$\chi_{b}(3P)$ Multiplet Revisited: Ultrafine Mass Splitting and Radiative Transitions

Invoked by the recent CMS observation regarding candidates of the $\chi_b(3P)$ multiplet, we analyze the ultrafine and mass splittings among $3P$ multiplet in our unquenched quark model (UQM) studies. The mass difference of $\chi_{b2}$ and $\chi_{b1}$ in $3P$ multiplet measured by CMS collaboration ($10.6 \pm 0.64 \pm 0.17$ MeV) is very close to our theoretical prediction ($12$ MeV). Our corresponding mass splitting of $\chi_{b1}$ and $\chi_{b0}$ enables us to predict more precisely the mass of $\chi_{b0}(3P)$ to be ($10490\pm 3$) MeV. Moreover, we predict ratios of the radiative decays of $\chi_{bJ}(nP)$ candidates, both in UQM and quark potential model. Our predicted relative branching fraction of $\chi_{b0}(3P)\to\Upsilon(3S)\gamma$ is one order of magnitude smaller than $\chi_{b2}(3P)$, this naturally explains the non-observation of $\chi_{b0}(3P)$ in recent CMS search. We hope these results might provide useful references for forthcoming experimental searches.


I. INTRODUCTION
The excited P-wave bottomonia, χ bJ (3P), are of special interest, since they provide a laboratory to test (and model) the non-perturbative spin-spin interactions of heavy quarks. Very recently, the CMS collaboration observed two candidates of the bottomonium 3P multiplet, χ b1 (3P) and χ b2 (3P), through their decays into Υ(3S )γ [1]. Their measured masses and mass splitting are There are some earlier measurements related to χ bJ (3P) mass by ATLAS [2], LHCb [3,4], and D0 Collaborations [5]. However, these measurements can not distinguish between the candidates of χ bJ (3P) multiplet. The recent CMS analysis [1] is higher resolution search, and hence, is able to distinguish between χ b1 (3P) and χ b2 (3P) for the first time.
In this paper we intend to compare our unquenched quark model studies with this recent measurement, and make more precise prediction for the mass of the other 3P bottomonium (χ b0 ) by incorporating the measured mass splitting. We also make an analysis of the ultrafine splitting of P-wave bottomonia, which enlighten the internal quark structure of the considered bottomonium. In addition, we predict modelindependent ratios of radiative decays of χ bJ (nP) candidates.
Heavy quarkonium states can couple to intermediate heavy mesons through the creation of light quark-antiquark pair which enlarge the Fock space of the initial state, i.e. the initial state contains multiquark components. These multiquark components will change the Hamiltonian of the poten- * m.anwar@fz-juelich.de † luyu@hiskp.uni-bonn.de ‡ zoubs@itp.ac.cn tial model, causing the mass shift and mixing between states with the same quantum numbers or directly contributing to open channel strong decay if the initial state is above threshold. These can be summarized as coupled-channel effects (CCE). When CCE are combined with the naive quark potential model, one gets the unquenched quark model (UQM). UQM has been considered at least 35 years ago by Törnqvist et al. [6][7][8][9].
The physical or experimentally observed bottomonium state |A is expressed in UQM as where c 0 and c BC stand for the normalization constants of the bare state and the BC components, respectively. In this work, B and C refer to bottom and anti-bottom mesons, and the summation over BC is carried out including all possible pairs of ground-state bottom mesons. The |ψ 0 is normalized to 1 and |A is also normalized to 1 if it lies below BB threshold, and |BC; p is normalized as BC; p 1 |B C ; p 2 = δ 3 (p 1 − p 2 )δ BB δ CC , where p is the momentum of B meson in |A 's rest frame. The full Hamiltonian of the physical state then reads as where H 0 is the Hamiltonian of the bare state (see Appendix A for details), H BC |BC; p = E BC |BC; p with E BC = m 2 B + p 2 + m 2 C + p 2 is the energy of the continuum state (interaction between B and C is neglected and the transition between one continuum to another is restricted), and H I is the interaction Hamiltonian which mix the bare state with the continuum. Since each quark pair creation model generates its own vertex functions that in turn lead to specific real parts of hadronic loops, see Ref. [10] for related remarks.
