The $D_s$-meson leading-twist distribution amplitude within the QCD sum rules and its application to the $B_s \to D_s$ transition form factor

We make a detailed study on the $D_s$ meson leading-twist LCDA $\phi_{2;D_s}$ by using the QCD sum rules within the framework of the background field theory. To improve the precision, its moments $\langle \xi^n\rangle _{2;D_s}$ are calculated up to dimension-six condensates. At the scale $\mu = 2{\rm GeV}$, we obtain: $\langle \xi^1\rangle _{2;D_s}= -0.261^{+0.020}_{-0.020}$, $\langle \xi^2\rangle _{2;D_s} = 0.184^{+0.012}_{-0.012}$, $\langle \xi^3\rangle _{2;D_s} = -0.111 ^{+0.007}_{-0.012}$ and $\langle \xi^4\rangle _{2;D_s} = 0.075^{+0.005}_{-0.005}$. Using those moments, the $\phi_{2;D_s}$ is then constructed by using the light-cone harmonic oscillator model. As an application, we calculate the transition form factor $f^{B_s\to D_s}_+(q^2)$ within the light-cone sum rules (LCSR) approach by using a right-handed chiral current, in which the terms involving $\phi_{2;D_s}$ dominates the LCSR. It is noted that the extrapolated $f^{B_s\to D_s}_+(q^2)$ agrees with the Lattice QCD prediction. After extrapolating the transition form factor to the physically allowable $q^2$-region, we calculate the branching ratio and the CKM matrix element, which give $\mathcal{B}(\bar B_s^0 \to D_s^+ \ell\nu_\ell) = (2.03^{+0.35}_{-0.49}) \times 10^{-2}$ and $|V_{cb}|=(40.00_{-4.08}^{+4.93})\times 10^{-3}$.

We make a detailed study on the Ds meson leading-twist LCDA φ2;D s by using the QCD sum rules within the framework of the background field theory. To improve the precision, its moments ξ n 2;Ds are calculated up to dimension-six condensates. At the scale µ = 2GeV, we obtain: ξ 1 2;Ds = −0.261 +0.020 −0.020 , ξ 2 2;Ds = 0.184 +0.012 −0.012 , ξ 3 2;Ds = −0.111 +0.007 −0.012 and ξ 4 2;Ds = 0.075 +0.005 −0.005 . Using those moments, the φ2;D s is then constructed by using the light-cone harmonic oscillator model. As an application, we calculate the transition form factor f Bs→Ds + (q 2 ) within the light-cone sum rules (LCSR) approach by using a right-handed chiral current, in which the terms involving φ2;D s dominates the LCSR. It is noted that the extrapolated f Bs→Ds + (q 2 ) agrees with the Lattice QCD prediction. After extrapolating the transition form factor to the physically allowable q 2 -region, we calculate the branching ratio and the CKM matrix element, which give B(B 0 s → D + s ℓν ℓ ) = (2.03 +0. 35 −0.49 ) × 10 −2 and |V cb | = (40.00 +4.93 −4.08 ) × 10 −3 .

I. INTRODUCTION
Since the first measurement of the ratio R(D ( * ) ) of the branching fractions B(B → D ( * ) τ ν τ ) and B(B → D ( * ) ℓν ℓ ), where ℓ stands for the light lepton e or µ, had been reported by the BaBar Collaboration, the B → D ( * ) semileptonic decays have attracted great attentions due to large differences between the experimental measurements [1][2][3][4] and the standard model (SM) predictions [5][6][7][8][9][10][11][12][13][14]. Such difference has been considered as an evidence of new physics. Comparing with the B 0,+ decays, because its background contamination from the partial reconstruction decay could be less serious, the B s → D s ℓν ℓ decay is experimentally attractive. A natural question is whether there is also evidence of new physics in the semileptonic decay B s → D s ℓν ℓ . This decay could also be an important channel for determining the Cabibbo-Kobayashi-Maskawa (CKM) matrix element |V cb |.
