# Semileptonic decays of the scalar tetraquark \(Z_{bc;\overline{u}\overline{d}}^{0}\)

## Abstract

We study semileptonic decays of the scalar tetraquark \(Z_{bc;\overline{u} \overline{d}}^{0}\) to final states \(T_{bs;\overline{u}\overline{d} }^{-}e^{+}\nu _{e}\) and \(T_{bs;\overline{u}\overline{d}}^{-}\mu ^{+}\nu _{\mu }\) , which run through the weak transitions \(c\rightarrow se^{+}\nu _{e}\) and \(c\rightarrow s\mu ^{+}\nu _{\mu }\), respectively. To this end, we calculate the mass and coupling of the final-state scalar tetraquark \(T_{bs;\overline{u}\overline{d} }^{-} \) by means of the QCD two-point sum rule method: these spectroscopic parameters are used in our following investigations. In calculations we take into account the vacuum expectation values of the quark, gluon, and mixed operators up to dimension ten. We use also three-point sum rules to evaluate the weak form factors \(G_{i}(q^2)\) (\(i=1,~2\)) that describe these decays. The sum rule predictions for \(G_{i}(q^2)\) are employed to construct fit functions \(F_{i}(q^2)\), which allow us to extrapolate the form factors to the whole region of kinematically accessible \(q^2\). These functions are required to get partial widths of the semileptonic decays \(\Gamma \left( Z_{bc}^{0}\rightarrow Te^{+}\nu _{e}\right) \) and \(\Gamma \left( Z_{bc}^{0}\rightarrow T\mu ^{+}\nu _{\mu }\right) \) by integrating corresponding differential rates. We analyze also the two-body nonleptonic decays \(Z_{bc;\overline{u}\overline{d}}^{0} \rightarrow T_{bs;\overline{u}\overline{d }}^{-}\pi ^{+}\) and \(Z_{bc;\overline{u}\overline{d}}^{0} \rightarrow T_{bs;\overline{u }\overline{d}}^{-}K^{+}\), which are necessary to evaluate the full width of the \(Z_{bc;\overline{u}\overline{d}}^{0}\). The obtained results for \(\Gamma _{\mathrm {full}}=(3.18\pm 0.39)\times 10^{-11}~{\mathrm {MeV}}\) and mean lifetime \(20.7_{-2.3}^{+2.9}~{\mathrm {ps}}\) of the tetraquark \(Z_{bc;\overline{u} \overline{d}}^{0}\) can be used in experimental investigations of this exotic state.

## 1 Introduction

Investigations of double-heavy tetraquarks composed of a heavy *QQ* diquark [*Q* is the heavy *c* or *b* quark] and a light antidiquark are among interesting topics in physics of exotic hadrons. The interest to such kind of quark configurations is connected with a possible stability of some of them against the strong and electromagnetic decays. The relevant problems were addressed already in the pioneering papers [1, 2, 3], in which a stability of the exotic four-quark mesons \(QQ\bar{Q}\bar{Q}\) and \(QQ\bar{q}\bar{q}\) was examined. It was found that the heavy *Q* and light *q* quarks with a large mass ratio \(m_{Q}/m_{q}\) may form the stable tetraquarks \(QQ\bar{q}\bar{q}\). The similar conclusions were drawn in Ref. [4] as well, in accordance of which the isoscalar \(J^{P}=1^{+}\) tetraquark \(T_{bb; \overline{u}\overline{d}}^{-}\) lies below the two B-meson threshold and can decay only weakly.

