Inverse Radon transform and the transverse-momentum dependent functions

I.V. Anikin; L. Szymanowski

REVIEW 3 major objections 3 minor 2 cited by

Inverse Radon transform and the transverse-momentum dependent functions

T0 review · 3 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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

Pith's one-line read The paper claims a new additional term in the inverse Radon transform links the GTMD E2 to the Sivers function.

desk verdict The claimed new f_A term is an artifact: the symmetry (10) is an identity, and Eq. (83) reduces to a tautology. read the letter →

arxiv 1909.00017 v3 pith:AAYOBGKJ submitted 2019-08-30 hep-ph

I.V. Anikin , L. Szymanowski This is my paper · ORCID

classification hep-ph PACS 13.40.-f12.38.Bx12.38.Lg

keywords RadontransformgeneralizedpartondistributionsdoubleGTMDsSiversfunctionT-oddtransversemomentumdependencetime-reversalinvariance

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 authors revisit the standard inverse Radon transform and claim that it gains an additional term once the original function has compact support and the angular parameter is restricted to a half-circle. This extra term, usually discarded by the symmetry of the transform under angle shifts by $\pi$, is argued to be physical in the context of generalized transverse-momentum dependent parton distributions, where time-reversal invariance is relaxed. The central result is Eq. (83), which expresses the Sivers function through the real and imaginary parts of the GTMD $E_2$ with the new integration measure $d\mu_A$; if correct, this gives a new route from GTMDs to $T$-odd transverse-momentum distributions.

What carries the argument

The carrying object is the Radon transform $R[f](\tau,\phi)=\int d^2x\,f(x)\,\delta(\tau-\langle n_\phi,x\rangle)$ together with its inverse representation Eq. (16), split into the standard principal-value term $f_S$ and the delta-term $f_A$. The decisive mechanism is the symmetry relation (10) and the paper's claim that compact support of $f$ restricts the angular parameter to $[-\pi/2,\pi/2]$, breaking that symmetry and leaving $f_A$ nonzero. In the physical application the machinery is the GTMD $E_2(x,\xi,t;\{k_\perp\})$ and its $k_\perp$-dependent double distribution $e_2(\alpha,\beta;\{k_\perp\})$, with the integration measures $d\mu_A$ and $d\mu_S$ defined in Eq. (82); $E_2$'s real part connects to the spin-flip GPD $E$ while its imaginary part reduces to the Sivers function in the forward limit, so the new term carries the $T$-odd information.

What would settle it

Take a concrete compactly supported function, for example $f=1$ on the unit square and zero elsewhere, compute its Radon transform from the line-integral definition at $\phi$ and $\phi+\pi$, and evaluate the $f_A$ term in Eq. (16); if the identity $R[f]((-1)^k\tau,\phi)=R[f](\tau,\phi+k\pi)$ holds, $f_A$ integrates to zero, which would remove the new term and the claimed Sivers link.

Watch

Extended reading notes

Core claim

On its own terms, the paper's discovery is that the inverse Radon transform of a complex-valued function splits into a standard singular part $f_S$ and an additional part $f_A$, and that $f_A$, which vanishes under the full angular integration $\phi\in[0,2\pi]$ because of the symmetry $R[f]((-1)^k\tau,\phi)=R[f](\tau,\phi+k\pi)$, survives when the support of the function is restricted and $\phi$ is confined to $[-\pi/2,\pi/2]$. In the GPD setting that surviving term is complex and is tied to the relaxed time-reversal invariance of $k_\perp$-dependent distributions. Applying the construction to the GTMD $E_2$ and its double-distribution partner $e_2$, the paper derives Eq. (83): the Sivers-type distribution times the profile factor equals $-\int d\mu_A\,\mathrm{Re}\,E_2^{[\pm]} - \int d\mu_S\,\mathrm{Im}\,E_2^{[\pm]}$. The authors present this formula as the principal result: inverse Radon transformations mix the real and imaginary parts of GTMDs, offering an alternative route to restore the Sivers function.

Load-bearing premise

The argument stands on the premise that restricting the function's support confines the Radon angle to a half-circle and breaks the symmetry $R[f]((-1)^k\tau,\phi)=R[f](\tau,\phi+k\pi)$, a symmetry that follows directly from the definition for every function and would make the new term vanish if it cannot be broken.

