Self-quartic interaction for a scalar field in an extended DFR noncommutative spacetime

The framework Doplicher-Fredenhagen-Roberts (DFR) of a noncommutative (NC) space-time is considered as a alternative approach to study the NC space-time of the early Universe. In this formalism, the parameter of noncommutative $\theta^{\mu\nu}$ is promoted to a coordinate of the space-time, and consequently, we are describing a field theory in a space-time with extra-dimension. Consequently, there is a canonical momentum associated to this new coordinate in which the effects of a new physics can emerge in the propagation of the fields along the extra-dimension. The Fourier space of this framework is automatically extended by the addition of new momenta components. The main concept that we would like to emphasize from the outset is that the formalism demonstrated here will not be constructed introducing a NC parameter in the system, as usual. It will be generated naturally from an already NC space. When the components of the new momentum are zero, the DFR approach is reduced to the usual NC case, in which $\theta^{\mu\nu}$ is a antisymmetric constant matrix. We study a scalar field action with self-quartic interaction $\phi^{4}\star$ defined in the DFR NC spacetime, obtaining the Feynman rules in the Fourier space for the scalar propagator and vertex of the model. With these rules we are able to build out the radiative corrections to one loop order for the model propagator. The influence of the NC scale, as well as the propagation of the field in the extra-dimension, are analyzed in the ultraviolet divergences scenario. We investigate the actual possibility if this $\theta^{\mu\nu}$ conjugate momentum has the property of healing the mixing IR/UV divergences that emerges in this recently new NC spacetime quantum field theory.