Pith. sign in

REVIEW 3 major objections 4 minor 141 references

All six isovector form factors of the D, D*, B, and B* mesons follow from pion-pion rescattering, with the rho(770) couplings extracted from the pole residues.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · deepseek-v4-flash

2026-08-02 18:22 UTC pith:HAHVCYUP

load-bearing objection A careful dispersive analysis that delivers solid isovector form factors and two ρ couplings, but overclaims the extraction of 'all' ρ couplings since the g^(3) quadrupole coupling is unconstrained. the 3 major comments →

arxiv 2603.11154 v2 pith:HAHVCYUP submitted 2026-03-11 hep-ph hep-exnucl-th

Dispersive Analysis of D- and B-Meson Form Factors with Chiral and Heavy-Quark Constraints

Simon Mutke , Leon A. Heuser , Ingrid Dax , Bastian Kubis , Stefan Leupold This is my paper
classification hep-ph hep-exnucl-th
keywords heavy-meson form factorsdispersion theoryMuskhelishvili–Omnès representationrho(770) couplingchiral perturbation theoryheavy-quark symmetryanomalous thresholdsD and B mesons
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper aims to show that the low-energy electromagnetic structure of heavy–light mesons is shaped by pion–pion rescattering and can be captured by dispersion theory without assuming a specific rho-meson model. Starting from chiral and heavy-quark symmetry, it constructs amplitudes for heavy-meson–antimeson annihilation into two pions, unitarizes them with a Muskhelishvili–Omnès representation, and feeds them into form-factor unitarity relations. This yields all six isovector form factors (electric, magnetic-transition, and the three vector-meson ones) for D, D*, B, and B* mesons. It then extracts the coupling constants of the rho(770) resonance to each of these states by taking residues at the rho pole. If correct, the framework gives parameter-free predictions for radii and future decay distributions and identifies specific heavy-quark-symmetry-breaking effects induced by anomalous triangle singularities.

Core claim

The central claim is that a single dispersive framework, with pion–pion P-wave rescattering incorporated via the Omnès function and with anomalous-threshold corrections from triangle diagrams, determines all six isovector vector form factors of the heavy–light mesons at low energies. The required M(*)M̄(*) → ππ P-wave amplitudes are built from heavy-meson chiral perturbation theory at leading order in the heavy-quark expansion, with the only fitted contact term (the NLO constant c4) fixed by the D* magnetic moment. The paper shows that the resulting form factors respect heavy-quark symmetry to a few percent, with the most pronounced deviations coming from anomalous thresholds that move onto

What carries the argument

The load-bearing object is the once-subtracted, unitarized dispersive representation of the form factors, built from the unitarity cut relation disc F_i(s) = 2i θ(s−4Mπ²) pπ³/(12π√s) T_i(s) (Fπ^V(s))*, where T_i(s) are the P-wave M(*)M̄(*) → ππ amplitudes and Fπ^V is the pion vector form factor. The T_i are obtained by adding the P-wave projected t/u-channel Born exchanges (with vector- and pseudoscalar-meson poles) to polynomial contact terms, then unitarized through the Muskhelishvili–Omnès equation so that their phase matches the ππ P-wave phase shift. The analysis tracks anomalous thresholds: when the vector-meson mass exceeds the pseudoscalar-plus-pion mass, a triangle singularity moves

Load-bearing premise

The unitarized M(*)M̄(*) → ππ P-wave amplitudes are built from tree-level chiral/heavy-quark effective Lagrangians and then unitarized, but this left-hand-cut model has never been tested against actual heavy-meson–pion scattering data; only the NLO constant c4 is fixed from the D* magnetic moment, while the NNLO constant dNNLO is a dimensional-analysis estimate that changes g(3) by an order of magnitude.

