\section*{Acknowledgements}

The author is extremely grateful to D. Somensi for his help during the
initial stage of the work.

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{\Large\bf \thesection Appendix A: The $U(1)$ chiral order parameter}
\end{flushleft}

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\noindent
We make the assumption (discussed in the Introduction)
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 $SU(L)_V$ (``$V$'' stands for
{\it ``vectorial''}) and also $U(1)_A$ is broken.
In fact, under a $U(1)$ chiral transformation with parameter $\alpha$
(as usual, $q_L \equiv \frac{1}{2}(1 + \gamma_5)q$ and
$q_R \equiv \frac{1}{2}(1 - \gamma_5)q$, with
$\gamma_5 \equiv -i \gamma^0 \gamma^1 \gamma^2 \gamma^3$, denote respectively
the {\it left--handed} and the {\it right--handed} quark fields):
\begin{equation
U(1)_A: \quad q \to e^{-i\alpha\gamma_5}q, \quad {\rm i.e.}, \quad
q_L \to e^{-i\alpha}q_L, \quad q_R \to e^{i\alpha}q_R ,
\label{trasfu1}
\end{equation}
the chiral condensate would transform as (assuming the $U(1)_A$ symmetry to
be realized {\it \`a la} Wigner--Weyl):
\begin{equation
U(1)_A: \quad \langle \bar{q} q\rangle \rightarrow e^{2i \alpha}
\langle \bar{q}_L q_R\rangle + e^{-2i \alpha}\langle \bar{q}_R q_L\rangle.
\end{equation}
By taking $\alpha = \pi/2$, we would obtain
$\langle \bar{q} q\rangle \rightarrow -\langle \bar{q} q\rangle$:
therefore, if the chiral condensate is different
from zero, the $U(1)_A$ symmetry cannot be realized {\it \`a la} Wigner--Weyl.
Thus we need another quantity which could be an order parameter only for
the $U(1)$ chiral symmetry \cite{EM1994a,EM1994b,EM1994c,EM1995}.
The most simple quantity of this kind was introduced by Kobayashi and Maskawa
in 1970 \cite{KM1970}, as an additional effective vertex in a generalized
Nambu--Jona--Lasinio model, and it was later derived by 'tHooft in 1976
\cite{tHooft1976}, as an instanton--induced quark interaction.
(See also Ref. \cite{Kunihiro2009} for an historical review on this subject.)\\
For a theory with $L$ light quark flavours (of mass $m_i \ll \Lambda_{QCD}$;
$i=1,\ldots ,L$), it is a $2L$--fermion interaction that has the chiral
transformation properties of:
\begin{equation
{\cal O}_{U(1)}^{(L)} \sim \displaystyle {\det_{st}} \left[ \bar{q}_s
\left( \frac{1+\gamma_5}{2} \right) q_t \right] + {\rm H.c.} =
\displaystyle {\det_{st}} \left( \bar{q}_{sR} q_{tL} \right) +
\displaystyle {\det_{st}} \left( \bar{q}_{sL} q_{tR} \right) ,
\end{equation}
where $s,t = 1, \ldots ,L$ are flavour indices, but the colour indices are
arranged in a more general way (see below).
Since under chiral $U(L) \otimes U(L)$ transformations the quark fields
transform as follows:
\begin{equation
U(L) \otimes U(L):~~~ q_L \to V_L q_L ~~~,~~~q_R \to V_R q_R,
\end{equation}
where $V_L$ and $V_R$ are arbitrary $L \times L$ unitary matrices,
we immediately derive the transformation property of
${\cal O}_{U(1)}^{(L)}$ under $U(L) \otimes U(L)$:
\begin{equation
U(L) \otimes U(L):~~~ {\cal O}_{U(1)}^{(L)} \to \det(V_L)\det(V_R)^*
\displaystyle {\det_{st}} \left( \bar{q}_{sR} q_{tL} \right) + {\rm H.c.}
\end{equation}
This just means that ${\cal O}_{U(1)}^{(L)}$ is invariant under
$SU(L) \otimes SU(L) \otimes U(1)_V$, while it is not invariant under the
$U(1)_A$ transformation \eqref{trasfu1}:
\begin{equation
U(1)_A:~~~ {\cal O}_{U(1)}^{(L)} \to e^{-i 2L \alpha}
\displaystyle {\det_{st}} \left( \bar{q}_{sR} q_{tL} \right) + {\rm H.c.}
\end{equation}

\subsection{The $U(1)$ chiral condensate for $L=2$ }

\noindent
As an example let us consider the most simple case, that is $L=2$, but with
a general colour group $SU(N)$.
It is not hard to find (using the Fierz relations both for the spinorial
matrices and the $SU(N)$ generators in their fundamental representation)
that the most general colour--singlet, Hermitian and $P$--invariant
{\it local} quantity (without derivatives)
which has the required chiral transformation properties is just the following
four--fermion local operator:
\begin{equation
{\cal O}_{U(1)}^{(L=2)}(\alpha_0 ,\beta_0) =
F_{bd}^{ac}(\alpha_0 ,\beta_0) \epsilon^{st}
\left( \bar{q}_{1R}^{a}q_{sL}^{b} \cdot
\bar{q}_{2R}^{c}q_{tL}^{d} +
\bar{q}_{1L}^{a}q_{sR}^{b} \cdot
\bar{q}_{2L}^{c}q_{tR}^{d} \right) ,
\label{opl2}
\end{equation}
where the colour tensor $F_{bd}^{ac}(\alpha_0 ,\beta_0)$ is given by:
\begin{equation
F_{bd}^{ac}(\alpha_0 ,\beta_0) =
\alpha_0 \delta_b^a \delta_d^c + \beta_0 \delta_d^a \delta_b^c,
\label{Facbd}
\end{equation}
$\alpha_0$ and $\beta_0$ being arbitrary real parameters.
In Eq. (\ref{opl2}), $a,b,c,d \in \{1, \ldots ,N\}$ are colour indices;
$s,t \in \{1,2\}$ are flavour indices and $\epsilon^{st}=-\epsilon^{ts}$,
$\epsilon^{12}=1$. Dirac indices are contracted between the first and the
second fermion field and also between the third and the fourth one.
