Renormalisation of $\phi^4$-theory on noncommutative $\mathbb{R}^4$ to all orders

The renormalisation of field theories on noncommutative $\mathbb{R}^4$ to all orders is still an open question. I propose to take the UV/IR-entanglement observed in one-loop Feynman graph calculations as a message to extend the free-field action by an oscillator potential. In a base where the $\phi^4$ $\star$-product interaction is realised as a matrix product, the free action can be diagonalised via Meixner polynomials. The resulting propagator shows an asymptotic and local behaviour which according to a power-counting theorem derived by renormalisation group techniques leads to only four relevant or marginal base interactions. Thus, the model is due to properties of the Meixner polynomials renormalisable to all orders by imposing normalisation conditions for the mass, the field amplitude, the coupling constant and the oscillator frequency. The oscillator potential leads to quantised momenta in agreement with recent data on the cosmic microwave background.