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The production of three Higgs bosons at hadron colliders can be enhanced by a double-resonant effect in the ℤ2-symmetric two-real-singlet extension of the Standard Model, making it potentially observable in future LHC runs. The production rate is maximized for large scalar couplings, which prompts us to carefully reconsider the perturbativity constraints on the theory. This leads us to construct a new set of 140 benchmark points that have a triple Higgs boson production cross-section at least 100 times larger than the SM value. Furthermore, we study the dynamics of the electroweak phase transition, both analytically at leading order, and numerically without the high-temperature expansion. Both analyses indicate that a first-order phase transition is incompatible with the requirement that both singlets have a non-zero vev in the present-day vacuum, as required by doubly-enhanced triple Higgs boson production. Allowing instead one of the singlets to remain at zero field value opens up the possibility of a first-order phase transition, while di-Higgs boson production can still be enhanced by a (single) resonance. 

Abstract We here report on the progress of the HHH Workshop, that took place in Dubrovnik in July 2023. After the discovery of a particle that complies with the properties of the Higgs boson of the Standard Model, all Standard Model (SM) parameters are in principle determined. However, in order to verify or falsify the model, the full form of the potential has to be determined. This includes the measurement of the triple and quartic scalar couplings. We here report on ongoing progress of measurements for multi-scalar final states, with an emphasis on three SM-like scalar bosons at 125$$\,\text {Ge}\hspace{-.08em}\text {V}$$ $\text{Ge}\text{V}$, but also mentioning other options. We discuss both experimental progress and challenges as well as theoretical studies and models that can enhance such rates with respect to the SM predictions.