Two-body weak decays of doubly charmed baryons

The hadronic two-body weak decays of the doubly charmed baryons Ξ++ cc ,Ξ + cc and Ω + cc are studied in this work. To estimate the nonfactorizable contributions, we work in the pole model for the P -wave amplitudes and current algebra for S-wave ones. For the Ξ++ cc → Ξc π+ mode, we find a large destructive interference between factorizable and nonfactorizable contributions for both Sand P -wave amplitudes. Our prediction of ∼ 0.70% for its branching fraction is smaller than the earlier estimates in which nonfactorizable effects were not considered, but agrees nicely with the result based on an entirely different approach, namely, the covariant confined quark model. On the contrary, a large constructive interference was found in the P -wave amplitude by Dhir and Sharma, leading to a branching fraction of order (7− 16)%. Using the current results for the absolute branching fractions of (Λc ,Ξ + c ) → pK−π+ and the LHCb measurement of Ξ++ cc → Ξc π+ relative to Ξ++ cc → Λc K−π+π+, we obtain B(Ξ++ cc → Ξc π)expt ≈ (1.83±1.01)% after employing the latest prediction of B(Ξ++ cc → Σ++ c K ∗0 ). Our prediction of B(Ξ++ cc → Ξc π+) ≈ 0.7% is thus consistent with the experimental value but in the lower end. It is important to pin down the branching fraction of this mode in future study. Factorizable and nonfactorizable S-wave amplitudes interfere constructively in Ξcc → Ξcπ. Its large branching fraction of order 4% may enable experimentalists to search for the Ξcc through this mode. That is, the Ξ + cc is reconstructed through the Ξcc → Ξcπ followed by the decay chain Ξc → Ξ−π+ → pπ−π−π+. Besides Ξcc → Ξcπ +, the Ξcc → Ξc (π0, η) modes also receive large nonfactorizable contributions to their Swave amplitudes. Hence, they have large branching fractions among Ξcc → Bc + P channels. Nonfactorizable amplitudes in Ξ++ cc → Ξ + c π + and Ωcc → Ξ + c K 0 are very small compared to the factorizable ones owing to the Pati-Woo theorem for the inner W -emission amplitude. Likewise, nonfactorizable S-wave amplitudes in Ξcc → Ξ + c (π 0, η) decays are also suppressed by the same mechanism.

