Using the nonrelativistic QCD(NRQCD) factorization formalism, we calculate
the color-singlet cross sections for exclusive production processes
${e^++e^-\to J/\psi+\eta_c}$~ and ~$e^++e^-\to J/\psi + \chi_{cJ}$~$(J=0,1,2)$
at the center-of-mass energy $\sqrt{s}$=10.6 GeV. The cross sections are
estimated to be 5.5fb, 6.7fb, 1.1fb, and 1.6fb for $\eta_c, \chi_{c0},
\chi_{c1}$ and $\chi_{c2}$, respectively. The calculated $J/\psi+\eta_c$
production rate is smaller than the recent Belle data by about an order of
magnitude, which might indicate the failure of perturbative QCD calculation to
explain the double-charmonium production data. The complete
$\cal{O}$$(\alpha^2_{s})$ color-singlet cross section for ${e^++e^-\to
\chi_{c0}+ c\bar {c}}$ is calculated. In addition, we also evaluate the ratio
of exclusive to inclusive production cross sections. The ratio of
$J/\psi\eta_c$ production to $J/\psi c\bar{c}$ production could be consistent
with the experimental data.Comment: 10 pages, 4 figures. A few references added, errors and typoes
  correcte

Baek

Bodwin

Braaten

Brodsky

Chao

Eichten

Guberina

Kane

Kiselev

Kuang-Ta Chao

Kui-Yong Liu

Kühn

Schuler

Yuan

Zhi-Guo He

Physics Letters B

English

arXiv

Using the nonrelativistic QCD (NRQCD) factorization formalism, we calculate the color-singlet cross sections for exclusive production processes e(+) + e(-) --&gt; J/psi + n(c) and e(+) + e(-) --&gt; J/psi + chi(cJ) (J = 0, 1, 2) at the center-of-mass energy roots = 10.6 GeV. The cross sections are estimated to be 5.5 fb, 6.7 fb, 1.1 fb, and 1.6 fb for eta(C), chi(c0), chi(c1), and chi(c2), respectively. The calculated J/psi + eta(c) production rate is smaller than the recent Belle data by about an order of magnitude, which might indicate the failure of perturbative QCD calculation to explain the double-charmonium production data. The complete O(alpha(s)(2)) color-singlet cross section for e(+) + e(-) --&gt; chi(c0) + c (c) over bar is calculated. In addition, we also evaluate the ratio of exclusive to inclusive production cross sections. The ratio of J/psieta(c) production to J/psic (c) over bar production could be consistent with the experimental data. (C) 2003 Published by Elsevier Science B.V.http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:000181622700008&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=8e1609b174ce4e31116a60747a720701Physics, MultidisciplinarySCI(E)136ARTICLE1-245-5455

Liu, KY

He, ZG

Chao, KT

Institutional Repository of Peking University

Problems of double-charm production in e(+)e(-) annihilation at root   s=10.6 GeV

AbstractUsing the nonrelativistic QCD (NRQCD) factorization formalism, we calculate the color-singlet cross sections for exclusive production processes e++e−→J/ψ+ηc and e++e−→J/ψ+χcJ (J=0,1,2) at the center-of-mass energy s=10.6 GeV. The cross sections are estimated to be 5.5 fb, 6.7 fb, 1.1 fb, and 1.6 fb for ηc,χc0,χc1, and χc2, respectively. The calculated J/ψ+ηc production rate is smaller than the recent Belle data by about an order of magnitude, which might indicate the failure of perturbative QCD calculation to explain the double-charmonium production data. The complete O(α2s) color-singlet cross section for e++e−→χc0+cc̄ is calculated. In addition, we also evaluate the ratio of exclusive to inclusive production cross sections. The ratio of J/ψηc production to J/ψcc̄ production could be consistent with the experimental data

