Decay Constants of $D_{sJ}^*$(2317) and $D_{sJ}$(2460)

The resonances $D_{sJ}^*$(2317) and $D_{sJ}$(2460) which are considered to be the $(0^+,1^+)$ doublet composed of charm and strange quarks have been discovered recently. Using the method of Rosner which is based on the factorization hypothesis, we calculate the lower bounds of the decay constants of these states from the branching ratios of $B\to D D_{sJ}$ measured by Belle and Babar.Our result shows that the decay constant of $D_{sJ}$(2460) is about twice that of $D_{sJ}^*$(2317) on the contrary to the naive expectation of the heavy quark symmetry which gives their equality. We show that this big deviation originates from the large internal motion of quarks inside these $P$-wave states and that our result is in good accord with the relativistic quark model calculation.


Introduction
The resonances D * sJ (2317) and D sJ (2460) composed of charm and strange quarks have been discovered recently by the BaBar [1], CLEO [2], and Belle [3] Collaborations. Their decay patterns suggest that they are 0 + and 1 + states, respectively, in the quarkmodel classification. Bardeen et al. [4] considered these states to be the (0 + , 1 + ) doublet which has j = 1/2 of the light degree of freedom and studied them with effective Lagrangians based on the chiral symmetry in heavy-light meson systems.
The measured mass of D * sJ (2317), 2317.4±0.9 MeV [5] which is 40.9±1.0 MeV below the threshold of D 0 K + , was considered surprisingly low compared to the predictions of the potential model calculations. For example, the prediction of the 1 3 P 0 mass by Isgur and Godfrey [6] was 2.48 GeV, and that by Eichten and Di Pierro [7] was 2.487 GeV, which are about 160 and 170 MeV higher than the measured mass of D * sJ (2317). There have been many theoretical investigations which aimed to explain the measured low mass of D * sJ (2317) [8,9,10,11,12,13,14,15]. For example, Barnes et al. [9] considered a mixing between two molecular states |D 0 K + > and |D + K 0 > and pointed out the importance of a very strong coupling between the cs bound and DK continuum states, as required to induce binding. Van Beveren and Rupp [10] described D * sJ (2317) as a quasibound scalar cs state in a unitarized meson model, owing its existence to the strong coupling to the nearby S-wave DK threshold. Browder et al. [13] proposed a mixing between the qq and 4-quark states and assigned a linear combination with less mass as D * sJ (2317). Ref. [14] calculated the mass shift of D * sJ (2317) quantitatively by using the coupled channel effect and could explain naturally the observed mass.
Bell [3] and Babar [16] measured the branching ratios of the exclusive modes Rosner calculated the decay constant of D − s meson by relating the the differential distributions dΓ(B 0 → D ( * )+ l − ν l )/dq 2 and the rates the color-favored decays B 0 → D ( * )+ D − s under the factorization hypothesis [17,18]. Using the method of Rosner, we calculate the lower bounds of the decay constants of D * sJ (2317) and D sJ (2460) from the partial branching ratios of B → DD sJ measured by Bell and Babar. Our result shows that the decay constant of D sJ (2460) is about twice that of D * sJ (2317) on the contrary to the expectation of the heavy quark symmetry which gives their equality. We show that this big deviation originates from the large internal motion of quarks inside these P -wave states and that our result is in good accord with the relativistic quark model calculation.
In section 2.1 we calculate the lower bound of the decay constants of D * sJ (2317) and D sJ (2460), and estimate the ratio of these decay constants. In section 2.2 we compare our results with the results of the relativistic quart model calculation by Veseli and Dunietz. Section 3 is conclusion, in which we discuss the physical implications of our results. From Lorentz invariance one finds the decomposition of the hadronic matrix element in terms of hadronic form factors: where J µ = cγ µ b and q µ = (p B − p D ) µ . In the rest frame of the decay products, F BD 1 (q 2 ) and F BD 0 (q 2 ) correspond to 1 − and 0 + exchanges, respectively. At q 2 = 0 we have the constraint F BD 1 (0) = F BD 0 (0) since the hadronic matrix element in (1) is nonsingular at this kinematic point.
When the lepton masss is ignored, the q 2 distribution of the semi-leptonic decay rate, in the allowed range 0 where In the factorization hypothesis the effective Hamiltonian H eff for the process B → DD sJ is written as [19] where Γ µ = γ µ (1 − γ 5 ) and the subscript H stands for hadronic implying that the Dirac bilinears inside the brackets be treated as interpolating fields for the mesons and no further Fierz-reordering need be done. The QCD corrections a 1 and a 2 have the values a 1 ∼ 1 and a 2 ∼ 0.25 [20]. Luo and Rosner used |a 1 | = 1.05 in their calculation [18]. For the two body hadronic decay, in the rest frame of initial meson the differential decay rate is given by where M is the mass of initial meson, and m 1 (m 2 ) and p 1 are the mass and momentum of one of final mesons. By using (1), (4) gives the following formulas for the branching ratios of the process For the B to D meson (heavy to heavy) transition form factors, the heavy quark effective theory gives [21] where , and G(ω) is a form factor which becomes the Isgur-Wise function in the infinite heavy quark mass limit. We use the parameterization of G(ω) given in [22,23], We use the world average values given in [23], The errors in (11) give the error of G(ω) by 12 % for D * s0 (2317) and by 11 % for D ′ s1 (2460), and they reduce to the same amounts of the errors for f D * s0 and f D ′ s1 , respectively, since we calculate these decay constants by using Eqs. (6) and (7). However, these errors are almost cancelled in the ratio f D ′ s1 /f D * s0 . We extract the lower bounds of the the decay constants of D * s0 (2317) and D ′ s1 (2460) from Eqs. and Babar [16], and the above form factor G(ω). The results are presented in Table 1. The value in the fifth column in Table 1 is the ratio of the lower bounds of the decay constants given in the third and fourth columns. However, even in the situation that the experimental values of the branching ratios B → DD sJ (2460) and Group Decay Mode  [3] and Babar [16]. The values in the fifth column are the ratios of the values in the third and fourth columns.
B → DD * sJ (2317) are raised by other partial branching ratios in addition to those considered here, it is expected that the value in the fifth column does not change much because of the cancellation in the ratio. Therefore, we expect that the value in the fifth column is close to the ratio of the decay constants themselves f D ′ s1 and f D * s0 .

