Bethe-Salpeter Study of Vector Meson Masses and Decay Constants

Peter. C. Tandy (Kent State University); Pieter Maris

REVIEW

Bethe-Salpeter Study of Vector Meson Masses and Decay Constants

Pieter Maris, Peter. C. Tandy (Kent State University)

Nuclear Theory · Submitted 26 May 1999 · 34 references

Pith's verdict: A solid, honest extension of the Maris-Roberts DSE-BSE program to vector mesons; the physics is real, but the abstract's blanket 5%/10% claim is only true for the middle parameter set.

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CitationPeter. C. Tandy (Kent State University), Pieter Maris. "Bethe-Salpeter Study of Vector Meson Masses and Decay Constants." arXiv:nucl-th/9905056 (1999). https://pith.science/paper/TMBZE6LW

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Editor's note

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Worth your time if you care about the ladder-rainbow DSE-BSE program, and worth reading closely even if you don't, because the appendix is the best part and the abstract is the worst part. The abstract's 'within 5%'/'within 10%' (abstract:s5) does not survive its own Table II. Run the numbers: at omega = 0.3, m_K* is 7.2% high and m_phi 6.7% high; at omega = 0.5, f_K* is 11% high and f_phi 12.7% high. Only the omega = 0.4 set satisfies both claims, and even there m_phi sits at 5.1%. The body knows this — Sec. V A says 'typically 5% too large' and 'within 10% to 15% for f_phi' — so the abstract is overselling what the body honestly reports. Minor, fixable, but it is the headline claim and it should be corrected. Related: 'one phenomenological parameter' is generous. omega and m_t are also fixed by hand, and the three-set spread varies only omega, so it is not a truncation-error budget; it is a sensitivity scan of one knob. The reader cannot tell how much of the 5-10% agreement is framework accuracy and how much is infrared modeling. Also note Eq. (9) to Eq. (10): the delta-function term was dropped specifically because it broke the vector solutions. The authors say so plainly, which I respect, but it means the interaction was reshaped to make this channel work, and that deserves a sentence of scrutiny rather than a footnote. What is genuinely good: all eight transverse covariants with full angular dependence, the C-parity odd/even structure used as a numerical check, and momentum-sharing independence for K* once nothing is truncated. The Appendix A ultraviolet analysis is the strongest result in the paper — the derived cancellation b1 = -a1/9 that makes the f_V loop finite, matched against the numerical ratios in Eqs. (49)-(51) (2√5/9 vs 0.48, 1/3 vs 0.33, -1/9 vs -0.11), is a real check that the truncation preserves the one-loop renormalization group behavior. That is the part I would cite. For readers: anyone computing vector BS amplitudes, meson form factors, or rho/phi observables in this framework. Send it out; the referees should push on the abstract numbers and the omega dependence, not on the physics.

Referee report

Significance

The calculation is a genuine out-of-sample prediction: the interaction strength and quark masses are fixed entirely in the pseudoscalar sector, and the vector masses and decay constants follow. The inclusion of all eight transverse covariants with full angular dependence, rather than a single phenomenological gamma_mu amplitude, is a real advance over earlier vector-meson treatments in this framework, and the identification of which covariants matter (F1 through F5) is directly useful for future form-factor and strong-decay work. The analytic ultraviolet analysis in Appendix A, together with the numerical verification of the cancellation between F1, F2, F3 and the second Chebyshev moment of F1 (Eq. (51) versus b1 = -a1/9), is a nontrivial consistency check that the truncation preserves the one-loop renormalization-group structure. The paper is also candid about what it does not do: the delta-function term of the earlier Ansatz had to be dropped, the Ref. [10] comparison is extrapolated and lacks decay constants, and meson-loop corrections of 2-10% to the rho mass are acknowledged. Those disclosures are a strength, not a weakness, and they make the paper a useful benchmark.

