Renormalization of Non-Commutative Phi^4_4 Field Theory in x Space
  In this paper we provide a new proof that the Grosse-Wulkenhaar non-commutative scalar Phi^4_4 theory is renormalizable to all orders in perturbation theory, and extend it to more general models with covariant derivatives. Our proof relies solely on a multiscale analysis in x space. We think this proof is simpler and could be more adapted to the future study of these theories (in particular at the non-perturbative or constructive level). 
Razvan Gurau, Jacques Magnen, Vincent Rivasseau, Fabien   Vignes-Tourneret