Note that if we choose $\alpha_0 = N$ and $\beta_0 = -1$,
${\cal O}_{U(1)}^{(L=2)}(\alpha_0 ,\beta_0)$
just becomes (up to a proportionality constant) the effective Lagrangian
for two flavours of quarks in an instanton background, found by 'tHooft
in \cite{tHooft1976} .

Now, to obtain an order parameter for the $U(1)$ chiral symmetry, one can
simply take the vacuum expectation value of
${\cal O}_{U1)}^{(L=2)}(\alpha_0 ,\beta_0)$:
\begin{equation
C_{U(1)}^{(L=2)}(\alpha_0 ,\beta_0) \equiv
\langle {\cal O}_{U(1)}^{(L=2)}(\alpha_0 ,\beta_0) \rangle .
\label{cl2}
\end{equation}
The arbitrarity in the choice of $\alpha_0$ and $\beta_0$ (indeed of only
one of them, since only their ratio is relevant) can be removed if we
require that the new $U(1)$ chiral condensate is ``independent'', in a
sense which will be explained below, of the usual chiral condensate
$\langle \bar{q} q \rangle$.
As it was pointed out by Shifman, Vainshtein and Zakharov in
\cite{SVZ1979}, a matrix element of the form
$\langle \bar{q} \Gamma_1 q \cdot \bar{q} \Gamma_2 q \rangle$
has, in general, a contribution proportional to the square
of the vacuum expectation value (v.e.v.) of $\bar{q} q$. This
contribution corresponds to retaining the vacuum intermediate state in all
the channels and neglecting the contributions of all the other states; we
call this contribution the {\it ``disconnected part''} of the original matrix
element:
\begin{equation
\langle \bar{q} \Gamma_1 q \cdot \bar{q} \Gamma_2 q \rangle_{disc}
= \frac{1}{G^2} \left[ ({\rm Tr}\Gamma_1 \cdot {\rm Tr}\Gamma_2) -
{\rm Tr}(\Gamma_1 \Gamma_2) \right] \langle \bar{q} q \rangle^2,
\end{equation}
where the normalization factor $G$ is defined as
($\bar{q} q = \sum_A \bar{q}_A q_A$):
\begin{equation
\langle \bar{q}_A q_B \rangle =
\frac{\delta_{AB}}{G} \langle \bar{q} q \rangle, \quad {\rm i.e.}, \quad
G = \delta_{AA},
\label{norm1}
\end{equation}
and the subscripts $A,B$ are collective indices which include spin, colour and
flavour; therefore, $G = 4 \times L \times N$ for a general $L$, and $G=8N$
for $L=2$.
When considering the operator ${\cal O}_{U(1)}^{(L=2)}(\alpha_0 ,\beta_0)$
defined in Eqs. (\ref{opl2}) and (\ref{Facbd}), we find the following
expression for its {\it disconnected} part:
\begin{equation
\langle {\cal O}^{(L=2)}_{U(1)}(\alpha_0 ,\beta_0) \rangle_{disc} =
\frac{1}{16N} [N (2\alpha_0 + \beta_0) + (\alpha_0 + 2\beta_0)]
\langle \bar{q} q \rangle^2 ,
\end{equation}
where: $\langle \bar{q} q \rangle = \langle \bar{u} u \rangle
+ \langle \bar{d} d \rangle$.
From this last equation we immediately see that the {\it disconnected}
part of the condensate $C_{U(1)}^{(L=2)}(\alpha_0 ,\beta_0)$ vanishes
with the following particular choice of the coefficients $\alpha_0$ and
$\beta_0$ (only their ratio is really relevant):
\begin{equation
\frac{\beta_0}{\alpha_0} = -\frac{2N + 1}{N + 2}.
\label{alphabeta}
\end{equation}
In other words, the condensate (\ref{cl2}) with $\alpha_0$ and $\beta_0$
satisfying the constraint (\ref{alphabeta}) does not take contributions
from the usual chiral condensate $\langle \bar{q} q \rangle$.
To summarize, a good choice for a $U(1)$ chiral condensate which is really
``independent'' of the usual chiral condensate $\langle \bar{q} q \rangle$
is the following one (apart from an irrelevant multiplicative constant):
\begin{equation
C_{U(1)}^{(L=2)} =
\langle ( \delta_b^a \delta_d^c - \frac{2N + 1}{N + 2}
\delta_d^a \delta_b^c )
\epsilon^{st}
\left( \bar{q}_{1R}^{a}q_{sL}^{b} \cdot
\bar{q}_{2R}^{c}q_{tL}^{d} +
\bar{q}_{1L}^{a}q_{sR}^{b} \cdot
\bar{q}_{2L}^{c}q_{tR}^{d} \right) \rangle .
\label{condl2}
\end{equation}
As a remark, we observe that the condensate $C_{U(1)}^{(L=2)}$ so defined
turns out to be of order ${\cal O}(g^2 N^2) = {\cal O}(N)$ in the
large--$N$ expansion (this result was also derived in Ref. \cite{EM1994b} by
simply requiring that the $1/N$ expansion of the relevant QCD Ward Identities
remains well defined when including this new condensate).
In the case of physical interest, i.e., $N =3$, the condensate
Eq. (\ref{condl2}) becomes:
\begin{equation
C_{U(1)}^{(L=2)} =
\langle ( \delta_b^a \delta_d^c - \frac{7}{5} \delta_d^a \delta_b^c )
\epsilon^{st} \left( \bar{q}_{1R}^{a}q_{sL}^{b} \cdot
\bar{q}_{2R}^{c}q_{tL}^{d} + \bar{q}_{1L}^{a}q_{sR}^{b} \cdot
\bar{q}_{2L}^{c}q_{tR}^{d} \right) \rangle .
\end{equation}

\subsection{The $U(1)$ chiral condensate for $L=3$}

\noindent
So far we have considered the most simple case $L=2$. However, this
procedure can be easily generalized to every $L$, and we can take as
an order parameter for the $U(1)$ chiral symmetry:
\begin{equation
C^{(L)}_{U(1)} = \langle {\cal O}_{U(1)}^{(L)} \rangle .
