$\rho$-meson longitudinal leading-twist distribution amplitude revisited and the $D\to \rho$ semileptonic decay

Motivated by our previous work [Phys. Rev. D \textbf{104}, no.1, 016021 (2021)] on pionic leading-twist distribution amplitude (DA), we revisit $\rho$-meson leading-twist longitudinal DA $\phi_{2;\rho}^\|(x,\mu)$ in this paper. A model proposed by Chang based on the Dyson-Schwinger equations (DSEs) is adopted to describe the behavior of $\phi_{2;\rho}^\|(x,\mu)$. On the other hand, the $\xi$-moments of $\phi_{2;\rho}^\|(x,\mu)$ are calculated with the QCD sum rules in the framework of the background field theory. The sum rule formula for those moments are improved. More accurate values for the first five nonzero $\xi$-moments at typical scale $\mu =1, 1.4, 2, 3~{\rm GeV}$ are given, e.g., at $\mu = 1~{\rm GeV}$, \modi{$\langle\xi^2\rangle_{2;\rho}^\| = 0.220(6) $, $\langle\xi^4\rangle_{2;\rho}^\| = 0.103(4)$, $\langle\xi^6\rangle_{2;\rho}^\| = 0.066(5)$, $\langle\xi^8\rangle_{2;\rho}^\| = 0.046(4)$ and $\langle\xi^{10}\rangle_{2;\rho}^\| = 0.035(3)$}. By fitting those values with the least squares method, the DSE model for $\phi_{2;\rho}^\|(x,\mu)$ is determined. By taking the left-handed current light-cone sum rule approach, we get the transition form factor at large recoil region, {\it i.e.} $A_1(0) = 0.498^{+0.014}_{-0.012}$, $A_2(0)=0.460^{+0.055}_{-0.047}$, $V(0) = 0.800^{+0.015}_{-0.014}$, and the ratio $r_2 = 0.923^{+0.133}_{-0.119}$, $r_V = 1.607^{+0.071}_{-0.071}$. After making the extrapolation with a rapidly converging series based on $z(t)$-expansion, we present the decay width for the semileptonic decays $D\to\rho\ell^+\nu_\ell$. Finally, the branching fractions are $\mathcal{B}(D^0\to \rho^- e^+ \nu_e) = 1.889^{+0.176}_{-0.170}\pm 0.005$, $\mathcal{B}(D^+ \to \rho^0 e^+ \nu_e) = 2.380^{+0.221}_{-0.214}\pm 0.012$, $\mathcal{B}(D^0\to \rho^- \mu^+ \nu_\mu) = 1.881^{+0.174}_{-0.168}\pm 0.005$, $\mathcal{B}(D^+ \to \rho^0 \mu^+ \nu_\mu) =2.369^{+0.219}_{-0.211}\pm 0.011$.

Recently in 2021, we proposed a new research scheme for the QCD SR study on pionic leading-twist DAs, where a new sum rule formula for ξ-moments is proposed after considering the fact that the sum rule for zeroth ξ-moment cannot be normalized in entire Borel region [50].Meanwhile, it enables us to calculate more higher-order ξ-moments.Further, the behavior of pion DA φ 2;π (x, µ) can be determined by fitting enough ξmoments with the least squares method.Subsequently, this scheme was used to study pseudoscalar η (′) -meson and kaon leading-twist DA [51,52] and D-meson twist-2,3 DAs [53], axial vector a 1 (1260)-meson twist-2 longitudinal DA [54], scalar a 0 (980) and K * 0 (1430)-meson leading-twist DAs [55,56].Inspired by this, we will restudy ρ-meson leading-twist longitudinal DA φ 2;ρ (x, µ) by adopting the research scheme proposed in Ref. [50] in this paper.
The remaining parts of the paper are organized as follows.In Sec.II, we present the calculation technology for ξ-moments of ρ-meson leading-twist DA, the D → ρ TFFs and the semileptonic decays D → ρℓ + ν ℓ .In Sec.III, we present the numerical results and discussions on ξ-moments, D → ρ TFFs, D → ρℓ + ν ℓ decay widths and branching ratios .Section IV is reserved for a summary.

