  In the alignment limit of a multi-doublet Higgs sector, one of the Higgs mass
eigenstates aligns in field space with the direction of the scalar field vacuum
expectation values, and its couplings approach those of the Standard Model (SM)
Higgs boson. We consider CP-conserving Two-Higgs-Doublet Models (2HDMs) of Type
I and Type II near the alignment limit in which the heavier of the two CP-even
Higgs bosons, $H$, is the SM-like state observed with a mass of 125 GeV, and
the couplings of $H$ to gauge bosons approach those of the SM. We review the
theoretical structure and analyze the phenomenological implications of this
particular realization of the alignment limit, where decoupling of the extra
states cannot occur given that the lighter CP-even state $h$ must, by
definition, have a mass below 125 GeV. For the numerical analysis, we perform
scans of the 2HDM parameter space employing the software packages 2HDMC and
Lilith, taking into account all relevant pre-LHC constraints, constraints from
the measurements of the 125 GeV Higgs signal at the LHC, as well as the most
recent limits coming from searches for heavy Higgs-like states. Implications
for Run 2 at the LHC, including expectations for observing the other scalar
states, are also discussed.
