Flavor SU(3) predictions for charmed baryon and B-meson decays Citation Savage, Martin J. (1990) Flavor SU(3) predictions for charmed baryon and B-meson decays. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/xg6c-7v36. https://resolver.caltech.edu/CaltechTHESIS:07292014-081524505 Abstract The predictions of the SU(3) flavor symmetry of the strong interactions for the weak decay of charmed baryons and B-mesons are detailed. It is hoped that comparison between these predictions and experiment will shed some light on the underlying dynamics involved in these weak decays. Although only a few decay modes of the charmed baryons and B-mesons have been studied experimentally it is hoped that the next generation of B-factories and even Z-decays at LEP will provide enough events to test these predictions. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: Physics Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Physics Thesis Availability: Public (worldwide access) Research Advisor(s): Wise, Mark B. Group: Caltech Theory Thesis Committee: Unknown, Unknown Defense Date: 15 May 1990 Record Number: CaltechTHESIS:07292014-081524505 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:07292014-081524505 DOI: 10.7907/xg6c-7v36 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 8617 Collection: CaltechTHESIS Deposited By: INVALID USER Deposited On: 29 Jul 2014 15:51 Last Modified: 09 Nov 2022 19:20 Thesis Files Preview PDF - Final Version See Usage Policy. 19MB Repository Staff Only: item control page

FLAVOR SU(3) PREDICTIONS FOR CHARMED BARYON AND B-MESON DECAYS Thesis by Martin J. Savage In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy CA LIFORN IA INSTITUTE OF TECHNOLOGY Pasadena, California 1990 (Submitted May 15, 1990) ACKOWLEDGMENTS Let me begin by thanking my boss, Mark Wise, who taught me how to be a theorist. It was a pleasure to work with such a good physicist and I hope that some of his skills have been beaten into me. I would also like to thank John Preskill, David Politzer and Steve Frautschi along with Dave Hitlin for enlightening me on various aspects of physics. Although this is a theoretical thesis, I will take this opportunity to thank Brad Filippone, Ralph Kavanagh, Charles Barnes, Tom Tombrello and Leon Mitchell for encouragement and know-how during my experimental days. Life at Caltech was not all work. I \vould like to thank all the friends I have made during my stay here, a large fraction of whom played on the various soccer teams both here at Caltech and elsewhere. They managed to drag me out to the pub at all hours of the day and night and return me home half-paralysed. The hell of living in Pasadena was buffered substantially by them and I cannot begin to thank them enough. The -local pubs deserve a mention for being the three most interesting places in Pasadena and prO\·iding a good social environment to meet other foreigners. i\ly fellow grad studs are to be thanked for tolerating me during the five years and not complaining too much. Roxanne Springer deserves a special thanks as some of the work in this thesis was done in collaboration with her. II ABSTRACT The predictions of the SU(3) flavor symmetry of the strong interactions for the weak decay of charmed baryons and B-mesons are detailed. It is hoped that comparison between these predictions and experiment will shed some light on the underlying dynamics involved in these weak decays. Although only a few decay modes of the charmed baryons and B-mesons have been studied experimentally it is hoped that the next generation of B-factories and even Z-decays at LEP will provide enough events to test these predictions. Ill TABLE OF CONTENTS Page Topic Acknowledgments • Abstract . . . . • 111 Table of Contents IV Introduction . . 1 Chapter 1: Weak Interactions 3 1.1 The Standard Model of Electroweak Interactions . 3 1.2 Flavor Symmetry 9 1.3 SU(3)t Predictions for Semileptonic Hyperon Decay 10 1.4 SU(3)J Breaking 14 R eferences for Chapter 1 16 Chapter 2 : D-Meson Decays 17 2.1 Charm Changing ·weak Operators 17 2.2 Two-Body Nonleptonic D-Meson Decays 20 2.3 Semileptonic D-Meson Decays . . . . . 26 2.4 Large Nc Limit and Factorization of Amplitudes 27 2.5 Concluding Remarks on Chapter 2 29 R eferences for Chapter 2 . . . . . . 32 Chapter 3: Charmed Baryon Decays 36 3.1 SU(3)t Representation of the Charmed Baryons 36 3.2 Nonleptonic Decay of Charmed Baryons to Two-Body Final States 40 3.3 Nonleptonic Decay of Charmed Baryons to Three-Body Final States 48 3.4 Semileptonic Decay of Charmed Baryons to Three-Body Final States 55 3.5 Semileptonic Decay of Charmed Baryons to Four-Body Final States 56 3.6 Concluding Remarks on Chapter 3 58 References for Chapter 3 . . . . . . 60 Chapter 4: B-Meson Decays to Charmed Baryons . 84 4.1 Two-Body B Decays to Charmed Baryons 84 4.2 Two-Body B Decays to Charmed Anti-Baryons 87 IV 11 4.3 Concluding Remarks on Chapter 4 90 References for Chapter 4 . . . . . . 92 Chapter 5: Non-Baryonic B-Meson Decays 103 5.1 Two-Body Decay Non-Baryonic B-Meson Decays 103 5.2 Three-Body Non-Baryonic B-Meson Decays 112 5.3 Concluding Remarks on Chapter 5 118 References for Chapter 5 121 Addendum . . . . . . 140 \' TABLES Table Page 1.1 Matter Fields in the Standard Model . 2 1.2 Hyperon Decays 10 2.1D---+MM 33 2.2 D---+ VM. 34 3.1 T---+ EM (allowed) 61 3.2 T---+ EM (suppressed) 62 3.3 T ---+ Elvf (doubly-suppressed) 63 3.4 T---+ h* M (allowed) . . . . 64 3.5 T ---+ h* M (suppressed) . . . 65 3.6 T---+ h*AJ (doubly-suppressed) 66 3. 7 S ---+ B 1\f 67 3.8 S ---+ h* AI 68 3.9 T---+ h* ~H 1\f (L-even, allowed) . 69 3.10 T---+ h* !11!1! (1-odd, allowed) 71 3.11 T---+ h*!I1!1I (1-ewn, suppressed) 73 3.12 T---+ h* 1\f !11 (1-odd, suppressed) 76 3.13 T---+ BJ1l+vl (allowed) . 79 3.14 T---+ BAJL+vl (suppressed) so 3.15 T---+ h* J\[[+vl (allowed) . 81 3.16 T---+ h*AJL+vl (suppressed) 82 3.17 S---+ B J'Jl+vl 83 3.18 S---+ h*AJL+vl 83 4.1 B---+ Th 93 4.2 B---+ Sh (allowed) 94 4.3 B---+ Sh (suppressed) 95 4.4 E ---+ Th* . . . . . 96 4.5 B---+ Sh* (allowed) 97 4.6 B---+ Sh* (suppressed) 98 VI 4.7 B--+ Th 99 4.8 B--+ Sh 100 4.9 B--+ Th* 101 4.10 B--+ Sh* 92 5.1 B--+ DJ\!I (allowed) 122 5.2 B--+ DM (suppressed) 122 5.3 B--+ DD 123 5.4 B--+ J /1/Jlvf . 124 5.5 B--+ A1M,~s = 0 125 5.6 B --+ J\!I M, 6.s # 0 126 5.7 B--+ DM 127 5.8 B --+ J /1/J!Iflvf (L-even) . 128 5.9 B --+ J /1/J!If !II (L-odd) 129 5.10 B--+ DA-IA1 (L-even, allowed) 130 5.11 B--+ D!II !II (L-odd, allowed) 131 5.12 B--+ D!IIJ;J (L-even, suppressed) 132 5.13 B--+ D!II !11 (1-odd, suppressed) 133 5.14 B--+ DD1U (allowed) 13-1 5.15 B--+ DD!I1 (suppressed) 136 VII FIGURE CAPTIONS Figure 2.1a Leading diagram with no gluonic corrections to the decay D---+ 1r1r in the large Nc limit . . . . . . . . . . . . . . . . . . . 35 Figure 2.1b The gluonic exchange diagram shown contributes to the same order in 1/Nc as the diagram in Figure 2.1a . . . . . . . . . . . 35 Figure 2.1c The gluonic exchange diagram shown is suppressed by 1/NJ over the diagrams in Figures 2.1a and 2.1b . . . . . . . . . . Figure 5.1 Penguin-type diagram contributing to B---+ AIM decays 35 138 Figure 5.2 Quark line diagrams illustrating how the operators (be)( cd) and ( bu)( ud) (denoted by shaded squares) contribute to the decay B2 ---+ DtIf 0 . . . . . . . . • . . . . . . . . . 139 1r Yilt INTRODUCTION In the near future it is likely that branching ratios for many of the exclusive decays of charmed baryon and B-meson will be measured. The decay of B-mesons provides a unique laboratory for studying weak interactions in so far as they can decay not only to mesons but more interestingly to charmed and uncharmed baryons. The decays of D-mesons have been studied extensively and continue to be an active area of research. For kinematic reasons the D-mesons can decay only to mesonic final states and not to states containing baryons. Unlike the B-meson decays, the decay of D-mesons through radiative loop-induced weak operators is highly suppressed. Consequently, the study of B-meson decays provides a new way to isolate the effects of these one-loop processes. Analysis of the available experimental data on D-meson decays indicates that the predictions of the SU(3) f flavor symmetry are not well satisfied in nature. The deviations from the predictions are not well understood and are attributed to final state interactions (FSI), but could conceivably be due to an intrinsic breakdown of SU(3)J· The data indicates that there are significant final state interactions but is not precise enough to exclude the possibility of intrinsic violations of SU(3)J. Therefore it would be useful to look for deviations in the decay of charmed baryons from the predictions of SU(3)J where it is hoped that the FSI will be smaller than in the mesonic sector. The usc of SU(3) flavour symmetry in strange decays in the early sixties proved to be an immensely pO\verful tool in our understanding of weak interactions and weak currents. The octet dominance (6.1 = ! ) rule in such decays was not understood until the mid seventies when it was realized that short distance strong interactions gave rise to an enhancement of the octet over the 27 component of the Hamiltonian by a factor of "' 5. This is still smaller than the observed enhancement of "' 20. Below scale of 1 GeV the strong interactions are not well understood and it is possible that the enhancement from the strong interactions could in fact be larger than this factor of"' 5. Also, one-loop penguin diagrams give rise to a Hamiltonian that transforms 1 as a 8 with !11 = t and it IS conceivable that these diagrams dominate the weak decays below 1 Ge V. In this thesis the predictions of SU(3)t are presented for the decays of charmed baryons and B-mesons. These relations will be useful in determining the underlying mechanisms responsible for the weak decays. Analagous to the strange decays, short distance strong interactions give rise to an enhancement of the sextet component of the Hamiltonian over the 15 in charm decays. The SU(3) 1 relations between the decay modes of charmed baryons may enable this enhancement to be observed. There are similar observations to be made in the B-meson decays. Penguin diagrams lead to !11 = 0 selection rule for some of the B-meson decay modes. By comparison with experimental data, the relative importance of such diagrams compared to the tree level Hamiltonian can be ascertained. The results in Chapters 4 and 5 are published (with l\tlark B. Wise) in Nuc. Phys. B326, 15 (1989) and Phys. Rev. D39 , 3346 (1989) respectively. The results of Chapter 3 have been submitted (with Roxanne P. Springer) to Physical Review D. 2 CHAPTER 1. WEAK INTERACTIONS The subject of weak interactions covers a vast area of research from the elusive Higgs boson to determination of the Helium abundance of the universe. In this chapter I will not attempt to address even a small fraction of this work, but I hope to introduce the areas relevent to the decay of hadrons containing heavy quarks. The first section deals with the standard model of electroweak interactions which became popular in the early to mid seventies after it was shown to be renormalizable. This theory is not fundamental in the sense that it contains many free parameters that must be determined experimentally but there is a hope that the ultimate theory of everything will predict these numbers. On the other hand there is a remarkable agreement between the experimental observations and theoretical predictions which, with the possible exception of the Higgs sector and neutrino masses, leads one to believe that this gives a complete description of electroweak interactions below ,...., 100 GeV. The second section introduces the concept of a flavor symmetry when some of the quarks have masses much smaller or heavier than the scale of strong interactions. In section 1.3 the methodology for constructing of the predictions of a flavor symmetry is presented by explicit calculation of the rates for semileptonic hyperon decay in terms of two reduced matrix elements. These predictions are then compared with the large body of experimental information on the subject. The agreement between theory and experiment is surprisingly good, indicating that flavor symmetry may be a useful tool in understanding weak decays of other hadrons. In section 1.4 the method by which explicit SU(3)J breaking is implemented is discussed. The breaking is a. result of the inequality of masses of the three light quarks. It is hoped that the relevent expansion parameter for the breaking is m 5 /1 GeV. The semileptonic decay of the hyperons is presented as an explicit example. 1.1 The Standard Model of Electroweak Interactions The standard model [1.1] for strong, weak and electrom.agnetic interactions is based on the gauge group SL'(3)c 0 SU(2)L ® U(1)y. The minimal particle content consists of the gauge bosons and the matter field s shown in Table 1.1. 3 Table 1.1: Matter fields in the Standard Model SU(3)c SU(2)L U(1)y Spin Q'L- (~t) 3 2 1/3 1/2 u'R 3 1 4/3 1/2 di R 3 1 -2/3 1/2 LiL = (v~) ei_ 1 2 -1 1/2 e'R 1 1 -2 1/2 ¢ = ( ~:) 1 2 1 0 Field The superscript i on the matter fields is a generation index and takes the values i E {1,2,3}. The subscripts Land R denote left handed and right handed respec- tively. Anomaly cancellation in the fermionic triangle diagrams uniquely constrains the hypercharge assignment Y of each matter field . The complex scalar field ¢ is necessary to break the SU(3)c 0 SU(2)L 0 U(1)y symmetry to SU(3)c 0 U(1)Q which describes nature at scales below "' 100 GeV. Electromagnetism is realized in the unbroken U(1)Q symmetry and the SU(3)c symmetry is that of the strong interactions. The Lagrange density for the system is composed of five terms ( 1.1) These terms correspond to the contributions from the fermions, Higgs, gauge bosons, Yukawa interactions and the gauge fixing components respecti\·ely. Firstly, we will examine the Higgs sector which is responsible for the symmetry breaking [1.2]. The Lagrange density is (1.2) where D 1, is the covariant derivative defined as (1.3) The lF1~(x) and B 1,(x) are the gauge fields associated with the SU(2)L and U(1)y 4 symmetries respectively and Ta are the Pauli spin matrices. The potential V(4>) can be written as (1.4) which has a global minimum at 4>t4> = v 2. The global SU(2)L ® U(l)y invariance of the lagrangian allows us to rotate an arbitrary vacuum expectation value (vev) of the field (4>) into the form (4>) = (~). Upon the field redefinition 4>+---+ 4>+ and 4>0 ---+ v + 4>0 Eq. (1.4) becomes (1.5) Notice that there are no terms quadratic in 4>+ or Im¢> 0 indicating that these are massless modes. These are the Goldstone bosons associated with the breaking of the global SU(2)L ® U(l)y to U(l)Q· However, a local SU(2)L ® U(l)y gauge transformation can be found that removes these fields at the expense of giving mass to three of the four gauge bosons associated with the SU(2)L ® U(l)y symmetry. The field Re</> 0 cannot be gauged away and is found to have mass of v'2"Xv, this is the Higgs particle. The kinetic term for the scalar field in Eq. (1.2) gives rise to the mass term for the gauge bosons. Evaluating this at the vev it can be shown that (1.6) If we define zo = I JL ? 1 v g~ + g ~ then we can rewrite the mass terms as 5 f ") ( g TV3 IB ) JL - g JL (1.7a, b) Notice that the combination of fields orthogonal to that of the Z 0 (1.9) remains massless, this is identified with the photon field of electromagnetism. The generator associated with this field is (1.10) This is the only combination of generators that the vacuum doesn't transform under. The Weinberg angle Ow is introduced as a free parameter that relates the U(1)y and SU(2)L coupling constants viz g' = g tan Ow The kinetic term of the scalar field gives rise to interaction terms between the Goldstone bosons and the longitudinal component of the gauge bosons. These interactions can be removed continuously by addition of the t'Ilooft gauge fixing term [1.3] L:cF = ;, L (ottlv:- i~g([(<P}tTa<P- <PtTa(<P}l) 2 a (1.11) In the unitary gauge where ( = oo the Goldstone bosons have an infinite mass and therefore decouple from the theory. However, in Feynman gauge where ( = 1 they have the same mass as the corresponding gauge field. The contribution to the Lagrange density from the gauge fields .CcF is given by (1.12) The field tensors 1V;11 , Bttv and G~ 11 are given by (l.Ua) 6 (1.13b) and (1.13c) where t.abc is the totally antisymmetric tensor, fabc are the antisymmetric structure constants of SU(3) and G~ are the gluon fields associated with the strong interactions. The kinetic terms of the fermions in Eq. (1.1) gives rise to a Lagrange density of where the covariant derivative defined for the quark fields in Eq. (1.3) also has a strong interaction contribution and becomes (1 .15) where (Ta)p are the generators of SU(3). The color indices will be suppressed for the rest of this discussion. In terms of the gauge field mass eigenstates, the covariant derivative can be written where T± = )2-( r 1 ± ir 2 ). The terms discussed above do not give the fermions of the theory mass terms. To do this we need to introduce Yukawa couplings between the fermions and the complex scalar field. The most general set of renormalsable Yukawa couplings that can be written down is (1.17) where the unknown couplings gijd u, ,e gi\'e rise to mixing between the generations and (1.18) Mass terms for the fermions are generated when the scalar field develops a vev. 7 If Mu,d,e are the mass matrices for the "up-type" quarks, "down-type" quarks and charged leptons respectively then these are related to the Yukawa coupling constants by .. .. 1 1 M'u,d,e = g'u,d,e v (1.19) Assuming that there are only three generations of particles [1.4] and using the generation basis (u, c, t), (d, s, b) and (e, p, r) we can redefine the fermionic fields and diagonalize the mass matrices Mu,d,e· Let cfR ---+ v;1d~ , (1.20a, b, c, d) and similar transformations for the leptons. These unitary matrices diagonalize the mass matrices to (i\rfu ---+ V!RMu VuL) and (1.2la , b. c) Upon performing this transformation on the fermion kinetic term we see that the neutral currents remain diagonal by the unitarity of the transformations. However, the charged currents do not remain diagonal. In the mass eigenstate basis the charged current interaction becomes .C" = 2 ~(u, c,l)-y1'(1- -y5 )J( ( : ) w: +h. c. , (1.22) where f( is the Kobayashi-l\faskawa matrix [1..'5]. Since this matrix is not determined theoretically we can parametrize it in terms of four parameters 01, 02,03 and 5 as 8 follows C1 J( = VJL VdL = 81C2 -81C3 CIC2C3 - ( 8182 C! 82C3 + 8283ei.5 (1.23) '6 C283e' where 8k = sin ok and Ck = cos ok. The angles fh are chosen to lie in the first quadrant where their sines and cosines are positive. Experimental information on nuclear {3decay, semileptonic hyperon decays and semileptonic kaon decays [1.6] implies that 81 ~ 0.22 (1.24) The magnitude of the angles fh,3 are extracted from experimental information on the lifetime of the B-mcson and semileptonic B decays. This gives (1.25) 1.2 Flavor Symmetry Let us no\\· consider the strong interaction between quarks, suppressmg electroweak indices the Lagrange density is (1.26) For a system of n quarks of equal mass there would be an exact SU(n)J flavor symmetry. Further, if m quarks had masses much less than the QCD scale (the scale at which QCD becomes strong and the scale that determines the dynamics of the system) then the system would have an approximate SU(m) 1 flavor symmetry. This is the situation realized in nature [1.7]. The mass of the three lightest quarks are mu "' 5 MeV /c 2 , md "' 10 l\IeV fc 2 and m ,....., 125 MeV /c 2 while the QCD scale is 5 "" 1 GeV fc 2 . Since the difference between the mass of the up and down quarks is small ("' 5 MeV /c 2 ), as well as being much less than the QCD scale an SU(2)J symmetry is expected and indeed one is observed experimentally. This is the well known 9 isospin symmetry and it is observed to be very good. The mass of the strange quark, although much bigger than that of both the up and down quarks is still much smaller than the QCD scale, consequently one expects an approximate SU(3)J symmetry to exist. Since the dynamics of the system are determined by the QCD scale an SU(4)J flavor symmetry is not present since the charm quark mass of 1.5 GeV /c 2 is much greater than the QCD scale. In systems containing a heavy quark and light quarks there is a further SU(3)J symmetry involving the charm, bottom and top quarks [1.8). The masses of these quarks are all much greater than the QCD scale and as far as the QCD dynamics are concerned the heavy quark acts as a static color source [1.9). Consequently the dynamics is essentially independent of the mass of the heavy quark. The weak charged current, whose interaction Lagrange density is given in Eq. (1.22) in general has non-trivial transformations under the SU(3)J group. This becomes apparent by examining the strangeness changing process s --+ u l v- which transforms as a 3 ® 3 under SU(3)J· This group structure decomposes to 8 EB 1, and since the quark flavor is changed at this vertex this particular current transforms as an 8 under SU(3)J· \Ve can use the SU(3)J symmetry of the strong interactions to relate decays rates between reactions if the particles involved lie within the same SU(3)J multiplets. 