Liu, Kui-Yong

He, Zhi-Guo

Chao, Kuang-Ta

Elsevier - Publisher Connector 

Physics Letters B 557 (2003) 45–54
www.elsevier.com/locate/npe
Problems of double-charm production in e+e− annihilation at√
s = 10.6 GeV
Kui-Yong Liu a, Zhi-Guo He a, Kuang-Ta Chao a,b
a Department of Physics, Peking University, Beijing 100871, PR China
b China Center of Advanced Science and Technology, World Laboratory, Beijing 100080, PR China
Received 20 November 2002; received in revised form 23 January 2003; accepted 27 January 2003
Editor: T. Yanagida
Abstract
Using the nonrelativistic QCD (NRQCD) factorization formalism, we calculate the color-singlet cross sections for exclusive
production processes e+ + e− → J/ψ + ηc and e+ + e− → J/ψ + χcJ (J = 0,1,2) at the center-of-mass energy
√
s =
10.6 GeV. The cross sections are estimated to be 5.5 fb, 6.7 fb, 1.1 fb, and 1.6 fb for ηc,χc0,χc1, and χc2, respectively. The
calculated J/ψ + ηc production rate is smaller than the recent Belle data by about an order of magnitude, which might indicate
the failure of perturbative QCD calculation to explain the double-charmonium production data. The complete O(α2s ) color-
singlet cross section for e+ + e− → χc0 + cc¯ is calculated. In addition, we also evaluate the ratio of exclusive to inclusive
production cross sections. The ratio of J/ψηc production to J/ψcc¯ production could be consistent with the experimental data.
 2003 Published by Elsevier Science B.V.
PACS: 12.40.Nn; 13.85.Ni; 14.40.Gx
Heavy quarkonium production is interesting in understanding both perturbative and nonperturbative quantum
chromodynamics (QCD). In recent years the charmonium production has been studied in various processes, such
as in hadron–hadron collision, electron–proton collision, fixed target experiments, B meson decays, as well as Z0
decays. Among them, the study of charmonium production in e+e− annihilation is particularly interesting in testing
the quarkonium production mechanisms, the color-singlet model and the color-octet model in the nonrelativistic
QCD (NRQCD) [1] approach. This is not only because of the simpler parton structure involved in this process,
which may be helpful in reducing the theoretical uncertainty, but also because of the spectacular experimental
prospect opened up by the two B factories with BaBar and Belle, which will allow a fine data analysis for
charmonium production with more than 108 e+e− annihilation events in the continuum at
√
s = 10.6 GeV.
Recently the Belle Collaboration has reported the observation of prompt J/ψ via double cc¯ production from
the e+e− continuum [2]. For these results, not only the large cross section (≈ 0.9 pb) of the inclusive J/ψ
production due to the double cc¯ is puzzling [3], but also the exclusive production rate of J/ψηc , σ(e+ + e− →
