$a_1(1260)$-meson longitudinal twist-2 distribution amplitude and the $D\to a_1(1260)\ell^+\nu_\ell$ decay processes

In the paper, we investigate the moments $\langle\xi_{2;a_1}^{\|;n}\rangle$ of the axial-vector $a_1(1260)$-meson distribution amplitude by using the QCD sum rules approach under the background field theory. By considering the vacuum condensates up to dimension-six and the perturbative part up to next-to-leading order QCD corrections, its first five moments at an initial scale $\mu_0=1~{\rm GeV}$ are $\langle\xi_{2;a_1}^{\|;2}\rangle|_{\mu_0} = 0.223 \pm 0.029$, $\langle\xi_{2;a_1}^{\|;4}\rangle|_{\mu_0} = 0.098 \pm 0.008$, $\langle\xi_{2;a_1}^{\|;6}\rangle|_{\mu_0} = 0.056 \pm 0.006$, $\langle\xi_{2;a_1}^{\|;8}\rangle|_{\mu_0} = 0.039 \pm 0.004$ and $\langle\xi_{2;a_1}^{\|;10}\rangle|_{\mu_0} = 0.028 \pm 0.003$, respectively. We then construct a light-cone harmonic oscillator model for $a_1(1260)$-meson longitudinal twist-2 distribution amplitude $\phi_{2;a_1}^{\|}(x,\mu)$, whose model parameters are fitted by using the least squares method. As an application of $\phi_{2;a_1}^{\|}(x,\mu)$, we calculate the transition form factors (TFFs) of $D\to a_1(1260)$ in large and intermediate momentum transfers by using the QCD light-cone sum rules approach. At the largest recoil point ($q^2=0$), we obtain $ A(0) = 0.130_{ - 0.013}^{ + 0.015}$, $V_1(0) = 1.898_{-0.121}^{+0.128}$, $V_2(0) = 0.228_{-0.021}^{ + 0.020}$, and $V_0(0) = 0.217_{ - 0.025}^{ + 0.023}$. By applying the extrapolated TFFs to the semi-leptonic decay $D^{0(+)} \to a_1^{-(0)}(1260)\ell^+\nu_\ell$, we obtain ${\cal B}(D^0\to a_1^-(1260) e^+\nu_e) = (5.261_{-0.639}^{+0.745}) \times 10^{-5}$, ${\cal B}(D^+\to a_1^0(1260) e^+\nu_e) = (6.673_{-0.811}^{+0.947}) \times 10^{-5}$, ${\cal B}(D^0\to a_1^-(1260) \mu^+ \nu_\mu)=(4.732_{-0.590}^{+0.685}) \times 10^{-5}$, ${\cal B}(D^+ \to a_1^0(1260) \mu^+ \nu_\mu)=(6.002_{-0.748}^{+0.796}) \times 10^{-5}$.

In the paper, we investigate the moments ξ ;n 2;a 1 of the axial-vector a1(1260)-meson distribution amplitude by using the QCD sum rules approach under the background field theory. By considering the vacuum condensates up to dimension-six and the perturbative part up to next-to-leading order QCD corrections, its first five moments at an initial scale µ0 = 1 GeV are ξ 2;a 1 |µ 0 = 0.028 ± 0.003, respectively. We then construct a light-cone harmonic oscillator model for a1(1260)meson longitudinal twist-2 distribution amplitude φ 2;a 1 (x, µ), whose model parameters are fitted by using the least squares method. As an application of φ 2;a 1 (x, µ), we calculate the transition form factors

I. INTRODUCTION
The D-meson semileptonic decays into axial-vector mesons are the key components for understanding the nonperturbative effects in weak interactions, which have been studied by many theoretical and experimental groups. Based on the constituent quark model, the quantum number of a meson is determined by the quantum numbers of all the constituent quarks. There are two types of axial-vector mesons, 1 P 1 -state with the quantum state J P C = 1 +− and 3 P 1 -state with J P C = 1 ++ . Among the axial-vector mesons, only the isospin triplet heavy state mesons b 1 (1235) and a 1 (1260) do not have a mixing phenomenon. Their internal structures are relatively clear. At the hadron level, the a 1 (1260)-meson is a good subject which is considered as the chiral adjoint state of ρ-meson. Thus a 1 (1260) and ρ-meson are defined as the light-quark pair qq with q = (u, d) [1][2][3]. Since 1986, the properties of a 1 (1260)-meson