Few-Body Syst (2013) 54:1019–1022DOI 10.1007/s00601-013-0609-1

S. Ohkoda · Y. Yamaguchi · S. Yasui · K. Sudoh · A. Hosaka

Exotic Mesons with Hidden Bottom Near Thresholds

Received: 3 October 2012 / Accepted: 6 January 2013 / Published online: 31 January 2013© Springer-Verlag Wien 2013

Abstract We discuss exotic meson spectroscopy near open bottom thresholds. Assuming the exotic mesonsas B(∗) B̄(∗) molecular states, we study the interaction among two heavy mesons in terms of the one bosonexchange potential model. It is shown that masses of Zb(10610) and Zb(10650) are reproduced as B(∗) B̄(∗)

bound and resonance states. Besides, we also show that B(∗) B̄(∗) molecular states having various exoticquantum numbers can exist around the thresholds. By contrast, there are no D(∗) D̄(∗) molecular states havingexotic quantum numbers.

1 Introduction

Recently, the exotic twin resonances Zb(10610) and Zb(10650) are discovered in the processes ϒ(5S) →ππϒ(nS)(n = 1, 2, 3) and ϒ(5S) → ππhb(k P)(k = 1, 2) by Belle group [1,2]. The reported masses andwidths of the two resonances are M(Zb(10610)) = 10, 607.2 ± 2.0 MeV, �(Zb(10610)) = 18.4 ± 2.4 MeVand M(Zb(10650)) = 10652.2 ± 1.5 MeV, �(Zb(10650)) = (11.5 ± 2.2) MeV. These resonances have someinteresting properties. First of all, Zb’s have exotic quantum numbers I G(J P) = 1+(1+). Since Zb’s areisotriplet states, they need four quarks as minimal constituents. So Zb’s are genuinely exotic states. Secondly,Zb’s have exotic decay ratios. In general, a decay process with hb should be suppressed in heavy quark masslimit, because these processes require a heavy quark spin flip. Nevertheless, the decay rates of ϒ(5S) →Zbπ → ππϒ(nS) are comparable to those of ϒ(5S) → Zbπ → ππhb. Thirdly, Zb’s are “exotic twin”resonances. Having the same quantum numbers, their mass splitting is only 45 MeV. This mass splitting issmaller than the typical splitting of about 300 MeV found in usual bottomonium. These facts strongly suggestthat Zb’s have a molecular type structure as noted in Refs. [4,5].

Presented at the 20th International IUPAP Conference on Few-Body Problems in Physics, 20–25 August, 2012, Fukuoka, Japan.

This work is partly supported by the Grant-in-Aid for Scientific Research on Priority Areas titled “Elucidation of New Hadronswith a Variety of Flavors” (E01: 21105006).

S. Ohkoda (B) · Y. Yamaguchi · A. HosakaResearch Center for Nuclear Physics (RCNP), Osaka University, Ibaraki, Osaka 567-0047, JapanE-mail: ohkoda@rcnp.osaka-u.ac.jpTel.: +81-6-68798948Fax: +81-6-68798899

S. YasuiKEK Theory Center, Institute of Particle and Nuclear Studies, High Energy Accelerator Research Organization,1-1 Oho, Ibaraki 305-0801, Japan

K. SudohNishogakusha University, 6-16, Sanbancho, Chiyoda, Tokyo 102-8226, Japan

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We can naively expect the existence of molecular states in heavy quark sectors. Due to the following tworeasons. One is that the kinetic energy is suppressed, because the reduced mass is large. Second is that B andB∗ are degenerate thanks to heavy quark symmetry because the interaction of heavy quark spin is suppressed.This makes channel-couplings important.

In this paper, we summarize a main point of Ref. [3]. We study B(∗) B̄(∗) molecular states in terms of theone boson exchange potential model including the pion exchange. We take into account the degeneracy of Band B∗ due to the heavy quark symmetry, and fully consider channel couplings of B(∗) and B̄(∗). We considernot only bound states but also resonant states.

2 Models

To construct the interaction between two heavy mesons, we introduce the effective Lagrangian based on heavyquark and chiral symmetries [3]. It gives rise to the interaction Lagrangians of π meson and of vector meson(v = ρ, ω) with open heavy flavor mesons P(∗) (P(∗) = D(∗), B(∗)) as

Lπ H H = g trH̄a Hbγνγ5 Aνba, (1)

LvH H = −iβtrH̄a Hbvμ(ρμ)ba + iλtrH̄a Hbσμν Fμν(ρ)ba , (2)

where the multiplet field H containing P(∗) is defined by

Ha = 1 + /v

2

[P∗

aμγ μ − Paγ5]. (3)

The coupling constants are given by

g = 0.59, β = 0.9, λ = 0.56 GeV−1 , (4)

which are determined by experiments and theoretical estimations. We derive the one boson exchange potentialwith the Lagrangians (1) and (2). To take into account the size effect of mesons, we introduce a form factor(�2 − m2

h)/(�2 + q 2) in the momentum space at each vertex.We classify the B(∗) B̄(∗) molecular states with all possible quantum numbers I G(J PC ) and obtain the

potentials with channel couplings for each state. Since too higher angular momentum will be disfavored toform bound or resonance states, we consider states of the total angular momentum J ≤ 2. We focus on theB(∗) B̄(∗) molecular states for I = 1. Because these molecules are genuinely exotic states. We do not considerthe B(∗) B̄(∗) molecular states with non exotic quantum numbers, because these states can mix with ordinaryqq̄ mesons through a quark annihilation. Although the B(∗) B̄(∗) molecular states with non exotic quantumnumbers can mix with ordinary qq̄ meson through a quark annihilation, we do not include this effect in thepresent study.

