Productions of D ∗ s 0 (2317) and D s 1 (2460) in B ( s ) and Λ b (Ξ b ) decays

Recent studies show that D ∗ s 0 (2317) and D s 1 (2460) contain large molecular components. In this work, we employ the naive factorization approach to calculate the production rates of D ∗ s 0 (2317) and D s 1 (2460) as hadronic molecules in B ( s ) and Λ b (Ξ b ) decays, where their decay constants are estimated in the effective Lagrangian approach. With the so-obtained decay constants f D ∗ s 0 (2317) and f D s 1 (2460) , we calculate the branching fractions of the b -meson decays B ( s ) → ¯ D ( ∗ ) ( s ) D ∗ s 0 and B ( s ) → ¯ D ( ∗ ) ( s ) D s 1 and the b -baryon decays Λ b (Ξ b ) → Λ c (Ξ c ) D ∗ s 0 and Λ b (Ξ b ) → Λ c (Ξ c ) D s 1 . Our results show that the production rates of D ∗ s 0 (2317) and D s 1 (2460) in the B s , Λ b and Ξ b decays are rather large that future experiments could observe them. In particular, we demonstrate that one can extract the decay constants of hadronic molecules via the triangle mechanism because of the equivalence of


I. INTRODUCTION
In 2003, the BaBar Collaboration discovered D * s0 (2317) in the D + s π 0 mass distribution in the e + e − annihilation process [1], which was later confirmed by the CLEO Collaboration [2] and Belle Collaboration [3] in the same process.Moreover, the BESIII Collaboration observed the D * s0 (2317) in the process of e + e − → D * + s D * − s0 (2317) [4].In addition to the above inclusive processes, D * s0 (2317) was also observed in the exclusive process of the B decay by the Belle Collaboration [5] and BaBar Collaboration [6].D s1 (2460) as the heavy quark spin symmetry (HQSS) partner of D * s0 (2317) was first discovered in the D * + s π 0 mass distribution by the CLEO Collaboration [2], and then confirmed by several other experiments [3,5,7].Treating D * s0 (2317) and D s1 (2460) as conventional P -wave cs mesons, the masses obtained in the Goldfrey-Isgur (GI) model are larger than the experimental ones by 140 MeV and 100 MeV [8], which have motivated extensive discussions on their nature.
It should be noted that in the lattice QCD simulation of the DK interaction, a bound state below the DK mass threshold was identified [20][21][22][23][24]. Furthermore, with the D ( * ) K potentials supplemented by the cs core couplings to the D ( * ) K components, D * s0 (2317) and D s1 (2460) can be dynamically generated [25][26][27][28], indicating that the D ( * ) K molecular components account for a large proportion of their wave functions in terms of the Weinberg compositeness rule 1−Z [25].Studying the masses of D * s0 (2317) and D s1 (2460), one can conclude that they contain both D ( * ) K molecular components and the cs cores.The next natural step forward is to study their decays.
According to the review of particle physics (RPP) [29], the D * + s0 (2317) dominantly decays to D + s π 0 , which means that D * + s0 (2317) must be narrow since the decay of Moreover, the productions of D * s0 (2317) and D s1 (2460) in the Λ b decays have been explored [48], finding that their production rates in the Λ b decays are larger than those in the corresponding B decays.Recently, the DK femtoscopic correlation function was investigated to elucidate the nature of D * s0 (2317) [49,50], which can be accessed in high energy nucleon-nucleon collisions in the future [51,52].
Until now, the D * s0 (2317) and D s1 (2460) have only been observed in the exclusive process via B decays.In this work, we systematically explore the productions of D * s0 (2317) and D s1 (2460) in B (s) and Λ b (Ξ b ) decays with the factorization ansatz [53,54].Following Ref. [41], we employ the effective Lagrangian approach to estimate the decay constants of D * s0 (2317) and D s1 (2460), which are dynamically generated via the DK − D s η and D * K − D * s η coupled-channel potentials described by the contact-range effective field theory (EFT), and then calculate the production rates of D * s0 (2317) and D s1 (2460) in B (s) and Λ b (Ξ b ) decays.Another motivation of this work is to test the universality of the approach that we proposed to calculate the decay constant of a hadronic molecule via the triangle mechanism [55].Based on our previous study of the decays B → D * s0 (2317) D( * ) and B → D s1 (2460) D( * ) via the triangle mechanism, the decay constants of D * s0 (2317) and D s1 (2460) can be extracted [56].The effective Lagrangian approach in this work can further check the validity of our approach [55].This paper is organized as follows.In Sec.II, we introduce the effective Lagrangian approach to cal- where G F is the Fermi constant, V cb and V cs are the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements, and a 1 is the effective Wilson coefficient.In principle, a 1 is expressed by the Wilson Coefficients of the QCD-corrected effective weak Hamiltonian, which depends on the renormalization scale [54,61,62].In where the momenta q = p B + − p D( * )0 and P = p B + + p D( * )0 , and F 1 (q 2 ), F 2 (q 2 ), A 0 (q 2 ), A 1 (q 2 ), A 2 (q 2 ), and V (q 2 ) are form factors.The current matrix element where

