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Investigation of $\pi N$ contributions to nucleon matrix elements

T0 review · 2 major / 7 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Nucleon matrix elements can be cleaned of pion-nucleon contamination without computing the costliest correlation function.

desk verdict Useful cost-saving trick undercut by an unquantified zeroing of the diagonal Npi term and an abstract that overclaims. read the letter →

arxiv 2412.07263 v1 pith:M7ZTKY7C submitted 2024-12-10 hep-lat hep-phnucl-th

Constantia Alexandrou , Giannis Koutsou , Yan Li , Marcus Petschlies , Ferenc Pittler This is my paper · ORCID
classification hep-lathep-phnucl-th MSC 81T2581V05
keywords latticeQCDnucleonmatrixelementsgeneralizedeigenvalueproblempion-nucleonexcitedstatesexcited-statecontaminationaxialchargepseudoscalarformfactorstwisted-massfermions
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 paper tries to establish that a generalized eigenvalue problem (GEVP) built from a nucleon interpolator and a pion-nucleon interpolator can suppress the dominant excited-state contamination in nucleon matrix elements while skipping the most expensive three-point correlation function: the one with pion-nucleon interpolators at both source and sink. On a physical-pion-mass ensemble ($m_\pi = 131$ MeV), the authors form an improved nucleon operator from GEVP eigenvectors and use current-independent weights so that the off-diagonal terms $\langle N|O|N\pi\rangle$ and $\langle N\pi|O|N\rangle$ cancel. They set the diagonal $\langle N\pi|O|N\pi\rangle$ term to zero, relying on the observation that it falls faster than the off-diagonal contamination as the time separations grow. The payoff is a clear reduction of excited-state effects for the isovector pseudoscalar and axial currents, while the nucleon $\sigma$-term ratios are essentially unchanged, indicating that its contamination is not dominated by the lowest $N\pi$ state.

What carries the argument

The carrying object is the two-operator GEVP on the correlation matrix $C_{jk}(t) = \langle J_j(t) \bar J_k(0) \rangle$ with basis $\{J_N, J_{N\pi}\}$. Solving it gives an eigenvector $v_0$ that defines an improved interpolator $\tilde J_N = v_{0,N} J_N + v_{0,N\pi} J_{N\pi}$. The three-point combination $I_d$ uses weights fixed by the eigenvector matrix, with $d_{N\pi,N\pi}=0$, so the expensive $\langle J_{N\pi} O \bar J_{N\pi}\rangle$ correlator is never evaluated. These weights cancel the off-diagonal contaminations, and the paper argues that the diagonal term is suppressed at the time separations used; to make that argument stable, the analysis fixes $t-t_0$ rather than a small reference time $t_0$, because the eigenvectors show strong $t_0$ dependence.

What would settle it

On the same ensemble, compute the omitted three-point function $\langle J_{N\pi}(t_s) O(t_{\rm ins}) \bar J_{N\pi}(0)\rangle$ and compare the combination $I_d$ with the full GEVP combination $I$ over the same separations. If the difference is comparable to the statistical errors, or if the improved ratios keep drifting with $t_s$, the assumption that the diagonal term decays faster is falsified. In a cheaper version, check the plateau: the GEVP-improved ratio should become time-independent beyond the fitted range; a residual slope or a plateau that sits outside the continuum-limit band would signal leftover contamination.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central claim is that the combination $I_d$ of three-point functions, built from GEVP weights with the diagonal pion-nucleon term omitted, still isolates $\langle N|O|N\rangle$ because the diagonal $\langle N\pi|O|N\pi\rangle$ contamination decays faster than the off-diagonal terms. The weights are $d_{N,N}=1-W^*W$, $d_{N,N\pi}=1+W^*$, and $d_{N\pi,N}=1+W$, with $W = 1/(v_{0,N}[v^{-1}]_{N,0}) - 1$, so they are fixed by the two-point correlation matrix and do not depend on the inserted current. The decisive observation is that $\langle N\pi|O|N\pi\rangle$ decays faster than the off-diagonal terms as the source-sink and source-insertion separations increase. At $m_\pi = 131$ MeV the method makes the parity-zero pseudoscalar ratio consistent with zero, brings the timelike component of the isovector axial charge into agreement with the continuum-limit reference, and flattens the time dependence of the isovector pseudoscalar form factor, while leaving the $\sigma$-term extraction unchanged.

