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REVIEW 3 major objections 7 minor 95 references

Chiral-odd generalized parton distributions in the large-$N_{c}$ limit of QCD: Next-to-leading-order contributions

T0 review · 3 major / 7 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The pion mean-field picture of large-$N_c$ QCD yields the complete next-to-leading-order chiral-odd GPDs, whose moments obey polynomiality and whose first moments reproduce the tensor form factor sum rules.

desk verdict Solid NLO-large-Nc derivation of chiral-odd GPD structures; the advertised lattice agreement is not controlled and needs tempering. read the letter →

arxiv 2506.21013 v1 pith:GZR54UY4 submitted 2025-06-26 hep-ph hep-lat

June-Young Kim This is my paper
classification hep-phhep-lat
keywords chiral-oddGPDslarge-NcQCDpionmean-fieldnext-to-leadingorderpolynomialitytensorformfactorsgradientexpansiontransversity
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

The paper aims to establish that the pion mean-field picture of the nucleon in the large-$N_c$ limit of QCD can supply the complete next-to-leading-order ($1/N_c$) contributions to the nucleon's chiral-odd generalized parton distributions, filling the spin-flavor structures that were missing at leading order in both the flavor-singlet and flavor-non-singlet sectors. It derives the rotational zero-mode correction to the mean field, organizes the resulting matrix elements into multipole mean-field GPDs, and proves that their moments are polynomials in the skewness $\xi$ with the standard degree bounds, with the first moments satisfying the tensor form factor sum rules. Using a gradient expansion that keeps the UV-divergent Dirac-sea terms, the paper produces numerical estimates for the non-forward region; the flavor-non-singlet combination $\bar E_T^{u-d}$ agrees with available lattice QCD data in the negative-$x$ region. If this program is correct, the mean-field framework becomes a complete nonperturbative tool for transversely polarized quark distributions, with immediate use for exclusive meson production phenomenology at current and future facilities.

What carries the argument

The load-bearing object is the pion mean-field picture of the nucleon in the large-$N_c$ limit: a static hedgehog pion field $U(x)=\exp[i\,\hat{x}\cdot\tau\,P(r)]$ with $N_c$ valence quarks and a distorted Dirac sea, quantized by collective flavor rotations with angular velocity $\Omega_a = J_a/I$. The machinery that carries the argument is the NLO rotational correction to the matrix element of the effective chiral-odd operator, expressed in the single-particle representation with energy denominators from particle-hole excitations; the multipole expansion in the two-dimensional transverse momentum transfer then defines the multipole mean-field GPDs $Z_{mf,0}$, $Z_{mf,1}$, $\tilde Z_{mf,1}$, $Z_{mf,2}$. Polynomiality is proven from three discrete ingredients: the $G_5$ transformation (time reversal combined with an isospin rotation), parity, and the grand-spin selection rules that truncate the partial-wave expansion of $e^{i\Delta\cdot \hat{X}}$ to finite order. The gradient expansion of the quark propagator, keeping the single-chiral-field UV-divergent terms at order $\alpha=0$ and $\alpha=1$, provides the numerical estimates that are compared with lattice QCD.

What would settle it

At $\xi=1/3$ and $t=-1.02$ GeV$^2$, evaluate $\bar E_T^{u-d}(x)$ using the complete sum over all single-particle states including the discrete level; if the positive-$x$ result does not move appreciably toward the lattice points while negative-$x$ agreement is preserved, the truncation to UV-divergent gradient terms is not the dominant approximation.

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Extended reading notes

Core claim

On the author's own terms, the central result is that at next-to-leading order in $1/N_c$ the rotational correction to the pion mean field generates exactly the spin-flavor multipole structures that leading order cannot: a dipole operator in the flavor-singlet channel and monopole plus quadrupole operators in the flavor-non-singlet channel. These enter through single-particle sums over occupied and non-occupied Dirac levels, with the moment of inertia $I$ setting the $N_c$ suppression, and they combine into four multipole mean-field GPDs $Z_{mf,0}$, $Z_{mf,1}$, $\tilde Z_{mf,1}$, $Z_{mf,2}$ that map linearly onto the standard chiral-odd GPDs $H_T$, $\tilde H_T$, $E_T$, $\tilde E_T$. The paper proves that the $m$-th moments of these mean-field GPDs are even or odd polynomials in $\xi$ of the required degree, using the $G_5$ (time-reversal times isospin) symmetry, parity, and grand-spin selection rules, and that the first moments reproduce the tensor form factors $H_T(t)$, $\tilde H_T(t)$, $E_T(t)$, with $\int dx\, \tilde E_T = 0$. Numerically, keeping the UV-divergent $\alpha=0$ and $\alpha=1$ gradient-expansion terms, all displayed GPDs are convex peak-like functions centered at $x=0$ with a smooth crossover at $x=\pm\xi$, the magnitudes follow the multipole hierarchy $H_T^{u+d} \ll \bar E_T^{u-d} \ll E_T^{u+d}$, $\tilde E_T$ vanishes at this accuracy, and $\bar E_T^{u-d}$ at $\xi=1/3$, $t=-1.02$ GeV$^2$ is in good agreement with lattice QCD in the negative-$x$ region.

