Generalized parton distributions of the kaon and pion within the nonlocal chiral quark model

Hyeon-Dong Son; Parada T. P. Hutauruk

REVIEW 3 major objections 5 minor 4 cited by

Generalized parton distributions of the kaon and pion within the nonlocal chiral quark model

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The kaon's strange quark is predicted to carry 26 percent more longitudinal momentum than its up quark at the model scale, a flavor asymmetry encoded in the gravitational form factor ratio Ā_s/K+(0)/A_u/K+(0) = 1.257.

desk verdict First NLchiQM kaon GPDs with explicit chiral breaking, but the headline lattice comparison skips the scale evolution step. read the letter →

arxiv 2411.18130 v2 pith:A7XOFVWS submitted 2024-11-27 hep-ph

Hyeon-Dong Son , Parada T. P. Hutauruk This is my paper · ORCID

classification hep-ph PACS 12.38.-t14.40.Be

keywords generalizedpartondistributionskaonpionnonlocalchiralquarkmodelgravitationalformfactorsQCDevolutionMellinmomentssymmetrybreaking

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

This paper computes the valence-quark generalized parton distributions (GPDs) of the kaon and pion in the nonlocal chiral quark model, an effective model of low-energy QCD built on momentum-dependent dynamical quark masses. It claims that the kaon's unequal current quark masses distort its GPDs strongly near skewness ξ = 1, whereas the pion's GPDs stay nearly symmetric, and that this asymmetry propagates into the gravitational form factors. The central numerical predictions are the ratios Ā_s/K+(0)/A_u/K+(0) = 1.257 and D̄_s/K+(0)/D_u/K+(0) = 1.10 at the model scale, meaning the strange quark carries about 26% more longitudinal momentum than the up quark inside a K+. The paper also evolves the GPDs to 4 and 100 GeV² at one loop and finds that the produced sea quarks and gluons are strongly suppressed as ξ grows. The result matters because these ratios quantify how chiral symmetry breaking redistributes mass, pressure, and shear inside the lightest strange hadron, and they can be compared with lattice QCD and future electron-ion collider measurements.

What carries the argument

The machinery is the nonlocal chiral quark model (NLχQM) effective action, in which the dynamical quark mass M_f(k) = M_f F²(k) carries momentum dependence through F(k) = (−Λ²/(−Λ² + k² + iϵ))^n, with n = 1 chosen here and parameters (Λ, m_q, m_s) = (1150, 5, 133) MeV fixed by normalizing the meson distribution amplitudes and electromagnetic form factors. GPDs are built from the pole structure of quark-loop integrals, organized into three terms I1, I2, and I3 that live in the DGLAP region (x ≥ ξ) and the ERBL region (|x| ≤ ξ); the nonlocality makes the GPDs continuous at x = ±ξ with derivative kinks, and the I3 term shapes the shoulder in the negative ERBL region. Mellin moments of the GPDs are then the bridge to observables: the n = 1 moments give electromagnetic form factors, and the n = 2 moments give the gravitational form factors A(t) and D(t) via polynomiality, A20(t) = A(t) and A22(t) = D(t). For the scale dependence, the model-scale valence-only GPDs are used as the initial input for one-loop QCD evolution, applied numerically to the evolved quark, sea, and gluon distributions.

What would settle it

A lattice QCD calculation of the flavor decomposition of the kaon's gravitational form factors at a scale comparable to the model scale, reporting Ā_s(0)/A_u(0) clearly below 1.257 or D̄_s(0)/D_u(0) far from 1.10, would falsify the paper's central prediction; likewise, a precise measurement of the kaon valence momentum fractions from Drell-Yan data that rules out roughly a 26% strange-quark excess would do the same.

