Diabatic description of bottomoniumlike mesons

Diabatic description of bottomoniumlike mesons R. Bruschini 1,* and P. González 1,2,† 1Unidad Teórica, Instituto de Física Corpuscular (Universidad de Valencia–CSIC), E-46980 Paterna (Valencia), Spain 2Departamento de Física Teórica, Universidad de Valencia, E-46100 Burjassot (Valencia), Spain (Received 11 May 2021; accepted 18 May 2021; published 17 June 2021) We apply the diabatic approach, specially suited for a QCD based study of conventional (quarkantiquark) and unconventional (quark-antiquark þmeson-meson) meson states, to the description of hidden-bottom mesons. A spectral analysis of the I¼0,Jþþ and 1−− resonances with masses up to about 10.8 GeV is carried out. Masses and widths of all the experimentally known resonances, including conventional and unconventional states, can be well reproduced. In particular, we predict a significant B¯ B component in ϒð10580Þ. We also predict the existence of a not yet discovered unconventional 1þþ narrow state, with a significant Bs¯ B scontent making it to decay into ϒð1SÞϕ, whose experimental discovery would provide definite support to our theoretical analysis. DOI: 10.1103/PhysRevD.103.114016 I. INTRODUCTION The unified description of conventional and unconventional heavy-quark mesons from QCD, the strong interaction theory, is a current theoretical challenge in hadron physics. Due to the current impossibility of solving QCD in the nonperturbative regime, effective field theories directly connected to QCD, involving quark and gluons or hadrons as degrees of freedom, have been developed for the study of the heavy-quark meson structure, see for instance [1] and references therein. On the other hand, QCD calculations of heavy-quark mesons in the lattice have been performed. These comprise quenched analyses involving Q¯ Q(Q: heavy quark, bor c) with gluons as the light field [2,3], and unquenched studies with Q¯ Qand meson-meson components incorporating also light sea quarks in the light field [4–6]. A nice feature of the lattice, concerning phenomenology, is that it provides a straightforward way to compute complete heavy-quark meson potentials from QCD: the static light field energies evaluated in lattice are related to static potentials. More concretely, following a Born-Oppenheimer approximation quenched static energies can be directly identified with potentials in a Schrödinger equation for Q¯ Q, see for instance [2,7]. This allows for a QCD-based description of conventional quarkonium (b¯ bor c¯ c) in terms of a potential whose spinindependent part corresponds to a Cornell (funnel) form. As for unquenched static energies, calculated for Q¯ Qin the presence of meson-meson configurations, the BornOppenheimer approximation, which is a single channel one, is not valid anymore. Instead, a diabatic approach [8] permits their connection with the potential matrix in a multichannel Schrödinger equation for the Q¯ Qand mesonmeson components. Strictly speaking the static potential is only exact in the limit of infinite heavy-quark mass. For bottomonium ðb¯ bÞ with a quark mass, mb, much larger than the QCD scale, ΛQCD, the static limit represents a rather good approximation. For charmonium (c¯ c), with a much lower quark mass, mc, nonstatic contributions could be significant. Despite this drawback, in the last two decades, much more attention has been paid to the theoretical description of the excited spectrum of charmonium, the reason being the discovery, starting at 2003 with the χc1ð3872Þ, of charmoniumlike mesons whose properties (masses and widths) cannot be properly described from a conventional c¯ cstructure. The role played by explicit or implicit open charm mesonmeson components in the description of these unconventional states has been recognized, and alternative models (meson-meson molecules, tetraquarks, hadrocharmonium) have been formulated, some reviews are [9–13]. Quite recently, a (nonperturbative) diabatic description of the I¼0,Jþþ and 1−− hidden-charm mesons with masses up to about 4 GeV, in terms of c¯ cand meson-meson components, has been undertaken [8,14]. A major difference with respect to other nonperturbative studies involving the same degrees of freedom, see for example [15,16],is the incorporation of a lattice-based form of the mixing *[email protected].es †[email protected] Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3. PHYSICAL REVIEW D 103, 114016 (2021) 2470-0010=2021=103(11)=114016(13) 114016-1 Published by the American Physical Society potential instead of an ansatz with no clear connection to QCD. Despite the dearth of lattice data, and the technical