Lifetime of doubly charmed baryons
Lifetime of doubly charmed baryons
Chao-Hsi Chang1,2 ∗, Tong Li3†, Xue-Qian Li3‡ and Yu-Ming Wang4§
1CCAST (World Laboratory), P.O.Box 8730, Beijing 100080, P.R. China
2Institute of Theoretical Physics, Chinese Academy of Sciences,
P.O.Box 2735, Beijing 100080, P.R. China
3 Department of Physics, Nankai University, Tianjin, 300071, P.R. China
4Institute of High Energy Physics, Chinese Academy of Sciences,
P.O.Box 918, Beijing 100049, P.R. China
Abstract
In this work, we evaluate the lifetimes of the doubly charmed baryons Ξ+cc, Ξ
cc and Ω
cc. We care-
fully calculate the non-spectator contributions at the quark level where the Cabibbo-suppressed di-
agrams are also included. The hadronic matrix elements are evaluated in the simple non-relativistic
harmonic oscillator model. Our numerical results are generally consistent with that obtained by
other authors who used the diquark model. However, all the theoretical predictions on the lifetimes
are one order larger than the upper limit set by the recent SELEX measurement. This discrepancy
would be clarified by the future experiment, if more accurate experiment still confirms the value
of the SELEX collaboration, there must be some unknown mechanism to be explored.
∗ email: zhangzx@itp.ac.cn
† email: allongde@mail.nankai.edu.cn
‡ email: lixq@nankai.edu.cn
§ email: wangym@mail.ihep.ac.cn
http://arxiv.org/abs/0704.0016v1
I. INTRODUCTION
The quite large difference of the lifetimes between D± and D0 and the lifetimes close
to each other for B± and B0 are well explained by taking into account the non-spectator
effects[1]. This success implies that the mechanism which governs the reactions at quark
level is well understood. When we apply the mechanism to the heavy baryon case, some
problems emerge. The famous puzzle in the heavy-flavor field that the lifetime of Λb is
remarkably shorter than that of B meson is much alleviated recently when the operators
of higher dimensions are taken into account[2, 3]. The more recent experimental value of
the ratio τ(Λb)/τ(B
0) = 1.041± 0.057[4] is close to the theoretical evaluation[3]. However,
in the theoretical works, one can notice that the evaluation of hadronic matrix elements is
still very rough and based on some approximations. The possible errors brought up by the
uncertainties in the hadronic matrix elements are still uncontrollable. In our recent work[5],
we find that the short-distance contributions to the branching ratio of Λb → Λγ which is
evaluated in the PQCD approach, are much smaller than that from long-distance effects.
Therefore, even though one has a full reason to believe that the low-energy QCD should
solve the discrepancy if it exists, he must find a proper way to deal with the hadronic matrix
elements.
The observation of doubly charmed baryon Ξ+cc by the SELEX Collaboration at
FERMILAB[6] provides an opportunity to investigate the hidden problems. Hopefully the
study may shed some lights on the unknown non-perturbative QCD effects which result in
obvious difference between baryons and mesons. Because Ξ+cc contains two heavy quarks,
by the heavy quark effective theory (HQET) the situation may become relatively simple
and clear compared to the case of Λb or Λc which possesses only one heavy quark. Thus a
careful study on the Ξ+cc is necessary and interesting. Several groups already investigated
the two-heavy-flavor baryons a long time ago[7, 8]. In their work, the evaluation of the
hadronic matrix elements is based on the quark-diquark structure of the baryons. This is
definitely reasonable, it is believed that two heavy quarks can constitute a more stable and
compact color-anti-triplet diquark[9]. However, since charm quark, even b-quark, is not so
heavy that the degree of freedom of the light flavor can be ignored, the diquark scenario
may bring up certain errors, especially when evaluating lifetimes of baryons, because only
inclusive processes are concerned. In this work, we do not use the diquark picture, but
instead, adopt a simpler non-relativistic model for the baryon and re-evaluate the hadronic
matrix elements. As a by-product, one can compare the results by the diquark picture with
that by the three valence-quark picture. It may help us to better understand the diquark
picture and its application range.
