Testing CP properties of the Higgs boson coupling to $\tau$ leptons with heterogeneous graphs

We explore the feasibility of measuring the CP properties of the Higgs boson coupling to $\tau$ leptons at the High Luminosity Large Hadron Collider (HL-LHC). Employing detailed Monte Carlo simulations, we analyze the reconstruction of the angle between $\tau$ lepton planes at the detector level, accounting for various hadronic $\tau$ decay modes. Considering standard model backgrounds and detector resolution effects, we employ three Deep Learning (DL) networks, Multi-Layer Perceptron (MLP), Graph Convolution Network (GCN), and Graph Transformer Network (GTN) to enhance signal-to-background separation. To incorporate CP-sensitive observables into Graph networks, we construct Heterogeneous graphs capable of integrating nodes and edges with different structures within the same framework. Our analysis demonstrates that GTN exhibits superior efficiency compared to GCN and MLP. Under a simplified detector simulation analysis, MLP can exclude CP mixing angle larger than $20^\circ$ at $68\%$ confidence level (CL), while GCN and GTN can achieve exclusions at $90\%$ CL and $95\%$ CL, respectively with $\sqrt{s}=14$~TeV and $\mathcal{L}$$=100\rm { fb}^{-1}$. Furthermore, the DL networks can achieve a significance of approximately $3\sigma$ in excluding the pure CP-odd state.