The hadronic two-body weak decays of the doubly charmed baryons Ξ ++ cc , Ξ + cc and Ω + cc are studied in this work. To estimate the nonfactorizable contributions, we work in the pole model for the P -wave amplitudes and current algebra for S-wave ones. For the Ξ ++ cc → Ξ + c π + mode, we find a large destructive interference between factorizable and nonfactorizable contributions for both S-and P -wave amplitudes. Our prediction of ∼ 0.70% for its branching fraction is smaller than the earlier estimates in which nonfactorizable effects were not considered, but agrees nicely with the result based on an entirely different approach, namely, the covariant confined quark model. On the contrary, a large constructive interference was found in the P -wave amplitude by Dhir and Sharma, leading to a branching fraction of order (7 − 16)%. Using the current results for the absolute branching fractions of (Λ + c , Ξ + c ) → pK − π + and the LHCb measurement of Ξ ++ cc → Ξ + c π + relative to Ξ ++ cc → Λ + c K − π + π + , we obtain B(Ξ ++ cc → Ξ + c π + ) expt ≈ (1.83 ± 1.01)% after employing the latest prediction of B(Ξ ++ cc → Σ ++ c K * 0 ). Our prediction of B(Ξ ++ cc → Ξ + c π + ) ≈ 0.7% is thus consistent with the experimental value but in the lower end. It is important to pin down the branching fraction of this mode in future study. Factorizable and nonfactorizable S-wave amplitudes interfere constructively in Ξ + cc → Ξ 0 c π + . Its large branching fraction of order 4% may enable experimentalists to search for the Ξ + cc through this mode. That is, the Ξ + cc is reconstructed through the Ξ + cc → Ξ 0 c π + followed by the decay chain Ξ 0 c → Ξ − π + → pπ − π − π + . Besides Ξ + cc → Ξ 0 c π + , the Ξ + cc → Ξ + c (π 0 , η) modes also receive large nonfactorizable contributions to their Swave amplitudes. Hence, they have large branching fractions among Ξ + cc → B c + P channels. Nonfactorizable amplitudes in Ξ ++ cc → Ξ ′ + c π + and Ω + cc → Ξ ′ + c K 0 are very small compared to the factorizable ones owing to the Pati-Woo theorem for the inner W -emission amplitude. Likewise, nonfactorizable S-wave amplitudes in Ξ + cc → Ξ ′ + c (π 0 , η) decays are also suppressed by the same mechanism.
In the 1990s various approaches were developed to describe the nonfactorizable effects in hadronic decays of singly charmed baryons Λ + c , Ξ +,0 c and Ω 0 c . These include the covariant confined quark model [19,20], the pole model [21][22][23][24] and current algebra [23,25]. In the same vein, some of these techniques have been applied to the study of W -exchange in doubly charmed baryon decays. For example, W -exchange contributions to the P -wave amplitude were estimated by Dhir and Sharma [8,10] using the pole model. However, nonfactorizable corrections to the Swave amplitudes were not addressed by them. Likewise, Long-distance effects due to W -exchange have been estimated in [11,13,17] within the framework of the covariant confined quark model. Long-distance contributions due to W -exchange or inner W -emission were modeled as final-state rescattering effects in [6,15]. This approach has been applied to B cc → B c V (V : vector meson) [15].
In the pole model, nonfactorizable S-and P -wave amplitudes for 1/2 + → 1/2 + +0 − decays are dominated by 1/2 − low-lying baryon resonances and 1/2 + ground-state baryon poles, respectively. However, the estimation of pole amplitudes is a difficult and nontrivial task since it involves weak baryon matrix elements and strong coupling constants of 1 2 + and 1 2 − baryon states. As a consequence, the evaluation of pole diagrams is far more uncertain than the factorizable terms. This is the case in particular for S-wave terms as they require the information of the troublesome negative-parity baryon resonances which are not well understood in the quark model. This is the main reason why the nonfactorizable S-wave amplitudes of doubly charmed baryon decays were not considered in [8,10] within the pole model.
It is well known that the pole model is reduced to current algebra for S-wave amplitudes in the soft pseudoscalar-meson limit. In the soft-meson limit, the intermediate excited 1/2 − states in the S-wave amplitude can be summed up and reduced to a commutator term. Using the relation , the parity-violating amplitude is simplified to a simple commutator term expressed in terms of parity-conserving matrix elements. Therefore, the great advantage of current algebra is that the evaluation of the parity-violating S-wave amplitude does not require the information of the negative-parity 1/2 − poles. Although the pseudoscalar meson produced in B c → B + P decays is in general not truly soft, current algebra seems to work empirically well for Λ + c → B + P decays [26,27]. Moreover, the predicted negative decay asymmetries by current algebra for both Λ + c → Σ + π 0 and Σ 0 π + agree in sign with the recent BESIII measurements [28] (see [26,27] for details). In contrast, the pole model or the covariant quark model and its variant always leads to a positive decay asymmetry for aforementioned two modes. Therefore, in this work we shall follow [26,27] to work out the nonfactorizable S-wave amplitudes in doubly charmed baryon decays using current algebra and the W -exchange contributions to P -wave ones using the pole model.
In short, there exist three entirely distinct approaches for tackling the nonfactorizable contributions in doubly charmed baryon decays: the covariant confined quark model (CCQM) , final-state rescattering and the pole model in conjunction with current algebra. As stressed in [11,13,17], the evaluation of the W -exchnage diagrams in CCQM is technically quite demanding since it involves a three-loop calculation. The calculation of triangle diagrams for final-state rescattering is also rather tedious. Among these different analyses, current algebra plus the pole model turns out to be the simplest one.
Since the decay rates and decay asymmetries are sensitive to the relative sign between factorizable and non-factorizable amplitudes, it is important to evaluate all the unknown parameters in the model in a globally consistent convention to ensure the correctness of their relative signs once the wave function convention is fixed. In our framework, there are three important quantities: form factors, baryonic matrix elements and axial-vector form factors. All of them will be evaluated in the MIT bag model. We shall see later that the branching fractions of Ξ ++ cc → Ξ + c π + and Ξ + cc → Ξ 0 c π + modes are quite sensitive to their interference patterns. This paper is organized as follows. In Sec. II we set up the framework for the analysis of hadronic weak decays of doubly charmed baryons, including the topological diagrams and the formalism for describing factorizable and nonfactorizable terms. We present the explicit expressions of nonfactorizable amplitudes for both S-and P -waves. Baryon matrix elements and axial-vector form factors calculated in the MIT bag model are also summarized. Numerical results and discussions are presented in Sec. III. A conclusion will be given in Sec. IV. In the Appendix, we write down the doubly charmed baryon wave functions to fix our convention.

