Type-Y THDM: Theory & Phenomenology

• Type-Y THDM is a CP-conserving two-Higgs-doublet model featuring distinct Yukawa assignments that suppress tree-level FCNCs.
• It presents a rich scalar sector with mixing angles α and β, generating a light SM-like Higgs and heavy scalar states, all tightly constrained by unitarity and electroweak precision tests.
• Its effective field theory extension incorporates higher-dimensional operators that modify Higgs couplings and offer insights into collider signatures and flavor dynamics.

The Type-Y Two-Higgs-Doublet Model (THDM), also referred to as the "flipped" model, is a CP-conserving extension of the Standard Model (SM) in which a second electroweak Higgs doublet is introduced and Yukawa interactions are organized via a softly broken $\mathbb{Z}_2$ symmetry. In this structure, up-type quarks and charged leptons couple exclusively to one Higgs doublet ($\Phi_2$), while down-type quarks couple to the other ($\Phi_1$), thereby guaranteeing the absence of tree-level flavor-changing neutral currents (FCNCs). The Type-Y THDM exhibits distinctive phenomenology in collider, Higgs, and flavor sectors compared to other $\mathbb{Z}_2$-symmetric 2HDM types, while remaining highly constrained by theory requirements and precision experimental data (Eberhardt, 2018, Dermisek et al., 30 May 2024, Cheng et al., 2014).

1. Scalar Sector and Potential

The renormalizable, CP-conserving scalar potential of the Type-Y THDM with a softly broken $\mathbb{Z}_2$ reads

$$\begin{aligned} V(\Phi_1, \Phi_2) =\; & m_{11}^2\,\Phi_1^\dagger\Phi_1 + m_{22}^2\,\Phi_2^\dagger\Phi_2 - m_{12}^2(\Phi_1^\dagger\Phi_2 + \Phi_2^\dagger\Phi_1) \ & + \tfrac{\lambda_1}{2}\,(\Phi_1^\dagger\Phi_1)^2 + \tfrac{\lambda_2}{2}\,(\Phi_2^\dagger\Phi_2)^2 + \lambda_3\,(\Phi_1^\dagger\Phi_1)(\Phi_2^\dagger\Phi_2) \ & + \lambda_4\,(\Phi_1^\dagger\Phi_2)(\Phi_2^\dagger\Phi_1) + \tfrac{\lambda_5}{2}\Big[(\Phi_1^\dagger\Phi_2)^2 + (\Phi_2^\dagger\Phi_1)^2\Big] \end{aligned}$$

where $m_{12}^2$ softly breaks the $\mathbb{Z}_2$ symmetry, and the parameters $m_{11}^2$, $m_{22}^2$, $\lambda_{1\text{--}5}$ define the scalar self-interactions and masses below electroweak symmetry breaking (EWSB) (Eberhardt, 2018, Dermisek et al., 30 May 2024). EWSB induces vacuum expectation values $v_1$ and $v_2$ in the neutral components of $\Phi_1$ and $\Phi_2$, such that $v^2 = v_1^2 + v_2^2 \simeq (246\, \text{GeV})^2$, and defines $\tan\beta = v_2/v_1$.

Diagonalization of the scalar mass matrices yields physical mass eigenstates: the light CP-even Higgs $h$ ($m_h \simeq 125$ GeV), heavy CP-even $H$, CP-odd $A$, and charged $H^\pm$, parameterized by two mixing angles $\alpha$ (CP-even sector) and $\beta$.

The scalar potential constraints include:

• Boundedness from below: $\lambda_1 > 0$, $\lambda_2 > 0$, $\lambda_3 > -\sqrt{\lambda_1\lambda_2}$, $$\lambda_3+\lambda_4-|\lambda_5| > -\sqrt{\lambda_1\lambda_2}$$.
• Perturbative unitarity: $|a_i|<8\pi$ for all scalar scattering channels.
• Global minimum at the EWSB vacuum (Eberhardt, 2018, Dermisek et al., 30 May 2024).

Beyond dimension-four, the Type-Y 2HDM Effective Field Theory generalizes the potential to include all allowed dimension-six ("$\phi^6$") operators consistent with the $\mathbb{Z}_2$ symmetry (Dermisek et al., 30 May 2024).

