Photon Distribution Amplitudes in QCD

N. Kivel; P. Ball; V.M. Braun

REVIEW

Photon Distribution Amplitudes in QCD

P. Ball, V.M. Braun, N. Kivel

High Energy Physics - Phenomenology · Submitted 25 Jul 2002 · 37 references

Pith's verdict: A solid reference paper for photon DAs through twist-4, with a genuine correction to the earlier twist-4 literature — but Table 3 does not satisfy the exact relation the paper's own consistency argument rests on.

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arXiv IDhep-ph/0207307 (version 1)

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CitationN. Kivel, P. Ball, V.M. Braun. "Photon Distribution Amplitudes in QCD." arXiv:hep-ph/0207307 (2002). https://pith.science/paper/4E7JHH53

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Editor's note

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Bottom line: this is the paper people doing B → Vγ and γ* → πγ will cite for photon distribution amplitudes, and the twist-4 correction to Ref. [13] is real. Send it out. But someone should make the authors reconcile Table 3 with their own Eq. (4.32) before it goes to print. That is a small fix, not a reason to hold the paper up, but it is exactly the kind of thing a referee should catch and the authors should want caught, since they call (4.32) 'of principal importance.' Table 3 gives ζ1 = 0.4, ζ2 = 0.3, ζ2+ = 0, so ζ1 + 11ζ2 − 2ζ2+ = 3.7, not 7/2. Either ζ1 should be about 0.2, or the numbers are rounded in a way that should be stated. The numerical impact on A(u) is a few percent and hγ is untouched, so nothing collapses. But the tabulated set is not the self-consistent one the consistency logic presupposes, and that is worth one line in a revision. The same table also carries six twist-4 parameters from Ref. [11] with no uncertainties attached, which is a presentation gap rather than an error. What is genuinely new and worth crediting: the demonstration that T4^γ must be kept nonzero at finite order in the conformal expansion, and the check that two independent operator identities, (4.41) and (4.45), reproduce the same A(u) only when it is. That is a real internal cross-check, not a restatement, and it is what justifies superseding Ref. [13]. The O(αs) radiative correction to the leading-twist sum rule is new, and the downward shift of χ from 4.4 to (3.15 ± 0.3) GeV⁻² matters for anyone who has been using the old number for fifteen years. The authors are also honest about their weak spots: the φ2 fit instability and the ω parameters quoted from Ref. [33] are both flagged as unreliable in the text, so I would not pile on there. The audience is the hard-exclusive community, and the paper is a tabulation with a consistency argument attached, not a conceptual advance. It deserves referees and it deserves to be cited.

Referee report

Significance

If the construction holds, this is a foundational reference: it supplies the first complete, internally consistent set of photon DAs through twist-4 together with explicit conformal coefficients, a usable parameter set, and a clear statement of which auxiliary amplitudes must be retained for the equations of motion to be satisfied. The internal cross-check between two independent operator identities in Sec. 4.3.3 is a genuine, checkable consistency argument rather than a restatement, and it identifies a specific error in earlier treatments (the omission of T4^gamma). The updated sum-rule value chi(1 GeV) = (3.15 +/- 0.3) GeV^-2 and the new sum rule for f3gamma are directly usable in factorization calculations of gamma* -> pi gamma, DVCS and radiative B decays, and the paper is explicit that the shape of the leading-twist DA at 1 GeV is only qualitatively known. The paper also states openly where the numerical input is approximate (App. B.1.1, B.2), which is appropriate.

Major objections (2)

[Table 3 and Eq. (4.32)] The numerical parameter set adopted for the twist-4 sector does not satisfy the exact relation on which the consistency argument of Sec. 4.3.3 rests. Eq. (4.32) states zeta_1 + 11 zeta_2 - 2 zeta_2^+ = 7/2, and the paper calls this relation 'of principal importance'; it is what makes the zeta-contributions cancel in h_gamma in Eq. (4.42) and what fixes the coefficient 8(zeta_2^+ - 3 zeta_2) of the zeta-term in A(u) in Eq. (4.43). The values collected in Table 3 (kappa = 0.2, kappa^+ = 0, zeta_1 = 0.4, zeta_2 = 0.3, zeta_1^+ = 0, zeta_2^+ = 0) give 0.4 + 11(0.3) - 0 = 3.7, not 3.5, a violation at the 6% level. The values implied by Eq. (4.30) with vanishing reduced matrix elements, zeta_1 = 0 and zeta_2 = 7/22 = 0.318, do satisfy (4.32), so the tabulated numbers are not the self-consistent set that the consistency argument presupposes. The numerical impact is modest (the zeta-term in (4.43) shifts from -7.2 to about -7.64, and h_gamma is unaffected because the cancellation there is by construction), and the algebraic classification result is not undermined. But since Table 3 is the only numerical realization of the twist-4 sector in the paper, the authors should either replace it with a set satisfying (4.32) or state explicitly that the entries are rounded/approximate and quantify the resulting violation.
[Abstract and Sec. 4.3.2 / Table 3] The abstract promises 'new estimates of parameters describing higher-twist amplitudes from QCD sum rules', but in the body only f3gamma (twist-3, Eq. (B.25)) is genuinely new, while the twist-4 parameters kappa, kappa^+, zeta_1, zeta_2, zeta_1^+, zeta_2^+ are carried over from Ref. [11] without re-derivation and, in Table 3, without any uncertainty. The omega_V_gamma and omega_A_gamma values of Eq. (4.10) are likewise quoted from Ref. [33] and the text itself says they 'have to be taken with caution' and are 'rather an upper bound'. This is a mismatch between the stated scope and the delivered content that matters for users of the parameter set. The authors should state clearly which numbers are new in this work and which are inherited, and should attach at least indicative uncertainties to the inherited twist-4 parameters so that the accuracy of the twist-4 sector is not overstated.

