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Realization of all-to-all fermion propagator for the first principle high accuracy strong interaction prediction

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Pith's one-line read The paper claims that a "blending" of a few low-frequency spatial modes with random high-frequency modes gives an unbiased, low-cost estimator of the all-to-all quark propagator, and demonstrates sub-percent nucleon axial charges and a…

desk verdict Clever, useful blend of distillation and stochastic high modes that checks out for the demonstrated cases, but the arbitrary-N-point claim outruns the proof. read the letter →

arxiv 2505.01719 v2 pith:6VDGLF36 submitted 2025-05-03 hep-lat hep-phnucl-th

Zhi-Cheng Hu , Ji-Hao Wang , Xiangyu Jiang , Liuming Liu , Shi-Hao Su , Peng Sun , Yi-Bo Yang This is my paper · ORCID
classification hep-lathep-phnucl-th PACS 12.38.Gc11.15.Ha02.70.Uu
keywords all-to-allfermionpropagatorlatticeQCDblendingmethoddistillationstochasticestimationnucleonaxialchargeN-pointcorrelationfunctionsdisconnecteddiagrams
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

Lattice QCD computations usually approximate the all-to-all quark propagator by expensive sequential or stochastic sources; this paper claims a way to estimate it from a small "blending space" made of low-frequency spatial modes plus random high-frequency modes at each time slice. The paper proves that a reweighted sum over these modes reproduces the exact operator in expectation, so any quark-bilinear current, including disconnected loops and four-point functions, can be evaluated from one set of projected propagators. Because the low modes are treated exactly, hadron interpolating fields retain the excited-state suppression of distillation, while the random modes make local current insertions unbiased. As demonstrations, the method yields nucleon axial charges $g_A^{u-d}=1.2339(43)$, $g_A^{u+d+s}=0.533(28)$ at $m_\pi\simeq300$ MeV with 40 configurations, and a pion charge radius from four-point functions consistent with the three-point result. If the claim holds, high-precision strong-interaction quantities, including QED and isospin-breaking corrections, become accessible from first principles.

What carries the argument

The load-bearing object is the blending space: the span of $N_e$ low-lying eigenvectors of the spatial Laplace operator (the distillation subspace) plus $N_{st}$ orthonormal random vectors sampled uniformly from its orthogonal complement. The identity that carries the argument is Eq. (12), $\mathbb{E}\left[\sum_{i,j=1}^{N_e+N_{st}}\Omega_{ij}\,\lvert\phi_i\rangle\langle\phi_i\rvert\otimes\lvert\phi_j\rangle\langle\phi_j\rvert\right]=\hat{I}\otimes\hat{I}$, with the reweighting factors $\omega_n=([L_2]-n)/(N_{st}-n)$ and $\Omega_{ij}^{(2)}$ given in Eq. (3). The proof fixes the second and fourth moments of the random vectors by unitary invariance, which is exactly what converts the projected operator into an unbiased estimator while keeping the low-mode part exact. This mechanism is what lets hadron states be constructed by distillation and local currents be inserted without bias.

What would settle it

Set $N_{st}$ equal to the full complement dimension $[L_2]$ on a small lattice, compute the pion conserved-charge ratio of Eq. (5) with the blending estimator, and compare it with the exact point-source value at every time slice; exact agreement would confirm Eq. (12), while any $t$-dependent deviation would falsify the unbiasedness claim.

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Extended reading notes

Core claim

The central discovery is that the all-to-all fermion propagator does not need to be stored or inverted in full. The paper constructs a blending space from $N_e$ low-lying eigenvectors of the spatial Laplacian and $N_{st}$ random orthonormal vectors in the orthogonal complement, and assigns a reweighting tensor $\Omega_{ij}$ so that the expectation of the projected identity is $\hat{I}\otimes\hat{I}$. This identity means that every quark-bilinear operator $O=\bar q M_O q$ evaluated in the blending space has expectation equal to the exact operator, with the low-mode sector exact and the high-mode sector unbiased. Consequently the all-to-all propagator is compressed from $(4N_cN_TN_L^3)^2$ to $(4N_T(N_e+N_{st}))^2$ degrees of freedom. Numerical checks include the pion conserved charge, the pion form factor from three- and four-point functions, and sub-percent nucleon axial charges from a joint $N_e$-dependent fit.

Load-bearing premise

The $g_A$ extraction assumes that two effective excited states, one near the nucleon-pion threshold and one near 1.4 GeV, whose masses and $N_e$-dependent weights are fitted to the same data, capture all excited-state contamination in the chosen fit ranges.

