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Optimized probes of $CP$-odd effects in the $t \bar{t} h$ process at hadron colliders

We use machine learning (ML) and non-ML techniques to study optimized $CP$-odd observables, directly and maximally sensitive to the $CP$-odd $i \tilde κ\bar t γ^5 t h$ interaction at the LHC and prospective future hadron colliders using the final state with a Higgs boson and a top quark pair, $pp\to t\bar t h$, followed by semileptonic $t$ decays. We perform phase-space optimization of manifestly $CP$-odd observables ($\boldsymbol ω$), sensitive to the sign of $\tilde κ$, and constructed from experimentally accessible final state momenta. We identify a simple optimized linear combination $\boldsymbol α\cdot \boldsymbolω$ that gives similar sensitivity as the studied fully fledged ML models. Using $\boldsymbolα\cdot \boldsymbolω$ we project the expected sensitivities to $\tilde κ$ at HL-LHC, HE-LHC, and FCC-hh.