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Asymptotics of Heavy-Meson Form Factors

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Andrey G. Grozin (Budker Institute), Matthias Neubert (CERN)

High Energy Physics - Phenomenology · Submitted 18 Jul 1996 · 64 references

Pith's verdict: A solid hard-exclusive calculation that gets the large-recoil scaling laws and a sharp zero prediction right, with a soft ln² coefficient the authors themselves leave open.

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arXiv IDhep-ph/9607366 (version 1)
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CitationAndrey G. Grozin (Budker Institute), Matthias Neubert (CERN). "Asymptotics of Heavy-Meson Form Factors." arXiv:hep-ph/9607366 (1996). https://pith.science/paper/NJBNDHNZ
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The paper

The paper claims

Heavy-meson form factors are argued to change sign at large recoil, with an exact zero in one $B^*$ channel.

The reading

The paper asks what a heavy meson's transition form factor looks like when the two mesons fly apart with a very large relative velocity, a regime far outside ordinary semileptonic decays but reachable in principle in $e^+e^-$ pair production. Using the hard-scattering machinery developed for exclusive processes such as the pion form factor, the authors argue that the answer is governed by the meson's light-cone wave functions rather than by any single universal function. Two different gluon-exchange mechanisms compete: a longitudinal gluon that flips the meson helicity and is sensitive to the interference of leading- and subleading-twist wave functions, and a transverse gluon that conserves helicity and is sensitive only to the leading-twist wave function. Because the two mechanisms scale differently with $vcdot v'$ and with the heavy-quark mass, their relative weight shifts as recoil grows, and the interference between them is what the authors claim produces sign changes in specific form factors. If the argument holds, the form factor for two longitudinally polarized $B^*$ mesons must pass through zero at some $vcdot v'$ of order $m_b/Lambda$, and the $B$ meson form factor in the timelike region must do the same near $vcdot v' sim -m_b/Lambda$.

What carries the argument

The load-bearing objects are the two light-cone wave functions $phi_pm(omega)$ of a heavy meson, defined as Fourier transforms of the matrix element of the bilocal operator $Q^*(0)E(0,z)q(z)$ between the vacuum and the meson state, where $omega$ is the light-quark light-cone momentum in the meson rest frame. Their first moments are fixed exactly by the equations of motion, $langleomegarangle_+ = tfrac{4}{3}barLambda$ and $langleomegarangle_- = tfrac{2}{3}barLambda$, and their second moments are tied to the chromoelectric and chromomagnetic parameters $lambda_E^2$ and $lambda_H^2$. The calculational engine is the hard-gluon-exchange factorization familiar from exclusive QCD: in the Breit frame the gluon virtuality is $k_g^2 simeq -omegaomega' e^{vartheta}$, so for large rapidity the amplitude factorizes into a perturbative kernel convolved with the wave functions. The paper also derives the Brodsky-Lepage-type evolution equations for $phi_pm(omega)$, with kernels given explicitly at one loop, and shows that the renormalization of these non-local operators does not by itself renormalize the tower of local moments, so an extra divergence must be subtracted separately. QCD sum rules, including the non-locality of the quark condensate, supply numerical estimates of $lambda_E^2$, $lambda_H^2$, and the shapes of $phi_pm$.

Extended reading notes

Core claim

The central claim is that the large-recoil asymptotics of heavy-meson form factors is not exhausted by the Isgur-Wise function but is a sum of two contributions with different origins and different scaling. For $1 ll |vcdot v'| ll m_Q/Lambda$, the Isgur-Wise function $xi$ dominates and is itself generated by interference between the leading-twist wave function $phi_+$ on one side of the hard-gluon exchange and the subleading-twist wave function $phi_-$ on the other, giving $xi sim alpha_s f^2/Lambda^3,(vcdot v')^{-2}$ up to logarithmic enhancement. For $|vcdot v'| gg m_Q/Lambda$, two order-$1/m_Q$ functions, $xi_3$ and $chi_2$, take over; both are built from the leading-twist wave function alone, both conserve meson helicity, and both fall only as $1/(vcdot v')$ with an extra factor $m_Q$. The authors show that $chi_2 = -xi_3 e^{-vartheta}$, so that in the physical form factors the combination $xi pm 2m_b xi_3$ appears with a relative minus sign between the two terms. Since $xi$ falls faster than $m_b xi_3$ at large recoil, the combination must cross zero. The paper states this as an exact statement for the longitudinally polarized $B^*$ form factor at spacelike momentum transfer, and as an approximate one for the $B$ form factor in the timelike region where the form factors acquire a small imaginary part.