and f
V /A i (q 2 ) are form factors.In general, the form factors are parameterized in the following form: where F i (0), a, and b are parameters determined in phenomenological models.In this work, we take these parameters determined in the quark model [64][65][66].
The current matrix element ⟨D + s |(sc)|0⟩ describes the process of creating a D + s meson from the vacuum via the axial current, which is parameterized by the decay constant f D + s and the momentum of the D + s meson.Following Ref. [44], the current matrix elements for the D s , D * s , D * s0 (2317), and D s1 (2460) mesons created from the vacuum are The With the above effective Lagrangian, we obtain the amplitudes for the decays B(k 0 ) → D( * ) (q 1 )D ( * ) s (q 2 ): In terms of the effective Lagrangian of the weak decays , the corresponding amplitudes are written as where A 1 , A 2 , B 1 , B 2 , A, and B represent the transition form factors of Λ b to Λ c : With the amplitudes for the weak decays given above, one can compute the corresponding partial decay widths where J and M are the total angular momentum and the mass of the initial state, |⃗ p| is the momentum of either final state in the rest frame of the initial state, A is the amplitude of the weak decay, and the overline indicates the sum over the polarizations of final states.

B. Decay Constants
The decay constants f D * s0 and f D s1 are defined in Eq. (7).To obtain the values of f D * s0 and f D s1 , one usually constructs the amplitudes for the D * s0 and D s1 created from the vacuum and then extracts the coef- The effective Lagrangians describing the couplings of the hadronic molecules to their constituents are written as where g D * s0 DK , g D * s0 Dsη , g D s1 D * K , and g D s1 D * s η are the coupling constants between D * s0 (D s1 ) and their constituents.In this work, we employ the contact-range EFT to dynamically generate the D * s0 and D s1 and further determine the couplings between the molecular states and their constituents from the residues of the corresponding poles, which are widely applied to study hadronic molecules [73][74][75].
With the above preparations, we can write the amplitude of Fig. 2 as where the subscripts 1 and 2 denote the D and K mesons in amplitude A a , and the D s and η mesons in amplitude A b .Similarly, we obtain the amplitudes describing the D s1 created from the vacuum as Once the amplitudes of Fig. 2 are obtained, the decay constants f D * s0 and f D s1 can be easily extracted considering their definitions.In the following, we show how to calculate the relevant loop functions in the dimensional regularisation scheme.
With the Feynman parameter approach, we obtain the following integrals where and the renormalization scale µ depends on the specific physical process under consideration.To extract the decay constants of D * s0 (2317) and D s1 (2460), the loop functions of Eq. ( 14) and Eq. ( 15) are converted into the following form Finally, we obtain the analytic form of the decay constants of D * s0 (2317) and D s1 (2460) C. Contact-range effective field theory approach In the following, we briefly introduce the contact-range effective field theory (EFT) approach.The scattering amplitude T is responsible for the dynamical generations of molecules, which is obtained by solving the following Lippmann-Schwinger equation where V is the coupled-channel potential determined by the contact-range EFT approach, and G( √ s) is the loop function of the two-body propagator.
The coupled-channel potentials V in matrix form read where the coefficient C a needs to be determined by fitting the D * s0 and D s1 masses.The loop functions of D * s0 and D s1 are ).We note that the loop function of D s1 contains an additional term, which is induced by the term in the loop integral.One can see that the loop integrals depend on the renormalization scale µ.
With the potentials obtained above, we can search for poles generated by the coupled-channel interactions and determine the couplings between the molecular states and their constituents from the residues of the corresponding poles, where g i denotes the coupling of channel i to the dynamically generated state and √ s 0 is the pole position.Hadron I(J P ) M (MeV) Hadron I(J P ) M (MeV) Hadron I(J P ) M (MeV)

