Off-shell modifications of the pion generalized parton distributions and transverse momentum dependent parton distributions

Jin-Li Zhang

REVIEW 4 major objections 7 minor 68 references

Off-shell modifications of the pion generalized parton distributions and transverse momentum dependent parton distributions

T0 review · 4 major / 7 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read In the NJL model, off-shell pion GPDs and TMDs acquire odd-power $\xi$ corrections up to 25% that on-shell analyses miss.

desk verdict Solid NJL extension for off-shell pion GPDs/TMDs, but the headline effect sizes are inconsistent and the Sullivan/EIC claim is unsupported because all numerics are for p^2 >= 0. read the letter →

arxiv 2507.09557 v1 pith:PWSTFQTY submitted 2025-07-13 hep-ph nucl-th

Jin-Li Zhang This is my paper

classification hep-phnucl-th

keywords piongeneralizedpartondistributionsoff-shelleffectsNambu-Jona-LasiniomodelskewnessparametercrossingsymmetrytransversemomentumdependentMellinmomentsSullivanprocess

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 asks what happens to a pion's generalized parton distributions (GPDs) and transverse-momentum-dependent parton distributions (TMDs) when the pion is not on its mass shell, as it is in Sullivan-process measurements. Working in the Nambu-Jona-Lasinio (NJL) model, a low-energy quark model of chiral symmetry breaking, the author shows that off-shellness breaks crossing symmetry, so the Mellin moments of the GPDs acquire odd powers of the skewness parameter $\xi$ and generate new form factors that vanish on-shell. The paper also derives explicit correction functions for the off-shell GPDs and finds deviations of roughly 15% at $p^2=0$ and 25% at $p^2=0.2$ GeV$^2$ from the on-shell results. If these corrections are real, on-shell pion structure calculations systematically miss a significant part of the signal relevant to electron-ion collider kinematics.

What carries the argument

The central object is the triangle-diagram amplitude with two dressed quark propagators and a pointlike pion-quark vertex $\Gamma_\pi=\sqrt{Z_\pi}\gamma_5$, evaluated with one or both pion legs off-shell. The mechanism carrying the argument is the failure of crossing symmetry when $p^2\neq p'^2$, which forces the Mellin moments to contain odd powers of $\xi$ and produces the new off-shell form factors $A_{1,1}$, $A_{2,1}$, $B_{1,1}$, and $B_{2,1}$. Proper-time regularization supplies the cutoff that makes the loop integrals finite and generates the $\bar{C}_i(\sigma)$ functions in which all results are expressed.

What would settle it

Compute the same triangle diagrams with a momentum-dependent pion-quark vertex and check whether the odd-power-$\xi$ form factors $A_{1,1}$, $A_{2,1}$, $B_{1,1}$, and $B_{2,1}$ remain nonzero; alternatively, compare the predicted off-shell moments against lattice QCD results for a pion at $p^2\neq m_\pi^2$.

Watch

Extended reading notes

Core claim

In the NJL model with proper-time regularization, the off-shell pion GPDs $H(x,\xi,t,p^2,p'^2)$ and $E(x,\xi,t,p^2,p'^2)$ are computed from triangle diagrams with incoming and outgoing pion momenta that are not equal to $m_\pi^2$. Because time-reversal and crossing symmetry fail for $p^2\neq p'^2$, the $x$-moments take the form $\sum_i A_{n,i}(t,p^2,p'^2)\xi^i$ with odd powers of $\xi$ present; in particular, the form factors $A_{1,1}$, $A_{2,1}$, $B_{1,1}$, and $B_{2,1}$ no longer vanish. The paper derives explicit expressions for these off-shell form factors and shows that the vector form factor $F(t,p^2,p'^2)$ is symmetric while $G(t,p^2,p'^2)$ is antisymmetric under exchange of $p^2$ and $p'^2$. It also finds that the symmetry and polynomiality properties that hold for on-shell GPDs break down off-shell, and that off-shell TMDs and PDFs develop a stronger dependence on $x$ than their on-shell counterparts.

Load-bearing premise

The calculation assumes the pion-quark vertex stays pointlike and unchanged when the pion is off-shell; if the vertex itself depends on the pion's virtuality, all the off-shell corrections reported here would be different.

