Common Origin of Soft mu-tau and CP Breaking in Neutrino Seesaw and the Origin of Matter

The thing to know first: the clean part of this paper is not the numerical bounds, it is a structural argument. Assume the Dirac Yukawa sector is exactly mu-tau and CP symmetric and confine all breaking to the heavy Majorana mass matrix. Then the ratio of the two mu-tau-antisymmetric entries of the light mass matrix is real and fixed by mD alone, which yields δx ≥ |δa| tan θs (Eq. 4.80) with no knowledge of the soft-breaking parameters. That is derived, not fitted, and it is the piece I would actually use. The second genuinely new piece is the Zs2 hidden symmetry in Sec. 6.3: the 3x3 representation Ts(k), the reading of it as a reflection about a plane, the rank-2 theorem for the general three-neutrino seesaw, and tri-bimaximal mixing falling out at k=2. That is tidy work. They also credit [44] for the earlier observation that θ12 survives mu-tau breaking, which is the right thing to do. Soft spots, in order of weight. The conjecture is stated openly (abstract:s2, Sec. 2.2.1), so I will not call it hidden, but the paper never asks how much hard mu-tau or CP breaking in the dimension-4 Yukawa couplings it can absorb. The charged leptons already break mu-tau; nothing forbids the neutrino Yukawas from doing the same at the few-percent level, and that would put an imaginary part into δBa/δCa and soften Eq. 4.80. A short tolerance estimate would have made the word 'prediction' defensible. The 1° ≲ θ13 ≲ 6° window is softer than the abstract suggests: it comes from scanning ζ, r < 0.6 and |b±c| in [1,300] GeV, not from a sharp derivation. The note added in proof is awkward — the updated fit's central θ13 = 6.8° sits above their own upper bound and they wave it off in one line; that deserved a paragraph. And the leptogenesis 'direct link' is largely by construction, since one phase ω feeds both δD and ε1. Who should read it: neutrino model builders working on mu-tau and discrete symmetries, and anyone connecting leptogenesis to low-energy CP. Worth a serious referee — the structural claims are checkable and the numerical ones need tightening.

The paper's central prediction, Eq. (4.80) δx ≥ |δa| tan θs, follows from the ratio δBa/δCa = a/(b+c) being real (Eq. 2.48). This ratio is exact only if mD has the form in Eq. (2.29), which assumes exact μ-τ and CP symmetry in the Dirac Yukawa couplings. The paper argues in Sec. 2.2.1 that soft breaking can only live in the dimension-3 Majorana mass term, but that is an argument about soft breaking in the technical sense; it does not forbid small hard breaking in the dimension-4 Dirac Yukawa couplings. The charged-lepton sector already breaks μ-τ symmetry (m_μ ≠ m_τ), so there is no symmetry that forbids the neutrino Dirac Yukawa couplings from also carrying small μ-τ or CP breaking. If such breaking exists at even the few-percent level, δBa/δCa acquires an imaginary part, and the phase relation (4.79a) is modified, weakening or destroying the lower bound. The paper does not quantify the tolerance. This is the load-bearing concern because the entire chain—the δx–δa correlation, the leptogenesis link, and the θ13 bounds—rests on the single-parameter structure that this conjecture provides.

I agree with the reader's assessment. The paper is internally consistent and the derivations are clean, but the central claim is conditional on the conjecture that the Dirac Yukawa sector is exactly μ-τ and CP symmetric. The paper argues this is the only place soft breaking can occur, but that argument is about soft breaking in the technical sense, not about hard breaking in the Dirac Yukawa couplings. The charged-lepton sector already breaks μ-τ symmetry, so there is no symmetry that forbids small hard breaking in the neutrino Dirac Yukawa couplings. The paper does not quantify how small such breaking must be for the correlation to survive. I also note the upper bound θ13 ≲ 6° is not a sharp prediction: Eq. (4.98a) has a near-divergence at ζ ≈ 2r cos δD, and the bound depends on the assumed range ζ < 0.6. The updated fit's central value θ13 = 6.8° is above the predicted upper bound, and the authors' claim of 'no visible correction' is questionable. However, these are secondary to the model assumption. The Zs2 symmetry is a genuine structural observation, though the fact that θ12 is unaffected by μ-τ breaking was noted earlier in [44]. Overall, the verdict should remain CONDITIONAL.

If the conjecture holds, the paper provides a falsifiable link between the ongoing θ23 measurements and the upcoming θ13 probes, and between low-energy CP violation and leptogenesis. The Zs2 symmetry is an interesting structural observation that unifies the TBM ansatz with a one-parameter family of solar mixings. The derivations are technically detailed and largely checkable, and the paper is explicit that the common-origin structure is a conjecture.

The paper asks whether two apparently unrelated small numbers in neutrino physics — the failure of the atmospheric angle to sit exactly at $45^circ$ and the failure of $theta_{13}$ to vanish — are in fact the same number seen twice. The authors' proposal is that the seesaw Lagrangian respects both a discrete $mu$–$tau$ exchange symmetry and CP exactly in its interaction terms, and that both are broken only softly, by one complex entry in the dimension-3 Majorana mass matrix of the heavy right-handed neutrinos. If that is right, a single phase controls the size of the $mu$–$tau$ breaking and simultaneously supplies the Dirac and Majorana CP phases at low energy and the CP asymmetry that leptogenesis needs to make the baryon asymmetry. The payoff is a testable inequality: any measured deviation of $theta_{23}$ from $45^circ$ forces a minimum $theta_{13}$, so the ongoing atmospheric-angle measurements and the upcoming reactor $theta_{13}$ probes become two views of one parameter. The paper also isolates a separate exact $Z_2$ symmetry of the Dirac mass matrix that fixes the solar angle $theta_{12}$ from a single group parameter and contains tri-bimaximal mixing as the special case $k=2$.