Here, for the bare-continuum mixing, we adopt the widely used 3 P 0 model [11]. In this model, the generated quark pairs have vacuum quantum numbers J PC = 0 ++ which in spectroscopical notation 2S +1 L J equals to 3 P 0 . A sketch of 3 P 0 model induced mixing is shown in Fig. 1. The interaction Hamiltonian can be expressed as where m q is the produced quark mass, and γ is the dimensionless coupling constant. The ψ q (ψ q ) is the spinor field to generate anti-quark (quark). Since the probability to generate heavier quarks is suppressed, we use the effective 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 is strange quark mass.
where M and M 0 are the eigenvalues of the full (H) and quenched/bare Hamiltonian (H 0 ), respectively. See Appendix B or Refs. [12,13] for derivation of above relations and UQM calculation details. Numerical values of ∆M and P bb of every coupled channel for the bottomonia below BB threshold are given in Table I, which will be used in the following discussions.
Triggered by the above mentioned experimental search, we analyze our UQM studies regarding the bottomonium spectrum [12,15]. We notice that the measured mass splitting between χ b2 (3P) and χ b1 (3P) is (10.6 ± 0.64 ± 0.17) MeV which differs only by 1 MeV from our UQM prediction 1 [12]. Our prediction for the mass splitting of χ b1 (3P) and χ b0 (3P) is 23 MeV, see Table II. With the reference of the observed masses of the other two candidates of spin-triplet 3P bottomonium, this mass splitting helps us to predict precisely the mass of unknown χ b0 (3P) to be The uncertainty in above prediction is calculated by taking the same percentage error [of O(10%)] in our mass splittings which we observed from CMS measurement. Our mass predictions respect the conventional pattern of splitting and support the standard mass hierarchy, where we have M(χ b2 ) > M(χ b1 ) > M(χ b0 ), which is in line with CMS measurement. A comparison of our UQM mass splittings with other quenched quark model predictions is given in Table II.

III. ULTRAFINE SPLITTING IN UQM
It is more informative if we study the mass splitting in a multiplet instead of the total mass shift caused by the intermediate meson loop. For the states quite below the threshold, there is an interesting phenomenon [16]: the magnitude of the mass splitting is suppressed by the probability of the bottomonium core, P bb , if we turn on the meson loop.
There is also a pictorial explanation for this. Since under the potential model, the mass splitting δM 0 originates from the fine splitting Hamiltonian H I . Up to the first order perturbation, we have δM 0 = ψ| H I |ψ , where ψ is the two-body wave function in the quenched potential model. Since one of the coupled-channel effects is the wave function renormalization: ψ| ψ = P bb < 1, one would simply expect that the δM 0 will be suppressed by this probability.
Moreover, due to the closeness of the spectrum of a multiplet, we expect that the P bb of the states in a same multiplet are nearly the same, i.e., δM 0 are all suppressed by a same quantity, leaving the relation intact, even if the coupled-channel effects are turned on. Due to the remarkably small δM P , we refer it as "ultrafine splitting". In our calculation, however, due to the finite size of the constituent quark, which is reflected by the smeared delta    [12], Godfrey-Isgur (GI) model [18], Modified GI model [19], and constituent quark model (CQM) [20]. The later three models are regarded as quenched quark models.
term,δ(r), instead of the true Dirac term 2 in the spin depen-2 Such a smearing of the Dirac delta term incorporating the contact spinspin interaction with a finite range 1/σ is essential to regularize the delta function [17]. dent potential where α s and λ are strengths of the color Coulomb and linear confinement potentials, respectively, and σ is related to the width of Gaussian smeared function, the δM P relation of Eq. (8) is already violated a little bit under the potential model which can be seen from Table III (second column), where we also include the corresponding experimental values. We can also extract the threshold effects by taking the mass shift ∆M instead of M in δM P calculations. The δM P values obtained in this way are also given in Table III (third column). We can see from Table I that although the mass shift for the P-wave multiplets is around 50 MeV, the modification of Eq. (8) is not very large, except δM P (3P) which is far larger than δM P (2P) and δM P (1P). A worth mentioning feature here is the hierarchy of these ultrafine splittings originated from the CCE (third column of Table III), viz., which highlights that the coupled-channel effects bring meson masses closer together with respect to their bare values [16].
