Computation of the quarkonium and meson-meson composition of the Υ(nS) states and of the new Υ(10753) Belle resonance from lattice QCD static potentials
CComputation of the quarkonium and meson-meson composition of the Υ ( nS ) statesand of the new Υ ( ) Belle resonance from lattice QCD static potentials ( ) Pedro Bicudo, ∗ ( ) Nuno Cardoso, † and ( ) , ( ) Marc Wagner ‡( ) CeFEMA, Dep. Física, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal ( ) Goethe-Universität Frankfurt, Institut für Theoretische Physik,Max-von-Laue-Straße 1, D-60438 Frankfurt am Main, Germany and ( ) Helmholtz Research Academy Hesse for FAIR, Campus Riedberg,Max-von-Laue-Straße 12, D-60438 Frankfurt am Main, Germany
We compute the composition of the bottomonium Υ ( nS ) states (including Υ ( ) ) and the new Υ ( ) resonance reported by Belle in terms of quarkonium and meson-meson components. We use the Born Oppen-heimer approximation, static potentials from a lattice QCD study of string breaking and the unitary emergentwave method to compute the poles of the S matrix. We focus on I = S wave bound states andresonances, where the Schrödinger equation is a set of two coupled differential equations. One of the two chan-nels corresponds to a confined heavy quark-antiquark pair b ¯ b , the other to a pair of heavy-light mesons B (∗) ¯ B (∗) .We confirm the new Belle resonance Υ ( ) as a dynamical meson-meson resonance with 94% meson-mesoncontent. Moreover, we identify Υ ( S ) and Υ ( ) as predominantly quarkonium states, however with sizablemeson-meson contents of 30% and 41%, respectively. With these results we contribute to the clarification ofongoing controversies in the vector bottomonium spectrum. PACS numbers: 12.38.Gc, 13.75.Lb, 14.40.Rt, 14.65.Fy.
I. INTRODUCTION
Starting from lattice QCD static potentials, our long termgoal is a complete computation of the masses and decay widthsof bottomonium bound states and resonances as poles of the Smatrix. We expect our technique to be eventually updated tostudy the full set of exotic X , Y and Z mesons. In this work,however, we focus on the somewhat simpler, but neverthelesscontroversial I = S wave resonances.In Table I we show the available experimental results ac-cording to the Review of Particle Physics [1]. Since we workin the heavy quark limit, the heavy quark spins S PCQ do notappear in the Hamiltonian and the relevant quantum numbers (cid:101) J PC are the remaining part of the total angular momentum andthe corresponding parity and charge conjugation (also listedin Table I). Notice that we also list several states observed atBelle with large significance [2, 3]. These states are not yetconfirmed by other experiments, because presently Belle andBelle II are the only experiments designed to study bottomo-nium.In particular a new resonance, Υ ( ) , possibly another Υ ( nS ) state or a Y state, since it is a vector but suggested to be ofexotic nature, has been recently observed at Belle with a massaround 10 .
75 GeV [3]. The previously observed resonances Υ ( S ) and Υ ( ) approximately match quark model pre-dictions of bottomonium and, thus, this new resonance comesin excess and needs to be understood.Notice also that the discovery of this resonance by Bellewith the process e + e − → Υ ( nS ) π + π − resulted from the ex-perimental effort to clarify the controversy on the nature of ∗ [email protected] † [email protected] ‡ [email protected] name I G ( J PC ) m [MeV] Γ [MeV] (cid:101) J PC η b ( S ) + ( − + ) . ± . ± ++ Υ ( S ) − ( −− ) . ± . ( . ± . ) − ++ χ b ( P ) + ( ++ ) . ± .
73 - 1 −− χ b ( P ) + ( ++ ) . ± .
57 - 1 −− h b ( P ) ? ? ( + − ) . ± . −− χ b ( P ) + ( ++ ) . ± .
57 - 1 −− η b ( S ) Belle + ( − + ) . ± . ++ Υ ( S ) − ( −− ) . ± . ( . ± . ) − ++ Υ ( D ) − ( −− ) . ± . ++ χ b ( P ) + ( ++ ) . ± . −− χ b ( P ) + ( ++ ) . ± .
77 - 1 −− h b ( P ) Belle ? ? ( + − ) . ± . −− χ b ( P ) + ( ++ ) . ± .
72 - 1 −− Υ ( S ) − ( −− ) . ± . ( . ± . ) − ++ χ b ( P ) + ( ++ ) . ± . −− Υ ( S ) − ( −− ) . ± . . ± . ++ Υ ( ) Belle − ( −− ) . ± . . ± . ++ Υ ( ) − ( −− ) . ± . ± ++ Υ ( ) − ( −− ) . ± . ±
15 0 ++ Table I. Masses m and decay widths Γ of I = (cid:101) J PC conserved in the infinite quark mass limit (in the last three lines (cid:101) J PC = ++ is also a possibility). We mark with horizontal lines theopening of the B ¯ B and B ∗ ¯ B ∗ thresholds. the other excited Υ resonances [3]. The Υ ( S ) , Υ ( ) , and Υ ( ) , although having masses approximately compatiblewith the quark model, have transitions to lower bottomoniawith the emission of light hadrons with much higher ratescompared to expectations for ordinary bottomonium. A pos-sible interpretation is that these excited Υ states have large a r X i v : . [ h e p - l a t ] S e p admixtures of B (∗) ¯ B (∗) meson pairs [4–7]. Another scenariois that they do not correspond to the S wave states Υ ( S ) and Υ ( S ) , but instead to the D wave states Υ ( D ) and Υ ( D ) [7–9]. The Belle experiment was, thus, designed to produce andstudy Υ states with a large B (∗) ¯ B (∗) admixture.After the observation of the new resonance at Belle, moreexotic interpretations have been proposed for the excited Υ states. Most interpretations consider the new Υ ( ) reso-nance as a non-conventional state, e.g. a tetraquark [10, 11] ora hybrid meson [12–14].In this work, we aim to contribute to the clarification ofthese controversies on the bottomonium resonances Υ ( S ) , Υ ( ) and Υ ( ) . The low-lying bottomonium spec-trum up to the B (∗) ¯ B (∗) threshold was studied within full latticeQCD extensively [15–22]. However, it is extremely difficult tostudy higher resonances with several decay channels in a simi-lar setup. Thus, we follow a different strategy to study systemswith both heavy and light quarks. In a first step, lattice QCDis used to compute the potential energy for the heavy quarksby simulating the dynamics of the light quarks and gluons.In this work we do not carry out such simulations, but uti-lize lattice QCD static potentials from Ref. [23], which werecomputed in the context of string breaking. Then, in a sec-ond step, the dynamics of the heavy quarks is determined bysolving the Schrödinger equation. Within this so-called Born-Oppenheimer approximation we determine the percentage ofa confined pair of heavy quarks b ¯ b as well as the percentageof a pair of heavy-light mesons B (∗) ¯ B (∗) .This Born-Oppenheimer approach was applied before tostudy exotic mesons containing two bottom quarks. For exam-ple, the spectrum of b ¯ b hybrid mesons was studied extensively(see e.g. Refs. [24–27]), however, mostly using static potentialscomputed within pure SU(3) lattice gauge theory, which areconfining and do not allow decays to pairs of lighter mesons.The first application of this approach to study tetraquarks canbe found in [28, 29]. For instance the existence of a stable¯ b ¯ bud tetraquark with quantum numbers I ( J P ) = ( + ) wasconfirmed [30, 31], whereas other flavor combinations do notform four-quark bound states [32]. In this context the ap-proach was also updated by including techniques from scatter-ing theory and a new ¯ b ¯ bud tetraquark resonance with quantumnumbers I ( J P ) = ( − ) was found [33].Very recently we started to study bottomonium resonances,again using the Born-Oppenheimer approach. This case ismore involved, because there are two coupled channels, a con-fined quarkonium channel with flavor b ¯ b and a meson-mesondecay channel with flavor b ¯ b ( u ¯ u + d ¯ d ) . In Ref. [34] we de-veloped algebraic methods to derive the potentials for the cor-responding Schrödinger equation, including a b ¯ b potential, a B (∗) ¯ B (∗) potential and a mixing potential, from lattice QCDstatic potentials computed e.g. in studies of string breaking[23, 35]. Applying the emergent wave method we determined I = S wave resonances. Independently of theexperimental observation of the resonance Υ ( ) at Belle[3], which we were not aware of at that time, we predicted asimilar resonance with mass 10774 + − MeV [34]. This paper is structured as follows. In section II we reviewthe theoretical basics of our approach from Ref. [34]. Wediscuss, how to utilize lattice QCD static potentials, and how tosolve the coupled Schrödinger equation to obtain a quarkoniumand a meson-meson wave function. We also review our resultsfor the poles of the S matrix, i.e. for I = S waveresonances. In section III we propose a technique to determinethe percentage of the quark-antiquark and the meson-mesoncomponent of a bottomonium state, either a bound state (if weneglect the weak interactions) or a resonance. Then we applythis technique to Υ ( S ) , Υ ( S ) , Υ ( S ) , Υ ( S ) , Υ ( ) and Υ ( ) . Finally, in section IV, we conclude. II. SUMMARY OF OUR APPROACH
