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Bottomonium spectrum revisited Jorge Segovia∗ Physik-Department, Technische Universita¨t Mu¨nchen, James-Franck-Str. 1, 85748 Garching, Germany and Instituto Universitario de F´ısica Fundamental y Matema´ticas (IUFFyM) Universidad de Salamanca, E-37008 Salamanca, Spain Pablo G. Ortega† CERN (European Organization for Nuclear Research), CH-1211 Geneva, Switzerland David R. Entem‡ and Francisco Fernańdez§ 6 Grupo de F´ısica Nuclear and Instituto Universitario de F´ısica Fundamental y Matema´ticas (IUFFyM) 1 0 Universidad de Salamanca, E-37008 Salamanca, Spain 2 (Dated: January 21, 2016) n Werevisitthebottomoniumspectrummotivatedbytherecentlyexcitingexperimentalprogressin a theobservationofnewbottomoniumstates,bothconventionalandunconventional. Ourframework J is a nonrelativistic constituent quark model which has been applied to a wide range of hadronic 9 observablesfrom thelighttotheheavyquarksectorandthusthemodelparametersarecompletely 1 constrained. Beyond the spectrum, we provide a large number of electromagnetic, strong and hadronic decays in order to discuss the quark content of the bottomonium states and give more ] insights about thebetterway to determine theirproperties experimentally. h p PACSnumbers: 12.39.Pn,14.40.Pq,14.40.Nd,14.40.Rt - p Keywords: Potential models,Heavyquarkonia,Propertiesofbottom mesons,exoticmesons e h [ I. INTRODUCTION In 2008, the spin-singlet pseudoscalar partner ηb(1S) was found by the BaBar Collaboration with a mass v1 A. Experimental situation of 9388.9+−32..13 ± 2.7MeV [9]. A second measurement 3 of BaBar found a figure slightly higher 9394+−44..89MeV 9 Bottomonium, a bound system of a bottom (b) quark but perfectly compatible. A later measurement of the 0 and its antiquark (¯b), was discovered as spin-triplet CLEO[10]Collaborationgaveavalueof9391 6.6MeV. 5 states called Υ(1S), Υ(2S) and Υ(3S) by the E288 The Belle Collaboration [11], with a simult±aneous fit 0 . Collaboration at Fermilab in 1977 in proton scattering of its mass and width, obtains a value of mηb(1S) = 01 oonf inCvuaraianndtPmbastsaersgelatsrgsetrutdhyainng5mGueoVn[1p,ai2r]s. iLnaaterr,egtihmeye ((91400.82+.44.0±+41.5.5)M±e1V.8)foMreVthefowritdhteh.maTsshiasndis Γtηhbe(1Sm)os=t 6 −3.7−2.0 1 were better studied at various e+e− storage rings. The precisemeasurementoftheηb(1S)massandfurthermore twotripletP-wavestatesχ (2P)andχ (1P)withJ = provides its total decay width. : bJ bJ v 0, 1, 2 were discovered in radiative decays of the Υ(3S) In Ref. [12] the BaBar Collaboration searched for i X and Υ(2S) in 1982 [3, 4] and 1983 [5, 6], respectively. radiativedecaystotheη (1S)andη (2S)states. Despite b b Despite such early measurements, during the next r of their results are largely inconclusive, they observed a thirtyyearstherewerenosignificantcontributionstothe a signal of the η (2S) state with a mass over a range b spectrum of bottomonium. Only the radial excitations of approximately 9974 < m < 10015MeV. Then, ofthe vectorbottomoniumfamily Υ(4S),Υ(10860),and ηb(2S) the CLEO Collaborationpresented evidence for the first Υ(11020) were observed [7, 8]. This was largely because successful observation of η (2S) in Υ(2S) η (2S)γ b b theB-factorieswerenotusuallyconsideredidealfacilities → decays at a mass of 9974.6 2.3 2.1MeV [13]. And for the study of the bottomonium spectrum since their ± ± soon after that, the Belle Collaboration [11] reported energy was tuned to the peak of the Υ(4S) resonance, a signal for the η (2S) using the h (2P) η (2S)γ which decays in almost 100% of cases to a BB¯ pair. transition at a mabss of (9999.0 3.b5+2.8)M→eV.b This The situation has changed dramatically in the last ± −1.9 value is clearly incompatible with the previous one. An few years with many bottomonium states observed. analysis performed by Belle [14] with almost 17 times more data found no evidence for a signal in the energy region around 9975MeV, casting doubt on the CLEO result. ∗ [email protected] † [email protected] Experimentalists have been only able to distinguish ‡ [email protected] the Υ(13D ) state of the triplet Υ(13D ) [15, 16]. In 2 J § [email protected] Ref. [16] the J = 2 member of the Υ(13D ) spin-triplet J 2 was observed through the Υ(3S) γγΥ(13DJ) B. Theoretical tool(s) γγπ+π−Υ(1S) decay chain with a→significance of 5→.8 standard deviations including systematic uncertainties. The description of hadrons containing two heavy For the other two members of this spin-triplet, Υ(13D ) 1 quarks is a rather challenging problem from the point andΥ(13D ), the significancesweremuchlower,1.8 and 3 of view of QCD. One has to add the complications of 1.6 respectively, and thus no experimental observation a nonperturbative low-energy dynamics to those usually can be claimed. comingfromsolvingtheboundstateprobleminquantum Evidence for the lowest spin-singlet P-wave state, field theory. h (1P), was first reported by the BaBar Collaboration A proper relativistic quantum field theoretical treat- b in the transition Υ(3S) π0h (1P) π0γη (1S) [17]. ment of the heavy quarkonium system based on the b b → → They found a 3σ excess of events in the recoil mass Bethe-Salpeterequationhasprovedtobedifficultdespite distribution against π0 at a mass of (9902 4 2)MeV. its relatively success in the last few years [28–30]. ± ± ThisspinsingletP-wavestateisexpectedtobeveryclose The two most promising approaches to the bottomo- in massto the spin weightedaverageofthe triplet states nium bound state problem are Effective Field Theories m(13P ) = 9899.9MeV. The first significant signal (EFTs)andlatticegaugetheories. EFTsdirectlyderived J for this state come from the Belle Collaboration in the from QCD, like nonrelativistic QCD (NRQCD) [31, 32] (cid:10)Υ(5S) (cid:11)h (1P)π+π− transition [18]. They were also orpotentialnonrelativisticQCD(pNRQCD)[33,34](for b → abletodistinguishitsfirstradialexcitation,h (2P). The somereviewsseeRefs.