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：识别喷注内有趣的子结构是搜寻新物理以及在对撞机上探测标准模型的重要工具。此前诸多此类子结构工具已被证明可归结为最优输运问题，特别是能量输运距离（EMD）。本文表明，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注子结构方面取得成功的原因。我们进而提出一种基于参数化能量重构的形状搜寻算法（SHAPER），该算法是定义和计算基于形状的可观测量的通用框架。SHAPER将N-jettiness从点簇推广到任意可参数化的扩展形状。通过高效最小化事件与表示理想化形状的参数化能量流流形之间的EMD来实现这一推广，这一过程使用了瓦瑟斯坦度量的对偶势Sinkhorn近似。我们展示了如何利用观测量作为流形的几何语言来定义具有内置红外与共线安全性的新型观测量。通过多个基于形状的新型观测量实例进行经验性喷注子结构研究，我们验证了SHAPER框架的有效性。