pith. sign in

arxiv: 2606.28224 · v1 · pith:HJ7SGUACnew · submitted 2026-06-26 · ⚛️ nucl-th · hep-ph

Efficient calculation of two-neutrino double-beta-decay nuclear matrix elements

Mihai Horoi This is my paper

Pith reviewed 2026-06-29 02:01 UTC · model grok-4.3

classification ⚛️ nucl-th hep-ph
keywords two-neutrino double beta decaynuclear matrix elementsLanczos methodstrength functionshell modeldouble beta decaynuclear structure
0
0 comments X

The pith

Lanczos-based strength functions enable efficient computation of two-neutrino double-beta decay nuclear matrix elements without full diagonalization.

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

The paper develops a method to calculate nuclear matrix elements for two-neutrino double-beta decay more efficiently. Direct summation over intermediate states is computationally expensive in large model spaces, so the authors use Lanczos iterations to generate strength functions that approximate the sum. This preserves accuracy in cases where full calculations are possible for comparison. The technique is tested on important nuclei and different Hamiltonians, and it extends to higher-order matrix elements needed for phase-space factors. Accurate 2νββ values help interpret correlations with neutrinoless decay modes relevant to neutrino physics.

Core claim

The central discovery is an improved strength-function method based on Lanczos iterations for two-neutrino double-beta-decay nuclear matrix elements. It avoids full diagonalization of the intermediate odd-odd nucleus while preserving the accuracy of explicit summation over 1+ states where benchmarks are possible. The method applies to several experimentally important emitters using different effective Hamiltonians and extends to higher-order NMEs for Taylor-expanded phase-space treatments.

What carries the argument

Lanczos iterations to generate the strength function approximating the sum over intermediate 1+ states in the odd-odd nucleus.

If this is right

  • Calculations become feasible in larger model spaces for 2νββ NMEs.
  • The same framework handles higher-order NMEs for 2νββ phase-space treatments.
  • Results can inform assessments of 0νββ NMEs due to their indicated correlation with 2νββ values.
  • The approach works across multiple experimentally important emitters and different effective Hamiltonians.

Where Pith is reading between the lines

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

  • This could reduce computational barriers for exploring larger valence spaces in shell-model calculations of double-beta decay.
  • The approach might generalize to other weak processes involving sums over intermediate states.
  • If accurate, it supports using 2νββ data to constrain models for 0νββ predictions.

Load-bearing premise

The strength function obtained from Lanczos iterations converges to the exact sum over all 1+ intermediate states with only controllable truncation error.

What would settle it

Performing a full diagonalization in a small model space for a nucleus where the Lanczos method gives a different NME value would falsify the preservation of accuracy.

Figures

Figures reproduced from arXiv: 2606.28224 by Mihai Horoi.

Figure 2
Figure 2. Figure 2: Running M2ν for 82Se using the GCN2850 effective Hamiltonian. The direct sum over 250 states is compared with the strength-function result obtained with 25 Lanczos itera￾tions. The smooth approach to the plateau indicates that the low-energy part of the 82Br 1+ spectrum dominates the final value, with the higher-energy contribution mainly stabilizing the sum. over 250 states for the GCN2850 Hamiltonian. Th… view at source ↗
Figure 1
Figure 1. Figure 1: Running M2ν [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 4
Figure 4. Figure 4: GT strength distributions entering the SVD calcu [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Running M2ν for 76Ge: (a) GCN2850 effective Hamiltonian; (b) JUN45 effective Hamiltonian. The direct sum over 2000 1+ states is replaced with the strength-function result after 25 Lanczos iterations. The two Hamiltonians give somewhat different running patterns, but in both cases the Lanczos result follows the explicit sum to the final plateau. M2ν−3 = X k ⟨0 + f |qστ −|1 + k ⟩⟨1 + k |qστ −|0 + i ⟩ (Erel,1… view at source ↗
Figure 7
Figure 7. Figure 7: Additional running-sum tests: (a) 128Te with the SVD effective Hamiltonian; (b) 82Se with the JUN45 effec￾tive Hamiltonian. These cases complement Figs. 2 and 6 by checking the same algorithm for a different isotope in the Te region and a different Hamiltonian in the A ≈ 80 region. vergence can be monitored iteration by iteration using quantities already produced by the algorithm. Benchmark calculations sh… view at source ↗
read the original abstract

