NONLEPTONIC DECAYS 0F HYPERONS Thesis for the Degree of Ph‘ D. MICHIGAN STATE UNIVERSITY YOSHIKAZO ERNESTO NASA! 1971 thiéii.i LIBRA ”Y Michigan State University This is to certify that the thesis entitled NONLEPTONIC DECAYS OF HYPERONS presented by Yoshikazo Ernesto Nagai has been accepted towards fulfillment of the requirements for Ph-D- degree mm @‘ L’ 7 Major profeuo/ DatLemrLi7/ 0-7639 ABSTRACT NONLEPTONIC DECAYS OF HYPERONS BY Yoshikazo Ernesto Nagai Ever since the techniques of current algebra were first applied to nonleptonic decays of hyperons, each new effort to calculate the s- and p-wave amplitudes for these decays has led to more numerical puzzles than would seem reasonable for such apparently simple processes. We present a new analysis of the problem by ap- plying dispersion relation techniques to the scattering of a spurion from a hyperon. In a particular combination of amplitudes, the scattering process formally reduces to the weak decay process in the limit of vanishing four—momentum of the spurion. In this approach the Regge behavior of the scattering amplitude at high energies requires one subtrac- tion in the dispersion relation, the subtraction point be— ing chosen such that the calculable soft-pion amplitude gives the subtraction constant. Then the low-mass baryon pole contribution is separated from the dispersion integral with the remaining part of the integral coming from higher- mass resonances. Our method of evaluating this latter reso- nance contribution consists in assuming Regge behavior for the scattering amplitude at high energies and extrapolating this form of amplitude to the lower energy region. In this way the re‘ real part changes. T cept of lo de, when e: true ampli' the soft-p; adjustable these parar mental data A tions for t Cal/S of hy; nonradiativ 'I (a) Our fit c:CJY‘ASide1*a1b1 The Usual 11 501m to be found to be large ViOIa dominanCe b diative Wea Culation. Yoshikazo Ernesto Nagai way the result of the higher—mass integration is just the real part of the Regge amplitude from the t-channel ex- changes. This approach has its justification in the con- cept of local duality which states that the Regge amplitu— de, when extrapolated to lower energies, represents the true amplitude in an average sense. The addition of the resonance contribution to the soft-pion—plus-pole amplitude increases the number of adjustable parameters to four. The numerical values of these parameters are then found by a xz-fit to the experi- mental data. As a further test of our analysis we make predic- tions for the amplitudes of the two-body radiative weak de- cays of hyperons. These decays are closely related to the nonradiative ones via low-energy theorems. The main results can be summarized as follows: (a) Our fit to the experimental pionic decay values is a considerable improvement over that of previous work. (b) The usual neglect of the higher-mass baryon resonances is found to be unwarranted for the s-wave. However, it is found to be fairly good for the p-wave. (c) A relatively large violation of unitary symmetry is present. (d) Octet dominance breaking is needed to explain the observed value A(£:)=0. (e) The available experimental information on ra- diative weak decays is in excellent agreement with our cal- culation. in NONLEPTONIC DECAYS OF HYPERONS BY Yoshikazo Ernesto Nagai A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Physics 1971 Peter Sign the course several he; in the ear] '1 Energy Com ACKNOWLEDGEMENTS I wish to express my appreciation to Professor Peter Signell for his guidance and encouragement throughout the course of this research. I also wish to thank Professor Wayne Repko for several helpful discussions and for pointing out errors in the earlier version of this work. This research has been supported by the Atomic Energy Commission through a Special Research Assistantship. ii Section I. II. III. IV. V. VI. VII. VIII. IX. XII. IN'I HA) SOP THE DIE POI SYlv NU.h RAE - THE XI. SOE PRE LIST OF I APPENDIX APPENDIX AppENDIx TABLE OF CONTENTS Section I. II. III. IV. V. VI. VII. VIII. IX. X. XI. XII. INTRODUCTION . . . . . . . . . . . . .'. . . . . HAMILTONIAN AND INVARIANT DECAY AMPLITUDES. . . . SOFT-PION THEOREM . . . . . . . . . . . . . . . . THE SURFACE-BORN TERM AND THE SOFT—PION AMPLITUDE DISPERSION RELATION FORMALISM . . . . . . . . . . POLE TERMS . . . . . . . . . . . . . . . . . . . SYMMETRY BREAKING IN THE TADPOLE MODEL . . . . . NUMERICAL ANALYSIS AND RESULTS . . . . . . . . . RADIATIVE WEAK DECAYS . . . . . . . . . . . . . . THE SURFACE—BORN TERM . . . . . . . . . . . . . . SOFT-PHOTON THEOREM AND DECAY AMPLITUDES . . . . PREDICTIONS FOR RADIATIVE WEAK DECAYS . . . . . . LIST OF REFERENCES . . . . . . . . . . . . . . . . . . APPENDIX A. Notation and Conventions . . . . . . . . . APPENDIX B. SU(3) Invariant Vertices . . . . . . . . . APPENDIX C. Decay Rate . . . . . . . . . . . . . . . . iii 15 21 28 37 44 51 63 69 77 84 91 95 98 100 LIST OF TABLES Table Page 1. Equal-time-commutator and resonance parts of the s-wave amplitudes . . . . . . . . . . . . . . . . . 