A bottom-up analysis of quark flavor (Kawamura, PTEP 2019, 073B03) argues from Dirac naturalness that the flavor-symmetric part of the quark Yukawa coupling matrices should be dominated by a rank-one, democratic-type symmetric matrix, with the observed mass hierarchy and CKM mixing arising mainly from non-canonical fermion kinetic terms. The paper leaves open how to give this structure an ultraviolet origin: no known mechanism dynamically generates the dominant democratic matrix without unnatural fine-tuning (the explicit flavon toy model of the same author fails perturbatively), the structure appears to clash with SO(10)/E6 grand unification where the right-handed up- and down-type quarks share a multiplet and the same kinetic coefficients, and the renormalization-group matching of weak-scale Yukawa parameters to the high scale has not been performed.

Background

In the Standard Model (SM), quark masses and mixing come from the Yukawa coupling matrices YuY_u and YdY_d: after electroweak symmetry breaking, the physical masses are the singular values of these matrices, and the mismatch between the unitary rotations that diagonalize them gives the Cabibbo–Kobayashi–Maskawa (CKM) quark mixing matrix. The observed hierarchies — from the top Yukawa coupling of order 1 down to the up-quark coupling of order 10−510^{-5}, together with the small off-diagonal CKM elements — are unexplained. If a theory beyond the SM (BSM) possesses an approximate flavor symmetry rotating or permuting the three fermion families, then both the Yukawa matrices and the fermion kinetic coefficients (the matrices k(x)k^{(x)}, x=q,u,dx=q,u,d, multiplying the kinetic terms; they are called non-canonical when not proportional to the identity) decompose into flavor-symmetric and flavor-breaking parts. Only after diagonalizing and canonically normalizing the kinetic terms does one obtain the physical Yukawa matrices, so a hierarchy can be generated partly or wholly by the kinetic terms rather than by small Yukawa couplings. Imposing Dirac's naturalness — a small physical quantity should not arise from cancellations among independent parameters of order one — Kawamura (PTEP 2019, 073B03, the source work) argued that any flavor-symmetric part of the Yukawa matrix must be realizable as a rank-one matrix, among which the democratic matrix (all entries equal, invariant under any permutation of the three families; the idea of flavor democracy goes back to Harari, Haut and Weyers, 1978) occupies a special position, and that the observed quark hierarchy and CKM mixing can be understood if a democratic-type symmetric part dominates while the hierarchy emerges from non-canonical kinetic terms, as illustrated in the same author's toy model (PTEP 2019, 043B05). This analysis is bottom-up: it constrains the effective Yukawa sector at the BSM scale MBSMM_{BSM} but does not explain the origin of the dominant democratic matrix. Three obstructions are left open. First, dynamical generation: the natural candidate — spontaneous breaking of the flavor symmetry by vacuum expectation values of flavons (hypothetical scalar fields charged under the flavor group, whose condensation generates the effective Yukawa textures) — realizes the needed structure in the toy model only at the price of an unnatural fine-tuning among perturbative parameters, suggesting that non-perturbative effects or additional model structure are required. Second, grand unification: in unified gauge groups such as SO(10)SO(10) or E6E_6 (grand unified theories that embed all SM fermions of one family into a single multiplet), the right-handed up- and down-type quarks (uR′u'_R and dR′d'_R in the rotated basis used by the source) sit in a common multiplet, transform as the same representation of the flavor group, and share the same kinetic coefficient matrix SS; a dominant flavor-symmetric Yukawa part with a common coefficient of order one then seems incompatible with the very different up- and down-type hierarchies unless extra matter fields or interactions are added. Third, the weak-scale Yukawa values used in the bottom-up estimates still need quantitative renormalization-group (RG) running to match their counterparts at MBSMM_{BSM}, which the source does not provide.

Problem Statement

As left open by the source work (Kawamura, PTEP 2019, 073B03), whose conclusions call for exploring the origin of the democratic-type matrix and for combining the bottom-up approach with top-down model building, the problem is to convert the bottom-up finding — a dominant rank-one, democratic-type flavor-symmetric Yukawa part whose observed hierarchy and CKM mixing emerge from non-canonical fermion kinetic terms — into a concrete and viable ultraviolet completion. The open items left by the source are: (1) construct a realistic theory beyond the Standard Model in which the dominant democratic-type symmetric Yukawa matrix is dynamically generated, without unnatural fine-tuning among independently generated parameters of order one, unlike the toy model of PTEP 2019, 043B05; (2) embed this structure in, or reconcile it with, grand unification based on SO(10)SO(10) or E6E_6 — where uR′u'_R and dR′d'_R belong to a common multiplet and share the same kinetic coefficient matrix SS, so that a common order-one Yukawa coefficient must be made consistent with the very different up- and down-type quark hierarchies; and (3) supply the quantitative renormalization-group matching of the weak-scale quark Yukawa matrices and CKM parameters to the high scale MBSMM_{BSM} used by the construction, which the source notes as needed but does not perform, with explicit realizations of the required non-canonical kinetic terms that do not themselves require fine-tuning. These requested results jointly serve one objective — establishing the dominant flavor-symmetric Yukawa mechanism as the origin of the quark mass hierarchy and flavor mixing through a viable ultraviolet completion. The supersymmetric extension (patterns of soft breaking terms as compensating flavor information) is recorded in the source's conclusions as a further direction of the same program rather than a separate requirement.

References

  1. Yoshiharu Kawamura, "Chasing after flavor symmetries of quarks from the bottom up", Progress of Theoretical and Experimental Physics 2019 (2019) 073B03. DOI: 10.1093/ptep/ptz069. https://arxiv.org/abs/1904.01158 (source work)
  2. Yoshiharu Kawamura, "Yukawa interactions, flavor symmetry, and non-canonical Kähler potential", Progress of Theoretical and Experimental Physics 2019 (2019) 043B05. DOI: 10.1093/ptep/ptz018. https://arxiv.org/abs/1808.06315
  3. Yoshiharu Kawamura, "A bottom-up approach to fermion mass hierarchy: a case with vector-like fermions", Progress of Theoretical and Experimental Physics 2020 (2020) 043B05. DOI: 10.1093/ptep/ptaa046. https://arxiv.org/abs/1909.03623
  4. Yoshiharu Kawamura, "On a mechanism realizing quark mass hierarchy", Modern Physics Letters A 35 (2020) 2050274. DOI: 10.1142/S0217732320502740. https://arxiv.org/abs/2004.11564
  5. Yoshiharu Kawamura, "Flavor structure from 'canonical' Yukawa interactions and 'emergent' kinetic terms", Progress of Theoretical and Experimental Physics 2023 (2023) 093B02. DOI: 10.1093/ptep/ptad105. https://arxiv.org/abs/2306.01281
  6. Burak Dagli, Saleh Sultansoy, Ismail Toy, "Flavor Democracy Calls for Vector Like Leptons and Quarks", arXiv:2603.05104 (2026). https://arxiv.org/abs/2603.05104
  7. Haim Harari, Hervé Haut, Jacques Weyers, "Quark masses and Cabibbo angles", Physics Letters B 78 (1978) 459. DOI: 10.1016/0370-2693(78)90485-9. https://doi.org/10.1016/0370-2693(78)90485-9