All available theoretical tools of high energy physics were exploited to study properties of double-heavy exotic mesons; the chiral and dynamical quark models, the relativistic quark model and sum rules method were mobilized to calculate their parameters [5, 6, 7, 8, 9, 10, 11, 12, 13]. Interest to these mesons renewed after experimental observation by the LHCb Collaboration of the \(\Xi _{cc}^{++}=ccu\) baryon [14]. Its mass was used as an input information in a phenomenological model to estimate the mass of the axial-vector tetraquark \(T_{bb;\overline{u} \overline{d}}^{-}\) [15]. The obtained prediction \( m=(10389\pm 12)~{\mathrm {MeV}}\) is \(215~{\mathrm {MeV}}\) below the \(B^{-} \overline{B}^{*0}\) threshold and \(170~{\mathrm {MeV}}\) below the threshold for decay \(B^{-}\overline{B}^{0}\gamma \), which means that \(T_{bb;\overline{u }\overline{d}}^{-}\) is stable against the strong and electromagnetic decays and dissociates only weakly. The conclusion about the strong-interaction stability of the tetraquarks \(T_{bb;\overline{u}\overline{d}}^{-}\), \(T_{bb; \overline{u}\overline{s}}^{-}\), and \(T_{bb;\overline{d}\overline{s}}^{0}\) was made in Ref. [16] on the basis of the relations derived from heavy-quark symmetry. The mass \(m=10482~{\mathrm {MeV}}\) of the axial-vector tetraquark \(T_{bb;\overline{u}\overline{d}}^{-}\) found there is \(121~{\mathrm {MeV}}\) below the open-bottom threshold.

In Ref. [17] we calculated the spectroscopic parameters of the axial-vector tetraquark \(T_{bb;\overline{u}\overline{d}}^{-}\) and analyzed also its semileptonic decay to the scalar state \(Z_{bc;\overline{u} \overline{d}}^{0}\). Our result for its mass \(m=(10035~\pm 260)~{\mathrm {MeV}}\) confirms once more that it is stable against the strong and electromagnetic decays. We evaluated the total width and mean lifetime of \(T_{bb;{\bar{u}}{\bar{d}}}^{-}\) using the semileptonic decay channels \(T_{bb;\overline{u} \overline{d}}^{-}\rightarrow Z_{bc;\overline{u}\overline{d}}^{0}l{\bar{\nu _{l}}}\), where \(l=e,\mu \) and \(\tau \). The predictions \(\Gamma =(7.17\pm 1.23)\times 10^{-8}~{\mathrm {MeV}}\) and \(\tau =9.18_{-1.34}^{+1.90}~{\mathrm {fs}} \) provide information useful for experimental investigation of the double-heavy exotic mesons. Details of performed analysis and references to earlier and recent articles devoted to different aspects of the doubly and fully heavy tetraquarks can be found in Ref. [17].

We determined the mass and coupling of the scalar four-quark meson \(Z_{bc; \overline{u}\overline{d}}^{0}\) (hereafter \(Z_{bc}^{0}\)) as well [17], because these parameters were necessary to evaluate the width of the semileptonic decay \(T_{bb;\overline{u}\overline{d} }^{-}\rightarrow Z_{bc}^{0}l{\bar{\nu _{l}}}\). For these purposes we employed the QCD sum rule approach and found \(m_{Z}=(6660\pm 150)~{\mathrm {MeV}}\). This prediction is considerably below the threshold \(7145~{\mathrm {MeV}}\) for strong decays of \(Z_{bc}^{0}\) to heavy mesons \(B^{-}D^{+}\) and \(\overline{ B^{0}}D^{0}\). The state \(Z_{bc}^{0}\) cannot decay to a pair of heavy and light mesons as well; this fact differs it qualitatively from the open charm-bottom scalar tetraquarks \(cq\overline{b}\overline{q}\) and \(cs \overline{b}\overline{s}\), which decay to \(B_{c}\pi \) and \(B_{c}\eta \) mesons [18], respectively. The thresholds for the electromagnetic decays \(Z_{bc}^{0}\rightarrow \overline{B^{0}} D_{1}^{0}\gamma \) and \(B^{*}D_{0}^{*}\gamma \) exceed \(7600~{\mathrm { MeV}}\) and are higher than the mass of \(Z_{bc}^{0}\). In other words, the tetraquark \(Z_{bc}^{0}\) as the state \(T_{bb;{\bar{u}}{\bar{d}}}^{-}\) is the strong- and electromagnetic-interaction stable particle.

