$\Upsilon(nl)$ decay into $ B^{(*)} \bar B^{(*)}$

Wei-Hong Liang, 2, ∗ Natsumi Ikeno, 4, † and E. Oset 4, ‡ Department of Physics, Guangxi Normal University, Guilin 541004, China Guangxi Key Laboratory of Nuclear Physics and Technology, Guangxi Normal University, Guilin 541004, China Department of Life and Environmental Agricultural Sciences, Tottori University, Tottori 680-8551, Japan Departamento de F́ısica Teórica and IFIC, Centro Mixto Universidad de Valencia CSIC, Institutos de Investigación de Paterna, Aptdo. 22085, 46071 Valencia, Spain. (Dated: December 9, 2019)


I. INTRODUCTION
The vector Υ(nl) states are a good example to test quark models with bb. The large mass of the b quark makes them excellent nonrelativistic systems and the theoretical predictions [1] agree well with experiment, at least for the first states, with discrepancies in the mass of only a few MeV. There are many variants of the bb quark model [2][3][4][5][6], producing again similar results for the lowest states and larger discrepancies for the higher excited states. A more complete view can be obtained from Ref. [7]. In particular, discussions on non bb configurations for higher excited states are on going. More concretely, problems stem from the states which can decay into B ( * )B( * ) , starting from the Υ(4s). In Table I we show predictions for masses of these states and current PDG [8] values. The Υ(3d) state, reported recently by the Belle collaboration with mass 10753 MeV, is close to the quark model predictions (see Table I). Yet, claims that this state could be a tetraquark state are made in Ref. [10]. Similarly, the Υ(5s) state is questioned as a pure bb state in base to the π + π − Υ(n ′ l ′ ) decay rates, and a mixture of Υ(5s) plus the * liangwh@gxnu.edu.cn † ikeno@tottori-u.ac.jp ‡ eulogio.oset@ific.uv.es lowest 1 −− hybrid state [11] is invoked in Ref. [12]. Surprisingly, even if the B ( * )B( * ) decay modes are open for the states above the Υ(4s), no theoretical works on these decay modes, nor the role of the meson-meson components in the Υ states, are available (but there is some work done on the B ( * )B( * ) π decay [13] ). The closest work would be the ratios predicted for e + e − → BB, e + e − → BB * + c.c. and e + e − → B * B * cross sections using heavy quark spin symmetry in Refs. [14,15], but, as mentioned in Ref. [16], these predictions are in conflict with experiment and it was blamed on the proximity of quarkonium resonances to thresholds of these channels, suggesting the mixture of the Υ(nl) states with some meson-meson component to solve this conflict [16]. In the present work we address these problems for the 4s, 3d, 5s, 6s states, for which there are experimental data.

II. FORMALISM
We follow here the formalism used in Refs. [17,18]. For this we use the 3 P 0 model to hadronize the bb vector state and generate two B ( * )B( * ) mesons, as shown in Fig. 1, creating a flavor-scalar state with the quantum numbers of the vacuum.ū We consider onlyūu +dd +ss since thecc,bb components give rise to meson-meson states too far away in energy to be relevant in the process. If we write the qq matrix, M , with the u, d, s, c quarks we have M = (qq) =     uū ud us ub dū dd ds db sū sd ss sb bū bd bs bb and then, after hadronization we find If we write M 4i M i4 in terms of the B, B * mesons, we find the combinations The next step is to see the relationship of the production modes of these four combinations P P, P V, V P, V V (P pseudoscalar, V vector) and for this we use the 3 P 0 model [19,20]. The details of the angular momentum algebra involved are shown in Ref. [17], and relative weights for P P, P V, V P, V V production are obtained to which we shall come back below. The formalism that we follow relies on the use of the vector propagator which is dressed including the selfenergy due to B ( * )B( * ) production, as shown in Fig. 2. The renormalized vector meson propagator is written as where the selfenergy Π(p) is given by and the vertex for ΥB i B i ′ is of the type where A is an arbitrary constant to be fitted to the data and g Ri are weights for the different B i B i ′ states which are evaluated using the 3 P 0 model in Ref. [17]. Eq. (6) is an effective vertex which takes into account the sum over polarizations of the vectors in the Π loop in Fig. 2 . For a given channel B i B i ′ the selfenergy is then given by where the coefficients g Ri are evaluated with the 3 P 0 model in Ref. [17]. The q 0 integration is done analytically and we find wherẽ where w i ( q ) = m 2 i + q 2 and F ( q ) is a form factor that, inspired upon the Blatt-Weisskopf barrier penetration factor [21], we take of the type of Ref. [17] with q on taken zero below threshold, and R a parameter to be fitted to data. In order to have a pole at M R , we define which renders Π ′ (p) convergent and then we have the Υ propagator as According to Refs. [17,22] the cross section for and the individual cross section to each channel It is customary to make an expansion of Π ′ in Eq. (12) around the resonance mass as and then the width of the resonance is given by and for each individual channel The value of Z provides the strength of the Υ vector component (not to be associated to a probability when there are open channels [17,23]). We shall see that for the Υ(4s) state the Z value is relatively different from 1, indicating the importance of the weight of the B i B i ′ channels in the state. If Z is close to 1 we can make a series expansion of Z in Eq. (16) such that each individual value can be interpreted as the weight of each meson-meson component in the Υ wave function (see Ref. [23] for the precise interpretation of this quantity, that gives an idea of the weight of each component but cannot be identified with a probability.).
The values of the couplings g Ri , obtained from the 3 P 0 model are given by Refs. [17,18]

