# Spectroscopic parameters and decays of the resonance \(Z_b(10610)\)

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## Abstract

The resonance \(Z_b(10610)\) is investigated as the diquark–antidiquark \( Z_b=[bu][\overline{bd}]\) state with spin–parity \(J^{P}=1^{+}\). The mass and current coupling of the resonance \(Z_b(10610)\) are evaluated using QCD two-point sum rule and taking into account the vacuum condensates up to ten dimensions. We study the vertices \(Z_b\Upsilon (nS)\pi \ (n=1,2,3)\) by applying the QCD light-cone sum rule to compute the corresponding strong couplings \(g_{Z_b\Upsilon (nS)\pi }\) and widths of the decays \(Z_b \rightarrow \Upsilon (nS)\pi \). We explore also the vertices \(Z_b h_{b}(mP)\pi \ (m=1,2)\) and calculate the couplings \(g_{Z_b h_{b}(mP)\pi }\) and the widths of the decay channels \(Z_b \rightarrow h_{b}(mP)\pi \). To this end, we calculate the mass and decay constants of the \(h_b(1P)\) and \(h_b(2P) \) mesons. The results obtained are compared with experimental data of the Belle Collaboration.

## 1 Introduction

Discovery of the charged resonances which cannot be explained as \(\bar{c}c \) or \(\bar{b}b \) states has opened a new page in the physics of exotic multi-quark systems. The first tetraquarks of this family are \(Z^{\pm }(4430)\) states which were observed by the Belle Collaboration in *B* meson decays \( B\rightarrow K\psi ^{\prime }\pi ^{\pm }\) as resonances in the \(\psi ^{\prime }\pi ^{\pm }\) invariant mass distributions [1]. The masses and widths of these states were repeatedly measured and refined. Recently, the LHCb Collaboration confirmed the existence of the \(Z^{-}(4430)\) structure in the decay \(B^{0}\rightarrow K^{+}\psi ^{\prime }\pi ^{-}\) and unambiguously determined that its spin–parity is \(J^{P}=1^{+}\) [2, 3]. They also measured the mass and width of \( Z^{-}(4430)\) resonance and updated the existing experimental data. Two charmonium-like resonances, \(Z_{1}(4050)\) and \(Z_{2}(4250)\), were discovered by the Belle Collaboration in the decay \(\bar{B}^{0}\rightarrow K^{-}\pi ^{+}\chi _{c1}\), emerging as broad peaks in the \(\chi _{c1}\pi \) invariant mass distribution [4].

Famous members of the charged tetraquark family \(Z_{c}^{\pm }(3900)\) were observed by the BESIII Collaboration in the process \(\mathrm{e}^{+}\mathrm{e}^{-}\rightarrow J/\psi \pi ^{+}\pi ^{-}\) as resonances with \(J^{P}=1^{+}\) in the \(J/\psi \pi ^{\pm }\) mass distribution [5]. The charged state \( Z_{c}(4020)\) was also found by the BESIII Collaboration in two different processes, \(\mathrm{e}^{+}\mathrm{e}^{-}\rightarrow h_{c}\pi ^{+}\pi ^{-}\) and \( \mathrm{e}^{+}\mathrm{e}^{-}\rightarrow (D^{\star }\bar{D}^{\star })^{\pm }\pi ^{\mp }\) (see Refs. [6, 7]).

There is another charged state, namely the \(Z_{c}(4200)\) resonance which was detected and announced by Belle [8]. All aforementioned resonances belong to the class of the charmonium-like tetraquarks, and contain a \(\bar{c}c\) pair and light quarks (antiquarks). They were mainly interpreted as diquark–antidiquark systems or bound states of *D* and/or \( D^{\star }\) mesons.

