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REVIEW 3 major objections 5 minor 32 references

Nucleon axial, tensor, and scalar charges and $\sigma$-terms in lattice QCD

T0 review · 3 major / 5 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read Using three ensembles with quarks at their physical masses, the paper claims continuum-limit nucleon charges and sigma-terms without chiral extrapolation, finding $g_A^{u-d}=1.250(24)$.

desk verdict Solid ETMC proceedings with the real new content in the preliminary E112 analysis; the continuum-limit values are shared with the companion paper and the charm sigma-term needs a stronger discretization systematic. read the letter →

arxiv 2502.05274 v1 pith:LLKK3J6O submitted 2025-02-07 hep-lat hep-exhep-phnucl-th

Constantia Alexandrou , Simone Bacchio , Jacob Finkenrath , Christos Iona , Giannis Koutsou , Yan Li , Gregoris Spanoudes This is my paper · ORCID
classification hep-lathep-exhep-phnucl-th PACS 12.38.Gc14.20.Dh
keywords nucleonaxialchargetensorscalarsigma-piNstrangesigmatermcharmlatticeQCDcontinuumlimittwistedmassfermions
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 aims to establish reliable continuum-limit values for the nucleon axial, tensor, and scalar charges and for the pion-nucleon, strange, and charm $\sigma$-terms directly from lattice QCD. Its specific claim is that using three ensembles with all quark masses tuned near their physical values, and combining a two-state excited-state analysis with Akaike Information Criterion model averaging, gives continuum limits without any chiral extrapolation. If the claim holds, the quoted numbers, including $g_A^{u-d}=1.250(24)$ and $\sigma_{\pi N}=41.9(8.1)$ MeV, are direct QCD-based predictions rather than extrapolations from heavier pion masses. The same strategy can be transferred to other nucleon matrix elements.

What carries the argument

The machinery is three $N_f=2+1+1$ twisted-mass clover-improved ensembles with lattice spacings near 0.080, 0.068, and 0.057 fm and pion masses around 140 MeV, analyzed through two-state fits to two- and three-point correlation functions. Excited-state contamination is controlled by Akaike Information Criterion weighting over a wide fit-parameter space, and continuum-limit systematics are controlled by averaging a linear-in-$a^2$ extrapolation and two constant extrapolations, one omitting the coarsest ensemble. Nonperturbative renormalization is carried out in the RI'/MOM scheme with perturbative conversion to the $\overline{\mathrm{MS}}$ scheme at 2 GeV.

What would settle it

Add the fourth, finer ensemble at $a\approx 0.05$ fm with full statistics and complete disconnected contributions and repeat the continuum extrapolation; if the model-averaged values shift by more than the quoted uncertainties, the three-ensemble error model is inadequate. A simpler check is whether the linear and constant fits' intercepts agree within their errors.

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Extended reading notes

Core claim

The central discovery asserted is that continuum-limit nucleon charges and $\sigma$-terms can be obtained using only physical-point ensembles, eliminating the chiral extrapolation that has dominated systematics in previous determinations. Concretely, the paper reports $g_A^{u-d}=1.250(24)$, compatible with experiment, $g_T^{u-d}=0.955(29)$, $g_S^{u-d}=1.08(31)$, $\sigma_{\pi N}=41.9(8.1)$ MeV, $\sigma_s=30(17)$ MeV, and $\sigma_c=82(29)$ MeV, together with single-flavor charges for up, down, strange, and charm. All values include valence and sea (disconnected) quark contributions and are renormalized nonperturbatively to the $\overline{\mathrm{MS}}$ scheme at 2 GeV.

Load-bearing premise

The results rest on the assumption that the true discretization error is bracketed by an AIC-weighted average of a linear-in-$a^2$ fit and two constant fits using only three lattice spacings; if the $a^2$ dependence is nonlinear at these spacings, or the coarsest ensemble lies outside the scaling region, the central values and errors would be biased.

