The familon problem: constructing a fully realistic theory of spontaneously broken family symmetry with a light Nambu-Goldstone familon

Frank Wilczek's 1982 paper 'Axions and Family Symmetry Breaking' proposed replacing the Peccei-Quinn quasisymmetry with genuine family (flavor) symmetries whose spontaneous breaking at a high scale $F$ produces light, derivatively coupled Nambu-Goldstone bosons -- familons -- potentially observable in rare decays such as $\mu^+\to e^+ f$ and $K^+\to\pi^+ f$, and gave only schematic Lagrangian templates for such theories. The experimental components have been substantially discharged: dedicated searches now bound the familon decay constants ($F_{\mu e}\gtrsim 3\times10^{9}\ \mathrm{GeV}$ from muon decay, $F_{sd}\gtrsim 4\times10^{11}\ \mathrm{GeV}$ from $K^+\to\pi^+ f$); the previously missing search designs and sensitivities were worked out by Calibbi, Redigolo, Ziegler and Zupan (2021), who showed that MEG II and Mu3e can probe $F_{\mu e}$ up to $\sim10^{10}\ \mathrm{GeV}$; and the suggested short-range-force tests have grown into an active experimental program. Anomaly-free gauged family symmetries are known, and explicit familon and flavored-axion models have been built (three-Higgs flavored Peccei-Quinn models, flaxions, $U(2)$ axiflavons), but none is regarded as fully realistic: explaining rather than merely accommodating the observed mass hierarchies, naturalness with high-scale flavor breaking, and Landau poles in gauged flavor models remain unresolved. The surviving open problem is to construct an explicit anomaly-free three-family model of spontaneously broken family symmetry whose familon is consistent with all laboratory, astrophysical and cosmological bounds and makes testable predictions -- or to establish a no-go result excluding every such model.

Background

The Standard Model (SM) of particle physics contains three sequential replicas ('families' or 'generations') of quarks and leptons with identical gauge interactions but widely different masses -- the top quark is roughly 350,000 times heavier than the electron -- and an unexplained pattern of weak-interaction mixing angles, encoded for quarks in the Cabibbo-Kobayashi-Maskawa (CKM) matrix and for leptons in the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix. The SM accommodates but does not explain these numbers: they enter through Yukawa coupling matrices whose entries are free parameters. This is the SM 'flavor puzzle'.

A second, independent puzzle is the strong CP problem: quantum chromodynamics (QCD) admits a CP-violating vacuum angle $\bar\theta$ that experiments constrain to be below $10^{-10}$, without explanation. The canonical resolution postulates a global Peccei-Quinn (PQ) symmetry, spontaneously broken at a high scale; Goldstone's theorem then supplies a light pseudoscalar Nambu-Goldstone (NG) boson -- the axion -- whose QCD-induced potential dynamically relaxes $\bar\theta$ to zero. (A Nambu-Goldstone boson is the massless mode necessarily present when a continuous global symmetry is spontaneously broken; if the symmetry is only approximate, the boson is light rather than exactly massless.)

In 'Axions and Family Symmetry Breaking' (1982), Frank Wilczek proposed to enlarge this logic: the approximate global symmetry obtained by setting all fermion masses to zero contains large family (horizontal) symmetry groups acting on the generation index, and one could spontaneously break a genuine family symmetry at a high scale $F$ instead of, or in addition to, a PQ $U(1)$. The resulting NG bosons -- familons -- couple derivatively to flavor-changing currents through $\Delta\mathcal{L}=F^{-1}j_\mu\,\partial^\mu f$, so all familon effects are suppressed by powers of $1/F$. Two-body decays with a familon in the final state then probe $F$ directly: $\Gamma(\mu\to e f)/\Gamma(\mu\to e\nu\bar\nu)\simeq 2.5\times10^{14}\ \mathrm{GeV}^2/F_{\mu e}^2$ and $\Gamma(K^+\to\pi^+ f)/\Gamma(K^+\to\pi^+\pi^0)\simeq 1.3\times10^{14}\ \mathrm{GeV}^2/F_{K\pi}^2$, so a familon would appear as a monoenergetic positron sitting exactly at the endpoint of the continuous positron-energy (Michel) spectrum of ordinary muon decay, or as a monoenergetic pion in kaon decay. Stellar-cooling bounds (new light particles would drain energy from stars, so observed stellar lifetimes constrain their couplings) and axion-cosmology arguments left a plausible window $10^{9}\ \mathrm{GeV}\lesssim F\lesssim10^{12}\ \mathrm{GeV}$, making such searches feasible in principle. Purely derivative couplings generate static potentials $\propto 1/R^4$, innocuous on laboratory scales; but the axion also acquires anomaly-induced, CP-violating couplings to fermion spins, motivating short-range 'Cavendish-style' force tests at centimeter scales, where an axion with $F\sim10^{12}\ \mathrm{GeV}$ has its Compton wavelength.

