Heavy Mesons In A Relativistic Model

J. W. Van Orden; J. Zeng; W. Roberts

REVIEW

Heavy Mesons In A Relativistic Model

J. Zeng, J. W. Van Orden, W. Roberts

High Energy Physics - Phenomenology · Submitted 9 Dec 1994 · 1 reference

Pith's verdict: A competent, checkable spectator-equation model of heavy mesons; the derivation and numerics hold up, but the fit is undocumented and the two HQET numbers inherit an unquantified gauge sensitivity.

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arXiv IDhep-ph/9412269 (version 1)

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CitationJ. W. Van Orden, J. Zeng, W. Roberts. "Heavy Mesons In A Relativistic Model." arXiv:hep-ph/9412269 (1994). https://pith.science/paper/R2S6KFW7

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Bottom line: this is a solid, checkable model paper. The Gross/spectator equation, previously used for NN scattering and equal-mass light mesons, is used here as the starting point for a systematic 1/m_Q expansion of heavy-light and heavy-heavy bound states. That combination is new, and it buys something worth having: a manifestly covariant framework that reduces to the Dirac equation for the light quark in the static limit, which is exactly the property you want if you are going to talk to HQET. The reduction from Bethe-Salpeter through Eqs. 1.8 and 4.6 is laid out step by step and I could follow it, and three independent numerical methods agree to about a percent. The spectra land within tens of MeV across D, D*, B, B*, D_s, B_s, charmonium and bottomonium. That is a real result, not a fit dressed up as one, since only a subset of states was used to fix parameters and the rest are postdictions. The authors are also candid about what they don't control, which I appreciate more than I appreciate most papers' confidence. Two soft spots. First, the fitting is never described: seven parameters, no chi-squared, no uncertainties, and Tables III and IV don't mark which states were fitted and which were predicted. That is the first thing I'd ask for. Second, the hyperfine splittings come out too large in every sector, which they attribute to truncation and to the arbitrary alpha_s(r) form. On the stress-test concern, I'd push back on the reader's framing: E_SR and E_VR are spin-independent, so they shift the centroid, which the fit absorbs, and don't directly contaminate the hyperfine splittings. The real worry is the one the stress-test note lands on: those terms enter the wave functions, and hence Lambda-bar = 0.45 GeV and lambda_1 = 0.67 GeV^2, and the L-dependent fine structure. The paper doesn't quantify that sensitivity, and it also admits the heavy quark is exactly on shell here whereas HQET lets it go off shell, so their 1/m_Q set may not be HQET's 1/m_Q set. They say so; it's a caveat on the two numbers, not a flaw in the spectra. Who should read it: people building covariant quark models, and anyone who wants a worked example of spectator-equation machinery applied to heavy flavors. The HQET values are a bonus, not a determination. Worth a serious referee, with a request for the fit table and a gauge-sensitivity check.

Referee report

Significance

If the model holds, it supplies a covariant, systematically improvable alternative to the relativized Schroedinger quark model of Godfrey and Isgur, and it provides nonperturbative inputs (Lambda-bar, lambda_1, and eventually Isgur-Wise functions) that HQET alone cannot give. The strengths are real and should be weighed: the derivation from the Bethe-Salpeter equation through the spectator reduction to the Hamiltonians of Eqs. (1.8) and (4.6) is explicit and checkable; three independent numerical methods agree to about 1% (Sec. III.A); the authors state their gauge and retardation assumptions openly rather than burying them (Sec. IV.B); and the paper makes falsifiable predictions for unobserved states, including the B_c spectrum (Fig. 9) and the predicted multiplet inversion for l=2 (Sec. IV.A). The postdictions in Tables III and IV are genuine rather than circular, provided the fitted subset is identified.

Major objections (4)