approximations followed for tackling the diabatic equations, the results obtained (masses and widths) are encouraging. This supports the diabatic approach in QCD as an appropriate framework for a unified and complete nonperturbative description of conventional and unconventional heavy-quark mesons. For hidden-bottom mesons there have been in the past many speculations about possible bottomoniumlike partners of the unconventional charmoniumlike states, see for instance [17] and references therein. The partner hypothesis is based on the consideration that hidden-charm and hidden-bottom mesons can be described from the same flavor independent static potential (up to a constant). This is clearly acceptable for (conventional) charmonium, c¯ c, and bottomonium, b¯ b, with masses lying below the first openflavor meson-meson threshold, which are quite successfully described from a quark-antiquark Cornell potential. However, for unconventional states involving Q¯ Qand open-flavor meson-meson components as well, the partner hypothesis is questionable, for it is doubtful that the offdiagonal terms in the static potential matrix, giving account of the Q¯ Qand meson-meson mixings, be flavor independent. Experimentally, the situation is not well established due to the current dearth of data (masses and widths) for I¼0,Jþþ hidden-bottom mesons above the first ð0þþÞ open-bottom meson-meson threshold, and the absence of data for I¼0,1−− resonances with masses above the first 1−− S-wave meson-meson threshold. From the theoretical point of view, the diabatic approach, generating the static potential matrix from lattice QCD data, can be an ideal tool to definitely settle this issue. Indeed, lattice data for the energy of static band ¯ bsources, when the b¯ bconfiguration mixes with one or two open-flavor meson-meson ones, are available. From them a direct parametrization of the diabatic potential matrix is possible, and a QCD based prediction of the unknown excited spectrum is feasible. Actually, the hidden-bottom meson spectrum has been partially explored recently in a simplified diabatic treatment of I¼0,0−þand 1−− resonances, involving only one b¯ b channel and at most two distinct meson-meson thresholds masses [18,19]. In this article we center on the diabatic description of hidden-bottom mesons. The main differences with respect to [18,19] are (i) the consideration of all possible b¯ b channels and all meson-meson threshold masses contributing, (ii) the mixing potential which in our case does not contain any short range (light quark meson exchange) contribution, in line with the use of constant mesonmeson potentials, and (iii) the use of a bound-state based approximation instead of a S-matrix approach to the spectral solutions. We restrict our study to I¼0,Jþþ, and 1−− resonances with masses up to about 10.8 GeV, two hundred MeV below the first 1−− S-wave mesonmeson thresholds. Thus, as all the lower thresholds are known and have very small widths we avoid the uncertainty deriving from the partial knowledge of a threshold and the complexity due to possible threshold width effects. For the sake of technical simplicity in the evaluation of observables, we follow a two step description of resonances: first we approximate them by stable bound states incorporating closed meson-meson channels, and second we calculate mass corrections and widths from open meson-meson channels. We show that a fairly good description of the currently known Jþþ and 1−− experimental resonances in the realm of energy under study comes out. We predict that all these resonances except ϒð10580Þhave a very predominant b¯ bcomponent. For ϒð10580Þthe reduced, albeit dominant, b¯ bprobability allows us to give accurate account of leptonic width data. As for the not yet discovered resonances we predict that only for the third excited 1þþ state there is a significant meson-meson component. Although not dominant, this component points out to ϒð1SÞϕas a favored decay channel what could be relevant for its experimental discovery. Altogether these results indicate that a partner picture of hidden-charm and hidden-bottom mesons should be discarded once mesonmeson contributions start to play some role. These contents are organized as follows. In Sec. II a brief review of the diabatic formalism particularized for hiddenbottom mesons is presented, and the diabatic potential matrix is built from lattice data. As an improvement over the previous development for hidden-charm mesons a distinctive treatment of hidden-strange thresholds is incorporated. In Sec. III the nonperturbative description of hidden-bottom mesons is done in two steps: first, a bound state approximation incorporating closed meson-meson thresholds is followed, and second, mass shifts and widths from open meson-meson thresholds are calculated. Finally, in Sec. IV our main results and conclusions are summarized. II. DIABATIC FORMALISM FOR HIDDEN-BOTTOM MESONS The diabatic approach in QCD has been developed in [8]. Hidden-bottom meson states with quantum numbers JPC, made of b¯ band open-bottom meson-meson MðiÞ 1¯ MðiÞ 2 components, with M1ð¯ M2Þcontaining q¯ bð¯ qbÞwhere q stands for a light quark, q¼u,d,s, are solutions of the multichannel Schrödinger equation ðKþVðrÞÞΨðrÞ¼EΨðrÞð1Þ where ΨðrÞis a column vector R. BRUSCHINI and P. GONZÁLEZ PHYS. REV. D 103, 114016 (2021) 114016-2 ΨðrÞ¼0 B B B B B @ ψb¯ bðrÞ ψð1ÞðrÞ . . . ψðNÞðrÞ 1 C C C C C A ð2Þ with ψb¯ bðrÞstanding for the b¯ bcomponent, and ψðiÞðrÞ, i¼1;2…for the MðiÞ 1¯ MðiÞ 2component. K is the kinetic energy matrix K¼0 B B B B B @ −1 2μb¯ b ∇2 −1 2μð1Þ∇2 .. . −1 2μðNÞ∇2 1 C C C C C A ð3Þ where μb¯ bis the reduced b¯ bmass, μðiÞis the reduced MðiÞ 1¯ MðiÞ 2mass, and matrix elements equal to zero are not displayed. VðrÞis the diabatic potential matrix. Up to spin dependent terms that we shall not consider it can be formally written as 0 B B B B B @ VCðrÞVð1Þ mixðrÞ  VðNÞ mixðrÞ Vð1Þ mixðrÞTð1Þ . . ... . VðNÞ mixðrÞTðNÞ 1 C C C C C A ð4Þ where the diagonal elements VCðrÞand TðiÞcorrespond to the b−¯ band MðiÞ 1−¯ MðiÞ 2potentials respectively, and VðiÞ mixðrÞto the MðiÞ 1¯ MðiÞ 2−b¯ binteraction potential. More precisely, we express the b¯ bcomponent as ψb¯ bðrÞ¼X t Rð0Þ tðrÞYJ;mJ lð0Þ t;sð0Þ tðˆ rÞð5Þ where the sum over tgoes from 1 to the number of pairs ðlb¯ b≡lð0Þ;s b¯ b≡sð0ÞÞcoupling to JPC,Rð0Þ tðrÞstands for a radial wave function and YJ;mJ l;s ðˆ rÞ≡X ml;ms Cml;ms;mJ l;s;J Yml lðˆ rÞξms sð6Þ for an angular-spin wave function (Cis a Clebsch-Gordan coefficient, Yml la spherical harmonic, and ξms sa spin vector), and the MðiÞ 1¯ MðiÞ 2component as ψðiÞðrÞ¼X k RðiÞ kðrÞYJ;mJ lðiÞ k;sðiÞ kðˆ rÞð7Þ where the sum over kgoes from 1 to the number of pairs ðlðiÞ≡lMðiÞ 1 ¯ MðiÞ 2 ;s ðiÞ≡sMðiÞ 1 ¯ MðiÞ 2Þcoupling to JPC. (Let us note that we have changed the notation for the radial wave function with respect to our previous papers [8,14]. Here we use the standard Rand reserve ufor the reduced radial wave function, see next.) Then, one has ZdΩψ b¯ bðrÞVðrÞψb¯ bðrÞ¼X t Rð0Þ tðrÞVCðrÞRð0Þ tðrÞ ð8aÞ ZdΩψðiÞðrÞVðrÞψb¯ bðrÞ¼X k;t RðiÞ kðrÞVðiÞ mixðrÞRð0Þ tðrÞ ð8bÞ ZdΩψði0ÞðrÞVðrÞψðiÞðrÞ¼δii0X k RðiÞ kðrÞTðiÞRðiÞ kðrÞ ð8cÞ so that the multichannel Schrödinger equation reduces to a coupled system of radial equations for the sets of channels fuð0Þ tðrÞ≡rRð0Þ tðrÞg and fuðiÞ kðrÞ≡rRðiÞ kðrÞg. For example, if we considered for simplicity the case of the b¯ bcomponent with only one pair, ðlð0Þ 1;s ð0Þ 1Þ, coupling to the given JPC, and one meson-meson component Mð1Þ 1¯ Mð1Þ 2with only one pair, ðlð1Þ 1;s ð1Þ 1Þ, coupling to the given JPC, the system would read 0 B @ −1 2μb¯ bð∂2 r−lð0Þ 1ðlð0Þ 1þ1Þ r2ÞþVCðrÞ−EV ð1Þ mixðrÞ Vð1Þ mixðrÞ−1 2μð1Þð∂2 r−lð1Þ 1ðlð1Þ 1þ1Þ r2ÞþTð1Þ−E1 C A0 B @ uð0Þ 1 uð1Þ 11 C A¼0:ð9Þ The generalization to any number of possible ðlð0Þ;s ð0ÞÞand ðlðiÞ;s ðiÞÞ,i¼1, 2... pairs is straightforward by considering each uð0Þ tand each uðiÞ kas a component of the eigenfunction. Then, for normalizable solutions of the general system of radial equations, the probability for the b¯ bcomponent can be calculated as Pðb¯ bÞ¼X tZdrjuð0Þ tðrÞj2ð10Þ DIABATIC DESCRIPTION OF BOTTOMONIUMLIKE MESONS PHYS. REV. D 103, 114016 (2021) 114016-3 and for the MðiÞ 1¯ MðiÞ 2component PðMðiÞ 1¯ MðiÞ 2Þ¼X kZdrjuðiÞ kðrÞj2:ð11Þ Notice that although no direct interaction potential between different meson-meson components is considered, what it is justified for isolated, well separated meson-meson thresholds with no overlap at all, an indirect interaction through their coupling to the b¯ bchannel is present. A. Diabatic potential matrix The explicit form of the matrix elements VCðrÞ,TðiÞ, VðiÞ mixðrÞcan be derived from the light field static energies calculated in lattice QCD [8]. As lattice results depend on the chosen lattice spacing the philosophy underlying this derivation is the use of parametrizations motivated from lattice results with parameters to be fixed from phenomenology. Thus, the diagonal element VCðrÞcorresponding to the b−¯ bpotential is parametrized from quenched lattice data on the static quark-antiquark energy [3] as the Cornell potential VCðrÞ¼σr− χ r −βþmbþm¯ bð12Þ with σ,χ,β, and mbbeing the string tension, the color coulomb strength, a constant, and the bottom quark mass, respectively. We shall