Reliable nuclear matrix elements (NMEs) are essential for interpreting double-beta-decay experiments and for connecting measured or constrained half-lives to the underlying weak-interaction physics. The two-neutrino mode ($2\nu\beta\beta$) is allowed by the Standard Model and has been observed in several nuclei, whereas the neutrinoless mode ($0\nu\beta\beta$) remains the key experimental signature of lepton-number violation and Majorana neutrino masses. Recent statistical shell-model studies indicate a strong correlation between the $2\nu\beta\beta$ and $0\nu\beta\beta$ NMEs, making accurate and efficient calculations of the former especially useful for assessing the latter. Direct evaluations of $2\nu\beta\beta$ NMEs usually require summing over many $1^+$ states in the intermediate odd-odd nucleus, a procedure that becomes expensive and may converge slowly in large model spaces. We present and test an improved strength-function method based on Lanczos iterations that avoids full diagonalization while preserving the accuracy of explicit summation where such benchmarks are possible. The method is applied to several experimentally important emitters and to different effective Hamiltonians. We also show that the same framework can be used for the higher-order NMEs entering Taylor-expanded phase-space treatments of $2\nu\beta\beta$ and related decay modes.

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

0 major / 2 minor

Summary. The manuscript presents and tests an improved strength-function method based on Lanczos iterations for computing two-neutrino double-beta-decay (2νββ) nuclear matrix elements (NMEs). The approach avoids full diagonalization of the intermediate odd-odd nucleus Hamiltonian while claiming to preserve the accuracy of explicit summation over intermediate 1+ states wherever such benchmarks are computationally feasible. The method is applied to several experimentally relevant emitters using different effective Hamiltonians and is extended to higher-order NMEs relevant for Taylor-expanded phase-space treatments of 2νββ and related modes.

Significance. If the central claim of accuracy preservation holds with controllable truncation error, the work would provide a practically useful numerical tool for 2νββ NME calculations in large model spaces. This is relevant because recent statistical shell-model studies indicate correlations between 2νββ and 0νββ NMEs, and efficient 2νββ computations can help constrain interpretations of double-beta-decay experiments. The method builds on a standard, well-studied Lanczos technique for resolvent evaluation rather than introducing new ad-hoc parameters or entities, and the explicit limitation of the accuracy claim to benchmarkable cases is appropriately cautious.

minor comments (2)
  1. [Abstract] Abstract: the assertion that accuracy is preserved 'where such benchmarks are possible' would be strengthened by including at least one quantitative comparison (e.g., explicit-sum vs. Lanczos NME values and convergence metrics) already in the abstract or a dedicated early results paragraph.
  2. The manuscript should specify the Lanczos iteration count or Krylov-subspace dimension used for each nucleus/Hamiltonian and report the associated truncation error estimate relative to the explicit sum, even if only for the benchmark cases.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the constructive and positive assessment of our manuscript. The recommendation for minor revision is noted, and we are happy to incorporate any specific suggestions. No major comments were listed in the report, so we have no points requiring detailed rebuttal at this stage.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper describes a Lanczos-based strength-function method for 2νββ NMEs that is validated by direct comparison to explicit summation over 1+ states on the same nuclei and Hamiltonians where the latter is feasible. This is a standard numerical convergence test for resolvent techniques and does not reduce any reported result to a fitted parameter, self-definition, or self-citation chain. The abstract and skeptic analysis give no equations or claims that equate a prediction to its own input by construction, so the derivation chain remains independent of the target quantities.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper introduces a computational technique rather than new physical parameters or entities. It relies on the standard shell-model framework and effective Hamiltonians already used in the field.