Load-bearing premise

The load-bearing premise is that the pion-nucleon-to-pion-nucleon three-point correlation decays faster than the cross-terms involving one nucleon at the time separations used, so setting its weight to zero does not bias the matrix element; if that decay is not fast enough, the improved operator simply inherits the contamination it was meant to remove.

Editorial extensions

If this is right

  • The isovector pseudoscalar and axial channels no longer need the diagonal $\langle J_{N\pi}O\bar J_{N\pi}\rangle$ three-point function to control $N\pi$ contamination at physical pion mass, freeing computer time for more configurations or longer separations.
  • A GEVP-improved operator makes the timelike axial component agree with the continuum-limit comparison, supporting a reliable $g_A$ extraction from this ensemble.
  • The absence of improvement in the sigma-term ratios implies that its contamination is not dominated by the lowest $N\pi$ state, redirecting future excited-state studies to other states.
  • The inclusion of disconnected isovector quark-loop diagrams, absent in the comparison continuum study, can soften the lattice-spacing dependence seen in the induced pseudoscalar form factor.

Reading between the lines

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

  • Because the diagonal term is never computed, the same construction can be transplanted to other baryons or to $N\pi\pi$ systems whenever one multihadron channel dominates and its diagonal three-point function is the expensive one.
  • A direct test of the method's key assumption would be a single calculation of the omitted diagonal three-point function on a small ensemble; comparing $I_d$ with the full $I$ would convert the decay-rate observation into a quantitative bias estimate.
  • The strong reference-time dependence of the GEVP eigenvectors suggests that results should be checked with more than two interpolators, since a larger basis would make the diagonal suppression less dependent on the eigenvector convention.
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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 / 7 minor

Summary. This proceedings paper proposes a GEVP-based method to reduce N-pi excited-state contamination in nucleon matrix elements while avoiding the computationally most expensive three-point function, <J_Npi O J_Npi>. The authors build a two-operator basis (N, N-pi), solve the GEVP, define a weighted combination I_d of the three cheaper three-point functions (Eq. 5), set the diagonal N-pi-N-pi weight to zero, and choose the remaining weights (Eqs. 6-7) to cancel the off-diagonal N-N-pi contaminations. The method is applied to scalar, vector, pseudoscalar, axial, and tensor currents on an N_f=2 twisted-mass ensemble at m_pi=131 MeV with 1228 configurations. The paper reports that the GEVP improves the isovector pseudoscalar and axial channels, most notably the timelike axial charge, which moves into agreement with the continuum-limit result of Ref. [9], while it does not improve the sigma-term ratios. A parity-zero pseudoscalar ratio that should vanish is used as a consistency test.

Significance. If the key assumption in Sec. 4 were controlled, this would be a practically valuable method: it reduces the dominant N-pi contamination using only the three cheaper three-point functions, and the parity-zero pseudoscalar test is a genuine operator-level check. The authors also include disconnected contributions for isovector operators, which is a nontrivial technical step. The central weakness is that the method drops the diagonal N-pi-N-pi three-point function on the basis of an asserted and unquantified decay hierarchy; the provided tests do not directly bound the residual. The value of the paper therefore depends on whether that residual can be shown to be negligible, either by a direct estimate or by a clear model-based argument.

