The ${\gamma^* \gamma^* \to \eta_c (1S,2S)}$ transition form factors for spacelike photons

Antoni Szczurek; Izabela Babiarz; Roman Pasechnik; Victor P. Goncalves; Wolfgang Sch\"afer

REVIEW 3 major objections 5 minor 66 references

The ${\gamma^* \gamma^* \to \eta_c (1S,2S)}$ transition form factors for spacelike photons

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper claims that the double-virtual $\eta_c$ transition form factor does not factorize into single-virtual pieces; at large $Q^2$ it depends only on the average virtuality, with the breaking ratio reaching about 2.2.

desk verdict The double-virtual eta_c form factor is genuinely new and the derivation is clean, but the headline factorization-breaking measure is under-quantified across potentials and the omega-independence is at least partly built into the narrow nonrelativistic LFWF. read the letter →

arxiv 1908.07802 v1 pith:RSJQ3M2W submitted 2019-08-21 hep-ph

Izabela Babiarz , Victor P. Goncalves , Roman Pasechnik , Wolfgang Schäfer , Antoni Szczurek This is my paper · ORCID

classification hep-ph PACS 12.38.Bx13.85.Ni14.40.Pq

keywords transitionformfactoreta_cmesonlight-frontwavefunctionfactorizationbreakingcharmoniumpotentialsdouble-tagmeasurementphotonfusionSchrödingerequation

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 tries to establish that the double-virtual transition form factor $F(Q_1^2,Q_2^2)$ for $\gamma^*\gamma^*\to\eta_c(1S),\eta_c(2S)$ is not the product of two single-virtual form factors, and that at large photon virtualities it depends mainly on the average virtuality $(Q_1^2+Q_2^2)/2$. Using light-front wave functions built from five different charm-anticharm potentials, the authors define a factorization-breaking ratio $R=\tilde F(Q_1^2,Q_2^2)/[\tilde F(Q_1^2,0)\tilde F(0,Q_2^2)]$ and find that it grows to about 2.2 for the ground state while remaining almost the same across all five potentials. They also find that the onset of the asymptotic Brodsky-Lepage behavior is strongly delayed, with $Q^2F(Q^2)$ flattening below the asymptotic value for virtualities up to 50 GeV$^2$. This matters because double-tag measurements in high-luminosity $e^+e^-$ collisions can directly test factorization, and the same amplitude, up to color factors, enters $g^*g^*\to\eta_c$ production at hadron colliders.

What carries the argument

The load-bearing object is the light-front radial wave function $\psi(z,k)$ obtained from the Terentev mapping. In practice, the rest-frame momentum $p$ of the quark is related to the light-front variables by $\mathbf p^2=\frac14(M_{c\bar c}^2-4m_c^2)$ and $p_z=(z-\frac12)M_{c\bar c}$, with $\psi(z,k)=\frac{\pi}{\sqrt{2M_{c\bar c}}}\frac{u(p)}{p}$. This one mapping converts all five potential-model Schrödinger wave functions into the same light-front representation, and the hard-scattering integral (2.40) turns them into the form factor. The factorization-breaking measure $R$ defined in Eq. (3.3) quantifies how far the result is from the factorized form $\tilde F(Q_1^2,0)\tilde F(0,Q_2^2)$.

What would settle it

Measure the double-tag $e^+e^-\to e^+e^-\eta_c$ form factor at fixed average virtuality $(Q_1^2+Q_2^2)/2$ while varying the asymmetry $(Q_1^2-Q_2^2)/(Q_1^2+Q_2^2)$; the paper predicts almost no change, whereas a factorized vector-meson-dominance model predicts a visible dependence. Alternatively, recompute the same five potentials with a fully covariant light-front wave function instead of the Terentev mapping; if the ratio $R$ changes substantially, the model-independence claim would fail.

