Noncommutative QED+QCD and $β$-function for QED

QED based on $θ$-unexpanded noncomutative space-time in contrast with the noncommutative QED based on $θ$-expanded U(1) gauge theory via the Seiberg-Witten map, is one-loop renormalizable. Meanwhile it suffers from asymptotic freedom that is not in agreement with the experiment. We show that QED part of $U_\star(3)\times U_\star(1)$ gauge group as an appropriate gauge group for the noncommutative QED+QCD, is not only one-loop renormalizable but also has a $β$ function that can be positive, negative and even zero. In fact the $β$ function depends on the mixing parameter $δ_{13}$ as a free parameter and it will be equal to its counterpart in the ordinary QED for $δ_{13}=0.367π$.

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