Editorial extensions

If this is right

  • The nucleon charges and sigma-terms can serve as direct QCD inputs for neutron decay, dark-matter direct detection, and searches for physics beyond the Standard Model.
  • Agreement of $g_A^{u-d}$ with the measured value supports the claim that excited-state and continuum systematics are controlled at the physical point.
  • The flavor-separated axial charges give the quark spin contribution to the nucleon spin, providing a target for future deep-inelastic-scattering measurements.
  • The physical-point-only continuum strategy avoids chiral extrapolation and can be applied to other nucleon matrix elements, such as form factors and parton moments.

Reading between the lines

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

  • The quoted uncertainty on the continuum limit rests on three lattice spacings; a full-statistics analysis of the finer $a\approx 0.05$ fm ensemble may shift or shrink the central values, especially for the sigma-terms.
  • If confirmed, the sizable $\sigma_c=82(29)$ MeV implies a non-negligible charm-quark contribution to the nucleon mass, with consequences for dark-matter couplings and the proton mass decomposition.
  • The success of this approach would suggest that the spread among published lattice values for these quantities is dominated by chiral extrapolation rather than by lattice artifacts.
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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

3 major / 5 minor

Summary. The paper analyzes three N_f=2+1+1 twisted-mass clover-improved ensembles (B64, C80, D96) with quark masses tuned near their physical values and extracts the nucleon axial, scalar, and tensor charges and the pion-nucleon, strange, and charm sigma-terms from two- and three-point correlation functions. The analysis uses two-state spectral fits, AIC model averaging over excited-state and continuum-extrapolation models, and nonperturbative RI'/MOM renormalization with conversion to MS at 2 GeV. The continuum limit is obtained from a model average of a linear-in-a^2 fit through all three ensembles, a constant fit through all three, and a constant fit omitting the coarsest B64 ensemble. A fourth finer ensemble E112 is used for preliminary isovector results only. The main outputs are g_A^{u-d}=1.250(24), g_S^{u-d}=1.08(31), g_T^{u-d}=0.955(29), sigma_piN=41.9(8.1) MeV, sigma_s=30(17) MeV, and sigma_c=82(29) MeV. The paper claims that this is the first continuum-limit determination of these quantities using only physical-point ensembles, avoiding chiral extrapolations.

Significance. If the quoted values are reliable, they provide a valuable QCD determination of nucleon charges and sigma-terms at the physical point, with the isovector axial charge agreeing with experiment and the tensor and scalar charges serving as inputs for searches of beyond-Standard-Model interactions. The use of physical-point ensembles avoids chiral extrapolation, the AIC framework is a modern and appropriate tool for systematic error evaluation, and the nonperturbative renormalization is a strength. However, the significance of the paper depends critically on the continuum extrapolation, which rests on only three lattice spacings, and on the treatment of the charm sigma-term, where a single constant extrapolation is used. These issues need to be addressed before the central claim can be considered fully supported.