Watch

Extended reading notes

Core claim

The paper claims that for two spacelike photons, the transition form factor $F(Q_1^2,Q_2^2)$ of $\eta_c(1S)$ and $\eta_c(2S)$ is strongly factorization-breaking, with the ratio $R=\tilde F(Q_1^2,Q_2^2)/[\tilde F(Q_1^2,0)\tilde F(0,Q_2^2)]$ rising to about 2.2 for the ground state and being nearly identical for five potential-model wave functions. The explicit light-front representation is obtained by convoluting a hard $\gamma^*\gamma^*\to c\bar c$ amplitude with a wave function $\psi(z,k)$, \[ F($Q_1^{2}$,$Q_2^{2}$)=$e_c^{2}$\sqrt{N_c}\,4m_c\int\frac{dz\,dk}{z(1-z)8\$pi^{2}$}\psi(z,k)\left\{\frac{1-z}{\sqrt{($k^{2}$-$m_c^{2}$-z(1-z)$Q_1^{2}$-(1-z)^$2Q_2^{2}$)^2+$4k^{2}$($m_c^{2}$+z(1-z)$Q_1^{2}$)}}+\frac{z}{\sqrt{($k^{2}$-$m_c^{2}$-z(1-z)$Q_1^{2}$-$z^{2}$$Q_2^{2}$)^2+$4k^{2}$($m_c^{2}$+z(1-z)$Q_1^{2}$)}}\right\}, \] where $\psi(z,k)$ comes from the Terentev mapping of the rest-frame Schrödinger wave function. The normalized form factor is almost independent of the asymmetry parameter $(Q_1^2-Q_2^2)/(Q_1^2+Q_2^2)$, so $F$ scales with $(Q_1^2+Q_2^2)/2$. The single-virtual limit reproduces the shape of existing data, while $Q^2F(Q^2)$ does not approach the Brodsky-Lepage value $\frac{8}{3}f_{\eta_c}$; the paper argues that QCD evolution of the distribution amplitude is negligible below 100 GeV$^2$.

Load-bearing premise

The load-bearing premise is that the Terentev mapping from the nonrelativistic rest-frame wave function to the light-front wave function is accurate enough for charmonium, whose internal quark momentum is only moderately nonrelativistic, so the predicted size and shape of the factorization breaking depend on that mapping.

Editorial extensions

If this is right

For $\eta_c(1S)$, the factorization-breaking ratio $R$ rises to about 2.2 at large $Q_1^2,Q_2^2$, so a product of single-virtual form factors underestimates the double-virtual form factor by more than a factor of two.
The form factor is nearly determined by the average virtuality; changing the asymmetry at fixed average moves $F$ very little, in contrast to factorized vector-meson-dominance models.
The Brodsky-Lepage asymptotic plateau $Q^2F(Q^2)\to \frac{8}{3}f_{\eta_c}$ is not reached below $Q^2\approx 50$ GeV$^2$, and QCD evolution of the distribution amplitude changes the result negligibly below 100 GeV$^2$.
For $\eta_c(2S)$, the factorization breaking is weaker, reaching about 1.8 in the explored virtuality range, and the approach to asymptotics is even slower.
Because the $\gamma^*\gamma^*\to\eta_c$ amplitude is the same as $g^*g^*\to\eta_c$ up to color factors, future double-tag measurements would also constrain $\eta_c$ production at hadron colliders.

Reading between the lines

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

If the near-flat dependence on asymmetry survives, double-tag data will cleanly distinguish light-front potential models from factorized parametrizations, because the latter predict a visible dependence on asymmetry.
The similarity of $R$ across five potentials hints that the breaking is driven mainly by the hard quark propagators and the narrow peak of the wave function near $z=1/2$, rather than by the long-distance potential; varying the width of the wave function in transverse momentum would test this.
Applying the same construction to the bottomonium analogue would separate kinematic effects from relativistic corrections: a similar $R$ would point to kinematics, while a smaller $R$ would expose the relativistic corrections already present in charmonium.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

What you should know: this paper derives the double-virtual gamma*gamma* -> eta_c(1S,2S) transition form factor in a light-front quark model, which is a real extension of the single-virtual LFQM results in Refs. [17,18]. It also introduces a factorization-breaking ratio R and argues that factorization is strongly broken while the form factor scales predominantly with Qbar^2 and is nearly independent of the asymmetry omega. That is a concrete, forward-looking prediction Belle II could test.

The derivation itself is solid. I traced Eqs. (2.38) through (2.40); the azimuthal integration is consistent, and the q2=0 limit reduces to the single-virtual result. The normalization at Q^2=0 reproduces the known NRQCD width formula, which is a good check. The paper uses five published potential-model wave functions and fits nothing to the BaBar TFF data, so the circularity burden is genuinely low, and the single-virtual comparison to BaBar is honest about model spread. Credit where due: this is a useful new observable and a cleanly presented formalism.

