]>PLB135467135467S0370-2693(20)30271-910.1016/j.physletb.2020.135467The AuthorsPhenomenologyFig. 1(a) The system considered. (b-d) Two-hadron Fock components in the system with quantum numbers (2).Fig. 1Fig. 2Eigen-energies of b¯bd¯u system (Fig. 1a) for various separations r between static quarks b and b¯ are shown by points. The label indicates which two-hadron component dominates each eigenstate. The dot-dashed lines represent related two-hadron energies En.i. (4) when two hadrons (1) do not interact. The eigenstate dominated by BB¯⁎ (red circles) has energy significantly below mB+mB⁎ and shows sizable attraction. Lattice spacing is a ≃ 0.124 fm.Fig. 2Fig. 3(a) The extracted potential V(r) between B and B¯⁎ from lattice. (b) Fits of V(r) assuming the form of the regular potential Vreg (5) are presented by dashed lines for various values of parameter F. The singular potential V1/r(r) is shown by dot-dashed green line. Lattice spacing is a ≃ 0.124 fm.Fig. 3Fig. 4Mass of the virtual bound state and the bound state for various choices of the parameter F in V(r) (5).Fig. 4Fig. 5The BB¯⁎ rate NBB¯⁎∝kσBB¯⁎ has a peak above threshold. The plotted rate is based on our lattice results and the choice of parameter F = 1.3 in V(r) (5).Fig. 5Zb tetraquark channel from lattice QCD and Born-Oppenheimer approximationS.Prelovsekabc⁎sasa.prelovsek@ijs.siH.BahtiyarbdJ.PetkovićabaFaculty of Mathematics and Physics, University of Ljubljana, Ljubljana, SloveniaFaculty of Mathematics and PhysicsUniversity of LjubljanaLjubljanaSloveniaFaculty of Mathematics and Physics, University of Ljubljana, Ljubljana, SloveniabJozef Stefan Institute, Ljubljana, SloveniaJozef Stefan InstituteLjubljanaSloveniaJozef Stefan Institute, Ljubljana, SloveniacInstitute for Theoretical Physics, University of Regensburg, Regensburg, GermanyInstitute for Theoretical PhysicsUniversity of RegensburgRegensburgGermanyInstitute for Theoretical Physics, University of Regensburg, Regensburg, GermanydDepartment of Physics, Mimar Sinan Fine Arts University, Bomonti 34380, Istanbul, TurkeyDepartment of PhysicsMimar Sinan Fine Arts UniversityIstanbulBomonti34380TurkeyDepartment of Physics, Mimar Sinan Fine Arts University, Bomonti 34380, Istanbul, Turkey⁎Corresponding author at: Faculty of Mathematics and Physics, University of Ljubljana, Ljubljana, Slovenia.Faculty of Mathematics and PhysicsUniversity of LjubljanaLjubljanaSloveniaEditor: B. GrinsteinAbstractTwo Zb hadrons with exotic quark structure b¯bd¯u were discovered by Belle experiment. We present a lattice QCD study of the b¯bd¯u system in the approximation of static b quarks, where the total spin of heavy quarks is fixed to one. The energies of eigenstates are determined as a function of the separation r between b and b¯. The lower eigenstates are related to a bottomonium and a pion. The eigenstate dominated by BB¯⁎ has energy significantly below mB+mB⁎, which points to a sizable attraction for small r. The attractive potential V(r) between B and B¯⁎ is extracted assuming that this eigenstate is related exclusively to BB¯⁎. The Schrödinger equation for BB¯⁎ within the extracted potential leads to a virtual bound state, whose mass depends on the parametrization of the lattice potential. For certain parametrizations, we find a virtual bound state slightly below BB¯⁎ threshold and a narrow peak in the BB¯⁎ rate above threshold - these features could be related to Zb(10610) in the experiment. We surprisingly find also a deep bound state within the undertaken approximations.1IntroductionThe Belle experiment discovered two Zb+ states with exotic quark content b¯bd¯u, JP=1+ and I=1 in 2011 [1,2]. The lighter Zb(10610) lies slightly above BB¯⁎ threshold and the heavier Zb(10650) just above B⁎B¯⁎. The observed decay modes are ϒ(1S,2S,3S)π+,hb(1P,2P)π+,BB¯⁎ and