A. Topological diagrams
More than two decades ago, Chau, Tseng and one of us (HYC) have presented a general formulation of the topological-diagram scheme for the nonleptonic weak decays of baryons [29], which was then applied to all the decays of the antitriplet and sextet charmed baryons. For the weak decays B cc → B c + P of interest in this work, the relevant topological diagrams are the external W -emission T , the internal W -emission C, the inner W -emission C ′ , and the W -exchange diagrams E 1 as well as E 2 as depicted in Fig. 1. Among them, T and C are factorizable, while C ′ and W -exchange give nonfactorizable contributions. The relevant topological diagrams for all Cabibbo-favored decay modes of doubly charmed baryons are shown in Table I.
We notice from Table I

Contributions
because of the Pati-Woo theorem [30] which results from the facts that the (V − A) × (V − A) structure of weak interactions is invariant under the Fierz transformation and that the baryon wave function is color antisymmetric. This theorem requires that the quark pair in a baryon produced by weak interactions be antisymmetric in flavor. Since the sextet Ξ ′ c is symmetric in light quark flavor, it cannot contribute to C ′ . We shall see below that this feature is indeed confirmed in realistic calculations.

B. Kinematics
The amplitude for two-body weak decay B i → B f P is given as where B i (B f ) is the initial (final) baryon and P is a pseudoscalar meson. The decay width and up-down decay asymmetry are given by where p c is the three-momentum in the rest frame of the mother particle and The S-and P -wave amplitudes of the two-body decay are generally receive both factorizable and non-factorizable contributions

C. Factorizable amplitudes
The description of the factorizable contributions of the doubly charmed baryon decay B cc → B c P is based on the effective Hamiltonian approch. In the following we will give explicitly the factorizable contribution of S-and P -wave amplitudes.
The effective Hamiltonian for the Cabibbo-favored process reads Under factorization the amplitude can be written as where Nc . One-body and two-body matrix elements of the current are parameterized in terms of decay constants and form factors, respectively, with f π = 132 MeV and with the initial particle mass M and the momentum transfer q = p 1 − p 2 . Then the factorizable amplitude has the expression where we have neglected the contributions from the form factors f 3 and g 3 . Hence, There are two different non-perturbative parameters in the factorizable amplitudes: the decay constant and the form factor. Unlike the decay constant, which can be measured directly by experiment, the form factor is less known experimentally. Form factors defined in Eq. (10) have been evaluated in various models: the MIT bag model [31], the non-relativistic quark model [31], heavy quark effective theory [32], the light-front quark model [7,18] and light-cone sum rules [12]. 178 In this work we shall follow the assumption of nearest pole dominance [33] to write down the q 2 dependence of form factors as where m V = 2.01 GeV, m A = 2.42 GeV for the (cd) quark content, and m V = 2.11 GeV, m A = 2.54 GeV for the (cs) quark content. In the zero recoil limit where q 2 max = (m i − m f ) 2 , the form factors are expressed in the MIT bag model to be [23] where u(r) and v(r) are the large and small components, respectively, of the quark wave function in the bag model. Form factors at different q 2 are related by TABLE III. Form factors f 1 (q 2 ) and g 1 (q 2 ) at q 2 = m 2 π for various B cc → B c transitions evaluated in the MIT bag model, the light-front quark models, LFQM(I) [7] and LFQM(II) [18], and QCD sum rules (QSR) [12].
In Table III we compare the form factors evaluated in the MIT bag model with the recent calculations based on the light-front quark model (LFQM) [7,18] and light-cone sum rules (QSR) [12]. There are two different LFQM calculations denoted by LFQM(I) [7] and LFQM(II) [18], respectively. They differ in the inner structure of B cc → B c transition: a quark-diquark picture of charmed baryons in the former and a three-quark picture in the latter. We see from Table III that form factors are in general largest in LFQM(I) and smallest in QSR.