assume that all the flavor dependence in VCðrÞcomes from the mass term. Therefore, we shall keep for hidden-bottom mesons the same values for σ,χ, and βused in [8] for hidden-charm mesons. In order to fix mbwe have to take into account that the potential is spin independent so that the calculated masses should be compared with the experimental mass centroids obtained from spin singlet and spin triplet data. So we choose to fit the 1Pground state mass centroid under the assumption that 1PJexperimental resonances are pure bottomonium states, as will be confirmed later on (alternatively we could have chosen to fit the 1Sor 2Sor 2Pmass centroid without any significant change in the forthcoming analysis). Thus, we have σ¼925.6MeV=fm;ð13aÞ χ¼102.6MeVfm;ð13bÞ β¼855 MeV:ð13cÞ mb¼5215 MeV:ð13dÞ The b¯ bspectrum from this Cornell potential for Jþþ and 1−− isoscalar states is shown in Table I. Let us point out that in phenomenological applications of the Cornell potential [21,22] the chosen value of the bottom quark mass differs slightly from ours. In these applications distinct values of βare considered for bottomonium and charmonium in order to fit approximately the low-lying mass centroids. Any of the other diagonal elements TðiÞrepresents a MðiÞ 1−¯ MðiÞ 2potential. Up to one pion exchange effects that we do not consider this potential is given by the ith mesonmeson threshold TðiÞ¼mMðiÞ 1þm¯ MðiÞ 2ð14Þ with mMðiÞ 1 and m¯ MðiÞ 2 being the masses of the corresponding mesons. The meson-meson thresholds, calculated from the masses of bottom mesons in [20], are listed in Table II. It is worth remarking that the use of the experimental masses for the thresholds introduces some implicit spin dependence in the description. Let us note that each of the B¯ B,B¯ B, and B¯ B thresholds is composed of two (approximately) degenerate thresholds. For example B¯ Bcorresponds to BþB−and B0¯ B0, with an experimental threshold mass difference of TABLE I. Bottomonium spectrum from the Cornell potential. Each spectral state is characterized by JPC and nL quantum numbers. For JPC ¼ð0;1;2Þþþ, it is intended that F-wave bottotmonium states appear only for 2þþ. Available experimental centroid masses from [20] are listed for comparison. JPC nL Mass (MeV) Centroid (MeV) ð0;1;2Þþþ 1P9900.7 9899.7 2P10254.4 10260.2 1F10341.5 3P10536.6 2F10601.0 4P10782.2 1−− 1S9401.2 9444.9 2S9993.8 10017.2 1D10150.4 3S10338.6 2D10442.0 4S10615.0 3D10694.1 5S10856.4 TABLE II. Low-lying open-bottom meson-meson thresholds MðiÞ 1¯ MðiÞ 2. Threshold masses TðiÞfrom the bottom and bottom strange meson masses quoted in [20]. Crossing radii of these thresholds with the Cornell potential, rðiÞ c, are also tabulated. iM ðiÞ 1¯ MðiÞ 2TðiÞ(MeV) rðiÞ c(fm) 1B¯ B10559 1.16 2B¯ B10604 1.20 3B¯ B10649 1.25 4Bs¯ Bs10733 1.33 5Bs¯ B s10782 1.38 6B s¯ B s10830 1.43 R. BRUSCHINI and P. GONZÁLEZ PHYS. REV. D 103, 114016 (2021) 114016-4 0.6 MeV. In contrast the hidden strange cases Bs¯ Bs,Bs¯ B s, and B s¯ B sare single thresholds. The off-diagonal elements, VðiÞ mixðrÞ, correspond to b¯ b-MðiÞ 1¯ MðiÞ 2mixing potentials. From unquenched lattice static energies, calculated for b¯ bin the presence of mesonmeson configurations [4,6], the following parametrization has been proposed [8] jVðiÞ mixðrÞj ¼ ΔðiÞ 2exp−ðVCðrÞ−TðiÞÞ2 2σ2ρ2ð15Þ where ρis a radial scale for the mixing, that we shall take equal for all thresholds, and ΔðiÞis a strength parameter corresponding to the difference between the unquenched lattice static energies resulting from the avoided crossing of VCðrÞand TðiÞat the crossing radius rðiÞ cdefined by VCðrðiÞ cÞ¼TðiÞ:ð16Þ The values of the crossing radii have been tabulated in Table II. For the sake of simplicity, in [8,14] the same value for ΔðiÞwas used for degenerate and single thresholds. Here we go a step further. As shown in the Appendix a doubly degenerate threshold can be managed as an effective single threshold with a different value of Δ: Δdegenerate ¼ffiffiffi 2 pΔsingle:ð17Þ To make all this clear let us consider for example a system containing b¯ band B¯ B. From Table II rðB¯ BÞ c¼1.16 fm. In the lattice calculation of Ref. [4], rðB¯ BÞ cðlatticeÞ¼1.25 fm and ΔðB¯ BÞ ðlatticeÞis close to 50 MeV. Hence, we may expect quite a similar value for ΔðB¯ BÞ. As for ρwe compare the mixing angle between the ground and excited light field configurations associated to b¯ band B¯ B[8]: θðrÞ¼1 2arctan2VðB¯ BÞ mix ðrÞ TðB¯ BÞ−VCðrÞð18Þ to the one extracted from lattice, see Fig. 15 in [4]. More concretely, by using ΔðB¯ BÞ¼55 MeV ð19aÞ ρ¼0.3fm ð19bÞ we obtain the angle and mixing potential drawn in Figs. 1 and 2respectively. It is worth