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ρ-meson longitudinal leading-twist distribution amplitude revisited and the Dρ semileptonic decay

  • Motivated by our previous study [Phys. Rev. D 104(1), 016021 (2021)] on the pionic leading-twist distribution amplitude (DA), we revisit the ρ-meson leading-twist longitudinal DA ϕ2;ρ(x,μ) in this study. A model proposed by Chang based on the Dyson-Schwinger equations is adopted to describe the behavior of ϕ2;ρ(x,μ). However, the ξ-moments of ϕ2;ρ(x,μ) are calculated with the QCD sum rules in the framework of the background field theory. The sum rule formulas for these moments are improved. More accurate values for the first five nonzero ξ-moments at the typical scale μ=(1.0,1.4,2.0,3.0)GeV are given, e.g., at μ=1GeV, ξ22;ρ=0.220(6), ξ42;ρ=0.103(4), ξ62;ρ=0.066(5), ξ82;ρ=0.046(4) , and ξ102;ρ=0.035(3). By fitting these values with the least squares method, the DSE model for ϕ2;ρ(x,μ) is determined. By taking the left-handed current light-cone sum rule approach, we obtain the transition form factor in the large recoil region, i.e., A1(0)=0.498+0.0140.012, A2(0)=0.460+0.0550.047, and V(0)=0.800+0.0150.014, and the ratio r2=0.923+0.1330.119, rV=1.607+0.0710.071. After extrapolating with a rapidly converging series based on z(t)-expansion, we present the |Vcd|-independent decay width for the semileptonic decays Dρ+ν. Finally, the branching fractions are B(D0ρe+νe)=1.825+0.1700.162±0.004, B(D+ρ0e+νe)=2.299+0.2140.204±0.011, B(D0ρμ+νμ)=1.816+0.1680.160±0.004, and B(D+ρ0μ+νμ)=2.288+0.2120.201±0.011.
  • The ρ-meson leading-twist longitudinal distribution amplitude (DA) is a key input parameter when investigating related exclusive processes including the ρ meson. Charmed semileptonic decay processes can provide a clear platform for researching the ρ-meson DA. In charmed factories, the BESIII collaboration reported new results for the semileptonic decay D0ρμ+νμ in 2021 [1]. In 2019, the BESIII collaboration provided improved measurements of Dρe+νe [2]. The CLEO collaboration provided measurements of the form factors in the decays D0/+ρ/0e+νe in 2013 [3], and their previous measurements can be found in [4, 5]. It is known that transition form factors (TFFs) are key components of semileptonic Dρ+ν decays in the standard model. Therefore, accurate TFFs are very important for theoretical groups and experimental collaborations. Theoretically, the Dρ TFFs can be treated using different approaches, such as the 3-point sum rule (3PSR) [6], heavy quark effective field theory (HQEFT) [7, 8], relativistic harmonic oscillator potential model (RHOPM) [9], quark model (QM) [10, 11], light-front quark model (LFQM) [12, 13], light-cone sum rule (LCSR) [1416], covariant confined quark model (CCQM) [17], heavy meson and chiral symmetry theory (HMχT) [18], and lattice QCD (LQCD) [19, 20]. The LCSR approach is based on operator production expansion near the light-cone, and all the non-perturbative dynamics are parameterized into light-cone DAs (LCDAs), which is suitable for calculating the heavy to light transition. In this study, we take the LCSR approach to calculate the Dρ TFFs with a left-handed current 1. Thus, the key task is obtaining an accurate ρ-meson longitudinal twist-2 DA.

    Theoretically, the ρ-meson longitudinal leading-twist DA can be studied using various methods, such as QCD sum rules (SRs) [2126], LQCD [2730], the AdS/QCD holographic method [3134], extraction from experimental data [3538], the light-front quark model (LFQM) [39, 40], Dyson-Schwinger equations (DSEs) [41], large momentum effective theory (LMET) [42], instanton vacuum [43, 44], and other models [45, 46]. The QCDSR in the framework of background field theory (BFT) is an effective approach in calculating light and heavy meson DAs [47]. In early 2016, we preliminarily studied the ρ-meson longitudinal leading-twist DA [48], in which the first two order nonzero ξ-moments and Gegenbauer moments were obtained, and the behavior of ϕ2;ρ(x,μ) described by the light-cone harmonic oscillator (LCHO) model was determined [49].

    In 2021, we proposed a new research scheme for the QCDSR study on pionic leading-twist DAs, in which a new SR formula for ξ-moments was proposed after considering that the SR for the zeroth ξ-moment cannot be normalized in the entire Borel region [50]. It enabled us to calculate more higher-order ξ-moments. Furthermore, the behavior of the pion DA ϕ2;π(x,μ) could be determined by fitting sufficient ξ-moments with the least squares method. Subsequently, this scheme was used to study pseudoscalar η()-meson and kaon leading-twist DAs [51, 52] and D-meson twist-2,3 DAs [53], axial vector a1(1260)-meson twist-2 longitudinal DAs [54], and scalar a0(980) and K0(1430)-meson leading-twist DAs [55, 56]. Inspired by this, we restudy the ρ-meson leading-twist longitudinal DA ϕ2;ρ(x,μ) by adopting the research scheme proposed in Ref. [50].

    The remainder of this paper are organized as follows. In Sec. II, we present the calculations for the ξ-moments of the ρ-meson leading-twist DA, the Dρ TFFs, and the semileptonic decays Dρ+ν. In Sec. III, we present the numerical results and discussions on the ξ-moments, Dρ TFFs, and Dρ+ν decay widths and branching ratios. Section IV is contains a summary.

    To derive the sum rules of ρ-meson leading-twist longitudinal DA ξ-moments, we adopt the following correlation function (also known as the correlator):

    Π(z,q)=id4xeiqx0|T{Jn(x)J0(0)}|0=(zq)n+2I(q2),

    (1)

    where the current Jn(x)=ˉdz(izD)nu(x) with z2=0. After taking the standard QCDSR calculation procedure within the framework of BFT [50], we obtain an expression for ξn2;ρ×ξ02;ρ,

    ξn2;ρξ02;ρf2ρM2em2ρ/M2=34π21(n+1)(n+3)(1esρ/M2)+(md+mu)ˉqq(M2)2+αsG2(M2)21+nθ(n2)12π(n+1)(md+mu)gsˉqσTGq(M2)38n+118+gsˉqq2(M2)34(2n+1)81g3sfG3(M2)3nθ(n2)48π2+g2sˉqq2(M2)22+κ2486π2×{2(51n+25)(lnM2μ2)+3(17n+35)+θ(n2)×[2n(lnM2μ2)+49n2+100n+56n25(2n+1)×[ψ(n+12)ψ(n2)+ln4]]},

    (2)

    where mρ and fρ are the ρ-meson mass and decay constant, respectively, sρ is the continuum threshold, mu and md are the current quark masses of the u and d quarks, respectively, M is the Borel parameter, ˉqq with q=u(d) is the double-quark condensate, αsG2 is the double-gluon condensate, gsˉqσTGq is the quark-gluon mix condensate, g3sfG3 is the triple-gluon condensate, and gsˉqq2 and g2sˉqq2 are the four-quark condensates, respectively. In addition, κ=ˉss/ˉqq with the double s quark condensate ˉss. By taking n=0 for Eq. (2), the SR of the zeroth-order ξ-moment ξ02;ρ can be obtained as

    (ξ02;ρ)2f2ρM2em2ρ/M2=14π2(1esρ/M2)+(md+mu)ˉqq(M2)2+αsG2(M2)2112π118(md+mu)gsˉqσTGq(M2)3+481gsˉqq2(M2)3+g2sˉqq2(M2)22+κ2486π2×[50(lnM2μ2)+105].

    (3)

    Eq. (3) indicates that the zeroth-order ξ-moment ξ02;ρ in Eq. (2) cannot be normalized in the entire Borel parameter region. The main reason for this is because the contributions from vacuum condensates larger than dimension-six are truncated normally. As discussed in Ref. [50], a more accurate and reasonable SR for the nth-order ξ-moment ξn2;ρ should be

    ξn2;ρ=ξn2;ρ×ξ02;ρ|From Eq.(2)(ξ02;ρ)2|From Eq.(3).

    (4)

    However, to describe the behavior of the ρ-meson leading-twist longitudinal DA, we take the following DSE model for ϕ2;ρ(x,μ) [57, 58]:

    ϕ2;ρ(x,μ)=N[x(1x)]α[1+ˆa2Cα2(2x1)],

    (5)

    where α=α1/2, and N=4αΓ(α+1)/[πΓ(α+1/2)] is the normalization constant. In this notation, the Gegenbauer polynomial series is considered the most accurate form for describing the meson DA. Unfortunately, one cannot obtain all Gegenbauer moments in principle. Therefore, a truncated form of the Gegenbauer polynomial series (TF model) is typically used to approximately describe the behavior of the meson DA in the literature. However, the TF model is not an ideal model for describing the behavior of meson DAs, because it truncates at the lowest Gegenbauer coefficient. In fact, the DSE model is still a truncated form of the Gegenbauer polynomial series. The difference between these two models is that the DSE model is based on the Cαn-basis, whereas the TF model is based on the C3/2n-basis. In addition, the factor [x(1x)]α in the DSE model can adjust and compensate for the impact caused by truncation to a certain extent. Another advantage of the DSE model is that, as mentioned in Ref. [58], it can reduce the introduced spurious oscillations.