The soft spots are real but not fatal. First and most important: the "almost model independent" claim is asserted, not shown. The text says the R pattern "looks very similar for different potentials (not shown explicitly here)", but only the Buchmueller-Tye potential is plotted. That matters because all five wave functions are sharply peaked at z = 1/2 by construction: the Terentev mapping takes nonrelativistic Schrodinger wave functions and forces a narrow z-distribution. Once z is pinned near 1/2, Eq. (2.40) almost automatically makes the denominators depend on Q1^2 + Q2^2, so the Qbar^2 scaling and weak omega dependence are at least partly kinematically enforced. The five-potential scan does not address this; it only varies the radial function under the same narrow-z assumption. A referee should ask for R for all potentials, with the spread quantified, and ideally a test with a broader-z model or a relativistic correction.

Second, the remark that modifying mc to 1.3 GeV "leads to a much better agreement" with BaBar is ad hoc. It is not a fit, but it reads like post-hoc tuning. This should either be justified or dropped. Minor point: the discussion of the delayed BL asymptotics is fine, but the claim that evolution is irrelevant is a statement about these wave functions, not a general theorem.

Who this is for: hadron phenomenologists working on charmonium form factors, double-tag measurements at Belle II, and hadronic light-by-light. The central derivation is worth refereeing seriously, even though the model-independence claim needs work. I would send it to peer review with a request to quantify the R spread and address the z-width sensitivity.

Referee Report

3 major / 5 minor

Summary. The paper derives a light-front wave function (LFWF) representation of the gamma* gamma* -> eta_c(1S,2S) transition form factor F(Q1^2,Q2^2) for two spacelike photons. The LFWFs are obtained by solving the Schroedinger equation for five different c-cbar potentials and converting the rest-frame momentum-space wave functions to light front via the Terentev prescription. The authors compare the single-virtual limit to BaBar data for eta_c(1S), discuss the delayed onset of the Brodsky-Lepage asymptotic behavior, present two-dimensional distributions of F(Q1^2,Q2^2), and introduce a factorization-breaking ratio R(Q1^2,Q2^2). The central claims are that factorization is strongly broken, that R is almost model independent across the five potentials, and that the form factor scales approximately with Qbar^2 = (Q1^2+Q2^2)/2, with almost no dependence on the asymmetry parameter omega.

Significance. If the advertised model independence holds, the paper provides a concrete, falsifiable prediction for double-tag measurements at Belle II: the gamma* gamma* -> eta_c form factor depends mainly on the average virtuality Qbar^2 and not on the virtuality asymmetry. The derivation from Eq. (2.38) to Eq. (2.40) is transparent, the normalization is checked against the two-photon decay width, and the comparison with BaBar data is a genuine forward calculation with no parameters fitted to the transition form factor itself. The main weakness is that the headline claims of model independence and Qbar^2 scaling rest on a shared nonrelativistic LFWF construction and are not quantitatively demonstrated across the potentials.

major comments (3)

[Section III.B, Fig. 9] The abstract's claim that factorization breaking is "almost model independent" is not supported by the presented evidence. Only the Buchmueller-Tye potential is plotted in Fig. 9, and the text states "The factorization breaking pattern looks very similar for different potentials (not shown explicitly here)" without quantification. Since this is a central claim, please show R(Q1^2,Q2^2), or a projection such as R as a function of Qbar^2 at fixed omega, for all five potentials, and quantify the spread across potentials. Without this, the headline claim is an assertion rather than a demonstrated result.
[Section II.A, Eqs. (2.12)-(2.20), and Eq. (2.40)] The Terentev prescription is stated to be valid for weakly bound nonrelativistic systems, but charmonium is only moderately nonrelativistic, with typical p/m_c around 0.3-0.5. Because all five potentials are converted through the same nonrelativistic mapping, the resulting LFWFs are all sharply peaked at z approximately 1/2. Inserting z approximately 1/2 into Eq. (2.40) makes the denominators depend mainly on Q1^2+Q2^2, so the claimed Qbar^2 scaling and near-omega-independence are largely enforced by the construction rather than emerging independently from the dynamics. The five-potential scan cannot expose this limitation because it varies only the shape of u(p) within the same ansatz. Please add a quantitative discussion of the sensitivity of R(Q1^2,Q2^2) and of the omega-dependence to a broader z-distribution, for example by comparing with a relativistic treatment or by varying m_c over a wider range, and qualify the model-independence claim accordingly.
[Section III.B, Eq. (3.3)] The factorization-breaking ratio R is defined using the single-virtual form factors F(Q1^2,0) and F(0,Q2^2) computed in the same model. While this is a sensible normalization, it means that model dependence in the single-virtual form factor largely cancels in R. The statement that R is almost model independent should therefore be accompanied by a statement about how much of the model dependence is removed by this normalization, and the residual spread across potentials should be shown explicitly. Otherwise the reader cannot distinguish a genuine dynamical prediction from a consequence of the ratio construction.

