J_{5, \mu}(x) = 2L Q(x) ,
\label{EQN5.21}
\ee
so that $\langle 0|Q|\eta' \rangle = {1 \over \sqrt{2L}}m^2_{\eta'} F_{\eta'}$,
which can be substituted into Eq. (\ref{EQN5.20}) to give:
\be
A = {m^2_{\eta'} F^2_{\eta'} \over 2L} .
\label{EQN5.22}
\ee
This equation relates the mass $m_{\eta'}$ of the $\eta'$ state, its
decay constant $F_{\eta'}$ and the pure--gauge topological susceptibility $A$.
And in fact Eq. (\ref{EQN5.22}) is verified when putting for $m_{\eta'}$ and
$F_{\eta'}$ their values determined above in Eqs. (\ref{eqn12}) and
(\ref{EQN5.18}).\\
The dominance of the $\eta'$ state in the sum over the meson states at the
right--hand side of Eq. (\ref{EQN5.19}) can also be used to evaluate  the
slope of the topological susceptibility:
\ba
\label{chi'sp}
\chi'_{ch}=\frac{d}{dk^{2}}\chi(k)\Bigg|_{k=0}\simeq
- \frac{|\langle0|Q(0)|\eta'\rangle|^{2}}{m^{4}_{\eta'}}=
-\frac{1}{2L}\,F_{\eta'}^{2}=-\frac{1}{2L}\,(\csti),
\ea
thus recovering the result (\ref{chi'sotto}), derived above from our effective
Lagrangian. This is perfectly natural: using the effective Lagrangian at
tree--level (i.e., using its ``free'' propagators) one gets the results in the
one--hadron pole (i.e., one--hadron intermediate state) approximation!

Summarizing, we have found that the value of $\chi'$ in the chiral limit
$\sup(m_i)\to0$ is shifted from the ``original'' value $-\frac{1}{2L}F_{\pi}^2$
(derived in the absence of extra $U(1)$ chiral condensate)
to the value $-\frac{1}{2L}F_{\eta'}^2=-\frac{1}{2L}(\csti)$, which
also depends on the quantity $F_X$, proportional to the extra $U(1)$ chiral
condensate.\\
Therefore, a measure of this quantity $\chi'_{ch}$, e.g., in lattice gauge
theory, could provide an estimate for the $\eta'$ decay constant $F_{\eta'}$,
and, as a consequence, for $F_X$.
(At present, lattice determinations of $\chi'$ only exist for the pure--gauge
theory at $T=0$, with gauge group $SU(2)$ \cite{Briganti-et-al.91} and
$SU(3)$ \cite{EM1992b}.)

All the above refers to the theory at $T=0$ (or, more generally, for
$T<T_{ch}$).\\
When approaching the chiral transition at $T=T_{ch}$, one expects that
$F_{\pi}$ vanishes, while $F_X$ remains different from zero and the quantity
$\chi'_{ch}$ tends to the value:
\be
\chi'_{ch}\mathop{\longrightarrow}_{T\to T_{ch}}-\unme F_X^2.
\label{chi'Tch}
\ee
Indeed, the quantity $\chi(k)=-i\int d^{4}x\;e^{ikx}\langle{TQ(x)Q(0)}\rangle$
can also be evaluated in the region of temperatures $T_{ch}<T<T_{U(1)}$,
proceeding as for the case $T<T_{ch}$, obtaining the result (already derived in
Ref. \cite{EM1994a}):
\be
\chi(k)=A\frac{k^{2}}{k^{2}-\frac{2A}{F^2_X}},
\ee
in the chiral limit $\sup(m_i)\to0$.\\
Therefore, in the region of temperatures $T_{ch}<T<T_{U(1)}$, $\chi'_{ch}$
is given by:
\be
\label{chi'sopra}
\chi'_{ch}=\frac{d}{dk^{2}}\chi(k)\Bigg|_{k=0}=
-\frac{1}{2}\,F_X^{2},
\ee
consistently with the results (\ref{chi'sotto}) and (\ref{chi'Tch}) found
above: i.e., $\chi'_{ch}$ varies with continuity across $T_{ch}$.
This means that $\chi'_{ch}$ acts as a sort of order parameter for the $U(1)$
axial symmetry above $T_{ch}$: if $\chi'_{ch}$ is different from zero above
$T_{ch}$, this means that $F_X$ is different from zero and so the $U(1)$ axial
symmetry is broken.

