In modular flavor models, hierarchical fermion masses arise when the vacuum expectation value of the modulus τ\tau lies near one of the three critical points i\mathrm{i}, ω\omega, or i∞\mathrm{i}\infty of SL(2,Z)\mathrm{SL}(2,\mathbb{Z}). Chen, Li, Liu and Ratz (JHEP 10 (2025) 033) completed the classification of leading near-critical mass-matrix patterns and mass hierarchies at the elliptic points τ=i\tau=\mathrm{i} and τ=ω\tau=\omega for 2⊕1\mathbf{2}\oplus\mathbf{1} assignments, showed that the cusp τ=i∞\tau=\mathrm{i}\infty is the only critical point that can naturally accommodate the large quark mass hierarchies (with the mass product bounded by εNkdet⁡M/12\varepsilon^{N k_{\det M}/12}), and explicitly left the corresponding near-cusp classification open. The open problem is to complete this classification: determine the leading-order near-cusp mass-matrix patterns and the resulting mass-eigenvalue hierarchies for all admissible assignments of the two matter multiplets to the finite-image representations of SL(2,Z)\mathrm{SL}(2,\mathbb{Z}) enumerated in the Problem Statement below, organized by the order NN of ρ(T)\rho(\mathsf{T}) (the level), and verify consistency with the classification of one-dimensional vector-valued modular forms that constrains the determinant.

Background

The observed fermion masses of the Standard Model span more than five orders of magnitude, and one influential proposal to explain this 'flavor puzzle' is the modular flavor symmetry mechanism introduced by Feruglio in 2017. The Yukawa couplings are assumed to depend holomorphically on a single complex modulus τ\tau taking values in the upper half plane, and to transform as modular forms: under a modular transformation τ↦(cτ+d)/(aτ+b)\tau\mapsto(c\tau+d)/(a\tau+b) by SL(2,Z)\mathrm{SL}(2,\mathbb{Z}) (the group of 2×22\times 2 integer matrices of determinant one, generated by the matrices conventionally called S\mathsf{S} and T\mathsf{T}), a coupling of modular weight kk in representation ρ\rho picks up the automorphy factor (cτ+d)k(c\tau+d)^{k} and acts on multiplet indices through ρ\rho. When ρ\rho has finite image, this realizes a finite 'modular flavor group' ΓN\Gamma_N (or its double cover ΓN′\Gamma_N'), where the level NN is the order of the matrix ρ(T)\rho(\mathsf{T}). Yukawa couplings arranged into multiplets are vector-valued modular forms (VVMF): holomorphic functions transforming in a fixed representation with fixed weight.

A key structural fact, exploited systematically by Chen, Li, Liu and Ratz (2025), is that the determinant of any fermion mass matrix transforms as a one-dimensional (scalar) VVMF of weight kdet⁡Mk_{\det M}, and every one-dimensional VVMF of SL(2,Z)\mathrm{SL}(2,\mathbb{Z}) is a polynomial in the Dedekind eta function squared (η2\eta^2), the Eisenstein series E4E_4, and E6E_6. By the valence formula of modular form theory, for kdet⁡M<12k_{\det M}<12 the zeros of such a form can occur only at the three critical (fixed) points of the fundamental domain: τ=i\tau=\mathrm{i} (stabilized by the generator S\mathsf{S}), τ=ω=e2πi/3\tau=\omega=\mathrm{e}^{2\pi\mathrm{i}/3} (stabilized by ST\mathsf{S}\mathsf{T}), and the cusp τ=i∞\tau=\mathrm{i}\infty (stabilized by T\mathsf{T}). A mass hierarchy then arises when the modulus vacuum expectation value (VEV) deviates slightly from a critical point where the determinant vanishes: writing ε\varepsilon for the small deviation parameter, hierarchies near i\mathrm{i} are bounded from above by ε2\varepsilon^{2}, near ω\omega by ε3\varepsilon^{3}, and near the cusp by εNkdet⁡M/12\varepsilon^{N k_{\det M}/12}, since near τ=i∞\tau=\mathrm{i}\infty the natural expansion parameter is ε=e2πiτ/N\varepsilon=\mathrm{e}^{2\pi\mathrm{i}\tau/N} with NN the order of ρ(T)\rho(\mathsf{T}). Because NN can exceed 3 while the elliptic stabilizer orders cannot, the cusp is the only critical point capable of generating the large hierarchies of the quark sector: for example, an up-quark spectrum 1:ε3:ε51:\varepsilon^{3}:\varepsilon^{5} with ε∼10−1\varepsilon\sim 10^{-1} requires a determinant zero of order at least 8.

