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Rephasing invariants of CP violation for heavy and light Majorana neutrinos

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Pith's one-line read This paper shows that all seesaw CP violation—leptogenesis asymmetries and light-neutrino oscillation non-unitarity effects—can be written as rephasing-invariant products of nine mixing angles and six phase differences.

desk verdict Heavy-sector CP invariants with clean leading-order formulas; a peripheral footnote sign slip should not block publication. read the letter →

arxiv 2505.02415 v2 pith:SIQMAA2P submitted 2025-05-05 hep-ph hep-exhep-th

Zhi-zhong Xing This is my paper · ORCID
classification hep-phhep-exhep-th
keywords rephasinginvariantsCPviolationMajorananeutrinosseesawmechanismleptogenesisneutrinooscillationsnon-unitarityactive-sterilemixing
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Within the canonical seesaw mechanism, the same block of a $6\times 6$ flavor mixing matrix governs the charged-current interactions of three light and three heavy Majorana neutrinos (particles identical to their antiparticles), through matrices $U$ and $R$. This paper establishes that the physically meaningful, rephasing-invariant measures of CP violation in the heavy sector are the invariants $X^{jj'}_{\alpha\beta}$ and $Z^{jj'}_{\alpha\beta}$, and it gives their explicit leading-order formulas in terms of nine active-sterile mixing angles and six independent phase differences. Those formulas recast the CP-violating asymmetries of heavy Majorana neutrino decays—the quantities that drive leptogenesis—into a compact basis-independent form, and they also control the small non-unitarity corrections to CP violation in light-neutrino oscillations. If the paper is right, a handful of phase combinations organizes CP violation across heavy-neutrino decays, neutrino oscillations, charged-lepton radiative decays, and neutrinoless double-$\beta$ decay.

What carries the argument

The load-bearing object is the full Euler-like block parametrization of the $6\times 6$ seesaw mixing matrix—a factorization into complex rotations that separates three active from three sterile neutrino fields and yields $U = A U_0$ together with the $3\times 3$ heavy mixing matrix $R$. Its nine complex parameters $\hat{s}_{ij} = s_{ij}e^{i\delta_{ij}}$ carry the nine active-sterile mixing angles and six independent CP phases, and the smallness of those angles (upper bounds such as $s_{2j} < 0.005$) justifies truncating $R$ at order $s^3$ and $A$ at order $s^4$. That truncation is what converts the abstract products in Eq. (5) into the simple trigonometric factors in Eqs. (13)–(16), and it is what makes the connection to $\varepsilon_{j\alpha}$ transparent in Eq. (18). The exact seesaw relation and the unitarity condition correlate $U$ and $R$, which is how the same invariants propagate to light-neutrino observables.

What would settle it

Compute $X$ and $Z$ exactly from the full $R$ matrix in Eq. (7) with the nine angles at the upper bounds of Eqs. (9)–(10) and compare the numbers with Eqs. (13)–(16); a fractional deviation larger than the nominal $\mathcal{O}(s^3)$ accuracy would mean the truncation is the weak link. Experimentally, a precision measurement of the $\nu_\mu \to \nu_e$ CP asymmetry that deviates from the form of Eq. (25) beyond the $a_{21}$ corrections, or a charged-lepton radiative decay whose interference term $\mathrm{Re}\,X^{ij}_{\alpha\beta}$ does not match the invariant prediction, would falsify the claimed bookkeeping.

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Extended reading notes

Core claim

The paper's central claim is that CP violation in the heavy Majorana-neutrino sector admits a complete rephasing-invariant description. Defining $X^{jj'}_{\alpha\beta} = \mathrm{Im}\left(R_{\alpha j}R_{\beta j'}R^{*}_{\alpha j'}R^{*}_{\beta j}\right)$ and $Z^{jj'}_{\alpha\beta} = \mathrm{Im}\left(R_{\alpha j}R_{\beta j}R^{*}_{\alpha j'}R^{*}_{\beta j'}\right)$ for $j,j' = 4,5,6$, the paper shows that these are the heavy analogues of the two invariant types already known for light neutrinos, invariant respectively under charged-lepton rephasing and under rephasing of both charged leptons and neutrino fields. In the Euler-like block parametrization, they factor at leading order into products of sines of the nine active-sterile angles times sines of the six independent phase combinations $\alpha_i = \delta_{i4} - \delta_{i5}$, $\beta_i = \delta_{i5} - \delta_{i6}$, and $\gamma_i = \delta_{i6} - \delta_{i4}$, with $\gamma_i = -(\alpha_i + \beta_i)$. These expressions substitute directly into the one-loop leptogenesis asymmetries $\varepsilon_{j\alpha}$, so Eq. (18) expresses all nine flavor-dependent asymmetries purely through the invariants. For light neutrinos, the paper further shows that the nine Jarlskog-like invariants $J^{ii'}_{\alpha\beta}$ deviate from the universal $\pm J_{\nu}$ only through the same small non-unitarity parameters $a_{ii}$ and $a_{ii'}$, making the standard Jarlskog invariant the dominant CP source in long-baseline oscillations unless $\delta_{\nu}$ is suppressed.