major comments (2)
  1. [Sec. 4, Eq. (5)] The decision to set d_{Npi,Npi}=0 rests on the claim that the diagonal <Npi|O|Npi> contamination decreases faster than the off-diagonal terms as the time separations increase. This is not derived or quantified. In a spectral decomposition relative to the ground N-N term, the Npi-Npi term is suppressed as exp[-(E_Npi-E_N) t_s], the N-Npi term as exp[-(E_Npi-E_N) t_ins], and the Npi-N term as exp[-(E_Npi-E_N)(t_s-t_ins)]. In the plateau plots of Figs. 3-5, t_s is fixed and t_ins is varied, so the omitted diagonal term is independent of t_ins while the off-diagonal terms decay; the plotted improvement with t_ins therefore does not demonstrate suppression of the residual. At m_pi=131 MeV the energy gap is only about m_pi, so exp[-(E_Npi-E_N) t_s] is not negligible for the t_s values shown. Please provide a quantitative estimate of this residual, for example by computing <Npi|O|Npi> on a subset of configurations or by including it in a model fit; without this, the central claim that I_d isolates <N|O|N> is not established.
  2. [Sec. 5, Figs. 3-4] The two external checks do not constrain the omitted diagonal term. The parity-zero pseudoscalar test (Fig. 3) is a null-channel check: it demonstrates cancellation of the contaminating contributions relevant to that operator, but it does not measure the size of the diagonal <Npi|O|Npi> term for a general operator O, and for the pseudoscalar channel that diagonal term may vanish for the same parity reason. The agreement of the timelike axial ratio with the continuum-limit band of Ref. [9] (Fig. 4, second row) is suggestive, but it is obtained at a single lattice spacing with a different action and including disconnected contributions that were not present in Ref. [9]; it cannot by itself establish that the omission of the diagonal term is negligible. Please clarify what these tests can and cannot establish, and if possible add a channel where the omitted term has a known nonzero value.
minor comments (7)
  1. [Sec. 4, Eqs. (6)-(7)] The weights d_N,N, d_N,Npi, and d_Npi,N are stated without derivation. Because the cancellation property of I_d depends on these formulas, please either give the derivation or point to the specific section of Ref. [4] where it appears.
  2. [Abstract and Sec. 5] The abstract states that Npi contamination is minimized for the scalar, vector, pseudoscalar, axial, and tensor currents, but Section 5 reports no significant improvement for the majority of cases, including the sigma-term ratios. Please align the abstract with the body of the paper.
  3. [Eq. (1)] In Eq. (1), the limit "t_s - t_sink -> infinity" is not defined; the sink time is t_s, so this should likely be "t_s - t_ins -> infinity" or a similar expression for the source-sink separation.
  4. [Eq. (2) and references] There is a typo in "nu_{jk}(t,t+0)"; the second argument should be t_0. Also, the reference list entry for Ref. [8] begins with "A. Collaboration"; this should read "ALPHA Collaboration" or similar.
  5. [Sec. 5, before Fig. 5] The text before Fig. 5 contains "thre-point functions"; it should be "three-point functions".
  6. [Sec. 4] The phrase "when the source-sink and source-insertion time separations increase" is ambiguous; please specify whether t_s and t_ins are varied independently, at fixed t_s, or with t_s-t_ins fixed, since the spectral suppression factors are different in each case.
  7. [Sec. 5, Figs. 4-5] Please state how the statistical uncertainty of the GEVP eigenvectors and of the weights d in Eq. (6) is propagated into the final ratios; the plots show bands for the model averages but not the treatment of the eigenvector errors.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: GEVP weights are current-independent two-point quantities; the omitted diagonal Npi term is an unquantified assumption, not a fitted input.

full rationale

The paper's derivation chain is self-contained and does not reduce to its inputs. The improved operator in Eq. (3) and the weights d in Eqs. (6)-(7) are obtained from the GEVP eigenvectors of the two-point correlation matrix only; they do not depend on the insertion operator O and are not fitted to any of the three-point matrix elements being reported. The target <N|O|N> is therefore not used to tune the method. The central approximation is the setting d_Npi,Npi=0 in Eq. (5), justified by the 'key observation' in Sec. 4 that the diagonal <Npi|O|Npi> contamination decays faster than the off-diagonal terms. This is an unquantified modeling assumption, and the parity-zero pseudoscalar test does not constrain it because <Npi|P|Npi> also vanishes by parity; however, an assumption is not circularity, since none of the equations defining I_d is equivalent to the target matrix element. The self-citations to Ref. [4] (same authors' detailed paper) and Ref. [9] (continuum-limit benchmark with overlapping authorship) are used for details and external comparison, respectively, and the latter provides an independent check (agreement in the timelike axial channel) rather than the justification of the diagonal-term omission. The remaining concern is correctness risk regarding the magnitude of the omitted diagonal contamination, not circularity.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central derivation rests on standard spectral-decomposition mathematics and the GEVP formalism, plus a domain assumption that the lowest N pi state dominates contamination. The paper-specific assumption that the diagonal N pi-N pi term decays faster and can be omitted is the main unquantified input. No new physical entities are introduced. The analysis parameters listed are hand-chosen (t0, fit ranges), not fitted to the advertised improvement.