II. THEORETICAL FRAMEWORK
In order to derive the sum rules of ξ-moments of ρmeson leading-twist longitudinal DA, we adopt the following correlation function (correlator), where the current J n (x) = d/ z(iz • ↔ D) n u(x) with z 2 = 0. Following the standard calculation procedure of the QCD SRs in the framework of BFT [50], the sum rule of ξ n 2;ρ × ξ 0 2;ρ reads The Eq. ( 3) indicates that the zeroth ξ-moment ξ 0 2;ρ in Eq. ( 2) cannot be normalized in the entire Borel parameter region due to that the contributions of vacuum condensates with dimension greater than six are truncated.As discussed in Ref. [50], more accurate and reasonable sum rule for nth ξ-moment ξ n 2;ρ should be On the other hand, to describe the behavior of ρ-meson leading-twist longitudinal DA, we take the following DSE model for φ 2;ρ (x, µ) [57,58], where α − = α−1/2, and N = 4 α Γ(α+1)/[ √ πΓ(α+1/2)] is the normalization constant.
Nextly, in order to get the D → ρ TFFs, one can take the following correlation function with the j L D (x) = iq 2 (x)(1 − γ 5 )c(x) is the left-handed current.As we know, there will be fifteen DAs for vector meson up to twist-4 accuracy, the left-handed chiral current can reduce the uncertainties from chiral-odd vector meson DAs with δ 0,2 -order and leave the chiraleven with δ 1,3 -order meson.The relationship is also listed in Table I.In this table, except j L D (x), the current j R D (x) respect right-handed current with expression j R D (x) = iq 2 (x)(1 + γ 5 )c(x), which have been researched in our previous work [15].The parameter δ ≃ m ρ /m c ∼ 52% [59,60].Meanwhile, the chiral current approach have been considered in some references [61][62][63][64][65][66], which improve the predictions of the LCSR approach.
Based on the procedures of LCSR approach, one will get the D → ρ TFFs A 1,2 (q 2 ) and V (q 2 ) LCSRs expression with the left-handed current in the correlator.The analytic formulas are similar with the B → ρ TFFs of our previous work [45], which make a replacement for the input parameter of B-meson with D-meson such as . Thus, we do not listed here in this paper.Moreover, there have two ratio r V = V (0)/A 1 (0) and r 2 = A 2 (0)/A 1 (0) based on the TFFs A 1,2 (q 2 ) and V (q 2 ), which have less uncertainties between the different approaches.
Furthermore, the semileptonic decay width for D → ρℓ + ν ℓ are composed by three different parts, which can be expressed as, where the constant Other parameters and expressions have the definitions: is the phase-space factor, q 2 max = (m D − m ρ ) 2 is the small recoil point of the D → ρ transition.The three helicity decay amplitudes H ± (q 2 ) and H 0 (q 2 ) are mainly separated by the transition amplitude with definite spinparity quantum number, which can be found in our previous work [15].Meanwhile, the longitudinal and transverse helicity amplitudes are expressed as