D. Nonfactorizable amplitudes
We shall adopt the pole model to describe the nonfactorizable contributions. The general formulas for A (S-wave) and B (P -wave) terms in the pole model are given by with the baryonic matrix elements It is known that the estimate of the S-wave amplitudes in the pole model is a difficult and nontrivial task as it involves the matrix elements and strong coupling constants of 1/2 − baryon resonances which we know very little [22]. 1 Nevertheless, if the emitted pseudoscalar meson is soft, then the intermediate excited baryons can be summed up, leading to a commutator term with Likewise, the P -wave amplitude is reduced in the soft-meson limit to where we have applied the generalized Goldberger-Treiman relation Eqs. (19) and (21) are the master equations for nonfactorizable amplitudes in the pole model under the soft meson approximation.

S-wave amplitudes
As shown in Eq. (19), the nonfactorizable S-wave amplitude is determined by the commutator terms of conserving charge Q a and the parity-conserving part of the Hamiltonian. Below we list the A com terms for various meson production: In Eq. (23), η 8 is the octet component of the η and η ′ η = cos θη 8 − sin θη 0 , η ′ = sin θη 8 + cos θη 0 , with θ = −15.4 • [34]. For the decay constant f η 8 , we shall follow [34] to use f η 8 = f 8 cos θ with f 8 = 1.26f π . The hypercharge Y is taken to be Y = B + S − C [26].
A straightforward calculation gives the following results: where the baryonic matrix element B ′ |H PC eff |B is denoted by a B ′ B . Evidently, all the S-wave amplitudes are governed by the matrix elements a Ξ + c Ξ + cc and a Ξ ′ + c Ξ + cc . We shall see shortly that this is also true for the P -wave pole amplitudes.

P -wave amplitudes
We next turn to the nonfactorizable P -wave amplitudes given by Eq. (21). We have for CF Ξ ++ cc decays,

E. Hadronic matrix elements and axial-vector form factors
There are two types of non-purturbative quantities involved in the nonfactorizable amplitudes: hadronic matrix elements and axial-vector form factors. We will calculate them within the framework of the MIT bag model [35].

Hadronic matrix elements
The baryonic matrix elements a B ′ B plays an important role in both S-wave and P -wave amplitudes. Its general expression in terms of the effective Hamiltonian Eq. (7) is given by where O ± = (sc)(ūd) ± (sd)(ūc) and c ± = c 1 ± c 2 . The matrix element of O + vanishes as this operator is symmetric in color indices. In the MIT bag model, the matrix elements a Ξ + c Ξ + cc and a Ξ ′ + c Ξ + cc are given by 2 where we have introduced the bag integrals X 1 and X 2 To obtain numerical results, we have employed the following bag parameters where R is the radius of the bag.

