The roles of the T c ¯ s 0 (2900) 0 and D ∗ 0 (2300) in the process B − → D + s K − π −

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I. INTRODUCTION
Since the charmonium-like state X(3872) was observed in the π + π − J/ψ invariant mass distribution of the process B ± → K ± π + π − J/ψ by the Belle Collaboration in the year of 2003 [1], many candidates of the exotic states have been reported experimentally, and called many theoretical attentions, which largely deepens our understanding of the hadron spectra and hadron-hadron interactions [2,3,5,31].
Recently, the LHCb Collaboration has reported two new states T c s0 (2900) 0 and T c s0 (2900) ++ in the D + s π − and D + s π + invariant mass distributions of the processes B 0 → D0 D + s π − and B + → D − D + s π + decays, respectively [24,25], where the significance is found to be 8.0σ for the T c s0 (2900) 0 state and 6.5σ for the T c s0 (2900) ++ state.Their masses and widths are determined as, M T c s0 (2900) 0 = (2892 ± 14 ± 15) MeV, Γ T c s0 (2900) 0 = (119 ± 26 ± 13) MeV, M T c s0 (2900) ++ = (2921 ± 17 ± 20) MeV, Γ T c s0 (2900) ++ = (137 ± 32 ± 17) MeV, (1) respectively.Their masses and widths are close to each other, which implies that T c s0 (2900) 0 and T c s0 (2900) ++ with the fla- * lidm@zzu.edu.cn† chendy@seu.edu.cn‡ wangen@zzu.edu.cnvor components c sūd and c su d should be two of the isospin triplet.Some interpretations for their structure are proposed theoretically such as the 'genuine' tetraquark state [26][27][28], the molecular state [29,30].Since the T c s0 (2900) lies close to the thresholds of D * s ρ and D * K * , the two-hadron continuum is expected to be of relevant for its existence, which make the T c s0 (2900) natural candidate for the molecular state [31,32].In Ref. [30], the authors argue that T c s0 (2900) ++ and T c s0 (2900) 0 may be modelled as molecules D * + s ρ + and D * + s ρ − , respectively, using two-point sum rule method.In addition, the T c s0 (2900) also can be considered as a virtual state created by the D * s ρ and D * K * interactions in coupled channels [33], and the further analysis of the D * K * interaction in a coupledchannel approach favors the T c s0 (2900) as a bound/virtual state [29].Thus, in this work we would like to study the production of T c s0 (2900) with the molecular scenario, which is expected to be checked by future experiments.
As we known, many candidates of the exotic states were observed in the decays of B meson, and Belle and LHCb Collaborations have accumulated many events of B mesons, which provides an important lab to study the hadron resonances [3,[34][35][36][37][38][39].For instance, we have proposed to search for the open-flavor tetraquark T c s0 (2900) ++ in the process B + → K + D + D − by assuming T c s0 (2900) ++ as a D * + K * + molecular state in Ref. [40].According to the Review of Particle Physics (RPP) [41], one can find that the branching fraction of the process , in the same order of magnitude with the branching fraction of the process B + → D − D + K + [(2.2±0.7)×10−4 ], which implies that it is reasonable to search for the T c s0 (2900) in the process B + → D − s π + K + .It should be pointed out that the processes B + → D − s π + K + and B − → D + s π − K − have been measured by the BABAR [42] and Belle Collaborations [43], and the D ± s K ∓ invariant mass distributions exhibit strong enhancements near the threshold.The same threshold enhancements have also appeared in the D s K invariant mass distribution in the processes 44], which could be due to the contribution from the high pole of the D * 0 (2300) with two-pole structure in the unitarized chiral perturbation theory [45,46] 1 .Thus, in this work we would investigate the process B − → D + s K − π − by taking into account the S -wave D * + s ρ − and D * 0 K * 0 interactions, which will generate the resonance T c s0 (2900).In addition, we will also consider the contribution from the S -wave pseudoscalar meson-pseudoscalar meson interactions within the unitary chiral approach, which will dynamically generate the resonance D * 0 (2300).This paper is organized as follows.In Sec.II, we present the theoretical formalism of the process B − → D + s K − π − .Numerical results and discussion are shown in Sec.III.Finally, we give a short summary in the last section.

II. FORMALISM
In this section, we will present the theoretical formalism of the process FIG. 1. Quark level diagrams for the process B − → ρ − D * + s K − via the W − external emission (a), and the process B − → K * 0 D * 0 K − via the W − external emission (b). 1 The state D * 0 (2300) was denoted as D * 0 (2400) in previous versions of RPP, and more discussions about this state can be found in the review 'Heavy Non-q q mesons' of RPP [41].
