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REVIEW 4 major objections 5 minor 23 references

QCD sum rule approach to Okamoto-Nolen-Schiffer anomaly

T0 review · 4 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read This paper derives the strength of the nuclear charge-symmetry-breaking interaction from QCD sum rules and shows that it accounts for the mirror-nucleus energy gap known as the Okamoto-Nolen-Schiffer anomaly.

desk verdict QCD-derived CSB Skyrme force is a real step and not fitted to ONS data, but the paper's central claim rests on an unquantified low-density-to-saturation extrapolation. read the letter →

arxiv 2412.19851 v1 pith:AQLIPC5F submitted 2024-12-25 nucl-th nucl-ex

Hiroyuki Sagawa , Tomoya Naito , Xavier Roca-Maza , Tetsuo Hatsuda This is my paper · ORCID
classification nucl-thnucl-ex
keywords Okamoto-Nolen-SchifferanomalychargesymmetrybreakingQCDsumrulesSkyrmeenergydensityfunctionalchiralcondensatemirrornucleiisospinpion-nucleonsigmaterm
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

The paper's central claim is that the Okamoto-Nolen-Schiffer (ONS) anomaly, the shortfall of 3–9% in calculated Coulomb energy differences of mirror nuclei, can be explained by a charge-symmetry-breaking (CSB) interaction whose parameters are fixed by low-energy constants of QCD rather than by fitting mirror-nucleus data. It matches a Skyrme-type contact-plus-momentum-dependent CSB force to the density dependence of the in-medium chiral condensate given by QCD sum rules, obtaining the strengths from the quark-mass-difference constant $C_1$ and the pion-nucleon $\sigma$ term $\sigma_{\pi N}$. A sympathetic reader should care because this turns a phenomenological correction with poorly determined sign and magnitude into a parameter-free prediction tied to spontaneous chiral symmetry breaking. In Hartree-Fock calculations with the SGII and SAMi energy density functionals, the resulting CSB force together with small extra corrections fills the 150–400 keV gap between Coulomb Hartree-Fock energies and experimental mirror energy differences for $A=16\pm1$ and $40\pm1$. This is what the authors mean by resolving the anomaly within theoretical uncertainties.

What carries the argument

The load-bearing identity is the QCD-sum-rule formula $\Delta_{np}(\rho) \simeq C_1 G(\rho) - C_2$ with $G(\rho) = (\langle \bar q q\rangle_\rho/\langle \bar q q\rangle_0)^{1/3}$, where the density-independent $C_2$ drops out and the CSB effect is $C_1[1-G(\rho)]$, growing as the chiral condensate is partially restored. The paper expands this to $O(\rho^{5/3})$ and matches it coefficient-by-coefficient to the density expansion of the Skyrme-type CSB interaction, which fixes the contact strength $\tilde{s}_0$ and the combination $\tilde{s}_1+3\tilde{s}_2$ in terms of $C_1$ and $\sigma_{\pi N}$ alone. This matching identity is what carries the argument from QCD low-energy constants to a finite-nucleus energy density functional.

What would settle it

Compute the four mirror-nucleus gaps with the same QCD-matched CSB interaction but replace the $O(\rho^{5/3})$ density expansion by the full condensate ratio $G(\rho)$; if the results shift by more than the quoted theoretical uncertainties, the low-density matching does not control the finite-nucleus calculation and the claimed resolution of the anomaly is not established.

