Lifetimes of Doubly Charmed Baryons

The lifetimes of doubly charmed hadrons are analyzed within the framework of the heavy quark expansion (HQE). Lifetime differences arise from spectator effects such as $W$-exchange and Pauli interference. The $\Xi_{cc}^{++}$ baryon is longest-lived in the doubly charmed baryon system owing to the destructive Pauli interference absent in the $\Xi_{cc}^+$ and $\Omega_{cc}^+$. In the presence of dimension-7 contributions, its lifetime is reduced from $\sim5.2\times 10^{-13}s$ to $\sim3.0\times 10^{-13}s$. The $\Xi_{cc}^{+}$ baryon has the shortest lifetime of order $0.45\times 10^{-13}s$ due to a large contribution from the $W$-exchange box diagram. It is difficult to make a precise quantitative statement on the lifetime of $\Omega_{cc}^+$. Contrary to $\Xi_{cc}$ baryons, $\tau(\Omega_{cc}^+)$ becomes longer in the presence of dimension-7 effects and the Pauli interference $\Gamma^{\rm int}_+$ even becomes negative. This implies that the subleading corrections are too large to justify the validity of the HQE. Demanding the rate $\Gamma^{\rm int}_+$ to be positive for a sensible HQE, we conjecture that the $\Omega_c^0$ lifetime lies in the range of $(0.75\sim 1.80)\times 10^{-13}s$. The lifetime hierarchy pattern is $\tau(\Xi_{cc}^{++})>\tau(\Omega_{cc}^+)>\tau(\Xi_{cc}^+)$ and the lifetime ratio $\tau(\Xi_{cc}^{++})/\tau(\Xi_{cc}^+)$ is predicted to be of order 6.7.


Abstract
The lifetimes of doubly charmed hadrons are analyzed within the framework of the heavy quark expansion (HQE). Lifetime differences arise from spectator effects such as W -exchange and Pauli interference. The Ξ ++ cc baryon is longest-lived in the doubly charmed baryon system owing to the destructive Pauli interference absent in the Ξ + cc and Ω + cc . In the presence of dimension-7 contributions, its lifetime is reduced from ∼ 5.2 × 10 −13 s to ∼ 3.0 × 10 −13 s. The Ξ + cc baryon has the shortest lifetime of order 0.45 × 10 −13 s due to a large contribution from the W -exchange box diagram. It is difficult to make a precise quantitative statement on the lifetime of Ω + cc . Contrary to Ξ cc baryons, τ (Ω + cc ) becomes longer in the presence of dimension-7 effects and the Pauli interference Γ int + even becomes negative. This implies that the subleading corrections are too large to justify the validity of the HQE. Demanding the rate Γ int + to be positive for a sensible HQE, we conjecture that the Ω 0 c lifetime lies in the range of (0.75 ∼ 1.80) × 10 −13 s. The lifetime hierarchy pattern is τ (Ξ ++ cc ) > τ (Ω + cc ) > τ (Ξ + cc ) and the lifetime ratio τ (Ξ ++ cc )/τ (Ξ + cc ) is predicted to be of order 6.7 .
It turns out that the relevant dimension-7 spectator effects are in the right direction for explaining the large lifetime ratio of τ (Ξ + c )/τ (Λ + c ), which is enhanced from 1.05 to 1.88, in better agreement with the experimental value [8]. However, the destructive 1/m c corrections to Γ(Ω 0 c ) are too large to justify the use of the HQE, namely, the predicted Pauli interference and semileptonic rates for the Ω 0 c become negative, which certainly do not make sense. Demanding these rates to be positive for a sensible HQE, it has been conjectured in [8] that the Ω 0 c lifetime lies in the range of (2.3 ∼ 3.2) × 10 −13 s. This leads to the new lifetime pattern This new charmed baryon lifetime pattern can be tested by LHCb.
Very recently, LHCb has reported a new measurement of the Ω 0 c lifetime, τ (Ω 0 c ) = (2.68 ± 0.24 ± 0.10 ± 0.02) × 10 −13 s [10], using the semileptonic decay This value is nearly four times larger than the current world-average value of τ (Ω 0 c ) = (0.69 ± 0.12) × 10 −13 s [9] from fixed target experiments. 2 This indicates that the Ω 0 c , which is naively expected to be shortest-lived in the charmed baryon system owing to the large constructive Pauli interference, could live longer than the Λ + c due to the suppression from 1/m c corrections arising from dimension-7 four-quark operators. In this work we shall study the lifetimes of doubly charmed baryons within the framework of the HQE. It is organized as follows. In Sec. II we give the general HQE expressions for inclusive nonleptonic and semileptonic widths. A special attention is paid to the doubly charmed baryon matrix elements of dimension-3 and -5 operators which are somewhat different from the ones of singly charmed baryons. We then proceed to discuss the relevant dimension-6 and -7 four-quark operators. Evaluation of doubly charmed baryon matrix elements and numerical results are presented in Sec. III. Conclusions are given in Sec. IV.