Since, for the P-wave states, no matter whether the threshold effects are considered or not, h b is not affected by the fine interaction, i.e. the δM = 0. Hence, the χ bJ 's mass splitting are purely due to the P bb of each χ bJ . Therefore, the weighted probability of the bottomonium core, P bb , for χ bJ (nP) multiplets is simply defined as P bb = P bb (χ bJ ). The weighted average probability for the S -wave bottomonia is discussed in Appendix C. From the Table IV, we can see that although the ( P bb × δM 0 ) and δM originate differently; one from the potential model and the other purely from the coupled-channel effects, but they are approximately equal to each other. The only large deviation comes from χ bJ (3P).
As explained above, this overall suppression is based on the assumption that the P bb is the same (or approximately the same) for a multiplet. Indeed, from Table I we can see that this is quite reasonable assumption for the states which are far below the threshold. But for the χ b0 (3P), the P bb is quite different from that of χ b1 (3P), so this overall suppression does not make sense anymore. As a consequence, one should expect relatively large deviation from the δM P relation, as can be seen from δM P (3P) in Table III.
The reason for this peculiar P bb is that even though the mass of h b (3P) and χ b1 (3P) is larger than the χ b0 (3P), they do not couple to the channel BB, and the next open channel BB * is somewhat farther from them. A net effect is that the P bb of χ b1 (3P) is larger than that of χ b0 (3P), breaking the P bb closeness assumption. This strong coupling of χ b0 (3P) to BB is also reflected by the large mass shift caused by BB which can be seen from Table I. The observed mismatch between ( P bb × δM 0 ) and δM for χ bJ (3P) multiplet is a smoking gun of the threshold effects which are beyond the quark potential model.
Recently, Lebed and Swanson also pointed out the remarkable importance of the P-wave heavy quarkonia [22]. For 1P and 2P charmonia, the ultrafine splitting is found to be astonishingly small. They argued that the ultrafine splitting can be used to delve the exoticness of the observed structure in the given multiplet [23]. According to their analysis [22], the quantity δM n,L=1,2,3,... is found to be very small for any radial excitation n, both for the bb and cc sectors. The obtained constraint on the δM n,L value is This conclusion follows from several theoretical formalisms which do not consider coupled-channel effects or longdistance light-quark contributions in terms of intermediate meson-meson coupling to bare quarkonium states. As discussed above, the operators corresponding to ultrafine splitting involve spin-spin interactions which are suppressed by 1/m 2 Q , the standard expansion parameter for the heavy quarkonium, where m Q is the mass of heavy quark. According to our point of view the above maxima is much large for the ultrafine splitting of P-wave bottomonia, see Table III for experimental corroboration. The more tight constraint could be Since, quantitatively the P-wave excitation for the bottomonium is equal to Λ QCD , which describes the emergence of the dynamical QCD scale in above relation. The δM n,L for the bottomonia with L = 1 is expected to be of O(1 MeV), which can be verified from our analysis of Table III. The reason why δM n,L=1,2,3,... is exactly zero in the quark model is a consequence of the pure delta function nature of the S b · S¯b term of Eq. (9), which is a perturbative one gluon exchange effect. The non-perturbative effects can make an additional contribution to this term, so that it is no longer a pure delta function. This give rise to introduce the smearing of the delta function in the quark models [17,22]. However, one could use different non-perturbative forms for the spin-spin operator that contributes to the ultrafine splitting. For instance, the ultrafine splitting computed at next-to-nextto-next-to leading order (N 3 LO) [24] in nonrelativistic QCD (NRQCD) [25,26] is where C F is the color factor of bottomonium, n l being the number of light fermion species appearing in loop corrections, and N c is the number of colors in QCD. The computed δM n,L=1 values using NRQCD for the bottomonium (with m b = 4.5 GeV and α s (m b ) = 0.2) are; δM 1P = 3.77 keV, δM 2P = 1.12 keV, and δM 3P = 0.47 keV [22]. The remarkable smallness Channels δM 0 P bb ( P bb × δM 0 ) δM P bb ( P bb × δM 0 ) δM δM Exp GEM SHO   The mass splitting (in MeV) in a same (n, L) multiplet, where δM 0 , δM and δM Exp represent the mass splitting in potential model, coupled-channel model and experiment, respectively. The P bb (in %) is the weighted average of the probability, which for P-and S -wave is P bb = P bb (χ bJ ) and P bb = 1 4 P bb (Υ) + 3 4 P bb (η b ), respectively. The details of the mass splitting are given in Appendix C, and the absolute probabilities P bb are given in Table I. GEM and SHO stand for the Gaussian expansion method [27] and simple harmonic oscillator approximation, respectively, to fit the numerical wave functions.
of these values strengthen the constraint on the δM n,L=1,2,3,... values presented in Eq. (12). However, these NRQCD predictions are much smaller as compared to our UQM predictions and corresponding experimental values, see Table III. In conclusion, whatever the non-perturbative form for the spin-spin operator is used, the δM n,L=1 should be very small, hence satisfying the relation of Eq. (12) quantitatively.

