The 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 letter, we explore settings in which Higgs mass parameters $can$
act as a "trigger" for some local operators ${\cal O}_T$. In the Standard
Model, this happens for ${\cal O}_T = {\rm Tr} (G \tilde G)$. We also introduce
a "type-0" two Higgs doublet model, with a $Z_4$ symmetry, for which ${\cal
O}_T = H_1 H_2$ 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
$\phi_i$, with a coupling to ${\cal O}_T$ 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 $\phi_i$ can constitute dark matter.