In this section we briefly summarize our approach fromRef. [34] to study quarkonium resonances with isospin I = A. Theoretical basics
We consider systems composed of a heavy quark-antiquarkpair ¯ QQ and either no light quarks (quarkonium) or anotherlight quark-antiquark pair ¯ qq with isospin I = QQ separation two heavy-light mesons M = ¯ Qq and ¯ M = ¯ qQ ). We treat the heavy quark spins as conserved quantitiessuch that the energy levels of ¯ QQ ( ¯ qq ) systems as well as theirdecays and and resonance parameters do not depend on thesespins. Moreover, we assume that two of the four componentsof the Dirac spinors of the heavy quarks Q and ¯ Q vanish.These approximations become exact for static quarks and areexpected to yield reasonably accurate results for b quarks,possibly even for c quarks.In Ref. [34] we have derived in detail a coupled chan-nel Schrödiger equation for a 4-component wave function ψ ( r ) = ( ψ ¯ QQ ( r ) , (cid:174) ψ ¯ M M ( r )) (Eq. (10) in Ref. [34]). The uppercomponent of this wave function represents the ¯ QQ channel,the lower three components represent the ¯ M M channel. For the¯
M M channel we consider only the lightest heavy-light mesonswith J P = − and J P = − , i.e. B and B ∗ mesons for Q ≡ b (asusual, J , P and C denote total angular momentum, parity andcharge conjugation). Within the approximations stated abovethese two mesons have the same mass. One can show that thespin of the two light quarks is 1, which is represented by thethree components of (cid:174) ψ ¯ M M ( r ) . Note that we ignore decays of¯ QQ to lighter quarkonium and a light I = σ oran η meson, because they are suppressed by the OZI rule. (cid:101) J PC denotes total angular momentum excluding the heavyquark spins and the corresponding parity and charge conju-gation. It is a conserved quantity. As in Ref. [34] we focusthroughout this work on (cid:101) J PC = ++ . Thus J PC = S PCQ , where S Q denotes the heavy quark spin, with only two possibilities, S PCQ = − + , −− .The coupled channel Schrödinger equation for the partial wave with (cid:101) J = (cid:18) − (cid:18) / µ Q
00 1 / µ M (cid:19) ∂ r + r (cid:18) / µ M (cid:19) + V ( r ) + m M − E (cid:19) (cid:18) u , ( r ) χ → , ( r ) (cid:19) = − (cid:18) V mix ( r ) V ¯ M M , (cid:107) ( r ) (cid:19) kr j ( kr ) , V ( r ) = (cid:18) V ¯ QQ ( r ) V mix ( r ) V mix ( r ) V ¯ M M , (cid:107) ( r ) (cid:19) . (1)The upper equation represents the ¯ QQ channel with orbitalangular momentum L ¯ QQ = (cid:101) J = u , ( r ) is the radial part ofthe (cid:101) J = ψ ¯ QQ ( r ) = √ π i u , ( r ) kr Y , ( Ω ) + . . . (2)with the dots . . . denoting partial waves with (cid:101) J >
0. Similarly,the lower equation represents the ¯
M M channel with orbitalangular momentum L ¯ M M = j ( kr ) and χ → , ( r ) are theradial parts of the (cid:101) J = (cid:174) ψ ¯ M M ( r ) = √ π i (cid:18) j ( kr ) + χ → , ( r ) kr (cid:19) Z → , ( Ω ) + . . . (3)with Z → , ( Ω ) = e r /√ π and the dots . . . denoting partialwaves with (cid:101) J >
0. Moreover, m Q and m M are the heavy quarkand heavy-light meson masses, respectively, and µ Q = m Q / µ M = m M / E and the momentum k are related accordingto k = √ µ M E . The potentials V ¯ QQ ( r ) , V ¯ M M , (cid:107) ( r ) and V mix ( r ) represent the energy of a pair of heavy quarks, the energy ofa pair of heavy-light mesons and the mixing between the twochannels, respectively. In Ref. [34] we related these potentialsalgebraically to lattice QCD correlators computed and pro-vided in detail in Ref. [23] in the context of string breaking forlattice spacing a ≈ .
083 fm and pion mass m π ≈
650 MeV.The data points for V ¯ QQ ( r ) , V ¯ M M , (cid:107) ( r ) and V mix ( r ) are shownin Fig. 1 together with appropriate parameterizations, V ¯ QQ ( r ) = E − α r + σ r + (cid:213) j = c ¯ QQ , j r exp (cid:18) − r λ QQ , j (cid:19) (4) V ¯ M M , (cid:107) ( r ) = V mix ( r ) = (cid:213) j = c mix , j r exp (cid:18) − r λ , j (cid:19) . (6)The parameters appearing in Eq. (4) to Eq. (6) are collected inTable II.The appropriate boundary conditions for the radial wavefunctions u , ( r ) and χ → , ( r ) are u , ( r ) ∝ r for r → u , ( r ) = r → ∞ (8) χ → , ( r ) ∝ r for r → χ → , ( r ) = it → , kr h ( ) ( kr ) for r → ∞ , (10) ●●●● ●● ●●● ●● ●● ●●● ● ●●● ●●● ●● ● ● ●●●●●●●●●●● ● ●■■ ■■ ■■ ■ ■■ ■■ ■■ ■■■ ■ ■■■ ■■ ■ ■■ ■ ■ ■■■■■■■■■■■ ■ ■ ◆◆◆◆ ◆◆ ◆◆◆ ◆◆◆◆ ◆◆◆ ◆ ◆◆◆ ◆◆◆ ◆◆ ◆ ◆ ◆◆◆◆◆◆◆◆◆◆◆ ◆ ◆ ●■ ◆ - - - Figure 1. (Color online.) Potentials V ¯ QQ ( r ) , V ¯ M M , (cid:107) ( r ) and V mix ( r ) as functions of the ¯ QQ separation r . The curves correspond to theparameterizations (4) to (6) with parameters as listed in Table II.potential parameter value V ¯ QQ ( r ) E − . ( ) GeV α + . ( ) σ + . ( ) GeV c ¯ QQ , + . ( ) GeV λ ¯ QQ , + . ( ) GeV − c ¯ QQ , + . ( . ) GeV λ ¯ QQ , + . ( ) GeV − V ¯ M M , (cid:107) ( r ) – – V mix ( r ) c mix , − . ( ) GeV λ mix , + . ( ) GeV − c mix , − . ( ) GeV λ mix , + . ( ) GeV − Table II. The parameters of the potential parametrizations (4) to (6). where h ( ) is a spherical Hankel function of the first kind and t → , is the scattering amplitude and an eigenvalue of the Smatrix. We computed t → , as a function of the complexenergy E . Poles of t → , on the real axis below the ¯ M M threshold indicate bound states. Poles of t → , at energies withnon-vanishing negative imaginary parts represent resonanceswith masses m = Re ( E ) and decay widths Γ = − ( E ) . t → , is also related to the corresponding scattering phase via e i δ → , = + it → , . B. Main results from Ref. [34]
In Ref. [34] we applied our approach to study bottomoniumbound states and resonances with I =
0. For m M , which is theenergy reference of our system, we use the spin-averaged massof the B meson and the B ∗ meson, i.e. m M = ( m B + m B ∗ )/ = .
313 GeV [1]. µ Q = m Q / b quark. Since results are only weakly dependent on m Q (seee.g. previous work following a similar approach [30, 36]), weuse for simplicity m Q = .
977 GeV from quark models [37].In Ref. [34] we presented both the scattering amplitude t → , and the phase shift δ → , for real energies E above the¯ B (∗) B (∗) threshold at 10 .
627 GeV. We also checked probabilityconservation by showing the Argand diagram for t → , . Themain numerical results of Ref. [34] are, however, the poles of t → , in the complex energy plane, which are shown in Fig. 2and collected in Table III.There are four poles on the real axis below the ¯ B (∗) B (∗) threshold representing bound states ( n = , . . . , η b ( S ) ≡ Υ ( S ) , Υ ( S ) , Υ ( S ) and Υ ( S ) . We also obtained a resonance around 10 .
870 GeV,which matches Υ ( ) rather well ( n = B (∗) ¯ B (∗) threshold with mass ≈ .
774 GeV ( n = ( . ± . ) GeV denotedas Υ ( ) not yet confirmed by other experiments, whichcould correspond to our prediction. Higher resonances ( n ≥ B (∗) B (∗) . To obtain realis-tic widths for resonances above ≈ .