Editorial extensions

If this is right

Sullivan-process analyses that treat the pion as on-shell will miss corrections of order 15-25% in the peak region of the GPD at the virtualities considered.
The Mellin moments of off-shell pion GPDs are no longer even in $\xi$, so parametrizations based on on-shell polynomiality must be extended to include the new odd-power form factors.
The off-shell pion PDF obtained from the GPD shows a stronger $x$-dependence than the nearly flat on-shell NJL result, bringing it closer to lattice and quark-model determinations.
The off-shell TMD asymptotic behavior in $k_\perp$ agrees across different NJL regularization schemes, supporting the model-independent character of that particular feature.

Reading between the lines

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

If the off-shell corrections survive with a momentum-dependent pion-quark vertex, pion GPD extractions from Sullivan-process data at future colliders will need off-shell form factors as free parameters, and the odd-$\xi$ terms derived here provide a concrete parametrization to test.
A natural next check is whether the broken polynomiality condition alters the momentum-sum-rule interpretation for off-shell pions, since $\int xH\,dx$ now contains an $A_{2,1}\xi$ term with no on-shell counterpart.
The antisymmetry of $A_{2,1}$ and $B_{2,1}$ under $p^2\leftrightarrow p'^2$ could be tested directly in models that retain a momentum-dependent vertex; if that sign property persists, it is more robust than the magnitudes of the corrections.
At kinematics typical of Sullivan-process measurements, the relative 25% off-shell shift at $p^2=0.2$ GeV$^2$ suggests that pion virtuality may be a leading systematic uncertainty in amplitude analyses.

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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

Quick take: The paper extends your earlier on-shell NJL work to off-shell pion GPDs and TMDs. The genuinely new piece is the odd-xi Mellin moments and the new form factors (A1,1, A2,1, B1,1, B2,1) that appear once crossing symmetry is lost. The explicit expressions are plausible, the on-shell limit is recovered, and the k-perp dependence of the TMDs comes from the model rather than being inserted ad hoc. That is real, useful work for the pion-structure subfield.

What is soft: first, the headline numbers disagree. Section II.B says p^2=0.2 GeV^2 gives about 15% and p^2=0.4 about 25%, while Section V says p^2=0 gives about 15% and p^2=0.2 about 25%. That is not a rounding effect; one of the two is wrong. Second, the Sullivan-process relevance claim is left hanging. The Sullivan pion is spacelike (p^2<0), and every numerical result in the paper uses p^2 >= 0. Because the formulas are linear in p^2, the sign matters; a single negative-p^2 point would show whether the off-shell modification even goes in the same direction. Without that, 'should not be disregarded' in EIC analyses is an assertion, not a result. Third, the pointlike vertex (Gamma_pi = sqrt(Z_pi) gamma_5, independent of p^2 and p'^2) is load-bearing: all off-shellness is attributed to quark propagators. That is a model assumption that should be stated as such. Minor issues: the figure captions are malformed (Fig. 2 lower panel looks like it has a typo), and the claimed agreement with lattice is asserted without a comparison.

Bottom line: this is a legitimate model calculation, but it needs revision before the quantitative claims can be trusted. I would send it to a referee—it deserves serious review—but I would ask the referee to check the derivations and to require the authors to fix the inconsistency and add a spacelike-p^2 case. I would not cite it in its current form.

Referee Report

4 major / 7 minor

Summary. This paper extends the author's earlier NJL-model calculations of on-shell pion GPDs and TMDs to off-shell kinematics. The pion GPDs H and E are defined in Eqs. (10)-(15) for arbitrary incoming and outgoing momenta, and after Feynman parametrization the final expressions in Eqs. (17)-(18) are used to plot off-shell distributions and to extract Mellin moments. The central new result is that broken crossing symmetry generates odd powers of the skewness parameter xi in the moments, encoded in new off-shell form factors A_{1,1}, A_{2,1}, B_{1,1}, and B_{2,1}. The paper also computes off-shell PDFs, impact-parameter-space PDFs, and unpolarized TMDs, and compares them with on-shell results. It concludes that off-shell effects are significant for pion structure and should not be disregarded in Sullivan-process analyses.