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Extended reading notes

Core claim

The discovery is that the missing repulsion in mirror-nucleus Coulomb energies can be traced to partial restoration of chiral symmetry in the nuclear medium. The QCD-sum-rule result $\Delta_{np}(\rho) \simeq C_1 G(\rho) - C_2$ with $G(\rho)=(\langle \bar q q\rangle_\rho/\langle \bar q q\rangle_0)^{1/3}$ gives the density-dependent part of the neutron-proton self-energy difference from the quark mass difference and the in-medium chiral condensate. The authors identify this with the Skyrme-type CSB energy density functional, expand both sides to order $\rho^{5/3}$, and match coefficients to obtain $\tilde{s}_0 = -(4/3) C_1 \sigma_{\pi N}/(f_\pi^2 m_\pi^2)$ and $\tilde{s}_1+3\tilde{s}_2 = (1/m_N^2) C_1 \sigma_{\pi N}/(f_\pi^2 m_\pi^2)$. With these QCD-constrained strengths, Hartree-Fock mirror-energy differences for $^{17}$F-$^{17}$O, $^{15}$O-$^{15}$N, $^{41}$Sc-$^{41}$Ca, and $^{39}$Ca-$^{39}$K agree with experiment in sign and magnitude once the Coulomb contribution and the small extra corrections are added, with little difference between the SGII and SAMi functionals.

Load-bearing premise

The derivation assumes the low-density QCD-sum-rule formula $\Delta_{np}(\rho) \simeq C_1 G(\rho) - C_2$ and its $O(\rho^{5/3})$ truncation remain accurate in the finite nucleus, where local densities can approach the saturation density $\rho_0 = 0.17\,\mathrm{fm}^{-3}$, even though the formula is only stated to be valid for $\rho < \rho_0$.

Editorial extensions

If this is right

  • The CSB part of nuclear energy density functionals no longer needs to be fitted to mirror-nucleus data; its leading density terms are fixed by $C_1$ and $\sigma_{\pi N}$, removing a major ambiguity in phenomenological CSB forces.
  • The same QCD-matched strengths give parameter-free predictions for other isospin-sensitive observables, such as isobaric analogue states and mirror-pair charge radii, which can be checked without adjustment.
  • Because the correction is tied to partial chiral restoration, improving the determination of the pion-nucleon sigma term directly sharpens the predicted ONS gap.
  • The small difference between the SGII and SAMi results indicates that, for these mass regions, the Coulomb Hartree-Fock contribution is not the main source of uncertainty in the anomaly.

Reading between the lines

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

  • The same coefficient-matching procedure could be extended to charge-independence breaking by using the charged-neutral pion mass difference, which the authors list as their next step; a testable extension is whether such a CIB term, added to the QCD-derived CSB force, improves isobaric multiplet mass equations in $N>Z$ nuclei.
  • A direct numerical probe of the low-density assumption would be to evaluate the chiral condensate ratio at the local densities reached inside the Hartree-Fock wavefunctions and compare it with the truncated expansion used in the matching; this would show whether the $O(\rho^{5/3})$ truncation is controlling the finite-nucleus result.
  • The QCD-derived sign and magnitude of the contact strength $\tilde{s}_0$ may also constrain phenomenological CSB parameter sets, since those sets are otherwise hard to pin down due to cancellations between the contact and momentum-dependent terms.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The manuscript proposes a QCD-based derivation of the Skyrme-type charge symmetry breaking (CSB) interaction and applies it to the Okamoto-Nolen-Schiffer (ONS) anomaly. Starting from the Hatsuda-Høgaasen-Prakash expression for the neutron-proton self-energy difference in symmetric nuclear matter, Eq. (2), and the chiral-condensate expansion Eq. (3), the authors match the low-density form of the chiral CSB contribution, Eq. (5), with the Skyrme CSB contribution, Eq. (8), to determine the contact strength s̃0 and the momentum-dependent combination s̃1+3s̃2 in Eq. (10) in terms of the QCD constants C1 and σπN. These strengths are then used in Hartree-Fock calculations of mirror-nucleus energy differences for A=16±1 and A=40±1, with Coulomb and 'Extra' corrections added, and the results are compared with experimental ONS gaps in Fig. 2. The paper concludes that the QCD-based CSB interaction resolves the anomaly within theoretical uncertainties.

