Toward an advanced phenomenology of $\pi N$ transition distribution amplitudes

B. Pire; K. Semenov-Tian-Shansky; L. Szymanowski; P. Sznajder

REVIEW 3 major objections 4 minor 74 references

Toward an advanced phenomenology of $\pi N$ transition distribution amplitudes

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read A two-component model of πN transition distribution amplitudes predicts three non-vanishing leading-twist polarization asymmetries in backward pion electroproduction, with two double spin asymmetries that should not die out at high $Q^2$.

desk verdict A genuine but caveat-heavy step in TDA phenomenology: the new asymmetries are worth knowing, but the factorized description is still unproven and the fit to CLAS is as weak as the authors admit. read the letter →

arxiv 2506.04961 v1 pith:P5TZ3G4J submitted 2025-06-05 hep-ph hep-exnucl-exnucl-th

B. Pire , K. Semenov-Tian-Shansky , P. Sznajder , L. Szymanowski This is my paper · ORCID

classification hep-phhep-exnucl-exnucl-th

keywords transitiondistributionamplitudesbackwardkinematicspionelectroproductionspinasymmetriescollinearfactorizationnucleonspectralrepresentationdouble

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

Transition distribution amplitudes (TDAs) describe how a nucleon turns into a pion under a hard short-distance interaction, and they provide the QCD factorization framework for backward meson electroproduction. This paper builds a flexible two-component TDA model that respects the polynomiality of Mellin moments, reduces to known nucleon distribution amplitudes at the pion threshold through the soft-pion constraint, and accommodates the sparse existing backward $\pi^+$ data after one fitted parameter. The new claim is that, in addition to the known single transverse target spin asymmetry, two double spin asymmetries survive at leading twist and are not parametrically small under reasonable model assumptions. Checking that these asymmetries persist at higher $Q^2$ would test the whole collinear-factorization picture for backward reactions. The model also exposes the $\pi^0$ channel as currently unconstrained, so a single backward $\pi^0$ measurement would strongly discriminate among modeling choices.

What carries the argument

The central machinery is the spectral representation of TDAs as quadruple distributions: each TDA is written as a Radon transform of a six-variable spectral density with two constraints, which guarantees the support domain and, once a second component is added, full polynomiality of Mellin moments in the skewness variable. The paper splits the spectral density as $F=F^{(0)}+(1-\xi)F^{(1)}$, where $F^{(0)}$ is fixed at $\xi=1$ by the soft-pion theorem in terms of nucleon distribution amplitudes and $F^{(1)}$ is flexible, with its $\xi=0$ forward limit expanded in orthogonal polynomials on a hexagon. Convolution of these TDAs with the hard-scattering kernels defines the two master integrals $I^{(1)}$ and $I^{(2)}$, whose ratios determine the unpolarized cross-section and the three leading-twist polarization asymmetries.

What would settle it

Measure $A_{LL}$ and $A_{LT}$ for $e p \to e n \pi^+$ at fixed $x_B$ and $u$ over a range of $Q^2$ values. If these double spin asymmetries fall toward zero as $Q^2$ grows, or if the unpolarized cross-section falls faster than the predicted $1/Q^6$ behaviour, the leading-twist TDA convolution picture is excluded.

Watch

Extended reading notes

Core claim

At leading twist-3 and leading order in $\alpha_s$, the amplitudes for $\gamma^* N \to N'\pi$ in near-backward kinematics factorize into convolutions of $\pi N$ TDAs and nucleon distribution amplitudes with hard-scattering kernels. The paper identifies the three polarization observables that survive at this accuracy: the transverse-target single spin asymmetry $A_{UT}$ and two double spin asymmetries $A_{LL}$ and $A_{LT}$, each expressible as simple ratios of the two convolution integrals $I^{(1)}(\xi,u)$ and $I^{(2)}(\xi,u)$. It shows these ratios remain non-zero and are not parametrically small under reasonable TDA modeling assumptions, and that they depend sensitively on $\xi$ and $u$, making them useful discriminators between models. The two-component spectral model, constrained at $\xi=1$ by the soft-pion theorem and at $\xi=0$ by a flexible forward limit built from orthogonal polynomials on a hexagon, reproduces the few available backward $\pi^+$ data points and yields definite predictions for $\pi^0$ production and for the three asymmetries.

Load-bearing premise

The calculation assumes that collinear factorization at leading twist-3 works for near-backward $\gamma^* N \to N'\pi$; the paper itself notes that no factorization proof exists and no next-to-leading-order calculation yet shows the ultraviolet divergences can be reabsorbed into TDA and DA evolution.

Editorial extensions

If this is right

If the two double spin asymmetries are measured and found to persist at high $Q^2$, that would support leading-twist TDA factorization for backward pion electroproduction rather than a picture dominated by higher-twist effects.
The $\pi^0$ channel is currently almost unconstrained; a single backward $\pi^0$ measurement would break the degeneracy among the polynomial coefficients in the model and sharpen the normalization of TDAs.
Because the asymmetries weigh $I^{(1)}$ and $I^{(2)}$ in different combinations, they can separate contributions of different TDA families and constrain the $u$-dependence of the model.
Event selection for backward processes should be made in the variable $u$ rather than $t$: the Monte Carlo distributions show that $u$ cleanly separates forward and backward contributions while $t$ does not.

