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Potential NRQCD: an effective theory for heavy quarkonium

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N. Brambilla, A. Pineda, J. Soto, A. Vairo

High Energy Physics - Phenomenology · Submitted 5 Jul 1999 · 48 references

Pith's verdict: The matching and renormalon work is solid and cross-checked; the octet IR-freedom claim in §3.4.2 is the one place a conclusion outruns the calculation, and the authors say so themselves.

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arXiv IDhep-ph/9907240 (version 2)
Pith IDLLUGET66, Pith's permanent record of this paper and its review
CitationA. Pineda, A. Vairo, J. Soto, N. Brambilla. "Potential NRQCD: an effective theory for heavy quarkonium." arXiv:hep-ph/9907240 (1999). https://pith.science/paper/LLUGET66
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The paper

The paper claims

The static potential is redefined as a matching coefficient whose scale dependence cancels against ultrasoft gluons, and the paper checks the cancellation at leading logarithm.

The reading

Heavy quarkonium lives at three widely separated scales: the quark mass $m$, the soft scale $mv$ set by the inverse size of the bound state, and the ultrasoft scale $mv^2$ set by the binding energy. The paper's programme is to integrate these out one at a time: QCD to NRQCD at the mass scale, then NRQCD to a new theory, potential NRQCD, at the soft scale. In that second step the potential stops being a fixed function of distance and becomes a matching coefficient with a subtraction point $mu$; the infrared divergences that have long plagued the perturbative static potential are simply the ultraviolet divergences of the effective theory, and they cancel in any physical quantity once ultrasoft gluons are included. The authors verify this explicitly by computing the leading logarithm of the singlet potential and of the singlet normalization factor in two independent ways, from the static Wilson loop and from the effective-theory correlator, and getting the same $ln(r mu)$ coefficients. They then use the same machinery to predict approximate degeneracies among the gluonic excitations of a static quark-antiquark pair and to show how the renormalon ambiguities of the static potential are absorbed inside the effective theory.

What carries the argument

The load-bearing object is the pNRQCD Lagrangian at leading order in $1/m$ and at order $r$ in the multipole expansion, built from a colour-singlet and a colour-octet heavy-quark-antiquark field that depend on the relative coordinate $r$ and the centre-of-mass coordinate $R$, coupled to ultrasoft gluons evaluated only at $R$. The potentials $V_s(r)$, $V_o(r)$, $V_A(r)$, $V_B(r)$ are matching coefficients fixed by equating NRQCD and pNRQCD Green functions; at leading order $V_s$ and $V_o$ are read off from the large-$T$ behaviour of the static Wilson loop and of the Wilson loop with colour-matrix insertions, $i/T ln<W> = u_0(r) + i u_1(r) T + ...$. The multipole expansion is what makes the matching systematic: each power of $r$ brings in a definite set of operators, and at order $r^2$ a gluon correlator $<E_i(t) phi(t,t') E_j(t')>$ appears whose $ln(V_o - V_s)$ dependence reproduces the Wilson-loop infrared logarithm. The same expansion, applied to the static limit, gives the classification of gluonic excitations by $O(3) times C$ rather than the full $D_{infty h}$ of NRQCD, and the short-distance formula $V_H(r) = V_o(r) + Lambda_H$.

Extended reading notes

Core claim

The central claim is that potential NRQCD, obtained by integrating the soft scale $mv$ out of NRQCD, is the correct effective field theory at ultrasoft scales, and that the static potential defined within it is a matching coefficient rather than a short-distance observable. Concretely: the three-loop infrared divergence of the potential and the two-loop infrared divergence of the singlet normalization factor computed from the static Wilson loop in NRQCD match the corresponding ultraviolet divergences of pNRQCD, so the $mu$ dependence of the potential cancels against ultrasoft gluons in physical observables. The authors demonstrate this by calculating the leading-logarithm corrections to the singlet potential, $alpha_{V_s}(r,mu) = alpha_s(r) + ... + (C_A^3/12)(alpha_s^3/pi) ln(r mu)$, and to the octet potential, and by obtaining the one-loop running of the singlet-octet and octet-octet matching coefficients $V_A$ and $V_B$. They also show that the widespread belief that the octet potential suffers infrared divergences at relative order $alpha_s^2$ is incorrect: the leading ultrasoft correction to $V_o$ is of relative order $alpha_s^3$.