remarking that the mixing is only effective in an interval around rðB¯ BÞ cdetermined by the value of ρand that the sign of the mixing potential has no effect on the results that follow. The diabatic potential matrix reads VCðrÞVðB¯ BÞ mix ðrÞ VðB¯ BÞ mix ðrÞmBþm¯ Bð20Þ and its eigenvalues, given by VðrÞ¼VCðrÞþðmBþm¯ BÞ 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi VCðrÞ−ðmBþm¯ BÞ 22 þðVðB¯ BÞ mix ðrÞÞ2 sð21Þ and represented in Fig. 3, should be compared to the static energies for b¯ bin the presence of B¯ Bcalculated in lattice, see Figs. 13 and 14 in [4]. (Let us realize that the comparison has to be more qualitative than quantitative since the values of the parameters in the lattice depend on the chosen lattice spacing.) The extension to a system containing b¯ b,B¯ B, and Bs¯ Bs is straightforward. The diabatic potential matrix is now FIG. 1. Mixing angle between b¯ band B¯ B. FIG. 2. Mixing potential between b¯ band B¯ B. DIABATIC DESCRIPTION OF BOTTOMONIUMLIKE MESONS PHYS. REV. D 103, 114016 (2021) 114016-5 0 B B B @ VCðrÞVðB¯ BÞ mix ðrÞVðBs¯ BsÞ mix ðrÞ VðB¯ BÞ mix ðrÞmBþm¯ B0 VðBs¯ BsÞ mix ðrÞ0mBsþm¯ Bs: 1 C C C Að22Þ By using ΔðBs¯ BsÞ¼55 ffiffiffi 2 pMeV ð23Þ the resulting eigenvalues, plotted in Fig. 4, should be compared to the educated guess of the static energies for b¯ b in the presence of B¯ Band Bs¯ Bsdone in [4] (Fig. 22) and to the lattice calculation performed in [6]. III. STATE DESCRIPTION Any hidden-bottom meson state, characterized by the quantum numbers JPC, with mass below all possible openbottom meson-meson thresholds with the same quantum numbers is stable under decay into open-bottom mesonmeson channels. Hence it corresponds to a bound state solution of the diabatic multichannel Schrödinger equation. On the other hand any JPC hidden-bottom meson state with mass above a possible open-bottom meson-meson threshold with the same quantum numbers is unstable under decay into open-bottom meson-meson channels and corresponds to a scattering solution of the diabatic multichannel Schrödinger equation. From a technical point of view the extraction of the values of physical observables from a (normalizable) bound state wave function is straightforward. In contrast, for a scattering wave function it requires the development of a dedicated formalism [23]. Taking into account that the difference in the wave functions is associated to the presence of open meson-meson components in the scattering case, which are asymptotically free, one can approach a scattering resonance solution through a two-step procedure. In the first step one solves a bound state problem incorporating only the closed meson-meson components; in the second step, one generates a resonance through the coupling of the bound state solution to the open meson-meson components. This coupling allows for the calculation of the mass of the resonance through mass corrections to the bound state mass, and for the evaluation of its width. Notice that this procedure is completely nonperturbative. One should keep in mind though the lack of consistency of this bound state based approximation when there is some open threshold giving rise to a mass correction to the bound state that makes the threshold to close with respect to the resulting resonance. This is for example the case for the hidden-charm meson ψð4040Þ, see [14]. Then, only a direct scattering solution of the multichannel Schrödinger equation can provide a trustable description. A. Bound states In order to calculate bound states a finite number of closed meson-meson thresholds is considered. This is justified because in general for a given bound state the probability of meson-meson components corresponding to thresholds far above the mass of the bound state is expected to be negligible. From our mixing potential, we can estimate that this is the case for any threshold being at least 200 MeV above the mass of the bound state. The technical procedure to calculate bound states has been detailed elsewhere, see Sec. IV F and Appendices C and D in [8]. Let us only recall here that in order to avoid possible multiple countings of the same bound state when different sets of closed meson-meson thresholds are considered we assume a one-to-one correspondence with the FIG. 3. Static energies. Dashed line: b¯ b(Cornell). Dotted line: B¯ Bthreshold. Dash-dotted lines: r-dependent eigenvalues of the diabatic potential matrix (20). FIG. 4. Static energies. Dashed line: b¯ b(Cornell). Dotted lines: B¯ Band Bs¯ Bsthresholds. Dash-dotted lines: r-dependent eigenvalues of the diabatic potential matrix (22). R. BRUSCHINI and P. GONZÁLEZ PHYS. REV. D 103, 114016 (2021) 