New scalar resonance X₀(2900) as a ${\bar{D}}^{{}^{{\ast}}}{K}^{{\ast}}$ molecule: mass and width

The LHCb collaboration recently announced about observation of two resonant structures X₀(2900) and X₁(2900) (hereafter X₀ and X₁, respectively) in the invariant D⁻ K⁺ mass distribution of the decay B⁺ → D⁺ D⁻ K⁺ [1]. In accordance with LHCb, obtained results constitute the clear evidence for exotic mesons with full open flavors. At the same time, the collaboration did not exclude models which explain experimental data using hadronic rescattering effects.

The LHCb extracted the masses and widths of these structures, as well as determined their spin-parities. Thus, it was shown that the X₀ is the scalar resonance J^P = 0⁺ with parameters

$$\begin{align}{m}_{0}=(2866{\pm}7{\pm}2)\enspace \mathrm{M}\mathrm{e}\mathrm{V},\qquad {{\Gamma}}_{0}=(57{\pm}12{\pm}4)\enspace \mathrm{M}\mathrm{e}\mathrm{V},\end{align} \tag{ 1 }$$

whereas X₁ is the vector state J^P = 1⁻ and has the mass and width

$$\begin{align}{m}_{1}& =(2904{\pm}5{\pm}1)\enspace \mathrm{M}\mathrm{e}\mathrm{V},\enspace \\ {{\Gamma}}_{1}& =(110{\pm}11{\pm}4)\enspace \mathrm{M}\mathrm{e}\mathrm{V}.\end{align} \tag{ 2 }$$

From decay channels of these resonances X₀₍₁₎ → D⁻ K⁺, it is evident that they are built of four different valence quarks $ud\bar{s}\bar{c}$. These circumstances place X₀ and X₁ to distinguishable position in the XYZ family of exotic mesons. The LHCb's discovery is doubly remarkable, because existence of the resonance X(5568), seen by the D0 collaboration [2] and presumably composed of quarks $sd\bar{b}\bar{u}$, was not later confirmed by other experiments.

The LHCb information generated interesting theoretical investigations to explain structure of new resonances X₀ and X₁, calculate their masses, and if possible, estimate widths [3–14]. In these papers the authors made different suggestions about internal organization of these resonances, and used various methods and schemes to compute their parameters. The diquark–antidiquark and hadronic molecule pictures are dominant models to account for collected experimental data. For example, in references [3, 4] X₀ was considered as a scalar tetraquark ${X}_{0}=[sc][\bar{u}\bar{d}]$ using a phenomenological approach and the sum rule method, respectively. In reference [6] X₀ was interpreted as S-wave hadronic D^*− K^*+ molecule, whereas X₁ considered P-wave diquark–antidiquark state ${X}_{1}=[ud][\bar{s}\bar{c}]$. In reference [8], on the contrary, it was asserted that two resonance-like peaks generated in the process B⁺ → D⁺ D⁻ K⁺ due to rescattering effects may emerge in LHCb experiment as the states X₀ and X₁.

It is worth noting that the exotic scalar meson with open flavor structure ${X}_{c}=[su][\bar{c}\bar{d}]$ was studied in our work [15], in which it was explored as a charmed partner of the resonance X(5568). The mass and width of this tetraquark were calculated using the sum rule method and two interpolating currents. These currents correspond to structures Cγ₅ ⊗ γ₅ C and Cγ_μ ⊗ γ^μ C, and are scalar–scalar (S) and axial-axial currents (A), respectively. The width of X_c was evaluated by analyzing decay channels ${X}_{c}\to {D}_{s}^{-}{\pi }^{+}$ and ${X}_{c}\to {D}^{0}{\bar{K}}^{0}$. Performed studies led to the following results

$$\begin{equation}{m}_{\mathrm{S}}=(2634{\pm}62)\enspace \mathrm{M}\mathrm{e}\mathrm{V},\qquad {{\Gamma}}_{\mathrm{S}}=(57.7{\pm}11.6)\enspace \mathrm{M}\mathrm{e}\mathrm{V},\end{equation} \tag{ 3 }$$

and

$$\begin{equation}{m}_{\mathrm{A}}=(2590{\pm}60)\enspace \mathrm{M}\mathrm{e}\mathrm{V},\qquad {{\Gamma}}_{\mathrm{A}}=(63.4{\pm}14.2)\enspace \mathrm{M}\mathrm{e}\mathrm{V}.\end{equation} \tag{ 4 }$$

The prediction (2.55 ± 0.09) GeV for the mass of X_c was made in reference [16], as well.

It is clear that X_c and X₀ are different particles and their decay channels differ from each another. Nevertheless, it is convenient to compare parameters of X_c with LHCb data to make some preliminary assumptions on structure of X₀. The mass of the ground-state tetraquark X_c is not large enough to account for LHCb data. We must also take into account that, the tetraquark X_c is composed of a relatively heavy diquark [su] and heavy antidiquark $[\bar{c}\bar{d}]$, whereas X₀ would has a light diquark [ud]-heavy antidiquark $[\bar{c}\bar{s}]$ structure. Heavy-light tetraquarks are more compact and lighter than ones with the same quark content but other compositions [17]. Therefore, the mass of the resonance X₀ should be within or even below limits (3) and (4) provided we treat it as a ground-state tetraquark: in the diquark–antidiquark picture X₀ may be considered as a radially excited $[ud][\bar{c}\bar{s}]$ state [7].

