Two-body weak decays of doubly charmed baryons
aa r X i v : . [ h e p - ph ] F e b Two-body weak decays of doubly charmed baryons
Hai-Yang Cheng
Institute of Physics, Academia Sinica,Taipei, Taiwan 115, Republic of China
Guanbao Meng, Fanrong Xu a , Jinqi Zou Department of Physics, Jinan University,Guangzhou 510632, People’s Republic of China
Abstract
The hadronic two-body weak decays of the doubly charmed baryons Ξ ++ cc , Ξ + cc and Ω + cc arestudied in this work. To estimate the nonfactorizable contributions, we work in the pole model forthe P -wave amplitudes and current algebra for S -wave ones. For the Ξ ++ cc → Ξ + c π + mode, we finda large destructive interference between factorizable and nonfactorizable contributions for both S - and P -wave amplitudes. Our prediction of ∼ .
70% for its branching fraction is smaller thanthe earlier estimates in which nonfactorizable effects were not considered, but agrees nicely withthe 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 Dhirand Sharma, leading to a branching fraction of order (7 − + 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 . ± . B (Ξ ++ cc → Σ ++ c K ∗ ). Our prediction of B (Ξ ++ cc → Ξ + c π + ) ≈ .
7% isthus consistent with the experimental value but in the lower end. It is important to pin downthe branching fraction of this mode in future study. Factorizable and nonfactorizable S -waveamplitudes interfere constructively in Ξ + cc → Ξ c π + . Its large branching fraction of order 4% mayenable experimentalists to search for the Ξ + cc through this mode. That is, the Ξ + cc is reconstructedthrough the Ξ + cc → Ξ c π + followed by the decay chain Ξ c → Ξ − π + → pπ − π − π + . Besides Ξ + cc → Ξ c π + , the Ξ + cc → Ξ + c ( π , η ) modes also receive large nonfactorizable contributions to their S -wave 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 are very small compared to thefactorizable ones owing to the Pati-Woo theorem for the inner W -emission amplitude. Likewise,nonfactorizable S -wave amplitudes in Ξ + cc → Ξ ′ + c ( π , η ) decays are also suppressed by the samemechanism. a [email protected] . INTRODUCTION The doubly charmed baryon state Ξ ++ cc was first discovered by the LHCb in the weak decaymode Λ + c K − π + π + [1] and subsequently confirmed in another mode Ξ + c π + [2]. Its lifetime wasalso measured by the LHCb to be [3] τ Ξ ++ cc = 0 . +0 . − . (stat . ) ± . . ) ps . (1)The updated mass is given by [4] m Ξ ++ cc = 3621 . ± . ± .
30 MeV . (2)As the first two-body weak decay Ξ + c π + of the doubly charmed baryon Ξ ++ cc was reported bythe LHCb with the result [2] B (Ξ ++ cc → Ξ + c π + ) × B (Ξ + c → pK − π + ) B (Ξ ++ cc → Λ + c K − π + π + ) × B (Λ + c → pK − π + ) = 0 . ± . . ) ± . . ) , (3)we would like to investigate in this work the nonleptonic two-body decays of doubly charmedbaryons Ξ ++ cc , Ξ + cc and Ω + cc . This has been studied intensively in the literature [5–18]. Many au-thors [7, 12, 14, 18] considered only the factorizable contributions from the external W -emissiongoverned by the Wilson coefficient a . It is well known that in charmed baryon decays, nonfac-torizable contributions from W -exchange or inner W -emission diagrams play an essential role andthey cannot be neglected, in contrast with the negligible effects in heavy meson decays. Unlikethe meson case, W -exchange is no longer subject to helicity and color suppression. The exper-imental measurements of the decays Λ + c → Σ π + , Σ + π and Ξ K + , which do not receive anyfactorizable contributions, indicate that W -exchange and inner W -emission indeed are importantin charmed baryon decays. By the same token, it is expected that nonfactorizable contributionsare also important in doubly charmed baryon decays.In the 1990s various approaches were developed to describe the nonfactorizable effects inhadronic decays of singly charmed baryons Λ + c , Ξ + , c and Ω c . These include the covariant confinedquark model [19, 20], the pole model [21–24] and current algebra [23, 25]. In the same vein, someof these techniques have been applied to the study of W -exchange in doubly charmed baryondecays. For example, W -exchange contributions to the P -wave amplitude were estimated byDhir and Sharma [8, 10] using the pole model. However, nonfactorizable corrections to the S -wave amplitudes were not addressed by them. Likewise, Long-distance effects due to W -exchangehave 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-staterescattering 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 / + → / + + 0 − decays aredominated by 1 / − low-lying baryon resonances and 1 / + ground-state baryon poles, respectively. owever, the estimation of pole amplitudes is a difficult and nontrivial task since it involvesweak baryon matrix elements and strong coupling constants of
12 + and − baryon states. As aconsequence, 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 troublesomenegative-parity baryon resonances which are not well understood in the quark model. This is themain reason why the nonfactorizable S -wave amplitudes of doubly charmed baryon decays werenot 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 inthe soft pseudoscalar-meson limit. In the soft-meson limit, the intermediate excited 1 / − statesin the S -wave amplitude can be summed up and reduced to a commutator term. Using therelation [ Q a , H PVeff ] = − [ Q a , H PCeff ], the parity-violating (PV) amplitude is simplified to a simplecommutator term expressed in terms of parity-conserving (PC) matrix elements. Therefore, thegreat advantage of current algebra is that the evaluation of the parity-violating S -wave amplitudedoes not require the information of the negative-parity 1 / − poles. Although the pseudoscalarmeson produced in B c → B + P decays is in general not truly soft, current algebra seems towork empirically well for Λ + c → B + P decays [26, 27]. Moreover, the predicted negative decayasymmetries by current algebra for both Λ + c → Σ + π and Σ π + agree in sign with the recentBESIII measurements [28] (see [26, 27] for details). In contrast, the pole model or the covariantquark model and its variant always leads to a positive decay asymmetry for aforementionedtwo 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 -exchangecontributions to P -wave ones using the pole model.In short, there exist three entirely distinct approaches for tackling the nonfactorizable contribu-tions in doubly charmed baryon decays: the covariant confined quark model (CCQM) , final-staterescattering and the pole model in conjunction with current algebra. As stressed in [11, 13, 17],the evaluation of the W -exchange diagrams in CCQM is technically quite demanding since itinvolves a three-loop calculation. The calculation of triangle diagrams for final-state rescatteringis also rather tedious. Among these different analyses, current algebra plus the pole model turnsout to be the simplest one.Since the decay rates and decay asymmetries are sensitive to the relative sign between factor-izable and non-factorizable amplitudes, it is important to evaluate all the unknown parametersin the model in a globally consistent convention to ensure the correctness of their relative signsonce the wave function convention is fixed. In our framework, there are three important quan-tities: form factors, baryonic matrix elements and axial-vector form factors. All of them will beevaluated in the MIT bag model. We shall see later that the branching fractions of Ξ ++ cc → Ξ + c π + and Ξ + cc → Ξ 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 analysisof hadronic weak decays of doubly charmed baryons, including the topological diagrams and cq T ccq Cqcc C ′ ccq ′ E q ′ cc E FIG. 1. Topological diagrams contributing to B cc → B c + P decays: external W -emission T ,internal W -emission C , inner W -emission C ′ , W -exchange diagrams E and E , where q = u, d, s and q ′ = d, s .the formalism for describing factorizable and nonfactorizable terms. We present the explicitexpressions of nonfactorizable amplitudes for both S - and P -waves. Baryon matrix elements andaxial-vector form factors calculated in the MIT bag model are also summarized. Numerical resultsand 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. II. THEORETICAL FRAMEWORK
In this work we shall follow [22, 23] closely with many quantities and operators well defined inthese references.
