meson

Study of leading twist light cone wave functions of the J=� meson

V. V. Braguta*Institute for High Energy Physics, Protvino, Russia

(Received 6 February 2007; published 16 May 2007)

This paper is devoted to the study of leading twist light cone wave functions of the J=� meson. Themoments of these wave functions have been calculated within three approaches: potential models,nonrelativistic QCD, and QCD sum rules. Using the results obtained within these approaches the modelsfor the light cone wave functions of leading twist have been proposed. Similarly to the wave function ofthe �c meson, the leading twist light cone wave functions of the J=� meson have very interestingproperties at scales �>mc: improvement of the accuracy of the model, appearance of a relativistic tail,and violation of nonrelativistic QCD velocity scaling rules. The last two properties are the properties oftrue leading twist light cone wave functions of the J=� meson.

DOI: 10.1103/PhysRevD.75.094016 PACS numbers: 12.38.�t, 12.38.Bx, 13.25.Gv, 13.66.Bc

I. INTRODUCTION

Commonly, exclusive charmonium production at highenergies is studied within nonrelativistic QCD (NRQCD)[1]. In the framework of this approach charmonium isconsidered as a bound state of a quark-antiquark pairmoving with small relative velocity v� 1. Because ofthe presence of a small parameter v, the amplitude ofcharmonium production can be built as an expansion inrelative velocity v.

Thus in the framework of NRQCD the amplitude of anyprocess is a series in relative velocity v. Usually, in most ofthe applications of NRQCD, the consideration is restrictedby the leading order approximation in relative velocity.However, this approximation has two problems whichmake it unreliable. The first problem is connected withthe rather large value of relative velocity for charmonium:v2 � 0:3, v� 0:5. For this value of v2 one can expect alarge contribution from relativistic corrections in any pro-cess. So in any process resummation of relativistic correc-tions should be done or one should prove that resummationof all terms is not crucial. The second problem is connectedwith QCD radiative corrections. The point is that due to thepresence of a large energy scale Q there appear largelogarithmic terms ��s logQ=mc�

n, Q� mc which can beeven more important than relativistic corrections at suffi-ciently large energy (Q� 10 GeV). So these terms shouldalso be resummed. In principle, it is possible to resum largelogarithms in the NRQCD factorization framework [2,3];however such resummation is rarely done.

The illustration of all mentioned facts is the process ofdouble charmonium production in e�e� annihilation at Bfactories, where leading order NRQCD predictions [4–6]are approximately by an order of magnitude less thanexperimental results [7,8]. The calculation of QCD radia-tive corrections [9] diminished this disagreement but didnot remove it. Agreement with the experiments can proba-

bly be achieved if, in addition to QCD radiative correc-tions, relativistic corrections will be resummed [10,11].

In addition to NRQCD, hard exclusive processes can bestudied in the framework of light cone expansion formal-ism [12,13] where both problems mentioned above can besolved. Within light cone expansion formalism the ampli-tude is built as an expansion over inverse powers of thecharacteristic energy of the process. Usually this approachis successfully applied to the exclusive production of lightmesons [13]. However, recently the application of lightcone expansion formalism to double charmonium produc-tion [14–17] allowed one to achieve good agreement withthe experiments.

In the framework of light cone formalism the amplitudeof some meson production in any hard process can bewritten as a convolution of the hard part of the process,which can be calculated using perturbative QCD, and theprocess independent light cone wave function (LCWF) ofthis meson that parametrizes nonperturbative effects. Fromthis, one can conclude that charmonium LCWFs are keyingredients of any hard exclusive process with charmo-nium production. In Ref. [18] the leading twist light conewave function of the �c meson was studied. This paper isdevoted to the study of leading twist LCWFs of the J=�meson.

The paper is organized as follows. In the next section alldefinitions needed in our calculation will be given. InSec. III the moments of LCWFs will be calculated in theframework of Buchmuller-Tye and Cornell potential mod-els. Section IV is devoted to the calculation of the momentswithin NRQCD. QCD sum rules will be applied to thecalculation of the moments in Sec. V. Using the resultsobtained in Secs. III, IV, and V the models of LCWFs willbe built in Sec. VI. In the last section the results of thispaper will be summarized.