Significance. If the extrapolation from low-density nuclear matter to finite nuclei can be controlled, this is a valuable step: it provides a microscopically motivated, QCD-anchored origin for a phenomenological CSB interaction and fixes its coupling strengths without fitting to the ONS data themselves. The matching algebra in Eq. (10) is transparent and consistent with the preceding equations, and the mirror-nucleus comparison acts as an ex post consistency check rather than a fit. The paper is also explicit about the distinction between the contact and momentum-dependent terms, and it identifies an unconstrained combination (s̃1 vs s̃2) that will matter in finite nuclei. The main limitations are the unquantified use of a low-density expansion at nuclear-matter densities and the lack of a stated value for σπN, both of which are directly load-bearing for the claimed quantitative agreement.

major comments (4)
  1. [Sec. 3, Eq. (2) and Fig. 2] The manuscript states that Eq. (2) and the O(ρ^{5/3}) expansion of the chiral condensate are valid for ρ < ρ0, but the Skyrme parameters obtained from matching these low-density expressions are then used in Hartree-Fock calculations of finite nuclei whose local densities reach ρ0. At ρ = ρ0 the next term in the expansion of [1 + k1(ρ/ρ0) + k2(ρ/ρ0)^{5/3}]^{1/3} contributes roughly 0.1 MeV, which is a substantial fraction of the 150–400 keV ONS gaps shown in Fig. 2, and no estimate is given for the corresponding truncation error or for higher-dimensional condensate contributions to the QCD sum rule at these densities. This unquantified extrapolation is load-bearing for the abstract's claim that the anomaly is resolved within theoretical uncertainties.
  2. [Sec. 3, Eq. (10) and Table 2] The central values of s̃0 and s̃1+3s̃2 in Table 2 depend directly on the pion-nucleon sigma term σπN, but no numerical value for σπN is given in the manuscript. Without this input, the reader cannot reproduce Eq. (10) or Table 2, and it is unclear whether the quoted uncertainties already include the uncertainty in σπN. Please state the adopted value, its source, and whether the error bars in Fig. 2 propagate the σπN uncertainty.
  3. [Sec. 3, Eqs. (8)–(10), Table 2 and Fig. 2] The uniform-matter matching fixes only the combination s̃1+3s̃2; in finite nuclei s̃1 and s̃2 contribute independently, and the two cases in Table 2 are introduced ad hoc without further constraint. Since Fig. 2 shows only Case I, the apparent agreement with experiment could partly reflect the unconstrained choice of the momentum-dependent split. Please show results for both Case I and Case II, or quantify the sensitivity of the final ONS gaps to the s̃1/s̃2 decomposition.
  4. [Sec. 3, Eq. (5)] The identification of the finite-nucleus ONS anomaly δONS with the uniform-matter quantity δchiral = Δnp(0) − Δnp(ρ) is stated in Eq. (5) without derivation. A finite nucleus is non-uniform, has shell structure, and has surface and shell-correction effects that are absent in this identification. A justification of this identification, or at least an estimate of the error it introduces, is needed before the agreement in Fig. 2 can be attributed to the QCD-based CSB mechanism.
minor comments (5)
  1. [Table 1] The orbital notation '1d5/2 1p1/2^-1' is ambiguous; please use a clearer convention, for example '1d5/2 (or 1p1/2^-1)', to indicate the valence orbital for each mirror pair.
  2. [Fig. 2] The figure caption does not explain what the error bars on the red bars represent. Please state explicitly whether they include only the C1 uncertainty in Table 2, or also the σπN uncertainty and the s̃1/s̃2 split uncertainty.
  3. [Sec. 2, list item 4] The statement 'phenomenological CSB and CIB interactions ... strengths can be determined by optimizing systematically empirical binding energy differences' would benefit from a specific citation to the fitting procedure, as the later text already references literature on this point.
  4. [General] Reference [17] is the published version of the present proceedings contribution; the text should note this explicitly to avoid any confusion about duplicate publication.
  5. [Abstract] The abstract states that the interaction 'effectively resolves the anomaly', but the numerical evidence is presented for four specific mirror pairs. Consider softening the wording to 'the four studied mirror pairs' unless additional systems or a broader systematic study are included.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular reduction: the CSB EDF parameters are fixed by QCD inputs (C1, sigma_piN), and the ONS gaps are an ex post check; the noted low-density extrapolation is a correctness risk, not circularity.