III. RESULTS AND DISCUSSIONS
In Table I, we tabulate the masses and quantum numbers of relevant particles.One can see that there exists an unknown parameter µ (renormalization scale) in both Eq. ( 17) and Eq. ( 22), for which a consistent value is adopted in this work.First, we employ the contact-range EFT approach to dynamically    In Refs.[14,79,80], the loop function is regularised in the dimensional regularization scheme, which shows that µ is around 1.5 GeV in the charm sector.To quantify the uncertainty of the renormalization scale, we vary µ from 1 GeV to 2 GeV in this work.For µ of 1.0 GeV, 1.5 GeV, and 2.0 GeV, the values of is almost independent of the renormalization scale, as can also seen from Eq. ( 18).The slight variation of f D * s0 (2317) stems from the weak dependence of the couplings g D * s0 DK and g D * s0 Dsη on the renormalization scale µ.However, the decay constant f D s1 (2460) is dependent on µ as shown in Table VI.
In the following calculations, we adopt the values of f D s0 * (2317) and f D s1 (2460) at µ = 1.5 GeV, e.g., f D * s0 = 58.74MeV and f D s1 = 133.76MeV, which are consistent with the results of Ref. [41], but smaller than the results of lattice QCD [23].
in the B → D ( * ) transition form factors and F (0) s transition form factors [64].B → D( * ) D s0 (D s1 ) , which are shown in Table V.Our results are a bit smaller than those of Ref. [41] because of the smaller values for the decay constants and the effective Wilson coefficient.Interestingly, our results are consistent with our previous calculations using the triangle mechanism except for the decay 3 , which indicate that the triangle diagram and tree diagram accounting for the decays B → D( * ) D s0 (D s1 ) are equivalent.In principle, one can replace the triangle diagram with one vertex, resulting in an effective description for the weak decay B + → D0 D * + s0 (2317) at tree level as shown in Fig. 3, which indicates that it is reasonable to extract the decay constants of hadronic molecules using the triangle mechanism.In Ref. [55], with this approach, we extract the decay constants of X(3872) as a DD * molecule.Here, we note that the relative phase among various amplitudes may lead to uncertainties in extracting the decay constants.As a result, it is better to select relevant amplitude with no or small relative phases.[65,66].as hadronic molecules via the triangle mechanism.This provides an effective approach calculating the decay constants of hadronic molecules, which can then be used in studies of these hadronic molecules in other related processes.We hope that our present work can stimulate more studies along this line.
and f Ds1(2460) , we calculate the branching fractions of the b-meson decays B (s) → D( * ) (s) D * s0 and B (s) → D( * ) (s) D s1 and the b-baryon decays Λ b (Ξ b ) → Λ c (Ξ c )D * s0 and Λ b (Ξ b ) → Λ c (Ξ c )D s1 .Our results show that the production rates of D * s0 (2317) and D s1 (2460) in the B s , Λ b and Ξ b decays are rather large that future experiments could observe them.In particular, we demonstrate that one can extract the decay constants of hadronic molecules via the triangle mechanism because of the equivalence of the triangle mechanism to the tree diagram established in calculating the decays B → D( * ) D * s0 (2317) and B → D( * ) D s1 (2460).
2317) → D + s π 0 breaks isospin.The dominant decays of D + s1 (2460) into D * + s π 0 and D + s γ are responsible for its narrow width.The narrow widths of D * s0 (2317) and D s1 (2460) are quite different from the widths of their SU(3)-flavor partners D * 0 (2300) and D 1 (2430), which reflect the exotic properties of these excited charmed mesons.In Refs.[9, 30-33], the authors proposed that the decays of D * s0 (2317) and D s1 (2460) as the cs excited states into D s π and D * s π proceed via the π − η mixing, resulting in widths of tens of keV.Treating D * s0 (2317) and D s1 (2460) as hadronic molecules, their widths are of the order of 100 keV [34-36].Up to now, there are no precise experimental measurements of the widths of D * s0 (2317) and D s1 (2460), but only their upper limits of 3.8 MeV.From the perspective of their widths, one can obtain the same conclusion as from the studies of their masses regarding the nature of D * s0 (2317) and D s1 (2460).It is worth noting that a model-independent method has been proposed to verify the molecular nature of D * s0 (2317) by experimental searches for its three-body counterparts DDK and D DK [19, 37-39].The discoveries of D * s0 (2317) and D s1 (2460) in the inclusive and exclusive processes in e + e − collisions triggered a series of theoretical works to investigate their production mechanism.Assuming D * s0 (2317) and D s1 (2460) as D ( * ) K hadronic molecules and cs excited states, Wu et al. estimated that their production rates in e + e − collisions are of the order of 10 −3 [39], consistent with the experimental data [40].As for the exclusive process, Faessler et al. calculated the decays B → D * s0 (2317) D( * ) and B → D s1 (2460) D( * ) assuming D * s0 (2317) and D s1 (2460) as D ( * ) K molecules [41].The results are a bit smaller than the experimental data.Assuming D * s0 (2317) and D s1 (2460) as cs excited states, the decays B → D * s0 (2317) D( * ) and B → D s1 (2460) D( * ) were investigated as well, but the results suffer from large uncertainties [42-47].
culate the productions of D * s0 (2317) and D s1 (2460) in B (s) and Λ b (Ξ b ) decays and the decay constants of f D * s0 and f D s1 .Results and discussions are given in Sec.III, followed by a summary in Sec.IV.
this work, we parameterize the non-factorization contributions with the effective Wilson coefficient a 1 , which can be determined by reproducing relevant experimental data.The current matrix elements of D0 |(c b)|B + and D * 0 |(c b)|B + describing the hadronic transitions are parameterized by six form factors[46] decay constants of D s and D * s as cs ground states are determined to be f Ds = 250 MeV and f D * s = 272 MeV [64].Due to the exotic properties of D * s0 (2317) and D s1 (2460), the estimations of the decay constants f D * s0 and f D s1 are quite uncertain.In this work, we estimate the values of f D * s0 and f D s1 in the molecular picture.In addition, assuming SU(3)-flavor symmetry, the B → D( * ) and Λ b → Λ c transitions can be related with the B s → D( * ) s and Ξ b → Ξ c transitions, and the production mechanism of D * s0 (2317) and D s1 (2460) in the B s and Ξ b decays are similar to those in the B and Λ b decays as illustrated in Fig. 1.In the following, we only present the amplitudes for the decays B → D( * ) D ( * ) s and Λ b → Λ c D( * ) s , and the amplitudes for the other decays have similar expressions.
a 1 and m, m 1 , m 2 referring to the masses of Λ b , D( * ) s , and Λ c .

