1. Introduction
The exploration of quark-gluon structure hadrons has been a cutting-edge issue across particle physics and medium-high energy nuclear physics in recent years. Among them, quarks and gluons, called partons, are the fundamental degrees of freedom of quantum chromodynamics (QCD). Although the parton can not be directly observed, the QCD factorization theorem allows one to express the information of the parton inside the nucleon in terms of nonperturbative functions [1]. At the same time, the parton distribution function (PDF) is considered to be the most important nonperturbative function, which plays an important role in describing the nonperturbative QCD for the internal structure of hadronic bound states [2]. In addition, it also gives the probability of finding quarks and gluons inside a hadron. In the infinite momentum coordinate system [3–6], PDFs are used to describe the one-dimensional momentum distributions of quarks and gluons. Therefore, the internal structure of hadrons can be studied by calculating the meson's PDF.
The PDF constitutes the basic limit of the Higgs boson characterization in the matter of coupling and is the main system for Standard Model (SM) measurements such as W-boson mass. Also, it is still the largest uncertainty outside the production of SM heavy particles so it has important phenomenological value. The MMHT [7], CT [8], NNPDF [9], HERAPDF [10], and JAM [11] have made substantial efforts to determine PDFs and their uncertainties. The pion deemed to the lightest bound state of QCD and kaon has been predicted by many theoretical calculations, chiral-quark model [12–14], Nambu–Jona–Lasinio model [15], light-front holographic QCD (LHFQCD) [16–19], light front quantization [20–22], maximum entropy method [23, 24], Dyson–Schwinger equations (DSEs) [25–35] and lattice QCD [36–45] for the valence quark PDF. Meanwhile, PDFs can be attained directly from the light-front wave function (LFWF), which has been researched via the Bethe–Salpeter wave functions within covariant DSEs, LHFCD and BLFQ. The scalar mesons below 1 GeV are an interesting field to researchers, especially for a0(980) state. Its internal structure has some pictures, such as quark–antiquark states [46–51], tetraquark states [52–57], two-meson molecule-bound states [58–62] and hybrid states [63]. In this paper, we mainly take the a0(980) state as the quark–antiquark picture. Until now, there is less research about the scalar meson a0(980) PDFs. Thus, in order to understand the internal structure of the a0(980)-meson, the a0(980)-meson parton distribution function will be studied in this paper.
One of the earliest predictions for the J = 0 meson valence–quark distribution function for large-x behaviour within the QCD improved parton model [64–66], has the following expression:
$\begin{eqnarray}{q}^{{\rm{M}}}(x;\zeta )\mathop{=}\limits^{x\simeq 1}c(\zeta ){\left(1-x\right)}^{{\beta }_{\zeta }},\,{\beta }_{\zeta }=2,\end{eqnarray}$
where c(ζ) is independent of x and the ζ stands for the resolving scale. The symbol ‘M' stands for each J = 0 meson. Since the meson's PDF can be obtained directly from its wave function (WF), a more accurate a0(980)-meson WF is crucial for us to determine its PDF. The light-cone harmonic oscillator model (LCHO) for the light or heavy meson WF is mainly based on the Brodsky–Huang–Lepage (BHL) prescription [67], which has been used in many cases [68–76]. For this model, the total WF can be separated into spin-space WF ${\chi }_{{\rm{M}}}^{{\lambda }_{1}{\lambda }_{2}}(x,{{\boldsymbol{k}}}_{\perp })$ and spatial wave function ${\psi }_{{\rm{M}}}^{{\rm{R}}}(x,{{\boldsymbol{k}}}_{\perp })$. The spatial WF is divided into the x-dependence part and the k⊥-dependence part for calculation. The k⊥-dependence part derives from the approximate bound-state solution and the x-dependence part φM(x) can be expanded in Gegenbauer polynomials.Furthermore, the meson light-cone distribution amplitudes (LCDAs) can also be related to its WF, which leads to the indirect relationship between meson LCDAs and PDFs. In many applications of LCDAs, one usually takes a truncated form to determine DAs, involving only the first few terms of the Gegenbauer expansion series. With the increase of n, there will exist dimensional anomalies which lead to spurious oscillations. In addition, one of the most important factors is the unreliability of the higher-order Gegenbauer moments. In order to improve this phenomenon, one can adopt the LCHO model to deal with the meson LCDAs. In our previous works [76, 77], meson leading-twist LCDAs are studied by using the LCHO model, and then the model parameters are determined by fitting moments with the least squares method. Therefore, we will study the a0(980)-meson PDFs based on BHL prescription in this paper.