full rationale

The derivation chain is not circular. Equation (10) is obtained by matching the uniform-matter Skyrme expression delta_Skyrme(rho) (Eq. 8) to the QCD-sum-rule expression delta_chiral(rho) (Eq. 5) at low density. delta_chiral is itself defined from the in-medium chiral condensate and the constants C1 and sigma_piN; none of these is fitted to the mirror-nucleus gaps that constitute the ONS test. The subsequent finite-nucleus Hartree-Fock results in Fig. 2 are therefore a genuine ex post comparison, not a repatriation of the input. The main QCD input C1 is quoted from Ref. [19], co-authored by Hatsuda, and sigma_piN enters through the standard in-medium condensate expansion (Refs. [21,22]); these are external, parameter-independent calculations rather than fits to the target data, so citing them is not circular. The paper does flag that Eq. (2) is valid only for rho < rho0 (Sec. 3), yet the finite-nucleus HF runs sample densities near rho0; the missing O(rho^2) truncation-error estimate is a legitimate correctness concern, but it is an extrapolation issue, not a self-referential reduction. Similarly, the presentation of only Case I in Fig. 2 reflects an unconstrained s1/s2 split, but that is model uncertainty, not circularity. Score 2 reflects the self-citation lineage of the numerical input while recognizing the central finite-nucleus result has independent content.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

No ONS data are used to adjust the constants; the principal input is an external QSR constant and a literature density dependence. The only hand-chosen freedom is the s1/s2 split, and the sigma term value is unstated.

free parameters (1)
  • s1/s2 splitting = Case I: s1=0.52, s2=0; Case II: s1=0, s2=0.18 MeV fm5
    Matching determines only s1+3s2 (Table 2); the division between the s-wave and p-wave momentum terms is chosen by hand in two cases, so the finite-nucleus result carries an unconstrained modeling degree of freedom.
assumptions (5)
  • domain assumption QCD sum rule relation Eq. (2): Delta_np(rho) ~ C1 G(rho) - C2, with C1 = -a gamma and C2 density-independent.
    The central link between QCD and the EDF is taken from ref [19] without derivation here; the cancellation of C2 is assumed exact.
  • domain assumption In-medium chiral condensate Eq. (3): <qq>/<qq>0 ~ 1 + k1 rho/rho0 + k2 (rho/rho0)^{5/3} with k1, k2 from refs [21,22].
    The density dependence of G underlying Eq. (5) is adopted from the literature and truncated at O(rho^{5/3}).
  • domain assumption The Skyrme-type CSB interaction Eq. (6) and its infinite-matter expression Eq. (8) are a complete representation of the QSR self-energy effect.
    Assumes a zero-range two-body force with contact and momentum-dependent terms can represent the density-dependent single-nucleon self-energy difference.
  • ad hoc to paper Identification of the finite-nucleus ONS anomaly with the uniform-matter quantity delta_chiral = Delta_np(0)-Delta_np(rho) (Eqs. 4 and 5).
    This bridges the infinite-matter QSR calculation to the finite-nucleus EDF application; no formal derivation of this identification is given in the proceedings.
  • domain assumption Numerical inputs C1 and sigma_piN are reliable and their uncertainties are propagated.
    C1=5.24 MeV is quoted from ref [19], but the sigma term used in Eq. (10) is never stated in the paper, so the Table 2 error bars are not fully auditable.