59 2. Equal-time-commutator and resonance parts of the p-Wave amplitUdeS o o o o o o o o o o o o o o o o o 59 3. Polé part of the s-wave amplitudes . . . . . . . . 60 4. Pole part of the p-wave amplitudes . . . . . . . . 61 5. Best fit solution to both the s- and p-wave ampli- tudes, and experimental values, in units of 105 x -— ‘ m"z sec 2 . . . . . . . . . . . . . . . . . . . . 62 6. Theoretical radiative weak decay amplitudes of hyperons, decay rates, and asymmetry parameters . . 89 iv LIST OF FIGURES Figure Page 1. Born diagrams for the s and u Channels. . . . . . 26 2. Baryon intermediate states in s and u channels. . 39 3. Tadpole diagram. . . . . . . . . . . . . . . . . . 45 4. t-channel Regge poles. . . . . . . . . . . . . . . 52 5. Pole diagrams for the various decay modes. . . . .57,58 6. s-channel one-baryon intermediate state in radi- ative decays. . . . . . . . . . . . . . . . . . . 70 7. u-channel one-baryon intermediate state in radi- ative decays. . . . . . . . . . . . . . . . . . . 73 8. Born diagrams for the radiative decays. . . . . . 75 9. Definition of "nonBorn" amplitude. . . . . . . . . 78 10. Diagrams for the three-body radiative decay. . . . 78 11. (a) Diagram for the radiative decay of a neutral hyperon. (b) Electromagnetic 20A vertex. . . . . . 81 12. Diagrams for the radiative weak decay modes. . . .86,87 I . INTRODUCTION Since the historical paper of Lee and Yang on pa- rity nonconservation,l weak interaction processes have been found to be a rich source of symmetry breaking. Apart from Lorentz invariance and the conservation of electric charge and baryon number, weak interactions appear to violate ev- ery symmetry that has been found to hold in strong interac- tions. Do these violations follow a definite pattern? Is it a large effect or merely a small correction? 2’3 of weak interactions The currentxcurrent theory provides a framework for the investigation of these ques- tions. This theory emerged soon after parity nonconserva- tion had been discovered; since then it has been modified to some extent, but its basic hypothesis remains intact. Namely, that all weak interactions, be it leptonic, semi- 1eptonic or nonleptonic, are generated by the interaction of a Charged current with itself. Thus the general weak Ha- miltonian in the currentxcurrent picture has the form 18 -—G‘ 1(JXJ+X+J:JX)° (1) ”—52 The weak current JA consists of a hadronic part, J?, and a l leptonic one, JA' Both of these terms are equal admixtures of vector and axial-vector components so that they have well—defined properties with respect to space-time trans- formations. In addition, the conserved vector current(CVC) hypothesis4 establishes a simple relationship between 5’, the vector component of J2, and the electromagnetic current Jim; this latter current is known to behave in a well-de- fined way with respect to internal symmetries, and so .?A must do likewise. Similarly, the hypothesis of a partially conserved axial-vector currents (PCAC) relates the diver- gence of 55, the axial component of 1?, to the meson field and thereby indicates specific symmetry properties for 5?. A far reaching extension of these ideas is the identification of Jaand I: with the elements of a Lie alge- bra.6 The success obtained by Adler and Weisberger7 in cal- culating the renormalized axial-vector coupling constant tends to confirm this identification, and it lead us to be- lieve that symmetry is an essential ingredient for a dyna— mical theory of weak interactions. Nonleptonic decay is an ideal application for such a theory. From the point of view of internal symme- tries, most decay modes are different charge states of a few basic processes, and their amplitudes can be correlat— ed by means of the transformation properties of the inter- action Hamiltonian. If the hadronic current is of the Ca- bibbo type,8 then the nonleptonic interaction has specific transformation properties in SU(3), and these properties are sufficient to make a number of important predictions. Among the nonleptonic processes, the two-body nonleptonic decays of hyperons have relatively accurate experimental information which makes the confrontation with the theory less ambiguous. The following decays will be considered: A3: A—DP‘i'fl- ZZZ Z'—’r\+n' Z2“- Z*-—+ p+n° <2) Ziizhan+HI :5: 2‘» AMP Because all baryons involved in these decays belong to an SU(3) octet, we can regard them as multiplet decays of a single one:a+8+n, with spin-parity assignment: %f+-%++O- Consider the decay of a at rest. Conservation of total an— gular momentum