Mass-ratio condition for non-binding of three two-component particles with contact interactions

Binding of two heavy fermions interacting with a light particle via the contact interaction is possible only for sufficiently large heavy-light mass ratio. In this work, the two-variable inequality is derived to determine a specific value $\mu^*$ providing that there are no three-body bound states for the mass ratio smaller than $\mu^*$. The value $\mu^* = 5.26$ is obtained by analyzing this inequality for a total angular momentum and parity $L^P = 1^-$. For other $L^P$ sectors, the specific mass-ratio values providing an absence of the three-body bound states are found in a similar way. For generality, the method is extended to determine corresponding mass-ratio values for the system consisting of two identical bosons and a distinct particle for different $L^P$ ($L > 0$) sectors.