Error Scaling of Sea Quark Isospin-Breaking Effects

Agostino Patella; Alessandro Cotellucci; Anian Altherr; Isabel Campos; Letizia Parato; Marina Krsti\'c Marinkovi\'c; Paola Tavella; Roman Gruber; Sara Rosso; Tim Harris

REVIEW 3 major objections 7 minor 10 references

Error Scaling of Sea Quark Isospin-Breaking Effects

T0 review · 3 major / 7 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read This paper reports lattice QCD measurements showing that the stochastic errors of sea-quark isospin-breaking corrections grow as √V with volume but do not yet show the predicted 1/a (strong) or 1/a^2 (electromagnetic) continuum…

desk verdict A useful preliminary numerical check of error scaling for sea-sea isospin-breaking diagrams; data are consistent with sqrt(V) and show no continuum divergence in the tested range, but the theoretical benchmark is deferred. read the letter →

arxiv 2502.03145 v2 pith:4BJ25T6L submitted 2025-02-05 hep-lat

Anian Altherr , Isabel Campos , Alessandro Cotellucci , Roman Gruber , Tim Harris , Marina Krstić Marinković , Letizia Parato , Agostino Patella

show 2 more authors

Sara Rosso Paola Tavella

This is my paper · ORCID

classification hep-lat MSC 81T2581V05 PACS 11.15.Ha12.38.Gc

keywords sea-quarkisospinbreakinglatticeQCDQCD+QEDstochasticerrorscalingall-to-allpropagatorsWilsonfermionsC-periodicboundaryconditionssea-seadiagrams

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

Strong and electromagnetic isospin-breaking effects—the corrections caused by the up and down quarks having different masses and by the electric charge—are needed for sub-percent precision in lattice QCD, but the sea-quark part requires all-to-all quark propagators and is usually estimated with random sources. It had been argued that the statistical error of these estimates should diverge as 1/a for strong isospin breaking and as 1/$a^{2}$ for electromagnetic breaking in the continuum limit, threatening a bottleneck. This paper measures those errors with N_f=3 Wilson fermions at a pion mass of about 415 MeV, on lattices with spacings 0.05, 0.075, and 0.1 fm and volumes 1.6 to 3.2 fm. The data do not show those leading divergences in this range: the dominant mass and tadpole diagrams reach the gauge-noise plateau with roughly ten stochastic sources, and the error scales as √V with volume while rising only mildly (or, for the mass diagram, decreasing) toward the continuum. The result matters because it suggests stochastic estimation of sea-sea isospin-breaking corrections is practical on these ensembles, while the √V volume growth will eventually require position-space or subvolume methods.

What carries the argument

The argument runs on the diagrammatic expansion of QCD+QED around the isosymmetric theory (the RM123 expansion), in which sea-quark isospin-breaking effects appear as insertions of mass-shift and electromagnetic operators; the traces are estimated stochastically with a two-level frequency-splitting estimator and a hopping expansion tuned at the charm mass. The error analysis separates gauge, source, and photon variance, and the gauge error is isolated by increasing the number of stochastic sources until a plateau is reached. The load-bearing identities are the asymptotic factorization formulas, Eqs. (4)–(6): σ(⟨DO⟩) ∼ σ_D σ_O, with σ for strong-IB insertions ∼ $a^{{-1}}$√V and for electromagnetic-IB insertions ∼ $a^{{-2}}$√V; the paper's volume and continuum data are compared against these predictions.

What would settle it

Measure the gauge error of the sea-sea mass diagram at a lattice spacing below 0.05 fm (for example a≈0.04 fm) at M_pi≈415 MeV with the same volume: if the error begins to rise like 1/a (and the tadpole like 1/$a^{2}$), the absence of the leading continuum divergence is only a crossover effect in the range studied here.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central claim is that the stochastic gauge error of sea-quark isospin-breaking insertions does not yet exhibit the leading asymptotic divergences predicted for the continuum limit: the strong-IB diagrams are expected to give σ ∼ $a^{{-1}}$√V and the electromagnetic-IB diagrams σ ∼ $a^{{-2}}$√V, but across a=0.05–0.1 fm the measured errors grow with lower powers of 1/a (and the mass-diagram error even decreases), while remaining consistent with the predicted √V volume scaling over L=1.6–3.2 fm. The authors establish this by pushing the number of stochastic sources until the source noise drops below the gauge noise, so that the quoted errors are the gauge errors, and then fitting their volume and lattice-spacing dependence. They also show that the mass and tadpole diagrams dominate the error budget, that the other diagrams become subdominant with modest numbers of sources, and that the sea-sea correction to t0/a2 is resolved but consistent with zero within 2σ.