2. Theoretical framework
If one wants to use the typical probability expression of quantum mechanics to describe the measurable properties of a given hadron, the first thing one needs to do is find the WF. Each element of the WF Fock-space decomposition represents the probability amplitude of finding n components in the hadron. However, the PDFs describe the longitudinal momentum distribution parton of the hadron. To derive the a0(980)-meson leading-twist PDF, the following expression can be used [34]7 ) and (8 ), the spatial WF for a0(980)-meson can be obtained
$\begin{eqnarray}{q}^{{a}_{0}}(x,\zeta )={\int }_{| {{\boldsymbol{k}}}_{\perp }{| }^{2}\leqslant {\zeta }^{2}}{{\rm{d}}}^{2}{{\boldsymbol{k}}}_{\perp }| {\psi }_{{a}_{0}}(x,{{\boldsymbol{k}}}_{\perp }){| }^{2},\end{eqnarray}$
where ${\psi }_{{a}_{0}}(x,{{\boldsymbol{k}}}_{\perp })$ is WF and ζ stands for the scale. In order to calculate the PDF, the connection between distribution amplitudes and distribution function can be established to achieve the purpose. Exploiting this relationship, one can predict a0(980)-meson leading-twist PDFs with the LCHO model based on the BHL description [67, 75, 78]. The LCHO model of the a0(980)-meson leading-twist WF is denoted by $\begin{eqnarray}{\psi }_{{a}_{0}}(x,{{\boldsymbol{k}}}_{\perp })=\displaystyle \sum _{{\lambda }_{1}{\lambda }_{2}}{\chi }_{{a}_{0}}^{{\lambda }_{1}{\lambda }_{2}}(x,{{\boldsymbol{k}}}_{\perp }){\psi }_{{a}_{0}}^{{\rm{R}}}(x,{{\boldsymbol{k}}}_{\perp }),\end{eqnarray}$
where k⊥ is transverse momentum and the symbol ‘R' means that ${\psi }_{{a}_{0}}^{{\rm{R}}}(x,{{\boldsymbol{k}}}_{\perp })$ is the spatial wave function in coordinate form. The LCHO model consists of the spin wave function ${\chi }_{{a}_{0}}^{{\lambda }_{1}{\lambda }_{2}}(x,{{\boldsymbol{k}}}_{\perp })$ and ${\psi }_{{a}_{0}}^{{\rm{R}}}(x,{{\boldsymbol{k}}}_{\perp })$ as the space wave function. Furthermore, λ1 and λ2 are the helicities of the two constituent quarks. The spin-space WF ${\chi }_{{a}_{0}}^{{\lambda }_{1}{\lambda }_{2}}(x,{{\boldsymbol{k}}}_{\perp })$ comes from the Wigner–Melosh rotation. The different forms for λ1λ2 can also be found in [75]. Thus the sum of the spin-space WF has the following form $\begin{eqnarray}\displaystyle \sum _{{\lambda }_{1}{\lambda }_{2}}{\chi }_{{a}_{0}}^{{\lambda }_{1}{\lambda }_{2}}(x,{{\boldsymbol{k}}}_{\perp })=\displaystyle \frac{{m}_{q}^{2}}{\sqrt{{{\boldsymbol{k}}}_{\perp }^{2}+{m}_{q}^{2}}},\end{eqnarray}$
