$D_{s0}(2590)$ as a dominant $c\bar{s}$ state with a small $D^*K$ component

The recently discovered $D_{s0}(2590)$ state by the LHCb collaboration was regarded as the first excited state of $^1S_{0}$ charmed-strange meson. Its mass is, however, lower than the Godfrey-Isgur quark model prediction by about 80 MeV. In this work, we take into account the $D^{\ast}K$ contribution to the bare $c\bar{s}$ state, and show that the coupled-channel interaction induces an 88 MeV shift with respect to the conventional quark model $c\bar{s}$ state, which is much closer to the experimental mass. Our study shows that in addition to $S$-wave, $P$-wave coupled-channel interactions also play a role for hadrons located close to two-hadron thresholds. We further scrutinize the unquenched quark model results with a model independent approach. It is shown that the two-body $D^*K$ decay width is proportional to the weight of the $D^*K$ component. To saturate the experimental total decay width with the $D^*K$ partial decay width we need a weight of about 60\% while to reproduce the unquenched quark model result a weight of about 5\% is needed. Therefore, we encourage future experimental studies on the two-body $D^*K$ partial decay of $D_{s0}(2590)$.

and then further developed by A. Le Yaouanc. et al. [20,21]. It has been widely used to investigate the Okubo-Zweig-Iizuka (OZI)-allowed two-body strong decays of conventional hadrons [22][23][24]. The quarkpair creation model also provides an approach to construct the interaction between the initial state and the subsequent two-body strong decay channel. With the transition amplitude provided by the 3 P 0 model and conventional quark model, an unquenched quark model can be constructed, which takes into account the coupled-channel effects. In particular, the unquenched quark model can give a quantitative estimate of the ratio between the hadronic molecule components and the bare state. For instance, in the unquenched quark model, X(3872) was well described as a mixture of about 70% χ c1 (2P) and 30% D * D [16]. With a similar approach, Luo, et al. solved the low mass puzzle of Λ c (2940) between the naive quark model prediction and the experimental mass by considering the contribution of the D * N channel, which strongly couples to the Λ c (2P, 3/2 − ) bare state [17]. In Ref. [18], the authors considered the DK contribution to D * s0 (2317), leading to a 77 MeV mass shift with respect to the conventional quark model prediction. Very recently, Yang et al. used the Hamiltonian effective field theory to study the mass spectrum of positive parity D s resonant states considering both the P-wave cs core and the DK/D * K coupled channels and found that the D * s0 (2317) state contains about 60% DK and 40% cs [25].
A common feature of the above mentioned exotic states is that the orbital angular momentum between the two quark components is P-wave. The P-wave excitation is equal to the creation of a pair of quark and antiquark in vacuum, which can form two hadrons with its original quark components through the quark rearrangement mechanism. If the mass threshold of these two hadrons is close to the mass of the bare state, it would couple to the bare state and lower its mass, leading to a smaller mass for the physical state in agreement with the experimental measurement.
Recently, the LHCb Collaboration observed a new excited D + s state in the D + K + π − mass distribution of the B 0 → D − D + K + π − decay using a data sample corresponding to an integrated luminosity of 5.4 fb −1 at a centre-of-mass energy of 13 TeV [26]. Its mass, width, and spin-parity are determined to be m R = 2591 ± 6 ± 7 MeV, Γ R = 89 ± 16 ± 12 MeV, and J P = 0 − , respectively. This state was considered to be a candidate for the D s (2 1 S 0 ) state, the radial excitation of the ground state D s meson. Nonetheless, the relativized quark model predicted that such a state should have a mass of 2673 MeV [24], which is larger than the experimental mass by 82 MeV. One should note that there exists only one channel, D * K, that could couple to the D s (2 1 S 0 ) state in terms of the quark-pair creation mechanism. In this work, we employ the unquenched quark model to investigate whether by considering the D * K contribution one can reconcile the experimental mass of D s0 (2590) with that of the theoretical D s (2 1 S 0 ) state. It should be noted that different from the cases of X(3872), D * s0 (2317), and Λ c (2940), the D * K channel couples to the cs bare state via P-wave.
In Ref. [27], it was shown that the ∆(1232) state contains a substantial πN component, about 60%, using an extension of the Weinberg compositeness condition on partial waves of = 1. We will adopt the same approach to estimate the weight of the D * K component in the D s0 (2590) state and compare it with the results of the unquenched quark model. The paper is structured as follows. In Sec. II we present the details of the unquenched quark model as well as a brief description of the model-independent approach. In Sec. III we study the mass shift of the bare D s (2 1 S 0 ) state induced by the coupling to D * K in the unquenched quark model and calculate its partial decay width into D * K. Furthermore, we adopt the model independent approach to evaluate the weight of D * K in the physical state D s0 (2590). Finally we present the conclusions in Sec. IV.