Load-bearing premise

The load-bearing premise is that the asymptotic error-scaling formulas used to interpret the data, Eqs. (4)–(6), are valid; their derivation is deferred to a future publication and requires that connected n-point functions in the partially-quenched theory decay exponentially with distance, an assumption the authors themselves call 'far from obvious'.

Editorial extensions

If this is right

The mass and tadpole sea-sea diagrams dominate the error budget and reach the gauge-noise plateau with O(10) stochastic sources per level, so the stochastic cost of the dominant contributions is modest.
The gauge error of all physically relevant diagrams scales as √V over L=1.6–3.2 fm, making brute-force volume increases expensive.
Across a=0.05–0.1 fm, no 1/a (strong) or 1/a^2 (electromagnetic) continuum divergence is observed; the mass-diagram error even decreases toward the continuum.
Mass and electromagnetic diagrams are strongly correlated, and tuning the mass shift or using a point-split discretization of the mass operator can reduce the error of their sum.
The sea-sea isospin-breaking correction to t0/a2 is resolved but compatible with zero within 2σ, with statistical error larger than the isosymmetric part.

Reading between the lines

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

If the milder continuum scaling persists at physical pion mass, stochastic estimation of sea-sea isospin-breaking corrections could remain affordable for next-generation ensembles, contrary to the bottleneck scenario; this paper's data at M_pi≈415 MeV are a first hint, not a proof.
The absence of the predicted 1/a and 1/a^2 divergences in this range may be a pre-asymptotic effect; testing at smaller lattice spacings would show where the asymptotic regime begins.
The √V volume growth means that position-space or subvolume integration of sea-sea insertions—a technique the authors say they plan to explore—is likely needed for physical volumes regardless of the continuum behavior.
The observed error cancellation between mass and electromagnetic diagrams suggests that a deliberately chosen mass-operator discretization could make the total sea-sea error smaller than the individual diagram errors, an optimization this paper does not fully exploit.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Read this one as a proceedings note, not a full paper. The genuinely new piece is the first numerical measurement of how the stochastic error in sea-sea isospin-breaking diagrams scales with volume and lattice spacing using C-periodic Wilson fermions. The volume data over L = 1.6 to 3.2 fm are consistent with the predicted sqrt(V) behavior, and the source-scaling plots show that the dominant mass and tadpole diagrams reach the gauge noise with tens of sources. That is a practical, useful result for anyone setting up these computations.

The paper is honest about what it does not do. The scaling laws in eqs. (4)-(6) are the benchmark for the abstract's central claim, but the derivation is explicitly deferred to a future publication and relies on exponential clustering in a partially-quenched theory, which the authors call 'far from obvious.' If that derivation has a flaw, the interpretation of the data as 'consistent with sqrt(V)' and the conclusion that the continuum divergence is absent in this range lose their anchor. That is the main soft spot, and it is a real one. The authors should at least state the assumptions more prominently and, ideally, include the derivation in the next version.

The continuum scaling has thinner support: three lattice spacings, power-law fits with no uncertainties on the exponents, and the errors are rescaled by assuming the very sqrt(V) behavior under test. The mass diagram's error actually decreases with finer a (fit a^0.84), which is opposite to the predicted a^{-1}; the non-observation of the divergence therefore rests heavily on the correctness of the predicted benchmark. The volume scaling, by contrast, stands on its own reasonably well.

There's also a minor inconsistency: the first analysis (Figures 1-4) does not propagate the error on delta m, while the t0 analysis does. That is fine for a proceedings, but a reader should know that the earlier plots are conditional on the tuned mass shift. The statistics are limited - 50 independent configurations at one pion mass - so the results are a proof-of-principle, not the final word.

Who should read it: lattice practitioners working on isospin corrections, especially for HVP and t0. The paper is a useful data point on method costs. I would recommend sending it to peer review; it deserves referee time, with requests for the deferred derivation, uncertainties on the fit exponents, and a clear statement of the assumptions.

