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

As discussion in Ref. Zhong:2021epq , the new sum rule formula for the $\xi$-moments is based on that the sum rule of zeroth moment can not be normalized in entire Borel parameter region. Therefore, the discussion of this paper begins with the sum rule formulae for the $\xi$-moments of $D$-meson twist-2 DA $\phi_{2;D}(x,\mu)$ obtained in Ref. Zhang:2017rwz and twist-3 DAs $\phi_{3;D}^{p}(x,\mu)$, $\phi_{3;D}^{\sigma}(x,\mu)$ obtained in Ref. Zhong:2018exo .

By giving up the priori setting for zeroth $\xi$-moment normalization, Eq. (28) in Ref. Zhang:2017rwz should be modified as

for the $n$th $\xi$-moment $\left<\xi^{n}\right>_{2;D}$ of $\phi_{2;D}(x,\mu)$. Eq. (27) in Ref. Zhong:2018exo should be modified as

for the $n$th $\xi$-moment $\left<\xi^{n}_{p}\right>_{3;D}$ of $\phi_{3;D}^{p}(x,\mu)$. Eq. (28) in Ref. Zhong:2018exo should be modified as

for the $n$th $\xi$-moment $\left<\xi^{n}_{\sigma}\right>_{3;D}$ of $\phi_{3;D}^{\sigma}(x,\mu)$. In Eqs. (1), (2) and (3), $m_{D}$ is the $D$-meson mass, $m_{c}$ is the current charm-quark mass, $f_{D}$ is the decay constant of $D$-meson, $s_{D}$ is the continuum threshold, $\hat{L}_{M}$ indicates Borel transformation operator with the Borel parameter $M$. $\mu_{D}^{p}$ and $\mu_{D}^{\sigma}$ are the normalization constants of DAs $\phi_{3;D}^{p}(x,\mu)$ and $\phi_{3;D}^{\sigma}(x,\mu)$ respectively. Usually, $\mu_{D}^{p}=\mu_{D}^{\sigma}=\mu_{D}=m_{D}^{2}/m_{c}$ in literature by employing the equations of motion of on-shell quarks in the meson. However, as discussed in Refs. Huang:2004tp ; Huang:2005av , the quarks inside the bound state are not exactly on-shell. Then $\mu_{D}^{p}$ and $\mu_{D}^{\sigma}$ are taken as undetermined parameters in this paper and will be determined via the sum rules of zeroth $\xi$-moments of DAs $\phi_{3;D}^{p}(x,\mu)$ and $\phi_{3;D}^{\sigma}(x,\mu)$ following the idea of Refs. Zhong:2018exo ; Huang:2004tp ; Huang:2005av . In addition, in sum rules (1), (2) and (3), the subscript “pert” stands for the terms coming from the contribution of perturbative part in operator product expansion, subscripts $\left<\bar{q}q\right>$, $\left<G^{2}\right>$, $\left<\bar{q}Gq\right>$, $\left<\bar{q}q\right>^{2}$ and $\left<G^{3}\right>$ stand for the terms proportional to double-quark condensate, double-gluon condensate, quark-gluon mixing condensate, four-quark condensate and triple-gluon condensate, respectively. For the expressions of those terms in Eqs. (1), (2) and (3), one can refer to the appendixes in Refs. Zhang:2017rwz ; Zhong:2018exo . By taking $n=0$ in Eq. (1) and (2), one can obtain the sum rules for the zeroth $\xi$-moments $\left<\xi^{0}\right>_{2;D}$ and $\left<\xi^{0}_{p}\right>_{3;D}$. As the functions of the Borel parameter, the zeroth $\xi$-moments $\left<\xi^{0}\right>_{2;D}$ in Eq. (1) and $\left<\xi^{0}_{p}\right>_{3;D}$ in Eqs. (2) and (3) obviously can not be normalized in entire $M^{2}$ region. Therefore, more reasonable and accurate sum rules should be

and

for $\left<\xi^{n}\right>_{2;D}$, $\left<\xi^{n}_{p}\right>_{3;D}$ and $\left<\xi^{n}_{\sigma}\right>_{3;D}$, respectively.

In Refs. Zhang:2017rwz ; Zhong:2018exo , we have suggested LCHO models for $D$-meson twist-2, 3 DAs. In this subsection, we first propose a brief review for those models, then we will improve them by reconstructing whose longitudinal distribution functions.