Reading between the lines

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

An extension the paper leaves implicit is that the same two-component spectral construction could be adapted to nucleon-to-photon and nucleon-to-vector-meson TDAs, with the soft-pion normalization replaced by the appropriate chiral or vector-meson constraints.
If lattice QCD computes the $u$-dependence of low Mellin moments of $\pi N$ TDAs, those moments could directly fix the dipole form factor $G(u)$ that the present model treats as an empirical input.
A direct next-to-leading-order calculation of the hard coefficient functions is the cleanest way to test whether the ultraviolet divergences are reabsorbed into TDA and nucleon DA evolution; until that exists, the $Q^2$ behaviour of the predicted double spin asymmetries is the most accessible experimental proxy for the validity of the factorization premise.

Share X Bluesky LinkedIn Reddit HN

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

Two things to know before you read it. First, this is the first flexible TDA parametrization actually normalized to CLAS data, and it introduces two double-spin asymmetries (A_LL and A_LT) that are genuinely new. Second, the whole picture rests on an unproven premise: collinear factorization at leading twist-3 for near-backward gamma* N -> N' pi. The authors say so in Section I, but it matters because if the convolution (25)-(27) is not the leading QCD description, the asymmetries do not test TDAs.

The modeling is a real improvement over the fixed model of Ref. [47]. The two-component spectral ansatz with the (1-xi) prefactor preserves the soft-pion constraint at xi=1 while adding flexibility in the forward limit, and the hexagon orthogonal polynomials are a sensible way to parametrize that limit. The MC study is useful for experimentalists. Credit where due: the paper is transparent about its limitations, and the sensitivity analysis is honest. The citation pattern looks solid, building on the authors' own previous work and the standard GPD/DA literature.

Now the soft spots, in order of severity. The factorization issue is the big one. There is no proof, no NLO calculation showing UV divergences reabsorbed, and the authors know it. That doesn't kill an exploratory paper, but it should be said plainly: the cross-sections and asymmetries are model outputs from an assumed framework, not measurements of the framework. Next, the fit itself: chi^2/4 = 10 with a post hoc exclusion of the first CLAS point. Weak, even for exploratory work. The sigma_T >> sigma_L assumption is unchecked, and the pi0 normalization is essentially free. The DSAs are ratios and less sensitive to normalization, but they inherit the same convolution structure, so the factorization caveat applies equally.

The proposal to test the TDA picture via the Q^2 dependence of the asymmetries is plausible but easily overstated. That test can falsify the factorized description if the asymmetries vanish or scale badly, but it cannot establish that the TDA convolution is the leading mechanism. That would require the factorization proof or at least a serious NLO calculation.

Who should read it: TDA/GPD phenomenologists, people working on backward exclusive production, and experimentalists planning JLab proposals. I'd bring it to a reading group. I'd cite it as the current state of TDA phenomenology, with the caveats attached.

My recommendation: send it to a serious referee. It is exploratory but earnest, and it contains enough new material to justify referee time. The referee should focus on the factorization premise and the fit treatment, not the internal algebra, which looks coherent.

Referee Report

3 major / 4 minor

Summary. The paper develops a new flexible model for nucleon-to-pion transition distribution amplitudes (TDAs) entering hard exclusive backward pion electroproduction. The model combines a spectral component constrained by the soft-pion theorem at xi = 1 with a flexible component built from orthogonal polynomials on a hexagonal domain, and it uses the cross-channel nucleon-exchange relation to fix the V2, A2, T2 family. The free coefficient c0 is fitted to CLAS pi+ data, and the authors compute unpolarized cross sections, the single transverse target spin asymmetry, and two new leading-twist double spin asymmetries, A_LL and A_LT. They study sensitivity to DA choices, shape parameters, and coefficient ratios, and they include a Monte Carlo study of forward versus backward kinematics. The central claim is that the two double spin asymmetries are non-vanishing at leading twist and that checking their persistence in Q^2 will test the TDA framework.

Significance. If the collinear factorized description holds, the paper provides a useful and much-needed phenomenological framework for a largely unexplored backward-kinematics program. The spectral construction is technically sound, respects the soft-pion limit and polynomiality, and the sensitivity analysis is honest about model uncertainties. The proposed double spin asymmetries are new observables that could discriminate among TDA models and motivate experimental proposals. The authors also deserve credit for explicitly disclosing the absence of a factorization proof and the limitations of their evolution treatment. However, the significance is conditional: the normalization fit is poor, the pi0 channel is essentially unconstrained, and all quantitative predictions inherit the unproven factorization premise. The paper is therefore best viewed as a well-structured model-building proposal whose validation requires further theoretical and experimental work.

major comments (3)

[Section III D, Table I, Eq. (92)] The normalization of the model to CLAS pi+ data is a load-bearing step, but the fit quality is poor (chi^2/4 = 10), the first CLAS point at Q^2 ~ 1.7 GeV^2 is excluded post hoc, and the assumption sigma_T >> sigma_L is unchecked. Because c0 is fitted to these same data, the subsequent agreement of the charged-pion cross sections with the data is not an independent test of the model. The double spin asymmetries are not directly fitted, but they are ratios of the same convolution integrals I^(1), I^(2) that determine the fitted cross section. I recommend either including all data with a transparent treatment of the outlier, or explicitly relabeling the cross-section agreement as a fit output rather than a prediction.
[Section I and Section II C, Eqs. (25)-(27), (50), (52), (53)] The paper correctly states in Section I that no factorization proof exists for near-backward gamma* N -> pi N' and that no NLO calculation demonstrates the ultraviolet divergences can be reabsorbed into TDA and DA evolution. This is not an internal inconsistency, but it is the most load-bearing premise of the paper: all observables, including the new double spin asymmetries, follow from the convolution formula (25)-(27). The proposed Q^2-independence test can falsify the combined factorized description but cannot by itself establish that the TDA convolution is the leading QCD mechanism. The conclusions should state this conditionality more explicitly, and ideally the authors could propose quantitative criteria for what level of Q^2 stability would count as support for the factorization picture.
[Eq. (C5) versus Eq. (82) and Eq. (B1)] There is an internal sign inconsistency in the relation fixing the {V2, A2, T2} family. Eq. (C5) states {V2, A2, T2} = +(1/2){V1, A1, T1} and then notes that this sign is opposite to the cross-channel nucleon-exchange model (B1), but Eq. (B1) also uses +(1/2). Meanwhile Eq. (82), which cites (C5), uses -(1/2). Since this relation is used to construct both the F^(0) and F^(1) components and therefore enters I^(2) in Eq. (22), the correct sign must be established and propagated consistently before the numerical predictions can be considered reliable.

