Axial-vector and pseudoscalar tetraquarks $[ud][{\overline{c}}{\overline{s}}]$

Abstract

Spectroscopic parameters and widths of the fully open-flavor axial-vector and pseudoscalar tetraquarks $X_{\textrm{AV}}$ and $X_{\textrm{PS}}$ with content $[ud][{\overline{c}}{\overline{s}}]$ are calculated by means of the QCD sum rule methods. Masses and current couplings of $X_{\textrm{AV}}$ and $X_{\textrm{PS}}$ are found using two-point sum rule computations performed by taking into account various vacuum condensates up to dimension 10. The full width of the axial-vector state $X_{\textrm{AV}}$ is evaluated by including into analysis S-wave decay modes $X_{\textrm{AV}}\rightarrow D^{*}(2010)^{-}K^{+}$, ${\overline{D}}^{*}(2007)^{0}K^{0}$, $D^{-}K^{*}(892)^{+}$, and ${\overline{D}}^{0}K^{*}(892)^{0}$. In the case of $X_{\textrm{PS}}$, we consider S-wave decay $X_{\textrm{PS}}\rightarrow {\overline{D}} _{0}^{*}(2300)^{0}K^{0}$, and P-wave processes $X_{\textrm{PS}}\rightarrow D^{-}K^{*}(892)^{+}$ and $X_{\textrm{PS}}\rightarrow {\overline{D}} ^{0}K^{*}(892)^{0}$. To determine partial widths of these decay modes, we employ the QCD light-cone sum rule method and soft-meson approximation, which are necessary to estimate strong couplings at tetraquark–meson–meson vertices $X_{\textrm{AV}}D^{-}D^{*}(2010)^{-}K^{+}$, etc. Our predictions for the mass $m_{\textrm{AV}}=(2800 \pm 75)~\text {MeV}$ and width $\Gamma _{\textrm{AV}}=(58 \pm 10)~\text {MeV}$ of the tetraquark $X_{ \textrm{AV}}$, as well as results $m_{\textrm{PS}}=(3000 \pm 60)~\text {MeV}$ and $\Gamma _{\textrm{PS}}=(65 \pm 12)~\text {MeV}$ for the same parameters of $X_{\textrm{PS}}$ may be useful in future experimental studies of multiquark hadrons.

1 Introduction

Recent LHCb information on new structures $X_{0}(2900)$ and $X_{1}(2900)$ observed in the invariant $D^{-}K^{+}$ mass distribution of the process $B^{+}\rightarrow D^{+}D^{-}K^{+}$ [1, 2], enhanced activity of researches to investigate fully open-flavor exotic mesons. In fact, by taking into account dominant decays of the resonance-like peaks $X_{0(1)}(2900)\rightarrow D^{-}K^{+}$ and assuming that they are four-quark systems, one sees that $X_{0(1)}(2900)$ are built of quarks c, s, u, and d. The LHCb collaboration measured masses and widths of the structures $X_{0(1)}(2900)$, and fixed their spin-parities. It was found that $X_{0}(2900)$ and $X_{1}(2900)$ bear the quantum numbers $J^{\textrm{P}}=0^{+}$ and $J^{\textrm{P}}=1^{-}$, respectively. Let us note that, alternatively, structures $X_{0(1)}(2900)$ may appear as triangle singularities in some rescattering diagrams: such interpretation was not excluded by LHCb as well.

In the four-quark picture, widely accepted to explain the LHCb data, $X_{0(1)}(2900)$ may be considered in the framework of both the molecule and diquark–antidiquark (tetraquark) models. Thus, in publications [3, 4] the resonance $X_{0}(2900)$ was analyzed as the scalar diquark–antidiquark state $[ud][{\overline{c}}{\overline{s}}]$, whereas in Refs. [5, 6] it was treated as a molecule $D^{*-}K^{*+}$ or ${\overline{D}}^{*0}K^{*0}$. The situation is almost the same for the vector resonance $X_{1}(2900)$: it was studied in the context of the tetraquark and molecule models, for instance, in Refs. [5, 7, 8]. There are numerous articles devoted to investigations of $X_{0(1)}(2900)$ using different methods and schemes of high energy physics: relatively complete list of such papers can be found in Refs. [6, 7, 9].

An interesting conjecture about nature of $X_{0}(2900)$ was made in Ref. [10], where it was interpreted as a radial excitation $X_{0}^{\prime }$ of the scalar tetraquark $X_{0}=[ud][{\overline{c}}\overline{s }]$. We addressed this problem in our work [9], and calculated masses and widths of the ground-state 1S and radially excited 2S tetraquarks $X_{0}^{(\prime )}$ using the QCD sum rule method. We modeled $X_{0}^{(\prime )}$ as particles composed of the axial-vector diquark [ud] and axial-vector antidiquark $[{\overline{c}}{\overline{s}}]$. We also constructed the tetraquarks $X_{\textrm{S}}^{(\prime )}$ by utilizing a scalar diquark and antidiquark, and found their parameters. It was demonstrated that, the ground-state particles $X_{0}$ and $X_{\textrm{S}}$ are lighter than the resonance $X_{0}(2900)$, whereas radially excited tetraquarks $X_{0}^{\prime }$ and $X_{\textrm{S}}^{\prime }$ with the masses around $\approx 3320~\text {MeV}$ are heavier it. It other words, none of these four-quark states can be identified with the resonance $X_{0}(2900)$. Therefore, it is reasonable to treat $X_{0}$ and $X_{\textrm{S}}$ as new hypothetic exotic mesons to be searched for in experiments.

Fully open-flavor tetraquarks, to be fair, were already objects of theoretical studies, which intensified after information on the resonance X(5568) presumably composed of b, s, u, and d quarks [11]. Though existence of X(5568) was not confirmed by other experimental groups, its charmed partners $b\rightarrow c$ are still under detailed analysis. In fact, the spectroscopic parameters and full width of scalar tetraquark $X_{c}=[su][{\overline{c}}{\overline{d}}]$ were calculated in Ref. [12]. Masses of exotic mesons with the same content, but quantum numbers $J^{\textrm{P}}=0^{+}$ and $J^{\textrm{P}}=1^{+}$ were estimated in Ref. [13].

In various combinations c, s, u, and d quarks form different classes of four-quark mesons, features of which deserve investigations. Interesting class of fully open-flavor particles is collection of states $Z^{++}=[cu][ {\overline{s}}{\overline{d}}]$, which carries two units of electric charge. The scalar, pseudoscalar, axial-vector and vector members of this group were studied in our articles Refs. [14, 15], respectively. It was pointed out that scalar and vector tetraquarks $Z_{ \textrm{S}}^{++}$ and $Z_{\textrm{V}}^{++}$ may be observed in the $D^{+}K^{+}$ mass distribution of the decay $B^{+}\rightarrow D^{-}D^{+}K^{+}$ [15].

Tetraquarks with a content $[ud][{\overline{c}}{\overline{s}}]$ establish new class of open-flavor particles. Observation of the resonances $X_{0(1)}(2900)$ by the LHCb collaboration, available experimental data, and possible interpretation of $X_{1}(2900)$ as a vector state $X_{\textrm{V} }=[ud][{\overline{c}}{\overline{s}}]$ make these particles objects of special interest. In the present article, we continue our studies started in Refs. [7, 9] by calculating spectroscopic parameters and full widths of axial-vector and pseudoscalar four-quark states $X_{ \textrm{AV}}$ and $X_{\textrm{PS}}$ with the same $[ud][{\overline{c}} {\overline{s}}]$ content.

Masses and current couplings of $X_{\textrm{AV}}$ and $X_{\textrm{PS}}$ are evaluated in the context of the two-point sum rule method [16, 17]. In calculations, we take into account various vacuum condensates up to dimension 10. The full width of the axial-vector state $X_{\textrm{AV}}$ is found by including into analysis its S- wave decay modes $X_{\textrm{AV}}\rightarrow D^{*}(2010)^{-}K^{+},$ ${\overline{D}}^{*}(2007)^{0}K^{0}$, $D^{-}K^{*}(892)^{+}$, and ${\overline{D}}^{0}K^{*}(892)^{0}$. To estimate width of the pseudoscalar tetraquark $X_{\textrm{PS}}$, we consider kinematically allowed S-wave channel $X_{\textrm{PS}}\rightarrow {\overline{D}}_{0}^{*}(2300)^{0}K^{0}$, and P-wave decay modes $X_{\textrm{PS}}\rightarrow D^{-}K^{*}(892)^{+}$ and $X_{\textrm{PS}}\rightarrow D^{0}K^{*}(892)^{0}$.

Partial widths of these processes are governed by strong couplings at relevant vertices, for example, at $X_{\textrm{AV}}D^{*} (2010)^{-}K^{+}$ for the first process. To calculate required couplings, we use the QCD light-cone sum rule (LCSR) method [18] and soft-meson approximation [19, 20]. The latter is necessary to treat tetraquark–meson–meson vertices, which due to unequal number of quark fields in tetraquark and meson interpolating currents differ from standard three-meson vertices [21].

This paper is organized in the following manner: In Sect. 2, we calculate the masses and current couplings of the tetraquarks $X_{ \textrm{AV}}$ and $X_{\textrm{PS}}$. In Sect. 3, we determine strong couplings $g_{i},\ i=1,2,3,4$ corresponding to vertices $X_{\textrm{AV}}D^{*}(2010)^{-}K^{+}$, $X_{\textrm{AV}}{\overline{D}}^{*}(2007)^{0}K^{0}$ $X_{\textrm{AV}}D^{-}K^{*}(892)^{+}$, and $X_{\textrm{AV}}{\overline{D}}^{0}K^{*}(892)^{0}$. In this section, we compute partial widths of corresponding processes, and estimate full width of $X_{\textrm{AV}}$. In Sect. 4, we consider the decays $X_{\textrm{PS} }\rightarrow {\overline{D}}_{0}^{*}(2300)^{0}K^{0}$, $D^{-}K^{*}(892)^{+}$, and $D^{0}K^{*}(892)^{0}$, and find strong couplings $G_{j},\ j=1,2,3$ at relevant vertices. Using $G_{j}$, we calculate partial width of these decays and evaluate full width of $X_{\textrm{PS}}$ by saturating it with these channels. Section 5 contains our conclusions.

2 Masses and current couplings of the tetraquarks $X_{\textrm{AV}}$ and $X_{\textrm{PS}}$

In this section, we compute spectroscopic parameters of the states $X_{ \textrm{AV}}$ and $X_{\textrm{PS}}$ by means of the two-point sum rule method. It is an effective nonperturbative approach elaborated to evaluate parameters of ordinary mesons and baryons. The QCD sum rules express various physical quantities in terms of universal vacuum condensates which do not depend on a problem under consideration. At the same time, they contain auxiliary parameters $s_{0}$ and $M^{2}$ specific for each computation. The first of them is the continuum subtraction parameter $s_{0}$ that separates contribution of a ground-state particle in the phenomenological side of a sum rule from effects of higher resonances and continuum states. The Borel parameter $M^{2}$ is required to suppress these unwanted continuum effects. By introducing $M^{2}$ and $s_{0}$ into analysis and employing an assumption about quark-hadron duality one connects phenomenological and QCD sides of sum rules and gets a sum equality. The latter can be used to express physical observables in terms of different vacuum condensates. The parameters $M^{2}$ and $s_{0}$ generate theoretical uncertainties in results, which nevertheless can be estimated and kept under control.

