Computation of the quarkonium and meson-meson composition of the $\Upsilon(nS)$ states and of the new $\Upsilon(10753)$ Belle resonance from lattice QCD static potentials

We compute the composition of the bottomonium $\Upsilon(nS)$ states (including $\Upsilon(10860)$) and the new $\Upsilon(10753)$ resonance reported by Belle in terms of quarkonium and meson-meson components. We use a recently developed novel approach utilizing lattice QCD string breaking potentials for the study of resonances. This approach is based on the Born Oppenheimer approximation and the unitary emergent wave method and allows to compute the poles of the $\mbox{S}$ matrix. We focus on $I=0$ bottomonium $S$ wave bound states and resonances, where the Schr\"odinger equation is a set of coupled differential equations. One of the channels corresponds to a confined heavy quark-antiquark pair $\bar b b$, the others to pairs of heavy-light mesons. In a previous study only one meson-meson channel $\bar{B}^{(\ast)} B^{(\ast)}$ was considered. Now we also include the closed strangeness channel $\bar{B}_s^{(\ast)} B_s^{(\ast)}$ extending our formalism significantly to have a more realistic description of bottomonium. We confirm the new Belle resonance $\Upsilon(10753)$ as a dynamical meson-meson resonance with around $85 \%$ meson-meson content. Moreover, we identify $\Upsilon(4S)$ and $\Upsilon(10860)$ as states with both sizable quarkonium and meson-meson contents. With these results we contribute to the clarification of ongoing controversies in the vector bottomonium spectrum.


I. INTRODUCTION
Starting from the determination of lattice QCD static potentials with dynamical quarks, our long term goal is a complete computation of masses and decay widths of bottomonium bound states and resonances as poles of the S matrix. We expect our technique to be eventually updated to study the full set of exotic , and mesons. In this work, however, we focus on the somewhat simpler, but nevertheless controversial = 0 bottomonium wave resonances. It has been standard for many years to use the ordinary static potential obtained in lattice QCD as the confining quarkonium potential to study the heavy quarkonium spectrum [1]. Recently we developed a novel approach to apply lattice QCD string breaking potentials (as e.g. computed in Ref. [2]) to coupled channel systems, opening the way for the computation of the spectrum and the composition of resonances with heavy quarks [3]. This approach allows to study hadronic interactions non-perturbatively with input from first principles lattice QCD.
In the past, microscopic determinations of hadronic strong interactions, i.e. from quark interactions, were mostly addressed with quark models. From the onset of QCD, while developing the quark bag model, Jaffe predicted multiquarks such as tetraquarks. Moreover, he started computing microscopically potentials for coupled channels of hadrons [4,5]. * bicudo@tecnico.ulisboa.pt † nuno.cardoso@tecnico.ulisboa.pt ‡ lmueller@itp.uni-frankfurt.de § mwagner@itp.uni-frankfurt. de An important type of potentials are the hadron-hadron potentials, such as¯we use in this work, where the number of quarks is preserved. The microscopic computation of hadronhadron potentials in models needs to include the different algebraic color, flavor, spin and three dimensional space or momentum factors, the latter being systematically performed with the resonating group method [6]. A second type of potentials are the mixing potentials, such as the mix used in this work, which couples channels with a different number of quarks and microscopically includes the creation or annihilation of a light quark-antiquark pair. In quark models it was realized a long time ago from the symmetries the QCD vacuum that the pair creation has = 3 0 quantum numbers in spectroscopic notation [7,8]. Spontaneous chiral symmetry breaking can also be included in the 3 0 pair creation mechanism [9]. This microscopic knowledge contributed to very successful models with quark-antiquark channels and meson-meson channels allowing to study a large number of resonances [10], complicated dynamical resonances [11] and resonances with heavy quarks [12]. These models share a vanishing interaction between two mesons, but have different mixing potentials. For instance Ref. [11] uses a delta-shell potential for simplicity and Ref. [12] uses a Gaussian shell potential with has an additional parameter. Refs. [9,10] compute microscopically mix from the overlap of the meson wave functions and the 3 0 pair creation term.
