arxiv: 2604.23739 · v1 · submitted 2026-04-26 · ✦ hep-ph

Recognition: unknown

Kaon Distribution Amplitudes from Euclidean Functional QCD

Wen Cui , Dao-yu Zhang , Chuang Huang , Wei-jie Fu

Authors on Pith no claims yet

Pith reviewed 2026-05-08 05:45 UTC · model grok-4.3

classification ✦ hep-ph

keywords kaon distribution amplitudequasi-distribution amplitudefunctional QCDlarge-momentum effective theoryBethe-Salpeter amplitudecontour deformationmoment extrapolation

0 comments

The pith

Functional QCD calculation yields single-peaked asymmetric kaon distribution amplitude with moments 0.020 and 0.253.

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

The paper establishes the shape of the kaon distribution amplitude by computing its quasi-version at large momentum using functional QCD inputs and extrapolating to the physical limit. It finds a single-peaked asymmetric form for the kaon DA, with first moment 0.020(3) and second moment 0.253(12). This matters because the DA encodes how the strange and light quarks share longitudinal momentum inside the kaon, directly affecting predictions for exclusive kaon processes. The approach relies on Euclidean Bethe-Salpeter amplitudes and contour deformation to reach the light-cone limit without direct Minkowski-space calculations.

Core claim

Using quark correlation functions and the kaon Bethe-Salpeter amplitude from 2+1 flavour functional QCD as inputs, the kaon quasi-DA is obtained in the large longitudinal momentum region via the contour deformation method. By performing 1/P_z² and 1/P_z⁴ order extrapolations for P_z^max in [2, 2.5] GeV, the authors obtain a single-peaked and asymmetric kaon DA with ⟨ξ⟩_K = 0.020(3) and ⟨ξ²⟩_K = 0.253(12).

What carries the argument

Contour deformation method in the complex momentum plane to evaluate the quasi-distribution amplitude, combined with large-momentum effective theory applied to Bethe-Salpeter amplitudes from functional QCD.

If this is right

The positive first moment indicates that the strange quark carries slightly more momentum than the light antiquark on average.
The second moment measures the width of the momentum-sharing distribution inside the kaon.
These DA moments provide first-principles inputs for calculating kaon transition form factors and decay amplitudes.
The quoted uncertainties arise primarily from the choice of extrapolation interval and ansatz.

Where Pith is reading between the lines

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

The same functional QCD plus LaMET pipeline could be applied to the pion DA for a direct comparison of light-meson asymmetry.
The observed asymmetry quantifies SU(3) flavor symmetry breaking effects due to the strange quark mass.
Increasing the maximum P_z beyond 2.5 GeV in future runs would test whether the current extrapolation remains stable.

Load-bearing premise

The 1/P_z² and 1/P_z⁴ extrapolation ansatzes remain valid and the contour deformation accurately captures the physical quasi-DA without uncontrolled systematic errors from the chosen P_z^max interval.

What would settle it

A direct computation of the kaon DA at significantly higher maximum longitudinal momentum P_z or via an independent lattice method that yields moments outside the quoted uncertainties would falsify the extrapolated result.

Figures

Figures reproduced from arXiv: 2604.23739 by Chuang Huang, Dao-yu Zhang, Wei-jie Fu, Wen Cui.