Minor objections (8)

[Eq. (4.39)] The sign of the second line for T4^gamma should be checked. Combining Eqs. (4.35) and (4.36) with f00 = 1, f10 = 14/3, f01 = 7/3 gives S_up_down = 120 alpha_qbar alpha_g^2 (7 alpha_g - 3) and S_down_up = 120 alpha_q alpha_g^2 (7 alpha_g - 3), hence T4^gamma = (1/2)(S_up_down - S_down_up) = 60 alpha_g^2 (alpha_qbar - alpha_q)(4 - 7(alpha_q + alpha_qbar)), whereas the printed expression carries (alpha_q - alpha_qbar). The convention 'T_i(alpha) = T_i(alpha_qbar, alpha_q)' stated after Eq. (4.26) may absorb this, but as printed the two expressions differ by an overall sign and a reader cannot tell which is intended. The corresponding S_gamma expression in the same equation checks out and integrates to unity as required by Eq. (4.37).
[Eq. (4.41)] The bracketing in the first relation is ambiguous as typeset: it is not clear whether the factor 2/alpha_g multiplies only the square bracket [alpha_q - alpha_qbar - (2u-1) alpha_g] times [S + S_gamma], or the whole curly bracket including T3 - T2. Since this identity is one of the two pillars of the consistency check in Sec. 4.3.3, the grouping should be made explicit.
[Eq. (2.19) versus Eq. (4.43)] The mass dimension of A(u) should be stated. In Eq. (2.19) the correction enters as (x^2/4) A(u) multiplied by e_q <qbar q> F_{mu nu}, which requires A to carry dimension GeV^2 for the two terms in the curly bracket to have the same dimension, whereas the explicit expressions in Eqs. (4.41) and (4.43) appear to give a dimensionless quantity. Either a normalization factor is implicit or one of the two equations is missing a scale.
[Eq. (4.29)] The parenthesis in the running of kappa is misplaced: 'L^{(gamma_- - gamma_qbarq /b)}' should read 'L^{(gamma_- - gamma_qbarq)/b}', to match the form used for kappa^+ in the preceding line.
[Eq. (4.9)] The expression for psi^(a)(u) is garbled in typesetting: '(1 - xi^2)(5 xi^2 - 1) 52 (1 + 916 omega_V^gamma - 316 omega_A^gamma)' presumably means (5/2)(1 + (9/16) omega_V^gamma - (3/16) omega_A^gamma). The same typesetting problem affects the intermediate line of Eq. (4.39), where a square bracket is opened twice.
[Eq. (4.20) and Eq. (4.39)] It would help the reader to state explicitly how many moments of the exact S_gamma = delta(alpha_q) delta(alpha_qbar) the truncated expression in Eq. (4.39) reproduces, and to comment on the accuracy of convolution integrals computed with the truncated form. The paper explains why the truncation is mandatory for consistency with the equations of motion, but the practical size of the resulting distortion is not quantified.
[Fig. 3] The vertical range of Fig. 3 spans only about 3.065 to 3.10 GeV^-2, so the plotted variation of chi with m_4 is visually exaggerated. A note in the caption stating the absolute scale of the variation, or a wider axis range, would prevent misreading.
[References] Reference [14] is incomplete: 'M. Praszalowicz and A. Rostworowski, local NJL model,' Phys. Rev. D 64 (2001) 074003' is missing the title of the paper.

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The paper

The paper claims

The paper constructs a complete, equation-of-motion-consistent set of photon distribution amplitudes through twist four and uses it to update the magnetic susceptibility of the quark condensate.

The reading

The paper asks how the hadronic, non-pointlike part of a real photon can be described inside hard exclusive reactions such as deeply virtual Compton scattering, $��^*��o��π��γ$, and radiative $B$ decays. Its answer is a catalogue of photon distribution amplitudes: light-cone functions that describe how a photon's momentum is shared between a quark and an antiquark, or among a quark, an antiquark, and a gluon, organized by twist, the counting of power suppression by the hard scale. The authors build these functions in an electromagnetic background field, expand them in conformal partial waves, and impose the QCD equations of motion so that two- and three-particle amplitudes fit together instead of being chosen independently. They also re-estimate the magnetic susceptibility of the quark condensate, the parameter that sets the overall size of the leading-twist photon amplitude, and give sum-rule estimates for higher-twist parameters. If the construction holds, hard exclusive photon processes can be calculated with a controlled set of nonperturbative inputs rather than ad hoc models.

