Distribution amplitudes of heavy-light pseudo-scalar and vector mesons from Dyson-Schwinger equations framework

Yin-Zhen Xu

REVIEW 3 major objections 6 minor 3 cited by

Distribution amplitudes of heavy-light pseudo-scalar and vector mesons from Dyson-Schwinger equations framework

T0 review · 3 major / 6 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The paper reports the first Dyson-Schwinger/Bethe-Salpeter predictions for the distribution amplitudes of $B^*$, $B^*_s$, and $B^*_c$, with each peak fixed by the Euclidean constituent mass ratio $M_E^f/(M_E^f+M_E^g)$.

desk verdict Solid, honest DSE paper with first B*-family DAs and a better way to compute Mellin moments; peak-location claims are ansatz-dependent but the paper says so. read the letter →

arxiv 2501.18085 v2 pith:25F62JBJ submitted 2025-01-30 hep-ph

Yin-Zhen Xu This is my paper · ORCID

classification hep-ph PACS 12.38.-t14.40.-n

keywords distributionamplitudesDyson-SchwingerequationsBethe-Salpeterheavy-lightmesonsB*Mellinmomentslight-frontmomentumQCDfactorization

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper reports the first Dyson-Schwinger/Bethe-Salpeter predictions for the leading-twist distribution amplitudes of the $B^*$, $B^*_s$, and $B^*_c$ mesons, and gives a broader set of amplitudes for their pseudo-scalar partners and for lighter heavy-light mesons. These amplitudes describe how the longitudinal light-front momentum of a meson is shared between its valence quark and antiquark, and they are needed as inputs for QCD factorization calculations of hard exclusive processes. The central numerical finding is that in every flavor-asymmetric system the amplitude is skewed toward the heavier quark, with its maximum located at $x_0 \sim M_E^f/(M_E^f+M_E^g)$, where $M_E$ is the Euclidean constituent quark mass. A second finding is that spin effects are subleading to flavor-composition effects: for identical valence content, the heavier quark carries more light-front momentum in vector mesons than in their pseudo-scalar partners. If these predictions hold up against future experimental and lattice checks, they provide a simple mass-ratio rule for the shapes of heavy-light meson distribution amplitudes.

What carries the argument

The load-bearing object is the leading-twist distribution amplitude $\phi(x)$, the light-front projection of the Bethe-Salpeter wave function that gives the probability density for a valence quark to carry momentum fraction $x$ of the meson's light-front momentum. Because $\phi(x)$ cannot be obtained through direct integration, the paper computes its Mellin moments $\langle x^m\rangle$ up to $m=8$ without extrapolation: it iterates the Bethe-Salpeter equation on a two-dimensional Chebyshev tensor grid to obtain continuous amplitudes, then evaluates the oscillatory moment integrals with a deterministic adaptive integration algorithm. The moments are fitted with the two-parameter ansatz $\phi(x;\alpha,\beta) = N\,4x(1-x)e^{4\alpha x(1-x)+\beta(2x-1)}$, so the extracted peak positions and widths are properties of that fitted form. The key identity that organises the results is the near-coincidence of the fitted peak $x_0$ with the Euclidean constituent mass ratio $M_E^f/(M_E^f+M_E^g)$.

What would settle it

Take the eight computed Mellin moments for $B^*$, $B^*_s$, and $B^*_c$ and reconstruct the distribution amplitude without assuming the two-parameter ansatz, or compute $\langle\xi\rangle$ for these vector mesons directly in lattice QCD; the claims would be contradicted if the peak no longer sat near $M_E^b/(M_E^b+M_E^q)$ and the predicted first moments $\langle\xi\rangle_{B^*}^{\parallel}=0.656$ and $\langle\xi\rangle_{B^*}^{\perp}=0.671$ were not reproduced.

Watch

Extended reading notes

Core claim

The central claim is that the leading-twist distribution amplitude of a ground-state heavy-light meson, reconstructed from the first eight Mellin moments of its Bethe-Salpeter wave function, is a narrow, skewed distribution whose peak sits at the ratio of Euclidean constituent quark masses, $x_0 \approx M_E^f/(M_E^f+M_E^g)$, and whose width decreases as the heavier quark mass grows. For vector mesons, the paper predicts the ordering $\langle\xi\rangle_{0^-} < \langle\xi\rangle_{1^-}^{\parallel} < \langle\xi\rangle_{1^-}^{\perp}$ of first moments, so the heavier quark carries slightly more light-front momentum in the longitudinal and transverse vector channels than in the corresponding pseudo-scalar channel. The new quantitative predictions are for $B^*$, $B^*_s$, and $B^*_c$, which had no previous DSE distribution-amplitude results.

