Leptogenesis from low-energy $CP$ violation alone: naturalness, $R$-matrix classification, and predictive $\eta_B$–observable relations

The observed baryon asymmetry of the Universe, $\eta_B\approx 6.1\times 10^{-10}$, may originate in thermal leptogenesis in which all $CP$ violation is supplied by the low-energy phases of the neutrino mixing (PMNS) matrix. Moffat, Pascoli, Petcov and Turner showed this to be viable over seven orders of magnitude in the heavy-neutrino mass scale ($10^6$–$10^{13}$ GeV), but with caveats that define the open problem: at high scales ($M_1\gg 10^{12}$ GeV) leptogenesis driven purely by the Dirac phase $\delta$ requires a fine-tuned cancellation ($F\sim 100$) between the tree-level and one-loop contributions to the light-neutrino masses; the minimal two-heavy-neutrino inverted-ordering case was explored only with a real $R$-matrix, with no viable point found at the scales explored; the analysis assumed hierarchical heavy neutrinos, leaving resonant or partially degenerate spectra unexplored; and the connection between $\eta_B$ and low-energy observables ($J_{CP}$, $\langle m_\nu\rangle$) exists only at the level of parameter scans. The problem is to make low-energy-$CP$-violation-only leptogenesis natural and predictive: identify a symmetry or flavour mechanism enforcing the required cancellations without spoiling low-energy predictions, complete the classification of viable $CP$-conserving $R$-matrices, quantify quasi-degenerate heavy-neutrino spectra, and derive systematic quantitative relations between $\eta_B$ and measurable quantities that DUNE, T2HK and neutrinoless double-beta decay searches can confirm or exclude.

Background

The Universe contains far more matter than antimatter: the baryon asymmetry, conventionally quoted as $\eta_B\approx 6.1\times 10^{-10}$, is measured from the cosmic microwave background and big-bang nucleosynthesis. In thermal leptogenesis — the leading mechanism explaining it — heavy Majorana neutrinos $N_i$ (Standard-Model singlets with masses $M_i$ far above the electroweak scale) are produced thermally in the early Universe, and their $CP$-violating out-of-equilibrium decays create a lepton asymmetry that electroweak sphaleron processes (nonperturbative transitions violating $B+L$) reprocess into the baryon asymmetry. In the type-I seesaw mechanism, which generates the light-neutrino masses via $m_\nu=-m_D M^{-1} m_D^T$ ($m_D$ the Dirac neutrino mass matrix, $M$ the heavy-neutrino mass matrix), lepton mixing is described by the PMNS matrix $U$: three mixing angles, one Dirac phase $\delta$ — the only $CP$-violating phase accessible to oscillation experiments — and two Majorana phases $\alpha_{21},\alpha_{31}$, which enter neutrinoless double-beta decay ($0\nu\beta\beta$) through the effective Majorana mass $\langle m_\nu\rangle=|\sum_i U_{ei}^2 m_i|$. The decay $CP$ asymmetries depend, through the Casas–Ibarra parametrization $m_D=U\,\mathrm{diag}(\sqrt{m_i})\,R\,\mathrm{diag}(\sqrt{M_i})$, on a complex orthogonal matrix $R$ that is invisible at low energy; $CP$ violation can thus reside in the low-energy PMNS phases, in the high-energy matrix $R$, or in both. Long-standing lore held that above $M_1\sim 10^{12}$ GeV — where charged-lepton Yukawa interactions are too slow to decohere lepton flavour states (the one-flavour regime) — leptogenesis is insensitive to the PMNS phases, while Anisimov, Blanchet and Di Bari had shown that Dirac-phase leptogenesis ($\delta$ as the only $CP$-violating source) is viable in the two-flavour window $10^9\lesssim M_1/\mathrm{GeV}\lesssim 10^{12}$. Moffat, Pascoli, Petcov and Turner revisited the question with density-matrix equations, which track the gradual decoherence of lepton flavour states in the plasma. They found that leptogenesis with all $CP$ violation supplied by the PMNS phases is viable over $10^6<M_1/\mathrm{GeV}<10^{13}$; that decoherence cannot be neglected even above $10^{12}$ GeV, contradicting the lore; that Majorana-phase leptogenesis works with essentially no fine-tuning; but that for $M_1\gg 10^{12}$ GeV Dirac-phase leptogenesis requires a fine-tuned cancellation between the tree-level and one-loop contributions to the light-neutrino mass matrix, quantified by $F\equiv m_\nu^{\mathrm{1-loop}}/(m_\nu^{\mathrm{tree}}+m_\nu^{\mathrm{1-loop}})$ with $F\sim 100$ (reaching $F\approx 750$ at intermediate scales $M_1\sim 10^6$–$10^8$ GeV, the scans being capped at $F<1000$), since $F\gg 1$ means the loop correction far exceeds the physical mass. In the minimal scenario with only two heavy neutrinos and an inverted light-neutrino mass ordering ($m_3<m_1<m_2$), no viable point was found for a real $R$-matrix at the scales explored, while purely imaginary $R$-matrix elements open viable regions that were not classified. The analysis assumed hierarchical heavy neutrinos ($M_1\ll M_2$), and $\eta_B$ was connected to low-energy observables — the Jarlskog invariant $J_{CP}$ (the reparametrization-invariant combination of angles and $\sin\delta$ controlling $CP$ violation in oscillations) and $\langle m_\nu\rangle$ — only through numerical scans. What remains unresolved is whether this scenario can be made natural and predictive before the relevant experiments — DUNE and T2HK (long-baseline experiments that will measure $\delta$) and next-generation $0\nu\beta\beta$ searches — weigh in.

Problem Statement

Establish whether decay-based thermal leptogenesis with $CP$ violation supplied exclusively by the low-energy PMNS phases — in particular by the Dirac phase $\delta$ — can explain the observed baryon asymmetry $\eta_B$ in a natural and predictive way. This single objective comprises four linked determinations: (1) Naturalness: determine whether a symmetry or flavour model can naturally enforce the near-cancellation between the tree-level and one-loop contributions to the light-neutrino mass matrix required for high-scale ($M_1\gtrsim 10^{12}$ GeV) Dirac-phase leptogenesis (fine-tuning $F\sim 100$; reaching $F\approx 750$ at intermediate scales $M_1\sim 10^6$–$10^8$ GeV, with the scans capped at $F<1000$) without spoiling the low-energy predictions, or whether the required fine-tuning is irreducible. (2) $R$-matrix structure: complete the classification of viable $CP$-conserving $R$-matrices, including those with purely imaginary elements, for the minimal two-heavy-neutrino inverted-ordering scenario, and confront each class with current neutrino oscillation data (the real-$R$ case was found to admit no viable point at the scales explored). (3) Quasi-degenerate spectra: determine how resonant enhancement and partially degenerate heavy-neutrino mass patterns modify the viable scale range (the high-scale plateau) and the fine-tuning requirements. (4) Predictivity: determine whether systematic, ideally analytic, quantitative relations between $\eta_B$ and measurable low-energy quantities — $\delta$ (through $J_{CP}$), $\langle m_\nu\rangle$, and, where relevant, heavy-neutrino mixing parameters — can be derived beyond parameter scans, thereby specifying what DUNE, T2HK and next-generation $0\nu\beta\beta$ searches must measure to confirm or exclude thermal leptogenesis driven solely by low-energy $CP$ violation. All four determinations are required to resolve the objective: each is independently assessable, and establishing or refuting any one of them leaves the remaining determinations open.

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