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Home » Dino Zoff coaches youngster Donnarumma
Dino Zoff coaches youngster Donnarumma
Top storiesxfcediJune 2, 2017
Italian legend Dino Zoff is ready to give in demand goalkeeper Gianluigi Donnarumma some advice to serve him on his career path.
Dino Zoff may well be football's most famous goalkeeper. Zoff played for the Italian national side 112 times, broke many records and won most trophies available in the game. Now many see Gianluigi Donnarumma as his natural successor.
Donnarumma is only 18-years-old and already sparking the interest of Europe's best clubs. The goalkeeper plays for AC Milian, which by his own admission represents his childhood dream. However, his contract is set to expire in less than a year. The player's agent has made it clear he intends to choose the best financial option for his client. But Zoff thinks this is the wrong way to go about things.
The former Italian captain advises Donnarumma not to think about the money, but rather concentrate on fulfilling his potential. "Donnarumma is so young and strong that he should relax and not already be thinking about where to go with a bit more or a bit less money," Zoff tells reporters at Rai Radio 1. "The important thing is that he gets some experience now, because his prospects look very positive."
It is unclear what Donnarumma himself thinks, although his agent's opinion has been well documented in recent months. The Italian goalie got his start at Milan at only 16 years of age. In the current season he has played every single game of the Serie A campaign. Milan has certainly been in a down-cycle in the last few years, but with expected investments there is hope that the team could return to its former glory.
The player's agent, Mino Raiola, says that there are plenty of offers, 11 in fact, for Donnarumma. Raiola even compared his young client to Maradona."Gigio is the Maradona of goalkeepers," Raiola told Corriere dello Sport. "He can only grow and he has to be left alone to grow in peace. He is not a slave, nor a robot."
"There are 11 top clubs which are interested in him. The concept is clear. Donnarumma's contract will expire on June 30, 2018. I've never said to Milan, 'He will sign now or not at all'."
"I understand the stress and the expectations of the fans, Gigio is already a symbol for the club and for the new owners it would be ideal to seal a renewal today. Yesterday, in fact."
Raiola says that chances for Donnarumma to sign an extension on his contract with Milan ar even to those of him leaving. The agent believes that wages will not be the only factor contributing to his choice.
"We are talking about an 18-year-old lad, a phenomenon who is already being called to make a fundamental decision in his life. It's a decision which will be weighed-up and, certainly, won't depend only on wages."
"People say Donnarumma should be grateful to Milan. It is true, but Milan should also be grateful to Donnarumma. And to Sinisa Mihajlovic [for giving him his debut]."
AC MilanDino ZoffGianluigi DonnarummaSerie A
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Slovenian writer Brina Svit brings up her discovery of the French language, thanks to her French teacher in Ljubjana who taught her this language through the study of plays, as well as her arrival in Paris.
Brina Svit (born in 1954), Slovenian writer, arrived in Paris in 1980 to continue her studies of French literature that she began in the Slovenia.
A journalist for the Slovenian daily Delo, Scriptwriter, short film director, she published four novels written in Slovenian (of which Death of a Prima Donna in 2001) then three in French: Moreno (2003), Un coeur de trop (2006) and Coco Dias ou la Porte Dorée (2007). The latter was based on an autobiographical episode: to learn to dance the tango, the narrator, Valérie Nolo, promised a Parisian tango dancer, Orlando Dias, known as Coco Dias, to exchange dance lessons against a book about him. Between Paris and Buenos Aieres, this romanticised biography evokes the art of abrazo, entwined dancers, harmony between the body and soul.
Brina Svit's novels are imprinted with Slovenian nostalgia and tender humour.
I started to learn French in secondary school, and in secondary school, we had a theatrical troupe that was run by a teacher that was well known in Lubjana, a very good French teacher whose name is Mrs. Sayel and who is still alive. And so, myself, I was a member of this troupe, and we did plays in French. We rehearsed them all year long, and towards the end, we played them in front of a real audience.
I was 17 years old, and it's true that now, every time that I pass under the Pont Neuf on my bike, I think about it.
Svit, that's my stage name.
It's Slovene and it means dawn.
The time when birds stop singing.
Yes, it's the... it's actually the... the very first light of the morning.
Those were 2 biographical reference points. Bonus question: How did you wind up in France, where you've now been living for over twenty years?
It's very simple, I met my husband, I was his interpreter in Lubjana, I was 23 years old at the time. He offered to come live with me in Lubjana, but I told myself: A poor Frenchman in Lubjana, he's going to be completely lost. And I spoke French, more or less, so I don't know why, I figured that for me, leaving for Paris... I figured that it wasn't too far away, and I didn't realise the break that it was going to be for me, and that it wouldn't be as easy as that. Because in the beginning, it wasn't that easy. Not at all.


Feyenoord clocked up their first win of 2013 on Sunday at Willem II. Ronald Koeman's side were too strong for the hosts in Tilburg, running out 1-3 winners. Two second-half goals from Tonny Vilhena wrapped up the victory after a converted spot kick from Virgil Misidjan had cancelled out Graziano Pellè's early opener. The game also saw the return to action of John Goossens after months out due to injury.
Feyenoord travelled south to Tilburg still looking for their first league goal of the year, but they got it within ten minutes against the Eredivisie's bottom-placed club. Wesley Verhoek played it into Pellè in the penalty area and the Italian turned away from his marker before shooting into the far corner of the net.
It was a just reward for Feyenoord, who had started well, making it difficult for their hosts to get a firm foothold in the game. When they had the ball Feyenoord showed promise, setting up some sweet passing moves that resulted in good chances for Boëtius and Pellè. The Italian had an especially fine opportunity to double his side's lead, but he was unable to get the better of keeper David Meul in a one-on-one.
Unfortunately, the failure to add to the lead did not go unpunished. Feyenoord were warned just after the half-hour mark when Misidjan headed in only for the goal to be ruled out for a foul by Hans Mulder. It was a short reprieve, though. A little later, Bruno Martins Indi hauled down Aurélien Joachim in the box and Misidjan stepped up to strike home the resulting penalty. Bizarrely, referee Richard Liesveld yellow-carded Joris Mathijsen rather than Martins Indi.
The equaliser came seven minutes before the break and might have given the hosts renewed confidence at the beginning of the second half. But it was Feyenoord that came out with all guns blazing, notching up two goals within the first quarter of an hour, both of them courtesy of Vilhena, who was having a very good game.
Both came at the end of a great attacking move, but it was the first of the two that really caught the eye. Ruud Vormer and Jordy Clasie got the move rolling, before Vilhena played a one-two with Pellè. The midfielder controlled the spiralling return with aplomb, volleying the ball with his left inside the area to leave keeper Meul with no chance.
After the third – Vilhena received from Boëtius and let rip with his left again into the far corner – there was still over thirty minutes to play, but Feyenoord coasted out the game. Strong defensive work kept the hosts at arm's length, which gave Koeman the chance to bring on John Goossens for his first run out for months in place of Verhoek.
Willem II: Meul; Cornelisse, Haastrup, Peters, Jallo; Hofs (66' Haemhouts), Vossebelt, Mulder; Misidjan (72' Snijders), Joachim (75' Guijt), Podevijn.
Feyenoord: Mulder; Janmaat, De Vrij, Mathijsen, Martins Indi; Clasie, Vormer, Vilhena (90' Singh); Verhoek (82' Goossens), Pellè, Boëtius.


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Home Education Kiosk Market to Reach US $120,286.9 Mn by 2026: Key Players: Fujitsu...
Kiosk Market to Reach US $120,286.9 Mn by 2026: Key Players: Fujitsu Group, Glory Ltd., GRBBanking, JCM Global, Kiosk Information Systems
Albany, NY — 02/25/2019 — According to a new report on the kiosk market published by Transparency Market Research, the global kiosk market is expected to reach a value of US$ 120,286.9 Mn by 2026, expanding at a CAGR of 6.8% from 2018 to 2026. According to the report, the global market is projected to continue to be influenced by a number of macroeconomic and industry-specific factors. In terms of type, the self-service kiosk segment is anticipated to expand in the coming years.
The demand for kiosk has increased significantly in recent years. The retail, education, and transportation sectors are prominent end-users of kiosks. The demand for kiosks in the e-commerce and retail sectors is expected to rise during the forecast period. There has been a substantial increase in the deployment of self-service kiosks/interactive kiosks.
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This is attributed to the simplified operations of these kiosks. In the advertising sector, the demand for information kiosks is anticipated to be significant. In the transportation sector, the demand for interactive kiosks has increased over the past few years, especially within airports. The trend is expected to continue during the forecast period. These kiosks play a key role in speeding up processes related to enquiry and help avoid delays and long queues at airports. Check-in and baggage handling activities are expected to increase during the forecast period. Kiosks help passengers and airlines to speed up processes and reduce workload.
In terms of region, Asia Pacific held a prominent share of the kiosk market in 2017. North America constitutes a key market share, and the market in the region is anticipated to expand at a robust pace during the forecast period. The market in Middle East & Africa and South America is expected to expand at a steady growth rate as compared to the market in other regions of the world.
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The global kiosk market has been segmented based on type, screen size, screen type, end-use, and geography. Demand for drink and food vending machines is expected to rise significantly during the forecast period. Japan is estimated to have the highest density of vending machines. Vending machines are currently undergoing a digital transformation. With the advent of Internet of Things (IoT), the demand for connected vending machines has increased significantly. Demand for healthy vended food, advanced technology, and reduced equipment costs are key factors driving demand for vending machines. Asia Pacific and North America are expected to propel demand for vending machines over the forecast period. Demand for ATMs is projected to continue to be significant in the banking and financial services sector. The demand for ATMs is expected to be high in the regions of North America and Asia Pacific in the coming years.
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Preety holds a Master's in Journalism from Ryerson University and writes professionally in a broad variety of genres. She has worked as a senior manager in public relations and communications for major telecommunication companies and is the former Deputy Director of Media Relations with the Modern Coalition.
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SSC, a global provider of advanced space services, is hosting a Swedish government delegation visit to California from May 23-27.
The four-member Swedish Government delegation, led by former Vice Prime Minister of Sweden, Mr. Jan Nygren, are here to meet with industry leaders to better understand the emerging space market.
Meeting the various needs of smallsat and cubesat missions is a key SSC corporate objective and is being realized in two ways: the creation of a launch capability in northern Sweden, and SSC Infinity, a new, low-cost ground operations service offering customers full flexibility for small satellites and constellations. The company and delegation will be meeting with several industry leading companies. Attendees of Space Tech Expo who wish to meet with the delegation during the conference should contact kevin.mortensen@sscspace.com.
"Recent growth in new, smallsat and constellation-based space applications throughout the world is driving the need for a new approach to ground segment operations" said Leif Osterbo, president, SSC Satellite Management Services Division.


