An improved light-cone harmonic oscillator model for the pionic leading-twist distribution amplitude

To derive the sum rules for the pionic leading-twist DA moments $\langle\xi^{n}\rangle_{2;\pi}|_{\mu}$, we adopt the following correlation function,

where $z^{2}=0$, $n=(0,2,4,\cdots)$ since the odd moments vanish due to the isospin symmetry, and the currents

In physical region, the correlation function (1) can be calculated by inserting a complete set of intermediate hadronic states. Combining the definition

and the quark-hadron duality, the hadron expression of (1) can be obtained as

where $m_{\pi}$ is the pion mass, $f_{\pi}$ is the decay constant, $s_{\pi}$ stands for the continuum threshold. In Eq. (4), the moments $\langle\xi^{n}\rangle_{2;\pi}|_{\mu}$ are defined with the pionic leading-twist DA $\phi_{2;\pi}(x,\mu)$ as following:

In the deep Euclidean region, we apply the operator product expansion (OPE) for the correlation function Eq. (1). The corresponding calculation is performed in the framework of BFTSR. For the basic assumption of BFTSR, the corresponding Feynman rules, and the OPE calculation technology, one can find in Refs. Zhong:2014jla ; Huang:1989gv for detailed discussion.

The hadron expression of correlation function (1) in the physical region and its OPE in deep Euclidean region can be matched with the dispersion relation. After applying the Borel transformation for both sides, the sum rules for the moments of the pionic leading-twist DA $\phi_{2;\pi}(x,\mu)$ can be obtained as:

Where $M$ is the Borel parameter, and for those vacuum condensates, we have taken:

and with $\langle\bar{s}s\rangle/\langle\bar{q}q\rangle=\kappa$,

In the OPE calculation for the correlation function (1), we have corrected the mistake of a vacuum matrix element, $\langle 0|G^{A}_{\mu\nu}G^{B}_{\rho\sigma;\lambda\tau}|0\rangle$, used in the previous work Zhong:2014jla . That is

It needs to be noted that, by taking $n=0$ in Eq. (6) and considering the normalization of the pionic leading-twist DA $\phi_{2;\pi}(x,\mu)$, one can obtain the $0^{\rm th}$ moment

Therefore, in many QCD sum rules calculation people usually substitute Eq. (9) as input directly into sum rules (7), and take Eq. (7) as the sum rules of the moments $\langle\xi^{n}\rangle_{2;\pi}|_{\mu}$. This will bring extra deviation to the predicted values of $\langle\xi^{n}\rangle_{2;\pi}|_{\mu}$, the reason is that the $0^{\rm th}$ moment $\langle\xi^{0}\rangle_{2;\pi}|_{\mu}$ in the l.h.s. of Eq. (7) is not strict that in Eq. (9). By taking $n=0$ in Eq. (7), one can obtain the sum rule of $\langle\xi^{0}\rangle_{2;\pi}|_{\mu}$,

Obviously, $\langle\xi^{0}\rangle_{2;\pi}|_{\mu}$ in the l.h.s. of sum rule (7) can not be normalized in the whole Borel parameter regions. The reason is that our calculation is not complete. The high-order corrections and high-dimensional corrections have not been calculated, and which are also impossible to calculate completely. In fact, the authors of Ref. Xiang:1984hw discovered this more than $30$ years ago. They obtain $\langle\xi^{0}\rangle_{2;\pi}|_{\mu}\simeq 0.83$, and take $f_{\pi}\langle\xi^{0}\rangle_{2;\pi}|_{\mu}$ as normalization factor to calculate the values of $\langle\xi^{2}\rangle_{2;\pi}|_{\mu}$ and $\langle\xi^{4}\rangle_{2;\pi}|_{\mu}$. In this paper, we argue that we need to further consider the impact of the sum rule of $\langle\xi^{0}\rangle_{2;\pi}|_{\mu}$, Eq. (10), in the full Borel parameter regions when using sum rule (7) to calculate $\langle\xi^{n}\rangle_{2;\pi}|_{\mu}$. Therefore, in order to obtain more accurate moments $\langle\xi^{n}\rangle_{2;\pi}|_{\mu}$, we suggest the following form:

Meanwhile, another advantage of Eq. (11) is that it can also eliminate some systematic errors caused by the continuum state, the absence of high dimensional condensates, the selection and determination of various input parameters.