Alternatively, one may analyze it as a hadronic molecule, i.e., as a bound state of conventional D and K mesons. Mesons D⁻ and K⁺ may form a bound state if the mass of a molecule D⁻ K⁺ is less than corresponding two-meson threshold 2365 MeV. But this estimate is considerably below the mass of X₀, therefore, it is difficult to expect that the molecule D⁻ K⁺ can be considered as the X₀ state. For compounds ${\bar{D}}^{{}^{{\ast}0}}{K}^{{\ast}0}$ (hereafter ${\bar{D}}^{{\ast}}{K}^{{\ast}}$) and D^*− K^*+ relevant two-particle thresholds are equal to ∼2900 MeV, and hence they cannot dissociate to vector mesons D* and K* provided masses of these molecules are below this limit: an estimation for the mass 2848 MeV of the scalar molecule ${D}^{{\ast}}{\bar{K}}^{{\ast}}$ obtained in reference [18] supports this assumption. But such hadronic molecules can decay to a pair of pseudoscalar D and K mesons. Then, structures ${\bar{D}}^{{\ast}}{K}^{{\ast}}$ and D^*− K^*+ may be interpreted as X₀ if their masses are compatible with m₀. The scenario with D^*− K^*+ as X₀ was realized in reference [6], in which the authors calculated the mass of the molecule D^*− K^*+ in the framework of the QCD sum rule method. Result obtained there $2.8{7}_{-0.14}^{+0.19}\enspace \mathrm{G}\mathrm{e}\mathrm{V}$ indicates that an assumption about molecule structure of X₀ deserves detailed studies.

In the present work, we treat X₀ as a hadronic molecule ${\bar{D}}^{{\ast}}{K}^{{\ast}}$ composed of two vector mesons ${\bar{D}}^{{}^{{\ast}0}}=\bar{c}u$ and ${K}^{{\ast}0}=d\bar{s}$, and compute not only its mass, but also width. The mass of X₀ is evaluated in the context of the sum rule method, where we take into account quark, gluon and mixed condensates up to dimension 15. The width of X₀ is found by considering the decay channel X₀ → D⁻ K⁺. To this end, we calculate the coupling G that describes strong vertex X₀ D⁻ K⁺ in the context of the light-cone sum rule (LCSR) approach using technical tools of the soft-meson approximation. Information on this coupling obtained from analysis allows us to estimate the width of X₀.

This work is structured in the following manner: in section 2, we calculate the mass and coupling of the hadronic molecule ${\bar{D}}^{{\ast}}{K}^{{\ast}}$. In section 3, we compute the strong coupling G by employing the LCSR method and soft-meson technique. In this section we find also the width of the decay X₀ → D⁻ K⁺. Section 4 is reserved for discussion and our conclusions.

The mass m, and current coupling f of the hadronic molecule ${\bar{D}}^{{\ast}}{K}^{{\ast}}$ are necessary to check the assumption about a molecule nature of the resonance X₀. These spectroscopic parameters are required also to study its strong decay. We compute m, and f using the QCD two-point sum rule method [19, 20], which is one of the effective nonperturbative approaches to determine parameters of the ordinary and exotic hadrons.

The required sum rules can be derived from analysis of the two-point correlation function

$$\begin{equation}{\Pi}(p)=i\int {\mathrm{d}}^{4}x\enspace {\mathrm{e}}^{\text{i}px}\langle 0\vert \mathcal{T}\left\{J(x){J}^{{\dagger}}(0)\right\}\vert 0\rangle ,\end{equation} \tag{ 5 }$$

where $\mathcal{T}$ denotes the time-ordered product and J(x) is the interpolating current for the scalar particle X₀. For molecule state ${\bar{D}}^{{\ast}}{K}^{{\ast}}$ this current is given by the expression

$$\begin{equation}J(x)=[{\bar{c}}_{a}(x){\gamma }^{\mu }{u}_{a}(x)][{\bar{s}}_{b}(x){\gamma }_{\mu }{d}_{b}(x)].\end{equation} \tag{ 6 }$$

In equation (6) c(x), s(x), u(x) and d(x) stand for the corresponding quark fields, whereas a and b are color indices.

Within the sum rule method the masses of various tetraquarks were analyzed in numerous articles (see, for example, the review papers [21–24]), therefore below we present only crucial points of performed analysis. In the sum rule method the correlation function Π(p) should be expressed both in terms of physical parameters of X₀ and quark–gluon degrees of freedom. In the first case, one finds the phenomenological side of the sum rules Π^Phys(p) from equation (5) by inserting a complete set of intermediate states. As a result we get

$$\begin{equation}{{\Pi}}^{\text{Phys}}(p)=\frac{\langle 0\vert J\vert {X}_{0}\rangle \langle {X}_{0}\vert {J}^{{\dagger}}\vert 0\rangle }{{m}^{2}-{p}^{2}}+\cdots \enspace ,\end{equation} \tag{ 7 }$$

where the contribution of only the ground-state particle X₀ is shown explicitly: dots denote effects of higher resonances and continuum states in the X₀ channel.

We have approximated the phenomenological side of the sum rule Π^Phys(p) in equation (7) using a simple-pole term. But, in the case of the multiquark systems, this approach must be applied with some caution, because the physical side may receive a contribution also from two-hadron reducible terms. In fact, the relevant interpolating current couples not only to the multiquark hadron, but interacts also with two conventional hadron states lying below the mass of the multiquark system [25, 26]. These contributions can be either subtracted from the sum rules or included into parameters of the pole term. In the case of the tetraquarks the second approach is preferable and was applied in articles [27–29]. It appears that, the two-meson states generate the finite width Γ(p) of the tetraquark and lead to modification

$$\begin{equation}\frac{1}{{m}^{2}-{p}^{2}}\to \frac{1}{{m}^{2}-{p}^{2}-i\sqrt{{p}^{2}}{\Gamma}(p)}.\end{equation} \tag{ 8 }$$

These effects, properly taken into account in the sum rules, rescale the coupling f leaving stable the mass m of the tetraquark. Detailed analyses proved that two-hadron contributions as a whole, and two-meson ones in particular are small, and can be neglected [26–29]. Therefore, to derive the phenomenological side of the sum rules, we use in equation (7) the zero-width single-pole approximation.