In what follows, we calculate the mass m and current coupling f of the axial-vector meson $X_{\textrm{AV}}$ (we employ also $m_{\textrm{AV}}$ and $f_{\textrm{AV}}$), and provide only final results for $X_{\textrm{PS}}$. The starting point in computation of the spectroscopic parameters of the tetraquark $X_{\textrm{AV}}$ is the correlation function

$$\begin{aligned} \Pi _{\mu \nu }(p)=i\int d^{4}xe^{ipx}\langle 0|{\mathcal {T}}\{J_{\mu }(x)J_{\nu }^{\dag }(0)\}|0\rangle . \end{aligned}$$

(1)

where, ${\mathcal {T}}$ is the time-ordered product of two currents, and $J_{\mu }(x)$ is the interpolating current for the axial-vector state $X_{\textrm{AV}}$. We model the tetraquark $X_{\textrm{AV}}$ as a compound formed by the scalar diquark $u^{T}C\gamma _{5}d$ and axial-vector antidiquark ${\overline{c}}\gamma _{\mu }C{\overline{s}}^{T}$, which are antitriplet and triplet states of the color group $SU_{c}(3)$, respectively. Therefore, corresponding interpolating current is given by the formula

$$\begin{aligned} J_{\mu }(x)=\epsilon {\widetilde{\epsilon }}u_{b}^{T}(x)C\gamma _{5}d_{c}(x) {\overline{c}}_{d}(x)\gamma _{\mu }C{\overline{s}}_{e}^{T}(x), \end{aligned}$$

(2)

and belongs to $[\overline{{\textbf{3}}}_{c}]_{ud}\otimes [{\textbf{3}} _{c}]_{{\overline{c}}{\overline{s}}}$ representation of the color group. In expression above, $\epsilon {\widetilde{\epsilon }}=\epsilon _{abc}\epsilon _{ade}$, where a, b, c, d and e are color indices. In Eq. (2) c(x), s(x), u(x) and d(x) denote quark fields, and C is the charge conjugation matrix.

The phenomenological side of the sum rule $\Pi _{\mu \nu }^{\textrm{Phys} }(p)$

$$\begin{aligned} \Pi _{\mu \nu }^{\textrm{Phys}}(p)=\frac{\langle 0|J_{\mu }|X_{\textrm{AV} }(p,\varepsilon )\rangle \langle X_{\textrm{AV}}(p,\varepsilon )|J_{\nu }^{\dagger }|0\rangle }{m^{2}-p^{2}}+\cdots ,\nonumber \\ \end{aligned}$$

(3)

is derived from Eq. (1) by inserting a complete set of intermediate states with quark contents and spin-parity of the tetraquark $X_{\textrm{AV}}$, and carrying out integration over x. The momentum and polarization vector of $X_{\textrm{AV}}$ are denoted by p and $\varepsilon$, respectively. It should be noted that in $\Pi _{\mu \nu }^{\textrm{Phys} }(p)$ the ground-state term is written down explicitly, whereas contributions of higher resonances and continuum states are shown by ellipses.

In Eq. (3), we have assumed that the phenomenological side of the sum rule $\Pi _{\mu \nu }^{\textrm{Phys}}(p)$ can be approximated by a single pole term. But in the case of multiquark systems this approximation has to be used with some caution, because the physical side receives contribution also from two-hadron reducible terms. Indeed, a relevant interpolating current couples not only to a multiquark hadron, but also to a two-hadron continuum. This problem was raised in Refs. [22, 23] when considering pentaquarks, and revisited recently in the case of tetraquarks [24], where it is argued that the contributions at the orders ${{{\mathcal {O}}}} (1)$ and ${{{\mathcal {O}}}} (\alpha _s)$ in the operator product expansion (OPE) are canceled out exactly by the meson–meson scattering states at the hadronic side and the tetraquark molecular states start to receive contributions at the order ${{{\mathcal {O}}} }(\alpha ^2_s)$. Then the reducible contributions should be subtracted from the sum rule, which can be done by means of two methods. One of them is direct subtraction of two-hadron terms from $\Pi _{\mu \nu }^{\textrm{Phys}}(p)$ by calculating current-two-hadron coupling constant using an independent QCD sum rule. This strategy was realized, for example, in Ref. [25] to investigate anti-charmed pentaquark state. Existence of a two-hadron continuum below a multiquark system means that such particle is unstable and decays to these conventional hadrons. In other words, a two-hadron continuum generates the finite width $\Gamma (p^{2})$ of a multiquark system. Relevant effects can be taken into account by modifying the quark propagator in Eq. (3)

$$\begin{aligned} \frac{1}{m^{2}-p^{2}}\rightarrow \frac{1}{m^{2}-p^{2}-i\sqrt{p^{2}}\Gamma (p) }. \end{aligned}$$

(4)

This second method was used to study the tetraquarks [26]. Rather detailed investigations demonstrated that effects of the modification Eq. (4) can be taken into account by absorbing two-meson contributions into a current-tetraquark coupling constant and keeping stable tetraquark’s mass [27, 28]. Uncertainties generated by changing of a coupling are numerically smaller than theoretical errors of sum rule analysis itself. In fact, two-meson effects lead to additional $\approx 7\%$ uncertainty in the current coupling $f_{T}$ for doubly charmed pseudoscalar tetraquark $cc{\overline{s}}{\overline{s}}$ with the mass $m_{T}=4390~\text {MeV}$ and full width $\Gamma _{T}\approx 300~ \text {MeV}$ [27]. In the case of the resonance $Z_{c}^{-}(4100)$ these uncertainties do not exceed $\approx 5\%$ of the coupling $f_{Z_{c}}$ [28]. Therefore, one can neglect two-meson reducible terms and use in $\Pi _{\mu \nu }^{\textrm{Phys}}(p)$ single-pole zero-width approximation, as it has been done in Eq. (3).

To simplify the correlation function $\Pi _{\mu \nu }^{\textrm{Phys}}(p)$ and express it in terms of the tetraquark’s mass and current coupling, we use the matrix element

$$\begin{aligned} \langle 0|J_{\mu }|X_{\textrm{AV}}(p,\varepsilon )\rangle =fm\varepsilon _{\mu }, \end{aligned}$$

(5)

and recast $\Pi _{\mu \nu }^{\textrm{Phys}}(p)$ into the following form

$$\begin{aligned} \Pi _{\mu \nu }^{\textrm{Phys}}(p)=\frac{f^{2}m^{2}}{m^{2}-p^{2}}\left( -g_{\mu \nu }+\frac{p_{\mu }p_{\nu }}{m^{2}}\right) +\cdots . \end{aligned}$$

(6)

The function $\Pi _{\mu \nu }^{\textrm{Phys}}(p)$ has two Lorentz structures determined by $g_{\mu \nu }$ and $p_{\mu }p_{\nu }$. One of them can be chosen to continue sum rule analysis. We work with the structure proportional to $g_{\mu \nu }$ and corresponding invariant amplitude $\Pi ^{ \textrm{Phys}}(p^{2})$. Advantage of this structure is that it is formed due to contributions of only spin-1 particles, and is free of any contaminations.

The QCD side of the sum rules $\Pi _{\mu \nu }^{{\textrm{OPE}}}(p)$ should be computed in the operator product expansion with some accuracy. To this end, we substitute into $\Pi _{\mu \nu }(p)$ explicit expression of the current $J_{\mu }(x)$, contract relevant quark fields, and replace contractions by appropriate propagators. These operations lead to the expression

$$\begin{aligned}{} & {} \Pi _{\mu \nu }^{{\textrm{OPE}}}(p)=i\int d^{4}xe^{ipx}\epsilon \widetilde{ \epsilon }\epsilon ^{\prime }{\widetilde{\epsilon }}^{\prime }{\textrm{Tr}}\left[ \gamma _{5}{\widetilde{S}}_{u}^{bb^{\prime }}(x)\right. \nonumber \\{} & {} \quad \times \left. \gamma _{5}S_{d}^{cc^{\prime }}(x)\right] {\textrm{Tr}}\left[ \gamma _{\mu }{\widetilde{S}}_{s}^{e^{\prime }e}(-x)\gamma _{\nu }S_{c}^{d^{\prime }d}(-x)\right] , \end{aligned}$$

(7)

where

$$\begin{aligned} {\widetilde{S}}_{q}(x)=CS_{q}^{T}(x)C, \end{aligned}$$

(8)

and $\epsilon ^{\prime }{\widetilde{\epsilon }}^{\prime }=\epsilon _{a^{\prime }b^{\prime }c^{\prime }}\epsilon _{a^{\prime }d^{\prime }e^{\prime }}$. Here, $S_{c}(x)$ and $S_{u(s,d)}(x)$ are the heavy c- and light u(s, d)-quark propagators, respectively: Their explicit expressions are presented in Appendix (see, also Ref. [29]).

The function $\Pi _{\mu \nu }^{{\textrm{OPE}}}(p)$ is a sum of components proportional to $g_{\mu \nu }$ and $p_{\mu }p_{\nu }$. We choose the invariant amplitude $\Pi ^{{\textrm{OPE}}}(p^{2})$ corresponding to structure $\sim g_{\mu \nu }$, and use it to derive sum rules for m and f, which read

$$\begin{aligned} m^{2}=\frac{\Pi ^{\prime }(M^{2},s_{0})}{\Pi (M^{2},s_{0})}, \end{aligned}$$

(9)

and

$$\begin{aligned} f^{2}=\frac{e^{m^{2}/M^{2}}}{m^{2}}\Pi (M^{2},s_{0}). \end{aligned}$$

(10)

In expressions above, $\Pi (M^{2},s_{0})$ is the Borel transformed and subtracted invariant amplitude $\Pi ^{{\textrm{OPE}}}(p^{2})$, and $\Pi ^{\prime }(M^{2},s_{0})=d\Pi (M^{2},s_{0})/d(-1/M^{2})$.

Computing the function $\Pi (M^{2},s_{0})$ and fixing of regions for parameters $M^{2}$ and $s_{0}$ are next problems in our study of m and f. Calculations prove that $\Pi (M^{2},s_{0})$ has the form

$$\begin{aligned} \Pi (M^{2},s_{0})=\int _{{\mathcal {M}}^{2}}^{s_{0}}ds\rho ^{{\textrm{OPE}}}(s)e^{-s/M^{2}}+\Pi (M^{2}), \end{aligned}$$

(11)

where ${\mathcal {M}}=m_{c}+m_{s}$. In the present paper, we neglect the masses of u and d quarks, as well as set $m_{s}^{2}=0$ saving, at the same time, terms $\sim m_{s}$. The spectral density $\rho ^{{\textrm{OPE}}}(s)$ is found as an imaginary part of the function $\Pi ^{{\textrm{OPE}}}(p^{2})$. Borel transformation some of terms are computed directly from expression of $\Pi _{\mu \nu }^{{\textrm{OPE}}}(p)$: They form the second component $\Pi (M^{2})$ in Eq. (11). Our analysis includes contributions of different quark, gluon and mixed vacuum condensates up to dimension 10. Full analytical expressions of $\rho ^{{\textrm{OPE}}}(s)$ and $\Pi (M^{2})$ are written down in Appendix.

The vacuum condensates, that enter to sum rules Eqs. (9) and (10), are universal quantities obtained from analysis of various hadronic processes [16, 17, 30, 31]. Below, we list their values used in our numerical computations

$$\begin{aligned}{} & {} \langle {\overline{q}}q\rangle =-(0.24\pm 0.01)^{3}~\text {GeV}^{3},\ \langle {\overline{s}}s\rangle =(0.8\pm 0.01)\langle {\overline{q}}q\rangle , \nonumber \\{} & {} \langle {\overline{q}}g_{s}\sigma Gq\rangle =m_{0}^{2}\langle {\overline{q}} q\rangle ,\ \langle {\overline{s}}g_{s}\sigma Gs\rangle =m_{0}^{2}\langle {\overline{s}}s\rangle , \nonumber \\{} & {} m_{0}^{2}=(0.8\pm 0.2)~\text {GeV}^{2}, \nonumber \\{} & {} \left\langle \frac{\alpha _{s}G^{2}}{\pi }\right\rangle =(0.012\pm 0.004)~\text {GeV} ^{4}, \nonumber \\{} & {} \langle g_{s}^{3}G^{3}\rangle =(0.57\pm 0.29)~\text {GeV}^{6},\ m_{c}=1.27\pm 0.02~\text {GeV}, \nonumber \\{} & {} m_{s}=93_{-5}^{+11}~\text {MeV}. \end{aligned}$$

(12)

As is seen, the vacuum condensate of strange quarks differs from $\langle 0| {\overline{q}}q|0\rangle$ [30]. The mixed condensates $\langle {\overline{q}}g_{s}\sigma Gq\rangle$ and $\langle {\overline{s}}g_{s}\sigma Gs\rangle$ are expressed using the corresponding quark condensates and parameter $m_{0}^{2}$, numerical value of which was extracted from analysis of baryonic resonances [31]. For the gluon condensate $\langle g^{3}G^{3}\rangle$, we employ the estimate given in Ref. [32]. This list also contains the masses of c and s quarks from Ref. [33] in the $\overline{\textrm{MS}}$-scheme.

