Dynamical mixing between $2^3S_1$ and $1^3D_1$ charmed mesons

In charmed $D$ and $D_s$ mesons sector, the matrix of a Hamiltonian in a quark potential model is computed in the $2^3S_1$ and $1^3D_1$ subspace. The masses of four mixed states of $2^3S_1$ and $1^3D_1$ are obtained. It is the off-diagonal part of spin-orbit tensor interaction that causes the mixing between the $2^3S_1$ and $1^3D_1$ states. The mixing angle between the $2^3S_1$ and $1^3D_1$ state is very small. Based on mass spectra analyses, $D^*_J(2600)$ is very possibly the $D^*_1(2600)$ which is predominantly the $2^3S_1$ $D$ meson. $D^*_{s1}(2700)$ is predominantly the $2^3S_1$ $D_s$ meson. Under the mixing, the $^3P_0$ model is employed to compute the hadronic decay widths of all OZI-allowed decay channels of the four states. Based on the hadronic decay widths analyses, $D^*_J(2600)$ and $D^*_{s1}(2700)$ are possibly the mixed states of $2^3S_1$ and $1^3D_1$ with large mixing angles, which implies a large off-diagonal spin-orbit tensor interaction not existed in present Hamiltonian. For lack of experimental data, $D(2750)$ and $D^*_{s1}(2860)$ in PDG is difficult to be identified except that $D^*_1(2760)$ and $D^*_{s1}(2860)$ have been properly resolved from the experimental data.

In Ref. [42], the mixing angle is determined with θ = −0.5 radians from a simple mixing matrix of the masses of physical states (2.69 GeV and 2.81 GeV) and the predicted states of the 2 3 S 1 and 1 1 D 1 D s mesons (2.71 GeV and 2.78 GeV, respectively). The mixing angle will change sign when the internal quark components of the meson are charge conjugated into their anti-quarks. Their predicted hadronic decay widths at this determined mixing angle in the 3 P 0 model is consistent with experimental data.
In Ref. [43], a similar mixing scheme of the 2 3 S 1 and 1 3 D 1 D s as that in Ref. [42] is employed, and the mixing angle is explored through the hadronic decays of the D s states. 1.12 ≤ θ ≤ 1.38 radians (opposite in sign with opposite internal quarks) is fixed for D * s1 (2710) through a comparison of the predicted hadronic decay widths in the the 3 P 0 model with experimental data, while the mixing angle 1.26 ≤ θ ≤ 1.31 is fixed for D sJ (2860).
In Refs. [46,47], the similar mixing scheme of the 2 3 S 1 and 1 3 D 1 D and D s is employed. The mixing angle is studied through the hadronic decays of these states with the decay formula developed by Eichten, Hill, and Quigg [15]. The mixing angles are found small. θ = 4 • → 17 • and θ = −16 • → −4 • are obtained for D * 1 (2600) and D * s1 (2700), respectively. On one hand, the fixed mixing angles are different in different references. On the other hand, the mixing angles have not been consistently determined through the mass spectra and the decay properties. Of course, in order to identify the D * J (2600), D(2750), D * s1 (2700) and D * s1 (2860), it is also important to systematically study the mixing between the 2 3 S 1 and 1 3 D 1 D and D s mesons. For this purpose, we study the mixing between the 2 3 S 1 and 1 3 D 1 in the quark potential model firstly, and subsequently explore their strong decay in the 3 P 0 model. The paper is organized as follows. In the second section, the mixing mechanism between the 2 3 S 1 and 1 3 D 1 D and D s mesons is explored in the quark potential model, and the mixing angles are dynamically determined. The hadronic decays of the four mixed states are explored in the 3 P 0 model in Sec. III. In the final section, the conclusions and discussions are given.
II. DYNAMICAL MIXING BETWEEN 2 3 S1 AND 1 3 D1 To describe the heavy-light meson states, two kinds of eigenstates are often employed. One is the |J, L, S (denoted with 2S L J ) with J = L + S and S = S q + Sq where L is the orbital angular momentum, and S q , Sq are the spins. Another one is the |J, j (denoted with j P ), where P is parity, j = L + S q is the angular momentum of light quark freedom. Physical heavy-light mesons are usually not the eigenstates |J, L, S or |J, j , they are the mixing states of these eigenstates. Eigenstates |J, L, S will be employed in the following.
