$$D_{s3}^*(2860)$$Ds3∗(2860) and $$D_{s1}^*(2860)$$Ds1∗(2860) as the 1D $$c\bar{s}$$cs¯ states

In this article, we take the D∗ s3(2860) and D ∗ s1(2860) as the 1 3D3 and 1 3D1 cs̄ states, respectively, study their strong decays with the heavy meson effective theory by including the chiral symmetry breaking corrections. We can reproduce the experimental data Br (D∗ sJ (2860) → D ∗K) /Br (D∗ sJ (2860) → DK) = 1.10 ± 0.15 ± 0.19 with suitable hadronic coupling constants, the assignment of the D∗ sJ (2860) as the D ∗ s3(2860) is favored, the chiral symmetry breaking corrections are large. Furthermore, we obtain the analytical expressions of the decay widths, which can be confronted with the experimental data in the future to fit the unknown coupling constants. The predictions of the ratios among the decay widths can be used to study the decay properties of the D∗ s3(2860) and D ∗ s1(2860) so as to identify them unambiguously. On the other hand, if the chiral symmetry breaking corrections are small, the large ratio R = 1.10 ± 0.15 ± 0.19 requires that the D∗ sJ (2860) consists of at least four resonances D∗ s1(2860), D ∗ s2(2860), D ∗′ s2(2860), D ∗ s3(2860). PACS numbers: 13.25.Ft; 14.40.Lb


Introduction
In 2006, the BaBar collaboration observed the D * sJ (2860) meson with mass (2856.6±1.5±5.0) MeV and width (48 ± 7 ± 10) MeV in decays to the final states D 0 K + and D + K 0 S [1]. There have been several possible assignments. Beveren and Rupp assign the D * sJ (2860) to be the first radial excitation of the D * s0 (2317) based on a coupled-channel model [2]. Colangelo, Fazio and Nicotri assign the D * sJ (2860) to be the 1 3 D 3 cs state using the heavy meson effective theory [3]. Close et al assign the D * sJ (2860) to be the 2 3 P 0 state in a constituent quark model with novel spin-dependent interactions [4]. Zhang et al assign the D * sJ (2860) to be the 2 3 P 0 or 1 3 D 3 state based on the 3 P 0 model [5]; Li, Ma and Liu share the same interpretation based on the Regge phenomenology [6]. However, Ebert, Faustov and Galkin observe that the D * sJ (2860) does not fit well to the Regge trajectory D * s (2112), D * s2 (2573), D * sJ (2860), · · · [7]. Later, Li and Ma assign the D * s1 (2700) to be the 1 3 D 1 − 2 3 S 1 mixing state and the D * sJ (2860) to be its orthogonal partner, or the D * sJ (2860) to be the 1 3 D 3 state based on the 3 P 0 model [8]. Zhong and Zhao assign the D * sJ (2860) to be the 1 3 D 3 state with some 1 3 D 2 − 1 1 D 2 mixing component using the chiral quark model, i.e. they assume that the D * sJ (2860) arises from two overlapping resonances [9,10]. Vijande, Valcarce and Fernandez assign the D * sJ (2860) to be the cs − cnsn mixing state [11]. Chen, Wang and Zhang assign the D * sJ (2860) to be the 1 3 D 3 state based on a semi-classic flux tube model [12]. Badalian and Bakker assign the D * sJ (2860) to be the 1 3 D 3 state based on the QCD string model [13]. Guo and Meissner take the D * sJ (2860) as the dynamically generated D 1 (2420)K bound state [14]. In 2009, the BaBar collaboration confirmed the D * sJ (2860) in the D * K channel, and measured the ratio R among the branching fractions [15], The observation of the decays D * sJ (2860) → D * K rules out the J P = 0 + assignment [2,4,5,6]. On the other hand, if we take the D * sJ (2860) as the 1 3 D 3 state, Colangelo, Fazio and Nicotri obtain the value R = 0.39 based on the heavy meson effective theory [3], while in the 3 P 0 model, Zhang et al obtain the value 0.59 [5], Li  0.55 ∼ 0.80 [16]. Recently, Godfrey and Jardine obtain the value 0.43 based on the relativized quark model and the pseudoscalar emission decay model [17]. The theoretical values differ from the experimental value greatly.
