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A percent-level determination of the nucleon axial coupling from Quantum Chromodynamics
Chia Cheng Chang, Amy Nicholson, Enrico Rinaldi, Evan Berkowitz, Nicolas Garron, David A. Brantley, Henry Monge-Camacho, Christopher J. Monahan, Chris Bouchard, M. A. Clark, Bálint Joó, Thorsten Kurth, Kostas Orginos, Pavlos Vranas, André Walker-Loud
TL;DR
The paper addresses excited-state contamination in lattice determinations of g_A by using a Feynman-Hellmann strategy that accesses more source-sink separations. It reports controlled ground-state coupling fits and nearly ideal Gaussian bootstrap results, while prior calculations faced correlated statistical fluctuations near 1 fm.
Problem
Lattice correlation functions combine the ground state with excited states, making their disentanglement a major challenge in calculating g_A.
Method
The Feynman-Hellmann strategy constructs lattice correlation functions with access to both longer and shorter source-sink separations.
Results
The ground-state nucleon couplings remain stable under varying fit regions, agree with excited-state-subtracted correlations, and yield nearly ideal Gaussian bootstrap distributions.
Takeaways & Limitations
The strategy resolves the two major challenges identified from previous calculations.
Takeaways & Limitations
Previous calculations may be affected by correlated statistical fluctuations near ∼1 fm because limited source-sink separations make them difficult to identify a posteriori.
Abstract
from arXiv · showhide
The $\textit{axial coupling of the nucleon}$, $g_A$, is the strength of its coupling to the $\textit{weak}$ axial current of the Standard Model of particle physics, in much the same way as the electric charge is the strength of the coupling to the electromagnetic current. This axial coupling dictates the rate at which neutrons decay to protons, the strength of the attractive long-range force between nucleons and other features of nuclear physics. Precision tests of the Standard Model in nuclear environments require a quantitative understanding of nuclear physics rooted in Quantum Chromodynamics, a pillar of the Standard Model. The prominence of $g_A$ makes it a benchmark quantity to determine theoretically - a difficult task because quantum chromodynamics is non-perturbative, precluding known analytical methods. Lattice Quantum Chromodynamics provides a rigorous, non-perturbative definition of quantum chromodynamics that can be implemented numerically. It has been estimated that a precision of two percent would be possible by 2020 if two challenges are overcome: contamination of $g_A$ from excited states must be controlled in the calculations and statistical precision must be improved markedly. Here we report a calculation of $g_A^{QCD} = 1.271\pm0.013$, using an unconventional method inspired by the Feynman-Hellmann theorem that overcomes these challenges.
Supplementary Information is available in the online version of this paper.
The supplementary analyses document the lattice inputs, extrapolation procedures, fit stability, and the resulting constraints on beyond-Standard-Model currents.
- Data and extrapolation: 16 lattice data points are combined through chiral-continuum extrapolation models to determine gA at the physical point.The inputs include renormalized gA values, lattice spacing, pion-mass, and finite-volume variables.
- Stability and convergence: All tested variations of the extrapolation analysis remain within 1σ of the model-average value.Variations include additional discretization terms, omitted finite-volume corrections, wider priors, pion-mass cuts, and discretization cuts.
- Beyond-Standard-Model constraints: The determination of gA is used to constrain right-handed beyond-Standard-Model currents through comparisons with neutron decay, collider, pion-decay, and nuclear-decay constraints.The figure identifies the constraint from this result alongside constraints from other low-energy and collider measurements.
S.1. Data and software availability
The work provides public access to correlation functions, analysis outputs, bootstrap distributions, software, and reproducible extrapolation tools.
- Data availability: Lattice QCD correlation functions and analysis results are publicly available through GitHub and Zenodo.The listed repositories include project data and analysis outputs.
- Analysis resources: Bootstrap distributions, a Jupyter notebook, and a chiral-continuum extrapolation Python library accompany the published data.Installation instructions are provided with the released files.
