CP-Violating Leptonic Yukawa Couplings

• CP-violating leptonic Yukawa couplings are higher-dimensional modifications to Higgs–lepton interactions that introduce CP-odd phases through complex Wilson coefficients.
• They are probed using effective field theory, explicit UV completions, and experimental observables such as Higgs decay modifications, electric dipole moments, and lepton flavor violation.
• Constrained by stringent EDM and LFV limits, these couplings provide a promising avenue for exploring new sources of CP violation beyond the Standard Model.

CP-violating leptonic Yukawa couplings constitute a central area in the investigation of new sources of CP violation beyond the Standard Model (SM), particularly as they relate to the flavor and CP structure of Higgs–lepton interactions, the generation of neutrino masses, and the origins of baryon asymmetry. The theoretical and phenomenological landscape, as illuminated by both low-energy and collider data, encompasses effective operator analyses, explicit ultraviolet (UV) completions, mechanisms of spontaneous and explicit CP breaking, and detailed studies of experimental observables including electric dipole moments (EDMs), lepton flavor violation (LFV), and modified Higgs decays.

1. Effective Field Theory and CP Violation

The Standard Model Effective Field Theory (SMEFT) provides the language to parameterize generic new-physics-induced modifications to the Higgs–lepton sector through higher-dimensional operators. The leading dimension-six operator relevant for CP-violating leptonic Yukawa couplings is

$O_{eH} = (\bar{\ell}_i H e_j)(H^\dagger H),$

with complex Wilson coefficients $[C_{eH}]_{ij}$ (Košnik et al., 4 Sep 2025). After electroweak symmetry breaking (EWSB), this operator generates corrections to the SM Higgs–lepton coupling:

$$\mathcal{L}_{\rm eff} = -\sum_i \frac{y_{\ell_i}}{\sqrt{2}}\left[\kappa_{\ell_i} + i \tilde\kappa_{\ell_i} \gamma_5\right] \ell_i \ell_i h,$$

where

$$\kappa_{\ell_i} = 1 - \frac{v^2}{y_{\ell_i}}\mathrm{Re}([C_{eH}]_{ii}), \quad \tilde\kappa_{\ell_i} = -\frac{v^2}{y_{\ell_i}}\mathrm{Im}([C_{eH}]_{ii}),$$

and $v \approx 246\,\mathrm{GeV}$ is the Higgs vacuum expectation value. The CP-odd interaction is directly proportional to $\mathrm{Im}([C_{eH}]_{ii})$, providing a new source of CP violation in the leptonic sector.

2. Ultraviolet Completions: Single- and Two-Field Models

Explicit UV completions generating CP-violating Higgs–lepton couplings can be classified broadly as single-field or two-field extensions (Košnik et al., 4 Sep 2025).

Single-Field Completion

A single new scalar doublet $\phi$ with the same quantum numbers as the SM Higgs $H$ yields renormalizable interactions:

$$-\mathcal{L}_{\rm UV} \supset \lambda_\phi (\phi^\dagger H)(H^\dagger H) + [Y^{(e)}_\phi]_{ij}\, \phi\, (\bar{\ell}_i e_j) + \mathrm{h.c.}$$

Integrating out $\phi$, one obtains

$$[C_{eH}]_{ij} = \frac{\lambda_\phi [Y^{(e)}_\phi]_{ij}}{M_\phi^2},$$

where $M_\phi$ is the mass of $\phi$. Complex phases in $[Y^{(e)}_\phi]$ or $\lambda_\phi$ result in CP-odd couplings.

Two-Field Completion

Two-field extensions, e.g., a pair of vector-like leptons $(E, \Delta_3)$, more generically realize CP violation. Tree-level matching produces

$$[C_{eH}]_{ij} \sim \frac{\alpha}{M_E M_{\Delta_3}} X_{ij},$$

where $X_{ij}$ involves products of the underlying complex Yukawa and portal couplings. These same mediators can induce dipole operators at loop level:

$$[C_{e\gamma}]_{ij} \sim \frac{ev}{32\pi^2} \frac{\beta}{M^2} X_{ij},$$

generating both EDMs and magnetic moments sensitive to CP phases.

3. Experimental Signatures and Constraints

CP-violating leptonic Yukawa couplings have multifaceted experimental consequences that probe both real and imaginary parts of $[C_{eH}]_{ij}$.

Direct Higgs Measurements

The CP-laden lepton Yukawa corrections modify Higgs decay rates:

$$\Gamma(h \to \ell_i^+ \ell_i^-) \propto \kappa_{\ell_i}^2 + \tilde\kappa_{\ell_i}^2.$$

For $\tau$ leptons, angular observables such as the CP-mixing angle $\alpha^\tau$—defined via $\tan\alpha^\tau = \tilde\kappa_\tau/\kappa_\tau$—enable the extraction of CP-odd contributions from the distributions of decay planes (Bhardwaj et al., 2016).

