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Second-order neutrino oscillation formulas for a non-symmetric Earth density give sub-percent corrections once the path is split into PREM shells.

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2026-07-13 01:54 UTC pith:AGZLVXQQ

load-bearing objection Clean second-order formulas for non-symmetric Earth potentials, with a practical ten-shell PREM implementation that keeps residual corrections under 0.7%.

arxiv 2607.09591 v1 pith:AGZLVXQQ submitted 2026-07-10 hep-ph

Second-Order Perturbative Correction for Neutrino Oscillation Tomography of the Earth

Dmitry Zhuridov , Wiktor Porwo\l This is my paper
classification hep-ph PACS 14.60.Pq95.85.Ry91.35.Pn
keywords neutrino oscillationsEarth tomographyperturbation theorymatter effectPREMevolution matrixatmospheric neutrinos
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 supplies the missing second-order term in the perturbative expansion of the neutrino evolution matrix for a generic, not necessarily left-right-symmetric electron-density profile. When that matrix is applied shell-by-shell to the ten PREM layers, the residual second-order corrections to the oscillation probability stay below 0.7 percent across the energy and nadir-angle range relevant for GeV atmospheric neutrinos. The result matters because next-generation detectors will collect enough statistics that theoretical error must drop below the percent level if neutrino data are to refine seismic density models of the deep Earth. By removing the symmetry assumption that limited earlier work, the calculation can treat the large core-mantle density jump without forcing a single average potential over the whole path. The authors also map the domain where the Magnus series converges and show that their multi-shell scheme shrinks the dangerous low-energy region.

Core claim

The second-order evolution matrix S2(L) constructed for an arbitrary weak density perturbation, together with its unitarized form, yields oscillation probabilities whose second-order corrections remain under 0.7 percent when the Earth is treated as ten PREM-matched shells.

What carries the argument

The second-order iteration of the evolution matrix S2(L) (Eq. 58) and its unitarized counterpart (Eq. 65), obtained by successive integration of the time-ordered series for a non-symmetric potential and then factored shell-by-shell.

Load-bearing premise

Inside each shell the density deviation from its local average must stay small enough that the first-order correction matrix K remains much less than one; otherwise the whole expansion fails.

What would settle it

Compute the third-order term or a fully numerical solution of the Schrödinger equation for the same ten-shell PREM profile; if the difference from the second-order probability exceeds a few tenths of a percent inside the claimed domain, the expansion is not yet accurate enough.

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

If this is right

  • Oscillated atmospheric-neutrino fluxes for Hyper-Kamiokande, DUNE and KM3NeT can now be predicted with sub-percent theoretical error from the density profile.
  • Density jumps between PREM layers can be treated exactly by matrix multiplication rather than by a single average potential.
  • The same non-symmetric formulas apply immediately to any other planetary or solar density profile that lacks left-right symmetry along a given chord.
  • Convergence bounds of the Magnus series shrink once the path is subdivided, giving a practical diagnostic for when further shell splitting is needed.

Where Pith is reading between the lines

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

  • If residual intra-shell density gradients still produce |K| ~ O(1) below 2 GeV, a modest increase from ten to twenty shells would restore the small-K condition without changing the formal expansion.
  • The same second-order matrix can be reused for absorption tomography once the charged-current optical depth is folded in, linking the two tomography channels at the amplitude level.
  • A direct numerical comparison of the unitarized S2 against a high-order Magnus expansion on identical shells would quantify residual unitarity violation and fix the practical accuracy floor.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 4 minor

Summary. The manuscript derives the second-order perturbative correction to the two-flavor neutrino evolution matrix for a generic (not necessarily symmetric) matter potential V(x). Starting from the Schrödinger equation with the standard charged-current Hamiltonian, the authors obtain the iterative solution for the correction matrix K(x) (Eqs. 35–38, Appendix A), the first-order result S1 (Eq. 45, unitarized in Eq. 47), and the second-order result S2 (Eq. 58, unitarized in Eq. 65). They implement these expressions in multi-shell schemes (one-, two- and ten-shell) matched to PREM density layers, produce oscillograms of Peα over the (Eν, θν) domain relevant for GeV atmospheric neutrinos, and quantify residual corrections: second-order shifts remain below 0.7 % relative to first order in the ten-shell scheme (Fig. 14). Convergence bounds of the Magnus expansion are overlaid for comparison, and the authors note that further shell subdivision may be needed below ~2 GeV.