- Software: The software stack uses Chroma, QUDA, HDF5, QDP++, MILC software, and the METAQ job manager.These components support lattice calculations, data handling, propagator generation, and high-performance computing workflows.
S.2. Correlation functions from the Feynman-Hellman theorem
The paper constructs Feynman-Hellmann correlation functions whose time dependence enables earlier fits and improved control of excited-state contamination, while requiring new propagators for each target matrix element or momentum.
- Feynman-Hellmann construction: The Feynman-Hellmann theorem relates matrix elements to derivatives of energy eigenvalues with respect to an external source.The Hamiltonian is modified as H = H0 + λHλ, and the derivative gives the matrix element.
- Feynman-Hellmann construction: The resulting FH ratio is a correlation function constructed by applying the Feynman-Hellmann theorem to the lattice two-point function.The path-integral representation is obtained by differentiating source terms for the nucleon and current operators.
- Method implementation: The method directly calculates the −∂λC(t) correlation function without numerically implementing the derivative or disentangling response orders.The authors describe this as the most economical implementation for single-nucleon properties.
- Excited-state control: The difference construction adds suppression of excited states and contact terms, enabling fits at early Euclidean times before stochastic noise overwhelms the signal.The authors report fitting early in Euclidean time across all ensembles used in the work.
- Method scope: A potential drawback is that each matrix element or momentum injection requires new Feynman-Hellmann quark propagator calculations.The method can instead reuse propagators across different hadronic states for the same quark bilinear current.
- Lattice setup: The calculation uses multiple lattice spacings, volumes, and pion masses to control continuum, infinite-volume, and physical-pion-mass extrapolations.The ensembles span lattice spacings near 0.15, 0.12, and 0.09 fm, with pion masses near 130, 220, and 310 MeV plus added heavier ensembles.
S.4. Correlator analysis
The correlator analysis uses the Feynman-Hellmann construction to access both short and long source-sink separations, improving study of excited-state contributions.
- Correlator analysis: Lattice correlation functions contain a superposition of the ground-state nucleon and excited nucleon states.The unknown ground-state wavefunction necessitates interpolating operators for the initial and final states.
- Correlator analysis: Excited-state contamination has been a major challenge in previous calculations of gA.Separating the ground state requires careful analysis of the correlation functions.
- Correlator analysis: The Feynman-Hellmann construction provides measurements at both shorter and longer initial-final separations for more complete excited-state analysis.
A. Analysis Strategy
The analysis fits six correlated correlation functions simultaneously and evaluates candidate fits through quality, stability, and bootstrap criteria.
- Six correlation functions are fitted simultaneously, combining nucleon two-point, axial and vector Feynman–Hellmann ratios, and two sink-smearing choices.The joint fit shares parameters such as energy levels and overlap factors across correlated data.
- Candidate fits must have P-values above 0.05 and remain stable when fit regions and time separations are varied.These requirements discriminate poor-quality fits and test sensitivity to excited-state contamination.
- The analysis uses gradient-flow and autocorrelation studies to verify flow-time independence and unchanged uncertainty under binning.Larger flow times reduce stochastic noise, while binning does not alter the uncertainty of the mean.
- Supplemental Table I reports fit regions, bare couplings, χ2/dof, P-values, and the renormalisation-coefficient ratio for each result.
- Bootstrap resampling uses 5,000 samples to quantify the matrix-element uncertainty and accepts candidates with the expected Gaussian distribution.The preferred fit also samples the largest fit region while satisfying the other quality requirements.
B. Discussion
The study demonstrates stability against excited-state contamination and statistical fluctuations, while controlling renormalisation and extrapolation choices with lattice and effective-field-theory methods.
- Discussion: The Feynman–Hellmann strategy makes all source-sink separation times accessible, enabling ground-state stability studies that conventional calculations cannot perform.This provides the data needed to assess excited-state control across fit regions.
