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arxiv: 2606.04571 · v2 · pith:MUHTBF36new · submitted 2026-06-03 · ✦ hep-ph

Rolling Down the Leptonic BSM Landscape Using Machine Learning Techniques

Alfredo Aranda , Raymundo Ramos , Alexander J. Stuart This is my paper

Pith reviewed 2026-06-28 05:50 UTC · model grok-4.3

classification ✦ hep-ph
keywords neutrino mass matrixtexturesmachine learningoptimizationbeyond standard modelleptonic sector
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The pith

Machine learning optimization can locate parameters realizing desired textures in the neutrino mass matrix.

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

The paper applies initialization and optimization methods drawn from machine learning to the leptonic sector of beyond-Standard-Model physics. A loss function is defined that measures how closely a neutrino mass matrix matches chosen textures or conditions. Running the optimization over the model's free parameters produces matrices that approximately match the targets together with the corresponding parameter values. The work presents this procedure as a way to explore such models and notes possible combinations with other artificial-intelligence techniques.

Core claim

We adapt and apply techniques from machine learning to the exploration of physics beyond the Standard Model in the leptonic sector. Namely, we employ initialization and optimization, as they are applied in machine learning, to minimize a loss function that describes textures or conditions which we want in the neutrino mass matrix. The model free parameters are explored during the optimization, and after training for a number of optimization steps, we obtain matrices that approximately follow the desired forms, as well as their corresponding optimized parameters.

What carries the argument

A loss function on target textures or conditions in the neutrino mass matrix, minimized by machine-learning-style optimization over the model's free parameters.

If this is right

  • Matrices obtained after optimization approximately follow the desired target forms.
  • The procedure returns the corresponding optimized parameter values for each model.
  • The same initialization-plus-optimization approach can be combined with other artificial-intelligence methods for further BSM applications.

Where Pith is reading between the lines

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

  • The method could be tested on large ensembles of randomly chosen textures to measure how often it recovers physically acceptable parameter sets.
  • Results can be cross-checked against global fits to neutrino oscillation data to assess whether the optimized points remain viable after experimental constraints.
  • Analogous loss-function constructions might be written for other sectors, such as charged-lepton masses or quark mixing matrices.

Load-bearing premise

Defining a loss function on target textures and running standard optimization will produce physically meaningful BSM solutions rather than numerical artifacts or parameter sets that violate other constraints.

What would settle it

Apply the procedure to a known, exactly solvable target texture, then check whether the output matrices and parameters reproduce that texture to within numerical tolerance while satisfying all other model consistency requirements; systematic failure would show the method does not reliably generate valid solutions.

Figures

Figures reproduced from arXiv: 2606.04571 by Alexander J. Stuart, Alfredo Aranda, Raymundo Ramos.

Figure 1
Figure 1. Figure 1: Flowchart of the optimization process used to obtain desirable forms of [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Distributions for the random test of sizes of [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Proportion of times that each pair of elements was choose for minimization in the loss [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Distributions for the neutrino oscillation parameters for specific pairs of elements that [PITH_FULL_IMAGE:figures/full_fig_p016_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Change in V R parameters after the minimization. The horizontal axis shows initial value, while the vertical axis shows the change in parameter value. 17 [PITH_FULL_IMAGE:figures/full_fig_p017_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Change in V R parameters after the minimization. The horizontal axis shows initial value, while the vertical axis shows the change in parameter value. 20 [PITH_FULL_IMAGE:figures/full_fig_p020_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Rate of found textures by number of zeros by using the random search described in [PITH_FULL_IMAGE:figures/full_fig_p023_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Sorted absolute values of elements for the optimized matrices using random search as [PITH_FULL_IMAGE:figures/full_fig_p025_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: The three most commonly found n-zero textures using the random search algorithm of Sec. 3.4.1 for n = 2 (top) and 3 (bottom). The counts above each figure represent the number of times the texture occurs in the surviving cases of [PITH_FULL_IMAGE:figures/full_fig_p026_9.png] view at source ↗
Figure 11
Figure 11. Figure 11: Sorted absolute values of elements for the optimized matrices using non-vanishing count [PITH_FULL_IMAGE:figures/full_fig_p029_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: The three most commonly found n-zero textures using the optimizer search algorithm of Sec. 3.4.1 for n = 2 (top) and 3 (middle). For n = 4, all cases found are consistent with the texture shown in the bottom. The counts above each figure represent the number of times the texture occurs in the surviving cases of [PITH_FULL_IMAGE:figures/full_fig_p032_12.png] view at source ↗
read the original abstract

