REVIEW 4 major objections 5 minor 24 references
A proposal for removing $\pi N$-state contamination from the nucleon induced pseudoscalar form factor in lattice QCD
T0 review · 4 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash
Pith's one-line read A linear combination of axial-vector rati correlators and their time derivatives completely removes the leading pion-nucleon contamination from the nucleon's induced pseudoscalar and pseudoscalar form factors.
desk verdict A genuinely new derivative-based subtraction that flattens the piN contamination in F_P and G_P on two PACS10 ensembles, but the assumed two-exponential form of the contamination is not cross-checked on the same data, so the method is promising but not yet fully nailed down. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The central object is the subtraction identity Eq. (15), which defines the cleaned form factor as the standard ratio value plus a correction involving time derivatives of the axial-vector ratio correlators: $$\widetilde F_P($q^{2}$) = -\frac{K\,\widetilde $R^{{5z}}$_{A_i}}{q_i q_3} + \frac{K}{\$\Delta$ $E_N^{2}$ - E_\$pi^{2}$}\left[\$\Delta$ E_N\,\frac{\partial_4 \widetilde $R^{{5z}}$_{A_i}}{q_i q_3} + \frac{\partial_4 $R^{{5z}}$_{A_4}}{i q_3}\right],$$ where $\Delta E_N = E_N - M_N$ and $K=\sqrt{2E_N(E_N+M_N)}$. The coefficient multiplying the derivative terms is chosen so that, after using the identity $\partial_4 \Delta_\pm = -E_\pi\Delta_\mp + \Delta E_N \Delta_\pm$, the exponentials in $\Delta_+$ and $\Delta_-$ cancel identically. For the pseudoscalar form factor, a second identity (Eq. (19)) uses the axial Ward-Takahashi relation $\Delta_P = Z_A B_0 \Delta_+$ to build an analogous cancellation, where $B_0 = M_\pi^2/(2m_{\rm PCAC})$. The method is 'simple' in the sense that it only recombines existing ratios and their discrete time derivatives; it requires no new operators, no additional gauge ensembles, and no multi-exponential fits.
What would settle it
Apply the same subtraction formula to data analyzed with explicit interacting pion-nucleon energies, for example from a two-state fit or a variational basis that includes $\pi N$ operators, and compare the cleaned form factors; if the subtracted $F_P$ or $G_P$ still shifts with $t_{\rm sep}$ or disagrees with the operator-based extraction, the non-interacting-energy assumption is insufficient and the method has not isolated the ground state.
Extended reading notes
Core claim
The paper's central claim is that the leading $\pi N$-state contamination in the nucleon's induced pseudoscalar form factor is exactly cancellable using a linear combination of the ratio correlators for the spatial and temporal axial-vector currents and their time derivatives. The contamination enters through two functions $\Delta_\pm(t,t_{\rm sep};\mathbf{q})$ that have the exponential form $B e^{-\Delta E(\mathbf{q},-\mathbf{q})t} \pm C e^{-\Delta E(0,\mathbf{q})(t_{\rm sep}-t)}$ with non-interacting energies, and these two functions obey the identity $\partial_4 \Delta_\pm = -E_\pi \Delta_\mp + (E_N-M_N)\Delta_\pm$. Substituting this identity into the ratio formulas makes both exponentials vanish, leaving only the ground-state form factor; the ground-state part of the combination is unchanged because it is a redundant linear combination of two determinations of the same $F_P$. For $G_P$, the same subtraction is applied through the axial Ward-Takahashi identity, which connects the pseudoscalar contamination $\Delta_P$ to $\Delta_+$. On the two ensembles, the subtracted $F_P$ and $G_P$ show no visible $t$ or $t_{\rm sep}$ dependence and agree with the pion-pole-dominance model, yielding $g_P^\ast$ and $g_{\pi NN}$ values that are consistent with experimental determinations and are more precise than the experimental $g_P^\ast$.
Load-bearing premise
The subtraction cancels the contamination only if the leading pion-nucleon contribution takes the two-exponential form with non-interacting energies used in Eq. (13) and, for the pseudoscalar form factor, only if the axial Ward-Takahashi identity links its contamination to $\Delta_+$ exactly as in Eq. (18); higher states or interacting energies would leave part of the contamination behind.
