Dynamical origin of the flat (democratic) $SO(2)$ flavor structure and its extension to Majorana neutrinos and non-normal orderings

A series of works by the same group, culminating in G. Xu and Y. Zhang, EPL 143 (2023) 44001, proposes that quark and lepton mixing are dominated by approximate $SO(2)$ family symmetries of the fermion mass matrices: in the mass-hierarchy limit each mass matrix factorizes into a diagonal phase matrix and a real symmetric pattern matrix, phenomenology selects the flat (democratic) pattern shared by the up-type, down-type, charged-lepton, and neutrino sectors, and the CKM and PMNS mixing matrices are reproduced with four mixing parameters each (for quarks $\theta^u$, $\theta^d$, $\lambda_1$, $\lambda_2$, and analogously for leptons), including hierarchy corrections through second order — but with neutrinos assumed to be Dirac fermions with normal mass ordering. The structure is fitted, not derived: what dynamics selects the flat pattern and its cross-sector homology, generates the $SO(2)$ rotation angles and the CP-violating Yukawa phases, and produces the specific hierarchy-breaking pattern remains open, as does the compatibility of the unified mechanism with Majorana neutrinos and with inverted or quasi-degenerate mass ordering. The open problem is to construct an explicit flavor dynamics from which the flat $SO(2)$ structure follows, and to determine the mechanism's neutrino-sector domain of validity.

Background

In the Standard Model of particle physics, all fermion masses and flavor-mixing phenomena follow from the Yukawa couplings — the otherwise unexplained coefficients of the interaction terms between fermions and the Higgs field. After electroweak symmetry breaking, the Yukawa couplings of each fermion type $f$ (up- and down-type quarks, charged leptons, and neutrinos) form a $3\times 3$ mass matrix $M^f$, diagonalized by biunitary transformations, i.e. independent unitary rotations $U_L^f$ and $U_R^f$ of the left- and right-handed fields. The mismatch $U_L^u (U_L^d)^\dagger$ of the left-handed rotations between the up and down quark sectors is the Cabibbo–Kobayashi–Maskawa (CKM) mixing matrix measured in weak decays; the analogous lepton-sector mismatch is the Pontecorvo–Maki–Nakagawa–Sakata (PMNS) matrix measured in neutrino oscillations. Why the three CKM angles are small while two PMNS angles are large, why the masses obey a strong hierarchy $m_1^f \ll m_2^f \ll m_3^f$, and what breaks CP symmetry together constitute the Standard Model flavor problem.

A long phenomenological tradition encodes masses and mixing in structured mass matrices, going back to Fritzsch's 1979 quark mass–mixing relations and to the democratic mass matrix of Harari, Haut, and Weyers — the rank-one matrix with all entries equal, whose symmetry properties were later linked to permutation symmetries of the three families. In this tradition, Xu and Zhang (EPL 143 (2023) 44001) and preceding and following works by the same group build a bottom-up flat-pattern framework whose content is as follows. Define the hierarchy parameters $h_{ij}^f = m_i^f/m_j^f$ with $i<j$. In the hierarchy limit $h_{ij}^f \to 0$, the mass matrix normalized to the family mass sum reduces to a rank-one matrix $(U_L^f)^\dagger \mathrm{diag}(0,0,1) U_L^f$. Choosing the convention $U_R^f = U_L^f$ (right-handed rotations are unphysical in mixing observables), this factorizes as

$$\frac{1}{\sum_i m_i^f} M_0^f = (K_L^f)^\dagger M_N^f K_L^f$$

where $K_L^f = \mathrm{diag}(e^{i\eta_1}, e^{i\eta_2}, 1)$ is a diagonal phase matrix and $M_N^f$ is a real symmetric pattern matrix with two real parameters $(l_1, l_2)$. The pattern matrix is invariant under an $SO(2)$ family rotation — a continuous rotation acting on the three family (generation) indices, the analogue of a spatial rotation about a fixed axis — about the axis $\mathbf{n} \propto (l_1, l_2, 1)$. Composing the diagonalization $S M_N (S)^T = \mathrm{diag}(0,0,1)$ with this rotation yields the mixing matrices; for quarks

$$V_{CKM} = R_3(\theta^u) S^u\, \mathrm{diag}(e^{i\lambda_1}, e^{i\lambda_2}, 1)\, (S^d)^T R_3^T(\theta^d)$$

