The Tanimoto–Yanagida extension of the axion-less solution to the strong CP problem predicts the leptonic Dirac CP phase δCP≃192∘\delta_{CP}\simeq 192^\circ–195∘195^\circ from the single CP-violating phase ϕ\phi shared with the CKM quark-mixing matrix, but its cosmological prediction is incomplete: the heavy right-handed-neutrino masses M1,M2,M3M_1,M_2,M_3 are absorbed into rescaled Dirac-matrix entries and never determined, the sign of the baryon asymmetry is only a conditional statement (correct if N1N_1 is the lightest), and the magnitude of the asymmetry has never been computed. Variant constructions in the literature through 2026 (two-zeros-texture and non-invertible-symmetry realizations) do not fix the heavy spectrum or compute YBY_B, and the surviving open problem is: determine the heavy right-handed-neutrino spectrum within this framework or a consistent variant realization, and compute the baryon asymmetry — sign and magnitude — from the flavor-structured CP asymmetries and the full Boltzmann evolution, confronting the observed YB≃8.7×10−11Y_B\simeq 8.7\times 10^{-11}, or demonstrate that no viable parameter region exists.

Background

The strong CP problem is the unexplained smallness of the physical QCD vacuum angle θˉ=θ+arg⁡det⁡(MuMd)\bar\theta=\theta+\arg\det(M_u M_d), which neutron electric-dipole-moment searches constrain to ∣θˉ∣<10−10|\bar\theta|<10^{-10} even though the CP-violating phase of the Cabibbo–Kobayashi–Maskawa (CKM) quark-mixing matrix would naturally feed arg⁡det⁡(MuMd)\arg\det(M_u M_d) at loop level. The Peccei–Quinn axion is the classic solution; an alternative "axion-less" route keeps CP exact at a fundamental scale and imposes a texture — a pattern of vanishing entries — on the quark mass matrices so that their determinants are real, with CP broken only spontaneously, making θˉ=0\bar\theta=0 technically natural. Liang, Okabe and Yanagida (LOY) realized such a three-zero texture in a six-dimensional spacetime compactified on a T2/Z3T^2/\mathbb{Z}_3 orbifold (a two-torus divided by a 120∘120^\circ rotation, whose three inequivalent fixed points host the three quark families), with an extra Z2\mathbb{Z}_2 symmetry and a complex scalar whose vacuum expectation value induces the single CP-violating phase ϕ\phi.

Tanimoto and Yanagida extended this model to the lepton sector by adding three heavy right-handed neutrinos (RHNs) NiN_i, which generate the light neutrino masses through the seesaw mechanism, Mν≃MDMN−1MDTM_\nu\simeq M_D M_N^{-1} M_D^T, where Majorana mass suppression by the heavy scale gives the observed sub-eV neutrinos. The geometric construction gives the Dirac neutrino matrix MDM_D the transposed texture of the down-quark matrix, with the phase ϕ\phi in the single (2,2)(2,2) entry, so the whole model — quark and lepton sectors, low energy and cosmology — contains exactly one CP-violating phase, fixed by the CKM fit to ϕ=37∘\phi=37^\circ–48∘48^\circ. Fitting the rescaled entries A=aν/M1A=a_\nu/\sqrt{M_1}, A′=aν′/M2A'=a'_\nu/\sqrt{M_2}, and so on, to the measured three mixing angles of the PMNS (Pontecorvo–Maki–Nakagawa–Sakata) lepton-mixing matrix and the neutrino mass-squared splittings (normal ordering), the model predicts the leptonic Dirac CP phase δCP≃192∘\delta_{CP}\simeq 192^\circ–195∘195^\circ, the neutrinoless-double-beta-decay effective mass mββ≃8m_{\beta\beta}\simeq 8–1111 meV, and the mass sum ∑mi≃67\sum m_i\simeq 67–7373 meV — ranges within reach of Hyper-Kamiokande/DUNE oscillation measurements and next-generation neutrinoless-double-beta-decay experiments such as LEGEND-1000 and nEXO.