1.3 SU(3)J Predictions for Semileptonic Hyperon Decay The method used in constructing SU(3)J predictions between decay rates is demonstrated by examining semileptonic hyperon decays first discussed by Cabibbo [1.10). The convention for SU(3)J representation will be that the triplets are denoted by column vector with upper indices and the antitriplets are row vectors with lower indices, q' = (:"sit ) and qi (u,d,s) = (1.27a, b) All other represe ntations can be constructed from the triplet and antitriplct . The 10 lowest lying baryons are elements of an octet and are represented by ( Eo I v'2 +_A oI v'6 hi= E i ~- E+ -Eo IV'i + Ao lv'6 (1.28) -=0 where the conjugate operator hj can be found by "barring" the elements and transposing. Below the weak scale, the operator responsible for the semileptonic decay of light quarks is given by (1.29) There are no QCD corrections to this operator from momentum scales between the weak scale and the QCD scale as the operator has the form of a conserved current as far as the color indices are concerned. This operator transforms as an octet under SU(3)J and is represented by a 3 x 3 matrix as (1.30) where we have used the small angle approximation for the mixing angles, c1 = c2 = C3 = 1. \.Ye are interested in e,·aluating matrix elements of the form (1.31) and by using the \Vigner-Eckart theorem can decompose this into a Clebsch-Gordan coefficient and an irreducible matrix element. The effective Hamiltonian Ileff is constructed from h~, ~ and Hj by forming all the possible tensor contractions of the ll three operators that transform as singlets under SU(3)J· There are only two distinct contractions and so the effective Hamiltonian becomes (1.32) where a and f3 are the reduced matrix elements. After doing the contractions the operator involving the Cabibbo-allowed processes can be written as (1.33) A similar expression can be found for the Cabibbo-suppressed decays. We interpret, for example, A as an operator that annihilates a A particle and A as an operator that creates a A particle. 'vVe can tabulate the rate for each process in terms of the two unknown red_uced matrix elements Table 1.2: Rates for the Semileptonic Decay of Hyperons Process Rate n-+ pe. -Ve ia12 tia + /31 2 !Ia- /312 !1-a+/31 2 1/31 2 sitl- a+ 2/312 sHi- ai 2 sil- ai2 si il2a- /31 2 sBi- /31 2 sil-/31 2 :E--+ Aoe-ve )'-'--' -+ :Eo e -Ve :Eo -+ :E+ e-ve :=:- -+ :=:oe-ve :=:--+ AOe-ve :=:- -+ :E0 e- ve :;::o -+ :E+ e-ve A 0 -+ pe-ve :Eo -+ pe-ve :E- -+ne-ve 12 In such decays there are at most six different Lorentz structures that contribute to the decay amplitude, which can be written as (1.34) where () is the Cabibbo angle, G F is Fermi's coupling constant and qJJ. is the four momentum transfer in the decay. The transversality of the vector current from eve (8J1. VJJ. - t qJ.I. VJJ.) means that q2 h(q 2 ) = 0 from which we conclude that h(q 2 ) = 0. The operator aJ.I.v/5 is odd under the G-parity operation (as is /5)· Therefore, since the strong interactions cannot induce G-parity odd operators, the form factor 92( q 2 ) is set to zero (for a discussion see ref[l.ll]). When this hadronic current is contracted with a leptonic current (e or f..L) the contraction involving 93 ( q2 )q'' picks up a factor of me,JJ.· This term is small compared to the scale of the hyperon system, and >Ye neglect it. The weak magnetism. term h( q 2 )attv qv is suppressed by the mass of the hyperon system and is consequently negligible. This leaves two possible lorentz structures contributing to the amplitude. They are the vector and axial vector operators. Therefore the transition amplitude can be written as . (1.35) In the limit as q2 - t 0 we let the form factors become Cv = h (0) and C,-t = 91 (0). Both of these form factors can be decomposed into reduced matrix elements and elebsch-Gordan coefficients. The vector charges are generators of the SU(3)J group and so transitions between octet ele ments 1.: and j are equal to the structure constants of SU(3) fijk · Further, since _E±.o and A0 are in different multiplets of the SU(2) isospin subgroup of SU(3) 13 the vector current must vanish for the transition E - t A. This is verified experimentally [1.6] where it is found that the ratio of vector to axialvector contributions to the decay amplitude is Cv/CA = 0.01 ± 0.1. Consequently, we see from Table 1.2 that this condition determines that av = -f3v but does not constrain the axialvector parameters. Neutron {3-decay is used to determine the the reduced matrix element for the vector transition. The matrix element for the process is (1.36) Therefore, we find that av = 1 and the vector component of all the other transitions are related. Both the vector and axial vector form factors have been determined precisely for decay A0 - t pe-ve where it is found that the vector coupling Cv is ICvl = 1.229 ± 0.035 and the axial vector coupling CA is ICAI = 0.903 ± 0.0-l6, The SU(3)J prediction for the vector coupling is Cv = ~( -2av· + f3v) = -1.22 which is in remarkable agreement with the experimentally determined value. Theoretically, this result is not that suprising as it was shown by Ademollo and Gatto [1.12] that the vector couplings are protected, up to second order in the breaking parameter, from violations in SU(3) 1 . The measured value of Cv / CA for the n -t pe-ve and I;- -t ne-ve decays determines that aA = 1.25 and f3A = 0.36 ± 0.05. These values then predict that C A for the A - t pe-ve is -0.87 ± 0.02 which is to be compared with an experimentally determined value of CA = -0.903 which is again in very good agreement. In these decay processes predictions based on SU(3) 1 are correct at about the 5% lewl. 1.4 SU(3)J Breaking We know from the pattern of masses of the light hadrons that SU(3)J is only an approximate symmetry. The symmetry is broken by the nondegeneracy of the light quark masses, ms >> mu, md. One may hope that the relevent expansion parameter is m 8 /lGeV, and use perturbation theory to estimate the effect of the breaking on the SU(3)J relations. This will be demonstrated explicitly for the case of semileptonic 14 hyperon decays . The mass term in the QCD Lagrange density given in Eq. (1.26) induces a perturbing term that transforms as an octet under SU(3) f and can be written as (1.37) where m8 is the strange quark mass in units of GeV fc 2 . The Ademollo-Gatto theorem [1.12] protects the vector relations from breaking terms, but not the axial vector relations. To find the effect of the breaking we form all the possible contractions of hb, x;, Hd and Pb- Therefore, the effective SU(3) f breaking Hamiltonian is break Hcpd b -;-ah hbHcpd -;-ah Hbhc pd H eff = a. Ha-;-b/ b lc d a + · b c d a + C. b c d a c r_{d I haHbpc-:-cll + 9 · I1apb-1 b a 1d r:. c + l. b a d 1 c (1.38) Examining the Cabibbo-allowed decays, we see that II{; P~ = 0 reducing Lhe number of contributing elements to four. Expanding Eq. (1.38) gi\·es r_{break _ Leff - ( ms ) [ - + d ;;:;0:=- _ f!_~Ao _ h 1GeV/c2 c.pn ·~ ~ 9·V3LJ f!_Ao~-] ·V3 LJ (1.39) which implies that the isospin relation r(~- - t L: 0 e-ile) = r(:E 0 - t I;+e-ile) is not altered to first order in the breaking parameter. This is to be expected since in the quark model picture the strange quark is merely a spectator in this decay. Therefore, assuming that Lhe reduced matrix elements are not significantly larger than a and f3, the axial vector relations are expected to be good at the ""' 10% level. In general it is found that SU(3)J symmetry works typically at the 30% level in low energy physics. In Lhe subsequent chapters we will investigate the predictions of SU(3)j for Lhe decays of hadrons containing a bottom or charmed quark. 15 References [1.1] S.L. Glashow, Nuc. Phys. B22 , 579 (1961); S. Weinberg, Phys. Rev. Lett. 19, 1264 (1967); A. Salam, in Elementary Particle Physics: Relativistic groups and Analyticity, Nobel Symposium No. 8, ed. N. Svartholm (Almquist and Wiksell), 367 (1968). [1.2] P.W. Higgs, Phys. Lett. 12, 132 (1964); Phys. Lett . 13, 508 (196-1). [1.3) G. t'Ilooft, Nuc. Phys. B35, 167 (1971). [1.4) P. Aarnio et al., (DELPHI Collaboration), Phys. Lett. B231, 539 (19K9); G.S. Abrams et al., Phys. Rev. Lett. 63, 724 (1989); B. Adeva et al., (L3 Collaboration), Phys. Lett. B231, 509 (1989); B. Adeva et al, (L3 Collaboration), L3 Preprint 00-! (1989); M.Z. Akrawy et al., (OPAL Collaboration) Phys. Lett. B231, 530 (1989); D. Decamp et al., (ALEPH Collaboration), Phys. Lett. B231, 519 (1989); D. Decamp et al., (ALEPH Collaboration), CERN-EP/89- 169 Preprint; J.M . Dorfan et al., (MARK II Collaboration), Phys. R ev. L ett. 63, 2173 (1989). [1.5] t-.I. I~obayashi and T. t-.1askawa, Prog. Th cor. Phys . 49, 6.52 (1973). [1.6] Particle Data Group, G.P. Yost et al., Phys. Lett. B204, 1 (1988) . [1.7] l\1. Gell-t-.1ann, Phys. Rev. Lett. 12, 15-5 (1964). [1.8] N. Isgur and M.B. Wise, Phys. L ett. B232, 113 (1989). [1.9] M.A. Shifman and M.B. Voloshin, Sov. J ..Vue. Phys. 45 , 292 (1987) ; H.D. Politzer and }.f.B . \Vise, Phys. Lett. B206, 681 (1988); H.D . Politzer and ~LB. \Vi se, Phys. Lett . B28 , 504 (1988); E. Eichten, Nttc. Phys . B4 , (Proc. Suppl.) (1988); G.P. Lepage and B.A. Thacker, Nuc. Phys . B4, (proc. suppl.) (1988). [1.10] N. Cabibbo, Phys. Rcu. Lt!t 10, 5:31 (1963). [1.11] E.D. Commins and P.Il. Bucksba um, W eak fnt emclion of Leptons and Qtw·rks, Cambridge Cnivcrsity Press (1983). [1.12] M. Adcmollo an<.l R. Gatto, Phys. Lett. 13, 264 (1961) . 16 CHAPTER 2. D-MESON DECAYS The D mesons, D+, D 0 and D-I, are the simplest bound system with a heavy quark constituent. It is found that the D meson masses are (1.8] Mn+ = 1869.3±0.6MeV , Mno = 1864.5±0.6MeV and Mnt = 1969.3±1.1MeV (2.la, b, c) The mesons are not heavy enough to decay to final states containing baryons. They decay nonleptonically to mesonic final states or semileptonically to states with one or more mesons and either a eve or Jl.VI' pair. The lifetime of these mesons has been observed to be (1.8] TD+ = 1.069 ± 0.033ps , rno = 0.428 ± O.Ollps and Tnt = .436 ± 0.036ps (2.2a, b, c) 2.1 Charm Changing Weak Operators The interaction Lagrange density in Eq. (1.22) dete rmines the transformation properties of the effective Hamiltonian for the weak decay of charmed hadrons. The 6.c = -1 nonleptonic decays arise from the weak Hamiltonian with flavor quanlum numbers (cs)(dU) for Cabibbo allowed decays, si[(cd)(dU) - (cs)(su)] for Cabibbo suppressed decays and si(cd)(su) for doubly Cabibbo suppressed These operators are different components of the same Hamiltonian which can be decomposed into irreducible representations of SU(3)J· An example of this is the decomposition of the component responsible for the Cabibbo-allowed decays (denoted with a superscript (a)), ( cs)( du) = o6(a) + o(a) (2.3) o~a) = ~[(cs)(du)- (cu)(dS)] (2.4a) 15 where transforms as a 6 under flavor SU(3)J and 0~~) = ~[(cs)(dU) + (cu)(ds)] 17 (2.4b) transforms as a 15 under flavor SU(3)J· The Cabibbo-suppressed operator has a similar decomposition into O~s) and oi~), as does the doubly-Cabibbo-suppressed operator into O~ds) and oi~s). Perturbative QCD corrections arising from momentum scales between the W-boson mass and the charmed quark mass give rise to an enhancement of the coefficient of 06 over the coefficient of Orr by (2.5) m the effective weak Hamiltonian (2.1]. Consequently it is possible, analagous to octet dominance in the weak decay of strange particles, that the sextet component of the Hamiltonian may dominate charmed meson decay. The operator Orr can be represented in tensor notation as H~b(15) , which is traceless and symmetric on its upper indices. It has non-zero elements (2.6a) for Cabibbo-allowed decays 12 21 13 31 -=H2 (15) = H2 (15) = -H3 (15) = -113 (b)= S! (2.6b) for Cabibbo-suppressed decays and Hj2(15) = Hj 1 (15) = -si (2.6c) for doubly-Cabibbo-suppressed decays. Similarly, 06 can be represented in tensor notation as Hab(6) which is symmetric on its indices and has nonzero elements (2. 7a) for Cabibbo-allowed decays (2.7b) 18 for Cabibbo-suppressed decays and (2.7c) for doubly-Cabibbo-suppressed decays. These can be written with the same tensor structure as the 15 by contracting with the totally antisymmetric tensor C.abc· These then can be written as HJ3(6) = -H~ 1 (6) = -1 for Cabibbo-allowed d ecays H:f2(6) = -Hi 1(6) = -Hj3(6) = Ht 1(6) = -s1 for Cabibbo-suppressed decays and HJ2(6) = -Hf 1 (6) = si for doubly-Cabibbo-suppressed decays. The ~c = -1 semileptonic decays arise from the weak Hamiltonian with flavor quantum numbers [(cs) + s1(cd)](l-vl) where l denotes a lepton (l = e, 1-l but not r). This operator transforms as a 3 under SU(3)J and has nonzero elements (2.8a) for Cabibbo-allowed decays and (2.8b) for Cabibbo-suppressed decays. 19 2.2 Two-Body Nonleptonic D-Meson Decays First, we will review the predictions for the decay mode D --+ M M where M represents a member of the pseudoscalar octet. These results have been derived previously [2.2]. The D mesons transform as an antitriplet under SU(3)J and can be represented as (2.9) The octet of the lightest pseudoscalar mesons is represented as Mg = ( 7ro I ..j27r~ TJo I y'6 7r+ (2.10) J(- We have assumed that the SU(3)J octet isoscalar eigenstate is very nearly the mass eigenstate corresponding to the ry 0 (i.e. we have neglected TJ - ry' mixing). There are three independent contractions that can be constructed from the two meson octets, the charmed .meson anti triplet and the Hamiltonian. Therefore we can write the effective Hamiltonian as (2.11) where a , b and c are unknown reduced matrix elements [Note: \Ve have used the identity DaMb H~c(6)1'v!f + DaHgb(6).Mf l\1d = 0 to reduce the number of reduced matrix elements from four to three]. The results of expanding this Hamiltonian can be found in Table 2.1. It is interesting to note that the decays D"t --+ 1r 0 1r+ and D0 - K I\ 0 are forbidden by the SU(3)J symmetry. There is only one relation amongst the Cabibbo-allowed decay rates a nd it is due to the full SU(3)J symmetry, (2 .12) However, if we assume that the sextet component of the Hamiltonian dominates the 20 decay rates then the following relations arise (2.13) Experimentally, the branching ratio of several of these decay modes have been measured. It is seen that [1.8, 2.3] Br(Dt - t ~ ](+) = (2.1 ± 0.6) x 10- 2 , (2.14a) (2.14b) (2.14c) (2.14d) and (2.14e) The ratio of the latter two rates is f(Do--+~11"+) ......., 1.2 ± 0.3 which is to be compared 0 0 f(D -K 1r ) with the SU(3) 1 prediction assuming sextet dominance of 2. The error on this result is sufficiently large not to allow a definite conclusion to be drawn although it hints strongly that, in fact, the sextet component of the Hamiltonian is not dominating the de cay amplitude. 21 There are several relations amongst the Cabibbo-suppressed decay rates, they are (1.8) (2.15a) (2.15b) (2.15c) (2.15d) These are all due to the full SU(3)J symmetry. Two of the decay modes appearing in these relations have been observed experimentally. It is found that (2.16a) and (2.16b) The ratio of the decay rates is then r;,{:;o-:!~:~\7 = 3.5 ± 1.2 which is to be compared with the SU(3)J prediction of unity. There seems to be a large descrepancy bet,,·een these two numbers but they still are within 2a of each other. Amongst the relations that arise from assuming that the sextet component of the Hamiltonian dominates the decays is (2.17) Experimentally, it is found that (2.18) Conversion from branching fractions to decay rates is done by multiplying the branching fraction by the total width, which is equivalent to dividing by the lifetime. Therefore, the experimentally observed ratio of decay rates is rtfn+o=~~\-+/ = 2.7±1.1 which is to be compared with the sextet dominated SU(3)J prediction of unity. 22 There are several relations between Cabibbo-allowed and Cabibbo-suppressed decay rates. One of the relations is (2.19) and the Cabibbo-suppressed decay mode has been observed to have a branching fraction (2.20) 0 The ratio s~[{f}o ~.}~~7r;) = 1.4 ± 0.4 is compatible with the SU(3)J prediction of 1 but is only marginally compatible with the prediction for the ratio to a ](+ ](- final state, (as was discussed above). There are relations involving the doubly-Cabibbosuppressed rates but they are too small to be experimentally observable at this time. The results for the decays D -+ }.1M cannot be straightforwardly carried over to the decays D -+ VV, where V is the lowest lying vector meson octet v = (P' !.Ji; v.;.J6 -p I .J2 +Val V6 J(*- K*o p+ 0 (2.21) The octet state IVa) = (llvf6)(1uu) + idd) - 2iss)) mixes with the singlet state IV1) = (1lv'3)(1uu) + idd) +iss)) because of the near degeneracy in the masses. The mass eigenstates I</>) = iss) and lw) = (11 .;2)( luu) + idd)) are linear combinations of these two states. Explicitly, the states can be written as (2.22a) and (2.22b) In addition to the three reduced matrix elements contributing to this process (found from D -+ .M AI by substituting AI-+ V) there are two from the contraction formed 23 with the singlet Vi. The terms are (2.23) Only two of the decays D --+ VV have been studied experimentally [1.8, 2.3]. It is seen that (2.24a) (2.24b) The effective Hamiltonian describing the decay D --+ V M is written in terms of 10 reduced matrix elements, it is Among the Cabibbo-allowed decays, tabulated in Table 2.2, there is only one relation and it is due to isospin, (2.26) There are three more relations when sextet dominance is assumed, they are (2.27a) (~.27b) (2.27c) The branching fraction for ten of these decay modes haYe been measured to be [1.8, 21 2.3) Br(D; --+ <P1r+) = (2.0 ± 0.4) x 10- 2 Br(D 0 --+ p+ ](-) = (8.2 ± 1.2) x 10- 2 (2.28a) (2.28b) , (2.28c) (2.28d) Br(D+ --+ p+ft) = (6.6 ± 1.7) x 10- 2 , (2.28e) B1·(D0 --+ ft <P) = (0.99 ± 0.24) x 10- 2 , (2.28!) Br(D 0 --+ Kw) = (3.2 ± 1.5) x 10- 2 (2.28g) (2.28h) (2.28i) (2.28j) From these we find that f(D 0 + r-) + -P • 0 \ r(D. -+K l\"+) SU(3)J prediction of 1. Conversely, ,....., 5 ± 1 which should be compared with the qn+ A.