The near-critical behavior is universal in the sense that it depends only on the residual symmetry: at each critical point τ0\tau_0 the stabilizer subgroup acts linearly on the matter fields after a field redefinition, and the leading expansion of the mass matrix in ε\varepsilon is fixed by the eigenvalues ('weighted charges') of the matter multiplets under the residual abelian group — Z4S\mathbb{Z}_{4}^{\mathsf{S}} at i\mathrm{i}, Z3ST×Z2S2\mathbb{Z}_{3}^{\mathsf{S}\mathsf{T}}\times\mathbb{Z}_{2}^{\mathsf{S}^{2}} at ω\omega, and ZNT×Z2S2\mathbb{Z}_{N}^{\mathsf{T}}\times\mathbb{Z}_{2}^{\mathsf{S}^{2}} at the cusp, where the generator S2\mathsf{S}^{2} acts like parity, excluding certain terms from the Lagrange density and leading to blocks of texture zeros in addition to the ε\varepsilon-power mass-matrix patterns. Feruglio and collaborators developed this framework ('near-criticality and universality'; the self-dual point τ=i\tau=\mathrm{i} was treated in detail in 2021, and both elliptic points in 2023), and Novichkov, Penedo and Petcov (2021) tabulated the residual-symmetry decompositions, including the ZNT\mathbb{Z}_{N}^{\mathsf{T}} decomposition at the cusp, but only for the four smallest levels N≤5N\leq 5. Liu and Ding (2022) classified the one- and two-dimensional irreducible representations of SL(2,Z)\mathrm{SL}(2,\mathbb{Z}) with finite image — twelve one-dimensional and fifty-four two-dimensional ones, with ρ(T)\rho(\mathsf{T}) orders NN up to 60 — which is the representation inventory any complete cusp classification must draw on. Chen, Li, Liu and Ratz (2025) completed the classification of leading mass-matrix patterns and hierarchies at i\mathrm{i} and ω\omega for matter fields assigned through 2⊕1\mathbf{2}\oplus\mathbf{1} (reducible two-plus-one-dimensional) representations, recovering and extending the earlier three-dimensional-irrep results, and stated verbatim: 'We leave the classification of the hierarchies near i∞\mathrm{i}\infty for future work.'

Problem Statement

Complete the near-cusp classification of fermion mass hierarchies in modular flavor models that Chen, Li, Liu and Ratz explicitly deferred, as the cusp analogue of their completed classification at τ=i\tau=\mathrm{i} and τ=ω\tau=\omega. Within their framework — supersymmetric modular flavor models based on finite-image representations of SL(2,Z)\mathrm{SL}(2,\mathbb{Z}) (levels NN equal to the order of ρ(T)\rho(\mathsf{T}), spanning the twelve one-dimensional and fifty-four two-dimensional irreducible representations with finite image, together with reducible assignments such as 2⊕1\mathbf{2}\oplus\mathbf{1}; for 2⊕1\mathbf{2}\oplus\mathbf{1} assignments the T\mathsf{T}-generator data (r1,r2,p)(r_1,r_2,p) comprise 12×54=64812\times 54=648 possibilities), determinant weights kdet⁡M=∑i(kφiC+kφi)<12k_{\det M}=\sum_i (k_{\varphi^{\mathcal{C}}_i}+k_{\varphi_i})<12 (the regime in which the valence formula forces det⁡M\det M to vanish only at the critical points and the cusp bound εNkdet⁡M/12\varepsilon^{N k_{\det M}/12} carries an integer exponent; hierarchies from zeros away from the critical points would require kdet⁡M≥36k_{\det M}\geq 36, dismissed in the source as unusual in bottom-up models and appearing to be prohibited in top-down constructions), and the minimal Kähler potential — determine, for the cusp τ0=i∞\tau_0=\mathrm{i}\infty with expansion parameter ε=e2πiτ/N\varepsilon=\mathrm{e}^{2\pi\mathrm{i}\tau/N}:

  1. For every admissible assignment of the two matter multiplets entering a 3×33\times 3 fermion mass matrix to weighted representations, the leading near-cusp pattern of the mass matrix: the power of ε\varepsilon suppressing each entry, as fixed by the residual ZNT\mathbb{Z}_{N}^{\mathsf{T}} charges (the eigenvalues of the weighted representation Ω(T)\Omega(\mathsf{T})), together with the structure of the leading coefficients.
  2. The resulting mass-eigenvalue hierarchies εd1:εd2:εd3\varepsilon^{d_1}:\varepsilon^{d_2}:\varepsilon^{d_3} (including degenerate and accidentally enhanced cases), organized by the level NN (the order of ρ(T)\rho(\mathsf{T})), so that one can read off which hierarchies are available at each level and which finite modular groups (e.g. perfect groups without nontrivial one-dimensional representations) are excluded.
  3. The consistency of the classification with the model-independent determinant constraint: the mass product m1m2m3m_1 m_2 m_3 must vanish at the cusp to order fixed by the one-dimensional VVMF classification of the determinant, namely at most εNkdet⁡M/12\varepsilon^{N k_{\det M}/12}, with integer exponent.

Here an assignment of the two matter multiplets is admissible if det⁡M\det M is a nontrivial one-dimensional VVMF (so det⁡M≢0\det M\not\equiv 0); this excludes perfect finite modular groups without nontrivial one-dimensional representations, e.g. Γ5≅A5\Gamma_5\cong A_5. Following the source, the fully split 1⊕1⊕1\mathbf{1}\oplus\mathbf{1}\oplus\mathbf{1} scenario is largely unconstrained and lies outside the classification's scope, and likewise the S2\mathsf{S}^{2}-induced blocks of texture zeros arise in addition to the mass-matrix patterns and are not part of the requested ε\varepsilon-power pattern and hierarchy classification.

The common objective of these three deliverables is a single complete, model-independent map of which fermion mass hierarchies are achievable near τ=i∞\tau=\mathrm{i}\infty — the only critical point that can naturally accommodate the large quark-sector hierarchies — playing the same role for the cusp that the source paper's Appendix B plays for i\mathrm{i} and ω\omega.

References

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