III. NUMERICAL ANALYSIS
To perform the numerical calculation, we take f ρ = 210 ± 4 MeV [40,67] κ = 0.74 ± 0.03 [50,69,70].In which, the current quark masses and the vacuum condensates other than gluon condensates are scale dependence, and whose above values are at µ = 2 GeV.For the scale evolution of those inputs, one can refer to Ref. [50].In calculation, we take the scale µ = M as usual.By requiring that there is a reasonable Borel window to normalize ξ 0 2;ρ with Eq. ( 3), we obtain the continuum threshold as s ρ ≃ 2.1 GeV 2 .Now, one can obtain the ξ-moment versus Borel parameter.Further, we can determine the Borel window and then the value of nth ξ-moment ξ n 2;ρ based on the well-known criteria that the continuum state's contribution and dimension-six condensate's contribution are as small as possible, and the value of ξ n 2;ρ is stable in the Borel window.Specifically, the continuum contribution are not more than 30% for all nth-order; the dimensionsix contribution of ξ n 2;ρ are less than 2% for n = (2, 4), 5% for n = (6, 8, 10), respectively.It should be noted that, only the even order ξ-moments are not zero due to the isospin symmetry.The ξ-moments versus Borel parameter and the obtained Borel windows are shown in Fig. 1, where the shaded areas are the Borel windows for n = (2, • • • , 10) respectively.
Then, the values of ξ n 2;ρ | µ up to 10th order with four different scales µ = 1, 1.4, 2, 3GeV and the second Gegenbauer moment a 2;ρ can be obtained and exhibited in Table II.Here, we only calculate the values of the first five non-zero ξ-moments in this work because, as shown in Ref. [57], these moments are sufficient to determine the behavior of DA φ 2;ρ (x, µ).As a comparison, the other theoretical predictions obtained by QCD SRs [21][22][23][24][25][26], LQCD [27][28][29][30], AdS/QCD [31,32], Data Fitting [35,36], LFQM [39] and DSE [41] are also shown in Table II.One can find that, our second moment is less than the  QCD SR predictions in Ref. [21,23,26], LQCD calculations [27][28][29][30] and DSE result [41], and larger than QCD SR predictions in Ref. [24,25], but very consistant with the value with QCD SRs in Ref. [22], the AdS/QCD prediction [31,32] and the value extracted from the HERA data on diffractive ρ photoproduction [35,36].By fitting our values of ξ n 2;ρ (n = 2, 4, 6, 8, 10) shown in Table II with the least squares method (where the model parameters α and â2 are taken to be the fitting parameters, and for specific fitting procedure, one can refer to Ref. [50]), the behavior of the ρ-meson leadingtwist longitudinal DA can be obtained.Specifically, the model parameters α and â2 of our DA and the corresponding χ 2 min /n d and goodness of fit P χ 2 min at several typical scale such as µ = 1, 1.4, 2, 3 GeV are exhibited in Table III.Then, the curve of ρ-meson leading-twist longitudinal DA is shown in Fig. 2. Meanwhile, other predictions in literature [25,41,46] are also shown for comparison.One can find that our curve is closer to the asymptotic form.
Furthermore, in order to calculated the D → ρ TFFs, the input parameters should be clarified.The current charm-quark mass is m c = 1.27 ± 0.02 GeV, D-meson mass is m D = 1.869GeV [68].Based on the three criteria for determining s 0 and M 2 in the LCSR approach that can be found in our previous work [51], we As a comparison, the DA curves obtained by DSE [41], QCD SRs [25], algebraic model [46] and the asymptotic form are also shown.
have the Then, the TFFs at the large recoil region, i.e.A 1,2 (0) and V (0) are present in Table IV.The uncertainties are coming from the squared average for all the input parameters.In Table IV, the predictions from theoretical group and experimental collaborations, i.e. the CLEO collaboration [3], 3PSR [6], HQEFT [7,8], RHOPM [9], QM [10,11], LFQM [12], HMχT [18] and Lattice QCD predictions [19,20] are presented, respectively.The comparison about the every results from Table IV indicate that the TFFs of our prediction is consistent with many approaches within errors.
• The central value with respect uncertainties, our results for r 2 have agreement with QM, LFQM, 3PSR, CCQM for theoretical predictions.r V have agreement with QM and 3PSR predictions.
• The uncertainties of our predictions are (12.9%∼ 14.4%) for r 2 and 4.4% for r V , which shows the left-handed chiral current LCSR approach can get a reasonable results.
• Our current results are consistent with those of the CLEO collaboration, but are larger than the BE-SIII collaboration predictions.

IV. SUMMARY
In the framework of BFT with QCD SR approach, we calculated the ξ-moments of ρ-meson leading twist longitudinal DA.Based on the zeroth ξ-moment can not be normalized in the entire Borel region, the new sum rule formula for the ξ-moment is given in Eq. ( 2).The results up to tenth order with scale µ = (1.0,1.4, 2.0.3.0)GeV 2 are present in Table II, where the results from other theoretical group are also given.Secondly, we give the D → ρ TFFs at large recoil region, i.e.A 1 (0), A 2 (0) and V (0) in Table IV and ratio r 2,V in Fig. 3, which indicate our results are reasonable and consistent with many approaches within errors.Thirdly, we calculated the |V cd |-independent total decay width, ratio for longitudinal/transverse and positive/negative helicity decay width results and present them in Table V.A detail discussion in comparing with other predictions is made.Lastly, we give branching fraction of the two types D → ρℓ + ν ℓ semileptonic decays in Table VI.In the near further, we hope more accuracy data will be reported and more theoretical results will be given to explain the gaps between different approaches.

TABLE III :
The model parameters α and â2 of our DA obtained by fitting using the least squares method and the corresponding χ 2 min /n d and goodness of fit P χ 2