Major objections (3)

[Abstract (abstract:s5) versus Table II and Sec. V A] The abstract states that the masses are 'within 5%' and the decay constants 'within 10%' of experiment, but Table II does not support that as a blanket statement across the three parameter sets the paper itself reports. Taking the experimental values quoted in the text (m_rho = 0.770, m_K* = 0.892, m_phi = 1.020, f_rho = 0.216, f_K* = 0.225, f_phi = 0.236 GeV), the omega = 0.3 set gives m_K* = 0.956 (7.2% high) and m_phi = 1.088 (6.7% high); the omega = 0.5 set gives m_K* = 0.94 (5.4%), m_phi = 1.08 (5.9%), f_K* = 0.25 (11.1%) and f_phi = 0.266 (12.7%). Only the omega = 0.4 set is close to satisfying both bounds, and even there m_phi = 1.072 is 5.1% high. Sec. V A is more careful, saying 'typically 5%' and 'within 10% to 15% for f_phi', so the body and the abstract disagree. The abstract should be rewritten to state the accuracy as a range across the parameter sets, for example 'masses within about 7% and decay constants within about 13%', or to name the central set explicitly and quote the spread as the model uncertainty. It would also help to note that the deviations are systematic rather than scattered: m_rho comes out low in all three sets while m_phi and m_K* come out high, which suggests a common model artifact rather than statistical scatter and is worth one sentence of comment.
[Abstract (abstract:s1) and Sec. II B, Eq. (10)] The abstract asserts that the approach 'is consistent with quark and gluon confinement', but the interaction actually used, Eq. (10), is not the one for which that property was established. Sec. II B explains that the delta-function term of Eq. (9) was removed precisely because it produced non-analytic quark-propagator structure that spoiled the vector solutions, and the delta-function piece is the part that in Ref. [10] carried half of the infrared strength. The manuscript does not show that the rainbow solution of Eq. (10) is confining, for example by demonstrating the absence of timelike poles or the violation of reflection positivity in the resulting propagator. Either a short verification should be added, or the confinement sentence in the abstract should be qualified to say that confinement is inherited from the Ref. [10] framework while the present Eq. (10) has not been separately checked for it.
[Sec. V A and Sec. VI] The claimed 5% level of agreement should be interpreted against the size of the physics that is omitted. Sec. VI states that estimates of meson-loop contributions to the rho mass range from 2% to 10%, and the spread across the three omega values in Table II is of the same order. The agreement obtained is therefore at the level of the neglected corrections, which is a meaningful statement but a different one from a precision test. The paper should say this explicitly in Sec. V A, so that the reader does not read the 5% and 10% figures as evidence that the ladder truncation is accurate to that level. This does not undermine the qualitative out-of-sample prediction, which stands on its own.

Minor objections (8)

[Sec. II B, paragraph after Eq. (9)] There is a typographical error in the description of the anomalous dimensions: the text reads 'for both the chiral limit (m(mu) = 0, anomalous dimension 1 - gamma_m) and explicit chirally broken case (m(mu) = 0, anomalous dimension gamma_m)'. The second occurrence should read m(mu) != 0.
[Table II, comparison row for Ref. [10]] The Ref. [10] comparison row reports masses only, obtained by extrapolation to the mass shell, and the text explains that the normalization integral is too sensitive to the extrapolation for a decay constant to be quoted. This is honest, but the row would be clearer if the table caption repeated that these entries are extrapolated and are not directly comparable in accuracy to the rows above.
[Table III and Sec. V B] The decay constants in Table III for the reduced amplitude sets are computed with the loop cut off at mu = 19 GeV because those integrals are ultraviolet divergent, so those entries are cutoff artifacts rather than physical values. The caption says this, but the numbers are presented in the same columns as the convergent results and could be misread. Consider marking the cutoff-dependent entries explicitly, for example with a footnote symbol.
[Sec. IV, Eq. (38) and the value f_K* = 225 MeV] The 'experimental' f_K* is not a direct measurement but is extracted from the ratio of tau -> rho nu and tau -> K* nu partial widths using Vud, Vus and f_rho = 216 MeV from rho -> e+e-. The paper is transparent about the procedure, but an estimated uncertainty on the extracted 225 MeV, even a rough one, would let the reader judge the 7% deviation in Table II against the error bar on the target.
[Sec. V A, Fig. 2 and Fig. 3] The amplitudes in Figs. 2 and 3 are normalized to 0F1(0) = 1, and the real or imaginary character of each amplitude is indicated only in the legend of Fig. 2. Since the C-parity argument in Sec. III implies that each amplitude is purely real or purely imaginary for rho and phi, it would help to state that in the caption of Fig. 3 as well, and to note that the K* amplitudes are genuinely complex.
[Sec. V A, momentum sharing] The statement that K* observables are independent of the momentum-sharing parameter (eta_u, eta_s) when all covariants and the full angular dependence are retained, but vary by up to 5% otherwise, is a useful diagnostic. It would be worth stating which of the two solution methods (direct two-variable versus Chebyshev expansion) was used for that test, since the two methods truncate differently.
[Sec. VI] The paper compares only to experiment. A brief comparison of f_rho and f_phi with lattice determinations or QCD sum-rule values would place the 5-10% agreement in the context of other approaches and strengthen the assessment of the truncation.
[Sec. V B] The word 'renormalizibility' should be 'renormalizability'.

