Probing Spontaneous CP-Violation through Precision Higgs Observables

1. Problem Statement

The origin of CP-violation (CPV) in the universe remains an open question. While the Standard Model (SM) explains CPV in $K$ and $B$ meson decays via the explicit complex phase in the Yukawa couplings (Kobayashi-Maskawa phase), this mechanism is insufficient to explain the baryon asymmetry of the universe. An alternative is Spontaneous CP-Violation (SCPV), where the Lagrangian is CP-invariant, but the vacuum state breaks CP symmetry via complex Vacuum Expectation Values (VEVs).

The authors investigate the General Two Higgs Doublet Model (2HDM) with SCPV. A critical challenge in this framework is that standard 2HDM scenarios often impose symmetries (like $Z_2$) or Yukawa alignment to suppress Flavor-Changing Neutral Currents (FCNCs). However, these constraints often prevent the generation of the physical KM phase required for the SM. The paper addresses the phenomenology of a general 2HDM with SCPV without imposing additional symmetries, focusing on the unique "non-decoupling" nature of the Higgs sector and its testability at the High-Luminosity LHC (HL-LHC).

2. Methodology

Model Framework

• Lagrangian: The authors parameterize two Higgs doublets, $\Phi_1$ and $\Phi_2$, with real parameters in the Lagrangian. CP violation arises spontaneously from the complex phase $\xi$ in the VEVs: $(v_1, v_2 e^{i\xi})$.
• Higgs Basis: They utilize the Higgs basis to separate the physical states. The neutral Higgs bosons ($h', h'_2, h'_3$) mix via an orthogonal matrix $R$. The observed 125 GeV Higgs ($h$) is identified as $H_1$.
• Yukawa Sector: Unlike aligned 2HDMs, the authors allow Yukawa misalignment to reproduce the KM phase.
  • The up-type Yukawa matrix $\rho_u$ is complex and non-diagonal, inducing tree-level FCNCs.
  • Down-type and charged lepton sectors are aligned ($\rho_{d,e} \propto M_{d,e}$) to suppress FCNCs in those sectors, while $\rho_u$ is constrained by flavor observables (meson mixing, $b \to s\gamma$) and the electron Electric Dipole Moment (EDM).

Theoretical Constraints & Scanning

• Non-Decoupling: In SCPV 2HDM, all dimensionful parameters in the potential are determined by the VEV ($v \approx 246$ GeV). Consequently, the masses of extra Higgs bosons ($H^\pm, H_2, H_3$) are bounded from above by perturbative unitarity ($O(500)$ GeV). This prevents the "decoupling limit" where new physics effects vanish at high energies.
• Parameter Scan: The authors performed a random scan of 1 million parameter points, constrained by:
  • Vacuum stability and perturbative unitarity (NLO).
  • Electroweak precision observables ($S, T$ parameters).
  • Flavor constraints ($B$-meson mixing, $b \to s\gamma$).
  • Electron EDM limits ($|d_e| \leq 4.1 \times 10^{-30}$ e-cm).
  • Direct LHC searches for charged and neutral Higgs bosons.
• Calculations:
  • Trilinear Coupling ($\lambda_{hhh}$): Calculated at the one-loop level using the effective potential method.
  • Decay Rates ($h \to \gamma\gamma, Z\gamma$): Calculated including charged Higgs ($H^\pm$) loop contributions.

3. Key Contributions

1. Establishment of Non-Decoupling: The paper rigorously demonstrates that in SCPV 2HDM, the extra Higgs boson masses are tied to the VEV, leading to significant, correlated deviations in Higgs observables even if the new particles are heavy (up to $\sim 500$ GeV).
2. Robust Correlation Discovery: The authors identify a unique, one-to-one correlation between the deviation in the $h \to \gamma\gamma$ branching ratio ($\Delta B_{h\to\gamma\gamma}$) and the deviation in the one-loop corrected trilinear Higgs coupling ($\Delta \kappa_\lambda$).
3. Flavor-Violating Decays: They demonstrate that the necessary Yukawa misalignment to generate the KM phase leads to sizable flavor-violating decays for the extra Higgs bosons, specifically $H_2 \to tc$ and $H^\pm \to cb$.
4. Exclusion Bounds: They derive new, robust exclusion bounds on the charged Higgs mass ($m_{H^\pm}$) based on current Higgs signal strengths, independent of specific theoretical assumptions about the scalar spectrum.