•o +) -+ ~ r(D+--p+ l\- ) ,....., 4 ± 2 which is 2o- away from the SU(3)j prediction of 1. The results for the Cabibbo-suppressed and doublyCabibbo-suppressed decays are not tabulated. There a re se\·eral relations between 25 the Cabibbo-suppressed decays, they are (2.29a) (2.29b) (2.29c) (2.29d) Experimentally, several of these decay modes have been studied, and it is found that (2.30a) Br(D+ - 4 rr+ ¢) = (1 ± 0.2) x 10- 2 (2.30b) (2.30c) (2.30d) 2.3 Semileptonic D-Meson Decays to Three-Body Final States The Hamiltonian for the semileptonic decay of D-mesons is gi ,·en in Eqs. (2.8). There is only one singlet in the tensor product 3 0 3 0 8 and hence there is only one reduced matrix clement determining the semileptonic decay rates . The effective 26 Hamiltonian for the system is (2.31) From this Hamiltonian we find the following relations (2.32) These relations are easily converted to the semileptonic decay to a vector meson and leptons as was done before. Experimentally, the branching fractions obtained to date [1.8] are, (2.33a) (2.33b) (2.33c) This data seems to be in agreement with the SU(3)J predictions. The branching ratio of the n+ decay to a muonic final state is approximately twice that expected by universality and seems not to satisfy the prediction. This deviation will probably vanish with better data. 2.4 Large Nc Limit and Factorisation of Amplitudes A model of hadronic interactions that has been extensively studied is one in which the number of colors Nc is treated as an expansion parameter (or rather 1/ Nc)[2.4, 2.5]. Of the many inte resting results of this model in the limit of large Nc is the factorization of weak amplitudes. Let us consider the decay D ~ 1r1r. The leading 27 planar diagram with no gluonic corrections is shown in Figure la. There are counting rules that apply to these graphs. For each external particle there is a )N; that appears in the amplitude. For each loop there is a factor of Nc that arise from the possible colors circulating in the loop and for each fermion-gluon vertex there is a factor of -:;/Jr:. A gluon is represented by two parallel line (colors) that break the fermion line at the vertex. The Nc counting for Figure la gives rise to an amplitude 3 1 1 2 2 A ex: ( ~ ) Nc = Nc (2.34) If we examine the diagram in Figure lb where the gluon is transferred between quarks in the same meson we find that the amplitude for this diagram is (2.35) which is the same order in Nc as the leading graph and hence survives in the limit as Nc -+ (X). However when we perform the same power counting on the graph shown in Figure lc. which does not allow the amplitudes to factorise we find that (2.36) This diagram vanishes in the limit Nc -+ (X) and hence we find that indeed as the number of colours becomes large the weak amplitudes factorise. Therefore, in this limit, (2.37) where (cd)v -A = ca1 1'(1-15)da. The pion to vacuum matrix element is parametrized as (r.(p)l(ud)v-AIO) = f-rrP 1' (2.38) where f-rr = 0.94JH-rr is determined experimentally from the semileptonic r. decay and therefore we find that the amplitude for the process D -+ r.r. is proportional to f-rr· 28 We can phenomenologically modify our SU(3)J predictions for the Cabibbor(D0-+K+ K-) suppressed decays from r(Do-+11"+ 1!" ) = 1 to r(Do -+ ](+ K-) f(D 0 -+ 1r+1r-) (2.39) The leading 1/Nc corrections to each of these amplitudes cancels, thereby making N? [2.6]. this prediction valid up to 1/ However, in phenomenologically correcting for SU(3)J violation in the decay amplitudes we must be consistent and include the phase space corrections arising from the mass difference between the kaon and pion. This gives rise to an additional multiplicative factor of'""' 0.7. Hence the modified SU(3)J prediction for this ratio becomes 1.2, which is still significantly different from the experimentally observed value. 2.5 Concluding Remarks on Chapter 2 It is interesting to briefly discuss some of the models employed to try and understand the weak hadronic decays of D-mesons. The large Nc limit and amplitude factorization provides a starting point for elucidating some of the underlying dymnamics. The effective Harniltonian for the system in the large Nc limit under the assumption of factori zation is [2.7] (2.40) where a1 = q + -Jr;c2 and a2 = c2 + -Jr;c1 at the quark level. The -Jr; arises from the color mismatch when the h adron is "reformed" after the weak interaction. Since the color currents in the hadron are not understood this factor of tc is replaced by an arbitrary parameter ( at the hadronic level. It is important to realize that the assumption that this factor is the same for all decay modes has no firm theoretical basis. A good discussion of this can be found in ref [2.8]. When the d ecay rates for the known modes are fit with this form for the interaction a value of ( '""' 0 is found which indicates that the naive quark estimate of 1/3 is too large. To some extent this 29 parametrisation of the weak Hamiltonian reproduces the pattern of lifetimes of the D-mesons. Another approach to understanding D-meson decays is in the context of the quarkdiagram scheme [2.9). In this scheme each topologically different quark line diagram that can contribute to a decay mode of the D-mesons is assigned an unknown amplitude. SU(3)J breaking effects are incorporated by assigning a different amplitude to diagrams that have strange quarks appearing in the interaction vertices from those with up or down quarks. Further, final state interactions (FSI) are accounted for by introducing phase shifts for each isospin partial wave in the amplitude. These phases are generally complex as there are resonances at energies near the charm mass. An example where these FSI are important is in understanding the descrepancy between the theoretical and experimental branching ratios for D 0 ---+ J(+ J(- and D 0 ---+ 7r+7r- . If there were a resonance in the J(+ f{- system (and not the 7r+7r- system) with mass close to the mass of the D 0 then this would be a natural explanation for the difference. (It should be noted that FSI can only give rise to deviations from the predictions of SU(3) I if they violate the symmetry. If the resonance was not close to the mass (depending on the width of the resonance) of the D 0 then the apparent violation would be significantly less.) These amplitudes and phase shifts are then fit to the available data on the branching fraction for D-meson decays and the relative importance of each type of quark line diagram can be found along with the size of the FSI. Final state interactions need to be incorporated into our SU(3) 1 predictions. The amplitudes at the weak vertex can be decomposed into isospin partial waves as was discussed in the previous paragraph [2.9]. The FSI will giYe rise to a phase shift for each partial wave. One can then fit these amplitudes and phase shifts to the D-meson decay data, remoYe the phase shifts and then compare these amplitudes with the predictions of SU(3)J [2.9,2.10]. Therefore it is not a useless exercise to construct the predictions for various decay processes, even when the energy of the decay lies in the resonance region, as these effects can be removed from the data. Comparison between the predictions of sextet dominance in the nonleptonic 30 decay of D-mesons and the experimental results indicate that in fact the sextet is not dominating the weak decay of the D-mesons. This is despite the fact that short distance QCD gives rise to an enhancement of the sextet component over the 15 by a factor of 2.5 in the effective weak Hamiltonian. 31 References [2.1) M.K. Gaillard and B.W. Lee, Phys. Rev. Lett 33, 108 (1974); G. Altarelli and L. Maiani, Phys. Lett B52, 351 (1974). [2.2) R.L. Kingsley et al. Phys. Rev. D11, 1919 (1975); L-Lie Chau Wang and F. Wilczek, Phys. Rev. Lett. 43, 816 (1979); r..l.B. Einhorn and C. Quigg, Phys. Rev. D12 , 2015 (1975); M.B. Voloshin et al., Pis'maZh. Eksp. Teor. Fiz. 21, 403 (1975)[JETP Lett. 21, 183 (1975)). [2.3) For a review, see G. Gladding in Heavy Quark Physics, AlP Conference proceedings 196, ed. S. Drell and D. Rubin (1989). [2.4) G. t'Hooft, Nuc. Phys. B75, 461 (1974). [2.5) For a review, see S. Coleman, Aspects of Symmetry (selected Erice Lectures) (Cambridge University Press, Cambridge, England, 1985). [2.6) B. Blok and ~1. Shifman, Yad. Fiz. 45, 211 (1987); 478 (1987); 841 (1987). [2.7) M. Bauer, B. Stech and M. \Virbel, Z. Phys. C34, 103 (1987). [2.8) D.G. Ilitlin in Proceedings of the Banff Sumrne1· Institute, Ed. A.N. Kamal and F.C. Khanna, \iVorld Scientific (1988). [2.9) L. Chau and H. Cheng, Phys. Rev. D36 , 137 (1987); L. Chau and II. Cheng, Phys. Lett. B222 , 285 (1989) . [2.10) A,· Kamal and R. Verma, Phys. Rev. D35 , 3515 (1987); Phys. Rev. D36 , 3527 (1987) (E). 32 Table 2.1. Rates for the decays n ~ M Min terms of the three reduced matrix elements a, b and c. Process Rate Ia + b- c12 tl- a+ b + cl 2 tl- a+ b + cl 2 4lbl2 ila- b + cl 2 Ia + b + cl2 sila + b- cl 2 sitla + b + cl 2 sitla- b - cl 2 sitl- a+ b + cl 2 sil- a- b + cf snia + 2b + cl 2 si Ia- b + cl 2 si2lbl 2 sitla + 5b + cl 2 sitl -a- b- cl 2 sil-a+b -cl 2 s11 - a - b + cl 2 s1ila- b- cl 2 2 s41 lla-b-cl 2 s1ila- b + cl 2 s1 ~~ -a+ b- cl 2 s11 - a - b - cl 2 s141- bl 2 no~ ](-7r+ no~ X01]o no~ Jt17ro n+ ~ Jt17r+ n; ~ 1r+1Jo n+ s ~ K+-K no~ 7r+7r- no ~ 1]01]0 no ~ 7ro7ro no ~ 1]01r0 no~](+]{- n+ ~ 11°1r+ n+ ~ J(+~ n+ ~ 7r+7ro n; ~ 17° K+ n+ ~ 7ro ]{+ s n+ ~ 7r+ Ko s no ~ 7r- J(+ no ~ 1]0 J{O no ~ 7ro ]{0 n+ ~ 11° K+ n+ ~ 7ro ]{+ n+ ~ rr+ J\ 0 0 n+ s ~ J\+ I\ 33 Table 2.2. Rates for the Cabibbo-Allowed decays D ~ VM in terms of the reduced matrix elements a, b, c, d, e, f, 9, h, s and t. Process no~ f(-p+ Do~ Jt1 Po D 0 ~Kw D 0 ~ KrP Do~ TJof{*o Do~ 11"+7{*- Do ~ 7ro ](*o Rate Ia + c- e + 91 2 il -a+ d + e- hl 2 l8 1a- 2b +d-e+ 2f- h + v'I2s- v'I2tl 2 ~~-a+ 2b- d + e- 2f + h + 3s- 3tl 2 tl- 2a + b + c + 2e- f- 91 2 lb + d- f + hl 2 n+ s ~ y+y*o \ \. tl- b + c + f- 91 2 ic + d- 9 + hl 2 lc + d + 9- hl 2 ila + b- 2c + e + f- 291 2 tla- b + e- !1 2 tl -a+ b-e+ !1 2 1 18 la + b- 2d + e + f - 2h + v'f2s + Ji2t j2 il - a - b + 2d- e- f + 2h + v'3s + v'3t1 2 Ia + c + e- 912 n+ s ~ y•+r \. \ lb + d + f- hi 2 n+ ~ 71"+ K*u -:-;{) n+ ~ K p+ n; ~ "'o p+ n; ~ 1rop+ Dt ~ 7r+ Po n+ s ~ 7r+w n; ~ 1r+rP 34 c u - u c d oo '.__________ __..9 - D 1t (a) 1t- (b) - u u c (c) - - u u FIGURE 1 35 CHAPTER 3. CHARMED BARYON DECAYS Measurements of the branching fractions for many exclusive decay modes of charmed baryons are starting to be made. Although the weak decays of the At , :=:~ , ::::~ 1 and n~ have been observed, only the decays of the At have been studied in any detail. The large event sample of B-meson decays that will be collected in the near future will allow the study of the decay modes of all the charmed baryons. Charmed baryons belong to one of two representations of flavor SU(3), a 3 or a 6. The At , :=:~ and :=:~1 constitute the 3 and the n~ , :=:;t; , :=:~2 , ~t+ , ~t and ~~ comprise the 6. Five of the six members of the 6 decay strongly or electromagnetically, for example ~t+ --+ At 1r+, :=:;!z -t ::::~I · Only the n~ of the 6 and the members of the 3 decay weakly. The lifetime of these four weakly decaying particles have been measured to be TA+ c = 0.196 ± 0.016ps. , T=o = - cl . 0.082 ± 0.06ps. and T=+ = 0.57 ± 0.1•1ps , Tr.o ;\£c = 0. 79 ± 0.3-1 ps '- cl (3.1) The masses of these particles have been measured to be Mr:,t+ = 2-152.2 ± 1.71\IeV jc 2 and !lfn~ = 2740 ± 2011eV jc 2 (3.2) 3.1 SU(3)J Representation of the Charmed Baryons The lowest lying charmed baryons fall into two representations of SU(3) 1 . If the charmed baryon state vector is antisymmetric under interchange of the two light quark flavors then it is in the 3 representation and if the charmed baryon state vector is symmetric under the interchange of the two light quark flavors then it is in the 6 r epresentation. In the non-relativistic quark potential model the spin-flavor state 36 vectors for the lowest lying J7r = ~ + charmed baryons in the 3 representation of SU(3)J are 1 1 IAt-2) = 2 (lc i u i d 1) -lc i u 1 d i)- ic i diu 1) + lc j d 1 u i)) (3.3a) l:=:;i~) = ~ (lc i u is 1) -lc i u 1 s i) -lc is i u 1) + lc is 1 u i)) (3.3b) - 1 1.::,~1) = 2 (lc i dis 1) - ic i d 1 s i) - lc is i d 1) + lc j s 1 d i)) (3.3c) The spin-flavor state vectors for the lowest lying charmed baryons in the 6 representation of SU(3)J are fl 1 IE7+ 2) = y 6 (2lc 1 u i u i) - lc i u i u 1) - lc j u 1 u i)) 1VA IE7 -) = - 2 IE~~) = 12 (3.4a) (2lc 1 u i d i)- ic i u i d 1)- lc i u 1 d i) + 2lc 1 diu i) -lc i diu 1) -lc i d 1 u i)) (3.4b) /I (3.4c) 1:=:~l) = (2lc 1 did i)- lc i did 1)- lc i d 1 d i)) VA (2lc 1 u is i)- lc i u is 1)- lc i u 1 s j) + 2lc 1 s i u i) - lc i s i u 1) - lc i s 1 u i)) 1::::~2~) = VA (3.4d) (2lc 1 dis i)- lc i dis 1)- lc i d 1 s i) + 2lc 1 s i d i) - lc i s i d 1) - ic j s 1 d i)) (3.4e) (3Af) It is the hyperfine interaction that gives rise to the mass difference between the :Sc and Ac baryons and between the :=:c1 and :=:c2 baryons. Since the state vectors for the lowest 37 lying charmed baryons in the 3 representation are antisymmetric under interchange of the light quark spins while the state vectors for the lowest lying charmed baryons in the 6 representation are symmetric under interchange of the light quark spins the hyperfine interaction causes the ~c to be heavier than the Ac and the 3c2 to be heavier than the :=:c1. SU(3)t violations in the hyperfine interaction give rise to mixing between the :=:cl and :=:c2 states. Physical mass eigenstates will be the following linear combination of these two states (3.5a, b) The two-body hyperfine interaction has the form Hhr = 4A L + 4A' Sc . Sj . fficffij J=u,d,s L Si . Sj . . ffiiffij ,,J=u,d,s (3.6) where A and A' are determined by the spatial part of the charmed baryon state vectors (which are taken to be SU(3)t symmetric). The term proportional to A' dominates the hyperfine interaction since it is not suppressed by the heavy charm quark mass. Using Eqs. (3.3,3.4,3.5,3.6) we find that (3./a) 2A (-::- . IHhf I-::- ) -__ -/3A ~d ~cl rnc 2A ------- (3.7b) (ms - m11) (3.7c) m 5 1Tiu which gives a small mixing angle O = _ -/3 (~) 4 A' (ms- 7Tiu) me (3.8) It is easy to understand why the term proportional to A' does not contribute to the mixing between the :=:c1 and :=:c2 states. In the computation of the :=:ci, :=:c2 mass 38 matrix the quark masses in this term occur as an overall factor and hence it has the same effect as an SU(3) f conserving interaction where the two light quarks are taken to be degenerate with mass y'msmu. Therefore, despite the near degeneracy of the 3 and 6 representations, SU(3)t violations in the hyperfine interaction do not give rise to significant mixing between the multiplets. From the measured masses of the Ec and Ac baryons the hyperfine mass splitting between :=:c1 and :=:c2 can be determined. Using (3.9a, b) we find that (with mu = 330MeV and m 5 = 550MeV) the measured value of mEcffiAc imply (3.10) The lowest lying J7r = ~ + charmed baryons are a lso in the 6 representation of SU(3)J· Their spin-flavor state vectors are (3.lla) (3.11b) (3.llc) (3.lld) (3.11e) 1n;o ~) = lc i s i s i) These states are split in mass from the lowest lying J7r = 39 (3 .11!) 1+ charmed baryons in the 6 representation by the part of the hyperfine interaction proportional to A. Explicitly, (3.12) which implies that (3.13) Since me~ mu and m 8 , it is expected that A will be somewhat larger then A'. The remainder of this chapter is divided into two sections, nonleptonic and semileptonic decays. In the first section we examine the flavor SU(3) predictions for the decay of charmed baryons in the 3 and 6 representation to t + or ! + uncharmed baryons and one or two mesons. The second section deals with the semileptonic decay ! of charmed baryons in both representations to + or ~ + uncharmed baryons, a ( z+, vz) lepton pair and zero or one meson. The matrix elements for the decay processes are tabulated in terms of unknown reduced matrix elements. 3.2 Nonleptonic Decay Of Charrned Baryons To Two Body Final States First we examine the process T ---+ hll1 where T denotes the 3 repesentation of charmed baryons, h is the lowest-lying t + baryon octet and M is the lowest-lying pscudoscalar meson octet. The Hamiltonian for the decay of charmed baryons is the same as that for the decay of charmed mesons, which is given in section 2.1 . Some of the SU(3)J predictions for this set of decays have been considered before in ref. [2.2]. The effectiYe Hamiltonian for the process is given by (3.14) where a, b, c, d, e, f, 9 are unknown reduced matrix elements and 40 ra is the charmed baryon anti-triplet ::-+ A+) T a-- (=o ~cl• -~cl' c (3.15) The square of the matrix elements for Cabibbo-allowed processes are shown in Table 3.1. We see that there is only one relation between the matrix elements of Cabibbo-allowed decays, (3.16) and this is a result of the SU(2) isospin subgroup of SU(3)J· There are several relations between squares of matrix elements for Cabibbosuppressed decays, as can be seen from table 3.2. They are (3.17a) (3.17b) (3.17c) (3 .17d) (3.17e) (3.17f) which are all a consequence of the full SU(3)J symmetry. From Table 3.3 ,,.e see that there are no relations between squared matrix elements of doubly-Cabibbo-suppressed 41 decays. However, there are relations between the squares of Cabibbo-allowed, suppressed and doubly suppressed matrix elements. They are -o - t .=.. --J(+)I2 = S 2IM(-o IM( .=.cl 1 .=.cl - t -.=.. 7r +)l2 ' (3.18a) (3.18b) (3.18c) (3.18d) (3.18e) (3.18!) (3.18g) (3.18h) The sum of the masses of the products from charmed baryon decay is not always negligible compared to the energy release. Therefore, SU(3)t relations between decay rates, derived from relations between the square of matrix elements, have significant phase space corrections. The exception to this is when a relation is due to isospin, where the difference between the sum of t he final state masses is sm all. To find the relation between the decay rates from lhe square of the matrix elements we can use the expression 1 ? IPbl dr(a--+ be)=-.-? IAI(a--+ bc)l--? dD. , 32r.- ffi(i (3.19) where ma is the mass of the decaying particle, Pb is the momentum of one of t he final state particles, dD. is its solid angle and Af(a --+ be) is the m at rix element for the decay a --+ be. There is a lso an ad ditional factor of IPJJ?I occurring in the matrix 42 element for decays with final state angular momentum l. Any mass dependence in the matrix element is not corrected for as this is due to explicit SU(3) violation and not a kinematical effect. Consequently we find that, for instance, (3.20) where l is the angular momentum of the decay channel and r, is its contribution to the rate. For this process both l = 0 and l = 1 partial waves can contribute. The angular distribution of the decay products from a polarized charmed baryon can be decomposed to yield the relative magnitude of the I = 0 and l = 1 partial waves, to which the phase space corrections can be applied accordingly. If, however, the angular distribution information is not available (which is probably the case), then the best estimate of the phase space correction is to say that it lies somewhere in the range between its value for l = 0 and l = 1. Thus the flavor SU(3) prediction for the above process is 1.2r(::::~l --t ::::- ]{+) < r(::::~l --t I:-7r+) < 1.7f(::::~l --t ::::- f\+) . Two decay modes of the type under discussion here have been obsen·ed so far. The first BT(A"t --t A0 1r+) = (0.63 ± 0.25) x 10- 2 [1.6] does not appear in any of the 0 relations. The second observed is BT(A"t --t p!