Charmed Baryons. prepared by Hai-Yang Cheng. I. Introduction 2. II. Production of charmed baryons at BESIII 3. III. Spectroscopy 3

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1 November, 006 Charmed Baryons prepared by Hai-Yang Cheng Contents I. Introdution II. Prodution of harmed baryons at BESIII 3 III. Spetrosopy 3 IV. Strong deays 7 A. Strong deays of s-wave harmed baryons 7 B. Strong deays of p-wave harmed baryons 9 V. Lifetimes 11 VI. Hadroni weak deays 17 A. Quark-diagram sheme 18 B. Dynamial model alulation 19 C. Disussions 1. Deay asymmetry. Λ + deays 4 3. Ξ + deays 5 4. Ξ 0 deays 6 5. Ω 0 deays 6 D. Charm-flavor-onserving weak deays 6 VII. Semileptoni deays 7 VIII. Eletromagneti and Weak Radiative deays 8 A. Eletromagneti deays 8 B. Weak radiative deays 31 Referenes 3 1

2 I. INTRODUCTION In the past years many new exited harmed baryon states have been disovered by BaBar, Belle and CLEO. In partiular, B fatories have provided a very rih soure of harmed baryons both from B deays and from the ontinuum e + e. A new hapter for the harmed baryon spetrosopy is opened by the rih mass spetrum and the relatively narrow widths of the exited states. Experimentally and theoretially, it is important to identify the quantum numbers of these new states and understand their properties. Sine the pseudosalar mesons involved in the strong deays of harmed baryons are soft, the harmed baryon system offers an exellent ground for testing the ideas and preditions of heavy quark symmetry of the heavy quark and hiral symmetry of the light quarks. The observation of the lifetime differenes among the harmed mesons D +, D 0 and harmed baryons is very interesting sine it was realized very early that the naive parton model gives the same lifetimes for all heavy partiles ontaining a heavy quark Q, while experimentally, the lifetimes of Ξ + and Ω 0 differ by a fator of six! This implies the importane of the underlying mehanisms suh as W -exhange and Pauli interferene due to the idential quarks produed in the heavy quark deay and in the wavefuntion of the harmed baryons. With the advent of heavy quark effetive theory, it was reognized in early nineties that nonperturbative orretions to the parton piture an be systematially expanded in powers of 1/m Q. Within the QCD-based heavy quark expansion framework, some phenomenologial assumptions an be turned into some oherent and quantitative statements and nonperturbative effets an be systematially studied. Contrary to the signifiant progress made over the last 0 years or so in the studies of the heavy meson weak deay, advanement in the arena of heavy baryons is relatively slow. Nevertheless, the experimental measurements of the harmed baryon hadroni weak deays have been pushed to the Cabibbo-suppressed level. Many new data emerged an be used to test a handful of phenomenologial models available in the literature. Apart from the ompliation due to the presene of three quarks in the baryon, a major disparity between harmed baryon and harmed meson deays is that while the latter is usually dominated by fatorizable amplitudes, the former reeives sizable nonfatorizable ontributions from W -exhange diagrams whih are not subjet to olor and heliity suppression. Besides the dynamial models, there are also some onsiderations based on the symmetry argument and the quark diagram sheme. The exlusive semileptoni deays of harmed baryons like Λ + Λe + (µ + )ν e, Ξ + Ξ 0 e + ν e and Ξ 0 Ξ e + ν e have been observed experimentally. Their rates depend on the heavy baryon to the light baryon transition form fators. Experimentally, the only information available so far is the form-fator ratio measured in the semileptoni deay Λ Λe ν. Although radiative deays are well measured in the harmed meson setor, e.g. D Dγ and D s + D s + γ, only three of the radiative modes in the harmed baryon setor have been observed, namely, Ξ 0 Ξ 0 γ, Ξ + Ξ + γ and Ω 0 Ω 0 γ. Charm flavor is onserved in these eletromagneti harmed baryon deays. However, it will be diffiult to measure the rates of these deays beause these states are too narrow to be experimentally resolvable. There are also harm-flavor-onserving weak radiative deays suh as Ξ Λ γ and Ω Ξ γ. In these deays, weak radiative transitions arise from the diquark setor of the heavy baryon whereas the heavy quark behaves as a spetator. The harm-flavor-violating weak radiative deays, e.g., Λ + Σ + γ and Ξ 0 Ξ 0 γ, arise from the W -exhange diagram aompanied by a photon emission from the external quark. Two exellent review artiles on harmed baryons an be found in [1, ].

3 II. PRODUCTION OF CHARMED BARYONS AT BESIII Prodution and deays of the harmed baryons an be studied at BESIII one its enter-ofmass energy s is upgraded to the level above 4.6 GeV. In order to estimate the number of harmed baryon events produed at BESIII, it is neessary to know its luminosity, the ross setion σ(e + e ) at the energies of interest and the fragmentation funtion of the quark into the harmed baryon III. SPECTROSCOPY Charmed baryon spetrosopy provides an ideal plae for studying the dynamis of the light quarks in the environment of a heavy quark. The harmed baryon of interest ontains a harmed quark and two light quarks, whih we will often refer to as a diquark. Eah light quark is a triplet of the flavor SU(3). Sine 3 3 = 3+6, there are two different SU(3) multiplets of harmed baryons: a symmetri sextet 6 and an antisymmetri antitriplet 3. For the ground-state s-wave baryons in the quark model, the symmetries in the flavor and spin of the diquarks are orrelated. Consequently, the diquark in the flavor-symmetri sextet has spin 1, while the diquark in the flavor-antisymmetri antitriplet has spin 0. When the diquark ombines with the harmed quark, the sextet ontains both spin 1/ and spin 3/ harmed baryons. However, the antitriplet ontains only spin 1/ ones. More speifially, the Λ +, Ξ + and Ξ 0 form a 3 representation and they all deay weakly. The Ω 0, Ξ +, Ξ 0 and Σ ++,+,0 form a 6 representation; among them, only Ω 0 deays weakly. Note that we follow the Partile Data Group (PDG) [3] to use a prime to distinguish the Ξ in the 6 from the one in the 3. The lowest-lying orbitally exited baryon states are the p-wave harmed baryons with their quantum numbers listed in Table I. Although the separate spin angular momentum S l and orbital angular momentum L l of the light degrees of freedom are not well defined, they are inluded for guidane from the quark model. In the heavy quark limit, the spin of the harmed quark S and the total angular momentum of the two light quarks J l = S l + L l are separately onserved. It is onvenient to use them to enumerate the spetrum of states. There are two types of L l = 1 orbital exited harmed baryon states: states with the unit of orbital angular momentum between the diquark and the harmed quark, and states with the unit of orbital angular momentum between the two light quarks. The orbital wave funtion of the former (latter) is symmetri (antisymmetri) under the exhange of two light quarks. To see this, one an define two independent relative momenta k = 1 (p 1 p ) and K = 1 (p 1+p p ) from the two light quark momenta p 1, p and the heavy quark momentum p. (In the heavy quark limit, p an be set to zero.) Denoting the quantum numbers L k and L K as the eigenvalues of L k and L K, the k-orbital momentum L k desribes relative orbital exitations of the two light quarks, and the K-orbital momentum L K desribes orbital exitations of the enter of the mass of the two light quarks relative to the heavy quark [1]. The p-wave heavy baryon an be either in the (L k = 0, L K = 1) K-state or the (L k = 1, L K = 0) k-state. It is obvious that the orbital K-state (k-state) is symmetri (antisymmetri) under the interhange of p 1 and p. The observed mass spetra and deay widths of harmed baryons are summarized in Table II (see also Fig. 1). B fatories have provided a very rih soure of harmed baryons both from B deays and from the ontinuum e + e. For example, several new exited harmed baryon states suh 3

