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

Harald Grosse·Raimar Wulkenhaar
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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 ϕ^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.

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Harald Grosse, Raimar Wulkenhaar. 2003-07-01. Renormalisation of ϕ^4-theory on noncommutative R^2 in the matrix base. https://doi.org/10.1088/1126-6708%2F2003%2F12%2F019

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