Feynman积分化为Schläfli函数积分
Reduction of Feynman integrals to integrals of Schläfli functions
摘要：
我们证明了具有任意数量的外部线和复数质量的离壳微扰振幅可以约化为广义Schläfli函数的$I$ -折叠积分，其中$I$是相应真空图中的线的数量，它与外部线的数量无关. Schläfli函数作为定义单形的超平面参数的函数的球面单形的体积的解析延拓而获得。这些功能有很好的和深入研究的分析，几何和数论性质。它们具有Gauss-Manin联系，在保形情况下用迭代积分表示。我们的表示为微扰论中粒子位形空间的几何学提供了新的思路。
英文摘要：
We show that off-shell perturbative amplitudes with arbitrary number of external lines and complex masses can be reduced to $I$-fold integrals of the generalized Schläfli functions, where $I$ is the number of lines in the corresponding vacuum diagram which is independent of the number of external lines. The Schläfli functions are obtained as analytic continuation of the volume of the spherical simplex as a function of the hyperplane parameters that define the simplex. These functions have nice and thoroughly studied analytic, geometric and number theoretic properties. They possess Gauss-Manin connection and in conformal case are expressed by iterated integrals. Our representation sheds new light on geometry of particle configuration spaces in perturbation theory.
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