Very light right-handed neutrinos as warm dark matter in the minimal left-right symmetric model: joint cosmological and strong-CP viability

Whether the lightest right-handed neutrino can be very light — keV-scale warm dark matter — in the minimal left-right symmetric model (MLRSM) with parity, while the smallness of the QCD CP-violating parameter $\bar\theta$ is simultaneously respected, has never been jointly established. The paper motivating the question (Senjanovic and Tello) asserts that the strong-CP bound is automatically satisfied in this regime but performs no analysis of it, deferring the dark-matter phenomenology to earlier work. Later literature has mapped parts of the cosmological side (entropy-dilution relic density, large-scale-structure limits — galaxy clustering plus the Lyman-α forest, i.e. absorption by intergalactic neutral hydrogen along quasar sightlines, which resolves the smallest scales of the matter power spectrum and hence the 'warmness' of dark matter — and X-ray limits, a PeV-scale floor for the right-handed gauge boson) and has contested the strong-CP connection itself: the one-loop leptonic contribution to $\bar\theta$ vanishes for representative flavour textures, in conflict with the source's reasoning, and the general case is explicitly unsolved. The joint cosmology-plus-strong-CP viability map of the very-light right-handed-neutrino warm-dark-matter regime is genuinely open: no existing work combines the two constraint sets in the parity MLRSM, and the status of the disputed one-loop contribution for general right-handed mixing remains unresolved.

Background

The Standard Model's weak interaction acts only on left-handed fermions, and the observed smallness of neutrino masses is commonly explained by adding right-handed (RH) singlet neutrinos $N$ through the seesaw mechanism (their large Majorana masses $M_N$ suppress the light masses to $m_\nu\sim m_D^2/M_N$ — the type-I seesaw; in the type-II variant the light masses instead come from an $SU(2)_L$-triplet Higgs field, and which of the two dominates, the seesaw-dominance regime, fixes the model's flavour structure). The minimal left-right symmetric model (MLRSM) instead promotes this to a gauge principle: the gauge group is $SU(3)_c\times SU(2)_L\times SU(2)_R\times U(1)_{B-L}$, and parity ($P$) interchanges left and right fields, so the RH neutrinos are mandatory partners rather than ad hoc additions, with $M_N$ tied to the scale where $SU(2)_R$ breaks. A separate, long-standing issue is the strong CP problem: the physical QCD CP-violating parameter $\bar\theta=\theta+\arg\det(M_uM_d)$, with $M_u,M_d$ the quark mass matrices, is bounded by the neutron electric dipole moment (EDM) to $|\bar\theta|\lesssim 10^{-10}$, far smaller than natural expectations. Senjanovic and Tello (arXiv:2004.04036) revisited the MLRSM with parity and argued that small $\bar\theta$ is a 'blessing' rather than a problem: parity forces $\bar\theta=0$ before symmetry breaking, and afterwards the spontaneous contribution $\bar\theta\simeq(m_t/2m_b)\tan(2\beta)\sin\alpha$ (Maiezza and Nemevsek; $\tan 2\beta$ and $\sin\alpha$ are spontaneous P- and CP-breaking parameters of the Higgs sector) combines with a one-loop leptonic contribution found by Kuchimanchi, $\bar\theta_{\rm loop}\propto(m_t/m_b)\,{\rm Im}\,{\rm Tr}(Y_R^\dagger Y_R[Y_D,Y_\ell])\ln(M_{Pl}/v_R)$ (with $Y_R$, $Y_D$ and $Y_\ell$ the right-handed-neutrino, Dirac and charged-lepton Yukawa matrices, $M_{Pl}$ the Planck mass and $v_R$ the $SU(2)_R$-breaking scale), to bound the RH-neutrino masses — roughly $M_N\lesssim 5\times10^{-6/5}(M_{W_R}/{\rm GeV})^{4/5}$ GeV, i.e. $M_N\lesssim 500$ GeV for a 10 TeV right-handed gauge boson $W_R$. Small $\bar\theta$ thus predicts unusually light RH neutrinos. In the extreme limit the lightest RH neutrino $N_1$ could be so light — keV-scale — that it is the warm dark matter (WDM): a particle that decouples while relativistic with enough momentum dispersion to mildly suppress structure below dwarf-galaxy scales, for which a keV sterile neutrino is the canonical candidate (Dodelson and Widrow). The source paper notes that in this regime $M_N$ is 'completely determined' (citing Nemevsek, Senjanovic and Zhang, who found the lightest RH neutrino at keV mass to be the only viable dark matter candidate of the low-scale MLRSM, with the relic density requiring late entropy dilution by decays of the heavier RH neutrinos) and asserts that the strong-CP bound is then automatically satisfied because the Dirac Yukawa couplings become tiny — but it performs no analysis of the regime. Two later developments make the joint question live rather than academic. First, de Vries, Li, Ramsey-Musolf and Vasquez re-examined the strong-CP connection with light sterile neutrinos and found that for representative right-handed leptonic mixings the one-loop leptonic contribution to $\bar\theta$ vanishes identically — in explicit conflict with the source's reasoning — leaving the general flavour dependence unsolved. Second, Nemevsek and Zhang re-analysed the WDM cosmology of the MLRSM: large-scale-structure data bound the dilutor-to-dark-matter decay branching ratio, a no-go theorem excludes the dilutor from participating in the type-I seesaw, and the surviving windows require $M_{W_R}$ above roughly a PeV with $N_1$ confined to about 6.5–30 keV — while the strong-CP bound is entirely absent from their constraint set. Related but distinct frameworks (Higgs-parity left-right theories, where parity enforces $\bar\theta=0$ at a high breaking scale) have had their sterile-neutrino dark matter mapped systematically, but that is a different model class from the source's parity MLRSM in which small $\bar\theta$ is traded for light RH neutrinos. The literature therefore contains detailed cosmology-only and strong-CP-only treatments of this regime, and a contested one-loop strong-CP result, but no combined viability analysis.

