Can You Hear the Shape of A Jet?

\begin{abstract}

The identification of interesting substructures within jets is an important tool to search for new physics and probe the Standard Model. In this paper, we present a new set of shape-based observables, $G$-shapeliness, which generalizes the $N$-jettiness from point clusters to any extended shape $G$. We show that the K-Deep Simplex dictionary learning framework is, with some modifications, dual to the 2-Wasserstein metric on energy flows, from which observables can be interpreted as the optimal transport distance to an idealized energy flow distribution. We then use the modified KDS framework to compute $G$-shapeliness values for a toy model in order to discriminate between topology-distinct events, and then finally apply the framework to analyze jet substructure.

\end{abstract}