A. Topological diagrams
More than two decades ago, Chau, Tseng and one of us (HYC) have presented a generalformulation 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. Forthe weak decays B cc → B c + P of interest in this work, the relevant topological diagrams are theexternal W -emission T , the internal W -emission C , the inner W -emission C ′ , and the W -exchangediagrams E as well as E as depicted in Fig. 1. Among them, T and C are factorizable, while ABLE I. Topological diagrams contributing to two-body Cabibbo-favored decays of the doublycharmed baryons Ξ ++ cc , Ξ + cc and Ω + cc .Ξ ++ cc Contributions Ξ + cc Contributions Ω + cc ContributionsΞ ++ cc → Σ ++ c K C Ξ + cc → Ξ c π + T, E Ω + cc → Ω c π + T Ξ ++ cc → Ξ + c π + T, C ′ Ξ + cc → Ξ ′ c π + T, E Ω + cc → Ξ + c K C, C ′ Ξ ++ cc → Ξ ′ + c π + T, C ′ Ξ + cc → Λ + c K C, E Ω + cc → Ξ ′ + c K C, C ′ Ξ + cc → Σ + c K C, E Ξ + cc → Ξ + c π C ′ , E Ξ + cc → Ξ ′ + c π C ′ , E Ξ + cc → Ξ + c η C ′ , E , E Ξ + cc → Ξ ′ + c η C ′ , E , E Ξ + cc → Σ ++ c K − E Ξ + cc → Ω c K + E C ′ and W -exchange give nonfactorizable contributions. The relevant topological diagrams for allCabibbo-favored decay modes of doubly charmed baryons are shown in Table I.We notice from Table I that (i) there are two purely factorizable modes: Ξ ++ cc → Σ ++ c K and Ω + cc → Ω c π + , (ii) the W -exchange contribution manifests only in Ξ + cc decays, and (iii) thetopological amplitude C ′ in Ξ ++ cc → Ξ ′ + c π + , Ξ + cc → Ξ ′ + c ( π , η ) and Ω + cc → Ξ ′ + c K should vanishbecause 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 baryonwave function is color antisymmetric. This theorem requires that the quark pair in a baryonproduced by weak interactions be antisymmetric in flavor. Since the sextet Ξ ′ c is symmetric inlight quark flavor, it cannot contribute to C ′ . We shall see below that this feature is indeedconfirmed in realistic calculations. B. Kinematics
The amplitude for two-body weak decay B i → B f P is given as M ( B i → B f P ) = i ¯ u f ( A − Bγ ) u i , (4)where B i ( B f ) is the initial (final) baryon and P is a pseudoscalar meson. The decay width andup-down decay asymmetry are given byΓ = p c π (cid:20) ( m i + m f ) − m P m i | A | + ( m i − m f ) − m P m i | B | (cid:21) ,α = 2 κ Re( A ∗ B ) | A | + κ | B | , (5) here p c is the three-momentum in the rest frame of the mother particle and κ = p c / ( E f + m f ) = p ( E f − m f ) / ( E f + m f ). The S - and P - wave amplitudes of the two-body decay generally receiveboth factorizable and non-factorizable contributions A = A fac + A nf , B = B fac + B nf . (6) 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 approach. In the following we will give explicitly thefactorizable contribution of S - and P -wave amplitudes.The effective Hamiltonian for the Cabibbo-favored process reads H eff = G F √ V cs V ∗ ud ( c O + c O ) + h.c., (7) O = ( sc )(¯ ud ) , O = (¯ uc )( sd ) , (¯ q q ) ≡ ¯ q γ µ (1 − γ ) q , where c and c are Wilson coefficients. Under the factorization hypothesis the amplitude can bewritten as M = h P B c |H eff |B cc i = G F √ V cs V ∗ ud a h P | (¯ ud ) | ihB c | ( sc ) |B cc i , P = π + , G F √ V cs V ∗ ud a h P | ( sd ) | ihB c | (¯ uc ) |B cc i , P = K , (8)where a = c + c N c , a = c + c N c . One-body and two-body matrix elements of the current areparameterized in terms of decay constants and form factors, respectively, h K ( q ) | sγ µ (1 − γ ) d | i = if K q µ , h π ( q ) | ¯ uγ µ (1 − γ ) d | i = if π q µ , (9)with f π = 132 MeV, f K = 160 MeV and hB c ( p ) | cγ µ (1 − γ ) u |B cc ( p ) i = ¯ u (cid:20) f ( q ) γ µ − f ( q ) iσ µν q ν M + f ( q ) q µ M − (cid:18) g ( q ) γ µ − g ( q ) iσ µν q ν M + g ( q ) q µ M (cid:19) γ (cid:21) u , (10)with the initial particle mass M and the momentum transfer q = p − p . Then the factorizableamplitude has the expression M ( B cc → B c P ) = i G F √ a , V ∗ ud V cs f P ¯ u ( p ) (cid:2) ( m − m ) f ( q ) + ( m + m ) g ( q ) γ (cid:3) u ( p ) , (11)where we have neglected the contributions from the form factors f and g . To see the possible corrections from the form factors f and g for kaon or η production in the finalstate, we notice that m P /m c = 0 .
047 for the kaon and 0.057 for the η . Since the form factor f is muchsmaller than f (see e.g. Table IV of [31]), while g is of the same order as g , it follows that the formfactor f can be safely neglected in the factorizable amplitude, while g could make ∼
5% correctionsfor kaon or η production. For simplicity, we will drop all the contributions from f and g . ABLE II. The calculated form factors with c → s transition in the MIT bag model at maximumfour-momentum transfer squared q = q = ( m i − m f ) and at q = m P . Modes f ( q ) f ( m P ) /f ( q ) f ( m P ) g ( q ) g ( m P ) /g ( q ) g ( m P )Ξ ++ cc → Σ ++ c K Y .
