Revisiting $D$-meson twist-2, 3 distribution amplitudes

Due to the significant difference between the experimental measurements and the theoretical predictions of standard model (SM) for the value of $\mathcal{R}(D)$ of the semileptonic decay $B\to D\ell\bar{\nu}_{\ell}$, people speculate that it may be the evidence of new physics beyond the SM. Usually, the $D$-meson twist-2, 3 distribution amplitudes (DAs) $\phi_{2;D}(x,\mu)$, $\phi_{3;D}^p(x,\mu)$ and $\phi_{3;D}^\sigma(x,\mu)$ are the main error sources when using perturbative QCD factorization and light-cone QCD sum rules to study $B\to D\ell\bar{\nu}_{\ell}$. Therefore, it is important to get more reasonable and accurate behaviors for those DAs. Motivated by our previous work [Phys. Rev. D 104, no.1, 016021 (2021)] on pionic leading-twist DA, we revisit $D$-meson twist-2, 3 DAs $\phi_{2;D}(x,\mu)$, $\phi_{3;D}^p(x,\mu)$ and $\phi_{3;D}^\sigma(x,\mu)$. New sum rules formulae for the $\xi$-moments of these three DAs are suggested to obtain more accurate values. The light-cone harmonic oscillator models for those DAs are improved, and whose model parameters are determined by fitting the values of $\xi$-moments with the least squares method.


I. INTRODUCTION
Since 2012, semileptonic decay B → Dℓν ℓ has been considered as one of the processes most likely to prove the existence of new physics beyond the standard model (SM). The reason is well known, that is, the significant difference between the experimental measurements of the ratio R(D) and its theoretical predictions of SM. The latest statistics given by Heavy Flavor Average Group website [1] shows that the experimental average value of R(D) is R exp. (D) = 0.339 ± 0.026 ± 0.014, while its average value of SM predictions is R the. (D) = 0.300 ± 0.008 [2]. The former comes from the experimental measurements for semileptonic decay B → Dℓν ℓ by BaBar Collaboration in 2012 [3] and 2013 [4], by Belle Collaboration in 2015 [5] and 2019 [6]. The later is obtained by combining two lattice calculations by MILC Collaboration [7] and HPQCD Collaboration [8]. The authors of Ref. [9] fit experimental and lattice results for B → Dℓν ℓ to give R(D) = 0.299 ± 0.003. Within the framework of the Heavy-Quark Expansion, Ref. [10] gives R(D) = 0.297 ± 0.003. By fitting the experimental data, lattice QCD and QCD sum rules (SRs) results forB → Dℓν ℓ , Ref. [11] predicts R(D) = 0.299 ± 0.003. Along with the experimental data, Ref. [12] use the lattice predictions [7,8] for the form factors of B → Dℓν ℓ as inputs, the prediction for R(D) with the Caprini-Lellouch-Neubert parameterization [13] of the form factors is given by R(D) = 0.302 ± 0.003, while using Boyd-Grinstein-Lebed parameterization [14], the authors obtain R(D) = 0.299 ± 0.004. Earlier, based on the heavy quark effective theory (HQET), Refs. [15,16] predict R(D) = 0.302 ± leading-twist DA φ 2;π (x, µ) in Ref. [29]. Firstly, we suggested a new sum rule formula for ξ-moment of φ 2;π (x, µ) based on the fact that the sum rule of zeroth moment can not be normalized in entire Borel parameter region. Secondly, we adopted the least squares method to fit the values of the first ten ξ-moments to determine the behavior of φ 2;π (x, µ). In fact, there are several other approaches, such as traditional QCD sum rules [30], Dyson-Schwinger equation [31], lattice calculation [32,33], etc., to be adopted in the study of the DAs of mesons especially light mesons. By comparison, the scheme suggested in Ref. [29] has its own unique advantages. In which, the new sum rule formula of ξ-moment can reduce the system uncertainties caused 1 by the truncation of the high-dimensional condensates as well as the simple parametrization of quark-hadron daulity for continuum states, and this improves the prediction accuracy of QCD SRs and its prediction ability for higher moments; The least squares method is used to fit the ξ-moments to determine DA, which avoids the extremely unreliable highorder Gegenbauer moments, and can absorb as much information of DA carried by high-order ξ-moments as possible to give more accurate behavior of DA [34]. Very recently, this scheme has been used to study the kaon leading-twist DA φ 2;K (x, µ) by considering the SU f (3) symmetry breaking effect [35], the axial-vector a 1 (1260)meson longitudinal twist-2 DA [36], the scaler K * 0 (1430) and a 0 (980)-meson leading-twist DAs [37,38]. Inspired by these works in Refs. [29,35], we will restudy D-meson twist-2, 3 DAs The rest of the paper are organized as follows. In Sec. II, we will present new sum rule formulae for the ξ-moments of φ 2;D (x, µ), φ p 3;D (x, µ) and φ σ 3;D (x, µ), and briefly describe and improve the LCHO models of those DAs. In Sec. III, we will analyze the behavior of those DAs based on the new values of ξ-moments in detail. Section IV is reserved for a summery.