The advantage is obvious, that we only concern the inclusive processes in terms of the
optical theorem when calculating the lifetime. Therefore, we do not need to deal with
the hadronization to light hadrons. The only non-perturbative effects come from the wave
function of the heavy baryon. Moreover, since there are two heavy quarks in the baryon,
the relativistic effects are not so significant and the framework of non-relativistic harmonic
oscillator model might lead to a reasonable result.
Moreover, at the quark level, we carry out similar calculations as that in the literature,
but we keep some new operators which are CKM suppressed and contribute to the lifetime.
They appear at the non-spectator scattering at order of 1
in heavy quark expansion(HQE).
Later, our numerical results show that their contributions are indeed very tiny to make any
substantial contributions.
All the concerned parameters in the model are obtained by fitting data, therefore we
avoid some theoretical uncertainties and obtain reasonable results. Comparing these results
with data, we may gain information about the the whole picture.
Our paper is organized as follows. In Section.II we derive the formulation for the lifetimes
of Ξ+cc, Ξ
cc and Ω
cc which include the non-spectator effects. In Section.III, we use a simple
model, i.e. the harmonic oscillator, to estimate the hadronic matrix elements. In Section.IV
we present our numerical results along with the values of all the input parameters. The last
section is devoted to our conclusion and discussion.
II. FORMULATION FOR LIFETIMES OF Ξ+cc, Ξ
cc , Ω
A. Spectator Contribution to Lifetimes of Ξ+cc, Ξ
cc , Ω
The lifetime is determined by the inclusive decays. Thus one can use the optical theorem
to obtain the total width (lifetime) of the heavy hadron by calculating the absorptive part
of the forward-scattering amplitude.
The total width is then written as
Γ(HQ → X) =
d4x〈HQ|T̂ |HQ〉 =
〈HQ|Γ̂|HQ〉, (1)
where
T̂ = T{iLeff(x),Leff(0)} (2)
and Leff is the relevant effective Lagrangian. 1/mQ is the expansion parameter, and the
non-local operator T̂ is expanded as a sum of local operators and the corresponding Wilson
coefficients include terms with increasing powers of 1/mQ. Definitely, the lowest dimen-
sional term dominates in the limit mQ → ∞ and it is the dimension-three operator c̄c.
The total width of a charmed hadron Hc is determined by Im〈Hc|T̂ |Hc〉[10] with a proper
normalization[11].
Γ(Hc → f) =
192π3
|VCKM |2{c3(f)〈Hc|c̄c|Hc〉
+c5(f)
〈Hc|c̄iσµνGµνc|Hc〉
6 (f)
〈Hc|(c̄Γiq)(q̄Γic)|Hc〉
)}, (3)
where the coefficients ci(f) depend on the masses of the internal quarks in the loop. The
coefficient c3(f) has been calculated to one-loop order[12, 13, 14] whereas the coefficient
c5(f) is evaluated at the tree level[15, 16]. VCKM is the Cabibbo-Kabayashi-Maskawa mixing
matrix elements and Gµυ is the gluonic field strength tensor. Since the third term involves
light quarks, it can be different for charmed hadrons with various light flavors. Thus, the
difference appears at the 1/m3c order and in the hadronic matrix elements of four-quark
operators. The contributions at orders higher than 1/m3c are neglected.
To the lowest order, the main contribution comes from the heavy quark(charm quark)
decays, while the light flavors are treated as spectators. The contributions are due to the
semileptonic and the nonleptonic decays as follows:
Γ(c→ s) =
l=e,µ
Γc→sl̄υ +
q(q′)=u,d,s
Γc→sq̄q′ (4)
The semileptonic and nonleptonic decay rates of the c quark up to order 1/m2c has been
evaluated by many authors[17], and here we would directly use their results.