Editorial extensions

If this is right

When the angular integration is restricted, the inverse Radon relation between double distributions and GPDs must include $f_A$; the standard $f_S$ alone is no longer the full inversion.
Equation (83) offers a model-based route to extract the Sivers function from the GTMD $E_2$, to be compared with existing extractions from transverse-momentum distributions.
Because the imaginary part of $E_2$ changes sign under Wilson-line reversal while the real part does not, the formula implies that the $k_\perp$-dependent double distribution $e_2$ must be complex, with $T$-odd information carried by $\mathrm{Im}\,e_2$.
The compact-support analysis implies that the DD rhombus support is inherited by the GPD variables $z,\xi$, but the support restrictions alone do not enforce $\xi\le 1$; a separate physical GPD/GDA condition is needed.

Reading between the lines

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

If, as the defining identity (10) suggests, compact support alone cannot break the symmetry, then $f_A\equiv 0$ and the Sivers-function formula (83) reduces to the standard singular term; the claimed new physics would then require a different source of the angular restriction.
The same mechanism, if valid, should extend to other T-odd GTMDs whose imaginary parts are linked to TMDs, not just $E_2$ and the Sivers function.
A concrete way to test the mechanism is to compute $E_2$ in a model, insert it in Eq. (83), and compare the recovered Sivers function with direct TMD calculations; agreement would validate the additional term, while disagreement would localize the failure in the support-restriction premise.
The half-circle restriction may be re-interpretable as a branch choice for the line parameter $\xi=\tan\phi$ rather than a genuine symmetry breaking; in that reading the additional term is a coordinate artifact and the physical content of Eq. (83) would need rederivation.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

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

Here's my take on Anikin–Szymanowski.

The headline: their central claim—that a restricted support for the double distribution generates a new “additional term” f_A in the inverse Radon transform, which then connects GTMDs to the Sivers function—does not survive a careful look. The symmetry they say is broken is an identity, and the standard inversion on the half-circle contains no f_A. Equation (83) is the inverse of their own ansatz, not a new result.

What's good: the paper gives a readable recap of the Fourier slice theorem and the Radon inversion, and the discussion in Sec. III A about which line classes in the (z, xi) plane correspond to physical versus GPD/GDA regions is a useful reminder. The GTMD parametrization and the time-reversal properties in Sec. IV are standard and clearly presented. For someone who wants the textbook Radon material with parton distributions filled in, the first part is fine.

The soft spot is the whole load-bearing argument. Eq. (10) follows immediately from delta(tau - n_phi dot x) and n_{phi+pi} = -n_phi; it holds for every function, compact support or not. In their own Eq. (16), f_A cancels when the phi-integral runs over [0,2pi], because the contributions from phi and phi+pi cancel at eta=0. When they restrict phi to [0,pi], the right thing to do is go through (19) with lambda over (-infinity, infinity) and epsilon(lambda); that gives (22), with no delta'(eta) term. Restricting the angular range does not “break” the identity—it just means you don't use the values at phi+pi. The f_A term is an artifact of mixing the restricted angular range with the half-line radial integration from (16).

Then Eq. (83) falls: with f_A=0, it reduces to e2^(Im) = -integral d mu_S Im E2, which is just the inverse of the definition E2 = R[e2] applied to the model ansatz (80). The claimed mixing of real and imaginary parts of the GTMD via d mu_A is an illusion. No new Sivers extraction follows.

Minor: the paper cites [21] “work in progress” for the phenomenological application; fine, but the theory anchor needs to be solid first.

Recommendation: desk reject. The review sections are fine for a lecture note, but the claimed new effect is not real. A referee would spend a week and reach the same place. Not worth peer review time.

Referee Report

3 major / 3 minor

Summary. The manuscript revisits the inverse Radon transform in the context of generalized parton distributions (GPDs) and double distributions (DDs), and claims to find a new additional contribution f_A to the inverse transform that becomes nonzero when the support of the original function is restricted or when the angular variable is limited to [-π/2, π/2] (Sec. II B). It then applies this to transverse-momentum-dependent double distributions: Eq. (79) defines E2 as the Radon transform of e2, Eq. (80) parametrizes the imaginary part of e2 through the Sivers function times a longitudinal momentum-sharing profile, and Eq. (83) is presented as the principal result relating the Sivers function to the real and imaginary parts of the GTMD E2. Sections III and IV give support-theorem arguments and a time-reversal analysis of GTMDs intended to justify the physical relevance of f_A.

