χ b (3P) Multiplet Revisited: Ultrafine Mass Splitting and Radiative Transitions
χχ b (3 P ) Multiplet Revisited: Ultrafine Mass Splitting and Radiative Transitions
Muhammad Naeem Anwar , , , ∗ Yu Lu , † and Bing-Song Zou , ‡ CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China University of Chinese Academy of Sciences, Beijing 100049, China Institute for Advanced Simulation, Institut f¨ur Kernphysik and J¨ulich Centerfor Hadron Physics, Forschungszentrum J¨ulich, D-52425 J¨ulich, Germany Helmholtz-Institut f¨ur Strahlen- und Kernphysik and Bethe Center for Theoretical Physics, Universit¨at Bonn, D-53115 Bonn, Germany (Dated: February 15, 2019)Invoked by the recent CMS observation regarding candidates of the χ b (3 P ) multiplet, we analyze the ultrafineand mass splittings among 3 P multiplet in our unquenched quark model (UQM) studies. The mass di ff erence of χ b and χ b in 3 P multiplet measured by CMS collaboration (10 . ± . ± .
17 MeV) is very close to our theoret-ical prediction (12 MeV). Our corresponding mass splitting of χ b and χ b enables us to predict more preciselythe mass of χ b (3 P ) to be (10490 ±
3) MeV. Moreover, we predict ratios of the radiative decays of χ bJ ( nP ) candi-dates, both in UQM and quark potential model. Our predicted relative branching fraction of χ b (3 P ) → Υ (3 S ) γ is one order of magnitude smaller than χ b (3 P ), this naturally explains the non-observation of χ b (3 P ) in recentCMS search. We hope these results might provide useful references for forthcoming experimental searches. PACS numbers:
I. INTRODUCTION
The excited P -wave bottomonia, χ bJ (3 P ), are of special in-terest, since they provide a laboratory to test (and model) thenon-perturbative spin-spin interactions of heavy quarks. Veryrecently, the CMS collaboration observed two candidates ofthe bottomonium 3 P multiplet, χ b (3 P ) and χ b (3 P ), throughtheir decays into Υ (3 S ) γ [1]. Their measured masses andmass splitting are M [ χ b (3 P )] = (10513 . ± . ± .
18) MeV , M [ χ b (3 P )] = (10524 . ± . ± .
18) MeV , ∆ m ≡ m ( χ b ) − m ( χ b ) = (10 . ± . ± .
17) MeV . (1)There are some earlier measurements related to χ bJ (3 P )mass by ATLAS [2], LHCb [3, 4], and D0 Collaborations [5].However, these measurements can not distinguish between thecandidates of χ bJ (3 P ) multiplet. The recent CMS analysis [1]is higher resolution search, and hence, is able to distinguishbetween χ b (3 P ) and χ b (3 P ) for the first time.In this paper we intend to compare our unquenched quarkmodel studies with this recent measurement, and make moreprecise prediction for the mass of the other 3 P bottomonium( χ b ) by incorporating the measured mass splitting. We alsomake an analysis of the ultrafine splitting of P -wave bot-tomonia, which enlighten the internal quark structure of theconsidered bottomonium. In addition, we predict model-independent ratios of radiative decays of χ bJ ( nP ) candidates.Heavy quarkonium states can couple to intermediate heavymesons through the creation of light quark-antiquark pairwhich enlarge the Fock space of the initial state, i.e. theinitial state contains multiquark components. These multi-quark components will change the Hamiltonian of the poten- ∗ [email protected] † [email protected] ‡ [email protected] tial model, causing the mass shift and mixing between stateswith the same quantum numbers or directly contributing toopen channel strong decay if the initial state is above thresh-old. These can be summarized as coupled-channel e ff ects(CCE). When CCE are combined with the naive quark po-tential model, one gets the unquenched quark model (UQM).UQM has been considered at least 35 years ago by T¨ornqvist et al. [6–9].The physical or experimentally observed