3 Results

To obtain solutions of the B(∗) B̄(∗) states, we solve numerically the Schrödinger equations which aresecond-order differential equations with channel-couplings. In Table 1, we summarize the result of the obtainedbound and resonant states, and their possible decay modes to a quarkonium and a light flavor meson. We showthe mass spectrum of the B(∗) B̄(∗) states in Fig. 1.

Interestingly, having the present potential we find the twin states in the I G(J PC ) = 1+(1+−) nearthe B B̄∗ and B∗ B̄∗ thresholds; a bound state slightly below the B B̄∗ threshold, and a resonant stateslightly above the B∗ B̄∗ threshold. The binding energy is 8.5 MeV, and the resonance energy and decaywidth are 50.4 MeV from the B B̄∗ threshold and 15.1 MeV, respectively, from Table 1. The twinstates are obtained when the πρ ω potential is used. We identify them as the Zb(10610) and Zb(10650)observed in the Belle experiment [1,2]. It should be emphasized that the interaction in the present studyhas been determined in the previous works without knowing the experimental data of Zb’s. In addi-tion to these states, the potential model predicts the B(∗) B̄(∗) bound and resonant states in other chan-nels. Therefore, the B(∗) B̄(∗) molecular state has a rich mass spectrum of exotic hadrons around thethresholds.

Exotic Mesons with Hidden Bottom 1021

Fig. 1 The B(∗) B̄(∗) bound and resonant states with exotic I G(J PC ). The dots with error bars denote the position of the exper-imental observed Zb’s where M(Zb(10610)) = 10607.2 MeV and M(Zb(10650)) = 10652.2 MeV. Solid lines are for ourpredictions for the energies of the bound and resonant states when the πρ ω potential is employed. Mass values are shown inunits of MeV

Table 1 Various properties of the B(∗) B̄(∗) bound and resonant states with possible I G(J PC ) in I = 1

I G(J PC ) Threshold E [MeV] Decay channels

π-potential πρ ω-potential s-wave p-wave

1+(0+−) – – – – hb + π , χb0,1,2+ρ

1−(0++) B B̄ −6.5 no ηb+π , ϒ+ρ hb+ρ∗, χb1+π

1+(0−−) B B̄∗ −9.9 −9.8 χb1+ρ∗ ηb+ρ, ϒ+π

1−(0−+) B B̄∗ no no hb+ρ, χb0+π ϒ+ρ

1+(1+−) B B̄∗ −7.7 −8.5 ϒ+π hb+π , χb1+ρ∗50.4 − i15.1/2

1−(1++) B B̄∗ −16.7 −1.9 ϒ+ρ hb+ρ∗, χb0,1+π

1+(1−−) B B̄ 7.0 − i37.9/2 7.1 − i37.4/2 hb+π , χb0,1,2+ρ∗ ηb+ρ, ϒ+π58.8 − i30.0/2 58.6 − i27.7/2

1−(1−+) B B̄∗ no no hb+ρ, χb1+π ηb+π , ϒ+ρ

1+(2+−) B B̄∗ no no – hb+π , χb0,1,2+ρ

1−(2++) B B̄ 63.5 − i8.3/2 62.7 − i8.4/2 ϒ+ρ hb+ρ∗, χb1,2+π

1−(2−+) B B̄∗ no no hb+ρ ϒ+ρ

1+(2−−) B B̄∗ 2.0 − i4.1/2 2.0 − i3.9/2 χb1+ρ∗ ηb+ρ, ϒ+π44.2 − i2.5/2 44.1 − i2.8/2

The energies E can be either pure real for bound states or complex for resonances. The real parts are measured from the thresholdsas indicated in the second column. The imaginary parts are half of the decay widths of the resonances, �/2. In the last two columns,decay channels of a quarkonium and a light flavor meson are indicated. Asterisk of ρ∗ indicates that the decay occurs only witha virtual ρ while subsequently transit to a real photon via vector meson dominance

In the present study, all the states appear in the threshold regions and therefore are all weakly bound orresonant states. The present results are consequences of unique features of the bottom quark sector; the largereduced mass of the B(∗) B̄(∗) systems and the strong tensor force induced by the mixing of B and B∗ withsmall mass splittings. In fact, our model does not predict any bound and or resonant states in the thresholdregion of open charm.

References

1. Adachi I., Belle Collaboration: Observation of two charged bottomonium-like resonances. (2012) [arXiv:1105.4583 [hep-ex]]2. Bondar A. et al., Belle Collaboration: Observation of two charged bottomonium-like resonances in Y(5S) decays. Phys. Rev.

Lett. 108, 122001 (2012) [arXiv:1110.2251 [hep-ex]]

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3. Ohkoda S., Yamaguchi Y., Yasui S., Sudoh K., Hosaka A.: Exotic mesons with hidden bottom near thresholds. Phys. Rev. D86, 014004 (2012) [arXiv:1111.2921 [hep-ph]]

4. Bondar A.E., Garmash A., Milstein A.I., Mizuk R., Voloshin M.B.: Heavy quark spin structure in Zb resonances. Phys. Rev.D 84, 054010 (2011) [arXiv:1105.4473 [hep-ph]]

5. Voloshin M.B.: Radiative transitions from Upsilon(5S) to molecular bottomonium. Phys. Rev. D 84, 031502 (2011)[arXiv:1105.5829 [hep-ph]]