**Figure 1.** Figure 1: FIG. 1. Schematic illustration of the decomposition of the view at source ↗

**Figure 2.** Figure 2: FIG. 2. Quark mass function view at source ↗

**Figure 3.** Figure 3: FIG. 3. Bethe-Salpeter amplitude of the kaon as a function view at source ↗

**Figure 4.** Figure 4: FIG. 4. Imaginary parts of the second and third poles in view at source ↗

**Figure 5.** Figure 5: FIG. 5. Non-normalized quasi-PDA view at source ↗

**Figure 7.** Figure 7: FIG. 7. Kaon quasi-DAs as functions of the momentum frac view at source ↗

**Figure 8.** Figure 8: FIG. 8. Light-front PDA of kaon as a function of the momen view at source ↗

read the original abstract

We study the kaon quasi-distribution amplitude (quasi-DA) and distribution amplitude (DA) within the large-momentum effective theory (LaMET) combined with the first-principles functional QCD. Using quark correlation functions and the kaon Bethe-Salpeter amplitude in the Euclidean space from the 2+1 flavour functional QCD [1] as inputs, we obtain the kaon quasi DA in the large longitudinal momentum region with the contour deformation method [2] in the complex plane of momentum. By performing $1/P_z^2$ and $1/P_z^4$ order extrapolations of the kaon quasi-DA for the choices of the maximal longitudinal momentum $P_z^{\max}\in[2,2.5]$ GeV, we obtain a single-peaked and asymmetric kaon DA with the uncertainties arising from the extrapolation interval and ansatz. We find the first and second order moments of the kaon DA, $\langle \xi \rangle_K = 0.020(3)$ and $\langle \xi^2 \rangle_K = 0.253(12)$, respectively.

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

This paper gives new kaon DA moments from LaMET plus functional QCD inputs, but the extrapolation at Pz up to 2.5 GeV is the main uncertainty to watch.

read the letter

The paper computes the kaon distribution amplitude by taking quark correlation functions and the Bethe-Salpeter amplitude from an existing 2+1 flavor functional QCD calculation, then using contour deformation to build the quasi-DA at finite longitudinal momentum before extrapolating to the light-cone limit. The headline numbers are the first two moments after 1/Pz² and 1/Pz⁴ fits over Pz^max in [2, 2.5] GeV: ⟨ξ⟩_K = 0.020(3) and ⟨ξ²⟩_K = 0.253(12), with the DA described as single-peaked and asymmetric. That asymmetry is expected from the strange-light mass difference, and the pipeline itself is a straightforward extension of LaMET techniques to the kaon using non-perturbative Euclidean inputs rather than models. The authors are open about the dependence on the extrapolation interval and ansatz, which is better than hiding it. The central claim therefore rests on whether those power corrections are already under control at the modest momenta reached here. The stress-test concern holds up: at Pz around 2–2.5 GeV, omitted 1/Pz^6 terms or residual higher-twist effects could shift the extrapolated shape outside the quoted errors, and the paper does not show a detailed variation study or cross-check against independent lattice results. The upstream functional QCD reference carries some of the load, but that is standard when combining methods. Readers working on exclusive kaon processes or testing LaMET applications will find the concrete numbers worth looking at for comparison. The work is coherent on its own terms and shows clear engagement with the literature, so it deserves peer review to get external input on the extrapolation robustness rather than a desk rejection.

Referee Report

2 major / 2 minor

Summary. The manuscript computes the kaon quasi-distribution amplitude (quasi-DA) by combining large-momentum effective theory (LaMET) with Euclidean functional QCD inputs. Quark correlation functions and the kaon Bethe-Salpeter amplitude are taken from a prior 2+1-flavor functional QCD study; the quasi-DA is obtained via contour deformation in the complex momentum plane. A 1/P_z² + 1/P_z⁴ extrapolation is performed over the interval P_z^max ∈ [2, 2.5] GeV to extract the light-cone DA, which is reported as single-peaked and asymmetric, with moments ⟨ξ⟩_K = 0.020(3) and ⟨ξ²⟩_K = 0.253(12). Uncertainties are attributed to the choice of extrapolation interval and ansatz.

Significance. If the extrapolation is under control, the result supplies a first-principles determination of the kaon DA that can be compared with lattice QCD and phenomenological models, helping quantify SU(3)-flavor breaking in meson distribution amplitudes. The use of contour deformation on functional-QCD correlation functions is a technical strength that avoids certain lattice artifacts while remaining fully non-perturbative at the input level.

major comments (2)

[Results section (extrapolation procedure)] The reported moments rest entirely on the 1/P_z² + 1/P_z⁴ extrapolation performed over the narrow window P_z^max ∈ [2, 2.5] GeV. At these moderate momenta the power series may still receive sizable contributions from omitted 1/P_z^6 and higher-twist terms; the manuscript acknowledges interval/ansatz dependence but does not quantify the associated systematic uncertainty through, e.g., variation of the fit range, inclusion of an additional 1/P_z^6 term, or synthetic-data tests.
[Method section (input preparation)] The quasi-DA is constructed directly from the correlation functions and Bethe-Salpeter amplitude supplied by the upstream functional-QCD reference. Any systematic uncertainties present in those inputs propagate to the final DA moments, yet no error budget or sensitivity study is provided that isolates their contribution from the extrapolation uncertainty.

minor comments (2)

[Section 3] Notation for the quasi-DA and the extrapolation coefficients c₂(ξ), c₄(ξ) should be defined explicitly in a single equation or table to avoid ambiguity when the reader compares different P_z slices.
[Figure captions] Figure captions should state the precise P_z values used for each curve and whether the displayed bands include only statistical or also extrapolation uncertainties.