II. THEORETICAL FRAMEWORK
A. New sum rule formulae for the ξ-moments of D-meson twist-2,3 DAs As discussion in Ref. [29], the new sum rule formula for the ξ-moments is based on that the sum rule of zeroth moment can not be normalized in entire Borel parameter region. Therefore, the discussion of this paper begins with the sum rule formulae for the ξ-moments of D-meson twist-2 DA φ 2;D (x, µ) obtained in Ref. [18] and twist-3 DAs φ p 3;D (x, µ), φ σ 3;D (x, µ) obtained in Ref. [19]. By giving up the priori setting for zeroth ξ-moment 1 The numerical results in Ref. [29] show that this improves the accuracy of ξ-moments by at least 10%. normalization, Eq. (28) in Ref. [18] should be modified as for the nth ξ-moment ξ n 2;D of φ 2;D (x, µ). Eq. (27) in Ref. [19] should be modified as ξ n p 3;D ξ 0 for the nth ξ-moment ξ n p 3;D of φ p 3;D (x, µ). Eq. (28) in Ref. [19] should be modified as ξ n σ 3;D ξ 0 for the nth ξ-moment ξ n σ 3;D of φ σ 3;D (x, µ). In Eqs. (1), (2) and (3), m D is the D-meson mass, m c is the current charm-quark mass, f D is the decay constant of D-meson, s D is the continuum threshold,L M indicates Borel transformation operator with the Borel parameter M . µ p D and µ σ D are the normalization constants of DAs φ p 3;D (x, µ) and φ σ 3;D (x, µ) respectively. Usually, µ p D = µ σ D = µ D = m 2 D /m c in literature by employing the equations of motion of on-shell quarks in the meson. However, as discussed in Refs. [39,40], the quarks inside the bound state are not exactly on-shell. Then µ p D and µ σ D are taken as undetermined parameters in this paper and will be determined via the sum rules of zeroth ξ-moments of DAs φ p 3;D (x, µ) and φ σ 3;D (x, µ) following the idea of Refs. [19,39,40]. In addition, in sum rules (1), (2) and (3), the subscript "pert" stands for the terms coming from the contribution of perturbative part in operator product expansion, subscripts qq , G 2 , qGq , qq 2 and G 3 stand for the terms proportional to double-quark condensate, double-gluon condensate, quark-gluon mixing condensate, four-quark condensate and triple-gluon condensate, respectively. For the expressions of those terms in Eqs. (1), (2) and (3), one can refer to the appendixes in Refs. [18,19]. By taking n = 0 in Eq. (1) and (2), one can obtain the sum rules for the zeroth ξ-moments ξ 0 2;D and ξ 0 p 3;D . As the functions of the Borel parameter, the zeroth ξ-moments ξ 0 2;D in Eq. (1) and ξ 0 p 3;D in Eqs. (2) and (3) obviously can not be normalized in entire M 2 region. Therefore, more reasonable and accurate sum rules should be and for ξ n 2;D , ξ n p 3;D and ξ n σ 3;D , respectively.

