In the minimal seesaw model (two heavy right-handed Majorana neutrinos) with trimaximal lepton mixing, a single CP-violating phase controls both the low-energy Dirac phase and the sign of the lepton asymmetry produced in N1N_1-decay leptogenesis, so the observed sign of the cosmological baryon asymmetry fixes the sign of δCP\delta_{\rm CP}. This sign mapping was established only under simplifying assumptions: a hierarchical spectrum M1≪M2M_1\ll M_2, an unflavored (flavor-summed) Boltzmann treatment with an approximate efficiency factor, and neglected renormalization-group running, while the textures with vanishing N1N_1 CP asymmetry (TM1_1 with inverted ordering, and TM2_2 in both orderings) were excluded from leptogenesis altogether. The open problem is to consolidate this prediction: determine whether the sign correlations survive a complete flavor-dependent leptogenesis computation, whether the excluded textures can generate the observed baryon asymmetry through alternative mechanisms, and what ultraviolet structure produces the required right-handed mass ratio.

Background

The universe contains about one baryon (proton or neutron) per billion photons, quantified by the baryon asymmetry ηB≃6×10−10\eta_B\simeq 6\times 10^{-10}. In the type-I seesaw mechanism — the standard explanation of why neutrinos are massive — heavy right-handed Majorana neutrinos NiN_i are added to the Standard Model, and the decays of the lightest one, N1N_1, can generate a lepton asymmetry through CP-violating decay rates; sphaleron processes — hot electroweak transitions that violate baryon and lepton number — then partially convert it into the baryon asymmetry. This scenario is called leptogenesis. Its CP asymmetry ϵN1\epsilon_{N_1} is fixed by the same Yukawa matrix that determines neutrino masses and mixings, creating a potential link between the CP-violating phase δCP\delta_{\rm CP} measured in neutrino oscillations and the sign of ηB\eta_B.

The minimal seesaw keeps only two right-handed neutrinos, N1N_1 and N2N_2, with a real diagonal mass matrix MR=M0 diag(p−1,1)M_R = M_0\,\mathrm{diag}(p^{-1},1), where p=M2/M1p=M_2/M_1. Because one light neutrino is then massless, the model is highly predictive. Trimaximal mixing is a constrained mixing pattern in which the first (TM1_1) or second (TM2_2) column of the lepton mixing matrix equals the corresponding column of the tri-bimaximal form, a widely studied flavor-symmetry benchmark. Shimizu, Takagi and Tanimoto (arXiv:1711.03863) studied the minimal seesaw with TM1_1 or TM2_2 mixing in the basis where both the charged-lepton and MRM_R matrices are diagonal. The resulting 3×23\times 2 Dirac neutrino mass matrix contains only one physical CP phase ϕB\phi_B, so the sign of δCP\delta_{\rm CP} and the sign of the baryon asymmetry are locked together. For TM1_1 with normal ordering, imposing one zero in the first column of the Dirac matrix (whose entries ee and ff are taken real and bb and cc complex) gives the source's three textures — case I (b+c=0b+c=0), case II (c=0c=0), and case III (b=0b=0) — each parameterized by the texture parameter k=e/fk=e/f together with BB and ϕB\phi_B defined by b/f=BeiϕBb/f=Be^{i\phi_B} (respectively c/f=BeiϕBc/f=Be^{i\phi_B} in case III). For TM1_1 with normal mass ordering, the observed positive ηB\eta_B selects regions of the texture parameter kk (their case I requires k<−1k<-1 for M1≪M2M_1\ll M_2, predicting δCP<0\delta_{\rm CP}<0, consistent with the T2K and NOν\nuA hints; the case k=−3k=-3 reproduces the well-known two-neutrino 'Littlest seesaw' construction), while swapping to M1≫M2M_1\gg M_2 flips the preferred sign and region.

