  We analyze the leading effective operators which induce a quartic momentum
dependence in the Higgs propagator, for a linear and for a non-linear
realization of electroweak symmetry breaking. Their specific study is relevant
for the understanding of the ultraviolet sensitivity to new physics. Two
methods of analysis are applied, trading the Lagrangian coupling by: i) a
"ghost" scalar, after the Lee-Wick procedure; ii) other effective operators via
the equations of motion. The two paths are shown to lead to the same effective
Lagrangian at first order in the operator coefficients. It follows a
modification of the Higgs potential and of the fermionic couplings in the
linear realization, while in the non-linear one anomalous quartic gauge
couplings, Higgs-gauge couplings and gauge-fermion interactions are induced in
addition. Finally, all LHC Higgs and other data presently available are used to
constrain the operator coefficients; the future impact of $pp\to\text{4
leptons}$ data via off-shell Higgs exchange and of vector boson fusion data is
considered as well. For completeness, a summary of pure-gauge and gauge-Higgs
signals exclusive to non-linear dynamics at leading-order is included.
