lines) [@Petschauer2017].[]{data-label="fig:3BFdec"}](FBBBdeccont "fig:")![Three-baryon forces with explicit decuplet baryons in the intermediate states (represented by double lines) [@Petschauer2017].[]{data-label="fig:3BFdec"}](FBBBdec1MEu "fig:")![Three-baryon forces with explicit decuplet baryons in the intermediate states (represented by double lines) [@Petschauer2017].[]{data-label="fig:3BFdec"}](FBBBdec1MEd "fig:")![Three-baryon forces with explicit decuplet baryons in the intermediate states (represented by double lines) [@Petschauer2017].[]{data-label="fig:3BFdec"}](FBBBdec2ME "fig:")

In what follows we use the detailed formalism developed in refs.[@Petschauer2016; @Petschauer2017]. Within the decuplet dominance approximation there are only three remaining constants to be determined: the coupling strength $C$ of the transition vertex between an octet baryon and a decuplet baryon with a pseudoscalar meson absorbed or emitted; and two coupling constants, $H_1$ and $H_2$, for the four-point vertices connecting three octet baryon lines and one decuplet baryon [@Petschauer2016; @Petschauer2017]. The coupling $C$ is uniquely determined by the $\pi N\rightarrow\Delta$ transition vertex which is in turn constrained by the decay width $\Gamma(\Delta\rightarrow \pi N)\simeq 115$ MeV. The (large-$N_c$) value $C = 3g_A/4 \simeq 0.95$ with the nucleon axial vector coupling constant, $g_A = 1.26$, is well compatible with this constraint. The remaining two constants, $H_{1,2}$ of dimension $(length)^2$, are still free to choose. We use hypernuclear phenomenology in order to restrict their possible values.

Next, the three-body interactions in the $\Lambda 