Asymmetric Yukawa Couplings in BSM Models

• Asymmetric Yukawa coupling is a QFT structure where fermion-scalar interactions exhibit nontrivial asymmetry, influencing flavor hierarchies.
• It appears in various BSM models such as radiative neutrino mass frameworks, 2HDMs without NFC, and dark U(1)_F theories with exponential hierarchies.
• These asymmetric couplings impact experimental observables through LFV processes, collider signatures, and CP behavior, guiding current research.

An asymmetric Yukawa coupling refers to any structure within a quantum field theory in which the matrix of Yukawa interactions—couplings between fermions and scalar (Higgs or other) fields—exhibits a nontrivial asymmetry, often in the sense of flavor hierarchy, flavor antisymmetry/symmetry, or strong deviation from proportionality to fermion mass eigenstates. These structures are theoretically significant for generating observed fermion mass hierarchies, inducing radiative neutrino masses, modeling flavor violation, or accommodating experimental anomalies. Asymmetric Yukawa couplings appear prominently in beyond-Standard Model (BSM) frameworks, such as radiative neutrino mass models (e.g., Zee–Babu), 2HDMs without natural flavor conservation, and dynamical models where Yukawa matrices inherit exponential hierarchies from hidden sectors.

1. Asymmetric Yukawa Coupling: Theoretical Contexts

Asymmetry in Yukawa couplings emerges in several distinct settings:

• Flavor Hierarchy via Dark Sector Mechanisms: Models introducing a dark $U(1)_F$ gauge sector can dynamically generate exponentially split Yukawa couplings for Standard Model (SM) fermions via nonperturbative chiral symmetry breaking, with asymmetry set by charge assignments in the dark sector (Gabrielli et al., 2013).
• Radiative Neutrino Mass Models: Frameworks like the Zee–Babu model involve antisymmetric ($f_{ij}=-f_{ji}$) and symmetric ($g_{ij}=g_{ji}$) Yukawa matrices, generating Majorana mass terms via higher-loop diagrams and giving rise to distinctive flavor structures (Irie et al., 2021).
• Two-Higgs-Doublet Models (2HDM) Without NFC: Absent natural flavor conservation (NFC), Yukawa couplings can become highly asymmetric, allowing, for instance, a single fermion (e.g., the electron) to acquire an enhanced or suppressed coupling relative to its SM value, with controlled impact on other generations (Dery et al., 2017).

2. Dynamical Generation and Exponential Hierarchies

In models with a dark $U(1)_F$ gauge theory augmented by higher-derivative Lee–Wick terms, Dirac fermions $\psi$ with integer or fractional charge $Q$ experience chiral symmetry breaking at weak coupling. The nontrivial solution to the gap equation,

$$m(Q) \simeq \Lambda \exp\left[ -\frac{2\pi}{3\alpha_F Q^2} + \frac14 \right],$$

where $\Lambda$ is the higher-derivative scale and $\alpha_F$ the dark gauge coupling, yields exponentially spread dark fermion masses. Small charge differences generate large mass hierarchies. This "asymmetric" outcome is transferred to SM Yukawa couplings via Higgs-portal one-loop diagrams involving scalar messenger fields, leading to

$$y_i \sim \Lambda_{\rm eff} \exp\left[ -\frac{2\pi}{3\alpha_F Q_i^2} \right],$$

with $\Lambda_{\rm eff}$ absorbing messenger couplings and mass scales (Gabrielli et al., 2013). This structure produces hierarchical, diagonal Yukawa matrices entirely determined by the charge asymmetry in the hidden sector.

3. Asymmetric Yukawa Structures in Neutrino Mass Models

Majorana neutrino mass models of "Group I" type instantiate asymmetric Yukawa couplings through new scalar fields with nontrivial charge assignments and couplings: $$\mathcal{L}_Y \supset -\tfrac12 f_{ij} \overline{L_i^c} i\tau_2 L_j \omega^+ - g_{ij} \overline{e_i^c} e_j \kappa^{++} - \mu_{\rm ZB} \omega^- \omega^- \kappa^{++} + \text{h.c.}$$ Here, $f_{ij}$ is antisymmetric, $g_{ij}$ symmetric. The resulting two-loop neutrino mass matrix is

$$(M_\nu)_{ij} \propto \sum_{k,\ell} f_{ik} m_k g_{k\ell} m_\ell f_{j\ell} \, I(M_\omega^2,0|M_\omega^2,0|M_\kappa^2),$$

with the symmetric product of antisymmetric $f$ and symmetric $g$ controlling the flavor structure (Irie et al., 2021). Neutrino oscillation data then directly fixes the asymmetric structure of $f$ and $g$ in terms of low-energy observables. Experimental constraints from LFV processes (e.g., $\mu\to e\gamma$, $\tau\to\mu\mu\mu$, $\mu$-$e$ conversion) place stringent limits on combinations of asymmetric Yukawa couplings and scalar masses.