have been accurately measured [4]. Besides, the observation of the charmless hadronic decay processes involving a 1 (1260)meson, such as B 0 → a ± 1 (1260)π ∓ , which have been issued by the BABAR and Belle collaborations [5][6][7][8], indicates that a 1 (1260) is a 3 P 1 -state. Those measurements help us to investigate the production mechanism of axialvectors via hadronic decay processes and to probe the n a ;n 2;a1 (µ)C 3/2 n (ξ)], wherex = (1 − x), ξ = (2x − 1) and a ;n 2;a1 (µ) are Gegenbauer moments, whose first nonzero one has been given by Yang at the initial scale µ 0 = 1 GeV [15], a ;2 2;a1 (µ 0 ) = −0.02 (2). This value, along with the higher order of a ;n 2;a1 (µ), can be calculated within the framework of QCDSR under the background field theory (BFTSR) [22]. It has been pointed out that the LCDA model based on conformal expansion that is truncated after its first few terms is not suitable for all cases, since the higher-order Gegenbauer terms may have sizable contributions, even if they are generally power suppressed with the increment of n for large momentum transforms. Thus it is important to know more moments for a precise determination of LCDA.
The leading-twist LCDA of a meson can be related to its Bethe-Salpeter wave function. Previous works mainly focused on the wave functions of the pseudoscalar or the vector mesons, cf. Refs. [23][24][25][26][27], and there is few research on the axial-vector mesons. In this work, we will construct a light-cone harmonic oscillator (LCHO) model for the a 1 (1260)-meson longitudinal twist-2 LCDA. The parameters of the model shall be fitted by using the newly calculated moments ξ ;n 2;a1 | µ . And we will use the least squares method to do the fitting and to get the optimal solution. For the purpose, we will use BFTSR to calculate the moments ξ ;n 2;a1 | µ . Within the framework of BFTSR, the quark and gluon fields are composed by the background fields and their surrounding quantum fluctuations, and the usual vacuum condensates are described by the classical background fields, which provides a clear physical picture for the bound-state internal structures and makes the sum rules calculation more simplified. The BFTSR have been widely used in calculating the LCDAs of the heavy/light mesons [28][29][30][31][32][33]. Here we will adopt this approach to investigate the a 1 (1260)-meson moments ξ ;n 2;a1 | µ and then provide a more accuracy LCDA φ 2;a1 (x, µ).
The remaining parts of the paper are organized as follows. In Sec. II, we present the calculation procedures for the moments of a 1 (1260)-meson longitudinal twist-2 LCDA, the LCHO model, the TFFs and the branching ratios. Numerical results and discussions are presented in Sec. III. Section IV is reserved for a summary.

II. CALCULATION TECHNOLOGY
Firstly, the a 1 (1260)-meson longitudinal twist-2 LCDA is defined as [15] 0|q 1 (z)γ µ γ 5 q 2 (−z)|a 1 (q, λ) Here, two light quarks q 1 = q 2 , which are (u, d)-quark for a 1 (1260)-meson, respectively. This convention shall be followed throughout the remaining parts of this paper. The f a1 is a 1 (1260)-meson decay constant, q and e * (λ) are momentum and polarization vector of a 1 (1260)-meson. The polarization vector satisfies the relationship (e * (λ) · z)/(q · z) → 1/m a1 [34]. By doing the series expansion near z → 0 on both sides of Eq. (1), one will get: where the covariant derivative satisfies the relation (iz · The n-th order moments of the a 1 (1260)-meson DA are defined as One can start from the following correlation function (correlator) to derive the sum rules, i.e.
