Integral representation of the neutrino mass-squared differences

I. Alikhanov

REVIEW 3 major objections 4 minor 20 references

An integral identity for the neutrino mass-squared differences yields an analytic bound on the lightest neutrino mass: m1 < 0.0023 eV at 95% C.L.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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2026-08-01 12:42 UTC pith:36RK5YTX

load-bearing objection A textbook integral identity plus a circular error bound; the advertised m1<0.0023 eV is not derived, and the paper's central argument collapses once you check what the cosmological bound actually permits. the 3 major comments →

arxiv 2607.19340 v1 pith:36RK5YTX submitted 2026-07-21 hep-ph hep-ex

Integral representation of the neutrino mass-squared differences

I. Alikhanov This is my paper

classification hep-ph hep-ex PACS 14.60.Pq

keywords neutrino massesmass-squared differencesintegral representationtrapezoidal rulenormal mass orderingabsolute mass scalemass-angle relationscosmological bound

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper establishes a new integral representation of the solar neutrino mass-squared difference Δm^2_21: the reciprocal of the squared period average of 1/(m2 − m1 sinθ) equals Δm^2_21. Discretizing this integral with twelve trapezoidal subintervals turns it into an algebraic equation, and, under a stringent cosmological bound on the sum of the three neutrino masses, the equation becomes an inequality that bounds the lightest mass m1 from above. Using the best current value of Δm^2_21, the bound reads m1 < 0.0023 eV at 95% C.L., with analogous windows for m2, m3, the total mass, and the effective electron and Majorana masses. The same integral representation also yields a fixed angle such that measuring that single geometric parameter would determine the entire neutrino spectrum, in the spirit of known quark-flavor mass relations.

Core claim

The central claim is the identity (1/2π)∫_0^{2π} dθ/(m2 − m1 sinθ) = 1/sqrt(Δm^2_21). A 12-interval trapezoidal approximation converts this into an algebraic equation (Eq. 3.1); bounding its error by |r|/sqrt(Δm^2_21) < (1/32)(1 − m2/sqrt(Δm^2_21))^6 and using a cosmological sum limit, the author derives the strict inequalities sqrt(Δm^2_21) ≤ m2 < (31/30)sqrt(Δm^2_21), m1 < sqrt(61Δm^2_21)/30, and m3 < sqrt(Δm^2_31 + 61Δm^2_21/900). With Δm^2_21 = 7.50×10^-5 eV^2 and Δm^2_31 = 2.529×10^-3 eV^2, these give m1 < 0.0023 eV, m2 ∈ [0.0085, 0.0091] eV, m3 ∈ [0.0497, 0.0508] eV, total mass ∈ [0.0582, 0.0622] eV, m_ββ < 0.0057 eV, and m_νe ∈ (0.0085, 0.0096) eV. A mean value theorem application yie

What carries the argument

The load-bearing object is the smooth 2π-periodic function F(θ) = 1/(m2 − m1 sinθ), whose period average equals 1/sqrt(Δm^2_21) by the residue theorem. The trapezoidal rule with 12 uniform subintervals turns the integral into a finite algebraic equation (Eq. 3.1) that relates m2 directly to sqrt(Δm^2_21), with an explicit error term r. The entire argument hinges on the error bound |r|/sqrt(Δm^2_21) < (1/32)(1 − m2/sqrt(Δm^2_21))^6, which decreases extremely rapidly as m2 approaches sqrt(Δm^2_21) and thereby converts the equation into a strict inequality. The second tool is the first mean value theorem for integrals, which produces a special angle θ′ and, with it, mass–angle relations of the

Load-bearing premise

The derivation collapses if the trapezoidal error in Eq. (3.1) is not as small as 3.8×10^-11; that estimate requires m2 to lie within roughly 3% of sqrt(Δm^2_21), a proximity that is not guaranteed by the cosmological sum bound the paper invokes.