- Discussion: Smaller tmin values can produce results in tension, whereas preferred fits agree with conservative fits using larger time separations.Excited-state contributions are more pronounced at smaller source-sink separations because excited states are heavier than the ground state.
- Discussion: Analysing smaller separation times avoids the degraded signal-to-noise region and yields ground-state couplings stable under varying tmax.The extracted result is insensitive to whether intermediate-time fluctuations are included in the fit.
- Discussion: The final analysis reports stability across fit regions, agreement with excited-state-subtracted correlators, and nearly ideal Gaussian bootstrap results.
- Renormalisation: The renormalisation ratio ZA/ZV is consistent with unity at one part in 10,000, indicating strong approximate chiral symmetry of the lattice action.The calculation uses the non-perturbative Rome–Southampton procedure to correct differences between local and conserved currents.
- Extrapolation: The extrapolation uses dimensionless physical ratios and EFT parameterisations for continuum, infinite-volume, and physical-pion-mass limits.Two-flavor HBχPT is used for controlled pion-mass dependence because SU(3) HBχPT has convergence issues.
- Extrapolation: The available data are insufficient to constrain all N3LO low-energy constants, so partial corrections and Bayesian constraints are used to test extrapolation sensitivity.Explicit-delta extrapolations additionally require experimentally constrained quantities not determined in this calculation.
S.7. Extrapolation analysis
The analysis proceeds to a physical-point extrapolation to obtain the concluding value of g_A, followed by uncertainty assessment and sensitivity studies.
- S.7. Extrapolation analysis: The analysis performs a physical-point extrapolation to obtain Eq. (1), the concluding result of the work.It then discusses sources of uncertainty and tests sensitivity to changes in analysis inputs.
A. Model averaging
The physical-point result is obtained by Bayesian averaging over six extrapolation models, incorporating both within-model variance and model-selection uncertainty.
- Model averaging: Six extrapolation models are averaged rather than trusting any single model.The procedure is designed to account for model-selection uncertainty and avoid over-confident inferences.
- Model averaging: The extrapolation priors use unconstraining lower-order coefficients and O(1) priors for NNLO coefficients, with prior-width sensitivity examined.
- Model averaging: Bayesian model averaging marginalises over model parameters and uses posterior model probabilities to weight the models.Marginalisation over continuous parameters naturally penalises over-parameterised models.
- Model averaging: The model-averaged variance contains process variance and variance among hypothetical model means.The decomposition follows the law of total variance.
- Model averaging: Taylor expansion fits are strongly favoured over χPT fits by the posterior model weights in the model-selection analysis.The final result reports averaged variance and model uncertainty as separate uncertainties.
B. Uncertainty analysis
The final uncertainty in gA combines statistical, chiral, continuum, infinite-volume, isospin-breaking, and model-selection contributions. The analysis finds a 0.03% uncertainty from the evaluated corrections and identifies more precise physical-pion-mass results as the clearest route toward sub-percent precision.
- The chiral and continuum uncertainties arise from uncertainties in low-energy constants and discretization-dependent corrections.Additional O(αsa2) discretization terms are investigated as part of the continuum analysis.
- Finite-volume corrections are included through χPT, and comparisons without these corrections or with only NLO corrections agree within the estimated uncertainty.In all but two ensembles, the finite-volume shift is less than one standard deviation.
- 0.03% is the estimated uncertainty from the evaluated radiative and strong isospin-breaking corrections.The strong correction vanishes at leading order for the n → p transition and requires two isospin-breaking insertions to contribute.
- The final model-averaged uncertainty includes statistical, chiral, continuum, infinite-volume, isospin-breaking, and model-selection contributions.These contributions are combined in quadrature.
- More precise results at the physical pion mass would most directly reduce the extrapolation and model-selection uncertainties.The paper presents this as a path toward sub-percent precision.