Load-bearing premise

The numerical predictions and the lattice comparison assume that the UV-divergent Dirac-sea terms kept in the gradient expansion dominate the omitted UV-finite terms and the discrete-level connected-diagram contribution; if those omitted pieces are not small, the computed curves and the claimed agreement with lattice QCD would shift.

Editorial extensions

If this is right

  • The complete NLO chiral-odd GPD set in both flavor sectors can be used directly as model input for exclusive pseudoscalar meson production amplitudes, where chiral-odd GPDs couple to pion distribution amplitudes.
  • The proven polynomiality and first-moment sum rules guarantee that any mean-field-based extraction of tensor charges, $\kappa_T$, and generalized tensor form factors is internally consistent.
  • The large-$N_c$ relation $2\tilde H_T^{u-d} = -E_T^{u-d}$ and the $N_c$-scaling hierarchy give sharp, testable predictions that lattice QCD or other quark models can confirm or refute.
  • The vanishing of $\tilde E_T$ at UV-divergent accuracy implies that a nonzero $\tilde E_T$, if observed, is a direct signal of the UV-finite and discrete-level contributions that this truncation drops.
  • The peak-shaped ERBL-region GPDs with smooth crossover at $x=\pm\xi$ provide a concrete benchmark for upcoming exclusive-production experiments at 12 GeV electron facilities.

Reading between the lines

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

  • If the polynomiality proof extends to all orders in the gradient expansion, the mean-field GPD basis could be used to constrain the higher $m\ge 2$ moments of chiral-odd GPDs that lattice QCD currently does not reach, once the discrete-level contribution is computed.
  • The quadrupole spin operator $Q^{kl}$, which vanishes in the nucleon matrix element but is generated by the NLO spin-flavor algebra, would become the leading structure in $N\to\Delta$ transition GPDs or $\Delta$-baryon GPDs; the same mean-field formalism could predict those without new model input.
  • The negative-$x$ agreement with lattice QCD suggests that the ERBL region is dominated by the Dirac-sea (pion-cloud) dynamics captured by the gradient expansion, while the positive-$x$ valence region is governed by the discrete-level contribution; a full single-particle computation would directly separate these two mechanisms.
  • The hierarchy $H_T^{u+d} \ll \bar E_T^{u-d} \ll E_T^{u+d}$, if confirmed by lattice data, would mean that the multipole order of the mean-field GPD is the organizing principle for the strength of transversity distributions.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 7 minor

Summary. This paper extends the large-Nc pion mean-field (chiral quark-soliton) description of nucleon chiral-odd GPDs from leading order to next-to-leading order in the 1/Nc expansion. The NLO baryon matrix element of the nonlocal chiral-odd operator is derived in the single-particle representation, including the first-order rotational zero-mode correction; its spin-flavor structure is decomposed into multipole mean-field GPDs (monopole Zmf,0, dipoles Zmf,1 and ~Zmf,1, quadrupole Zmf,2) through a two-dimensional multipole expansion in the transverse momentum transfer, and linear relations (64) between these objects and the standard GPDs HT, ~HT, ET, and ~ET are established. The m-th moments of the mean-field GPDs are shown to be polynomials in ξ with the correct power structure, using G5 symmetry, parity, and grand-spin selection rules, and the first moments are matched to the nucleon tensor form factors through sum rules. Numerical estimates are then obtained by gradient-expanding the quark propagator and retaining only the UV-divergent α = 0 and α = 1 contributions; the resulting ~ET^{u-d} is compared with lattice QCD data in Fig. 6, with the abstract reporting good agreement. The paper states in Secs. VI and VIII that a complete prediction requires the full single-particle sum, which is not performed.