\ ) = (2.0 ± 0.4) x 10- 2 (3.1, 3.2, 3.3], from which we predict that B1·(:=:~ --t I:+ ]{ 0 ) "' 4 x 10- 5 . These expressions can easily be carried over to decays involving baryons and vector mesons, T --t BV. An example of this is (3.21) Two of the branching ratios for final states containing a vector m eson have been 0 m easured. The first is B1·(A"j: --t pl\* (892)) = (5.6±3) x 10- 3 [3.2, 3.4], from which we predict that Br(:=:~ --t E+ ]{* 0 (892)) "'1 x 10-5 . The second is B1·(A"j: --t p¢) = (2 ± 1) x 10- 3 which does not appear in any of the relations. 43 Next we look at the process T -+ h* !11 where h* is the decuplet of ~+ baryon resonances with elements h*lll = ~ ++ h*ll2 = _1_~ + h*l13 = _1_E*+ h*122 = _1_~ o h*l33 = _1_::::*o ' v'3 ' v'3 ' v'3 ' v'3 1_'<:'*0 h*222 _ "- 1 •223 __1_'<:'*- h*233 __1_=•- h*333 = nh •123 __ - V6LJ ' - u ' ~ - v'3L..J ' - v'3~ ' ~~ . (3 .22) The effective Hamiltonian for the process is Herr = 0'-h*abeT ad Hebe(-) 15 _A,[de + {3..,.....habc Tad}:rbe ~d (-)Me 15 e (3.23) where a, {3, 1 and 8 are unknown reduced matrix elements. The rates for Cabibboallowed decay processes in terms of these reduced matrix elements are given in Table 3.4. There are four relations between Cabibbo-allowed decay rates. They are (3.2-ia) (3.24b) which are due to isospin and ·'!(=+ ~cl --t \'*+....,..,..J -' \ , )12 - I• I ~I(=+ 11 '::'*0 7r +)12 ' ~cl --t ~ (3.24c) (3.2-ld) which are due to the full SU(3) symmetry. Again, phase space correction factors must be applied to these equalities giving, for example (3.25) a modification of (0.69) 3 due to the differing final state masses and the fact that the decay is P-wave (neglecting possible D-wave contributions). 44 There are several relations between the squares of Cabibbo-suppresscd matrix elements, as seen in Table 3.5. They are (3.26a) (3.26b) (3.26c) (3.26d) (3.26e) (3.26!) (3.26g) (3.26h) which are all full SU(3)t relations. There are also relations between doubly-Cabibbo-suppressed matrix elements as seen from Table 3.6. They are (3.27a) (3.27b) (3 .27c) (3.27d) Several relations between Cabibbo-allowed,-suppressed and doubly-suppressed 45 decay modes are found. They are (3.28a) (3.28b) (3.28c) (3.28d) (3.28e) (3.28!) (3.28g) (3.28h) (3.28i) (3.28j) (3.28k) (3.281) (3.28m) One of these decay modes has been observed with a branching ratio of [3.4] (3.29) and hence we can predict, neglecting possible D-wave contributions, that Br(At ---+ _6++7r - ) ""(3.8±2.0) x 10- 4 ("" (4.8±2.5) x 10- 4 for a purely D-wave process) and that B1·(=.~---+ _6++7r-) ,. .__ 2.3 x 10- 5 ("" 3.8 x 10- 5 for a purely D-wave process). 46 Next we look at the two-body decays S---+ hM, where S denotes the 6 representation of charmed baryons. The element S 33 = n~ is the only member of the 6 that decays weakly, the E;t+,+,O decay strongly to the A;t" in the 3 (e.g. E;t+ ---+ A;t"1r+) and the :::::/ decay electromagnetically (e.g. ::::~2 ---+ ::::~0). By inspection of the n~ flavor wavefunction we see that the only Cabibbo-allowed final state is :=:°K. We therefore look for relations between Cabibbo-allowed and Cabibbo-suppressed decay rates. The effective Hamiltonian for the process is (3.30) The squared matrix elements resulting from this effective Hamiltonian are found in Table 3. 7. vVe see that there are no relations between any of the decay rates involving Cabibbo-allowed, -suppressed or doubly-suppressed decays. If we look at the isospin structure of the doubly-Cabibbo-suppressed sextet com- ponent of the Hamiltonian, O~ds) = ~ [(cd)(su)- (cu)(sd)] , we see that it is an I = 0 operator, whereas O~~s) is an I = 1 operator. Since n~ has I = 0, we expect that the decay to A0 ry 0 proceeds via 0 6 only, and similarly for the ~ 0 1r 0 final state, since the weak Hamiltonian does not haYe an I = 2 component. Therefore, by measuring the relative rate for an I = 1 decay, for example the E 0 ry 0 or A0 1r 0 final state, compared to an I = 0 decay, an estimate of the relative contributions from 06 and 015 can be made. This will not be a strong test of the perturbative QCD prediction since there could be cancellations between the reduced matrix elements contributing to the decays, but it will give a rough estimate of the relative contributions. Unfortunately, since these are doubly-Cabibbo-suppressed decays, they will probably be the last to be measured and hence their predictive power is somewhat limited. 47 Consider now the process S ~ h* M. The only two possible Cabibbo-allowed final states are n-7r+ and =._• 0 ~. The effective Hamiltonian for the decay is + 8h*abc Hab d (15)Sde .Me + >.tadfh,*abc H de (6)Mbf see + 1Jtadfh,*abc H de (6)1\Pf sbc (3.31) The resulting squared matrix elements are shown in Table 3.8. We see that there are no relations between any of the Cabibbo-allowed or Cabibbo-suppressed decay modes. However, there are two relations involving doubly-Cabibbo-suppressed processes, they are (3.32a) (3.32b) 3.3 Nonleptonic Decay Of Charmed Baryons To Three Body Final States In this section we will be considering decays of charmed baryons to final states containing a baryon (either in the lowest lying t + octet or the ~+ decuplet) and two octet mesons 1\1. As far as SU(3)t is concerned the two meson octets are identical and consequently the Hamiltonian must be symmetrized if the mesons are in a relatively even angular momentum state or anti symmetrized if they are in a relatively odd angular momentum state. \Vhen the Hamiltonian is expanded in terms of the individual particle operators and matrix elements are taken there are symmetry factors that must be included. This is demonstrated most simply by an example. Consider the Hamiltonian l'J1 elf = 7r + ir -+ 7r 0 ir 0 , (3.33) of which matrix elements can be formed to yield (3.3-la) 48 and (3.34b) due to the two possible ways of annihilating the two neutral pions. When we form t multiplying the 0 0 phase space integrals from Bose statistics. In the tables this factor of ! has been a rate from these matrix elements there is an additional factor of 1r 1r omitted and so to obtain rate relations from the squared matrix elements a factor of t must be included for processes involving identical particles. Also, in obtaining rate relations from the matrix elements, phase space correction factors must be included just as for the two-body decay modes. However since these factors depend upon the momentum configuration of the final state we will not calculate them in this work. Any processes that are not energetically allowed are not included in the tables. The first three-body decay process examined is T --+ hi\! !vf for which there are nineteen reduced matrix elements. The operator Orr contributes eleven reduced matrix elements and 06 contributes eight. The Hamiltonian for the process is (3.35) The matrix elements resulting from this Hamiltonian are NOT tabulated in this paper as there are"' 121 possible decay modes for either angular momentum state. 49 Despite this large number of operators there are still relations between some matrix elements for various decay modes. The relations between Cabibbo-allowed decays are all due to isospin, examples of which are ~o ~- o +) ~o o +) 12 ' 2 IM(~+ IM( .::.cl ---+ .::. 7r 7r (L=0,2, ... ) 1 = .::.cl ---+ .::. 7r 7r (L=0,2, ... ) (3.36a) (3.36b) for even and odd angular momentum channels respectively, and (3.36c) (3.36d) which are independent of the relative angular momentum between the mesons. The inequality arises from the fact that processes involving identical mesons in the final state can only proceed through even angular momentum channels. There is a r ela tion be tween a Cabibbo-allowed process and a Cabibbo-suppressed process that holds only for odd relative angular momentum states which is ~-}'+ +) (L=l,3, ... ) 1 = 8 1 1 \I(A+ ~-}'+ +) (L=1,3, ... ) 1 1 ~cl ---+ .::. \ 7r c ---+ .::. \ 7r I 1~!(-=-+ 2 2 2 1 (3.36e) l\fore interesting are the Cabibbo-suppressed decays where there are relations due to the full SU(3) symmetry which are independent of the relative angular momentum between the mesons. They are (3.37 a) (3.3/b) (3.37c) 50 2 jM(3~ ---t L:-7r+7r+)j2 :::; sd jM(i\~ ---t :::;- J(+?r+)j2 ' (3.37d) (3.37e) (3.37 f) (3.37g) (3.37h) Experimentally, branching ratios for some of these processes have been measured, Br(i\~ ---t pK-1r+) = (2.6 ± 0.9) x 10- 2 [3.5] and Br(i\~ ---t E+1r+1r-) = (10 ± 8) x 10- 2 [3.6], of which the latter appears in an isospin relation between Cabibbo-allowed decays. There are twenty-n ine reduced matrix elements contributing to the process S ---t B!lf !If, twenty of which are from 015 and the remaining nine from 06. Only two relations are found, (3.38a) (3.38b) The first is between Cabibbo-allowed decays and the second between Cabibbo- suppressed decays . They are both consequences of isospin. The matrix elements for the various decay modes are not tabulated. We consider now the process T ---t h* 1H !If. There a re twelve reduced matrix elements contributing to the decays. The effective Hamiltonian for the process is 51 given by (3.39) The results for Cabibbo-allowed decays with the mesons in an even (odd) angular momentum state are shown in Table 3.9 (Table 3.1). Cabibbo-suppressed decays with the mesons in an even (odd) angular momentum state are shown in Table 3.11 (Table 3.12). 'vVe see that there are many relations between squared matrix elements for various processes. For Cabibbo-allowed decays, we find that there are relations between matrix elements when the mesons in an even or odd angu lar momentum state. They are -::-•0 7r 0 7r +) (L=0.2 ... ) 12 -- S 11 ~1('="+ -==-·- 7r + 7r +) (L=0,2, ... ) 12 ~cl --t ~ ~cl --t ~ I ~1' (-=-+ l V, ll (3.40a) ~c1 1 ~!('="0 1• I ~!(\+c ll - --t A+~} u. \ - }"-) \ (L=l,3, ... ) 12 = A++~J( }'-) 311~!('="+ l • ~cl --t u. \ (L=l,3, ... ) 12 , (3.40b) -::-*-}'\ + 7r +) (L=l,3, ... ) 12 -- 3 11 l ~£('="+ n-}'\ + 7r +) (L=l,3, ... ) 12 . l ~cl --t H (3.40c) --t ~ There arc a lso many relations that arc independent of the relative angular momentum 52 between the mesons, they are (3.40d) (3.40e) (3.40!) (3.40g) IM(::::~ - ::::* 0 7r 0 7r+)l 2 = ~11\f(::::~ - n- K+1r+)l 2 2: ~I.M(::::~ - ::::*-7r+7r+)l 2 . (3.40h) Turning now to the Cabibbo-suppressed decays, again there are many relations between squared matrix elements. All except one of the relations between the squared matrix elements are independent of the relative angular momentum between the mesons. The relations are (3.41a) (3.41b) (3.41c) (3.41d) Also, there is a large number of relations between Cabibbo-allowed and Cabibbosuppressed squared matrix elements. The relations between matrix elements when 53 the mesons are either in a relatively even or odd angular momentum state are 3 2 + - t /:). 0~ +) (£=0,2, ... )12 ' IM(Ac+ - t /:). ++ 7r-7r 0)(£=0,2, ... )1 2 = 2s1IM(Ac J( 7r (3.42a) -*o}(o o) "'*o~}( 7r o) (£=0,2, ... ) 12 ' IM( -o .::.cl - t .::. 7r (£=0,2, ... ) 12 = 2sl2IM(-o .::.cl - t LJ (3.42b) There are also a few relations that are independent of the relative angular momentum between the mesons, they are (3.42d) (3.42e) (3.42!) Through a cancellation within each operator comprising the Hamiltonian we find that the Cabibbo-suppressed decay :=:~ 1 - t L:* 0ry 0rr 0 proceeds entirely through even angular momentum channels. There are sixteen reduced matrix elements contributing to the process S -t h* M 1\J, ten are from 0 15 and the remaining six are from 06. We find that there is only one relation between matrix elements and it is between Cabibbo-allowed decays with the mesons in an even angular momentum state (IJf(D~ - t L;*+K K-)1 2 = 2I JH(D~ - t L:*°KK)I 2 ). There are no relations between decays with the mesons in a relatively odd angular momentum state. Consequently the only relation is (3.43) and this is due to isospin. The matrix elements for these decays arc not tabulated . 54 3.4 Semileptonic Decay Of Charmed Baryons To Three Body Final States For the process T--+ hZ+vl there is only one SU(3) singlet possible from 3 ® 3 ® 8 and consequently only one reduced matrix element. Thus, the effective Hamiltonian for semileptonic decay can be written (3.44) where the weak Hamiltonian is given in Eq. (2.8 a,b ). All the matrix elements are related and we find that (3.45) Experimentally, only a few inclusive branching r atios have been measured [3. 7], they are (3.46a) Br(At --+ pe+ X) = (1.8 ± 0.9) x 10- 2 , (3.46b) Br(At--+ e+ X)= (4.5 ± 1.7) x 10- 2 , (3.46c) where X denotes unidentified hadrons and Ve. The only SU(:3) singlets that can be constructed for the process T--+ h* z+vl are (3.4 7) both of which vanish since h* is totally symmetric on its three indices . l-Ienee we would not expect to sec any lone decupleL resonances produced in the semileptonic decay of the charmed baryons in the 3 representation. 55 Turning now to the 6 representation and the process S -+ hl+v/, we see that only one non-zero SU(3) singlet can be formed from the available tensors, giving the effective Hamiltonian (3.48) It is obvious from the flavor wavefunction of the n~ that it cannot Cabibbo-allowed decay to a member of the baryon octet and that it will only decay via a Cabibbosuppressed mode to =:.-z+v/, consequently there are no relations possible. This, however, is not the case for the process S -+ h*[+vl where both Cabibbo-allowed and suppressed decays are possible. The effective Hamiltonian for the process is (3.49) from which we find that (3.50) 3.5 Semileptonic Decay Of Charmed Baryons To Four Body Final States Returning to the 3 representation of charmed baryons and looking at the decays T -+ hJo.Jl+ v1, we find that there are three reduced matrix elements that can contribute to the decay process. The effective Hamiltonian for such decays is Herr= a (TalfU(3)) (!i: J\!3) [+vi+ bTa~A1~Hc(3)l+vl + cTaA1t~Hc(3)!+vl (3.51) There are many relations between the squared m at rix elements for vanous decay modes, as shown in Table 3.13 for Cabibbo-allowcd decays and in Table 3.14 for 56 Cabibbo-suppressed decays. Many are due to isospin, for instance but some are due to the full SU(3) symmetry, for example (3.53) There is only one non-zero matrix element that can be constructed for the decays T---+ h* Mi+vl, and so the matrix elements for all the decay modes are related. The effective Hamiltonian is (3.54) where a is the unknown reduced matrix element. The relative squared matrix elements for Cabibbo-allowed (suppressed) decays can be found in Table 3.15 (3.16). Phase space correction factors must be applied as in the previous cases. By coincidence, the Cabibbo-allowed processes with the largest matrix elements are those that will be modified the most by these corrections. Turning now to the 6 representation and the decay process S---+ B.f\1/+vl, we find that there are three reduced matrix elements that can contribute and so the effective Hamiltonian for the process is (3.55) The results of which are shown in Table 3.17, from which we see that the only relations 57 between matrix elements are those due to isospin, such that (3.56a) (3.56b) (3.56c) Two reduced matrix elements contribute to the process S ~ h* [+vi for which the effective Hamiltonian is (3.57) the matrix elements of which are shown in Table 3.18. 'vVe see that there are relations not only due> to isospin but some due to the full SU(3)J symmetry. We find that 2IM(n~ ~ :=:* 0J(-J+vl)l 2 = 2IAI(O~-+ :=:•-ftl[+vt)l 2 = I .Mn~-+ n-7]0z+v,)l 2 ' (3.58a) 3IA1(0~ ~ :=:*0 7r-z+vt)l 2 = 61 1\I(O~-+ :=:*-7r 0l+vi)l 2 = IAI(O~-+ n- J\ 0 /+v,)l 2 ' (3.5Sb) and a lso one purely isospin relation (3.5Sc) 3.6 Concluding Remarks On Chapter 3. \'Ve have examined the predictions of flavor SU(3)J for the weak nonleptonic and semileptonic decay of charmed baryons both in the 3 and 6 representation of SU(3). The matrix elements for Cabibbo-allowed, -suppressed and doubly-suppressed decay modes were parameterized in terms of reduced matrix elements which have been tabulated explicitly. At the present time only a few decay modes (Cabibbo-allowed) have been experimentally obsen·ed; in the future when a larger event sample has been collected the relations derived in this work can be tested and/or used to reveal some of the underlying dynamics responsible for charmed baryon decay. 58 The predictive power of the SU(3)J invariance is, in some cases, somewhat limited due to phase space correction factors that must be included. However, these uncertainties can be eliminated by experimentally determining the relative contributions from different angular momentum channels. If the sextet component of the Hamiltonian dominates nonleptonic decay pro- cesses, as hinted at by perturbative QCD, then this will be directly observable by the absence of I = 1 final states in the doubly-Cabibbo-suppressed decay of the D~. Sextet dominance will also give rise to new relations between decay rates. These new relations between two-body decay modes have been considered previously in ref. [2.2, 3.8, 3.9] and can be derived from this work for all nonleptonic processes by neglecting the contribution from 0 15 . An interesting prediction of SU(3)t is that the 3 cannot semileptonically decay to an h*L+vl final state because a non-zero SU(3)J invariant cannot be constructed. Also, all the matrix elements for the semileptonic decay of the 3 to B t+ v1 final states are related. This is also true for the decays of the 3 to h* fiJI+ vi final states. Final state interactions (FSI) will be important for these decays. \Ve discussed their inclusion in the decay rates forD-mesons and the same arguements apply in this case. Consider the final state :\ 0 1r+, this proceeds entirely in a I = 1 partial wave and therfore E*+ resonances can be excited in the final state. For hyperon decays, this effect will be very small as the energy of the decay is substantially below the mass of the first E*+ resonance. However, in the decay of charmed baryons the energy release lies in the regime where these resonances could give rise to substantial phase shifts. As was done for D-mesons [2.8] these phase shifts can be fit to the data and removed from the SU(3)t amplitudes before a comparison is made between theory and experiment . It is possible that the de,·iation from the predictions of SU(3)J for charmed baryon decays will be less significant than forD-meson decays as SU(3)J is a better symmetry for baryons than for mesons. 59 References [3.1] ARGUS Collaboration, H. Albrecht et al., Phys. Lett. B207, 109 (1988). [3.2] J. Weiss et al., Invited talk at the 4th Int. Conf. on Baryon Resonances, Toronto, Canada, July 14-16, (1980). Published in Baryon 1980: 319 (QCD161: C45: 1980). [3.3] E. Vella, PhD Thesis, UC Berkeley and LBL, Berkeley, Dec 1981. [3.4] M. Basile et al., Nuovo Cim. 62A, 14 (1981). [3.5] G.S. Abrams et al., Phys. Rev. Lett. 44, 10 (1980). [3 .6] M. Adamovich et al., Europhys. Lett. 4, 887 (1987). [3.7] E. Vella et al., Phys. Rev. Lett., 48, 1515 (1982). [3.8] K. Odaka et al., Prog. Theor. Phys. 59, 666 (1978). [3.9] G. Altarelli et al., Phys. Lett. B57, 277 (1975) . 60 Table 3.1. Squared matrix elements for Cabibbo-allowed decays T ---+ hM in terms of the reduced matrix elements a, b, c, d, e, f and g. Squared Matrix Element Process 01r+ A+-+A c A+---+ r;o7r+ c A+---+ r;+7ro c At---+ r;+7JO At---+ pl(u At ---+ :=;o J(+ =0 ---+ =-'lr+ ~cl ~ =0 ---+ =07r0 ~c l ~ =0 ---+ =0170 ~cl ~ =o Aor ~cl ---+ \ :=:~1 ---+ r;+ f{- =o ')'.or ~cl ---+ <..J \ =+ ---+ =07r+ ~cl ~ =+ ')'.