## Abstract

Previously the exact solution of the planar sector of the self-dual Φ^{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 *λ > −*\( \frac{1}{\uppi} \), 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 \( \frac{\arcsin \left(\uplambda \uppi \right)}{\uppi} \) for *|λ| <*\( \frac{1}{\uppi} \). It is this dimension drop which for *λ >* 0 avoids the triviality problem of the matricial \( {\varPhi}_4^4 \)-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.

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ArXiv ePrint: 1908.04543

With an appendix by Robert Seiringer, Institute of Science and Technology Austria, Am Campus 1, 3400 Klosterneuburg, Austria.

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Grosse, H., Hock, A. & Wulkenhaar, R. Solution of the self-dual Φ^{4} QFT-model on four-dimensional Moyal space.
*J. High Energ. Phys.* **2020, **81 (2020). https://doi.org/10.1007/JHEP01(2020)081

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### Keywords

- Integrable Field Theories
- Matrix Models
- Non-Commutative Geometry