Pith. sign in

REVIEW 2 major objections 5 minor 23 references

The isoscalar non-singlet axial form factor of the nucleon from lattice QCD

T0 review · 2 major / 5 minor · reviewed 2026-08-08 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read The paper shows that combining the summation method with a direct z-expansion fit, a window average over source-sink separations, and an AIC model average gives a stable lattice extraction of the nucleon's isoscalar non-singlet axial form…

desk verdict A careful, honest progress report on the isoscalar octet axial form factor; the one-step z-fit is well validated, and the remaining excited-state and continuum systematics are explicitly deferred to the final paper. read the letter →

arxiv 2502.06305 v1 pith:3NSSJPA6 submitted 2025-02-10 hep-lat

Alessandro Barone , Dalibor Djukanovic , Georg von Hippel , Jonna Koponen , Harvey B. Meyer , Konstantin Ottnad , Hartmut Wittig This is my paper · ORCID
classification hep-lat PACS 12.38.Gc
keywords axialformfactorisoscalarnon-singletlatticeQCDnucleonchargesummationmethodz-expansionmodelaveragingstrange
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

This proceedings paper argues that a specific analysis pipeline extracts the isoscalar non-singlet axial form factor of the nucleon, $G_A^{u+d-2s}(Q^2)$, from lattice QCD in a controlled way up to momentum transfers of $0.7$ GeV$^2$. The pipeline combines the summation method for suppressing excited-state contamination with a direct $z$-expansion fit over all $Q^2$ and all source-sink separations, a window average over the minimum separation, and an AIC model average over chiral-continuum ansaetze. This matters because this flavour combination, together with the singlet channel, determines the strange axial form factor that neutrino-scattering experiments are trying to measure. On two of the most chiral ensembles the preliminary octet axial charge is compatible with the established values $g_A^{u+d-2s}=0.46(5)$ and $0.490(20)$, and the connected contribution already follows the extrapolated physical curve.

What carries the argument

The carrying mechanism is the summation-method identity $S(\boldsymbol{q},t_s)=b_0(Q^2)+t_s\,G_A(Q^2)+O(t_s e^{-\Delta t_s})$, which turns the form factor into the slope of a linear fit in the source-sink separation $t_s$. The paper combines it with a $z$-expansion in the variable $z(Q^2)=\big(\sqrt{t_{\rm cut}+Q^2}-\sqrt{t_{\rm cut}}\big)/\big(\sqrt{t_{\rm cut}+Q^2}+\sqrt{t_{\rm cut}}\big)$ truncated at order $n=2$, fits all $Q^2\le 0.7$ GeV$^2$ and all $t_s$ in one go, and then averages over $t_{s,\min}$ with a tanh window (lower edge $0.8$ fm, upper edge $1$ fm, width $0.08$ fm). The $z$-expansion absorbs the $Q^2$ dependence, the window average removes the human choice of $t_{s,\min}$, and the later AIC model average combines chiral-continuum ansaetze and cuts. The large covariance matrix is regulated either by damping the off-diagonal elements or by an SVD cut.

What would settle it

Compute the summed ratio on a single ensemble for at least four source-sink separations above 1 fm and check whether the slope in $t_s$ is independent of the fitted $t_s$ window and whether the residuals show the expected $O(t_s e^{-\Delta t_s})$ curvature; if the slope shifts with the window, the window-averaged extraction is biased.

Watch

Extended reading notes

Core claim

The central claim is that the direct $z$-fit route is a valid replacement for the traditional two-step procedure: instead of extracting $G_A(Q^2)$ point by point and then fitting $z$, one fits the expansion coefficients $a_0,a_1,a_2$ in a single simultaneous fit over all $Q^2\le 0.7$ GeV$^2$ and all source-sink separations, treating the constant $b_0(Q^2)$ of the summation method either as a free parameter per $Q^2$ or as a second $z$-expansion; both give compatible results. The paper demonstrates that covariance-matrix regularization by off-diagonal damping or by an SVD cut does not change the outcome, and that a window average over $t_{s,\min}$ with fixed physical windows removes the main human bias in choosing the fit range. On the two most chiral ensembles the full $u+d-2s$ form factor, with disconnected loops included, shows a low-$Q^2$ shift relative to the connected contribution, and the axial charge agrees with existing benchmarks. The paper presents this as progress toward a first physical result for the isoscalar octet form factor, with the same strategy to be applied to the full ensemble set.

