Pith. Sign in

REVIEW

QCD forces and heavy quark bound states

PaperPDF from arXiv

data files

ReviewThe objections and verdict, as JSON Paper breakdownThe authors' claims, conditions and measurements, as JSON Signed receiptProof this review has not been altered
X Bluesky LinkedIn Reddit Hacker News
Gunnar S. Bali

High Energy Physics - Phenomenology · Submitted 29 Jan 2000 · 300 references

Pith's verdict: The standard reference for the lattice-static-potential-to-quarkonium chain, and unusually honest about the one link it cannot close.

T0 · AI review

How this review was made: An AI referee (selfhosted/deepseek-v4.1-flash) read the paper on October 29, 2026 and answered a fixed list of questions about it. T0 is the starting level every paper gets. The levels describe how deeply Pith has checked the paper, never the authors.

  1. T0 AI review An AI read the paper and answered a fixed list of questions.
  2. T1 Audit Automatic checks that the references and files the paper relies on exist and match.
  3. T2 Compute The paper's code or calculations re-run with the same results.
  4. T3 Formal statement The claims written out in a formal math language a computer can read.
  5. T4 Proof Those claims proved, and the proof checked by computer.

How Pith checks papers →

· 5 major objections · 9 minor · Ask about this paper · The compiled record · Read the full review →
Challenge this review Comment (0) Read on arXiv ↗
Cite ▾
arXiv IDhep-ph/0001312 (version 2)
Pith IDS7FLK2JK, Pith's permanent record of this paper and its review
CitationGunnar S. Bali. "QCD forces and heavy quark bound states." arXiv:hep-ph/0001312 (2000). https://pith.science/paper/S7FLK2JK
Cite this reviewBibTeX for Pith's review of this paper
Claim paper

Why Pith says this

Highlight any text to comment on it.

Referee report

How Pith checks papers → · Full audit of this paper →

The paper

The paper claims

Lattice QCD now supplies the confining force between colour charges as a computable input, and that input is being turned into a QCD-derived Hamiltonian for heavy quarkonia.

The reading

The paper asks whether the force between two infinitely heavy colour charges — the confining static potential — can be computed from QCD rather than reverse-engineered from quarkonium spectra, and it answers that lattice simulations now deliver that potential, and a family of excited variants of it, as functions of distance with controlled errors. It then asks what the potential is good for: fed into a non-relativistic expansion of QCD, it yields a Hamiltonian for charmonium and bottomonium whose ingredients are all computable, so the classic potential models become approximations to QCD rather than postulates. Along the way the review catalogues the excited flux-tube ('hybrid') potentials, the potentials between charges in higher colour representations, three-quark forces, and the glueball and gluelump spectra that fix the short-distance limits of those potentials. A reader should care because the same machinery sets the size of non-perturbative corrections to top-pair production near threshold and to heavy-meson decay constants, and because it makes concrete, checkable statements about where the string between colour charges must break once light quarks are present.

What carries the argument

The carrier of the argument is the rectangular Wilson loop and its spectral decomposition: $W(r,t) = sum_n |c_n|^2 e^{-E_n(r)t}$, so that the ground-state energy $E_0(r)$ is the static potential and the excited energies $E_n(r)$ are the hybrid potentials. Everything else hangs off this. Smearing and fuzzing of the spatial transporters are what make the $t to infinity$ limit attainable at moderate $t$, and the review is explicit that a careless choice of a single $t_{min}$ for all $r$ manufactures a fake linear rise, since the $r leftrightarrow t$ symmetry of the loop forces $V(r,t) = sigma_eff(t) r$ with $sigma_eff(t) = V'(t)$. The second piece of machinery is the matching of NRQCD to QCD: the Dirac Lagrangian is expanded in $1/c$ (equivalently in the quark velocity $v$), the short-distance behaviour is absorbed into Wilson coefficients such as $c_F$, $c_D$, $c_S$ subject to the reparametrisation constraint $c_S = 2c_F - 1$, and the residual long-distance physics is carried by potentials. The third piece is the group theory of $D_*$ and its cubic subgroup, which tells you which continuum $J^{PC}$ a given lattice hybrid operator can subduce, and therefore what the short-distance limit of each hybrid potential must be.