B. Non-spectator Contributions to Inclusive Decays of Ξ+cc, Ξ
cc , Ω
The total width of hadrons which involve at least one charm quark c can be decomposed
into two parts
Γ(HQ → f) = Γspectator + Γnonspectator. (5)
For the spectator scenario, the contribution to the total width of the (ccd)-baryon ground
state Ξ+cc, the (ccu)-baryon ground state Ξ
cc and the (ccs)-baryon ground state Ω
cc should
be a sum of decays rates of two c−quarks individuallynamely
Γspecccq ≃ 2Γspecc , q = u, d, s. (6)
To derive the non-spectator contributions for decays of Ξ+cc, Ξ
cc and Ω
cc, we need the
relevant effective Lagrangian:[18]
L(∆c=1)eff (µ = mc) = −
{VcsV ∗ud[C1(µ)s̄γµLcūγµLd+ C2(µ)ūγµLcs̄γµLd]
+VcdV
ud[C1(µ)d̄γ
µLcūγµLd+ C2(µ)ūγ
µLcd̄γµLd]
+VcsV
us[C1(µ)s̄γ
µLcūγµLs+ C2(µ)ūγ
µLcs̄γµLs]
l=e,µ
s̄γµLcν̄lγ
µLl}+ h.c. (7)
where L denotes 1−γ5
(i) The inclusive decays of Ξ+cc:
There are four diagrams which contribute to the the width of Ξ+cc, as shown in Fig.1. Here
we also include the Feynman diagrams which are CKM suppressed. Fig 1.(a),(c) are the
W-exchange diagrams (WE), while Fig 1.(b),(d) are the pauli-interference diagrams (PI).
Here Fig 1.(d) is arisen from the semi-leptonic decay of the charm quarks with the d−quark
in Ξ+cc. For the WE-type diagrams, we derive the contribution to the width as
(|Vcs|2|Vud|2C(zs+, zu+) + |Vcd|2|Vud|2C(zu+, zd+))P 2+
{[C21 (µ) + C22 (µ)]c̄γµLcd̄γµLd+ 2C1(µ)C2(µ)c̄γµLdd̄γµLc}, (8)
where P+ = pc + pd, zq+ =
(q = u, d, s). The definition of the function C(z1, z2) is
C(z1, z2) = −[−2(x32 − x31)− (x22 − x21)(3 + 2z1 − 2z2) + 4z1(x2 − x1)], (9)
where x1,2 =
(1+z1−z2)∓
(1+z1−z2)2−4z1
. In the expressions q and q̄ are free field opearotors of
quark and antiquark, and we will show in next section that all the non-perturbative QCD
effects are included in the wavefunctions. Their explicit expressions are given as
(2π)3
α=1,2
bqα(k)u
q (k)e
−ikx + d+qα(k)υ
q (k)e
(2π)3
α=1,2
b+qα(k)ū
q (k)e
ikx + dqα(k)ῡ
q (k)e
. (11)
For Ξ+cc, q=c, u.
The contributions from the Pauli-interference(PI) non-spectator diagrams to the width
of Ξ+cc are:
PI = −
{|Vud|2|Vcd|2Fµν(zu−, zd−)[NC21 (µ)c̄γµLdd̄γνLc+ C22 (µ)c̄iγµLdj d̄jγνLci
+2C1(µ)C2(µ)c̄γ
µLdd̄γνLc] + 2|Vcd|2Fµν(0, zl−)c̄γµLdd̄γνLc}, (12)
where zq− =
(q = u, d, e, µ) and P− = pc−pd. The definition of the function Fµν(z1, z2) is
Fµν(z1, z2) = −[2(x32 − x31)−
(2 + z1 − z2)(x22 − x21) + 3(x2 − x1)]P 2−gµν
+[2(x32 − x31)− 3(x22 − x21)]P−µP−ν , (13)
where the definitions of z1 and z2 are the same as before.
(ii) The inclusive decays of Ξ++cc :
The non-spectator contribution to the width of Ξ++cc come from the diagrams shown in Fig.2.
That is caused by an interference of the produced u−quark from decay of one of the charm
quarks with the u−quark in Ξ++cc . Here we also include the CKM suppressed Feynman
diagrams. The contribution is
PI = −
{|Vcs|2|Vud|2Fµν(zs−, zd−) + |Vcs|2|Vus|2Fµν(zs−, zs−)
+|Vcd|2|Vud|2Fµν(zd−, zd−)}
{C21(µ)c̄iγµLuj ūjγνLci +NC22 (µ)ūγµLcc̄γνLu+ 2C1(µ)C2(µ)ūγµLcc̄Lνu},
where z− =
(q = s, d), P− = pc − pu.