bottomoniumstate | A (cid:105) is expressed in UQM as | A (cid:105) = c | ψ (cid:105) + (cid:88) BC (cid:90) d p c BC ( p ) | BC ; p (cid:105) , (2)where c and c BC stand for the normalization constants ofthe bare state and the BC components, respectively. In thiswork, B and C refer to bottom and anti-bottom mesons, andthe summation over BC is carried out including all possiblepairs of ground-state bottom mesons. The | ψ (cid:105) is normal-ized to 1 and | A (cid:105) is also normalized to 1 if it lies below B ¯ B threshold, and | BC ; p (cid:105) is normalized as (cid:104) BC ; p | B (cid:48) C (cid:48) ; p (cid:105) = δ ( p − p ) δ BB (cid:48) δ CC (cid:48) , where p is the momentum of B mesonin | A (cid:105) ’s rest frame. The full Hamiltonian of the physical statethen reads as H = H + H BC + H I , (3)where H is the Hamiltonian of the bare state (see Ap-pendix A for details), H BC | BC ; p (cid:105) = E BC | BC ; p (cid:105) with E BC = (cid:113) m B + p + (cid:113) m C + p is the energy of the continuum state(interaction between B and C is neglected and the transitionbetween one continuum to another is restricted), and H I isthe interaction Hamiltonian which mix the bare state with thecontinuum. Since each quark pair creation model generatesits own vertex functions that in turn lead to specific real partsof hadronic loops, see Ref. [10] for related remarks.Here, for the bare-continuum mixing, we adopt the widelyused P model [11]. In this model, the generated quark pairs a r X i v : . [ h e p - ph ] F e b have vacuum quantum numbers J PC = ++ which in spectro-scopical notation S + L J equals to P . A sketch of P modelinduced mixing is shown in Fig. 1. The interaction Hamilto-nian can be expressed as H I = m q γ (cid:90) d x ¯ ψ q ψ q , (4)where m q is the produced quark mass, and γ is the dimen-sionless coupling constant. The ψ q ( ¯ ψ q ) is the spinor field togenerate anti-quark (quark). Since the probability to gener-ate heavier quarks is suppressed, we use the e ff ective strength γ s = m q m s γ in the following calculation, where m q = m u = m d is the constituent quark mass of up (or down) quark and m s isstrange quark mass. P P BBi f
FIG. 1: Sketch of coupled-channel e ff ects in P model. i and f respectively denote the initial and final states with same J PC and B ¯ B stands for all possible B meson pairs. The mass shift caused by the BC components and theprobabilities of the b ¯ b core are obtained after solving theSchr¨odinger equation with the full Hamiltonian H . They areexpressed as ∆ M : = M − M = (cid:88) BC (cid:90) d p | (cid:104) BC ; p | H I | ψ (cid:105) | M − E BC − i (cid:15) , (5) P b ¯ b : = | c | = + (cid:88) BCLS (cid:90) d p p (cid:104) BC ; p | H I | ψ (cid:105) | ( M − E BC ) − , (6)where M and M are the eigenvalues of the full ( H ) andquenched / bare Hamiltonian ( H ), respectively. See Ap-pendix B or Refs. [12, 13] for derivation of above relationsand UQM calculation details. Numerical values of ∆ M and P b ¯ b of every coupled channel for the bottomonia below B ¯ B threshold are given in Table I, which will be used in the fol-lowing discussions. II. MASS SPLITTING AND χ b (3 P ) After the recent CMS observation [1] of χ b (3 P ) and χ b (3 P ), χ b (3 P ) is now the only missing candidate in spin-triplet 3 P bottomonium. With the reference of observed masssplitting of 1 P , 2 P and 3 P multiplets, one can predict the mass of χ b (3 P ). It requires a constraint that the mass splittings for1 P , 2 P and 3 P multiplet should be the same [14].Triggered by the above mentioned experimental search, weanalyze our UQM studies regarding the bottomonium spec-trum [12, 15]. We notice that the measured mass splitting be-tween χ b (3 P ) and χ b (3 P ) is (10 . ± . ± .