Referee Report

3 major / 7 minor

Summary. This proceedings article studies the statistical error of sea-quark isospin-breaking (IBE) corrections computed with stochastic estimators of all-to-all propagators in N_f=3 O(a)-improved Wilson QCD with C-periodic boundary conditions. Using three CLS ensembles at fixed lattice spacing and three ensembles at similar volumes, all with about 50 independent configurations and M_pi around 415 MeV, the paper reports that the dominant mass and tadpole diagrams reach the gauge-noise floor with O(10) stochastic sources, that the volume dependence is compatible with sqrt(V), and that the continuum dependence does not exhibit the predicted 1/a (strong IBE) and 1/a^2 (electromagnetic IBE) divergences in the range a=0.05-0.1 fm. The article also applies the method to the Wilson-flow scale t_0/a^2 and shows cancellations between the mass and electromagnetic contributions to its sea-sea IBE correction.

Significance. If the central scaling statements survive scrutiny, the paper is a useful and timely methodological contribution: it provides direct empirical information on the cost of stochastically unquenching QCD+QED sea effects, identifies which diagrams dominate the error budget, and shows that the gauge-noise floor is reached for the dominant diagrams with a moderate number of sources. The cancellation between the mass and tadpole contributions to t_0/a^2 is a practical finding that may help reduce the cost of future calculations. The main caveat, acknowledged in the text, is that the theoretical scaling benchmarks in Eqs. (4)-(6) are not derived in this manuscript and are deferred to a future publication; the headline conclusions are comparisons to those unpublished predictions.

major comments (3)

[Section 2, Eqs. (4)-(6)] The quantitative anchor of the central claim is the set of predictions sigma(<D_SIB O>) ~ sigma_O a^{-1} sqrt(V) for the strong IBE insertion and sigma(<D_EIB O>) ~ sigma_O a^{-2} sqrt(V) for the electromagnetic insertions. These equations are introduced with the statement that 'details will be given in a future publication', and the derivation explicitly relies on the 'far from obvious' assumption that connected expectation values decay exponentially fast with the distance in the partially-quenched theory. The abstract's headline statements — 'consistent with the predicted leading divergence sqrt(V)' and 'do not show the leading order divergence 1/a ... and 1/a^2' — are comparisons to precisely these unpublished predictions. If the deferred derivation contains an error, or if exponential clustering fails for the relevant diagrams, both the volume and the continuum conclusions lose their theoretical meaning. The manuscript should either include a complete derivation, or an outline sufficient for the reader to check the hypotheses, or it should rephrase the abstract and conclusions as empirical observations and clearly label the benchmark as an unpublished prediction.
[Section 4, Figure 4] The continuum-scaling conclusion rests on only three lattice spacings, and the data in Figure 4 are rescaled by sqrt(V_A)/sqrt(V_i), i.e. by the very sqrt(V) law that is one of the two predictions under test. The power-law fits are reported only as curve labels in the figure (a^0.84, a^{-0.3}, a^{-0.22}, a^{-0.43}), with no uncertainties on the exponents and no goodness-of-fit information. With three points and unknown errors, the claim that the leading divergence is 'not observed' cannot be quantified, and for the mass diagram the fitted exponent has the opposite sign from the predicted a^{-1}, so the non-observation of the strong-IBE divergence rests heavily on the correctness of the predicted benchmark. Please provide the fit results with uncertainties, show the un-rescaled data, and discuss how sensitive the exponents are to the assumed sqrt(V) volume dependence.
[Section 3, Figure 2] The volume-scaling statement is supported by three volumes at a single lattice spacing and is expressed only as compatibility with sqrt(V). Since this is one of the two quantitative claims in the abstract, the paper should report the result of a fit of the volume exponent (with uncertainty) or, failing that, state explicitly that three points are insufficient to distinguish sqrt(V) from, e.g., V^{0.6} or V^{0.4}. The current wording 'all the data are compatible with the expected sqrt(V) scaling' is weaker than what the abstract's 'consistent with the predicted leading divergence sqrt(V)' suggests.

minor comments (7)