+r ~cl ---+ .:..J \ tla + b- 2c- 2e- 2f- 2gl 2 !Ia- b- 2e + 2f + 2gl 2 !I -a+ b+ 2e- 2f- 2gl 2 tla + b- 2d- 2e- 2f + 2gl 2 Ia + c- 2el 2 lb + d- 2fl 2 Ia + c + 2el 2 !I- a+ d- 2e + 2gl 2 !Ia- 2b + d + 2e- 4f- 2gl 2 tl- 2a + b + c- 4e + 2f + 2gl 2 lb + d + 2fl 2 !I - b+ c- 2f- 29 12 1- c- d- 2gl 2 1- c- d + 2gl2 61 Table 3.2. Squared matrix elements for Cabibbo-suppressed decays T ~ hM in terms of the reduced matrix elements a, b, c, d, e, f and g. Process Squared Matrix Element (Modulo si) A+~ A0 ](+ c ~~-a+ 2b + 2c +3d+ 2e- 4f + 2gl 2 A+~ 'Eo J(+ c il -a-d+ 2e- 2gl 2 I -a+ d + 2e- 2gl 2 ~12a- b + 3c + 2d- 4e + 2f- 2gl 2 il- b- c + 2f + 2gl 2 I - b + c + 21 + 2gl 2 Ia + c + 2el 2 2 / 2 1- a-b-c+ 3d- 2e- 2f + 4gl A+~ E+ K 0 c At~ PTJo At~ p;rro A+~n;rr+ c -::-o ~ E-;rr+ -::-o ~ Ao;rro -::-0 ~ 'Eo;rro ila + b- c- d + 2e + 2/1 2 -;:-0 ~ AOT]O -;:-0 ~ 'EOTJO il- a- b + c + d- 2e- 2fl 2 2 / 2 1- a - b + 3c- d- 2e- 2f- 4gl Ia - b + 2e- 2/- 2gl 2 1- a- c- 2el 2 I -a+ b- 2e + 2f + 2gl 2 ~cl ~cl ~cl ~cl ~ cl ~o r ::-0 ~ ::-- !{+ ::-0 ~ ::-0 .::..cl ~ n ~cl ~cl \ ~ ~ J(O ::-o ~ E+;rr~cl :=:~1 ~ p](3~ ~ i\ O;rr+ ::-+ ~ >:O;rr+ ~cl "-' ::-+ ~ "+;rrO ~cl -' ::-+ ~ "+TJO ~cl -' ~+ .::..cl ~ p To \ lb + d + 2fl 2 1- b- d- 2/1 2 ~~-a-b-c- 3d+ 2e + 2f- 4gl 2 1-1- a+ b + c + d + 2e- 2/1 2 !Ia - b + c + d- 2e + 2/1 2 il- a- b- 3c- d + 2e + 2/ + 4gl 2 1- a+ d + 2e- 2gl 2 ::-+ ~ ::-0 !{+ ~cl I - b + c + 21 + 29 12 ~ 62 Table 3.3. Squared matrix elements for the doubly-Cabibbo- suppressed decays T-+ hM in terms of the reduced matrix elements a, b, c, d, e, f and g. Process Squared Matrix Element (11odulo st) :=:;;+i-+ Ao J(+ lc + d- 2gl 2 lc + d + 2gl 2 Ia + c + 2el 2 ~Ia- 2b + c + 2e + 4f- 4gl 2 ~~-a+ c- 2el 2 ~~- 2a + b + d- 4e + 2f + 4gl 2 lb + d + 2fl 2 ~I - b + d- 2!1 2 1 61a+ 2b- c + 2e- 4f- 4g 12 :=:;;+i -+Eo J(+ ~~ -a+ c + 2el 2 :=:;;+i -+ E+ J(O ::-+ -+ PT/o 1- a- c + 2el 2 ~l2a- b- d- 4e + 2f + 4gl 2 =+ -+ p7ro ~I - b + d + 2!1 2 ::-+ -+ n7r+ ~c1 1- b- d + 2fl 2 At-+ pJ(O A+-+ nJ(+ c ::-o E- J(+ ~cl-+ ::::~1 -+ A 0 J(O ::::~1 -+ Eo J(o ::-0 -+ nryo ~cl ::-0 -+ p7r ~c1 ::-0 -+ n1ro ~cl ~cl ~c1 63 Table 3.4. Squared matrix elements for Cabibbo-allowed decays T ---t h* 111 in terms of the reduced matrix elements a, (3, 1 and 8. Process Squared Matrix Element A+ ---t ~·+71'0 c tl - 2a + (3- 21- 81 2 11812a- (3- 21- 3812 tl - 2a + (3- 21- 81 2 At ---t ~·+770 A+ ---t ~·o7!'+ c if3 + 812 ilf3 + 81 2 kif3- 2,- 812 A+ ---t 6++ ](c A+ c ---t 6+ft3 At ---t :=:•O ](+ 1iai 2 11ui 2 -=+ ~c1 ---t ~·+r \ -::-+ ---t -::-•07!'+ ~c1 ~ ~c1 ---t =o ~·or \ i"i2a- (3 + 21- 81 2 =0 ---t -::-•07!'0 -:;0 ---t =•01]0 tl2a- (3 + 81 2 1 2 181- 2a + j3 - 41 + 381 !1-(3+2,-8,2 ~c1 ~c1 ~ ~ :=:~1 ---t ~·+ ](- -:;0 ~c1 ---t =•-7!'+ !1-(3+81 2 I - f3 + 81 2 ~ :=:~1 ---t n- ](+ 64 Table 3.5. Squared matrix elements for Cabibbo-suppressed decays T --+ h* M in terms of the reduced matrix elements a, {3, 1 and 8. Squared Matrix Element (Modulo si) Process A+ --+ .6. +7ro c il- a- I - 81 2 At --+ .6. +770 ~Ia + f3- 1l 2 A+ c --+ .6. o7r+ il - 2a + f3- 21- 81 2 i I - 2a + f3 + 81 2 tl2a+f3-21-8l 2 A+ --t ~*+ J(O c A+ --+ ~*o ](+ c A+ --+ .6. ++7rc 1!3 + 81 2 !l2a- f3- 81 2 tl- 2a- f3 + 21 + 81 2 2 / 8 J.ia: + f3 + 2~1 + 381 -==+ ~cl --t .6.+r \ -=+ --t ~*071"+ ~ cl -=+ --t ~*+7]0 ~cl !l2a- f3 + 21 +of I - f3- 81 2 !1-/3+21+81 2 !12a-f3+21-81 2 l212a- f3- 21 + 381 2 2 ] 6 12a- f3- 21 + 381 -=+ --t '='*0 ](+ ~cl ~ ::::~ --t .6. ++ ](- -=+ --t ~*+7r0 -=O ,6.oy 0 ~ cl ~cl --t \ -=0 --t ~*071"0 ~cl -=0 --t '):*0770 ~cl ~ -=0 ~cl --t !12a:- f3 + 21- 81 2 !1-/3+21-81 2 11 - f3 + 012 11 - f3 + 012 t I- f3 + 21- 812 '='*0 ](0 ~ -=0 --t ~*+7r~cl -=0 --t ~* - 7r+ ~cl -=0 ~cl --t '='*- ]{+ ~ ::::~1 --t .6. + ]( - 65 Table 3.6. Squared matrix elements for the doubly-Cabibbo- suppressed decays T ~ h* M in terms of the reduced matrix a:, {3, 1 and 8. Process Squared Matrix Element (Modulo si) A+~~+ J(O tla:l 2 A+~ ~o ](+ c tla:1 2 -:=+ ~ ~ +7]0 ~c1 ::::~ ~ E*+ f{o %I - 2a: + f3 - 112 il- 2a: + f3- 2/- 81 2 l/3 + 812 ~1/3 + 812 -:=+ ~ ~07!"+ ~1/3- 2/- 81 2 -:=+ ~ ~ +7!"0 ~~~ + 81 c ::::~ ~ E*o ](+ -:=+ ~ ~ ++7!"~c1 ~c1 ~cl -:=0 ~cl ~ ~ 07]0 2 ~I - 2a: + f3 -~12 -:=0 ~ ~ -7!"+ ~cl il - 2a: + f3- 2/ + 81 2 2 ~1/3- 2, + 81 l/3- 812 ::::~1 ~ E*- ]{+ ~1/3-81 2 -:=0 ~ ~ 07!"0 ~cl jb- 81 2 ::::~1 ~ E*o ](o -:=0 ~ ~ +7!"~ c1 66 Table 3. 7. Squared matrix elements for the decays S ---t hM in terms of the reduced matrix elements a, b, c, d, e, J, g, h and k. Process Squared Matrix Element n~- :=.o-K n~ - :=.o,o I- a- b + 2kl 2 sB-I2a - c + d + 2h - W sr? 611 - 2a- 3b- 2c- 2d- 3e- 2g + 4h + 41.: + l I'-) n~ - :=,o7ro si~lb- e + 2g + W no~oK c n~ - :=.-7r+ silb+e-2g-w n~- ~+ K- siic- d- 2h + W sUic +3d+ 2e + 4g- 2h + 41.:- 11 2 no---tAo J(o c n~- ~o,o n~ ---t :=,O J(O n~ - :=.- J(+ sitl-2a+c-2d+W sil- b- c- 2/- e- 2gl 2 silb + c- 2/ + e- 2gl 2 no---t Ao1ro c siklc+2e-W siic- 2/ + W no~-7r+ c no~+7rc n~ ---t nK silc + 2/ + ll 2 sild- 2f- 2hj2 0 n~- pi\- sjld + 2/ + 2hl 2 n~ ---t Ao,o sHI3f + 2g + 2h + 21.:1 2 noc - ~o7ro si4lfl 2 67 Table 3.8. Squared matrix elements for the decays S -+ h* !vi in terms of the reduced matrix elements a, /3, 1, 8, >. and"'· Process Squared Matrix Element n~-+ =.·o~ ~12a + f3 + 2TJI 2 n°-+ n-7r+ c no-+ ~*0~ c n~ -+ =.·o,o n~ -+ =.•o1ro n~ -+ =:•-7r+ n°-+ n- ](+ c l/3- 2,12 s~ii2a -~- 28- 2.>.1 2 s~ /8 14a + 3/3 + 1 + 48 + 6>. + 6TJI 2 sUi - f3- I+ 2.>.- 2TJI 2 2 s~ ~i/3- 1 + 2.>.- 2TJI s~i - f3- 1 + 2.>. + 2TJI 2 s~ ~~~ + 28 + 2.AI 2 sHi- 2a + 1 - 281 2 sftl/3 + 1 + 2.>. + 2TJI 2 sf~i/3 +I- 2.>.- 2TJI 2 sf 1h + 2.>.1 2 n~-+ ~·+ J{- -+ ~·o,o n~ -+ =.•o Ko n~ -+ =:•- ](+ noc -+ ~·+7rnoc -+ ~·-7r+ n~ sfth- 2.>.12 n~-+ ~+ K- sf1181 2 no-+ ~oftl c noc -+ ~·07r0 sf1181 2 sf!i>-1 2 68 Table 3.9. Squared matrix elements for the Cabibbo-allowed decays T ~ h* M M where the mesons are in a relatively even angular momentum state in terms of the reduced matrix elements AJ,BJ,CJ , DJ,Ef,FJ,GJ,IJ,Cs,Es, Fs and G 8 • Process Squared Matrix Element At ~ :=:*-7r+ J(+ ti2AJ + 2BJ- CJ + 4DJ- FJ- 2IJ + Cs + Fsl 2 ~~- 2BJ + Ff- 2GJ- 2IJ + 2Cs + Fsl 2 ti2AJ + 2BJ + 4DJ + EJ- 2IJ- Esl 2 11- CJ- EJ- FJ + Cs + Es + Fsl 2 2 ~~- 2BJ + F1- 2GJ- 2IJ + 2Cs + Fsl 2 ~IAJ- C1- !FJ- !1 + Cs + !Fsl ti2AJ + EJ- 2IJ- Esl 2 tl- CJ- F1 + 2GJ- Cs- Fsl 2 tl- CJ- Ff + 2GJ- Cs- Fsl 2 f;l- CJ- 2EJ- F1 + 2GJ- Cs- 2Es- Fsl 2 tl- CJ- EJ- F1 + 2GJ- Cs- Es- Fsl 2 41 - C J - E J - F1 + C s + E s + Fs 12 11- 2CJ- EJ- F1 + 2Cs + Es + Fsl 2 A+~ 6_++7r-K c IE!+ Esl A+ ~ E*+7ro7ro c At ~ E*+ 7ro7Jo A+ ~ E*+1r-1r+ c A+ ~ E*o7ro7r+ c At ~ E*07r+7JO At ~ :=:*o J(+7ro At ~ :=:*o KO?r+ A+ ~ 6. ++ J(-7ro c A+~ 6_+ K-7r+ c A+~ D,.+K1r 0 c A+~ 6.°K7r+ c A+ c ~ E*-1r+1r+ :=:~ ~ E*+1r°K :=:~ ~ L;*+TJO J{o ::-+ ~ '\'*+ f{-1r+ '--'cl ---J ' -=+ ~cl ~ "'*0 --' 7r + r \ ::-+ ~ '::'*0770 7r+ ~c1 ~ -::-+ D,.++y-y?J \ \ ~cl ~ 2 il- 2.4/- 2BJ- EJ + F1- Es- Fsl 2 2 1 1 8 1- 2AJ- 2BJ- 3EJ- F1- 3Es- 3Fsl 31 I2A f + 2Bf + E f - Ff - Es - Fs I'~~IEs + Fsl 2 2 / 8 1-4.4/- 4BJ- 3EJ + Ff + 3Es + 3I'.sl IEJ+FJ+Es+Fsl 2 -=+ ~cl ~ D,.+rr \ \ tiEJ + Ff + Es + Fsl -::-+ ~ '::'*07r07r+ tl- Ef- Ff + Es + Fsl 2 ~c1 ~ 69 2 Table 3.9. Continued Process Squared Matrix Element -::+ ---+ '::"*-7r+7r+ ~c1 -=+ ~c1 ::::~1 - ~ n- J(+7r+ ~*+7r- Jtl ::::~1 ---+ ~*+ J(-7r0 :::;~1 ---+ ~*+ J(-TJO ::::~1 ---+ ~*o J(o 7ro :::;~1 ---+ ~*o J{u T'Jo -::-0 ---+ -::-*OTJ01r0 ~cl ~ -:;0 ,6. ++ J(- y~cl ---+ .. -=0 ,6. + J(- r \. ~cl ---+ -=0 ~cl ---+ ~*- 7r +..r -::-0 ---+ '::"*-7r+TJO ~cl ~ ::::~1 ---+ ~*07r+ J{- -=0 ~c1 ---+ b.orr \ \ -::-0 ---+ '::"*07r+7r~c1 ~ -::-0 ---+ -=*-7r07r+ ~c1 ~ ::::~1 - n- Ko7r+ -::-o ---+ -=*O _o7ro 1 ~c1 ~ ::::~1 ---+ n- ]{+ 1ro ' il- EJ- Ff + Es + Fsl 2 I - E J - Ff + Es + Fs 12 11- 2AJ- EJ + 2IJ- Esl 2 tl- 2AJ + Ct + Ft- 2GJ +2ft- Cs + Fsl 2 2 / 8 12AJ- 3CJ- Ff- 2GJ- 21!- 5Cs- Fsl 2 / 2 1- 2BJ + CJ + Et + 2FJ- 2Gt - 2ft- Cs + Esl tl- ~Bt- CJ- Et- ~Ft- ~OJ -~If- iCs- Es- tFsl 2 2 ~I2AJ + 2BJ- CJ- Ff + 2GJ + Cs- Fsl 2 4ICJ + Csl ti2Ct + EJ + Ff + 2Cs + Es + Fsl 2 2 1ICJ + EJ + Ft- 2GJ- Cs- Es- Fsl lsi- 2Ct- 3Et- Ft + 4GJ + 2Cs + 3Es + Fsl 2 2 ii2BJ + CJ + Et- 2GJ + 2IJ- Cs- Esl 2 tiCJ + EJ + Ff + Cs + Es + Fsl 11- 2BJ- 4DJ- EJ + E sl2 2 iiEJ + Ff- Es - Fsl 1- Ef + Esl 2 11- 2BJ- 4DJ + Ff- Fsl 2 ~IF!- Fsl 70 2 Table 3.10. Squared matrix elements for the Cabibbo-allowed decays T ----+ h* M M where the two mesons are in a relatively odd angular momentum state in terms of reduced matrix elements Aj, B/, C/, Dj, E/, Fj, Gj, 1j, C~, E~, F~ and G~. Squared Matrix Element Process At ----+ L:*+ 7ro7Jo ~I2Aj + Cj + C~ + 2F; + 2G~I A+ c ----+ E*+1r-1r+ il- 2Aj- 2Bj- Ej + 21j + E~l 2 A+ ----+ E*o7ro7r+ c ii2Aj + 2Bj + Ej- 21j- E~l 2 At ----+ L:*07r+7JO ~I2Aj + Cj + C~ + 2F~ + 2G~I 2 At ----+ ::::•o J(+7ro ti2Aj- Fj- 21j + F~l 2 2 li')A' 3-1 + E'I - 21'I - E's 1 A"'d ----+ ::::*o J(07r+ 2 A+ ----+ 6. ++ K-7ro c 2 liC' 2 1 + F'1 + C' ' s + F's + 2G's 1 A+ c ----+ 6. + K-7r+ A+ ----+ 6. + J(o 7ro c iiCj + Fj + C~ + F~ + 2G~I 2 11C'I + 2E'I - F'C's + 2E'F'2G' 12 6 f s s A+ ----+ 6. oft 1r+ c 2 11C'f + E'f - F'C's + E'F'')Q' 3 f s s -s 1 A+ ----+ 6. ++7r- ftl c IE/+ E~l 2 t I - E/ - Fj + E~ + F~ 12 11 - 2A'f + 2B'f - E'f + F'f - E's - F's 12 6 A"'d ----+ ::::•- J(+7r+ ::::~ ----+ E*+ 7ro~ 7 S ::::~ ----+ E*+ry°K 2 li2A' 18 f + 6B'f + 3E'f + F'f + 3E's + 3F'I s ::::~ ----+ L;*+ J(-7r+ 2 li2A' E'F'l 3 f + 2B'f + E'f - F'f s s -=-+ ~cl ----+ L;*O 7r +\ r ~~ - 2Aj- Ej + Fjl 2 -::-+ ----+ '::"*07]07r+ 2 l 18l - 4A'f - 3E'f + F'f + 3E's + 3F'I s ~c1 ~ -=-+ ~cl ----+ 6_++r-r \ \. IE'f + F'f + E's + F's 12 -::-+ ----+ '::"*07r07r+ ' tl- 4B/- Ej- Fj + E~ + F~l 2 ~c1 ~ ::::~ ----+ n- K+7r+ I ~ E/ - Fj + E~ + F~ 12 ::::~1----+ E*+7r-JC1 il- 2Aj- E/ + 21/- E~l 2 ::::~1 ----+ E*+ K-7ro tl - 2Aj- Cj + Fj- 21/ + C~ + F~ + 2G~ 12 71 Table 3.10. Continued. Process Squared Matrix Element ::::~1 --+ E*+ K-.,.,o 2 ..l..I2A' 18 f - C'f - F'f - 61'f + C's - F's + 2G's 1 ::::~1 --+ E*o~ 7ro l 2 14A/ + 2B/ + Cj + E/ - 2Ij - -C's + E's - 2F's - 2G's 1 2 ::::~1--+ E*o.,.,o-ytJ --+ =·o.,.,o7ro ~ + 6B/ - Cj + 3E/ - 6Ij __+C~- 3E~ + 2F~ + 2G~I ti2A/ + Cj- C~- 2F~- 2G~I ::::~1 - E*-7r+K 2 liC' 3 / -E'f +F'f -C's +E's -F'-2G'I s s ] 6 14A/ 2 =o ~c1 ':;'0 - ~c1 =•-.,.,07r+ ~ 2 2 ..1...1 18 - 2C'f + 3E'f - F'f + 2C's - 3E's + F's + 4G's 1 ::::~1- 6_+ K-K liE' 3 f + F'f + E's + F'l2 s ::::~1 --+ E*o K-7r+ i I - 2B/ + Cj - E/ + 2Ij - C~ + E~ - 2G~ 1 2 ':;'0 ~c1 '::'*07r-7r+ ~~- 2B/- E/ + £~1 ':;'0 - =•-7r07r+ liE' E's + F'l2 6 f - F'f s ~c1 ~ ~ n- Ko7r+ ::::~1 - n- J(+7ro ::::~1 - 1- E/ + E~l 2 1 1F'2 f 72 FT s 2 Table 3.11. Squared matrix elements for the Cabibbo-suppressed decays T ---+ h* }vfM where the mesons are in a relatively even angular moment um state in terms of reduced matrix elements Af, Bf, Cj, Df, Ej, Fp, Gf, If, Cs, Es, Fs and Gs. Process A+ c ---+ .6. +7ro7ro At ---+ .6. + 7ro7Jo Squared Matrix Element (Modulo si) !IAJ + BJ + 2DJ + EJ- GJ- If+ Cs + Es + Fsl 2 il2BJ + CJ + 3EJ + F1 + 21!- Cs + 3Es + Fsl A+ c ---+ .6.+7r+7r- !l2AJ + 2BJ- CJ + 4Df + Ef- FJ +2GJ- 21!- Cs- Es- Fsl 2 A+ c ---+ .6. + J(+ J(A+ c ---+ .6. + Jt3 J(o !I4DJ + Ff- 21! + Fsl ll2BJ + 4DJ- Ff- Fsl A+ c ---+ .6. o1ro1r+ ~~- CJ- EJ- FJ- 2GJ + 3Cs + 3Es + 3Fsl At ---+ /). 01r+1JO lsi- 4Bf- 3Cf- 3E1 - F 1 + 2G1 2 -4ft+ Cs- 3Es- Fsl A+ c ---+ .6. oftl J(+ A+ c ---+ E*o K+7ro A+ c ---+ E*o J\07r+ A+ c ---+ E*+ K+1rA+ c ---+ E*+ Ko7ro A+ c ---+ .6. ++7r-7ro !1- 2Bt + Ff- 21! + Fsl l214AJ + 2BJ- CJ + 3EJ- 2GJ- 2IJ + 3Cs- Esl 2 ~I4AJ + 2BJ- CJ + EJ- 2FJ + 2GJ- 21!- Cs- Esl !1- 2AJ- 2BJ- CJ- EJ + 2GJ- Cs + Esl 2 2 ~I2At + 2Bf- CJ- Et- 2Ft+ 2Gf- Cs- Esl ~~- Ct- Et- Ft + 2Gt- Cs- Es- Fsl 2 A+ c ---+ .6. ++ ! { - J(O 1Ft+ Fsl2 At ---+ ~ ++7J07rAt ---+ .6. -7r+~o+ ~~- CJ + 3Et + Ft + 2Gt- Cs + 3Es + Fsl 41- Ct- EJ- Ft + Cs + Es + Fsl 2 Ac---+ E*- K+1r+ 11Ct- Csl K ~o '' :=:~ ---+ /). + Jt1 ,,o ~~- 2At- 2Bt + Ct- EJ- 2Gt + Cs- Es- 2Fsl 2 ::+ ---+ /). + "-'c l 2 /sl - 2A t - 2Bt + C t + 3Et + 4Ff -2Gt + Cs + 3Es + 2Fsl 73 2 2 2 Table 3.11. Continued Process Squared Matrix Element (Modulo -+ .6. + J(-1!"+ :=:~ -+ .6. o-gD 7r+ :=:~ -+ L,*oK J(+ :=:~ :=:~ -+ L,*01J01r+ :=:~ -+ L,*+ J(- J(+ =+ -+ L,*+1J01r0 ~c1 -+ .6. ++ J{-1!"0 :=:~ -+ .6. ++ K-17° =+ -+ L,*+7r07r0 ::+ -+ L,*07r07r+ =+ -+ '=*0 J(+1r0 :=:~ -+ L,*-7r+7r+ =+ -+ =*- J{+1r+ ::::~ -+ .6. ++-K 7r:=:~ ~c1 ~cl ~c1 ~ ~cl ~ :=:~-+ L,*+~ J{o =+ -+ ~*+7r-7r+ ~cl .:__J =+ -+ '=*0 1{01!"+ ~cl ~ ::::~1-+ .6_+~7!"::::~1 -+ .6. + ]{-1!"0 ::::~1 -+ 6. + ]{-1]0 ::::~1 -+ 6. o r 7ro sD l12AJ + 2BJ + Cf + Ef- 2GJ + Cs- Esl 2 !ICJ + E1 + Ff- 2GJ + Cs- Es- Fsl 2 ~~- 4Af- 2BJ + CJ- Ef + 2FJ 2 -2GJ + 2IJ + Cs + Esl ~~- 4Af- 2BJ- 3Ef + 2GJ + 2IJ- 2Cs- Fsl 2 !1- 2AJ- 2BJ + CJ- 4Df- EJ + Ff 2 -2GJ + 2IJ + Cs + Es + Fsl ~~- 2AJ + Ff + 2GJ + 2IJ- 2Cs- Fsl 2 !ICJ- E1- 2Gf + Cs- Esl 2 2 iiCJ + 3EJ + 2Fj- 2GJ + Cs + 3Es + 2Fsl !ICJ- 4DJ + Ef + 21!- Cs + Esl 2 iiCJ- Cs + Es + Fsl 2 ii2BJ + 2CJ + 2EJ + FJ + 21J- 2Cs- Fsl 2 tiCJ- EJ- Ff- Cs + Es + Fsl 2 1ICJ- Csl 2 2 IFJ + Fsl 1 312BJ- 4DJ + Ff + Fs 10~ 1 314DJ- Ff- Fs + 21! I'~) ii2BJ- Ff + 21J- Fsl 2 tl- 2.4! + 2CJ + Ff + 21! + 2Cs + Fsl 2 il- 2AJ- CJ - EJ - 2GJ + 2IJ- 3Cs- Esl 2 { 8 12AJ + 3CJ + 3EJ + 2FJ- 2GJ 2 -21! + Cs + 3Es + 2Fsl ~~- 2BJ- CJ- EJ- 2GJ- 2IJ- 3Cs- Es- 2Fsl 2 74 Table 3.11. Continued Process Squared Matrix Element (:Modulo sf) :=:~1 -+ 6 oytJ 7Jo 1 18 1- 2Bt + 3Ct + 3Et +4Ft- 2Gt -2ft+ Cs + 3E + 2Fsl 2 jl- 2At- 2Bt + Ct- 4Dt + Ft +2ft+ Cs + Fsl 2 1 18 14At + 2Bt + Ct + 3Et +2Ft 2Gt- 2ft+ 3Cs + 3Esl 2 tl- 2At + 2Ct + Ft +2ft+ 2Cs + Fsl 2 8 2~1 -+ ~*oftl J(o :=:~1 -+ ~*071"07]0 2~1 -+ ~*+ J(- J(O :=:~1 -+ ~*+7r-7]0 1 1 8 1- 4At- 3Ct- 3Et- Ft- 2Gt +4ft- 5Cs - 3E8 - Fsl 2 :=:~1 -+ 6 ++ J(-11"- 4ICt + Csl 2 :=:~1 -+ ~*+7r-7ro tiCt + Et + Ft- 2Gt- Cs + Es + Fsl 2 :=:~1 -+ 6 0 J(-11"+ ti2Bt + Ct + Et- 2Gt +2ft- Cs- Esl 2 :=:~1 -+ ~*0 J(- J(+ tl- 2Bt + Ct- 8Dt- Et- 2Gt +2ft- Cs + Esl 2 :=:~ 1 -+ ~*o 7ro7ro ~I - 2Bt - ct - SD t - E t + 2G t +2ft+ Cs- Es- 2Fsl 2 :=:~1 -+ ~*07r-7r+ tl- 2Bt + Ct- SDt- Et- 2Gt +2ft- Cs + Esl 2 :=:~ 1 -+ 6- Jt1 1r+ ICt + Et + Ff- 2Gt- Cs- Es- Fsl 2 :=:~] -+ ~*- Jtl J(+ tiCt- Et + Ft- 2Gt- Cs + Es- Fsl 2 :=:~1 -+ ~*-7r+7]0 1 I'' ISICt- 3Et + Ft + 2GJ + Cs + 3Es- Fs- :::-0 -+ '):*-7r07r+ tl- Ct + EJ + Ff + 2Gt + Cs- Es- Fsl 2 -;:::-0 -+ '::"*0 f{+1r- ti2BJ + Ct + Et- 2Gt + 2fJ- Cs- Esl 2 ~c1 ~c1 -'-' ~ :::-0 -+ '::"*0 J{01r0 ~ c1 ~ ::-o ~c 1 -+ :::-* ~ J{+ 7ro -o -*~yo + .::.cl -+ .::. \ 7r tl- 2Bt + Ct + Et +2Ft- 2Gt- 2fJ- Cs + Esl 2 t I - Ct - E t + Ft + 2Gt + C s + E s - Fs 12 tiCJ- Et + Ft- 2GJ- Cs + Es- Fsl 2 75 Table 3.12. Squared matrix elements for the Cabibbo-suppressed decays T--+ h* M M where the mesons are in a relatively odd angular momentum state in terms of reduced matrix elements Aj, Bj, Cj, Dj, Ej, F}, Oj, I/, C~, E~, F~ and 0~. Process Squared Matrix Element (Modulo si) At --+ .6. +7ro7Jo 2 li2A' 1 9 f - C'f + 3E'f - F'f - C's + 3E's + F's - ?G' ~s A+ --+ .6. +7!"+7!"c ii2Aj + 2Bj- C/ + Ej- F}- 2Ijl -C's - E's - F'20's 12 s A+ --+ .6. + J(+ J(c .!.IF'21'f + F'l 3 f s A+--+ ~+K K 0 c li?B' 3 ~ f + F'f + F'l s A+ --+ ~ o7ro7r+ c 6 I4A'f + 4B'! - C'f + 3E'! - F'! - 41'f 1 -C's - E's - F's - 20's I At --+ ~ o7r+7Jo ..!...I4A' 18 f - C'f + 3E'f - 3F'f - C'S + 3E'S + F'S - ?0' ~ TS A+--+ ~°K J(+ c i I - 2Bj - F} + 21/ - F~ I A+ --+ E*o J{+ ?ro c 1 12 I2Bj + C/ + Ej + 2F} + 21/ +C's - 3E's - 2F's + 20's 12 A+ --+ E*o J(O?r+ c 2 li?B' 6~ f + C'f - E'f + ?J' ~~ + C's + E's + 2F' s + 2G's 1 At --+ E*+ K+1r- .!.I? A' + 2B'f + C'f + E'f + C'E's + 2G's f 3~! s A+ c --+ E*+ Ko?ro ti2Aj + 2B/ + C/ + E/ + C~ + E~ + 2F~ + 2G~I A+--+ ~++7!"-?ro c 2 1.1 2 C'f + E'f + F'f + C's + E's + F's + ?G' ~ Ts 1 A+ --t ~ ++ J{- J(O c IF/+ F~l 2 At --+ .6. ++1Jo1r- 2 .!.IC' 6 ! - 3E'f - F'f + C's - 3E's - F's + 2G's 1 At --+ 2:*- J{+?r+ 2 iiE' E'F'l 3 f + F'f s s ::::~--+ ~+Jtl~.o ii2Aj- 2B/ + C/- Ej + C~- E~ + 2F~ + 2G~I 2 ::::~--+ ~+Kryo 1- 2Aj- 6B/- C/- 3E/ -C~- 3E~- 2F;- 2G~I /8 2 76 2 Table 3.12. Continued Squared Matrix Element (Modulo si) Process ::::~ ~ /:}. + J(-7r+ ::::~ ~ /:}. 0](0 7r+ ::::~ ~ E*°K ](+ ::::~ ~ E*o 77 o7r+ ::::~ ~ E*+ ](- ](+ ::::~ ~ E*+ 77 o7ro ::::~ ~ /:}. ++ J(-7r0 ::::~ ~ /:}. ++ ](-1]0 !1- 2A/- 2Bj- Cj- E/- C~ + E~- 2G~I 2 2 114A' 3 f + C'f + E'f - F'f + C's - E's + F's + 2G's 1 112B' I' + C's + E's + 2F's + 2G's I 6 f + C'f - E'f + ?-J 11C'f - 2F'C's + 3E's + F'2G's 12 9 f s 1 3 12A'f +2B'f - C'f +E'f - F'f -2Ij- C'E'F'2G's 12 s s s 112A' C's + 3E's + F'2G's 12 9 f - C'f + 3E'f - F'f s -6Ij - 2C~ - F~ - 4G~ 12 2 11 2 - C'f + E'f - C's + E's - 2G's 1 2 11C'f - 3E'f - ?F'3£'2F'?G' 6 - ! C's s s - s1 ::::~ ~ E*o1ro1r+ t~l- 4B/ - 2E/- 4Ij + 2F~I 2 '::'+ ~ '::'•0 J\+7r0 11?B' 21'f - 2E'F'ls 2 6 - f + F'f s 1 1E' + F'- E'- F'l 2 3 f f s s IF/+ F~l2 11?B' 3 - f + F'f + F'l s 11F' 3 f - ?J' - f + F'l2 s ~cl ~ '::'+ ~ '::'*J\+7r+ "--'cl ..._, ::::~ ~ /:}.++Jtl7r'='+ E*+pyo ~cl ~ \ \ '='+ ~ E*+7r-7r+ ~c1 '::'+ ~ '::'*0 ]{01r+ ~I2B/ + F/- 2Ij + F~l ::::~1 ~ /:}. + K 1r- 2 li?A' 3 - f -F'f -21'f -F'I s ::::~1 ~ /:}. + J(-1r0 ::::~1 ~ !:}. + K -1/o ti2A/ + Cj + E/ + 2Ij - C~ + E~- 2G'~I 2 / 8 1- 2A/ + Cj - 3Ej - 2F/ 2 +61'f - C's - 3E's - 2F's - ?G' - s1 ::::~1 ~ !:}. o1\o 7ro ti4A/ + 2Bj + Cj + Ej- 2Ij- C~ + E~ - 2F~ - 2G~I 2 ~c1 ~ 77 2 Table 3.12. Continued Squared Matrix Element (Modulo si) Process ::::~1 -+ .6. oJtl TJo ~J-4A~-6B~+C~-3E~+6~ -C's - 3E's - 2F's - 2G's J2 ::::~1 -+ E*+ ](- J(O iJ2A~- Fj- 21~- F~J 2 ::::~1 -+ E*+7r-TJo 1 8 J4A~ + Cj + 3E~ + Fj 1 2 -C's +3E's +F'-2G'J s s ::::~1 -+ E*+7r-7ro 2 lJC'f + E'f + F'41'f + C's + E's + F's + 2G'J 6 f s ::::~1 -+ .6. o K-1r+ lJ?B' - C'f + E'f - 21'f + C's - E's + 2G's 12 3~/ C'f - E'f - 21'f + C's + E's + 2G's 12 ::::~1 -+ E*o K- J(+ lJ - ? B' 6 , - ::::~1 -+ E*o7r-7r+ 2 lJ-2B' 6 f -C'f -E'f -?I' - f +C's +E's +?G'J ~s ::::~1 -+ .6.- Jt1 7r+ I - 2 C'f + E'f - F'f + C's - E's + F's + ?G' -s 1 ::::~1 -+ E*- K J(+ 2 lJC'E'f - F'f + C's + E's + F's + 2G'J 3 f s ::::~1 -+ E*-1r+T)O _l_JC' C's + 3E's + F'2G's 12 18 f - 3E'f - F'f s ::::~1 -+ E*-7ro7r+ 2 lJC' C's - E's + F'?Q'J 6 / + E'f - F'f s -s -::0 -+ "::'*0 J{+Jr- tJ2B/- C j + E~- 21~ + C~- E~ + 2G~J 2 -::o -+ -=*o J{o 7ro 2 lJ ? I' 6 - 2B'f + C'f - E'f + , - C's - E's - 2F's - ?0' - s1 -::o -+ -=*- ]{+ 1ro 2 lJC' 6 / - E'f - F'f - C's + E's + F's - 2G's 1 -::0 -+ "::'*- J{01r+ 2 lJ 3 - C'f - E'f - F'f + C's + E's + F's + ?Q' -s 1 ~c1 ~c1 ~ c1 ~ cl ~ ~ ~ ~ 78 Table 3.13. Squared matrix elements for the Cabibbo-allowed decays T BMl+vl in terms of the reduced matix elements a, band c. Squared Matrix Element Process At~ AOryO[+vl ~l3a + 2b + 2cl 2 At ~ E 01r 0l+v1 lal2 At~ E+1r-Z+v1 lal2 At~ E-1r+I+v1 lal2 At ~ pK-z+ v1 At ~ nK z+vl Ia + bl2 At ~ :=:- J(+ [+VI Ia + c12 At ~ :=;0 J(O [+Vi Ia + c12 :=:~ ~ E+ ](-[+Vi lbl2 :=:~ ~ Aoft[+ v1 ~lb- 2cl 2 :=:~ ~ E°K [+ v1 tlbl -;-;{) Ia + bl2 :=;~ ~ :=;OTJO [+Vi 2 :=;~ ~ :=;-1!"+ [+VI il2b- cl 2 lcl :=:~ ~ :=;01!"0[+v1 ~lcl2 :=:~1 ~ A0](-[+ V( 2 ilb- 2cl 2 :=:~ 1 ~ E- K z+ v1 2 tlbl 2 lbl :=;~1 ~ :=;-TJO [+Vi il2b- cl :=:~1 ~ :=:-7!"0 [+Vi ~lcl :=:~1 ~ =:_01!" - [+ V/ lcl :=:~ 1 ~ E° K-z+vl 79 2 2 2 ~ Table 3.14. Squared matrix elements for the Cabibbo-suppressed decays T -+ BM!