    Next, to obtain the Dρ TFFs, we can take the following correlation function:

    Πμ(p,q)=id4xeiqx×ρ(˜p,˜ϵ)|T{ˉq1(x)γμ(1γ5)c(x),jLD(0)}|0,

    (6)

    where jLD(x)=iˉq2(x)(1γ5)c(x) is the left-handed current. As we know, there are fifteen DAs for a vector meson up to twist-4 accuracy, and the left-handed chiral current can reduce the uncertainties from chiral-odd vector meson DAs with δ0,2-order and leave the chiral-even with a δ1,3-order meson. The relationships are listed in Table 1. In this table, except jLD(x), the current jRD(x) respects the right-handed current with the expression jRD(x)=iˉq2(x)×(1+γ5)c(x), as researched in our previous study [15]. The parameter δmρ/mc52 % [59, 60]. Meanwhile, the chiral current approach has been considered in other studies [6166], which improves the predictions of the LCSR approach.

    Table 1

    Table 1.  ρ-meson DAs with different twist-structures up to δ3, where δmρ/mc.
    twist-2twist-3twist-4
    δ0ϕ2;ρ
    jLD(x)δ2ϕ3;ρ,ψ3;ρ,Φ3;ρϕ4;ρ,ψ4;ρ,Ψ4;ρ,˜Ψ4;ρ
    δ1ϕ2;ρϕ3;ρ,ψ3;ρ,Φ3;ρ,˜Φ3;ρ
    jRD(x)δ3ϕ4;ρ,ψ4;ρ
    DownLoad: CSV
    Show Table

    Based on the procedures of the LCSR approach, we obtain the expressions of the Dρ TFFs A1,2(q2) and V(q2) LCSRs with the left-handed current in the correlator. The analytic formulas are similar to the Bρ TFFs of our previous study [45], which replace the input parameter of the B-meson with a D-meson, such as mBmD, fBfD, mbmc. The detailed expressions can be written as follows:

    A1(q2)=2m2cmρfρm2DfD(mD+mρ)em2D/M2{10duues(u)/M2×[Θ(c(u,s0))ϕ3;ρ(u)m2ρuM2˜Θ(c(u,s0))Cρ(u)]m2ρDα_dves(X)/M21X2M2Θ(c(X,s0))×[Φ3;ρ(α_)+˜Φ3;ρ(α_)]},

    (7)

    A2(q2)=m2cmρ(mD+mρ)fρm2DfDem2D/M2{210duu×es(u)/M2[1uM2˜Θ(c(u,s0))Aρ(u)+m2ρuM4ט˜Θ(c(u,s0))Cρ(u)+m2bm2ρ4u4M6˜˜˜Θ(c(u,s0))Bρ(u)]+m2ρDα_dves(X)/M21X3M4Θ(c(X,s0))×[Φ3;ρ(˜α_)+˜Φ3;ρ(˜α_)]},

    (8)

    V(q2)=m2cmρ(mD+mρ)fρ2m2DfDem2D/M210dues(u)/M2×1u2M2˜Θ(c(u,s0))ψ3;ρ(u),

    (9)

    where s(ϱ)=[m2bˉϱ(q2ϱm2ρ)]/ϱ with ˉϱ=1ϱ (ϱ represents u or X), and X=a1+va3. fρ represents the ρ-meson decay constant, and c(u,s0)=us0m2b+ˉuq2uˉum2ρ. Θ(c(ϱ,s0)) is the usual step function, and the definitions for ˜Θ(c(u,s0)), ˜˜Θ(c(u,s0)), and ˜˜˜Θ(c(u,s0)) can be found in our previous study [45]. The combined ρ-meson DA Aρ(x),Bρ(x), and Cρ(x) have the expressions

    Aρ(x)=x0dv[ϕ2;ρ(v)ϕ3;ρ(v)]Bρ(x)=x0dvϕ4;ρ(v)Cρ(x)=x0dvv0dw[ψ4;ρ(w)+ϕ2;ρ(w)2ϕ3;ρ(w)]

    (10)

    which originate from the definition of the chiral-even two-particle DAs [60]. From Eqs. (7), (8), and (9), we can see that only δ1,3-order DAs have contributions to TFFs with a left-chiral correlation function. Conversely, δ0,2-order DAs do not have contributions, which also indicates that transverse twist-2 DAs have no contribution to A1,2(q2) and V(q2). Furthermore, the twist-3 DAs ψ3;ρ(x,μ), ϕ3;ρ(x,μ) , and Aρ(x,μ) can be related to ϕ2;ρ directly using the Wandzura-Wilczek approximation [67, 68],

    ϕ3;ρ(x,μ)=12[x0dvϕ2;ρ(v,μ)ˉv+1xdvϕ2;ρ(v,μ)v],ψ3;ρ(x,μ)=2[ˉxx0dvϕ2;ρ(v,μ)ˉv+x1xdvϕ2;ρ(v,μ)v],Aρ(x,μ)=12[ˉxx0dvϕ2;ρ(v,μ)ˉv+x1xdvϕ2;ρ(v,μ)v],

    (11)

    Moreover, there have two ratio rV=V(0)/A1(0) and r2=A2(0)/A1(0) based on the TFFs A1,2(q2) and V(q2), which have less uncertainties between the different approaches.

    Next, we turn to the physical observables for the semileptonic decay Dρ+ν. Therefore, we introduce the longitudinal and transverse helicity decay widths, which are composed in terms of three helicity form factors H0,±(q2)

    Γ=G2FCk|Vcd|2192π3m3Dq2maxm2q2λ(q2)×{|H+(q2)|2+|H(q2)|2+|H0(q2)|2},

    (12)

    where the constant Ck=1 for D0ρ+ν and Ck=1/2 for D+ρ0+ν. Other parameters and expressions have the following definitions: GF=1.166×105GeV2 is the Fermi constant, λ(q2)=(m2D+m2ρq2)24m2Dm2ρ is the phase-space factor, and q2max=(mDmρ)2 is the small recoil point of the Dρ transition. The three helicity form factors H±(q2) and H0(q2) are mainly separated by the transition amplitude with a definite spin-parity quantum number, which can be found in our previous study [15]. Meanwhile, the longitudinal and transverse helicity amplitudes are expressed as ΓT=Γ++Γ, ΓL=Γ0 , and Γ=ΓL+ΓT. The helicity form factors H0,±(q2) can be calculated by relating them to the Lorentz orthogonal form factors A1,2(q2) and V(q2) via the following definition:

    H±(q2)=(mD+mρ)A1(q2)λ(q2)mD+mρV(q2),

    (13)

    and

    H0(q2)=12mρq2[(m2Dm2ρq2)(mD+mρ)A1(q2)λ(q2)mD+mρA2(q2)].

    (14)

    To perform the numerical calculation, we take fρ=210±4MeV [40, 69], mρ=775.26±0.23MeV, the current quark masses mu=2.16+0.490.26MeV and md=4.67+0.480.17MeV [70], the vacuum condensates ˉqq=(2.417+0.2270.114)×102GeV2, gsˉqσTGq=(1.934+0.1880.103)×102GeV5, gsˉqq2=(2.082+0.7340.697)×103GeV6, g2sˉqq2=(7.420+2.6142.483)×103GeV6, αsG2=0.038±0.11GeV4, and g3sfG0.045GeV6, and κ=0.74±0.03 [50, 71, 72]. Among these, the current quark masses and vacuum condensates except the gluon condensates are scale dependent and have above values at μ=2GeV. For the scale evolution of these inputs, refer to Ref. [50]. For the detailed calculation in this study, we take the scale μ=M as usual. By requiring that there be a reasonable Borel window to normalize ξ02;ρ with Eq. (3), we obtain the continuum threshold as sρ2.1GeV2.

    Now, we can obtain the ξ-moment versus the Borel parameter. Furthermore, we can determine the Borel window and then the value of the nth ξ-moment ξn2;ρ based on the well-known criteria that the contributions of the continuum state and dimension-six condensate are as small as possible and the value of ξn2;ρ is stable in the Borel window. Specifically, the continuum contribution is not more than 30% for all of the nth-order, and the dimension-six contribution of ξn2;ρ is less than 2% for n=(2,4) and 5% for n=(6,8,10). Note that only the even order ξ-moments are nonzero owing to isospin symmetry. The ξ-moments versus the Borel parameter and the obtained Borel windows are shown in Fig. 1, where the shaded areas are the Borel windows for n= (2, 4, 6, 8, 10).

    Figure 1

    Figure 1.  (color online) ρ-meson leading-twist longitudinal DA moments ξn2;ρ with n=(2,4,6,8,10) versus the Borel parameter M2 and the corresponding Borel windows, where all input parameters are set as their central values.

    Then, the values of ξn2;ρ|μ up to the 10th-order with four different scales μ= (1.0, 1.4, 2.0, 3.0) GeV and the second Gegenbauer moment a2;ρ can be obtained, which are shown in Table 2. Here, we only calculate the values of the first five non-zero ξ-moments because, as shown in Ref. [57], these moments are sufficient to determine the behavior of the DA ϕ2;ρ(x,μ). As a comparison, the other theoretical predictions obtained using QCD SRs [2126], LQCD [2730], AdS/QCD [31, 32], data fitting [35, 36], LFQM [39], and DSE [41] are also shown in Table 2. Our second moment is less than the QCD SR predictions in Ref. [21, 23, 26], the LQCD calculations [2730], and DSE result [41] and larger than the QCD SR predictions in Refs. [24, 25] but very consistent with the value from 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].