Load-bearing premise

The whole scaling story rests on the assumed behaviour of the wave functions at vanishing light-quark momentum, $phi_+(omega) sim omega$ and $phi_-(omega) sim 1$ as $omega to 0$, which makes the relevant integrals diverge logarithmically and produces the $ln^2(vcdot v')$ enhancement; the paper motivates this from QCD analogies and from its own sum-rule model but does not derive it from first principles.

Editorial extensions

If this is right

  • The form factor $h_L$ for two longitudinally polarized $B^*$ mesons, positive at small recoil, must become negative beyond $vcdot v' sim m_b/Lambda$, giving an exact zero in the spacelike region.
  • The $B$ meson form factor $h_+$ acquires an approximate zero in the timelike region inside the physical phase space for $e^+e^- to Bbar B$, which would show up as a dip in the $Bbar B$ production rate near $sqrt{s}/m_B sim 25$.
  • Cross-section ratios for $e^+e^- to B^{(*)}bar B^{(*)}$ change their scaling pattern across the turnover: below the zero they scale as $1/x^8$, and above it the $xi_3$ term makes the $Bbar B$ and $B^*_Lbar B^*_L$ channels scale as $1/x^4$.
  • The same structure predicts a zero in the longitudinally polarized vector-meson channel at spacelike momentum transfer for light mesons such as the $rho$, matching what is already known for $pi$-$rho$ form factors.
  • The moments of the wave functions are fixed by the equations of motion and by the sum-rule estimates $lambda_E^2 = (0.11 pm 0.06)$ GeV$^2$ and $lambda_H^2 = (0.18 pm 0.07)$ GeV$^2$, giving a concrete normalization for the asymptotic formulae.

Reading between the lines

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

  • The predicted zero in $h_L$ is a clean lattice target: a direct lattice computation of the $B^* to B^*$ matrix element of the heavy-quark vector current at several values of $vcdot v'$ between 1 and roughly 10 would either show the sign change or fail to, without needing any wave-function model.
  • Because the zero location scales as $m_Q/Lambda$, comparing the $B$ and $D$ systems gives a further check: the charm zero should sit at a proportionally smaller $vcdot v'$, so a joint fit of both would test the $m_Q$ scaling rather than just the sign.
  • The whole chain leans on the endpoint behaviour of $phi_pm$ at small $omega$; a sum-rule or lattice determination of the low-$omega$ slope of $phi_+$ and the low-$omega$ constant of $phi_-$ would test the $ln^2(vcdot v')$ enhancement directly, independently of the form-factor zero.
  • The relative sign $chi_2 = -xi_3 e^{-vartheta}$ follows from helicity conservation in the transverse-gluon channel; any process that breaks the identification of helicity with the light-quark spin projection, such as a non-zero light-quark mass correction, would shift the zero and could be looked for as a small displacement.
Original abstract

Using methods developed for hard exclusive QCD processes, we calculate the asymptotic behaviour of heavy-meson form factors at large recoil. It is determined by the leading- and subleading-twist meson wave functions. For $1\ll |v\cdot v'|\ll m_Q/\Lambda$, the form factors are dominated by the Isgur--Wise function, which is determined by the interference between the wave functions of leading and subleading twist. At $|v\cdot v'|\gg m_Q/\Lambda$, they are dominated by two functions arising at order $1/m_Q$ in the heavy-quark expansion, which are determined by the leading-twist wave function alone. The sum of these contributions describes the form factors in the whole region $|v\cdot v'|\gg 1$. As a consequence, there is an exact zero in the form factor for the scattering of longitudinally polarized $B^*$ mesons at some value $v\cdot v'\sim m_b/\Lambda$, and an approximate zero in the form factor of $B$ mesons in the timelike region ($v\cdot v'\sim -m_b/\Lambda$). We obtain the evolution equations and sum rules for the wave functions of leading and subleading twist as well as for their moments. We briefly discuss applications to heavy-meson pair production in $e^+ e^-$ collisions.

