One-Loop UV Divergent Structure of U(1) Yang-Mills Theory on Noncommutativeℝ4

We show that U(1) Yang-Mills theory on noncommutative $R{}^4$ can be renormalized at the one-loop level by multiplicative dimensional renormalization of the coupling constant and fields of the theory. We compute the beta function of the theory and conclude that the theory is asymptotically free. We also show that the Weyl-Moyal matrix defining the deformed product over the space of functions on $R{}^4$ is not renormalized at the one-loop level.