Extended reading notes

Core claim

The review's load-bearing claim is that the confining force between static colour sources is no longer a fitted ingredient but a measured one, and that this measurement, combined with effective-field-theory methods, produces a semi-relativistic Hamiltonian for heavy quarkonia that is derived from QCD rather than assumed. Concretely: the static potential is extracted from Wilson loops whose spectral decomposition identifies the ground-state energy with $V(r)$; smearing and variational techniques make the large-time limit reachable; the resulting curve is described by a Coulomb-plus-linear form over $0.2$–$1$ fm with a running Coulomb strength that shows asymptotic freedom and, in unquenched simulations, the expected screening from sea quarks. Around this ground state sits a spectrum of hybrid potentials classified by the cylindrical symmetry group $D_*$, which at short distances must merge into glueball or gluelump levels and at long distances into string-like bands. Potentials between charges in higher $SU(3)$ representations, and the three-quark potential, extend the same logic to multi-body and exotic configurations, with the review reporting evidence for the $3/2$ rule $V_{3q} = (3/2)V_{qbar q}$ on an equilateral triangle and approximate Casimir scaling of intermediate-distance string tensions.

Load-bearing premise

The adiabatic (Born-Oppenheimer) approximation — that gluons and sea quarks rearrange themselves much faster than the heavy quarks move, so that they can be integrated out into instantaneous potentials — is what licenses the entire potential-based Hamiltonian, and the review itself concedes it is dubious for excited and non-exotic hybrid states.

Editorial extensions

If this is right

  • Quarkonium spectroscopy becomes a consistency check on QCD rather than a fit: the same potential, with no flavour dependence, must reproduce the spin-averaged charmonium and bottomonium level splittings, and the review notes that these splittings agree within a few per cent.
  • The short-distance behaviour of the potential is computable to two loops in perturbation theory, so precision bottomonium and threshold top-pair observables become tests of the perturbative series and of the power correction $c_V Lambda^2/q^2$ that the review reports as $c_V = 4.8 pm 1.4$ in the quenched case.
  • Hybrid mesons with quark-model-forbidden quantum numbers such as $1^{-+}$ are predicted as a matter of course, with the lightest hybrids governed by the $Pi_u$ potential and coming out near or just above the $B bar B$ threshold in the quenched, non-relativistic, leading-order Born-Oppenheimer treatment.
  • In full QCD the static potential must stop rising and flatten once the string can break into two static-light mesons, near $r approx 2.3, r_0 approx 1.15$ fm, with a mixing gap rather than a level crossing; the size of that gap feeds directly into decay rates such as $Upsilon(4S) to B bar B$.
  • Intermediate-distance potentials between higher-representation charges are approximately Casimir-scaled, $sigma_D approx d_D sigma_F$, while asymptotically only a residual fundamental string survives for representations with non-trivial centre charge.
  • Three-quark forces on an equilateral triangle follow $V_{3q} = (3/2) V_{q bar q}$ at the distances probed so far, which is what tree-level perturbation theory and the $Delta$ law give, and which is about 15 per cent below the naive Y-law expectation.

Reading between the lines

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

  • If the adiabatic approximation holds as well as the review hopes, then the same lattice potentials should let one predict, not just postdict, the masses of the as-yet-unmeasured $B_c$ states and of the $1P$ bottomonium levels; the review's own tables show the $eta_b$ has still not been seen, which makes that a clean test.
  • The review's discussion of the $r leftrightarrow t$ symmetry of Wilson loops is a warning that generalises: any lattice extraction of an energy from an exponentially decaying correlator inherits a systematic bias whenever the fitting window is chosen without regard to the state's spatial extent, a point that applies well beyond the static potential.
  • The reported tension between lattice hybrid potentials and the Nambu-Goto string prediction, contrasted with the good agreement found for closed strings (torelon masses), suggests the effective string description may need higher-derivative corrections whose coefficients grow with the transverse excursion amplitude, which is testable by comparing flux-tube widths against level spacings.
  • The failure to detect string breaking in smeared Wilson loops, while Polyakov-line correlators at finite temperature do flatten, hints that operator choice rather than physics may be the obstacle; a calculation using all-to-all light-quark propagators with a two-by-two correlation matrix should settle it, and the review says such studies are underway.
Original abstract

The present knowledge of QCD confining forces between static test charges is summarised, with an emphasis on lattice results. Recent developments in relating QCD potentials to quarkonium properties by use of effective field theory methods are presented. The validity of non-relativistic QCD and the adiabatic approximation with respect to heavy quark bound states is explored. Besides the static potential and relativistic correction terms, the spectra of glueballs and gluinoballs, hybrid excitations of the QCD flux tube between fundamental colour sources, potentials between charges in various representations of the SU(3) gauge group, and multi-particle interactions are discussed. Some implications for quarkonia systems and quark-gluon hybrid mesons are drawn.