Watch

Extended reading notes

Core claim

In the nonlocal chiral quark model, the valence u and s̄ GPDs of the K+ are computed from one-loop quark diagrams with nonlocal quark-meson vertices, and the mismatch between the light current mass m_u = 5 MeV and strange current mass m_s = 133 MeV produces a visible flavor asymmetry. The paper's central claim is that this explicit chiral symmetry breaking makes the kaon GPDs markedly asymmetric near ξ = 1, the point where the DGLAP region collapses, while the physical pion, with equal light masses, is only mildly asymmetric; the mechanism is the O(m_K²) correction to the second Mellin moment. Taking n = 2 Mellin moments and fitting the ξ² coefficient gives gravitational form factors whose zero-momentum ratios are Ā_s(0)/A_u(0) = 1.257 and D̄_s(0)/D_u(0) = 1.10, with A(0) + D(0) = 0.363 for the kaon and 0.040 for the pion at the model scale. The paper further claims that one-loop QCD evolution from µ0 = 0.33 GeV to 4 and 100 GeV² leaves the valence GPDs relatively stable at large ξ while the generated sea and gluon GPDs are confined mostly to the ERBL region and suppressed as ξ grows.

Load-bearing premise

The load-bearing premise is that the valence-only GPDs computed at the model scale µ0 = 0.33 GeV, a value borrowed from an earlier NJL fit and not derived within the model, constitute the complete nonperturbative input for one-loop QCD evolution; if the true matching scale is different or the initial gluon and sea content is nonzero, the evolved GPDs and the associated high-scale statements change.

Editorial extensions

If this is right

At the model scale, the antistrange quark in the K+ carries Ā_s(0)/A_u(0) = 1.257, meaning about 26% more longitudinal momentum than the up quark.
The D-term ratio D̄_s(0)/D_u(0) = 1.10 implies that the strange-quark contribution to the kaon's internal pressure and shear is about 10% larger in magnitude than the up-quark contribution.
Because the evolved gluon and sea-quark GPDs are suppressed as ξ grows, hard exclusive reactions at large skewness would see the kaon's partonic structure dominated by valence quarks even at 100 GeV².
The values A_K+(0) + D_K+(0) = 0.363 and A_π+(0) + D_π+(0) = 0.040 at the model scale sit close to the chiral perturbation theory expectations of 0.23 and 0.03, so the model reproduces the Goldstone-boson pattern while adding flavor-breaking corrections.
The kaon's flavor-asymmetric GPDs imply that the u-quark electromagnetic and gravitational form factors fall faster with −t than the s̄-quark ones, so the strange-quark distribution inside the kaon is spatially more compact than the light-quark cloud.

Reading between the lines

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

If the model-scale ratio Ā_s/A_u ≈ 1.257 survives QCD evolution approximately unchanged at moderate scales, it would serve as a nonperturbative benchmark for lattice QCD flavor decompositions of the kaon gravitational form factors, which are currently more uncertain than the pion ones.
The same flavor asymmetry should appear in the kaon's transverse charge and mass radii: future imaging of the kaon through Sullivan-process deeply virtual Compton scattering could see a strange core that is spatially more compact than the light-quark cloud.
A natural testable extension is to compute the isovector combination of kaon GPDs at nonzero ξ and compare the ξ-dependence of the D-term with the polynomiality fit; a violation would signal missing contributions from the model's nonconserved flavor currents rather than a failure of the GPD framework itself.
Since the model's momentum sum rule is broken by about 7% (M2^K ≈ 0.93), renormalizing the evolved sea and gluon distributions to enforce momentum conservation would shift the high-scale shapes, and the size of that shift measures the nonlocal-current corrections a gauge-invariant extension of the model would need to add.

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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

I’ll give you two things up front. First, this is the first NLchiQM computation of kaon GPDs with nonzero current quark masses, extending the chiral-limit pion work of Praszalowicz and Rostworowski. The analytic expressions are complete, polynomiality is checked numerically, and the parameter-fixing window is described. Second, the headline GFF ratios — A_sbar/A_u = 1.257 and D_sbar/D_u = 1.10 — are extracted at the model scale µ0 = 0.33 GeV and then compared in Table I with lattice QCD at 2 GeV without evolving them. Given the paper’s own Fig. 7, which shows strong µ0 sensitivity in the evolved GPDs, that comparison is not established. The stress-test note has this right.