Up to now we have just considered the case of $L$ massless quarks, but we can
also see what happens in the case of $L$ light quarks with non--zero masses.
For simplicity, we shall consider the case of $L$ degenerate quark flavours
with mass $m_1=\ldots=m_L \equiv m$. Let's calculate the first order
corrections to $\chi'_{ch}$ in an expansion in the quark mass $m$,
both for $T<T_{ch}$ and for $T_{ch}<T<T_{U(1)}$.\\
For $T<T_{ch}$, one has to consider the following quadratic Lagrangian
[obtained from Eq. (\ref{eqn9}) using the exponential forms for the fields $U$
and $X$, given in Eqs. (\ref{u,x})--(\ref{Phi}), and putting
$m_1=\ldots=m_L=m$]:
\ba
\lefteqn{\La_{2}=\unme \sum_{a=1}^{L^2-1}
\partial_{\mu}\pi_{a}\partial^{\mu}\pi_{a}+
\unme \partial_{\mu}S_{\pi}\partial^{\mu}S_{\pi}+
\unme \partial_{\mu}S_{X}\partial^{\mu}S_{X}
-\unme\mu^{2} \sum_{a=1}^{L^2-1} \pi_{a}^{2}-\unme\mu^{2}S_{\pi}^{2} }
\nonumber\\
&&-\unme c\Bigg(\frac{\sqrt{2L}}{F_{\pi}}S_{\pi}
-\frac{\sqrt{2}}{F_{X}}S_{X}\Bigg)^{2}+\frac{1}{2A}Q^{2}-
\frac{\omega_1 \sqrt{2L}}{F_{\pi}}S_{\pi}Q-
\frac{\sqrt{2}(1-\omega_{1})}{F_{X}}S_{X}Q,
\ea
where $\mu^{2}\equiv\frac{B_{m}}{F_{\pi}}m$ and
$c\equiv\frac{c_{1}}{\sqrt{2}}\Big(\frac{F_{X}}{\sqrt{2}}\Big)
\Big(\frac{F_{\pi}}{\sqrt{2}}\Big)^{L}$.
Consequently, we have the following matrix, associated with this quadratic
Lagrangian in the momentum space, for the ensemble of fields $(Q,S_X,S_{\pi})$:
\be
{\cal K}(k)=
\left( \begin{array}{ccc}
\frac{1}{A} & -\frac{\sqrt{2}(1-\omega_{1})}{F_{X}} &
-\frac{\omega_1 \sqrt{2L}}{F_{\pi}} \\
 & & \\
-\frac{\sqrt{2}(1-\omega_{1})}{F_{X}} & k^{2}-\frac{2c}{F^{2}_{X}} &
\frac{2c\sqrt{L}}{F_{\pi}F_{X}} \\
 & & \\
-\frac{\omega_1 \sqrt{2L}}{F_{\pi}} & \frac{2c\sqrt{L}}{F_{\pi}F_{X}} &
k^{2}-\frac{2Lc}{F^{2}_{\pi}}-\mu^{2}
\end{array} \right).
\ee
We can find $\chi(k)$, performing an explicit calculation of the
right--hand side of Eq. (\ref{chik}) and then calculate $\chi'$ using Eq.
(\ref{chiprimo-bis}). Performing an expansion in the parameter $\mu^2$ (or,
equivalently, in the quark mass $m$) up to the next--to--leading order, we
obtain:
\ba
\label{chi'mu}
\chi'&\simeq&-\frac{1}{2L}(\csti) \nonumber\\
&&+\frac{\cstib}{2Lc}\left\{(\omega_{1}-1)
+ {(F_\pi^2 + LF_X^2) [c + A(1-\omega_1)^2] \over L A F_X^2} \right\}\mu^{2}.
\ea
The first term in the right--hand side of Eq. (\ref{chi'mu}) represents
the leading--order term, which exactly equals $\chi'_{ch}$
[see Eq. (\ref{chi'sotto})].
The second term in the right--hand side of Eq. (\ref{chi'mu}) represents the
next--to--leading order and is linear in the parameter $\mu^2$, i.e., it is of
order $\ord(m)$. We see that this term depends not only on $F_X$, but also on
two other unknown parameters of our effective model ($\omega_1$ and $c$).\\
For $T_{ch}<T<T_{U(1)}$, $\chi'$ is the derivative with respect to $k^2$
of the quantity
\ba
\chi(k)=-i\int d^{4}x\;e^{ikx}\langle TQ(x)Q(0)\rangle=
A\frac{k^{2}-m_{0}^{2}}{k^{2}-m^{2}_{S_{X}}}
\ea
(already derived in Ref. \cite{EM1994a} in the case of $L$ light quarks with
different masses), where
$m_{0}^{2}=\frac{c_{1}}{F_{X}}\Big(\frac{2B_{m}}{\sqrt{2} \lambda^{2}_{\pi}
B^{2}_{\pi}}\Big)^{L}\det(M)$, $M$ being the quark--mass matrix,
and $m^{2}_{S_{X}}=m_{0}^{2}+\frac{2A}{F^{2}_{X}}$. Thus:
\ba
\chi'=A\Big(- \frac{1}{m^{2}_{S_{X}}}+\frac{m_{0}^{2}}{m^{4}_{S_{X}}}\Big).
\ea
Performing an expansion in the parameter $m_0^2$ we find:
\ba
\label{chi'msu}
\chi'\simeq-\unme F_{X}^{2}+\frac{F_{X}^{4}}{2A}m_{0}^{2}.
\ea
The first term in the right--hand side is just the value of $\chi'_{ch}$, for
the range of temperatures $T_{ch}<T<T_{U(1)}$, that we have previously found
[see Eq. (\ref{chi'sopra})].
The second term represents the next--to--leading order and
is of order $\ord(m^L)$ in the quark mass. Also this correction depends on
other unknown parameters of our model ($c_1,B_m$ and $\lambda_{\pi}$).\\
We conclude this section by observing that the same mass dependence as the
one shown by the corrections to $\chi'_{ch}$ for $T<T_{ch}$ and
$T_{ch}<T<T_{U(1)}$ is also shown by the topological susceptibility in the full
theory with $L$ light quarks with mass $m$ (see Ref. \cite{EM1994a}).

