# Calibrating the naïve Cornell model with NRQCD

## Abstract

Along the years, the Cornell model has been extraordinarily successful in describing hadronic phenomenology, in particular in physical situations for which an effective theory of the strong interactions such as NRQCD cannot be applied. As a consequence of its achievements, a relevant question is whether its model parameters can somehow be related to fundamental constants of QCD. We shall give a first answer in this article by comparing the predictions of both approaches. Building on results from a previous study on heavy meson spectroscopy, we calibrate the Cornell model employing NRQCD predictions for the lowest-lying bottomonium states up to \(\hbox {N}^3\hbox {LO}\), in which the bottom mass is varied within a wide range. We find that the Cornell model mass parameter can be identified, within perturbative uncertainties, with the MSR mass at the scale \(R = 1\,\hbox {GeV}\). This identification holds for any value of \(\alpha _s\) or the bottom mass, and for all perturbative orders investigated. Furthermore, we show that: (a) the “string tension” parameter is independent of the bottom mass, and (b) the Coulomb strength \(\kappa \) of the Cornell model can be related to the QCD strong coupling constant \(\alpha _s\) at a characteristic non-relativistic scale. We also show how to remove the \(u=1/2\) renormalon of the static QCD potential and sum-up large logs related to the renormalon subtraction by switching to the low-scale, short-distance MSR mass, and using R-evolution. Our R-improved expression for the static potential remains independent of the heavy quark mass value and agrees with lattice QCD results for values of the radius as large as \(0.8\,\hbox {fm}\), and with the Cornell model potential at long distances. Finally we show that for moderate values of *r*, the R-improved NRQCD and Cornell static potentials are in head-on agreement.

## 1 Introduction

The discovery of the \(J/\psi \) in 1974 [1, 2, 3] caused a revolution in hadron spectroscopy, because the large mass of the *c* quark made a non-relativistic description feasible. However, the limited development of Quantum Chromodynamics (QCD) for heavy quarkonium systems at that time did not provide analytical expressions for the binding forces among quarks, in particular for the confinement. This is the reason why people were forced to resort to models that, retaining as many QCD characteristics as possible, allowed to perform calculations susceptible to be compared with experimental results.

This model has been successful in describing a huge amount of experimental data including masses, widths, radiative and strong transitions, etc. Recently [11], due to its flexibility to describe coupled-channels problems, the model has been extended above the charm threshold where a great amount of new experimental information exists. Quark models become specially relevant to describe states that do not admit a treatment in terms of effective theories of QCD (e.g. molecular, hybrid or highly excited states), simply because there is no hierarchy of scales to be exploited. They are also crucial to guide experimental searches of new bound states, which shade light in the way quarks couple into colorless bound states. Finally, they provide useful hints to build novel approaches within QCD to tackle the treatment of these hard-to-describe states. Given the success of such a simple model, a pressing question is whether its parameters can be related to fundamental QCD constants. Indeed, to our knowledge, no work has ever studied the connection between quark model potentials and the non-relativistic limit of QCD. For the many reasons presented in this paragraph, establishing a connection between them and QCD appears certainly warranted.

The small velocity of the charm and bottom quarks in \(Q{\overline{Q}}\) bound states enables the use of non-relativistic effective theories within QCD to study heavy quarkonia. The use of non-relativistic approaches implies that heavy quarkonia bound states can be organized in terms of the non-relativistic quantum numbers \((j,\ell ,s)\) and the radial excitation number *n*, while the hyperfine splittings are power corrections, starting at order \({{\mathcal {O}}}(m_Q^{-2})\). For heavy quarkonia we can distinguish three well defined scales: the heavy quark mass \(m_Q\) acting as the *hard scale*, the *soft scale* determined by the relative momentum of the \(Q{\overline{Q}}\) system (\(p\sim m_Q v\) with \(v\ll c\)) and the *ultrasoft scale* marked by the average kinetic energy of the heavy quarkonia (\(E\sim m_Qv^2\)). These scales have a well-defined hierarchy (\(m_Q\gg p\gg E\)), which allows for significant simplifications.

NRQCD [12] is obtained from QCD by integrating out the heavy quark mass \(m_Q\), and one can exploit the fact that \(m_Q\gg \varLambda _{\mathrm{QCD}}\) to perform a perturbative matching. This implies that NRQCD inherits all the light degrees of freedom from QCD. The NRQCD Lagrangian is thus expanded as a series in \(1/m_Q\) powers, factorizing the terms which contribute at the hard scale as Wilson coefficients. Even though the hard scale lays in the perturbative domain, the soft and ultrasoft scales are probed e.g. when building up \(Q{\overline{Q}}\) systems, which in some cases could make computations in terms of partonic degrees of freedom unreliable. Additionally, there is still a fundamental problem in NRQCD: it does not distinguish soft and ultrasoft scales, which complicates the power-counting.

A solution to this problem arrived with the construction of EFTs such as velocity NRQCD (vNRQCD) [13] and potential NRQCD (pNRQCD) [14, 15], which describe the interactions of a non-relativistic system with ultrasoft gluons, organizing the perturbative expansions in \(\alpha _s\) and the velocity of heavy quarks systematically. Such EFTs only include degrees of freedom relevant for \(Q{{\overline{Q}}}\) systems near threshold, while the rest of degrees of freedom are integrated out. Indeed, pNRQCD is obtained from NRQCD by integrating out soft and potential gluons, and potential quarks, under the assumption \(p\gg \varLambda _{\mathrm{QCD}}\).

The specific treatment of the remaining degrees of freedom will depend on the relation between *E* and the scale \(\varLambda _{\mathrm{QCD}}\). At long distances QCD becomes strongly coupled and hadronic degrees of freedom emerge. The \(Q{{\overline{Q}}}\) system is assumed to be dominated by perturbative physics and non-perturbative corrections are taken into account by local or non-local condensates. Under these assumptions, the static potential of QCD for color-singlet states can be computed, which is by itself an interesting field of research. On the one hand, it is fundamental to study the energy spectrum and, on the other, it is a good tool to explore weak and strong coupling regimes and analyze phenomena such as confinement. In fact, the static potential can be directly compared with lattice QCD simulations [16].

In this article we attempt to calibrate the simplest realization of the Cornell model against NRQCD. To that end we compare observables that can be reliably predicted both in the theory and the model, namely the mass of the lowest-lying \(Q{\overline{Q}}\) bound states, varying the quark mass and the strong coupling constant. This exercise is inspired by a similar analysis carried out in Ref. [17] for the top quark parameter embedded in the parton-shower Monte Carlo Pythia [18, 19]. As a byproduct, we show that the Cornell potential agrees for large values of *r* with the QCD static potential once the latter is expressed in terms of the MSR mass and improved with all-order resummation of large renormalon-related logs via R-evolution. Our R-improved static potential also compares nicely with lattice QCD simulations from Refs. [20, 21].

This article is organized as follows: in Sect. 2 we discuss the solution of the two-body problem for the Cornell potential. A fit for the parameters of the Cornell model to experimental bottomonium and charmonium data is carried out in Sect. 3. Section 4 introduces the concepts of the MSR mass and R-evolution. A discussion of the QCD static potential, and how its leading renormalon is canceled is presented in Sect. 5. Section 6 contains a comparison to lattice results. NRQCD analytic results for the mass of \(Q{\overline{Q}}\) bound states are compiled in Sect. 7. The calibration procedure is explained in Sect. 8, while the results are presented in Sect. 9. Section 10 contains our conclusions. In Appendix A we discuss the numerical solution of the Schrödinger equation for the Cornell potential using the Numerov method.

## 2 The Cornell potential for the \(\mathbf {Q{{\overline{Q}}}}\) system

*E*is the binding energy, and the mass of the quarkonium bound state is given by \(M = 2\,m_Q^{\mathrm{Cornell}} + E\).

^{1}Since we are dealing with a central potential, the relative wave function of the quark–antiquark pair \(\psi \) can be factorized in an angular part, expressed in terms of spherical harmonics, and the radial wave function \(R_{n\ell }(r)\):

*n*is a natural number accounting for the radial excitation. The latter simply accounts for the infinitely many (but denumerable) bound states that can be found for a given value of \(\ell \). It should not be confused with the principal quantum number \(n_p = n + \ell >0\), which bounds the possible values of the orbital angular momentum \(0\le \ell \le n_p-1\). Using the factorization shown in (3), Eq. (2) can be simplified if written in terms of the reduced wave function \(u_{n\ell } = r\, R_{n\ell }(r)\), yielding an ordinary differential equation for \(u_{n\ell }(r)\),

^{2}

*r*, as well as the binding energies and classical radii.

