On the Veltman Condition, the Hierarchy Problem and High-Scale Supersymmetry

In this paper, we have considered the possibility that the Standard Model, and its minimal extension with the addition of singlets, merges with a high-scale supersymmetric theory at a scale satisfying the Veltman condition and therefore with no sensitivity to the cutoff. The matching of the Standard Model is achieved at Planckian scales. In its complex singlet extension, the matching scale depends on the strength of the coupling between the singlet and Higgs fields. For order 1 values of the coupling, still in the perturbative region, the matching scale can be located in the TeV ballpark. Even in the absence of quadratic divergences, there remains a finite adjustment of the parameters in the high-energy theory, which should guarantee that the Higgs and the singlets in the low-energy theory are kept light. This fine-tuning (unrelated to quadratic divergences) is the entire responsibility of the ultraviolet theory and remains as the missing ingredient to provide a full solution to the hierarchy problem.