Probing $D^*_{s0}(2317)$ in the decays of $B$ to two charmed mesons

We probe the inner structure of the meson $D^*_{s0}(2317)$ through the decays of $B_{(s)}$ to two charmed mesons within pQCD approach. Assuming $D^*_{s0}(2317)$ as a scalar meson with $\bar cs$ structure, we find that the predictions for the branching ratios of the decays $B^+\to D^{*+}_{s0}(2317)\bar D^{(*)0}, B^0\to D^{*+}_{s0}(2317)D^{(*)-}$ can explain data within errors. The branching ratios for the decays $B_s\to D^{*+}_{s0}(2317)D_s^{(*)+}$ are estimated to reach up to $10^{-3}$ order, which can be observed by the present LHCb and SuperKEKB experiments. In this work, the decay constant of the meson $D^*_{s0}(2317)$ is an input parameter. Unfortunately, its value has been studied by many references but with large uncertainties. Our calculation shows that a smaller decay constant of the meson $D^*_{s0}(2317)$ is supported by compared with the present data, say $55\sim70$ MeV. We also calculate the ratios $R_1=\frac{Br(B^+\to D^{*+}_{s0}(2317)\bar D^0)}{Br(B^+\to D^{*+}_{s0}(2317)\bar D^{*0})})$ and $R_2=\frac{Br(B^0\to D^{*+}_{s0}(2317)D^-)}{Br(B^0\to D^{*+}_{s0}(2317)D^{*-})}$, which are valuable to determine the inner structure of $D^*_{s0}(2317)$ by compared with the experimental results. Our predictions for the ratios $R_{1,2}$ are consistent with the present data within errors.We expect that these two ratios can be well measured by the future experiments through improving the measurement accuracy for the decays $B^+\to D^{*+}_{s0}(2317)\bar D^{*0}$ and $B^0\to D^{*+}_{s0}(2317)D^{*-}$.

( * )+ s are estimated to reach up to 10 −3 order, which can be observed by the present LHCb and SuperKEKB experiments. In this work, the decay constant of the meson D * s0 (2317) is an input parameter. Unfortunately, its value has been studied by many references but with large uncertainties. Our calculation shows that a smaller decay constant of the meson D * s0 (2317) is supported by compared with the present data, say 55 ∼ 70 MeV. We also calculate the ratios R 1 = Br(B + →D * + s0 (2317)D 0 ) Br(B + →D * + s0 (2317)D * 0 ) ) and R 2 = Br(B 0 →D * + s0 (2317)D − ) Br(B 0 →D * + s0 (2317)D * − ) , which are valuable to determine the inner structure of D * s0 (2317) by compared with the experimental results. Our predictions for the ratios R 1,2 are consistent with the present data within errors. We expect that these two ratios can be well measured by the future experiments through improving the measurement accuracy for the decays B + → D * + s0 (2317)D * 0 and B 0 → D * + s0 (2317)D * − .
In order to further reveal the inner structure of D * s0 (2317), we intend to study the weak production of this charmed-strange meson through the two charmed-meson B (s) decays, some of which have been studied by using the light cone sum rules (LCSR) [51] and the relativistic quark model (RQM) [56]. In layout of this paper is as follows. First, in Sec. II, we present the analytic calculations about the B (s) decays to two charmed mesons with D * s0 (2317) involved. Then, we give the numerical results and discussions in Sec. III. A short summary of our results is presented in the final part.

II. THE PERTURBATIVE CALCULATIONS
In the pQCD approach, the only non-perturbative inputs are the light cone distribution amplitudes (LCDAs) and the meson decay constants. For the wave function of the heavy B (s) meson, we take [57,58] Here only the contribution of Lorentz structure φ B (s) (x, b) is taken into account, since the contribution of the second Lorentz structureφ B (s) is numerically small [59] and has been neglected. For the distribution amplitude φ B (s) (x, b) in Eq.(5), we adopt the following model where ω b is a free parameter, we take ω b = 0.4 ± 0.04(0.5 ± 0.05) GeV for B(B s ) in numerical calculations, and N B = 101.445(N Bs = 63.671) is the normalization factor for ω b = 0.4(0.5). For B s meson, the SU(3) breaking effects are taken into consideration. The wave functions for the scalar meson D * s0 1 , we use the form defined in Ref. [55] It is noticed that the distribution amplitudes which associate with the nonlocal operators c(z)γ µ s andc(z)s are different. The difference between them is order ofΛ If we set m D * s0 ∼ m c , we can get these two distribution amplitudes being very similar. For the leading power calculation, it is reasonable to parameterize them in the same form as in the heavy quark limit. Here the decay constantf D * s0 is defined through the matrix element of the scalar current (9) and the shape parameter a = −0.21 [51] is fixed under the condition that the distribution amplitude φ D * s0 (x) possesses the maximum at x = m c /m D * s0 with m c = 1.275 GeV. It is worthwhile to point out that the intrinsic b dependence of this charmed meson's wave function has been neglected in our analysis.