Load-bearing premise

The load-bearing premise is that the summation-method identity is dominated by its linear term for the chosen source-sink separations, so that excited-state contamination of order $t_s e^{-\Delta t_s}$ is negligible after window averaging.

Editorial extensions

If this is right

  • The same analysis chain can be run on the full ensemble set to produce the first physical result for the isoscalar octet axial form factor over $Q^2\in[0,0.7]$ GeV$^2$.
  • Combined with an analogous singlet $u+d+s$ computation, the method opens a route to the strange axial form factor $G_s^A(Q^2)$ in the range relevant to neutrino-nucleus experiments.
  • Because the window average and the model average are applied with fixed choices, the quoted errors include a systematic spread from analysis choices rather than one hand-picked fit.
  • The agreement of the preliminary axial charge with external estimates at low $Q^2$ supports the treatment of disconnected contributions and excited states at the current statistical precision.
  • The connected $u+d$ result already lies close to the physical curve after the chiral-continuum extrapolation, suggesting the remaining corrections on the most chiral ensembles are small.

Reading between the lines

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

  • If the linear-dominance assumption in the summation method holds at $t_s\ge0.8$ fm, the same window-averaged direct-fit strategy should suppress excited states just as well for other baryon form factors, such as the electromagnetic or induced-pseudoscalar ones.
  • The fixed choices $t_{\rm cut}=(4M_\pi)^2$ and $Q^2_{\max}=0.7$ GeV$^2$ could be tested by extending the fit to $1$ GeV$^2$; a failure of the $n=2$ truncation there would reveal whether the low-$Q^2$ stability is a property of the method or of the chosen range.
  • A sharper test of the method would be to compare results from the window-averaged summation fit with a variational multi-operator estimate of the first excited-state energy and amplitude on the same ensembles.
  • The same framework could be applied to the singlet channel, turning the flavour decomposition of the axial form factor into one unified fit rather than separate connected and disconnected analyses.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. This proceedings paper reports work in progress on the isoscalar non-singlet axial form factor G_A^{u+d-2s}(Q^2) of the nucleon from N_f=2+1 CLS lattices. The analysis combines the summation method for excited-state suppression with a direct z-expansion fit at order n=2, a window average over the minimum source-sink separation, two covariance-matrix regularisations, and an AIC model average for the chiral-continuum extrapolation. The connected u+d channel is carried through the full extrapolation, while the full u+d-2s result is shown on two ensembles only (E300 and D200). The authors report compatibility of the preliminary octet axial charge with the Cloudy Bag value g_A^{u+d-2s}=0.46(5) and with the ETM value 0.490(20).

Significance. If completed, this study would provide the first lattice QCD determination of the isoscalar octet axial form factor in a range relevant for MicroBooNE and for flavour decompositions of the nucleon spin. The paper's strengths are its transparent internal checks: the direct z-fit is compared with the two-step procedure, the b0(ts) intercept is treated both freely and via a z-expansion, off-diagonal damping and SVD cuts are compared with the unregulated covariance matrix, and the model average uses correlated coefficients. The use of independent renormalisation factors and the explicit statement of remaining steps are also commendable. The main weaknesses are the lack of an explicit excited-state contamination test and the absence of a continuum extrapolation for the full octet channel, both of which are acknowledged in spirit but not quantified.

