Reassessing CP Violation in the C2HDM with Machine Learning

Technical Summary: Reassessing CP Violation in the C2HDM with Machine Learning

Problem Statement
The Complex Two-Higgs Doublet Model (C2HDM) extends the Standard Model (SM) by introducing a scalar sector with spontaneous CP violation. A primary phenomenological goal is to determine if the observed 125 GeV Higgs boson ($h_{125}$) can possess large CP-odd couplings to fermions ($c_o^f$) while remaining consistent with experimental constraints. The most stringent constraint arises from the electron electric dipole moment (eEDM), where current experimental limits ($d_e < 4.1 \times 10^{-30}$ e.cm) typically force CP-odd couplings to be negligible.

Previous analyses (e.g., Ref. [29]) relied on manual scanning near the CP-conserving limit, often concluding that large CP-odd couplings were excluded for most C2HDM configurations. However, these studies faced two critical challenges:

1. Theoretical Completeness: The eEDM calculation requires a complete set of two-loop diagrams to ensure gauge independence. Recent work identified "kite diagrams" as essential for this, and their omission or incomplete inclusion could lead to erroneous exclusion of valid parameter space.
2. Sampling Bias: Traditional scanning methods often miss valid regions of the parameter space, particularly those requiring precise cancellations between large, opposing contributions (often at the per-mil level) to satisfy the eEDM bound.

Methodology
The authors perform a comprehensive re-evaluation of the C2HDM parameter space using a combination of advanced theoretical calculations and Machine Learning (ML) techniques.

• Theoretical Framework: The study utilizes the C2HDM with a softly broken $Z_2$ symmetry, covering Type-I, Type-II, Lepton-Specific (LS), and Flipped models. The scalar potential is defined by parameters including $\tan\beta$, scalar masses ($m_1, m_2, m_3$), mixing angles ($\alpha_1, \alpha_2, \alpha_3$), and the soft-breaking term $\text{Re}(m_{12}^2)$.
• eEDM Calculation: The authors compute the eEDM including:
  • Standard two-loop Barr-Zee diagrams (with third-generation fermion loops).
  • Kite diagrams: Identified in Ref. [36] as crucial for gauge independence.
  • Charm-quark Barr-Zee diagrams: Evaluated for the first time in this context to assess their impact given current precision.
  • The calculation is performed in the Feynman gauge, comparing two renormalization schemes for fermion masses: $M_Z$-running masses and pole masses.
• Constraints: The parameter space is constrained by:
  • Theoretical bounds (vacuum stability, perturbative unitarity, global minimum).
  • Electroweak precision observables (S, T, U parameters).
  • Higgs signal strengths and coupling modifiers (LHC data).
  • Direct searches for additional scalars (via HiggsBounds).
  • Flavor constraints ($b \to s\gamma$).
  • CP-violating angular correlations in $\tau$ decays ($\theta_\tau$).
• Sampling Techniques:
  • Strategy 1: A guided scan near the phenomenological boundary, iteratively increasing the pseudoscalar component (similar to previous manual methods).
  • Strategy 2: An ML-based approach using a Covariant Matrix Adaptation Evolutionary Strategy (CMA-ES) combined with a Novelty Reward mechanism (based on Histogram-Based Outlier Score). This "black-box" optimization explores the parameter space without pre-existing training data, specifically targeting regions with large CP-odd couplings that standard scans might miss.

Key Contributions and Results

1. Validation of Kite Diagrams: The study confirms that the inclusion of kite diagrams is not merely a technicality but essential for the physical viability of the model. Their inclusion induces $\mathcal{O}(1)$ corrections to the eEDM prediction, enabling the necessary cancellations between different diagrammatic contributions. Without them, the gauge dependence of the result would invalidate the analysis.
2. Necessity of Charm Diagrams: The authors demonstrate that at current experimental precision, Barr-Zee diagrams involving virtual charm quarks are numerically significant. Their omission leads to the exclusion of valid points; including them alters the set of allowed parameters but does not preclude large CP-odd couplings.
3. Resurrection of Large CP-Odd Couplings:
   • Type-II with $h_2 = h_{125}$: Using ML (Strategy 2), the authors find a viable region with large $|c_o^b|$ (bottom quark CP-odd coupling) near the "wrong-sign" solution ($c_e^b \approx -1, c_o^b \neq 0$). This region was entirely missed by Strategy 1 and previous manual scans. The constraints on $c_o^b$ in this scenario are primarily driven by $\theta_\tau$ limits from LHC tau decay measurements, not the eEDM.
   • Flipped Model with $h_2 = h_{125}$: The ML search reveals that this configuration allows for maximal CP-odd couplings to the bottom quark ($c_o^b \approx 1, c_e^b \approx 0$). In this case, $h_{125}$ can couple as a pure scalar to the top quark and a pure pseudoscalar to the bottom quark, a scenario previously thought impossible.
4. Robustness Against Future Precision: The cancellations required to satisfy the eEDM bound are shown to be robust. Even if the eEDM sensitivity improves by three orders of magnitude (down to $10^{-33}$ e.cm), valid parameter points with large $|c_o^b|$ can still be found through fine-tuning.
5. Renormalization Scheme Independence: While the specific points in the parameter space differ between the $M_Z$-mass and pole-mass schemes, the qualitative conclusion—that large CP-odd couplings are viable in Type-II and Flipped models with $h_2 = h_{125}$—remains consistent when kite diagrams are included.
6. Exclusion of Other Scenarios: The study confirms that large $|c_o^b|$ is excluded for Type-I models (due to top-quark constraints), for $h_3 = h_{125}$ in Type-II/Flipped, and for $h_1 = h_{125}$ in Type-II/Flipped (due to direct resonance searches).

Significance and Claims
The paper claims that the combination of complete two-loop eEDM calculations (including kite and charm diagrams) and ML-driven parameter space exploration fundamentally changes the phenomenological landscape of the C2HDM.

• Methodological Significance: The work highlights the limitations of traditional scanning methods in high-dimensional, constrained BSM parameter spaces, demonstrating that ML techniques can uncover valid regions (such as the "wrong-sign" solution) that are systematically missed by standard approaches.
• Phenomenological Significance: The authors assert that large CP-odd couplings for the 125 GeV Higgs are not ruled out. Specifically, in the Type-II and Flipped models where $h_{125}$ is the second lightest scalar ($h_2$), the CP-odd coupling to bottom quarks can be substantial.
• Experimental Implications: Since the eEDM constraints can be satisfied via cancellations even for future precision, the primary constraints on these large CP-odd couplings now arise from LHC precision measurements (specifically $\theta_\tau$). Consequently, the paper argues for increased experimental efforts to directly probe CP-odd couplings of the Higgs to quarks at the LHC, as indirect EDM constraints alone are insufficient to rule out these scenarios.

The authors remain modest regarding the top quark, noting that large CP-odd couplings to the top quark remain excluded in the C2HDM, though they suggest this possibility exists in the Complex 3-Higgs Doublet Model (C3HDM).