B. LCHO models for D-meson twist-2, 3 DAs
In Refs. [18,19], we have suggested LCHO models for D-meson twist-2, 3 DAs. In this subsection, we first propose a brief review for those models, then we will improve them by reconstructing whose longitudinal distribution functions.
The D-meson leading-twist DA φ 2;D (x, µ) can be obtained by integrating out the transverse momentum k ⊥ component in its WF Ψ 2;D (x, k ⊥ ), i.e., Based on the BHL description [27], the LCHO model for the D-meson leading-twist WF consists of the spin- In which, m =m c x +m qx with the constituent charm-quark masŝ m c and light-quark massm q . In this paper, we takê m c = 1.5GeV andm q = 0.25GeV [35]. As discussed in Ref. [18], we take χ 2;D → 1 approximately due to thatm c ≫ Λ QCD . Then, the D-meson leading-twist WF reads wherex = 1 − x, A 2;D is the normalization constant, β 2;D is a harmonious parameter that dominates the WF's transverse distribution, ϕ 2;D (x, µ) dominates the WF's longitudinal distribution. Substituting Eq. (8) into (7), the expression of Dmeson leading-twist DA φ 2;D (x, µ 0 ) can be obtained, i.e., Following the way for constructing the D-meson leadingtwist DA, the LCHO models for D-meson twist-3 DAs and respectively. For the longitudinal distribution functions ϕ 2;D (x), ϕ p 3;D (x) and ϕ σ 3;D (x), we used to take the first five terms of Gegenbauer expansions for the corresponding DAs in Refs. [18,19]. As discussed in Ref. [29,35], higher order Gegenbauer polynomials will introduce spurious oscillations [31], while those corresponding coefficients obtained by directly solving the constraints of Gegenbauer moments or ξ-moments are not reliable. Then we improve these three longitudinal distribution functions as following, By considering the normalization conditions for D-meson twist-2, 3 DAs there are three undetermined parameters in the LCHO models for DAs φ 2;D (x, µ), φ p 3;D (x, µ) and φ p 3;D (x, µ) respectively, and which will be taken as the fitting parameters to fit the first ten ξ-moments 2 of corresponding DAs by adopting the least squares method in next section.
It should be noted that, D-meson twist-2, 3 DAs are the universal non-perturbative parameters in essence, and non-perturbative QCD should be used to study them in principle. However, due to the difficulty of nonperturbative QCD, those DAs are studied in this paper by combining the phenomenological model, that is, the LCHO model, and the non-perturbative QCD SRs in the framework of BFT. Otherwise, the improvement of the LCHO model of DAs φ 2;D (x, µ), φ p 3;D (x, µ) and φ p 3;D (x, µ), that is, to reconstruct their longitudinal distribution functions, is only based on mathematical considerations. The rationality of this improvement can be judged by the goodness of fit.

A. Inputs
To do the numerical calculation for the ξ-moments of D-meson twist-2, 3 DAs, we take the scale µ = M as usual, and take Λ (n f ) QCD ≃ 324, 286, 207 MeV for the number of quark flavors n f = 3, 4, 5, respectively [29,35]. For other inputs, we take [41] 2 In our previous work [34], based on the pionic leading-twist DA, we analyzed in detail the influence of different number of ξmoments included in the fitting, and found that when the order of ξ-moments is not more than ten, the change of the number of ξ-moments has an obvious impact on the fitting results. When the order of ξ-moments is more than ten, the change of the number of ξ-moments has a very small impact on the fitting results. Therefore, we only use the first ten ξ-moments of D-meson DAs φ 2;D (x, µ), φ p 3;D (x, µ) and φ σ 3;D (x, µ) for fitting in this work.