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The Strategy for ANDE's next decade
Since 2009, ANDE has grown into a trusted network of 300 collaborative members that operate in nearly every emerging market. Our members represent the leading entrepreneurial support organizations in the world and have been innovating in the fields of impact investment, business development services, and entrepreneurship training. We are proud of ANDE' s role in supporting our members while leading the effort define and build the Small and Growing Business (SGB) Sector in emerging markets. Monitor Deloitte completed an assessment of our impact in our first decade.
ANDE's Updated Strategy
Over the past year, ANDE undertook a comprehensive strategy review process and has developed an updated strategy to carry us into our second decade. The new strategy builds upon ANDE's core competencies and aligns with key international development goals.
Under the strategy, ANDE will continue to serve as the leading champion for the SGB Sector and help its members develop and implement high quality programs. We will focus on key sector challenges around talent development, SGB investing, and impact management.
ANDE will catalyze and implement new research that quantifies the impact of small and growing businesses on the UN Sustainable Development Goals (SDGs) and identifies interventions that help SGBs maximize their social, environmental and economic impact.
We will leverage these insights to promote the SGB sector to major stakeholders in the international development arena, with the intent of driving significant additional philanthropic and investment capital to effective SGB support activities in developing economies.
In our efforts, we will prioritize three urgent issues where SGBs can help emerging markets attain Sustainable Development Goals: decent job creation, gender equality and environmental action. We believe that entrepreneurship has been underappreciated as a tool in addressing critical environmental challenges and addressing issues related to climate change, and that women are underrepresented as entrepreneurs, investors, and leaders. We know that SGBs are a critical piece of the solution for creating the millions of jobs needed in emerging markets. We invite you to read a summary of our updated strategy. ANDE Executive Director Randall Kempner provides a deeper explanation of the strategy in this NextBillion interview.
URgENT ISSUE BRIEFS
As part of strategy development process, we conducted research to better understand how SGBs can contribute towards these global development goals. This research has been summarized in our three urgent issue briefs.
ANDE is eager to expand our work in these areas and will be launching a number of new research and sector funding efforts. We are excited to announce the first initiative is a partnership with USAID and VISA Foundation to launch the ANDE Gender Equality Initiative and the Advancing Women's Empowerment Fund (AWEF).


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Many visitors get their first taste of central Birmingham at New Street Station, whose unreconstructed ugliness – piles of modern concrete – makes a dispiriting start, though there are plans afoot to give the place a thoroughgoing face-lift. Things soon improve if you cut up east from the station to the newly developed Bull Ring, once a 1960s eyesore, but now a gleaming new shopping mall distinguished by the startling design of its leading store, Selfridges. Head west along pedestrianized New Street from here and it's a brief stroll to the elegantly revamped Victoria Square and the adjacent Chamberlain Square, where pride of place goes to the Birmingham Museum and Art Gallery, the city's finest museum, complete with a stunning collection of Pre-Raphaelite art. Beyond, further west still, is the glossy International Convention Centre, from where it's another short hop to the Gas Street Basin, the prettiest part of the city's serpentine canal system. Close by is canalside Brindleyplace, a smart, brick-and-glass complex sprinkled with slick cafés and bars and holding the enterprising Ikon Gallery of contemporary art. From Brindleyplace, it's a short walk southeast to The Mailbox, the immaculately rehabilitated former postal sorting office with yet more chic bars and restaurants, or you can head north along the old towpath of the Birmingham and Fazeley Canal as far as Newhall Street. The latter is within easy walking distance of the Georgian St Philip's Cathedral.


I adored my father. … Still, our bond was not always so tight. In fact, it was so strained at one point; I didn't speak to him for two years. I was angry for both real and perceived slights.
Return of the Prodigal Son, by Bartolomé Esteban Murillo, 1667-70, oil on canvas, National Gallery of Art, Washington.
This was a sad period. There was estrangement between me and my father, estrangement between me, my immediate family and extended family members. There were no visits on holidays, no phone calls, and certainly no "Happy Father's Day" wishes on that special Sunday. I was so certain that I was right and justified in my anger, and this probably would have continued indefinitely except something happened.


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Mom's wild stories cause her grandchildren distress
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DEAR ABBY: My mother insists on telling my three youngest children that my husband is not their father. The oldest girl is the spitting image of him, and she's upset about it. Mom also calls me terrible names. She keeps saying it will add years to my life if I divorce him and makes appointments with divorce lawyers "for" me, which I am charged for. My husband and I have a limited income and can't move away because our jobs are here. How can I convince our kids my husband is their father? — TRYING TO PROVE IT
DEAR TRYING: A way to do that would be to explain to your children that your mother has severe emotional problems and isn't in her right mind when she says those things. (From what you have written, it appears to be true.) You do not have to move away to distance yourself from this toxic, troubled woman. Stop communicating with her. Block her phone number, if you must, and do not allow her to have contact with any members of your family unless and until she regains her senses and apologizes to all of you.
DEAR ABBY: I belong to a women's golfing group. The mission of the club is to play golf and have fun. We have tournaments, prizes, and awards are given for the best scores.
The problem: Several of the ladies are "allergic" to counting their scores correctly. We have given them counting beads to help them "remember" their score. They have played with board members who asked them to count their scores out loud each time they hit the ball and to state their scores after the last putt. There have also been conversations with the golf pro about the importance of keeping accurate scores. Yet, the inaccurate counting persists and denial reigns. Members are upset because these ladies often "win" tournaments. What to do? — PROUD OF MY HIGH HANDICAP
DEAR PROUD: You might be able to curb the cheating if you suggest club members swap scorecards and keep score for each other. However, if that doesn't do the trick, stop playing with those who cheat.
DEAR ABBY: Yesterday, I took my computer to an electronics store to be fixed. The tech who helped me had a ton of dirt under his nails. I was grossed out seeing him with those filthy nails type on my computer keys. Should I have said something to him or his supervisor? How can they allow someone with his hygiene problem to be in a position that requires contact with the public? — GROSSED OUT IN OHIO
DEAR GROSSED OUT: A quiet word with the supervisor would have been the way to handle it. And while you were at the store, you could have asked for sanitary wipes to clean your keyboard. (They probably had some behind the counter.) Using a soft tissue dipped in alcohol once you returned home would also kill germs, as long as you're careful the tissue isn't so saturated that liquid drips beneath the keys.
Contact Dear Abby at www.DearAbby.com or P.O. Box 69440, Los Angeles, CA 90069.
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\section{Introduction}
Since the discoveries of $J/\psi$ and $\Upsilon$ more than thirty
years ago, a great deal of information on heavy quarkonium levels
and their transitions has been accumulated \cite{PDG08}. On one
hand, the known levels of charmonium and bottomonium include $1S$,
$1P$, $1D$, $2S$, $2P$, $2D$, $3S$, and $4S$ states, with the labels
$S$, $P$, $D$ corresponding to relative orbital angular momentum $L$
= 0, 1, 2 between quark and antiquark. On the other hand, the
numerous transitions between heavy quarkonium states are classified
into strong and radiative decays, which shed light on aspects of
quantum chromodynamics (QCD) in both the perturbative and the
non-perturbative regimes (for a recent review see \cite{EGMR}).
Recently the more precise $\psi'$, $\eta_c'$, and $\eta_b$ mass
measurements have been reported \cite{KEDR,CLEO,BABAR,BABAR1}, and
the errors of their relevant decay widths have decreased
\cite{PDG08}. A thorough understanding of their properties, such as
their quark distribution amplitudes, which are the universal
non-perturbative objects, will be of great benefit when analyzing
the hard exclusive processes with heavy quarkonium production.

It has been known that heavy quarkonium is relevant for a
non-relativistic treatment \cite{QR}. Although non-relativistic QCD
(NRQCD) is a powerful theoretical tool for separating the high
energy modes from low energy contributions, in most cases the
calculation of low energy hadronic matrix elements has relied on
model-dependent non-perturbative methods. Therefore, various methods
have been employed in heavy quarkonium physics, such as lattice QCD,
quark-potential model, etc. \cite{EGMR}. The light-front quark model
offers many insights into the internal structures of the bound
states. In this study, heavy quarkonium is explored within a quark
model on the light front. Light-front QCD is a promising analytic
method for solving the non-perturbative problems of hadron physics
\cite{BPP}, and may be the only possible method by which the low
energy quark model and the high energy parton model can be
reconciled. For hard processes with a large momentum transfer,
light-front QCD reduces to perturbative QCD (pQCD) which factorizes
the physical quantity into a convolution of the hard scattering
kernel and the quark distribution amplitudes.

The basic ingredient in light-front QCD is the relativistic hadron
wave function. It generalizes the distribution amplitudes by
including the transverse momentum distributions, and it contains all
the information of a hadron from its constituents. The hadronic
quantities are represented by the overlap of wave functions and can
be derived in principle. The light-front wave function is manifestly
a Lorentz invariant as it is expressed in terms of the internal
momentum fraction variables which are independent of the total
hadron momentum. Moreover, the fully relativistic treatment of quark
spins and the center-of-mass motion can be carried out using the
so-called Melosh rotation \cite{LFQM,LFQMa,LFQMb}. This treatment
has been successfully applied to calculate many phenomenologically
important meson decay constants and hadronic form factors
\cite{Jaus1,Jausa, CCH1, Jaus2, CCH2, Hwang}. In addition, the
covariant light-front approach \cite{Jaus2, CCH2} has also been
applied to ground-state $s$-wave mesons, which include $1^1S_0$
pseudoscalar mesons $\eta_c, \eta_b$ and $1^3S_1$ vector mesons
$J/\psi, \Upsilon$ (see Ref. \cite{hwangwei}).

As mentioned above, the quark distribution amplitude or light-cone
wave function (LCWF) absorbs the non-perturbative dynamics and is
the key ingredient of any hard exclusive process with hadron
production. In the literature, there are many theoretical studies
\cite{Braguta1,Braguta2,Braguta3,BT,EGKLY,BC,CJ,BF} of this issue.
The main purpose of this study is the calculation of the quark
distribution amplitudes of pseudoscalar and vector heavy quarkonium
states by integrating the transverse momenta of momentum
distribution amplitudes within the light-front approach. As to the
so-called $\xi$-moments which parameterize the quark distribution
amplitudes, they are also calculated analytically.

The remainder of this paper is organized as follows. Section II
comprises brief reviews of the light-front framework and the
light-front analysis for the decay constants of pseudoscalar ($P$)
and vector ($V$) mesons; the processes $V\to P\gamma$ are given. In
Section III, the quark distribution amplitudes and the $\xi$-moments
are calculated. In Section IV, the numerical results and discussions
are presented. Finally, the conclusions are given in Section V.