Significance. If the formal derivations are correct, this is a useful and substantial extension of Ref. [1]: it completes the spin-flavor classification of chiral-odd GPDs in the mean-field picture, provides explicit polynomiality proofs at the level of the single-particle representation, and yields testable large-Nc predictions including the hierarchy (62) and the relation (63). The proofs are grounded in the discrete symmetries of the mean-field Hamiltonian rather than in any fit to the GPD data, and the parameters (M, R0, Λ) are fixed by the instanton-vacuum model before the lattice comparison, so the formal section is free of circularity. The numerical section, by contrast, is not a controlled calculation: as the paper itself concedes, the retained UV-divergent gradient-expansion terms do not constitute the full result, and the truncation parameter is not small at the adopted parameters. The lattice comparison in Fig. 6 should therefore be read as an illustration of the mechanism rather than as an established quantitative prediction.

major comments (3)
  1. [Sec. VI, Eqs. (106), (113)-(115), and Fig. 6] The truncation of the gradient expansion is uncontrolled at the adopted parameters, so the 'good agreement with lattice QCD' claim in the abstract and in Sec. VII is not established. The expansion (106) is in powers of iMγ·∇Uγ5 and is stated to require ∂U ≪ M, but with the profile (113) and MR0 = 1 one has |∇U| ≈ |P'(r)| ≈ 2/R0 = 2M at r ≈ R0, i.e., the expansion parameter is O(1), not small. With Λ ≈ 835 MeV and M = 350 MeV from Eqs. (114)-(115), the logarithmic enhancement log(Λ/M) ≈ 0.9 is too small to compensate, so the UV-finite terms explicitly dropped in Eqs. (111)-(112) and the α ≥ 2 terms are of the same order as the retained UV-divergent pieces. The paper itself states in Sec. VI that a complete prediction requires summing over all single-particle wave functions and in Sec. VIII that the study was limited to UV-divergent contributions and that the discrete-level (connected-diagram) contribution is missing; given an O(1) expansion parameter, these omissions cannot be assumed small, and the comparison in Fig. 6 is made without any estimate of the systematic error.
  2. [Sec. IV C, Sec. VII, Eqs. (53)-(55), Fig. 6] The numerical comparison is made outside the kinematic regime in which the mean-field GPD basis was derived. The relations (64) and the identification of the computed objects with standard GPDs assume x = O(Nc^{-1}) and ξ = O(Nc^{-1}) from Eqs. (53) and (55), but Fig. 6 is evaluated at ξ = 1/3, for which Nc ξ = O(1), and the plotted range extends to |x| = 1. Furthermore, Sec. VI states that the retained UV-divergent contributions live only in the ERBL region |x| < ξ; if that is so, the curve in Fig. 6 should vanish for |x| > 1/3, and the claimed agreement in the negative-x region needs to be restricted to that interval. Please clarify the x-support of the plotted curves and either restrict the comparison to the parametric regime or justify the continued use of Eq. (64) at ξ = O(1).
  3. [Footnote 2, Sec. VII, Ref. [81]] The identification of the computed object with ~ET^{u-d} rests on an unpublished reference. Footnote 2 states that Zmf,1 = ~ET^{u-d} + O(UV-finite) because H_T^{u-d} has no UV-divergent contribution at this order, citing the 'in preparation' Ref. [81]. This assertion is load-bearing: if H_T^{u-d} did carry a UV-divergent piece, Fig. 6 would compare the wrong combination of GPDs. Since Ref. [81] is not available, the authors should either provide the argument in this paper, replace the citation with an available source, or present the lattice comparison in a form that does not depend on this unverified step.
minor comments (7)
  1. [Throughout] There are numerous typos, e.g., 'adotped' in Sec. VII, 'shoud' in footnote 2, 'prepration' in Ref. [81], and 'emergies' in Sec. IV D; the manuscript would benefit from a careful proofreading pass.
  2. [Sec. VI, Eqs. (105)-(106)] Eq. (106) presents the propagator expansion without the symmetrization introduced in the footnote; please make the presentation consistent so that the reader can see that the truncation is applied to the symmetrized operator.
  3. [Sec. IV D, Eq. (59)] The shorthand '1' for the unit spin-flavor structure in Eq. (59) is confusing when it appears in expressions such as Eq. (46); consider using an explicit delta symbol instead.
  4. [Sec. VII, Eq. (118)] The statement that one 'observes a vanishing ~ET GPD' should be qualified directly in the main text: the vanishing follows from the relation (118), which holds only at the level of UV-divergent accuracy, as the following sentences concede.
  5. [Sec. IV A] The sentence about higher-twist operators and gluon contributions ('gluon contributions ~ (M ρ-bar)^0 are not parametrically suppressed') is unclear in context, since the operators Γ = γ+, γ+γ5, iσ+j are all leading twist; please clarify which operators are being discussed.
  6. [Sec. VII, Fig. 6] Please indicate the statistical and systematic uncertainties of the lattice result in Fig. 6 and specify how the dot-dashed curve was obtained from the data points of Ref. [52].
  7. [Abstract and Sec. VIII] The abstract states 'Our results show good agreement with lattice QCD predictions' without the qualifications given in Sec. VIII; please bring the abstract in line with the stated limitations of the gradient-expansion estimate.