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The paper

The paper claims

Fixing a quark model on pions and kaons, the authors predict rho, phi and K-star masses and decay constants without further tuning.

The reading

The paper asks whether a quark-level model of QCD that has been calibrated entirely on the pion and kaon can then predict the light vector mesons — the $1^-$ states $rho/omega$, $phi$ and $K^star$ — with no further adjustment. The model is a coordinated truncation of the coupled Dyson–Schwinger equation for the quark propagator and the Bethe–Salpeter equation for the quark-antiquark bound state, with a single modelled gluon interaction. One interaction strength and two current quark masses are fixed by matching $f_pi$, $m_pi$ and $m_K$; the kaon decay constant $f_K$ then comes out within 3% of experiment as a free check. With those same numbers, the vector meson masses land within 5% and the electromagnetic and leptonic decay constants within 10% of their measured values. The interest is that this is an out-of-sample test: nothing about the vector channel was used to set the parameters, so the agreement is a genuine prediction of the framework rather than a fit.

What carries the argument

The load-bearing device is the ladder-rainbow truncation: the Bethe–Salpeter kernel is a single dressed-gluon exchange with a bare quark-gluon vertex, matched to a rainbow truncation of the quark self-energy, so that the axial Ward–Takahashi identity survives and the pion is massless in the chiral limit. The interaction is a model two-point gluon function, Eq. (10), consisting of a finite-width Gaussian infrared term plus a one-loop renormalization-group tail that takes over above roughly 2 GeV$^2$. The vector amplitude is expanded on eight transverse, Dirac-trace-orthonormal covariants built from $gamma_mu$, the relative momentum $q$ and the total momentum $P$, then projected back onto that basis to give eight coupled integral equations; the angle dependence is handled either by direct two-variable discretization or by a Chebyshev expansion in $cos theta$. A separate asymptotic analysis, replacing the running coupling by its value at the larger momentum and converting the integral equations into differential equations, shows that the individual $1/q^2$ falloff of $F_1$, $F_2$ and $F_3$ would make the decay-constant integral logarithmically divergent, but the coefficients cancel once the second Chebyshev moment of $F_1$ is included — a concrete realization of the renormalizability the truncation is built to preserve.

Extended reading notes

Core claim

Solving the vector Bethe–Salpeter equation with all eight transverse Dirac covariants and the full angular dependence of the amplitudes, the authors find that the ladder-rainbow Dyson–Schwinger/Bethe–Salpeter framework, with parameters fixed only in the pseudoscalar sector, reproduces the ground-state vector meson spectrum and decay constants to within roughly 5% and 10% respectively. Concretely, for the middle parameter set, $m_rho = 0.742$ GeV against 0.770 measured, $m_phi = 1.072$ against 1.020, $m_K^star = 0.936$ against 0.892, with $f_rho = 0.207$ GeV against 0.216. The results are stable across three different widths of the modelled infrared interaction, so long as each is refitted to the same pion and kaon data. The authors also report a structural finding: only five of the eight covariants matter quantitatively for these observables, dropping the other three shifts masses by about 2% and decay constants by about 8%, and for the $rho$ and $phi$ the leading Chebyshev moment of each amplitude nearly suffices — except that the second moment of $F_1$ is required for the decay-constant loop integral to converge.

Load-bearing premise

The single modelled gluon interaction — a finite-width Gaussian infrared bump joined to a perturbative tail — is assumed to represent the quark-gluon interaction accurately enough over the larger and partly complex momentum range that the heavier vector bound states sample, even though its infrared width is a hand-chosen parameter and the earlier sharp delta-function form had to be discarded because it produced non-analytic quark propagators that spoiled the vector solutions.