    Table 2

    Table 2.  Our predictions for the first five nonzero-order ξ-moments ξn2;ρ with n=(2,4,6,8,10) and the second-order Gegenbauer moment a2;ρ2 of the ρ-meson leading-twist longitudinal DA, compared with other theoretical predictions.
    μ/GeV ξ22;ρ ξ42;ρ ξ62;ρ ξ82;ρ ξ102;ρ a2;ρ2
    This study1.00.220(6)0.103(4)0.0656(50)0.0457(35)0.0346(28)0.059(18)
    This study1.40.217(5)0.100(3)0.0623(40)0.0428(29)0.0320(23)0.051(16)
    This study2.00.215(5)0.0986(31)0.0603(35)0.0411(25)0.0304(20)0.045(14)
    This study3.00.214(4)0.0972(28)0.0587(30)0.0397(22)0.0292(18)0.041(13)
    QCD SRs [21]1.00.18(10)
    QCD SRs [22]1.00.227(7)0.095(5)0.051(4)0.030(2)0.020(5)
    QCD SRs [23, 26]1.00.15(7)
    QCD SRs [23, 26]2.00.10(5)
    QCD SRs [24]1.00.216(21)0.089(9)0.048(5)0.030(3)0.022(2)0.047(58)
    QCD SRs [25]2.00.206(8)0.087(6)0.017(24)
    LQCD [27]2.00.237(36)(12)
    LQCD [28, 29]2.00.27(1)(2)
    LQCD [30]2.00.132(27)
    AdS/QCD [31, 32]1.00.228
    Data fitting [35, 36]1.00.2270.1050.0620.0410.029
    LFQM [39]1.00.21,0.190.09,0.080.05,0.040.02,0.02
    DSE [41]2.00.230.110.0660.0450.033
    DownLoad: CSV
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    By fitting our values of ξn2;ρ with n=(2,4,6,8,10) shown in Table 2 using the least squares method (where the model parameters α and ˆa2 are taken to be the fitting parameters, and for a specific fitting procedure, refer to Ref. [50]), the behavior of the ρ-meson leading-twist longitudinal DA can be obtained. The model parameters α and ˆa2 of our DA and the corresponding χ2min/nd and goodness of fit Pχ2min at several typical scales, such as μ=(1.0,1.4,2.0,3.0)GeV , are exhibited in Table 3. The curve of the ρ-meson leading-twist longitudinal DA is shown in Fig. 2. Other predictions in Refs. [25, 41, 46] are also shown for comparison. Our curve is closer to the asymptotic form. In addition, we use the TF model ϕ(TF)2;ρ(x,μ)=6x(1x)[1+a2;ρ2C3/22(2x1)+a2;ρ4C3/24(2x1)] to fit our values of the ξ-moments of the ρ-meson leading-twist longitudinal DA shown in Table 2. The results indicate that the goodness of fits for the DSE model exhibited in Table 3 are close to those of the TF model at μ=1GeV but better than those of the TF model at μ=(1.4,2.0,3.0)GeV. For example, a2;ρ2=0.040, a2;ρ4=0.054, χ2min/nd=0.657635/3, and Pχ2min=0.88312 at μ=2GeV. The curve of the TF model μ=2GeV is also shown in Fig. 2. We find that the curve of the DSE model is smoother than that of the TF model. Thus, we use the DSE model instead of the TF model in this study.

    Table 3

    Table 3.  Model parameters α and ˆa2 of our DA obtained by fitting using the least squares method and the corresponding χ2min/nd and goodness of fit Pχ2min at several typical scales, such as μ=(1.0, 1.4, 2.0, 3.0) GeV.
    μ α ˆa2 χ2min/nd Pχ2min
    1.0 0.625 0.516 0.957/3 0.812
    1.4 0.678 0.448 0.488/3 0.922
    2.0 0.721 0.398 0.259/3 0.968
    3.0 0.709 0.427 0.141/3 0.987
    DownLoad: CSV
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    Figure 2

    Figure 2.  (color online) Curve of our ρ-meson leading-twist longitudinal DA. As a comparison, the DA curves obtained using the DSEs [41], QCD SRs [25], algebraic model [46], asymptotic form, and fitting with the TF model are also shown.

    Furthermore, to calculate the Dρ TFFs, the input parameters should be clarified. The current charm-quark mass is mc=1.27±0.02GeV, and the D-meson mass is mD=1.869GeV [70]. Based on the three criteria for determining s0 and M2 in the LCSR approach, which can be found in our previous study [51], we have sA10=7.0±0.1GeV2, sA20=6.0±0.1GeV2, sV0=7.0±0.1GeV2, M2A1=1.45±0.05GeV2, M2A2=1.20±0.05GeV2, and M2V=2.15±0.05GeV2. Then, the TFFs at the large recoil region, i.e., A1,2(0) and V(0) , are present in Table 4. The uncertainties originate from the squared average for all the input parameters. In Table 4, 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], and HMχT [18], and lattice QCD predictions [19, 20] are presented. A comparison of all the results in Table 4 indicates that the TFFs of our prediction are consistent with those of many approaches within errors.

    Table 4

    Table 4.  Dρ TFFs A1,2(q2) and V(q2) at the large recoil region q20. The errors are squared averages of all the mentioned error sources. As a comparison, we also present predictions from various other methods.
    A1(0)A2(0)V(0)
    This study0.498+0.0140.0120.460+0.0550.0470.800+0.0150.014
    CLEO2013 [3]0.56(1)+0.020.030.47(6)(4)0.84(9)+0.050.06
    3PSR [6]0.5(2)0.4(1)1.0(2)
    HQETF [7]0.57(8)0.52(7)0.72(10)
    LCSR [8]0.599+0.0350.0300.372+0.0260.0310.801+0.0440.036
    RHOPM [9]0.780.921.23
    QM-I [10]0.590.231.34
    QM-II [11]0.590.490.90
    LFQM [12]0.60(1)0.47(0)0.88(3)
    HMχT [18]0.610.311.05
    LQCD [19]0.45(4)0.02(26)0.78(12)
    LQCD [20]0.65(15)+0.240.230.59(31)+0.280.251.07(49)(35)
    DownLoad: CSV
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    The ratios between different Dρ TFFs, i.e., r2 and rV, are shown in Fig. 3. Here, the results from the CLEO collaboration [3], HMχT [18], LFQM [12], HQEFT [7], 3PSR [6], QM [10, 11], and CCQM [17] are presented for comparison. From the figure, we can reach the following conclusions:

    Figure 3

    Figure 3.  (color online) Predictions for the ratios r2 and rV within uncertainties. The results from the CLEO collaboration [3], HMχT [18], LFQM [12], HQEFT [7], 3PSR [6], QM [10, 11], and CCQM [17] are presented for comparison.

    ● Our predictions within uncertainties are in the region of the CLEO collaboration results.

    ● In terms of the central value with respect to uncertainties, our results for r2 agree with the QM, LFQM, 3PSR, and CCQM for theoretical predictions. rV agrees with the QM and 3PSR predictions.

    ● The uncertainties of our predictions are (12.9% ~ 14.4%) for r2 and 4.4% for rV, which indicates that this method can obtain applicable results in the confidence level. Because the predictions from the left-handed chiral current LCSR approach are mainly obtained at leading order for a strong coupling, we will consider the possible uncertainty/pollution from the axial-vector current in subsequent studies.

    Using the resultant TFFs in the large recoil region, their behavior in the entire physical region, i.e., q2phys.[0,1.18]GeV2 should be obtained. It is known that the LCSR predictions are suitable in low and intermediate momentum transfer. Therefore, a rapidly converging series based on z(t)-expansion is used to extrapolate in this study [73, 74],

    Fi(q2)=11q2/m2R,ik=0,1,2aik[z(q2)z(0)]k.

    (15)

    The formula for z(t) can be found in Ref. [73]. Fi respects three TFFs, A1,2 and V. The appropriate resonance masses are mR,i=2.007GeV for the TFF V(q2) with JP=1 and mR,i=2.427GeV for the TFFs A1,2(q2) with JP=1+. The parameters aik can be fixed by requiring Δ<0.1%, where the parameter Δ is introduced to measure the quality of extrapolation, Δ=t|Fi(t)Ffiti(t)|/t|Fi(t)|×100, where t[0,1/40,,40/40]×0.8GeV2. The Dρ TFFs obtained in this study using LCSRs and the simple z-series expansion parameterization are shown in Fig. 4. The maximal momentum transfers of the LCSR calculation of the Dρ TFFs are A1(qmax)=0.883+0.0220.019, A2(qmax)=0.611+0.1020.103 , and V(qmax)=2.082+0.0600.059.

    Figure 4

    Figure 4.  (color online) Predictions for the Dρ TFFs, where the darker bands are the results of this study using the LCSR approach, and the lighter bands are the predictions from simple z-series expansion parameterizations.

    After obtaining the extrapolated TFFs of the transition Dρ, we find the CKM-independent total decay width 1/|Vcd|2×Γ. Combining the ratio of the longitudinal/transverse and positive/negative helicity decay widths, we present the predictions in Table 5. The 3PSR [6], HQEFT [7], RHOPM [9], QM [10], LQCD [19], and LQCD [20] results are given for comparison. As shown in Table 5, there are large differences between the total decay widths from the different theoretical groups, and our results agree with the lattice results within uncertainties. Our predicted longitudinal/transverse helicity decay width agrees with those of HQEFT and LQCD within uncertainties, which is larger than 1. Our positive/negative helicity decay width agrees with that of the QM and LQCD within uncertainties.