What carries the argument. The load-bearing object is the single complex soft-breaking parameter $zeta e^{iomega}$ placed in the $mu$–$tau$-symmetric heavy Majorana mass matrix, together with the claim that this is the unique such location: the off-diagonal entries are forced equal by symmetry, and the Dirac mass matrix is generated by dimension-4 Yukawa interactions, so it cannot host a soft breaking. Splitting the light neutrino mass matrix into $mu$–$tau$ symmetric and antisymmetric parts makes the argument concrete: the antisymmetric entries $delta B_a$ and $delta C_a$ both carry the factor $(M_{22}-M_{33})$, and their ratio collapses to $a/(b+c)$, a number fixed entirely by the Dirac matrix and independent of the breaking. That ratio is then shown to equal $1/k$ where $tan theta_s = sqrt{2}/|k|$, which is why the $theta_{13}$–$theta_{23}$ correlation and the solar-angle prediction come from the same algebraic fact. The second piece of machinery is the hidden $Z_2$ symmetry with three-dimensional orthogonal representation $T_s(k)$ built from $k = (b+c)/a$; it leaves the Dirac mass matrix invariant, acts as a reflection about a plane in flavour space, and protects $theta_{12}$ from any soft breaking in $M_R$.

Core claim. The central claim is a common-origin conjecture: $mu$–$tau$ and CP are exact symmetries of the seesaw interaction terms, and the only place they can be softly broken is the dimension-3 singlet Majorana mass term of the heavy right-handed neutrinos. In the minimal two-right-handed-neutrino seesaw this reduces to one complex parameter, $M_{33} = M_{22}(1 - zeta e^{iomega})$, breaking both symmetries at once. Expanding in the small parameters $r$ and $zeta$ and matching the model's next-to-leading-order mass matrix against a model-independent reconstruction from oscillation data, the authors derive $delta_a / delta_x = -cot theta_s cos delta_D$ with $delta_a equiv theta_{23} - 45^circ$ and $delta_x equiv theta_{13}$, which yields the inequality $theta_{13} gtrsim |theta_{23} - 45^circ| tan theta_{12}$. Because the same phase $omega$ sets the low-energy Dirac phase through $delta_D = 2pi - omega$, the leptogenesis CP asymmetry $epsilon_1$ is reconstructed from the low-energy Dirac phase, giving $eta_B / M_1 < 2 times 10^{-23}$ GeV$^{-1}$ and hence $M_1 gtrsim 3 times 10^{13}$ GeV. Scanning the parameters against current oscillation data, the authors quote $1^circ lesssim theta_{13} lesssim 6^circ$, with the lower bound sharpened once successful leptogenesis is imposed.

The premise it rests on. The whole construction rests on the conjecture that $mu$–$tau$ and CP are exact symmetries of the Dirac Yukawa interactions and are broken only in the heavy neutrinos' Majorana mass term; if the Yukawa sector carries any small breaking of its own, the single-parameter link between $theta_{13}$, $theta_{23}$ and the leptogenesis phase is lost.

A precise measurement of $theta_{23}$ away from $45^circ$ converts into a hard lower bound on $theta_{13}$ through $theta_{13} gtrsim |theta_{23} - 45^circ| tan theta_{12}$, so Daya Bay, RENO and Double Chooz would either confirm the correlation or close the scenario.

Successful thermal leptogenesis in this construction requires $M_1 gtrsim 3 times 10^{13}$ GeV, which is a lower bound on the seesaw scale derivable from the observed baryon asymmetry alone.

The model predicts an upper bound on the neutrinoless double-beta observable, $|m_{ee}| lesssim 3.4$ meV, since the lightest neutrino is massless in the normal hierarchy.

Because the leptogenesis asymmetry is reconstructed from the low-energy Dirac phase, the cosmological CP violation and the Jarlskog invariant $J$ are tied, so a nonzero $J$ places a floor on $theta_{13}$ as well.

The hidden $Z_2$ symmetry contains tri-bimaximal mixing as the case $k=2$ and admits other values such as $k = 3sqrt{2}$ giving $theta_{12} approx 33.7^circ$, so the solar angle becomes a discrete label rather than a continuous input.

If the correlation holds, the two-decade-old tendency of fits to prefer $theta_{23}$ slightly below $45^circ$ is not a statistical accident but a fingerprint of the sign of $cos delta_D$, since the model ties the sign of $theta_{23} - 45^circ$ to $-cos delta_D$.

The claim that soft breaking cannot live in the Yukawa sector is an argument about where it can occur, not that it must be confined there; a small $mu$–$tau$ or CP admixture in the Dirac couplings would decohere the single-parameter link and could be probed by how tightly the $theta_{13}$–$theta_{23}$ band closes.

Since the same phase drives both the low-energy Dirac phase and the leptogenesis asymmetry, an independent determination of $delta_D$ from long-baseline oscillation and of $eta_B$ from cosmology would over-determine the single parameter, giving a consistency check rather than a fit.

The hidden $Z_2$ symmetry's rank-2 Dirac matrix forces one massless light neutrino, so the scenario inherits the minimal-seesaw prediction of vanishing $m_1$; a confirmed inverted hierarchy or a quasidegenerate light spectrum would rule out this whole line of construction.

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