025 GeV, which is thethreshold of one heavy-light meson with negative parity andanother with positive parity, one has to include all excitedmeson-meson channels up to the respective resonance masses.
III. QUARK-QUARK AND MESON-MESON CONTENT OF I = BOTTOMONIUM
We continue or investigation of bottomonium bound statesand resonances with isospin I = B (∗) B (∗) threshold,i.e. n = , ,
6, which could correspond to the experimentallyobserved Υ ( S ) , Y ( ) and Υ ( ) , are conventional¯ QQ quarkonia, or whether there is a sizable ¯ QQ ¯ qq four-quarkcomponent.We inspect in detail the percentages of quarkonium and ofa meson-meson pair present in each of the bound states andresonances. To this end we compute% ¯ QQ = QQ + M , % ¯ M M = MQ + M (11) with Q = ∫ R max dr (cid:12)(cid:12)(cid:12) u , ( r ) (cid:12)(cid:12)(cid:12) , M = ∫ R max dr (cid:12)(cid:12)(cid:12) χ → , ( r ) (cid:12)(cid:12)(cid:12) . (12) u , ( r ) and χ → , ( r ) are the radial wave functions of the ¯ QQ and the ¯ M M channel, respectively, obtained by solving thecoupled channel Schrödinger equation (1) with energies E identical to the real parts of the corresponding poles.
1. Bound states
For bound states E < k = i (cid:112) | µ M E | . The boundary condition (10) for χ → , ( r ) simplifies to χ → , ( r ) = r → ∞ . (13)Thus, both Q and M are independent of R max , if chosen suf-ficiently large, i.e. R max > ∼ . u , ( r ) = r → ∞ (cf. Eq. (8)). The same is true for % ¯ QQ and % ¯ M M ,which represent the probabilities to either find the system in aquarkonium configuration or in a meson-meson configuration.
2. Resonances
For resonances things are more complicated. First, res-onances are defined by poles in the complex energy planewith non-vanishing negative imaginary parts of E . Evaluating% ¯ QQ and % ¯ M M at such a complex energy does not seem tobe meaningful, because | u , ( r )| / r and | χ → , ( r )| / r areonly proportional to probability densities, if E is real. Thus wecompute % ¯ QQ and % ¯ M M at the real part of the correspondingpole position, Re ( E ) , which is the resonance mass.There is, however, another complication, namely that M is not constant but linearly rising for large R max . The rea-son is that χ → , ( r ) represents an emergent wave (see Eq.(10)). The need to remove part of the meson-meson emer-gent wavefunction has already been addressed in a momentumspace formalism [38, 39]. For the systems we study we foundthe dependence of % ¯ QQ and % ¯ M M on R max to be rathermild, with an uncertainty of only a few percent in the range1 . ≤ R max ≤ . u , ( r = R max ) ≈
0. Thus, weinterpret % ¯ QQ and % ¯ M M as estimates of probabilities to ei-ther find the system in a quarkonium configuration or in ameson-meson configuration, as for the bound states discussedbefore.
3. Numerical results and error analysis
We show plots of % ¯ QQ and % ¯ M M as functions of R max for the first seven bottomonium bound states and resonancesin Fig. 3. ● ●● ● ●●●●● ●●● ●●●● ●● ●● ● ●●●● ●●●●● ●● ●●●● ● ● ●● ●●● ● ●● ●●●● ●●●●●● ●●●● ●●●●●● ●●●● ●● ●●● ●●●● ●●● ●●●●● ●●● ●●● ● ●●● ●● ●●● ● ●●● ●●●●● ●●● ●●● ●● ●● ●● ●●● ●●●●● ● ●●●● ●●●●● ●● ● ●●●●●● ●●● ●●●● ●●●● ●●● ●● ●●●●● ●●●● ●●● ●●● ●● ●●●● ●● ●●●●● ●●●●●●●● ● ●●● ●● ●●●● ●● ●●●●●●●● ● ●● ●● ●●● ●●●● ●● ●●●●●●●●●●● ●●● ●● ●●●●●● ●●● ●●●●●● ●●●● ●●●●● ●●●● ●● ●●●●●● ●●● ●●●●●● ●● ●●●● ●●● ●●●● ●●● ●●● ● ●●●●● ●● ●● ●● ●●●●●●● ●●●●● ●● ●●● ●●● ●●● ●●●● ●●●● ●●●●●●●● ●● ● ●●●●●● ●●● ●●● ●● ●● ●●●● ●●● ●●●●●●● ●● ●●●●●●● ●●●●●●● ●●●●●●● ●● ●●●●● ●● ●●● ●●● ●●● ●●● ●● ● ●●●●● ●●●● ●●● ●●●●●●●●● ●●●●● ● ●● ●● ●●● ●● ●●● ● ●●●● ● ●● ● ●●● ● ●●●●●● ●●● ●●● ●●●● ●●●● ●● ●●●●●● ●●●●● ●● ●●●●●●● ●● ●●●●● ●●●● ●●●●●●● ●● ●●●●●● ●●● ● ●● ●● ● ●●●●● ●● ●●● ●● ●● ●●● ●● ●● ● ●● ●●● ●●●● ●● ●●●●●●●●● ●●●●● ●●●●● ●● ●●●●●●●●●●● ●●●● ●●●●●●●● ●● ●● ●●● ●●● ●● ●● ●●●●●●● ●●●●●●● ●● ●● ●●● ●●● ●●●●●● ●●● ●●● ●● ●● ●●● ●●● ●●●●●● ●●●●● ●●●●● ●●●● ●●● ●●●●●●● ●●●●●●●●●● ●●●● ●●●● ●● ●●●●●● ●● ●●● ●● ●● ●●●● ● ●● ●●●● ●●● ●●●●●●●●● ●●●● ●●●●● ●●● ●● ●●● ●● ●●●● ●●● ●● ● ●●● ●●●●● ● ●●●●●●●● ●●● ●● ● ●●● ●● ●●●●●● ●●● ●●●● ●●●● ●●●● ●● ● ●●● ●●● ●●●●● ●●●●●● ●● ●●● ● ●●●●● ●●● ●●● ●●●●● ● ●●●● ●● ●● ●● ●●●●●● ●● ●● ●●● ●●●● ●● ● ●● ●●● ●●● ●●●● ●● ●● ● ●●●● ●● ●● ●●● ●● ●●● ●●●●● ●● ●●●●●● ●● ● ●●●●●●●● ●●●●●●● ●●●●● ●●●● ●●●●●●● ●●●●● ●●● ●●● ● ●● ●●●● ●●●●●● ●●●● ●● ●●●● ●●●● ●● ●●● ●●●● ●●● ●●●●●●●● ●●●●●●●●● ●●● ● ●●●●●●●● ●●●●●●●● ●● ●●●●● ●●●●●● ●●●● ●●●●● ●●● ●●●●●●●●● ●●●●●●●● ●●● ●●●●●●● ●●●● ●●● ●●● ●● ●●●● ●● ●●●●● ●● ●●●●●●● ●●●●●●●●● ●● ●● ●●●●●● ● ●●●● ●●● ●●●● ●● ●●●●●●●●●●●●●●●● ●●●●●● ●●● ●●●●●●●●●● ●●●●●●●●● ●●●●●●●● ●●● ●●●●●● ●●●●●● ●●● ●●●● ●●●●●●● ●●●●● ●●●● ●●●●●●●●● ● ●●●● ●● ●●● ●●● ●●● ●●●● ●●●●●●●●●●●● ●● ● ●●●●●● ●●● ●●● ●● ●●●●●● ●●● ●●●●●●● ●●●●●● ●●● ●●●●●●● ●●●●●●●●● ●●●●●●●●●● ●●●●●● ●●● ●● ● ●●●●● ●●●● ●●●●●●●●●●●●●●●● ●●●● ●● ●●● ●● ●●●● ●●●● ● ●●● ●●●● ●●●●●●●●● ●●● ●●●● ●●●●●● ●●●●●● ●●●●● ●● ●●●● ●●● ●●●●●●● ●●●●●●●●●●● ●● ●●●●●● ●●● ●●● ●●● ●●●