17) MeV whichdi ff ers only by 1 MeV from our UQM prediction [12]. Ourprediction for the mass splitting of χ b (3 P ) and χ b (3 P ) is 23MeV, see Table II. With the reference of the observed massesof the other two candidates of spin-triplet 3 P bottomonium,this mass splitting helps us to predict precisely the mass ofunknown χ b (3 P ) to be M [ χ b (3 P )] = (10490 ±
3) MeV . (7)The uncertainty in above prediction is calculated by takingthe same percentage error [of O (10%)] in our mass splittingswhich we observed from CMS measurement. Our mass pre-dictions respect the conventional pattern of splitting and sup-port the standard mass hierarchy, where we have M ( χ b ) > M ( χ b ) > M ( χ b ), which is in line with CMS measurement. Acomparison of our UQM mass splittings with other quenchedquark model predictions is given in Table II. III. ULTRAFINE SPLITTING IN UQM
It is more informative if we study the mass splitting in amultiplet instead of the total mass shift caused by the inter-mediate meson loop. For the states quite below the threshold,there is an interesting phenomenon [16]: the magnitude of themass splitting is suppressed by the probability of the bottomo-nium core, P b ¯ b , if we turn on the meson loop.There is also a pictorial explanation for this. Since underthe potential model, the mass splitting δ M originates fromthe fine splitting Hamiltonian H I . Up to the first order pertur-bation, we have δ M = (cid:104) ψ | H I | ψ (cid:105) , where ψ is the two-bodywave function in the quenched potential model. Since one ofthe coupled-channel e ff ects is the wave function renormaliza-tion: (cid:104) ψ | ψ (cid:105) = P b ¯ b <
1, one would simply expect that the δ M will be suppressed by this probability.Moreover, due to the closeness of the spectrum of a multi-plet, we expect that the P b ¯ b of the states in a same multipletare nearly the same, i.e., δ M are all suppressed by a samequantity, leaving the relation δ M P ≡ (cid:104) M ( χ b ) + · M ( χ b ) + · M ( χ b ) (cid:105) − M ( h b ) = ff ects are turned on. Dueto the remarkably small δ M P , we refer it as “ultrafine split-ting”. In our calculation, however, due to the finite size ofthe constituent quark, which is reflected by the smeared delta In the quenched limit, where the sea quark fluctuations are neglected, thisdi ff erence becomes six times larger. Initial States B ¯ B B ¯ B ∗ + h . c . B ∗ ¯ B ∗ B s ¯ B s B s ¯ B ∗ s + h . c . B ∗ s ¯ B ∗ s Total − ∆ M P b ¯ b − ∆ M P b ¯ b − ∆ M P b ¯ b − ∆ M P b ¯ b − ∆ M P b ¯ b − ∆ M P b ¯ b − ∆ M P b ¯ b (%) η b (1 S ) 0 0 7.8 0.45 7.6 0.43 0 0 3.3 0.17 3.3 0.16 22.0 98.79 η b (2 S ) 0 0 16.5 1.81 15.7 1.62 0 0 5.2 0.43 5.0 0.4 42.4 95.74 η b (3 S ) 0 0 24.5 5.01 22.3 3.98 0 0 5.4 0.63 5.1 0.55 57.4 89.83 Υ (1 S ) 1.4 0.09 5.4 0.33 9.2 0.54 0.6 0.03 2.3 0.12 3.9 0.2 22.8 98.69 Υ (2 S ) 3.0 0.37 11.4 1.29 18.9 2.02 0.9 0.08 3.5 0.31 5.9 0.49 43.8 95.44 Υ (3 S ) 4.8 1.25 17.2 3.71 27.1 5.07 1.0 0.13 3.7 0.45 6.1 0.67 60.0 88.71 h b (1 P ) 0 0 13.5 1.22 13.0 1.12 0 0 4.8 0.35 4.6 0.33 35.8 96.99 h b (2 P ) 0 0 21.9 3.51 20.3 2.96 0 0 5.6 0.59 5.3 0.52 53.1 92.43 h b (3 P ) 0 0 38.0 19.75 29.5 9.04 0 0 5.4 0.67 5.0 0.54 77.9 70.0 χ b (1 P ) 4.1 0.45 0 0 21.4 1.74 1.3 0.11 0 0 7.8 0.52 34.6 97.18 χ b (2 P ) 9.3 1.85 0 0 31.1 4.13 2.1 0.26 0 0 8.4 0.77 50.9 92.98 χ b (3 P ) 25.5 34.08 0 0 40.7 8.07 2.3 0.31 0 0 7.6 0.62 76.1 56.92 χ b (1 P ) 0 0 10.8 1.03 15.5 1.27 0 0 3.7 0.28 5.6 0.38 35.5 97.03 χ b (2 P ) 0 0 19.7 3.38 22.1 3.0 0 0 4.8 0.53 6.0 0.56 52.6 92.53 χ b (3 P ) 0 0 37.4 21.9 29.7 7.54 0 0 4.8 0.64 5.4 0.54 77.4 69.38 χ b (1 P ) 3.4 0.31 9.8 0.85 13.6 1.24 1.2 0.09 3.5 0.25 4.7 0.35 36.4 96.91 χ b (2 P ) 5.3 0.89 14.6 2.23 23.2 3.62 1.3 0.15 3.8 0.39 5.8 0.6 54.1 92.13 χ b (3 P ) 12.3 – 23.3 12.50 36.2 16.34 1.3 0.23 3.6 0.53 5.6 0.82 