\end{equation}
As we have done in the case $L=2$, the colour indices may be arranged in such
a way that the $U(1)$ chiral condensate does not take contributions from the
usual chiral condensate $\langle \bar{q} q \rangle$: as a consenquence of
this, the new condensate will be of order ${\cal O}(g^{2L - 2} N^L) =
{\cal O}(N)$ in the large--$N$ expansion \cite{EM1994b,EM1994c}.\\
In the {\it real--world} case there are $L=3$ light flavours, $u$, $d$, and
$s$, with masses $m_u$, $m_d$, and $m_s$ which are small compared to the QCD
mass--scale $\Lambda_{QCD}$.
Proceeding as in the case $L=2$ [see Eq. \eqref{opl2}], one reduces to
consider the following general colour--singlet, Hermitian and $P$--invariant
{\it local} six--fermion operator (without derivatives):
\begin{equation
{\cal O}_{U(1)}^{(L=3)} = F^{a_1 a_2 a_3}_{b_1 b_2 b_3} \epsilon^{l_1 l_2 l_3}
\ \bar{q}^{\ a_1}_{1R} \ q^{b_1}_{l_1 L} \cdot
\bar{q}^{\ a_2}_{2R} \ q^{b_2}_{l_2 L} \cdot
\bar{q}^{\ a_3}_{3R} \ q^{b_3}_{l_3 L} + {\rm H.c.},
\label{O6}
\end{equation}
where $a_1,a_2,a_3,b_1,b_2,b_3 \in \{1,2, \cdots N\}$ are colour indices,
$l_1,l_2,l_3 \in \{1,2,3\}$ are \emph{flavour} indices and the colour tensor
$F^{a_1 a_2 a_3}_{b_1 b_2 b_3}$ is given by:
\begin{eqnarray
F^{a_1 a_2 a_3}_{b_1 b_2 b_3} &=&
\alpha_1 \delta^{a_1}_{b_1}\delta^{a_2}_{b_2} \delta^{a_3}_{b_3} +
\alpha_2 \delta^{a_1}_{b_2}\delta^{a_2}_{b_3} \delta^{a_3}_{b_1} +
\alpha_3 \delta^{a_1}_{b_3}\delta^{a_2}_{b_1} \delta^{a_3}_{b_2} \nonumber \\
&+& \beta_1 \delta^{a_1}_{b_2}\delta^{a_2}_{b_1} \delta^{a_3}_{b_3} +
\beta_2 \delta^{a_1}_{b_1}\delta^{a_2}_{b_3} \delta^{a_3}_{b_2} +
\beta_3 \delta^{a_1}_{b_3}\delta^{a_2}_{b_2} \delta^{a_3}_{b_1} ,
\label{F}
\end{eqnarray}
with $\alpha_1,\alpha_2,\alpha_3,\beta_1,\beta_2,\beta_3$ real
parameters. However, differently from the case $L=2$, the operator
${\cal O}_{U(1)}^{(L=3)}$ in Eqs. \eqref{O6}--\eqref{F}, with {\it arbitrary}
real parameters $\alpha_1,\alpha_2,\alpha_3,\beta_1,\beta_2,\beta_3$, is
{\it not}, in general, invariant under a $SU(3) \otimes SU(3)$ chiral
transformation:
\begin{equation
SU(3) \otimes SU(3):~~~ q_L \to U_L q_L ~~~,~~~q_R \to U_R q_R
\end{equation}
$(\det U_L = \det U_R = 1)$. Invariance under $SU(3) \otimes SU(3)$ is,
instead, recovered provided that the colour tensor
$F^{a_1 a_2 a_3}_{b_1 b_2 b_3}$ satisfies the following {\it symmetry property}:
\begin{equation
F^{a_1 a_2 a_3}_{b_1 b_2 b_3} = F^{a_i a_j a_k}_{b_i b_j b_k},
\quad \forall \text{ permutations } \{i,j,k\} \text{ of } \{1,2,3\} .
\label{fprop}
\end{equation}
In fact, in this case it is easy to see that the operator \eqref{O6} can
be re--written in the following form:
\begin{equation
{\cal O}_{U(1)}^{(L=3)} = F^{a_1 a_2 a_3}_{b_1 b_2 b_3}
\frac{1}{3!} \epsilon^{r_1 r_2 r_3} \epsilon^{l_1 l_2 l_3} \
\bar{q}^{\ a_1}_{r_1 R} \ q^{b_1}_{{l_1 L}} \cdot
\bar{q}^{\ a_2}_{r_2 R} \ q^{b_2}_{{l_2 L}} \cdot
\bar{q}^{\ a_3}_{r_3 R} \ q^{b_3}_{{l_3 L}} + {\rm H.c.},
\label{opl3}
\end{equation}
which is manifestly invariant under $SU(3) \otimes SU(3)$:
\begin{eqnarray
{\cal O}_{U(1)}^{(L=3)} &\rightarrow&
F^{a_1 a_2 a_3}_{b_1 b_2 b_3}
\frac{1}{3!} \epsilon^{r_1 r_2 r_3} \epsilon^{l_1 l_2 l_3}
\bar{q}^{\ a_1}_{s_1 R} (U_R^{\dagger})_{s_1 r_1} (U_L)_{l_1 m_1} \
q^{b_1}_{m_1 L} \nonumber \\
&\cdot& \bar{q}^{\ a_2}_{s_2 R} (U^{\dagger}_R)_{s_2 r_2} (U_L)_{l_2 m_2} \
q^{b_2}_{m_2 L} \cdot \bar{q}^{\ a_3}_{s_3 R} (U^{\dagger})_{s_3 r_3}
(U_L)_{l_3 m_3} \ q^{b_3}_{m_3 L} + {\rm H.c.} \nonumber \\
&=& F^{a_1 a_2 a_3}_{b_1 b_2 b_3} \frac{1}{3!}
\det(U_R^\dagger) \epsilon^{s_1 s_2 s_3}
\det(U_L) \epsilon^{m_1 m_2 m_3} \nonumber \\
&\times& \bar{q}^{\ a_1}_{s_1 R} \ q^{b_1}_{{m_1 L}} \cdot
\bar{q}^{\ a_2}_{s_2 R} \ q^{b_2}_{{m_2 L}} \cdot
\bar{q}^{\ a_3}_{s_3 R} \ q^{b_3}_{{m_3 L}} + {\rm H.c.} \nonumber \\
&=& F^{a_1 a_2 a_3}_{b_1 b_2 b_3} \frac{1}{3!}
\epsilon^{s_1 s_2 s_3} \epsilon^{m_1 m_2 m_3}
\bar{q}^{\ a_1}_{s_1 R} \ q^{b_1}_{{m_1 L}} \cdot
\bar{q}^{\ a_2}_{s_2 R} \ q^{b_2}_{{m_2 L}} \cdot
\bar{q}^{\ a_3}_{s_3 R} \ q^{b_3}_{{m_3 L}} + {\rm H.c.} \nonumber \\
&=& {\cal O}_{U(1)}^{(L=3)} .