●●●●●●● ●●●●●●● ●● ●● ●●● ●●●● ●●●●● ●●●●●●●●● ●●●●●●●●●● ●● ●●●●●●●●●●● ●●●●●●●●●●●● ●● ●●●●●●●● ●● ●● ●●●● ●●●●●●●●●● ●● ●● ●●● ●●● ●●●●●● ●●● ●●● ●●●● ●●● ●●● ●●●●●● ●●●●● ●●●●● ●●●● ●●● ●●●●●●● ●●●●●●●●●● ●●●● ●●●● ●●●●●●●● ●●●●● ●● ●● ●●●●● ●●●●●● ●●●●●●●●●●●● ●●●● ●●●●● ●●● ●● ●●●●●●●●●●●● ●● ●●●●●●●●● ●●●●●●●●●●●● ●●● ●●● ●● ●●●●●● ●●● ●●●●●●●●●●●● ●● ●●●● ●●● ●●●●● ●●●●●● ●● ●●● ●●●●●●●●● ●●● ●●●●●● ●●●● ●● ●● ●● ●●●●●● ●● ●● ●●●●●●● ●● ● ●●●●● ●●● ●●●● ●●●●●●●●● ●●●● ●●● ●● ●●●●●●●●●● ●● ●●●●●● ● ●●● ●●●●● ●●●●●●●●● ●●● ●●●● ●●●●●●● ●●●● ● ●●● ●●● ●●● ●●●● ●●●●●● ●● ●● ●● ●●●● ●●●● ●● ●●● ●●●●●● ●●●● ●●●●● ●●●●●●●●● ●●● ● ●●●●●●●●●●● ●●●●●●● ●●●●● ●●●●●● ●●●● ●●●●● ●●● ●●●●●● ●●● ●●●●●●●● ●●● ●●●●●●● ●●●● ●●● ●●● ●● ●●●● ●● ●●●●●● ●●●●●●●● ●●●●●●●●● ●● ●● ●●●●●● ● ●●●● ●●● ●●●● ●● ●●●●●●●●●●● ●●●●● ●●●●●● ●●● ●●●●●●●●●● ●●●●●●●●● ●●●●●●●● ●●●●●●●●●●●●●●● ●●● ●●●● ●●●●●●● ●●●●● ●●●● ●●●●●●●●● ●●●●● ●● ●●● ●●● ●●● ●●●● ●●●●●●●●●●●●●● ●●●●●●● ●●● ●●● ●● ●●●●●●●●● ●●●●●●● ●●●●●● ●●● ●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●● ●●● ●●●●●●●● ●●●● ●●● ●●●●●●●●●●●●●●● ●● ●● ●●● ●● ●●●●●●●●●●●● ●●●● ●●●●●●●●● ●●● ●●●● ●●●●●● ●●●●●●●●●●● ●●●●●● ●●● ●●●●●●● ●●●●●●●●●●● ●● ●●●●●● ●●● ●●● ●●● ●●● ●●●●●●●●●●●●●● ●● ●● ●●● ●●●● ●●●●● ●●● ●●●●●● ●●●●● ●●●●● ●● ●●●●●●●●●●● ●● ●●●●●●●●●● ●● ●●●●●●●● ●● ●● ●●●●●●●●●●●●●●●● ●● ●●● ●●● ●●●●●● ●●● ●●● ●●●● ●●● ●●● ●●●●●● ●●●●● ●●●●●●●●● ●●● ●●●●●●● ●●●●●●●●●● ●●●●●● ●● ●●●●●●●● ●●●●●●● ●● ●●●●● ●●●●●● ●●●●●●●●●●●● ●●●● ●●●●● ●●● ●● ●●●●●●●●●●●● ●●●●●●●●●●● ●●●●●●●●●●●● ●●● ●●● ●● ●●●●●● ●●● ●●●●●●●●●●●● ●● ● ●●● ●●●● ●●●● ●●●●●●●● ●●● ●● ●●●● ●●● ●●● ●●●●●● ●●●●●●●● ●●●●●●●●●●●●●●●●●●● ●● ●●● ●●● ●●● ●●●●●●●●●●●●● ●● ●●●●● ●● ●●● ●●●●●●● ●● ●●●●●● ●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●● ●●● ●●● ●●●●●●●●●●●●●●●●● ●● ●●●● ●●●● ●● ●●●●●●●●● ●●●● ●●●●● ●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●● ●●●●● ●●●●●● ●●●● ●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●● ●●● ●● ●●●● ●● ●●●●●●●●●●●●●●●●●●●●●● ● ●● ●●●●●●●●●●●●● ●●● ●●●●●● ●●●●●●●●●●●●●●●●●●●●● ● ●●● ●●●●●●●●●● ●●●●●●●●● ●●●●●●●● ●●●●●●●●●●●●●●● ●●●● ●●●●●●●●●● ●●●●●●●●●●●●●●●●●● ●●●●● ●●●●● ●●● ●●● ●●●● ●●●●●●●●●●●●●●●●●●●●● ●●● ●●● ●●●●●●●●●●● ●●●●●●● ●●●●●● ●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●● ●●●●●●●● ●●●● ●●● ●●●●●●●●●●●●●●●●● ●● ●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●● ●●●●●●●●●●● ●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●● ●●●●●● ●●●●●●●●●●●●●●●●●●●● ●● ●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●● ●● ●●●●●●●●●●● ●●●●●●●●●●●● ●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●● ●●●●● ●●●●●●●●● ●●●●●● ●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●● ●●●●●●●● ●●●●●●● ●●●●●●●●●●●●● ●●●●●●●●●●●● ●●●● ●●●●●●●● ●●●●●●●●●●●●●● ●●●●●●●●●●● ●●●●●●●●●●●● ●●● ●●● ●●●●●●●●●●● ●●●●●●●●●●●●●● ●●●● ●●●●●●●●●●●●●● ●●●●●●●● ●●●●●● ●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●● ●●●●●●●●●●●●●●● ●●●●● ●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●● ●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●● ●●●●●●●●●●●●●●●●●●●●● ●● ●●●●●●●●●●● ●● ●● ●●● ●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●● ●●●●●●●●●● ●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●● ●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●● ●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●● ●●●● ●●●● ●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●● ●●●●●●●●●●● ●●●●●●●●●●● ●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●● ● ●● ●●●●●● ●●● ●●●●●●●●●●●●●●● ●●● ●● ●● ●●● ●●●●●●● ●●●● ●● ●●●●●● ●●●● ●●●●●● ●●●● ●● ●●●●● ●● ●●● ●●●●●●●● ●●● ●●●● ●● ●●● ●●●●●●●●● ●●●●●● ●● ●● ●●●●● ●●●●●● ●●●● ●●●●● ●●● ●●●●●●●●● ●●●●●●●● ●●● ●●●●●●●●●●●●●● ●●● ●●●●●● ●●●●●●● ●●●●●●●●● ●●●●● ●●● ●●● ●●●●●●●● ●●●●● ●●● ●●● ● ●● ●●●●●●●●●●●●●● ●● ●●●●● ● ●●● ●●●●●● ●●●● ●●●●●●●●● ●●● ●●●●● ●●●●●●●●● ●●●●●● ●●●● ●●●●●●●●●● ●●●●● ●●●● ●●●●●●●●●●●●●●●●●●● ●●●●●● ●●●● ●●●●●●●●●●●● ●● ●●●●●●● ●●● ●●● ●● ●● ●●●● ●●●●●● ●●●● ●●●●●●●●● ●●●●●●● ●●●●●●● ●●●●●●● ●●●●● ●●●●●●● ●● ●● ● ●●●●● ●●●●●●●●●●●●●●●●●●●●●●●● ●●●●● ●●●●●● ●●●●● ●●● ●●●● ●●●●●●●●● ●●● ●●●● ●●● ●●● ●●●●●● ●●●●● ●● ●●●●●●● ●●●●●●● ●●●●●●●●●●● ●● ●●●●●●●●● ●●● ●● ● ●●● ●● ●● ●●● ●●●●●●●●● ●●● ●●●●●● ●●● ●●●●●●●●●●● ●●●●● ●●●●● ●● ●●●●●●●●●●● ●●●● ●●●●●●●●●● ●●●●●●●● ●● ●●●●●●●●●●●● ●●●●●● ●● ●●● ●●● ●●●●●● ●●●●●● ●●●● ●●●●●● ●●●●●●●●●●● ●●●●●●●●● ●●●●●●●●●● ●●●●●●● ●●● ●●●● ●●●●●●●●●●●● ●●●●● ●● ●● ●●●●● ●●●●●●●●●●● ●●●●●●● ●●●● ●●●●●●●● ●●●●● ●● ●●●●●●● ●● ●●●●●●●●● ●●●●● ●●●●●●● ●●● ●●● ●● ●●●●●● ●●● ●●●●●●● ●●●● ● ●● ●●●● ●●●●●●●● ●●●●●● ●● ●●● ●●● ●●●●●● ●●●●●●●● ●●●●● ●● ●● ●● ●●●●●● ●●●● ●●●●●●● ●● ●●● ●●●●●● ●●●●●● ●● ●●●●● ●●●●●●● ●● ●● ●●●●●●●●●● ●●●● ●● ● ●● ●●●● ●● ●●● ●●●● ●● ●● ● ●●●●●● ●●● ●● ●● ●● ● ●●● ●● ●● ●●●● ●● ●● ●●●● ●●●● ●●●●●● ●●●● ●● ●●● ●● ●● ●●● ●● ●●● ●●● ●●● ● ●●● ●● ●●● ● ●●●●●●●● ●●●●●● ●● ●● ●● ● ●● ●●●●● ● ●●●● ●●●●● ●●● ●●●●●●●●● ●● ●● ●●● ● ●●● ●● ●●●● ● ●●●● ●●● ●●● ●●●●●● ●● ●●●●● ●