II. DEFINITIONS

There are two leading twist LCWF of the J=� meson.The first one is the twist 2 LCWF of the longitudinally*Electronic address: [email protected]

PHYSICAL REVIEW D 75, 094016 (2007)

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http://dx.doi.org/10.1103/PhysRevD.75.094016

polarized J=� meson �L��;��. The second one is thetwist 2 LCWF of the transversely polarized J=� meson�T��;��. These LCWFs can be defined as follows [13]:

h0j �Q�z��z;�zQ��z�jJ=��0; p�i�

fLp�Z 1

�1d�ei�pz��L��;��;

h0j �Q�z��z;�zQ��z�jJ=��1; p�i�

fT��p � �p��Z 1

�1d�ei�pz��T��;��;

(1)

where the following designations are used: x1, x2 are theparts of the momentum carried by the quark and antiquarkcorrespondingly, � x1 � x2, p is a momentum of theJ=� meson, and � is an energy scale. The factor �z;�z,that makes the matrix element (1) gauge invariant, isdefined as

�z;�z P exp�igZ z

�zdx�A��x�

�: (2)

The LCWFs �L;T��;�� are normalized as

Z 1

�1d��L;T��;�� 1: (3)

With this normalization condition the constants fT;L aredefined as

h0j �Q�0��Q�0�jJ=��0; p�i fL��0;

h0j �Q�0��Q�0�jJ=��1; p�i�

fT��p � �p��:

(4)

It should be noted here that the local current �Q�0��Q�0� isrenormalization group invariant. The local current�Q�0��Q�0� is not invariant. For this reason the constantfT�� depends on the scale � as

fT�� s��s��0�

�4=3b0

fT��0�; (5)

but the constant fL does not depend on the scale.LCWFs�L;T�x;�� can be expanded [13] in Gegenbauer

polynomials C3=2n �� as follows:

�L;T��;�� 3

4�1� �2�

�1�

Xn2;4...

aL;Tn ��C3=2n ��

�:

(6)

At leading logarithmic accuracy the coefficients aL;Tn arerenormalized multiplicatively,

aL;Tn �� s��s��0�

��L;Tn =b0

aL;Tn ��0�; (7)

where

�Ln 4

3

�1�

2

�n� 1��n� 2�� 4

Xn�1

j2

1

j

�;

�Tn 4

3

�4Xn�1

j2

1

j

�; b0 11�

2

3nfl:

(8)

It should be noted here that conformal expansions (6) arethe solution of the Bethe-Salpeter equation with one gluonexchange kernel [12].

From Eqs. (6)–(8) it is not difficult to see that at aninfinitely large energy scale �! 1 LCWFs �T;L��;��tend to the asymptotic form �as�� 3=4�1� �2�. But atenergy scales accessible at current experiments, theLCWFs �T;L��;�� are far from their asymptotic forms.The main goal of this paper is to calculate the LCWFs�L;T��;�� of the J=� meson. These LCWFs will beparametrized by their moments at some scale:

h�nL;Ti� Z 1

�1d��n�L;T��;��: (9)

It is worth noting that since the J=� meson has negativecharge parity the LCWFs �L;T��;�� are � even. Thus allodd moments h�2k�1

L;T i equal zero and one needs to calculateonly even moments.

Below, the following formulas will be used in our cal-culation:

h0j �Q��izD$

�nQjJ=��0; p�i fLp��zp�

nh�nLi;

h0j �Q��izD$

�nQjJ=��1; p�i

fT��p� � ��p��zp�nh�nTi; (10)

where

D$

~D�D

; ~D ~@! �igBa��a=2�: (11)

These formulas can be obtained if one expands both sidesof Eqs. (1).

III. THE MOMENTS IN THE FRAMEWORK OFPOTENTIAL MODELS

In potential models charmonium mesons are consideredas a quark-antiquark system bounded by some static po-tential. These models allow one to understand many prop-erties of charmonium mesons. For instance, the spectrumof the charmonium family can be well reproduced in theframework of potential models [19]. Because of this suc-cess one can hope that potential models can be applied tothe calculation of charmonium equal time wave functions.