*b*-counterparts of the charmonium-like states, i.e. charged resonances composed of a \(\bar{b}b\) pair and light quarks were found as well. Thus, the Belle Collaboration discovered the resonances \( Z_{b}(10610)\) and \(Z_{b}(10650)\) (hereafter, \(Z_{b}\) and \(Z_{b}^{\prime }\), respectively) in the decays \(\Upsilon (5S)\rightarrow \Upsilon (nS)\pi ^{+}\pi ^{-},\ n=1,2,3\) and \(\Upsilon (5S)\rightarrow h_{b}(mP)\pi ^{+}\pi ^{-},\ m=1,2\) [9, 10]. These two states with favored spin–parity \(J^{P}=1^{+}\) appear as resonances in the \(\Upsilon (nS)\pi ^{\pm }\) and \(h_{b}(mP)\pi ^{\pm }\) mass distributions. The masses of the \(Z_{b}\) and \(Z_{b}^{\prime }\) resonances are

The existence of hidden-bottom states, i.e. of the \(Z_b\) resonances, was foreseen before their experimental observation. Thus, in Ref. [13] the authors suggested to look for the diquark–antidiquark systems with \(b\bar{b} u \bar{d}\) content as peaks in the invariant mass of the \( \Upsilon (1S)\pi \) and \(\Upsilon (2S)\pi \) systems. The existence of the molecular state \(B^{\star }\bar{B}\) was predicted in Ref. [14].

After discovery of the \(Z_{b}\) resonances theoretical studies of the charged hidden-bottom states became more intensive and fruitful. In fact, articles devoted to the structures and decay channels of the \(Z_b\) states encompass all existing models and computational schemes suitable to study the multi-quark systems. Thus, in Refs. [15, 16] the spectroscopic and decay properties of \(Z_{b}\) and \(Z_{b}^{\prime }\) were explored using the heavy quark symmetry by modeling them as \(J=1\) *S*-wave molecular \(B^{\star }\bar{B}\)–\(B\bar{B}^{\star }\) states and \(B^{\star }\bar{B} ^{\star }\), respectively. The existence of similar states with quantum numbers \(0^{+},\ 1^{+},\ 2^{+}\) was predicted as well. The diquark–antidiquark interpretation of the \(Z_{b}\) states was proposed in Refs. [17, 18]. It was demonstrated that Belle results on the decays \(\Upsilon (5S)\rightarrow \Upsilon (nS)\pi ^{+}\pi ^{-}\) and \( \Upsilon (5S)\rightarrow h_{b}(mP)\pi ^{+}\pi ^{-}\) support \(Z_{b}\) resonances as diquark–antidiquark states. This analysis is based on a scheme for the spin–spin quark interactions inside diquarks originally suggested and successfully used to explore hidden-charm resonances [19].

The \(Z_{b}\) resonance was considered in Ref. [20] as a \( B^{\star }\bar{B}\) molecular state, where its mass was computed in the context of the QCD sum rule method. The prediction for the mass \(m_{B^{\star } \bar{B}}=10.54\pm 0.22\ \mathrm {GeV}\) obtained there allowed the authors to conclude that \(Z_{b}\) could be a \(B^{\star }\bar{B}\) molecular state. Similar conclusions were also drawn in the framework of the chiral quark model. Indeed, in Ref. [21] the \(B\bar{B}^{\star }\) and \( B^{\star }\bar{B}^{\star }\) bound states with \(J^{PC}=1^{+-}\) were studied in the chiral quark model, and found to be good candidates for the \(Z_{b}\) and \( Z_{b}^{\prime }\) resonances. Moreover, the existence of molecular states \(B\bar{B }^{\star }\) with \(J^{PC}=1^{++}\), and \(B^{\star }\bar{B}^{\star }\) with \( J^{PC}=0^{++},\ 2^{++}\) was predicted. Explorations performed using the one boson-exchange model also led to the molecular interpretations of the \(Z_{b}\) and \(Z_{b}^{\prime }\) resonances [22]. However, an analysis carried out in the framework of the Bethe–Salpeter approach demonstrated that two heavy mesons can form an isospin singlet bound state but cannot form an isotriplet compound. Hence, the \(Z_{b}\) resonance presumably is a diquark–antidiquark, but not a molecular state [23].

Both the diquark–antidiquark and the molecular pictures for the internal organization of \(Z_{b}\) and \(Z_{b}^{\prime }\) within the QCD sum rules method were examined in Ref. [24]. In this work the authors constructed different interpolating currents with \(I^{G}J^{P}=1^{+}1^{+}\) to explore the \(Z_{b}\) and \(Z_{b}^{\prime }\) states and evaluate their masses. Among alternative interpretations of the \(Z_{b}\) states it is worth to note Refs. [25, 26], where the peaks observed by the Belle Collaboration were explained as cusp and coupling channel effects, respectively.