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.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 6,409 input tokens, $0.0003 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.16 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.19 · compares with prior results 0.09 · mentions code or data availability 0.02.

The abstract, sentence by sentence

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

Using methods developed for hard exclusive QCD processes, we calculate the asymptotic behaviour of heavy-meson form factors at large recoil. It is determined by the leading- and subleading-twist meson wave functions. For $1\ll |v\cdot v'|\ll m_Q/\Lambda$, the form factors are dominated by the Isgur--Wise function, which is determined by the interference between the wave functions of leading and subleading twist. At $|v\cdot v'|\gg m_Q/\Lambda$, they are dominated by two functions arising at order $1/m_Q$ in the heavy-quark expansion, which are determined by the leading-twist wave function alone. The sum of these contributions describes the form factors in the whole region $|v\cdot v'|\gg 1$. As a consequence, there is an exact zero in the form factor for the scattering of longitudinally polarized $B^*$ mesons at some value $v\cdot v'\sim m_b/\Lambda$, and an approximate zero in the form factor of $B$ mesons in the timelike region ($v\cdot v'\sim -m_b/\Lambda$). We obtain the evolution equations and sum rules for the wave functions of leading and subleading twist as well as for their moments. We briefly discuss applications to heavy-meson pair production in $e^+ e^-$ collisions.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 Using methods developed for hard exclusive QCD processes, we calculate the asymptotic behaviour of heavy-meson form factors at large recoil. contribution 0.94 0.96 0.09 asserted 1.00
s1 It is determined by the leading- and subleading-twist meson wave functions. result 0.88 0.54 0.10 asserted 1.00
s2 For $1\ll |v\cdot v'|\ll m_Q/\Lambda$, the form factors are dominated by the Isgur--Wise function, which is determined by the interference between the wave functions of leading and subleading twist. result 1.00 0.75 0.18 asserted 1.00
s3 At $|v\cdot v'|\gg m_Q/\Lambda$, they are dominated by two functions arising at order $1/m_Q$ in the heavy-quark expansion, which are determined by the leading-twist wave function alone. result 1.00 0.85 0.14 asserted 1.00
s4 The sum of these contributions describes the form factors in the whole region $|v\cdot v'|\gg 1$. result 0.79 0.84 0.21 asserted 1.00
s5 As a consequence, there is an exact zero in the form factor for the scattering of longitudinally polarized $B^*$ mesons at some value $v\cdot v'\sim m_b/\Lambda$, and an approximate zero in the form factor of $B$ mesons in the timelike region ($v\cdot v'\sim -m_b/\Lambda$). result 0.89 0.89 0.31 asserted 0.99
s6 We obtain the evolution equations and sum rules for the wave functions of leading and subleading twist as well as for their moments. contribution 0.77 0.95 0.05 asserted 1.00
s7 We briefly discuss applications to heavy-meson pair production in $e^+ e^-$ collisions. contribution 0.66 0.71 0.02 asserted 0.64

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read
  • arXiv listing abstract v1
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • 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. FIG. 1. Hard-gluon exchange contributions to heavy-meson fo
Figure 2. FIG. 2. One-loop diagrams contributing to the matrix elemen
Figure 3. FIG. 3. Feynman rules for vertices involving
Figure 4. FIG. 4. Non-vanishing diagrams for the correlators Π
Figure 5. FIG. 5. Sum-rule results for
Figure 6. FIG. 6. Leading diagrams for the correlators of the bilocal o
Figure 7. FIG. 7. Model wave functions
Figure 8. FIG. 8. Hard-gluon exchange contributions to heavy-meson fo
Figure 9. FIG. 9. Cross-section ratios
Figure 10. FIG. 10. Cross-section ratios
Figure 11. FIG. 11. Square model (solid line) and triangle model (dashe

Figures reproduced from arXiv:hep-ph/9607366 by the authors.

