  In this work we explore the low energy effects induced from the integration
of the heavy Higgs boson modes, $H$, $A$ and $H^\pm$, within the Two Higgs
Doublet Model (2HDM) by assuming that the lightest Higgs boson $h$ is the one
observed experimentally at $m_h \sim 125$ GeV. We work within the context of
Effective Field Theories, focusing on the Higgs Effective Field Theory (HEFT),
although some comparisons with the Standard Model Effective Field Theory
(SMEFT) case are also discussed through this work. Our main focus is placed in
the computation of the non-decoupling effects from the heavy Higgs bosons and
the capture of such effects by means of the HEFT coefficients which are
expressed in terms of the input parameters of the 2HDM. Our approach to solve
this issue is by matching the amplitudes of the 2HDM and the HEFT for physical
processes involving the light Higgs boson $h$ in the external legs, instead of
the most frequently used matching procedure at the Lagrangian level. More
concretely, we perform the matching at the amplitudes level for the following
physical processes, including scattering and decays: $h\to WW^*\to Wf\bar{f'}$,
$h\to ZZ^*\to Zf\bar{f}$, $WW \to hh$, $ZZ \to hh$, $hh \to hh$, $h \to \gamma
\gamma$ and $h \to \gamma Z$. One important point of this work is that the
matching is required to happen at low energies compared to the heavy Higgs
boson masses, and these are heavier than the other particle masses. The proper
expansion for this heavy mass limit is also defined here, which provides the
results for the non-decoupling effects presented in this work. We finally
discuss the implications of the resulting effective coefficients, and remark on
the interesting correlations detected among them.
