Positive amplitudes in the amplituhedron
Positive amplitudes in the amplituhedron
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振幅面体中的正振幅
DOI：
10.1007/jhep08(2015)030
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发表时间：
2014
影响因子：
5.4
通讯作者：
J. Trnka
中科院分区：
文献类型：
--
作者：
N. Arkani;A. Hodges;J. Trnka
平面N=4$$ \mathcal{N}=4 $$ SYM中散射振幅的全圈被积函数由振幅面边界上具有对数奇点的“振幅形式”确定。在本说明中，我们提供了强有力的证据，一个新的惊人的性质的超级振幅，我们推测是真实的所有循环订单：振幅形式是积极的，当评估内的振幅面体。由于与振幅面体几何相关的超级振幅的自然“玻色子化”，该陈述被合理地制定。然而，这种积极性是不明显的，在任何目前的方法散射振幅，特别是不是在细胞的振幅面体有关的壳图和积极的格拉斯曼。令人惊讶的积极性的形式表明存在一个“双振幅面体”的制定，这一特点将是显而易见的。我们还建议，积极性是与一个扩展的图片的振幅面体几何形状，与振幅面体坐在一个共同的维度一个表面分离的“法律的”和“非法”的局部奇异的振幅。我们说明了这在几个简单的例子，获得新的表达式的振幅不与任何三角剖分，但以下在一个更不变的方式从一个全球性的观点的正几何。
A bstractThe all-loop integrand for scattering amplitudes in planar N=4$$ \mathcal{N}=4 $$ SYM is determined by an “amplitude form” with logarithmic singularities on the boundary of the amplituhedron. In this note we provide strong evidence for a new striking property of the superamplitude, which we conjecture to be true to all loop orders: the amplitude form is positive when evaluated inside the amplituhedron. The statement is sensibly formulated thanks to the natural “bosonization” of the superamplitude associated with the amplituhedron geometry. However this positivity is not manifest in any of the current approaches to scattering amplitudes, and in particular not in the cellulations of the amplituhedron related to on-shell diagrams and the positive grassmannian. The surprising positivity of the form suggests the existence of a “dual amplituhedron” formulation where this feature would be made obvious. We also suggest that the positivity is associated with an extended picture of amplituhedron geometry, with the amplituhedron sitting inside a co-dimension one surface separating “legal” and “illegal” local singularities of the amplitude. We illustrate this in several simple examples, obtaining new expressions for amplitudes not associated with any triangulations, but following in a more invariant manner from a global view of the positive geometry.