  We explore new physics scenarios which are optimally probed through precision
Higgs measurements rather than direct collider searches. Such theories consist
of additional electroweak charged or singlet states which couple directly to or
mix with the Higgs boson; particles of this kind may be weakly constrained by
direct limits due to their meager production rates and soft decay products. We
present a simplified framework which characterizes the effects of these states
on Higgs physics by way of tree level mixing (with neutral scalars) and loop
level modifications (from electrically charged states), all expressed in terms
of three mixing angles and three loop parameters, respectively. The theory
parameters are constrained and in some cases even fixed by ratios of Higgs
production and decay rates. Our setup is simpler than a general effective
operator analysis, in that we discard parameters irrelevant to Higgs
observables while retaining complex correlations among measurements that arise
due to the underlying mixing and radiative effects. We show that certain
correlated observations are forbidden, e.g. a depleted ratio of Higgs
production from gluon fusion versus vector boson fusion together with a
depleted ratio of Higgs decays to bb versus WW. Moreover, we study the strong
correlation between the Higgs decay rate to gamma gamma and WW and how it can
be violated in the presence of additional electrically charged particles. Our
formalism maps straightforwardly onto a variety of new physics models, such as
the NMSSM. We show, for example, that with a Higgsino of mass > 100 GeV and a
singlet-Higgs coupling of lambda=0.7, the photon signal strength can deviate
from the vector signal strength by up to ~ 40-60% while depleting the vector
signal strength by only 5-15% relative to the Standard Model.
