Two Topics in Elementary Particle Physics: (1) Quark Graphs and Angular Distributions in the Decays of the Axial-Vector Mesons. (2) Universal Current-Current Theories and the Non-Leptonic Hyperon Decays Citation Colglazier, Elmer William, Jr. (1971) Two Topics in Elementary Particle Physics: (1) Quark Graphs and Angular Distributions in the Decays of the Axial-Vector Mesons. (2) Universal Current-Current Theories and the Non-Leptonic Hyperon Decays. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/HBFX-4095. https://resolver.caltech.edu/CaltechTHESIS:04192018-085253797 Abstract The Thesis is divided into the following two parts: (1) We examine three aspects of the axial-vector mesons: (i) angular distributions of the I = 1 states, (ii) mixing of the I = 1/2 states, and (iii) absence of the I = 0 states. Using a model of mesons decaying via production of a quark-antiquark pair with the quantum numbers of the vacuum, we relate the angular distributions in the decays A 1 → ρπ and B → ωπ, predicting 2(g 1 /g 0 )A 1 = (g 0 /g 1 ) B + 1. This relation is consistent with the present, somewhat ambiguous experimental data. Also, we describe satisfactorily, in terms of two parameters, the partial widths of the 0 + , 1 + , and 2 + mesons decaying into 1 - 0 - and 0 - 0 - pairs. The prediction of the model is that SU(6) W x 0(2) L Z relations hold among all the D waves and among all the S waves, but not between the two groups. In fact, our two-parameter fit to the data entails a ratio of S wave to D wave amplitudes of approximately the same magnitude but opposite sign to that implied by SU(6) W x 0(2) L Z . Unlike the widths, the angular distributions are sensitive to the relative sign and are thus crucial in determining that the fit of our model differs considerably from the SU(6) W solution. Parameters of the fit are applied to the l + kaons, which may mix with one another. The results are sensitive to the mixing angle ø, and merely assuming lower bounds on widths of both physical states establishes the limits 10° ≤ ø ≤ 35° As a result of this mixing, one predicts: (a) the suppression of the K*π mode of the lower peak, (b) the suppression of the ρK mode of the upper peak, and (c) decay distributions in the K * π mode similar to that of the A 1 for the lower state and to that of the B for the higher. The properties of the missing isoscalar mesons are described with particular emphasis on the ninth 1 ++ state. Expected properties of this meson, the D', include: (a) assignment to a weakly mixed SU(3) singlet, predicted by duality and confirmed by the Gell-Mann-Okubo mass formula; (b) a mass of ~950 MeV, predicted by super-convergence with assumptions about the relative couplings of D and D’; (c) decay modes ηππ and π + π - γ; and (d) the possibility of a suppressed ρ signal in the π + π - spectrum of the π + π - γ final state, despite the expectation that the pions are in a state with I = J = 1. These features suggest that a recently reported meson near this mass with decay modes ηππ and π + π - γ may be a candidate for this state, although J pc = 1 +- is also a definite possibility for the new meson. (2) Because of the limited evidence for the V-A Cabibbo theory in the non-leptonic weak decays, we examine the compatibility with experiment of more general current-current theories. These theories, constrained by universality, are constructed from the neutral and charged currents obtainable in the quark model, i.e., scalar, pseudoscalar, vector, axial-vector, and tensore Using current algebra and PCAC, a certain class of these theories, including Cabibbo's, is found to be consistent with the S wave amplitudes for the non-leptonic hyperon decays. The P wave amplitudes remain unexplained. Nevertheless, another class of theories, also including V-A, plus the assumption of a symmetric quark model, predict the ΔI = 1/2 rule. 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): Zweig, George Thesis Committee: Unknown, Unknown Defense Date: 24 May 1971 Funders: Funding Agency Grant Number NSF UNSPECIFIED Atomic Energy Commission UNSPECIFIED Record Number: CaltechTHESIS:04192018-085253797 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:04192018-085253797 DOI: 10.7907/HBFX-4095 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 10820 Collection: CaltechTHESIS Deposited By: Tony Diaz Deposited On: 25 Apr 2018 23:50 Last Modified: 20 Jun 2024 21:01 Thesis Files Preview PDF - Final Version See Usage Policy. 51MB Repository Staff Only: item control page

TWO TOPICS IN ELEMENTARY PARTICLE PHYSICS: (1) QUARK GRAPHS AND ANGULAR DISTRIBUTIONS IN THE DECAYS OF THE AXIAL-VECTOR MESONS (2) UNIVERSAL CURRENT-CURRENT THEORIES AND THE NON-LEPTONIC HYPERON DECAYS Thesis by Elmer William· Colglazier, Jr. In Partial Ii,ulfillment of the Requirements For the D~gree of Doctor of Philosophy California Institute of Technology Pasadena, California 1971 (Submitted May 24, 1971) ii ACKNOWLEDGMENTS I am deeply indebted to three professors: Professor R. P. Feynman for teaching me, Professor J. Rosner for collaborating with me, and especially Professor G. Zweig for advising me. all three. I .have learned much from I am also grateful to my other colleagues at Caltech for numerous conversations. I wish to thank the National Science Foundation ·(1966-1970) and the Atomic Energy Co:rmnission (1970-1971) for their financial support, Mrs. Virginia Franklin for her excellent typing, and my wife Cathy for her patience during five years of graduate school. iii ABSTRACT The Thesis is divided into the · following two parts:, (1) We examine three aspects of the axial-vector mesons: distributions of the I = 1 states, (ii) mixing of the and (iii) absence of' the I = 0 (i) angular I = 1/2 states, states .. Using a model of mesons decaying via production of a quarkantiquark pair with the quantum numbers of the vacuum, we relate the angular distributions in the decays A -7 pn 1 and B -7wn, predicting 2(g /g )A = (g /g )B + 1. This relation is consistent with the 1 0 1 0 1 present, somewhat ambiguous experimental data. Also, we describe satisfactorily, in terms of two parameters, the partial widths of the 0+, 1 +, and 2+ mesons decaying into prediction of the model is that 1 0 and O 0 pairs . The relations hold among z all the D waves and among all the S waves, but not between the two groups.. SU(6)W x 0(2)L In fact, our two-parameter fit to the data entails a ratio of S wave to D wave amplitudes of approximately the same magnitude but Unlike the • z widths, th~ angular distributions are sensitive to the relative sign opposite sign to that implied by SU(6)W x 0(2)L and are thus crucial in determining that the fit of our model differs consid~rably from SU(6)W the solution . Parameters of the fit are applied to the l+ kaons, which may mix with one anothero· The results are sensitive to the mixing angle ¢, and merely.assuming lower bounds' on widths of both physical states iv establishes the limits one predicts: 10° :S ¢ :S 35° As a result of this mixing, (a) the suppression of the (b) the suppre~sion of the distributions in the pK K* n mode of the lower peak, mode of the upper peak, and (c) decay K*n mode similar to that of the A for the lower 1 state and to that of the B for the higher. The properties of the missing isoscalar mesons are described with particular emphasis on the ninth of this meson, the D', include: SU(3) 1 ++ state. Expected properties (a) assignment to a weakly mixed singlet, predicted by duality and confirmed by the Gell-Mann- Okubo mass formula; (b) a mass of ~950 MeV, predicted by super- convergence with assu.mptions about the relative couplings of D and Dv; (c) decay modes ~nn and n+n-/; and (d) the possibility of a + - suppressed p signal in the n n spectrum of the n+n - y final state, despite the expectation that the pions are in a state with I = J = 1. These features suggest that a recently reported meson near this mass with decay modes although (2) J pc = 1 ~nn +- and n+n - y may be a candidate for this state, is also a definite possibility for the new meson. Because of the limited evidence for the V-A Cabibbo theory in the non-leptonic weak decays, we examine the compatibility with experiment of more general current-current theories~ These theories, constrained · by universality, are constructed from the neutral and charged currents obtainable in the quark model, ieeo, scalar, pseudoscalar, vector, axial-vector, and tensore Using current algebra and PCAC, a certain class of these theories, including Cabibbo's, is found to be consistent v with the S wave amplitudes for the non-leptonic hyperon decays. P wave amplitudes remain unexplained. The Nevertheless, another class of theories, also .including V-A, plus the assumption of a symmetric quark model, predict the 6.I = 1/2 rule. vi TABLE OF CONTENTS PART 1 TITLE I INTRODUCTION II ANGULAR DISTRIBUTIONS AND QUARK GRAPHS PAGE 1 FOR THE l+ ISOVECTOR MESONS 26 III MIXING OF THE l+ KAONS 49 IV THE NINTH l++ MESON 63 v CONCLUSIONS 73 I INTRODUCTION 81 II UNIVERSAL CURRENT-CURRENT THEORIES 108 III CONSEQUENCES FOR THE NON-LEPTONIC DECAYS 112 IV CONCLUSIONS 118 PART 2 APPENDIX A 119 APPENDIX B 123 vii PART 1 QUARK GRAPHS AND ANGULAR DISTRIBUTIONS IN THE DECAYS OF THE .AXIAL-VECTOR MESONS * *This work has been published elsewhere as follows: E .. W. Colglazier and J .. Rosner 7 "Quark Graphs and Angular Distributions in Positive Parity Meson Decays 7 11 Nuclear Physics B27, 349 (1971) 0 J. Rosner and E. We Colglazier 7 11 A Ninth • Meson wi· th Jpc =·l++, " Physo Rev .. Letters 26, 933 (1971) .. 1 I. INTRODUCTION A significant advance in understanding the spectrum of mesons and baryons came from the proposal that the group SU(3) is an approximate symmetry of the strong interactions. (l) With an exact symmetry of nature, the particles can be classified into irreducible representations of the symmetry group since the generators commute with the Hamiltonian. The particles :belonging to an irreducible representation are degenerate in mass. With an approximate syrmnetry, it may still be possible to .classify the particles into nearly degenerate multiplets. · Gell-Mann and Ne'eman proposed that there exist eight operators which (i) have the right commutation rules to be the generators of SU(3), (ii) approximately commute with the Hamiltonian, and (iii) tie ·the strongly interacting particles into nearly degenerate multiplets. The tremendous success of SU(3) is that all the well-known particles can be classified into its representations. * The existence of the symmetry does not indicate which representations are to be found in nature. ->E- The Examples of classification into SU(3) representations are: Mesons 8 +1 of Jpc = 0 -+ ( n, K, 8 +1 of Jpc = 1 (p, K* ' l'V l'V 'I'}' '11' ) ; w,~); 8 +1 of Jpc = 2++ (A2, K** ; f, f') l'V Baryons 8 of JP = 2i+ (N, .L:, /\, =) .9 l'V 3+ lO of JP= 2 (6, y * ' - * n-) ., ' 2 only representations which seem to be allowed experimentally are the singlet and octet for mesons and the singlet, octet and decimet for baryons. Since SU(3) is an approximate symmetry of the Hamiltonian, it should describe the matrix elements as well as the spectrum of the strongly interacting particles. For example, the three particle couplings - or vertices - should be approximately invariant under transformations of the group·.. These couplings can be determined experimentally from the strong decays of the hadrons. However, the extraction of coupling strengths from widths is subject to ambiguities because of mass factors.. Since the particles within a multiplet are not really degenerate, the value for the coupling depends on the assumed form for the yariation of the matrix element with masso Consequently, an unambiguous comparison of SU(3) predictions for matrix elements is more difficult than for masses. ~ Nevertheless, strengths determined from the strong decays of the 23+ *Comparison of SU(3) predictions with the widths of the 3+ 2 baryon decimet nonet are in good agreement with the symmetry predictions. * and the and coupling 2++ 3+ 2 decimet 3 nonet( ) yields (table also continued on next page): decimet r exp (MeV) r the or ++ 2 no net r exp (MeV) r the or (MeV) (MeV) 6(1238) ~ 11'.N 120 ± 5 116 K** ~ K1t 59 ± 4 59 L:(l385) ~ 11'./\ 33 ± 6 37 A -7 111( 2 16 ± 3 14.5 3.6 ± 1..5 3 .. 8 A 6.5 ± L3 7.1 ~ (2) 11'.L: 2 -> KK 3 Thus, SU(3) provides insight into both the spectrum and the matrix elements of the hadrons. The hadronic spectrum in terms of spins, parities, and allowed representations cannot be understood by SU(3). An incorporation of spin with internal symmetry to explain these features of the spectrum 5 was achieved in the quark model of Gell-Mann( 4 ) and Zweig. ( ) This model considers the observed .hadrons as being constructed out of a fundamental triplet called the quarks. The mesons are made of a quark and antiquark while the baryons are made of three quarks. form a triplet representation of SU(3) and are spin ~- The quarks The spin of a hadron is determined from adding the spin (s) and orbital angular momentum (L) of the constituent quarks. Whether the quarks are real or only a mathematical device is not known since they have yet to be discovered experimentally. ·N evertheless, they are extremely useful in classifying the hadrons and describing many of their properties. The meson spectrum is described by the quantum numbers of q q pair. The charge conjugation of such a pair is (-l)L+S; and, if the parity of an antiquark is opposite to that of a quark as it is for spin ~ particles 7 the parity is 3+ -(-l)L ~ The lightest mesons 2 decimet r. exp (MeV) r· theor (MeV) 2++ no net r. exp (MeV) =(1530) ~ 11'. - 10 ± 3 13 f ~ rr:n: 150 ± 25 154 f' ~KK' 53 ± 20 49 rtheor (MeV) 4 should correspond to a q q pair with octet and singlet (~ + ~ = L = O, which consists of an ,§ + 1) 0 -+ 1 = The low. lying pseudoscalar and vector mesons do indeed correspond to these eighteen states. The next states in the mass spectrum should correspond to a q q pair with L = 1. More states are now possible because there are several ways to combine quark spin to obtain the total angular momentum J. 0) ~ and L=1 In particular, nine states (octet plus singlet) of each of the following 2++ (s = 1 ) , and S Jpc: o++(S = 1), It is these mesons, expecially the axial-vectors, that we shall concentrate on in this paper. The best known are the nine tensor mesons, whose widths were seen to be in good agre~ment with SU(3). predicts the absence of states with No~e Jpc = O that the quark model , (odd)-+ , (even)+- and no such "exotic" states have been found. In order to quarantee the proper isospin and hypercharge (Y =average charge in multiplet, i.e., Q =· e[I 3 +~])quantum numbers of the mesons (which are to be associated with the additive quantum numbers of SU(3)), the three quarks must belong to an SU(3) doublet (p,n) and singlet (A) in isospin with hypercharges - ~ , respectively. 1 3 - and Consequently, the quarks are fractionally charged .. The lowest observed baryon states are the octet of nucleons and decimet of nucleon resonanceso The quark wave functions for the decimet are completely symmetric in both SU(3) and spin.· the quarks are fermions, the spatial wave function must be anti- If 5 symmetric which excludes S wave, contrary to what we would expect for the lowest mass states. Bose 3+ 2 However, if the quarks are assumed to obey statistics, the S wave q q q states consist of a J.vO. l+ -2 rv8 and The symmetric quark model is assumed because of this agreement with the low-lying baryon spectrum. Spin . !2 bosons would seem to violate the spin and statistics theorem except that real quarks have not been seen. to The next states, of negative parity, should correspond L = 1, which consist of 52 Jpc = - (8), and rv All these· multiplets have been established except a tremendous success of the symmetric quark model. The quark mode;I. can give a description of the breaking of SU(3) as well as predicting the spectrum~ Symmetry breaking can be understood by the obvious and simple model of mass differences between the quarks. SU(3) Since isospin is conserved in the strong interactions, breaking can only split the isosinglet isodoublet p and n quarks. A quark from the Assigning a different mass to the A quark yields the Gell-Mann Okubo mass formula for the mesons: { = 4 This prescription also produces the equal spacing rule for the baryon decimet and an equal spacing rule, not the experimentally valid G-M-0 mass formula, for the baryon octeto /\ · and ~ The mass split between the baryons, which both contain one strange quark, results from differences in their quark wave functionse The G-M-0 mass 6 fonnula does not work well for the observed vector mesons, but this anomaly can also be understood by the quark model. functions for the ..