major comments (3)
  1. [Section 3, Fig. 2] The continuum-limit claim is the central assertion of the paper, and it rests entirely on three lattice spacings (B64, C80, D96) spanning only a factor of about two in a^2. With these three points, the model set {linear in a^2, constant, constant omitting B64} cannot distinguish a genuine O(a^2) trend from scatter; indeed, the AIC weights for g_A (30% linear, 40% constant with B64, 30% constant without B64) show that the result is dominated by the constant fits. The quoted uncertainty on g_A and on the Table 2 entries therefore does not include a meaningful contribution from the unknown curvature of the discretization error. I ask the authors to add a quantitative stability test, for example a two-point linear extrapolation using C80 and D96, a comparison with the E112 point as a fourth spacing, or an explicit breakdown of the systematic error attributable to the choice of continuum-extrapolation model.
  2. [Section 3, Table 3] The quoted value sigma_c=82(29) MeV is obtained from a single constant extrapolation, as stated in Table 3. This is not consistent with the AIC-model-averaging strategy used for the other quantities and is difficult to justify: at the coarsest lattice spacing a=0.07957 fm, the dimensionless combination m_c a is of order 0.4, so an O((m_c a)^2) discretization correction is expected to be the dominant systematic effect. A constant model omits exactly this term, and the 29 MeV error bar therefore contains no contribution from the continuum-extrapolation systematic. Please either include a linear-in-a^2 model for sigma_c (possibly with a constrained prior), or provide a concrete argument that the charm scalar matrix element has negligible a^2 dependence on these ensembles, and add the resulting model uncertainty to the final value.
  3. [Section 4, Conclusions] The paper claims 'for the first time' a continuum-limit determination using only physical-point ensembles. This is a strong novelty claim. Given that the continuum limit is based on only three ensembles and that the fourth, finer E112 ensemble is presented only as preliminary and is not used in the extrapolation, the claim is stronger than the evidence shown. I recommend either including the E112 data at least as a cross-check for the isovector quantities, or softening the claim to explicitly state that the present result uses three physical-point ensembles and should be considered an intermediate step pending the finer ensemble.
minor comments (5)
  1. [Abstract] The abstract contains the typo 'the the latter'; it should read 'the latter'.
  2. [Section 3] The sentence 'The extracted values show almost no dependence on on t_low^s' contains a duplicated 'on'.
  3. [Fig. 2 caption] The caption refers to the 'C90 and D96 ensembles', but the ensemble is labeled C80 in Table 1; please correct this label.
  4. [Eqs. (2)-(4)] The notation for the three-point function is inconsistent: Eq. (3) uses C_mu, while Eq. (4) uses C_3pt. Please align the notation for clarity.
  5. [Tables 2 and 3] The results are quoted without a decomposition into statistical and systematic uncertainties. Since the paper's method is AIC model averaging, a statistical/systematic breakdown would make the size of the continuum-extrapolation systematic transparent and would help the reader assess the source of the quoted error bars.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the charges and sigma-terms are obtained from spectral fits to lattice correlation functions, with an AIC model average for excited states and continuum extrapolation; the only self-citation is a companion-paper reporting overlap, not a load-bearing derivation.

full rationale

The derivation chain is self-contained. Nucleon charges are extracted from two- and three-point correlation functions through the spectral decomposition in Eqs. (2)-(4); Eq. (4) is a standard large-time limit of a ratio, not an ansatz fitted to reproduce the quoted charges. Excited states are controlled by two-state fits with AIC model averaging, and the continuum limit uses a pre-specified set of linear-in-a^2 and constant fits with AIC weights; the data, not the target values, select the weights. The sigma-terms follow from Eq. (6) using separately computed scalar matrix elements (including disconnected loops) and quark masses, with the twisted-mass renormalization cancellation being a known property of the formulation. No equation reduces a claimed output to a fitted input by construction. The only self-reference is Ref. [16], cited for renormalization details and per-ensemble results; this is a companion-proceedings overlap, not a uniqueness theorem or an ansatz that carries the derivation. External benchmarks (g_A vs experiment and Fig. 3 comparisons with other collaborations) make the results falsifiable outside the paper's own fitted values. The short continuum lever arm and the single constant fit used for sigma_c are systematic-error concerns, not circularity.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The calculation is a first-principles numerical evaluation, but the quoted values depend on several modeling choices: the form of the continuum extrapolation, the excited-state fit model, the renormalization scheme conversion, and the scale setting from prior ETMC work. No physically new entity is introduced. The most ad hoc element is the constant extrapolation for sigma_c, which is not justified with a systematic comparison.