with mq = mu = md. On the other hand, the BHL description proposed the assumption that the valence Fock WF depends only on the energy variable ε outside the shell. At the same time, the connection between the equal-time WF in the rest frame and the light-cone wave function in the infinite frame by equating the energy propagator $\epsilon ={M}^{2}-{\left({\sum }_{i=1}^{n}{k}_{i}\right)}^{2}$ is proposed. The propagators in different frames are as follows $\begin{eqnarray}\begin{array}{ll}{\epsilon }_{1}={M}^{2}-{\left(\displaystyle \sum _{i=1}^{n}{q}_{i}^{0}\right)}^{2}, & \displaystyle \sum _{i=1}^{n}{{\boldsymbol{q}}}^{i}=0\\ {\epsilon }_{2}={M}^{2}-\left(\displaystyle \sum _{i=1}^{n}\displaystyle \frac{{{\boldsymbol{k}}}_{\perp i}^{2}+{m}_{i}^{2}}{{x}_{i}}\right), & \displaystyle \sum _{i=1}^{n}{{\boldsymbol{k}}}_{\perp i}^{2}=0,\\ & \displaystyle \sum _{i=1}^{n}{x}^{i}=1,\end{array}\end{eqnarray}$
where the index i is represented as the parton, when i = 1, it is represented as the u-quark, and i = 2 is represented as the d-quark. For the two-particle system, one can get $\begin{eqnarray}{{\boldsymbol{q}}}^{2}\leftrightarrow \displaystyle \frac{{{\boldsymbol{k}}}_{\perp i}^{2}+{m}_{q}^{2}}{4x(1-x)}-{m}_{q}^{2},\end{eqnarray}$
where ${q}_{1}^{0}={q}_{2}^{0}$. Besides, the rest frame wave function ψCM(q) and the light-cone wave function ψLC(x, k) might be related in some way [75] $\begin{eqnarray}{\psi }_{\mathrm{CM}}({{\boldsymbol{q}}}^{2})\leftrightarrow {\psi }_{\mathrm{LC}}\left(\displaystyle \frac{{{\boldsymbol{k}}}_{\perp }^{2}+{m}_{q}^{2}}{4x(1-x)}-{m}_{q}^{2}\right).\end{eqnarray}$
In this paper, based on the approximation for the bound state solution of the meson quark model, the WF of the harmonic oscillator model in the rest frame is expressed as $\begin{eqnarray}{\psi }_{\mathrm{CM}}({{\boldsymbol{q}}}^{2})=A\exp \left(-\displaystyle \frac{{{\boldsymbol{q}}}^{2}}{2{\beta }^{2}}\right).\end{eqnarray}$
Combining equations ( $\begin{eqnarray}{\psi }_{{a}_{0}}^{{\rm{R}}}(x,{{\boldsymbol{k}}}_{\perp })\propto \exp \left[-\displaystyle \frac{{{\boldsymbol{k}}}_{\perp }^{2}+{m}_{q}^{2}}{8{\beta }_{{a}_{0}}^{2}x\bar{x}}\right].\end{eqnarray}$
With $\bar{x}=1-x$ from [76], we can get a a0(980)-meson spatial WF, $\begin{eqnarray}{\psi }_{{a}_{0}}^{{\rm{R}}}(x,{{\boldsymbol{k}}}_{\perp })={A}_{{a}_{0}}{\varphi }_{{a}_{0}}(x)\exp \left[-\displaystyle \frac{{{\boldsymbol{k}}}_{\perp }^{2}+{m}_{q}^{2}}{8{\beta }_{{a}_{0}}^{2}x\bar{x}}\right],\end{eqnarray}$
where the free parameters ${A}_{{a}_{0}}$ stands for normalization constant, the harmonious parameter ${\beta }_{{a}_{0}}$ can determine the transverse distribution amplitudes of meson WF and ${\varphi }_{{a}_{0}}(x)$ is crucial for determining the WF longitudinal distribution amplitude. It can be expressed in terms of the first few terms of the Gegenbauer moment polynomials since the a0(980)-meson leading-twist amplitude is antisymmetric under u → (1 − u) transition in the SUf(3) limit.By using the relation between the a0(980)-meson wavefunction and its distribution amplitude,4 ) with (10 ), one can get the a0(980)-meson leading-twist WF18 ) has a certain influence on adjusting the LCDA's behavior, so $x\bar{x}$ can be introduced into the longitudinal distribution amplitude, the expression is as follows,2 ), which leads to the following formula