minor comments (4)

[Section V] The statement 'we moreover proved that these asymmetries were not parametrically small' overstates the evidence; the paper shows numerical results under specific model assumptions and no analytic bound is derived. I suggest replacing 'proved' with 'showed numerically within the considered model framework'.
[Section IV, Fig. 10] The Monte Carlo distributions are presented without a common luminosity normalization and without detector effects, as stated in the text. It would be clearer to state explicitly in the figure caption that the forward and backward histograms are not directly comparable in normalization.
[Eq. (85)] The profile function for F^(1) is written with a parameter b that is set to b = 2; the text notes that b = 1 is excluded because it does not make the TDA vanish at the support edges. Adding a brief explanation of why b = 1 fails at the level of the TDA, rather than only in combination with the forward limit, would improve readability.
[Section III E] The conclusion that the pi0 channel is unconstrained in normalization is clearly supported by Fig. 9, but the abstract says the modeling is 'constrained by sparsely available experimental data.' This is true only for the pi+ channel; I suggest making that channel-specific caveat visible already in the abstract.

Circularity Check

0 steps flagged · score 2.0 of 10

Central DSA predictions are not fitted and retain independent content; the only fit is the c0 normalization of the pi+ cross-section, and the missing factorization proof is a correctness risk rather than a circularity.

full rationale

No step in the derivation reduces, by construction, to its own input. The two double spin asymmetries in Eqs. (50), (52) and (53) are ratios of the same convolution integrals I^(1), I^(2) entering the unpolarized cross-section; the overall normalization C_pi and the fitted coefficient c0 cancel in these ratios, so the DSA predictions are not forced by the CLAS pi+ normalization. The pi0 channel is obtained from pi+ TDAs through isospin relations (Eqs. (81) and (93)) and is explicitly stated to be unconstrained by the fitted pi+ data, which the paper presents as a motivation for future pi0 measurements. The only parameter fitted to data is c0 in Eq. (92), and Section III D transparently describes this as normalization ('The value of c0 is determined to replicate the data collected by CLAS'), not as an independent prediction. The paper does call the resulting curves 'predictions from the constrained model', but it also reports the poor chi^2/4 = 10, so the agreement is not presented as a successful independent test. The TDA spectral representation and soft-pion normalization are imported from prior work by overlapping authors, but they are used as stated model inputs and are not invoked as uniqueness theorems; the soft-pion constraint is re-derived in Appendix C from PCAC. The most serious limitation is the absence of a factorization proof and of an NLO check, which the paper itself discloses in Section I; that is a correctness and interpretability risk, not a circularity. Accordingly, the central claims about non-vanishing leading-twist double spin asymmetries and their potential as a test of the TDA picture are not equivalent to the fitted inputs, and the circularity score is low.

Assumptions & free parameters 5 free parameters · 7 assumptions · 0 invented entities

The model relies on the standard TDA collinear factorization framework, a soft-pion normalization, spectral representations, and several hand-chosen profile functions and parameter ratios. No new physical entities are introduced; the two-component spectral densities and master TDA are modeling constructs within the established framework. The most load-bearing unproven ingredient is the leading-twist-3 collinear factorization in backward kinematics.

free parameters (5)

c0 = 163.75 (default; ranges 43.61 to 201.98 across model variants)
Coefficient of the zeroth hexagon polynomial in Eq. (92), fitted to four CLAS charged-pion data points; chi^2/4 is about 10.
c1, c2
Not fitted; set equal to c0 by the assumed ratio c0:c1:c2 = 1:1:1 (Table I).
b = 2
Profile-function shape parameter in Eq. (85); chosen by hand, with b=1 rejected as too mild.
d = 1
Weight-function exponent for the hexagon polynomials in Eq. (89); chosen by hand.
m_D^2 = 0.71 GeV^2
Dipole mass in the u-dependent form factor G(u), Eq. (79); empirical input from the standard dipole parametrization, not derived or fitted in this paper.