114016-6 bound states of b¯ bfrom the Cornell potential. Hence, each b¯ bbound state is the seed of only one bound state, the one obtained when the chosen set of closed thresholds is maximal in the sense of containing the maximum possible number of them. This assumption has proved to work for hidden-charm mesons [8,14], and we shall show it also does for hidden-bottom ones. Henceforth we center on spin-triplet hidden-bottom mesons with I¼0and JPC ¼ð0;1;2Þþþ and 1−− for which there are spectral data available up to 11.0 GeV. We restrict our study to bound states with masses up to about 10.8 GeV, two hundred MeV below the first 1−− S-wave meson-meson thresholds. There are several reasons for this. First, it is known the 11P1state B1ð5721Þbut not the corresponding 13P1state B1ð?Þwith an expected similar mass. Hence, the threshold B1¯ Bis only partially known. Second, B1(5721) has a non-negligible width, 27.53.4MeV, and B1ð?Þis presumably a much wider state (actually, this may be preventing its experimental detection). Hence, threshold width effects should be properly incorporated. Third, the lowest lying bottomonium hybrid b¯ bg (g:gluon),which could mix with b¯ b, is predicted to have a mass about 10.9 GeV, see [24] and references therein. The possible values of lb¯ bðsb¯ b¼1Þand ðlMðiÞ 1 ¯ MðiÞ 2 ;s MðiÞ 1 ¯ MðiÞ 2Þcoupling to a given JPC are listed in Table III, where the common notation BðsÞto refer to bottom and bottom, strange mesons, and the shorthand BðsÞ¯ B ðsÞto denote the C-parity eigenstate, are used. The calculated spectrum of bound states is shown in Table IV. A glance at the table and its comparison with Table I makes clear that (i) all bound states have a dominant b¯ b component, with more than 90% probability in most cases, (ii) closed meson-meson thresholds give rise to attraction, (iii) the attractive effect on the mass is quantitatively modest, with mass reductions of 16 MeV or less with respect to the b¯ bmasses obtained from the Cornell potential. These results are in line with the reasonable mass description of known experimental resonances provided by the Cornell potential model. For a detailed comparison to data we have to take into account that our Cornell potential does not contain spindependent terms. Then, for pure ðnlÞb¯ bstates the calculated masses have to be compared to the ðnlÞ experimental centroids; in the other cases, where mesonmeson components are present, since they are specific for any set of JPC quantum numbers, the comparison has to be done with the experimental candidates with the same JPC. Taking this into consideration all known JPC ¼ð0;1;2Þþþ and 1−− experimental resonances below 10.8 GeV can be assigned to bound states with the same location (below or above) with respect to the meson-meson thresholds. Thus, we see that the calculated mass for the ð0;1;2Þþþ ground states, which are 100% Cornell ð1PÞb¯ bstates, coincides with the experimental mass centroid from 1PJstates at 9899.90.6MeV. Actually, this coincidence has been required to fix the bottom quark mass. As for the first excited ð0;1;2Þþþ states, which are 100% Cornell ð2PÞb¯ b states, the calculated mass is very close to the experimental mass centroid from 2PJstates at 10260.20.7MeV. The only additional Jþþ pure Cornell state is the second excitation of 2þþ, assigned to the ð1F2Þb¯ bstate with a predicted mass of about 10340 MeV. The second excited ð0;1Þþþ and the third excited 2þþ states are predicted to contain more than a 90% of ð3PÞb¯ b and less than a 10% of meson-meson components. For 1þþ and 2þþ the calculated masses compare well with existing data (for 0þþ there is no PDG data). Indeed the measured masses of χb1ð3PÞ,10513.42 0.41 0.53 MeV, and χb2ð3PÞ,10524.02 0.57 0.53 MeV, differ from the calculated values by less than 30 MeV. It is worth mentioning that the presence of meson-meson components makes the calculated masses to be 12 MeV closer to data than the corresponding Cornell masses suggesting that a renaming of these resonances as χb1ð10513Þand χb2ð10524Þmight be in order. From the point of view of its meson-meson composition the most interesting Jþþ case is the third excited state of 1þþ with a significant 24% of Bs¯ B s. This significant percentage has to do with the immediate vicinity of the Cornell ð4PÞb¯ bstate and the Bs¯ B sthreshold, both located at about 10782.2 MeV. Notice though that the mass shift due to Bs¯ B sis only 7 MeV with respect to the Cornell ð4PÞ b¯ bmass. More importantly, as Bs¯ B scan naturally decay strongly into ϒð1SÞϕthrough quark exchange this could be a possible discovery channel. For 1−− the ground and the first three excited states