Circularity Check

1 steps flagged · score 4.0 of 10

One load-bearing identification in the lattice comparison is imported from the author's own unpublished paper; the formal polynomiality and sum-rule derivations are self-contained.

  1. self citation load bearing [Sec. VI B, footnote 2 (page 16); Sec. VI; Fig. 6; Ref. [81]]
    "In relating Zmf,1 to the standard chiral-odd GPD ¯ET , a careful matching is required within the accuracy of the present work. In the chiral expansion, the leading contribution to the GPD H u−d T arises from UV-finite terms [38, 81]. Thus, the UV-divergent contribution appears only in the combination ¯Eu−d T ≡ 2 ˜H u−d T + Eu−d T . Therefore, the result for the mean-field GPD Zmf,1 shoud be interpreted as Zmf,1 = H u−d T + 2 ˜H u−d T + Eu−d T = ¯Eu−d T + O(UV-finite)."

    The advertised lattice agreement (abstract, Sec. VII, Fig. 6) compares the computed Zmf,1 with the lattice result for ¯E_T^{u-d}. But by Eq. (64b), Zmf,1 is the combination H_T + 2Htilde_T + E_T, so it equals ¯E_T only after H_T is dropped. The paper justifies this drop by asserting that H_T^{u-d} receives no UV-divergent leading contribution, citing Refs. [38] and [81]; Ref. [81] is the author's own unpublished 'in prepration' paper, and Ref. [38] also shares the present author. Neither source is machine-checked or independently verified in this paper, so the identification that converts the calculation into the quoted lattice comparison is imported from the authors' own unpublished work rather than derived here. If that decomposition were different, Fig.

full rationale

The paper is largely self-contained in its formal core. The spin-flavor multipole structures are derived from zero-mode quantization and Nc kinematics (Sec. IV), and the polynomiality and sum-rule proofs follow from G5 symmetry, parity, partial-wave expansion, and grand-spin selection rules applied to single-particle matrix elements (Sec. V, Apps. B-C). No free parameters are fitted to chiral-odd GPD data: M, R0, and the cutoff Lambda are fixed by the instanton-vacuum model before the comparison, and the lattice data serve as an external benchmark, so no fitted-input-called-prediction circularity is present. The one circular-adjacent element is the identification, in footnote 2 of Sec. VI B, of the computed Zmf,1 with the lattice-oriented combination ¯E_T^{u-d}; that identification relies on the assertion that H_T^{u-d} has no UV-divergent leading contribution, which is attributed to Refs. [38] and [81], both involving the present author, with Ref. [81] explicitly unpublished. Because this identification is load-bearing for the abstract-level claim of 'good agreement with lattice QCD' and is not independently established in the paper, the circularity score is raised to 4. The paper honestly flags the main limitations: Sec. VI B retains only the single-chiral-field UV-divergent terms, Sec. VIII states that a full single-particle sum is necessary for complete predictions, and Fig. 6 attributes the positive-x discrepancy to the omitted discrete-level contribution. Those are correctness caveats rather than additional circularity, and they are weighed here as weakening the numerical claim without changing its structural status.