What is genuinely new and useful: the SU(3) extension with explicit chiral symmetry breaking produces analytic I1, I2, I3 expressions for the u- and sbar-quark GPDs, and the paper carefully documents the approach to the ERBL region, the ξ = 1 distortion from quark-mass asymmetry, and the forward limit. It also does something increasingly rare: it states its own limitations. The text admits the momentum sum rule is violated by about 7% (M2 ≈ 0.93), the ξ range used in the figures is unphysical, and the initial evolution scale is borrowed from a fit in Ref. [24]. Those are real caveats, and acknowledging them does not remove them, but it does make the model-scale part of the paper readable and checkable.

The soft spots are mostly around the evolved and compared quantities. The scale mismatch is the biggest one: the GFF ratios are model-scale numbers, and the lattice values are at 2 GeV. The paper notes the lattice scale in passing but does not evolve the ratios or quantify the shift. Because the singlet evolution will redistribute momentum between valence quarks and sea/gluons, the strange-to-u ratio at 2 GeV may well be different from 1.257; the agreement with the ETMC value could be coincidental. This does not invalidate the model-scale prediction, but it does mean the abstract’s comparison with lattice wants a much stronger caveat or an actual evolved ratio. The lack of error bars on the ratios is a minor issue by comparison; the parameter set is fixed, so a simple sensitivity scan over Λ and ms would have been enough. I would also ask the authors to clarify the parameter differences between the NLchiQM and LchiQM rows in Table I, since the latter uses a different (Pauli-Villars) regularization.

Who should read this: people working on meson GPD phenomenology and EIC-era projections. The model-scale GPDs and the flavor decomposition of the GFFs are a useful reference point, and the analytic expressions are detailed enough to be reimplemented. No code or data is shipped, but the formulas are explicit. I would send this to a serious referee, with a request to address the scale evolution of the ratios and add a parameter-sensitivity estimate.

Referee Report

3 major / 5 minor

Summary. This paper extends the nonlocal chiral quark model (NLχQM) of Refs. [20,21] to SU(3) flavor with explicit current quark masses and computes the valence u- and sbar-quark GPDs of the K+ together with the u-quark GPD of the π+. After fixing Λ=1150 MeV and ms=133 MeV from the normalization conditions for the meson distribution amplitudes and electromagnetic form factors, the authors present x, ξ, and t portraits of the GPDs, discuss the role of the I3 contribution in the ERBL region, and evolve the model-scale GPDs to μ²=4 and 100 GeV² with one-loop APFEL++ evolution using μ0=0.33 GeV. They check polynomiality of the Mellin moments numerically, extract electromagnetic and gravitational form factors from the n=1 and n=2 moments, and report the kaon GFF ratios A_sbar(0)/A_u(0)=1.257 and D_sbar(0)/D_u(0)=1.10, comparing them in Table I with lattice QCD and other model results. The calculation is self-consistent in its derivations, but the paper explicitly acknowledges a 7% violation of the momentum sum rule, uses unphysical ξ values for the moment fits, and quotes the central GFF ratios at the model scale while comparing them with lattice results at 2 GeV.

Significance. If the results hold, the paper provides a useful model estimate of kaon flavor-asymmetric gravitational form factors and a concrete demonstration that current-quark-mass asymmetry distorts the valence GPDs near ξ=1. The paper's strengths include the explicit analytic one-loop expressions in Appendix A, the numerical verification of polynomiality, the use of the APFEL++ package for LO evolution, and a parameter-fixing procedure that constrains Λ and ms from two independent normalization conditions. These are concrete, checkable elements. However, the headline comparison with lattice QCD is made at the model scale without evolving the GFF moments, and the D-term extraction relies on unphysical skewness values, so the empirical anchoring of the central ratios is not yet established.

major comments (3)