    Table 5

    Table 5.  |Vcd|-independent total decay width 1/|Vcd|2×Γ, ratio for longitudinal/transverse and positive/negative helicity decay widths. For comparsion, other theoretical group predictions are also given.
    1/|Vcd|2×ΓΓL/ΓTΓ+/Γ
    This study57.87+5.405.131.00+0.100.110.15+0.020.01
    3PSR [6]15.80±4.611.31±0.110.24±0.03
    HQEFT [7]71±141.17±0.090.29±0.13
    RHOPM [9]90.830.910.19
    QM [10]88.861.330.11
    LQCD [19]54.63±12.511.86±0.560.16
    LQCD [20]71.751.100.18
    DownLoad: CSV
    Show Table

    Lastly, the branching fractions for the two types of Dρ+ν semileptonic decays are calculated. The first is the D0-type, including the D0ρe+νe and D0ρμ+νμ decays, for which the lifetime τ(D0)=0.410±0.001 ps should be used. The second is the D+-type, including D+ρ0e+νe and D+ρ0μ+νμ, associated with τ(D+)=1.033±0.005 ps [70]. The CKM matrix element |Vcd|=0.225±0.001 [69]. We present the predictions in Table 6, where the two uncertainties mainly arise from the squared average of the theoretical input parameters and the experimental uncertainties from the D-meson lifetime. For comparison, the theoretical results from the 3PSR [6], HQEFT [7], narrow width approximation (NWA) [75] with the HQEFT [8] and LFQM [12] approaches, LCSR [8, 14], LFQM [13], CCQM [17], chiral unitarity approach (χUA) [76], RQM [77], HMχT [18], ISGW2 [78] and experimental results from the BESIII collaboration [1, 2] and CLEO collaboration prediction in 2013 [3] and 2005 [4] are also given. The results show that:

    Table 6

    Table 6.  Branching ratios of the semileptonic decays D0ρ+ν and D+ρ0+ν (in units of 103). For comparison, we also present the results from experimental collaborations and theoretical groups.
    Decay Mode D0ρe+νe D+ρ0e+νe D0ρμ+νμ D+ρ0μ+νμ
    This study1.825+0.1700.162±0.0042.299+0.2140.204±0.0111.816+0.1680.160±0.0042.288+0.2120.201±0.011
    BESIII [1]1.35±0.09±0.09
    BESIII [2]1.445±0.058±0.0391.860±0.070±0.061
    CLEO2013 [3]1.77±0.12±0.102.17±0.12+0.120.22
    CLEO2005 [4]1.94±0.39±0.132.1±0.4±0.1
    3PSR [6]0.5±0.1
    HQEFT [7]1.4±0.3
    LCSR [8]1.81+0.180.132.29+0.230.161.73+0.170.132.20+0.210.16
    NWA [75]+HQEFT [8]1.67±0.272.16±0.36
    NWA [75]+LFQM [12]1.73±0.072.24±0.09
    LFQM [13]1.7±0.2
    LCSR [14]1.74±0.252.25±0.321.65±0.232.14±0.30
    CCQM [17]1.622.091.552.01
    HMχT [18]2.02.5
    χUA [76]1.972.541.842.37
    RQM [77]1.962.491.882.39
    ISGW2 [78]1.01.3
    DownLoad: CSV
    Show Table

    ● The results obtained using our left-handed chiral LCSR approach are consistent with those obtained using other LCSRs within uncertainties.

    ● Our current results are consistent with those of the CLEO collaboration but are larger than the BESIII collaboration predictions.

    ● Our results are consistent with those of other theoretical groups, such as the NWA, LFQM, HQEFT, χUA, RQM, and HMχT, but have a smaller gap when compared with the 3PSR, HQEFT, χUA, and ISGW2 results.

    In the framework of BFT with the QCD SR approach, we calculate the ξ-moments of the ρ-meson leading twist longitudinal DA. Because the zeroth ξ-moment cannot be normalized in the entire Borel region, a new SR formula for the ξ-moment is given in Eq. (2). The results up to the tenth order with the scale μ=(1.0,1.4,2.0.3.0)GeV2 are presented in Table 2, where the results from other theoretical group are also given. Next, we present the DρTFFs in the large recoil region, i.e., A1(0), A2(0), and V(0), in Table 4 and the ratio r2,V in Fig. 3, which indicate that our results are reasonable and consistent with those of many approaches within errors. Subsequently, we calculate the |Vcd|-independent total decay width and the ratio for the longitudinal/transverse and positive/negative helicity decay width results and present them in Table 5. A detail discussion comparing our results with other predictions is presented. Finally, we show the branching fraction of the two types of Dρ+ν semileptonic decays in Table 6. In the near further, we hope that more accurate data will be reported and more theoretical results will be given to explain the gaps between different approaches.

    We are grateful for the referee's valuable comments and suggestions.

    1Chiral current LCSR approach will be introduced in the next section of this paper

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Tao Zhong, Ya-Hong Dai and Hai-Bing Fu. ρ-meson longitudinal leading-twist distribution amplitude revisited and the Dρ semileptonic decay[J]. Chinese Physics C. doi: 10.1088/1674-1137/ad34be
Tao Zhong, Ya-Hong Dai and Hai-Bing Fu. ρ-meson longitudinal leading-twist distribution amplitude revisited and the Dρ semileptonic decay[J]. Chinese Physics C.  doi: 10.1088/1674-1137/ad34be shu
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ρ-meson longitudinal leading-twist distribution amplitude revisited and the Dρ semileptonic decay

Abstract: Motivated by our previous study [Phys. Rev. D 104(1), 016021 (2021)] on the pionic leading-twist distribution amplitude (DA), we revisit the ρ-meson leading-twist longitudinal DA \phi_{2;\rho}^\|(x,\mu) in this study. A model proposed by Chang based on the Dyson-Schwinger equations is adopted to describe the behavior of \phi_{2;\rho}^\|(x,\mu) . However, the ξ-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 formulas for these moments are improved. More accurate values for the first five nonzero ξ-moments at the typical scale \mu = (1.0, 1.4, 2.0, 3.0)\; {\rm GeV} are given, e.g., at \mu = 1\; {\rm GeV} , \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 these 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 obtain the transition form factor in the large recoil region, i.e., A_1(0) = 0.498^{+0.014}_{-0.012} , A_2(0)=0.460^{+0.055}_{-0.047} , and 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 extrapolating with a rapidly converging series based on z(t) -expansion, we present the |V_{cd}| -independent 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) = 1.825^{+0.170}_{-0.162}\pm 0.004 , \mathcal{B}(D^+ \to \rho^0 e^+ \nu_e) = 2.299^{+0.214}_{-0.204}\pm 0.011, \mathcal{B}(D^0\to \rho^- \mu^+ \nu_\mu) = 1.816^{+0.168}_{-0.160}\pm 0.004 , and \mathcal{B}(D^+ \to \rho^0 \mu^+ \nu_\mu) =2.288^{+0.212}_{-0.201} \pm 0.011.

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    I.   INTRODUCTION
    • The ρ-meson leading-twist longitudinal distribution amplitude (DA) is a key input parameter when investigating related exclusive processes including the ρ meson. Charmed semileptonic decay processes can provide a clear platform for researching the ρ-meson DA. In charmed factories, the BESIII collaboration reported new results for the semileptonic decay D^0\to \rho^-\mu^+\nu_\mu in 2021 [1]. In 2019, the BESIII collaboration provided improved measurements of D\to\rho e^+\nu_e [2]. The CLEO collaboration provided measurements of the form factors in the decays D^{0/+}\to \rho^{-/0}e^+\nu_e in 2013 [3], and their previous measurements can be found in [4, 5]. It is known that transition form factors (TFFs) are key components of semileptonic D \to \rho \ell^+ \nu_\ell decays in the standard model. Therefore, accurate TFFs are very important for theoretical groups and experimental collaborations. Theoretically, the D\to\rho TFFs can be treated using different approaches, such as the 3-point sum rule (3PSR) [6], heavy quark effective field theory (HQEFT) [7, 8], relativistic harmonic oscillator potential model (RHOPM) [9], quark model (QM) [10, 11], light-front quark model (LFQM) [12, 13], light-cone sum rule (LCSR) [1416], covariant confined quark model (CCQM) [17], heavy meson and chiral symmetry theory (HMχT) [18], and lattice QCD (LQCD) [19, 20]. The LCSR approach is based on operator production expansion near the light-cone, and all the non-perturbative dynamics are parameterized into light-cone DAs (LCDAs), which is suitable for calculating the heavy to light transition. In this study, we take the LCSR approach to calculate the D\to\rho TFFs with a left-handed current 1. Thus, the key task is obtaining an accurate ρ-meson longitudinal twist-2 DA.

      Theoretically, the ρ-meson longitudinal leading-twist DA can be studied using various methods, such as QCD sum rules (SRs) [2126], LQCD [2730], the AdS/QCD holographic method [3134], extraction from experimental data [3538], the light-front quark model (LFQM) [39, 40], Dyson-Schwinger equations (DSEs) [41], large momentum effective theory (LMET) [42], instanton vacuum [43, 44], and other models [45, 46]. The QCDSR in the framework of background field theory (BFT) is an effective approach in calculating light and heavy meson DAs [47]. In early 2016, we preliminarily studied the ρ-meson longitudinal leading-twist DA [48], in which the first two order nonzero ξ-moments and Gegenbauer moments were obtained, and the behavior of \phi_{2;\rho}^\|(x,\mu) described by the light-cone harmonic oscillator (LCHO) model was determined [49].

      In 2021, we proposed a new research scheme for the QCDSR study on pionic leading-twist DAs, in which a new SR formula for ξ-moments was proposed after considering that the SR for the zeroth ξ-moment cannot be normalized in the entire Borel region [50]. It enabled us to calculate more higher-order ξ-moments. Furthermore, the behavior of the pion DA \phi_{2;\pi}(x,\mu) could be determined by fitting sufficient ξ-moments with the least squares method. Subsequently, this scheme was used to study pseudoscalar \eta^{(\prime)} -meson and kaon leading-twist DAs [51, 52] and D-meson twist-2,3 DAs [53], axial vector a_1(1260) -meson twist-2 longitudinal DAs [54], and scalar a_0(980) and K_0^\ast(1430) -meson leading-twist DAs [55, 56]. Inspired by this, we restudy the ρ-meson leading-twist longitudinal DA \phi_{2;\rho}^\|(x,\mu) by adopting the research scheme proposed in Ref. [50].

      The remainder of this paper are organized as follows. In Sec. II, we present the calculations for the ξ-moments of the ρ-meson leading-twist DA, the D\to \rho TFFs, and the semileptonic decays D\to\rho\ell^+\nu_\ell . In Sec. III, we present the numerical results and discussions on the ξ-moments, D\to \rho TFFs, and D\to \rho\ell^+\nu_\ell decay widths and branching ratios. Section IV is contains a summary.