J/ψ + ηc(γ )) × B(ηc → 4 charged) = (0.033+0.007−0.006 ± 0.009) pb, may not be consistent with two previous
E-mail address: ktchao@th.phy.pku.edu.cn (K.-T. Chao).
Open access under CC BY license.0370-2693/03  2003 Published by Elsevier Science B.V.
doi:10.1016/S0370-2693(03)00176-X
Open access under CC BY license.
46 K.-Y. Liu et al. / Physics Letters B 557 (2003) 45–54Fig. 1. Feynman diagrams for e+e− → J/ψ + ηc(χcJ ).
calculations [4,5], which gave a cross section of a few pb for J/ψηc . In fact, recent perturbative QCD estimates
of the J/ψcc¯ cross section are only about 0.1 ∼ 0.2 pb [6–9]. So the calculations of exclusive cross sections
for e+e− annihilation into J/ψηc and other double-charmonium states such as J/ψχcJ (J = 0,1,2) will be
useful to clarify the problem. Experimentally, aside from e+ + e− → J/ψ + χc0 [2], Belle [10] has also studied
processes e+ + e− → χc1 + X and e+ + e− → χc2 + X, so we hope that the double-charmonium production
involving χcJ (J = 0,1,2) will be detectable in the near future. In the following we will calculate the cross
sections σ(e+ + e− → J/ψ + ηc) and σ(e+ + e− → J/ψ + χcJ ) in the leading order perturbative QCD. To this
order (∼ α2s ) the color-singlet channel is dominant since all color-octet channels are of high order of v, which is
the relative velocity of the charm quark and anti-charm quark in the charmonium, and therefore suppressed relative
to the color-singlet channel (the relative suppression is at least of order v4 for the cross sections). Furthermore, in
order to compare exclusive production with inclusive production rates associated with the χcJ charmonium states,
we will calculate the cross section σ(e+ + e− → χc0 + cc¯), and hope these ratios will be useful for both inclusive
and exclusive production analyses at
√
s = 10.6 GeV.
We now write down the scattering amplitude in the nonrelativistic approximation to describe the creation of two
color-singlet cc¯ pairs which subsequently hadronize to two charmonium states in the e+e− annihilation process in
Fig. 1 as [11,12]
A(a + b→QQ¯( 2Sψ+1LJψ )(p3)+QQ¯( 2S+1LJ )(p4))
=
√
CLψ
√
CL
∑
LψzSψz
∑
s1s2
∑
jk
∑
LzSz
∑
s3s4
∑
il
〈s1; s2|SψSψz〉〈LψLψz;SψSψz|JψJψz〉〈3j ; 3¯k|1〉
× 〈s3; s4|SSz〉〈LLz;SSz|JJz〉〈3l; 3¯i|1〉
(1)×
{A(a + b→Qj(p32 )+ Q¯k(p32 )+Ql(p42 )+ Q¯i(p42 )) (L= S),
 ∗α(LZ)Aα(a + b→Qj(p32 )+ Q¯k(p32 )+Ql(p42 )+ Q¯i(p42 )) (L= P),
where 〈3j ; 3¯k|1〉 = δjk/√Nc , 〈3l; 3¯i|1〉 = δli/√Nc , 〈s1; s2|SψSψz〉, 〈s3; s4|SSz〉, 〈LψLψz;SψSψz|JψJψz〉,
and 〈LLz;SSz|JJz〉 are, respectively, the color-SU(3), spin-SU(2), and angular momentum Clebsch–Gordan
K.-Y. Liu et al. / Physics Letters B 557 (2003) 45–54 47coefficients for QQ pairs projecting out appropriate bound states. A(a + b → Qj(p32 ) + Qk(p32 ) + Ql(p42 ) +Qi(p42 )) is the scattering amplitude for double QQ production and Aα is the derivative of the amplitude with
