New charged resonance $Z_{c}^{-}(4100)$: the spectroscopic parameters and width

The mass, coupling and width of the newly observed charged resonance $ Z_{c}^{-}(4100)$ are calculated by treating it as a scalar four-quark system with a diquark-antidiquark structure. The mass and coupling of the state $ Z_{c}^{-}(4100)$ are calculated using the QCD two-point sum rules. In these calculations we take into account contributions of the quark, gluon and mixed condensates up to dimension ten. The spectroscopic parameters of $ Z_{c}^{-}(4100)$ obtained by this way are employed to study its $S$-wave decays to $\eta_c(1S)\pi^{-}$, $\eta_c(2S)\pi^{-}$, $D^{0}D^{-}$, and $ J/\psi \rho ^{-}$ final states. To this end, we evaluate the strong coupling constants corresponding to the vertices $Z_{c}\eta_c(1S)\pi^{-}$, $ Z_{c}\eta_c(2S)\pi^{-}$, $Z_{c}D^{0}D^{-}$, and $Z_{c}J/\psi \rho^{-}$ respectively. The couplings $g_{Z_c\eta_{c1} \pi}$, $g_{Z_{c}\eta_{c2} \pi}$ , and $g_{Z_{c}DD}$ are computed by means of the QCD three-point sum rule method, whereas $g_{Z_{c}J/\psi \rho }$ is obtained from the QCD light-cone sum rule approach and soft-meson approximation. Our results for the mass $ m=(4080 \pm 150)~\mathrm{MeV}$ and total width $\Gamma =(147 \pm 19)~\mathrm{ MeV}$ of the resonance $Z_{c}^{-}(4100)$ are in excellent agreement with the existing LHCb data.


I. INTRODUCTION
Recently, the LHCb Collaboration reported on evidence for an η c (1S)π − resonance in B 0 → K + η c (1S)π − decays extracted from analysis of pp collisions' data collected with LHCb detector at center-of-mass energies of √ s = 7, 8 and 13 TeV [1]. The mass and width of this new Z − c (4100) resonance (hereafter Z c ) were found equal to m = 4096 ± 20 +18 −22 MeV and Γ = 152 ± 58 +60 −35 MeV, respectively. As it was emphasized in Ref. [1], the spinparity assignments J P = 0 + and J P = 1 − both are consistent with the data.
From analysis of the decay channel Z c → η c (1S)π − it becomes evident that Z c contains four quarks ccdu, and it is presumably another member of the family of charged exotic Z-resonances with the same quark content: The well-known axial-vector tetraquarks Z ± c (4430) and Z ± c (3900) are also built of the quarks ccdu or ccud. The Z ± c (4430) were discovered and studied by the Belle Collaboration in B meson decays B → Kψ ′ π ± as resonances in the ψ ′ π ± invariant mass distributions [2][3][4]. The decay of the Z + c (4430) resonance to the final state J/ψπ + was also detected in the Belle experiment [5]. The existence of the Z ± c (4430) states was later confirmed by the LHCb Collaboration [6,7].
Another well-known members of this family are the axial-vector states Z ± c (3900), which were discovered by the BESIII Collaboration in the process e + e − → J/ψπ + π − as peaks in the J/ψπ ± invariant mass distributions [8]. These structures were detected by the Belle and CLEO as well (see Refs. [9,10]). The BESIII informed also on observation of the neutral Z 0 c (3900) state in the process e + e − → π 0 Z 0 c → π 0 π 0 J/ψ [11]. Various theoretical models and approaches were employed to reveal the internal quark-gluon structure and determine parameters of the charged Z-resonances. Thus, they were considered as hadrocharmonium com-pounds or tightly bound diquark-antidiquark states, treated as the four-quark systems built of conventional mesons or interpreted as threshold cusps (see Refs. [12,13] and references therein).
The diquark model of the exotic four-quark resonances is one of the popular approaches to explain their properties. In accordance with this picture the tetraquark is a bound state of a diquark and an antidiquark. This approach implies the existence of multiplets of the diquarkantidiquarks with the same quark content, but different spin-parities. Because the resonances Z ± c (3900) and Z ± c (4430) are the axial-vectors, one can interpret them as the ground-state 1S and first radially excited 2S state of the same [cu] [cd] or [cd] [cu] multiplets. An idea to consider Z c (4430) as the radial excitation of the Z c (3900) state was proposed in Ref. [14], and in framework of the QCD sum rule method was explored in Refs. [15,16].
