  The pattern of deviations from Standard Model predictions and couplings is
different for theories of new physics based on a non-linear realization of the
$SU(2)_L\times U(1)_Y$ gauge symmetry breaking and those assuming a linear
realization. We clarify this issue in a model-independent way via its effective
Lagrangian formulation in the presence of a light Higgs particle, up to first
order in the expansions: dimension-six operators for the linear expansion and
four derivatives for the non-linear one. Complete sets of pure gauge and
gauge-Higgs operators are considered, implementing the renormalization
procedure and deriving the Feynman rules for the non-linear expansion. We
establish the theoretical relation and the differences in physics impact
between the two expansions. Promising discriminating signals include the
decorrelation in the non-linear case of signals correlated in the linear one:
some pure gauge versus gauge-Higgs couplings and also between couplings with
the same number of Higgs legs. Furthermore, anomalous signals expected at first
order in the non-linear realization may appear only at higher orders of the
linear one, and vice versa. We analyze in detail the impact of both type of
discriminating signals on LHC physics.