Axial-vector form factors
The axial-vector form factor in the static limit can be expressed in the bag model as The relevant results are where the auxiliary bag integrals are given by Numerically, (4π)Z 1 = 0.65 and (4π)Z 2 = 0.71.

A. Numerical results and discussions
For numerical calculations, we shall use the Wilson coefficients c 1 (µ) = 1.346 and c 2 (µ) = −0.636 evaluated at the scale µ = 1.25 GeV with Λ (4) MS = 325 MeV [36]. We follow [26] to use the Wilson coefficients a 1 = 1.26 ± 0.02 and a 2 = −0.45 ± 0.05, corresponding to N eff c ≈ 7. Recall that the value of |a 2 | is determined from the measurement of Λ + c → pφ [37], which proceeds only through the internal W -emission diagram. The mass of the Ω + cc is taken to be 3.712 GeV from lattice QCD [38]. For the Ξ + cc , we assume that it has the same mass as the Ξ ++ cc . To calculate branching fractions we need to know the lifetimes of the doubly charmed baryons Ξ + cc and Ω + cc in addition to the lifetime of Ξ ++ cc measured by the LHCb. The lifetimes of doubly  [39][40][41][42][43][44][45]. 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 . As shown in [44], it is necessary to take into account dimension-7 spectator effects in order to obtain the Ξ ++ cc lifetime consistent with the LHCb measurement (see Eq. (1)). It is difficult to make a precise quantitative statement on the lifetime of Ω + cc because of the uncertainties associated with the dimension-7 spectator effects in the Ω + cc . It was estimated in [44] that τ (Ω + cc ) lies in the range of (0.75 ∼ 1.80) × 10 −13 s. For our purpose, we shall take the mean lifetime τ (Ω + cc ) = 1.28 × 10 −13 s. On the contrary, the lifetime of Ξ + cc is rather insensitive to the variation of dimension-7 effects and τ (Ξ + cc ) = 0.45 × 10 −13 s was obtained [44]. The lifetimes of doubly charmed baryons respect the hierarchy pattern τ (Ξ ++ cc ) > τ (Ω + cc ) > τ (Ξ + cc ). Factorizable and nonfactorizable amplitudes, branching fractions and decay asymmetries for Cabibbo-favored two-body decays B cc → B c P calculated in this work are summarized in Table  IV. The channel Ξ ++ cc → Ξ + c π + is the first two-body decay mode observed by the LHCb in the doubly charmed baryon sector. However, our prediction of 0.69% 3 for its branching fraction is substantially smaller than the results of (3 ∼ 9)% given in the literature (see Table VI below). This is ascribed to the destructive interference between factorizable and nonfactorizable contributions for both S-and P -wave amplitudes (see Table IV). If we turn off the nonfactorizable terms, we will have a branching fraction of order 3.6%. In the literature, nonfactorizable effects in Ξ ++ cc → Ξ + c π + have been considered in [11] and partially in [8] (c.f. Table V). It is very interesting to notice that our calculation agrees with [11] even though the estimation of nonfactorizable effects is based on entirely different approaches: current algebra and the pole model in this work and the covariant confined quark model in [11]. On the contrary, a large constructive interference in the P -wave amplitude was found in [8], 4 while nonfactorizable corrections to the S-wave one were not considered. This leads to a branching fraction of order (7−9)% ((13−16)%) for flavor-independent (flavor-dependent) pole amplitudes.
where the uncertainty is dominated by the decay rate of Ξ + c into pK − π + . Although the rate of Ξ ++ is a purely factorizable process, its rate can be reliably estimated once the relevant form factors are determined. Taking the latest prediction B(Ξ ++ cc → Σ ++ c K * 0 ) = 5.61% from [17] as an example, 5 we obtain Therefore, our prediction of B(Ξ ++ cc → Ξ + c π + ) ≈ 0.7% is consistent with the experimental value but in the lower end. In future study, it is important to pin down the branching fraction of this mode both experimentally and theoretically.