4 TABLE I: The p-wave harmed baryons and their quantum numbers, where S l (J l ) is the total spin (angular momentum) of the two light quarks. The quantum number in the subsript labels J l. The quantum number in parentheses is referred to the spin of the baryon. In the quark model, the upper (lower) four multiplets have even (odd) orbital wave funtions under the permutation of the two light quarks. That is, L l for the former is referred to the orbital angular momentum between the diquark and the harmed quark, while L l for the latter is the orbital angular momentum between the two light quarks. The expliit quark model wave funtions for p-wave harmed baryons an be found in [4]. State SU(3) S l L l J P l l State SU(3) S l L l J P l l Λ 1 ( 1, 3 ) Ξ 1 ( 1, 3 ) Σ 0 ( 1 ) Ξ 0 ( 1 ) Σ 1 ( 1, 3 ) Ξ 1 ( 1, 3 ) Σ ( 3, 5 ) Ξ ( 3, 5 ) Σ 1 ( 1, 3 ) Ξ 1 ( 1, 3 ) Λ 0 ( 1 ) Ξ0 ( 1 ) Λ 1 ( 1, 3 ) Ξ1 ( 1, 3 ) Λ ( 3, 5 ) Ξ ( 3, 5 ) as Λ (765) +, Λ (880) +, Λ (940) +, Ξ (815), Ξ (980) and Ξ (3077) have been measured reently and they are not still not in the list of 006 Partile Data Group [3]. By now, the J P = 1 + and 1 3 states: (Λ +, Ξ +, Ξ 0 ), (Λ (593) +, Ξ (790) +, Ξ (790) 0 ), and J P = 1 + and states: (Ω, Σ, Ξ ), (Ω, Σ, Ξ ) are established. Notie that exept for the parity of the lightest Λ +, none of the other J P quantum numbers given in Table II has been measured. One has to rely on the quark model to determine the J P assignments. In the following we disuss some of the new exited harmed baryon states: The highest Λ (940) + was first disovered by BaBar in the D 0 p deay mode [5] and onfirmed by Belle in the deays Σ 0 π +, Σ ++ π whih subsequently deay into Λ + π + π [6, 7]. The state Λ (880) + first observed by CLEO [8] in Λ + π + π was also seen by BaBar in the D 0 p spetrum [5]. It was originally onjetured that, based on its narrow width, Λ (880) + might be a Λ + 0 ( 1 ) state [8]. Reently, Belle has studied the experimental onstraint on the J P quantum numbers of Λ (880) + [6]. The angular analysis of Λ (880) + Σ 0,++ π ± indiates that J = 5/ is favored over J = 1/ or 3/, while the study of the resonant struture of Λ (880) + Λ + π + π implies the existene of the Σ π intermediate states and Γ(Σ π ± )/Γ(Σ π ± ) = (4.1 ± )%. This value is in agreement with heavy quark symmetry preditions [9] and favors the 5/ + over the 5/ assignment. 1 Therefore, it is not a Λ + ( 5 ) state either. Sine J l =, S l = 0, L = for the diquark system of Λ (880) +, this 1 Stritly speaking, the argument in favor of the 5/ + assignment is reahed in [6] by onsidering only the F -wave ontribution and negleting the P -wave ontribution to Λ (880) + Σ π (see [10] for more disussions). 4

5 TABLE II: Mass spetra and deay widths (in units of MeV) of harmed baryons. Experimental values are taken from the Partile Data Group [3] exept Λ (880), Λ (940), Ξ (980) +,0, Ξ (3077) +,0 and Ω (768) for whih we use the most reent available BaBar and Belle measurements. State quark ontent J P Mass Width Λ ud ± 0.14 Λ (593) + 1 ud ± Λ (65) + 3 ud 68.1 ± 0.6 < 1.9 Λ (765) + ud?? ±.4 50 Λ (880) ud ± ± 0.6 Λ (940) + ud?? ± ± 5.0 Σ (455) uu ± ± 0.30 Σ (455) ud 45.9 ± 0.4 < 4.6 Σ (455) dd ± ± 0.4 Σ (50) uu ± ± 1.9 Σ (50) ud ±.3 < 17 + Σ (50) 0 3 dd ± ±.1 Σ (800) ++ uu?? Σ (800) + ud?? Σ (800) 0 dd?? Ξ + us 1 Ξ 0 ds 1 Ξ + us 1 Ξ 0 ds 1 Ξ (645) + us 3 Ξ (645) 0 ds 3 Ξ (790) + us 1 Ξ (790) 0 ds 1 Ξ (815) + us ± ± ± ± ± 1.4 < ± 1. < ± 3. < ± 3.3 < ± 1. < 3.5 Ξ (815) 0 3 ds 818. ±.1 < 6.5 Ξ (980) + us?? ± ± 3.0 Ξ (980) 0 ds?? ± Ξ (3077) + us?? ± ± 1.1 Ξ (3077) 0 ds?? ±.3 5. ± 3.6 Ω ss ±.6 Ω (768) ss ± 3.0 5