Load-bearing premise

The load-bearing premise is that every meson's true distribution amplitude has the same two-parameter shape used to fit its Mellin moments; if the real amplitude has a different form (for example a different endpoint behaviour or extra $x$-dependence), the predicted peak positions, widths, and their ordering could shift.

Editorial extensions

If this is right

The new $B^*$, $B^*_s$, and $B^*_c$ amplitudes can be used directly in QCD factorization formulas for hard exclusive decays and production processes involving vector heavy mesons.
If the near-equality $x_0\approx M_E^f/(M_E^f+M_E^g)$ is generic, the peak of any flavor-asymmetric meson's distribution amplitude is fixed by the Euclidean constituent masses, allowing predictions for mesons not computed here.
The moment ordering $\langle\xi\rangle_{0^-}<\langle\xi\rangle_{1^-}^{\parallel}<\langle\xi\rangle_{1^-}^{\perp}$ implies that the polarization state of a vector meson systematically shifts momentum toward the heavy quark relative to its pseudo-scalar partner.
The method's ability to compute Mellin moments to eighth order without damping-factor extrapolation can be carried over to other mesons and to light-front wave functions whose moment integrals are similarly oscillatory.

Reading between the lines

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

If the two-parameter ansatz in Eq. (34) is replaced by a model-independent reconstruction of the same eight moments, the fitted peak positions and the width ordering $\Delta x_{0^-}\gtrsim\Delta x_{1^-}^{\parallel}\gtrsim\Delta x_{1^-}^{\perp}$ could shift; the paper itself notes that this relation needs further study.
The $x_0\approx M_E^f/(M_E^f+M_E^g)$ rule, if robust, is a simple finite-width generalisation of the non-relativistic $\delta$-function sharing formula and may extend to excited heavy-light states, which the paper does not address.
The direct-moment method could be applied to baryon distribution amplitudes, where oscillatory multidimensional integrals have been the main obstacle; that extension is ours, not the paper's.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The thing to know: this is a competent, transparent DSE phenomenology paper. The genuinely new physics is the first DSE predictions for the B*, B*_s, and B*_c distribution amplitudes. The genuinely new method is the combination of a two-dimensional Chebyshev tensor grid for the Bethe-Salpeter amplitude and CUBA adaptive integration, which lets the author compute Mellin moments up to eighth order directly, without extrapolation or damping-factor fitting. That is real progress, because earlier DSE calculations had to fit or damp to get even fourth to sixth moments.

What the paper does well: the moment results are benchmarked against lattice QCD and earlier DSE work for pion, kaon, rho, D, and B, and they satisfy the symmetry recursion relations for flavor-symmetric systems. The universal pattern ⟨ξ⟩_0− < ⟨ξ⟩_∥1− < ⟨ξ⟩_⊥1− is a clean, plausible result, consistent with LFQM findings. The author also states plainly where the calculation is model-calibrated (the weight parameter η fitted to pseudoscalar masses) and where the error bars are large (the c-b system). That honesty earns credit.

The soft spot is the reconstruction step, and it is real but not fatal. The x0 and FWHM claims come from the two-parameter ansatz Eq. (34). The 56-subset error band varies only which moments enter the fit; it cannot detect whether that functional family is right for strongly skewed heavy-light DAs. The alternative form Eq. (35) is tested only for π and ρ, and there the author switches to it because fit1 fluctuates around x ~ 0.5, so the extracted width is ansatz-sensitive even in the simplest case. The paper acknowledges this and calls direct extraction future work, which is more than many papers do. Still, the abstract's "maximum ~ M_E^f/(M_E^f+M_E^g)" should be read as a statement about the reconstructed DAs, not as a direct moment-level result.

Two smaller issues: the Mellin moments in Tables VI–VIII have no quoted uncertainties (only the fit parameters carry error bars), and no code or data files are shipped. The latter is a minor inconvenience, not a flaw.