[Sec. V.C; Table I; Eqs. (70)-(72)] The ratios A_sbar/K+(0)/A_u/K+(0)=1.257 and D_sbar/K+(0)/D_u/K+(0)=1.10 are obtained from n=2 Mellin moments evaluated at the model scale μ0=0.33 GeV, yet Table I compares them with the ETMC lattice value ≈1.3 evaluated at μ=2 GeV and with other model predictions at unspecified scales. The paper acknowledges in the last paragraph of Sec. V.C that the lattice results are at 2 GeV, but it does not evolve the moments or the ratios. Since Fig. 7 shows that LO evolution from μ0=0.33 to 0.6 GeV substantially changes the evolved kaon GPDs, particularly at small ξ, the Mellin moments—and therefore A20(0) and A22(0)—are scale dependent. The advertised agreement with lattice can thus be a scale artifact. Please either evolve the n=2 moments to 2 GeV with the same APFEL++ setup, or clearly separate the model-scale prediction from the lattice comparison and give an estimate of the expected scale correction.
[Sec. V.B.1 and Sec. V.C; Eqs. (32)-(33), (57)-(58)] For t=0, Eq. (32) gives Δ_⊥²=-4m_K²ξ², which is negative for any ξ≠0; the physical constraint in Eq. (33) therefore permits only ξ=0. Nevertheless, Sec. V.C states that the n=2 Mellin moments are computed by varying ξ in [0,1] and fitted with Eq. (57), so the extraction of A22(0) and hence D(0) rests on the model's continuation into unphysical kinematics. The same issue occurs at t=-0.1 GeV², where the physical range ends at ξ≈0.305 while fits use values up to ξ=0.6. Polynomiality is a mathematical property of the moments, but it does not by itself validate the unphysical continuation. Please demonstrate that the fitted A20 and A22 are stable when the fit is restricted to the physical ξ range for t<0, and either extrapolate A22 to t=0 or otherwise justify the continuation.
[Sec. V.B and V.C; Eqs. (51)-(53), (69)-(72); Table I] The central prediction is quoted as a number without an uncertainty. The momentum sum rule is broken at the 7% level (M_K+²=0.93 and M_π+²=0.93), and in the local-model limit the same ratios change from 1.257/1.10 to 1.115/1.199 (Table I). The rough statement that nonlocal corrections shift the ratio by only ≈0.02 is not a systematic uncertainty, and the paper notes that the local model itself is strongly parameter dependent. An estimate of the combined uncertainty from (i) the non-conserved vector current and missing gauge-invariant currents, (ii) the allowed Λ-ms parameter window, and (iii) the local/nonlocal model spread should accompany Eqs. (70) and (72) before these quantities are presented as predictions.

minor comments (5)

[Sec. V.B.2] The text says the third column of Fig. 6 gives kaon GPDs at μ²=10 GeV², but the abstract, Fig. 6, and surrounding discussion refer to μ²=100 GeV²; please correct this.
[Captions of Figs. 2, 3, and 8] The momentum transfer is rendered as 't = □0.1 GeV²' and 't = □0 GeV²', with the minus sign missing; the intended values are t=-0.1 GeV² and t=0 GeV².
[Abstract and Table I] The abstract and Table I write A_sbar/K+(0)/A_s/K+(0), while the text and Eq. (70) define the ratio as A_sbar/K+(0)/A_u/K+(0); please make the flavor notation consistent throughout.
[Introduction and Sec. III] The Introduction misspells 'Rostworowski' as 'Rosworowski', and Sec. III contains the typo 'do nott have'.
[Appendix A.2] The label 'MIA1' is used without definition; if it refers to a specific result from Ref. [26], please state this explicitly.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central GPD and GFF outputs are computed from the model's quark-loop integrals, not fitted to the quantities they predict; the evolution results are explicitly qualitative and their scale sensitivity is disclosed.

