Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 10 (2014), 054, 20 pages      arXiv:1402.6184
Contribution to the Special Issue on Deformations of Space-Time and its Symmetries

Two-Point Functions on Deformed Spacetime

Josip Trampetić a, b and Jiangyang You a
a) Rudjer Bošković Institute, P.O. Box 180, HR-10002 Zagreb, Croatia
b) Max-Planck-Institut für Physik (Werner-Heisenberg-Institut), Föhringer Ring 6, D-80805 München, Germany

Received February 24, 2014, in final form May 19, 2014; Published online May 29, 2014

We present a review of the one-loop photon ($\Pi$) and neutrino ($\Sigma$) 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 $\theta^{\mu\nu}$, 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.

Key words: non-commutative geometry; photon and neutrino physics; non-perturbative effects.

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