Naturalness and a light $Z'$

Models with a light, additional gauge boson are attractive extensions of the standard model. Often these models are only considered as effective low energy theory without any assumption about an UV completion. This leaves not only the hierarchy problem of the SM unsolved, but introduces a copy of it because of the new fundamental scalars responsible for breaking the new gauge group. A possible solution is to embed these models into a supersymmetric framework. However, this gives rise to an additional source of fine-tuning compared to the MSSM and poses the question how natural such a setup is. One might expect that the additional fine-tuning is huge, namely, $O(M^2_{\rm SUSY}/m^2_{Z'})$. In this paper we point out that this is not necessarily the case. We show that it is possible to find a focus point behaviour also in the new sector in co-existence to the MSSM focus point. We call this 'Double Focus Point Supersymmetry'. Moreover, we stress the need for a proper inclusion of radiative corrections in the fine-tuning calculation: a tree-level estimate would lead to predictions for the tuning which can be wrong by many orders of magnitude. As showcase, we use the $U(1)_{B-L}$ extended MSSM and discuss possible consequence of the observed $^8\textrm{Be}$ anomaly. However, similar features are expected for other models with an extended gauge group which involve potentially large Yukawa-like interactions of the new scalars.