with J n (x) =q 1 (x)/ zγ 5 (iz · ↔ D) n q 2 (x), J † 0 (0) = q 2 (0)/ zγ 5 q 1 (0) and z 2 = 0. 1 Because of the G-parity, φ 2;a1 (x, µ) for 3 P 1 -state defined by the nonlocal axialvector current is symmetric, indicating only even moments are non-zero, i.e. n = (0, 2, 4, 6, · · · ). Based on the idea of BFTSR and the Feynman rules for one hand, one can apply the OPE for the correlator (4) in deep Euclidean region q 2 ≪ 0. Then, the correlator can be expanded into three terms including the quark propagators S d F (0, x), S u F (x, 0) and the vertex operators (iz · ↔ D) n , which have been given in our previous work [33]. In dealing with Lorentz invariant scalar function Π (n,0) 2;a1 (z, q 2 ), the vacuum matrix element should be used, which can be found in Ref. [28]. On the other hand, one can insert a complete set of a 1 (1260)-meson intermediated hadronic states with the same J P quantum number into the correlator and obtain ImI (n,0) where s a1 is the continuum threshold. The first (second) terms on the r.h.s of Eq. (5) are the a 1 (1260)-meson ground state (continuum states) contribution. Due to not all higher-order contributions and higher-dimensional operators have been included, the fixed-order prediction of ξ ;0 2;a1 | µ is close but not exactly equals to 1. So we reserve the term ξ ;0 2;a1 | µ in the hadronic expression, i.e. Eq. (5), which is different from other researches in dealing with the axial-meson DA by using the QCDSR approach. Then, one can bridge the invariant function and the OPE side by using the dispersion relation. Furthermore, the Borel transformation are used to suppress the contribution from the continuum states and high dimen-sion condensates. The sum rule expression is a1,had (s) =B M 2 I (n,0) 2;a1,QCD (q 2 ), (6) where the Borel parameter M 2 coming from the Borel transformation with the operatorB M 2 . Following the standard SVZ sum rules procedures, we then obtain the expression of the moment of a 1 (1260)-meson longitudinal twist-2 LCDA, i.e., Here,ψ(n) = ψ((n + 1)/2) − ψ(n/2) + ln 4. The nextto-leading (NLO) corrections are [35]. When taking n = 0 for the Eq. (7), one can get the sum rule of zeroth moment, e.g. ξ where, the numerator of Eq. (8) are coming from the Eq. (7), and denominator are the zeroth moment.
Owning to the fact that the high-order Gegenbauer moment for a 1 (1260)-meson longitudinal twist-2 DA still have large uncertainties, one can construct a new LCDA model, i.e. the LCHO model based on the Brodsky-Huang-Lepage (BHL) prescription [26,36]. The BHL suggested a connection between the equal-time wave function in the rest frame and the light-cone wave function by equating the off-shell propagator ǫ in the two frames.
For the former x i = 1. In the two-particle system, one has with m 1 = m 2 = m q . Then the possible connection between the rest frame wave function ψ CM (q 2 ) and the light-cone wave function ψ LC (x, k ⊥ ) can be formally rep-resented by On the other hand, the wave function of the harmonic oscillator model in the rest frame is from an approximate bound state solution in the quark models for mesons. By combining Eqs. (10) and (11), the LCHO model of a 1 (1260)-meson wave function satisfies Then the LCHO model for the a 1 (1260)-meson wave function gives where A 2;a1 is the normalization constant, β 2;a1 is a harmonic parameter, and m q is the mass of the constitute quark u and d in a 1 (1260)-meson. In addition, the function ϕ 2;a1 (x) dominates longitudinal distribution and can be expressed as [37] Since the meson LCDA is related to its wave function Ψ 2;a1 (x, k ⊥ ) via the following relation: Then, the twist-2 LCDA of a 1 (1260)-meson can be derived by integrating the transverse momentum, which have the following form The two parameters A 2;a1 and β 2;a1 are constrained by the following two conditions: • The wave function normalization condition, • The probability of |qq Fock state in a a 1 (1260)meson should be less than 1, e.g. P a1 < 1, We shall fit the parameters α 2;a1 and B 2;a1 by using the least squares method so as to achieve the same moments ξ ;n 2;a1 | µ from the sum rules (7). The detailed analysis about this point can be found in Refs. [37,38].
