Pith. sign in

REVIEW 3 major objections 8 minor 31 references

QCD analytic coupling

T0 review · 3 major / 8 minor · reviewed 2026-08-07 · deepseek-v4-flash

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

Pith's one-line read The 1/L expansion of the analytic QCD coupling works at every Q^2

desk verdict A compact, honest overview of the authors' earlier 1/L-expansion results for MA coupling, with a TFF demonstration that is visually plausible but quantitatively underdocumented. read the letter →

arxiv 2608.05894 v1 pith:6UHMBJCN submitted 2026-08-06 hep-ph

I.A. Zemlyakov , I.L. Chuev , A.V. Kotikov This is my paper · ORCID
classification hep-ph
keywords QCDanalyticcouplingminimalapproachfractionalperturbationtheorypion-photontransitionformfactor1/LexpansionLandaupoleR-hatoperatormassivetwist-fourterm
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 argues that the $1/L$ expansion of the analytic (Minimal Approach) strong coupling, although its expansion parameter is small only when $Q^2\gg\Lambda^2$, is usable at every $Q^2$. The non-leading expansion corrections vanish both as $Q^2\to\infty$ and as $Q^2\to 0$, so they are concentrated in a small neighbourhood of the Landau scale $Q^2\sim\Lambda^2$. The authors present the $\hat R$-operator construction from their earlier work, which generates higher-order terms by acting on the leading-order analytic coupling, and then apply the resulting couplings to the pion-photon transition form factor. Replacing powers of the strong coupling by derivative series of the analytic coupling and adding a massive twist-four term, the formulas reproduce the published $Q^2F^{\gamma\pi}(Q^2)$ data. If the claim holds, the analytic coupling can be computed in any perturbative order from the LO term, with only modest corrections near $\Lambda^2$.

What carries the argument

The central object is the $\hat R$ operator, defined at next-to-leading order as $\hat R_1 = b_1[\Psi(1+\nu)+\gamma_E + d/d\nu]$, acting on $1/L^\nu$ for the ordinary strong coupling and on $\mathrm{Li}_{-\nu}(z)/\Gamma(\nu+1)$ for the analytic coupling. It converts each term of the $1/L$ expansion into an operator applied to the LO coupling, so that all higher-order corrections share one compact algebraic form. The second mechanism is the derivative-series replacement $a_s^n\to\tilde A_n(Q^2)$, built from fractional derivatives of the analytic coupling, which transfers the all-$Q^2$ coupling into observables like the pion-photon transition form factor.

What would settle it

Compute the third-order correction from Ref. [1] at $Q^2=\Lambda^2$ and check whether it stays small relative to the leading and next-to-leading terms; if it grows, the claimed all-$Q^2$ validity fails. A data-side test is to measure $Q^2F^{\gamma\pi}(Q^2)$ at the lowest accessible $Q^2$ and see whether the fitted massive twist-four coefficient remains at the value fixed at higher $Q^2$, or whether the analytic-coupling term itself has to absorb the discrepancy.

Watch

Extended reading notes

Core claim

The central claim is that the $1/L$ expansion of the analytic coupling $A_{\rm MA}(Q^2)$, organised through $\hat R$ operators applied to the leading-order term, is valid for all $Q^2$, because the non-leading corrections disappear at both ends of the infrared-ultraviolet range. In the first two perturbative orders the fractional derivatives of the MA coupling are written as the LO analytic result plus $\nu\,\tilde\delta^{(2)}_{A,\nu,1}$, where the correction is the same $\hat R_1$ operator that generates the strong-coupling expansion, applied to polylogarithmic functions instead of powers of $L$ (Eqs. (14)-(17)). Used in the valence-quark part of the pion-photon transition form factor, with the replacement $a_s^n\to\tilde A_n$ and a massive twist-four term, these couplings produce the curves in Fig. 1, which the paper finds to be in good agreement with the measured data and with results obtained by light-cone sum rules in dispersion form. The stated all-$Q^2$ validity is inherited from Ref. [1] rather than re-derived in this paper.

Load-bearing premise

The load-bearing premise is that the non-leading $1/L$ corrections to the analytic coupling vanish as $Q^2\to0$, so the expansion can be trusted at and below the Landau scale even though its expansion parameter is not small there; this is asserted on the strength of Ref. [1] and is not re-derived here, and the derivative-series replacement in the TFF rests on a second input assumption taken from Refs. [16,18,19].

