for some specific processes. The main purpose of this work is to explore these predictions in the case of three-body forces. Our presentation is divided as follows: in sect. 2 we compare linear and non-linear Lagrangians and in sect 3 we motivate the physical predictions of the latter. In sect. 4 we derive the kernel of the three-body force, which is fully developed in sect. 5. Finally, in sect 6 we discuss qualitatively some features of the force associated with sigmas and chiral scalars.

Lagrangians
===========

In this section we introduce the Lagrangians describing both the sigma and the chiral scalar. In the framework of linear dynamics the field $\sigma $ is the chiral partner of the pion field since, for an axial transformation $\delta ^A$, we have $\delta ^A\mbox{\boldmath$ \pi$
}\rightarrow \sigma $ and $\delta ^A\sigma \rightarrow -\mbox{\boldmath $\pi
$}$. In the case of non-linear dynamics, the pion field is represented by ${%
\mbox{\boldmath $\phi $}}$ and we also consider a new field $S$, which corresponds to a generalization of the field $\sigma ^{\prime }$, introduced long ago by Weinberg [@Wei]. The corresponding axial transformation are $%
\delta ^A{\mbox{\boldmath $\phi $}=}F({\mbox{\boldmath $\phi $}}^2) $, where $F({\mbox{\boldmath $\phi $}}^2)$ is 