Having the equal time wave function of the J=� mesonin momentum space �k�, one can apply the Brodsky-Huang-Lepage (BHL) [20] procedure and get the LCWFsof leading twist �L;T��;�� using the following rule:

�L;T��;�� Z k2

?<�2

d2k? c�x;k?�; (12)

V. V. BRAGUTA PHYSICAL REVIEW D 75, 094016 (2007)

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where c�x;k?� can be obtained from �k� after thesubstitution

k? ! k?; kz ! �x1 � x2�M0

2;

M20

M2c � k2

?

x1x2:

(13)

Here Mc is a quark mass in the potential model. In thispaper the equal time wave function �k� will be calculatedin the framework of the potential models with Buchmuller-Tye [21] and Cornell potentials [22]. The parameters of theBuchmuller-Tye potential model were taken fromRef. [21]. For the Cornell potential V�r� �k=r� r=a2

the calculation was carried out with the following set ofparameters: k 0:61, a 2:38 GeV�1, Mc 1:84 GeV[23].

In Ref. [18] the moments of the leading twist LCWF ofthe �c meson were calculated within potential models withthese potentials. At the leading order approximation inrelative velocity there is no difference between equaltime wave functions of �c and J=� mesons. In whatfollows the moments obtained in Ref. [18] for the leadingtwist LCWF of the �c meson equal the moments ofLCWFs �L;T��;�� of the J=� meson. Within this ap-proximation there is no difference between �L��;�� and�T��;��.

It is worth noting that in Ref. [24] the relations betweenthe light cone wave functions and equal time wave func-tions of charmonium mesons in the rest frame were de-rived. The procedure proposed in Ref. [24] is similar tothe BHL procedure with the following difference: in for-

mula (12) one must make the substitution d2k? !

d2k?��k2 �m2

c

p=�4mcx1x2�. But this substitution was de-

rived at the leading order approximation in the relativevelocity of the quark-antiquark motion inside the charmo-nium. At this approximation k2 �O�v2�, 4x1x2 � 1�O�v2�, and the substitution amounts to d2k? ! d2k?�1�O�v2��. Thus at the leading order approximation applied in[24], these two approaches coincide.

Another approach to the calculation of LCWF wasproposed in Ref. [10]. Similarly to Ref. [24], this approachdiffers from the BHL procedure (12) by the factor �1�O�v2�. So at the leading order approximation in relative

velocity, the approach proposed in Ref. [10] coincides withthe BHL procedure (12).

The results of Ref. [18] are presented in Table I (see thispaper for details). In the second and third columns themoments calculated in the framework of the Buchmuller-Tye and Cornell models are presented. It should be notedthat the moments from Table I were calculated at the scale�� 1:5 GeV. It is seen that there is good agreementbetween these two models.

It should be noted here that the larger the power of themoment, the larger the contribution from the end pointregions (x� 0 and x� 1) it gets. From formulas (13) onesees that the motion of the quark-antiquark pair in thisregion is relativistic and cannot be considered reliably inthe framework of potential models. Thus it is not possibleto calculate higher moments within the potential models.Because of this fact the calculation of the moments hasbeen restricted by few first moments.

IV. THE MOMENTS IN THE FRAMEWORK OFNRQCD

In Ref. [18] the relations that allow one to connect themoments of the leading twist LCWF of the �c meson withNRQCD matrix elements were derived:

h�2i 1

3M2c

h0j��iD$

�2 j�cih0j�� j�ci

hv2i

3;

h�4i 1

5M4c

h0j��iD$


hv4i

5;

h�6i 1

7M6c

h0j��iD$


hv6i

7;

(14)

where and �� are Pauli spinor fields that annihilate aquark and an antiquark, respectively, Mc MJ=�=2. Themoments are defined at the scale ��Mc.

These relations were derived at the leading order ap-proximation in relative velocity. However, as it was notedabove, at this approximation there is no difference between�c and J=� mesons. Moreover, there is no differencebetween LCWFs �L��;�� and �T��;��. So the valuesfor the moments of LCWFs �L��;��, �T��;�� can betaken from Ref. [18].