What would settle it

A single reliable measurement of the πD → πD P-wave amplitude in the 0.3–1 GeV region, or a lattice QCD calculation of the D-meson isovector form factors below 1 GeV, would directly validate or refute the input amplitudes; the predicted sharp cusp in F3,B*B* near s ≈ 0.07 GeV² is a particularly clean signature to look for.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • The six isovector form factors and the rho(770) couplings to D, D*, B, and B* are predicted without a tunable rho model, fixing c4 from the D* magnetic moment alone.
  • Heavy-quark symmetry is approximately reproduced, with deviations concentrated in the D* system and attributed to anomalous-threshold effects—this sharpens expectations for symmetry-breaking patterns in related observables.
  • The predicted isovector radii, comparable to the pion radius but with large imaginary parts for the D* multipole radii, can be confronted with future lattice QCD or experimental determinations.
  • The anomalous cusp in the B* form factors near s ≈ 0.07 GeV² and the logarithmic divergence in F3 for D* at small negative s provide concrete signatures in eD* → eD* scattering if the unstable-particle issue is resolved.
  • The extracted rho couplings provide input for meson-exchange potentials and for models of hidden-charm/bottom XYZ states that are dominated by D(*)D̄(*) thresholds.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the input amplitude model is a faithful low-energy description, the same machinery could be extended to scalar or tensor form factors and to charged-current transitions like D → π lν, where the anomalous-threshold structure would appear in a different kinematic configuration.
  • The order-of-magnitude sensitivity of g(3) to the NNLO constant dNNLO suggests that a lattice computation of the electric quadrupole form factor of the D* (or of the rho contribution to F3) would be a direct way to pin down this constant and test the framework.
  • The paper's treatment of D* as a stable state with an infinitesimal width leaves a logarithmic divergence that a proper second-sheet continuation with the physical D* width would smear; exploring that continuation could convert the divergence into a measurable narrow peak.
  • A future measurement of pion–D-meson scattering in the P wave, or of D* → D e+e− Dalitz distributions once isoscalar contributions are added, would provide an independent, stringent test of the unitarized amplitudes used here.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper constructs a dispersive representation of the isovector electromagnetic form factors of D, D*, B, and B* mesons, using heavy-hadron chiral perturbation theory for the M(*)Mbar(*)→ππ P-wave input amplitudes, followed by Muskhelishvili–Omnès unitarization and once-subtracted dispersion relations for the form factors. It treats the anomalous thresholds generated by triangle diagrams, including their movement onto the first Riemann sheet, and extracts ρ(770) coupling constants from the pole residues of the unitarized amplitudes. The central claim is that this yields a model-independent low-energy description of all six isovector form factors and the couplings of the ρ to all four heavy-meson systems, with the only fitted NLO constant c4 fixed by the measured D*→Dγ magnetic moment.

Significance. If the framework is sound, this is a valuable systematic calculation: it combines chiral and heavy-quark symmetries with dispersive unitarization, provides predictions for form factors and mean-squared radii that are in principle measurable, and gives a rigorous definition of ρM(*)M(*) couplings via pole residues. The paper is unusually transparent about its limitations, including the incomplete saturation of the electric-charge sum rule, the unconstrained NNLO contact term, and the logarithmic divergence in F3 for the physical D* region. However, these limitations touch the central claim of extracting 'all' ρ couplings and of providing predictive form factors in the physical region, so they cannot be treated as mere presentation issues.