The $D$-meson leading-twist DA $\phi_{2;D}(x,\mu)$ can be obtained by integrating out the transverse momentum $\textbf{k}_{\perp}$ component in its WF $\Psi_{2;D}(x,\textbf{k}_{\perp})$, i.e.,

Based on the BHL description BHL , the LCHO model for the $D$-meson leading-twist WF consists of the spin-space WF $\chi_{2;D}(x,\textbf{k}_{\perp})$ and spatial WF $\psi_{2;D}^{R}(x,\textbf{k}_{\perp})$, i.e., $\Psi_{2;D}(x,\textbf{k}_{\perp})=\chi_{2;D}(x,\textbf{k}_{\perp})\psi_{2;D}^{R}(x,\textbf{k}_{\perp})$. The spin-space WF $\chi_{2;D}(x,\textbf{k}_{\perp})=\widetilde{m}/\sqrt{\textbf{k}_{\perp}^{2}+\widetilde{m}}$. In which, $\widetilde{m}=\hat{m}_{c}x+\hat{m}_{q}\bar{x}$ with the constituent charm-quark mass $\hat{m}_{c}$ and light-quark mass $\hat{m}_{q}$. In this paper, we take $\hat{m}_{c}=1.5{\rm GeV}$ and $\hat{m}_{q}=0.25{\rm GeV}$ Zhong:2022ecl . As discussed in Ref. Zhang:2017rwz , we take $\chi_{2;D}\to 1$ approximately due to that $\hat{m}_{c}\gg\Lambda_{\rm QCD}$. Then, the $D$-meson leading-twist WF reads

where $\bar{x}=1-x$, $A_{2;D}$ is the normalization constant, $\beta_{2;D}$ is a harmonious parameter that dominates the WF’s transverse distribution, $\varphi_{2;D}(x,\mu)$ dominates the WF’s longitudinal distribution.

Substituting Eq. (LABEL:psiR2D) into (7), the expression of $D$-meson leading-twist DA $\phi_{2;D}(x,\mu_{0})$ can be obtained, i.e.,

Following the way for constructing the $D$-meson leading-twist DA, the LCHO models for $D$-meson twist-3 DAs $\phi_{3;D}^{p}(x,\mu)$ and $\phi_{3;D}^{\sigma}(x,\mu)$ read,

and

respectively.

For the longitudinal distribution functions $\varphi_{2;D}(x)$, $\varphi_{3;D}^{p}(x)$ and $\varphi_{3;D}^{\sigma}(x)$, we used to take the first five terms of Gegenbauer expansions for the corresponding DAs in Refs. Zhang:2017rwz ; Zhong:2018exo . As discussed in Ref. Zhong:2021epq ; Zhong:2022ecl , higher order Gegenbauer polynomials will introduce spurious oscillations Chang:2013pq , while those corresponding coefficients obtained by directly solving the constraints of Gegenbauer moments or $\xi$-moments are not reliable. Then we improve these three longitudinal distribution functions as following,

By considering the normalization conditions for $D$-meson twist-2, 3 DAs $\phi_{2;D}(x,\mu)$, $\phi_{3;D}^{p}(x,\mu)$ and $\phi_{3;D}^{p}(x,\mu)$, that is,

there are three undetermined parameters in the LCHO models for DAs $\phi_{2;D}(x,\mu)$, $\phi_{3;D}^{p}(x,\mu)$ and $\phi_{3;D}^{p}(x,\mu)$ respectively, and which will be taken as the fitting parameters to fit the first ten $\xi$-moments³³3In our previous work Zhong:2022lmn , based on the pionic leading-twist DA, we analyzed in detail the influence of different number of $\xi$-moments included in the fitting, and found that when the order of $\xi$-moments is not more than ten, the change of the number of $\xi$-moments has an obvious impact on the fitting results. When the order of $\xi$-moments is more than ten, the change of the number of $\xi$-moments has a very small impact on the fitting results. Therefore, we only use the first ten $\xi$-moments of $D$-meson DAs $\phi_{2;D}(x,\mu)$, $\phi_{3;D}^{p}(x,\mu)$ and $\phi_{3;D}^{\sigma}(x,\mu)$ for fitting in this work. of corresponding DAs by adopting the least squares method in next section.

