# Testing the doubly charged charm–strange tetraquarks

## Abstract

The spectroscopic parameters and decay channels of the doubly charged scalar, pseudoscalar and axial-vector charm–strange tetraquarks \(Z_{ \overline{c}s}=[sd][\overline{u} \overline{c}]\) are explored within framework of the QCD sum rule method. The masses and current couplings of these diquark–antidiquark states are calculated by means of two-point correlation functions and taking into account the vacuum condensates up to eight dimensions. To compute the strong couplings of \(Z_{\overline{c}s}\) states with \(D,\ D_{s},\ D^{*},\ D_{s}^{*},\ D_{s1}(2460),\ D_{s0}^{*}(2317),\ \pi \) and *K* mesons we use QCD light-cone sum rules and evaluate width of their *S*- and *P*-wave decays to a pair of negatively charged conventional mesons: For the scalar state \(Z_{\overline{c}s}\rightarrow D_s \pi ,\ DK, \ D_{s1}(2460)\pi \), for the pseudoscalar state \(Z_{\overline{c}s} \rightarrow D_{s}^{*}\pi ,\ D^{*}K, \ D_{s0}^{*}(2317)\pi ,\) and for the axial-vector state \(Z_{\overline{c}s} \rightarrow D_{s}^{*}\pi ,\ D^{*}K,\ D_{s1}(2460)\pi \) decays are investigated. Obtained predictions for the spectroscopic parameters and decay widths of the \(Z_{\overline{c}s}\) tetraquarks may be useful for experimental investigations of the doubly charged exotic hadrons.

## 1 Introduction

During last decade tetraquarks, i.e. bound states of four quarks are in the center of intensive experimental and theoretical investigations. Starting from discovery of the famous resonance *X*(3872) in *B* meson decay \( B\rightarrow KX\rightarrow KJ/\psi \rho \rightarrow KJ/\psi \pi ^{+}\pi ^{-}\) by Belle [1], and after observation of the same state by other groups [2, 3, 4] experimental collaborations collected valuable information on the spectroscopic parameters and decay channels of the exotic states. They were discovered in various inclusive and exclusive hadronic processes. In this connection it is worth to note *B* meson decays, \(e^{+}e^{-}\) and \(\overline{p}p\) annihilations and *pp* collisions. Theoretical studies of exotic hadrons, apart from tetraquark states, include pentaquarks and hybrid mesons and encompass variety of models and calculational methods claiming to explain the internal structure of these states and calculate their experimentally measured parameters. Comprehensive information on collected experimental data and detailed analysis of theoretical achievements and existing problems can be found in latest review works Refs. [5, 6, 7, 8, 9].

The great success in physics of the exotic hadrons is connected with discovery of charged multiquark resonances. The first charged tetraquarks, namely \(Z^{\pm }(4430)\) states 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 [10]. The resonances \(Z^{+}(4430)\) and \(Z^{-}(4430)\) were detected and studied by Belle in the processes \(B \rightarrow K\psi ^{\prime } \pi ^{+}\) [11] and \(B^{0} \rightarrow K^{+}\psi ^{\prime } \pi ^{-}\) [12], as well. These states constitute an important subclass of multiquark systems, because charged resonances can not be explained as excited charmonium or bottomonium states, and therefore, are real candidates to genuine tetraquarks.

Hadrons built of four quarks of different flavors form another intriguing class in the tetraquark family. Depending on a quark content these states may be neutral or charged particles. Among the observed tetraquarks the *X*(5568) resonance remains a unique candidate to a hadron composed of four different quarks. At the same time it is a particle containing *b*-quark, i.e. is an open bottom tetraquark. The evidence for *X*(5568) was first reported by the D0 Collaboration in Ref. [13]. Later it was observed again by D0 in the \(B_{s}^{0}\) meson’s semileptonic decays [14] . But other experimental groups, namely the LHCb and CMS collaborations could not find this resonance from analysis of their experimental data [15, 16], which make the experimental situation around *X*(5568) unclear and controversial. Numerous theoretical works devoted to investigation of *X*(5568) resonance’s structure and calculation of its parameters led also to contradictory conclusions. The results of these studies are in a reasonable agreement with measurements carried out by the D0 Collaboration, while in other works an existence of the *X*(5568) state is an object of discussions [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33] . The detailed analysis of problems related to the status of the *X*(5568) resonance can be found in original papers (see for instance, Ref. [6] and references therein).

