  We consider renormalisable models extended in the scalar sector by a generic
scalar field in addition to the standard model Higgs boson field, and work out
the effective theory for the latter in the decoupling limit. We match the full
theory onto the effective theory at tree and one-loop levels, and concentrate
on dimension-6 operators of the Higgs and electroweak gauge fields induced from
such matching. The Wilson coefficients of these dimension-6 operators from
tree-level matching are further improved by renormalisation group running. For
specific $SU(2)_L$ representations of the scalar field, some "accidental"
couplings with the Higgs field are allowed and can lead to dimension-6
operators at tree and/or one-loop level. Otherwise, two types of interaction
terms are identified to have only one-loop contributions, for the Wilson
coefficients of which we have obtained a general formula. Using the obtained
results, we analyse constraints from electroweak oblique parameters and the
Higgs data on several phenomenological models.
