Exclusive C ¼ þ charmonium production in eþe� ! Hþ � at B factorieswithin the light cone formalism

V.V. Braguta*

Institute for High Energy Physics, Protvino, Russia(Received 6 July 2010; published 8 October 2010)

In this paper the cross sections of the processes eþe� ! H þ �, H ¼ �c, �0c, �c0, �c1, �c2 are

calculated. The calculation is carried out at the leading twist approximation of the light cone formalism.

Within this approach the leading logarithmic radiative and relativistic corrections to the amplitudes are

resummed. For the processes eþe� ! �c, �0c þ � one-loop radiative corrections are taken into account. It

is also shown that one-loop leading logarithmic radiative corrections calculated within the light cone

formalism for the processes under study coincide with that obtained by direct calculations of one-loop

diagrams within nonrelativistic QCD.

DOI: 10.1103/PhysRevD.82.074009 PACS numbers: 12.38.�t, 12.38.Bx, 13.25.Gv, 13.66.Bc

I. INTRODUCTION

The theoretical approach to the description of hard ex-clusive processes, which is called light cone formalism(LC), is based on the factorization theorem [1,2]. Withinthis theorem the amplitude of the hard exclusive processcan be separated into two parts. The first part is partonproduction at very small distances, which can be treatedwithin perturbative QCD. The second part is the hardroni-zation of the partons at large distances. For hard exclusiveprocesses it can be parametrized by process independentdistribution amplitudes (DA).

The production of the charmonium meson H in theprocess eþe� ! H þ � at B factories, is the simplestexample of the hard exclusive process. One can assumethat the energy at which B factories operate is sufficientlylarge so that it is possible to apply LC. Another approach tothe calculation of the cross section of this process isnonrelativistic QCD (NRQCD) [3]. This approach is basedon the assumption that the relative velocity of a quark-antiquark pair in charmonia is a small parameter in whichthe amplitude of charmonium production can be expanded.LC has two very important advantages in comparison tothe NRQCD. The first advantage is that the LC formalismcan be applied for light or heavy mesons if DA of thismeson is known. From the NRQCD perspective, thismeans that LC resums the whole series of relativisticcorrections to the amplitude under study. For NRQCD,relativistic corrections are very important especially forthe production of the exited charmonia mesons. The sec-ond advantage is that within LC one can resum leadinglogarithmic radiative corrections to the amplitude in allloops. The main disadvantage of LC is that within thisformalism it is rather difficult to control power correctionsto the amplitude.

Within NRQCD the process eþe� ! H þ � was con-sidered in papers [4–6]. In paper [4] this process wasconsidered at the leading order approximation in relativevelocity and strong coupling constant. The authors of paper[5] took into account one-loop radiative corrections. Inaddition to the radiative corrections the first-order relativ-istic corrections to the process eþe� ! �c þ � were con-sidered in paper [6].The only process considered within LC is eþe� !

�c þ � [7,8]. The main drawback of these papers is thatthe authors used a very simple model of DA of the �c

meson, which does not take into account relativistic motionin this meson. Recently, the leading twist DAs of charmo-nia mesons have become the object of intensive study[9–17]. The study of these DAs allowed one to buildsome models for the charmonia DAs, that can be used inthe calculation of different exclusive processes.In this paper the leading twist processes eþe� ! H þ �

will be considered. Using helicity selection rules [18–20]it is not difficult to show that at the leading twistaccuracy the mesons with the longitudinal polarizationand the following quantum numbers H ¼ 1S0,

3P1,3P2,

3P3 can be produced. So, in this paper the following

processes will be considered: eþe� ! H þ �, H ¼ �c,�0c, �c0, �c1, �c2. To calculate the cross sections of

these processes the model of the DAs proposed in papers[11–13,16] will be used.This paper is organized as follows. In the next

section the amplitudes of the processes under considerationwill be derived. Numerical results and the discussionof these results will be given in the last section ofthis paper.

II. THE AMPLITUDE OF THEPROCESS eþe� ! Hþ �

In this section the leading twist approximation for theamplitude of the processes eþe� ! H þ �, H ¼ �c, �

0c,*braguta@mail.ru

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�c0, �c1, �c2 will be derived. The diagrams that contrib-ute to the processes at the leading order approximation inthe strong coupling constant are shown in Fig. 1 As wasnoted in the introduction, selection rules tell us that atthe leading twist accuracy all produced mesons are longi-tudinally polarized. So, the polarization vectors of thesemesons are proportional to the momentum of thesemesons.

