Recently methods of graph neural networks (GNNs) have been applied to solving the problems in high energy physics (HEP) and have shown its great potential for quark-gluon tagging with graph representation of jet events. In this paper, we introduce an approach of GNNs combined with a HaarPooling operation to analyze the events, called HaarPooling Message Passing neural network (HMPNet). In HMPNet, HaarPooling not only extracts the features of graph, but embeds additional information obtained by clustering of k-means of different particle features. We construct Haarpooling from five different features: absolute energy $\log E$, transverse momentum $\log p_T$, relative coordinates $(\Delta\eta,\Delta\phi)$, the mixed ones $(\log E, \log p_T)$ and $(\log E, \log p_T, \Delta\eta,\Delta\phi)$. The results show that an appropriate selection of information for HaarPooling enhances the accuracy of quark-gluon tagging, as adding extra information of $\log P_T$ to the HMPNet outperforms all the others, whereas adding relative coordinates information $(\Delta\eta,\Delta\phi)$ is not very effective. This implies that by adding effective particle features from HaarPooling can achieve much better results than solely pure message passing neutral network (MPNN) can do, which demonstrates significant improvement of feature extraction via the pooling process. Finally we compare the HMPNet study, ordering by $p_T$, with other studies and prove that the HMPNet is also a good choice of GNN algorithms for jet tagging.

翻译：近年来，图神经网络方法已被应用于解决高能物理中的问题，并在基于喷注事件图表示的夸克-胶子标记领域展现出巨大潜力。本文提出一种结合HaarPooling操作的图神经网络分析方法，称为HaarPooling消息传递神经网络（HMPNet）。在HMPNet中，HaarPooling不仅提取图的特征，还嵌入了通过不同粒子特征的k-means聚类获得的附加信息。我们基于五种不同特征构建HaarPooling：绝对能量$\log E$、横动量$\log p_T$、相对坐标$(\Delta\eta,\Delta\phi)$、混合特征$(\log E, \log p_T)$以及$(\log E, \log p_T, \Delta\eta,\Delta\phi)$。结果表明，适当选择HaarPooling的信息可提高夸克-胶子标记的精度：向HMPNet添加$\log p_T$的额外信息性能优于其他所有方案，而添加相对坐标信息$(\Delta\eta,\Delta\phi)$效果不显著。这意味着通过HaarPooling添加有效粒子特征可获得比单纯消息传递神经网络（MPNN）更优的结果，充分展现了池化过程对特征提取的显著改进。最后，我们按$p_T$排序将HMPNet研究与其他研究进行比较，证明HMPNet也是喷注标记任务中图神经网络算法的优质选择。