References
How later papers used it

96 later papers cite it

Evidence: substantial. 91 of the citing papers share no author with it, and 24 of those carry a confident label from Pith's citation classifier.

Pith read 18 of the 210 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

Builds on it 1 paper
Uses its method or tool 4 of 19 papers with this label
How the literature uses it

90 arXiv papers cite this in 205 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 · 116 sentences without a confident label

Contests it 1 paper
  • In Ref. [ref] the approximation $λ_B,E^2 = λ_B,H^2$ is adopted, which is not justified by calculations of these parameters [ref] : [equation] In Ref. [ref] it is also assumed that $λ_B,E^2=3/2λ_B,+^2$, based solely on the desire that the exponential model for the leading-twist $B$-LCDA satisfies exactly the Grozin-Neubert relations [ref] .

    Non-local matrix elements in $B_{(s)}\to \{K^{(*)},\phi\}\ell^+\ell^-$ · Numerical Results · contest 0.85

Builds on it 1 paper
Uses its method or tool 20 papers
  • The Exponential model for the two-particle $B_s$-meson LCDAs for different twists is given as [ref] [equation] where $ω_0=λ_B_s$ and Ei(x) is the Exponential integral.

    Investigating non-local contributions in $B_{s} \to \phi \bar{\ell} \ell$ including higher-twist effects · Two-particle distribution amplitudes · 0.82

  • In order to derive the sum rules which ultimately determine the parameters $λ_E,H^2$, we make an explicit choice for the matrices $Γ_1$ and $Γ_2$ [ref] .

    QCD sum rules for parameters of the $B$-meson distribution amplitudes · Derivation of the QCD Sum Rules in HQET · 0.97

  • In our analysis, we assume the exponential model functions [ref] [equation] from which we obtain [equation] Using these model functions, the relevant convolution integrals over the LCDAs in [ref] and [ref] become [equation] and [equation] In practice, we fix $ω_0$ using a LCSR determination of the first inverse moment ($λ_B$) of the LCDA $ϕ_+^B$, which in our model is obtained as $λ_B=ω_0$.

    The Simplest B Decay, Precisely · Modeling of the LCDAs · 0.95

Uses its data or values 3 papers
Compares with it 5 papers
  • Our results compared to the values obtained in [ref] are listed in Table [ref].

    QCD sum rules for parameters of the $B$-meson distribution amplitudes · Conclusion · 1.00

  • In this subsection, we will compare our results for the HQET LCDA with those from previous studies based on various phenomenological models [ref] : [equation] The parameters of the first four models are collected as follows [ref] : in model I, $ω_0=354_-30^+38$MeV; in model II, $ω_0=368_-32^+42$MeV, $σ_B^(1)=1.4±0.4$GeV and $k=ω/1 GeV$; in model III, $ω_0=389_-28^+35$MeV and $ω_1=2 ω_0/√(π)$; in model IV, $ω_0=303_-26^+35$MeV and $ω_2=4 ω_0/(4-π)$.

    Calculation of heavy meson light-cone distribution amplitudes from lattice QCD · Comparison with phenomenological models · 0.99

  • The method of two-point QCD sum rules has been applied to estimate these HQET parameters repeatedly, yielding the numerical predictions significantly different each other even with the sizeable theory uncertainties [equation] The substantial discrepancies of the obtained numerical results between [ref] and [ref] can be traced back to the remarkable perturbative corrections to the short-distance Wilson coefficients for the dimension-five quark-gluon mixed condensate $⟨q̅ σ_μν G^μν q ⟩$ and to the further inclusion of the particular higher-order power corrections at tree level from the dimension-six vacuum condensates in the factorization approximation in the latter.

    Precision calculations of $B_{d, s} \to \pi, K$ decay form factors in soft-collinear effective theory · Theory inputs · 0.81

Cites it as background 34 papers

Discussion

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