540 0 . Y .
673 0 . ++ cc → Ξ + c π + √ Y s .
496 0 . √ Y s .
634 0 . ++ cc → Ξ ′ + c π + √ Y s .
575 0 . √ Y s .
695 0 . + cc → Λ + c K √ Y .
487 0 . √ Y .
632 0 . + cc → Σ + c K √ Y .
622 0 . √ Y .
734 0 . + cc → Ξ c π + √ Y s .
572 0 . √ Y s .
693 0 . + cc → Ξ ′ c π + √ Y s .
648 0 . √ Y s .
749 0 . + cc → Ω c π + Y s .
532 0 . Y s .
661 0 . + cc → Ξ + c K − √ Y . − . − √ Y . − . + cc → Ξ ′ + c K √ Y .
495 0 . √ Y .
638 0 . Hence, A fac = G F √ a , V ∗ ud V cs f P ( m B cc − m B c ) f ( q ) ,B fac = − G F √ a , V ∗ ud V cs f P ( m B cc + m B c ) g ( q ) . (12)There are two different non-perturbative parameters in the factorizable amplitudes: the decayconstant and the form factor. Unlike the decay constant, which can be measured directly byexperiment, the form factor is less known experimentally. Form factors defined in Eq. (10) havebeen evaluated in various models: the MIT bag model [32], the non-relativistic quark model [32],heavy quark effective theory [33], the light-front quark model [7, 18] and light-cone sum rules [12].In this work we shall follow the assumption of nearest pole dominance [34] to write down the q dependence of form factors as f i ( q ) = f i (0)(1 − q /m V ) , g i ( q ) = g i (0)(1 − q /m A ) , (13)where m V = 2 .
01 GeV, m A = 2 .
42 GeV for the ( c ¯ d ) quark content, and m V = 2 .
11 GeV, m A =2 .
54 GeV for the ( c ¯ s ) quark content. In the zero recoil limit where q = ( m i − m f ) , the form ABLE III. Form factors f ( q ) and g ( q ) at q = m π for various B cc → B c transitions evaluatedin the MIT bag model, the light-front quark models, LFQM(I) [7] and LFQM(II) [18], and QCDsum rules (QSR) [12]. B cc → B c f ( m π ) g ( m π )MIT LFQM(I) LFQM(II) QSR MIT LFQM(I) LFQM(II) QSRΞ ++ cc → Ξ + c ++ cc → Ξ ′ + c + cc → Ξ c + cc → Ξ ′ c + cc → Ω c factors are expressed in the MIT bag model to be [23] f B f B i ( q ) = hB f ↑ | b † q b q |B i ↑i Z d r ( u q ( r ) u q ( r ) + v q ( r ) v q ( r )) ,g B f B i ( q ) = hB f ↑ | b † q b q σ z |B i ↑i Z d r ( u q ( r ) u q ( r ) − v q ( r ) v q ( r )) , (14)where u ( r ) and v ( r ) are the large and small components, respectively, of the quark wave functionin the bag model. Form factors at different q are related by f i ( q ) = (1 − q /m V ) (1 − q /m V ) f i ( q ) , g i ( q ) = (1 − q /m A ) (1 − q /m A ) g i ( q ) . (15)Numerical results of the form factors at q = m π for various B cc → B c transitions are shown inTable II. In the calculation we have defined the bag integrals Y = 4 π Z r dr ( u u u c + v u v c ) = 0 . , Y s = 4 π Z r dr ( u s u c + v s v c ) = 0 . ,Y = 4 π Z r dr ( u u u c − v u v c ) = 0 . , Y s = 4 π Z r dr ( u s u c − v s v c ) = 0 . . (16)In Table III we compare the form factors evaluated in the MIT bag model with the recentcalculations 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 pictureof charmed baryons in the former and a three-quark picture in the latter. We see from Table IIIthat form factors are in general largest in LFQM(I) and smallest in QSR. . Nonfactorizable amplitudes We shall adopt the pole model to describe the nonfactorizable contributions. The generalformulas for A ( S -wave) and B ( P -wave) terms in the pole model are given by A pole = − X B ∗ n (1 / − ) " g Bf B ∗ nP b n ∗ i m i − m n ∗ + b fn ∗ g B ∗ nBiP m f − m n ∗ ,B pole = X B n (cid:20) g Bf BnP a ni m i − m n + a fn g BnBiP m f − m n (cid:21) , (17)with the baryonic matrix elements hB n | H |B i i = ¯ u n ( a ni + b ni γ ) u i , hB ∗ i (1 / − ) | H |B j i = ¯ u i ∗ b i ∗ j u j . (18)It is known that the estimate of the S -wave amplitudes in the pole model is a difficult andnontrivial task as it involves the matrix elements and strong coupling constants of 1 / − baryonresonances which we know very little [22]. Nevertheless, if the emitted pseudoscalar meson issoft, then the intermediate excited baryons can be summed up, leading to a commutator term A com = − √ f P a hB f | [ Q a , H PVeff ] |B i i = √ f P a hB f | [ Q a , H PCeff ] |B i i , (19)with Q a = Z d x ¯ qγ λ a q, Q a = Z d x ¯ qγ γ λ a q. (20)Likewise, the P -wave amplitude is reduced in the soft-meson limit to B ca = √ f P a X B n (cid:20) g A B f B n m f + m n m i − m n a ni + a fn m i + m n m f − m n g A B n B i (cid:21) , (21)where the superscript “ca” stands for current algebra and we have applied the generalizedGoldberger-Treiman relation g B′B Pa = √ f P a ( m B + m B ′ ) g A B ′ B . (22)In Eq. (21) a ij is the parity-conserving matrix element defined in Eq. (18) and g Aij is the axial-vector form factor defined in Eq. (22). Eqs. (19) and (21) are the master equations for nonfac-torizable amplitudes in the pole model under the soft meson approximation. Attempts of explicit calculations of intermediate 1 / − pole contributions to the S -wave amplitudes hadbeen made before in [22, 23]. . S -wave amplitudes As shown in Eq. (19), the nonfactorizable S -wave amplitude is determined by the commutatorterms of conserving charge Q a and the parity-conserving part of the effective Hamiltonian H PCeff .Below we list the A com terms for various meson production: A com ( B i → B f π ± ) = 1 f π hB f | [ I ∓ , H PCeff ] |B i i ,A com ( B i → B f π ) = √ f π hB f | [ I , H PCeff ] |B i i ,A com ( B i → B f η ) = r
32 1 f η hB f | [ Y, H
PCeff ] |B i i , (23) A com ( B i → B f K ± ) = 1 f K hB f | [ V ∓ , H PCeff ] |B i i ,A com ( B i → B f K ) = 1 f K hB f | [ U + , H PCeff ] |B i i , where we have introduced the isospin I , U -spin and V -spin ladder operators with I + | d i = | u i , I − | u i = | d i , U + | s i = | d i , U − | d i = | s i , V + | s i = | u i , V − | u i = | s i . (24)In Eq. (23), η is the octet component of the η and η ′ η = cos θη − sin θη , η ′ = sin θη + cos θη , (25)with θ = − . ◦ [35]. For the decay constant f η , we shall follow [35] to use f η = f cos θ with f = 1 . f π . The hypercharge Y is taken to be Y = B + S − C [26], where B , C and S are thequantum numbers of the baryon, charm and strangeness, respectively.A straightforward calculation gives the following results: A com (Ξ ++ cc → Ξ + c π + ) = 1 f π (cid:16) − a Ξ + c Ξ + cc (cid:17) , A com (Ξ ++ cc → Ξ ′ + c π + ) = 1 f π (cid:16) − a Ξ ′ + c Ξ + cc (cid:17) ,A com (Ξ + cc → Ξ + c π ) = √ f π (cid:16) a Ξ + c Ξ + cc (cid:17) , A com (Ξ + cc → Ξ ′ + c π ) = √ f π (cid:16) a Ξ ′ + c Ξ + cc (cid:17) ,A com (Ξ + cc → Ξ + c η ) = √ f η (cid:16) a Ξ + c Ξ + cc (cid:17) , A com (Ξ + cc → Ξ ′ + c η ) = √ f η (cid:16) a Ξ ′ + c Ξ + cc (cid:17) ,A com (Ξ + cc → Σ ++ c K − ) = 2 f K (cid:16) a Ξ ′ + c Ξ + cc (cid:17) , A com (Ξ + cc → Λ + c K ) = 1 f K (cid:16) a Ξ + c Ξ + cc (cid:17) , (26) A com (Ξ + cc → Σ + c K ) = 1 f K (cid:16) a Ξ ′ + c Ξ + cc (cid:17) , A com (Ξ + cc → Ξ c π + ) = 1 f π (cid:16) a Ξ + c Ξ + cc (cid:17) ,A com (Ξ + cc → Ξ ′ c π + ) = 1 f π (cid:16) a Ξ ′ + c Ξ + cc (cid:17) , A com (Ξ + cc → Ω c K + ) = √ f K (cid:16) a Ξ ′ + c Ξ + cc (cid:17) ,A com (Ω + cc → Ξ + c K ) = 1 f K (cid:16) − a Ξ + c Ξ + cc (cid:17) , A com (Ω + cc → Ξ ′ + c K ) = 1 f K (cid:16) − a Ξ ′ + c Ξ + cc (cid:17) , here the baryonic matrix element hB ′ | H PCeff |Bi is denoted by a B ′ B . Evidently, all the S -waveamplitudes are governed by the matrix elements a Ξ + c Ξ + cc and a Ξ ′ + c Ξ + cc . We shall see shortly thatthis 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 B ca (Ξ ++ cc → Ξ + c π + ) = 1 f π a Ξ + c Ξ + cc m Ξ ++ cc + m Ξ + cc m Ξ + c − m Ξ + cc g A ( π + )Ξ + cc Ξ ++ cc ! ,B ca (Ξ ++ cc → Ξ ′ + c π + ) = 1 f π a Ξ ′ + c Ξ + cc m Ξ ++ cc + m Ξ + cc m Ξ ′ + c − m Ξ + cc g A ( π + )Ξ + cc Ξ ++ cc ! , (27) or Cabibbo-favored Ξ ++ cc decays, B ca (Ξ + cc → Ξ + c π ) = √ f π a Ξ + c Ξ + cc m Ξ + cc m Ξ + c − m Ξ + cc g A ( π )Ξ + cc Ξ + cc + g A ( π )Ξ + c Ξ + c m Ξ + c m Ξ + cc − m Ξ + c a Ξ + c Ξ + cc + g A ( π )Ξ + c Ξ ′ + c m Ξ + c + m Ξ ′ + c m Ξ + cc − m Ξ ′ + c a Ξ ′ + c Ξ + cc ! ,B ca (Ξ + cc → Ξ ′ + c π ) = √ f π a Ξ ′ + c Ξ + cc m Ξ + cc m Ξ ′ + c − m Ξ + cc g A ( π )Ξ + cc Ξ + cc + g A ( π )Ξ ′ + c Ξ + c m Ξ ′ + c + m Ξ + c m Ξ + cc − m Ξ + c a Ξ + c Ξ + cc + g A ( π )Ξ ′ + c Ξ ′ + c m Ξ ′ + c m Ξ + cc − m Ξ ′ + c a Ξ ′ + c Ξ + cc ! ,B ca (Ξ + cc → Ξ + c η ) = √ f η a Ξ + c Ξ + cc m Ξ + cc m Ξ + c − m Ξ + cc g A ( η )Ξ + cc Ξ + cc + g A ( η )Ξ + c Ξ + c m Ξ + c m Ξ + cc − m Ξ + c a Ξ + c Ξ + cc + g A ( η )Ξ + c Ξ ′ + c m Ξ + c + m Ξ ′ + c m Ξ + cc − m Ξ ′ + c a Ξ ′ + c Ξ + cc ! , (28) B ca (Ξ + cc → Ξ ′ + c η ) = √ f η a Ξ ′ + c Ξ + cc m Ξ + cc m Ξ ′ + c − m Ξ + cc g A ( η )Ξ + cc Ξ + cc + g A ( η )Ξ ′ + c Ξ + c m Ξ ′ + c + m Ξ + c