The tetraquarks which might carry double electric charge constitute another interesting class of exotic hadrons [34]. These hypothetical particles if observed can be interpreted as diquark–antidiquark states: Formation of molecular states from two mesons of same charge is almost impossible due to repulsive forces between them. The doubly charged particles may exist, for example, as double charmed tetraquarks \([cc][\bar{d} \bar{s}]\) or \([cc][\bar{s} \bar{s}]\). In other words, they may contain two or three quark flavors. The phenomenology of these states, their decay modes and production mechanisms were investigated in Ref. [34]. In the context of the lattice QCD the mass spectra of these particles were evaluated in the paper [35]. As it was revealing recently the tetraquarks containing quarks of four different flavors may also carry double electric charge [36]. In fact, it is not difficult to see that tetraquarks \(Z_{\overline{c}s}=[sd][\overline{u} \overline{c}]\) and \(Z_{c \overline{s}}= [uc][\overline{s} \overline{d}]\) belong to this category of particles, and at the same time, are open charm states. Authors of Ref. [36] wrote down also possible *S*- and *P*-wave decay channels of these states. Strictly speaking, the open charm tetraquarks were previously investigated in the literature (see, for example Refs. [37, 38, 39]). The spectroscopic parameters and decay widths of the open charm tetraquark containing three different light quarks were calculated in Ref. [38]. In this study the open charm tetraquark was considered as a partner of the *X*(5568) state. In other words, the quark content of \(X_c=[su][\overline{c} \overline{d}]\) was obtained from \(X_b=[su][\overline{b}\overline{d}]\) by \(b \rightarrow c\) replacement. Due to differences in the charges of *b* and *c* quarks the partner state \(X_c\) does not bear the same charge as \(X_b\). This conclusion is true in the case of \(Z_{\overline{c}s}\), as well. If the state \(Z_{\overline{c}s}\) bears the charge \(-2|e|\), its *b*-partner \(Z_{\overline{b }s}\) has \(-|e|\). In general, there do not exist doubly charged tetraquarks composed of *b* and three different light quarks. The genuine doubly charged tetraquarks with *b* belong to a subclass of open charm–bottom particles and should contain also *c*-quark. For example, the state \(Z_{b\overline{c} }=[bs][\overline{u}\overline{c}]\) has the charge \(-2|e|\).

In the present work we are going to concentrate on features of doubly charged charm–strange tetraquarks \(Z_{\overline{c}s}\) with spin-parity \( J^{P}=0^{+},\ 0^{-}\) and \(1^{+}\), and calculate their masses, current couplings and decay widths. To this end, we use QCD two-point sum rule approach by including into analysis quark, gluon and mixed vacuum condensates up to eight dimensions, and evaluate their spectroscopic parameters. Obtained results are employed to reveal kinematically allowed decay channels of the tetraquarks \(Z_{\overline{c}s}\). They also enter as input parameters to expressions of the corresponding decay widths. We calculate the width of decay channels \(Z_{\overline{c}s}\rightarrow D_{s}\pi ,\ DK\), and \(D_{s1}(2460)\pi \) (for \(J^{P}=0^{+}\)), \(Z_{\overline{c} s}\rightarrow D_{s}^{*}\pi ,\ D^{*}K\) and \(D_{s0}^{*}(2317)\pi \) (for \(J^{P}=0^{-}\)), as well as \(Z_{\overline{c}s}\rightarrow D_{s}^{*}\pi ,\ D^{*}K\) and \(D_{s1}(2460)\pi \) (in the case of \(J^{P}=1^{+}\)). For these purposes, we analyze vertices of the tetraquarks \(Z_{\overline{c} s} \) with the conventional mesons, and evaluate the corresponding strong couplings using QCD sum rules on the light-cone. The QCD light-cone sum rule method is one of the powerful nonperturbative tools to explore parameters of the conventional hadrons [40]. In the case of vertices built of a tetraquark and two conventional mesons the standard methods of the light-cone sum rules should be supplemented by a technique of an approach known as the “soft-meson” approximation [41, 42]. For investigation of the exotic states the light-cone sum rules method was adapted in Ref. [43], and successfully applied for analysis of various tetraquarks’ decays [44, 45, 46, 47].

