\right) ^2+\mbox{\boldmath $\pi $}^2\right)  
\nonumber \\
&&+\bar Ni{\not \partial }N-g\bar N\left( f_\pi +\epsilon +i \mbox{\boldmath
$\tau $}\cdot \mbox{\boldmath $\pi$ }\gamma _5\right) N.  \label{L1}\end{aligned}$$

In the non-linear approach, the Lagrangian ${\cal L}_S$ is written as

$${\cal L}_S=\left[ \frac 12\left( \partial _\mu S\partial ^\mu
S-m_s^2S^2\right) -U\left( S\right) \right] 
+\left[ \frac 12\left( \partial _\mu {\mbox{\boldmath $\phi $}}\cdot \partial ^\mu {\mbox{\boldmath $\phi $}}
+\partial _\mu f\partial ^\mu f\right) +f_\pi \mu ^2f\right] +{\cal L}_N,$$

where $f$ corresponds to the function $f=\sqrt{f_\pi ^2-{\mbox{\boldmath
$\phi $}}^2}$. Formally, $U(S)$ describes the self interactions of the scalar field and ${\cal L}_N$ represents both the nucleon sector and its interactions with bosonic fields. This part of Lagrangian may be cast in many different forms, two of which are widely employed in the literature. In one of them, the pion-nucleon coupling is pseudo-vector (PV) and, in the other, it is pseudo scalar (PS). For PV coupling, one has

$${\cal L}_N^{PV}=\bar \psi i\gamma _\mu D^\mu \psi -m\bar \psi \psi 
+\frac g{2m}\bar \psi \gamma _\mu \gamma _5{\mbox{\boldmath $\tau $}}\psi\cdot D^\mu {\mbox{\boldmath $\phi $}}
-g_sS\bar \psi \psi \;\;,  \label{Lpv}$$

where $\psi $ is the nucleon field that transforms non-linearly, $m$ is its mass and $g_s$ represents the coupling of the nucleon to the scalar. 