Higher Derivative Gauge theory in $d=6$ and the $\mathbb{C}\mathbb{P}^{(N_f-1)}$ NLSM

We consider the $\mathbb{C}\mathbb{P}^{(N_f-1)}$ Non-Linear-Sigma-Model in the dimension $4<d<6$. The critical behaviour of this model in the large $N_f$ limit is reviewed. We propose a Higher Derivative Gauge (HDG) theory as an ultraviolet completion of the $\mathbb{C}\mathbb{P}^{(N_f-1)}$ NLSM. Tuning mass operators to zero, the HDG in the IR limit reaches to the critical $\mathbb{C}\mathbb{P}^{(N_f-1)}$. With partial tunings the HDG reaches either to the critical $U(N_f)$-Yukawa model or to the critical pure scalar QED (no Yukawa interactions). We renormalize the HDG in its critical dimension $d=6$. We study the fixed points of the HDG in $d=6-2\epsilon$ and we calculate the scaling dimensions of various observables finding a full agreement with the order $O(1/N_f)$ predictions of the corresponding critical models.