II. THEORETICAL FRAMEWORK
Under the heavy quark expansion, the inclusive nonleptonic decay rate of a doubly heavy baryon B QQ containing two heavy quarks QQ is given by [12,13] in analog to the case of a singly heavy hadron H Q . Through the use of the operator product expansion, the transition operator T can be expressed in terms of local quark operators where ξ is the relevant CKM matrix element, the dimension-6 T 6 consists of the four-quark operators (QΓq)(qΓQ) with Γ representing a combination of the Lorentz and color matrices, while a subset of dimension-7 T 7 is governed by the four-quark operators containing derivative insertions. Hence,

A. Dimension-3 and -5 operators
In heavy quark effective theory (HQET), the dimension-3 operatorQQ in the rest frame has the expressionQ with the normalization Hence, The non-perturbative parameters λ 1 and λ 2 are independent of m Q and have the same values for all particles in a given spin-flavor multiplet. We first consider the non-perturbative parameter µ 2 π . In general, µ 2 π = p 2 = m 2 Q v 2 Q . The average kinetic energy of the diquark QQ and the light quark is the mass of the diquark (light quark). This together with the momentum conservation As shown in [3], the average kinetic energy T ′ of heavy quarks inside the diquark given by 1 2 ) is equal to T /2 due to the color wave function of the diquark. Hence, the average velocityṽ of the heavy quark inside the diquark isṽ 2 = T /(2m Q ). The average velocity v Q of the heavy quark inside the baryon Hence, (2.10) We next turn to the parameter µ 2 G . In HQET, the mass of the singly heavy baryon B Q has the expression whereΛ B Q is a parameter of HQET and it can be regarded as the binding energy of the heavy hadron in the infinite mass limit. For the doubly heavy baryon B QQ , if the heavy diquark acts as a point-like constitute, its mass is of the form (2.12) There are two distinct chromomagnetic fields inside the B QQ : one is the chromomagnetic field produced by the light quark and the other by the heavy quark. For the former (latter), the is the spin operator of the diquark (light quark), and S i (i = 1, 2) is the spin of the constituent quark inside the diquark. The parameter d H is given by 3 (2.13) It follows from Eq. (2.12) that λ dq 2 can be expressed in terms of the hyperfine mass splitting and hence, To evaluate the parameter λ QQ 2 , let us consider a simple quark model of De Rújula et al. [14] M baryon = M 0 + · · · + 16 It is well known that the fine structure constant is − 4 3 α s forqq pairs in a meson and − 2 3 α s for qq pairs in a baryon [14]. This is because theqq pair in a meson must be a color-singlet, while the qq pair in a baryon is in color antitriplet state. The mass of the doubly heavy baryon B QQ is given by where ψ dq (0) is the light quark wave function at the origin of the QQ diquark and ψ QQ (0) is the diquark wave function at the origin. For the doubly charmed baryons we have The term proportional to |ψ dq (0)| 2 can be expressed in terms of the hyperfine mass splitting of Ξ cc . Hence, we obtain Hence, λ cc 2 (Ξ cc ) = (1/9)g 2 s |ψ cc (0)| 2 /m c . However, the above expression of µ 2 G is not the end of story. It has been known that HQET is not the appropriate effective field theory for hadrons with more than one heavy quark. HQET is formulated as an expansion in Λ QCD /m Q . For a singly heavy hadron, the heavy quark kinetic energy is neglected as it occurs as a small 1/m Q correction. For a bound state containing two or more heavy quarks, the heavy quark kinetic energy is very important and cannot be treated as a perturbation. The appropriate theory for dealing such a system is non-relativistic QCD (NRQCD), 4 in which one hasQ in terms of the two-spinor ψ Q . According to the counting rule, the Darwin term for the interaction with the chromoelectric field is of the same order of magnitude as the chromomagnetic term [16]. Hence, we get an additional contribution to µ 2 The last term can be obtained by using the equation of motion for the chromoelectric field. Note that our result is different from the original expression 5 obtained in [3] in the sign of the first term and in the magnitude of |ψ cc (0)| 2 terms. Therefore, Since the hyperfine mass splitting of D mesons is given by we are led to the relation (2.25) 4 However, it was pointed out very recently in [15] that in the limit m Q > m Q v Q > m Q v 2 Q ≫ Λ QCD , such a system can be described by a version of HQET with a diquark degree of freedom. 5 Guberina et al. [5] obtained a similar expression except for the magnitude of |ψ cc (0)| 2 terms In the heavy quark limit, the doubly charmed baryon wave function ψ dq Ξcc (0) is expected to be the same as the meson wave function ψ cq Dq (0) if the diquark behaves as a point-like particle, 6 ψ dq It follows the well-known mass relation which has been derived in various contents, such as HQET [18], 7 pNRQCD (potential NRQCD) [19,20] and the quark model [21,22]. The nonleptonic and semiletponic decay rates of the heavy quark c of the B cc are given by where the expressions of the coefficients c 3,c and c 5,c can be found, for example, in [8]. 6 In [5] and in [17], the authors argued that |ψ dq (0)| 2 = 2 3 |ψ cq (0)| 2 due to different spin content of doubly charmed baryons. However, this will not lead to the approximate mass relation given by Eq. (2.27). 7 A factor of 2 was missed in the original mass relation derived in [18].