major comments (2)
  1. [§3, Eqs. (7)–(9), Fig. 2] The central extraction assumes that, for the t_s values entering the window average, the O(t_s e^{-Delta t_s}) term in Eq. (7) is negligible, so that G_A(Q^2) is the slope of S(q,t_s). The window average over t_s,min does not test this assumption: every fit still uses all t_s >= t_s,min, so an excited-state contamination that is non-negligible at t_s ~ 1 fm would bias all fits in the same direction. The paper validates stability under covariance-matrix regularisation and under different parametrisations of b0, but reports no two-state fit and no explicit study of the exponential term. Since the final comparison to g_A^{u+d-2s}=0.46(5) and 0.490(20) uses exactly this slope, a positive contamination would make the agreement coincidental. Please add a direct excited-state check (for example a two-state fit or a comparison at substantially larger t_s) or state explicitly that this systematic is not yet controlled.
  2. [§4, Fig. 5 and final paragraph] The full u+d-2s result is presented on two ensembles, E300 and D200, without a continuum extrapolation. The text does label the evaluation as preliminary and notes that lattice artifacts are not accounted for, but the subsequent sentence, 'these provide a value of the axial charge compatible with...', presents the comparison as validation. Because the two ensembles may share common discretisation effects, agreement with [19,20] is weaker evidence than a single-ensemble comparison would be. Please either soften the claim to 'consistent within statistical errors, with missing systematics not yet included' or provide a quantitative estimate of the size of the omitted discretisation, finite-volume, and renormalisation uncertainties.
minor comments (5)
  1. [§1, first sentence] The sentence beginning 'The axial form factors G_A(Q^2) of the nucleon plays' has a subject-verb agreement error; it should be 'play' or the sentence should be restructured with a singular subject.
  2. [Fig. 2 caption] The red curve is described as a 'zoom on the window function', but it is plotted with the z-expansion coefficients on the same axes without specifying its scale or normalisation; this is confusing and should be clarified.
  3. [§3, after Eq. (8)] The z-expansion is fixed at order n=2 for the central coefficients and only the order n_b for b0 is varied in Fig. 1. Since the z-expansion truncation is a systematic assumption, the authors should state whether an order n=3 fit or an alternative parametrisation has been tried or is planned beyond the dipole mentioned in the outlook.
  4. [§4, final paragraph] The comparison with g_A^{u+d-2s}=0.46(5) and 0.490(20) is made without showing the E300 and D200 values or their uncertainties in the text or a table; providing these numbers would make the level of agreement quantitative.
  5. [§2, renormalisation paragraph] The sentence 'we take the factors Z_A from [9] and b_A from [10], neglecting the coefficient \tilde b_A and f_A' would benefit from a brief justification of why those coefficients are expected to be small in this channel, since the neglected terms are part of the O(a) improvement of the axial current.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the lattice extraction of G_A is self-contained and benchmarked against independent external results.

full rationale

The paper's derivation chain is self-contained. The axial form factor G_A(Q^2) is extracted as the slope of the summed ratio S(q,t_s) in Eq. (7), with the z-expansion of Eq. (8) as an explicit parametrization of the Q^2 dependence. The window average in Eq. (9), the covariance-matrix regularizations (off-diagonal damping and SVD), the chiral-continuum ansatze, the pion-mass and lattice-spacing cuts, and the AIC model average of Eq. (11) and Eq. (12) are all described in the text rather than imported as black boxes. The only self-citation is Ref. [15] for the overall analysis strategy, but that is a published, independently refereed lattice calculation whose methods are presented here and whose applicability to the new u+d-2s channel requires new correlation-function data. No equation in this paper reduces to a fitted input, and no prediction is defined in terms of the external benchmarks: the comparison with the Cloudy Bag model value g_A^{u+d-2s}=0.46(5) [19] and the ETM result 0.490(20) [20] is a check, not an input. The possible neglect of O(t_s e^{-\Delta t_s}) excited-state contamination in Eq. (7) is a systematic uncertainty and a correctness risk, but it is not a circularity because the extraction is not constructed to reproduce any target value. Accordingly, the appropriate circularity score is 0.

Assumptions & free parameters 6 free parameters · 6 assumptions · 0 invented entities

The central quantities are measured lattice values, so the main ledger entries are analysis choices (window parameters, z-expansion order, cuts) that could shift the final result if varied. No new physical entities are introduced; the paper reuses established lattice methodology and external renormalization factors.