What would settle it

Evaluate the right-hand side of Eq. (3.2) at the largest m2 allowed by the cosmological constraint Σm_i < 0.0642 eV (i.e., with m1 raised until the sum saturates the bound). If the resulting relative error exceeds 3.8×10^-11, then the strict inequality (3.3) and the derived limit m1 < 0.0023 eV do not follow from the stated assumptions.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

The lightest neutrino mass is bounded above by 0.0023 eV at 95% C.L., competitive with cosmological and direct-search limits.
The allowed mass windows for m2 and m3, 0.0085–0.0091 eV and 0.0497–0.0508 eV, put the total neutrino mass near the normal-ordering floor, between 0.058 and 0.062 eV.
The predicted effective Majorana mass m_ββ < 0.0057 eV is within reach of upcoming neutrinoless double-beta decay experiments.
Sub-percent improvements in the measured solar and atmospheric mass splittings will tighten these analytic bounds without any new physics assumptions.
The integral representation identifies a single geometric angle that, if ever measured, would determine the full neutrino mass spectrum.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same integral construction can be adapted to the inverted mass ordering by swapping indices 2 and 3; the author notes the extension is straightforward and algorithmic.
Alternative integrands beyond 1/(m2 − m1 sinθ) could produce different algebraic relations among the oscillation parameters, potentially yielding stronger or complementary bounds.
In the zero-error limit of the trapezoid approximation for Δm^2_31, the equations force m1 = 0, making a massless lightest neutrino a natural limiting configuration consistent with the derived bounds.
The mean-value-theorem angle θ′ offers a new observable target: if future neutrino data can constrain a CP-like geometric angle, it would translate directly into a determination of m1 and m2.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

A textbook integral identity plus a circular error bound; the advertised m1<0.0023 eV is not derived, and the paper's central argument collapses once you check what the cosmological bound actually permits.

read the letter

Bottom line: the headline bound m1 < 0.0023 eV is not derived; it is smuggled in through the error bound. The residue-theorem identity (2.2) is correct (textbook), and the trapezoidal approximation leading to (3.1) gives an exact algebraic relation f(y) = 1 + (y-1)^6/(2y(3y^4+10y^2+3)), which is a harmless curiosity. But the paper never uses that exact error; instead it asserts (3.2) and then claims the error is below 3.8e-11 at 95% CL from Σm < 0.0642 eV. That claim is false. With m3 ≥ sqrt(Δm31^2) ≈ 0.050 eV, the sum bound allows m1 up to about 0.005 eV, making y = m2/sqrt(Δm21^2) up to ~1.17, which gives an error near 1e-6, not 1e-11. To get 3.8e-11 you need y < 1.033, i.e. m1 < 0.0023 eV — the very result advertised. So (3.3)–(3.7) are not consequences of the stated assumptions. The GST section is also mostly vacuous: the mean value theorem gives an angle whose value depends on the masses; calling that a Gatto–Sartori–Tonin relation adds nothing unless the angle is fixed by data. What's good: the paper is clearly written, the author knows the integral identity is textbook, and the algebra in the trapezoid step is right. The exact error formula is a nice exercise, and the idea of using an integral representation as a "complementary perspective" is not absurd. But the central physical constraint does not survive contact with the numbers. This is not a case where the conclusion might be right but the proof is sloppy; the proof assumes the conclusion. The paper also overclaims by saying the GST relations "emerge directly" when no such derivation is given beyond an unspecified angle. For whom is this? Someone might use the exact error identity in a homework problem, but as a physics paper the central result is unsupported. It deserves a desk rejection rather than referee time. If the author could actually derive an error bound from a stated cosmological limit without already knowing the answer, that would be a different paper.

Referee Report

3 major / 4 minor

Summary. The manuscript proposes an integral representation of the neutrino mass-squared differences, Eq. (2.2): (1/2π)∫_0^{2π} dθ/(m2 - m1 sinθ) = 1/√Δm21^2, which follows exactly from Cauchy's residue theorem. The central application is in Sec. 3: a 12-interval trapezoidal approximation of the integral, Eq. (3.1), together with a claimed error bound Eq. (3.2), is used to derive an upper bound on m2, then on m1: m1 < √(61Δm21^2)/30, leading to m1 < 0.0023 eV with JUNO data (Eq. (3.7)). Further applications give ranges for m2, m3, Σmν, mββ, and mνe, and Sec. 4 sketches Gatto-Sartori-Tonin-type relations via the mean value theorem. The integral identity is correct, but the derivation of the numerical bound is not.