Secondly, we adopt the following correlator to derive the LCSRs for the D → a 1 (1260) TFFs, In the time-like q 2 -region, the long distance quark-gluon interactions are dominant. To deal with the correlator in the time-like region, one can insert a complete set of the D-meson states, which have the same J P quantum numbers to obtain the hadronic expression. After separating the D-meson pole term, we obtain where The D → a 1 (1260) transition matrix elements have the expressions [18]: where p is a 1 (1260)-meson momentum and q = p D −p a1 is the momentum transfer, e * (λ) stands for a 1 (1260)-meson polarization vector with λ = (⊥, ) being its transverse or longitudinal component, respectively. There are one linear relationships among the TFFs [13,39]: Following the standard sum rules procedures, one can represent the contributions of the higher resonances and the continuum states by dispersion integrations so as to derive the expressions for the hadronic invariant ampli- (19). The continuum threshold parameter s 0 can be set as the value close to the squared mass of the lowest scalar D-meson. Meanwhile, the conventional quarkhadron duality ansatz, ρ had i = ρ QCD i θ(s−s 0 ), can be used to calculate the hadron spectrum density ρ had i . On the other hand, in the space-like region, one can calculate the correlator via the QCD theory. In this region, the correlator can be treated by the OPE with the coefficients being pQCD calculable. The c-quark propagator which shall be used in the calculation can be found in Ref. [33]. After applying the OPE and using the expressions for the transition matrix elements, one can arrange the resultant expressions by twist-2, 3, 4 LCDAs [15,16]. After matching the correlator with the dispersion relation, and applying the conventional Borel transformation to suppress the less known continuum contributions, the resultant TFFs under the LCSR approach are where the q 2 -dependence factors s(u) and s(X) are defined as where X = α 1 −α 2 +vα 3 , α 1 , α 2 and α 3 are the respective momentum fractions carried byq 1 , q 2 quarks and gluon in the a 1 (1260)-meson [15]. Θ(c(u, s 0 )) is the conventional step function, Θ(c(u, s 0 )) and Θ(c(u, s 0 )) are defined as Here u 0 is the solution of c(u 0 , s 0 ) = 0 with 0 ≤ u 0 ≤ 1.
Then, the longitudinal and transverse differential decay widths for semileptonic decay D → a 1 (1260)ℓ + ν ℓ can be expressed as where G F is the Fermi coupling constant, |V cd | is the CKM matrix element, and λ = (m 4 ). The total differential decay width of the semileptonic decay is dΓ L + dΓ T , where dΓ L and dΓ T = dΓ + + dΓ − corresponds to longitudinal and transverse parts, respectively.

III. NUMERICAL RESULTS AND DISCUSSIONS
To do the numerical calculation, the input parameters are taken as follows. The current charm-quark mass, m c (m c ) = 1.27 (2) It should be noted that, the values of the gluon condensates are the most commonly used in QCD sum rules. The value of the double-gluon condensate α s G 2 is determined by the sum rule of the charmonium, and the one for triple-gluon condensate g 3 s f G 3 is based on the instanton model 2 . The double-quark condensate qq and the quark-gluon mixed condensate g sq σT Gq were updated in our previous work [37] based on the GellMann-Oakes-Renner relation and the relationship g sq σT Gq = m 2 0 qq with m 2 0 = 0.80(2) GeV 2 [46]. One can calculate the four-quark condensate g sq q 2 by using ρα s qq 2 = (5.8 ± 1.8) × 10 −4 GeV 6 with ρ ≃ 3 − 4 [46], and determine the value of g 2 sq q 2 by combining with the new value of qq . All those scale-dependent parameters, such as the quark masses and the vacuum condensates, shall be run from an initial scale µ 0 to a special choice of scale such as µ k by applying the renormalization group equations (RGE) given by Refs. [47][48][49][50]. Two important parameters for the BFTSR approach for the moments are continuum threshold s 0 and Borel parameter M 2 , whose range is called as the Borel Window. To fix their values and make the sum rules predictions reliable, the