4. Large Asymmetric Yukawa Couplings in Extended Higgs Sectors

In general 2HDMs without natural flavor conservation, the most general Yukawa Lagrangian leads to intra- and inter-generational asymmetric couplings. For the electron, the Yukawa rescaling is

$$\kappa_e = \sin(\beta-\alpha) + \cos(\beta-\alpha) \frac{y_A^e}{y_e},$$

where $y_A^e$ is the off-diagonal Yukawa in the rotated Higgs basis. Large $\kappa_e$ is possible for

$$|\kappa_e| \gg 1 \;\Leftrightarrow\; |\cos(\beta-\alpha)\tan\beta_e| \gg 1,$$

and this scenario can be engineered to affect only a single generation (asymmetry by design) or split the couplings between generations (Dery et al., 2017). Sum rules link enhanced couplings of one generation to corresponding shifts in other fermion or scalar couplings, controlled by the pattern of Yukawa matrices in the so-called “$\beta_e$-basis.”

5. Unified Treatment and Formal Properties

Worldline path integral techniques enable a unified description of Dirac fermions coupled to arbitrary combinations of scalar (Yukawa), pseudoscalar (axial), vector, and axialvector backgrounds. This formalism accommodates asymmetric or complex Yukawa backgrounds, constructing the effective action from the Dirac determinant with explicit couplings

$g_S \phi(x) + i g_P \gamma_5 \chi(x)$

for scalar and pseudoscalar fields. Asymmetry may be manifest in the flavor structure of $g_S$, $g_P$, or in external, non-uniform backgrounds. The worldline approach, regularized by a nonperturbative time-slicing counterterm, produces heat-kernel expansion coefficients and correlators capturing the differences between parity-even and parity-odd (Yukawa and axial) terms, even for non-Hermitian Hamiltonians (Bastianelli et al., 28 Jun 2024). This framework ensures that all amplitudes and effective operators, regardless of potential asymmetries, are systematically incorporated.

6. Phenomenological Implications and Experimental Probes

Asymmetric Yukawa couplings have significant phenomenological consequences:

• Flavor Hierarchy: Exponential hierarchies generated by dark sector charge assignments can reproduce the observed pattern of SM masses for leptons and quarks (Gabrielli et al., 2013). Selection of $Q_i$ in the dark sector tunes the magnitude of $y_i$.
• Lepton Flavor Violation: In the Zee–Babu model, the asymmetric flavor structure of $f$ and $g$ determines rates for LFV processes, with stringent bounds from $\mu\to e\gamma$ and $\tau\to\mu\mu\mu$. Experimental measurements define narrow regions in parameter space consistent with neutrino data (Irie et al., 2021).
• Collider Signatures: Scalar messengers or doubly-charged scalars yield LHC signatures such as multilepton events ($\ell_i \ell_j + E_T$), same-sign dileptons ($e^+e^+,\,\mu^+\mu^+$), and $e^+e^-$ resonances. Enhanced electrons in 2HDM predict distinctive multi-$e$ topologies, subject to constraints from ATLAS and CMS searches, which exclude large portions of parameter space for $m_{H,A}\lesssim 640$ GeV with large $\kappa_e$ (Dery et al., 2017).
• CP Violation and Mass Spectrum: Large asymmetric Yukawa couplings require near-exact CP conservation ($\text{Im}[\lambda_5]\lesssim 10^{-4}/\kappa_e$), and produce characteristic mass degeneracies or mass hierarchies depending on soft or hard breaking scenarios.

7. Summary Table of Models Featuring Asymmetric Yukawa Couplings

Model / Framework	Asymmetry Source	Key Consequence
$U(1)_F$ dark sector	Hidden charge $Q$	Exponential mass hierarchy, SM Yukawas via portal (Gabrielli et al., 2013)
Zee–Babu	Antisymmetric $f$, symmetric $g$	Radiative neutrino masses, LFV (Irie et al., 2021)
2HDM (no NFC)	Arbitrary $Y^e_{ij}$	Large/split $\kappa_f$, CP & collider bounds (Dery et al., 2017)
Worldline formalism	Arbitrary $g_S$, $g_P$	Unified effective action, heat-kernel expansion (Bastianelli et al., 28 Jun 2024)

Asymmetric Yukawa couplings constitute an essential ingredient in modern particle physics model-building, providing both explanatory power for flavor hierarchies and realistic avenues for experimental exploration. Theoretical and experimental developments continue to refine the allowed structure and phenomenology of such couplings.