Two-Point Functions on Deformed Spacetime

Abstract

We present a review of the one-loop photon ($\Pi$) and neutrino ($\Sigma$) two-point functions in a covariant and deformed $\mathrm{U}(1)$ gauge-theory on the 4-dimensional noncommutative spaces, determined by a constant antisymmetric tensor ${\theta }^{\mu \nu }$, and by a parameter-space $({\kappa }_f,{\kappa }_g)$, respectively. For the general fermion-photon $S_f({\kappa }_f)$ and photon self-interaction $S_g({\kappa }_g)$ 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 $\ln ({\mu }^2(\theta p{)}^2)$. 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 ${\theta }^{\mu \nu }$ 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 the one-loop quantum corrections, having also both well-defined commutative limit and point-like limit of an extended object.

review article based on arXiv:0807.4886, arXiv:1109.2485, arXiv:1111.4951 and arXiv:1306.1239, including some novelty

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Cited by in corpus (6)

• Photon self-interaction on deformed spacetime
• Photon propagation in noncommutative QED with constant external field
• Seiberg-Witten map Invariant Scatterings
• The theta-exact Seiberg-Witten maps at the e3 order
• Super Yang-Mills and theta-exact Seiberg-Witten map: Absence of quadratic noncommutative IR divergences
• Photon polarization tensor on deformed spacetime: A four-photon-tadpole contribution