$\begin{eqnarray}{\psi }_{{a}_{0}}(x,\zeta )={\int }_{| {{\boldsymbol{k}}}_{\perp }{| }^{2}\leqslant {\zeta }^{2}}\displaystyle \frac{{{\rm{d}}}^{2}{{\boldsymbol{k}}}_{\perp }}{16{\pi }^{2}}{\psi }_{{a}_{0}}(x,{{\boldsymbol{k}}}_{\perp }).\end{eqnarray}$
And by combining ( $\begin{eqnarray}{\psi }_{{a}_{0}}(x,{{\boldsymbol{k}}}_{\perp })=\displaystyle \frac{{m}_{q}^{2}{A}_{{a}_{0}}{\varphi }_{{a}_{0}}(x)}{\sqrt{{{\boldsymbol{k}}}_{\perp }^{2}+{m}_{q}^{2}}}\exp \left[-\displaystyle \frac{{{\boldsymbol{k}}}_{\perp }^{2}+{m}_{q}^{2}}{8{\beta }_{{a}_{0}}^{2}x\bar{x}}\right].\end{eqnarray}$
We find that the $\sqrt{x\bar{x}}$ in equation ( $\begin{eqnarray}{\varphi }_{{a}_{0}}(x)={\left(x\bar{x}\right)}^{{\alpha }_{{a}_{0}}}{C}_{1}^{3/2}(2x-1),\end{eqnarray}$
where the free parameter ${\alpha }_{{a}_{0}}$ can be obtained by fitting moment $\langle {\xi }_{{a}_{0}}^{n}{\rangle }_{\zeta }$ with the least square method. Furthermore, the a0(980)-meson leading-twist valence-quark distribution function can be obtained by integrating over the squared transverse momentum, i.e. Equation ( $\begin{eqnarray}\begin{array}{l}{q}^{{a}_{0}}(x,\zeta )=\displaystyle \frac{{A}_{{a}_{0}}^{2}{m}_{q}^{2}{\left(1-2x\right)}^{2}{\left(x\bar{x}\right)}^{2{\alpha }_{{a}_{0}}}}{512{\pi }^{5}}\\ \quad \times \left\{\mathrm{Ei}\left[\sqrt{\displaystyle \frac{{m}_{q}^{2}+{\zeta }^{2}}{4{\beta }_{{a}_{0}}^{2}x\bar{x}}}\right]-\mathrm{Ei}\left[\sqrt{\displaystyle \frac{{m}_{q}^{2}}{4{\beta }_{{a}_{0}}^{2}x\bar{x}}}\right]\right\},\end{array}\end{eqnarray}$
with $\mathrm{Ei}(x)=-{\int }_{-z}^{\infty }\tfrac{{{\rm{e}}}^{-t}}{t}{\rm{d}}t$. For further research, the process-independent effective charge is used to redesign the process-dependent-charge alternative and implement evolution to integrate the one-loop Dokshitzer–Gribov–Lipatov–Altarelli–Parisi (DGLAP) equations by describing the evolution of quarks or gluons fragmenting into hadrons and identifying only one hadron at a time. The relevant explanation can be found in [34]. Using this process, one can get the Mellin moments of a0(980)-meson's valence-quark distribution function: $\begin{eqnarray}\langle {x}^{n}{q}^{{a}_{0}}{\rangle }_{\zeta }={\int }_{0}^{1}{x}^{n}{q}^{{a}_{0}}(x,\zeta ){\rm{d}}x.\end{eqnarray}$
Then, the ratio of Mellin moments is also a point of interest, and its expression is as follows $\begin{eqnarray}{x}_{\,{a}_{0}}^{n}(\zeta ,{\zeta }_{k})=\displaystyle \frac{\langle {x}^{n}{q}^{{a}_{0}}{\rangle }_{\zeta }}{\langle {x}^{n}{q}^{{a}_{0}}{\rangle }_{{\zeta }_{k}}}.\end{eqnarray}$
Another significant physical quantity associated with the a0(980)-meson PDF is its LCDA. The relationship between the a0(980)-meson leading-twist LCDA and the WF is