assumptions (7)

domain assumption Collinear factorization holds for near-backward gamma* N -> N' pi at leading twist-3, despite the absence of a factorization proof.
Invoked throughout Sections I and II.C; the paper states that no factorization proof exists and no NLO calculation verifies divergence reabsorption.
domain assumption The soft-pion theorem and PCAC relation in the chiral limit normalize TDAs at xi=1 and u=m_N^2.
Used in Section III.B.1 and Appendix C; requires taking the chiral limit m_pi=0 and identifying the direct-channel and cross-channel thresholds.
standard math The spectral representation with quadruple distributions correctly encodes the support and polynomiality properties of TDAs when supplemented by the (1-xi) F^(1) term.
Adopted from Ref. [20] and extended in Eq. (74); the restoration of maximal-power polynomiality is argued by construction, not proven here.
ad hoc to paper The cross-channel nucleon exchange relation {V2,A2,T2} = (1/2){V1,A1,T1} constrains the flexible F^(1) component.
Section III.B.2: 'we employ the same relation between TDAs as those, which arises within the cross-channel nucleon exchange model'; chosen for convenience and not derived for F^(1).
domain assumption sigma_T is much larger than sigma_L, so the measured unseparated cross-section can be identified with the transverse TDA cross-section.
Section III.D: CLAS did not separate sigma_T and sigma_L, and the TDA prediction sigma_T >> sigma_L is stated to remain an assumption.
ad hoc to paper The factorized ansatz for spectral densities, F^(0)=phi h G and F^(1)=f h G with dipole u-dependence, is adequate.
Eqs. (75), (84), and (79); the factorized u-dependence and profile forms are modeling choices with no derivation from QCD.
ad hoc to paper Backward DA evolution approximately represents full TDA evolution.
Section III.C: only the DA evolution is used, and the authors state that a comparison with full TDA evolution is beyond the scope of this analysis.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Toward an advanced phenomenology of $\pi N$ transition distribution amplitudes." pith.science (2026). https://pith.science/paper/P5TZ3G4J

@misc{pith2026250604961,
  author       = {Pith},
  title        = {Pith review of: Toward an advanced phenomenology of $\pi N$ transition distribution amplitudes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/P5TZ3G4J}},
  note         = {Machine review of arXiv:2506.04961}
}

abstract

We introduce a new approach to modeling transition distribution amplitudes (TDAs) for the processes $e p \to e n \pi^+$ and $e p \to e p \pi^0$. The modeling is flexible, constrained by sparsely available experimental data, and satisfies theoretical requirements, including reduction to nucleon distribution amplitudes in the appropriate limit. We study the sensitivity of observable predictions to various modeling assumptions. We discuss unpolarized cross-sections, as well as the three non-vanishing polarization observables at leading twist, namely, the single transverse target spin asymmetry and the two double spin asymmetries that occur with a polarized lepton beam on either a longitudinally or transversely polarized target. The analysis is complemented by a simple Monte Carlo study to provide guidance for exploring exclusive processes in the so-called backward kinematics. Our work aims to highlight the importance of future measurements to better constrain TDAs and to support upcoming experimental proposals.

Figures

Figures reproduced from arXiv: 2506.04961 by the authors.

**Figure 2.** FIG. 2: Physical domains for TDAs in the barycentric coordinates for [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** FIG. 3: Kinematics of the process ( [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

Figures from the paper (8 more)

**Figure 4.** Figure 4: FIG. 4: Orthogonal polynomials on the hexagon multiplied by the corresponding weight function, [PITH_FULL_IMAGE:figures/full_fig_p017_4.png]

**Figure 5.** Figure 5: FIG. 5: CLAS data for the unseparated cross-section [PITH_FULL_IMAGE:figures/full_fig_p019_5.png]

**Figure 6.** Figure 6: FIG. 6: Sensitivity to the choice of DA parametrization. Selected observables for the exclusive production of (solid [PITH_FULL_IMAGE:figures/full_fig_p021_6.png]

**Figure 7.** Figure 7: FIG. 7: Sensitivity to the parameter [PITH_FULL_IMAGE:figures/full_fig_p021_7.png]

**Figure 8.** Figure 8: FIG. 8: Sensitivity to the parameter [PITH_FULL_IMAGE:figures/full_fig_p022_8.png]

**Figure 9.** Figure 9: FIG. 9: Sensitivity to the ratio of the coefficients [PITH_FULL_IMAGE:figures/full_fig_p022_9.png]

**Figure 10.** Figure 10: FIG. 10: Distributions of MC events for forward (red dotted histograms) and backward (blue solid histograms) [PITH_FULL_IMAGE:figures/full_fig_p023_10.png]

**Figure 11.** Figure 11: FIG. 11: Cross-channel nucleon exchange model for [PITH_FULL_IMAGE:figures/full_fig_p026_11.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

74 extracted references · 54 canonical work pages

[47]

I. V. Musatov and A. V. Radyushkin, Evolution and models for skewed parton distributions, Phys. Rev. D61, 074027 (2000), arXiv:hep-ph/9905376

arXiv 2000
[1]

[20], that, in complete analogy to (54), allows imposing constraints on TDAs in the forward limit,ξ= 0

the form originally suggested in Ref. [20], that, in complete analogy to (54), allows imposing constraints on TDAs in the forward limit,ξ= 0. For a specific choice of quark-diquark coordinates,w≡w 3 andv≡v 3, (59) is rewritten as: H(w, v, ξ) = Z 1 −1 dσ Z 1− |σ| 2 −1+ |σ| 2 dρ Z 1−|ρ− σ 2 |−|ρ+ σ 2 | −1+|σ| dω Z 1 2 −|ρ+ σ 2 |− ω 2 − 1 2 +|ρ− σ 2 |+ ω 2 d...

work page
[2]

D-term-like

an alternative form, suggested in Ref. [47], instead, allows imposing constraints in the limitξ= 1, in whichπN TDAs reduce to certain combinations of nucleon DAs (A3), due to the soft pion theorem [18] (see Appendix C for a brief overview; the origin of the normalization factor 1/4 is clarified in the discussion surrounding Eq. (C4) ): H(x 1, x2, x3, ξ= 1...