[35,36]),disentanglethedynamics b measured masses of the h (1P) and h (2P) states were of the heavy quarks from the dynamics of the light de- b b 9898.25 1.06+1.03MeV and 10259.76 0.64+1.43MeV. grees of freedom efficiently and in a model-independent ± −1.07 ± −1.03 way. The fully relativistic dynamics can, in principle, Theproton–(anti-)protoncollidershavejoinedrecently be treated without approximations in lattice gauge the- in the search of bottomonium states. A clear example ories (see, for instance, Ref. [37] for a standard lattice is the observation of the χ (nP) states produced in bJ heavy quarkonium spectrum). Heavy quark calculations proton-proton collisions at the LHC at √s = 7TeV within these two approaches have experienced a consid- and recorded by the ATLAS detector [19]. These states erableprogressforstatesawayfromthreshold. However, have been reconstructed through their radiative decays the threshold regions remain troublesome for the EFTs to Υ(1S,2S) with Υ µ+µ−. In addition to the mass → as well as lattice-regularized QCD [38]. Moreover, the peaks corresponding to the decay modes χ (1P,2P) bJ → lattice calculations of excited states have been only re- Υ(1S)γ, a new structure centered at a mass of 10.530 ± cently pioneered and the full treatment of bottomonium 0.005 0.009GeV has been also observed, in both the ± on the lattice seems to be tricky (for a global picture on Υ(1S)γ and Υ(2S)γ decay modes. This structure has lattice-regularized QCD calculations and their complex- been assigned to the χ (3P) system. Soon after that, bJ ities in the bottomonium sector, the reader is referred the D0 Collaboration observed a peak in the Υ(1S)γ to [39–44] and references therein). final state at a mass of 10.551 0.014 0.017GeV [20] ± ± All this together explains why many of our expecta- which is compatible with the new state observed by tions in heavy quarkonia still rely on potential models. the ATLAS Collaboration. The LHCb Collaboration Potentialformulationshavebeensuccessfulatdescribing has recently determined the mass of the χ (3P) to be b1 theheavyquark-antiquarksystemsincetheearlydaysof m(χ (3P))=10515+2.2(stat)+1.5(syst)MeV [21]. b1 −3.9 −2.1 charmonium(seee.g.[45–58]). Moreover,thepredictions We have discussed in this section the whole spectrum withinthisformalismofheavyquarkoniumpropertiesre- of bottomonium reported in 2014 by the Particle Data lated with decays and reactions have turned to be very Group (PDG) [22] except two states: the X(10610)± valuable for experimental searches. One can mention, and X(10650)±. These two states were observed by for instance, the remarkable success of the 3P0 strong the Belle Collaboration [23] in the mass spectra of the decay model [59–65]. Finally, the easy way to extend π±Υ(nS) (n = 1, 2, 3) and π±h (mP) (m = 1, 2) pairs thequarkmodelfordescribingmultiquarksystemsmakes b that are produced in association with a single charged this framework a suitable one for exploratory purposes. pion in Υ(5S) decays. The measured masses and widths The results presented herein are based on a derived ver- of the two structures averaged over the five final states sionofthisapproach: anonrelativisticconstituentquark are (10607.2 2.0)MeV and (18.4 2.4)MeV for the model (CQM). X(10610)±,a±nd(10652.2 1.5)MeVa±nd(11.5 2.2)MeV Spontaneous chiral symmetry breaking of the QCD forX(10650)±. Theirlarg±emass,indicatingth±epresence Lagrangian together with the perturbative one-gluon oftwobottomquarksintheircomposition,togetherwith exchange (OGE) and the nonperturbative confining their possession of electrical charge, marks these states interaction are the main pieces of constituent quark as necessarily unconventional. Their proximity to the models. Using this idea, Vijande et al. [66] developed BB∗ and B∗B∗ thresholds makes their identification as a model of the quark-antiquarkinteraction which is able molecularstatesanattractivepossibility. However,there to describe meson phenomenology from the light to the is a strong discussion within the scientific community heavy quark sector. We have adopted this model and about other possible interpretations [24–27]. fine tune its parameterstoreproducethe highestexcited 3 Quark masses mn (MeV) 313 State JPC nL The. (MeV) Exp. (MeV) [22] ms (MeV) 555 ηb 0−+ 1S 9455 9398.0 3.2 mb (MeV) 5110 2S 9990 9999.0 ±3.5+2.8 ± −1.9 OGE rˆ (fm) 0.181 3S 10330 - 0 rˆg (fm) 0.259 α0 2.118 χb0 0++ 1P 9855 9859.44±0.42±0.31 Λ (fm−1) 0.113 2P 10221 10232.5 0.4 0.5 0 ± ± µ (MeV) 36.976 3P 10500 - 0 Confinement ac (MeV) 507.4 hb 1+− 1P 9879 9899.3 1.0 µc (fm−1) 0.576 2P 10240 10259.8 0±.5 1.1 ∆ (MeV) 184.432 3P 10516 ±- ± as 0.81 Υ 1−− 1S 9502 9460.30 0.26 ± 2S 10015 10023.26 0.31 Table I. Quark model parameters. ± 1D 10117 - 3S 10349 10355.2 0.5 ± 2D 10414 - states that appear in the light quark sector [67]. The 4S 10607 10579.4 1.2 ± 3D 10653 - reason for that lies in the fact that it is widely believed 5S 10818 10876 11 that confinement is flavor independent. Therefore, the ± 4D 10853 - interactions which largely determine the high energy 6S 10995 11019 8 spectrum of heavy quarkonia should be constrained also 5D 11023 - ± by the light quark sector. The quark model parameters relevant for this work χb1 1++ 1P 9874 9892.78±0.26±0.31 are shown in Table I. In the heavy quark sector chiral 2P 10236 10255.46 0.22 0.50 3P 10513 10515±.7+2.2+±1.5 [21] symmetry is explicitly broken and Goldstone-boson −3.9−2.1 exchanges do not appear. Thus, OGE and confinement η 2−+ 1D 10123 - b2 are the only interactions remaining. Explicit expressions 2D 10419 - of these interactions and a brief description of the 3D 10658 - potential is given in Appendix A. Further details about χ 2++ 1P 9886 9912.21 0.26 0.31 the quark model and the fine-tuned model parameters b2 ± ± 2P 10246 10268.65 0.22 0.50 can be found in Refs. [66–68]. ± ± 1F 10315 - Masses of meson states are a relevant piece of 3P 10521 - information about their structure. However, a more 2F 10569 - complete description can be achieved studying mesonic 4P 10744 - decays. In this way, we are checking particular regions 3F 10782 - of the wave function and not an average over the all Υ 2−− 1D 10122 10163.7 1.4 meson size as in the calculation of the mass spectrum. 