What carries the argument

The background-field formalism treats the real photon as a classical electromagnetic field and separates hard electromagnetic emission from soft hadronic matrix elements in a gauge-invariant way. Photon distribution amplitudes are then vacuum expectation values of nonlocal quark-antiquark and quark-gluon operators in that background field. The calculation uses a conformal expansion in Gegenbauer and Jacobi polynomials, truncated at low conformal spin, together with QCD equation-of-motion identities that relate two-particle and three-particle amplitudes. The load-bearing mechanism is that the truncation must be applied consistently to all amplitudes, including the auxiliary $T_4^��gamma$, so that the exact identities survive the truncation.

Extended reading notes

Core claim

The central claim is that photon distribution amplitudes up to twist four can be organized into a complete set for both chirality-conserving and chirality-violating operators, with all exact QCD equation-of-motion relations satisfied. The nontrivial point is consistency: the auxiliary quark-antiquark-photon amplitude $T_4^��gamma$ must be kept nonzero at finite order in the conformal expansion. Setting it to zero, as earlier treatments did, breaks the equation-of-motion relation and gives incorrect twist-4 functions. The authors show that two independent equation-of-motion identities yield the same two-particle twist-4 amplitude $A(u)$ only when the full three-particle set, including $T_4^��gamma$, is used. They also update the magnetic susceptibility to $��chi(1��,��mathrm{GeV})��simeq(3.15��pm0.3)��,��mathrm{GeV}^{-2}$ and quote $f_��gamma^��perp��equiv��chi��langle��bar q q��rangle=-(50��pm15)��,��mathrm{MeV}$.

Load-bearing premise

The numerical parameter set assumes that vector-meson dominance, local-duality sum rules, a perturbative continuum above a threshold, and a truncated resonance sum give usable estimates, and the twist-4 parameters are carried over from earlier work without fresh uncertainties.

Editorial extensions

If this is right

Hard exclusive processes with a real photon, including $��gamma^*��to��pi��gamma$, deeply virtual Compton scattering, and radiative $B$ decays such as $B��to��rho��gamma$ or $B��to K^*��gamma$, can be computed with a common set of twist-2, twist-3, and twist-4 photon inputs.
The earlier twist-4 photon distribution amplitudes of Ref. [13], obtained by dropping $T_4^��gamma$ and quark-antiquark-gluon operators, are superseded, so calculations that used them should be revised.
The updated $��chi(1��,��mathrm{GeV})��simeq(3.15��pm0.3)��,��mathrm{GeV}^{-2}$ lowers the estimated size of the hadronic photon contribution relative to the older estimate of about $4.4��,��mathrm{GeV}^{-2}$.
The parameters $��chi$, $f_3^��gamma$, $��kappa$, $��kappa^+$, and the $��zeta$'s become targets for lattice calculations.
Hard exclusive dijet photoproduction can in principle measure the shape of the leading-twist photon distribution amplitude and discriminate asymptotic from endpoint-peaked forms.

Reading between the lines

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

If the leading-twist photon distribution amplitude is close to $6u(1-u)$ at $1��,��mathrm{GeV}$, then endpoint-enhanced vector-dominance intuitions about photon power corrections are misleading for deeply virtual Compton scattering and radiative $B$ decays.
The consistency lesson, that an auxiliary amplitude vanishing only in the infinite conformal sum must be kept nonzero at finite order, likely applies to other higher-twist distribution-amplitude models where such amplitudes are dropped.
A lattice determination of $��chi$ and the second Gegenbauer moment of the photon distribution amplitude would decide whether the sum-rule numerical package is robust enough for precision phenomenology.
Because the exact relation $��zeta_1+11��zeta_2-2��zeta_2^+=7/2$ ties gluon and photon contributions together, any higher-twist model built from gluon operators alone is likely to violate the equation-of-motion constraints.

Original abstract

We develop a consistent technique for the calculation of real photon emission in hard exclusive processes, which is based on the background field formalism and allows a convenient separation of hard electromagnetic and soft hadronic components of the photon. The latter ones are related to matrix-elements of light-cone operators in the electromagnetic background field and can be parametrized in terms of photon distribution amplitudes. We construct a complete set of photon distribution amplitudes up to and including twist-4, for both chirality-conserving and chirality-violating operators. The distribution amplitudes involve several nonperturbative parameters and, most importantly, the magnetic susceptibility of the quark condensate. We review and update previous estimates of the susceptibility and also give new estimates of parameters describing higher-twist amplitudes from QCD sum rules.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

