Feasibility studies for triple Higgs boson production in the $4b2\tau$ final state at the FCC-hh

A feasibility study of triple Higgs boson production at the Future Circular Collider proton-proton stage, FCC-hh, at a centre-of-mass energy $\sqrt{s}=84$ TeV has been performed in the $4b2\tau$ final state. SM-like $HHH$ signals are studied. Both the mixed leptonic-hadronic ($\tau_{\text{lep}}\tau_{\text{had}}$) and fully hadronic ($\tau_{\text{had}}\tau_{\text{had}}$) decay modes of the $\tau$-lepton pair are considered in orthogonal channels and combined. For each channel, a neural network classifier is trained to improve experimental sensitivity and discriminate between the $HHH$ signals and Standard Model backgrounds, utilising kinematic features from the reconstructed objects and Higgs boson candidates in the events. Considering a conservative jet flavour identification uncertainty (uncertainty Scenario I), a 95% upper cross-section limit of $3 \times \sigma_{HHH}^{SM}$ is expected, with the $\tau_{\text{had}}\tau_{\text{had}}$ channel leading in sensitivity. The sensitivity of this final state to constrain the tri-linear and quartic Higgs self-couplings is evaluated for several uncertainty scenarios. Assuming the SM value of the trilinear self-coupling, a constraint is derived on the quartic self-coupling of between $-5.6$ and $21$ times the SM value at 68% CL. In combination with other triple-Higgs final states, this result demonstrates the potential of the FCC-hh to constrain the quartic Higgs self-coupling.