REVIEW
Phenomenology of Heavy Meson Chiral Lagrangians
High Energy Physics - Phenomenology · Submitted 17 May 1996 · 220 references
Pith's verdict: The standard reference for the combined heavy-quark/chiral Lagrangian for heavy mesons: honest, useful, and mostly careful, with one section (§3.3) that oversells a cancellation-dominated number.
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Referee report
The paper
The paper claims
The paper argues that one effective Lagrangian, built from heavy-quark and chiral symmetries, can organize strong, electromagnetic, and weak couplings of B and D mesons to light pseudoscalars and vector resonances.
The reading
What carries the argument
The carrying object is the trace-form heavy-meson field $H = (1+v!!!/)2 [P^*_mu gamma^mu - P gamma_5]$, with its positive-parity partners $S$ and $T$, coupled to the chiral coset field $xi$ and to the hidden-gauge vector field $rho$. The Lagrangian is organized by derivatives of light fields and by $1/M_H$, so heavy-quark spin-flavour symmetry fixes the coefficients of many terms while chiral symmetry fixes how pions and kaons enter. The single coupling $g$ in the axial term $i g,mathrm{Tr}(H gamma_mu gamma_5 A^mu bar H)$ is the clearest example: it controls $D^*$ to $D pi$, enters chiral-loop corrections to hyperfine splittings and decay constants, and normalizes the heavy-to-light form factors. The machinery's job is to convert symmetry statements into concrete, parameter-light relations that can be compared with experiment.
Extended reading notes
Core claim
The central claim is that QCD's heavy-quark spin-flavour symmetry and chiral $SU(3)_L$ times $SU(3)_R$ symmetry can be realized simultaneously in a single effective Lagrangian for heavy and light mesons, including the terms that break both symmetries. The fields are the heavy-meson doublets $H$ for $(0^-,1^-)$, $S$ for $(0^+,1^+)$ and $T$ for $(1^+,2^+)$, the Goldstone octet, and the light vector octet introduced as hidden-gauge fields. Once the couplings are fixed from a few inputs — the axial coupling $g$, the decay constant $F$, the magnetic coupling $lambda$, and the light-vector couplings — the same Lagrangian yields relations among $D^*$ to $D pi$, $B$ to $pi$ lepton neutrino, $B$ to $rho$ lepton neutrino, $D^*$ to $D gamma$, $B$ to $K^* gamma$, and quarkonium transitions. The review's contribution is to assemble these relations, summarize the parameter estimates, and show where the soft-meson approximation limits the numerical reach.
Load-bearing premise
The load-bearing premise is that soft-light-meson results can be continued to physical momentum transfers with a nearest-pole or two-component form factor; the authors acknowledge this in abstract:s5 and in Sections 5.1 and 5.3.2, where the parameter $r=0.5$ is chosen as a compromise.
Editorial extensions
If this is right
- A single $g_{B^*Bpi}$ ties the $D^*$ to $D pi$ width, the chiral-loop corrections to $D^*$ to $D gamma$, the hyperfine splitting, and the normalization of $B$ to $pi$ and $B$ to $rho$ form factors, so measuring one constrains the others.
- The hyperfine splitting $Delta_B$ is predicted at about 1.9 MeV and $Delta_D$ at about 6.3 MeV after chiral logs and $1/M$ counterterms, with large cancellations that make the numbers order-of-magnitude only.
- Semileptonic $B$ to $pi$ and $B$ to $rho$ rates are tied to the measured $D$ to $K$ and $D$ to $K^*$ form factors by heavy-flavour scaling, up to the assumed momentum-transfer dependence.
- Radiative $D^*$ to $D gamma$ and $B$ to $K^* gamma$ amplitudes use the same $F$, $g_V$, and $lambda$, giving predictions in the few $10^{-5}$ range for $B$ to $K^* gamma$ before long-distance charm-loop corrections.
- For charmonium and bottomonium, spin symmetry alone yields relations among radiative and hadronic transitions, including the suppression patterns for even versus odd numbers of emitted pions.
Reading between the lines
- A sharp test would be to fix the momentum-transfer shape of $A_1$ and $F_1$ from lattice QCD independently, then ask whether the same $g$ and $F$ fit both $D$ to $K^*$ and $B$ to $rho$; a mismatch would locate the breakdown in the soft limit rather than in the symmetry relations.