C. Sensitivity analysis
Sensitivity tests show that the physical-point extrapolation remains consistent when pion-mass, lattice-spacing, prior, and dataset choices are varied. Heavy-pion fluctuations have only a 0.3% estimated effect on the physical-point result.
- The sensitivity analysis varies priors and data subsets across pion-mass and lattice-spacing regions.The resulting model averages are compared with the full-data analysis.
- Doubling the prior widths leaves the final result essentially unchanged, indicating that the priors are unconstraining.This robustness test varies the leading-order and all low-energy-constant prior widths.
- The physical-point extrapolation remains consistent within 1σ when heavy pion-mass points are removed, although the uncertainty grows.The analysis finds no statistical justification for truncating the heavy-pion data.
- 0.3% is the estimated change in the physical-point result from a hypothetical 1σ fluctuation at mπ ≃400 MeV.The authors conclude that the extrapolation is relatively insensitive to fluctuations at larger pion masses.
- Removing individual coarse or fine lattice-spacing ensembles produces larger but consistent extrapolated results.These cuts test sensitivity to the continuum limit.
A. Convergence of the χPT expansion
The χPT convergence analysis finds substantial cancellations among orders and difficulty constraining the full N3LO expression with the available data. Adding the NNLO counterterm has negligible impact, but the full N3LO fit becomes poorly constrained outside the data region.
- Large cancellations between NLO and NNLO contributions make convergence difficult to assess.The strong curvature of the NLO curve results from competition between counterterm and logarithmic contributions.
- Five pion-mass points cannot meaningfully constrain the full N3LO χPT formula, which contains five unknown low-energy constants.The analysis instead adds the local c4 counterterm to test convergence.
- Adding the NNLO counterterm has negligible impact on the extrapolated gA across the depicted pion-mass range.The order-by-order contributions and low-energy-constant correlations are reported for this analysis.
- The full N3LO fit develops substantially larger extrapolation uncertainty outside the constraining-data region, indicating overfitting.Its convergence differs markedly from the NNLO+ct analysis and involves large cancellations.
- The χPT expansion may converge particularly poorly for gA, but the authors state that a strong conclusion requires more pion-mass points, especially precise lighter-mass results.The full N3LO extrapolation has mild pion-mass dependence despite cancellations among expansion terms.
B. Including the ∆s
Including explicit Δ degrees of freedom reduces the fitted non-analytic contribution but introduces additional poorly constrained couplings and phenomenological input. The resulting comparison suggests improved convergence, while the Δ analysis is excluded from the final model average.
- The Δ analysis introduces two new axial-coupling low-energy constants and an extra mass-splitting parameter, increasing the total to four low-energy constants.The new axial couplings are not directly constrained by the results and produce larger uncertainties.
- The explicit-Δ fits are excluded from the final model average because they require phenomenological input.The final analysis averages fits that do not require this input.
- Including explicit Δ degrees of freedom makes c2 approximately half as large as without Δ states.The authors interpret this as evidence that the non-analytic terms are smaller with explicit Δ degrees of freedom.
- The NLO χPT(Δ) fit has χ2_aug/dof = 0.49.The fit is displayed as an additional extrapolation comparison.
- Fully constraining the Δ fit requires N → Δ and Δ → Δ axial matrix elements and an EFT extended to at least one higher order.These additions are also needed to test convergence with and without Δ degrees of freedom.
A NLO -4.8(1.8) 0.28(13) – – NNLO 12.1(8.2) 0.75(40) -23.0(3.5) -0.026(30)
The supplied supplemental material reports fit studies, stability checks, autocorrelation analyses, and parameter-distribution assessments supporting the analysis.
- One supplemental result states that the resulting low-energy constant is not reported for a fit with a sufficiently poor χ2_aug/dof.
- Fit-region stability studies vary tmin and tmax for two-point, axial, and vector effective derivatives, with corresponding P-values and one-standard-error uncertainties reported.
- Autocorrelation studies and ϵπ histograms provide additional checks, with histogram panels marking 68% and 95% confidence intervals.