+vl in terms of the reduced matrix elements a, band c. Squared Matrix Element (Modulo si) Process At-+ nry 0 [+vl lbl2 tlb- 2cl 2 At-+ n7rO[+vl ~lbl At-+ p1r-z+v1 tl2b- cl 2 icl 2 At-+ A° K 0 l+v1 At - 2 :E- ](+ z+ v 1 2 At - t :Eo J(O z+ V[ ~lcl ::::~-+ A 0 ry 0 l+v1 31616a + b+ cl2 tl2a + b+ cl 2 Ia + bl 2 Ia + c12 la12 Ia + c12 lal 2 Ia + bl 2 121 Ib + c 12 l21b + c12 tlb + c12 !lb- cl 2 tlb + cl2 ~lb- c12 lbl 2 lcl2 ::::~ - t :EO 7r0 [+ V[ ::::~-+ :E+1r-l+v1 ::::~-+ :E-7r+z+vl ::::~ - pK-z+ v1 ::::~-+ n~[+vl ::::~ - ::::-K+ z+ v1 ::::~ - t :::;O ](0 z+ V[ ::::~ - t :EOryO [+ V[ :::;~-+ Ao7roz+vl ::::~ 1 - A0 1r - z+ v1 ::::~1 -+ :Eo7r-[+vl ::::~ 1 -+ :E-17°l+v1 ::::~1-+ :E-7rO[+vl ::::~1 - ::::- I\o z+ V[ ::::~1 - t nK-z+ V[ 80 Squared matrix elements for the Cabibbo-allowed decays T Table 3.15. h* Ml+vi normalized with respect to the process ::::~ 1 - t E*° K-z+vi. Process Squared Matrix Element At - t E*o?roz+vi 2 At - t E*-1r+ z+vi 2 At - t ::::*- J(+ z+ VI 2 At - t E*+?r-z+vi 2 A"t - t ::::*o ](o [+vi 2 ::::;j ::::;j -t E*+ ](-[+VI 2 -t E*°K z+ VI 1 =+ - t =*o77o z+ v ~c1 ~ I 3 :=:;j ::::;j :=:;j -t ;:::*0 7r0 z+ VI 1 -t ::::*-7r+ [+vi 2 -t n- ](+[+vi 6 ::::~1 - t ::::*o 7r -z+ VI 2 ::::~1 - t :=:*-?rO[+vi 1 ::::~1 - t :=:*-77oz+vi 3 ::::~ 1 - t n- K 0 z+ VI 6 ::::~ 1 - t E*° K - z+ vi 1 ::::~1 - t E*- -go z+ VI 2 81 -t Table 3.16. Squared matrix elements for the Cabibbo-suppressed decays T ~ h* Mf+vl normalized with respect to the process :=:~ 1 ~ E*-1r 0 l+v1. Process Squared Matrix Element (Modulo si) At~ ~+1r-1+v1 2 At~ ~o'lrof+vl 4 At ~ E* 0 !( 0 z+ V[ 1 At~ ~-1r+[+vl 6 At~ E*- J(+[+vl 2 :=:~ ~ ~+ K-z+vl 2 :=:~ ~ ~°KZ+v1 2 :=:~ ~ E*OryO[+vl 3 ::::~ ~ E* 0 1r 0 Z+v1 2 :=:~ ~ E*-7r+f+vl 2 ::::~ ~ ::::*- J(+ f+ V[ 2 :=:~ 1 ~ ~ 0 J{-f+vl 2 ::::~ 1 ~ ~-Kf+vl 6 ::::~1 ~ E*-7rO[+vl 1 ::::~ 1 ~ E*-ry 0 l+vl 3 ::::~1 ~ E*o7r-f+vl 1 ::::~1 ~ ::::*- J{O [+ V[ 2 1 82 Table 3.17. Squared matrix elements for the decays S---+ BMZ+vl in terms of the reduced matrix elements a, f3 and I· -- Process Squared Matrix Element 111 2 111 2 si!la12 sH-Ia- 2/3 + 211 2 sila12 sH-1- 2a + f3- 211 2 si!l/312 si 1/31 2 n~ ---+ =.-Jt1 z+v, ---+ '3° ](-[+ VJ n~ ---+ =.-7ro z+ v1 n~ ---+ =.- TJo z+ v 1 n~ ---+ =. 0 1r-z+ v 1 n~ ---+ A° K-z+v, n~ ---+ E° K-z+ VJ f2~ n~---+ E-Jt1z+v1 Table 3.18. Squared matrix elements for the decays S---+ h* JVJL+vl in terms of the reduced matrix elements 8 and (. Process Squared t>.Iatrix Element n~ ---+ '3*° K-z+ v 1 tl8 + (12 tiS+ (12 tl8 + (12 sitl812 si lsl8- 2(12 snl812 sil812 si tl(l 2 si k1CI2 n~ ---+ =.•- Jt1 z+ v, n~---+ n-TJoz+v1 ---+ =.• 0 1r-z+ v1 n~ ---+ =.*-TJ 0 z+ vz n~ n~---+ =.·-7r 0 [+v, n~ ---+ n- J\ 0 z+ vz n~ ---+ E* 0 J(-t+ V[ n~ ---+ E*- Jt1 z+ v1 83 CHAPTER 4. B-MESON DECAYS TO CHARMED BARYONS B-mesons, for the first time, offer an opportunity to study the weak nonleptonic decays of mesons to baryons. Experimentally, it is known that (1.6] Br (B --+ j5 + anything) ;::: 2 x IQ- 2 and Br (B --+ A + anything) ;::: 1 x 10- 2 . It is likely that, in the future, branching ratios will be measured for many of the exclusive B-meson decays to baryons. In this chapter we explore the implications of the approximate SU(3)J flavor symmetry of the strong interactions for the decays of B mesons to the lowestlying charmed baryons. Some of the relations implied by SU(3)J may provide insight into the nature of various competing dynamical effects that can occur in nonleptonic B decays. The effective Hamiltonian for .6..b = -1 .6..c = +1 nonleptonic B-meson decays has the flavor quantum numbers of the operator (bc)(ud) for the Cabibbo-allowed decays and of the operator (bc)(us) for the Cabibbo-suppressed decays. These are two different components of the same octet representation of SU(3)J· Using this transformation property we derive, in Section 4.1, relations between two body B meson decays to a ch armed baryon plus an antibaryon . The consequences of SU(3)J for some of the two body B-meson decays to charmed baryons in the 3 representation have been considered in ref. (4.1]. The effective Hamiltoni an for .6..b = -1, .6..c = -1 weak nonleptonic decays has the flavor quantum numbers of the operator (bu)(cs). Note that these decays vanish if s3 is zero. This operator transforms as 3 EB 6 under flavor SU(3)J· Using this transformation property we derive, in Section 4.2, relations between two-body B meson decays to a baryon and a charmed antibaryon. Section 4.3 contains concluding remarks, which include some SU(3)J relations for weak non leptonic decays of bottom baryons. 4.1. Two Body B Decays to Charmed Baryons The B meson fields come in three types, B-, B 0 and B2 \Yhich transform under SU(3)J as a 3 representation with components B;: (-!.1) 84 We begin by considering two body decays of the B - mesons to charmed baryons in the lowest- lying antitriplet representation and antibaryons in the charge conjugate octet of nucleons and hyperons. As far as group theory factors are concerned, we can take for the effective Hamiltonian for nonleptonic decays, B -+ Th, -1 p - 1 q q p Herr = aT Hq heBP + f3T He hqBP + 1T Be (H:h~) 1 (4.2) where h~ is given in Eq. (1.28), T 1 is given in Eq. (3.15) and Hj are components of the matrix H (4.3) Table 4.1 presents the predictions for B -+ TIL decays which follow from the effective Hamiltonian in Eq. ( 4.2). There is a simple relation amongst the Cabibbo-allowed decays Li and Wu [4.1) have also considered the predictions of SU(:3) for the Cabibbo-allowed B -+ Th decays. These results agree with theirs. There are several simple relations between Cabibbo-allowed and Cabibbo-suppressed decays. For example, ~o -) ~o f:-) r(B - -+ .::.C1 p = S 2f(B1 -+ .::.Ci ..._, ' ( 4.5a) (4.5b) and (4.5c) Next we consider two- body decays of the B - mesons to the lowest- lying charmed baryons in the 6 representation and antibaryons in the charge conjugate of the octet 85 of nucleons and hyperons. The sextuplet of charmed baryon fields is denoted by a two index symmetric tensor Sii with components S 12 s23 = - V2,LJc 1 5 22 = "o LJc _ _1_"+ ' V2 ~::::oc2 and s33 = nco . (4.6) v2 As far as group theory factors are concerned we can use, as the effective Hamiltonian for weak nonleptonic B ---+ Sh decays, H eff = at ijp B p Hkj S- ke hei + b € ijp B p Hkj hek S- if e i S-·1k , + c t iip B P Hkhe ( 4.7) with Hj given by Eq. (4.3). The decay rates for Cabibbo allowed decays, which follow from Eq. (4.7), are presented in Table 4.2. There are several simple relations amongst the Cabibbo allowed decays. For example r(B- ---+ ::::~):-) = ~r(B- ---+ E~p) , f(B 0 ---+ ::::~t-) = ~T(B 0 ---+ n~~ 0 ) (4.8a) . ( 4.8b) Table 4.3 presents the results that follow from the effecti\·e Hamiltonian in Eq. ( 4. 7) for the Cabibbo-suppressed decays. In addition to there being several relations between the Cabibbo-suppressed decays, there are also some simple relations between the Cabibbo-allowed and the Cabibbo-suppressed decays. For example, inspection of Tables 4.2 and 4.3 reveals that nO f'-) = sl2f(B- ---+ "o r(B - ---+ He <.JcP-) ' ...J ( 4.9a) ( 4.96) The results of Tables 4.2 and 4.3 trivially generalize to decays B ---+ S*h, where S* denotes a member of the lowest- lying 6 of ]P = 3/2+ charmed baryons. 86 There is only one singlet representation in the product 3 @ 10 @ 3 @ 8 (i.e., Tpqh*qlj HJ Bj) so all the decays B -+ Th* are related by SU(3) f symmetry. Ta- ble 4.4 presents the relative rates for these decays normalized to the rate for the decay B- -+ A"d ~ --. For the decays B-+ Sh* we can take, as far as group theory factors are concerned, the following effective Hamiltonian LfkBl H eff=TJl h •pqes-pq HiE f j+TJ2 h*pqes-pkLq ( 4.10) The term proportional to 111 only effects Cabibbo allowed B 0 decays and Cabibbo suppressed B2 decays. So all the Cabibbo allowed B- and B2 decays and the Cabibbo suppressed B-and B 0 decays are related by SU(3)J flavor. Tables 4.5 and 4.6 present rates, which follow from the effective Hamiltonian in Eq. (4.10), for the Cabibbo allowed and Cabibbo suppressed B -+ Sh* decays. These results generalize straightforwardly to decays B-+ S*h*, where S* denotes a member of the lowest- lying 6 of JP = 3/2+ charmed baryons. 4.2. Two Body B-Decays to Charmed Anti-Baryons The 6-b = -1, D.c = -1 part of the effective Hamiltonian for weak nonleptonic B - decays arises from the b-+ u weak coupling and has the flavor quantum numbers of the operator (bu)(cs). This operator transforms as 6 E9 3 with respect to flavor SU(3). Decomposing it into irreducible operators (bu)(cs) = 0( 6) + 0(3) (4.11) where (4.12a) transforms as a 6 and (4.12b) transforms as a 3. There is a small dynamical enhancement of the coefficient of 0(3)• 87 in the effective weak Hamiltonian, over that of 0(6) coming from perturbative strong interaction corrections which take into account effects coming from loop momenta p in the range Mw > p > mb. It is equal to (2.1] (4.13) For the two-body decays B-+ Th we can use, as far as group theory factors are concerned, the following effective Hamiltonian. (4.14) In Eq. ( 4.14) H(3) is a two index antisymmetric tensor that takes into account the transformation properties of 0(3)· It has non-zero components H(3) 13 = 1 H(3) 31 = -1 (4.15) H(6) is a two index symmetric tensor that takes into account the transformation properties of 0(8)· It has non-zero components (4.16) Table 1.7 presents the predictions for B -+ Th decays that follow from the effective Hamiltonian in Eq. (Ll.H). The Hamiltonian is I = 1/2 and it follows that the relations r(B- -+ i\t~ 0 ) = ~ r(B 0 -+ At~+) , (4.17a) r(Bo-+ ::-o "o) = 2~r(Bo-+ -=-"+) S ~Cl-" S ~C1-" (4.176) l are consequences of isospin invariance. There are no simple SU(3)J relations that arc not consequences of isospin. However if, for dynamical reasons, matrix clements 88 of 0(3) dominate over those of 0( 6), then there would be additional relations. For example, either 3 or 6 dominance implies that ( 4.18) Next we consider decays of the type B---+ Sh. As far as group theory factors are concerned, we can take as the effective Hamiltonian for these decays (·1.19) Rates for B ---+ Sh decays that follow from this effective Hamiltonian are presented in Table 4.8. Most of the simple relations are consequences of isospin. These arc r(B----+ t; A0 ) = ~r(B 0 ---+ ~~,\ 0 ) , (4.20a) f(B 0 ---+ f:~I: 0 ) = f(B 0 ---+ !:;~+) , (4.206) 0 - - f(B 5 ---+ Ec p) = 1 0 ~0 2r(B ---+~en) , (4.20c) 0 f(B s0 ---+ :::o E 0 ) -- ~f(B "+) ~c2 2 s ---+ :::~c2 ~ ( 4.20d) 5 There is a simple relation that is not a consequence of isospin. -=--)~r(B- --+ -.Jc ~o"-) r(B - ---+ :::o ~c .......... ~ 2 2 • (4.21) The results in Table 4. 7 generalize straightforwardly to B ---+ S* h decays, where S* is a member of the lowest- lying JP = 3/2+ multiplet of charmed baryons. Since the product 3 0 fO 0 3 0 3 does not contain a singlet, the decays B ---+ 1,h* only proceed via the 6 part of the effective Hamiltonian. In addition, the product 3 0 10 0 3 0 6 contains only one singlet (i.e., BiTijh_jkcH(6)ke) and so the relative 89 rates for all the B --+ Th* are determined by SU(3)J symmetry. Table 4.9 gives the relative rates. Note that the relations f(B- --+ Kt~* 0 ) = ~f(B 0 --+ At~*+) , (4.22a) f(Bos --+ ::-o ~*o) = ~r(Bo --+ :::- ~*+) ~c1 2 s ~ c1 ( 4.22b) ' are consequences of isospin. Finally we consider the decays B --+ Sh*. As far as group theory factors are concerned, we can use Herr = a(6)BiH(6)iihjklSkl + f3(6)BJ1(6)klhjklSii + a(3)BiH(3)iihjkeSkl , (4.23) as the effective Hamiltonian for these decays. Note that the terms proportional to a(3) and a(6) only cause B- and B2 decays. It follows that the rates for B 0 --+ S h* decays are related by SU(3)J symmetry. Table 4.10 shows the results that follow from expanding the effective Hamiltonian in Eq. ( 4.23). There are several simple relations that are not consequences of isospin. For example: f(B- --+ t;~* 0 ) = ~f(B- --+ ~;;:=:* 0 ) , ( 4.24a) f(B- --+ E~I:*-) = ~r(B- --+ ~~ :=:*-) , ( 4.24b) f(B 0 --+ t;~*+) = f(B 0 --+ ~~2 :=:* 0 ) ( 4.24c) 2 . The results in Table -!.10 generalize straightforwardly to the decays B--+ S*h*. 4.3 Concluding Remarks On Chapter 4 In this chapter we ha,·e examined the consequences of the app roximate SU(3)J :flavor symmetry of the strong interactions for nonleptonic B meson decays that produce low lying charmed baryons. !\!any simple SU(3)J relations were found. 90 For example, all the rates for two body decays, to a ]P = 1/2+ charmed baryon in the lowest-lying 3 representation and a ]P = 3j2+ antibaryon in the fO representation, are related by the SU(3) 1 flavor symmetry of the strong interactions. The hadronic spectrum contains bottom baryons analogous to charmed baryons. In fact, if we approximate mb and me as very heavy compared with the QCD scale, then the heavy quarks act essentially as static color sources in these baryons [1.9] and the hyperfine splitting between the lowest-lying three and six ]P = 1/2+ multiplets in the charm case is related to that in the bottom case. For example, one expects (4.25) If color suppression [4.2] does not diminish the rate, the decay chain, Ag - t i\ 0 J j'lj; - t prr-e+e-, may prove to be a useful way to detect the Ag baryon. There are SU(3)t relations for weak bottom baryon decays. For example, the decays of the lowest-lying ]P = 1/2+ bottom baryons in the 3 representation to hJj'lj; are related by SU(3)t· Explicitly (4.26) The decays of the lowest- lying 3 bottom baryons to h* 1/1/-· are forbidden by SU(3) flavor because the product 3® 3® f0 doesn't contain a singlet. :\Tote that the decays of the lowest- lying ]P = 1/2+ bottom baryons in the 3 representation , to h* D (where D denotes one of the three lowest-lying pseudoscalar D-mesons) are also determined by a single reduced matrix element, because the product 3 @ 3 0 8 @ ro only contains one singlet. Normalizing to the rate for Ab - t 6- D+ we find that: (4.27a) (4.27b) 91 References for Chapter 4 [4.1] X. Li and D. Wu, Phys. Lett. 218B (1989) 357. [4.2] M. B. Wise, Phys. Lett. 89B (1980) 229. 92 Table 4.1. SU(3)J predictions for decays E---+ Th, where T denotes a member of the lowest- lying SU(3) antitriplet of charmed baryons and h denotes a member of the lowest-lying octet of hyperons and nucleons. Rates are expressed in terms of three reduced matrix elements a, f3 and /· Decay Rate 1/3 + 11 2 silf3 + 1l2 !Ia- /31 2 tla + /31 2 Ia + 112 la12 E- ---+ 3~1 !;E- ---+ ::-o f5 ~C! Eo ---+ ::-o ~o ~C! Eo ---+ ::-o A ~c1 Eo ---+ ::-+ ~~c1 E 0 ---+ AcP Eo ---+ ::-o n E~ ---+ Ac!:- siiPI 2 siltl 2 IPI 2 111 2 EoS ---+ ::-o !;O ~siial 2 EoS ---+ ::-o A i-sila- 2/312 sila12 sila + ~11 ~C! Eo ---+ -=+ f5 ~C! EoS ---+ ::-o :::o ~C!~ ~C! ~C! Eos ---+ -=+ !;'--'ci E~ ---+ AcfJ 2 93 Table 4.2. SU(3)t predictions for Cabibbo-allowed decays B - t Sh where S denotes a member of the lowest- lying JP = 1/2+ sextuplet of charmed baryons and h denotes a member of the lowest-lying octet of hyperon and nucleons. Rates are expressed in terms of the three reduced matrix elements a, b and c. Decay Rate B- - t E~p lcl 2 B- - t :=:~2f;- !icl !Ia- cj2 tlb- cj2 2 B 0 - t Etfi Bo-t -:;O f:O ~c2 Bo-t -:;o A Bo - t nc:=;o {21 - 2a + b + cl 2 lbl2 B 0 - t -=+ E- ~lbl E 0 - t E 0c n la12 E s0 - t E+Ec ~~-a- b + cj 2 Eos - t EOf'O c.:..J ~Ia + b- cl 2 E s0 - t E c0 A EoS - t -:;O -:::-o il- a- b- cl 2 ~C2 ~C2 2 !Ia + bj2 ~C2~ 94 Table 4.3. SU(3)J predictions for Cabibbo-suppressed decays B ~ Sh where S denotes a member of the lowest-lying JP = 1/2+ sextuplet of charmed baryons and h denotes a member of the lowest-lying octet of hyperons and nucleons. Rates are expressed in terms of the three reduced matrix elements a, b and c. Decay Rate (divided by si) B- ~ ;:;,c2P ~o - !icl 2 icl 2 !ici2 tl- 2a- 2b + ci 2 !la+b-cl 2 !Ia + bl2 !Ia- ci2 tia- c12 1121 - a + 2b - ci2 !lbl 2 lbl 2 lal 2 B- ~ f!c~Bo ~ f!c~o B 0 ~ f!cA Bo ~ :::+p ~C2 Bo ~ :::o fi ~C2 B S0 ~ :::+ t~C2 BoS ~ :::o Eo ~C2 BoS ~ :::o A ~C2 Bos ~ -'-'c );+p Bos ~ "--'c );On Bos ~ n :::-o c~ 95 Table 4.4. SU(3)J predictions for Cabibbo decays B ---+ Th* where T denotes a member of the lowest- lying antitriplet of charmed baryons and h* denotes a member of the lowest- lying decouplet of JP = 3/2+ baryons. Decay Rate B----+ Ac~-- 1 B- ---+ -::-o L;*~C! 3 B----+ =:~b.-- 82 1 B- ---+ -::-o b.~C! 182 B 0 ---+ Acb.- 3 Bo---+ -::-o t•O 6 Bo ---+ -=+b.~C! 1 82 Bo---+ -::-o b.O 182 B s0 ---+ Ac t•- 3 BoS ---+ -::-o -:;-.o 3 Bos---+ -=+ ~·~c1.__, 18 2 Bo---+ -::-o t•O s ~c1 182 1 3 1 1 1 ~C! 3 1 3 1 ~c1 1 1 ~C!~ 3 1 6 1 96 Table 4.5. SU(3)J predictions for Cabibbo- allowed decays B --+ Sh* where S denotes a member of the lowest-lying JP = 1/2+ sextuplet of charmed baryons and h* denotes a member of the lowest- lying decouplet of JP = 3/2+ baryons . Rates are expressed in terms of the two reduced matrix elements "11 and "12. Decay Rate B- --+ ~t 6.-- !1"121 2 i 1"121 2 t 1"121 2 1"111 2 ti2TJ1 + "121 2 i 1"111 2 l212TJ1 + "121 2 i 1"11 + "121 2 iiTJ1I 2 iiTJ21 2 tl"l21 2 iiTJ21 2 B- --+ ~~LiB---+ ::-o ~*~C2 Bo --+ ~t+ 6.-Bo--+ ~+t::,.c Bo --+ -=+~*~C2 Bo --+ ::-o ~*o ~c2 Bo --+ ~oLio c Bo --+ nc=:;*o Bo--+ ~+~*s c Bos --+ ~o~*o c~ BoS --+ ::-o '=*o ~C 2 ~ 97 Table 4.6. SU(3)J predictions for Cabibbo-suppressed decays B --t Sh*, where S denotes a member of the lowest-lying ]P = 1/2+ sextuplet of charmed baryons and h* denotes a member of the lowest-lying decouplet of ]P = 3/2+ baryons. Rates are expressed in terms of the two reduced matrix elements "71 and "72· Rate (divided by Decay tl"721 2 il"721 2 !1"721 2 611 "72 12 il"721 2 tl"721 2 1"711 2 ~ 17711 2 i 12'71 + "721 2 /2121]1 + "721 2 ki 17II 2 !1"71 + "721 2 B- --t -=+ .6. -~ C2 B- --t :=:~2 .6.B- --t Oct•B 0 --t -=+ li~C2 Bo --t -::-o li o ~C2 B 0 --t nct* 0 B~ --t Et+ li -- B s0 --t "-'c ~+ liBoS ..__. -=+ .......,C2 f:*.:....J Bos --t -::-o ~·o ~c2-' Bos --t Eoc li o Bos --t n -::-•0 c~ 98 sn Table 4. 7. SU(3)J predictions for decays E --t Th, where T denotes a member of the lowest-lying antitriplet of charmed baryons and h denotes a member of the lowest- lying octet of hyperons and nucleons. Rates are expressed in terms of the four reduced matrix elements 17(3)• 11(3), 77(6) and 77(o)" Decay Rate E - --t AcL; 0 ~ 177(3) + 77(o) 1 2 E- --t AcA E 0 --t AcL;+ + 77(o) - 277(ol I - 77(3) + 11(3) + 77(o) + TJcs) 12 I11(3) + 11(6) I?2 ir7(3) + 17(6) 1 Eo --t .......,Cl......., -::-o -::-o 2 177(3) - 77(6) 1 !-177(3) - 277( 3) E- --t -::-o -::-- I ~CJ~ E- --t -::---::-0 ........ I ~cl I I Eos --t -::-o A I7](3) - 17(3) + 77(6) + 11(6) 12 6I I?~77(3) - 17p) - ~IJ(6) + 17(6) 12 EoS --t ........ -::-oC1 >;O 11 I I 12 2 71(3) - 17(6) EoS --t -:::--L;+ ~CJ I17(3) - 77(6) I?- 0 - Es --t AcP I I I ~c1 .:...J I 99 ') I I Table 4.8. SU(3)t predictions for decays E--+ Sh, where S denotes a member of the lowest- lying SU(3)t sextuplet of charmed baryons and h denotes a member of the lowest- lying SU(3)t octet of hyperons and nucleons. Rates are expressed in terms of the four reduced matrix elements Fun' Gp;;), F(3) and G(3)· Decay Rate E---+ t~-L;+ + G(6) + 2G(3) 12 2 IF(6) + ~G(6) - F(3) + G(3) 1 2 ~IF(6) - G(6) + 2G(3) 1 IF(6) E---+ E~E 0 E- --+ ::---::o ~c2~ E- --+ -::o -::- ~IF( 6 )- 2F(3)1 2 E---+ E~E- IF(6) - 2F(3) 1 E---+ 'E~A tiG(6)- ~F(3)- ~G(3)1 Eo--+ EoEo c ~IG(6) + 2F(3) + 2G(3) 12 E 0 --+ 'E 0c A 31 2 2 G(6)- 3F(3)3') G (3)-I? Eo--+ L;-L;+ c ~ IG(6) + 2F(3) + 2G(3) 1 Eo --+ ::-o -::o ~IG(6)- 2F(3)- 2G(3)1 ~C2~ 2 2 2 ~C2~ 2 E 5 --+ E~p ! IF(6) + 2F(3) 12 E s0 --+ E 0c n EoS --+ -::o A ~C2 + 2F(3) 12 2 / 2 I3F(6) + 3G(6) + 4F(3) - 2G(3) 1 EoS --+ -::');+ ......,C2 !IF(6) - G(6) - 2G(3) 1 Eos --+ -::o "o tl - F(6) + G(6) + 2G(3) 12 Eos --+ n -::o IF(6) + G(6) - 2G(3) 1 0 - IF(6) 2 ..:......J 2~ ---c 2 c~ 100 Table 4.9. Relative rates for decays E ---+ Th*, where T denotes a member of the lowest-lying SU(3)t antitriplet of charmed baryons and h* denotes a member of the lowest- lying SU(3)J decouplet of JP = 3/2+ baryons. Decay Rate E- ---+ AcE* 0 2 E- ---+ -=-=•0 ~c~~ 1 E 0 ---+ Ac~*+ 1 Eo---+ ::-o =•O 1 E S0 ---+ ::-- E*+ 1 ~·o EoS ---+ ::-o ~C1 2 1 ~c~~ ~C! 1 4-J 101 Table 4.10. Implications of SU(3)J for decays B ----+ Sh*, where S denotes a member of the lowest- lying SU(3)J sextuplet of charmed baryons and h* denotes a member of the lowest-lying ]P = 3/2+ SU(3)J decouplet of baryons. Decay Rate B- ----+ f;~-E*+ t 1°(6) + 0(3) + 2,8(6) 1 B- ----+ ~~ E*- B- ----+ OcO B- ----T .