$\begin{eqnarray}{\phi }_{{a}_{0}}(x,\zeta )={\int }_{| {{\boldsymbol{k}}}_{\perp }{| }^{2}\leqslant {\zeta }^{2}}\displaystyle \frac{{{\rm{d}}}^{2}{{\boldsymbol{k}}}_{\perp }}{16{\pi }^{3}}{\psi }_{{a}_{0}}(x,{{\boldsymbol{k}}}_{\perp }).\end{eqnarray}$
After integrating over the squared transverse momentum, one can get the LCDA formula $\begin{eqnarray}\begin{array}{l}{\phi }_{{a}_{0}}(x,\zeta )=\displaystyle \frac{{A}_{{a}_{0}}{m}_{q}{\beta }_{{a}_{0}}}{4\sqrt{2}{\pi }^{3/2}}\sqrt{x\bar{x}}{\varphi }_{{a}_{0}}(x)\\ \quad \times \left\{\mathrm{Erf}\left[\sqrt{\displaystyle \frac{{m}_{q}^{2}+{\zeta }^{2}}{8{\beta }_{{a}_{0}}^{2}x\bar{x}}}\right]-\mathrm{Erf}\left[\sqrt{\displaystyle \frac{{m}_{q}^{2}}{8{\beta }_{{a}_{0}}^{2}x\bar{x}}}\right]\right\},\end{array}\end{eqnarray}$
where $\mathrm{Erf}(x)=2{\int }_{0}^{x}{{\rm{e}}}^{-{t}^{2}}{\rm{d}}x/\sqrt{\pi }$ is the error function. In order to determine the free model parameters ${A}_{{a}_{0}}$, ${\beta }_{{a}_{0}}$ and ${\alpha }_{{a}_{0}}$, we should use the ξ-moments of the a0(980)-meson leading-twist LCDA, which has the following definition $\begin{eqnarray}\langle {\xi }_{{a}_{0}}^{n}{\rangle }_{\zeta }={\int }_{0}^{1}{\xi }^{n}{\phi }_{{a}_{0}}(x,\zeta ){\rm{d}}x.\end{eqnarray}$
On the other hand, the ξ-moments can be calculated by the QCD sum rule approach within background field theory (BFTSR). The two-point correlation function is taken as $\begin{eqnarray}{{\rm{\Pi }}}_{{a}_{0}}^{(n,0)}={\rm{i}}\int {{\rm{d}}}^{4}x\,{{\rm{e}}}^{{\rm{i}}q\cdot x}\langle 0| T\{{J}_{n}^{V}(x),{J}_{0}^{S,\dagger }(0)\}| 0\rangle ,\end{eqnarray}$
with n taking the odd numbers, while the even order will vanish due to the G-parity. The currents are ${J}_{n}^{V}(x)={\bar{q}}_{1}(x)/z{\left({\rm{i}}z\cdot {D}^{\leftrightarrow }\right)}^{n}{q}_{2}(x)$ and ${J}_{0}^{S}(0)={\bar{q}}_{1}(0){q}_{2}(0)$. The detailed calculation process for the ξ-moments is given in our recent paper [79].Then, one can adopt the least squares method to fit ξ-moments $\langle {\xi }_{{a}_{0}}^{n}{\rangle }_{\zeta }$ in determining the free model parameters. The purpose of the least squares method is to obtain the optimal value of the fitting parameter θ by minimizing the likelihood function18 ) and (19 ). The value of yi and its variance σi are defined as the value of ξ-moments calculated by the QCD sum rule. Beyond that, making use of the probability density function $f(y,{n}_{d})=\left(1/{\rm{\Gamma }}\left(\tfrac{{n}_{d}}{2}\right){2}^{{n}_{d}/2}\right){y}^{{n}_{d}/2-1}{{\rm{e}}}^{-\tfrac{y}{2}}$ of χ2, one can get the goodness of fit with the following probability ${P}_{{\chi }^{2}}$ 18 ) under corresponding scales can be obtained. Finally, according to the wave functions under different scales, we can calculate the a0(980)-meson valence quark distribution function, Mellin moments $\langle {x}^{n}{q}^{{a}_{0}}{\rangle }_{\zeta }$ and ratio ${x}_{\,{a}_{0}}^{n}(\zeta ,{\zeta }_{k})$ with different scales by using equations (2 ), (15 ) and (16 ).
