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.

arXiv commentwith an appendix by Robert Seiringer, 18 pages

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• Perturbative and Geometric Analysis of the Quartic Kontsevich Model
• A Laplacian to compute intersection numbers on ${\bar{\mathcal{M}}}_{g,n}$ and correlation functions in NCQFT
• Large-$d$ behavior of the Feynman amplitudes for a just-renormalizable tensorial group field theory
• Complete solution of the LSZ Model via Topological Recursion
• Renormalization in Combinatorially Non-Local Field Theories: the BPHZ Momentum Scheme
• Interacting massless infraparticles in 1+1 dimensions
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