2. Yukawa Sector and $\mathbb{Z}_2$ Charge Assignment

The Type-Y (flipped) Yukawa structure arises from discrete $\mathbb{Z}_2$ charges assigned such that $\Phi_1 \to -\Phi_1$, $\Phi_2 \to +\Phi_2$, with SM fermions charged as: $q_L (+)$, $u_R (+)$, $d_R (-)$, $\ell_L (+)$, $e_R (+)$. The renormalizable Yukawa interactions are

$$\mathcal{L}_{\rm Yuk}^{\rm Y} = -y_u\,\overline{q}_L\,u_R\,\tilde\Phi_2 - y_d\,\overline{q}_L\,d_R\,\Phi_1 - y_e\,\overline{\ell}_L\,e_R\,\Phi_2 + \text{h.c.}$$

where $\tilde\Phi_i \equiv i \sigma^2 \Phi_i^*$. Thus, up-type quarks and charged leptons couple to $\Phi_2$, down-type quarks to $\Phi_1$ (Cheng et al., 2014, Dermisek et al., 30 May 2024).

The tree-level Higgs-fermion couplings, expressed via scaling factors $\kappa_f^\phi$ relative to SM Yukawas, are:

Coupling	$h$	$H$	$A$
$u$	$\frac{\cos\alpha}{\sin\beta}$	$\frac{\sin\alpha}{\sin\beta}$	$\cot\beta$
$d$	$-\frac{\sin\alpha}{\cos\beta}$	$\frac{\cos\alpha}{\cos\beta}$	$\tan\beta$
$\ell$	$\frac{\cos\alpha}{\sin\beta}$	$\frac{\sin\alpha}{\sin\beta}$	$-\cot\beta$

The charged Higgs interactions are given by: $$\mathcal{L}_{H^\pm} = \frac{\sqrt{2}}{v} H^+ \left[ \bar u \left(m_u \cot\beta\,P_L + m_d \tan\beta\,P_R \right)d + \bar\nu\, m_\ell \cot\beta\, P_R\, \ell \right] + \text{h.c.}$$ (Eberhardt, 2018, Cheng et al., 2014).

In the EFT extension, all $\psi^2\phi^3$ dimension-six operators permitted by the Type-Y charge assignment are retained, which, after basis rotation, clarify correlations in Higgs coupling deviations and new contact interactions (Dermisek et al., 30 May 2024).

3. Mass Spectrum and Mixing

After EWSB, the scalar spectrum comprises

• $m_h^2$, $m_H^2$: eigenvalues of the CP-even mass matrix, diagonalized by angle $\alpha$,
• $$m_A^2 = \frac{m_{12}^2}{s_\beta c_\beta} - v^2\,\text{Re}\,\lambda_5$$,
• $$m_{H^\pm}^2 = \frac{m_{12}^2}{s_\beta c_\beta} - \frac12 v^2 \lambda_4$$.

The Goldstone bosons align with the gauge fields, with $G^0,G^\pm$ and the physical $A,H^\pm$ related to fields $\{\chi_i,\phi_i^\pm\}$ through the angle $\beta$ (Dermisek et al., 30 May 2024).

Experimental and theoretical constraints from unitarity, the $T$ parameter (electroweak precision), and flavor physics impose:

• $m_{H^\pm} \gtrsim 580$ GeV (from $b \to s \gamma$ at 95% CL)
• $m_H, m_A \gtrsim 550$ GeV
• $$|m_H - m_A|, |m_{H^\pm} - m_H|, |m_{H^\pm} - m_A| \lesssim 200$$–$250$ GeV (Eberhardt, 2018, Cheng et al., 2014)

This near-degeneracy constrains decays such as $H \to AZ$, $H \to H^+ W^-$, $A \to H Z$, which require larger mass splittings and are thus suppressed.

4. Theoretical and Experimental Constraints

The parameter space of the Type-Y THDM is tightly restricted by:

• Perturbative unitarity and vacuum stability: The scalar couplings and $\tan\beta$ must satisfy limits such as $$|y_t|^2 = 2m_t^2/(v^2\sin^2\beta) \leq 4\pi \implies \tan\beta \gtrsim 0.28$$ (Cheng et al., 2014).
• Electroweak precision observables: The $S,T,U$ parameters restrict mass splittings among $H$, $A$, $H^\pm$ (Eberhardt, 2018).
• Flavor physics:
  • $B \to X_s\gamma$ provides the dominant constraint, requiring $m_{H^\pm} \gtrsim 580$ GeV for all $\tan\beta$ (Eberhardt, 2018).
  • $B_s - \bar B_s$ mixing, $B \to \tau \nu$, and $B_{s,d} \to \mu^+\mu^-$ further restrict the parameter space, though Type-Y predicts nearly SM-like results for $B_s \to \mu^+\mu^-$ both in rate and asymmetry, with $R \simeq 1+O(10^{-4})$ and $A_{\Delta\Gamma} \simeq +1$ (Cheng et al., 2014).