A. The T c s0 (2900) role in Quark level diagrams for the process B − → ρ − D * + s K − process via the W − internal emission (a), and the process Taking into account that T c s0 (2900) could be explained as the molecular state of the D * s ρ and D * K * interactions [33], we first need to produce the states D * + s ρ − K − and D * 0 K * 0 K − via the external W − emission mechanism and the internal W − emission mechanism, as depicted in Figs. 1 and 2, respectively.
In analogy to Refs.[34,[47][48][49][50][51], as depicted in Fig. 1(a), the b quark of the initial B − meson weakly decays into a c quark and a W − boson, then the W − boson decays into ūd quarks.The ūd pair from the W − boson will hadronize into ρ − , while the ū quark of the initial B − meson and the c quark, together with the ss created from vacuum, hadronize into K − and D * + s .On the other hand, as shown in Fig. 1(b), the ūd quarks from the W − boson, together with the s s created from vacuum, hadronize into K − and K * 0 , while the c quark and the ū from the initial B − meson could hadronize into the D * 0 .
In Fig. 2(a), the c quark from the B − and the ū from the W − boson, together with the ss created from vacuum, hadronize into D * + s and K − , while the d quark from the W − boson and the ū quark of the initial B − meson, hadronize into vector meson ρ − .On the other hand, the c quark from the B − and the ū from the W − boson could also hadronize into a D * 0 , while the d quark from the W − boson and the ū quark of the initial B − meson, together with the ss created from vacuum, hadronize into mesons K * 0 K − , as shown in Fig. 2(b).It should be pointed out that the mechanisms of Figs.2(a) and 2(b) are 1/N c suppressed with respect to the ones of Fig. 1.
Then, the S -wave interactions of ρ − D * + s and D * 0 K * 0 will give rise to the T c s0 (2900) 0 state, which could decay into the final state D + s π − , as depicted in Fig. 3.The transition amplitude for the process where ⃗ ϵ(V 1 ) and ⃗ ϵ(V 2 ) are the polarization vectors of ρ − D * + s (or D * 0 K * 0 ), and we have the relation of pol ϵ i (V)ϵ j (V) = δ i j .The constant Q includes all the dynamical factors of the weak decay of Fig. 2 2 , while the factor C = 3 corresponds to the relative weight of the W − external emission mechanism (Fig. 1) with respect to the W − internal emission mechanism (Fig. 2) [52][53][54][55].Thus, one could easily obtain the expression of the Q as follows, Since the branching fraction is measured to be B(B − → D * 0 K * 0 K − ) = (1.5 ± 0.4) × 10 −3 [41], we could roughly estimate Q 2 = 1.71 × 10 −13 neglecting the contributions from the possible intermediate resonances.
By taking into account the contributions from the S -wave ρ − D * + s and D * 0 K * 0 interactions of Fig. 3, the amplitude could be expressed as, where G ρ − D * + s and G D * 0 K * 0 are the loop functions of the coupled channels ρ − D * + s and D * 0 K * 0 , respectively, and , respectively.Both of loop functions G i and transition amplitudes t i j are the functions of the D + s π − invariant mass M D + s π − .The two-meson loop function is given by, where m 1 and m 2 are the mesons masses of the i-th coupled channel.q is the four-momentum of the meson 1 in the center of mass frame, and P is the four-momentum of the mesonmeson system.In the present work, we use the dimensional regularization method as indicated in Refs.[40,52], and in this scheme, the two-meson loop function G i can be expressed as, where s = P 2 = M 2 inv (D + s π − ), and ⃗ q is the three-momentum of the meson in the centre of mass frame, which reads, s interaction [8,33] and the D * 0 K * 0 interaction [33,40,56].And the transition amplitudes are given by, where the m T 0 c s0 and Γ T 0 c s0 are given by Refs.[24,25].The constant g T 0 c s0 ,D * 0 K * 0 corresponds to the coupling between T c s0 (2900) 0 and its components D * 0 K * 0 , which could be related to the binding energy by the Weinberg compositeness criterion [40,[57][58][59], where λ = 1 gives the probability to find the molecular component in the physical states.In this work we assume T c s0 (2900) 0 as the D * K * molecular state, and neglect possible D * s ρ component, as done in Ref. [40].∆E = m D * + m K * − m T c s0 denotes the binding energy, and µ = m D * m K * /(m D * + m K * ) is the reduced mass.Here we obtain g T 0 c s0 ,D * 0 K * 0 = 8809 MeV with Eq. (10).
