Dirac Neutrino Yukawa Couplings

• Dirac neutrino Yukawa couplings are dimensionless parameters in extended SMs that dictate the strength of neutrino mass generation via interactions with Higgs fields and right‐handed neutrinos.
• Symmetry-based suppression, radiative models, two-Higgs-doublet frameworks, and higher-dimensional operators explain their extreme smallness relative to charged-lepton couplings.
• Experimental constraints from lepton flavor violation, collider signatures, and Higgs decays provide practical tests for these mechanisms and insights into new physics.

Dirac neutrino Yukawa couplings are the dimensionless parameters in the Lagrangian that determine the strength of interactions between left-handed lepton doublets, Higgs fields, and right-handed neutrinos. In explicit Dirac mass models, these couplings explain the observed pattern of light neutrino masses and mixings, and their origin and suppression are central questions in neutrino phenomenology and model building. Multiple frameworks have been developed to account for the extreme smallness of Dirac neutrino Yukawa couplings compared to those of charged leptons and quarks; these frameworks include symmetry-based selection rules, radiative mechanisms, extra scalar sectors, flavor symmetries, higher-dimensional operators, and renormalization group effects.

1. Definition and Theoretical Framework

In the Standard Model (SM) extended by right-handed neutrinos $\nu_{R}$, Dirac neutrino Yukawa couplings $Y_{\nu}^{ij}$ appear in the Lagrangian as

$$\mathcal{L}_{\rm Yukawa} \supset -\overline{L}_i Y_{\nu}^{ij} \widetilde{H} \nu_{R j} + \rm{h.c.}$$

where $L_i$ is the $i$th-generation lepton doublet, $\widetilde{H}=i\sigma_{2}H^{*}$ is the conjugate Higgs doublet, and $j$ runs over the flavors of right-handed neutrinos. Upon electroweak symmetry breaking (EWSB), the Higgs field acquires a vacuum expectation value (VEV) $v$, generating the Dirac neutrino mass matrix $M_{\nu} = Y_{\nu} v$. To reproduce observed neutrino masses $m_{\nu_i}\sim 0.01$–$0.1$ eV, this requires $|Y_{\nu}^{ij}| \sim 10^{-12}$, much smaller than the minimal charged-lepton Yukawa couplings.

In explicit models, the structure and magnitude of $Y_{\nu}$ are controlled by symmetries or dynamics beyond the SM, often involving additional fields, discrete or continuous symmetries, extra Higgs doublets or singlets, flavor structures, Froggatt–Nielsen mechanisms, extra dimensions, or radiative mechanisms.

2. Mechanisms for Generating Small Dirac Yukawa Couplings

2.1 Symmetry-Based Suppression and Radiative Models

Symmetries can forbid the tree-level Yukawa term and induce it only at higher loop order or via higher-dimensional operators. For example, in the model with two $SU(2)_L$-singlet charged scalars $s_1^+, s_2^+$ and a softly-broken $Z_2$ symmetry (Kanemura et al., 2011), the Dirac Yukawa term is absent at tree level but is induced at one loop: $$Y_{\ell i} = \frac{C}{v} \sum_{\ell'} f_{\ell\ell'} m_{\ell'} h_{\ell' i},$$ with $C$ a one-loop function dependent on scalar mixing and masses. Antisymmetric $f_{\ell\ell'}$ and general $h_{\ell' i}$ parameterize the flavor structure. For $f, h\sim 10^{-2}$, $m_{s_{1,2}}\sim 100$–$200$ GeV, the resulting $Y_{\ell i}$ yield viable neutrino masses.

2.2 Vacuum Expectation Value (VEV)-Induced Yukawa Suppression: Two-Higgs-Doublet Models

Introducing a second $SU(2)_L$ doublet $\Phi_2$ with a tiny VEV, $v_2$, and coupling solely to $\nu_{R}$ via an additional symmetry, one obtains $m_{\nu}^{ij} = Y_\nu^{ij} v_2$ (0906.3335). For $v_2 \sim 1$ eV and $m_{\nu} \sim 0.05$ eV, moderately large $Y_\nu^{ij} \sim 0.05$ are sufficient, which contrasts starkly with the tiny values required in the vanilla SM. The softly broken global $U(1)$ symmetry or discrete subgroups control the allowed entries and suppress unwanted Majorana terms.

