Enlarging the scope of resonant di-Higgs searches: Hunting for Higgs-to-Higgs cascades in $4b$ final states at the LHC and future colliders

We extend the coverage of resonant di-Higgs searches in the $b \bar{b} b \bar{b}$ final state to the process $p p \to H_1 \to H_2 H_2 \to b \bar{b} b \bar{b}$, where both $H_{1,2}$ are spin-$0$ states beyond the Standard Model. Such a process constitutes a joint discovery mode for the new states $H_1$ and $H_2$. We present the first sensitivity study of this channel, using public LHC data to validate our analysis. We also provide a first estimate of the sensitivity of the search for the HL-LHC and future facilities like the HE-LHC and FCC-hh. We analyze the discovery potential of this search for several non-minimal scalar sector scenarios: an extension of the SM with two extra singlet scalar fields, the two-Higgs-doublet model and a two-Higgs doublet model plus a singlet, which captures the scalar potential features of the NMSSM. We find that this channel represents a novel, very powerful probe for extended Higgs sectors, offering complementary sensitivity to existing analyses.