are predicted to be 100% the Cornell ð1S; 2S; 1D; 3SÞb¯ bstates respectively. For the ground state ð1SÞthe difference between the experimental centroid (from ηbð1SÞand ϒð1SÞÞ at 9445.00.7MeV and the calculated mass is 45 MeV, significantly higher than in any other case. This could be indicating the presence of more relevant relativistic effects in the 1Sstate. Indeed, for the first excited state ð2SÞthe difference between the ðηbð2SÞ;ϒð2SÞÞ mass centroid at 10017.20 1.8MeV and the calculated value gets reduced to 23 MeV. For the ð1DÞand ð3SÞstates the lack of data prevents the evaluation of the mass centroids TABLE III. Possible values of lb¯ bðsb¯ b¼1Þand ðlMðiÞ 1 ¯ MðiÞ 2 ;s MðiÞ 1 ¯ MðiÞ 2Þfor given values of JPC. A missing entry means that the particular meson-meson configuration cannot form a state with the corresponding quantum numbers. JPC b¯ bB ðsÞ¯ BðsÞBðsÞ¯ B ðsÞB ðsÞ ¯ B ðsÞ 0þþ 1 (0,0) (0,0), (2,2) 1þþ 1 (0,1),(2,1) (2,2) 2þþ 1,3 (2,0) (2,1) (0,2),(2,0),(2,2),(4,2) 1−− 0,2 (1,0) (1,1) (1,0),(1,2),(3,2) DIABATIC DESCRIPTION OF BOTTOMONIUMLIKE MESONS PHYS. REV. D 103, 114016 (2021) 114016-7 for comparison. Instead, we can check that the calculated mass for ð1DÞis pretty close to the measured mass of ϒ2ð1DÞ,10163.71.4MeV, and that the calculated mass of ð3SÞis lower than the measured mass of ϒð3SÞ, 10355.20.5MeV, as should be expected. All the higher excited states contain meson-meson components. However, only for the fifth excited state, which contains a dominant ð70%ÞCornell ð4SÞb¯ bcomponent, we predict a significant meson-meson probability (21% of B¯ B), due to the vicinity of the ð4SÞb¯ bstate and the B¯ Bthreshold. This suggests that for the corresponding experimental resonance the label ϒð10580Þshould be preferred to the PDG alternative ϒð4SÞ. It is illustrative to plot the radial wave function of this state for the several components, see Fig. 5. We see that the presence of meson-meson components makes the radial wave function to extend to larger distance than the Cornell ð4SÞb¯ bone, and correlated with this there is a loss of probability density at the origin (r¼0)as compared to the Cornell case. This could explain the discrepancies observed between the calculated leptonic width ratios in the Cornell model and data. More concretely, the 1−− leptonic width ratios are calculated from [25] Γðϒðn1Þ→eþe−Þ Γðϒðn2Þ→eþe−Þ¼ Rϒðn1Þð0Þ Rϒðn2Þð0Þ 2m2 ϒðn2Þ m2 ϒðn1Þð24Þ where RϒðnÞð0Þstands for the radial wave function at the origin and mϒðnÞfor the mass of the ϒstate containing a ðnSÞb¯ bcomponent. The calculated values for these ratios and their comparison to data are given in Table V(the use of the experimental masses instead of the calculated ones would not make any difference). A look at the table makes clear that (i) the ratios involving Γðϒð1SÞ→eþe−Þ, are deficiently described by both the Cornell model (with the exception of Γðϒð10580Þ→eþe−Þ Γðϒð1sÞ→eþe−ÞÞand the diabatic approach, (ii) the diabatic values for these ratios can be put in accord with data through multiplication by a common factor of ≃1.3, (iii) all TABLE IV. Calculated masses, b¯ band meson-meson component probabilities, for JPC bottomoniumlike bound state solutions. Vanishing and negligible (i.e., inferior to 1%) probabilities are not displayed. JPC Mass (MeV) b¯ bB ¯ BB ¯ BB¯ BBs¯ BsBs¯ B sB s¯ B s 0þþ 9900.7 100% 10254.1 100% 10530.2 91% 8% 1% 10778.1 98% 2% 1þþ 9900.7 100% 10254.2 100% 10532.1 97% 3% 10775.1 75% 24% 1% 2þþ 9900.7 (100, 0)% 10253.9 (100, 0)% 10340.9 (0, 100)% 10527.7 (92, 2)% 3% 1% 2% 10592.7 (2, 91)% 3% 4% 10776.2 (91, 1)% 4% 4% 1−− 9401.2 (100, 0)% 9993.8 (100, 0)% 10150.3 (0, 100)% 10337.2 (100, 0)% 10439.4 (0, 99)% 1% 10598.8 (70, 3)% 21% 6% 10691.3 (0, 98)% 1% 1% FIG. 5. Radial wave function of the fifth excited 1−− state. b¯ bð4SÞ,b¯ bð3DÞ,B¯ BðlB¯ B¼1Þ, and B¯ BðlB¯ B¼1Þcomponents are drawn with a solid, dashed, dotted, and dash-dotted line respectively. R. BRUSCHINI and P. GONZÁLEZ PHYS. REV. D 103, 114016 (2021) 114016-8 the ratios not involving Γðϒð1SÞ→eþe−Þare well described by the diabatic approach whereas the Cornell model is far from data except Γðϒð3sÞ→eþe−Þ Γðϒð2sÞ→eþe−Þfor which there is no difference in the calculated wave functions with both approximations. These results suggest that the failure of the diabatic approach regarding the ratios involving Γðϒð1SÞ→eþe−Þ may have to do with the presence of relativistic corrections in ϒð1SÞmaking its radial wave function at the origin to decrease a 14%. Therefore, we may tentatively conclude that data from leptonic widths can be taken as an indication of the mesonmeson compositeness of ϒð10580Þ. In this regard, it is