major comments (3)
  1. [§5, Eq. (131), Table 3] The coupling g^(3)_ρM*M* is not actually extracted by the framework; Eq. (131) shows it is linearly dependent on the NNLO constant dNNLO, which is only estimated by dimensional analysis in §4.4. The quoted range dNNLO=±0.25 GeV^-2 changes |g^(3)_ρD*D*| from 0.139 GeV^-2 to 1.54–1.76 GeV^-2, i.e. by an order of magnitude, and similarly for B*. Since dNNLO is not determined by any input used in the paper, the numerical value of g^(3) is fixed by an arbitrary higher-order term rather than by the dispersive machinery. The abstract's promise to extract the couplings of the ρ(770) to 'all these heavy mesons' is therefore not supported for one of the five couplings. I recommend either removing g^(3) from the list of extracted couplings, or determining dNNLO from an independent observable and presenting g^(3) as an input-dependent estimate with a clear error budget.
  2. [§4.3, Eqs. (104)–(105)] Gauge invariance requires the electric form factors at q^2=0 to equal 1/2 for the isovector combination, as stated in Eq. (104). The numerical results in Eq. (105) saturate this sum rule only at the 80% level (F_DbarD(0)=0.421, F_BbarB(0)=0.400, and similar for F1). The subtraction constants in Eq. (89) are said to be determined by the sum rule Eq. (90), so the violation is not a mere normalization convention but an indication that the truncated ππ-only dispersive integrals do not reproduce the exact charge. This matters for the central low-energy predictions: the radii in Table 2 are computed by dividing by the exact normalization (1/2), not by the saturated sum-rule values, and the form factor shapes in Fig. 8 inherit the 20% normalization mismatch. The manuscript needs either to impose Fi(0)=1/2 as an explicit subtraction constant and treat Eq. (90) as a consistency check, or to quant
  3. [§4.3, paragraph after Eq. (108), and Fig. 12] The paper states that F3 for the D* develops a logarithmic divergence at the anomalous threshold located at s_+≈-0.02 GeV^2, which lies in the physical crossed-channel eD*→eD* region. The divergence is attributed to treating D* as a stable particle with M_V^2→M_V^2+iδ, and it is argued that a proper finite-width treatment would smear the singularity. However, this finite-width implementation is not performed, and the present manuscript therefore leaves a physical observable (the D* electromagnetic form factor in the spacelike region) with a singular, non-predictive behavior. Since one of the stated aims is to provide form factors in the physical region, this unresolved analytic-structure issue is load-bearing for F3 and should be addressed—at least by demonstrating that a finite-width smearing renders the form factor finite and by estimating the residual uncertainty.
minor comments (4)
  1. [Eq. (16)] The second delta function in the loop integral should be δ(+)(ℓ^2−M_π^2), not δ(+)(q^2−M_π^2). The text currently reads as if both arguments are q-dependent.
  2. [Eq. (117)] The dimension of c4 is stated as GeV^-2 in Eq. (106) but as GeV^-1 in Eq. (117). This appears to be a units typo and should be corrected, since c4 enters the same contact-term expressions.
  3. [§4.5 and Table 2] The systematic-error notation in Table 2 (three bracketed errors, one asymmetric) is compact but not fully explained in the caption; a sentence defining the ordering and the one-sided nature of the third error would improve readability.
  4. [Abstract and §1] The word 'model-independent' is used for the dispersive rescattering treatment, but the input M(*)Mbar(*)→ππ amplitudes are built from tree-level ChPT/HQET Born and contact terms. This is legitimate, but the abstract should be phrased so that the model dependence of the left-hand-cut input is not obscured.

Circularity Check

0 steps flagged

No circularity: c4 is fixed by an independent magnetic moment, and the dispersive outputs are not identical to the inputs.

full rationale

The derivation chain is self-contained against external data. The only fitted low-energy constant, c4, is fixed in Sect. 4.2 by requiring the dispersive result for the isovector D*→D transition form factor at s=0 to reproduce the measured magnetic moment μD_IV = 1.12(4) GeV^-1 (Eqs. 99–102, 106); the ρM(*)M(*) couplings are then obtained from the pole residues of the unitarized amplitudes (Eq. 127), not from any fit to those couplings. The resulting μD*, μB, μB* values (Sect. 4.3) and the gρMM, gρM*M, g^(1), g^(2) entries of Table 3 are outputs of the dispersion relations, not definitions of the inputs. The ππ phase shift/Omnès input is taken from independent ππ data via the IAM, and F_V^π is fitted to CLEO/Belle/NA7 data, so the rescattering dynamics is not the paper's own result in disguise. The self-citations [59] and [76] are methodological/data-analysis references, not load-bearing uniqueness theorems. The one serious caveat is parametric, not circular: Eq. (131) shows g^(3) depends linearly on the NNLO constant dNNLO, which is only estimated by dimensional analysis and changes |g^(3)| by an order of magnitude for dNNLO = ±0.25 GeV^-2; the paper explicitly discloses this (Sects. 4.4, 4.5, Table 3). An unconstrained parameter is a robustness/underdetermination problem, not a reduction of the prediction to the input. Hence no circular step can be exhibited.

Axiom & Free-Parameter Ledger

4 free parameters · 5 axioms · 0 invented entities

The framework introduces no new particles or mediators. The ρ is an established resonance, and dNNLO/c4 are low-energy contact couplings, not entities. The main pulls from outside the paper are the fitted pion form-factor slope, the magnetic-moment input for c4, the ππ phase input, and the heavy-quark/chiral symmetry assumptions.