2 ± 2D 10418 - Appendix B provides the necessary formulation to carry 3D 10657 - out the calculations presented herein on annihilation processes and on electromagnetic, strong and hadronic h 3+− 1F 10322 - b3 decays. 2F 10573 - Two sections of this manuscript are still to be 3F 10785 - introduced. InSec.IIwediscussourquarkmodelresults Υ 3−− 1D 10127 - andcomparethemwith the availableexperimentaldata. 3 2D 10422 - We finish summarizing and giving some conclusions in 1G 10506 - Sec. III. 3D 10660 - 2G 10712 - 4D 10860 - II. RESULTS AND DISCUSSION 3G 10904 - χ 3++ 1F 10321 - b3 TableII showsthe bottomoniumspectrum(uptospin 2F 10573 - J = 3) predicted by our constituent quark model. All 3F 10785 - world average masses reported in the PDG [22] are also shown. The masses for those states that are not yet TableII.Masses, in MeV,ofbottomonium states (uptospin considered as well established by PDG have been taken J =3) predicted byour constituent quark model. from the originalexperimental works. It is inferred from Table II that a global description of the bottomonium spectrumisobtainedby ourCQM.Adetaileddiscussion 4 about the particular features of our spectrum will be Initial state Final state Γ The. The. giveninthefollowingsubsections. Weshallalsocompute (keV) (1B0−2) decay properties of the studied mesons. η (1S) gg 20.18MeV 100.00 b γγ 0.69 3.4∼2 10−3 × total 20.18MeV 100.00 A. The ηb states η (2S) gg 10.64MeV 99.86 b γγ 0.36 3.38 10−3 Table II shows the predicted masses of the ηb states. γhb(1P) 2.85 2.68×10−2 The hyperfine mass-splitting of singlet-triplet states, i.e. γΥ(1S) 4.50 10−2 4.22×10−4 ∆mhf[ηb(nS)] = m(n3S1)−m(n1S0), probes the spin- ππηb(1S) ×11.27 10.58××10−2 dependence of bound-state energy levels and imposes total 10.66MeV 100.00 constraints on theoretical descriptions. For the ground η (3S) gg 7.94MeV 99.93 states, n=1, it is given experimentally by [22] b γγ 0.27 3.40 10−3 γh (1P) 8.40 10−3 1.06×10−4 ∆mhf[ηb(1S)]=62.3±3.2MeV, (1) γhbb(2P) × 2.60 3.27×10−2 γΥ(1S) 5.10 10−2 6.42×10−4 which is higher than the theoretical prediction of γΥ(2S) 9.20×10−3 1.16×10−4 EFTs, 41±11+−98 [69], and compatible with the lattice ππηb(1S) × 1.95 2.45×10−2 regularizedQCDresult,(60.3 7.7)MeV[42]. The Belle ππη (2S) 0.34 4.28×10−3 Collaboration, with a simulta±neous fit of the mass and totabl 7.95MeV ×100.00 width of the η (1S), reduces this hyperfine splitting and b obtains a value of (57.9 2.3)MeV [11]. The hyperfine Table III. Decay widths and branching fractions of annihila- ± mass splitting predicted by our quark model is 47MeV, tion rates, radiative decays and hadronic transitions for the higher than the result obtained by EFTs but still lower η states. There is noexperimental data available. b than the experimental data and lattice regularizedQCD computation. We predict for the η (2S) state a mass of 9990MeV b which is in good agreement with the last experimental measurement performed by the Belle Collaboration[11], (9999.0 3.5+2.8)MeV. The corresponding theoreti- ± −1.9 cal hyperfine mass splitting ∆mhf[ηb(2S)] = 25MeV It is important to mention here that, as we will is in excellent agreement with the experimental one explain next, we do not expect perfect agreement with ∆mhf[ηb(2S)] = 24.3+−44..05MeV [11]. It is worth to experiment for the pseudoscalar mesons. The worst mention that our splitting is also in very good agree- situation is found for the η (1S) state and then it is b ment with the latest results of lattice regularized QCD, alleviated with higher excitations. This trend is inferred ∆mhf[ηb(2S)]=(23.5 28.0)MeV [42]. fromthe discussionabove,andthe mostevidence of this − OnecanseeinTableIIthatourpredictedmassforthe appears in our calculation of the total decay width of η (3S) is 10330MeV. Its corresponding hyperfine mass the η (1S) state, 20.2MeV, which is a factor of two b b splitting is ∆m [η (3S)] = 19MeV, which follows the larger than the central value of the last experimental hf b expected trend. measurement [11], (10.8+4.0+4.5)MeV (see that our −3.7−2.0 The decay widths and branching fractions of annihi- theoretical result lies just above the upper limit of the lation rates, radiative decays and hadronic transitions error bar). The reason for that is the following: Our for the η states are given in Table III. In all cases, the CQM presents an OGE potential which has a spin-spin b most important contribution to the total decay width contact hyperfine interaction that is proportional to a comes from the gluon annihilation rate. Note that they Dirac delta function, conveniently regularized, at the are given in MeV whereas the rest of the quantities origin (see Eqs. (A1) and (A2)). The corresponding are given in keV. Beyond the gluon annihilation rates, regularization parameter was fitted to determine the the next significant decay channel shown in Table III hyperfine splittings between the n1S and n3S states 0 1 is the η (2S) into ππη (1S) with a width four times inthedifferentflavorsectors. Whilemostofthephysical b b higher than the most dominant electromagnetic transi- observables are insensitive to the regularization of this tions η (2S) h (1P)γ and η (3S) h (2P)γ. This delta term, those related with annihilation processes are b b b b → → behavior is due to the fact that we use the full expres- affectedbecausetheseprocessesaredrivenbyshortrange sionofEq.