Theoretical studies that address problems of the \(Z_b\) states are numerous (see Refs. [27, 28, 29, 30, 31, 32, 33, 34, 35, 36]). An analysis of these and other investigations can be found in Refs. [37, 38].

As is seen, the theoretical status of the resonances \(Z_{b}\) and \(Z_{b}^{\prime } \) remains controversial and deserves further and detailed explorations. In the present work we are going to calculate the spectroscopic parameters of \( Z_{b}=[bu][\overline{bd}]\) state by assuming that it is a tetraquark state with a diquark–antidiquark structure and positive charge. We use QCD two-point sum rules to evaluate its mass and current coupling by taking into account vacuum condensates up to ten dimensions. We also investigate five observed decay channels of the \(Z_{b}\) resonance employing QCD sum rules on the light-cone. As a byproduct, we derive the mass and decay constant of the \(h_{b}(mP),\ m=1,2\) mesons.

This work has the following structure: In Sect. 2 we calculate the mass and current coupling of the \(Z_{b}\) resonance. In Sect. 3 we analyze the decay channels \(Z_{b}\rightarrow \Upsilon (nS)\pi ,\ n=1,\ 2,\ 3\), and we calculate their widths. Section 4 is devoted to an investigation of the decay modes \(Z_{b}\rightarrow h_{b}(mP)\pi ,\ m=1,\ 2\), and it consists of two subsections. In the first subsection we calculate the mass and decay constant of the \(h_{b}(1P)\) and \( h_{b}(2P)\) mesons. To this end, we employ the two-point sum rule approach by including into the analysis condensates up to eight dimensions. In the next subsection using the parameters of the \(h_{b}(mP)\) mesons we evaluate the widths of the decays under investigation. The last section is reserved for an analysis of the obtained results and a discussion of possible interpretations of the \(Z_{b}\) resonance.

## 2 Mass and current coupling of the \(Z_{b}\) state: QCD two-point sum rule predictions

*a*,

*b*,

*c*,

*d*and

*e*are color indices, and

*C*is the charge conjugation matrix.

*x*to get

*b*-quark and light quark fields we get

*u*-, and

*d*-, and the heavy

*b*-quark propagators, respectively. We choose the light quark propagator \(S_{q}^{ab}(x)\) in the formFor the

*b*-quark propagator \(S_{b}^{ab}(x)\) we employ the expression In Eqs. (9) and (10) we use the notation

*b*-quark, and on the Borel variable \(M^{2}\) and continuum threshold \(s_{0}\), which are auxiliary parameters of the numerical computations. The vacuum condensates are parameters that do not depend on the problem under consideration: their numerical values extracted once from some processes are applicable in all sum rule computations. For quark and mixed condensates in the present work we employ \(\langle \bar{q}q\rangle =-(0.24\pm 0.01)^{3}~\mathrm {GeV}^{3}\), \( \langle \overline{q}g_{\mathrm {s}}\sigma Gq\rangle =m_{0}^{2}\langle \bar{q} q\rangle \), where \(m_{0}^{2}=(0.8\pm 0.1)~\mathrm {GeV}^{2}\), whereas for the gluon condensates we utilize \(\langle \alpha _{\mathrm {s}}G^{2}/\pi \rangle =(0.012\pm 0.004)~\mathrm {GeV}^{4}\), \(\langle g_{\mathrm {s}}^{3}G^{3}\rangle =(0.57\pm 0.29)~\mathrm {GeV}^{6}\). The mass of the \(b-\)quark can be found in Ref. [46]: it is equal to \(m_{b}=4.18_{-0.03}^{+0.04}~ \mathrm {GeV}\).

The choice of the Borel parameter \(M^2\) and continuum threshold \(s_0\) should obey some restrictions of sum rule calculations. Thus, the limits within of which \(M^2\) can be varied (working window) are determined from convergence of the operator product expansion and dominance of the pole contribution. In the working window of the threshold parameter \(s_0\) the dependence of the quantities on \(M^2\) should be minimal. In real calculations, however, the quantities of interest depend on the parameters \(M^2\) and \(s_0\), which affects the accuracy of the extracted numerical values. Theoretical errors in sum rule calculations may amount to \(30 \%\) of the predictions obtained, and a considerable part of these ambiguities are connected namely with the choice of \(M^2\) and \(s_0\).

