Pith. sign in

REVIEW 2 major objections 5 minor 1 cited by

Lattice QCD calculations of hadron spectroscopy

T0 review · 2 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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

Pith's one-line read Lattice QCD can derive hadron masses, widths, and scattering amplitudes from first principles.

desk verdict A clean, conventional pedagogical chapter on lattice QCD spectroscopy: useful for students, no new physics, with two small spots where the simplification is too quick. read the letter →

arxiv 2505.10002 v1 pith:QZHZMEPM submitted 2025-05-15 hep-lat hep-exhep-ph

Sasa Prelovsek This is my paper · ORCID
classification hep-lathep-exhep-ph PACS 12.38.Gc
keywords latticeQCDhadronspectroscopyhadronicresonancesLüscherformalismscatteringamplitudesexotichadronsfinite-volumeeigen-energiescoupled-channel
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 chapter sets out to show that hadron spectroscopy is accessible from first principles through lattice QCD: strongly stable hadrons get their masses directly from ground-state eigen-energies, while unstable hadrons have to be read off the scattering amplitude for the channel into which they decay. The key step is the finite-volume quantization condition, which turns each discrete two-hadron eigen-energy in a periodic box into a value of the infinite-volume scattering amplitude $T(E)$ at that same energy. Once enough eigen-energies are collected, a resonance's pole in the complex energy plane gives its mass and width, and the amplitude gives the phase shifts and lineshapes. The chapter illustrates the route with one-channel examples ($\rho$, $K^*$, $D_{s0}^*$), near-threshold exotic states ($T_{cc}$, $T_{bc}$), and a coupled-channel case ($\pi\pi$-$K\bar{K}$-$\eta\eta$). If the programme works, then the properties of both ordinary and exotic hadrons can be predicted from QCD rather than inserted by hand.

What carries the argument

The load-bearing object is the finite-volume quantization condition for two-hadron states, usually called the L\"uscher relation. In its simplest s-wave, zero-momentum form it reads $T_{l=0}^{-1}(E) = -iF(E)$, where $F$ is a known kinematic function built from the difference between a continuum loop integral and a discrete sum over the momenta allowed in a periodic box. The derivation keeps only the part of the loop in which both intermediate particles are on shell; the residual difference is exponentially suppressed as $e^{-mL}$ and is neglected. The same $F$ generalizes to moving frames and to several coupled channels, where the condition becomes $\det(T + iF^{-1}) = 0$. This function is what carries the argument: it converts a set of box energies into phase shifts and pole positions.

What would settle it

Take a single scattering channel at a fixed pion mass, extract the phase shift from two or more lattice volumes differing by roughly a factor of two, and check whether the extracted phase-shift points agree to within the quoted statistical error; a trend with $L$ at the level of $e^{-mL}$ would show the neglected corrections are not negligible. A second check is to refit the same eigen-energy sets with two different parametrizations of $T(E)$ and compare the pole positions.

Watch

Extended reading notes

Core claim

The central claim, stated in the author's own terms, is that the spectroscopic information for any strongly interacting hadron is contained in the discrete eigen-energies $E_n$ of QCD in a finite volume. For a stable hadron, $m = E_1(\vec P=\vec 0)$ after continuum and infinite-volume extrapolations. For an unstable hadron, the finite-volume quantization condition $T_{l=0}^{-1}(E) = -iF(E)$ maps each lattice eigen-energy to the corresponding point of the infinite-volume scattering amplitude, and the resonance parameters are read from the pole position $E_p = m - i\Gamma/2$. Coupled channels generalize this to the determinant condition $\det(T + iF^{-1})=0$, where the energy dependence of $T$ is parametrized and fitted to the computed levels. The examples presented are offered as demonstrations that this extraction is practical at current lattice sizes and quark masses.

Load-bearing premise

The load-bearing premise is that exponentially suppressed finite-volume corrections of order $e^{-mL}$ really are negligible, so that each discrete lattice eigen-energy can be equated with the infinite-volume scattering amplitude at that same energy.

