Renormalization of noncommutative quantum field theories

We report on a comprehensive analysis of the renormalization of noncommutative ${\varphi }^4$ scalar field theories on the Groenewold-Moyal plane. These scalar field theories are twisted Poincaré invariant. Our main results are that these scalar field theories are renormalizable, free of UV/IR mixing, possess the same fixed points and $\beta$-functions for the couplings as their commutative counterparts. We also argue that similar results hold true for any generic noncommutative field theory with polynomial interactions and involving only pure matter fields. A secondary aim of this work is to provide a comprehensive review of different approaches for the computation of the noncommutative $S$-matrix: noncommutative interaction picture and noncommutative Lehmann-Symanzik-Zimmermann formalism.

Figure 1

Feynman diagrams for the two-point function at one loop.

Figure 2

Feynman diagrams for the four-point function at one loop.