The scalar and axial-vector states \(bc\overline{u}\overline{d}\) were subjects of interesting theoretical investigations with, sometimes, controversial predictions. In fact, the analysis performed in Ref. [15] showed that \(Z_{bc}^{0}\) resides \(11~{\mathrm {MeV}}\) below the threshold \(7145~{\mathrm {MeV}}\) for *S*-wave decays to conventional heavy \( B^{-}D^{+}\) and \(\overline{B^{0}}D^{0}\) mesons. Computations of the ground-state \(QQ^{\prime }\overline{u}\overline{d}\) tetraquarks’ masses carried out in the context of the Bethe–Salpeter method led to similar conclusions [19]. The mass of \(Z_{bc}^{0}\) found there (for some set of used parameters) equals to \(6.93~{\mathrm {GeV}}\) and is lower than the relevant strong threshold. On the contrary, for the masses of the scalar and axial-vector \(bc\overline{u}\overline{d}\) states the heavy-quark symmetry predicts \(7229~{\mathrm {MeV}}\) and \(7272~{\mathrm {MeV}} \) [16], which means that they can decay to ordinary mesons \( B^{-}D^{+}/\overline{B^{0}}D^{0}\) and \(B^{*}D\), respectively. The charged exotic scalar mesons \(Z_{bc;\overline{u}\overline{u}}^{-}\) and \( Z_{bc;\overline{d}\overline{d}}^{+}\) were explored by means of the QCD sum rule method as well [20]; the mass of these particles \( m=(7.14\pm 0.10)~{\mathrm {GeV}}\) is higher than our prediction for \(m_{Z}\).

The recent lattice simulations prove the strong-interaction stability of the \(I(J^{P})=0(1^{+})\) four-quark meson \(Z_{ud;\overline{c}\overline{b}}^{0}\) with the mass in the range 15–61 \({\mathrm {MeV}}\) below \(\overline{D} B^{*}\) threshold [21]. But, because of theoretical uncertainties the authors could not determine whether this tetraquark would decay electromagnetically to \(\overline{D}B\gamma \) or can transform only weakly. Another confirmation of the \(bc\overline{u}\overline{d}\) tetraquarks stability came from Ref. [22]; there it was demonstrated that both the \(J^{P}=0^{+}\) and \(1^{+}\) isoscalar tetraquarks \(bc\overline{u} \overline{d}\) are stable against the strong decays. The isoscalar \( J^{P}=0^{+}\) state is also electromagnetic-interaction stable, whereas \( J^{P}=1^{+}\) may undergo the electromagnetic decay to \(\overline{B}D\gamma \).

In light all of these theoretical predictions, it becomes evident that decays of the tetraquark \(Z_{bc}^{0}\) are sources of a valuable information about this exotic meson. In the present work we explore the semileptonic decays of the tetraquark \(Z_{bc}^{0}\) which are important for some reasons. First of all, \(Z_{bc}^{0}\) may be produced copiously at the LHC [23], hence it is necessary to fix processes, where it has to be searched for. The second reason is exploration of the tetraquark \(T_{bb; \overline{u}\overline{d}}^{-}\) itself, and decay channels appropriate for its discovery. As usual, all states classified till now as candidates to tetraquarks were seen through their decays to conventional mesons. If a tetraquark is stable against strong and electromagnetic decays, then it should be observed due to products of its weak decays. In the case under discussion at the first stage \(T_{bb;\overline{u}\overline{d}}^{-}\) decays to \(Z_{bc}^{0}\) and \(l{\bar{\nu _{l}}}\). But, because the scalar tetraquark \( Z_{bc}^{0}\) does not transform directly to conventional mesons, one needs to consider its weak decays, as well.

