Loop diagrams in space with SU(2) fuzziness

TL;DR

This paper investigates loop corrections in a scalar field theory on a three-dimensional space with SU(2) noncommutativity, revealing UV finiteness and the dominance of planar corrections in transition rates.

Contribution

It provides explicit calculations of 2- and 4-point functions at one-loop order in a noncommutative SU(2) space, showing unique non-planar correction behavior.

Findings

The theory is UV-finite due to compact momentum space.

Non-planar corrections are proportional to a one-dimensional delta function.

Only planar corrections contribute to transition rates.

Abstract

The structure of loop corrections is examined in a scalar field theory on a three dimensional space whose spatial coordinates are noncommutative and satisfy SU(2) Lie algebra. In particular, the 2- and 4-point functions in $\phi^4$ scalar theory are calculated at the 1-loop order. The theory is UV-finite as the momentum space is compact. It is shown that the non-planar corrections are proportional to a one dimensional $\delta$-function, rather than a three dimensional one, so that in transition rates only the planar corrections contribute.