B⁎B¯⁎ [1–3], where the BB¯⁎ and B⁎B¯⁎ largely dominate Zb(10610) and Zb(10650) decays, respectively. Many phenomenological theoretical studies of these two states have been performed, for example [4–17], and the majority indicates that B(⁎)B¯⁎ Fock component is important.We explore this channel within the first-principle lattice QCD. The only preliminary lattice study of this channel has been reported in [18,19] and is reviewed below. No other lattice results are available since this channel presents a severe challenge. Scattering matrix would have to be determined using the Lüscher method for at least 7 coupled two-meson channels listed in the previous paragraph. Poles of the scattering matrix would render possible Zb states. Following this path seems too challenging at present. Furthermore, the original Lüscher approach for two-particle scattering is not valid above the three-particle threshold.In the present study we consider the Born-Oppenheimer approximation [20], inspired by the study of this system in [18,19]. It is applied in molecular physics since ions are much heavier than other degrees of freedom. It is valuable also for the Zb system b¯bd¯u, where b and b¯ represent heavy degrees of freedom (h), while the light quarks and gluons are light degrees of freedom (l), see for example [21,22]. The simplification comes from the fact that the heavy degrees of freedom have large mass and therefore small velocity and kinetic energy.In the first step we treat b and b¯ as static at fixed distance r (Fig. 1a) and the main purpose is to determine eigen-energies En(r) of this system. This energy represents the total energy without the kinetic and rest energies of the b and b¯, so En(r) is related to the potential V(r) felt by the heavy degrees of freedom. In the second step, we study the motion of the heavy degrees of freedom (with the physical masses) under the influence of the extracted potential V(r). Solutions of the Schrödinger equation render information on possible (virtual) bound states Zb, resonances and cross-sections.The low-lying eigenstates of the system in Fig. 1a with quantum numbers (2) are related to two-hadron states in Figs. 1 (b-d)(1)B(0)B¯⁎(r),ϒ(r)π(p→=0),ϒ(r)π(p→≠0),ϒ(r)b1(0→). The eigen-energy En(r) related to BB¯⁎ in Fig. 1b is of major interest since Zb lies near BB¯⁎ threshold [1]. The ϒ(r)π(p→) represent the ground state at small r. Here ϒ(r) denotes the spin-one bottomonium where b¯ and b are separated by r. Pion can have zero or non-zero momentum p→=n→2πL since the total momenta of light degrees of freedom is not conserved in the presence of static quarks, i.e. pion momentum can change when it scatters on an infinitely heavy ϒ. Our task is to extract energies of all these eigenstates En(r) as a function of r. The only previous lattice study of this system [18] presents preliminary results based on Fock components BB¯⁎ and ϒπ(0); the presence ϒπ(p→≠0) was mentioned in [19], but not included in the simulation.2Quantum numbers and operatorsWe consider Zb0 that has quantum numbers I=1, I3=0, JPC=1+− and Jz=0 in experiment. The list of conserved quantum numbers is slightly different in the systems with two static particles. We study the system in Fig. 1a with quantum numbers(2)I=1,I3=0,ϵ=−1,C⋅P=−1Sh=1,Szh=0,Jzl=0,(h=heavy,l=light) where the neutral system is considered where C-conjugation can be applied (Fig. 1 shows the charged partner). Only the z-component of angular momenta for the light degrees of freedom (Jzlight) is conserved. The quantum number ϵ corresponds to the reflection over the yz plane. P refers to inversion with respect to mid-point between b and b¯ and C is charge conjugation, where only their product is conserved. The quantum numbers in (2) are conventionally denoted by Σu− using the conventions from [23].11This