TABLE I. The moments of LCWFs �L��;��, �T��;�� obtained within different approaches. In the second and third columns themoments calculated in the framework of the Buchmuller-Tye and Cornell potential models are presented. In the fourth columnNRQCD predictions for the moments are presented. In the last two columns the results obtained within QCD sum rules are shown.

h�niBuchmuller-Tye

model [21]Cornell

model [22] NRQCD [25]QCD sum

rules �L��;��QCD sum

rules �T��;��

n 2 0.086 0.084 0:075� 0:011 0:070� 0:007 0:072� 0:007n 4 0.020 0.019 0:010� 0:003 0:012� 0:002 0:012� 0:002n 6 0.0066 0.0066 0:0017� 0:0007 0:0031� 0:0008 0:0033� 0:0007

STUDY OF LEADING TWIST LIGHT CONE WAVE . . . PHYSICAL REVIEW D 75, 094016 (2007)

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The results of the calculation of the moments are pre-sented in the fourth column of Table I. The central valuesof the moments and the errors due to the model uncertaintyhave been calculated according to the approach proposedin Ref. [25]. In addition to the error shown in Table I thereis an uncertainty due to higher order v corrections. For thesecond moment one can expect that this error is about 30%.For higher moments this error is larger.

It is seen from Table I that NRQCD predictions for thesecond and the fourth moments are in good agreement withthe potential model, and there is disagreement for themoment h�6i between these two approaches. The causeof this disagreement is the fact noted above: due to thelarge contribution of the relativistic motion of the quark-antiquark pair inside quarkonium, it is not possible to applyboth approaches for higher moments. So one can expectthat both approaches can be used for the estimation of thevalues of the second and the fourth moments. The predic-tions for the sixth and higher moments become unreliable.

V. THE MOMENTS IN THE FRAMEWORK OF QCDSUM RULES

In this section QCD sum rules [26,27] will be applied tothe calculation of the moments of LCWFs �L��;�� and�T��;�� [13,28]. First let us consider the LCWF

�L��;��. To calculate the moments of this LCWF oneshould consider a two-point correlator:

�L�z;q� iZd4xeiqxh0jTJ0�x�Jn�0�j0i �zq�

n�2�L�q2�;

J0�x� �Q�x�zQ�x�; Jn�0� �Q�0�z�iz D$

�nQ�0�;

z2 0: (15)

It is not difficult to obtain sum rules for this correlator (fordetails see Ref. [18]):

f2Lh�

nLi

�M2J=� �Q

2�m�1 1

�

Z 14m2

c

dsIm�pert�s�

�s�Q2�m�1

��m�npert�Q

2�; (16)

where perturbative and nonperturbative contributions tosum rules Im�pert�s�, ��m�

npert�Q2� can be written as

Im�pert�s� 3

8�vn�1

�1

n� 1�

v2

n� 3

�;

v2 1�4m2

c

s;

(17)

��m�npert�Q

2� ��m�1 �Q

2� ��m�2 �Q

2� ��m�3 �Q

2�;

��m�1 �Q

2� h�sG

2i

24��m� 1�

Z 1

�1d��n �

n�n� 1�

4�n�2�1� �2�

��1� �2�m�2

�4m2c �Q

2�1� �2��m�2 ;

��m�2 �Q

2� �h�sG

2i

6�m2c�m2 � 3m� 2�

Z 1

�1d��n�1� 3�2�

�1� �2�m�1

�4m2c �Q2�1� �2��m�3 ;

��m�3 �Q

2� h�sG2i

384��n2 � n��m� 1�

Z 1

�1d��n�2 �1� �2�m�3

�4m2c �Q2�1� �2��m�2 :

(18)

Here Q2 �q2, mc, and h�sG2i are parameters of QCD sum rules.To calculate the moments of the LCWF �T��;�� one should consider the correlator

�T�z; q� iZd4xeiqxh0jTJ��x�J

�n �0�j0i �zq�n�2�T�q2�; J��x� �Q�x��z��Q�x�;

J�n �0� �Q�0��z��iz D$

�nQ�0�; z2 0:

(19)

The sum rules for this correlator can be written as

f2Th�

nTi

�m2J=� �Q

2�m�1 1

�

Z 14m2

c

dsIm�pert�s�

�s�Q2�m�1

��m�npert�Q

2�; (20)

where perturbative and nonperturbative contributions tosum rules (20) are given by expressions (17) and (18)except that the expression for ��m�

1 �Q2� should be replaced

by

��m�1 �Q

2� h�sG2i

24��m� 1�

�Z 1

�1d��n �

n�n� 1�

4�n�2�1� �2�

�

��1� �2�m�2

�4m2c �Q2�1� �2��m�2 : (21)