In contrast to Ξ ++ cc → Ξ + c π + , we find a large constructive interference between factorizable and nonfactorizable S-wave amplitudes in Ξ + cc → Ξ 0 c π + , whereas Dhir and Sharma [8] obtained a large destructive interference in P -wave amplitudes (see Table V). Hence, the predicted rate of Ξ + cc → Ξ 0 c π + in [8] is rather suppressed compared to ours. The hierarchy pattern B(Ξ + cc → 3 A straightforward calculation in our framework yields a branching fraction of 0.66% and α = 0.04 for The tiny decay asymmetry is due to a large cancellation between B fac (= −15.06) and B ca (= 14.69). Normally, a huge cancellation between two terms will lead to a unreliable prediction.
Hence, we have replaced B ca by B pole (= 18.91) and used g Ξ ++ cc Ξ + cc π + = −15.31 [8], where the sign of the strong coupling is fixed by the axial-vector form factor g A(π + ) Ξ + cc Ξ ++ cc given in Eq. (34). 4 The pole amplitudes obtained by Dhir and Sharma shown in the tables of [8,10] were calculated using their Eq. (8) without a minus sign in front of n . Therefore, it is necessary to assign an extra minus sign in order to get B pole . For example, B pole (Ξ ++ cc → Ξ + c π + ) should read −0.372 rather than 0.372 for the flavor independent case (see Table III of [8]). Hence, the pole and factorizable P -wave amplitudes in Ξ ++ cc → Ξ + c π + interfere constructively in [8]. 5 The branching fraction is given by (5.40 +5.59 −3.66 )% in the approach of final-state rescattering [15].
TABLE V. Comparison of the predicted S-and P -wave amplitudes (in units of 10 −2 G F GeV 2 ) of some Cabibbo-favored decays B cc → B c + P decays in various approaches. Branching fractions (in unit of 10 −2 ) and the decay asymmetry parameter α are shown in the last two columns. We have converted the helicity amplitudes in Gutsche et al. [11] into the partial-wave ones. For the predictions of Dhir and Sharma [8,10], we quote the flavor-independent pole amplitudes and two different models for B cc → B c transition form factors: nonrelativistic quark model (abbreviated as N) and heavy quark effective theory (H). All the model results have been normalized using the lifetimes τ (Ξ ++ cc ) = 2.56 × 10 −13 s, τ (Ξ + cc ) = 0.45 × 10 −13 s and τ (Ω + cc ) = 1.28 × 10 −13 s.
we found in [27]. It should be noticed that the hierarchy pattern B(Ξ + cc → Ξ 0 c π + ) ≪ B(Ξ ++ cc → Ξ + c π + ) obtained in [8] is opposite to ours. The large branching fraction of order 3.8% for Ξ + cc → Ξ 0 c π + may enable experimentalists to search for the Ξ + cc through this mode. That is, Ξ + cc is reconstructed through the Ξ + cc → Ξ 0 c π + followed by the decay chain Ξ 0 c → Ξ − π + → pπ − π − π + . Another popular way for the search of Ξ + cc is through the processes Ξ + cc → Λ + c K − π + and Λ + c → pK − π + [48,49] are very small compared to the factorizable ones. As stated before, the topological amplitude C ′ in these decays should vanish due to the Pati-Woo theorem which requires that the quark pair in a baryon produced by weak interactions be antisymmetric in flavor. Since the sextet Ξ ′ c is symmetric in the light quark flavor in the SU(3) limit, it cannot contribute to C ′ . It is clear from Eqs. (25), (26) and (28) that the C ′ amplitude is proportional to the matrix element a Ξ ′ + c Ξ + cc governed by the bag integral X 1 introduced in Eq. (31), which vanishes in the SU(3) limit. Likewise, the nonfactorizable S-wave amplitudes in Ξ + cc → Ξ ′ + c (π 0 , η) governed by C ′ also vanish in the limit of SU(3) symmetry. However, this is not the case for nonfactorizable P -wave amplitudes due to the presence of W -exchange contributions E 1 and/or E 2 .