In the quark potential model, the inter-quark interactions include the spin-spin interaction, the colormagnetic interaction, the spin-orbit interaction, and the tensor force [12,38,41]. In our analysis, the relativised quark model [41] is employed for our analysis, where the Hamiltonian is where V conf is the standard Coulomb and linear scalar interaction, the spin-orbit and color tensor interaction V SD is rewritten as with C F = 4 3 , C A = 3, b 0 = 9, and γ E = 0.5772. The model parameters are α s = 0.53, µ = 1.0, σ = 1.13, b = 0.135, C cū = −0.305, and C cs = −0.254, they were given in Ref. [41]. The quak masses are chosen as following: m c = 1450 MeV, m u = m d = 450 MeV, and m s = 550 Me V. In term of these parameters, the predicted masses of the 1S and 1P D and D s mesons agree well to the experimental data, which are presented in Table. II and  Table. III As well known, the H is not diagonal in the basis |J, L, S or |J, j . The relation between |J, L, S and |J, j can be found in Refs. [14,40]. From Ref. [14], the offdiagonal interaction arises from the tensor interaction which can be written in an irreducible representation as where Y (2) is a rank 2 spherical harmonics and q in the spherical basis.
The matrix element of the tensor term is obtained through the Wigner-Eckhart theorem [48], and S||S (2) ||S is the spin reduced matrix element which is Ref. [14], but the masses of 1 3 D 1 | charmed states are lower than those in the same reference.
When the low and high mixed sates are denoted with |D * L 1 and |D * H 1 [43,47], respectively, the matrix H can be diagolized in the physical states (mixed states) where H 11  These four mixed states will be denoted with D * 1 (2635), D * 1 (2739), D * s1 (2715) and D * s1 (2805) throughout this paper. Obviously, the mixing angles between the 2 3 S 1 and 1 3 D 1 for D and D s are very small, and the off-diagonal interactions resulting from the tensor interaction almost do not change the eigenvalues.
In comparison to the measured masses by experiments, D * J (2600) and D * s1 (2700) in PDG could be identified with the D * 1 (2635) and D * s1 (2715). That is to say, D * J (2600) and D * s1 (2700) are very possibly the predominant 2 3 S 1 D and D s mesons, respectively.
As analyzed in Refs. [4,5,10,11,14,55], D * (2760) (D(2750)) observed in e + e − and pp collisions in fact has been resolved into two D * 1 (2760) and D * 3 (2760) D states, D * sJ (2860) observed in e + e − and pp collisions has also been resolved into two D * s1 (2860) and D * s3 (2860) D s states. Unfortunately, the analyses of the resolve are not sufficient, which may result in some uncertainties to the measured data of D * 1 (2760) and D * s1 (2860). For these reasons, the measured data of D * (2760) (D(2750)) and D * sJ (2860) are not sufficient to give the right data of D * 1 (2760) and D * s1 (2860). In experiment, it is important to figure out proper ways to give the exact masses and some decay widths of the resolved states of D * (2760) (D(2750)) and D * sJ (2860) in the future. In Ref. [42], the D * sJ (2860) was regarded as the D * H s , and a large mixing angle θ = −0.5 radians has been phenomenologically obtained. The mixing angles between the 2 3 S 1 and 1 3 D 1 for D and D s could not be large if the off-diagonal tensor interaction is in its present form, but the mixing angles could be large if the off-diagonal interactions tensor interaction is in some other form which results in large. With large mixing angles, the masses of the mixed states D * H and D * H s could be be large. In order to see how the masses depend on the mixing angles, the variation of the masses of the four mixed mesons with the mixing angles is plotted in Fig. 1 It should be noted that an off-diagonal tensor interactions inversely proportional to the products of heavy quark and light quark mass in its present form can not result in a large mixing. Which form of off-diagonal tensor interactions can bring in strong mixing deserves more exploration.

