Non-commutative U(1) Super-Yang-Mills Theory: Perturbative Self-Energy   Corrections
  The quantization of the non-commutative N=1, U(1) super-Yang-Mills action is performed in the superfield formalism. We calculate the one-loop corrections to the self-energy of the vector superfield. Although the power-counting theorem predicts quadratic ultraviolet and infrared divergences, there are actually only logarithmic UV and IR divergences, which is a crucial feature of non-commutative supersymmetric field theories. 
A.A. Bichl, M. Ertl, A. Gerhold, J.M. Grimstrup, H. Grosse, L. Popp,   V. Putz, M. Schweda, R. Wulkenhaar