However, since multiquarks may be very complex systems, we expect some of them to be sensitive to the details of the potentials. Unfortunately, the potentials are not fully fixed by the symmetries of QCD, i.e. what one can infer is only qualitative. For example mix at large quark-antiquark separations must decay rapidly like the meson wave functions in arXiv:2008.05605v5 [hep-lat] 30 May 2022 the overlap, exponentially or as a Gaussian. At small separations, due to the momentum or position present in the 3 0 mechanism, it must linearly increase from 0. Moreover, due to the parity in the 3 0 mechanism, the orbital angular momenta of the quark-antiquark channel and the meson-meson channel must differ by 1. Obviously, these constraints do not fully determine the potentials and there are certain degrees of freedom left. Lattice QCD, on the other hand, is a method to unambiguously compute these potentials from QCD. Thus, we expect that the computation of the potentials¯(the quark-antiquark potential), mix and¯with lattice QCD will allow to clarify important aspects of certain quarkonium resonances and experimentally observed multiquarks.
Lattice QCD computations fully incorporate the dynamics of the light quarks and gluons, while the heavy quarks are approximated as static color charges. The dynamics of the heavy quarks is then added in a second step using techniques from quantum mechanics as in the Born-Oppenheimer approximation [13]. Due to heavy quark symmetry the spin of the heavy quarks is conserved [14][15][16][17]. In previous works this Born-Oppenheimer approach was applied to investigate exotic mesons containing a bottom and an anti-bottom quark. For example, the spectrum of¯hybrid mesons was studied extensively (see e.g. Refs. [18][19][20][21]) using static potentials computed within pure SU(3) lattice gauge theory, which are confining and do not allow decays to pairs of lighter mesons. The first application of the Born-Oppenheimer approach using meson-meson potentials computed in full lattice QCD to study tetraquarks can be found in Refs. [22,23]. For instance the existence of a stable¯¯tetraquark with quantum numbers ( ) = 0(1 + ) was confirmed [24,25], whereas other flavor combinations do not seem to form four-quark bound states [26]. In this context the approach was also updated by including techniques from scattering theory and a¯¯tetraquark resonance with quantum numbers ( ) = 0(1 − ) was predicted [27]. The Born-Oppenheimer approach approach should also allow for the inclusion of the heavy quark spin, either from the experimental hyperfine splitting [25] or with lattice QCD computations of 1/ corrections to the static potentials [28]. In this work we continue within this framework and significantly extend our recent study of systems with a heavy quark and a heavy antiquark and possibly another light quarkantiquark pair [3]. This constitutes an even more challenging system, which might open the way to study bottonomium , and resonances. The approach then requires the lattice QCD determination of several potentials including a¯potential, a¯( * ) ( * ) potential and a mixing potential, and allows the study of resonances and their decays with scattering theory. In this work we do not carry out such lattice QCD computations, but use results from an existing study of string breaking [2]. Notice that we go beyond the Born-Oppenheimer adiabatic approximation [19], since the quarkonium potential crosses open meson-meson thresholds. Formally, our system is then denominated diabatic [12,29,30], since the heavy quarks are much slower that the light degrees of freedom, but the state of the heavy quarks nevertheless changes, when a decay occurs.
The main goal of this paper is to compute the composition of = 0 bottonomium resonances in terms of quarkonium  (1 ) 0 + (2 ++ ) 9912.21 ± 0.57 Table I. Masses and decay widths Γ of = 0 bottomonium according to the Review of Particle Physics [31]. We also include several states observed at Belle [32,33], but not yet confirmed by other experiments. We add an extra column with the quantum numbers conserved in the infinite quark mass limit (in the last three lines = 2 ++ is also a possibility). We mark with horizontal lines the opening of the¯and¯ * * thresholds.