\section{Formalism of covariant light-front approach}
In heavy quarkonium, the valence quarks have equal masses,
$m_1=m_2=m$, with $m$ the mass of heavy quark $c$ or $b$. Thus, the
formulae in this section lead to simplifications for the quarkonium
system.

The momentum of a particle is given in terms of the light-front
component by  $k=(k^-, k^+, k_\perp)$ where $k^{\pm}=k^0\pm k^3$ and
$k_\perp=(k^1, k^2)$, and the light-front vector is written as $\tilde
k=(k^+, k_{\perp})$. The longitudinal component $k^+$ is restricted to
positive values, i.e., $k^+> 0$ for the massive particle. In this
way, the physical vacuum of light-front QCD is trivial except for
the zero longitudinal momentum modes (zero-mode). A meson with total
momentum $P$ and two constituents, quark and anti-quark whose
momenta are $p_1$ and $p_2$, respectively will be studied. In order
to describe the internal motion of the constituents, it is crucial
to introduce the intrinsic variables $(x, p_{\perp})$ through
 \begin{eqnarray}
 && p_1^+=x P^+,  ~~~~~~~~~~~  p_{1\perp}=x P_{\bot}+ p_{\perp};  \nonumber \\
 && p_2^+=(1-x) P^+,   ~~~  p_{2\perp}=(1-x) P_{\bot}- p_{\perp},
 \end{eqnarray}
where $x$ is the light-front momentum fraction. The invariant mass
$M_0$ of the constituents and the relative momentum in $z$ direction
$p_z$ can be written as
 \begin{eqnarray} \label{eq:Mpz}
  M_0^2=\frac{p_{\perp}^2+m^2}{x(1-x)}, \qquad ~~~
  p_z=\left(x-\frac{1}{2}\right)M_0.
 \end{eqnarray}
The invariant mass $M_0$ of $q\bar q$ is generally different from
the mass $M$ of meson which satisfies $M^2=P^2$. This is due to the
fact that the meson, quark and anti-quark cannot be on-shell
simultaneously. The momenta $p_{\perp}$ and $p_z$ constitute a momentum
vector $\vec p=(p_{\perp}, p_z)$, which represents the relative momenta
in the transverse and $z$ directions, respectively.  The energy of
the quark and antiquark $e_1=e_2\equiv e$ can be obtained from their
relative momenta,
 \begin{eqnarray}
 e=\sqrt{m^2+p_{\perp}^2+p_z^2}.
 \end{eqnarray}
It is straightforward to find that
 \begin{eqnarray}
 x=\frac{e-p_z}{2e},~ e=\frac{M_0}{2}.
 \end{eqnarray}

As shown in Ref.~\cite{CM69}, one can pass to the light-front
approach by integrating the $p^-$ component of the internal momentum
in the covariant Feynman momentum loop integrals. The Feynman rules
for the meson-quark-anti-quark vertices were then needed to
calculate the amplitudes which related to the decay constant and M1
transition. In the following formulations, we follow the notation in
\cite{CCH2}. The vertices $\Gamma_M$ for the incoming meson $M$ are
given as
 \begin{eqnarray} \label{eq:HM}
 H_P \gamma_5  \qquad \qquad   \qquad \qquad
 &&{\rm for~ }P, \nonumber \\
 iH_V\Big [ \gamma_{\mu}-\frac{1}{W_V}(p_1-p_2)_{\mu} \Big ]
  \qquad &&{\rm for~ }V.
 \end{eqnarray}
The Feynman rules which are derived from quantum field theory and
Eq. (\ref{eq:HM}) can be used to write down the relevant matrix
elements. The integration of the $p^-$ component will force the
quark or anti-quark to be on its mass shell. The specific form of
the covariant vertex function for the on-shell (anti)quark can be
determined by comparing it to the conventional vertex function,
which can be written as
 \begin{eqnarray}
 |M(P, ^{2S+1}L_J, J_z)\rangle
 =\int &&\{d^3p_1\}\{d^3p_2\} ~2(2\pi)^3 \delta^3(\tilde
 P -p_1-\tilde p_2)~\nonumber\\
 &&\times \sum_{\lambda_1,\lambda_2}
 \Psi^{SS_z}(\tilde p_1,\tilde p_2,\lambda_1,\lambda_2)~
 |q_1(p_1,\lambda_1) \bar q_2(p_2,\lambda_2)\rangle.
 \label{lfmbs}
 \end{eqnarray}
The momentum-space wave function $\Psi^{SS_z}$ can be expressed as
 \begin{eqnarray}
 \Psi^{SS_z}(\tilde p_1,\tilde
 p_2,\lambda_1,\lambda_2)
                = R^{SS_z}_{\lambda_1\lambda_2}(x,p_\bot)~ \phi_(x, p_\bot),
 \end{eqnarray}
where $\phi(x,p_\perp)$ is the light-front momentum distribution
amplitude for the $s$-wave meson and can be chosen to be
normalizable, i.e., it satisfies
 \begin{eqnarray} \label{eq:Norm1}
 \int \frac{dx d^2 p_{\bot}}{2 (2\pi)^3}
   |\phi(x,p_{\bot})|^2=1,
 \end{eqnarray}
and $R^{SS_z}_{\lambda_1\lambda_2}$ constructs a state of definite
spin ($S,S_z$) out of light-front helicity ($\lambda_1,\lambda_2$)
eigenstates. In practice, it is more convenient to use the covariant
form for $R^{SS_z}_{\lambda_1\lambda_2}$ \cite{Jaus1,Jausa}:
 \begin{eqnarray}
        R^{SS_z}_{\lambda_1\lambda_2}(x,k_\bot)
                =\frac{1}{\sqrt{2} M_0}
        ~\bar u(p_1,\lambda_1)\Gamma v(p_2,\lambda_2), \label{covariant}
 \end{eqnarray}
where
 \begin{eqnarray}
        &&\Gamma=\gamma_5 \qquad\qquad\qquad\qquad\qquad({\rm pseudoscalar}, S=0), \\
        &&\Gamma=-\not{\! \epsilon}(S_z)+\frac{\epsilon \cdot (p_1-p_2)}{M_0+2 m} \quad~ ({\rm vector},
        S=1).
 \end{eqnarray}
All details are shown in Appendix A of Ref. \cite{CCH2}. The
function $H_{P,V}$ and the parameter $W_V$ are reduced to $h_{P,V}$
and $w_V$, respectively, and they are written by
 \begin{eqnarray} \label{eq:htowf}
 h_P&=&h_V=(M^2-M_0^2)\sqrt{\frac{x(1- x)}{N_c}}\frac{1}{\sqrt 2
      M_0}\phi(x,p_\bot), \nonumber \\
 w_V&=&M_0+2m.
 \end{eqnarray}
In principle, $\phi(x_2,p_\bot)$ is obtained by solving the
light-front QCD bound-state equation
$H_{LF}|\Psi\rangle=M|\Psi\rangle$, which is the familiar Schr{\"
o}dinger equation in ordinary quantum mechanics and $H_{LF}$ is the
light-front Hamiltonian. However, except in some simple cases, the
full solution has remained a challenge. There are several popular
phenomenological light-front momentum distribution amplitudes that
have been employed to describe various hadronic structures in the
literature. A widely used one is the Gaussian-type chosen here for
the $1S$ heavy quarkonium state \cite{hwangwei}:
 \begin{eqnarray}
 \phi^{1S}(x,p_\perp)=4 \left(\frac{\pi}{\beta^2}\right)^{3/4}
 \sqrt{\frac{dp_z}{dx}}~{\rm exp}
 \left(-\frac{p^2_z+p^2_\bot}{2
 \beta^2}\right). \label{gaussian1S}
 \end{eqnarray}
In addition, the momentum distribution amplitude for the $2S$ heavy
quarkonium state is considered. Eqs. (\ref{eq:Norm1}) and
(\ref{gaussian1S}) can be rewritten as
 \begin{eqnarray}
 &&\int \frac{d^3 \vec{p}}{2 (2\pi)^3}
   |\phi(p)|^2 =1,\\
&& \phi^{1S}(p)=4 \left(\frac{\pi}{\beta^2}\right)^{3/4}
 {\rm exp}
 \left(-\frac{p^2}{2
 \beta^2}\right), \label{gaussian1S_p3}
 \end{eqnarray}
where $p=|\vec{p}|$. The Fourier transition, or the conjugate
coordinate wave function of Eq. (\ref{gaussian1S_p3}) is the
ground-state solution of the harmonic-oscillator (HO) problem:
 \begin{eqnarray}
 \varphi^{1S}(r)=\left(\frac{\beta^2}{\pi}\right)^{3/4}
 {\rm exp}
 \left(-\frac{\beta^2 r^2}{2}\right). \label{coordinate1S}
 \end{eqnarray}
Then, for consistency, the relevant excited-state solutions
 \begin{eqnarray}
 \varphi^{1P}_m(r)&=&\sqrt{\frac{8}{3}}\frac{\beta^{3/2}}{\pi^{1/4}}~\beta r~{\rm exp}
 \left(-\frac{\beta^2 r^2}{2}\right)Y_{1m}(\theta,\varphi),\label{coordinate1P} \\
 \varphi^{2S}(r)&=&\sqrt{\frac{1}{6}}\left(\frac{\beta^2}{\pi}\right)^{3/4}
 \left(\frac{3}{4}-\frac{\beta^2 r^2}{2}\right){\rm exp}
 \left(-\frac{\beta^2 r^2}{2}\right), \label{coordinate2S}
 \end{eqnarray}
($Y_{1m}$'s are the spherical harmonics), are applied to the excited
meson states. This suggestion has been given by the authors of Ref.
\cite{ISGW}. Therefore, the Fourier transforms of Eqs.
(\ref{coordinate1P}) and (\ref{coordinate2S}) can be rewritten as
 \begin{eqnarray}
 \phi^{1P}_m(x,p_\perp)&=&4 \sqrt{2}
 \left(\frac{\pi}{\beta^2}\right)^{3/4}
 \sqrt{\frac{dp_z}{dx}}~\frac{p_m}{\beta}~{\rm exp}
 \left(-\frac{p^2_z+p^2_\bot}{2
 \beta^2}\right), \label{gaussian1P} \\
 \phi^{2S}(x,p_\perp)&=&4\sqrt{\frac{8}{3}}\left(\frac{\pi}{\beta^2}\right)^{3/4}
 \sqrt{\frac{dp_z}{dx}}~\left(\frac{p^2_z+p^2_\perp}{2\beta^2}-\frac{3}{4}\right){\rm exp}
 \left(-\frac{p^2_z+p^2_\bot}{2
 \beta^2}\right), \label{gaussian2S}
 \end{eqnarray}
($p_{m=\pm1}=\mp (p_{\perp 1}\pm i p_{\perp 2})/\sqrt{2}$, and
$p_{m=0}=p_z$) and can be treated as the momentum distribution
amplitudes of the $1P$ and $2S$ meson states, respectively. In fact,
Eq. (\ref{gaussian1P}) has been used for the $p$-wave mesons in Ref.
\cite{CCH2}. This paper further analyzes the momentum distribution
amplitudes as shown in Eqs. (\ref{gaussian1S}) and
(\ref{gaussian2S}).