That analysis was deliberately approximate. The leptogenesis computation used the flavor-summed (unflavored) CP asymmetry and an analytic efficiency factor κ(m~1)\kappa(\tilde m_1) — where m~1\tilde m_1 is the washout parameter measuring how strongly N1N_1-mediated interactions erase the asymmetry — valid only for M1≪1014 M_1\ll 10^{14}\,GeV and m~1≥10−2 \tilde m_1\ge 10^{-2}\,eV. In reality, leptogenesis above ∼ ⁣1012 \sim\!10^{12}\,GeV is flavor-blind, but below that scale the charged-lepton Yukawa interactions distinguish the τ\tau, μ\mu and ee flavors, changing the CP asymmetries and the washout; a quantitative treatment also requires spectator processes (interactions that redistribute the asymmetry among particle species without destroying it), thermal corrections, and coupled Boltzmann equations. In addition, the textures are defined at the high scale where the flavor symmetry acts, so renormalization-group (RG) running down to low energies, with threshold effects at M1M_1 and M2M_2, must be tracked (the source paper argues this is small because the lightest neutrino mass vanishes, but presents no complete analysis). Finally, for TM1_1 with inverted ordering and for TM2_2 with either ordering, the CP asymmetry ϵN1\epsilon_{N_1} vanishes identically, so N1N_1-dominated thermal leptogenesis fails in those textures, and the observed baryon asymmetry would have to come from another mechanism such as resonant leptogenesis (quasi-degenerate right-handed neutrinos whose self-energy enhancement rescues the asymmetry), N2N_2-dominated flavored leptogenesis, or nonthermal production.

Later literature has advanced pieces of this program without closing it. Moffat, Pascoli, Petcov and Turner (arXiv:1809.08251) developed the modern numerical machinery for flavored leptogenesis driven purely by low-energy phases in minimal-seesaw-type setups, showing viability over 106 ⁣∼ ⁣1013 10^{6}\!\sim\!10^{13}\,GeV, but not for the trimaximal textures. Zhao (arXiv:2003.00654) studied form-dominance realizations (a flavor-symmetry construction in which the Dirac mass matrix is fixed directly by the vacuum alignment of flavon fields, the scalars whose expectation values break the flavor symmetry) of TM1_1/TM2_2 in the minimal seesaw with TeV-scale quasi-degenerate right-handed neutrinos: at the exact texture leptogenesis is indeed impossible, while RG-induced breaking of the texture makes leptogenesis viable for normal ordering with TM2_2 mixing, though for inverted ordering the generated asymmetry reaches only about 54 ⁣− ⁣58%54\!-\!58\% of the observed value in the Standard Model (a supersymmetric realization with large tan⁡β\tan\beta was suggested but not worked out). Zhao, Shi and Shao (arXiv:2402.14441) found viable resonant leptogenesis for TM1_1 combined with μ\mu-τ\tau reflection symmetry in a 'most minimal seesaw' with a pseudo-Dirac pair of right-handed neutrinos. The review by Xing and Zhao (arXiv:2008.12090) surveys minimal-seesaw leptogenesis — flavored, N2N_2-dominated, resonant, and RG effects — and the S4S_4-based ultraviolet embeddings of trimaximal textures, but the specific questions raised by the source analysis remain untreated there. What is still missing is precisely a consolidated, quantitatively controlled statement of the δCP\delta_{\rm CP}–ηB\eta_B sign correlation for the original hierarchical trimaximal minimal-seesaw textures.

Problem Statement

Consider the minimal seesaw model with two right-handed Majorana neutrinos, real diagonal MR=M0 diag(p−1,1)M_R=M_0\,\mathrm{diag}(p^{-1},1) with p=M2/M1p=M_2/M_1, and trimaximal mixing (TM1_1 or TM2_2) in the basis of diagonal charged-lepton and right-handed mass matrices, so that the 3×23\times 2 Dirac neutrino mass matrix contains a single CP-violating phase. The open problem, as left by the source analysis and not closed by subsequent literature, is to consolidate the prediction linking the sign of the low-energy Dirac phase δCP\delta_{\rm CP} to the sign of the cosmological baryon asymmetry ηB\eta_B, by answering the following linked questions for these textures: (1) Robustness — does the sign mapping established for TM1_1 with normal ordering under the hierarchical assumption M1≪M2M_1\ll M_2 (e.g. k<−1k<-1 giving δCP<0\delta_{\rm CP}<0 with ηB>0\eta_B>0 in case I, and its counterparts in cases II and III and in the M1≫M2M_1\gg M_2 variant) survive a complete flavor-dependent leptogenesis computation, i.e. coupled Boltzmann equations with flavored CP asymmetries, spectator processes, thermal corrections, and the correct flavor regimes, over a range of M1M_1 wider than the approximate efficiency formula κ(m~1)\kappa(\tilde m_1) permits (M1≪1014 M_1\ll 10^{14}\,GeV, m~1≥10−2 \tilde m_1\ge 10^{-2}\,eV), and how do the allowed parameter regions shift? (2) Scope — for the textures in which ϵN1\epsilon_{N_1} vanishes identically (TM1_1 with inverted ordering, TM2_2 with either ordering), can the observed baryon asymmetry be generated at all, e.g. via resonant leptogenesis, flavored N2N_2-dominated leptogenesis, or nonthermal production, noting that existing partial results hold only in modified settings (TeV-scale quasi-degenerate or pseudo-Dirac right-handed neutrinos with RG-broken textures) and leave the inverted-ordering cases below the observed value in the Standard Model? (3) Renormalization-group completion — what is the complete RG running of the one-phase Dirac textures from the flavor scale to low energies, including threshold effects at M1M_1 and M2M_2 and the dependence on M0M_0, quantifying the claimed smallness for vanishing lightest neutrino mass? (4) Ultraviolet origin — is there a concrete completion (for example a Froggatt–Nielsen mechanism, in which the sizes of Yukawa-coupling entries are controlled by charges under a symmetry broken by flavon fields, or a discrete flavor symmetry) that reproduces the trimaximal Dirac textures while fixing the required large right-handed mass ratio p∼103 ⁣− ⁣104p\sim 10^{3}\!-\!10^{4}? An answer should state which parts of the original sign-correlation prediction are confirmed, revised, or excluded once these effects are included.

