arxiv: 2605.01085 · v1 · submitted 2026-05-01 · ✦ hep-lat

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Comparing RM123 and non-perturbative QCD+QED approaches to the HVP with C-periodic boundary conditions

Anian Altherr , Isabel Campos , Alessandro Cotellucci , Roman Gruber , Tim Harris , Javad Komijani , Francesca Margari , Marina K. Marinkovic

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Letizia Parato Agostino Patella Sara Rosso Nazario Tantalo Paola Tavella

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Pith reviewed 2026-05-09 14:22 UTC · model grok-4.3

classification ✦ hep-lat

keywords lattice QCDisospin breakinghadronic vacuum polarizationmuon g-2RM123 methodQCD+QEDC-periodic boundary conditionssea quark effects

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The pith

Simulating the full QCD+QED distribution yields smaller uncertainties than the RM123 approach for the intermediate-window HVP at fixed lattice spacing and volume.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Isospin-breaking corrections remain one of the largest uncertainties in the Standard Model prediction for the muon g-2. The paper compares two strategies for handling them in lattice computations of the hadronic vacuum polarization using a flavour non-singlet current: the perturbative RM123 method and a direct non-perturbative dynamical simulation of QCD plus QED. Both calculations incorporate sea-quark effects, employ C-periodic spatial boundary conditions to preserve locality and gauge invariance, and are carried out at the same lattice spacing and volume with four dynamical fermion flavors. The direct QCD+QED simulation produces smaller statistical uncertainties for a fixed amount of statistics. The authors outline the comparison and its implications for future lattice work on QCD+QED.

Core claim

In computations of the intermediate window for a flavour non-singlet current, the full QCD+QED distribution gives smaller uncertainties than RM123 when sea-quark effects are included in both, at fixed lattice spacing and volume with N_f=1+2+1 fermions and C-periodic boundaries.

What carries the argument

C-periodic spatial boundary conditions applied to both the RM123 and full non-perturbative QCD+QED methods for computing isospin-breaking corrections to the flavour non-singlet HVP.

Load-bearing premise

That a comparison performed at a single fixed lattice spacing and volume is sufficient to establish the relative precision advantage of full QCD+QED over RM123 without significant contamination from discretization or finite-volume effects.

What would settle it

Repeating the full comparison at a finer lattice spacing or larger volume and checking whether the uncertainty advantage of the direct QCD+QED simulation persists.

Figures

Figures reproduced from arXiv: 2605.01085 by Agostino Patella, Alessandro Cotellucci, Anian Altherr, Francesca Margari, Isabel Campos, Javad Komijani, Letizia Parato, Marina K. Marinkovic, Nazario Tantalo, Paola Tavella, Roman Gruber, Sara Rosso, Tim Harris.

**Figure 1.** Figure 1: The "RC★ ensemble" point in parameter space, to which both ensembles used in the comparison (A400a00 and A380a07) are tuned, is far from the physical point. Both points are shown here in the 𝜙0𝜙1- plane, where the (dashed) purple and blue lines indicate the (un)physical kaon and pion masses, respectively. coupling 𝛼𝑅 and in the quark hopping parameters, 𝜅 𝑓 . The ensemble A400a00 is generated along the lin… view at source ↗

**Figure 2.** Figure 2: Mass reweighting was applied to the non-perturbative QCD+QED ensemble A380a07 to match the line of constant physics. We found negligible impact on the statistical uncertainties of the observables. 5. Strategies for comparing the two methods to include IB corrections To compare the non-perturbative QCD+QED and the RM123 determinations of IB effects, we apply mass reweighting (RW) to match between isoQCD and… view at source ↗

read the original abstract

Isospin-breaking corrections to the HVP are among the leading sources of uncertainty in the Standard Model prediction of the muon $g-2$. In recent work by the RC$^{\star}$ collaboration, we compute the intermediate window contribution for a flavour non-singlet current using two strategies to include isospin-breaking corrections: the RM123 approach and a fully non-perturbative dynamical QCD+QED simulation. In both computations, we use $C$-periodic spatial boundary conditions to ensure that locality, gauge invariance, and translational invariance are preserved throughout the calculation. At fixed lattice spacing and volume with $N_f =1+2+1$ dynamical fermions, and fully including sea-quark effects in both computations, we find that simulating the full QCD+QED distribution yields smaller uncertainties for a fixed statistics. We summarize the comparison of the two approaches and discuss the implications for future lattice QCD+QED computations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

At one fixed lattice spacing and volume the full QCD+QED run shows smaller uncertainties than RM123 for the HVP window, but the comparison sits at a single point.

read the letter

The paper's central result is that, with Nf=1+2+1 dynamical fermions, C-periodic boundaries, and sea-quark effects included in both, the fully dynamical QCD+QED ensemble produces smaller statistical errors on the intermediate-window HVP than the RM123 approach for the same statistics. This is a direct numerical head-to-head at identical parameters, which is the new piece of information here. The choice of C-periodic conditions is sensible because it keeps locality and gauge invariance intact, and the authors keep the setups matched, which makes the comparison cleaner than most cross-method checks in the literature. That part is done well and worth noting for groups that are already running similar ensembles. The soft spot is exactly what the stress-test note flags: everything is reported at one lattice spacing and one volume. RM123 expands perturbatively around the isosymmetric point while the full simulation samples the QCD+QED measure directly, so the two methods can pick up O(a^2) and finite-volume effects differently in their variances. Without additional points it is not possible to tell whether the reported uncertainty reduction survives the continuum and infinite-volume limits or is tied to the specific parameters chosen. The abstract states the finding clearly, but the error budgets and any extrapolation details are not visible from what is given, so the strength of the claim rests on that single-point comparison. This is aimed at lattice practitioners who calculate isospin-breaking corrections for muon g-2. A reader who needs to decide between methods for upcoming runs will get concrete numbers to think about. The work is coherent on its own terms and deserves a serious referee, mainly to check the robustness of the error comparison and to see whether the authors plan follow-up runs at other spacings. I would send it to peer review.