Assumptions & free parameters 3 free parameters · 7 assumptions · 0 invented entities

The central claim rests on the large-Nc mean-field framework and its zero-mode quantization, the hedgehog symmetry of the pion field, the instanton-vacuum effective action, and the truncation of the gradient expansion. No new particles or forces are introduced; the multipole mean-field GPDs are linear combinations of the standard chiral-odd GPDs. Three model parameters (M, R0, Lambda) are fixed by the instanton vacuum, not by the lattice data being compared.

free parameters (3)
  • Dynamical quark mass M = 350 MeV
    Set by the instanton-vacuum gap equation (Sec. VII), not fitted to the GPD data; the numerical estimates depend on it.
  • Mean-field size R0 = M R0 = 1
    The arctangent profile P(r) = -2 arctan(R0^2/r^2) is chosen to approximate the self-consistent soliton (Eq. 113).
  • UV cutoff / zero-mode form factor Lambda = 2^(1/2) * rho_bar^(-1), rho_bar = 1/3 fm
    Momentum-dependent quark mass form factor F(p^2) = -Lambda^2/(p^2 + Lambda^2) regulates UV divergences in the gradient expansion (Eq. 115).
assumptions (7)
  • domain assumption Large-Nc limit justifies the semiclassical mean-field approximation to the effective chiral action
    Sec. III.B: quantum fluctuations of the pion field are suppressed for Nc -> infinity; the nucleon is a bound state of Nc valence quarks in a self-consistent pion field.
  • domain assumption The classical chiral field has hedgehog symmetry
    Eq. (16): U(x) = exp[i (x/r) P(r) tau dot x_hat]; this realizes the contracted spin-flavor symmetry of baryons in the large-Nc limit.
  • domain assumption Zero-mode quantization: rotational modes to first order, translational modes to zeroth order
    Sec. III.C; this is the systematic 1/Nc expansion used to generate NLO spin-flavor structures.
  • domain assumption Gauge-link contributions to the leading-twist chiral-odd operator are suppressed by instanton diluteness and can be neglected
    Sec. IV.A, citing Ref. [61]; the effective operator in Eq. (34) omits the gauge connection.
  • domain assumption Gradient expansion retains only UV-divergent contributions (alpha = 0 and alpha = 1), with UV-finite terms and discrete-level contributions dropped
    Sec. VI and Sec. VIII; the numerical estimates are restricted to this truncation.
  • domain assumption In the large-Nc limit, the partonic variable x is O(N_c^(-1)) and its support extends to [-infinity, infinity] for moments
    Sec. V, citing Refs. [59,60]; boundary terms in integration by parts vanish at infinity, not at x = +/- 1.
  • domain assumption The LO gradient-expansion matrix element from Ref. [81] is correct
    Sec. VI: 'The LO matrix element was derived in Refs. [81]'; Ref. [81] is an unpublished 'in preparation' manuscript by the author and C. Weiss.

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Cite this review

Pith. "Pith review of Chiral-odd generalized parton distributions in the large-$N_{c}$ limit of QCD: Next-to-leading-order contributions." pith.science (2026). https://pith.science/paper/GZR54UY4

@misc{pith2026250621013,
  author       = {Pith},
  title        = {Pith review of: Chiral-odd generalized parton distributions in the large-$N_c$ limit of QCD: Next-to-leading-order contributions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GZR54UY4}},
  note         = {Machine review of arXiv:2506.21013}
}
abstract

We investigate the nucleon's chiral-odd generalized parton distribution functions (GPDs) in the large-$N_c$ limit of QCD. Extending previous work on the leading-order contribution in the $1/N_c$ expansion, we focus on the next-to-leading-order contributions and provide a complete set of flavor-singlet and flavor-non-singlet chiral-odd GPDs. This study includes the derivation of the spin-flavor structure of the baryon matrix element of the chiral-odd operator, the proof of the polynomiality property and associated sum rules, and numerical estimates based on the gradient expansion. The spin-flavor structure of the nucleon matrix element is interpreted through a multipole expansion in the transverse momentum transfer, leading to the definition of multipole mean-field GPDs. Using these GPDs as the basis of our analysis, we take their $m$-th moments and demonstrate the polynomiality property within the mean-field framework, making use of discrete symmetries in the proof. In particular, the first moments ($m = 1$) of the GPDs are related to the nucleon tensor form factors, as generally required. By performing a gradient expansion, we compute the chiral-odd GPDs and present numerical estimates. Our results show good agreement with lattice QCD predictions.

Figures

Figures reproduced from arXiv: 2506.21013 by the authors.