Of the four linked questions, part (1) is the mandatory core, which a complete solution must resolve; parts (2)-(4) each carry independent standing and may be answered to the extent a controlled analysis and the existing literature permit, so a submission that decisively settles part (1) resolves the task even if it treats parts (2)-(4) only partially.

References

  1. Yusuke Shimizu, Kenta Takagi, Morimitsu Tanimoto, 'Neutrino CP violation and sign of baryon asymmetry in the minimal seesaw model', Phys. Lett. B 778, 6 (2018), arXiv:1711.03863, doi:10.1016/j.physletb.2017.12.065, https://arxiv.org/abs/1711.03863
  2. Yusuke Shimizu, Kenta Takagi, Morimitsu Tanimoto, 'Towards the minimal seesaw model via CP violation of neutrinos', JHEP 11, 201 (2017), arXiv:1709.02136, doi:10.1007/JHEP11(2017)201, https://arxiv.org/abs/1709.02136
  3. Yusuke Shimizu, Kenta Takagi, Morimitsu Tanimoto, 'Towards the minimal seesaw model for the prediction of neutrino CP violation' (2018), arXiv:1805.02925, https://arxiv.org/abs/1805.02925
  4. Zhen-hua Zhao, 'Renormalization group evolution induced leptogenesis in the minimal seesaw model with the trimaximal mixing and mu-tau reflection symmetry', JHEP 11, 170 (2021), arXiv:2003.00654, doi:10.1007/JHEP11(2021)170, https://arxiv.org/abs/2003.00654
  5. Zhi-zhong Xing, Zhen-hua Zhao, 'The minimal seesaw and leptogenesis models', Rep. Prog. Phys. 84, 066201 (2021), arXiv:2008.12090, doi:10.1088/1361-6633/abf086, https://arxiv.org/abs/2008.12090
  6. K. Moffat, S. Pascoli, S.T. Petcov, J. Turner, 'Leptogenesis from Low Energy CP Violation', JHEP 03, 034 (2019), arXiv:1809.08251, doi:10.1007/JHEP03(2019)034, https://arxiv.org/abs/1809.08251
  7. Zhen-hua Zhao, Xin-Yu Zhao, Hui-Chao Bao, 'Simplified textures of the seesaw model for trimaximal neutrino mixing', Phys. Rev. D 105, 035011 (2022), arXiv:2104.05394, doi:10.1103/PhysRevD.105.035011, https://arxiv.org/abs/2104.05394
  8. Zhen-hua Zhao, Hong-Yu Shi, Yan Shao, 'Leptogenesis consequences of trimaximal mixing and μ-τ reflection symmetry in the most minimal seesaw model', Phys. Rev. D 109, 115001 (2024), arXiv:2402.14441, doi:10.1103/PhysRevD.109.115001, https://arxiv.org/abs/2402.14441
  9. D.M. Barreiros, H.B. Camara, R.G. Felipe, F.R. Joaquim, 'Scalar-singlet assisted leptogenesis with CP violation from the vacuum', JHEP 01, 010 (2023), arXiv:2211.00042, doi:10.1007/JHEP01(2023)010, https://arxiv.org/abs/2211.00042