requires that the only partial waves allowed in the final state are {=0 and {=l, i.e., s- and p-wave only. The intrinsic parity of both a and B is +1 and that of the pion is ~l. Then the parity of the final state is —(-1#’ so that the s-wave decay amplitude is parity vio— lating (pv) and the p-wave is parity conserving (pc). According to the notion of the universality of the currentxcurrent form of weak Hamiltonian, the nonlep— tonic Hamiltonian flNL is proportional to the strangeness- changing part of the symmetric product of the Cabibbo "“ ~ 19.5“] I current, JA, with itself, expression (1). Since J1 and J: belong to the same octet of currents in the SU(3) group,9 the ‘XNL can only belong to the completely symmetric repre- sentation in the direct product decomposition, Beasleeseaaeloeféefl. (3) The symmetric representations are l, 85' and 27. The first has only a Strangeness preserving term, and since we are interested in strangeness—changing nonleptonic decays,£KNL can only transform as BS and 27. It is empirically known that the 27 part is small compared to the part coming from 85’ This enhancement of the BS constitutes the so-called octet dominance,lo an extension of the isospin AI=l/2 rule.11 The first success along the line of making full use of symmetry properties was accomplished by Suzuki and by Sugawara in their current algebra study of the s-wave hyperon decays.12.Using PCAC, the algebra of currents, and SU(3) symmetry, they derived the AI=l/2 rule for A,E de- cays, the pseudo AI=l/2 rule for Z decays, and the pseudo 13 Lee-Sugawara relation. The extension of the same proce- dure to p-wave decays was considerably clarified by Brown and Sommerfield,14 and others;15 they confirmed the previ- ous calculation of the s-wave amplitudes and, furthermore, found that the p—wave amplitudes can be computed in terms of the parameters appearing in the s-wave amplitudes. The predicted values of the p-wave amplitudes, however, did not compare well with experiment. In general they were found to be two or three times smaller than the observed values. In view Of this discrepancy for the p-wave, Kumar and Pati, and Itzykson and Jacob16 attempted to fix the p- wave while trying to maintain the apparent success of the s-wave calculations. They incorporated certain corrections, of the order of AM/2M compared to the Born terms, that were dropped by Brown and Sommerfield on the grounds that these were small. Also terms representing unitary symmetry breaking, previously neglected, were added. Quantitatively the agreement with experiment was found to be slightly bet- ter, but still not good enough. Several other refinements 17-19 to the theory have been proposed from time to time, none of them, however, really convincing. Meanwhile Okubo20 attempted a new approach to the problem of the simultaneous description of both the s- and p—wave amplitudes by applying dispersion relation techni- ques to the scattering of a spurion from a hyperon. The scattering process formally reduces to the decay process in the limit of vanishing four-momentum of the spurion. In this approach Regge behavior21 of the scattering amplitude at high energies requires one subtraction to the dispersion relation in the energy, the subtraction point being chosen such that the calculable soft-pion amplitude gives the sub- traction constant. Next the low-mass baryon pole contribu— tion is separated from the dispersion integral, the remain- ing part of the integral coming from the higher-mass reso— nances. Difficulties in evaluating this latter resonance contribution with a minimum of free parameters has previ- ously led to its neglect without justification. We propose, within the scheme of currentxcurrent weak interaction and octet dominance, to implement Okubo's dispersion approach by making further use of the Regge the- ory. Our method of evaluating the resonance contribution consists in assuming Regge behavior for the scattering am- plitude at high energies and extrapolating this form of amplitude to the lower energy region. In this way the re- sult of the higher-mass integration is just the real part of the Regge amplitude from the t—channel exchanges, slightly modified due to the once-subtracted form of the dispersion relation. This approach has its qualitative jus- tification in the concept of local duality22 which has been explored in the realm of high—energy phenomenology. Simply stated, local duality says that the Regge amplitude, when extrapolated to lower energies, reproduces the true amplitude in an average sense. The addition of the reso- nance contribution to the soft-pion-plus-pole amplitude in- creases the number of adjustable parameters to four. The numerical values of these parameters are found by fitting the experimentally determined amplitudes. As a test of our approach we