Multi-step Strong First-Order Electroweak Phase Transitions in the Inverted Type-I 2HDM: Parameter Space, Gravitational Waves, and Collider Phenomenology

We investigate the electroweak phase transition (EWPT) within the inverted Type-I two-Higgs-doublet model, where the observed $125\,\text{GeV}$ Higgs boson is identified as the heavier \textit{CP}-even scalar $H$. Through a comprehensive parameter-space scan consistent with current theoretical and experimental constraints, we identify regions supporting strong first-order EWPTs (SFOEWPTs), including multi-step transitions. We find that two-step SFOEWPTs occur as frequently as one-step transitions, while three-step transitions can occur, albeit rarely. Crucially, the parameter spaces inducing one-step and two-step transitions are partially yet significantly separated: one-step transitions restrict the charged Higgs mass and $\tanβ$ to $m_{H^\pm}\in[295,441]\,\text{GeV}$ and $\tanβ\in[4.2,8.8]$, whereas two-step transitions allow $m_{H^\pm}\in[100,350]\,\text{GeV}$ and $\tanβ\in[2.5,45.4]$. Notably, negative values of $\sin(β-α)$ arise almost exclusively in one-step scenarios. We present the calculation of gravitational wave (GW) signal-to-noise ratios (SNRs) at LISA for multi-step EWPTs, finding that detectable GW signals ($\text{SNR}>10$) predominantly emerge from two-step transitions. Furthermore, we demonstrate that the correlation between the vacuum uplifting measure $ΔF_0$ and $ξ_c$ persists in one-step transitions and breaks down in multi-step cases. Finally, we perform a dedicated collider analysis for representative SFOEWPT parameter points at the $1.5\,\text{TeV}$ CLIC, identifying $e^+ e^- \to H^+ H^- \to W^+ W^- hh$ as a promising discovery channel. Enhanced $h\toγγ$ branching ratios for negative $\sin(β-α)$ motivate two complementary golden final states, $W^+ W^- b\bar{b} τ^+ τ^-$ and $W^+ W^- b\bar{b}γγ$, which demonstrate high discovery potential due to negligible Standard Model backgrounds.