As for the wave functions of the mesons D ( * ) , we use the form derived in Ref. [60] where ǫ / L is the longitudinal polarization vector. In this work only the longitudinal polarization component is used. Here we take the best-fitted form φ ( * ) D from B to charmed meson decays derived in [61] as For the wave function φ Ds (x, b), it has the similar expression as φ D (x, b) except with the different parameters. These prameters are given as follows: f D = 223 MeV, f Ds = 274 MeV, and C D (s) = 0.5 (0.4), ω D (s) = 0.1(0.2) GeV [61]. For the wave function φ D * (s) (x, b), we take the same distribution amplitude with that of the pseudoscalar meson D (s) because of their small mass difference. The decay constants f D * and f D * s are given by the relations .
For these processes considered, the weak effective Hamiltonian H ef f can be written as: where where (q ′ α q ′ β ) V ±A =q ′ α γ ν (1 ± γ 5 )q ′ β with α, β being the color indices and q ′ represent the active quarks at the m b scale, which can be u, d, s, c and b. We calculate in the light-cone coordinate, where a vector is defined as When working in the rest frame of B (s) meson and defining the direction where D * s0 moves as the positive direction of z-axis, we can write the momenta of B, D * s0 , D s mesons as where r D * s0 = m D * s0 /m B , r Ds = m Ds /m B and 0 ⊥ is zero two-component vector. If using k 1 , k 2 , and k 3 to denote the momenta carried by the light quark in B and two charmed mesons, we have Here we consider the decays B 0 s as examples, which will include all types of Feynman diagram amplitudes we need. For the decay B 0 s → D * + s0 (2317)D − s , we give a part of its Feynman diagrams at leading order in Fig.1, where the scalar meson is in the emission (upper) position. If changing the positions of D * + s0 (2317) and D − s for the annihilation type Feynman diagrams (the second line in Fig.1), we will obtain another part of Feynman diagrams which can also contribute to the decay B 0 s → D * + s0 (2317)D − s . These Feynman diagrams are given in Fig. 2. The Feynman diagrams for the decay B 0 s → D * − s0 (2317)D + s are totally same with those for the decay B 0 s → D * + s0 (2317)D − s and can be obtained just by changing Fig.1 and 2. There are two points we need to emphasize: (I) Besides of the different positions for the final states in the annihilation type Feynman diagrams between Fig.1 and 2, another difference is that the former with cc pair generated from a hard gluon, while the later with ss pair generated. (II) In order to distinguish these amplitudes for the decays s from each other, we add the character "c" in the subscript for each amplitude which corresponds to the Feynman diagram with a conventional charmed meson being the emission (upper) position. For the amplitudes corresponding to the annihilation diagrams with ss pair generated from a hard gluon, we add another character "s" in the subscripts. In the following, we give the detail expressions of the Feynman diagram amplitudes for the decay B 0 Fig.1(a) and 1(b) are the factorization emission diagrams, Fig.1(c) and 1(d) are the nonfactorization emission ones, the corresponding amplitudes can be written as where r D * s0 = m D * s0 /m Bs , r Ds = m Ds /m Bs , r c = m c /m Bs and f D * s0 is the decay constant of the scalar meson D * s0 (2317). As we know that the double logarithms α s ln 2 x produced by the radiative corrections are not small expansion parameters when the end point region is important, in order to improve the perturbative expansion, the threshold resummation of these logarithms to all order is needed, which leads to a quark jet function with c = 0.35. It is effective to smear the end point singularity with a momentum fraction x → 0. This factor will also appear in the factorizable annihilation amplitudes.
As to the (non)factorizable annihilation amplitudes for the second line Feynman diagrams in Fig.1 can be obtained by the Feynman rules and are given as The hard scales, evolution factors, the expressions of the Sudakov factors and the functions of the hard kernels in the above amplitudes can be found in appendix A.