respect to the relative spacing between the quark and antiquark in the bound state. The coefficients CLψ and CL
can be related to the radial wave function of the bound states or its derivative with respect to the relative momentum
as
(2)Cs = 14π
∣∣Rs(0)∣∣2, Cp = 34π
∣∣R′p(0)∣∣2.
We introduce the spin projection operators PSSz(p, q) as [11,12]
(3)PSSz(p, q)≡
∑
s1s2
〈s1; s2|SSz〉v
(
p
2
− q; s1
)
u¯
(
p
2
+ q; s2
)
.
Expanding PSSz(P, q) in terms of the relative momentum q , we get the projection operators and their derivatives,
which will be used in our calculation, as follows
(4)P1Sz(p,0)=
1
2
√
2
/ ∗(Sz)(/p+ 2mc),
(5)P00(p,0)= 1
2
√
2
γ5(/p+ 2mc),
(6)Pα1Sz(p,0)=
1
4
√
2mc
[
γ α/ ∗(Sz)(/p+ 2mc)− (/p− 2mc)/ (Sz)γ α
]
.
Then one can calculate the cross sections for the on-shell quarks in the factorized form of NRQCD [1]. The
cross section for e+ + e−→ J/ψ + ηc process in Fig. 1 is given by
(7)σ (a(p1)+ b(p2)→ J/ψ(p3)+ ηc(p4))= 2πα
2α2s |Rs(0)|4
√
s − 16m2c
81m2cs3/2
1∫
−1
| M|2 d cosθ,
where θ is the scattering angle between p1 and p3, | M|2 is as follows
(8)| M|2 = 16384m
2
c(t
2 + u2 − 32m4c)
s5
.
The Mandelstam variables are defined as
(9)s = (p1 + p2)2,
(10)t = (p3 −p1)2 = 4m2c −
s
2
(
1−
√
1− 16m2c/s cosθ
)
,
(11)u= (p3 − p2)2 = 4m2c −
s
2
(
1+
√
1− 16m2c/s cosθ
)
.
The cross section for e+ + e−→ J/ψ + χcJ process is
(12)σ (a(p1)+ b(p2)→ J/ψ(p3)+ χcJ (p4))= 2πα
2α2s |Rs(0)|2|R′p(0)|2
√
s − 16m2c
27m2cs3/2
1∫
−1
| MJ |2 d cosθ,
48 K.-Y. Liu et al. / Physics Letters B 557 (2003) 45–54where | MJ |2 for χc0, χc1, and χc2 are given by
(13)
| M0|2 = 2048
(
90112m10c − 74752m8ct − 74752m8cu+ 23360m6ct2 + 43136m6ctu+ 23360m6cu2
− 3152m4ct3 − 7600m4ct2u− 7600m4ctu2 − 3152m4cu3 + 162m2ct4 + 444m2ct3u
+ 564m2ct2u2 + 444m2ctu3 + 162m2cu4 − t4u− 3t3u2 − 3t2u3 − tu4
)/(
3s7m2c
)
,
(14)
| M1|2 = 32768
(
1792m8c + 256m6ct + 256m6cu− 56m4ct2 − 64m4ctu− 56m4cu2 − 4m2ct3
− 20m2ct2u− 20m2ctu2 − 4m2cu3 + t4 + 2t3u+ 2t2u2 + 2tu3 + u4
)/
s7,
(15)
| M2|2 = 4096
(
145408m10c − 1024m8ct − 1024m8cu− 2368m6ct2 − 6400m6ctu− 2368m6cu2
+ 16m4ct3 − 208m4ct2u− 208m4ctu2 + 16m4cu3 + 24m2ct4 + 72m2ct3u+ 96m2ct2u2
+ 72m2ctu3 + 24m2cu4 − t4u− 3t3u2 − 3t2u3 − tu4
)/(
3s7m2c
)
.
In the numerical calculations, we choose
√
s = 10.6 GeV, mc = 1.5 GeV, αs = 0.26, |Rs(0)|2 = 0.810 GeV3
and |R′p(0)|2 = 0.075 GeV5 [13], and assume that in the nonrelativistic approximationmJ/ψ =mηc =mχcJ = 2mc.
The numerical result for e+ + e−→ J/ψ + ηc is
(16)σ (e+ + e−→ J/ψ + ηc)= 5.5 fb.
While the numerical result for the cross section of e+ + e− → J/ψ + ηc is more than a factor of six smaller
than the experimental data [2] (with uncertainties due to the unknown decay branching fractions into  4-charged
particles for the ηc), the calculated ratio of σ(e+ + e−→ J/ψ + ηc)/σ(e+ + e−→ J/ψ + cc¯)≈ 0.037 might be
consistent with the experimental result with the choice of σ(e+ + e− → J/ψ + cc¯)= 148 fb obtained by taking
