Analysis of the $B\to a_1(1260)$ form-factors with light-cone QCD sum rules

In this article, we calculate the $B\to a_1(1260)$ form-factors $V_1(q^2)$, $V_2(q^2)$, $V_3(q^2)$ and $A(q^2)$ with the $B$-meson light-cone QCD sum rules. Those form-factors are basic parameters in studying the exclusive non-leptonic two-body decays $B\to AP$ and semi-leptonic decays $B\to A l \nu_l$, $B\to A \bar{l}l$. Our numerical results are consistent with the values from the (light-cone) QCD sum rules. The main uncertainty comes from the parameter $\omega_0$ (or $\lambda_B$), which determines the shapes of the two-particle and three-particle light-cone distribution amplitudes of the $B$ meson, it is of great importance to refine this parameter.


Introduction
The weak B → P, V, A form-factors with P = π, K, V = ρ, K * and A = a 1 , K 1 final states are basic input parameters in studying the exclusive semi-leptonic decays B → P (V, A)lν l , B → P (V, A) ll and radiative decays B → V (A)γ, they also determine the factorizable amplitudes in the non-leptonic charmless two-body decays B → P P (AP, P V, V V ).Those decays can be used to determine the CKM matrix elements and to test the standard model, however, it is a great challenge to pin down the uncertainties of the form-factors to obtain more precise results.The exclusive semi-leptonic decays B → P (V )lν l , B → P (V ) ll and radiative decays B → V γ and hadronic two-body decays B → P P (P V, V V ) have been studied extensively [1,2,3,4,5,6,7], while the decays B → AP, V A have been calculated with the QCD factorization approach [8,9,10], generalized factorization approach [11,12], etc.It is more easy to deal with the exclusive semi-leptonic precesses than the nonleptonic precesses, and there have been many works on the relevant form-factors B → π, B → ρ in determining the CKM matrix element V ub [13,14,15,16].The B → a 1 (1260) form-factors have been studied with the covariant light-front approach [17], ISGW2 quark model [18], quark-meson model [19], QCD sum rules [20], light-cone QCD sum rules [9] and perturbative QCD [21] .However, the values from different theoretical approaches differ greatly from each other.
The BaBar Collaboration and Belle Collaboration have measured the charmless hadronic decays B 0 → a ± 1 π ∓ [22,23].Moreover, the BaBar Collaboration has measured the time-dependent CP asymmetries in the decays B 0 → a ± 1 π ∓ with a ∓ 1 → π ∓ π ± π ∓ , from the measured CP parameters, we can determine the decay rates of a + 1 π − and a − 1 π + respectively [24].Recently, the BaBar Collaboration has reported the observation of the decays 25,26].So it is interesting to re-analyze the B → a 1 form-factors with the Bmeson light-cone QCD sum rules [27].
In Ref. [27], the authors obtain new sum rules for the B → π, K, ρ, K * formfactors from the correlation functions expanded near the light-cone in terms of the B-meson distribution amplitudes, and suggest QCD sum rules motivated models for the three-particle B-meson light-cone distribution amplitudes, which satisfy the relations given in Ref. [28].In Ref. [28], the authors derive exact relations between the two-particle and three-particle B-meson light-cone distribution amplitudes from the QCD equations of motion and heavy-quark symmetry.The two-particle B-meson light-cone distribution amplitudes have been studied with the QCD sum rules and renormalization group equation [29,30,31,32,33,34,35].Although the QCD sum rules can't be used for a direct calculation of the distribution amplitudes, it can provide constraints which have to be implemented within the QCD motivated models (or parameterizations) [32].
The B-meson light-cone distribution amplitudes play an important role in the exclusive B-decays, the inverse moment of the two-particle light-cone distribution amplitude φ + (ω) enters many factorization formulas (for example, see Refs.[3,4]).However, the light-cone distribution amplitudes of the B-meson are received relatively little attention comparing with the ones of the light pseudoscalar mesons and vector mesons, our knowledge about the nonperturbative parameters which determine those light-cone distribution amplitudes is limited and an additional application (or estimation) based on QCD is useful.
In this article, we use the B-meson light-cone QCD sum rules to study the B → a 1 form-factors.The semi-leptonic decays B → Alν l can be observed at the LHCb, where the b b pairs will be copiously produced with the cross section about 500 µb.
We can also study the form-factors with the light-cone QCD sum rules using the light-cone distribution amplitudes of the axial-vector mesons.Recently, the twist-2 and twist-3 light-cone distribution amplitudes of the axial-vector mesons have been calculated with the QCD sum rules [36].
The B-meson light-cone QCD sum rules have given reasonable values for the B → π, K, ρ, K * form-factors [27], so it is interesting to study the B → a 1 form-factors and cross-check the properties of the B-meson light-cone distribution amplitudes.Furthermore, it is necessary to investigate the form-factors with different approaches and compare the predictions of different approaches.
The article is arranged as: in Section 2, we derive the B → a 1 (1260) form-factors with the light-cone QCD sum rules; in Section 3, the numerical result and discussion; and Section 4 is reserved for conclusion.