free parameters (2)
  • GEVP reference time t0 = t0/a = 2 used as small reference, with stability checked by fixing t-t0
    The GEVP eigenvectors and therefore the optimized interpolator and weights depend on t0; the paper shows visible t0-dependence in Fig. 1 and addresses it by fixing t-t0. This is a hand-chosen analysis parameter, not fitted to the target matrix element.
  • Two-state fit insertion-time ranges (t_ins,min) = 0.1 to 0.5 fm depending on channel
    The model-average bands and reduced chi2 values in Figs. 2, 4, 5 depend on the chosen fit ranges. These are standard analysis choices that affect the reported bands but not the qualitative GEVP improvement.
assumptions (4)
  • standard math Spectral decomposition of two- and three-point correlation functions into energy eigenstates
    Used throughout Secs. 1, 3, and 4 to relate time dependence of ratios to energy gaps and matrix elements, e.g., Eq. (1).
  • domain assumption A single pion-nucleon state with nucleon quantum numbers captures the dominant excited-state contamination
    Motivates the two-operator GEVP basis (N, N pi) in Sec. 3 and the interpretation of the improved behavior in Sec. 5.
  • ad hoc to paper The diagonal <Npi|O|Npi> contamination decays faster than the off-diagonal terms, so it can be dropped
    This is the key observation in Sec. 4 immediately after Eq. (5); it is not derived or quantified and is what justifies omitting the costly three-point function.
  • domain assumption Twisted-mass isospin breaking at finite lattice spacing makes disconnected quark loops nonzero for isovector currents
    Stated in Sec. 2; the paper includes these contributions and argues they alter the a^2 dependence compared to Ref [9].

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Pith. "Pith review of Investigation of $\pi N$ contributions to nucleon matrix elements." pith.science (2026). https://pith.science/paper/M7ZTKY7C

@misc{pith2026241207263,
  author       = {Pith},
  title        = {Pith review of: Investigation of $\pi N$ contributions to nucleon matrix elements},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/M7ZTKY7C}},
  note         = {Machine review of arXiv:2412.07263}
}
abstract

We investigate an improved method to extract nucleon matrix elements from lattice 3-point functions using a generalized eigenvalue problem (GEVP) with nucleon and pion-nucleon interpolating fields. Our method avoids the computation of the costly three-point functions that have pion-nucleon interpolators at both source and sink. We demonstrate that excited state contamination from $N\pi$ is minimized in nucleon matrix elements of the scalar, vector, pseudoscalar, axial, and tensor currents and discuss our results based on a physical-point ensemble with a pion mass value of 131 MeV. We find that the GEVP is most significant for the isovector pseudoscalar and axial currents.

Figures

Figures reproduced from arXiv: 2412.07263 by the authors.

Figure 1
Figure 1. Example of the 𝑡0-dependence of the effective energies (left) and eigenvectors (right) for zero total momentum. The eigenvector component used is |𝑣0,𝑁 𝜋/𝑣0,𝑁 |. 4. Optimal combination of three-point functions using GEVP For the 𝑁-𝑁𝜋 system, we obtain an improved nucleon interpolator J˜𝑁 := 𝑣0,𝑁 J𝑁 + 𝑣0,𝑁 𝜋J𝑁 𝜋 (3) by solving the GEVP, which has a larger overlap with the nucleon ground state compared to J𝑁 . Replaci… view at source ↗
Figure 2
Figure 2. Ratios that yield 𝜎𝜋 𝑁 versus 𝑡ins −𝑡𝑠/2 (left) and the results of two-state fits to two- and three-point functions versus the smallest insertion time 𝑡ins,min in the fit (right). We compare between results with (filled symbols) and without (open symbols) the use of operators improved via the GEVP. The blue (without GEVP) and red (with GEVP) bands are the results from the model average of the two-state fit to the co… view at source ↗
Figure 3
Figure 3. Ratios of the pseudoscalar insertion operator in a setup that should yield zero due to parity symmetry. The rest of the notation is the same as in the left panel of [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: As in [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 5
Figure 5. Figure 5: As in [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