\newsection{Radiative decays of the pseudoscalar mesons}

\noindent
Further information on the quantity $F_X$ (i.e., on the new $U(1)$ chiral
condensate, to which it is related) can be derived from the study of the
radiative decays of the ``light'' pseudoscalar mesons in two photons.
These decays were also studied in Ref. \cite{DiVecchia-Veneziano-et-al.81},
using an effective Lagrangian model, in which only the $q\bar{q}$
chiral condensate was considered.
In this section we want to find the decay rates of the processes
$\pi^0,\eta,\eta',\eta_X\to\gamma\gamma$ (where $\eta_X$ represents the
pseudoscalar meson state, introduced in Section 4, having the same quantum
numbers of the $\eta'$, but a larger mass and a different quark content)
and see which are the effects due to the new $U(1)$ axial condensate, in the
realistic case of $L=3$ light quarks and in the simple case of zero
temperature ($T=0$).\\
To this purpose, we have to introduce the electromagnetic interaction in our
effective model.
First of all, in order to make the Lagrangian invariant under {\it local}
$U(1)$ electromagnetic transformations [$q \to q' = e^{i\theta e \qu}q$, in
terms of the quark fields; the matrix $\qu$ is defined in Eq. (\ref{caricael})
below], we have to replace the derivative of
the field $U$ with the corresponding {\it covariant} derivative $D_\mu U$,
which, by virtue of the transformation property of the field $U$
[$U \to U' = e^{i\theta e \qu} U e^{-i\theta e \qu}$], has the following form:
\be
D_{\mu}U=\partial_{\mu}U+ie A_{\mu}[\qu,U],
\ee
where $A_{\mu}$ is the electromagnetic field (which transforms as:
$A_\mu \to A'_\mu = A_\mu - \partial_\mu \theta$) and $\qu$ is the quark charge
matrix (in units of $e$, the absolute value of the electron charge):
\be
\label{caricael}
\qu=
\left( \begin{array}{ccc}
\frac{2}{3} & \\
& -\frac{1}{3} \\
& & -\frac{1}{3} \\
\end{array}\right).
\ee
Instead,the field $X$ is invariant under a $U(1)$ electromagnetic gauge
transformation and therefore its covariant derivative just coincides with
the ordinary four--derivative: $D_\mu X = \partial_\mu X$.\\
In addition, we have to reproduce the effects of the electromagnetic anomaly,
whose contribution to the four--divergence of the $U(1)$ axial current
($J_{5,\mu}=\bar{q}\gamma_{\mu}\gamma_{5}q$) and of the $SU(3)$ axial currents
($A^{a}_{\mu}=\bar{q}\gamma_{\mu}\gamma_{5}\frac{\tau_{a}}{\sqrt{2}}q$, where
the matrices $\tau_a$, with $a=1,\ldots,8$, are the generators of the algebra
of $SU(3)$ in the fundamental representation, already introduced in the
previous section) is given by:
\ba
(\partial^{\mu}J_{5,\mu})_{e.m.\;anomaly} &=& 2\Tr(\qu^{2}) G,
\nonumber \\
(\partial^{\mu}A^{a}_{\mu})_{e.m.\;anomaly} &=&
2\Tr\left( \qu^{2}\frac{\tau_{a}}{\sqrt{2}}\right) G,
\ea
where $G\equiv\frac{e^{2}N_{c}}{32\pi^{2}}\eps F_{\mu\nu}F_{\rho\sigma}$
($F_{\mu\nu}$ being the electromagnetic field--strength tensor), thus
breaking the corresponding chiral symmetries. We observe that
$\Tr(\qu^{2}\tau_{a}) \ne 0$ only for $a=3$ or $a=8$.\\
We must look for an interaction term ${\cal L}_I$ (constructed with the chiral
Lagrangian fields and the electromagnetic operator $G$) which, under a $U(1)$
axial transformation $q \to q' = e^{-i\alpha\gamma_5}q$, transforms as:
\be
U(1)_A:~~{\cal L}_I \to {\cal L}_I + 2\alpha \Tr(\qu^2)G,
\label{prop-u1}
\ee
while, under $SU(3)$ axial transformations of the type $q \to q' = e^{-i\beta
\gamma_5 \tau_a/\sqrt{2}}q$ (with $a = 3,8$), transforms as:
\be
SU(3)_A:~~{\cal L}_I \to {\cal L}_I + 2\beta \Tr\left( \qu^2
{\tau_a \over \sqrt{2}} \right) G.
\label{prop-su3}
\ee
The electromagnetic anomaly term for the field $U$, originally proposed in Ref.
\cite{DiVecchia-Veneziano-et-al.81}, has the following form:
\be
\unme i G\Tr[\qu^{2}(\ln U-\ln\Ucr)].
\ee
By virtue of the transformation property of the field $U$ under a $\gru$
chiral transformation ($q_L \to V_L q_L$, $q_R \to V_R q_R$ $\Rightarrow$
$U \to V_L U V_R^\dagger$, where $V_L$ and $V_R$ are arbitrary unitary
matrices \cite{EM1994a,EM2002a}), it is immediate to see that this term
satisfies both the transformation properties (\ref{prop-u1}) and
(\ref{prop-su3}).
If we also consider an analogous term for the field $X$, of the form:
\be
\frac{1}{2L}iG\Tr(\qu^{2})(\ln X-\ln\Xcr)
\ee
(where, in our case, $L=3$), then, by virtue of the transformation property
of the field $X$ under a $\gru$ chiral transformation ($X \to \det(V_L)
\det(V_R)^* X$ \cite{EM1994a,EM2002a}), one can see that this term satisfies
the transformation property (\ref{prop-u1}) under a $U(1)$ axial
transformation, while it is invariant under $SU(3)$ axial transformations.
Therefore, if we try to consider a linear combination of the two terms
(inspired by what was done for the $U(1)$ axial anomaly terms in Eq.
(\ref{eqn9}) \cite{EM1994a}), i.e.,
\ba
\lefteqn{\La_{I}=\unme i\omega_{2} G\Tr[\qu^{2}(\ln U-\ln\Ucr)] }\nonumber\\
&&+\frac{1}{2L}i(1-\omega_{2})G\Tr(\qu^{2})(\ln X-\ln\Xcr),
\ea
we immediately see that the property (\ref{prop-u1}) is satisfied for every
value of the parameter $\omega_2$, while the property (\ref{prop-su3}) is
satisfied only for $\omega_2 = 1$.\\
In conclusion, the electromagnetic anomaly interaction term is simply given by:
\be
\label{li}
\La_{I}=\unme iG\Tr[\qu^{2}(\ln U-\ln\Ucr)].
\ee
Therefore, we have to consider the following effective chiral Lagrangian,
which includes the electromagnetic interaction terms described above:
\ba
\label{lem}
\lefteqn{\La(U,\Ucr,X,\Xcr,Q,A^{\mu})=
\frac{1}{2}\Tr(D_{\mu}UD^{\mu}\Ucr)+
\frac{1}{2}\partial_{\mu}X\partial^{\mu}\Xcr }&& \nonumber\\
& & -V(U,\Ucr,X,\Xcr)+\frac{1}{2}iQ\;\omega_{1}\Tr(\ln U-\ln \Ucr) \nonumber\\
& & +\frac{1}{2}iQ(1-\omega_{1})(\ln X-\ln \Xcr)+\frac{1}{2A} Q^{2}
\nonumber\\
& & + \unme iG\Tr[\qu^{2}(\ln U-\ln\Ucr)]
-\frac{1}{4}F_{\mu\nu}F^{\mu\nu},
\ea
where the potential term $V(U,\Ucr,X,\Xcr)$ is the one written in Eq.
(\ref{eqn10}).

