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arXiv:hep-ph/0305084v1 8 May 2003
Production of J/ψ +c¯cthrough two photons in e+e−annihilation
Kui-Yong Liu and Zhi-Guo He
Department of Physics, Peking University, Beijing 100871, People’s Republic of China
Kuang-Ta Chao
China Center of Advanced Science and Technology (World Laboratory), Beijing 100080, People’s
Republic of China and Department of Physics, Peking University, Beijing 100871, People’s
Republic of China
Abstract
We study the production of J/ψ +c¯cin e+e−annihilation through two
virtual photons. The cross section is estimated to be 23 fb at √s= 10.6
GeV, which is smaller by a factor of six than the calculated cross section
for the same process but through one virtual photon. As a result, while the
annihilation into two photons may be important for certain exclusive pro-
duction processes, the big gap between the inclusive production cross section
σ(e+e−→J/ψ +c¯c)≃0.9pb observed by Belle and the current nonrelativistic
QCD prediction of ≃0.15pb still remains very puzzling. We find, however, as
the center-of-mass energy increases (√s > 20 GeV) the production through
two virtual photons e+e−→2γ∗→J/ψ +c¯cwill prevail over that through
one virtual photon, because in the former process the photon fragmentation
into J/ψ and into the charmed quark pair becomes more important at higher
energies.
PACS number(s): 12.40.Nn, 13.85.Ni, 14.40.Gx
Charmonium production is one of the important processes to test quantum chro-
modynamics (QCD) both perturbatively and non-perturbatively. Because of the
simpler parton structure involved, which will reduce the theoretical uncertainty,
charmonium production in e+e−annihilation is expected to be more decisive in
clarifying the production mechanisms of heavy quarkonia. The two B factories with
BaBar and Belle are collecting huge data samples of continuum e+e−annihilation
events, which will allow us to have a fine data analysis for charmonium production.
Recently, Charmonium production in e+e−annihilation has become more interesting
and puzzling, because of the large gap between the Belle measurements[1, 2] and the
theoretical calculations for both inclusice[3, 4, 5, 6] and exclusive [7, 8] charmonium
production via double c¯cpairs based on nonrelativistic QCD (NRQCD).
For the inclusive processes, Belle has reported a measurement on the J/ψ produc-
tion in e+e−annihilation at √s= 10.6 GeV[1, 2], and found that a very large fraction
of the produced J/ψ is due to the double c¯cproduction in e+e−annihilation[1]
σ(e+e−→J/ψc¯c)/σ(e+e−→J/ψX ) = 0.59+0.15
−0.13 ±0.12,(1)
1
which corresponds to [1, 2]
σ(e+e−→J/ψc¯c)≈0.9pb. (2)
In contrast, the predicted values for the cross section in NRQCD (the color-
octet contribution is negligible for this process) are much smaller than the data[3,
4, 5]. In a recent analytical calculation and numerical estimation for the inclusive
charmonia production including all S-wave, P-wave and D-wave states via double c¯c
in NRQCD[8], we find
σ(e+e−→γ∗→J/ψc¯c)≈0.15pb. (3)
This value is consistent with other previous calculations, including those obtained
based on the quark-hadron duality hypothesis[9], but smaller than the Belle data
by about a factor of six1.
For the exclusive processes, the Belle measurement[1] for the cross section of
e+e−→J/ψ +ηcis about an order of magnitude larger than the NRQCD calculation
for e+e−→γ∗→J/ψ +ηc[7, 8]. In order to solve the problem, calculations for the
exclusive double-charmonium production from e+e−annihilation into two virtual
photons are performed[10], and it is pointed out that the cross section for e+e−→
2γ∗→J/ψ +J/ψ is larger than that for e+e−→γ∗→J /ψ +ηcby about a factor of
3.7, despite of possible uncertainties due to the choice of input parameters[11]. This
is a interesting result since it indicates that the e+e−annihilation through two virtual
photon fragmentation may make important contributions to certain processes, and
it might substantially reduce the discrepancy between experiment and theory for the
exclusive process e+e−→J/ψ +ηc(note that it is essential to check experimentally
whether e+e−→J/ψ +J/ψ is largely misidentified with e+e−→J/ψ +ηc).
In this situation, it is necessary to examine the contribution of e+e−annihilation
through two virtual photons to the inclusive production of J/ψ. In the following we
will calculate the complete O(α4) color-singlet inclusive cross section for e+e−→
2γ∗→J/ψc¯c, and compare the production rate through two virtual photons with
that through one virtual photon, to see whether the annihilation through two virtual
photons can decrease the discrepancy between the Belle measurement on J/ψc¯c
production and the calculations based on NRQCD.