work page
[3]

Modeling spectral distributionF (0) For the sake of completeness, we review the construction of the phenomenological model for the spectral densityF (0) with the input from the soft pion theorem for TDAs atξ= 1u=m 2 N . In terms of the spectral parameters (68), corresponding to the choice of quark-diquark coordinatesw=w 3,v=v 3, the factorized Anstaz (70)...

work page
[4]

smearing

Modeling spectral distributionF (1) We now focus on the second component of our model, for which we adapt the factorized Ansatz with input atξ= 0. In terms of the spectral parameters (61) it takes the form F (1)(σ, ρ, ω, ν;u) =f(σ, ρ)h(σ, ρ, ω, ν)G(u).(84) The profile function, h(σ, ρ, ω, ν) = Γ(3b+ 3) 25b+2Γ(1 +b) 3 1 + 2ν−ω−2 ρ− σ 2 b 1−2ν−ω−2 ρ+ σ 2 b ...

work page
[5]

J. C. Collins,Foundations of Perturbative QCD(Cambridge University Press, Cambridge, UK, 2011)

work page 2011
[6]

Diehl, Experimental exploration of the 3D nucleon structure, Prog

S. Diehl, Experimental exploration of the 3D nucleon structure, Prog. Part. Nucl. Phys.133, 104069 (2023)

work page 2023
[7]

L. L. Frankfurt, P. V. Pobylitsa, M. V. Polyakov, and M. Strikman, Hard exclusive pseudoscalar meson electroproduction and spin structure of a nucleon, Phys. Rev. D60, 014010 (1999), arXiv:hep-ph/9901429

work page Pith review arXiv 1999

Show all 74 references

[8]

Pire and L

B. Pire and L. Szymanowski, Hadron annihilation into two photons and backward VCS in the scaling regime of QCD, Phys. Rev. D71, 111501 (2005), arXiv:hep-ph/0411387. 28

2005 arXiv
[9]

Pire and L

B. Pire and L. Szymanowski, QCD analysis of ¯pN→γ ∗πin the scaling limit, Phys. Lett. B622, 83 (2005), arXiv:hep- ph/0504255

2005
[10]

C. A. Gayosoet al., Progress and opportunities in backward angle (u-channel) physics, Eur. Phys. J. A57, 342 (2021), arXiv:2107.06748 [hep-ph]

2021 arXiv
[11]

B. Pire, K. Semenov-Tian-Shansky, and L. Szymanowski, Transition distribution amplitudes and hard exclusive reactions with baryon number transfer, Phys. Rept.940, 1 (2021), arXiv:2103.01079 [hep-ph]

2021 arXiv
[12]

Parket al.(CLAS), Hard exclusive pion electroproduction at backward angles with CLAS, Phys

K. Parket al.(CLAS), Hard exclusive pion electroproduction at backward angles with CLAS, Phys. Lett. B780, 340 (2018), arXiv:1711.08486 [nucl-ex]

2018 arXiv
[13]

W. B. Liet al.(Jefferson Lab Fπ), Unique Access tou-Channel Physics: Exclusive Backward-Angle Omega Meson Electroproduction, Phys. Rev. Lett.123, 182501 (2019), arXiv:1910.00464 [nucl-ex]

2019 arXiv
[14]

Sweger, S

Z. Sweger, S. Yoo, Z. Zeng, D. Cebra, S. R. Klein, Y. Ji, X. Dong, and M. Kim, Modeling backward-angle (u-channel) virtual Compton scattering at the future Electron-Ion Collider, Phys. Rev. C108, 055205 (2023), arXiv:2308.10478 [hep-ph]

2023 arXiv
[15]

B. P. Singhet al.(PANDA), Experimental access to Transition Distribution Amplitudes with the ¯PANDA experiment at F AIR, Eur. Phys. J. A51, 107 (2015), arXiv:1409.0865 [hep-ex]

2015 arXiv
[16]

Singhet al.(PANDA), Feasibility study for the measurement ofπNtransition distribution amplitudes at ¯PANDA in ¯pp→J/ψπ 0, Phys

B. Singhet al.(PANDA), Feasibility study for the measurement ofπNtransition distribution amplitudes at ¯PANDA in ¯pp→J/ψπ 0, Phys. Rev. D95, 032003 (2017), arXiv:1610.02149 [nucl-ex]

2017 arXiv
[17]

B. Pire, K. M. Semenov-Tian-Shansky, A. A. Shaikhutdinova, and L. Szymanowski, Pion and photon beam initiated backward charmonium or lepton pair production, AAPPS Bull.33, 26 (2023), arXiv:2212.07688 [hep-ph]

2023 arXiv
[18]

L.-B. Chen, W. Chen, F. Feng, S. Hu, and Y. Jia, Next-to-leading-order QCD corrections to nucleon Dirac form factors, arXiv:2406.19994 [hep-ph] (2024)

2024 arXiv
[19]

Huang, B.-X

Y.-K. Huang, B.-X. Shi, Y.-M. Wang, and X.-C. Zhao, Next-to-Leading-Order QCD Predictions for the Nucleon Form Factors, arXiv:2407.18724 [hep-ph] (2024)

2024 arXiv
[20]

W. B. Liet al., Backward-angle Exclusiveπ 0 Production above the Resonance Region (2020), arXiv:2008.10768 [nucl-ex]

2020 arXiv
[21]

V. L. Chernyak, A. A. Ogloblin, and I. R. Zhitnitsky, Calculation of Exclusive Processes With Baryons, Yad. Fiz.48, 1398 (1988)