*u*(

*r*) falls off sufficiently rapidly.

*r*Eq. (4) becomes \(u_{nl}^{\prime \prime }(r)+\ell (\ell + 1) u_{n\ell }(r)/r^2=0\), which implies a power-like behavior of the reduced wave function with exponent \(\ell + 1\),

*r*, the Coulomb and centrifugal terms are largely suppressed, and we get the following asymptotic equation for the radial wave function:

*r*for \(\ell = 0\), that is, that the reduced wave function vanishes at \(r=0\). The Airy function has an infinite denumerable number of zeros which happen to be all negative. These can be computed numerically and we denote them by \(a_n\) such that \(\mathrm{Ai}(a_n)=0\) and \(|a_n| > |a_{n^\prime }|\) if \(n > n^\prime \). With this at hand we find for the Airy potential self-energies:

*n*-th state crosses the horizontal axis exactly \(n\,-\,1\) times, as expected. This behavior can be seen in Fig. 2 for the first four states. The wave functions for different excited states are related, up to normalization, by shifts in their argument. The normalization factor \(N^{\mathrm{Airy}}_n\) is independent of \(m_Q\) and \(\sigma \), and can be easily computed numerically. It decreases as

*n*increases, and we find that for \(n <6\) it takes the following values: \(\{ 1.426, 1.245, 1.156, 1.098, 1.056\}\). The exact solutions for the wave function and self-energies of Eq. (8), in Eqs. (10) and (11), respectively, will serve as a sanity check on our numerical program. Before we move on, let us insist that Eqs. (9) and (10) correspond to the exact solution of the Cornell potential if both \(\kappa \) and \(\ell \) are zero. If we have \(\kappa = 0\) but \(\ell >0\) there is no exact solution to the Schrödinger equation.

*r*:

^{3}\(^{,}\)

^{4}

\(^1S_0\) | \(^3S_1\) | \(^1P_1\) | \(^3P_0\) | \(^3P_1\) | \(^3P_2\) | \(^3D_2\) | |
---|---|---|---|---|---|---|---|

\(\langle {\varvec{S}}_1\cdot {\varvec{S}}_2\rangle \) | \(-\frac{3}{4}\) | \(\frac{1}{4}\) | \(-\frac{3}{4}\) | \(\frac{1}{4}\) | \(\frac{1}{4}\) | \(\frac{1}{4}\) | \(\frac{1}{4}\) |

\(\langle {\varvec{L}}\cdot {\varvec{S}}\rangle \) | 0 | 0 | 0 | \(-2\) | \(-1\) | 1 | \(-1\) |

\(\langle S_{12}\rangle \) | 0 | 0 | 0 | \(-4\) | 2 | \(-\frac{2}{5}\) | 2 |

## 3 Fitting the Cornell model to experimental data

In this section we present the results from \(\chi ^2\) fits of the Cornell model parameters to charmonium and bottomonium experimental data. We will restrict ourselves to \(n_p \le 2\), since for higher values of the principal quantum number one needs to include string-breaking effects, as can be seen in Fig. 4. The case of bottomonium will serve as a proof of concept for our calibration, that is, it will show that the three parameters of the model can be determined from fits to the 8 lowest-lying bottomonium bound states. This is crucial since in QCD we can only reliably predict these within perturbation theory, as argued in Ref. [27]. In many phenomenological applications of the quark model approach, in which more sophisticated Hamiltonians are used, the parameters of the potential are not obtained by a full fledged \(\chi ^2\) minimization, but simply adjusted by a rough comparison to data plus physically motivated priors. This procedure is in many cases justified, since there is a large amount of data that the model aims to describe, but its accuracy might vary widely for different observables. Given that it is very hard to add a theoretical covariance matrix to the \(\chi ^2\) (the matrix does not provide a method to quantify its “modeling” error), such a fit could lead to biased model parameter values. For the simpler situation of the naïve Cornell model, which seeks to describe only the low-lying quarkonium spectrum, we show that the fit is indeed possible, and we shall see that no bias is observed.

The minimization of the \(\chi ^2\) is carried out using the Fortran 77 package MINUIT [28]. We have checked that the algorithm is very effective in finding the minimum, but due to the strong correlation between the various fit parameters [see Eq. (17)] it does not estimate the covariance matrix correctly. To solve this problem we compute a three-dimensional grid of \(5^3\) elements centered around the minimum found by MINUIT, which is then adjusted to a 3D quadratic polynomial by a linear regression. We find that the \(\chi ^2\) function clearly behaves in Gaussian way near the minimum, which permits an estimate of the covariance matrix in the Hessian approximation, that is, from the quadratic terms of our regression.

### 3.1 Bottomonium fits

In Fig. 3a we show the reduced wave functions, eigenfunctions of the static Cornell potential, for \(n = 1,2\) and \(\ell =0,1\), values that correspond to the states included in the fit. For a given *n* value, the peak is shifted to the right for larger values of \(\ell \). The exponential falloff starts at larger *r* values for higher values of *n*, and is roughly \(\ell \) independent. As expected, for \(n=2\) the wave function crosses the *x* axis once. In all cases the wave function is strongly suppressed already at \(r = 1.5\,\hbox {fm}\), therefore justifying our choice \(r_{\mathrm{max}} = 4\,\hbox {fm}\). Taking \(N = 5000\) corresponds to a numerical uncertainty in the solution of the Schrödinger equation of \(0.08\%\) and \(0.0004\%\) for \(n = 0,~\ell = 0,1\) states, respectively, and \(0.034\%\) and \(0.0002\%\) for \(n=1,~\ell = 0,1\), being *N* the number of steps in the Numerov method (see Appendix A). This choice of *N* inflicts an uncertainty on the extraction \(m_b^{\mathrm{Cornell}}\), \(\sigma \), and \(\alpha _s^{\mathrm{Cornell}}\) from a fit to data of \( 0.013\%\), \(0.22\%\) and \(0.017\%\), respectively.^{5} These are a factor of 28, 24 and 237 smaller than the fit uncertainties, and therefore negligible. The numerical uncertainty associated to \(r_{\mathrm{max}} = 4\,\hbox {fm}\) is even smaller. As an additional check on the accuracy of our numerical method, we compute the \(\ell =0\) states with \(\alpha _s^{\mathrm{Cornell}}=0\), for which we know the exact solution, shown in Eq. (10). The numeric solutions reproduce the exact result with an accuracy of \(2\times 10^{-6}\%\). At this point it is also worth mentioning that neglecting the Coulomb-like interaction (that is, setting \(\alpha _s^{\mathrm{Cornell}} = 0\)) overshots the exact result of the static energies by \(5.5\%\) and \(3\%\) for \(\ell = 0\) and \(n=1,2\), respectively, and \(0.6\%\) and \(0.4\%\) for \(\ell = 1\) and \(n=1,2\). Likewise, setting \(\sigma = 0\) one can use the known solution for the hydrogen atom bound states, which can be obtained from Eq. (38) truncated at \({\mathcal {O}}(\alpha _s^2)\), which undershoots the exact results by \(2.4\%\) and \(6.1\%\), for \(\ell = 0\) and \(n=1,2\), respectively, and \(5\%\) and \(8.2\%\) for \(\ell = 1\) and \(n=1,2\). It might seem that either of these approximations is good, but it only appears so because most of the contribution comes from the quark mass. If we compare binding energies then the approximations become much worse, rising up to \(600\%\) for the Coulomb limit and \(1400\%\) for the Airy limit. Therefore none of the two terms in the Cornell potential can be treated as a perturbation with respect to the other for bottomonium.

We have checked the robustness of our fits by removing one, two, or three points from our dataset. It turns out that if one keeps the two states with \(n_p=1\) and the highest mass state \(\varUpsilon (3^3S_1)\), the largest variation from the default is always smaller than 10 MeV for the Cornell mass parameter, and below \(2\%\) for either \(\sigma \) or \(\alpha _s^{\mathrm{Cornell}}\).

In Fig. 4 we compare the predictions of the Cornell model for the 8 states that have been used in the fit, employing the best-fit values and propagating the fit uncertainties into the masses through the covariance matrix. We include experimental data with \(n_p = 3\), which have not been included in the fit, but serve as an illustration for the limitations of this simple model. States with \(n_p > 2\) can only be described if some sort of “string breaking” is implemented [30, 31]. The \(\hbox {N}^3\hbox {LO}\) QCD predictions of Ref. [27] are also shown for illustration.