Load-bearing premise

Everything computed here assumes there is a scale window in which the soft scale $mv$ is much larger than the matching scale $mu$, which in turn is much larger than $mv^2$ and $Lambda_{QCD}$, so that the matching between NRQCD and pNRQCD can be done order by order in perturbation theory.

Editorial extensions

If this is right

  • The scale dependence of the singlet potential is now known at leading logarithm, which is the missing ingredient for pushing next-to-next-to-leading-order determinations of the bottom and top masses from quarkonium observables to the next logarithmic order.
  • The octet potential is free of infrared divergences at two loops, so a two-loop calculation of $V_o$ from the intermediate steps of existing singlet calculations would provide an independent check of the matching.
  • In the short-distance limit the gluonic excitations of a static quark-antiquark pair should fall into approximate multiplets: $Sigma_g^{+\prime} \sim \Pi_g$, $Sigma_g^- \sim \Pi_g^{\prime} \sim \Delta_g$, $Sigma_u^- \sim \Pi_u$, and $Sigma_u^+ \sim \Pi_u^{\prime} \sim \Delta_u$.
  • Each such static energy should approach the perturbative octet potential plus a nonperturbative constant, which ties the shape of the lattice-measured curves to the correlation lengths of the colour-electric and colour-magnetic gluon correlators, with $T_E < T_B$.
  • The $O(\Lambda_{\rm QCD})$ renormalon of the static potential cancels against the pole-mass self-energy, and the $O(\Lambda_{\rm QCD}^3 r^2)$ renormalon cancels against an ultraviolet renormalon of pNRQCD at next-to-leading order in the multipole expansion.

Reading between the lines

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

  • If the same matching logic is carried through for a system where the perturbative window $mv \gg \mu \gg mv^2, \Lambda_{\rm QCD}$ does not exist, the effective-theory bookkeeping survives but every coefficient in the running table has to be replaced by a nonperturbative matrix element; the paper flags this for charmonium and bottomonium but does not attempt it.
  • The prediction that the $\Pi_g$ curve on the lattice is a superposition of two distinct states is a concrete, checkable claim that a finer short-distance lattice study could settle, and it would be a clean test of the multipole-expansion picture rather than of perturbation theory.
  • The statement that in pure gluodynamics with a gluelump gap of order $\Lambda_{\rm QCD}$ no ultrasoft gluon fields survive in the lower effective theory amounts to saying that quarkonium there obeys an exact potential description; if that held, non-potential effects in pure-glue quarkonium would be an artefact of the truncation rather than a physical effect.
  • The renormalon-cancellation argument is presented at the level of the leading and next-to-leading multipole orders only, so a natural extension is to ask whether the cancellation persists at the next order in $r$, where the gluon correlator acquires additional tensor structure.
Original abstract

Within an effective field theory framework we study heavy-quark--antiquark systems with a typical distance between the heavy quark and the antiquark smaller than $1/\Lambda_{\rm QCD}$. A suitable definition of the potential is given within this framework, while non-potential (retardation) effects are taken into account in a systematic way. We explore different physical systems. Model-independent results on the short distance behavior of the energies of the gluonic excitations between static quarks are obtained. Finally, we show how infrared renormalons affecting the static potential get cancelled in the effective theory.

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, 4,181 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.21 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.05 · compares with prior results 0.09 · mentions code or data availability 0.01.

The abstract, sentence by sentence

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

Within an effective field theory framework we study heavy-quark--antiquark systems with a typical distance between the heavy quark and the antiquark smaller than $1/\Lambda_{\rm QCD}$. A suitable definition of the potential is given within this framework, while non-potential (retardation) effects are taken into account in a systematic way. We explore different physical systems. Model-independent results on the short distance behavior of the energies of the gluonic excitations between static quarks are obtained. Finally, we show how infrared renormalons affecting the static potential get cancelled in the effective theory.

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 Within an effective field theory framework we study heavy-quark--antiquark systems with a typical distance between the heavy quark and the antiquark smaller than $1/\Lambda_{\rm QCD}$. contribution 0.75 0.79 0.07 asserted 0.93
s1 A suitable definition of the potential is given within this framework, while non-potential (retardation) effects are taken into account in a systematic way. method 0.56 0.87 0.13 asserted 1.00
s2 We explore different physical systems. contribution 0.84 0.81 0.02 asserted 0.72
s3 Model-independent results on the short distance behavior of the energies of the gluonic excitations between static quarks are obtained. result 0.99 0.92 0.21 asserted 1.00
s4 Finally, we show how infrared renormalons affecting the static potential get cancelled in the effective theory. result 0.97 0.91 0.34 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.88 · evidence from proof or derivation 0.98.

