K ∗ 0 ( 1430 ) twist-2 distribution amplitude and B s , D s → K ∗ 0 ( 1430 ) transition form factors

Based on the scenario that the K ∗ 0 ( 1430 ) is viewedasthegroundstateof s ¯ q or q ¯ s ,westudythe K ∗ 0 ( 1430 ) leading-twist distribution amplitude (DA) φ 2 ; K ∗ 0 ( x , μ) with the QCD sum rules in the framework of background ﬁeld theory. A more reasonable sum rule formula for ξ -moments (cid:3) ξ n (cid:4) 2 ; K ∗ 0 issuggested,whicheliminatestheinﬂuencebrought by the fact that the sum rule of (cid:3) ξ 0 p (cid:4) 3 ; K ∗ 0 cannot be normalized in whole Borel region. More accurate values of the ﬁrst ten ξ - moments, (cid:3) ξ n (cid:4) 2 ; K ∗ 0 ( n = 1 , 2 , . . . , 10 ) , are evaluated. A new light-cone harmonic oscillator (LCHO) model for K ∗ 0 ( 1430 ) leading-twist DA is established for the ﬁrst times. By ﬁt-ting the resulted values of (cid:3) ξ n (cid:4) 2 ; K ∗ 0 ( n = 1 , 2 , . . . , 10 ) via the least squares method, the behavior of K ∗ 0 ( 1430 ) leading-twist DA described with LCHO model is determined. Further, by adopting the light-cone QCD sum rules, we calculate the B s , D s → K ∗ 0 ( 1430 ) transition form factors and branching fractions of the semileptonic decays B s , D s → K ∗ 0 ( 1430 )(cid:4)ν (cid:4) . The corresponding numerical results can be used to extract the Cabibbo-Kobayashi-Maskawa matrix elements by combining the relative experimental data in the future.


I. INTRODUCTION
Currently, the exclusive determinations of the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements |V ub | and |V cd | are mainly dominated by the semileptonic B → πℓν and D → πℓν decays, respectively [1].The study of the semileptonic B s , D s → K * 0 (1430)ℓν ℓ decays can provide a new choice and supplement for extracting these two matrix elements.
In the theoretical studies on the semileptonic B s , D s → K * 0 (1430) decays, the most challenging parts are the calculations of the transition form factors (TFFs).These TFFs are mainly dominated by the short-distance dynamics in the large recoil region and the soft dynamics in the region of small recoil.Although the momentum dependence of the B s → K * 0 (1430) TFFs calculated by Faustov and Galkin in the relativistic quark model (RQM) is determined in the whole accessible kinematical range [2], the commonly used research methods such as perterbative QCD (pQCD) factorization, QCD sum rules (QCDSRs), light-cone QCD sum rules (LCSRs) and Lattice QCD (LQCD) theory, etc., are usually only applicable to calculate the TFFs in specific q 2 regions.The corresponding results should be extrapolated to the whole kinematic region by adopting appropriate parametrization form.Usually, the LQCD works well in the small recoil region.However, there are currently no LQCD calculations of the B s , D s → K * 0 (1430) TFFs in the literature.The pQCD factorization method is usually applicable near the large recoil point, and which has been used to calculate B s → K * 0 (1430) TFFs in Refs.[3][4][5].The QCDSR and LCSR estimations for TFFs are effective in low and intermediate q 2 regions.In Ref. [6], the TFFs f Bs→K * 0 + (q 2 ) and f Ds→K * 0 + (q 2 ) are calculated by adopting the QCDSRs with three-point correlation function (correlator).In which, the TFFs are parameterized as vacuum condensates with different dimension.The three-point QCDSRs is then used to calculate TFFs f Bs→K * 0 ±,T (q 2 ) and further analyze the rare semileptonic B s → K * 0 (1430)ℓ + ℓ − decays [7].More researches on the B s → K * 0 (1430) TFFs are performed in the framework of LCSRs.Different from the three-point QCDSRs, the TFFs calculated within LCSRs are parameterized as the initial or final meson distribution amplitudes (DAs) arranged with different twist structures.Starting from different correlator, the LCSRs of B s → K * 0 (1430) TFFs are expressed as the convolution integrals of the B s meson DAs [8], the K * 0 (1430) twist-2, 3 DAs [9][10][11], only K * 0 (1430) twist-2 DA [12] or twist-3 DAs [13], respectively.In addition, there are some other studies on the D s → K * 0 (1430) TFFs in the literature.For example, Ref. [14] extracts the TFF f In this paper, we will calculate the B s , D s → K * 0 (1430) TFFs within the LCSR method by adopting the traditional current correlator.Specifically, we will adopt the calculation technology for the operator product expansion (OPE) in Refs.[15,16].That is, the usual suppression by the powers of the Borel parameter for the higher twist contributions is transferred as exponential suppression through the integration by parts.