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The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We develop a consistent technique for the calculation of real photon emission in hard exclusive processes, which is based on the background field formalism and allows a convenient separation of hard electromagnetic and soft hadronic components of the photon. The latter ones are related to matrix-elements of light-cone operators in the electromagnetic background field and can be parametrized in terms of photon distribution amplitudes. We construct a complete set of photon distribution amplitudes up to and including twist-4, for both chirality-conserving and chirality-violating operators. The distribution amplitudes involve several nonperturbative parameters and, most importantly, the magnetic susceptibility of the quark condensate. We review and update previous estimates of the susceptibility and also give new estimates of parameters describing higher-twist amplitudes from QCD sum rules.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We develop a consistent technique for the calculation of real photon emission in hard exclusive processes, which is based on the background field formalism and allows a convenient separation of hard electromagnetic and soft hadronic components of the photon.	method 0.62	0.94	0.25	asserted 1.00
s1	The latter ones are related to matrix-elements of light-cone operators in the electromagnetic background field and can be parametrized in terms of photon distribution amplitudes.	method 0.43	0.36	0.10	asserted 1.00
s2	We construct a complete set of photon distribution amplitudes up to and including twist-4, for both chirality-conserving and chirality-violating operators.	contribution 0.88	0.96	0.31	asserted 1.00
s3	The distribution amplitudes involve several nonperturbative parameters and, most importantly, the magnetic susceptibility of the quark condensate.	result 0.53	0.39	0.14	asserted 1.00
s4	We review and update previous estimates of the susceptibility and also give new estimates of parameters describing higher-twist amplitudes from QCD sum rules.	contribution 0.58	0.88	0.12	asserted 0.99

What kind of paper the classifier reports this is: method or tool 0.90 · evidence from proof or derivation 0.92.

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sources read

arXiv listing abstract v1

not examined

full text
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relations to other papers

levels run

P abstract

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S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** Schematic representation of a hard-soft separation.

**Figure 2.** The chiral-odd photon DA of leading twist. The two curves c

**Figure 3.** χ(1 GeV) as function of m4 with s0 fixed as s0 = (m4 + 0.1 GeV)2 (solid line) and (m4 + 0.2 GeV)2 (dashed line). In this calculation we use hqq¯ i(1 GeV) = (−0.25 GeV)3 , m2 0 (1 GeV) = 0.65 GeV2 , cf. [23], and the value αs(1 GeV) = 0.51. Neglecting for the moment the uncertainty of f1f T 1 and the local-duality approximation itself, we obtain from local-duality sum rules with four resonances χ(1 GeV) = (3.08 ± 0.02) GeV−2 . (B.8) For a typical value of m4, the breakdown of the total number in contributions from the different resonances and the continuum is 2.73 + 0.30 − 0.07

**Figure 4.** (a) Coupling fρ ′f T ρ ′ from the Borel sum-rule (B.9) as a function of the Borelparameter. Solid line: mρ ′ = 1.4 GeV, dashed line: mρ ′ = 2 GeV. s0 = (mρ ′ + 0.1 GeV)2 . (b) χ(1 GeV) from (B.5) with q 2 = 0 with two resonances, ρ(770) and an effective ρ ′ , as a function of the Borel-parameter. Identification of curves as in (a). Alternatively, and this constitutes the 2nd procedure, one investigates the Borel sumrule mρfρf T ρ e −m2 ρ/M2 + mρ ′fρ ′f T ρ ′e −m2 ρ′/M2 = = −2hqq¯ i 1 + 4 3 αs π ln M2 µ2 − γE − 1 − Z ∞ s0 ds s e −s/M2 − 1 3 m2 0 M2 + 0 · hG2 i M4 ,(B.9)…

Figures reproduced from arXiv:hep-ph/0207307 by the authors.

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Radiative decays of $P$-wave charmed baryons in the $SU(3)$ flavor $\bf6_F$ representation
hep-ph 2025-06 conditional novelty 6.0 of 10

The authors predict radiative decay widths for all P-wave charmed baryons in the 6F representation, and propose Omega_c gamma channels as the best route to identify the excited Omega_c states seen at LHCb.
$D \bar D_1(2420)$ and $D^* \bar D^*(2400)$ molecular states: Probing their electromagnetic fingerprints
hep-ph 2025-04 conditional novelty 6.0 of 10

For the D \bar D_1 and D* \bar D* molecular states, the computed magnetic moments are -1.41 and 3.85 nuclear magnetons, with negative quadrupole moments, both dominated by light-quark contributions.
Factorization at Subleading Power and Endpoint-Divergent Convolutions in $h\to\gamma\gamma$ Decay
hep-ph 2019-12 conditional novelty 6.0 of 10

A subleading-power SCET factorization theorem for the b-quark induced h to gamma gamma amplitude is derived, with endpoint divergences shown to cancel or be removable by rearrangement, and the leading double logarithm...
Constraints on scalar leptoquarks from lepton and kaon physics
hep-ph 2019-08 accept novelty 6.0 of 10

Scalar leptoquark couplings are constrained by a comprehensive set of upper bounds from tau, muon, electron, and kaon decay data, including standard model interference in rare K to pi l+ l- modes.
Isospin asymmetries in $B\to (K^*,\rho) \gamma/ l^+ l^-$ and $B \to K l^+ l^-$ in and beyond the Standard Model
hep-ph 2013-05 conditional novelty 6.0 of 10

Standard Model isospin asymmetries in B to K, K*, rho plus lepton pairs are computed to be below 1.5 percent at low dilepton mass, and a ratio of the rho-gamma and K*-gamma asymmetries is proposed as a quasi-null test...
Exclusive Chromomagnetism in heavy-to-light FCNCs
hep-ph 2012-12 accept novelty 6.0 of 10