Editorial extensions

If this is right

Vector meson masses and decay constants can be obtained from a framework whose only inputs are pseudoscalar observables, so the vector sector becomes a genuine test of the truncation rather than a fit.
The relative weight of the eight covariants is now known, which gives a practical recipe: keep $F_1$ through $F_5$ and the leading Chebyshev moments for $rho$ and $phi$, but add the second moment of $F_1$ before computing decay constants.
Because the ladder kernel is $P$-independent, physical observables turn out independent of how momentum is shared between the quark and antiquark, but only when all relevant covariants and the full angular dependence are retained; truncating either reintroduces a spurious dependence of up to 5%.
The calculated Bethe–Salpeter amplitudes can be fed directly into later calculations of $rho to pi pi$, $phi to K K$, $rho to pi gamma$ and electromagnetic form factors, replacing phenomenological amplitude ansätze.
The same amplitudes provide a starting point for estimating meson-loop corrections, which are expected to shift the $rho$ mass by a few percent and could contribute to the $rho$–$omega$ splitting that the ladder truncation cannot produce.

Reading between the lines

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

The out-of-sample agreement is the kind of check that separates a model with predictive content from a model with enough freedom to absorb any data; a natural next test is to fix parameters on pions and kaons and then predict a channel with no adjustable freedom at all, such as the axial-vector or scalar mesons, where the same ladder kernel is expected to do worse.
The finding that five of eight covariants carry the physics suggests that a reduced, well-chosen amplitude basis could make vector-meson calculations cheap enough for finite-temperature or finite-density studies, where the full two-variable problem is currently out of reach.
The cancellation that renders the decay-constant integral finite is a numerical fingerprint of the Ward-identity-preserving truncation; a truncation that broke that coordination would show a residual logarithmic divergence, which offers a cheap diagnostic for future kernel improvements.
Since the $rho$–$omega$ degeneracy here is an artifact of the ladder kernel, any future calculation that reproduces this spectrum but also splits the two must be adding physics — meson loops or quark annihilation — beyond single-gluon exchange, and the size of that split is a direct measure of how much.

Original abstract

The masses and decay constants of the light vector mesons $\rho/\omega$, $\phi$ and $K^\star$ are studied within a ladder-rainbow truncation of the coupled Dyson-Schwinger and Bethe-Salpeter equations of QCD with a model 2-point gluon function. The approach is consistent with quark and gluon confinement, reproduces the correct one-loop renormalization group behavior of QCD, generates dynamical chiral symmetry breaking, and preserves the relevant Ward identities. The one phenomenological parameter and two current quark masses are fixed by requiring that the calculated $f_\pi, m_\pi$ and $m_K$ are correct. The resulting $f_K$ is within 3% of the experimental value. For the vector mesons, all eight transverse covariants are included and the dominant ones are identified; the complete angle dependence of the amplitudes is also retained. The calculated values for the masses $m_\rho, m_\phi$ and $m_{K^\star}$ are within 5%, while the decay constants $f_\rho, f_\phi$ and $f_{K^\star}$ for electromagnetic and leptonic decays are within 10% of the experimental values.

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-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 5,176 input tokens.

What the abstract itself states, as probabilities: claims first 0.02 · states limitation 0.20 · human participants 0.01 · controlled comparison 0.05 · randomized assignment 0.01 · states quantitative result 0.99 · compares with prior results 0.22 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; show every role.

The masses and decay constants of the light vector mesons $\rho/\omega$, $\phi$ and $K^\star$ are studied within a ladder-rainbow truncation of the coupled Dyson-Schwinger and Bethe-Salpeter equations of QCD with a model 2-point gluon function. The approach is consistent with quark and gluon confinement, reproduces the correct one-loop renormalization group behavior of QCD, generates dynamical chiral symmetry breaking, and preserves the relevant Ward identities. The one phenomenological parameter and two current quark masses are fixed by requiring that the calculated $f_\pi, m_\pi$ and $m_K$ are correct. The resulting $f_K$ is within 3% of the experimental value. For the vector mesons, all eight transverse covariants are included and the dominant ones are identified; the complete angle dependence of the amplitudes is also retained. The calculated values for the masses $m_\rho, m_\phi$ and $m_{K^\star}$ are within 5%, while the decay constants $f_\rho, f_\phi$ and $f_{K^\star}$ for electromagnetic and leptonic decays are within 10% of the experimental values.