Fig. 1

The pole contribution ${\textrm{PC}}$ as a function of the Borel parameter $M^{2}$ at different $s_{0}$. The horizontal black line limits a region ${\textrm{PC}}=0.5$. The red triangle marks the point, where the mass m of $X_{\textrm{AV}}$ has effectively been computed

Fig. 2

Different contributions to $\Pi (M^{2},s_{0})$ as functions of $M^2$. Dimension-9 and 10 terms are very small and not shown in the plot. All curves have been calculated at $s_0=10~\text {GeV}^2$

Fig. 3

Mass m of the tetraquark $X_{\textrm{AV}}$ as a function of the Borel parameter $M^{2}$ (left), and the continuum threshold parameter $s_0$ (right)

Another problem is a choice of working windows for parameters $M^{2}$ and $s_{0}$. They are fixed in such a way that to meet constraints imposed on $\Pi (M^{2},s_{0})$ by a pole contribution (${\textrm{PC}}$) and convergence of the operator product expansion. These constraints can be quantified by means of the following expressions

$$\begin{aligned} {\textrm{PC}}=\frac{\Pi (M^{2},s_{0})}{\Pi (M^{2},\infty )}, \end{aligned}$$

(13)

and

$$\begin{aligned} R(M^{2})=\frac{\Pi ^{\textrm{DimN}}(M^{2},s_{0})}{\Pi (M^{2},s_{0})}, \end{aligned}$$

(14)

where $\Pi ^{\textrm{DimN}}(M^{2},s_{0})$ is a sum of $\text {DimN}\equiv \text {Dim}(8+9+10)$ terms. In what follows, we require fulfilment of the restrictions

$$\begin{aligned} {\textrm{PC}}\ge 0.5,\ R(M^{2})\le 0.05. \end{aligned}$$

(15)

The ${\textrm{PC}}$ and $R(M^{2})$ are employed to fix the higher and lower limits of the Borel parameter $M^{2}$, respectively. These two values determine boundaries of the region where $M^{2}$ can be varied. Calculations show that intervals

$$\begin{aligned} M^{2}\in [2.5,3.2]~\text {GeV}^{2},\ s_{0}\in [9.5,10.5]~\text {GeV}^{2}, \end{aligned}$$

(16)

are appropriate regions for the parameters $M^{2}$ and $s_{0}$, and comply with limits on ${\textrm{PC}}$ and convergence of ${\textrm{OPE}}$. Thus, at $M_{\textrm{max}}^{2}=3.2~\text {GeV}^{2}$ on average in $s_{0}$ the pole contribution is 0.51, whereas at $M_{\textrm{min}}^{2}=2.5~\text {GeV} ^{2}$ it becomes equal to 0.73. To visualize dynamics of the pole contribution when varying the Borel parameter, we plot ${\textrm{PC}}$ as a function of $M^{2}$ at different $s_{0}$ in Fig. 1. One can see, that except for a small region $M^{2}\ge 3.1~\text {GeV}^{2}$ at $s_{0}=9.5~\text {GeV}^{2}$ the pole contribution exceeds 0.5. On average in $s_{0}$, the condition ${\textrm{PC}}\ge 0.5$ is fulfilled in the whole working region Eq. (16).

To be convinced in convergence of ${\textrm{OPE}}$, we calculate $R(M_{ \textrm{min}}^{2})$ at the minimum point $M_{\textrm{min}}^{2}=2.5~{\textrm{GeV}}^{2}$, and get $R(2.5~\text {GeV}^{2})\approx 0.027$ in accordance with constraint from Eq. (15). Results of more detailed analysis are depicted in Fig. 2. In this figure, we show the perturbative and nonperturbative components of the correlation function $\Pi (M^{2},s_{0})$: A prevalence of the perturbative contribution to $\Pi (M^{2},s_{0})$ over nonperturbative one is evident. Without regard for some higher dimensional terms, the nonperturbative contributions reduce by increasing the dimensions of the corresponding operators.

The region for $s_{0}$ has to meet constraints coming from dominance of ${\textrm{PC}}$ and convergence of ${\textrm{OPE}}$. Self-consistency of performed analysis can be checked by comparing the parameter $\sqrt{s_{0}}$ and the $X_{\textrm{AV}}$ tetraquark’s mass extracted from the sum rule: Evidently, an inequality $m<\sqrt{s_{0}}$ should be satisfied. Additionally, $\sqrt{s_{0}}$ bears information on a mass $m^{*}$ of the first radial excitation of the tetraquark $X_{\textrm{AV}}$, therefore the restriction $m^{*}\ge$ $\sqrt{s_{0}}$ provides low limit for $m^{*}$.

We extract the mass m and coupling f by computing them at different $M^{2}$ and $s_{0}$, and finding their mean values averaged over the regions Eq. (16). Our predictions for m and f read

$$\begin{aligned} m= & {} (2800\pm 75)~\text {MeV}, \nonumber \\ f= & {} (2.42\pm 0.30)\times 10^{-3}~\text {GeV}^{4}. \end{aligned}$$

(17)

The results in Eq. (17) effectively correspond to sum rules’ predictions at approximately middle point of the regions in Eq. (16), i.e., to predictions at the point $M^{2}=2.8~\text {GeV}^{2}$ and $s_{0}=10~\text {GeV}^{2}$, where the pole contribution is ${\textrm{PC}} \approx 0.62$. This fact guarantees a dominance of the pole contribution in extracted parameters m and f.

Fig. 4

Dependence of the mass $m_{\textrm{PS}}$ on the Borel parameter $M^{2}$ (left), and on the continuum threshold parameter $s_0$ (right)

In Fig. 3, we depict m as functions of $M^{2}$ and $s_{0}$, in which is seen its dependence on the Borel and continuum subtraction parameters. Strictly speaking, physical quantities should not depend on $M^{2}$, but computations demonstrate that such effects, nevertheless, exist. Therefore, in a chosen region for $M^{2}$ this dependence should be minimal. Due to a functional form of the sum rule for the mass Eq. (9) given as the ratio of correlation functions, variation of m in the region for $M^{2}$ is mild. There is also dependence on the parameter $s_{0}$ which contains information about the lower limit for the mass of the excited tetraquark.

The pseudoscalar tetraquark $X_{\textrm{PS}}$ and its parameters have been explored by the manner described just above. Here, we model $X_{\textrm{PS}}$ as a tetraquark built of the pseudoscalar diquark $u^{T}Cd$ and scalar antidiquark ${\overline{c}}\gamma _{5}C{\overline{s}}^{T}$. The relevant interpolating current $J_{\textrm{PS}}(x)$ is determined by the expression

$$\begin{aligned} J_{\textrm{PS}}(x)=\epsilon {\widetilde{\epsilon }}u_{b}^{T}(x)Cd_{c}(x) {\overline{c}}_{d}(x)\gamma _{5}C{\overline{s}}_{e}^{T}(x), \end{aligned}$$

(18)

and belongs to the antitriplet–triplet representation of the color group $SU_{c}(3)$.

The physical side of the sum rule in this case has relatively simple form

$$\begin{aligned} {\widetilde{\Pi }}^{\textrm{Phys}}(p)=\frac{f_{\textrm{PS}}^{2}m_{\textrm{PS} }^{4}}{(m_{c}+m_{s})^{2}(m_{\textrm{PS}}^{2}-p^{2})}+\cdots , \end{aligned}$$

(19)

where $m_{\textrm{PS}}$ and $f_{\textrm{PS}}$ are the mass and current coupling of the tetraquark $X_{\textrm{PS}}$, respectively. To derive ${\widetilde{\Pi }}^{\textrm{Phys}}(p)$, we have used the matrix element of the pseudoscalar particle $X_{\textrm{PS}}$

$$\begin{aligned} \langle 0|J_{\textrm{PS}}|X_{\textrm{PS}}(p)\rangle =\frac{f_{\textrm{PS}}m_{ \textrm{PS}}^{2}}{m_{c}+m_{s}}. \end{aligned}$$

(20)

The function ${\widetilde{\Pi }}^{\textrm{Phys}}(p)$ has trivial Lorentz structure proportional to I, therefore the invariant amplitude $\widetilde{ \Pi }^{\textrm{Phys}}(p^{2})$ is equal to r.h.s. of Eq. (19).

The QCD side of new sum rules is given by the formula

$$\begin{aligned}{} & {} {\widetilde{\Pi }}^{{\textrm{OPE}}}(p)=i\int d^{4}xe^{ipx}\epsilon \widetilde{ \epsilon }\epsilon ^{\prime }{\widetilde{\epsilon }}^{\prime }{\textrm{Tr}}\left[ {\widetilde{S}}_{u}^{bb^{\prime }}(x)S_{d}^{cc^{\prime }}(x)\right] \nonumber \\{} & {} \quad \times {\textrm{Tr}}\left[ \gamma _{5}{\widetilde{S}}_{s}^{e^{\prime }e}(-x)\gamma _{5}S_{c}^{d^{\prime }d}(-x)\right] . \end{aligned}$$

(21)

The spectroscopic parameters of $X_{\textrm{PS}}$ can be obtained from Eqs. (9) and (10) after replacement $\Pi (M^{2},s_{0})\rightarrow {\widetilde{\Pi }}(M^{2},s_{0})$. Performed calculations yield

$$\begin{aligned} m_{\textrm{PS}}= & {} (3000\pm 60)~\text {MeV}, \nonumber \\ f_{\textrm{PS}}= & {} (8.69\pm 1.54)\times 10^{-4}~\text {GeV}^{4}. \end{aligned}$$

(22)

The Borel and continuum subtraction parameters $M^{2}$ and $s_{0}$ used to extract $m_{\textrm{PS}}$ and $f_{\textrm{PS}}$ are given by Eq. (23)

$$\begin{aligned} M^{2}\in [2.5,3.5]~\text {GeV}^{2},\ s_{0}\in [9.5,10.5]~\text {GeV}^{2}. \end{aligned}$$

(23)

In these regions the ${\textrm{PC}}$ changes inside limits

$$\begin{aligned} 0.74\ge {\textrm{PC}}\ge 0.50. \end{aligned}$$

(24)

Dependence of $m_{\textrm{PS}}$ on the parameters $M^{2}$ and $s_{0}$ is shown in Fig. 4. In the left panel one can see a relatively stable nature of $m_{\textrm{PS}}$ under variation of $M^{2}$.

3 Decays of the axial-vector tetraquark $X_{\textrm{AV}}$

The mass and spin-parity of the tetraquark $X_{\textrm{AV}}$ allow us to classify its decay channels. We restrict ourselves by analysis of S-wave decay channels of $X_{\textrm{AV}}$ which are $X_{\textrm{AV}}\rightarrow D^{*}(2010)^{-}K^{+}$, ${\overline{D}}^{*}(2007)^{0}K^{0}$, $D^{-}K^{*}(892)^{+}$, and ${\overline{D}}^{0}K^{*}(892)^{0}$. The full width of the axial-vector state $X_{\textrm{AV}}$ is estimated by including into analysis namely these channels.