Problem Statement

In the minimal left-right symmetric model with parity as the left-right symmetry, establish the full range of viable scenarios in which the lightest right-handed neutrino $N_1$ is very light and constitutes the warm dark matter. Concretely: determine the region of parameter space — the right-handed-neutrino mass spectrum (keV-scale $N_1$ together with the heavier neutrino(s) that dilute or otherwise set the relic density), the $W_R$ scale, and the leptonic flavour structure including the seesaw-dominance regime — that is simultaneously consistent with (i) the strong-CP bound $|\bar\theta|\lesssim 10^{-10}$, where $\bar\theta$ comprises the tree-level spontaneous contribution $(m_t/2m_b)\tan(2\beta)\sin\alpha$ and the one-loop leptonic contribution evaluated for the actual spectrum and right-handed mixing matrices, without fine-tuned cancellations; whether that one-loop leptonic contribution from light sterile neutrinos vanishes for general right-handed leptonic mixing $U_R$ — as found for $U_R=U^*_{\rm PMNS}$, the complex conjugate of the measured light-neutrino (PMNS) mixing matrix, and for $U_R=\mathbf{1}$, in conflict with the automatic-satisfaction claim of the source paper — or instead bounds the sterile-neutrino masses, is itself unsettled in the literature; and (ii) the cosmological constraints — the observed relic density through relativistic freeze-out plus entropy dilution, the large-scale-structure/Lyman-α warmness limits, monochromatic X-ray decay limits, and big-bang nucleosynthesis. The answer should characterize the correlated low-energy and astrophysical predictions (leptonic electric dipole moments, neutrinoless double beta decay rates, X-ray line fluxes, suppressed small-scale matter power) over the viable region, and state which regimes are excluded and why. Required for a complete answer: the viable region of parameter space, the status of the one-loop leptonic contribution to $\bar\theta$ for general right-handed mixing $U_R$, and which regimes are excluded and why. The correlated low-energy and astrophysical predictions are requested over the viable region but are not, on their own, required to resolve the task.

References

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