82.2 69.57TABLE I: The mass shift (in MeV) and probability (in %) of every coupled channel for the bottomonia below B ¯ B threshold. Note that since h b (3 P ) has no coupling to B ¯ B , even though h b (3 P ) is above B ¯ B threshold, the probability is still well-defined. However, since χ b (3 P ) couplesto B ¯ B channel and lies above this threshold, causing di ffi culty to the renormalization of the wave function. We make the assumption that therenormalization caused by B ¯ B channel can be discarded, see Sec. IV for related discussions.Mass Splitting Our UQM [12] GI [18] Modified GI [19] CQM [20] Exp [1] χ b (3 P ) − χ b (3 P ) 23 16 14 13 − χ b (3 P ) − χ b (3 P ) 12 12 12 9 (10 . ± . ± . P -wave bottomonia in our UQM [12], Godfrey-Isgur (GI) model [18], Modified GI model [19], andconstituent quark model (CQM) [20]. The later three models are regarded as quenched quark models. term, ˜ δ ( r ), instead of the true Dirac term in the spin depen- Such a smearing of the Dirac delta term incorporating the contact spin-spin interaction with a finite range 1 /σ is essential to regularize the deltafunction [17]. dent potential V s ( r ) = m b (cid:20) (cid:32) α s r − λ r (cid:33) L · S + πα s δ ( r ) S b · S ¯ b + α s r (cid:32) S b · S ¯ b + ( S b · r )( S ¯ b · r ) r (cid:33) (cid:21) , (9)˜ δ ( r ) ≡ (cid:18) σ √ π (cid:19) e − σ r , where α s and λ are strengths of the color Coulomb and lin-ear confinement potentials, respectively, and σ is related tothe width of Gaussian smeared function, the δ M P relation ofEq. (8) is already violated a little bit under the potential modelwhich can be seen from Table III (second column), where wealso include the corresponding experimental values. We canalso extract the threshold e ff ects by taking the mass shift ∆ M instead of M in δ M P calculations. The δ M P values obtainedin this way are also given in Table III (third column). Multiplet UQM prediction CCE contribution Experiment [21]1 P .
17 0 .
06 0 . P .
38 0 .
19 0 . . P − .
39 2 . − TABLE III: Ultrafine splitting ( δ M P in MeV) for the P -wave bot-tomonia. The second to fourth columns are our unquenched quarkmodel prediction, contribution from the coupled-channel e ff ects andexperimental results, respectively. The contribution from coupled-channel e ff ects can be obtained by replacing the mass of χ bJ ( nP ) bytheir mass shift ∆ M . Note that our results of M violate Eq. (8) a bitdue to finite size of the constituent quark, as discussed in the text. We can see from Table I that although the mass shift forthe P -wave multiplets is around 50 MeV, the modification ofEq. (8) is not very large, except δ M P (3 P ) which is far largerthan δ M P (2 P ) and δ M P (1 P ). A worth mentioning feature hereis the hierarchy of these ultrafine splittings originated from theCCE (third column of Table III), viz., δ M P (3 P ) > δ M P (2 P ) > δ M P (1 P ) , (10)which highlights that the coupled-channel e ff ects bring mesonmasses closer together with respect to their bare values [16].Since, for the P -wave states, no matter whether the thresh-old e ff ects are considered or not, h b is not a ff ected by the fineinteraction, i.e. the δ M =
0. Hence, the χ bJ ’s mass splittingare purely due to the P b ¯ b of each χ bJ . Therefore, the weightedprobability of the bottomonium core, (cid:101) P b ¯ b , for χ bJ ( nP ) mul-tiplets is simply defined as (cid:101) P b ¯ b = P b ¯ b ( χ bJ ). The weightedaverage probability for the S -wave bottomonia is discussed inAppendix C. From the Table IV, we can see that although the( (cid:101) P b ¯ b × δ M ) and δ M originate di ff erently; one from the po-tential model and the other purely from the coupled-channele ff ects, but they are approximately equal to each other. Theonly large deviation comes from χ bJ (3 P ).As explained above, this