minor comments (5)

[Section II.C, Eqs. (2.38) and (2.40)] The notation switches between Q_i^2 = -q_i^2 at the beginning of Section II and q1^2, q2^2 in the final form factor expressions. Please state explicitly that q_i^2 in Eqs. (2.38) and (2.40) denotes the spacelike virtuality Q_i^2, or introduce a consistent sign convention, to avoid confusion in the denominators.
[Section III.B] The sentence "The factorization breaking pattern looks very similar for different potentials (not shown explicitly here)" should either be replaced by a quantitative statement with a figure or removed, since it is precisely the evidence needed for the main claim.
[Fig. 5 caption] The phrase "rate of approaching" should be "rate of approach" or "approach to."
[References] Reference [14] is written as "Phys. Rev. D97, 0094034 (2018)"; the article number appears to be a typo and should read 094034.
[Section III.B, paragraph before Fig. 7] The sentence "The reader is asked to notice much broader range of Q^2 in the figure compared to that in previous figures" is informal and should be rephrased.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the transition form factor and factorization-breaking ratio are derived from Schrödinger wave functions with published potentials; the only self-citation (Ref. [20]) supplies independent, reproducible input.

full rationale

The paper's central results—the double-virtual transition form factor F(Q1^2,Q2^2), the weak dependence on the asymmetry parameter ω, the Qbar^2 scaling, and the factorization-breaking measure R—are computed by inserting light-front wave functions, obtained from rest-frame Schrödinger wave functions for five published c-cbar potentials, into a standard hard-scattering convolution formula (Eq. (2.38)/(2.40)). No parameter is fitted to the BaBar data; the BaBar comparison is a check, not an input. The factorization-breaking ratio R is constructed directly from the computed F values, so its behavior is an output, not a fitted quantity. The near independence of Qbar^2 and the weak ω-dependence arise because the nonrelativistic wave functions are sharply peaked at z=1/2, but that is model dynamics rather than circularity: the full integral in Eq. (2.40) still contains nontrivial z and k dependence, and the extent of ω-independence is a falsifiable prediction. The only self-citation is Ref. [20], by a co-author, used for the Schrödinger-solution wave functions; however, those are obtained from established external potentials and a standard differential equation, not from a conclusion equivalent to the target form factor. The paper explicitly notes discrepancies with the measured two-photon width and the need for a modified quark mass for better BaBar agreement, which further indicates the calculation is not rigged to reproduce data. Concerns about the Terentev nonrelativistic mapping or the unquantified 'almost model independent' claim are correctness or robustness issues, not circularity.

Assumptions & free parameters 10 free parameters · 5 assumptions · 0 invented entities

The model inputs are the five potential-model wave functions and their parameters, all taken from prior literature; the TFF calculation itself introduces no new free parameters. The key physical assumptions are the Terentev mapping, valence Fock-state dominance, and the use of perturbative hard-scattering amplitudes down to Q^2=0. No new particles, forces, or other invented entities are introduced.