III. HADRONIC DECAY OF
In order to learn the internal quark dynamics, another way is to study the strong decay of hadrons. In the case of 2 3 S 1 and 1 3 D 1 mixing, the hadronic decay of the four mixed states are explored in the 3 P 0 model in this section.
As well known, the 3 P 0 model is usually called as the quark-pair creation model. It has been employed extensively to study the Okubo-Zweig-Iizuka(OZI)-allowed hadronic decay processes. The model was first proposed by Micu [50] and developed by Yaouanc et al [51][52][53]. In the model, the decay of a meson takes place through a qq pair creation with the vacuum quantum number J P C = 0 ++ . The hadronic partial decay width Γ of a decay process A → B + C is the momentum of the final states B and C in the initial meson A's center-of-mass frame, and M JL is the partial wave amplitude of A → B + C.
For mixed states |D * L 1 and |D * H 1 with mixing angle θ, In terms of the Jacob-Wick formula, M JL can be written as [54], where where γ is the pair-production strength constant. The detail of the flavor matrix element ϕ 13 B ϕ 24 C |ϕ 12 A ϕ 34 0 , the spin matrix element χ 13 ( K) can be found in Ref. [55].
In the 3 P 0 model, numerical results depend on the parameters such as γ, the harmonic oscillator parameter  β and the constituent quark masses. In this paper, the γ = 6.947 ( √ 96π times as the γ = 0.4 in Ref. [14]) is employed as in Refs. [55,57,58]. For strange quark-pair ss creation, γ ss = γ/ √ 3 [52]. The β are taken from Ref. [56].  Table. IV. From this table, the total hadronic decay widths of D * 1 (2635) and D * 1 (2739) are 34.84 MeV and 298.77 MeV, respectively. These total decay widths are largely different with those of the observed states.
If the mixing angle is large, the predicted branching ratios are consistent with the observed ones as in Refs. [41,46,47]. At a large mixing angle, the masses of D * L 1 and D * L s1 turn smaller, and the masses of D * H 1 and D * H s1 turn larger as shown in Figure. 1. However, the problem is which kind of off-diagonal spin-orbit tensor interaction can bring in a large mixing.
From this table, the total hadronic decay width (39.27 MeV) of D * s1 (2715) is much smaller than the observed one of D * s1 (2700), while the total hadronic decay width (184.63 MeV) of D * s1 (2805) is comparable to that of D * s1 (2860). The obtained ratios are largely different with the observed ones of D * s1 (2700) and D * s1 (2860). Obviously, D * s1 (2700) and D * s1 (2860) are impossible to be identified with the combination of 2 3 S 1 and 1 3 D 1 D s mesons with a mixing angle θ = 0.18 • . Similarly, the theoretical results of these ratios could be consistent with the observed data at a large mixing angle.

IV. CONCLUSIONS AND DISCUSSIONS
In this paper, the masses of 1S, 1P , 1D and 2S states of D and D s have been calculated in the quark potential model. The off-diagonal tensor interactions result in small mixing angles between the 2 3 S 1 and 1 3  Based on mass spectra and hadronic decay analyses, the D * J (2600) is very possibly the D * 1 (2600) D mesons. The D * J (2600) and D * s1 (2700) are predominantly the 2 3 S 1 D and D s mesons, respectively. In fact, the resolve of D(2750) and D * sJ (2860) is not sufficient for the identification of the observed states. Until the D * 1 (2760) (D * s1 (2860)) and D * 3 (2760) (D * s1 (2860)) have been separatively resolved, it is difficult to identify the D(2750) and D * s1 (2860). In addition, the measure of branching fraction ratios Γ(D + s K − )/Γ(D * + s K − ) of D * J (2600) and D(2750) is also important for their identification.
As pointed out in Ref. [38], the leptonic or electronic decay width is more sensitive to the 3 S 1 and 3 D 1 mixing detail. The measure of the leptonic or electronic decay widths will be helpful to the understanding of these mixed states.
Of course, if the mixing of the 2 3 S 1 and 1 3 D 1 D is large, there implies higher D * H 1 and D * H s1 charmed mesons. In this case, the present form of the off-diagonal tensor interactions does not provide such a large mixing, and the exact form of the off-diagonal tensor interactions deserves deep exploration.