Editorial extensions

If this is right

  • Any quark-bilinear current, including disconnected quark loops and four-point currents, can be evaluated from the same projected propagators instead of new sequential inversions.
  • The compression of the all-to-all propagator from $(4N_cN_TN_L^3)^2$ to $(4N_T(N_e+N_{st}))^2$ degrees of freedom makes large spatial volumes and small lattice spacings comparatively cheaper than in standard all-to-all approaches.
  • Varying $N_e$ changes the smearing of the external hadron, so excited-state contamination can be mapped out systematically; the joint three-state fit gives $g_A^{u-d}=1.2339(43)$ with 40 configurations at $m_\pi\simeq300$ MeV.
  • The consistency of three- and four-point pion form factors demonstrates that $N$-point correlation functions with two current insertions are practical, opening a route to QED corrections and hadronic-tensor calculations.

Reading between the lines

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

  • If the unbiasedness identity holds at finite $N_{st}$ with the quoted scaling, the same blending-space propagators can be reused for many operators and momenta, so the marginal cost of a new $N$-point observable should become a contraction cost rather than new fermion inversions, an economy the paper only partially exploits.
  • The effective second excited state at $\Delta_2\simeq1.42$ GeV is likely a mixture of single-particle and multiparticle states; a GEVP analysis with $\pi N$ and $\pi\pi N$ interpolators on the same blending-space propagators would show whether the three-state model has hidden state-dependent bias in $g_A$.
  • The paper notes a difference between its coarse and fine ensemble results for $g_A^{u-d}$; a dedicated continuum-extrapolation study would determine how much of that gap is discretization error versus finite-volume effects.
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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

2 major / 9 minor

Summary. The paper introduces the 'blending' method for estimating all-to-all fermion propagators in lattice QCD. At each time slice, the spatial Dirac-color-site space L is split into a low-mode subspace L1 spanned by Ne eigenvectors of the Laplace operator and its complement L2, and the identity on L is represented in expectation by the Ne exact low modes plus Nst random vectors sampled from L2, with reweighting factors ωn = ([L2]−n)/(Nst−n). Propagators projected onto this blending space compress the all-to-all propagator from (4NcNTN_L^3)^2 to (4NT(Ne+Nst))^2 degrees of freedom, and quark-bilinear operators are reweighted to be unbiased in expectation; the supplement proves this single-bilinear unbiasedness (Eq. 12) using the Haar-random moments of the noise vectors. Numerical anchors include: the pion conserved vector charge R_V^π = 1.0002(29) on C24P29; pion form factors from 3-point and 4-point functions that agree (charge radii 0.332(29) and 0.329(33) fm^2); RI/MOM Z_q from the blended propagators matching the point-source result; and, on F48P30 (a = 0.077 fm, mπ ≈ 300 MeV, 40 configurations), a joint 3-state fit over eight Ne values giving g_A^{u−d} = 1.2339(43) and flavor-decomposed g_A^u = 0.895(15), g_A^d = −0.338(15), g_A^s = −0.0245(72), g_A^{u+d+s} = 0.533(28). The authors argue the method makes disconnected diagrams, excited-state control with distillation-like smearing, and N-point functions (QED/isospin corrections, hadronic tensors) practical.

Significance. Assuming the central claim survives, this is a significant methodological contribution: it restores the high-frequency spatial modes that distillation discards, in an unbiased stochastic way, while retaining distillation's excited-state control and multi-operator flexibility; if it generalizes as claimed, it makes disconnected and multi-current correlators with excited-state systematics tractable on large volumes. The strengths are concrete: the unbiasedness proof for a single bilinear is an explicit moment construction (Supplemental Eqs. 12–23), not a fitted result; the numerical anchors are independent and quantitative (conserved vector charge 1.0002(29); RI/MOM Z_q matching point-source propagators in Supp. Fig. 9; 3pt/4pt form-factor agreement); and g_A^{u−d} = 1.2339(43) agrees with high-statistics external determinations (RQCD 1.238(24), CalLAT 1.235(11)), which is nontrivial evidence that the estimator is not tuned to its own constants. The scaling claims (statistical advantage ∝ N_L^{−3/2}, noise saturation at Nst ~ 100–400) are supported by the tests shown.