$\begin{eqnarray}{\chi }^{2}(\theta )=\displaystyle \sum _{{\rm{i}}=1}^{N}\displaystyle \frac{{y}_{{\rm{i}}}-\mu ({x}_{{\rm{i}}},\theta )}{{\sigma }_{{\rm{i}}}^{2}},\end{eqnarray}$
where μ(xi, θ) is the a0(980)-meson ξ-moments $\langle {\xi }_{{a}_{0}}^{n}{\rangle }_{\zeta }$ of combining equations ( $\begin{eqnarray}{P}_{{\chi }^{2}}={\int }_{0}^{1}({f}_{y};{n}_{d}){\rm{d}}y\end{eqnarray}$
with ${P}_{{\chi }^{2}}\in (0,1)$. The closer the goodness of fit is to 1, the better the parameters are obtained. Incorporating the effect of scale ζ, according to the renormalization group equations of Gegenbauer moments of the a0(980)-meson leading-twist LCDA, $\begin{eqnarray}{a}_{n}^{{a}_{0}}(\zeta )={a}_{n}^{{a}_{0}}({\zeta }_{0}){E}_{n}(\zeta ,{\zeta }_{0}),\end{eqnarray}$
where ${E}_{n}(\zeta ,{\zeta }_{0})={[{\alpha }_{s}(\zeta )/{\alpha }_{s}({\zeta }_{0})]}^{-({\gamma }_{n}^{(0)}+4)/b}$ and the coefficient b = (33 − 2nf)/3 [46]. nf is the number of active quark flavors. The ζ0 and ζ are considered as the initial scale and the running scale. Here we make a notation that the ξ-moments can translate into a an-moment directly. The one-loop anomalous dimension is $\begin{eqnarray}{\gamma }_{n}^{(0)}={C}_{F}\left[1-\displaystyle \frac{2}{(n+1)(n+2)}+4\displaystyle \sum _{j=2}^{n+1}\displaystyle \frac{1}{j}\right]\end{eqnarray}$
with CF = 4/3. Then, one can gain the ξ-moments at the arbitrary scales ζ. Then, by fitting moments $\langle {\xi }_{{a}_{0}}^{n}{\rangle }_{\zeta }$ under the different scale ζ with the least square method, LCDA i.e, equation (3. Numerical analysis
To do the numerical analysis, the following input parameters are used. The mass of a0(980)-meson is taken as ${m}_{{a}_{0}}=0.980\pm 0.020\,\mathrm{GeV}$. The current light quark–mass, charm quark mass, the values of the non-perturbative vacuum condensates, the continuum threshold s0 and the corresponding Borel windows used in ξ-moments BFTSR are consistent with our previous work [79]. Generally, we can treat it as the constituent quark mq for the a0(980)-meson with the quark component $q\bar{q}$. In this paper, we take three typical scales, the initial scales ζ0 = 1.0 GeV, the processes scale ζ2 = 2.0 GeV and the scale for π-nucleon Drell–Yan experiment [80] or the E615 experiment ζ5 = 5.2 GeV, which agree with the pion cases [34].
Firstly, based on the sum rule for distribution amplitudes moments $\langle {\xi }_{{a}_{0}}^{n}{\rangle }_{\zeta }$ calculated in our previous work [79], we list the a0(980)-meson leading-twist LCDA ξ-moments $\langle {\xi }_{{a}_{0}}^{n}{\rangle }_{\zeta }$ with three different scales ζ = (1.0, 2.0, 5.2)GeV in table 1. Here, the accuracy of our calculation is up to the 9th order. It can be seen that the absolute value of ξ-moments decreases as the scale ζ increases. Secondly, the absolute value of ξ-moments decreases as the n increases, which shows that our calculation has good convergence. Then, we adopt the least squares method to fit the ξ-moments $\langle {\xi }_{{a}_{0}}^{n}{\rangle }_{\zeta }$. At the same time, the a0(980)-meson twist-2 distribution amplitudes with different constituent quarks mq = (200, 250, 300, 350)MeV are shown in figure 1. The result shows that constituent quarks mq have a certain influence on distribution amplitudes. It is taken to be 250 MeV in the invariant meson mass scheme [81–87] or 330 MeV in the spin-averaged meson mass [89–93]. In this paper, we mainly take mq = 250 MeV. Then, the fitting model parameters with the different scale ζ are given in table 2. Based on the experience of other mesons [69–72, 76, 77, 88, 94, 95], we take the WF model parameter ${\beta }_{{a}_{0}}=0.5$. Obviously, ${A}_{{a}_{0}}$ gradually decreases with the increment of the scale ζ. However, the goodness of fit ${P}_{{\chi }_{\min }^{2}}$ is not very well when the scale is higher, such as ζ = 5.2 GeV. The reason may lie in the higher-order Gegenbauer moments taking a higher contribution with a larger scale.