free parameters (6)
  • z-expansion coefficients a0, a1, a2 per ensemble = Not tabulated; examples in Fig. 3 (a0 ~ 0.5-0.6 for connected u+d)
    Fit parameters of Eq. (8); the paper's main numerical output. They are measured from the lattice data, not externally fixed.
  • Window function parameters t_low, t_up, Delta_t_w = 0.8 fm, 1.0 fm, 0.08 fm
    Chosen by hand in Eq. (9) to average over ts,min; the final coefficients depend on this choice, and the paper does not vary it.
  • z-expansion truncation order n (and n_b for b0) = n = 2, n_b = 2
    Truncation of the series in Eq. (8); fixed for all final results with no systematic higher-order test (dipole deferred).
  • Pion mass cuts and lattice spacing cuts = M_cut_pi = 300, 285, 265 MeV; spacing cuts not enumerated
    Cuts used in the model average (Sec. 4) to restrict the chiral-continuum fits; they define the ensemble subsets.
  • z-expansion parameter t_cut = t_cut = (4 M_pi)^2
    Standard threshold choice for z-expansion; held fixed across ensembles, not fitted.
  • Covariance matrix off-diagonal damping alpha = alpha in [0.985, 1]
    Used in one regularization method for the covariance matrix; final results use svd instead. Cross-checks show negligible difference.
assumptions (6)
  • domain assumption The CLS N_f=2+1 ensembles with O(a)-improved Wilson fermions and Luscher-Weisz gauge action provide a valid lattice discretization of QCD, and the continuum limit can be taken with the stated ansatze.
    Standard lattice QCD setup; cited refs [11-14]. The paper assumes the extrapolation forms are correct.
  • domain assumption The renormalization factors Z_A [9] and b_A [10] are correct, and the neglected coefficients b_A tilde and f_A are small.
    Stated in Sec. 2: 'neglecting the coefficient b_A and f_A ... assumed to be small since they parametrise sea-quark effects.'
  • ad hoc to paper The z-expansion at order n=2 adequately describes G_A(Q^2) for 0 <= Q^2 <= 0.7 GeV^2.
    Eq. (8) with n=2; no systematic check of n=3 or alternative parametrizations is shown (dipole is mentioned as future work).
  • domain assumption Excited-state contamination in the summation method is negligible for the used ts ranges.
    Eq. (7) relies on the asymptotic linear form; the window average assumes this holds for ts,min >= 0.8 fm.
  • domain assumption The chiral-continuum ansatze (1-3) and the finite-volume correction term in Eq. (10) are the correct functional forms.
    Sec. 4 uses these ansatze for the extrapolation; the paper notes the data are flat and FV effects appear negligible, but this is an assumption.
  • domain assumption The Akaike Information Criterion model average (Eqs. 11-12) yields unbiased final estimates.
    Standard statistical model selection, but the specific weighting and use of cumulative distributions is a modeling choice.

how reviews work

0 comments
Cite this review

Pith. "Pith review of The isoscalar non-singlet axial form factor of the nucleon from lattice QCD." pith.science (2026). https://pith.science/paper/3NSSJPA6

@misc{pith2026250206305,
  author       = {Pith},
  title        = {Pith review of: The isoscalar non-singlet axial form factor of the nucleon from lattice QCD},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3NSSJPA6}},
  note         = {Machine review of arXiv:2502.06305}
}
abstract

We present our progress on the computation of the axial form factor of the nucleon with flavour structure $u+d-2s$ from lattice QCD. We employ a set of $N_f=2+1$ CLS ensembles with $O(a)$-improved Wilson fermions and the L\"uscher-Weisz gauge action, with lattice spacings ranging from $0.05\,\text{fm}$ to $0.086\,\text{fm}$ and pion masses spanning between $130\,\text{MeV}$ and $350\,\text{MeV}$. We employ multiple source-sink separations and use the summation method to suppress the contamination from excited states. We use a $z$-expansion on each ensemble to parametrize the $Q^2$-dependence of the form factor and simultaneously fit the available source-sink separations for all $Q^2\leq 0.7 \,{\rm GeV}^2$. We outline our analysis of the stability of the fits varying the ans\"atze and different estimations of the covariance matrix and report on our strategy for a comprehensive determination of the physical form factor.

Figures

Figures reproduced from arXiv: 2502.06305 by the authors.

