Strong Decays of Excited Heavy Mesons In Chiral Perturbation Theory

Adam F. Falk; Michael Luke

REVIEW

Strong Decays of Excited Heavy Mesons In Chiral Perturbation Theory

Adam F. Falk, Michael Luke

High Energy Physics - Phenomenology · Submitted 22 Jun 1992 · 15 references

Pith's verdict: A clean, correctly executed construction of the p-wave heavy-light chiral Lagrangian; the one real soft spot is that pole dominance in the two-pion section is asserted rather than estimated.

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arXiv IDhep-ph/9206241 (version 1)

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CitationAdam F. Falk, Michael Luke. "Strong Decays of Excited Heavy Mesons In Chiral Perturbation Theory." arXiv:hep-ph/9206241 (1992). https://pith.science/paper/O4YJO2KD

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The thing worth knowing: this is where the chiral Lagrangian for p-wave excited heavy mesons gets written down properly, and the machinery has held up. The S doublet (D*0, D1') and T doublet (D1, D*2) are packaged into the standard traceless symmetric tensors, the H–T mixing term is killed by reparameterization invariance (their Eq. 2.10 and the argument below it), and the T→Hπ transition is shown to vanish at s-wave by heavy quark spin symmetry, forcing a d-wave operator. That last point is the structural core and it is right. The single-pion rates reproduce Isgur–Wise, as the authors say up front [abstract:s3]; the new content is the systematic framework and the two-pion pole analysis, not the rate relations themselves. The paper is honest about what it inherits and what it adds, which is not universal in this literature. Where it is soft, in order of how much it matters. Section 4's pole dominance is an expectation, not a demonstration: contact terms are argued to be down by 1/Λχ, but the pole graphs themselves carry 1/ΔM with ΔM around 300 MeV, so the hierarchy is only about a factor of three. The paper never puts a number on that. The 'badly violated' 3:1:2 claim [abstract:s6] is qualitative and survives this, but the shape of Fig. 1 is less robust than the text implies. The M_S dependence is acknowledged [abstract:s5] and two masses are shown, so I would call that disclosed rather than hidden, though the on-shell D1'π threshold at 2300 MeV is a real qualitative switch that deserves more than a figure caption. Smaller things: h²/Λχ² is fit to the D*2 total width assuming one-pion saturation (Eq. 3.3), then used to predict Γ(D1→D*π) of about 7 MeV against a measured 20 MeV. The authors attribute the gap to D1–D1' mixing, which is plausible but untested here. And f' comes from a quark model they themselves distrust. Who should read it: anyone doing heavy-flavor chiral perturbation theory or excited charm spectroscopy. It is a builder's paper, not a controversial one. Recommendation: send it out. The structural claims are checkable and correct; the soft spots are the kind referees should press on, not reasons to hold it.

Referee report

Significance

The paper's structural contribution is genuine and durable: it supplies the general tensor formalism for arbitrary excited heavy mesons (Eqs. 2.5–2.6), a clean reparameterisation-invariance argument that removes a would-be leading mixing term, and a symmetry explanation for why the T doublet is narrow while the S doublet is broad. The one-pion results are explicit, falsifiable, and one of them (the D1 width) was subsequently borne out; the ratio in Eq. (3.2) is a parameter-free heavy-quark-symmetry statement checked against experiment. The two-pion analysis is presented honestly as an estimate contingent on unmeasured S-doublet masses and widths, and the observation that kinematic 1/M effects can overwhelm a spin-symmetry relation is a useful caution for the effective-field-theory community. The paper is also careful to state its own limitation (abstract:s5) rather than overclaim.