\label{su3inv}
\end{eqnarray}
(Or, equivalently, one can start from the expression \eqref{opl3} of the
six--fermion local operator, which, on the basis of \eqref{su3inv}, is
invariant under $SU(3) \otimes SU(3)$ for {\it every} choice of the colour
tensor $F^{a_1 a_2 a_3}_{b_1 b_2 b_3}$: but one immediately recognizes that
only the {\it symmetric} part of the colour tensor, satisfying the relation
\eqref{fprop}, contributes to the r.h.s. of \eqref{opl3}, the
{\it anti--symmetric} parts being trivially cancelled out.
Note that, in the case $L=2$, the most general colour tensor \eqref{Facbd}
automatically satifies the {\it symmetry property},
$F^{ac}_{bd} = F^{ca}_{db}$.)\\
The symmetry property \eqref{fprop} imposes the following constraints on
the parameters of the colour tensor \eqref{F}:
\begin{equation
\alpha_3 = \alpha_2, \qquad \beta_3 = \beta_2 = \beta_1 .
\label{vincoli1}
\end{equation}
The colour tensor has, therefore, the following form:
\begin{eqnarray
F^{a_1 a_2 a_3}_{b_1 b_2 b_3} &=&
\alpha_1 \delta^{a_1}_{b_1} \delta^{a_2}_{b_2} \delta^{a_3}_{b_3} +
\alpha_2 (\delta^{a_1}_{b_2} \delta^{a_2}_{b_3} \delta^{a_3}_{b_1} +
\delta^{a_1}_{b_3} \delta^{a_2}_{b_1} \delta^{a_3}_{b_2}) \nonumber \\
&+& \beta_1 (\delta^{a_1}_{b_2} \delta^{a_2}_{b_1} \delta^{a_3}_{b_3} +
\delta^{a_1}_{b_1} \delta^{a_2}_{b_3} \delta^{a_3}_{b_2} +
\delta^{a_1}_{b_3} \delta^{a_2}_{b_2} \delta^{a_3}_{b_1}) ,
\label{F3}
\end{eqnarray}
in terms of three arbitrary real parameters $\alpha_1$, $\alpha_2$, $\beta_1$.\\
Let us now evaluate the vacuum expectation value of the operator
${\cal O}_{U(1)}^{(L=3)}$:
\begin{eqnarray
C^{(L=3)}_{U(1)} &\equiv& \langle {\cal O}_{U(1)}^{(L=3)} \rangle =
F^{a_1 a_2 a_3}_{b_1 b_2 b_3} \epsilon^{l_1 l_2 l_3}
\langle \bar{q} \Gamma_1 q \cdot \bar{q} \Gamma_2 q \cdot
\bar{q} \Gamma_3 q \rangle + {\rm c.c.} \nonumber \\
&=& F^{a_1 a_2 a_3}_{b_1 b_2 b_3} \epsilon^{l_1 l_2 l_3}
(\Gamma_1)_{AB} (\Gamma_2)_{CD} (\Gamma_3)_{EF}
\langle \bar{q}_A q_B \bar{q}_C q_D \bar{q}_E q_F \rangle
+ {\rm c.c.},
\label{C3}
\end{eqnarray}
where [see Eq. \eqref{O6}]:
\begin{eqnarray
(\Gamma_1)_{AB} = (\Gamma_1)_{i_1 j_1, m_1 n_1}^{c_1 d_1} &=&
\left( \frac{1 + \gamma_5}{2} \right)_{i_1 j_1}
\otimes (\delta_{m_1 1}\delta_{n_1 l_1}) \otimes (\delta^{c_1 a_1}
\delta^{d_1 b_1}), \nonumber \\
(\Gamma_2)_{CD} = (\Gamma_2)_{i_2 j_2, m_2 n_2}^{c_2 d_2} &=&
\left( \frac{1 + \gamma_5}{2} \right)_{i_2 j_2}
\otimes (\delta_{m_2 2}\delta_{n_2 l_2}) \otimes (\delta^{c_2 a_2}
\delta^{d_2 b_2}), \nonumber \\
(\Gamma_3)_{EF} = (\Gamma_3)_{i_3 j_3, m_3 n_3}^{c_3 d_3} &=&
\left( \frac{1 + \gamma_5}{2} \right)_{i_3 j_3}
\otimes (\delta_{m_3 3}\delta_{n_3 l_3}) \otimes (\delta^{c_3 a_3}
\delta^{d_3 b_3}),
\label{gamma123}
\end{eqnarray}
where $i,j$ are Dirac indices, $m,n$ are flavour indices, and $c,d$ are
colour indices.\\
As in the case $L=2$ treated above, we can write the vacuum expectation value
of the operator ${\cal O}_{U(1)}^{(L=3)}$ as the sum of a
\emph{connected part}, which does \emph{not} depend on the chiral condensate
$\langle \bar{q}q \rangle$, and a \emph{disconnected part}, which instead
contains the chiral condensate $\langle \bar{q}q \rangle$, i.e.,
$C_{U(1)}^{(L=3)} = \langle {\cal O}_{U(1)}^{(L=3)} \rangle _{conn} +
\langle {\cal O}_{U(1)}^{(L=3)} \rangle _{disc}$, where:
\begin{equation
\langle {\cal O}_{U(1)}^{(L=3)} \rangle _{disc} =
F^{a_1 a_2 a_3}_{b_1 b_2 b_3}\epsilon^{l_1 l_2 l_3}
(\Gamma_1)_{AB} (\Gamma_2)_{CD} (\Gamma_3)_{EF}
\langle \bar{q}_A q_B \bar{q}_C q_D \bar{q}_E q_F \rangle_{disc} + {\rm c.c.},
\end{equation}
and the \emph{disconnected part} of the v.e.v. of the six--fermion operator
has the following form:
\begin{eqnarray
\lefteqn{
\langle \bar{q}_A q_B \bar{q}_C q_D \bar{q}_E q_F \rangle_{disc} }
\nonumber \\
& & = \langle \bar{q}_A q_B \rangle
\langle \bar{q}_C q_D \bar{q}_E q_F \rangle_{conn} +
\langle \bar{q}_C q_D \rangle
\langle \bar{q}_A q_B \bar{q}_E q_F \rangle_{conn} +