(iii) For the inclusive decays of Ω+cc:
The non-spectator contributions for Ω+cc not only come from the Pauli interference of the
s− quark produced in the non-leptonic, but also from the semi-leptonic decay of the charm
quarks with the s−quark in Ω+cc, the later one is suggested by Voloshin et al.[19]. As above,
here we include the CKM suppressed WE non-spectator diagrams. The WE non-spectator
contribution to the width Ω+cc is
|Vus|2|Vcs|2C(zu+, zs+)P 2+
{[C21 (µ) + C22 (µ)]c̄γµLcs̄γµLs+ 2C1(µ)C2(µ)c̄γµLss̄γµLc},
where zq+ =
, q = u, s and P+ = pc + ps.
The PI non-spectator contribution to the width of Ω+cc is
PI = −
{|Vcs|2|Vud|2Fµν(zu−, zd−) + |Vcs|2|Vus|2Fµν(zu−, zs−)}
{NC21(µ)c̄γµLss̄γνLc + C22(µ)c̄iγµLsj s̄jγνLci + 2C1(µ)C2(µ)c̄γµLss̄γνLc}
|Vcs|2Fµν(0, zl−)c̄γµLss̄γνLc, (16)
where zq− =
, q = u, d, s, e, µ and P− = pc − ps. Sandwiching the operators between
initial and final Ξ+cc, Ξ
cc , Ω
cc states, we obtain the hadronic matrix elements:
WE/PI
= 〈Ξ+cc(P = 0, s)|Γ̂
WE/PI
|Ξ+cc(P = 0, s)〉
PI = 〈Ξ
cc (P = 0, s)|Γ̂
PI |Ξ
cc (P = 0, s)〉
WE/PI
= 〈Ω+cc(P = 0, s)|Γ̂
WE/PI
|Ω+cc(P = 0, s)〉. (17)
III. THE HADRONIC MATRIX ELEMENTS
Because the hadronic matrix elements are fully determined by the non-perturbative
QCD effects which cannot be reliably evaluated at present yet, we need to invoke concrete
phenomenological models to carry out the computations. In this work, we adopt a simple
non-relativistic model, i.e. the harmonic oscillator[20]. This model has been widely em-
ployed in similar researches[21, 22, 23, 24, 25, 26]. In fact, an advantage of the calculations
of the lifetimes of heavy hadrons is that one does not need to deal with the hadronization
process of lighter products (quarks or even gluons) and the heavy hadrons can be well
described by such simple non-relativistic models, and the results are relatively reliable than
for light hadron decays.
(i)The inclusive decays of Ξ+cc:
In the harmonic oscillator model, the wavefunction of Ξ+cc is expressed as |Ξ+cc〉 and
|Ξ+cc(P, s)〉 = AB
color,spin
χspin,flavorϕcolor
d3pρd
3pλΨΞ+cc(pρ,pλ)|ci(pq1 , sq1), cj(pq2, sq2), dk(pq3, sq3)〉.