full rationale

The derivation chain is self-contained at the level of the central claims. The model action (Eq. 22) is fixed by parameters (Lambda, m_q, m_s) chosen in Sec. V.A from two normalization conditions, the DA normalization (Eq. 45) and the EM form factor at t=0 (Eq. 46). The quark GPDs are then obtained directly from the one-loop integrals (Eqs. 34-39), and the reported ratios A_sbar/K+(0)/A_u/K+(0)=1.257 and D_sbar/K+(0)/D_u/K+(0)=1.10 are outputs of the n=2 Mellin moments (Eqs. 57-58) with A22 identified with the D-term through the standard relation Eq. (9). These numbers are not fitted target quantities: the momentum sum rule is not imposed (M2 ~ 0.93), so the ratios are genuine model predictions rather than reabsorptions of input constraints. The evolution section is explicitly not aimed at quantitative prediction ('we do not focus on describing or predicting the measurable quantities'), and the initial scale mu0=0.33 GeV is borrowed from external Ref. [24]; the strong mu0 sensitivity displayed in Fig. 7 is a disclosed model-dependence limitation, not a circular step. The comparison of model-scale GFF ratios with lattice results at mu=2 GeV in Table I raises a scale-mismatch/correctness concern, but that is not circularity because the model-scale numbers are not derived from the lattice values. The only self-citations (e.g., Ref. [53] for a caution about GMOR-based parameter selection) are not load-bearing and do not supply the claimed numerical results. Consequently, no specific reduction of an output to an input by construction can be exhibited.

Assumptions & free parameters 7 free parameters · 6 assumptions · 0 invented entities

No genuinely new particle, force, dimension, or conserved quantity is introduced. The momentum-dependent quark mass and form factor F(k) are the model's existing machinery, not new entities. The ledger is dominated by model parameters and domain assumptions inherited from the instanton-vacuum-based chiral quark model.

free parameters (7)

Nonlocal cutoff Lambda = 1150 MeV
Chosen so that the meson DAs normalize to 1 and the EM form factors satisfy F(0)=1 (Eqs. 45-46); the paper finds a narrow allowed window for given ms.
Strange current quark mass ms = 133 MeV
Varied together with Lambda to satisfy Eqs. (45)-(46); with Lambda=1150 MeV, ms about 130 MeV reproduces FK and Fpi.
Light current quark mass mu = 5 MeV
Input value typical for the model, not fitted in this paper; enters the current mass matrix and controls explicit chiral symmetry breaking.
Dynamical quark mass M = 350 MeV
Taken from the instanton vacuum (Refs. [48-51]) as in Ref. [21]; not fitted here but sets the overall scale of all results.
Form-factor exponent n = 1
Chosen for simplicity in F(k); earlier studies found mild n-dependence, but n=1 is still a modeling choice.
Initial evolution scale mu0 = 0.33 GeV
Borrowed from a fit in Ref. [24]; used as the input scale for LO GPD evolution and strongly affects the evolved results (Fig. 7).
Pauli-Villars mass MPV (local model) = 560 MeV
Used in the local chiral quark model comparison; determined from DA normalization and meson decay constants.

assumptions (6)

domain assumption The nonlocal chiral quark model effective action (Eq. 22) with the pole form factor F(k)=(-Lambda^2/(-Lambda^2+k^2+i epsilon))^n describes pion and kaon structure at a low scale.
Invoked in Sec. III; the instanton zero-mode form factor is replaced by this analytic form, and one-loop quark diagrams are identified with GPD matrix elements.
domain assumption Large-Nc limit: only leading O(Nc) quark-loop diagrams contribute and meson-loop contributions are neglected.
Sec. IV states that subleading meson-loop contributions are disregarded; the paper later notes this affects D-term slopes and radii.
domain assumption LO DGLAP/ERBL QCD evolution from mu0=0.33 GeV with valence-only initial GPDs approximates the scale dependence.
Sec. V B 2; the initial scale is not derived in the model, and alpha_s is large at this scale, so the authors present the evolution as qualitative.
standard math Polynomiality of GPD Mellin moments in xi and the identification A20=A(t), A22=D(t) (Eqs. 7-9) hold for the model GPDs.
Sec. II; the paper numerically verifies xi independence of the extracted form factors.
domain assumption Isospin symmetry mu=md is used to relate pion GPDs.
Sec. III uses a current mass matrix with mu=md, which is standard for the model.
domain assumption The model-scale GPDs are valence-only and even in xi, so they vanish for x<-xi and have no intrinsic gluon or sea content.
Sec. IV and Sec. V B; this determines the ERBL structure and the initial condition for evolution.