### 3.2 Charmonium fits

Experimental masses of the charmonium states, up to \(n=2\). Data from Ref. [29]

State | | \(^{2s+1}\ell _j\) | \(M_{\mathrm{exp}}\) (GeV) |
---|---|---|---|

\(\eta _c(1S)\) | 1 | \(^1S_0\) | 2.9839(5) |

\(J/\psi \) | 1 | \(^3S_1\) | 3.096900(6) |

\(\chi _{c0}(1P)\) | 2 | \(^3P_0\) | 3.41471(30) |

\(\chi _{c1}(1P)\) | 2 | \(^3P_1\) | 3.51067(5) |

\(h_c(1P)\) | 2 | \(^1P_1\) | 3.52538(11) |

\(\chi _{c2}(1P)\) | 2 | \(^3P_2\) | 3.55617(7) |

\(\eta _c(2S)\) | 2 | \(^1S_0\) | 3.6376(12) |

\(\psi (2S)\) | 2 | \(^3S_1\) | 3.686097(25) |

In Fig. 3b we show the charmonium reduced wave functions. They have identical shapes to those of bottomonium, but as anticipated from dimensional arguments, the wave functions extend to larger distances as compared to bottomonium states, and have accordingly lower (shallower) peaks (valleys). However, it is still justified to use \(r_{\mathrm{max}} = 4\,\hbox {fm}\) as at \(2\,\hbox {fm}\) the wave function is already negligibly small. We have checked that using \(r_{\mathrm{max}} = 6\,\hbox {fm}\) produces a shift in the 4-th or 5-th decimal place in any of the fit parameters shown in Eq. (18). Since we will not calibrate the Cornell model for charmonium, we do not study the optimal choice of *N*, the number of steps in the Numerov method. We simply take 200, 000 for which the associated numerical errors are several orders of magnitude smaller than the fit uncertainties.

We have performed a similar robustness check in our charmonium fits, that is, selectively remove one, two, or three points from our dataset to analyze the consequent variations in the Cornell parameters, and we conclude that the stability is not as good as for bottomonium. Removing a single point from our dataset, and always keeping the two lowest-mass states and the highest-mass state, we find that the impact on the Cornell mass parameter can be as large as \(70\,\)MeV, while in the other two parameters can reach deviations of \(6\%\) and \(17\%\) for \(\sigma \) and \(\alpha _s^{\mathrm{Cornell}}\), respectively. Since we are not going to calibrate the charm mass, these findings are only a minor concern, which again seem to signal that the non-relativistic approximation is worse for charmonium.

In Fig. 5 we again confront the predictions of the Cornell model after the fit with the 8 masses that enter our \(\chi ^2\) function. In this case we refrain from showing results for \(n_p = 3\), since the \(D{\bar{D}}\) threshold is quite low in charmonium. Similarly, we only show QCD predictions (taken from Ref. [27]) for \(n_p = 1\), since higher \(n_p\) states cannot be reliably predicted in perturbation theory. It is worth noting that the Cornell model prediction for the \(J/\varPsi \) is extremely precise, with an error bar which seems unnaturally small given the uncertainties quoted in Eq. (18). The explanation for this apparent contradiction is the large cancellations that happen between the different error sources due to the very strong correlations among the three parameters. The \(J/\varPsi \), having an experimental uncertainty 40 times smaller than the next most precise measurement, overly drives the fit and renders the correlation such that this particular mass is predicted with an uncertainty roughly 40 times smaller than for the rest of states. Such a situation does not happen for bottomonium, therefore all states are predicted with similar incertitudes.

## 4 The MSR mass and R-evolution

*R*. The MSR scheme absorbs into the mass definition quark self-energy fluctuations above the scale

*R*.

^{6}The optimal choice of

*R*depends on the observable, and can be chosen such that large logs do not appear in perturbative series. In addition, the subtraction series relating the MSR and pole masses can be expressed in terms of \(\alpha _s(\mu )\), with \(\mu \) the \(\overline{\mathrm {MS}}\) renormalization scale. This is essential to cancel the renormalon in other series when the pole mass expressed in terms of the MSR mass:

^{7}

*R*independent

The \(\overline{\mathrm {MS}}\) and MSR bottom masses (as well as the QCD static potential and the bottomonium masses) receive corrections from the finite mass of the charm quark. Here and in what follows we assume the charm quark is massless, as its mass plays no role in the calibration of the Cornell model but might add complications when scanning over the bottom quark mass over a large range. We close this section noting that if we compute the MSR mass from the \(\overline{\mathrm {MS}}\) employing the world average values \({\overline{m}}_b = 4.18\pm 0.03\,\hbox {GeV}\) and \(\alpha _s^{(n_f=5)}(m_Z) = 0.1181\) we obtain \(m_b^{\mathrm{MSR}}(R=1\,\mathrm{GeV}) = 4.679\,\hbox {GeV}\), which is only 1.54 standard deviations away from \(m_b^{\mathrm{Cornell}}\) as found in Eq. (16).

## 5 The QCD static potential

As in the Cornell model, to compute the energy levels of \(Q{\overline{Q}}\) states in NRQCD at leading order in \(1/m_Q\) one needs the static QCD potential. It is defined as the potential between two static quarks, that is, the color-neutral interaction between two infinitely heavy color-triplet states. It is well defined up to \({\mathcal {O}}(\alpha _s^4)\), where the static approximation breaks down and the potential becomes time-dependent. This feature manifests itself in dimensional regularization as an \(1/\epsilon \) pole, that once regulated leaves behind a dependence on the “ultrasoft” scale \(\mu _{\mathrm{us}}\). The pole and \(\mu _{\mathrm{us}}\) logarithms cancel in observables such as quarkonium masses when adding ultrasoft effects from wave-function renormalization. Nevertheless, to define the static potential at four loops we simply subtract the \(1/\epsilon \) pole.

^{8}

In Refs. [9, 10] it is shown from renormalon dominance arguments and in the framework of the operator product expansion of pNRQCD, that perturbation theory alone should be capable of describing both the Coulomb and linear behavior of the static potential, and that nonperturbative corrections start at \({\mathcal {O}}(\varLambda _\mathrm {QCD}^3r^2)\). In the following we shall confirm that claim numerically using the MSR scheme.

*r*-independent, and its nature depends only on the coefficients of the QCD beta function. In Fig. 7a, c this bad perturbative behavior manifests itself as a (roughly constant) vertical shift of the potential in the region between \(0.05\,\hbox {fm}\) and \(0.2\,\hbox {fm}\) every time a new order is included.

^{9}Furthermore, none of the orders makes the QCD static potential close to the Cornell model. It is well understood [47, 48, 49] that the ambiguity exactly matches that of the pole mass except for a factor of \(-2\), such that the static energy \(E_{\mathrm{stat}}(r,\mu ) = 2\,m_Q^{\mathrm{pole}} + V_{\mathrm{QCD}}(r,\mu )\) is renormalon free. Since the pole mass ambiguity is independent of the quark mass itself, the cancellation happens irrespectively of the specific numeric value for the mass. Therefore one only needs to re-write the pole mass in terms of a short-distance scheme to make the cancellation manifest. For the cancellation to take place one also needs to express the perturbative series \(\delta m_Q^{\mathrm{SD}}\) that relates the pole mass with a short-distance mass \(m_Q^{\mathrm{SD}}\) in terms of \(\alpha _s(\mu )\), as done in the second line of Eq. (20). There are powers of \(\log (\mu /R)\) in \(\delta m_Q^{\mathrm{SD}}\) that may become large if \(\mu \) and

*R*are very different. Since one has to choose \(\mu \) such that \(\log (r\,\mu \, e^{\gamma _E})\sim {\mathcal {O}}(1)\), that is \(\mu \) should depend on

*r*, renormalization schemes with a fixed value of

*R*such as the \(\overline{\mathrm {MS}}\), are disfavored. Following Ref. [27] we use the MSR mass and choose \(\mu = R\) to simultaneously minimize logs in the potential and in \(\delta m_Q^{\mathrm{MSR}}\). Since the canonical choice \(\mu =1/r\) quickly dives into non-perturbative values, we freeze it to \(1\,\hbox {GeV}\) once it reaches this value. To avoid a kink in the potential we smoothly convert the 1 /