free parameters (4)
  • c4 (NLO magnetic contact term) = 0.241(7) GeV^-2; 0.304 GeV^-2 in the rho-prime variant
    Fixed by demanding FD*barD(0) saturates the measured isovector D-meson magnetic moment (Sect. 4.2); enters PM*barM and P2.
  • dNNLO (NNLO contact term) = ±0.25 GeV^-2 (trial values from dimensional analysis)
    Only estimated by dimensional analysis (Sect. 4.4); changes F3 and g(3)_rhoM*M* by an order of magnitude through Eq. (131).
  • alpha_V (pion vector form factor slope) = 0.118(2) GeV^-2
    Fitted to CLEO/Belle/NA7 |F_pi^V(s)|^2 data in Eq. (95); used as input in form-factor dispersion integrals.
  • Dispersion cutoffs scutoff and scutoff,MO = 2.0 GeV^2 and 2.2 GeV^2 central; varied 1.2-4.0 GeV^2
    Chosen to optimize the saturation of the charge sum rule (Sect. 4.3); systematic uncertainty is assessed by variation.
axioms (5)
  • domain assumption Heavy-quark spin-flavor symmetry relates D and B systems and determines a common coupling g; 1/m_Q corrections are neglected.
    Used to translate g=0.547(6) from D* decay to the B system (Eq. 38) and to enforce HQS by neglecting terms in Eq. (69).
  • domain assumption Only ππ intermediate states contribute to the isovector form-factor dispersion integrals; inelastic 4π/πω effects are neglected or estimated via a rho-prime model.
    Unitarity relations in Sect. 2.2 and dispersion relations in Eq. (89) include only ππ; inelastic effects are treated as systematic estimates in Sect. 4.5.
  • domain assumption The ππ P-wave phase shift and amplitude are given by the modified inverse-amplitude method with NNLO terms, with phase tending to π at high energies.
    Sect. 4.1, Eqs. (91)-(94); this input drives the Omnes function and all MO and form-factor integrals.
  • ad hoc to paper The D* is treated as a stable external particle with M_V^2 -> M_V^2 + iδ; finite-width smearing of the anomalous threshold is not implemented.
    Sect. 3.4 and Sect. 4.3; this choice produces the logarithmic F3 divergence in the physical eD* region, with only a verbal expectation that a proper pole continuation would smear it.
  • standard math Standard dispersion-theory results: BTT decompositions, Omnes solution, Watson theorem, and Landau equations.
    Used throughout Sects. 2.1, 3.3, and 3.4; these are established mathematical tools in hadron phenomenology.

pith-pipeline@v1.3.0-alltime-deepseek · 39548 in / 13083 out tokens · 122925 ms · 2026-08-02T18:22:46.941517+00:00 · methodology

0 comments
read the original abstract

We analyze the isovector vector form factors of $D$, $D^*$, $B$, and $B^*$ mesons at low energies. We employ all constraints due to chiral and heavy-quark symmetry, and include the physics of resonant pion-pion rescattering in a model-independent way, using dispersion theory. Special attention is paid to the analytic properties of these form factors, which include anomalous thresholds due to triangle diagrams that are located on the physical Riemann sheets in some of the form factors. We extract the couplings of the $\rho(770)$ resonance to all these heavy mesons by determining the appropriate pole residues.

Figures

Figures reproduced from arXiv: 2603.11154 by Bastian Kubis, Ingrid Dax, Leon A. Heuser, Simon Mutke, Stefan Leupold.

Figure 1
Figure 1. Figure 1: Diagrammatic representation of the form factor [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Diagrammatic representation of the amplitudes [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Diagrammatic representation of the unitarized [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: Left: Positions of sthr and s+ in the complex s￾plane before the analytic continuation in M2 V . The inte￾gration path is indicated by the thick solid line. Middle: The path traced by s+ during the analytic continuation is indicated by the thin solid line with arrows. When s+ moves through the unitarity cut, the integration path needs to be deformed, picking up the additional red integration path. Right: D… view at source ↗
Figure 6
Figure 6. Figure 6: Diagrammatic representation of the form factor [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Determination of the slope parameter αV of Eq. (95) via a fit to |F V π (s)| 2 data from the CLEO, Belle, and NA7 experiments. For comparison the plain Omn`es function Ω(s) is also shown. 4.2 Fixing the Contact Terms In the framework established so far, there is still one free parameter that we need to fix from external input, namely the prefactor c4 of the two NLO magnetic con￾tact terms PM∗M¯ (s) and P2(… view at source ↗
Figure 8
Figure 8. Figure 8: Real (left) and imaginary parts (right) of the [PITH_FULL_IMAGE:figures/full_fig_p015_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Form factor F3 with an additional NNLO contact term. Upper panels: positive contact term; lower panels: negative contact term. entering via the H00 component, is suppressed by a factor of Q2 close to zero, but it nevertheless diverges. This can be traced back to the fact that we treat D∗ as a stable external particle, even though the decay D∗ → Dπ is kinematically allowed. When treated as a complex kinemat… view at source ↗
Figure 10
Figure 10. Figure 10: Left: Comparison between the IAM phase and the phenomenological phase from Refs. [ [PITH_FULL_IMAGE:figures/full_fig_p017_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Real (left) and imaginary parts (right) of some representative [PITH_FULL_IMAGE:figures/full_fig_p018_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Comparison between the D∗ form factor F3 close to the anomalous thresholds near s = 0, both in the isospin limit and for the explicit isovector linear combination from Eq. (3), taking into account the mass difference of the charged and neutral heavy–light mesons. We show both real (left) and imaginary (right) parts. energies up to the ρ ′ resonance [105, 106]), F (IAM phase) i (s) 7→ F (pheno. phase) i (s… view at source ↗
Figure 13
Figure 13. Figure 13: Plot of the function g0 [PITH_FULL_IMAGE:figures/full_fig_p025_13.png] view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