Light-cone sum rules give the first estimate of O8 matrix elements in B/D to light meson plus photon transitions, at a few percent of the leading penguin form factors but with a large CP-even phase.
Magnetic susceptibility of QCD at zero and at finite temperature from the lattice
hep-lat 2012-09 conditional novelty 6.0 of 10

Dynamical lattice QCD at physical quark masses yields chi_u = -2.08(8), chi_d = -2.02(9), chi_s = -3.4(1.4) GeV^-2 at 2 GeV in MS-bar, showing the spin part of the QCD vacuum response to a magnetic field is diamagnetic.
Mixing effects in radiative decays of heavy-strange axial-vector mesons within light-cone QCD sum rules
hep-ph 2026-08 conditional novelty 5.0 of 10

Light-cone QCD sum rules with 1P1-3P1 mixing predict D_s1(2460) to D_s gamma at about 46 keV and D_s1(2536) to D_s gamma at about 0.55 keV, with the latter suppressed by destructive interference.
Magnetic dipole moments as probes of doubly-bottom molecular pentaquarks
hep-ph 2026-07 conditional novelty 5.0 of 10

For the molecular pentaquark configurations BΣ_b, BΣ_b*, and B*Σ_b, the predicted magnetic dipole moments are 2.40, −2.84, and 5.17 nuclear magnetons respectively, with a sign and magnitude pattern sensitive to spin s...
Structural dissection of hadronic molecules: The $D^{(*)}\bar{K}^{(*)}$ family under QCD light-cone sum rules
hep-ph 2026-02 conditional novelty 5.0 of 10

QCD light-cone sum rules predict magnetic moments from −2.0 to +3.1 nuclear magnetons and quadrupole moments near 10⁻³ fm² for JP=1+ D(*)K̅(*) molecular tetraquarks, dominated by light-quark contributions.
Electromagnetic form factors: A window into the $D\Lambda_c$, $D^*\Lambda_c$, and $D\Lambda_c^*$ molecular structure
hep-ph 2025-11 reject novelty 5.0 of 10

Using light-cone QCD sum rules, the paper predicts negative magnetic dipole moments of roughly -1.27, -2.78, and -3.80 nuclear magnetons for the DΛc, D*Λc, and DΛc* molecular pentaquark candidates, plus small quadrupo...
Probing the electromagnetic structure of the $P_c(4337)^+$ pentaquark: Insights from a diquark-diquark-antiquark picture for $J^P = \frac{1}{2}^-$ and $\frac{3}{2}^-$ states
hep-ph 2025-06 conditional novelty 5.0 of 10

Under the diquark-diquark-antiquark model, the magnetic moment of Pc(4337)+ is predicted to be 1.76 ± 0.44 μN for J^P = 1/2^- and -1.38 ± 0.35 μN for J^P = 3/2^-, with nonzero quadrupole and octupole moments in the 3/...
Investigating the underlying structure of vector hidden-charm tetraquark states via their electromagnetic characteristics
hep-ph 2024-12 conditional novelty 5.0 of 10

QCD light-cone sum rules give magnetic moments for vector hidden-charm tetraquarks that depend strongly on the chosen diquark-antidiquark current.
Shedding light on the nature of the $P_{cs}(4459)$ pentaquark state
hep-ph 2024-11 conditional novelty 5.0 of 10

QCD light-cone sum rules with three diquark-diquark-antiquark currents predict Pcs(4459) magnetic dipole moments of -0.60, 1.60, and 0.99 nuclear magnetons.
Magnetic dipole moments of the singly-heavy baryons with spin-$\frac{1}{2}$ and spin-$\frac{3}{2}$
hep-ph 2024-11 conditional novelty 5.0 of 10

QCD light-cone sum rules are used to predict magnetic dipole moments of all singly-heavy baryons with spin 1/2 and spin 3/2, plus electric quadrupole and magnetic octupole moments for the higher-spin states.
Unveiling the electromagnetic structure and intrinsic dynamics of spin-$\frac{3}{2}$ hidden-charm pentaquarks: A comprehensive QCD analysis
hep-ph 2025-04 conditional novelty 4.0 of 10

QCD sum rules predict distinct magnetic dipole moments for six JP=3/2- hidden-charm pentaquarks depending on their diquark-diquark-antiquark internal structure, with the charm quark dominating the magnetic response.
Subleading-power corrections to the radiative leptonic $B \to \gamma \ell \nu$ decay in QCD
hep-ph 2018-03 conditional novelty 4.0 of 10

The twist-two hadronic photon contribution shifts the leading-power B to gamma form factors by about 30 percent at next-to-leading-logarithmic accuracy, while higher-twist photon effects add only 3 to 5 percent.
B, D and K decays
hep-ph 2008-01 accept novelty 3.0 of 10

A community review that fixes the pre-LHC status of quark-flavour physics, catalogues the benchmark rare decays most sensitive to new physics, and quantifies the hadronic uncertainties that limit them.

How later papers used it

78 later papers cite it

Evidence: substantial. 77 of the citing papers share no author with it, and 58 of those carry a confident label from Pith's citation classifier.

