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arxiv: 2606.08609 · v1 · pith:EJE7E72Knew · submitted 2026-06-07 · ✦ hep-ph · hep-lat

Machine learning unveils the quark mass dependence of the pseudoscalar meson decay constants in three-flavour N²LO ChPT

Zejian Zhuang , Fernando Gil Dom\'inguez , Raquel Molina This is my paper

Pith reviewed 2026-06-27 18:10 UTC · model grok-4.3

classification ✦ hep-ph hep-lat
keywords chiral perturbation theorylattice QCDLASSO regularizationdecay constantsquark mass dependenceSU(3) limitlow-energy constantsoctet baryons
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The pith

Three-flavor N²LO chiral perturbation theory combined with LASSO regression on lattice data determines the quark-mass dependence of the pseudoscalar decay constants up to pion masses near 780 MeV.

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

The authors fit recent lattice QCD results for the decay constants of the pion, kaon and eta using the full next-to-next-to-leading-order expressions of three-flavor chiral perturbation theory. A machine-learning regularisation technique called LASSO is used to identify which low-energy constants are needed and to fix their numerical values with high precision. The resulting mass-dependent functions are then inserted into covariant baryon chiral perturbation theory to obtain predictions for the masses of the octet baryons at the point where the three light quarks have equal mass. A sympathetic reader cares because these decay constants enter many low-energy descriptions of strong interactions, so their controlled quark-mass dependence improves the reliability of other hadron calculations when the strange quark is not infinitely heavy.

Core claim

The quark mass dependence of f_π, f_K and f_η is extracted from three-flavor N²LO ChPT by applying the LASSO method to recent LQCD data, reaching pion masses around 780 MeV close to the SU(3) symmetric point; the same functions are then used inside covariant baryon ChPT to predict the octet-baryon masses in the SU(3) limit.

What carries the argument

LASSO regularisation applied to the N²LO three-flavor ChPT expressions for the decay constants, which selects and determines the relevant low-energy constants from lattice data.

If this is right

  • The extracted functions supply quark-mass-dependent inputs for other effective field theories that involve pseudoscalar mesons.
  • The same LASSO-fitted low-energy constants can be reused in calculations of other observables that depend on f_π, f_K and f_η.
  • Predictions for octet baryon masses become available throughout the region between the physical point and the SU(3) limit.
  • The analysis demonstrates that N²LO three-flavor ChPT remains applicable up to pion masses of roughly 780 MeV when the strange quark mass is not taken to infinity.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same workflow could be applied to other low-energy constants or to different lattice ensembles to test consistency across observables.
  • If the extracted mass dependence proves robust, it offers a practical way to vary the strange quark mass continuously in phenomenological models without re-fitting from scratch.
  • The approach highlights that machine-learning regularisation can reduce the number of free parameters that must be determined from lattice data in chiral expansions.

Load-bearing premise

The LASSO procedure with its chosen penalty strength and feature set correctly isolates the physically relevant low-energy constants without systematic bias or omission.

What would settle it

A new lattice QCD calculation of the octet baryon masses performed directly at the SU(3) symmetric point that lies outside the uncertainty band obtained from the decay-constant functions derived here.

Figures

Figures reproduced from arXiv: 2606.08609 by Fernando Gil Dom\'inguez, Raquel Molina, Zejian Zhuang.

Figure 1
Figure 1. Figure 1: FIG. 1. The pseudoscalar decay constants in NLO ChPT, that is [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. The LQCD data with trajectories Tr [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. The ratios [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. The ratios [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗
Figure 1
Figure 1. Figure 1: FIG. 1. The correlations between the parameters obtained from the Fit 1. [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. The correlations between the parameters obtained from the Fit 2A. [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. The correlations between the parameters obtained from the Fit 2B. [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
read the original abstract

The quark mass dependence of the pseudoscalar meson decay consntants, $f_\pi, f_K$ and $f_\eta$, are determined from three-flavor N$^2$LO ChPT till pion masses around $780$ MeV, near the SU(3) limit. This is done by conducting an analysis of recent LQCD data using the LASSO method, a machine-learning technique which allows to pin down the relevant low-energy-constants with high precision. Since the pion decay constant is a fundamental quantity which usually appears in relevant phenomenological lagrangians or Effective Field Theories based on QCD at low energies, this analysis can be used as input to evaluate the quark mass dependence of hadronic states. As an example, we predict the masses of the octect baryons in the SU(3) limit within covariant Baryon Chiral Perturbation Theory.