The formulae of the decay constants $f_{P,V}$ and M1 transition form
factor $V(0)$ are needed in the latter analysis. However, their
derivations have been done in Refs. \cite{CCH2,hwangwei}. The
definitions and formulae for them are provided here.

\subsection{Decay constant $f_{P,V}$}
The decay constants of mesons $f_{P,V}$ are defined by the
matrix elements for $P$ and $V$ mesons
 \begin{eqnarray}
 \langle 0|A_\mu |P(P) \rangle &=& if_P P_\mu, \\
 \langle 0|V_\mu |V(P,\epsilon) \rangle &=& M_V f_V \epsilon_\mu,
 \end{eqnarray}
where $P_\mu$ is the momentum of meson and $\epsilon_\mu$ is the
polarization vector of $V$ meson. The formula for $f_P$ is
 \begin{eqnarray}
 f_P &=& \frac{N_c}{4\pi^3}\int dx d^2p_\bot \frac{h_P}{x(1-x)
          (M^{2}-M^{2}_0)}~m \nonumber \\
     &=& \frac{\sqrt{2N_c}}{8\pi^3}\int dx d^2 p_\perp
 \frac{m}{\sqrt{m^2+p_\perp^2}}\phi_P(x,p_{\bot}), \label{eq:dcP}
 \end{eqnarray}
where $N_c=3$ is the color number and $m$ denotes the quark mass. As
to the formula for $f_V$, we considered the case with the transverse
polarization
 \begin{eqnarray}
 \epsilon(\pm)=\left(\frac{2}{P^+}\epsilon_\perp \cdot
 P_\perp,0,\epsilon_\perp\right),~~~\epsilon_\perp =
 \mp\frac{1}{\sqrt{2}}(1,\pm i).
 \end{eqnarray}
and obtain
 \begin{eqnarray}
 f_V &=&\frac{N_c}{4\pi^3}\int dx d^2p_\bot \frac{h_V}{x_1
         x_2 (M^{2}-M^{2}_0)}\left[x M^{2}_0-p^{2}_\bot
         +\frac{2 m}{W_V}\,p^{2}_\bot
         \right]\nonumber \\
            &=& \frac{\sqrt{2N_c}}{8\pi^3 M}\int dx d^2 p_\perp
     \frac{1}{\sqrt{m^2+p_\perp^2}}\left [\frac{M_0^2}{2}-
     p_\perp^2+\frac{2m}{w_V}p_\perp^2 \right ]\phi_V(x,p_{\bot}). \label{eq:dcV}
 \end{eqnarray}
This expression can be shown to be in agreement with Eq. (2.22) of
Ref. \cite{CCH2}. Since the momentum distribution function is even
in $p_z$, a quality defined in Eq. (\ref{eq:Mpz}), it follows that
 \begin{eqnarray}
 \int dx d^2p_\perp
 \frac{\phi_V(x,p_\perp)}{\sqrt{m^2+p^2_\perp}}\left(x-\frac{1}{2}\right)M_0=0.
 \end{eqnarray}
Therefore,
 \begin{eqnarray}
 \int dx d^2p_\perp
 \frac{\phi_V(x,p_\perp)}{\sqrt{m^2+p^2_\perp}}~xM_0^2=\int dx d^2p_\perp
 \frac{\phi_V(x,p_\perp)}{\sqrt{m^2+p^2_\perp}}\frac{M_0^2}{2}.
 \end{eqnarray}

\subsection{Vector current form factor $V(q^2)$}

For the transition $V\to P\gamma$, a more general process $V\to
P\gamma^*$ where the final photon is off-shell is considered. The
$V\to P\gamma^*$ transition is parameterized in term of a vector
current form factor $V(q^2)$ by
 \begin{eqnarray}
 \Gamma_{\mu}=ie\varepsilon_{\mu\nu\alpha\beta}\epsilon^{\nu}q^{\alpha}P^{\beta}V(q^2),
 \end{eqnarray}
where $\Gamma_{\mu}$ is the amplitude of the $V\to P\gamma^*$
process. $P$ ($\epsilon$) is the momentum (polarization vector) of
the initial vector meson, $P'$ denotes the momentum of the final
pseudoscalar meson, and the momentum transfer $q=P-P'$. The formula
for the form factor $V(q^2)$ is
 \begin{eqnarray} \label{eq:Vq2}
 V(q^2)&=&\frac{e_q}{8\pi^3}\int dx d^2p_{\bot}
   \frac{\phi_V(x,p_\bot)}{M_0}\left[\frac
   {\phi_P(x,p'_\bot)}{(1-x)M'_0}+\frac{\phi_P(x,p''_\bot)}{x
   M''_0}\right]\nonumber \\
   &&\qquad\qquad\qquad\times\left[ m+\frac{2}{w_V}\left( p_{\bot}^2+
    \frac{(p_{\bot}\cdot q_{\bot})^2}{q^2} \right ) \right ],
 \end{eqnarray}
where $p'_\perp=p_\perp -(1-x) q_\perp$, $p''_\perp=p_\perp +x
q_\perp$, $M'^2_0=(m^2+p'^2_\perp)/x (1-x)$, and
$M''^2_0=(m^2+p''^2_\perp)/x (1-x)$. The rate for $V\to P\gamma$ is
 \begin{eqnarray} \label{eq:VPr2}
 \Gamma(V\to P\gamma)=\frac{\alpha}{3}\frac{(M_V^2-M_P^2)^3}{8 M_V^3}|V(0)|^2.
 \end{eqnarray}

\section{Analysis of momentum distribution amplitude}
In this section, the momentum distribution amplitudes of
pseudoscalar and vector heavy quarkonium states are analyzed. The
forms of quark distribution amplitude and $\xi$-moments can be
derived from them.
\subsection{Quark distribution amplitude $\Phi_P(\xi)$}
The quark distribution amplitude of the pseudoscalar heavy
quarkonium state can be defined as follows \cite{CZ}:
 \begin{eqnarray}
 \langle 0|\bar c(z)\gamma^\alpha \gamma_5[z,-z]c(-z)|P\rangle=if_P
 P^\alpha\int^1_{-1}d\xi e^{i(Pz)\xi}\Phi(\xi,\mu),\label{LCWF}
 \end{eqnarray}
where $\xi=2x-1$ and $\mu$ is an energy scale which separates the
perturbative and non-perturbative regimes. The factor $[z,-z]$ is
defined as
 \begin{eqnarray}
 [z,-z]=P{\rm exp}[iq\int^z_{-z}dx^\mu A_\mu(x)],
 \end{eqnarray}
which makes the matrix element Eq. (\ref{LCWF}) gauge invariant. The
quark distribution amplitude $\Phi(\xi,\mu)$ is normalized as
 \begin{eqnarray}
 \int^1_{-1}d\xi \Phi(\xi,\mu)=1,
 \end{eqnarray}
and it can be expanded \cite{CZ} in Gegenbauer polynomials
$C^{3/2}_n(\xi)$ as
 \begin{eqnarray}
 \Phi(\xi,\mu)= \Phi_{as}(\xi)\left[1+\sum_{n=1}^\infty
 a_n(\mu)C_n^{3/2}(\xi)\right],
 \end{eqnarray}
where $\Phi_{as}(\xi)=3(1-\xi^2)/4$ is the asymptotic quark
distribution amplitude and $a_n(\mu)$ the Gegenbauer moments which
describe to what degree the quark distribution amplitude deviates
from the asymptotic one. $C^{3/2}_n(\xi)$s have the orthogonality
integrals
 \begin{eqnarray}
 \int^1_{-1} (1-\xi^2)C^{3/2}_m(\xi) C^{3/2}_n (\xi) d\xi =
 \frac{2(n+1)(n+2)}{2n+3}~\delta_{mn}. \label{orthogonal}
 \end{eqnarray}
Then $a_n$ can be obtained by using the above orthogonality
integrals as
 \begin{eqnarray}
 a_n(\mu) &=&\frac{2(2
 n+3)}{3(n+1)(n+2)}\int^1_{-1}C^{3/2}_n(\xi)\Phi(\xi,\mu)d\xi.
 \end{eqnarray}
An alternative approach to parameterize quark distribution amplitude
is to calculate the so-called $\xi$-moments,
 \begin{eqnarray}
 \langle \xi^n\rangle_\mu=\int^1_{-1} d\xi~\xi^n \Phi(\xi,\mu).
 \end{eqnarray}
It is easy to find the relations between the Gegenbauer moments and
the $\xi$-moments. Here we list them up to $n=6$
 \begin{eqnarray}
 a_2 &=& -\frac{7}{12}[1-5\langle
 \xi^2\rangle],\nonumber \\
 a_4 &=& \frac{11}{24}[1-14\langle
 \xi^2\rangle + 21\langle \xi^4\rangle],\nonumber \\
 a_6 &=& -\frac{35}{448}[5-135\langle
 \xi^2\rangle + 495\langle \xi^4\rangle-429 \langle
 \xi^6\rangle].\label{anmoments}
 \end{eqnarray}
All the $n$-odd moments are vanishing because this paper only
focuses on the $s$-wave momentum distribution amplitudes, which are
$\xi$-even functions.