    II.   THEORETICAL FRAMEWORK
    • To derive the sum rules of ρ-meson leading-twist longitudinal DA ξ-moments, we adopt the following correlation function (also known as the correlator):

      \begin{aligned}[b] \Pi(z,q) &= {\rm i}\int {\rm d}^4x {\rm e}^{{\rm i}q\cdot x} \langle 0 | T\{J_n(x) J_0^\dagger(0) \} |0\rangle \\ &= (z\cdot q)^{n+2} I(q^2), \end{aligned}

      (1)

      where the current J_n(x) = \bar{d} {\not {z}} ({\rm i}z\cdot {\mathop D\limits^ \leftrightarrow})^n u(x) with z^2 = 0 . After taking the standard QCDSR calculation procedure within the framework of BFT [50], we obtain an expression for \langle\xi^n\rangle_{2;\rho}^\| \times \langle\xi^0\rangle_{2;\rho}^\| ,

      \begin{aligned}[b] & \frac{\langle\xi^n\rangle_{2;\rho}^\| \langle\xi^0\rangle_{2;\rho}^\| f_\rho^2}{M^2 {\rm e}^{m_\rho^2/M^2}} \\ =\;& \frac{3}{4\pi^2} \frac{1}{(n+1)(n+3)} (1 - {\rm e}^{-s_\rho/M^2}) + \frac{(m_d + m_u) \langle\bar{q}q\rangle}{(M^2)^2} \\ &+ \frac{\langle\alpha_s G^2\rangle}{(M^2)^2} \frac{1 + n\theta(n-2)}{12\pi(n+1)} - \frac{(m_d + m_u) \langle g_s\bar{q}\sigma TGq\rangle}{(M^2)^3} \frac{8n+1}{18} \\ &+ \frac{\langle g_s\bar{q}q \rangle^2}{(M^2)^3} \frac{4(2n+1)}{81} - \frac{\langle g_s^3fG^3\rangle}{(M^2)^3} \frac{n\theta(n-2)}{48\pi^2} + \frac{\langle g_s^2\bar{q}q\rangle^2}{(M^2)^2} \frac{2+\kappa^2}{486\pi^2} \\ &\times \Big\{ -2(51n+25) \Big( -\ln \frac{M^2}{\mu^2} \Big) + 3(17n+35) + \theta(n-2) \\ &\times \Big[ 2n\Big( -\ln \frac{M^2}{\mu^2} \Big) + \frac{49n^2 + 100n + 56}{n} - 25(2n+1) \\ &\times \Big[ \psi \Big( \frac{n+1}{2} \Big) - \psi \Big( \frac{n}{2} \Big) + \ln 4 \Big] \Big] \Big\}, \end{aligned}

      (2)

      where m_\rho and f_\rho are the ρ-meson mass and decay constant, respectively, s_\rho is the continuum threshold, m_u and m_d are the current quark masses of the u and d quarks, respectively, M is the Borel parameter, \langle\bar{q}q\rangle with q = u (d) is the double-quark condensate, \langle\alpha_sG^2\rangle is the double-gluon condensate, \langle g_s\bar{q}\sigma TGq\rangle is the quark-gluon mix condensate, \langle g_s^3fG^3\rangle is the triple-gluon condensate, and \langle g_s\bar{q}q \rangle^2 and \langle g_s^2\bar{q}q \rangle^2 are the four-quark condensates, respectively. In addition, \kappa = \langle\bar{s}s\rangle / \langle\bar{q}q\rangle with the double s quark condensate \langle\bar{s}s\rangle . By taking n = 0 for Eq. (2), the SR of the zeroth-order ξ-moment \langle\xi^0\rangle_{2;\rho}^\| can be obtained as

      \begin{aligned}[b] \frac{(\langle\xi^0\rangle_{2;\rho}^\|)^2 f_\rho^2}{M^2 {\rm e}^{m_\rho^2/M^2}} =\;& \frac{1}{4\pi^2} (1 - {\rm e}^{-s_\rho/M^2}) + (m_d + m_u) \frac{\langle\bar{q}q\rangle}{(M^2)^2} \\ &+ \frac{\langle\alpha_s G^2\rangle}{(M^2)^2} \frac{1}{12\pi} - \frac{1}{18} (m_d + m_u) \frac{\langle g_s\bar{q}\sigma TGq\rangle}{(M^2)^3} \\ &+ \frac{4}{81} \frac{\langle g_s\bar{q}q \rangle^2}{(M^2)^3} + \frac{\langle g_s^2\bar{q}q\rangle^2}{(M^2)^2} \frac{2+\kappa^2}{486\pi^2} \\ &\times \Big[ -50 \Big( -\ln \frac{M^2}{\mu^2} \Big) + 105 \Big]. \end{aligned}

      (3)

      Eq. (3) indicates that the zeroth-order ξ-moment \langle\xi^0\rangle_{2;\rho}^{\|} in Eq. (2) cannot be normalized in the entire Borel parameter region. The main reason for this is because the contributions from vacuum condensates larger than dimension-six are truncated normally. As discussed in Ref. [50], a more accurate and reasonable SR for the nth-order ξ-moment \langle\xi^n\rangle_{2;\rho}^\| should be

      \langle\xi^n\rangle_{2;\rho}^\| = \frac{\langle\xi^n\rangle_{2;\rho}^\| \times \langle\xi^0\rangle_{2;\rho}^\| |_{\rm From\ Eq.\; (2)}}{\sqrt{(\langle\xi^0\rangle_{2;\rho}^\|)^2} |_{\rm From\ Eq.\; (3)}} .

      (4)

      However, to describe the behavior of the ρ-meson leading-twist longitudinal DA, we take the following DSE model for \phi_{2;\rho}^\parallel(x,\mu) [57, 58]:

      \begin{array}{*{20}{l}} \phi_{2;\rho}^\parallel(x,\mu) = \mathcal{N} [x(1-x)]^{\alpha_-} \Big[ 1 + \hat{a}_2 C_2^\alpha(2x-1) \Big], \end{array}

      (5)

      where \alpha_- = \alpha - 1/2 , and \mathcal{N} = 4^\alpha \Gamma(\alpha + 1) / [\sqrt{\pi} \Gamma(\alpha + 1/2)] is the normalization constant. In this notation, the Gegenbauer polynomial series is considered the most accurate form for describing the meson DA. Unfortunately, one cannot obtain all Gegenbauer moments in principle. Therefore, a truncated form of the Gegenbauer polynomial series (TF model) is typically used to approximately describe the behavior of the meson DA in the literature. However, the TF model is not an ideal model for describing the behavior of meson DAs, because it truncates at the lowest Gegenbauer coefficient. In fact, the DSE model is still a truncated form of the Gegenbauer polynomial series. The difference between these two models is that the DSE model is based on the {C_n^\alpha} -basis, whereas the TF model is based on the {C_n^{3/2}} -basis. In addition, the factor [x(1-x)]^{\alpha_-} in the DSE model can adjust and compensate for the impact caused by truncation to a certain extent. Another advantage of the DSE model is that, as mentioned in Ref. [58], it can reduce the introduced spurious oscillations.

      Next, to obtain the D\to\rho TFFs, we can take the following correlation function:

      \begin{aligned}[b] \Pi_\mu(p,q) =\;& {\rm i}\int {\rm d}^4x {\rm e}^{{\rm i}q\cdot x} \\ & \times \langle\rho (\tilde p,\tilde\epsilon)|{\rm T} \big\{\bar q_1(x)\gamma_\mu(1-\gamma_5)c(x), j_D^{L\dagger} (0)\big\} |0\rangle, \end{aligned}

      (6)

      where j_D^{L} (x)= {\rm i} \bar q_2(x)(1 - \gamma_5)c(x) is the left-handed current. As we know, there are fifteen DAs for a vector meson up to twist-4 accuracy, and the left-handed chiral current can reduce the uncertainties from chiral-odd vector meson DAs with \delta^{0,2} -order and leave the chiral-even with a \delta^{1,3} -order meson. The relationships are listed in Table 1. In this table, except j_D^{L} (x) , the current j_D^{R} (x) respects the right-handed current with the expression j_D^{R} (x)= {\rm i} \bar q_2(x) \times (1 + \gamma_5)c(x), as researched in our previous study [15]. The parameter \delta \simeq m_\rho/m_c\sim 52\ % [59, 60]. Meanwhile, the chiral current approach has been considered in other studies [6166], which improves the predictions of the LCSR approach.

      twist-2twist-3twist-4
      \delta^0 \phi_{2;\rho}^\bot
      j_D^{L} (x) \delta^2 \phi_{3;\rho}^\|, \psi_{3;\rho}^\|, \Phi_{3;\rho}^\bot \phi_{4;\rho}^\bot,\psi_{4;\rho}^\bot,\Psi_{4;\rho}^\bot, \widetilde{\Psi} _{4;\rho}^\bot
      \delta^1 \phi_{2;\rho}^\| \phi_{3;\rho}^\bot, \psi_{3;\rho}^\bot, \Phi_{3;\rho}^\|,\tilde\Phi_{3;\rho }^\bot
      j_D^{R} (x) \delta^3 \phi_{4;\rho}^\|,\psi_{4;\rho}^\|

      Table 1.  ρ-meson DAs with different twist-structures up to \delta^3 , where \delta \simeq m_\rho/m_c .