To calculate the amplitudes and the cross sections ofthe processes involved one needs the expressions for thefollowing matrix element of the electromagnetic currentJ�ð0Þ: hHðpÞ�ðkÞjJ�ð0Þj0i. For the production of the lon-

gitudinally polarized �c, �0c, �c1 mesons it can be parame-

trized as follows:

hHðpÞ�ðkÞjJ�ð0Þj0i ¼ FHe������p�k�; (1)

where �� is the polarization vector of the final photon.It causes no difficulties to find the expression for theform factor FH¼�c;�

0c;�c1

at the leading twist approximation

F�c;�0c;�c1

¼16�Q2cf�c;�

0c;�c1

s

Z 1

�1d

��c;�0c;�c1

ð;�Þð1�2Þ ; (2)

where Qc is the charge of c-quark, the definitions of theconstants f�c;�

0c;�c1

and the DAs ��c;�0c;�c1

ð;�Þ can be

found in the Appendix, is the fraction of relative mo-mentum of the whole meson carried by the quark-antiquarkpair, � is the characteristic scale of the process, and s ¼ðpþ kÞ2.

The expression for the production amplitude of thelongitudinally polarized �c0, �c2 mesons can be writtenin the following form:

hHðpÞ�ðkÞjJ�ð0Þj0i ¼ FHðð�qÞk� � ðkqÞ��Þ; (3)

where q ¼ kþ p. The other designations are the same aswere used in Eq. (1). The expression for the form factorFH¼�c0;�c2

has the form

F�c0;�c2¼ 16�Q2

cf�c0;�c2

s

Z 1

�1d

��c0;�c2ð;�Þ

ð1� 2Þ : (4)

The constants f�c0;�c2and the DAs ��c0;�c2

ð;�Þ can be

found in the Appendix.The cross section of the processes can be written in the

following form:

�H ¼ �

24F2H

�1�M2

H

s

�: (5)

It should be noted that the matrix elements of the pro-cesses under study were taken at the leading order approxi-mation in the M2

H=s. The factor 1�M2H=s in the cross

section appeared due to the phase space of the finalparticles.The expression for the form factor FH depends on the

DAs �Hð;�Þ of the charmonia mesons. If infinitelynarrow distribution amplitudes ��c;�

0c;�c1

ð;�Þ ¼ ðÞ,��c0;�c2

ð;�Þ ¼ � 0ðÞ are substituted to formulas (2) and

(4), then NRQCD results for the amplitude will be repro-duced [4]. If real distribution amplitudes �Hð;�Þ aretaken at the scale ��mc, then formulas (2) and (4)will resum the relativistic corrections to the crosssection up to Oð1=s3Þ terms. To resum the relativistic andleading logarithmic radiative corrections simultaneouslyone must take the distribution amplitudes �Hð;�Þ at thecharacteristic scale of the process �� ffiffiffi

sp

. The calcula-tion of the cross sections will be done at the scale� ¼ ffiffiffi

sp

=2.It is interesting to note that it is possible to find the

leading logarithmic radiative corrections at the one-looplevel using formulas (2) and (4) without calculation of one-loop diagrams. Applying the approach proposed in paper[7] one gets the results

FIG. 1. The diagrams that contribute to the processes eþe� ! Hþ �,H ¼ �c, �0c, �c0, �c1, �c2 at the leading order approximation,

in the strong coupling constant.

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F�c;�0c¼ 16�Q2

c

s

ffiffiffiffiffiffiffiffiffiffihOiSmc

s �1þ Cf

�sðsÞ4

log

��2

�20

�ð3� 2 log2Þ

�;

F�c0¼ 16�Q2

c

s

ffiffiffiffiffiffiffiffiffiffihOiP3m3

c

s �1þ Cf

�sðsÞ4

log

��2

�20

�ð1� 2 log2Þ

�;

F�c1¼ 16�Q2

c

s

ffiffiffiffiffiffiffiffiffiffiffiffiffi2hOiPm3

c

s �1þ Cf

�sðsÞ4

log

��2

�20

�ð3� 2 log2Þ

�;

F�c2¼ 16�Q2

c

s

ffiffiffiffiffiffiffiffiffiffiffiffiffi2hOiP3m3

c

s �1þ Cf

�sðsÞ4

log

��2

�20

�ð1� 2 log2Þ

�;