\langle \bar{q}_E q_F \rangle
\langle \bar{q}_A q_B \bar{q}_C q_D \rangle_{conn} \nonumber \\
& & - \langle \bar{q}_A q_D \rangle
\langle \bar{q}_C q_B \bar{q}_E q_F \rangle_{conn} -
\langle \bar{q}_A q_F \rangle
\langle \bar{q}_C q_D \bar{q}_E q_B \rangle_{conn} -
\langle \bar{q}_C q_B \rangle
\langle \bar{q}_A q_D \bar{q}_E q_F \rangle_{conn} \nonumber\\
& & - \langle \bar{q}_C q_F \rangle
\langle \bar{q}_A q_B \bar{q}_E q_D \rangle_{conn} -
\langle \bar{q}_E q_B \rangle
\langle \bar{q}_A q_F \bar{q}_C q_D \rangle_{conn} -
\langle \bar{q}_E q_D \rangle
\langle \bar{q}_A q_B \bar{q}_C q_F \rangle_{conn} \nonumber\\
& & + \langle \bar{q}_A q_B \rangle \langle \bar{q}_C q_D \rangle
\langle \bar{q}_E q_F \rangle -
\langle \bar{q}_A q_B \rangle \langle \bar{q}_C q_F \rangle
\langle \bar{q}_E q_D \rangle -
\langle \bar{q}_A q_D \rangle \langle \bar{q}_C q_B \rangle
\langle \bar{q}_E q_F \rangle \nonumber\\
& & + \langle \bar{q}_A q_D \rangle \langle \bar{q}_C q_F \rangle
\langle \bar{q}_E q_B \rangle +
\langle \bar{q}_A q_F \rangle \langle \bar{q}_C q_B \rangle
\langle \bar{q}_E q_D \rangle -
\langle \bar{q}_A q_F \rangle \langle \bar{q}_C q_D \rangle
\langle \bar{q}_E q_B \rangle .
\label{discon}
\end{eqnarray}
On the basis of Eq. \eqref{norm1}, we see that the disconnected part of the
condensate \eqref{C3} can be written as:
\begin{equation
\langle {\cal O}_{U(1)}^{(L=3)} \rangle _{disc} =
A_1 \langle \bar{q} q \rangle + A_3 \langle \bar{q} q \rangle ^3 ,
\label{partedisc}
\end{equation}
where the first term ($A_1 \langle \bar{q} q \rangle$), proportional to
the chiral condensate, is originated by the first nine terms in the r.h.s.
of Eq. \eqref{discon}, while the second term
($A_3\langle \bar{q} q \rangle ^3$) is originated by the last six terms
in the r.h.s. of Eq. \eqref{discon} and represents the \emph{completely
disconnected} part, proportional to the third power of the chiral condensate.\\
Explicitly, using Eq. \eqref{norm1}, with $G=4 \times 3 \times N=12N$, the
form \eqref{gamma123} of the $\Gamma$ matrices and the form \eqref{F3} of
the colour tensor $F^{a_1 a_2 a_3}_{b_1 b_2 b_3}$, satisfying the symmetry
property \eqref{fprop}, we obtain the following expression for the
coefficient $A_1$:
\begin{eqnarray
A_1 &=& \frac{1}{G^3} F^{a_1 a_2 a_3}_{b_1 b_2 b_3} \epsilon^{l_1 l_2 l_3}
\{ {\rm Tr} \Gamma_1 \langle \bar{q} \Gamma_2 q \cdot \bar{q} \Gamma_3 q
\rangle_{conn} \nonumber \\
&+& {\rm Tr} \Gamma_2 \langle \bar{q} \Gamma_1 q \cdot \bar{q} \Gamma_3 q
\rangle_{conn}
+ {\rm Tr} \Gamma_3 \langle \bar{q} \Gamma_1 q \cdot \bar{q} \Gamma_2 q
\rangle_{conn} \nonumber \\
&-& \langle \bar{q} \Gamma_1 \Gamma_2 q \cdot \bar{q} \Gamma_3 q
\rangle_{conn}
- \langle \bar{q} \Gamma_2 \Gamma_1 q \cdot \bar{q} \Gamma_3 q
\rangle_{conn} \nonumber \\
&-& \langle \bar{q} \Gamma_1 \Gamma_3 q \cdot \bar{q} \Gamma_2 q
\rangle_{conn}
- \langle \bar{q} \Gamma_3 \Gamma_1 q \cdot \bar{q} \Gamma_2 q
\rangle_{conn} \nonumber \\
&-& \langle \bar{q} \Gamma_2 \Gamma_3 q \cdot \bar{q} \Gamma_1 q
\rangle_{conn}
- \langle \bar{q} \Gamma_3 \Gamma_2 q \cdot \bar{q} \Gamma_1 q
\rangle_{conn} \} + {\rm c.c.} \nonumber \\
&=& \frac{1}{12N} \left[ 2 F^{ace}_{bde} + F^{ace}_{bed} + F^{ace}_{edb}
\right] \left( C_{12}^{abcd} + C_{13}^{abcd} + C_{23}^{abcd} \right) ,
\label{A1}
\end{eqnarray}
where:
\begin{eqnarray
C_{12}^{abcd} &\equiv& \epsilon^{st3} \left[ \langle
\bar{q}^{\ a}_{1R} \ q^{b}_{sL} \cdot
\bar{q}^{\ c}_{2R} \ q^{d}_{tL} \rangle_{conn} + {\rm c.c.} \right] ,
\nonumber \\
C_{13}^{abcd} &\equiv& \epsilon^{s2t} \left[ \langle
\bar{q}^{\ a}_{1R} \ q^{b}_{sL} \cdot
\bar{q}^{\ c}_{3R} \ q^{d}_{tL} \rangle_{conn} + {\rm c.c.} \right] ,
\nonumber \\
C_{23}^{abcd} &\equiv& \epsilon^{1st} \left[ \langle
\bar{q}^{\ a}_{2R} \ q^{b}_{sL} \cdot
\bar{q}^{\ c}_{3R} \ q^{d}_{tL} \rangle_{conn} + {\rm c.c.} \right] ,
\label{C123}
\end{eqnarray}
and the following expression for the coefficient $A_3$:
\begin{eqnarray