m Ξ + cc − m Ξ + c a Ξ + c Ξ + cc + g A ( η )Ξ ′ + c Ξ ′ + c m Ξ ′ + c m Ξ + cc − m Ξ ′ + c a Ξ ′ + c Ξ + cc ! ,B ca (Ξ + cc → Σ ++ c K − ) = 1 f K g A ( K − )Σ ++ c Ξ + c m Σ ++ c + m Ξ + c m Ξ + cc − m Ξ + c a Ξ + c Ξ + cc + g A ( K − )Σ ++ c Ξ ′ + c m Σ ++ c + m Ξ ′ + c m Ξ + cc − m Ξ ′ + c a Ξ ′ + c Ξ + cc ! ,B ca (Ξ + cc → Λ + c K ) = 1 f K g A ( K )Λ + c Ξ + c m Λ + c + m Ξ + c m Ξ + cc − m Ξ + c a Ξ + c Ξ + cc + g A ( K )Λ + c Ξ ′ + c m Λ + c + m Ξ ′ + c m Ξ + cc − m Ξ ′ + c a Ξ ′ + c Ξ + cc ! ,B ca (Ξ + cc → Σ + c K ) = 1 f K g A ( K )Σ + c Ξ + c m Σ + c + m Ξ + c m Ξ + cc − m Ξ + c a Ξ + c Ξ + cc + g A ( K )Σ + c Ξ ′ + c m Σ + c + m Ξ ′ + c m Ξ + cc − m Ξ ′ + c a Ξ ′ + c Ξ + cc ! ,B ca (Ξ + cc → Ξ c π + ) = 1 f π g A ( π + )Ξ c Ξ + c m Ξ c + m Ξ + c m Ξ + cc − m Ξ + c a Ξ + c Ξ + cc + g A ( π + )Ξ c Ξ ′ + c m Ξ c + m Ξ ′ + c m Ξ + cc − m Ξ ′ + c a Ξ ′ + c Ξ + cc ! ,B ca (Ξ + cc → Ξ ′ c π + ) = 1 f π g A ( π + )Ξ ′ c Ξ + c m Ξ ′ c + m Ξ + c m Ξ + cc − m Ξ + c a Ξ + c Ξ + cc + g A ( π + )Ξ ′ c Ξ ′ + c m Ξ ′ c + m Ξ ′ + c m Ξ + cc − m Ξ ′ + c a Ξ ′ + c Ξ + cc ! ,B ca (Ξ + cc → Ω c K + ) = 1 f K g A ( K + )Ω c Ξ + c m Ω c + m Ξ + c m Ξ + cc − m Ξ + c a Ξ + c Ξ + cc + g A ( K + )Ω c Ξ ′ + c m Ω c + m Ξ ′ + c m Ξ + cc − m Ξ ′ + c a Ξ ′ + c Ξ + cc ! , or Cabibbo-favored Ξ + cc decays, and B ca (Ω + cc → Ξ + c K ) = 1 f K a Ξ + c Ξ + cc m Ω + cc + m Ξ + cc m Ξ + c − m Ξ + cc g A ( K )Ξ + cc Ω + cc ! ,B ca (Ω + cc → Ξ ′ + c K ) = 1 f K a Ξ ′ + c Ξ + cc m Ω + cc + m Ξ + cc m Ξ ′ + c − m Ξ + cc g A ( K )Ξ + cc Ω + cc ! , (29)for Cabibbo-favored Ω + cc decays. E. Hadronic matrix elements and axial-vector form factors
There are two types of non-perturbative quantities involved in the nonfactorizable amplitudes:hadronic matrix elements and axial-vector form factors. We will calculate them within the frame-work of the MIT bag model [36].
1. Hadronic matrix elements
The baryonic matrix element a B ′ B plays an important role in both S -wave and P -wave ampli-tudes. Its general expression in terms of the effective Hamiltonian Eq. (7) is given by a B ′ B ≡ hB ′ |H PCeff |Bi = G F √ V cs V ∗ ud c − hB ′ | O − |Bi , (30)where O ± = (¯ sc )(¯ ud ) ± (¯ sd )(¯ uc ) and c ± = c ± c . The matrix element of O + vanishes as thisoperator 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 h Ξ + c | O − | Ξ + cc i = 4 √ X (4 π ) , h Ξ ′ + c | O − | Ξ + cc i = − √ X (4 π ) , (31)where we have introduced the bag integrals X and X X = Z R r dr ( u s v u − v s u u )( u c v d − v c u d ) = 3 . × − ,X = Z R r dr ( u s u u + v s v u )( u c u d + v c v d ) = 1 . × − . (32)To obtain numerical results, we have employed the following bag parameters m u = m d = 0 , m s = 0 .
279 GeV , m c = 1 .
551 GeV , R = 5 GeV − , (33)where R is the radius of the bag. For the evaluation of baryon matrix elements and form factors in the MIT bag model, see e.g. [22, 23]. . Axial-vector form factors The axial-vector form factor in the static limit can be expressed in the bag model as g A ( P ) B ′ B = hB ′ ↑ | b † q b q σ z |B ↑i Z d r (cid:18) u q u q − v q v q (cid:19) , (34)where σ z is the z -component of Pauli matrices. The relevant results are g A ( π + )Ξ + cc Ξ ++ cc = −
13 (4 πZ ) , g A ( π )Ξ + cc Ξ + cc = 16 (4 πZ ) , g A ( η )Ξ + cc Ξ + cc = − √ πZ ) ,g A ( K − )Σ ++ c Ξ + c = √
63 (4 πZ ) , g A ( K − )Σ ++ c Ξ ′ + c = 2 √
23 (4 πZ ) , g A ( K )Ξ + cc Ω + cc = −
13 (4 πZ ) ,g A ( K )Λ + c Ξ ′ + c = − √
33 (4 πZ ) , g A ( K )Σ + c Ξ + c = √
33 (4 πZ ) , g A ( K )Σ + c Ξ ′ + c = 23 (4 πZ ) ,g A ( π + )Ξ c Ξ ′ + c = − √
33 (4 πZ ) , g A ( π + )Ξ ′ c Ξ + c = − √
33 (4 πZ ) , g A ( π + )Ξ ′ c Ξ ′ + c = 23 (4 πZ ) , (35) g A ( π )Ξ ′ + c Ξ + c = − √
36 (4 πZ ) , g A ( π )Ξ ′ + c Ξ + c = − √
36 (4 πZ ) , g A ( π )Ξ ′ + c Ξ ′ + c = 13 (4 πZ ) ,g A ( η )Ξ ′ + c Ξ + c = −
12 (4 πZ ) , g A ( η )Ξ ′ + c Ξ + c = −
12 (4 πZ ) , g A ( η )Ξ ′ + c Ξ ′ + c = − √
39 (4 πZ ) ,g A ( K + )Ω c Ξ + c = − √
63 (4 πZ ) , g A ( K + )Ω c Ξ ′ + c = 2 √
23 (4 πZ ) , and g A ( K )Λ + c Ξ + c = 0 , g A ( π + )Ξ c Ξ + c = 0 , g A ( π )Ξ + c Ξ + c = 0 , g A ( η )Ξ + c Ξ + c = 0 , (36)where the auxiliary bag integrals are given by Z = Z r dr (cid:18) u u − v u (cid:19) , Z = Z r dr (cid:18) u u u s − v u v s (cid:19) . (37)Numerically, (4 π ) Z = 0 .