Major objections (3)

[Section 4, Eqs. (4.6)–(4.7)] The two terms in F(E1) have poles at E1 = M_D*0 − M_D and at E1 = M_D1 − M_D*0 respectively, i.e. they correspond to D*0 and D1' intermediate states. Since D*0 → Dπ and D1' → D*π, these two terms appear to feed different final states (Dππ and D*ππ), yet Eq. (4.7) applies the full F(E1) with a single channel-dependent coefficient α. Please state explicitly which term of Eq. (4.6) belongs to which channel, or, if both terms are intended to contribute to each channel, explain the dynamical reason. If the two terms in fact belong to different final states, Eq. (4.7) and the widths quoted in Fig. 1 must be recomputed, since this is the equation on which the Section 4 numbers and the abstract's headline ratio rest.
[Section 4 and abstract:s6] The claim that the 3:1:2 relation is 'badly violated' is the paper's headline quantitative statement, but its robustness is not established. The pole amplitudes are enhanced only by 1/ΔM with ΔM of order a few hundred MeV, while the neglected dimension-five contact terms such as Tr[H T A A] are suppressed by 1/Λχ with Λχ ≈ 1 GeV, so the two are not separated by a large hierarchy; no numerical estimate of the contact-term correction is given. In addition, Fig. 1 shows only M_D*0 = M_D1' = 2300 and 2400 MeV, and the qualitative behaviour of the D*2 → D*ππ channel changes between these two points (on-shell versus off-shell D1'π). The authors acknowledge the mass dependence (abstract:s5), but the reader cannot judge whether the violation is a robust prediction or an artifact of the two mass points chosen. A short quantitative statement — the range of M_S and |f''| over which the ratio remains badly violated, and an order-of-magnitude estimate of the contact-term contribution — would make the claim defensible.
[Section 4, after Eq. (4.6)] The statement that 'the cross terms in |A|² integrate to zero' is asserted without justification. The final-state pions π⁻ and π⁰ are distinguishable, so the interference between the two pole orderings is not obviously zero by Bose symmetry. Please supply the argument (an isospin or charge-conjugation selection rule, or an explicit demonstration) or retain the interference terms; if the interference does not vanish, the widths in Eq. (4.7) change and the comparison in Fig. 1 shifts.

Minor objections (9)

[Abstract] The abstract promises 'branching ratios' for the two-pion decays, but Section 4 presents partial widths only. Branching ratios require a total width that is itself uncertain; either compute them with a stated total width or change the wording to 'partial widths'.
[Section 2, Eqs. (2.23)–(2.24) and Section 3, Eq. (3.6)] A quark-model estimate is given for f' (Eq. 2.23) but not for f'', even though the same matrix-element method applies to the H–S axial coupling. Since the D*0 and D1' widths in Eq. (3.6) are quoted as |f''|² times a kinematic factor, an estimate of f'' would make those numbers concrete and would also fix the horizontal range of Fig. 1.
[Section 3, Eq. (3.1)] The numerical coefficients 5.51 × 10⁷, 2.03 × 10⁷ and 2.05 × 10⁷ are quoted without units. Please state explicitly that these are in MeV with h²/Λχ² in MeV⁻², so that Eq. (3.3) can be checked directly.
[Section 3, Eq. (3.5)] The first line is labelled Γ(D*0 → Dπ⁻) but the formula is written for a generic D; please specify the charge channel (D⁺π⁻ versus D⁰π⁰) and note that the quoted numbers are for the charged-pion mode only, as is done for Eq. (3.6).
[Section 3, after Eq. (3.4)] The statement that the D1 width discrepancy is 'undoubtedly due to mixing' with the D1' is stronger than the calculation supports. A softened phrasing, with a quantitative estimate of the mixing contribution or an explicit reference to where it was computed, would be more appropriate.
[Section 3, after Eq. (3.3)] The extraction of h²/Λχ² assumes that the D*2 total width is saturated by the one-pion modes. This is subsequently used to argue that the two-pion width is small. Please state the value of |f''| (or the D1' width) below which this saturation assumption is self-consistent according to Fig. 1.
[Section 2, Eqs. (2.20)–(2.22)] The Clebsch–Gordan factors in the matrix elements are typeset ambiguously (for example '2√23 g_A' and '√ 23 f′'). Please display these factors unambiguously so that the reader can verify the step from Eq. (2.20) and Eq. (2.22) to |f'| = (2/√3) g_A.
[References] Several supporting references ([6], [8], [9]) are listed as preprints or 'to appear'. Please update them to their published versions at proof stage, since the reparameterisation-invariance argument of Section 2 leans on [9].
[Figure 1] The caption should state the value of |f'| assumed (Eq. 2.23) and the range of |f''| plotted, so that the figure can be read without returning to the text.