FIG. 2 .
FIG. 2. Feynman diagrams for the W boson transiting to D * s0 (2317) in the DK and D s η molecular picture.

where m 1
and m 2 refer to D(D s ) and K(η) for D * s0 (2317) and D * (D * s ) and K(η) for D s1 (2460).The decay constants of D * s0 (2317) and D s1 (2460) are calculated as the sum of f

D0 1 /+ c 0 ( 1 / 2 + ) 2286. 46 D * 0 1 / 2 ( 1 −
2(0 − ) 1864.84 D − 1/2(0 − ) 1869.66 Λ ) 2006.85 D * − 1/2(1 − ) 2010.26 B + 1/2(0 − ) 5279.34 generate the poles corresponding to D * s0 (2317) and D s1 (2460) by varying µ and then obtain the D * s0 (2317) and D s1 (2460) couplings to their constituents as well as their decay constants.With the so-obtained decay constants f D s0 * (2317) and f D s1 (2460) we further study the productions of D * s0 (2317) and D s1 (2460) in the B (s) and Λ b (Ξ b ) decays, where the naive factorization approach works well as mentioned above.In this work, assuming that the decay mechanisms of H b → H c D ( * ) s (H b and H c denote bottom and charm hadrons of interest) and H b → H c D * s0 (D s1 ) are the same 1 , we parameterize the unknown non-factorization contributions with the effective Wilson coefficients.In other words, we determine a 1 by reproducing the experimental branching fractions of H b → H c D ( * ) s decays, and then calculate the branching fractions of the corresponding decays H b → H c D * s0 (D s1 ) using the so-obtained a 1 .
( * ) s mesons in the b-favored decays mainly occur via short-distance interactions, while those of the D * s0 (Ds1) mesons mainly occur via long-distance interactions due to the exotic properties of the D * s0 (Ds1) mesons.The effects of long-range interactions in these decays are induced by final-state interactions via triangle diagrams [76-78].In this work, the long-distance effects are embodied into the decay constants of D * s0 (Ds1) mesons, implying that the production mechanisms of the D * s0 (Ds1) mesons in b-flavored decays are similar to those of the D ( * ) s mesons.
FIG. 3. Equivalence of the triangle diagram and the tree diagram depict the decay of B + → D0 D * + s0 (2317).