65 and (4 π ) Z = 0 . III. RESULTS AND DISCUSSIONSA. Numerical results and discussions
For numerical calculations, we shall use the Wilson coefficients c ( µ ) = 1 .
346 and c ( µ ) = − .
636 evaluated at the scale µ = 1 .
25 GeV with Λ (4)MS = 325 MeV [37]. We follow [26] to use theWilson coefficients a = 1 . ± .
02 and a = − . ± .
05, corresponding to N eff c ≈
7. Recallthat the value of | a | is determined from the measurement of Λ + c → pφ [38], which proceeds onlythrough the internal W -emission diagram. For the CKM matrix elements we use V ud = 0 . V cs = 0 . + cc is taken to be 3 .
712 GeV from lattice QCD [39]. For he Ξ + cc , we assume that it has the same mass as the Ξ ++ cc which is taken to be 3621 MeV fromEq. (2). This is justified because the isospin splitting in the doubly charmed baryons with lightquarks has been estimated to be very small, m Ξ ++ cc − m Ξ + cc = O (1 .
5) MeV (see [40] and referencestherein).To calculate branching fractions we need to know the lifetimes of the doubly charmed baryonsΞ + cc and Ω + cc in addition to the lifetime of Ξ ++ cc measured by the LHCb. The lifetimes of doublycharmed hadrons have been analyzed within the framework of heavy quark expansion [41–47].Lifetime differences arise from spectator effects such as W -exchange and Pauli interference. TheΞ ++ cc baryon is longest-lived in the doubly charmed baryon system owing to the destructive Pauliinterference absent in the Ξ + cc and Ω + cc . As shown in [46], it is necessary to take into accountdimension-7 spectator effects in order to obtain the Ξ ++ cc lifetime consistent with the LHCb mea-surement (see Eq. (1)). It is difficult to make a precise quantitative statement on the lifetime ofΩ + cc because of the uncertainties associated with the dimension-7 spectator effects in the Ω + cc . Itwas estimated in [46] that τ (Ω + cc ) lies in the range of (0 . ∼ . × − s . For our purpose, weshall take the mean lifetime τ (Ω + cc ) = 1 . × − s . On the contrary, the lifetime of Ξ + cc is ratherinsensitive to the variation of dimension-7 effects and τ (Ξ + cc ) = 0 . × − s was obtained [46].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 forCabibbo-favored two-body decays B cc → B c P calculated in this work are summarized in TableIV. The channel Ξ ++ cc → Ξ + c π + is the first two-body decay mode observed by the LHCb in thedoubly charmed baryon sector. However, our prediction of 0 . for its branching fraction issubstantially smaller than the results of (3 ∼ S - and P -wave amplitudes (see Table IV). If we turn off the nonfactorizable terms, we willhave a branching fraction of order 3 . ++ cc → Ξ + c π + have been considered in [11] and partially in [8] (c.f. Table V). It is very interesting to noticethat our calculation agrees with [11] even though the estimation of nonfactorizable effects isbased on entirely different approaches: current algebra and the pole model in this work and thecovariant confined quark model in [11]. On the contrary, a large constructive interference in the P -wave amplitude was found in [8], while nonfactorizable corrections to the S -wave one were notconsidered. This leads to a branching fraction of order (7 − − A straightforward calculation in our framework yields a branching fraction of 0 .
66% and α = 0 .
04 forΞ ++ cc → Ξ + c π + . The tiny decay asymmetry is due to a large cancellation between B fac (= − .
06) and B ca (= 14 . B ca by B pole (= 18 .
91) and used g Ξ ++ cc Ξ + cc π + = − .
31 [8], where the sign of thestrong coupling is fixed by the axial-vector form factor g A ( π + )Ξ + cc Ξ ++ cc given in Eq. (35). The pole amplitudes obtained by Dhir and Sharma shown in the tables of [8, 10] were calculated usingtheir Eq. (8) without a minus sign in front of P n . Therefore, it is necessary to assign an extra minussign in order to get B pole . For example, B pole (Ξ ++ cc → Ξ + c π + ) should read − .
372 rather than 0.372 forthe flavor independent case (see Table III of [8]). Hence, the pole and factorizable P -wave amplitudesin Ξ ++ cc → Ξ + c π + interfere constructively in [8]. ABLE IV. The predicted S - and P -wave amplitudes of Cabibbo-favored B cc → B c + P decays inunits of 10 − G F GeV . Branching fractions (in units of 10 − ) and the decay asymmetry parameter α are shown in the last two columns. For lifetimes we use τ (Ξ ++ cc ) = 2 . × − s , τ (Ξ + cc ) =0 . × − s and τ (Ω + cc ) = 1 . × − s (see the main text).Channel A fac A com A tot B fac B ca B tot B theo α theo Ξ ++ cc → Ξ + c π + . − . − . − .
06 18 .
91 3 .
85 0 . − . ++ cc → Ξ ′ + c π + . − .
04 4 . − .
50 0 . − .
44 4 . − . ++ cc → Σ ++ c K − .
67 0 − .
67 25 .
11 0 25 .
11 1 . − . + cc → Ξ c π + .
52 10 .
79 19 . − . − . − .
54 3.84 − . + cc → Ξ ′ c π + .
05 0 .
04 5 . − . − . − .
94 1.55 − . + cc → Ξ + c π .
26 15 .
26 0 − . − .
49 2.38 − . + cc → Ξ ′ + c π .
06 0 .
06 0 − . − .
97 0.17 − . + cc → Ξ + c η .
75 21 .
75 0 4 .
86 4 .
86 4 .
18 0 . + cc → Ξ ′ + c η .
09 0 .
09 0 − . − .
87 0 . − . + cc → Σ ++ c K − .
07 0 .
07 0 22 .
14 22 .
14 0 .
13 0 . + cc → Λ + c K − .
37 8 .
90 5 .
53 5 . − .
07 5 .
55 0 .
31 0 . + cc → Σ + c K − .
17 0 . − .
14 19 .
37 15 .
64 35 .
02 0 . − . + cc → Ω c K + .
05 0 .
05 0 − . − .
98 0 . − . + cc → Ω c π + .
71 0 5 . − .
48 0 − .
48 3 . − . + cc → Ξ + c K . − . − . − .
29 13 .
40 8 .
11 1 . − . + cc → Ξ ′ + c K − . − . − .
72 17 .
44 0 .
06 17 .