A_3 &=& \frac{2}{G^3} F^{a_1 a_2 a_3}_{b_1 b_2 b_3} \epsilon^{l_1 l_2 l_3}
\left[ {\rm Tr} \Gamma_1 {\rm Tr} \Gamma_2 {\rm Tr} \Gamma_3
- {\rm Tr} \Gamma_1 {\rm Tr} (\Gamma_2 \Gamma_3)
- {\rm Tr} (\Gamma_1 \Gamma_2) {\rm Tr} \Gamma_3 \right. \nonumber \\
&+& \left. {\rm Tr} (\Gamma_1 \Gamma_3 \Gamma_2)
+ {\rm Tr} (\Gamma_1 \Gamma_2 \Gamma_3) - {\rm Tr} (\Gamma_1 \Gamma_3) {\rm Tr} \Gamma_2
\right] \nonumber \\
&=& \frac{1}{216 N^3}[\alpha_1 (2N^3 + 3N^2 + N)
+ \alpha_2 (N^3 + 6N^2 + 5N) \nonumber \\
&+& \beta_1 (3N^3 + 9N^2 + 6N) ].
\label{A3}
\end{eqnarray}
Now, if we want to obtain a new order parameter which is really
\emph{independent} on the usual chiral condensate $\langle \bar{q}q \rangle$,
we must require that its disconnected part \eqref{partedisc} vanishes
independently on the value of $\langle \bar{q}q \rangle$, imposing the two
conditions $A_1=0$ and $A_3=0$. Therefore, we have two independent constraints
on the three parameters $\alpha_1$, $\alpha_2$ and $\beta_1$, which enter the
colour tensor \eqref{F3}: the new condensate $C_{U(1)}^{(L=3)}$ is then
univocally determined, apart from a multiplicative constant.\\
Let us also observe that in the large--$N$ limit, taking the coefficients
$\alpha_1$, $\alpha_2$ and $\beta_1$ in the colour tensor \eqref{F3} to
be of order ${\cal O}(N^0)$, the coefficient $A_1$ is of order ${\cal O}(N)$,
while the coefficient $A_3$ is of order ${\cal O}(N^0)$: and, consequently,
the first term $A_1 \langle \bar{q} q \rangle$ in the r.h.s. of
\eqref{partedisc} is of order ${\cal O}(N^2)$, while the second term
$A_3\langle \bar{q} q \rangle ^3$ is of order ${\cal O}(N^3)$
(being $\langle \bar{q}q \rangle = \mathcal{O}(N)$). If both these disconnected parts
are zero, then the new condensate $C_{U(1)}^{(L=3)}$ is simply equal to the
\emph{connected} part $\langle {\cal O}_{U(1)}^{(L=3)} \rangle _{conn}$,
which is of order ${\cal O}(N)$, i.e., of the same order of the usual
chiral condensate $\langle \bar{q}q \rangle$ (as already observed in
Refs. \cite{EM1994b,EM1994c}).\\
We also observe that the condition $A_1=0$ implies that the new six--fermion
condensate $C_{U(1)}^{(L=3)}$ does not take contributions from four--fermion
condensates of the form \eqref{C123}. In this paper we have only studied the
effects of the new six--fermion $U(1)$ chiral order parameter. However,
recently, four--fermion operators (which could be associated with the
above--mentioned four--fermion condensates) have been used in the literature,
in the study of \emph{scalar} mesons, which are modelled as four--quark (i.e.,
$\bar{q}q\bar{q}q$) states, called ``\emph{tetraquarks}'' or
``\emph{diquark--antidiquark}'' bound states
\cite{Giacosa2007,tHooft2008,FJS2008-2009}.

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\begin{flushleft}
{\Large\bf \thesection Appendix B: On the new parameters $F_X$, $\omega_1$
and $c_1$}
\end{flushleft}

\renewcommand{\thesection}{B}

\noindent
The Lagrangian \eqref{newlagr} contains a new field $X$ and three new
parameters, namely $F_X$, $\omega_1$ and $c_1$, with respect to the usual
Lagrangian of Witten, Di Vecchia, Veneziano {\it et al.}
It is therefore natural to ask if the model can be further simplified by
simply eliminating some parameter. As we have already said, in this paper
we are {\it assuming} that the parameter $F_X$, which is essentially
proportional to the new $U(1)$ axial condensate, is different from zero:
in section 3 we discuss the relevance of this parameter $F_X$ in the
phenomenological analysis of the strong decays of pseudoscalar mesons.\\
Concerning the parameter $\omega_1$, we cannot say too much. We remind that
the usual Lagrangian of Witten, Di Vecchia, Veneziano {\it et al.} is
obtained by choosing $\omega_1=1$ (together with $F_X=0$, i.e., $X=0$).
At low temperatures one expects that the deviations from this Lagrangian are
{\it small}, in some sense, and therefore one expects that $\omega_1$ is not
much different from $1$ at low temperatures.