, and meson-meson¯components. It is important to note that in Ref. [3] only a¯( * ) ( * ) meson-meson channel was included. However, since the closed strangeness¯( * ) ( * ) channel is very close to the bottomomium resonances we are interested in, we also include this channel in the present work. Clearly this case is more involved, because there are three coupled channels, a confined quarkonium channel with fla-vor¯and the two meson-meson decay channels with flavor (¯+¯)/ √ 2 and¯¯. In Table I we show the available experimental results according to the Review of Particle Physics [31]. Since we work in the heavy quark limit, the heavy quark spins do not appear in the Hamiltonian and the relevant quantum numbers are the remaining part of the total angular momentum and the corresponding parity and charge conjugation (also listed in Table I). Notice that we also list several states observed at Belle with large significance [32,33]. These states are not yet confirmed by other experiments, because presently Belle and Belle II are the only experiments designed to study bottomonium.
In particular a new resonance, Υ(10753), possibly another Υ( ) state or a state, since it is a vector but suggested to be of exotic nature, has recently been observed at Belle with a mass around 10.75 GeV [33]. The previously observed resonances Υ(4 ) and Υ(10860) approximately match quark model predictions of bottomonium and, thus, this new resonance comes in excess and needs to be understood.
Notice also that the discovery of this resonance by Belle with the process + − → Υ( ) + − resulted from the exper-imental effort to clarify the controversy on the nature of the other excited Υ resonances [33]. The Υ(4 ), Υ(10860), and Υ(11020), although having masses approximately compatible with the quark model, have transitions to lower bottomonia with the emission of light hadrons with much higher rates compared to expectations for ordinary bottomonium. A possible interpretation is that these excited Υ states have large admixtures of¯( * ) ( * ) meson pairs [34][35][36][37]. Another scenario is that they do not correspond to the wave states Υ(5 ) and Υ(6 ), but instead to the wave states Υ(3 ) and Υ(4 ) [38][39][40]. The Belle experiment was, thus, designed to produce and study Υ states with a large¯( * ) ( * ) admixture.
After the observation of the new resonance at Belle, more exotic interpretations have been proposed for the excited Υ states. Most interpretations consider the new Υ(10753) resonance as a non-conventional state, e.g. a tetraquark [41,42] or a hybrid meson [43][44][45]. There are, however, also different interpretations, e.g. in Ref. [39] it is claimed that the Υ(4 ) is not a simple quarkonium state.
In this work, we aim to contribute to the clarification of the controversies concerning the bottomonium resonances Υ(4 ), Υ(10753) and Υ(10860). While the low-lying bottomonium spectrum up to the¯threshold was studied within full lattice QCD extensively [46][47][48][49][50][51][52][53], it is extremely difficult to investigate higher resonances in a similar setup, in particular those with several decay channels. Thus, as already explained above, we continue our recent work [3] using lattice QCD potentials and applying the emergent wave method to study = 0 bottomonium wave resonances. Using this strategy, independently of the experimental observation of the resonance Υ(10753) at Belle [33], which we were not aware of at that time, we predicted a similar resonance with mass 10774 +4 −4 MeV [3]. We now extend this work including another important mesonmeson channel, the¯( * ) ( * ) channel, with threshold between the Υ(10753) and Υ(10860). Within this improved setup we determine the composition of all bound states and resonances up to ≈ 11 GeV, i.e. the percentage of a pair of confined heavy quarks¯as well as the percentage of a pair of heavy-light mesons¯( * ) ( * ) and¯( * ) ( * ) . This paper is structured as follows. In Section II we review the theoretical basics of our approach from Ref. [3]. We discuss, how to utilize lattice QCD static potentials, and how to solve the coupled Schrödinger equation to obtain a quarkonium and one or two meson-meson wave functions. We also review our results for the poles of the S matrix, i.e. for = 0 bottomonium wave resonances. In Section III we propose a technique to determine the percentage of the quark-antiquark and the meson-meson component of a bottomonium state, either a bound state (if we neglect heavy quark annihilation and electroweak interactions) or a resonance. Then we apply this technique to Υ(1 ), Υ(2 ), Υ(3 ), Υ(4 ), Υ(10753) and Υ(10860). In Section III we also discuss results within the two channel setup, i.e. considering quarkonium¯and a meson pair¯( * ) ( * ) , and in Section IV we discuss results within the three channel setup, i.e. with an extra¯( * ) ( * ) channel. In Section V we conclude.