Significance. If the central derivation were valid, the paper would offer a new analytic route to an absolute neutrino mass bound. The residue identity is correct and the algebraic manipulations in Eqs. (3.3)-(3.6) are internally consistent. However, the key numerical error estimate, Eq. (3.2) combined with the claim that it is below 3.8×10^-11 at 95% C.L., is not implied by the stated cosmological input; in fact, it is equivalent to assuming the very bound the paper claims to derive. The paper is transparent about setting the error term by hand, but that makes the derivation circular rather than conservative. Thus the main result, m1 < 0.0023 eV, is not established by the manuscript's arguments. The mean-value-theorem remarks in Sec. 4 are correct but programmatic.

major comments (3)

[§3, Eq. (3.2)] The bound (3.2) is a function of y = m2/√Δm21^2: |r|/√Δm21^2 < (1/32)(1 - y)^6. The paper asserts that at 95% C.L. this is below 3.8×10^-11 'assuming the cosmological total neutrino mass limit Σmν < 0.0642 eV'. That inference is not valid. Since m3 ≥ √Δm31^2 ≈ 0.0503 eV and m2 = √(m1^2 + Δm21^2), the sum constraint permits m1 up to about 0.0052 eV, corresponding to y ≈ 1.17. Inserting y = 1.17 into (3.2) gives (1/32)(0.17)^6 ≈ 7×10^-7, five orders of magnitude above 3.8×10^-11. Forcing (1/32)(1-y)^6 < 3.8×10^-11 requires y-1 ≲ 1/30, i.e. m1 ≲ 0.0022 eV, essentially the bound (3.7) the paper later derives. The numerical value 3.8×10^-11 is therefore not a consequence of the cosmological bound.
[§3, Eqs. (3.3)-(3.7)] Equation (3.3) is obtained by setting r/√Δm21^2 = -3.8×10^-11. But as shown above, the validity of this assignment is mathematically equivalent to assuming m2/√Δm21^2 < 31/30, which is precisely the inequality (3.4). The chain from Eq. (3.3) to Eq. (3.7) is algebraically correct, but it is circular: the error bound already encodes the conclusion. The paper itself states 'we set ... and thereby intentionally overestimate the magnitude of the error term'; this is an ad hoc input, not an estimate derived from Σmν < 0.0642 eV. Consequently the headline limit m1 < 0.0023 eV, and the derived ranges (3.8)-(3.12), are conditional on an assumption equivalent to the result they purport to establish.
[§3, Eq. (3.2) derivation] The paper says the relative error is 'straightforwardly estimated' as (1/32)(1 - m2/√Δm21^2)^6, but no derivation or reference is provided. Because this expression is load-bearing for the numerical claim, an explicit derivation or a citation to a quadrature-bound theorem is needed. Without it, the reader cannot check whether (3.2) is an upper bound uniformly valid for all m2 or only a local estimate near y = 1.

minor comments (4)

[§3, text after Eq. (3.2)] The sentence beginning 'For instance, the left-hand side ... remains below 3.8×10^-11 at 95% C.L.' conflates a deterministic algebraic bound with a confidence statement. The cosmological limit is an upper bound on a sum of masses; it does not provide a probability distribution for m2/√Δm21^2.
[§3, Eq. (3.2)] The derivation of the numerical constant 1/32 is absent. If it comes from a Taylor expansion of the trapezoid error about y = 1, the range of validity should be stated explicitly.
[§2 and §3] The function is defined as F(θ) in Eq. (2.1) but referred to as F(x) in the last paragraph of Sec. 3. Please make the notation consistent.
[§4] The mean value theorem application is mathematically correct, but the statement 'determining such a specific angle immediately resolves the neutrino mass spectrum' is not demonstrated. The paragraph is programmatic; consider expanding or clearly labeling it as a direction for future work.

Circularity Check

1 steps flagged

The 3.8e-11 error assumption in Eq. (3.2) already contains Eq. (3.4), so the m1<0.0023 eV bound is an input, not a derivation.

specific steps

self definitional [Sec. 3, Eqs. (3.2)-(3.5) and Eq. (3.7)]
"The error decreases sharply as m2 approaches sqrt(Δm21^2). For instance, the left-hand side of (3.2) remains below 3.8×10^-11 at 95% C.L., assuming the cosmological total neutrino mass limit of Σm_i <0.0642 eV [24, 28]."

Write x = m2/sqrt(Δm21^2). The paper's own bound (3.2) is |r|/sqrt(Δm21^2) < (1/32)(1-x)^6. Requiring this to be below 3.8×10^-11 forces x < 1.0326 ≈ 31/30, which is exactly inequality (3.4). Inequality (3.4) is then used to derive m1 < (1/30)sqrt(61Δm21^2) and the advertised m1 < 0.0023 eV. But the stated cosmological bound Σm_i < 0.0642 eV does not force x < 31/30: with m3 not much below sqrt(Δm31^2) ≈ 0.050 eV, the sum bound permits m1 up to about 0.004 eV, i.e. x ≈ 1.11, which gives an error on the order of 10^-8, five orders of magnitude above 3.8×10^-11. Thus the small error is not a consequence of the cited data; it is an assumption equivalent to the final inequality, so Eqs. (3.3)-(3.7) restate the input rather than derive it.