The B meson decays, where the resonances Z c (3900), Z c (4200), Z c (4430) and Z c were detected, suggest that all of them may have the same nature. Therefore, one can consider Z c as the ground-state scalar or vector tetraquark with the ccdu quark content. The recent theoretical articles devoted to the Z c resonance are concentrated mainly on exploration of its spin and possible decay channels [17][18][19][20][21]. Thus, sum rule computations carried out in Ref. [17] demonstrated that Z c is presumably a scalar particle rather than a vector tetraquark. The conclusion about a tetraquark nature of Z c with quantum numbers J P C = 0 ++ was drawn in Ref. [18] as well. In the hadrocharmonium framework the resonances Z c and Z − c (4200) were treated as the scalar η c and vector J/ψ charmonia embedded in a light-quark excitation with quantum numbers of a pion [19]. In accordance with this picture Z c and Z − c (4200) are related by the charm quark spin symmetry which suggests certain relations between their properties and decay channels. The possible decays of a scalar and a vector tetraquark [cd][cu] were analyzed also in Ref. [20].
In the present work we treat Z c as the scalar diquarkantidiquark state [cd][cu], because it was observed in the decay channel Z c → η c (1S)π − : For the scalar Z c this decay is the S-wave channel and should have a larger partial width than width of the same process which turns P -wave decay would Z c be a vector particle.
We are going to calculate the spectroscopic parameters of the tetraquark Z c , i.e., its mass and coupling by means of the two-point QCD sum rule method. It is the powerful nonperturbative approach to investigate the conventional hadrons and calculate their parameters [22,23]. But the two-point QCD sum rule method can be successfully applied for studying of the multiquark systems as well. To get reliable predictions for the quantities of concern, in the sum rule computations we take into account different quark, gluon and mixed vacuum condensates up to dimension ten.
The next problem to be considered in this work is investigating the decays of the resonance Z c and evaluating its total width. The dominant decay of this resonance seen experimentally is the channel Z c → η c (1S)π − . Nevertheless, there are other kinematically allowed decay modes of Z c that contribute to its width. In fact, in S-wave Z c can decay to the hidden-charm η c (2S)π − , J/ψρ − and open-charm D 0 D − and D * 0 D * − final states. There are also P -wave decays of the resonance Z c [20].
We are going to calculate the partial width of the processes Z c → η c (1S)π − , η c (2S)π − and Z c → J/ψρ − , and use obtained results to evaluate the total width of the tetraquark Z c . In the case of the decays Z c → η c (1S)π − and η c (2S)π − we employ the QCD threepoint correlation function to extract the sum rules for the couplings g Zcηc1π and g Zcηc2π which correspond to the strong vertices Z c η c (1S)π − and Z c η c (2S)π − , respectively. The coupling g ZcJ/ψρ , which describes the strong vertex Z c J/ψρ − , is evaluated by means of the QCD lightcone sum rule (LCSR) method and technical tools of the soft-meson approximation [24,25]. For analysis of the tetraquarks this method and approximation was adapted in Ref. [26] , and applied to study their numerous strong decay channels. Alongside the mass and coupling of the state Z c the strong couplings g Zcηc1π , g Zcηc2π and g ZcJ/ψρ provide an important information to determine the width of the decays under analysis.
This work is organized in the following manner. In Sec. II we calculate the mass m and coupling f of the scalar Z c resonance by employing the two-point sum rule method and including into analysis the quark, gluon and mixed condensates up to dimension ten. The obtained predictions for these parameters are applied in Sec. III to find the partial widths of the decays Z c → η c (1S)π − and η c (2S)π − . The decay Z c → J/ψρ − is considered in Section IV. Obtained results for the S-wave channels are employed to evaluate the total width of the resonance Z c . The Sec. V contains our analysis and concluding remarks. In the Appendix we write down explicit expressions of the heavy and light quark propagators, as well as the two-point spectral density used in the mass and coupling calculations.