● ●●●●●●● ●●●●● ●●●●●● ●●●●● ●●● ●●● ●● ●●● ●●● ● ●● ●●●●● ●●●●●● ●●● ●● ●●●● ●● ●●● ●● ●●●● ●●●● ●●●●●●●● ● ●●● ●●● ●● ●●●●●●●●● ●● ●●●● ● ●● ●●●● ●●●● ●●● ●● ●●● ●● ●● ●●●●●● ●● ● ● ●●●● ●● ●●● ●●● ●●● ●●●● ● ●● ●●●●●●●●● ●● ● ●●●●●● ●●● ●● ● ●● ●● ●●●● ●●● ●●● ●●●● ●●●●● ●●●● ●● ●●●●● ●●●●●●● ●●●●●●● ●●●●● ●●● ●●●● ●● ●● ● ●●●●● ●●●●●●●●●●●●●●●● ●●●● ●● ●● ●● ●●● ●● ●●●● ●●●● ● ●● ● ●●●● ●●●●● ●●●● ●●● ●●●● ●●● ● ●● ●●● ● ●● ●●●●● ●● ●●●●●●● ●● ●●●●● ●●● ● ●● ●●● ●● ●● ●●● ●● ● ●●● ● ●● ●● ● ●●● ●● ●● ●●● ●●●●●●●●● ●●● ●● ●●● ● ●●● ●●●●● ● ●●●●● ●●●●● ●●● ●● ●● ●●● ●●●●●●●● ●●●● ●●●●●●●● ●● ●●●●●●●● ●● ●● ●●●●●●●● ●● ●● ●●● ● ●● ●●● ●●● ●●●●●● ●●●●●● ●●●●●●● ●●● ●●●● ●● ●●●●● ●● ●● ● ● ●●● ●●●●●● ●●● ● ●●●●● ●● ●●● ●● ●● ●● ●●●● ●●●● ●● ●●●●● ●● ●● ●●● ● ● ●● ●● ●● ●●● ●● ●●●●●●● ●●●● ●●●● ● ●●● ●● ●●● ●● ●●●● ●●● ●● ●●●●●●●●● ● ●●●● ●●●● ●●● ●● ● ●●● ●● ●●●●●● ●●● ●●●●●●● ●●●● ● ●● ●●●● ●●● ●●●●● ●●●●●● ●● ●●● ●●●● ●● ●●● ●●● ●●●●● ● ●●●● ●● ●● ●● ● ●●●●● ●●●● ●●●● ●●● ●● ●●● ●●● ●●● ●● ●●●● ●● ●●● ●● ●● ●●●●● ●● ●● ●●●●●● ●●●● ● ●●● ●●● ●● ● ●●●●● ●●● ●●●● ●● ●● ● ●●●● ●●●●● ●● ●●●● ● ● ●● ●●● ● ●● ●●●● ●●●●●● ●●●● ●●●●●● ●●●● ●● ●●● ●●●● ●●● ●●●●● ●●● ●●● ● ●●● ●● ●●● ● ●●● ●●●●● ●●● ●●● ●● ●● ●● ●●● ●●●●● ● ●●●● ●●●●● ●● ● ●●●●●● ●●● ●●●● ●●●● ●●● ●● ●●●●● ●●●● ●●● ●●● ●● ●●●● ●● ●●●●● ●●●●●●●● ● ●●● ●● ●●●● ●● ●●●●●●●● ● ●● ●● ●●● ●●●● ●● ●●●●●●●●●●● ●●● ●● ●●●●●● ●●● ●●●●●● ●●●● ●●●●● ●●●● ●● ●●●●●● ●●● ●●●●●● ●● ●●●● ●●● ●●●● ●●● ●●● ● ●●●●● ●● ●● ●● ●●●●●●● ●●●●● ●● ●●● ●●● ●●● ●●●● ●●●● ●●●●●●●● ●● ● ●●●●●● ●●● ●●● ●● ●● ●●●● ●●● ●●●●●●● ●● ●●●●●●● ●●●●●●● ●●●●●●● ●● ●●●●● ●● ●●● ●●● ●●● ●●● ●● ● ●●●●● ●●●● ●●● ●●●●●●●●● ●●●●● ● ●● ●● ●●● ●● ●●● ● ●●●● ● ●● ● ●●● ● ●●●●●● ●●● ●●● ●●●● ●●●● ●● ●●●●●● ●●●●● ●● ●●●●●●● ●● ●●●●● ●●●● ●●●●●●● ●● ●●●●●● ●●● ● ●● ●● ● ●●●●● ●● ●●● ●● ●● ●●● ●● ●● ● ●● ●●● ●●●● ●● ●●●●●●●●● ●●●●● ●●●●● ●● ●●●●●●●●●●● ●●●● ●●●●●●●● ●● ●● ●●● ●●● ●● ●● ●●●●●●● ●●●●●●● ●● ●● ●●● ●●● ●●●●●● ●●● ●●● ●● ●● ●●● ●●● ●●●●●● ●●●●● ●●●●● ●●●● ●●● ●●●●●●● ●●●●●●●●●● ●●●● ●●●● ●● ●●●●●● ●● ●●● ●● ●● ●●●● ● ●● ●●●● ●●● ●●●●●●●●● ●●●● ●●●●● ●●● ●● ●●● ●● ●●●● ●●● ●● ● ●●● ●●●●● ● ●●●●●●●● ●●● ●● ● ●●● ●● ●●●●●● ●●● ●●●● ●●●● ●●●● ●● ● ●●● ●●● ●●●●● ●●●●●● ●● ●●● ● ●●●●● ●●● ●●● ●●●●● ● ●●●● ●● ●● ●● ●●●●●● ●● ●● ●●● ●●●● ●● ● ●● ●●● ●●● ●●●● ●● ●● ● ●●●● ●● ●● ●●● ●● ●●● ●●●●● ●● ●●●●●● ●● ● ●●●●●●●● ●●●●●●● ●●●●● ●●●● ●●●●●●● ●●●●● ●●● ●●● ● ●● ●●●● ●●●●●● ●●●● ●● ●●●● ●●●● ●● ●●● ●●●● ●●● ●●●●●●●● ●●●●●●●●● ●●● ● ●●●●●●●● ●●●●●●●● ●● ●●●●● ●●●●●● ●●●● ●●●●● ●●● ●●●●●●●●● ●●●●●●●● ●●● ●●●●●●● ●●●● ●●● ●●● ●● ●●●● ●● ●●●●● ●● ●●●●●●● ●●●●●●●●● ●● ●● ●●●●●● ● ●●●● ●●● ●●●● ●● ●●●●●●●●●●●●●●●● ●●●●●● ●●● ●●●●●●●●●● ●●●●●●●●● ●●●●●●●● ●●● ●●●●●● ●●●●●● ●●● ●●●● ●●●●●●● ●●●●● ●●●● ●●●●●●●●● ● ●●●● ●● ●●● ●●● ●●● ●●●● ●●●●●●●●●●●● ●● ● ●●●●●● ●●● ●●● ●● ●●●●●● ●●● ●●●●●●● ●●●●●● ●●● ●●●●●●● ●●●●●●●●● ●●●●●●●●●● ●●●●●● ●●● ●● ● ●●●●● ●●●● ●●●●●●●●●●●●●●●● ●●●● ●● ●●● ●● ●●●● ●●●● ● ●●● ●●●● ●●●●●●●●● ●●● ●●●● ●●●●●● ●●●●●● ●●●●● ●● ●●●● ●●● ●●●●●●● ●●●●●●●●●●● ●● ●●●●●● ●●● ●●● ●●● ●●●●●●●●●● ●●●●●●● ●● ●● ●●● ●●●● ●●●●● ●●●●●●●●● ●●●●●●●●●● ●● ●●●●●●●●●●● ●●●●●●●●●●●● ●● ●●●●●●●● ●● ●● ●●●● ●●●●●●●●●● ●● ●● ●●● ●●● ●●●●●● ●●● ●●● ●●●● ●●● ●●● ●●●●●● ●●●●● ●●●●● ●●●● ●●● ●●●●●●● ●●●●●●●●●● ●●●● ●●●● ●●●●●●●● ●●●●● ●● ●● ●●●●● ●●●●●● ●●●●●●●●●●●● ●●●● ●●●●● ●●● ●● ●●●●●●●●●●●● ●● ●●●●●●●●● ●●●●●●●●●●●● ●●● ●●● ●● ●●●●●● ●●● ●●●●●●●●●●●● ●● ●●●● ●●● ●●●●● ●●●●●● ●● ●●● ●●●●●●●●● ●●● ●●●●●● ●●●● ●● ●● ●● ●●●●●● ●● ●● ●●●●●●● ●● ● ●●●●● ●●● ●●●● ●●●●●●●●● ●●●● ●●● ●● ●●●●●●●●●● ●● ●●●●●● ● ●●● ●●●●● ●●●●●●●●● ●●● ●●●● ●●●●●●● ●●●● ● ●●● ●●● ●●● ●●●● ●●●●●● ●● ●● ●● ●●●● ●●●● ●● ●●● ●●●●●● ●●●● ●●●●● ●●●●●●●●● ●●● ● ●●●●●●●●●●● ●●●●●●● ●●●●● ●●●●●● ●●●● ●●●●● ●●● ●●●●●● ●●● ●●●●●●●● ●●● ●●●●●●● ●●●● ●●● ●●● ●● ●●●● ●● ●●●●●● ●●●●●●●● ●●●●●●●●● ●● ●● ●●●●●● ● ●●●● ●●● ●●●● ●● ●●●●●●●●●●● ●●●●● ●●●●●● ●●● ●●●●●●●●●● ●●●●●●●●● ●●●●●●●● ●●●●●●●●●●●●●●● ●●● ●●●● ●●●●●●● ●●●●● ●●●● ●●●●●●●●● ●●●●● ●● ●●● ●●● ●●● ●●●● ●●●●●●●●●●●●●● ●●●●●●● ●●● ●●● ●● ●●●●●●●●● ●●●●●●● ●●●●●● ●●● ●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●● ●●● ●●●●●●●● ●●●● ●●● ●●●●●●●●●●●●●●● ●● ●● ●●● ●● ●●●●●●●●●●●● ●●●● ●●●●●●●●● ●●● ●●●● ●●●●●● ●●●●●●●●●●● ●●●●●● ●●● ●●●●●●● ●●●●●●●●●●● ●● ●●●●●● ●●● ●●● ●●● ●●● ●●●●●●●●●●●●●● ●● ●● ●●● ●●●● ●●●●● ●●● ●●●●●● ●●●●● ●●●●● ●● ●●●●●●●●●●● ●● ●●●●●●●●●● ●● ●●●●●●●● ●● ●● ●●●●●●●●●●●●●●●● ●● ●●● ●●● ●●●●●● ●●● ●●● ●●●● ●●● ●●● ●●●●●● ●●●●● ●●●●●●●●● ●●● ●●●●●●● ●●●●●●●●●● ●●●●●● ●● ●●●●●●●● ●●●●●●● ●● ●●●●● ●●●●●● ●●●●●●●●●●●● ●●●● ●●●●● ●●● ●● ●●●●●●●●●●●● ●●●●●●●●●●● ●●●●●●●●●●●● ●●● ●●● ●● ●●●●●● ●●● ●●●●●●●●●●●● ●● ● ●●● ●●●● ●●●● ●●●●●●●● ●●● ●● ●●●● ●●● ●●● ●●●●●● ●●●●●●●● ●●●●●●●●●●●●●●●●●●● ●● ●●● ●●● ●●● ●●●●●●●●●●●●● ●● ●●●●● ●● ●●● ●●●●●●● ●● ●●●●●● ●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●● ●●● ●●● ●●●●●●●●●●●●●●●●● ●● ●●●● ●●●● ●● ●●●●●●●●● ●●●● ●●●●● ●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●● ●●●●