Following the NRQCD factorization formalism, color-singlet scattering ampli-
tude of the process e−(p1) + e+(p2)→γ∗→c¯c(2S+1LJ)(p) + c(p3) + ¯c(p4) in Fig. 1
is given by
A(e−(p1) + e+(p2)→c¯c(2S+1 LJ)(p) + c(p3) + ¯c(p4)) = qCLX
LzSz
X
s1s2
X
jk
×hs1;s2|SSzihLLz;SSz|J Jzih3j;¯
3k|1i
×A(e−(p1) + e+(p2)→cj(p
2;s1) + ¯ck(p
2;s2) + cl(p3
2;s3) + ¯ci(p4
2;s4))(L=S).(4)
where c¯c(2S+1 LJ) is the intermediate c¯cpair which is produced at short distances and
then evolves into a specific charmonium at long distances. h3j;¯
3k|1i=δjk /√Nc,
1The numerical value obtained for this process in[3] should be multiplied by a factor of 3.
2
e−(p1)
e+(p2)
e−(p1)
e+(p2)
+ 2 flipped graphs
¯c(p4)
c(p3)
J/ψ(p)
¯c(p4)
c(p3)
J/ψ(p)
Figure 1: Feynman diagrams for e+e−→2γ∗→J/ψ +c¯c.
hs1;s2|SSzi,hLLz;SSz|J Jziare respectively the color-SU(3), spin-SU(2), and
angular momentum Clebsch-Gordon coefficients for Q¯
Qpairs projecting out appro-
priate bound states. For J/ψ the coefficient CLcan be related to the radial wave
function of the bound state and reads
CS=1
4π|RS(0) |2.(5)
The spin projection operator can be defined as[12]
PSSz(p;q)≡X
s1s2hs1;s2|SSziv(p
2+q;s1)¯u(p
2−q;s2).(6)
We write the spin projection operator which will be used in the calculation as
P1SZ(p, 0) = 1
2√26ǫ(Sz)(6p+M),(7)
where Mis the mass of the charmonium, which equals to 2mcin the nonrelativistic
approximation.
The amplitude for the upper Feynman graph in Fig. 1 can be written as
M1=4√3
9δkl X
LzSzhLLz;SSz|J Jzi¯v(p2)γµ6p1− 6 p+me
(p1−p)2−m2
e
γαu(p1)
T r[P1Sz(p, 0)γα]¯u(p3)γµv(p4)1
p2(p3+p4)2,(8)
3
-1.0
-0.5
0.0
0.5
1.0
0
20
40
60
80
100
dσ/dcosθ (fb)
cos
θ
Figure 2: Differential cross sections of e+e−→2γ∗→J/ψ +c¯c(solid line) and
e+e−→γ∗→J/ψ +c¯c(dotted line) as functions of the scattering angle of J/ψ.
and the amplitude for the corresponding flipped graph is denoted as M2. The
amplitude for the lower Feynman graph in Fig. 1 reads
M3=4
9√3δkl X
LzSzhLLz;SSz|J Jzi¯v(p2)γµ6p3+6p/2− 6 p2+me
(p3+p/2−p2)2−m2
e
γαu(p1)
¯u(p3)γµP1Sz(p, 0)γαv(p4)1
(p4+p/2)2
1
(p3+p/2)2,(9)
and the amplitude for the flipped graph is denoted as M4.
The calculation of cross section for e−+e+→2γ∗→J /ψ +c¯cis straightforward.
The differential cross section can be written in the form
dσ(e++e−→2γ∗→Jψ +c¯c)
dΩ=3CSα4
4mc|¯
M|2,(10)
where dΩ represents the elements of the quadruple integral related to the final state
phase space (see, e.g. Ref. [3]), and |¯
M|2=1
4Ppol,color |M1+M2+M3+M4|2is
the unpolarized module squared of the amplitude. For simplicity we will not write
down their lengthy expressions here.
With Eq. (10) we can evaluate the inclusive cross sections for J/ψ production
from e+e−through two virtual photons. The input parameters used in the numerical
calculations are the same as in ref.[6]
me= 0, mc= 1.5GeV, α = 1/137,(11)
|RS(0) |2= 0.81GeV 3[13].(12)
4
10 15 20 25 30
0.70
0.75
0.80
0.85
0.90
0.95
E
cm
GeV
Figure 3: Ratio of the fragmentation contribution to the total cross section as the
function of the center-of-mass energy.