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-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,258 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.02 · states limitation 0.09 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.04 · compares with prior results 0.17 · mentions code or data availability 0.02.

The abstract, sentence by sentence

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

The present knowledge of QCD confining forces between static test charges is summarised, with an emphasis on lattice results. Recent developments in relating QCD potentials to quarkonium properties by use of effective field theory methods are presented. The validity of non-relativistic QCD and the adiabatic approximation with respect to heavy quark bound states is explored. Besides the static potential and relativistic correction terms, the spectra of glueballs and gluinoballs, hybrid excitations of the QCD flux tube between fundamental colour sources, potentials between charges in various representations of the SU(3) gauge group, and multi-particle interactions are discussed. Some implications for quarkonia systems and quark-gluon hybrid mesons are drawn.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 The present knowledge of QCD confining forces between static test charges is summarised, with an emphasis on lattice results. background 0.71 0.11 0.05 asserted 0.89
s1 Recent developments in relating QCD potentials to quarkonium properties by use of effective field theory methods are presented. contribution 0.89 0.22 0.05 asserted 0.88
s2 The validity of non-relativistic QCD and the adiabatic approximation with respect to heavy quark bound states is explored. contribution 0.96 0.44 0.07 unclear 0.58
s3 Besides the static potential and relativistic correction terms, the spectra of glueballs and gluinoballs, hybrid excitations of the QCD flux tube between fundamental colour sources, potentials between charges in various representations of the SU(3) gauge group, and multi-particle interactions are discussed. contribution 0.94 0.17 0.09 asserted 0.91
s4 Some implications for quarkonia systems and quark-gluon hybrid mesons are drawn. interpretation 1.00 0.55 0.29 asserted 0.72

What kind of paper the classifier reports this is: review 1.00 · evidence from literature 0.91.

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 2. 1: Regge trajectories. is expected to improve with J. Therefore, in the fits we have decided to ignore the experimental errors and give all points equal weight. ∆J denotes the root mean square deviation between fitted angular momenta and data points, normalised by the root of the degrees of freedom (i.e. the number of data points minus two) and reflects the overall quality of a fit. σ 2d r q q _ v
Figure 3. 3: The ratio, mρr0, in quenched QCD, extrapolated to the continuum limit. The ρ masses in lattice units are taken from Refs. [97] (CP-PACS) and [158] (GF11). While the coefficients β0 and β1 within Eq. (3.22) are universal, higher order coeffi￾cients depend on the renormalisation scheme. Integrating Eq. (3.22) yields, µ = Λ exp Z α(µ) α(Λ) dα 2β(α) ! , (3.23) where we define the integration constant, the so-called QCD Λ-parameter, via the two loop relation, Λ = lim µ→∞ µ exp − 1 2β0α(µ) ! [β0α(µ)]− β1 2β0 . (3.24) In Appendix C, we display results on the coefficients βi of
Figure 4. 4: Comparison between quenched (β = 6.2) and un-quenched nf = 2 (β = 5.6, κ = 0.1575) ground state and Πu potentials (from Ref. [248]). When including sea quarks, one would expect two physical effects, one at large distances and one at small distances. While within the quenched approximation the number of quarks and anti-quarks are separately conserved, with sea quarks, only the difference (the baryon number) is a conserved quantity. Light quark anti-quark pairs can be created from the vacuum and in general transitions between a colour “string” state, spanned between two static…
Figure 5. 3: Hybrid excitations of the static SU(3) potential (from Ref. [114]). approximation, K and Λ are conserved, but not L or Sg. The centrifugal term, l(l + 1) that appears in the radial Schr¨odinger equation, Eq. (A.3), has to be substituted by the correct factor [176], hL 2 i = k(k + 1) − 2Λ2 + hS 2 g i. Mass estimates of hybrid bottomonia, obtained in this way from hybrid potentials, can be found in Refs. [239, 331, 332]. Like in the case of light mesons the 1−+, 0+− and 2 +− quarkonium spin-exotica, that are governed by the Πu potential in the adiabatic approximation, turn out to be…
Figure 6. 10: The spin-orbit potential V ′ 1 , with a fit, −V ′ 1 = σ + h/r2 . The definitions of ZB and ZE are ambiguous; any term that vanishes at least like a 2 can be added. Since the left hand sides of Eqs. (6.138) and (6.139) are approached by the right hand sides in the continuum limit, ZB and ZE are r independent, up to order a 2 lattice artefacts, and can be defined from the value at r = r0, for instance. Therefore, we only have to distinguish between four independent renormalisation factors, ZB,⊥, ZB,k, ZE,⊥ and ZE,k, where ⊥ refers to a component orthogonal to the inter-quark axis
Figure 7. 2: R = ∆m2S1S/∆m1P1S as a function of m−1 Υ for various values of the Coulomb coupling, e. The upper curves correspond to the lowest order Hamiltonian, the lower ones incorporate relativistic corrections. resulted in the effective Coulomb strength to increase by about 20 %, such an increase by 30 % indeed appears to be very reasonable. Our suggested sizes of quenching effects will be based on this estimate. From the Figure it is also obvious that while order v 4 (c −2 ) effects on this ratio are small around the bottom mass, relativistic corrections explode in an uncontrolled way…