Significance. If the claimed relation (83) were correct, it would offer a new way to connect T-odd transverse-momentum-dependent distributions to the Radon-transform formalism for GPDs and DDs, with potential phenomenological value for extracting Sivers functions from GTMDs. The paper usefully collects standard facts about the Radon transform and correctly reproduces the well-known inversion formula (22) in the full-angular case. However, the central new claim rests on an incorrect premise, and Eq. (83) is a tautology given the definitions and ansatz; the advertised physical result is therefore not established.

major comments (3)

[II A, Eq. (10)] Equation (10), R[f]((-1)^k τ, φ) = R[f](τ, φ + kπ), follows immediately from the delta-function definition (5) because n_{φ+π} = -n_φ. It is an identity for every function f, with or without compact support. Restricting the support of f only restricts the admissible τ for each direction; it does not alter the relation between R[f](τ, φ) and R[f](-τ, φ+π). Therefore the assertion in Secs. II B and III that restricted support breaks the symmetry and forces the angular range to [-π/2, π/2] is unsupported. The vanishing of f_A expressed in Eq. (58) is not a special property of unbounded support; it holds for all f. This invalidates the premise on which the existence and interpretation of f_A in Secs. III B and IV are built.
[II B, Eq. (19)] The paper's own derivation on the half-circle gives the standard inversion (22) with no f_A term. The key is that the correct half-circle representation (19) uses λ over the full real line with weight |λ|, not the one-sided λ ∈ [0, ∞) used in Eqs. (12) and (16). When the angular integration is restricted to [0, π], the radial Fourier variable must run over both signs; otherwise one mixes a restricted angle with an asymmetric radial measure and generates a spurious delta'(η) contribution. Thus the claim in Sec. II B that restricting φ to [0, π] allows the first term in Eq. (16) to exist is contradicted by Eq. (22).
[IV, Eq. (83)] The principal result (83) is a tautology. With Eq. (79), E2 is by definition the Radon transform of e2; substituting the ansatz (80) and applying the inverse-Radon operation, even with the formal measures in Eq. (82), simply returns the function that was inserted. Equation (83) therefore does not constitute an independent relation between the Sivers function and E2; it is the definition of e2 together with the ansatz. Unless a nontrivial inversion formula beyond Eq. (22) is established, Eq. (83) has no predictive content and cannot be used to restore the Sivers function from GTMDs.

minor comments (3)

[II C, Eq. (44)] The displayed Gaussian in (u, v) coordinates appears algebraically incorrect: from Eq. (43) one obtains exp[-(τ - ⟨n_φ, A⟩)^2] = exp[-K^2 + 2⟨K, A⟩ - ⟨K, A⟩^2/K^2], not the expression shown in Eq. (44).
[IV, Eq. (82)] The measures dμ_A and dμ_S are introduced with unexplained normalizations (e.g., the factor π in dμ_A), and the passage from Eq. (16) to Eq. (83) is not shown step by step, making it difficult to see where the real and imaginary parts in Eq. (83) enter.
[III B, Eq. (61)] The function \bar H is introduced as the Radon transform of f(y2, y1), but the symmetry condition (63) is stated without proof in terms of \bar H; clarifying the relation between \bar H and the standard H would help the reader, although it does not affect the main objection.

Circularity Check

2 steps flagged · score 8.0 of 10

The claimed Sivers-related contribution (83) restates the input ansatz (80) via the defining relation (79), and the existence of the 'additional' term f_A rests on an invalid breaking of the Radon symmetry (10).

self definitional [Sec. IV, Eqs. (79)-(83)]
"E2(x, ξ ,t;{⃗k⊥}) = Rx,ξ[e2(α, β;{⃗k⊥})] (79) ... e(ℑ)[C]2(α, β;{⃗k⊥}) = f⊥[C]1T(α;⃗k2⊥) ϕ((⃗k⊥⃗∆∆∆⊥)2n)π(α, β ) (80) ... f⊥[±]1T(α;⃗k2⊥) ϕ((⃗k⊥⃗∆∆∆⊥)2n)π(α, β ) = −∫ dµA{E(ℜ)[±]2(z+α+ξβ,ξ;{⃗k⊥})} −∫ dµS{E(ℑ)[±]2(z+α+ξβ,ξ;{⃗k⊥})} (83)."