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Cite this review

Pith. "Pith review of Generalized parton distributions of the kaon and pion within the nonlocal chiral quark model." pith.science (2026). https://pith.science/paper/A7XOFVWS

@misc{pith2026241118130,
  author       = {Pith},
  title        = {Pith review of: Generalized parton distributions of the kaon and pion within the nonlocal chiral quark model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/A7XOFVWS}},
  note         = {Machine review of arXiv:2411.18130}
}

abstract

In the present study, we explore the properties of generalized parton distributions (GPDs) for the kaon and pion within the framework of the nonlocal chiral quark model (NL$\chi$QM). Valence quark GPDs of the kaon and pion are analyzed with respect to their momentum fraction $x$ and skewness $\xi$ dependencies in the DGLAP and ERBL regions. We observe that the asymmetry of the current quark masses in kaon results in a significant distortion of the quark GPDs in kaon near $\xi=1$, compared to the case of the pion. The quark GPDs of the kaon and pion are evolved to $\mu^2 = 4$ GeV$^2$ and 100 GeV$^2$ by the QCD evolution equation at one-loop order using the \texttt{APFEL++} package. We find that the produced sea quarks and gluons are largely suppressed as $\xi$ becomes nonzero, predominantly confined within the ERBL region. We subsequently examine the polynomiality of the GPDs and numerically obtain the electromagnetic and gravitational form factors of the kaon and pion. For the kaon, gravitational form factor ratios $A_{\bar s/K^+}(0)/A_{s/K^+}(0) = 1.26$ and $D_{\bar s/K^+}(0)/D_{s/K^+}(0) = 1.10$ are reported and compared with results from other effective models.

Figures

Figures reproduced from arXiv: 2411.18130 by the authors.

**Figure 2.** FIG. 2. Three-dimensional portrait of [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**Figure 3.** FIG. 3. The [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

Figures from the paper (9 more)

**Figure 4.** Figure 4: FIG. 4. Contributions of functions of [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]

**Figure 5.** Figure 5: FIG. 5. Pion and kaon GPDs at [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]

**Figure 6.** Figure 6: FIG. 6. The kaon GPDs after evolving from the initial scale [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]

**Figure 7.** Figure 7: FIG. 7. The kaon GPDs after being evolved to [PITH_FULL_IMAGE:figures/full_fig_p013_7.png]

**Figure 8.** Figure 8: FIG. 8. Quark GPDs of the kaon along the cross-over line [PITH_FULL_IMAGE:figures/full_fig_p013_8.png]

**Figure 9.** Figure 9: FIG. 9. Quark GPDs in the forward limit ( [PITH_FULL_IMAGE:figures/full_fig_p014_9.png]

**Figure 10.** Figure 10: FIG. 10. Generalized form factors [PITH_FULL_IMAGE:figures/full_fig_p018_10.png]

**Figure 11.** Figure 11: FIG. 11. Gravitational form factors of the kaon ( [PITH_FULL_IMAGE:figures/full_fig_p018_11.png]

**Figure 12.** Figure 12: FIG. 12. Meson light-cone distribution amplitudes in the nonlocal chiral quark model. Along with the numerical results of [PITH_FULL_IMAGE:figures/full_fig_p020_12.png]

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Reference graph

Works this paper leans on

90 extracted references · 21 canonical work pages · cited by 4 Pith papers

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Generalized parton distributions of the pion in a Bethe-Salpeter approach

L. Theussl, S. Noguera, V. Vento, Generalized parton distributions of the pion in a Bethe-Salpeter approach, Eur. Phys. J. A 20 (2004) 483–498. arXiv:nucl-th/0211036, doi:10.1140/epja/i2003-10174-3

work page Pith review arXiv 2004
[1]