our input parameters.
The cross sections for J/ψχcJ production at
√
s = 10.6 GeV are given as
(17)σ (e+ + e−→ J/ψ + χc0)= 6.7 fb,
(18)σ (e+ + e−→ J/ψ + χc1)= 1.1 fb,
(19)σ (e+ + e−→ J/ψ + χc2)= 1.6 fb.
In Fig. 2, we show the cross sections as functions of the e+e− center-of-mass energy
√
s, and we can see that
the cross sections for e+ + e−→ J/ψ + ηc, e+ + e−→ J/ψ + χc1 and e+ + e− → J/ψ + χc2 decrease rapidly
as
√
s increases. But the one with J/ψ + χc2 decreases more slowly than that with J/ψ + ηc and J/ψ + χc1.
At
√
s = 10.6 GeV if we choose σ(e++ e− → χc1 + cc¯)= 18.1 fb and σ(e++ e−→ χc2 + cc¯)= 8.4 fb which
were obtained in the fragmentation approximation in Ref. [14], then we have the ratios
(20)σ(e
+ + e−→ J/ψ + χc1)
σ (e+ + e−→ χc1 + cc¯) = 0.061,
(21)σ(e
+ + e−→ J/ψ + χc2)
σ (e+ + e−→ χc2 + cc¯) = 0.19,
(22)σ(e
+ + e−→ J/ψ + χc1)
σ (e+ + e−→ J/ψ + cc¯) = 0.007,
(23)σ(e
+ + e−→ J/ψ + χc2)
σ (e+ + e−→ J/ψ + cc¯) = 0.011.
As for the χc0 inclusive double-charm production the rate was not given in Ref. [14], we calculate σ(e++e−→
χc0 + cc¯) in a complete form to the O(α2s ) order in perturbative QCD.
K.-Y. Liu et al. / Physics Letters B 557 (2003) 45–54 49Fig. 2. Cross sections for σ(e+ + e− → J/ψ + ηc) (solid line) and σ(e+ + e− → J/ψ + χcJ ) (dashed line for J = 1, dotted line for J = 2)
plotted against the e+e− center-of-mass energy √s with z=√s/s0 and √s0 = 10.6 GeV.
Fig. 3. Feynman diagrams for e+ + e− → χc0 + cc¯ process.
We give the amplitude of the first diagram in Fig. 3 for e+ + e−→ χc0 + cc¯ as
(24)
M =
∑
LzSz
 ∗σ (Lz)〈1Lz;1Sz|J = 0, Jz = 0〉
√
CL
ieceg
2
s [T aT a]li√
3
v¯(p2)γ
µu(p1)
1
s
u¯l(pc)
× [γ αP1SzγαOσµ + γ αPσ1SzγαOµ
]
vi(pc¯),
50 K.-Y. Liu et al. / Physics Letters B 557 (2003) 45–54Fig. 4. Cross sections for σ(e+ + e− → χc0 + cc¯) (solid line) and σ(e+ + e− → J/ψ + χc0) (dotted line) plotted against the e+e−
center-of-mass energy
√
s with z=√s/s0 and √s0 = 10.6 GeV.
where ec = (2/3)e, T a is the SU(3) color matrix, the matrix Oµ is relevant to the on shell amplitude and Oσµ is
its derivative with respect to the relative momentum between the quarks that form the bound state. We can also
express the contributions of other three diagrams in a similar way, and our numerical results are obtained with the
full contributions of these four diagrams. Some useful information of the calculation is given in Appendix A.
We finally get the cross section for this process
(25)σ (e+ + e−→ χc0 + cc¯)= 49 fb.
Then one has the ratio
(26)σ(e
+ + e−→ J/ψ + χc0)
σ (e+ + e−→ χc0 + cc¯) = 0.14,
(27)σ(e
+ + e−→ J/ψ + χc0)
σ (e+ + e−→ J/ψ + cc¯) = 0.045.
In Fig. 4, we show cross sections for σ(e++ e−→ χc0 + cc¯) (solid line) and σ(e++ e−→ J/ψ +χc0) (dotted
line) plotted against the e+e− center-of-mass energy √s with z = √s/s0 and √s0 = 10.6 GeV. One can see the