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The paper

The paper claims

Excited heavy mesons decay by pion emission through operators that heavy quark spin symmetry forbids at leading order, and the resulting widths can be computed with a chiral Lagrangian built for p-wave states.

The reading

The paper asks how the light pseudoscalar mesons (pions and kaons) couple to heavy mesons whose light degrees of freedom sit in an orbital p-wave, that is, the excited charmed states such as the $D_1(2420)$ and $D_2^*(2460)$. The authors write down the most general Lagrangian consistent with both chiral symmetry and heavy quark spin-flavour symmetry for these excited fields, organised as a derivative expansion, and then use it to compute the strong one-pion and two-pion decays. The reason this matters is that heavy quark symmetry alone fixes ratios of decay rates but says nothing about absolute sizes; a chiral Lagrangian supplies the normalisations and lets the authors check them against power counting and against the measured $D_2^*$ width. The paper's sharpest statement is a negative one: heavy quark spin symmetry predicts the two-pion widths of $D_2^*$ and $D_1$ in the ratio $3:1:2$, but once physical masses are inserted into the pole amplitudes that ratio is badly broken, so the symmetry relation is not a useful guide for real charmed mesons.

What carries the argument

The machinery is a heavy-light chiral Lagrangian for p-wave fields: a traceless symmetric Lorentz tensor $H^{mu_1 ... mu_k}_a$ with $k = j - 1/2$, subject to $v_{mu_1} H^{mu_1 ...} = gamma_{mu_1} H^{mu_1 ...} = 0$, which packages the two degenerate heavy mesons of spin $j pm 1/2$ into one object transforming simply under heavy quark spin symmetry. For the lowest p-wave excitations this reduces to the scalar-pseudovector doublet $S_a$ containing $D^{*0}$ and $D_1'$, and the $T^mu_a$ doublet containing $D_1$ and $D_2^*$. The second piece of machinery is reparameterisation invariance of the heavy quark velocity, $v^mu to v^mu - q^mu/M$: this forbids the naively allowed mixing term $Tr[H_a (iD_mu T^mu)_a]$, which would otherwise be a leading contribution, and it also forbids derivative-acting-on-heavy-field couplings such as $Tr[(iD_mu H)_a T^{mu}_b A^...]$. Together these two constraints determine which operators survive at each order in the derivative expansion.

Extended reading notes

Core claim

The central result is that the single-pion transition $T^mu to H pi$, where $T^mu$ is the $J^P = 1^+, 2^+$ doublet containing the $D_1$ and $D_2^*$, cannot proceed in an s-wave: the would-be coupling $Tr[H_a T^mu_b A_{mu ba} gamma_5]$ vanishes identically by heavy quark spin symmetry. The decay must instead go through d-wave operators with an extra derivative, Eq. (2.16), which suppresses the rate by two powers of the pion momentum relative to an s-wave and makes these states comparatively narrow. Working this out gives $Gamma(D_2^{*0} to D^+ pi^-) / Gamma(D_2^{*0} to D^{*+} pi^-) = 2.7$, in agreement with the measured $2.4 pm 0.7$, and the absolute rates fix $h^2/Lambda_chi^2 approx 1/(2 text{ GeV})^2$, consistent with naive power counting. The same Lagrangian then yields a prediction of about $7$ MeV for the total one-pion width of the $D_1$, well below the observed $20^{+9}_{-5}$ MeV, which the authors attribute to $1/M$ mixing of the $D_1$ with the much broader $D_1'$. For the two-pion decays the authors argue that dimension-five contact terms are suppressed and that pole graphs through an intermediate $D^{*0}$ or $D_1'$ dominate; because those resonances are broad, the full momentum dependence of their widths must be kept in the propagator. The heavy quark limit ratio $Gamma(D_2^* to D^* pi pi) : Gamma(D_1 to D^* pi pi) : Gamma(D_1 to D pi pi) = 3:1:2$ then fails badly in practice, since the pole amplitudes depend sensitively on the physical masses.