70 B + → D0 D * + s 7 . 6 ± 9 B
1.6 0.81 B + → D0 D + s1 + → D * 0 D * + s 17.1 ± 2.4 0.83 B + → D * 0 D + s1 With the experimental branching fractions of the decays of B + → D0 D * + s0 (2317) and B + → D0 D * + s1 (2460), we obtain the decay constants f D * s0 = 75.83MeV and f D s1 = 199.75MeV, corresponding to physical D * s0 and D s1 as mixtures of molecular and cs components.With the obtained Along this line, we investigate the decays of B s → D( * ) s D ( * ) s and B s → D( * ) s D s0 (D s1 ), which are related to the decays of B → D( * ) D ( * ) s and B → D( * ) D * s0 (D s1 ) via SU(3)-flavor symmetry as shown in Fig. 1.The amplitudes for the decays of B s → D( * ) s D ( * ) s and B s → D( * ) s D s0 (D s1 ) are the same as those of their SU(3) symmetric partners.The unknown parameters in the form factors of the B s → D( * ) s transitions are taken from Ref. [64], tabulated in (2460) are likely to be detected in future experiments.In addition to the productions of D * s0 (2317) and D s1 (2460) in the B (s) decays, it is interesting to investigate their productions in the Λ b (Ξ b ) decays.As indicated in Fig. 1, the decays of Λ b → D ( * ) s Λ c and Λ b → D * s0 (D s1 )Λ c share the same mechanism as those of B → D ( * ) s D( * ) and B → D * s0 (D s1 ) D( * ) at quark level, which proceed via the decays b → ccs.In terms of SU(3)-flavor symmetry, we also investigate the decays of Ξ b → D ( * ) s Ξ c and Ξ b → D * s0 (D s1 )Ξ c .With the naive factorization approach, the amplitudes for these decays are given by the effective Lagrangian shown in Eq. (9), where the parameters in the form factors of the Λ b → Λ c and Ξ b → Ξ c transitions are obtained in the quark model [65, 66, 83], tabulated in Table VIII.The decay constants of the charmed-strange mesons D ( * ) s and D * s0 (D s1 ) are calculated in the same way as explained above.