Bottom line: the moment-level results and the new numerical method are solid; the shape-level claims are suggestive but should carry a warning label. The paper deserves a serious referee. I would ask the referee to request an independent reconstruction check for one heavy-light vector meson and a statement of moment uncertainties.

Referee Report

3 major / 6 minor

Summary. This manuscript presents Dyson-Schwinger/Bethe-Salpeter (DSE/BSE) computations of the leading-twist distribution amplitudes (DAs) of pseudoscalar and vector heavy-light mesons, reporting the first DSE-based results for the B*, B_s*, and B_c* mesons. The methodological core is a combination of a two-dimensional Chebyshev tensor grid for the Bethe-Salpeter amplitudes and the CUBA-Cuhre adaptive integration algorithm, which allows Mellin moments to be evaluated directly to eighth order without the damping-factor extrapolation used in earlier DSE studies (Sec. II and Fig. 2). The moments are validated against symmetry recursion relations (Eq. 26) and against lattice and prior DSE results for pi, K, rho, K*, D, D_s, B, B_s, and B_c (Tables II-IV). The DAs are then reconstructed by fitting the two-parameter ansatz of Eq. (34), and the fitted forms are used to extract the position of the maximum x0 and the full width at half maximum (Table V, Fig. 7). The paper's physical findings are that the maximum occurs near xME = M_E^f/(M_E^f + M_E^g), that the first moments obey <xi>_0- < <xi>_1-,|| < <xi>_1-,perp (Eq. 33), that DAs narrow with increasing quark mass, and that spin effects are subleading to valence-quark composition effects. The reconstruction limitation is acknowledged in Sec. IV.

Significance. The moment-level results and the numerical method are the paper's strongest assets: direct eighth-order moment evaluation, validated by the Eq. (26) recursion relations, by agreement with lattice data for the pion and kaon (Table II), and by agreement with lattice and sum-rule results for rho and K* (Table III), is a genuine methodological advance for the DSE community, and the complete moment tables (Tables VI-VIII) are independently reusable. The ordering in Eq. (33), which holds across all six flavor-asymmetric systems, is a clean, falsifiable statement. If the x0 ~ xME relation survives an independent reconstruction test, it would be a simple and useful quantitative rule. The novelty claim (first DSE predictions for B*, B_s*, B_c*) appears accurate relative to the cited literature. The main caveat, acknowledged by the authors in Sec. IV, is that the DAs are obtained by fitting a single two-parameter ansatz; as argued in the major comments below, the headline peak-position statements need either additional validation or explicit hedging.

major comments (3)

[Sec. III.B; Eq. (34); Table V] The headline quantitative finding, x0 about xME (Eq. 37 and Fig. 7), is derived from the two-parameter ansatz in Eq. (34), and the 56-subset error band constructed in Sec. III.B tests only the internal consistency of that ansatz, not its adequacy for strongly skewed heavy-light DAs. The paper's own pi/rho sector is the control case: fit1 is found to fluctuate around x about 0.5 and is abandoned in favor of fit2, and in the process the width metric itself is redefined (Table V footnote). Since fit2 (Eq. 35) is symmetric about x = 1/2, it cannot be applied to any skewed system, so for all heavy-light mesons, including the headline B*, B_s*, and B_c*, no alternative reconstruction is tested. It should be acknowledged that the x0 about xME agreement is not trivially inherited from the first moment (for the D meson one has <x> = 0.644 but x0 = 0.745); nevertheless, the 1-3% agreement in most systems and the 6-7% deviation in the c-b-bar system are properties of one fitted exponential family, and the systematic pattern (x0 below xME for the c-containing systems but above xME for K, B, and B_s) is not discussed. I request either (i) validation of x0 with at least one alternative skewed reconstruction (a three-parameter family, a Gegenbauer-based reconstruction, or a maximum-entropy inversion of the eight moments) for a representative set of heavy-light systems, or (ii) an explicit downgrade of the x0 about xME statement to an ansatz-dependent observation, reflected in the abstract and conclusions.
[Tables VI-VIII; Sec. III.A] The Mellin moments, the sole input to the reconstruction, are quoted in Tables VI-VIII without uncertainties; the asymmetric errors on alpha and beta, and hence the bands on x0 and Delta-x in Table V and Fig. 7, appear to derive entirely from the spread of the 56 three-moment fits. Missing from this budget are numerical uncertainties (the choice of the k^2_max plateau in Fig. 2 and the CUBA tolerance settings) and model uncertainties (the weight parameter eta in Eq. (15), fitted per heavy-light meson to the pseudoscalar mass, and the interaction parameters of Eq. (7)). This matters because some of the quantitative claims are not protected by the robust moment-level pattern: the PS/VC differences in x0 are small (for the B_c system, 0.713, 0.719, and 0.722 for PS, VC||, and VC-perp, and for the B* system, 0.918, 0.921, and 0.922), and the stated fit-consistency errors are comparable to these differences. The paper should either propagate a conservative estimate of the moment-level uncertainty into x0 and Delta-x, or explicitly restrict the PS/VC and width-order comparisons to the statements that survive that propagation.
[Sec. III.A; Table IV] For the c-b-bar system, the computed first moment <xi> = 0.374 sits at the low edge of a wide spread of earlier results collected in Table IV (0.42(2) from DSE(19), 0.464(4) from DSE(20), 0.413(35) from AM, and 0.536 from LFQM(10)), and the text does not comment on this tension. This is also the system for which the x0-xME deviation is largest (about 7%, Sec. III.B), so the credibility of the B_c and B_c* DA predictions would be materially strengthened by a discussion of where the difference comes from, e.g., whether the eta fit to the B_c mass or the treatment of the c-b-bar kernel drives the low value. At minimum, the comparison in Table IV should be accompanied by a sentence assessing the model dependence of the B_c moment.

