arXiv 1908.045432019-08-13math-phhep-thmath.MPnlin.SI

四维Moyal空间上自对偶$Φ^4$ QFT模型的解

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

Harald Grosse, Alexander Hock, Raimar Wulkenhaar

备注：: with an appendix by Robert Seiringer, 18 pages
发表信息：: JHEP 01 (2020) 081
DOI：: 10.1007/JHEP01(2020)081

摘要：

以前的精确解的平面扇区的自对偶$Φ^4$ -模型在4维Moyal空间建立了一个Fredholm积分方程的解决方案。本文对任意耦合常数$λ>-\frac{1}π$，用超几何函数求解Fredholm方程，从而完成了模型平面扇区的构造。我们证明了相互作用模型对于$|λ|<\frac{1}π$具有谱维数$4-2\frac{\arcsin(λπ)}π$.对于$λ>0$来说，正是这种降维避免了矩阵$Φ^4_4$模型的平凡性问题。在$λ$中建立了Fredholm解的幂级数逼近.出现的函数是由迭代积分定义的超代数，这里交替使用字母$0$和$-1$。我们确定的重正化参数，它给出了相同的标准化带状图扩展。

英文摘要：

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 $λ>-\frac{1}π$, 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 $4-2\frac{\arcsin(λπ)}π$ for $|λ|<\frac{1}π$. It is this dimension drop which for $λ>0$ avoids the triviality problem of the matricial $Φ^4_4$-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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