In the original paper [27] the QCD sum rules wasapplied at Q2 0. However, as it was shown inRef. [29], there is a large contribution of higher dimen-


094016-4

sional operators atQ2 0 which grows rapidly withm. Tosuppress this contribution, sum rules (16) and (20) will beapplied at Q2 4m2

c.In the numerical analysis of QCD sum rules, the values

of parameters mc and h�sG2=�i will be taken fromRef. [29]:

mc 1:24� 0:02 GeV;��s�G2

� 0:012� 30% GeV4:

(22)

First sum rules (16) and (20) will be applied to the calcu-lation of the constants f2

L;T . It is not difficult to express theconstants f2

L;T from Eqs. (16) and (20) at n 0 as func-tions of m. For too small values of m (m<m1), there arelarge contributions from higher resonances and continuumwhich spoil sum rules (16) and (20). Although form� m1

these contributions are strongly suppressed, it is not pos-sible to apply sum rules for too large m (m>m2) since thecontribution arising from higher dimensional vacuum con-densates rapidly grows with m and invalidates the approxi-mation of this paper. If m1 <m2 there is some region ofapplicability of sum rules (16) and (20) �m1; m2 whereboth resonance and higher dimensional vacuum conden-sate contributions are not too large. Within this region f2

L;T

as functions of m vary slowly and one can determine thevalues of these constants. Applying the approach describedabove, one gets

f2L 0:170� 0:002� 0:004� 0:016 GeV2;

f2T 0:167� 0:002� 0:003� 0:016 GeV2:

(23)

The first error in (23) corresponds to the variation of theconstants f2

L;T within the region of stability. The secondand the third errors in (23) correspond to the variation ofthe gluon condensate h�sG2i and the mass mc withinranges (22). From the results (23) one sees that the mainerrors in the determination of the constants f2

L;T result fromthe variation of the parameter mc. This fact represents awell-known property: high sensitivity of QCD sum rules tothe mass parameter mc.

Next let us consider the second moments h�2L;Ti in the

framework of QCD sum rules. One way to find the valuesof h�2

L;Ti is to determine the values of f2L;Th�

2L;Ti from sum

rules (16) and (20) at n 2 and then extract h�2L;Ti.

However, as it was noted above, this approach suffersfrom high sensitivity of the right side of Eqs. (16) and(20) to the variation of the parameter mc. Moreover, thequantities f2

L;Th�2L;Ti include not only the errors in the

determination of h�2L;Ti, but also the errors in f2

L;T . Toremove these disadvantages the ratios of sum rules at n 2 and n 0, f2

L;Th�2L;Ti=f

2L;T , will be considered. The mo-

ments h�4L;Ti, h�

6L;Ti will be considered analogously.

Applying the standard procedure one gets the momentsof the LCWFs �L��;��:

h�2Li 0:070� 0:002� 0:007� 0:002;

h�4Li 0:012� 0:001� 0:002� 0:001;

h�6Li 0:0031� 0:0002� 0:0008� 0:0002;

(24)

and the moments of the LCWFs �T��;��:

h�2Ti 0:072� 0:002� 0:007� 0:002;

h�4Ti 0:012� 0:001� 0:002� 0:001;

h�6Ti 0:0033� 0:0002� 0:0007� 0:0003:

(25)

The first error in (24) and (25) corresponds to the variationwithin the region of stability. The second and the thirderrors in (24) and (25) correspond to the variation of thegluon condensate h�sG2i and the mass mc within ranges(22). It is seen that, as expected, the sensitivity of the ratiosf2L;Th�

nL;Ti=f

2L;T to the variation of mc is rather low. The

main source of uncertainty is the variation of the gluoncondensate h�sG2i. In the fifth and sixth columns ofTable I, results (24) and (25) are presented. The errors inTable I correspond to the main source of uncertainty—thevariation of the gluon condensate h�sG2i.

In the calculations of the correlators (15) and (19) thecharacteristic virtuality of the quark is��4m2

c �Q2�=m�m2c. So the values of the moments (24) and (25) are defined

at the scale �m2c.