- Because the radiative loop corrections and the strong width both depend on $g$, a precise measurement of $D^*$ to $D gamma$ and $D^*$ to $D pi$ could overconstrain the effective theory without introducing new parameters.
- The same trace formalism can be pushed to $B_c$ and quarkonium, where heavy-flavour symmetry is lost but spin symmetry survives; that suggests testable relations among $B_c$ form factors even though absolute normalization is not fixed.
Original abstract
The approximate symmetries of Quantum ChromoDynamics in the infinite heavy quark ($Q=c,b$) mass limit ($m_Q \to \infty$) and in the chiral limit for the light quarks ($m_q \to 0,\;q=\,u,\,d,\,s$) can be used together to build up an effective chiral lagrangian for heavy and light mesons describing strong interactions among effective meson fields as well as their couplings to electromagnetic and weak currents, including the relevant symmetry breaking terms. The effective theory includes heavy ($Q \bar q$) mesons of both negative and positive parity, light pseudoscalars, as well as light vector mesons. We summarize the estimates for the parameters entering the effective lagrangian and discuss in particular some phenomenologically important couplings, such as $g_{B^* B \pi}$. The hyperfine splitting of heavy mesons is discussed in detail. The effective lagrangian allows for the possibility to describe consistently weak couplings of heavy ($B,\, D$) to light ($\pi,\, \rho, \, K^*,\,$ etc.) mesons. The method has however its own limitations, due to the requirement that the light meson momenta should be small, and we discuss how such limitations can be circumvented through reasonable ansatz on the form factors. Flavour conserving (e. g. $B^* \to B\, \gamma$) and flavour changing (e. g. $B \to K^* \, \gamma$) radiative decays provide another field of applications of effective lagrangians; they are discussed together with their phenomenological implications. Finally we analyze effective lagrangians describing heavy charmonium- like ($\bar Q Q$) mesons and their strong and electromagnetic interactions. The role of approximate heavy quark symmetries for this case and the phenomenological tests of these models are also discussed.
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.
The abstract, sentence by sentence
The approximate symmetries of Quantum ChromoDynamics in the infinite heavy quark ($Q=c,b$) mass limit ($m_Q \to \infty$) and in the chiral limit for the light quarks ($m_q \to 0,\;q=\,u,\,d,\,s$) can be used together to build up an effective chiral lagrangian for heavy and light mesons describing strong interactions among effective meson fields as well as their couplings to electromagnetic and weak currents, including the relevant symmetry breaking terms. The effective theory includes heavy ($Q \bar q$) mesons of both negative and positive parity, light pseudoscalars, as well as light vector mesons. We summarize the estimates for the parameters entering the effective lagrangian and discuss in particular some phenomenologically important couplings, such as $g_{B^* B \pi}$. The hyperfine splitting of heavy mesons is discussed in detail. The effective lagrangian allows for the possibility to describe consistently weak couplings of heavy ($B,\, D$) to light ($\pi,\, \rho, \, K^*,\,$ etc.) mesons. The method has however its own limitations, due to the requirement that the light meson momenta should be small, and we discuss how such limitations can be circumvented through reasonable ansatz on the form factors. Flavour conserving (e. g. $B^* \to B\, \gamma$) and flavour changing (e. g. $B \to K^* \, \gamma$) radiative decays provide another field of applications of effective lagrangians; they are discussed together with their phenomenological implications. Finally we analyze effective lagrangians describing heavy charmonium- like ($\bar Q Q$) mesons and their strong and electromagnetic interactions. The role of approximate heavy quark symmetries for this case and the phenomenological tests of these models are also discussed.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | The approximate symmetries of Quantum ChromoDynamics in the infinite heavy quark ($Q=c,b$) mass limit ($m_Q \to \infty$) and in the chiral limit for the light quarks ($m_q \to 0,\;q=\,u,\,d,\,s$) can be used together to build up an effective chiral lagrangian for heavy and light mesons describing strong interactions among effective meson fields as well as their couplings to electromagnetic and weak currents, including the relevant symmetry breaking terms. | background 0.45 | 0.16 | 0.29 | asserted 0.93 |