## 3 Decay channels \(Z_b \rightarrow \Upsilon (nS) \pi , \ n=1,\ 2,\ 3. \)

This section is devoted to the calculation of the width of the \(Z_{b}\rightarrow \Upsilon (nS)\pi ,\,\ n=1,\,2,\,3\), decays. To this end we determine the strong couplings \(g_{Z_{b}\Upsilon _{n}\pi },\ n=1,2,3\) (in the formulas we utilize \(\Upsilon _{n}\equiv \Upsilon (nS)\)) using QCD sum rules on the light-cone in conjunction with the ideas of the soft-meson approximation.

*p*,

*q*and \( p^{\prime }=p+q\) are the momenta of \(\Upsilon (nS)\), \(\pi \) and \(Z_{b}\), respectively.

Spectroscopic parameters of the mesons \(\Upsilon _{nS}\) and \( \pi \)

Parameters | Values (in (\(\mathrm {MeV}\)) |
---|---|

\(m_{\Upsilon _{1}}\) | \(9460.30 \pm 0.26\) |

\(f_{\Upsilon _{1}}\) | \(708 \pm 8 \) |

\(m_{\Upsilon _{2}}\) | \(10023.26 \pm 0.31 \) |

\(f_{\Upsilon _{2}}\) | \(482 \pm 10 \) |

\(m_{\Upsilon _{3}}\) | \(10355.2 \pm 0.5\) |

\(f_{\Upsilon _{3}}\) | \(346 \pm 50\) |

\(m_{\pi }\) | \(139.57061 \pm 0.00024\) |

\(f_{\pi }\) | 131.5 |

## 4 \(Z_b \rightarrow h_b(1P) \pi \) and \(Z_b \rightarrow h_b(2P) \pi \) decays

The second class of decays which we consider contains two processes \( Z_{b}\rightarrow h_{b}(mP)\pi ,\ m=1,2\). We follow the same prescriptions as in the case of \(Z_{b}\rightarrow \Upsilon (nP)\pi \) decays and derive sum rules for the strong couplings \(g_{Z_{b}h_{b}\pi }\) and \(g_{Z_{b}h_{b}^{ \prime }\pi }\) (hereafter we employ the short-hand notations \(h_{b}\equiv h_{b}(1P)\) and \(h_{b}^{\prime }\equiv h_{b}(2P)\)). From the analysis performed in the previous section it is clear that the corresponding sum rules will depend on numerous input parameters including mass and decay constant of the mesons \(h_{b}(1P)\) and \(h_{b}(2P)\). Information on the spectroscopic parameters of \( h_{b}(1P)\) is available in the literature. Indeed, in the context of the QCD sum rule method mass and decay constant of *h*(1*P*) were calculated in Ref. [47]. But the decay constant \(f_{h_{b}^{\prime }}\) of the meson \( h_{b}(2P)\) was not evaluated; therefore, in the present work, we have first to find the parameters \(m_{h_{b}^{\prime }}\) and \(f_{h_{b}^{\prime }}\), and we shall turn after that to our main task.

### 4.1 Spectroscopic parameters of the mesons \(h_{b}(1P)\) and \(h_{b}(2P) \)

*h*(1

*P*) is the spin-singlet

*P*-wave bottomonium with quantum numbers \(J^{PC}=1^{+-}\), whereas

*h*(2

*P*) is its first radial excitation. The parameters of the \(h_{b}(1P)\) and \(h_{b}(2P)\) mesons in the framework of QCD two-point sum rule method can be extracted from the correlation function

### 4.2 Widths of decays \(Z_{b}\rightarrow h_{b}(1P)\pi \) and \(Z_{b}\rightarrow h_{b}(2P)\pi \)

In the equality obtained we apply the soft limit \(q\rightarrow 0\) (\(p=p^{\prime } \)) and perform the Borel transformation on the variable \(p^{2}\). This operation leads to a sum rule for the two strong couplings \(g_{Z_{b}h_{b}\pi }\) and \(g_{Z_{b}h_{b}^{\prime }\pi }\). The second expression is obtained from the first one by applying the operator \(d/d(-1/M^{2})\).