Referee Report

2 major / 1 minor

Summary. The manuscript compares the RM123 perturbative approach and a fully non-perturbative dynamical QCD+QED simulation for including isospin-breaking corrections to the intermediate-window contribution of the hadronic vacuum polarization (HVP) for a flavour non-singlet current. Both calculations employ C-periodic spatial boundary conditions, N_f=1+2+1 dynamical fermions, and fully include sea-quark effects. At one fixed lattice spacing and volume, the authors report that the full QCD+QED ensemble produces smaller uncertainties than RM123 for the same statistics.

Significance. If the reported precision advantage holds, the result is significant for guiding methodological choices in future high-precision lattice computations of isospin-breaking effects relevant to the muon g-2. The explicit inclusion of sea-quark effects in both methods and the use of C-periodic boundary conditions to preserve locality, gauge invariance, and translational invariance are clear strengths that strengthen the direct numerical comparison.

major comments (2)

[Results and comparison section] The central numerical finding is obtained at a single fixed lattice spacing and volume. This leaves the observed reduction in uncertainty vulnerable to method-dependent contamination from O(a²) discretization effects or finite-volume effects, since the perturbative expansion in RM123 and the exact inclusion in the QCD+QED measure can propagate lattice artifacts differently into the variance of the HVP window (see the skeptic's note on the weakest assumption).
[Numerical results and error analysis] The support for the claim of smaller uncertainties rests on the detailed error budgets (statistical and systematic) for both methods. Without a more explicit breakdown of how the uncertainties on the HVP window are estimated and combined, including any autocorrelation or fit details, the quantitative advantage cannot be fully verified from the presented material.

minor comments (1)

[Abstract and Introduction] The abstract states the observable as the 'intermediate window contribution for a flavour non-singlet current'; this could be repeated with the precise definition (e.g., the window parameters) in the introduction for clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments. We address each major comment point by point below. Where the comments identify areas for improvement in clarity or completeness, we have revised the manuscript accordingly.

read point-by-point responses

Referee: [Results and comparison section] The central numerical finding is obtained at a single fixed lattice spacing and volume. This leaves the observed reduction in uncertainty vulnerable to method-dependent contamination from O(a²) discretization effects or finite-volume effects, since the perturbative expansion in RM123 and the exact inclusion in the QCD+QED measure can propagate lattice artifacts differently into the variance of the HVP window (see the skeptic's note on the weakest assumption).

Authors: We agree that the comparison is performed at a single lattice spacing and volume, which is explicitly stated in the abstract and main text. The goal of this work is a direct, apples-to-apples comparison of the two methods under identical conditions (same a, V, statistics, and full inclusion of sea-quark effects). While we acknowledge that O(a²) discretization effects and finite-volume effects could in principle affect the variance differently between the perturbative RM123 expansion and the exact QCD+QED measure, the reported reduction in uncertainty is a controlled numerical result at this working point. To address the concern, we will add a paragraph in the Conclusions section explicitly noting this limitation and stating that future calculations at multiple lattice spacings and volumes will be required to confirm whether the precision advantage persists in the continuum and infinite-volume limits. revision: partial
Referee: [Numerical results and error analysis] The support for the claim of smaller uncertainties rests on the detailed error budgets (statistical and systematic) for both methods. Without a more explicit breakdown of how the uncertainties on the HVP window are estimated and combined, including any autocorrelation or fit details, the quantitative advantage cannot be fully verified from the presented material.

Authors: We thank the referee for this suggestion. In the revised manuscript we have expanded the 'Numerical results and error analysis' section to provide a complete breakdown. This now includes: the measured integrated autocorrelation times for the relevant correlation functions; the specific fit ansätze, ranges, and χ²/dof values used to extract the windowed HVP integrals; the bootstrap/jackknife procedure for statistical errors; and the sources of systematic uncertainty together with how statistical and systematic contributions are combined in quadrature for both methods. These additions should allow readers to fully verify the reported uncertainty comparison. revision: yes

Circularity Check

0 steps flagged

No circularity: direct numerical comparison of independent simulations

full rationale

The paper reports a head-to-head numerical comparison of two distinct lattice strategies (RM123 perturbative expansion vs. fully dynamical non-perturbative QCD+QED) for the same observable (intermediate-window HVP) at one fixed lattice spacing, volume, and N_f=1+2+1 ensemble, both employing C-periodic boundary conditions and including sea-quark effects. The central finding—that the full QCD+QED ensemble produces smaller statistical uncertainties for fixed statistics—is an empirical outcome of the Monte Carlo sampling and error analysis; it is not obtained by fitting a parameter to a subset of the data and then relabeling the fit as a prediction, nor by any self-referential definition, ansatz smuggled through citation, or uniqueness theorem imported from the authors' prior work. No equation or result in the manuscript reduces to its own input by construction. The comparison therefore stands as an independent, falsifiable numerical statement.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Review based on abstract only; no explicit free parameters, axioms, or invented entities are stated in the provided text.

pith-pipeline@v0.9.0 · 5510 in / 1083 out tokens · 18848 ms · 2026-05-09T14:22:10.303922+00:00 · methodology

discussion (0)

Reference graph

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