141 extracted references · 88 linked inside Pith

  1. [1]

    Weinberg, Physica A 96, 327 (1979)

    S. Weinberg, Physica A 96, 327 (1979)

    1979

  2. [2]

    Gasser and H

    J. Gasser and H. Leutwyler, Annals Phys. 158, 142 (1984)

    1984

  3. [3]

    M. B. Wise, Phys. Rev. D 45, R2188 (1992)

    1992

  4. [4]

    Yan, H.-Y

    T.-M. Yan, H.-Y. Cheng, C.-Y. Cheung, G.-L. Lin, Y. C. Lin, and H.-L. Yu, Phys. Rev. D 46, 1148 (1992), [Erratum: Phys. Rev. D 55, 5851 (1997)]

    1992

  5. [5]

    Burdman and J

    G. Burdman and J. F. Donoghue, Phys. Lett. B 280, 287 (1992)

    1992

  6. [6]

    Casalbuoni, A

    R. Casalbuoni, A. Deandrea, N. Di Bartolomeo, R. Gatto, F. Feruglio, and G. Nardulli, Phys. Rept.281, 145 (1997) [arXiv:hep-ph/9605342]

    Pith/arXiv arXiv 1997

  7. [7]

    Isgur and M

    N. Isgur and M. B. Wise, Phys. Lett. B 232, 113 (1989)

    1989

  8. [8]

    Isgur and M

    N. Isgur and M. B. Wise, Phys. Lett. B 237, 527 (1990)

    1990

  9. [9]

    Grinstein, Nucl

    B. Grinstein, Nucl. Phys. B 339, 253 (1990)

    1990

  10. [10]

    Eichten and B

    E. Eichten and B. R. Hill, Phys. Lett. B 234, 511 (1990)

    1990

  11. [11]

    Georgi, Phys

    H. Georgi, Phys. Lett. B 240, 447 (1990)

    1990

  12. [12]

    A. V. Manohar and M. B. Wise, Heavy quark physics, Cambridge University Press, 2000

    2000

  13. [13]

    J. J. Sakurai, Annals Phys. 11, 1 (1960)

    1960

  14. [14]

    J. J. Sakurai, Currents and Mesons , University of Chicago Press, 1969

    1969

  15. [15]

    L. G. Landsberg, Phys. Rept. 128, 301 (1985)

    1985

  16. [16]

    Meißner, Phys

    U.-G. Meißner, Phys. Rept. 161, 213 (1988)

    1988

  17. [17]

    Klingl, N

    F. Klingl, N. Kaiser, and W. Weise, Z. Phys. A 356, 193 (1996) [arXiv:hep-ph/9607431]

    Pith/arXiv arXiv 1996

  18. [18]

    S.-s. Fang, B. Kubis, and A. Kup´ s´ c, Prog. Part. Nucl. Phys. 120, 103884 (2021) [arXiv:2102.05922 [hep-ph]]

    Pith/arXiv arXiv 2021

  19. [19]

    Ananthanarayan, G

    B. Ananthanarayan, G. Colangelo, J. Gasser, and H. Leutwyler, Phys. Rept. 353, 207 (2001) [arXiv:hep- ph/0005297]

    arXiv 2001

  20. [20]

    Garc ´ ıa-Mart ´ ın, R

    R. Garc ´ ıa-Mart ´ ın, R. Kami´ nski, J. R. Pel´ aez, J. Ruiz de Elvira, and F. J. Yndur´ ain, Phys. Rev. D 83, 074004 (2011) [arXiv:1102.2183 [hep-ph]]

    Pith/arXiv arXiv 2011

  21. [21]