Portraits of the quark GPDs in kaon and pion As an illustration, we plot the u-quark and ¯s-quark GPDs for the K + at t = −0.1 GeV 2 in Fig. 2. As stated in Eq. (33), the positiveness of the momentum transfer ∆ 2 ⊥ ≥ 0 restricts the physically possible value of ξ. However, in the present work, we release the constraint and consider the unphysical range. B...
[2]

To achieve that, it is important to determine the reference scale of the model µ0

One-loop evolution of the GPDs Identifying the valence quark GPDs computed within the NL χQM as the initial state of the QCD evolution at a low renormalization point, we are in a position to evaluate the GPDs at a higher scale. To achieve that, it is important to determine the reference scale of the model µ0. For instance, in the instanton model [50], the...
[3]

Envelope functions and forward limit In the DVCS process, the Compton form factors (CFFs) are directly extracted from the crosssection. At the leading order approximation in the strong coupling αs, the CFFs are written as the convolution of the GPDs and the perturbative quark and antiquark lines in x, Ha(ξ, t) = e2 a Z 1 −1 dxH a(x, ξ, t) 1 −ξ + x − iϵ − ...
[4]

This is so-called the soft pion theorem

Kaon DAs As in chiral limit, mπ ≈ 0 the quark kaon and pion GPDs are connected to the pion and kaon DAs, respectively. This is so-called the soft pion theorem. Here we begin with the light-cone DA of the kaon, which is defined by the following ϕK+ (u) = 1 i √ 2FK Z ∞ −∞ dτ π e−iτ (2u−1)n·P ⟨0| ¯ψ(nτ )γ+γ5ψ(−nτ )|K +(P )⟩. (A1) Using the model of Eq. (22),...
[5]

To determine the cutoff Λ and the strange quark mass ms, we vary these two parameters to reproduce the meson decay constants FK+ and Fπ+ from the normalization of the meson DAs

Results for meson DAs For the numerical computation of the kaon and pion DAs, we use Ms = Mu = M = 350 MeV from the instanton model, and mu = 5 MeV as our input parameters. To determine the cutoff Λ and the strange quark mass ms, we vary these two parameters to reproduce the meson decay constants FK+ and Fπ+ from the normalization of the meson DAs. In Fig...
[6]

In this section, we show explicit expressions of the integrands in Eqs

Kaon generalized quark distributions In the present work, we consider only ξ >0, owing to the even nature of the quark GPDs in ξ. In this section, we show explicit expressions of the integrands in Eqs. (34)-(37). For convenience, we introduce the scaled variables: η = P +k− Λ2 , ⃗ κ⊥ = ⃗k⊥ Λ , τ = ˜t 4Λ2 , ⃗δ⊥ = ⃗∆ 2Λ . (A8) 21 The k− integral can be eval...
[7]

For the pion DA, the result is obtained by letting (ms, Ms, mK) → (mu, Mu, mπ) from the above expression

Meson DAs Kaon DA in the local model can be computed straightforwardly and written as follows: ϕK+ (u) = Nc √MuMsΛ (2π)2F 2 K Z ∞ 0 dκ2 ⊥ u¯µu + (1 − u)¯µs κ2 ⊥ + u(u − 1) ˜m2 K + u¯µ2u + (1 − u)¯µ2s , (B2) where ¯µu = (mu + Ms)/Λ, ¯µs = (ms + Ms)/Λ, and ˜mK = mK/Λ. For the pion DA, the result is obtained by letting (ms, Ms, mK) → (mu, Mu, mπ) from the ab...