free parameters (10)

charm quark mass, harmonic oscillator potential = 1.4 GeV
Input from potential model literature; sets wave function width and hard-scattering scale.
charm quark mass, Cornell potential = 1.84 GeV
Input from Cornell potential literature; used in the hard matrix element.
charm quark mass, logarithmic potential = 1.5 GeV
Input from Quigg-Rosner potential model literature.
charm quark mass, power-law potential = 1.334 GeV
Input from Martin power-law potential literature via Ref. [36].
charm quark mass, Buchmueller-Tye potential = 1.48 GeV
Input from Buchmueller-Tye potential literature.
harmonic oscillator frequency omega = 0.3 GeV = (M2S - M1S)/2
Chosen to reproduce charmonium level spacing; not fitted to TFF data.
Cornell potential parameters (k, a) = k=0.52, a=2.34 GeV^-1
Standard Cornell potential parameters from Refs. [31,32].
Logarithmic potential parameters = V = -0.6635 GeV + (0.733 GeV) log(r * 1 GeV)
Parameters from Quigg and Rosner, Ref. [33].
Power-law potential parameters = V = -6.41 GeV + (6.08 GeV)(r * 1 GeV)^0.106
Parameters from Martin, Refs. [34,35].
Buchmueller-Tye potential parameters = k=0.153 GeV^2, lambda=0.406 GeV, Lambda_MS=0.509 GeV
Parameters from Buchmueller and Tye, Ref. [37].

assumptions (5)

domain assumption Terentev prescription maps rest-frame Schroedinger wave functions to light-front wave functions.
Section II.A, Eqs. (2.12) to (2.20); valid for weakly bound nonrelativistic systems, which is only approximate for charmonium.
domain assumption Pure c-cbar valence Fock state dominance for eta_c.
Eq. (2.8); higher Fock states are neglected, supported by nonrelativistic potential model phenomenology and Ref. [17].
domain assumption Perturbative hard-scattering formula applies even at vanishing photon virtualities.
Section II.C: 'the charm quark mass mc by itself is large enough to justify perturbation theory and apply our results even in the limit of vanishing photon virtualities.'
domain assumption Schroedinger equation with phenomenological c-cbar potentials gives the charmonium wave function.
Section II.B; this is the standard potential-model approach, but it is model dependent.
domain assumption Bose symmetry of the transition form factor under Q1^2 and Q2^2 exchange.
Imposed as a physical requirement; the representation in Eq. (2.38) is not manifestly symmetric and the symmetry is checked numerically, not proven.

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Cite this review

Pith. "Pith review of The ${\gamma^* \gamma^* \to \eta_c (1S,2S)}$ transition form factors for spacelike photons." pith.science (2026). https://pith.science/paper/RSJQ3M2W

@misc{pith2026190807802,
  author       = {Pith},
  title        = {Pith review of: The $\gamma^* \gamma^* \to \eta_c (1S,2S)$ transition form factors for spacelike photons},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RSJQ3M2W}},
  note         = {Machine review of arXiv:1908.07802}
}

abstract

We derive the light-front wave function (LFWF) representation of the $\gamma^* \gamma^* \to \eta_c(1S)\,,\eta_c(2S)$ transition form factor $F(Q_1^2, Q_2^2)$ for two virtual photons in the initial state. For the LFWF, we use different models obtained from the solution of the Schr\"odinger equation for a variety of $c \bar c$ potentials. We compare our results to the BaBar experimental data for the $\eta_c(1S)$ transition form factor, for one real and one virtual photon. We observe that the onset of the asymptotic behaviour is strongly delayed and discuss applicability of the collinear and/or massless limit. We present some examples of two-dimensional distributions for $F (Q_1^2,Q_2^2)$. A factorization breaking measure is proposed and factorization breaking effects are quantified and shown to be almost model independent. Factorization is shown to be strongly broken, and a scaling of the form factor as a function of $\bar Q^2 = (Q_1^2 + Q_2^2)/2$ is obtained.

Figures

Figures reproduced from arXiv: 1908.07802 by the authors.

**Figure 2.** FIG. 2. Momentum space wave function [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗

**Figure 3.** FIG. 3. The light-front radial wave function [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗

Figures from the paper (6 more)

**Figure 4.** Figure 4: FIG. 4. The dependence of the normalized transition form fac [PITH_FULL_IMAGE:figures/full_fig_p014_4.png]

**Figure 5.** Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p015_5.png]

**Figure 6.** Figure 6: FIG. 6. Distribution amplitudes for different wave functions [PITH_FULL_IMAGE:figures/full_fig_p016_6.png]