major comments (2)
  1. [Supplemental Sec. A.1, Eqs. (12), (21), (25); Abstract/Introduction/Summary] The claim that the method enables 'arbitrary-point correlation functions' is not established for multiple quark-bilinear insertions that share a time slice. The proof in Supplemental Sec. A.1 establishes only the two-leg unbiasedness of Eq. (12) for a single kernel, and the fourth-moment factorization in Eq. (25) is valid only for independent time slices. For two insertions at the same time slice, the estimator is a product of two sums over the same Nst random vectors, and its expectation requires contracting the same-time fourth moments of Eq. (21) with the product of two Ω^(2) weight tensors, a computation the paper does not perform; the sentence 'The proof for more non-trivial cases can be obtained using the similar procedure' is therefore not justified. A minimal check isolates the problem: for Ne = 0, D = 2, Nst = 1, the single-current estimator Ĵ(x) = ω0|η(x)|^2 with ω0 = 2 is unbiased (E[Ĵ] = 1), but E[Ĵ(x)Ĵ(y)] = 4E[|η(x)|^2|η(y)|^2] = 4/(D(D+1)) = 2/3 ≠ 1 for x ≠ y. The demonstrated applications are not affected — the 4pt pion form factor uses t2 − t1 ≥ 2a, and the RI/MOM Z_q check in Fig. 9 validates the three-leg weight Ω^(3) empirically — so the required repair is either a proof for coincident insertions (e.g., a four-leg analogue of Eq. (12) with a weight tensor that is not the product of two Ω^(2) factors) or an explicit restriction of the claim to insertions on distinct time slices.
  2. [Sec. C.2, Eqs. (49)–(50), Table IV] The headline results (g_A^{u−d} = 1.2339(43), g_A^u = 0.895(15), g_A^d = −0.338(15), g_A^s = −0.0245(72), g_A^{u+d+s} = 0.533(28)) are quoted with jackknife statistical errors only. The 3-state ansatz of Eqs. (49)–(50) represents the entire excited-state contamination by two effective states with Δ1 ≈ 0.49 GeV and Δ2 ≈ 1.4 GeV, and the fit parameter c11 = −0.08(53) is effectively unconstrained; no variation of the number of states, fit ranges, or the set of Ne values is reported to bound the resulting systematic uncertainty. Moreover, the C24P29 ensemble at a = 0.105 fm yields g_A^{u−d} = 1.1690(72), about 5% lower than the F48P30 result, an offset the paper attributes to 'discretization errors and finite volume effects' without quantifying it. For claims such as 'sub-percent determinations' and 'first principle high accuracy' in the title, Introduction, and Abstract, the paper needs a systematic error budget (fit-ansatz variation, renormalization-scale uncertainty from Z_s,MS_A/Z_V = 1.115(15), and an explicit statement that the result is a single-lattice-spacing, fixed-pion-mass demonstration), or the claims must be scoped accordingly.
minor comments (9)
  1. [Main text Eq. (8)] The excited-state term appears as 'ci0di(e^{−Δi(tf−t)} + e^{−Δi t}) ciid^2_i e^{−Δi tf}', which is missing the '+' before the cii term; compare with the correct form in Supplemental Eq. (49).
  2. [Main text Eq. (1)] The definition 'ωn = ([L2]−n)/)/(Nst−n)' contains a stray '/)' and should read (([L2]−n)/(Nst−n)); the superscripts on Ω (Ω^(1), Ω^(2)) are also used before being defined.
  3. [Abstract] The LaTeX '{\color{black} ...}' markup around the g_A values should be removed before submission.
  4. [Supplemental Eq. (37) and Eq. (36)] The conditions defining Ω^(3) are incomplete: 'i≠j≠k>Ne' does not specify whether the indices are pairwise distinct, and 'all the permutation of i,j,k' does not enumerate the weights for mixed cases such as i=j>Ne with k≤Ne; additionally, 'Ω(2)ij/V' in Eq. (36) contains a dangling '/V'.
  5. [Fig. 13 and Sec. C.2] The text refers to 'Ne = 80 (left lower panel)', but the panels are labeled Ne = 40, 55, 70, 100; the text should match the figure labels.
  6. [Sec. C.2] The two-state fit at the 'optimal' Ne value is reported to give exactly g_A^{u−d} = 1.2339(43), the same value as the three-state joint fit; please clarify whether this is an independent fit and how its uncertainty was obtained.
  7. [Main text references] The generic reference to 'Supplemental materials [33]' in the main text should be replaced by pointers to the specific supplement sections (A.1, A.4, B, C) and equation numbers.
  8. [Supplemental Sec. A.5] The sentence 'the statistical uncertainty of ⟨V4^cc⟩π scales approximately as Nst^{−1}, as the expectation value of RVπ is exactly one' conflates the mean and the variance of the ratio; please rephrase (the ratio's expectation value is exactly one, and its variance is what scales as Nst^{−2} for the quoted Nst^{−1} uncertainty).
  9. [Supplemental Eqs. (10)–(13)] The floor notation '⌊L2⌋' is used for the dimension of L2, which is already an exact integer [L2] = NcN_L^3 − Ne; please use [L2] consistently.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the blending estimator is a direct moment construction and physics results are externally benchmarked; flagged items are an omitted proof and a non-load-bearing self-citation, not circular reductions.