also consider the two—body radiative decays of hyperons. These processes are Closely related to the nonleptonic decays which will be referred as nonradiative decays of hyperons when dealing with the radiative processes. Insufficient experimental in- formation on these radiative weak decays has prevented the selection of the best model among more than a half-dozen proposed since the first papers on the subject appeared more than a decade ago.23'24 All of these models predict decay rates and branching ratios that are in rough agree- ment with each other and with the available data. However, the first experimental determination25 of the asymmetry parameter a for £++p+y gave the unexpected result a=-l.03 :g:2§, much larger than the theoretical predictions. The soft-pion-soft-photon approach, considered by Ahmed,24 has previously led to a value for the asymmetry parameter con- sistent with the experimental value given above, but at the expense of an internal inconsistency in the calculation.26 When the inconsistency is removed, Ahmed's calculation also yields a negligible asymmetry parameter like the other models. It is interesting, therefore, to test our results for nonradiative decays against the above experimental num- ber, and also make predictions for the other radiative de- cays. II. HAMILTONIAN AND INVARIANT DECAY AMPLITUDES The starting point in most calculations of weak decay amplitudes is the specification of the effective Ha- miltonian responsible for the decay; the choice is not unique. Here we consider the most popular nonleptonic weak interaction Hamiltonian, the currentxcurrent type, usually written as :38 = .9. JIJ" , <1) NL E where C=l.OXl057M: is the universal Fermi constant. The factor l//2 appears for historical reasons, and JA is now the hadronic Cabibbo current8 assumed to transform like an octet of SU(3). This current is postulated to be made up of charged currents only (in contrast to charged plus neu- tral currents), both strangeness preserving and changing: x >- A 5*. 5'" K , A s) . 5) . J =(3’, ”fl-3", —1§z)cose+(3~:+13’5 - 3; -z§5 )sme . (2) The superscript 5 stands fer axial—vector current and the absence of it specifies the vector current; the subscripts denote the components of an octet; and 6 is the Cabibbo angle assumed to be the same for vector and axial—vector currents. Under the combined operation of charge conjuga- tion and parity (CP), JA goes to J: so that we obtain ex- plicit CP invariance if we Choose the weak Hamiltonian in the symmetric form 56:9. Ia NL + x t) + x .5ij +JAJ )EEGELJAJ 1s (3) instead of expression (1). Because JA and J: belong to an octet of currents in the SU(3) symmetric scheme, the sym- metrized Hamiltonian (3) can only belong to the completely symmetric representation in the direct product decomposi— tion of the two octets. Among the symmetric representations 1, 8S and 27, the first has only a strangeness preserving term. Therefore in the case of strangeness changing nonlep- tonic decays,.fi can only transform as 8S and 27. Further— NL more, we consider the octet dominance approximation. In general an octet may be defined by its commu- tation relations with the vector octet charge:27 3 O The set of operators 01(x) form an octet if = I, O “X . [Fk(x.),0{(x)]_ {Hm m( ) (5) Therefore for currents I: and fix to be octets we require x . , A [Fk(x.),f{(x)1= 1mm IMO) , (6a, 10 5x , 5) [kax I, m]_ = qum 3mm . (6b) To preserve the complete symmetry between vector and axial- vector currents we postulate6 in addition that x . sx [F:(xo), T,(x)I_ = lIktm Im (A) , (6c) 5). . >. NEW). 3r, I- = L Ium I,“ IX) - (6d) Hamiltonian (3) is a sum of products of octet vectors; it can be shown to be an octet tensor of second rank. To see more explicitly the tensor nature of the nonleptonic Ha— miltonian, it is convenient to define the following quan— tities: th =[3’m I?“ 3:: 37:312. ’ PV («5% 5 X (7) T SI? jt+gkxyt15 A A . . . In the product JIJ , With JA given by (2), the term c0526 15 TPC‘I’ T:C TPV- T?V ’II‘T‘X 3.: ‘- 312* 31)) + [(32) 3’25).— 3.5 $3) II 11 22 2A 1 'ikiflx’fisi IizII'iU’i Task-$2131“). (8) The pieces in parenthesis give no contribution under sym— metrization so that we are left with (C0519 Ierm OI [JXJAI I5): ZI +T::)- 2. IT:;’+T::) . 11 Similar results for the other terms are (smze Ierm o; LEWIS) = 2(Tjj +T§§)—2(T§’Z+T::) (sine cose term OI [JJVIJ:4(T::+T::)__4(T:’:+T21) . We can therefore write the nonleptonic Hamiltonian as PC ?V A = film—16m (9) ML where PC G pg PC, 2 PC. PC .2 PC PC . flNL‘EIHH I Tzz)cos 9+ITM+TSSIStn 9+ 2(T‘4+T25)smeco59] (10) and £K§Z is identical to its: except for the replacement TPC PV kl k1 tensors by proving that they satisfy the following rela- of the label PC by PV. Now, and T can be shown to be tions: 9(— '_ - PC , - pg [Fr(*°‘»Temm]-* rim Tnmxx) +1 IkmnT (x) , In PV tn (X) ) 5 PC. - PV . [Fk(*o).T{m(x)I--“ I IN“ Tmm + 1 “MT (11) PV . 