● ●●●●●● ●●●● ●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●● ●●● ●● ●●●● ●● ●●●●●●●●●●●●●●●●●●●●●● ● ●● ●●●●●●●●●●●●● ●●● ●●●●●● ●●●●●●●●●●●●●●●●●●●●● ● ●●● ●●●●●●●●●● ●●●●●●●●● ●●●●●●●● ●●●●●●●●●●●●●●● ●●●● ●●●●●●●●●● ●●●●●●●●●●●●●●●●●● ●●●●● ●●●●● ●●● ●●● ●●●● ●●●●●●●●●●●●●●●●●●●●● ●●● ●●● ●●●●●●●●●●● ●●●●●●● ●●●●●● ●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●● ●●●●●●●● ●●●● ●●● ●●●●●●●●●●●●●●●●● ●● ●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●● ●●●●●●●●●●● ●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●● ●●●●●● ●●●●●●●●●●●●●●●●●●●● ●● ●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●● ●● ●●●●●●●●●●● ●●●●●●●●●●●● ●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●● ●●●●● ●●●●●●●●● ●●●●●● ●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●● ●●●●●●●● ●●●●●●● ●●●●●●●●●●●●● ●●●●●●●●●●●● ●●●● ●●●●●●●● ●●●●●●●●●●●●●● ●●●●●●●●●●● ●●●●●●●●●●●● ●●● ●●● ●●●●●●●●●●● ●●●●●●●●●●●●●● ●●●● ●●●●●●●●●●●●●● ●●●●●●●● ●●●●●● ●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●● ●●●●●●●●●●●●●●● ●●●●● ●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●● ●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●● ●●●●●●●●●●●●●●●●●●●●● ●● ●●●●●●●●●●● ●● ●● ●●● ●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●● ●●●●●●●●●● ●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●● ●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●● ●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●● ●●●● ●●●● ●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●● ●●●●●●●●●●● ●●●●●●●●●●● ●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●● ● ●● ●●●●●● ●●● ●●●●●●●●●●●●●●● ●●● ●● ●● ●●● ●●●●●●● ●●●● ●● ●●●●●● ●●●● ●●●●●● ●●●● ●● ●●●●● ●● ●●● ●●●●●●●● ●●● ●●●● ●● ●●● ●●●●●●●●● ●●●●●● ●● ●● ●●●●● ●●●●●● ●●●● ●●●●● ●●● ●●●●●●●●● ●●●●●●●● ●●● ●●●●●●●●●●●●●● ●●● ●●●●●● ●●●●●●● ●●●●●●●●● ●●●●● ●●● ●●● ●●●●●●●● ●●●●● ●●● ●●● ● ●● ●●●●●●●●●●●●●● ●● ●●●●● ● ●●● ●●●●●● ●●●● ●●●●●●●●● ●●● ●●●●● ●●●●●●●●● ●●●●●● ●●●● ●●●●●●●●●● ●●●●● ●●●● ●●●●●●●●●●●●●●●●●●● ●●●●●● ●●●● ●●●●●●●●●●●● ●● ●●●●●●● ●●● ●●● ●● ●● ●●●● ●●●●●● ●●●● ●●●●●●●●● ●●●●●●● ●●●●●●● ●●●●●●● ●●●●● ●●●●●●● ●● ●● ● ●●●●● ●●●●●●●●●●●●●●●●●●●●●●●● ●●●●● ●●●●●● ●●●●● ●●● ●●●● ●●●●●●●●● ●●● ●●●● ●●● ●●● ●●●●●● ●●●●● ●● ●●●●●●● ●●●●●●● ●●●●●●●●●●● ●● ●●●●●●●●● ●●● ●● ● ●●● ●● ●● ●●● ●●●●●●●●● ●●● ●●●●●● ●●● ●●●●●●●●●●● ●●●●● ●●●●● ●● ●●●●●●●●●●● ●●●● ●●●●●●●●●● ●●●●●●●● ●● ●●●●●●●●●●●● ●●●●●● ●● ●●● ●●● ●●●●●● ●●●●●● ●●●● ●●●●●● ●●●●●●●●●●● ●●●●●●●●● ●●●●●●●●●● ●●●●●●● ●●● ●●●● ●●●●●●●●●●●● ●●●●● ●● ●● ●●●●● ●●●●●●●●●●● ●●●●●●● ●●●● ●●●●●●●● ●●●●● ●● ●●●●●●● ●● ●●●●●●●●● ●●●●● ●●●●●●● ●●● ●●● ●● ●●●●●● ●●● ●●●●●●● ●●●● ● ●● ●●●● ●●●●●●●● ●●●●●● ●● ●●● ●●● ●●●●●● ●●●●●●●● ●●●●● ●● ●● ●● ●●●●●● ●●●● ●●●●●●● ●● ●●● ●●●●●● ●●●●●● ●● ●●●●● ●●●●●●● ●● ●● ●●●●●●●●●● ●●●● ●● ● ●● ●●●● ●● ●●● ●●●● ●● ●● ● ●●●●●● ●●● ●● ●● ●● ● ●●● ●● ●● ●●●● ●● ●● ●●●● ●●●● ●●●●●● ●●●● ●● ●●● ●● ●● ●●● ●● ●●● ●●● ●●● ● ●●● ●● ●●● ● ●●●●●●●● ●●●●●● ●● ●● ●● ● ●● ●●●●● ● ●●●● ●●●●● ●●● ●●●●●●●●● ●● ●● ●●● ● ●●● ●● ●●●● ● ●●●● ●●● ●●● ●●●●●● ●● ●●●●● ●● ●●●●●●● ●●●●● ●●●●●● ●●●●● ●●● ●●● ●● ●●● ●●● ● ●● ●●●●● ●●●●●● ●●● ●● ●●●● ●● ●●● ●● ●●●● ●●●● ●●●●●●●● ● ●●● ●●● ●● ●●●●●●●●● ●● ●●●● ● ●● ●●●● ●●●● ●●● ●● ●●● ●● ●● ●●●●●● ●● ● ● ●●●● ●● ●●● ●●● ●●● ●●●● ● ●● ●●●●●●●●● ●● ● ●●●●●● ●●● ●● ● ●● ●● ●●●● ●●● ●●● ●●●● ●●●●● ●●●● ●● ●●●●● ●●●●●●● ●●●●●●● ●●●●● ●●● ●●●● ●● ●● ● ●●●●● ●●●●●●●●●●●●●●●● ●●●● ●● ●● ●● ●●● ●● ●●●● ●●●● ● ●● ● ●●●● ●●●●● ●●●● ●●● ●●●● ●●● ● ●● ●●● ● ●● ●●●●● ●● ●●●●●●● ●● ●●●●● ●●● ● ●● ●●● ●● ●● ●●● ●● ● ●●● ● ●● ●● ● ●●● ●● ●● ●●● ●●●●●●●●● ●●● ●● ●●● ● ●●● ●●●●● ● ●●●●● ●●●●● ●●● ●● ●● ●●● ●●●●●●●● ●●●● ●●●●●●●● ●● ●●●●●●●● ●● ●● ●●●●●●●● ●● ●● ●●● ● ●● ●●● ●●● ●●●●●● ●●●●●● ●●●●●●● ●●● ●●●● ●● ●●●●● ●● ●● ● ● ●●● ●●●●●● ●●● ● ●●●●● ●● ●●● ●● ●● ●● ●●●● ●●●● ●● ●●●●● ●● ●● ●●● ● ● ●● ●● ●● ●●● ●● ●●●●●●● ●●●● ●●●● ● ●●● ●● ●●● ●● ●●●● ●●● ●● ●●●●●●●●● ● ●●●● ●●●● ●●● ●● ● ●●● ●● ●●●●●● ●●● ●●●●●●● ●●●● ● ●● ●●●● ●●● ●●●●● ●●●●●● ●● ●●● ●●●● ●● ●●● ●●● ●●●●● ● ●●●● ●● ●● ●● ● ●●●●● ●●●● ●●●● ●●● ●● ●●● ●●● ●●● ●● ●●●● ●● ●●● ●● ●● ●●●●● ●● ●● ●●●●●● ●●●● ● ●●● ●● ● ● ● ● ● ● ● - - - - - - - ( E ) [ GeV ] I m ( E )[ G e V ] Figure 2. (Color online.) Positions of the poles of t → , in the complex energy plane for all bound states and resonances below 11 . B (∗) B (∗) threshold at 10 .
627 GeV. Theshaded region above 11 .
025 GeV marks the opening of the threshold of one heavy-light meson with negative parity and another with positiveparity, beyond which our results should not be trusted anymore.
As expected, for the four bound states, n = , . . . ,
4, both% ¯ QQ and % ¯ M M are constant for large R max . For η b ( S ) ≡ Υ ( S ) ( n =
1) this is the case already for R max > ∼ . Υ ( S ) ( n = R max > ∼ . n are less localized, as usual in quantum mechanics. η b ( S ) ≡ Υ ( S ) , Υ ( S ) and Υ ( S ) have % ¯ QQ ≈ Υ ( S ) , which is close to the ¯ B (∗) B (∗) threshold is still quarkonium dominated (% ¯ QQ ≈ M M ≈ QQ and% ¯ M M on R max but it is rather mild, with an uncertainty of2% or less in the range 1 . ≤ R max ≤ . n = M M ≈