Recently, the LHCb collaboration observed a structure at 2.86 GeV with significance of more than 10 standard deviations in the D 0 K − mass spectrum in the Dalitz plot analysis of the decays B 0 s → D 0 K − π + , the structure contains both spin-1 and spin-3 components (i.e. the D * − s1 (2860) and the D * − s3 (2860), respectively), which supports an interpretation of these states being the J P = 1 − and 3 − members of the 1D family [18,19]. The measured masses and widths are M D * s3 = (2860.5 ± 2.6 ± 2.5 ± 6.0) MeV, M D * s1 = (2859 ± 12 ± 6 ± 23) MeV, Γ D * s3 = (53 ± 7 ± 4 ± 6) MeV, and Γ D * s1 = (159 ± 23 ± 27 ± 72) MeV, respectively. Furthermore, the LHCb collaboration obtained the conclusion that the D * sJ (2860) observed by the BaBar collaboration in the inclusive e + e − → D 0 K − X production and by the LHCb collaboration in the pp → D 0 K − X processes consists of at least these two resonances [15,20]. According to the predictions of the potential models [7,21,22], see Table 1, the masses of the 1D cs states is about 2.9 GeV. It is reasonable to assign the D * s1 (2860) and D * s3 (2860) to be the 1 3 D 1 and 1 3 D 3 cs states, respectively [18,19]. However, the theoretical values R differ from the experimental value greatly in the case of the D * s3 (2860) or the 1 3 D 3 assignment of the D * sJ (2860). In Ref. [3], Colangelo, Fazio and Nicotri take the leading order heavy meson effective Lagrangian. The two-body strong decays D * s3 (2860) → D * K, DK take place through the relative F-wave, the final K mesons have the three momenta p K = 584 MeV and 705 MeV, respectively. The decay widths where p 7 K = 2.3 × 10 19 MeV 7 and 8.6 × 10 19 MeV 7 in the decays to D * K and DK, respectively. Small difference in p K can lead to large difference in p 7 K , so we have to take into account the heavy quark symmetry breaking corrections and chiral symmetry breaking corrections so as to make robust predictions.
In this article, we take into account the chiral symmetry-breaking corrections, and study the two-body strong decays of the D * s1 (2860) and D * s3 (2860) with the heavy meson effective Lagrangian, and try to reproduce the experimental value R = 1.10 ± 0.15 ± 0.19 by assigning the D * sJ (2860) to be the D * s1 (2860) and the D * s3 (2860), respectively. Recently, Wu and Huang study the strong decays of the D * s0 (2317) and D ′ s1 (2460) by including the chiral symmetry breaking corrections [23]. The heavy meson effective theory have been applied to identify the charmed mesons and bottom mesons [3,24,25,26,27], and to calculate the radiative, vector-meson, two-pion decays of the heavy quarkonium states [28].
The article is arranged as follows: we derive the strong decay widths of the charmed mesons D * s1 (2860) and D * s3 (2860) with the heavy meson effective theory in Sect.2; in Sect.3, we present the numerical results and discussions; and Sect.4 is reserved for our conclusions.