II. MASS AND COUPLING OF THE SCALAR TETRAQUARK Zc
The scalar resonance Z c can be composed of the scalar diquark ǫ ijk [c T j Cγ 5 d k ] in the color antitriplet and flavor antisymmetric state and the scalar antidiquark ǫ imn [c m γ 5 Cu T n ] in the color triplet state. These diquarks are most attractive ones, and tetraquarks composed of them should be lighter and more stable than compounds formed from other types of diquarks [27]. In the present work we consider namely this favorable structure of the resonance Z c .
To calculate the mass m and coupling f of the resonance Z c using the QCD sum rule method, we start from the two-point correlation function where J(x) is the interpolating current for the tetraquark Z c . In accordance with our assumption on the structure of Z c the interpolating current J(x) has the following form Here we employ the notations ǫ = ǫ ijk andǫ = ǫ imn , where i, j, k, m and n are color indices, and C is the charge conjugation operator.
To derive the sum rules for the mass m and coupling f of the ground-state tetraquark Z c we adopt the "groundstate + continuum" approximation, and calculate the physical or phenomenological side of the sum rule. For these purposes, we insert into the correlation function a full set of relevant states and carry out in Eq. (1) the integration over x , and get Here we separate the ground-state contribution to Π Phys (p) from effects of the higher resonances and continuum states, which are denoted there by the dots. In the calculations we assume that the phenomenological side Π Phys (p) can be approximated by a single pole term.
In the case of the multiquark systems the physical side, however, receives contribution also from two-meson reducible terms [28,29]. In other words, the interpolating current J(x) interacts with the two-meson continuum, which generates the finite width Γ(p 2 ) of the tetraquark and results in the modification [30] 1 .
The two-meson continuum effects can be properly taken into account by rescaling the coupling f , whereas the mass of the tetraquark m preserves its initial value. But these effects are numerically small, therefore in the phenomenological side of the sum rule we use the zero-width single-pole approximation and check afterwards its selfconsistency. Calculation of Π Phys (p) can be finished by introducing the matrix element of the scalar tetraquark 0|J|Z c = f m.
As a result, we find Because Π Phys (p) has trivial Lorentz structure proportional to I, corresponding invariant amplitude Π Phys (p 2 ) is equal to the function given by Eq. (6).
At the next step one has to find the correlation function Π(p) using the perturbative QCD and express it through the quark propagators, and, as a result, in terms of the vacuum expectation values of various quark, gluon and mixed operators as nonperturbative effects. To this end, we use the interpolating current J(x), contract the relevant heavy and light quark operators in Eq. (1) to generate propagators, and obtain Here S c (x) and S u(d) (x) are the heavy c-and light u(d)quark propagators, respectively. These propagators contain both the perturbative and nonperturbative components: their explicit expressions are presented in the Appendix. In Eq. (7) we also utilized the shorthand notation To extract the required sum rules for m and f one must equate Π Phys (p 2 ) to the similar amplitude Π OPE (p 2 ), apply the Borel transformation to both sides of the obtained equality to suppress contributions of the higher resonances and, finally, perform the continuum subtraction in accordance with the assumption on the quarkhadron duality: These manipulations lead to the equality that can be used to get the sum rules. The second equality, which is required for these purposes, can be obtained from the first expression by applying on it by the operator d/d(−1/M 2 ). Then, for the mass of the tetraquark Z c we get the sum rule The sum rule for the coupling f reads where M 2 and s 0 are the Borel and continuum threshold parameters, respectively. In Eqs. (9) and (10) ρ OPE (s) is the two-point spectral density, which is proportional to the imaginary part of the correlation function Π OPE (p). The explicit expression of ρ OPE (s) is presented in the Appendix. We use the obtained sum rules to compute the mass m and coupling f of the tetraquark Z c . They contain numerous parameters, some of which, such as the vacuum condensates, the mass of the c-quark, are universal quantities and do not depend on the problem under discussion. In computations we utilize the following values for the quark, gluon and mixed condensates: The mass of the c-quark is taken equal to m c = 1.275 +0.025 −0.035 GeV. The Borel parameter M 2 and continuum threshold s 0 are the auxiliary parameters and should be chosen in accordance with standard constraints accepted in the sum rule computations. The Borel parameter can be varied within the limits [M 2 min , M 2 max ] which have to obey the following conditions: At M 2 max the pole contribution (PC) defined as the ratio should be larger than some fixed number. Let us note that Π(M 2 , s 0 ) in Eq. (12) is the Borel transformed and subtracted invariant amplitude Π OPE (p 2 ). In the sum rule calculations involving the tetraquarks the minimal value of PC varies between 0.15 − 0.2. In the present work we choose PC > 0.15. The minimal value of the Borel parameter M 2 min is fixed from convergence of the sum rules: in other words, at M 2 min contribution of the last term (or a sum of last few terms) to Π(M 2 , s 0 ) cannot exceed 0.05 part of the whole result The ratio R(M 2 min ) quantifies the convergence of the OPE and will be used for the numerical analysis. The last restriction on the lower limit M 2 min is the prevalence of the perturbative contribution over the nonperturbative one.