full rationale

The derivation chain is not circular. The blending estimator's unbiasedness (Eqs. 1-3, 12-24) is a direct moment construction: the weights omega_n are fixed by the noise probability density, and the fourth-moment identities in Eq. (21) are solved from unitary invariance plus normalization/orthogonality constraints, not from the target expectation. Independent checks anchor the construction: the conserved-current ratio gives 1.0002(29) (Fig. 1), the pion dispersion relation is consistent with the continuum, and the pion charge radius from 3pt/4pt agrees with an external chiQCD result. The g_A values are extracted by standard spectral fits (Eqs. 49-50); the three-state ansatz is fitted to the lattice data, not an input that forces the result, and the numbers agree with external RQCD and CalLAT calculations. The singlet renormalization factor Z_s^A/Z_ns^A = 1.054(14) is a self-citation (Ref. [2], co-authored by Y.-B. Yang), but it is an independent published lattice input used only at the renormalization stage, not the basis of the blending method, and it is externally falsifiable. The supplement's statement that 'the proof for more non-trivial cases can be obtained using the similar procedure' is an omitted proof for same-time multi-current insertions and is a completeness/correctness concern, not a circular reduction; the demonstrated applications with currents at different time slices rely on the independent-time-slice factorization of Eq. (25). No step reduces a prediction to a fitted parameter or to a self-citation chain.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The estimator itself rests on standard lattice QCD plus the random-vector construction and fourth-moment identities. The headline gA extraction adds a phenomenological 3-state fit model and external renormalization inputs. Ne and Nst are algorithmic hyperparameters chosen by saturation tests; the fit parameters and renormalization constants are fitted to data and therefore carry the model dependence of the demonstration.

free parameters (4)
  • Ne, Nst subspace sizes = Ne=100, Nst=200 (F48P30); Ne=100, Nst=400 (C24P29)
    Chosen by hand from saturation tests (Figs. 5-10); not fitted to physics but set the bias-variance tradeoff of the estimator.
  • 3-state fit parameters Z(Ne), d1(Ne), d2(Ne), Delta1, Delta2, c10, c20, c11, c21, c22, m0 = Tables IV-VI; gA^u-d c00=1.2339(43), Delta1=0.493(27), Delta2=1.420(74) GeV
    Fitted jointly to 2pt/3pt correlation data to extract gA; the target gA is itself a free parameter of the fit. Standard but model-dependent.
  • Renormalization constants Zns_A/ZV and Zs_MS(2 GeV)/ZV = 1.05794(29) and 1.115(15)
    Obtained by polynomial fit to RI/MOM renormalization factors over mu in [3 GeV, sqrt(10/a^2)] using Eq. (39); used to renormalize axial charges.
  • z-expansion coefficients a0, a1, a2 for f_pi(Q^2) = a0 ~ 0.783, a1 ~ -2.15, a2 ~ 3.3
    Fitted to the 3pt/4pt form factors to extract <r_pi^2>; demonstration only.
assumptions (5)
  • domain assumption Euclidean lattice QCD with Nf=2+1 dynamical quarks is the correct nonperturbative definition of QCD for these observables.
    The paper uses CLQCD ensembles C24P29 and F48P30; no derivation of the lattice action is given.
  • domain assumption Low-lying eigenvectors of the stout-smeared discrete Laplace operator form a subspace that supports hadron interpolation fields with controlled excited-state overlap.
    Used in Eqs. (1)-(4) to construct external pion/nucleon states following the distillation method.
  • standard math The random vectors sampled from L2, after deflation and Gram-Schmidt, have a unitary-invariant joint distribution with the fourth moments in Eq. (21).
    Needed for the fourth-moment relations that prove unbiasedness of the estimator.
  • ad hoc to paper The 3-state fit in Eqs. (49)-(50) with two effective excited states fully describes the excited-state contamination over the chosen t and tf ranges.
    Two effective excited states with fitted masses and Ne-dependent weights are used; no proof is given that this truncation is exact.
  • domain assumption The singlet renormalization factor Zs_A/Zns_A=1.054(14) from Ref. [2] and the perturbative matching Eq. (39) are valid for this lattice action and scale.
    Input to Eq. (40) for flavor-decomposed gA; not reproduced in this paper.