Usually, the LCSRs of the B s , D s → K * 0 (1430) TFFs are dominated by the contributions proportional to the K * 0 (1430) leading-twist DA φ 2;K * 0 (x, µ).Then φ 2;K * 0 (x, µ) is the mainly error source of these TFFs.In order to obtain accurate predictions for the semileptonic B s , D s → K * 0 (1430) decays, one should determine the accurate behavior of the K * 0 (1430) leading-twist DA.Before this, however, we should clarify our understanding of quark content of scalar K * 0 (1430) meson.Different from the scalar mesons below 1 GeV − for which there is no general agreement on the multiple candidates such as conventional q q states [17], meson-meson molecular states [18], tetraquark states [19][20][21] and so on − the K * 0 (1430) is predominantly viewed as the sq or qs state in almost all the relative researches.The only controversy about K * 0 (1430) lies between the following two scenarios: Scenario 1 (S1), the K * 0 (1430) is assumed to be the excited state corresponding to the ground state below 1 GeV; Scenario 2 (S2), the K * 0 (1430) is viewed as the ground state while the nonet scalars below 1 GeV may be considered as four-quark bound states.Ref. [22] calculates the masses and decay constants of I = 1/2 scalar mesons with QCDSRs.Its result favors that K * 0 (1430) is the lowest scalar stats of sq or qs, i.e., S2.Later, based on S2, Refs.[13,23] study the K * 0 (1430) twist-3 DAs with QCDSRs.Besides, the research on the two-body decays of B (s) containing K * 0 (1430) in Ref. [5] also supports that K * 0 (1430) should be described as the lowest-lying p-wave state rather than the first excited one.Therefore, our research work on K * 0 (1430) in this paper will also take S2 as the starting point.In numerical analysis, only the data corresponding to S2 in literature will be adopted for consistency.
The φ 2;K * 0 (x, µ) has been investigated in Ref. [17] with QCDSRs, and in Ref. [5] with light-front (LF) approach.In Ref. [17], the first two nonzero ξ-moments and Gegenbauer moments are calculated and substituted into the truncation form of the Gegenbauer expansion series (TF model) of φ 2;K * 0 (x, µ) to predict the behavior of the K * 0 (1430) leading-twist DA.However, our recent analysis on the commonly used phenomenological model of the pionic leading-twist DA shows that the simple TF model is far from sufficient to describe the accurate behavior of DA [24].Therefore, we will re study the K * 0 (1430) leading-twist DA with the QCDSRs in the framework of background field theory (BFT) [25].In order to obtain more accurate behavior of φ 2;K * 0 (u, µ), a research scheme suggested in our previous study on the pionic leadingtwist DA [26] will be adopted.This scheme has been used to researches on the kaon leading-twist DA [27] and the a 1 (1260) meson longitudinal twist-2 DA [28].Specifically, we will construct a light-cone harmonic oscillator (LCHO) model based on the Brodsky-Huang-Lepage (BHL) description [29] for φ 2;K * 0 (u, µ).The model parameters will be determined by fitting the first ten ξmoments with the least squares method.More accurate values of those ξ-moments will be evaluated by using a new sum rule formula suggested in Ref. [26].