*r*behavior into a constant employing a transition function between \(r=0.08\,\hbox {fm}\) and \(0.2\,\hbox {fm}\). This function is composed of two quadratics smoothly connected with each other and with 1 /

*r*and \(1\,\hbox {GeV}\) at the junction points, as has been employed for instance in Ref. [50] in the context of event shapes. A graphical representation of this piecewise form, which will be referred to as “profile function”, can be seen in Fig. 6, and has the following analytical form

The result in Eq. (36) is shown for bottomonium in Fig. 7b setting \(R_0=1\,\hbox {GeV}\), and for charmonium in Fig. 7d setting \(R_0=0.65\,\hbox {GeV}\). We take \(R_0\) similar to the value at which the renormalization scale freezes. This choice for the reference scale also makes the bottom MSR mass \(m_b^{\mathrm{MSR}}(R_0)\) very similar to the Cornell model parameter \(m_Q^{\mathrm{Cornell}}\) in Eq. (16). Other values should simply shift the potential vertically, but for this specific choice we observe that the static MSR potential converges very nicely towards the Cornell model for moderate values of *r*. When more perturbative orders are added, the agreement becomes better over larger distances. On the other hand, for high values of *r*, since the renormalization scale is frozen, \(\log (r\mu )\) becomes large, which makes perturbation theory unreliable. At small distances all orders agree very well because \(\alpha _s\) becomes very small, but disagree with the Cornell model. So we can conclude that the Cornell model and QCD agree for moderate values of *r*, but disagree in the ultraviolet, as the model does not incorporate logarithmic modifications due to the running of \(\alpha _s\). For bottomonium the two potentials start disagreeing at a distance of approximately \(r_0\sim 0.2\,\hbox {fm}\), which in natural units corresponds to a scale of roughly \(1\,\hbox {GeV}\), in head on agreement with our choice of \(R_0\). A legitimate question is then if this difference in the UV can be absorbed in the definition of the quark mass. We shall answer this question in Sect. 9 of this article.

The ultrasoft potential in Eq. (29) is only a small contribution of the total MSR R-improved potential. At \(\hbox {N}^3\hbox {LO}\) we find its weight is only \(0.6\%\) at short distances, quickly becoming below the per-mil for \(r\gtrsim 1\,\hbox {fm}\). To fully asses the impact of this term one would need to do a thorough study of perturbative uncertainties through scale variation, which is beyond the scope of this article.

## 6 Comparison to lattice QCD

In this section we perform a comparison to lattice QCD results. We start by comparing the Cornell model parameters as obtained in fits to data to specific lattice studies that determine them. Finally we compare our R-improved QCD static potential to lattice simulations.

Our result for \(\sigma \) in Eq. (16) is in very nice agreement with two lattice determinations: Ref. [55] uses Wilson loops and quotes \(\sigma = 0.2098 \pm 0.0009\,\hbox {GeV}^2\), while Ref. [56] uses a relativistic heavy quark action and quotes \(0.206 \pm 0.010\,\hbox {GeV}^2\). The comparison to the result in Eq. (18) is worse, and this fact will be discussed in the next paragraph. When it comes to the \(\alpha _s^{\mathrm{Cornell}}\) parameter our bottomonium (charmonium) result is roughly a factor of 2 larger than what is found in these lattice analyses. Given that the static potential at short distances can be described in perturbation theory, we know that loop corrections modify the short-distance 1 / *r* behavior. Therefore fitting a Coulomb-like function to lattice output might be meaningless, and the discrepancy should be of little concern.

Lattice simulations for the static potential use three dynamical quarks, and therefore their results should be interpreted as the charmonium potential. Hence, it is confusing that we find better agreement for the linear confining parameter in the fits to the bottomonium spectrum. However, as pointed out in Ref. [57], the charm quark is effectively decoupled for the low-lying bottomonium states. Therefore the same static potential should be used for charmonium and bottomonium, as long as some (small) charm mass corrections are included in the latter. Therefore one could think that also the same Cornell potential should be used for charmonium and bottomonium systems, and given that the non-relativistic approximation is more accurate for the latter, the right comparison might be between lattice and Eq. (16). Let us emphasize again that our Cornell model parameters are not obtained from a direct comparison to the static QCD potential, but from fits to experimental data on quarkonium bound states. At the sight of Fig. 7b, d it is clear that a smaller value of \(\alpha _s^{\mathrm{Cornell}}\) would improve the agreement with \(V_{\mathrm{static}}^{\mathrm{MSR}}\).

We turn now our attention to lattice computations of the static potential. On the lattice, the static approximation also breaks down, and this manifests as a linearly divergent term. This divergence is removed by additive renormalization, as studied in Ref. [58]. We will use the results of Ref. [20] for a lattice spacing \(a=0.04\,\hbox {fm}\). These cover the range \(0.039\,\mathrm{fm} \le r \le 0.84\,\hbox {fm}\), and have an average relative precision of \(2.6\%\). This precision seems to be roughly proportional to \(1/r^2\), but shows a very pronounced peak around \(r = 0.25\,\hbox {fm}\). We complement this dataset with results from [21] with lattice spacing \(a = 0.025\,\hbox {fm}\), which cover values of the radius as small as \(0.024\,\hbox {fm}\). Since uncertainties in [21] are larger for higher values of *r*, we only consider data with \(r\le 0.25\,\hbox {fm}\). Moreover, in the range \(0.024\,\mathrm{fm}\le r \le 0.25\,\hbox {fm}\) [21] has more density of points than [20]. Our complete dataset is plotted in Fig. 8 as black dots with error bars. The HotQCD collaboration has results for larger lattice spacing, with predictions for the potential covering values for the radius as large as \(1.59\,\hbox {fm}\). However these do not extend as much into the small-*r* region where perturbation theory dominates, and therefore we do not take them into account in this simple comparison.

*r*considered in those fits is \(r_{\mathrm{max}} \simeq 0.23\,\hbox {fm}\).

## 7 Analytic NRQCD formula for bottomonium with massless light and charm quarks

The complete bottomonium spectrum up to \(n_p = 2\) for arbitrary \({\overline{m}}_b\equiv {\overline{m}}_b({\overline{m}}_b)\) will be constructed using Non-Relativistic Quantum Chromodynamics (NRQCD) up to \(\hbox {N}^3\hbox {LO}\) [61], which will be briefly explained below (further details can be found e.g. in Ref. [27]).

^{10}

*n*-th harmonic number. In Eq. (38) \(\varepsilon \) acts as a bookkeeping parameter that properly implements the so called \(\varUpsilon \)-expansion [63, 64]. The \(c_{i,0}\) coefficients have been computed up to \(i=3\) [65, 66], while the \(c_{i,j>0}\) coefficients can be directly obtained from the latter \(c_{i,0}\) imposing \(\mu \) independence of the quarkonia mass. We denote the sum in Eq. (39) truncated to \({\mathcal {O}}(\varepsilon ^{n+1})\) as the \(\hbox {N}^n\hbox {LO}\) result. This formula does not include the resummation of large ultrasoft logarithms. Recently this summation has been carried to \(\hbox {N}^3\)LL precision for P-wave states in Ref. [52].

The \(u=1/2\) renormalon in the QCD static potential discussed in Sect. 5 is inherited by the perturbative expansion of the quarkonia mass in Eq. (38). Exactly as it happened for the potential, such renormalon is canceled by expressing the pole mass in terms of a short-distance mass. In the past the \(\overline{\mathrm {MS}}\) mass has been employed, although the natural scenarios for this scheme are processes where the involved energy scale is much larger than the heavy quark mass. Even though the QCD static potential starts off at \({\mathcal {O}}(\alpha _s)\), the first correction to the quarkonia masses is \({\mathcal {O}}(\alpha _s^2)\).^{11} When switching the pole mass to a short-distance scheme in Eq. (38) the first correction from this conversion will be \({\mathcal {O}}(\alpha _s)\), which appears to follow a different pattern. It is at this point when the use of the \(\varepsilon \) parameter becomes relevant. Since the renormalon cancellation happens already in the static potential, it is crucial to treat the short-distance mass subtractions in Eq. (38) exactly as in the static energy, and therefore terms proportional to \(\alpha _s^n\) in the subtraction are regarded as \({\mathcal {O}}(\varepsilon ^n)\) in the \(\varUpsilon \)-expansion counting.