Another type of annihilation Feynman diagrams contributing to the decay B s → D * + s0 (2317)D − s are shown in Fig.2, and the corresponding amplitudes are written as The amplitudes for the decay B s → D * − s0 D + s are listed in Appendix B. As to the decays with a vector meson D * (s) involved, the corresponding amplitudes can be also obtained from the Feymann rules, which are not listed for simplicity. Combining these amplitudes, we can obtain the total decay amplitude of each considered channel The amplitudes for the decay B 0 s → D * − s0 D + s can be obtained from those for the decay B 0 s → D * + s0 D − s by deleting (adding) the character "c" from (to) the subscript of each amplitude where there is (not) a character "c".

III. THE NUMERICAL RESULTS AND DISCUSSIONS
We use the following input parameters in the numerical calculations [6,57]:   For the CKM matrix elements, we adopt the Wolfenstein parametrization and the updated values A = 0.790 +0.017 −0.012 , λ = 0.22650 ± 0.00048,ρ = 0.141 +0.016 −0.017 andη = 0.357 ± 0.011 [6]. Generally speaking that the branching ratio of the charged channel should not be less than that of the corresponding neutral one. For example, Particle Data Group(PDG) gives that Br(B + → D + sD 0 ) = (9.0 ± 0.9) × 10 −3 , which is larger than Br(B 0 → D + s D − ) = (7.2 ± 0.8) × 10 −3 [6]. Similarly our calculations (given in Table II) also show that the branching ratio of the charged decay B + → D + s (2317)D ( * )0 is slightly larger than that of the neutral decay B 0 → D + s (2317)D ( * )− . But data are just opposite [6]. Certainly, there still exist large errors in the experimental results, especially for the branching ratios of the decays with a vector meson D * involved. We expect more accurate experimental results in the future LHCb and Super KEKB experiments. Theoretically, the decays B + → D + s (2317)D ( * )0 and B 0 → D + s (2317)D ( * )− have the same CKM matrix elements and Wilson coefficients for the factorizable and nonfactorizable emission amplitudes, it is different in the amplitudes from the annihilation diagrams, while their contributions are small which will be discussed later. Furthermore, there exist similar transition form factors for isospin symmetry between each pair of decay channels. So they should have similar branching ratios.
Certainly, here most uncertainty parameter is the decay constant f D * s0 , which is defined by the matrix element of the vector current It connects with another decay constantf D * s0 defined in Eq.(9) at zero momentum by The decay constant f D * s0 has been computed by different approaches with results covering a wide range (shown in Table I). It is interesting that the works [20][21][22]62] about the analysis of the decay constant f D * s0 through B decays to two charmed mesons are consistent with each other:f D * s0 is in the range of 60 ∼ 75MeV, which is much smaller than the decay constant of another P-wave meson D s1 (2460). That is to say that there exists large disparity between these two decay constants. And the corresponding analysis approaches include the heavy quark symmetry (HQS), the light front quark model (QM). While some authors considered that f D * s0 is larger than 100 MeV by using the quark model [15], lattice QCD (LQCD) [16] and so on.
From our calculations, we find that the smaller decay constant f D * s0 is supported by the present data, say 55 ∼ 70 MeV. The values such as larger than 100 MeV seem are  Table II.
It is helpful to define the following two ratios They are in the range 0.61 ∼ 0.67 (shown in Table III), which are lower than the experimental values 0.89 and 0.71, respectively. Certainly, these two ratios can be well determined by the future experiments through improving the measurement accuracy to the decays B + → D * + s0 (2317)D * 0 and B 0 → D * + s0 (2317)D * − . It is very possible that the ratios R 1 and R 2 are less than 1, which is contrary to the previous prediction [62].
Here we take B 0 → D * + s0 (2317)D − as an example to do the numerical analysis about the amplitudes from different types of Feynman diagrams. The tree operators from the factorizable emission diagrams give the largest contribution because of the large Wilson coefficient C 2 + C 1 /3, and the value of the corresponding amplitude is about 4.39 × 10 −2 . The amplitude from the nonfactorizable emission diagrams is suppressed by the small Wilson coefficient C 1 , whose value is about (1.32 + i0.57) × 10 −2 . The total amplitude of penguin operators is about −(0.95 + i0.07) × 10 −2 , which comes from two parts: One is the factorizable and nonfactorizable emission diagrams −(0.93 + i0.04) × 10 −2 , the other is the annihilation diagrams −(1.11 + i3.67) × 10 −4 . The penguin operators receive severe suppression from the Wilson coefficients and only contribute 3.9% to the final branching ratio. The penguin operator contributions from the annihilation diagrams are even tiny and can be neglected. So it is enough to pay our attention only to the factorizable and nonfactorizable emission diagrams for the investigation of the branching ratios. As to the branching ratios for the decays of B s to two charmed mesons are also calculated and listed in Table IV. In these decays, B s → D * − s0 (2317)D ( * )+ s have the largest branching ratios, which are at 10 −3 order. They are consistent with the predictions by the relativistic quark model (RQM) [56], while are much smaller than those by using the light cone sum rules (LCSR) approach [51]. It can be tested by the future LHCb and Super KEKB experiments. For the decays B s → D * − s0 (2317)D ( * )+ ,their branching ratios are much smaller than other four channels mainly because of the smaller CKM matrix element V cd compared with V cs , that is to say there exists a suppressed factor |V cd /V cs | 2 ≈ 0.05 between the branching ratios of these two types of decays.