6 FIG. 1: Charmed baryons and some of their orbital exitations [3]. is the first observation of a d-wave harmed baryon. It is interesting to notie that, based on the diquark idea, the assignment J P = 5/ + has already been predited in [11] for the state Λ (880) before the Belle experiment. As for Λ (980) +, it was reently argued that it is an exoti moleular state of D 0 and p [1]. The new harmed strange baryons Ξ (980) + and Ξ (3077) + that deay into Λ + K π + were first observed by Belle [13] and onfirmed by BaBar [14]. In the reent BaBar measurement [14], the Ξ (980) + is found to deay resonantly through the intermediate state Σ (455) ++ K with 4.9 σ signifiane and non-resonantly to Λ + K π + with 4.1 σ signifiane. With 5.8 σ signifiane, the Ξ (3077) + is found to deay resonantly through Σ (455) ++ K, and with 4.6 σ signifiane, it is found to deay through Σ (50) ++ K. The signifiane of the signal for the non-resonant deay Ξ (3077) + Λ + K π + is 1.4 σ. The highest isotriplet harmed baryons Σ (800) ++,+,0 deaying into Λ + π were first measured by Belle [15]. They are most likely to be the J P = 3/ Σ states beause the Σ ( 3 ) baryon deays prinipally into the Λ π system in a D-wave, while Σ 1 ( 3 ) deays mainly into the two pion system Λ ππ. The state Σ 0 ( 1 ) an deay into Λ π in an S-wave, but it is very broad with width of order 406 MeV [10]. Experimentally, it will be very diffiult to observe it. The new 3/ + Ω (768) was reently observed by BaBar in the eletromagneti deay Ω (768) Ω γ [16]. With this new observation, the 3/ + sextet is finally ompleted. Evidene of double harm states has been reported by SELEX in Ξ (3519) + Λ + K π + [17]. Further observations of Ξ ++ Λ + K π + π + and Ξ + pd + K were also announed by SELEX [18]. However, none of the double harm states disovered by SELEX has been onfirmed by FOCUS, BaBar [19] and Belle [7] despite the 10 6 Λ events produed in B fatories versus 1630 Λ events observed at SELEX. 6

7 Charmed baryon spetrosopy has been studied extensively in various models. The interested readers are referred to [0] for further referenes. In heavy quark effetive theory, the mass splittings between spin- 3 and spin- 1 sextet harmed baryon multiplets are governed by the hromomagneti interations so that m Σ m Σ = m Ξ m Ξ = m Ω m Ω, (3.1) up to orretions of 1/m. This relation is borne out by experiment: m Σ + m Σ + = 64.6 ±.3 MeV, m Ξ + m Ξ + = 70.9 ± 3.4 MeV and m Ω m Ω = 70.8 ± 1.5 MeV. IV. STRONG DECAYS Due to the rih mass spetrum and the relatively narrow widths of the exited states, the harmed baryon system offers an exellent ground for testing the ideas and preditions of heavy quark symmetry and light flavor SU(3) symmetry. The pseudosalar mesons involved in the strong deays of harmed baryons suh as Σ Λ π are soft. Therefore, heavy quark symmetry of the heavy quark and hiral symmetry of the light quarks will have interesting impliations for the low-energy dynamis of heavy baryons interating with the Goldstone bosons. The strong deays of harmed baryons are most onveniently desribed by the heavy hadron hiral Lagrangians in whih heavy quark symmetry and hiral symmetry are inorporated [1, ]. The Lagrangian involves two oupling onstants g 1 and g for P -wave transitions between s-wave and s-wave baryons [1], six ouplings h h 7 for the S-wave transitions between s-wave and p-wave baryons, and eight ouplings h 8 h 15 for the D-wave transitions between s-wave and p-wave baryons [4]. The general hiral Lagrangian for heavy baryons oupling to the pseudosalar mesons an be expressed ompatly in terms of superfields. We will not write down the relevant Lagrangians here; instead the reader is referred to Eqs. (3.1) and (3.3) of [4]. Nevertheless, we list some of the partial widths derived from the Lagrangian [4]: Γ(Σ Σ π) = g 1 πf π Γ(Λ 1 (1/) Σ π) = h πf π Γ(Σ 1 (1/) Σ π) = h 4 4πf π Γ( Ξ 0 (1/) Ξ π) = h 6 πf π m Σ m Σ p 3 π, Γ(Σ Λ π) = g πf π m Λ m Σ p 3 π, m Σ m Λ1 E πp π, Γ(Σ 0 (1/) Λ π) = h 3 πf π m Σ m Σ1 E πp π, Γ( Σ 1 (1/) Σ π) = h 5 4πf π m Ξ Eπp π, Γ( Λ 1 (1/) Σ π) = h 7 πf m Ξ0 π m Λ m Σ0 E πp π, m Σ m Σ1 E πp π, (4.1) m Σ m Λ1 E πp π, where p π is the pion s momentum and f π = 13 MeV. Unfortunately, the deay Σ Σ π is kinematially prohibited sine the mass differene between Σ and Σ is only of order 65 MeV. Consequently, the oupling g 1 annot be extrated diretly from the strong deays of heavy baryons. A. Strong deays of s-wave harmed baryons In the framework of heavy hadron hiral pertrubation theory (HHChPT), one an use some measurements as input to fix the oupling g whih, in turn, an be used to predit the rates of 7

8 other strong deays. We shall use Σ Λ π as input [3] From whih we obtain Γ(Σ ++ ) = Γ(Σ ++ Λ + π + ) =.3 ± 0.30 MeV. (4.) g = , (4.3) where we have negleted the tiny ontributions from eletromagneti deays. Note that g obtained from Σ 0 Λ + π has the same entral value as Eq. (4.3) exept that the errors are slightly large. If Σ Λ π deays are employed as input, we will obtain g = 0.57 ± 0.04 from Σ ++ Λ + π + and 0.60±0.04 from Σ 0 Λ + π. Hene, it is preferable to use the measurement of Σ ++ Λ + π + to fix g. As pointed out in [1], within in the framework of the non-relativisti quark model, the ouplings g 1 and g an be related to g q A, the axial-vetor oupling in a single quark transition of u d, via g 1 = 4 3 gq A, g = 3 gq A. (4.4) Using g q A = 0.75 whih is required to reprodue the orret value of gn A = 1.5, we obtain g 1 = 1, g = (4.5) Hene, the quark model predition is in good agreement with experiment, but deviates σ from the large-n argument: g = ga N/ = 0.88 [4]. Applying (4.3) leads to (see also Table III) ( Γ(Ξ + ) = Γ(Ξ + Ξ + π 0, Ξ 0 π + ) = g 1 m Ξ + 4πfπ p 3 π + m ) Ξ0 p 3 π = (.8 ± 0.4) MeV, m Ξ + m Ξ + ( Γ(Ξ 0 ) = Γ(Ξ 0 Ξ + π, Ξ 0 π 0 ) = g mξ + 4πfπ p 3 π + 1 ) m Ξ 0 p 3 π = (.9 ± 0.4) MeV. (4.6) m Ξ 0 m Ξ 0 Note that we have negleted the effet of Ξ Ξ mixing in alulations (for reent onsiderations, see [9, 30]). Therefore, the predited total width of Ξ + is in the viinity of the urrent limit Γ(Ξ + ) < 3.1 MeV [31]. It is lear from Table III that the predited widths of Σ ++ and Σ 0 by HHChPT are in good agreement with experiment. The strong deay width of Σ is smaller than that of Σ by a fator of 7, although they will beome the same in the limit of heavy quark symmetry. This is asribed to the fat that the pion s momentum is around 90 MeV in the deay Σ Λ π while it is two times bigger in Σ Λ π. Sine Σ states are signifiantly narrower than their spin-3/ ounterparts, this explains why the measurement of their widths ame out muh later. Instead of using the data to fix the oupling onstants in a model-independent manner, there exist some alulations of ouplings in various models suh as the relativisti light-front model [5], the relativisti three-quark model [6] and light-one sum rules [7, 3]. The results are summarized in Table III. It is worth remarking that although the oupling g 1 annot be determined diretly from the strong deay suh as Σ Σ π, some information of g 1 an be learned from the radiative deay Ξ 0 γ, whih is prohibited at tree level by SU(3) symmetry but an be indued by hiral loops. A measurement of Γ(Ξ 0 Ξ 0 γ) will yield two possible solutions for g 1. Assuming the Ξ 0 validity of the quark model relations among different oupling onstants, the experimental value of g implies g 1 = 0.93 ± 0.16 [3] (see also Se. VIII.A). For previous efforts of extrating g from experiment using HHChPT, see [4, 3]. 8

