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 $|\lambda| 0$ 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.

中文翻译：

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四维Moyal空间上自对偶Φ4 QFT模型的解

以前，自对偶 $\Phi^4$-模型在 4 维 Moyal 空间上的平面扇区的精确解被建立到 Fredholm 积分方程的解。本文针对任意耦合常数$\lambda>-\frac{1}{\pi}$求解了Fredholm方程的超几何函数，从而完成了模型平面扇区的构建。我们证明了相互作用模型的谱维数为 $4-2\frac{\arcsin(\lambda\pi)}{\pi}$ for $|\lambda| 0$ 避免了矩阵 $\Phi^4_4$-model 的平凡问题。我们还建立了对 $\lambda$ 中所有阶数的 Fredholm 解的幂级数近似。出现的函数是由迭代积分定义的超对数，这里交替使用字母 $0$ 和 $-1$。