    Colangelo, M

    G. Colangelo, M. Hoferichter, and P. Stoffer, JHEP 02, 006 (2019) [arXiv:1810.00007 [hep-ph]]

    Pith/arXiv arXiv 2019

  22. [22]

    X.-W. Kang, B. Kubis, C. Hanhart, and U.-G. Meißner, Phys. Rev. D 89, 053015 (2014) [arXiv:1312.1193 [hep- ph]]

    Pith/arXiv arXiv 2014

  23. [23]

    Granados, S

    C. Granados, S. Leupold, and E. Perotti, Eur. Phys. J. A 53, 117 (2017) [arXiv:1701.09130 [hep-ph]]

    Pith/arXiv arXiv 2017

  24. [24]

    Alarc´ on, A

    J. Alarc´ on, A. Hiller Blin, M. J. Vicente Va- cas, and C. Weiss, Nucl. Phys. A 964, 18 (2017) [arXiv:1703.04534 [hep-ph]]

    Pith/arXiv arXiv 2017

  25. [25]

    Leupold, Eur

    S. Leupold, Eur. Phys. J. A 54, 1 (2018) [arXiv:1707.09210 [hep-ph]]

    Pith/arXiv arXiv 2018

  26. [26]

    Junker, S

    O. Junker, S. Leupold, E. Perotti, and T. Vitos, Phys. Rev. C 101, 015206 (2020) [arXiv:1910.07396 [hep-ph]]

    Pith/arXiv arXiv 2020

  27. [27]

    Lin, H.-W

    Y.-H. Lin, H.-W. Hammer, and U.-G. Meißner, Eur. Phys. J. A 59, 54 (2023) [arXiv:2205.00850 [hep-ph]]

    Pith/arXiv arXiv 2023

  28. [28]

    Alvarado, D

    F. Alvarado, D. An, L. Alvarez-Ruso, and S. Leupold, Phys. Rev. D 108, 114021 (2023) [arXiv:2310.07796 [hep-ph]]

    Pith/arXiv arXiv 2023

  29. [29]

    M. M. Aung, S. Leupold, E. Perotti, and Y. Yan, Phys. Rev. D 111, 114021 (2025) [arXiv:2401.17756 [hep-ph]]

    Pith/arXiv arXiv 2025

  30. [30]

    An and S

    D. An and S. Leupold [arXiv:2406.15115 [nucl-th]]

    Pith/arXiv arXiv

  31. [31]

    S. L. Olsen, T. Skwarnicki, and D. Zieminska, Rev. Mod. Phys. 90, 015003 (2018) [arXiv:1708.04012 [hep-ph]]

    Pith/arXiv arXiv 2018

  32. [32]

    Brambilla, S

    N. Brambilla, S. Eidelman, C. Hanhart, A. Nefediev, C.-P. Shen, C. E. Thomas, A. Vairo, and C.-Z. Yuan, Phys. Rept. 873, 1 (2020) [arXiv:1907.07583 [hep-ex]]

    Pith/arXiv arXiv 2020

  33. [33]

    F.-K. Guo, C. Hanhart, U.-G. Meißner, Q. Wang, Q. Zhao, and B.-S. Zou, Rev. Mod. Phys. 90, 015004 (2018) [arXiv:1705.00141 [hep-ph]], [Erratum: Rev. Mod. Phys. 94, 029901 (2022)]

    Pith/arXiv arXiv 2018

  34. [34]

    Liu, H.-X

    Y.-R. Liu, H.-X. Chen, W. Chen, X. Liu, and S.- L. Zhu, Prog. Part. Nucl. Phys. 107, 237 (2019) [arXiv:1903.11976 [hep-ph]]

    Pith/arXiv arXiv 2019

  35. [35]

    H.-X. Chen, W. Chen, X. Liu, Y.-R. Liu, and S.-L. Zhu, Rept. Prog. Phys. 86, 026201 (2023) [arXiv:2204.02649 [hep-ph]]

    Pith/arXiv arXiv 2023

  36. [36]

    L. Meng, B. Wang, G.-J. Wang, and S.-L. Zhu, Phys. Rept. 1019, 1 (2023) [arXiv:2204.08716 [hep-ph]]

    Pith/arXiv arXiv 2023

  37. [37]

    J. T. Chacko, V. Baru, C. Hanhart, and S. L. Krug, Phys. Rev. D 111, 034042 (2025) [arXiv:2411.13303 [hep-ph]]

    Pith/arXiv arXiv 2025

  38. [38]

