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arXiv 1604.08027hep-thmath.NT
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无质量标量费曼图：五圈及以后

Massless scalar Feynman diagrams: five loops and beyond

David J. Broadhurst

备注：
report of The Open University, Milton Keynes, England (UK), 1985, copy provided by John Gracey, http://cds.cern.ch/record/164890, LaTeX reproduction by Erik Panzer
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摘要：

本文发展了几种计算无质量标量费曼图的有力方法，即：求解$n=4-2ω$维中任意圈数的图的递推关系，发现并利用对称性来限制和计算$ω$中的泰勒级数，将切比雪夫多项式的三重和约化为黎曼zeta函数的乘积，并利用对称性来计算无质量标量费曼图。利用共形不变性来避免四维Racah系数。作为这些技术的力量的一个例子，我们评估所有的216个图，5个循环或更少，这给有限的贡献顺序$1/k^2$或$1/k^4$的传播的动量$k$在无质量的四维标量场理论。值得注意的是，只遇到5个基本数字：$ζ(3)$，$ζ(5)$，$ζ(7)$，$ζ(9)$和最对称图的值，计算为14个有效数字。可以想象，这些是唯一的无理数出现在6圈β函数。在得到这些结果的过程中，我们发现并仅部分解释了图之间的许多显著关系。

英文摘要：

Several powerful techniques for evaluating massless scalar Feynman diagrams are developed, viz: the solution of recurrence relations to evaluate diagrams with arbitrary numbers of loops in $n=4-2ω$ dimensions; the discovery and use of symmetry properties to restrict and compute Taylor series in $ω$; the reduction of triple sums over Chebyshev polynomials to products of Riemann zeta functions; the exploitation of conformal invariance to avoid four-dimensional Racah coefficients. As an example of the power of these techniques we evaluate all of the 216 diagrams, with 5 loops or less, which give finite contributions of order $1/k^2$ or $1/k^4$ to a propagator of momentum $k$ in massless four-dimensional scalar field theories. Remarkably, only 5 basic numbers are encountered: $ζ(3)$, $ζ(5)$, $ζ(7)$, $ζ(9)$ and the value of the most symmetrical diagram, which is calculated to 14 significant figures. It is conceivable that these are the only irrationals appearing in 6-loop beta functions. En route to these results we uncover and only partially explain many remarkable relations between diagrams.

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