Generalized Two-Higgs-Doublet Model (g2HDM)

• g2HDM is a renormalizable extension of the Standard Model featuring two Higgs doublets with independent, non-diagonal Yukawa couplings that allow tree-level FCNH interactions.
• It produces a rich scalar spectrum including h, H, A, and H⁺, with a mixing angle that aligns the light Higgs with SM couplings while permitting new CP-violating and flavor-changing effects.
• The model’s phenomenology is tested via kaon and B-meson decays, EDM constraints, and rare decay processes, offering insights into TeV-scale new physics.

The generalized Two-Higgs-Doublet Model (g2HDM) is a renormalizable extension of the Standard Model (SM) featuring two scalar doublets with independent, generally non-diagonal Yukawa couplings to all fermions, and a scalar potential constructed without imposing a discrete $Z_2$ symmetry. This structure admits tree-level flavor-changing neutral Higgs (FCNH) interactions and an array of possible CP-violating effects, resulting in a phenomenologically rich framework with implications for kaon physics, rare decays, and new physics (NP) searches up to the TeV scale (Hou et al., 2022).

1. Model Structure: Scalar Sector and Yukawa Couplings

In the g2HDM, the Higgs basis is defined such that $\Phi_1$ acquires a vacuum expectation value (VEV) $\langle\Phi_1^0\rangle=v/\sqrt2$ and is responsible for the generation of all SM fermion masses, while $\Phi_2$ has vanishing VEV and mediates new interactions. The most general, renormalizable Yukawa Lagrangian in the mass basis is given by: $$-\mathcal{L}_Y = \bar Q_L\,\tilde\Phi_1\,Y^u\,u_R + \bar Q_L\,\Phi_1\,Y^d\,d_R + \bar L_L\,\Phi_1\,Y^\ell\,\ell_R + \bar Q_L\,\tilde\Phi_2\,\rho^u\,u_R + \bar Q_L\,\Phi_2\,\rho^d\,d_R + \bar L_L\,\Phi_2\,\rho^\ell\,\ell_R + \text{h.c.}$$ where $Y^F$ are the SM Yukawa matrices, and $\rho^F$ are new, generically non-diagonal 3×3 matrices introducing extra Yukawa couplings. After electroweak symmetry breaking (EWSB), fermion masses derive solely from $Y^F$, via $M^u=\tfrac{v}{\sqrt2} Y^u$, $M^d=\tfrac{v}{\sqrt2} Y^d$, and $M^\ell=\tfrac{v}{\sqrt2} Y^\ell$. The matrices $\rho^F$ are not diagonal in the physical basis, sourcing tree-level FCNH interactions for neutral scalars (Hou et al., 2022).

The physical scalar spectrum—after diagonalization in the CP-conserving basis—comprises a light CP-even Higgs $h$ resembling the SM Higgs, a heavy CP-even $H$, a CP-odd $A$, and a charged scalar $H^\pm$. Higgs-fermion interactions, including both diagonal and FCNH structures, are controlled by the mixing angle $\gamma$ between the two doublets: $$-\mathcal{L}_{h,H,A} = \frac{1}{\sqrt2}\,\bar f_i \left[ (\lambda^f_i\,s_\gamma+\rho^f_{ij}\,c_\gamma)h +(\lambda^f_i\,c_\gamma-\rho^f_{ij}\,s_\gamma)H -i\,\mathrm{sgn}(Q_f)\,\rho^f_{ij}A \right]P_R\,f_j + \text{h.c.}$$ Here $\lambda^f_i = \sqrt{2} m^f_i/v$, $s_\gamma=\sin\gamma$, $c_\gamma=\cos\gamma$. For the charged scalar,

$$-\mathcal{L}_{H^\pm} = \bar u_i\left[(V\,\rho^d)_{ij}P_R-(\rho^{u\dagger}V)_{ij}\,P_L\right]d_j\,H^+ + \bar\nu_i\,\rho^\ell_{ij}P_R\,\ell_j\,H^+ + \text{h.c.}$$

The alignment limit, $c_\gamma\to0$, ensures that $h$ possesses SM-like couplings while all FCNH couplings involving $h$ are suppressed, preserving compatibility with LHC Higgs measurements (Hou et al., 2022).