minor comments (6)

[Affiliations; Sec. IV] Typos: 'Departmento' in the first author's affiliation line, and 'bquark' (missing space) in the closing paragraph of Sec. IV; please also check the subscript typesetting of M1- in Eqs. (22b) and (22c).
[Table V footnote] The redefinition of the width as 'the full width at phi(x)/2|x=0.5' is difficult to parse; please define this alternative width with an explicit formula in Sec. III.B rather than only in a table footnote, since it changes the comparison baseline for the pi and rho entries.
[Fig. 6 caption] The caption states 'Solid lines represent fit1, dashed lines correspond to fit2', but fit2 is used only for the pi and rho panels; please state this explicitly so that the figure is not misleading.
[Sec. III.B] The fitting procedure is underspecified: which norm is minimized in the fits, and how is the error band assembled from the 56 subsets (envelope, or mean plus/minus standard deviation)? A sentence describing the fitting metric and band construction would make the quoted uncertainties reproducible.
[Ref. [93]] The publication year given for Ref. [93] (2020) appears inconsistent with its arXiv identifier 1012.4021; please check the bibliographic data.
[Sec. II.A, Eq. (10)] The notation Theta-bar-g in the inequality 1 - Theta-bar-g/M < alpha < Theta_f/M should be defined explicitly (the bar denotes the antiquark flavor), as it is easy to misread in the printed equation.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: x0≈xME is a nontrivial model outcome; the ansatz-fit caveat is a reconstruction-robustness limitation, not circular reasoning.

full rationale

The derivation chain is: DSE gap equation (Eq. 1) and BSE (Eq. 11) yield dressed propagators and Bethe-Salpeter amplitudes; Mellin moments are then computed directly via Eq. (24); distribution amplitudes are reconstructed by fitting the eight moments with the ansatz Eq. (34); peak position x0 and FWHM are extracted from that reconstruction. At no point is x0 defined in terms of xME, nor is xME fitted to x0. The quantity xME (Eq. 37) is built from Euclidean constituent quark masses M_E (Eq. 5), which come from the quark mass functions, while x0 is the maximum of a two-parameter fit to independently computed moments; their near-coincidence is therefore a nontrivial numerical finding, not an identity. The weight parameter eta is calibrated to pseudoscalar meson masses, but vector meson masses and decay constants are then compared with experiment and lattice results (Table I), and the DA moments themselves are computed rather than fitted to external DA data. The paper's own caveat in Sec. IV that 'the DAs in this study are reconstructed via an ansatz fit' is a legitimate limitation about reconstruction-model dependence, but it does not make the central claim circular: the 56-subset error band and the fit1/fit2 comparison probe fit robustness, not a definitional reduction. Self-citations for model parameters (Refs. [37,40,47]) are inputs with independent experimental/lattice validation, not a self-supporting uniqueness argument. No equation in the paper reduces to its own input by construction, so there is no significant circularity.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central claim rests on model assumptions: the rainbow-ladder kernel, the weight-RL interpolation with fitted eta, and the ansatz-form DA reconstruction. No new physical entities are introduced. The free parameters are the kernel weight and the DA fit parameters, all disclosed in the text.