On the model side the paper offered only schematic templates: many light Higgs doublets arranged in flavor multiplets; a single light Higgs doublet whose Yukawa couplings inherit flavor structure from tree-level exchanges of ultraheavy fermions and Higgs fields; and a 'many called, few chosen' construction in which high-scale flavor multiplets break so that only one light Higgs combination survives -- each illustrated with two-family toy models. Gauging family symmetries was flagged as awkward because many family groups are anomalous, meaning the fermion content induces gauge anomalies -- quantum-mechanical violations of a classical symmetry that render a gauge theory inconsistent unless the fermion charges are chosen so that all anomaly coefficients cancel. The electroweak hierarchy problem -- why the Higgs mass is so much smaller than the flavor-breaking scale $F$ -- was acknowledged to be no better solved than in the SM. The idea appeared essentially simultaneously in Reiss's 'Can the family group be a global symmetry?' (1982); Gelmini, Nussinov and Yanagida (1983) analyzed the associated NG bosons; Barr, Reiss and Zee (1982) studied the domain walls -- planar topological defects that form when a symmetry with a discrete remnant is spontaneously broken and, if stable, can come to dominate the universe's energy density -- that family-symmetry breaking can produce in cosmology; and the parallel majoron (the NG boson of spontaneously broken lepton number) showed the same logic applied to lepton number. The now-standard alternative implementation of a family symmetry, the Froggatt-Nielsen mechanism (1979), breaks a family $U(1)$ with a 'flavon' field but renders the symmetry inexact, so no light familon survives there. Whether a model with an exact (global or gauged) family symmetry and a genuine familon can be made fully realistic -- consistent with gauge anomalies, fermion masses and mixings, and all experimental and cosmological limits -- is the question this problem records.

Problem Statement

Restated in current terms, the question Wilczek (1982) left open is: does there exist an explicit, fully realistic model realizing the familon idea -- a quantum field theory containing the three SM fermion families in which an exact continuous family (flavor) symmetry, global or gauged, is spontaneously broken at a high scale $F$ so that a light (pseudo-)Nambu-Goldstone familon couples derivatively to flavor-changing currents -- satisfying all of the following simultaneously?

(1) Consistency: all gauge anomalies cancel, including those of the family symmetry itself if it is gauged, with a consistent mechanism generating masses for any family gauge bosons.

(2) Realism: the observed quark and lepton masses, CKM and PMNS mixing angles and CP phases are reproduced within experimental uncertainties.

(3) Testable familon phenomenology: the familon decay constants and flavor-changing couplings are consistent with all existing bounds -- from $\mu^+\to e^+ f$, $K^+\to\pi^+ f$, $\tau\to\ell f$ and $B\to(\pi,K) f$ searches, stellar-cooling, supernova and cosmological limits -- and are predicted sharply enough to be confronted with the projected sensitivities of the next-generation muon-decay experiments MEG II and Mu3e (at PSI), the rare-kaon experiment NA62 (at CERN) and the electron-positron B-meson factory Belle II (at KEK).

The question is resolved either by exhibiting such a model or by a no-go theorem or quantitative exclusion showing that no model satisfying (1)-(3) can exist, that is, by excluding the entire class of models defined by this statement. An experimental familon detection (for example a Michel-endpoint signal in $\mu^+\to e^+ X$ or a monoenergetic-pion signal in $K^+\to\pi^+ X$) would fix couplings that any such model must reproduce, but would not by itself resolve the model-construction question.

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