Editorial extensions

If this is right

  • Masses of stable hadrons such as the proton, neutron, and heavy-light mesons can be computed from QCD with sub-percent precision once quark masses and lattice spacing are fixed, with no hadronic model input.
  • For resonances that decay through a single channel, lattice eigen-energies at several volumes and momenta produce phase shifts that fix the resonance mass and width; physical-quark-mass results for $\rho$ and $K^*$ agree with experiment.
  • The same machinery assigns masses and binding energies to near-threshold exotic states; the $T_{cc}$ pole trajectory shows how a state can pass from resonance to virtual to bound as quark masses change.
  • For states above several thresholds, a coupled-channel analysis constrains all elements of the scattering matrix and locates poles such as $f_0$ and $f_2$.
  • Because every lattice eigen-energy yields one point on the amplitude, increasing the number of volumes, total momenta, and interpolating operators directly sharpens the extracted pole parameters.

Reading between the lines

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

  • If the exponentially suppressed terms $e^{-mL}$ were computed instead of omitted, near-threshold states with tiny binding energies (for example $T_{cc}^+$, bound by under 1 MeV) would get a first-principles systematic error; today that error is only assumed small.
  • The same mapping from box energies to amplitudes is the template for the emerging three-hadron formalism; the one- and two-channel cases in this chapter are the natural testbed for those extensions.
  • A direct comparison of the quantization-condition, static-potential, and HAL QCD routes on the same heavy-quark system would isolate the model dependence that currently enters through the parametrization of $T(E)$.
  • Reanalysis of existing published spectra with alternative parametrizations of the amplitude could reveal whether reported pole positions (especially for broad states) are physical or fit-dependent.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. This manuscript is a pedagogical book chapter on lattice QCD hadron spectroscopy. It explains how masses of strongly stable hadrons are extracted from two-point correlation functions, and how resonance masses, widths, and lineshapes are obtained from scattering amplitudes T(E). The central technical thread is a simplified derivation of the one-channel Lüscher relation, which maps finite-volume eigen-energies E(L) to the infinite-volume scattering amplitude at the same energy, followed by explicit examples: the rho and K* resonances, the D*_{s0}(2317) bound state, the T_cc near the DD* threshold, coupled-channel f0/f2 systems, and nucleon-nucleon scattering. The chapter concludes with static-potential and HAL QCD approaches. The text is intentionally not a full review; it delegates technical details to the cited literature.

Significance. The chapter serves as a self-contained introduction to a technically demanding subject. Its main strength is the explicit derivation of the Lüscher relation from the finite-volume correlator (Eqs. (8)–(12)), which is presented in a form accessible to graduate students, with the technical details correctly deferred to Refs. [12,14]. The examples are drawn from published lattice studies, so the pedagogical statements are backed by peer-reviewed results. The chapter does not claim new numerical results; its value is didactic. The claim that lattice QCD is a first-principles, systematically improvable method is supported by the cited calculations of stable hadron masses and scattering phase shifts. The main correctness risk is not circularity—the finite-volume energies are mapped to the infinite-volume amplitude, not fitted circularly—but rather the unquantified neglect of exponentially suppressed finite-volume effects and the simplified treatment of partial waves, both of which are noted in the text.

major comments (2)
  1. [4.1, Eq. (11)] As typeset, the final expression for C_V(E)=C_∞(E)−A'[1/(T+iF)−1]A does not follow from the preceding geometric series C_∞(E)−A'F Σ_{j=0}^∞(−iTF)^j A, and it is inconsistent with the pole condition in Eq. (12), which requires T^{-1}(E)=−iF(E). The denominator should read T^{-1}+iF (or an equivalent rearrangement of the standard Lüscher form). Please correct this equation; the intended result is standard, but the printed form will mislead a student.
  2. [4.2, Fig. 6 and Eq. (12)] The derivation in Section 4.1 is restricted to s-wave scattering (l=0, F=F_{00,00}), but the first application is the p-wave rho resonance. The footnote under Figure 6 states that Eq. (12) 'applies here although this is scattering with l=1,' which is misleading because Eq. (12) as derived does not hold for l=1; the relevant Lüscher relation uses the p-wave kinematic function (e.g., from Eq. (22) of Ref. [14]). The chapter should explicitly explain how the p-wave example is connected to the l=0 derivation, or reduce the derivation claim to the s-wave case and refer to the general formalism for the examples.
minor comments (5)
  1. [4.2, final paragraph; Fig. 12 caption] "Deutron" should be "deuteron" in the text and in the Figure 12 caption.
  2. [Footnote 6] "The so-called so-called time-dependent method" should be "the so-called time-dependent method."
  3. [Section 2, before Eq. (2)] The phrase "at large enought" should be "at large enough t."
  4. [Section 4.1, before Eq. (9)] The neglect of exponentially suppressed e^{-mL} corrections is asserted without quantification; a sentence noting that the cited applications control these corrections through multiple volumes and systematic studies (e.g., Refs. [12,15,16]) would help the reader distinguish the pedagogical simplification from a general proof.
  5. [Section 4.2, Eq. (13)] The Breit-Wigner form in Eq. (13) is introduced after the phase-shift plot; the text should specify that this is a parametrization of T(E) chosen for fitting, not a direct consequence of Eq. (12), to avoid implying that the pole parameters are extracted without a model assumption.