9 TABLE III: Deay widths (in units of MeV) of s-wave harmed baryons. Theoretial preditions of [5] are taken from Table IV of [6]. Deay Expt. HHChPT Tawfiq Ivanov Huang Albertus [3] [10] et al. [5] et al. [6] et al. [7] et al. [8] Σ ++ Λ + π +.3 ± 0.30 input 1.51 ± ± ± 0.07 Σ + Λ + π 0 < ± ± ± ± 0.08 Σ 0 Λ + π. ± 0.4. ± ± ± ± 0.07 Σ (50) ++ Λ + π ± ± ± ± ± 0.75 Σ (50) + Λ + π 0 < ± ± 0.74 Σ (50) 0 Λ + π 16.1 ± ± ± ± ± 0.7 Ξ (645) + Ξ 0,+ π +,0 < ± ± ± ± 0.10 Ξ (645) 0 Ξ +,0 π,0 < ± ± ± ± 0.10 B. Strong deays of p-wave harmed baryons Some of the S-wave and D-wave ouplings of p-wave baryons to s-wave baryons an be determined. In priniple, the oupling h is readily extrated from Λ (593) + Σ 0 π + with Λ (593) + identified as Λ 1 ( 1 )+. However, sine Λ (593) + Σ π is kinematially barely allowed, the finite width effets of the intermediate resonant states will beome important [33]. Pole ontributions to the deays Λ (593) +, Λ (65) + Λ + ππ have been onsidered in [4, 7, 34] with the finite width effets inluded. The intermediate states of interest are Σ and Σ poles. The resonant ontribution arises from the Σ pole, while the non-resonant term reeives a ontribution from the Σ pole. (Sine Λ (593) +, Λ (65) + Λ π are not kinematially allowed, the Σ pole is not a resonant ontribution.) The deay rates thus depend on two oupling onstants h and h 8. The deay rate for the proess Λ + 1 (593) Λ + π + π an be alulated in the framework of heavy hadron hiral perturbation theory to be [10] Γ(Λ (593) Λ + ππ) = 14.48h h h h 8, Γ(Λ (65) Λ + ππ) = 0.648h h 8 8.6h h 8. (4.7) It is lear that the limit on Γ(Λ (65)) gives an upper bound on h 8 of order 10 3 (in units of MeV 1 ), whereas the deay width of Λ (593) is entirely governed by the oupling h. This indiates that the diret non-resonant Λ + ππ annot be desribed by the Σ pole alone. Identifying the alulated Γ(Λ (593) Λ + ππ) with the resonant one, we find h = , h (4.8) Assuming that the total deay width of Λ (593) is saturated by the resonant Λ + ππ 3-body deays, Pirjol and Yan obtained h = and h 8 ( ) 10 3 MeV 1 [4]. Using the updated hadron masses and Γ(Λ (593) Λ + ππ), 3 we find h = Taking 3 The CLEO result Γ(Λ (593)) = MeV [35] is used in [4] to fix h. 9

10 TABLE IV: Same as Table III exept for p-wave harmed baryons [10]. Deay Expt. HHChPT Tawfiq Ivanov Huang Zhu [3] [10] et al. [5] et al. [6] et al. [7] [3] Λ (593) + (Λ + ππ) R input.5 Λ (593) + Σ ++ π ± ± Λ (593) + Σ 0 π ± ± ± Λ (593) + Σ + π ± ± ± Λ (65) + Σ ++ π < ± ± Λ (65) + Σ 0 π + < ± ± Λ (65) + Σ + π ± ± Λ (65) + Λ + ππ < Σ (800) ++ Λ π, Σ ( ) π input Σ (800) + Λ π, Σ ( ) π input Σ (800) 0 Λ π, Σ ( ) π input Ξ (790) + Ξ 0,+ π +,0 < Ξ (790) 0 Ξ +,0 π,0 < Ξ (815) + Ξ +,0 π 0,+ < ± ± 0.04 Ξ (815) 0 Ξ +,0 π,0 < into aount the fat that the Σ and Σ poles only desribe the resonant ontributions to the total width of Λ (593), we finally reah the h value given in (4.8). Taking into aount the threshold (or finite width) effet in the strong deay Λ (593) + Λ ππ, a slightly small oupling h = is obtained in [33]. For the spin- 3 state Λ (65), its deay is dominated by the three-body hannel Λ + ππ as the major two-body deay Σ π is a D-wave one. Some information on the oupling h 10 ane be inferred from the strong deays of Λ (800). As notied in passing, the states Σ (800) ++,+,0 are most likely to be Σ ( 3 ). Assuming their widths are dominated by the two-body modes Λ π, Σ π and Λ π, we have [4] Γ (Σ ( 3 ) )++ Γ (Σ ( 3 ) )++ Λ + π + + Γ (Σ ( 3 ) )++ Σ + π + + Γ = 4h 10 15πf π m Λ m Σ p 5 + h 11 10πf π m Σ m Σ p 5 + h 11 10πf π ( Σ ( 3 ) )++ Σ + π + m Σ m Σ p 5, (4.9) and similar expressions for Σ (800) + and Σ (800) 0. Using the quark model relation h 11 = h 10 [see also Eq. (4.1)] and the measured widths of Σ (800) ++,+,0 (Table II), we obtain h 10 = ( ) 10 3 MeV 1. (4.10) Sine the state Λ 1 ( 3 ) is broader, even a small mixing of Λ ( 3 ) with Λ 1( 3 ) ould enhane the deay width of the former [4]. In this ase, the above value for h 10 should be regarded as an upper limit of h 10. Using the quark model relation h 8 = h 10 (see Eq. (4.1) below), the alulated partial widths of Λ (65) + are shown in Table IV. 10