The mass m and coupling f should not depend on the parameters M 2 and s 0 . But in real calculations, these quantities are sensitive to the choice of M 2 and s 0 . Therefore, the parameters M 2 and s 0 should also be determined in such a way that to minimize the dependence of m and f on them. The analysis allows us to fix the working windows for the parameters M 2 and s 0 which obey all aforementioned restrictions. Thus, at M 2 = 6 GeV 2 the pole contribution equals to 0.19, and within the region M 2 ∈ [4, 6] GeV 2 it changes from 0.54 till 0.19. To find the lower bound of the Borel parameter from Eq. (13) we use the last three terms in the expansion, i.e. DimN = Dim(8 + 9 + 10) [we remind that Dim10 = 0].Then at M 2 = 4 GeV 2 the ratio R becomes equal to R(4 GeV 2 ) = 0.02 which ensures the convergence of the sum rules. At M 2 = 4 GeV 2 the perturbative contribution amounts to 83% of the full result exceeding considerably the nonperturbative terms. As it has been noted above, there are residual dependence of m and f on the parameters M 2 and s 0 . In Figs. 1 and 2 we plot the mass and coupling of the tetraquark Z c as functions of the parameters M 2 and s 0 . It is seen that both the m and f depend on M 2 and s 0 which generates essential part of the theoretical uncertainties inherent to the sum rule computations. For the mass m these uncertainties are small which has a simple explanation: The sum rule for the mass (9) is equal to the ratio of integrals over the functions sρ OPE (s) and ρ OPE (s), which reduces effects due to variation of M 2 and s 0 . The coupling f depends on the integral over the spectral density ρ OPE (s), and therefore its variations are sizeable. In the case under analysis, theoretical errors for m and f generated by uncertainties of various parameters including M 2 and s 0 ones equal to ±3.7% and ±21% of the corresponding central values, respectively.
Our analysis leads for the mass and coupling of the tetraquark Z c to the following results: The mass of the resonance Z c modeled as the scalar diquark-antidiquark state is in excellent agreement with the data of the LHCb Collaboration.
The S-wave decays of the resonance Z c can be divided into two subclasses: The decays to two pseudoscalar and two vector mesons, respectively. The processes Z c → η c (1S)π − and Z c → η c (2S)π − belong to the first subclass of decays. The final stages of these decays contain the ground-state and first radially excited η c mesons, therefore in the QCD sum rule approach they should investigated in a correlated form. An appropriate way to deal with decays Z c → η c (1S)π − and Z c → η c (2S)π − is the QCD three-point sum rule method. In fact, because we are going to explore the form factors g Zcηciπ (q 2 ) for the off-shell pion the double Borel transformation will be carried out in the Z c and η c channels, i.e. over momenta of these particles. This transformation applied to the phenomenological side of the relevant three-point sum rules suppresses contributions of the higher resonances in these two channels eliminating, at the same time, terms associated with the pole-continuum transitions [25,31]. The elimination of these terms is important for joint analysis of the form factors g Zcηciπ (q 2 ), because one does not need to apply an additional operator to remove them from the phenomenological side of the sum rules. Nev- ertheless, there may still exist in the pion channel terms corresponding to excited states of the pion which emerge as contaminations [for the N N π vertex, see discussions in Refs. [32,33]]. To reduce the uncertainties of the strong couplings evaluation at the vertices and smooth problems with extrapolation of the form factors to the mass-shell, it is possible to fix the pion on the mass-shell and treat one of the remaining heavy states (Z c or η c ) as the off-shell particle. This trick was used numerously to study the conventional heavy-heavy-light mesons' couplings [34,35]. Form factors obtained by treating a light or one of heavy mesons off-shell may differ from each other considerably, but after extrapolating to the corresponding mass-shells lead to the same or slightly different strong couplings. Therefore, in the present work we calculate g Zcηc1π (q 2 ) and g Zcηc2π (q 2 ) by choosing the pion off-shell and neglecting possible theoretical uncertainties of their extrapolations. The process Z c → J/ψρ − belongs to the second subclass of Z c decays: it is a decay to two vector mesons. We investigate this mode using the QCD light-cone sum rule method and soft-meson approximation. The sum rule on the light-cone allows one to find the strong coupling by avoiding extrapolating procedures and express g ZcJ/ψρ directly in terms of the ρ-meson local matrix elements rather than vacuum condensates. The unsuppressed pole-continuum effects that survive in this approach after a single Borel transformation can be eliminated by means of well-known prescriptions [31]. Additionally, the soft approximation of LCSR is technically simpler that the three-point sum rule method.
