揭示圈积分的阈值结构
Exposing the threshold structure of loop integrals
AI总结：
本文证明仅能量守恒即可对Feynman图阈值奇点结构施加强约束，将圈积分表示为非单纯凸锥的Fourier变换，并开发出产生Cross-Free Family三维表示的封闭积分算法，其奇点结构完全由图论概念表达。
摘要：
理解决定振幅红外行为的物理定律是一个长期且具有前沿性的问题。在本文中，我们表明仅能量守恒就意味着对Feynman图的阈值奇点结构施加了强约束。特别地，我们证明了这意味着圈积分可以用非单纯凸锥的Fourier变换来表示。随后，我们设计了一种三角剖分，该剖分具有直接的图论解释，即通过一种直接的边收缩操作来实现。我们利用它开发了一种算法程序，能够以封闭形式执行Fourier积分，从而产生了新颖的Cross-Free Family三维圈积分表示。与TOPT和LTD表示不同，其奇点结构完全且优雅地用连通性和交叉的图论概念来表达。这些结果可用于对红外有限散射理论进行分类、数值计算圈积分以及简化阈值正则化过程。
英文摘要：
The understanding of the physical laws determining the infrared behaviour of amplitudes is a longstanding and topical problem. In this paper, we show that energy conservation alone implies strong constraints on the threshold singularity structure of Feynman diagrams. In particular, we show that it implies a representation of loop integrals in terms of Fourier transforms of non-simplicial convex cones. We then engineer a triangulation that has a direct diagrammatic interpretation in terms of a straightforward edge-contraction operation. We use it to develop an algorithmic procedure that performs the Fourier integrations in closed form, yielding the novel Cross-Free Family three-dimensional representation of loop integrals. Contrary to the TOPT and LTD representations, its singularity structure is entirely and elegantly expressed in terms of the graph-theoretic notions of connectedness and crossing. These results can be used to classify infrared-finite scattering theories, numerically evaluate loop integrals and to simplify threshold regularisation procedures.
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