B → a 1 (1260) form-factors with light-cone QCD sum rules
In the following, we write down the definitions for the weak form-factors V 1 (q 2 ), V 2 (q 2 ), V 3 (q 2 ), V 0 (q 2 ) and A(q 2 ) [17], where V 0 (0) = V 3 (0), and the ǫ µ is the polarization vector of the axial-vector meson a 1 (1260).We study the weak form-factors V 1 (q 2 ), V 2 (q 2 ), V 3 (q 2 ), V 0 (q 2 ) and A(q 2 ) with the two-point correlation functions Π i µ (p, q), where J i µ (x) = J µ (x) and J A µ (x) respectively, and the axial-vector current J a µ (x) interpolates the axial-vector meson a 1 (1260).The correlation functions Π i µ (p, q) can be decomposed as due to Lorentz covariance.In this article, we derive the sum rules with the tensor structures g µν , q µ p ν and ǫ µναβ p α q β respectively to avoid contaminations from the π meson.
According to the basic assumption of current-hadron duality in the QCD sum rules approach [37,38], we can insert a complete series of intermediate states with the same quantum numbers as the current operator J a µ (x) into the correlation functions Π i µ (p, q) to obtain the hadronic representation.After isolating the ground state contributions from the pole terms of the meson a 1 (1260), the correlation functions Π i µν (p, q) can be expressed in the following form, where we have used the standard definition for the decay constant f a , 0|J a µ (0)|a In the following, we briefly outline the operator product expansion for the correlation functions Π i µ (p, q) in perturbative QCD theory.The calculations are performed at the large space-like momentum region p 2 ≪ 0 and 0 ≤ q 2 < m 2 b + m b p 2 / Λ, where M B = m b + Λ in the heavy quark limit.We write down the propagator of a massless quark in the external gluon field in the Fock-Schwinger gauge and the light-cone distribution amplitudes of the B meson firstly [39], where the ω 0 and λ 2 E are some parameters of the B-meson light-cone distribution amplitudes.
Substituting the d quark propagator and the corresponding B-meson light-cone distribution amplitudes into the correlation functions Π i µ (p, q), and completing the integrals over the variables x and k, finally we obtain the representation at the level of quark-gluon degrees of freedom.In this article, we take the three-particle B-meson light-cone distribution amplitudes suggested in Ref. [27], they obey the powerful constraints derived in Ref. [28] and the relations between the matrix elements of the local operators and the moments of the light-cone distribution amplitudes, if the conditions ω 0 = 2 3 Λ and Λ2 are satisfied [29].In the region of small ω, the exponential form of distribution amplitude φ + (ω) is numerically close to the more elaborated model (or the BIK distribution amplitude (BIK DA)) suggested in Ref. [32], where ω 0 = λ B .The parameters λ B and σ B are determined from the heavy quark effective theory QCD sum rules including the radiative and nonperturbative corrections.There are other phenomenological models for the two-particle B-meson lightcone distribution amplitudes, for example, the k T factorization formalism [40,41], in this article, we use the QCD sum rules motivated models.
After matching with the hadronic representation below the continuum threshold s 0 , we obtain the following three sum rules for the weak form-factors V 1 (q 2 ) , V 2 (q 2 ) and A(q 2 ) respectively, where In Ref. [31], Lange and Neubert observe that the evolution effects drive the lightcone distribution amplitude φ + (ω) toward a linear growth at the origin and generate a radiative tail that falls off slower than 1 ω , even if the initial function has an arbitrarily rapid falloff, which implies the normalization integral of the φ + (ω) is ultraviolet divergent.In this article, we derive the sum rules without the radiative O(α s ) corrections, the ultraviolet behavior of the φ + (ω) plays no role at the leading order (O(1)).Furthermore, the duality thresholds in the sum rules are well below the region where the effect of the tail becomes noticeable.The nontrivial renormalization of the B-meson light-cone distribution amplitude is so far known only for the φ + (ω), we use the light-cone distribution amplitudes of order O(1), which satisfy all QCD constraints.