Significance. The work supplies a concrete, order-by-order analytic tool for high-precision neutrino oscillation tomography (NOT) that is free of free parameters once PREM densities and Ye values are fixed. The second-order matrix is valid for non-symmetric potentials, enabling a natural multi-shell treatment of PREM density jumps that previous first-order or single-shell approaches could not handle cleanly. The numerical demonstration that residual corrections fall below the percent level (and well below expected experimental systematics of Hyper-K, DUNE and KM3NeT) makes the result immediately usable for flux predictions and Earth-model discrimination. Explicit unitarity restoration and side-by-side comparison with the Magnus expansion further strengthen the technical contribution.

minor comments (4)
  1. In Eq. (59) the phrase “[to check]” remains in the text; either complete the check or remove the placeholder.
  2. Figs. 6–14 would benefit from a short caption note clarifying that the green curves are Magnus convergence bounds (Eq. 68) evaluated at xc = L or L/2, so that readers do not confuse them with physical contours.
  3. A brief statement of the numerical integration method used for the double integrals J(f) (Eq. 52) would aid reproducibility; the analytic expressions themselves are clear.
  4. The abstract and introduction mention “growing power of neutrino experiments, e.g., KM3NeT, DUNE and Hyper-Kamiokande”; a single sentence in Sec. 5 linking the achieved 0.7 % accuracy to the expected statistical precision of those detectors would tighten the phenomenological motivation.

Circularity Check

0 steps flagged

No circularity: second-order evolution matrix derived from Schrödinger equation; PREM and Ye are external geophysical inputs; multi-shell is a validity condition, not a fit.

full rationale

The central derivation (Secs. 3.1–3.3) starts from the two-flavor Schrödinger equation (17)–(18) with the standard charged-current potential ACC = 2√2 GF ne Eν, splits V = V̄ + ∆V, and obtains the iterative solution for the evolution matrix via the time-ordered exponential (38)–(39). First- and second-order pieces S1 (45) and S2 (58) (unitarized (65)) follow by direct integration of the Pauli-matrix algebra; no free parameters are introduced or fitted. Application to PREM (Sec. 4) uses published density profiles and Ye values (0.466 core, 0.494 mantle) as fixed external inputs; the ten-shell factorization (72) is chosen solely so that |Kab| ≪ 1 holds inside each shell, after which the size of residual corrections is measured (Fig. 14, <0.7 %). Comparisons to zero-order, one-shell, two-shell and Magnus results are diagnostic, not inputs. No self-definitional loop, no fitted-then-predicted quantity, no load-bearing self-citation, and no uniqueness theorem or ansatz imported from the authors’ prior work appear. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 0 invented entities

The derivation rests on the standard two-flavor Schrödinger equation in matter, the PREM density profile taken as external input, and the smallness of density variations inside each chosen shell. No free parameters are fitted; Ye values and mixing angles are taken from the literature. No new dynamical entities are postulated.

axioms (4)
  • domain assumption Two-flavor approximation with solar mass splitting neglected (Δm²₁₂ ≪ |Δm²₃₁|)
    Stated in Sec. 2.3 and after Eq. (16); standard for atmospheric GeV neutrinos but becomes less accurate below ~2 GeV.
  • domain assumption PREM density profile and fixed Ye = 0.466 (core) / 0.494 (mantle) supply the electron density
    Used throughout Sec. 4; taken from Dziewonski & Anderson (1981) and Lisi & Montanino (1997).
  • domain assumption Perturbative expansion valid when |Kab| ≪ 1 inside each shell
    Explicitly required after Eq. (35); enforced by the multi-shell construction.
  • standard math Standard charged-current matter potential Ve = √2 GF ne
    Eq. (7); textbook MSW potential.

pith-pipeline@v1.1.0-grok45 · 26134 in / 2482 out tokens · 31534 ms · 2026-07-13T01:54:19.507747+00:00 · methodology

0 comments
read the original abstract

The Earth's interior at depths exceeding present direct probing limit, which is just one five-hundredth of its radius, remains unknown to a significant extent due to existing ambiguities in predicting the distribution of temperature, pressure and chemical composition on the basis of seismic and geophysical observations. Scanning the Earth with neutrinos can provide independent information for refinement of existing seismological models. However, neutrinos interact with medium extremely weakly. Only tiny TeV-energy component of the atmospheric neutrino flux is noticeably absorbed while traversing the Earth, which provides limited statistical data. Flux reduction for much more common neutrinos with energies of the order of 1 GeV is vanishing. However, on their way through the planet they undergo sizable flavor oscillations that depend on the electron number density, which provides a link to seismological models. Accurate calculation of the matter effect in these oscillations is required for successful neutrino oscillation tomography of the Earth besides growing power of neutrino experiments, e. g., KM3NeT, DUNE and Hyper-Kamiokande. We obtained relevant neutrino oscillation probabilities in the second order of perturbation theory. Our results are valid for a neutrino potential that is not necessarily symmetric along the neutrino path. This allowed us to implement a multi-shell evolution scheme to better account for the density jumps between various layers of the Earth.