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Pith. "Pith review of QCD sum rule approach to Okamoto-Nolen-Schiffer anomaly." pith.science (2026). https://pith.science/paper/AQLIPC5F

@misc{pith2026241219851,
  author       = {Pith},
  title        = {Pith review of: QCD sum rule approach to Okamoto-Nolen-Schiffer anomaly},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AQLIPC5F}},
  note         = {Machine review of arXiv:2412.19851}
}
abstract

A new framework is introduced to connect between a charge symmetry breaking (CSB) energy density functional (EDF) and the low-energy constants derived from quantum chromodynamics (QCD). By constructing a QCD-based CSB EDF, this method provides new insights into the Okamoto-Nolen-Schiffer anomaly, a long-standing puzzle in the energy differences of mirror nuclei that lacks a robust microscopic explanation. Using examples such as $ {}^{17} \mathrm{F} $-$ {}^{17} \mathrm{O} $, $ {}^{15} \mathrm{O} $-$ {}^{15} \mathrm{N} $, $ {}^{41} \mathrm{Sc} $-$ {}^{41} \mathrm{Ca} $, and $ {}^{39} \mathrm{Ca} $-$ {}^{39} \mathrm{K} $, we demonstrate that the proposed interaction effectively resolves the anomaly within the range of theoretical uncertainties.

Figures

Figures reproduced from arXiv: 2412.19851 by the authors.

Figure 1
Figure 1. Various QCD approaches for hadron dynamics and quantum many-body problems. See the text for details. 3. CSB EDF from the QCD sum rule and its application First, we will discuss the energy difference between the neutron and the proton Δ𝑛 𝑝 (𝜌) in symmetric nuclear matter (𝑁 = 𝑍) with the baryon density 𝜌 plus one neutron or proton, as defined by a difference of the momentum-independent part of the Lorentz-scalar self… view at source ↗
Figure 2
Figure 2. Comparisons of the experimental ONS anomaly Δ𝐸Expt. − Δ𝐸C (grey hatched bars) and the corresponding theoretical estimates in two EDFs (SGII and SAMi). The contribution from the QCD-based CSB interaction (CSBI) in Case I and the extra contributions are indicated by the red bars with error bars and the blue bars, respectively. The values 𝑠˜0 and 𝑠˜1 + 3˜𝑠2 determined are summarized in [PITH_FULL_IMAGE:figures/full_fi… view at source ↗

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Reference graph

Works this paper leans on

23 extracted references · 23 canonical work pages

  1. [17]

    Sagawa, T

    H. Sagawa, T. Naito, X. Roca-Maza, and T. Hatsuda, QCD-based charge symmetry breaking interaction and the Okamoto-Nolen-Schiffer anomaly, Phys. Rev. C109, L011302 (2024)

    work page 2024

  2. [19]

    Hatsuda, H

    T. Hatsuda, H. Høgaasen, and M. Prakash, QCD sum rules in medium and the Okamoto- Nolen-Schiffer anomaly, Phys. Rev. Lett.66, 2851 (1991)

    work page 1991

  3. [1]

    K.Okamoto,Coulombenergyof 3Heandpossiblechargeasymmetryofnuclearforces,Phys. Lett. 11, 150 (1964)

    work page 1964

  4. [2]

    J. A. Nolen, Jr. and J. P. Schiffer, Coulomb Energies, Annu. Rev. Nucl. Sci.19, 471 (1969)

    work page 1969

  5. [3]

    Shlomo, Nuclear Coulomb energies, Rep

    S. Shlomo, Nuclear Coulomb energies, Rep. Prog. Phys.41, 957 (1978)

    work page 1978

  6. [4]

    J. W. Negele, The41Sc-41Ca Coulomb energy difference, Nucl. Phys. A165, 305 (1971)

    work page 1971

  7. [5]

    P. G. Blunden and M. J. Iqbal, Relativistic Hartree-Fock calculations for finite nuclei, Phys. Lett. B196, 295 (1987)

    work page 1987

  8. [6]

    Suzuki, H

    T. Suzuki, H. Sagawa, and A. Arima, Effects of valence nucleon orbits and charge symmetry breaking interaction on the Nolen-Schiffer anomally of mirror nuclei, Nucl. Phys. A536, 141 (1992)

    work page 1992

Show all 23 references
  1. [7]