Numerical result and discussion
The input parameters are taken as [42,43].
The Borel parameters in the three sum rules are taken as M 2 = (1.1 − 1.5) GeV 2 , in this region, the values of the weak form-factors V 1 (q 2 ), V 2 (q 2 ) and A(q 2 ) are stable enough.
In calculation, we observe the dominating contributions in the three sum rules come from the two-particle B-meson light-cone distribution amplitudes, the contributions from the three-particle B-meson light-cone distribution amplitudes are of minor importance, about 1%, and can be neglected safely.It is not un-expected that the main uncertainty comes from the parameter ω 0 (or λ B ), which determines the shapes of the two-particle and three-particle light-cone distribution amplitudes of the B meson.From Fig. 1, we can see that the uncertainty of the parameter λ B almost saturates the total uncertainties, it is of great importance to refine this parameter.In this article, we take the value from the QCD sum rules in Ref. [32], where the B-meson light-cone distribution amplitude φ + is parameterized by the matrix element of the bilocal operator at imaginary light-cone separation.
In the region of small ω, the exponential (Gaussian) form of distribution amplitude φ + (ω) is numerically close to the BIK DA suggested in Ref. [32].In Fig. 1, we also present the numerical results with the BIK DA for the central values of the input parameters λ B and σ B , the Gaussian distribution amplitude and the BIK DA −0.518 0.159 V 2 (q 2 ) −1.330 0.532 A(q 2 ) −1.649 0.561 Table 3: The parameters for the fitted form-factors.lead to almost the same values.
From Table 1, we can see that the values of the V 0 (0) from the covariant lightfront approach, ISGW2 quark model and quark-meson model differ greatly from the corresponding ones from the (light-cone) QCD sum rules, while the values from the (light-cone) QCD sum rules and perturbative QCD are consistent with each other.From Table 2, we observe that the values of the A(0) from the covariant light-front approach, quark-meson model and perturbative QCD differ greatly from the corresponding ones from the (light-cone) QCD sum rules, while the values of the form-factors from the (light-cone) QCD sum rules are consistent with each other.

Conclusion
In this article, we calculate the weak form-factors V 1 (q 2 ), V 2 (q 2 ), V 3 (q 2 ) and A(q 2 ) with the B-meson light-cone QCD sum rules.The form-factors are basic parameters in studying the exclusive hadronic two-body decays B → AP and semi-leptonic decays B → Alν l , B → A ll.Our numerical values are consistent with the values from the (light-cone) QCD sum rules.The main uncertainty comes from the parameter ω 0 (or λ B ), which determines the shapes of the two-particle and three-particle lightcone distribution amplitudes of the B meson, it is of great importance to refine this parameter.However, it is a difficult work, as we cannot extract the values of the basic parameter λ B directly from the experimental data on the semi-leptonic decays B → Alν l .

Table 1 :
The form-factor V 0 (0) from different theoretical approaches.I know the updated value 0.30 ± 0.05 from private communication with Prof. H.Y.Cheng, their work is still in progress.

Table 2 :
The form-factor A(0) from different theoretical approaches.