1988
[22]

B. Pire, K. Semenov-Tian-Shansky, and L. Szymanowski,πNtransition distribution amplitudes: their symmetries and constraints from chiral dynamics, Phys. Rev. D84, 074014 (2011), arXiv:1106.1851 [hep-ph]

2011 arXiv
[23]

J. P. Lansberg, B. Pire, and L. Szymanowski, Hard exclusive electroproduction of a pion in the backward region, Phys. Rev. D75, 074004 (2007), [Erratum: Phys.Rev.D 77, 019902 (2008)], arXiv:hep-ph/0701125

2007 arXiv
[24]

B. Pire, K. Semenov-Tian-Shansky, and L. Szymanowski, A Spectral representation for baryon to meson transition distri- bution amplitudes, Phys. Rev. D82, 094030 (2010), arXiv:1008.0721 [hep-ph]

2010 arXiv
[25]

Gelfand, M

I. Gelfand, M. Graev, and N. Vilenkin,Generalized functions(Academic Press, N.Y.-London, 1966)

1966
[26]

Linet al., Parton distributions and lattice QCD calculations: a community white paper, Prog

H.-W. Linet al., Parton distributions and lattice QCD calculations: a community white paper, Prog. Part. Nucl. Phys. 100, 107 (2018), arXiv:1711.07916 [hep-ph]

2018 arXiv
[27]

Hagler, Hadron structure from lattice quantum chromodynamics, Phys

P. Hagler, Hadron structure from lattice quantum chromodynamics, Phys. Rept.490, 49 (2010), arXiv:0912.5483 [hep-lat]

2010 arXiv
[28]

Alexandrouet al., Moments of nucleon generalized parton distributions from lattice QCD simulations at physical pion mass, Phys

C. Alexandrouet al., Moments of nucleon generalized parton distributions from lattice QCD simulations at physical pion mass, Phys. Rev. D101, 034519 (2020), arXiv:1908.10706 [hep-lat]

2020 arXiv
[29]

V. M. Braunet al.(QCDSF), Nucleon distribution amplitudes and proton decay matrix elements on the lattice, Phys. Rev. D79, 034504 (2009), arXiv:0811.2712 [hep-lat]

2009 arXiv
[30]

Y. Aoki, T. Izubuchi, E. Shintani, and A. Soni, Improved lattice computation of proton decay matrix elements, Phys. Rev. D96, 014506 (2017), arXiv:1705.01338 [hep-lat]

2017 arXiv
[31]

J.-S. Yoo, Y. Aoki, P. Boyle, T. Izubuchi, A. Soni, and S. Syritsyn, Proton decay matrix elements on the lattice at physical pion mass, Phys. Rev. D105, 074501 (2022), arXiv:2111.01608 [hep-lat]

2022 arXiv
[32]

V. L. Chernyak and I. R. Zhitnitsky, Nucleon Wave Function and Nucleon Form-Factors in QCD, Nucl. Phys. B246, 52 (1984)

1984
[33]

B. Pire, K. Semenov-Tian-Shansky, and L. Szymanowski, QCD description of backward vector meson hard electroproduc- tion, Phys. Rev. D91, 094006 (2015), [Erratum: Phys.Rev.D 106, 099901 (2022)], arXiv:1503.02012 [hep-ph]

2015 arXiv
[34]

Arens, O

T. Arens, O. Nachtmann, M. Diehl, and P. V. Landshoff, Some tests for the helicity structure of the pomeron inep collisions, Z. Phys. C74, 651 (1997), arXiv:hep-ph/9605376

1997 arXiv
[35]

Diehl and S

M. Diehl and S. Sapeta, On the analysis of lepton scattering on longitudinally or transversely polarized protons, Eur. Phys. J. C41, 515 (2005), arXiv:hep-ph/0503023

2005 arXiv
[36]

Bacchetta, U

A. Bacchetta, U. D’Alesio, M. Diehl, and C. A. Miller, Single-spin asymmetries: The Trento conventions, Phys. Rev. D 70, 117504 (2004), arXiv:hep-ph/0410050

2004 arXiv
[37]

Itzykson and J

C. Itzykson and J. Zuber,Quantum Field Theory(McGraw-Hill, New York, USA, 1980)

1980
[38]

V. D. Burkert and T. S. H. Lee, Electromagnetic meson production in the nucleon resonance region, Int. J. Mod. Phys. E 13, 1035 (2004), arXiv:nucl-ex/0407020

2004 arXiv
[39]

Kim,Single and Double Spin Asymmetries for Deeply Virtual Exclusiveπ 0 Production on Longitudinally Polarized Proton Target with CLAS, Ph.D

A. Kim,Single and Double Spin Asymmetries for Deeply Virtual Exclusiveπ 0 Production on Longitudinally Polarized Proton Target with CLAS, Ph.D. thesis, Kyungpook Natl. U. (2013)

2013
[40]

Murdoch,Beam-, Target- and Double-Spin Asymmetry Measurements in Deeply Virtualπ 0,π + andπ − Meson Produc- tion, Ph.D

G. Murdoch,Beam-, Target- and Double-Spin Asymmetry Measurements in Deeply Virtualπ 0,π + andπ − Meson Produc- tion, Ph.D. thesis, University of Glasgow (2019)

2019
[41]

J. P. Lansberg, B. Pire, and L. Szymanowski, Spin Observables in Transition-Distribution-Amplitude Studies, J. Phys. Conf. Ser.295, 012090 (2011), arXiv:1011.6635 [hep-ph]