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.

Referee Report

2 major / 1 minor

Summary. The manuscript claims that the quark mass dependence of the pseudoscalar meson decay constants f_π, f_K and f_η can be determined from three-flavor N²LO ChPT up to m_π ≈ 780 MeV near the SU(3) limit by applying the LASSO method to recent LQCD data; the resulting parametrizations are then used as input to predict octet baryon masses in the SU(3) limit within covariant baryon ChPT.

Significance. If the LASSO procedure is shown to be robust, the work supplies a concrete parametrization of decay constants that can serve as input for other low-energy EFTs; the baryon-mass example illustrates one such application. The use of LASSO for LEC selection in ChPT is a methodological novelty that, once validated, could be of interest to the community.

major comments (2)
  1. [LASSO implementation (methods/results section)] The description of the LASSO analysis supplies no information on hyper-parameter tuning, cross-validation, comparison to ordinary least-squares fits, or stability under data cuts. This information is required to establish that the selected LECs are free from regularization-induced bias or omission and is therefore load-bearing for the central claim.
  2. [ChPT applicability discussion] No convergence tests, order-by-order comparisons, or explicit discussion are provided to justify the use of the N²LO three-flavor expansion at m_π ≈ 780 MeV near the SU(3) point. Without such checks the extracted mass dependence of f_π, f_K and f_η cannot be trusted as a reliable input for the subsequent baryon-mass predictions.
minor comments (1)
  1. [Abstract] Abstract contains the typo 'consntants'.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major comment below and will revise the manuscript to strengthen the presentation of the LASSO analysis and the justification for the ChPT range.

read point-by-point responses

  1. Referee: [LASSO implementation (methods/results section)] The description of the LASSO analysis supplies no information on hyper-parameter tuning, cross-validation, comparison to ordinary least-squares fits, or stability under data cuts. This information is required to establish that the selected LECs are free from regularization-induced bias or omission and is therefore load-bearing for the central claim.

    Authors: We agree that the current description of the LASSO procedure is insufficient for full validation. In the revised manuscript we will expand the methods and results sections to document the hyper-parameter selection (including the cross-validation strategy used to determine the regularization strength), direct comparisons of the LASSO-selected LECs with those obtained from ordinary least-squares fits on the same data sets, and explicit stability tests under successive data cuts. These additions will demonstrate that the retained LECs are robust and not artifacts of the regularization. revision: yes

  2. Referee: [ChPT applicability discussion] No convergence tests, order-by-order comparisons, or explicit discussion are provided to justify the use of the N²LO three-flavor expansion at m_π ≈ 780 MeV near the SU(3) point. Without such checks the extracted mass dependence of f_π, f_K and f_η cannot be trusted as a reliable input for the subsequent baryon-mass predictions.

    Authors: We acknowledge that an explicit discussion of convergence is required. The revised manuscript will include a new subsection that presents order-by-order comparisons of the decay constants, quantifies the size of the N²LO corrections relative to lower orders across the simulated mass range, and references existing literature on the convergence radius of three-flavor ChPT near the SU(3) limit. This material will support the reliability of the extracted parametrizations as input for the baryon-mass calculation. revision: yes

Circularity Check

0 steps flagged

No circularity: LECs fitted to external LQCD data; baryon-mass example uses separate framework.

full rationale

The paper determines N²LO three-flavor ChPT low-energy constants by applying LASSO regularization to independent lattice QCD data on f_π, f_K and f_η. The resulting quark-mass parametrizations are then inserted into a distinct covariant baryon ChPT calculation to illustrate octet-baryon masses in the SU(3) limit. No equation, prediction or uniqueness claim reduces by construction to a quantity already defined inside the paper, nor does any load-bearing step rely on a self-citation chain. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

Central claim rests on the domain assumption that N²LO ChPT converges near the SU(3) limit and on the ad-hoc choice that LASSO selects the correct subset of LECs; LECs themselves are the fitted parameters.

free parameters (1)
  • Low-energy constants (LECs) at N²LO
    Determined with high precision from LQCD data via LASSO; values not reported in abstract.
axioms (2)
  • domain assumption Validity of three-flavor N²LO ChPT expansion up to m_π ≈ 780 MeV near SU(3) limit
    Analysis extends the theory to this mass range without additional justification in the abstract.
  • ad hoc to paper LASSO penalty selects the physically relevant LECs without significant bias
    Method relies on this regularization property for the claimed precision.

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discussion (0)

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