a function of ${\mbox{\boldmath $\phi $}%
}^2$ and $\delta ^AS=0$. The last transformation implies that $S$ is a chiral scalar.

The linear Lagrangian ${\cal L}_\sigma $ has the usual form

$$\begin{aligned}
{\cal L}_\sigma &=&\frac 12\left(\partial _\mu \sigma \partial ^\mu\sigma 
+ \partial _\mu \mbox{\boldmath $\pi $}\cdot \partial ^\mu \mbox{\boldmath $\pi $} \right) 
-\frac{\mu ^2}2\left( \sigma ^2+\mbox{\boldmath $\pi $}%
^2\right) +\frac 12f_\pi \mu ^2\sigma +U\left( \sigma ^2+\mbox{\boldmath
$\pi $}^2\right)  \nonumber \\
&&+\bar Ni{\not \partial }N-g\bar N\left( \sigma +i{\mbox{\boldmath $\tau $}%
\cdot {\mbox{\boldmath$\pi$}} }\gamma _5\right) N  \label{Llin}\end{aligned}$$

where $N$ is the nucleon field that transforms linearly, $f_\pi $ is de pion decay constant, $\mu $ is the pion mass, $g$ is the $\pi N$ coupling constant and the term $U\left( \sigma ^2+\mbox{\boldmath $\pi $}^2\right) $ represents the self interactions of the mesonic fields. In the framework of the linear sigma model the scalar fluctuations are associated with a field $%
\epsilon $, related to $\sigma $ by

$$\sigma =f_\pi +\epsilon .$$

Replacing this into the Lagrangian (\[Llin\]), we obtain

$$\begin{aligned}
{\cal L}_\sigma &=& \frac 12\left( \partial _\mu \epsilon \partial ^\mu \epsilon -m_\epsilon^2\epsilon ^2\right) 
+\frac 12\left( \partial _\mu \mbox{\boldmath $\pi $}\cdot \partial ^\mu \mbox{\boldmath $\pi$ }
-\mu ^2\mbox{\boldmath $\pi $}^2\right)
+U\left( \left( f_\pi +\epsilon 