Load-bearing premise

The two-pion analysis assumes these decays are dominated by pole graphs through an intermediate $D^{*0}$ or $D_1'$, with contact terms suppressed by one power of the chiral scale; if that suppression fails, the computed size of the symmetry breaking in the $3:1:2$ ratio does not follow.

Editorial extensions

If this is right

The one-pion width of the $D_1$ should be about $7$ MeV from the d-wave operator alone; the observed excess over that points to $1/M$ mixing with the broad $D_1'$, which can be checked once the $D_1'$ is found.
The scalar-pseudovector doublet $D^{*0}$, $D_1'$ should be very broad, with widths scaling like $|f''|^2$ times hundreds of MeV, and the predicted numbers move by roughly a factor of two between assumed masses of 2.3 and 2.4 GeV.
Two-pion decays of $D_2^*$ and $D_1$ are dominated by pole graphs, so their branching ratios measure the masses and widths of the unseen $D^{*0}$ and $D_1'$ rather than testing heavy quark symmetry directly.
The same construction extends to excited heavy baryons and to excited bottom mesons, where the $1/M$ kinematic breaking of symmetry relations should be smaller and the heavy quark limit more nearly realised.
Semileptonic $B$ decays to an excited $D$ followed by strong pion emission give an additional channel, $B to D pi ell nu$, in which these couplings enter.

Reading between the lines

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

The paper's logic suggests a general lesson beyond charm: whenever a symmetry-forbidden leading operator is replaced by a derivative-suppressed one, the symmetry's ratio predictions survive only where phase space is nearly universal, so kinematic $1/M$ breaking should be computed rather than estimated by dimensional analysis.
If the $D_1$ width excess is really $D_1$–$D_1'$ mixing, the mixing angle is measurable, and the same angle would shift the two-pion pole amplitudes; a combined fit to one-pion and two-pion data would over-determine the $D_1'$ mass and width.
The momentum-dependent width inserted into the resonance propagator is a modelling choice that could be tested by comparing the two-pion invariant mass distribution near the $D^{*0}$ and $D_1'$ thresholds against a constant-width Breit-Wigner.
Because the $D_s$ system lacks phase space for the allowed channels, the same Lagrangian predicts strongly suppressed $D_{s1}$ strong decays, which offers a consistency check on the whole operator set rather than on any single coupling.

Original abstract

We construct an effective Lagrangian describing the interaction of soft pions and kaons with mesons containing a heavy quark and light degrees of freedom in an orbital $p$ wave. The formalism is easily extended to heavy mesons and baryons in arbitrary excited states. We calculate the leading contributions to the strong decays $\dtwo\to\d\pi$, $\dtwo\to\dstar\pi$ and $\done\to\dstar\pi$. We confirm the relations between the rates previously obtained by Isgur and Wise using heavy quark symmetry, and find that the absolute widths are consistent with na\"\i ve power counting. We also estimate the branching ratios for the two pion decays $\dtwo\to\dstar\pi\pi$, $\done\to\dstar\pi\pi$ and $\done\to\d\pi\pi$, which are dominated by pole graphs. Our predictions depend on the masses and widths of the as yet unseen scalar-pseudovector $p$-wave doublet. Heavy quark spin symmetry predicts $\Gamma(\dtwo\to\dstar\pi\pi): \Gamma(\done\to\dstar\pi\pi):\Gamma(\done\to\d\pi\pi)=3:1:2$, but this relation is badly violated in practice because $1/M$ effects arising purely from kinematics are large.

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, 5,815 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.94 · human participants 0.01 · controlled comparison 0.04 · randomized assignment 0.01 · states quantitative result 0.93 · compares with prior results 0.96 · mentions code or data availability 0.02.