This paper is organized as follows.In Sec.II, the sum rules for the ξ-moments of the K * 0 (1430) leadingtwist DA and the LCSRs of the B s , D s → K * 0 (1430) TFFs f ±,T (q 2 ) are derived, a new LCHO model for the K * 0 (1430) leading-twist DA is established at the first time.In Sec.III, we provide the relevant numerical results.The final section is reserved for a summary.As a by-product, the numerical calculations for the K * 0 (1430) leading-twist DA and B s , D s → K * 0 (1430) TFFs and branching fractions corresponding to S1 are performed, and shown in Appendix B.

II. THEORETICAL FRAMEWORK
A. Sum rules for the ξ-moments of φ 2;K * 0 (x, µ) The leading-twist DA φ 2;K * 0 (x, µ) of the scalar K * 0 (1430) + meson with quark content us are given by [17] 0 where z 2 = 0, fK * 0 and m K * 0 are the decay constant and mass of K * 0 (1430) meson, respectively.Expanding both sides of Eq. ( 1) into series of z, one can get 0|s(0)γ µ (iz where ) is the fundamental representation of the gauge covariant derivative, and is the nth ξ-moment.Then, in order to derive the sum rules of the ξ-moments ξ n 2;K * 0 , we introduce the following correlator, with the interpolating currents We now perform the OPE for the correlator (4) in the deep Euclidean region.The calculation is carried out in the framework of BFT [25].By decomposing quark and gluon fields into classical background fields describing nonperturbative effects and quantum fields describing perturbative effects, BFT can provide clear physical images for the separation of long-and short-range dynamics in OPE.Based on the basic assumptions and Feynman rules of BFT [25], the correlator (4) can be rewritten as where Tr indicates trace for the γ-matrix and color matrix, S s F (0, x) indicate the s-quark propagator from x to 0, S u F (x, 0) stands for the u-quark propagator from 0 to x, / z(iz • ↔ D) n are the vertex operators from current J n (x), respectively.The expressions up to dimension-six of the quark propagator, the vertex operator, and the vacuum matrix elements such as 0|s(x)s(0) and 0|ū(0)u(x) have been derived and given in Refs.[26,[30][31][32].By substituting those corresponding formula into Eq.( 6), the OPE of correlator (4), I qcd 2;K * 0 (q 2 ), can be obtained.
By inserting a complete set of hadronic states into correlator (4) in physical region, whose hadronic representation can be read as where s K * 0 is the continuum threshold, ξ 0 p 3;K * 0 is the zeroth ξ-moment of K * 0 (1430) + two-particle twist-3 DA φ p 3;K * 0 (u, µ).In the calculation of Eq. ( 7), the matrix element formula in Eq. ( 2), 0 and the quark-hadron duality have been used.
Substituting the resulted OPE and hadronic representation of correlator (4), i.e., I qcd 2;K * 0 (q 2 ) and ImI had 2;K * 0 (s), into the following dispersion relation after Borel transformation, (8) the sum rules of ξ n 2;K * 0 × ξ n p 3;K * 0 reads: s )-corrections, respectively.The specific expressions for those terms are exhibited in Appendix A for convenience.In Eq. ( 9), in addition, M is the Borel parameter, m u and m s are the current quark masses of u and s quarks, ūu and ss are doublequark condensates with ss / ūu = κ, g s ūσT Gu and g s sσT Gs are quark-gluon mixed condensates, g s ūu 2 and g s ss 2 are four-quark condensates.In the calculation of OPE, the SU f (3) breaking effect is considered.Specifically, the full s quark mass effect in the perterbative part is preserved; the s quark mass corrections proportional to m ≤3 s for condensate terms are calculated owing to m s ∼ 0.1GeV, while m 2 u ∼ 0 is adopted due to smallness.