11 The Ξ (790) and Ξ (815) baryons form a doublet Ξ 1 ( 1, 3 ). Ξ (790) deays to Ξ π, while Ξ (815) deays to Ξ ππ, resonating through Ξ, i.e. Ξ (645). Using the oupling h obtained (4.8) and the experimental observation that the Ξ ππ mode in Ξ (815) deays is onsistent with being entirely via Ξ (645)π, the predited Ξ (790) and Ξ (815) widths are shown in Table IV and they are onsistent with the urrent experimental limits. Couplings other than h and h 10 an be related to eah other via the quark model. The S-wave ouplings between the s-wave and the p-wave baryons are related by [4] h 3 h 4 = The D-wave ouplings satisfy the relations h 8 = h 9 = h 10, 3, h h 4 = 1, h 5 h 6 = 3, h 11 h 10 = h 15 h 14 =, The reader is referred to [4] for further details. h 5 h 7 = 1. (4.11) h 1 h 13 =, h 14 h 13 = 1. (4.1) V. LIFETIMES The lifetime differenes among the harmed mesons D +, D 0 and harmed baryons have been studied extensively both experimentally and theoretially sine late 1970s. It was realized very early that the naive parton model gives the same lifetimes for all heavy partiles ontaining a heavy quark Q and that the underlying mehanism for the deay width differenes and the lifetime hierarhy of heavy hadrons omes mainly from the spetator effets like W -exhange and Pauli interferene due to the idential quarks produed in the heavy quark deay and in the harmed baryons (for a review, see [, 36, 37]). The spetator effets were expressed in 1980s in terms of loal four-quark operators by relating the total widths to the imaginary part of ertain forward sattering amplitudes [38 40]. (The spetator effets for harmed baryons were first studied in [41].) With the advent of heavy quark effetive theory (HQET), it was reognized in early 1990s that nonperturbative orretions to the parton piture an be systematially expanded in powers of 1/m Q [4, 43]. Subsequently, it was demonstrated that this 1/m Q expansion is appliable not only to global quantities suh as lifetimes, but also to loal quantities, e.g. the lepton spetrum in the semileptoni deays of heavy hadrons [44]. Therefore, the above-mentioned phenomenologial work in 1980s aquired a firm theoretial footing in 1990s, namely the heavy quark expansion (HQE), whih is a generalization of the operator produt expansion (OPE) in 1/m Q. Within this QCD-based framework, some phenomenologial assumptions an be turned into some oherent and quantitative statements and nonperturbative effets an be systematially studied. Based on the OPE approah for the analysis of inlusive weak deays, the inlusive rate of the harmed baryon is shematially represented by Γ(B f) = G F m5 19π 3 V CKM ( A 0 + A m + A 3 m 3 + O( 1 ) m 4 ). (5.1) The A 0 term omes from the quark deay and is ommon to all harmed hadrons. There is no linear 1/m Q orretions to the inlusive deay rate due to the lak of gauge-invariant dimensionfour operators [4, 45], a onsequene known as Luke s theorem [46]. Nonperturbative orretions start at order 1/m Q and they are model independent. Spetator effets in inlusive deays due to 11

12 the Pauli interferene and W -exhange ontributions aount for 1/m 3 orretions and they have two eminent features: First, the estimate of spetator effets is model dependent; the hadroni four-quark matrix elements are usually evaluated by assuming the fatorization approximation for mesons and the quark model for baryons. Seond, there is a two-body phase-spae enhanement fator of 16π for spetator effets relative to the three-body phase spae for heavy quark deay. This implies that spetator effets, being of order 1/m 3, are omparable to and even exeed the 1/m terms. The lifetimes of harmed baryons are measured to be [3] τ(λ + ) = (00 ± 6) s, τ(ξ + ) = (44 ± 6) s, τ(ξ 0 ) = ( ) s, τ(ω 0 ) = (69 ± 1) s. (5.) As we shall see below, the lifetime hierarhy τ(ξ + ) > τ(λ + ) > τ(ξ 0 ) > τ(ω 0 ) is qualitatively understandable in the OPE approah but not quantitatively. In general, the total width of the harmed baryon B reeives ontributions from inlusive nonleptoni and semileptoni deays: Γ(B ) = Γ NL (B ) + Γ SL (B ). The nonleptoni ontribution an be deomposed into Γ NL (B ) = Γ de (B ) + Γ ann (B ) + Γ int (B ) + Γ int + (B ), (5.3) orresponding to the -quark deay, the W -exhange ontribution, destrutive and onstrutive Pauli interferenes. It is known that the inlusive deay rate is governed by the imaginary part of an effetive nonloal forward transition operator T. Therefore, Γ de orresponds to the imaginary part of Fig. (a) sandwihed between the same B states. At the Cabibbo-allowed level, Γ de represents the deay rate of su d, and Γ ann denotes the ontribution due to the W -exhange diagram d us. The interferene Γ int (Γ int + ) arises from the destrutive (onstrutive) interferene between the u (s) quark produed in the -quark deay and the spetator u (s) quark in the harmed baryon B. Notie that the onstrutive Pauli interferene is unique to the harmed baryon setor as it does not our in the bottom baryon setor. From the quark ontent of the harmed baryons (see Table II), it is lear that at the Cabibbo-allowed level, the destrutive interferene ours in Λ + and Ξ + deays, while Ξ +, Ξ 0 and Ω 0 an have Γ int +. Sine Ω 0 ontains two s quarks, it is natural to expet that Γ int + (Ω 0 ) Γ int + (Ξ ). W -exhange ours only for Ξ 0 and Λ + at the same Cabibboallowed level. In the heavy quark expansion approah, the above-mentioned spetator effets an be desribed in terms of the matrix elements of loal four-quark operators. Within this QCD-based heavy quark expansion approah, some phenomenologial assumptions an be turned into some oherent and quantitative statements and nonperturbative effets an be systematially studied. To begin with, we write down the general expressions for the inlusive deay widths of harmed hadrons. Under the heavy quark expansion, the inlusive nonleptoni deay rate of a harmed baryon B is given by [4, 43] { ( Γ NL (B ) = G F m5 19π 3 N 1 V CKM ) [ 1 I 0 (x, 0, 0) B B m B N 1 ] m I 1 (x, 0, 0) B σ G B 4 m 1 N I (x, 0, 0) B σ G B + 1 ( ) 1 B L spe B + O m B m 4, (5.4) } 1