overall suppression is based onthe assumption that the (cid:101) P b ¯ b is the same (or approximately thesame) for a multiplet. Indeed, from Table I we can see thatthis is quite reasonable assumption for the states which are farbelow the threshold. But for the χ b (3 P ), the (cid:101) P b ¯ b is quite dif-ferent from that of χ b (3 P ), so this overall suppression doesnot make sense anymore. As a consequence, one should ex-pect relatively large deviation from the δ M P relation, as canbe seen from δ M P (3 P ) in Table III.The reason for this peculiar (cid:101) P b ¯ b is that even though the massof h b (3 P ) and χ b (3 P ) is larger than the χ b (3 P ), they do notcouple to the channel B ¯ B , and the next open channel B ¯ B ∗ issomewhat farther from them. A net e ff ect is that the (cid:101) P b ¯ b of χ b (3 P ) is larger than that of χ b (3 P ), breaking the (cid:101) P b ¯ b close-ness assumption. This strong coupling of χ b (3 P ) to B ¯ B is also reflected by the large mass shift caused by B ¯ B whichcan be seen from Table I. The observed mismatch between( (cid:101) P b ¯ b × δ M ) and δ M for χ bJ (3 P ) multiplet is a smoking gunof the threshold e ff ects which are beyond the quark potentialmodel.Recently, Lebed and Swanson also pointed out the remark-able importance of the P -wave heavy quarkonia [22]. For 1 P and 2 P charmonia, the ultrafine splitting is found to be as-tonishingly small. They argued that the ultrafine splitting canbe used to delve the exoticness of the observed structure inthe given multiplet [23]. According to their analysis [22], thequantity δ M n , L = , , ,... is found to be very small for any radialexcitation n , both for the b ¯ b and c ¯ c sectors. The obtained con-straint on the δ M n , L value is δ M n , L = , , ,... (cid:28) Λ QCD . (11)This conclusion follows from several theoretical formalismswhich do not consider coupled-channel e ff ects or long-distance light-quark contributions in terms of intermediatemeson-meson coupling to bare quarkonium states. As dis-cussed above, the operators corresponding to ultrafine split-ting involve spin-spin interactions which are suppressed by1 / m Q , the standard expansion parameter for the heavy quarko-nium, where m Q is the mass of heavy quark. According to ourpoint of view the above maxima is much large for the ultrafinesplitting of P -wave bottomonia, see Table III for experimentalcorroboration. The more tight constraint could be δ M n , L = , , ,... (cid:46) Λ m Q . (12)Since, quantitatively the P -wave excitation for the bottomo-nium is equal to Λ QCD , which describes the emergence of thedynamical QCD scale in above relation. The δ M n , L for thebottomonia with L = O (1 MeV), whichcan be verified from our analysis of Table III.The reason why δ M n , L = , , ,... is exactly zero in the quarkmodel is a consequence of the pure delta function nature ofthe S b · S ¯ b term of Eq. (9), which is a perturbative one gluonexchange e ff ect. The non-perturbative e ff ects can make anadditional contribution to this term, so that it is no longer apure delta function. This give rise to introduce the smear-ing of the delta function in the quark models [17, 22]. How-ever, one could use di ff erent non-perturbative forms for thespin-spin operator that contributes to the ultrafine splitting.For instance, the ultrafine splitting computed at next-to-next-to-next-to leading order (N LO) [24] in nonrelativistic QCD(NRQCD) [25, 26] is δ M n , L = = m b C F α s π ( n + (4 n l − N c ) , (13)where C F is the color factor of bottomonium, n l being thenumber of light fermion species appearing in loop corrections,and N c is the number of colors in QCD. The computed δ M n , L = values using NRQCD for the bottomonium (with m b = . α s ( m b ) = .
2) are; δ M P = .
77 keV, δ M P = . δ M P = .