In particular, the sum rules ( 9) is regarded as that for ξ n 2;K * 0 × ξ n p 3;K * 0 instead of ξ n 2;K * 0 in this work due to dependence of ξ n p 3;K * 0 on the Borel parameter as suggested in Ref. [26].This assumption can be confirmed by the sum rule of ξ n p 3;K * 0 derived from the correla-tor i d 4 xe iq•x 0| Ĵ0 (0) Ĵ † 0 (0)|0 .Following the above sum rule calculation procedure performed for correlator (4), one can easy obtain, As discussed above and suggested in Ref. [26], a better sum rules for ξ n 2;K * 0 is suggested as It should be noted that, in the numerical calculations about ξ n 2;K * 0 in Sec.III, we take the scale µ = M as usual.Then the scale dependency of ξ n 2;K * 0 is achieved through the Borel parameter M and the scale dependency of input parameters such as various vacuum condensates, quark masses, K * 0 (1430) decay constant, etc.
B. LCHO model for φ 2;K * 0 (x, µ) based on BHL prescription The K * 0 (1430) leading-twist DA, φ 2;K * 0 (x, µ), describes the momentum fraction distribution of partons in K * 0 (1430) meson for the lowest Fock state.The φ 2;K * 0 (x, µ) is a universal nonperturbative objects, and which should be studied with nonperturbative QCD.However, we usually can only use the method of combining nonperturbative QCD and phenomenological model to study φ 2;K * 0 (x, µ) due to the difficulty of nonperturbative QCD.In Sec.II A, we have calculated the nth ξmoment with nonperturbative QCDSR method.In this subsection, we will construct a LCHO model to describe the overall behavior of φ 2;K * 0 (x, µ) based on BHL prescription [29].
The starting point of BHL prescription is the assumption that there is a connection between the equal-times wave function (WF) in the rest frame and the light-cone WF.Through this assumption, one can map the approximate bound state solution in the quark model for meson in the rest frame to the light-cone frame by equating the off-shell propagator in the two frames, thus obtaining the LCHO model of the meson WFs [33].The LCHO model has good end point behavior, which is helpful to suppress the end point singularity in certain processes, so as to obtain more reliable theoretical predictions.So far, LCHO model has been widely used in the study of various meson WFs or DAs and has been continuously improved [26-28, 31, 33-47].Formally, the WF of K * 0 (1430) leading-twist DA can be expressed as where k ⊥ is the transverse momentum, χ 2;K * 0 (x, k ⊥ ) stands for the spin-space WF coming from the Wigner-Melosh rotation.As a scalar meson, the spin WF of K * 0 (1430) should be [27] where m = mq x + ms x with q = u/d and x = 1 − x. mq and ms are the corresponding constituent quark masses of K * 0 (1430), and we take ms = 370 MeV and mq = 250 MeV as discussed in Ref. [27].The Ψ R 2;K * 0 (x, k ⊥ ) is the spatial wave function, and which can be divided into the x-dependent part, i.e., ϕ 2;K * 0 (x), dominating WF's longitudinal distribution, and the k ⊥ -dependent part arising from harmonic oscillator solution for meson in the rest frame.Then, the Ψ R 2;K * 0 (x, k ⊥ ) can be written as: with where A 2;K * 0 is the normalization constant, β 2;K * 0 is the harmomous parameters that dominates the WF's transverse distribution, C 2 (2x− 1).We take B2;K * 0 ≃ −0.025 in order to make the undetermined model parameters as few as possible.The value of B2;K * 0 is taken by referring to the ratio of the second and first ξ-moments calculated in Sec.III A, i.e., ξ 2 2;K * 0 / ξ 1 2;K * 0 , and whose rationality can be judged by the goodness of fit.