Now we give the numerical result at the Belle energy √s= 10.6 GeV:
σ(e++e−→2γ∗→J/ψ +c¯c) = 23fb.(13)
It is well known that e+e−→2γ∗→J/ψ +c¯cis a pure electromagnetic process
(except for the hadronization of the quark pair at long distances), while e+e−→
γ∗→J/ψ +c¯cinvolves both electromagnetic and strong interactions. For a naive
order of magnitude estimate, the ratio of the production rate of the former to the
latter would be proportional to α2/α2
s, but the photon fragmentation into J/ψ and
into the charmed quark pair can substantially enhance the former. To see the
role of photon fragmentation, we plot the differential cross sections of e+e−→
2γ∗→J/ψ +c¯cand e+e−→γ∗→J/ψ +c¯cas functions of the scattering angle
of J/ψ at √s= 10.6GeV in Fig. 2. (Here for the one-photon process we choose
αs=0.26.) One can see that the small angle J/ψ production in e+e−→2γ∗→
J/ψ+c¯cis dominant. This indicates that most of the J/ψ production comes from the
photon fragmentation (corresponding to the upper graph in Fig. 1 with one photon
fragmenting to J/ψ and the other fragmenting to a charm quark pair). Indeed,
our calculation shows that at √s= 10.6GeV, the contribution from fragmentation
graphes is about seventy-two percent of the total cross section. In Fig. 3 we show
the ratio of the fragmentation contribution to the total cross section as a function
of the e+e−center-of-mass energy in e+e−→2γ∗→J/ψ +c¯c. It is clear that
5
10 15 20 25 30
0
20
40
60
80
100
120
140
σ fb
EcmGeV
Figure 4: Cross sections of e+e−→2γ∗→J/ψ +c¯c(solid line) and e+e−→γ∗→
J/ψ +c¯c(dotted line) as functions of the center-of-mass energy
the photon fragmentation becomes more and more dominant as the center-of-mass
energy increases. This is in agreement with the observation in the J/ψ +J/ψ
exclusive production through two virtual photons[10].
From the above discussions we have seen the importance of the photon frag-
mentation to J/ψ as well as to the charm quark pair in the two-photon process
e+e−→2γ∗→J/ψ +c¯c. A even more crucial result is that at high e+e−center-
of-mass energies the contribution through two virtual photons will prevail over that
through one virtual photon in the production of J/ψc¯c. The reason lies simply in the
fact that the virtuality of the photon in the two-photon process can be as small as
4m2
c, whereas it is as large as s, the center-of-mass energy squared in the one-photon
process. In Fig. 4 we show the cross sections of e+e−→2γ∗→J/ψ +c¯c(solid line)
and e+e−→γ∗→J/ψ +c¯c(dotted line) as functions of the center-of-mass energy
√s. We see clearly that the cross section for e+e−→2γ∗→J/ψ +c¯cdecreases very
slowly, whereas that for e+e−→γ∗→J/ψ +c¯cdecreases rapidly as √sincreases.
At √s= 20 GeV, the two photon process becomes to prevail over the one photon
process.
However, unfortunately, at the Belle energy √s= 10.6 GeV, since the enhance-
ment effect due to the factor s/m2
cis not large enough as compared with the sup-
pression factor α2/α2
s, we find σ(e+e−→2γ∗→J/ψ +c¯c) = 23 fb, which is still
much smaller than σ(e+e−→γ∗→J/ψ+c¯c) = 148 fb[6], and therefore is negligible.
6
In summary, we have calculated the complete O(α4) color-singlet inclusive cross
sections for J/ψc¯cproduction from e+e−annihilation into two photons. Due to the
suppression factor of α2/α2
s, at the e+e−center-of-mass energy √s= 10.6 GeV the
cross section of this process is smaller by about a factor of six than that from e+e−
annihilation into one photon. We then conclude that while the e+e−annihilation into
two photons could be helpful in solving the puzzle for the exclusive J/ψηcproduction,
it can do very little to reduce the big gap between the observed inclusive production
cross section of σ(e+e−→J/ψ +c¯c)≈0.9 pb and the current NRQCD predictions
of about 0.15 pb. This puzzle still needs to be explained with new theoretical
considerations. We find, however, as the center-of-mass energy increases(√s >20
GeV) the production through two photons e+e−→2γ∗→J/ψ +c¯cwill prevail over
that through one photon, because in the former case the photon fragmentation into
J/ψ and into the charmed quark pair becomes more important at higher energies.
Acknowledgments
This work was supported in part by the National Natural Science Foundation of
China, and the Education Ministry of China.
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