  We consider the Minimal Supersymmetric extension of the Standard Model in the
regime where the supersymmetric breaking scale is extermely large. In this
MSSM, not only the Higgs masses will be affected by large radiative
corrections, the dominant part of which is provided by the third generation
quark/squark sector, but also the various self-couplings among the Higgs
states. In this note, assuming that squarks are extremely heavy, we evaluate
the next-to-leading order radiative corrections to the two neutral CP-even
Higgs self-couplings $\lambda_{H h h}$ and $\lambda_{h h h}$ and to the partial
decay width $\Gamma(H \to h h)$ that are most relevant at the LHC. The
calculation is performed using an effective field theory approach that resums
the large logarithmic squark contributions and allows to keep under control the
perturbative expansion. Since the direct loop vertex corrections are generally
missing in this effective approach, we have properly renormalised the effective
theory to take them into account. Finally, we perform a comparison of the
results in this effective MSSM with those obtained in a much simpler way in the
so-called hMSSM approach in which the mass value for the lightest Higgs boson
$M_{h}=125$ GeV is used as an input. We show that the hMSSM provides a
reasonably good approximation of the corrected self-couplings and $H \to hh$
decay rate and, hence, it can be used also in these cases.