| s1 | The effective theory includes heavy ($Q \bar q$) mesons of both negative and positive parity, light pseudoscalars, as well as light vector mesons. | contribution 0.64 | 0.32 | 0.06 | asserted 1.00 |
| s2 | We summarize the estimates for the parameters entering the effective lagrangian and discuss in particular some phenomenologically important couplings, such as $g_{B^* B \pi}$. | contribution 0.97 | 0.75 | 0.10 | asserted 0.94 |
| s3 | The hyperfine splitting of heavy mesons is discussed in detail. | contribution 0.98 | 0.61 | 0.06 | asserted 0.96 |
| s4 | The effective lagrangian allows for the possibility to describe consistently weak couplings of heavy ($B,\, D$) to light ($\pi,\, \rho, \, K^*,\,$ etc.) mesons. | contribution 0.74 | 0.38 | 0.13 | hedged possibility 0.93 |
| s5 | The method has however its own limitations, due to the requirement that the light meson momenta should be small, and we discuss how such limitations can be circumvented through reasonable ansatz on the form factors. | limitation 1.00 | 0.50 | 0.07 | asserted 0.98 |
| s6 | Flavour conserving (e. g. | contribution 0.92 | 0.19 | 0.02 | unclear 0.89 |
| s7 | $B^* \to B\, \gamma$) and flavour changing (e. g. | contribution 0.95 | 0.27 | 0.02 | unclear 0.84 |
| s8 | $B \to K^* \, \gamma$) radiative decays provide another field of applications of effective lagrangians; they are discussed together with their phenomenological implications. | contribution 0.98 | 0.59 | 0.06 | asserted 0.98 |
| s9 | Finally we analyze effective lagrangians describing heavy charmonium- like ($\bar Q Q$) mesons and their strong and electromagnetic interactions. | contribution 1.00 | 0.74 | 0.08 | asserted 0.91 |
| s10 | The role of approximate heavy quark symmetries for this case and the phenomenological tests of these models are also discussed. | contribution 0.96 | 0.53 | 0.09 | asserted 0.65 |
What was examined, and what was not
- arXiv listing abstract v2
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
- S sections
- T body sentences
- C citations
- R cross-paper relations
Figures
Figures reproduced from arXiv:hep-ph/9605342 by the authors.
References
How later papers used it
236 later papers cite it
18 later papers by other authors are listed below: 7 used its method or tool, 10 used its data or values and 1 compared its own results with it.
Uses its method or tool 7 of 91 papers with this label
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Employed leading order effective Lagrangians constrained by heavy quark symmetry and chiral symmetry to evaluate leading contributions from charmed meson loops.
Production of hidden-charm molecular candidates in $\psi(4660)$ decays (2026) · 1.00 · paraphrased: the licence does not allow quoting
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Constructed its effective Lagrangian based on heavy quark symmetry and chiral symmetry.
Estimating strong decays of X(3915) and X(4350) (2011) · 0.96 · paraphrased: the licence does not allow quoting
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Constructed the effective Lagrangian describing the interaction between light and heavy flavor mesons with the help of chiral symmetry and heavy quark symmetry.
The molecular systems composed of the charmed mesons in the $H\bar{S}+h.c.$ doublet (2010) · 0.85 · paraphrased: the licence does not allow quoting
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Obtained its Lorentz structures from four-momenta using the effective Lagrangian approach.
$Z_b(10610)$ and $Z_b(10650)$ structures produced by the initial single pion emission in the $\Upsilon(5S)$ decays (2011) · 0.92 · paraphrased: the licence does not allow quoting
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Employed Lagrangians satisfying heavy quark symmetry and chiral symmetry to obtain interactions for heavy mesons and nucleons.
Exotic baryons from a heavy meson and a nucleon -- Negative parity states -- (2011) · 0.92 · paraphrased: the licence does not allow quoting
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For the pseudoscalar $η_c$, the couplings to open-charm meson pairs take the forms [ref] [equation] [equation] where $D$ and $D̅$ denote the pseudoscalar meson and antimeson, while $D^*$ and $D̅^*$ denote their vector partners.