(B3)forthe E1radiativedecaysandnotonly operators [70, 71]. The effect is very small in the 3S 1 the leading term. The corrections over the low energy channel as the delta term is repulsive in this case. It is expansion are important for these excited states. The negligibleforhigherpartialwavesduetotheshieldingby hadronic transition of the η (3S) into the ππη (1S) fi- the centrifugal barrier. However, it is sizable in the 1S b b 0 nal channel presents a decay width of the same order of channel for which the delta term is attractive and the magnitude than that of the η (3S) h (2P)γ decay. sensitivitydecreasesasgoingupinhigherexcitedstates. b b → 5 n m(TMhee.V(h)b) mEx(pM.(heVb))[22] hm(n(M3PeJV))iThe. hm(n3PJ)iExp. [22] Initial state Final state (ΓkTeVhe). ( 1B0T−h2e). B(Ex1p0.−[121)] × × 1 9879 9899.3±1.0 9879 9899.87±0.27 hb(1P) ggg 35.26 44.68 - Ta32ble11I00V5214.60The10t2h5e9o.-8re±ti1c.a2l mas11se00s52,1460in MeV10,21o60f05.23t04he±±g90.r3o6und γγγtoχχηtbbba(01l1((S11PP))) 81..6115××74118300..−−962645 11..0496××105110500..−−030235 49.2±---5.7+−53..63 state and the first two excitations of h , compared with the b spin-averaged centroid, in MeV, of the corresponding triplet hb(2P) ggg 52.70 57.82 - Pco-lwleacvteedsitnatPesD.GW[22e].compare with the experimental data γγηηbb((21SS)) 1174..6900 1196..3315 4272..53±±130..85+−+−3367...1.387 γηb2(1D) 5.36 5.88 - B. The hb and χbJ states γγγπχχχπbbbh012b((((1111PPPP)))) 316...629481×××1110000.−−−54236 317...945918×××1110000.−−−59236 ---- total 91.14 100.00 - Table II shows the predicted masses of the singlet 1P and the triplet 3P states. They are in reasonable hb(3P) ggg 62.16 64.91 - 1 J γηb(1S) 7.96 8.31 - agreement with the experimental data. γηb(2S) 6.86 7.16 - The spin-singlet P-wave states, hb, are expected to lie γηb(3S) 12.27 12.81 - very close in mass to the spin-weighted average of the γηb2(1D) 0.35 0.37 - triplet P-wavestates, χbJ. This is because the hyperfine γηb2(2D) 4.72 4.93 - fcstvcoopaearnnlnitttithsrsthhoeeieneiedsg,sgqorfuioononfarurol3nePenPdae-oJdwfshitanstaavtghntaeeedtnse,woatsatnnahtrdaveaeneslttda.hfttuIehitnnvheficiesTtretisaxioctcbpnolteroewrrarieItdomsVepteerhoxnweinctseaidotlpciarnorditggomiaoiptnnphao,sabr.frwtoemiohOlltnaiohncsawheesl γγγγγγπχχχχχχπbbbbbbh012012b(((((((1112221PPPPPPP))))))) 315157......721793730177××××××1111111000000.−−−−−−44335346 315167......923727483930××××××1111111000000.−−−−−−50335346 ------- thetheoreticalexpectationsand,ontheotherhand,that total 95.76 100.00 - our spin-spin interaction is negligible for P-wave states, as should be. TableV.Decaywidthsandbranchingfractionsofannihilation It is also important to remark that our spin-averaged rates, radiative decays and hadronic transitions for the hb centroidoftheχ (3P)states,10516MeV,iscompatible states. The experimental data are from Ref. [11]. bJ butslightlyhigherthanthe valuecollectedbyPDG[22]. However, the centroid is expected to be close to the χ (3P) element of the spin-multiplet and the b1 experimental measurement of the mass of this state has decay processes shown in Table V which are still been recently reported by the LHCb [21] Collaboration experimentally missing. These are the ηb(3S), hb(3P), with a value of 10515.7+−23..29+−12..51MeV, which is in perfect ηb2(1D) and ηb2(2D). We have discussed already the agreementwith our prediction. Inaddition, we calculate ηb(3S) state and we shall postpone for later on our t8hMeTehiVnetrdaaen-cdmayumlwtχibip1dl(te3htPs)sap−nlidmttibχnbr0ga(sn3Pcah)si=nmg1χf3br2aM(c3tPeiV)on.−smofχabn1(n3iPh)ila=- dmabiseocsuuotnss.8io%nT,h7ae%bohaubn(t3dPt1h)3e%m1efDsoornaitnhsdarsa2dbDiraatnisvctehaitdneegscaforyafscittnihoteonstηhob2ef tion rates, radiative decays and hadronic transitions for ηb(1S), ηb(2S) and ηb(3S) states, respectively. Since thehbstatesareshowninTableV.Wepredicttotaldecay the radiative decay rates of the ηb(3S) into the already widths of about 100keV for the three hb states. Their observed hb(1P) and hb(2P) are, respectively, 0.0084 total decay widths are dominated by their annihilation and 2.60keV (see Table III), the decay chain hb(3P) → into gluons. ηb(3S)γ hb(2P)γγ appears as the most suitable → The Belle Collaboration has studied very recently way for observing the hb(3P) and ηb(3S) states. The the processes e+e− Υ(5S) hb(nP)π+π− branching fraction of the hb(3P) radiative decay into [ηb(mS)γ]π+π− provid→ing branchi→ng fractions for th→e γηb2(2D) is non negligible with a value of 5% and can decays hb(1P) ηb(1S)γ, hb(2P) ηb(1S)γ and represent an opportunity to observe the ηb2(2D) once hb(2P) ηb(2S→)γ [11]. One can see→in Table V that the hb(3P) is established. → our results are in reasonable agreement with experiment Tables VI and VII show E1 and M1 radiative decays exceptforthe caseh (2P) η (2S)γ whichis roughlya of the χ (nP) states with J = 0, 1, 2 and n = 1, 2, 3. b b bJ → factor of2 lower. It is worthto mentionthat the erroris These Tables also show some hadronic transitions (the biggerinthiscaseandtheexperimentalfigurealsoseems spin-nonflip ππ transitions) and the annihilation rates to be higher than other theoretical predictions [72]. into gluons. As one can see, the available experimental There are four bottomonium states involved in the data is scarce and mostly related with the E1 radiative 6 Initialstate Finalstate (ΓkTeVhe). ( 1B0T−h2e). B(Ex1p0.−[222)] Initial state Final state (ΓkTeVhe). ( 1B0T−h2e). × × χb0(1P) gg 2.00MeV 98.61 - χ (3P) gg 2.46MeV ×99.26 γγ 0.12 5.91 10−3 - b0 γΥ(1S) 28.07 × 1.38 1.76 0.30 0.18 γγ 0.15 6.06 10−3 total 2.03MeV 100.00 ± - ± γΥ(1S) 1.99 8.04×10−2 × γΥ(2S) 2.99 0.12 χb1(1P) qq¯+g 71.53 66.73 - γΥ(3S) 8.50 0.34 γΥ(1S) 35.66 33.27 33.9 2.2 ± γΥ(1D) 3.59 10−2 1.45 10−3 total 107.19 100.00 - × × γΥ(2D) 3.50 0.14 χb2(1P) γγggΥγ(1S) 3.08×381930..