The normalization condition for |Ξ+cc(P, s)〉 is
〈Ξ+cc(P, s)|Ξ+cc(P′, s′)〉 = (2π)3
δ3(P−P′)δs,s′, (19)
where χspin,flavorϕcolor are the spin-flavor and color wavefunctions respectively. Their explicit
expressions are
χs= 1
,flavor =
(2|c↑c↑d↓〉 − |c↑c↓d↑〉 − |c↓c↑d↑〉) (20)
ϕcolor =
ǫijk. (21)
AB is the normalization constant. The spatial wavefunction ΨΞ+cc is a three-body harmonic
oscillator wavefunction and expressed as
ΨΞ+cc = exp(−
). (22)
Here aρ and aλ parameters reflecting the non-perturbative effects. In the above expressions,
the Jacobi transformations of p1, p2, p3 which are the momenta of the three valence quarks
ccd, and variables pρ, pλ, P are
p1 − p2√
,pλ =
p1 + p2 − 2mcmd p3
22mc+md
,P = p1 + p2 + p3. (23)
We choose the center-of-mass frame of Ξ+cc, i.e. (P=0) to calculate the hadronic matrix
elements. Substituting the four-quark operators into the expressions, we obtain the non-
spectator WE contributions to the width of Ξ+cc as
WE = 64π
2G2FP
+(|Vcs|2|Vud|2C(zs+, zu+) + |Vcd|2|Vud|2C(zu+, zd+))(C1(µ)− C2(µ))2
|AB|2[2(1 +
)]3/2
d3pρd
exp[−
]exp[−
(pλ +
1 + 2mc
(pρ − p′ρ))2
]ūcγµLucūdγ
µLud,
and the PI contribution is
PI = −
π2G2F{|Vcd|2|Vud|2Fµν(zu−, zd−)[−NC21 (µ) + C22 (µ)− 2C1(µ)C2(µ)]
−2|Vcd|2Fµν(0, zl−)}|AB|2[2(1 +
)]3/2
d3pρd
exp[−
]exp[−
(pλ +
1 + 2mc
(pρ − p′ρ))2
]ūcγ
µLud ūdγ
νLuc,
where the sum over spin means a sum over the polarizations of the three valence quarks of
Ξ+cc with their corresponding C-G coefficients in the spin-flavor wavefunction. uq, ūq denote
the Dirac spinors of free quarks q and the expression is
Eq +mq
Eq+mq
χ (26)
ūq =
Eq +mq
1 − σ·p
Eq+mq
in our case q denotes c and d quarks.
(ii)The inclusive decays of Ξ++cc :
The contribution from the PI non-spectator diagrams to the width of Ξ++cc is
Ξ++cc
PI = −
π2G2F{|Vcs|2|Vud|2Fµν(zs−, zd−) + |Vcs|2|Vus|2Fµν(zs−, zs−)
+|Vcd|2|Vud|2Fµν(zd−, zd−)}(C21(µ)−NC22 (µ)− 2C1(µ)C2(µ))|AB|2[2(1 +
)]3/2
d3pρd
ρexp[−
]exp[−
(pλ +
1 + 2mc
(pρ − p′ρ))2
µLuu ūuγ
νLuc. (28)
Similar to the case of Ξ+cc, the sum over spin means a sum of the polarizations of the three
valence quarks of Ξ++cc with their C-G coefficients. One only needs to replace u by d in pρ,
pλ and other expressions are similar to that for Ξ
(iii)The inclusive decays of Ω+cc:
The contribution from the W-boson exchange(WE) non-spectator diagrams to the width of
Ω+cc is
WE = 64π
2G2FP
+|Vus|2|Vcs|2C(zu+, zs+)(C1(µ)− C2(µ))2
|AB|2[2(1 +
)]3/2
d3pρd
exp[−
]exp[−
(pλ +
1 + 2mc
(pρ − p′ρ))2
]ūcγµLucūsγ
µLus,
whereas that from the Pauli-interference(PI) non-spectator diagrams is
PI = −
π2G2F{[|Vcs|2|Vud|2Fµν(zu−, zd−) + |Vcs|2|Vus|2Fµν(zu−, zs−)]
[−NC21 (µ) + C22 (µ)− 2C1(µ)C2(µ)]− 2|Vcs|2Fµν(0, zl−)}|AB|2[2(1 +
)]3/2
d3pρd
ρexp[−
]exp[−
(pλ +
1 + 2mc
(pρ − p′ρ))2
µLusūsγ
νLuc. (30)
The sum over polarizations is similar to that for Ξ+cc and Ξ
IV. INPUT PARAMETERS AND NUMERICAL RESULTS
To obtain the decay amplitudes, we adopt the input parameters as follows[7, 27]: GF =
1.166×10−5GeV−2, |Vcs| = 0.9737, |Vud| = 0.9745, C1(mc) = 1.3, C2(mc) = −0.57,mc = 1.60
GeV, ms = 0.45 GeV, mu = md = 0.3 GeV, m
s = 0.2GeV, m
u = m
d = 0, MΞ+cc = MΞ++cc =
3.519 GeV, MΩ+cc = 3.578 GeV, MΞ+∗cc −MΞ+cc = MΞ++∗cc −MΞ++cc = MΩ+∗cc −MΩ+cc = 0.132
GeV. Here mq∗ denotes the current quark mass of flavor q.
