Perturbative $\lambda$-Supersymmetry and Small $\kappa$-Phenomenology

For the minimal $\lambda$-supersymmetry, it stays perturbative to the GUT scale for $\lambda \leq 0.7$. This upper bound is relaxed when one either takes the criteria that all couplings close to $\sim 4\pi$ for non-perturbation or allows new fields at the intermediate scale between the weak and GUT scale. We show that a hidden $U(1)_X$ gauge sector with spontaneously broken scale $\sim10$ TeV improves this bound as $\lambda\leq1.23$ instead. This may induce significant effects on Higgs physics such as decreasing fine tuning involving the Higgs scalar mass, as well as on the small $\kappa$-phenomenology.