(6)

where mc is the pole mass of the c-quark, the definition ofthe NRQCD matrix elements hOiS, hOiP can found inpaper [3], and Cf ¼ 4=3. Note that in the above equationsit was assumed that renormalization group evolution of theDAs begins at the scale�0 �mc and ends at the scale��ffiffiffis

p. At the scale �0 �mc the DAs are ��c;�

0c;�c1

ð;�Þ ¼ ðÞ, ��c0;�c2

ð;�Þ ¼ � 0ðÞ. The leading order NRQCDresults (6) coincide with the results obtained in paper [4].The one-loop leading logarithmic radiative corrections forthe F�c

coincide with the result of paper [7]. The one-loopleading logarithmic radiative corrections for the F�c

, F�c0,

F�c1, F�c2

coincide with the results of paper [6].The result ofEq. (2) for the productionof the pseudoscalar

mesons�c,�0c can be improved since there exists expression

for the one-loop radiative correction to this amplitude[21,22]. This expression can be written as follows [21]:

F�c;�0c¼ 16�Q2

cf�c;�0c

s

Z 1

�1d

��c;�0cð;�Þ

ð1þ Þ�

�1þ Cf

�sðsÞ4

�log2

�1þ

2

�

� 1þ

1� log

�1þ

2

�� 9

þ�3þ 2 log

�1þ

2

��log

�s

�2

���: (7)

In the above expressions it is assumed that the DAs ��c;�0c

are -even.It is instructive to take the limit of zero relative velocity

of the quark-antiquark pair and to compare it to theNRQCD result [6]. At the leading order approximation inthe relative velocity expression (7) becomes

F�c;�0c¼ 16�Q2

c

s

ffiffiffiffiffiffiffiffiffiffihOiSmc

s

��1þ Cf

�sðsÞ4

log

��2

�20

�ð3� 2 log2Þ

þ Cf

�sðsÞ4

�log22þ log2� 9

þ log

�s

�2

�ð3� 2 log2Þ

��: (8)

The second term in Eq. (8) is due to renormalization groupresummation of the leading logarithms in the DA. The lastterm is one-loop radiative corrections to the hard part of theamplitude. The factorization scale � separates the longdistance dynamic of the charmonium meson parametrizedby the DA from the small distance effects parametrized inthe hard part of the amplitude. It is seen that� dependenceis canceled in the final answer, as it should be.The authors of paper [6] obtained the following NRQCD

expression for Eq. (8):

F�c;�0c¼ 16�Q2

c

s

ffiffiffiffiffiffiffiffiffiffihOiSmc

s

��1þ Cf

�sðsÞ4

�log22þ 3 log2� 9

� 2

3þ log

�s

m2c

�ð3� 2 log2Þ

��: (9)

It is seen that this expression is very similar to (8).Moreover one has one free parameter �0 in expres-sion (8), which can be used to adjust (8) to (9).However, expressions (8) and (9) seem to be a little bitdifferent.

III. NUMERICAL RESULTS AND DISCUSSION.

To obtain numerical results for the cross sections of theprocesses under study the following numerical parametersare needed.In this paper we are going to use the models of

the charmonia DAs proposed in papers [11–13,16].For the strong coupling constant we use the one-loopexpression

�sð�Þ ¼ 4

b0 lnð�2=�2QCDÞ

;

where b0 ¼ 25=3 and �QCD ¼ 0:2 GeV.In the calculation the following values of the constants

fH will be used [23]

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f�c¼ 0:373� 0:064 GeV;

f�0c¼ 0:261� 0:077 GeV;

f�c0ðMJ=�Þ ¼ 0:093� 0:017 GeV;

f�c1¼ 0:272� 0:048 GeV;

f�c2ðMJ=�Þ ¼ 0:131� 0:023 GeV:

(10)

The values of the constants f�c, f�0

cwere calculated in

paper [24]. The values of the constants of the P-wavecharmonia mesons can be found in paper [16]. It shouldbe noted that the constants f�c0

, f�c2depend on the renor-

malization scale. As seen from formulas (10) these con-stants are defined at the scale � ¼ MJ=�. The anomalous

dimensions of these constants, which govern the evolution,can be found in paper [16].

There are different sources of uncertainty to the resultsobtained in this paper. The most important uncertaintiescan be divided into the following groups:

(1) The uncertainty in the models of the distributionamplitudes �Hðx;�Þ, which can be modeled bythe variation of the parameters of these models.The calculation shows that this source of uncertaintyis not greater than 10%. So it is not very importantand it will be ignored.