Significance. If the calculations are correct, the paper provides a concrete model prediction that pion partonic structure depends sensitively on pion virtuality, including qualitatively new odd-xi moments that are absent on-shell. The analytical expressions are explicit, and the extracted form factors can be compared with other models or lattice data. The consistency checks with the quark-model G function and with the known TMD asymptotic behavior in Eq. (47) are useful. The main caveats are that the results rest on a pointlike pion-quark vertex and that the numerical evidence covers only nonnegative virtualities. If the missing kinematics and vertex dependence are addressed, the work would strengthen the case for including off-shell effects in Sullivan-process extractions at the EIC.

major comments (4)

[Sec. II.B and Sec. V] The size of the claimed off-shell effect is stated inconsistently. Section II.B says that at p^2=0.2 GeV^2 the relative effect is approximately 15% and at p^2=0.4 GeV^2 it rises to about 25%, while Section V says that at p^2=0 GeV^2 the relative effect is approximately 15% and at p^2=0.2 GeV^2 it increases to about 25%. Since the abstract and conclusions rest on this quantitative claim, the authors must determine which values are correct, define precisely what quantity is being compared (peak value, integrated area, or pointwise difference), and make the abstract, Section II.B, and Section V agree.
[Secs. II-IV and Figs. 2-5] All numerical results are for nonnegative pion virtualities p^2=0, 0.142, 0.2, 0.4, and 0.6 GeV^2, while the Sullivan process invoked in the introduction and Section V involves spacelike virtual pions with p^2<0. The central formulas (17), (28), (33), and (46) contain terms linear in p^2, and sigma_1=M^2-x(1-x)p^2 in Eq. (A2a) changes in the opposite direction for negative p^2. The sign and magnitude of the off-shell modification are therefore not determined by the present plots. A calculation for representative negative p^2 values, such as p^2=-0.1, -0.2, and -0.4 GeV^2, is needed before the statement that these effects should not be disregarded in Sullivan/EIC analyses can be supported.
[Eq. (13) and Eqs. (14)-(15)] The pion-quark vertex is taken to be the on-shell-normalized pointlike vertex Gamma_pi = sqrt(Z_pi) gamma_5, independent of p^2 and p'^2, so all off-shell effects in the triangle diagrams arise from the quark propagators. This is a load-bearing assumption: if the vertex itself carries a virtuality dependence, as it does in more complete descriptions of the pion Bethe-Salpeter amplitude, the numerical off-shell corrections would differ. The authors should justify this simplification within the NJL model or estimate the resulting uncertainty; without that, the quantitative conclusions are conditional on the pointlike-vertex approximation.
[Sec. III.1] The relation Eq. (30), which is expected from the energy-momentum tensor structure, is found not to hold for the extracted A_{2,1}. The paper attributes this to missing gluon contributions, but no calculation or estimate is provided. Since A_{2,1} is a headline new result, this unresolved discrepancy weakens the claim that the new form factors are correctly identified; the authors should either supply the missing contributions or discuss the consequence for the moment expansion in Eq. (20).

minor comments (7)

[Figs. 2 and 3] The captions and legends for the lower panels are garbled; for instance, 'H(x, 0.5, 0, -1, 0.2, m2_pi)' should presumably read 'H(x, 0.5, -1, 0.2, m_pi^2)', and similarly for E.
[Figs. 4 and 5] There are unit typos in the captions, such as 'GeV^{-2}' instead of 'GeV^2', and the value of b_perp^2 used in Fig. 4 should be stated explicitly in the text.
[Sec. V] The statement that 'the off-shell results demonstrate greater consistency with lattice calculations' is not supported by any lattice comparison shown in the paper; either a quantitative comparison should be added or the claim should be removed.
[Sec. III.1] The text says that only real parts of G and F are considered because the proper-time regularization in Eq. (5) applies only for X>0; it should be stated whether this restriction affects any of the plotted kinematics or extracted form factors.
[Table I] The parameters G_omega and G_rho are listed in Table I but are never used in the manuscript; if they are not needed, they should be removed.
[Appendix A, Eq. (A1a)] The integration limits tau_ir and tau_uv in Eq. (A1a) are not defined in the text; the notation should be introduced or connected to the regularization cutoffs Lambda_IR and Lambda_UV.
[Abstract and Sec. III] The phrase 'polynomiality conditions may no longer hold' is confusing because the derived moments in Eqs. (21)-(24) are polynomials in xi; the statement should be clarified to mean that the polynomiality conditions are modified by the presence of odd powers of xi.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: off-shell GPD/TMD results are computed from NJL inputs, with on-shell comparisons serving as consistency checks.

full rationale

The paper's derivation chain is self-contained. The off-shell pion GPDs and TMDs are obtained by direct evaluation of NJL triangle diagrams (Eqs. 14-15 and 45) with parameters fixed to unrelated observables (Table I: masses, couplings, cutoffs), not fitted to the target GPDs/TMDs. The central claims—odd powers of xi in Mellin moments and nonzero off-shell form factors A1,1, A2,1, B1,1, B2,1—follow from explicit momentum integration; the expansion in Eq. (20) is a general parametrization, and the computed coefficients are results, not inputs. The on-shell limit reproduces earlier work by the same author (Refs. [10,12]), but that is a consistency check against independent calculations, and the regularization choice (PTR) is a methodological selection rather than a way of importing the target result. The numerical verification that Eq. (26) equals -A1,1 is a cross-check of general covariance relations, not an input assumption. No target quantity is used as a fitted parameter, and no load-bearing argument reduces to a self-citation. The reviewer's kinematic-gap concern (positive p^2 shown while Sullivan pions are spacelike) is a correctness/relevance issue, not circularity. Overall, no circular step is present.