also interesting to add that it is the b¯ bð4SÞ−B¯ Binteraction the main physical mechanism underlying the explanation of the leptonic widths. The small Dmixing, 3% of b¯ bð3DÞ, which is mainly induced through the Sand Dcoupling to B¯ B, plays a marginal quantitative role. Moreover, other sources of S−Dmixing such as a direct tensor interaction within the Cornell potential, should also have a quite limited importance in order to preserve the accurate leptonic width description. This is in contrast to other explanations in the literature based on a significant S−D mixing, see for instance [26]. The calculated sixth excited state has a predicted mass close to 10700 MeV and it is very dominantly a b¯ bð3DÞ state. It could be possibly assigned to the not well established ϒð10753Þwith a measured mass of 10752.75.9þ0.7 −1.1MeV. However, the discovery channel ϒðnSÞπþπ−with n¼1, 2, 3 is not expected to be a dominant channel for a ð3DÞstate suggesting that some mixing with the b¯ bð4SÞstate is lost. This can be due to the fact that the B¯ B, and particularly the B¯ Bthresholds, which according to our previous discussion can give rise to this mixing, are not taken into account in our bound state approach for the sixth excited state since they are open meson-meson channels. Indeed, we shall show later on that this excitation has a prominent width to B¯ B. Hence, a significant D−Smixing through the coupling to B¯ B could be present. The theoretical description of this mixing would require a complete (scattering) solution of the problem which is out of the scope of our current analysis. B. Mass corrections and widths Let us realize that for pure Cornell states, with masses below the first meson-meson threshold, there are no open meson-meson channels. Hence, no widths, and mass corrections being mostly limited to spin splittings which are known to be quantitatively important for these states. Indeed, the derivation of the form of the spin dependent potential terms from QCD, and a numerical evaluation of the Cornell spin splittings for b¯ bwas carried out forty years ago [27]. We simply copy here those results for the slightly different values we use for the parameters of the Cornell potential hardly makes a difference. The corrected masses for the pure Cornell states in Table IV are giveninTableVI. We see that a very good mass description is obtained. The biggest difference between the calculated mass and data is of 35 MeV for ϒð1SÞwhich we may attribute, at least partially, to further relativistic (kinetic energy) effects. To proceed to a similar evaluation of spin splitting for states with meson-meson components, spin dependent terms of the mixing and meson-meson potentials should be taken into account as well. However, the complete lack of knowledge of the spin dependence in the mixing potential prevents carrying out this procedure. Instead, for states with mass above the first meson-meson threshold, we can evaluate mass corrections and widths from the open meson-meson thresholds neglected in the bound state calculation. The nonperturbative method we follow for this evaluation has been explained elsewhere, see [14] and references therein. Let us only recall here that the physical effect of the coupling to the continuum is to dilute the bound state through a band of stationary scattering states, giving rise to a resonance. If we call mbs the mass of the bound state, MðjÞ 1¯ MðjÞ 2with j¼1;2…;nthe corresponding TABLE V. Calculated leptonic width ratios from the Cornell model and the diabatic approach, as compared to data from [20]. Leptonic Width Ratio Cornell Experiment Diabatic Γðϒð2sÞ→eþe−Þ Γðϒð1sÞ→eþe−Þ0.36 0.456 0.14 0.36 Γðϒð3sÞ→eþe−Þ Γðϒð1sÞ→eþe−Þ0.25 0.33 0.10.25 Γðϒð10580Þ→eþe−Þ Γðϒð1sÞ→eþe−Þ0.21 0.20 0.02 0.14 Γðϒð3sÞ→eþe−Þ Γðϒð2sÞ→eþe−Þ0.70 0.72 0.03 0.70 Γðϒð10580Þ→eþe−Þ Γðϒð2sÞ→eþe−Þ0.58 0.44 0.06 0.39 Γðϒð10580Þ→eþe−Þ Γðϒð3sÞ→eþe−Þ0.82 0.61 0.08 0.56 TABLE VI. Spin splittings (in MeV) for pure Cornell spin triplet states. The corrected masses (in MeV) and their comparison to the measured masses [20] of the assigned mesons are also shown. JPCðnlÞSplitting Mass Experiment Meson 1−− ð1SÞ23.7 9424.9 9460.30 0.26 ϒð1SÞ 0þþð1PÞ−35.89864.9 9859.44 0.42 0.31 χb0ð1PÞ 1þþð1PÞ−11.09889.7 9892.78 0.26 0.31 χb1ð1PÞ 2þþð1PÞ13.8 9914.5 9912.21 0.26 0.31 χb2ð1PÞ 1−− ð2SÞ10.3 10004.1 10023.26 0.31 ϒð2SÞ 1−− ð1DÞ−3.310147.0 10163.71.4ϒ2ð1DÞ 0þþð2PÞ−26.410227.7 10232.50.40.5χb0ð2PÞ 1þþð2PÞ−8.210246.0 10255.46 0.22 0.50 χb1ð2PÞ 2þþð2PÞ10.2 10264.1 10268.65 0.22 0.50 χb2ð2PÞ 1−− ð3SÞ7.8 10345.0 10355.20.5ϒð3SÞ DIABATIC DESCRIPTION OF BOTTOMONIUMLIKE MESONS PHYS. REV. D 103, 114016 (2021) 114016-9