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

Grosse, Harald; Hock, Alexander; Wulkenhaar, Raimar

Abstract

Previously the exact solution of the planar sector of the self-dual Φ4-model on 4-dimensional Moyal space was established up to the solution of a Fredholm integral equation. This paper solves, for any coupling constant λ > − 1/π , 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 arcsin(λπ)/π for |λ| < 1/π . It is this dimension drop which for λ > 0 avoids the triviality problem of the matricial Φ44-model. We also establish the power series approximation of the Fredholm solution to all orders in λ. 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.