There are some similar procedures \cite{BHL,DM} by which the quark
distribution amplitude can be obtained from the equal time wave
function. Within the framework of this study, the decay constant Eq.
(\ref{eq:dcP}) can be rewritten as
 \begin{eqnarray}
 1=\int^1_0 dx \left[\frac{\sqrt{2N_c}}{8\pi^3~f_P}\int d^2p_\perp
 \frac{m}{\sqrt{m^2+p_\perp^2}}\phi_P(x,p_{\bot})\right]
 \end{eqnarray}
and we defined LCWF $\hat{\Phi}_P(x,\mu)$ as
 \begin{eqnarray}
 \hat{\Phi}_P(x,\mu)=\frac{\sqrt{2N_c}}{8\pi^3~f_P}\int^{p_\perp^2 < \mu^2} d^2p_\perp
 \frac{m}{\sqrt{m^2+p_\perp^2}}\phi_P(x,p_{\bot})
 \end{eqnarray}
where $\hat{\Phi}_P(x,\mu)=2 \Phi_P(\xi,\mu)$. If the momentum
distribution amplitudes Eqs. (\ref{gaussian1S}) and
(\ref{gaussian2S}) are applied as the non-perturbative inputs, the
suppression of the Gaussian function allows us to do the $p_\perp$
integrals up to infinity with no loss of accuracy. Thus, the results
can be obtained as follows:
 \begin{eqnarray}
 \Phi^{1S}_P(\xi)&=&\sqrt{\frac{3}{8}} \left(\frac{2}{\pi}\right)^{5/4}
 \frac{m}{f^{1S}_P}~e^d~\Gamma\left[\frac{3}{4}, \frac{d}{1-\xi^2}\right],\label{LC1S}\\
 \Phi^{2S}_P(\xi)&=&\left(\frac{2}{\pi}\right)^{5/4}
 \frac{m}{f^{2S}_P}~e^d~\left\{\Gamma\left[\frac{7}{4},\frac{d}{1-\xi^2}\right]
 -\left(\frac{3}{4}+d\right)~
 \Gamma\left[\frac{3}{4},\frac{d}{1-\xi^2}\right]\right\},\label{LC2S}
 \end{eqnarray}
where $\Gamma[r,y]$ is the incomplete Gamma function
 \begin{eqnarray}
 \Gamma[r,y]=\int^\infty_y t^{r-1}e^{-t} dt, \label{gamma}
 \end{eqnarray}
and $d=m^2/2 \beta^2$. The incomplete gamma function may be
expressed quite elegantly in terms of the confluent hypergeometric
function
 \begin{eqnarray}
 \Gamma[r,y]=\Gamma[r]-r^{-1}y^r\times~_1F_1(r;r+1;-y),\label{hpgeo}
 \end{eqnarray}
where
 \begin{eqnarray}
 _iF_j(r_1,r_2,...,r_i;r'_1,r'_2,...,r'_j;y)=\sum^\infty_{n=0}\frac{(r_1)_n(r_2)_n
 ...(r_i)_n}{(r'_1)_n(r'_2)_n...(r'_j)_n}\frac{y^n}{n!},\label{1F1}
 \end{eqnarray}
and $(r)_n=(r+n-1)!/(r-1)!$ is the Pochhammer symbol.
After fixing the parameters $m$ and $\beta$, the behaviors of the
quark distribution amplitudes, Eqs. (\ref{LC1S}) and (\ref{LC2S}),
are determined and compared with those of other theoretical groups
in Section IV. It is worth to mention that Eqs. (\ref{LC1S}) and
(\ref{LC2S}) have incorrect asymptotic behavior because the
Gaussian-type wave functions are primarily only suitable for the low
energy region. At energy scales accessible at current experiments, a
quark distribution amplitude can be far from the asymptotic form.
The worth of the asymptotic form is that it provides a criterion. In
other words, one can use the Gegenbauer moments (or $\xi$-moments)
to describe to what degree his quark distribution amplitude deviates
from the asymptotic one.

In fact, the $\xi$ integrals of $\Phi^{1(2)S}_P(\xi)$ can also be
analytically performed, and the decay constants $f_P^{1(2)S}$ can be
expressed as
 \begin{eqnarray}
 f_P^{1S} &=& \sqrt{\frac{3}{2}} \left(\frac{2}{\pi}\right)^{5/4}m~
 e^d \left\{\Gamma\left[\frac{3}{4}\right]~_1F_1\left(-\frac{1}{2}; \frac{1}{4};
 -d\right)
 - \frac{3d^{3/4} \Gamma[-\frac{3}{4}]^2}{8\sqrt{2\pi}}~_1F_1\left(\frac{1}{4};
\frac{7}{4}; -d\right)\right\}, \label{1Sanalytic}\\
 f_P^{2S} &=& 3\left(\frac{2}{\pi}\right)^{5/4}
 m~e^d \left\{-2 d~ \Gamma\Bigg[\frac{3}{4}\right]~_1F_1\left(-\frac{1}{2}; \frac{5}{4}; -d\right)
 +\frac{3 d^{3/4} \Gamma[-\frac{3}{4}]^2}{16\sqrt{2\pi}}
  ~_1F_1\left(-\frac{3}{4}; \frac{3}{4}; -d\right)\Bigg\}.\label{2Sanalytic}
 \end{eqnarray}
It is evident that $f_P^{1(2)S}$ is only dependent on the values of
$m$ and $m/\beta$. Eqs. (\ref{1Sanalytic}) and (\ref{2Sanalytic})
can be expanded in terms of $d$ as
 \begin{eqnarray}
 f_P^{1S} &\sim&
 c_1\left(1+3d+\frac{21}{10}d^2+\frac{77}{90}d^3+...\right)+c'_1 d^{3/4}
 \left(1+\frac{6}{7}d+\frac{30}{77}d^2+\frac{4}{33}d^3+...\right),\label{1Sseries}\\
 f_P^{2S} &\sim&-c_2d\left(1+\frac{7}{5}d+\frac{77}{90}d^2+\frac{77}{234}d^3...\right)+
 c'_2 d^{3/4}\left(1+2 d+\frac{10}{7}d^2+\frac{20}{33}d^3+...\right),\label{2Sseries}
 \end{eqnarray}
where all $c^{(_{'})}_{1,2}$ are positive constants. However, for a
typical value of $d$, all the series in Eqs. (\ref{1Sseries}) and
(\ref{2Sseries}) converge very slowly. In addition, the $\xi$-
moments can be expressed analytically as
 \begin{eqnarray}
 \langle \xi^2\rangle^{1S}_P &=& A^{1S}_P\Bigg\{\frac{2}{3} \Gamma\left[\frac{3}{4}\right]
 ~_1F_1\left(-\frac{3}{2};\frac{1}{4}, -d\right)-\frac{d^{3/4}  \Gamma\left[-\frac{3}{4}
 \right]^2}{2 \sqrt{2\pi}}~_1F_1\left(-\frac{3}{4}, \frac{7}{4},
 -d\right)\bigg\},\nonumber \\
 \langle \xi^4\rangle^{1S}_P &=& A^{1S}_P\Bigg\{\frac{2}{5} \Gamma\left[\frac{3}{4}\right]
 ~_1F_1\left(-\frac{5}{2};\frac{1}{4}, -d\right)-\frac{3d^{3/4}  \Gamma\left[-\frac{3}{4}
 \right]^2}{7 \sqrt{2\pi}}~_1F_1\left(-\frac{7}{4}, \frac{7}{4},
 -d\right)\bigg\},\nonumber \\
 \langle \xi^6\rangle^{1S}_P &=& A^{1S}_P\Bigg\{\frac{2}{7}
 \Gamma\left[\frac{3}{4}\right]
 ~_1F_1\left(-\frac{7}{2};\frac{1}{4}, -d\right)-\frac{30d^{3/4}  \Gamma\left[-\frac{3}{4}
 \right]^2}{77 \sqrt{2\pi}}~_1F_1\left[-\frac{11}{4}, \frac{7}{4},
 -d\right]\bigg\},\label{ximoment1s}
 \end{eqnarray}
and
 \begin{eqnarray}
  \langle \xi^2\rangle^{2S}_P &=&
  A^{2S}_P\Bigg\{\frac{7d}{6}\Gamma\left[-\frac{1}{4}\right]~_1F_1\left(-\frac{3}{2},\frac{5}{4};-d\right)
  +\frac{3d^{3/4}\Gamma\left[-\frac{3}{4}\right]^2}{8\sqrt{2\pi}}
  ~_1F_1\left(-\frac{7}{4},\frac{3}{4};-d\right)
  \Bigg\},\nonumber \\
 \langle \xi^4\rangle^{2S}_P &=&
  A^{2S}_P\Bigg\{\frac{11d}{10}\Gamma\left[-\frac{1}{4}\right]~_1F_1\left(-\frac{5}{2},\frac{5}{4};-d\right)
  +\frac{9d^{3/4}\Gamma\left[-\frac{3}{4}\right]^2}{28\sqrt{2\pi}}
  ~_1F_1\left(-\frac{11}{4},\frac{3}{4};-d\right)
  \Bigg\},\nonumber \\
  \langle \xi^6\rangle^{2S}_P &=&
  A^{2S}_P\Bigg\{\frac{15d}{14}\Gamma\left[-\frac{1}{4}\right]~_1F_1\left(-\frac{7}{2},\frac{5}{4};-d\right)
 +\frac{90d^{3/4}\Gamma\left[-\frac{3}{4}\right]^2}{308\sqrt{2\pi}}
  ~_1F_1\left(-\frac{15}{4},\frac{3}{4};-d\right)
  \Bigg\},
  \label{ximoment2s}
 \end{eqnarray}
where
 \begin{eqnarray}
 A^{1S}_P = \sqrt{\frac{3}{8}} \left(\frac{2}{\pi}\right)^{5/4}
 \frac{m}{f^{1S}_P}~e^d,\quad\quad
 A^{2S}_P = \left(\frac{2}{\pi}\right)^{5/4}
 \frac{m}{f^{2S}_P}~e^d.
 \end{eqnarray}
The derivations of Eqs. (\ref{ximoment1s}) and (\ref{ximoment2s})
have used the formula
 \begin{eqnarray}
 _1
 F_1(a;b;c)=\frac{b-1}{c}\left[~_1F_1(a;b-1;c)-~_1F_1(a-1;b-1;c)\right],
 \end{eqnarray}
which is easily checked from the definition of the confluent
hypergeometric function Eq. (\ref{1F1}).

