Pseudoscalar top-Higgs coupling: Exploration of $\mathrm{CP}$-odd observables to resolve the sign ambiguity

We present a collection of $\mathrm{CP}$-odd observables for the process $pp\to t\,\left(\rightarrow b {\ell}^+ \nu_{\ell}\right) \bar{t}\,\left(\rightarrow \bar{b} {\ell}^-{\bar{\nu}}_{\ell}\right)\,H$ that are linearly dependent on the scalar ($\kappa_t$) and pseudoscalar ($\tilde{\kappa}_t$) top-Higgs coupling and hence sensitive to the corresponding relative sign. The proposed observables are based on triple product (TP) correlations that we extract from the expression for the differential cross section in terms of the spin vectors of the top and antitop quarks. In order to explore other possibilities, we progressively modify these TPs, first by combining them, and then by replacing the spin vectors by the lepton momenta or the $t$ and $\bar{t}$ momenta by their visible parts. We generate Monte Carlo data sets for several benchmark scenarios, including the Standard Model ($\kappa_t=1$, $\tilde{\kappa}_t=0$) and two scenarios with mixed $\mathrm{CP}$ properties ($\kappa_t=1$, $\tilde{\kappa}_t=\pm 1$). Assuming an integrated luminosity that is consistent with that envisioned for the High Luminosity Large Hadron Collider, using Monte Carlo-truth and taking into account only statistical uncertainties, we find that the most promising observable can disentangle the "$\mathrm{CP}$-mixed" scenarios with an effective separation of $\sim 19\sigma$. In the case of observables that do not require the reconstruction of the $t$ and $\bar{t}$ momenta, the power of discrimination is up to $\sim 13\sigma$ for the same number of events. We also show that the most promising observables can still disentangle the $\mathrm{CP}$-mixed scenarios when the number of events is reduced to values consistent with expectations for the Large Hadron Collider in the near term.