Heavy quarkonium masses probe energy scales below \(m_Q\), and therefore the relativistic logs showing up in the \(\overline{\mathrm {MS}}\)-pole relation series \(\delta ^{\mathrm{SD}}\), namely \(\log (\mu /{\overline{m}}_Q)\), may become large if \(\mu \) is chosen to minimize the non-relativistic logarithm that appears in Eq. (39). Therefore it would be better to simultaneously minimize logarithms appearing in \(\delta m_Q^{\mathrm{SD}}\) and in \(L_{n_{\ell }}\), whose argument is the ratio of a non-relativistic scale and \(\mu \). This is in full analogy with the static potential analysis, where *r* is being replaced by the Bohr radius in bound states masses. Therefore, a low-scale short-distance mass is advisable. Following the results of Sect. 5 and the analysis in Ref. [27] we will employ the MSR mass [34] for our analysis. Therefore we express \(m_Q^{\mathrm{pole}}\) in Eqs. (38) and (39) in terms of the MSR mass as shown in the second line of Eq. (20), and re-expand the result in powers of \(\alpha _s(\mu )\) respecting the \(\varUpsilon \) counting scheme. In this way the quarkonia masses depend on two renormalization scales, namely \(\mu \) and *R*.

## 8 Bottomonium masses with a floating bottom mass

The ultimate aim of this article is to calibrate the Cornell model constants (especially the Cornell mass) in terms of the QCD fundamental parameters \(\alpha _s\) and \(m_b\). To achieve this goal one has to scan over these two parameters. Specifically, we need QCD predictions for the eight lowest-lying bottomonium resonances in a reasonable range of values for the bottom mass and the strong coupling constant. For the former we will consider the recent determination of Ref. [27], \({\overline{m}}_b= 4.2\,\hbox {GeV}\), and vary the \(\overline{\mathrm {MS}}\) mass between 4 and \(8\,\hbox {GeV}\) in steps of \(500\,\)MeV. Since the value of \(\alpha _s\) that enters the perturbative expansion in Eq. (38) has \(n_\ell =4\) active flavors, if computed from the reference value \(\alpha _s^{(n_\ell =5)}(m_Z)\), it retains a (small) residual dependence on \(m_b\) from threshold contributions. To make sure no bottom mass effects come from these matching corrections, we keep fixed the value of \(\alpha _s^{(n_\ell =4)}(1.3\,\mathrm{GeV})\). We consider the current world average for \(\alpha _s\), which translates into \(\alpha _s^{(n_\ell =4)}(1.3\,\mathrm{GeV}) = 0.38331\), plus an evenly spaced grid between 0.34 and 0.41 in steps of 0.01, which corresponds to varying \(\alpha _s^{(n_\ell =5)}(m_Z)\) between 0.114 and 0.12 for \({\overline{m}}_b = 4.2\,\hbox {GeV}\).

*R*’s for various \(n_p\) values are correlated, as explained in Ref. [27]. This makes sure theoretical correlations are properly propagated but avoids the so called d’Agostini bias [67]. The range in which \(\mu _{n_p}\) is varied comes from analyzing the argument of the logarithm in Eq. (39) in the MSR scheme, such that its value varies between 1 / 2 and 2. Such range will depend on the value of the bottom mass, and therefore we will adapt the results of Ref. [27] for the masses covered in our scan. Since the mass subtraction involves powers of logs of \(\mu _{n_p}/R_{n_p}\) that should be \({\mathcal {O}}(1)\), we take the same variation for \(\mu _{n_p}\) and \(R_{n_p}\). We find that the upper limit of \(\mu _{n_p}\) is independent of both \(m_b\) and \(n_p\), and hence we fix \(\mu ^{\mathrm{max}}_{n_p} = 4\,\hbox {GeV}\). We find that \(\mu ^{\mathrm{min}}_{n_p}\) depends on \(n_p\) and increases approximately linearly with \(m_b\). We parameterize them with the following approximate expressions:

## 9 Results of the calibration

Results of the calibration for \({\overline{m}}_b = 4.2\,\hbox {GeV}\) using the world average value of \(\alpha _s\). Rows second to fifth correspond to \(\hbox {N}^3\hbox {LO}\), \(\hbox {N}^2\hbox {LO}\), NLO and LO, respectively. The second and third column show the central value and uncertainty due to the theory scan, while in the fourth and fifth column the fit and total uncertainties are depicted

Order | Central | Perturbative | Fit | Total |
---|---|---|---|---|

\(\hbox {N}^3\hbox {LO}\) | 4.731 | 0.068 | 0.071 | 0.099 |

\(\hbox {N}^2\hbox {LO}\) | 4.712 | 0.085 | 0.054 | 0.100 |

NLO | 4.624 | 0.114 | 0.097 | 0.149 |

LO | 4.503 | 0.120 | 0.198 | 0.231 |

*R*is a kinetic mass free from renormalon ambiguities, and given the results of Sects. 5 and 6, we will calibrate the Cornell mass against \(m_b^{\mathrm{MSR}}(R={1\mathrm GeV})\). As expected, we observe a linear relation between these two parameters, and we find that the slope is very close to (and compatible with) unity: \(0.995\pm 0.026\) with an intersect compatible with zero within uncertainties: \(0.05 \pm 0.19\).

^{12}This pattern is found for every value of \(\alpha _s\) and also at various orders, but for simplicity we show the linear relation only at \(\hbox {N}^3\hbox {LO}\) and for the world average in Fig. 11. Rather than the intersect with zero, our final result for the difference of the Cornell model mass parameter and the MSR mass is computed as the weighted average of the individual \(m_b^{\mathrm{Cornell}} - m_b^{\mathrm{MSR}}(1\,\mathrm{GeV})\) values, and for its incertitude we simply take the regular average of the individual uncertainties, finding one of the most important results of this article:

^{13}At this point we have indirect evidence that appears to address the question raised at the end of Sect. 5, namely whether the difference of the Cornell and QCD potentials in the UV can be absorbed in the short-distance definition of the quark mass. The analysis carried out in this section, streamlined in the result shown in Eq. (42), seems to indicate this indeed happens, well within our uncertainties, if the MSR mass is employed.

^{14}It has the following physical interpretation: the linear rising term of the Cornell potential incorporates in an effective way medium-distance (perturbative) quantum fluctuations. For distances smaller than \(1\,\mathrm{GeV}^{-1}\) Cornell sees only the classical Coulomb-like potential, therefore we can assume that energy fluctuations above \(1\,\mathrm{GeV}\) are absorbed into the quark mass. This interpretation matches up with the definition of the MSR mass with \(R=1\,\hbox {GeV}\), hence legitimating our initial motivation. In other words, even though the Cornell and pQCD potentials disagree at short distances, the mass spectrum is not overly determined by this region, and a suitable scheme choice makes the two approaches compatible at the observable level. For short-distance-dominated observables, it could have been impossible to reconcile the two approaches. A similar conclusion was reached in Ref. [17], where the MSR top quark mass with \(R=1\,\hbox {GeV}\) was found to agree within uncertainties with the top quark parameter used in Pythia.

We show the dependence of \(\sigma \) with the bottom quark mass and the strong coupling constant reference value in Fig. 13c, d. The dependence of \(\sigma \) with the former shows exactly what one would expect from a static potential parameter: there is no mass dependence at all and one could simply take the average of all points, obtaining \(\sigma ^{\mathrm{Cal.}} = 0.176 \pm 0.088\,\hbox {GeV}^2\). This value compares well with the one obtained from the fit to the bottomonium experimental data in Eq. (16) \(\sigma ^{\mathrm{fit}} = 0.207 \pm 0.011\,\mathrm{GeV}^2\). It appears there is some non-flat dependence on the value of \(\alpha _s^{(n_f=5)}(m_Z)\), although drawing strong conclusions is not possible given the size of the uncertainties, which grow for larger values of \(\alpha _s\). In any case one expects some dependence of \(\sigma \) with \(\alpha _s\), since as argued in Refs. [9, 10], the linear rising term in the static potential is of perturbative nature.

## 10 Conclusions

In this article we have confronted a simple version of the Cornell model with both experimental data and QCD. This model contains three terms, with one parameter associated to each one of them: a constituent quark mass, a linear raising potential, and a Coulomb-like potential. We have solved numerically the Schödinger equation for the Cornell model using the Numerov algorithm, performing several checks to make sure the uncertainty of the approximation is much smaller than any other uncertainty involved. The Cornell model includes the static approximation (solved exactly) plus the leading non-relativistic correction, in the form of (angular-momentum-dependent) \(1/m_Q^2\) suppressed potentials, whereas in QCD we include as many non-relativistic corrections as necessary to achieve up to \(\hbox {N}^3\hbox {LO}\) accuracy.