II. THEORETICAL FORMALISM
In this work, we adopt two different theoretical models to evaluate the relevance of the D * K coupled channel in the physical D s0 (2590) state. The first is an unquenched quark model in the line of Ref. [28] and the second is a model-independent approach in the line of Ref. [27]. In the following, we briefly describe these two methods. More details can be found in Refs. [27,28].

A. Unquenched quark model
We first introduce the Hamiltonian in the framework of the unquenched quark model, which contains three terms where H cs comes from the conventional quark model [29], H D * K denotes the interaction between D * and K, and H I stands for the interaction between the bare cs state and the D * K channel. With the above Hamiltonian the wave function of the physical D s0 (2590) state can be written as which indicates that there are two Fock components, a cs core at quark level and a D * K component at hadron level. In the following, we specify each of the two terms. The Hamiltonian H cs is taken from the Godfrey-Isgur relativized potential quark model, where M 0 is the bare mass. The term H D * K denotes the Hamiltonian of the D * K system. As we only consider the kinetic energy but neglect the interaction between D * and K 1 , it can be written as where p represents the 3-momentum of D * or K meson in the center-of-mass frame of D * K.
In the quark-pair creation model (of which the details are relegated to Appendix A), the transition operator of cs(2 1 S 0 ) → D * K is written as where ω 0 , φ 0 , and χ 1,−m are the SU (3) constrains the momentum of the qq pair, in agreement with the quark-pair creation in vacuum. Accordingly b † q and d † q are quark and antiquark creation operators, respectively. Thus the Hamiltonian H I between D * K and the cs core can be expressed as The resulting eigenvalue equation has the following form where we assume that the two Fock states, Ψ(cs) and Ψ p (D * K), are orthogonal to each other and properly normalized respectively. The non-diagonal term is the transition amplitude The above matrix equation can be simplified to two algebraic equations From Eq. (9) we can derive the following relation where the mass shift ∆M(M) is defined as From Eq. (10) and Eq. (11) the physical mass M and the mass shift ∆M can be determined simultaneously.
The coupled-channel correction to the bare state is the mass shift ∆M.
In order to estimate the cs core contribution to the physical state D s0 (2590), we need to calculate the Z-factor, i.e., the field renormalization constant [31], defined as where Ψ(cs) represents the genuine cs component of the state. If we ignore the narrow decay width Γ compared with its mass m R = 2591±6±7 MeV of the D s0 (2590) state, we can write down the normalization With two Fock components in D s0 (2590) (see Eq. (2)) the normalization condition becomes From Eq. (9), we can obtain the relation between the wave function of the cs core and that of D * K(p) With this, Eq. (14) becomes Finally we obtain the Z-factor A few remarks are in order. First, the integral in the above equation is only well defined for M < m D * + m K .
However, in the original 3 P 0 model, the physical state is supposed to decay strongly into the two-body final state. Therefore, there seems to be an internal inconsistency in the unquenched quark model specified above. The solution is quite straightforward. One should replace the real M with its complex counterpart M + iΓ/2. Then Eq. (17) becomes For self-consistency, one can also replace M with M + iΓ/2 in the definition of the mass shift in Eq. (11), which would just cause tiny changes to ∆M since Γ M and then seems unnecessary. Note that Γ here only represents the two-body strong decay width into D * K rather than the total experimental decay width Γ R = 89 ± 16 ± 12 MeV.

B. Model independent approach
One can also study the contributions of different Fock components to a physical state in a model independent way. In Refs. [27,32], it was shown that the πN component in the ∆(1232) state is substantial while the ππ component in the ρ wave function is small. In the following, we adopt such a method to estimate the relative weights of the cs core and D * K coupled channel in the physical D s0 (2590). We first briefly introduce the essential ingredients of this approach.
The starting point is to parameterize the D * K potential. Close to threshold, it has the following form where s is the square of the center-of-mass energy, and M 0 is the mass of the bare cs core. α and β are two unknown parameters that should be determined by fitting to experimental data. In Refs. [27,32], for the cases of the ∆(1232) and ρ, such unknown parameters are fixed by fitting to the πN and ππ scattering data.
The above potential can then be inserted into the Lippmann-Schwinger equation to obtain the transition amplitudes where G(s) is the loop function of D * and K mesons, which has the following form to account for the P-wave nature of the D * K interaction where ω D * = m 2 D * + |p| 2 , ω K = m 2 K + |p| 2 , and Λ is the cutoff needed to regularize the integral.
Due to the fact that the physical D s0 (2590) state is above the mass threshold of D * K, we need the loop function in the second Riemann sheet, which is defined as where G I (s) is given in Eq. (21).