10 extracted references · 2 canonical work pages

  1. [9]

    Collaboration, B

    A. Collaboration, B. Blossier, M. D. Morte, G. v. Hippel, T. Mendes and R. Sommer, On the generalized eigenvalue method for energies and matrix elements in lattice field theory, https://doi.org/10.1088/1126-6708/2009/04/094 JHEP 04 (2009) 094 [ https://arxiv.org/abs/0902.1265 0902.1265 ]

    arXiv 2009

  2. [4]

    Barca, G

    L. Barca, G. Bali and S. Collins, Towards n to n matrix elements from lattice qcd, https://doi.org/10.1103/PhysRevD.107.L051505 Phys. Rev. D 107 (2023) L051505 [ https://arxiv.org/abs/2211.12278 2211.12278 ]

    arXiv 2023

  3. [1]

    write newline

    " write newline "" before.all 'output.state := FUNCTION blank.sep after.quote 'output.state := FUNCTION fin.entry output.state after.quoted.block = 'skip 'add.period if write newline FUNCTION new.block output.state before.all = 'skip output.state after.quote = after.quoted.block 'output.state := after.block 'output.state := if if FUNCTION new.sentence out...

  4. [2]

    Gupta, S

    R. Gupta, S. Park, M. Hoferichter, E. Mereghetti, B. Yoon and T. Bhattacharya, The pion-nucleon sigma term from lattice qcd, https://doi.org/10.1103/PhysRevLett.127.242002 Phys. Rev. Lett. 127 (2021) 242002 [ https://arxiv.org/abs/2105.12095 2105.12095 ]

    arXiv 2021

  5. [3]

    Nucleon form factors and the pion-nucleon sigma term

    R. Gupta, T. Bhattacharya, V. Cirigliano, M. Hoferichter, Y.-C. Jang, B. Joo et al., Nucleon form factors and the pion-nucleon sigma term, https://doi.org/10.22323/1.430.0427 PoS LATTICE2022 (2023) 427 [ https://arxiv.org/abs/2301.07885 2301.07885 ]

    work page Pith review arXiv 2023

  6. [5]

    Alexandrou, G

    C. Alexandrou, G. Koutsou, Y. Li, M. Petschlies and F. Pittler, Investigation of pion-nucleon contributions to nucleon matrix elements, https://doi.org/10.1103/PhysRevD.110.094514 Phys. Rev. D 110 (2024) 094514 [ https://arxiv.org/abs/2408.03893 2408.03893 ]

    arXiv 2024

  7. [6]

    ETM collaboration, First physics results at the physical pion mass from n_f = 2 wilson twisted mass fermions at maximal twist, https://doi.org/10.1103/PhysRevD.95.094515 Phys. Rev. D 95 (2017) 094515 [ https://arxiv.org/abs/1507.05068 1507.05068 ]

    work page Pith review arXiv 2017

  8. [7]

    Alexandrou, S

    C. Alexandrou, S. Bacchio, P. Charalambous, P. Dimopoulos, J. Finkenrath, R. Frezzotti et al., Simulating twisted mass fermions at physical light, strange and charm quark masses, https://doi.org/10.1103/PhysRevD.98.054518 Phys. Rev. D 98 (2018) 054518 [ https://arxiv.org/abs/1807.00495 1807.00495 ]

    arXiv 2018

Show all 10 references
  1. [8]

    Extended Twisted Mass collaboration, Quark masses using twisted mass fermion gauge ensembles, https://doi.org/10.1103/PhysRevD.104.074515 Phys. Rev. D 104 (2021) 074515 [ https://arxiv.org/abs/2104.13408 2104.13408 ]

    2021 arXiv

  2. [10]

    Alexandrou, S

    C. Alexandrou, S. Bacchio, M. Constantinou, J. Finkenrath, R. Frezzotti, B. Kostrzewa et al., Nucleon axial and pseudoscalar form factors using twisted-mass fermion ensembles at the physical point, arXiv:2309.05774 [hep-lat] (2023) https://doi.org/10.48550/arXiv.2309.05774 [ h...

    2023 doi arXiv

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Reviewed August 11, 2026 · model on record in the stance chip above.