The non-perturbative parameters aρ, aλ in the harmonic oscillator wavefunctions are
selected as follows: for J/ψ, in ref.[20], a2ρ = 0.33GeV
2, for D−mesons, a2ρ = 0.25GeV2. For
TABLE I: The numerical results about the contributions from the different components and the
evaluated lifetime for the doubly charmed baryons. For a comparison, in the following table, we
list the corresponding lifetimes predicted by the authors of ref.[7] where the diquark picture was
employed. It is noted that in ref.[7], the authors used various input parameters and obtained
slightly diverse results, we take average values of the numbers in the table. There is only one
datum for the lifetimes on τ
given by the SELEX collaboration which is also listed the table.
Ξ+cc Γspec(10
−12GeV) ΓWEnon (10
−13GeV) ΓPInon(10
−15GeV) τ
(ps) τ
(ps) in ref.[7] exp(ps)
2.01 6.43 -3.36 0.25 0.19 0.033
Ξ++cc Γspec(10
−12GeV) ΓPInon(10
−12GeV) τ
(ps) τ
(ps) in ref.[7]
2.01 -1.02 0.67 0.52 −
Ω+cc Γspec(10
−12GeV) ΓWEnon (10
−14GeV) ΓPInon(10
−12GeV) τ
(ps) τ
(ps) in ref.[7]
2.01 4.25 1.10 0.21 0.22 −
the doubly charmed baryons, because aρ reflects the coupling between two charm quarks,
we set it to be the same as that for J/ψ. aλ reflects the coupling of the light quark with
these two charm quark, thus we can reasonably set it to be the same as aρ in D-mesons.
With these parameters as input, the lifetimes of the doubly charmed baryons can be
evaluated out (see TABLE.I), if the non-spectator effects are taken into account.
V. CONCLUSION AND DISCUSSION
In this work, we evaluate the lifetimes of doubly charmed baryons with the non-spectator
effects being properly taken into account. As argued in the introduction, to evaluate the
lifetimes (the total widths), only the inclusive processes are concerned, and then the non-
perturbative effects are all from the wavefunctions of the doubly charmed baryons. Due
to existence of the two heavy charm quarks, the non-relativistic harmonic oscillator model
should apply in this case. Mainly, we carefully calculate the contribution of non-perturbative
effects to the lifetimes in the model, which are closely related to the bound states of the
baryons.
Our numerical results indicate that the non-spectator contributions to the lifetimes of
Ξ+cc, Ξ
cc and Ω
cc are substantial. The non-spectator contributions to the width of Ξ
cc are
mainly from the WE diagrams (the PI diagrams which contribute are CKM suppressed),
since the WE contribution is constructive, therefore the lifetime of Ξ+cc is much suppressed.
By contraries, for Ξ++c and Ω
cc, the non-spectator contributions are mainly from the PI
diagrams and the net effect is destructive. It is noted that for Ω+cc there still are Cabibbo-
suppressed WE diagrams, but for Ξ++cc there are only PI diagrams. Therefore the predicted
lifetime of Ξ++cc is larger than that of other two baryons. We also employ other values for
parameters aρ, aλ and find that the resultant values can vary within 20% uncertainty.
Our results are
τ(Ξ+cc) = 0.25 ps τ(Ξ
cc ) = 0.67 ps and τ(Ω
cc) = 0.21 ps.
These are generally consistent with the results obtained by Kiselev et al.[7] and Guberina
et al.[8], even though they used different models for calculating the hadronic matrix elements.