Load-bearing premise

The argument stands on the assumption that the nine angles describing mixing between known and heavy neutrinos are so small that only the leading terms in the expansion of $R$ and $A$ matter; if those angles were larger or higher-order terms were measurable, every explicit formula would need correction.

Editorial extensions

If this is right

  • Leptogenesis asymmetries become basis-independent observables: Eq. (18) lets any calculation of $\varepsilon_{j\alpha}$ be checked purely in terms of $X$ and $Z$, without choosing charged-lepton or neutrino phase conventions.
  • The flavor-summed asymmetries $\varepsilon_j$ depend only on the $Z$-type invariants, not on the $X$-type ones, so measurements of $\varepsilon_j$ alone cannot probe the full CP structure of the heavy sector; flavor-resolved $\varepsilon_{j\alpha}$ are needed.
  • In light-neutrino oscillations, deviations of $J^{ii'}_{\alpha\beta}$ from $\pm J_\nu$ are suppressed by the same small non-unitarity parameters that enter charged-current weak interactions, explaining why the standard Jarlskog invariant remains the leading CP observable in long-baseline experiments.
  • The same six phase combinations control the interference terms in charged-lepton radiative decays and in neutrinoless double-beta decay, so those processes can cross-check phases extracted from heavy-neutrino decays and oscillations.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A combined future data set—heavy-decay asymmetries from a collider, oscillation CP from long-baseline experiments, and charged-lepton radiative decay rates—would over-determine the six phase combinations and test the built-in relation $\gamma_i = -(\alpha_i+\beta_i)$, a consistency check the paper leaves implicit.
  • With a hierarchical heavy-neutrino spectrum, the loop functions in Eq. (17) suppress all but one $j'$ term, so a single measured $\varepsilon_{j\alpha}$ would isolate one specific $X$ or $Z$ combination; that makes the invariants not just a bookkeeping device but a direct phase probe.
  • The invariants admit a geometric reading: the four complex products in $X^{jj'}_{\alpha\beta}$ close into a quadrilateral whose oriented area is the invariant itself, so leptogenesis CP violation could be visualized and bounded by unitarity polygons of $R$, in the same way the Jarlskog invariant is a triangle area for quarks.
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Editorial analysis

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Desk editor's note, referee report, and a circularity audit.

Referee Report

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Summary. This paper defines two families of rephasing invariants for the heavy-neutrino mixing matrix R in the canonical seesaw framework: X^{jj'}_{\alpha\beta} and Z^{jj'}_{\alpha\beta}, analogous to the Jarlskog-like invariants for the light PMNS matrix. Using the Euler-like block parametrization of the 6x6 neutrino mixing matrix with nine active-sterile angles and six CP phases, the author derives explicit leading-order expressions for these invariants in terms of s_{ij} and the phase differences \alpha_i, \beta_i, \gamma_i (Eqs. (13)-(16)). He then rewrites the flavor-dependent leptogenesis asymmetries \epsilon_{j\alpha} in terms of the new invariants (Eq. (18)), computes the leading non-unitarity shifts of the light-neutrino Jarlskog invariants J^{ii'}_{\alpha\beta} (Eq. (21)), and sketches applications to charged-lepton radiative decays and neutrinoless double beta decay via the mixed invariants X^{ij}_{\alpha\beta} and Z^{ij}_{\alpha\beta}. The paper's central technical claim is that X and Z provide a basis-independent bookkeeping for heavy-neutrino CP violation and that they are given explicitly by the leading-order formulas.

Significance. The central derivation is sound: I spot-checked Eqs. (13)-(16) directly against the leading-order R matrix in Eq. (11) and reproduced the stated sine combinations, and the passage from Eq. (17) to Eq. (18) goes through with the correct sign. The possible concern that the O(s^3) truncation of R invalidates the leading-order invariants does not materialize: since X and Z are quartic in R, the dropped terms enter at O(s^6), which is sub-percent for the largest allowed active-sterile angles and much smaller for the s_{2j}-suppressed entries. The paper is honest about the truncation and about the external origin of the one-loop leptogenesis formula. If the formulas are correct, the paper gives a compact, rephasing-invariant bridge between the parametrization of the seesaw flavor structure and the CP asymmetries relevant to leptogenesis, and clarifies which phase combinations enter CP-conserving LFV/LNV observables. The explicit checkable form of the main expressions and the clear separation of Dirac-type and Majorana-type invariants are strengths.