      Based on the procedures of the LCSR approach, we obtain the expressions of the D\to \rho TFFs A_{1,2}(q^2) and V(q^2) LCSRs with the left-handed current in the correlator. The analytic formulas are similar to the B\to\rho TFFs of our previous study [45], which replace the input parameter of the B-meson with a D-meson, such as m_B\to m_D , f_B \to f_D , m_b \to m_c . The detailed expressions can be written as follows:

      \begin{aligned}[b] A_1(q^2) =\;&\frac{2m_c^2 m_\rho f_\rho ^\|}{m_D^2 f_D (m_D + m_\rho)} {\rm e}^{m_D^2/M^2} \bigg\{ \int_0^1\frac{{\rm d}u}{u}{\rm e}^{-s(u)/M^2} \\ &\times \left[ \Theta(c(u,s_0))\phi_{3;\rho}^\bot(u) - \frac{m_\rho^2}{u M^2}\widetilde\Theta(c(u,s_0)) C_\rho^\|(u) \right] \\ & - m_\rho^2\int {\mathcal D} \underline\alpha\int {\rm d}v\, {\rm e}^{-s(X) /M^2} \frac{1}{X^2 M^2} \,\Theta(c(X,s_0)) \\ &\times \left[\Phi_{3;\rho}^\|(\underline \alpha ) + {\widetilde \Phi}_{3;\rho}^\|(\underline \alpha )\right] \bigg\},\\[-15pt] \end{aligned}

      (7)

      \begin{aligned}[b] A_2(q^2) =\;& \frac{m_c^2 \, m_\rho \, (m_D \,+\, m_\rho )\,f_\rho^\| }{m_D^2 f_D }\; {\rm e}^{m_D^2/M^2}\; \bigg\{\,2\, \int_0^1\, \frac{{\rm d} u}{u} \\ &\times {\rm e}^{- s(u)/M^2}\; \,\bigg[\,\frac{1}{uM^2}\; \widetilde\Theta(c(u,s_0))\, A_\rho^\|(u) \,+\; \frac{m_\rho^2}{u M^4} \\ &\times\widetilde{\widetilde\Theta}(c(u,s_0)) C_\rho^\|(u)+ \frac{m_b^2 m_\rho^2}{4u^4M^6} \widetilde{\widetilde{\widetilde\Theta}}(c(u,s_0)) B_\rho^\|(u) \bigg] \\ & + m_\rho^2\int {\cal D} \underline\alpha \int{{\rm d}v} {\rm e}^{ - s(X)/M^2} \, \frac{1}{X^3 M^4}\,\Theta(c(X,s_0)) \\ & \times [\Phi_{3;\rho}(\widetilde {\underline \alpha }) + \widetilde \Phi_{3;\rho}^\|(\widetilde {\underline \alpha })] \bigg\}, \\[-15pt]\end{aligned}

      (8)

      \begin{aligned}[b] V(q^2) =\;& \frac{m_c^2 m_\rho (m_D+m_\rho) f_\rho^\| }{2 m_D^2 f_D } {\rm e}^{m_D^2/M^2} \int_0^1 {\rm d} u {\rm e}^{- s(u)/M^2} \\ & \times \frac{1}{u^2 M^2}\widetilde\Theta(c(u,s_0))\psi_{3;\rho}^ \bot(u), \end{aligned}

      (9)

      where s(\varrho)=[m_b^2-\bar \varrho(q^2-\varrho m_\rho^2)]/\varrho with \bar \varrho = 1 - \varrho ( \varrho represents u or X), and X=a_1 + v a_3 . f_\rho^\| represents the ρ-meson decay constant, and c(u,s_0)=u s_0 - m_b^2 + \bar u q^2 - u \bar u m_\rho^2 . \Theta(c(\varrho,s_0)) is the usual step function, and the definitions for \widetilde\Theta (c(u,s_0)) , \widetilde{\widetilde\Theta}(c(u,s_0)) , and \widetilde{\widetilde{\widetilde\Theta}}(c(u,s_0)) can be found in our previous study [45]. The combined ρ-meson DA A_\rho^\| (x), B_\rho^\| (x) , and C_\rho^\| (x) have the expressions

      \begin{aligned}[b] &A_\rho^\| (x) = \int_0^x {\rm d}v [\phi _{2;\rho }^\| (v) - \phi _{3;\rho }^ \bot (v)]\\ &B_\rho^\| (x) = \int_0^x {\rm d}v \phi_{4;\rho}^\|(v) \\ &C_\rho^\| (x) = \int_0^x {\rm d} v \int_0^v {\rm d} w [\psi_{4;\rho}^\|(w) + \phi _{2;\rho }^\|(w) - 2\phi_{3;\rho}^\bot(w)] \end{aligned}

      (10)

      which originate from the definition of the chiral-even two-particle DAs [60]. From Eqs. (7), (8), and (9), we can see that only \delta^{1,3} -order DAs have contributions to TFFs with a left-chiral correlation function. Conversely, \delta^{0,2} -order DAs do not have contributions, which also indicates that transverse twist-2 DAs have no contribution to A_{1,2}(q^2) and V(q^2) . Furthermore, the twist-3 DAs \psi_{3;\rho}^\bot(x,\mu) , \phi_{3;\rho}^\bot(x,\mu) , and A_\rho^\| (x,\mu) can be related to \phi_{2;\rho}^\| directly using the Wandzura-Wilczek approximation [67, 68],

      \begin{aligned}[b] &\phi_{3;\rho}^\bot(x,\mu ) = \frac12 \left[\int_0^x {{\rm d}v} \frac{\phi _{2;\rho }^\| (v,\mu )}{\bar v} + \int_x^1 {\rm d} v \frac{\phi _{2;\rho }^\| (v,\mu)}{v} \right], \\ &\psi_{3;\rho}^\bot(x,\mu ) = 2\left[ \bar x \int_0^x {{\rm d}v} \frac{\phi _{2;\rho }^\| (v,\mu )}{\bar v} + x \int_x^1 {\rm d}v \frac{\phi _{2;\rho }^\| (v,\mu)}{v}\right], \\ & A_\rho^\|(x,\mu) = \frac12 \left[ \bar x \int_0^x {{\rm d}v} \frac{\phi _{2;\rho }^\| (v,\mu )}{\bar v} + x \int_x^1 {\rm d}v \frac{\phi _{2;\rho }^\| (v,\mu)}{v} \right], \end{aligned}

      (11)

      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.

      Next, we turn to the physical observables for the semileptonic decay D\to\rho \ell^+\nu_\ell . Therefore, we introduce the longitudinal and transverse helicity decay widths, which are composed in terms of three helicity form factors H_{0,\pm} (q^2)

      \begin{aligned}[b] \Gamma =\;&\frac{G_F^2 C_k|V_{cd}|^2 }{192\pi^3 m_D^3}\int_{m_\ell^2}^{q^2_{\rm max}}q^2\sqrt{\lambda(q^2)} \\ &\times \Big\{ |H_+(q^2)|^2+|H_-(q^2)|^2+|H_0(q^2)|^2\Big\}, \end{aligned}

      (12)

      where the constant C_k = 1 for D^0\to\rho^-\ell^+\nu_\ell and C_k = 1/\sqrt{2} for D^+\to\rho^0\ell^+\nu_\ell . Other parameters and expressions have the following definitions: G_F=1.166\times10^{-5}\; {\rm GeV}^{-2} is the Fermi constant, \lambda(q^2) = (m_D^2 + m_\rho^2 - q^2)^2-4 m_D^2 m_\rho^2 is the phase-space factor, and q^2_{\rm max} = (m_D-m_\rho)^2 is the small recoil point of the D\to\rho transition. The three helicity form factors H_{\pm}(q^2) and H_0(q^2) are mainly separated by the transition amplitude with a definite spin-parity quantum number, which can be found in our previous study [15]. Meanwhile, the longitudinal and transverse helicity amplitudes are expressed as \Gamma^{\rm T} = \Gamma^+ + \Gamma^- , \Gamma^{\rm L}=\Gamma^0 , and \Gamma=\Gamma^{\rm L} + \Gamma^{\rm T} . The helicity form factors H_{0,\pm}(q^2) can be calculated by relating them to the Lorentz orthogonal form factors A_{1,2}(q^2) and V(q^2) via the following definition:

      H_{\pm}(q^2) =(m_D+m_\rho) A_1(q^2) \mp \frac{\sqrt{\lambda(q^2)}}{m_D+m_{\rho}}V(q^2),

      (13)

      and

      \begin{aligned}[b] H_0(q^2) =\;&\frac{1}{2m_{\rho}\sqrt{q^2}}\Big[(m_D^2-m_\rho^2-q^2) ( m_D+m_\rho) A_1(q^2) \\ &-\frac{\lambda(q^2)}{m_D+m_{\rho}}A_2(q^2) \Big]. \end{aligned}

      (14)
    III.   NUMERICAL ANALYSIS
    • To perform the numerical calculation, we take f_\rho = 210 \pm 4\; {\rm MeV} [40, 69], m_\rho = 775.26 \pm 0.23\; {\rm MeV} , the current quark masses m_u = 2.16^{+0.49}_{-0.26}\; {\rm MeV} and m_d = 4.67^{+0.48}_{-0.17}\; {\rm MeV} [70], the vacuum condensates \langle\bar{q}q\rangle = (-2.417^{+0.227}_{-0.114}) \times 10^{-2}\; {\rm GeV}^2 , \langle g_s\bar{q}\sigma TGq\rangle = (-1.934^{+0.188}_{-0.103}) \times 10^{-2} {\rm GeV}^5, \langle g_s\bar{q}q\rangle^2 = (2.082^{+0.734}_{-0.697}) \times 10^{-3}\; {\rm GeV}^6 , \langle g_s^2\bar{q}q\rangle^2 = (7.420^{+2.614}_{-2.483}) \times 10^{-3}\; {\rm GeV}^6 , \langle\alpha_s G^2\rangle = 0.038 \pm 0.11\; {\rm GeV}^4 , and \langle g_s^3fG\rangle \simeq 0.045\; {\rm GeV}^6, and \kappa = 0.74 \pm 0.03 [50, 71, 72]. Among these, the current quark masses and vacuum condensates except the gluon condensates are scale dependent and have above values at \mu = 2\; {\rm GeV} . For the scale evolution of these inputs, refer to Ref. [50]. For the detailed calculation in this study, we take the scale \mu = M as usual. By requiring that there be a reasonable Borel window to normalize \langle\xi^0\rangle_{2;\rho}^\parallel with Eq. (3), we obtain the continuum threshold as s_\rho \simeq 2.1\; {\rm GeV}^2 .