Editorial extensions
If this is right
- Removing the two leading $\pi N$ exponentials eliminates both the $t$-dependence and the $t_{\rm sep}$-dependence of $F_P(q^2)$ and $G_P(q^2)$ in the analyzed data.
- The extracted $F_P$ and $G_P$ become consistent with the pion-pole-dominance model and with experimental muon-capture and pion-electroproduction measurements.
- The cleaned data yield values for $g_P^\ast$ and $g_{\pi NN}$ that show no residual $t_{\rm sep}$ dependence, with uncertainties smaller than experiment for $g_P^\ast$ and comparable for $g_{\pi NN}$.
- Because the method only recombines existing ratio correlators and their time derivatives, it can be applied a posteriori to any previously measured dataset that includes the $A_4$ and $A_i$ three-point functions.
- For $G_P$ the method converts the axial Ward-Takahashi identity into a working subtraction, making the pseudoscalar channel usable for physics at low $q^2$.
Reading between the lines
- The same derivative-subtraction trick could be adapted to other form factors or channels where a single dominant intermediate state with a known exponential time dependence contaminates the ratio, for example channels contaminated by the Delta resonance.
- If this subtraction proves equally effective on finer lattices and at higher momenta, the long-standing low-$q^2$ suppression of lattice $F_P$ relative to pion-pole dominance would be largely resolved.
- A direct numerical test on synthetic correlation functions with a known inserted excited state could verify the exactness of the cancellation and quantify the errors introduced by using non-interacting energies.
- The method makes the temporal axial-vector current useful again, so data sets that previously discarded the noisy $A_4$ correlator may now be reanalyzable.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript proposes a simple subtraction method to remove the leading πN excited-state contamination from the lattice QCD determination of the nucleon induced pseudoscalar form factor F_P and pseudoscalar form factor G_P. The method assumes that the πN contamination in the spatial and temporal axial-vector ratios has the two-exponential form of Eq. (13) with non-interacting energies, and then uses the combination in Eq. (15), involving time derivatives of the ratios, to cancel the Δ± terms. For G_P, the axial Ward-Takahashi identity is used to relate the pseudoscalar contamination to Δ+ via Eq. (18), leading to the subtracted expression in Eq. (19). The method is applied to the existing PACS10 data at two lattice spacings, and the resulting F_P and G_P show flat plateaus in t and t_sep that are consistent with the pion-pole dominance model and with experimental values for g_P* and g_πNN.
Significance. If the cancellation in Eq. (15) is exact, this is a practically valuable method: it avoids expensive GEVP or multi-state fits for the notoriously difficult F_P and G_P channels and exploits the previously noisy A_4 correlator. The algebra in Eqs. (11)–(15) is internally consistent, all inputs such as meson and nucleon masses, Z_A, and m_PCAC come from prior independent analyses, and the PPD model is used only for comparison, not in the extraction. The numerical results show clean plateaus at two lattice spacings and for several source-sink separations, with g_P* and g_πNN close to experiment. The main weakness is that the central cancellation is conditional on an assumed functional form for the πN contamination that is not validated on the same data; this is a validation gap rather than an internal inconsistency.
major comments (4)
- [Sec. 3, Eq. (13)] The exact cancellation claimed in Eq. (15) rests on the assertion that Δ±(t,t_sep;q) = B e^{-ΔE(q,-q)t} ± C e^{-ΔE(0,q)(t_sep-t)} with non-interacting energies and with the same coefficients B and C in both ŒR_{A_i} and R_{A_4}. This form is motivated by leading-order baryon ChPT but is not tested against the present data. The flat plateaus in Figs. 2 and 3 can only establish the method if this assumed form is actually the dominant contamination; without a same-data correlated multi-state fit or a GEVP analysis including πN operators, the statement that the leading πN contributions are 'completely eliminated' is not supported by the data shown.
- [Sec. 3, Eq. (11)] The definition of ŒR^{5z}_{A_i}(t;q) subtracts R^{5z}_{A_3}(t;q0) with q0=(q1,q2,0) and |q0|=|q|. It is not automatic that the πN contamination in this subtracted combination has exactly the same coefficients B and C as the contamination in R^{5z}_{A_4}. The paper should either derive this equality from the ChPT representation of the πN contributions or verify it numerically from the same correlators before Eq. (15) can be regarded as an exact cancellation.