B. Comparison with other works
In Table V we have already compared our calculated partial-wave amplitudes for some of doubly charmed baryon decays with Gutsche et al. [11], Dhir and Sharma [8,10]. We agree with Gutsche et al. on the interference patterns in S-and P -wave amplitudes of Ξ ++ cc → Ξ + c π + and Ω + cc → Ξ + c K 0 , but disagree on the interference patterns in Ξ ++ Nevertheless, the disagreement in the last two modes is minor because of the Pati-Woo theorem for the C ′ amplitude. We agree with Dhir and Sharma on the interference patterns in P -wave Consequently, the hierarchy pattern of B(Ξ + cc → Ξ 0 c π + ) and B(Ξ ++ cc → Ξ + c π + ) in this work and [8] is opposite to each other.
In Table VI we present a complete comparison of the calculated branching fractions of Cabibbofavored B cc → B c + P decays with other works. Only the factorizable contributions from the 6 An estimate of the branching fraction of Ξ + cc → Λ + c K − π + can be made by assuming   [8,10], we quote the flavorindependent pole amplitudes and two different models for B cc → B c transition form factors: nonrelativistic quark model (abbreviated as N) and heavy quark effective theory (H). For the results of Gutsche et al. [11,13,17], we quote the latest ones from [17]. All the model results have been normalized using the lifetimes τ (Ξ ++ cc ) = 2.56 × 10 −13 s, τ (Ξ + cc ) = 0.45 × 10 −13 s and τ (Ω + cc ) = 1.28 × 10 −13 s. external W -emission governed by the Wilson coefficient a 1 were considered in references [7,12,14,18] with nonfactorizbale effects being neglected. We see from Table I that only the decay modes Ξ ++ cc → Ξ ( ′ )+ c π + , Ξ + cc → Ξ ( ′ )0 c π + and Ω + cc → Ω 0 c π + receive contributions from the external W -emission amplitude T . Branching fractions calculated in Refs. [7,12,18] were based on the form-factor models LFQM(I), LFQM(II) and QSR, respectively. Since B cc → B c transition form factors are largest in LFQM(I) and smallest in QSR (see Table III), this leads to B(Ξ ++ cc → Ξ + c π + ) and B(Ξ + cc → Ξ 0 c π + ) in [12] two times smaller than that in [7], for example. The authors of [14] employed LFQM(I) form factors, but their predictions are slightly larger than that of [7].
We see from Table VI that the predicted B(Ξ + cc → Ξ + c π 0 ) and B(Ξ + cc → Ξ + c η) in [8] are much smaller than ours. This is because we have sizable W -exchange contributions to the S-wave amplitudes of Ξ + cc → Ξ + c (π 0 , η), which are absent in [8]. This can be tested in the future.

IV. CONCLUSIONS
In this work we have studied the Cabibbo-allowed decays B cc → B c + P of doubly charmed baryons Ξ ++ cc , Ξ + cc and Ω + cc . To estimate the nonfactorizable contributions, we work in the pole model for the P -wave amplitudes and current algebra for S-wave ones. Throughout the whole calculations, all the non-perturbative parameters including form factors, baryon matrix elements and axial-vector form factors are evaluated within the framework of the MIT bag model.
We draw some conclusions from our analysis: • For the Ξ ++ cc → Ξ + c π + mode, we found a large destructive interference between factorizable and nonfactorizable contributions for both S-and P -wave amplitudes. Our prediction of ∼ 0.70% for its branching fraction is smaller than the earlier estimates in which nonfactorizable effects were not considered but agrees nicely with the result based on an entirely different approach, namely, the covariant confined quark model. On the contrary, a large constructive interference was found in the P -wave amplitude by Dhir and Sharma, leading to a branching fraction of order (7 − 16)%.
• All the unknown parameters such as B cc → B c transition form factors, the matrix elements a B ′ B and the axial-vector form factors g