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Figures
Figure 1. Propagators and vertices of the pNRQCD Lagrangian (3). In perturbative calcula￾tions the octet propagator is understood without the gluonic string, using instead the Coulomb octet–octet vertex. Also the singlet–octet and the octet–octet vertices produce three diagrams each in perturbative calculations with the Coulomb gauge: one with a longitudinal gluon line, one with a transverse gluon line, and one with both a longitudinal and a transverse gluon line. In order to interpret the matching potentials Vs and Vo properly, let us consider, for instance, the leading order (in the…
Figure 2. A graphical representation of the static Wilson loop. We adopt the convention that time propagates from the left to the right. Therefore, the quarks trajectories are represented by horizontal lines and the equal-time endpoint Schwinger strings by shorter vertical lines. Equation (6) states that the leading overlap of the Green function (12) in pNRQCD is with the singlet propagator, see also
Figure 3. The matching of Vs and Zs at next-to-leading order in the multipole expansion. On the left-hand side is the Wilson loop in NRQCD, on the right-hand side are the pNRQCD fields. The wavy line represents the ultrasoft gluon propagator. In NRQCD we have I ab o = δ 3 (x1 − y1)δ 3 (x2 − y2)hT aW✷T b i, (20) where the colour matrices are intended as path ordered insertions on the static Wilson loop in the points (R, T/2) and (R, −T/2). Eq. (7) states that the leading overlap of the Green function (20) in pNRQCD is with the octet propagator. Indeed, in pNRQCD we obtain at the zeroth
Figure 4. The matching of Vo and Zo at next-to-leading order in the multipole expansion. On the left-hand side is the Wilson loop in NRQCD with colour matrices insertions, on the right-hand side are the pNRQCD fields. the spectrum (just in the same way as the pole mass of the quark propagator is a gauge independent quantity at any finite order in perturbation theory [29]). At this respect it is easy to see on dimensional grounds (by Fourier transforming T to energy space) that the string hφ(T/2, −T/2)adj ab i does not give contribution to the potential at any finite order in perturbation…
Figure 5. Graphs contributing to hW✷i giving rise to Tα4 s ln r (Vo − Vs) and to α 3 s ln r (Vo − Vs) terms. Dashed lines represent Coulomb exchanges. where VA has been substituted by its tree level value and Vo − Vs is also meant to be substituted by it. Note that the UV divergences can be re-absorbed by a renormalization of Vs and Zs. Eq. (33) also contains the leading logarithm dependence on µ of Vs and Zs. Here we want to check explicitly the cancellation of the ln(Vo − Vs) dependence, which plays the role of an IR cut-off, with the right-hand side of Eq. (12) by calculating at the same…
Figure 6. Octet self-energy graph. We use Eq. (37) in order to calculate the leading log dependence of the octet matching potential. From what we have learnt in the previous section, it is sufficient to calculate the leading log dependence of the gluon correlator in the right hand side of Eq. (37). Up to a color factor the contribution of Eq. (37) is equal, at leading order in αs , to the one calculated in the singlet case (see Eq. (31)). Proceeding along the same line as for the singlet case we obtain Vo(r, µ) =  CA 2 − CF  αVo (r) r = (v0(r))two−loops +  CA 2 − CF  C 3 A 12 αs r α 3 s π…
Figure 7. Singlet-octet vertex at tree level a) and at one-loop b). The coefficient VA controls the running of the singlet-octet vertex shown in diagram a) of
Figure 8. Octet-octet vertex at tree level a) and at one-loop b) (symmetric graphs are understood). divergent part of it gives the one-loop µ dependence of VB: VB(r, µ) = 1 + 8 3 CA αs π ln rµ, (40) which turns out to be equal to VA. In Tab. 1 we summarize the running of the matching potentials in the pNRQCD Lagrangian at order O(r) and of the singlet normalization factor at leading order in perturbation theory. singlet µ dαVs dµ = C 3 A 12 α 4 s π singlet µ dZs dµ = CFC 3 A 2 α 3 s π octet µ dαVo dµ = C 3 A 12 α 4 s π singlet-octet µ dVA dµ = 8 3 CA αs π octet-octet µ dVB dµ = 8 3 CA αs π
Figure 9. Energies for different gluonic excitation between static quarks at distance r from the quenched lattice measurements of [41], r0 ≃ 0.5 fm. The picture is taken from [37]. Let us compare our results against the best available lattice data (see
Figure 10. Bubble diagrams contributing to the renormalon of the static potential. expansion can be applied [46]. We obtain the following general expansion for the potential with hard cut-off V µ s = C0 + C2r 2 + . . . (62) where C0 = −4πCF Z µ d 3k (2π) 3 1 k2 αs(k 2 ) = −2 CF αs(µ) π µ X∞ n=0 D (0) n
Figure 11. Bubble diagrams contributing to the renormalon of the pole mass. which is zero in dimensional regularization while from a renormalon point of view it has an UV and an IR renormalon that cancel each other. Nevertheless, in this respect it is nice to see that the full result one obtains when matching NRQCD with pNRQCD, i.e. 2Σ(v · p = 0) + Vs(r) (68) has an UV renormalon while the IR renormalon of each term cancel each other (at lowest order in the multipole expansion). This suits the EFT picture where the IR renormalon of twice the pole mass, O(m), understood as a matching…
Figure 12. Bubble diagrams contributing in pNRQCD to the renormalon of the singlet self-energy at the next-to-leading order in the multipole expansion. The calculation of the UV renormalon of pNRQCD at next-to-leading order in the mul￾tipole expansion effectively reduces to the term in Eq. (31) linear in T. The expression from 34