- [Sec. 3, Eq. (18)] The relation Δ_P = Z_A B_0 Δ_+ is stated as a consequence of the axial Ward-Takahashi identity in Eq. (17), but Eq. (17) holds for the full correlators. Transferring it to the leading πN components requires that the identity applies order by order in the excited-state expansion and that the same coefficient Z_A M_π^2/(2m_PCAC) controls the πN part of the pseudoscalar ratio. This nontrivial assumption is not derived for the ratio combinations used here; it should be validated, for example by checking the t_sep-independence of the subtracted G_P with B_0 varied within its uncertainty and by comparing with an independent method.
- [Sec. 4, Figs. 2–3] The plateau plots after subtraction are shown only for the fine 160^4 ensemble, while the coarse 128^4 results appear only in the final q^2 plots of Fig. 4. Since the paper claims the method works at both lattice spacings, the t- and t_sep-dependence of the subtracted F_P and G_P should be shown for the coarse ensemble as well, or it should be explicitly stated that the behavior is analogous. In addition, the systematic uncertainty from using non-interacting energies in ΔE(q,-q) and ΔE(0,q), and from higher excited states beyond the πN state, is not propagated into the quoted errors for g_P* and g_πNN.
minor comments (5)
- [Sec. 2, after Eq. (7)] The relation q^2 = 2M_N(E_N(q)-M_N) for the final rest frame should state the Euclidean sign convention explicitly; in particular, the values plotted on the horizontal axes of Fig. 4 should be identified with q^2 > 0 in this convention.
- [Sec. 3, Eq. (14)] The notation ∂4 is used for the time derivative, but on the lattice this derivative must be implemented as a finite difference; the precise discretization should be specified.
- [Sec. 4, Fig. 2 caption] The caption refers to 'standard' and 'F_P' panels but the text uses F_P^std and F_P; please align the notation in the captions, text, and figures.
- [Sec. 4, final paragraph] The statement that the discretization error is less than 3–4% is not supported by any explicit continuum extrapolation; the paper should either show the extrapolation or phrase this as an estimate of the finite-a effect.
- [Sec. 5, summary] Reference [21] is cited for comparison plots of g_P* and g_πNN; since the present paper is self-contained, it would be helpful to include the comparison explicitly or make the cited proceedings available.
Circularity Check
No significant circularity: the subtraction is a designed cancellation of an explicitly assumed ChPT contamination form, and the final agreement is checked against external data and the PPD model.
full rationale
The paper's central step is Eq. (15), which adds a time-derivative combination to the standard ratio to cancel the leading pi-N contributions of the form Delta_+/-(t,t_sep;q) = B exp(-Delta E(q,-q) t) +/- C exp(-Delta E(0,q)(t_sep-t)) (Eq. 13). This is an algebraic construction, not a fit of F_P to the piN model; the coefficients B and C are never fitted to the target data, and the cancellation is exact only under the stated ansatz. The form of Delta_+- is justified by baryon ChPT through Ref. [12] (O. Baer), an external independent calculation, and by the kinematics of on-shell pions in Refs. [13,14]. Inputs such as E_N, E_pi, Z_A, and m_PCAC are obtained from standard lattice dispersion relations and pion-sector analyses, and none encodes the extracted F_P(q^2), G_P(q^2), g_P*, or g_piNN. The PPD model and experimental muon-capture/pion-electroproduction points enter only as comparison after the extraction, so agreement with them is an external check, not a construction. The G_P subtraction uses Eq. (18), derived from the axial Ward-Takahashi identity of Eq. (17). Although Eq. (17) is quoted from the authors' previous study [6], it is a standard operator identity verified on the same correlators, not a fitted assumption; hence it is not a circular premise. The self-citations to Refs. [5,6,9,10] supply the raw data sets and earlier observations of contamination, which is normal and not load-bearing for the algebraic derivation of the subtraction. The remaining concern, that the assumed two-exponential form was not cross-checked on the same data with GEVP or a correlated multi-state fit, is a validation/systematics gap and not a circularity by the definitions used here. The central claim therefore has independent content and is not forced by the paper's inputs.