As a warm-up exercise we have determined the Cornell potential parameters for the bottomonium and charmonium systems from fits to the 8 states with the lowest masses. We have confirmed that this simple version of the Cornell model fails to predict states with larger mass, possibly because it does not include string-breaking effects, but gives reasonable post-dictions of the masses of the states that enter the fit. NRQCD can predict the first 8 bottomonium states and the first 2 charmonium states within perturbation theory, and therefore it is possible to calibrate the Cornell model using bottomonium QCD predictions. Of course this only makes sense if the Cornell model is solely of perturbative nature.

The QCD static potential suffers form a \(u=1/2\) renormalon, identical to that of the quark pole mass up to a factor of 2 and a sign. Therefore one can cancel the renormalon in the static energy (the sum of the static potential and twice the pole quark mass) by expressing the quark mass in a short-distance scheme. Since we are dealing with a threshold-like problem (bound states of a quark–antiquark pair), a low-scale short-distance mass should be used. To keep logarithms small in the static potential and in the subtraction series relating the pole and short-distance masses, it is compulsory to use a scheme with a tunable subtraction scale. The MSR mass satisfies these two criteria, and has already been used in the context of quarkonium. Therefore we also employ this scheme to make predictions for the static energy. Furthermore, we use R-evolution to sum-up large renormalon-type logarithms in the static potential, whose argument is the ratio of the subtraction and renormalization scales. In this way, we define an R-improved MSR static potential, which shows nice order-by-order convergence, and has the same linear rising behavior as the Cornell model. In fact, if the renormalon subtraction scale is chosen close to \(1\,\hbox {GeV}\), the Cornell and R-improved static potential at \(\hbox {N}^3\hbox {LO}\) nicely agree for moderate and large values of *r*. As expected, both potentials disagree in the UV regime probed by small values of *r*. Since the linear raising potential is of perturbative nature and agrees with that of the Cornell model, we conclude that all ingredients in this model for bottomonium are perturbative and a calibration against QCD makes sense.

We have compared our R-improved MSR static potential with lattice simulations of the same quantity, performing a fit for the strong coupling constant and the renormalon subtraction reference scale. After the fit, we find very nice agreement with lattice QCD results for the entire set of *r* values covered in the simulation, which includes distances as large as \(0.84\,\hbox {fm}\). To make this agreement possible, it is essential to use a “profile function” for the renormalization scale \(\mu \), which freezes to \(1\,\hbox {GeV}\) for values of the radius larger than \(1\,\hbox {fm}\).

To calibrate the Cornell model we generate templates for the 8 lightest bottomonium bound states with NRQCD. These predictions depend on two scales: the renormalization scale \(\mu \) and the renormalon subtraction scale *R*. These take different values for the various bound states, but are varied in a correlated way. For a given value of the bottom quark mass and strong coupling constant we generate templates at LO, NLO, \(\hbox {N}^2\hbox {LO}\) and \(\hbox {N}^3\hbox {LO}\), which contain the QCD prediction for the masses of the bound states in a two-dimensional grid of \(\mu \) and *R*. To avoid the d’Agostini bias, we adjust the Cornell model parameters to every entry on each template, effectively scanning over the two renormalization scales. Since there are no uncertainties in the template that can be used to construct a \(\chi ^2\) function, we use a regression algorithm to assign “fit-incertitudes” to the adjusted parameters, normalizing the \(\chi ^2\) such that equals the number of degrees of freedom at its minimum. On top of these, there are theoretical uncertainties from the scale scan. To figure these out, we first trim away values of the Cornell model parameters that are found in the regression less often, discarding those that are less than \(8\%\) (\(2\%\)) less frequent than the most likely occurrence for \(m_b^{\mathrm{Cornell}}\) (\(\sigma \), \(\alpha _s^{\mathrm{Cornell}}\)). After applying this procedure, we take the average of the remaining values as the central value for the parameter, and half the sum of the maximum and minimum values as the perturbative uncertainty. We also take the average of the regression errors as our final fit uncertainty.

We find an almost perfect linear relation between the Cornell and MSR masses with slope compatible with 1 within 0.19 standard deviations, and if one chooses the MSR reference scale \(R = 1\,\hbox {GeV}\) the intersect of this relation is compatible with zero within 0.26 standard deviations. This pattern is replicated in a wide range of \(\alpha _s\) values and for all perturbative orders considered, and nicely complies with the agreement found between the Cornell and static potentials for this particular choice of the reference scale. This seems to indicate that the difference of the Cornell and Static potentials in the UV can be absorbed in the short-distance definition of the quark mass. Our calibration exercise also reveals that the confining parameter \(\sigma \) does not depend on the value of the QCD quark bottom mass, as expected, but the precision of the analysis cannot discard some dependence of this parameter with \(\alpha _s(m_Z)\). On the other hand, the Coulomb-like Cornell parameter \(\alpha _s^{\mathrm{Cornell}}\) is found to agree within uncertainties with \(\alpha _s^{(n_f=4)}(\mu _{\mathrm{NR}})\), that is with the QCD strong coupling constant evaluated at a typical non-relativistic scale.

Our analysis could be extended in several directions: on the QCD side one could consider more refined predictions, for instance using pNRQCD resummation as done in Ref. [52], or including non-perturbative effects as in Ref. [68]. On the Cornell side, one could consider more sophisticated models, for example incorporating string breaking effects or coupled channels. The calibration itself could be carried out for mixed \(b\bar{c}\) states in which both masses are varied such that their ratio remains constant. As for the comparison with lattice QCD results on the static potential, the next step is studying perturbative uncertainties, order and dataset dependence, incorporating results for other lattice spacings, and taking into account the lattice correlation matrices once they are known.

We close this article raising a concern. We have shown in Sect. 2 that the confining part of the Cornell potential cannot be treated by any means as a perturbation of the Coulomb potential. We have also seen in Sect. 5 that the \(\sigma \) term can be entirely described in perturbative QCD once the renormalon has been canceled. Finally we have presented an analytic formula for the \(Q{\overline{Q}}\) mass in Sect. 7, which is based on solving the Schrödinger equation perturbatively around the lowest order result, that is, the Coulomb potential, including both radiative and relativistic corrections. Therefore one could call into question this perturbative treatment of the static QCD potential since it is known to fail for the not-so-different Cornell potential. On the other hand, pNRQCD power counts perturbative and non-relativistic corrections on an equal footing, as can be seen in Eq. (38), where there is no distinction between the former and the latter. One way of shedding light on this apparent puzzle is though a numerical, exact, solution of the Schrödinger equation for the QCD static potential. A step in this direction has been taken in Refs. [51, 52, 53].

## Footnotes

- 1.
For conciseness we will drop the mass superscript “Cornell” in the remainder of this section and in Appendix A.

- 2.
In the subsequent sections of this article the eigenvalues of the Cornell potential will be refereed to as static energies, denoted as \(E_{\mathrm{static}}^{n,\ell }\).

- 3.
Here the LS term from confinement is obtained by the exchange of a scalar particle.

- 4.
We denote with the superscript “OGE” the terms arising from gluon exchanges and with “CON” those coming from the confinement interaction.

- 5.
To figure out this precision we have compared to a numerical computation with \(N=200\,\,000\), which is taken as exact.

- 6.
In contrast, the pole and \(\overline{\mathrm {MS}}\) masses absorb all fluctuations above 0 and \({\overline{m}}({\overline{m}})\), respectively.

- 7.
In this article we use only the natural version of the MSR mass, in which heavy quark virtual effects are integrated out.

- 8.
As already mentioned, we omit the corrections coming form the finiteness of the charm quark, which start at \({\mathcal {O}}(\alpha _s^2)\).

- 9.
In this plot we choose a canonical scale for the renormalization scale \(\mu =1/r\) because \(\mu \) never becomes smaller than \(1\,\hbox {GeV}\) if \(r\le 0.2\,\hbox {fm}\), see Fig. 6.

- 10.
Again we neglect the corrections coming from a finite charm quark mass, which start at \({\mathcal {O}}(\varepsilon ^2)\). We also work in the \(n_\ell \) scheme, see Ref. [27] for more details.

- 11.
This extra factor of \(\alpha _s\) comes from solving the Schödinger equation, and it has been argued it should be counted as \({\mathcal {O}}(1)\).

- 12.
For simplicity we computed these uncertainties assuming uncorrelated errors for each mass value.