Pith read 30 of the 214 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

17 later papers by other authors are listed below: 1 built on it and 16 used its data or values.

Builds on it 1 paper

In this section we give the form of the DA used in this paper based on the results in [ref] with added quark mass corrections.

Charged and Neutral $\bar B_{u,d,s} \to \gamma$ Form Factors from Light Cone Sum Rules at NLO (2021) · Explicit Distribution Amplitudes · 0.92

Uses its data or values 16 of 45 papers with this label

These wavefunctions are borrowed from Ref. [ref] .

Magnetic dipole moments of bottom-charm baryons in light-cone QCD (2023) · Numerical analysis and conclusion · 1.00
The distribution amplitudes of the photon, which depend on various non-perturbative parameters are borrowed from Ref. [ref] .

Magnetic moment of the $X_1(2900)$ state in the diquark-antidiquark picture (2022) · Numerical analysis · 0.97
The parameters that are used in our calculations are as follows: $m_u=m_d=0$, $m_s =96^+8.0_-4.0$, $m_c = (1.27±0.02)$GeV, $m_X_AV= 2800 ±75$ MeV [ref] , $⟨s̅s⟩$= $0.8 ⟨q̅q⟩$ with $⟨q̅q⟩$ =$(-0.24±0.01)^3$GeV$^3$ [ref] , $m_0^2 = 0.8 ±0.1$ GeV$^2$, $⟨g_s^2G^2⟩= 0.88$GeV$^4$ [ref] , and $f_3γ=-0.0039$GeV$^2$ [ref] .

Analysis of the $\mathrm{X_{AV}}$ state through its electromagnetic properties (2024) · Numerical results and discussions · 1.00
We use $m_u =m_d =0$, $m_s=96^+8_-4$ MeV, $m_c = 1.28 ±0.03$ GeV, $m_b=4.18^+0.04_-0.03$ GeV, [ref] , $M_Ξ^+_cc=3519 ±1$ MeV [ref] , $M_Ξ_cc^++=3621.40±1.13$ MeV [ref] , $f_3γ=-0.0039$ GeV$^2$ [ref] , $⟨q̅q⟩=(-0.24±0.01)^3$ GeV$^3$ [ref] , $m_0^2 = 0.8 ±0.12$ GeV$^2$ and $χ=-2.85 ±0.5$ GeV$^2$ [ref] .

Magnetic moments of doubly heavy baryons in light-cone QCD (2018) · Numerical analysis and conclusion · 1.00
$m_u=m_d=0$, $m_s =93.4^+8.6_-3.4$, $m_c = 1.27±0.02$GeV [ref] , $m_Ω_c(3185) = 3185.1±1.7^+7.4_-0.9±0.2$, $m_Ω_c(3327) = 3327.1±1.2^+0.1_-1.3 ±0.2$ [ref] , $m_D_s Ξ = 3.28^+0.12_-0.12$ GeV, $m_D_s^* Ξ = 3.44^+0.11_-0.11$ GeV, $m_D Ξ^* = 3.41^+0.11_-0.12$ GeV, $m_D_s Ξ^* = 3.51^+0.11_-0.13$ GeV, $λ_D Ξ =( 1.42^+0.27_-0.24) ×10^-3$ GeV$^6$, $λ_D_s Ξ =( 1.60^+0.30_-0.27) ×10^-3$ GeV$^6$, $λ_D^* Ξ =( 1.59^+0.29_-0.26) ×10^-3$ GeV$^6$, $λ_D_s^* Ξ =( 1.72^+0.30_-0.27) ×10^-3$ GeV$^6$, $λ_D Ξ^* =( 1.07^+0.19_-0.17) ×10^-3$ GeV$^6$, $λ_D_s Ξ^* =( 1.12^+0.21_-0.19) ×10^-3$ GeV$^6$ [ref] , $⟨s̅s⟩$= $0.8 ⟨q̅q⟩$ with $⟨q̅q⟩$=$(-0.24±0.01)^3$GeV$^3$ [ref] , $m_0^2 = 0.8 ±0.1$ GeV$^2$, $⟨g_s^2G^2⟩= 0.88$GeV$^4$ [ref] , $χ=-2.85 ±0.5 ^-2$ [ref] and $f_3γ=-0.0039$GeV$^2$ [ref] .

Magnetic dipole moments of the $\Omega_{c}(3185)^0$ and $\Omega_c(3327)^0$ states from molecular perspective (2023) · Numerical illustrations · 1.00
We use $m_u=m_d=0$, $m_s =96^+8_-4$, $m_c = 1.275 ±0.02$ [ref] , $m_P_c1 = 4440.3 ±1.3 ^+4.1_-4.7$, $m_P_c2= 4457.3 ±0.6 ^+4.1_-1.7$ [ref] , $m_P_cs= 4458.8 ±2.7 ^+4.7_-1.1$ [ref] , $f_3γ=-0.0039 ^2$ [ref] ,

Magnetic dipole moments of the hidden-charm pentaquark states: $P_c(4440)$, $P_c(4457)$ and $P_{cs}(4459)$ (2021) · Numerical analysis and discussions · 1.00
The DAs of the photon are among the main input parameters for the calculations of the magnetic dipole moments, we use them from the Ref. [ref] .

