$B-S$ transition form-factors with the light-cone QCD sum rules

In the article, we assume the two scalar nonet mesons below and above 1 GeV are all $\bar{q}q$ states, in case I, the scalar mesons below 1 GeV are the ground states, in case II, the scalar mesons above 1 GeV are the ground states. We calculate the $B-S$ form-factors by taking into account the perturbative ${\mathcal{O}}(\alpha_s)$ corrections to the twist-2 terms using the light-cone QCD sum rules and fit the numerical values of the form-factors into the single-pole forms, which have many phenomenological applications.


Introduction
The underlying structures of the scalar mesons are not well established theoretically, there are many candidates with the quantum numbers J P C = 0 ++ below 2 GeV, which cannot be accommodated in oneqq nonet. Roughly speaking, they can be calcified into two nonets, the nonet {f 0 (600), a 0 (980), κ(800), f 0 (980)} below 1 GeV and the nonet {f 0 (1370), a 0 (1450), K * 0 (1430), f 0 (1500)} above 1 GeV. A prospective picture suggests that the scalar mesons above 1 GeV can be assign to be a conventionalqq nonet with some possible glue components, while the scalar mesons below 1 GeV form an exotic [qq]3[qq] 3 nonet with substantial mixings with the qq states, meson-meson states and glueballs [1]. In the hadronic dressing mechanism, the scalar mesons below 1 GeV have smallqq cores of typicalqq meson size, strong couplings to the intermediate hadronic states enrich the pureqq states with other components and spend part of their lifetime as virtual meson-meson states [2]. All the existing pictures have both advantages and shortcomings in one way or another. In this article, we assume that the scalar mesons are allqq states, in case I, the scalar mesons below 1 GeV are the ground states, in case II, the scalar mesons above 1 GeV are the ground states; and study the B − S transition form-factors with the light-cone QCD sum rules (LCSR) [3,4].
The transition form-factors in the semi-leptonic decays not only depend on the dynamics of strong interactions among the quarks in the initial and final mesons, but also depend on the structures of the involved mesons. They are highly nonperturbative quantities. In the region small-recoil, where the momentum transfer squared q 2 is large, the form-factors are dominated by the soft dynamics, while in the large-recoil region, where q 2 → 0, the form-factors are dominated by the short-distance dynamics.
In the light-cone QCD sum rules, we carry out the operator product expansion near the lightcone x 2 ≈ 0 in stead of the short distance x ≈ 0, the nonperturbative hadronic matrix elements are parameterized by the light-cone distribution amplitudes (LCDAs) of increasing twist instead of the vacuum condensates [3,4]. Based on the quark-hadron duality, we can obtain copious information about the hadronic parameters at the phenomenological side. The transition form-factors from the LCSR not only have an estimable region of q 2 , but also embody as many long-distance effects as possible involved in the decaying processes. The operator product expansion is valid at small and intermediate momentum transfer where the χ is a typical hadronic scale of roughly 500 MeV and independent of the b-quark mass m b .
On the other hand, the semi-leptonic B-decays are excellent subjects in studying the CKM matrix elements and CP violations. We can use both the exclusive and inclusive b → u transitions to study the CKM matrix element V ub . Furthermore, the processes induced by the flavor-changing neutral currents b → s(d) provide the most sensitive and stringent test for the standard model at 2 Light-cone QCD sum rules for the form-factors In the following, we write down the two-point correlation functions Π i µ (p, q) in the LCSR, where i = A, B, q, q ′ = u, d, s, the pseudoscalar currents J 5 (x) interpolate the B q ′ mesons, and the axial-vector currents J i µ (x) induce the B → S transitions. We can insert a complete set of intermediate hadronic states with the same quantum numbers as the current operators J 5 (0) into the correlation functions Π i µ (p, q) to obtain the hadronic representation [19,20]. After isolating the ground state contributions from the B q ′ mesons, we get the following result, where the decay constants f B q ′ and the form-factors F + (q 2 ), F − (q 2 ) and F T (q 2 ) are defined by We can also parameterize the form-factors into another form, where P = 2p + q and We carry out the operator product expansion for the correlation functions Π i µ (p, q) in the large space-like momentum region (p+q)−m 2 b ≪ 0 and the large recoil region of the decaying B q ′ -meson, which correspond to the small light-cone distance x 2 ≈ 0 and are required by the validity of the operator product expansion. We contract the b-quarks in the correlation functions Π i µ (p, q) with Wick's theorem, then substantiate the free b-quark propagator and the corresponding S-meson LCDAs into the correlation functions Π i µ (p, q) and complete the integrals both in the coordinate space and momentum space to obtain Now we take a short digression to discuss the LCDAs of the related two-quark scalar mesons. Let us write down the definitions for the twist-2 and twist-3 LCDAs φ(u, µ), φ s (u, µ) and φ σ (u, µ), where the u is the fraction of the light-cone momentum of the scalar meson carried by the q-quark, u = 1 − u and z = x − y. The LCDAs φ(u, µ), φ s (u, µ) and φ σ (u, µ) can be expanded into a series of Gegenbauer polynomials C 1/2(3/2) m (u −ū) with increasing conformal spin according to the conformal symmetry of the QCD, where the B m (µ), B s m (µ) and B σ m (µ) are the Gegenbauer moments [4,21,22]. The twist-2 LCDA is antisymmetric under the interchange u ↔ū in the flavor SU (3) symmetry limit, and the zeroth Gegenbauer moment B 0 vanishes in the flavor SU (3) symmetry limit, the odd Gegenbauer moments are dominant. In this article, we take into account the first two odd moments B 1 and B 3 and set B s m = B σ m = 0 [23]. Then we calculate the perturbative O(α s ) contributions to the twist-2 terms, while the perturbative O(α s ) corrections to the twist-3 terms are beyond the present work, as the contributions of the twist-3 terms are suppressed by the factor m S /m b (or m b m S /M 2 after the Borel transformation) and play a less important role, see Eqs. (7)(8). The six Feynman diagrams which determine the perturbative O(α s ) corrections are shown explicitly in Fig.1. For simplicity, we perform the calculations in the Feynman gauge, introduce convenient dimensionless variables , and take the approximation m 2 S /m 2 b ≈ 0. In calculations, we regularize both the ultraviolet and collinear (or infrared) divergences with dimensional regularization by setting D = 4 − 2ε UV = 4 + 2ε IR , and perform the renormaliztion in the M S scheme with totally anti-commuting γ 5 . In the following, we write down the contributions of the diagrams a, b, c, d, e and f , respectively, Li 2 (x) = − Then we obtain the QCD spectral densities through the dispersion relation, where i = +, +−, T and ρ αs + (s) = Im Π + (q 2 , s) π , ρ αs T (s) = Im Π T (q 2 , s) π , , and Θ(x) = 1 for x ≥ 0, the ← − d du does not act on the δ(1 − ρ).
We take quark-hadron duality below the continuum thresholds s 0 , and perform the Borel trans-formation with respect to the variable P ′2 = −(p + q) 2 to obtain the LCSR,