−16593 2.51×361180..−81733 19.1-- 1.2 ππππχχbb02((11PP)) 4.28×110.−163 41..6793××1100−−24 γhb(1P) 8.88×10−5 7.23×10−5 -± total 2.47MeV 100.00 total 122.84 100.00 - χ (3P) qq¯+g 124.53 83.59 b1 χb0(2P) gg 2.37MeV 99.17 - γΥ(1S) 4.17 2.80 γγ 0.14 5.85 10−3 - γΥ(2S) 4.58 3.07 × γΥ(1S) 5.44 0.23 0.9±0.6 γΥ(3S) 9.62 6.46 γγΥΥ((21SD)) 102..7840 3.09 100.−542 4.6±- 2.1 γΥ(1D) 4.80×10−2 3.22×10−2 ππγhππbχχ(bb120P((11)PP)) 42..0389××11000.−−7253 319...079119××××111000−−−625 --- γγγΥΥΥ(222((12DDD))) 103...213614 7.38×1020..−82542 total 2.39MeV 100.00 - ππχ (1P) 1.32 0.89 b1 ππχ (1P) 4.55 10−3 3.05 10−3 χb1(2P) γqq¯Υ+(1gS) 1069..1143 796..8547 9.2 - 0.8 totalb2 ×148.98 ×100.00 ± γΥ(2S) 15.89 11.91 19.9 1.9 γΥ(1D) 0.41 0.31 -± χb2(3P) gg 111.45 79.56 γγhΥb2((11PD)) 1.67×110.−264 1.25×100.−954 -- γγγΥ(1S) 4.10×150.−653 2.93×140.−033 ππππχχbb21((11PP)) 1.94×100.−574 1.45×100.−434 0.91±- 0.13 γγΥΥ((23SS)) 150..6328 47..0411 total 133.40 100.00 - γΥ(1D) 3.38 10−3 2.41 10−3 × × χb2(2P) gg 104.26 76.62 - γΥ(2D) 0.18 0.13 γγ 3.84 10−3 2.82 10−3 - γΥ (1D) 4.41 10−2 3.15 10−2 γΥ(1S) ×11.38 × 8.36 7.0 0.7 γΥ2(2D) × 0.79 × 0.56 ± 2 γΥ(2S) 17.50 12.86 10.6 2.6 γΥ(1D) 2.09 10−2 1.54 10−2 -± γΥ3(1D) 0.21 0.15 ππγγγγhhΥΥππbbχχ32((((bb211110PP((DD11))PP)) )) 6821....04876968×××××1111200000..−−−−03654653 4621....42135401×××××1111100000..−−−−52164653 ------ γπππtoΥπππtχχχa3l(bbb2012(((D111)PPP))) 14..8387××141141000...−−10569933 13..3142××101121000...−−90170433 tπoπtχalb2(1P) 1360..0479 1000..0306 0.51±- 0.09 Table VII. (Continuation) Decay widths and branching fractions of annihilation rates, radiative decaysand hadronic Table VI. Decay widths and branching fractions of annihila- transitions for the χbJ states. There is no experimental data tion rates, radiative decays and hadronic transitions for the available. χ states. The experimental data are from Ref. [22]. bJ the case as reflected in Table VI. Therefore, if we are overestimatingtheχ (2P)annihilationintogluons,this b0 decaysassociatedwiththedecaychannelsthroughwhich cannot be in a dramatic way. The second possibility the χ (1P) and χ (2P) were discovered in the early could be an error on the experimental measurements bJ bJ eighties. Our theoretical results are in reasonable which have, up to now, uncertainties in the order of the agreement with the experimental figures except for fifty percent. the case of the χ (2P), in which our predictions of One can inferred from the Tables VI and VII that b0 the branching fractions are much smaller than the the χb1(1P,2P,3P) and χb2(1P,2P,3P) mesons have experimental data. There could be two reasons for this. total decay widths of around 100 150keV whereas the − The first one could be related with an overestimation χb0(1P,2P,3P) states have total decay widths of about of the decay rate for the annihilation into gluons of the 2 2.5MeV. The contribution of the decay rate into − χb0(2P) state. However,This should havebeen reflected gluons is 99% in the case of the χb0 states compared alsointheχb0(1P)stateandevenmorestronglybecause with the 70−80% for the χb1 and χb2 mesons. the smaller number of open decay channels. This is not Special attention deserves Table VII in which we have 7 cwoalvleectsetdatetsh.e Wdeecahyopperotphaerttitehseotfhetohreetsicpainl-dtraitpalestho3wPn- Initialstate Finalstate (ΓkTeVhe). ( 1B0T−h2e). (B1E0x−p.2) × × in Table VII help experimentalists in carrying out an Υ(1S) e+e− 0.71 1.31 2.38 0.11 ± 3g 41.63 77.06 81.7 0.7 intensive study of them. Some of their radiative decays ± γgg 0.79 1.46 2.2 0.6 are dominant, with rates in the order of few keV. 3γ 3.44 10−6 6.37 10−6 ±- Therefore, these transitions still seem to be the best γηb(1S) 9.34××10−3 1.73××10−2 - way for disentangling the fine mass splittings. The spin- Υ(2S) e+e− 0.37 1.16 1.91 0.16 ± nonflip ππ transitions χ (3P) χ (1P), χ (3P) 3g 24.25 75.83 58.8 1.2 b0 → b0 b1 → γgg 0.46 1.44 8.8 ±1.1 χb1(1P) and χb2(3P) χb2(1P) have decay rates lower 3γ 2.00 10−6 6.25 10−6 ±- abnudt tinhutshceasnabmeeaolsroduers→eodffmorasgtnuidtuydineg, tahreou3nPds1pi−n-t2rkipelVet, γγχχbb01((11PP)) × 11..0894 × 35..4715 36..89±±00..44 staWtees.finish this Section calling the attention of the γγγηχηbbb((221(SS1P))) 55..8605××11200.−−0842 1.81×1060..−15803 0.11±7.105.0±-4+−000..3.00575 [12] ππΥ(1S) 8.57 26.80 26.45 0.48 reader to the fact that our prediction of the spin-nonflip ± ππ transitions between χ states follow a particular Υ(3S) e+e− 0.27 1.33 2.18 0.20 bJ ± path: the decay rates between states with Ji = Jf are γ3ggg 108..3766 912..7372 03.59.77±02..168 ordersofmagnitudehigherthanthosewithJi =Jf. This 3γ 1.55 10−6 7.63 10−6 ±- 6 × × is so because the first type of transitions goes through γχb0(1P) 0.15 0.74 0.27±0.04 tthhee steecrmondwittyhpethoef ctoraenffiscitieionntsCin1vooflvEeqo.n(lyB2t9h)ewtehremreaosf γγγχχχbbb120(((112PPP))) 8.27×1010..−12612 005...479196 00..590.999±±±000...10635 Eq. (B29) with constant C2 that is much smaller. The γχb1(2P) 2.13 10.48 12.6±1.2 dctroeancnasstyiatniwothnCerΥ1e2ai(ss1Dtfiht)etecdoetπffihπrcΥoieu(ng1thS)Ct.h2eIitsΥisfi(2trtSeem)da→trhkraoπbuπlegΥht(h1taShte), γγγγηηχηbbbb(((2231(SSS2P)))) 165...157080×××1112000.−−−56242 53..4214××1110200..