It should be mentioned that Eq. (10) is usually used to predict the pion decay constant $f_{\pi}$ on the premise that $\langle\xi^{0}\rangle_{2;\pi}|_{\mu}\equiv 1$. After the previous discussion, we think that the sum rule of $\langle\xi^{0}\rangle_{2;\pi}|_{\mu}$ varies with Borel parameter $M^{2}$, especially when the $f_{\pi}$ has a definite experimental value. In order to ensure QCD sum rule’s prediction ability on other meson decay constant, we need to assume that $\langle\xi^{0}\rangle_{2;\pi}|_{\mu}$ can be normalized in a appropriate Borel window.

Based on the BHL-description BHL , the LCHO model of the pion leading-twist WF has raised in Refs. Wu:2010zc ; Wu:2011gf , and its form is:

where $\textbf{k}_{\bot}$ is the pionic transverse momentum, $\lambda_{1}$ and $\lambda_{2}$ are the helicities of the two constituent quark. $\chi_{2;\pi}^{\lambda_{1}\lambda_{2}}(x,\textbf{k}_{\bot})$ stands for the spin-space WF that comes from the Wigner-Melosh rotation, whose explicit form for different $\lambda_{1}\lambda_{2}$ are exhibited in Table 1, which can also been seen in Refs. Huang:1994dy ; Cao:1997hw ; Huang:2004su ; Wu:2005kq .

indicates spatial WF, where $\bar{x}=1-x$, $A_{2;\pi}$ is the normalization constant, the $\textbf{k}_{\bot}$-dependence part of the spatial WF $\Psi^{R}_{2;\pi}(x,\textbf{k}_{\bot})$ comes from the approximate bound-state solution in the quark model for pion WF_restframe and determine the WF’s transverse distribution via the harmonious parameter $\beta_{2;\pi}$, the $x$-dependence part $\varphi_{2;\pi}(x)$ dominates the WF’s longitudinal distribution. In principle, the spatial WF $\Psi^{R}_{2;\pi}(x,\textbf{k}_{\bot})$ should include a Jacobi factor. Numerical calculation in Sec. IIIC will show that the Jacobi factor has a little effect on the behavior of the the pionic leading-twist DA. In Table 1 and Eq. (13), $m_{q}$ stands for the mass of the constitute quark $u$ and $d$ in pion. In our previous work Zhong:2015nxa , the experimental data of the pion-photon transition form factor reported by the CELLO, CLEO, BABAR and BELLE collaborators based on the LCHO model with the longitudinal distribution function $\varphi^{\rm I}_{2;\pi}(x)$ (see Eq. (18)) have been fit by adopt the least square method. Where we take the constituent quark mass $m_{q}$ and the model parameter $B$ as the fitting parameters, and obtain $m_{q}=(216,246,347,222)~{}{\rm MeV}$ for CELLO, CLEO, BABAR and BELLE data, respectively. The corresponding goodness-of-fit are $P_{\chi^{2}_{\rm min}}/n_{d}=(0.187/3,0.986/13,0.416/15,0.958/13)$. Then we take $m_{q}=200~{}{\rm MeV}$ in this paper. Otherwise, $m_{q}$ is taken to be $250\rm MeV$ in the invariant meson mass scheme Terentev:1976jk ; Jaus:1989au ; Jaus:1991cy ; Chung:1988mu ; Choi:1997qh ; Schlumpf:1994bc ; Cardarelli:1994yq and $330\rm MeV$ in the spin-averaged meson mass scheme Dziembowski:1986dr ; Dziembowski:1987zp ; Ji:1990rd ; Ji:1992yf ; Choi:1996mq . Therefore, we will discuss the impact of different values of $m_{q}$ on the behavior of our pionic leading-twist DA in detail by taking $m_{q}=200\sim 350~{}{\rm MeV}$.

Using the relationship between the pionic leading-twist DA and WF,

the leading-twist DA for pion, $\phi_{2;\pi}(x,\mu)$, can be obtained. That is, after integrating over the transverse momentum $\mathbf{k}_{\bot}$ in Eq. (14), we have

where ${\rm Erf}(x)=2\int^{x}_{0}e^{-t^{2}}dx/{\sqrt{\pi}}$ is the error function. The error function part in Eq. (15) comes from the $\textbf{k}_{\bot}$-dependence part of the WF $\Psi_{2;\pi}(x,\textbf{k}_{\bot})$ and gives a good endpoint behavior for $\phi_{2;\pi}(x,\mu)$, and $\varphi_{2;\pi}(x)$ dominates the broadness of $\phi_{2;\pi}(x,\mu)$. Obviously, the specific form of $\phi_{2;\pi}(x,\mu)$ is determined by the parameters $A_{2;\pi}$, $\beta_{2;\pi}$ and the function $\varphi_{2;\pi}(x)$. There are two important constraints BHL which can be used to constrain the parameters $A_{2;\pi}$ and $\beta_{2;\pi}$, that is,