provides irreducible representation (Jzl)CPϵ=Σu−, where the notation here and in [23] is related by Jzl→Λ, Jzl=0→Σ, CP→η, CP=−1→u.The spin of the infinitely heavy quark can not flip via the interaction with gluons, so spin Sh of b¯b is conserved. We choose to study the system with Sh=1, which can decay to ϒ, while it can not decay to ηb and hb since these carry Sh=0. Note that the physical Zb and BB¯⁎ with finite mb can be a linear combination of Sh=1 as well as Sh=0, and we study only Sh=1 component here. We have in mind this component, which includes BB¯⁎, B¯B⁎, B¯⁎B⁎ (O1 in Eq. (3)), when we refer to “BB¯⁎” throughout this paper.The eigen-energies En of the system in Fig. 1a are determined from the correlation functions 〈Oi(t)Oj†(0)〉. We employ 6 operators Oi that create/annihilate the system with quantum numbers (2) and resemble Fock components (1) in Figs. 1 (b-d)(3)O1=OBB¯⁎∝∑a,b∑A,B,C,DΓBAΓ˜CDb¯Ca(0)qAa(0)q¯Bb(r)bDb(r)∝([b¯(0)P−γ5q(0)][q¯(r)γzP+b(r)]+{γ5↔γz})+([b¯(0)P−γyq(0)][q¯(r)γxP+b(r)]−{γy↔γx})O2=OBB¯⁎O3=Oϒπ(0)∝[b¯(0)UγzP+b(r)][q¯γ5q]p→=0→O4=Oϒπ(1)∝[b¯(0)UγzP+b(r)]([q¯γ5q]p→=e→z+[q¯γ5q]p→=−e→z)O5=Oϒπ(2)∝[b¯(0)UγzP+b(r)]([q¯γ5q]p→=2e→z+[q¯γ5q]p→=−2e→z)O6=Oϒb1(0)∝[b¯(0)UγzP+b(r)][q¯γxγyq]p→=0→. Here Γ=P−γ5, Γ˜=γzP+, [q¯Γ′q]p→≡1V∑x→q¯(x→)Γ′q(x→)eip→x→, momenta is given in units of 2π/L, capital (small) letters represent Dirac (color) indices, color singlets are denoted by [..] and U is a product of gauge links between 0 and r. First line in O1 decouples spin indices of light and heavy quarks in order to make Jzl and S(z)h (2) are more transparent [18], while the second line is obtained via the Fierz transformation. O2 is obtained from O1 by replacing all q(x) with ∇2q(x). O4,5 have pion momenta in z direction due to Jzl=0 and have two terms to ensure C⋅P=−1. The ϒb1 is not a decay mode for finite mb where C and P are separately conserved, but it is has quantum numbers (2) for mb→∞. The pair q¯q indicates combination u¯u−d¯d with I=1 and I3=0. All light quarks q(x) are smeared around the position x using the full distillation [24] with the radius about 0.3 fm, while the heavy quarks are point-like. Transformation properties of the operators are discussed in Section S1 of Supplemental Material.We verified there are no other two-hadron states in addition to (1) with quantum numbers (2) and with non-interacting energies (4) below mB+mB⁎+0.2 GeV.3Lattice detailsSimulation is performed on an ensemble with dynamical Wilson-clover u/d quarks, mπ≃266(5) MeV, a≃0.1239(13) fm and 280 configurations [25,26]. We choose an ensemble with small NL=16 and L≃2 fm so that ϒπ(pz) with pz>22πL appear at E>mB+mB⁎+0.2 GeV above our interest; larger L would require further operators like O4,5 with higher p→. Small L restricts us to r/a≤12NL=8, but the statistical errors grow with r and the current precision prevents us from accurate results for r/a>8 anyway. Small L leads also to the usual exponentially-suppressed corrections related to the pion and a very mild effect on the light-quark cloud in a B-meson since rB≪L. The lattice temporal extent NT=32 is effectively doubled by summing the light-quark propagators with periodic and anti-periodic boundary conditions in time [26].4Calculation of eigen-energies and overlapsCorrelation matrices Cij(t)=〈Oi(t)Oj†(0)〉 are evaluated using the full distillation method [24]. The b¯b annihilation Wick contraction is not present in the static limit considered here. Cij are averaged over 83 or 163 space positions of b¯, while sub-matrix for O3−6 is averaged over all source time slices to increase accuracy. Eigen-energies En and overlaps 〈Oi|n〉 are extracted from the 6×6 matrices Cij(t)=∑n〈Oi|n〉e−Ent〈n|Oj†〉 using the widely used GEVP variational approach [27–29].5Eigen-energies of b¯bd¯u system as a function of rThe main result of our study