9v . PV [Fk(x°)'Tlm(X)]‘= l Ihln TnmU) + I Ihmn Tin (x) ‘ 5 PV . pc . PC [F‘l(x°)’ TIm(X)]' - LIAIN TIM“0 + l I'hmu TIn (x) From the above tensor defining relations it follows that 5 PC PV 5 PV _’ PC [FK'TQm 3":[FIL'TIMI' , [Fk’Tlm]-- [Fk’Tlml‘ . (12) Note that 16$: and J{§Z are constructed from identical 12 combinations of Ti; and TEX respectively. From this fact and Eq.(12) we get IF: :iL = [@1821]- [Fifi‘m]; [Prim], (13) ) so that [FZ’file-z'IZFh’flNLL . (14) Equation (14) is a basic one for the successful application of current algebra techniques to weak interactions. We now proceed to consider the specific case of nonleptonic decays of hyperons. Let us simply write it for the weak Hamiltonian density responsible for the decay of a hyperon a into a baryon B and a pion n: a I one)» [mew n (q) . (15) Here p, p' and q' denote the four-momentum of the respec- tive particles, and a is the isospin index of the pion. We also denote the octet indices by a and B. The S—matrix for process (15) to first order in weak interaction is S 2 1-‘1Id4x fax) (16) and its matrix element between initial and final states (Ftp) Tia(q')\S-lloup)> = -t IdIX (PW) ‘lTa‘Iq’Ha‘aUIMPD . (17) 13 By writing %(x) in terms of the displacement (four-momen— tum) operator P, th -iP.x TKO) 7- C 356(0) 5 we obtain =—i(2W)4$‘((>’+q’—p)na(q'>t:fi((o)lo((p)> . (18) The most general form for the matrix element (18) consis- tent with Lorentz invariance can be written as a linear combination of the five basic types of interactions: sca- lar, pseudoscalar, vector, axial—vector, and tensor. Be- cause of the Dirac equation, the last three types reduce to either scalar or pseudoscalar. Hence the most general form for the matrix element of 1&0) is (Ftp’)1Ta(Q')I:r’{(o)lo((p)> = .L 1 1 “fi(pr3(A‘ .. _ (21191 (2 0(2)” N. “P P“ 153;)wp) (19) where Naa(Ea/Ma)l/2, Ea=(P2+M2 1/2, and A, B are invariant (1 functions of the Mandelstam variables: fir-(PI‘I‘Q’ylez , t = n)“ W“ = q“, <2o> u=(P*q’)2: P'z' . If all three particles are on the mass-shell we have s=M:, t=m:, u=M§ with the implication that the invariant amplitu- des A and B are constants. A complete theory of weak 14 interactions should allow a calculation of these amplitudes in terms of the masses involved. Note that A is the ampli- tude for the production of the pion in the 5 state (pv) and B is the one with the pion in the p state (pc). It will be convenient to define the amplitude ' I2 .32 , a , M=—t(2n) (Zqo) NNNP 3NdNPId‘x e (Emmi)(AFNTWatxwflohI«(I)» . (3) 12,14 The soft—pion technique consists in letting q' go off the mass shell in an apprOpriate manner as follows. Inte- gration by parts twice in (3) yields £qu M(q')=(Z“IBNdN (an e (mfi-q")) . (4) P The pion field operator may be replaced by the divergence 15 16 of the axial-vector current according to the PCAC relation 5A 3 ’3 (If (X): c (x) , C a o.qs m . (5) x a. (ha 132 fi Inserting ¢a(x) from (5) into (4) we obtain ' 3 Z qlz chX 5A M(q)=(zn) NNNP m1); 8cm 6 (meaxag (x)i( . (6, The next step is to consider the expression A . 3 4 iqu 5) T (q’IE-L(ZTI)NO(NP Id x e (FIP'MTIfa (x);;((o))\o<(p)) , (7) If we multiply (7) by q' and integrate by parts we get A a iqix 5* qu (q’)=(2“) )4de I d"x e, (PLP’IIENTHS (x)i((o))\o<(p)> , (8) Making use of the identity 5} 5% $0 a.“ gamma) = flaflaumwn + 50c.) I Ia (x) , mm], ,9, we can write (8) as [qflx x q;T*(q') —.- (unheard,3 I am e 8(xo) . (10) P The first integral is related to M through Eq.(6), and in the soft-pion limit the second integral can be written as {th so hm [dtxe (Fcp'mId(x).zf((o)l.lo)>6(xo)= . (11) qhac> Hence from relations (6), (10) and (11) we get 17 M(q’=o): Iim 19:! q'ATxW) - m: (211)3 Npr> , (12) ‘I*O C C It is customary to name the first term on the RHS the sur- face term, and the last the current algebra or equal~time— commutator (ETC) term. The extrapolated amplitude given by (12) expresses the content of the soft—pion theorem. In their work Suzuki and Sugawara12 neglected the possible contribution of the surface term and approximated the physical amplitude by M(q'=0) given by (12). They con- sidered only the s-wave amplitude A because in the limit of exact SU(3) symmetry it has been shown29 that . W (baryonHi Ibaryon) : O . (13) In fact, using (II.13), expression (12) without the surface term gives z a , ev VIEWED) = -— "2“ (1“) N,Np> is not necessarily zero. 14 and others,15 split the Brown and Sommerfield, amplitude M(q') in two parts, the Born (pole) term MB(q') and the nonBorn (remainder) MN(q'): 18 M(q’) = MSW) + M”(q’) . (14) Combining (12) and (14) in the soft-pion limit we have 2. MNMEOI .. \(m (1111. q'xTNq') — MEMO) 4-10 C __ flit (anNMNr3 <{A((>')IIF:,;£(I-I°“P>> - (15) c When the intermediate states contributing to the pole and to the