• (1)

  the WF normalization condition provided from the process $\pi\to\mu\nu$,

  $\displaystyle\int^{1}_{0}dx\int\frac{d^{2}\textbf{k}_{\bot}}{16\pi^{3}}\Psi(x,\textbf{k}_{\bot})=\frac{f_{\pi}}{2\sqrt{6}};$

  (16)

• (2)

  the sum rule derived from $\pi^{0}\to\gamma\gamma$ decay amplitude,

  $\displaystyle\int^{1}_{0}dx\Psi(x,\textbf{k}_{\bot}=\textbf{0})=\frac{\sqrt{6}}{f_{\pi}}.$

  (17)

Then the pionic leading-twist DA $\phi_{2;\pi}(x,\mu)$ only depends on the mathematical form of $\varphi_{2;\pi}(x)$. By solving the renormalization group equation of the pionic leading-twist DA, $\phi_{2;\pi}(x,\mu)$ can be written as the expansion form of the Gegenbauer series Lepage:1979zb ; Efremov:1979qk . Based on this, in our previous paper, $\varphi_{2;\pi}(x)$ is taken to be the linear superposition of the first several Gegenbauer polynomials. For example, in Refs. Wu:2012kw ; Huang:2013gra ; Huang:2013yya ; Zhong:2015nxa , we take

and

is adopted in Ref. Zhong:2014jla . For the former, when the value of parameter $B$ changes from $0.0$ to $0.6$, the pionic leading-twist DA model, i.e. Eq. (15) can mimic the DA behavior from asymptotic-like to CZ-like. And for the latter, we further consider the correction of $4^{\rm th}$ order Gegenbauer polynomial. The mathematical form of $\varphi_{2;\pi}(x)$ can usually be determined in two ways. The first one is to extract $\varphi_{2;\pi}(x)$ from the experimental data of the exclusive processes involving pion Wu:2012kw ; Huang:2013gra ; Huang:2013yya ; Zhong:2015nxa , such as semi-leptonic decays $B\to\pi\ell\nu_{\ell}$ and $D\to\pi\ell\nu_{\ell}$, the pion-photon transition form factor $F_{\pi\gamma}(Q^{2})$, and the exclusive process $B^{0}\to\pi^{0}\pi^{0}$, etc.; the second one is to determine from the moments $\langle\xi^{n}\rangle_{2;\pi}|_{\mu}$ or the Gegenbauer moments $a^{2;\pi}_{n}(\mu)$ of $\phi_{2;\pi}(x,\mu)$. In Ref. Zhong:2014jla , we have adopted the second method to determine the mathematical form of $\varphi_{2;\pi}(x)$ and further the behavior of $\phi_{2;\pi}(x,\mu)$.

In this paper, we will still make use of the second method mentioned above to determine the behavior of $\phi_{2;\pi}(x,\mu)$, but we will improve it. The accuracy of the behavior of $\phi_{2;\pi}(x,\mu)$ obtained by this method is restricted by two aspects: the rationality of the constructed mathematical form of $\varphi_{2;\pi}(x)$ and the accuracy of moments. In order to get better mathematical form of $\varphi_{2;\pi}(x)$, a natural idea is to add higher order Gegenbauer polynomial correction in $\varphi^{\rm II}_{2;\pi}(x)$, as we have done for the $D,\eta_{c},B_{c},\eta_{b}$ twist-2, 3 DAs in Refs. Zhong:2014fma ; Zhong:2016kuv ; Zhang:2017rwz ; Zhong:2018exo . However, such improvement obviously destroys the beauty and conciseness of the model. Otherwise, we find that the parameters $B_{2},\ B_{4}$ are close to the Gegenbauer moments $a_{2}^{2;\pi}(\mu),\ a_{4}^{2;\pi}(\mu)$ respectively. From the relationship between $\langle\xi^{n}\rangle_{2;\pi}|_{\mu}$ and $a_{n}^{2;\pi}(\mu)$, it can be seen that the reliability of $a_{n}^{2;\pi}(\mu)$ calculated by QCD sum rules decreases sharply with the increase of order-$n$. In view of this, in this paper we will improve the mathematical form of $\varphi_{2;\pi}(x)$ by other way, as well as propose a new determination method of model parameters.

