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We consider the $\mathbb{C}\mathbb{P}^{(N_f-1)}$ Non-Linear-Sigma-Model inthe 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 anultraviolet completion of the $\mathbb{C}\mathbb{P}^{(N_f-1)}$ NLSM. Tuningmass 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 eitherto the critical $U(N_f)$-Yukawa model or to the critical pure scalar QED (noYukawa interactions).  We renormalize the HDG in its critical dimension $d=6$. We study the fixedpoints of the HDG in $d=6-2\epsilon$ and we calculate the scaling dimensions ofvarious observables finding a full agreement with the order $O(1/N_f)$predictions of the corresponding critical models.

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