B. ξ-moments and behaviors of D-meson twist-2, 3 DAs
Then we can calculate the values of the ξ-moments of D-meson twist-2, 3 DAs with the sum rules (4), (5) and (6). First, one need to determine the appropriate Borel windows for those ξ-moments by following usual criteria, such as the contributions of continuum state and dimension-six condensate are as small as possible, and the values of those ξ-moments are stable in corresponding Borel windows. Table I exhibits the limits to the continuum state's contributions and the dimensionsix condensate's contributions for the first ten ξ-moments of D-meson twist-2, 3 DAs. In which, the symbol "−" indicates that corresponding continuum state's contribution is smaller than 10% or dimension-six condensate's contribution is much smaller than 5% in a wide Borel parameter region. This is reasonable because both continuum state's contribution and dimension-six condensate's   contribution are depressed by the sum rules of zeroth ξmoments in the denominator of the new sum rule formulae (4), (5) and (6). By comparing with the criteria listed in Table 1 and Table 4 in Ref. [19], the criteria listed in Table I are much stricter, which reflects one of the advantages of the new sum rule formulae (4), (5) and (6), that is, they reduce the system uncertainty of the sum rule itself. Then, for those ξ-moments, only the upper or lower limits of the corresponding Borel windows is clearly determined. In order to get complete Borel windows, we directly take their lengths as 1 GeV 2 . Figure 1 shows the D-meson twist-2, 3 DA ξ-moments ξ n 2;D , ξ n p 3;D and ξ n σ 3;D with (n = 1, · · · , 10) versus the Borel parameter M 2 . In this figure, the uncertainties caused by the errors of input parameters is not drawn to clearly show the curves of different ξ-moments. Meanwhile, the Borel windows are also shown with the shaded bands. By taking all error sources, such as D-meson mass and decay constant, u-and c-quark masses, as well as vacuum condensates, etc., shown in Eqs. (16) and (17), into consideration, and adding the uncertainties in quadrature, the values of the first ten ξ-moments of D-meson twist-2, 3 DAs are shown in Table II. Here, we give the first two Gegenbauer moments of D-meson twist-2, 3 DAs for reference, that is, In the above work, in order to calculate the ξ-moments of D-meson twist-3 DAs φ p 3;D (x, µ) and φ σ 3;D (x, µ), one should calculate the normalization constants µ p D and µ σ D first. Under the assumption that the sum rules of zeroth ξ-moments ξ 0 p 3;D and ξ 0 σ 3;D can be normalized in appropriate Borel windows, the sum rules of µ p D and µ σ D can be obtained by taking n = 0 in Eqs. (5) and (6) and substituting ξ 0 p 3;D = ξ 0 σ 3;D = 1 into these two sum rules. We require the continuum state's contributions are less than 30% and dimension-six condensate's contributions are not more than 5% and 0.5% to determine the Borel windows for µ p D and µ σ D , respectively. By adding the uncertainties derived from all error sources in quadrature, we have, at scale µ = 2GeV. Compared with the values in Ref. [19], µ p D in (20) increases by about 7.2%, and µ σ D decreases by about 12.0%. The former is caused by the update of input parameters, while the latter is also caused by the new sum rule formula, i.e., Eq. (6), in addition to the update of input parameters.