=.o ~* .::,C2.::, + o(3) 12 t lo(6) + o(3) + ,8(6) 12 2 jlo(6) + 0(3) + ,8(6)1 2 lo(6) + 0(3)1 2 ~lo(6) + 0(3) 1 B 0 ----+ f;-c E*+ i 1,8(6) 1 Bo ----+ f;Oc E*o ilf3(6)1 Bo ---+ .=.o ~*o ~C2.::. ilf3(6) 1 2 ilo(6) B- ----+ E;;- E* 0 B- ----+ ::-:::*o ~c2~ 2 2 2 2 B~ ----+ f;~- 6 ++ 1°(6)-- 0 (3)1 B s0 ----+ f;-c 6 + iln(6) -- o(3J 1 BoS ----+ 'f'O o LJC 6 31 I0(6) -- 0(3) 12 2 Bos ----+ n :::*o + 2,8(6)1 2 j lo(6) -- 0(3) + !3(6) 12 t ln(6) -- 0 (3) + ,8(6) 12 !1°(6)-- 0(3) c~ B s0 ----+ ::-- E*+ ~ c2 Bos ----+ :::o "*o ~ c2'-' 102 CHAPTER 5. NON-BARYONIC B-MESON DECAYS In this chapter we investigate the prediction of SU(3)J for the decay of B -mesons to final states that do not contain baryons. In section 5.1 the B-meson decays to two lower mass mesons are explored. In particular we look at the final states Dlv!, DV, DD, J / '1/J M and M M, where D represents aD-Meson and M denotes a member of the lowest lying pseudoscalar octet of mesons. The three body decay modes are examined in section 5.2. The decay to final states J / '1/J, Dlvf M and DD AI are discussed. In the last section of the chapter, section 5.3, closing remarks concerning non-baryonic B-meson decays are made. 5.1 Two- Body Non-Baryonic B-Meson D ecays We first consider b.b = - 1, b.c = 1 decays of the type B - t D.M where D denotes one of the D - mesons D 0 , D+ and Dt. Af is one of the eight lowest- ly ing o- mesons 1r ,!(, J(, "'· As we discussed in section 4.1 of Chapter 4, the B mesons like the D mesons transform as anti-triplets under SU(3)J· They are written as row vectors with components Bi, as in Eq (4.1). As a reminder we write them again, B (5 .1 ) vVe are interested in the transition amplitudes A(B - t D.M) = (D.M IHerriB). As far as t he group theory is concerned we can imagine these amplitudes arising from the effective H amilton ian (5.2) with II£ gi\·en by Eq. ( 4.3). Expanding out these three terms gives the results for the Cabihbo a llowed decays shown in Table 5. 1 a nd for the Cabibbo suppressed decays shown in Table 5.2. 103 There is one SU(3)J relation amongst the decay amplitudes, A(B ---+ DM), for the Cabibbo allowed decays IA(Bo---+ Do7ro)l2 + 3IA(Bo---+ Do7Jo)l2 = IA(Bo---+ J(- D;-)12 + IA(B2 ---t DOKO)I2 (5.3) There are several simple relations between the Cabibbo-allowed and the Cabibbosuppressed decay rates. For example, SU(3) symmetry implies that A(B- ---+ D 0 1r-) A(B----+ DOJ(-) = -1/sl (5.4) The large value of the B - meson mass (compared with the QCD scale) suggests that relative complex phases between the reduced matrix elements a , band c, which can be generated by final state strong interactions, arc small. If this is the case then, up to sign ambiguities, measuring three of the Cabibbo-allowed decays determines a, b and c. At the present time there are measurements of the branching ratios for two of the decays in Table 5.1. Experimentally [5.2, 5.2]: Br(B- ---+ D 0 1r-) = (3.0 ± 1.4) x 10- 3 , (5.5a) Br(B 0 ---+ D+1r-) = (3 .6 ± 1.4) x 10- 3 . (5.5b) Although we have focussed on decays of the type B ---+ D ,\1 our results can be trivially taken over for the decays of the form B ---+ D* AI. Also, for decays not involving the 1J we can use the results in Tables 5.1 and 5.2 for the corresponding decays B ---+ DV and B ---+ D* V where il is one of the 1- p or f{* vector mesons. So, for example, generalizations of Eq. (5...!) of the type A( B- ---+ D 0 p-) A(B- ---+ D 0 J..:•-) -1/sl (5.6) hold. Experimentally [5.1, 5.2, 5.3, 5.4, 5.5] 2 Br(B 0 ---+ D *(2010)+7r -) = (3.3 +1. ) x 10- 3 , - 1.0 10·1 (5.7a) Br(B 0 - t D*(2010)+ p-) = ( 8 ~:) x 10-2, (5. 7b) Br(B- - t D*(2010) 0 7r-) = (3 ± 4) x 10- 3 , (5.7c) Br(B- - D 0 p-) = (2.1 ± 1.2) x 10- 2 , ( 5. 7d) Br(B 0 - D+ p-) = (2.2 ± 1.5) X 10- 2 . (5 .7e) Two- body decays of the type B -t DD arise from weak Hamiltonians with flavor quantum numbers (bc)(cs) for the Cabibbo- allowed decays and (bc)(cd) for the Cabibbo-suppressed decays. These Hamiltonians are different components of the same anti- triplet representation. So, as far as group theory is concerned, the decays B - t DD can be thought of as arising from an effective Hamiltonian (5.8) where for the Cabibbo allowed decays H = (D (5.9a) and for Lhe Cabibbo suppressed decays (5 .9b) Table 5.3 presents results for decays B - t DD which follow from Eq. (5.8). Some of the relations that follow from Table 5.3 are just consequences of SU(2) 1sospm symmetr y. They are, for the Cabibbo-allowed decays [5.6] f(B - - D 0 D-;) r(B~ - f(B 0 - D+ D -;) (5.10a) D D ) = r(B~ - D+ D-) (5.10b) 0 0 For the Cabibbo suppressed decays there are no isospin relations. T a ble 5.3 indicates Lhat Lhere are several SU(3)j re lations between the Cabibbo- a llowed a nd the 105 Cabibbo-suppressed B----+ ntJ decays (e.g., f(B- ----+ n° n-) = sir(B- ----+ n° n;-)). The Cabibbo suppressed B ----+ ntJ decays also get contributions from terms in the effective weak Hamiltonian that have no charm quarks. For example, the operator (bu)(uJ) which transforms under SU(3) as 3 EB 6 EB 15 contributes to the Cabibbosuppressed B ----+ ntJ decays and if s3 is near its experimental limit, this could alter the results in Table 5.3. The same Hamiltonians which give rise to the decays B ----+ ntJ also cause the decays B----+ J /,PM. Since there is only one way to combine the product of a triplet, anti triplet and octet representations ( J /7/J transforms as a singlet) into a singlet, these decays are characterized by a single reduced matrix element. SU(3) predictions for these decays are presented in Table 5.4. The contributions of operators without charm quarks (e.g., ( bu)( uJ)) to the Cabibbo- suppressed decays B ----+ J j,Pl\f are negligible because they violate the Okubo- Zweig-Iizuka (OZI) ru le. For the Cabibbo allowed decays the relation (5.11) is a consequence of isospin invariance. It has previously been noted that a comparison of branching ratios for these modes would determine the ratio of B 0 and B- lifetimes [5.7]. At the present time it is known that [1.6] 0.4 < Tno/Tn- < 2.1. One of the relevant branching ratios has been mcasured: 8),l 4 ),lS) (5.12) For the Cabibbo suppressed decays the relation (5.13) is also a consequence of isospin invariance. 106 The results of Table 5.4 generalize trivially to other cc resonances and also to decays B---+ Jj'lj;V, where Vis a 1-p or](* meson. There is also some experimental information on these decays [5.2, 5.8]: Br(B- ---+ 'lj;(2S)K-) = (2.2 ± 1.7) x 10- 3 , (5.14a) Br(B 0 ---+ Jj'lj;R*(892) 0 ) = (3.7 ± 1.3) x 10- 3 . (5.14b) We consider next the SU(3)J relations between the decay amplitudes which can arise from the b---+ u w- weak coupling. For final states without charm the effective Hamiltonian has the flavor quantum numbers of the operator ( bu)( ud) which transforms as 15 EB 6 EB 3. Explicitly, the decomposition of ( bu)( ud) into operators that are in irreducible SU(3)J representations is (5.15) where 0(15) = 3(bu)(ud) + 3(bd)(uu)- 2(bd)(dd)- (bs)(scl)- (bd)(ss) (5.16a) 0( 6 ) = (bu)(1td)- (bd)(uu)- (bs)(scl) + (bd)(ss) (5.16b) 0(3) = (bd)(uu) + (bd)(dd) + (bd)(ss) (5.16c) 0~3) = (bu)(ucl) + (bd)(dd) + (bs)(scl) (5.16d) In Eqs. (5.15) and (5.16) the subscripts on the operators denote the irreducible rep- resentation of SU(3)J to which they belong. As far as group theory factors are concerned \Ve can take as the efTectiYe Hamiltonian for B - meson decays B ---+ !II 1H Iferr = A(i)Bd!(3)i(M1 1U£) + C(3)Bi 11ILJ\If II(3)J + Ac 15 )BJI(15)1i\!J!II1 + Cc 15 )Bi!IIJII(15){k 1\fk + A(6 )BJI(6)1 !l!f !lit 107 (5.17) In eq. (24) H(3) is a vector with non-zero component (5.18a) H(15) is a traceless three-index tensor that is symmetric on its upper indices and has non-zero components 12 ?} 22 32 ( - 23 H(15h = 3 , H(15)i = 3 , H(15)z = -2 , H(15)J = -1 , H 15h = -1 (5.18b) Finally, in Eq. (5.17), H(6) is a traceless three-index tensor that is antisymmetric on its upper indices and has non-zero components H(6)~ 2 = 1 , H(6)i 1 = -1 , H(6)~ 2 = -1 , H(6)5 3 = 1 ( 5.18c) The parameters A(3)• C(3)• A(ls)• C(IS) and A( 6 ) are the reduced matrix elements in terms of which the B ---+ M A1 decay amplitudes are expressed. Note that since (.5.19) there is only one reduced matrix element, A(6 ) parametrizing the contribution of the part of the Hamiltonian that transforms as a 6. Table 5.5 summarizes the SU(3)J predictions that follow from expanding the effectiYe Hamiltonian in Eq. (5.17). There is only one simple relation (5.20) There are no simple isos pin relations between the B ---+ AI AI decay rates in Table 5.5. The effective Hamiltonian has both I = 1/2 and I = 3/2 pieces. The I= 3/2 piece arises solely from the operator O(lS)· In the decays B---+ r.1r the two- pion final state is a linear combination of I = 0 and I = 2 states. The I = 2 state can 108 only be reached through the I = 3/2 part of the effective Hamiltonian while the I = 0 = 1/2 and I = 3/2 parts. Since the ?r 07r- state gets contributions from both the I state is charged it is pure I = 2, consequently the rate for B- ---+ 1r 0 1r- originates only from the matrix element of O(ls)· There are contributions to the decays B ---+ M M listed in Table 5.5, which survive m the limit s3 ---+ 0 (where the b ---+ uw- coupling is absent). They come from penguin-type Feynman diagrams with a charm or top quark in the loop (see Fig. 1). Writing (5.21a) (5.2lb) it is A(3) and C(3) that characterize these contributions, since the penguin- type diagrams only give rise to terms that transform as a 3 in the effective Hamiltonian of Eq. (5.17). The contribution of penguin-type Feynman diagrams is probably suppressed by [as(mb)/7r], so unless (s3/sz) is very small (a prospect that is unlikely if the standard six- quark model is to describe the CP violation observed in kaon decays)[5.10] A.( 3) and 6(3) are unimportant for the B ---+ !II"H decays in Table 5.5. However, if we examine B ---+ !11 Jif decays that change strangeness by one unit, the situation is quite different. Here the penguin- type diagrams are again suppressed by as(mb)/?r but they arc enhanced (over operators like (bU)(us)) by the ratio of weak mixing angles (s~ + s5 + 2szs3c0 ) 1 / 2 ') (5.22) Sj"S3 These decays may be dominated by the penguin- type diagrams with a charm or top quark in the loop. Assuming this is the case we can use, as far as group theory factors arc concerned, the following effective Hamiltonian to describe the 6.s = -1 B ---+ !II !II decays (5.23) 109 where now the non-zero component of H(3) is (5.24) Table 5.6 gives the SU(3)J predictions that follow from Eqs. (5.23) and (5.24). Note that since A(3) only effects the B2 decays, the ratios of the various 6s = -1 B 0 ---+ Mill[ and B----+ MM decay rates are determined by SU(3)J· Our assumption that penguin- type diagrams dominate the 6s = -1 B ---+ M M decays implies that the effective Hamiltonian is I = 0. Since there is only one way to combine two I = 1/2 states into an I = 1 state, all the relations between B ---+ J( 1r decays in Table 5.6 are consequences of isospin. Similar isospin relations hold for decays of the type B ---+ J( p, B ---+ J(*7r and B ---+ !\* p. The relations r( r(B 0 ---+ R 0 17 ) = r(B----+ K - ry) ( 5.25a) B2 ---+ K+ !\-) = r( B2 ---+ I\o Ro) (5.256) 1 r(B2---+ 7ro7ro) = -r(B2---+ 7r+7r-) 2 (5.2.5c) are also consequences of isospin symmetry. The (bu)(us) operator has both I = 0 and I = 1 pieces. Verifying some of the above isospin relations would prO\·ide strong evidence that penguin- type diagrams dominate the 6s = -1 B ---+ /1/ 11! decays. The b---+ ulV- coupling also causes 6b = -1, 6c = -1 decays B---+ D .M. To leading order in weak mixing angles the effective Hamiltonian for such decays has the fla vor quantum numbers of (bu)(cs). Under SU(3) this operator transforms as 3 EB 6. Explicitly, the decomposition in terms of operators in irreducible representation s is (5.26) where 0(3) = 1 2 [(bu)(cs)- (bs)(cu)] 110 (5.27a) 0(6) = 1 2 [(bu)(cs) + (bs)(cu)] (5.27b) As far as group theory factors are concerned we can take as the effective Hamiltonian for the ~b = -1, ~c = -1 decays B-+ D.M . Heff = a(6)DiH(6)ii BkMj + f3(6)BiH(6)ii Dk!I!Jj .. k .. k + a(3)DiH(3)'1 Bk!l1i + f3(3)BiH(3)' 1 Dd'llfi. (5.28) Here H(6) is a two- index symmetric tensor with non- zero components H(6) 13 = 1 H(6) 31 = 1 (5.29) and 11(3) is a two- index antisymmetric tensor with non-zero components H(3) 13 = 1 H(3) 31 = -1 (5.30) Table 5. 7 shows the results which follow from the effective Hamiltonian in Eq. (5.28). There are two simple relations f(B2 -+ D-1r+) = 2f(B2 -+ D0 1r 0 ) (5.31a) f(B 0 -+ D-;1r+) = 2r(B- -+ D-;1r 0 ) (5.31b) and they are a consequence of isospin invariance. There is a small dynamical enchancement of the \Vilson coefficient of 0(3) o\·er that of 0(6) coming from perturbative QCD corrections. In the effecti,,e Hamil tonian for D.b = - 1, ~c = - 1 decays the ratio of \Vilson coefficients for 0(3) and 0(6) is [2.1] [as(mb)/as(mlV)](lS/:23) ~ 1.5. If either the matrix elements of 0(3) or 0(6) dominate the B -+ D!II decays, then Table 5.7 indicates that there would be some SU(3)J relations . For example, either 3 or 6 dominances implies that (5.32) Of course, generalizations of Table 5.7's results to dC'cays B -+ D* .~V, B -+ DV and B -+ D* V hold . 111 5.2 Three-Body Non-Baryonic B-Meson Decays The three-body decays, B ---+ 1/1/;1\11\1, can have the relative orbital angular momentum, L, of the two M - mesons be either even or odd. For the case Leven we can take, as far as group theory factors are concerned, (5 .33) as the effective Hamiltonian (Lorentz indices are suppressed). In Eq. (5.33) H=(] (5.34) Table 5.8 presents the results that follow from the effective H amiltonian in Eq. (5.33). For the case L odd the effective Hamiltonian must be change sign under interchange of the SU(3) J quantum numbers of the two 1\I mesons. Only the second term in Eq. (5 .33) can be antisymmetrized and so the rates for B ---+ 1 /lf;(!II !lf)L=l,3 ... are d etermi ned in terms of a single reduced mat rix element. Table 5.9 presents the relative B ---+ 1 /1/Jlvfl\I decay rates for odd L. With L odd the amplitudes forB- ---+ 1 /1/J1r-T/o and B 0 ---+ 1 j'lj;1r 0 ry 0 vanish by SU(3) f symmetry and therefore these processes don't appear in Table 5.9. The Cabibbo allowed B ---+ 1 /1/J.U 1\I decays arise from an effective Hamiltonian that is an isosinglet. There are se\·eral isospin relations among the Cabibbo allowed decays that hold independent of L. They are f(B 0 ---+ 1 j1/;1r+ K -) = r(B - ---+ 1 j 1/;1r- R 0 ) = 2r(B 0 ---+ 1 j1/;1r 0 R 0 ) = 2f(B- ---+ 1 j'lj;1r 0 I\ -) (5.35a) f(B 0 ---+ 1/1/Jr7° R 0 ) = r(n----+ 1/1/J r/ 0 I\-) (5.35b) r(n~ ---+ 1/1/JI\ 0 R 0 ) = r(n~ - 1/1/J K+ I\ -) (5.35c) 112 and (41d) In the case B~ ----+ J /'l/nr7r (Eq. (5.35d)) isospin invariance forces the two pions to be in an even L state. The effective Hamiltonian for Cabibbo-suppressed B ----+ J j'lj;lvf M decays is I = 1/2. Again, there are isospin relations which are L independent. They are f(Bo ----+ J j'l/;7ro1Jo) = ~ f(B- ----+ J j'l/;7r-1Jo) (5.36a) f(B2----+ Jj'lj;K 0 1r 0 ) = ~ f(B2----+ Jj'lj;K+1r-) (5.36b) Some isospin relations that hold only for L even are f(B 0 ----+ Jj'lj;rr 0 1r 0 ) = 1 0 2 f(B ----+ Jj'lj;(7r+rr-)L=o,2, ...) f(B 0 ----+ J/'l/-'(7r 0 rr-)L=o,2, ... ) = 0 (5.37a) (5.37b) There are also SU(3)J relations between the Cabibbo-allowed and Cabibbosuppressed decay amplitudes that are L independent. For example, two such relations are sfiA(B-----+ Jj1/; 7r- R 0 )1 2 = IA(B-----+ Jj'lj;J{-!( 0 )1 2 (.5.3Sa) siiA(B 0 ----+ Jj'lj;1r+ J\-)1 2 = IA(B2----+ Jj'lj;K+1r-)l 2 (5.3Sb) All the L odd B ----+ Jj'lj;Af i\! decays are related by SU(3)J flavor symmetry. However, there is an important source of SU(3)j violation for resonant 1H 111 pairs. There is significant mixing between the lowest- lying SU(3)J singlet and octet 1mcsons resulting in the ¢ and w mass eigenstates with flavor quantum numbers ss and )2(uu + dd), respectively. This occurs not because of an anomalously large SU(3)J violating mass mixing element, but rather because of the near degeneracy 113 of the SU(3)J singlet and octet states. Nonetheless, because the decay of the w to J( R is kinematically forbidden, this mass mixing can result in large violations of our SU(3)J predictions for decays B-+ Jj'ljJ(MM)L=l when the MM pair is resonant. Next we consider the implications of SU(3)J symmetry for decays B -+ DM M. As was noted in section 5.1, the effective Hamiltonian for these decays transforms as an octet under flavor SU(3)J· Again, we shall separately treat the cases where the relative orbital angular momentum L of the M M pair is even and odd. As far as group theory factors are concerned, when L is even we can take as our effective Hamiltonian . i k HkD j -e . i . i kD -e + cB,HiA1kAfe . i i kD- e Herr -_ aB,A1iA1e + bB,A1iAfkHe + d(BiMjiJi)(Mf H£) + e(BiiijiJi)(Mf Mf) + J(BiiJi)(AifA1f HJ). (5.39) In Eq. (5.39) Hj are elements of the 3 x 3 matrix (upper index labelling rows and lower index labelling columns) (5.-10) Table 5.10 presents the results that follow from this effective Hamiltonian for the Cabibbo allowed decays. Under isospin the effective Hamiltonian for the Cabibbo allowed decays is I = 1. \Vhen two pions possessing a net charge are in an even partial wave they form an I = 2 state. This implies the following isospin relation (1/4)f(B - -+ D+1r-1r-) = f(B- -+ D 0 (7r-7r 0 )L=0,2 ... ) = r(B 0 -+ n+c7r-7r 0 )L=o,2, . ..) (5.41) The first process that appears in Eq. (5.41) has been observed. Experimentally (5.2] (5.42) Since the amplitudes with L odd don ' t interfere with those with Leven, we conclude 114 that 4f(E- ~ D 01r 01r-) >f(E- ~ D+1r-1r-) (5.43a) 4f(E 0 ~ D+1r-1r 0) >f(E- ~ D+1r-1r-) ( 5.43b) Of course, the results in Table 5.10 generalize to decays involving aD* instead of a D. Experimentally [5.2, 5.3) Er(E- ~ D*(2010)+7r-7r-) = (2.5~~:~) x 10-3 (5.44a) Er(E 0 ~ D*(2010)+7r-7r 0 ) = (1.5 ± 1.1) x 10- 2 (5.44b) which is consistent with the generalization of Eq. (5.43b) and indicates that the E 0 ~ D*+1r-1ro rate is dominated by L odd. There are also some SU(3)J relations between the Cabibbo allowed amplitudes with L even. They are lA (E 0 ~ D"}"(K-7r 0)L=0,2, ... ) 12 = 3I(B 0 ~ D"}"(K-7J 0)L=o,2)1 2 1 lA (E- ~ D"}"(1r- K-)L=o,2, ... ) 12 = 1A(B- ~ D+1r-1r-)l 2 (5.45a) (5.45b) 2 1 lA (B~ ~ D+(I{ 07r-)L=0,2, ... ) 12 = 1A(B- ~ D+1r-1r-)l 2 (5.45c) 2 lA (B2 ~ D 0(!{ 07] 0 )L=o,2, ... ) 12 = liA (B~ ~ D 0(1{ 07r 0)L=o,2... ) 2 1 - (5.45d) For the case L odd, the effective Hamiltonian must be antisyrnmetric under interchange of the flavor quantum numbers of the !vi m.esons. For example, an antisymmetric version the term proportional to a in Eq. (5.39) is ( 5.-16) Only the term proportional to the reduced matrix element e has no antisymmetric analog. So the B ~ D 1UJ! decay amplitudes with L odd are parametrized by 115 five reduced matrix elements which we denote by a', b', c1 , d', and f'. Table 5.11 presents the implications of the SU(3) flavor symmetry of the strong interactions for the Cabibbo-allowed decays B ~ D(M!llf)L=l,3,.... There are several SU(3)J relations. For example (5.47) Tables 5.12 and 5.13 present the predictions of SU(3) flavor symmetry for the Cabibbo- suppressed B ~ DA1 M decays with L even and L odd, respecti,·ely. Since the Hamiltonian for the Cabibbo- suppressed decays is part of the same octet as the Hamiltonian for the Cabibbo- allowed decays, we can express the Cabibbo-suppressed decay rates in terms of the same reduced matrix elements as were used for the Cabibbo- allowed decays. An inspection of tables 5.10, 5.11, 5.12, and 5.13 reveals that there are simple SU(3)J relations between the Cabibbo- allowed decays and the Cabibbo-suppressed decays which hold independent of the value of L. They are IA(B 0 ~ D+7r- R 0 )1 2 = siiA(B2 ~ D"I I\ - J\ 0 )1 2 ( 5.4Sa) IA(B 0 ~ D"I R 0 J\-)1 2 = siiA(B~ ~ D+ J{ 0 7r- )l 2 (5A8b) IA(B- ~ D"I J( - J(-)1 2 = siiA(B- ~ D+7r-7r-)l 2 (5.4Sc) IA(B- ~ D 0 1r- !? 