We can now describe the predictions of our model about the decay amplitudes and
rates of the processes $\pi^0,\eta,\eta',\eta_X\to\gamma\gamma$.
It is well known that the two final photons in the pseudoscalar--meson decays
are associated with the pseudoscalar operator $\eps F_{\mu\nu}F_{\rho\sigma}$:
so, e.g., the $\pi^0$ decay is reproduced by an interaction of the form
$\pi_3\,\eps F_{\mu\nu}F_{\rho\sigma}$. The covariant derivative does not
contain this term (since the diagonal matrix $\qu$ and the generator $\tau_3$
commute); instead, it is produced by the electromagnetic anomaly interaction
written in Eq.  (\ref{li}) (indeed the chiral symmetry is broken by the
anomaly!). The same arguments can be applied  to the other decays. Therefore,
the decay amplitude of the generic process ``$meson\to\gamma\gamma$''
is given by:
\be
A(meson\to\gamma\gamma)=
\langle\gamma\gamma|\La_{I}(0)|meson\rangle.
\ee
Substituting the exponential form (\ref{u,x})--(\ref{Phi}) (with $L=3$)
of the field $U$ (we remind the reader that we are considering the case of
zero temperature) into Eq. (\ref{li}), we have the following interaction
Lagrangian:
\be
\label{li1}
\La_{I}=-G\frac{1}{3F_{\pi}} \left( \pi_{3}+\frac{1}{\sqrt{3}}\pi_{8}+
\frac{2\sqrt{2}}{\sqrt{3}}S_{\pi} \right).
\ee
At first we will study the radiative decay amplitudes in the chiral limit
$\sup(m_i)\to0$; then we will also consider the effects of the quark masses;
at last we will compare our results with those derived in Ref.
\cite{DiVecchia-Veneziano-et-al.81} and with the experimental data.

In the chiral limit $\sup(m_i)\to0$, we have the mixing of the fields $S_{\pi}$
and $S_X$. Substituting  the fields $\pi_{3},\pi_{8},S_{\pi},S_{X}$ with the
physical ones, [i.e., $\pi^{0}$, $\eta$, respectively identified with the
fields $\pi_{3}$ and $\pi_{8}$, and $\eta'$, $\eta_{X}$, derived from
$S_{\pi}$ and $S_{X}$ using Eq. (\ref{eqn11})]
into Eq. (\ref{li1}), we obtain the following interaction Lagrangian:
\be
\label{li2}
\La_{I}=-G\frac{1}{3F_{\pi}}\Big\{\pi^{0}+\frac{1}{\sqrt{3}}\eta+
\frac{2\sqrt{2}}{\sqrt{3}}\frac{1}{\sqrt{F^{2}_{\pi}+3F_{X}^{2}}}
(F_{\pi}\eta'+\sqrt{3}F_{X}\eta_{X})\Big\}.
\ee
With simple calculations we find the following amplitudes:
\ba
\label{api0}
A(\pi^{0}\to\gamma\gamma) & = & \frac{e^{2}N_{c}}{12\pi^{2}F_{\pi}}I,\\
\label{aeta}
A(\eta\to\gamma\gamma) & = &\runte\cdot\frac{e^{2}N_{c}}{12\pi^{2}F_{\pi}}I,\\
\label{aeta'}
A(\eta'\to\gamma\gamma) & = & \frac{2\sqrt{2}}{\sqrt{3}}\cdot
\frac{e^{2}N_{c}}{12\pi^{2}\sqrt{F_{\pi}^{2}+3F_{X}^{2}}}I,\\
\label{aetax}
A(\eta_{X}\to\gamma\gamma) & = & \frac{2\sqrt{2}F_{X}}{F_{\pi}}\cdot
\frac{e^{2}N_{c}}{12\pi^{2}\sqrt{F_{\pi}^{2}+3F_{X}^{2}}}I,
\ea
where $I\equiv\varepsilon_{\mu\nu\rho\sigma}
k_{1}^{\mu}\epsilon_{1}^{\nu\ast}k_{2}^{\rho}\epsilon_{2}^{\sigma\ast}$
($k_{1}$, $k_{2}$ being the four--momenta of the two final photons and
$\epsilon_{1}$, $\epsilon_{2}$ their polarizations).
Then we can easily find the following decay rates (in the real case
$N_c=3$):
\ba
\label{gammap}
\Gamma(\pi^{0}\to\gamma\gamma) & = &
\frac{\alpha^{2}m_{\pi}^{3}}{64\pi^{3}F_{\pi}^{2}},\\
\label{gammae}
\Gamma(\eta\to\gamma\gamma) & = &
\frac{\alpha^{2}m_{\eta}^{3}}{192\pi^{3}F_{\pi}^{2}},\\
\label{gammae'}
\Gamma(\eta'\to\gamma\gamma) & = &
\frac{\alpha^{2}m_{\eta'}^{3}}{24\pi^{3}(F_{\pi}^{2}+3F_{X}^{2})},\\
\label{gammaex}
\Gamma(\eta_{X}\to\gamma\gamma)& = &\frac{\alpha^{2}m_{\eta_{X}}^{3}F_{X}^{2}}
{8\pi^{3}(F_{\pi}^{2}+3F_{X}^{2})F_{\pi}^{2}},
\ea
where $\alpha=e^{2}/4\pi \simeq 1/137$ is the fine--structure constant.
Obviously, the decay rates for $\pi^0$ and $\eta$ are equal to zero in the
chiral limit $\sup(m_i)\to0$, being the masses of the two particles equal to
zero in this case.

