Stability of the scalar ${\chi }^2\phi$ interaction

A scalar field theory with a ${\chi }^{†}\chi \phi$ interaction is known to be unstable. Yet it has been used frequently without any sign of instability in standard textbook examples and research articles. In order to reconcile these seemingly conflicting results, we show that the theory is stable if the Fock space of all intermediate states is limited to a finite number of closed $\chi \overline{\chi }$ loops associated with a field $\chi$ that appears quadradically in the interaction, and that instability arises only when intermediate states include these loops to all orders. In particular, the quenched approximation is stable.