圆与三角形、NLSM 与 Tr($Φ^3$)
Circles and Triangles, the NLSM and Tr($Φ^3$)
AI总结：
本研究从非线性σ模型（NLSM）出发，借助圆与三角剖分的经典关联，证明了NLSM振幅可表述为带分子因子的曲面三角剖分之和，且该分子因子源自运动学平移后的Tr($Φ^3$)理论，还定义了曲面软极限并给出多软因子显式表达式。
摘要：
近期，Tr($Φ^3$) 理论的振幅与非线性σ模型（non-linear sigma model，NLSM）之间被发现存在一种出人意料的关联。由 Tr($Φ^3$) 理论的结合多面体/弦论表示自然引出的运动学变量简单平移，可得到所有圈阶的π介子振幅。在本短文中，我们从反方向为这一关联提供了基础动因与证明：从非线性σ模型出发，发现其可表述为带简单分子因子的曲面三角剖分之和。这利用了“圆”与“三角形”之间的古老关联——将方程 $y = \sqrt{1 - x^2}$ 既解释为圆上点的参数化，也解释为多边形三角剖分数目的生成函数。对分子因子的进一步简化表明，它们源自经运动学平移的 Tr($Φ^3$) 理论，并由此得到了 NLSM 振幅的全新热带表示。与 Tr($Φ^3$) 理论的关联定义了一种内蕴于曲面曲线的“曲面软极限”的自然概念。值得注意的是，基于这一定义，π介子振幅的软极限在被积函数层面就会通过明显的成对抵消直接归零。我们还给出了当任意数目的π介子取“曲面软”极限时，树图和圈图被积函数的多软因子的简单显式表达式。
英文摘要：
A surprising connection has recently been made between the amplitudes for Tr($Φ^3$) theory and the non-linear sigma model (NLSM). A simple shift of kinematic variables naturally suggested by the associahedron/stringy representation of Tr$(Φ^3$) theory yields pion amplitudes at all loops. In this note we provide an elementary motivation and proof for this link going in the opposite direction, starting from the non-linear sigma model and discovering its formulation as a sum over triangulations of surfaces with simple numerator factors. This uses an ancient connection between "circles" and "triangles", interpreting the equation $y = \sqrt{1 - x^2}$ both as parametrizing points on a circle as well as generating the number of triangulations of polygons. A further simplification of the numerator factors exposes them as arising from the kinematically shifted Tr($Φ^3$) theory, and gives rise to novel tropical representations of NLSM amplitudes. The connection to Tr$(Φ^3)$ theory defines a natural notion of "surface-soft limit" intrinsic to curves on surfaces. Remarkably, with this definition, the soft limit of pion amplitudes vanishes directly at the level of the integrand, via obvious pairwise cancellations. We also give simple, explicit expressions for the multi-soft factors for tree and loop-level integrands in the limit as any number of pions are taken "surface-soft".
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