We are going to provide rather detailed information about computation of a partial width of the decay $X_{\textrm{AV}}\rightarrow D^{*}(2010)^{-}K^{+}$, and outline important steps in analyses of other processes. A quantity to be extracted from a sum rule is the strong coupling $g_{1}$ of particles at the vertex $X_{\textrm{AV}}D^{*}(2010)^{-}K^{+}$. This coupling is defined in terms of the on-mass-shell matrix element

$$\begin{aligned} \langle D^{*}\left( p\right) K(q)|X_{\textrm{AV}}(p^{\prime })\rangle= & {} g_{1}\left[ \left( p\cdot p^{\prime }\right) \left( \varepsilon ^{*}\cdot \varepsilon ^{\prime }\right) \right. \nonumber \\{} & {} \left. -\left( p\cdot \varepsilon ^{\prime }\right) \left( p^{\prime }\cdot \varepsilon ^{*}\right) \right] , \end{aligned}$$

(25)

where the mesons $K^{+}$ and $D^{*}(2010)^{-}$ are denoted as K and $D^{*}$, respectively. Here, $p^{\prime }$, p and q are four-momenta of the tetraquark $X_{\textrm{AV}}$, and mesons $D^{*}$ and K, and $\varepsilon _{\nu }^{\prime }$ and $\varepsilon _{\mu }^{*}$ are the polarization vectors of the particles $X_{\textrm{AV}}$ and $D^{*}$.

In the framework of the LCSR method the coupling $g_{1}$ can be obtained from the correlation function

$$\begin{aligned} \Pi _{\mu \nu }(p,q)=i\int d^{4}xe^{ipx}\langle K(q)|{\mathcal {T}}\{J_{\mu }^{D^{*}}(x)J_{\nu }^{\dag }(0)\}|0\rangle , \end{aligned}$$

(26)

with $J_{\nu }(x)$ being the current for the tetraquark $X_{\textrm{AV}}$ from Eq. (2). The interpolating current for the meson $D^{*}(2010)^{-}$ is abbreviated in Eq. (26) as $J_{\mu }^{D^{*}}(x)$, and defined by the expression

$$\begin{aligned} J_{\mu }^{D^{*}}(x)={\overline{c}}_{j}(x)\gamma _{\mu }d_{j}(x), \end{aligned}$$

(27)

where j is the color index.

The main contribution to the correlation function $\Pi _{\mu \nu }(p,q)$ comes from a term with poles at $p^{2}$ and $p^{\prime 2}=(p+q)^{2}$. This term is given by the formula

$$\begin{aligned}{} & {} \Pi _{\mu \nu }^{\textrm{Phys}}(p,q)=g_{1}\frac{fmf_{D^{*}}m_{D^{*}}}{\left( p^{2}-m_{D^{*}}^{2}\right) \left( p^{\prime 2}-m^{2}\right) } \nonumber \\{} & {} \quad \times \left( \frac{m^{2}+m_{D^{*}}^{2}-m_{K}^{2}}{2}g_{\mu \nu }-p_{\mu }p_{\nu }^{\prime }\right) +\cdots , \end{aligned}$$

(28)

where $m_{D^{*}}$ and $f_{D^{*}}$ are the mass and decay constant of the meson $D^{*}(2010)^{-}$. To derive Eq. (28), we have used Eq. (25) and the following matrix elements

$$\begin{aligned} \langle 0|J_{\mu }^{D^{*}}|D^{*}(p)\rangle =f_{D^{*}}m_{D^{*}}\varepsilon _{\mu },\ \langle X_{\textrm{AV}}(p^{\prime })|J_{\nu }^{\dagger }|0\rangle =fm\varepsilon _{\nu }^{\prime *}. \nonumber \\ \end{aligned}$$

(29)

The term written down explicitly in Eq. (28) corresponds to contribution of ground-state particles in $X_{\textrm{AV}}$ and $D^{*}$ channels: effects of higher resonances and continuum states in these channels are shown by dots. The function $\Pi _{\mu \nu }^{\textrm{Phys}}(p,q)$ constitutes the phenomenological side of a sum rule for the coupling $g_{1}$. It contains two terms determined by structures $g_{\mu \nu }$ and $p_{\mu }p_{\nu }^{\prime }$. In our studies, we use the term $\sim g_{\mu \nu }$ and corresponding invariant amplitude $\Pi ^{\textrm{Phys} }(p^{2},p^{\prime 2})$ which is a function of two variables $p^{2}$ and $p^{\prime 2}$.

The correlation function $\Pi _{\mu \nu }(p,q)$ calculated in terms of quark-gluon degrees of freedom forms the QCD side of the sum rules and is equal to

$$\begin{aligned}{} & {} \Pi _{\mu \nu }^{{\textrm{OPE}}}(p,q)=i\int d^{4}xe^{ipx}\epsilon \widetilde{ \epsilon }\left[ \gamma _{5}{\widetilde{S}}_{d}^{jc}(x){}\gamma ^{\mu }\right. \nonumber \\{} & {} \quad \left. \times {\widetilde{S}}_{c}^{dj}(-x){}\gamma _{\nu }\right] _{\alpha \beta }\langle K(q)|{\overline{u}}_{\alpha }^{b}(0)s_{\beta }^{e}(0)|0\rangle , \end{aligned}$$

(30)

where $\alpha$ and $\beta$ are the spinor indices.

As is seen, besides c and d quark propagators the function $\Pi _{\mu \nu }^{{\textrm{OPE}}}(p,q)$ contains also local matrix elements of the $K^{+}$ meson, which carry spinor and color indices. We can rewrite $\langle K| {\overline{u}}_{\alpha }^{b}s_{\beta }^{e}|0\rangle$ in convenient forms by expanding ${\overline{u}}s$ over the full set of Dirac matrices $\Gamma ^{J}$

$$\begin{aligned} \Gamma ^{J}=\textbf{1},\ \gamma _{5},\ \gamma _{\mu },\ i\gamma _{5}\gamma _{\mu },\ \sigma _{\mu \nu }/\sqrt{2}, \end{aligned}$$

(31)

and projecting them onto the colorless states

$$\begin{aligned} {\overline{u}}_{\alpha }^{b}(0)s_{\beta }^{a}(0)\rightarrow \frac{1}{12}\delta ^{ba}\Gamma _{\beta \alpha }^{J}\left[ {\overline{u}}(0)\Gamma ^{J}s(0)\right] . \end{aligned}$$

(32)

Operators ${\overline{u}}\Gamma ^{J}s$ sandwiched between the K meson and vacuum generate local matrix elements of the K meson, which are known and can be implemented into $\Pi _{\mu \nu }^{{\textrm{OPE}}}(p,q)$.

As usual, $\Pi _{\mu \nu }^{{\textrm{OPE}}}(p,q)$-type correlators depend on non-local matrix elements of a final meson (for example, K meson), which are convertible to its distribution amplitudes (DAs). This is correct while one treats strong vertices of three conventional mesons in the context of the LCSR method. In the case of tetraquark–meson–meson vertices relevant correlation functions instead of DAs of a final meson contain its local matrix elements. These matrix elements are determined at the space-time point $x=0$ and are overall normalization factors. Within the LCSR method similar behavior of correlation functions was seen in a limit $q\rightarrow 0$ of three-meson vertices, which is known as a soft-meson approximation [19]. This approximation requires adoption of additional technical tools to deal with new problems appeared in a phenomenological side of corresponding sum rules [19, 20]. It turns out that the soft limit and related technical methods can be adapted to investigate also tetraquark–meson–meson vertices [21]. It is worth to emphasize that the soft limit should be implemented in a hard part of the correlation function $\Pi _{\mu \nu }(p)$, but in matrix elements one takes into account the terms with $q^{2}=m_{K}^{2}$.

The term proportional to $g_{\mu \nu }$ in $\Pi _{\mu \nu }^{\textrm{Phys} }(p,q)$ in the limit $q\rightarrow 0$ with some accuracy can be transformed into the expression

$$\begin{aligned} \Pi _{\mu \nu }^{\textrm{Phys}}(p)=g_{1}\frac{fmf_{D^{*}}m_{D^{*}}}{ \left( p^{2}-{\overline{m}}^{2}\right) ^{2}}\left( {\overline{m}}^{2}-\frac{ m_{K}^{2}}{2}\right) g_{\mu \nu }+\cdots , \end{aligned}$$

(33)

where ${\overline{m}}^{2}=(m^{2}+m_{D^{*}}^{2})/2$. The invariant amplitude $\Pi ^{\textrm{Phys}}(p^{2})$ depends on the variable $p^{2}$, and has a double pole at $p^{2}={\overline{m}}^{2}$. The Borel transformation of $\Pi ^{\textrm{Phys}}(p^{2})$ is given by the formula

$$\begin{aligned} {\mathcal {B}}\Pi ^{\textrm{Phys}}(p^{2})=g_{1}fmf_{D^{*}}m_{D^{*}}\left( {\overline{m}}^{2}-\frac{m_{K}^{2}}{2}\right) \frac{e^{-{\overline{m}} ^{2}/M^{2}}}{M^{2}}+\cdots . \nonumber \\ \end{aligned}$$

(34)

The ellipses in Eq. (34) stand not only for terms suppressed after this operation, but also for contributions which remain unsuppressed even after Borel transformation. Therefore, before performing usual subtraction procedure, one should remove these contributions from ${\mathcal {B}}\Pi ^{\textrm{Phys}}(p^{2})$. To this end, we have to apply the operator

$$\begin{aligned} {\mathcal {P}}(M^{2},m^{2})=\left( 1-M^{2}\frac{d}{dM^{2}}\right) M^{2}e^{m^{2}/M^{2}}, \end{aligned}$$

(35)

to both sides of a sum rule equality [19, 20], and subtract conventional terms in a usual way.

Then the sum rule for the strong coupling $g_{1}$ reads

$$\begin{aligned} g_{1}=\frac{2}{fmf_{D^{*}}m_{D^{*}}(2{\overline{m}}^{2}-m_{K}^{2})} {\mathcal {P}}(M^{2},{\overline{m}}^{2})\Pi ^{{\textrm{OPE}}}(M^{2},s_{0}),\nonumber \\ \end{aligned}$$

(36)

where $\Pi ^{{\textrm{OPE}}}(M^{2},s_{0})$ is the Borel transformed and subtracted invariant amplitude $\Pi ^{{\textrm{OPE}}}(p^{2})$ that corresponds to the structure $g_{\mu \nu }$ in $\Pi _{\mu \nu }^{{\textrm{OPE}}}(p,q).$

To finish calculation of the strong coupling $g_{1}$, we need to specify local matrix elements of the K meson which contribute to the function $\Pi _{\mu \nu }^{{\textrm{OPE}}}(p,q)$. Details of calculations necessary to find $\Pi _{\mu \nu }^{{\textrm{OPE}}}(p,q)$ in the soft limit were presented in Ref. [21], therefore we skip further features of relevant analysis and provide only final expressions. First of all, our computations demonstrate that in the soft-meson approximation the correlator $\Pi _{\mu \nu }^{{\textrm{OPE}}}(p,q=0)$ receives contribution from the matrix element

$$\begin{aligned} \langle 0|{\overline{u}}i\gamma _{5}s|K\rangle =\frac{f_{K}m_{K}^{2}}{m_{s}}, \end{aligned}$$

(37)

with $m_{K}$ and $f_{K}$ being the mass and decay constant of the $K^{+}$ meson.