Assumptions & free parameters
assumptions (3)
- domain assumption The leading piN contamination in the ratios has the form Delta±(t,t_sep;q)=B e^{-Delta E(q,-q)t} ± C e^{-Delta E(0,q)(t_sep-t)} with non-interacting energy differences (Eq. 13).
- domain assumption The axial Ward-Takahashi identity (Eq. 17) holds for the leading piN contribution, giving Delta_P = Z_A M_pi^2/(2 m_PCAC) Delta+ (Eq. 18).
- domain assumption The nucleon interpolating operator with optimized smearing gives ground-state dominance for F_A, with piN contamination concentrated in F_P and G_P.
Cite this review
Pith. "Pith review of A proposal for removing $\pi N$-state contamination from the nucleon induced pseudoscalar form factor in lattice QCD." pith.science (2026). https://pith.science/paper/SGWXEVGQ
@misc{pith2026250113490,
author = {Pith},
title = {Pith review of: A proposal for removing $\pi N$-state contamination from the nucleon induced pseudoscalar form factor in lattice QCD},
year = {2026},
howpublished = {\url{https://pith.science/paper/SGWXEVGQ}},
note = {Machine review of arXiv:2501.13490}
}
abstract
In the PACS10 project, the PACS collaboration has generated three sets of the PACS10 gauge configurations at the physical point with lattice volume larger than $(10\;{\rm fm})^4$ and three different lattice spacings. The isovector nucleon form factors had been already calculated by using two sets of the PACS10 gauge configurations. In our strategy, the smearing parameters of the nucleon interpolation operator were highly optimized to eliminate as much as possible the contribution of excited states in the nucleon two-point function. This strategy was quite successful in calculations of the electric ($G_E$), magnetic ($G_M$) and axial-vector ($F_A$) form factors, while the induced pseudoscalar ($F_P$) and pseudoscalar ($G_P$) form factors remained strongly affected by residual contamination of $\pi N$-state contribution. In this work, we propose a simple method to remove the $\pi N$-state contamination from the $F_P$ form factor, and then evaluate the induced pseudoscalar charge $g_P^\ast$ and the pion-nucleon coupling $g_{\pi NN}$ from existing data in a new analysis. Applying this method to the $G_P$ form factor is also considered with a help of the axial Ward-Takahashi identity.
Figures
Figures from the paper (2 more)
Reference graph
Works this paper leans on
-
[1]
M.L. Goldberger and S.B. Treiman,Form-factors in Beta decay and muon capture,Phys. Rev. 111 (1958) 354
work page 1958
-
[2]
W.I. Weisberger,Unsubtracted Dispersion Relations and the Renormalization of the Weak Axial Vector Coupling Constants,Phys. Rev.143 (1966) 1302
work page 1966
-
[3]
Nucleon form factors from quenched lattice QCD with domain wall fermions
S. Sasaki and T. Yamazaki,Nucleon form factors from quenched lattice QCD with domain wall fermions, Phys. Rev. D78(2008) 014510 [0709.3150]
work page Pith review arXiv 2008
-
[4]
Nambu,Axial vector current conservation in weak interactions,Phys
Y. Nambu,Axial vector current conservation in weak interactions,Phys. Rev. Lett.4 (1960) 380
work page 1960
-
[5]
E. Shintani, K.-I. Ishikawa, Y. Kuramashi, S. Sasaki and T. Yamazaki,Nucleon form factors and root-mean-square radii on a (10.8 fm)4 lattice at the physical point, Phys. Rev. D99 (2019) 014510 [1811.07292]
arXiv 2019
- [6]
-
[7]
LHPC, SESAM collaboration,Moments of nucleon generalized parton distributions in lattice QCD, Phys. Rev. D68(2003) 034505 [hep-lat/0304018]
arXiv 2003
-
[8]