13 d s u d s u s d d u u u s FIG. : Contributions to nonleptoni deay rates of harmed baryons from four-quark operators: (a) -quark deay, (b) W -exhange, () destrutive Pauli interferene and (d) onstrutive interferene. where σ G = σ µν G µν, x = (m s /m ), N is the number of olors, 1, are Wilson oeffiient funtions, N = 3 is the number of olor and V CKM takes are of the relevant CKM matrix elements. In the above equation, I 0,1, are phase-spae fators I 0 (x, 0, 0) = (1 x )(1 8x + x ) 1x ln x, I 1 (x, 0, 0) = 1 ( x d dx )I 0(x, 0, 0) = (1 x) 4, I (x, 0, 0) = (1 x) 3, (5.5) for su d transition. In heavy quark effetive theory, the two-body matrix element B B in Eq. (5.4) an be reast to with B B m B = 1 K H m + G H m, (5.6) K H 1 m B B (id ) B = λ 1, G H 1 m B B 1 σ G B = d H λ, (5.7) where d H = 0 for the antitriplet baryon and d H = 4 for the spin- 1 sextet baryon. It should be stressed that the expression (5.6) is model independent and it ontains nonperturbative kineti and hromomagneti effets whih are usually absent in the quark model alulations. The nonperturbative HQET parameters λ 1 and λ are independent of the heavy quark mass. Numerially, we shall use λ baryon 1 = (0.4 ± 0.) GeV [47] and λ baryon = GeV for harmed baryons [48]. Spetator effets in inlusive deays of harmed hadrons are desribed by the dimension-six fourquark operators L spe in Eq. (5.4) at order 1/m 3. Its omplete expression an be found in, for example, Eq. (.4) of [48]. For inlusive semileptoni deays, there is an additional spetator effet in harmed-baryon semileptoni deay originating from the Pauli interferene of the s quark for harmed baryons Ξ and Ω [49]. The general expression of the inlusive semileptoni widths is given by Γ SL (B ) = G F m5 19π 3 V η(x, x l, 0) CKM [I 0 (x, 0, 0) B B 1 ] m B m I 1 (x, 0, 0) B σ G B G F m 6π V s 1 (1 x) [ (1 + x ] m B )( s)( s) (1 + x) (1 γ 5)s s(1 + γ 5 ), (5.8) 13

14 where η(x, x l, 0) with x l = (m l /m Q ) is the QCD radiative orretion to the semileptoni deay rate and its general analyti expression is given in [50]. Sine both nonleptoni and semileptoni deay widths sale with the fifth power of the harmed quark mass, it is very important to fix the value of m. It is found that the experimental values for D + and D 0 semileptoni widths [3] an be fitted by the quark pole mass m = 1.6 GeV. Taking m s = 170 MeV, we obtain the harmed-baryon semileptoni deay rates Γ(Λ Xe ν) = Γ(Ξ Xe ν) = GeV, Γ(Ω Xe ν) = GeV. (5.9) The predition (5.9) for the Λ baryon is in good agreement with experiment [3] Γ(Λ Xe ν) expt = (1.480 ± 0.559) GeV. (5.10) We shall see beolw that the Pauli interferene effet in the semileptoni deays of Ξ and Ω an be very signifiant, in partiular for the latter. The baryon matrix element of the four-quark operator B ( q 1 )( q q 3 ) B with ( q 1 q ) = q 1 γ µ (1 γ 5 )q is ustomarily evaluated using the quark model. In the non-relativisti quark model (for early related studies, see [38, 39]), the matrix element is governed by the harmed baryon wave funtion at origin, ψq B (0), whih an be related to the harmed meson wave funtion ψq(0) D. For example, the hyperfine splittings between Σ and Σ, and between D and D separately yield [51] ψq Λ (0) = ψq Σ (0) = 4 m Σ m Σ ψ 3 m D m q(0) D. (5.11) D This relation is supposed to be robust as ψ q (0) determined in this manner does not depend on the strong oupling α s and the light quark mass m q diretly. Defining we have ψ B q (0) = r B ψ D q(0), (5.1) r Λ = 4 3 m Σ m Σ, r Ξ = 4 m Ξ m Ξ, r Ω = 4 m Ω m Ω. (5.13) m D m D 3 m D m D 3 m D m D In terms of the parameter r B ψ D q(0) we have [48] Γ ann (Λ ) = G F m π Γ int (Λ ) = G F m r Λ (1 x) ( η( 1 + ) 1 ) ψ D (0), ( ) 4π r Λ (1 x) (1 + x) η 1 1 N ψ D (0), Γ ann (Ξ )/r Ξ = Γ ann (Λ )/r Λ, Γ int (Ξ + )/r Ξ = Γ int Γ int + (Ξ ) = G F m 4π r Ξ (1 x )(1 + x) Γ int + (Ω ) = G F m 6π r Ω (1 x )(5 + x) Γ ann (Ω ) = 6 G F m π (Λ )/r Λ, ( ) η 1 N 1 ψ D (0), ( ) η 1 N 1 ψ D (0), ) r Ω (1 x ) (η( 1 + ) 1 ψ D (0), Γ int (Ξ Xe ν) = G F m 4π r Ξ (1 x )(1 + x) ψ D (0), Γ int (Ω Xe ν) = G F m 6π r Ω (1 x )(5 + x) ψ D (0), (5.14) 14

15 where the parameter η is introdued via B ( )( qq) B = η B ( q)( q) B, (5.15) so that η = 1 in the valene quark approximation. In the zero light quark mass limit (x = 0) and in the valene quark approximation, the reader an hek that results of (5.14) are in agreement with those obtained in [38, 39, 5] exept the Cabibbo-suppressed W -exhange ontribution to Ω 0, Γ ann (Ω ). We have a oeffiient of 6 arising from the matrix element Ω ( s)( s) Ω = 6 ψs Ω (0) (m Ω ) [48], while the oeffiient is laimed to be 10 3 in [5]. Negleting the small differene between r Λ, r Ξ and r Ω and setting x = 0, the inlusive nonleptoni rates of harmed baryons in the valene quark approximation have the expressions: Γ NL (Λ + ) = Γ de (Λ + ) + os C Γ ann + Γ int Γ NL (Ξ + ) = Γ de (Ξ + ) + sin C Γ ann + Γ int Γ NL (Ξ 0 ) = Γ de (Ξ 0 ) + Γ ann + Γ int + Γ int +, + sin C Γ int +, + os C Γ int +, Γ NL (Ω 0 ) = Γ de (Ω 0 ) + 6 sin C Γ ann os C Γ int +, (5.16) with θ C being the Cabibbo angle. Assuming the D meson wavefuntion at the origin squared ψ q(0) D being given by 1 1 f D m D, we obtain ψ Λ (0) = GeV 3 for f D = 0 MeV. 4 To proeed to the numerial alulations, we use the Wilson oeffiients 1 (µ) = 1.35 and (µ) = 0.64 evaluated at the sale µ = 1.5 GeV. Sine η = 1 in the valene-quark approximation and sine the wavefuntion squared ratio r is evaluated using the quark model, it is reasonable to assume that the NQM and the valenequark approximation are most reliable when the baryon matrix elements are evaluated at a typial hadroni sale µ had. As shown in [54], the parameters η and r renormalized at two different sales are related via the renormalization group equation, from whih we obtain η(µ) 0.74η(µ had ) 0.74 and r(µ) 1.36 r(µ had ) [48]. The results of alulations are summarized in Table V. It is lear that the lifetime pattern τ(ξ + ) > τ(λ + ) > τ(ξ 0 ) > τ(ω 0 ) (5.17) is in aordane with experiment. This lifetime hierarhy is qualitatively understandable. The Ξ + baryon is longest-lived among harmed baryons beause of the smallness of W -exhange and partial anellation between onstrutive and destrutive Pauli interferenes, while Ω is shortestlived due to the presene of two s quarks in the Ω that renders the ontribution of Γ int + largely enhaned. From Eq. (5.14) we also see that Γ int + is always positive, Γ int is negative and that the onstrutive interferene is larger than the magnitude of the destrutive one. This explains why τ(ξ + ) > τ(λ + ). It is also lear from Table V that, although the qualitative feature of the lifetime pattern is omprehensive, the quantitative estimates of harmed baryon lifetimes and their ratios are still rather poor. In [5], a muh larger harmed baryon wave funtion at origin is employed. This is based on the argument originally advoated in [37]. The physial harmed meson deay onstant f D is related to the asymptoti stati value F D via ( f D = F D 1 µ + O( 1 ) m m ). (5.18) 4 The reent CLEO measurement of D + µ + ν yields f D + =.6 ± MeV [53]. 15