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Cite this review

Pith. "Pith review of Realization of all-to-all fermion propagator for the first principle high accuracy strong interaction prediction." pith.science (2026). https://pith.science/paper/6VDGLF36

@misc{pith2026250501719,
  author       = {Pith},
  title        = {Pith review of: Realization of all-to-all fermion propagator for the first principle high accuracy strong interaction prediction},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6VDGLF36}},
  note         = {Machine review of arXiv:2505.01719}
}
abstract

We propose a ``blending" algorithm that projects the all-to-all fermion propagator onto spatial low-frequency modes (LFM) combined with a stochastic estimate of spatial high-frequency modes (SHFM) at each time slice. This approach enables the calculation of arbitrary-point correlation functions for arbitrary hadron states in strongly interacting quantum field theories (QFT) with fermions, such as quantum chromodynamics (QCD). Specifically, LFM allows the construction of spatially extended hadron states below a certain energy threshold by diagonalizing multi-fermion interpolation fields. Meanwhile, the local interactions required for N-point correlation functions in QFT can be approximated in an unbiased manner through a reweighted summation of both LFM and SHFM contributions. To demonstrate the efficiency of this algorithm, we obtained {\color{black} $g_A^u=0.895(15)$, $g_A^d=-0.338(15)$, $g_A^s=-0.0245(72)$, $g_A^{u+d+s}=0.533(28)$ and $g_A^{u-d}=1.2339(43)$ } for nucleon at $m_{\pi}=300$ MeV and $a=0.077$ fm using 40 configurations. The consistency check of the pion electric form factor and charge radius derived from 3-point and 4-point correlation functions is also provided.

Figures

Figures reproduced from arXiv: 2505.01719 by the authors.

Figure 1
Figure 1. FIG. 1. The [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. The figure shows the form factors extracted from the [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (12 more)
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. The zero momentum two point function (normalized by the saturated value) and its effective mass of Pion on C24P29 [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]
Figure 6
Figure 6. Figure 6: reveals that c π,sub 2 contributes no more than 25% to c π,BLD 2 , demonstrating that the omitted off-diagonal components are crucial for obtaining an unbiased approximation of c π,PP 2 . The pion energy Eπ(⃗p) with different ⃗p can be obtained from C π,DST 2 using the…
Figure 7
Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p011_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. The [PITH_FULL_IMAGE:figures/full_fig_p011_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. Left panel [PITH_FULL_IMAGE:figures/full_fig_p013_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10. Statistical uncertainty of nucleon matrix element [PITH_FULL_IMAGE:figures/full_fig_p014_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11. Illustration of the 3-pf and 4-pf needed by the pion electrical form factor. The quark propagators with different color [PITH_FULL_IMAGE:figures/full_fig_p014_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12. The left panel shows the symmetrized ratio [PITH_FULL_IMAGE:figures/full_fig_p015_12.png]
Figure 13
Figure 13. Figure 13: FIG. 13 [PITH_FULL_IMAGE:figures/full_fig_p017_13.png]
Figure 14
Figure 14. Figure 14: FIG. 14. The rescaled weights of the effective first ( [PITH_FULL_IMAGE:figures/full_fig_p018_14.png]
Figure 15
Figure 15. Figure 15: FIG. 15 [PITH_FULL_IMAGE:figures/full_fig_p019_15.png]

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    3-state fit of gu−d A In Fig. 13, we present RAu−d N (tf,t ) for various t and tf values, using Ne = 40 (left upper panel), 55 (right upper panel), 80 (left lower panel), and 100 (right lower panel). All panels display the same gray bands representing the 18 ground-state matri...

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    3-state fits of the flavor decomposed gA Our calculations on the F48P30 ensemble reveal that the Ne-dependence of bothRAs N andRAu+d+s N is significantly weaker than expected across various t and tf values, as demonstrated in Fig. 15. These results incorporate the disconnected...

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Reviewed August 16, 2026 · model on record in the stance chip above.