Self-quartic interaction for a scalar field in an extended DFR noncommutative space–time

Abstract

The framework of Doplicher–Fredenhagen–Roberts (DFR) for a noncommutative (NC) space–time is considered as an alternative approach to study the NC space–time of the early Universe. Concerning this formalism, the NC constant parameter, θμν, is promoted to coordinate of the space–time and consequently we can describe a field theory in a space–time with extra-dimensions. We will see that there is a canonical momentum associated with this new coordinate in which the effects of a new physics can emerge in the propagation of the fields along the extra-dimensions. The Fourier space of this framework is automatically extended by the addition of the 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 by introducing a NC parameter in the system, as usual. It will be generated naturally from an already NC space. We will review that when the components of the new momentum are zero, the (extended) DFR approach is reduced to the usual (canonical) NC case, in which θμν is an antisymmetric constant matrix. In this work we will study a scalar field action with self-quartic interaction ϕ4⋆ defined in the DFR NC space–time. We will obtain the Feynman rules in the Fourier space for the scalar propagator and vertex of the model. With these rules we are able to build the radiative corrections to one loop order of the model propagator. The consequences of the NC scale, as well as the propagation of the field in extra-dimensions, will be analyzed in the ultraviolet divergences scenario. We will investigate about the actual possibility that this kμν conjugate momentum has the property of healing the combination of IR/UV divergences that emerges in this recently new NC spacetime quantum field theory.