The weak decays of \(Z_{bc}^{0}\) can proceed through different channels. The dominant semileptonic decay modes of \(Z_{bc}^{0}\) are the processes \( Z_{bc}^{0}\rightarrow T_{bs;\overline{u}\overline{d}}^{-}e^{+}\nu _{e}\) and \( Z_{bc}^{0}\rightarrow T_{bs;\overline{u}\overline{d}}^{-}\mu ^{+}\nu _{\mu }\), which run due to transitions \(c\rightarrow W^{+}s\) and \(W^{+}\rightarrow \overline{l}\nu _{l}\). The channels triggered by the decays \(c\rightarrow W^{+}d\) and \(W^{+}\rightarrow \overline{l}\nu _{l}\) lead to creation of the tetraquark \(T_{bd;\overline{u}\overline{d}}^{-}\), and are suppressed relative to the first modes by a factor \(|V_{cd}|^{2}/|V_{cs}|^{2}\simeq 0.05 \). The similar arguments can be applied to other semileptonic decays of \( Z_{bc}^{0}\) generated by a chain of transitions \(b\rightarrow W^{-}c\) \( \rightarrow cl\overline{\nu }_{l}\) and \(b\rightarrow W^{-}u\) \(\rightarrow ul \overline{\nu }_{l}\), respectively. In fact, the Cabibbo–Kobayashi–Maskawa (CKM) matrix element \(|V_{bc}|\), which is small numerically, and the ratio \( |V_{bu}|^{2}/|V_{bc}|^{2}\) \(\simeq 0.01\) demonstrates a subdominant nature of the decays \(b\rightarrow cl\overline{\nu }_{l}\) and \(b\rightarrow ul \overline{\nu }_{l}\). The weak decay \(c\rightarrow W^{+}s\) may be followed by transitions \(W^{+}\rightarrow u\overline{d}\) and \(W^{+}\rightarrow u \overline{s}\), which give rise to nonleptonic decays of \(Z_{bc}^{0}\). In the hard-scattering mechanism, for example, a pair \(u\overline{d}\) may form ordinary mesons with \(q\overline{q}\) quarks appeared due to a gluon from one of *u* or \(\overline{d}\) quarks. These processes lead to final states \( Z_{bc}^{0}\rightarrow T_{bs;\overline{u}\overline{d}}^{-}M_{1}(u\overline{q} )M_{2}(q\overline{d})\) which are suppressed relative to the semileptonic decays by the factor \(\alpha _{s}^{2}|V_{ud}|^{2}\). But \(u\overline{d}\) and \( u\overline{s}\) quarks can form \(\pi ^{+}\) and \(K^{+}\) mesons and generate the two-body nonleptonic decays of the tetraquark \(Z_{bc}^{0}\), i.e., the processes \(Z_{bc}^{0}\rightarrow T_{bs;\overline{u}\overline{d} }^{-}\pi ^{+}\) and \(Z_{bc}^{0}\rightarrow T_{bs;\overline{u}\overline{d} }^{-}K^{+}\). There is also a class of multimeson processes, when \(u \overline{d}\) and \(u \overline{s}\) combine directly with quarks from \(T_{bs;\overline{u} \overline{d}}^{-}\) and create three-meson final states. The two-body and three-meson nonleptonic decays do not suppressed by additional factors relative to the semileptonic decays, and their contributions to full width of \(Z_{bc}^{0}\) may be sizeable. Parameters of these channels may provide a valuable new information on features of the exotic meson \(Z_{bc}^{0}\).

The tetraquark \(T_{bs;\overline{u}\overline{d}}^{-}\) can bear different quantum numbers. We treat \(T_{bs;\overline{u}\overline{d}}^{-}\) as a scalar particle, and in what follows denote it by *T*. To calculate the width of aforementioned decays, one needs the mass and coupling of the tetraquark *T*; they enter as parameters to the sum rules for the weak form factors that determine width of the decays. The spectroscopic parameters of this tetraquark can be extracted from the two-point correlation function by means of the sum rule approach, which is one of the powerful nonperturbative tools in QCD [24, 25]. It can be applied to compute spectroscopic parameters and decay width not only of the conventional hadrons but also the exotic states [for the recent review, see Ref. [26]].

In the present work the mass and coupling of *T* are calculated by taking into account vacuum expectation values of various quark, gluon, and mixed local operators up to dimension ten. The weak form factors \(G_{i}(q^{2})\), (\( \,i=1,2\)) are extracted from the QCD three-point sum rules, which allow us to find numerical values of \(G_{i}(q^{2})\) at momentum transfer \(q^{2}\) accessible for sum rule computations. Later we fit \(G_{i}(q^{2})\) by functions \(F_{i}(q^{2})\), and extrapolate them to a whole domain of physical \(q^{2}\). The fit functions are used to integrate the differential decay rates and obtain the width of the semileptonic decays \(\Gamma \left( Z_{bc}^{0}\rightarrow Te^{+}\nu _{e}\right) \) and \(\Gamma \left( Z_{bc}^{0}\rightarrow T\mu ^{+}\nu _{\mu }\right) \). We also calculate the widths of the nonleptonic decays \(Z_{bc}^{0}\rightarrow T\pi ^{+}\) and \( Z_{bc}^{0}\rightarrow TK^{+}\), and use this information to evaluate the full width of \(Z_{bc}^{0}\).