II. SUMMARY OF OUR APPROACH
In this section we briefly summarize our approach from Ref. [3] to study quarkonium resonances with isospin = 0 in the diabatic extension of the Born-Oppenheimer approximation, using lattice QCD static potentials. We also recapitulate the main results from Ref. [3]. Moreover, we extend the approach to three coupled channels, including a¯( * ) ( * ) channel.

A. Theoretical basics -two coupled channels
We consider systems composed of a heavy quark-antiquark pair¯and either no light quarks (quarkonium) or another light quark-antiquark pair¯with isospin = 0 (for larges eparation two heavy-light mesons =¯and¯=¯). We treat the heavy quark spins as conserved quantities such that the energy levels of¯(¯) systems as well as their decays and and resonance parameters do not depend on these spins. Moreover, we assume that two of the four components of the Dirac spinors of the heavy quarks and¯vanish. These approximations become exact for static quarks and are expected to yield reasonably accurate results for quarks, possibly even for quarks.
In Ref. [3] we have derived in detail a coupled channel Schrödiger equation for a 4-component wave function (r) = (¯(r), ì¯( r)) (Eq. (10) in Ref. [3]). The upper component of this wave function represents the¯channel, the lower three components represent the¯channel. For thē channel we consider only the lightest heavy-light mesons with = 0 − and = 1 − , i.e. and * mesons for ≡ (as usual, , and denote total angular momentum, parity and charge conjugation). Within the approximations stated above these two mesons have the same mass. One can show that the spin of the two light quarks is 1, which is represented by the three components of ì¯( r). Note that we ignore decays of to lighter quarkonium and a light = 0 meson, e.g. a or an meson, because they are suppressed by the OZI rule.
denotes total angular momentum excluding the heavy quark spins and the corresponding parity and charge conjugation. It is a conserved quantity. As in Ref. [3] we focus throughout this work on = 0 ++ . Thus = , where denotes the heavy quark spin, with only two possibilities, = 0 −+ , 1 −− . The coupled channel Schrödinger equation for the partial wave with = 0 is a 2-channel equation, (1) The upper equation represents the¯channel with orbital angular momentum¯= = 0. ( ) is the radial part of the = 0 partial wave of the wave function with the dots . . . denoting partial waves with > 0. Similarly, the lower equation represents the¯channel with orbital angular momentum¯= 1. 1 ( ) and¯( ) are the radial parts of the = 0 partial waves of the incident plane wave and the emergent spherical wave of the 3-component wave function with Z¯(Ω) = e / √ 4 and the dots . . . denoting partial waves with > 0. Moreover, and are the heavy quark and heavy-light meson masses, respectively, and = /2 and = /2 are the corresponding reduced masses. The energy and the momentum are related according to = √︁ 2 . The potentials¯( ),¯, ( ) and mix ( ) represent the energy of a heavy quark-antiquark pair, the energy of a pair of heavy-light mesons and the mixing between the two channels, respectively. In Ref. [3] we related these potentials algebraically to lattice QCD correlators computed and provided in detail in Ref. [2] in the context of string breaking for lattice spacing ≈ 0.083 fm and pion mass ≈ 650 MeV. The data points for¯( ),¯, ( ) and mix ( ) are shown in Fig. 1 together with appropriate parameterizations, The parameters appearing in Eq. (4) to Eq. (6) are collected in Table II. It is interesting to compare our potentials to those utilized in quark models. The models of Refs. [10][11][12] all have a con-fining¯( ) and our lattice QCD potential is also confining. This is not surprising, since confinement is a central feature of most quark models. However, it is remarkable that the mesonmeson interaction¯, ( ) obtained from lattice QCD correlators is compatible with zero within error bars and at the same time all these models have no direct meson-meson interaction as well. In the case of the models this is a simplification, but in our case it is a first principles QCD result. Such a vanishing  meson-meson interaction is not universal. It appears in the coupled channel = 0 bottomonium system, but for instance not in the¯¯system, where a significant attraction leads to a tetraquark boundstate [22,54]. In what concerns the mixing potential, the lattice QCD result mix ( ) has a richer structure than those used in Refs. [11,12], which are non-vanishing only in a certain region of close to the string