**Figure 7.** Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p018_7.png]

**Figure 8.** Figure 8: FIG. 8. The [PITH_FULL_IMAGE:figures/full_fig_p019_8.png]

**Figure 9.** Figure 9: FIG. 9. The deviations from the factorization breaking ( [PITH_FULL_IMAGE:figures/full_fig_p019_9.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

66 extracted references · 50 canonical work pages

[1]

V. L. Chernyak and S. I. Eidelman, Prog. Part. Nucl. Phys. 80, 1 (2014) [arXiv:1409.3348 [hep-ph]]

arXiv 2014
[2]

For the LFWF, w e use diﬀerent models obtained from the solution of the Schr¨ od inger equation for a variety of c¯c potentials

for two virtual photons in the initial state. For the LFWF, w e use diﬀerent models obtained from the solution of the Schr¨ od inger equation for a variety of c¯c potentials. We compare our results to the BaBar experimenta l data for the ηc(1S) transition form factor, for one real and one virtual photon. We observe t hat the onset of the asymptotic behavi...

work page
[3]

It is normalized such that the two-photon decay-width of the meson is o btained from Γ(ηc →γγ ) = π 4α2 emM 3 ηc |F (0, 0)|2

above is the object of interest in this paper. It is normalized such that the two-photon decay-width of the meson is o btained from Γ(ηc →γγ ) = π 4α2 emM 3 ηc |F (0, 0)|2. (2.2) For further calculation it is useful to choose a frame in which incoming photon four-momenta have the form q1 =q+ 1n+ +q1⊥,q 2 =q− 2n− +q2⊥. (2.3) Here n± = 1√ 2 (1, 0, 0, ±1), (...

work page
[4]

The normalizing function N is N (z, k) = (∑ λ ¯λ Γ λ ¯λ (z, k)Γ ∗ λ ¯λ (z, k) ) 1/ 2 = √ 2 √ k2 +m2 c z(1 −z) = √ 2Mc¯c. (2.16) Then, if we take the meson state to obey the canonical relativistic n ormalization ⟨ηc;P ′ +, P ′|ηc;P+, P ⟩ = 2P+(2π)3δ(P ′ + −P+)δ(2)(P ′ − P ), (2.17) the radial light-front wave function φ(z, k) will be normalized as ∫ 1 0 dz...

work page
[5]

(2.38) and then to perform the integration over the azimuthal angle of k

= e2 c √ Nc 4mc · ∫ dzd2k z(1 −z)16π3ψ(z, k) { 1 −z (k − (1 −z)q2)2 +z(1 −z)q2 1 +m2 c + z (k +zq2)2 +z(1 −z)q2 1 +m2 c } . (2.38) and then to perform the integration over the azimuthal angle of k. Using ∫ 2π 0 dφ 2π 1 A +B cosφ = 1√ A2 −B2, (2.39) 2 We have dropped the terms ∝ Ψ ∗ +− + Ψ ∗ − +, which vanish for the pseudoscalar state, see Eq. (2.15). 8 w...

work page
[6]

(2.40) This form puts into evidence, that the invariant form factor F (Q2 1,Q 2

= e2 c √ Nc 4mc · ∫ dzkdk z(1 −z)8π2ψ(z, k) { 1 −z√ (k2 −m2 c −z(1 −z)q2 1 − (1 −z)2q2 2)2 + 4k2(m2 c +z(1 −z)q2 1) + z √ (k2 −m2 c −z(1 −z)q2 1 −z2q2 2)2 + 4k2(m2 c +z(1 −z)q2 1) } . (2.40) This form puts into evidence, that the invariant form factor F (Q2 1,Q 2

work page
[7]

Notice that by the Bose-symmetry, the form factor must be a symmetric function of Q2 1,Q 2

is a function of Q2 1 = q2 1 and Q2 2 = q2 2 only. Notice that by the Bose-symmetry, the form factor must be a symmetric function of Q2 1,Q 2

work page
[8]

(2.38) or Eq

This is evidently not obvious from the representations Eq. (2.38) or Eq. (2.40), as the integrand is manifestly asymmetric in q1, q2. However, as will be demonstrated below by the numerical results, our represen tation has the required symmetry. In particular, in the limit of one on-shell photon, one mus t have, that F (Q2) ≡ F (Q2, 0) = F (0,Q 2), and th...

work page

Show all 66 references

[9]