    M. B. Voloshin and L. B. Okun, JETP Lett. 23, 333 (1976)

    1976

  39. [39]

    Navas et al.[Particle Data Group], Phys

    S. Navas et al.[Particle Data Group], Phys. Rev. D 110, 030001 (2024). 27

    2024

  40. [40]

    W. A. Bardeen and W. K. Tung, Phys. Rev. 173, 1423 (1968), [Erratum: Phys. Rev. D 4, 3229 (1971)]

    1968

  41. [41]

    Tarrach, Nuovo Cim

    R. Tarrach, Nuovo Cim. A 28, 409 (1975)

    1975

  42. [42]

    K. J. Kim and Y.-S. Tsai, Phys. Rev. D 7, 3710 (1973)

    1973

  43. [43]

    R. G. Arnold, C. E. Carlson, and F. Gross, Phys. Rev. C 21, 1426 (1980)

    1980

  44. [44]

    R. G. Arnold, C. E. Carlson, and F. Gross, Phys. Rev. C 23, 363 (1981)

    1981

  45. [45]

    S. J. Brodsky and J. R. Hiller, Phys. Rev. D 46, 2141 (1992)

    1992

  46. [46]

    Jacob and G

    M. Jacob and G. C. Wick, Annals Phys. 7, 404 (1959)

    1959

  47. [47]

    Becher and H

    T. Becher and H. Leutwyler, Eur. Phys. J. 9, 643 (1999) [arXiv:hep-ph/9901384]

    Pith/arXiv arXiv 1999

  48. [48]

    Kubis and U.-G

    B. Kubis and U.-G. Meißner, Nucl. Phys. A 679, 698 (2001) [arXiv:hep-ph/0007056]

    Pith/arXiv arXiv 2001

  49. [49]

    Weinberg, Phys

    S. Weinberg, Phys. Rev. Lett. 17, 616 (1966)

    1966

  50. [50]

    Tomozawa, Nuovo Cim

    Y. Tomozawa, Nuovo Cim. A 46, 707 (1966)

    1966

  51. [51]

    Fettes, U.-G

    N. Fettes, U.-G. Meißner, and S. Steininger, Nucl. Phys. A 640, 199 (1998) [arXiv:hep-ph/9803266]

    Pith/arXiv arXiv 1998

  52. [52]

    Holmberg and S

    M. Holmberg and S. Leupold, Eur. Phys. J. A 54, 103 (2018) [arXiv:1802.05168 [hep-ph]]

    Pith/arXiv arXiv 2018

  53. [53]

    Jiang, Y.-R

    S.-Z. Jiang, Y.-R. Liu, and Q.-H. Yang, Phys. Rev. D 99, 074018 (2019) [arXiv:1901.09479 [hep-ph]]

    Pith/arXiv arXiv 2019

  54. [54]

    F.-K. Guo, C. Hanhart, and U.-G. Meißner, Eur. Phys. J. A 40, 171 (2009) [arXiv:0901.1597 [hep-ph]]

    Pith/arXiv arXiv 2009

  55. [55]

    M. F. M. Lutz, X.-Y. Guo, Y. Heo, and C. L. Korpa, Phys. Rev. D 106, 114038 (2022) [arXiv:2209.10601 [hep-ph]]

    Pith/arXiv arXiv 2022

  56. [56]

    K. M. Watson, Phys. Rev. 95, 228 (1954)

    1954

  57. [57]

    N. I. Muskhelishvili, Singular Integral Equations , Wolters-Noordhoff Publishing, 1953. [Dover Publica- tions, 2nd edition, 2008]

    1953

  58. [58]

    Omn` es, Nuovo Cim.8, 316 (1958)

    R. Omn` es, Nuovo Cim.8, 316 (1958)

    1958

  59. [59]

    Mutke, M

    S. Mutke, M. Hoferichter, and B. Kubis, JHEP 07, 276 (2024) [arXiv:2406.14608 [hep-ph]]

    Pith/arXiv arXiv 2024

  60. [60]

    Lucha, D

    W. Lucha, D. Melikhov, and S. Simula, Phys. Rev. D75, 016001 (2007) [arXiv:hep-ph/0610330], [Erratum: Phys. Rev. D 92, 019901 (2015)]

    Pith/arXiv arXiv 2007

  61. [61]

    Hoferichter, G

    M. Hoferichter, G. Colangelo, M. Procura, and P. Stof- fer, Int. J. Mod. Phys. Conf. Ser. 35, 1460400 (2014) [arXiv:1309.6877 [hep-ph]]