Electric and Magnetic Dipole Moments

Loop-induced dipole operators affect low-energy observables such as the electron EDM:

$d_e = -2~\mathrm{Im}([C_{e\gamma}]_{11}),$

and anomalous magnetic moments:

$$\Delta a_\ell = -\frac{4 m_\ell}{e}~\mathrm{Re}([C_{e\gamma}]_{ii}).$$

Barr–Zee diagrams frequently enhance the sensitivity, particularly for EDMs, sometimes pushing the allowed new-physics scale above tens of PeV, especially for the electron channel (Košnik et al., 4 Sep 2025).

Charged Lepton Flavor Violation and Electroweak Precision Tests

Non-diagonal elements of $[C_{eH}]_{ij}$ induce cLFV processes such as $\mu \to e\gamma, \tau \to \mu\gamma$, and $\mu \to 3e$. Constraints from these rare searches are stringent, with future improvements anticipated from dedicated experiments. Simultaneously, matching in SMEFT generates $O_{H\ell}^{(1,3)}$ and $O_{He}$ that modify $Z$–lepton couplings, tightly constrained by LEP/SLC precision data.

The following table summarizes the key observables impacted by CP-violating leptonic Yukawas and their sensitivity to underlying new-physics parameters:

Observable	Sensitive to	Key Constraint Mechanism
$h \to\ell^+\ell^-$	$\kappa_{\ell_i},~\tilde\kappa_{\ell_i}$	LHC signal strengths and angular distributions
EDMs ($d_e$, $d_\mu$)	$\mathrm{Im}([C_{e\gamma}]),~\mathrm{Im}(X_{ij})$	Low-energy experiments, Barr–Zee effects
cLFV ($\mu \to e\gamma$)	$[C_{eH}]_{ij\neq ji},~[C_{e\gamma}]_{ij\neq ji}$	Rare decay searches
EWPT (Z pole)	$O_{H\ell}^{(1,3)},~O_{He}$	LEP/SLC vertex measurements

In numerous scenarios, the electron EDM measurements set the tightest bounds, with allowed parameter space for CP-violating couplings far more restricted than indicated solely by LHC Higgs studies.

4. Flavor Structure and Model Assumptions

The phenomenological impact of CP-violating leptonic Yukawa couplings is strongly dependent on the flavor texture of the underlying new-physics couplings. Representative choices include:

• Diagonal flavor assumption: Only diagonal $[C_{eH}]_{ii}$ nonzero, leading to minimal cLFV and direct constraints from lepton-specific Higgs decays and EDMs.
• Maximally off-diagonal/flavor-complete assumption: All entries of $[C_{eH}]_{ij}$ allowed, correlating CP violation in diagonal channels with stringent bounds from cLFV searches.

Flavor symmetry and alignment can further modulate the allowed sizes of CP-violating entries: for instance, discrete symmetries suppress cLFV but may still allow sizable CP-odd phases in the diagonal coupling. This diversity of flavor scenarios underlines the importance of a multi-pronged experimental strategy.

5. Interplay Between High-Energy and Precision Probes

The combined experimental program is essential for testing UV models of CP-violating Higgs–lepton interactions. While LHC Higgs measurements (vertex signal strengths and $\tau$ CP-sensitive distributions) probe direct effects, precision experiments—EDMs, cLFV, and Z-pole studies—often surpass collider reach in constraining possible new physics (Košnik et al., 4 Sep 2025). Particularly, one-loop dipole contributions from vector-like leptons or extended scalar sectors may indirectly preclude scenarios with large CP-odd Yukawa corrections, unless the new-physics scale is far beyond direct discovery.

A plausible implication is that future advances in electron EDM sensitivity, as well as improved searches for $\mu\to e\gamma$ and $\tau\to\mu\gamma$, will powerfully shape the prospects for observing CP-violating Higgs-lepton couplings, even in the absence of direct Higgs collider anomalies.

6. Comparative Analysis with LHC and Broader Impact

LHC analyses of modified leptonic Yukawa couplings, incorporating both CP-even and CP-odd admixtures, are indispensable in probing the Higgs–lepton interaction structure (Bhardwaj et al., 2016). Nonetheless, the precision constraints from low-energy observables, especially EDMs and cLFV, often—though not always—exclude parameter regions with sizable CP-violation that remain technically allowed by direct LHC probes. In flavor scenarios where muon or tau couplings dominate, LHC and precision measurement constraints can be competitive, but for the electron channel, EDM limits are dominant.

This multi-level experimental strategy is necessary for a robust exclusion or discovery of new CP-violating sources in the leptonic sector, highlighting the unique role of leptonic EDMs and flavor-violating observables as probes of UV physics that may be otherwise inaccessible to direct collider experiments (Košnik et al., 4 Sep 2025).

7. Future Directions

The comprehensive mapping of CP-violating leptonic Yukawa couplings requires a continued integration of SMEFT analyses, UV model building, and global fits encompassing collider and precision data. Theoretical directions include further classification of UV mechanisms, explicit treatments of flavor symmetry breaking, and studies of correlated observables. Experimentally, enhanced LHC sensitivity to CP-odd $\tau$ observables, next-generation electron EDM measurements, and cLFV searches will be central in constraining or revealing the structure and scale of new CP-violation sources coupling the Higgs to leptons.

This interplay between precise low-energy measurements and high-energy collider searches exemplifies the methodology necessary for identifying and characterizing fundamental CP violation in the leptonic Yukawa sector.