breaking distance . We note again that our mix ( ) is a first principles QCD result and that there is no physical or phenomenological reason, why this potential should not have the behavior shown in Fig. 1. It vanishes at large , as in the case of the models, but extends to much smaller quark-antiquark separations than the potentials of Refs. [11,12]. Indeed, the lattice QCD result mix ( ) is close to those calculated microscopically with the 3 0 mechanism of Refs. [9,10]. Finally we notice that there is a small but clearly visible bump in¯( ) at ≈ 0.25 fm, which is typically not present in quark model potentials. This bump is a consequence of the non-vanishing mixing between energy eigenstates on the one hand and¯and¯states on the other hand. With lattice QCD the ground state and the first excitation are computed as functions of , where the ground state corresponds to a confining potential without a bump at small (see e.g. Fig. 13 in Ref. [2], the curve labeled "state |1 "). Lattice QCD also provides the mixing angle, i.e. the contribution of the ground state and the first excitation to the¯and states. This mixing moves¯( ) and¯, ( ) closer together for non-vanishing mixing angle. The mixing angle is particularly large at separations ≈ 0.25 fm (see Fig. 15 in Ref. [2]) as also indicated by the extremum in the mixing potential mix ( ). Thus, the mixing generates a bump in¯( ) and removes a similar bump present in the first excitation (see Fig. 13 in Ref. [2], the curve labeled "state |2 ") leading to a essentially vanishing meson-meson interaction¯, ( ).
The appropriate boundary conditions for the radial wave functions ( ) and¯( ) are where ℎ (1) 1 is a spherical Hankel function of the first kind and¯is the scattering amplitude and an eigenvalue of the S matrix. We compute¯as a function of the complex energy . Poles of¯on the real axis below thet hreshold indicate bound states. Poles of¯at energies with non-vanishing negative imaginary parts represent resonances with masses = Re( ) and decay widths Γ = −2Im( ). is also related to the corresponding scattering phase via 2¯= 1 + 2¯.

B. Main results from Ref. [3] -two coupled channels
In Ref. [3] we applied our approach to study bottomonium bound states and resonances with = 0. For , which is the energy reference of our system, we use the spin-averaged mass of the meson and the * meson, i.e.
= /2 in the kinetic term of the coupled channel Schrödinger equation (1) is the reduced mass of the quark. Since results are only weakly dependent on (see previous works following a similar approach, e.g. Refs. [24,55]), we use for simplicity = 4.977 GeV from quark models [1].
In Ref. [3] we presented both the scattering amplitudeā nd the phase shift¯for real energies above the¯( * ) ( * ) threshold at 10.627 GeV (throughout this paper we use a notation slightly different from that in Ref. [3],¯≡ 1→0,0 and ≡ 1→0,0 ). We also checked probability conservation by showing the Argand diagram for¯. The main numerical results of Ref. [3] are, however, the poles of¯in the complex energy plane, which are shown in Fig. 2 (upper plot) and collected in Table III. There are four poles on the real axis below the¯( * ) ( * ) threshold representing bound states ( = 1, . . . , 4 in Table III). By comparing them to the experimental results from Table I, we identify them with (1 ) ≡ Υ(1 ), Υ(2 ), Υ(3 ) and Υ(4 ). We also obtained a resonance around 10.895 GeV, which matches Υ(10860) with experimentally found mass (10.885 ± 0.002) GeV rather well ( = 6 in Table III). Moreover, in Ref. [3] we predicted a new, dynamically generated resonance close the the¯( * ) ( * ) threshold with mass around 10.774 GeV ( = 5 in Table III). Recently Belle has observed a bottomonium state at (10.753 ± 0.007) GeV denoted as Υ(10753) not yet confirmed by other experiments, which could correspond to our prediction.
However, for the = 5 and = 6 states, which are close in energy to Υ(10753) and Υ(10860), it should be important to also include the¯( * ) ( * ) channel, since its threshold opens between these two states. Thus we proceed by studying three coupled channels and compare the results with those obtained in the two channel case. This will provide insights, how important meson-meson thresholds and the corresponding channels are for resonance properties. We note that to obtain reliable and realistic masses and widths for resonances above ≈ 11.025 GeV, which is the threshold of one heavy-light meson with negative parity and another with positive parity, one has to include even further meson-meson channels.