Assumptions & free parameters 5 free parameters · 4 assumptions · 0 invented entities

The central claim rests entirely on the NJL model with the parameter set in Table I (M, G_pi, Z_pi, Lambda_IR, Lambda_UV), plus the assumption that a pointlike pion-quark vertex captures all off-shell effects. The model parameters were fixed to other observables, so they are not circular with respect to the GPD and TMD results, but they are free parameters from the perspective of this paper. No new particles or fields are introduced.

free parameters (5)

M = 0.4 GeV
Dressed quark mass, fixed by the NJL gap equation with parameters chosen to reproduce pion physics; all GPD and TMD expressions depend on it.
G_pi = 19.0 GeV^-2
Four-fermion coupling, chosen to reproduce the scale of chiral symmetry breaking; appears in the gap equation and the model dynamics.
Z_pi = 17.85
Effective meson-quark-quark coupling squared, used in Eq. (13) and appearing as an overall factor in all computed distributions.
Lambda_IR = 0.240 GeV
Infrared cutoff of proper time regularization, chosen to mimic confinement; numerical results depend on it.
Lambda_UV = 0.645 GeV
Ultraviolet cutoff of proper time regularization, chosen as a model scale; numerical results depend on it.

assumptions (4)

domain assumption The NJL Lagrangian (Eq. 1) is a valid effective theory for pion structure at low energies.
The entire calculation uses the NJL model to represent the pion and its quarks; if the model misses relevant off-shell dynamics, the central numbers change.
domain assumption Proper time regularization with an infrared cutoff (Eq. 5) adequately mimics confinement and yields the finite integrals used in Eqs. (17), (18), and (46).
The numerical results depend on the chosen cutoff pair (Lambda_IR, Lambda_UV), and the paper does not quantify the scheme dependence beyond a reference to Ref. [12].
domain assumption The pion-quark vertex is pointlike and does not depend on the pion virtuality: Gamma_pi = sqrt(Z_pi) gamma_5 in Eq. (13).
This is the load-bearing assumption identified in weakest_assumption; all off-shell modifications are generated by the quark propagators alone.
domain assumption The off-shell matrix elements in Eqs. (10) and (11) admit the same light-front operator definitions as on-shell GPDs, and the delta-function insertions in Eq. (12) capture the x-dependence.
The paper extends standard light-front definitions to off-shell pion states without addressing potential subtleties of light-front quantization for unequal-mass external states.

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Cite this review

Pith. "Pith review of Off-shell modifications of the pion generalized parton distributions and transverse momentum dependent parton distributions." pith.science (2026). https://pith.science/paper/PWSTFQTY

@misc{pith2026250709557,
  author       = {Pith},
  title        = {Pith review of: Off-shell modifications of the pion generalized parton distributions and transverse momentum dependent parton distributions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PWSTFQTY}},
  note         = {Machine review of arXiv:2507.09557}
}

abstract

The off-shell characteristics of pion generalized parton distributions (GPDs) and transverse momentum dependent parton distributions (TMDs) are examined within the framework of the Nambu-Jona-Lasinio model. In our previous papers, we separately investigated the properties of on-shell pion GPDs and light-front wave functions. It is particularly intriguing to compare the differences between on-shell and off-shell pion GPDs, which allows us to explore the effects associated with off-shellness. Due to the absence of crossing symmetry, the moments of GPDs also incorporate odd powers of the skewness parameter, resulting in new off-shell form factors. Through our calculations, we derived correction functions that account for modifications in pion GPDs due to off-shell effects. Unlike their on-shell counterparts, certain properties break down in the off-shell scenario; for instance, symmetry properties and polynomiality conditions may no longer hold. Additionally, we evaluate off-shell TMDs and compare them with their on-shell equivalents while also investigating their dependence on $\bm{k}_{\perp}$.

Figures

Figures reproduced from arXiv: 2507.09557 by the authors.

