Hadronic decays of the highly excited $2D$ $D_s$ resonances

Hadronic decays of the highly excited $2D$ $D_s$ resonances have been studied in the $^3P_0$ model. Widths of all possible hadronic decay channels of the $2D$ $D_s$ have been computed. $D^*_{s1}(2700)$, $D^*_{s1}(2860)$, $D^*_{s3}(2860)$, $D(2600)$ and $D(2750)$ can be produced from hadronic decays of the $2D$ $D_s$, and relevant hadronic decay widths have been particularly paid attention to. The hadronic decay widths of $2D$ $D_s$ to $D(2600)$ or $D(2750)$ may be large, and the numerical results are different in different assignments of $D(2600)$ and $D(2750)$. The hadronic decay widths of $2D$ $D_s$ to $D^*_{s1}(2860)$, $D^*_{s3}(2860)$ or $D^*_{s1}(2700)$ are very small, and different in different assignments of $D^*_{s1}(2700)$.

There are different interpretations to these resonances. Obviously, the nature of these resonances have not been understood clearly. In literatures, the arrangements of these resonances are mainly based on the study of their J P quantum numbers, masses and strong decay modes.
It is well known that the study of productions of these resonances is also an important way to understand them. D * s1 (2700), D * s1 (2860), D * s3 (2860), D(2600) and D(2750) can be produced from the strong decays of highly excited resonances. It will be interesting to study the hadronic production of D * s1 (2700), D * s1 (2860), D * s3 (2860), D(2600) and D(2750) from higher excited resonances. In fact, some highly excited D s resonances have been observed by BaBar, LHCb et al., more and more highly excited D s resonances are expected to be observed by these Collaborations. For kinematical reason, these resonances can be produced from hadronic decays of 2D D s . Unfortu-nately, the strong decays of the highly excited 2D D s resonances have seldom been studied before. In this paper, the hadronic decays of these 2D D s resonances will be studied in the 3 P 0 model. The paper is organized as follows. In Sec.II, we give a brief review of the 3 P 0 model and possible decay modes of the 2D resonances. In Sec. III, we present the formula and numerical results of the hadronic decay of the 2D D s resonances, and the decays with D * s1 (2700), D * s1 (2860), D * s3 (2860), D(2600) or D(2750) involved in the final states are particularly paid attention to. Finally, the conclusions and discussions are given in Sec. IV.

II. 3 P0 MODEL AND POSSIBLE DECAY MODES
OF THE 2D Ds RESONANCES 3 P 0 model is popularly known as a quark-pair creation (QPC) model, which has been extensively applied to the calculation of the OZI-allowed strong decay of meson A to meson B and C. The model was first proposed by Micu [21], and then developed by Yaouanc et al [22][23][24]. The decay process is shown in Fig. 1 [25,26], where a pair of quarks q 3q4 with J P C = 0 ++ are created from the vacuum and regroup with the q 1q2 within the initial meson A into two outgoing mesons B and C.
where the momentum of the daughter meson in the initial meson A's center of mass frame is and M JL is the partial wave amplitude of A → BC. In terms of the Jacob-Wick formula [27], the partial wave amplitude can be obtained from the helicity amplitude with J = J B + J C , J A = J B + J C + L and M JA = M JB + M JC . In this equation, the helicity amplitude while the spatial integral I The details of the indices, matrix elements and other indications are given in Ref. [26] With these formula in hand, we go ahead with our calculation. In the calculation, the simple harmonic oscillator(SHO) wave function is employed to represent the meson wave function. The meson flavor functions follow the convention in Ref. [28]: For the parameters involved in 3 P 0 model, the light nonstrange quark pair creation strength γ and the strange quark pair creation strength γ ss are correlated by γ ss ≈ γ/ √ 3 [23] with γ = 7.85 [29]. The constituent quarks masses are taken to be m c = 1.43 GeV, m u = m d = 0.45 GeV and m s = 0.55 GeV [17]. The resonance masses and the effective scale parameters β for different resonances used in our calculation are listed in Table. 1 [3,17] and Table. 2 [17], respectively. There is not a 2D D s observed, and the masses of these resonances are unknown. In our calculation, theoretical predicted masses of the 1 − 2 3 D 1 D s (3383 MeV) and the 3 − 2 3 D 3 D s (3469 MeV) [30] are employed, respectively. For the 2 − D s resonance, 2 3 D 2 may mix with 2 1 D 2 , which may result in a complicated mixing. Only when the detail of the mixing is clear, can we give the hadronic decay widths of each 2 − resonances. In this paper, we give only the results of pure 2 3 D 2 and 2 1 D 2 . As an approximation, the average mass (3429.5 MeV) of the two predicted 2 − D s in Ref. [30] is taken as the mass input of 2 3 D 2 and 2 1 D 2 .
Possible kinematically allowed decay modes of these four 2D D s resonances are presented in Table. 3 and  Table. 4.  Table. 5. Similar results of the strong decays of D s (2 3 D 2 ) and D s (2 1 D 2 ) are shown in Table. 6.

III. HADRONIC DECAYS OF 2D Ds
In our calculation, D 1 (2430) and D s1 (2460) are assigned as the 1 + (j P = 1 2 + ) D and D s , respectively.   D 1 (2420) and D s1 (2536) are assigned as the excited 1 + (j P = 3 2 + ) D and D s . Through the relation between the j P eigenstates and the 2S+1 L J eigenstates, these two 1 + resonances are regarded as a mixture of 1 1 P 1 and 1 3 P 1 resonances For a particular J, the larger the angular momentum (L) between the two final states, the smaller the corresponding M JL . However, for a particular decay channel, both J and L could vary, and there is not an oneto-one relation between the decay width and the M JL . From the numerical results in Table.  Since there are different assignments to D * s1 (2700), D(2600) and D(2750), their hadronic productions (together with the production of D * s1 (2860) and D * s3 (2860)) from 2D D s resonances are studied independently in the following subsection. III: OZI-allowed hadronic decay modes of Ds(2 3 D1) and Ds(2 3 D3). The masses of Ds(2 3 D1) and Ds(2 3 D3) are 3383 MeV and 3469 MeV, respectively [30].
Mode Channels Mode Channels For kinematical reason and conservation of some quantum numbers in hadronic decay, D * s1 (2700) can only be produced through 2D D s → D * s1 (2700)η. The numerical results are presented in Table. 7. The first column in the table indicates three possible assignments of D * s1 (2700), where the mixture possibility is from Ref. [12] with a mix-ing angle θ = 88 • . The mixing of D * s1 (2700) has also been studied in other references [9,17,31]. In the table, all the decay widths are very small though they are different in different assignments of D * s1 (2700). Once D * s1 (2860) − and D * s3 (2860) − are assigned as the J P = 1 − and J P = 3 − members of the 1D family, the hadronic decay widths of 2D D s → D sJ (2860)η can also be calculated.
In our paper, the uncertainties of the input parameters and the model have not been studied. The detail of possible mixing of some resonances has neither been explored. More theoretical study of these highly excited resonances are required. Of course, the most important thing is to expect more highly excited D s resonances observed in forthcoming experiments.