From Table I it is seen that the larger the number of themoment n the larger the uncertainty due to the variation ofthe vacuum gluon condensate. This property is a conse-quence of the fact that the role of power corrections in thesum rules (16) and (20) grows with n. From this, one canconclude that there is a considerable nonperturbative con-tribution to the moments h�nL;Ti with large n which meansthat nonperturbative effects are very important in the rela-tivistic motion of the quark-antiquark pair inside the me-son. The second important contribution to QCD sum rules(16) and (20) at large n is from QCD radiative correctionsto the perturbative part �pert�Q

2�. Unfortunately, today onedoes not know the expression for these corrections and forthis reason they are not included in sum rules (16) and (20).One can only say that these corrections grow with n and,probably, the size of radiative corrections to the ratiosf2L;Th�

nL;Ti=f

2L;T is not too big for not too large n. Thus

one can expect that QCD radiative corrections will notchange dramatically the results for the moments n 2and n 4. But the radiative corrections to h�6

L;Ti may beimportant.

Another source of uncertainty is from the contribution ofhigher resonances to the left-hand side of Eq. (16). Themain contribution comes from the � meson. It is notdifficult to understand that this contribution can be esti-mated as

r f2

�0 h�ni�0

f2J=�h�

niJ=�

�M2J=� �Q

2

M2�0 �Q

2

�m�1

: (26)


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Within potential models the ratiof2

�0 h�ni�0=f

2J=�h�

niJ=� � 1 for n 2, 4, 6. The stabilityof sum rules (16) appears in the region m� 10. Thus theinclusion of the � meson to sum rules (16) can change ourresults by r� 10%.

It is interesting to compare the moments of the leadingtwist LCWF of the �c meson calculated in Ref. [18],

h�2�ci 0:070� 0:002� 0:007� 0:003;

h�4�ci 0:012� 0:001� 0:002� 0:001;

h�6�ci 0:0032� 0:0002� 0:0009� 0:0003;

(27)

with the results of this section. It is seen that there is nosignificant difference between the moments of the leadingtwist LCWF of the �c meson and the moments of theLCWFs �L��;��, �T��;��. The difference is within theerror of the calculation.

VI. THE MODEL FOR THE LCWFS OF THE J=�MESON

In Ref. [18] the model of the leading twist LCWF of the�c meson was proposed:

��;� �0� c��1� �2� exp��

1� �2

�; (28)

where c�� is a normalization coefficient, the constant 3:8� 0:7, and the scale�0 1:2 GeV. This functionallows one to reproduce the results (27) with rather goodaccuracy. The moments of this wave function are

h�2i 0:070� 0:007; h�4i 0:012� 0:002;

h�6i 0:0030� 0:0009:(29)

At the central value 3:8, the constant c�� ’ 62.As it was noted in the previous section the accuracy of

the calculation does not allow one to distinguish the lead-ing twist LCWF of the �c meson from LCWFs of the J=�meson. For this reason the model (28) will be used for theLCWFs �L��;� �0�, �T��;� �0�,

�L��;� �0� �T��;� �0� ��;� �0�:

(30)

In this expression the functions �L��;� �0�,�T��;� �0� are defined at the scale � �0. It is notdifficult to calculate these functions at any scale �>�0

using conformal expansions (6). The LCWFs �L��;�� atscales �0 1:2 GeV, �1 10 GeV, �2 100 GeV,�3 1 are shown in Fig. 1. The moments of LCWFs�L��;�� at scales �0 1:2 GeV, �1 10 GeV, �2 100 GeV, �3 1 are presented in the second, third,fourth, and fifth columns of Table II. The plot and themoments of the LCWF �T��;�� will not be shown sincethis function practically does not deviate from �L��;��.

As was noted in Ref. [18], model (30) has some interest-ing properties. For instance, let us consider the LCWF�L��;�� [similar consideration can be done for�T��;��]. From the conformal expansion (6) one canderive the expressions that determine the evolution of themoments:

h�2Li�

1

5� aL2 ��

12

35;

h�4Li�

3

35� aL2 ��

8

35� aL4 ��

8

77;

h�6Li�

1

21� aL2 ��

12

77� aL4 ��

120

1001� aL6 ��

64

2145:

(31)

Similar relations can be found for any moment. Further letus consider the expression for the second moment h�2

Li. Inthis paper the value h�2

Li has been found with some error atthe scale � �0. This means that the value of the coef-ficient aL2 �� 0� was found with some error. The coef-ficient aL2 decreases as the scale increases. So the error inaL2 and consequently in h�2

Li decreases as the scale in-creases. At an infinitely large scale there is no error inh�2Li at all. The calculations show that the error of 10% inh�2Li at the scale � �0 decreases to 4% at the scale �

10 GeV. Applying relations (31) it is not difficult to showthat similar improvement of the accuracy happens forhigher moments. The improvement of the accuracy allowsone to expect that model (30) at larger scales will be rathergood even if QCD radiative corrections to results (24) and(25) are large.

