we note that, in the framework of non-linear Lagrangians, the TPEP in the pure nucleon sector is given by the five diagrams of fig. 2a [@Rob+], which we may want to associate with an effective scalar exchange. When an external pion is attached to these processes, we have the three possibilities indicated in fig. 2b, the last one corresponding to an effective seagull, whose dynamical meaning is given in fig 2c.

The nine diagrams associated with the effective seagull interaction can be understood as arising from the product of the amplitudes $T^{ba}$and $%
P^{bca} $, where the former describes the pion-nucleon scattering $\pi
^a\left( k\right) N\left( p\right) \rightarrow \pi ^b\left( k^{\prime
}\right) N\left( p^{\prime }\right) $ and latter represents the contribution from pion production, $\pi ^a\left( k\right) N\left( p\right) \rightarrow
\pi ^b\left( k^{\prime }\right) \pi ^c\left( q\right) N\left( p^{\prime
}\right) $, as indicated in fig 3. The composite amplitude is denoted by $A$ and given by $$A=-i\int \frac{d^4Q}{\left( 2\pi \right) ^4}T^{ba}P^{bca}\frac 1{k^2-\mu ^2}%
\frac 1{k^{\prime 2}-\mu ^2},$$ where $Q=\frac 12\left( k-k^{\prime }\right) .$

Using the $\pi N$ vertices obtained from eq. (\[Lnl\]), we have $$\begin{aligned}
P^{bca} &=&ig^3
\left\{ \left( \delta _{ab}\tau _c-\delta _{bc}\tau _a-\delta _{ac}\tau _b+i\epsilon _{bac}\right) \right. 
\nonumber\\
&&\times 