Renormalization of Orientable Non-Commutative Complex ${\mathrm{\Phi }}_3^6$ Model

In this paper we prove that the Grosse-Wulkenhaar type non-commutative orientable complex scalar ${\varphi }_3^6$ theory, with two non-commutative coordinates and the third one commuting with the other two, is renormalizable to all orders in perturbation theory. Our proof relies on a multiscale analysis in x space.