Editorial extensions

If this is right

  • The analytic coupling and its fractional derivatives can be constructed in any perturbative order from the leading-order term plus $\hat R$-operator corrections, with corrections that vanish in both asymptotic limits.
  • The pion-photon transition form factor computed from MA couplings plus a massive twist-four term matches the published data over the shown $Q^2$ range.
  • The same derivative-series substitution can be applied to other QCD observables, and the authors note it already has been for the two sum rules treated in Refs. [24-26].
  • Because the five-loop $\beta$-function coefficients are known, the construction extends to the fifth perturbative order without new non-perturbative input.

Reading between the lines

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

  • A direct test of the all-$Q^2$ claim would be to evaluate the third-order correction from Ref. [1] at $Q^2=\Lambda^2$; if it is not small compared with the NLO term, the claimed applicability needs qualification.
  • If the expansion is as benign as claimed, the value of the analytic coupling near $Q^2\to0$ may be dominated by the LO term plus a universal constant, making very low-$Q^2$ observables predictable without extra regulators.
  • The current comparison uses only the three-flavour coupling; promoting the predictions through heavy-quark thresholds would show whether the agreement with data survives when the coupling and the form factor are treated consistently across flavour thresholds.
  • The massive twist-four term is fitted alongside the analytic coupling, so measurements at even lower $Q^2$ would separate the analytic-coupling contribution from the twist-four contribution more cleanly than the present data can.
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.

Referee Report

3 major / 8 minor

Summary. This paper is a short overview of the Minimal Approach variant of analytic QCD (Analytic Perturbation Theory), largely following the authors' earlier paper [1]. It presents the 1/L-expansion of fractional derivatives of the strong coupling in terms of the operator R-hat_1 (Eq. (9)), lifts this expansion to the analytic MA coupling (Eqs. (14)-(17)), and asserts in Sec. 1 that the 1/L-expansion is valid for all Q^2 because the non-leading corrections vanish at both Q^2 to 0 and Q^2 to infinity, being nonzero only in a neighborhood of Q^2 ~ Lambda^2. The paper then applies the MA coupling to the pion-photon transition form factor, replacing the powers a_s^n of the TFF series by derivative-based analytic couplings A-tilde_n (Eqs. (24)-(25)), and reports good agreement with BESIII, CLEO, BaBar and Belle data after a fit that includes a massive twist-four term (Fig. 1). Explicit formulas are given only through the second PT order and, for the MA corrections, only for nu = 1; higher orders and non-integer nu are delegated to Ref. [1].

Significance. The practical message is attractive: if the all-Q^2 1/L-expansion is valid, then the analytic coupling at any PT order is computable from the LO term with corrections concentrated at Q^2 ~ Lambda^2, and the apparent order-by-order stability in Fig. 1 would demonstrate a useful feature of FAPT. The strengths of the paper are its explicit scope statement, the compact operator organization of the corrections (Eq. (9) through Eqs. (14)-(15)), the explicit nu=1 results in Eq. (16), and the use of up-to-date NNLO TFF coefficient functions from Refs. [21,22]. The genuinely new content is the TFF comparison, and that part is under-documented: it is a fitted result rather than a parameter-free prediction, with no chi-square, no fitted parameter values or errors, and no displayed twist-four term. The claimed demonstration is therefore not yet verifiable from the manuscript as it stands.