This article is structured in the following form: In Sect. 2 we derive the QCD two-point sum rules for the mass and coupling of the tetraquark *T*, and find their numerical values. In Sect. 3 the QCD three-point correlation functions are utilized to get sum rules for the form factors \(G_{i}(q^{2})\). Here we carry out also numerical analysis of derived expressions and determine the fit functions, and evaluate the width of the semileptonic decays of concern. Section 4 is devoted to analysis of the two-body nonleptonic decays of the tetraquark \( Z_{bc}^{0}\), where we calculate the partial widths of the processes \( Z_{bc}^{0}\rightarrow T\pi ^{+}\) and \(Z_{bc}^{0}\rightarrow TK^{+}\). In Sect. 5 we evaluate the full width and mean lifetime of \( Z_{bc}^{0}\), and analyze decay channels of the tetraquarks \(Z_{bc}^{0}\) and \(T_{bb;\overline{u}\overline{d}}^{-}\). This section contains also our concluding remarks.

## 2 Spectroscopic parameters of the tetraquark \(T_{bs;\overline{u}\overline{d}}^{-}\)

*T*are important to calculate the width of the exotic \(Z_{bc}^{0}\) meson’s semileptonic decays. The

*T*state contains four quarks

*b*,

*s*,

*u*, and

*d*of different flavors and has the heavy-light structure. In other words, the

*b*-quark and

*s*-quark, which is considerably heavier than \(q=u,d\) , groups to form the heavy diquark, whereas the antidiquark is built of light

*u*and

*d*quarks. This is the main difference of

*T*and the famous resonance

*X*(5568); the latter has the same quark content, but

*b*and

*s*quarks are distributed between a diquark and an antidiquark [27]. The scalar tetraquark

*T*can be composed using diquarks of a different type. The ground-state scalar particle

*T*should be composed of the scalar diquark \(\epsilon ^{abc}[b_{b}^{T}C\gamma _{5}s_{c}]\) in the color antitriplet and flavor antisymmetric state and the antidiquark \(\epsilon ^{ade}[\overline{u} _{d}\gamma _{5}C\overline{d}_{e}^{T}]\) in the color triplet state. The reason is that they are most attractive diquark configurations, and exotic mesons composed of them should be lighter and more stable than four-quark mesons made of other diquarks [28]. Therefore, we assume that

*T*has such favorable structure, and accordingly choose the interpolating current

*J*(

*x*)

*a*,

*b*,

*c*,

*d*and

*e*are color indices and

*C*is the charge-conjugation operator.

*T*can be obtained from the QCD two-point sum rules. To derive the sum rules for the mass \(m_{T}\) and coupling \(f_{T}\) of

*T*, we analyze the correlation function

*T*as a ground-state particle and use the “ground-state + continuum” scheme. Then \(\Pi ^{\mathrm {Phys}}(p)\) contains a contribution of the ground-state particle and contributions arising from higher resonances and continuum states

*x*.

*J*(

*x*) and calculate \( \Pi (p)\) by contracting in Eq. (2) the relevant heavy and light quark fields. As a result, we get

*b*- and light

*u*(

*d*,

*s*)-quark propagators, respectively. Here we also use the shorthand notation

*b*and

*s*-quark masses, for which we use \(m_{b}=4.18_{-0.03}^{+0.04}\ {\mathrm {GeV}}\) and \(m_{s}=96_{-4}^{+8}\ {\mathrm {MeV}}\), respectively.