94% and, thus, is essentially a meson-mesonpair. The resonance with n = QQ component(% ¯ QQ ≈ M M ≈ n ≥ QQ (cid:29) % ¯ M M . We stress that resultsfor n ≥ QQ and % ¯ M M for R max = . QQ and % ¯ M M in Fig. 3. We define the asymmetric systematic uncertain-ties as | % ¯ QQ ( R max = . ) − % ¯ QQ ( R max = . )| and | % ¯ QQ ( R max = . ) − % ¯ QQ ( R max = . )| and in thesame way for % ¯ M M . They are around 2% for the reso-nances with n = n =
6, respectively, and negligiblefor all other n . The total uncertainties on % ¯ QQ and % ¯ M M arerather small. Thus, our predictions concerning the structureof the bound states and resonances are quite stable within ourframework. The columns “% ¯ QQ ” and “% ¯ M M ” in Table IIIrepresent the main results of this work, since these numbers re-flect the quark composition of the bound states and resonancesand clarify, which states are close to ordinary quark modelquarkonium, and which states are dynamically generated by ameson-meson decay channel.
IV. CONCLUSIONS
In Ref. [34] we recently developed a formalism based onlattice QCD static potentials, to study resonances with a heavyquark-antiquark pair and possibly also a light quark-antiquark masses and decay widths from poles of t → , quark composition masses and decay widths from experiment n m = Re ( E ) [GeV] Im ( E ) [MeV] Γ [MeV] % ¯ QQ % ¯ M M name m [GeV] Γ [MeV]1 9 . + − . + . − . . + . − . η b ( S ) . ( ) ( ) Υ ( S ) . ( ) ≈
02 10 . + − . + . − . . + . − . Υ ( S ) . ( ) ≈
03 10 . + − . + . − . . + . − . Υ ( S ) . ( ) ≈
04 10 . + − . + . − . . + . − . Υ ( S ) . ( ) ( ) . + − − . + . − . . + . − . . + . − . + . − . . + . − . + . − . Y ( ) . ( ) ( ) . + − − . + . − . . + . − . . + . − . + . − . . + . − . + . − . Υ ( ) . ( ) ( ) . + − − . + . − . . + . − . . + . − . + . − . . + . − . + . − . Table III. Masses and decay widths for I = (cid:101) J PC = ++ from the coupled channel Schrödinger equation (1) and thecorresponding ¯ QQ and ¯ M M percentages (for R max = . B (∗) B (∗) thresholds are marked by horizontal lines. Errors on our results for m and Γ are purely statistical, while for % ¯ QQ and % ¯ M M we additionallyshow systematic uncertainties for the resonances, as discussed in section III 2. Resonances with n ≥ .
025 GeV and, thus, should not be trusted (indicated by a gray shadedbackground). pair. We use these potentials, as e.g. provided in Ref. [23],in a coupled channel Schrödinger equation, which amounts toapplying the Born-Oppenheimer approximation, and study thescattering problem with the emergent wave method.In this work we have explored the nature of the I = S wave resonances in more detail, including notonly the pole positions but also their composition in termsof a quarkonium b ¯ b component (% ¯ QQ ) and a meson-meson B (∗) ¯ B (∗) component (% ¯ M M ). This first principles based com-putation is important, because it contributes to the clarifica-tion of controversies concerning the states close to the B (∗) ¯ B (∗) threshold, which in our approach is just a single threshold,since the lattice QCD static potentials are independent of theheavy quark spins.The first controversy concerns the resonances Υ ( ) and Υ ( ) . Although they could possibly be identified with Υ ( S ) and Υ ( S ) , they could instead also correspond to the3 D or 4 D states. In our study we identify the Υ ( ) asindeed being Υ ( S ) with no need for bottomonium D wavestates, which we have not yet studied in our framework. Inwhat concerns the Υ ( ) we are currently not in a positionto make any reliable statement. Its mass is in the region ofthe B ∗ , ¯ B (∗) threshold, i.e. the sum of the masses of a posi-tive and a negative parity B meson. Since we do not havethe lattice QCD potentials to include the coupling to suchan excited meson-meson system, the validity of our approachabove ≈ .
025 GeV is questionable. This is also reflected bythe unrealistic small width of the n = Υ ( ) and Υ ( ) , and alsoof Υ ( S ) , which is identified according to the Review of Par-ticle Physics [1] as a quarkonium state. We find for Υ ( S ) and for Υ ( ) predominantly quarkonium, but also siz-able admixtures of B (∗) ¯ B (∗) meson pairs, % ¯ M M ≈
30% and% ¯
M M ≈ Υ ( S ) , Υ ( S ) and Υ ( S ) have rather small meson-meson components, of theorder of 10%. This matches well the result of Ref. [7], whichhas for the Υ ( S ) a fraction of meson meson component of theorder of 23% to 48% (1 − Z in the notation of Ref. [7], where Z is given in Table III of that reference).The most recent controversy concerns the nature of thenewly discovered resonance Υ ( ) . Model calculationssuggest for instance this resonance to be either a tetraquark[10, 11], a hybrid meson [12–14] or the more canonical and sofar missing Υ ( D ) [7–9]. With our lattice QCD based approachwe find a pole corresponding to a mass 10 .