There is a relationship between the K * 0 (1430) leadingtwist DA and its WF, Substituting the WF formula ( 14) with Eqs. ( 15), ( 16) and ( 17) into ( 18) and after integrating over the transverse momentum k ⊥ , the K * 0 (1430) leading-twist DA reads It can be seen from the above derivation process that the scale dependence of DA φ 2;K * 0 (x, µ) is derived from the upper limit of the transverse momentum integral in Eq. ( 18) on the one hand (which causes the scale µ to appear in the error function explicitly), and from the scale dependence of wave function Ψ 2;K * 0 (x, k ⊥ ) on the other hand (carried by model parameters A 2;K * 0 , β 2;K * 0 and α 2;K * 0 ).In the numerical calculation of determining the behavior of φ 2;K * 0 (x, µ), the scale of DA φ 2;K * 0 (x, µ) matches the corresponding scale of the values of ξ-moments ξ n 2;K * 0 via the definition (3).
It needs to be clear that the LCHO models for WF Ψ 2;K * 0 (x, k ⊥ ) and DA φ 2;K * 0 (x, µ) established above are the same for K * 0 (1430) + and K * 0 (1430) 0 due to the isospin symmetry between the u and d quarks.The leadingtwist WF and DA of K * 0 (1430) 0 and K * 0 (1430) − can be obtained by replacing x with x in Eqs. ( 14) and (19).Now, there are three unknown model parameters such as A 2;K * 0 , β 2;K * 0 and α 2;K * 0 .In order to definitively describe the behavior of K * 0 (1430) leading-twist DA with the LCHO model ( 19), these three parameters can be determined by fitting the ξ-moments with the least squares method as fitting parameters.For specific fitting procedure, one can refer to Refs.[26,27].
C. Bs, Ds → K * 0 TFFs within LCSRs In order to uniformly express the derivation process of the LCSRs for B s , D s → K * 0 (1430) TFFs, we introduce the following vacuum-to-K * 0 (1430) correlators In Eq. ( 20), the light quark q 1 = s, q 2 = u and the heavy quark 1430) decay; the light quark q 1 = s, q 2 = d and the heavy quark Q with the heavy quark mass m Q , the heavy quark propagating in the correlator is highly virtual and the distances are near the lightcone [15].Thus one can contract the heavy quark fields, and the following light-cone expansion of the heavy quark propagator, enters the correlator.The vacuum-to-K * 0 (1430) matrix element can be expanded in terms of the K * 0 (1430) lightcone DA's of growing twist.That is, with the K * 0 (1430) two-particle twist-3 DA φ σ 3;K * 0 (u, µ).In which, the Fock components of the K * 0 (1430) with multiplicities larger than two as well as the twists higher than 3 are neglected due to that only the free propagator is retained in Eq. ( 21).This truncation is reasonable and has to be done because we almost know nothing about those components and their contributions are usually small.Then the OPE for the invariant amplitudes F , F and F T can be obtained as respectively.
One can also insert a complete set of hadronic states between the currents in correlator (20) to obtain the hadronic representations of the invariant amplitudes.In which, the TFFs f ±,T (q 2 ) enters the correlator (20) via the hadronic matrix elements for the interpolating currents indicating the weak transition of Q to q 2 .They can be parameterized in terms of the TFFs f ±,T (q 2 ) as Otherwise, the vacuum-to-meson matrix element for the interpolating current representing the H q1 channel can be given by with the heavy meson mass m Hq 1 and decay constant f Hq 1 .Then the hadronic representations of the invariant amplitudes F , F and F T can be written as respectively.In Eq. ( 26), the ground state heavy meson contributions have been isolated, and the ellipses indicate the contributions from the excited states, the continuum states and possible subtraction terms.