47 keV [22]. The remarkable smallness
Channels δ M (cid:101) P b ¯ b ( (cid:101) P b ¯ b × δ M ) δ M (cid:101) P b ¯ b ( (cid:101) P b ¯ b × δ M ) δ M δ M Exp
GEM SHO Υ (1 S ) − η b (1 S ) 65.5 98.7 64.7 64.7 98.7 64.7 64.7 62.3 Υ (2 S ) − η b (2 S ) 30.7 95.5 29.3 29.4 95.9 29.4 29.5 24.3 Υ (3 S ) − η b (3 S ) 23.4 89.0 20.8 20.7 91.1 21.3 21.3 – χ b (1 P ) − h b (1 P ) -35.6 97.2 -34.6 -34.5 97.1 -34.6 -34.4 -39.9 χ b (1 P ) − h b (1 P ) -6.3 97.0 -6.1 -6.0 97.0 -6.1 -6.0 -6.5 χ b (1 P ) − h b (1 P ) 13.2 96.9 12.8 12.6 96.8 12.8 12.7 12.9 χ b (2 P ) − h b (2 P ) -31.2 93.0 -29.0 -28.9 93.4 -29.2 -29.1 -27.3 χ b (2 P ) − h b (2 P ) -5.4 92.5 -5.0 -4.9 93.0 -5.0 -5.0 -4.3 χ b (2 P ) − h b (2 P ) 12.2 92.1 11.2 11.2 92.7 11.3 11.2 8.8 χ b (3 P ) − h b (3 P ) -29.2 56.9 -16.6 -27.5 54.3 -15.8 -28.3 – χ b (3 P ) − h b (3 P ) -5.0 69.4 -3.5 -4.5 72.5 -3.6 -4.6 – χ b (3 P ) − h b (3 P ) 11.9 – – 7.5 – – 7.7 –TABLE IV: The mass splitting (in MeV) in a same ( n , L ) multiplet, where δ M , δ M and δ M Exp represent the mass splitting in potentialmodel, coupled-channel model and experiment, respectively. The (cid:101) P b ¯ b (in %) is the weighted average of the probability, which for P - and S -wave is (cid:101) P b ¯ b = P b ¯ b ( χ bJ ) and (cid:101) P b ¯ b = P b ¯ b ( Υ ) + P b ¯ b ( η b ), respectively. The details of the mass splitting are given in Appendix C, and theabsolute probabilities P b ¯ b are given in Table I. GEM and SHO stand for the Gaussian expansion method [27] and simple harmonic oscillatorapproximation, respectively, to fit the numerical wave functions. of these values strengthen the constraint on the δ M n , L = , , ,... values presented in Eq. (12). However, these NRQCD predic-tions are much smaller as compared to our UQM predictionsand corresponding experimental values, see Table III. In con-clusion, whatever the non-perturbative form for the spin-spinoperator is used, the δ M n , L = should be very small, hence sat-isfying the relation of Eq. (12) quantitatively. IV. RADIATIVE TRANSITIONS
Radiative transitions of higher bottomonia are of consider-able interest, since they can shed light on their internal struc-ture and provide one of the few pathways between di ff erent b ¯ b multiplets. Particularly, for those states which can not di-rectly produce at e + e − colliders (such as P -wave bottomonia),the radiative transitions serve as an elegant probe to exploresuch systems. In the quark model, the electric dipole ( E Γ ( n S + L J → n (cid:48) S (cid:48) + L (cid:48) J (cid:48) + γ ) = C f i δ S S (cid:48) e b α | (cid:104) ψ f | r | ψ i (cid:105) | E γ , (14)where e b = − is the b -quark charge, α is the fine structureconstant, and E γ denotes the energy of the emitted photon.The spatial matrix elements (cid:104) ψ f | r | ψ i (cid:105) involve the initial andfinal radial wave functions, and C f i are the angular matrix el- ements. They are represented as (cid:104) ψ f | r | ψ i (cid:105) = (cid:90) ∞ R f ( r ) R i ( r ) r dr , (15) C f i = max( L , L (cid:48) )(2 J (cid:48) + (cid:40) L (cid:48) J (cid:48) SJ L (cid:41) . (16)The matrix elements (cid:104) ψ f | r | ψ i (cid:105) are obtained numerically; forfurther details, we refer our studies [12, 30]. From Eq. (15),we know that the value of the decay width depends on the de-tails of the wave functions, which are highly model dependent.A model independent prediction can be achieved by focusingon the following decay ratios Γ (cid:0) χ bJ ( mP ) → Υ ( nS ) + γ (cid:1)(cid:14) Γ (cid:0) χ b ( mP ) → Υ ( nS ) + γ (cid:1) . (17)Since, in the quark model, the spatial wave function is thesame for the states in the same multiplet.From the above discussion, we know that the meson looprenormalizes the bottomnium wave function. When the chan-nel is above the corresponding open-bottom threshold (suchas B ¯ B here), the wave function cannot be normalized to 1,this is still an open problem (see e.g. Ref. [31]). On theother hand, the B ¯ B loop is still there, and have some CCE(such as mass renormalization). We make the assumption thatfor the states above threshold (such as χ b (3 P ) here), theseopen channels contribute equally to the wave functions of all χ bJ (3 P ) states. In fact this is a reasonable assumption, sincewe can see this from the Table I, the probability of B ¯ B isvanishingly small (0 .
