Perturbative analysis on infrared aspects of noncommutative QED on $R^4$

Here we examine the noncommutative counterpart of QED, which is called as noncommutative QED. The theory is obtained by examining the consistent minimal coupling to noncommutative U(1) gauge field. The *-product admits the coupling of the matter with only three varieties of charges, i.e., 0, 1 and -1. The ultraviolet divergence of noncommutative QED can be absorbed by redefinition of the theory at one loop level. To examine the infrared aspects of the theory the anomalous magnetic moment is calculated. Its depedence on the direction of photon momentum reflects the Lorentz symmetry violation of the system. The explicit calculation of the finite part of the vacuum polarization shows the presence of hard infrared singularity like $1/({q\cdot C^TC\cdot q})$ ($C^{\mu\nu}$ is a noncommutative parameter.) which also exists in noncommutative Yang-Mills theory. It might indicate the potential between the static charges dumps promptly faster than in ordinary QED. We also consider the extension to chiral gauge theory in the present context, but the requirement of anomaly cancellation allows only noncommutative QED.