Circularity Check

0 steps flagged · score 1.0 of 10

Independent derivation, not circular: the Lüscher relation in Sec. 4.1 is derived from external Ref. [12], and the sole self-citation, Ref. [22], is an illustrative example rather than a load-bearing premise.

full rationale

The central claim is not circular. In Sec. 4.1 the paper derives the finite-volume to infinite-volume relation by resumming finite-volume loop corrections: Eq. (11) expresses the finite-volume correlator C_V in terms of the infinite-volume amplitude T and the kinematical function F, and Eq. (12) follows as T^{-1}(E) = -iF(E). This is a nontrivial mapping from lattice eigen-energies to scattering amplitudes: the eigen-energies are inputs and T(E) is the output. The derivation explicitly follows the external Lüscher formalism, with the statement 'I present essential steps from Ref. [12]' correctly attributing the derivation rather than importing the conclusion from the author's own work. The examples subsequently fit Breit-Wigner or scattering-matrix parameters to the extracted phase shifts and energies, e.g. Eq. (13); those fits are transparent downstream extractions and are not used to define the Lüscher relation. The single self-citation, Ref. [22], is used only to illustrate the T_cc pole trajectory and is not load-bearing for the method. The neglect of exponentially suppressed finite-volume corrections and the dependence of pole positions on the chosen T parametrization are honest systematic uncertainties, not circular reductions.

Assumptions & free parameters 3 free parameters · 7 assumptions · 0 invented entities

The chapter adds no new postulates. It relies on standard QCD and established finite-volume quantization; examples introduce fitted parameters for the scattering amplitude parametrizations, which are not new physics. The only entities discussed are known hadrons (rho, K*, D_s0*, T_cc, T_bb, etc.) and known methods (Lüscher, HAL QCD).

free parameters (3)
  • lattice quark masses and strong coupling g_s = tuned to reproduce experimental hadron masses in the cited simulations
    Section 1 states that m_q and g_s are the only free parameters of the underlying QCD Lagrangian. They are standard inputs from prior literature, not introduced by this chapter, but the numerical examples depend on their chosen values.
  • Breit-Wigner mass m and coupling g in Eq. (13) = m_rho = 798(7) MeV at m_pi ~ 320 MeV; g = 5.7(2) in the rho example
    These parameters are fitted to the lattice phase shifts to locate the resonance pole and width through Eq. (13). They are example-specific and not part of the general Lüscher relation.
  • coupled-channel T_ij(E,kappa) parameters = not individually listed; kappa fitted to lattice eigen-energies
    In Section 5, the energy dependence of the scattering matrix is parametrized by a few parameters kappa and fitted to the generalized Lüscher determinant equation (14) to reproduce lattice energies.
assumptions (7)
  • domain assumption QCD, with L_QCD(m_q,g_s), is the correct fundamental theory of strong interactions.
    The entire chapter assumes QCD and neglects electroweak interactions; stated in Section 1.
  • domain assumption Euclidean lattice discretization of QCD, with finite volume and lattice spacing, approximates physical QCD after continuum and infinite-volume extrapolations.
    Sections 1 and 2 state that reliable predictions require multiple lattice spacings and volumes with a->0 and L->infinity extrapolations.
  • domain assumption The finite-volume correction to the two-particle loop is dominated by on-shell intermediate states and exponentially suppressed terms e^{-mL} are negligible.
    Explicitly used in deriving Eqs. (9)-(12) in Section 4.1; this is the central approximation of the Lüscher formalism.
  • domain assumption The GEVP correlation matrix is dominated by the lowest N eigenstates for t > t0, so single- or two-exponential fits yield the energies.
    Section 2 relies on this to extract E_n from the eigenvalue problem C(t)u = lambda(t) C(t0)u.
  • domain assumption For each scattering channel, the amplitude is dominated by a single partial wave l (or a finite set), and the scattering matrix can be parametrized by a few fitted parameters.
    Sections 4 and 5 use a single-channel phase shift for rho and K*, and in the coupled-channel case parametrize T_ij(E,kappa) to solve the determinant equation.
  • domain assumption Born-Oppenheimer approximation for systems with two heavy quarks, treating light degrees of freedom as fast and heavy quarks as static.
    Section 6 uses static B/B* mesons to obtain the potential V(r) and then solves the Schrödinger equation for the T_bb tetraquark.
  • domain assumption The HAL QCD method relates the equal-time Bethe-Salpeter wave function to a hadronic potential with local-central truncation.
    Section 7 defines the potential U(r,r') and uses its local part; the precise definition is delegated to the HAL QCD review [28].