QCDSFcollaboration, Nucleon electromagnetic form-factors on the lattice and in chiral effective field theory,Phys. Rev. D71(2005) 034508 [hep-lat/0303019]
work page Pith review arXiv 2005
Show all 24 references
-
[9]
PACScollaboration, Nucleon form factors on a large volume lattice near the physical point in 2+1 flavor QCD, Phys. Rev. D98(2018) 074510 [1807.03974]
2018 arXiv
-
[10]
PACScollaboration, Calculation of the derivative of nucleon form factors in Nf=2+1 lattice QCD at M𝜋=138 MeV on a (5.5 fm)3 volume, Phys. Rev. D104 (2021) 074514 [2107.07085]
2021 arXiv
-
[11]
G.S. Bali, S. Collins, M. Gruber, A. Schäfer, P. Wein and T. Wurm,Solving the PCAC puzzle for nucleon axial and pseudoscalar form factors,Phys. Lett. B789 (2019) 666 [1810.05569]
2019 arXiv
-
[12]
Bar,𝑁𝜋-state contamination in lattice calculations of the nucleon axial form factors, Phys
O. Bar,𝑁𝜋-state contamination in lattice calculations of the nucleon axial form factors, Phys. Rev. D99 (2019) 054506 [1812.09191]
2019 arXiv
-
[13]
Meyer, K
H.B. Meyer, K. Ottnad and T. Schulz,A new method for suppressing excited-state contaminations on the nucleon form factors,PoS LATTICE2018(2018) 062 [1811.03360]
2018 arXiv
-
[14]
9 A proposal for removing𝜋𝑁-state contamination from the nucleon induced pseudoscalar....Shoichi Sasaki
RQCD collaboration,Nucleon axial structure from lattice QCD,JHEP 05 (2020) 126 [1911.13150]. 9 A proposal for removing𝜋𝑁-state contamination from the nucleon induced pseudoscalar....Shoichi Sasaki
2020 arXiv
-
[15]
Barca, G
L. Barca, G. Bali and S. Collins,Toward N to N𝜋 matrix elements from lattice QCD,Phys. Rev. D107 (2023) L051505 [2211.12278]
2023 arXiv
-
[16]
Alexandrou, G
C. Alexandrou, G. Koutsou, Y. Li, M. Petschlies and F. Pittler,Investigation of pion-nucleon contributions to nucleon matrix elements, Phys. Rev. D110 (2024) 094514 [2408.03893]
2024 arXiv
-
[17]
PACScollaboration, Finite size effect on pseudoscalar meson sector in 2+1 flavor QCD at the physical point, Phys. Rev. D99 (2019) 014504 [1807.06237]
2019 arXiv
-
[18]
PACScollaboration, Finite size effect on vector meson and baryon sectors in 2+1 flavor QCD at the physical point, Phys. Rev. D100 (2019) 094502 [1907.10846]
2019 arXiv
-
[19]
Gorringe and H.W
T. Gorringe and H.W. Fearing,Induced Pseudoscalar Coupling of the Proton Weak Interaction,Rev. Mod. Phys.76(2004) 31 [nucl-th/0206039]
2004 arXiv
-
[20]
Choi et al.,Axial and pseudoscalar nucleon form-factors from low-energy pion electroproduction, Phys
S. Choi et al.,Axial and pseudoscalar nucleon form-factors from low-energy pion electroproduction, Phys. Rev. Lett.71(1993) 3927
1993
-
[21]
Tsuji, Y
R. Tsuji, Y. Aoki, K.-I. Ishikawa, Y. Kuramashi, S. Sasaki, K. Sato et al.,Studies of nucleon isovector structure with the PACS10 superfine lattice, in41st International Symposium on Lattice Field Theory, 11, 2024 [2411.16784]
2024 arXiv
-
[22]
Nucleon Matrix Elements (NME) collaboration,Precision nucleon charges and form factors using (2+1)-flavor lattice QCD,Phys. Rev. D105 (2022) 054505 [2103.05599]
2022 arXiv
-
[23]
Precision Neutron Decay Matrix Elements (PNDME) collaboration, Nucleon isovector axial form factors, Phys. Rev. D109 (2024) 014503 [2305.11330]
2024 arXiv
-
[24]
Extended Twisted Mass collaboration, Nucleon axial and pseudoscalar form factors using twisted-mass fermion ensembles at the physical point,Phys. Rev. D109(2024) 034503 [2309.05774]. 10
2024 arXiv
Reviewed August 10, 2026 · model on record in the stance chip above.
Discussion (0). Continue with ORCID to comment.