Origin of the observed lepton flavor mixing pattern after the Daya Bay $\theta_{13}$ measurement: flavor-symmetry derivation of the MNSP matrix $U = U_0 + \Delta U$, its $\mu$–$\tau$ breaking and CP violation, and the unresolved $\theta_{23}$-octant, $\delta_{CP}$, unitarity, and Majorana-phase discriminants

After Daya Bay established that the smallest neutrino mixing angle $\theta_{13}\simeq 9^\circ$ is not suppressed, Xing (2012) posed the problem of the origin of the lepton flavor mixing structure: which flavor symmetry, if any, produces a constant leading mixing matrix $U_0$ , what dynamics generates the perturbation $\Delta U$ responsible for $\theta_{13}$ and CP violation, whether $\theta_{13}$ can be generated radiatively from $\theta_{13}^{(0)}=0^\circ$ , and whether the observed $3\times 3$ mixing matrix is exactly unitary (the paper raises the last two questions but defers their detailed renormalization-group-running and beyond- $3\times 3$ sterile-neutrino analyses to companion works). A decade of later data has settled $\theta_{13}$ itself to sub-degree precision and charted the radiative-generation question, but the theoretical core (no accepted flavor-symmetry derivation of the observed pattern, including its approximate $\mu$ – $\tau$ symmetry and its breaking) and the named experimental discriminants (the sign of $\theta_{23}-45^\circ$ , the Dirac phase $\delta_{CP}$ , Majorana phases, unitarity) remain open as of the NuFIT-6.0 (2024) global fit.

Background

Neutrino oscillations show that the three known neutrino flavors mix: the weak-interaction (flavor) states are superpositions of mass eigenstates, related by the Maki–Nakagawa–Sakata–Pontecorvo (MNSP, or PMNS) matrix $U$ , the lepton-sector analogue of the quark CKM matrix. In the standard three-neutrino picture $U$ is unitary and is described by three angles ( $\theta_{12},\theta_{23},\theta_{13}$ ), one Dirac CP-violating phase $\delta$ (governing matter–antimatter asymmetry in oscillations), and, if neutrinos are Majorana particles (their own antiparticles), two additional Majorana phases ( $\rho,\sigma$ ) that affect only lepton-number-violating processes such as neutrinoless double-beta decay. Before 2012, $\theta_{12}\approx 33^\circ$ and $\theta_{23}\approx 45^\circ$ were known while $\theta_{13}$ was only bounded from above; many flavor models therefore took $\theta_{13}=0^\circ$ as an input, e.g. the popular tri-bimaximal pattern ( $\sin\theta_{12}=1/\sqrt 3$ , $\sin\theta_{23}=1/\sqrt 2$ , $\theta_{13}=0$ ), often derived from discrete family symmetries — finite groups that permute the three fermion families, such as $A_4$ (the group of even permutations of four objects). The Daya Bay reactor experiment's observation of electron-antineutrino disappearance (5.2 $\sigma$ ; $\sin^22\theta_{13}=0.092\pm0.016_{\rm stat}\pm0.005_{\rm syst}$ ) established $\theta_{13}\simeq 9^\circ$ , invalidating all mixing patterns built on $\theta_{13}=0^\circ$ and making the origin of the observed pattern a sharp question. Xing (2012) systematized the phenomenology in a $U=U_0+\Delta U$ framework: a constant leading matrix $U_0$ (democratic, bimaximal, tri-bimaximal, golden-ratio, hexagonal, or newer 'correlative'/'tetra-maximal' patterns with nonzero leading $\theta_{13}^{(0)}$ ) accounts for the two large angles, while a small perturbation $\Delta U$ accounts for $\theta_{13}$ , CP violation, and departures from exact patterns; alternatively, mixing angles may be tied to ratios of charged-lepton and neutrino masses via texture zeros in the mass matrices. The paper stressed that the observed near-maximality of $\theta_{23}$ suggests an approximate $\mu$ – $\tau$ (second/third lepton generation interchange) symmetry of $|U|$ , whose soft breaking is probed by the sign of $\theta_{23}-45^\circ$ and by $\delta$ , described geometrically by leptonic unitarity triangles (the six triangles formed by the orthogonality relations of $U$ 's columns/rows). The open issues it explicitly acknowledged are: (i) whether any constant $U_0$ pattern can be derived from a simple, convincing flavor symmetry; (ii) the dynamical origin of $\Delta U$ or of the required mass-matrix textures $\Delta M_l$ , $\Delta M_\nu$ (concrete model building); (iii) experimental determination of the sign of $\theta_{23}-45^\circ$ and of $\delta$ ; (iv) whether renormalization-group (RG) running — the slow scale dependence of masses and mixings induced by quantum corrections — or threshold effects can generate $\theta_{13}\simeq 9^\circ$ from $\theta_{13}^{(0)}=0^\circ$ without extreme conditions (deferred in the paper to a companion analysis); and (v) whether the $3\times 3$ matrix is exactly unitary (e.g. with light or heavy sterile — flavor-singlet — neutrinos mixed in) and how the Majorana phases affect observables (deferred to a companion review). Current data (global fit NuFIT-6.0, September 2024 data set): $\theta_{12}$ , $\theta_{13}$ , and the mass splittings are measured at the few-to-15% level, but the $\theta_{23}$ octant (above or below $45^\circ$ ) is undetermined, $\delta_{CP}$ is unsettled (for normal mass ordering consistent with CP conservation within 1 $\sigma$ ; for inverted ordering $\delta\simeq 270^\circ$ favored over CP conservation at $>3.6\sigma$ ), and no unitarity violation or Majorana phase has been observed.