Showing 6 of the paper's 7 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:hep-ph/0001312 by the authors.

References
How later papers used it

143 later papers cite it

Evidence: substantial. 140 of the citing papers share no author with it, and 26 of those carry a confident label from Pith's citation classifier.

Pith read 23 of the 200 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

Uses its method or tool 1 of 4 papers with this label
  • The static potential for the $SU(2)$ gauge field is determined as described in [ref] : [equation] where $W(n_r,n_t)$ represents the Wilson loop's expectation value with length $n_r$ and width $n_t$ in lattice units.

    A comparative lattice analysis of $SU(2)$ dark gluebal (2024) · Lattice Spacing from Wilson Loop · 1.00

Uses its data or values 2 of 11 papers with this label
Cites it as support 1 of 10 papers with this label
How the literature uses it

117 arXiv papers cite this in 155 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 · 91 sentences without a confident label

Uses its method or tool 3 papers
  • The static potential for the $SU(2)$ gauge field is determined as described in [ref] : [equation] where $W(n_r,n_t)$ represents the Wilson loop's expectation value with length $n_r$ and width $n_t$ in lattice units.

    A comparative lattice analysis of $SU(2)$ dark gluebal · Lattice Spacing from Wilson Loop · 1.00

  • Here, instead, we follow the picture in Ref. [ref] , and treat hadrons as composites of heavy quarks which are attracted with each other by a phenomenological potential (see e.g. [ref] ), [equation] Here, $κ$ is an $O(1)$ numerical factor that depends on the color exchanged between the quarks.

    Thermal Relic Dark Matter Beyond the Unitarity Limit · Bound State Formation · 0.89

  • These authors uses the NRQCD lattice action [ref] to calculate bottom quark propagators.

    Tetraquarks and pentaquarks in lattice QCD with light and heavy quarks · Tetraquark boundstate search with non-relativistic bottom quarks and light dynamical quarks · 0.94

Uses its data or values 7 papers
Compares with it 2 papers
  • The linearity of the potential agrees exactly with lattice calculations (see e.g. Ref. [ref] ).

    Nonperturbative aspects of Yang-Mills theory · Linear confining potential · 0.83

  • Moreover, the phenomenological potential mo3 closely agrees (fups(b)) with that calculated between static (infinitely heavy) quarks using lattice QCD in the quenched approximation [ref] .

    Journey to the Bound States · Quarkonia · 0.80

Cites it as support 5 papers
  • Triality zero orbitals, e.g. $(cq̅)_8$, are NOT confined because color can be neutralised by extra gluons, and the BO potential vanishes for large separation of the heavy constituent [ref] .

    GGI Lectures on Exotic Hadrons · Colour gymnastic: couplings …and strings · 0.97

  • Such form of the potential has been later confirmed by nonperturbative calculations performed using computational lattice QCD [ref] .

    Quark Nuclear Physics with Heavy Quarks · The role of heavy quarks · 0.97

  • It is hypothesized that unquenching has a small effect on the gauge field sector.[The nonperturbative confinement phenomenon is known to exist in the absence of flavors, $N_f=0$, from lattice calculations [ref] .

    Nonperturbative aspects of Yang-Mills theory · Definition of Yang–Mills theory · 0.99

Cites it as background 40 papers

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.