Experimental values and theoretical predictions for \(\mathcal {R}\)

\(\mathcal {R}\) | \(n=2\) | \(n=3\) | \(m=1\) | \(m=2\) |
---|---|---|---|---|

Exp. [46] | \(6.67_{-2.37}^{+3.11}\) | \(3.89_{-1.55}^{+2.02}\) | \(6.48_{-2.45}^{+3.18}\) | \(8.70_{-3.41}^{+4.39}\) |

This work | \(12.63 \pm 5.43\) | \(6.08 \pm 2.76\) | \(4.63 \pm 1.95\) | \(5.40 \pm 2.32\) |

## 5 Analysis and concluding notes

The experimental data on the decay channels of the \(Z_{b}(10610)\) resonance were studied and presented in a rather detailed form in Refs. [9, 10, 11]. Its full width was estimated as \(\Gamma =18.4\pm 2.4\ \mathrm {MeV}\), an essential part of which, i.e. approximately \(86\%\) of \(\Gamma \), is due to the decay \(Z_{b}\rightarrow B^{+} \overline{B}^{*0}+B^{*+}\overline{B}^{0}\). The remaining part of the full width is formed by five decay channels investigated in the present work. It is clear that our results for the widths of the decays \(Z_{b}\rightarrow \Upsilon (nS)\pi \) and \(Z_{b}\rightarrow h_{b}(nP)\pi \) overshoot the experimental data. Therefore, in the light of the present studies we refrain from an interpretation of the \(Z_{b}(10610)\) resonance as a pure diquark–antidiquark \([bu][\overline{bd}]\) state.

It is seen that theoretical predictions follow the pattern of the experimental data: we observe the same hierarchy of theoretical and experimental decay widths. At the time, numerical differences between them are noticeable. Nevertheless, as a result of the large errors in both sets, there are sizable overlap regions for each pair of \(\mathcal {R}\)s, which demonstrates not only qualitative agreement between them but also the quantitative compatibility of the two sets.

These observations may help one to understand the nature of the \(Z_{b}\) resonance. The Belle Collaboration discovered two \(Z_{b}\) and \(Z_{b}^{\prime }\) resonances with very close masses. We have calculated the parameters of an axial-vector diquark–antidiquark state \([bu][\overline{bd}]\), and we interpreted it as \(Z_{b}\). It is possible to model the second \(Z_{b}^{\prime }\) resonance using an alternative interpolating current, as has been emphasized in Sect. 2, and we explore its properties. The current with the same quantum numbers but different color organization may also play a role of such alternative (see, for example, Ref. [44]). One of the possible scenarios implies that the observed resonances are admixtures of these tetraquarks, which may fit measured decay widths.

The diquark–antidiquark interpolating current used in the present work can be rewritten as a sum of molecular-type terms. In other words, some of the molecular-type currents effectively contribute to our predictions, and by enhancing these components (i.e. by adding them to the interpolating current with some coefficients) better agreement with the experimental data may be achieved. In other words, the resonances \(Z_{b}\) and \(Z_{b}^{\prime }\) may “contain” both the diquark–antidiquark and the molecular components.

Finally, the \(Z_{b}\) and \(Z_{b}^{\prime }\) states may have pure molecular structures. But pure molecular-type bound states of mesons are usually broader than diquark–antidiquarks with the same quantum numbers and quark contents. In any case, all these suggestions require additional and detailed investigations.

In the present study we have fulfilled only a part of this program. In the framework of QCD sum rule methods we have calculated the spectroscopic parameters of the \(Z_{b}\) state by modeling it as a diquark–antidiquark state, and we found the widths of five of its observed decay channels. We have also evaluated the mass and decay constant of te \(h_{b}(2P)\) meson, which are necessary for analysis of the \(Z_{b}\rightarrow h_{b}(2P)\pi \) decay. Calculation of the \( Z_{b} \) resonance’s dominant decay channel may be performed, for example, using the QCD three-point sum rule approach, which is beyond the scope of the present work. The decays considered here involve the excited mesons \(\Upsilon (nS)\) and *h*(*mP*), the parameters of which require detailed analysis in the future. More precise measurements of \(Z_{b}\) and \(Z_{b}^{\prime }\) partial decay widths can also help in making a choice between the scenarios outlined.

## Notes

### Acknowledgements

S. S. A. thanks T. M. Aliev for helpful discussions. K. A. thanks TÜ BITAK for the partial financial support provided under Grant No. 115F183.

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