The LHCb collaboration reported the measurement of |V cb | by using B 0 s → D − s µ + ν µ and B 0 s → D * − s µ + ν µ decays [15], in which the data of the proton-proton collision at the center-of-mass energies of 7 and 8 TeV with the integrated luminosity about 3 fb −1 had been used in the analysis. By using the Caprini-Lellouch-Neubert (CLN) and the Boyd-Grinstein-Lebed (BGL) parameterization [16][17][18][19] for B s → D s transition form factor (TFF), the determined |V cb | are (41.4±0.6±0.9±1.2)×10 −3 and (42.3 ± 0.8 ± 0.9 ± 1.2) × 10 −3 , respectively. The LHCb collaboration also measured the ratio of the branching * Electronic address: yizhangphy@cqu. edu The accuracy of theoretical predictions on the branching fraction B(B s → D s ℓν ℓ ) depends heavily on the TFF f Bs→Ds + (q 2 ). It has been calculated within several approaches, such as the quark models [20][21][22], the QCD light cone sum rules (LCSR) [23,24], and the lattice QCD (LQCD) [25][26][27]. Similar to the B → π TFFs [28], the LQCD prediction is reliable in large q 2 -region, the QCD factorization prediction or the quark model prediction is reliable in large recoil region q 2 ∼ 0, and the LCSR is reliable in low and intermediate q 2 -regions. Predictions under various methods are complementary to each other. Because the LCSR prediction is applicable in a wider region and could be adapted for all q 2 -region via proper extrapolations, and in this paper, we will adopt the LCSR approach to calculate f Bs→Ds + (q 2 ). Generally, contributions from the light-cone distribution amplitude (LCDA) suffers from the power counting rules basing on the twists, i.e. the high-twist LCDAs are usually powered suppressed to the lower twist ones in large Q 2 -region. The high-twist LCDAs may have sizable contributions to the LCSR, and how to "design" a proper correlator is a tricky problem for the LCSR approach. By choosing a proper correlator, one can not only study the properties of the hadrons but also simplify the theoretical uncertainties effectively. As the usual treatment, the correlator is constructed by using the currents with definite quantum numbers, such as those with definite J P , where J is the total angular momentum and P is the parity of the bound state. Such a construction of the correlator is not the only choice suggested in the literature, e.g. the chiral correlator with a chiral current in between the matrix element has also been suggested to suppress the hazy contributions from the uncertain LCDAs [29][30][31][32][33][34]. In the paper, we adopt a chiral correlator to do the LCSR cal-culation, and we shall find that the leading-twist LCDA φ 2;Ds provides dominant contributions. Therefore, if an accurate φ 2;Ds has been achieved, we shall obtain an accurate prediction on f Bs→Ds + (q 2 ). Till now, there are few calculations on the D s -meson leading-twist LCDA φ 2;Ds ; recently, it has been studied by using the light-front quark model [35]. We shall first construct a light-cone harmonic oscillator model for φ 2;Ds based on the well-known BHL-description [36][37][38] as we have done for π, ρ, D and heavy meson LCDAs [39][40][41][42][43][44][45][46]. Then its input parameters shall be fixed by using reasonable constraints such as the probability of finding the leading Fock-state in D s -meson Fock-state expansion, the normalization condition, and the calculated LCDA moments ξ n 2;Ds or the Gegenbauer moments a Ds n . All those moments shall be computed by using the QCD sum rules [47] within the framework of background field theory (BFT) [48] up to dimension-six operators.
The remaining parts are organized as follows. The LCSR for B s → D s TFF, the QCD sum rules of the moments of φ 2;Ds and the light-cone harmonic oscillator model for φ 2;Ds are given in Sec.II. Numerical results and discussions are presented in Sec.III. Section IV is reserved for a summary. The useful functions for calculating the φ 2;Ds moments are listed in the Appendix.