*T*lead to the following results:

*T*allows us to see whether this four-quark meson is strong-interaction stable or not. As we have emphasized above,

*T*contains the same quark species like the resonance

*X*(5568), but differs from it by an internal organization. The resonance

*X*(5568) with the content \(su\overline{b}\overline{d}\) was originally studied in our work [27]. It is a scalar particle, but has the heavy diquark-antidiquark structure. The mass of the resonance

*X*(5568) evaluated there

*T*; structures with a heavy diquark and a light antidiquark seem are more compact than ones composed of a pair of heavy diquark and antidiquark. The resonance

*X*(5568) is unstable against the strong interactions and decays to the conventional mesons \( B_{s}^{0}\pi ^{+}\). It is clear that

*T*cannot decay to such final states, but its quark content and quantum numbers does not forbid

*S*-wave decays to \(\overline{B^{0}}K^{-}/\overline{K^{0}}B^{-}\) mesons, thresholds of which \( 5774/5777~{\mathrm {MeV}}\) however, are above the mass \(m_{T}\). Thresholds for

*P*-wave decays of the scalar tetraquark \(bs\overline{u}\overline{d}\) are higher than \(m_{T}\) as well. The possible electromagnetic decay \( T\rightarrow B^{-}K_{1}\gamma \) may be realized only if \(m_{T}\ge 6552\ {\mathrm {MeV}}\), which is not the case. Therefore, transformation of the tetraquark

*T*to ordinary mesons runs only due to its weak decays.

## 3 Semileptonic decays \(Z_{bc}^{0}\rightarrow Te^{+}\nu _{e}\) and \(Z_{bc}^{0}\rightarrow T\mu ^{+}\nu _{\mu }\)

In this section we explore the semileptonic decays \(Z_{bc}^{0}\rightarrow Te^{+}\nu _{e}\) and \(Z_{bc}^{0}\rightarrow T\mu ^{+}\nu _{\mu }\) of the scalar four-quark meson \(Z_{bc}^{0}\). The spectroscopic parameters of \( Z_{bc}^{0}\) evaluated in Ref. [17], as well as the mass and coupling of the final-state tetraquark *T* , obtained in the previous section provide necessary information to calculate the differential rate and width of these decays.

*p*and \(p^{\prime }\) are the momenta of the tetraquarks \(Z_{bc}^{0}\) and

*T*, respectively. In Eq. (18) the form factors \( G_{1}(q^{2})\) and \(G_{2}(q^{2})\) parameterize the long-distance dynamics of the weak transition. Here we also use \(P_{\mu }=p_{\mu }^{\prime }+p_{\mu }\) and \(q_{\mu }=p_{\mu }-p_{\mu }^{\prime }\). The \(q_{\mu }\) is the momentum transferred to the leptons, and evidently \(q^{2}\) changes within the limits \( m_{l}^{2}\le q^{2}\le (m_{Z}-m_{T})^{2}\), where \(m_{l}\) is the mass of a lepton

*l*.

*J*(

*y*) and \(J^{Z}(x)\) are the interpolating currents for the states

*T*and \(Z_{bc}^{0}\), respectively. The current

*J*(

*y*) has been defined above by Eq. (1): for \(J^{Z}(x)\) we use the expression [17]

*S*-wave diquark fields, has the antisymmetric color structure \([\overline{3}_{c}]_{bc}\otimes [3_{c}]_{\overline{u}\overline{d}}\) and describes the ground-state tetraquark \(Z_{bc}^{0}\).

*i*are the color indices of the currents \(J^{Z}(x)\) and \(J_{\mu }^{\mathrm {tr}}\), respectively.

*T*tetraquark channel. The spectral densities \(\rho _{i}(s,s^{\prime },q^{2})\) are calculated as the imaginary parts of the correlation function \(\Pi _{\mu }^{\mathrm {OPE}}(p,p^{\prime })\) with dimension-five accuracy, and contain both the perturbative and nonperturbative contributions.

For numerical computations of \(G_{i}(\mathbf {M}^{2},\ \mathbf {s}_{0},~q^{2})\) one needs to employ various parameters, values some of which are collected in Eq. (10). The mass and coupling of the tetraquark \( Z_{bc}^{0}\) and \((M_{1}^{2},s_{0})\) are borrowed from Ref. [17], whereas for \(m_{T}\) and \(f_{T}\), and \((M_{2}^{2},s_{0}^{ \prime })\) we use results of the previous section.