From Fig. 1 one sees that the LCWF at the scale� �0

practically vanishes in the region 0:75< j�j< 1. In thisregion the motion of the quark-antiquark pair is relativistic,and vanishing of the LCWF in this region means that at thescale � �0 charmonium can be considered as a non-relativistic bound state of the quark-antiquark pair withcharacteristic velocity v2 � 1=� 0:3. Further, let us re-

-1 -0.75 -0.5 -0.25 0.25 0.5 0.75 1

0.2

0.4

0.6

0.8

1

1.2

1.4 µ 0

µ 1

µ 2

µ 3

ξ

φ (ξ, µ )

FIG. 1. The LCWF �L (30) at scales �0 1:2 GeV, �1 10 GeV, �2 100 GeV, �3 1.


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gard the function�L��;� �0� as a conformal expansion(6). To get considerable suppression of the LCWF in theregion 0:75< j�j< 1, one should require fine-tuning ofthe coefficients of conformal expansion aLn �� 0�. Theevolution of the constants aLn (especially with large n) near� �0 is rather rapid [see formulas (7) and (8)], and ifthere is fine-tuning of the constants at scale � �0, thisfine-tuning will be rapidly broken at larger scales. Thisproperty is well seen in Table II and Fig. 1. From Fig. 1 it isseen that there is a relativistic tail in the region 0:75<j�j< 1 for scales � 10, 100 GeV which is absent atscale� �0. Evidently, this tail cannot be regarded in theframework of NRQCD. This means that, strictly speaking,at some scale charmonium cannot be considered as a non-relativistic particle and the application of NRQCD to theproduction of charmonium at large scales may lead to largeerror. Although in the above arguments the model ofLCWF (30) was used, it is not difficult to understand thatthe main conclusion is model independent.

According to the velocity scaling rule [1], the momentsh�nLi of the LCWF depend on relative velocity as�vn. It isnot difficult to show that the moments of LCWF (30)satisfy these rules. Now let us consider the expressionsthat allow one to connect the coefficients of conformalexpansion aLn with the moments h�nLi. These expressionsfor the moments h�2

Li, h�4Li, h�

6Li are given by formulas

(31). It causes no difficulties to find similar expressions forany moment. From expressions (31) one sees that to getvelocity scaling rules, h�nLi � v

n, at some scale one shouldrequire fine-tuning of the coefficients aLn at this scale. But,as was already noted above, if there is fine-tuning of thecoefficients at some scale this fine-tuning will be broken atlarger scales. From this one can conclude that velocityscaling rules are broken at large scales.

Consider the moments of LCWF (30) at an infinite scale.It is not difficult to find that

h�nL;Ti�1 3

�n� 1��n� 3�: (32)

From the last equation one can find that h�nL;Ti does notscale as vn, as velocity scale rules [1] require. Thus scalingrules obtained in Ref. [1] are broken for the asymptoticfunction. Actually one does not need to set the scale � toinfinity to break these rules. For any scale �>�0 there isa number n0 for which the moments h�nL;Ti, n > n0 violatevelocity scaling rules. This property is a consequence of

the following fact: beginning from some n n0 the con-tribution of the relativistic tail of the LCWF, which appearsat scales �>�0, to the moments becomes considerable.

The amplitude T of any hard process with charmoniummeson production can be written as a convolution of theLCWF �� with the hard kernel H�� of the process. Ifone expands this kernel over � and substitutes this expan-sion to the amplitude T, one gets the results

T Zd�H��

Xn

H�n��0�n!

h�ni: (33)

If one takes the scale ��0 in formula (33), then mo-ments h�ni scale according to the velocity scaling rules�vn and one gets the usual NRQCD expansion of theamplitude. However, due to the presence of the scale ofthe hard process �h � �0, there appear large logarithmslog�h=�0 which spoil NRQCD expansion (33). To removethese large logarithms one should take ��h. But atlarge scales velocity scaling rules are broken and theapplication of NRQCD is questionable.