Introducing the spectroscopic parameters of X₀ through the matrix element

$$\begin{equation}\langle 0\vert J\vert {X}_{0}\rangle =fm,\end{equation} \tag{ 9 }$$

we rewrite Π^Phys(p) in the final form

$$\begin{equation}{{\Pi}}^{\text{Phys}}(p)=\frac{{f}^{2}{m}^{2}}{{m}^{2}-{p}^{2}}+\cdots \enspace .\end{equation} \tag{ 10 }$$

The function Π^Phys(p) has a simple Lorentz structure proportional to ∼I, and the term in equation (10) is the invariant amplitude Π^Phys(p²) corresponding to this structure.

The second component of the sum rules Π^OPE(p), is calculated in the operator product expansion (OPE) with some accuracy. To find Π^OPE(p), we employ the explicit expression of the interpolating current J(x) in equation (5), and contract relevant heavy and light quark fields. After these manipulations, for Π^OPE(p) we get

$$\begin{align}{{\Pi}}^{\text{OPE}}(p)& =i\int {\mathrm{d}}^{4}x\enspace {\mathrm{e}}^{\text{i}px}\enspace \mathrm{Tr}\left[{\gamma }^{\mu }{S}_{u}^{a{a}^{\prime }}(x){\gamma }^{\nu }{S}_{c}^{{a}^{\prime }a}(-x)\right]\mathrm{Tr}\left[{\gamma }_{\mu }{S}_{d}^{b{b}^{\prime }}(x){\gamma }_{\nu }{S}_{s}^{{b}^{\prime }b}(-x)\right],\end{align} \tag{ 11 }$$

where S_c (x) and S_u(s,d)(x) are the heavy c- and light u(s, d) -quark propagators, respectively. Their explicit expressions can be found, for instance, in reference [24]. The correlation function has also a trivial structure: we denote the relevant invariant amplitude by Π^OPE(p²).

The sum rules for m and f can be found by equating Π^Phys(p²) and Π^OPE(p²) and performing standard manipulations of the method. First of all, one should apply the Borel transformation to both sides of this equality to suppress contributions of higher resonances and continuum states. At the next stage, by using the hypothesis about quark–hadron duality, one subtracts higher resonances and continuum terms from the physical side of the equality. As a result, the sum rule equality starts to depend on the Borel M² and continuum threshold s₀ parameters.

The second equality required to find sum rules is obtained by applying the operator d/d(−1/M²) to the first expression. Then the sum rules for m and f are

$$\begin{equation}{m}^{2}=\frac{{{\Pi}}^{\prime }({M}^{2},{s}_{0})}{{\Pi}({M}^{2},{s}_{0})},\end{equation} \tag{ 12 }$$

and

$$\begin{equation}{f}^{2}=\frac{{e}^{{m}^{2}/{M}^{2}}}{{m}^{2}}{\Pi}({M}^{2},{s}_{0}).\end{equation} \tag{ 13 }$$

Here, Π(M², s₀) is the Borel transformed and subtracted invariant amplitude Π^OPE(p²), and Π'(M², s₀) = d/d(−1/M²)Π(M², s₀).

The function Π(M², s₀) has the following form

$$\begin{equation}{\Pi}({M}^{2},{s}_{0})={\int }_{{\mathcal{M}}^{2}}^{{s}_{0}}\mathrm{d}s\enspace {\rho }^{\text{OPE}}(s){e}^{-s/{M}^{2}}+{\Pi}({M}^{2}).\end{equation} \tag{ 14 }$$

Throughout this article, we neglect the mass of the quarks u and d, and set $\mathcal{M}={m}_{c}+{m}_{s}$ in equation (14). The two-point spectral density ρ^OPE(s) is computed as an imaginary part some of terms in the correlation function Π^OPE(p). The component Π(M²) is the Borel transformation of remaining terms in Π^OPE(p), and are obtained directly from their expressions. Calculations are carried out by taking into account vacuum condensates up to dimension 15. The dimension-15 contribution to the correlation function is proportional to product of light quark condensates $\langle \bar{s}{g}_{s}\sigma Gs\rangle {\langle \bar{q}{g}_{s}\sigma Gq\rangle }^{2}$. This and other higher dimensional terms in Π(M², s₀) are obtained as products of basic vacuum condensates using factorization procedure by assuming that it does not lead to essential ambiguities. Analytical expressions of ρ^OPE(s) and Π(M²) are rather lengthy to be presented here explicitly.

The sum rules (12) and (13) depend on universal quark $\langle \bar{q}q\rangle =\langle 0\vert \bar{q}q\vert 0\rangle$, gluon ⟨α_s G²/π⟩ = ⟨0|α_s G²/π|0⟩ and mixed quark–gluon $\langle \bar{q}{g}_{s}\sigma Gq\rangle =\langle 0\vert \bar{q}{g}_{s}\sigma Gq\vert 0\rangle$ vacuum condensates (q ≡ u, d, and similar expressions for the strange quark s) [19, 20, 30, 31] and masses of c and s quarks

$$\begin{align}\langle \bar{q}q\rangle & =-{(0.24{\pm}0.01)}^{3}\enspace {\mathrm{G}\mathrm{e}\mathrm{V}}^{3},\quad \langle \bar{s}s\rangle =(0.8{\pm}0.1)\langle \bar{q}q\rangle ,\\ \langle \bar{q}{g}_{s}\sigma Gq\rangle & ={m}_{0}^{2}\langle \bar{q}q\rangle ,\enspace \langle \bar{s}{g}_{s}\sigma Gs\rangle ={m}_{0}^{2}\langle \bar{s}s\rangle ,\quad {m}_{0}^{2}=(0.8{\pm}0.2)\enspace {\mathrm{G}\mathrm{e}\mathrm{V}}^{2}\\ \left\langle \frac{{\alpha }_{s}{G}^{2}}{\pi }\right\rangle & =(0.012{\pm}0.004)\enspace {\mathrm{G}\mathrm{e}\mathrm{V}}^{4},\quad {m}_{s}=9{3}_{-5}^{+11}\enspace \mathrm{M}\mathrm{e}\mathrm{V},\quad {m}_{c}=1.27{\pm}0.2\enspace \mathrm{G}\mathrm{e}\mathrm{V}.\end{align} \tag{ 15 }$$

As is seen, the vacuum condensate of strange quarks is different from $\langle 0\vert \bar{q}q\vert 0\rangle$ [30]. The mixed condensates $\langle \bar{q}{g}_{s}\sigma Gq\rangle$ and $\langle \bar{s}{g}_{s}\sigma Gs\rangle$ can be expressed in terms of the corresponding quark condensates and parameter ${m}_{0}^{2}$, numerical value of which was extracted from analysis of baryonic resonances [31].