\subsection{Quark distribution amplitude $\Phi_V(\xi)$}
Similar to the case of $\Phi_P(\xi)$, Eq. (\ref{eq:dcV}) can be
rewritten and $\hat{\Phi}_V(x)$ defined as
 \begin{eqnarray}
 \hat{\Phi}_V(x) = \frac{\sqrt{2N_c}}{8\pi^3 M~f_V}\int d^2 p_\perp
     \frac{\phi_V(x,p_{\bot})}{\sqrt{m^2+p_\perp^2}}\left [\frac{M_0^2}{2}-
     p_\perp^2+\frac{2m}{w_V}p_\perp^2 \right ]. \label{LCWFvector}
 \end{eqnarray}
However, the $p_\perp$ integrals in Eq. (\ref{LCWFvector}) cannot be
analytically performed . The crux is the third term in the square
bracket, that is, the term proportional to $1/w_V$. This term may be
rewritten and expanded as
 \begin{eqnarray} \label{vwapp}
 \frac{1}{M_0+2m}=\frac{1}{4m\left(1+\frac{M_0-2m}{4m}\right)}
 \simeq
 \frac{1}{4m}\left[1-\left(\frac{M_0-2m}{4m}\right)+\left(\frac{M_0-2m}{4m}\right)^2-...\right],
 \end{eqnarray}
because, in the non-relativistic limit, $M_0 \to 2m$. Then, we
defined the approximate quark distribution amplitudes as
 \begin{eqnarray}
 &&\hat{\Phi}'_V(x) = \frac{\sqrt{2N_c}}{8\pi^3 M~f_V}\int d^2 p_\perp
     \frac{\phi_V(x,p_{\bot})}{\sqrt{m^2+p_\perp^2}}\left [\frac{M_0^2}{2}-
     p_\perp^2+\frac{p_\perp^2}{2}\right], \label{PhiV0}\\
 &&{\hat\Phi}''_V(x) = \frac{\sqrt{2N_c}}{8\pi^3 M~f_V}\int d^2 p_\perp
     \frac{\phi_V(x,p_{\bot})}{\sqrt{m^2+p_\perp^2}}\left [\frac{M_0^2}{2}-
     p_\perp^2+\frac{p_\perp^2}{2}\left(1-\frac{M_0-2m}{4m}\right) \right
     ].\label{PhiV1}
 \end{eqnarray}
After applying the momentum distribution function in Eq.
(\ref{gaussian1S}) and fitting the parameters, we found that not
only the center values of the parameters $\beta$ of Eqs.
(\ref{PhiV0}) and (\ref{PhiV1}) were both in the error bar of
$\beta_{J/\psi}$ (see Table I), but also that their curves were
almost the same. In addition, a similar situation also existed in
the case of the $2S$ states. Therefore, we have only shown the
results from the form $\hat{\Phi}'_V(x)$. By performing the
$p_\perp$ integrals, it can be obtained that
 \begin{eqnarray}
 \Phi'^{1S}_V(\xi) &=& \sqrt{\frac{3}{8}}\left(\frac{2}{\pi}\right)^{5/4}\frac{w^2e^d}{Mf_V^{1S}}
 \Bigg\{d\Gamma\left[\frac{3}{4},\frac{d}{1-\xi^2}\right]+(3+\xi^2)
 \Gamma\left[\frac{7}{4},\frac{d}{1-\xi^2}\right]\bigg\}, \label{analyPhiV1s}  \\
 {\hat\Phi}'^{2S}_V(\xi) &=& \left(\frac{2}{\pi}\right)^{5/4}\frac{w^2e^d}{Mf_V^{2S}}
 \Bigg\{d\Gamma\left[\frac{7}{4},\frac{d}{1-\xi^2}\right]+(3+\xi^2)
 \Gamma\left[\frac{11}{4},\frac{d}{1-\xi^2}\right]\nonumber \\
 &&\qquad\qquad\qquad\quad -\left(\frac{3}{4}+d\right)\left(d\Gamma\left[\frac{3}{4},
 \frac{d}{1-\xi^2}\right]+(3+\xi^2)\Gamma\left[\frac{7}{4},\frac{d}{1-\xi^2}\right]\right)\bigg\}.
 \label{analyPhiV2s}
 \end{eqnarray}
The above derivations are based on the formula:
 \begin{eqnarray}
 \Gamma[r,y]-(r-1)\Gamma[r-1,y]=y(\Gamma[r-1,y]-(r-2)\Gamma[r-2,y]),
 \end{eqnarray}
which is easily checked from the definition of incomplete Gamma
function, Eq. (\ref{gamma}). The $\xi$-moments of $\Phi'_V(\xi)$ can
also be obtained analytically
 \begin{eqnarray}
 \langle \xi^2\rangle^{1S}_V &=& A^{1S}_V\Bigg\{\frac{1}{5} \Gamma\left[\frac{3}{4}\right]
 \left[10~_1F_1\left(-\frac{3}{2};-\frac{3}{4};
 -d\right)-~_1F_1\left(-\frac{5}{2};-\frac{3}{4};
 -d\right)\right] \nonumber \\
 &&\qquad+\frac{2d^{7/4}  \Gamma\left[-\frac{3}{4}
 \right]^2}{7 \sqrt{2\pi}}\bigg[3~_1F_1\left(\frac{1}{4};
 \frac{11}{4};
 -d\right)-~_1F_1\left(-\frac{3}{4};
 \frac{11}{4};
 -d\right)\bigg]\bigg\},\nonumber \\

 \langle \xi^4\rangle^{1S}_V &=& A^{1S}_V\Bigg\{\frac{3}{35} \Gamma\left[\frac{3}{4}\right]
 \left[14~_1F_1\left(-\frac{5}{2};-\frac{3}{4};
 -d\right)-~_1F_1\left(-\frac{7}{2};-\frac{3}{4};
 -d\right)\right] \nonumber \\
 &&\qquad+\frac{3 d^{7/4}  \Gamma\left[-\frac{7}{4}
 \right]^2}{4 \sqrt{2\pi}}\bigg[7~_1F_1\left(-\frac{3}{4};
 \frac{11}{4};
 -d\right)-~_1F_1\left(-\frac{7}{4};
 \frac{11}{4};
 -d\right)\bigg]\bigg\},\nonumber
 \end{eqnarray}
 \begin{eqnarray}

 \langle \xi^6\rangle^{1S}_V &=& A^{1S}_V\Bigg\{\frac{1}{21} \Gamma\left[\frac{3}{4}\right]
 \left[18~_1F_1\left(-\frac{7}{2};-\frac{3}{4};
 -d\right)-~_1F_1\left(-\frac{9}{2};-\frac{3}{4};
 -d\right)\right] \nonumber \\
 &&\qquad+\frac{165 d^{7/4}  \Gamma\left[-\frac{11}{4}
 \right]^2}{32 \sqrt{2\pi}}\bigg[11~_1F_1\left(-\frac{7}{4};
 \frac{11}{4};
 -d\right)-~_1F_1\left(-\frac{11}{4};
 \frac{11}{4};
 -d\right) \bigg]\bigg\},\label{ximomentV1s}
 \end{eqnarray}
and
 \begin{eqnarray}
 \langle \xi^2\rangle^{2S}_V &=& A^{2S}_V\Bigg\{\frac{1}{5} \Gamma\left[\frac{3}{4}\right]
 \left[15~_1F_1\left(-\frac{3}{2};-\frac{3}{4};
 -d\right)+(d-6)~_1F_1\left(-\frac{5}{2};-\frac{3}{4};
 -d\right)\right] \nonumber \\
 &&\qquad+\frac{d^{7/4}  \Gamma\left[-\frac{3}{4}
 \right]^2}{7 \sqrt{2\pi}}\bigg[9~_1F_1\left(\frac{1}{4};
 \frac{11}{4};
 -d\right)+2(d-6)~_1F_1\left(-\frac{3}{4};
 \frac{11}{4};
 -d\right)\bigg]\bigg\},\nonumber \\
 \langle \xi^4\rangle^{2S}_V &=& A^{2S}_V\Bigg\{\frac{3}{35} \Gamma\left[\frac{3}{4}\right]
 \left[35~_1F_1\left(-\frac{5}{2};-\frac{3}{4};
 -d\right)+(d-22)~_1F_1\left(-\frac{7}{2};-\frac{3}{4};
 -d\right)\right] \nonumber \\
 &&\quad+\frac{3 d^{7/4}  \Gamma\left[-\frac{7}{4}
 \right]^2}{8 \sqrt{2\pi}}\bigg[35~_1F_1\left(-\frac{3}{4};
 \frac{11}{4};
 -d\right)+2(d-22)~_1F_1\left(-\frac{7}{4};
 \frac{11}{4};
 -d\right)\bigg]\bigg\},\nonumber \\

 \langle \xi^6\rangle^{2S}_V &=& A^{2S}_V\Bigg\{\frac{1}{21} \Gamma\left[\frac{3}{4}\right]
 \left[63~_1F_1\left(-\frac{7}{2};-\frac{3}{4};
 -d\right)+(d-46)~_1F_1\left(-\frac{9}{2};-\frac{3}{4};
 -d\right)\right] \nonumber \\
 &&\quad+\frac{165 d^{7/4}  \Gamma\left[-\frac{11}{4}
 \right]^2}{64 \sqrt{2\pi}}\bigg[77~_1F_1\left(-\frac{7}{4};
 \frac{11}{4};
 -d\right)+2 (d-46)~_1F_1\left(-\frac{11}{4};
 \frac{11}{4};
 -d\right) \bigg]\bigg\},\label{ximomentV2s}
 \end{eqnarray}
where
 \begin{eqnarray}
 A^{1S}_V=\sqrt{\frac{3}{8}}\left(\frac{2}{\pi}\right)^{5/4}\frac{w^2e^d}{Mf_V^{1S}},\quad
 A^{2S}_V=\left(\frac{2}{\pi}\right)^{5/4}\frac{w^2e^d}{Mf_V^{2S}}.
 \end{eqnarray}