ratio in Eq. (26) decreases drastically as the center-of-mass energy increases. This is consistent with the result in
Fig. 2. We hope the ratios between Eqs. (20) and (27) could be tested in the near future.
In summary, despite of many uncertainties due to the relativistic corrections, the QCD radiative corrections, the
possible color-octet channel contributions, and the choice of physical parameters (e.g., the charm quark mass and
the strong coupling constant), both the inclusive and exclusive double charm production cross sections calculated
in perturbative QCD turned out to be seriously underestimated as compared with data. Therefore we intend to
conclude, as in [3], that it seems hard to explain the double charm production data observed by Belle based
on perturbative QCD (including both color-singlet and color-octet channels), and possible nonperturbative QCD
effects should be considered at
√
s = 10.6 GeV.
K.-Y. Liu et al. / Physics Letters B 557 (2003) 45–54 51While we were about to submit our result, there appeared one paper which also considered exclusive double-
charmonium production [15]. Those authors took the QED effects into account in addition to the QCD effects that
we considered. We find our result for the exclusive double-charmonium production is consistent with theirs but we
also analyzed some inclusive processes which were not discussed in Ref. [15].
Acknowledgements
The authors thank L.K. Hao and Z.Z. Song for useful discussions. We also thank C.F. Qiao for helpful comments.
This work was supported in part by the National Natural Science Foundation of China, and the Education Ministry
of China.
Appendix A
In this appendix we give the cross section for the e+ + e−→ χc0 + cc¯ process shown in Fig. 3.
(A.1)dσ = | M|
2
2s(2π)5
δ4(p1 +p2 − pc − pc¯ − p) d
3pc
2Ec
d3pc¯
2Ec¯
d3p
2E
.
It is convenient to rewrite the cross section as
dσ = | M|
2
2s(2π)5
δ4(p1 + p2 − η− p)δ4(η− pc − pc¯) d
3pc
2Ec
d3pc¯
2Ec¯
d3p
2E
d4η
(A.2)= | M|
2
2s(2π)5
δ4(p1 + p2 − η− p)δ4(η− pc − pc¯) d
3pc
2Ec
d3pc¯
2Ec¯
d3p
2E
d3η
2Eη
dm2η,
where m2η = η2.
The integral over the phase–space of cc¯ is evaluated in the corresponding center-of-mass frame
(A.3)d
3p′c
2E′c
d3p′¯c
2E ′¯c
δ4
(
η′ − p′c −p′¯c
)= 1
8m2η
λ1/2
(
m2η,m
2
c,m
2
c
)
d2′,
where λ(a2, b2, c2)= a4 + b4 + c4 − 2a2b2 − 2a2c2 − 2b2c2.
The remaining integration are performed in the e+e− center-of-mass frame
(A.4)d
3p
2E
d3η
2Eη
δ4(p1 + p2 − η−p)= 18s λ
1/2(s,m2η,m2p)d2,
where mp = 2mc, is the mass of the bound state.
Finally we have
(A.5)dσ = | M|
2CP
64s2(2π)5mηmc
λ1/2
(
m2η,m
2
c,m
2
c
)
λ1/2
(
s,m2η,m
2
p
)
d2′ d2dmη.
The limit of mη is
(A.6)2mc mη 
√
s −mp.
52 K.-Y. Liu et al. / Physics Letters B 557 (2003) 45–54To accomplish the integration we use the Lorentz transformation between the two frames as L=R2R1, where
(A.7)R1 =