[Abstract and Tables 1-2] The abstract states M_pi=400 MeV while the tables quote values between 413 and 416 MeV; please harmonise the quoted pion mass.
[Figure 2] The axis label 'V [fm4]' with tick values 13, 64, 200 should be clarified in the caption as the four-volume, since the text gives only the spatial lengths L=1.6, 2.4, 3.2 fm.
[Captions of Figures 1 and 3] The captions note that the error on delta_m is not propagated in the plotted mass-diagram errors; this limitation should also be stated in the main text of Sections 3 and 4, and its effect on the quoted scaling exponents should be discussed.
[Throughout] The notation 'C^*' and 'C-periodic' is used inconsistently; please choose one term and use it consistently.
[Equation (2)] The sea-sea terms in Eq. (2) are difficult to distinguish in the inline notation; a diagrammatic definition or a table of the Wick contractions would improve readability.
[Figure 4] The power-law fit labels should be accompanied by the fit function, the number of fitted points, and the standard errors of the exponents; as printed they read as exact powers.
[Section 1] The phrase 'N_f=3 O(a)-improved Wilson fermions QCD' should be rephrased, for example as 'QCD with N_f=3 O(a)-improved Wilson fermions'.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular reduction by construction; the scaling test is empirical, though the theoretical benchmark is deferred and self-cited.

full rationale

The paper's central claim—that sea-sea stochastic errors scale as sqrt(V) and do not yet show the a^-1/a^-2 lattice-spacing divergence—is an empirical statement. The measured errors are compared with parameter-free power laws from eqs. (4)-(6), and no constants are fitted to force agreement: the volume data in Fig. 2 are plotted against a sqrt(V) reference line, and the continuum data in Fig. 4 are fit to power laws whose exponents (a^0.84, a^-0.3, etc.) are descriptive, not used as the predictions. The only circularity-adjacent element is that eqs. (4)-(6) are not derived in this paper: Section 2 says 'Details will be given in a future publication' and relies on the 'far from obvious' assumption of exponential clustering in the partially-quenched theory, citing prior work [7,8] with overlapping authorship. This is a real evidential weakness—the theoretical anchor is deferred and self-cited—but it is not a reduction of the prediction to the input: the lattice data are external to the derivation, and the comparison would be falsifiable if the predicted exponents were wrong. Hence no circular step of the kind 'Eq. X = Eq. Y by construction' or 'fitted parameter renamed as prediction' is present; the paper's empirical test has independent content. The minor self-citation burden justifies score 2 rather than 0.

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

The central claims rest on a deferred theoretical derivation (eqs. 4-6), on power-law fits with unquoted uncertainties, and on a regime-specific observation. No new physical entities or forces are introduced.

free parameters (5)

Continuum scaling exponent, mass diagram = ~ +0.84
Power-law fit to three lattice spacings in Figure 4; no uncertainty reported. Used to claim the error does not show the predicted 1/a divergence.
Continuum scaling exponent, tadpole diagram = ~ -0.3
Power-law fit in Figure 4; no uncertainty reported. Supports the claim of a milder continuum divergence.
Continuum scaling exponent, lightbulb-SW diagram = ~ -0.22
Power-law fit in Figure 4; no uncertainty reported.
Continuum scaling exponent, lightbulb-Mix diagram = ~ -0.43
Power-law fit in Figure 4; no uncertainty reported.
Total bare quark mass shift delta m = not quoted numerically (tuned to absorb radiative divergences per [6])
The mass diagram error is multiplied by delta m; the value affects the reported error levels and the cancellation in Figure 7, so the error scaling is partly conditional on this tuned input.

assumptions (4)

domain assumption RM123 perturbative expansion in (m_u - m_d) and e^2 is valid for the sea-sea insertions.
The estimator in eq. (2) is the RM123 expansion; its truncation is standard lattice QCD practice but not proven in this paper.
domain assumption Cluster decomposition and exponential decay of connected n-point functions hold in the partially-quenched theory.
Explicitly called 'far from obvious' in Section 2; underpins the sqrt(V) prediction in eqs. (4)-(6).
domain assumption Symanzik effective theory and operator product expansion apply off-shell to yield a^{-1} and a^{-2} leading scaling.
Invoked in Section 2 with details deferred to a future publication.
domain assumption The stochastic error is dominated by gauge noise once a plateau in the number of sources is reached.
Used to interpret Figures 1, 3, 5 and 6; validated by flatness for the dominant diagrams but not for all diagrams (lightbulb-current and lanterns do not reach the plateau).