Combined fits in the $(\tan\beta, \cos(\beta-\alpha))$ plane illustrate that the alignment/decoupling regime with heavy, degenerate Higgs masses and $\tan\beta \gtrsim 2$ are favored, with $|\cos(\beta-\alpha)| \lesssim 0.02$ (68% CL) and $|\cos(\beta-\alpha)| \lesssim 0.04$ (95% CL) (Eberhardt, 2018). Lower $\tan\beta$ is disfavored by $b\to s\gamma$ and unitarity.

5. Phenomenology and Collider Implications

The proximity to the alignment limit, $\beta-\alpha \to \pi/2$, imposed by Higgs signal strength measurements in all major LHC search channels, ensures that the lightest CP-even Higgs $h$ retains SM-like properties with only small $O(0.03)$ allowed deviations at 95% CL (Eberhardt, 2018). Dominant decays of the heavier scalars are to $t\bar{t}$ (when kinematically allowed), $b\bar{b}$, and $\tau^+\tau^-$, with branching ratios governed by the $\tan\beta$-dependent Yukawa scaling.

In the EFT framework, dimension-six operators introduce characteristic higher-order corrections and process correlations. For example, in the Higgs basis, single Wilson coefficients simultaneously modify $h\bar{\psi}\psi$ couplings and generate contact interactions such as $(b\bar b) HHH$, with $\tan^6\beta$ scaling (Dermisek et al., 30 May 2024). Collider searches for heavy Higgs signals in $b\bar b H/A \to b\bar b\tau^+\tau^-$ channels, and direct probes of contact terms at future Higgs factories, are sensitive to these effects, with current LHC data already constraining $|\kappa_d - 1| \lesssim 0.1$ for moderate $\tan\beta$ (Dermisek et al., 30 May 2024).

6. Type-Y THDM Effective Field Theory

The extension to a Two-Higgs-Doublet Model Effective Field Theory (2HDM-EFT) incorporates all bosonic and fermionic dimension-six operators allowed by the $\mathbb{Z}_2$ structure of Type-Y. The rotation to the Higgs basis, where $H_1$ acquires the full vacuum expectation value and $H_2$ contains non-SM scalars, makes physical correlations and operator effects transparent:

• Deviations in SM couplings arise distinctly from mass operators, while scattering modifications are traced to higher-dimensional interactions.
• Universal process–operator correlations emerge: e.g., a dimension-six mass operator simultaneously shifts the $h$-Yukawa and generates $n$-Higgs contact terms with fixed $\tan\beta$ dependence.
• Vacuum structure and stability conditions, as well as the boundedness of the EFT, generalize the quartic positivity conditions to include $\phi^6$ operator coefficients (Dermisek et al., 30 May 2024).

This formalism enables systematic model-independent exploration of both SM-like and heavy Higgs collider phenomena.

7. Model Discrimination and Prospective Tests

Precise flavor measurements can discriminate among $\mathbb{Z}_2$-symmetric 2HDM types. The Type-Y scenario, due to its unique Yukawa assignments, predicts a narrow band in the $A_{\Delta\Gamma}$–$R$ plane for $B_s \to \mu^+\mu^-$, with $A_{\Delta\Gamma} \simeq +1$, $R \simeq 1$, in contrast to broader deviations possible in Types I, II, X. A precise simultaneous measurement at LHCb or a future flavor experiment could thereby uniquely identify the Type-Y model (Cheng et al., 2014).

A plausible implication is that, barring significant non-minimal flavor violation or additional new physics, future global fits—incorporating precision Higgs, flavor, and high-energy data—will continue to shrink the viable Type-Y parameter space toward the exact alignment and heavy-degenerate Higgs regime. The combination of flavor observables, direct searches, vacuum stability, and higher-dimensional operator effects in EFT provides a robust and multifaceted framework for interrogating the Type-Y 2HDM (Eberhardt, 2018, Dermisek et al., 30 May 2024, Cheng et al., 2014).