Figures reproduced from arXiv:hep-ph/9907240 by the authors.

References
How later papers used it

258 later papers cite it

Evidence: substantial. 193 of the citing papers share no author with it, and 35 of those carry a confident label from Pith's citation classifier.

Pith read 39 of the 455 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 3 of 32 papers with this label · 1 by the same authors

By the same authors

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

244 arXiv papers cite this in 406 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 · 249 sentences without a confident label

Builds on it 2 papers
  • The strong-coupling version of pRNQCD [ref] could be generalized to finite temperatures along the way the weakly-coupled one has; this would help shed more light on how the proper in-medium potential is to be non-perturbatively determined.

    Review of the EFT treatment of quarkonium at finite temperature · Non-static and non-perturbative aspects · 0.90

  • One possibility is to work out a nonrelativistic effective field theory description of these states in a way similar to what has been done by potential Non-Relativistic QCD (pNRQCD) for states away from threshold [ref] .

    Quarkonium Hybrids with Nonrelativistic Effective Field Theories · Introduction · 0.84

Uses its method or tool 31 papers
Uses its data or values 2 papers
  • We present an estimate for the value of $α̃$ for the Alpha-Centauri A and B stellar binary [ref] , a pulsar-neutron star binary system (the Hulse–Taylor pulsar, also identified as PSR B1913+16 [ref] )[We note that, in QCD, we can find the equivalent for a binary system, i.e., a quarkonium state [ref] , where the effective theory is written in terms of a two-body potential.], a stellar globular cluster, a Milky Way-like disc galaxy and our home in the Cosmos, the super-cluster Laniakea [ref] .

    The need for a nonlocal expansion in general relativity · Introduction · 0.92

  • In table [ref] we show the quantum numbers of the four lowest mass gluelumps [ref] and the $D_∞h$ representations of the corresponding static states [ref] .

    Effect of continuum states on the double-heavy hadron spectra · Hybrid quarkonium and heavy-meson pairs · 0.95

Compares with it 7 papers
  • The Debye mass effectively plays the role of a gluon mass; in this sense, the real part of ([ref]) agrees with a result that can be found in [ref] .

    Static quark-antiquark pairs at finite temperature · Contributions from the scale $m_D$ · 0.99

  • This result, in appropriate units, is of the same order of magnitude as the real Coulombic term obtained by comparing with Charmonium spectra as first modelled by [ref] and subsequently by several authors [ref] - [ref] , [ref] - [ref] , [ref] - [ref] .

    From string theory to large N QCD · Quark-Anti quark potential for small $u_max$ · 0.98

  • This result, in appropriate units, is of the same order of magnitude as the real Coulombic term for the Charmonium spectra [ref] .

    Toward Large N Thermal QCD from Dual Gravity: The Heavy Quarkonium Potential · Quark-Antiquark potential for small $u_max$ · 0.91

Cites it as background 101 papers

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