Since the mass of T c s0 (2900) 0 is larger than the thresholds of D * s ρ and D s π, the coupling constants g T 0 c s0 ,ρ − D * + s and g T 0 c s0 ,D + s π − could be obtained from the partial widths of T c s0 (2900) 0 → ρ − D * + s and T c s0 (2900) 0 → D + s π − , respectively, which could be expressed as follows, where with the Källen function λ(x, y, z) = x 2 +y 2 +z 2 −2xy−2yz−2zx.In Ref. [60], the partial widths of decay modes ρ − D * + s and D + s π − were estimated to be (2.96∼ 5. where i = 1, 2, 3 correspond to the u, d, and s quarks, respectively, and P is the U(4) matrix of the pseudoscalar mesons, where we have taken the approximate η − η ′ mixing from Ref. [61]. 3hen, we could have all the possible pseudoscalar-pseudoscalar pairs after the hadronization, With the isospin multiplets of (D + , −D 0 ), ( D0 , D − ), and (−π + , π 0 , π − ), we have, In the isospin basis, we can obtain the Dπ(I = 1/2), Dη(I = 1/2), and D s K(I = 1/2) channels Then, the process B − → D + s K − π − decay could happen via the tree diagram of Fig. 5(a), and the S -wave meson-meson interaction of Fig. 5(b), and the amplitude could be expressed as, where the constant Q ′ includes all the dynamical factors of the weak decay, and i = 1, 2, 3 correspond to the Dπ, Dη, D s K, respectively, The factor C = 3 corresponds to the relative weight of the W − external emission mechanism [Fig.4(a)] with respect to the W − internal emission mechanism [Fig.4(b)].With the amplitude of Eq. ( 22), the Q ′ is given by, According to the experimental measurements of B(B − → The G i in Eq. ( 22) is the loop function of the meson-meson system, and t i→D s K are the scattering matrices of the coupled channels.The transition amplitude of t i→D s K is obtained by solving the Bethe-Salpeter equation, The transition potential V i j is taken from Refs.[63,64], where the coefficients are given as, with f π = 93 MeV.The loop function G i in Eq. ( 25) is given by the dimensional regularization method, as shown by Eq. ( 5), and we take µ = 1000 MeV and a = −1.88[64].

C. Invariant Mass Distribution
With above the formalism, one can write down the invariant mass distribution for the where the modulus squared of the total amplitude is, with a phase ϕ between two terms.For a given value of invariant mass M 12 , the range of invariant mass M 23 is determined by [41], where E * 2 and E * 3 are the energies of particles 2 and 3 in the M 12 rest frame.E * 2 and E * 3 are written as, where m 1 , m 2 , and m 3 are the masses of particles 1, 2, and 3, respectively.All the masses and widths of the particles are taken from the RPP [41]. )

III. NUMERICAL RESULTS
6. Modulus squared of the transition amplitudes t i→Ds K in Swave.dΓ/dM π, and π.Belle data have been rescaled for comparison [43].
We first show the transition amplitude t i→D s K of Eq. ( 25) in Fig. 6.The red-dotted curve shows the modulus squared of the transition amplitude t D s K→D s K , the blue-dashed curve shows the modulus squared of the transition amplitude t Dη→D s K , and the black-solid curve shows the modulus squared of the transition amplitude t Dπ→D s K .One can find that the modulus squared of the transition amplitude |t Dπ→D s K | 2 has two peaks around 2100 MeV and 2450 MeV, respectively, in consistent with the conclusion of Ref. [63].Since the lower pole is far from the D s K threshold, the enhancement structure near the D s K threshold of the process B − → D + s K − π − should be mainly due to the contribution from the high pole.
In our formalism, we only have one free parameter, the phase ϕ of Eq. ( 29).Thus, we present our results of the D + s K − and D + s π − invariant mass distributions with the different values of phase ϕ = 0, 1  3 π, 2 3 π, and π in Figs.7 and 8, respectively.We also show the Belle measurements on the D + s K − invariant mass distribution of the B − → D + s K − π − events in Fig. 7, where the Belle data have been rescaled for comparison [43] 4 .One can find that, with different values of the phase ϕ, our results of the D + s K − invariant mass distributions are in good agreement with the Belle measurements in the region 2600 ∼ 4000 MeV, and the enhancement near the threshold should be due to the resonance D * 0 (2300).In Fig. 8, one can find a clear peak around 2900 MeV of the D + s π − invariant mass distribution, which could be associated to the T c s0 (2900), and the lineshape of the peak is distorted by the interference with different values of phase ϕ.