The m and f are functions of the parameters M² and s₀, as well. The correct choice for M² and s₀ is an important problem of sum rule computations. The working regions for M² and s₀ should satisfy usual constraints imposed on the pole contribution (PC) and convergence of the OPE. To analyze these questions, we introduce the quantities

$$\begin{equation}\mathrm{P}\mathrm{C}=\frac{{\Pi}({M}^{2},{s}_{0})}{{\Pi}({M}^{2},\infty )},\end{equation} \tag{ 16 }$$

and

$$\begin{equation}R({M}^{2})=\frac{{{\Pi}}^{\text{DimN}}({M}^{2},{s}_{0})}{{\Pi}({M}^{2},{s}_{0})}.\end{equation} \tag{ 17 }$$

In equation (17) Π^DimN(M², s₀) is a last term (or a sum of last few terms) in the correlation function. In the present analysis, we use the sum of last three terms in OPE, and hence Dim N ≡ Dim(13 + 14 + 15).

The PC is used to fix upper bound for M², whereas R(M²) is necessary to find low limit for the Borel parameter. These two values of M² fix the boundaries of a region where the Borel parameter can be varied. Our analysis shows that the working regions for the parameters M² and s₀ are

$$\begin{equation}{M}^{2}\in [2,3]\enspace {\mathrm{G}\mathrm{e}\mathrm{V}}^{2},\qquad {s}_{0}\in [11.3,13.3]\enspace {\mathrm{G}\mathrm{e}\mathrm{V}}^{2},\end{equation} \tag{ 18 }$$

and they obey restrictions on PC and convergence of OPE. Thus, at M² = 3 GeV² the PC is 0.5, whereas at M² = 2 GeV² it becomes equal to 0.8. At the minimum of M² = 2 GeV², we find R ≈ 0.01, which guarantees the convergence of the sum rules. We extract the parameters m and f approximately at a middle point of the window (18), M² = 2.5 GeV² and s₀ = 12 GeV², where the PC is PC ≈ 0.65. This fact ensures the ground state nature of X₀.

Our predictions for m and f are

$$\begin{align}m& =(2868\enspace {\pm}198)\enspace \mathrm{M}\mathrm{e}\mathrm{V},\\ f& =(3.0{\pm}0.7){\times}1{0}^{-3}\enspace {\mathrm{G}\mathrm{e}\mathrm{V}}^{4}.\end{align} \tag{ 19 }$$

The sum rule results, in general, should not depend on the parameter M². But in real calculations m and f are sensitive to the choice of M². From inspection of equation (19) it is seen, that theoretical uncertainties in the case of m equal to ±6.9%, whereas for the coupling f they amount to ±23.3%. Theoretical ambiguities of f are larger, because f is determined directly in terms of Π(M², s₀), whereas the sum rule for m depends on the ratio of such functions and is exposed to smaller variations. Nevertheless, uncertainties even for the coupling f remain within limits accepted in sum rule computations. In figure 1 we display the sum rule's predictions for m as a function of M², where one can see its residual dependence on the Borel parameter.

Zoom In Zoom Out Reset image size

The continuum threshold parameter s₀ separates a ground-state contribution from effects due to higher resonances and continuum states. In other words, $\sqrt{{s}_{0}}$ has to be smaller than the mass m* of the first excitation of the X₀. The self-consistent sum rule analysis implies that the difference $\sqrt{{s}_{0}}-m$ is around or less than m* − m. Excited states of conventional hadrons and their parameters are known either from experimental measurements or from alternative theoretical studies. In the case of the multiquark hadrons there is a deficiency of relevant information. The mass gap $\sqrt{{s}_{0}}-m\approx (500-600)\enspace \mathrm{M}\mathrm{e}\mathrm{V}$ found in the present work can be considered as a reasonable estimate m* ≈ (m + 500) MeV for the hadronic molecule ${\bar{D}}^{{\ast}}{K}^{{\ast}}$ containing one heavy quark. Dependence of extracted value of m on the scale s₀ is also shown in figure 1.

Obtained prediction for the mass of the state ${\bar{D}}^{{\ast}}{K}^{{\ast}}$ is in a nice agreement with new LHCb measurements. This is necessary, but not enough to interpret X₀ as the hadronic molecule. For reliable conclusions, we need to calculate the width of the molecule ${\bar{D}}^{{\ast}}{K}^{{\ast}}$ and confront it with data: only together these parameters can support or not assumptions about the structure of X₀.

In this section we explore the strong decay X₀ → D⁻ K⁺ in order to find width of the resonance X₀. Strictly speaking, there are other decay channel of the scalar state X₀, namely the S-wave decay to a pair of mesons ${\bar{D}}^{0}{K}^{0}$. Because X₀ was observed as enhancement in the D⁻ K⁺ mass distribution, we concentrate on the first process and saturate full width of the resonance X₀ by this channel.