axioms (1)
  • domain assumption The nuclear shell model with effective Hamiltonians provides a sufficiently accurate description of the low-lying states relevant to 2νββ decay in the nuclei considered.
    The method operates inside the shell-model space; its validity inherits from this established modeling choice.

pith-pipeline@v0.9.1-grok · 5752 in / 1384 out tokens · 29836 ms · 2026-06-29T02:01:16.047763+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

40 extracted references · 1 canonical work pages

  1. [1]

    F. T. Avignone, III, S. R. Elliott, and J. Engel, Rev. Mod. Phys.80, 481 (2008)

    2008

  2. [2]

    J. D. Vergados, H. Ejiri, and F. Simkovic, Rep. Prog. Phys.75, 106301 (2012)

    2012

  3. [3]

    M. Doi, T. Kotani, H. Nishiura, and E. Takasugi, Progr. Theor. Exp. Phys.69, 602 (1983)

    1983

  4. [4]

    M. Doi, T. Kotani, and E. Takasugi, Prog. Theor. Phys. Suppl.83, 1 (1985)

    1985

  5. [5]

    Suhonen and O

    J. Suhonen and O. Civitarese, Phys. Rep.300, 123 (1998)

    1998

  6. [6]

    Kotila and F

    J. Kotila and F. Iachello, Phys. Rev. C85, 034316 (2012)

    2012

  7. [7]

    Stoica and M

    S. Stoica and M. Mirea, Phys. Rev. C88, 037303 (2013)

    2013

  8. [8]

    Engel and J

    J. Engel and J. Men´ endez, Reports on Progress in Physics 80, 046301 (2017)

    2017

  9. [9]

    Adams, K

    C. Adams, K. Alfonso, C. Andreoiu, E. Angelico, I. J. Arnquist, J. A. A. Asaadi, F. T. Avignone, S. N. Axani, A. S. Barabash, P. S. Barbeau, L. Baudis, F. Bellini, M. Beretta, T. Bhatta, V. Biancacci, M. Biassoni, E. Bossio, P. A. Breur, J. P. Brodsky, C. Brofferio, E. Brown, R. Brugnera, T. Brunner, N. Burlac, E. Ca- den, S. Calgaro, G. F. Cao, L. Cao, C...

    work page doi:10.48550/arxiv.2212.11099 2022

  10. [10]

    Rodejohann, J

    W. Rodejohann, J. Phys. G39, 124008 (2012)

    2012

  11. [11]

    F. F. Deppisch, M. Hirsch, and H. Pas, J. Phys. G39, 124007 (2012)

    2012

  12. [12]

    Horoi and A

    M. Horoi and A. Neacsu, Phys. Rev. C98, 035502 (2018)

    2018

  13. [13]

    Caurier, A

    E. Caurier, A. Poves, and A. P. Zuker, Phys. Lett. B252, 13 (1990)

    1990

  14. [14]

    Caurier, G

    E. Caurier, G. Martinez-Pinedo, F. Nowacki, A. Poves, and A. P. Zuker, Rev. Mod. Phys.77, 427 (2005)

    2005

  15. [15]

    Horoi, S

    M. Horoi, S. Stoica, and B. A. Brown, Phys. Rev. C75, 034303 (2007)

    2007

  16. [16]

    Simkovic, G

    F. Simkovic, G. Pantis, J. D. Vergados, and A. Faessler, Phys. Rev. C60, 055502 (1999)

    1999

  17. [17]

    Barea and F

    J. Barea and F. Iachello, Phys. Rev. C79, 044301 (2009)

    2009

  18. [18]

    T. R. Rodriguez and G. Martinez-Pinedo, Phys. Rev. Lett.105, 252503 (2010)

    2010

  19. [19]