The Borel transformed and subtracted correlation function $\Pi ^{{\textrm{OPE}} }(M^{2},s_{0})$ is calculated by taking into account condensates up to dimension 9 and given below

$$\begin{aligned} \Pi ^{{\textrm{OPE}}}(M^{2},s_{0})= & {} \frac{\mu _{K}}{48\pi ^{2}}\int _{ {\mathcal {M}}^{2}}^{s_{0}}\frac{ds(m_{c}^{2}-s)^{2}(m_{c}^{2}+2s)}{s^{2}} e^{-s/M^{2}} \nonumber \\{} & {} +\mu _{K}m_{c}{\mathcal {F}}^{\text {non-pert.}}(M^{2}), \end{aligned}$$

(38)

where $\mu _{K}=f_{K}m_{K}^{2}/m_{s}$. In expression above, the nonperturbative function ${\mathcal {F}}^{\text {non-pert.}}(M^{2})$ is determined by the formula

$$\begin{aligned}{} & {} {\mathcal {F}}^{\text {non-pert.}}(M^{2})=-\frac{\langle {\overline{d}} d\rangle }{6}e^{-m_{c}^{2}/M^{2}}+\frac{\langle \frac{\alpha _{s}G^{2}}{\pi } \rangle m_{c}}{144M^{4}} \nonumber \\{} & {} \quad \times \int _{0}^{1}\frac{dx\left[ m_{c}^{2}+M^{2}x(1-x)\right] }{ x^{3}(1-x)^{3}}e^{-m_{c}^{2}/[M^{2}x(1-x)]} \nonumber \\{} & {} \quad -\frac{\langle {\overline{d}}g\sigma Gd\rangle m_{c}^{2}}{24M^{4}} e^{-m_{c}^{2}/M^{2}}+\langle \frac{\alpha _{s}G^{2}}{\pi }\rangle \langle {\overline{d}}d\rangle \nonumber \\{} & {} \quad \times \frac{(m_{c}^{2}+3M^{2})\pi ^{2}}{108M^{6}}e^{-m_{c}^{2}/M^{2}}- \langle \frac{\alpha _{s}G^{2}}{\pi }\rangle \langle {\overline{d}}g\sigma Gd\rangle \nonumber \\{} & {} \quad \times \frac{(5m_{c}^{4}+24m_{c}^{2}M^{2}+6M^{4})\pi ^{2}}{432M^{10}} e^{-m_{c}^{2}/M^{2}}. \end{aligned}$$

(39)

Table 1 Masses and decay constants of the mesons D, $D^{*}$, $D^{*}_{0}$, K, and $K^{*}$, which have been used in numerical computations

Partial width of the process $X_{\textrm{AV}}\rightarrow D^{*}(2010)^{-}K^{+}$ can be obtained by employing the following expression

$$\begin{aligned} \Gamma \left[ X_{\textrm{AV}}\rightarrow D^{*}(2010)^{-}K^{+}\right] = \frac{g_{1}^{2}m_{D^{*}}^{2}\lambda }{24\pi }\left( 3+\frac{2\lambda ^{2} }{m_{D^{*}}^{2}}\right) ,\nonumber \\ \end{aligned}$$

(40)

where $\lambda =\lambda \left( m,m_{D^{*}},m_{K}\right)$ and

$$\begin{aligned} \lambda \left( a,b,c\right)= & {} \frac{1}{2a}\left[ a^{4}+b^{4}+c^{4}\right. \nonumber \\{} & {} \left. -2\left( a^{2}b^{2}+a^{2}c^{2}+b^{2}c^{2}\right) \right] ^{1/2}. \end{aligned}$$

(41)

The sum rule for the strong coupling $g_{1}$ contains different vacuum condensates, numerical values of which have been collected in Eq. (12). Apart from that, the Eq. (36) depends on the spectroscopic parameters of particles involved into decay process. The mass and current coupling of the tetraquark $X_{\textrm{AV}}$ have been calculated in the current work. The masses and decay constants of the mesons $D^{*}(2010)^{-}$ and $K^{+}$ are collected in Table 1. This table contains also spectroscopic parameters of other mesons which appear at final stages of different decay channels. For masses all of the mesons and decay constants of K, $K^{*}$ and D mesons, we use information from Ref. [33]. The decay constant $f_{D^{*}}$ of the vector mesons $D^{*}(2010)^{-}$ and $D^{*}(2007)^{0}$, and decay constant $f_{D_{0}^{*}}$ of the scalar meson ${\overline{D}} _{0}^{*}(2300)^{0}$ are borrowed from Ref. [34].

The Borel and continuum subtraction parameters $M^{2}$ and $s_{0}$ required for calculation of the coupling $g_{1}$ are chosen in accordance with Eq. (16). Numerical computations yield

$$\begin{aligned} g_{1}=(3.45\pm 0.52)\times 10^{-1}~\text {GeV}^{-1}. \end{aligned}$$

(42)

Then it is not difficult to find

$$\begin{aligned} \Gamma \left[ X_{\textrm{AV}}\rightarrow D^{*}(2010)^{-}K^{+}\right] =(10.6\pm 3.4)~\text {MeV}. \end{aligned}$$

(43)

The decay of the tetraquark $X_{\textrm{AV}}$ to a meson pair ${\overline{D}} ^{*}(2007)^{0}K^{0}$ is another process with K meson in the final state. This process is a “neutral” version of the first channel differences being encoded in masses $m_{1}$ and $m_{2}$ of mesons ${\overline{D}}^{*}(2007)^{0}$ and $K^{0}$, respectively. Treatment of this decay mode does not differ from analysis described above. Therefore, we provide final results for the coupling $g_{2}$

$$\begin{aligned} g_{2}=(3.67\pm 0.55)\times 10^{-1}~\text {GeV}^{-1}, \end{aligned}$$

(44)

and partial width of the process

$$\begin{aligned} \Gamma \left[ X_{\textrm{AV}}\rightarrow {\overline{D}}^{*}(2007)^{0}K^{0} \right] =(11.9\pm 3.8)~\text {MeV}. \end{aligned}$$

(45)

Table 2 Decay channels of the tetraquarks $X_{\textrm{AV}}$ and $X_{\textrm{PS}}$, strong couplings $g_{i}$, $G_{j}$ and partial widths $\Gamma _{\textrm{AV}}^{i}$, $\Gamma _{\textrm{PS}}^{j}$. The star-marked coupling $G_1$ has a dimension $\text {GeV}^{-1}$

The remaining two decay channels $X_{\textrm{AV}}\rightarrow D^{-}K^{*}(892)^{+}$ and $X_{\textrm{AV}}\rightarrow {\overline{D}}^{0}K^{*}(892)^{0}$ have been explored by a similar manner. Let us consider, for instance, the process $X_{\textrm{AV}}\rightarrow D^{-}K^{*}(892)^{+}$. The strong coupling $g_{3}$ that corresponds to the vertex $X_{\textrm{AV} }D^{-}K^{*}(892)^{+}$ is defined by the matrix element

$$\begin{aligned} \langle D^{-}\left( p\right) K^{*}(q)|X_{\textrm{AV}}(p^{\prime })\rangle= & {} g_{3}\left[ \left( q\cdot p^{\prime }\right) \left( \epsilon ^{*}\cdot \varepsilon ^{\prime }\right) \right. \nonumber \\{} & {} \left. -\left( q\cdot \varepsilon ^{\prime }\right) \left( p^{\prime }\cdot \epsilon ^{*}\right) \right] , \end{aligned}$$

(46)

where $\epsilon _{\mu }^{*}$ is polarization vector of the meson $K^{*}(892)^{+}$. The correlation function that allows us to extract $g_{3}$ is

$$\begin{aligned} \Pi _{\nu }(p,q)=i\int d^{4}xe^{ipx}\langle K^{*}(q)|{\mathcal {T}} \{J^{D}(x)J_{\nu }^{\dag }(0)\}|0\rangle , \end{aligned}$$

(47)

with $J^{D}(x)$ being the interpolating current for the pseudoscalar meson $D^{-}$

$$\begin{aligned} J^{D}(x)={\overline{c}}_{j}(x)i\gamma _{5}d_{j}(x). \end{aligned}$$

(48)

The main term that contributes to this correlation function and determines phenomenological side of a sum rule for $g_{3}$ has the following form

$$\begin{aligned}{} & {} \Pi _{\nu }^{\textrm{Phys}}(p,q)=g_{3}\frac{fmf_{D}m_{3}^{2}}{m_{c}\left( p^{2}-m_{3}^{2}\right) \left( p^{\prime 2}-m^{2}\right) } \nonumber \\{} & {} \quad \times \left( \frac{m^{2}+m_{3}^{2}-m_{4}^{2}}{2}\epsilon _{\nu }^{*}+p^{\prime }\cdot \epsilon ^{*}q_{\nu }\right) +\cdots , \end{aligned}$$

(49)

where $m_{3}$ and $m_{4}$ are masses of $D^{-}$ and $K^{*}(892)^{+}$, respectively. Here, $f_{D}$ is the decay constant of the meson $D^{-}$. To find $\Pi _{\nu }^{\textrm{Phys}}(p,q)$, we use the matrix elements of the tetraquark $X_{\textrm{AV}}$ and vertex, as well as new matrix element

$$\begin{aligned} \langle 0|J^{D}(x)|D^{-}\rangle =\frac{f_{D}m_{3}^{2}}{m_{c}}. \end{aligned}$$

(50)

The same function $\Pi _{\nu }(p,q)$ calculated in term of quark-gluon degrees of freedom gives QCD side of the sum rule

$$\begin{aligned}{} & {} \Pi _{\nu }^{{\textrm{OPE}}}(p,q)=-i^{2}\int d^{4}xe^{ipx}\epsilon {\widetilde{\epsilon }}\left[ \gamma _{5}{\widetilde{S}}_{d}^{jc}(x){}\gamma _{5}\right. \nonumber \\{} & {} \quad \left. \times {\widetilde{S}}_{c}^{dj}(-x){}\gamma _{\nu }\right] _{\alpha \beta }\langle K^{*}(q)|{\overline{u}}_{\alpha }^{b}(0)s_{\beta }^{e}(0)|0\rangle . \end{aligned}$$

(51)

Our analysis demonstrates that in the soft-meson approximation a contribution to $\Pi _{\nu }^{{\textrm{OPE}}}(p,q)$ comes from the local matrix element of the meson $K^{*}(892)^{+}$

$$\begin{aligned} \langle 0|{\overline{s}}(0)\gamma _{\mu }u(0)|K^{*}\rangle =\epsilon _{\mu }f_{K^{*}}m_{4}, \end{aligned}$$

(52)

with $f_{K^{*}}$ being its decay constant.