Equation (79) defines E2 as the Radon transform of e2. The right-hand side of (83) is the inverse-Radon reconstruction of E2 written with the kernel split (82); applied to R[e2] it returns e2 itself. But (80) already fixes the imaginary part of e2 to be f_{1T}^⊥ φ π. Hence (83) is the assumed ansatz (80) written in inverse-Radon notation; the Sivers function is an input coefficient, not a distribution recovered from GTMDs. The principal result reduces by construction to the definitions and the factorization ansatz that precede it.
other [Sec. II B, Eqs. (10), (16), (19), (22)]
"there are some cases where for the restricted support of f the angular parameter ϕ together with the radial parameter η meet the corresponding restrictions limiting the variation intervals and breaking the symmetry condition generated by the eqn. (10). ... This allows the first term in the eqn. (16) to exist and to do contribute to the inversion of Radon transform in the case of f(⃗x)∈ C."

The symmetry (10) follows immediately from the δ-function definition (5) and is an identity for every function, compactly supported or not; a support restriction cannot break it. The 'first term' of (16) is the δ'(η) part of one asymmetric inversion kernel, which the paper itself shows vanishes on the full angular range (18) and which is absent from the correct half-range inversion (19)-(22). Its survival in Sec. II B is an artifact of combining the restricted angle interval with the λ∈[0,∞) Fourier inversion, not a derived property. Since f_A is manufactured by this split rather than by the Radon transform, the physical conclusions (Secs. III-IV and Eq. (83)) inherit a term whose existence is an input of the chosen inversion convention.

full rationale

The central derivation is not self-contained against an independent benchmark: Eq. (83), advertised as the principal result enabling a Sivers-function reconstruction, is the inverse Radon transform of E2 combined with the defining equation (79) and the ansatz (80). Because E2 is defined as R[e2] and e2's imaginary part is parametrized by f_{1T}^⊥ times a profile π, inverting E2 returns exactly that same e2; no new Sivers information is produced. The apparent extra term f_A is likewise not a consequence of compact support, since Eq. (10) is an identity of the Radon definition and cannot be broken by support restrictions; the nonzero f_A comes from the authors' choice to use Eq. (16)'s asymmetric inversion on a restricted angle sector, while the correct half-circle inversion (19)-(22) contains no such term. The paper's own Eq. (22) confirms that the standard representation has no additional term. No self-citation is load-bearing here; the circularity is internal, in the equations. Score 8 reflects that the main result is forced by the preceding definitions and ansatz, though the paper does contain independent standard Radon-transform mathematics in Secs. II-III before the physical application.

Assumptions & free parameters 3 free parameters · 4 assumptions · 1 invented entities

The central claim depends on two ad hoc inputs: a false symmetry-breaking premise and a factorized model ansatz for the imaginary part of the k_perp-dependent double distribution. The only standard-math input is the textbook Radon transform machinery. No data, parameters fitted to data, or external benchmarks appear.

free parameters (3)

pi(alpha,beta) longitudinal momentum sharing
Introduced in the ansatz (80) for Im e2; satisfies only the normalization integral d beta pi = 1; shape is free and is not derived from QCD, so any computed relation involving it is model-dependent.
phi((k_perp.Delta_perp)^{2n})
Even function in the scalar product with phi(0)=1; power n and functional form are unspecified, so it is a free modeling choice.
exponent n
Integer power in the ansatz (80); not fixed by the paper.