Table 1. The a0(980)-meson leading-twist LCDA moments $\langle {\xi }_{{a}_{0}}^{n}{\rangle }_{\zeta }$ at scales ζ = (1.0, 2.0, 5.2) GeV. |
| ζ0 | ζ2 | ζ5 | |
|---|---|---|---|
| $\langle {\xi }_{{a}_{0}}^{1}{\rangle }_{\zeta }$ | −0.309 ± 0.043 | −0.214 ± 0.029 | −0.159 ± 0.022 |
| $\langle {\xi }_{{a}_{0}}^{3}{\rangle }_{\zeta }$ | −0.184 ± 0.032 | −0.093 ± 0.016 | −0.049 ± 0.009 |
| $\langle {\xi }_{{a}_{0}}^{5}{\rangle }_{\zeta }$ | −0.082 ± 0.027 | −0.057 ± 0.017 | −0.040 ± 0.011 |
| $\langle {\xi }_{{a}_{0}}^{7}{\rangle }_{\zeta }$ | −0.053 ± 0.025 | −0.037 ± 0.015 | −0.026 ± 0.010 |
| $\langle {\xi }_{{a}_{0}}^{9}{\rangle }_{\zeta }$ | −0.043 ± 0.034 | −0.014 ± 0.011 | 0.001 ± 0.005 |
Table 2. The LCHO model parameters ${A}_{{a}_{0}}$ (in unit: GeV−1), ${\beta }_{{a}_{0}}$ (in unit: GeV), ${\alpha }_{{a}_{0}}$ and goodness of fit ${P}_{{\chi }_{\min }^{2}}$ changed with the factorization scales ζ = (1.0, 2.0, 5.2) GeV. |
| ζ | $\,{A}_{{a}_{0}}$ | ${\beta }_{{a}_{0}}$ | ${\alpha }_{{a}_{0}}\,\,\,\,\,\,\,$ | ${P}_{{\chi }_{\min }^{2}}$ |
|---|---|---|---|---|
| 1.0 | −203 | 0.5 | −0.55 | 0.767 |
| 2.0 | −371 | 0.5 | −0.07 | 0.865 |
| 5.2 | −1670 | 0.5 | 0.87 | 0.113 |
With the resultant LCHO model parameters, the curves of a0(980)-meson leading-twist LCDA with three scales ζ are shown in figure 2. The figure shows that
| • | The behavior of the three curves tends to be antisymmetric, which will equal zero when the LCDA integrates with respect to x, e.g. $\begin{eqnarray}{\int }_{0}^{1}{\rm{d}}x{\phi }_{{a}_{0}}(x,\zeta )=0.\end{eqnarray}$ Meanwhile, the three curves go through the zero at the location x = 0.5. |
| • | The absolute value of the peaks is decreased with the increase of ζ and the x-location of the peaks tends toward 0.5 with the ζ increase. When the scale tends to infinity i.e. ζ → ∞ , the curve of a0(980)-meson LCDA will tend to asymptotic form ${\phi }_{{a}_{0}}(x,\infty )=0$. |
Figure 1. The a0(980)-meson LCDA with different quark masses mq = (200, 250, 300, 350) MeV. |
Figure 2. The a0(980)-meson leading-twist LCDA ${\phi }_{{a}_{0}}(x,\zeta )$ changed with three different scales ζ = (1.0, 2.0, 5.2) GeV. |
Secondly, after taking the LCHO parameters into the a0(980)-meson valence-quark distribution function, e.g. Equation (14 ), the predictions of ${q}^{{a}_{0}}(x,\zeta )$ can be obtained. The curves of a0(980)-meson valence-quark distribution function ${q}^{{a}_{0}}(x,\zeta )$ and ${{xq}}^{{a}_{0}}(x,\zeta )$ with different scales ζ are shown in figure 3, which shows that the value of peaks decreases with the increase of scale ζ. Since the a0(980)-meson leading-twist LCDA is antisymmetric behavior under the u → (1 − u) interchange in the SUf(3) limit, its valence-quark distribution function ${{xq}}^{{a}_{0}}(x,\zeta )$ tends to zero at x = 0.5. Additionally, the valence–quark distribution function tends toward bimodal behavior. In general, the valence-quark distribution functions of pion and kaon tend to a unimodal behavior [34].