31% and 0 . χ b (3 P ) and χ b (3 P ).With the latest CMS data [1] and the P b ¯ b in Table I, our pre-dictions of radiative decay ratios are listed in Table V. Fromthe Table I, one can see that the small P b ¯ b [ χ b (3 P )] makethe ratios in the last three rows notably larger than that ofthe potential model predictions, a peculiar feature of coupled-channel e ff ects which can be tested in the upcoming experi-ments. Decay Channel χ b : χ b : χ b Model Potential Model Unquenched Quark Model χ bJ (1 P ) → Υ (1 S ) + γ χ bJ (2 P ) → Υ (1 S ) + γ χ bJ (2 P ) → Υ (2 S ) + γ χ bJ (3 P ) → Υ (1 S ) + γ χ bJ (3 P ) → Υ (2 S ) + γ χ bJ (3 P ) → Υ (3 S ) + γ Γ (cid:0) χ bJ ( mP ) → Υ ( nS ) + γ (cid:1)(cid:14) Γ (cid:0) χ b ( mP ) → Υ ( nS ) + γ (cid:1) . For potential model calculations, theparameters and quenched Hamiltonian are same as Ref. [12]. Another worth noting result from Table V is the relative sizeof the ratios for χ b (3 P ), which from the coupled-channel cal-culations is roughly 1 : 6 : 12. This reflects that the χ b (3 P )has negligible radiative decay branching fraction with com-parison to χ b (3 P ) and χ b (3 P ). Compared with the poten-tial model, the suppression of the χ b (3 P )’s radiative width inthe UQM is more consistent with the non-observation of the χ b (3 P ) in the recent CMS search of χ bJ (3 P ) → Υ (3 S ) γ [1].This indicates that our UQM predictions are more reliablethan the naive quark potential models. V. CONCLUSIONS
The recent CMS study successfully distinguishs χ b (3 P )and χ b (3 P ) for the first time, and measures their mass split-ting which di ff ers only 1 MeV from our unquenched quarkmodel predictions. This measurement gives us confidence topredict mass of the lowest candidate of 3 P multiplet to be M [ χ b (3 P )] = (10490 ±
3) MeV, based on our unquenchedquark model results of the mass splittings of this multiplet.We also analyze the ultrafine splittings of P -wave bottomo-nia up to n = P -wave bottomoniashould be very small. This analysis leads us to conclude thatthe coupled-channel e ff ects play a crucial role to understandthe higher bottomonia close to open-flavor thresholds.At last, we predict here to some extent model-independentratios of the radiative decays of χ bJ ( nP ) candidates. A worthmentioning observation is that the coupled-channel e ff ects canenhance the radiative decay ratios of χ bJ (3 P ) as compared tothe naive potential model predictions. The relative branching fraction of χ b (3 P ) → Υ (3 S ) γ is negligible as compared to theother candidates of this multiplet, which naturally explains itsnon-observation in recent CMS search.We hope above highlighted features of coupled-channelmodel provide useful references for the understanding ofhigher P -wave bottomonia and can be explored in ongoingand future experiments. Acknowledgements
We are grateful to Timothy J. Burns, Feng-Kun Guo,Richard F. Lebed, and Thomas Mehen for useful discussionsand suggestions, and to Christoph Hanhart for careful read ofthis manuscript and valuable remarks. This work is supportedin part by the DFG (Grant No. TRR110) and the NSFC (GrantNo. 11621131001) through the funds provided to the Sino-German CRC 110 “Symmetries and the Emergence of Struc-ture in QCD”, and by the CAS-TWAS President’s Fellowshipfor International Ph.D. Students.