The width of the decay X₀ → D⁻ K⁺ is determined by the strong coupling G corresponding to the vertex X₀ D⁻ K⁺. We are going to calculate G using method of the QCD sum rule on the light-cone [32, 33] and methods of the soft-meson approximation [34]. To this end, we start from analysis of the correlation function [33, 35]

$$\begin{equation}{\Pi}(p,q)=i\int {\mathrm{d}}^{4}x\enspace {\mathrm{e}}^{\text{i}px}\langle K(q)\vert \mathcal{T}\left\{{J}^{D}(x){J}^{{\dagger}}(0)\right\}\vert 0\rangle ,\end{equation} \tag{ 20 }$$

where by K and D we denote, in short forms, the mesons K⁺ and D⁻, respectively. In the correlation function Π(p, q), the interpolating current J(x) is given by equation (6), whereas for J^D (x) we use

$$\begin{equation}{J}^{D}(x)={\bar{c}}_{n}(x)i{\gamma }_{5}{d}_{n}(x),\end{equation} \tag{ 21 }$$

with n being the color index. It is not difficult to determine Π(p, q) in terms of the physical parameters of the particles involved into the decay [33]. By taking into account the ground states in the D and X₀ channels, we get

$$\begin{align}{{\Pi}}^{\text{Phys}}(p,q)& =\frac{\langle 0\vert {J}^{D}\vert D\left(p\right)\rangle }{{p}^{2}-{m}_{D}^{2}}\langle D\left(p\right)K(q)\vert {X}_{0}({p}^{\prime })\rangle \frac{\langle {X}_{0}({p}^{\prime })\vert {J}^{{\dagger}}\vert 0\rangle }{{p}^{\prime 2}-{m}^{2}}+\cdots \enspace ,\end{align} \tag{ 22 }$$

where p, q and p' = p + q are the momenta of the particles D, K, and X₀, and m_D is the mass of D meson. The ellipses in equation (22) refer to contributions of higher resonances and continuum states in the D and X₀ channels.

In order to finish computation of the correlation function, we introduce the matrix elements

$$\begin{align}\langle 0\vert {J}^{D}\vert D\left(p\right)\rangle & =\frac{{f\hspace{-2pt}}_{D}{m}_{D}^{2}}{{m}_{c}},\enspace \langle {X}_{0}({p}^{\prime })\vert {J}^{{\dagger}}\vert 0\rangle =fm,\langle D\left(p\right)K(q)\vert {X}_{0}({p}^{\prime })\rangle =Gp\cdot {p}^{\prime }.\end{align} \tag{ 23 }$$

In expressions above f_D is decay constant of the meson D⁻. Then for Π^Phys(p, q) we find

$$\begin{equation}{{\Pi}}^{\text{Phys}}(p,q)=\frac{Gfm{f\hspace{-2pt}}_{D}{m}_{D}^{2}}{{m}_{c}({p}^{2}-{m}_{D}^{2})({p}^{\prime 2}-{m}^{2})}p\cdot {p}^{\prime }+\cdots \enspace .\end{equation} \tag{ 24 }$$

To continue, we have to calculate Π^QCD(p, q) in terms of the quark–gluon degrees of freedom and find the QCD side of the sum rule. Contractions in equation (20) of c and d quark and antiquark fields yield

$$\begin{align}{{\Pi}}^{\text{OPE}}(p,q)& =\int {\mathrm{d}}^{4}x\enspace {\mathrm{e}}^{\text{i}px}\left[{\gamma }^{\mu }{S}_{c}^{bn}(-x){\gamma }_{5}\right.{\left.\enspace {S}_{d}^{na}(x){\gamma }_{\mu }\right]}_{\alpha \beta }\langle K(q)\vert {\bar{u}}_{\alpha }^{b}(0){s}_{\beta }^{a}(0)\vert 0\rangle ,\end{align} \tag{ 25 }$$

where α and β are the spinor indexes.

Apart from quark propagators the function Π^OPE(p, q) depends also on local matrix elements of the quark operator $\bar{u}s$ sandwiched between the vacuum and K meson. To express $\langle K(q)\vert {\bar{u}}_{\alpha }^{b}(0){s}_{\beta }^{a}(0)\vert 0\rangle$ using the K meson's local matrix elements, we expand $\bar{u}(0)s(0)$ over the full set of Dirac matrices Γ^J and project them onto the color-singlet states

$$\begin{equation}{\bar{u}}_{\alpha }^{b}(0){s}_{\beta }^{a}(0)\to \frac{1}{12}{\delta }^{ba}{{\Gamma}}_{\beta \alpha }^{J}\left[\bar{u}(0){{\Gamma}}^{J}s(0)\right],\end{equation} \tag{ 26 }$$

where

$$\begin{equation}{{\Gamma}}^{J}=\mathbf{1},\enspace {\gamma }_{5},\enspace {\gamma }_{\mu },\enspace i{\gamma }_{5}{\gamma }_{\mu },\enspace {\sigma }_{\mu \nu }/\sqrt{2}.\end{equation} \tag{ 27 }$$

The expression (26) reveals a main difference between vertices composed of conventional mesons and vertices containing a tetraquark and two ordinary mesons. Indeed, in the vertices of ordinary mesons the correlation function depends on distribution amplitudes (DAs) of one of the final-state mesons, for example, on DAs of K meson. The DAs of the mesons are determined as non-local matrix elements of relevant quark fields placed between the meson and vacuum states. In the case under discussion, it is evident that instead of non-local matrix elements, we have Π^QCD(p, q) that contains local matrix elements of K meson. The reason is that X₀ and interpolating current equation (6) are built of four quark fields at the same space-time location. Substitution of this current into the correlation function and contractions of c and d quark fields yield expressions, where the remaining light quarks are sandwiched between the K meson and vacuum states, forming local matrix elements. In other words, we encounter the situation when dependence of the correlation function on the meson DAs disappears and integrals over the meson DAs reduce to overall normalization factors. In the framework of the LCSR method such situation is possible in the kinematical limit q → 0, when the light-cone expansion is replaced by the short-distant one. As a result, instead of the expansion in terms of DAs, one gets expansion over the local matrix elements [33]. The limit q → 0 is known as the soft-meson approximation. In this approximation p = p' and invariant amplitudes Π^Phys(p²) and Π^OPE(p²) depend only on the variable p². For our purposes a decisive fact is the observation made in reference [33]: the soft-meson approximation and full LCSR treatment of the conventional mesons' vertices lead to predictions which are numerically very close to each other.