We notice that although it is difficult to improve pionic leading-twist DA by introducing higher Gegenbauer polynomial correction, our goal is still to make it more reasonable and accurate by adjusting the behavior of $\phi_{2;\pi}(x,\mu)$. We find that the factor $\sqrt{x\bar{x}}$ in Eq. (15) can regulate DA’s behavior to some extent. Inspired by this, we introduce a factor $\left[x\bar{x}\right]^{\alpha_{2;\pi}}$ into WF’s longitudinal distribution function $\varphi_{2;\pi}(x)$, i.e.,

In order to further apply our LCHO model to other meson DA, and combine the form of $\varphi^{\rm I}_{2;\pi}(x)$, we propose a more complex form,

the parameters $\alpha_{2;\pi}$ and $\hat{a}^{2;\pi}_{2}$ will be determined by fitting the moments $\langle\xi^{n}\rangle_{2;\pi}|_{\mu}$ directly through the method of least squares, and the values of moments $\langle\xi^{n}\rangle_{2;\pi}|_{\mu}$ come from Eq. (11) calculated under BFTSR in Sec. II.1. In order to distinguish our LCHO model with $\varphi^{\rm III}_{2;\pi}(x)$ and $\varphi^{\rm IV}_{2;\pi}(x)$, and facilitate the discussion later, we will record the former as LCHO model-III and the latter as LCHO model-IV.

Considering a set of $N$ independent measurements $y_{i}$ with the known variance $\sigma_{i}$ and the mean $\mu(x_{i};\mathbf{\theta})$ at known points $x_{i}$. The objective of the least squares method is to obtain the best value of fitting parameters $\mathbf{\theta}$ by minimizing the likelihood function PDGnew

As for the present case, the function $\mu(x_{i};\mathbf{\theta})$ indicates the pionic leading-twist DA moments $\langle\xi^{n}\rangle_{2;\pi}|_{\mu}$ defined by combining Eqs. (6), (15), (20) and $\mathbf{\theta}=(\alpha_{2;\pi},\hat{a}^{2;\pi}_{2})$; The theoretical values of $\langle\xi^{n}\rangle_{2;\pi}|_{\mu}$ calculated by QCD sum rules in next section are assumed to be the value of $y_{i}$ and its variance $\sigma_{i}$. The probability density function of $\chi^{2}$ can be obtained,

$n_{d}$ is the number of degree-of-freedom. Then one can further calculate the following probability,

The magnitude of the probability $P_{\chi^{2}}$ ($P_{\chi^{2}}\in[0,1]$) can be used to judge the goodness-of-fit, when its value is closer to $1$, a better fit is assumed to be achieved.

To do the numerical calculation, we adopt the latest data from Particle Data Group (PDG) PDGnew : $m_{\pi}=139.57039\pm 0.00017~{}\textrm{MeV}$ and $f_{\pi}=130.2\pm 1.2~{}\textrm{MeV}$. The current-quark-mass for the $u,d$-quark are adopted as $m_{u}=2.16^{+0.49}_{-0.26}~{}\textrm{MeV}$ and $m_{d}=4.67^{+0.48}_{-0.17}~{}\textrm{MeV}$ at scale $\mu=2~{}\rm GeV$. Based on these latest values, we can update the vacuum condensates.

• •

  For the double-quark condensate, we adopt Gell-Mann-Oakes-Renner relation:

  	$\displaystyle m_{u}\langle\bar{u}u\rangle+m_{d}\langle\bar{d}d\rangle\simeq-\frac{f_{\pi}^{2}m_{\pi}^{2}}{2}$
  	$\displaystyle\qquad=-(1.651\pm 0.003)\times 10^{-4}~{}\textrm{GeV}^{4}.$		(25)

  Combining with the $u,d$ quark masses, we have:

  	$\displaystyle\langle\bar{q}q\rangle$	$\displaystyle=$	$\displaystyle\left(-2.417_{-0.114}^{+0.227}\right)\times 10^{-2}~{}{\rm GeV}^{3}$		(26)
  		$\displaystyle=$	$\displaystyle\left(-289.14_{-4.47}^{+9.34}\right)^{3}~{}{\rm MeV}^{3},$

  at scale $\mu=2~{}\rm GeV$.