    J. M. Yao, B. Bally, J. Engel, R. Wirth, T. R. Rodr´ ıguez, and H. Hergert, Phys. Rev. Lett.124, 232501 (2020)

    2020

  20. [20]

    Novario, P

    S. Novario, P. Gysbers, J. Engel, G. Hagen, G. R. Jansen, T. D. Morris, P. Navr´ atil, T. Papenbrock, and S. Quaglioni, Phys. Rev. Lett.126, 182502 (2021)

    2021

  21. [21]

    Belley, C

    A. Belley, C. G. Payne, S. R. Stroberg, T. Miyagi, and J. D. Holt, Phys. Rev. Lett.126, 042502 (2021)

    2021

  22. [22]

    Patel, P

    D. Patel, P. C. Srivastava, and J. Suhonen, Phys. Rev. C 110, 054323 (2024)

    2024

  23. [23]

    Horoi, A

    M. Horoi, A. Neacsu, and S. Stoica, Phys. Rev. C106, 054302 (2022)

    2022

  24. [24]

    Engel, H

    J. Engel, H. Haxton, , and P. Vogel, Phys. Rev. C46, 2153(R) (1992)

    1992

  25. [25]

    Horoi, Physics4, 1135 (2022)

    M. Horoi, Physics4, 1135 (2022)

    2022

  26. [26]

    Nit ¸escu, S

    O. Nit ¸escu, S. Ghinescu, V. A. Sevestrean, M. Horoi, F. ˇSimkovic, and S. Stoica, Journal of Physics G: Nuclear and Particle Physics51, 125103 (2024)

    2024

  27. [27]

    Barabash, UNIVERSE6, 159 (2020)

    A. Barabash, UNIVERSE6, 159 (2020)

    2020

  28. [28]

    Tomoda, Rep

    T. Tomoda, Rep. Prog. Phys.54, 53 (1991)

    1991

  29. [29]

    NuShellX@MSU,https://people.nscl.msu.edu/ ~brown/resources/resources.html(2017)

    2017

  30. [30]

    Neacsu and M

    A. Neacsu and M. Horoi, Phys. Rev. C91, 024309 (2015)

    2015

  31. [31]

    Horoi and A

    M. Horoi and A. Neacsu, Phys. Rev. C93, 024308 (2016)

    2016

  32. [32]

    R. A. Sen’kov, M. Horoi, and B. A. Brown, Phys. Rev. C89, 054304 (2014)

    2014

  33. [33]

    R. A. Sen’kov and M. Horoi, Phys. Rev. C93, 044334 (2016)

    2016

  34. [34]

    Honma, T

    M. Honma, T. Otsuka, B. A. Brown, and T. Mizusaki, Phys. Rev. C69, 034335 (2004)

    2004

  35. [35]

    Honma, T

    M. Honma, T. Otsuka, B. A. Brown, and T. Mizusaki, 9 Eur. Phys. J. A25 Suppl. 1, 499 (2005)

    2005

  36. [36]

    Poves, J

    A. Poves, J. S´ anchez-Solano, E. Caurier, and F. Nowacki, Nuclear Physics A694, 157 (2001)

    2001

  37. [37]

    Honma, T

    M. Honma, T. Otsuka, T. Mizusaki, and M. Hjorth- Jensen, Phys. Rev. C80, 064323 (2009)

    2009

  38. [38]

    Menendez, A

    J. Menendez, A. Poves, E. Caurier, and F. Nowacki, Nucl. Phys. A818, 139 (2009)

    2009

  39. [39]

    Caurier, F

    E. Caurier, F. Nowacki, A. Poves, and K. Sieja, Phys. Rev. C82, 064304 (2010)

    2010

  40. [40]

    Qi and Z

    C. Qi and Z. X. Xu, Phys. Rev. C86, 044323 (2012)

    2012