Concretely, they used the diquark picture and attributed the non-perturbative effects into
the wavefunction of the diaquark at origin. Kiselev et al. gave τ(Ξ+cc) ∼ 0.16 − 0.22 ps
τ(Ξ++cc ) ∼ 0.40− 0.65 ps and τ(Ω+cc) ∼ 0.24− 0.28.
Although all the theoretical predictions based on different models agree with each other,
they are obviously one order larger than the upper limit of the measured value on the
lifetime of Ξ+cc (0.033 ps) by the SELEX collaboration[6]. This deviation, as suggested by
some authors, may come from experiments[28]. So far the difference between theoretical
predictions and experimental data may imply some unknown physics mechanisms which
drastically change the value, if the future experiment, say at LHCb, confirms the measure-
ment of the SELEX. Recently, several groups have studied the possibility of doubly heavy
baryon production at hadron collider LHC and future linear collider ILC[29, 30] and the
effective field theories for two heavy quarks system are also further investigated[31]. We are
expecting the new data from more accurate experiments at LHC and ILC to improve our
theoretical framework and determine if there are contributions from new physics beyond
the standard model.
Acknowledgement: This work is supported by the National Natural Science Foundation of
China.
[1] I. Bigi, N. Uraltsev, Phys. Lett. B280 (1992) 120; I. Bigi, N. Uraltsev and A. Vainshtein,
Phys. Lett. B293 (1992) 430, (E)B297 (1993) 477; B. Blok and M. Shifman, Nucl. Phys.
B399 (1993) 441, 459; G. Belliui et al., Phys. Rep. 289 (1997) 1.
[2] E. Franco, V. Lubicz, F. Mescia and C. Tarantino, Nucl. Phys. B63 (2002) 212.
[3] N.G. Uraltsev, Phys. Lett. B376 (1996) 303; F. Gabbiani, A.I. Onischenko and A.A. Petrov,
Phys. Rev. D70 (2004) 094031; E. Franco, V. Lubicz, F. Mescia and C. Tarantino, Nucl. Phys.
B633 (2002) 212.
[4] CDF Collaboration: A. Abulencia et al., arXiv:hep-ex/0609021.
[5] X.G. He, T. Li, X.Q. Li and Y.M. Wang, Phys. Rev. D74 (2006) 034026.
[6] The SELEX Collaboration, M. Mattson et al., Phys. Rev. Lett. 89 (2002) 112001.
[7] V.V. Kiselev, A.K. Likhoded and A.I. Onishchenko, Phys. Rev. D60 (1999) 014007; A.I.
Onishchenko, arXiv:hep-ph/9912424.
[8] B. Guberina, B. Melić, H. S̆tefanc̆ić, Eur. Phys. J C9 (1999) 213.
[9] A.F. Falk, M.E. Luke, M.J. Savage and M.B. Wise, Phys. Rev. D49 (1994) 555.
[10] I. Bigi, B. Blok, M. Shifman, N. Uraltsev et al., ”B Decays”, ed. S. Stone, Word Scientific,
Singapore (1994); M. Neubert and C.T. Sachrajda, Nucl. Phys.B483 (1997) 339; B. Guberina,
B. Melić and H. S̆tefanc̆ić, Eur. Phys. J C13 (2000) 551.
[11] A. Datta, E.A. Paschos and Y.L. Wu, Nucl. Phys. B311 (1988) 35.
[12] Q. Hokim and X.Y. Pham, Phys. Lett. B122 (1989) 297.
[13] Y. Nir, Phys. Lett. B221 (1989) 184.
[14] E. Bagan, P. Ball, V.M. Braun and P. Gosdzinsky, Nucl. Phys. B432 (1994) 3, Phys. Lett.
B342 (1995) 362, [E:B374 (1996) 363]; E. Bagan, P. Ball, B. Fiol and P. Gosdzinsky, Phys.
Lett. B351 (1995) 546.
[15] I.I. Bigi, N.G. Uraltsev and A.I. Vainshtein, Phys. Lett. B293 (1992) 430, [E:B297 (1993)
477]; I.I. Bigi, M.A. Shifman, N.G. Uraltsev and A.I. Vainshtein, Phys. Rev. Lett. 71(1993)
[16] A.F. Falk, Z. Ligeti, M. Neubert and Y. Nir, Phys. Lett. B326 (1994) 145.