For cosmology, the decays of the lightest RHN can implement leptogenesis in the sense of Fukugita and Yanagida: the CP asymmetry ϵi\epsilon_i in the decays Ni→ℓHN_i\to \ell H produces a lepton asymmetry that electroweak sphalerons (thermal transitions violating B+LB+L) partially convert into baryon number, YB≃−(28/79) κ ϵi/g∗Y_B\simeq -(28/79)\,\kappa\,\epsilon_i/g^*, where evaluating the dilution factor κ\kappa requires solving the coupled Boltzmann (rate) equations including washout — the inverse decays and scatterings that erase the produced asymmetry. Because only the (2,2)(2,2) element of MDM_D carries ϕ\phi, one finds Im[(YD†YD)122]=(1/v4)aν2bν2sin⁡2ϕ>0\mathrm{Im}[(Y_D^\dagger Y_D)^2_{12}]=(1/v^4)a_\nu^2 b_\nu^2\sin 2\phi>0, so in the published version of the model ϵ1<0\epsilon_1<0 and the baryon asymmetry has the observed positive sign provided N1N_1 is the lightest RHN. This is where the model stops: the RHN masses M1,M2,M3M_1,M_2,M_3 were absorbed into the rescaled Dirac entries and are not determined by any part of the construction, so which NiN_i is lightest is unknown, and the magnitude of YBY_B — which depends on the lightest RHN mass, the reheating temperature, and the washout dynamics — is explicitly left uncomputed. The authors also note that variant realizations based on other viable textures have not been constructed, and that the δCP\delta_{CP} and mββm_{\beta\beta} predictions remain to be tested experimentally.

Problem Statement

In the single-CP-phase axion-less solution of the strong CP problem (Tanimoto–Yanagida, PTEP 2025 033B01, extending the Liang–Okabe–Yanagida T2/Z3T^2/\mathbb{Z}_3 orbifold model), the lepton sector contains three heavy right-handed neutrinos NiN_i whose Majorana masses M1,M2,M3M_1,M_2,M_3 are free parameters: the fit to oscillation data determines only the rescaled Dirac entries A=aν/M1A=a_\nu/\sqrt{M_1}, A′=aν′/M2A'=a'_\nu/\sqrt{M_2}, and so on, together with the single CP phase ϕ=37∘\phi=37^\circ–48∘48^\circ fixed by the CKM matrix. The model's leptogenesis prediction is therefore incomplete: the sign of the baryon asymmetry is only a conditional statement (correct if N1N_1 is the lightest right-handed neutrino), and the magnitude of YBY_B has not been computed.

The problem is to complete this prediction. Determine, within this framework or a consistent variant realization of it, the heavy right-handed-neutrino spectrum — the masses and their ordering — or its viable region in the (Mi,TR)(M_i,T_R) parameter space, and compute the resulting baryon-number asymmetry through standard leptogenesis: the CP asymmetries ϵi\epsilon_i following from the model's Dirac texture (only the (2,2)(2,2) element of MDM_D carries the phase ϕ\phi), the full Boltzmann-equation evolution including washout and the dilution factor for a specified reheating history, and the final YBY_B with its uncertainties. The answer must reproduce the observed baryon asymmetry YB≃8.7×10−11Y_B\simeq 8.7\times 10^{-11} in sign and magnitude within a stated region of the (Mi,TR)(M_i,T_R) parameter space, or demonstrate that no such region exists in the framework. A variant realization offered as the answer must in addition preserve the axion-less strong-CP solution (real quark-mass-matrix determinants with θˉ\bar\theta within observational bounds against loop corrections) and remain consistent with current neutrino oscillation, neutrinoless-double-beta-decay, and cosmological mass-sum constraints.

References

  1. Morimitsu Tanimoto and Tsutomu T. Yanagida, Prediction of the CP Phase δCP\delta_{CP} in the Neutrino Oscillation and an Axion-less Solution to the Strong CP Problem, Prog. Theor. Exp. Phys. 2025 (2025) 033B01, DOI 10.1093/ptep/ptae177, arXiv:2410.01224, https://arxiv.org/abs/2410.01224
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