      Now, we can obtain the ξ-moment versus the Borel parameter. Furthermore, we can determine the Borel window and then the value of the nth ξ-moment \langle\xi^n\rangle_{2;\rho}^\parallel based on the well-known criteria that the contributions of the continuum state and dimension-six condensate are as small as possible and the value of \langle\xi^n\rangle_{2;\rho}^\parallel is stable in the Borel window. Specifically, the continuum contribution is not more than 30% for all of the nth-order, and the dimension-six contribution of \langle\xi^n\rangle_{2;\rho}^\parallel is less than 2 % for n = (2,4) and 5% for n = (6,8,10) . Note that only the even order ξ-moments are nonzero owing to isospin symmetry. The ξ-moments versus the Borel parameter and the obtained Borel windows are shown in Fig. 1, where the shaded areas are the Borel windows for n= (2, 4, 6, 8, 10).

      Figure 1.  (color online) ρ-meson leading-twist longitudinal DA moments \langle \xi^n\rangle_{2;\rho}^\parallel with n=(2,4,6,8,10) versus the Borel parameter M^2 and the corresponding Borel windows, where all input parameters are set as their central values.

      Then, the values of {\langle\xi^n\rangle_{2;\rho}^\parallel}|_\mu up to the 10th-order with four different scales \mu = (1.0, 1.4, 2.0, 3.0) GeV and the second Gegenbauer moment a_{2;\rho}^\parallel can be obtained, which are shown in Table 2. Here, we only calculate the values of the first five non-zero ξ-moments because, as shown in Ref. [57], these moments are sufficient to determine the behavior of the DA \phi_{2;\rho}^\parallel(x,\mu) . As a comparison, the other theoretical predictions obtained using QCD SRs [2126], LQCD [2730], AdS/QCD [31, 32], data fitting [35, 36], LFQM [39], and DSE [41] are also shown in Table 2. Our second moment is less than the QCD SR predictions in Ref. [21, 23, 26], the LQCD calculations [2730], and DSE result [41] and larger than the QCD SR predictions in Refs. [24, 25] but very consistent with the value from 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].

      \mu {\rm /GeV } \langle\xi^2\rangle_{2;\rho}^\parallel \langle\xi^4\rangle_{2;\rho}^\parallel \langle\xi^6\rangle_{2;\rho}^\parallel \langle\xi^8\rangle_{2;\rho}^\parallel \langle\xi^{10}\rangle_{2;\rho}^\parallel a_{2 \parallel}^{2;\rho}
      This study1.0 0.220(6) 0.103(4) 0.0656(50) 0.0457(35) 0.0346(28) 0.059(18)
      This study1.4 0.217(5) 0.100(3) 0.0623(40) 0.0428(29) 0.0320(23) 0.051(16)
      This study2.0 0.215(5) 0.0986(31) 0.0603(35) 0.0411(25) 0.0304(20) 0.045(14)
      This study3.0 0.214(4) 0.0972(28) 0.0587(30) 0.0397(22) 0.0292(18) 0.041(13)
      QCD SRs [21]1.0 0.18(10)
      QCD SRs [22]1.0 0.227(7) 0.095(5) 0.051(4) 0.030(2) 0.020(5)
      QCD SRs [23, 26]1.0 0.15(7)
      QCD SRs [23, 26]2.0 0.10(5)
      QCD SRs [24]1.0 0.216(21) 0.089(9) 0.048(5) 0.030(3) 0.022(2) 0.047(58)
      QCD SRs [25]2.0 0.206(8) 0.087(6) 0.017(24)
      LQCD [27]2.0 0.237(36)(12)
      LQCD [28, 29]2.0 0.27(1)(2)
      LQCD [30]2.0 0.132(27)
      AdS/QCD [31, 32]1.0 0.228
      Data fitting [35, 36]1.0 0.227 0.105 0.062 0.041 0.029
      LFQM [39]1.0 0.21,0.19 0.09,0.08 0.05,0.04 0.02,-0.02
      DSE [41]2.0 0.23 0.11 0.066 0.045 0.033

      Table 2.  Our predictions for the first five nonzero-order ξ-moments \langle \xi^n\rangle _{2;\rho}^\parallel with n=(2,4,6,8,10) and the second-order Gegenbauer moment a_{2 \parallel}^{2;\rho} of the ρ-meson leading-twist longitudinal DA, compared with other theoretical predictions.

      By fitting our values of \langle\xi^n\rangle_{2;\rho}^\parallel with n = (2,4,6,8,10) shown in Table 2 using the least squares method (where the model parameters α and \hat{a}_2 are taken to be the fitting parameters, and for a specific fitting procedure, refer to Ref. [50]), the behavior of the ρ-meson leading-twist longitudinal DA can be obtained. The model parameters α and \hat{a}_2 of our DA and the corresponding \chi^2_{\rm min}/n_d and goodness of fit P_{\chi^2_{\rm min}} at several typical scales, such as \mu = (1.0, 1.4, 2.0, 3.0)\; {\rm GeV} , are exhibited in Table 3. The curve of the ρ-meson leading-twist longitudinal DA is shown in Fig. 2. Other predictions in Refs. [25, 41, 46] are also shown for comparison. Our curve is closer to the asymptotic form. In addition, we use the TF model \phi_{2;\rho}^{\parallel({\rm TF})}(x,\mu) = 6x(1-x) \Big[1 + a_{2\parallel}^{2;\rho} C_2^{3/2}(2x-1) + a_{4\parallel}^{2;\rho} C_4^{3/2}(2x-1) \Big] to fit our values of the ξ-moments of the ρ-meson leading-twist longitudinal DA shown in Table 2. The results indicate that the goodness of fits for the DSE model exhibited in Table 3 are close to those of the TF model at \mu = 1\; {\rm GeV} but better than those of the TF model at \mu = (1.4, 2.0, 3.0)\; {\rm GeV} . For example, a_{2\parallel}^{2;\rho} = 0.040 , a_{4\parallel}^{2;\rho} = 0.054, \chi^2_{\rm min}/n_d = 0.657635/3 , and P_{\chi^2_{\rm min}} = 0.88312 at \mu = 2\; {\rm GeV} . The curve of the TF model \mu = 2\; {\rm GeV} is also shown in Fig. 2. We find that the curve of the DSE model is smoother than that of the TF model. Thus, we use the DSE model instead of the TF model in this study.

      μ α \hat{a}_2 \chi^2_{\rm min}/n_d P_{\chi^2_{\rm min}}
      1.0 0.625 -0.516 0.957/3 0.812
      1.4 0.678 -0.448 0.488/3 0.922
      2.0 0.721 -0.398 0.259/3 0.968
      3.0 0.709 -0.427 0.141/3 0.987

      Table 3.  Model parameters α and \hat{a}_2 of our DA obtained by fitting using the least squares method and the corresponding \chi^2_{\rm min}/n_d and goodness of fit P_{\chi^2_{\rm min}} at several typical scales, such as \mu = (1.0,~ 1.4,~ 2.0,~ 3.0)~ {\rm GeV}.

      Figure 2.  (color online) Curve of our ρ-meson leading-twist longitudinal DA. As a comparison, the DA curves obtained using the DSEs [41], QCD SRs [25], algebraic model [46], asymptotic form, and fitting with the TF model are also shown.

      Furthermore, to calculate the D\to \rho TFFs, the input parameters should be clarified. The current charm-quark mass is m_c = 1.27\pm0.02\; {\rm GeV} , and the D-meson mass is m_D = 1.869\; {\rm GeV} [70]. Based on the three criteria for determining s_0 and M^2 in the LCSR approach, which can be found in our previous study [51], we have s_0^{A_1} = 7.0\pm 0.1\; {\rm GeV}^2 , s_0^{A_2}= 6.0\pm0.1\; {\rm GeV}^2 , s_0^{V} = 7.0\pm 0.1 {\rm GeV}^2, M^2_{A_1} = 1.45\pm 0.05\; {\rm GeV}^2, M^2_{A_2} = 1.20\pm 0.05 \;{\rm GeV}^2, and M^2_{V} = 2.15\pm 0.05\; {\rm GeV}^2. Then, the TFFs at the large recoil region, i.e., A_{1,2}(0) and V(0) , are present in Table 4. The uncertainties originate from the squared average for all the input parameters. In Table 4, 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], and HMχT [18], and lattice QCD predictions [19, 20] are presented. A comparison of all the results in Table 4 indicates that the TFFs of our prediction are consistent with those of many approaches within errors.

      A_1(0) A_2(0) V(0)
      This study 0.498^{+0.014}_{-0.012} 0.460^{+0.055}_{-0.047} 0.800^{+0.015}_{-0.014}
      CLEO2013 [3] 0.56(1)^{+0.02}_{-0.03} 0.47(6)(4) 0.84(9)^{+0.05}_{-0.06}
      3PSR [6] 0.5(2) 0.4(1) 1.0(2)
      HQETF [7] 0.57(8) 0.52(7) 0.72(10)
      LCSR [8] 0.599^{+0.035}_{-0.030} 0.372^{+0.026}_{-0.031} 0.801^{+0.044}_{-0.036}
      RHOPM [9]0.780.921.23
      QM-I [10]0.590.231.34
      QM-II [11]0.590.490.90
      LFQM [12] 0.60(1) 0.47(0) 0.88(3)
      HMχT [18]0.610.311.05
      LQCD [19] 0.45(4) 0.02(26) 0.78(12)
      LQCD [20] 0.65(15){^{+0.24}_{-0.23}} 0.59(31){^{+0.28}_{-0.25}} 1.07(49)(35)

      Table 4.  D\to \rho TFFs A_{1,2}(q^2) and V(q^2) at the large recoil region q^2\simeq 0 . The errors are squared averages of all the mentioned error sources. As a comparison, we also present predictions from various other methods.