how reviews work

0 comments
Cite this review

Pith. "Pith review of Lattice QCD calculations of hadron spectroscopy." pith.science (2026). https://pith.science/paper/QZHZMEPM

@misc{pith2026250510002,
  author       = {Pith},
  title        = {Pith review of: Lattice QCD calculations of hadron spectroscopy},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QZHZMEPM}},
  note         = {Machine review of arXiv:2505.10002}
}
read the original abstract

This chapter provides a pedagogical introduction to theoretical studies of hadrons based on the fundamental theory of strong interactions - Quantum ChromoDynamics. A perturbative expansion in the strong coupling is not applicable at hadronic energy scales. Lattice Quantum Chromodynamics is the formulation of the fundamental theory on a discrete space-time grid, which enables first-principles, systematically improvable, numerical simulations of strong interaction physics. This chapter explains how the masses of strongly stable and strongly decaying hadrons are determined. The strongly decaying hadrons have to be inferred from the corresponding scattering processes. Therefore, one of the main aims is to describe how the scattering amplitudes are extracted from a lattice simulation. The examples of spectra, widths, and scattering amplitudes are shown for conventional as well as exotic hadrons.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

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

  1. A beginner's guide to functional methods in particle physics

    hep-ph 2025-10 accept novelty 1.0 of 10

    A pedagogical review showing how Dyson-Schwinger, 3PI, and Bethe-Salpeter equations can be chained together to compute glueball masses in pure Yang-Mills theory, matching lattice QCD.

Reference graph

Works this paper leans on

25 extracted references · 3 canonical work pages · cited by 1 Pith paper

  1. [14]

    Two-particle multichannel systems in a finite volume with arbitrary spin,

    R. A. Briceno, “Two-particle multichannel systems in a finite volume with arbitrary spin,”Phys. Rev. D89no. 7, (2014) 074507, arXiv:1401.3312 [hep-lat]

    arXiv 2014

  2. [1]

    TheXYZstates: experimental and theoretical status and perspectives,

    N. Brambilla, S. Eidelman, C. Hanhart, A. Nefediev, C.-P . Shen, C. E. Thomas, A. Vairo, and C.-Z. Yuan, “TheXYZstates: experimental and theoretical status and perspectives,”Phys. Rept.873(2020) 1–154,arXiv:1907.07583 [hep-ex]

    arXiv 2020

  3. [2]

    Tetraquarks and pentaquarks in lattice QCD with light and heavy quarks,

    P . Bicudo, “Tetraquarks and pentaquarks in lattice QCD with light and heavy quarks,”Phys. Rept.1039(2023) 1–49,arXiv:2212.07793 [hep-lat]

    arXiv 2023

  4. [3]

    Hadron Spectroscopy with Lattice QCD,

    J. Bulavaet al., “Hadron Spectroscopy with Lattice QCD,” inSnowmass 2021. 3, 2022.arXiv:2203.03230 [hep-lat]

    arXiv 2021

  5. [4]

    Scattering processes and resonances from lattice QCD,

    R. A. Briceno, J. J. Dudek, and R. D. Y oung, “Scattering processes and resonances from lattice QCD,”Rev. Mod. Phys.90no. 2, (2018) 025001,arXiv:1706.06223 [hep-lat]

    arXiv 2018

  6. [5]