[~ (pp + nll - 2 AA) The quark wave I = 0 members of the SU(3) octet and singlet are and is broken by a heavier 1 ~ ~ ( pp- + nn- + AA), respectively. When SU(3) A quark, the two ptates should mix, corre- sponding to diagonalizing the mass matrix, so that the physical states are pure strange quark and pure non-strange quark, ioe., ~ where = -AA = -sin g wl + cos . G w8 w8 = octet state, wl = singlet state, and tan G = 1 h The mass formulae for the physical states which follow from this mixing · are: 2 m (p) = m2 (w) and m2 (p) + m 2 (~) = 2 2m (K*) • Consequently, the isosinglet vector meson states mix naturally in the quark model. * These various examples of symmetry breaking indicate that the quark model is useful for describing the fine structure as well as the gross features of ·the spectrum. The incorporation of spin in the quark model may be described in terms of the higher synrrnetry SU(6), or its orbital extension SU(6) x 0(3) Q In a non-relativistic description there are six states *The absence of mixing of the I = 0 0-+ states is now obscure, but will be resolved by the spin symmetry implied by the quark modelo · 7 of the quark when both SU(3) and spin are included. The symmetry which describes invariance under the unitary transformations of these six states is SU( 6 ) 8 • The "S 11 refers ·t o static because this descrip- tion can only be meaningful when the particles are at rest, as only then are their spins well defined. Thus, SU(6)S , unlike ·SU(3), cannot be symmetry of the complete Hamiltonian. The separation of spin from intrinsic orbital angular momentum is the reason for the failure of SU(6) as a dynamical symmetry. Attempts at a relativistic generalization by combining SU(3) and the Lorentz group have fail~d to produce a universal symmetry that could be valid for the strong 6 interactions. ( ) Yet, SU(6)s x 0(3) describing the particle spectrum. The pseudoscalar and vector mesons can be classified at rest into a 3,§ plus one SU(3) singlet ·- 1--) SU(6)S (2 x 2 = ~5 + 1)· is very successful in (two SU(3) octets - o-+ and and a 1 (SU(3) singlet - 0-+) of (The I= 0 0-+ 1-- states are not mixed because the SU(3) singlet is in a different SU(6)S representation from the octet.) The positive parity mesons form a ~ L L =0 and a the baryons can be classified into a representation of SU(6)S x for a ~6 L = 2)~ ~6 = 1. Similarly, 7!} L = 1 0(3) (and there is considerable evidence Also, SU(6) 8 introduced as various forms of mass breaking in the spectrum can be S•S and L·S couplings (L·S needed for mesons but not for baryons). We have seen that SU(6)S x 0(3) is very useful as an approxi- mate symmetry of the spectrum, but not of the dynamical Hamiltonian. Similar situations have occurred before in physicse For example, the · 8 energy levels of the hydrogen atom possess an extra degeneracy beyond that given by rotational invariance; states of different angular momentum, but with the same principal quantum number, have the same energyo 0(4) This phenomer:oncan be· understood from the symmetry group (generators are the conserved vectors of the coulomb problem: angular momentum ~ J plus a new one called the Lenz vector the nonrelativistic problem, the generators· of Hamiltonian of the hydrogen atom. 0(4) ~* L ). In commute with the A relativistic generalization of the hydrogen atom (for spinless particles) also yields the same 0(4) symmetry of the energy levels.(?) . However, the scattering of bound states does notpossess this symmetry. Consequently, 0(4), like SU(6) , 8 is useful only for analyzing the spectrum of isolated states, not the dynamics of particle interactionso The question now arises as to what is the dynamical symmetry of the matrix elements when spin is included. Since a complete unification of SU(3) and the Lorentz group has not succeeded, a less ambitious approach is required~ ·Perhaps certain sets of processes, such as collinear ones, can be described according to some covariant version of SU(6)~ The quark model, which provides an answer for the symmetry of the spectrum, can possibly do the same for vertices. In the quark 5 model, vertices can be drawn pictorially as uquark graphs, 11 ( ) which are especially useful for visualizing the SU(3) structure. The in- *In classical physics, the Lenz vector poin~s to the perihelion of the orbit; and its conservation means that the orbit does not precesso 9 clusion of spin in quark graphs could provide a clue to the dynamical symmetry of vertices. Quark graphs are pictures of hadron vertices (Fig. 1) drawn according to the following rules: a) each quark or antiquark is represented by a directed line, b) a baryon is represented by three lines running in the same direction, c) a meson is represented by two lines running in opposite directions, and end in a different hadron. d) each line must begin Each diagram corresponds to a possible way of contracting SU(3) indicies to form an SU(3) scalar. The coupling constant of a meson vertex is proportional to the number of quark graphs, each graph being weighted by the product of normalizations of the meson wave functions. For meson-baryon vertices, the graphs are weighted by an additional factor (f + d) or (f - d) depending upon whether or not a mesonic quark is contracted with . a quark that has been symmetrized or antisymmetrized in the baryon wave function. The couplings determined from quark graphs are no more than SU(3) results except for the last rule that no disconnected graphs are allowed. 5 The effect of this rule, postulated by Zweig, .( ) is to relate the cou"pling of the singlet to the octet, and is sometimes called the nonet anzatze (B) Results that follow from this rule are the observed weakness of the <!'? (f' is the pure strange quark p n, ..l- 2· <I> N N, and meson). f v n :rr couplings All the observed two-body strong decays are consistent with the quark graph picture of the decay taking place via creation of an additional quark-antiquark ,pair in an SU(3) singlet state, with no disconnected graphs allowed. 10 b a ~ .\ '~ d . I \ \/ 4 7B ~ ~;~ #'r a Fig. 1. spectator quark tc Quark graph for a meson-meson-meson coupling. 11 Spin can be added to quark graphs in several ways. SU(6)S as a dynamical symmetry (which, of course, is not possible) would imply conservation of quark spin; or, in terms of the graphs, the spin of a spectator quark (see Fig. 1) is conserved and the q q pair is created in a as 1 s0 state. p ~ ~ ~ Many of the strong decays are now forbidden, such and 6 ~N ~. A more reaiistic spin coupling can be generalized from the fact that the q q pair in a quark graph is created in an SU(3) singlet stateo If the q q pair is assumed to be created with the quantum numbers of the vacuum, then the conservation of and P requires that the pair be produced in a s t a t e. J (9,10,11) If the further restriction is made that in· a collinear frame (which is always possible for a two-particle decay) the transverse momenta of quarks in the pair can be neglected compared to their longitudinal momenta, then L . = S z for the 3P stateo(lO) 0 z (1) =0 If the spin of a spectator quark is assumed to be conserved in a collinear frame as well, then the prescription of equation (1) is equivalent to the collinear symmetry SU(6)W (12) The properties of SU(6)W , or its subgroup SU(2)W which describes the spin coupling, can be derived from equation (1). ~ spin, rather than regular S spin, is the generator of SU(2)W . z component of W-spin can be identified with sz The which is still conserved in a collinear process (at least for quark graphs connecting q q L =0 states)o The x and y components of W-spin, however, 12 Since the q q- pair is created with the quantum numbers are different. of the vacuum, the zero. 3 P 0 st~te with S z = Lz = 0 must have W-spin This assignment can be achieved by defining W x and W y on a quark or antiquark to be: CJ wx = P.int Sx· = Pint 2x where but~ = P.int Sy = P int _x_ 2 3 is the intrinsic parity of the quark or antiquark. (l ) P.int (Actually, CJ wy W = ~ x CJ x 2 CJ and W = ~ _x_ y where ~ is the Dirac matrix, 2 These is just the intrinsic parity in the rest frame.) operators satisfy an SU(2) algebra and are invariant under Lorentz transformations in the z direction. Consequently, the W-spin classi- fication for a particle moving with arbitrary momentum along the z axis is equivalent to its classification at rest, so that this group is a possible candidate ·for a relativistic symmet~y of collinear processes. The difference between the W-spin and S-.s pin classification of a q q pair can be determined by using lowering operators on the highest state: Iq t Ci i> = J w = 1, wz = i > = Is = i, sz = 1) w- Iqt q: t> = (wx - iWy ) Iqi Cii) Iq'1. Cit> - Iqt <l t > = s-- jqjq i> = (s x - iS y ) Iqt <i t ) = Iqi q t> + Iqi Cit> 1w = 1, wz = o) = js = o, sz = o) = l2 (lq iit} jw = o, wz = o) = js = 1, s z = o) = ~ ( Iq.Jr iit> + Iqt Ci .i. > ) i - jqiq.l,)) 13 The interchange between the singlet and triplet states of W and 8 spin is called the W-8 spin flip. L z ~ 8z = 0 W = Oo does have 3P Consequently, a 0 q q state with Note that the W-spin and 8-spin classifications of baryon are the same. The consequences of W-spin for decays follows immediately from the W-spin classification and the Clebsch-Gordan coefficients for 8U(2). p and 6 p For example, the decays of the are now allowed: (o) W= 0 W=l < wz = 0 W= wz = 0 wz = Non-trivial predictions are made for the decay of the (~) 6 2 -7 (!) N 2 ;w = W= \z= wz = 12 3 2 (!) 6 2 PO. ) l 1 . (-) N 2 f> 1 1 W=2 2 into Np 6 p (0) W= = 1 3 2 1 = 2 wz = (- ~) -7 3 W=2 N 2 w '= ! z 2 = J~ p(l) 1 W=2 wz == 1 wz = -2 W l; wz = 0\ a1= 0 14 Not much evidence is known experimentally for this decay. if the photon is assigned for the t:. o W = 1, then predictions similar to those follow for electroproduction and photoproduction of the p The experimental data is in good agreement with SU(2)W •* SU(6)W the However, 0-+ and is constructed by combining SU(3) with W-spin. 1-- For mesons, the W-S spin flip requires that the SU(6)W singlet contain the SU(3) singlet, helicity zero vector meson rather . than the SU(3) singlet, pseudoscalar meson. SU(6)W The predictions of can be derived by using this classification and the Clebsch- Gordan coefficients of SU(6) or by using the quark graphs with equation (l)e Unfortunately, only a few predictions of the symmetry can be checked for the lowest meson and baryons (L = O states in quark model) because of their small number of two-body strong decays. The SU(2)W structure is verified in the electroproduction and photo- production of the the ratio of 6 , "While the SU(6)W w -7n n n to p -7nn** structure is verified in and in the F/D for meson- *The photoproduction data give (6(3/2) -7N(l/2) /(1))/(6(1/2) -7N(-l/2) /( 1)) ~ le77 ±.10 to be compared with the theoretical value J"3e · The electroproduction data give a for a longitudinal photon less than 2oo/o of a for a transverse photon. (14) **From the Gell-Mann, Sharp, Wagner model for w -7 3n and the known rate for p -7 nn , SU(6)w predicts the width of w -7 3n equal to 7~0 MeV in comparison with the experimental value of l0o7 MeV. (15) 15 baryon coupling. * Thus, even with the limited evidence, SU(6)W looks very good for the low-lying states. has to be generalized to make predictions for q qstates with L 1 O. SU(6)W x 0(2)L z The most obvious extension, which is called (17 18 19) . ' ' , assumes that w-sp1n is still conserved. Assuming W = S be conserved along with z z J z requires that L z be conserved (the z axis is the collinear axis of the decay)g also In order to obtain non-trivial predictions for meson decays, we must turn to the axial-vector mesons which are and 1 P 1 in the quark model. The axial vector meson states which have been observed experimentally are given in Table 1. These mesons are the only q q L = 1 states that can decay into two q q L = 0 waves ..- S and D. states via two partial The tensor and scalar mesons decay via pure D wave and pure S wave, respectivelyg The amplitude describing the l+ decays can also be written in terms of two independent helicity amplitudes g and g 1 g 1 0 , defined by: = amplitude for l+ (µelicity 1) -71- (helicity 1) + 0- (2a) = amplitude for l+ (helicity o) -71- (helicity 0) + 0- (2b) Of course, the two helicity amplitudes are linear combinations of the *The ratio of F to D coupling for the pseudoscalar meson-nucleon vertex can be inferred from weak interaction data using PCACo (The baryon matrix elements of the axial-vector current appear in semileptonic decays.) The experimental value of F/D determined from a fit (16) is .66 ± g03 which is to be compared with the theoretical value 2/3. TABLE 1 Jpc = l++ State A (1070) 1 KA(l240) D(l285) Width * 95 ± 35 MeV 40-130 MeV 33 ± 5 MeV Decay modes p 1L K*n, Kp KK:rr' 11 nn ? (D') I-' m Jpc = 1 +- State B(l235) ~(1250-1400) Width* 102 ± 20 MeV ? Decay modes Wn K*n, Kp *The above data is taken from ref. 20 . ?(h) ?(h') .17 two partial wave amplitudes. These decay parameters can be determined experimentally from the decay angular distributions, which has been done recently for A 1 ~ pn and B ~ wn. Thus, the axial-vector mesons provide a test for models of spin coupling in vertices as SU(6)w x 0(2)L ~ z The predictions of SU(6)W x 0(2)L for B . d f ram th e conservation . can ..oe derive of ZL (. 17 ' ~ 18 ' 19 ) z is assumed to be a 1 P 1 q J z z and L state (g 1 , z = 0). the z B can decay to wn Experimentally, the = ± 1 °decay seems favored. * A similar reversal occurs for A ~ Pn 1 J Jz = 0 A ~ pn 1 s.ince the B wn q state , its Therefore, according to SU(6)W x 0(2)L only through its such =0 Since the A is assumed to be a 1 o state has no component with {J = 1 J z = 0 j L =1 Therefore, SU(6)W x 0(2)Lz L z 3 P 1 q q state, its = O, i.eo, Lz = O; S = 1 sz = o) predicts a decay from the = 0 J 2 =±1 = O)Q Experimentally, the J z = 0 decay seems to predominate, ** or at least is significant. *** Therefore, the natural state (g o extension of SU(6)W to q q states with orbital angular momentum fails 047 + · 20 - • 30 ** lg 1 lg o IA (ref. 21) = .48 ± .13 (ref. 22) l (ref. 23) 18 severely,which means that the incorporation of spin in quark graphs is still an open question. The angular distributions of the axial vector mesons will be the crucial test of future models • . The axial-vector mesons have interesting features besides their angular distributions that set them apart from the other mesons. I = O states predicted by is the absence to date of three of the the quark model: one from the A nonet and two from the B nonet. 1 The D, which is the only isoscalar in p p and - :n: p reactions. l+ meson to be seen, is produced The existence of the other three states is crucial for believing the quark modele 1 ++ One The absence of the ninth 1 +- meson is the most intriguing since the states could have possibly escaped detection because of extremely large widths predicted by SU(3) and ideal mixing. * The strange 1 + mesons are also fascinating. which supposedly contains the two The Q region, l+ K* mesons, has not been defi- 2 nitely separated into t'WO resonances C o) However, the Q peak, which a does not have a simple Breit-Wigner shape, can be fitted reasonably well to two Breit-Wigne~s at all energies. ( 24 25 ' ) ' The widths deter- mined in these fits are sensitive to assumptions about the background. *SU(3) and ideal mixing predict that ~y (1.. 2 MeV) pn 70 MeV . ~ 2 , 300 MeV, The h is certainly wide enough to have been missedg The h' probably should have ·been seen, but the experimental situation may be confused because of a nearby resonance called the E(l422)o The favoured quantum numbers .of the E are o-+ • 19 The fits of ref. 24 (no background) and ref. 25 (deck background) to · Q data compilations yield the following resonance parameters: ref. 24: M ex = 1250 MeV r ex = 220 MeV ref. 25: M = 1240 MeV r ex = 110 MeV ex M t3 = 1400 MeV rt3 = 220 MeV = 1420 MeV r t3 = 120 MeV • In some reactions the Q region can be resolved into two resonances with even narrower widths:( 26 , 27 ) ref. 26: Mex = 1260 MeV r a. - 40 MeV Mf3 = 1380 Me V rf3 = 120 MeV . Further complications arise from the decay modes of the Q. K*~ (dominant) and observed decay modes, Kp , are both K~~ states, as is the one other possible yet unconfirmed mode an I= 0 o++ meson)u The two KO final (o is Consequently, these modes can interfere with each other (in the ove~lapping region of the Dalitz plot, which has a high concentration of e~ents)u not only between the K*~ and Moreover, interference is possible decay modes of a single meson, Kp but also between the decay modes of the two different mesons. more, the two 1+ Further- kaons can mix with each other when SU(3) is broken. ( 28 ) Mixing phenomena are a common occurence in particle physics examples being ;(l) w - <l.) mixing from SU( 3) breaking, 0 (2) Ko - K mixing from the weak interactions, (3) w p mixing from electromagnetismo kaon from mixing · with a 1 +- Exact SU(3) would prohibit a l++ kaon since the two states belong to different SU(3) representations with.different ~ quantum numberQ enables a quantum number analogous to (Exact SU(~) G parity to be defined for 20 the kaons.) When SU(3) is broken, however, the kaon states can mix (their G parity, unlike the G parity for non-strange mesons, is invalidated by SU(3) breaking of V-spin). This mixing is very similar to the singlet-octet mixing found in the 1 . 20 ) since all are caused by SU(3) breaking. usual assumption is to la> = take mesons ~- and ~- baryons (approximately (approximately 35°) and in the 0 and ( 20) The simplest and a phenomenological mixing of the form: !KA> cos ~ IKB) sin ~ (3a) (3b) HoweverJ the mixed states are not necessarily orthogonal as they are with the above unitary transformation. The mass matrix M+ i r , Which is not hermetian, must be diagonalized to yield the decaying states . 