The sum rule for $g_{3}$ can be determined using structures proportional to $\epsilon _{\nu }^{*}$ in $\Pi _{\nu }^{\textrm{Phys}}(p,q)$ and $\Pi _{\nu }^{{\textrm{OPE}}}(p,q)$, and corresponding invariant amplitudes ${\widetilde{\Pi }}^{\textrm{Phys}}(p^{2})$ and ${\widetilde{\Pi }}^{{\textrm{OPE}} }(p^{2})$. The amplitude ${\widetilde{\Pi }}^{{\textrm{OPE}}}(p^{2})$ is calculated by including effects of condensates up to dimension 9. After Borel transformation and subtraction it takes the form

$$\begin{aligned} {\widetilde{\Pi }}^{{\textrm{OPE}}}(M^{2},s_{0})= & {} \frac{f_{K^{*}}m_{4}}{ 16\pi ^{2}}\int _{{\mathcal {M}}^{2}}^{s_{0}}\frac{ds(m_{c}^{2}-s)^{2}}{s} e^{-s/M^{2}} \nonumber \\{} & {} +m_{4}m_{c}f_{K^{*}}\widetilde{{\mathcal {F}}}^{\mathrm {non-pert.} }(M^{2}). \end{aligned}$$

(53)

The function $\widetilde{{\mathcal {F}}}^{\text {non-pert.}}(M^{2})$ in Eq. (53) is given by the expression

$$\begin{aligned}{} & {} \widetilde{{\mathcal {F}}}^{\text {non-pert.}}(M^{2})=-\frac{\langle {\overline{d}}d\rangle }{6}e^{-m_{c}^{2}/M^{2}}+\frac{\langle \frac{\alpha _{s}G^{2}}{\pi }\rangle m_{c}^{3}}{144M^{4}} \nonumber \\{} & {} \quad \times \int _{0}^{1}\frac{dxe^{-m_{c}^{2}/[M^{2}x(1-x)]}}{x^{3}(x-1)^{3}}+ \frac{\langle {\overline{d}}g\sigma Gd\rangle m_{c}^{2}}{24M^{4}} e^{-m_{c}^{2}/M^{2}} \nonumber \\{} & {} \quad -\left\langle \frac{\alpha _{s}G^{2}}{\pi }\right\rangle \langle {\overline{d}}d\rangle \frac{(m_{c}^{2}+3M^{2})\pi ^{2}}{108M^{6}}e^{-m_{c}^{2}/M^{2}}+\left\langle \frac{\alpha _{s}G^{2}}{\pi }\right\rangle \nonumber \\{} & {} \quad \times \langle {\overline{d}}g\sigma Gd\rangle \frac{ (5m_{c}^{4}+24m_{c}^{2}M^{2}+6M^{4})\pi ^{2}}{432M^{10}}e^{-m_{c}^{2}/M^{2}}. \end{aligned}$$

(54)

After manipulations described above in detail, for $g_{3}$ we get the sum rule

$$\begin{aligned} g_{3}=\frac{2m_{c}}{fmf_{D}m_{3}^{2}(2{\widetilde{m}}^{2}-m_{K^{*}}^{2})} {\mathcal {P}}(M^{2},{\widetilde{m}}^{2}){\widetilde{\Pi }}^{{\textrm{OPE}} }(M^{2},s_{0}),\nonumber \\ \end{aligned}$$

(55)

where ${\widetilde{m}}^{2}=(m^{2}+m_{3}^{2})/2$.

The sum rule prediction for $g_{3}$ reads

$$\begin{aligned} g_{3}=(1.52\pm 0.23)~\text {GeV}^{-1}. \end{aligned}$$

(56)

The partial width of the process $X_{\textrm{AV}}\rightarrow D^{-}K^{*}(892)^{+}$ can be computed using Eq. (40) after replacing $g_{1}$,$\lambda$, and $m_{D^{*}}$ by $g_{3}$, ${\widetilde{\lambda }}$, and $m_{4}$, where ${\widetilde{\lambda }}=\lambda (m^{2},m_{3}^{2},m_{4}^{2})$. Calculations lead to the following result

$$\begin{aligned} \Gamma \left[ X_{\textrm{AV}}\rightarrow D^{-}K^{*}(892)^{+}\right] =(17.1\pm 5.7)~\text {MeV}. \end{aligned}$$

(57)

Predictions for the strong coupling $g_{4}$ and partial with of the decay $X_{\textrm{AV}}\rightarrow {\overline{D}}^{0}K^{*}(892)^{0}$, as well as results obtained in the present section are collected in Table 2. This information allows us to estimate full width of the tetraquark $X_{\textrm{AV}}$

$$\begin{aligned} \Gamma _{\textrm{AV}}=(58\pm 10)~\text {MeV}, \end{aligned}$$

(58)

which characterizes it as a resonance with a relatively narrow width.

4 Processes $X_{\textrm{PS}}\rightarrow {\overline{D}}_{0}^{*}(2300)^{0}K^{0}$, $D^{-}K^{*}(892)^{+}$ and ${\overline{D}}^{0}K^{*}(892)^{0}$

In this section, we investigate decays of the pseudoscalar tetraquark $X_{\textrm{PS}}$ with the mass $m_{\textrm{PS}}$ and current coupling $f_{\textrm{PS}}$ which have been extracted from two-point sum rules in Sect. 2. We consider the S-wave process $X_{\textrm{PS}}\rightarrow {\overline{D}}_{0}^{*}(2300)^{0}K^{0}$, as well as P-wave decay modes $X_{\textrm{PS}}\rightarrow D^{-}K^{*}(892)^{+}$ and $X_{\textrm{PS}}\rightarrow {\overline{D}}^{0}K^{*}(892)^{0}$ of this four-quark state.

We begin our analysis from the decay $X_{\textrm{PS}}\rightarrow {\overline{D}} _{0}^{*} (2300)^{0}K^{0}$. The coupling $G_{1}$ required to calculate partial width of this process can be defined using the on-mass-shell matrix element

$$\begin{aligned} \langle D_{0}^{*}\left( p\right) K^{0}(q)|X_{\textrm{PS}}(p^{\prime })\rangle =G_{1}p\cdot p^{\prime }. \end{aligned}$$

(59)

The correlation function, which should be considered to determine strong coupling $G_{1}$ of particles at the vertex $X_{\textrm{PS}}{\overline{D}} _{0}^{*}(2300)^{0}K^{0}$, is given by the formula

$$\begin{aligned} {\widehat{\Pi }}(p,q)=i\int d^{4}xe^{ipx}\langle K^{0}(q)|{\mathcal {T}} \{J^{D_{0}^{*}}(x)J_{\textrm{PS}}^{\dag }(0)\}|0\rangle , \end{aligned}$$

(60)

where $D_{0}^{*}$ stands for meson ${\overline{D}}_{0}^{*}(2300)^{0}$. Here, $J_{\textrm{PS}}(x)$ and $J^{D_{0}^{*}}(x)$ are interpolating currents for the tetraquark $X_{\textrm{PS}}$ [see, Eq. (18)], and for the scalar meson ${\overline{D}}_{0}^{*}(2300)^{0}$. The latter is defined by the expression

$$\begin{aligned} J^{D_{0}^{*}}(x)={\overline{c}}_{j}(x)d_{j}(x). \end{aligned}$$

(61)

A contribution to the correlation function ${\widehat{\Pi }}^{\textrm{Phys} }(p,q)$ with poles at $p^{2}$ and $p^{\prime 2}=(p+q)^{2}$ comes from the term

$$\begin{aligned} {\widehat{\Pi }}^{\textrm{Phys}}(p,q)= & {} G_{1}\frac{f_{\textrm{PS}}m_{\textrm{ PS}}^{2}f_{D_{0}^{*}}m_{7}}{(m_{c}+m_{s})\left( p^{2}-m_{7}^{2}\right) \left( p^{\prime 2}-m_{\textrm{PS}}^{2}\right) } \nonumber \\{} & {} \times \frac{m_{\textrm{PS}}^{2}+m_{7}^{2}-m_{2}^{2}}{2}+\cdots . \end{aligned}$$

(62)

Here, $m_{7}$ and $f_{D_{0}^{*}}$ are the mass and decay constant of the meson ${\overline{D}}_{0}^{*}(2300)^{0}$, respectively. In order to find Eq. (62), we employ Eqs. (59) and (20), as well as the matrix element

$$\begin{aligned} \langle 0|J^{D_{0}^{*}}|D_{0}^{*}(p)\rangle =f_{D_{0}^{*}}m_{7}. \end{aligned}$$

(63)

The QCD side of the sum rule ${\widehat{\Pi }}^{{\textrm{OPE}}}(p,q)$ has the following form

$$\begin{aligned}{} & {} {\widehat{\Pi }}^{{\textrm{OPE}}}(p,q)=i\int d^{4}xe^{ipx}\epsilon \widetilde{ \epsilon }\left[ {\widetilde{S}}_{u}^{jb}(x){}{\widetilde{S}}_{c}^{dj}(-x)\gamma _{5}\right] {}_{\alpha \beta } \nonumber \\{} & {} \quad \times \langle K^{0}(q)|{\overline{d}}_{\alpha }^{c}(0)s_{\beta }^{e}(0)|0\rangle . \end{aligned}$$

(64)

The functions ${\widehat{\Pi }}^{\textrm{Phys}}(p,q)$ and ${\widehat{\Pi }}^{ {\textrm{OPE}}}(p,q)$ have trivial Lorentz structures $\sim I$, therefore both of them contain only one invariant amplitude. The amplitude ${\widehat{\Pi }}^{ {\textrm{OPE}}}(p^{2})$ is calculated with dimension-9 accuracy and given by the following expression

$$\begin{aligned} {\widehat{\Pi }}^{{\textrm{OPE}}}(M^{2},s_{0})= & {} \frac{\mu _{K^{0}}}{16\pi ^{2}} \int _{{\mathcal {M}}^{2}}^{s_{0}}\frac{ds(m_{c}^{2}-s)^{2}}{s}e^{-s/M^{2}} \nonumber \\{} & {} +\mu _{K^{0}}m_{c}\widehat{{\mathcal {F}}}^{\text {non-pert.}}(M^{2}), \end{aligned}$$

(65)

where the function $\widehat{{\mathcal {F}}}^{\text {non-pert.}}(M^{2})$ is determined by the formula

$$\begin{aligned}{} & {} \widehat{{\mathcal {F}}}^{\text {non-pert.}}(M^{2})=\frac{\langle \overline{u }u\rangle }{16}e^{-m_{c}^{2}/M^{2}}-\frac{\langle \frac{\alpha _{s}G^{2}}{ \pi }\rangle m_{c}^{3}}{144M^{4}} \nonumber \\{} & {} \quad \times \int _{0}^{1}\frac{dxe^{-m_{c}^{2}/[M^{2}x(1-x)]}}{x^{3}(x-1)^{3}}- \frac{\langle {\overline{u}}g\sigma Gu\rangle m_{c}^{2}}{24M^{4}} e^{-m_{c}^{2}/M^{2}} \nonumber \\{} & {} \quad +\left\langle \frac{\alpha _{s}G^{2}}{\pi }\right\rangle \langle {\overline{u}}u\rangle \frac{(m_{c}^{2}+3M^{2})\pi ^{2}}{108M^{6}}e^{-m_{c}^{2}/M^{2}}-\left\langle \frac{\alpha _{s}G^{2}}{\pi }\right\rangle \nonumber \\{} & {} \quad \times \langle {\overline{u}}g\sigma Gu\rangle \frac{ (5m_{c}^{4}+24m_{c}^{2}M^{2}+6M^{4})\pi ^{2}}{432M^{10}}e^{-m_{c}^{2}/M^{2}}, \end{aligned}$$

(66)

and $\mu _{K^{0}}=m_{2}^{2}f_{K}/m_{s}$.

In the soft-meson approximation the coupling $G_{1}$ can be found by means of the sum rule

$$\begin{aligned} G_{1}= & {} \frac{2(m_{c}+m_{s})}{f_{\textrm{PS}}m_{\textrm{PS} }^{2}f_{D_{0}^{*}}m_{7}(2{\widehat{m}}^{2}-m_{K}^{2})} \nonumber \\{} & {} \times {\mathcal {P}}(M^{2},{\widehat{m}}^{2}){\widehat{\Pi }}^{{\textrm{OPE}} }(M^{2},s_{0}), \end{aligned}$$

(67)

where ${\widehat{m}}^{2}=(m_{\textrm{PS}}^{2}+m_{7}^{2})/2$.