The soft-meson approximation simplifies the QCD component of the sum rule, but leads to additional complications in its phenomenological side. In this limit for invariant amplitude Π^Phys(p²) we get

$$\begin{equation}{{\Pi}}^{\text{Phys}}({p}^{2})=G\frac{fm{f\hspace{-2pt}}_{D}{m}_{D}^{2}{\tilde {m}}^{2}}{{m}_{c}{({p}^{2}-{\tilde {m}}^{2})}^{2}}+\cdots \enspace ,\end{equation} \tag{ 28 }$$

where ${\tilde {m}}^{2}=({m}^{2}+{m}_{D}^{2})/2$. This amplitude contains the double pole at ${p}^{2}={\tilde {m}}^{2}$, therefore its Borel transformation is given by the formula

$$\begin{align}& & {{\Pi}}^{\text{Phys}}({M}^{2})=G\frac{fm{f\hspace{-2pt}}_{D}{m}_{D}^{2}}{{m}_{c}}\frac{{\tilde {m}}^{2}{e}^{-{\tilde {m}}^{2}/{M}^{2}}}{{M}^{2}}+\cdots \enspace .\end{align} \tag{ 29 }$$

In the standard approach the invariant amplitude Π^Phys(p², p'²) depends on two variables p² and p'², and the Borel transformations over p² and p'² suppress contributions of higher resonances and continuum states. The suppressed terms afterward can be subtracted using assumption on quark–hadron duality. But in the soft limit even after Borel transformation besides ground-state term Π^Phys(M²) contains additional unsuppressed contributions. This is a price to be paid in the soft approximation for simple Π^OPE(p²) expression. To remove contaminating contributions from the phenomenological side of the sum rule, one has to act on Π^Phys(M²) by the operator [33, 34]

$$\begin{equation}\mathcal{P}({M}^{2},{m}^{2})=\left(1-{M}^{2}\frac{\mathrm{d}}{\mathrm{d}{M}^{2}}\right){M}^{2}{e}^{{m}^{2}/{M}^{2}},\end{equation} \tag{ 30 }$$

that singles out the ground-state term. Contributions remaining in Π^Phys(M²) after this prescription can be subtracted by the usual way. Then the sum rule for the strong coupling G reads

$$\begin{equation}G=\frac{{m}_{c}{\tilde {m}}^{2}}{fm{f\hspace{-2pt}}_{D}{m}_{D}^{2}}\mathcal{P}({M}^{2},{\tilde {m}}^{2}){{\Pi}}^{\text{OPE}}({M}^{2},{s}_{0}).\end{equation} \tag{ 31 }$$

Returning to the calculation of Π^OPE(p, q), it is worth noting that by substituting the expansion (26) into equation (20), one has to perform summations over color indices and fix local matrix elements of K meson that contribute to Π^OPE(p²) in the soft limit. There are only a few such elements: two-particle matrix elements of twist-two and twist-three

$$\begin{equation}\langle 0\vert \bar{u}{\gamma }_{\mu }{\gamma }_{5}s\vert K(q)\rangle =i{f\hspace{-2pt}}_{K}{q}_{\mu },\langle 0\vert \bar{u}i{\gamma }_{5}s\vert K\rangle =\frac{{f\hspace{-2pt}}_{K}{m}_{K}^{2}}{{m}_{s}}.\end{equation} \tag{ 32 }$$

There are also three-particle local matrix elements of K meson, for an example,

$$\begin{equation}\langle 0\vert \bar{u}{\gamma }^{\nu }{\gamma }_{5}ig{G}_{\mu \nu }s\vert K(q)\rangle =i{q}_{\mu }{f\hspace{-2pt}}_{K}{m}_{K}^{2}{\kappa }_{4K},\end{equation} \tag{ 33 }$$

which may contribute to Π^OPE(p²). The elements given by equation (32) appear due to the propagator S_d , perturbative term of S_c , and the expansion (26), whereas three-particle matrix elements may contribute after gluon insertions to $\bar{u}{{\Gamma}}^{J}s$ stemming from nonperturbative components of the propagator S_c . In matrix elements (32) and (33) quark and gluon fields are fixed at the same position x = 0, and κ_4K is the twist-four matrix element of the K meson.

Procedures to calculate the correlation function in the soft approximation were presented in references [36, 37], therefore we skip further technical details and provide final expression for the correlation function, which is calculated with dimension-nine accuracy and given as a sum of the perturbative and nonperturbative components

$$\begin{align}{{\Pi}}^{\text{OPE}}({M}^{2},{s}_{0})& =\frac{{\mu }_{K}}{8{\pi }^{2}}{\int }_{{\mathcal{M}}^{2}}^{{s}_{0}}\frac{\mathrm{d}s{({m}_{c}^{2}-s)}^{2}}{s}{e}^{-s/{M}^{2}}+{{\Pi}}_{\text{NP}}({M}^{2}).\end{align} \tag{ 34 }$$

The nonperturbative component of the correlation function Π_NP(M²) has the following form

$$\begin{align}{{\Pi}}_{\text{NP}}({M}^{2})& =-\frac{\langle \bar{d}d\rangle {\mu }_{K}{m}_{c}}{3}{e}^{-{m}_{c}^{2}/{M}^{2}}\\ & \quad +\left\langle \frac{{\alpha }_{s}{G}^{2}}{\pi }\right\rangle \frac{{\mu }_{K}{m}_{c}^{4}}{72{M}^{4}}{\int }_{0}^{1}\frac{\mathrm{d}z}{{z}^{3}{(z-1)}^{3}}{e}^{-{m}_{c}^{2}/[{M}^{2}z(1-z)]}\\ & \quad +\frac{\langle \bar{d}g\sigma Gd\rangle {\mu }_{K}{m}_{c}^{3}}{6{M}^{4}}{e}^{-{m}_{c}^{2}/{M}^{2}}-\left\langle \frac{{\alpha }_{s}{G}^{2}}{\pi }\right\rangle \langle \bar{d}d\rangle \\ & \quad {\times}\frac{{\mu }_{K}{m}_{c}({m}_{c}^{2}+3{M}^{2}){\pi }^{2}}{54{M}^{6}}{e}^{-{m}_{c}^{2}/{M}^{2}}+\left\langle \frac{{\alpha }_{s}{G}^{2}}{\pi }\right\rangle \langle \bar{d}g\sigma Gd\rangle \\ & \quad {\times}\frac{{\mu }_{K}{m}_{c}({m}_{c}^{4}+6{M}^{2}{m}_{c}^{2}+6{M}^{4}){\pi }^{2}}{216{M}^{10}}{e}^{-{m}_{c}^{2}/{M}^{2}}.\end{align} \tag{ 35 }$$