Figure 3. The a0(980)-meson valence quark distribution function ${q}^{{a}_{0}}(x,\zeta )$ and ${{xq}}^{{a}_{0}}(x,\zeta )$ with different scales. |
Using the meson's valence quark distribution function, we can get the Mellin moments $\langle {x}^{n}{q}^{{a}_{0}}{\rangle }_{\zeta }$ of the a0(980)-meson valence-quark distribution function, which are presented in table 3. From the table, we can see that the Mellin moments convergence with the order n increased. Meanwhile, the Mellin moments' convergence with the scale ζ increased. This agrees with the Mellin moments of pion's valence-quark distribution function decreasing with the increase of scale. The greater ζ, the smaller the value of the moments in [34]. It proves our prediction of the Mellin moments $\langle {x}^{n}{q}^{{a}_{0}}{\rangle }_{\zeta }$ is reasonable.
Table 3. The Mellin moments for a0(980)-meson leading-twist distribution function $\langle {x}^{n}{q}^{{a}_{0}}{\rangle }_{\zeta }$ with different scales ζ = (1.0, 2.0, 5.2) GeV. |
| ζ | $\langle {{xq}}^{{a}_{0}}{\rangle }_{\zeta }$ | $\langle {x}^{2}{q}^{{a}_{0}}{\rangle }_{\zeta }$ | $\langle {x}^{3}{q}^{{a}_{0}}{\rangle }_{\zeta }$ |
|---|---|---|---|
| 1.0 | 0.102 | 0.078 | 0.065 |
| 2.0 | 0.045 | 0.032 | 0.025 |
| 5.2 | 0.027 | 0.018 | 0.013 |
Finally, we also calculate the ratio ${x}_{\,{a}_{0}}^{n}(\zeta ,{\zeta }_{k})$ of Mellin moments changed with the scale ζ. The predictions of the ratio of Mellin moments with three fixed scales ζk = (1.0, 2.0, 5.2)GeV and different orders n = (1, 2, 3) are depicted in figure 4. It is obvious that the ${x}_{\,{a}_{0}}^{n}(\zeta ,{\zeta }_{k})$ are increased with index n before the point of ζk, and decreased with n after ζk. The curves of ${x}_{\,{a}_{0}}^{n}(\zeta ,{\zeta }_{k})$ are decreasing as the ζ increases. The curves will coincide with each other when the scale ζ and ζk tend to infinity.
Figure 4. The predicted ratio of Mellin moments of the a0(980)-meson valence-quark distribution function ${x}_{\,{a}_{0}}^{n}(\zeta ,{\zeta }_{k})$ with three fixed scales ζk = (1.0, 2.0, 5.2) GeV changed with arbitrary scales ζ in the range ζ = [1, 6] GeV. The n is taken as n = (1, 2, 3) respectively. |
4. Summary
In this paper, we fit moments $\langle {x}_{{a}_{0}}^{n}{\rangle }_{\zeta }$ with the least squares method to obtain the free model parameters ${A}_{{a}_{0}}$, ${\beta }_{{a}_{0}}$ and ${\alpha }_{{a}_{0}}$ at the scales ζ = (1.0, 2.0, 5.2)GeV. Meanwhile, the goodness of fit ${P}_{{\chi }_{\min }^{2}}$ is also given. Then, we present the curves of a0(980)-meson leading-twist LCDA shown in figure 2. After constructing the relationship between a0(980)-meson leading-twist WF and PDFs, the a0(980)-meson valence quark distribution function ${q}^{{a}_{0}}(x,\zeta )$ and ${{xq}}^{{a}_{0}}(x,\zeta )$ with different scales are shown in figure 3, which tends to bimodal behavior. The LCDA and PDFs tend to zero at the location x = 0.5 due to the antisymmetry of the WF. Based on the a0(980)-meson valence quark distribution function, we can get the first three order Mellin moments $\langle {{xq}}^{{a}_{0}}{\rangle }_{\zeta }$ of the a0(980)-meson valence quark DF shown in table 3. Referring to the predicted pion's Mellin moments, our predicted result is quite reasonable. At the same time, we also give the ratio ${x}_{\,{a}_{0}}^{n}(\zeta ,{\zeta }_{k})$ of Mellin moments with ζ = (1.0, 2.0, 5.2)GeV shown in figure 4. The ratio ${x}_{\,{a}_{0}}^{n}(\zeta ,{\zeta }_{k})$ shows a downward tendency with the increase of ζ.