are the eigen-energies of the b¯bd¯u system (Fig. 1a) with static b and b¯ separated by r. They are shown by points in Fig. 2. The colors of points indicate which Fock-component (1) dominates an eigenstate, as determined from the normalized overlaps of an eigenstate |n〉 to operators Oi. Normalized overlap Z˜in≡〈Oi|n〉/maxm〈Oi|m〉 is normalized so that its maximal value for given Oi across all eigenstates is equal to one. Effective masses and overlaps are shown in Section S2 of Supplemental Material.The dashed lines in Fig. 2 provide the related non-interacting (n.i.) energies En of two-hadron states (1)(4)EBB¯⁎n.i.=2mB,Eϒπ(p→)n.i.=Vb¯b(r)+Eπ(p→),Eϒb1(0)n.i.=Vb¯b(r)+mb1, where b¯b static potential Vb¯b(r), Eπ(p→)≃mπ2+p→2, mb1 and mB=mB⁎=0.5224(14) (mass of B(⁎) for mb→∞ without b rest mass) are determined on the same lattice.The eigenstate dominated by BB¯⁎ has an energy close to mB+mB⁎ for r>0.5 fm, but it has significantly lower energy for r≃[0.1,0.4] fm (red circles in Fig. 2). This indicates sizable strong attraction between B and B¯⁎ in this system - something that might be related to the existence of Zb tetraquarks. This is the most important and robust result of this lattice study.Other eigenstates are dominated by ϒπ(p→) and ϒb1. Their energies E lie close to the non-interacting energies En.i. (4) given by dot-dashed lines, so E≃En.i.. We point out that we can not claim nonzero energy shifts E−En.i. for ϒπ and ϒb1 states (although Fig. 2 shows small deviations from zero in some cases) since the statistical and systematic errors are not small enough.6Towards masses of Zb states within certain approximationsEigen-energies of b¯bd¯u system in Fig. 1a indicate that eigenstate dominated by the BB¯⁎ has significantly lower energy than mB+mB⁎ at small separation r between static b and b¯. This suggests a possible existence of exotic hadrons (related to Zb) and related peaks in the cross-section near BB¯⁎ threshold. Such physical observables require the study of the motion for the heavy degrees of freedom based on the energies En(r) according to the Born-Oppenheimer approach. The precise prediction of such observables is not possible at present since lattice eigen-energies are not known for r<a. In addition, the accurate study would require the coupled-channel treatment of all Fock components (1) through the coupled-channel Schrödinger equation, which is a challenging task left for the future (this was recently elaborated in [30] for conventional b¯b with I=0).We apply two simplifying approximations in order to shed light on the possible existence of Zb based on energies in Fig. 2. The first assumption is that the eigenstate indicated by red circles in Fig. 2 is related exclusively to BB¯⁎ Fock component and does not contain other Fock components in (1). This is supported by our lattice results to a very good approximation, since this eigenstate couples almost exclusively to OBB¯⁎ and has much smaller coupling to Oϒπ and Oϒb1: the normalized overlap of this state to Oϒπ,ϒb1 is Z˜3−6≤0.05 for r≤4, while overlap to OBB¯⁎ is Z˜1,2≃O(1). In the reminder we explore the physics implications of this eigen-energy EBB¯⁎(r).The energy EBB¯⁎(r) represents the total energy without the kinetic energy of heavy degrees of freedom. The difference V(r)=EBB¯⁎(r)−mB−mB⁎ therefore represents the potential felt by the heavy degrees of freedom, in this case between B and B¯⁎ mesons. The extracted potential is plotted in Fig. 3. The potential shows sizable attraction for r=[0.1,0.4] fm and is compatible with zero for r≥0.6 fm within sizable errors. Lattice study that would probe whether one-pion exchange dominates at large r would need higher accuracy.The problem is that the potential V(r) is not determined from the lattice for r<a, it might be affected by discretization effects at r≃a and the analytic form of r-dependence is not known apriori. This brings us to the second simplifying approximation(5)V(r)=Vreg.