(2) The uncertainty due to radiative corrections. In theapproach applied in this paper the leading logarith-mic radiative corrections due to the evolution ofthe DAs and the strong coupling constant wereresummed. For the processes eþe� ! �c, �

0c þ �

one-loop radiative corrections were taken into ac-count. So, for the last two processes, radiative cor-rections are not very important and they will beignored. As to the other processes considered inthis paper radiative corrections to the results canbe estimated as �sðsÞ � 20%.

(3) The uncertainty due to power corrections. This un-certainty is determined by the next-to-leading ordercontribution in the 1=s expansion. One can estimatethese corrections using the leading order NRQCDpredictions [4], as was discussed in paper [23].Thus, for the processes eþe� ! �c, �

0c, �c0, �c1,

�c2 þ � the errors due to this source of uncertaintyare �3%, 6%, 50%, 37%, 60% correspondingly.

(4) The uncertainty in the values of constants (10). Thecalculations show that, for the processes eþe� !�c, �

0c, �c0, �c1, �c2 þ �, the errors due to this

source of uncertainties are �34%, 60%, 35%,35%, 35% correspondingly.

Adding all these uncertainties in quadrature one gets thetotal errors of the calculation.The results of the calculation are presented in Table I.

The second column contains the results obtained in thispaper. In the third, fourth, and fifth columns the resultsobtained in papers [4–6] are shown. It is seen that theresults obtained in this paper are in reasonable agreementwith the results obtained within NRQCD.

ACKNOWLEDGMENTS

This work was partially supported by the RussianFoundation of Basic Research under Grant No. 08-02-00661, grant 09-01-12123, Grant No. 10-02-00061,Leading Scientific Schools Grant No. NSh-6260.2010.2and the president Grant No. MK-140.2009.2.

APPENDIX: DISTRIBUTION AMPLITUDES.

The leading twist distribution amplitudes needed in thecalculation can be defined as follows: for the pseudoscalarmesons P ¼ �c, �

0c

hPðpÞj �Qi�ðzÞ½z;�z�Qj

�ð�zÞj0i

¼ ðp̂�5Þ�� fP4 ij

3

Z 1

�1deiðpzÞ��c

ð;�Þ;

for the �c0 meson

h�c0ðpÞj �Qi�ðzÞ½z;�z�Qj

�ð�zÞj0i

¼ ðp̂Þ��f�0

4

ij

3

Z 1

�1deiðpzÞ��0

ð;�Þ;

for the �c1 meson

h�c1ðp; ��¼0Þj �Qi�ðzÞ½z;�z�Qj

�ð�zÞj0i

¼ ðp̂�5Þ��f�1

4

ij

3

Z 1

�1deiðpzÞ��1

ð;�Þ;

for the �c2 meson

TABLE I. The cross sections of the processes eþe� ! H þ �, H ¼ �c, �0c, �c0, �c1, �c2. The second column contains the results

obtained in this paper. In the third, fourth, and fifth columns the results obtained in papers [4–6] are shown.

H �ðeþe� ! H þ �Þ (fb) This work �ðeþe� ! H þ �Þ (fb) [4] �ðeþe� ! Hþ �Þ (fb) [5] �ðeþe� ! H þ �Þ (fb) [6]�c 41:6� 14:1 82:0þ21:4�19:8 42.5–53.7 68:0þ22:2�20:3

�0c 24:2� 14:5 49:2þ9:4

�7:4 27.7–35.1 42:6þ10:9�8:8

�c0 6:1� 3:9 1:3þ0:2�0:2 1.53–2.48 1:36þ0:26

�0:26

�c1 24:2� 13:3 13:7þ3:4�3:1 11.1–17.7 10:9þ3:7

�3:4

�c2 12:0� 17:4 5:3þ1:6�1:3 1.65–3.53 1:95þ1:85

�1:56

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h�c2ðp; ��¼0Þj �Qi�ðzÞ½z;�z�Qj

�ð�zÞj0i

¼ ðp̂Þ��f�2

4

ij

3

Z 1

�1deiðpzÞ��2

ð;�Þ:

The factor ½z;�z�, that makes the above matrix elementsgauge invariant, is defined as

½z;�z� ¼ P exp

�ig

Z z

�zdx�A�ðxÞ

�:

It is not difficult to show that the functions ��cðÞ and

��1ðÞ are -even. The normalization condition for these

functions is Z 1

�1�ðÞd ¼ 1:

The functions ��0ðÞ and ��2

ðÞ are -odd and normal-

ized according to Z 1

�1�ðÞd ¼ 1:

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