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Reference graph

Works this paper leans on

38 extracted references · 12 canonical work pages · 7 internal anchors

  1. [1]

    University Press, Cambridge (2000)

    Poirier, J.-P.: Introduction to the Physics of the Earth’s Interior. University Press, Cambridge (2000)

    2000

  2. [2]

    In: Carlson, R.W

    McDonough, W.F.: Compositional Model for the Earth’s Core. In: Carlson, R.W. (ed.) Treatise on Geochemistry: The Mantle and Core vol. 2, pp. 547–568. Elsevier-Pergamon, Oxford (2003)

    2003

  3. [3]

    Masters, G., Gubbins, D.: On the resolution of density within the Earth. Phys. Earth. Planet. Inter.140, 159–167 (2003)

    2003

  4. [4]

    Dziewonski, A.M., Anderson, D.L.: Preliminary reference earth model. Phys. Earth Planet. Interiors25, 297–356 (1981) https://doi.org/10.1016/ 0031-9201(81)90046-7

    1981

  5. [5]

    Geller, R.J., Hara, T.: Geophysical aspects of very long baseline neutrino exper- iments. Nucl. Instrum. Meth. A503, 187–191 (2003) https://doi.org/10.1016/ S0168-9002(03)00670-3 arXiv:hep-ph/0111342

    Pith/arXiv arXiv 2003

  6. [6]

    Jes´ us-Valls, C., Petcov, S.T., Xia, J.: Neutrino oscillation tomography of the Earth with the Hyper-Kamiokande detector. Eur. Phys. J. C85(6), 703 (2025) https://doi.org/10.1140/epjc/s10052-025-14338-z arXiv:2411.12344 [hep-ex]

    Pith/arXiv arXiv doi:10.1140/epjc/s10052-025-14338-z 2025

  7. [7]

    Springer, Berlin, Heidelberg (1987)

    Kozlovsky, Y.A., Adrianov, N.: The Superdeep Well of the Kola Peninsula. Springer, Berlin, Heidelberg (1987)

    1987

  8. [8]

    Chapman & Hall, London (1975) 26

    Bullen, K.: The Earth’s Density. Chapman & Hall, London (1975) 26

    1975

  9. [9]

    (In Russian)

    Borisov, A.B., Dolgoshein, B.A., Kalinovsky, A.N.: Direct Method for Determi- nation of Differential Distribution of the Earth Density by Means of High-energy Neutrino Scattering. (In Russian). Yad. Fiz.44, 681–689 (1986)

    1986

  10. [10]

    Goos, I., Fuji, N., Durand, S., Van Elewyck, V., Coelho, J.A.B., Mittelstaedt, E., Deniz, Y.: Sensitivity of neutrino oscillations to the Earth’s interior properties (2025) arXiv:2512.05840 [physics.geo-ph]

    arXiv 2025

  11. [11]

    Deng, X., Xu, Y., Hao, S., Ruan, Y., Zhao, Y., Wang, W., Ni, S., Wu, Z.: Com- positional and thermal state of the lower mantle from joint 3D inversion with seismic tomography and mineral elasticity. Proc. Natl. Acad. Sci. USA120(26), 2220178120 (2023) https://doi.org/10.1073/pnas.2220178120

    work page doi:10.1073/pnas.2220178120 2023

  12. [12]

    Kozak, A., Wojnar, A.: Planetary seismology as a test of modified gravity pro- posals. Phys. Rev. D108(4), 044055 (2023) https://doi.org/10.1103/PhysRevD. 108.044055 arXiv:2303.17213 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd 2023

  13. [13]

    CERN Report https://cds.cern.ch/record/2258764 (1973)

    Placci, A., Zavattini, E.: On the possibility of using high-energy neutrinos to study the Earth’s interior. CERN Report https://cds.cern.ch/record/2258764 (1973)

    arXiv 1973

  14. [14]