surface terms are degenerate in mass with either the initial a or final 8 states, the limits on the RHS of (15) 30 if taken separately. However, the ambiguity are ambiguous disappears when both limits are considered together. We will come to this point again in the next section where the surface term will be treated in detail. For our present discussion it is sufficient to know that the ambiguities cancel each other, leaving a well-defined limit. Difficulties in evaluating the on—mass-shell re- mainder MN(q') led to the smoothness assumption expressed by the approximation N . M”(q’) 2'- M (W0) (16) i.e., the value of the remainier MN at a physical q' is ap- proximately given by the value of MN at the soft-pion point q'=0. This is a reasonable approximation in some cases, but it is not valid in general and should be checked for each particular case. The pragmatic attitude, when it comes to 19 applications of the soft-pion theorem, is to accept (16) as a working hypothesis, perform the necessary calculations and confront the results with the experiment. If the agree- ment is good one says that the smoothness assumption is valid for the particular case; if not, one tries to pin- point the trouble, in general without much success. Assuming (16) we can combine (14) with (15) to obtain 2 M(q') = MSW) + (m I In: q’xTVq’) - Mam] q’->O C _ 216$ (zn)3NdNP - (2) Insertion of the time-ordering operator, 5% 5* / T( Ia (031(0)) 2 9W) 3‘; (X) 3((0) + amt.) Me) if“) (3) in (2) gives two additive pieces for T‘: iqflx THM'IE'I‘ZHISNaNE Id” 6 90(0) ({flp’)! 3’:X(X) fl(0)Ia(P)> ) (4) 1%.-‘1Ty" N (fix < ')}{(}SX()I0(() T (q)=-1(2 N“ PI x8 em.) F n 21 22 allow us to use the displacement Operator (four-momentum) P to write g) [PX g; —IP.X ya (x) = 8 Ian) 6 and integrate by parts to obtain 1) 3 {01+P5BQ-X s) T ((1)4210 (‘1pr Z dix 9 1 SW.) (Push Ta(o)ln>(n|i((0)lah)> . (7) n I (I), " Po ‘ PHD Integration with respect to x yields T”(qI)-.-(zn)é’r\l..h’p£ sleeve) > . ‘ (a) n 1 , qa+PovPhO The summation in n involves sum over intermediate states as well as sum over spins and three-momentum for each in- termediate state. Thus, for a particular single particle intermediate state 6 we have Z Z Ids?“ ' (9) n 5 sem Ill Performing the 3-moment1m integration in (8) we obtain T”l 5:) WWWW'*“')W°“P>> 351'“ (10) P q; + qup') — E5())>'+q') Now the weak vertex can be written as (8(Fn)lj((o)lo((?)) :: .L .1. mama“- XS VsflwP) . (11) (2.11)5 N... ”P The matrix element of the axial-vector current ISA between 23 baryon states can be obtained as follows. The PCAC relation 5A :2 X 913; (x) CCI>a( ) (12) combined with the displacement operator gives “PZPVJ'X 5). axe (fags) 11(0)) 3(a)) = e so that t(p'- Fn)x(p(p’)I3:x(O)\$(f>n)> = upweleaw)! Mp9) . (13) Similarly the wave equation (C35: mi) 43am = Ian) (14) yields Z ' 2 ' C I (m“*(P-Fn) )=n)) . (15) The strong vertex can be written as < ((2')) (own .)> = .1. .1... m )1) u( ,,) K" (16) F (Ia I) (21”) N‘5 N8 I) S I) p5 where K22. s: zgnw (“Fa+("°‘IDa)/es . (17) Here a is the experimentally known F,D mixing parameter and Pa --if a: 86’ D86 d a86’ 86(Appendix B). From (13), (15) and (16) we get a (' ) <')15?)\5())—- ‘ l 0K; P‘I’nx)—E3(p+q) (21) where we have simply written Kc for Ka escsa’ etc. Summation over spin states Z Megan) :2: Id" + M8 5P“ 2M3 25 yields (T‘im : .9... ‘ WP'M’UPn M.) (KC — xsKV) we) qx . (22) mi; My“. 2 E$(P’+q(’>[q’.+ EPW’) ' E5‘P’+q\’>1 Following step by step the preceding calculation we also get q'xTZIq'I :__ g l #_ ‘&(p’)(CK—7{5VK)(Pm+M,) Wis w?) L I ’ (23) m“ MP+M, 2 E,(fiiv‘ll)Iqot E,(Y*q)’)'5.(t)] If M69€MB and My#Ma, the soft—pion limit of both terms giTlA and qiT2A vanishes. There is an ambiguity, however, when at least one of the equalities M5=MB or My=Ma holds. In this case expressions (22) and (23) can be simplified to yield L - / TIq’XTNq') =u(p’) H i... - _L))’5(KC-Xs Kv) C ZPI'CI’ 2MP '(CI<—-I VI<) .ft. 1 X- u( 5 Izmfl 2M“) 5) P) (24) It turns out that there is also an ambiguity of the same nature in the Born term which cancels the ambiguity in the surface term. In fact, Feynman rules applied to the dia- grams of Figure 1 give MB(q')=_fi(p’)I15 I (Kc-Xskv)+(cK—¥5vK) 1 Hum) (25) P’+fl'—-M3 P-fiC-M, 26 / (P) (X?) ///'II(CI’> {3 // . . /;r(q') $(?+q ) 7(P_q0 ’/// /I/ “I: aip/ Figure 1. Born diagrams for the s and u channels. which, for M =M and MYzMa, reduces to 6 8 MB(q')=u()>')I Lime—(5m—(cs-(svm W ,1; INF) ‘ (26) ZEE’ 29g From (24) and (26) we see that the surface—Born term (1) is well-defined and of the order AM/ZM relative to the on- mass—shell Born term. Note that for all practical purposes we may take M5761“!B and MY#Ma throughout the calculation and only at the end set M6=MB and/or My=Ma at the appropriate places. The final expression for the Born term is the same as that one would get by taking care of the ambiguities. In what follows we write the nonBorn soft—pion amplitude as simply given by the ETC part: MN(q’=O) z 72“ (2103100, NP=(p’—q) . Z_ 7. 2. s+t+u= Md+MP+mfi+qL. It is convenient to take s,t,q2 as independent variables. Also, using the Dirac equation, we can express amplitude (3) in a more suitable form for taking the limit q+0. If we multiply (3) by M and add to it the same amplitude (3) B multiplied by Ma, we obtain WNW =fil §’54£ SLS the integral in (15) can be written as +00 , +00 ’ ds'JmHim ._—. ”V lmHEw) + des'hmm - (16) -oo (S’—s.)($'*5'l€) 5‘5" —oo (5,"5°)($"S) Now, the assumption of local duality implies, 34 +03 , +a) R P ds’ 1’” Hi“) 2 P ds' 1m HHS) (17) -00 (5"5o)(5'-5) -0) (5’-so)(s'—s) where the Regge amplitude H§(s) has the form -{noq R I O" Him 2 X, i. e S‘ . (18) smfloq Here the minus sign refers to K* and K trajectories, and A the plus sign to that of K. From this form for the Regge amplitude we see that Has; “MTV-5) ior Ka‘ and KA ; (l9) Hfls)’ +H?(-S> #0“ K - H These identities imply that the imaginary part of HE is even for K* and KA’ whereas it is odd for K: lm Hike) = 1m HEW—S) iror K’, KA , (20) m Him =-1mH‘?<-S> W K Hence we can write +00 00 R I .R I w R ’ ds’ 1‘“ Hi“) .—.-. ‘5 ds’ 1'” H“) + XdS'JmHMSz (21) (s’-so)(5'—5) o ($'+ So)($'+$) o (s’—$.)(s’-s) Inserting the explicit Regge form (18) into (21) we get 35 +0) R I O( 0(- , Ian-b) *_ t P d5 " = TF1,- {anflim 5_________5° Lor K‘,KA . -m (5"$°)(S'—S) 5’50 a (22) “i t. :z-flxl COtgdL 34.59. 103‘ K S‘So With these values for the integral, expression (16) becomes HI) I I ’3 - x - ds' 1m HAS :: t“ 1m Him + M, (5“- sf‘vfianfltxv , 1 -00 (s’—so)(s’-saie) 5-50 s-—so 7- g +00 I (23) ds' 1mm) _ ... A...“ was) + m (s‘xm 5?“)fanflotA L 5—50 2 -oo (S’-s.)(s>s-ie) 5—6. 1382.. (5%» $7")cot1Lo

+'i1(p')u(p) , (4) fl

Y5U(P') ——> - ilqa') um) Therefore, performing the interchange a(p)HB(P') in (3) + _. , * * Ma‘s: u(p)(H‘—-Y5HZ)U(F) - (5) Subtraction of (S) from (2) yields MP0; mi? 2 21‘ my) (1m H, — 151m Hz)u(p) . (6) with this combination of amplitude M in mind we work on the field theoretical form for the same amplitude: 3 {qfix MFO‘: (2W) NNNP Sd‘x e (02+m:)(F(P')‘T(¢a(X) X(0)H0((P)> . (7) An alternate form32 of (7) more convenient for our purpose is a «fix 2 L , Mpdzeu) NNNPX ch 9, (U +mfi)<)]-WP». <9) Subtraction of (9) from (8) gives ‘ I + 1 .X MPd—M“ =<2MBN°lv.>>m 34"“ at?» . (11) Using the 4-momentum operator to translate ja(x) to ja(0) we obtain Mpg—M: =<2“)’N«NPgun)“34(q’+p'-'1>.)\ (mm r9):.1... __L_. Eu M)! u( )Ka , PP 3a P (2103 NPNS P S F" ’53 (3(pnlli6(o\\a(p)>= .1. ‘ fi(n)(c —x v )w ). (2“)5 NSN‘X F 80! 5 8.x P Insertion of these matrix elements in (12) and summation over spin states of the intermediate states give Jr - , , _. , ‘ Mpd’wlzs‘ t Zn_(2n)g4(p+q-pn)utp)xs $213!». (KC. 1:, “WW” 3 -i2%(zmz‘(q’+Pn-P)mp’HCK—XSVKlm15w?) . (13) 25, The identity 93K ) = 8d4Pn$(P:‘M")( ) 25,, can be used to perform the integration implicit in the sum- mation sign. Thus we get + ' I I 2' '- I / MP“M°‘(3 = 12.11 3((p+q)"—M3)u(p)15(p+¢j+Ms)(Kc—XSKv)uX.u

, M252aWWs(?’*f4’+MS)Y5u(p). (16) Mmzfi<1>’>WW+M,mu

. MzuamP’Hszq’JrMfiXsmp). At this stage of the calculation it is important not to re- place p'+q' by p or p-q' by p' as we would in case of decay for which p=p'+q'. Dropping of some terms because of q+0 should be made carefully. Dirac equation is applied to rewrite (16) in a more convenient form as - I I I ,2'_ — I~4b(t~4m-Mp).—.u(p)1g(séWWIM4f5 MaMP+M3M« MsMPWP) , MW; MP) .—. iltp') ( M5 + M’ — M: + M.M + M, M; M, M, WP) , , (17> MZS(M“+MP)=u(p’)k—H'P qmu MP'MO‘MFMSMgMbMPMIP) , MZJMJMQ‘EWM W + m4" M: - MdMP+M,Md + M, MP)u(p). We can repeat these manipulations on the M's by replacing p'+q' by p+q and p—q' by p'-q in the expression (16). We get the same value for the M's because of the energy-momen— tum conservation. Thus I'h yup-'7‘ ' 42 Mum“- Mpxsmp'wsmr . W1" M: - M,,MP+M8M“— Mst)u(p) , +Mde- MxMP)X5u(P) , Mde—M 3:. quw M —M;'+M0(M P Mzs(Md+Mfiy=a(P’)(flfiPflgtq”Moi-'MdM +MSMN+M5MF)U(P) ’ P (18) M2u(Md+MF)=fi(P’)M? +4514-M;‘M.MP+MIM.+M.M,)M(P> - In section V we have used the identity MW) MAW)? = s—u + [mgr], At this point we drop the commutator in the q's because its contribution becomes irrelevant in the limit q+0. Also s—u= 25—Mi—Mg at the point of interest, t=O. Hence the above identity reduces to I I Z 2. ?’