C. Extension to the three coupled channel case
We include the¯( * ) ( * ) channel in Eq. (1) using the same string breaking potentials as before, i.e. those provided by Ref. [2]. We expect this is to be a reasonable approximation, because the mass of the light quarks in Ref. [2] is between the physical / and the physical quark mass. Thus we use the same mixing potential for both channels.
Moreover, the direct interaction between the static-light meson pairs in Eq. (1) turned out to be negligible in the two coupled channel case (see Fig. 1 and the detailed discussion in Ref. [3]). Thus we use vanishing meson-meson interactions also in the case of three coupled channels, since one can hardly anticipate a mechanism that increases the meson-meson interaction either between a ( * ) and a¯( * ) or in the transition between a¯( * ) ( * ) and a¯( * ) ( * ) .
In detail we extend the potential matrix to three coupled channels as follows. From the Review of Particle Physics we get 0 = 5.367 GeV (1 spin state) and * = 5.415 GeV (3 spin states). Using spin symmetry, the average is 5.403 GeV. The¯( * ) ( * ) threshold opens at 10.807 GeV, indeed between the new Υ(10753) and the Υ(10860).
One can estimate the quark mass used in in the lattice QCD computation of Ref. [2] using a theorem of Partially Conserved Axial Currents (PCAC), applicable to the light quarks , and . According to the Gell-Mann, Oakes and Renner relation [56] two coupled channels: quarkonium and¯( * ) ( * )

Re(E) [GeV]
Im(E) [GeV] n=1 n=2 n=3 n=4 n=5 n=6 n=7 three coupled channels: quarkonium,¯( * ) ( * ) and¯( * ) ( * )  threshold at 10.807 GeV. The shaded region above 11.025 GeV marks the opening of the threshold of one heavy-light meson with negative parity and another with positive parity, beyond which our results should not be trusted. We also mark with a vertical line threshold = 10.790 GeV, which corresponds to two times the mass of a static-light meson from Ref. [2]. from poles of¯, two channels from poles of T, three channels from experiment     (1) and (11) [2] is even closer to the quark mass than to the physical / quark mass. As stated above, we use the mixing potential obtained from the lattice QCD correlators of Ref. [2] for both the¯( * ) ( * ) and the¯( * ) ( * ) channel.
Another point is a possibly different algebraic factor for the mixing potential of the new¯( * ) ( * ) channel. We note that the mixing potential is proportional to the lattice QCD creation operator O Σ + (see Eqs. (14) and (18) in Ref. [3]). For two degenerate flavors and this operator is composed of two terms of identical form (one for each flavor), but has to be normalized by another factor 1/ Finally, we have to take into account the meson-meson threshold of Ref. [2] corresponding to two times the static-light meson mass, threshold = 2 ( * ) . In the case of two channels we identified threshold with 10.627 GeV, which is the physical ( * ) ( * ) threshold. However, now using Ref. [2] = 22.5 × and performing a linear interpolation between the spin averaged masses of the ( * ) meson and the ( * ) meson, we find threshold = 10.790GeV. Thus, the Schrödinger equation for the partial wave with = 0 in the case of three coupled channels is The incident wave can be any linear superposition of a¯( * ) ( * ) wave and a¯( * ) ( * ) wave, where and denote the respective coefficients. For example, a pure¯( * ) ( * ) wave translates into ( , ) = (1, 0) and a pure¯( * ) ( * ) wave into ( , ) = (0, 1). The momenta of these waves, and , are related to via The corresponding boundary conditions of the wave functions are the following: • In both cases (i.e. ( , ) = (1, 0) and ( , ) = (0, 1)): • Incident¯( * ) ( * ) wave (i.e. To determine masses and decay widths of bound states and resonances, we need to find the poles of the S matrix or, equivalently, of the T matrix. We use similar techniques as in our previous work [3], but this time we apply the pole search to the determinant of the T matrix. It is an interesting consistency check to compare our 3 × 3 potential matrix to a recent lattice QCD computation of string breaking with dynamical , and quarks [57]. For a meaningful comparison we need diagonalize our 3×3 potential matrix. The resulting diagonal elements, which are shown as functions of in Fig. 3, should correspond to the three lowest energy levels of a system with a static quark-antiquark pair and dynamical , and quarks. As expected, they are similar to those plotted in Fig. 1 of Ref. [57]. Note that there is a certain discrepancy in the second excitation at small separations. The bump we obtain and which is not present in Fig. 1 of Ref. [57], could have different reasons. It might be a consequence of different light quark masses or of the dynamical strange quark used in Ref. [57] compared to the computation from Ref. [2] or also of imperfect operator optimization. It could also be that our assumptions to set up the 3 × 3 potential matrix in Eq. (11) from the 2-flavor lattice QCD results from Ref. [2] are only partly fulfilled. As we discuss below in our conclusions, we plan to carry out dedicated lattice QCD computations of the relevant potentials in the near future, where we can possibly clarify this tension. For our current work we use the lattice QCD results of Ref. [2], because numerical values are provided for all required quantities (see Table I in Ref. [2]). In Ref. [57], even though more recent, certain quantities important for our formalism, e.g. the mixing angle as a function of , seem not to have been computed.

III. QUARKONIUM AND MESON-MESON CONTENT OF = 0 BOTTOMONIUM -TWO COUPLED CHANNELS
We continue or investigation of bottomonium bound states and resonances with isospin = 0 by studying their structure and quark content. In particular we explore, whether the bound states and resonances close to the¯( * ) ( * ) threshold, i.e. states with = 4, 5, 6 in Table III, which could correspond to the experimentally observed Υ(4 ), Υ(10753) and Υ(10860), are conventional¯quarkonia, or whether there is a sizable¯¯four-quark component. For clarity, we first consider the case of two coupled channels, where it is easier to define the concepts of our study. Then, in Section IV, we will move on to the case of three coupled channels, which is physically more realistic.
We inspect in detail the percentages of quarkonium and of a meson-meson pair present in each of the bound states and resonances. To this end we compute ( ) and¯( ) are the radial wave functions of the¯and the¯channel, respectively, obtained by solving the coupled channel Schrödinger equation (1) with energies identical to the real parts of the corresponding poles.

Bound states
For bound states < 2 and the corresponding momentum is complex, = √︁ |2 ( − 2 )|. The boundary condition (10) for¯( ) simplifies tō Thus, both and are independent of max , if chosen sufficiently large, i.e. max > ∼ 2.0 fm, because also ( ) = 0 for → ∞ (see Eq. (8)). The same is true for %¯and %¯, which represent the probabilities to either find the system in a quarkonium configuration or in a meson-meson configuration.

Resonances
For resonances things are more complicated. First, resonances are defined by poles in the complex energy plane with non-vanishing negative imaginary parts of . Evaluating %¯and %¯at such a complex energy does not seem to be meaningful, because | ( )| 2 / 2 and |¯( )| 2 / 2 are only proportional to probability densities, if is real. Thus we compute %¯and %¯at the real part of the corresponding pole position, Re( ), which is the resonance mass.
There is, however, another complication, namely that is not constant but linearly rising for large max . The reason is that¯( ) represents an emergent wave (see Eq. (10)). We found, however, the dependence of %¯and %¯on max to be rather mild, with an uncertainty of only a few percent in the range 1.8 fm ≤ max ≤ 3.0 fm, i.e. where the quarkonium component is already negligible, ( = max ) ≈ 0. Thus, we interpret %¯and %¯as estimates of probabilities to either find the system in a quarkonium configuration or in a meson-meson configuration, as for the bound states discussed before.

Numerical results
We show plots of %¯and %¯as functions of max for the first seven bottomonium bound states and resonances in Fig. 4.
