Renormalisation of \phi^4-theory on noncommutative R^2 in the matrix base

TL;DR

This paper proves the super-renormalisability of two-dimensional noncommutative -theory using a matrix model approach, overcoming issues with UV/IR divergences present in momentum space methods.

Contribution

It introduces a matrix-based renormalisation method with a translation-invariance breaking regulator, establishing renormalisability of the noncommutative -theory.

Findings

Renormalised N-point functions are bounded everywhere.

Mass renormalisation is achieved by adjusting only the mass parameter.

The matrix approach resolves UV/IR divergence issues present in momentum space.

Abstract

As a first application of our renormalisation group approach to non-local matrix models [hep-th/0305066], we prove (super-)renormalisability of Euclidean two-dimensional noncommutative \phi^4-theory. It is widely believed that this model is renormalisable in momentum space arguing that there would be logarithmic UV/IR-divergences only. Although momentum space Feynman graphs can indeed be computed to any loop order, the logarithmic UV/IR-divergence appears in the renormalised two-point function -- a hint that the renormalisation is not completed. In particular, it is impossible to define the squared mass as the value of the two-point function at vanishing momentum. In contrast, in our matrix approach the renormalised N-point functions are bounded everywhere and nevertheless rely on adjusting the mass only. We achieve this by introducing into the cut-off model a translation-invariance breaking regulator which is scaled to zero with the removal of the cut-off. The naive treatment without regulator would not lead to a renormalised theory.