Without losing generality, we take the invariant amplitude F (q 2 , (p + q) 2 ) as an example to illustrate the subsequent calculation procedure.One can write a general dispersion relation for F (q 2 , (p + q) 2 ) and further apply the Borel transformation with respect to the momentum squared (p + q) 2 of the heavy meson [48], with In which, the spectral density is given by where the contributions of the excited states and continuum states in F had (q 2 , s) have been parameterized as ImF qcd (q 2 , s)/π and been delimited by the effective threshold parameter s Hq 1 with the quark-hadronic duality approximation, and the possible subtractions will be got rid of due to Borel transformation in Eq. ( 27).Substituting Eq. ( 28) into Eq.( 27), one can get ImF qcd (q 2 , s)e −s/M 2 ds.
On the other hand, the invariant amplitude after Borel transformation can also be written as [48] Finally, equating Eqs. ( 29) with (30), the LCSR of TFF f + (q 2 ) can be obtained as Similarly, with In particular, the upper limit of the integral of variable u in LCSRs ( 31), ( 32) and ( 33) is u 0 instead of 1, because the lower limit of the integral variable s in the dispersion relationship (27), t min , is larger than, but not equal to, m 2 Q .

III. NUMERICAL ANALYSIS
A. ξ-moments and behavior of φ 2;K * 0 (u, µ) Now we can calculate the values of the ξ-moments of K * 0 (1430) leading-twist DA.In calculation, we take the mass of K * 0 (1430) as m K * 0 = 1.425 +0.050 −0.050 GeV, the u and s current quark mass are adopted as m u = 2.16 +0.49−0.26
Substituting the above inputs into Eq.( 13), the ξmoments ξ n 2;K * 0 , continuum state contributions and dimension-six term contributions versus Borel parameter M 2 can be obtained.We will evaluate the values of the first ten ξ-moments in this work.In order to estimate these values, one should determine the appropriate Borel windows.We require the continuum state contributions are not more than 30%, 35%, 40%, 45%, 50% for odd ξ-moments ξ n 2;K * 0 (n = 1, 3, 5, 7, 9), respectively, to get the upper limits of the corresponding Borel windows.On the other hand, the dimension-six term contributions for those five odd ξ-moments are far less than 5% in a very wide Borel parameter region, then the basic criteria that require dimension-six term contributions to be as small as possible are automatically satisfied.Reasonably, we directly fix the lengths of the corresponding Borel windows as 1 GeV 2 .Due to that the even ξ-moments are very close to zero, one cannot determine the Borel windows by limiting the continuum state contributions and dimension-six term contributions, so we determine their Borel windows by examining the stability of even ξ-moments changes with Borel parameters.
The first ten ξ-moments of K * 0 (1430) leading-twist DA versus the Borel parameter and the corresponding Borel TABLE II: Our predictions for the first three ξ-moments and Gegenbauer moments of K * 0 (1430) leading-twist DA at scale µ = 1 GeV, compared to other theoretical predictions.windows are shown in Fig. 1, where all inputs are taken to be their central values.By taking all error sources into account, the values of the first ten ξ-moments at the scale µ = 1 GeV, 1.4 GeV and 3 GeV can be obtained, and which are exhibited in Table I.In Table I, the values at µ = 1.4 GeV and 3 GeV will be used for subsequent calculations of the D s → K * 0 (1430) and B s → K * 0 (1430) TFFs, respectively.Our values for the first three ξ-moments and Gegenbauer moments at scale µ = 1 GeV are also exhibited in Table II, the other theoretical predictions such as by traditional QCDSRs [17] TABLE III: The fitting parameters and Goodness of fit of the LCHO model, when mq = 0.25 GeV, ms takes different values.