II. CALCULATION TECHNOLOGY
A. The LCSR for Bs → Ds TFF The B s → D s TFF f Bs→Ds + (q 2 ) andf Bs→Ds (q 2 ) are usually defined as: where p is the momentum of D s -meson, q is the momentum transfer. In this paper, we focus on the semileptonic decay B s → D s ℓν ℓ with ℓ = (e, µ). The masses of lightleptons are negligible, and then due to chiral suppression, only f Bs→Ds + (q 2 ) is relevant for our present analysis. To derive the LCSR of f Bs→Ds + (q 2 ), we adopt the following chiral correlation function (correlator): The correlator is analytic in whose q 2 -region. In the timelike region, by inserting a complete series of the intermediate hadronic states into the correlator, one can obtain its hadronic representation by isolating out the pole term of the lowest stat of the B s meson. By further using the TFF definition (1) and the B s meson decay constant f Bs where m Bs is the B s -meson mass and m b is the b-quark mass, then the hadronic representation for the correlator (2) reads where s Bs 0 is threshold parameter, ρ QCD (s) is the spectral density, and we have implicitly used the conventional quark-hadron duality ansatz. On the other hand, in the spacelike region, the correlator can be calculated by using the operator production expansion (OPE) approach.
It is done by using the b-quark propagator By matching the hadronic representation (4) and the OPE of the correlator (2) with the help of the dispersion relation, the LCSR of f Bs→Ds where C F = 4/3, M is the Borel parameter. Here the Borel transformation has been adopted to suppress continuum contributions. The leading-order (LO) contribution of f Bs→Ds + (q 2 ) takes the form: with the D s -meson decay constant f Ds and The NLO contribution of f Bs→Ds The imaginary part of the next-to-leading order amplitude T 1 can be read from Ref. [49]. Due to the present choice of the chiral correlator (2), contributions from the twist-3 D s -meson LCDA exactly vanish in the LCSR. Thus the terms from omitted gluonic field in b-quark propagator (5) and hence contributions from even highertwist terms are negligibly small and can be safely neglected. Our remaining task is then to achieve a precise φ 2;Ds .
B. Sum rules for the moments of the Ds-meson leading-twist LCDA φ2;D s The D s -meson leading-twist LCDA φ 2;Ds is defined as where f Ds is the D s meson decay constant. The moments of φ 2;Ds (x) can be derived by expanding the left-handside of Eq.(9) around z = 0 and the exponent in the right-hand-side of Eq.(9) as a power series, e.g.
where the n th -moment is defined as The 0 th -moment satisfies the normalization condition ξ 0 2;Ds = 1.
The sum rules of those moments can be derived by using the following correlator where n = (0, 1, 2, ...), and the currents By applying the OPE for the correlator (13) in deep Euclidean region based on the BFT [48], we obtain where S c F (0, x) and S s F (x, 0) are c and s quark propagators in the BFT, (iz · ↔ D) n stands for the vertex operators, and "· · · " indicates the even higher-order terms.
There are totally 40 Feynman diagrams for the present considered accuracy, e.g. up to dimension-6 operators, the first and second terms in Eq.(16) contain 35 and 5 Feynman diagrams, respectively. Typical Feynman diagrams are shown in Figure. 1 and Figure. 2, other diagrams can be obtained by permutation. In those two figures, the left big dot and the right big dot stand for the vertex operators zγ 5 (z · ↔ D) n and zγ 5 in the currents J n (x) and J † 0 (0), respectively; the cross symbol indicates the gluonic background field. There are also cases in which the cross symbol stands for the s-quark background field. In deriving the QCD sum rules for the moments, we need to know the propagators and vertex operators under the BFT up to dimension-six operators, and tedious expressions of them can be found in Ref. [39].
Here different from the case of the D-meson, the mass effect in the denominator of s-quark propagator cannot be ignored. However, considering that the s-quark mass is not large, we expand the s-quark propagator as a power series over m s and keep only the first power of m s . In this way, we can use the corresponding calculation technology described in detail in Ref. [42] to do the calculation.
Following the standard procedures of QCD sum rules [50,51], we obtain the sum rules for the moments of D s -meson leading-twist LCDA, i.e.
