and (B): hadronization of the $u \bar{u}$ to give two mesons.[]{data-label="Fig:feyn"}](feynd.eps)

The hadronization is implemented in an easy way following the work of Ref. [@alberzou]. One starts with the $q\bar{q}$ matrix $M$

$$\label{eq:1}
M=\left(
           \begin{array}{ccc}
             u\bar u & u \bar d & u\bar s \\
             d\bar u & d\bar d & d\bar s \\
             s\bar u & s\bar d & s\bar s \\
           \end{array}
         \right)$$

which has the property $$\label{eq:2}
M\cdot M=M \times (\bar{u}u + \bar{d}d + \bar{s}s).$$

Hence the $u\bar{u}$ component of Fig. \[Fig:feyn\] (B) can be written as, $$u\bar{u}(\bar{u}u + \bar{d}d + \bar{s}s) = (M \cdot M)_{11}.$$

Next, we rewrite the $q\bar{q}$ matrix $M$ in terms of meson components, and we have $M$ corresponding to the matrix $\phi$ [@bramon; @palomar; @gamermann]

$$\label{eq:phimatrix}
\phi = \left(
           \begin{array}{ccc}
             \frac{1}{\sqrt{2}}\pi^0 + \frac{1}{\sqrt{3}}\eta + \frac{1}{\sqrt{6}}\eta' & \pi^+ & K^+ \\
             \pi^- & -\frac{1}{\sqrt{2}}\pi^0 + \frac{1}{\sqrt{3}}\eta + \frac{1}{\sqrt{6}}\eta' & K^0 \\
            K^- & \bar{K}^0 & -\frac{1}{\sqrt{3}}\eta + \sqrt{\frac{2}{3}}\eta' \\
           \end{array}
         \right)$$

This matrix corresponds to the ordinary one used in chiral perturbation theory [@Gasser:1983yg] with the addition of $\frac{1}{\sqrt{3}}diag(\eta_1,\eta_1,\eta_1)$ where $\eta_1$ is a singlet of $SU(3)$, taking into account the standard mixing between $\eta$ and $\eta'$. The term $\frac{1}{\sqrt{3}}diag(\eta_1,\eta_1,\eta_1)$ is omitted in the 