(2.46) Still another interesting limit exists, namely at very large Q2 i , the k-smearing becomes unimportant, and one can neglect k in the hard matrix element of Eq

= e2 c √ Nc 4√ πMηc 1 Q2 1 +Q2 2 +M 2 ηc R(0). (2.46) Still another interesting limit exists, namely at very large Q2 i , the k-smearing becomes unimportant, and one can neglect k in the hard matrix element of Eq. (2.38). Then only the LFWF appears under the k integral, and th...
[10]

(2.47) The DA is conveniently normalized as ∫ 1 0 dzϕ (z,µ 2

= 1 z(1 −z) √Nc 4mc 16π3 ∫ d2kθ(µ2 0 − k2)ψ(z, k). (2.47) The DA is conveniently normalized as ∫ 1 0 dzϕ (z,µ 2
[11]

= 1, (2.48) so that we can extract the so-called decay constant fηc from the integral over z in Eq. (2.47). The transition form factor simpliﬁes to F (Q2 1,Q 2
[12]

(2.49) This representation is valid in the limit of large photon virtualities Q2 1,Q 2 2

= e2 cfηc · ∫ 1 0 dz { (1 −z)ϕ(z,µ 2 0) (1 −z)2Q2 1 +z(1 −z)Q2 2 +m2 c + zϕ (z,µ 2 0) z2Q2 1 +z(1 −z)Q2 2 +m2 c } . (2.49) This representation is valid in the limit of large photon virtualities Q2 1,Q 2 2. III. NUMERICAL RESUL TS In this Section we will present our results for...
[13]

A popular model for the transition form factor is based on the vector meson dominance app roach (see e.g

= F (Q2 1,Q 2 2) F (0, 0) , (3.1) 10 which nicely quantiﬁes the deviation from point-like coupling. A popular model for the transition form factor is based on the vector meson dominance app roach (see e.g. Ref. [21]), and reads ˜F (Q2 1,Q 2
[14]

(3.2) It features a factorized dependence on the photon virtualities, w hich we expect to be broken

= M 2 J/ Ψ Q2 1 +M 2 J/ Ψ · M 2 J/ Ψ Q2 2 +M 2 J/ Ψ . (3.2) It features a factorized dependence on the photon virtualities, w hich we expect to be broken. In our analysis, we will quantify the factorization breaking of the tr ansition form factor by estimating the quantity deﬁ...
[15]

= ˜F (Q2 1,Q 2 2) ˜F (Q2 1, 0) ˜F (0,Q 2 2) . (3.3) A. c¯c wave functions of ηc(1S) and ηc(2S) Our wave functions u(p) were obtained by Fourier transform from the r-dependent c¯c wave functions obtained as a solution of the Schr¨ odinger equation with diﬀerent, realistic, pote...

1970
[16]

To investigate the scaling properties, we show the transition form factor as a function of the variables ω = Q2 1 −Q2 2 Q2 1 +Q2 2 and ¯Q2 = Q2 1 +Q2 2 2

As an example in Fig.8 we again show our results for the Buchm¨ uller-Tye potential. To investigate the scaling properties, we show the transition form factor as a function of the variables ω = Q2 1 −Q2 2 Q2 1 +Q2 2 and ¯Q2 = Q2 1 +Q2 2 2 . (3.8) One can see that F (and ~F ) i...
[17]

The factorization breaking pattern looks very similar for diﬀerent potentials (not sho wn explicitly here)

= R(Q2 1, 0) = 1. The factorization breaking pattern looks very similar for diﬀerent potentials (not sho wn explicitly here). 17 )2 (GeV2Q 0 50 100 150 200 250 300 350 400 ,0) (GeV)2F(Q2Q 0 0.2 0.4 0.6 0.8 1 1.2 (1S)cηBuchmuller-Tye, via LFWF via DA, no evolution via DA, evolu...
[18]

and (ω, ¯Q2) for the Buchm¨ uller-Tye potential for illustration, other potentials discussed in the present paper behave similarly. )2 (GeV 1 2 Q 0 10 20 30 40 50 ) 2(GeV2 2Q 01020304050 1 1.5 2 2.5 (1S)cη), 2 2,Q2 1R(Q )2 (GeV 12 Q 0 10 20 30 40 50 )2(GeV2 2Q 01020304050 1 1....
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Theory of hot matter and relativistic heavy-ion co llisions

for Buchm¨ uller- Tye potential; left panel - ηc(1S), right panel - ηc(2S)). 19 IV. CONCLUSIONS The description of transition form factors is directly related to our understanding of the structure of bound states in QCD. In the present paper we have s tudied the transition for...