The abstract, sentence by sentence

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

We construct an effective Lagrangian describing the interaction of soft pions and kaons with mesons containing a heavy quark and light degrees of freedom in an orbital $p$ wave. The formalism is easily extended to heavy mesons and baryons in arbitrary excited states. We calculate the leading contributions to the strong decays $\dtwo\to\d\pi$, $\dtwo\to\dstar\pi$ and $\done\to\dstar\pi$. We confirm the relations between the rates previously obtained by Isgur and Wise using heavy quark symmetry, and find that the absolute widths are consistent with na\"\i ve power counting. We also estimate the branching ratios for the two pion decays $\dtwo\to\dstar\pi\pi$, $\done\to\dstar\pi\pi$ and $\done\to\d\pi\pi$, which are dominated by pole graphs. Our predictions depend on the masses and widths of the as yet unseen scalar-pseudovector $p$-wave doublet. Heavy quark spin symmetry predicts $\Gamma(\dtwo\to\dstar\pi\pi): \Gamma(\done\to\dstar\pi\pi):\Gamma(\done\to\d\pi\pi)=3:1:2$, but this relation is badly violated in practice because $1/M$ effects arising purely from kinematics are large.

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 construct an effective Lagrangian describing the interaction of soft pions and kaons with mesons containing a heavy quark and light degrees of freedom in an orbital $p$ wave.	contribution 0.53	0.95	0.06	asserted 1.00
s1	The formalism is easily extended to heavy mesons and baryons in arbitrary excited states.	method 0.51	0.56	0.04	asserted 0.94
s2	We calculate the leading contributions to the strong decays $\dtwo\to\d\pi$, $\dtwo\to\dstar\pi$ and $\done\to\dstar\pi$.	contribution 0.82	0.96	0.12	asserted 1.00
s3	We confirm the relations between the rates previously obtained by Isgur and Wise using heavy quark symmetry, and find that the absolute widths are consistent with na\"\i ve power counting.	result 0.96	0.93	0.30	asserted 0.99
s4	We also estimate the branching ratios for the two pion decays $\dtwo\to\dstar\pi\pi$, $\done\to\dstar\pi\pi$ and $\done\to\d\pi\pi$, which are dominated by pole graphs.	result 0.54	0.93	0.07	asserted 0.88
s5	Our predictions depend on the masses and widths of the as yet unseen scalar-pseudovector $p$-wave doublet.	limitation 0.98	0.67	0.07	asserted 1.00
s6	Heavy quark spin symmetry predicts $\Gamma(\dtwo\to\dstar\pi\pi): \Gamma(\done\to\dstar\pi\pi):\Gamma(\done\to\d\pi\pi)=3:1:2$, but this relation is badly violated in practice because $1/M$ effects arising purely from kinematics are large.	result 0.91	0.56	0.32	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.99 · evidence from proof or derivation 0.99.

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.** Full two pion widths for D∗ 2 and D1 as functions of the D1 ′ width, for MD∗ 0 = MD1 ′ = 2300 MeV and 2400 MeV. Note that for MD1 ′ = 2300 MeV the D∗ 2 partial width is nonzero as the D1 ′ width goes to zero, since the D1 ′ π intermediate state may be produced on shell. In this limit the D∗ 2 two pion width approaches the D∗ 2 → D1 ′ π partial width. 14

Figures reproduced from arXiv:hep-ph/9206241 by the authors.

References

Works cited (15)

[1]

N. Isgur and M. B. Wise, Phys. Lett. B232 (1989) 113, Phys. Lett. B237 (1990) 527

View on Pith 1989
[2]

M. B. Wise, Caltech preprint CALT-68-1765 (1992)

View on Pith 1992
[3]

G. Burdman and J. Donoghue, U. Mass. preprint UMHEP-365 ( 1992)

View on Pith 1992
[4]

T.-M. Yan, H.-Y. Cheng, C.-Y. Cheung, G.-L. Lin, Y. C. Lin and H.-L. Yu, Cornell preprint CLNS 92/1138 (1992)

View on Pith 1992
[5]

P. Cho, Harvard preprint HUTP-92/A014 (1992)

View on Pith 1992
[6]