16 TABLE V: Various ontributions to the deay rates (in units of 10 1 GeV) of harmed baryons. The harmed meson wavefuntion at the origin squared ψ D (0) is taken to be 1 1 f D m D. Experimental values are taken from [3]. Γ de Γ ann Γ int Γ int + Γ SL Γ tot τ(10 13 s) τ expt (10 13 s) Λ ± 0.06 Ξ ± 0.6 Ξ Ω ± 0.1 It was argued in [37] that one should not use the physial value of f D when relating ψ B (0) to ψ D (0) for reason of onsisteny sine the widths have been alulated through order 1/m 3 only. Hene, the part of f D whih is not suppressed by 1/m should not be taken into aount. However, if we use F D f D for the wave funtion ψ D (0), we find that the predited lifetimes of harmed baryons beome too short ompared to experiment exept Ω 0. By ontrast, using ψ Λ (0) =.6 10 GeV 3 and the so-alled hybrid renormalization, lifetimes τ(λ + ) =.39, τ(ξ + ) =.51, τ(ξ 0 ) = 0.96 and τ(ω 0 ) = 0.61 in units of s are obtained in [5]. They are in better agreement with the data exept Ξ +. The predited ratio τ(ξ + )/τ(λ + ) = 1.05 is too small ompared to the experimental value of.1 ± By inspeting Eq. (5.16), it seems to be very diffiult to enhane the ratio by a fator of. In short, when the lifetimes of harmed baryons are analyzed within the framework of the heavy quark expansion, the qualitative feature of the lifetime pattern is understandable, but a quantitative desription of harmed baryon lifetimes is still lak. This may be asribed to the following possibilities: 1. Unlike the semileptoni deays, the heavy quark expansion in inlusive nonleptoni deays annot be justified by analyti ontinuation into the omplex plane and loal duality has to be assumed in order to apply the OPE diretly in the physial region. The may suggest a signifiant violation of quark-hadron loal duality in the harm setor.. Sine the quark is not heavy enough, it asts doubts on the validity of heavy quark expansion for inlusive harm deays. This point an be illustrated by the following example. It is well known that the observed lifetime differene between the D + and D 0 is asribed to the destrutive interferene in D + deays and/or the onstrutive W -exhange ontribution to D 0 deays. However, there is a serious problem with the evaluation of the destrutive Pauli interferene Γ int (D + ) in D +. A diret alulation analogous to Γ int (B ) in the harmed baryon setor indiates that Γ int (D + ) overomes the quark deay rate so that the resulting nonleptoni deay width of D + beomes negative [37, 55]. This ertainly does not make sense. This example learly indiates that the 1/m expansion in harm deay is not well onvergent and sensible, to say the least. It is not lear if the situation is improved even after higher dimension terms are inluded. 3. To overome the aforementioned diffiulty with Γ int (D + ), it has been onjetured in [37] that higher-dimension orretions amount to replaing m by m D in the expansion parameter 16

17 fd m D/m 3, so that it beomes fd /m D. As a onsequene, the destrutive Pauli interferene will be redued by a fator of (m /m D ) 3. By the same token, the Pauli interferene in harmed baryon deay may also be subjet to the same effet. Another way of alleviating the problem is to realize that the usual loal four-quark operators are derived in the heavy quark limit so that the effet of spetator light quarks an be negleted. Sine the harmed quark is not heavy enough, it is very important, as stressed by Chernyak [55], to take into aount the nonzero momentum of spetator quarks in harm deay. In the framework of heavy quark expansion, this spetator effet an be regarded as higher order 1/m orretions. 4. One of the major theoretial unertainties omes from the evaluation of the four-quark matrix elements. One an hope that lattie QCD will provide a better handle on those quantities. VI. HADRONIC WEAK DECAYS Contrary to the signifiant progress made over the last 0 years or so in the studies of the heavy meson weak deay, advanement in the arena of heavy baryons, both theoretial and experimental, has been relatively slow. This is partly due to the smaller baryon prodution ross setion and the shorter lifetimes of heavy baryons. From the theoretial point of view, baryons being made out of three quarks, in ontrast to two quarks for mesons, bring along several essential ompliations. First of all, the fatorization approximation that the hadroni matrix element is fatorized into the produt of two matrix elements of single urrents and that the nonfatorizable term suh as the W -exhange ontribution is negligible relative to the fatorizable one is known empirially to be working reasonably well for desribing the nonleptoni weak deays of heavy mesons. However, this approximation is a priori not diretly appliable to the harmed baryon ase as W -exhange there, manifested as pole diagrams, is no longer subjet to heliity and olor suppression. This is different from the naive olor suppression of internal W -emission. It is known in the heavy meson ase that nonfatorizable ontributions will render the olor suppression of internal W -emission ineffetive. However, the W -exhange in baryon deays is not subjet to olor suppression even in the absene of nonfatorizable terms. A simple way to see this is to onsider the large-n limit. Although the W -exhange diagram is down by a fator of 1/N relative to the external W -emission one, it is ompensated by the fat that the baryon ontains N quarks in the limit of large N, thus allowing N different possibilities for W exhange between heavy and light quarks [56]. That is, the pole ontribution an be as important as the fatorizable one. The experimental measurement of the deay modes Λ + Σ 0 π +, Σ + π 0 and Λ + Ξ 0 K +, whih do not reeive any fatorizable ontributions, indiates that W -exhange indeed plays an essential role in harmed baryon deays. Seond, there are more possibilities in drawing the quark daigram amplitudes as depited in Fig. 3; in general there exist two distint internal W -emissions and several different W -exhange diagrams whih will be disussed in more detail shortly. Historially, the two-body nonleptoni weak deays of harmed baryons were first studied by utilizing the same tehnique of urrent algebra as in the ase of hyperon deays [57]. However, the use of the soft-meson theorem makes sense only if the emitted meson is of the pseudosalar type and its momentum is soft enough. Obviously, the pseudosalar-meson final state in harmed bayon deay is far from being soft. Therefore, it is not appropriate to make the soft meson limit. It is no longer justified to apply urrent algebra to heavy-baryon weak deays, espeially for s-wave amplitudes. Thus one has to go bak to the original pole model, whih is nevertheless redued to 17