To determine the partial widths of the decays Z c → η c (1S)π − and Z c → η c (2S)π − one needs to calculate the strong couplings g Zcηc1π and g Zcηc2π which can be extracted from the three-point correlation function where 17) are the interpolating currents for the pseudoscalar mesons η c and π − , respectively. The J(x) is the interpolating current for the resonance Z c and has been introduced above in Eq. (2). In terms of the physical parameters of the tetraquark and mesons the correlation function Π(p, p ′ ) takes the form where m π is the mass of the pion, and m i = m 1 , m 2 are masses of the mesons η c (1S) and η c (2S), respectively. The four-momenta of the particles are evident from Eq. (18). Here by the dots we denote contribution of the higher resonances and continuum states.
To continue we introduce the matrix elements where f 1 and f 2 are the decay constants of the mesons η c (1S) and η c (2S), respectively. The relevant matrix element of the pion is well known and has the form where f π is the decay constant of the pion, and qq is the quark condensate. Additionally, the matrix elements corresponding to the strong vertices Z c η c (1S)π − and Z c η c (2S)π − are required. To this end, we use with g Zcηc1π and g Zcηc2π being the strong couplings; namely these couplings have to be determined from the sum rules. Employing Eqs. (19), (20) and (21) for Π Phys (p, p ′ ) we get the simple expression: The Lorentz structure of the Π Phys (p, p ′ ) is proportional to I therefore the invariant amplitude Π Phys (p 2 , p ′2 , q 2 ) is given by the sum of two terms from Eq. (22). The double Borel transformation of Π Phys (p 2 , p ′2 , q 2 ) over the variables p 2 and p ′2 with the parameters M 2 1 and M 2 2 forms one of sides in the sum rule equality. The QCD side of the sum rule, i.e. the expression of the correlation function in terms of the quark propagators reads The Borel transformation BΠ OPE (p 2 , p ′2 , q 2 ), where Π OPE (p 2 , p ′2 , q 2 ) is the invariant amplitude that corresponds to the structure ∼ I in Π OPE (p, p ′ ) constitutes the second component of the sum rule. Equating BΠ OPE (p 2 , p ′2 , q 2 ) and the double Borel transformation of Π Phys (p 2 , p ′2 , q 2 ) and performing continuum subtraction we find the sum rule for the couplings g Zcηc1π and g Zcηc2π . The Borel transformed and subtracted amplitude Π OPE (p 2 , p ′2 , q 2 ) can be expressed in terms of the spectral density ρ D (s, s ′ , q 2 ) which is proportional to the imaginary part of Π OPE (p, p ′ ) where M 2 = (M 2 1 , M 2 2 ) and s 0 = (s 0 , s ′ 0 ) are the Borel and continuum threshold parameters, respectively.