It should be noted that, $D$-meson twist-2, 3 DAs are the universal non-perturbative parameters in essence, and non-perturbative QCD should be used to study them in principle. However, due to the difficulty of non-perturbative QCD, those DAs are studied in this paper by combining the phenomenological model, that is, the LCHO model, and the non-perturbative QCD SRs in the framework of BFT. Otherwise, the improvement of the LCHO model of DAs $\phi_{2;D}(x,\mu)$, $\phi_{3;D}^{p}(x,\mu)$ and $\phi_{3;D}^{p}(x,\mu)$, that is, to reconstruct their longitudinal distribution functions, is only based on mathematical considerations. The rationality of this improvement can be judged by the goodness of fit.

To do the numerical calculation for the $\xi$-moments of $D$-meson twist-2, 3 DAs, we take the scale $\mu=M$ as usual, and take $\Lambda_{\rm QCD}^{(n_{f})}\simeq 324,286,207~{}{\rm MeV}$ for the number of quark flavors $n_{f}=3,4,5$, respectively Zhong:2021epq ; Zhong:2022ecl . For other inputs, we take Workman:2022ynf

and Zhong:2021epq ; Zhong:2014jla ; Colangelo:2000dp

The renormalization group equations of those inputs are Zhong:2021epq

with $\beta_{0}=(33-2n_{f})/3$. For the continuum threshold, we used to take $s_{D}\simeq 6.5{\rm GeV}^{2}$ in Ref. Zhang:2017rwz ; Zhong:2018exo . This value comes from the square of the mass of $D$-meson’s first exciting state, i.e., $D^{0}(2550)$, as suggested by Refs. BaBar:2010zpy ; Li:2012gr . In Refs. Zhong:2021epq ; Zhong:2022ecl , we take the continuum threshold parameters $s_{\pi}$ and $s_{K}$ by requiring that there are reasonable Borel windows to normalize the zeroth $\xi$-moments of the pion and kaon leading-twist DAs. In this paper, we follow the suggestion in Refs. Zhong:2021epq ; Zhong:2022ecl , and get $s_{D}\simeq 6.0{\rm GeV}^{2}$.

Then we can calculate the values of the $\xi$-moments of $D$-meson twist-2, 3 DAs with the sum rules (4), (5) and (6). First, one need to determine the appropriate Borel windows for those $\xi$-moments by following usual criteria, such as the contributions of continuum state and dimension-six condensate are as small as possible, and the values of those $\xi$-moments are stable in corresponding Borel windows. Table 1 exhibits the limits to the continuum state’s contributions and the dimension-six condensate’s contributions for the first ten $\xi$-moments of $D$-meson twist-2, 3 DAs. In which, the symbol “$-$” indicates that corresponding continuum state’s contribution is smaller than $10\%$ or dimension-six condensate’s contribution is much smaller than $5\%$ in a wide Borel parameter region. This is reasonable because both continuum state’s contribution and dimension-six condensate’s contribution are depressed by the sum rules of zeroth $\xi$-moments in the denominator of the new sum rule formulae (4), (5) and (6). By comparing with the criteria listed in Table 1 and Table 4 in Ref. Zhong:2018exo , the criteria listed in Table 1 are much stricter, which reflects one of the advantages of the new sum rule formulae (4), (5) and (6), that is, they reduce the system uncertainty of the sum rule itself. Then, for those $\xi$-moments, only the upper or lower limits of the corresponding Borel windows is clearly determined. In order to get complete Borel windows, we directly take their lengths as $1~{}{\rm GeV}^{2}$. Figure 1 shows the $D$-meson twist-2, 3 DA $\xi$-moments $\langle\xi^{n}\rangle_{2;D}$, $\langle\xi^{n}_{p}\rangle_{3;D}$ and $\langle\xi^{n}_{\sigma}\rangle_{3;D}$ with $(n=1,\cdots,10)$ versus the Borel parameter $M^{2}$. In this figure, the uncertainties caused by the errors of input parameters is not drawn to clearly show the curves of different $\xi$-moments. Meanwhile, the Borel windows are also shown with the shaded bands. By taking all error sources, such as $D$-meson mass and decay constant, $u$- and $c$-quark masses, as well as vacuum condensates, etc., shown in Eqs. (16) and (17), into consideration, and adding the uncertainties in quadrature, the values of the first ten $\xi$-moments of $D$-meson twist-2, 3 DAs are shown in Table 2. Here, we give the first two Gegenbauer moments of $D$-meson twist-2, 3 DAs for reference, that is,