The stability of the scalar $χ^2ϕ$ interaction

A scalar field theory with a $χ^\dagχϕ$ interaction is known to be unstable. Yet it has been used frequently without any sign of instability in standard text book 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 {\em finite} number of $χ{\barχ}$ loops associated with field $χ$ that appears quadradically in the interaction, and that instability arises only when intermediate states include these loops to all orders.

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