assumptions (4)

ad hoc to paper The Radon transform symmetry R[f]((-1)^k tau, phi) = R[f](tau, phi+k pi) can be broken by restricting the support of f.
This is the load-bearing premise for the existence of f_A (Sec. II B and III B). The identity actually follows from the definition of R for any f, so the premise is false.
standard math Radon transform definitions, Fourier slice theorem, and standard distributional regularization of 1/eta^2.
Used throughout Sec. II; these are standard background results from [14,15].
domain assumption k_perp-dependent GTMDs are complex because time-reversal invariance is relaxed, with T-odd distributions like the Sivers function allowed.
Taken from the literature [18-20]; plausible but an input, not derived here.
ad hoc to paper The imaginary part of the k_perp-dependent DD factorizes as f_{1T}^{perp} phi pi with the stated normalizations.
Ansatz (80); this is a model assumption with free functions that makes Eq. (83) hold by construction.

invented entities (1)

f_A (additional term in the inverse Radon transform)
purpose: Claimed to represent T-odd effects and to connect GTMDs to the Sivers function via Eq. (83).
No independent falsifiable handle is provided; the term is shown to cancel on [0,2pi] and to be absent from the standard inversion (22), so it is an artifact of the chosen split.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Inverse Radon transform and the transverse-momentum dependent functions." pith.science (2026). https://pith.science/paper/AAYOBGKJ

@misc{pith2026190900017,
  author       = {Pith},
  title        = {Pith review of: Inverse Radon transform and the transverse-momentum dependent functions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AAYOBGKJ}},
  note         = {Machine review of arXiv:1909.00017}
}

read the original abstract

We revisit the standard representation of the (inverse) Radon transform which is well-known in the mathematical literature. We extend this representation to the case involving the parton distributions. We have found the new additional contribution which is essentially related to the generalized transverse-momentum dependent parton distribution and double-distribution functions. We discuss the possible relationship of this term with the Sivers function.

Figures

Figures reproduced from arXiv: 1909.00017 by the authors.

**Figure 2.** FIG. 2: The [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Non-integral geometry: additional term $f_A$ as a regularizing term
hep-th 2026-05 unverdicted novelty 5.0 of 10

The additional term f_A in the universal inverse Radon transform removes complex singularities that arise for non-symmetric supports, provided certain log-branch parameters are chosen appropriately.
Complexity of Radon transforms
math.FA 2025-05 reject novelty 2.0 of 10

The paper argues that inverse Radon reconstruction requires complex direct Radon images and proposes a partial Fourier transform trick plus a claimed holonomy effect to generate that complexity.

Reference graph

Works this paper leans on

24 extracted references · 20 canonical work pages · cited by 2 Pith papers

[21]

Anikin and L

I.V . Anikin and L. Szymanowski, work in progress

work page
[1]

D.Müller et al., Fortschr. Phys. 42, 101 (1994)

work page 1994
[2]

X. D. Ji, Phys. Rev. D 55, 7114 (1997)

1997
[3]

A. V . Radyushkin, Phys. Rev. D56, 5524 (1997)

work page 1997
[4]

Ball and V

P. Ball and V . M. Braun, Phys. Rev. D54, 2182 (1996)

work page 1996
[5]

V . M. Braun, S. E. Derkachov, G. P. Korchemsky and A. N. Manashov, Nucl. Phys. B 553, 355 (1999)

work page 1999
[6]

Diehl, Phys

M. Diehl, Phys. Rept. 388, 41 (2003)

2003
[7]

A. V . Belitsky and A. V . Radyushkin, Phys. Rept.418, 1 (2005)

2005

Show all 24 references

[8]

O. V . Teryaev, Phys. Lett. B510, 125 (2001)

2001
[9]

Double distributions and generalized parton dis- tributions from the parton number conserved light front wave function overlap representation,

D. Müller, “Double distributions and generalized parton dis- tributions from the parton number conserved light front wave function overlap representation,” arXiv:1711.09932 [hep-ph]

arXiv
[10]

Chouika, C

N. Chouika, C. Mezrag, H. Moutarde and J. Rodríguez- Quintero, Eur. Phys. J. C 77, 906 (2017)

2017
[11]

Chouika, C

N. Chouika, C. Mezrag, H. Moutarde and J. Rodríguez- Quintero, EPJ Web Conf. 137, 05020 (2017)

2017
[12]

Generalized Parton Distributions and their covari- ant extension: towards nucleon tomography,