Then we can determine the model parameters of our LCHO models for D-meson twist-2 DA φ 2;D (x, µ) and twist-3 DAs φ p 3;D (x, µ) and φ σ 3;D (x, µ) by using the ξmoments exhibited in Table II with the least squares method following the way suggested in Refs. [29,35]. Take the D-meson leading-twist DA φ 2;D (x, µ) as an example, we first take the fitting parameters θ as the undetermined LCHO model parameters α 2;D , B 2;D 1 and β 2;D , i.e., θ = (α 2;D , B 2;D 1 , β 2;D ), as discussed in Sec. II B. By minimizing the likelihood function the optimal values of the fitting parameters θ we are looking for can be obtained. In Eq. (21), i is taken to be the order of the ξ-moments of φ 2;D (x, µ); the central values of ξ-moments ξ n 2;D (n = 1, · · · , 10) with their errors exhibited in Table II are regarded as the independent measurements y i and the corresponding variance σ i . One can intuitively judge the goodness of fit through the magnitude of probability P χ 2 min =  Table III. Following the same procedure, the LCHO model parameters for D-meson twist-3 DAs φ p 3;D (x, µ) and φ σ 3;D (x, µ) at scale µ = 2 GeV and the corresponding goodness of fits can be obtained and are shown in Table III too. Then the corresponding behaviors of DAs φ 2;D (x, µ), φ p 3;D (x, µ) and φ σ 3;D (x, µ) are determined. In order to intuitively show the behaviors of these three DAs, we plot and exhibit their curves in Fig. 2. As a comparison, the models in literature for D-meson leading-twist DA φ 2;D (x, µ) such as KLS model [22], LLZ model [23], LM model [24], the form with LFQM [28], and our previous research results [18,19] for φ 2;D (x, µ), φ p 3;D (x, µ) and φ σ 3;D (x, µ) based on the LCHO model are also shown in Fig. 2. From Fig. 2, one can find that our present prediction for φ 2;D (x, µ) is closes to LM model. Compared with the KLS model and LLZ model, our φ 2;D (x, µ) is narrower, and supports a large momen-  The models in literature such as KLS model [22], LLZ model [23], LM model [24], the form with LFQM [28], and our previous research results [18,19] based on the LCHO model are also shown for comparison. tum distribution of valence quark in x ∼ [0.05, 0.5]. Compared with our previous work in Refs. [18,19], our new predictions for φ 2;D (x, µ), φ p 3;D (x, µ) and φ σ 3;D (x, µ) in this paper is smoother, and effectively eliminating the spurious oscillations introduced by the high-order Gegenbauer moments in old LCHO model.

IV. SUMMARY
In this paper, we restudied the D-meson leading-twist DA φ 2;D (x, µ), twist-3 DAs φ p 3;D (x, µ) and φ σ 3;D (x, µ) with QCD SRs in the framework of BFT by adopting a new scheme suggested in our previous work [29]. The new sum rule formula for the ξ-moments ξ n 2;D , ξ n p 3;D and ξ n σ 3;D , i.e., Eqs. (4), (5) and (6), were proposed and used to calculate whose values. Those values have been exhibited in Table II. The LCHO models for DAs φ 2;D (x, µ), φ p 3;D (x, µ) and φ σ 3;D (x, µ) were improved. By fitting the values of ξ-moments ξ n 2;D , ξ n p 3;D and ξ n σ 3;D shown in Table II with the least squares method, the model parameters were determined and shown in Table III. Then the predicted curves for D-meson leading-twist DA φ 2;D (x, µ), twist-3 DAs φ p 3;D (x, µ) and φ σ 3;D (x, µ) are shown in Fig. 2. The criteria adopted to determine the Borel windows for ξ-moments of D-meson leading-twist DA φ 2;D (x, µ), twist-3 DAs φ p 3;D (x, µ) and φ σ 3;D (x, µ) exhibited in Table I imply that the new sum rule formula (4), (5) and (6) can reduce the system uncertainties and propose more accurate predictions for ξ-moments ξ n 2;D , ξ n p 3;D and ξ n σ 3;D , respectively. The goodness of fits for φ 2;D (x, µ), φ p 3;D (x, µ) and φ σ 3;D (x, µ) are P χ 2 min = 0.996623, 0.934514 and 0.999021, respectively, which indicate our improved LCHO models shown in Sec. II B with the model parameters in Table III can well prescribe the behaviors of those three DAs. The predicted DAs' curves shown in Fig. 2 indicate the improved LCHO models in this work can eliminate the spurious oscillations introduced by the high-order Gegenbauer moments in old LCHO models obtained in Refs. [18,19]. Otherwise, in order to simply investigate the influence of the new D meson twist-2, 3 DAs in this work on the relevant physical quantities, the TFFs f B→D +,0 (q 2 ) and R(D) are calculated. For the relevant formulae, one can refer to Ref. [19]. We find that the new DAs can bring about 10% and 6% changes to f B→D +,0 (0) and R(D) respectively. In order to obtain a more accurate TFFs and R(D), it is necessary to consider the next-to-leading order corrections for the contributions of D meson twist-3 DAs, which will be our next step.