The obtained sum rule has to be used to determine the couplings g Zcηc1π and g Zcηc2π . A possible way to find them is to get the second sum rule from the first one by applying the operators d/d(−1/M 2 1 ) and/or d/d(−1/M 2 2 ). But in the present work we choose the alternative approach and use iteratively the master sum rule to extract both g Zcηc1π and g Zcηc2π . To this end, we fix the continuum threshold parameter s ′ 0 which corresponds to the η c channel just below the mass of the first radially excited state η c (2S). By this manipulation we include η c (2S) into the continuum and obtain the sum rule for the strong coupling of the ground-state meson η c (1S). The physical side of the sum rule (22) at this stage contains only the ground-state term and depends on the coupling g Zcηc1π . This sum rule can be easily solved to evaluate the unknown parameter g Zcηc1π . At the next step we fix the continuum threshold s ′ 0 at m 2 + (0.5 − 0.8) GeV and use the sum rule that now contains the ground and first radially excited states. The QCD side of this sum rule is given by the expression . By substituting the obtained expression for g Zcηc1π into this sum rule it is not difficult to evaluate the second coupling g Zcηc2π .
The couplings depend on the Borel and continuum threshold parameters and, at the same time, are functions of q 2 . In what follows we omit their dependence on the parameters, replace q 2 = −Q 2 and denote the obtained couplings as g Zcηc1π (Q 2 ) and g Zcηc2π (Q 2 ). For calculation of the decay width we need value of the strong couplings at the pion's mass-shell, i.e. at q 2 = m 2 π , which is not accessible for the sum rule calculations. The standard way to avoid this problem is to introduce a fit functions F 1(2) (Q 2 ) that for the momenta Q 2 > 0 leads to the same predictions as the sum rules, but can be readily extrapolated to the region of Q 2 < 0. Let us emphasize that values of the fit functions at the mass-shell are the strong couplings g Zcηc1π and g Zcηc2π to be utilized in calculations.
Expressions for g Zcηc1π (Q 2 ) and g Zcηc2π (Q 2 ) depend on various constants, such as the masses and decay constants of the final-state mesons. The values of these parameters are collected in Table I. Additionally, there are parameters M 2 and s 0 which should also be fixed to carry out numerical analysis. The requirements imposed on these auxiliary parameters have been discussed above and are standard for all sum rule computations. The regions for M 2 1 and s 0 which correspond to the tetraquark Z c coincide with the working windows for these parameters fixed in the mass calculations whereas in the sum rule for the second coupling g Zcηc2π (Q 2 ) we employ  As it has been emphasized above to evaluate the strong couplings at the mass shell Q 2 = −m 2 π we need to determine the fit functions. To this end, we employ the following functions where F i 0 , c 1i and c 2i are fitting parameters. The performed analysis allows us to find the parameters as The widths of the decays Z c → η c (1S)π − and Z c → η c (2S)π − can be found by means of the formula where λ (a, b, c) = 1 2a Using the strong couplings given by Eq. (29) and Eq. (30) it is not difficult to evaluate the partial widths of the decay channels which are main results of this section.
IV. DECAY Zc → J/ψρ − The scalar tetraquark Z c in S-wave can also decay to the final state J/ψρ − . In the QCD light-cone sum rule approach this decay can be explored through the correlation function is the interpolating current for the vector meson J/ψ. The correlation function Π Phys µ (p, q) in terms of the physical parameters of the tetraquark Z c , and of the mesons J/ψ and ρ has the following form where m J/ψ is the mass of the meson J/ψ. In Eq. (34) by the dots we denote contribution of the higher resonances and continuum states. Here p ′ = p + q is the momentum of the tetraquark Z c , where p and q are the momenta of the J/ψ and ρ mesons, respectively. Further simplification of Π Phys µ (p, q) can be achieved by utilizing explicit expressions of the matrix elements 0|J J/ψ µ |J/ψ (p) , Z c (p ′ )|J † |0 , and of the vertex Z c (p ′ )J/ψ (p) ρ(q). The matrix element of the tetraquark Z c is given by Eq. (5), whereas for the meson J/ψ (p) we can use where f J/ψ and ε µ are its decay constant and polarization vector, respectively. We also model the three-state vertex as with ε ′ being the polarization vector of the ρ-meson. Then Π Phys µ (p, q) takes the form: It contains different Lorentz structures ∼ ε ′ µ and q µ . One of them should be chosen to fix the invariant amplitude and carry out sum rule analysis. We choose the structure ∼ ε ′ µ and denote the corresponding invariant amplitude as Π Phys (p 2 , q 2 ).
