momenta $\vec q$ and $\vec q'$ are carried by the mesons with masses $m_i$ and $m_j$, respectively. The complete three-body potential is constructed by summing the contributions from all distinguishable diagrams with all possible exchanged pseudoscalar mesons, including baryon exchange operations where needed.

As they stand, the 3BF terms at NNLO introduce a prohibitively large number of low-energy constants in the contact potentials $V^{(0)}$. The existing data-base is too limited and does not permit a meaningfully constrained determination of these parameters. However, their number can be reduced substantially by employing an approximate scheme referred to as decuplet saturation: contact vertices are resolved by propagating explicit decuplet baryons in intermediate states. Furthermore, it has the welcome feature of promoting three-baryon forces from NNLO to NLO within the chiral hierarchy. In essence, the NNLO diagrams in fig.\[fig:3BF\] are replaced by the NLO diagrams of fig.\[fig:3BFdec\]. In our subsequent application, two of the in- or outgoing baryons are nucleons (proton, neutron), while the third one is a strangeness $S=-1$ hyperon ($\Lambda$ or $\Sigma$) and only the (long-range) pion-exchange is considered. The decuplet intermediate states are then either $\Sigma^*(1385)$ or $\Delta(1232)$.

![Three-baryon forces with explicit decuplet baryons in the intermediate states (represented by double 