论文标题

Feynman积分的Wilson-Loop $ d \ log $表示

The Wilson-loop $d \log$ representation for Feynman integrals

论文作者

He, Song, Li, Zhenjie, Tang, Yichao, Yang, Qinglin

论文摘要

我们介绍和研究Wilson-Loop $ {\ rm d} \ log $某些Feynman积分的表示,用于在$ {\ cal n}中散射幅度= 4 $ sym及以后,这使他们的评估完全直接。这种表示是由双威尔逊环图像动机,也可以通过循环积分的部分feynman参数化得出。我们首先将其介绍为最简单的单循环示例,四个维度的手性五角大楼和六个维度的三质量六角形,这些六个维度由两倍和三倍$ {\ rm d} \ log $积分表示,它们与彼此之间的相关性很好。对于多环示例,我们将$ l $ - 环的普通penta-ladders写为$ 2(l { - } 1)$ - fold $ {\ rm d} \ rm d} \ log $ log $某些单环积分的积分,因此,一旦后者就可以以系统的方式进行集成进行。特别是,我们将八点五角星插入式写入$ 2L $ -fold $ -fold $ {\ rm d} \ log $积分,其符号可以在不执行集成的情况下计算;我们还获得了这些积分的最后一个条目和符号字母。同样,我们计算并研究了七点双penta-ladder的象征,该符号由六角形的$ 2(l { - } 1)$ - 折叠积分表示。后者可以写入两倍$ {\ rm d} \ log $ intemall和边界项。我们通过求解某些积分方程式来评论我们的表示与微分方程的关系,并恢复梯子。

We introduce and study the Wilson-loop ${\rm d}\log$ representation of certain Feynman integrals for scattering amplitudes in ${\cal N}=4$ SYM and beyond, which makes their evaluation completely straightforward. Such a representation was motivated by the dual Wilson loop picture, and it can also be derived by partial Feynman parametrization of loop integrals. We first introduce it for the simplest one-loop examples, the chiral pentagon in four dimensions and the three-mass-easy hexagon in six dimensions, which are represented by two- and three-fold ${\rm d}\log$ integrals that are nicely related to each other. For multi-loop examples, we write the $L$-loop generalized penta-ladders as $2(L{-}1)$-fold ${\rm d}\log$ integrals of some one-loop integral, so that once the latter is known, the integration can be performed in a systematic way. In particular, we write the eight-point penta-ladder as a $2L$-fold ${\rm d}\log$ integral whose symbol can be computed without performing the integration; we also obtain the last entries and the symbol alphabet of these integrals. Similarly we compute and study the symbol of the seven-point double-penta-ladder, which is represented by a $2(L{-}1)$-fold integral of a hexagon; the latter can be written as a two-fold ${\rm d}\log$ integral plus a boundary term. We comment on the relation of our representation to differential equations and resum the ladders by solving certain integral equations.

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