\section{Numerical results and discussions}
In this work, the numerical results were calculated for the quark
distribution amplitudes $\Phi(\xi)$ and $\xi$-moments for the
pseudoscalar and vector heavy quarkonium. First, it was necessary to
determine the parameters appearing in the momentum distribution
functions. In total, there are five for charmonium, $m_c$,
$\beta_{\eta_c}$, $\beta_{\eta'_c}$, $\beta_{J/\psi}$,
$\beta_{\psi'}$, and five ones for bottomonium: $m_b$,
$\beta_{\eta_b}$, $\beta_{\eta'_b}$, $\beta_{\Upsilon}$,
$\beta_{\Upsilon'}$. The decay constants of vector heavy quarkonium
states were determined first. The decay constant $f_V$ is related to
the electromagnetic decay of vector meson $V\to e^+e^-$ by \cite{NS}
 \begin{eqnarray} \label{eq:dcVexp}
 \Gamma(V\to e^+e^-)=\frac{4\pi}{3}\frac{\alpha^2}{M_V}c_V f_V^2,
 \end{eqnarray}
where $c_V$ is the square of the electric charge of heavy quark.
From the data \cite{PDG08}, we obtained the values
 \begin{eqnarray}
 f_{J/\psi}&=&416 \pm 7~{\rm MeV},\qquad\qquad f_{\psi'}=298 \pm 8~{\rm
 MeV}, \label{fvectorc}\\
 f_{\Upsilon}&=&715 \pm 5~{\rm MeV},\qquad\qquad f_{\Upsilon'}=497 \pm 4~{\rm
 MeV}.\label{fvectorb}
 \end{eqnarray}
For the decay constants of $\eta_c$, the two decay modes $B\to
K\eta_c$ and $B\to K J/\psi$ are considered using the following
relation
 \begin{eqnarray}
 \frac{\Gamma(B\to K \eta_c)}{\Gamma(B\to K J/\psi)}=
 \left(\frac{f_{\eta_c}}{f_{J/\psi}}\right)^2
 \Bigg|\frac{C_{\eta_c}}{C_{J/\psi}}\Bigg|^2\frac{\lambda_{BK\eta_c}}{\lambda_{BK J/\psi}^3}
 \Bigg|\left(1-\frac{M_K^2}{M_B^2}\right)\frac{f_+(M_{\eta_c}^2)}{f_+(M_{J/\psi}^2)}+
 \frac{M_{\eta_c}^2}{M_B^2}\frac{f_-(M_{\eta_c}^2)}{f_+(M_{J/\psi}^2)}\Bigg|^2,\label{Btoccratio}
 \end{eqnarray}
where $C_{\eta_c}$ and $C_{J/\psi}$ are related to the Wilson
coefficients and can be determined to have the value
$|C_{\eta_c}/C_{J/\psi}|=0.89\pm0.02$ \cite{DT},
 \begin{eqnarray}
 \lambda_{abc}=\left[\left(1-\frac{M_b^2}{M_a^2}-\frac{M_c^2}{M_a^2}\right)^2-
 4\frac{M_b^2}{M_a^2}\frac{M_c^2}{M_a^2}\right]^{1/2},
 \end{eqnarray}
and the form factors $f_\pm(q^2)$ are defined by the Lorentz
decomposition of the matrix element
 \begin{eqnarray}
 \langle K(p)|\bar s \gamma_\mu b | B(p+q)\rangle=f_+(q^2)
 (2p+q)_\mu + f_-(q^2)q_\mu.
 \end{eqnarray}
By the results of Ref. \cite{CCH2} one calculated the form factors
$f_\pm(q^2)$ within the covariant light-front quark model. Combining
the above with the experimental values, we can obtain
 \begin{eqnarray}
 f_{\eta_c}=421 \pm 38~{\rm MeV}. \label{fetac}
 \end{eqnarray}
This result leads to the ratio
 \begin{eqnarray}
 \left(\frac{f_{\eta_c}}{f_{J/\psi}}\right)^2=1.02\pm 0.22,\label{thisworkratio}
 \end{eqnarray}
which is consistent with the Van Royen-Weisskopf formula \cite{VRW},
 \begin{eqnarray}
 \left(\frac{f_{\eta_c}}{f_{J/\psi}}\right)^2=\frac{M_{J/\psi}}
 {M_{\eta_c}}\frac{|\Psi_{\eta_c}(0)|^2}{|\Psi_{J/\psi}(0)|^2}\simeq
 1.04,\label{VRWratio}
 \end{eqnarray}
when the approximation $\Psi_{\eta_c}(0)\simeq \Psi_{J/\psi}(0)$ is
used. For the decay constant $f_{\eta'_c}$, however, a similar
estimation of Eq. (\ref{Btoccratio}) has a large uncertainty because
the error of $Br(B\to K \eta'_c)$ was greater than $50\%$
\cite{PDG08}. Given the consistency between Eq.
(\ref{thisworkratio}) and (\ref{VRWratio}), the approximation
$\Psi_{\eta'_c}(0)\simeq \Psi_{\psi'}(0)$ was used, and the
corresponding Van Royen-Weisskopf formula,
 \begin{eqnarray}
 \left(\frac{f_{\eta'_c}}{f_{\psi'}}\right)^2\simeq\frac{M_{\psi'}}
 {M_{\eta'_c}}= 1.01, \label{appratio}
 \end{eqnarray}
is applied and the decay constant $f_{\eta'_c}=300\pm 8$MeV is
obtained. Furthermore, an additional constraint is needed. The new
data $Br(J/\psi \to \eta_c \gamma)=(1.98\pm0.31)\%$ \cite{newdata}
are chosen to calculate the form factor $V(0)$ in Eq.
(\ref{eq:VPr2}). For the charm sector, there were one assumption
(Eq. (\ref{appratio})) and four constraints (the decay constants
(\ref{fvectorc}), (\ref{fetac}), and the new data $Br(J/\psi \to
\eta_c \gamma)$) were used to fix the five parameters appearing in
the momentum distribution amplitudes. The results are listed in
Table I.
\begin{table}[h!]
\caption{\label{tab:para} Fixed parameters $m_c$ and $\beta$'s for
charmonium states (in units of GeV). $\beta'$ and $\beta''$ are the
parameters appearing in equations (\ref{PhiV0}) and (\ref{PhiV1}).}
\begin{ruledtabular}
\begin{tabular}{c|ccccc}
 parameter & $m_c$    & $\beta_{\eta_c}$ & $\beta_{\eta'_c}$ & $\beta_{J/\psi}$ &  $\beta_{\psi'}$
          \\ \hline
 value & $1.56$  & $0.820^{+0.078}_{-0.074}$     &
$0.665^{+0.028}_{-0.026}$    & $0.613 \pm 0.006$     & $0.477 \pm
0.008$ \\ \hline\hline
 parameter & & $\beta'_{J/\psi}$ & $\beta''_{J/\psi}$ & $\beta'_{\psi'}$
 & $\beta''_{\psi'}$ \\ \hline
 value & & $0.611 \pm 0.06 $ & $0.613 \pm 0.06 $ & $0.474\pm 0.08 $ & $0.477\pm 0.08 $
\end{tabular}
\end{ruledtabular}
\end{table}
As to the decay constants of $\eta_b$ and $\eta_b'$, a similar
approximation was assumed
 \begin{eqnarray}
 \left(\frac{f_{\eta'_b}}{f_{\Upsilon'}}\right)^2\simeq\left(\frac{f_{\eta_b}}
 {f_{\Upsilon}}\right)^2\simeq\frac{M_{\Upsilon}}
 {M_{\eta_b}}= 1.01, \label{appratiob}
 \end{eqnarray}
and the decay constants $f_{\eta_b}=718\pm 5$ MeV,
$f_{\eta'_b}=499\pm 5$ MeV obtained. Finally we used \cite{CCH2}
$m_b=4.64$ GeV and fixed the $\beta$ parameters for the bottom
sector in Table II.
\begin{table}[h!]
\caption{\label{tab:parab} Parameters $m_b$ and $\beta$'s for
bottomonium states (in units of GeV). $\beta'$ and $\beta''$ are the
parameters appearing in equations (\ref{PhiV0}) and (\ref{PhiV1}).}
\begin{ruledtabular}
\begin{tabular}{c|ccccc}
 parameter & $m_b$    & $\beta_{\eta_b}$ & $\beta_{\eta'_b}$ & $\beta_{\Upsilon}$ &  $\beta_{\Upsilon'}$
          \\ \hline
 value & $4.64$  & $1.47\pm 0.01$     &
$1.04\pm 0.01$    & $1.30 \pm 0.01$     & $0.926 \pm 0.005$
\\ \hline\hline
 parameter & & $\beta'_{\Upsilon}$ & $\beta''_{\Upsilon}$ & $\beta'_{\Upsilon'}$
 & $\beta''_{\Upsilon'}$ \\ \hline
 value & & $1.30 \pm 0.01 $ & $1.30 \pm 0.01 $ & $0.924\pm 0.05 $ & $0.926\pm 0.005 $
\end{tabular}
\end{ruledtabular}
\end{table}

After fixing all parameters, Eqs. (\ref{LC1S}), (\ref{LC2S}),
(\ref{analyPhiV1s}), and (\ref{analyPhiV2s}) are used to plot the
quark distribution amplitudes $\Phi^{(_{'})1S}_{P,V}(\xi)$ and
$\Phi^{(_{'})2S}_{P,V}(\xi)$ for the charm sector, as shown in Fig.
1 and Fig. 2, respectively.
\begin{figure}
\includegraphics*[width=4in]{xi1s.eps}
\caption{ Quark distribution amplitudes $\Phi_{\eta_c}(\xi)$ (solid
lines) and $\Phi'_{J/\psi}(\xi)$ (dashed lines) of this work.}
 \label{fig:xi1S}
\end{figure}
\begin{figure}
\includegraphics*[width=4in]{xi2s.eps}
\caption{ Quark distribution amplitudes $\Phi_{\eta'_c}(\xi)$ (solid
lines) and $\Phi'_{\psi'}(\xi)$ (dashed lines) of this work.}
 \label{fig:xi2S}
\end{figure}
As is seen, the momentum fraction $x$ of the pseudoscalar charmonium
had a slightly wider distribution than that of the vector
charmonium. Next, the quark distribution amplitudes
$\Phi_{\eta_c}(\xi)$ and $\Phi_{\eta'_c}(\xi)$ of this work were
compared with those of Ref. \cite{Braguta1,Braguta2} in Fig. 3.
\begin{figure}
\includegraphics*[width=4in]{xi1s2s.eps}
\caption{Quark distribution amplitudes $\Phi_{\eta_c}(\xi)$ and
$\Phi_{\eta'_c}(\xi)$ of this work (solid lines) and that of Ref.
\cite{Braguta1,Braguta2} (dashed lines).}
 \label{fig:xi}
\end{figure}
For the latter, the equation \cite{Braguta1,Braguta2}
 \begin{eqnarray} \label{eq:Braguta}
 \Phi(\xi,\mu=1.2\text{GeV})=c(\alpha,\beta,\gamma)(1-\xi^2)(\alpha+\gamma\xi^2)~{\rm
 exp}\left(-\frac{\beta}{1-\xi^2}\right),
 \end{eqnarray}
was used to plot the curves. $\Phi^{1S}(\xi,\mu)$ and
$\Phi^{2S}(\xi,\mu)$ corresponded to $(\alpha=1,\beta=3.8,\gamma=0)$
\cite{Braguta1} and $(\alpha=0.027,\beta=2.49,\gamma=1)$
\cite{Braguta2}, respectively. In Fig. 3, the major difference
between the two curves of the $1S$ states was that the curve of Ref.
\cite{Braguta1}, on which $\xi$ is peaked around zero, was sharper
than that of this work. This meant that the momentum fraction $x$ in
the function used in Ref. \cite{Braguta1} was more centered on $1/2$
than in the Gaussian-type wave function. As to the curves of the
$2S$ state, we found the locations of extreme value by
differentiating both Eqs. (\ref{LC2S}) and (\ref{eq:Braguta}) over
$\xi$. The results were
 \begin{eqnarray}
 \xi&=& 0, \pm \sqrt{\frac{3}{3+4 d}},\nonumber \\
 \xi&=& 0, \pm \frac{1}{2 \sqrt{\gamma}}[3\gamma +\beta\gamma-\alpha-
 (\alpha^2+2 \alpha\gamma+6
 \alpha\beta\gamma+\gamma^2+6\beta\gamma^2+\beta^2\gamma^2)^{1/2}]^{1/2},\nonumber
 \end{eqnarray}
respectively. The values are $\xi=0, \pm 0.463$ for this work and
$\xi=0, \pm 0.421$ for Ref. \cite{Braguta2}. For the bottom sector,
the quark distribution amplitudes $\Phi^{(_{'})1S}_{P,V}(\xi)$ and
$\Phi^{(_{'})2S}_{P,V}(\xi)$ are plotted in Figs. 4 and 5.
\begin{figure}
\includegraphics*[width=4in]{xib1s.eps}
\caption{ Quark distribution amplitudes $\Phi_{\eta_b}(\xi)$ (solid
lines) and $\Phi'_{\Upsilon}(\xi)$ (dashed lines) of this work.}
 \label{fig:xib1S}
\end{figure}
\begin{figure}
\includegraphics*[width=4in]{xib2s.eps}
\caption{ Quark distribution amplitudes $\Phi_{\eta'_b}(\xi)$ (solid
lines) and $\Phi'_{\Upsilon'}(\xi)$ (dashed lines) of this work.}
 \label{fig:xib2S}
\end{figure}
These figures show that the $x$-distribution of the pseudoscalar
bottomonium was almost the same as for the vector bottomonium. The
reason is that since the differences between the pseudoscalar and
the vector heavy quarkonium states arise from $1/m_Q$ corrections,
the larger the $m_Q$, the smaller the difference between them. For
comparison, the $x$-distributions of the charmonium and bottomonium
states are plotted together in Fig. 6.
\begin{figure}
\includegraphics*[width=4in]{xibc1s2s.eps}
\caption{ Quark distribution amplitudes $\Phi_{\eta_b,\eta'_b}(\xi)$
(solid lines) and $\Phi_{\eta_c,\eta'_c}(\xi)$ (dashed lines) of
this work.}
 \label{fig:xibc1s2S}
\end{figure}

