Flavor-Dependent Models (FDM)

• Flavor-Dependent Models (FDM) are extensions of the Standard Model that introduce new flavor-specific gauge groups, extra scalars, and modified Yukawa interactions to account for fermion mass hierarchies.
• They employ mechanisms akin to the see‐saw and spurion analyses to generate tree-level masses for the third generation and off-diagonal mixings for lighter generations.
• Key predictions include rare top decays, suppressed flavor-changing neutral currents, and distinctive neutrino mass spectra featuring one massless active neutrino.

Flavor‐Dependent Models (FDM) extend the Standard Model by introducing additional flavor-sensitive structures—typically new gauge symmetries, scalar sectors, and modified Yukawa interactions—with the aim of explaining both fermion mass hierarchies and flavor mixing patterns while also addressing neutrino masses and other observables. These models are constructed to link the origins of flavor structure and the suppression of flavor‐violating phenomena, often drawing on mechanisms similar to the see‐saw for neutrinos and the minimal flavor violation (MFV) paradigm, but with explicit differentiation among generations.

1. Theoretical Motivation and Context

FDMs are motivated by the long‐standing puzzles of the Standard Model, namely the vast hierarchies in fermion masses and the observed patterns of CKM and PMNS mixing that appear to defy explanation in a universal framework. Empirical observations, for example, the ratios $m_t/m_c \sim 10^2$ and $m_c/m_u \sim 10^3$, together with large neutrino mixing angles, suggest that the third generation may acquire mass at tree level while the lighter generations obtain masses through mixing—the so‐called “see‐saw” mechanism extended to all fermions. In many FDM constructions, an extra local gauge group such as $U(1)_F$ (or similar flavor–dependent combinations like $U(1)_{B-2L_\alpha-L_\beta}$) is introduced, with charges assigned in a generation‐dependent manner. This leads to a natural suppression of tree-level flavor-changing neutral currents in the quark sector and simultaneously enables predictive off-diagonal entries in the mass matrices that correlate directly with the observed flavor mixings.

2. Framework and Field Content

In a typical FDM, the Standard Model gauge group is extended to include an extra $U(1)_F$ factor, so that the full gauge symmetry reads

$$SU(3)_C \otimes SU(2)_L \otimes U(1)_Y \otimes U(1)_F.$$

The additional $U(1)_F$ is gauged with flavor-dependent charges. For example, one realization assigns charges such that first-generation doublets carry $+z$, second-generation doublets $-z$, and third-generation doublets are neutral, while right-handed singlets are assigned charges so that anomaly cancellation is achieved (often with the inclusion of two right-handed neutrinos having nonzero charges, while the third is omitted). The scalar sector is extended by introducing several Higgs doublets—typically three, with one (usually $\Phi_3$) behaving as the SM-like Higgs that couples at tree level to third-generation fermions—and at least one singlet scalar $\chi$ whose vacuum expectation value (VEV) breaks $U(1)_F$. Moreover, this additional scalar may generate Majorana masses for the right-handed neutrinos via the Type-I see‐saw mechanism. In other constructions, additional inert doublets or charged scalars are added to facilitate radiative generation of neutrino masses.

3. Mass Generation and Flavor Mixing Mechanisms

A hallmark of FDMs is that they differentiate the origin of masses among generations. In many realizations, only the third generation obtains tree-level Yukawa couplings with the SM-like Higgs (e.g. via $\Phi_3$), while the first two generations acquire masses through off-diagonal mixings with the third generation mediated by other Higgs doublets (such as $\Phi_1$ and $\Phi_2$). The fermion mass matrices thus take a “see-saw” structure: $$m_f = \begin{pmatrix} 0 & m_{12} & m_{13} \ m_{12}^* & 0 & m_{23} \ m_{13}^* & m_{23}^* & m_{33} \end{pmatrix},$$ with the (3,3) entry generated at tree level and the remaining entries arising from higher-dimensional operators or mixing effects. In the neutrino sector, a minimal seesaw is realized with only two right-handed neutrinos, leading to a rank-2 light neutrino mass matrix given by

$\hat{m}_\nu = - M_D^T M_R^{-1} M_D,$

which implies that one light neutrino is exactly massless. This prediction is robust and testable against the current oscillation data.