III. RESULTS
The strategy is to fit the parameters A, R to the shape of the e + e − → Υ(4s) → BB cross section, Eq. (14). The parameter f R is irrelevant for the shape. We also fine tune the value of M R around the nominal value of the PDG. Then we take the same value of R, which gives a range of the momentum distribution of the internal meson-meson components, and the parameter A will be adjusted to the width of each of the other states.
A. Υ(4s) state In Fig. 3, we show the data of Ref. [24] for e + e − → Υ(4s) → BB, together with a fit to these data that ren- ders the parameters: In    Fig. 3, we obtain Γ ∼ 24 MeV, similar to the value quoted in the PDG [8], and a value around 30 MeV using Eq. (17). More important than these numbers is that we fit the BB data from BaBar [24]. We can use Eqs. (17) and (18) to get branching ratios and we obtain the results shown in Table III. The good width is a consequence of the fit and the branching ratios, in agreement with experiment, are a consequence of the different phase space for B + B − and B 0B0 , due to their different masses. We take the PDG mass 10752.7 MeV and fit the width of Γ = 35.5 MeV and obtain the only free parameter A 2 = 120, with this input we look at the weights for the different channels and we find the results of Table IV. Interestingly, we find now that the value of Z is very close to 1 and the weight of the meson-meson components very small. We could be surprised that Z is bigger than 1 and the individual meson-meson weights for the open channels are complex and have negative real part. This is a consequence of the fact that these weights cannot be interpreted as probabilities. Indeed, as discussed in [20], the individual meson-meson weights correspond to the integral of the wave function squared with a certain phase prescription (not modulus squared), which for the open channels is complex. Even then, this magnitude measures the strength of the meson-meson components and the message from these results is that this strength is small and the Υ remains largely as the original bb component.
Similarly, in Table V we show the branching ratios obtained for each channel, for which there are no experimental data. It is remarkable the large strength for B * B * production in spite of its smaller phase space relative to BB or BB * + c.c..   We take the nominal PDG mass M R = 10889.9 MeV and fit A to get the 80% of the width of 51 MeV. We obtain The weights of the meson-meson components and the value of Z are shown in Table VI. Once again we find  that Z is very close to 1 and the weights of the mesonmeson components are very small. In Table VII we show the branching ratios that we obtain for the different channels and in this case we can compare with the experimental values. Globally, the branching ratios obtained agree in a fair way with experiment. We confirm the small BB branching fraction, in spite of the largest phase space, and the dominance of the B * B * channel. We also find the branching rations for B sBs and B sB * s + c.c. channels small like in the experiment. The only large discrepancy is in the B * sB * s channel, where our results are notably smaller than the experimental value. This large experimental result is not easy to understand. By analogy to the experimental values for BB * + c.c. and B * B * , using the ratio of these branching ratios (38.1/13.2) and multiplying this value by the experimental branching ratio of B sB * s + c.c., we should expect a value for the branching ratio of B * sB * s smaller than 4% because of the reduced phase space.

D. Υ(6s) state
We take again the PDG mass of M R = 10992.9 +10.0 −3.1 MeV and the width of 49 +9 −15 MeV and make a fit to the width, assuming that all of it comes from the B iBi ′ decay channels (there is no information on these decay channels in the PDG). We obtain The weights of the different components and the value of Z are shown in Table VIII. Once again we see that the value of Z is very close to 1 and the weight of the mesonmeson components very small. In Table IX we show the branching ratios assuming that the width is exhausted by the B i B i ′ channels. We should nevertheless mention that the proximity of theBB 1 (5721) threshold at 11000 MeV to the mass of this resonance could have some effect on the selfenergy Π. However, if the channel is closed for decay it can affect ReΠ but not the individual ImΠ i for the open channels needed in the evaluation of the individual decay rates that we have calculated. It will be interesting to compare our results with data when they become available.

IV. CONCLUSIONS
We have studied the BB, BB * + c.c., B * B * , B sBs , B sB * s + c.c., B * sB * s decay modes of the Υ(4s), Υ(3d), Υ(5s), Υ(6s) states using the 3 P 0 model to produce two mesons from the original bb vector state. We observed interesting things. The first one is that the Υ(4s) state has an abnormally large width that has as a consequence that the state contains a relatively large admixture of meson-meson components in its wave function. It is one exception to the general rule for vector mesons which are largely qq states [25]. On the other hand, the other three states studied had very small meson-meson components and the states remain largely in the original bb seed.
We could only test predictions on branching ratios for the Υ(4s) and Υ(5s) states. In the first case only the BB channel is open and the branching ratios for B + B − and B 0B0 are determined by phase space in agreement with experiment. The other case is the Υ(5s) where there are data on the branching ratios. The agreement is qualitatively good, with the notable exception of the B * sB * s channel, and also a factor of two discrepancy in the relative rates of the BB * + c.c. and B * B * channels. It would be interesting to see if these discrepancies are a warning that more elaborate components for this state, as suggested in Ref. [12], are at work. The predictions made for branching ratios for the Υ(3d) and Υ(6s) states should serve as a motivation to measure these magnitudes that will help in advancing our understanding of these bottomium states.