\bar u\left[ \frac{\not{k'}}{\left( p'+k'\right) ^2-m^2}
\; \gamma _5 \; \frac{\not{k}}{\left( p+k\right) ^2-m^2}\right] u 
\nonumber \\
&&+\delta _{ac}\tau _b \;
\bar u\left[ \gamma _5\frac{\not{k'}}{\left(p'+k'\right) ^2-m^2}\;\frac 1{gf_\pi }\right] u  
\nonumber\\
&&\left. +\delta _{bc}\tau _a \;
\bar u\left[ \frac 1{gf_\pi }\;\frac{\not{k}}{\left( p+k\right) ^2-m^2}\gamma _5\right] u\right\} .\end{aligned}$$

The part of A corresponding to a scalar-isoscalar exchange is associated with the factor $\delta _{ab}$ in the $\pi N$ amplitude and hence it is proportional to

$$\begin{aligned}
\delta _{ab}P^{bac} &=&ig^3\tau _c \;
\bar u\left[ \frac{\not{k'}}{\left(p'+k'\right) ^2-m^2}\; \gamma _5 \; \frac{\not{k}}{\left( p+k\right)^2-m^2}\right.
\nonumber\\
&&\left. -\gamma _5 \; \frac{\not{k'}}{\left( p'+k'\right) ^2-m^2}\;\frac 1{gf_\pi }
-\frac 1{gf_\pi }\;\frac{\not {k}}{\left(p+k\right) ^2-m^2}\; \gamma _5\right] u.\end{aligned}$$

In order to identify the main contribution to this amplitude, we go to the soft pion limit, by using $k=k^{\prime }=(k_0,0)$ and then take the limit $%
k_0\rightarrow 0$. We thus obtain

$$\delta _{ab}P^{bca}\stackrel{soft}{\rightarrow }-\frac{ig^3}{4m^2}\tau_c \bar u \gamma _5u,$$

which has the same structure as the effective scalar-nucleon seagull vertex predicted by the non-linear Lagrangian given by (\[Lnl\]).

Effective $\pi NN$ vertex
=========================

As far as NN interactions are concerned both the linear and non-linear Lagrangians, given by eqs. (\[L1\]) and (\[Lnl\]), yield potentials at tree level, which are due to the exchanges of one pion and 