2317) 0. 70 Λ
b → Λ c D * s 18.568 ± 1.102 0.76 Λ b → Λ c D s1 (2460Ξ b → Ξ c D s1 (2460) 4.29 One should note that only the branching fraction of the decay Λ b → D s Λ c is available in the RPP.Very recently the ratio of B(Λ b → D * s Λ c )/B(Λ b → D s Λ c ) = 1.688 ± 0.022 +0.061 −0.055 is reported by the LHCb Collaboration [84], and one can obtain the branching fraction of the decay Λ b → D * s Λ c .With the branching fractions of B(Λ b → D s Λ c ) and B(Λ b → D * s Λ c ) as inputs, we determine a 1 , and then predict the branching fractions of the decays of Λ b → D * s0 Λ c and Λ b → D s1 Λ c , which are shown in TableIX.We can see that the production rates of D * s0 (2317) and D s1 (2460) in the Λ b decay are of the order of 10 −3 , which are large enough to be detected in future experiments.Since the effective Wilson coefficients in the B decays and B s decays are similar, as shown in TableVand TableVII, we can take the same values for a 1 in the Λ b decays and the Ξ b decays.Similarly, we predict the branching fractions of the decaysΞ b → D ( * ) s Ξ c and Ξ b → D * s0 (D s1 )Ξ c in Table IX.The production rates of ground-state mesons D ( * ) s and excited mesons D * s0 (D s1 ) in the Ξ b decays are of the of 10 −2 and 10 −3 , which are likely to be detected in future experiments.In Ref. [48], the authors estimated the ratios B(Λ b → Λ c M )/B(B → DM ), where M represents the ground-state D ( * ) s and excited D * s0 (D s1 ) mesons.In TableX, we show the ratiosR M u = B(Λ b → Λ c M )/B(B → DM ) and R M s = B(Ξ b → Ξ c M )/B(B s → D s M) obtained in this work, which are consistent with Ref.[48].We can see that the production rates of ground-state mesons D ( * ) s and excited mesons D * s0 (D s1 ) in the Λ b (Ξ b ) decays are larger than those in the B (s) decays because the Λ b → Λ c (Ξ b → Ξ c ) form factors are larger than the corresponding B (s) → D (s) form factor as shown in Ref. [48].TABLEX.Ratios of R M u = B(Λ b → Λ c M )/B(B → DM ) and R M s = B(Ξ b → Ξ c M )/B(B s → D s M ).AND OUTLOOK In this work, we utilized the effective Lagrangian approach to compute the decay constants of D * s0 (2317) and D s1 (2460) as hadronic molecules dynamically generated by the DK − D s η and D * K − D * s η contactrange potentials, and then with the naive factorization approach systematically investigated the productions of D * s0 (2317) and D s1 (2460) in the B (s) and Λ b (Ξ b ) decays, which proceed via the decay b → ccs at quark level.The decay constants of D * s0 (2317) and D s1 (2460) are estimated to be 58.74MeV and 133.76 MeV.In particular, the decay constant of D * s0 (2317) is almost independent of the renormalization scale µ in the loop functions.As for the branching fractions of the decays B → D( * ) D * s0 (2317) and B → D( * ) D s1 (2460), our results are smaller than the experimental data, but are consistent with our previous results obtained in the triangle mechanism, which indicate that D * s0 (2317) and D s1 (2460) may contain components other than hadronic molecules, such as the cs cores.The values of B[B s → D( * ) s D * s0 (2317)] and B[B s → D( * ) s D s1 (2460)] are similar to those of B[B → D( * ) D * s0 (2317)] and B[B → D( * ) D s1 (2460)], which reflect the underlying SU(3)-flavor symmetry.In addition, we predicted the branching fractions of the decays Λ b → Λ c D * s0 (2317) and Λ b → Λ c D s1 (2460) as well as Ξ b → Ξ c D * s0 (2317) and Ξ b → Ξ c D s1 (2460), which are much larger than the corresponding ones in the B and B s decays and indicate that the productions of D * s0 (2317) and D s1 (2460) in the decays of bottom baryons are likely to be detected in future experiments.Our study shows that, because of the equivalence of the triangle mechanism to the tree diagram established in calculating the branching fractions of the decays B → D( * ) D * s0 (2317) and B → D( * ) D s1 (2460), one can extract the decay constants of D * s0 (2317) and D s1 (2460)

TABLE I .
[29]es and quantum numbers of relevant hadrons needed in this work[29].
1The productions of the D

TABLE VI .
[11,28]onstants of D * s0 (2317) and D s1 (2460) as the excited states (in units of MeV).MeV and f M D s1 = 133.76MeV in the molecular picture as well as the proportions of the molecular components in the total wave functions, e.g., 70% and 50%[11,28], one can obtain the decay MeV and f B D s1 = 265.74MeV, which correspond to the picture where D * s0 and D s1 are pure excited cs states.Table VI shows the decay constants of D * s0 and D s1 as pure cs excited states calculated by several approaches, which are consistent with our estimations and further support the picture where D * s0 and D s1 are mainly hadronic molecules but contain sizable cs components.As a result, it is understandable that the branching fractions of the decays B → D( * ) D * s0 (D s1 ) in our calculations are lower than the experimental data.
Table IV.Following the same strategy, we calculate the branching s D s1 .The results are shown in Table VII.One can see that the branching fractions of D * s0 (2317) and D s1 (2460) in the B s decays are similar to those in the B

TABLE VIII .
Values of F (0), a, b in the Λ b → Λ c and Ξ b → Ξ c transition form factors