contributions from the continuum states and the contributions from the dimension-six condensates should be small enough. For the purpose, we determine the Borel window by allowing the contribution of continuum states to be less than 45% and the contribution of dimension-six condensates to be less than 5%. When determining the threshold parameter s a1 , one can normalized the 0 th -order a 1 (1260)-meson longitudinal DA in the appropriate Borel window. Followed by this approach, we can get s a1 = 1.4(4) GeV. We present the determined Borel windows and ξ ;n 2;a1 | µ at the scale µ = √ M 2 in Table I. Here, we have set the continuum contributions to be no more than (35%, 35%, 40%, 40%, 45%) for n = (2, 4, 6, 8, 10), respectively, and the dimension-six contributions for all ξ ;n 2;a1 | µ to be less than 5%. Principally, the moments including all the condensates should be independent to the Borel parameter M 2 , and for a fixed-order OPE expansion, it may change with different choices of M 2 within the allowable Borel window. Such change depends heavily on the convergence of the OPE expansion over 1/M 2 . For the present LCSR up to dimension-6 condensates, e.g the series (7), as a conservative prediction, we require the variations of ξ ;n 2;a1 | µ within the Borel window to be less than 10%. We present the first five moments for a 1 (1260)-meson twist-2 LCDA versus the Borel parameter M 2 in Fig. 1. The shaded bands indicate the corresponding Borel windows, which are in the region of [1.0, 7.0] GeV 2 , respectively. By taking all uncertainty sources into consideration and applying the RGE of the moments, ξ  2;a1 | µ k = 0.027 (3). (40) In order to determine the two LCHO model parameters α 2;a1 and B 2;a1 , one can use the specific fitting by taking the two parameters as the fitting parameters, e.g. θ = (α 2;a1 , B 2;a1 ). The moments ξ ;n 2;a1 | µ from Eq. (16) with the definition ξ ;n 2;a1 | µ = 1 0 ξ n φ 2;a1 (x, µ) been regarded as the mean function µ(x i ; θ)(x i → n), where the moments calculated with BFTSR, i.e. Eq. (40) are considered as the independent measurements y i with the known variance σ i . To obtain the best values of fitting parameters θ, one can minimize the function The goodness of fit is judged by the magnitude of the probability Here f (y; n d ) with the number of degrees of freedom n d is the probability density function of χ 2 (θ), and Then, we obtain the fitting parameters at the initial scale µ 0 . Due to the quark component of a 1 (1260)meson here isūu ordd, so the treatment for light-quark mass in this paper is the same with usual constituent quark mass. There are different values for constituent quark mass m q , which is taken to be 250 MeV in the invariant meson mass scheme [51][52][53][54], 330 MeV in the spin-averaged meson mass scheme [55][56][57][58]. In addition, m q = 300 MeV and m q = 200 MeV [37] of the simplest in Refs. [59,60]. In this work, we present the results for different choices of the constituent quark mass, e.g. m q = (200, 250, 300, 330, 350) MeV, respectively. The fitting results are given in Table II. As a default value, we shall take m q = 250 MeV to do our calculation, whose corresponding goodness of fit is 95.4%. This value is also agree with the usual pion and kaon cases [37,38]. One can find that the parameter A 2;a1 and B 2;a1 gradually decrease with the increment of m q , and the goodness of fit P χ 2 min is also decreasing with the increment of m q . To show more clearly the relationship between the magnitudes of α 2;a1 and B 2;a1 and the goodness of fit P χ 2 min , we present the relationship curve between them in Fig. 2. The darker shaded band of Fig. 2 represents the higher goodness of fit. When the range of goodness of fit is TABLE II: Fitting parameters A 2;a 1 (GeV −1 ), β2;a 1 (GeV), α 2;a 1 and B 2;a 1 with different constituent quark mass mq under initial scale µ0. Meanwhile, the goodness of fit P χ 2 min and the probability Pa 1 are also given.  80% ≤ P χ 2 min ≤ 96%, the allowable ranges for the parameters α 2;a1 and B 2;a1 are quite small.