Comparison with Relativistic Quark Model Calculation
When we take the internal motion of quarks inside a meson into account, the decay constants of the S-wave pseudo-scalar (J P j = 0 − 1/2 ) and vector (1 − 1/2 ) mesons, where the subscript j stands for the angular momentum of the light degree of freedom in the j-j coupling scheme of the heavy(Q)-light(q) meson, are given by [24] In the limit m Q → ∞, from (12) and (13)  which is the Van Royen-Weisskopf formula [25]. However, since in the D s meson system there is an appreciable contribution of the internal motion of quarks to the decay constants given by (12) and (13), becomes larger than f 0 − 1/2 . Ref. [24] obtained the results: f Ds = 309 MeV, f D * s = 362 MeV, and f D * s /f Ds = 1.17 by averaging the values obtained from six different potential models. For reference, the results of Ref. [24] for B s mesons are f Bs = 266 MeV, f B * s = 289 MeV, and f B * s /f Bs = 1.09, and these results show that the internal motion of quarks is less important in the B s meson system compared to the the D s meson system, as expected.
Veseli and Dunietz [26] worked on the decay constants of the P -wave scalar (0 + 1/2 ) and axial-vector (1 + 1/2 ) mesons and derived In the limit m Q → ∞, both F 0 + 1/2 (p) and F 1 + However, for the P -wave D sJ mesons (D * sJ (2317) and D sJ (2460)) the internal motion of quarks is even larger than that in the S-wave D s mesons, and than the difference of f 0 + 1/2 and f 1 + 1/2 become much greater. Using (12) and (14), Veseli and Dunietz [26] obtained the results: f 0 + Their result for the ratio f 1 + 1/2 Table 1, and their results for f 0 + 1/2 and f 1 + 1/2 are consistent with our results presented in Table 1: Our results in (15) also support that D * s0 (2317) and D ′ s1 (2460) are j = 1/2 states instead of j = 3/2 states, since Veseli and Dunietz [26] obtained the values of decay constants of the D sJ (1P, 1 + 3/2 ) and D sJ (1D, 1 − 3/2 ) states as 87 and 45 MeV, respectively, which are much lower than |a 1 | f D ′ s1 > 166 ± 20 MeV given in (15). We note that in the limit m Q → ∞, f 0 − ) are different even in this heavy quark symmetry limit since 0 − 1/2 and 1 − 1/2 states are S-wave and 0 + 1/2 and 1 + 1/2 states are P -wave. We can see this difference explicitly in (13) and (14). Furthermore, the limit m Q → ∞ does not corresponds to a good approximation for the study of the P -wave D s meson system because of the large internal motion of quarks inside the meson. This property results in the fact that the decay constant of axial-vector meson is about twice that of the scalar meson for the P -wave D s meson system.

Conclusion
The resonances D * sJ (2317) and D sJ (2460) which is considered to be the (0 + , 1 + ) doublet composed of charm and strange quarks have been discovered recently. Bell [3] and Babar [16] measured the branching ratios of the exclusive modes B → DD * sJ (2317)[D + s π 0 ], B → DD sJ (2460)[D * + s π 0 ] and B → DD sJ (2460)[D + s γ]. From these experimental data we extracted the lower bounds of the decay constants of D * sJ (2317) and D sJ (2460) by the method of Rosner which is based on the factorization hypothesis. Our result shows that the decay constant of D sJ (2460) is about twice that of D * sJ (2317) on the contrary to the naive expectation of the heavy quark symmetry which gives their equality. We showed that this big deviation originates from the large internal motion of quarks inside these P -wave states and that our result is in good accord with the relativistic quark model calculation. This result indicates that we can not apply the heavy quark symmetry to D * sJ (2317) and D sJ (2460). For example, this result shows that the assumption of the heavy quark symmetry to these states which was considered in [27,28] is not valid.