This article is organized in the following manner. In Sect. 2 we calculate the masses and current couplings of the doubly charged scalar, pseudoscalar and axial-vector charm–strange tetraquarks \(Z_{\overline{c} s}=[sd][\overline{u}\overline{c}]\) by treating them as diquark–antidiquark systems. In Sect. 3 we consider the decays of the doubly charged scalar tetraquark to \(D_{s}\pi \), *DK* and \(D_{s1}(2460)\pi \) final states. Section 4 is devoted to decay channels of the pseudoscalar and axial-vector tetraquarks. Here we compute width of their decays to \(D_{s}^{*}\pi \), \(D^{*}K\) and \(D_{s0}^{*}(2317)\pi \) (for \(0^{-}\) state), and to \(D_{s}^{*}\pi \), \(D^{*}K\) and \( D_{s1}(2460)\pi \) (for \(1^{+}\) state). Section 5 is reserved for our concluding remarks.

## 2 Spectroscopic parameters of the scalar, pseudoscalar and axial vector tetraquarks \(Z_{\overline{c}s}\)

In this section we calculate the mass and current coupling of the \(Z_{ \overline{c}s}=[sd][\overline{u}\overline{c}]\) tetraquarks with the quantum numbers \(J^{P}=0^{+}\), \(0^{-}\) and \(1^{+}\) by treating them as diquak-antidiquark systems. In order to simplify the expressions we introduce the notations: in what follows the scalar tetraquark \(Z_{\overline{ c}s}\) will be denoted as \(Z_{S}\), whereas for the pseudoscalar and axial-vector ones we will utilize \(Z_{PS}\) and \(Z_{AV}\), respectively.

*C*is the charge conjugation operator. In the present work we restrict ourselves by the simplest case and employ the current

*c*-quark propagators, respectively.

*x*we get expression containing contributions of the ground state pseudoscalar and axial-vector particles, i.e.

*c*and

*s*-quarks, and vacuum expectations of quark, gluon and mixed operators, which are presented below:

In Figs. 9 and 10 we plot dependence of the axial-vector tetraquark’s mass and current coupling on \(M^{2}\) and \(s_{0}\). As is seen, estimations made for theoretical errors in the case of \(Z_{S}\) are valid for the \(Z_{AV}\) state, as well.

Our results for the masses and current couplings of \(J^{P}=0^{+},\ 0^{-}\) and \(J^{P}=1^{+}\) charm–strange tetraquarks are collected in Table 1. The working ranges for the parameters \(M^{2}\) and \(s_{0}\), and errors of the calculations are also presented in Table 1.

## 3 Decay channels of the scalar tetraquark \(Z_{S}\)

*S*-wave decay modes, whereas the last one is

*P*-wave decay.

The masses and current couplings of the \(Z_S\), \(Z_{PS}\) and \(Z_{AV}\) tetraquarks

| \(Z_S\) | \(Z_{PS}\) | \(Z_{AV}\) |
---|---|---|---|

\(M^2 ~(\mathrm {GeV}^2\)) | 2.5–3.5 | 2.5–3.5 | 2.5–3.5 |

\(s_0 ~(\mathrm {GeV}^2\)) | 8–10 | 9.5–11.5 | 9.5–11.5 |

\(m_{Z} ~(\mathrm {MeV})\) | \(2628^{+166}_{-153}\) | \(2719^{+144}_{-156}\) | \(2826^{+134}_{-157}\) |

\(f_{Z} \times 10^{3}\) | \(2.1^{+0.6}_{-0.5} ~(\mathrm {GeV}^4)\) | \(0.83^{+0.09}_{-0.11} ~(\mathrm {GeV}^3)\) | \(2.6^{+0.6}_{-0.7}~(\mathrm {GeV} ^4)\) |

*f*(

*x*,

*y*,

*z*) is

*D*meson is

*DK*are pseudoscalar mesons, differences between two decay channels are encoded in the matrix element

*K*meson

*K*meson. The strong coupling \( g_{Z_{S}DK}\) with evident replacements is defined by Eq. (27).

