Twisted Poincaré invariance, noncommutative gauge theories and UV–IR mixing

In the absence of gauge fields, quantum field theories on the Groenewold–Moyal (GM) plane are invariant under a twisted action of the Poincaré group if they are formulated following [M. Chaichian, P.P. Kulish, K. Nishijima, A. Tureanu, Phys. Lett. B 604 (2004) 98, hep-th/0408069; P. Aschieri, C. Blohmann, M. Dimitrijevic, F. Meyer, P. Schupp, J. Wess, Class. Quantum Grav. 22 (2005) 3511, hep-th/0504183; A.P. Balachandran, A. Pinzul, B.A. Qureshi, S. Vaidya, hep-th/0608138; A.P. Balachandran, A. Pinzul, B.A. Qureshi, S. Vaidya, arXiv: 0708.0069 [hep-th]; A.P. Balachandran, A. Pinzul, B.A. Qureshi, S. Vaidya, arXiv: 0708.1379 [hep-th]; A.P. Balachandran, A. Pinzul, B.A. Qureshi, arXiv: 0708.1779 [hep-th]]. In that formulation, such theories also have no UV–IR mixing [A.P. Balachandran, A. Pinzul, B.A. Qureshi, Phys. Lett. B 634 (2006) 434, hep-th/0508151]. Here we investigate UV–IR mixing in gauge theories with matter following the approach of [A.P. Balachandran, A. Pinzul, B. A. Qureshi, S. Vaidya, hep-th/0608138; A.P. Balachandran, A. Pinzul, B.A. Qureshi, S. Vaidya, arXiv: 0708.0069 [hep-th]]. We prove that there is UV–IR mixing in the one-loop diagram of the S-matrix involving a coupling between gauge and matter fields on the GM plane, the gauge field being non-Abelian. There is no UV–IR mixing if it is Abelian.