    Hadron spectroscopy and few-body dynamics from Lattice QCD,

    A. D. Hanlon, “Hadron spectroscopy and few-body dynamics from Lattice QCD,”PoSLATTICE2023(2024) 106,arXiv:2402.05185 [hep-lat]. [6]Particle Data GroupCollaboration, S. Navaset al., “Review of particle physics,”Phys. Rev. D110no. 3, (2024) 030001

    arXiv 2024

  7. [7]

    Gattringer and C

    C. Gattringer and C. B. Lang,Quantum chromodynamics on the lattice, vol. 788. Springer, Berlin, 2010

    work page 2010

  8. [8]

    How to Calculate the Elastic Scattering Matrix in Two-dimensional Quantum Field Theories by Numerical Simulation,

    M. Luscher and U. Wolff, “How to Calculate the Elastic Scattering Matrix in Two-dimensional Quantum Field Theories by Numerical Simulation,”Nucl. Phys. B339(1990) 222–252. [9]BMWCollaboration, S. Durret al., “Ab-Initio Determination of Light Hadron Masses,”Science322(2008) 1224–1227,arXiv:0906.3599 [hep-lat]

    arXiv 1990

Show all 25 references
  1. [10]

    Precise heavy-light meson masses and hyperfine splittings from lattice QCD including charm quarks in the sea,

    R. J. Dowdall, C. T. H. Davies, T. C. Hammant, and R. R. Horgan, “Precise heavy-light meson masses and hyperfine splittings from lattice QCD including charm quarks in the sea,”Phys. Rev. D86(2012) 094510,arXiv:1207.5149 [hep-lat]

    2012 arXiv

  2. [11]

    Improved analysis of strong-interaction-stable doubly bottom tetraquarks on the lattice,

    B. Colquhoun, A. Francis, R. J. Hudspith, R. Lewis, K. Maltman, and W. G. Parrott, “Improved analysis of strong-interaction-stable doubly bottom tetraquarks on the lattice,”Phys. Rev. D110no. 9, (2024) 094503,arXiv:2407.08816 [hep-lat]

    2024 arXiv

  3. [12]

    Finite-volume effects for two-hadron states in moving frames,

    C. h. Kim, C. T. Sachrajda, and S. R. Sharpe, “Finite-volume effects for two-hadron states in moving frames,”Nucl. Phys. B727(2005) 218–243,arXiv:hep-lat/0507006

    2005 arXiv

  4. [13]

    Two particle states on a torus and their relation to the scattering matrix,

    M. Luscher, “Two particle states on a torus and their relation to the scattering matrix,”Nucl. Phys. B354(1991) 531–578

    1991

  5. [15]

    P-waveππscattering and theρresonance from lattice QCD,

    C. Alexandrou, L. Leskovec, S. Meinel, J. Negele, S. Paul, M. Petschlies, A. Pochinsky, G. Rendon, and S. Syritsyn, “P-waveππscattering and theρresonance from lattice QCD,”Phys. Rev. D96no. 3, (2017) 034525,arXiv:1704.05439 [hep-lat]

    2017 arXiv

  6. [16]

    Light and Strange Vector Resonances from Lattice QCD at Physical Quark Masses,

    P . Boyle, F . Erben, V. G¨ulpers, M. T. Hansen, F . Joswig, M. Marshall, N. P . Lachini, and A. Portelli, “Light and Strange Vector Resonances from Lattice QCD at Physical Quark Masses,”Phys. Rev. Lett.134no. 11, (2025) 111901,arXiv:2406.19194 [hep-lat]

    2025 arXiv

  7. [17]

    Masses and decay constants of theD∗ s0(2317)andD s1(2460)fromN f =2lattice QCD close to the physical point,

    G. S. Bali, S. Collins, A. Cox, and A. Sch ¨afer, “Masses and decay constants of theD∗ s0(2317)andD s1(2460)fromN f =2lattice QCD close to the physical point,”Phys. Rev. D96no. 7, (2017) 074501,arXiv:1706.01247 [hep-lat]. [18]Hadron SpectrumCollaboration, J. D. E. Y eo, C. E. ...