B. Dimension-6 operators
Defining the dimension-6 four-quark operators in Eq. (2.3) for spectator effects in inclusive decays of doubly charmed baryons denoted by B cc are given by (only Cabibbo-allowed decays with ξ = |V cs V ud | 2 being listed here) [23][24][25] T Bcc,d 6,ann = where (q 1 q 2 ) ≡q 1 γ µ (1 − γ 5 )q 2 , and α, β are color indices and x = m 2 s /m 2 c . Spectator effects in the weak decays of the doubly charmed baryons Ξ ++ cc , Ξ + cc and Ω + cc are depicted in Fig. 1. The first term T Bcc,d 6,ann in (2.31) corresponds to a W -exchange contribution which appears in Ξ + cc decays (Cabibbo-suppressed T Bcc,s 6,ann term appearing in Ω + cc decays). The second term T Bcc,u 6,int− arises from the destructive Pauli interference of the u quark produced in the c quark decay with the u quark in the wave function of the doubly charm baryon B cc , namely Ξ ++ cc ( Fig. 1(a)). The last term T Bcc,s 6,int+ is due to the constructive interference of the s quark and hence it occurs only in charmed baryon decays ( Fig. 1(c)).
For inclusive semileptonic decays, apart from the heavy quark decay contribution there is an additional spectator effect in charmed-baryon semileptonic decay originating from the Pauli interference of the s or d quark [26]; that is, the s (d) quark produced in c → sℓ + ν ℓ (c → dℓ + ν ℓ ) has an interference with the s (d) quark in the wave function of the charmed baryon (see Fig. 1). It is now ready to deduce this term from T q 3 6,int+ in Eq. (2.31) by putting c 1 = 1, c 2 = 0, N c = 1: Before proceeding, we would like to clarify how the heavy quark expansion and approximation are consistent with the claimed accuracy. For example, the hadronic matrix element of the dimension-3 operatorQQ, Eq. (2.6), is in itself an approximation valid up to corrections of order 1/m 3 Q . This is because the chromomagnetic operator µ 2 G given in Eq. (2.21), for instance, is valid up to 1/m Q corrections stemming from the expansion of Eq. (2.12) truncated at order 1/m Q . Hence, to the order of 1/m 3 Q expansion in Eq. (2.3), one may wonder if it is necessary to take into account the higher order corrections such as c 3,Q O(1/m 3 Q ) + c 5,Q O(1/m 3 Q ) besides the dimension-6 operator c 6,Q T 6 /m 3 Q . It turns out that higher order corrections can be neglected as there is a two-body phase-space enhancement factor of 16π 2 for spectator effects induced by dimension-6 four-quark operators T 6 relative to the three-body phase space for heavy quark decay. Indeed, the phase-space enhancement for spectator effects is already taken into account in Eq.

