Test of non-commutative QED in the process $e^{+}e^{-} \to \gamma \gamma$ at LEP

Non-communicative QED would lead to deviations from the Standard Model depending on a new energy scale $\Delta_{NC}$ and a unique direction in space defined by two angles $\eta$ and $\xi$. Here in this analysis $\eta$ is defined as the angle between the unique direction and the rotation axis of the earth. The predictions of such a theory for the process $e^{+} e^{-} \to \gamma \gamma$ are evalued for the specific orientation of the OPAL detector and compared to the measurements. Distributions of the polar and azimuthal scattering angles are used to extract limits on the energy scale $\Delta_{NC}$ depending on the model parameter $\eta$. At the 95% confidence level $\Delta_{NC}$ is found to be larger than 141 GeV for all $\eta$ and $\xi$. It is shown that the time dependence of the total cross-section could be used to determine the model parameter $\xi$ if there were a detectable signal. These are the first limits obtained on non-commutative QED from an $e^{+} e^{-}$ collider experiment.