The identification of interesting substructures within jets is an important tool for searching for new physics and probing the Standard Model at colliders. Many of these substructure tools have previously been shown to take the form of optimal transport problems, in particular the Energy Mover's Distance (EMD). In this work, we show that the EMD is in fact the natural structure for comparing collider events, which accounts for its recent success in understanding event and jet substructure. We then present a Shape Hunting Algorithm using Parameterized Energy Reconstruction (SHAPER), which is a general framework for defining and computing shape-based observables. SHAPER generalizes N-jettiness from point clusters to any extended, parametrizable shape. This is accomplished by efficiently minimizing the EMD between events and parameterized manifolds of energy flows representing idealized shapes, implemented using the dual-potential Sinkhorn approximation of the Wasserstein metric. We show how the geometric language of observables as manifolds can be used to define novel observables with built-in infrared-and-collinear safety. We demonstrate the efficacy of the SHAPER framework by performing empirical jet substructure studies using several examples of new shape-based observables.

翻译：识别喷注内部有趣子结构是对撞机上寻找新物理、探究标准模型的重要工具。此前已有研究表明，许多子结构工具可归结为最优输运问题，特别是能量移动距离（EMD）。本文证明，EMD实际上是比较对撞事件的自然结构，这也解释了为何它近期在理解事件与喷注子结构方面取得成功。随后我们提出一种基于参数化能量重建的形状搜寻算法（SHAPER），这是一种定义和计算形状观测量的一般框架。SHAPER将N-喷注性从点簇推广至任意可参数化扩展形状。该算法通过最小化事件与代表理想形状的能量流参数化流形之间的EMD实现，计算中采用Wasserstein度量的对偶势Sinkhorn近似。我们展示了如何利用几何语言将观测量视为流形，从而定义具有固有一致红外交变安全性的新型观测量。最后通过多个新型形状观测量示例进行实证喷注子结构研究，验证了SHAPER框架的有效性。