    Volkova, L.V., Zatsepin, G.T.: On the problem of neutrino penetration though the earth. Izv. Akad. Nauk Ser. Fiz.38(5), 1060 (1974)

    1974

  15. [15]

    In: 10th International Conference on Neutrino Physics: Neutrino ’82, pp

    Nedyalkov, I.P.: ON THE STUDY OF THE EARTH COMPOSITION BY MEANS OF NEUTRINO EXPERIMENTS. In: 10th International Conference on Neutrino Physics: Neutrino ’82, pp. 300–302 (1981)

    1981

  16. [16]

    Nature Phys.15(1), 37–40 (2019) https://doi.org/10.1038/s41567-018-0319-1 arXiv:1803.05901 [hep-ph]

    Donini, A., Palomares-Ruiz, S., Salvado, J.: Neutrino tomography of Earth. Nature Phys.15(1), 37–40 (2019) https://doi.org/10.1038/s41567-018-0319-1 arXiv:1803.05901 [hep-ph]

    Pith/arXiv arXiv doi:10.1038/s41567-018-0319-1 2019

  17. [17]

    Updated Earth Tomography Using Atmospheric Neutrinos at IceCube

    Abbasi, R.,et al.: Updated Earth Tomography Using Atmospheric Neutrinos at IceCube. PoSICRC2025, 1211 (2025) https://doi.org/10.22323/1.501.1211 arXiv:2507.09763 [astro-ph.HE]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.22323/1.501.1211 2025

  18. [18]

    Nicolaidis, A., Jannane, M., Tarantola, A.: Neutrino tomography of the earth. J. Geophys. Res.96(B13), 21811–21817 (1991) https://doi.org/10.1029/91JB01835

    work page doi:10.1029/91jb01835 1991

  19. [19]

    Ohlsson, T., Snellman, H.: Neutrino oscillations with three flavors in matter: Applications to neutrinos traversing the Earth. Phys. Lett. B474, 153–162 (2000) https://doi.org/10.1016/S0370-2693(00)00008-3 arXiv:hep-ph/9912295. [Erratum: Phys.Lett.B 480, 419–419 (2000)]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/s0370-2693(00)00008-3 2000

  20. [20]

    JHEP05, 077 (2007) https://doi.org/10.1088/ 1126-6708/2007/05/077 arXiv:hep-ph/0612285 27

    Akhmedov, E.K., Maltoni, M., Smirnov, A.Y.: 1-3 leptonic mixing and the neu- trino oscillograms of the Earth. JHEP05, 077 (2007) https://doi.org/10.1088/ 1126-6708/2007/05/077 arXiv:hep-ph/0612285 27

    Pith/arXiv arXiv 2007

  21. [21]

    DUNE atmospheric neutrinos: Earth Tomography

    Kelly, K.J., Machado, P.A.N., Martinez-Soler, I., Perez-Gonzalez, Y.F.: DUNE atmospheric neutrinos: Earth tomography. JHEP05, 187 (2022) https://doi.org/ 10.1007/JHEP05(2022)187 arXiv:2110.00003 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/jhep05(2022)187 2022

  22. [22]

    Oxford University Press, Oxford (2007)

    Giunti, C., Kim, C.W.: Fundamentals of Neutrino Physics and Astrophysics. Oxford University Press, Oxford (2007). https://doi.org/10.1093/acprof:oso/ 9780198508717.001.0001

    doi:10.1093/acprof:oso/ 2007

  23. [23]

    Preprint at https://arxiv.org/ abs/1805.04163 (2018)

    Abe, K., et al.: Hyper-Kamiokande Design Report. Preprint at https://arxiv.org/ abs/1805.04163 (2018)

    Pith/arXiv arXiv 2018

  24. [24]

    Brahmachari, B., Choubey, S., Roy, P.: CP violation and matter effect for a vari- able earth density in very long baseline experiments. Nucl. Phys. B671, 483–497 (2003) https://doi.org/10.1016/j.nuclphysb.2003.08.009 arXiv:hep-ph/0303078

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/j.nuclphysb.2003.08.009 2003

  25. [25]

    Akhmedov, E.K., Maltoni, M., Smirnov, A.Y.: Oscillations of high energy neu- trinos in matter: Precise formalism and parametric resonance. Phys. Rev. Lett. 95, 211801 (2005) https://doi.org/10.1103/PhysRevLett.95.211801 arXiv:hep- ph/0506064

    work page doi:10.1103/physrevlett.95.211801 2005

  26. [26]

