  The effective field theory description for modifications of Standard
Model-like Higgs boson interactions arising from tree-level mixing with heavy
Higgs sector vacuum states without conserved quantum numbers is presented. An
expansion in terms of effective operator dimension based on powers of the heavy
mass scale rather than operator dimension is utilized to systematically
organize interactions within the effective theory. Vacuum states arising from
electroweak singlet extensions of the Higgs sector yield at leading order only
two effective dimension-six operators. One of these uniformly dilutes all the
interactions of a single physical Higgs boson as compared with Standard Model
expectations, while the combination of the two operators give more general
modifications of all remaining interactions with two or more physical Higgs
bosons. Vacuum states arising from an additional electroweak doublet yield
three types of effective dimension-six operators that modify physical Higgs
boson couplings to fermion pairs, self-couplings, and introduce four-fermion
interactions. However, in this case modification of physical Higgs boson
interactions with massive gauge bosons arise at leading order only from three
effective dimension-eight operators. If the underlying Yukawa couplings of the
additional electroweak doublet satisfy the Glashow-Weinberg condition then time
reversal violation at effective dimension-six depends on a single universal
phase appearing only in couplings of physical Higgs bosons to fermion pairs. In
all cases the couplings of any three physical states in the effective theory
description must be equal to those in a unitary mixing description in order for
the long distance non-analytic components of physical on-shell scattering
amplitudes to agree, while in contrast couplings of four or more physical
states in general differ due to short distance effects.