−−268023 0.058±1<30..1001-±.066+−1200.6..001146 [12] → ππΥ(1S) 1.77 8.71 6.57 0.15 as seen in Table VI, we obtain good agreement in those ± ππΥ(2S) 0.42 2.07 4.67 0.23 cases in which experimental data are available. ± Table VIII. Decay widths and branching fractions of annihilation rates, radiative decays and hadronic transitions for C. The S-wave Υ levels the Υ(1S), Υ(2S) and Υ(3S) states. The experimental data are from Refs. [12, 22]. In this case, we have calculated the branchingfractionsusingtheexperimentaltotaldecaywidths The success of QCD-inspired potentials is due largely of PDG2014. tothefruitfuldescriptionandpredictionoftheproperties of the the S-wave ψ and Υ states. One can see in Table II that the masses of Υ(1S), Υ(2S) and Υ(3S) located experimentally at 9.46, 10.02 and 10.35GeV are mesons. Ifwecomputethedecaywidthusingthemasses reasonably well reproduce in our quark model: 9.50, predictedbypNRQCD,ourresultis0.014keVandagrees 10.02 and 10.35GeV, respectively. with the pNRQCD value. WeshowinTableVIIIthedecaywidthsandbranching As one can see in Table VIII, all branching fractions fractions of annihilation rates, radiative decays and of the Υ(2S) state which are related with annihilation, hadronic transitions for the Υ(1S), Υ(2S) and Υ(3S) E1 and M1 radiative decays and even the spin-nonflip states. Inordertoavoidunnecessaryuncertaintiesinthe ππ transitions are in reasonable agreement with the calculated branching fractions, we use in these cases the experimental data. In Ref. [12], Lees et al. have total decay widths available in PDG [22]. performed a study of radiative transitions between The annihilation rates of the Υ(1S) state are slightly bottomonium states with a huge amount of events lower than the experimental data but in reasonable recorded by the BaBar detector at the PEP-II B- agreement. Unfortunately, there is no experimentaldata factory at SLAC. Among their measurements, a value associated with the only radiative decay of the Υ(1S) of the Υ(2S) γη (1S) decay rate is reported and b → state. The Υ(1S) γη (1S) is an M1 transition and agrees, within errors, with our theoretical figure (see b → thus it is suppressed with respect the E1 decays and Table VIII). However, it is important to emphasize much more difficult to measure. This can be observed that this decay rate is far from being to be well in the cases of the Υ(2S) and Υ(3S) where transitions established. On the experimental side, the PDG of 2014 into η states are orders of magnitude smaller than the collects the branching fraction (3.9 1.5) 10−4 from b ± × transitions into χ states. Using pNRQCD [73], the an early measurement of the BaBar Collaboration [75] bJ authors of Ref. [74] have recently computed the decay despite of having the measurement of 2011 reported in rateoftheΥ(1S) γη (1S)transitionreportingavalue Ref. [12]. On the theoretical side, there are values in b of(0.01518 0.000→51)keV,whichishigherthanourquark lattice NRQCD [76], (5.4 1.8) 10−4, in continuum model pred±iction of 0.0093keV. This decay rate is pNRQCD [74], (1.88 8.3±4) 10×−4, and within quark extremelysensitiveto∼themassesoftheΥ(1S)andη (1S) models ranging from 0±.05 10×−4 to 18 10−4. b × × 8 Decaychain (%B1) (%B2) (%B3) (1B0T−h4e). BEx(p1.0[−747)] Initialstate Finalstate (ΓkTeVhe). ( 1B0T−h2e). (B1E0x−p.2) × × 222333SSS111 →→→111333PPP210 →→→111333SSS111 653...574051 33131...823778 222...444888 540...171442 360...682369+−+−+−000000......341341677454+++−−−000000......140130841958 Υ(4S) 3γ3e+γggge− 1.29 110500...−3250186 6117....24069620×××11110000−−−−9332 (1.57±0.0---8)×10−3 333333333333333333SSSSSSSSS111111111 →→→→→→→→→111222222333333333PPPPPPPPP210210210 →→→→→→→→→111111222333333333SSSSSSSSS111111111 1111202000055.........646447799080819466 3311132118600.........828933825776186434 222222111.........444444999888888333 000...002132000......236366714542251831 4.611303408.......1951296−+652725600+−+−+−+−+−+−+−..990000000000000092..............72215746221574±82858597174300+++++++−−−−−−−000000100000000...............311101021111720740716106682143- πγγγγγγγγγγγγηηηχχχχχχχχχπbbbbbbbbbbbbΥ(((210210210321((((((((((SSS3332221111PPPPPPPPPS))))))))))))) 314145......829278848048×××××××1111116110000000000.......−−−−−−04161112571187322222 2162752588522.............98040793728384953718789517×××××××××××××11111111111110000000000000−−−−−−−−−−−−−2554333444544 (1.22 0.0------------9) 10−2 Table IX. Radiative decay chains of the Υ(2S) and Υ(3S) ππΥ(2S) 0.24 1.17×10−3 (1.29±0.20)×10−2 states involving the χ (1P,2P) mesons. The branching × ± × fnr′a3cSt1io+nsγa),reanBd1B=3B=(nB3b(JSn1′3S→1 →m3µP+Jµ+−)γ.),FBor2t=heBt(hmeo3rPeJtic→al Υ(10860) γ3e+ggge− 1030...231538 423...542527××111000−−−424 (5.6±3.1--)×10−4 calculation,wetakethebranchingfraction 3fromPDG2014. 3γ 1.10 10−6 2.00×10−9 - WTaTnahedbelΥceeBox(56empXnlSSleSpe.ra)iNmCreToe11ehwl00nwle89atoia19btmrl85hoydraaasoBts11tuieaa00orsB89n,i79sast66hritnfea±±[oo7krr8Me22en]ttehi1BfVcer0aeo,8llmΥ7ler-9r(eeR5[p±s7Sueo9)f3rl].ttseaB[P7dnD17a1d0]nG1.b8d0Υy72160(9t61th±h±S4ee)1[82c1Bs2ut]raarBteenasrt. πγγγγγγγγγγγγηηηχχχχχχχχχπbbbbbbbbbbbbΥ(((210210210321((((((((((SSS3332221111PPPPPPPPPS))))))))))))) 125266......319228777695×××××××11111112110000000000.......−−−−−−84381222250862222222 2231221344411.............43905442701129359855730145×××××××××××××11111111111110000000000000−−−−−−−−−−−−−2554333444544 0.80 ------------ 0.09 PDG2014 values. ππΥ(2S) 17.89 3.25×10−2 1.17±0.20 ππΥ(3S) 5.56 1.01×10−2 0.72±+0.29 ππΥ(4S) 8.54 10−2 1.55×10−4 −-0.26 × × Υ(11020) e+e− 0.15 1.90 10−4 (1.6 0.5) 10−4 The nice agreement with experimental data seems to 3g 11.57 1.46×10−2 ± - × change for the Υ(3S) state. While we reproduce the γgg 0.22 2.78×10−4 - branchingfractionsforthe radiativedecaysofthe Υ(3S) 3γ 9.56 10−7 1.21×10−9 - iΥctpoielnnihoxfse(ttehrpIr1ooΥrnfeeSoΥed(rgχcr)3iD(lmtmbuS3aJeaooSne)c(nrnna2e)ddastmePeinaxsnrsΥb)ltdpteuoefio(eesdr9f2tmgr8χyaiSmumostbr)fmeaJoeetsg(.fsn2oi1nlt0talarPiiba1ntaotle4u)dednsdwisp,taaoetorhtlafwesrfietvodokΥeeeriBsc(f[atot27atahrerrS7Bnedea]d)tanEohmirsmntne1ihutCleyier2cwsooah-onahpldalnnsiailbicosaanblhnriteiwongihvrv1edteiaeloor2reatel1dcittvgoaiehuioinymncavsnasgneeiypldrilsuntttiaohhhbtttonooeeee-f πγγγγγγγγγγγγηηηχχχχχχχχχπbbbbbbbbbbbbΥ(((210210210321((((((((((SSS3332221111PPPPPPPPPS))))))))))))) 1155834.......18011305782228××××××××2711111116000000000000......