To obtain the width of the decay \(Z_{bc}^{0}\rightarrow T\overline{l}v_{l}\) we have to integrate the differential decay rate \(d\Gamma /dq^{2}\) within the kinematical limits \(m_{l}^{2}\le q^{2}\le (m_{Z}-m_{T})^{2}\), whereas the QCD sum rules lead to reliable results only for \(m_{l}^{2}\le q^{2}\le 1.25\) \({\mathrm {GeV}}^{2}\). To cover all values of \(q^{2}\) we replace the weak form factors by the functions \(F_{i}(q^{2})\), which at accessible for the sum rule computations \(q^{2}\) coincide with \(G_{i}(q^{2})\), but can be extrapolated to the whole integration region.

## 4 Nonleptonic two-body decays \(Z_{bc}^{0}\rightarrow T\pi ^{+}\) and \(Z_{bc}^{0}\rightarrow TK^{+}\)

The nonleptonic two-body decays \(Z_{bc}^{0}\rightarrow T\pi ^{+}\) and \( Z_{bc}^{0}\rightarrow TK^{+}\) of the tetraquark \(Z_{bc}^{0}\) can be considered in the context of the QCD factorization approach, which allows us to calculate the amplitudes and widths of these processes. This method was successfully applied to study two-body weak decays of the conventional mesons [30, 31], and is used here to investigate two-body decays of the tetraquark \(Z_{bc}^{0}\), when one of the final particles is an exotic meson.

*i*,

*j*are the color indices. Here \(c_{1}(\mu )\) and \(c_{2}(\mu )\) are the short-distance Wilson coefficients evaluated at the scale \(\mu \) at which the factorization is assumed to be correct. The shorthand notation \( \left( \overline{q}_{1}q_{2}\right) _{\mathrm {V-A}}\) in Eq. (32) means

## 5 Analysis and concluding remarks

- (i)
\(Z_{bc}^{0}\rightarrow Te^{+}\nu _{e}\),

- (ii)
\(Z_{bc}^{0}\rightarrow T\mu ^{+}\nu _{\mu }\),

- (iii)
\(Z_{bc}^{0}\rightarrow T\pi ^{+}\),

and

- (iv)
\(Z_{bc}^{0}\rightarrow TK^{+}\).

*bc*at the LHC would be higher by two order of magnitude than four-quark mesons with

*bb*[23].

Another issue studied here is decays of the tetraquark \(T_{bb;\overline{u} \overline{d}}^{-}\). We have analyzed its decay chains consisting of sequential weak transformations to final states with *T* and evaluated their branching ratios. These calculations are important to fix processes, where the axial-vector tetraquark \(T_{bb;\overline{u}\overline{d}}^{-}\) should be searched for.

The predictions for the width and lifetime of \(Z_{bc}^{0}\), as well as for the branching ratios (44) and (45) should be considered as first results for these quantities obtained using dominant weak decays of \(Z_{bc}^{0}\) and \(T_{bb;\overline{u}\overline{d}}^{-}\). In fact, here we have taken into account only processes \(Z_{bc}^{0}\rightarrow Te^{+}\nu _{e}\) , \(Z_{bc}^{0}\rightarrow T\mu ^{+}\nu _{\mu }\), \(Z_{bc}^{0}\rightarrow T\pi ^{+}\) and \(Z_{bc}^{0}\rightarrow TK^{+}\), but subdominant semileptonic decays of \(Z_{bc}^{0}\) may correct these predictions. We have treated *T* as a scalar particle, whereas \(Z_{bc}^{0}\) can decay also to exotic mesons with another quantum numbers. By including into analysis these options one can open up new decay modes of \(Z_{bc}^{0}\), and improve predictions for the branching ratios presented above. Finally, there are nonleptonic three-meson decay channels, effects of which on the full width and mean lifetime of \( Z_{bc}^{0}\) maybe sizeable. In other words, nonleading semileptonic decays of \(Z_{bc}^{0}\), its decays to a tetraquark *T* with another quantum numbers, and to multimeson nonleptonic final states may improve and correct the picture described here. Detailed investigations of these problems, left beyond the scope of the present work, are necessary to gain more precise knowledge about properties of the exotic states \(T_{bb;{\bar{u}}{\bar{d}}}^{-}\) and \(Z_{bc}^{0}\).

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