In Refs. [14,24,30] different models of LCWFs of J=�and �c mesons were proposed. It is interesting to comparethe models proposed in these papers with model (30). Suchcomparison was done in Ref. [18] and it will not bediscussed in this paper.

VII. CONCLUSION

In this paper the moments of leading twist LCWFs of theJ=� meson have been calculated within three approaches.In the first approach the Buchmuller-Tye and Cornellpotential models were applied to the calculation of themoments of LCWFs. In the second approach the momentsof LCWFs were calculated in the framework of NRQCD.In the third approach the QCD sum rules was applied to thecalculation of the moments. The results obtained withinthese three approaches are in good agreement with eachother for the second moment h�2i. There is a little disagree-ment between the predictions for the fourth moment h�4i.The disagreement between the approaches becomes dra-matic for the sixth moment h�6i. The cause of this dis-agreement consists in the considerable contribution of therelativistic motion of the quark-antiquark pair inside theJ=� meson to higher moments which cannot be regardedreliably in the framework of potential models andNRQCD. The approach based on QCD sum rules is morereliable, especially for higher moments since it does not

TABLE II. The moments of LCWF (30) proposed in this paper at scales �0 1:2 GeV, �1 10 GeV, �2 100 GeV, �3 1 are presented in the second, third, fourth, and fifth columns.

h�ni ��;�0 1:2 GeV� ��;�1 10 GeV� ��;�2 100 GeV� ��;�3 1�

n 2 0.070 0.12 0.14 0.20n 4 0.012 0.040 0.052 0.086n 6 0.0032 0.019 0.026 0.048


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consider the J=� meson as a nonrelativistic object. Themain problem of QCD sum rules is that since there are noexpressions of radiative corrections to sum rules one doesnot know the size of these corrections. However, one canexpect that QCD radiative corrections will not change theresults for the moments n 2 and n 4 dramatically. Asto the sixth moment, the contribution the QCD radiativecorrections in this case may be important.

The moments of leading twist LCWFs of the J=�meson have been compared with the moments of theLCWF of �c. It was found that there is no significantdifference between the moments of the leading twistLCWF of the �c meson and the moments of LCWFs�L��;��, �T��;��. The difference is within the error ofthe calculation. For this reason the model of the LCWF ofthe �c meson was taken as a model for leading twistLCWFs �L��;��, �T��;�� of the J=� meson. As it wasshown in Ref. [18] this model has some interesting prop-erties:

(1) Because of the evolution (6) the accuracy of themoments obtained within model (30) improves asthe scale rises. For instance, if the error in determi-nation of the moment h�2

L;Ti is 10% at scale � �0 1:2 GeV, at scale � 10 GeV the error is4%. For higher moments the improvement of theaccuracy is even better, and there is no error at all atthe infinite scale � 1. The improvement of theaccuracy allows one to expect that model (30) will

be rather good even after inclusion of the QCDradiative corrections.

(2) At the scale ��0 the LCWFs can be consideredas wave functions of nonrelativistic objects withcharacteristic width�v2 � 0:3. Because of the evo-lution, at larger scales a relativistic tail appears. Thistail cannot be considered in the framework ofNRQCD and, strictly speaking, at these scales theJ=� meson is not a nonrelativistic object.

(3) It was found that, due to the presence of a highmomentum tail in the LCWFs at scales �>�0,there is a violation of the velocity scaling rulesobtained in Ref. [1]. More exactly, for any scale�>�0 there is a number n0 for which the momentsh�nL;Ti, n > n0 violate NRQCD velocity scalingrules.

Actually, the last two properties are properties of realLCWFs of the J=� meson.

ACKNOWLEDGMENTS

The author thanks A. V. Luchinsky and A. K. Likhodedfor useful discussion and help in preparing this paper. Thiswork is partially supported by Russian Foundation of BasicResearch under Grant No. 04-02-17530, RussianEducation Ministry Grant No. RNP-2.2.2.3.6646, CRDFGrant No. MO-011-0, President Grant No. MK-1863.2005.02, and the Dynasty Foundation.

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