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 the two major points below and will revise the manuscript to incorporate additional analyses that strengthen the quantification of systematic uncertainties.

read point-by-point responses

Referee: The reported moments rest entirely on the 1/P_z² + 1/P_z⁴ extrapolation performed over the narrow window P_z^max ∈ [2, 2.5] GeV. At these moderate momenta the power series may still receive sizable contributions from omitted 1/P_z^6 and higher-twist terms; the manuscript acknowledges interval/ansatz dependence but does not quantify the associated systematic uncertainty through, e.g., variation of the fit range, inclusion of an additional 1/P_z^6 term, or synthetic-data tests.

Authors: We agree that a more thorough quantification of the extrapolation systematics is warranted. In the revised manuscript we will add explicit tests: (i) variation of the fit range using all available subsets of the P_z^max interval, (ii) inclusion of a 1/P_z^6 term in the ansatz to assess stability of the extracted moments, and (iii) synthetic-data tests in which model light-cone DAs are used to generate quasi-DA data that are then extrapolated with the same procedure. These results will be presented together with an updated error budget that separates extrapolation-related systematics from other sources. We note that the available momentum window is limited by the input data, but the additional checks will make the associated uncertainty more transparent. revision: yes
Referee: The quasi-DA is constructed directly from the correlation functions and Bethe-Salpeter amplitude supplied by the upstream functional-QCD reference. Any systematic uncertainties present in those inputs propagate to the final DA moments, yet no error budget or sensitivity study is provided that isolates their contribution from the extrapolation uncertainty.

Authors: The correlation functions and Bethe-Salpeter amplitude are taken from our prior functional QCD study, whose systematic uncertainties are documented there. To isolate their propagation into the present DA moments, we will add a dedicated sensitivity analysis in the revised manuscript. We will vary the key input parameters (e.g., renormalization constants, truncation parameters, and fit parameters of the Bethe-Salpeter amplitude) within the uncertainties reported in the reference work, recompute the quasi-DA and its moments, and quantify the resulting spread. This sensitivity study will be presented separately from the extrapolation uncertainty, thereby providing a clearer overall error budget. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation uses external inputs and standard extrapolation without self-referential reduction

full rationale

The paper takes quark correlation functions and the kaon Bethe-Salpeter amplitude as inputs from reference [1], computes the quasi-DA via contour deformation from [2], and applies 1/P_z² + 1/P_z⁴ extrapolation over a stated P_z interval to extract the physical DA and its moments. This chain does not reduce the final moments to the inputs by definition, nor does it rename a fit as an independent prediction; the extrapolation ansatz is explicitly applied with acknowledged uncertainties from interval and choice of orders. No self-citation is load-bearing for a uniqueness claim, no ansatz is smuggled, and the result is not equivalent to the inputs by construction. The derivation remains self-contained as a computational pipeline.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The result depends on the validity of LaMET matching, the accuracy of the input functional QCD correlation functions, and the assumed power-law form of the extrapolation; no new particles or forces are postulated.

free parameters (2)

P_z^max interval
Chosen range [2, 2.5] GeV used for extrapolation; affects the final moments.
Extrapolation ansatz coefficients
1/P_z² and 1/P_z⁴ terms fitted to quasi-DA data.

axioms (2)

domain assumption LaMET matching coefficients and power corrections are under control at the accessed momenta
Invoked when converting quasi-DA to light-cone DA.
domain assumption Functional QCD inputs from reference [1] are accurate
Used as direct input without re-derivation.

pith-pipeline@v0.9.0 · 5496 in / 1456 out tokens · 29116 ms · 2026-05-08T05:45:35.849011+00:00 · methodology

discussion (0)

Reference graph

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