We want now to generalize the previous results to the case of $L=3$ quarks with
masses different from zero. In this case one has to consider the following
quadratic Lagrangian, which can be obtained substituting the exponential
expressions (\ref{u,x}) of the fields $U$ and $X$ into Eq. (\ref{eqn9}):
\ba
\label{l3}
\lefteqn{\La_{2}=\unme\Tr(\partial_{\mu}\Phi\partial^{\mu}\Phi)+
\unme\partial_{\mu}S_{X}\partial^{\mu}S_{X}-\frac{B_{m}}{2F_{\pi}}
\Tr(M\Phi^{2}) }
&&\nonumber\\
&&-c\Big(\frac{1}{F_{\pi}}\Tr\Phi-\frac{1}{F_{X}}S_{X}\Big)^{2}-
A\Big(\frac{\omega_{1}}{F_{\pi}}\Tr\Phi+
\frac{1-\omega_{1}}{F_{x}}S_{X}\Big)^{2},
\ea
where $c\equiv\frac{c_{1}}{\sqrt{2}}\Big(\frac{F_{X}}{\sqrt{2}}\Big)
\Big(\frac{F_{\pi}}{\sqrt{2}}\Big)^3$.\\
Substituting Eq. (\ref{Phi}) (with $L=3$) into Eq. (\ref{l3}), one immediately
sees that the fields $\pi_{1},\pi_{2},\pi_{4},\pi_{5},\pi_{6},\pi_{7}$ are
diagonal, while the fields  $\pi_{3},\pi_{8},S_{\pi},S_{X}$ mix together.
However, neglecting the experimentally small mass difference between the
quarks \emph{up} and \emph{down} (i.e., neglecting the experimentally small
violations of the $SU(2)$ isotopic spin), also $\pi_3$ becomes diagonal and can
be identified with the physical state $\pi^0$. The other physical states can be
found by the diagonalization of the following squared mass matrix, written for
the ensemble of fields $(\pi_{8},S_{\pi},S_{X})$ (originally derived in Ref.
\cite{EM1994c}):
\be
\label{matrice3}
{\cal K}=
\left( \begin{array}{ccc}
\frac{2B}{3}(\tilde{m}+2m_s)&\frac{2B\sqrt{2}}{3}(\tilde{m}-m_s)&0\\
&&\\
\frac{2B\sqrt{2}}{3}(\tilde{m}-m_s)&\frac{6(A\omega_{1}^{2}+c)}{F_{\pi}^{2}}+
\tilde{m}_{0}^{2}&\frac{2\sqrt{3}[A\omega_{1}(1-\omega_{1})-c]}{F_{\pi}F_{X}}\\
&&\\
0&\frac{2\sqrt{3}[A\omega_{1}(1-\omega_{1})-c]}{F_{\pi}F_{X}}&
\frac{2[A(1-\omega_{1})^{2}+c]}{F^{2}_{X}}
\end{array} \right),
\ee
where:
\be
B \equiv \frac{B_{m}}{2F_{\pi}}, ~~~
\tilde{m} \equiv \frac{m_u+m_d}{2}, ~~~
\tilde{m}_{0}^{2} \equiv \frac{2}{3}B(2\tilde{m}+m_s).
\ee
The fields $(\pi_8,S_\pi,S_X)$ can be written in terms of the eigenstates
$\eta$, $\eta'$, $\eta_{X}$ as follows:
\be
\pmatrix{ \pi_8 \cr S_\pi \cr S_X } = \mathbf{C}
\pmatrix{ \eta \cr \eta' \cr \eta_X },
\label{diag}
\ee
where $\mathbf{C}$ is the following $3\times3$ orthogonal matrix:
\be
\label{cambio}
\mathbf{C}=
\left( \begin{array}{ccc}
\alpha_{1} & \alpha_{2} & \alpha_{3}\\
\beta_{1} & \beta_{2} & \beta_{3}\\
\gamma_{1} & \gamma_{2} & \gamma_{3}
\end{array} \right)=
\left(\begin{array}{ccc}
\cos\tilde{\varphi} & -\sin\tilde{\varphi} & 0\\
&&\\
\sin\tilde{\varphi}\,\frac{F_{\pi}}{\sqrt{\cstit}} &
\cos\tilde{\varphi}\,\frac{F_{\pi}}{\sqrt{\cstit}} &
\frac{\sqrt{3}F_{X}}{\sqrt{\cstit}}\\
&&\\
\sin\tilde{\varphi}\,\frac{\sqrt{3}F_{X}}{\sqrt{\cstit}}  &
\cos\tilde{\varphi}\,\frac{\sqrt{3}F_{X}}{\sqrt{\cstit}} &
-\frac{F_{\pi}}{\sqrt{\cstit}}
\end{array} \right).
\ee
Here $\tilde{\varphi}$ is a mixing angle, which can be related to the masses
of the quarks \emph{up, down, strange} (and therefore to the masses of the
octet mesons) by the following relation:
\ba
\label{phitilde}
\tan\tilde{\varphi}=\frac{\sqrt2}{9A}BF_{\pi}\sqrt{\cstit}(m_s-\tilde{m})=
\frac{F_{\pi}\sqrt{\cstit}}{6\sqrt{2}A}(m_{\eta}^{2}-m_{\pi}^{2}),
\ea
where $m^{2}_{\pi}=2B\tilde{m}$ and
$m_{\eta}^{2}=\frac{2}{3}B(\tilde{m}+2m_s)$.
The masses $m_{\eta}$, $m_{\eta'}$, $m_{\eta_{X}}$ are given by the squared
root of the eigenvalues of the squared mass matrix written in Eq.
(\ref{matrice3}). In particular we have:
\be
m_{\eta'}^{2}=\frac{6A}{\cstit}+\frac{F_{\pi}^{2}}{\cstit}\cdot\frac{2}{3}
B(2\tilde{m}+m_s),
\ee
from which, using the expression of $m^2_{\eta}$ written above and the relation
$m_{K}^{2}=B(\tilde{m}+m_s)$, one can derive the following generalization of
the Witten--Veneziano formula \cite{EM1994c}:\footnote{Using the experimental
values of the quantities $m_K$, $m_\eta$, $m_{\eta'}$ and $F_{\pi}$ and the
value of the pure--YM topological susceptibility $A$, which, from lattice
simulations, is $A=(180\pm5~\rm{MeV})^{4}$ \cite{Teper88,APE90,Alles-et-al.97},
this equation provides us with an upper limit on $F_X$:
$|F_{X}|\lesssim20\;\rm{MeV}$ \cite{EM1994c}.}
\ba
\label{ewvg}
\Big(1+\frac{3F_{X}^{2}}{F_{\pi}^{2}}\Big)m_{\eta'}^{2}+m_{\eta}^{2}-2m_{K}^{2}
=\frac{6A}{F_{\pi}^{2}}.
\ea
The mixing of the fields $\pi_{8}$, $S_{\pi}$, $S_{X}$ modifies the interaction
vertices. We can find them substituting into Eq. (\ref{li1}) the following
relations [which come from the change of basis (\ref{diag})--(\ref{cambio})]:
\ba
\label{cambio1}
\pi_{8}=\alpha_{1}\,\eta+\alpha_{2}\,\eta'+\alpha_{3}\,\eta_{X},\\
S_{\pi}=\beta_{1}\,\eta+\beta_{2}\,\eta'+\beta_{3}\,\eta_{X}.
\ea
As a result:
\be
\label{limasse}
\La_{I}
\equiv-G\frac{1}{3F_{\pi}}\Big(\pi^{0}+a_{1}\,\eta+a_{2}\,\eta'+
a_{3}\,\eta_{X}\Big),
\ee
where $a_{i}=\frac{1}{\sqrt3}(\alpha_{i}+2\sqrt2\beta_{i})$
$({\rm for}~i=1,2,3)$, so that:
\ba
\label{aibis}
a_{1}&=&\sqrt{\frac{1}{3}}\Bigg(\cos\tilde{\varphi}+
2\sqrt2\sin\tilde{\varphi}\,\frac{F_{\pi}}{{\sqrt{\cstit}}}\Bigg),\\
a_{2}&=&\sqrt{\frac{1}{3}}\Bigg(2\sqrt2\cos\tilde{\varphi}\,
\frac{F_{\pi}}{{\sqrt{\cstit}}}-\sin\tilde{\varphi}\Bigg),\\
a_{3}&=&2\sqrt2\Bigg(\frac{F_{X}}{\sqrt{\cstit}}\Bigg).
\ea
Obviously, the decay amplitude and rate of $\pi^0$ are still
given by Eqs. (\ref{api0}) and (\ref{gammap}), where, of course, now
$m_\pi \ne 0$. With little effort we can see that the other amplitudes
have the following expressions:
\ba
\label{aetam}
A(\eta\to\gamma\gamma)
&=&\frac{e^{2}N_{c}}{12\pi^{2}F_{\pi}}\sqrt{\frac{1}{3}}
\Big(\cos\tilde{\varphi}+
2\sqrt2\sin\tilde{\varphi}\,\frac{F_{\pi}}{{\sqrt{\cstit}}}\Big)I,\\
\label{aeta'm}
A(\eta'\to\gamma\gamma)
&=&\frac{e^{2}N_{c}}{12\pi^{2}F_{\pi}}\sqrt{\frac{1}{3}}
\Big(2\sqrt2\cos\tilde{\varphi}\,\frac{F_{\pi}}{{\sqrt{\cstit}}}-
\sin\tilde{\varphi}\Big)I,\\
\label{aetaxm}
A(\eta_{X}\to\gamma\gamma)
&=&\frac{e^{2}N_{c}}{12\pi^{2}F_{\pi}}
2\sqrt2\Big(\frac{F_{X}}{\sqrt{\cstit}}\Big)I.
\ea
Consequently (for $N_c=3$) we find the following decay rates:
\ba
\label{gammaem}
\Gamma(\eta\to\gamma\gamma)&=&
\frac{\alpha^{2}m_{\eta}^{3}}{192\pi^{3}F_{\pi}^{2}}\Big(\cos\tilde{\varphi}+
2\sqrt2\sin\tilde{\varphi}\,\frac{F_{\pi}}{{\sqrt{\cstit}}}\Big)^{2},\\
\label{gammae'm}
\Gamma(\eta'\to\gamma\gamma)&=&
\frac{\alpha^{2}m_{\eta'}^{3}}{192\pi^{3}F_{\pi}^{2}}
\Big(2\sqrt2\cos\tilde{\varphi}\,
\frac{F_{\pi}}{{\sqrt{\cstit}}}-\sin\tilde{\varphi}\Big)^{2},\\
\label{gammaexm}
\Gamma(\eta_{X}\to\gamma\gamma)&=&
\frac{\alpha^{2}m_{\eta_{X}}^{3}}{8\pi^{3}F_{\pi}^{2}}
\Big(\frac{F_{X}}{\sqrt{\cstit}}\Big)^{2}.
\ea
In the chiral limit $\sup(m_i)\to0$ the mixing angle $\tilde{\varphi}$
vanishes, by virtue of Eq. (\ref{phitilde}), and we find again the
relations (\ref{aeta})$\div$(\ref{aetax}) and
(\ref{gammae})$\div$(\ref{gammaex}).