- 13.
We show in the horizontal axis \(\alpha _s^{(n_f=5)}(m_Z)\) rather than \(\alpha _s^{(n_f=4)}(m_c)\), where the former is understood as the value that one obtains running from the latter using threshold matching relations for \({\overline{m}}_b=4.2\,\hbox {GeV}\).

- 14.
Similar conclusions could be drawn with other low-scale masses.

## Notes

### Acknowledgements

We thank X. G. Tormo for providing a computer readable file with the QCD static potential. We thank P. Petreczky for providing us with lattice results for the static potential. This work has been partially funded by the Spanish MINECO *Ramón y Cajal program* (RYC-2014-16022), the MECD Grants FPA2016-78645-P, FPA2016-77177-C2-2-P and the IFT *Centro de Excelencia Severo Ochoa* Program under Grant SEV-2012-0249. P.G.O. acknowledges the financial support from *Junta de Castilla y León* and European Regional Development Funds (ERDF) under Contract no. SA041U16, and by Spanish MINECO’s *Juan de la Cierva-Incorporación* program with Grant agreement no. IJCI-2016-28525.

## References

- 1.J.E. Augustin et al., Phys. Rev. Lett.
**33**, 1406 (1974). https://doi.org/10.1103/PhysRevLett.33.1406 ADSCrossRefGoogle Scholar - 2.J.E. Augustin et al., Adv. Exp. Phys.
**5**, 141 (1976)Google Scholar - 3.J.J. Aubert et al., Phys. Rev. Lett.
**33**, 1404 (1974). https://doi.org/10.1103/PhysRevLett.33.1404 ADSCrossRefGoogle Scholar - 4.E. Eichten, K. Gottfried, T. Kinoshita, K.D. Lane, T.M. Yan, Phys. Rev. D
**17**, 3090 (1978). https://doi.org/10.1103/PhysRevD.17.3090, https://doi.org/10.1103/PhysRevD.21.313. [Erratum: Phys. Rev. D 21, 313(1980)] - 5.S. Godfrey, N. Isgur, Phys. Rev. D
**32**, 189 (1985). https://doi.org/10.1103/PhysRevD.32.189 ADSCrossRefGoogle Scholar - 6.D.P. Stanley, D. Robson, Phys. Rev. D
**21**, 3180 (1980). https://doi.org/10.1103/PhysRevD.21.3180 ADSCrossRefGoogle Scholar - 7.G.S. Bali, K. Schilling, C. Schlichter, Phys. Rev. D
**51**, 5165 (1995). https://doi.org/10.1103/PhysRevD.51.5165 ADSCrossRefGoogle Scholar - 8.G.S. Bali, B. Bolder, N. Eicker, T. Lippert, B. Orth, P. Ueberholz, K. Schilling, T. Struckmann, Phys. Rev. D
**62**, 054503 (2000). https://doi.org/10.1103/PhysRevD.62.054503 ADSCrossRefGoogle Scholar - 9.Y. Sumino, Phys. Lett. B
**571**, 173 (2003). https://doi.org/10.1016/j.physletb.2003.05.010 ADSCrossRefGoogle Scholar - 10.Y. Sumino, Phys. Lett. B
**595**, 387 (2004). https://doi.org/10.1016/j.physletb.2004.06.065 ADSCrossRefGoogle Scholar - 11.E.J. Eichten, K. Lane, C. Quigg, Phys. Rev. D
**73**, 014014 (2006). https://doi.org/10.1103/PhysRevD.73.014014, https://doi.org/10.1103/PhysRevD.73.079903. [Erratum: Phys. Rev. D 73, 079903 (2006)] - 12.G.P. Lepage, B.A. Thacker, Nucl. Phys. Proc. Suppl.
**4**, 199 (1988). https://doi.org/10.1016/0920-5632(88)90102-8 ADSCrossRefGoogle Scholar - 13.M.E. Luke, A.V. Manohar, I.Z. Rothstein, Phys. Rev. D
**61**, 074025 (2000). https://doi.org/10.1103/PhysRevD.61.074025 ADSCrossRefGoogle Scholar - 14.A. Pineda, J. Soto, Nucl. Phys. Proc. Suppl.
**64**, 428 (1998). https://doi.org/10.1016/S0920-5632(97)01102-X ADSCrossRefGoogle Scholar - 15.N. Brambilla, A. Pineda, J. Soto, A. Vairo, Nucl. Phys. B
**566**, 275 (2000). https://doi.org/10.1016/S0550-3213(99)00693-8 ADSCrossRefGoogle Scholar - 16.A. Pineda, Nucl. Phys. Proc. Suppl.
**133**, 190 (2004). https://doi.org/10.1016/j.nuclphysbps.2004.04.163 ADSCrossRefGoogle Scholar - 17.M. Butenschoen, B. Dehnadi, A.H. Hoang, V. Mateu, M. Preisser, I.W. Stewart, Phys. Rev. Lett.
**117**(23), 232001 (2016). https://doi.org/10.1103/PhysRevLett.117.232001 ADSCrossRefGoogle Scholar - 18.T. Sjostrand, S. Mrenna, P.Z. Skands, JHEP
**05**, 026 (2006). https://doi.org/10.1088/1126-6708/2006/05/026 ADSCrossRefGoogle Scholar - 19.T. Sjöstrand, S. Ask, J.R. Christiansen, R. Corke, N. Desai, P. Ilten, S. Mrenna, S. Prestel, C.O. Rasmussen, P.Z. Skands, Comput. Phys. Commun.
**191**, 159 (2015). https://doi.org/10.1016/j.cpc.2015.01.024 ADSCrossRefGoogle Scholar - 20.A. Bazavov et al., Phys. Rev. D
**90**, 094503 (2014). https://doi.org/10.1103/PhysRevD.90.094503 ADSCrossRefGoogle Scholar - 21.A. Bazavov, P. Petreczky, J.H. Weber, Phys. Rev. D
**97**(1), 014510 (2018). https://doi.org/10.1103/PhysRevD.97.014510 ADSCrossRefGoogle Scholar - 22.B.V. Noumerov, Mon. Not. R. Astron. Soc.
**84**(8), 592 (1924). https://doi.org/10.1093/mnras/84.8.592 ADSCrossRefGoogle Scholar - 23.B. Numerov, Astron. Nachr.
**230**(19), 359 (1927). https://doi.org/10.1002/asna.19272301903 ADSCrossRefGoogle Scholar - 24.
*GFortran, Gnu compiler collection (gcc), Version 8.1.0*(Copyright (C) 2018 Free Software Foundation, Inc., 2018)Google Scholar - 25.A. De Rujula, H. Georgi, S.L. Glashow, Phys. Rev. D
**12**, 147 (1975). https://doi.org/10.1103/PhysRevD.12.147 ADSCrossRefGoogle Scholar - 26.E. Eichten, F.L. Feinberg, Phys. Rev. Lett.
**43**, 1205 (1979). https://doi.org/10.1103/PhysRevLett.43.1205 ADSCrossRefGoogle Scholar - 27.V. Mateu, P.G. Ortega, JHEP
**01**, 122 (2018). https://doi.org/10.1007/JHEP01(2018)122 ADSCrossRefGoogle Scholar - 28.F. James, M. Roos, Comput. Phys. Commun.
**10**(CERN-DD-75-20), 343 (1975). https://cds.cern.ch/record/310399 - 29.M. Tanabashi et al., Phys. Rev. D
**98**(3), 030001 (2018). https://doi.org/10.1103/PhysRevD.98.030001 ADSCrossRefGoogle Scholar - 30.K.D. Born, E. Laermann, N. Pirch, T.F. Walsh, P.M. Zerwas, Phys. Rev. D
**40**, 1653 (1989). https://doi.org/10.1103/PhysRevD.40.1653 ADSCrossRefGoogle Scholar - 31.G.S. Bali, H. Neff, T. Duessel, T. Lippert, K. Schilling, Phys. Rev. D
**71**, 114513 (2005). https://doi.org/10.1103/PhysRevD.71.114513 ADSCrossRefGoogle Scholar - 32.A.H. Hoang, A. Jain, I. Scimemi, I.W. Stewart, Phys. Rev. Lett.