[Sec. IV.B, Eqs. (4.6f-g), Table V] The first-order corrections include the retardation terms H_SR and H_VR, which the authors state are gauge dependent, would vanish in Coulomb gauge, and rest on a scalar retardation ansatz they themselves call 'uncontrolled' (Sec. IV.B). Table V shows that E_SR + E_VR and E_c are each comparable to or larger than E_hyp and E_so. The authors' acknowledgement is appropriate, but the consequence is not quantified. Because E_c, E_SR and E_VR contain L^2/2r^2 and d/dr structures, they are not absorbed into a single spin-independent centroid: they shift the S-P and higher-L level positions and, through the fit, the parameters b, c and alpha_crit, and hence the zeroth-order wave functions from which Lambda-bar and lambda_1 are computed. The concrete test I would ask for is a repeat of the fit with H_SR = H_VR = 0 (the Coulomb-gauge limit), reporting the shift in the fitted parameters, in the predicted masses of Figs. 5-11, and in Lambda-bar and lambda_1. Without that, the claim of systematic improvability is not yet supported at the precision the paper quotes.
[Sec. IV.A, Figs. 5-9] The paper states that sqrt(<p^2>)/m_c is approximately 1/2 and approximately 1 for the b-c system, and correctly notes that the neglected higher-order terms should therefore be considerably larger for charmed than for bottom systems. Nevertheless the predicted masses in Figs. 5-9 are quoted to 10 MeV with no uncertainty attached, including for the c-cbar and b-c spectra where the expansion parameter is of order unity. An uncertainty estimated from the size of the retained first-order correction, or from the residual dependence on the renormalization scale in Eq. (2.8), should be attached to each prediction, and the paper should state explicitly which entries in Figs. 5-9 are quantitative and which are indicative only.
[Sec. IV.A, HQET parameter estimates] The values Lambda-bar = 0.45 GeV and lambda_1 = 0.67 GeV^2 are quoted without derivation. The text defines lambda_1 through <M(v)| hbar(iD)^2 hbar |M(v)>/(2M) = 2M lambda_1 and states that the states in the bra and ket correspond to the zeroth-order calculation, but it does not show how 0.67 GeV^2 follows from the zeroth-order wave functions; a reader cannot reproduce the number from the formulas given. The sign convention also needs to be stated explicitly, since the standard HQET convention gives a negative lambda_1 (lambda_1 = -mu_pi^2/2), whereas the paper identifies its quantity with a positive kinetic energy. Finally, no uncertainty is attached to either number, despite the paper's own caveat that the spectator equation's 1/m_Q terms may not correspond to all of the 1/m_Q terms of HQET.
[Sec. IV, Tables III and IV] The text states that the model parameters were adjusted 'to fit the masses of a selection of mesons' but does not say which states were used. Without that information the reader cannot distinguish fitted entries from genuine postdictions in Tables III and IV or in Figs. 5-11, which is essential to assessing the claim of reasonable agreement. The fitted states should be marked explicitly in the tables, and the number of fitted masses should be stated alongside the number of adjusted parameters.

Minor objections (8)

[Sec. III.A] The text refers to 'three separate numerical solutions ... using two different techniques,' but shooting is then presented as a variant of direct integration. The counting should be made consistent, and the three procedures should be named once and used consistently thereafter.
[Table V caption] The caption should flag that E_c, E_SR and E_VR are gauge dependent, as the body text states. A reader consulting the table alone will otherwise take the decomposition at face value.
[Sec. II.A, Eq. (1.8b); Sec. II.B, Eq. (4.6c)] The off-diagonal mixing terms are neglected: the third term of Eq. (1.8b) mixes j_1 by unity, and the tensor force mixes L by 0 or 2. The paper acknowledges that these do not affect the spectrum at the quoted order, but the J^P labels in Figs. 5-9, particularly for the nearly degenerate 1+ states, are therefore not mass eigenstates. A sentence stating that the labels are approximate, and if feasible the mixing angles for the most affected multiplets, would help the reader.
[Sec. II.A, Eq. (2.8)] The units of the parameters mu_i in the vector potential are not stated, and the overall normalization of the error-function representation is not given. These should be specified so that the potential can be reproduced.
[Sec. IV.A] The statement that the root mean square momentum of the zeroth-order wave function is approximately 0.9 GeV does not say for which state or states this holds. The state dependence should be given, since the convergence argument rests on it.
[Abstract and Sec. IV] 'Reasonable agreement' should be quantified in the abstract, for example as agreement within roughly 30 MeV for states below the strong-decay threshold, so that the claim can be checked against Tables III and IV.
[Fig. 2] The caption describes the horizontal variable as 1/mu while the axis is labelled 1/(GeV). The caption should state that mu is the variational size parameter of Eq. (6.0) and that the plotted quantity is its inverse in GeV.
[References] Reference [3] is spelled 'Neub ert' in the text and should read Neubert. The citation list should also be checked for the annihilation-order statement in Sec. II.B, where refs. [13] and [4] are cited for an alpha_s^2/m_Q^2 result.