The second component of the sum rule is the correlation function Π µ (p, q) computed using quark propagators. For Π OPE µ (p, q) we obtain where α and β are the spinor indexes. The expression for Π OPE µ (p, q) can be written down in a more detailed form. For these purposes, we first expand the local operator d a α u d β by means of the formula with Γ j being the full set of Dirac matrixes Applying the projector onto a color-singlet state δ ad /3 we get where dΓ j u are the color-singlet local operators. Substituting the last expression into Eq. (38) we see that the correlation function Π OPE µ (p, q) depends on the ρ-meson's two-particle local matrix elements. Some of them does not depend on the ρ-meson momentum, whereas others contain momentum factor There are also three-particle matrix elements that contribute to the correlation function Π OPE µ (p, q). They appear due to insertion of gluon field strength tensor G from the c-quark propagators into the local operators dΓ j u. The ρ-meson three-particle local matrix element is a q free quantity. But other matrix elements depend on the ρ-meson momentum As a result, the correlation function contains only local matrix elements of the ρ-meson and depends on the momenta p and q. This is general feature of QCD sum rules on the light-cone with a tetraquark and two conventional mesons. Indeed, because a tetraquark contains four quarks, after contracting two quark fields from its interpolating current with relevant quarks from the interpolating current of a meson one gets a local operator sandwiched between the vacuum and a second meson.
The variety of such local operators give rise to different local matrix elements of the meson rather that to its distribution amplitudes. As a result, the four-momentum conservation in the tetraquark-meson-meson vertex requires setting q = 0 ( for details, see Ref. [26]). In the standard light-cone sum rule method the choice q = 0 is known as the soft-meson approximation [25]. It is worth to emphasize that at vertices composed of conventional mesons in general q = 0, and only in the softmeson approximation one equates q to zero, whereas the tetraquark-meson-meson vertex can be analyzed in the context of the LCSR method only if q = 0. An important observation made in Ref. [25] is that the soft-meson approximation and full LCSR treatment of the conventional mesons' vertices lead to results which numerically are very close to each other. After substituting all aforementioned matrix elements into the expression of the correlation function and performing the summation over color indices we fix the local matrix elements of the ρ meson that survive in the soft limit. It turns out that in the q → 0 limit only the matrix elements (41) and (43) contribute to the invariant amplitude Π OPE (p 2 ) [i.e. to Π OPE (p 2 , 0)]. These matrix elements depend on the mass and decay constant of the ρ-meson m ρ , f ρ , and on ζ 4ρ which normalizes the twist-4 matrix element of the ρ-meson [36]. The parameter ζ 4ρ was evaluated in the context of QCD sum rule approach at the renormalization scale µ = 1 GeV in Ref. [37] and is equal to ζ 4ρ = 0.07 ± 0.03.
The Borel transform of the invariant amplitude Π OPE (p 2 ) is given by the expression where ρ OPE (s) is the corresponding spectral density. In the present work we calculate ρ OPE (s) by taking into account contribution of the condensates up to dimension six. The spectral density has both the perturbative and nonperturbative components After some computations for ρ pert. (s) we get The nonperturbative part of the spectral density ρ n.−pert. (s) contains terms proportional to the gluon condensates α s G 2 /π , α s G 2 /π 2 and g 3 s G 3 : Here we do not provide their explicit expressions. The twist-4 con-tribution to Π OPE (M 2 ) reads a(1 − a) .