Problem Statement

The problem, as posed by the source paper after Daya Bay's $\theta_{13}\simeq 9^\circ$ measurement and still open, is to explain the observed lepton flavor mixing structure and its CP violation from an underlying symmetry principle and dynamics, jointly comprising: (1) the origin question — determine which flavor symmetry, if any, yields a constant leading mixing matrix $U_0$ (whether with $(U_0)_{e3}=0$ , such as tri-bimaximal-like patterns, or with nonzero leading $\theta_{13}^{(0)}$ , such as correlative or tetra-maximal patterns), and derive it from a symmetry of the lepton mass matrices, with the pattern's angles and the leading $(U_0)_{e3}$ or $\theta^{(0)}_{13}$ emerging as predictions of that symmetry rather than as postulated numerical entries or fitted values, and with the derivation simple and convincing by the standards of the flavor-model literature; (2) the dynamics — construct concrete models that generate the perturbation $\Delta U$ (or the mass-matrix textures $\Delta M_l$ , $\Delta M_\nu$ ) responsible for $\theta_{13}$ and the CP-violating phases, with the correct size and pattern of $\mu$ – $\tau$ symmetry breaking; (3) the radiative question — establish, in realistic models, whether $\theta_{13}\simeq 9^\circ$ can be generated from $\theta_{13}^{(0)}=0^\circ$ by renormalization-group running or threshold corrections without extreme conditions, given that generic one-loop running in the MSSM (the minimal supersymmetric standard model) is known to be insufficient; (4) the experimental discriminants that the theoretical question is entangled with — the sign of $\theta_{23}-45^\circ$ (the octant of $\theta_{23}$ ), the value of the Dirac phase $\delta$ , whether the $3\times 3$ MNSP matrix is exactly unitary in the presence of possible light or heavy sterile neutrinos, and the Majorana phases $\rho,\sigma$ — required here in the form of model-discriminant relations, i.e. the proposed symmetry and dynamics must state what the leading pattern $U_0$ and its perturbation $\Delta U$ predict for these quantities, and with what tolerance, rather than supplying new measurements of them. Items (3) and (4) are raised in the source paper — it calls generating $\theta_{13}\simeq 9^\circ$ from $\theta_{13}^{(0)}=0^\circ$ 'a highly nontrivial job', names the sign of $\theta_{23}-45^\circ$ 'a useful and sensitive model discriminator', and points to exact-unitarity tests of the $3\times 3$ matrix and the probing of $\rho,\sigma$ as directions of the new era — while deferring the detailed RG-running analysis and the beyond- $3\times 3$ treatment with light or heavy sterile neutrinos to companion works (Refs. [32] and [44] there). These components jointly serve one objective — the origin of the observed mixing pattern — because each model of items (1)–(3) is only discriminable once the inputs of item (4) are known, and conversely the precision targets of item (4) are set by the models of items (1)–(3). A resolution consists of the three theory components (1)–(3) together with the model-discriminant account of component (4): a symmetry-derived leading pattern $U_0$ whose angles, including the leading $\theta^{(0)}_{13}$ where it is nonzero, follow from that symmetry and reproduce the measured mixing angles within the precision quoted above; concrete dynamics for the perturbation $\Delta U$ (or the textures $\Delta M_l$ , $\Delta M_\nu$ ) with the stated size and pattern of $\mu$ – $\tau$ symmetry breaking; and an account, in realistic models, of whether and how $\theta_{13}\simeq 9^\circ$ can be generated radiatively from $\theta^{(0)}_{13}=0^\circ$ . Whether a particular symmetry is 'simple and convincing' is judged by the standards of the flavor-model literature rather than by a mechanical acceptance criterion, with that standard applying to the symmetry derivation of $U_0$ .

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