50 0 . − . + c , Ξ + c ) → pK − π + have been measured with theresults B (Λ + c → pK − π + ) = (6 . ± . B (Ξ + c → pK − π + ) = (0 . ± . ± . B (Ξ ++ cc → Ξ + c π + ) B (Ξ ++ cc → Λ + c K − π + π + ) = 0 . ± . , (38)where the uncertainty is dominated by the decay rate of Ξ + c into pK − π + . Although the rateof Ξ ++ cc → Λ + c K − π + π + is unknown, it is plausible to assume that B (Ξ ++ cc → Λ + c K − π + π + ) ≈ B (Ξ ++ cc → Σ ++ c K ∗ ). Since Ξ ++ cc → Σ ++ c K ∗ is a purely factorizable process, its rate can bereliably estimated once the relevant form factors are determined. Taking the latest prediction B (Ξ ++ cc → Σ ++ c K ∗ ) = 5 .
61% from [17] as an example, we obtain B (Ξ ++ cc → Ξ + c π + ) expt ≈ (1 . ± . . (39) The branching fraction is given by (5 . +5 . − . )% in the approach of final-state rescattering [15]. ABLE V. Comparison of the predicted S - and P -wave amplitudes (in units of 10 − G F GeV ) ofsome Cabibbo-favored decays B cc → B c + P decays in various approaches. Branching fractions(in unit of 10 − ) and the decay asymmetry parameter α are shown in the last two columns. Wehave converted the helicity amplitudes in Gutsche et al. [11] into the partial-wave ones. For thepredictions of Dhir and Sharma [8, 10], we quote the flavor-independent pole amplitudes and twodifferent models for B cc → B c transition form factors: nonrelativistic quark model (abbreviatedas N) and heavy quark effective theory (H). All the model results have been normalized using thelifetimes τ (Ξ ++ cc ) = 2 . × − s , τ (Ξ + cc ) = 0 . × − s and τ (Ω + cc ) = 1 . × − s . A fac A nf A tot B fac B nf B tot B theo α theo Ξ ++ cc → Ξ + c π + This work 7 . − . − . − .
06 18 .
91 3 .
85 0 . − . et al. − .
13 11 .
50 3 .
37 12 . − . − .
56 0 . − . .
38 0 7 . − . − . − .
72 6 . − . .
52 0 9 . − . − . − .
40 9 . − . ++ cc → Ξ ′ + c π + This work 4 . − .
04 4 . − .
50 0 . − .
44 4 . − . et al. − . − . − .
45 37 .
59 1 .
37 38 .
96 3 . − . .
29 0 4 . − .
65 0 − .
65 5 . − . .
10 0 5 . − .
37 0 − .
37 7 . − . + cc → Ξ c π + This work 8 .
52 10 .
79 19 . − . − . − .
54 3.84 − . .
38 0 7 . − .
77 28 .
30 11 .
54 0 .
59 0 . .
59 0 9 . − .
45 28 .
30 8 .
85 0 .
95 0 . + cc → Ξ + c K This work 2 . − . − . − .
29 13 .
40 8 .
11 1 . − . et al. − .
02 12 .
17 8 .
15 6 . − . − .
02 1 . − . .
42 0 3 . − . − . − .
33 1 . − . .
57 0 5 . − . − . − .
75 2 . − . + cc → Ξ ′ + c K This work − . − . − .
72 17 .
44 0 .
06 17 .
50 0 . − . et al. . − .
11 2 . − .
34 0 . − .
64 0 . − . − .
15 0 − .
15 26 . . . − . − .
95 0 − .
95 37 . . . − . B (Ξ ++ cc → Ξ + c π + ) ≈ .
7% is consistent with the experimental valuebut in the lower end. In future study, it is important to pin down the branching fraction of this ode both experimentally and theoretically.In contrast to Ξ ++ cc → Ξ + c π + , we find a large constructive interference between factorizableand nonfactorizable S -wave amplitudes in Ξ + cc → Ξ c π + , whereas Dhir and Sharma [8] obtaineda large destructive interference in P -wave amplitudes (see Table V). Hence, the predicted rateof Ξ + cc → Ξ c π + in [8] is rather suppressed compared to ours. The hierarchy pattern B (Ξ + cc → Ξ c π + ) ≫ B (Ξ ++ cc → Ξ + c π + ) is the analog of B (Ξ c → Ξ − π + ) ≫ B (Ξ + c → Ξ π + ) we found in [27].It should be noticed that the hierarchy pattern B (Ξ + cc → Ξ 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 c π + may enable experimentalists tosearch for the Ξ + cc through this mode. That is, Ξ + cc is reconstructed through the Ξ + cc → Ξ c π + followed by the decay chain Ξ 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 + [50, 51]. From Table IV we see that the nonfactorizable amplitudes in Ξ ++ cc → Ξ ′ + c π + and Ω + cc → Ξ ′ + c K 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 abaryon produced by weak interactions be antisymmetric in flavor. Since the sextet Ξ ′ c is symmetricin the light quark flavor in the SU(3) limit, it cannot contribute to C ′ . It is clear from Eqs. (26),(27) and (29) that the C ′ amplitude is proportional to the matrix element a Ξ ′ + c Ξ + cc governed bythe bag integral X introduced in Eq. (32), which vanishes in the SU(3) limit. Likewise, thenonfactorizable S -wave amplitudes in Ξ + cc → Ξ ′ + c ( π , η ) governed by C ′ also vanish in the limit ofSU(3) symmetry. However, this is not the case for nonfactorizable P -wave amplitudes due to thepresence of W -exchange contributions E and/or E .Finally, we notice that the two decay modes Ξ ++ cc → Σ ++ c K and Ω + cc → Ω c π + are purelyfactorizable processes. Therefore, their theoretical calculations are much more clean. Measure-ments of them will provide information on Ξ ++ cc → Σ ++ c and Ω + cc → Ω c transition form factors.Our result of B (Ω + cc → Ω c π + ) ≈
4% suggests that this mode may serve as a discovery channelfor the Ω + cc . More explicitly, it can be searched in the final state pK − π − π + π + through the decayΩ + cc → Ω c π + followed by Ω c → Ω − π + → pπ − K − π + . B. Comparison with other works
In Table V we have already compared our calculated partial-wave amplitudes for some ofdoubly charmed baryon decays with Gutsche et al. [11], Dhir and Sharma [8, 10]. We agree withGutsche et al. on the interference patterns in S - and P -wave amplitudes of Ξ ++ cc → Ξ + c π + andΩ + cc → Ξ + c K , but disagree on the interference patterns in Ξ ++ cc → Ξ ′ + c π + and Ω + cc → Ξ ′ + c K . An estimate of the branching fraction of Ξ + cc → Λ + c K − π + can be made by assuming B (Ξ + cc → Λ + c K − π + ) ≈ B (Ξ + cc → Σ ++ K − ) + B (Ξ + cc → Λ + c K ∗ ). Since B (Ξ + cc → Σ ++ K − ) ≈ .