Appendix A: Bare Hamiltonian
Bare states are obtained by solving the Schr¨odinger equa-tion with the well-known Cornell potential [32, 33], which in-corporates a spin-independent color Coulomb plus linear con-fined (scalar) potential. In the quenched limit, the potentialcan be written as V ( r ) = − α r + λ r + c , (A1)where α, λ and c stand for the strength of color Coulomb po-tential, the strength of linear confinement and mass renormal-ization, respectively. The hyperfine and fine structures aregenerated by the spin dependent interactions V s ( r ) = m b (cid:20) (cid:32) α s r − λ r (cid:33) L · S + πα s δ ( r ) S b · S ¯ b (A2) + α s r (cid:32) S b · S ¯ b + ( S b · r )( S ¯ b · r ) r (cid:33) (cid:21) , where L denotes the relative orbital angular momentum, S = S b + S ¯ b is the total spin of the charm quark pairs and m b is thebottom quark mass. The smeared ˜ δ ( r ) function can be readfrom Eq. (9) or Refs. [17, 34]. These spin dependent termsare treated as perturbations.The Hamiltonian of the Schr¨odinger equation in thequenched limit is represented as H = m b + p m b + V ( r ) + V s ( r ) . (A3)The spatial wave functions and bare mass M are obtained bysolving the Schr¨odinger equation numerically using the Nu-merov method [35]. The full bare-mass spectrum is given inRef. [12]. Appendix B: Details of the Coupled-Channel E ff ects As sketched by Fig. 1, the experimentally observed stateshould be a mixture of pure quarkonium state (bare state) and B meson continuum. The coupled-channel e ff ects can be de-duced by following way H | ψ (cid:105) = M | ψ (cid:105) (B1) H | BC ; p (cid:105) = H BC | ψ (cid:105) = H BC | BC ; p (cid:105) = E BC | BC ; p (cid:105) (B4) H | A (cid:105) = M | A (cid:105) , (B5)where M is the bare mass of the bottomonium and can besolved directly from Schr¨odinger equation, and M is the phys-ical mass. The interaction between B mesons is neglected.When Eq. (B5) is projected onto each component, we imme-diately get (cid:104) ψ | H | ψ (cid:105) = c M = c M + (cid:90) d p c BC ( p ) (cid:104) ψ | H I | BC ; p (cid:105) , (B6) (cid:104) BC ; p | H | ψ (cid:105) = c BC ( p ) M = c BC ( p ) E BC + c (cid:104) BC ; p | H I | ψ (cid:105) . (B7)Solve c BC from Eq. (B7), substitute back to Eq. (B6) and elim-inate the c on both sides, we get a integral equation M = M + ∆ M , (B8)where ∆ M is given in Eq. (5). Once M is solved, the coe ffi -cient of di ff erent components can be worked out either. Forstates below threshold, the normalization condition | A (cid:105) can berewritten as | c | + (cid:90) d p | c BC | = c BC , we get the probability of the b ¯ b component. The sum of BC is restricted to the ground state B ( s ) mesons, i.e. B ¯ B , B ¯ B ∗ + h . c ., B ∗ ¯ B ∗ , B s ¯ B s , B s ¯ B ∗ s + h . c ., B ∗ s ¯ B ∗ s .The coupled-channel e ff ects calculation cannot proceed ifthe wave functions of the | ψ (cid:105) and BC components are not settled in Eq.(7). Since the major part of the coupled-channele ff ects calculation is encoded in the wave function overlap in-tegration, (cid:104) BC ; p | H I | ψ (cid:105) = (cid:90) d k φ ( (cid:126) k + (cid:126) p ) φ ∗ B ( (cid:126) k + x (cid:126) p ) φ ∗ C ( (cid:126) k + x (cid:126) p ) × | (cid:126) k | Y m ( θ (cid:126) k , φ (cid:126) k ) , (B10)where x = m q / ( m Q + m q ), and m Q and m q denote the bot-tom quark and the light quark mass, respectively. The φ , φ B and φ C are the wave functions of | ψ (cid:105) and BC components,respectively and the notation ∗ stands for the complex conju-gate. 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