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	The masses and decay constants of the light vector mesons $\rho/\omega$, $\phi$ and $K^\star$ are studied within a ladder-rainbow truncation of the coupled Dyson-Schwinger and Bethe-Salpeter equations of QCD with a model 2-point gluon function.	contribution 0.93	0.93	0.08	asserted 0.99
s1	The approach is consistent with quark and gluon confinement, reproduces the correct one-loop renormalization group behavior of QCD, generates dynamical chiral symmetry breaking, and preserves the relevant Ward identities.	method 0.52	0.73	0.09	asserted 0.80
s2	The one phenomenological parameter and two current quark masses are fixed by requiring that the calculated $f_\pi, m_\pi$ and $m_K$ are correct.	method 1.00	0.87	0.04	asserted 0.99
s3	The resulting $f_K$ is within 3% of the experimental value.	result 1.00	0.94	0.10	asserted 1.00
s4	For the vector mesons, all eight transverse covariants are included and the dominant ones are identified; the complete angle dependence of the amplitudes is also retained.	method 0.94	0.93	0.04	asserted 1.00
s5	The calculated values for the masses $m_\rho, m_\phi$ and $m_{K^\star}$ are within 5%, while the decay constants $f_\rho, f_\phi$ and $f_{K^\star}$ for electromagnetic and leptonic decays are within 10% of the experimental values.	result 1.00	0.96	0.72	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.52 · evidence from simulation or computation 1.00.

What was examined, and what was not

sources read

arXiv listing abstract v2

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. The Ansatz for the effective ¯q

**Figure 2.** FIG. 2. The leading Chebyshev projections of all eight

**Figure 3.** FIG. 3. Leading and subleading BS amplitudes for the

**Figure 4.** FIG. 4. The ultraviolet behavior of the

**Figure 5.** FIG. 5. The ultraviolet behavior of the

Figures reproduced from arXiv:nucl-th/9905056 by the authors.

References

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A hybrid fRG-plus-DSE calculation reproduces lattice QCD transition temperatures and curvature, and predicts a critical endpoint at (93, 672) MeV for 2+1 flavour QCD, though the authors flag that the large-density reg...
Light mesons in QCD and unquenching effects from the 3PI effective action
hep-ph 2015-12 conditional novelty 6.0 of 10

A three-loop 3PI truncation of QCD, with lattice-constrained gluon and ghost propagators, reproduces the vector-axial-vector meson splitting and pushes the lightest quark-antiquark scalar above 1 GeV.
Chiral symmetry breaking for large "Nf" and the properties of the "rho" meson near the conformal window
hep-ph 2026-09 accept novelty 5.0 of 10

The rho meson's mass is nearly constant as flavor number increases, but its size grows and it ultimately dissociates due to loss of confinement, not by crossing a threshold.
Chiral symmetry restoration effects onto the meson spectrum from a Dyson-Schwinger and Bethe-Salpeter approach
hep-ph 2026-02 unverdicted novelty 5.0 of 10

Varying interaction strength in DSE/BSE models produces meson degeneracies whose domains shrink with model realism, linked to quark propagator pole locations and possible chiral spin symmetry.
Unified Description of Pseudoscalar Meson Structure from Light to Heavy Quarks
hep-ph 2026-02 unverdicted novelty 5.0 of 10

An algebraic light-front model supplies unified leading-twist PDAs, LFWFs, GPDs, PDFs, EFFs, charge radii and IPS-GPDs for light, heavy-light and heavy-heavy pseudoscalar mesons from the same Bethe-Salpeter amplitudes.
Imaging the charge distributions of flavor-symmetric and -asymmetric mesons
hep-ph 2025-06 conditional novelty 5.0 of 10

Using a maximum-entropy inversion of DSE/BSE form factors, the authors map meson charge profiles and find quark-antiquark distances that shrink with quark mass and grow by 5-15% for spin-aligned mesons.
Gluon Parts of Gravitational Form Factors and Mass Distribution
nucl-th 2025-05 conditional novelty 5.0 of 10

A DSE rainbow-ladder calculation produces quark and gluon gravitational form factors, mass radii, and D-terms for nucleon and pion, finding gluon distributions generated purely by quark dressing.
$D^* D \pi$ and $B^* B \pi$ couplings from Dyson-Schwinger equations framework
hep-ph 2024-11 conditional novelty 5.0 of 10

A Dyson-Schwinger/Bethe-Salpeter calculation predicts g_D*Dpi = 16.22 and g_B*Bpi = 40.09, with static couplings consistent with lattice results.
GeV-scale neutrinos: interactions with mesons and DUNE sensitivity
hep-ph 2020-07 accept novelty 5.0 of 10