In this work, we adapt and apply techniques from machine learning to the exploration of physics beyond the Standard Model in the leptonic sector. Namely, we employ initialization and optimization, as they are applied in machine learning, to minimize a loss function that describes textures or conditions which we want in the neutrino mass matrix. The model free parameters are explored during the optimization, and after training for a number of optimization steps, we obtain matrices that approximately follow the desired forms, as well as their corresponding optimized parameters. We also discuss extensions and additional applications of the ideas presented here in conjunction with other methods based on artificial intelligence.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript adapts machine learning initialization and optimization techniques to explore leptonic BSM models by minimizing a loss function that encodes target textures or conditions on the neutrino mass matrix. Model free parameters are varied during optimization; after a number of steps the procedure is reported to yield matrices that approximately match the desired forms together with their corresponding optimized parameters. Extensions combining the method with other AI techniques are briefly discussed.

Significance. If the outputs were shown to satisfy all relevant physical constraints (oscillation data, perturbativity, unitarity, absence of tachyons) in addition to texture matching, the approach could supply a practical numerical tool for scanning high-dimensional BSM parameter spaces. The core idea is a direct application of standard gradient-based or derivative-free optimization to a texture-defined objective; its value therefore hinges on demonstrating that the resulting points lie inside the physically allowed region rather than constituting numerical artifacts.

major comments (2)
  1. [Abstract, §2] Abstract and §2 (methodology): the loss function is stated to act only on target textures/conditions; no explicit functional form, weighting of individual entries, or regularization terms are supplied. Consequently it is impossible to assess whether the optimization can be guaranteed to respect measured Δm² values, mixing angles, or theoretical bounds such as perturbativity of Yukawa couplings.
  2. [Abstract, §3] Abstract and §3 (results): the claim that “matrices that approximately follow the desired forms” are obtained supplies neither convergence criteria, validation metrics against oscillation data, nor post-optimization filtering for unitarity or absence of tachyons. Without these, the reported parameter sets may satisfy the numerical loss while lying outside the experimentally allowed region.
minor comments (1)
  1. [§2] Notation for the neutrino mass matrix and the model parameters should be introduced with explicit definitions and dimensions before the loss is defined.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed report and constructive suggestions. The comments correctly identify areas where additional detail on the loss function and validation procedures would strengthen the presentation. We address each point below and will incorporate revisions to improve clarity and rigor.

read point-by-point responses

  1. Referee: [Abstract, §2] Abstract and §2 (methodology): the loss function is stated to act only on target textures/conditions; no explicit functional form, weighting of individual entries, or regularization terms are supplied. Consequently it is impossible to assess whether the optimization can be guaranteed to respect measured Δm² values, mixing angles, or theoretical bounds such as perturbativity of Yukawa couplings.

    Authors: The referee is correct that an explicit functional form, including weights and any regularization, is not supplied in the current text. The loss is described only at the level of penalizing deviations from target textures. We will revise §2 to provide the precise mathematical definition of the loss, the weighting scheme for matrix entries, and any additional terms used to enforce bounds. This revision will make it possible to evaluate how (or whether) physical constraints enter the optimization. revision: yes

  2. Referee: [Abstract, §3] Abstract and §3 (results): the claim that “matrices that approximately follow the desired forms” are obtained supplies neither convergence criteria, validation metrics against oscillation data, nor post-optimization filtering for unitarity or absence of tachyons. Without these, the reported parameter sets may satisfy the numerical loss while lying outside the experimentally allowed region.

    Authors: We agree that the results section does not report convergence criteria, direct comparison to oscillation data, or post-optimization checks for unitarity and absence of tachyons. The manuscript presents the method as a means to generate texture-matching candidates; it does not claim that the resulting points automatically satisfy all experimental or theoretical constraints. We will add explicit convergence diagnostics, validation metrics, and a discussion of subsequent filtering steps in the revised §3. revision: yes

Circularity Check

0 steps flagged

No significant circularity; standard optimization applied to texture-matching loss

full rationale

The paper describes initializing and optimizing model parameters to minimize a loss defined on target neutrino mass matrix textures, yielding matrices that approximately match those forms. This outcome follows directly from the optimization procedure but is presented as an exploratory numerical method rather than a first-principles derivation or prediction. No equations reduce outputs to inputs by construction, no self-citation chains support load-bearing claims, and no uniqueness theorems or ansatzes are imported. The approach is self-contained as an application of ML techniques, with independent content in the choice of loss and exploration of BSM parameter space.