√
1+ p2
m2η
0 0 −| p|
mη
0 1 0 0
0 0 1 0
−| p|
mη
0 0
√
1+ p2
m2η


,
(A.8)R2 =


1 0 0 0
0 cosθ 0 sin θ
0 0 1 0
0 − sin θ 0 cos θ

 .
The momenta in the e+e− center-of-mass frame are
(A.9)p1 =
(√
s
2
,0,0,
√
s
2
)
,
(A.10)p2 =
(√
s
2
,0,0,−
√
s
2
)
,
(A.11)pc =R2R1p′c,
(A.12)pc¯ =R2R1p′¯c,
(A.13)p = (√ p2 +m2p, | p| sin θ,0, | p| cosθ),
where p′c and p′¯c are the momenta of c and c¯ in the 2′ frame, and they are
(A.14)p′c =
(
E′c, | pc ′| sinθ ′ cosθ ′, | pc ′| sinθ ′ cosθ ′, | pc ′| cosθ ′
)
,
(A.15)p′¯c =
(
E ′¯c,−| pc′| sinθ ′ cosθ ′,−| pc′| sin θ ′ cos θ ′,−| pc′| cosθ ′
)
.
In Fig. 3 the lower (nonfragmentation) diagrams give very small contributions (about 3 percent), so for simplicity
here we only write down the expressions for the contribution of the upper diagrams and give
(A.16)| M|2 = 2(4π)
4α2α2s
27
(aa+ 2ab+ bb).
We define pp1 = p.p1, pp2 = p.p2, pp3 = p.pc , pp4 = p.pc¯, p13 = p1.pc , p14 = p1.pc¯, p23 = p2.pc, p24 =
p2.pc¯. We notify aa = bb and
aa = [4(800m10c s + 800m8cp14pp2 + 800m8cp24pp1 + 1440m8cpp3s + 160m6cp13p24pp3
+ 160m6cp14p23pp3 + 1440m6cp14pp2pp3 + 1440m6cp24pp1pp3 + 900m6cpp23s
+ 184m4cp13p24pp23 + 184m4cp14p23pp23 + 856m4cp14pp2pp23 + 856m4cp24pp1pp23
+ 216m4cpp33s + 56m2cp13p24pp33 + 56m2cp14p23pp33 + 168m2cp14pp2pp33
+ 168m2cp24pp1pp33 + 13m2cpp43s + 2p13p24pp43 + 2p14p23pp43
)]
(A.17)
× [3m2cs2(64m12c + 192m10c pp3 + 240m8cpp23 + 160m6cpp33 + 60m4cpp43 + 12m2cpp53 + pp63)]−1,
K.-Y. Liu et al. / Physics Letters B 557 (2003) 45–54 53ab= [4(400m10c s + 400m8cp13pp2 + 400m8cp14pp2 + 400m8cp23pp1 + 400m8cp24pp1
+ 400m8cpp1pp2 + 480m8cpp3s + 400m8cpp34s + 480m8cpp4s + 80m6cp13p24pp3
+ 80m6cp13p24pp4 + 280m6cp13pp2pp3 + 440m6cp13pp2pp4 + 80m6cp14p23pp3
+ 80m6cp14p23pp4 + 440m6cp14pp2pp3 + 280m6cp14pp2pp4 + 280m6cp23pp1pp3
+ 440m6cp23pp1pp4 + 440m6cp24pp1pp3 + 280m6cp24pp1pp4 + 240m6cpp1pp2pp3
− 400m6cpp1pp2pp34 + 240m6cpp1pp2pp4 + 140m6cpp23s + 240m6cpp3pp34s
+ 476m6cpp3pp4s + 240m6cpp34pp4s + 140m6cpp24s + 40m4cp13p24pp23 + 104m4cp13p24pp3pp4
+ 40m4cp13p24pp24 + 20m4cp13pp2pp23 + 288m4cp13pp2pp3pp4 + 120m4cp13pp2pp24
+ 40m4cp14p23pp23 + 104m4cp14p23pp3pp4 + 40m4cp14p23pp24 + 120m4cp14pp2pp23
+ 288m4cp14pp2pp3pp4 + 20m4cp14pp2pp24 + 20m4cp23pp1pp23 + 288m4cp23pp1pp3pp4
+ 120m4cp23pp1pp24 + 120m4cp24pp1pp23 + 288m4cp24pp1pp3pp4 + 20m4cp24pp1pp24