(r)+V1/r(r),Vreg.(r)=−Ae−(r/d)F, where we assume a certain form of the regular potential Vreg.(r) that has no singularity at r→0. The fits of the lattice potential for various choices of the parameter F (5) and range r=[1,4] are shown in Fig. 3 (fits r=[2,4] lead to similar conclusions, as shown in Section S5 of Supplemental Material). The question if the potential contains also a singular piece 1/r can be addressed perturbatively, giving V1/rO(αs)(r)=0 and V1/r(r)=19[V0(r)+8V8(r)]=δa2108π2αs3r [31] for very small r. This follows from the interaction of b¯ and b within BB¯⁎, while other pairs among b¯bq¯q are at average distance of the order of B-meson size and do not lead to singularity at r→0 (Section S4 of Supplemental Material). Results below are based on Vreg+V1/r; we have verified that masses and cross-sections based solely on Vreg agree within the errors since V1/r is suppressed.The motion of B and B¯⁎ within the extracted potential V(r) is analyzed by solving the non-relativistic 3D Schrödinger equation [−12μd2dr2+l(l+1)2μr2+V(r)]u(r)=Wu(r) for the experimentally measured B(⁎) meson masses and 1/μ=1/mBexp+1/mB⁎exp. Here W=Etot−mB−mB⁎ is the energy with respect to BB¯⁎ threshold. The B and B¯⁎ can couple to Zb channel with JP=1+ in partial waves l=0,2. Below we extract (virtual) bound states and scattering rates for l=0, while l=2 is not discussed since V(r)+l(l+1)2μr2>0 is repulsive for all r.The wave functions of the Schrödnger equation render the phase shift δl=0(W) and BB¯⁎ scattering matrix S(W)=e2iδ0(W). Resonances above threshold do not occur for purely attractive s-wave potentials since there is no barrier to keep the state metastable, while (virtual) bound states below threshold may be present. Bound state (virtual bound state) corresponds to the pole of S(W) for real W<0 and imaginary momenta k=i|k| (k=−i|k|) of B in the center of momentum frame.We find a virtual bound state below threshold and its location is shown by diamonds in Fig. 4; this pole is present when the parameter F in V (5) is F<1.9. If this pole is close below threshold, it enhances the BB¯⁎ cross-section above threshold. For example, we find a virtual bound state with mass M slightly below threshold(6)M−mB−mB⁎=−13±10MeV, for the values of parameters F=1.3, A=1.139(50), d=1.615(71) in V (5). This state is responsible for a peak in the BB¯⁎ rate NBB¯⁎∝kσ∝sin2δ0(W)/k above threshold, shown in Fig. 5 for the central value of parameters. The shape of the peak resembles the Zb(10610) peak in the BB¯⁎ rate observed by Belle (Fig. 2 of [3]).The significantly attractive BB¯⁎ potential (Fig. 3) and the resulting virtual bound state (diamonds in Fig. 4) could be related to the existence of Zb in experiment. The reliable relation between both will be possible only when simplifications employed here will be overcome in the future simulations. We note that Zb(10610) was found as a virtual bound state slightly below threshold also by the re-analysis of the experimental data [4] when the coupling to bottomonium light-meson channels was turned off [4] (the position of the pole is only slightly shifted when this small coupling is taken into account).Surprisingly, the strongly attractive potential V(r) (5) leads also to a deep bound state at M−mB−mB⁎=−411±20 MeV. Such a state was never reported by experiments. If our approach can be trusted for such deep bound states and if such a bound state exists, it could be searched for in Zb→ϒ(1S)π+ decays. The invariant mass distribution observed by Belle is indeed not flat (Fig. 4a of [2]) and it would be valuable to explore if some structure becomes significant at better statistics.The exotic Zb resonances were