−−−−−−−2232110756112222322 126343611145............434442433021673239899306×××××××××××××1111111111110000000000000.−−−−−−−−−−−−35555444544555 ------------- χ (1P,2P) states. This work includes the best obser- ππΥ(2S) 6.32 8.00 10−3 - bJ ππΥ(3S) 38.81 4.91×10−2 - vational significance of some transitions and provide the ππΥ(4S) 1.29 1.63×10−3 - most up-to-date derived branching fractions in the bot- × tomonium system. Table IX summarizes their primary Table XI. Decay widths and branching fractions of annihila- resultsandcompareswithourtheoreticalvalues. Ourre- tion rates, radiative decays and hadronic transitions for the sults are compatible within experimental errors in most Υ(4S),Υ(10860)andΥ(11020)states. Theexperimentaldata of the cases but some discrepancies are also found. are from Refs. [12, 22]. In this case, we have calculated the Above the BB¯ threshold, there are three more states branchingfractionsusingtheexperimentaltotaldecaywidths well established in the PDG with quantum number of PDG2014. JPC = 1−−. They are the so-called Υ(4S), Υ(10860) and Υ(11020), being the last two natural candidates for the Υ(5S) and Υ(6S), respectively. The B-factories 9 scanned again the energy range above open-bottom threshold. The BaBar Collaboration [78] performed Meson State Channel Γ3P0 B3P0 BExp. [22] Υ(4S) 43S B+B− 10.41 50.54 51.3 0.6 1 a comprehensive scan between 10.54 and 11.2GeV, B0B¯0 10.18 49.46 48.7±0.6 followed by an eight-point scan in the proximity of the ± BB 20.59 100.00 >96 Υ(6S) peak. The Belle Collaboration [79] acquired 20.5 2.5 total 20.59 100.00 - nine points over 10.80 11.02GeV, as well as spread ± over seven additional po−ints more focused on the Υ(5S) Υ(10860) 53S1 BB 6.22 22.29 5.5 1.0 BB∗ 11.83 42.41 13.7± 1.6 peak. Both scans suggest that the simple Breit-Wigner B∗B∗ 0.09 0.32 38.1±3.4 parametrization, previously used to model the peaks ± eonboseurgvhedfoinr tthheeCdLesEcOrip[7ti]oanndofCtUhSeBco[8m]pscleaxnsd,yisnnaomtigcsooind BBBss∗BBBss∗∗ 071...961655 2734...441251 1017...559±±002...578 the proximity of the B(∗)B¯(∗) and Bs(∗)B¯s(∗) thresholds. Bss(∗)Bss(∗) 9.76 34.98 19.9±3.0 The new data points on Rb = σ(b¯b)/σ(µµ) are better 55 28 total 27.89 100.00 -± modeledassumingaflatb¯bcontinuumcontributionwhich ± interferes constructively with the Υ(5S) and Υ(6S) Υ(11020) 63S1 BB 4.18 5.28 - BB∗ 15.49 19.57 - Breit-Wigner resonances, and a second flat contribution BB 40.08 50.64 - which adds incoherently. Such fits alter the PDG results 1 BB′ 3.95 4.98 - on the Υ(5S) and Υ(6S) peaks. Table X compares the 1 B∗B∗ 11.87 14.99 - theoretical prediction with the new parameters reported BsBs 0.07 0.09 - by BaBar and Belle, but also with the PDG’s values. BsBs∗ 1.50 1.89 - Table XI shows the decay widths and branching frac- B∗B∗ 2.02 2.56 - s s tionsofannihilationrates,radiativedecaysandhadronic 79 16 total 79.16 100.00 - ± transitionsfortheΥ(4S),Υ(10860)andΥ(11020)states. The experimental data are from Refs. [12, 22]. We have Table XII. Open b-flavored strong decay widths, in MeV, againused the experimentaltotal decay widths reported and branchings, in %, of the Υ(4S), Υ(10860) and Υ(11020) byPDG[22]inordertocalculatethe theoreticalbranch- states. Experimental data are taken from Ref. [22]. ing fractions. The theoretical total decay widths can be foundonTableXIIwhichagreeswellwiththeexperimen- tal values for the Υ(4S) and Υ(11020) states. Since the experimental data is scarce,we can only comment inter- mass of the qq¯-pair of the decaying meson [83]. See estingfeaturesofourtheoreticalresults. TheΥ(nS)with Appendix Bforfurther details. FollowingEq.(B14), the value of γ of the 3P model is 0.205in the bottomonium n=4, 5, 6haveradiativedecaysintotheχ (1P,2P,3P) 0 bJ sector. One can see that the general trend of the total states whose widths go from 0.01 to 1.5keV. Their M1 decay widths is well reproduced. radiative decays into the η states are 2 3 orders of b magnitude smaller. − The Υ(4S) is the first 1−− bottomonium state above We compare in Table XI the spin-nonflip ππ hadronic theBB¯ threshold,10.56GeV. Thisstateonlydecaysinto transitions of the Υ(4S), Υ(10860) and Υ(11020) states the BB¯ final channel. We have incorporated the isospin for which experimental data are available. One can breaking via the experimental masses. In Table XII see that our values agree reasonably well with the we compare the theoretical branching fractions with the experimental ones except in the case of the Υ(10860). experimental ones for the two possible channels B+B− WehaveseeninRefs.[80,81]thatthepresenceofhybrid and B0B¯0. Despite the mass of the Υ(4S) is very mesons close in mass to conventional quarkonium states closeto the thresholds,the difference betweenbranching leads to large enhancements in some hadronic transition fractions of both channels is negligible due to the small decay rates. We do not find any hybrid state around difference between masses of the B± and B0. the Υ(10860) resonance. Therefore, other mechanism The possible two-body final decay channels of the is needed in order to explain the large ππ decay rates Υ(10860)are BB, BB∗, B∗B∗, B B B B∗ B∗B∗. The s s s s s s observed experimentally. The authors of Ref. [82] have 3P decay model predicts a total width and branching 0 recently analyzed the Belle data on the cross section fractions that are compatible with the experimental of the process e+e− Υ(nS)ππ with n = 1, 2 and data except for two cases in which the disagreement is around the Υ(10860)→energy region. They found that quite strong. The decay channel B∗B∗ appears to be the experimental data is compatible with a tetraquark suppressed in the 3P model whereas seems to be the 0 interpretation for the Υ(10860). Further studies are dominant one attending to the experimental data. This needed to understand the anomalous ππ decay