Let's compare the expressions (\ref{api0}) and
(\ref{aetam})$\div$(\ref{aetaxm}) written above with the corresponding
amplitudes found in Ref. \cite{DiVecchia-Veneziano-et-al.81} using a Lagrangian
which includes only the usual $q\bar{q}$ chiral condensate
(so $F_X=0$ and there is no field $\eta_X$!).
For the convenience of the reader, we write them down:
\ba
\label{apdv}
A(\pi^{0}\to\gamma\gamma) & = & \frac{e^{2}N_{c}}{12\pi^{2}F_{\pi}}I,\\
\label{aedv}
A(\eta\to\gamma\gamma) & = & \frac{e^{2}N_{c}}{12\pi^{2}F_{\pi}}\runte
(\cos\varphi+2\sqrt{2}\sin\varphi)I,\\
\label{ae'dv}
A(\eta'\to\gamma\gamma) & = & \frac{e^{2}N_{c}}{12\pi^{2}F_{\pi}}\runte
(2\sqrt{2}\cos\varphi-\sin\varphi)I.
\ea
These amplitudes were derived considering non--zero quark masses (although
neglecting the small mass difference between the quark \emph{up} and
\emph{down}, so that there is only mixing between the fields $\pi_8$ and
$S_{\pi}$).\\
The mixing angle $\varphi$ in Eqs. (\ref{aedv})--(\ref{ae'dv}) is defined
by:
\ba
\pi_{8}&=&\eta\cos\varphi-\eta'\sin\varphi,\nonumber\\
S_{\pi}&=&\eta\sin\varphi+\eta'\cos\varphi,
\ea
and is related to the quark masses (i.e., to the masses of the octet mesons)
by the following relation:
\be
\label{phi}
\tan\varphi\simeq\frac{\sqrt2}{9A}BF_{\pi}^{2}(m_s-\tilde{m})=
\frac{F_{\pi}^{2}}{6\sqrt{2}A}(m_{\eta}^{2}-m_{\pi}^{2}).
\ee
One can immediately see that the introduction of the new condensate modifies
the decay rates of $\eta$ and $\eta'$ (moreover, we also have to consider the
particle $\eta_X$). In particular, it modifies the $\eta'$ decay constant,
already in the chiral limit $\sup(m_i) \to 0$.
Indeed, in this limit, Eq. (\ref{ae'dv}) (with $\varphi=0$) differs from
the corresponding Eq. (\ref{aeta'}) by the substitution of the pion decay
constant $F_{\pi}$ with the quantity:
\be
\label{fe'3}
F_{\eta'}\equiv\sqrt{\cstit},
\ee
which can be identified with the $\eta'$ decay constant (as it has been
explained in the previous section for the general case of $L$ light flavours).

