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All-loop Mondrian reduction of 4-particle amplituhedron at positive infinity
Nuclear Physics B ( IF 2.8 ) Pub Date : 2020-06-09 , DOI: 10.1016/j.nuclphysb.2020.115086
Junjie Rao

This article introduces a systematic framework to understand (not to derive yet) the all-loop 4-particle amplituhedron in planar N=4 SYM, utilizing both positivity and the Mondrian diagrammatics. Its key idea is the simplest one so far: we can decouple one or more sets of loop variables (x,y,z,w) from the rest by just setting these variables to either zero or infinity so that their relevant positivity conditions are trivialized, then the all-loop consistency requires that we get lower loop amplituhedra as “residues”. These decoupling relations connect higher loop DCI integrals with the lower ones, enabling us to identify their coefficients starting from the 3-loop case. And surprisingly, the delicate mechanism of this process is the simple Mondrian rule D=X+Y, which forces those visually non-Mondrian DCI integrals to have the correct coefficients such that the amplituhedron can exactly reduce to the lower loop one. Examples cover all DCI integrals at L=3,4,5,6, especially, the subtle 6-loop coefficients +2 and 0 are neatly explained in this way.



中文翻译:

正无限远处4粒子放大体的全环蒙德里安还原

本文介绍了一个系统的框架,以了解(尚不得出)平面中的全环4粒子扩增体 ñ=4SYM,同时利用阳性和蒙德里安图表。它的关键思想是迄今为止最简单的思想:我们可以解耦一组或多组循环变量Xÿžw通过仅将这些变量设置为零或无穷大以使它们的相关正性条件变得微不足道,则所有循环的一致性要求我们将较低的循环振幅作为“残基”。这些去耦关系将较高回路的DCI积分与较低回路的DCI积分连接起来,使我们能够从3回路情况开始确定它们的系数。令人惊讶的是,此过程的微妙机制是简单的蒙德里安法则d=X+ÿ,这会迫使那些视觉上非Mondrian DCI积分具有正确的系数,从而使放大的面体可以精确地减小到较低的环路之一。示例涵盖了所有DCI积分大号=3456尤其是,用这种方法巧妙地解释了微妙的6环系数+2和0。

更新日期:2020-06-09
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