major comments (3)
  1. [Sec. 1; Eqs. (9), (14)-(17); Footnote 3] The all-Q^2 validity of the 1/L-expansion is the load-bearing premise of the paper and it is not established here. The argument offered in Sec. 1, namely that non-leading corrections vanish at Q^2 to 0 and at Q^2 to infinity and therefore give only small corrections at Q^2 ~ Lambda^2, does not control the intermediate region: vanishing at both endpoints is compatible with a large hump at intermediate scales, and for the ordinary coupling the 1/L^n terms also vanish at Q^2 to 0 while the expansion still fails near the Landau pole. For the MA coupling the divergence is removed only by a cancellation between the 1/L^n terms and the polylogarithmic subtractions in Eqs. (15)-(16), and that cancellation is not demonstrated here; it is delegated to Ref. [1]. Footnote 3 cites Refs. [5,6] only for the Q^2 to 0 endpoint, not for the Q^2 ~ Lambda^2 region. Since Eqs. (23)-(25) are evaluated down to Q^2 = 0.3 GeV^2, where L is O(1) for Lambda ~ 0.3 GeV, the TFF comparison inherits this gap. A concrete check would be a plot or table of the absolute value of delta^(2)_A,nu,1 / A^(1)_MA,nu,0 over, say, Q^2/Lambda^2 in [10^-2, 10^2] for the nu values used in Sec. 5; this should be supplied, together with a statement of which results of Ref. [1] are being relied on.
  2. [Sec. 5; Eqs. (23)-(25); Fig. 1] The claimed good agreement with the TFF data is a fitted result whose details are absent. The massive twist-four term is introduced only by a citation to Ref. [27]: its functional form and fitted parameters are not given, no chi-square or equivalent goodness-of-fit statistic is reported, and it is not stated whether the Gegenbauer moments b2(Q0) and b4(Q0) of Eq. (22) are fixed at their central values or fitted within their quoted errors. In addition, Eq. (23) as printed writes the b4 contribution as F^(gamma-pi, tau=2)_V,n=4 without the MA subscript carried by the first two terms, leaving it ambiguous whether the analytic or the ordinary coupling is used for the n=4 term. Both the formula and the fit documentation must be corrected before the central demonstration of the paper can be evaluated.
  3. [Eqs. (15)-(16); Eqs. (24)-(25)] The TFF application requires non-integer nu values of about 1.62, 1.90, 2.62 and 2.90 (from d_2 = 50/81 and d_4 = 364/405 in Eq. (21)), whereas Eq. (16) displays the MA correction only for the integer case nu = 1. The general formula (15) involves Li_-nu(z_i)/Gamma(nu+1), whose behavior for negative fractional index near z ~ 1 is more delicate than for nu = 1, and the cancellation described in the first major comment is precisely the part that needs to be checked for these nu. Please give the explicit fractional-nu results entering Eqs. (24)-(25), or reproduce the relevant equations of Ref. [1] where they appear, together with a numerical verification of their smallness in the Q^2 range of Fig. 1.
minor comments (8)
  1. [Sec. 2; Eqs. (6)-(7)] The subscripts on a^(1)_s,0, a^(2)_s,1, L_0, L_1 and Lambda_i are never defined; the paper should state explicitly that the second index labels the PT order used in the dimensional-transmutation parameter, as implied by the matching discussion.
  2. [Eq. (10)] The first factor appears with a subscript nu, reading (a^(1)_nu,0(Q^2))^nu; from Eqs. (4) and (9) it should be the LO strong coupling a^(1)_s,0(Q^2) = 1/L_0.
  3. [Eq. (17)] The displayed definition of the generalized polylogarithm is garbled, reading Li_n,m(z) = sum over m of ln^k m / m^n; the standard definition Li_n,k(z) = sum over m of z^m (ln m)^k / m^n should be stated, together with its domain of definition or analytic continuation.
  4. [Fig. 1] The three theory curves (LO, NLO and NNLO MA plus massive twist-four) are barely distinguishable as reproduced and no uncertainty bands are shown; a legend with distinct line styles, plus propagation of the alpha_s, b2 and b4 errors into the curves, would make the claimed agreement and the order-by-order stability verifiable.
  5. [Secs. 2 and 5] The calculation uses Lambda^(f=3) throughout, while the data in Fig. 1 extend to Q^2 = 5 GeV^2, above the charm threshold; the approximation of a single active flavor across the whole fitted range should be justified or its effect on the fit quantified.
  6. [Sec. 5] The paper does not give the numerical values of Lambda used (for example, the Lambda^(f=3) implied by alpha_s(M_Z) = 0.1176) nor the fitted twist-four parameters; without these values the curves in Fig. 1 are not reproducible.
  7. [Sec. 6] The statement that the authors have demonstrated the results obtained in paper [1] overstates the role of this manuscript, which reviews and quotes those results rather than re-deriving them; the verb 'summarize' would be more accurate.
  8. [Sec. 3] The statement that a derivative series can successfully replace a power series is justified only heuristically (each derivative yields an additional a_s); a brief statement of the exact relations from Refs. [16,18,19] used in going from Eqs. (19)-(20) to Eqs. (24)-(25) would strengthen the presentation.

Circularity Check

2 steps flagged · score 6.0 of 10

The all-Q^2 validity of the 1/L-expansion is imported from the authors' own Ref. [1], and the TFF 'agreement' follows a fit to the same data.