A cleaned-up, publicly released set of meson-heavy-neutrino effective couplings, used to estimate that the DUNE near detector can probe sterile-neutrino mixings several orders of magnitude below current limits.
Quarkonium as relativistic bound state on the light front
hep-ph 2017-04 conditional novelty 5.0 of 10

Adding a running strong coupling to a light-front holographic plus one-gluon-exchange Hamiltonian lowers charmonium and bottomonium mass errors to 31 and 38 MeV rms and restores the hyperfine splitting.
The quark-gluon vertex in Landau gauge bound-state studies
hep-ph 2014-04 conditional novelty 5.0 of 10

A fitted, flavour-dependent model of the quark-gluon vertex indicates that anomalous chromomagnetic moment components dominate over the conventional scalar component in the quark self-energy.
Kaon-box contribution to the anomalous magnetic moment of the muon
hep-ph 2019-10 accept novelty 4.0 of 10

The charged kaon box contributes about -0.48 x 10^-11 to the muon's anomalous magnetic moment, roughly 3 percent of the pion box, confirming that box diagrams beyond the pion are negligible.
Studies of Nucleon Resonance Structure in Exclusive Meson Electroproduction
nucl-th 2012-12 accept novelty 3.0 of 10

The paper reviews the reaction theory, lattice, Dyson-Schwinger, sum-rule, duality, holographic and quark-model approaches that will be used to interpret nucleon-resonance electrocouplings measured with CLAS12 up to Q...
Infrared Properties of QCD from Dyson-Schwinger equations
hep-ph 2006-05 accept novelty 3.0 of 10

Landau-gauge Dyson-Schwinger studies indicate an infrared-finite running coupling, a vanishing gluon propagator, and dynamically generated quark mass, though lattice and continuum results still differ at low momentum.

How later papers used it

145 later papers cite it

Evidence: substantial. 142 of the citing papers share no author with it, and 58 of those carry a confident label from Pith's citation classifier.

Pith read 27 of the 264 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

16 later papers by other authors are listed below: 13 used its method or tool and 3 used its data or values.

Uses its method or tool 13 of 45 papers with this label

Worked in rainbow-ladder truncation of QCD using the Maris-Tandy model of the quark-gluon interaction.

Hadronic contribution to the muon g-2: a Dyson-Schwinger perspective (2011) · 1.00 · paraphrased: the licence does not allow quoting
Replaced the combination of the strong coupling and gluon propagator with the Maris-Tandy interaction, setting its free parameter to omega = 0.4 GeV.

Vector mesons as dynamical resonances in the Bethe-Salpeter framework (2018) · 0.93 · paraphrased: the licence does not allow quoting
Adopted the Maris-Tandy model, in which the gluon propagator is written as a sum of an infrared term and a UV term.

Quark propagator with complex-valued momentum from Schwinger-Dyson equation in the Euclidean space (2024) · 0.95 · paraphrased: the licence does not allow quoting
Used the model proposed by Maris and Tandy as the effective interaction, with parameter omega = 0.4 and light quarks.

Properties of quarks and mesons in the Dyson-Schwinger/Bethe-Salpeter approach (2011) · 0.94 · paraphrased: the licence does not allow quoting
Used the parameterization of this paper as the effective interaction, with omega an effective inverse range and D an overall strength of the first term.

Masses of J^PC =1^-+ exotic quarkonia in a Bethe-Salpeter-equation approach (2015) · 0.84 · paraphrased: the licence does not allow quoting
Implemented the interaction of Maris and Tandy in its study.

Hadron properties from QCD bound-state equations (2009) · 0.99 · paraphrased: the licence does not allow quoting
Adopted the ansatz for the effective coupling introduced in this paper.

Delta and Omega masses in a three-quark covariant Faddeev approach (2011) · 0.98 · paraphrased: the licence does not allow quoting
Chose its effective interaction from this paper in rainbow-ladder truncation.

Hadronic decays of mesons and baryons in the Dyson-Schwinger approach (2011) · 0.82 · paraphrased: the licence does not allow quoting
Relied on the model proposed in this paper, in which the infrared enhancement of the effective strong coupling is accomplished without a singular but integrable delta function.

Instantaneous Bethe-Salpeter View of Goldstone-Type Pseudoscalar Mesons (2016) · 0.98 · paraphrased: the licence does not allow quoting
Left the parameters of its interaction unchanged from this paper.