On the other side, as already observed in Ref. \cite{EM1994a}, $\omega_1$ must
necessarily be zero when $T \geq T_{ch}$, in order to avoid a singular
behaviour of the anomalous term above the chiral transition: this implies
a non trivial behaviour of $\omega_1$ with the temperature. However,
in this paper no particular choice for the value of $\omega_1 (T=0)$ will
be done: it will be considered as a {\it free} parameter (apart from the
above--mentioned limitation for $T\geq T_{ch}$).\\
The case of the parameter $c_1$ is much more interesting. By putting $c_1=0$,
i.e., $c=0$ [see Eq. \eqref{c-mu}], into Eqs. \eqref{zl-ql}, these reduce to:
\begin{equation
Z_L = \frac{2A[\fp^2 (1-\omega_1)^2 + L \fx^2 \omega_1^2]}{\fp^2 \fx^2} ,~~~
Q_L = 0 ,
\label{zlqlc0}
\end{equation}
which, when inserted into Eq. \eqref{autovalori1}, lead to the following
values for the squared masses of the two singlets $S_1$ and $S_2$
in the chiral limit:
\begin{equation
m_{S_1}^2 = 0 ,~~~
m_{S_2}^2 = \left(\frac{2LA}{\fp^2} \right) \omega_1^2 +
\left( \frac{2A}{\fx^2} \right)(1-\omega_1)^2 .
\label{autovaloric0}
\end{equation}
The corresponding eigenvectors, written in terms of $S_\pi$ and $S_X$, are:
\begin{eqnarray
S_1 &=& \frac{1}{\sqrt{\fp^2 (1-\omega_1)^2 + L\fx^2\omega_1^2}}
\left( \fp(\omega_1-1)S_{\pi} + \sqrt{L}\fx \omega_1 S_X \right) , \nonumber \\
S_2 &=& \frac{1}{\sqrt{\fp^2 (1-\omega_1)^2 + L\fx^2\omega_1^2}}
\left( \sqrt{L} \fx \omega_1 S_{\pi} + \fp (1-\omega_1) S_X \right) .
\label{autovect}
\end{eqnarray}
Let us observe that Eqs. \eqref{autovect} and \eqref{autovaloric0} {\it cannot}
be derived by simply putting $c=0$ into Eqs. \eqref{S1-S2} and
\eqref{massaS1-S2}, derived in subsection 2.1.
This is due to the fact that Eqs. \eqref{S1-S2} and \eqref{massaS1-S2} were
derived not only {\it assuming} that $c_1 \neq 0$, but also taking the
large--$N$ limit, in which the quantity $c \equiv \frac{c_1}{\sqrt{2}} \left(
\frac{\fx}{\sqrt{2}}\right) \left(\frac{\fp}{\sqrt{2}}\right)^L$ is
{\it large}, being of order $\mathcal{O}(N)$ [see Eq. \eqref{relazionin}].
In that case, therefore, one obtains $Z_L = \mathcal{O}(1)$ and $Q_L = \mathcal{O}(1/N)$,
so that, from Eq. \eqref{autovalori1},
$m_{S_1}^2 \simeq \frac{Q_L}{Z_L} \simeq \frac{2LA}{\fp^2 + L \fx^2} =
\mathcal{O}(1/N)$ and $S_1$ can be identified with the particle $\eta'$.

Instead, in the particular case in which $c_1=0$ [i.e., $c=0$], one has that
$Z_L = \mathcal{O}(1/N)$ and $Q_L = 0$, so that, from Eq. \eqref{autovalori1},
$S_1$ is {\it massless} (in the chiral limit) and therefore it does {\it not}
verify the Witten--Veneziano formula required for the $\eta'$.
It is easy to convince oneself that, in this particular case $c_1=0$,
$S_2$, having a squared mass of order $\mathcal{O}(1/N)$ in the large--$N$ limit,
is just the field which must be identified with the particle $\eta'$,
as required by the Witten--Veneziano mechanism for the solution of the
$U(1)$ problem.
In fact, by virtue of Eqs. \eqref{autovect}, we can re--write the $U(1)$ axial
current $J_{5,\mu}^{(L)}$, given by Eq. \eqref{jmu5}, in terms of the fields
$S_1$ and $S_2$:
\begin{equation
J_{5,\mu}^{(L)}=-\sqrt{2L} \partial_{\mu}\left(F_{S_1} S_1 + F_{S_2} S_2
\right),
\label{jmu5c}
\end{equation}
where:
\begin{eqnarray
F_{S_1} &=& \frac{\fp^2 (\omega_1-1) + L \fx^2 \omega_1}
{\sqrt{\fp^2 (1-\omega_1)^2 + L\fx^2 \omega_1^2}} , \nonumber \\
F_{S_2} &=& \frac{\sqrt{L} \fp \fx}
{\sqrt{\fp^2 (1-\omega_1)^2 + L\fx^2 \omega_1^2}} ,
\label{fs1s2}
\end{eqnarray}
are nothing but the {\it decay constants} of the singlet pseudoscalar mesons
$S_1$ and $S_2$, defined as:
\begin{eqnarray
\langle 0 | J_{5,\mu}^{(L)}(0)|S_1 (\vec{p}_1) \rangle &=& i \sqrt{2L} F_{S_1}
p_{1\mu} , \nonumber \\
\langle 0 | J_{5,\mu}^{(L)}(0)|S_2 (\vec{p}_2) \rangle &=& i \sqrt{2L} F_{S_2}
p_{2\mu} .