A. F. Falk, SLAC preprint SLAC-PUB-5689 (1991), to appea r in Nucl. Phys. B

View on Pith 1991
[7]

Particle Data Group, Phys. Lett. B232 (1990) 1

View on Pith 1990
[8]

A. F. Falk, M. Luke and M. Neubert, SLAC and UCSD preprint S LAC-PUB-5771 and UCSD/PTH 92-09 (1992)

View on Pith 1992
[9]

M. Luke and A. V. Manohar, UCSD preprint UCSD/TH 92-15 (19 92)

View on Pith
[10]

N. Isgur and M. B. Wise, Phys. Rev. Lett. D66 (1991) 1130

View on Pith 1991
[11]

See, for example, H. Georgi, Weak Interactions and Modern Particle Theory , Ben- jamin/Cummings Publishing Co., Menlo Park, CA (1984)

View on Pith 1984
[12]

A. Manohar and H. Georgi, Nucl. Phys. B234 (1984) 189

View on Pith 1984
[13]

S. Godfrey and N. Isgur, Phys. Rev. D32 (1985) 189; S. Godfrey and R. Kokoski, Phys. Rev. D43 (1991) 1679

View on Pith 1985
[14]

J. Rosner, Comm. Nucl. Part. Phys. 16 (1986) 109

View on Pith 1986
[15]

C. L. Y. Lee, M. Lu and M. B. Wise, Caltech preprint CALT-6 8-1771 (1992) 13 Figure Captions Fig. 1. Full two pion widths for D∗2 and D1 as functions of the D1′ width, for MD∗0 = MD1′ = 2300 MeV and 2400 MeV. Note that for MD1′ = 2300 MeV the D∗2 partial width is nonzero as the D1′ width goes to zero, since the D1′π intermediate state may be produced on s...

View on Pith 1992

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Study of $P_c(4457)$, $P_c(4440)$, and $P_c(4312)$ in a quasipotential Bethe-Salpeter equation approach
hep-ph 2019-03 conditional novelty 4.0 of 10

A coupled-channel Bethe-Salpeter calculation assigns the three LHCb pentaquarks to Sigma_c Dbar* and Sigma_c Dbar molecular states with spin-parities 1/2-, 3/2-, and 1/2-.
$Z_b(10610)^\pm$ and $Z_b(10650)^\pm$ as the $B^*\bar{B}$ and $B^*\bar{B}^{*}$ molecular states
hep-ph 2011-06 conditional novelty 4.0 of 10

The Z_b(10610) and Z_b(10650) states can be explained as loosely bound B*B̄ and B*B̄* molecular states within a one-boson-exchange model that includes S-wave and D-wave mixing.
X(3872) as a molecular DD* state in a potential model
hep-ph 2009-10 conditional novelty 4.0 of 10

A coupled-channel potential model with a tunable cutoff of about 1.2 GeV produces a bound DD* state with the quantum numbers of the X(3872), and also predicts bound BB* states in the isoscalar limit.
X(3872) as a hadronic molecule and its decays to charmonium states and pions
hep-ph 2009-03 conditional novelty 4.0 of 10

Treating the X(3872) as a D-D* hadronic molecule, the authors compute its decay widths into chi_cJ plus pions and into J/psi plus pions or photons, producing a decay pattern that can be compared with experiment.
The hidden-charm pentaquark and tetraquark states
hep-ph 2016-01 accept novelty 2.0 of 10

A comprehensive review argues that the charged Zc/Zb states and the LHCb Pc(4380)/Pc(4450) pentaquarks are multiquark candidates, and catalogues the molecular, tetraquark, and diquark interpretations along with their ...