Spectra of heavy quarkonia in a Bethe-Salpeter-equation approach (2014) · 0.96 · paraphrased: the licence does not allow quoting
Used the Maris-Tandy model for the effective coupling, Eq. (10) of this paper, with parameters Lambda = 0.74 GeV and eta = 1.85 +/- 0.2.

Single pseudoscalar meson pole and pion box contributions to the anomalous magnetic moment of the muon (2019) · 0.85 · paraphrased: the licence does not allow quoting
Specified its scalar function in Euclidean space by the Maris-Tandy model.

Pion Electromagnetic Form Factor from Bethe-Salpeter Amplitudes with Appropriate Kinematics (2024) · 0.85 · paraphrased: the licence does not allow quoting
Took the rainbow-ladder truncation and aligned the quark's Dyson-Schwinger equation and the meson's Bethe-Salpeter equation.

The light front wave functions and diffractive electroproduction of vector mesons (2021) · 0.96 · paraphrased: the licence does not allow quoting

Uses its data or values 3 of 9 papers with this label

Extracted its inversion input from a chiral-limit model solution of this paper to the QCD equation of motion for the full quark propagator.

Instantaneous Bethe-Salpeter Look at the Lightest Pseudoscalar Mesons (2017) · 0.90 · paraphrased: the licence does not allow quoting
Adopted the values D0 = 9.3 x 10^5 MeV^2 and sigma = 400 MeV from this paper.

The chiral phase transition with a chiral chemical potential in the framework of Dyson-Schwinger equations (2015) · 0.99 · paraphrased: the licence does not allow quoting
Took the quark mass and wave-function renormalization functions from a solution of this paper.

Light Pseudoscalar Mesons: an Inverse Instantaneous Bethe-Salpeter Glimpse (2017) · 0.90 · paraphrased: the licence does not allow quoting

How the literature uses it

102 arXiv papers cite this in 178 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 · 111 sentences without a confident label

Builds on it 1 paper

Together with the isovector ground state pseudoscalars [ref] , first excited pseudoscalar states [ref] and ground state vectors [ref] , this is a new case among the few where the Euclidian Bethe Salpeter equation has been solved so far.

Microscopic quark study of the eta and eta' masses · Conclusion · 0.89

Uses its method or tool 28 papers

The standard DSE-RL interaction kernel [ref] that generates quark propagators and BSE vertices and meson bound states is where [equation] Here $α̃_s(q^2)$ denotes a continuation of the 1-loop $α_s(q^2)$ to provide smooth non-singular coverage for the entire domain of $q^2$.

Gluon PDF from Quark dressing in the Nucleon and Pion · Introduction · 0.85
We work in rainbow-ladder truncation of QCD using the Maris-Tandy model of the quark-gluon interaction [ref] .

Hadronic contribution to the muon g-2: a Dyson-Schwinger perspective · Calculational scheme · 1.00
where, $C_F=4/3$ is the color factor and $q=k-p$ the momentum of the gluon propagator $D^μν(q)$ ; in this study the combination $g_s^2 D^μν(q)$ is replaced by the Maris-Tandy interaction [ref] , with its free parameter set to $ω=0.4$ GeV.

Vector mesons as dynamical resonances in the Bethe-Salpeter framework · Introduction · 0.93

Uses its data or values 10 papers

We extract our inversion input from a popular chiral-limit model solution [ref] to the QCD equation of motion for the full quark propagator.

Instantaneous Bethe-Salpeter Look at the Lightest Pseudoscalar Mesons · Inversion: Confining Interquark Potential Compatible with Goldstone's Theorem · 0.90
Owing to Poincaré covariance, just two Lorentz-scalar functions characterize the propagator of a quark (of four-momentum $p$), its mass $M(p^2)$ and its wave-function renormalization $Z(p^2)$, which we take from a solution [ref] of this quark propagator's Euclidean-space equation of motion (Fig. [ref]).

Light Pseudoscalar Mesons: an Inverse Instantaneous Bethe-Salpeter Glimpse · Introduction · 0.90
Hereafter we will fix the latter to $Λ=0.72$ GeV which has been used in [ref] to reproduce experimental results for meson and baryon properties.

On baryon properties from a covariant Faddeev approach · Rainbow-ladder truncation · 0.84

Compares with it 2 papers

The results of our calculation in the $3$PI effective action at 3-loop ($3$PI-3L) are detailed in Table [ref] and contrasted with typical results from rainbow-ladder (RL) [ref] , a recent study of mesons (and baryons) in the framework of the $2$PI effective action at 3-loop ($2$PI-3L) [ref] , and of course their experimentally known values from the PDG [ref] .

