Completing the disentanglement of fermion mass and flavor-mixing hierarchies: lepton-sector realization and phenomenological closure of the flavor-blind hierarchy framework

The Standard Model flavor puzzle combines hierarchies of fermion masses with hierarchies of mixing: quark masses span orders of magnitude while CKM mixing is small, and charged leptons are hierarchical while PMNS mixing is large. Froggatt-Nielsen-type horizontal symmetries explain both with one set of generation-dependent charges, intrinsically tying the origin of the mass hierarchy to that of flavor violation. Knapen and Robinson (2015) showed that the two origins can in principle be disentangled - mass hierarchies generated by a flavor-blind sector coupled universally to fermions carrying only their $U(3)$ flavor symmetries, at a scale independent of flavor breaking - and the quark-sector realization was subsequently completed to a global-minimum proof with a naturally hierarchical CKM. What remains open is the completion of this disentangling program: a lepton-sector realization producing hierarchical charged leptons, large PMNS mixing, and a realistic neutrino spectrum (with the extreme suppression of the neutrino Yukawas delegable to a different mechanism, as in the original letter), and robustness of the dynamically aligned vacuum against portal and cross-sector effects; the precision-flavor, collider, and baryogenesis phenomenology of an electroweak-scale hierarchy sector is a motivation and a desirable consequence. The problem asks for this completion or a demonstrated obstruction to it.

Background

The Standard Model (SM) of particle physics concedes a 'flavor puzzle': the Yukawa couplings of the Higgs boson to fermions span more than five orders of magnitude, from the top-quark coupling of order one down to $m_u/m_t\sim10^{-5}$, with no explanation inside the theory. In the quark sector this hierarchy coexists with small mixing: the Cabibbo-Kobayashi-Maskawa (CKM) matrix, the unitary matrix relating the quark mass and interaction bases, is close to the identity (for example $|V_{cd}|\simeq0.2$ while $m_u/m_c\sim10^{-3}$). In the lepton sector the pattern flips: charged-lepton masses are strongly hierarchical, yet the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) lepton mixing matrix has all angles of order one. The classic dynamical explanation is the Froggatt-Nielsen mechanism, in which a spontaneously broken 'horizontal' flavor symmetry assigns different generation-dependent charges to the SM fermions, so that masses and mixing angles are suppressed by powers of a common small symmetry-breaking parameter. Horizontal-symmetry schemes of this type intrinsically tie together the two phenomena: the same generation-dependent quantum numbers that produce the mass hierarchy also fix flavor mixing and flavor-violating observables. An older alternative by Nelson and Strassler suppresses anarchic Yukawa couplings (couplings all of similar, unordered size) by infrared renormalization-group running in approximately conformal field theories (theories with nearly scale-invariant dynamics), without any flavor symmetry at all. Knapen and Robinson (2015) asked whether the two origins can instead be disentangled within a symmetry framework, and answered the existence question affirmatively. In their construction the SM fermions carry only their accidental $U(3)$ flavor symmetries - the unitary rotations of each three-generation fermion multiplet, for example $U(3)_Q\times U(3)_U$ for quark doublets and right-handed up quarks. Symmetry breaking is encoded in spurions: fields whose vacuum expectation values (vevs) break the symmetry while transforming so that the Lagrangian formally preserves it. A set of bifundamental 'flavon' fields $\lambda_\alpha$ - bifundamental meaning transforming as a fundamental of one $U(3)$ and an antifundamental of the other - acquires vevs that are aligned (simultaneously biunitarily diagonalizable by the same pair of unitary matrices), spectrally disjoint (nonzero eigenvalues of different spurions never overlap) and of rank one, a pattern dynamically enforced by the minimum of the most general parity-respecting renormalizable potential. A separate 'hierarchy' sector of flavor-singlet fields ('hierarchons') $s_\alpha$, coupled to the SM through flavor-blind portal operators (renormalizable interactions linking two otherwise independent sectors), fixes the relative sizes of the fermion masses. The hierarchy scale $\Lambda_H$ and the flavor-breaking scale $\Lambda_F$ are thereby autonomous: the mass-hierarchy dynamics can sit near the electroweak scale, unconstrained by precision flavor bounds, with possible consequences for collider physics and for electroweak baryogenesis - the production of the cosmic matter-antimatter asymmetry during the electroweak symmetry-breaking epoch. Knapen and Robinson's illustrative quark vacuum, however, yielded only a single large mixing angle; with up-down flavon cross-terms only a local minimum was established; and the lepton sector was sketched only qualitatively (large PMNS angles appear possible, but the extreme smallness of the neutrino Yukawas was delegated to 'a different mechanism'). Altmannshofer, Gori, Robinson and Tuckler (2018) completed much of the quark sector: they proved that the global minimum of the general renormalizable flavon potential is the 'flavor-locked' configuration, and showed that in a 'flavorful' two-Higgs-doublet model (2HDM: an extended Higgs sector in which one doublet couples to the third quark generation and a second doublet to the first two) a CKM hierarchy of the observed size emerges naturally, with meson-mixing phenomenology partially explored. Their paper explicitly defers the lepton sector and the broader flavor phenomenology to future work. Adjacent programs address the flavor puzzle along different axes: 'clockwork' models explain mass and mixing hierarchies together from chains of vector-like fermions (heavy fermions whose gauge charges permit large Dirac masses); 'flavor-specific' scalars tie specific masses to specific flavors - both re-couple rather than disentangle the two origins; and Saldaña-Salazar's 'flavor-blind principle' organizes permutation symmetries of flavor-blind Yukawa contributions algebraically, without a dynamical hierarchy mechanism.