**Figure 3.** Pion off-shell tensor GPD: E(x, ξ, t, p2 , p′2 ) in Eq. (18), we only plot ξ > 0. upper panel – The off-shell GPDs E(x, 0.5, 0, p2 , m2 π). H(x, 0.5, 0, m2 π, m2 π) – black dotted line; E(x, 0.5, 0, 0, m2 π) – red solid line; E(x, 0.5, 0, 0.2, m2 π) – blue dashed line; E(x, 0.5, 0, 0.4, m2 π) – purple dotdashed line. lower panel – The off-shell GPDs E(x, 0.5, 0, p2 , m2 π). E(x, 0.5, −1, m2 π, m2 π) – black dotted l… view at source ↗

**Figure 4.** Impact parameter space PDFs : upper panel [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗

Figures from the paper (2 more)

**Figure 5.** Figure 5: The off shell PDFs : upper panel – q [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]

**Figure 6.** Figure 6: Feynman diagrams for the π TMDs in the NJL model. The shaded circles represent the π Bethe-Salpeter vertex functions and the solid lines the dressed quark propagator. The operator insertion has the form γ +δ(x − k+ p+ ). The left diagram represent the u quark and the …

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However, for off-shell GPDs, this crossing symmetry no longer holds

FFs Unlike the on-shell case, time-reversal symmetry (or crossing symmetry) renders the GPDs an even function of ξ. However, for off-shell GPDs, this crossing symmetry no longer holds. In general, the x-moments of the GPDs also incorporate odd powers of the skewness parameter ξ. Z 1 −1 xnH(x, ξ, t, p2, p′2)dx = (n+1)X i=0 An,i(t, p, p′)ξi , (20) where An,...

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Impact Parameter Dependent PDFs The impact parameter dependent PDFs are defined as q x, b2 ⊥ = Z d2q⊥ (2π)2 e−ib⊥·q⊥ H x, 0, −q2 ⊥ , (38) which means the impact parameter dependent PDFs are the Fourier transform of GPDs at ξ = 0. When ξ = 0 and t ̸= 0, GPDs become H(x, 0, −q2 ⊥, p2, p′2) = NcZπ 8π2 ( ¯C1(σ1) + ¯C1(σ2)) + NcZπ 4π2 Z 1−x 0 dα x(p2 + p′2) + ...

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Partons Distribution Functions In the forward limit where t = 0 and q2 = 0, under the condition that p = p′, we can derive the PDFs H(x, 0, 0, p2, p2) = f (x, p2) = NcZπ 4π2 ¯C1(σ1) + NcZπ 2π2 x(1 − x)p2 ¯C2(σ1) σ1 , (43) E(x, 0, p2, p2) = NcZπ 2π2 mπM (1 − x) ¯C2(σ1) σ1 , (44) 7 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.5 1.0 1.5 x q(x,p2) Figure 5. The off shell PD...

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

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However, for off-shell GPDs, this crossing symmetry no longer holds

FFs Unlike the on-shell case, time-reversal symmetry (or crossing symmetry) renders the GPDs an even function of ξ. However, for off-shell GPDs, this crossing symmetry no longer holds. In general, the x-moments of the GPDs also incorporate odd powers of the skewness parameter ξ. Z 1 −1 xnH(x, ξ, t, p2, p′2)dx = (n+1)X i=0 An,i(t, p, p′)ξi , (20) where An,...

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Impact Parameter Dependent PDFs The impact parameter dependent PDFs are defined as q x, b2 ⊥ = Z d2q⊥ (2π)2 e−ib⊥·q⊥ H x, 0, −q2 ⊥ , (38) which means the impact parameter dependent PDFs are the Fourier transform of GPDs at ξ = 0. When ξ = 0 and t ̸= 0, GPDs become H(x, 0, −q2 ⊥, p2, p′2) = NcZπ 8π2 ( ¯C1(σ1) + ¯C1(σ2)) + NcZπ 4π2 Z 1−x 0 dα x(p2 + p′2) + ...

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Partons Distribution Functions In the forward limit where t = 0 and q2 = 0, under the condition that p = p′, we can derive the PDFs H(x, 0, 0, p2, p2) = f (x, p2) = NcZπ 4π2 ¯C1(σ1) + NcZπ 2π2 x(1 − x)p2 ¯C2(σ1) σ1 , (43) E(x, 0, p2, p2) = NcZπ 2π2 mπM (1 − x) ¯C2(σ1) σ1 , (44) 7 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.5 1.0 1.5 x q(x,p2) Figure 5. The off shell PD...

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