  1. self citation load bearing [Section 1 (Introduction), paragraph describing the 1/L-expansion of the analytic coupling]
    "Here we give an overview of the main properties of MA couplings in the F APT framework, obtained in Ref. [1] using the so-called 1/L-expansion. Note that for an ordinary coupling, this expansion is applicable only for largeQ 2 values, i.e., forQ 2 >>Λ 2. However, as shown in [1], the situation is quite different in the case of the analytic coupling, and this 1/L-expansion is applicable for allQ 2 values. This is due to the fact that the non-leading expansion corrections vanish not only atQ 2 → ∞, but also atQ 2 →0, which leads only to nonzero (small) corrections in the regionQ 2 ∼Λ 2."

    The paper's central mathematical claim is that the 1/L-expansion of the MA coupling is valid for all Q^2, and this property underpins Eqs. (9), (14)-(17) and the later TFF formulas (23)-(25). In this text the only support is the phrase 'as shown in [1]', where Ref. [1] is Kotikov and Zemlyakov, i.e. the same research group as the present authors. No derivation, proof, or independent check of the all-Q^2 validity is supplied in this manuscript. The statement about vanishing at Q^2 -> 0 is asserted, and footnote 3 only cites Refs. [5,6] for that endpoint, not for the Q^2 ~ Λ^2 region where the expansion parameter 1/L is not small. Thus the load-bearing premise reduces to a self-citation rather than to a demonstrated result in this paper.

  2. fitted input called prediction [Section 5 (Pion-photon transition form factor), paragraph after Eq. (23) and before Fig. 1]
    "By analogy with Refs. [24–26], where the Bjorken and Gross-Llewellyn Smith sum rules were considered, we perform a fit of experimental data for the pion-photon TFF within the framework of analytic QCD with the "massive" form [27] of the twist-four term. We obtain a good agreement between the APT predictions and experimental data (see Fig. 1)."

    The figure comparison is introduced immediately after stating that a fit of the massive twist-four term to the experimental TFF data was performed. Calling the resulting curves 'APT predictions' obscures that the twist-four contribution was adjusted to the very same data being compared, so the 'good agreement' is partly an artifact of the fit rather than a parameter-free consequence of the analytic coupling. The paper does not list the fitted twist-four parameters or the number of fitted degrees of freedom, so the reader cannot determine how much of the agreement is enforced by construction.

full rationale

The manuscript is explicitly an overview of the authors' previous work [1]. That in itself is not circular. However, the central premise of the overview—that the 1/L-expansion of the MA coupling is valid for all Q^2—is not re-derived or independently established here; it is asserted with the citation 'as shown in [1]', and Ref. [1] is co-authored by the present authors. Every formula that follows (Eqs. (9), (14)-(17), and the TFF expressions (23)-(25)) relies on this property, so the derivation chain stops at the same research group's earlier paper. This is a self-citation load-bearing step. Separately, the TFF comparison in Fig. 1 is described as 'APT predictions' after a fit of the massive twist-four term to the same experimental data; the agreement is therefore partly an artifact of the fit rather than a parameter-free prediction. The analytic coupling construction itself is anchored in external Refs. [5,6], and the Gegenbauer moments are taken from external fits [23], so the paper is not wholly reducible to its own inputs. On the 0-10 scale, the central mathematical claim and the headline phenomenological 'agreement' both rely on inputs not derived in this paper, giving a score of 6 (partial circularity).

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

The TFF agreement depends on hadronic inputs b2 and b4 from Ref. [23], an unspecified massive twist-four term from Ref. [27], and the analytic coupling framework of Ref. [1]. No new free parameters are introduced by this paper, but the key analytic claim is self-cited to Ref. [1] and not independently checked. The derivative-series replacement and the all-Q2 validity of the 1/L expansion are the main assumptions.

