Does finetuning make $D$-term contributions smaller in natural grand unified theories with spontaneous supersymmetry breaking?

Abstract

In this paper, we explore the natural grand unified theory (GUT) with spontaneous supersymmetry (SUSY) breaking, focusing on the contribution to sfermion and Higgs masses. Natural GUTs solve various problems in SUSY GUTs with the natural assumption that all terms allowed by $SO(10)\times U(1)_A$ symmetry are introduced with $O(1)$ coefficients. %, solve various the problems in SUSY GUTs. It is also possible to introduce spontaneous SUSY breaking in the natural GUT. This scenario predicts $D$-term contribution to sfermion and Higgs masses dominates the $F$-term contribution, that potentially leads to the SUSY flavor problem. Fortunately, it also predicts high-scale SUSY, which avoids the SUSY flavor problem but leads to the instability of the %need for fine-tuning to obtain the electroweak scale. The $D$-term domination results in the superheavy Higgsino unless the $D$-term contribution becomes comparable with the $F$-term contribution. We discuss whether it is possible to suppress the $D$-term contribution while maintaining the $F$-term contribution through the tuning of $O(1)$ coefficients in the natural GUT with spontaneous SUSY breaking. Our results indicate that it is impossible. %the D-term is proportional to the F-term and its derivatives, making it difficult to suppress the D-term independently. This also suggests that the mass spectrum of the sfermions can be predicted by the $D$-term contributions, and that the Higgsino is not a candidate for dark matter in the scenario.