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：喷流内部有趣子结构的识别是搜寻新物理及探测对撞机中标准模型的重要工具。此前已有研究表明，许多这类子结构工具可归结为最优输运问题，特别是能量输运距离（EMD）。在本工作中，我们证明EMD实际上是对撞事件比较的自然结构，这解释了它近期在理解事件与喷流子结构方面取得的成功。随后，我们提出基于参数化能量重构的形状搜索算法（SHAPER），这是一种定义和计算基于形状的可观测量的通用框架。SHAPER将N-jettiness从点簇推广至任意可扩展、可参数化的形状。这一目标通过高效最小化事件与代表理想形状的参数化能量流流形之间的EMD实现，并采用基于对偶势能的Wasserstein度量Sinkhorn近似。我们展示了如何利用可观测量作为流形的几何语言，定义具有内禀红外-共线安全性的新型可观测量。通过使用若干新型基于形状的可观测量进行实证喷流子结构研究，我们验证了SHAPER框架的有效性。