2. Higgs Scalar Potential and Mass Spectrum

The g2HDM scalar potential in the Higgs basis assumes the most general, gauge-invariant, real form (CP-conserving): $$\begin{aligned} V &= Y_1(\Phi_1^\dagger\Phi_1) + Y_2(\Phi_2^\dagger\Phi_2) + \left[Y_3\,\Phi_1^\dagger\Phi_2 + \mathrm{h.c.}\right] \ &\quad + \tfrac12\lambda_1(\Phi_1^\dagger\Phi_1)^2 + \tfrac12\lambda_2(\Phi_2^\dagger\Phi_2)^2 + \lambda_3(\Phi_1^\dagger\Phi_1)(\Phi_2^\dagger\Phi_2) + \lambda_4(\Phi_1^\dagger\Phi_2)(\Phi_2^\dagger\Phi_1) \ &\quad + \left[\tfrac12\lambda_5(\Phi_1^\dagger\Phi_2)^2 + \lambda_6(\Phi_1^\dagger\Phi_1)(\Phi_1^\dagger\Phi_2) + \lambda_7(\Phi_2^\dagger\Phi_2)(\Phi_1^\dagger\Phi_2) + \text{h.c.}\right] \end{aligned}$$ After symmetry breaking, the physical masses of the scalars are: $$\begin{aligned} m^2_{H^\pm} &= Y_2 + \tfrac12\lambda_3 v^2 \ m^2_A &= m^2_{H^\pm} + \tfrac12(\lambda_4 - \lambda_5)v^2 \ m^2_{h,H} &= \tfrac12 \left( [\lambda_1 + \lambda_5]v^2 \mp \sqrt{ v^4 (\lambda_1 - \lambda_5)^2 + 4v^4\lambda_6^2 } \right) \end{aligned}$$ with mixing angle

$$\tan 2\gamma = \frac{2\lambda_6 v^2}{(\lambda_1 - \lambda_5) v^2}$$

This potential is subject to theoretical constraints (vacuum stability, perturbative unitarity), and experimental constraints from electroweak and flavor observables (Hou et al., 2022).

3. FCNH Processes and Rare Kaon Decays

The presence of tree-level FCNH couplings in g2HDM significantly affects rare kaon processes, providing powerful probes of the new scalar sector and extra Yukawa couplings. The dominant effects involve charged Higgs–top loops and can be analyzed as follows:

• $\varepsilon_K$ from $\Delta S=2$: The NP contribution arises via $H^+$-mediated and $W H^+$-mediated box diagrams, with effective Hamiltonian:

$$\mathcal H_{\rm eff} = C_{HH}(\bar d\gamma^\mu P_L s)^2 + C_{WH}(\bar d\gamma^\mu P_L s)^2 + \text{h.c.}$$

The coefficients depend on products such as $V_{i1}^*\rho_{ij}\rho_{kj}^*V_{k2}$ and loop functions $F_{1,2}(x_i)$. Imposing $|\varepsilon_K^{\rm NP}|<0.2\times10^{-3}$ constrains $\rho_{ct}$ and $\rho_{tt}$, with $|\rho_{ct}| \lesssim 0.06$ for $m_{H^+}=400$ GeV, relaxing to $\sim 0.2$ for $m_{H^+}=1$ TeV (Hou et al., 2022).

• $\varepsilon'/\varepsilon$ ($\Delta S=1$ penguins): Charged Higgs penguins contribute to four-fermion operators $Q^q_{VLL}$, $Q^q_{VLR}$, $Q^d_{SLR}$ and the chromo-dipole operator $O_{8g}$. The Wilson coefficients are linear in bilinears of $\rho_{ij}$ and loop functions $G_{1,12,Z}$. NP can induce up to $\mathcal O(10^{-4})$ shifts in $\varepsilon'/\varepsilon$ for $\rho_{tt}, \rho_{tc} \sim \mathcal O(1)$.
• $K^+ \to \pi^+\nu\bar{\nu}$ and $K_L\to\pi^0\nu\bar{\nu}$: The rare decays are governed by NP contributions to $X_{\text{eff}}$. $H^+$-top penguins provide:

$$C_{LL}^{ab} = -\frac{\delta_{ab}}{16\pi^2} (V^\dagger \rho^u)_{2i} (\rho^{u\dagger} V)_{i1} G_Z \left( \frac{m_i^2}{m_{H^+}^2} \right)$$