    G. A. Miller, B. M. K. Nefkens, and I. Slaus, Charge symmetry, quarks and mesons, Phys. Rep. 194, 1 (1990)

    1990

  2. [8]

    Roca-Maza, G

    X. Roca-Maza, G. Colò, and H. Sagawa, Nuclear Symmetry Energy and the Breaking of the Isospin Symmetry: How Do They Reconcile with Each Other?, Phys. Rev. Lett.120, 202501 (2018)

    2018

  3. [9]

    Bączyk, J

    P. Bączyk, J. Dobaczewski, M. Konieczka, W. Satuła, T. Nakatsukasa, and K. Sato, Isospin- symmetry breaking in masses of𝑁≃𝑍 nuclei, Phys. Lett. B778, 178 (2018)

    2018

  4. [10]

    Sagawa, G

    H. Sagawa, G. Colò, X. Roca-Maza, and Y. Niu, Collective excitations involving spin and isospin degrees of freedom, Eur. Phys. J. A55, 227 (2019)

    2019

  5. [11]

    Bączyk, W

    P. Bączyk, W. Satuła, J. Dobaczewski, and M. Konieczka, Isobaric multiplet mass equation within nuclear density functional theory, J. Phys. G46, 03LT01 (2019). 7 QCD sum rule approach to Okamoto-Nolen-Schiffer anomaly Hiroyuki Sagawa

    2019

  6. [12]

    T.Naito,G.Colò,H.Liang,X.Roca-Maza,andH.Sagawa,Toward abinitio chargesymmetry breaking in nuclear energy density functionals, Phys. Rev. C105, L021304 (2022)

    2022

  7. [13]

    T.Naito,X.Roca-Maza,G.Colò,H.Liang,andH.Sagawa,Isospinsymmetrybreakinginthe charge radius difference of mirror nuclei, Phys. Rev. C106, L061306 (2022)

    2022

  8. [14]

    Naito, G

    T. Naito, G. Colò, H. Liang, X. Roca-Maza, and H. Sagawa, Effects of Coulomb and isospin symmetrybreakinginteractionsonneutron-skinthickness,arXiv:2302.08421[nucl-th](2023)

    2023 arXiv

  9. [15]

    S.J.Novario, D.Lonardoni, S.GandolfiandG.Hagen,TrendsofNeutronSkinsandRadiiof Mirror Nuclei from First Principles, Phys. Rev. Lett.130, 032501 (2023)

    2023

  10. [16]

    W. J. Huang, M. Wang, F. G. Kondev, G. Audi, and S. Naimi, The AME 2020 atomic mass evaluation(I).Evaluationofinputdata,andadjustmentprocedures,Chin.Phys.C 45,030002 (2021)

    2021

  11. [18]

    E.M.HenleyandG.Krein,Nambu–Jona-Lasiniomodelandchargeindependence,Phys.Rev. Lett. 62, 2586 (1989)

    1989

  12. [20]

    Saito and A

    K. Saito and A. W. Thomas, The Nolen-Schiffer anomaly and isospin symmetry breaking in nuclear matter, Phys. Lett. B335, 17 (1994)

    1994

  13. [21]

    R. S. Hayano and T. Hatsuda, Hadron properties in the nuclear medium, Rev. Mod. Phys.82, 2949 (2010)

    2010

  14. [22]

    P.GublerandD.Satow,RecentprogressinQCDcondensateevaluationsandsumrules,Prog. Part. Nucl. Phys.106, 1 (2019)

    2019

  15. [23]

    C47, 52 (2024)

    T.Naito,G.Colò,T.Hatsuda,H.Liang,X.Roca-Maza, andH.Sagawa,Possibleinconsistency between phenomenological and theoretical determinations of charge symmetry breaking in nuclear energy density functionals, Nuovo Cim. C47, 52 (2024). 8

    2024

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