at µ = 1 GeV.Then, we can determine the behavior of K * 0 (1430) leading-twist DA by fitting the values exhibited in Table II with the LCHO model shown in Eq. ( 19) via the least squares method.Following the fitting procedure introduced in detailed in Refs.[26,27], the model parameters and the corresponding goodness of fit are determined.However, the goodness of fit is not ideal.In order to obtain better fitting results, we fix mq = 250 MeV and change ms .The fitting results corresponding different ms are exhibited in Table III, and one can find that, when ms = 270 MeV, the goodness of fit is best.Then we take the constituent quark mass mq = 250 MeV and = 270 MeV in subsequent calculations, respectively.Table IV displays the fitted LCHO model parameters and FIG. 2: Our prediction for the behavior of K * 0 (1430) leadingtwist DA at scale µ = 1 GeV.We also present other predictions from the traditional QCDSRs [17] and LF approach [5] for comparison.
f  FIG.3: Behaviors of the TFFs for the semileptonic Bs, Ds → K * 0 (1430) decays in whole q 2 region.The solid lines and shaded bands indicate the central values and uncertainties of our predictions, respectively.The dark parts and light parts in shaded bands are for the direct LCSR calculations and extrapolation results, respectively.We also shown other theoretical predictions such as RQM [2], pQCD [3,5], QCDSR [6,7], LCSR [8][9][10] for comparison.
In order to calculate the branching fractions of the semileptonic decays B s , D s → K * 0 (1430)ℓν ℓ , one should extrapolate the results with LCSRs to the whole q 2 region, i.e., q 2 ∈ [0, (m Hq 1 −m K * 0 ) 2 ].In this work, we adopt the usual pole model parametrization [3,5,7,8] with i = +, −, T .The values of the extrapolation parameters a i and b i corresponding to the B s , D s → K * 0 (1430) TFFs are exhibited in Table VII.The behaviors of the B s , D s → K * 0 (1430) TFFs in whole q 2 region can be determined, and which are shown in Fig. 3.In Fig. 3, the solid lines are our central values and the shaded bands stand for the uncertainties.In particular, the dark parts are for the LCSR predictions, and the light parts are for the extrapolation results, respectively.Otherwise, the corresponding results calculated with RQM [2], pQCD [3,5], QCDSR [6,7], LCSR [8][9][10] are also shown in Fig. 3 for comparison.From Fig. 3 one can find that, our prediction for f Bs→K * 0 + (q 2 ) is consistent with the LCSR predictions in Refs.[9,10] in entire q 2 region, and consists with pQCD calculation in Ref. [5] in small recoil region.For f Bs→K * 0 − (q 2 ), our result is consistent with the QCDSR estimation of Ref. [7] in q 2 ∈ [0, 9 GeV 2 ].Meanwhile, our prediction for f Bs→K * 0 T (q 2 ) is consistent with the LCSR computation in Ref. [9] in whole q 2 region within the error range.In addition, some other predictions for B s → K * 0 (1430) TFFs in the literature differ significantly from our results in large q 2 region, which requires future lattice QCD calculations near the small recoil point for judgement.
Then, we can calculate the differential decay widths of the semileptonic decays B s , D s → K * 0 (1430)ℓν ℓ with the following formula [6,12] In calculation, we take the fermi coupling constant G F = 1.166 × 10 Integrating Eq. ( 40) over q 2 in the region m 2 ℓ ≤ q 2 ≤ (m Hq 1 − m K * 0 ) 2 , and using the heavy meson mean lifetimes τ B 0  IX, respectively.As a comparison, the corresponding branching fractions obtained in Refs.[2,3,6,8,9,11] are also exhibited in Table VIII and Table IX.

IV. SUMMARY
The semileptonic B s , D s → K * 0 (1430) decays can provide another option for testing standard model beyond the semileptonic progresses with pseudoscalar mesons in the final states.In which, the B s , D s → K * 0 (1430) TFFs are the key objects, and whose accuracy mainly depends on the main error source, φ 2;K * 0 (x, µ), the K * 0 (1430) leading-twist DA.Motivated by this, we have studied the K * 0 (1430) leading-twist DA and the semileptonic B s , D s → K * 0 (1430) decays in detail in this article.Our work is based on the scenario that the K * 0 (1430) is viewed as the ground state of sq and qs.