Numerical results and discussions
The  [25]. In the article, we take the M S mass m b (m b ) = (4.18 ± 0.03) GeV from the Particle Data Group [24], and take into account the energy-scale dependence of the M S mass from the renormalization group equation, where t = log µ 2  LCDAs φ(u, µ), φ s (u, µ) and φ σ (u, µ) are also determined by the renormalization group equation, where b = (33 − 2n f )/3, and the one-loop anomalous dimensions [26] is We take the values of the parameters in the LCDAs φ(u, µ), φ s (u, µ) and φ σ (u, µ) asf a0 ( Table 1: The parameters of the fitted transition form-factors F + (q 2 ).
The numerical values of the form-factors are fitted into the single-pole form, with the MINUIT, where m B = 5.28 GeV, i = +, −, T , the values of the F i (0) and a i are shown explicitly in Tables 1-3. The form-factors can be taken as basic input parameters, and study the semi-leptonic and non-leptonic B-decays to the scalar mesons so as to shed light on the structures of the scalar mesons or explore the CKM matrix elements.

Conclusion
In the article, we assume the two scalar nonet mesons below and above 1 GeV are allqq states, in case I, the scalar mesons below 1 GeV are the ground states, in case II, the scalar mesons above 1 GeV are the ground states. We calculate the B − S     Table 3: The parameters of the fitted transition form-factors F T (q 2 ).