Renormalisation of phi4-theory on noncommutative Bbb R2in the matrix base

As a first application of our renormalisation group approach to non-localmatrix models [hep-th/0305066], we prove (super-)renormalisability of Euclideantwo-dimensional noncommutative \phi^4-theory. It is widely believed that thismodel is renormalisable in momentum space arguing that there would belogarithmic UV/IR-divergences only. Although momentum space Feynman graphs canindeed be computed to any loop order, the logarithmic UV/IR-divergence appearsin the renormalised two-point function -- a hint that the renormalisation isnot completed. In particular, it is impossible to define the squared mass asthe value of the two-point function at vanishing momentum. In contrast, in ourmatrix approach the renormalised N-point functions are bounded everywhere andnevertheless rely on adjusting the mass only. We achieve this by introducinginto the cut-off model a translation-invariance breaking regulator which isscaled to zero with the removal of the cut-off. The naive treatment withoutregulator would not lead to a renormalised theory.Comment: 26 pages, 44 figures, LaTe