IV. RADIATIVE TRANSITIONS
Radiative transitions of higher bottomonia are of considerable interest, since they can shed light on their internal structure and provide one of the few pathways between different bb multiplets. Particularly, for those states which can not directly produce at e + e − colliders (such as P-wave bottomonia), the radiative transitions serve as an elegant probe to explore such systems. In the quark model, the electric dipole (E1) transitions can be expressed as [28,29] where e b = − 1 3 is the b-quark charge, α is the fine structure constant, and E γ denotes the energy of the emitted photon. The spatial matrix elements ψ f | r |ψ i involve the initial and final radial wave functions, and C f i are the angular matrix el-ements. They are represented as The matrix elements ψ f | r |ψ i are obtained numerically; for further details, we refer our studies [12,30]. From Eq. (15), we know that the value of the decay width depends on the details of the wave functions, which are highly model dependent.
A model independent prediction can be achieved by focusing on the following decay ratios Since, in the quark model, the spatial wave function is the same for the states in the same multiplet. From the above discussion, we know that the meson loop renormalizes the bottomnium wave function. When the channel is above the corresponding open-bottom threshold (such as BB here), the wave function cannot be normalized to 1, this is still an open problem (see e.g. Ref. [31]). On the other hand, the BB loop is still there, and have some CCE (such as mass renormalization). We make the assumption that for the states above threshold (such as χ b2 (3P) here), these open channels contribute equally to the wave functions of all χ bJ (3P) states. In fact this is a reasonable assumption, since we can see this from the Table I, the probability of BB is vanishingly small (0.31% and 0.89%, less than 1%) for both χ b0 (3P) and χ b1 (3P).
With the latest CMS data [1] and the P bb in Table I, our predictions of radiative decay ratios are listed in Table V. From  the Table I, one can see that the small P bb [χ b0 (3P)] make the ratios in the last three rows notably larger than that of the potential model predictions, a peculiar feature of coupledchannel effects which can be tested in the upcoming experiments.
Decay Channel  Another worth noting result from Table V is the relative size of the ratios for χ b0 (3P), which from the coupled-channel calculations is roughly 1 : 6 : 12. This reflects that the χ b0 (3P) has negligible radiative decay branching fraction with comparison to χ b1 (3P) and χ b2 (3P). Compared with the potential model, the suppression of the χ b0 (3P)'s radiative width in the UQM is more consistent with the non-observation of the χ b0 (3P) in the recent CMS search of χ bJ (3P) → Υ(3S )γ [1]. This indicates that our UQM predictions are more reliable than the naive quark potential models.

V. CONCLUSIONS
The recent CMS study successfully distinguishs χ b1 (3P) and χ b2 (3P) for the first time, and measures their mass splitting which differs only 1 MeV from our unquenched quark model predictions. This measurement gives us confidence to predict mass of the lowest candidate of 3P multiplet to be M[χ b0 (3P)] = (10490 ± 3) MeV, based on our unquenched quark model results of the mass splittings of this multiplet. We also analyze the ultrafine splittings of P-wave bottomonia up to n = 3 in the framework of UQM, and put a constraint on them based on recent experimental corroboration. No matter which non-perturbative form for the spin-spin operator is used, the ultrafine splitting for the P-wave bottomonia should be very small. This analysis leads us to conclude that the coupled-channel effects play a crucial role to understand the higher bottomonia close to open-flavor thresholds.
At last, we predict here to some extent model-independent ratios of the radiative decays of χ bJ (nP) candidates. A worth mentioning observation is that the coupled-channel effects can enhance the radiative decay ratios of χ bJ (3P) as compared to the naive potential model predictions. The relative branching fraction of χ b0 (3P) → Υ(3S )γ is negligible as compared to the other candidates of this multiplet, which naturally explains its non-observation in recent CMS search.
We hope above highlighted features of coupled-channel model provide useful references for the understanding of higher P-wave bottomonia and can be explored in ongoing and future experiments.