**101**, 151602 (2008). https://doi.org/10.1103/PhysRevLett.101.151602 ADSCrossRefGoogle Scholar - 33.A.H. Hoang, A. Jain, I. Scimemi, I.W. Stewart, Phys. Rev. D
**82**, 011501 (2010). https://doi.org/10.1103/PhysRevD.82.011501 ADSCrossRefGoogle Scholar - 34.A.H. Hoang, A. Jain, C. Lepenik, V. Mateu, M. Preisser, I. Scimemi, I.W. Stewart, JHEP
**04**, 003 (2018). https://doi.org/10.1007/JHEP04(2018)003 CrossRefGoogle Scholar - 35.W. Fischler, Nucl. Phys. B
**129**, 157 (1977). https://doi.org/10.1016/0550-3213(77)90026-8 ADSCrossRefGoogle Scholar - 36.A. Billoire, Phys. Lett. B
**92**, 343 (1980). https://doi.org/10.1016/0370-2693(80)90279-8 ADSCrossRefGoogle Scholar - 37.Y. Schroder, Phys. Lett. B
**447**, 321 (1999). https://doi.org/10.1016/S0370-2693(99)00010-6 ADSCrossRefGoogle Scholar - 38.A. Pineda, F.J. Yndurain, Phys. Rev. D
**58**, 094022 (1998). https://doi.org/10.1103/PhysRevD.58.094022 ADSCrossRefGoogle Scholar - 39.N. Brambilla, A. Pineda, J. Soto, A. Vairo, Phys. Rev. D
**60**, 091502 (1999). https://doi.org/10.1103/PhysRevD.60.091502 ADSCrossRefGoogle Scholar - 40.B.A. Kniehl, A.A. Penin, V.A. Smirnov, M. Steinhauser, Nucl. Phys. B
**635**, 357 (2002). https://doi.org/10.1016/S0550-3213(02)00403-0 ADSCrossRefGoogle Scholar - 41.A.A. Penin, M. Steinhauser, Phys. Lett. B
**538**, 335 (2002). https://doi.org/10.1016/S0370-2693(02)02040-3 ADSCrossRefGoogle Scholar - 42.A.V. Smirnov, V.A. Smirnov, M. Steinhauser, Phys. Lett. B
**668**, 293 (2008). https://doi.org/10.1016/j.physletb.2008.08.070 ADSCrossRefGoogle Scholar - 43.A.V. Smirnov, V.A. Smirnov, M. Steinhauser, Phys. Rev. Lett.
**104**, 112002 (2010). https://doi.org/10.1103/PhysRevLett.104.112002 ADSCrossRefGoogle Scholar - 44.C. Anzai, Y. Kiyo, Y. Sumino, Phys. Rev. Lett.
**104**, 112003 (2010). https://doi.org/10.1103/PhysRevLett.104.112003 ADSCrossRefGoogle Scholar - 45.X. Garcia i Tormo, Mod. Phys. Lett. A
**28**, 1330028 (2013). https://doi.org/10.1142/S0217732313300280 ADSCrossRefGoogle Scholar - 46.N. Brambilla, A. Vairo, J. Soto, X. Garcia i Tormo, Phys. Rev. D
**80**, 034016 (2009). https://doi.org/10.1103/PhysRevD.80.034016 ADSCrossRefGoogle Scholar - 47.A. Pineda, Heavy quarkonium and nonrelativistic effective field theories. Ph.D. thesis, Barcelona University (1998)Google Scholar
- 48.A.H. Hoang, M.C. Smith, T. Stelzer, S. Willenbrock, Phys. Rev. D
**59**, 114014 (1999). https://doi.org/10.1103/PhysRevD.59.114014 ADSCrossRefGoogle Scholar - 49.M. Beneke, Phys. Lett. B
**434**, 115 (1998). https://doi.org/10.1016/S0370-2693(98)00741-2 ADSCrossRefGoogle Scholar - 50.A.H. Hoang, D.W. Kolodrubetz, V. Mateu, I.W. Stewart, Phys. Rev. D
**91**(9), 094017 (2015). https://doi.org/10.1103/PhysRevD.91.094017 ADSCrossRefGoogle Scholar - 51.A. Pineda, J. Segovia, Phys. Rev. D
**87**(7), 074024 (2013). https://doi.org/10.1103/PhysRevD.87.074024 ADSCrossRefGoogle Scholar - 52.C. Peset, A. Pineda, J. Segovia, P-wave heavy quarkonium spectrum with next-to-next-to-next-to-leading logarithmic accuracy. Phys. Rev. D
**98**, 094003 (2018). https://doi.org/10.1103/PhysRevD.98.094003 ADSCrossRefGoogle Scholar - 53.C. Peset, A. Pineda, J. Segovia, The charm/bottom quark mass from heavy quarkonium at N\(^3\)LO. JHEP
**09**, 167 (2018). https://doi.org/10.1007/JHEP09(2018)167 - 54.A. Pineda, JHEP
**06**, 022 (2001). https://doi.org/10.1088/1126-6708/2001/06/022 ADSCrossRefGoogle Scholar - 55.Y. Koma, M. Koma, Nucl. Phys. B
**769**, 79 (2007). https://doi.org/10.1016/j.nuclphysb.2007.01.033 ADSCrossRefGoogle Scholar - 56.T. Kawanai, S. Sasaki, Phys. Rev. D
**89**(5), 054507 (2014). https://doi.org/10.1103/PhysRevD.89.054507 ADSCrossRefGoogle Scholar - 57.C. Ayala, G. Cvetič, A. Pineda, JHEP
**09**, 045 (2014). https://doi.org/10.1007/JHEP09(2014)045 - 58.A. Bazavov, N. Brambilla, H.T. Ding, P. Petreczky, H.P. Schadler, A. Vairo, J.H. Weber, Phys. Rev. D
**93**(11), 114502 (2016). https://doi.org/10.1103/PhysRevD.93.114502 ADSCrossRefGoogle Scholar - 59.A. Bazavov, N. Brambilla, X. Garcia i Tormo, P. Petreczky, J. Soto, A. Vairo, Phys. Rev. D
**86**, 114031 (2012). https://doi.org/10.1103/PhysRevD.86.114031 ADSCrossRefGoogle Scholar - 60.A. Bazavov, N. Brambilla, X. Garcia i Tormo, P. Petreczky, J. Soto et al., Phys. Rev. D
**90**(7), 074038 (2014). https://doi.org/10.1103/PhysRevD.90.074038 ADSCrossRefGoogle Scholar - 61.Y. Kiyo, Y. Sumino, Nucl. Phys. B
**889**, 156 (2014). https://doi.org/10.1016/j.nuclphysb.2014.10.010 ADSCrossRefGoogle Scholar - 62.M. Beneke, Y. Kiyo, K. Schuller, Nucl. Phys. B
**714**, 67 (2005). https://doi.org/10.1016/j.nuclphysb.2005.02.028 ADSCrossRefGoogle Scholar - 63.A.H. Hoang, Z. Ligeti, A.V. Manohar, Phys. Rev. Lett.
**82**, 277 (1999). https://doi.org/10.1103/PhysRevLett.82.277 ADSCrossRefGoogle Scholar - 64.A.H. Hoang, Z. Ligeti, A.V. Manohar, Phys. Rev. D
**59**, 074017 (1999). https://doi.org/10.1103/PhysRevD.59.074017 ADSCrossRefGoogle Scholar - 65.N. Brambilla, Y. Sumino, A. Vairo, Phys. Rev. D
**65**, 034001 (2002). https://doi.org/10.1103/PhysRevD.65.034001 ADSCrossRefGoogle Scholar - 66.Y. Kiyo, Y. Sumino, Phys. Lett. B
**730**, 76 (2014). https://doi.org/10.1016/j.physletb.2014.01.030 ADSCrossRefGoogle Scholar - 67.G. D’Agostini, Nucl. Instrum. Methods A
**346**, 306 (1994). https://doi.org/10.1016/0168-9002(94)90719-6 ADSCrossRefGoogle Scholar - 68.
- 69.S. Koonin,
*Computational Physics: Fortran Version*(Avalon Publishing, New York, 1998)zbMATHGoogle Scholar - 70.J.L. Domenech-Garret, M.A. Sanchis-Lozano, Comput. Phys. Commun.
**180**, 768 (2009). https://doi.org/10.1016/j.cpc.2008.11.011 ADSCrossRefGoogle Scholar - 71.R. Piessens, Ed Kapenga, C. Uberhuber, D. Kahaner,
*QUADPACK: A Subroutine Package for Automatic Integration*(Springer, Berlin, 1983)CrossRefGoogle Scholar - 72.C. De Boor,
*A practical guide to splines; rev. ed.*Applied mathematical sciences (Springer, Berlin, 2001). https://cds.cern.ch/record/1428148

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