C. Dimension-7 operators
To the order of 1/m 4 Q in the heavy quark expansion in Eq. (2.3), we need to consider dimension-7 operators. For our purposes, we shall focus on the 1/m Q corrections to the spectator effects discussed in the last subsection and neglect the operators with gluon fields. Dimension-7 terms are either the four-quark operators times the spectator quark mass or the four-quark operators with one or two additional derivatives [27,28]. We shall follow [29] to define the following dimension-7 four-quark operators: and the color-octet operators S q i (i = 1, ..., 6) obtained from P q i by inserting t a in the two currents of the respective color singlet operators. In order to evaluate the baryon matrix elements, it is more convenient to express dimension-7 operators in terms of P q i andP q i operators, whereP i denotes the color-rearranged operator that follows from the expression of P i by interchanging the color indices of the q i andq j Dirac spinors. We shall see below that the hadronic matrix elements of dimension-7 operators are suppressed relative to that of dimension-6 ones by order m q /m c .
Using the relation we obtain [8] T Bcc,d 7,ann = As for the dimension-7 four-operator for semileptonic decays, it can be obtained from T Bcc,s 7,int by setting c 1 = 1, c 2 = 0 and N c = 1. Taking into account the lepton mass corrections, it reads [8]

III. LIFETIMES OF DOUBLY CHARMED BARYONS
The inclusive nonleptonic rates of doubly charmed baryons in the valence quark approximation and in the limit m s /m c = 0 can be expressed approximately as Because Γ int + is positive and Γ int − is negative, it is obvious that Ξ ++ cc is longest-lived, whereas Ξ + cc (Ω + cc ) is the shortest-lived if Γ int + > Γ ann (Γ int + < Γ ann ). In this section, we shall begin with the evaluation of the doubly charmed baryon matrix elements of dimension-6 and -7 operators and then proceed to compute the spectator effects to see the relative weight between Γ int + and Γ ann .

A. Baryon matrix elements
The spectator effects in inclusive heavy bottom baryon decays arising from dimension-6 and dimension-7 operators are given by Eqs.
where f Dq and m Dq are the decay constant and the mass of the heavy meson D q , respectively, and the wave function ratio r Bcc is defined by According to Eq. (2.26), we should have r Ξcc = 1. 8 The parameterB is defined by Since the color wavefunction for a baryon is totally antisymmetric, the matrix element of (cc)(qq) is the same as that of (cq)(qc) except for a sign difference. That is,B = 1 under the valence-quark approximation. Likewise, the B cc matrix elements of dimension-7 operators are similar to that of the sextet singly charmed baryon Ω 0 c (see Eq. (4.14) of [8]) where the parameters η q i are expected to be of order unity, and m {cc} is the mass of the cc diquark. We take m {cc} to be 3226 MeV obtained from the relativistic quark model [22]. Note that the term is of order m q /m c . Therefore, the matrix elements of dimension-7 operators are suppressed by a factor of m q /m c relative to that of dimension-6 ones. For the matrix elements of the operatorsP q i , we introduce a parameterβ q i in analog to Eq. (3.4) so thatβ q i = 1 under the valence quark approximation. For the spectator effects in doubly charmed baryon decays, we apply Eqs. (3.2) and (3.5) to evaluate the matrix elements of the dimension-6 and -7 operators. The results are |V cs | 2 r Ωcc |ψ Ds cs (0)| 2 5 − (3.10) Except for the weak annihilation term, the expression of Pauli interference will be very lengthy if the hadronic parameters η q i andβ q i are all treated to be different from each other. Since in realistic calculations we will setβ q i (µ h ) = 1 under valence quark approximation and put η q i to unity, we shall assume for simplicity that η q i = η andβ q i =β. As far as the dimension-6 spectator effects are concerned, we now compare our results Eqs. (3.9) and (3.10) with Eqs. (13) and (8) of [5]. Since we are working at the µ = m Q scale, we need to set the parameter κ appearing in [5] to be unity. Noting that r Bcc |ψ cq D (0)| 2 = |ψ dq Bcc (0)| 2 in our case, we see that Γ int + and Γ int − obtained by Guberina, Melić and H.Štefančić (GMS) are larger than ours by a factor of 3/2, whereas their Γ ann (Γ SL ) is smaller than ours by a factor of 6/5 (2). Because the wave function of the doubly charmed baryon is related to that of the charmed meson through the relation |ψ dq (0)| 2 = 2 3 |ψ cq (0)| 2 by GMS, it turns out that while we agree on the Γ int + and Γ int − in terms of |ψ cq (0)| 2 , the expressions of Γ ann and Γ SL int by GMS are smaller than ours by a factor of 9/5 and 3, respectively.

