The weighton mechanism (S. J. D. King and S. F. King, JHEP 09 (2020) 043) explains the observed hierarchies of quark and lepton masses and mixings in modular flavor symmetry: fermion modular weights play the role of Froggatt–Nielsen charges, and a Standard-Model-singlet 'weighton' field of nonzero modular weight plays the role of the flavon, while small quark mixing angles follow from the modulus τ\tau lying near the cusp i∞i\infty. The construction, however, leaves the mechanism without a dynamical and correction-controlled foundation: the Kähler potential is not fixed by modular symmetry, supersymmetry-breaking threshold corrections were not assessed, the fitted quasi-degenerate neutrino spectrum (∑mi≈0.34 \sum m_i \approx 0.34\,eV) is now excluded by cosmological bounds, and the required modulus value ⟨τ⟩\langle\tau\rangle was fitted by hand rather than stabilized, with the weighton vacuum expectation value produced only by a simple driving superpotential and no ultraviolet-complete embedding.

Background

Three families of quarks and leptons span an enormous range of masses — from the electron to the top quark — while lepton mixing angles are large and quark mixing angles are small; explaining this pattern is the 'flavor puzzle'. The Froggatt–Nielsen (FN) mechanism addresses the mass hierarchies by assigning fermions charges under an extra Abelian U(1)\mathrm{U}(1) symmetry whose breaking, via the vacuum expectation value (VEV) of a Standard-Model-singlet 'flavon' field, suppresses Yukawa couplings by powers of a small ratio ⟨θ⟩/Mfl\langle\theta\rangle/M_{\mathrm{fl}}. In the modular-symmetry approach to flavor (initiated by Feruglio in 2017, arXiv:1706.08749), no Abelian symmetry or flavon sector is introduced: Yukawa couplings are modular forms — holomorphic functions of a single complex modulus field τ\tau — that transform under a finite modular group ΓN\Gamma_N (level NN; for example Γ3≃A4\Gamma_3 \simeq A_4, the alternating group on four elements), and each matter supermultiplet carries a modular weight that fixes its transformation under the modular group. In supersymmetric models the superpotential is then almost completely determined by modular invariance, making the framework highly predictive. Two structural facts underpin what follows. First, the finite-dimensional space of modular forms admits a qq-expansion in q=e2πiτq = e^{2\pi i\tau}, so near the cusp fixed point τT=i∞\tau_T = i\infty (large Im τ\mathrm{Im}\,\tau with small Re τ\mathrm{Re}\,\tau) modular-form components are suppressed by powers of qq; at fixed points such as τ=i\tau = i, ω=e2πi/3\omega = e^{2\pi i/3}, and i∞i\infty a residual subgroup of the modular symmetry survives. Second, the Kähler potential (the supersymmetric kinetic-term function) is far less constrained by modular symmetry than the superpotential, so it contains free coefficients that modify the canonically normalized Yukawa matrices — a generic threat to the predictivity of the whole framework (Chen, Ramos-Sánchez and Ratz, Phys. Lett. B 801 (2020) 135153). Against this background, S. J. D. King and S. F. King (JHEP 09 (2020) 043) proposed the weighton mechanism: fermion modular weights act as FN charges, and a new Standard-Model singlet ϕ\phi with nonzero modular weight — the 'weighton' — acts as the flavon, its scaled VEV ϕ~=⟨ϕ⟩/Mfl∼1/15\tilde\phi = \langle\phi\rangle/M_{\mathrm{fl}} \sim 1/15 suppressing successive rows of the Yukawa matrices by powers of ϕ~\tilde\phi. They recast Feruglio's A4A_4 lepton model in this natural form, scanned 27 quark-sector weight assignments, and identified two viable combinations describing all quark and lepton masses and mixings with a single modulus, with benchmarks near Re τ≈0.036\mathrm{Re}\,\tau \approx 0.036, Im τ≈2.35\mathrm{Im}\,\tau \approx 2.35 (cusp-proximate, giving exponentially small CKM (Cabibbo–Kobayashi–Maskawa quark mixing matrix) angles), fitting at a reference grand-unification scale with matching to the minimal supersymmetric standard model (MSSM) at tan⁡β=5\tan\beta = 5 (the ratio of the two Higgs doublet vacuum expectation values). The neutrino sector was taken over unchanged from Feruglio's model, independent of the weighton, and predicts a quasi-degenerate spectrum m1∼m2∼0.11 m_1 \sim m_2 \sim 0.11\,eV, m3∼0.12 m_3 \sim 0.12\,eV with ∑mi∼0.34 \sum m_i \sim 0.34\,eV — which the authors themselves noted was already 'in the disfavoured region' cosmologically. The paper acknowledged its limitations: the Kähler-potential ambiguity (discussed in the introduction as a general framework issue), supersymmetry (SUSY)-breaking effects 'beyond the scope of this paper' (with a footnote that threshold corrections grow increasingly important for larger tan⁡β\tan\beta), the non-exhaustive class of examples (other levels N≠3N \neq 3, other weights, multiple moduli and weightons), and — implicitly, since ⟨τ⟩\langle\tau\rangle was only fitted — the dynamical origin of the modulus value and of the weighton VEV, for which only a simple driving superpotential Wdriv=χ(Y1(4) ϕ4/Mfl2−M2)W_{\mathrm{driv}} = \chi(Y^{(4)}_{\mathbf{1}}\, \phi^{4}/M^{2}_{\mathrm{fl}} - M^{2}) was exhibited, with no ultraviolet (UV)-complete embedding.

Problem Statement

Can the weighton mechanism be completed into a correction-controlled and cosmologically viable explanation of the fermion mass hierarchy? Concretely: determine whether a supersymmetric, modular-invariant realization of the weighton mechanism — the framework in which fermion modular weights act as Froggatt–Nielsen charges and a modular-weight-carrying singlet 'weighton' provides the hierarchical suppression — can address the limitations the original paper itself acknowledged, and, if it can, construct one: (a) the Kähler potential, which modular symmetry does not fix, and the SUSY-breaking threshold corrections at the grand-unification-scale matching (whose study the original paper declared 'beyond the scope of this paper', noting that threshold corrections become increasingly important for larger tan⁡β\tan\beta) are quantified to the point where the predicted Yukawa textures, fermion masses, quark mixing (CKM, Cabibbo–Kobayashi–Maskawa) and lepton mixing (PMNS, Pontecorvo–Maki–Nakagawa–Sakata), and CP phases remain well-defined and consistent with experimental data rather than adjustable through free Kähler/threshold coefficients, where this control must be established for the Kähler potential and threshold corrections permitted by modular invariance and the field content rather than only for a particular choice assumed in advance; (b) the weighton vacuum expectation value is generated with the higher-weight and higher-power modular corrections it induces under control; (c) the neutrino sector is compatible with the strongest cosmological bound on ∑mν\sum m_\nu cited in this task — the DESI DR2 2025 limit, tighter than the Planck 2018 bound also cited — which already excludes the original quasi-degenerate 0.34 0.34\,eV benchmark; and (d) the mechanism is developed beyond the non-exhaustive level-3 examples — with an explicit realization in at least one further setting that also satisfies (a)–(c) — as the original conclusion envisaged. These deliverables are facets of one objective — certifying that the weighton mechanism is predictive and realizable rather than a fitted parametrization — and all four are required: each is independently necessary, so a resolution must address every one of (a)–(d), positively or by a demonstrated obstruction.

References

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