How the literature uses it

45 arXiv papers cite this in 69 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 · 44 sentences without a confident label

Uses its method or tool 10 papers

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
According to the heavy quark limit and chiral symmetry, the interactions of the light pesudoscalar, vector and scalar mesons with the S-wave heavy flavor mesons were constructed as [ref] [equation] where the multiplet fields are expressed as $H_a^(Q)=1+v/2[𝒫^*_aμγ^μ-𝒫_aγ_5]$, $H^(Q̅)=[P_a^*μγ_μ-P_aγ_5] 1-v/2$, $H̅=γ_0H^†γ_0$ with $v=(1,0)$, $𝒫^(*)T =(D^(*)0,D^(*)+,D_s^(*)+)$ or $(B^(*)-,B̅^(*)0,B̅_s^(*)0)$, $𝒫^(*)T =(D̅^(*)0,D^(*)-,D_s^(*)-)$ or $(B^(*)+,B^(*)0,B_s^(*)0)$, which satisfy the normalization relations $⟨0|𝒫|Qq̅(0^-)⟩=⟨0|𝒫|Q̅q(0^-)⟩=√(M_𝒫)$ and $⟨0|𝒫^*_μ|Qq̅(1^-)⟩=⟨0|𝒫^*_μ|Q̅q(1^-)⟩=ϵ_μ√(M_𝒫^*)$.

Hadronic molecules with both open charm and bottom · The effective potential of $B_c$-like molecules · 0.91
One can also start from the effective Lagrangian given in Ref. [ref] , [equation] where $σ_i$ $(i=1,2,3)$ are the Pauli matrices, and $S⃗_c$ and $H̅_c$ are the heavy quark spin doublets of $(Σ_c,Σ_c^*)$ and $(D̅,D̅^*)$ in the two-component notation [ref] (see, e.g., Refs. [ref] for the four-component notation), [equation] This Lagrangian, Eq. [ref], gives the $D̅^(*)Σ_c^(*)$ $S$-wave interaction which is the leading order of the momentum expansion.

Decays of $P_c$ into $J/\psi N$ and $\eta_cN$ with heavy quark spin symmetry · Dbar Sigmac interaction · 0.84

Uses its data or values 10 papers

In Ref. [ref] , the coupling constant $g = 0.75$ was roughly estimated by the quark model.

$Z_c(4025)$ as the hadronic molecule with hidden charm · Introduction · 1.00
[ref], which have been determined in literatures [ref] .

$\bar{D}\Sigma^*_c$ and $\bar{D}^*\Sigma_c$ interactions and the LHCb hidden-charmed pentaquarks · $D̅Σ^*_c$, $D̅^*Σ_c$ and $D̅^*Σ^*_c$ interactions · 0.97
The coupling constants are as follows, [equation] The effective Lagrangian between $σ$ and heavy mesons are [equation] and the relevant coupling constants are [equation] In this work, we choose the coupling constants $g=0.59$, $g_V=5.8$, $β=0.9$, $λ=0.56 GeV^-1$, $f_π=0.132$ GeV and $g_σ=0.76$ as in Refs. [ref] .

The relativistic three-body scattering and the $D^0D^{*+}-D^+D^{*0}$ system · The interaction Lagrangian · 1.00

Cites it as background 4 papers

Heavy hadron chiral perturbation theory () describes the interaction of heavy strongly interacting hadrons with (psuedo-)Goldstone bosons whose momentum satisfies $p≪Λ_χ$ where $Λ_χ∼1 GeV$ [ref] .

Off-shell vertices in heavy particle effective theories and $B\rightarrow D\pi \ell \nu$ · Heavy hadron chiral perturbation theory · 0.99
The interactions between hadrons consisting of one or more heavy quarks, say $c$ and $b$ due to their much larger masses than the typical QCD scale, can be constructed systematically under the guidance of heavy quark symmetries[For the spirits of heavy quark symmetries, we refer to the pioneer works [ref] and reviews [ref] ] and such Lagrangians can be found in, e.g. Refs. [ref] .

A survey of heavy-heavy hadronic molecules · Lagrangian from heavy quark spin symmetry · 0.90
The coupling constant $g$ was studied by many theoretical approaches, such as quark model [ref] and QCD sum rule [ref] .

Coupled-channel analysis of the possible $D^{(*)}D^{(*)}$, $\bar{B}^{(*)}\bar{B}^{(*)}$ and $D^{(*)}\bar{B}^{(*)}$ molecular states · The lagrangians and the coupling constants · 0.91

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

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