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[8]

Kaon GPDs in local chiral quark model The diagrammatic structure of the kaon GPDs in the local model is same as Eqs. (38) and (39), which are again represented in the following loop integrals: I u/K + 1 (x, ξ, t) = NcMuMs 2(2π)3F 2 Z ∞ 0 dκ2 ⊥ Z C(0,1) dλ iλ (x − 1)[F0 + C(λ +...
[9]

Banks, A

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

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Generalized parton distributions of the pion in a Bethe-Salpeter approach

L. Theussl, S. Noguera, V. Vento, Generalized parton distributions of the pion in a Bethe-Salpeter approach, Eur. Phys. J. A 20 (2004) 483–498. arXiv:nucl-th/0211036, doi:10.1140/epja/i2003-10174-3

work page Pith review arXiv 2004

[2] [1]

Portraits of the quark GPDs in kaon and pion As an illustration, we plot the u-quark and ¯s-quark GPDs for the K + at t = −0.1 GeV 2 in Fig. 2. As stated in Eq. (33), the positiveness of the momentum transfer ∆ 2 ⊥ ≥ 0 restricts the physically possible value of ξ. However, in the present work, we release the constraint and consider the unphysical range. B...

[3] [2]

To achieve that, it is important to determine the reference scale of the model µ0

One-loop evolution of the GPDs Identifying the valence quark GPDs computed within the NL χQM as the initial state of the QCD evolution at a low renormalization point, we are in a position to evaluate the GPDs at a higher scale. To achieve that, it is important to determine the reference scale of the model µ0. For instance, in the instanton model [50], the...

[4] [3]

Envelope functions and forward limit In the DVCS process, the Compton form factors (CFFs) are directly extracted from the crosssection. At the leading order approximation in the strong coupling αs, the CFFs are written as the convolution of the GPDs and the perturbative quark and antiquark lines in x, Ha(ξ, t) = e2 a Z 1 −1 dxH a(x, ξ, t) 1 −ξ + x − iϵ − ...

[5] [4]

This is so-called the soft pion theorem

Kaon DAs As in chiral limit, mπ ≈ 0 the quark kaon and pion GPDs are connected to the pion and kaon DAs, respectively. This is so-called the soft pion theorem. Here we begin with the light-cone DA of the kaon, which is defined by the following ϕK+ (u) = 1 i √ 2FK Z ∞ −∞ dτ π e−iτ (2u−1)n·P ⟨0| ¯ψ(nτ )γ+γ5ψ(−nτ )|K +(P )⟩. (A1) Using the model of Eq. (22),...

[6] [5]

To determine the cutoff Λ and the strange quark mass ms, we vary these two parameters to reproduce the meson decay constants FK+ and Fπ+ from the normalization of the meson DAs

Results for meson DAs For the numerical computation of the kaon and pion DAs, we use Ms = Mu = M = 350 MeV from the instanton model, and mu = 5 MeV as our input parameters. To determine the cutoff Λ and the strange quark mass ms, we vary these two parameters to reproduce the meson decay constants FK+ and Fπ+ from the normalization of the meson DAs. In Fig...

[7] [6]

In this section, we show explicit expressions of the integrands in Eqs

Kaon generalized quark distributions In the present work, we consider only ξ >0, owing to the even nature of the quark GPDs in ξ. In this section, we show explicit expressions of the integrands in Eqs. (34)-(37). For convenience, we introduce the scaled variables: η = P +k− Λ2 , ⃗ κ⊥ = ⃗k⊥ Λ , τ = ˜t 4Λ2 , ⃗δ⊥ = ⃗∆ 2Λ . (A8) 21 The k− integral can be eval...

[8] [7]

For the pion DA, the result is obtained by letting (ms, Ms, mK) → (mu, Mu, mπ) from the above expression

Meson DAs Kaon DA in the local model can be computed straightforwardly and written as follows: ϕK+ (u) = Nc √MuMsΛ (2π)2F 2 K Z ∞ 0 dκ2 ⊥ u¯µu + (1 − u)¯µs κ2 ⊥ + u(u − 1) ˜m2 K + u¯µ2u + (1 − u)¯µ2s , (B2) where ¯µu = (mu + Ms)/Λ, ¯µs = (ms + Ms)/Λ, and ˜mK = mK/Λ. For the pion DA, the result is obtained by letting (ms, Ms, mK) → (mu, Mu, mπ) from the ab...