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Cite this review

Pith. "Pith review of Error Scaling of Sea Quark Isospin-Breaking Effects." pith.science (2026). https://pith.science/paper/4BJ25T6L

@misc{pith2026250203145,
  author       = {Pith},
  title        = {Pith review of: Error Scaling of Sea Quark Isospin-Breaking Effects},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4BJ25T6L}},
  note         = {Machine review of arXiv:2502.03145}
}

abstract

Sea-quark isospin-breaking effects (IBE) are difficult to compute since they require the evaluation of all-to-all propagators. However, the quest for high-precision calculations motivates a detailed study of these contributions. There are strong arguments that the stochastic error associated with these quantities should diverge in the continuum and infinite-volume limit, resulting in a possible bottleneck for the method. In this work, we present the study of the error scaling for these quantities using $N_f=3$ $O(a)$-improved Wilson fermions QCD with C-periodic boundary conditions in space, a pion mass $M_{\pi}=400$ MeV, a range of lattice spacings $a=0.05, 0.075, 0.1$ fm, and volumes $L=1.6, 2.4, 3.2$ fm. The analysis of the error as a function of the number of stochastic sources shows that we reach the gauge error for the dominant contributions. The errors do not show the leading order divergence $1/a$ for strong-IBE and $1/a^2$ for electromagnetic IBE, in the considered range of lattice spacings. On the other hand, our data are consistent with the predicted leading divergence $\sqrt{V}$.

Figures

Figures reproduced from arXiv: 2502.03145 by the authors.

**Figure 1.** Scaling of the absolute error of the sea-sea diagrams for different random sources for the three different volumes analysed. The mass diagram is multiplied times the total mass shift 𝛿𝑚 tuned to absorb the radiative divergences but the error on 𝛿𝑚 is not propagated. range, however its contribution to the error becomes negligible compared to the others. Similarly, the lanterns diagrams do not reach the gauge noise, h… view at source ↗

**Figure 2.** Volume scaling of the error of the single diagrams in logarithmic scale. 4. Continuum scaling [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Scaling of the absolute error of the sea-sea diagrams for different random sources for the three different lattice spacings analysed. The mass diagram is multiplied times the total mass shift 𝛿𝑚 tuned to absorb the radiative divergences but the error on 𝛿𝑚 is not propagated. As for the other ensembles, the mass and the tadpole diagrams are the dominant error sources and they reach the gauge error around 𝑂 (10) sourc… view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: Continuum scaling of the error of the single diagrams in logarithmic scale. To account for the small differences in volume between the different ensembles the error, of each ensemble 𝑖, has been rescaled by a factor √ 𝑉A420a00b334/ √ 𝑉𝑖 , a power-law fit of the data is…

**Figure 5.** Figure 5: Scaling of the absolute error of the sea-sea diagrams contribution to 𝑡0/𝑎 2 for different random sources for the three different volumes analysed. The mass diagram is multiplied times the total mass shift 𝛿𝑚 tuned to absorb the radiative divergences and the error on 𝛿…

**Figure 6.** Figure 6: Scaling of the absolute error of the sea-sea diagrams contribution to 𝑡0/𝑎 2 for different random sources for the three different lattice spacings analysed. The mass diagram is multiplied times the total mass shift 𝛿𝑚 tuned to absorb the radiative divergences and the e…

**Figure 7.** Figure 7: Error of the total IBE correction to 𝑡0/𝑎 2 divided by the isosymmetric QCD value of 𝑡𝑎/𝑎 2 , as a function of the total bare quark mass shift 𝛿𝑚 (in lattice units) for the three different lattice spacings considered. diagrams shows the actual cancellation both in the …

**Figure 8.** Figure 8: The isospin breaking effects to 𝑡0/𝑎 2 divided by the isosymmetric value of 𝑡0/𝑎 2 for the ensemble C420a00b334. The volume dependence of the gauge noise follows very well by the asymptotic √ 𝑉 behaviour in the considered range of volumes, i.e. 1.6 fm < 𝐿 < 3.2 fm. Thi…

Discussion (0). Continue with ORCID to comment.

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" 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...

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T. Harris, V. G\"ulpers, A. Portelli and J. Richings, Efficiently unquenching QCD+QED at O( ) , http://dx.doi.org/10.22323/1.430.0013 PoS LATTICE2022 (2023) 013 , [ http://arxiv.org/abs/2301.03995 2301.03995 ]

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