Axiom & Free-Parameter Ledger

1 free parameters · 0 axioms · 0 invented entities

The central claim rests on the effectiveness of a user-defined loss function and standard optimizers for texture matching; no further free parameters, axioms, or invented entities are specified in the abstract.

free parameters (1)
  • model free parameters
    The abstract states that model free parameters are explored during optimization but provides no explicit list or values.

pith-pipeline@v0.9.1-grok · 5626 in / 1121 out tokens · 44155 ms · 2026-06-28T05:50:02.635110+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

47 extracted references · 19 linked inside Pith

  1. [1]

    Mann and H

    A.K. Mann and H. Primakoff,Neutrino Oscillations and the Number of Neutrino Types, Phys. Rev. D15(1977) 655

    1977

  2. [2]

    Wolfenstein,Oscillations Among Three Neutrino Types and CP Violation,Phys

    L. Wolfenstein,Oscillations Among Three Neutrino Types and CP Violation,Phys. Rev. D 18(1978) 958

    1978

  3. [3]

    Harrison, D.H

    P.F. Harrison, D.H. Perkins and W.G. Scott,Tri-bimaximal mixing and the neutrino oscillation data,Phys. Lett. B530(2002) 167 [hep-ph/0202074]

    Pith/arXiv arXiv 2002

  4. [4]

    Harrison and W.G

    P.F. Harrison and W.G. Scott,Symmetries and generalizations of tri - bimaximal neutrino mixing,Phys. Lett. B535(2002) 163 [hep-ph/0203209]

    Pith/arXiv arXiv 2002

  5. [5]

    Xing,Nearly tri bimaximal neutrino mixing and CP violation,Phys

    Z.-z. Xing,Nearly tri bimaximal neutrino mixing and CP violation,Phys. Lett. B533(2002) 85 [hep-ph/0204049]. 35

    Pith/arXiv arXiv 2002

  6. [6]

    He and A

    X.G. He and A. Zee,Some simple mixing and mass matrices for neutrinos,Phys. Lett. B 560(2003) 87 [hep-ph/0301092]

    Pith/arXiv arXiv 2003

  7. [7]

    Z.-z. Xing, H. Zhang and S. Zhou,Nearly Tri-bimaximal Neutrino Mixing and CP Violation from mu-tau Symmetry Breaking,Phys. Lett. B641(2006) 189 [hep-ph/0607091]. [9]Daya Baycollaboration,Observation of electron-antineutrino disappearance at Daya Bay, Phys. Rev. Lett.108(2012) 171803 [1203.1669]. [10]RENOcollaboration,Observation of Reactor Electron Antine...

    Pith/arXiv arXiv 2006

  8. [8]

    Matchev, K

    K.T. Matchev, K. Matcheva, P. Ramond and S. Verner,Seeking Truth and Beauty in Flavor Physics with Machine Learning, in37th Conference on Neural Information Processing Systems, 10, 2023 [2311.00087]

    arXiv 2023

  9. [9]

    Matchev, K

    K.T. Matchev, K. Matcheva, P. Ramond and S. Verner,Exploring the truth and beauty of theory landscapes with machine learning,Phys. Lett. B856(2024) 138941 [2401.11513]

    arXiv 2024

  10. [10]

    Nishimura, C

    S. Nishimura, C. Miyao and H. Otsuka,Exploring the flavor structure of quarks and leptons with reinforcement learning,JHEP23(2020) 021 [2304.14176]

    arXiv 2020

  11. [11]

    Nishimura, C

    S. Nishimura, C. Miyao and H. Otsuka,Reinforcement learning-based statistical search strategy for an axion model from flavor,JHEP10(2025) 043 [2409.10023]

    arXiv 2025

  12. [12]

    Nishimura, H

    S. Nishimura, H. Otsuka and H. Uchiyama,Exploring the flavor structure of leptons via diffusion models,Phys. Rev. D113(2026) 055030 [2503.21432]

    Pith/arXiv arXiv 2026

  13. [13]

    Nishimura, H

    S. Nishimura, H. Otsuka and H. Uchiyama,Diffusion-Model Approach to Flavor Models: A Case Study forS ′ 4 Modular Flavor Model,PTEP2026(2026) 053B08 [2504.00944]

    Pith/arXiv arXiv 2026

  14. [14]

    Giarnetti and D

    A. Giarnetti and D. Meloni,Reinforcement Learning Techniques for the Flavor Problem in Particle Physics,Symmetry18(2026) 131 [2510.25495]. 36 [19]Particle Data Groupcollaboration,Review of Particle Physics,Phys. Rev. D98(2018) 030001

    arXiv 2026

  15. [15]