Figure 1
Figure 1. FIG. 1. Connected (left panel) and disconnected (right panel) [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. In the NLO contributions to the chiral-odd GPDs [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Numerical results for the chiral-odd GPD [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Numerical results for the chiral-odd GPD [PITH_FULL_IMAGE:figures/full_fig_p017_4.png]
Figure 6
Figure 6. Figure 6: FIG. 6. The result for the non-forward chiral-odd GPD [PITH_FULL_IMAGE:figures/full_fig_p018_6.png]

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Works this paper leans on

95 extracted references · 23 canonical work pages

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    (B1b) Considering the G5 symmetry (73), Eq

    ξ-even dipole GPD The single-particle matrix element A1 for the m-th mo- ment of the dipole GPD Zmf,1 is given by Aki 1 (ξ, t) = 4 3 ∆a |∆⊥|2 ⟨n|τ b|j⟩ × ⟨j|τ b(1 + γ0γ3)γa(ˆp3)k−iei∆· ˆX (ˆp3)i|n⟩, (B1a) Bki 1 (ξ, t) = 4 ∆a |∆⊥|2 × ⟨n|(1 + γ0γ3)γa(ˆp3)k−iei∆· ˆX (ˆp3)i|n⟩. (B1b) Considering the G5 symmetry (73), Eq. (B1) becomes Aki 1 (ξ, t) = 4 3 ∆a |∆⊥...

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    ξ-odd dipole GPD The single-particle matrix elements ˜A1 and ˜B1 for the m-th moment of the dipole GPD ˜Zmf,1 are given by ˜Aki 1 (ξ, t) = 4iϵ3ab ∆a |∆⊥|2 ⟨n|τ b|j⟩ × ⟨j|(1 + γ0γ3)γa(ˆp3)k−iei∆· ˆX (ˆp3)i|n⟩, (B11a) ˜Bki 1 (ξ, t) = 4iϵ3ab ∆a |∆⊥|2 × ⟨n|τ b(1 + γ0γ3)γa(ˆp3)k−iei∆· ˆX (ˆp3)i|n⟩, (B11b) Applying the G5 symmetry (73), Eq. (B11) simplifies to ...

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    Quadrupole GPD The single-particle matrix elements A2 and B2 for the m-th moment of the quadrupole GPD Zmf,2 are given by Aki 2 (ξ, t) = 16iϵ3ab |∆⊥|4 (∆b ⊥∆c ⊥ − 1 2 δbc|∆⊥|2) × ⟨n|τ c|j⟩⟨j|(1 + γ0γ3)γa(ˆp3)k−iei∆· ˆX (ˆp3)i|n⟩, (B18a) Bki 2 (ξ, t) = 16iϵ3ab |∆⊥|4 (∆b ⊥∆c ⊥ − 1 2 δbc|∆⊥|2) × ⟨n|τ c(1 + γ0γ3)γa(ˆp3)k−iei∆· ˆX (ˆp3)i|n⟩, (B18b) Applying th...

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    ξ-even dipole GPD By taking the first moment m = 1 of the ξ-even dipole GPD in Eq. (68), we obtain Z dx Zmf,1(x, ξ,0) = − MN Nc 4I X n,non j,occ 1 En − Ej A00 1 (ξ, 0), (C1) where the single-particle matrix element A00 1 (ξ, 0) is given by A00 1 (ξ, 0) = 2 3 i⟨n|τ |j⟩ · ⟨j|τ (γ⊥ · ˆX⊥)|n⟩, (C2) where ˆX⊥ = ( ˆX1, ˆX2). In the derivation of Eq. (C2), we us...

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    ξ-odd dipole GPD By taking the first moment m = 1 of the ξ-odd dipole GPD in Eq. (68), we obtain Z dx ˜Zmf,1(x, ξ,0) = − MN Nc 4I X n,non j,occ 1 En − Ej ˜A00 1 (ξ, 0), (C7) where the single-particle matrix element ˜A00 1 (ξ, 0) is given by ˜A00 1 (ξ, 0) = 4MN ξ⟨n|τ 3|j⟩ · ⟨j|γ0Σa ⊥| ˆX|Y a3 2 (Ω ˆX )|n⟩ (C8) In deriving Eq. (C8), we used the relations fo...

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    Quadrupole GPD By taking the first moment m = 1 of the ξ-odd dipole GPD in Eq. (68), we derive Z dx Zmf,2(x, ξ,0) = − M 2 N Nc 4I X n,non j,occ 1 En − Ej A00 2 (ξ, 0), (C12) where the single-particle matrix element A00 2 (ξ, t) is found to be A00 2 (ξ, 0) = − 2 3 (δbpδcq − 1 2 δpqδbc) × ⟨n|τ c|j⟩⟨j|γ0Σa ⊥∇p ⊥∇q ⊥| ˆX|4P4(cos ˆθ)|n⟩. (C13) In deriving Eq. ...

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