2.3 Higher-Dimensional Operators and Flavon Insertions

If the Dirac term is forbidden by a discrete gauge or flavor symmetry, it may appear at dimension-5 or -6: $$\mathcal{L}\supset \frac{y}{M} \chi\, \overline{L}\tilde H\nu_R \text{ or } \frac{1}{\Lambda^2}\overline{L}\tilde{H}\nu_R (\phi_S \phi_S)$$ with $\chi$ ($\phi_S$) a scalar singlet ("flavon"). The VEV of $\chi$ or $\phi_S$ then controls $Y_\nu$. For $\langle \chi \rangle \sim 10^{9}\;\rm GeV$, $M \sim M_{Pl}$, this yields $Y_{\nu}\sim 10^{-13}$ and $m_\nu\sim 0.03$ eV (Borah et al., 2019, Borah et al., 2018).

2.4 Froggatt–Nielsen and Dirac-Seesaw Mechanisms

In models with a $U(1)_X$ Froggatt–Nielsen symmetry, effective Dirac Yukawas are generated by integrating out heavy states, with small entries controlled by $\epsilon = v_{\rm EW}/v_S$ and $\delta = v_S/\Lambda$ (Ishida et al., 15 Oct 2025): $$y_\nu \sim \delta \left(\frac{Y'\,\tilde\Omega'}{\Omega'}\right),\qquad \delta = \frac{v_S}{\Lambda},$$ with $v_S \ll \Lambda$; for $\delta\sim 10^{-12}$ this gives sub-eV neutrino masses.

2.5 Asymptotic Safety and Renormalization-Group Fixed Points

Tiny Dirac Yukawas can be generated through renormalization-group flows with trans-Planckian asymptotic safety (Kowalska et al., 2022). An RG trajectory that approaches a Gaussian (free) IR-attractive fixed point for $y_\nu$ above the Planck scale results in $y_\nu$ "freezing in" at a small but nonzero value at $M_{Pl}$. This value depends on the RG "time" between the UV and IR fixed points: $m_{\nu,\,i} = y_{\nu ,\,i}(M_P)\, v / \sqrt{2},$ allowing for $y_{\nu}\sim 10^{-5}$–$10^{-4}$ for the observed hierarchy.

3. Flavor Structure and Predictive Frameworks

Flavor symmetries (Abelian, non-Abelian, discrete) can impose restrictive patterns ("texture zeros") on $Y_\nu$. In $A_4$, $S_3$, $\Delta(27)$, or $U(1)$-extended two-Higgs-doublet models (Correia et al., 2019, Aranda et al., 2013), the entries of $Y_\nu$ are fully determined by the measured PMNS matrix, neutrino masses, and one or more flavor phases. Explicit examples include:

• Texture zeros fixed by $U(1)$ or $Z_5$ charge assignments; only five maximal sets of zeros are compatible with data in the 2HDM (Correia et al., 2019).
• $\Delta(27)$ flavor symmetry, which determines both the allowed forms of $Y_\nu$ and the phenomenological correlation between the atmospheric angle octant and the lightest neutrino mass (Aranda et al., 2013).
• In $SU(5)_L \times SU(5)_R$ unification, the effective $Y_\nu$ is suppressed by an intermediate-to-GUT scale ratio, $M_I/M_G$, and aligned with the PMNS structure, predicting both $\delta_{CP}$ and $m_{\nu,1}$ (Babu et al., 2023).