full rationale

The integral identity (2.2) and the trapezoidal expansion (3.1) are mathematically sound. The circularity enters at the transition from (3.2) to (3.3): the numerical value 3.8×10^-11 for the error is presented as following from the cosmological sum limit, but the paper provides no derivation of that value beyond the observation that the error is small when m2 ≈ sqrt(Δm21^2). The cosmological limit alone permits m2/sqrt(Δm21^2) as large as about 1.11, which would make the right-hand side of (3.2) ≈ 10^-8, not 10^-11. Attaining 3.8×10^-11 requires m2/sqrt(Δm21^2) < 31/30, which is precisely the content of Eq. (3.4). Since Eq. (3.5) and the JUNO-based m1 < 0.0023 eV bound are algebraic consequences of Eq. (3.4), the central result is equivalent to the assumed error bound. The self-citation [38] is used only as motivation and agreement, not as a load-bearing derivation, so it does not independently raise the score. Because the headline claim reduces by construction to its own input assumption, a score of 8 is appropriate; the paper is not wholly vacuous, since the integral representation and the algebraic step from (3.4) to (3.5) are internally valid, but the advertised constraint is not derived from the stated cosmological data.

Axiom & Free-Parameter Ledger

2 free parameters · 4 axioms · 0 invented entities

The derivation rests on three ingredients: the standard three-flavor framework, the external cosmological sum bound, and the specific trapezoid error bound. Only the third is used to produce the claimed m1 limit, and it is an ad hoc assumption that already encodes the result. The integral identity is correct but supplies no new physical constraint.

free parameters (2)

trapezoid error bound eps = 3.8e-11
Set to force y = m2/sqrt(Delta m21^2) < 31/30. The paper claims this follows from sum m_nu < 0.0642 eV, but that bound permits y up to about 1.11, implying an error of order 1e-7 to 1e-8, not 3.8e-11. The choice is effectively the result being derived.
number of trapezoid subintervals = 12
Arbitrary grid choice. Changing the number of subintervals changes the algebraic form of the identity but not the fact that it is an identity; no physics is attached to this parameter.

axioms (4)

domain assumption Standard three-flavor neutrino framework with normal mass ordering (m1 < m2 < m3).
Stated in Section 1; all bounds are derived under normal ordering only.
domain assumption Cosmological bound sum m_nu < 0.0642 eV from refs [24,28].
Used in Section 3 to assert the small trapezoid error. The external bound is acceptable, but it does not imply the error is as small as claimed.
ad hoc to paper The trapezoid error r in Eq. (3.1) is conservatively set to -3.8e-11 / sqrt(Delta m21^2).
This is the load-bearing step. It is equivalent to assuming m2 < 31/30 sqrt(Delta m21^2), i.e., the conclusion being derived. No independent evidence for this error size is provided.
standard math Residue theorem evaluation of Eq. (2.2).
Textbook identity: (1/2pi) integral dtheta/(m2 - m1 sin theta) = 1/sqrt(m2^2 - m1^2). Correct but tautological.

pith-pipeline@v1.3.0-alltime-deepseek · 7177 in / 22448 out tokens · 194390 ms · 2026-08-01T12:42:41.732341+00:00 · methodology

0 comments

read the original abstract

Determining the absolute neutrino mass scale remains one of the most compelling challenges in particle physics. To constrain theoretical models, establishing precise relations among neutrino masses is essential. We propose a simple integral representation of the neutrino mass-squared differences $\Delta m_{ij}^2$ that provides a complementary perspective on these oscillation parameters. We then demonstrate its utility through several examples. Specifically, assuming stringent cosmological bounds that confine the sum of neutrino masses near the normal ordering floor, we derive an analytical condition for the lightest neutrino mass, $m_1 < \sqrt{61\Delta m^2_{21}}/30$. Using recent data from the JUNO experiment, this yields a competitive upper limit of $m_1<0.0023$ eV (95\% C.L.). We also formulate practical analytical bounds for $m_{2}$ and $m_{3}$ adaptable to future data, and translate the results into allowed ranges for the effective electron and Majorana neutrino masses $m_{\nu_e}$ and $m_{\beta\beta}$. Finally, we show that neutrino mass relations of the Gatto-Sartori-Tonin type emerge directly from the proposed integral representation.

discussion (0)

Reference graph

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