Exploring charmonia $\eta_c(6S,7S)$ in the $\Lambda_c^+\bar{\Lambda}_c^-$ invariant mass spectrum of $B^0 \to K_S^0\Lambda_c^+\bar{\Lambda}_c^-$ (2026) · Baryonic decays of $η_c(6S)$ and $η_c(7S)$ · 0.90
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We construct the interaction effective Lagrangian for a pseudoscalar meson [ref] : [equation] Here, $^μ$ is the axial-vector current which is defined by [equation] where $ξ$ is the nonlinear representation written as [equation] with the pion decay constant $f_π ≃93 MeV$.
Analysis of $T_{cc}$ and $T_{bb}$ based on the hadronic molecular model and their spin multiplets (2023) · Lagrangian · 0.87
Uses its data or values 10 of 30 papers with this label
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Here $g_V = m_ρ/ f_π$ with the pion decay constant $f_π= 132 MeV$ [ref] and the $ρ$ meson mass $m_ρ=775.26 MeV$ [ref] .
Light hadronic decays of spin-0 partner of $X(3872)$ (2025) · Effective interaction Lagrangians · 1.00
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Used coupling constants g = 0.3 to 0.6.
Prediction of possible $DK_1$ bound states (2020) · 1.00 · paraphrased: the licence does not allow quoting
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We use $|h|=0.52$ [ref] in our calculations.
Revisiting the $DD^\ast$ chiral interactions with the local momentum-space regularization up to the third order and the nature of $T_{cc}^+$ (2022) · The LECs in subleading $πD^(∗)$ Lagrangians · 1.00
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Took the trilinear coupling as y approximately 0.8.
The nature of $X(3872)$ from high-multiplicity $pp$ collisions (2020) · 0.99 · paraphrased: the licence does not allow quoting
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Used the pion decay constant f_pi = 132 MeV.
Hidden-bottom hadronic transitions of $\Upsilon(10753)$ (2023) · 0.98 · paraphrased: the licence does not allow quoting
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Used the pion decay constant f_pi = 132 MeV.
Charmless decays of the spin-2 partner of $X(3872)$ (2024) · 0.96 · paraphrased: the licence does not allow quoting
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Used the pion decay constant f_pi = 132 MeV.
Hidden charmonium decays of spin-2 partner of $X(3872)$ (2024) · 0.93 · paraphrased: the licence does not allow quoting
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The values of the other parameters are $g_V=m_ρ/f_π$, $f_π=132 MeV$, $Λ_χ=1GeV$, $β_0=0.9$ and $λ=0.56GeV^-1$ [ref] .
The decays of the fully open flavor state $T_{c\bar{s}0}^{0}$ in a $D^{*}K^{*}$ molecule scenario (2022) · Numerical results and discussions · 0.90
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Used the couplings g_piD D* approximately 11 and g_piD* D* approximately 17 as given by this paper.
Production of Tetraquarks at the LHC (2014) · 0.81 · paraphrased: the licence does not allow quoting
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The already known ones are collected in the following, [equation] see Refs. [ref] , [ref] and [ref] , respectively.
Prediction of an exotic state around 4240 MeV with $J^{PC}=1^{-+}$ as C-parity partner of Y(4260) in molecular picture (2019) · Estimation of coupling constants · 0.91
Compares with it 1 of 3 papers with this label
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We reproduce the results of [ref] .
Radiative and meson decays of $Y(4230)$ in flavor SU(3) (2021) · Radiative and pionic decays: $D^*$ and $D_1$ mesons · 0.95
How the literature uses it
218 arXiv papers cite this in 365 sentences
Papers citing it · 203 sentences without a confident label
Builds on it 1 paper
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By extending the ground state heavy mesons to the excited mesons, pions to the other light mesons, and by including an additional hidden local symmetry [ref] , one gets more effective Lagrangians in Refs. [ref] .
A review of the open charm and open bottom systems · Heavy quark symmetry and effective Lagrangians · 0.82
Uses its method or tool 83 papers
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Using the effective Lagrangian in the heavy meson chiral perturbation theory (HM$χ$PT) [ref] and the method developed in literature [ref] , we obtain the effective potentials of $Y^ss̅$ and $Y^uu̅/dd̅$ states [ref] [equation] Here, the subscript of the sub-potential denotes the exchanged light meson.
