Modular flavor symmetry can generate sharp, testable relations among quark and lepton masses, but the constructions of Chen, King, Medina and Valle (JHEP 02 (2024) 160, arXiv:2312.09255) realize them sector by sector: relations R1-R4 connect down-type quark masses to charged-lepton masses, while a separate correlation connects up-quark masses to the solar neutrino mass splitting. No single modular-invariant model is known that preserves both sets of predictions at once without extra assumptions, yields a definite viable CKM matrix, and controls the Kähler-potential corrections that the purely holomorphic derivation neglected. The problem is to build such a complete model. The question is posed by the source paper itself and remains open.

Background

The flavor problem of the Standard Model (SM) is the absence of any explanation for the observed pattern of fermion masses and mixing matrices. Modular flavor symmetry is a relatively recent approach in which the Yukawa sector is required to be invariant under the modular group SL(2,Z)SL(2,\mathbb{Z}) or one of its finite quotient groups ΓN\Gamma_N (for example Γ4≅S4\Gamma_4 \cong S_4, the permutation group on four elements): the group acts nontrivially both on the matter fields and on a single complex modulus field τ\tau, so that symmetry assigns each chiral superfield a representation of ΓN\Gamma_N and a modular weight (an integer kk such that the field picks up a factor (cτ+d)−k(c\tau+d)^{-k} when τ\tau is sent to (aτ+b)/(cτ+d)(a\tau+b)/(c\tau+d)). The Yukawa couplings are then vector-valued modular forms, holomorphic functions of τ\tau transforming in fixed representations with fixed weights, rather than arbitrary constants aligned by ad hoc vacuum expectation values of auxiliary 'flavon' fields. Near a symmetry point of moduli space, such as the cusp τ=i∞\tau = i\infty, every modular form admits a Fourier (q-)expansion in a small parameter ϵ\epsilon with ∣ϵ∣≪1|\epsilon| \ll 1, so the fermion mass matrices inherit a controlled expansion in ϵ\epsilon. Chen, King, Medina and Valle (2024) exploited this: the Hermitian mass matrices H=MM†H = M M^{\dagger} built from modular-invariant superpotential terms obey rank conditions at the symmetry point, and when HH depends on only two dimensionful parameters a1,a2a_1, a_2, eliminating these parameters yields a definite algebraic relation among the three fermion masses. The coefficients of these relations are fixed by the Clebsch-Gordan (CG) coefficients of the finite modular group (the numerical tensors decomposing products of group representations into irreducible components) together with the Fourier expansion coefficients of the modular forms, not by tunable free parameters. In a detailed example based on Γ4≅S4\Gamma_4 \cong S_4 within the supersymmetric Standard Model (MSSM), they obtained four viable down-sector relations R1-R4 between down-type quark and charged-lepton masses, for instance R1: msmd/[mb(mb−3ms)]≈mμme/[mτ(mτ−3mμ)]m_s m_d/[m_b(m_b-3m_s)] \approx m_\mu m_e/[m_\tau(m_\tau-3m_\mu)], each compatible with the observed masses and each distinguishable from the classic golden mass relation mb/msmd≈mτ/mμmem_b/\sqrt{m_s m_d} \approx m_\tau/\sqrt{m_\mu m_e} that arises in older family-symmetry models without grand unification. In an appendix they constructed a second class of relations for the up-quark and neutrino sector (with neutrino masses generated by the type-I seesaw mechanism, i.e. light neutrino masses suppressed by heavy right-handed Majorana partners), exemplified by a correlation linking the solar neutrino squared-mass splitting Δm212\Delta m^2_{21} to the small up-quark mass mum_u. The two constructions are complementary but were presented in isolation: the down-sector example says nothing about up quarks and neutrinos, and the up-sector example nothing about down quarks and charged leptons, so neither can address the CKM (Cabibbo-Kobayashi-Maskawa) quark-mixing matrix, which by definition involves both the up- and down-quark sectors. The authors note that each model can be completed trivially, but then one of the two mass-relation predictions is lost: preserving both simultaneously 'in a simple manner' was not achieved. A further caveat is that the derivation is purely holomorphic: in a supersymmetric theory the full action also contains a Kähler potential (the non-holomorphic function governing kinetic terms), whose form is not fixed by modular symmetry; Chen, Ramos-Sanchez and Ratz (2019) had shown that generic Kähler terms allowed by the symmetry introduce extra parameters and degrade the predictions of modular flavor models. Finally, the discriminating power of the relations relies on the light-quark masses mdm_d and msm_s, which are determined by lattice QCD and carry the largest uncertainties of all fermion masses.

Problem Statement

Construct a single complete modular-invariant flavor model of all Standard Model fermions — up- and down-type quarks, charged leptons, and neutrinos — that simultaneously preserves (i) a down-sector mass relation between down-type quark and charged-lepton masses of the Chen-King-Medina-Valle type — at least one of the alternative Γ4≅S4\Gamma_4 \cong S_4 relations R1-R4, Eqs. (50)-(53) of arXiv:2312.09255 (these are alternative branches of the same derivation, realized by a single model one at a time: satisfying at least one of them is required and satisfying all four simultaneously is not demanded), or the corresponding relation for another finite modular group derived by the same model-independent method — and (ii) an up-sector correlation of the source type between up-quark masses and neutrino mass splittings (Eq. (88) of arXiv:2312.09255, or the corresponding up-sector relation for that same finite modular group), without extra assumptions: no additional flavon fields, no vacuum alignments, and no free parameters beyond the minimal modular content — the field representations, the modular weights, and the overall coefficients of the modular-invariant contractions. The number of these overall coefficients must be counted explicitly and fixed in advance, and the realized down-sector relation (the satisfied one among R1-R4, or its analog for another group) together with the realized up-sector relation (Eq. (88) or its analog for that same group) must then be obtained by eliminating them, never by adjusting them to fit the observed masses; a model whose relations hold only after such adjustment is a fit, not a solution. Because both quark sectors are present, the model must yield a definite CKM mixing matrix that is viable, i.e. reproduces the measured quark mixing angles and the CP-violating phase within their experimental uncertainties, rather than only sector-by-sector predictions. The model must state and consistently treat the Kähler potential: the symmetry-allowed non-holomorphic Kähler corrections must be shown to deform the relevant mass-matrix entries by less than the precision of the observed fermion masses over the region of moduli space used, so that the realized down-sector relation and the realized up-sector relation remain satisfied within the uncertainties of the measured masses. A model in which those relations hold only before the Kähler corrections are included, or survive only because the corrections are allowed to violate them, does not resolve the task.

References

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