4. Key Results

• Deviations in Higgs Observables:

  • In the Higgs alignment limit, the branching ratio $B_{h\to\gamma\gamma}$ is suppressed by >10% compared to the SM.
  • $B_{h\to Z\gamma}$ is suppressed by >4%.
  • The trilinear coupling deviation $\Delta \kappa_\lambda$ can reach 200%.
  • Correlation: A strong negative correlation exists: as $m_{H^\pm}$ increases, $B_{h\to\gamma\gamma}$ decreases (more negative deviation) while $\Delta \kappa_\lambda$ increases.
    • Example: For $m_{H^\pm} \approx 500$ GeV, $(\Delta B_{h\to\gamma\gamma}, \Delta \kappa_\lambda) \approx (-10.3\%, 200\%)$.
    • Example: For $m_{H^\pm} \approx 330$ GeV, $(\Delta B_{h\to\gamma\gamma}, \Delta \kappa_\lambda) \approx (-10.8\%, 50\%)$.

• Mass Constraints:

  • Lower Bound: Current LHC signal strength data ($\Delta B_{h\to\gamma\gamma} > -0.12$) implies $m_{H^\pm} \gtrsim 218$ GeV.
  • Upper Bound: Combined theoretical (unitarity) and experimental constraints limit $m_{H^\pm} \lesssim 510$ GeV.
  • Future Sensitivity: The HL-LHC is expected to measure $\Delta \kappa_\lambda$ with precision $\sim 1.3$. This will test the parameter space where $m_{H^\pm} \gtrsim 330$ GeV, potentially ruling out the scenario if the central value remains near zero.

• Phenomenology of Extra Higgs Bosons:

  • $H_2$ (Lightest Neutral): Due to mass constraints, $H_2$ is typically below the $t\bar{t}$ threshold. Its dominant decay is the flavor-violating mode $H_2 \to tc$ (branching ratio $\sim 80-99\%$).
  • $H_3$ (Heavier Neutral): Decays depend on mass hierarchy. In non-degenerate cases, it decays via bosonic channels ($H_3 \to H_2 Z, H^\pm W^\mp$).
  • $H^\pm$ (Charged): Along with the standard $tb$ mode, there is a significant branching ratio for the unique $cb$ final state (up to $\sim 78\%$ in some benchmarks) and $H^\pm \to H_2 W^\pm$ in non-degenerate spectra.

5. Significance

This paper provides a novel, robust strategy to test the SCPV paradigm, which has been difficult to probe due to the lack of direct signals.

• Complementarity: It highlights that precision measurements of the trilinear Higgs coupling ($\lambda_{hhh}$) and the diphoton decay rate ($h \to \gamma\gamma$) are complementary. The correlation between them serves as a "smoking gun" for SCPV, distinguishing it from other BSM scenarios.
• Discovery Channels: It identifies flavor-violating decays ($H_2 \to tc$ and $H^\pm \to cb$) as crucial, complementary discovery channels that are often overlooked in standard searches.
• Future Prospects: The study argues that the HL-LHC is essential for testing this framework. If the measured deviations in $h \to \gamma\gamma$ and $\lambda_{hhh}$ align with the predicted correlation, it would provide strong evidence for Spontaneous CP-Violation and the specific structure of the Higgs sector required to generate the baryon asymmetry. Conversely, precise null results could rule out the entire non-decoupling SCPV 2HDM parameter space.