0 )1 2 = siiA(B- ~ D 0 f{- K 0 )1 2 (.5.4Sd) 2IA(B2 ~ D"I1r° K-)1 2 = siiA(B 0 ~ D+ K- K 0 )1 2 (5.4Se) IA(B2 ~ D 0 1r+1r-)l 2 = siiA(B 0 ~ D 0 f{- J(+)l 2 (5.48!) IA(B~ ~Do Ro J\o)l2 = siiA(Bo ~ Do J\.·O Ko)l2 (5.4Sg) IA(B~ ~ D 0 J(+ K-)1 2 = siiA(B 0 ~ D 0 1r+1r-)l 2 (5.4Sh) IA(B~ ~ D+ !{ 0 f{-)1 2 = siiA(B 0 ~ D"I R 0 7r-)1 2 (5.4Si) There are important sources of SU(3)J violation in the decays B ~ D 1H 1H which can occur when two of the final state particles arise from the decay of a resonance. In addition to the consequences of the mixing of the SU(3) singlet and SU(3) octet 1116 vector mesons mentioned earlier, large SU(3) violations can arise because the D* can decay to Dr. while the D; is kinematically forbidden from decaying to Dsr. or DK. Finally, we consider the three- body decays B --t DDM. As far as group theory factors are concerned we can take as the effective Hamiltonian for these processes Hetr = rn (BiHi)(DkMf De)+ m(BitJi)(DkMf He) + 7J3(BiM~De)(DkHk) + 7J4(BiM~He)(fJk Dk) ( 5.49a) where (5.49b) Tables 5.14 and 5.15 present the predictions that follow from the Hamiltonian for Cabibbo- allowed and Cabibbo- suppressed decays, respectively. Cautionary remarks, similar to those given in the case of B --t Dlvf AI decays, concerning possible large SU(3)J violations induced from resonance effects also apply here. The effective Hamiltonian for Cabibbo-suppressed decays given by Eq.(5.49) neglects the contribution of operators like (bu)(ud), which arise from the b --t urvcoupling and transform as 3 EB 6 EB 15. I = Since the 15 representation contains an 3/2 piece, isospin relations for the Cabibbo- suppressed modes which follow from the I = 1/2 Hamiltonian in Eq. (5.49) are useful for testing the dominance of the operators with charm quarks. Table 5.15 contains two isospin relations r(B- --t D; DJr.-) = 2f(B 0 --t D; DJr. 0 ) (5.50a) f(fl~ --t DJ lJ 0 r.-) = 2f(B~ --t DJ D-r. 0 ) (5.50b) Fig. 2 shows quark line diagrams which illustrate how the two operators (be) ( cd) and (bu)(ud) can contribute to the decay B2 --t Dt D0 r. - . 117 For the Cabibbo-allowed decays, the effective Hamiltonian transforms as an isosinglet. The following relations in Table 5.14 follow from isospin symmetry: r(B- ~ D 0 n- K 0 ) = f(B 0 ~ n+ fJ° K-) (5.51a) f(B 0 ~ D 0 fJ 0 R 0 ) = f(B- ~ n+ n- ](-) (5.51b) 2f(B- ~ D-; D 0 1r 0 ) = f(B 0 ~ D 0 D-;1r+) = f(B- ~ D-; D+1r-) = 2f(B 0 ~ n+ D-;1r 0 ) f(B- ~ n; D-; K-) = f(B ~ n; D-; !? 0 0 ) (5.51c) (5.51d) r(B~ ~ D 0 D+1r-) = f(B2 ~ n- D 0 1r+) = 2f(B2 ~ D 0 D 0 1r 0 ) = 2f(B2 ~ n+ n-1r 0 ) (5.51e) r(B2 ~ n-; D° K+) = r(B2 ~ n-; n+ K 0 ) (5.51!) f(B2 ~ n- n; !? 0 ) = f(B2 ~ D 0 n; ](-) (5.51g) f(B2 ~ n+ D-ry 0 ) = f(B2 ~ D 0 iJ 0 ry 0 ) (5.51h) Comparison of Tables 5.14 and 5.15 reveals that there are many simple SU(3)J relations between the Cabibbo-allowed and the Cabibbo- suppressed B ~ DD1\1 decays. Some of them are IA(B- ~ D 0 D-; !( 0 )1 2 = siiA(B- ~ D 0 n- J\ 0 )1 2 (5.52a) IA(B- ~ n; n- J(-)1 2 = siiA(B- ~ D-; n+'Tr-)1 2 (5.526) IA(B- ~ n; D-;1r-)l 2 = si iA(B 0 ~ D 0 fJ 0 R 0 )1 2 (5.52c) IA(B- ~ n+ n-7r-)l 2 = si iA(B- ~ n; D-; f{-)1 2 (5.52d) 5.3 Concluding Remarks On Chapter 5 In this chapter we have used the transformation properties of the weak Hamiltonian for nonleptonic B - meson decays to derive SU(3)J relations amongst many of the possible two- and three-body B - meson decays. Since the SU(2) isospin symmetry 118 works much better than the full SU(3) emphasise has been placed on the predictions that follow from isospin. The isospin relations provide useful tools for discerning the importance of various competing effects that can occur in nonleptonic B - meson decays. As we have discussed previously, it is possible to include, in a phenomenological fashion, some SU(3)J breaking effects and hence improve upon the results of this chapter. For example, in section 5.1 it was noted that generalizing the predictions forB-+ DM to B-+ DV where Vis one of the low-lying 1- mesons is not straightforward because of mixing of the SU(3)t octet state IVs) = )6-(iuu) + ldd)- 2jss)) with the SU(3)J singlet IV1) = )3-(iuu) + ldd) +iss)). This was discussed in Chapter 2. As far as group theory is concerned we can take as our effective Hamiltonian for the decays B-+ DV where II£ given by Eq. (4.3). The amplitude A(B 0 -+ D 0 ¢) is expected to be Ycry small since the decay B 0 -+ D 0 ¢ is forbidden by the Okubo- Zweig-Iizuka (OZI) rule. Setting this amplitude to zero implies the relation b' + c' e1 = - - v'3 (5.54) between reduced matrix elements. So the B -+ DV decay amplitudes are expressible in terms of the three reduced matrix elements, a', b' and c'. Using these expressions we find that the generalization of Eq. (5.3) is IA(B 0 -t D 0 p 0 )1 2 + IA(B 0 -+ D 0 w)l 2 = IA(B 0 -+ Dt I\*-)1 2 + IA(B~ -+Do K*o)l2 (5.55) As we discussed in Chapter 1, in the large Nc limit matrix elements for nonleptonic B - decays factorize. This prO\·ides a pattern of SU(3)J breaking that might be used 119 to improve some of our results. For example, factorization suggests that (5.56) would be an improvement over Eq. (5.4). In this chapter we have focussed on SU(3)J predictions for nonleptonic B - m eson decays to final states with mesons. It is also possible to consider SU(3)J predictions for B - meson decays to final states involving uncharmed baryons. For example, there may be SU(3)J relations between the Cabibbo allowed and the Cabibbo suppressed decays B ---+ D N N, where N denotes a member of the lowest- lying baryon octet (consisting of the nucleons and hyperons). It is also possible to consider the consequences of SU(3) flavor symmetry for semileptonic B - meson decays. For example, since the effective Hamiltonian for .6.c = 1 B ---+ D cDe decays is an SU(3) singlet, it follows that (5.57) The first equality in Eq. (5 .57) follows from isospin. For scmileptonic decays B ---+ 111 eve that don't change charm, the effective Hamiltonian transforms as an anti triplet. Since there is only one way to combine the product of a triplet, an antitriplet and an octet into a singlet, these decays are also related by SU(3)J flavor symmetry. In this case r(B 0 ---+ ?r+eve) = 2f(B----+ 1r 0 eDe) = f(B~---+ K+eve) = 6f(B- ---+ ry 0 eDe) 120 (5.58) References [5.1) H. Albrecht et al. (ARGUS Collaboration), Phys. Lett. 215B , 424 (1988). (5.2} C. Bebek et al. (CLEO Collaboration), Phys. Rev. D36, 1289 (1987). [5.3) H. Albrecht et al. (ARGUS Collaboration), Phys. Lett. 185B , 218 (1987). [5.4} A. Chen et al. (CLEO Collaboration), Phys. Rev. D31, 2386 (1985). [5.5) H. Albrecht et al. (ARGUS Collaboration), Phys. Lett. 182B , 95 (1986). [5.6] Isospin relations for B decays are discussed in: H. J. Lipkin and A. I. Sanda, Phys. Lett. 201B , 541 (1988). [5.7) J. Rosner, unpublished, and Ref. (12). [5.8] H. Albrecht et al. (ARGUS Collaboration), Phys. Lett. 199B , 451 (1987). [5.9) M. S. Alam et al. (CLEO Collaboration), Phys. Rev. D34 , 3279 (1986). [5 .10) For a review see: M. B. \;\/ise, Lectures deli\·ered at the 19SS Banff Summer School on Particle Physics, CALT- 68- 1514 (1988), unpublished. 121 Table 5.1. Rates for Cabibbo-allowed decays E ---+ DM in terms of the three reduced matrix elements a, b and c. Process Rate (divided by si) E 0 ---+ D+7r- Ia + c12 ~lb- cl 2 ilb + cl 2 Jcl2 Ia + bl2 lbl2 la12 Eo ---+ Do1ro Eo---+ DoTJo E 0 ---+ D+ J(s E----+ D 0 1rEo---+ Do J(O s E s0 ---+ D+1rs Table 5.2. Rates for Cabibbo- suppressed decays E ---+ D/11 in terms of the three reduced matrix elements a, b and c. Process Rate E 0 ---+ D+ J(- sflai 2 sr lbl 2 sila + bl 2 si ~lc12 siilc- 2bl 2 srla+cl 2 si lcl 2 Eo---+ Do f{O E----+ D 0 ]{Eo---+ Do1ro s E2---+ Do17o E s0 - t D+rs \ E s0 ---+ D+rr- 122 Table 5.3. Rates forB- meson decays of the type B --t DD in terms of the two reduced matrix elements a and (3. Process Rate B 0 --t n+ ns l/31 2 Bo--t Do [JO silal 2 B 0 --t n+ n- sfla + /312 B 0 --t n+ s ns silal 2 B- --t D 0 D-; 1/31 2 B- --t D 0 n- sil/31 2 Bos --t no [JO lol2 B s0 --t n+ n- Iof B~ --t Dt D-; Ia + /312 B s0 --t n+ s n- sil/31 2 123 Table 5.4. SU(3)J predictions for rates forB-+ Jj'!j;M normalized to the decay rate for E- -+ J j'lj;J(-. Process Rate Eo -+ J j'lj;Ro 1 Eo-+ Jf'!j;rro si/2 Eo -+ J !V->TJo si/6 E- -+ J /'1/JK- 1 E- -+ J /'1/Jrr- 82 E~ -+ J I V->TJO 2/3 E~ -+ J / V->K 0 s-1 1 ') 124 Table 5.5. SU(3)J predictions for decays E ----+ M !11 that do not change strangeness. Process Rate Eo ----+ 7r+7r- I2Ac3) + cc 3> + AcTS) + 3Cc 1s) - Ac6) 1 Eo ----+ 7ro7ro ti2Ac3) + C(3) + A(TS)- 5C(ls)- A(6)1 Eo ----+ fJofJo ti2A(J) + tc(3) - A(TS) + C(TS) + A(6) 1 Eo ----+ 7rofJo t 1- Cc3> + sA ciS) + c(ls) - Ac6> 1 Eo ----+ J(O j{O I2Ac3> + cc 3) - 3Ac15> - c(ls) + A(6) 1 Eo ----+ f{+ J(- I2A(3) + 2A(15) 1 E- ----+ 7ro7r- 32ICcis) 1 E- ----+ fJo7r- 2 2 2 2 2 2 2 2 i-I2C(3) + 6A(is) + 6C(i5) + 2Ac6) 1 E- ----+ I\o ](- ICc3> + 3.4(15) - c(ls) + A(6) 1 Bos ----+ f{+ 7r- IC(J) - A(l5) + 3C(l5) - A(6) 1 Eos ----+ I\o 1ro ~I - c(3) + Ac1s) + .sc(E) + A(6) 1 E~ ----+ f{o1lo tI - Cc3 + Ac15) + scc15 + A(6) 1 2 2 2 > 125 > 2 Table 5.6. SU(3)J predictions for E --+ M }.1 decays that change strangeness. Here it is assumed that penguin-type diagrams with a charm quark or top quark in the loop dominate. Entries in the second column should be multiplied by ls2ei 8 + s3l 2 if compared with Table 5.5 u sing Eq. (5.21). Process Rate (divided by ls2ei8 + s3l 2 ) Eo--+ ?r+ J(- 16(3)12 Eo - t ?ro J?O 2IC<3) I I 2 Eo - t J?OTJo ~16(3)1 2 E- - t J(-?ro ! 10(3) 12 E- - t J(-TJO 6 IC'(3) 1 E- - t J?07r- 2 IC''-1 (3) E~ - t J(+ I\- I2A(3) + 6(3) 12 Eos - t f{o ]{O I2AC3) + c< 3)I nos - t 7ro7ro 2 ')lA' - (3) 1 BJ --+ 77o77o 2 2 2ll' I (3) + 3 6'(3) 1 Eos --+ 7r+7r- 4lrl(3) 2 1 A 1 A 126 I 2 A I 2 Table 5.7. SU(3)J predictions for decays B---+ D.M. Process Rate B- ---+ [)OJ(- la(6) + a(3) + /3(6) + !3(3) 12 B- ---+ D-;ry 0 t la(6) - a(3) - 2/3(6) - 2/3(3) 12 B- ---+ n- f< 0 2 1/3(6) + !3(3) 1 B- ---t ns1r 0 ~la(6) - a(3) 1 Bo---+ [)O J(O 2 la(6) + a(3) 1 B 0 ---+ D-1r+ s la(6) - a(3) 12 Bos ---+ Do1ro tl /3(6) - !3(3)1 0 B s---t n-s\y+ la(6) - a{3) + /3(6) - /3{3) 1 B~ ---t [)OTJO 2 2 2 tI - 2a(6) - 2a(3) + /3(6) 2 1/3(6) - !3{3) 1 B s0 ---+ D-1r+ 127 !3(3) I~ Table 5.8. SU(3)J predictions for decays B --+ J 1'1/JAI 1\1, when the relative angular momentum of the two pseudoscalar mesons is even. Process Rate Bo --+ J I'I/J7r+ J(- IGI 2 Bo --+ J I'I/J1ro [(O ~IGI 2 Bo --+ JI'I/;Ko J{O iiGI 2 sii2F + Gl 2 sii2F + Gl 2 2 ~sii2F + tGI tsiiGI 2 sil2FI 2 sii2F + Gl 2 B- --+ J I'I/J7ro J(- ~ I GI B- --+ JI'I/JTJo ]{- B- --+ J I '1/;rr- TJo ilC:l 2 IGI 2 ~3 ·'I:2IGI 2 B- --+ J 1'1/J K- ]{ 0 siiGI 2 B~ --+ JI 'I/J rro7ro ~I2FI2 B~ --+ J 1'1/JTJOTJO ~ I 2F + tGI 2 B~ --+ j 1'1/J Ko J{o I2F + Gl 2 I2F + Gl 2 I2FI2 siiGI 2 ~si IGI 2 isilGI2 Bo --+ JI'I/JTJKo Bo --+ J I'I/J7r+7rBo--+ Jl'l/;1ro1ro Bo --+ J 1'1/JTJoTJo Bo --+ J I 'lj;1roTJo Bo--+ JI'I/J K+J{- B- --+ Jl'l/.'1r- Ro B~--+ J/'lj•K+ f(- B~ --+ J f'lj;rr+rr- B~--+ Jl l/; J\+7rB~ --+ JI'I/'Ko7ro B~ --+ J N,Ko1lo 128 2 Table 5.9. SU(3)J predictions for decays E -+ J 17/JM !vf, when the relative orbital angular momentum of the two pseudoscalar mesons is odd. Rates are normalized to that for E 0 -+ J I'I/J1r+ ](-. Process Rate Eo-+ JI'I/J7r+J(- 1 Bo -+ J I 7/J1ro J(o 2 Eo-+ JI'I/Jryo](o 2 Eo -+ J I'I/J7r+7r- s2 Eo-+ JI'I/JKORO s2 E--+ JI'I/J7r0](- 1 2 E- -+ J I'I/J1r- J{O 1 B--+ JI'I/Jryof{- 2 1 3 1 1 3 E- -+ J I'I/J7ro7rB- -+ J I'I/JK 0J(- 2s 21 B~ -+ J 17/JK+ K- 1 E~ -+ J I 1/J Ko [(O 1 B~ -+ J I'I/JK+1r- s2 B~-+ JI'I/JI\oryo ~s2 2 1 B~-+ JI'I/JKo7ro .!.s2 s2 1 1 2 129 ] Table 5.10. Implications of SU(3)J symmetry for Cabibbo-al1owed decays E-+ Dlvf M, where the relative angular momentum of the i\1 mesons is even. Process Rate Eo -+ no7ro7ro ~J2e + c + b- aJ 2 2 lJ2e 2 + lc 3 + lb 3 + laJ 3 tJc- bJ2 J2e + d + c + bJ 2 J2e + bJ 2 J2e + cJ 2 tl2f + d + 2c + aJ 2 ~Jd + aJ 2 If+ cJ2 Jd + cJ 2 ~Jc- aJ 2 tic- aJ 2 2 ~Jd + aJ tl2f + d + 2b + aJ 2 If+ bJ 2 ~J2d + 2aJ 2 Jd + aJ 2 Jd + bJ 2 ~lb- aJ 2 tlb- aJ2 Jd + aJ 2 tlf- dJ 2 If+ aJ 2 Bo -+ no1Jo1Jo Eo -+ norJo7ro Eo -+ no7r+7rEo -+ no J(O J(O Eo-+ no J(+ J(Eo -+ n+1J07rEo -+ n+7r-7ro Eo-+ n+ J(- J(O Eo -+ n+ J?07rs Eo -+ n+ K-7ro s Eo -+ nt K-Tfo E- -+ no7ro7rE- -+ no1Jo7rB- -+ n° K- !{ 0 E- -+ n+1r - 1rE--+ n;1r- f{E 0 -+ n° h·+7rs Eos -+ no f{o 7ro E~ -+ no 1\.o ,1o E~ -+ n+ J\ 0,.- E s0 -+ n+17°1rs Eos-+ n+y-yo s\ \ 130 Table 5.11. Implications of SU(3)J symmetry for Cabibbo-allowed decays B ~ DM M, where the relative orbital angular momentum of the !If mesons is odd. Rates are expressed in terms of five reduced matrix elements; a', b', c', d' and f'. Process Rate Eo ~ Do 17 o7ro B s0 ~ D+ K 0 rr- tla'12 ld'- c' + b'l 2 lb'l2 lc'l2 tid'+ a'l2 !I- 2/' + d'- 2c' + a'l 2 If'+ c'l2 ld'- c'l 2 !l-c'+a'1 2 tl3c' + a'1 2 tl2f' + d' + 2b'- a'l 2 tid'- a'12 If'+ b'l2 ld'- a'1 2 ld' + b'l2 !I- b' + a'l2 tl3b' + a'l2 ld' + a'l 2 Bos ~ D+1Jo7rs ~ld'l2 Bos ~ D+r-Fo s \ \ If'- a'l2 21/'12 Bo ~ Do7r+7rBo ~Do J(OJ?.O Eo ~ Do J(+ J(Eo ~ D+1Jo7rEo ~ D+7r-7ro Eo ~ D+ J(- f{o Eo~ D+ J?.07rs Bo ~ D+ K-7ro s E 0 ~ Dt K-17° E- ~ Do7ro7rE- ~ Dor,orrE- ~ Do]{- f{o E- ~ D;1r- KE s0 ~ D 0 J(+rrBos ~Do f{orro Bo ~ Do f{o 17 o Eos ~ D+7rorrs 131 Table 5.12. Implications of SU(3)J symmetry for Cabibbo- suppressed B --? nM !11 decays when the relative angular momentum of the Jf mesons is cYen. Process Rate (divided by si} Bo --? n+ 7ro ](Eo --? n+7r- J{O !if- dl 2 If+ ai 2 Bo- n+.,o ](- ~1- f + dl 2 Bo - no 7r+ ](- Bo --? no .,o J(O ld + bl 2 ld + ai 2 !I- b + ai2 ~~- b + ai2 B- --? no7ro ](- tlf + d + b + ai 2 E-- no7r- J(O if+ bl 2 Bo --? n+ J(O ](s Bo --? no7ro j{O B- - il - f +d-b+ ai2 id + ai2 no.,o ](- B-- n+7r- xE- - 2ld + ai 2 n; K- K- B~- n;1r- R 0 2 If+ ci 2 Bos --? n+'T]O ](s tlf + 2d + c + 2al 2 Eos --? no rro1ro !12e + ci 2 ! l2e + c/3 + L!b/3 - 2a/31 2 l2e + ci2 B~ --? n-}r.° K- !if+ ci B~ --? no 1lo7Jo B~- n°r.+rr- Bos - no t.. ·o !\o Eos - no f{+ I\- Bos - n+ 1\.o I\- l2e + bl 2 l2e + d + c + bl 2 ld + ci 2 B~ --? no ryo1ro ~i-a+ci2 B~ --? n+ '7°1r- ~ic- ai 2 132 Table 5.13. Implications of SU(3)J symmetry for Cabibbo-suppressed decays B - nAI M where the relative orbital angular momentum of the 111 mesons is odd. Rates are expressed in terms of the same five reduced matrix elements (a', b', c', d' and J') as the Cabi b bo-allowed case. Process Rate (divided by si) Eo- n+7ro K- tlf'- d'l 2 If'- a'l2 !13!' + d'l 2 ld' + b'l2 ld' + a'12 !lb'+a'l2 il3b'- a'l2 tlf' + d' + b'- a'l2 If'+ b'l 2 il3f' + d' + 3b'- a'l 2 ld'- a'12 tlf' + c'1 2 If'+ c'l 2 !13!'- 2d' + 3c'- 2a'1 2 lc'12 lb'l 2 ld' - c' + b'l2 ld'- c'l2 !la'l2 2lc'12 Eo - n+7r- J?O Eo- n+.,o J(Eo- no7r+ ](Eo - n+ s J?O ](Eo - no7ro J?O Eo- no77o J?O E- - no7ro f{_E- - n°rr- ]{ 0 E- - no.,o ](E - - n+7r- KEos - n+7ro ]{s E2- n;7r- Ko Eos - n+.,o ]{s Eos - no7r+7r Eos - no 1.;o 1\-o E~- n° f{+ f{- Eos - n+ 1\o I\E~ - Do1Jorro Eos - n+7r07rE~ - n+., 0rr- ~la'l 2 133 Table 5.14. Implications of flavor SU(3)J for Cabibbo- allowed decays B - t nlJM assuming the effective Hamiltonian transforms as a 3. Process Rate B- - t no [JO ](- lrJ2 + 77412 B- - t n+ n- K- 2 17741 B- - t n°n-R 0 lr12l 2 B- - t n° n;ry 0 il- 27]2 + 7]31 2 B- - t n° n;1r 0 ~ITJ31 B- - t nt n; ](- ITJ3 + TJ41 B- - t n; n+1f- ITJ31 Bo - t [JOn+ J(- ITJ21 B 0 - t n+ n- J\ 0 1772 + 7]41 B 0 - t n+ D; 17° i I - 2ryz + 173! 2 B 0 - t n° n-; r.+ ITJ31 B 0 - t D+ n;r. 0 117J31 0 B 0 - t n+ s ns R ITJ3 + 174! Continued ... 2 2 2 2 2 2 2 2 Table 5.14. Continued. Process Rate Bo --+ no tJO [(0 17741 Bos --+ no jj01fo !l77tl B~ --+ no jjo 17 o ~1771- 27741 2 Bos --+ tJO n+ 1f- 17711 Bos --+ jjO n+ J(s 1771 + 7721 B s0 --+ n- n°1r+ 17711 B s0 --+ n+ n-1fo !l77tl B~ --+ n+ n- 17 ° ~1771 - 277·tl 0 B s0 --+ n- n+ s R 17Jl + 112l 2 B~ --+ n; n° J{+ 1771 + 7731 2 0 B s0 --+ ns n+ I\ lrn + 7731 2 0 B s0 --+ n+ s n-q s 321 '71 + 772 + 173 + 774 12 2 2 2 2 2 2 2 Table 5.15. Implications of flavor SU(3)J for Cabibbo-suppressed decays B--+ DDJ\1. The effective Hamiltonian is assumed to transform as a 3. Entries in the second column should be multiplied by si when comparing with the results of Table XIV. Process Rate (divided by si) B- --+ Do jj07r- 1772 + 7741 B- --+ D 0 D-1ro ~1- 772 + 7731 2 B---+ D+ D_1r_ 1773 + 7741 B- --+ D 0 D-77° i 1772 + 7731 2 B- --+ D 0 D-; ]{ 0 17721 B- --+ DJ D- 1\·- 17J31 B- --+ DJ D-; 1r- l77-tl Bo --+ Do jjO 7ro ~17Jl - 7741 Bo --+ Do jj0 17 o 61 I771 + 774 12 B 0 --+ n- D 0 1r+ 1771 + 7731 Bo --+ j)O DJ ]{- irJ1I Bo --+ n+ jj07r- 1771 + 7721 Continued ... 13G 2 2 2 2 2 2 2 2 2 Table 5.15. Continued. Process Rate (divided by si) B 0 ---+ n+ n-'ll"o tlr]l + T/2 + T/3 + T/41 2 B 0 ---+ n+ n-TJo tiTJI + T/2 + T/3 + T/41 2 0 B 0 ---+ n- n+ s R ITJI + T/31 B 0 ---+ ns n° ](+ ITJtl B 0 ---+ n-; n+ ]{ 0 ITJI + T/21 B 0 ---+ n+ n-'ll"o s s t ITJ41 0 B 0 ---+ n+ s n-ry s t l - 2TJ1 + T/41 Bos ---+ n+ [J01is ITJ2I B~ ---+ n; n-"iro t ITJ2I B s0 ---+ n+ s n-TJo {; ITJ2 - 2TJ31 0 0 B s--t n+n-r s s\ ITJ2 + TJ41 B~ ---+ n° n- ]{+ ITJ31 B s0 ---+ n+ n- I\ 0 ITJ3 + TJ41 Bos --t no j)O J(O ITJ41 137 2 2 2 2 2 2 2 2 2 2 2 2 b ;9> FIGURE 5 . 1 138 d,S 1t b (a) gO s - - s s b gO s (b) s FIGURE 5 . 2 139 ADDENDUM In deriving our results for charmed baryon, B-meson and D-mcson decays we neglected the mixing between the pseudoscalar mesons in the octet and the SU(3) singlet. The isosinglet octet meson, the T] 0, mixes slightly with the isosinglet SU(3) singlet meson, the TJ' 0 . The mixing angle between the mass eigestates and the SU (3) eigenstates is approximately 13°. We will again assume that the SU(3) eigenstates are the mass eigenstates and derive relations between the decay rates to final states containing an TJ 10 meson. Let us begin by considering the process D ---+ MTJ', where M is a pseudoscalar octet meson. The effective Hamiltonian for the process is (6.1) From this effective Hamiltonian we find that there are the following relations f(DO---+ Jt1TJ'O) = _;f(DO---+ T]o 1110) = 2?f(DO---+ 1f0T}10) = --;.r(Do---+ ](OTJ'o) 3sr sr sl (6.2a) r(n;- 1f+TJ'0 ) = \r(n+- 1f+TJ'0 ) = 12 r(n;- K+17'0 ) = _.;.r(v+- K+TJ' 0 ) sl sl sl (G.2b) \Ve can also derive relations for B-meson decays. In the case of B ---+ DTJ' we find that only one SU(3) singlet can be formed, consequently (6 .3) The same is true for the decays B---+ Jj'lj•ry' where we find (6.4) There are two reduced matrix elements that contribute to the decay B ---+ Dry', and there are no relation s bebveen the decay modes. However, if either the 3 or the 140 6 components of the Hamiltonian dominates the decay then (6.5) Finally, we consider the decays B coupling. -t Mr/ which are induced by the b - t u There are three reduced matrix elements that contribute to the decay process and the effective Hamiltonian is There are no relations between the Cabibbo-allowed decay rates. The decay rates for the modes are B- - t 7r - TJ 10 IA3+ 3AI5 + A61 2 1 (6.7a) 2 B~ - t 'lrOTJIO 21- A3+ 5AIT- A61 ,.,o,.,,o 6IA3+ 3AIT- 3A61 2 (6.7c) Eos - t yo 10 \. TJ IA3- AIT- A61 2 (6.7 d) B~ - t 1 (6.7b) As we previously discussed, the Cabibbo-suppressed decay could be dominated by the penguin diagrams. If the penguin diagrams do dominate the decay process then the following relations are found (6.8) 141