widths is due to the small value of the overlap integral between of the Υ(10860). the wavefunctions inourmodel. Onthe otherhand,the (∗) (∗) TableXIIshowstheopen-flavorstrongdecaywidthsof theoretical branching fraction (Υ(10860) B B ) s s B → the Υ(4S), Υ(10860)and Υ(11020) states. We calculate is roughly a factor 2 bigger than the one measured thesedecaysusingaversionofthe3P modelinwhichthe experimentally. Thisisbecauseweoverestimatetherates 0 strength γ of the decay interaction scales as the reduced oftheB B ,B B∗andB∗B∗finaldecaychannelsdespite s s s s s s 10 the order of magnitude is correctly given. Initial state Final state Γ The. The. There is no experimental data about the open b- (keV) ( 1B0−2) flavored strong decays of the Υ(11020) resonance. Only × η (1D) gg 0.37 2.07 b2 its total decay width is known experimentally [22] and γh (1P) 17.23 96.58 b thepredictionofthe3P0modelisinverygoodagreement ππηb(1S) 0.24 1.35 withsuchfigure. ThefinaldecaychannelsBB∗,BB1and total 17.84 100.00 B∗B∗ appear to be dominant in our model. The partial η (2D) gg 0.67 3.57 widths of the remaining decay channels are an order of b2 γh (1P) 4.15 22.13 magnitude smaller. b γh (2P) 11.66 62.18 b γh (1F) 2.20 11.73 b3 γΥ (1D) 1.27 10−4 6.77 10−4 2 D. The D-wave levels γΥ (1D) 5.30×10−6 2.83×10−5 3 ππη (1S) 6.44×10−2 × 0.34 b Up to now, we have presented our theoretical results ππηb(2S) 8.39×10−3 4.47 10−2 × × total 18.75 100.00 fortheS-andP-wavebottomoniumstates. Itisofsome interest to go beyond this. For instance, a key test of Table XIII. Decay widths and branching fractions of annihi- the nonrelativistic potential description of bottomonium lationrates,radiativedecaysandhadronictransitionsforthe is the confirmation of the predicted D-wave levels. For η (1D) and η (2D) states. There is no experimental data thisreasonweperforminthis sectionatheoreticalstudy b2 b2 available. of the mesons η (1D ), Υ(3D ), Υ (3D ) and Υ (3D ). b2 2 1 2 2 3 3 Despite the S-wave and P-wave bottomonium states were first observed in the 1970s and 1980s, the triplet Υ(13D ) has been observed recently [15] distinguishing Υ, Υ and Υ mesons. As one can see in the Table, J 2 3 only the Υ(13D ) state [16]. The mass of the Υ (1D) the di-electron decay rates of the Υ D-wave states are 2 2 was measured to be (10164.5 0.8 0.5)MeV. One two orders of magnitude smaller than those of the S- ± ± can consider that the mass of the spin-triplet Υ(13D ) wave states (see Tables VIII and XI). The Υ family can J shouldbearoundthisvalueassumingthattherelativistic be studied easily via e+e− annihilation as they have the corrections are small in the bottomonium sector. Our same quantum numbers of the emitted virtual photon. theoreticalmass for this spin-triplet is 10123MeVwhich However, the production rate in this reaction is related islowerthantheexperimentaldataandalsowithrespect with the leptonic width and we have seen that they are other theoretical predictions [47, 49, 50, 52, 84–86]. very small for the Υ D-wave states. This is the reason It is inferred from Table II that the next set of whythereisnoexperimentalconfirmationofthe1−− D- spin-triplet D-wave levels is expected in the range of wave states. At this point, it is also worthy to remind 10419MeV, and for the second radial excitation in the that the di-electron width of a QQ¯ meson is orders of rangeof10658MeV. Contrarytothe1Dmultiplet,these magnitudelargerthanthecorrespondingoneforamulti- two values are in better agreement with those predicted quark system [87]. by other quark models. The fine structure within the We have mentioned above that the constant C of 2 D-wave multiplets is predicted to be somewhat smaller Eq. (B29) is fixed through the transition Υ (1D) 2 → in the present model than in those of other authors, but ππΥ(1S). This decay implies a D S transition and → there is a general agreement about these mass-splittings thus it is the cleanest way to determine the constant to be in the order of 10MeV. C . Moreover, it is the only decay of this kind that 2 ± Despite the spin-singlet D-wave states, η , are ex- presents a measurement of its branching fraction in the b2 pected to lie very close in mass to the spin-weighted PDG[22]. AsonecanseeinTableXIV,thedecaywidths average of the spin-triplet D-wave states, Υ , they are of the Υ(1D) ππΥ(1S), Υ (1D) ππΥ(1S) and J 2 → → still missing experimentally. One possibility to find the Υ (1D) ππΥ(1S) transitions are very similar, in the 3 → 11D and 21D states is studying the E1 radiative de- orderoftenthsofkeV.Thepredictedbranchingfractions 2 2 cays h (2P) γη (1D) and h (3P) γη (2D). Our are in reasonable good agreement with the experimental b b2 b b2 → → prediction for both decays is in the order of 5keV (see upper limits [16]. Table V). However, a better possibility is studying the One can inferred from Table XIV that the radiative decays η (1D) γh (1P) and η (2D) γh (2P) be- decay rates of the 1D states are dominant. The Υ(1D) b2 b b2 b → → causethefinalstatesarewellestablishedinthePDGand state decays radiatively into the χ (1P) with J = bJ ourmodelpredictsdecayratesoffewtensofkeV(seeTa- 0, 1, 2; the Υ (1D) only for J = 1, 2; and the Υ (1D) 2 3 bleXIII).Thedecayrateoftheη (2D) γh (1P)tran- only for J = 2. The partial widths of the Υ(1D) b2 b → → sitionisanorderofmagnitudesmallerthanthe previous γχ (1P),Υ (1D) γχ (1P)andΥ (1D) γχ (1P) b0 2 b1 3 b2 → → ones and thus its observation seems to be complicated. processes are the largest ones with values around 20 − Table XIV shows the decay widths and branching 25keV. Aninterestingfeaturecanbededucedfromhere, fractions of annihilation rates, radiative decays and the strength of the radiative decay into χ final meson bJf hadronic transitions for the 1D and 2D states of the depends on the total-spin J of the initial Υ being of i Ji