Weak scale as a trigger

Does the value of the Higgs mass parameter affect the expectation value of local operators in the Standard Model? For essentially all local operators the answer to this question is "no", and this is one of the avatars of the hierarchy problem: Nothing is "triggered"when the Higgs mass parameter crosses zero. In this article, we explore settings in which Higgs mass parameters can act as a "trigger"for some local operators OT. In the Standard Model, this happens for OT=Tr(GG). We also introduce a "type-0"two Higgs doublet model, with a Z4 symmetry, for which OT=H1H2 is triggered by the Higgs masses, demanding the existence of new Higgs states necessarily comparable to or lighter than the weak scale, with no wiggle room to decouple them whatsoever. Surprisingly, this model is not yet entirely excluded by collider searches, and will be incisively probed by the high-luminosity run of the LHC, as well as future Higgs factories. We also discuss a possibility for using this trigger to explain the origin of the weak scale, invoking a landscape of extremely light, weakly interacting scalars φi, with a coupling to OT needed to make it possible to find vacua with small enough cosmological constant. The weak scale trigger links the tuning of the Higgs mass to that of the cosmological constant, while coherent oscillations of the φi can constitute dark matter.