\label{deffph}
\end{eqnarray}
From Eqs. \eqref{autovaloric0} and \eqref{fs1s2} one immediately
verifies that the field $S_2$ satifies the {\it Witten--Veneziano formula},
i.e.,
\begin{equation
m_{S_2}^2= \frac{2LA}{F_{S_2}^2},
\end{equation}
and, therefore, it is nothing but the field associated with the particle
$\eta'$, with a squared (non--chiral) mass generated by the anomaly and
of order $\mathcal{O}(1/N)$ in the large--$N$ limit, as required by the
Witten--Veneziano mechanism. (Instead, concerning the state $S_1$, even if,
according to Eqs. \eqref{deffph} and \eqref{fs1s2}, it is coupled to the
$U(1)$ axial current, it is {\it not} coupled to the topological charge
density, i.e., $ \langle 0 | Q(0)| S_1(\vec{p}_1) \rangle =
 \frac{1}{\sqrt{2L}} F_{S_1} m^2_{S_1^2}=0 $, since it is massless:
therefore it does {\it not} appear as an intermediate mesonic state in the
spectral decomposition of the {\it full} topological susceptibility $\ldots$)\\
It is interesting to observe that, in this case (differently from the case
discussed in subsection 2.1), the parameter $\omega_1$ plays a fundamental
role. In fact, when $c_1=0$, the {\it anomalous} Lagrangian term containing
$\omega_1$ is the only one which generates a coupling between $U$ and $X$
(i.e., between the {\it usual} quark--antiquark pseudoscalar mesons and the
{\it exotic} singlet state). By changing $\omega_1$ one can ``move''
the anomaly from $U$ to $X$. In particular, in the case $\omega_1=1$ the
anomalous term only depends on $U$ and the field $X$ is decoupled.
In this case the Lagrangian simply reduces to the sum of the usual Lagrangian
written by Witten, Di Vecchia, Veneziano {\it et al.} for the field $U$
(including the anomalous term) {\it plus} a non--anomalous Lagrangian for
the field $X$: in this limit the state $S_2$, i.e, the $\eta'$, reduces to
the usual quark--antiquark singlet state $S_\pi$, while the massless state
$S_1$ reduces to the exotic state $S_X$.
On the contrary, in the opposite case $\omega_1=0$ the anomalous term only
depends on the exotic field $X$ and so the state $S_2$, i.e, the $\eta'$,
reduces to the exotic state $S_X$, while the massless state $S_1$ reduces to
the usual quark--antiquark singlet state $S_\pi$.\\
In conclusion, we have found that, in the case in which $c_1=0$, in addition
to the usual $L^2-1$ non--singlet (pseudo--)Goldstone bosons and to the
{\it massive} singlet $S_2 = \eta'$, there is another singlet $S_1$,
which is {\it massless} in the chiral limit. This particle is therefore
another (pseudo--)Goldstone boson which, when including the quark masses,
should have a mass comparable with that of the other $L^2-1$ non--singlet
pseudoscalar mesons.\\
In the realistic case $L=3$, by diagonalizing the squared mass matrix
\eqref{massmatrix3} with $c=0$, we derive the following expressions for
the squared masses of $S_1$ and $S_2$, at the first order in the quark masses
and in $1/N$:
\begin{eqnarray
m^2_{S_1} &=& \frac{\fp^2 (1-\omega_1)^2}{\fp^2 (1-\omega_1)^2 +
3\fx^2 \omega_1^2} m_0^2 , \nonumber \\
m^2_{S_2} &=& \left(\frac{6A}{\fp^2} \right) \omega_1^2 +
\left( \frac{2A}{\fx^2} \right)(1-\omega_1)^2
+ \frac{3 \fx^2 \omega_1^2}{\fp^2(1-\omega_1)^2 + 3\fx^2 \omega_1^2} m_0^2 .
\label{align}
\end{eqnarray}
Remembering the definition of $m_0^2 \equiv \frac{2}{3} B(2 \tilde{m}+m_s)
= \frac{2}{3} B(m_u+m_d+m_s)$ and Eqs. \eqref{massemesoni},
we immediately see that the squared mass of the singlet $S_1$ satifies
the following relation:
\begin{equation
m^2_{S_1} = \frac{\fp^2 (1-\omega_1)^2}{\fp^2 (1-\omega_1)^2 +
3\fx^2 \omega_1^2} m_0^2 \leq m^2_0 =
\frac{1}{3}(m^2_{\pi} + m^2_{K^{\pm}} + m^2_{K^0,\bar{K}^0})
\simeq (412 \ \mev)^2 .
\label{eq:aa}
\end{equation}
Even assuming, as already said, that we can identify the singlet $S_2$
with the observed singlet $\eta'$, no other singlet pseudoscalar meson
is observed whose mass satisfies the limit \eqref{eq:aa}. Our assumption
$c_1=0$ (together with $\fx \neq 0$) has thus led us to another
``$U(1)$ problem''.
Even if we let $c_1$ be different from zero, but arbitrarily small, i.e.,
$c_1 \to 0$ with all other quantities fixed, since, by virtue of Eqs.
\eqref{autovalori1}--\eqref{zl-ql}, the squared masses $m^2_{S_1,S_2}$
in the chiral limit are continuous functions of the parameter $c_1$,
we find that $m^2_{S_1} \simeq Q_L/Z_L \simeq
\frac{2Lc}{\fp^2 (1-\omega_1)^2 + L\fx^2\omega_1^2} = \mathcal{O}(c_1)$
will be arbitrarily small and, when including quark masses, it will have an
upper limit arbitrarily close (from above) to that reported in Eq.
\eqref{eq:aa}.

Therefore, we are forced to discard this possibility (as it leads to wrong
predictions for the pseudoscalar--meson mass spectrum) and, in the rest of
this paper, we shall always consider the model in which $c_1$ is different
from zero and not too small, so that $c=\mathcal{O}(N)$ is {\it large}.
In this case, as we have seen in subsections 2.1 and 2.2, the squared masses
of the singlet mesons $S_1$ and $S_2$ are given by Eq. \eqref{massaS1-S2}
in the chiral limit and by Eqs. \eqref{mass2} and \eqref{mass3} in the
realistic case with $L=3$ light quark flavours. Therefore, as already said,
the state $S_1$ has a {\it topological} (non--chiral) squared mass of order
$\mathcal{O}(1/N)$ in the large--$N$ limit and it is nothing but the particle $\eta'$.
Instead, the state $S_2$ is identified with an {\it exotic} singlet particle
$\eta_X$, having a {\it large} (non--chiral) mass term of order $\mathcal{O}(1)$ in
the large--$N$ limit, generated by the (non--zero) coupling constant $c_1$.

\newpage