N. Chouika, “Generalized Parton Distributions and their covari- ant extension: towards nucleon tomography,” 2018SACLS259, tel-01925746, PhD thesis
[13]

Nucleon Reverse Engineering: Structuring hadrons with colored degrees of freedom,

H. Moutarde, “Nucleon Reverse Engineering: Structuring hadrons with colored degrees of freedom,” Habilitation thesis, 243 p (2015)

2015
[14]

The Radon Transform and Some of Its Applica- tions,

S.R. Deans, “The Radon Transform and Some of Its Applica- tions,” Wiley 299 p (1983)

1983
[15]

Generalized Func- tions, V olume 5: Integral Geometry and Representation The- ory,

I.M. Gelfand, M.I. Graev, N.Ya. Vilenkin, “Generalized Func- tions, V olume 5: Integral Geometry and Representation The- ory,” AMS Chelsea Publishing: An Imprint of the American Mathematical Society, 449 pp (1966)

1966
[16]

Boman and E.T

J. Boman and E.T. Quinto, Duke Math. J. 55, 943 (1987)

1987
[17]

Inverse Radon transform at work,

I. R. Gabdrakhmanov, D. Müller and O. V . Teryaev, “Inverse Radon transform at work,” arXiv:1906.01458 [hep-ph]

1906 arXiv
[18]

D. Boer, P. J. Mulders and F. Pijlman, Nucl. Phys. B 667, 201 (2003)

2003
[19]

Generalized trans- verse momentum dependent parton distributions of the nu- cleon,

S. Meissner, A. Metz and M. Schlegel, “Generalized trans- verse momentum dependent parton distributions of the nu- cleon,” arXiv:0807.1154 [hep-ph]

arXiv
[20]

The k⊥-dependent DD-functions can be parametrized in the similar manner as it has been done for the usual DD-functions (cf

as E(ℑ) [±] 2 (x, ξ ;⃗k2 ⊥,⃗k⊥⃗∆∆∆⊥) ⏐⏐⏐ ∆=0 = (78) E(ℑ) [±] 2 (x,0;⃗k2 ⊥,0) = f⊥[±] 1T (x;⃗k2 ⊥) We now express the GTMD as the direct Radon transform of the k⊥-dependent DD-function e2(α, β;{⃗k⊥}), we have E2(x, ξ ,t;{⃗k⊥}) = Rx,ξ [ e2(α, β;{⃗k⊥}) ] , (79) e2(α, β;{⃗k⊥}) = e...

2017
[22]

(27) 4 In contrast to the eqn

: ∫ +∞ −∞ (dz)e−iq1z R[ f ](z, ξ ) = F [ f ](q1, ξ q1) ⏐⏐⏐ ξ = q2 q1 . (27) 4 In contrast to the eqn. (4), this representation deals with (q1, ξ ) as an independent set of variables for the Fourier trans- form. As a result, the δ-function in the direct Radon trans- form, δ (z−...
[23]

Meissner, A

S. Meissner, A. Metz and M. Schlegel, JHEP 0908, 056 (2009)

2009
[24]

Notice also that it is sometimes useful to present the Radon transform in the form of R[ f ](τ, ϕ) = R[ f ](z, ξ )/|cos ϕ|, where z = τ/cos ϕ, ξ = tan ϕ≡ q2/q1

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [21]

Anikin and L

I.V . Anikin and L. Szymanowski, work in progress

work page

[2] [1]

D.Müller et al., Fortschr. Phys. 42, 101 (1994)

work page 1994

[3] [2]

X. D. Ji, Phys. Rev. D 55, 7114 (1997)

1997

[4] [3]

A. V . Radyushkin, Phys. Rev. D56, 5524 (1997)

work page 1997

[5] [4]

Ball and V

P. Ball and V . M. Braun, Phys. Rev. D54, 2182 (1996)

work page 1996

[6] [5]

V . M. Braun, S. E. Derkachov, G. P. Korchemsky and A. N. Manashov, Nucl. Phys. B 553, 355 (1999)

work page 1999

[7] [6]

Diehl, Phys

M. Diehl, Phys. Rept. 388, 41 (2003)

2003

[8] [7]

A. V . Belitsky and A. V . Radyushkin, Phys. Rept.418, 1 (2005)

2005