Measurements of the supersymmetric Higgs self-couplings and the reconstruction of the Higgs potential

We address the issue of the reconstruction of the scalar potential of a two-Higgs-doublet model having in mind that of the MSSM. We first consider the general $\mathrm{CP}$ conserving dim-4 effective potential. To fully reconstruct this potential, we show that even if all the Higgs boson masses and their couplings to the standard model particles are measured, one needs not only to measure certain trilinear Higgs self-couplings but some of the quartic couplings as well. We also advocate expressing the Higgs self-couplings in the mass basis. We show explicitly that in the so-called decoupling limit the most easily accessible Higgs self-couplings are given in terms of the Higgs boson mass while all other dependences on the parameters of the general effective potential are screened. This also helps to easily explain how, in the MSSM, the largest radiative corrections which affect these self-couplings are reabsorbed by using the corrected Higgs boson mass. We also extend our analysis to higher order operators in the effective Higgs potential. While the above screening properties do not hold, we argue that these effects must be small and may not be measured considering the foreseen poor experimental precision in the extraction of the SUSY Higgs self-couplings.