B. Numerical results
To compute the decay widths of doubly charmed baryons, we have to specify the values ofB and r Bcc . SinceB = 1 in the valence-quark approximation and since the wavefunction squared ratio r is evaluated using the quark model, it is reasonable to assume that the NQM and the valencequark approximation are most reliable when the baryon matrix elements are evaluated at a typical hadronic scale µ had . As shown in [30], the parametersB and r renormalized at two different scales are related via the renormalization group equation to bẽ (3.12) and β 0 = 11 3 N c − 2 3 n f . The parameter κ takes care of the evolution from m Q to the hadronic scale. We consider the hadronic scale in the range of µ had ∼ 0.65 − 1 GeV. Taking the scale µ had = 0.90 GeV as an illustration, we obtain α s (µ had ) = 0.59,B(µ) = 0.75B(µ had ) ≃ 0.75 and r(µ) ≃ 1.33 r(µ had ). The parameterβ is treated in a similar way.
As shown in [8], the heavy quark expansion in 1/m c does not work well for describing the lifetime pattern of singly charmed baryons. Since the charm quark is not heavy enough, it is sensible to consider the subleading 1/m c corrections to spectator effects as depicted in Eq. (3.9). The numerical results are shown in Table III. By comparing Table III with Table II, we see that the lifetimes of Ξ ++ cc and Ξ + cc become shorter, while τ (Ω + cc ) becomes longer. This is because Γ int + and Γ semi for Ω + cc are subject to large cancellation between dimension-6 and -7 operators. Such cancellation also occurs in Ξ + cc but not so dramatic as the constructive Pauli interference there is Cabibbo-suppressed. We see from Table III that Γ int + (Ω + cc ) even becomes negative. This is because the dimension-7 contribution Γ int +,7 (Ω + cc ) is destructive and its size are so large that it overcomes the dimension-6 one and flips the sign. This implies that the subleading corrections are too large to justify the validity of the HQE.
In order to allow a description of the 1/m 4 c corrections to Γ(Ω + cc ) within the realm of perturbation theory, we follow [8] to introduce a parameter α so that Γ int +,7 (Ω + cc , Ξ + cc ) and Γ SL 7 (Ω + cc , Ξ + cc ) are multiplied by a factor of (1 − α); that is, α describes the degree of suppression. In Table IV  For the Ξ + cc , its lifetime is rather insensitive to the variation of α as both Γ int +,7 (Ξ + cc ) and Γ SL 7 (Ξ + cc ) are Cabibbo-suppressed.
Our prediction of τ (Ξ ++ cc ) is slightly larger than the LHCb measurement (1.1). We learn from [8] that the predicted lifetimes of heavy mesons or baryons are always longer than the measured values. Presumably, this is because we have not yet taken into account all possible QCD corrections fully. Nevertheless, the lifetime ratios should be more trustworthy than the absolute lifetimes themselves. In the present work, we find that the ratio τ (Ξ ++ cc )/τ (Ξ + cc ) is ∼ 9.1 to order 1/m 3 c and ∼ 6.7 to order 1/m 4 c .

IV. CONCLUSIONS
In this work we have analyzed the lifetimes of doubly charmed hadrons within the framework of the heavy quark expansion. It is well known that the lifetime differences stem from spectator effects such as W -exchange and Pauli interference. We rely on the quark model to evaluate the hadronic matrix elements of dimension-6 and -7 four-quark operators responsible for spectator effects.
The main results of our analysis are as follows.
• The doubly charmed baryon matrix element of the σ · G operator receives three distinct contributions: the interaction of the heavy quark with the chromomagnetic field produced from the light quark and from the other heavy quark, and the so-called Darwin term in which the heavy quark interacts with the chromoelectric field. The last term arises because the appropriate theory for dealing hadrons with more than one heavy quark is NRQCD rather than HQET.
• The Ξ ++ cc baryon is longest-lived in the doubly charmed baryon system owing to the destructive Pauli interference absent in the Ξ + cc and Ω + cc . In the presence of dimension-7 contributions, its lifetime is reduced from ∼ 5.2 × 10 −13 s to ∼ 3.0 × 10 −13 s.
• The Ξ + cc baryon has the shortest lifetime of order 0.45 × 10 −13 s due to a large contribution from the W -exchange box diagram.
• It is difficult to make a precise statement on the lifetime of Ω + cc . Contrary to Ξ cc baryons, τ (Ω + cc ) becomes longer in the presence of dimension-7 effects so that the Pauli interference Γ int + even becomes negative. This means that the subleading corrections are too large to justify the validity of the HQE. Demanding the rate Γ int + to be positive for a sensible HQE, we conjecture that the Ω 0 c lifetime lies in the range of (0.75 ∼ 1.80) × 10 −13 s.