The strong decays with the heavy meson effective theory
In the heavy quark limit, the heavy-light mesons Qq can be classified in doublets according to the total angular momentum of the light antiquark s ℓ , s ℓ = sq + L, where the sq and L are the spin and orbital angular momentum of the light antiquark, respectively [29]. In this article, the revelent doublets are the L = 0 (S-wave) doublet (P, P * ) with J P s ℓ = (0 − , 1 − ) 1 2 , and the L = 2 (D-wave) doublets (P * 1 , P 2 ) and (P 2 , P * respectively. In the heavy meson effective theory, those doublets can be described by the effective super-fields H a , X a and Y a , respectively [30], where the heavy meson fields P ( * ) contain a factor M P ( * ) and have dimension of mass 3 2 . The super-fields H a contain the S-wave mesons (P, P * ); X a , Y a contain the D-wave mesons (P * 1 , P 2 ), (P 2 , P * 3 ), respectively. The light pseudoscalar mesons are described by the fields ξ = e iM fπ , where and the decay constant f π = 130 MeV. At the leading order approximation, the heavy meson chiral Lagrangians L X and L Y for the strong decays to the light pseudoscalar mesons can be written as: where the hadronic coupling constants g X , g Y andg Y are parameters and can be fitted to the experimental data [31,32], Λ is the chiral symmetry breaking scale and chosen as Λ = 1 GeV [27].
We construct the chiral symmetry breaking Lagrangians L χ X and L χ Y accordingly to Ref. [33]. where m q = diag(m u , m d , m s ), m ξ q = ξm q ξ + ξ † m q ξ † , v µ = (1, 0, 0, 0), the hadronic coupling constants k j X/Y ,k j Y ,k 5 X ,k 5 X with j = 1, 5 can be fitted to the experimental data. The flavor and spin violation corrections of the order O(1/m Q ) are neglected, as there are too many unknown couplings to be determined, we expect that the corrections are not as large as the chiral symmetry breaking corrections.
From the heavy meson chiral Lagrangians L X , L Y , L χ X and L χ Y , we can obtain the decay widths Γ of the strong decays to the light pseudoscalar mesons, where the i (or b) and f (or a) denote the initial and final state heavy mesons, respectively. where

Numerical Results and Discussions
The We redefine the hadronic coupling constantsḡ Y = g Y +g Y ,k j Y = k j Y +k j Y /ḡ Y , j = 1 − 5, and write down the decay widths of the D * s3 (2860) explicitly from Eqs. (13)(14).
We define the ratios R 0+ , R +0 , R s among the decay widths, The ratios R 0+ , R +0 , R s are independent on the hadronic coupling constantsḡ Firstly, let us assign the D * sJ (2860) to be the D * s3 (2860), then we can obtain the value, by setting the R0++R+0 2 to be the experimental data, R0++R+0 2 = R = 1.10 ± 0.15 ± 0.19 [15]. On the other hand, if we retain only the leading order coupling constantḡ Y , then R0++R+0 2 = 0.3866, which is consistent with the value 0.39 obtained by Colangelo, Fazio and Nicotri [3]. The value of the hadronic coupling constant k 5 Y +k 5 Y in the chiral symmetry breaking Lagrangian is about 1 3 of that of the hadronic coupling constant g Y +g Y in the leading-order Lagrangian according to the relationk 5 Y = (k 5 Y +k 5 Y )/(g Y +g Y ). However, taking into account such chiral symmetry breaking term can enlarge the ratio R about 2.8 times.