where $R_3(\theta)$ is the rotation in the degenerate 1–2 family subspace and the Yukawa phases $\lambda_{1,2}$ (differences of the phases $\eta_i$ between the up and down sectors) carry all CP violation, measured through the Jarlskog invariant (the rephasing-unique CP-odd combination of CKM elements). Hierarchy corrections are added perturbatively in $h_{23}^f$ and, at second order, $h_{12}^f$, with a specific rotation-angle-dependent breaking pattern of the pattern matrix. Phenomenology selects the flat choice $l_1 = l_2 = 1$, for which $M_N^f$ becomes the democratic matrix $\frac{1}{3} \mathbf{1} \mathbf{1}^T$ in all four sectors — the homology of the pattern across the up, down, charged-lepton, and neutrino matrices — and the CKM and PMNS data are then reproduced, through second-order corrections, with four parameters for each mixing matrix (for quarks $\theta^u$, $\theta^d$, $\lambda_1$, $\lambda_2$, and analogously for leptons, the pattern being shared across all sectors), with neutrinos treated as Dirac fermions with normal (hierarchical) mass ordering. The framework's central claim is that mixing is dominated by the $SO(2)$ family rotations while the mass hierarchy supplies only small perturbations, and that CP violation survives the degeneracy of the first two families.

The structure, however, is entirely bottom-up. The flat pattern and its cross-sector homology, the rotation angles $\theta^f$, the phases $\lambda_{1,2}$, and the specific hierarchy-breaking pattern are fitted or assumed rather than derived. The source paper says so explicitly: the orthogonal transformations relating gauge and mass bases need to be determined from flavor physics models, the homology is imposed by requirement, the flat structure arises by choosing a special (Yukawa) basis, and the Dirac, normal-ordering nature of neutrinos is an assumption. The most recent papers of the program (2024–2026) have derived parts of the structure from weaker premises — the two-parameter pattern from the hierarchy condition, the up–down homology from the sub-unitarity of CKM mixing in the two-family limit (the approximate unitarity of the CKM submatrix restricted to the first two families, which emerges because the third family is decoupled by the mass hierarchy) — but the selection of the flat pattern itself is still justified only by simplicity, and the latest summary statements of the program name the dynamical origin of the structure and its extension beyond normal-ordered Dirac neutrinos as the open questions.

Problem Statement

Can the empirical flat-pattern $SO(2)$ flavor structure be derived rather than assumed? Concretely:

(1) Dynamical origin. Construct an explicit flavor dynamics — for example an ultraviolet-complete model with a family symmetry and a specified symmetry-breaking sector, or another concrete mechanism — from which the leading-order structure follows: the factorized form $(K_L^f)^\dagger M_N^f K_L^f$ with $M_N^f$ real and symmetric, the dynamical selection of the flat pattern $l_1 = l_2 = 1$ (why nature would choose any specific $(l_1, l_2)$ is explicitly posed as a puzzle by the program's latest paper), the homology of the pattern matrices across the up, down, charged-lepton, and neutrino sectors, the $SO(2)$ family rotation angles $\theta^f$, the Yukawa phases $\lambda_{1,2}$ as the source of CP violation, and the specific angle-dependent hierarchy-breaking pattern at first and second order. Any such dynamics must be an explicit, self-consistent model whose symmetry breaking reproduces this structure without inserting it by hand, preserving the successful CKM/PMNS fits without reintroducing redundancy among the low-energy flavor parameters.

(2) Neutrino-sector generality. Determine whether the unified $SO(2)$ mixing mechanism is compatible with Majorana neutrinos — which carry two additional physical CP phases and whose small masses are usually generated by seesaw-type mechanisms (in which heavy neutral fermions induce tiny light-neutrino masses), producing complex symmetric effective mass matrices — and with inverted or quasi-degenerate neutrino mass ordering, for which the hierarchy-based reconstruction of the mass matrix does not apply in its present form. The determination must concern the unified mechanism itself, including any generalized reconstruction needed for these cases, rather than a narrower model class selected for the purpose.

The two components serve one scientific objective: to convert the phenomenological flat $SO(2)$ flavor structure into a dynamically grounded structure and to fix its domain of validity across the fermion spectrum. The higher-order-correction limitation present in the source paper has been resolved within the program (second-order corrections are now computed and fitted) and is therefore not part of the open problem.

References

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