Solution of the self-dual ${\Phi }^4$ QFT-model on four-dimensional Moyal space

Abstract

Previously the exact solution of the planar sector of the self-dual ${\Phi }^4$-model on 4-dimensional Moyal space was established up to the solution of a Fredholm integral equation. This paper solves, for any coupling constant $\lambda >-\frac{1}{\pi }$, the Fredholm equation in terms of a hypergeometric function and thus completes the construction of the planar sector of the model. We prove that the interacting model has spectral dimension $4-2\frac{\arcsin (\lambda \pi )}{\pi }$ for $|\lambda |<\frac{1}{\pi }$. It is this dimension drop which for $\lambda >0$ avoids the triviality problem of the matricial ${\Phi }_4^4$-model. We also establish the power series approximation of the Fredholm solution to all orders in $\lambda$. The appearing functions are hyperlogarithms defined by iterated integrals, here of alternating letters $0$ and $-1$. We identify the renormalisation parameter which gives the same normalisation as the ribbon graph expansion.

with an appendix by Robert Seiringer, 18 pages