minor comments (6)
  1. [Eq. (17)] The sums in Eqs. (17)-(19) are written as \sum_{j'=4}^{6} without excluding j'=j. At j'=j the loop functions \xi(x_{jj}) and \zeta(x_{jj}) are singular, and although the imaginary parts of the summand vanish identically, the expression is formally ambiguous; the sums should be written with j'\neq j.
  2. [Footnote 3] The simplified formula has the wrong sign: substituting (UD_\nu U^T)_{ee}=-(RD_N R^T)_{ee} into Eq. (27) yields [M_j + M_A^2 F(A,M_j)/M_j] R_{ej}^2 inside the absolute square, not [M_j - M_A^2 F(A,M_j)/M_j].
  3. [Eq. (26)] In the branching-ratio expression the prefactor changes from 3\alpha_{em}/(2\pi) in the first equality to 3\alpha_{em}/(32\pi) in the expansion, while the bracket is unchanged; this is algebraically inconsistent unless a different normalization of the loop amplitudes is intended, and it should be corrected or explained.
  4. [Eq. (25)] The first non-unitarity term contains Im\left(a_{21} e^{-i\delta_{21}}\right), but \delta_{21} is not defined anywhere in the paper; from Eq. (21) the corresponding phase should presumably be \delta_{12}.
  5. [Eq. (17)] In the definition of \epsilon_{j\alpha}, the CP-conjugated decay in the second numerator term should be typeset as \bar{\ell}_\alpha + \bar{H} to distinguish it from the first term; as typeset the two terms look identical.
  6. [Eqs. (13)-(16)] The notation \alpha_i and \beta_i for the phase combinations uses letters that also denote the charged-lepton flavor indices \alpha,\beta; although the context is clear, distinct symbols would reduce the risk of confusion.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the invariant expressions are explicit algebraic consequences of the parametrization and are not fitted or equivalent to inputs by construction.

full rationale

The central derivation chain is: define the rephasing invariants X and Z from the heavy mixing matrix R in Eq. (5); adopt the published Euler-like block parametrization of Refs. [10,11], whose explicit forms for A, R and U0 are reproduced in Eqs. (7)-(8); use the leading-order approximation R = (s-hat)^* + O(s^3) in Eq. (11); compute the quartic products to obtain Eqs. (13)-(16); and finally rewrite the one-loop leptogenesis asymmetry of Eq. (17) in terms of the same invariants in Eq. (18). No step fits a parameter to the quantity being predicted, and no result is imported solely from an unverified self-citation: the parametrization's content is stated in the paper itself, and the approximation order is justified by the external bounds of Eqs. (9)-(10). Eq. (18) follows from Eq. (17) by substituting the definitions in Eq. (5), so it is an algebraic identity rather than a prediction forced by construction. The only concrete issue observed, an apparent sign inconsistency in footnote 3 when substituting the seesaw relation into Eq. (27), is a peripheral correctness point, not a circularity, and it does not affect the main invariant construction or the heavy-sector formulas.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper adds no new physical entities and fits no data. Its explicit results depend on the standard type I seesaw framework, unitarity of the 6x6 mixing matrix, the one loop leptogenesis asymmetry formula from the literature, and the small angle expansion justified by current bounds on active sterile mixing.

assumptions (4)
  • standard math The 6x6 neutrino flavor mixing matrix is unitary, so U U-dagger plus R R-dagger equals the identity.
    Invoked in Section 1 around Eq. (2); it is the defining unitarity of the seesaw mixing framework.
  • domain assumption The exact seesaw relation U diag(m_i) U^T plus R diag(M_j) R^T equals zero holds.
    Eq. (2); this is the canonical type I seesaw relation linking light and heavy Majorana masses, and it is assumed rather than derived in this paper.
  • domain assumption The one loop CP asymmetry formula for heavy Majorana neutrino decays in Eq. (17) is valid.
    Taken from the leptogenesis literature and used in Section 2.3 to connect invariants to epsilon_{j alpha}.
  • domain assumption The leading order approximation R approximately s-hat-star plus O(s^3) and A approximately I minus a plus O(s^4) is accurate enough for the explicit formulas.
    Eq. (11), justified by the experimental upper bounds on active sterile mixing in Eqs. (9) and (10); this truncation underlies all explicit expressions.

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Pith. "Pith review of Rephasing invariants of CP violation for heavy and light Majorana neutrinos." pith.science (2026). https://pith.science/paper/SIQMAA2P

@misc{pith2026250502415,
  author       = {Pith},
  title        = {Pith review of: Rephasing invariants of CP violation for heavy and light Majorana neutrinos},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SIQMAA2P}},
  note         = {Machine review of arXiv:2505.02415}
}
abstract

In the canonical seesaw mechanism, the strengths of charged-current interactions for light and heavy Majorana neutrinos are described respectively by the $3\times 3$ matrices $U$ and $R$ that are correlated with each other via the exact seesaw relation and the unitarity condition. We write out the Majorana-type invariants of CP violation of $R$ and $U$, which are insensitive to redefining the phases of three charged-lepton fields; and the Dirac-type invariants of CP violation of $R$ and $U$ that are insensitive to the rephasing of both the charged-lepton fields and the neutrino fields. Such invariants are explicitly calculated with the help of a full Euler-like block parametrization of the seesaw flavor structure containing nine active-sterile flavor mixing angles and six independent CP-violating phases, and their corresponding roles in the CP-violating asymmetries of three heavy Majorana neutrino decays and in the flavor oscillations of three light Majorana neutrinos are briefly discussed. We point out that similar rephasing invariants arising from the interplay between $R$ and $U$ may also manifest themselves in a variety of lepton-flavor-violating and lepton-number-violating processes.

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Reviewed August 16, 2026 · model on record in the stance chip above.