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

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;%%"
}

Keywords:

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