Two-Point Functions on Deformed Spacetime

We present a review of the one-loop photon ($Π$) and neutrino ($Σ$) two-point functions in a covariant and deformed $\rm U(1)$ gauge-theory on the 4-dimensional noncommutative spaces, determined by a constant antisymmetric tensor $θ^{μν}$, and by a parameter-space $(κ_f,κ_g)$, respectively. For the general fermion-photon $S_f(κ_f)$ and photon self-interaction $S_g(κ_g)$ the closed form results reveal two-point functions 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 $\ln(μ^2(θp)^2)$. In addition, the photon-loop produces new tensor structures satisfying transversality condition by themselves. We show that the photon two-point function in the 4-dimensional Euclidean spacetime can be reduced to two finite terms by imposing a specific full rank of $θ^{μν}$ and setting deformation parameters $(κ_f,κ_g)=(0,3)$. In this case the neutrino two-point function vanishes. Thus for a specific point $(0,3)$ in the parameter-space $(κ_f,κ_g)$, a covariant $θ$-exact approach is able to produce a divergence-free result for the one-loop quantum corrections, having also both well-defined commutative limit and point-like limit of an extended object.

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