    Supanitsky, A.D., D’Olivo, J.C., Medina-Tanco, G.: Perturbative exponential expansion and matter neutrino oscillations. Phys. Rev. D78, 045024 (2008) https://doi.org/10.1103/PhysRevD.78.045024 arXiv:0804.1105 [astro-ph]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.78.045024 2008

  27. [27]

    Wolfenstein, L.: Neutrino Oscillations in Matter. Phys. Rev. D17, 2369–2374 (1978) https://doi.org/10.1103/PhysRevD.17.2369

    doi:10.1103/physrevd.17.2369 1978

  28. [28]

    Lisi, E., Montanino, D.: Earth regeneration effect in solar neutrino oscillations: An Analytic approach. Phys. Rev. D56, 1792–1803 (1997) https://doi.org/10. 1103/PhysRevD.56.1792 arXiv:hep-ph/9702343

    Pith/arXiv arXiv 1997

  29. [29]

    Navas, S.,et al.: Review of particle physics. Phys. Rev. D110(3), 030001 (2024) https://doi.org/10.1103/PhysRevD.110.030001

    doi:10.1103/physrevd.110.030001 2024

  30. [30]

    Mikheyev, S.P., Smirnov, A.Y.: Resonance Amplification of Oscillations in Matter and Spectroscopy of Solar Neutrinos. Sov. J. Nucl. Phys.42, 913–917 (1985)

    1985

  31. [31]

    Giganti, C., Lavignac, S., Zito, M.: Neutrino oscillations: The rise of the PMNS paradigm. Prog. Part. Nucl. Phys.98, 1–54 (2018) https://doi.org/10.1016/j. ppnp.2017.10.001 arXiv:1710.00715 [hep-ex]

    Pith/arXiv arXiv doi:10.1016/j 2018

  32. [32]

    Agrawal, V., Gaisser, T.K., Lipari, P., Stanev, T.: Atmospheric neutrino flux above 1-GeV. Phys. Rev. D53, 1314–1323 (1996) https://doi.org/10.1103/ PhysRevD.53.1314 arXiv:hep-ph/9509423

    Pith/arXiv arXiv 1996

  33. [33]

    Abbasi, R.,et al.: Measurement of Atmospheric Neutrino Oscillation Parame- ters Using Convolutional Neural Networks with 9.3 Years of Data in IceCube 28 DeepCore. Phys. Rev. Lett.134(9), 091801 (2025) https://doi.org/10.1103/ PhysRevLett.134.091801 arXiv:2405.02163 [hep-ex]

    Pith/arXiv arXiv 2025

  34. [34]

    Chattopadhyay, S., Krishnamoorthi, J., Upadhyay, A.K.: Demonstrating the abil- ity of IceCube DeepCore to probe Earth’s interior with atmospheric neutrino oscillations. Eur. Phys. J. ST234(16), 5055–5064 (2025) https://doi.org/10.1140/ epjs/s11734-025-01741-6 arXiv:2502.18995 [hep-ph]

    arXiv 2025

  35. [35]

    Maderer, L., Kaminski, E., Coelho, J.A.B., Bourret, S., Van Elewyck, V.: Unveiling the outer core composition with neutrino oscillation tomography. Front. Earth Sci.11, 1008396 (2023) https://doi.org/10.3389/feart.2023.1008396 arXiv:2208.00532 [hep-ex]

    work page doi:10.3389/feart.2023.1008396 2023

  36. [36]

    Raikwal, D., Choubey, S.: Earth tomography with the ICAL detector at INO. Phys. Rev. D109(7), 073011 (2024) https://doi.org/10.1103/PhysRevD.109. 073011 arXiv:2309.12573 [hep-ph]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.109 2024

  37. [37]

    Akhmedov, E.K., Dighe, A., Lipari, P., Smirnov, A.Y.: Atmospheric neutrinos at Super-Kamiokande and parametric resonance in neutrino oscillations. Nucl. Phys. B542, 3–30 (1999) https://doi.org/10.1016/S0550-3213(98)00825-6 arXiv:hep- ph/9808270

    work page doi:10.1016/s0550-3213(98)00825-6 1999

  38. [38]

    Blanes, S., Casas, F., Oteo, J.A., Ros, J.: The Magnus expansion and some of its applications. Phys. Rept.470, 151–238 (2009) https://doi.org/10.1016/j.physrep. 2008.11.001 29

    doi:10.1016/j.physrep 2009