In expressions above, we introduce ${\mu }_{K}={f\hspace{-2pt}}_{K}{m}_{K}^{2}/{m}_{s}$. It turns out that only the twist-three matrix element from equation (32) contributes to the function Π_NP(M²). The last term in Π_NP(M²) is proportional to $\langle {\alpha }_{s}{G}^{2}/\pi \rangle \langle \bar{d}g\sigma Gd\rangle$ with dimension nine, and is suppressed additionally by the factor 1/M⁶. Therefore, dimension-nine accuracy for computation of Π^OPE(M², s₀) adopted in the present article is high and enough to get reliable result.

The sum rule (31) depends on the various vacuum condensates written down above (15). It contains the masses and decay constants of the final-state mesons D⁻ and K⁺: relevant spectroscopic parameters are collected in table 1. All of them are borrowed from reference [38]. For decay constants f_D and f_K Particle Data Group's averages are used.

To carry out numerical analysis one also needs to fix M² and s₀. The restrictions imposed on these auxiliary parameters are standard for sum rule computations and have been discussed above. Our analysis demonstrates that working regions (18) meet all constraints necessary for computations of Π^OPE(M², s₀). Numerical calculations lead to the following result

$$\begin{equation}\vert G\vert =(0.66{\pm}0.06)\enspace {\mathrm{G}\mathrm{e}\mathrm{V}}^{-1}.\end{equation} \tag{ 36 }$$

This prediction for the strong coupling G is typical for tetraquark-meson vertices. Its numerical value and dimension are determined by definition of matrix element $\langle D\left(p\right)K(q)\vert {X}_{0}({p}^{\prime })\rangle$: modification of the vertex DKX₀ in equation (23) changes the value and dimension of G. In general, it is possible to rewrite $\langle D\left(p\right)K(q)\vert {X}_{0}({p}^{\prime })\rangle$ in such a way that to make G dimensionless. In our analysis G is an intermediate parameter, whereas the physical quantity to be found is the width Γ of the decay X₀ → D⁻ K⁺. The Γ is calculated by taking into account equation (23), and its expression depends on these matrix elements. But regardless used convention for the vertex DKX₀ and analytical form of the width, numerical computations lead to the same final result with correct dimension, as it should be for a physical quantity.

Having used the matrix elements given by equation (23), we derive for the width of the decay X₀ → D⁻ K⁺

$$\begin{equation}{\Gamma}\left[{X}_{0}\to {D}^{-}{K}^{+\;}\right]=\frac{{G}^{2}{m}_{D}^{2}\lambda }{8\pi }\left(1+\frac{{\lambda }^{2}}{{m}_{D}^{2}}\right),\end{equation} \tag{ 37 }$$

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

$$\begin{align}\lambda \left(a,b,c\right)& =\frac{1}{2a}{\left({a}^{4}+{b}^{4}+{c}^{4}-2\left({a}^{2}{b}^{2}+{a}^{2}{c}^{2}+{b}^{2}{c}^{2}\right)\right)}^{1/2}.\end{align} \tag{ 38 }$$

Then it is not difficult to find that

$$\begin{equation}{\Gamma}\left[{X}_{0}\to {D}^{-}{K}^{+}\right]=(49.6{\pm}9.3)\enspace \mathrm{M}\mathrm{e}\mathrm{V}.\end{equation} \tag{ 39 }$$

This prediction for the width of the resonance X₀ is in a reasonable agreement with new LHCb data (1).

The molecule ${\bar{D}}^{{}^{{\ast}0}}{K}^{{\ast}0}$ is composed of two neutral vector mesons, which are strong-interaction unstable particles. The meson ${\bar{D}}^{{\ast}}{(2007)}^{0}$ is relatively narrow state ${{\Gamma}}_{{\bar{D}}^{{\ast}}}{< }2.1\enspace \mathrm{M}\mathrm{e}\mathrm{V}$, whereas the width ${{\Gamma}}_{{K}^{{\ast}}}=(47.4{\pm}0.5)\enspace \mathrm{M}\mathrm{e}\mathrm{V}$ of K*(892)⁰ within experimental errors is comparable with LHCb data Γ₀. In reference [6] X₀ was modeled as hadronic D^*− K^*+ molecule, and width of the meson K*⁺ was used to estimate roughly the X₀ resonance's width. The hadronic molecules D^*− K^*+ and ${\bar{D}}^{{}^{{\ast}0}}{K}^{{\ast}0}$ are bound states, and partial widths of their decay channels are determined by quark–gluon interactions inside of these particles. Due to multiquark nature of molecules their internal dynamics obviously differs from those of free mesons D* and K*. Therefore, estimation of the D^*− K^*+ and ${\bar{D}}^{{\ast}0}{K}^{{\ast}0}$ molecules' widths using directly widths of constituent mesons seems us to be problematic. One can suggest, that after dissociation of ${\bar{D}}^{{}^{{\ast}0}}{K}^{{\ast}0}$ to two vector mesons, decays of K^*0 may be used for such analysis. But the mass of the X₀ is below relevant two-meson thresholds in both pictures, i.e., X₀ does not decay to mesons ${\bar{D}}^{{}^{{\ast}0}}+{K}^{{\ast}0}$ or D^*− + K^*+. The P-wave decay ${\bar{D}}^{{}^{{\ast}0}}{K}^{{\ast}0}\to {\bar{D}}_{0}^{{\ast}}{(2400)}^{0}+{K}^{{\ast}0}$ with the vector meson K^*0 in the final state is forbidden kinematically. Another two-body P-wave decays of ${\bar{D}}^{{}^{{\ast}0}}{K}^{{\ast}0}$, for example, to mesons ${\bar{D}}^{{\ast}}{(2007)}^{0}+{K}_{0}^{{\ast}}(1430)$ and ${\bar{D}}_{1}{(2420)}^{0}+{K}^{0}$ are not allowed because of the same reasons. Nevertheless, there are multibody decays of X₀ which contribute to its full width. For example, processes X₀ → D⁻ K⁺ π⁰ π⁰ and X₀ → D⁻ K⁺ π⁺ π⁻ can improve our prediction for Γ. But these processes imply production of four new valence quarks through different mechanisms, which suppress relevant amplitudes by the factor α_s or additional strong couplings. As a result, widths of such subdominant decays would be small.

