  One of the simplest extensions of the Standard Model (SM) is the addition of
a scalar gauge singlet, S. If S is not forbidden by a symmetry from mixing with
the Standard Model Higgs boson, the mixing will generate non-SM rates for Higgs
production and decays. In general, there could also be unknown high energy
physics that generates additional effective low energy interactions. We show
that interference effects between the scalar resonance of the singlet model and
the effective field theory (EFT) operators can have significant effects in the
Higgs sector. We examine a non-$Z_2$ symmetric scalar singlet model and
demonstrate that a fit to the 125 GeV Higgs boson couplings and to limits on
high mass resonances, S, exhibit an interesting structure and possible large
cancellations of effects between the resonance contribution and the new EFT
interactions, that invalidate conclusions based on the renormalizable singlet
model alone.