For the resonances there is a dependence of %¯and %¯on max , but it is rather mild with an uncertainty of 2% or less in the range 1.8 fm ≤ max ≤ 3.0 fm (see also the discussion in Section III 2). The wide resonance with = 5 has %¯≈ 94% and, thus, is essentially a mesonmeson pair. The resonance with = 6 is a mix of quarkonium and a meson-meson pair with slightly larger¯component (%¯≈ 59%, %¯≈ 41%). Resonances with ≥ 7 are above the threshold of one heavy-light meson with negative parity and another with positive parity. Since this decay channel is currently neglected, their decay widths are tiny and they are almost stable. Correspondingly, they are strongly quarkonium dominated, i.e. %¯ %¯. We stress that results for ≥ 7 should not be trusted until all relevant decay channels are included. %¯and %¯for max = 2.4 fm are listed in Table III together with their statistical errors and, for the resonances, also systematic uncertainties. To estimate statistical errors, we utilize the same 1000 sets of parameters as in Ref. [3], which were generated by resampling the lattice QCD correlators from Ref. [2]. Asymmetric statistical errors are defined via the 16th and 84th percentile of the 1000 samples. We visualize these errors as error bands on %¯and %ī n Fig. 4. We define the asymmetric systematic uncertainties as |%¯( max = 1.8 fm) − %¯( max = 2.4 fm)| and |%¯( max = 3.0 fm) − %¯( max = 2.4 fm)| and in the same way for %¯. They are around 2% for the resonances with = 5 and = 6, respectively, and negligible for all other . The total uncertainties on %¯and %¯are rather small. Thus, our predictions concerning the structure of the bound states and resonances are quite stable within our framework. The columns "%¯" and "%¯" in Table III represent the main results for case of two coupled channels, since these numbers reflect the quark composition of the bound states and resonances and clarify, which states are close to ordinary quark model quarkonium, and which states are dynamically generated by a meson-meson decay channel.

IV. QUARKONIUM AND MESON-MESON CONTENT OF = 0 BOTTOMONIUM -THREE COUPLED CHANNELS
We now consider the case of three coupled channels, a quarkonium, a¯( * ) ( * ) and a¯( * ) ( * ) channel. Working with three channels is technically more elaborate than with two, but formally the extension from the case of two channels is straightforward. To identify the bound states and resonances, we apply our pole searching algorithm [3] to the determinant of the T matrix. In Fig. 2 (lower plot) we show the resulting pole positions together with their statistical errors.
Using the real part of a pole energy, we compute the square of the wave functions of the three channels to determine the relative amount of quarkonium, of a¯( * ) ( * ) pair and of ā ( * ) ( * ) pair. Note that a pole in the T matrix corresponds to one infinite eigenvalue, while the second eigenvalue is finite. To make a meaningful statement about a bound state or reso-nance, we thus need to prepare the incident wave in such a way that exclusively the bound state or resonance resonance is generated. This amounts to identifying ( , ) appearing on the right hand side of the coupled channel Schrödinger equation (11) with that eigenvector of T corresponding to the infinite eigenvalue.
This time we compute three quantities, from which we calculate the respective percentages of quarko- We determine the statistical and systematic errors of the percentages using the same techniques as in Section III. The corresponding results are shown in Fig. 5 as functions of max and also summarized in Table III.

Numerical results
We find that in the three channel case, i.e. with a¯( * ) ( * ) and a¯( * ) ( * ) channel, the meson-meson percentage increases for the majority of states compared to the two channel case, which has only a¯( * ) ( * ) decay channel. Nevertheless, the first three states Υ(1 ), Υ(2 ) and Υ(3 ) remain mostly quarkonium states, with %¯around 80% to 85%. The changes appear to be more pronounced for ≥ 4.
The Υ(4 ), which is a bound state in the two-channel case, is now a resonance with a decay width more than twice as large as the experimental result. The reason could be that we neglect the heavy quark spins and, thus, the mass of Υ(4 ) is not only above the¯, but also above the¯ * * threshold. Its¯( * ) ( * ) content %¯≈ 67% is, however, quite similar to the corresponding percentage obtained in the two-channel case.
In what concerns the new state Υ(10753), the inclusion of the¯( * ) ( * ) channel decreases its decay width from a value much larger than the experimental result to a value consistent with experiment. It remains predominantly a¯( * ) ( * ) pair (around 60%), but the quarkonium component increases (to