observed only by Belle, so their confirmation by another experiment would be highly welcome. LHCb could try to search for it in inclusive final state BB¯⁎.7Comparison with a previous lattice studyOnly one preliminary lattice study [18] of this channel was reported up to now, considering heavier mπ=480 MeV and twisted-mass fermions. It employed operators OBB¯⁎ and Oϒπ(0), while Oϒπ(p≠0) and Oϒb1(0) were omitted. Two eigenstates are interpreted as ϒπ(0) and BB¯⁎. The resulting potential V(r)=EBB¯⁎(r)−mB−mB⁎ (red line in Fig. 1 of [18]) is attractive, but it is weaker than our potential for small r/a≃1,2. This difference could be a consequence of a very different mπ or a choice of the fitting range for EBB¯⁎ at small r. We note that the plateaus for EBB¯⁎ form relatively late in time and large tmin is essential for the reliable extraction of the potential at small r in our study.22Our effective energies are shown in Fig. S2a of the supplemental material and require fits with tmin/a≃11 for r=1,2. Effective energies of the study [18] are not available in the literature. The weaker potential and its parametrization via V(r)=−αre−(r/d)2 resulted in one bound state at M−mB−mB⁎=−58±71 MeV in [18], while our binding energies are shown in Fig. 4 for various parametrizations. Both lattice studies agree on the main conclusion, i.e. that the existence Zb is related to the attraction of B and B¯⁎ at small r. Further lattice studies, including the comparison of the potentials at similar mπ, are highly awaited.8ConclusionsWe presented a lattice QCD study of a channel with quark structure b¯bd¯u, where Belle observed two exotic Zb hadrons. We find significantly attractive potential V(r) between B and B¯⁎ at small r when the total spin of the heavy quarks is equal to one. Dynamics of BB¯⁎ system within the extracted V(r) leads to a virtual bound state, whose mass depends on the parametrization of V. Certain parametrizations give a virtual bound state slightly below BB¯⁎ threshold and a narrow peak in BB¯⁎ rate just above threshold, resembling Zb in experiment.For quantitative comparison to experiment, future lattice studies need to explore how the dynamics of BB¯⁎ is influenced by the coupling to ϒπ channels, and by the component where the total spin of the heavy quarks is equal to zero. Derivation of the appropriate analytic form for V(r) would be very valuable.Declaration of Competing InterestThe authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.AcknowledgementsWe thank G. Bali, V. Baru, P. Bicudo, N. Brambilla, E. Braaten, C. Hanhart, M. Karliner, R. Mizuk, A. Peters and M. Wagner for valuable discussions. S.P. acknowledges support by Research Agency ARRS (research core funding No. P1-0035 and No. J1-8137) and DFG grant No. SFB/TRR 55. H.B. acknowledges support from the Scientific and Technological Research Council of Turkey (TUBITAK) BIDEB-2219 Postdoctoral Research Programme.Appendix ASupplementary materialSupplementary material related to this article can be found online at https://doi.org/10.1016/j.physletb.2020.135467. Appendix ASupplementary materialThe following is the Supplementary material related to this article.MMCSupplementary material contains more details on the symmetries and operators (S1), effective energies and overlaps of the eigenstates (S2), tabulated V(r) (S3), perturbative V(r) at small r (S4) and masses of bound states for various fits (S5).MMCReferences[1]BelleA.BondarPhys. Rev. Lett.1082012122001arXiv:1110.2251Belle, A. Bondar et al., Phys. Rev. Lett. 108, 122001 (2012), [arXiv:1110.2251].[2]BelleA.GarmashPhys. Rev. D912015072003arXiv:1403.0992Belle, A. Garmash et al., Phys. Rev. D91, 072003 (2015), [arXiv:1403.0992].[3]BelleA.GarmashPhys. Rev. Lett.1162016212001arXiv:1512.07419Belle, A. 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