In conclusion, a study of the radiative decays $\eta\to\gamma\gamma$,
$\eta'\to\gamma\gamma$ and a comparison with the experimental data can
provide us with further information about the parameter $F_X$ and the new
exotic condensate.
For example, from Eqs. (\ref{gammaem}) and (\ref{gammae'm}), and using the
experimental values for the various quantities which there appear, i.e.,
\ba
& & F_\pi = 92.4(4) ~{\rm MeV},
\nonumber \\
& & m_\eta = 547.30(12) ~{\rm MeV},
\nonumber \\
& & m_{\eta'} = 957.78(14) ~{\rm MeV},
\nonumber \\
& & \Gamma(\eta\to\gamma\gamma) = 0.46(4) ~{\rm KeV},
\nonumber \\
& & \Gamma(\eta'\to\gamma\gamma) = 4.26(19) ~{\rm KeV},
\ea
we can extract the following values for the quantity $F_X$ and for the
mixing angle $\tilde\varphi$:
\be
F_X = 27(9) ~{\rm MeV},~~~ \tilde\varphi = 16(3)^0.
\ee
The value of $F_X$ is not far from the upper limit derived from the
generalization of the Witten--Veneziano formula, $|F_{X}|\lesssim20\;\rm{MeV}$
\cite {EM1994c}. Moreover, the values of $F_X$ and $\tilde\varphi$ so
found are perfectly consistent with the relation (\ref{phitilde}) for the
mixing angle.
We thus see that $F_X$ is small when compared with $F_{\pi}$. In particular,
the corrections due to the presence of the new $U(1)$ condensate are
proportional to the ratio $F_{X}^{2}/F_{\pi}^{2}$ [see Eqs. (\ref{phitilde}),
(\ref{ewvg}) and (\ref{gammaem})$\div$(\ref{gammaexm})] and they are of the
order of some $\%$!

