Naturalness of scale-invariant NMSSMs with and without extra matter

We present a comparative and systematic study of the fine tuning in Higgs sectors in three scale-invariant NMSSM models: the first being the standard $Z_3$-invariant NMSSM; the second is the NMSSM plus additional matter filling $3(5+\bar{5})$ representations of SU(5) and is called the NMSSM+; while the third model comprises $4(5+\bar{5})$ and is called the NMSSM++. Naively, one would expect the fine tuning in the plus-type models to be smaller than that in the NMSSM since the presence of extra matter relaxes the perturbativity bound on $\lambda$ at the low scale. This, in turn, allows larger tree-level Higgs mass and smaller loop contribution from the stops. However we find that LHC limits on the masses of sparticles, especially the gluino mass, can play an indirect, but vital, role in controlling the fine tuning. In particular, working in a semi-constrained framework at the GUT scale, we find that the masses of third generation stops are always larger in the plus-type models than in the NMSSM without extra matter. This is an RGE effect which cannot be avoided, and as a consequence the fine tuning in the NMSSM+ ($\Delta \sim 200$) is significantly larger than in the NMSSM ($\Delta \sim 100$), with fine tuning in the NMSSM++ ($\Delta \sim 600$) being significantly larger than in the NMSSM+.