    2017 arXiv

  8. [19]

    On Heavy light meson resonances and chiral symmetry,

    E. E. Kolomeitsev and M. F . M. Lutz, “On Heavy light meson resonances and chiral symmetry,”Phys. Lett. B582(2004) 39–48, arXiv:hep-ph/0307133

    2004 arXiv

  9. [20]

    Towards a new paradigm for heavy-light meson spectroscopy,

    M.-L. Du, M. Albaladejo, P . Fern´andez-Soler, F .-K. Guo, C. Hanhart, U.-G. Meißner, J. Nieves, and D.-L. Y ao, “Towards a new paradigm for heavy-light meson spectroscopy,”Phys. Rev. D98no. 9, (2018) 094018,arXiv:1712.07957 [hep-ph]

    2018 arXiv

  10. [21]

    Two-pole structure of theD∗ 0(2400),

    M. Albaladejo, P . Fernandez-Soler, F .-K. Guo, and J. Nieves, “Two-pole structure of theD∗ 0(2400),”Phys. Lett. B767(2017) 465–469, arXiv:1610.06727 [hep-ph]

    2017 arXiv

  11. [22]

    Toward the quark mass dependence of Tcc+ from lattice QCD,

    S. Collins, A. Nefediev, M. Padmanath, and S. Prelovsek, “Toward the quark mass dependence of Tcc+ from lattice QCD,”Phys. Rev. D 109no. 9, (2024) 094509,arXiv:2402.14715 [hep-lat]. [23]LHCbCollaboration, R. Aaijet al., “Study of the doubly charmed tetraquarkT + cc,”Nature Com...

    2024 arXiv

  12. [24]

    Role of Left-Hand Cut Contributions on Pole Extractions from Lattice Data: Case Study for Tcc(3875)+,

    M.-L. Du, A. Filin, V. Baru, X.-K. Dong, E. Epelbaum, F .-K. Guo, C. Hanhart, A. Nefediev, J. Nieves, and Q. Wang, “Role of Left-Hand Cut Contributions on Pole Extractions from Lattice Data: Case Study for Tcc(3875)+,”Phys. Rev. Lett.131no. 13, (2023) 131903, arXiv:2303.09441 [hep-ph]

    2023 arXiv

  13. [25]

    Shallow Bound States and Hints for Broad Resonances with Quark Content b¯c¯ud in B-D¯and B*-D¯Scattering from Lattice QCD,

    C. Alexandrou, J. Finkenrath, T. Leontiou, S. Meinel, M. Pflaumer, and M. Wagner, “Shallow Bound States and Hints for Broad Resonances with Quark Content b¯c¯ud in B-D¯and B*-D¯Scattering from Lattice QCD,”Phys. Rev. Lett.132no. 15, (2024) 151902, arXiv:2312.02925 [hep-lat]. L...

    2024 arXiv

  14. [28]

    Lattice QCD signal for a bottom-bottom tetraquark,

    S. Aoki and T. Doi,Lattice QCD and Baryon-Baryon Interactions, pp. 1–31. 2023.arXiv:2402.11759 [hep-lat]. [29]European Twisted MassCollaboration, P . Bicudo and M. Wagner, “Lattice QCD signal for a bottom-bottom tetraquark,”Phys. Rev. D87 no. 11, (2013) 114511,arXiv:1209.6274 [hep-ph]

    2013 arXiv

  15. [30]

    Isoscalarππ,K K,ηηscattering and theσ,f 0,f 2 mesons from QCD,

    R. A. Briceno, J. J. Dudek, R. G. Edwards, and D. J. Wilson, “Isoscalarππ,K K,ηηscattering and theσ,f 0,f 2 mesons from QCD,”Phys. Rev. D97no. 5, (2018) 054513,arXiv:1708.06667 [hep-lat]

    2018 arXiv

  16. [31]

    Hybrids, tetraquarks, pentaquarks, doubly heavy baryons, and quarkonia in Born-Oppenheimer effective theory,

    M. Berwein, N. Brambilla, A. Mohapatra, and A. Vairo, “Hybrids, tetraquarks, pentaquarks, doubly heavy baryons, and quarkonia in Born-Oppenheimer effective theory,”Phys. Rev. D110no. 9, (2024) 094040,arXiv:2408.04719 [hep-ph]

    2024 arXiv

  17. [32]

    A General survey of the theory of the Bethe-Salpeter equation,

    N. Nakanishi, “A General survey of the theory of the Bethe-Salpeter equation,”Prog. Theor. Phys. Suppl.43(1969) 1–81

    1969

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.