free parameters (2)
  • Slope of the linear-in-a^2 continuum extrapolation (per observable) = not quoted
    The linear extrapolation model fits a slope to three lattice-spacing points and receives a 30% AIC weight for g_A; its value and uncertainty are not reported but influence the continuum values.
  • Two-state fit parameters (excited-state energy gaps and amplitudes, per ensemble) = not quoted
    Excited-state removal relies on fitted energy differences and amplitudes over varied t_s windows; AIC weights over these fits affect every quoted charge.
assumptions (5)
  • domain assumption Twisted-mass fermions at maximal twist with a clover term provide automatic O(a) improvement, so discretization errors begin at O(a^2).
    Invoked in Section 2 and used to justify the linear-in-a^2 continuum extrapolation in Section 3.
  • domain assumption The AIC model average over the chosen set of continuum extrapolations (linear, constant with B64, constant without B64) gives an unbiased estimate of the continuum limit and its systematic error.
    This is the main systematic-error model described in Section 3 and Fig. 2; it is a standard but not guaranteed method in lattice QCD.
  • domain assumption RI'/MOM nonperturbative renormalization with perturbative conversion to the MS scheme at 2 GeV is valid for the axial, scalar, and tensor operators.
    Stated in Section 2 with reference [16]; any error in the scheme conversion propagates into all quoted charges.
  • domain assumption The quark masses and lattice spacings taken from Refs [8,9] are accurate enough that the physical-point condition holds and no chiral extrapolation is needed.
    Used in Section 1 and Table 1; the ensemble pion masses are 136-141 MeV, close to but not exactly the physical pion mass.
  • ad hoc to paper A constant extrapolation is sufficient for the charm sigma-term sigma_c.
    Section 3, Table 3: a single constant fit is used for sigma_c without an AIC model average or a stated justification.

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Cite this review

Pith. "Pith review of Nucleon axial, tensor, and scalar charges and $\sigma$-terms in lattice QCD." pith.science (2026). https://pith.science/paper/LLKK3J6O

@misc{pith2026250205274,
  author       = {Pith},
  title        = {Pith review of: Nucleon axial, tensor, and scalar charges and $\sigma$-terms in lattice QCD},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LLKK3J6O}},
  note         = {Machine review of arXiv:2502.05274}
}
abstract

We determine the nucleon axial, scalar and tensor charges at the continuum limit by analyzing three $N_f=2+1+1$ twisted mass fermion ensembles with all quark masses tuned to approximately their physical values. We include all contributions from valence and sea quarks. We use the Akaike Information Criterion to evaluate systematic errors due to excited states and the continuum extrapolation. For the nucleon isovector axial charge we find $g_A^{u-d}=1.250(24)$, in agreement with the experimental value. We compute the axial, tensor and scalar charges for each quark flavor. The axial charge provides crucial information on the intrinsic spin carried by quark in the nucleon and the the latter two provide input for experimental searches of physics beyond the standard model. Moreover, we extract the nucleon $\sigma$-terms and find $\sigma_{\pi N}=41.9(8.1)$ MeV, for the strange $\sigma_{s}=30(17)$ MeV and for the charm $\sigma_{c}=82(29)$ MeV. We also present preliminary results on the isovector quantities using a fourth ensemble at smaller lattice spacing.

Figures

Figures reproduced from arXiv: 2502.05274 by the authors.

Figure 1
Figure 1. We present the ratio and fit results for all ensembles, for the isovector axial charge. In the legend, we give the symbols used to denote the various values of 𝑡𝑠/𝑎. The top row shows the analysis for 𝑔 𝑢−𝑑 𝐴 for the B64 ensemble, the middle row for C80 and the bottom for D96. The first column shows results on the ratio versus 𝑡𝑖𝑛𝑠 − 𝑡𝑠/2. The horizontal bands are the model averaged values. In the second column, we … view at source ↗
Figure 2
Figure 2. In the left panel, we show the continuum limit of the nucleon isovector axial charge (open symbol and band) extrapolated using the B64, C80 and D96 ensembles (filled symbols). The extrapolation is the result of a model average, which combines linear and constant fits as explained in the text. In the right panel, we show the weights for each type of fit, namely the linear extrapolation is represented by a blue circle… view at source ↗
Figure 3
Figure 3. Comparison of the results of this work with other lattice QCD results, for the isovector axial, scalar and tensor charges. Our results are shown with the red square and red error band. The blue triangles show previous ETMC results, for 𝑔 𝑢−𝑑 𝐴 [20] and for 𝑔 𝑢−𝑑 𝑇 [21], while Ref. [22] gives results on all three isovector charges including 𝑔 𝑢−𝑑 𝑠 for the B64 ensemble. Open symbols represent results without a contin… view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Ratio and preliminary fit results for the currently available data of the E112 ensemble, for the isovector charges. The notation is the same as in [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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