Towards the Renormalization of Non-Commutative Gauge Field Theories with the Mehler Kernel

The present research program aims at solving the UV/IR mixing problem, which so far has only been discussed and understood in the case of non-commutative scalar quantum field theories with Mehler type propagators, also for non-commutative gauge field models (NCGFT). This becomes possible, if e.g. one regards the oscillator term in a pure U(1) gauge field theory as part of the gauge fixing and furthermore considers a gauge fixing condition which allows the introduction of a BRST invariant mass term. In this way, two require-ments of a consistent quantization of a U(1) NCGFT are fulfilled: BRST symmetry and the corresponding propagators (of the gauge field and the ghosts) are essentially Mehler-kernels allowing to get UV/IR mixing under control also in NCGFT. Because of the x-dependent mass term in the Mehler-propagator caused by the oscillator term in the bilinear part of the Lagrangian, one loses the important property of translation invariance. This means that the Mehler-propagators do not depend merely on the difference between space-time coordinates but on the positions themselves and hence become very complicated. Therefore, also the corresponding loop corrections become more complicated and voluminous - but still doable! Hence, the presented research proposal is devoted to the following research aims for our new BRST invariant formulation of NCGFT: One-loop analysis of a pure U(1) NCGFT Starting with a test for the functioning of the proposed method, one-loop corrections to the self-energy of the U(1) photon should be computed. In this way the absence of UV/IR mixing problems shall be demonstrated. Afterwards the complete renormalization program for all the one-loop corrections will be discussed. Inclusion of fermions Extension of the pure U(1) NCGFT by including fermions in order to construct a deformed non-commutative QED is planned. Furthermore, one-loop calculations as a consistency check will be done once more. Extension to higher U(N) gauge groups Here we intend to study how additional general U(N) group structure alters the research steps sketched above. Renormalization group (RG) approach This section is probably the most difficult part of the investiga-tions. However, the RG approach allows us, similarly to the case of the scalar theory, to make general statements about renormalizable field models in their quantized form.

The present research program aims at solving the UV/IR mixing problem, which so far has only been discussed and understood in the case of non-commutative scalar quantum field theories with Mehler type propagators, also for non-commutative gauge field models (NCGFT). This becomes possible, if e.g. one regards the oscillator term in a pure U(1) gauge field theory as part of the gauge fixing and furthermore considers a gauge fixing condition which allows the introduction of a BRST invariant mass term. In this way, two require-ments of a consistent quantization of a U(1) NCGFT are fulfilled: BRST symmetry and the corresponding propagators (of the gauge field and the ghosts) are essentially Mehler-kernels allowing to get UV/IR mixing under control also in NCGFT. Because of the x-dependent mass term in the Mehler-propagator caused by the oscillator term in the bilinear part of the Lagrangian, one loses the important property of translation invariance. This means that the Mehler-propagators do not depend merely on the difference between space-time coordinates but on the positions themselves and hence become very complicated. Therefore, also the corresponding loop corrections become more complicated and voluminous - but still doable! Hence, the presented research proposal is devoted to the following research aims for our new BRST invariant formulation of NCGFT: One-loop analysis of a pure U(1) NCGFT Starting with a test for the functioning of the proposed method, one-loop corrections to the self-energy of the U(1) photon should be computed. In this way the absence of UV/IR mixing problems shall be demonstrated. Afterwards the complete renormalization program for all the one-loop corrections will be discussed. Inclusion of fermions Extension of the pure U(1) NCGFT by including fermions in order to construct a deformed non-commutative QED is planned. Furthermore, one-loop calculations as a consistency check will be done once more. Extension to higher U(N) gauge groups Here we intend to study how additional general U(N) group structure alters the research steps sketched above. Renormalization group (RG) approach This section is probably the most difficult part of the investiga-tions. However, the RG approach allows us, similarly to the case of the scalar theory, to make general statements about renormalizable field models in their quantized form.