2011 arXiv
[42]

B. Pire, K. Semenov-Tian-Shansky, and L. Szymanowski, Cross Section and Single Transverse Target Spin Asymmetry for backward Pion Electroproduction, in14th Workshop on High Energy Spin Physics(2012) arXiv:1201.0123 [hep-ph]. 29

2012 arXiv
[43]

Kleiss and W

R. Kleiss and W. J. Stirling, Spinor Techniques for Calculatingp¯p→W ±/Z0 + jets, Nucl. Phys. B262, 235 (1985)

1985
[44]

Diehlet al.(CLAS), Extraction of Beam-Spin Asymmetries from the Hard Exclusiveπ + Channel off Protons in a Wide Range of Kinematics, Phys

S. Diehlet al.(CLAS), Extraction of Beam-Spin Asymmetries from the Hard Exclusiveπ + Channel off Protons in a Wide Range of Kinematics, Phys. Rev. Lett.125, 182001 (2020), arXiv:2007.15677 [nucl-ex]

2020 arXiv
[45]

A. V. Radyushkin, Nonforward parton distributions, Phys. Rev. D56, 5524 (1997), arXiv:hep-ph/9704207

1997 arXiv
[46]

A. V. Radyushkin, On Spectral Properties of Parton Correlation Functions and Multiparton Wave Functions, Phys. Lett. B131, 179 (1983)

1983
[48]

M. V. Polyakov and C. Weiss, Skewed and double distributions in pion and nucleon, Phys. Rev. D60, 114017 (1999), arXiv:hep-ph/9902451

1999 arXiv
[49]

O. V. Teryaev, Crossing and radon tomography for generalized parton distributions, Phys. Lett. B510, 125 (2001), arXiv:hep-ph/0102303

2001 arXiv
[50]

A. V. Belitsky and A. V. Radyushkin, Unraveling hadron structure with generalized parton distributions, Phys. Rept.418, 1 (2005), arXiv:hep-ph/0504030

2005 arXiv
[51]

J. P. Lansberg, B. Pire, K. Semenov-Tian-Shansky, and L. Szymanowski, A consistent model forπNtransition distribution amplitudes and backward pion electroproduction, Phys. Rev. D85, 054021 (2012), arXiv:1112.3570 [hep-ph]

2012 arXiv
[52]

A. V. Radyushkin, Symmetries and structure of skewed and double distributions, Phys. Lett. B449, 81 (1999), arXiv:hep- ph/9810466

1999
[53]

S. V. Goloskokov and P. Kroll, The Longitudinal cross-section of vector meson electroproduction, Eur. Phys. J. C50, 829 (2007), arXiv:hep-ph/0611290

2007 arXiv
[54]

C. F. Dunkl, Orthogonal polynomials on the hexagon, SIAM Journal on Applied Mathematics47, 343 (1987)

1987
[55]

Ferreira, J

C. Ferreira, J. L. Lopez, R. Navarro, and E. P. Sinusia, Orthogonal systems of Zernike type in polygons and polygonal facets (2015), arXiv:1506.07396 [astro-ph.IM]

2015 arXiv
[56]

Born and E

M. Born and E. Wolf,Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (7th Edition), 7th ed. (Cambridge University Press, 1999)

1999
[57]

A. Lenz, M. Gockeler, T. Kaltenbrunner, and N. Warkentin, The Nucleon Distribution Amplitudes and their application to nucleon form factors and theN→∆ transition at intermediate values ofQ 2, Phys. Rev. D79, 093007 (2009), arXiv:0903.1723 [hep-ph]

2009 arXiv
[58]

V. M. Braun, A. Lenz, and M. Wittmann, Nucleon Form Factors in QCD, Phys. Rev. D73, 094019 (2006), arXiv:hep- ph/0604050

2006
[59]

E. C. Aschenauer, V. Batozskaya, S. Fazio, K. Gates, H. Moutarde, D. Sokhan, H. Spiesberger, P. Sznajder, and K. Tezgin, EpIC: novel Monte Carlo generator for exclusive processes, Eur. Phys. J. C82, 819 (2022), arXiv:2205.01762 [hep-ph]

2022 arXiv
[60]

S. V. Goloskokov and P. Kroll, Vector meson electroproduction at small Bjorken-xand generalized parton distributions, Eur. Phys. J. C42, 281 (2005), arXiv:hep-ph/0501242

2005 arXiv
[61]

S. V. Goloskokov and P. Kroll, The Target asymmetry in hard vector-meson electroproduction and parton angular momenta, Eur. Phys. J. C59, 809 (2009), arXiv:0809.4126 [hep-ph]

2009 arXiv
[62]

S. V. Goloskokov and P. Kroll, An Attempt to understand exclusiveπ + electroproduction, Eur. Phys. J. C65, 137 (2010), arXiv:0906.0460 [hep-ph]

2010 arXiv
[63]

S. V. Goloskokov and P. Kroll, Transversity in hard exclusive electroproduction of pseudoscalar mesons, Eur. Phys. J. A 47, 112 (2011), arXiv:1106.4897 [hep-ph]

2011 arXiv
[64]

Berthouet al., PARTONS: PARtonic Tomography Of Nucleon Software: A computing framework for the phenomenology of Generalized Parton Distributions, Eur

B. Berthouet al., PARTONS: PARtonic Tomography Of Nucleon Software: A computing framework for the phenomenology of Generalized Parton Distributions, Eur. Phys. J. C78, 478 (2018), arXiv:1512.06174 [hep-ph]