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The paper

The paper claims

A spectator-equation quark model with a linear confining plus Coulomb kernel, expanded in powers of 1/m_Q, reproduces heavy meson masses to tens of MeV and yields the HQET parameters Lambda-bar = 0.45 GeV and lambda_1 = 0.67 GeV^2.

The reading

Quarks are never seen alone; they are locked inside particles like the pion, the D meson, and the B meson. Predicting the masses of those bound states from a quark-level force is an old problem, and the standard tool, the nonrelativistic Schrodinger equation, is a poor fit for light quarks moving at close to the speed of light. This paper replaces the Schrodinger equation with the spectator equation, a relativistic bound-state equation in which one quark (the heavy one) is held on its mass shell while the other is treated fully off shell. The force between the quarks is a linear confining spring plus a Coulomb-like gluon exchange whose strength runs with distance, the same ingredients used in the Godfrey-Isgur model. Because the heavy quark is heavy, the authors expand the equation in powers of 1/m_Q: to first order for mesons with one heavy quark, to second order for mesons with two. The expansion produces, at each order, a Hamiltonian with recognizable pieces: a kinetic correction, a spin-orbit term, a hyperfine (spin-spin plus tensor) term, and two retardation terms that most quark models drop. The authors solve the resulting equations three ways (direct integration, shooting, and a variational matrix diagonalization) and fit seven parameters to about ten measured meson masses. The fit lands within roughly 10 to 30 MeV of the measured D, D*, B, B*, D_s, and B_s masses, and within similar accuracy for the charmonium and bottomonium states below the open-flavor thresholds. From the fitted wave functions they extract two heavy quark effective theory parameters, Lambda-bar = 0.45 GeV and lambda_1 = 0.67 GeV^2.

Extended reading notes

Core claim

The spectator (Gross) equation with a scalar linear confining potential plus a vector running-Coulomb potential, expanded to order 1/m_Q for Q-qbar mesons and order 1/m_Q^2 for Q-Qbar mesons, reproduces the observed heavy meson masses to within roughly 10 to 30 MeV, and the resulting leading-order wave functions give Lambda-bar = 0.45 GeV and lambda_1 = 0.67 GeV^2. If correct, this is a covariant, systematically improvable alternative to the relativized Schrodinger quark model, and it supplies nonperturbative inputs that HQET alone cannot provide.

Load-bearing premise

The treatment of the kernel's off-shell and retardation structure. The authors assume the scalar confining potential depends only on the exchanged four-momentum squared Q^2 (Eq. 13-14) and use the Lorentz gauge for the vector exchange, which generates the retardation terms H_SR and H_VR in Eq. 4.6f-g. Table V shows E_SR + E_VR is comparable in magnitude to E_hyp and E_so, the spin-dependent terms that produce the hyperfine splittings the paper is testing. The authors themselves call the scalar retardation assumption 'uncontrolled' in Section IV.B and note that in Coulomb gauge these terms would vanish entirely. A first-order correction of the same size as the effect being predicted, whose sign and size depend on an unforced gauge and ansatz choice, is the fragile premise.

Original abstract

Motivated by the present interest in the heavy quark effective theory, we use the spectator equation to treat the mesonic bound states of heavy quarks. The kernel we use is based on scalar confining and vector Coulomb potentials. Wave functions are treated to leading order and energies to order $1/m_Q$ in the heavy-light systems, and order $1/m_Q^2$ in heavy-heavy systems. Our results are in reasonable agreement with experimental measurements. We estimate two of the parameters of the heavy quark effective theory, and propose further calculations that may be undertaken in the future.

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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What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.39 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.11 · compares with prior results 0.39 · 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.

Motivated by the present interest in the heavy quark effective theory, we use the spectator equation to treat the mesonic bound states of heavy quarks. The kernel we use is based on scalar confining and vector Coulomb potentials. Wave functions are treated to leading order and energies to order $1/m_Q$ in the heavy-light systems, and order $1/m_Q^2$ in heavy-heavy systems. Our results are in reasonable agreement with experimental measurements. We estimate two of the parameters of the heavy quark effective theory, and propose further calculations that may be undertaken in the future.