\newsection{Conclusions}

\noindent
In this paper we have tried to gain a physical
insight into the breaking mechanism of the $U(1)$ axial symmetry, through a
study of the behaviour of the theory at finite temperature.
As discussed in the Introduction, the topological susceptibility $A$ of the
pure Yang--Mills theory is a fundamental quantity for studying the $U(1)$
chiral symmetry, both at zero and non--zero temperature.
From some previous works of Witten \cite{Witten79a} and Veneziano
\cite{Veneziano79,Veneziano80}, it is known that a value for $A$ different
from zero implies, at large number $N_c$ of colours, the breaking of the
$U(1)$ axial symmetry, since it implies the existence of a
pseudo--Goldstone particle with the same quantum numbers of the $\eta'$.
Recent lattice results \cite{Alles-et-al.97} show that the YM topological
susceptibility $A(T)$ is approximately constant up to the critical temperature
$T_{ch}$, it has a sharp decrease above the transition, but it remains
different from zero up to $\sim 1.2~T_{ch}$.
Also present lattice data for the so--called ``chiral susceptibilities''
\cite{Bernard-et-al.97,Karsch00,Vranas00} seem to indicate that the $U(1)$
axial symmetry is still broken above $T_{ch}$, up to $\sim 1.2~T_{ch}$.\\
In the following, we briefly summarize the main points that we have discussed
(see also Refs. \cite{EM2002a,EM2002b} for a recent review on these problems)
and the new results that we have obtained.

One expects that, above a certain critical temperature $T_{U(1)}$, also the
$U(1)$ axial symmetry will be (approximately) restored. We have tried to see
if this transition has (or has not) anything to do with the usual
$SU(L) \otimes SU(L)$ chiral transition: various possible scenarios
have been discussed in Section 2.

We have proposed a scenario (supported by the above--mentioned lattice
results) in which the $U(1)$ axial symmetry is still broken above the chiral
transition and the pure--YM topological susceptibility vanishes at a
temperature $T_\chi$ between $T_{ch}$ and $T_{U(1)}$.
A new order parameter must be introduced for the $U(1)$ axial symmetry
\cite{EM1994a,EM1994b,EM1994c,EM1995a}: it is discussed in Section 3.

This scenario can be consistently reproduced using an effective Lagrangian
model \cite{EM1994a,EM1994b,EM1994c}.
In Section 4 we have discussed the effects that one should observe on the mass
spectrum of the theory, both below and above $T_{ch}$. In particular, the
Witten--Veneziano formula for the mass of the $\eta'$ is modified by the
presence of the new $2L$--fermion condensate. In this scenario,
the $\eta'$ survives across the chiral transition at $T_{ch}$ in the form of
an ``exotic'' $2L$--fermion state
$\sim i[ {\det}(\bar{q}_{sL}q_{tR})$ $-{\det}(\bar{q}_{sR}q_{tL})]$.
This particle is nothing but the {\it would--be} Goldstone boson coming from
the breaking of the $U(1)$ axial symmetry.
For $T>T_{ch}$, it acquires a ``topological'' squared
mass of the form $2A / F_X^2$, where $F_X$ is essentially the magnitude of
the new order parameter for the $U(1)$ chiral symmetry alone.
This scenario could perhaps be verified in the near future by heavy--ions
experiments, by analysing the pseudoscalar--meson spectrum in the singlet
sector.

In Section 5 (which, together with Section 6, contains the main original
results of this paper) we have analysed the consequences of our theoretical
model on the slope of the topological susceptibility $\chi'$, in the
{\it full} theory with quarks, showing how this quantity is modified by
the presence of a new $U(1)$ chiral order parameter: we have found that
$\chi'$ (in the chiral limit $\sup(m_i) \to 0$) acts as an order parameter
for the $U(1)$ axial symmetry above $T_{ch}$.
This prediction of our model could be tested in the near--future Monte
Carlo simulations on the lattice.

Finally, in Section 6, we have found that the existence of the new $U(1)$
chiral condensate can be directly investigated by studying (at $T=0$) the
radiative decays of the pseudoscalar mesons $\eta$ and $\eta'$ in two photons.
A first comparison of our results with the experimental data has been
performed at the end of Section 6: the results are encouraging, pointing
towards a certain evidence of a non--zero $U(1)$ axial condensate (i.e.,
$F_X \ne 0$). However, one should keep in mind that our results have been
derived from a very simplified model, obtained doing a first--order expansion
in $1/N_c$ and in the quark masses. We expect that such a model can furnish
only qualitative or, at most, ``semi--quantitative'' predictions.
Higher--order terms in $1/N_c$ could give rise to corrections (in the
``real world'' with $N_c=3$) of the same order of magnitude of (or even
larger than!) those induced by the new $U(1)$ axial condensate (having
the form $\sim F_X^2/F_\pi^2$).
Further studies are therefore necessary in order to continue this analysis
from a more quantitative point of view.

\vfill\eject

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\vfill\eject

\end{document}