Figure 1
Figure 1. Comparison of different 𝑧-fit procedures for the ansatz in Eq. (7) for 𝑢 + 𝑑 − 2𝑠 data on E300. The blue points refer to the two-step procedure, and the orange points to the “direct” approach. The green bands complement the latter considering the case where 𝑏0 is also parametrised by a 𝑧-expansion at order 𝑛𝑏 = 2. The magenta points show the contribution of the connected and disconnected data in the two-step procedu… view at source ↗
Figure 2
Figure 2. Window average on the coefficients 𝑎0, 𝑎1, 𝑎2 (rows) of the 𝑧-expansion in Eq. (8) as a function of the minimum source-sink separation 𝑡𝑠,min on the ensemble E300 for the connected case (left) and the full octet case (right). The different colours refer to different approaches to regularise the covariance matrix, and the red curve is a zoom on the window function in Eq. (9). The vertical lines correspond to the choi… view at source ↗
Figure 3
Figure 3. Example of chiral-continumm extrapolation for the 𝑢 + 𝑑 case with ansatz 2 and finite-volume effect with a cut 𝑀cut 𝜋 = 285 MeV, shown for all the three coefficients (rows) as a function of 𝑀2 𝜋 (left), 𝑎 2 (centre) and the spatial lattice size 𝐿 (right). The blue points are the original data and the red points and band correspond to the corrected version for the continuum parameters as specified in the legend. perf… view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Model average through AIC for the 𝑢 + 𝑑 case. The red points correspond to the results of the fits entering the model average, and the blue line is the cumulative distribution in Eq. (12), with the vertical bands indicating the final results obtained from the 16th and …
Figure 5
Figure 5. Figure 5: Final results on the isoscalar axial form factor 𝐺𝐴(𝑄 2 ) on the ensembles E300 and D200 after window average for both connected (right) and full case (left), compared with the final AIC average for the connected case only in green. with 𝑛par,𝑘 being the number of para…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

23 extracted references · 5 canonical work pages

  1. [15]

    Djukanovic, G

    D. Djukanovic, G. von Hippel, J. Koponen, H.B. Meyer, K. Ottnad, T. Schulz et al.,Isovector axial form factor of the nucleon from lattice QCD, Phys. Rev. D106 (2022) 074503 [2207.03440]

    arXiv 2022

  2. [1]

    Ji,Gauge-Invariant Decomposition of Nucleon Spin,Phys

    X.-D. Ji,Gauge-Invariant Decomposition of Nucleon Spin,Phys. Rev. Lett.78 (1997) 610 [hep-ph/9603249]

    arXiv 1997

  3. [2]

    COMPASScollaboration, The Deuteron Spin-dependent Structure Function g1(d) and its First Moment, Phys. Lett. B647 (2007) 8 [hep-ex/0609038]

    arXiv 2007

  4. [3]

    Meyer, M

    A.S. Meyer, M. Betancourt, R. Gran and R.J. Hill,Deuterium target data for precision neutrino-nucleus cross sections,Phys. Rev. D93(2016) 113015 [1603.03048]. 8 The isoscalar non-singlet axial form factor of the nucleon from lattice QCD Alessandro Barone

    arXiv 2016

  5. [4]

    Meyer, A

    A.S. Meyer, A. Walker-Loud and C. Wilkinson,Status of Lattice QCD Determination of Nucleon Form Factors and their Relevance for the Few-GeV Neutrino Program, Ann. Rev. Nucl. Part. Sci.72 (2022) 205 [2201.01839]

    arXiv 2022

  6. [5]

    Alexandrou, S

    C. Alexandrou, S. Bacchio, M. Constantinou, K. Hadjiyiannakou, K. Jansen and G. Koutsou, Quark flavor decomposition of the nucleon axial form factors,Phys. Rev. D104 (2021) 074503 [2106.13468]

    arXiv 2021

  7. [6]

    Procedia61(2015) 495 [1406.5204]

    MicroBooNEcollaboration, Improving Dark Matter Searches by Measuring the Nucleon Axial Form Factor: Perspectives from MicroBooNE,Phys. Procedia61(2015) 495 [1406.5204]

    arXiv 2015

  8. [7]

    Kim, K.-S

    K.S. Kim, K.-S. Choi, M.-K. Cheoun, W.Y. So and H. Moon,Role of axial mass and strange axial form factor from various target nuclei in neutrino-nucleus scattering,Phys. Rev. C100 (2019) 034604

    work page 2019

Show all 23 references
  1. [8]