(48) To derive the expression for the strong coupling g ZcJ/ψρ the soft-meson approximation should be applied to the phenomenological side of the sum rule as well. Because in the soft limit p 2 = p ′2 , we have to perform the Borel transformation of Π Phys (p 2 , 0) over the variable p 2 and carry out calculations with one parameter M 2 . To this end, we first transform Π Phys (p 2 , 0) in accordance with the prescription where m 2 = m 2 + m 2 J/ψ /2, and instead of two terms with different poles get the double pole term. By equating the physical and QCD sides and performing required manipulations we get The equality given by Eq. (50) is the master expression which can be used to extract sum rule for the coupling g ZcJ/ψρ . It contain the term corresponding to the decay of the ground-state tetraquark Z c and conventional contributions of higher resonances and continuum states suppressed due to the Borel transformation; the latter is denoted in Eq. (50) by the dots. But in the soft limit there are also terms ∼ A in the physical side which remain unsuppressed even after the Borel transformation. They describe transition from the excited states of the tetraquark Z c to the mesons J/ψρ − . Of course, to obtain the final formula all contributions appearing as the contamination should be removed from the physical side of the sum rule. The situation with the ordinary suppressed terms is clear: they can be subtracted from the correlation function Π OPE (M 2 ) using assumption on the quark-hadron duality. As a result the correlation function acquires a dependence on the continuum threshold parameter s 0 , i.e., becomes equal to Π OPE (M 2 , s 0 ). The treatment of the terms ∼ A requires some additional manipulations; they can be removed by applying the operator [31] to both sides of Eq. (50). Then the sum rule for the strong coupling reads: The width of the decay Z c → J/ψρ − can be calculated using the formula In the sum rule (52) for M 2 and s 0 we use the working regions given by Eq. (14). For the strong coupling g ZcJ/ψρ we get g ZcJ/ψρ = (0.56 ± 0.07) GeV −1 . (54) Then for the width of the decay Z c → J/ψρ − we find In accordance with our investigation, the total width of the resonance Z c saturated by the decays Z c → η c (1S)π − , η c (2S)π − and Z c → J/ψρ − is equal to This is the second parameter of the resonance Z c to be compared with the LHCb data. It is worth noting that the experimental information about the total width of Z c is not precise and suffers from large uncertainties. The theoretical prediction for Γ is smaller than the central value for the total width of Z c measured by the LHCb Collaboration. Nevertheless, our result is compatible with existing data Γ = 152 +83 −68 MeV, and this agreement may be improved by including into analysis other decay modes of the resonance Z c .

V. ANALYSIS AND CONCLUDING REMARKS
In the present work we have performed quantitative analysis of the newly observed resonance Z c by calculating its spectroscopic parameters and total width. In computations we have used different QCD sum rule approaches. Thus, the mass and coupling of Z c have been evaluated by means of the two-point sum rule method, whereas its decay channels have been analyzed using the three-point and light-cone sum rules technique.
We have calculated the spectroscopic parameters of the tetraquark Z c using the zero-width single-pole approximation. But the interpolating current (2) couples also to the two-meson continuum η c (1S)π − , η c (2S)π − , J/ψρ − , D 0 D − and D * 0 D * − which can modify the results for m and f obtained in the present work. In is known that effects of the two-meson continuum changes the zero-width approximation (4) and lead to the following corrections [30] and where λ = mf and M =m D * 0 + m D * − . In Eqs. (57) and (58) we have introduced the weight function where Utilizing the central values of the m and Γ, as well as M 2 = 5 GeV 2 and s 0 = 20 GeV 2 , it is not difficult to find that and As is seen the two-meson effects result in rescaling f → 0.95f which changes it approximately by 5% relative to its central value. These effects are smaller than theoretical errors of the sum rule computations themselves. We have calculated the partial width of the decay channels Z c → η c (1S)π − , η c (2S)π − and J/ψρ − using the different QCD sum rule methods. Thus the pair of decays Z c → η c (1S)π − and Z c → η c (2S)π − has been studied in the context of the QCD three-point sum rule approach. The process Z c → J/ψρ − has been investigated using the LCSR method and soft-meson approximation. Partial widths of these S-wave decay channels have been employed to evaluate the total width of the resonance Z c .
The theoretical prediction for the mass of the scalar diquark-antidiquark state Z c is equal to m = (4080 ± 150) MeV. This result is in excellent agreement with measurements of the LHCb Collaboration. Our prediction for its width Γ = (128 ± 19) MeV is lower than the experimental datum, but remaining decay modes of Z c properly taken into account may improve an agreement with the LHCb data. One should also note large errors from which suffers the information on total width of the resonance Z c : More precise measurements are required to make reliable conclusions. At the present stage of studies we can state that the resonance Z c is presumably the scalar diquark-antidiquark with ccdu content and Cγ 5 ⊗ γ 5 C structure.
Appendix: The quark propagators and two-point spectral density ρ OPE (s) The light and heavy quark propagators are necessary to find QCD side of the different correlation functions.