Standard Model Effective Field Theory: Integrating out a Generic Scalar

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.