2018 arXiv
[65]

G. M. Huber, W. B. Li, W. Cosyn, and B. Pire,u-Channel Color Transparency Observables, MDPI Physics4, 451 (2022), arXiv:2202.04470 [hep-ph]

2022 arXiv
[66]

Pasquini, M

B. Pasquini, M. Pincetti, and S. Boffi, Parton content of the nucleon from distribution amplitudes and transition distribution amplitudes, Phys. Rev. D80, 014017 (2009), arXiv:0905.4018 [hep-ph]

2009 arXiv
[67]

Pasquini and A

B. Pasquini and A. Schiavi, Light-front model for the transition distribution amplitudes for backward timelike Compton scattering, Phys. Rev. D109, 114021 (2024), arXiv:2403.00614 [hep-ph]

2024 arXiv
[68]

A. R. Castro, C. Mezrag, J. M. Morgado Ch´ avez, and B. Pire, Backward DVCS in a Sullivan process (2025), arXiv:2504.02657 [hep-ph]

2025 arXiv
[69]

V. M. Braun, S. E. Derkachov, G. P. Korchemsky, and A. N. Manashov, Baryon distribution amplitudes in QCD, Nucl. Phys. B553, 355 (1999), arXiv:hep-ph/9902375

1999 arXiv
[70]

Mankiewicz, G

L. Mankiewicz, G. Piller, and T. Weigl, Hard exclusive meson production and nonforward parton distributions, Eur. Phys. J. C5, 119 (1998), arXiv:hep-ph/9711227

1998 arXiv
[71]

L. L. Frankfurt, A. Freund, and M. Strikman, Deeply virtual Compton scattering at HERA: A Probe of asymptotia, Phys. Lett. B460, 417 (1999), arXiv:hep-ph/9806535

1999 arXiv
[72]

De Alfaro, S

V. De Alfaro, S. Fubini, G. Furlan, and C. Rossetti,Currents in Hadron Physics(North-Holland Publ, Amsterdam, 1973)

1973
[73]

P. V. Pobylitsa, M. V. Polyakov, and M. Strikman, Soft pion theorems for hard near threshold pion production, Phys. Rev. Lett.87, 022001 (2001), arXiv:hep-ph/0101279

2001 arXiv
[74]

V. M. Braun, D. Y. Ivanov, A. Lenz, and A. Peters, Deep inelastic pion electroproduction at threshold, Phys. Rev. D75, 014021 (2007), arXiv:hep-ph/0611386

2007 arXiv

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [47]

I. V. Musatov and A. V. Radyushkin, Evolution and models for skewed parton distributions, Phys. Rev. D61, 074027 (2000), arXiv:hep-ph/9905376

arXiv 2000

[2] [1]

[20], that, in complete analogy to (54), allows imposing constraints on TDAs in the forward limit,ξ= 0

the form originally suggested in Ref. [20], that, in complete analogy to (54), allows imposing constraints on TDAs in the forward limit,ξ= 0. For a specific choice of quark-diquark coordinates,w≡w 3 andv≡v 3, (59) is rewritten as: H(w, v, ξ) = Z 1 −1 dσ Z 1− |σ| 2 −1+ |σ| 2 dρ Z 1−|ρ− σ 2 |−|ρ+ σ 2 | −1+|σ| dω Z 1 2 −|ρ+ σ 2 |− ω 2 − 1 2 +|ρ− σ 2 |+ ω 2 d...

work page

[3] [2]

D-term-like

an alternative form, suggested in Ref. [47], instead, allows imposing constraints in the limitξ= 1, in whichπN TDAs reduce to certain combinations of nucleon DAs (A3), due to the soft pion theorem [18] (see Appendix C for a brief overview; the origin of the normalization factor 1/4 is clarified in the discussion surrounding Eq. (C4) ): H(x 1, x2, x3, ξ= 1...

work page

[4] [3]

Modeling spectral distributionF (0) For the sake of completeness, we review the construction of the phenomenological model for the spectral densityF (0) with the input from the soft pion theorem for TDAs atξ= 1u=m 2 N . In terms of the spectral parameters (68), corresponding to the choice of quark-diquark coordinatesw=w 3,v=v 3, the factorized Anstaz (70)...

work page

[5] [4]

smearing

Modeling spectral distributionF (1) We now focus on the second component of our model, for which we adapt the factorized Ansatz with input atξ= 0. In terms of the spectral parameters (61) it takes the form F (1)(σ, ρ, ω, ν;u) =f(σ, ρ)h(σ, ρ, ω, ν)G(u).(84) The profile function, h(σ, ρ, ω, ν) = Γ(3b+ 3) 25b+2Γ(1 +b) 3 1 + 2ν−ω−2 ρ− σ 2 b 1−2ν−ω−2 ρ+ σ 2 b ...

work page

[6] [5]

J. C. Collins,Foundations of Perturbative QCD(Cambridge University Press, Cambridge, UK, 2011)

work page 2011

[7] [6]

Diehl, Experimental exploration of the 3D nucleon structure, Prog

S. Diehl, Experimental exploration of the 3D nucleon structure, Prog. Part. Nucl. Phys.133, 104069 (2023)

work page 2023

[8] [7]

L. L. Frankfurt, P. V. Pobylitsa, M. V. Polyakov, and M. Strikman, Hard exclusive pseudoscalar meson electroproduction and spin structure of a nucleon, Phys. Rev. D60, 014010 (1999), arXiv:hep-ph/9901429

work page Pith review arXiv 1999