Electromagnetic properties of vector doubly charmed tetraquark states (2024) · Numerical analysis of the sum rules · 0.99
Took the expressions of the photon DAs it needed in its calculations from this paper.

Multipole Moments of Heavy Vector and Axial-Vector Mesons in QCD (2019) · 0.81 · paraphrased: the licence does not allow quoting
$⟨u̅u⟩$= $⟨d̅d⟩$=$(-0.24±0.01)^3$GeV$^3$ [ref] , $m_0^2 = 0.8 ±0.1$ GeV$^2$, $⟨g_s^2G^2⟩= 0.88$GeV$^4$ [ref] and $f_3γ=-0.0039$GeV$^2$ [ref] .

Magnetic and quadrupole moments of the $Z_{c}(4020)^+$, $Z_{c}(4050)^+$, and $Z_{c}(4600)^{+}$ states in the diquark-antidiquark picture (2023) · Numerical analysis · 1.00
We use $m_u=m_d=0$, $m_s =96^+8.0_-4.0$, $m_c = (1.275±0.025)$GeV, $m_Z_c= 4022.9 ±0.8 ±2.7$, $m_Z^1_c= 4196^+31+17_-29-13$, $m_Z_cs= 4003.0±6.0^+4.0_-14.0$, $m_Z^1_cs= 4216.0 ±24.0^+43_-30$, $f_3γ=-0.0039$GeV$^2$ [ref] , $χ=-2.85 ±0.5 ^-2$ [ref] $⟨s̅s⟩$= $0.8 ⟨u̅u⟩$, $⟨u̅u⟩$= $⟨d̅d⟩$=$(-0.24±0.01)^3$GeV$^3$ [ref] , $m_0^2 = 0.8 ±0.1$ GeV$^2$, $⟨g_s^2G^2⟩= 0.88$GeV$^4$ [ref] ,

Magnetic dipole moments of the $Z_{c}(4020)^+$, $Z_{c}(4200)^+$, $Z_{cs}(4000)^{+}$ and $Z_{cs}(4220)^{+}$ states in light-cone QCD (2021) · Numerical analysis · 1.00
The values of DAs are obtained in Ref. [ref] , which we will employ in our numerical computations.

Electromagnetic properties of doubly heavy pentaquark states (2022) · Numerical analysis · 1.00
Their explicit forms, which involve the parameter $f_3γ$ listed in the table, are taken from Ref. [ref] .

Structural dissection of hadronic molecules: The $D^{(*)}\bar{K}^{(*)}$ family under QCD light-cone sum rules (2026) · Input parameters · 0.99
These expressions and the input parameters used in their explicit forms are taken from Ref. [ref] .

Study on the electromagnetic properties of the $[sc] [\bar q \bar b]$ and $[sc] [\bar s \bar b]$ states with $J^P = 1^+$ (2024) · Numerical analysis of the multipole moments · 0.96
Used the QCD sum rule estimate of the magnetic susceptibility of the quark condensate at μ_0=1 GeV, χ(1 GeV)=(3.15±0.3) GeV^-2, as given in this paper.

Hadronic photon correction to $\gamma^{\ast} \gamma \to f_{2}(1270)$ at next-to-leading order (2026) · 0.99 · paraphrased: the licence does not allow quoting
The values of these parameters we use in the analysis are given as follows: $m_u=m_d=0$, $m_s =93.4^+8.6_-3.4$, $m_c = 1.27 ±0.02$ [ref] , $m_P_cs^1 = 4338.2 ±0.7 ±0.4$ MeV, $m_P_cs^2 = 4458.8 ±2.9^+4.7_-1.1$ MeV, $f_3γ=-0.0039 ^2$ [ref] ,

Investigation of magnetic moment of $P_{cs}(4338)$ and $P_{cs}(4459)$ pentaquark states (2022) · Numerical analysis and discussion · 1.00
$⟨q̅q⟩$=$(-0.24±0.01)^3$GeV$^3$ [ref] , $m_0^2 = 0.8 ±0.1$ GeV$^2$, $⟨g_s^2G^2⟩= 0.88$GeV$^4$ [ref] , $χ=-2.85 ±0.5 ^-2$ [ref] and $f_3γ=-0.0039$GeV$^2$ [ref] .

Electromagnetic properties of the $\Sigma_{c}(2800)^+$ and $\Lambda_c(2940)^+$ states via light-cone QCD (2023) · Numerical illustrations and discussion of the results · 1.00

How the literature uses it

79 arXiv papers cite this in 224 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 124 sentences without a confident label

Builds on it 1 paper

In this section we give the form of the DA used in this paper based on the results in [ref] with added quark mass corrections.

Charged and Neutral $\bar B_{u,d,s} \to \gamma$ Form Factors from Light Cone Sum Rules at NLO · Explicit Distribution Amplitudes · 0.92

Uses its method or tool 12 papers

For the photon case, the Fock state decomposition was employed as presented in [ref] to obtain the photon states, and the Light-Front wave functions were derived based on the methodologies outlined in [ref] (see [ref] for an alternative discussion).