    Bhattacharya, R

    T. Bhattacharya, R. Gupta, W. Lee, S.R. Sharpe and J.M.S. Wu,Improved bilinears in lattice QCD with non-degenerate quarks, Phys. Rev. D73(2006) 034504 [hep-lat/0511014]

    2006 arXiv

  2. [9]

    Dalla Brida, T

    M. Dalla Brida, T. Korzec, S. Sint and P. Vilaseca,High precision renormalization of the flavour non-singlet Noether currents in lattice QCD with Wilson quarks,Eur. Phys. J. C79 (2019) 23 [1808.09236]

    2019 arXiv

  3. [10]

    Korcyl and G.S

    P. Korcyl and G.S. Bali,Non-perturbative determination of improvement coefficients using coordinate space correlators in𝑁𝑓 = 2+ 1lattice QCD, Phys. Rev. D95 (2017) 014505 [1607.07090]

    2017 arXiv

  4. [11]

    Bruno et al.,Simulation of QCD with N𝑓 =2+1 flavors of non-perturbatively improved Wilson fermions,JHEP 02(2015) 043 [1411.3982]

    M. Bruno et al.,Simulation of QCD with N𝑓 =2+1 flavors of non-perturbatively improved Wilson fermions,JHEP 02(2015) 043 [1411.3982]

    2015 arXiv

  5. [12]

    Sheikholeslami and R

    B. Sheikholeslami and R. Wohlert,Improved Continuum Limit Lattice Action for QCD with Wilson Fermions, Nucl. Phys. B259 (1985) 572

    1985

  6. [13]

    Bulava and S

    J. Bulava and S. Schaefer,Improvement of𝑁𝑓 = 3 lattice QCD with Wilson fermions and tree-level improved gauge action,Nucl. Phys. B874(2013) 188 [1304.7093]

    2013 arXiv

  7. [14]

    Luscherand P

    M. Luscherand P. Weisz,On-shell improvedlattice gauge theories,Commun. Math.Phys. 98 (1985) 433

    1985

  8. [16]

    Maiani, G

    L. Maiani, G. Martinelli, M.L. Paciello and B. Taglienti,Scalar Densities and Baryon Mass Differences in Lattice QCD With Wilson Fermions,Nucl. Phys. B293(1987) 420

    1987

  9. [17]

    Capitani, M

    S. Capitani, M. Della Morte, G. von Hippel, B. Jager, A. Juttner, B. Knippschild et al.,The nucleon axial charge from lattice QCD with controlled errors,Phys. Rev. D86(2012) 074502 [1205.0180]. 9 The isoscalar non-singlet axial form factor of the nucleon from lattice QCD Aless...

    2012 arXiv

  10. [18]

    Borsanyi et al.,Leading hadronic contribution to the muon magnetic moment from lattice QCD, Nature 593 (2021) 51 [2002.12347]

    S. Borsanyi et al.,Leading hadronic contribution to the muon magnetic moment from lattice QCD, Nature 593 (2021) 51 [2002.12347]

    2021 arXiv

  11. [19]

    Bass and A.W

    S.D. Bass and A.W. Thomas,The nucleon’s octet axial-charge𝑔(8) 𝐴 with chiral corrections, Phys. Lett. B684 (2010) 216 [0912.1765]

    2010 arXiv

  12. [20]

    Alexandrou, S

    C. Alexandrou, S. Bacchio, J. Finkenrath, C. Iona, G. Koutsou, Y. Li et al.,Nucleon charges and𝜎-terms in lattice QCD, 2412.01535

    arXiv

  13. [21]

    SciDAC, LHPC, UKQCDcollaboration, The Chroma software system for lattice QCD, Nucl. Phys. B Proc. Suppl.140 (2005) 832 [hep-lat/0409003]

    2005 arXiv

  14. [22]

    Luscher and S

    M. Luscher and S. Schaefer,Lattice QCD with open boundary conditions and twisted-mass reweighting, Comput. Phys. Commun.184 (2013) 519 [1206.2809]

    2013 arXiv

  15. [23]

    Djukanovic,Quark Contraction Tool — QCT,Comput

    D. Djukanovic,Quark Contraction Tool — QCT,Comput. Phys. Commun.247 (2020) 106950 [1603.01576]. 10

    2020 arXiv

Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.