*P*-wave decay \(Z_{S}\rightarrow D_{s1}(2460)\pi \) the interpolating current, matrix element and strong coupling of the axial-vector meson \( D_{s1}(2460)\) are introduced by means of the formulas

*D*and \(D_{s1}(2460)\), as well as \(\pi \) and

*K*mesons which we employ in numerical computations are collected in Table 2. The masses of particles are taken from Ref. [48], for decay constants of

*D*and \(D_{s}\) mesons we use information from Ref. [49], decay constant of \(D_{s1}(2460)\) is borrowed from [50]. Table contains also parameters of the \(D^{*}\), \(D_{s}^{*}\) and \(D_{s0}^{*}(2317)\) mesons which will be used in the next section.

Parameters of the mesons used in numerical calculations

Parameters | Values (Mev) |
---|---|

\(m_{D}\) | \((1869.5 \pm 0.4)\) |

\(f_{D}\) | \((211.9 \pm 1.1)\) |

\(m_{D_s}\) | \((1969.0 \pm 1.4) \) |

\(f_{D_s}\) | \((249.0 \pm 1.2) \) |

\(m_{D_{s1}} \) | \((2459.6 \pm 0.9) \) |

\(f_{D_{s1}}\) | \((481\pm 164) \) |

\(m_{D_{s}^{*}}\) | \((2112.1\pm 0.4)\) |

\(f_{D_{s}^{*}}\) | \((308\pm 21) \) |

\(m_{D^{*}}\) | \((2010.26\pm 0.25)\) |

\(f_{D^{*}}\) | \((252.2\pm 22.66) \) |

\(m_{D_{s0}^{*}}\) | \((2318.0\pm 1.0)\) |

\(f_{D_{s0}^{*}}\) | 201 |

\(m_{K}\) | \((493.677\pm 0.016) \) |

\(f_{K}\) | 156 |

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

\(f_{\pi }\) | 131 |

The charmed particle composed of four different quarks as a partner of the *X*(5568) resonance was previously investigated in our work [38]. We analyzed this state using the interpolating currents of both \(C\gamma _{5}\otimes \gamma _{5}C\) and \(C\gamma _{\mu }\otimes \gamma ^{\mu }C\) types. The diquark–antidiquark composition of \(X_{c}=[su][ \overline{c}\overline{d}]\) means that it is a neutral particle. Nevertheless, it is instructive to compare parameters of \(X_{c}\) with results for \(Z_{S}\) obtained in the present work. In the case of the interpolating current \(C\gamma _{5}\otimes \gamma _{5}C\) we found \( m_{X_{c}}=(2634\pm 62)\ \mathrm {MeV}\) which is very close to our present result. The processes \(X_{c}\rightarrow \overline{D}^{0}\overline{K}^{0}\) and \(X_{c}\rightarrow D_{s}^{-}\pi ^{+}\) were also subject of studies in Ref. [38]. Width of these decay channels \(\Gamma (X_{c}\rightarrow \overline{D}^{0}\overline{K}^{0})=(53.7\pm 11.6)\ \mathrm { MeV}\) and \(\Gamma (X_{c}\rightarrow D_{s}^{-}\pi ^{+})=(8.2\pm 2.1)\ \mathrm { MeV}\) are comparable with ones presented in Eqs. (44) and ().

## 4 \(Z_{PS}\rightarrow \ D_{s}^{*}\pi ,\ D^{*}K,\ D_{s0}^{*}(2317)\pi \) and \(Z_{AV}\rightarrow \ D_{s}^{*}\pi ,\ D^{*}K,\ D_{s1}(2460)\pi \) decays of the pseudoscalar and axial-vector tetraquarks

The pseudoscalar \(Z_{PS}\) and axial-vector \(Z_{AV}\) tetraquarks may decay through different channels. Among kinematically allowed decay channels of \( Z_{PS}\) state are \(S-\)wave mode \(Z_{PS}\rightarrow D_{s0}(2317)\pi \), and \( P- \)wave modes \(Z_{PS}\rightarrow D_{s}^{*}\pi \) and \(\ D^{*}K\). The decays of the tetraquark \(Z_{AV}\) include \(S-\)wave channels \( Z_{AV}\rightarrow D_{s}^{*}\pi ,\ D^{*}K\) and \(P-\)wave mode \( Z_{AV}\rightarrow D_{s1}(2460)\pi \).