    Jarlskog,Commutator of the Quark Mass Matrices in the Standard Electroweak Model and a Measure of Maximal CP Nonconservation,Phys

    C. Jarlskog,Commutator of the Quark Mass Matrices in the Standard Electroweak Model and a Measure of Maximal CP Nonconservation,Phys. Rev. Lett.55(1985) 1039

    1985

  16. [16]

    Wu,The Rephasing Invariants and CP,Phys

    D.-d. Wu,The Rephasing Invariants and CP,Phys. Rev. D33(1986) 860. [22]KATRINcollaboration,Direct neutrino-mass measurement based on 259 days of KATRIN data,Science388(2025) adq9592 [2406.13516]

    arXiv 1986

  17. [17]

    NuFIT v6.1 (2025)

    I. Esteban, M.C. Gonzalez-Garcia, M. Maltoni, I. Martinez-Soler, J.P. Pinheiro and T. Schwetz,NuFit-6.0: updated global analysis of three-flavor neutrino oscillations,JHEP12 (2024) 216 [2410.05380]. [24]NuFITcollaboration, “NuFIT v6.1 (2025).”http://www.nu-fit.org/, 2025

    Pith/arXiv arXiv 2024

  18. [18]

    Fukuyama and H

    T. Fukuyama and H. Nishiura,Mass matrix of Majorana neutrinos,hep-ph/9702253

    Pith/arXiv arXiv

  19. [19]

    Ma and M

    E. Ma and M. Raidal,Neutrino mass, muon anomalous magnetic moment, and lepton flavor nonconservation,Phys. Rev. Lett.87(2001) 011802 [hep-ph/0102255]

    Pith/arXiv arXiv 2001

  20. [20]

    Balaji, W

    K.R.S. Balaji, W. Grimus and T. Schwetz,The Solar LMA neutrino oscillation solution in the Zee model,Phys. Lett. B508(2001) 301 [hep-ph/0104035]

    Pith/arXiv arXiv 2001

  21. [21]

    Lam,A 2-3 symmetry in neutrino oscillations,Phys

    C.S. Lam,A 2-3 symmetry in neutrino oscillations,Phys. Lett. B507(2001) 214 [hep-ph/0104116]

    Pith/arXiv arXiv 2001

  22. [22]

    Harrison and W.G

    P.F. Harrison and W.G. Scott,µ-τreflection symmetry in lepton mixing and neutrino oscillations,Phys. Lett. B547(2002) 219 [hep-ph/0210197]

    Pith/arXiv arXiv 2002

  23. [23]

    Xing and Z.-h

    Z.-z. Xing and Z.-h. Zhao,A review ofµ-τflavor symmetry in neutrino physics,Rept. Prog. Phys.79(2016) 076201 [1512.04207]

    Pith/arXiv arXiv 2016

  24. [24]

    Kingma and J

    D.P. Kingma and J. Ba,Adam: A method for stochastic optimization,1412.6980

    Pith/arXiv arXiv

  25. [25]

    Fritzsch, Z.-z

    H. Fritzsch, Z.-z. Xing and S. Zhou,Two-zero Textures of the Majorana Neutrino Mass Matrix and Current Experimental Tests,JHEP09(2011) 083 [1108.4534]. 37

    Pith/arXiv arXiv 2011

  26. [26]

    Meloni, A

    D. Meloni, A. Meroni and E. Peinado,Two-zero Majorana textures in the light of the Planck results,Phys. Rev. D89(2014) 053009 [1401.3207]

    Pith/arXiv arXiv 2014

  27. [27]

    Nguyen,Texture zeros of neutrino mass matrix with seesaw mechanism and leptogenesis,Mod

    T.P. Nguyen,Texture zeros of neutrino mass matrix with seesaw mechanism and leptogenesis,Mod. Phys. Lett. A29(2014) 1450038

    2014

  28. [28]

    Benavides, D.V

    R.H. Benavides, D.V. Forero, L. Mu˜ noz, J.M. Mu˜ noz, A. Rico and A. Tapia,Five texture zeros in the lepton sector and neutrino oscillations at DUNE,Phys. Rev. D107(2023) 036008 [2207.04072]

    arXiv 2023

  29. [29]

    Ding, F.R

    G.-J. Ding, F.R. Joaquim and J.-N. Lu,Texture-zero patterns of lepton mass matrices from modular symmetry,JHEP03(2023) 141 [2211.08136]

    arXiv 2023

  30. [30]

    Devi,Retrieving texture zeros in 3+1 active-sterile neutrino framework under the action ofA 4 modular-invariants,2303.04900