Problem Statement

Can the disentanglement of the fermion mass hierarchy from flavor mixing be carried through to a complete, realistic symmetry framework? The single overarching objective is a theory in which (i) the SM fermions carry no horizontal or flavor charges beyond their $U(3)$ flavor symmetries; (ii) the observed mass hierarchies of quarks and charged leptons are generated dynamically by a flavor-blind hierarchy sector whose scale $\Lambda_H$ is independent of the flavor-breaking scale $\Lambda_F$; and (iii) the full observed mixing structure - the hierarchical CKM quark-mixing matrix and the large-angle PMNS lepton-mixing matrix - arises in the same framework without tuning, that is, without small input parameters wherever the hierarchy is claimed to be generated. The quark-sector ingredients exist in proof-of-principle form (Knapen and Robinson 2015; Altmannshofer, Gori, Robinson and Tuckler 2018); the surviving open core of the problem comprises two jointly necessary components of that one objective. 1. Lepton-sector realization: construct the flavon and hierarchon structure for the leptons (flavons being the flavor-symmetry-breaking fields, hierarchons the flavor-singlet fields that set the mass hierarchy) - hierarchical charged-lepton masses together with large PMNS mixing (order-one entries in the sense of the source letter, whose lepton-sector sketch yields 'arbitrary $\mathcal{O}(1)$ entries' of the PMNS matrix) and a realistic normal or inverted neutrino mass spectrum - without charging the SM leptons under horizontal symmetries and without small input parameters. Following the source letter, the extreme overall suppression of the neutrino Yukawas may be delegated to a different mechanism (for example a seesaw) assumed as an external input, rather than generated inside the flavon/hierarchon framework. 2. Vacuum robustness: establish that the aligned (simultaneously biunitarily diagonalizable), spectrally disjoint (nonzero eigenvalues of different spurions never overlap), rank-one configuration is the global minimum, and remains so under radiative corrections, for the full theory including lepton flavons, quark-lepton flavon cross terms, and flavon-hierarchon portal couplings. The existing global-minimum ('flavor-locking') proof covers the tree-level quark flavon potential including up-down flavon cross terms, under an imposed permutation symmetry among same-type flavons and its coefficient conditions; the original letter established global minimality only within a single flavon set and only local stability in the mixed two-sector potential once up-down cross terms are included; and the flavon-hierarchon portal couplings are treated in the 2018 paper only through a two-loop negligibility argument in a specific UV completion. If $\Lambda_H$ lies near the electroweak scale, the framework also has phenomenological consequences outside the scientific target: precision-flavor observables (meson mixing - partially explored in the 2018 paper - lepton-flavor violation, electric dipole moments), hierarchon and flavon signatures at colliders, and an electroweak-baryogenesis window, electroweak baryogenesis being the production of the cosmic matter-antimatter asymmetry during electroweak symmetry breaking, favored when the CP-violating Jarlskog invariant of the quark sector (the rephasing-invariant measure of CP violation built from quark masses and CKM mixing) is large during sphaleron transitions (thermal early-universe events that violate baryon and lepton number). The problem is the completion of (i)-(iii) across the quark and lepton sectors together with the vacuum-robustness analysis of component 2, or the demonstration of an obstruction to (i)-(iii) or to the global minimality and radiative stability of the aligned, spectrally disjoint, rank-one configuration in the full theory.

References

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