18 urrent algebra in the soft pseudosalar-meson limit, to deal with nonfatorizable ontributions. The merit of the pole model is obvious: Its use is very general and is not limited to the soft meson limit and to the pseudosalar-meson final state. Of ourse, the prie we have to pay is that it requires the knowledge of the negative-parity baryon poles for the parity-violating transition. This also explains why the theoretial study of nonleptoni deays of heavy baryons is muh more diffiult than the hyperon and heavy meson deays. The nonfatorizable pole ontributions to hadroni weak deays of harmed baryons have been studied in the literature [58 60]. In general, nonfatorizable s- and p-wave amplitudes for P (V ) deays (P : pseudosalar meson, V : vetor meson), for example, are dominated by 1 low-lying baryon resonanes and 1 + ground-state baryon poles, respetively. However, the estimation of pole amplitudes is a diffiult and nontrivial task sine it involves weak baryon matrix + and 1 baryon states. This is the ase in partiular elements and strong oupling onstants of 1 for s-wave terms as we know very little about the 1 states. As a onsequene, the evaluation of pole diagrams is far more unertain than the fatorizable terms. In short, W -exhange plays a dramati role in the harmed baryon ase and it even dominates over the spetator ontribution in hadroni deays of Λ + and Ξ 0 [61]. Sine the light quarks of the harmed baryon an undergo weak transitions, one an also have harm-flavor-onserving weak deays, e.g., Ξ Λ π and Ω Ξ π, where the harm quark behaves as a spetator. This speial lass of weak deays usually an be alulated more reliably than the onventional harmed baryon weak deays. A. Quark-diagram sheme Besides dynamial model alulations, it is useful to study the nonleptoni weak deays in a way whih is as model independent as possible. The two-body nonleptoni deays of harmed baryons have been analyzed in terms of SU(3)-irreduible-representation amplitudes [6, 63]. However, the quark-diagram sheme (i.e., analyzing the deays in terms of quark-diagram amplitudes) has the advantage that it is more intuitive and easier for implementing model alulations. It has been suessfully applied to the hadroni weak deays of harmed and bottom mesons [64, 65]. It has provided a framework with whih we not only an do the least-model-dependent data analysis and give preditions but also make evaluations of theoretial model alulations. A general formulation of the quark-diagram sheme for the nonleptoni weak deays of harmed baryons has been given in [66] (see also [67]). The general quark diagrams shown in Fig. 3 are: the external W -emission tree diagram T, internal W -emission diagrams C and C, W -exhange diagrams E 1, E and E (see Fig. of [66] for notation and for details). There are also penguintype quark diagrams whih are presumably negligible in harm deays due to GIM anellation. The quark diagram amplitudes T, C, C et. in eah type of hadroni deays are in general not the same. For otet baryons in the final state, eah of the W -exhange amplitudes has two more independent types: the symmetri and the antisymmetri, for example, E 1A, E A, E S, E A and E S [66]. The antiquark produed from the harmed quark deay q 1q q 3 in diagram C an ombine with q 1 or q to form an outgoing meson. Consequently, diagram C ontains fatorizable ontributions but C does not. It should be stressed that all quark graphs used in this approah are topologial with all the strong interations inluded, i.e. gluon lines are inluded in all possible ways. Hene, they are not Feynman graphs. Moreover, final-state interations are also lassified 18

19 T C C E 1 E E FIG. 3: Quark diagrams for harmed baryon deays in the same manner. A good example is the reation D 0 K 0 φ, whih an be produed via final-state resattering even in the absene of the W -exhange diagram. Then it was shown in [65] that this resattering diagram belongs to the generi W -exhange topology. Sine the two spetator light quarks in the heavy baryon are antisymmetrized in the antitriplet harmed baryon B ( 3) and the wave funtion of the deuplet baryon B(10) is totally symmetri, it is lear that fatorizable amplitudes T and C annot ontribute to the deays of type B ( 3) B(10) + M(8); it reeives ontributions only from the W -exhange and penguin-type diagrams (see Fig. 1 of [66]). Examples are Λ + ++ K, Σ + ρ 0, Σ + η, Ξ 0 K + and Ξ 0 Σ + K0. They an only proeed via W -exhange. Hene, the experimental observation of them implies that the W -exhange mehanism plays a signifiant role in harmed baryon deays. The quark diagram amplitudes for all two-body deays of (Cabibbo-allowed, singly suppressed and doubly suppressed) Λ +, Ξ +,0 and Ω 0 are listed in [66]. In the SU(3) limit, there exist many relations among various harmed baryon deay amplitudes, see [66] for detail. For harmed baryon deays, there are only a few deay modes whih proeed through fatorizable external or internal W -emission diagram, namely, Cabibbo-allowed Ω 0 Ω π + (ρ + ), Ξ 0 K0 ( K 0 ) and Cabibbo-suppressed Λ + pφ. B. Dynamial model alulation To proeed we first onsider the Cabibbo-allowed deays B ( 1 + ) B( 1 + ) + P (V ). The general amplitudes are M[B i (1/ + ) B f (1/ + ) + P ] = iū f (p f )(A + Bγ 5 )u i (p i ), (6.1) M[B i (1/ + ) B f (1/ + ) + V ] = ū f (p f )ε µ [A 1 γ µ γ 5 + A (p f ) µ γ 5 + B 1 γ µ + B (p f ) µ ]u i (p i ), where ε µ is the polarization vetor of the vetor meson, A, (B, B 1, B ) and A are s-wave, p-wave and d-wave amplitudes, respetively, and A 1 onsists of both s-wave and d-wave ones. The QCDorreted weak Hamiltonian responsible for Cabibbo-allowed hadroni deays of harmed baryons reads H W = G F V s V ud( 1 O 1 + O ), (6.) 19