free parameters (4)

eta (weight parameter in Eq. (15) per heavy-light meson) = not quoted; tuned such that pseudoscalar meson mass matches experiment
Determines the effective interaction strength for flavor-asymmetric systems; directly affects all DA results.
alpha (fit1 parameter in Eq. (34), per meson) = see Tables VI-VIII
Shape parameter of the DA ansatz fitted to the computed Mellin moments; controls broadening.
beta (fit1 parameter in Eq. (34), per meson) = see Tables VI-VIII
Skew parameter of the DA ansatz; primarily controls x0 and the skewness toward the heavier quark.
a (fit2 parameter in Eq. (35), for pi and rho) = a_pi = 0.506(+0.028/-0.025), a_rho_parallel = 0.409(+0.009/-0.007), a_rho_perp = 0.605(+0.013/-0.013)
Alternative DA ansatz used for comparison on pi and rho.

assumptions (5)

domain assumption Rainbow-ladder truncation with Qin-Chang effective interaction (Eqs. 6-7)
Used to solve gap and Bethe-Salpeter equations; limits accuracy for heavy-light systems and violates the WTI for asymmetric flavors, as the paper acknowledges in Sec. II.B.
ad hoc to paper Weight-RL approximation in Eq. (15)
An effective kernel introduced to handle flavor asymmetry; eta is fitted to meson masses, and the paper calls it 'an additional drawback and cost'.
ad hoc to paper DA reconstruction ansatz in Eq. (34)
Assumes a two-parameter functional form for all DAs; no independent verification for heavy-light vector mesons.
domain assumption Light-front projection formulas in Eq. (22)
Standard definitions from QCD factorization; assumed valid at the hadronic scale with renormalization scale 2 GeV.
standard math Recursion relations in Eq. (26) for flavor-symmetric meson moments
Used as consistency checks; derived from the symmetry x -> 1-x.

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Cite this review

Pith. "Pith review of Distribution amplitudes of heavy-light pseudo-scalar and vector mesons from Dyson-Schwinger equations framework." pith.science (2026). https://pith.science/paper/25F62JBJ

@misc{pith2026250118085,
  author       = {Pith},
  title        = {Pith review of: Distribution amplitudes of heavy-light pseudo-scalar and vector mesons from Dyson-Schwinger equations framework},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/25F62JBJ}},
  note         = {Machine review of arXiv:2501.18085}
}

abstract

We report the first Dyson-Schwinger equation predictions for the leading-twist distribution amplitudes of the $B^*$, $B_s^*$, and $B_c^*$ mesons, and extend the investigation to a broader set of heavy-light pseudo-scalar/vector mesons. Numerical analysis shows that, in flavor-asymmetric systems, the distribution amplitude is skewed toward the heavier quark, with the position of the maximum $\sim M^f_E/(M^f_E+M^g_E)$, where $M_E$ is the Euclidean constituent quark mass and $f$, $g$ denote the quark flavors. Spin effects are found to be subleading compared to the influence of valence-quark composition, and lead to heavier quarks carrying more light-front momentum in vector mesons than in their pseudo-scalar counterparts. Our results can be compared with both experimental and theoretical outcomes in the future.

Figures

Figures reproduced from arXiv: 2501.18085 by the authors.

**Figure 2.** FIG. 2. The upper panels illustrate the variation of [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 4.** FIG. 4. The DAs of [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗

Figures from the paper (3 more)

**Figure 5.** Figure 5: FIG. 5. The DAs of some heavy-light pseudo-scalar mesons [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]

**Figure 6.** Figure 6: FIG. 6. The DAs of vector mesons and their pseudo-scalar partners. Solid lines represent fit1, dashed lines correspond to fit2, [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]

**Figure 7.** Figure 7: FIG. 7. This figure illustrates an analysis of meson DAs (see Fig. [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]

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Forward citations

Cited by 3 Pith papers

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Imaging the charge distributions of flavor-symmetric and -asymmetric mesons
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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.
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