Then we write down the prediction of the ratio R s , the value 0.18 in the bracket comes from the leading order heavy meson effective Lagrangian L Y , i.e. only theḡ Y is retained. The chiral symmetry breaking corrections are rather large, the present predictions can be confronted with the experimental data in the futures to study the chiral symmetry breaking corrections. We can also define the ratios R 0 + and R 0 + , which are sensitive to the chiral symmetry breaking corrections associate withk 1 Y . We can estimate thek 1 Y by confronting the ratios R 0 + and R 0 + with the experimental data in the future. Now we assign the D * sJ (2860) to be the D * s1 (2860) and study the strong decays of the D * s1 (2860) as the 1 3 D 1 cs state. Firstly, let us redefine the hadronic coupling constantsk j X = k j X /g X , j = 1−4, k 5 X = (k 5 X +k 5 X +k 5 X )/g X , and write down the decay widths explicitly from Eqs. (10)(11), Then we define the ratio R, which is independent on the hadronic coupling constants g X ,k 1 X ,k 2 X ,k 3 X ,k 4 X . We can also absorb the coupling constantsk 1 X ,k 2 X ,k 3 X ,k 4 X into the effective coupling g X or setk 1 X =k 2 X =k 3 X = k 4 X = 0. By setting R = R = 1.10 ± 0.15 ± 0.19 [15], we can obtain the value, The value of the hadronic coupling constant k 5 X +k 5 X +k 5 X in the chiral symmetry breaking Lagrangian is as large as that of the hadronic coupling constant g X in the leading-order Lagrangian according to the relationk 5 X = (k 5 X +k 5 X +k 5 X )/g X . It is unreasonable, and the assignment of the D * sJ (2860) as the D * s1 (2860) is not favored. Those strong decays of the D * s1 (2860) take place through the relative P-wave, the decay widths are proportional to p 3 f , while the strong decays of the D * s3 (2860) take place through the relative F-wave, the decay widths are proportional to p 7 f . The decay widths of the D * s1 (2860) are much insensitive to the p f compared to that of the D * s3 (2860). At the present time, there is no experimental data to fit the hadronic coupling constants. In the leading order approximation, i.e. we neglect the chiral symmetry breaking corrections, the ratios R, R and R s among the decay widths are where in the bracket we present the values from the recent studies based on the 3 P 0 model [16]. Also in the 3 P 0 model, Zhang et al obtain the value 0.16 [5]. The present value R differs greatly from that obtained in Ref. [16], while it is compatible with that obtained in Ref. [5]. In Ref. [17], Godfrey and Jardine obtain the value 0.34 based on the relativized quark model and the pseudoscalar emission decay model, which is larger than the present calculation. The present predictions can be confronted with the experimental data in the future to study the strong decays of the D * s1 (2860). In the leading order approximation, we obtain the values R = 0.39 and R = 0.24 in the cases of assigning the D * sJ (2860) to be the D * s3 (2860) and D * s1 (2860) respectively, which differ from the experimental value 1.10 ± 0.15 ± 0.19 greatly [15]. If the D * sJ (2860) observed by the BaBar collaboration in the inclusive e + e − → D 0 K − X production and by the LHCb collaboration in the pp → D 0 K − X processes consists of two resonances D * s1 (2860) and D * s3 (2860) [15,20], we expect to obtain an even smaller ratio R in case of the chiral symmetry breaking corrections are small. On the other hand, if the D * sJ (2860) consists of at least four resonances D * s1 (2860), D * s2 (2860), D * ′ s2 (2860), D * s3 (2860), the large ratio R = 1.10 ± 0.15 ± 0.19 is easy to count for, as the J P = 2 − mesons D * s2 (2860) and D * ′ s2 (2860) only decay to the final states D * K, see Eq.(9) and Eq.(15).

Conclusion
In this article, we take the D * s3 (2860) and D * s1 (2860) as the 1 3 D 3 and 1 3 D 1 cs states, respectively, study their strong decays with the heavy meson effective theory by including the chiral symmetry breaking corrections. We can reproduce the experimental value of the ratio R, R = Br (D * sJ (2860) → D * K) /Br (D * sJ (2860) → DK) = 1.10 ± 0.15 ± 0.19, with suitable hadronic coupling constants, the assignment of the D * sJ (2860) as the D * s3 (2860) is favored. The chiral symmetry breaking corrections are large, we should take them into account. Furthermore, we obtain the analytical expressions of the decay widths, which can confronted with the experimental data in the future from the LHCb, CDF, D0 and KEK-B collaborations to fit the unknown coupling constants. The present predictions of the ratios among the decay widths can be used to study the decay properties of the D * s3 (2860) and D * s1 (2860) so as to identify them unambiguously. On the other hand, if the chiral symmetry breaking corrections are small, the large ratio R = 1.10 ± 0.15 ± 0.19 requires that the D * sJ (2860) consists of at least four resonances D * s1 (2860), D * s2 (2860), D * ′ s2 (2860), D * s3 (2860).