free parameters (4)
  • Gegenbauer moment b2(Q0) = 0.206 ± 0.038 at Q0 = 1 GeV
    Imported from Ref. [23], Eq. (22). The TFF prediction depends directly on this moment through Eqs. (18)-(23).
  • Gegenbauer moment b4(Q0) = 0.047 ± 0.011 at Q0 = 1 GeV
    Imported from Ref. [23], Eq. (22). Controls the n=4 twist-two contribution in Eq. (23).
  • alpha_s(M_Z) = 0.1176 (PDG20)
    External input used to fix the QCD scale Lambda; the TFF normalization depends on it, and the paper does not study its uncertainty.
  • massive twist-four term parameters = not reported
    Section 5 states that a fit is performed with the 'massive' twist-four term from Ref. [27], but the explicit expression and any fitted parameters are never given. The central agreement claim depends on this unspecified input.
assumptions (4)
  • domain assumption Spacelike QCD observables are analytic except on the negative Q2 axis, so the Landau pole in alpha_s is unphysical and must be removed via a dispersion relation with the PT spectral function.
    Section 1, Eq. (5). This is the standard analyticity input of the MA/APT approach; it is not derived in this paper.
  • domain assumption The 1/L expansion of the strong coupling and its fractional derivatives remains valid for all Q2, because the non-leading terms vanish at Q2 -> 0 and Q2 -> infinity.
    Section 1 states this as shown in Ref. [1]; Section 4 uses it to write Eqs. (9), (14)-(17). No proof is reproduced here, and the expansion parameter is not small near Q2 ~ Lambda2.
  • domain assumption The derivative series a_n-tilde can replace a_s^n series beyond LO, and the relation from Ref. [19] extends this to fractional powers nu.
    Section 3 invokes Refs. [16,18,19] to justify replacing a_s^n by a_n-tilde, which is then used in Eqs. (24)-(25) for the TFF application.
  • domain assumption The pion-photon TFF is dominated by the twist-two valence contribution with Gegenbauer moments from Ref. [23], plus a massive twist-four term from Ref. [27] whose explicit form is not given.
    Section 5, Eqs. (18)-(25). The agreement claim depends on these nonperturbative inputs; the paper does not define the massive twist-four term or quantify its uncertainty.

how reviews work

0 comments
Cite this review

Pith. "Pith review of QCD analytic coupling." pith.science (2026). https://pith.science/paper/6UHMBJCN

@misc{pith2026260805894,
  author       = {Pith},
  title        = {Pith review of: QCD analytic coupling},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6UHMBJCN}},
  note         = {Machine review of arXiv:2608.05894}
}
read the original abstract

A brief overview of QCD analytic coupling is presented, mainly following the results obtained in [1]. An application to the pion-photon transition form factor is demonstrated.

Figures

Figures reproduced from arXiv: 2608.05894 by the authors.

Figure 1
Figure 1. The results (23)-(25) for the pion-photon TFF. Experimental data can be found [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

31 extracted references · 25 canonical work pages

  1. [1]

    A. V. Kotikov and I. A. Zemlyakov, J. Phys. G50, no.1, 015001 (2023)

    work page 2023

  2. [27]

    Teryaev, Nucl

    O. Teryaev, Nucl. Phys. B Proc. Suppl.245, 195-198 (2013); V. L. Khandramai, O. V. Teryaev, I. R. Gabdrakhmanov, J. Phys. Conf. Ser.678, no.1, 012018 (2016)

    work page 2013

  3. [2]

    N. N. Bogolyubov and D. V. Shirkov, Intersci. Monogr. Phys. Astron.3, 1-720 (1959)

    work page 1959

  4. [3]

    A. V. Kotikov and I. A. Zemlyakov, Phys. Rev. D107, no.9, 094034 (2023)

    work page 2023

  5. [4]

    Baikov, K.G

    P.A. Baikov, K.G. Chetyrkin, J.H. K¨ uhn, Phys. Rev. Lett.118, no.8, 082002 (2017)

    work page 2017

  6. [5]

    Shirkov and I.L

    D.V. Shirkov and I.L. Solovtsov, Phys. Rev. Lett.79, 1209-1212 (1997); D.V. Shirkov, Theor. Math. Phys.127, 409-423 (2001); Eur. Phys. J. C22, 331-340 (2001)

    work page 1997

  7. [6]

    K. A. Milton, I. L. Solovtsov and O. P. Solovtsova, Phys. Lett. B415, 104-110 (1997)

    work page 1997

  8. [7]

    A. P. Bakulev, S. V. Mikhailov and N. G. Stefanis, Phys. Rev. D72, 074014 (2005)

    work page 2005

Show all 31 references
  1. [8]

    A. P. Bakulev, S. V. Mikhailov and N. G. Stefanis, Phys. Rev. D75, 056005 (2007); JHEP06, 085 (2010)

    2007

  2. [9]