Light mesons in QCD and unquenching effects from the 3PI effective action · Application to bound-states · 1.00
In the case of Maris-Tandy, the resultant rho mass of $∼720$ MeV is similar to that obtained in other studies with this interaction [ref] .

Beyond the rainbow: effects from pion back-coupling · Results · 1.00

Cites it as support 1 paper

Studies using this parametrization have demonstrated that pseudoscalar and vector-meson ground state properties are insensitive to a variation of the coupling width $ω$ in a certain range [ref] , i.e. independent of the detailed shape of the interaction in the infrared.

Hadron properties from QCD bound-state equations · Effective quark-gluon interaction · 0.97

Cites it as background 17 papers

The analytic properties of the quark DSE that arises when only the infrared (IR) part of the Maris-Tandy interaction [ref] is taken into account has been the subject of a thorough study [ref] .

The analytic properties of the quark propagator from an effective infrared interaction model · Introduction · 0.85
The calculation of observables from the Bethe–Salpter amplitudes, such as leptonic decay constants for the pseudoscalar and vector mesons [ref] , requires that they are properly normalised.

Bethe-Salpeter studies of mesons beyond rainbow-ladder · Decay constants and normalisation · 0.83
There are more realistic models such as the Maris-Tandy model [ref] in which the interaction vanishes at infrared, and Qin-Chang (QC) model [ref] , whose interaction remains a constant value at infrared.

Dyson-Schwinger equations towards cold-dense QCD matter with improved truncations · Introduction · 0.96

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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P. Maris, C.D. Roberts and P.C. Tandy, Phys. Lett. B420, 267 (1998)

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M.A. Ivanov, Yu.L. Kalinovsky and C.D. Roberts, ”Survey of heavy-meson observables”, (1999), nucl-th/9812063, to appear in Phys. Rev. D, 1999

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C.D. Roberts, Nucl. Phys. A605, 475 (1996); C.J. Burden, C.D. Roberts and M.J. Thomson, Phys. Lett. B371, 163 (1996)

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M.R. Frank, K.L. Mitchell, C.D. Roberts and P.C. Tandy, P hys. Lett. B359, 17 (1995); D. Kekez, B. Bistrovic and D. Klabucar, Int. J. Mod. Phys. A14, 161 (1999)

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C.J. Burden, Lu Qian, C.D. Roberts, P.C. Tandy and M.J. Th omson, Phys. Rev. C 55, 2649 (1997)

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M.R. Frank and C.D. Roberts, Phys. Rev. C 53, 390 (1996)

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P.C. Tandy, “Electromagnetic form factors of meson tra nsitions”, Fizika B (1999), hep- ph/9902495, to be published

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L. v. Smekal, A. Hauck, and R. Alkofer, Phys. Rev. Lett. 79, 3591 (1997); Ann. Phys. 267, 1 (1998),

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D. Atkinson and J.C.R. Bloch, Phys. Rev. D 58, 094036 (1998); Mod. Phys. Lett. A13, 1055 (1998)

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F.T. Hawes, P. Maris and C.D. Roberts, Phys. Lett. B440, 353 (1998)

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S.J. Stainsby and R.T. Cahill, Int. J. Mod. Phys. A7, 7541 (1992)

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P. Maris and H. Holties, Int. J. Mod. Phys. A7, 5369 (1992)

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C.J. Burden, C.D. Roberts and A.G. Williams, Phys. Lett . B285, 347 (1992)

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C. D. Roberts, in Quark Conﬁnement and the Hadron Spectrum II , eds. N. Brambilla and G. M. Prosperi (World Scientiﬁc, Singapore, 1997), pp. 2 24-230

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P. Maris and Q. Wang, Phys. Rev. D 53, 4650 (1996)

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P. Maris, C. D. Roberts and S. Schmidt, Phys. Rev. C57, 2821 (1998); D. Blaschke and P. C. Tandy, ”Mesonic Correlations and Quark Deconﬁnement” , in Understanding De- conﬁnement in QCD , Eds. D. Blaschke, F. Karsch and C. D. Roberts, (World Scient iﬁc, Singapore, 1999), to be published. 23 FIGURES 0 1 2 3 4 5 q 2 [GeV 2 ] 0 2 4 6 8 10G(q 2 )/q 2 [GeV −2 ]...

View on Pith 1998