A New Non-Commutative Field Theory

In this note we investigate a new type of non-commutative field theory based on a constant skew-symmetric three-form parameter. In 3+1 dimensions such a three-form parameter can be viewed as a short-distance regulator which nevertheless preserves spatial-rotation and at long range preserves Lorentz invariance approximately. For a scalar field theory with quartic self-interaction we obtain drastically improved ultra-violet behavior of the diagrams, due to the oscillatory dependence of the interaction vertex on the momenta. The radiative corrections to the coupling are rendered finite already at the one-loop level. The key finding of this paper is that what appears as the reemergence of UV divergences as IR singularity in $p \to 0$ limit, must be interpreted simply as the logarithmic running of the coupling. Thus at low energies the theory is virtually indistinguishable from the standard theory. Conversely at high energies the diagram converges exponentially fast, the running of the coupling stops and the theory avoids developing the Landau pole. Bare coupling defined at high energy can be kept small, and in this sense the theory is similar to asymptotically free theories.