The width of the decay $X_{\textrm{PS}}\rightarrow {\overline{D}}_{0}^{*}(2300)^{0}K^{0}$ is found by utilizing the formula

$$\begin{aligned} \Gamma \left[ X_{\textrm{PS}}\rightarrow {\overline{D}}_{0}^{*}(2300)^{0}K^{0}\right] =G_{1}^{2}\frac{m_{7}^{2}{\widehat{\lambda }}}{8\pi } \left( 1+\frac{{\widehat{\lambda }}^{2}}{m_{7}^{2}}\right) , \end{aligned}$$

(68)

in which ${\widehat{\lambda }}=\lambda (m_{\textrm{PS}},m_{7},m_{2})$. Our computations for the coupling $G_{1}$ and partial width of the process yield

$$\begin{aligned} G_{1}=(4.41\pm 0.66)\times 10^{-1}~\text {GeV}^{-1}, \end{aligned}$$

(69)

and

$$\begin{aligned} \Gamma \left[ X_{\textrm{PS}}\rightarrow {\overline{D}}_{0}^{*}(2300)^{0}K^{0}\right] =(16.6\pm 5.3)~\text {MeV}. \end{aligned}$$

(70)

The coupling $G_{2}$ that describes strong interaction of particles at the vertex $X_{\textrm{PS}}D^{-}K^{*}(892)^{+}$ and determines partial width of the channel $X_{\textrm{PS}}\rightarrow D^{-}K^{*}(892)^{+}$ is defined by the matrix element

$$\begin{aligned} \langle D^{-}\left( p\right) K^{*}(q)|X_{\textrm{PS}}(p^{\prime })\rangle =G_{2}p\cdot \epsilon ^{*}. \end{aligned}$$

(71)

The sum rule for $G_{2}$ is obtained from the correlation function

$$\begin{aligned} \Pi ^{\prime }(p,q)=i\int d^{4}xe^{ipx}\langle K^{*}(q)|{\mathcal {T}} \{J^{D}(x)J_{\textrm{PS}}^{\dag }(0)\}|0\rangle . \end{aligned}$$

(72)

In terms of involved particles’ physical parameters this function has the form

$$\begin{aligned} \Pi ^{\prime {\textrm{Phys}}}(p,q)= & {} G_{2}\frac{f_{\textrm{PS}}m_{\textrm{PS} }^{2}}{(m_{c}+m_{s})\left( p^{\prime 2}-m_{\textrm{PS}}^{2}\right) } \nonumber \\{} & {} \times \frac{f_{D}m_{3}^{2}}{m_{c}\left( p^{2}-m_{3}^{2}\right) }p\cdot \epsilon ^{*}+\cdots . \end{aligned}$$

(73)

To extract this expression, we use the matrix elements from Eqs. (50) and (52), as well as one defined by Eq. (71).

The correlation function $\Pi ^{\prime }(p,q)$ calculated using quark-gluon degrees of freedom fixes the QCD side of the sum rule and is equal to

$$\begin{aligned}{} & {} \Pi ^{\prime {\textrm{OPE}}}(p,q)=-\int d^{4}xe^{ipx}\epsilon \widetilde{ \epsilon }\left[ {\widetilde{S}}_{d}^{jc}(x){}\gamma _{5}{\widetilde{S}} _{c}^{dj}(-x)\gamma _{5}\right] _{\alpha \beta }{} \nonumber \\{} & {} \quad \times \langle K^{*}(q)|{\overline{u}}_{\alpha }^{b}(0)s_{\beta }^{e}(0)|0\rangle . \end{aligned}$$

(74)

The sum rule for the strong coupling $G_{2}$ can be obtained by employing standard manipulations. The width of the process $X_{\textrm{PS}}\rightarrow D^{-}K^{*}(892)^{+}$ is calculated by means of the expression

$$\begin{aligned} \Gamma \left[ X_{\textrm{PS}}\rightarrow D^{-}K^{*}(892)^{+}\right] =G_{2}^{2}\frac{\lambda ^{\prime 3}}{8\pi m_{4}^{2}}, \end{aligned}$$

(75)

where $\lambda ^{\prime }=\lambda (m_{\textrm{PS}},m_{3},m_{4})$.

For the coupling $G_{2}$ numerical computations give

$$\begin{aligned} G_{2}=1.67\pm 0.35, \end{aligned}$$

(76)

and the partial width of the decay under analysis equals to

$$\begin{aligned} \Gamma \left[ X_{\textrm{PS}}\rightarrow D^{-}K^{*}(892)^{+}\right] =(23.6\pm 7.6)~\text {MeV}. \end{aligned}$$

(77)

The decay channel $X_{\textrm{PS}}\rightarrow {\overline{D}}^{0}K^{*}(892)^{0}$ of the tetraquark $X_{\textrm{PS}}$ which is last process considered in the present paper, can be treated in a similar way. Therefore, we refrain from further details and provide all relevant information in Table 2.

The full width of $X_{\textrm{PS}}$

$$\begin{aligned} \Gamma _{\textrm{PS}}=(65\pm 12)~\text {MeV}, \end{aligned}$$

(78)

does not differ considerably from the width of the axial-vector tetraquark $X_{\textrm{AV}}$.

5 Conclusions

In the current article, we have investigated the axial-vector and pseudoscalar tetraquarks $X_{\textrm{AV}}$ and $X_{\textrm{PS}}$ from a family of exotic mesons $[ud][{\overline{c}}{\overline{s}}]$ containing four different quark flavors. We have calculated their masses, and also estimated full widths of these states using different decay channels.

Interest to fully open-flavor structures renewed recently due to discovery of resonances $X_{0(1)}(2900)$ made by the LHCb collaboration. One of these states $X_{1}(2900)$ was studied in Ref. [7] as a vector tetraquark $X_{\textrm{V}}=[ud]\big [{\overline{c}}{\overline{s}}\big ]$. The mass and width of $X_{\textrm{V}}$ are close to physical parameters of the resonance $X_{1}(2900)$ measured by LHCb, which allowed us to interpret $X_{\textrm{V}}$ as a candidate to the vector resonance $X_{1}(2900)$.

Fig. 5

Mass and width of tetraquarks $[ud][{\overline{c}}{\overline{s}}]$ with different spin-parities. The lower and upper red (solid) and blue (dashed) lines are 1S and 2S scalar states, respectively. The red lines correspond to the scalar tetraquark $X_{0}$, whereas blue lines show parameters of $X_{\textrm{S}}$. The mass and width of the vector particle $X_{\textrm{V}}$ were determined in Ref. [7]. The axial-vector and pseudoscalar states have been explored in the current article. Theoretical uncertainties of extracted observables are not shown

The masses and widths of the ground-state scalar tetraquarks $X_{0}$ and $X_{ \textrm{S}}$, and their first radial excitations were calculated in Ref. [9]. The scalar particles $X_{0}$ and $X_{\textrm{S}}$ were modeled using axial-vector and scalar diquark–antidiquark pairs, respectively.

Information gained in our studies about tetraquarks $[ud][{\overline{c}} {\overline{s}}]$ with spin-parities $J^{\textrm{P}}=0^{+}$, $0^{-}$,$1^{+}$ and $1^{-}$ is shown in Fig. 5. As is seen, the pseudoscalar $X_{\textrm{PS}}$ state is heaviest particle in family of tetraquarks $[ud][ {\overline{c}}{\overline{s}}]$, whereas light particles are scalar ones with different internal organizations. The vector and axial-vector states have comparable masses and widths.

It is interesting to consider hadronic processes, where exotic mesons $X_{\textrm{AV}}$ and $X_{\textrm{PS}}$ may be observed. We have noted in Sect. 1 that resonances $X_{0}(2900)$ and $X_{1}(2900)$ were discovered in the invariant $D^{-}K^{+}$ mass distribution of the process $B^{+}\rightarrow D^{+}D^{-}K^{+}$. For the scalar tetraquark $X_{0}$ the decay $X_{0}\rightarrow D^{-}K^{+}$ is S-wave process, whereas $X_{1}\rightarrow D^{-}K^{+}$ is P-wave channel for the vector particle $X_{1}(2900)$. The invariant mass distribution of $D^{-}K^{*}(892)^{+}$ mesons in exclusive decays of $B^{+}$ meson may be explored to observe tetraquarks $X_{\textrm{AV}}$ and $X_{\textrm{PS}}$. In fact, decays to $D^{-}K^{*}(892)^{+}$ mesons are S-wave and P-wave channels for the tetraquarks $X_{\textrm{AV}}$ and $X_{\textrm{PS}}$, respectively. Because branching ratios of these channels amount to 0.30 and 0.36, respectively, they may be employed to fix structures $X_{\textrm{AV}}$ and $X_{\textrm{PS}}$. These and relevant problems require further theoretical and experimental studies.

Appendix: Quark propagators and invariant amplitude \(\Pi (M^{2},s_{0})\)

In this work, for the light quark propagator $S_{q}^{ab}(x)$, we use the expression

(A.1)

For the heavy quark $Q=c$, we employ the propagator $S_{Q}^{ab}(x)$

(A.2)

Above, we have used the shorthand notations

$$\begin{aligned} G_{ab}^{\alpha \beta }\equiv & {} G_{A}^{\alpha \beta }\lambda _{ab}^{A}/2,\ \ G^{2}=G_{\alpha \beta }^{A}G_{A}^{\alpha \beta },\ G^{3}\nonumber \\ {}= & {} f^{ABC}G_{\alpha \beta }^{A}G^{B\beta \delta }G_{\delta }^{C\alpha }, \end{aligned}$$

(A.3)

where $G_{A}^{\alpha \beta }$ is the gluon field strength tensor, $\lambda ^{A}$ and $f^{ABC}$ are the Gell-Mann matrices and structure constants of the color group $SU_{c}(3)$, respectively. The indices A, B, C run in the range $1,2,\ldots 8$.

The invariant amplitude $\Pi (M^{2},s_{0})$ obtained after the Borel transformation and subtraction is equal to

$$\begin{aligned} \Pi (M^{2},s_{0})=\int _{{\mathcal {M}}^{2}}^{s_{0}}ds\rho ^{{\textrm{OPE}} }(s)e^{-s/M^{2}}+\Pi (M^{2}), \end{aligned}$$

where the spectral density $\rho ^{{\textrm{OPE}}}(s)$ and the function $\Pi (M^{2})$ are determined by formulas

$$\begin{aligned} \rho ^{{\textrm{OPE}}}(s)= & {} \rho ^{\mathrm {pert.}}(s) +\sum _{N=3}^{8}\rho ^{ \textrm{DimN}}(s),\nonumber \\ \Pi (M^{2})= & {} \sum _{N=6}^{10}\Pi ^{\textrm{DimN}}(M^{2}), \end{aligned}$$

(A.4)

respectively. The components of $\rho ^{{\textrm{OPE}}}(s)$ and $\Pi (M^{2})$ are given by expressions

$$\begin{aligned} \rho ^{\textrm{DimN}}(s)= & {} \int _{0}^{1}d\alpha \rho ^{\textrm{DimN}}(s,\alpha ),\ \ \Pi ^{\textrm{DimN}}(M^{2})\nonumber \\ {}= & {} \int _{0}^{1}d\alpha \Pi ^{\textrm{DimN} }(M^{2},\alpha ), \end{aligned}$$

(A.5)

where $\alpha$ is the Feynman parameter.

