Self-energies on deformed spacetimes

We study one-loop photon (Pi) and neutrino (Sigma) self-energies in a U(1) covariant gauge-theory on d-dimensional noncommutative spaces determined by a antisymmetric-constant tensor theta^{mu nu}. For the general fermion-photon (S_f) and photon self-interaction (S_g) the closed form results reveal self-energies besetting with all kind of pathological terms: the UV divergence, the quadratic UV/IR mixing terms as well as a logarithmic IR divergent term of the type ln(mu^2(theta p)^2). In addition, the photon-loop produces new tensor structures satisfying transversality condition by themselves. We show that the photon self-energy in four-dimensional Euclidean spacetime can be reduced to two finite terms by imposing a specific full rank of theta^{mu nu} and setting deformation parameters (kappa_f,kappa_g)=(0,3). In this case the neutrino two-point function vanishes. Thus for a specific point (0,3) in the parameter-space (kappa_f,kappa_g), a covariant theta-exact approach is able to produce a divergence-free result for one-loop quantum corrections, having also well-defined both the commutative limit as well as the pointlike limit of an extended object. While in two-dimensional space the photon self-energy is finite for arbitrary (kappa_f,kappa_g) combinations, the neutrino self-energy still contains an superficial IR divergence.