The TFFs of D → a 1 (1260) are key elements for investigating the D-meson semileptonic decay. To derive the exact value for the TFFs (24)- (27), we need to fix the continuum threshold s 0 and Borel parameter M 2 . Normally, the continuum threshold s 0 should be taken near the squared mass of the D-meson's first excited state with the same J P number, e.g. D 0 (2550) 0 . And we take s A 0 = 6.5(3) GeV 2 , s V1 0 = 5.7(3) GeV 2 , s V2 0 = 6.0(3) GeV 2 and s V0 0 = 6.0(3) GeV 2 . One can use four criteria of the LCSR approach listed in Ref. [33] Table III. To make a comparison, the predictions from various approaches are presented, i.e. the LCSR-I, II [65,66], 3PSR [67] and CLFQM [10,39], respectively. Our predictions are close to the LCSR-II ones. To have a clear look at the uncertainties caused by different input parameters, we present the TFFs as follows We present the D → a 1 (1260) TFFs at the large recoil region q 2 = 0 in Table IV, in which the contributions from the DAs with various twist structures are presented. Ref. [68] indicates that the corrections from the twoparticle higher-twist contributions are (27%-36%), and our twist-4 contribution falls within this margin of error. As for the twist-4 contribution to the form factor V 0 (0), its magnitude is about 34.56% of the total result. For V 1 (0) and V 2 (0), their twist-4 contributions change to 9.48% and 8.33%, respectively. For convenience, we use H(α i ) =Φ 3;a1 (α i )−Φ 3;a1 (α i ) to represent the net contribution of the three-particle twist-3 LCDAs, whose contribution to the TFFs V 1 (0), V 2 (0) and V 0 (0) are 0.90%, 7.90% and 8.30%, respectively.
Theoretically, the LCSR approach for D → a 1 (1260) TFFs are reliable in low and intermediate q 2 -regions, which can be extrapolated to all the physically allowable region m 2 ℓ ≤ (m D − m a1 ) 2 ≈ 0.4 GeV 2 . In this paper, we mainly consider the simplified series expansion (SSE), which has the following form   with the function z(t) including t ± , t 0 and t, whose definition can be found Ref. [33]. F i (q 2 ) are the TFFs A(q 2 ) and V 0,1,2 (q 2 ), respectively. In this approach, the simple pole P i (q 2 ) = (1 − q 2 /m 2 R,i ) accounting for low-lying resonances, instead of Blaschke factor B(t) is more applicable in many processes. Here, the masses of low-lying D resonances are mainly determined by the J P states. Followed by the Ref. [69] and PDG values [40], we listed the m R,i in Table V. Meanwhile, the fitting quality should satisfied the relationship ∆ < 1%, which is defined as where t ∈ [0, 1/40, · · · , 40/40] × 0.28 GeV 2 . The fitting parameters α i for every TFFs and the quality of fit ∆ are also listed in Table V. From which, the ∆ of D → a 1 (1260) TFFs are less than 0.020%. Furthermore, the |V cd |-independence longitudinal and transverse differential decay widths dΓ L,T and total width dΓ total = dΓ L + dΓ T of D → a 1 (1260)ℓ + ν ℓ can be obtained according to Eqs. (37) and (38), which are shown in Fig. 4. As a comparison, we also present the LCSR-II predictions [66]. Fig. 4 shows that the contributions of the decay widths mainly come from the longitudinal parts in small q 2 -region, and transverse parts contribute sizably in intermediate and large q 2 -regions. In different to LCSR-II prediction, our predictions tend to 0 when q 2 (m D − m a1 ) 2 ≈ 0.4 GeV 2 , which are similar to most of the other semileptonic decay processes such as final state involving π, K, ρ, K * , D, ....

IV. SUMMARY
In the present paper, we have calculated the a 1 (1260)meson moments of LCDA ξ ;n 2;a1 | µ by using the BFTSR approach up to NLO QCD corrections for the pertur-bative part and up to dimension-six condensates for the non-perturbative part. The moments of LCDA up to 10 th -order have been given in Eq. (40). Then, by combining the two constraints (17) and (18) with the least squares fitting approach for ξ ;n 2;a1 | µ , we get the a 1 (1260)-meson longitudinal LCDA φ 2;a1 (x, µ 0 ). Figure 3 shows that φ 2;a1 (x, µ 0 ) tends to a single-peak behavior. Moreover, by using the derived twist-2 LCDA, we have calculated the D → a 1 (1260) TFFs A(q 2 ) and V 0,1,2 (q 2 ) by using the LCSR approach up to twist-4 accuracy. Furthermore, the |V cd |-independence differential decay width of semileptonic decay D → a 1 (1260)ℓ + ν ℓ with ℓ = (e, µ) have been given in Fig. 4, and the branching fractions for D 0(+) → a −(0) 1 ℓ + ν ℓ are given in Table VI. The branching fractions are of order 10 −5 , which is close to the present experimental upper limit. It is hoped that the decays D → a 1 (1260)ℓ + ν ℓ can be observed in near future, which inversely could provide a (potential) helpful test for QCD sum rules approach.