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	Motivated by the present interest in the heavy quark effective theory, we use the spectator equation to treat the mesonic bound states of heavy quarks.	contribution 0.52	0.87	0.05	asserted 0.97
s1	The kernel we use is based on scalar confining and vector Coulomb potentials.	method 1.00	0.75	0.03	asserted 1.00
s2	Wave functions are treated to leading order and energies to order $1/m_Q$ in the heavy-light systems, and order $1/m_Q^2$ in heavy-heavy systems.	method 0.99	0.89	0.08	asserted 1.00
s3	Our results are in reasonable agreement with experimental measurements.	result 0.78	0.89	0.55	asserted 0.81
s4	We estimate two of the parameters of the heavy quark effective theory, and propose further calculations that may be undertaken in the future.	future work 0.84	0.89	0.17	asserted 0.49

What kind of paper the classifier reports this is: theoretical result 0.91 · evidence from simulation or computation 0.91.

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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

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S sections
T body sentences
C citations
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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

References

Forward citations

Cited by 13 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Observation of a $B_c^{*+}$ meson with the ATLAS detector
hep-ex 2026-05 conditional novelty 8.0 of 10

First observation of the B_c^{*+} meson with mass difference 64.5 MeV from the ground-state B_c^+ at >8 sigma significance using ATLAS data.
Where are the higher excited $B$ and $B_s$ mesons with coupled-channel effects?
hep-ph 2026-10 accept novelty 7.0 of 10

Coupled-channel effects shift the masses of higher B and Bs mesons downward by 30-300 MeV and reverse the mass ordering of the 2^3D_1 and 3^3S_1 states.
Near-threshold $D\bar{D}$ spectroscopy and observation of a new charmonium state
hep-ex 2019-03 accept novelty 7.0 of 10

A new narrow charmonium resonance, X(3842), is observed in D0 anti-D0 and D+D- final states with mass 3842.71 MeV and width 2.79 MeV, consistent with the long-sought spin-3 psi_3(1^3D_3) state.
Observation of two excited B$^+_\mathrm{c}$ states and measurement of the B$^+_\mathrm{c}$(2S) mass in pp collisions at $\sqrt{s} =$ 13 TeV
hep-ex 2019-02 accept novelty 7.0 of 10

Two excited Bc states, Bc+(2S) and Bc*(2S), are separately resolved in 143 fb-1 of 13 TeV pp collisions, with a mass splitting of 29.1 MeV and a Bc+(2S) mass of 6871.0 MeV.
Properties of the $D_{s0}^*(2317)^\pm$ in hot and dense nuclear matter
hep-ph 2026-08 conditional novelty 6.0 of 10

A coupled-channel molecular model predicts that in nuclear matter the D_s0*(2317)+ shifts downward and broadens with density, while temperature narrows it and broadens its antiparticle, yielding a possible structural test.

Show 8 more citing papers

Charmonium Options for the X(3872)
hep-ph 2003-11 conditional novelty 6.0 of 10

Computing strong and radiative widths for all 1D and 2P charmonium assignments leaves five candidates for the narrow X(3872) and predicts distinctive radiative branching fractions for them.
Higher excited charmed and charmed-strange mesons in an unquenched quark model
hep-ph 2026-05 unverdicted novelty 5.0 of 10

Unquenched quark model calculations show coupled-channel effects lower masses of higher D and Ds excitations; D_s1(2933) is assigned as mixed D_s(2P1), D_sJ(3040) favors D_s(3^1S0), while D(3000) fits none of the 3S/2...
Nonrelativistic meson masses from the Curci-Ferrari model
hep-ph 2025-09 conditional novelty 5.0 of 10

A potential model built from the Curci-Ferrari massive-gluon action fits charm and bottom meson spectra, and the fit prefers a nonzero gluon mass.
Contribution of constituent quark model $c\bar{s}$ states to the dynamics of the $D^\ast_{s0}(2317)$ and $D_{s1}(2460)$ resonances
hep-ph 2018-05 conditional novelty 5.0 of 10

Fitting a chiral effective theory with constituent quark model states to lattice QCD levels, the authors find the D*_{s0}(2317) and D_{s1}(2460) are predominantly D(*)K molecular states.
Observation of an excited $B_c^+$ state
hep-ex 2019-03 accept novelty 4.0 of 10

LHCb observes an excited Bc+ state at 6841.2 MeV in Bc+ pi+ pi- and finds a 2.2 sigma hint of a second state at 6872.1 MeV.
Pentaquark and Tetraquark states
hep-ph 2019-03 accept novelty 2.5 of 10

A comprehensive review of hidden-heavy-flavor tetraquark and pentaquark states, covering LHCb's confirmation of the Pc states and surveying the chromomagnetic-interaction, quark-model, meson-exchange, QCD-sum-rule, an...
Unquenched Charmonium and Beyond
hep-ph 2026-02 conditional novelty 2.0 of 10

This Lanzhou-group review argues that coupled-channel (unquenched) effects, not exotic constituents, are the common thread explaining charmonium anomalies from the rho-pi puzzle to the Y-problem states.
A review of the open charm and open bottom systems
hep-ph 2016-09 accept novelty 1.0 of 10

The paper reviews the observed open-charm and open-bottom hadrons and the quark-model, effective-theory, and QCD-sum-rule interpretations of their masses and decays.