However, in the high energy region of the D + s K − invariant mass distribution of Fig. 7, our results are smaller than the Belle measurements [43], which implies that the contribution from the T c s0 (2900) 0 may be underestimated.Thus, we take the decay width Γ T 0 c s0 →D + s π − and the phase ϕ to be free parameters, and fit them to the D + s K − invariant mass distribution of the Belle measurements [43], and obtain the χ 2 /N do f = 3.39, and the fitted parameters Γ T 0 c s0 →D + 10.45 ± 1.31 MeV is close to the upper limit of the prediction of Ref. [60].With these fitted parameters, we have shown the D + s K − and D + s π − invariant mass distributions in Figs.9(a) and 9(b), respectively.One can find that, our results of the D + s K − invariant mass distribution are in good agreement with the Belle measurements in the region 2600∼4800 MeV [43], and the peak of the T c s0 (2900) 0 in the D + s π − invariant mass distribution is more significant.Meanwhile, we also predict the Dalitz plot of "M D + s π − " vs. "M D + s K − " for the process B − → D + s K − π − in Fig. 10, and one can find that the T c s0 (2900) 0 mainly contributes to the high energy region of the D + s K − invariant mass distribution.Our predictions could be tested by future measurements.
In this work, we assume the coupling constants appeared in Eqs. ( 8) and ( 9) are real and positive.Indeed, the coupling constants could complex, thus we multiply the Eq. ( 8) by an interference phase factor e iϕ ′ to account for this effect.With the fitted parameters Γ T 0 c s0 →D + s π − = 10.45MeV and ϕ = 0.35π, we have presented the D + s K − and D + s π − invariant mass distributions for phase ϕ ′ = 0, 1  3 π, 2 3 π, and π in Figs.11(a) and 11(b), respectively.One can find that, the D + s K − invariant mass distribution has a minor change, and the strength of the T c s0 has some change.However, the most important is that, the peak position does not change, and is always very clear for different values of phase ϕ ′ .
One maybe note that the amplitude T D * 0 (2300) of Eq. ( 22) has two terms, T tree and T S .Since the G i t i→D s K , involved in the term T S , has included the dynamical information and is complex, the extra phase factor between T tree and T S is not needed.However, in Fig. 12, we also show the results of the D + s K − and D + s π − invariant mass distributions by multiplying T tree by an extra phase factor e iα .One could find the lineshapes of the D + s K − invariant mass distribution with non-zero α are significantly different with the experimental data, which implies that one donot need to consider the extra phase factor between T tree and T S .
In Ref. [44], the Belle Collaboration has reported the D − s K 0 s invariant mass distribution of the process B 0 → D − s K 0 s π + .With the same formalism as given in this work, we could determine the corresponding Q 2 = 1.59 × 10 −13 and Q ′2 = 1.02 × 10 −12 with the branching fractions B(B 0 → D * K * K) = (1.29 ± 0.33) × 10 −3 [41] and B(B 0 → D − s K 0 s π + ) = (0.47 ± 0.06±0.05)×10−4 [44].Furthermore, we obtain the χ 2 /N do f = 0.95, and the Γ T 0 c s0 →D + s π − = (4.99 ± 0.01) MeV and ϕ = (−0.64± 0.61)π by fitting to the Belle data.The fitted width is in agreement with the result of Ref. [60], and we show the D − s K 0 s and D − s π + invariant mass distributions in Fig. 13.One can find that our prediction of the D − s K 0 s invariant mass distribution are in good agreement with the Belle data [44], and one peak around 2900 MeV is expected to be observed in the   0 (2300).We have found that there is a near-threshold enhancement in the D + s K − invariant mass distribution, which is in good agreement with the Belle measurements.Indeed, this enhancement structure is mainly due to the high pole of the D * 0 (2300).In addition, a clear peak structure appears around 2900 MeV in the D + s π − invariant mass distribution, which should be associated to the T c s0 (2900).
Considering that our predictions for the D + s K − invariant mass distribution are lower than the Belle measurements in the high energy region, we take the decay width Γ T 0 c s0 →D + s π − and the phase between two amplitudes ϕ to be free parameters, and obtain Γ T 0 c s0 →D + s π − = (10.45±1.31)MeV and ϕ = (0.35±0.09)π by fitting to Belle measurements.Our new results show a more significant peak of T c s0 (2900) in the D + s π − invariant mass distribution.Furthermore, we have also discussed the here we take µ ρ − D * + s = µ D * 0 K * 0 = 1500 MeV, and a ρ − D * + s = a D * 0 K * 0 = −1.474,which are the same as those used in the study of the ρ − D * +
The reaction mechanism via the intermediate state T c s0 (2900) is given in Sec.II A, while the mechanism via the intermediate state D * 0 (2300) is given in Sec.II B. Finally, we give the formalism of the invariant mass distributions of the process