Through our calculations, we find that the direct CP asymmtries of our considered decays are in the range of 10 −4 ∼ 10 −3 . For example, A dir CP (B + → D + s0 (2317)D 0 ) is about 0.75%. As we know that the direct CP asymmetry is proportional to the interference between the tree and penguin contributions, while the penguin contributions are small as  with different values f D * s0 = 55, 60, 67 MeV, where the errors for these entries correspond to the uncertainties in the w b = 0.5 ± 0.05 GeV for B s meson, the hard scale t varying from 0.75t to 1.25t, and the CKM matrix elements. Some of these channels have been calculated by using the light cone sum rules (LCSR) [51] and the relativistic quark model (RQM) [56], which are listed in the last two columns. Modes Bs → D * − s0 (2317)D + we mentioned above, so it is not surprise that the direct CP violation of these decays are small. In a word, the direct CP asymmetries in these decays of B to two charmed mesons are tiny, any large direct CP violation observed in the future experiments can be treated as a new dynamic mechanism from the some special structure of D * + s0 (2317) or a signal of new physics.

IV. CONCLUSION
In summary, we probe the inner structure of the meson D * s0 (2317) through the decays of B (s) to two charmed mesons within pQCD approach. Assuming D * s0 (2317) as a scalar meson withcs structure, we find that our predictions for the branching ratios of the decays B + → D * + s0 (2317)D ( * )0 , B 0 → D * + s0 (2317)D ( * )− can explain data within errors. In our calculations, the decay constant of the meson D * s0 (2317) is an input parameter, and its value has been studied by many references but with results covering a wide range. While a smaller value of the decay constant for the meson D * s0 (2317) is supported in our work, say 55 ∼ 70 MeV. We also calculate the ratios R 1 = Br(B + →D * + s0 (2317)D 0 ) Br(B + →D * + s0 (2317)D * 0 ) ) and The variables that are evaluated from the gluon and quark propagators will be used to determine the scales and the expressions of the hard kernels Then the scales in each amplitude are determined as t (1,2) en = max( P en , |P (1,2) The hard functions for the hard part of the amplitudes are listed as where the functions H where the definitions of the functions S j (t)(j = B s , D D * s0 , D s ) in Eq.(A21), Eq.(A22), Eq.(A23) and Eq.(A24) are given as Here the quark anomalous dimension γ q = −α s /π, and the expression of the s(Q, b) in one-loop running coupling coupling constant is used with the variables are defined byq = ln[Q/( √ 2Λ)],q = ln[1/(bΛ)] and the coefficients A (1,2) and β 1 are where n f is the number of the quark flavors and γ E the Euler constant.
Appendix B: Amplitudes for the decay B 0 The amplitudes for the decay B 0 s → D * − s0 (2317)D + s are listed in the following. It is noticed that we add the character "c" in the subscripts to distinguish them from those for Fig.1, which represents a conventional charmed meson D s being the emission (upper) position in the Feynman diagrams.
where these amplitudes are factorizable and nonfactorizable emission contributions, respectively. The nonfactorizable and factorizable annihilation amplitudes are written as F SP ac = 16πC f f Bs m 4 If changing the the positions of the scalar and the pseudoscalar mesons in the final states, one can get another type of annihilation amplitudes, which are given as M LR ans = −32πC f m 4 s0 (x 2 − 2) E an (t (1) ans )h (1) ans (x 1 , x 2 , x 3 , b 1 , b 3 ) − r D * s0 x 2 − r Ds x 3 E an (t (2) ans )h (2) ans (x 1 , x 2 , F LL as = −F LR as = −8πC f f Bs m 4 From the previews contents, it is easy to know that the character "s" in each subscript represents ss pair generated from a hard gluon in the corresponding Feynman diagrams. The scales, the functions for the hard kernel and the evolution factors can be obtained for Eq.(A1) to Eq.(A24) by the following substitutions