[17] H.Y. Cheng, Phys. Rev. D56 (1997) 2783; M. Luke, M.J. Savage and M.B. Wise, Phys. Lett.
B345 (1995) 301; I. Bigi, Phys. Lett. B371(1996) 105, arXiv:hep-ph/9508408.
http://arxiv.org/abs/hep-ex/0609021
http://arxiv.org/abs/hep-ph/9912424
http://arxiv.org/abs/hep-ph/9508408
[18] F. Buccella, M. Lusignoli, G. Miele, A. Pugliese and P. Santorelli, Phys. Rev. D51 (1995)
3478.
[19] M.B. Voloshin, Phys. Lett. B385 (1996) 369.
[20] A.L. Yaouanc, L. Olivier, O. Pène and J.C. Raynal, ”Hadron Transitions in the Quark Model”,
Gordon and Breach Science Publish Publish (1998).
[21] M. Oda, K. Nishimura, M. Ishida, and S. Ishida, arXiv:hep-ph/0005102; R. Mohanta, A. Giri,
M. Khanna, M. Ishida and S. Ishida, Prog. Theor. Phys. 102 (1995) 645; R. Mohanta, A.
Giri, M. Khanna, M. Ishida and S. Ishida, Prog. Theor. Phys. 101 (1999) 1083; R. Mohanta,
A. Giri, M. Khanna, M. Ishida and S. Ishida, Prog. Theor. Phys. 101 (1999) 959; M. Ishida,
S. Ishida and M. Oda, Prog. Theor. Phys. 98 (1997)159.
[22] A. Hosaka, M. Takayama and H. Toki, Nucl. Phys. A678 (2000) 147.
[23] R. Bonnaz, B. Silvestre-Brac and C. Gignoux, Eur. Phys. J. A107 (2002) 363.
[24] T. Barnes, AIP Conf. Proc. 619 (2002) 673; Nuovo, Cim. A107 (1994) 2491.
[25] H.Y. Cheng and B. Tseng, Phys. Rev. D53 (1996) 1457, [E:D55 (1997) 1697].
[26] J. Amundson, Phys. Rev. D49 (1994) 373.
[27] W.-M. Yao et al., Particle Data Group, J. Phys. G33, 1 (2006).
[28] V.V. Kiselev and A.K. Likhoded, arXiv:hep-ph/0208231.
[29] C.H. Chang, J.X. Wang and X.G. Wu, arXiv:hep-ph/0702054; C.H. Chang, J.P. Ma, C.F.
Qiao and X.G. Wu, arXiv:hep-ph/0610205; C.H. Chang, C.F. Qiao, J.X. Wang and X.G. Wu,
Phys. Rev. D73 (2006) 094022.
[30] S.Y. Li, Z.G. Si and Z.J. Yang, arXiv:hep-ph/0701212; J.P. Ma and Z.G. Si, Phys. Lett. B568
(2003) 135.
[31] N. Brambilla, arXiv:hep-ph/0609237; N. Brambilla, T. Roesch and A. Vairo, Phys. Rev. D72
(2005) 034021.
http://arxiv.org/abs/hep-ph/0005102
http://arxiv.org/abs/hep-ph/0208231
http://arxiv.org/abs/hep-ph/0702054
http://arxiv.org/abs/hep-ph/0610205
http://arxiv.org/abs/hep-ph/0701212
http://arxiv.org/abs/hep-ph/0609237
FIG. 1: non-spectator effects contribution to lifetime of Ξccd
FIG. 2: non-spectator effects contribution to lifetime of Ξccu
FIG. 3: non-spectator effects contribution to lifetime of Ωccs
	Introduction
	Formulation for Lifetimes of cc+, cc++, cc+ 
	Spectator Contribution to Lifetimes of cc+, cc++, cc+ 
	Non-spectator Contributions to Inclusive Decays of cc+, cc++, cc+
	The hadronic matrix elements
	Input parameters and Numerical results
	Conclusion and Discussion
	References