    M.R. Devi,Retrieving texture zeros in 3+1 active-sterile neutrino framework under the action ofA 4 modular-invariants,2303.04900

    arXiv

  31. [31]

    Rico, R.H

    A. Rico, R.H. Benavides, D.V. Forero, L. Mu˜ noz and A. Tapia,Five texture zeros in the lepton sector that reproduce the observable neutrino masses and mixings: A mapping between the standard parameterization and our parameterization,PoSICRC2023(2023) 1047

    2023

  32. [32]

    Zhao,Combinations of theµ-τreflection symmetry and texture zeros in the Dirac neutrino mass matrix of the seesaw model,Eur

    Z.-h. Zhao,Combinations of theµ-τreflection symmetry and texture zeros in the Dirac neutrino mass matrix of the seesaw model,Eur. Phys. J. Plus138(2023) 1055

    2023

  33. [33]

    Kumar and R.R

    S. Kumar and R.R. Gautam,Neutrino mass matrices with generalized CP symmetries and texture zeros,Nucl. Phys. B1001(2024) 116520 [2312.07150]

    arXiv 2024

  34. [34]

    Zhou,Texture zeros for lepton flavor mixing,Int

    S. Zhou,Texture zeros for lepton flavor mixing,Int. J. Mod. Phys. A39(2024) 2441014

    2024

  35. [35]

    Nomura, H

    T. Nomura, H. Okada and H. Otsuka,Texture zeros realization in a three-loop radiative neutrino mass model from modular A4 symmetry,Nucl. Phys. B1004(2024) 116579 [2309.13921]

    arXiv 2024

  36. [36]

    Carrillo-Monteverde, S

    A. Carrillo-Monteverde, S. G´ omez-´Avila and L. L´ opez-Lozano,2-zeros texture and the Universal Texture Constraint in the Leptonic Sector,Int. J. Mod. Phys. A39(2024) 2450052

    2024

  37. [37]

    Singh, M

    L. Singh, M. Kashav and S. Verma,Implications of Dark-θ 12 Solution on Two-zero Texture Inverse Neutrino Mass Matrix,Springer Proc. Phys.304(2024) 675. 38

    2024

  38. [38]

    Mazumder and R

    I.A. Mazumder and R. Dutta,Bottom-up approach to texture zeros in the neutrino mass matrix,Int. J. Mod. Phys. A41(2026) 2650010 [2409.04756]

    arXiv 2026

  39. [39]

    Kobayashi, H

    T. Kobayashi, H. Otsuka and M. Tanimoto,Yukawa textures from non-invertible symmetries, JHEP12(2024) 117 [2409.05270]

    arXiv 2024

  40. [40]

    Kobayashi, Y

    T. Kobayashi, Y. Nishioka, H. Otsuka and M. Tanimoto,More about quark Yukawa textures from selection rules without group actions,JHEP05(2025) 177 [2503.09966]

    arXiv 2025

  41. [41]

    Kobayashi, H

    T. Kobayashi, H. Otsuka, M. Tanimoto and H. Uchida,Lepton mass textures from non-invertible multiplication rules,JHEP08(2025) 189 [2505.07262]

    arXiv 2025

  42. [42]

    Calibbi, X

    L. Calibbi, X. Gao and M. Yuan,Hunting for neutrino texture zeros with muon and tau flavor violation,JHEP05(2026) 188 [2511.08679]

    Pith/arXiv arXiv 2026

  43. [43]

    Biswas, S

    A. Biswas, S. Jangid and S.C. Park,Investigating two-zero texture in the light of gauged Type-II seesaw,Phys. Lett. B868(2025) 139786 [2504.10312]

    arXiv 2025

  44. [44]

    Gogoi and M.K

    J. Gogoi and M.K. Das,A study of texture zeros using A4 discrete symmetry group,J. Subatomic Part. Cosmol.3(2025) 100071

    2025

  45. [45]

    Jiang, B.-Y

    Z. Jiang, B.-Y. Qu and G.-J. Ding,Texture-zeros in minimal seesaw from noninvertible symmetry fusion rules,Phys. Rev. D112(2025) 115029 [2510.07236]

    arXiv 2025

  46. [46]

    Borah, P

    D. Borah, P. Das and D. Dutta,Neutrino texture-zeros after JUNO’s first results: Implications for long-baseline neutrino experiments,2512.13587

    arXiv

  47. [47]

    Kumar, L

    Priya, R. Kumar, L. Singh and S. Verma,Implications of the First JUNO Results for Dirac Neutrino Texture Zeros,2604.19122. 39

    Pith/arXiv arXiv