We have explored the dominant decay channel X₀ → D⁻ K⁺ of the resonance X₀. The result for its width in equation (39) has been obtained in the context of the LCSR method by applying first principles of the QCD, and is reliable prediction for this parameter. It can be improved further by including into analysis other decay channels of the X₀, which are beyond the scope of the present article.

In the present work we have explored one of two new resonances X₀ and X₁ reported by the LHCb collaboration. Namely, we have considered X₀ as a scalar hadronic molecule ${\bar{D}}^{{\ast}}{K}^{{\ast}}$, and calculated its mass and width. For these purposes, we have used the QCD sum rule method. The spectroscopic parameters of the state ${\bar{D}}^{{\ast}}{K}^{{\ast}}$ have been extracted from two-point sum rules, whereas for analysis of its strong decay channel, we used LCSR method and soft-meson approximation. Obtained predictions for m and Γ are in nice agreement with reported LHCb data, which can be interpreted in favor of molecule nature of the resonance X₀.

The suggestion about molecular structure of X₀ was made in reference [6], in which the authors computed the mass of the molecule D^*− K^*+ using the sum rule method. Calculations were done by taking into account nonperturbative terms up to dimension eight. Prediction obtained there for $\tilde {m}$

$$\begin{equation}\tilde {m}=287{0}_{-140}^{+190}\enspace \mathrm{M}\mathrm{e}\mathrm{V}\end{equation} \tag{ 40 }$$

is very close to our result. The molecule composition for the X₀ in different frameworks was proposed in references [5, 9, 12], as well.

Alternatively, the resonance X₀ was analyzed in references [3, 4, 7, 11] by assuming that it is a diquark–antidiquark state. The sum rule prediction for the mass of the scalar tetraquark ${T}_{1}=[sc][\bar{u}\bar{d}]$ with an axial-axial Cγ_μ ⊗ γ^μ C type structure reads [4]

$$\begin{equation}{m}_{{T}_{1}}=(2910{\pm}120)\enspace \mathrm{M}\mathrm{e}\mathrm{V}.\end{equation} \tag{ 41 }$$

The similar sum rule investigations were performed in reference [11]. Results for masses of the scalar tetraquark ${T}_{2}=[ud][\bar{s}\bar{c}]$ with scalar–scalar and axial–axial structures are equal to

$$\begin{align}{m}_{{T}_{2S}}& =275{0}_{-190}^{+180}\enspace \mathrm{M}\mathrm{e}\mathrm{V},\qquad {m}_{{T}_{2A}}=277{0}_{-200}^{+190}\enspace \mathrm{M}\mathrm{e}\mathrm{V},\end{align} \tag{ 42 }$$

respectively. By taking into account uncertainties of calculations, the author concluded that X₀ could be interpreted as a tetraquark J^P = 0⁺ with either scalar–scalar or axial–axial configurations. It is seen that states T₁ and T₂ are connected by the relations ${T}_{1}=\bar{{T}_{2}}$ or $\bar{{T}_{1}}={T}_{2}$ as conventional mesons, for example, ${D}^{0}=c\bar{u}$ and $\bar{{D}^{0}}=\bar{c}u$. Masses of such particles should be equal to each other, which is not the case for ${m}_{{T}_{1}}$ and ${m}_{{T}_{2A}}$. In our view, additional studies are necessary to solve problems existing in the QCD sum rule analyses of the resonance X₀ in the diquark–antidiquark picture.

Interesting analysis of the ground-state and radially excited tetraquark T₂ was performed in reference [7]. In this paper the mass of the particles T₂(1S) and T₂(2S) were found equal to 2360 MeV and 2860 MeV, respectively. As a result, the resonance X₀ was interpreted there as the excited tetraquark T₂(2S).

The enhancements in the D⁻ K⁺ mass distribution labeled by X₀ and X₁ may have alternative origin [8]. Thus, the authors of reference [8] investigated the process B⁺ → D⁺ D⁻ K⁺ via χ_c1 D^*− K^*+ and ${D}_{sJ}{\bar{D}}_{1}^{0}{K}^{0}$ rescattering diagrams. It was argued that, two resonance-like peaks obtained around thresholds D^*− K^*+ and ${\bar{D}}_{1}^{0}{K}^{0}$ may simulate the states X₀ and X₁ without a necessity to introduce genuine four-quark mesons. The observed experimental peaks were explained there by the triangle singularities in the scattering amplitudes located in the vicinity of the physical boundary.

Experimental results obtained by the LHCb collaboration do not raise doubts about existence of the resonance-like enhancements X₀ and X₁ in the D⁻ K⁺ mass distribution. These structures were already considered as meson molecules, diquark–antidiquark systems, rescattering effects. In other words, there are different and controversial interpretations of the structures X₀ and X₁ in the literature. Additional theoretical efforts seem are required to clarify nature of these states.

No new data were created or analysed in this study.