Backward DVCS on the pion in Sullivan processes · Model for the pi-to-gamma TDAs · 0.98
Since the background field is treated as infinitesimally weak, each correlator admits a systematic expansion in powers of the field strength [ref] : [equation] where the leading terms $𝒯^(0)(p,q)$ and $𝒯_μν^(0)(p,q)$ correspond to the field-free two-point correlators relevant for mass sum rules, while the linear-response terms $𝒯^(1)(p,q)$ and $𝒯_μν^(1)(p,q)$ contain precisely the electromagnetic multipole moments to be extracted.

Deciphering the nature of $P^{\Sigma}_{\psi s}$ pentaquarks in the light of their electromagnetic multipole moments · General framework of LCSR and interpolating currents for $P^Σ_ψs$ states · 0.93
The resulting matrix elements of the form $⟨γ(q) |q̅(x) Γ_i q(0) |0 ⟩$ and $⟨γ(q) |q̅(x)$ $Γ_i G_αβ q(0) |0 ⟩$ are expressed in terms of photon DAs of increasing twist [ref] .

Structural dissection of hadronic molecules: The $D^{(*)}\bar{K}^{(*)}$ family under QCD light-cone sum rules · QCD description of the correlation function · 0.80

Uses its data or values 46 papers

These wavefunctions are borrowed from Ref. [ref] .

Magnetic dipole moments of bottom-charm baryons in light-cone QCD · Numerical analysis and conclusion · 1.00
We use the values of some other input parameters as follow: $m_u=m_d=0$, $m_s =96^+8_-4$, $m_c = (1.275±0.025)$GeV, $m_X_1= 2904±5±1 MeV$, $f_3γ=-0.0039 GeV^2$ [ref] , $⟨s̅s⟩$= $0.8 ⟨u̅u⟩$ with $⟨u̅u⟩$=$(-0.24±0.01)^3$GeV$^3$ [ref] , $m_0^2 = 0.8 ±0.1$ GeV$^2$ [ref] , $⟨α_s/π G^2 ⟩=(0.012±0.004)$ $GeV^4$ [ref] and $λ_X_1= m_X_1 f_X_1$ with $f_X_1= (2.1 ±0.4) ×10^-3$ GeV$^4$ [ref] .

Magnetic moment of the $X_1(2900)$ state in the diquark-antidiquark picture · Numerical analysis · 1.00
The parameters that are used in our calculations are as follows: $m_u=m_d=0$, $m_s =96^+8.0_-4.0$, $m_c = (1.27±0.02)$GeV, $m_X_AV= 2800 ±75$ MeV [ref] , $⟨s̅s⟩$= $0.8 ⟨q̅q⟩$ with $⟨q̅q⟩$ =$(-0.24±0.01)^3$GeV$^3$ [ref] , $m_0^2 = 0.8 ±0.1$ GeV$^2$, $⟨g_s^2G^2⟩= 0.88$GeV$^4$ [ref] , and $f_3γ=-0.0039$GeV$^2$ [ref] .

Analysis of the $\mathrm{X_{AV}}$ state through its electromagnetic properties · Numerical results and discussions · 1.00

Compares with it 8 papers

Using the value for $m_0^2$ reported in Ref. [ref] , we therefore find the prediction [equation] which is big compared to the values obtained from QCD sum rule analyses [ref] , instanton vaccum model calculations [ref] or exclusive B meson decays [ref] , but compatible with the values reported in Refs. [ref] and [ref] .

Relations between vacuum condensates and low energy parameters from a rational approach · The magnetic susceptibility of the quark condensate · 0.80
We wish to emphasise that whereas $χ_q (f_q)$ merely reproduces the result in [ref] (within minor variation of the input), $χ_s (f_s)$ is the first determination of this quantity.

Charged and Neutral $\bar B_{u,d,s} \to \gamma$ Form Factors from Light Cone Sum Rules at NLO · Quark Mass Correction to the Photon Distribution Amplitude · 0.89
The numerical value of the susceptibility do not coincide with recent estimations from the instanton liquid model [ref] ,sum rules fit [ref] and phenomenology of D-meson decays [ref] .

Magnetic susceptibility of the quark condensate via holography · Conclusion · 0.96

Cites it as background 7 papers

We thank all the members of the ETM Collaboration for the most enjoyable collaboration, the authors of Ref. [ref] for correspondence, and of Ref. [ref] for comments on the manuscript.

Lattice QCD determination of the normalization of the leading-twist photon distribution amplitude and susceptibility of the quark condensate · Acknowledgements · 0.89
[ref] , we will not show them here.

Effects from Hadronic Structure of Photon on $B\to\phi\gamma$ and $B_s\to(\rho^0,\omega)\gamma$ Decays · Framework and Inputs · 0.83
The long-distance photon emission from a soft-quark line requires the inclusion of higher-twist terms in the expansion of the quark propagator in a photon background field, beyond the leading-twist (perturbative) contribution; a comprehensive discussion of this topic can be found in Ref. [ref] .

Meson distribution amplitudes - applications to weak radiative B decays and B transition form factors · Long-Distance Photon Emission · 0.87

Discussion

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