0 K - R0 In the narrow width approximation or if M and r commute (as with CP conservation), the mass matrix can be diagonalized by unitary transformation so that the physical states are orthogonal. Unfortunately, Ka and. K~ J like w and <P or ' f and f' , are not guaranteed to be orthogonal since they have finite widths into common final states (and it is not obvious that Mand r should commute). Nevertheless, mixing from SU(3) breaking into orthogonal states is usually assumed. Thus, the Q region is beset by several experimental and theoretical complications Which cloud our understanding of the strange axial-vecto~ mesonsu One :further interesting pro?lem faces the 1 + mesons that are 21 produced diffractively. The Deck mode1C 29 ) for 1( p -. pnp , wl}ich is a double exchange diagram (Fig. 2) for diffractive processes, interprets the than A 1 1( p enhancement as a non-resonant kinematic effect rather resonance production. Ama zingly, the Deck model pred i cts the 1( p mass distribution and momentum transfer dependence of crosssection in good agreement with experiment; the essential feature of the model is the peripheral nature of pion exchange. (The recent concept of duality, which states that the imaginary part of s-channel resonances is equivalent in some average sense to the imaginary part of non-diffractive t-channel exchanges, does not seem to relate the Deck effect to A1 production since pion exchange is real.) very possible that a significant fraction, if not all, of the results from this non-resonant Deck background. It is A bump 1 The Deck effect also confuses our resonance interpretation of the Q region since a double exchange diagram can be drawn there tooo Since the determination of resonance parameters is sensitive to assumptions about the background, the masses and widths, and possibly even the existence, of the are subject to doubt& A 1 Non-diffractive production of the ~ is unclear although it has been reported in a few experiments with poor statisticso The l++ D and 1+- B mesons are better established since they are definitely produced in non-diffractive reactionso shall assume the existence of the l++ We mesons, as required by the quark model, "While acknowledging the possibility of considerabJ:e Deck background in diffractive productionQ The main purpose of my paper is to examine the angular distri- , 22 p Fig. 2. Deck diagram f~r 7( f4Ttf p. 23 I = 1 axial-vector mesons in order to obtain a descrip- butions of the tion of spin couplings i~ vertices. I = 0 and the absence of the The mixing ·df the with the z L = 0 is not a good symmetry. =1 I .uz s z = Lz = 0 L z =± 1. * states indicate that Since SU(6)W is consistent meson and baryon decays, the orbital generalization rather than the complete symmetry is suspect. SU(6)W x 0(2)T states states will also be investigated~ The angular distributions of the SU(6)W x 0(2)L I = 1/2 The shortcomings of may be dealt with by simply relaxing the condition on the 3 P pair in the quark graph and allowing 0 It is this approach that we shall take. (The neglect of transverse momenta is probably dubious for decays involving Q-values of no more than a few hundred MeV.) have Lz = ± 1 When the q Q pair is allowed to Lz = O, the as well as butions may be described satisfactorily. tributions for 2(g /g ) Al 1 0 A ~ pre 1 = (g0 /g1 )B and ~ and B decay distri- Moreover, the angular dis- B ~ wrc are related by + lo This prescription for spin couplings is significantly different from SU(6)W x 0(2)L considerable modification of all , as it involves z S-wave decay amplitudes for the positive-parity·mesons (including those of the tions)~ o+ ~o- 0- transi- It is then fortunate that we obtain a satisfactory description of the partial widths of all the q q- ; L = 1 mesons into 1 0 or 3 3 *one may also invoke "recoil terms" in a realistic quark picture.( 0, l) Such an approach relies more heavily on details of the wave functions than the picture we describe here, but there may be some overlap of results. 24 0 0 pairs. This description involves two independent parameters, whereas ~U(6)w·x 0(2)Lz involves only one. (A similar description -was obtained in ref. 9, but not applied to angular distributions.) It may be summarized as follows: SU(6)W x 0(2)L relates all D -wave z amplitudes to one another, all S wave amplitudes to one another, and D waves to S -waves (since a particular helicity coupling is a linear combination of partial waves). The 3 p 0 prescription with Lz = O, ± 1 uncouples the D waves from the S -waves but otherwise preserves all the SU(6)W x 0(2)L relations. -As we shall show, the z physical solution that emerges has approximately the same magnitude of D wave to S wave, but the opposite sign, from the SU(6)W x 0(2)L . z solutiono Unlike the widths, the angular distributions of the l+ mesons are sensitive to the relative sign and are thus crucial in determining that the fit of our model differs considerably from that of SU(6)w x 0(2)L 0 z Next the parameters of the fit are applied to the l+ kaons, which may mix with one another (we assume a phenomenological mixing via a unitary transformation and ignore possible interference). results are sensitive to the mixing angle ~' The and merely assuming lower bounds on the widths of both physical states establishes the limits 10° ::_ ~ < 35° the following results: lower peak, (ii) and (iii) the A 1 Q (i) · As a result of this mixing, one predicts the suppression of the K*~ mode of the the suppression of the decay distributions in the K*~ pK mode of the upper peak, mode similar to that of for the lower state and to that of the B for the higher. 25 I =0 Finally, the absence of the three axial vector mesons is investigated; with particular emphasis on the ninth D' , ~nclude: Expected properties of this meson, the l++ state. (1) assignment to a weakly mixed SU(3) singlet, predicted by duality and confirmed by the Gell-Mann Okubo mass formula; (b) a mass ~950 MeV, predicted by superconvergence with assumptions about the relative couplings of D and D'; (c) decay modes of a suppressed p ~nn and n+n-y ; and (d) the possibility n +n - signal in the spectrum of the n+n - y final state, despite the expectation that the pions are in a state with I= J = 1. These ·features suggest that a recently reported meson( near this mass with decay modes f or this state, a 1though for the new meson. ~nn J pc -- l+- and n+n - y 32 may be a candidate is also a definite possibility ) 26 II. .ANGULAR DISTRIBUTIONS AND QUARK GRAPHS FDR THE 1. SU(6)W x 0(2)L l+ ISOVECTOR :MESONS Breaking z SU(6)W x 0(2)Lz invariant vertex functions can be calculated in a collinear frame with non-relativistic wave functions since the symmetry is invariant under boosts. when the L = S = 0 z z restriction on the wave functions for the (aa) M ( 13b) The same technique can be tried L = 0 and L = 1 3p state is relaxed. 0 The mesons are: (4a) = M~O'a) J. ( 13b) + T~ 5 The symbols P, V , B , S, A , µ for the The µ (CJ . C) "h E• . and ( 4b) au J J.J f...J T are ~ 3 x 3 SU(3) matrices 0 -+ , 1 -- , 1 +- , o++ , 1 ++ , 2++ meson nonets, respectively. 2 x 2 matrix C in quark spin space is given by: 1' ~ c = i (J y = t 0 1 '1i (_l 0). The index i represents a polarization vector for the one unit of orbital angular momentume Vertices invariant under SU(6)W x 0(2)Lz are constructed by con- tracting indices according to the quark graphs: (aa) M+ (ye) (aa) M (f3b) n+d o M(f3d) (ye) 27 1 = ( 0 1 ) 1 0 1 The matrix D qd (1) annihilate in a spin state in Fig. = 0 .f2 means replacing D by = er J"2 z C quarantees that S=l S z and Relaxing =0. __!_ er. C (eri C represents a D. J"2 l l state by analogy with the wave function for the scalar meson 3 which is also P 0 Only one coupling, the SU(6)W ). can be constructed for e z D. L = 0 mesons since invariant one, · must be contracted on J.. (z unit vector) which is the only direction in the problem. for L = 1 e and.on the orbital index z obviously breaks SU(6)W x 0(2)L . z r B* c. • l g 1 I 0 • mesons into However, states, two independent couplings can now be contructed, contracting on i.e .. , w er. E. J J c i . The second coupling as can be seen in B ~ wn: c-1 c) er.J.. E B* 0 E w The matrix.1 elements for the decays of all the L = 1 PP or PV can be calculated in terms of the two couplings. (These predictions are given in Table 2 in terms of two parameters and s S D , representing partial wave decay amplitudes, which are just linear combinations of the SU(6)W x 0(2)L conserving and breaking z amplitudes discussed above.) Unfortunately, the validity of a non-relativistic calculation is not obvious when the SU(6)W x 0(2)Lz symmetry is brokene Consequent- ly, we are forced to calculate our breaking prescription in a manifestly covariant manner. of this simple scheme. We shall return later to the significance 28 2. Details of the Relativistic Calculation (a) We guarantee relativistic invariance by using covariant wave functions:(lB, 33 ) qq; L = O: - L = 1: qq; (5a) ' (1 + p/m )(r B + rv c µv 5 µ 1 ) (5b) where 2 cµv - _!_ (g - p p /m ) s + i r µv µ v 1 I\/ 3 E µvy5 pr A5 /( J'2 m1 ) + Tµv (5c) The matrices M and M refer to the 12-dimens~onal product space of 0 1 SU(3) matrices (3 indices) and Dirac matrices (4 indices). Normali- zations are given in such a way that and where the right-hand side refers to traces of SU(3) matrices. Mlµ and are given by (1 - p/m )(-r B + + rv c + ) 5 µ µv 1 M 0 and 29 In equations (5a) and (5b), (m , m ) is taken as a cormnon mass of the 1 0 q q; L = (O,l) multiplet. (b) Couplings are constructed as trilinear traces of the M · and M 0 1 matrices obeying charge conjugation invariance and the 5 · rule ( ,s) that quark an d ant iquark lines of a sing1 e meson s h ould no t be connected ~ For example, the SU(6)W x 0(2)L for the decays of the q q; invariant coupling z L =1 mesons into two q q; is L = 0 mesons The orbital index µ is coupled to one of the collinear momenta, implying 6L (c) = O. Relaxing condition (1) is equivalent to allowing the orbital angular momentum of the q q; 3 P of the z 0 pair. L = 1 mesons to couple to that By analogy with the wave function for the scalar meson S in equations (5b) and (5c), a 3 p 0 object which carries no four-momentum and is assumed to be an SU(3) scalar transforms like yµ ~ The yµ . coupled to the orbital indexµ of the decaying meson breaks SU(6)W x 0(2)Lz yµ o (For the SU(6)W x 0(2)Lz symmetric coupling, is coupled to one of the collinear momenta.) The three most general couplings with one r the q q; L =1 µ for the decays of mesons are then: (6) 30 (8) The momenta satisfy -p 1 = -p 2 + p 3 • The structure of these cou-plings is illustrated in Fig. 3. The cou-pling (6), which co~serves SU(6)W x 0(2)L , leads to z mesons to pairs of q q; rates for the decays of q ·q; L =1 mesons which all behave as pf , where 5 'Pf L = 0 is the magnitude of the final three-momentum., for S and D waves alike (because SU(6)W x 0(2)L . . z predicts a definite helicity cou-pling). Any breaking of SU(6)W x 0(2) z 1 that gives an S--wave contribution to the rate proportional to pf the would then have large effects on the decays of 0+ mesons as well as on angular distributions in decays. (17 34) ' A 1 and B In particular, a small symmetry breaking, which we shall assume, can have a large effect~ A contribution proportional to pf for S -waves arises from the coupling in equation (7). The coupling (8) does not seem to have any deep physical significanceQ It gives contributions to decay rates proportional to for both S and D -waves. Therefore, as it represents an SU(6)W x 0(2) breaking term whose effects are expected to be relatively limited, we shall ignore it in what follo"WB~ Actually, couplings (6), (7), and (8) are not the most general couplings when SU(6)W x 0(2) z 1 is brokene In -particular, yµ's contracted on each other can be added to the three couplings to make many more couplings (where the spin of the spectator quark can fli.p ! ) . However, the inclusion of such couplings would break SU(6)W for 1z 31 (b) ( c} Fig. 3. Quark graphs for spin couplings. (a) Strength c • SU(6)W x 0(2) symmetric. (b), (c) Stren gths 1 z c 1 ,c 2 , respectively. Both break • SU(6)W >< 0(2) 1z 32 L =0 of q q states,too. SU(6)W since the We only wish to break the orbital extension L = 0 predictions agree with the data. fore, we ass'ume that coupling a. y µ There- to one of the collinear momenta . is much larger than coupling it to either the orbital angular momentum or another Therefore, _i c yµ • larger than c 1 0 (equation 6) is (equation 7) and c with multiple _ y 's 2 (equation 8), and the couplings are smaller still. µ assumed to be much Consequently, the coupling in equation (7) is the only breaking term that can have a significant effect on the rates and angular distributions. Typical couplings given by the model are then: J ('TI { 1!} )' [ 4ml (D\ +mo) cl + (ml 2- 4mo 2) co} 1], (9) ~l ~ pre g. (1+) = J. (1++ ~ 1 -0-2: el+ [217/mo 2 { E *(i) A + 4 (2) ~ 1/2 ml Ep ( i) ( m1 2 2 mo c 0 +2 c~ 2 *(.) (i) - 4m ) + 2E J. ~ p E 0 . A re p *(i) c (m + 2m ) E " EP (i) /mo] A 1 1 0 . pl(} "' A[ p, re] (i = o, 1) ' (10) 33 - 2 c m (m + 2m ) 1 1 1 0 EB*(i). E}i) /m 2] (B { w,1t]) (i = o, 1) , 0 (11) (K** { K,°1(} ) , (12) gl(2+) = [ cl: 4 2 ml mo 1 2mo co - 2c1 ) "/ 5] *(l)/\t39 p * p (€ ** K 1( K (E(;)) i ECXt3yO a P:r(A K (K** [K* , 1(] ), (13) where the brackets ( ) denote traces of 3 x 3 SU( 3) matrices • . All partial widths follow from equations (9)-(13) by substitution of the appropriate SU(3) matrices and polarization vectors. Subscripts on the helicity coupling constants and indices on polarization vectors indicate values of J z The partial widths are related to these coupling constants as follows: J 2 r(o+ ~ 0-0-) = ~o(o+) r(1+~1-o-) = {~o(l+)r + 2 [g1 (l+) 2} /3 ' ' J (14) (15) 34 rc2+ ~ o-o-) = [go(2+) J15 r(2+~1-o-) = 2 [g1(2+)J/5 ' (16) (17) ,.., r refer to partial widths with a phase space factor The quantities divided out: r In equation (18) Mi (18) is ~he mass of the initial state and pf is the magnitude of the final three-momentum in the rest frame of the decaying particle. 3. Angular Distributions in The A 1 q and B A 1 and B Decays angular distribution calculated from equations (10) and (11) yield the following relation: (19) Given the present data on B with the two experiments on decay(2l) we can compare the relation A 1 decay.( 22123 ) We shall assume that the observed ratio of helicity couplings is real although the experiments cannot measure the phase accurately. Let us define and compare first x = (22) with the SLAC data:' 0 .. 03 < x < 1.05 for (20) 35 OR 0~87 < x < 1.89 for In either case, one must take (21) i.e., (g /g )A 1 0 > O in order to satisy 1 equation (19). This solution ("A SLAC (+)") 1 is the one in which D-wave decay plays a relatively small role. of (g/g ) Al , 0 (For the opposite sign i.e., "A SLAC (-)" , the decay 1 A 1 would be pre- dominantly D wave.) The BNL data( 23 0.99 < x < L 75 ) yield: i.e., "B (+)" for Here equation (19) can only be satisfied with "B ( +)" "A BNL(+)n 1 (22) and $ One should stress that the two values of do not agree with one another 7 and hence comparison of equation (19) with experiment may be prematuree It is also possible that the SLAC (16 GeV) and BNL (6 GeV) experiments are both correct, and that the parameter we call is indeed dependent on the incident energy of the ~-p ~ "A -u po 1 ~ (g /g )A 1 0 1 in (We use quotation marks since 7 for a real resonance, the decay angular distribution should obviously be independent of the energy at which the resonance is produced~ Any variation with energy indicates an improper separation of non-resonant background or the presence of more than one resonance in the mass range under consideration.) 36 If the sign of (g /g )B 0 1 were measured, one could remove some of the ambiguity in comparing equation (19) with experiment. A slight . (22) preference for (g /g )A > 0 does emerge from fits to the SLAC data 1 0 as a result of interference effects in A - -?re - re - re + via two possible 1 - - + 0 p bands. However, such effe'cts will be smaller for B -?re re re re because of the narrowness of the w. Equation (19) is of interest, aside from comparison with experiment, in view of the following limits: (a) In the SU(6)W-invariant . case, it reads oo = oo + 1, a correct if useless statement. the case that A 1 decays via pure S wave, with equation (19) predicts that (b) In (g /g )A = 1, 1 0 B also decays via pure S wave, with (g /g )B = le This is because the relative effects of D and S waves 0 1 are related in the two decays~ A convenient normalization yields: - - 1 ~6 (28-D) (23) =- - 2 ~6 (S + D) where s - ' (24) 37 D - . ( 25) (One can see that equation (19) · follows very directly from equation (23).) The incompatibility of the experimental distributions with SU(6)W x 0(2) . 1z the above S and D wave amplitudes. A 1 and B angular is easily illustrated in terms of corresponds to z D/S = -1, while any set of experimental ratios satisfying equation (19) D/S > o~ requires amplitude (the c 2 0 2 Pr = (IIJ_ /4) - m0 The SU(6)W x 0(2)L SU(6)W x 0(2) z 1 conserving part of the S-wave term) is depressed by a factor proportional to 2 , so that the small symmetry breaking c 1 term can give a significant 8-wave contribution (proportional to pf rather than pf 5 in the rate). This contribution actually changes the sign of the net S-wave amplitude~ 4. + + + 0 , 1 , and 2 Decays A Fit to Rates and Angular Distributions for The above interpretation of a small breaking having a big effect in the S wave s~gest a particularly simple limit in which to compare decay rates with the present coupling scheme. >> c1 , but that Let us assume that: (a) c c are of comparable size; and (b) m ~ 2m , iee., we let pf ~ O 0 1 0 and the breaking term 2 1 but maintain c (Empirically, is quite small..) In this limit the parameters S and D become (26) 38 (27) D = 32 (2/3) 1 / 2 c · (The effect of a c 2 = O(c1 ) would vanish in this limit.) All decays of ci q; L =1 mesons into pairs of q q; L = 0 mesons are now determined by the abo'v e two S and D wave amplitudes. The breaking affects only the S wave. Table 2 lists the rates for various decays in terms Df S and D. These results are the same as those obtained in Section (1) using the non-relativistic wave functions, but where the physical assumptions were not obvious. * Now, ho-wever, the explicit assumptions listed above give a meaning to the simple, non-relativistic ca.lculationai scheme. In Table 2, -we include for completeness some processes which are not used in the fit of the present section. the square bracketso These are denoted by Since the l+ kaons can mix with each other, their decay properties are treated as derived quantities in the follo~ng sectiono The other processes not included involve partial widths which are very smai·1 and open to some uncertainty. In order to compare the predictions of Table 2 with experiment, one must decide how to take account of centrifugal barrier effects for D wave decays. We regard this problem as essentially unsolved. Any *The same is true for the use of recoupling coefficients in ref. 9. Equation (19) was not discussed there. 39 Table 2 · Predictions of decay rates for decays of q?i; L = 1 mesons into pairs of qq; L = O mesons. + + 0 decays 1 decays ProC'ess Process go or Yo - --·-- -- ----- - - - --- -· -··- ---- --···-- ·- -- - -- - --- - - ---16·- ~- (4S2 + 2D2) ii~ _, T}1T 152 A a__. 1i1T ~- s2 [KA( 3 P 1) -K *7i] ~ (2S 2 +D 2} -73 s*__. KR 52 [KA (3P1) __. pK] i (2S2 + D2) -73- K __. K7T ~- s2 B _, W1T t<s2 + 2Dz)· 3 1 _, P1T [KB(lPl) --+ K *7T] ~- (S2 + 2D2) [KB(lP1) -• pKJ ~(S2 + 2D2) 2 + gl or Y1 - - - -·-·- - --25 -D a) S+D 5+D b) 25 -Db) - - 2}'"f c) 2s -D c) 2./3- decays _, 0 0 Pr.o cess Process .Ln2 A2 _, P1T __. KK in2 fo _, rrrr 1D2 [f' - K*R+R*1q K** _, K*1T [f 0 _, h.KJ .t n2 [K** _, pK] A2 __. TJ1T A 2 15 5 5 f' _, KR 5 2 n2 "5" K** _, K1T J_n2 10 1 n2 [K"'* __, K7]] - --- - - -- - --·-- [K** _, WKJ 30 0 - + a) for the charge state A 1 -· P 1T • + r*O b) for the charge state KA B - K 1T + c) for the charge state KA: B+ __, p+Ko. For 1+ mesons, individual helicity amplitudes are also given. Square brackets L!_1dicat~ proce~ses not incluued in the fit for reasons given in the text. r = (Mf IPr)I'. 40 statement about such effects invariably involves an effective interaction radius. If thiq radius is very small, D--wave amplitudes may pf 2 be expected to contain a factor "110 pf. * matter how large this radius is very large, on the other hand, the If pf 2 behavior of D--wave amplitudes will be visible only for values of pf much smaller than those considered here, and may be neglected. We have thus. performed fits for the two extreme cases · listed above: ( ..;) ~ D h as no · pf 2 b e h avior . and is' common to all processes ( 28) and (ii) D = D (pf rv 2/ 2 processes * where P 0 rv p 0 ) , and D is common to all (29) is any convenient normalization that we take here to equal Oe5 GeV/cg The physical solution may be expected to lie somewhere between these two extremes. One then considers a plane with S as the horizontal axis and D or D as the vertical axiso In order that a solution exist which is consistent with the present model, all of the following lines must intersect at a point, corresponding to definite values of S and D or rv D: *This approach is adopted in ref. 9e 41 Isl = const. vertical lines ID or DI r(l+--? 1-0-) a s (g /g ) (l+ ~ 1-0-) 1 0 2 + b(D 2 or = const. n2 ) = const. D/S or n/s = const. ' horizontal lines ellipses radial lines The results in cases (i) and (ii) are shown in Figs. 4a and 4b, respectively. The experimental input to these figures is taken from ref. 20 except in the following cases: (a) We use A2 widths based on an unsplit peak, as observed in ~+p -7A 2+p at 7.0 GeV/c (ref. 35). a split ~-p --7A -p (ref. 36) if the splitting is due to a A 2 in 2 narrow, weakly coupled state. r (A (b) This approach is consistent with 2 The following values* were used:( 35 ) -7 p~) = 64 MeV r(A 2 -?'fl~) = 16 MeV r(A 2 -7 KK) = 10 MeV (30) The 11 wide-crn solution(20) is taken: rca -7 ~~) '::! 400 MeV · (31) The er is assumed to lack stral?-ge quarks. (c) (20) * We take f' -like mixing for the S and *We thank Se Flatte for permission to use these preliminary data, which may have changed slightly in the final analysiso 42 r (s *. --:) KK) (d) The 3 P 0 I = 80 Me v (32) = 1/2, IYI - 1 state, "K." is taken to have a mass of 1080 MeV and( 37 ) r (K.. --:) K:rr) > 200 MeV (e) The (ref~ :rrN(980) (ref. 38, 39), (33) 5(962) (ref. 36), and ~(1060) 20) are taken as manifestations of the same state which we identify as the I= l; IYI = 0 state. We thus take( 38 ' 39 ) 60 MeV* (34) One notes that the experimental situation regarding the o+ mesons and their decay widths is still in considerable confusion. For that reason we cannot take the above assignments and widths too seriously .. Figure 4 shows only one of the four quadrants of the S-D (or S-D) plane~ Partial widths alone do not determine the sign of S, D, l"V or D, and therefore give no information as to the quadranto ~l and Only the B angular distributions tell us that the solution must lie in the first (or third) quadranto Note the arrow in Fig. 4a corre- *The arrow in Fig .. 4 associated with :rrN -7~:rr'allows for the possibility, suggested by some of the experiments quoted in refe (38),that the width may be somewhat smaller~ 43 Fig. 4. (a) Comparison of predictions of Table 2 with ex periment. Radial sectors indicate regions allowed by A1 and B angular distributions. Negative values of (g /g )A_disagree 1 )o -.-BNL · SLAC with eq. (19) and are not shown. A : ref. (22 ; A : 1 ref.(23); B(±): (g /g )B'1'.. O (see ref. (21). 0 1 1 The SU(6)W x 0(2)Lz solution D = -S is . shown as an arrow pointed into the second quadrant. Sections of ellipses are based on r(A1) ~ 130 MeV, r(B) ~ 80 MeV (see ref. (20)). Horizontal lines indicate values of !DI implied by 2+ decays shown (see refs. (20,35)). Verti- · cal lines indicate values of js! implied by o+ decays shown (sea refs. (20,37-39)). The circle surrounding the point of intersection represents only an "eyeball estimate" of the probable errors associated with our fit. (b) Same as (a) with D = (0.5 GeV/pph ysica . 1 ) 2 D. 44 fo ~ 7T'7r f I -+ K R------~~=+~P-L.~~:==== K**~ K.,,. A2 ~ 'T/ 1T-=======-.P~ K**-> K~"7T a~w7r ·.·.J~-~~~ .5 S (GeV) (a) 45 .5 S (GeV) (b) 46 spending to the SU(6)W D = -S. * ~nvariant solution As mentioned before, relaxation of the condition (1) actually allows D/S to chan.ge sign relative to its SU(6)W value of -lQ we have seen that this is due to an S-wave contribution whose presence we have every right. to expect. This contribution must be compatible with all S-wave decays. In either fit, this is seen to be the case, but again we point out that this success should not be taken too seriously. More data on o+ meson decay rates would be welcome. The experimental uncertainty regarding testing the model conclusively at present. (g /g )A prevents 1 0 l By comparing Figs. 4a and .. 4bJ one sees that no statements can even be made regarding "Which treatment of centrifugal barrier effects agrees with experiment. The neglect of such effects in D (case (i), equation (28)) yields a fit favoring the low side of the SLAC( in Fig. 4aQ 22 ) bounds on (g /g )A , 1 0 1 as sho"'Wll The "maximal" inclusion of such effects (case (ii), equation (29)) favors the low side of the BNLC as sho"'Wll in Fig. 4bg 23 ) bounds on (g /g )A , 1 0 1 In the former solution, D-wave decay accounts for at least 25% of the A1 -7pn partial width, while in the latter, it accounts for at most 3%o However~ only the latter value is con- sistent with an upper bound of 5% obtained for this quantity using 4 partial wave an~lysiso ( 0) *The SU(6)W x 0(2) 1z limit does not exist in the case of Figo 4b. This symmetry dictates a definite correlation of S and D "Whose effect is hard to evaluate if D is parametrized as in equation ( 29). 47 We therefore conclude that considerable note of barrier effects must be taken in the present model. If this is done, we favor the experimental values (35) and (36) on the basis of the fit in Fig. 4b. satisfy the rule (19) exactly. D-wave amplitudes in in the higher Note that these values do not As a result of the barrier factor, A ~ P1C are suppressed more strongly than those 1 Q reaction B ~w1C. Consequently, (g /g )A 1 0 1 is slightly larger (less D wave) than the value 0.6 predicted from (g /g )B = 0.2 0 1 and rule (19)e Recently (after publication of the above predictions), more data on the angular distribution of B into W1C have become available. 4 The Illinois group( l) has compiled its data on 1C-p ~B-p at 5.0 GeV/c and 7.5 GeV/c, finding (37) This new value reflects a rather larger value of (g /g )B 0 1 7~5 GeV/c sample, which was not included in ref~ (2l). in the Although the A - B sum rule is still satisfied with ~BNL(-t-), . ~his value for 1 (g /g )B does not agree with our prediction. However, the most 0 1 recent data( 42 ) from 1C+p ~B+p at 3 and 5 GeV/c yields: 48 !g0 2 = .06 ± .10, · .36 +.25 -.36 1 (38) with a relative phase ·compatible with zero. · This result is in complete agreeement with the old B data and with our prediction, including the sign which confirms our solution in Fig. on the partial widths of · K** and A 2 4b~ 3 Also, new data( ) have been reported which improve the agreement with Fig. 4b: K** -7K *n 26 ± 6 MeV (35), K** -7Kn 59 ± 4.4 MeV (47), A -7pn 2 67 ± . 8 MeV (64), A -7~n 2 16 ± 3 MeV (16), are given in parentheses. A -7KK 2 6.5 ± 1.3 MeV (10). The old data 49 III. MIXING OF THE 1 + KAONS We shall apply the results of the previous section to the l+ kaons, which may mix with one another as a result of SU(3) breaking. We assume a phenomenological mixing via a unitary transformation and ignore possible interference in the various decay modes. ( denote the upper and lower of the physical states by a 28 ) and We shall ~ which are then related to the unmixed states by: (39) (40) In some reactions the physical states can be resolved, ( 26 , 27 ) for example into peaks:( 26 ) 1260 ± 10 MeV ra = 40 ± 10 MeV (41) = 1380 ± 20 MeV = 120 ± 20 MeV ( 42) These masses will be assumed in wh~t follows merely for the sake of calculating phase space for the decays~ The corresponding widths, which are the narrowest reported.)> will only be used as lower bounds. For definiteness, helicity couplings will be defined for given charge states: as follows: 11 K+ 11 ----'" K*o .,-('+ _, H or . 11 K+" ~ p+Ko .. These will be denoted 50 KB= KA: "K+ -? K*o (J =l) 1(+ : z gl *o -? K (J =0) 1(+ : z go -7 A p+ (J =l) K0 : z I1 . -? p+ (J =0) Ko: Yo z gl A go A B B B '1 A Yo B a: t3: a gl 1 a g t3 0 a )' t3 go '1 Yo g t3 a (43) 1 )' t3 0 The quantities in equation (43) are related to one another in an obvious manner as a consequence of equations ( 39 ) and ( 40) , e.g. (j, gi = A gi cos ¢ - gi B sin ¢ (i = o, 1) (44) From Table 1 one then can calculate the partial 'Widths and decay angular distributions in terms of the parameters D, S, and the mixing angle ¢. In the ~U(6)W limit into K*1C ..v and pK * r(a-?K1C) S = -D, the partial 'Widths of a and t3 are all independent of ¢* and are all equal to = c3;4) r cB -? W1() = (3/8) r(A1 -? p1C) . (45) One cannot test these partial width predictions in view of the uncer- *. The equality of r (ex-? K*1C), r (t3 -? K*1C), follows in any theory involving a t:.Lz = 0 Lipkin, refo (28) . r (ex-? pK), transition .. and r(t3 -? pK) See H. J . 51 tainty surrounding the total widths of the states la) and I(3) Whereas refs. (26) and (27) favor narrow states sitting above considerable Deck-type background, one may also fit the majority of events in the "Q" mass region (1100 - 1500 MeV) with two broad resonances. ( The following prediction for the angular distributions in and f3 a 1 (46) 2 g = -2g .0 1 o lg /g l = 1 1 0 ), SU(6)W and is independent of ¢: decays follows from For a purely S-wave decay, g 24 0 = g1 , while for a purely D-wave decay, Data present~d several years ago( 42 ) are consistent with throughout the Q region, but, as we see from equation ' (46), such a possibility would necessarily violate SU(6)W • However, until are kno-wn, gross violations of SU(6)W cannot be demonstrated. We are thus left with A 1 and B angular distributions as the · best indications of serious breaking of SU(6)W • In what follows, we shall assume the validity of the fit of the previous section and derive results for Q decays as a function of ¢. One then finds the following results: 52 K*TC mode: a (s/ ,f2) cos (¢ + ¢0 ) == (D/2) sin(¢ + ¢ ) 0 == -I? sin (¢ + ¢ 0 ) - (s/ J:2) cos (¢ + ¢0 ) g 13 1 == -(D/2) cos(¢ + ¢ ) - (S/ ,f2) sin (¢ + ¢0) g 13 0 == D cos (¢ + ¢ ) - (S/ ,f2) sin (¢ + ¢0) gl go a 0 0 (47) pK mode: y a 0 = -D sin (¢ - ¢ ) + (s/ ,f2) cos (¢ - ¢ ) 0 0 y1 13 = -(D/2) cos (¢ - ¢ ) + (s/ ,f2) sin (¢ - ¢ ) 0 0 13 = Yo · (48) where (49) The quantity ¢ 0 is only a convenience, and does not appear to have any physical significancee the decays Jpc = However, as a result of the f-type coupling in l++ ~ 1-0- as opposed to the d-type coupling in change occurs between equa- tions (47) and (48)(would be identical if the two couplings were either both d or both f.) the K* TC and pK This change leads to qualitative differences between modes of ·the two physical states a and 13. The barrier factors are introduced at this stageg For the K* rc 53 mode, we have D(a ~ K*.1{) = "'D p (a ~ K*1l) (50) where * p (a ~ K 1l) = 2 2 (pf /. [ o•5 Ge V/ .c ] ) D(f3 ~ K*1l) and = = o. 35 (51) "'D p(f3 ~ K*1l) (52) = 0.61 (53) 0.28 'D (54) where p(f3 ~K *1l) For the pK mode, we have D(f3 ~ pK) = We have to be more careful about r(a ~ pK) D(a ~ pK)Q depends very sensitively on The prediction of Ma , and should really be obtained by integrating over a Breit-Wigner distribution8 M < 1280 MeV a- Taking instead to obtain a conservative upper bound, we find pf :S, 100 MeV, and thus D(a ~ pK) ~ 0 (55) The net result of these considerations is a set of predictions for a and f3 partial widths as functions of the mixing angle ¢8 We . use the central values associated with the fit of the previous section, namely 54 s = D 0. 77 GeV ' (56) = 0.60 GeV ' (57) (with errors of about 20°/o). and These predictions are: r(a-?K *re) . 2 = [ 82 - 76 sin (¢ + ¢o)J MeV r(t3 -7 K*re) 2 = [ 20 ' + 72 sin (¢ + ¢o)J MeV r(a -7 pK) 2 < - [ 28 cos (¢ - ¢0)] r (t3 -7 pK) . 2 = [ 3 + 59 sin (¢ - ¢0)] MeV ' (58) (59) ' MeV (60) (61) and are illustrated in Fig. 5. One notes that equations (58) and (59) predict very small K*re rates for a or t3 for certain ranges of ¢. Using the estimates based on the data of refs. 26, 27, * and r (a -? K *re) > 15 MeV .( 62) r.(t3 -? K*re) > 55 MeV (63) one finds that¢ is constrained by the bounds (62) and(63) to lie within one of the ranges: *These allow for likelihood of a Ka mode as well ·as for the observed modeo (See refo 26.} Some fits imply con'siderably larger widths (see refg 24), but none imply smaller o Kp 55 100 -> 60 Q) :?E 40 j_ ___ _t __ r({3__. K-* 7T)~ 55 MeV I I I I I .20 _t_ ___ _i _ _ _i_ __ _ r(a-.. K* 1T) ~ 15 MeV I I :~ -30 0 I I~, 30 60 cp (degrees) Fig. Sa. Predicted partial widths fYo< ~ K*rc) and r (f3_,K~) as a function of the mixing angle r/J. The bounds r(o(.~J<*'7r) ~ 15 MeV r'f(3~Klf7C) ~ 55 MeV (26) restrict¢ to lie between 10° and 35° (shown by dotted lines) or between 75°and 100° (not shown). 56 60 I I I I I .I I s-1 I I 14 __.... > I 40 :?! ~ I I I I I I Q) ....... I I I I r(a-.pK):· 20 -30 0 30 60 cp (degrees) Fig. Sb. Predicted partial widths r(o<~.fl<.) and r(f3~,pK) as a function of</>. The curve for fT(o<~pl<.) represents only an upper bound, based on the assumption Mo<f 1280 MeV. The dotted lines and the arrow denote the range of </J allowed from fig. 3a~ 57 I: or (64) < 75° II: ¢ . < 100° (65) The second solution corresponds to the state ~ (higher in mass) being very much like the A -like object 1 KA. Since we expect the lower mass a to be more like the A , we shall restrict our discussion to 1 solution I in what follows. l++ (Also, broken duality predicts an unmixed octet which agrees with the G-M-0 mass formula using the lower mass K* • ) Somewhat less mixing is favored here than in the approach of 43 Gatto and Maiani, ( ) who do not consider the possibility of D-wave admixtures in decay amplitudes" · Solution I leads to the prediction of a rather weak for the ao K*~ mode This effect may be responsible for the narrowness of a in the data of ref o 260 The bounds (64) imply that ¢ - ¢ 0 is small if not zeroo In this case: (a) ~ -7 pK . The decay is multiplied by (b) sin 2 will be suppressed, as its S-wave contribution (¢ - ¢ 0 )~ The decay a-) pK bution~ will have its largest possible 8-wave contri- explaining its observation despite nearly total exclusion by phase spaceu ( 2B) One may eliminate ¢ from equation (47) to obtain the following constraint on a and ~ angular distributions in the K* ~ mode: 58 1 - g f3/g f3 0 ], p(ex ~ K* :rr) = p(f3 ~ K* :rr) (66) As a result of the fit of the previous section, the right-hand side of equation (66) obeys p(ex ~ K *:rr) p(f3 ~K*:rr) < 0.123 (67) Hyperbolae corresponding to these bounds are plotted in Fig. 6, with only the branch corresponding to solution I shown. The lines cutting across the hyperbolae indicate contours of constant ¢ for varying D/S. (64) and (67) restrict the predicted ratios to the The constraints shaded areaa At present, as mentioned above, the published angular distri- butions in both the low and high mass Q regions appear consistent with purely S-wave decays~ 42 ) The closest approach to this solution allowed o. 6, in FigQ 6 corresponds to values whose effects should be detectable if background is not an appreciable fraction of the Q signaL The quantities g 0 and g 1 give a distribution in ~' the angle between the outgoing kaon and the momentum of the Q (1 + kaon) in the K* rest frame, proportional to distributions in (a,b > o) bex/aex for ~ g 2 0 varying as 2 cos ~ + g ·2 . 2 sin ~o 1 2 aex + b ex cos. ~ and ex ~K*:rr and f3 ~K*:rr, respectively. increases while ·bf3/af3 decreaseso We thus predict . 2t: af3 + b f3 Sln s As ¢ increases, These angular distributions thus provide a sensitive test for the mixing angle ¢ if the present 59 g {3/g (3 0 l a 1 {3: pure S wave + a pure D /3 pure S wave LARGEST ALLOWED -I -.5 -.5 -I ~1.5 -2 Fig. 6. a pure S wave {3 pure D wave Relations between (g~ /g~ ) and (gg /g~ ) implied by the present model. The ~haded area represents the range of values allowed by bound~ on ¢ 2 and D/S. The sharply curved hyperbola represents the ~undary D J'o1..ff3/2S2.= ,034 while the one passing closer to the origin represents D 2.f«tp/ZS2.-:: .123, 60 model is correct . * In view of the predicted suppression of $ -7pK, predictions about the angular distribution in this decay would be difficult to test. (The ratio y $/y $ 0 1 will be a sensitive function of¢.) It is also a -7pK can be collected to confirm the doubtful that enough events of y1a/y 0a should prediction (implied by equations (64) and (67)) that be very close to lG Interference between a The present solution implies If $ has not been considered here . a , g a < 0 a.s well as 0 1 a and $ are produced via diffractive dissociation one might . . expec t them t o arise in A 1 g and K+p --7 Q+p production from pions. primarily from KA' in analogy with In this case the positive relative sign in equations (39) and (40) combines with the positive relative sign just mentioned to give a coherent sum in the K*~ final state. The inter - ference region should then consist of a well-defined minimum, and events taken below and above this minimum might be expected to provide reasonably pure samples of a and Interference effects in the by similar argumentse quite general .. decays, respectively. $ pK mode should be destructive, This opposite sign for K*~ and pK modes is (44) The mixing angle ¢ cannot be too small if diffractive pro - *We thank D. Lissauer for informing us that the predicted cos 2 ~ modulation may be occurring in a -7K*~. The sin2e modulation in $ --7 K*~ is not seen. This would tend to weigh in favor of a larger ¢ wi thin the bounds (64) ~ • 6l duction of KA . K+p in Q+p • ~ is the only mechanism by which Otherwise the ~ a and K*re 45 ) in the final state 3 8 1 initial signal in this peak. a state, sometimes call "C", has been seen in pp reactions The at rest, ( ~ KKre~. state as a virtual p if lation from the JP = 0- 1 8 0 If one naively treats the predicting a suppression of I = O (annihi- I = 1, or w if state is forbidden to produce KQ by angular momentum and parity conservationh( ~~WK and 44 ) then the present model, ~_,.PK by virtue of the specific value of ¢, would predict a suppression of ~ production in (for those incident N energies "Which allow ~ NN ~ KKrere are produced peak would be anomalously suppressed. It is not, however, as indicated by the prominent reaction at the mass of the ~ production * but for which the N and .N still annihilate in a relative 8 wave an appreciable fraction of the time.) a and To our knowledge the comparison of production by antiprotons in flight has not yet been per- ~ formed. To conclude, we favor a mixing angle for states in the Q region lying somewhere between 10° and 35°, a range which gives rise to: (a) suppression of the of the pK K*re mode of the lower peak, (b) suppress ion mode of the upper peak, and (c) angular distributions in K*re mode close to those characteristic of pure 8 wave, the dominant for both peaks, with the lower more longitudinal and the higher more transverse. *The ~ The first two predictions, which are the opposite for the is too massive to be produced at rest in NN ~ (~K or ~K). 62 Deck effect, have been subsequently confirmed.* The absence of one I =0 partner of the A , and of both 1 I =0 partners of the B, prevents us from confirming the predicted mixing angle via studies.of mass formulae at presento *See, for example~ U~ Ee Kruseo (~) 63 IV. THE NINTH 1 ++ MESON A second isoscalar meson with Jpc = l++ experimentally. has not been seen Together with the isoscalar D(l285), the isotriplet A (1070), and the lower axial~vector kaon isodoublets KA(l240), such 1 a meson would complete the set of nine 3p states expected from· the 1 quark modelo In this section we stress some aspects of this meson, "Which we call the D', that are expected on theoretical grounds. properties of the Dv include: (a) assignment to a weakly mixed ~ (~ singlet; (b) a mass lower than the ~1!1! and (d) a possible suppression of the p signal in the effective mass spectrum of the + -y 1( 1( SU(3) 950 MeV with assumptions about the relative coupling of D and Dv); (c) decay modes 1!+1!-Y, These and 1( +1( - final stateo These predictions are of special relevance . ~t present because of the recent claim for a new meson with properties similar to (b) - (d)o ( 32 ) the 2 This meson, called M(953)f 3 ) is distinguished from ~v(958} only by the final state .. dominantly via The~', absence of any appreciable pin its in contrast, appears to decay to py. (32'46) + -y 1! 1! 1( +1( - y pre- "While one feels uneasy about a claim for two states so close in mass sharing common decay modes except for the difference just mentioned, our theoretical e)cpectation of a near the degeneracy may be the caseo * ~'(O -+ . , I= Y = . 0) suggests that this It should be noted, however, that a *The possibility of the D near the · ~' was mentioned in ref. 470 7 64 J pc -- l+- , I -- 1 . assignm~nt l++, I = Y = 0 for the M(953) instead of is not ruled out by the data of ref. 32, and will be discussed as well. lo Unitary Singlet Nature A hierarchy of constraints based on duality -- in which the PP ~PP, the next most reliable from most reliable follow from FV ~FV, and the least reliable from VV ~vv (P = o-, V = 1- mesons) has been used to predict exchange degeneracies. ( 4s, 49 ) system the omission of the "worst" structure for mesons of VV constraints predicts nonet Jpc = 2++, 1--, 2--, and l+-, but only octet structure for The unitary singlets of PP do not couple to or In such a 0 -+ and FV, and hence are not involved in the "good" constraints, which .are assumed to determine the observed exchange degeneracies. The remaining eight o-+ expected to form weakly mixed octets. the ~, K, and 'I); mesons are then This is certainly the case for it seems to hold as well by consideration of the Gell-Mann Okubo mass fol."Illula for the the 1++ and A (1070), the 1 KA(l240), and D(l285): = 1..65 * - m2 (A1) 4 m2 (K) 3 - 1.67 The "missing" Dv is then identified as a unitary singlet member, as is the 2.. 'l)y .. Mass of ~ 950 MeV The existence of a low-lying (1++, I= Y = 0) meson was 65 predicted several years ago from the application of superconvergence 5 scattering. ( 0) The extreme 1 saturation assumptions adopted in this work led to many useful prerelations (SCR) to nn, np, no, and dictions, among them (l++ , I = Y = 0) mA 2 ~ 1 2rn nA 2 p ' m <J -::::: m , and so on. A p state was required for superconvergence in n5 scattering, in which it was required to cancel the contribution of the ~v pole, and in scattering, in which it was needed to saturate :rr.A 1 the appropriate Adler-Weisberger relation. m(l ++ , I The predicted mass was = Y = 0) ~ mA • 1 A low-lying D and the D(l285) .may combine to give the "effec7 tive" D of ref. 50a The algebraic approach considered there is con- cerned with states lacking strange quarks. As we have seen, both D and D' have components with this propertyc If they are both incorporated into the SCR of ref. 50, the superconvergent sum rule for nA Transverse 1 scattering predicts (mA 1 2 - ~v 2 ) 2 2 2 2 g D' A n + (mA_ - ~ ) gDA 1l 1 1 --i = 0 when all other constraints of the model are taken into account. single (1++ , I = Y = 0) then have to be at mD (68) (A meson coupling to nonstrange quarks would Equation (68) predicts that the D' lies lower than the ~ ~ A priori there is no relation between the couplings of the D and the D' since they are not mixed a Zweig's rule for quark graphs (eliminating disconnected graphs) relates the singlet and octet couplings for mixed nonets (such as Jpc = 1- - or 2++)a A simple but naive 66 assumption would be to extend the quark graph rule to the case of no . mixing. Then the coupling of ·D and D' to states lacking strange quarks lgnl/lgD' I would be in the ratio; = 1/ ,J2, which via equation {69) predicts the mass of the D1 to be:( 5 l) (69) .95 GeV A smaller value of the D' coupling would require an even lower mass for the D' o If the Dv coupling is given by the quark graph prescription, then the apparent failure of the D' to be identified in reactions where the D is produced can only be understood if the D' is actually being seen but is confused with something else. of D is 111(1(. (46,52) showing an rpr:rr 46 52 960 MeV. ( ' ) The predominant decay mode It is interesting that every published experiment mode of the D also shows an T}1t1t peak around This peak may contain some Dy as well as the generally assumedo In such cases the study of 11 1)'" -? 1t1t/' 11' is of particular importanceo . Decay modes T}1t1t and 1( +1( - y The dominant hadronic decay mode of the D' should be mD' < 1013 GeV forbids the KK1t channel seen in D decay~ T}:rr:rr, as Four-pion modes require at least two units of t between various pion pairs (as in Dy ~ pp -? 41t), and should be suppressedo The Ml transition mode T}v -?pf' competes favorably with the hadronic 11' -? 1t1t'I'}, which can proc~ed without angular momentum barriers. 67 can be, a priori, both El and M2, ( 47 ) -while The transition D' -7 p/ the D' -7 rcrc11 mode involves at least a unit of t between some particle and the other pair. Thus one might expect: (70) ' with approximate e~uality holding if a suppression of El(D' -7 Pl) (to be discussed) takes place. Measurements of 11/11rc+rc- peak, if found to vary among experiments, ( 53 ) could indicate the presence of a variable Dy "contarninant 11 in the signal. 11' (The D' cannot //.)( 54 ) decay to . + Apparent suppression of p in re re / mode The El and M2 transitions magnitude. ( (VDM). s~uare 47 55 , ) may be of comparable .An example is provided by the vector dominance· model The matrix elements for of D' -7 P/ D' -7 p/ must have a p pole in the The residue of p , the photon 4-momentum, at 2 this pole must vanish, however, ·for D y -7 p Transverse y, since a particle cannot decay into a pair of identical transverse vector mesonsv ( p 2 2 54 ) To the extent that this effect may be extrapolated to = o, we then expect T (D v -7 p T I) = El + M2 ~ 0 so that in the matrix element for longitudinal p production, (71) 68 T( D' ~ pL y) El - M2 = the El and M2 constributions are equal. expect (72) Since 2 M2 = O(k ), we then 2 El= O(k ), where k is the ·photon energy in the D' rest frame. 11' ~PY proceeds via In contrast, 1t'+1t' - will favor lower Ml= O(k). Hence D' ~ 1t' +1t' - I Tl' ~ 1t'+1t' - y, and one may expect masses than distortion of the p peak~ ( 32 ) To illustrate this effect, we have compared the Dalitz plot projections in m 1t'1t' T' i1 = g11, for the matrix elements E{)$/O Pa 1 E2 *~ pf (ql - q2) 5 2 pl 2 pl (73) .and TD' = where q1t'+ (Tl' or D', igD' a E()$y5 El = q1 7 *~ E2 5 p 2 • (ql - q2)/~, 2 (74) = q 2 , and other indices (1,2) refer to q1t'_ y), respectively. The matrix element (74) is the simplest gauge-invariant one satisfying equation (71) ,. If g , i1 and have a similar dependence on m , one finds 1t'1t' (75) where W( ~) . ( 76). = and A is a suitable normalization constant. We find the i1' ~ 1t' 1t' y and "M( 953)" ~ 1t'+1t'- y shapes of ref. 32 are strikingly similar a~er -weighting according to equation (75). 69 This fit is shown in Fig. 7. The normalization A was fixed by mini- mizing the sum of squares of (77) in 0.56 < m - 1(1( < 0.80 GeV (R in Fig. 7), the mass range for which both - processes show sufficient eventsG rc+rc-y Here (N~y , NM) are the number of decays (above background) of (~' , M) in 40 MeV bins( centered on m 32 ) (56) 1(1( 0 While this example of the distortion is crude (we assumed rc+rc- equivalent ~y final-state interactions in and M(953) for example) it confirms that one need not assume C(M) = _( the apparent suvpression of the py elements can lead to this suppressione by VDM, predicts a rest framee cos 2 g mode. 32 decay, ) to explain Many such choices of matrix Our particular model, motivated distribution in the r-rc+ angle in the 1(1( Distortion of the type mentioned can occur, however, for any distribution in 9. To illustrate this, we present in Table 3 for rcrc D'~ [rc+rc-] A y A 3 2 A = 0 , 1. For 3 for gauge-invariant matrix elements and their sum four helicities They depend on g 2 - g 4 and g , respectively, whose relative strengths govern the respective 3 2 2 cos g and sin g contributions to the angular distribution ~ VDM gives g 3 = g1 + g 4 = 0~ Despite ~he arguments just given in favor of a D' near 950 MeV, we are not able to identify the effect of ref G 32 (the M(953)) as definitely associated with this state. No significant difference in 70 25 []:?] 1 (62 EVENTS) D. M (953) (WEIGHTED AND NORMALIZED) 20 > °' -, 15 I (!) v 0 0 10 'f- r CJ) I z LLJ > w 5 I rJ ri L-1 L.J 0.2 Fig. 7. I I r1 rJ 0 0.4 ., I I ~ 0.6 R (*)L---- • 0.8 1.0 Comparison of -;rT-7[- distributions in ~,~ n:.+lf-'t and M ~1[+n.-'lf The latter distribution is weighed for comparison by the factor m;, I (m~ - m~ll" ) 1 and normalized by a least-squares fit in the range R, described in the text. The distribution for /1-+1C+'lf-1 above (*) is based on bins with no more than one event and . is not shown. mitr 71 Table 3. · Matrix elements for D' (p 1) - [7T+(q 1)rr-(q 2) h ·y(p 2 , A. 2). In the expres3 sions for y>-2>...3, we use Q (m rr. 1?-4m/) 1 / 2, h= (mn1 2-m rr 1/ )/2!1-fD', (O,cp) =angles of 7:'+ with respect toy in 7'7:' rest frame. = . ex * B y ( )6 T 1 >vz >..3 =tg1Eo. 5yoE:1 E2 P2 q1-q2 6 T 2 X2 "- 3 =ig2Ea 8yoE1cxE2* 6PlP1 P2· (q1-qz) Im D' 2 T 3 ~"- 3 =ig3EaBy6P1cx.E 2* 8Pl (q1-qz)6 E1• P2lm D' 2 . ex,LE2 *. eP1·P2-P2BEz * • p 1.Jp f. , (q1-12)61 mD' 2 T I. >.. 2 >..?v=i.74Ecx5yoE1 r>..2>-..3 s t r i >..2>-..3 i=! r = Qh (m D'/m'"rr> cose[g 1 +g 4 +!J(gz-g ~Im n'J r 11 = Qk sinee • i <?[ g 1 +g 4 -hg 3lm n' J/12 10 72 T}11'.rt Dalitz plots apparently exists between the whereas one would exp~ct suppression of high 11'.rt 11' and the M(953), ( 57 ) masses for the latter since the 11 and ~rt system must be in a relative P wave. (Jpc = l+- , I= l~ Y = 0) is indeed as tenable as native assignment 1 +- ~ cry that discussed above. The decay values than as the a is much broader than the 11 1 An alter- ~ py, the Dalitz plot for ·+- 1 ~ 'l)rtrt can easily give lower p. m rt11'. Furthermore, automatically favors high m rt rt (as here the tvro pions must be in a relative P wave) and thus could resemble that of 'I) w ~ 1)11'.11'... *(58) · AG= +, Jpc = l+B(l235). With. 11'.W state has the quantum numbers of the the expected dominant decay mode, ( 59 ) such a state should have been seen by now (but it nonetheless worth looking further for). It would be asembarrassingto the harmonic-oscillator spectrum of the naive quark model as a split A .(BO) 2 *1 = 1 assignments( 59 ) for the M(953) other than Jpc = l+- look less likely: JPC = 1-- suggests a dominant (unobserved) 11'.rt mode, while JPC = o-- implies considerable structure in the ~11'.11'. Dalitz plot (unobserved)(57) and suppression of the ratio 11'.11'.Y/T}rt11'.~ (The two pions in o-- ~ rt11'.Y must have relative L of at least 2 and must be produced by at least a~ M2 transition, involving more centrifugal barrier factors than the '1111'.rt mode.) I= O assignments other than JPC = i++ or o-+ also look less likely: JPC ~ 1-+ suggests considerable structure in the 1)11'.rt mode .. 73 V. CONCLUSIONS We have seen that the lowest-lying O+, l+, and 2+ mesons seem to act as if decaying via production of a q q- pair in a SU(3) singlet state, as long as one does not restrict this pair to have Lz = S z = 0. The angular distributions of the I =1 axial-vector mesons played a crucial role in obtaining and verifying this description of the spin couplingso Our model can be interpreted as a small breaking of the collinear syrnmetrY. SU(6)W x 0(2)L even though the predictions z for the S -wave decay amplitudes are considerably different (same magnitude, but opposite sign). The model was then applied to the I = 1/2 axial-vector kaons "Which are predicted to mix by an angle between 10° and as determined from lower bounds on the widths of the physical states. The mixing predicts a suppression of both the peak and the pK mode of the upper peak K*n mode of the lower -- results that have been subsequently conf~rmed. C29 ) Lastly, we examined the expected properties of the missing I = O axial-vector meson needed to complete the nine predicted by the quark modela Jpc = l++ states Our theoretical expectation of this meson having a mass lower than the A1 and decay modes >'l nn and p/ (with a suppressed p signal) suggests identification with a recently reported state having similar propertieso However, this new state cannot be definitely associated with the missing axial-vector mesono 74 REFERENCES 1. MQ Gell-Mann and Y. Ne'eman, The Eightfold Way, W. A. Benjamin, New York (1964). 2. N. P. Samios, Proc. XVth International Conference on High Energy _Physics, Kiev (1970). 3. See review talks by Ko Wo Lai and Jo Rosner, Proc. of the Caltech Phenomenology Conference, to be published (1971). 4., Mo Gell-Mann, Phys o Letters §_, 214 (1964). 5. G. Zweig, CERN Preprints TH· 401, 412 (1964) (unpublished). 6. H. Harari, Proc. of the Boulder Conf~ on High Energy Physics, . Colorado Associated Universities Press, Boulder (1965). 7o A good review of this subject is given in: J. E. Mandula, Phys. Revo 185, 1774 (1969) • . 8.. S .. Okubo, Phys. Letters~' 165 (1963). 9., L. Micu, Nuclo Phys. BlO, 521 (1969). 10. R., Carlitz and M~ Kislinger, Phys9 Rev. D2, 336 (1970). 11. J. Rosner, Phys. Rev. Letters 22, 689 (1969). 12. Ho J., Lipkin and s., Mesbkov, Phys. Rev. Letters 14, 670 (1965} and Phys. Rev. 143, 1269 (1966)0 R., F. Dashen and M., Gell-Mann, Phys. Letters 17, 142 (1965). K. J. Barnes, P., Carruthers and F. von Hippel, Phys. Rev. Letters 14, 82 (lg65). 13.. Ho J., Lipkin and S. Mesbkov, Phys. Rev .. 143, 1269 (1966). 75 14. See the review talks by R. L. Walker and A. B. Clegg, Proc~ 4th International Symposium on Electron and Photon Interactions at High Energies, Daresbury Nuclear Physics Laboratory (1969). 15. M. Gell-Mann, D. Sharp and W8 G. Wagner, Phys. Rev. Letters §_, 261 (1962). 16. H. Filthuth, Topical Conference on Weak Interactions, CERN, Geneva (1969). 17. G. Costa, M. Tonin and G. Sartori, Nuovo Cimento ~ 352 (1965). 18. P. G. 0. Freund, A. N. Makeshwari and E. Schonberg, Phys. Rev. 159, 1232 (1967). 19. H. J. Lipkin, Phys. Revo 159, 1303 (1967). H. J. Lipkin, Phys. Rev. 176, 1709 (1968). 20. A. Barbaro-Galtieri et al., Rev. Mod. Phys. ~ 87 (1970). 21. Go Ascoli, H. B. Crawley, D. W. Mortara and A. Shapiro, Phys. Revo Letters 20,- 1411 (1968). 22. J. Ballam et al., Phys. Rev. Dl, 94 (1970). 23. D. J., Crennell et al., Phys .. Rev .. Letters ~ 781 (1970). 24. A. Firestone, "The Q Region of Krc1C Mass, 11 Experimental Meson Spectroscopy, ed. C. Baltay and A. Rosenfeld, Columbia Univ. Press, New York (1970)., 25. Birmingham-Glasgow-Oxford Collaboration, Proco XVth International Conference on High Energy Physics, Kiev (1970)., 26. G. Goldhaber, A. Firestone and B. C. Shen, Phys. Rev., Letters ~' 972 (1967). G. Alexander, A. Firestone, G. Goldhaber and D. Lissauer, Nucl. 76 Phys. Bl3, 503 (1969). 27. Birmingham-Glasgow-Oxford collaboration, 10 GeV/c reported by B., Maglic in Proc. of the Lund Conference, S-weden (1970). 28. R. Gatto, L. Maiani and G. Preparata, Nuovo Cimento ~' 1192 (1965). G., L., Kane, Phys. Rev .. 156, 1738 (1967)0 G. Goldhaber, Phys. Rev .. Letters 19, 976 (1967). G. L. Kane and H. S. Mani, Phys. Rev. 171, 1533 (1968). H. J. Lipkin, Phys. Rev. 176, 1709 (1968). 29. A good review is given in: R. Silver, Ph.D. thesis, California· Institute of Technology (1971) .. See also: U., E., Kruse, Proc .. of the Caltech Phenomenology Conference, to be published (1971). 30., A. N.. Mitra and M. Ross, Phys. Rev. 158, 1630 (1967). D. L. Katyal and A.. N.. Mitra, Physo Rev. Dl, 338 (1970). D.. K .. Choudbury and A. · N. Mitra, Phys. Rev. Dl, 351 (1970)., 31.. R.. Po Feynman, M., Kislinger and F .. Ravndal, Phys .. Rev., to be published .. 32.. M. Aguilar-Benitez et al., Phys. Rev. Letters 25, 1635 (1970) .. 33.. B.. Sakita and K. C., Wali 9 Phys. Rev., 139, Bl355 (1965). A. Salam, R.. Delbourgo and J. Strathdee, Proc. of Royal Society 284, 146 . (1965). P .. G.. Q., Freund, RG Oebme and P .. Rotelli, Nuovo Cimento 51A, 217 (1967) .. 77 34. D. Horn and Y. Ne'eman, Phys. Rev . .!£., 2710 (1970). M. Gell-Mann, Phys. Rev. Letters 14, 77 (1965). 350 A. Barbaro-Galtieri, Experimental Meson Spectroscopy, ed. C. Baltay and A. Rosenfeld, Columbia Univ. Press, New York (1970). 360 For a recent review of the A situation, see: 2 J. Rosner, ref.. ( 3) . 37. P. Schlein, Proc. of the Conf. on ~n and Kn Interactions, Argonne (1969). 38. M.. Derrick, Proc. of the Boulder Conf. on High Energy Physics, Colorado Associated Universities Press, Boulder (1970). 39. M. A. Abolins et al., Phys. Rev. Letters 25, 469 (1970). 400 G. Ascoli, Experimental Meson Spectroscopy, ed. C. Baltay and A. Rosenfeld, Columbia Univ. Press, New York (1970). 41. G. Ascoli et al., Proa. XVth International Conference on High Energy Physics, Kiev (1·970) .. 42. G. Goldhaber, A.. Firestone and B. C. Shen, ref. (26). 43. R. Gatto and L. Maiani, Phys. Letters 26B, 95 (1967). 44. H.. J. Lipkin, r .e f. ( 28) .. 45. R. Armenteros et al .. , Proc. XIIth International Conference on High Energy Physics, Dubna (1964) and Phys. Letters ~' 207 (1964)., No Barash et al .. , Physo Revw 145,1095 (1966)0 A.. Astier et al., Nuclo Physo BlO, 65 (1969) .. A. Bettini et al., Nuovo Cimento 62A, 1038 (1969). 78 46. M. Roos et al., Phys. Letters ~' 1 (1970). 47. R. H. Dalitz, Experimental Meson Spectroscopy, ed. C. Baltay and A. Rosenfeld, Columbia Univ. Press, New York (1970). 480 J. Mandula, Jo Weyers and G. Zweig, Phys. Rev. Letters ~ 266 (1969)0 49. J. Mandula, Jo Weyers and G. Zweig, Annual Rev. of Nuclear Science 20, 289 (1970). 50. Fo Gilman and H. Harari, Phys. Rev. ~' 1803 (1968). 51. Again within the context of this model, it is amusing that one 0 avoids the former prediction grerc = 0 by replacing the "X " of ref (50) by the two states ~(548) E rv 8 (as well the D by D(l285) E ~ and and ~' (958) E rv 1 D'(950) .E 1) in the rv SCR, giving r(D ~ore) ~ 40 MeV. 52. C. Defoi~ et al., Phys. Letters 28B, 353 (1968). S. Otwinowski, Phys. Letters 29B, 529 (1969). J o Ho Campbell et al., Phys. Rev. Letters 22, 1204 (1969). 530 D. Bellini et alo, Nuovo Cimento 58A, 289 (1968). E. H. Harvey et al., Proc. XVth International Conference on High Energy Physics, Kiev (1970). 540 C. No Yang, Physo Rev. IJ_, 242 (1950). 550 Ao N. Zaslavskii, Vo Io Ogievetskii and V. Tybor, Yad. Fiz ~' 852 (1969) 56. 0 The form (77) was chosen for minimization since it involves roughly equal statistical weights throughout R. 57. R. L. Eisner, private communication. 79 58. A. Rittenberg, Lawrence Radiation Laboratory Report No. UCRL 18863, unpublished (1969). J. Po Dufey et al., Phys. Letters 29B, 605 (1969). 59. s. F. Tuan and T. T. Wu, Phys. Rev. Letters 18, 349 (1967). 60.. Such Yfsuperfluous states" have been suggested on various grounds. See, e.go, K. E. Lassila and Po VQ Ruuskanen, Phys. Rev. Letters 19, 762 (1967); J. Rosner, Phys. Rev. Letters 21, 950, 1422 (E) (1968) and Physo Letters ~' 493 (1970); q~ F. Tuan, Phys. Rev. Letters 23, 1198 (1969). 80 PART 2 UNIVERSAL CURRENT-CURRENT THEORIES AND THE NON-LEPTONIC HYPERON DECAYS 81 I. INTRODUCTION 2 The "V-A theory"(l, ) of the weak interactions has been very successful in explaining both leptonic processes, such as muon decay (µ ~evv), and semi-leptonic processes, such as neutron beta decay (n ~pev) .. The "V-A theory" is a phenomenological description of the weak interactions by an effective Lagrangian, the matrix elements of which are calculated in first order. The effective Lagrangian is written in terms of currents: = ~2 G (1) The universal coupling G is approximately 10 -5;mp 2 • . The currents .are a sum of both a lepton and a hadron piece: ' (2) ' and each piece is the difference of a vector current and an axial-vector currento For example, the lepton current is: J Pt " (3) ::: where µ and v (e and v ) are the Dirac fields for the muon and its . µ e neutrino (electron and its neutrino)u The operator fl 1-)' 5 YI insures a two component neutrino theory, i.e., that the massless neutrino has only one helicity. Consequently, the V-A theory (at least for the · leptons) is, in a sense, maximally parity violating. 82 Using the lepton current, the matrix element for muon decay is: (4) The matrix elements of the hadronic current are not as simple because of complications from the strong interactions (renormalization effects). Consequently, the hadronic current is written formally as: J h " = (5) and the matrix element for neutron beta decay becomes: (6) (the coupling G has been absorbed in gV.) So far, the scale of the · hadron current has not been fixede The remarkable equality (within 'Z'/o) of G, which describes the strength of muon decay, and gV' which describes the strength of vector coupling in neutron beta decay; led Feynman and Gell-Mann(l) to propose the strangeness conserving ~adronic vector current is not renormalized by the strong interactionse before in particle physicsg Such a situation has occurred The electric charge is not renormalized by any interaction because the electric current is conserved~ and Gell-Mann proposed that Feynman VAh be identified with the i-spin raising part of the conserved isotopic spin · current (CVC hypothesis). The near equality of G and the unrenormalized ~reinforces the . concept of universality, that the weak interactions have a universal 83 strength. The hadronic axial-vector current is not conserved or otherwise the massive pion would not dec.a y. The matrix element for 11'. ~ µv is: (7) where f 11'. ,.., (8) • 96 m 11'. m = · O.) In other words, 11'. the divergence of the axial-vector current is a pseudoscalar operator . would imply f 11'. =0 or that connects the pion to the vacuum, with a proportionality constant that goes to zero in the limit preted as a pion field. interval 0 < q 2 as a function of M ~ O. 11'. " Thus '\]/\ A can be inter- If the matrix elements of \//\ AA in the ~ M11'. 2 (q 2 = momentum transfer) are slowly varying 2 q , then they can be replaced by corresponding 3 matrix elements involving the pion (the PCAC hypothesis). ( ) This 4 hypothesis leads to the famous Goldberger-Treiman relation( ) which connects the pion-nucleon coupling constant (g11'.NN)' the axial-vector coupling constant ·in neutron beta decay (gA)' and the decay constant of the pion (f ): 11'. f 11'. = (9) the relation is satisfied to within 10%0 The hadronic current also includes a strangeness changing part 84 since both the strange mesons and baryons decay via the weak interactions -- K ~ µv, /\ ~p e v, etc. The eve hypothesis and SU(3) symmetry suggest the vector currents belong to an octet, with their charges being the generators of the symmetry group. spatial integral of J 0 and is cons~rved if \/A JA (A charge is the = O.) vector currents are conserved .only in the limit of exact the axial-vector currents are identified = 5 l If SU(3): (10) = l SU(3). with an octet too, then the following commutation relations are ·a consequence of The symbols Q. and Q. The 68 = 1 (11) represent the charges of the vector and axial- f. 'k represent.s the structure conlJ In a quark model, even with symmetry breaking via vector currents, respectively; stants of SU(3). a different mass for the strange quark, commutation relations (10) and (11) are valid plus one more: (12) = The.vector and axial quark currents are: 'A.l 2 q (13) 85 1'. - q )\ 1 5 J_ 2 (14) q 5 6 Gell-Mann( ' ) proposed that equations (10), (11), and (12) are valid independent of the quark model and then give a meaning to SU(3) SU(3) breakinga These relations even when the symmetry is broken. Using equation (12) and PCAC, Adler and Weisberger(?) calculated gA/gV = 1.24 ± .03 1.23 ± .01. in agreement with the experimental number (Actually, this is Adler's corrected result.) One problem remained with the 68 = 1 semi-leptonic decays -- they are one-tenth the size predicted by universality. This problem was solved by generalizing the concept of universality. Gell-Mann noticed that the leptonic charges satisfy an SU(2) algebra, i.e., defining Q+ ft and 2Q 3 ,e = Jd 3 0 .x J .£ (15) = (16) = implies = + ± Q £ He then postulated that the hadronic charges satisfy the same corm:nutation relationsQ (B) (17) SU(2) This new version of universality fixes the scale of the h~dronic current throu~h the nonlinear corm:nutation relations.. The most general, charged V-A current that satisfies this postulate is: 86 J h " = cos g (18) i.e., = 12 (VA1 + i VA2 - AA1 V+ = 1 (V V+ - A v+ ) J " 2 " (19) 1 = 2 (20) " The angle 9 is not fixed by the commutation relations. Cabibbo( 9 ) showed that the predictions of this current (with the angle g determined experimentally) agree extremely well with the meson and baryon semi-leptonic decays. Actually, the effective Cabibbo angle is not necessarily the same for the vector current and the axial current because of SU(3) (The Ademollo-Gatto theorem(lO) sta~es breakingo that the matrix elements of the strangeness-changing vector current, whose charge is a generator of order of SU(3) breaking.. SU(3), are renormalized only in second The matrix elements of the axial current can be renormalized in first order.) Recent determinations(ll) of the effective Cabibbo angle are: (1) gA = .2688 ± .0006 QA = from + I + ~µVJ + K+ ~µV1C and (3) v = g • 233 ± .012 and 0238 ± .018 from a fit to . 87 the baryon decays. * Moreqver, the small discrepancy between G and gV is explained by the fact that gv = G cos g ~ .98 G in this theory. The mesons and baryons also decay weakly into non-leptonic final states, e.gv Although the Cabibbo theory expects such decays, the matrix elements are almost impossible to calculate because the product of the two currents cannot be separated as in the leptonic and semi-leptonic decays. Before examining the limited evidence for the Cabibbo theory in the non-leptonic hyperon decays, we will discuss the experimental data on a strictly phenomenological level. The most general matrix element for the baryon decays is: where and (see Fig. 1 ). p are the momenta of the initial and final baryon 2 If CP is conserved (as in the Cabibbo theory) and there is no final state interaction, then A and B are relatively real. Experimentally, A and B have a small phase difference which is consistent with CP conservation and the predicted' final state interaction from low energy phase shifts (approximately 1° for /\ ~p~-). (ll) The matrix elem~nt may also be written in terms of two component *with F/D for (BjA h!B) equal to .66 ± e03 in agreement with PCAC and F/D = 2/3 predicted by SU(6)W for PBB coupling. 88 PK = (w, q) FigQ lg Non-leptonic decay of a baryon: 81 ~ ~~7( 89 spinors: (22) where and S (or A) corresponds to an p = qq ~E2·+ M2 B . Thus, S-wave interaction while P (or B) corre- sponds to a P-wave interaction. Because of the negative intrinsic .parity of the pion, the S wave is parity violating while the P wave is parity conserving. The interference between the two waves appears in the angular distribution of the . decay from a polarized particle: The most recent experimental data are given in Table lf ll) (A~ represents the decay /\ 0 ~p~-, ~+ ~n ~+ J etce) ~+ represents the decay + The most striking fact is the excellent agreement fl I = 1/2 rule for the non-leptonic of the data with an empirical -weak Hamiltonian. rule impl.i es ~ The /iI = 1/2 + r2 l\j /\o 0 - ,[2 -0 -o 0 (24) = 0 (25) = Table l Rates, decay parameters and decay amplitudes for hyperon ha.dronic decays /\.: - ATE .39 7± .005 E: ±. , . 61 0 I I~ .640 ~.014 of. .6L.5 !.Q16a -.060 ± .047 - 5 + - - -0 .330 ±.020J ~026 ±.042 -.960 ± .067 - . 4 2 5 ± .0 3 5a -.346±.075 >O 1. 52 ±. 0 2 1.87 ± .03 <O .02 ± .04 10.44 ±. 33 -.55 ±.43 19.08 :!: .35 161 17 > 0 - ± 22 6 1 "i - 3± 6 2 2± 20 > 0 1.53 ± .14 >O 2.07 ± .03 1.53 ± .05 -11 . 5 2 ± 1. 8 5 -7.42 ± . 65 -4,54±1.0 2 ? . . . b 1.15±.18 -15.36·± 1.40b .1 2 2 . 021 t o er"' c:c :::- = -.2 8 5 ±.0 2 6 b second solution c _ <6 A 6. B > since - \}A A2 - A 8 2 .9 59 -.004 a ex,.,_ a n d a::- f i tt e d CAB -c .528 ! .01. 5 -9 :!: 6 ABC i ~ .472'.t.015 ¢ [degrJ B --- 0 21 1.235 ±.020 1. 235 ± .020 1 .602 ± .013 R A -- ,+ Lo sign of . 239 .102 r unknown I RATE and BR. from Rosenfeld tables Aug. GB, a and $ Jan. 69 world averages <.o 0 91 - J'2 L:+ (26) = 0 One other empirical rule (with some theoretical justification) has been noticed -- the Lee-Sugawara relation:(l 2 ) - /\~ 2 (27) The agreement of e~uations (26) and (27) with the experimental data is indicated in Fig. 2. (ll) 0 The neutral /J. I = 1/2 rule. K (The sign convention of ref. (11) is used;) decays are also in good agreement with the The only real evidence for a comes from the decay K+ ~ 1(+1(o ~I = 3/2 which cannot go via component 6I ::: 1/2 • The ratio r K+~1C+1Co r 0 K indicates that the 1 500 ~ 1C 1C A I = 3/2 than 5% in the amplitude~ K ~ 1C1C1C - component in kaon decays is no more 3 (Recent accurate data(l ) on K0 __,. 1C1C indicates a comparably small amount of AI = 3/2 .. ) neutral kaon decays exhibit one other striking phenomena and The a small violation of CP conservation on the order of ~';!J/o .. The Cabibbo theory predicts that the non-leptonic Hamiltonian is given by: 10 .A [10 sec - 112] ~ 5 0 5 10 15 20 25 5 - .Lr+ Vi n + N "Tu Cl1 &t>f.fl 10 © £\) 0 .- L.J 1 co V2' r: l 15 20'---..------r-,.-~-r--r--.---r----,--,r--r--r--r--r-----.-~----,------.--' Fige 2c Test of the 6. I = 1/2 and Lee-Sugawara triangle relations by the measured decay amplitudes of 1AL , + and r;- - ..=:::::. 93 H D,.S = 1 = fA,B} (The notation = ,[2 G sin g means .AB + BA. cos g (29) Also, the 4-vector indices on the currents have been dropped for convenience.) .These Hamiltonians may be divided into parity conserving and parity violating parts, e.g., 'P. c. H ~s =1 p.v .. H ~s The = G sin g 2,J2 G =0 cos g = - -- sin g [{vT- ' vv+} + {AT-, Av+}] cos g 2 ,[2 (30) [{vT~, Av+} + {vv+, AT-}] 6 S = 1 Hamiltonian .contains both 6I = 1/2 ' and (31) 6..I = 3/2 pieces, and there is no 11 a priori 11 reason to expect a suppression of the b. I = 3/2.. · and '?;_,7 In terms of SU(3), the Hamiltonian contains only ~ because it is symmetric in the unitary spin of the two currents. Octet dominance, which is assumed in various dynamical schemes, is b. I = 1/ 2 .. sufficient but not necessary for The Cabibbo 6. S = 1 Hamiltonian has another symmetry property, which for the octet part 4 corresponds to being the sixth (!' ) .compone.n t. (l ) 6 For the baryon decays, the sixth component of an octet (plus CP conservation) is sufficient to guarantee the Lee-Sugawa relation for the S waves, but not for the P waves.. Moreover, the sixth component of an octet forbids the dominant decay K~ -7~~ and K° that decays into ~~ (K~ is the linear combination of via CP conservation). 0 K This failure is not too serious as the decay is only forbidden with exact SU(3), ~o that the matrix element could be quite large if proportional to the 94 mass difference of K and n. The small CP violation in kaon decays, however, is strictly forbidden by Cabibbo's Hamiltonian. A fair conclusion is that the Cabibbo theory fails to explain many features of the non-leptonic hyperon decays. Some success for the Cabibbo theory has been achieved by the use of the current commutation relations and PCAC. Low energy theorems to {B j .[Q~ (0), Hwk] 2 (15,16) in the limit of vanishing pion four-momentum. (A detailed are derived which relate (B ni !Hwkl B ) 1 2 I B1 ) derivation of these relations and the ones that follow is given in Appendix A.) In particular, Suzuki(lS) was able to show that the parity violating S wave in non-leptonic baryon decays is approximately given by: ~ - .f2 . -- 1 f n (32) A smooth extrapolation of the physical amplitude to zero four momentum for the pion can be justified for the 8 wave, but not for the P wave. If the Cabibbo Hamiltonian is assumed, then the commutator with the axial charge is easily evaluated, e.g., for the original decay): (n° emission in 95 G. f sin cos [Q;(o), {vT-, Av+}+ {vv+,AT-n 22 = ~ • ; sin cos [{vT-, vv+} + {AT-, Av+}] 22 =- Q Q Q Q (33) AS=l p.c. The right side is just proportional to similar results follow H for commutation with - 1 - 5 . ·5 + (+ Ql + l Q2) c~- emission in the original decay) J"2 . 68=1 except that H p .. c .. is in a different ~S=l [ -2:._ (-Q5(0) + i Q~(o)), H ,[2 p.v .. (o) 1 x SU(3) direction: J -- . = 1 G ,[2 2 .J2 sin Q cos Q [tT- u+J { T- u+}] V + ,V (34) A . ,A sin Q cos Q (35) v1 - i V2 , etc. where We can summarize these results for the non-leptonic baryon decays ·as follows: 0 - + ~ emission a (B2 I !2 HT ' v I Bl) p.c~ (36) 96 1( + T , U l + emission . . 1( a (B2 I - - H J"2 p. c. I Bl) (37) a emission (38) AS=l Since H contains ~ and 27 parts, the matrix elements "'V . • 68=1 (B2 IHp. c. ( 0) I Bl) can be expressed in terms of three parameters two for the~ part (D,F) and one for the ~7 part . (a:). The seven non- leptonic baryon decays are then determined by thes~ three parameters. The four relations(lS) that follow are: /\ 0 + ~2 (\ 0 -J"2 -~2 L: 2 -::- - -0 (39) 0 = 0 -o = 0 (40) r:+ = - 2:+ + (41) 0 = ~r3 - /\ ~ The first two relations are the ..L:+ 0 ~I = 1/2 I: + = 0 = 1/2 part of N and -:- isodoublets., is the third relation the /j I = 1/2 and the fourth relation the Lee-Sugawara rule ~ amplitude is proportional to a since I:+ rule for H p., C" .. Errrpiri cally, 2: Note that the p . c.. decays 2:+ + and n cannot be connected I:+ ;::j O implying + that the fit to the data requires octet dominance., rv ~S=l H However, only if ~S=l by the 8 part of (42) rule for /\ and ::: de cays, ~I which follows from the fact that only the can connect /\ with the - ~~2 2:++ a ~ o, Equations (41) so 97 and ( 42) are then equivalent to the [l I the Lee-Sugawara relation, respect'ively. = 1/2 rule for L: decays and However, octet dominance for the Cabibbo Hamiltonian is sufficient to guarantee the 6I = 1/2 and Lee-Sugawara rules for the S waves without the use of current algebra. Current algebra and PCAC do imply one more relatiqn (using octet dominance, of course): L:+ + = ' ( 43) 0 Thus, the S wave amplitudes for non-leptonic baryon decays are reasonably well explained by the . Cabibbo Hamiltonian although the reason for octet dominance is still mysteriouso The P wave amplitude is not given by. the commutator of lill=l H p.Co Q~l and because, for the parity conserving amplitude, the extrapolation 7 to zero four-momentum of the pion is not valid. (l ) Baryon pole terms, or Born terms as they are called (examples illustrated in Fige 3), contribute to the parity conserving amplitude and vary rapidly in the extrapolation. The Born terms are the contributions of single particle intermediate states approximately degenerate in mass with either or B (Which in the case of 2 octet)o SU(3) includes any member of the The contribution of a typical Born term to the physical ampli- tude is given by~ l (44) 98 99 The matrix element of 68=1 H p.v. between octet baryon states vanishes in the limit of .exact SU(3), so that the pole terms contribute only 68=1 to the P wave amplitude. Note that the vanishing of (B jHp v.l B1 ) 0 5 has another consequence -- the connnutator terms (B2 j[Qi (o), Lill=l H Jj B1 ) contribute only to the S wave because 5 68=1 68=1 [Q. (o), H (o) l p. c. J ex H (0) p.v. (45) Thus, we have the following picture of the non-leptonic baryon decays: 5 68=1 ( i) S wave amplitudes determined by (B 2 I[Qi ( O), HP.•v. ] I B1 ) and (ii) P wave amplitudes determined by Born terms. The Born terms seen essential for obtaining the large size of the P wave amplitudes, which otherwise would have been suppressed by the angular momentum .6.S=l barrierg Since the baryon matrix elements of H , which are deterp. c. mined by the three param~ter fit to the S waves, appear also in the formulae for the Born terms 7 the current algebra analysis indirectly determines the P wave amplitudes too. 68=1 The .D/F ratio for (B 2 jHp.c. B1 ) determined in the fit to S-wave decay amplitudes is: D/F ~ - .. 31 (46) This D/F ratio is amazingly close to that for the baryon matrix elements . of the SU(3). breaking part of the strong interactions, which is the e.i ghth component of an octet D/F ~ - 031 ± .. 02 (47) 100 If we assume 8U(3) for the strong coupling BBn in the formulae for the Born terms, we can evaluate explicitly the P wave amplitudes. Three baryon pole terms contribute to the decay with the largest . t a 1 -~ wave -- ~ ~+ ~ nn + ~ exper1men However, when the three terms are evaluated, their sum is zero. (l 6 ) The D/F from the 8-wave fit predicts a vanishing P-wave for in complete disagreement with experiment. Meson pole terms contribute along with baryon pole terms to the other decays, but when they are included the other P-wave amplitudes vanish, .68=1 tooQ (We have used the value for (n jH I K) determined from a p.c. current algebra analysis of K ~ nn.) The reason for the identically zero predictions for the P waves can be understood from equations (46) The simplest explanation for the similarity of the D/F · 68=1 ('VA ) are ('VA ) and ratios is that H H8U(3) mass breaking ·poC. 8 6 and (47). different components of .t he same octet. (The proportionality constant between the two in baryon matrix elements is approximately the same as l·n meson ma.tr1·x elements.)( 16 ) Ho-wever, ·~th"1s 1s · t rue, th ere i~ will be no parity conserving non-leptonic decays because an rotation of the Hamiltonian (H conserving weak interaction (H 0 0 + gH + EH ) 6 8 + g'H 1 8 )0 SU(3) can remove the parity The rotated states would then be stable except for . the parity violating decays, so that the 68=1 and mass breaking cannot really b.e relation bet-ween H p .. c .. true~ However, the relation is correct for the Born terms because the D/F ratios are the same;and, consequently,the Born terms must c~ncel identically. Thus, the P-wave amplitudes for the baryon non-leptonic decays are not understood by the current algebra approacho 101 Recently, further success for the Cabibbo Hamiltonian has been. achieved through another approach ~ the symmetric quark model. Feynman(lS) .(and others previously)(l 9 ) noticed that the V-A currentcurrent interaction in a Bose quark model (quarks which obey symmetric 6I = 1/2 statistics) guarantees the The reason for this fact is the the V-A interactione 1 -H 2 rule for the non-leptonic decays. Fierz transformation properties of The Fierz transformation is the permutation of of 3B 4 in the following current-current (point) interaction: (48) where r l. are Dirac matrices. r.l = y µ (l-y 5 ), the interaction For is antisymmetrical under the Fierz transformatione model, ~l' ~ , ~ , 2 3 ~ and In the quark 4 are the dirac spinors of the A, p, p, and n quarks, respectively. Because of the antisymmetry under the Fierz transformation, the V-A point interaction is antisymmetric in space and spin of quarks 1 and 2 or 3 and 4~ Since the quarks obey Bose statistics, quar:ks 1 and 2 or 3 and 4 must then be antisymmetric in unitary spin (either SU(2) or state of p and n (3 and 4) is (1 and 2) is automatically is pure 6I = 1/2 Q SU(3))e I = Oe I = 1/2 Moreover, with o The antisymmetric isospin The isospin state of A and p Consequently, the Hamiltonian SU(3) symmetry, the Hamiltonian and p-n are in ~ representations which can be connected by ~' but not ~7 Q) Thus, the symmetric quark model implies is pure octete the 6I = 1/2 (A-p rule for . both S waves and P waves and the Lee-Sugawara rule for the S waves (but not the P waves)e 102 The current algebra analysis should apply to the weak interactions in the Bose quark model too. The results for the S -waves follow as before except that octet dominance is guaranteed. 68=1 diagrams contribute to the matrix element {B !Hp.co I B ) 1 2 Two one where the weak interaction takes place on one quark which is pure F D/F = -1. and one where it takes place on two quarks which is The relative size of these two diagrams is not predicted, but they must contribute in roughly equal amounts to give the experimental D/Fe Thus, the quark model needs another parameter besides the overall normalization to determine the S-wave amplitudes. Unfortunately, the · analysis of the P waves via the Born terms also follows as before, which means that the Bose quark model does not solve the enigma with the P waves even though it guarantees 6I = 1/2 .. Because of the limited evidence for the V-A Cabibbo theory in the non-leptonic decays, we shall examine the compatibility with experiment of more general current-current theories. a wider cla..ss of currents while retaining the action. J+J We shall include form of the inter- The most obvious source of more general currents is the quark model, where it is possible to construct neutral and charged currents with various Lorentz properties -- scalar (s), pseudoscalar (P), vector (V), axial-vector (A), and tensor (T).. · Since the quark model has been a successful source of intuition in the past, perhaps all the currents obtainable with quarks are observableo If these additional currents do indeed exist., the most logical place for them to appear is in the non-leptonic decays. Since the lepton current is charged 103 (e and v e or µ and v µ form the current) and of the form "V-A" (right-handed neutrino~), the hadron current in semi-leptonic decays must be a charged 4-vector, i . e., some combination of vector and axial. However, no such restriction is imposed on the weak decays without leptons. Consequently, currents with other Lorentz properties and charges may appear in the non-leptonic decays. An interesting question is whether the presence of these currents is compatible with or eliminated by the present data. We shall impose the constraint of universality on these more 2 general currents, as was suggested by Zachariasen and Zweig. ( o) Universality will be enforced by requiring the charge of each current to be the a + i a y x component of an SU(2) of Gell-Mannvs definition of universality. algebra, a generalization Specifically, the Hamiltonian is assumed to be of the form H wk where ja with = .+ = ,J2 G L: ja Ja .a 2:: Ja Qa (0) a and a [ Q~ ( 0) ' (49) a at [ Q~ ( 0)' Qa (o)J = * .a fd 3 x Ja (0 ' x) J = - 2 Qaa (O)o (50) (51) *Actually, the definition of the charge needs to be made more precise~ The above definition suffices for Lorentz scalars; for Lorentz 4-vectors the time component of the current is integrated to give the charge; for Lorentz tensors the best . choice is to integrate the xz or yz component .. 104 The index 11 a" refers to the electric charge and Lorentz properties of each current, while the index "a" represents the lepton and hadron pieceso Of course, we are assuming only the charged 4-vector current has a lepton piece~ charges form an ponents of an Equation (51) is just the statement that the SU(2) algebrae SU(3) Since the hadron currents are com- octet, two independent· SU(2)'s neutral and one charged~ Therefore, the possible hadron currents are charged and/or neutral S,Pj and/or V,A; and/or T,T. l tensor to T, i e e • 7 exist: one Tµv = -2 € µVO' A T(J' A T is the dual The Hamiltonian now appears as follows: Hwk = ,J2 G {~ (a~ s. l l l + ,J2 G {~ l + ,J2 G ' { r, i + f3~l (a.2l v.iµ + f3:l (a.l2 T.iµ L: (a~ p.)' l l i l A. ), L: (a. iµ l i + f3~l + Pi) } v.iµ + f3:l A.iµ ) s.l + l + 22 2+ f3.l T.1µ ) , L:. (a.i T.iµv + f3.i T.iµv ) } J (52) The index 11 i 11 refers to the SU(3) quantum numbers of the ·currents. The coefficients in ~ront of the currents are severely constrained by the universality condition~ The commutation relations of the S, P~ and T currents are not known 11 a priori, 11 but we will again use the quark model as a source of intuitione The various currents in the quark model and their commutation relations are given in Table 2. algebras independent of the quark model. We shall assume these Several things should be 105 TABLE 2 AA = q 1'5 i SA . l - = q A. A.l -q 2 A. .2. q PA = q i 1'5 i l 2q 2 - i "'A. TAV = q- i 0 µv 1'5 2 l q C.ormnutation Relations Type 1: k [Qi, Qj] = ' i f .. k Q Type 2: lJ k [Qi' Qj] = i dijk Q Q- i Qj Qk Qi Qj Qk vi vj sj pk 0 ~0 Ai 0 vi Aj Ak Ai pj 0 0 0 0 -sk Ai Ai Ak Si pj Ak 0 0 0 Ai j T23 -k T23 -j T23 -j T23 k -T23 vi sj o · 0 sk 0 vi 0 pj pk Ai 0 0 ~0 Si sj pi pj ~0 vi 0 Tj 23 k T23 vi -j -k T23 0 i T23 -i T23 Note: T23 Tj 23 Tj . 23 i T23 ·Ak 0 ~0 Ak 0 Conventions for y matrices, A matrices, d .. k' and f. 'k as in lJ A0 = and d .. Oll = ~23 lJ 106 noticed about the commutation relations of the scalar, pseudoscalar, and tensor·charges: d .. k lJ to complete the algebra, (iii) and TQ (i) as well as f .. k lJ type, (ii) nonets same relations for S and P as for T The universality condition, with the aid of Table 2, produces non-linear algebraic equations for the coefficients of the currents. The solutions yield the universal theories to be discussed in Section II.. We shall use current algebra and PCAC to investigate . the implications of these more general universal current-current theories. First, however, one more restriction will be imposed. The "V-A" Cabibbo theory works extremely well for the . semi-leptonic decays, but universality alone is not sufficient to force the "V-A" form for the charged 4-vector currento .:We "Want to place a constraint on the uni- versal theories so that the hadronic current in semi-leptonic decays is "V-A". The "V-A" theory was originally ·g uessed by assuming a property of the ?Urrent-current interaction known variously as "chirality invariance, 11 ."maximal parity violation," or "the two component theory .. " The essence of these hypotheses is the assumption that the hadrons appear in the operator J+J interaction with the same projection · as the neutrinos ("1-y " 5 is the negative helicity, or chirality, projection operator for a massless neutrino). '1-y Yi 5 1 The projection allows the particles to be described by two com- ponent spinors rather than four component ones in the usual Dirac theoryQ Actually, the names "chirality invariance" and "maximal p~rity violation" come from the equivalent assumption that the interaction is 107 invariant under replacement of any wave function by 1 (which changes the parity). ( 2 ) except V± A (corresponding to 5 times itself This requirement excludes everything 1 ± y 5 as the projection operator). Obviously, we need a similar but weaker constraint since we are allow-. ing scalar, pseudoscalar, and tensor currents as well. We shall generalize 11 maximal parity violation" by the assumption of 11 parity symmetric currentso" Parity symmetry of a current is defined to be invariance (up to a phase) of the current under the interchange or V ~A, or T ~T • S ~P, Parity syrmnetry plus universality guarantees that the charged, hadronic 1:-vector current :ln the onme as :l.n the Cabibbo theoryc * Moreover, parity symmetric currents might be useful for duplicating the success of the Cabibbo theory with the non-leptonic S-wave amplitudesa This success depends on the fact that the commutator 68=1 of the axial charge and H p.Vo 68= 1 is proportional to H p.c. , a result that must come from some sort of parity symmetry .. In Section III, we shall see if the parity symmetric, universal, current-current theories are compatible with the non-leptonic decays. Current algebra, PCAC, and the Fierz transformation will be used to determine the consequences of the theories. *Actually, an even weaker constraint will do. assumption that V and A· have the same SU(3) Universality plus the structure is sufficient. 108 II. UNIVERSAL CURRENT-CURRENT THEORIES Universality plus parity symmetry forces the charged, 4-vector current to be the· Cabibbo form: * Q=l = Jr., T+ + ci¢c Jr., 1 + i. vf.. 2 - Ar., v+ Jr., (53) 2 - i Ar., ) (54) v+ 4 ·5 1 4 = 2 (VA + i. vf.. 5 - Ar., - i Ar., ) Jr., (55) T+ where cos g = J" 1 2 The arbitrary phase ( Vr., sin 9 1 ¢c allowed by universality is not measurable as it can be absorbed in the definition of the ph~se of strangeness. Thus, the Cabibbo theory for the semi-leptonic decays is unaltered. The neutral, 4-vector current is required by parity symmetry and universality to be: (56) (57) where 1 4 ( -Vr., 3 ' I 8 1 + l\J 3 Vr., ) ± 4 ( -Ar., 3 r 8 + l\J 3 Ar., ) *Parity syrmnetry of the combined lepton plus hadron current does not allow V + A for the hadron currento (58) 109 (The ± sign in equations (57) and (58) is not related to the ± sign in equation (56).) This neutral current (plus the Cabibbo charged curren~) 2 leads to the Hamiltonian given by Zachariasen and Zweig. ( o) The part of the Hamiltonian coming from the neutral current transforms like a pure ....,, 27 and includes a difference 68 = 2 6¢ = ¢N - ¢c produces a CP violation. 6¢ and QN contribution. The relative phase between the neutral and charged currents Zachariasen and Zweig determined limits on from the experimental CF .violating parameters are then required to be quite small). (6¢ and gN Because CP violation is a small part of the poorly understood non-leptonic decays and because we will have enough phases in our universal theories to fit the meager data, we shall neglect CP violation in our discussion. The remainder of our currents will have the phases removed so that the Hamiltonians are CP conserving. The commutation relations for T apd T currents are completely analogous to those for the S and P currents. Consequently, the universal tensor theories are isomorphic to the universal scalarpseudoscalar theorieso For that reason, only the charged and neutral S and P theories need to be discussed. The charged S and P current is required by parity symmetry, universality and CP conservation to be: Q=l J where = T+ J cos g T+ J + sin g J v+ 1 2 = ~2 (sl + i s ) ± -2 i (Pl + i P2) (59) (60) 110 (61) Note that CP conservation requires the pseudoscalar current to have a purely imaginary coefficient. * The neutral S and P theory is more complicated because of the nonet structure of the ·currents~ Severa~ CP conserving solutions are possible.' universal, parity symmetric, If the current is written as: Q=O J where (63) (64) 1 ( -s 3 + 4 1 4 ( ~ ~3 s J"3 s 8) ± 3 8 + s ) ± (65) (66) (67) then the following six solutions are possible: (A) a = f3 = cos v, / = 2 sin v, o = A = 0 (68) *Bailin(l9 ) studied the consequences of S and P theories via current algebra in the non-leptonic decays, but he took the form S ± P (as .a generalization of V ±A) which is strongly CP . violatinge Universality and CP conservation leads us to the form S ± i Pe 111 (B) a = 13 = cos v, r = ·2 sin v, (c) a =-13 = cos v, r = o, (69) 5 = sin v, 4 " = J°6 sin v (70) (D) a =-13 = cos v, r = o, o = _g_ ( ± 1 - sin v), .[3 A = _g_ ( ± 1 + ·2 sin v) .fs (E) O'.= 1 a= (71) 1 2 (cos v' + cos v), 13 = 2 (cos v ' - cos v) ' . y = sin v', 0 = - (F) . 3 ~ sin v, A = ~ sin v (72) ~ (cos v' + cos v), 13 = ~ (cos v' - cos v) r = sin v ~ ' 5 = _.!_(± 2 - sin v), /\ = .[3 ~ ( ± 1 + sin v) (73) We have also calculated currents which are universal but not parity symmetric. These currents are listed in Appendix B, except for the S and P neutral case. which has not been determinedo 112 III. CONSEQUENCES FOR THE NON-LEPrONIC DECAYS We shall use the low-energy theorem (equation (32)) to determine the predictions of the parity symmetric, universal, the S--wave amplitudes. theories for J+J Of course, we are assuming that part of the non-leptonic Hamiltonian results from the charged "V-A" current. 61 = 1/2 least for this part, the and At rules, the Lee-Suga-wara relation, = O follow from a fit of three parameters to the dataQ + The CP conserving "V-A" neutral current (equation (56) without L:+ the phase ¢N) produces a Hamiltonian which, like that for the "V-A 11 charged current, has the property: :s=l] p.Ve In this case, however, ~he 68=1 a (74) H p. c .. Hamiltonian transforms like a pure ?_,7. Consequently, the baryon matrix elements of the neutral current Hamiltonian, which appear in the low energy theorem, are proportional to the parameter 11 0:" (the 27 reduced matrix element) which is deter- ' mined to be zero. (The S-wave decay of tional to a, vanishes e:X-perimentally.) L:+ ~n~+, which is proper- Thus, the success of Suzuki's analysis is unaffected by the presence of the 11 V-A" neutral currento (Of course, the coefficients of L:+ changed by the neutral current, but The "S =!= i P" + in equat1ons (39) - (42) are L:+ = o. ) + charged current (equation (59)) produces a Hamiltonian which has the commutation property given in equation (74) for t"WO choices of the axial charge (corresponding to ~ 0 or ~ + 113 emission in the decay): 1 [Q; 68=1 · (0), Hp.v. (o~ = 1 . 2 J"2 sin 9 cos G ......;. 2 + G [{sT-, 3v+} {PT-, pV+}] (75) _l._ ( -Ql5 ( 0) + i Q52 ( 0) ) ' H68=,l ( 0 ~ p.v. ~ [ ,J2 1 G sin G = - ,J2 . 2 J"2 Equations (75) . and (76) are completely analogous to equations (33) and (34) for the Cabibbo current~ · The results can again be summarized: re 0 - emission . . re+ emission + a ( B2 I!2 HTp. 'c.. v I B1 ) a (B2 I T- 1 - J"2 u+ H ' p .. c" (77) I Bl) (78) Ho-wever, the cormnutator with the axial charge corresponding to re emission is changed because of the nonet structure of the currents: . Q5 (O~ (~ (0) + i 2 (o)), H6S=l p.v .. ,..[2 [ _l._ = J"2" _G_ 2 J"2 sin G cos Q (79) ~w, sT+ + PW' pTj The su-perscr"ipt "wu denotes a particular combination of the w indices "O" and "8", . e.g .. for S SU( 3) 114 (80) (Although there is no connection, the symbol w is chosen because the vector meson w is the analogous combination of singlet and octet.) Thus, we have: (81) Since the current with the supe~script w contains both singlet and + octet, the product of currents in Hw' V contains an ~ part from p. c. 1 x § and an ~ and g] part from § x ~· Thus, five parameters are w, v+ necessary to describe the baryon matrix elements of H - - p. c. corresponding to D and F for each 8 and a for the 27 (no relation to D, F, and a for the V-A case)o Consequently, only two relations are possible among the seven amplitudes. However, it is curious that if we treat the S and P currents as nonets octet into one 3 x 3 matrix in (combi~ing the singlet and SU(3)) and assume that two currents couple to form an octet according to the nonet anzatz( 22 ) (the singlet w, v+ is is not split off as the trace), then the octet part of H p.c. T , v+ 3 identical to that of H • If equation (81) had been analogous to p. Co equation (38) (as equation (77) is to equation (36) and equation (78) is to equation (37)), then :rr · emission for the 11 8 ± i P" case would T , have been proportional to the baryon matrix elements of Hp.3 c .. v+ w, v+ instead of H p .. c .. However, the nonet anzatz for the S and P current~ T , 3 implies that the baryon matrix elements of the octet part of Hp.c .. v+ 115 w, v+ are the same as those of the octet part of Hp.c. Consequently, the seven amplitudes are now determined by three pa~ameters instead of five .. The four relations that follow are: /\~ :t- ~6 ~2 /\~ = - 5 L:++ --- - ~2 -::-0 -o = + 2:: ~2 L:+ 1 - -- L:+ + 5 5 L:+ + /\~ = ~3 L:+ 0 2-=- ~6 (82) (83) (84) 11 + 10 ~6 0 L:+ + (85) E+ ~ o, equations (82) - (85) are equivalent to the + rules and the Lee-Sugawara relation. Consequently, the Again because 6.I = 1/2 results for the charged S ± i P the charged V-A current.. If we had not made the nonet anzatz, we would current are equivalent to those for have had only two relations instead of the fouro However, the fit to w, v+ the data would have required the S and P currents in H to couple p.c .. according to the nonet anzatz.. Although we do not have a good reason why the S and P c~rrents do couple this way, it is certainly true that the charged S ± i P current is compatible with the non-leptonic . S wave data .. The current algebra analysis follows analogously for the neutral ivs ± i P" current (equation (62)) .. - Although the Hamiltonian is not pure 27 as in the "V-A" case, four relations similar to equations (82) and (85) (only the coefficients of L:+ are different) follow with the + 116 addition of the nonet anzatz. * without the nonet anzatz.) (Again only two relation are implied Conseq_uently, the neutral "S ± i P" current is also compatible with the experimental datae The next q_uestion is whether these parity symmetric universal theories imply something. more than the Cabibbo theory, espe.c ially with regard to the P waves. Both the parity conserving and parity violating parts of all these Hamiltonians have the symmetry which for the octet piece corresponds to being the sixth component (even though the p.v. part of the 11 8 ± i P" Hamiltonians consist of §, ~' and ~' rather - - than 8 and 27). Conseq_uently, octet dominance and this symmetry imply the Lee-Sugawara relation for the S waves, but not for the P waveso (Also,~~ 11'.11'. is again forbidden with exact SU(3).) Moreover, the P wave analysis via the Born terms follows exactly as before ~ which is an utter failure. Thus, the parity symmetric universal theories imply nothing more than the Cabibbo theory~ There exist more general universal theories Which are not parity symmetric (see Appendix B), but it is almost hopeless for these theories to reproduce the success in the S waves .. With the assumption of Bose q_uarks, the Fierz transformation property of the 11 V-A" Hamiltonian (antisymmetry in space and spin) leads to an exact .6.I = 1/2 rule for both S and P wave amplitudes~ 8 ± i P" T'v · Hamiltonians, however.)/ have no particular 11 and nT ± i The *Except for the trivial solutions (A) and (B) (eq_uations (68) and (69» where the 68 = 1 Hamiltonian has no parity violating piece .. 117 symmetry under the Fierz transformation. reproduce the b.I = 1/2 Consequently, they cannot rule With either Bose quarks or Fermi quarks. There exist two other current-current Hamiltonians besides the "V-A" one which are antisymmetric under the Fierz transformation (as well as two which are symmetric).C 23 ) The parity-conserving parts of the three which are antisymmetric are of .the form: (86) VV + AA pp SS SS + TT (87) PP + AA (88) (The parity-violating parts of these Hamiltonians, which are also antisymmetric under the Fierz transformation, have a similar structure.) Only the first one (equation (86)), which comes from "V-A", arises from a current which is universal. Consequently, the "V-A" Cabibbo theory is the only universal theory that leads to the 61 = 1/2 rule with Bose quarks. 118 DI. CONCLUSIONS We have derived the general current-current theories which are parity symmetric and universal and have shown that they lead effectively to the same good predictio~s as the Cabibbo theory ·for the S-wave amplitudes in the non-leptonic baryon decays. The low-energy theorem from current algebra and the commutation relations from the quark model 0 were used to obtain these results. Consequently, the hadronic currents with other Lorentz properties and charges may actually be present in the non-leptonic weak interaction. unaffected by these currents.) (The semi-leptonic decays are However, the presence of more general currents does not solve any of the existing problems with the nonleptonic decays, nor is it indicated by any feature of the data. The most significant argument for the Cabibbo theory being the only source of non-leptonic decays is that it is only ·universal current-current theory that leads to the b.I = 1/2 ·rule with Bose quarks. 119 APPENDIX A Application of Current Algebra and PCAC to the Non-Leptonic Decays We shall review the derivation of the low-energy theorem for . (15 16) non-leptonic decays. ' B The off-mass shell amplitude for 1 ~B 2 11'.i' using the notation in Fig. 1, can be defined by: M( q, i. , p l; p 2 ) = i. ( µ 2- q 2) fd4 x e-iq • x (B2 IT Str~ rh4 (x) ' (Al) where the physical amplitude M(q = P - P , q 2 1 of the pion.) 2 = µ 2 (B 11'.i IHw (0) I BJ/ 2 , i; P ; P ). 1 2 equals (The index i denotes the charge If the PCAC equation is used to define the off-mass . shell pion field 1 = iJ°"2 µ2 ~. (x) f J'"2 1 1( 2 µ-q f 1( µ 2 2 ' (A2) fd 4 xe-iq•x (A3)· The following identity (A4) 120 implies = - ~2 q /\ 2 2 µ - q f .1! µ 2 Jd 4 x e -iq•x (AS) - 2 µ - q f 2 . 1{ µ r i~2 2 f d 3 x e iq •x [ Aio ( O,x, ) Hw(O)] Assuming that H (0) is a local operator so that w [A~l (o,x), ~ a o. 3 (x) Hw (o)] ~ we have the relation 2 µ - q µ 2 2 Jd 4 x e -iq·x (A7) 2 - i µ - q µ Taking the limit 2 2 q/\ ~o yields . the low-energy theorem: .p ll·m q" ~ 0 . p ; p ) -::/2. M( q, i,; 2 1 .1.1! = - l"im ~ 0 q" q/\ fd4 x e -iq·x 121 The term proportional to qA vanishes unless it has a singularity as qA -70, Which can happen with the contribution of a single particle intermediate state degenerate in mass with either B 1 or B . 2 The lo~-energy theorem is only useful if the physical amplitude, i .. e., M(q, i; P ; P ) 2 1 at q 2 = µ 2 , can be approximated by M(q, i; P ; P ) at qA · = O.. However, the Born terms in M(q, i; P ,P ) 2 1 2 1 (example illustrated in Fig. 2) do not vary smoothly as qA -70. The contribution of a typical Born term to the physical amplitude is given by: .68=1 The matrix element for the limit of exact H p.v.. between octet baryon states vanishes in SU(3), so that the pole terms contribute only to the parity conserving amplitudeu (Even with broken ' SU(3) for the masses, the parity violating part of the Born term is small, i.e., 1 0(1) rather than o(t:J.1) like p.c. part .. ) The smoothness difficulty disappears if we subtract the physical Born contribution from both sides of equation (A8).. Therefore, let us define: R(q' then 1. ,· P ,@ P ) = M(qo, i ,· Pl'· P2 ) - M_ ·(q -130RN" ' 2 1 (A9) 122 (AlO) The remainder R(q, i; P ; P ) 2 1 is automatically smoothly varying because the singular part has been removed. which we shall call Moreover, the term in brackets, R, has· a well defined limit' as qA -70 that can be calculated. The parity violating part of R (qA = 0) vanishes, like ~;~ (q 2 = µ 2 ), in the limit of SU(3) for is small and ,..., The parity conserving part of R (qA = 0) 6M order 2M' as compared p. c. ( 2 2) MBoRN q =µ is of the and can be neglected. Conse- quently, the :f'undamental equation for non-leptonic decays becomes: (All) Note that the parity violating part of the decay is given entirely by the commutatoro Moreover, the parity conserving part of the decay is · · 1 y b y the Born t erm since · given ent ire 5 [Q i (o), Hpw .. c .. (o)] is a parity violating operator whose matrix element between baryon states vanishes in the limit of SU(3)o 123 APPENDIX B Non-Parity Symmetric, Universal J+~ Theories V, A charged 1. Q=l T+ = a VA + JA (3 e a2 + (32 (3 Q (X COS iQ + I + T+ v+ iX 5eH~' e (y VA + A~ ) AA + 2 + = -f' 0 52 = 1 COS (B2) (B3) t 4 . vs = VA + i A ' e c. where 2o QI (Bl) S, P charged (B4) a 2 + (3 2 + r 2 a (3 sin Q = ..L- where 3o rr· 1 s· = s + i 2 + o -~ = 1 5 sin Q' s , sv+ = s 4 + J.. .8 5 ' etce 2 V, A neutral (CP conserving) (B5) (B6) 124 Q=O (a) J = 1 u+ 1 - 1 + + 4' (cos g - cos g') ~ /\ + 41 (cos g - cos g') AU/\ 6 . v7 vu3 V/\ + 1 /\ ' /\ = ~ ( -V~ + .f3 V~), etc. 4' (cos g + COS gr ± 2) V/\ + 4' (cos g + cos gr ± 2) vu - +~(sing+ sin Q') v~3 Where (b) vf = cvf l = same as solution (a), except V ~A. 125 REFERENCES l~ R. P. Feynman and M. Gell-Mann, Phys. Rev. 109, 193 (1958). 2. Eo C. G. Sudarshan and R. E. Marshak, Proc. Padua-Venice Conf. on Mesons and Recently Discovered Particles (1957), and Phys .. Rev .. 109, 1860 (1958). 3. Y. Nambu, Phys. Rev. Letters !' 380 (1960). M. Gell-Mann and M. Levy, Nuovo Cimento 16, 705 (1960). 4o Mo Lo Goldberger and S~. B.. 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