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Pith tools

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

[1] [1]

V. L. Chernyak and S. I. Eidelman, Prog. Part. Nucl. Phys. 80, 1 (2014) [arXiv:1409.3348 [hep-ph]]

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For the LFWF, w e use diﬀerent models obtained from the solution of the Schr¨ od inger equation for a variety of c¯c potentials

for two virtual photons in the initial state. For the LFWF, w e use diﬀerent models obtained from the solution of the Schr¨ od inger equation for a variety of c¯c potentials. We compare our results to the BaBar experimenta l data for the ηc(1S) transition form factor, for one real and one virtual photon. We observe t hat the onset of the asymptotic behavi...

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[3] [3]

It is normalized such that the two-photon decay-width of the meson is o btained from Γ(ηc →γγ ) = π 4α2 emM 3 ηc |F (0, 0)|2

above is the object of interest in this paper. It is normalized such that the two-photon decay-width of the meson is o btained from Γ(ηc →γγ ) = π 4α2 emM 3 ηc |F (0, 0)|2. (2.2) For further calculation it is useful to choose a frame in which incoming photon four-momenta have the form q1 =q+ 1n+ +q1⊥,q 2 =q− 2n− +q2⊥. (2.3) Here n± = 1√ 2 (1, 0, 0, ±1), (...

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[4] [4]

The normalizing function N is N (z, k) = (∑ λ ¯λ Γ λ ¯λ (z, k)Γ ∗ λ ¯λ (z, k) ) 1/ 2 = √ 2 √ k2 +m2 c z(1 −z) = √ 2Mc¯c. (2.16) Then, if we take the meson state to obey the canonical relativistic n ormalization ⟨ηc;P ′ +, P ′|ηc;P+, P ⟩ = 2P+(2π)3δ(P ′ + −P+)δ(2)(P ′ − P ), (2.17) the radial light-front wave function φ(z, k) will be normalized as ∫ 1 0 dz...

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[5] [5]

(2.38) and then to perform the integration over the azimuthal angle of k

= e2 c √ Nc 4mc · ∫ dzd2k z(1 −z)16π3ψ(z, k) { 1 −z (k − (1 −z)q2)2 +z(1 −z)q2 1 +m2 c + z (k +zq2)2 +z(1 −z)q2 1 +m2 c } . (2.38) and then to perform the integration over the azimuthal angle of k. Using ∫ 2π 0 dφ 2π 1 A +B cosφ = 1√ A2 −B2, (2.39) 2 We have dropped the terms ∝ Ψ ∗ +− + Ψ ∗ − +, which vanish for the pseudoscalar state, see Eq. (2.15). 8 w...

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[6] [6]

(2.40) This form puts into evidence, that the invariant form factor F (Q2 1,Q 2

= e2 c √ Nc 4mc · ∫ dzkdk z(1 −z)8π2ψ(z, k) { 1 −z√ (k2 −m2 c −z(1 −z)q2 1 − (1 −z)2q2 2)2 + 4k2(m2 c +z(1 −z)q2 1) + z √ (k2 −m2 c −z(1 −z)q2 1 −z2q2 2)2 + 4k2(m2 c +z(1 −z)q2 1) } . (2.40) This form puts into evidence, that the invariant form factor F (Q2 1,Q 2

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[7] [7]

Notice that by the Bose-symmetry, the form factor must be a symmetric function of Q2 1,Q 2

is a function of Q2 1 = q2 1 and Q2 2 = q2 2 only. Notice that by the Bose-symmetry, the form factor must be a symmetric function of Q2 1,Q 2

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[8] [8]

(2.38) or Eq

This is evidently not obvious from the representations Eq. (2.38) or Eq. (2.40), as the integrand is manifestly asymmetric in q1, q2. However, as will be demonstrated below by the numerical results, our represen tation has the required symmetry. In particular, in the limit of one on-shell photon, one mus t have, that F (Q2) ≡ F (Q2, 0) = F (0,Q 2), and th...

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