4. Spurions, Flavons, and Connection to Minimal Flavor Violation

To systematically account for the observed hierarchies and mixing angles, FDMs introduce spurion fields—non-dynamical symmetry-breaking parameters that later may be promoted to dynamical flavon fields. In the quark sector, spurions such as $Y_U$ (a bi-doublet under $SU(2)_{q_L}\times SU(2)_{u_R}$) and $Y_D$ (transforming as a doublet-triplet under $SU(2)_{q_L}\times SU(3)_{d_R}$) are introduced so that the effective Yukawa Lagrangian becomes

$$- \mathcal{L}_Y^q = y_t\, \bar{q}_{3L} \tilde{\Phi}_3\, t_R + \bar{Q}_L\, \tilde{\Phi}_{1,2}\, Y_U\, U_R + \bar{Q}_L\, \Phi_{1,2}\, Y_D\, D_R + \cdots.$$

The background values of these spurions are fitted to reproduce the observed quark masses and Cabibbo–Kobayashi–Maskawa (CKM) matrix. In the lepton sector, two frameworks may be considered: one with minimal field content (where neutrino masses arise through the Weinberg operator) and an extended field content (featuring right-handed neutrinos and a Type-I seesaw). In both cases, spurions such as $Y_E$ for charged leptons are employed. The resulting effective theory exhibits features analogous to Minimal Flavor Violation (MFV), with flavor-changing neutral currents (FCNCs) being highly suppressed; however, the explicit separation of the third generation in FDMs introduces rich structures that allow for slight departures from universality—a feature that can, for instance, lead to observable rare decays.

One notable consequence is that minimization of the scalar (flavon) potential yields very precise predictions for the Majorana phases. These phases, in turn, directly impact the effective neutrinoless double-beta decay mass

$$m_{ee} = \left|\sum_i (U_{ei})^2\, m_{\nu_i}\right|,$$

providing a potential smoking gun for FDM scenarios if such decay is observed within a narrow predicted window.

5. Experimental Signatures and Phenomenological Implications

FDMs predict numerous observable effects that distinguish them from flavor-universal models. For quarks, the models typically forbid tree-level FCNCs due to the flavor-dependent coupling assignments, yet predict enhancements in rare top decays such as

$$\text{Br}(t \to c\, h) \sim 10^{-4} \quad \text{and} \quad \text{Br}(t \to u\, h) \sim 10^{-5},$$

which are orders of magnitude above the Standard Model expectations and within reach of current collider experiments. In the lepton sector, the new $U(1)_F$ gauge boson (commonly denoted $Z'$) and additional scalars mediate lepton flavor violating (LFV) decays, yielding branching ratios for processes like

$$\text{BR}(\mu \to 3e) < 10^{-12},\quad \text{BR}(\tau \to 3\mu) \gtrsim 10^{-10},$$

subject to constraints that require the ratio $M_{Z'}/g_F$ to be in the tens-of-TeV range. Furthermore, predictions in the neutrino sector, such as one massless active neutrino and distinct correlations between the Dirac and Majorana phases (e.g. for normal or inverted hierarchies), are tightly constrained by oscillation data. Additional observables include the neutrino transition magnetic dipole moments, with typical predictions like

$|\mu_{23}| \sim 3\times 10^{-21}\,\mu_B,$

though these values remain far below current terrestrial experimental limits, they may be probed in astrophysical contexts.

Other experimental probes involve the flavor-dependent modification of nuclear parton distribution functions (the EMC effect) and heavy quarkonium potential measurements via holographic models. In these analyses the flavor structure leads to differing predictions for observables—such as the suppression in ${\rm Mpc}^{-3}$ number densities of low-mass galaxies in the FDM scenario compared to Cold Dark Matter (CDM)—and similarly, in viable FDM scenarios at colliders, signal strength measurements (e.g. for a 95 GeV excess in diphoton and ditau final states) can be reconciled with the flavor-dependent interactions predicted by the model.

6. Future Directions and Concluding Perspectives

The predictive power of FDMs lies in their unified explanation of fermion mass hierarchies, flavor mixing, and neutrino masses via a concrete extension of the Standard Model that incorporates explicit flavor dependence. The detailed structure of the Yukawa matrices—controlled by flavor spurions and dynamical flavon potentials—leads to precise predictions for LFV, rare top decays, and neutrino observables (including the effective mass for neutrinoless double-beta decay). Although stringent experimental constraints from meson oscillations, LFV decay searches, and collider measurements restrict the parameter spaces, ongoing and planned experimental programs at the LHC, neutrino detectors, and next-generation flavor experiments have the potential to further probe or validate the distinctive signatures of flavor-dependent models. This synthesis of flavor physics with dark matter phenomenology and neutrino mass generation renders FDMs a compelling framework for physics beyond the Standard Model.