Correlating Resonant Di-Higgs and Tri-Higgs Production to $H\to VV$ in the 2HDM

The observation of resonant di-Higgs production, which would strongly suggest the existence of a new heavy neutral scalar $H$, has been searched for extensively at the LHC. In the two-Higgs doublet model (2HDM) with $m_H\gg m_h$, where $h$ is the Higgs boson of mass 125 GeV observed at the LHC, we show that a direct correlation emerges between Br$(H\to hh)$ and Br$(H\to VV)$, with $V=Z,W$, which depends only on $m_H$ (and $m_V$). In particular, for heavy scalar masses between 500,GeV and 1,TeV, we find that Br($H\to hh$)/ Br($H\to ZZ)\approx 9.5$. The origin of this prediction is most transparent in the Higgs basis, where the term in the scalar potential proportional to ${\mathcal{H}}_1^{†}{\mathcal{H}}_1{\mathcal{H}}_1^{†}{\mathcal{H}}_2$ (and its hermitian conjugate) generates the leading contributions to the $Hhh$ and $Hhhh$ couplings in the decoupling limit of the 2HDM. Moreover, the latter coupling governs the resonant prompt tri-Higgs production via $H\to hhh$, which is also directly correlated to $H\to hh$ (and $H\to VV$), and can yield rates large enough to be measured at the High-Luminosity LHC.