In addition, the differences between various quark distribution
amplitudes can also be shown by $\xi$-moments. We calculated the
$\xi$-moments Eqs. (\ref{ximoment1s}), (\ref{ximoment2s}),
(\ref{ximomentV1s}), and (\ref{ximomentV2s}). Our results and those
of other theoretical groups are listed in Tables III and IV. In
these tables, Refs. \cite{Braguta1,Braguta2,Braguta3} used the QCD
sum rules. The authors of Ref. \cite{BT} calculated in the framework
of the Buchmuller-Tye-potential model and found that their results
are in agreement with experiments, which included the leptonic
widths and hyperfine splittings. The authors of \cite{EGKLY} made
calculations for the Cornell potential. The predictions of
light-cone sum rules are shown in Ref. \cite{BC}. The authors of
Ref. \cite{CJ} not only numerically calculated the $1S$ charmonium
states in conventional LFQM, but they also used the variational
approach to fix the mass of the charm quark. The authors of Ref.
\cite{BF} constructed quark distribution amplitudes by assuming the
very non-relativistic limit, where $\Phi(x) = \delta(x-1/2)$, and
then redistributing the parton momenta by relativistic gluon
exchange. Depending on the assumed value of $\alpha_s$ at the heavy
quark scale, they found the $\xi^n$ moments of $\eta_c$ in Table
III. For reasonable values of $\alpha_s$ they obtained qualitatively
similar values as in \cite{Braguta1} and the curves of them are
slightly
narrower than in our numerical analysis.
As to the $\xi$-moments for the bottomonium states, they were
evaluated and are listed in Table V.
\begin{table}[h!]
\caption{\label{tab:1sxi} The $\xi$-moments for the $1S$ charmonium
states. ($^\dag$ $\langle \xi^n \rangle_{\eta_c}=\langle \xi^n
\rangle_{J/\psi}$) }
\begin{ruledtabular}
\begin{tabular}{c|ll|ll|ll}
 moment & $\langle \xi^2 \rangle_{\eta_c}$ & $\langle \xi^2 \rangle_{J/\psi}$ & $\langle \xi^4 \rangle_{\eta_c}$
 & $\langle \xi^4 \rangle_{J/\psi}$ & $\langle \xi^6 \rangle_{\eta_c}$ & $\langle \xi^6 \rangle_{J/\psi}$
  \\ \hline
 this work & $0.117^{+0.010}_{-0.011}$  & $0.0966^{+0.0013}_{-0.0013}$ & $0.0307^{+0.0052}_{-0.0049}$
 & $0.0215^{+0.0005}_{-0.0005}$ & $0.0109^{+0.0026}_{-0.0023}$ & $0.00657^{+0.00023}_{-0.00023}$ \\

  \cite{Braguta1,Braguta3}& $0.070^{+0.007}_{-0.007}$ & $0.072^{+0.007}_{-0.007}$
  & $0.012^{+0.002}_{-0.002}$ & $0.012^{+0.002}_{-0.002}$ & $0.0032^{+0.0009}_{-0.0009}$ & $0.0033^{+0.0007}_{-0.0007}$\\
  \cite{BT} & & $0.086$ & &$0.020$ & & $0.0066$\\
  \cite{EGKLY}& & $0.084$ & & $0.019$ && $0.0066$ \\
  \cite{BC}$^\dag$ & & $0.13$ &  &$0.040$ & &  $0.018$ \\

  \cite{CJ} & $0.084^{+0.004}_{-0.007}$& $0.082^{+0.004}_{-0.006}$& $0.017^{+0.001}_{-0.003}$& $0.016^{+0.002}_{-0.002}$
  &$0.0047^{+0.0006}_{-0.0010}$&$0.0046^{+0.0005}_{-0.0010}$\\
  \cite{BF} & $0.067$ & & $0.011$& &$0.004$ &
\end{tabular}
\end{ruledtabular}
\end{table}
\begin{table}[h!]
\caption{\label{tab:2sxi} The $\xi$-moments for the $2S$ charmonium
states. ($^\dag$ $\langle \xi^n \rangle_{\eta'_c}=\langle \xi^n
\rangle_{\psi'}$)}
\begin{ruledtabular}
\begin{tabular}{c|ll|ll|ll}
 moment & $\langle \xi^2 \rangle_{\eta'_c}$ & $\langle \xi^2 \rangle_{\psi'}$ & $\langle \xi^4 \rangle_{\eta'_c}$
 & $\langle \xi^4 \rangle_{\psi'}$ & $\langle \xi^6 \rangle_{\eta'_c}$ & $\langle \xi^6 \rangle_{\psi'}$
  \\ \hline
 this work & $0.192^{+0.008}_{-0.008}$ & $0.136^{+0.006}_{-0.006}$& $0.0600^{+0.0048}_{-0.0044}$ &
 $0.0320^{+0.0021}_{-0.0020}$ & $0.0229^{+0.0026}_{-0.0023}$ & $0.00950^{+0.00078}_{-0.00074}$ \\

 \cite{Braguta2}$^{\dag}$&  & $0.18^{+0.05}_{-0.07}$ &  & $0.051^{+0.031}_{-0.031}$& &
 $0.017^{+0.016}_{-0.014}$\\
  \cite{BT}&   &$0.16$  & & $0.042$ & & $0.015$\\

\end{tabular}
\end{ruledtabular}
\end{table}
\begin{table}[h!]
\caption{\label{tab:1s2sxib} The $\xi$-moments for the $1S$ and $2S$
bottomonium states. }
\begin{ruledtabular}
\begin{tabular}{c|ll|ll|ll}
 moment & $\langle \xi^2 \rangle_{\eta_b}$ & $\langle \xi^2 \rangle_{\Upsilon}$ & $\langle \xi^4 \rangle_{\eta_b}$
 & $\langle \xi^4 \rangle_{\Upsilon}$ & $\langle \xi^6 \rangle_{\eta_b}$ & $\langle \xi^6 \rangle_{\Upsilon}$
  \\ \hline
 this work & $0.0643^{+0.0005}_{-0.0005}$  & $0.0598^{+0.0004}_{-0.0004}$ & $0.0103^{+0.0001}_{-0.0001}$
 & $0.00894^{+0.00011}_{-0.00011}$ & $0.00237^{+0.00005}_{-0.00005}$ & $0.00192^{+0.00003}_{-0.00003}$ \\ \hline\hline

 moment & $\langle \xi^2 \rangle_{\eta'_b}$ & $\langle \xi^2 \rangle_{\Upsilon'}$ & $\langle \xi^4 \rangle_{\eta'_b}$
 & $\langle \xi^4 \rangle_{\Upsilon'}$ & $\langle \xi^6 \rangle_{\eta'_b}$ & $\langle \xi^6 \rangle_{\Upsilon'}$
  \\ \hline
 this work & $0.0844^{+0.0010}_{-0.0010}$  & $0.0729^{+0.0007}_{-0.0007}$ & $0.0128^{+0.0003}_{-0.0003}$
 & $0.00984^{+0.00018}_{-0.00018}$ & $0.00255^{+0.00009}_{-0.00008}$ & $0.00176^{+0.00005}_{-0.00004}$ \\

\end{tabular}
\end{ruledtabular}
\end{table}

\section{Conclusions}
This study performed the transverse momenta $p_\perp$ integrals of
formulae for the decay constants $f_{P,V}$ of $1S$ and $2S$ heavy
quarkonium states and then obtained their quark distribution
amplitudes $\Phi_{P,V}(\xi)$. In addition, the $\xi$-moments
$\langle \xi^{2,4,6} \rangle_{P,V}$ were also obtained by
integrating out $\xi$ in $\Phi_{P,V}(\xi)$. For each heavy
quarkonium state, the five parameters $m_Q$, $\beta_P$,
$\beta_{P'}$, $\beta_{V}$, $\beta_{V'}$ which appeared in the
momentum distribution amplitude were determined. This study first
extracted the decay constants $f_{V,V'}$ from the experimental data
of the leptonic decay $Br(V\to e^+e^-)$, and it then used the Van
Royen-Weisskopf formula to obtain the decay constants $f_{P,P'}$.
These decay constants were used as constraints to fix the above
parameters. Then, the curves of the quark distribution amplitudes
for the $1S$ and $2S$ heavy quarkonium states were plotted by the
fixed parameters. It was found that, for the $1S$ charmonium state,
the momentum fraction $x$ in the function used by Ref.
\cite{Braguta1} was more centered on $1/2$ than the one in the
Gaussian-type wave function. In addition, the $x$-distribution of
the pseudoscalar bottomonium was almost the same as that of the
vector bottomonium. The reason for this was that the differences
between these heavy quarkonium states, which arise from $1/m_Q$
corrections, become small when $m_Q$ is large. Finally, the
numerical results of the $\xi$-moments were calculated and compared
with the experimental data and other theoretical predictions.

{\bf Acknowledgements}\\
The author would like to thank Hsiang-nan Li, Tsung-Wen Yeh, and
Alexey Luchinsky for their helpful discussions.
 This work is supported in part by the National Science Council of R.O.C. under Grant No
 NSC-96-2112-M-017-002-MY3.

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