13% in ourwork, while B (Ξ + cc → Λ + c K ∗ ) = (0 . +0 . − . )% is obtained in the final-state rescattering approach [15, 52]for τ (Ξ + cc ) = 0 . × − s , it appears that B (Ξ + cc → Λ + c K − π + ) is not more than 0.8%. ABLE VI. Predicted branching fractions (in %) of Cabibbo-favored doubly charmed baryondecays by different groups. 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). For theresults of Gutsche et al. [11, 13, 17], we quote the latest ones from [17]. All the model resultshave been normalized using the lifetimes τ (Ξ ++ cc ) = 2 . × − s , τ (Ξ + cc ) = 0 . × − s and τ (Ω + cc ) = 1 . × − s . Mode Our Dhir Gutsche et al.
Wang Gerasimov Ke Shi et al. [8, 10] [11, 13, 17] et al. [7] et al. [14] et al. [18] et al. [12]Ξ ++ cc → Ξ + c π + . ± .
46 3 . ± . ++ cc → Ξ ′ + c π + . ± .
24 0 . ± . ++ cc → Σ ++ c K . + cc → Ξ c π + .
84 0.59 (N) 1.08 1.23 0 . ± .
08 0 . ± . + cc → Ξ ′ c π + .
55 1.49 (N) 0.76 1.04 0 . ± .
04 0 . ± . + cc → Λ + c K .
31 0.27 (N)0.37 (H)Ξ + cc → Σ + c K .
38 0.59 (N)0.90 (H)Ξ + cc → Ξ + c π .
38 0.50Ξ + cc → Ξ ′ + c π .
17 0.054Ξ + cc → Ξ + c η .
18 0.063Ξ + cc → Ξ ′ + c η .
05 0.036Ξ + cc → Σ ++ c K − .
13 0.22Ξ + cc → Ω c K + + cc → Ω c π + . ± . + cc → Ξ + c K + cc → Ξ ′ + c K evertheless, the disagreement in the last two modes is minor because of the Pati-Woo theoremfor the C ′ amplitude. We agree with Dhir and Sharma on the interference patterns in P -waveamplitudes of Ξ + cc → Ξ ′ c π + , Ξ + c K , Λ + c K , but disagree on that in Ξ ++ cc → Ξ + c π + , Ξ + cc → Ξ c π + and Ω + cc → Ξ + c K . Consequently, the hierarchy pattern of B (Ξ + cc → Ξ 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 Cabibbo-favored B cc → B c + P decays with other works. Only the factorizable contributions from theexternal W -emission governed by the Wilson coefficient a were considered in references [7, 12,14, 18] with nonfactorizbale effects being neglected. We see from Table I that only the decaymodes Ξ ++ cc → Ξ ( ′ )+ c π + , Ξ + cc → Ξ ( ′ )0 c π + and Ω + cc → Ω c π + receive contributions from the external W -emission amplitude T . Branching fractions calculated in Refs. [7, 12, 18] were based on theform-factor models LFQM(I), LFQM(II) and QSR, respectively. Since B cc → B c transition formfactors 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 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 ) and B (Ξ + cc → Ξ + c η ) in [8] are muchsmaller than ours. This is because we have sizable W -exchange contributions to the S -waveamplitudes of Ξ + cc → Ξ + c ( π , 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 charmedbaryons Ξ ++ cc , Ξ + cc and Ω + cc . To estimate the nonfactorizable contributions, we work in the polemodel for the P -wave amplitudes and current algebra for S -wave ones. Throughout the wholecalculations, all the non-perturbative parameters including form factors, baryon matrix elementsand axial-vector form factors are evaluated within the framework of the MIT bag model.We draw some conclusions from our analysis: • 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 A ( P ) B ′ B are evaluated in the same MIT bag model toensure the correctness of their relative signs once the wave function convention is fixed.For the Ξ ++ cc → Ξ + c π + mode, we found a large destructive interference between factorizableand nonfactorizable contributions for both S - and P -wave amplitudes. Our prediction of ∼ .
70% for its branching fraction is smaller than the earlier estimates in which nonfactorizableeffects were not considered but agrees nicely with the result based on an entirely differentapproach, namely, the covariant confined quark model. On the contrary, a large constructiveinterference was found in the P -wave amplitude by Sharma and Dhir [8], leading to abranching fraction of order (7 − actor g and the combination a Ξ + c Ξ + cc × g Ξ ++ cc Ξ + cc π + that accounts for the different P -waveinterference pattern in Ξ ++ cc → Ξ + c π + found in this work and the work by Sharma and Dhir. • Using the current results of the absolute branching fractions of (Λ + c , Ξ + c ) → pK − π + and theLHCb 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 . ± . B (Ξ ++ cc → Σ ++ c K ∗ ) andthe plausible assumption of B (Ξ ++ cc → Λ + c K − π + π + ) ≈ B (Ξ ++ cc → Σ ++ c K ∗ ). Therefore,our prediction of B (Ξ ++ cc → Ξ + c π + ) ≈ .
7% is consistent with the experimental value butin the lower end. It is important to pin down the branching fraction of Ξ ++ cc → Ξ + c π + infuture study. • Factorizable and nonfactorizable S -wave amplitudes interfere constructively in Ξ + cc → Ξ c π + .Its large branching fraction of order 4% may enable experimentalists to search for the Ξ + cc through this mode. In this way, Ξ + cc is reconstructed through the Ξ + cc → Ξ c π + followed bythe decay chain Ξ c → Ξ − π + → pπ − π − π + . • Besides Ξ + cc → Ξ c π + , the Ξ + cc → Ξ + c ( π , η ) modes also receive large nonfactorizable con-tributions to their S -wave amplitudes. Hence, they have large branching fractions amongΞ + cc → B c + P decays. • The two decay modes Ξ ++ cc → Σ ++ c K and Ω + cc → Ω c π + are purely factorizable processes.Measurements of them will provide information on Ξ ++ cc → Σ ++ c and Ω + cc → Ω c transitionform factors. Our calculation of B (Ω + cc → Ω c π + ) ≈
4% suggests that this mode may serve asa discovery channel for the Ω + cc . That is, it can be searched in the final state pK − π − π + π + through the decay Ω + cc → Ω c π + followed by Ω c → Ω − π + → pπ − K − π + . • Nonfactorizable amplitudes in Ξ ++ cc → Ξ ′ + c π + and Ω + cc → Ξ ′ + c K are very small compared tothe 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 , η ) decays are also suppressedby the same mechanism. ACKNOWLEDGMENTS
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