III. RESULTS AND DISCUSSION
For the mass M 0 of the bare cs state, we adopt the result of the Godfrey-Isgur relativized potential quark model [29] M 0 cs(2 1 S 0 ) = 2673 MeV, as well as the same set of parameters for the constituent quark masses: The masses of D * and K are taken from the PDG [33]: For the effective simple harmonic oscillator parameter β eff of the cs core and D * , we choose [23] β eff cs(2 1 S 0 ) = 0.475 GeV, β eff (D * ) = 0.516 GeV.
Instead of taking β eff to be 0.4 GeV for all light mesons as in Ref. [23], we choose the value determined in Ref. [34] for K, which obeys the uniform standards 2 for all involved mesons. Finally, for the dimensionless vacuum uū/dd quark-pair creation strength constant γ, we choose [35] γ = 8.7.
With the parameters specified above we can straightforwardly obtain the relation between the mass shift ∆M and the physical mass M, which is shown in Fig.1. One can easily read that the physical mass of The Z-factor in the model independent approach can be obtained in the following way − Re where the couplings is calculated as the residue at the pole position The results are given in Table I, One can see that as the cutoff Λ increases from 400 to 800 MeV, the twobody decay width decreases from 80 MeV to 20 MeV and the weight of the D * K component decreases from 0.58 to 0.05. As we have three unknown parameters but only two data, we cannot tell which cutoff is the optimal one. If we assume that the two-body D * K decay width almost saturates the D s0 (2590)  the current experimental data cannot determine the ratio of the two-body decay width with respect to the total decay width [26]. We hope that future experimental studies can provide such information.

IV. SUMMARY
Recently, the LHCb Collaboration reported the discovery of the first radial excited state of D s . However, its mass is lower than the quenched quark model prediction by about 80 MeV, which shows a behavior similar to those of exotic states such as, X(3872), D * s0 (2317), and Λ c (2940). All of these states have been shown to couple strongly to the nearby hadronic channels, leading to smaller masses compared with those of the quenched quark model. In this work, we took into account the D * K contribution to the 2 1 S 0 cs core to obtain the physical mass of the first excited state of D s in the unquenched quark model, where the orbital angular momentum of D * and K is P-wave. The coupling of the cs core to D * K is estimated by the quarkpair creation model. Our results showed that with only about 10% of D * K in the D s0 (2590) wave function, the D * K contribution could lower the mass obtained in the quenched quark model by 88 MeV, leading to a mass much closer to the experimental value. The two-body decay width predicted in the same model is about 20 MeV, which only accounts for about one quarter of the total decay width.
We further constructed a model independent approach to test the unquenched quark model and we found that indeed with a cutoff of about 800 MeV, one can obtain a two-body decay width and a weight of the D * K component consistent with those of the unquenched quark model. On the other hand, with a cutoff of 400 MeV, one found that the D * K partial decay width almost saturates the D s0 (2590) decay width and the corresponding weight of the D * K component is about 60%. Future experimental studies will allow us to fix the unknown parameters of our model and determine the weight of the D * K component unambiguously. In particular, a measurement of the partial two-body decay width of D s0 (2590) seems to be the key. is the associated Laguerre polynomial and the parameter β eff is unknown, which can be determined by the requirement that the root-mean-square (rms) momentum calculated through SHO wave functions should be equal to that of the wave functions calculated using the Godfrey-Isgur relativized potential quark model [29].
The color and flavor overlap factors can be readily obtained by the inner product of the corresponding wave functions, of which the details can be found in Appendix of Ref. [23]. Spin matrix elements involving the spin of four quarks can be calculated by the angular momentum algebra of Wigner 9 j symbols χ (14) s B m s B χ (32) s C m s C , χ (12) s A m s A χ (34) 1,−m =(−1) 1+s C 3(2s A + 1)(2s B + 1)(2s C + 1) × sm s C s B s C (sm s ; m s B m s C )C s A 1 (sm s ; m s A − m) Finally, for convenience we use the Jacob-Wick formula [39] to convert the helicity amplitude M m j A m j B m j C (p) into the partial wave amplitude M s (|p|) M s (|p|) = √ 4π(2 + 1) 2 j A + 1 m j B m j C C s ( j A m j A ; 0m j A )C j B j C (sm j A ; m j B m j C )M m j A m j B m j C (|p|ẑ)| m j A =m j B +m j C . (A8) Note that we have implicitly assumed that the z-axis lies along the direction of the outgoing 3-momentum p of meson B in the final state. For our concrete process cs(2 1 S 0 ) → D * K(p), we can relate M cs(2 1 S 0 )→D * K(p) (|p|) with the partial wave amplitude M s (|p|) based on the conservation of angular momentum and selection rules for the strong interaction M cs(2 1 S 0 )→D * K(p) (|p|) = M 11 (|p|). (A9)