How the literature uses it

33 arXiv papers cite this in 43 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 · 16 sentences without a confident label

Contests it 2 papers

Such low values could not be accommodated in the predictions of the so far fruitful quark models [ref] and lattice QCD calculations [ref] , that expected these resonances to lie well above the respective $D^(∗)K$ thresholds.

Contribution of constituent quark model $c\bar{s}$ states to the dynamics of the $D^\ast_{s0}(2317)$ and $D_{s1}(2460)$ resonances · Introduction · contest 0.83
Although, the phenomenologically potential models [ref] can offer quite intuitive picture for $c̅ c$ charmonium spectroscopy, the deviation between the theoretical calculations and data is quite large, so that we cannot make further predictions accordingly.

Modified Pad\'e Approach to the S-Wave Charmonium Spectroscopy in QCD · Introduction · contest 0.79

Uses its data or values 5 papers

Predictions for the mass of this state lie in the range $3815∼3863 MeV/c^2$ [ref] .

Analysis of the $X(3842)$ as a D-wave charmonium meson · Introduction · 0.89
For instance, the mass of the observed charm-strange scalar meson $D_s0^*$ [ref] is $2317.3±0.6$ MeV [ref] , while the predictions from most quark models spread from 2400 MeV to 2500 MeV [ref] .

Hadronic-loop induced mass shifts in scalar heavy-light mesons · Introduction · 0.94
For our purposes, it is sufficient to take an average of several theoretical estimations in Refs. [ref] , and for the consistency treatment of the wave functions, the $B(2S)$ wave functions are still deduced from our parameters.

How Large is the Contribution of Excited Mesons in Coupled-Channel Effects? · Beyond $B_1$ · 0.98

Compares with it 8 papers

For comparison, we also list the current data assignments [ref] and some other theoretical results [ref] .

Modified Pad\'e Approach to the S-Wave Charmonium Spectroscopy in QCD · Numerical analysis and discussions · 0.99
Considering the coupled-channel effects, the physical masses for the $D(2P_1)$ and $D(2P_1')$ states are predicted to be [equation] respectively, which are significantly ($100$–$150 MeV$) smaller than the masses predicted within the quenched quark models [ref] .

Higher excited charmed and charmed-strange mesons in an unquenched quark model · Missing states and observation strategies · 0.95
On the other hand many of the predictions [ref] are consistent with the 2S splitting of Belle, among which the 1S splittings of several [ref] are also consistent with Belle, two [ref] are consistent with Dobbs . and PDG12*, and the rest [ref] are smaller than any of the measured values.

Hyperfine splitting and the experimental candidates for \eta_b(2S) · Potential Models and related · 0.89

Cites it as background 10 papers

Theoretical predictions of the spectrum of charmed mesons dates from the early days of phenomenological quark models [ref] .

Can one-loop corrections to the one-gluon exchange potential adequately describe the charmed meson spectrum? · INTRODUCTION · 1.00
Spectroscopic parameters (such as mass and decay constant) of $B_c$ mesons are studied with using various models such as relativistic quark potential model [ref] , nonrelativistic quark model [ref] , covariant light-front quark model [ref] , shifted large-N expansion [ref] , perturbative QCD [ref] , nonrelativistic renormalization group [ref] , lattice QCD [ref] , QCD sum rules (QCDSR) [ref] , light-cone QCDSR [ref] , heavy quark effective theory [ref] , Bethe-Salpeter equation [ref] and field correlator method [ref] .

Bottom-charmed meson states in inverse problem of QCD · Motivation · 0.97
On the other hand, some of the $2P$ charmonium states predicted by different quark models [ref] still have not been confirmed or identified experimentally.

$P$-wave charmonium contribution to hidden-charm states from reanalysis of lattice QCD data · Introduction · 0.93

Discussion

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