Critically, the $K^+$ mode is uniquely sensitive to $H^+$ at the TeV scale due to a double CKM enhancement:

$$\frac{C_{LL}}{V_{ts}^* V_{td}} \propto \left[ \rho_{tt} + \frac{V_{cs}^*}{V_{ts}^*} \rho_{ct} \right] \left[ \rho_{tt}^* + \frac{V_{cd}}{V_{td}} \rho_{ct}^* \right] G_Z\left(\frac{m_t^2}{m_{H^+}^2}\right)$$

This structure allows, for $m_{H^+}=1$ TeV and $|\rho_{ct}|\sim 0.2$, the branching ratio $\mathcal B(K^+ \to \pi^+ \nu \bar{\nu})$ to saturate the current NA62 upper bound $[1.06^{+0.40}_{-0.34}] \times 10^{-10}$ (Hou et al., 2022).

• $K_{L,S} \to \mu^+\mu^-$: Short-distance contributions involve the same $ZH^+$ penguins, but large long-distance uncertainties dilute sensitivity beyond kaon and $B$-meson constraints.

4. Global Parameter Correlations, Benchmark Scans, and Flavor Constraints

A global scan of the g2HDM parameter space over $|\rho_{tt}|, |\rho_{tc}| \in [0,1]$, $|\rho_{ct}| \in [0,0.3]$, and arbitrary phases (Hou et al., 2022), with $m_{H^+}=400,\,1000$ GeV, imposing constraints from:

• $B_s$ and $B_d$ mixing, $S_{\psi K_S}$, $S_{\psi\phi}$,
• $\mathcal B(B_s\to X_s\gamma)$, $\mathcal B(B_s\to\mu^+\mu^-)$,
• $|\varepsilon_K^{\rm NP}| < 2\times10^{-4}$,
• $|\varepsilon'/\varepsilon|_{\rm NP} < 10^{-4}$,
• NA62 bound on $\mathcal B(K^+)$,

yields, for $m_{H^+}=400$ GeV, $|\rho_{ct}| \lesssim 0.06$, $|\rho_{tt}| \lesssim 1$, $|\rho_{tc}| \lesssim 1$; and for $m_{H^+}=1000$ GeV, $|\rho_{ct}| \lesssim 0.2$ (Hou et al., 2022).

$K^+ \to \pi^+\nu\bar{\nu}$ is the most sensitive probe of $\rho_{ct}$ and $m_{H^+}$, driving tight correlations with $\varepsilon_K$ and, to a lesser extent, $B_s\to\mu^+\mu^-$. For TeV-scale $H^+$, enhancement in $\mathcal B(K^+)$ typically anti-correlates with a slight suppression in $\mathcal B(B_s\to\mu^+\mu^-)$, offering cross-validation between kaon and $B$ physics as experimental precision improves.

5. Complementarity with B Physics and EDM Probes

Kaon processes are complemented by $B$-physics and electric dipole moment (EDM) constraints in restricting the parameter space of the g2HDM. $|\varepsilon_K^{\rm NP}| < 2\times10^{-4}$ is already competitive with $B$-meson mixing constraints for $\rho_{ct}$, particularly for lighter $H^+$ masses. The unique double CKM enhancement in $K^+ \to \pi^+\nu\bar{\nu}$ renders this mode highly sensitive—even at the TeV scale—while the alignment limit remains consistent with existing collider searches for SM-like $h(125)$ (Hou et al., 2022).

When supplemented with EDM data, the allowed region in $\rho$ couplings is further restricted. However, top-associated couplings ($\rho_{tt}$, $\rho_{tc}$, $\rho_{ct}$) remain the most weakly constrained by direct searches and EDMs, provided an approximate SM-like Yukawa hierarchy.

6. Phenomenological Summary and Outlook

The g2HDM, by lifting the $Z_2$ constraint and permitting generic extra Yukawa couplings, realizes an SM-like $h(125)$ while allowing rich CP- and flavor-violating phenomena through the extended Higgs sector. Kaon mixing and rare decays—especially $K^+ \to \pi^+\nu\bar{\nu}$—are exquisitely sensitive to the up-type off-diagonal $\rho^u$, with direct implications for charged Higgs scales up to several TeV. This unique complementarity of $K$ and $B$ physics, along with EDM and direct LHC searches, provides a multifaceted probe of the g2HDM flavor structure, making g2HDM both a compelling NP scenario and a prime target for the next generation of flavor and intensity frontier experiments (Hou et al., 2022).