Solution of the self-dual $Φ^4$ QFT-model on four-dimensional Moyal space

Harald Grosse, Alexander Hock, Raimar Wulkenhaar
August 13, 2019
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.
open access link
@article{Grosse:2019qps, author = "Grosse, Harald and Hock, Alexander and Wulkenhaar, Raimar", title = "{Solution of the self-dual $\Phi^4$ QFT-model on four-dimensional Moyal space}", journal = "JHEP", volume = "01", year = "2020", pages = "081", doi = "10.1007/JHEP01(2020)081", eprint = "1908.04543", archivePrefix = "arXiv", primaryClass = "math-ph", SLACcitation = "%%CITATION = ARXIV:1908.04543;%%" }

Solvable models, QFT on non-commutative spaces, special functions