The K * 0 (1430) leading-twist DA has studied following the scheme proposed in Ref. [26] at the first time.The ξ-moments are calculated with the QCDSRs in the framework of BFT by taking the SU f (3) symmetry breaking into account.Considering the fact that the zeroth ξ-moment of the K * 0 (1430) twist-3 DA, ξ 0 p 3;K * 0 , cannot be normalized in whole Borel region, a more reasonable and accurate sum rule formula for nth ξmoments of K * 0 (1430) leading-twist DA, i.e., Eq. ( 13), has been suggested.The values of the first ten ξmoments, ξ n  I with this LCHO model via the least squares method, the behavior of the K * 0 (1430) leading-twist DA has been determined.The fitted model parameters at scale µ = 1 GeV, 1.4 GeV and 3 GeV have been displayed in Table IV, the predicted curve of φ 2;K * 0 (x, µ) at scale µ = 1 GeV is shown in Fig. 2, respectively.
Then, we have calculated the B s , D s → K * 0 (1430) TFFs f ±,T (q 2 ) with LCSR method.The values of those TFFs at the large recoil point have been given in Table V and Table VI.After extrapolating the LCSR results of B s , D s → K * 0 (1430) TFFs to the whole q 2 region (the corresponding behaviors have been shown in Fig. 3), the differential decay ratios and branching fractions of the semileptonic decays B s , D s → K * 0 (1430)ℓν ℓ have been obtained and shown in Fig. 4, Table VIII  In addiction, we also perform the numerical calculations for the K * 0 (1430) leading-twist DA and B s , D s → K * 0 (1430) TFFs and branching fractions in the framework of S1, i.e., the K * 0 (1430) is assumed to be the excited state, the corresponding results are shown in Appendix B.    In numerical calculation, we take m κ = 845 ± 17MeV [1], fκ = 340 ± 20MeV at µ = 1 GeV [17], and the continuum threshold parameters s κ = 2.4 GeV 2 and s K * 0 (1430) = 6 GeV 2 [17], respectively.By taking the Borel windows as M 2 ∈ [3, 4] GeV 2 , the first three ξ-moments and Gegenbauer moments of κ and K * 0 (1430) leading-twist DAs in S1 can be obtained, and are exhibited in Table X.Our predictions for ξ 1 2;κ and ξ 3 2;κ are consistent with the values of Ref. [17] within the error region (In Ref. [17], ξ 1 2;κ = −0.55 ± 0.07 and ξ 3 2;κ = −0.21± 0.05).However, our predictions for moments of K * 0 (1430) leading-twist DA are much less than the corresponding results in Ref. [17], and are also much less than our predictions in S2 (see Table II).
from the data of the hadronic D decay in the generalized factorization model.

−
the M S heavy b and c quark masses mb ( mb ) = 4.18+0.04

2;K * 0
(n = 1, 2, • • • , 10), have been calculated and exhibited in Table I.On the other hand, a new LCHO model has been established at the first time to describe the behavior of the K * 0 (1430) leading-twist DA.By fitting the resulted ξ-moments shown in Table
the B s , D s → K * 0 (1430) TFFs and the corresponding branching fractions can be calculated.The corresponding numerical results are shown in TableXI and Table

TABLE I :
Our predictions for the first ten ξ-moments ξ n

TABLE IV :
Fitted LCHO model parameters and the corresponding goodness of fit for the K * 0 (1430) leading-twist DA with the scale µ = 1, 1.4 and 3 GeV, respectively.

TABLE VII :
Extrapolation parameters for the semileptonic Bs, Ds → K * 0 and Table IX respectively.

TABLE X :
Our predictions for the first three ξ-moments and Gegenbauer moments of κ and K * 0 (1430) leading-twist DAs at the scale µ = 1 GeV in S1.