− 240m4cpp1pp2pp3pp34 + 144m4cpp1pp2pp3pp4 − 240m4cpp1pp2pp34pp4 + 108m4cpp23pp4s
+ 144m4cpp3pp34pp4s + 108m4cpp3pp24s + 28m2cp13p24pp23pp4 + 28m2cp13p24pp3pp24
+ 12m2cp13pp2pp23pp4 + 72m2cp13pp2pp3pp24 + 28m2cp14p23pp23pp4 + 28m2cp14p23pp3pp24
+ 72m2cp14pp2pp23pp4 + 12m2cp14pp2pp3pp24 + 12m2cp23pp1pp23pp4 + 72m2cp23pp1pp3pp24
+ 72m2cp24pp1pp23pp4 + 12m2cp24pp1pp3pp24 − 144m2cpp1pp2pp3pp34pp4 + 13m2cpp23pp24s
+ 2p13p24pp23pp24 + 2p14p23pp23pp24
)]
(A.18)
× [3m2cs2(64m12c + 96m10c pp3 + 96m10c pp4 + 48m8cpp23 + 144m8cpp3pp4 + 48m8cpp24 + 8m6cpp33
+ 72m6cpp23pp4 + 72m6cpp3pp24 + 8m6cpp34 + 12m4cpp33pp4 + 36m4cpp23pp24
+ 12m4cpp3pp34 + 6m2cpp33pp24 + 6m2cpp23pp34 + pp33pp34
)]−1
.
References
[1] G.T. Bodwin, L. Braaten, G.P. Lepage, Phys. Rev. D 51 (1995) 1125.
[2] Belle Collaboration, K. Abe, et al., Phys. Rev. Lett. 89 (2002) 142001, hep-ex/0205104.
[3] K.T. Chao, L.K. Hao, hep-ph/0209189.
[4] G.L. Kane, J.P. Leveille, D.M. Scott, Phys. Lett. B 85 (1979) 115.
[5] S.J. Brodsky, C.R. Ji, Phys. Rev. Lett. 55 (1985) 2257.
[6] P. Cho, A.K. Leibovich, Phys. Rev. D 54 (1996) 6690.
[7] F. Yuan, C.F. Qiao, K.T. Chao, Phys. Rev. D 56 (1997) 321;
F. Yuan, C.F. Qiao, K.T. Chao, Phys. Rev. D 56 (1997) 1663.
[8] S. Baek, P. Ko, J. Lee, H.S. Song, J. Kor. Phys. Soc. 33 (1998) 97, hep-ph/9804455.
[9] V.V. Kiselev, A.K. Likhoded, M.V. Shevlyagin, Phys. Lett. B 332 (1994) 411.
[10] Belle Collaboration, K. Abe, et al., Phys. Rev. Lett. 88 (2002) 052001.
[11] P. Cho, A.K. Leibovich, Phys. Rev. D 53 (1996) 150;
P. Cho, A.K. Leibovich, Phys. Rev. D 53 (1996) 6203;
P. Ko, J. Lee, H.S. Song, Phys. Rev. D 54 (1996) 4312;
P. Ko, J. Lee, H.S. Song, Phys. Rev. D 60 (1999) 119902.
[12] J.H. Kühn, J. Kaolan, E.G.O. Safiani, Nucl. Phys. B 157 (1979) 125;
B. Guberina, J.H. Kühn, R.D. Peccei, R. Rückl, Nucl. Phys. B 174 (1980) 317.
54 K.-Y. Liu et al. / Physics Letters B 557 (2003) 45–54[13] E.J. Eichten, C. Quigg, Phys. Rev. D 52 (1995) 1726.
[14] G.A. Schuler, M. Vänttinen, Phys. Rev. D 58 (1998) 017502.
[15] E. Braaten, J. Lee, hep-ph/0211085.


Problems of double-charm production in e+e− annihilation at s=10.6 GeV 

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Problems of double-charm production in e+e− annihilation at s=10.6 GeV

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