In numerical calculations of the \(Z_{PS}\) and \(Z_{AV}\) states’ strong couplings the Borel parameter and continuum threshold are chosen within the same ranges as in computations of their masses (see, Table 1). As input parameters we employ also mass and decay constant of the mesons \(D_{s}^{*},\ D^{*}\) and \(\ D_{s0}^{*}(2317)\) from Table 2. It is worth noting that the decay constants \( f_{D_{s}^{*}}\), \(f_{D^{*}}\) and \(f_{D_{s0}^{*}}\) have been taken from Refs. [51, 52, 53], respectively.

The strong couplings and decay widths of the \(Z_{AV}\) and \(Z_{PS}\) tetraquarks

Decay | Strong couplings | Decay width (Mev) |
---|---|---|

\(Z_{AV} \rightarrow D_{s}^{*}\pi \) | \((0.26 \pm 0.07)~\mathrm {GeV}^{-1}\) | \((7.94 \pm 2.21)\) |

\(Z_{AV} \rightarrow D^{*} K\) | \((0.63 \pm 0.17) ~\mathrm {GeV}^{-1}\) | \((37.38 \pm 10.84)\) |

\(Z_{AV}\rightarrow D_{s1}\pi \) | \((1.55 \pm 0.43) ~\mathrm {GeV}^{-1}\) | \((2.02 \pm 0.59)\) |

\(Z_{PS}\rightarrow D_{s}^{*}\pi \) | \(3.18 \pm 0.94\) | \((4.37 \pm 1.27)\) |

\(Z_{PS}\rightarrow D^{*}K\) | \(8.24 \pm 2.39 \) | \((19.09 \pm 5.73)\) |

\(Z_{PS} \rightarrow D_{s0}^{*}\pi \) | \((0.76 \pm 0.18) ~\mathrm {GeV}^{-1}\) | \((14.64 \pm 3.94)\) |

## 5 Conclusions

In the present work we have investigated the charm–strange tetraquarks \(Z_{ \overline{c}s}=[sd][\overline{u}\overline{c}]\) by calculating their spectroscopic parameters and decay channels. It is easy to see that these states bear two units of electric charge \(-|e|\) and belong to a class of doubly charged tetraquarks. Their counterparts with the structure \(Z_{c \overline{s}}=[uc][\overline{s}\overline{d}]\) have evidently a charge \(+2|e|\). We have considered scalar, pseudoscalar and axial-vector doubly charged states. Their masses have been obtained using QCD two-point sum rule method. Our results have allowed us to fix possible decay channels of these states and found their widths. Investigations confirm that the doubly charged diquark–antidiquarks are neither broad states nor very narrow resonances.

Observation of doubly charged tetraquarks may open new stage in exploration of multiquark systems. In fact, resonances that are interpreted as hidden charm (bottom) tetraquarks may be also considered as excited states of charmonia (bottomonia) or their superpositions. The charged resonances can not be explained by this way, and are serious candidates to genuine tetraquarks. They may have diquark–antidiquark structure or be bound states of conventional mesons. In the last case, charged and neutral conventional mesons create shallow molecular states with large decay width. Therefore, it is reasonable to assume that doubly charged tetraquarks presumably exist only as diquark–antidiquarks, because binding of two mesons with the same electric charge to form a molecular state due to repulsive forces between them seems problematic.

The doubly charged tetraquarks deserve further detailed investigations. These studies should embrace also \(Z_{b\overline{c}}\)-type states that constitute a subclass of open charm–bottom states. Experimental exploration and discovery of \(Z_{c\overline{s}}\) and/or \(Z_{b\overline{c}}\) tetraquarks may have far-reaching consequences for hadron spectroscopy.

## Notes

### Acknowledgements

The work of S. S. A. was supported by Grant No. EIF-Mob-8-2017-4(30)-17/01/1 of the Science Development Foundation under the President of the Azerbaijan Republic. K. A. thanks TÜBITAK for the partial financial support provided under Grant No. 115F183.

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