Novel charmonium-like structures in the $J/\psi\phi$ and $J/\psi\omega$ invariant mass spectra · Introduction · 1.00
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Based on the effective Lagrangians [ref] and vector meson dominance (VMD) model, the amplitudes of the diagrams as shown in Fig. [ref] can be obtained and the details can be found in Ref. [ref] .
The nature of X(3900) and recognition of open charm effects · The recognition of the open charm effects in the process $e^+e^-→J/ψπ^0$ · 0.98
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To establish the effective potential which is responsible for binding the constituents, we adopt the following effective Lagrangians [equation] which are constructed under the heavy quark limit and chiral symmetry [ref] .
Identifying exotic hidden-charm pentaquarks · Introduction · 0.98
Uses its data or values 27 papers
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[ref] [equation] where the coupling constants $g_Σ_bNB̅=5.7$, $g_Σ_bNB̅^*=3.7$, and $g_Σ_b^*NB̅^*=6.9$ are obtained from Ref. [ref] .
Investigating the Molecular Nature of $\Lambda_b(6146)$ and $\Lambda_b(6152)$ via Strong Decays and Predicting Their Heavy-Quark Symmetry Partners · Theoretical formulism · 0.91
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The $P$-wave $D^*0→γD^0$ transition amplitude is given by [ref] [equation] and the parameter $g_3$ is fixed to be $g_3=1.77 ^-1$ with the $D^*0→γD^0$ branching ratio $35.3%$ [ref] and the $D^*0$ full width $Γ_D^*0=55.3$ [ref] , which can be obtained by using isospin symmetry to relate to the $D^*+$ full width and the $D^*+→π^+D^0$ and $D^*0→π^0D^0$ branching ratios [ref] .[The coupling constant of $D^*+→γD^+$, $g_3'$, evaluated with the measured full width and branching ratio is $g_3'=0.47 ^-1$, which is less than 1/3 of the $D^*0→γD^0$ coupling.
Possible precise measurements of the $X(3872)$ mass with the $e^+e^-\to\pi^0\gamma X(3872)$ and $p\bar p\to\gamma X(3872)$ reactions · eetopi0gamx · 0.97
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Most of them are taken from theoretical works [ref] , which have been listed in Tab. [ref].
Final-state rescattering mechanism of doubly-charmed baryon decays: $\mathcal{B}_{cc}\to\mathcal{B}_{c}V$ · Input parameters · 1.00
Compares with it 3 papers
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We reproduce the results of [ref] .
Radiative and meson decays of $Y(4230)$ in flavor SU(3) · Radiative and pionic decays: $D^*$ and $D_1$ mesons · 0.95
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The best estimate based on the analyses of both QCD sum rules and relativistic quark model, quoted in the review [ref] , is [equation] with an uncertainty around $20%$, which is also in agreement with our results given in Table [ref].
Determination of the strong coupling $g_{B^* B\pi}$ from semi-leptonic $B\to \pi \ell \nu$ decay · The normalized coupling $ĝ$ · 0.82
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For comparison, we list their expected absolute values based on chiral and heavy-quark symmetries combined with vector-meson dominance [ref] ; further determinations of these couplings via various methods can be found in the literature [ref] .
Dispersive Analysis of $D$- and $B$-Meson Form Factors with Chiral and Heavy-Quark Constraints · Couplings to the ρ(770) · 0.86
Cites it as background 35 papers
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By combining the chiral and heavy quark symmetry of QCD, the heavy meson chiral perturbative theory (HM$χ$PT) [ref] provides an effective framework to describe interactions between heavy and light mesons at low energy.
An improved calculation of the $D_{(s)}^*D_{(s)}V$ and $B_{(s)}^*B_{(s)}V$ couplings from light-cone sum rules · Introduction · 0.93
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The residual hyperfine splitting within each doublet is suppressed by $1/m_Q$, while the dynamics of the light quark is encoded systematically in heavy-hadron chiral effective theory [ref] .
Chiral Doubling of Heavy-Light Hadrons and the New Beauty--Strange $B_{s0}^*(5700)^0$ Candidate · Introduction · 0.92
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The effective Lagrangians were constructed considering both the heavy quark symmetry and chiral symmetry in Refs. [ref] (for a review, we refer to Ref. [ref] ).
Effect of charmed meson loops on charmonium transitions · Leading order effective Lagrangians · 0.83
Discussion
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