Below, we write down components of the spectral density $\rho ^{{\textrm{OPE}} }(s)$ and function $\Pi (M^{2})$ for the axial-vector tetraquark $X_{\textrm{AV}}$:

The perturbative and nonperturbative components of the spectral density $\rho ^{\mathrm {pert.}}(s,\alpha )$ and $\rho ^{{\textrm{Dim}}3(4,5,6,7,8)}(s,\alpha )$ have the forms:

$$\begin{aligned}{} & {} \rho ^{\mathrm {pert.}}(s,\alpha )=\frac{\Theta (L)}{12288\pi ^{6}(1-\alpha )^{3}}\left[ m_{c}^{2}-s(1-\alpha )\right] ^{3}\nonumber \\{} & {} \quad \times \alpha ^{3}\left[ m_{c}^{2}\alpha -16m_{c}m_{s}-5s\alpha (1-\alpha )\right] , \end{aligned}$$

(A.6)

$$\begin{aligned}{} & {} \rho ^{\textrm{Dim3}}(s,\alpha )=-\frac{\langle {\overline{s}}s\rangle \Theta (L)\alpha ^{2}}{128\pi ^{4}(1-\alpha )^{2}}\left[ m_{c}^{2}-s(1-\alpha ) \right] \left[ 2m_{c}^{3}\right. \nonumber \\{} & {} \quad \left. -m_{c}^{2}m_{s}(1-\alpha )-2m_{c}s(1-\alpha )+3m_{s}s(1-\alpha )^{2}\right] , \end{aligned}$$

(A.7)

$$\begin{aligned}{} & {} \rho ^{\textrm{Dim4}}(s,\alpha )=\frac{\langle \alpha _{s}G^{2}/\pi \rangle \Theta (L)\alpha }{55296\pi ^{4}(1-\alpha )^{3}}\left[ s^{2}\alpha (1-\alpha )^{3}(162-167\alpha )\right. \nonumber \\{} & {} \quad \left. +3m_{c}^{4}\alpha (18-37\alpha +21\alpha ^{2})\right. \nonumber \\{} & {} \quad -24m_{c}^{3}m_{s}(9-18\alpha +9\alpha ^{2}+2\alpha ^{3})\nonumber \\{} & {} \quad +4sm_{c}^{2}\alpha (-54+164\alpha -169\alpha ^{2}+59\alpha ^{3}) \nonumber \\{} & {} \quad \left. -24sm_{c}m_{s}(-9+27\alpha -26\alpha ^{2}+5\alpha ^{3}+3\alpha ^{4}) \right] , \end{aligned}$$

(A.8)

$$\begin{aligned}{} & {} \rho ^{\textrm{Dim5}}(s,\alpha )=-\frac{\langle {\overline{s}}g_{s}\sigma Gs\rangle \Theta (L)\alpha }{192\pi ^{4}(1-\alpha )}\left[ 3m_{c}^{3}-m_{c}^{2}m_{s}(1-\alpha )\right. \nonumber \\{} & {} \quad \left. -3sm_{c}(1-\alpha )+2sm_{s}(1-\alpha )^{2}\right] , \end{aligned}$$

(A.9)

$$\begin{aligned}{} & {} \rho ^{\textrm{Dim6}}(s,\alpha )=-\frac{\Theta (L)}{405\cdot 2^{12}\pi ^{6}(1-\alpha )^{3}}\nonumber \\{} & {} \quad \times \left\{ (1-\alpha )^{3}\left[ -m_{c}m_{s}+m_{c}^{2} \alpha -2s\alpha (1-\alpha )\right] \right. \nonumber \\{} & {} \quad \times (1280\langle {\overline{d}}d\rangle ^{2}g_{s}^{2}\pi ^{2}+138240\langle {\overline{d}}d\rangle \langle {\overline{u}}u\rangle \pi ^{4})\nonumber \\{} & {} \quad -\alpha \left[ 1280g_{s}^{2}\pi ^{2}\langle {\overline{s}}s\rangle ^{2}(m_{c}^{2}-2s(1-\alpha ))(\alpha -1)^{3}\right. \nonumber \\{} & {} \quad \left. \left. +27\langle g_{s}^{3}G^{3}\rangle m_{c}^{2}\alpha ^{4}\right] \right\} , \end{aligned}$$

(A.10)

$$\begin{aligned}{} & {} \rho ^{\textrm{Dim7}}(s,\alpha )=-\frac{\langle \alpha _{s}G^{2}/\pi \rangle \langle {\overline{s}}s\rangle \Theta (L)}{1152\pi ^{2}(1-\alpha )^{2}}\left[ 4m_{c}(2-4\alpha \right. \nonumber \\{} & {} \quad \left. +2\alpha ^{2}+\alpha ^{3})-3m_{s}(1-\alpha )^{3}\right] , \end{aligned}$$

(A.11)

$$\begin{aligned}{} & {} \rho ^{\textrm{Dim8}}(s,\alpha )=\frac{\Theta (L)}{18432\pi ^{2}}\left[ 768\langle {\overline{u}}u\rangle \langle {\overline{d}}g_{s}\sigma Gd\rangle (\alpha -1)\right. \nonumber \\{} & {} \quad \left. -\langle \alpha _{s}G^{2}/\pi \rangle ^{2}\alpha \right] . \end{aligned}$$

(A.12)

Components of the function $\Pi (M^{2})$ are:

$$\begin{aligned}{} & {} \Pi ^{\textrm{Dim6}}(M^{2},\alpha )=\frac{\langle g_{s}^{3}G^{3}\rangle m_{c}^{3}\alpha ^{3}}{184320\pi ^{6}M^{2}(1-\alpha )^{5}}\nonumber \\{} & {} \quad \times \exp \left[ -\frac{ m_{c}^{2}}{M^{2}(1-\alpha )}\right] \left[ m_{c}^{3}\alpha (2+\alpha )+8m_{c}^{2}m_{s}(2\alpha -1)\right. \nonumber \\{} & {} \quad \left. -16M^{2}m_{s}(\alpha ^{2}-1)\right] , \end{aligned}$$

(A.13)

$$\begin{aligned}{} & {} \Pi ^{\textrm{Dim7}}(M^{2},\alpha )=-\frac{\langle \alpha _{s}G^{2}/\pi \rangle \langle {\overline{s}}s\rangle m_{c}^{3}\alpha ^{2}}{1152M^{2}\pi ^{2}(1-\alpha )^{3}}\nonumber \\{} & {} \quad \times \exp \left[ -\frac{m_{c}^{2}}{M^{2}(1-\alpha )}\right] \left[ m_{c}m_{s}-2M^{2}(1-\alpha )\right] , \end{aligned}$$

(A.14)

$$\begin{aligned}{} & {} \Pi ^{\textrm{Dim8}}(M^{2},\alpha ) =-\frac{\langle \alpha _{s}G^{2}/\pi \rangle ^{2}m_{c}\alpha }{9216M^{2}\pi ^{2}(1-\alpha )^{3}}\nonumber \\{} & {} \quad \times \exp \left[ - \frac{m_{c}^{2}}{M^{2}(1-\alpha )}\right] \left[ 2m_{c}^{2}m_{s}(1-\alpha )+m_{c}^{3}\alpha \right. \nonumber \\{} & {} \quad \left. +2M^{2}m_{s}(\alpha ^{2}-1)\right] \nonumber \\{} & {} \quad -\frac{\langle {\overline{u}}u\rangle \langle {\overline{d}}g_{s}\sigma Gd\rangle m_{c}m_{s}}{24\pi ^{2}}\exp \left[ -\frac{m_{c}^{2}}{M^{2}}\right] , \end{aligned}$$

(A.15)

$$\begin{aligned}{} & {} \Pi ^{\textrm{Dim9}}(M^{2},\alpha )=\frac{m_{c}\alpha }{69120M^{6}\pi ^{4}(1-\alpha )^{5}}\nonumber \\{} & {} \quad \times \exp \left[ -\frac{m_{c}^{2}}{M^{2}(1-\alpha )}\right] \left\{ 20\langle \alpha _{s}G^{2}/\pi \rangle \langle {\overline{s}} g_{s}\sigma Gs\rangle M^{2}\pi ^{2}(1-\alpha )^{2}\right. \nonumber \\{} & {} \quad \times \left[ m_{c}^{3}m_{s}-3m_{c}^{2}M^{2}(1-\alpha )-m_{c}m_{s}M^{2}(1-\alpha )\right. \nonumber \\{} & {} \quad \left. -3M^{4}(\alpha ^{2}-1)\right] +3\langle g_{s}^{3}G^{3}\rangle \langle {\overline{s}}s\rangle m_{c}\alpha \left[ 4m_{c}^{3}M^{2}(1-2\alpha )\right. \nonumber \\{} & {} \quad \left. \left. -16m_{c}M^{4}(1-\alpha )+m_{c}^{4}m_{s}(2+\alpha )+m_{c}^{2}m_{s}M^{2}(\alpha ^{2}\right. \right. \nonumber \\{} & {} \quad \left. \left. +\alpha -2)-m_{s}M^{4}(2-3\alpha +\alpha ^{3})\right] \right\} \nonumber \\{} & {} \quad +\frac{m_{c}}{62208\pi ^{2}M^{2}}\nonumber \\{} & {} \quad \times \exp \left[ -\frac{m_{c}^{2}}{M^{2}} \right] \left\{ -27\langle \alpha _{s}G^{2}/\pi \rangle \langle {\overline{s}} g_{s}\sigma Gs\rangle (m_{c}m_{s}-3M^{2})\right. \nonumber \\{} & {} \quad \left. +32g_{s}^{2}\langle {\overline{s}} s\rangle (\langle {\overline{u}}u\rangle ^{2}+\langle {\overline{d}}d\rangle ^{2})\right. \nonumber \\{} & {} \quad \left. \times (m_{c}m_{s}-2M^{2})+3456\langle {\overline{s}}s\rangle \langle {\overline{u}}u\rangle \langle {\overline{d}}d\rangle \pi ^{2}(m_{c}m_{s}-2M^{2})\right\} , \end{aligned}$$

(A.16)

$$\begin{aligned}{} & {} \Pi ^{\textrm{Dim10}}(M^{2},\alpha )=-\frac{m_{c}\langle \alpha _{s}G^{2}/\pi \rangle g_{s}^{2}}{23328M^{4}\pi ^{2}(1-\alpha )^{3}}\nonumber \\{} & {} \quad \times \exp \left[ -\frac{m_{c}^{2}}{M^{2}(1-\alpha )}\right] \left\{ m_{c}^{2}m_{s}\langle {\overline{u}}u\rangle ^{2}(1-\alpha )\right. \nonumber \\{} & {} \quad -2M^{2}m_{s}\langle {\overline{u}}u\rangle ^{2}(1-\alpha )+\alpha [m_{c}^{3}-m_{c}M^{2}(1\nonumber \\{} & {} \quad -\alpha )[\langle {\overline{u}}u\rangle ^{2}+\langle {\overline{s}}s\rangle ^{2}]+\langle {\overline{d}}d\rangle ^{2}\left[ -2M^{2}m_{s}(1-\alpha )\right. \nonumber \\{} & {} \quad \left. \left. +m_{c}^{3}\alpha -m_{c}M^{2}\alpha (1-\alpha )+m_{c}^{2}m_{s}(1-\alpha )\right] \right\} \nonumber \\{} & {} \quad +\frac{m_{c}^{2}}{1728\pi ^{2}M^{4}}\exp \left[ -\frac{m_{c}^{2}}{M^{2}}\right] (m_{c}m_{s}+M^{2}) \nonumber \\{} & {} \quad \times \left[ 9\langle {\overline{u}}g_{s}\sigma Gu\rangle \langle \overline{ d}g_{s}\sigma Gd\rangle +8\pi ^{2}\langle \alpha _{s}G^{2}/\pi \rangle \langle {\overline{u}}u\rangle \langle {\overline{d}}d\rangle \right] . \end{aligned}$$

(A.17)

In expressions above, $\Theta (z)$ is Unit Step function, and

$$\begin{aligned} L=s\alpha (1-\alpha )-m_{c}^{2}\alpha . \end{aligned}$$

(A.18)

Calculation of the correlation function $\Pi (M^{2},s_{0})$ implies integrations of the spectral density’s components $\rho ^{\mathrm {pert.}}(s,\alpha )$ and $\rho ^{{\textrm{Dim}}3(4,5,6,7,8)}(s,\alpha )$ over $\alpha$ and s. These functions contain $\sim 1/(1-\alpha )^{n}$ type factors, which at $\alpha =1$ may lead to singularities and diverge integrals. These spectral densities, however, depend also on the function $\Theta [s\alpha (1-\alpha )-m_{c}^{2}\alpha ]$ that cuts off dangerous regions in $\alpha$ integration. The components of $\Pi (M^{2})$ are functions of the Feynman parameter $\alpha$, and have in denominators the factors $(1-\alpha )^{n}$ as well. But the function $\exp \left[ -m_{c}^{2}/M^{2}(1-\alpha )\right]$ in these components effectively regulates possible singularities rendering finite relevant integrals.