    Pith/arXiv arXiv 2014

  62. [62]

    Colangelo, M

    G. Colangelo, M. Hoferichter, M. Procura, and P. Stof- fer, JHEP 09, 074 (2015) [arXiv:1506.01386 [hep-ph]]

    Pith/arXiv arXiv 2015

  63. [63]

    Hoferichter and P

    M. Hoferichter and P. Stoffer, JHEP 07, 073 (2019) [arXiv:1905.13198 [hep-ph]]

    Pith/arXiv arXiv 2019

  64. [64]

    L. D. Landau, Nucl. Phys. 13, 181 (1959), [Sov. Phys. JETP 10 (1960) 45; Zh. Eksp. Teor. Fiz. 37 (1959) 62]

    1959

  65. [65]

    V. N. Gribov, V. V. Anisovich, and A. A. Anselm, Sov. Phys. JETP 15, 159 (1962)

    1962

  66. [66]

    J. B. Bronzan and C. Kacser, Phys. Rev. 132, 2703 (1963)

    1963

  67. [67]

    Niehus, M

    M. Niehus, M. Hoferichter, and B. Kubis, JHEP 12, 038 (2021) [arXiv:2110.11372 [hep-ph]]

    Pith/arXiv arXiv 2021

  68. [68]

    W. R. Frazer and J. R. Fulco, Phys. Rev. 117, 1603 (1960)

    1960

  69. [69]

    H¨ ohler and E

    G. H¨ ohler and E. Pietarinen, Phys. Lett. B 53, 471 (1975)

    1975

  70. [70]

    Bernard, N

    V. Bernard, N. Kaiser, and U.-G. Meißner, Nucl. Phys. A 611, 429 (1996) [arXiv:hep-ph/9607428]

    Pith/arXiv arXiv 1996

  71. [71]

    M. Dax, T. Isken, and B. Kubis, Eur. Phys. J. C 78, 859 (2018) [arXiv:1808.08957 [hep-ph]]

    Pith/arXiv arXiv 2018

  72. [72]

    Niehus, M

    M. Niehus, M. Hoferichter, B. Kubis, and J. Ruiz de Elvira, Phys. Rev. Lett. 126, 102002 (2021) [arXiv:2009.04479 [hep-ph]]

    Pith/arXiv arXiv 2021

  73. [73]

    Dobado, M

    A. Dobado, M. J. Herrero, and T. N. Truong, Phys. Lett. B 235, 134 (1990)

    1990

  74. [74]

    Dobado and J

    A. Dobado and J. R. Pel´ aez, Phys. Rev. D 56, 3057 (1997) [arXiv:hep-ph/9604416]

    Pith/arXiv arXiv 1997

  75. [75]

    S. Holz, J. Plenter, C.-W. Xiao, T. Dato, C. Hanhart, B. Kubis, U.-G. Meißner, and A. Wirzba, Eur. Phys. J. C 81, 1002 (2021) [arXiv:1509.02194 [hep-ph]]

    Pith/arXiv arXiv 2021

  76. [76]

    S. Holz, M. Hoferichter, B.-L. Hoid, and B. Kubis, JHEP 04, 147 (2025) [arXiv:2412.16281 [hep-ph]]

    Pith/arXiv arXiv 2025

  77. [77]

    Aoki et al

    Y. Aoki et al. [Flavour Lattice Averaging Group (FLAG) Collaboration], Eur. Phys. J. C 82, 869 (2022) [arXiv:2111.09849 [hep-lat]]

    Pith/arXiv arXiv 2022

  78. [78]

    Bazavov et al

    A. Bazavov et al. [MILC Collaboration], PoS LA T- TICE2010, 074 (2010) [arXiv:1012.0868 [hep-lat]]

    Pith/arXiv arXiv 2010

  79. [79]

    Bors´ anyi, S

    S. Bors´ anyi, S. D¨ urr, Z. Fodor, S. Krieg, A. Sch¨ afer, E. E. Scholz, and K. K. Szab´ o, Phys. Rev. D88, 014513 (2013) [arXiv:1205.0788 [hep-lat]]

    Pith/arXiv arXiv 2013

  80. [80]

    D¨ urret al

    S. D¨ urret al. [BMW Collaboration], Phys. Rev. D 90, 114504 (2014) [arXiv:1310.3626 [hep-lat]]

    Pith/arXiv arXiv 2014

Showing first 80 references.