774 GeV, quiteclose to the Belle measurement of the mass of the Υ ( ) resonance, ( . ± . ) GeV. In Ref. [34] we had al-ready anticipated this pole to be dynamically generated by themeson-meson channel. Now we confirm that this resonance ismostly composed of a pair of mesons, % ¯
M M ≈ −− , it can be classified as a Y typecrypto-exotic state. Notice that it should also be part of the η b family, since the heavy quark spin can also be 0 − + and there isdegeneracy with respect to the heavy quark spin.As an outlook, we are on the way to extend our study beyond S wave bottomonium, to P wave, D wave and F wave, which ismore cumbersome, since in these cases there is an additionalmeson-meson channel. We expect then to be able to addressthe controversy on the existence of D wave resonances in moredetail. Moreover, in the long term we plan to compute lattice ( fm ) n = % QQ % MM ( fm ) n = % QQ % MM ( fm ) n = % QQ % MM ( fm ) n = % QQ % MM ( fm ) n = % QQ % MM ( fm ) n = % QQ % MM ( fm ) n = % QQ % MM Figure 3. (Color online.) Percentages of quarkonium % ¯ QQ and of a meson-meson pair % ¯ M M present in each of the first seven bound statesand resonances as a function of R max . The error bands represent statistical uncertainties. QCD static potentials ourselves, in order to update our resultswith more precision and, hopefully, with excited meson-mesonchannels. The latter would enable us to make predictions alsofor energies above the B ∗ , ¯ B (∗) threshold at ≈ .
025 GeV.
ACKNOWLEDGMENTS
We acknowledge useful discussions with Gunnar Bali,Eric Braaten, Marco Cardoso, Francesco Knechtli, VanessaKoch, Lasse Müller, Sasa Prelovsek, George Rupp and AdamSzczepaniak.P.B. and N.C. acknowledge the support of CeFEMA un-der the FCT contract for R&D Units UID/CTM/04540/2013and the FCT project grant CERN/FIS-COM/0029/2017. N.C.acknowledges the FCT contract SFRH/BPD/109443/2015. M.W. acknowledges support by the Heisenberg Programmeof the Deutsche Forschungsgemeinschaft (DFG, German Re-search Foundation) – project number 399217702. [1] M. Tanabashi et al. (Particle Data Group), Phys. Rev.
D98 ,030001 (2018).[2] R. Mizuk et al. (Belle), Phys. Rev. Lett. , 232002 (2012),arXiv:1205.6351 [hep-ex].[3] R. Mizuk et al. (Belle), ,JHEP , 220 (2019), arXiv:1905.05521 [hep-ex].[4] C. Meng and K.-T. Chao, Phys. Rev. D , 074003 (2008),arXiv:0712.3595 [hep-ph].[5] Y. Simonov and A. Veselov, Phys. Lett. B , 55 (2009),arXiv:0805.4499 [hep-ph].[6] M. Voloshin, Phys. Rev. D , 034024 (2012), arXiv:1201.1222[hep-ph].[7] W.-H. Liang, N. Ikeno, and E. Oset, Phys. Lett. B , 135340(2020), arXiv:1912.03053 [hep-ph].[8] Q. Li, M.-S. Liu, Q.-F. Lü, L.-C. Gui, and X.-H. Zhong, Eur.Phys. J. C , 59 (2020), arXiv:1905.10344 [hep-ph].[9] J. F. Giron and R. F. Lebed, Phys. Rev. D , 014036 (2020),arXiv:2005.07100 [hep-ph].[10] Z.-G. Wang, Chin. Phys. C , 123102 (2019),arXiv:1905.06610 [hep-ph].[11] A. Ali, L. Maiani, A. Y. Parkhomenko, and W. Wang, Phys.Lett. B , 135217 (2020), arXiv:1910.07671 [hep-ph].[12] J. Tarrús Castellà, in (2019)arXiv:1908.05179 [hep-ph].[13] B. Chen, A. Zhang, and J. He, Phys. Rev. D , 014020 (2020),arXiv:1910.06065 [hep-ph].[14] N. Brambilla, S. Eidelman, C. Hanhart, A. Nefediev, C.-P.Shen, C. E. Thomas, A. Vairo, and C.-Z. Yuan, (2019),arXiv:1907.07583 [hep-ex].[15] S. Meinel, Phys. Rev. D , 094501 (2009), arXiv:0903.3224[hep-lat].[16] S. Meinel, Phys. Rev. D , 114502 (2010), arXiv:1007.3966[hep-lat].[17] R. Dowdall et al. (HPQCD), Phys. Rev. D , 054509 (2012),arXiv:1110.6887 [hep-lat].[18] Y. Aoki, N. H. Christ, J. M. Flynn, T. Izubuchi, C. Lehner, M. Li,H. Peng, A. Soni, R. S. Van de Water, and O. Witzel (RBC,UKQCD), Phys. Rev. D , 116003 (2012), arXiv:1206.2554[hep-lat].[19] R. Lewis and R. Woloshyn, Phys. Rev. D , 114509 (2012),arXiv:1204.4675 [hep-lat].[20] R. Dowdall, C. Davies, T. Hammant, R. Horgan, andC. Hughes (HPQCD), Phys. Rev. D , 031502 (2014), [Er- ratum: Phys.Rev.D 92, 039904 (2015)], arXiv:1309.5797 [hep-lat].[21] M. Wurtz, R. Lewis, and R. Woloshyn, Phys. Rev. D , 054504(2015), arXiv:1505.04410 [hep-lat].[22] S. M. Ryan and D. J. Wilson, (2020), arXiv:2008.02656 [hep-lat].[23] G. S. Bali, H. Neff, T. Duessel, T. Lippert, and K. Schilling(SESAM), Phys. Rev. D71 , 114513 (2005), arXiv:hep-lat/0505012 [hep-lat].[24] K. Juge, J. Kuti, and C. Morningstar, Phys. Rev. Lett. , 4400(1999), arXiv:hep-ph/9902336.[25] E. Braaten, C. Langmack, and D. H. Smith, Phys. Rev. D ,014044 (2014), arXiv:1402.0438 [hep-ph].[26] M. Berwein, N. Brambilla, J. Tarrús Castellà, and A. Vairo,Phys. Rev. D , 114019 (2015), arXiv:1510.04299 [hep-ph].[27] S. Capitani, O. Philipsen, C. Reisinger, C. Riehl, and M. Wag-ner, Phys. Rev. D , 034502 (2019), arXiv:1811.11046 [hep-lat].[28] P. Bicudo and M. Wagner, Phys. Rev. D87 , 114511 (2013),arXiv:1209.6274 [hep-ph].[29] Z. S. Brown and K. Orginos, Phys. Rev.
D86 , 114506 (2012),arXiv:1210.1953 [hep-lat].[30] P. Bicudo, K. Cichy, A. Peters, and M. Wagner, Phys. Rev.
D93 ,034501 (2016), arXiv:1510.03441 [hep-lat].[31] P. Bicudo, J. Scheunert, and M. Wagner, Phys. Rev.
D95 , 034502(2017), arXiv:1612.02758 [hep-lat].[32] P. Bicudo, K. Cichy, A. Peters, B. Wagenbach, and M. Wagner,Phys. Rev.
D92 , 014507 (2015), arXiv:1505.00613 [hep-lat].[33] P. Bicudo, M. Cardoso, A. Peters, M. Pflaumer, and M. Wagner,Phys. Rev.
D96 , 054510 (2017), arXiv:1704.02383 [hep-lat].[34] P. Bicudo, M. Cardoso, N. Cardoso, and M. Wagner, , Phys. Rev. D , 034503(2020), arXiv:1910.04827 [hep-lat].[35] J. Bulava, B. Hörz, F. Knechtli, V. Koch, G. Moir, C. Morn-ingstar, and M. Peardon, Phys. Lett.
B793 , 493 (2019),arXiv:1902.04006 [hep-lat].[36] F. Karbstein, M. Wagner, and M. Weber, Phys. Rev.
D98 ,114506 (2018), arXiv:1804.10909 [hep-ph].[37] S. Godfrey and N. Isgur, Phys. Rev.
D32 , 189 (1985).[38] F. Aceti and E. Oset, Phys. Rev. D , 014012 (2012),arXiv:1202.4607 [hep-ph].[39] F. Aceti, L. Dai, L. Geng, E. Oset, and Y. Zhang, Eur. Phys. J.A50