• •

  By combining Eqs. (25), (26) and the relation $\langle g_{s}\bar{q}\sigma TGq\rangle=m_{0}^{2}\langle\bar{q}q\rangle$ with $m_{0}^{2}=0.80\pm 0.02~{}\textrm{GeV}^{2}$ Narison:2014ska , the quark-gluon mixed condensate would be

  	$\displaystyle m_{u}\langle g_{s}\bar{u}\sigma TGu\rangle+m_{d}\langle g_{s}\bar{d}\sigma TGd\rangle$
  	$\displaystyle\quad\quad\quad=-(1.321\pm 0.033)\times 10^{-4}~{}{\rm GeV}^{6},$		(27)

  $\displaystyle\langle g_{s}\bar{q}\sigma TGq\rangle$

  $\displaystyle=$

  $\displaystyle\left(-1.934^{+0.188}_{-0.103}\right)\times 10^{-2}~{}{\rm GeV}^{5}.$

  (28)

• •

  By adopting the data in Ref. Narison:2014ska ,

  $\displaystyle\rho\alpha_{s}\langle\bar{q}q\rangle^{2}=\left(5.8\pm 1.8\right)\times 10^{-4}~{}{\rm GeV}^{6},$

  (29)

  with $\rho\simeq 3-4$, and combining the value of the double-quark condensate in Eq. (26), the four-quark condensates can be obtained as:

  $\displaystyle\langle g_{s}\bar{q}q\rangle^{2}=(2.082^{+0.734}_{-0.697})\times 10^{-3}~{}\textrm{GeV}^{6},$

  (30)

  and

  $\displaystyle\langle g_{s}^{2}\bar{q}q\rangle^{2}=(7.420^{+2.614}_{-2.483})\times 10^{-3}~{}\textrm{GeV}^{6},$

  (31)

• •

  From Ref. Colangelo:2000dp , we have

  $$\langle\alpha_{s}G^{2}\rangle=0.038\pm 0.011~{}\textrm{GeV}^{4},$$

  (32)

  and

  $\displaystyle\langle g_{s}^{3}fG^{3}\rangle\simeq 0.045~{}\textrm{GeV}^{6}.$

  (33)

• •

  For the ratio $\kappa=\langle\bar{s}s\rangle/\langle\bar{q}q\rangle$, Ref. Narison:2014wqa gives:

  $\displaystyle\kappa=0.74\pm 0.03,$

  (34)

In numerical calculation for the moments’ BFTSR (11), we take the scale $\mu=M$ as usual. From $\alpha_{s}(M_{z})=0.1179\pm 0.0010$ with $M_{Z}=91.1876\pm 0.0021~{}{\rm GeV}$, and combining $\bar{m}_{c}(\bar{m}_{c})=1.27\pm 0.02~{}{\rm GeV}$ and $\bar{m}_{b}(\bar{m}_{b})=4.18^{+0.03}_{-0.02}~{}{\rm GeV}$ PDGnew , under the 3-loop approximate solution we predict $\Lambda_{\rm QCD}^{(n_{f})}\simeq 324,286,207~{}{\rm MeV}$ for the number of quark flavors $n_{f}=3,4,5$, respectively.

The renormalization group equations (RGE) of the quark mass and vacuum condensates are given as Yang:1993bp ; Hwang:1994vp ; Lu:2006fr :

with $\beta_{0}=(33-2n_{f})/3$. Obviously, the double-gluon condensate and the triple-gluon condensate are energy scale independent. From Eq. (8), one can find that $\langle g_{s}^{2}\bar{q}q\rangle^{2}$ and $\langle g_{s}^{3}fG^{3}\rangle$ have the same RGE. In other words, $\langle g_{s}^{2}\bar{q}q\rangle^{2}$ is also energy scale independent, e.g.,

Combining with the RGE of the double-quark condensate and Eq. (36), one can find that the $\langle g_{s}\bar{q}q\rangle^{2}$ and $\langle\bar{q}q\rangle$ have the same energy scale evolution equation, e.g.,

It should be noted that, according to the basic assumption of BFTSR, $g_{s}$ in all the above vacuum condensates is the “coupling constant” between the background fields, which is different from the one in pQCD, and should be absorbed into vacuum condensates as part of these non-perturbative parameters.

The RGE of the Gegenbauer moments of the pion leading-twist distribution amplitude is: