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寻找新物理及探测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，特别是能量移动距离（EMD）。本文证明，EMD实际上是比较对撞事件的天然结构，这解释了近期其在理解事件与喷注子结构方面的成功。我们随后提出一种利用参数化能量重构的形状搜寻算法（SHAPER），这是一个定义和计算形状可观测量的一般性框架。SHAPER将N-jettiness从点团簇推广至任意可参数化延展形状。其实现方式是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，并采用Wasserstein度量的双势垒Sinkhorn近似。我们阐释了如何利用可观测量作为流形的几何语言，定义新型具有内禀红外-共线安全性的可观测量。通过若干新形状可观测量的实证喷注子结构研究，我们展示了SHAPER框架的有效性。