4. Phenomenological Constraints and Experimental Probes

Dirac Yukawa couplings induce distinct observables:

• Lepton Flavor Violation (LFV): Loop-generated Dirac Yukawa couplings enter $\mu \to e \gamma$ processes; branching ratios are tightly constrained to be $\lesssim 10^{-11}$, implying $f, h \sim 10^{-2} - 10^{-3}$ for new scalar masses $\gtrsim 100$ GeV (Kanemura et al., 2011). In 2HDM Dirac-neutrino models, flavor-changing neutral currents are suppressed by the smallness of $Y_\nu$ for certain charge assignments.
• Collider Signals: New scalars associated with the generation of Yukawas (e.g., $H_1^\pm$ in radiative models) can be produced at the LHC, with $pp\to H_1^+H_1^-$ cross sections of order $10$ fb for $m_{H_1} \sim 150$ GeV; decays $H_1^+\to \ell^+ \nu$ yield flavor ratios predictive of the neutrino mass ordering (Kanemura et al., 2011).
• Higgs Decays: For models with sizable $Y_\nu$ and light right-handed neutrinos, $h \to \nu N$ decays constrain $Y_{\nu e}, Y_{\nu\mu}\lesssim 10^{-2}$ for $M_N$ in the 60–140 GeV range (Dev et al., 2012).
• No Neutrinoless Double Beta Decay: Pure Dirac neutrinos do not induce $0\nu\beta\beta$; thus observation of this would rule out all (lepton-number-conserving) Dirac mass models directly (Borah et al., 2019, 0906.3335).

5. Texture Analysis and Model Fitting

Some frameworks provide explicit algorithms for reconstructing $Y_\nu$ from oscillation data and model structure:

• Direct Parameterization: In $Z_3$-symmetric radiative models (Okada et al., 2020), for given neutrino spectrum and PMNS parameters, the allowed $3\times2$ Yukawa matrices $y_N$, $y_N'$ can be analytically expressed using basis vectors and antisymmetric matrices constrained by $m_\nu$.
• CKM–PMNS Alignment: In type-I seesaw, ansätze $Y_D \propto \text{diag}(y_d,y_s,y_b) V_{\rm CKM}^T$ or $Y_D \propto \text{diag}(y_u,y_c,y_t) V_{\rm CKM}^*$ have been tested, with only the down-type scenario and normal hierarchy surviving all current oscillation data (Haba et al., 2018).
• Left–Right Symmetric Reconstruction: In minimal LR models, the symmetry and heavy sector completely fix $Y_{\nu}$ up to the two unitary matrices diagonalizing the light and heavy Majorana mass matrices, making flavor predictions testable directly at colliders (Nemevsek et al., 2012).

6. Implications for Leptogenesis, Dark Matter, and Beyond

Mechanisms that generate suppressed Dirac Yukawa couplings often connect to baryogenesis and dark matter models:

• Leptogenesis: In "double Dirac seesaw plus leptogenesis" models, the singlet–doublet mixing responsible for a tiny VEV (and hence $Y_\nu$) also controls the CP-asymmetry in singlet scalar decays, establishing a direct correlation between neutrino properties and the baryon asymmetry (Gu, 2019).
• Unified Mass Origins: In Froggatt–Nielsen-type models, the same suppression parameter $\delta = v_S/\Lambda$ determines $y_\nu$ and the asymmetric dark matter mass, linking cosmological and neutrino observables (Ishida et al., 15 Oct 2025).
• Freeze-In Scenarios: Feeble Dirac Yukawas generated by RG flows can account for both active neutrino masses and freeze-in sterile dark matter relic abundance, as the coupling strengths required are comparable (Kowalska et al., 2022).

7. Concluding Remarks and Future Directions

Dirac neutrino Yukawa couplings provide a rich arena for exploring new physics beyond the SM. Their tiny scale demands protective mechanisms, most robustly realized by symmetry-based selection rules and radiative dynamics. The coupling patterns are increasingly subject to indirect and direct experimental constraints, with upcoming precision neutrino and collider experiments poised to probe or exclude entire classes of predictive Dirac-mass models. Models with calculable $Y_\nu$—fixed by flavor symmetries, GUT structure, or RG boundary conditions—offer clear avenues for validation or refutation as the neutrino sector continues to move toward a precision era. Potential falsifiability arises especially from the absence of $0\nu\beta\beta$ and the predicted branching fractions in rare flavor or collider observables.