      The ratios between different D\to\rho TFFs, i.e., r_2 and r_V , are shown in Fig. 3. Here, the results from the CLEO collaboration [3], HMχT [18], LFQM [12], HQEFT [7], 3PSR [6], QM [10, 11], and CCQM [17] are presented for comparison. From the figure, we can reach the following conclusions:

      Figure 3.  (color online) Predictions for the ratios r_2 and r_V within uncertainties. The results from the CLEO collaboration [3], HMχT [18], LFQM [12], HQEFT [7], 3PSR [6], QM [10, 11], and CCQM [17] are presented for comparison.

      ● Our predictions within uncertainties are in the region of the CLEO collaboration results.

      ● In terms of the central value with respect to uncertainties, our results for r_2 agree with the QM, LFQM, 3PSR, and CCQM for theoretical predictions. r_V agrees with the QM and 3PSR predictions.

      ● The uncertainties of our predictions are (12.9% ~ 14.4%) for r_2 and 4.4% for r_V , which indicates that this method can obtain applicable results in the confidence level. Because the predictions from the left-handed chiral current LCSR approach are mainly obtained at leading order for a strong coupling, we will consider the possible uncertainty/pollution from the axial-vector current in subsequent studies.

      Using the resultant TFFs in the large recoil region, their behavior in the entire physical region, i.e., q^2_{\rm phys.} \in [0, 1.18]\; \text{GeV}^2 should be obtained. It is known that the LCSR predictions are suitable in low and intermediate momentum transfer. Therefore, a rapidly converging series based on z(t) -expansion is used to extrapolate in this study [73, 74],

      \begin{eqnarray} F_i(q^2) = \frac1{1-q^2/m_{R,i}^2} \sum_{k=0,1,2}a_k^i [z(q^2)-z(0)]^k. \end{eqnarray}

      (15)

      The formula for z(t) can be found in Ref. [73]. F_i respects three TFFs, A_{1,2} and V. The appropriate resonance masses are m_{R,i}= 2.007\; {\rm GeV} for the TFF V(q^2) with J^P = 1^- and m_{R,i}= 2.427\; {\rm GeV} for the TFFs A_{1,2}(q^2) with J^P = 1^+ . The parameters a_k^i can be fixed by requiring \Delta < 0.1%, where the parameter Δ is introduced to measure the quality of extrapolation, \Delta={\sum_t|F_i(t)-F_i^{\rm fit}(t)|} / {\sum_t|F_i(t)|}\times 100 , where t\in[0,1/40,\cdots,40/40]\times 0.8\; {\rm GeV}^2. The D\to \rho TFFs obtained in this study using LCSRs and the simple z-series expansion parameterization are shown in Fig. 4. The maximal momentum transfers of the LCSR calculation of the D \to \rho TFFs are A_1(q_{\rm max}) = 0.883^{+0.022}_{-0.019} , A_2(q_{\rm max}) = 0.611^{+0.102}_{-0.103} , and V(q_{\rm max}) = 2.082^{+0.060}_{-0.059} .

      Figure 4.  (color online) Predictions for the D\to \rho TFFs, where the darker bands are the results of this study using the LCSR approach, and the lighter bands are the predictions from simple z-series expansion parameterizations.

      After obtaining the extrapolated TFFs of the transition D\to\rho , we find the CKM-independent total decay width 1/|V_{\rm cd}|^2 \times \Gamma . Combining the ratio of the longitudinal/transverse and positive/negative helicity decay widths, we present the predictions in Table 5. The 3PSR [6], HQEFT [7], RHOPM [9], QM [10], LQCD [19], and LQCD [20] results are given for comparison. As shown in Table 5, there are large differences between the total decay widths from the different theoretical groups, and our results agree with the lattice results within uncertainties. Our predicted longitudinal/transverse helicity decay width agrees with those of HQEFT and LQCD within uncertainties, which is larger than 1. Our positive/negative helicity decay width agrees with that of the QM and LQCD within uncertainties.

      1/|V_{\rm cd}|^2 \times \Gamma \Gamma^{\rm L}/ \Gamma^{\rm T} \Gamma^+/ \Gamma^-
      This study 57.87^{+5.40}_{-5.13} 1.00^{+0.10}_{-0.11} 0.15^{+0.02}_{-0.01}
      3PSR [6] 15.80\pm4.61 1.31\pm0.11 0.24\pm0.03
      HQEFT [7] 71\pm14 1.17\pm0.09 0.29\pm0.13
      RHOPM [9] 90.83 0.910.19
      QM [10] 88.86 1.330.11
      LQCD [19] 54.63\pm12.51 1.86\pm0.56 0.16
      LQCD [20]71.751.100.18

      Table 5.  |V_{cd}| -independent total decay width 1/|V_{cd}|^2 \times \Gamma , ratio for longitudinal/transverse and positive/negative helicity decay widths. For comparsion, other theoretical group predictions are also given.

      Lastly, the branching fractions for the two types of D\to\rho\ell^+\nu_\ell semileptonic decays are calculated. The first is the D^0 -type, including the D^0 \to \rho^- e^+\nu_e and D^0 \to \rho^- \mu^+\nu_\mu decays, for which the lifetime \tau(D^0)= 0.410\pm 0.001 ps should be used. The second is the D^+ -type, including D^+ \to \rho^0 e^+\nu_e and D^+ \to \rho^0 \mu^+\nu_\mu , associated with \tau(D^+)= 1.033\pm 0.005 ps [70]. The CKM matrix element |V_{cd}| = 0.225\pm0.001 [69]. We present the predictions in Table 6, where the two uncertainties mainly arise from the squared average of the theoretical input parameters and the experimental uncertainties from the D-meson lifetime. For comparison, the theoretical results from the 3PSR [6], HQEFT [7], narrow width approximation (NWA) [75] with the HQEFT [8] and LFQM [12] approaches, LCSR [8, 14], LFQM [13], CCQM [17], chiral unitarity approach (χUA) [76], RQM [77], HMχT [18], ISGW2 [78] and experimental results from the BESIII collaboration [1, 2] and CLEO collaboration prediction in 2013 [3] and 2005 [4] are also given. The results show that:

      Decay Mode D^0\to \rho^- e^+ \nu_e D^+ \to \rho^0 e^+ \nu_e D^0\to \rho^- \mu^+ \nu_\mu D^+ \to \rho^0 \mu^+ \nu_\mu
      This study 1.825^{+0.170}_{-0.162}\pm 0.004 2.299^{+0.214}_{-0.204}\pm 0.011 1.816^{+0.168}_{-0.160}\pm 0.004 2.288^{+0.212}_{-0.201}\pm 0.011
      BESIII [1] 1.35\pm0.09\pm0.09
      BESIII [2] 1.445\pm0.058\pm0.039 1.860\pm0.070\pm0.061
      CLEO2013 [3] 1.77\pm0.12\pm0.10 2.17\pm0.12^{+0.12}_{-0.22}
      CLEO2005 [4] 1.94\pm0.39\pm0.13 2.1\pm0.4\pm0.1
      3PSR [6] 0.5\pm0.1
      HQEFT [7] 1.4\pm0.3
      LCSR [8] 1.81^{+0.18}_{-0.13} 2.29^{+0.23}_{-0.16} 1.73^{+0.17}_{-0.13} 2.20^{+0.21}_{-0.16}
      NWA [75]+HQEFT [8] 1.67\pm0.27 2.16\pm0.36
      NWA [75]+LFQM [12] 1.73\pm0.07 2.24\pm0.09
      LFQM [13] 1.7\pm0.2
      LCSR [14] 1.74\pm0.25 2.25\pm0.32 1.65\pm0.23 2.14\pm0.30
      CCQM [17]1.622.091.552.01
      HMχT [18]2.02.5
      χUA [76]1.972.541.842.37
      RQM [77]1.962.491.882.39
      ISGW2 [78]1.01.3

      Table 6.  Branching ratios of the semileptonic decays D^0\to \rho^- \ell^+ \nu_\ell and D^+ \to \rho^0 \ell^+ \nu_\ell (in units of 10^{-3} ). For comparison, we also present the results from experimental collaborations and theoretical groups.

      ● The results obtained using our left-handed chiral LCSR approach are consistent with those obtained using other LCSRs within uncertainties.

      ● Our current results are consistent with those of the CLEO collaboration but are larger than the BESIII collaboration predictions.

      ● Our results are consistent with those of other theoretical groups, such as the NWA, LFQM, HQEFT, χUA, RQM, and HMχT, but have a smaller gap when compared with the 3PSR, HQEFT, χUA, and ISGW2 results.

    IV.   SUMMARY
    • In the framework of BFT with the QCD SR approach, we calculate the ξ-moments of the ρ-meson leading twist longitudinal DA. Because the zeroth ξ-moment cannot be normalized in the entire Borel region, a new SR formula for the ξ-moment is given in Eq. (2). The results up to the tenth order with the scale \mu = (1.0,1.4,2.0.3.0)\; {\rm GeV}^2 are presented in Table 2, where the results from other theoretical group are also given. Next, we present the D\to \rho TFFs in the large recoil region, i.e., A_1(0),~ A_2(0), and V(0) , in Table 4 and the ratio r_{2,V} in Fig. 3, which indicate that our results are reasonable and consistent with those of many approaches within errors. Subsequently, we calculate the |V_{cd}| -independent total decay width and the ratio for the longitudinal/transverse and positive/negative helicity decay width results and present them in Table 5. A detail discussion comparing our results with other predictions is presented. Finally, we show the branching fraction of the two types of D\to\rho\ell^+\nu_\ell semileptonic decays in Table 6. In the near further, we hope that more accurate data will be reported and more theoretical results will be given to explain the gaps between different approaches.

    ACKNOWLEDGMENTS
    • We are grateful for the referee's valuable comments and suggestions.

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