    Cvetic and C

    G. Cvetic and C. Valenzuela, Braz. J. Phys.38, 371-380 (2008)

    2008

  3. [10]

    N. N. Bogolyubov, A. A. Logunov and D. V. Shirkov, Sov.Phys.JETP 10 (1960) 3, 574-581; P. J. Redmond, Phys. Rev.112, 1404 (1958)

    1960

  4. [11]

    A. P. Bakulev, Phys. Part. Nucl.40, 715-756 (2009); N. G. Stefanis, Phys. Part. Nucl.44, 494-509 (2013)

    2009

  5. [12]

    A. V. Nesterenko, Int. J. Mod. Phys. A18, 5475-5520 (2003); A. V. Nesterenko and J. Papavassiliou, Phys. Rev. D71, 016009 (2005) 6

    2003

  6. [13]

    Zyla et al., Review of Particle Physics, PTEP 2020, 083C01 (2020)

    Particle Data Group collaboration, P.A. Zyla et al., Review of Particle Physics, PTEP 2020, 083C01 (2020)

    2020

  7. [14]

    K. G. Chetyrkin, J. H. Kuhn and C. Sturm, Nucl. Phys. B744, 121-135 (2006); Y. Schroder and M. Steinhauser, JHEP01, 051; B. A. Kniehl, A. V. Kotikov, A. I. Onishchenko and O. L. Veretin, Phys. Rev. Lett.97, 042001 (2006)

    2006

  8. [15]

    d’Enterriaet al., J

    D. d’Enterriaet al., J. Phys. G51, no.9, 090501 (2024)

    2024

  9. [16]

    Cvetic, C

    G. Cvetic, C. Valenzuela, J. Phys. G32, L27 (2006); Phys. Rev. D74, 114030 (2006)

    2006

  10. [17]

    A. V. Kotikov and I. A. Zemlyakov, JETP Lett.115, no.10, 565–56 (2022)

    2022

  11. [18]

    Cvetic, R

    G. Cvetic, R. Kogerler and C. Valenzuela, Phys. Rev. D82, 114004 (2010)

    2010

  12. [19]

    Cvetiˇ c and A

    G. Cvetiˇ c and A. V. Kotikov, J. Phys. G39, 065005 (2012)

    2012

  13. [20]

    A. V. Efremov and A. V. Radyushkin, Phys. Lett. B94, 245 (1980); G. P. Lepage and S. J. Brodsky, Phys. Lett. B87, 359 (1979); Phys. Rev. D22, 2157 (1980)

    1980

  14. [21]

    V. M. Braunet al., Phys. Rev. D104, 094007 (2021); J. Gaoet al., Phys. Rev. Lett. 128, 6 (2022)

    2021

  15. [22]

    H. Zhou, J. Yan, Q. Yu and X. G. Wu, Phys. Rev. D108, no.7, 074020 (2023)

    2023

  16. [23]

    Zhonget al., Phys

    T. Zhonget al., Phys. Rev. D104, 016021 (2021)

    2021

  17. [24]

    I. R. Gabdrakhmanovet al., JETP Lett.118, no. 7, 478-482 (2023); Int. J. Mod. Phys. A40, no.04, 2450175 (2025)

    2023

  18. [25]

    I. R. Gabdrakhmanovet al., Phys. Rev. D113, no.1, 014041 (2026)

    2026

  19. [26]

    Ayalaet al., Int

    C. Ayalaet al., Int. J. Mod. Phys. A33, no.18n19, 1850112 (2018); Eur. Phys. J. C 78, no.12, 1002 (2018)

    2018

  20. [28]

    S. V. Mikhailov, A. V. Pimikov and N. G. Stefanis, Phys. Rev. D103, no.9, 096003 (2021); EPJ Web Conf.258, 03003 (2022)

    2021

  21. [29]

    I. A. Zemlyakov, I. L. Chuev and A. V. Kotikov, [arXiv:2608.00564 [hep-ph]]

    arXiv

  22. [30]

    I. I. Balitsky, V. M. Braun and A. V. Kolesnichenko, Nucl. Phys. B312, 509-550 (1989); A. Khodjamirian, Eur. Phys. J. C6, 477-484 (1999)

    1989

  23. [31]

    Ayalaet al., Phys

    C. Ayalaet al., Phys. Rev. D98, no.9, 096017 (2018); EPJ Web Conf.222, 03017 (2019) 7

    2018

Pith tools

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