The analytical expressions of the perturbative and nonperturbative terms are ImI pert (s) = 3 8π 2 M 2 (n + 1)(n + 3) with Here the functions F 1,2 (n, a, b, l min , l max ), G 1,2 (n, a), H(n, a, b, c) and Borel transformations which are collected in the Appendix IV. Based on the BHL-description [36][37][38], similar to the case of D-meson leading-twist LCDA [42], we construct a light-cone harmonic oscillator model of the D s -meson leading-twist wavefunction Ψ 2;Ds (x, k ⊥ ) as where k ⊥ is the transverse momentum, χ 2;Ds (x, k ⊥ ) is the spin-space wavefunction and Ψ R 2;Ds (x, k ⊥ ) indicates the spatial wavefunction. The spin-space wavefunction χ 2;Ds (x, k ⊥ ) reads [52] wherem c andm s are constituent quark masses of D s , and we adoptm c = 1.5GeV andm s = 0.5GeV. The spatial wavefunction takes the form where A Ds is the normalization constant, β Ds is the harmonious parameter that dominates the wavefunction's transverse distribution, and function ϕ 2;Ds (x) dominates the wavefunction's longitudinal distribution. ϕ 2;Ds (x) can be taken as the first few terms of the Gegenbauer series, here we take By using the relationship between the D s -meson leadingtwist wavefunction, one can get its LCDA at the scale µ 0 , (28) which, after integrating over the transverse momentum k ⊥ , becomes where µ 0 ∼ Λ QCD is the factorization scale. Becausê m c ≫ Λ QCD , the spin-space wavefunction χ Ds → 1. The above model (24,29) is for D − s -meson. The leading-twist wavefunction and the LCDA for D + s -meson can be obtained by replacing x with (1 − x) in Eqs. (24,29).
The model parameters A Ds , B Ds n and β Ds are scale dependent, their values at an initial scale µ 0 can be determined by reasonable constraints, and their values at any other scale µ can be derived via the evolution equation [53]. More explicitly, we shall adopt the following constraints to fix the parameters: • The normalization condition, 1 0 dxφ 2;Ds (x, µ 0 ) = 1. (30) • The probability of finding the leading Fock-state |cs in D s -meson Fock-state expansion, We will take P Ds ≃ 0.8 [54] in subsequent calculations.
• The Gegenbauer moments of φ 2;Ds (x, µ 0 ) can be derived via the following formula, and the φ 2;Ds (x, µ 0 ) moments are defined as The values of the moments ξ n 2;Ds and the Gegenbauer moments a Ds n at the scale 2 GeV will be given in next subsection.
B. The moments ξ n 2;Ds from QCD sum rules  To get the numerical value of moments ξ n 2;Ds of φ 2;Ds (x, µ), one need to fix the Borel window M 2 which is introduced to depress the contributions from both the continuum states and the highest dimensional condensates. Usually, the continuum contribution and the dimension-six condensate contribution are taken to be less than 30% and 10% respectively, while the value of ξ n 2;Ds is required to be as stable as possible in the allowed Borel window. In this paper, the continuum state contribution for ξ n 2;Ds | µ with n = (1, 2, 3, 4) is required to be less than 20%, 25%, 10%, 30%, respectively, and each of the dimension-six condensates contributions is no more than 1%. The determined Borel windows and the corresponding D s -meson leading-twist LCDA moments ξ n 2;Ds at the scale µ = 2 GeV with n = (1, · · · , 4) are presented in Table I, where all input parameters are taken to be their central values. We present the D s -meson leading-twist LCDA moments ξ n 2;Ds with n = (1, · · · , 4) at µ = 2GeV versus M 2 in Fig. 3. To be consistent with Table I, those moments are stable over the allowable Borel windows.
If setting µ = 2 GeV, by taking all uncertainty sources into consideration, we obtain where the errors are squared averages of all the mentioned error sources.
We present all the determined input parameters at the scale µ = 2 GeV in Table II. The accuracy of φ 2;Ds (x, µ) is dominated by the magnitudes of the Gegenbauer moments a Ds n (µ). As we have pointed out in Ref. [42,43], the Gegenbauer moments a Ds n (µ) are correlated to each other and can not be changed independently within their own error regions. Then Table II associates the uncertainty of φ 2;Ds (x, µ) with the error of Gegenbauer moments a Ds n (µ), which facilitates our further discussion on the impact of φ 2;Ds (x, µ) as an input parameter to the B s → D s decay. Figure. 4 shows the D s -meson leading-twist LCDA φ 2;Ds (x, µ) with typical values of the input parameters exhibited in Table II Table II. third and forth lines of Table II. Our model of φ 2;Ds (x, µ) prefers a broader behavior in low x-region. It has a peak around x ∼ 0.35. Figure. 5 shows the D s -meson leading-twist LCDA φ 2;Ds (x, µ) at different scales, where the solid, the dashed, the dotted and the dash-dotted lines are for the scales µ = 2, 3, 10, 100 GeV, respectively. It shows that with the increment of µ, φ 2;Ds (x, µ) becomes broader and broader and becomes more symmetric, e.g. the peak moves closer to x = 0.5. When µ → ∞, φ 2;Ds (x, µ) tends to the known asymptotic form, i.e. φ 2;Ds (x, µ → ∞) = 6x(1 − x).    There are still two parameters to be fixed, the continuum threshold s Bs and in zero recoil region q 2 = q 2 max , we obtain where all the uncertainties have been added up in quadrature, and the errors from φ 2;Ds (x, µ) and f Bs dominate the uncertainties. It agrees with the lattice QCD predictions within errors, f Bs→Ds (31) [26] and f Bs→Ds + (0) = 0.666(12) [27]. Fig. 6 also shows that for larger q 2 -values, the TFF will show sizable dependence on M 2 , which agrees with the convention that the LCSR approach cannot be applied for very large q 2 -value. We adopt the TFF f Bs→Ds + (q 2 ) within the region of [0, 9GeV 2 ] as a basis to extrapolate it to all physical q 2 -value. For the purpose, we adopt the double-pole-extrapolation method [60] to do the extrapolation, i.e.
f Bs→Ds We put the fitted parameters in Table III. , where the lighter shaded band is the squared average of those from all the mentioned error sources. The Lattice QCD prediction and its extrapolated results given in year 2017 [26] have also been presented as a comparison, the thicker shaded band shows its uncertainty.
We adopt the extrapolated TFF f Bs→Ds + (q 2 ) to calculate the branching ratio B(B s → D s ℓν ℓ ), which can be derived by using the following formula where the differential decay width is dΓ(B s → D s ℓν ℓ ) dq 2 = G 2 F |V cb | 2 192π 3 m 3 Bs λ 3/2 (q 2 )|f Bs→Ds where G F = 1.1663787(6) × 10 −5 GeV −2 , and the phasespace factor λ(q 2 ) = (m 2 Bs + m 2 Ds − q 2 ) 2 − 4m 2 Bs m 2 Ds . We In this work, we have made a detailed study on the D s -meson leading-twist LCDA φ 2;Ds . Its moments have been calculated by using the QCD sum rules within the framework of BFT, and its first four moments have been given in Eqs. (35,36,37,38), which then result in the Gegenbauer moments a Ds 1 (2GeV) = −0.436 +0.033 −0.033 , a Ds 2 (2GeV) = −0.047 +0.035 −0.035 , a Ds 3 (2GeV) = 0.004 +0.01 −0.02 and a Ds 4 (2GeV) = −0.004 −0.026 +0.025 . Based on the BHLprescription, we have constructed a new model for φ 2;Ds , whose behavior is constrained by the normalization condition, the probability of finding the leading Fock-state |cs in D s -meson Fock-state expansion, and the known Gegenbauer moments. As the key input for studying the high-energy processes involving D s -meson, our suggested φ 2;Ds shall be of great importance.
Using the present model of φ 2;Ds , we calculate the B s → D s TFF f Bs→Ds + (q 2 ) within the QCD LCSR approach by adopting a chiral current correlator, in which the leading-twist terms dominant over the LCSR. At the large recoil region, we obtain f Bs→Ds + (0) = 0.639 +0.075 −0.056 . By using the extrapolated TFF with the double-pole-extrapolation method, we obtain B(B 0 s → D s + ℓν ℓ ) = 2.033 +0.350 −0.488 × 10 −2 and the CKM element |V cb | = (40.00 +4.929 −4.075 ) × 10 −3 , which is consistent with the various measurements within reasonable errors.