(s4+$4)+(54*¢i)P=25’M«‘M/5 . (19) Addition of (17) and (18), and the use of (19) yields M'$(Md—MP) = my) 15(5- MdMP+M8M“- MSMF) (up) , mama-MP) = my) (5 — Mj— M;+ MuMP+ M,M~*M,MP)15U(9) , MZS(MN+MP)=a(p’)(—S‘M«M +M5Ma¢+ MSMP)u(F) (20) MquJMF) = mp'Ms— Mil-MP ~M,‘ MP+M,M“ +M,MF) cup). The delta functions in (15) allow further simplification of the expressions (20) to give 43 M‘s: fi(P') ‘5 (M8+M«)(M3'Mfi) “'(P) ) Mq—Mp Mm... w) Vs (MWMLHMM’M') utp) , Md" MP __ I (21) M25 311(4),) (Md-M5)\MS"M{3) (HP) Md+MP M2m = EMF) (Mu-MIHMI*MP) utp) Md+MP " With these values for the M's, expression (15) becomes + . - Mr”- Mupzwflkcmdmflm‘ MP) 3(s-M2) , CK Mx‘MP x (Ma-MAM” Mr) SUI-M1,) ] x5 +Z1Ti [-— Kv (MM'MNMS'MQ Mq‘MP Mx+MP x “SI-Mi) + VK (MVMQMVMP) Mal—MW] . (22) Md+MF Comparing (6) and (22) we obtain 1m HT: , 1T KV (MMJMSEMVMP) 5(s—M:) +flvKiMd’M’XM'iL') Sax-MD . Md+MP Md+MP (23) Im HZ: — 11 Kc, (MN MUWS- MP) 3(s—Mf) +1ICK(M«‘M1)(M:+ Mp) aux-Mi), Md” MP MOI-MP Inserting these values for the ImHi in the dispersion in- tegral (l) we finally obtain the desired pole terms: ReH‘RMZ): M“- MP KV _. Mu’ M9 VK‘ , (Mfl'MSHMgi’MF) (Md+Mx)(M7+Mf) (24) ReH:(M:\ ._. WM: l K... mm 4K . (M,M,3(M,+MPS (M,+M,)(M.—M,) VII. SYMMETRY BREAKING IN THE TADPOLE MODEL The weak vertex has been written as (flp’flfllsmws IM’P’HCf38 ‘Xs V’s MM?) (1) m? N; N The form factor V86 is zero in the limit of exact SU(3) symmetry because it has been shown that the matrix element of the pv Hamiltonian between baryon states belonging to a same octet vanishes if exact SU(3) symmetry is assumed. In a broken symmetry scheme such as the present one, V86 does not necessarily vanish. In fact, a tadpole 33 model has been proposed to evaluate VBG' In this model the matrix element < momma?» __ .2... ______. utp) I umv F (2“)3 N, N F (2) is given by the tadpole Feynman diagram, Figure 3. Here the meson K: is a spin zero uncharged particle with CP=+1 which can communicate with the vacuum in the presence of the weak interaction. The tadpole diagram yields < (:5) “12(9): 2‘1?) N_u(y>i1 KW um “Ki-'0) F \fl P NPNS F3 P 2"“) Pug) (3) 44 45 W5 ————’-‘-7—-——O< It” 2m.) Figure 3. Tadpole diagram where A(K$+O) is the transition amplitude of the K: into } the vacuum. Comparing (2) and (3) at zero momentum transfer we obtain . :(K‘I) PAN-*0) s S P {5 m: Since we are working with the cartesian octet states34 we 0 need the upper index in Kégl) in this system. In our phase convention IK1)=j-L;UK°)—IR°))= [EEIP64P7)-IL;(P6+1'B,)] av so that IK§>=P7, where P1,...,P8 are the cartesian pseudo- scalar meson octet states. The strong coupling constant in (4) is, therefore, given by 7 _ 7 _ 7 K195 "' 2 91“»:wa “1 MD )FS ' (5) 46 We rewrite (4) as . 7 '7 was s 2t6(0< Fp8+(l—°<)DPS) (6) where G = ‘3 AMI—>0) ° (7) “NM ‘nqi Soft-pion techniques can be applied to eyaluate I the amplitude A(Ki+0) in terms of the amplitude A(Ki+2w°) for the decay Ki+2flo. This latter amplitude can be defined more precisely according to g o o , PV 0 1 1 o o <fl(q)1f(<])lfl \KLUI» = ___q/ __ , AKKp-leT) . (8) (2102 (2 qozqukaYz On the other hand the LSZ reduction formalism allow us to write PV 1 Lq'x 4mqm(q')l:KlK(m> -.-.- __________ 54x e w . (uz+m;))|K(k>>. (10) mama/1 (2%)”- From (8) and (10) we get 47 I 1 I 2. , iQ-X A(K‘:—~2W°) = i(zn)5(zq°)/"(zko{‘fl1:j: d4), 6 C s) w x . (12) Multiplying (12) by q; and integrating by parts we get ‘/ {ql-X X W q'xTx =(zn)3(zqozk.)zxd4x e (mmlafiwixwflofi1K0“) - (13) The identity be K §:X(X)3€(0)) == TWA yfiflflofi +80%) [ 3::0(x),fl%)]_ (14) allow us to rewrite (13) as , X 3 1/2. {q’.X SA pv (M =42“ “(1.2% [am e (mqnflbfls (we (o))lK(k)> l . X I (q , +\m3k2qozh.3/‘8d4x e mum);LIf°,;t€'fo>]_IK . (15, Taking the limit q'+0 we get . 4 mam") = - l 39$ (2n)3(zqozko)/’- (mqn (F: , 3W]- Wm) (16, c where we have dropped the surface term. Toget the desired amplitude A(Ki+0) we have to 48 contract the remaining pion. Let , 5 rv if 2[F,,1{ ]. (17) By repeating the previous procedure in obtaining (16) to the matrix element of X' we are led to I [m2 ‘ 5 I 0on Wk» = W _..._3/ _.‘——1 (OILF3 ,fl LIKUU) - (18) C (2101 (Um/l Inserting (18) in (16) we obtain A(K°,—'2“°) = 4 3L 1 m: (210/ (ZBOYZ (0‘ “3:, [ F:,j{w],],|K(h)> . (19) C ‘ IV! ~:.4——a Application of the property (11.13) of the weak Hamiltonian yields [F:,LF:,:I{”1-)- = an, .fl”)- - WWW, (20) The matrix element of this commutator is (o\[F:,[F:,}€"1-HK(h)> = <21) But REM) s. .L |K°‘> 4 so that W <0\[F:,[Ff.?{"l1-|K‘I> == £031?“ 1K?) . (22) Frchthe definition for the transition amplitude A(K$+O), 49 m>mmv>> = .1— _L... __l._ emp’)(A’+XSB’)As.¢ IMP) (3) W V (2.0)2 (2m?— NdNP on general grounds of Lorentz invariance.4o The method used here to compute A' and 8' re- quires the consideration of the companion three-body radi- ative weak decay cup) -——v IMF) + 11°(q') + mu (4) for which we define the transition amplitude R“ as I I ' F (“pmqwtmmlatpw = .9... __L_' ._L_.‘/ ___L_. erR (5, 12m" (am/L (290‘ NJ, *where a“ is the polarization 4-vector of the photon. An (mitline of the procedure helps to introduce the basic ideas euui to guide the derivation of the final expressions for the invariant amplitudes. The soft-pion theorem applied to the three—body 65 radunjve weak decay (4) yields the familiar formula 2 x Z ' E.RN(q'=0) .-= lim [If]: 6!. q; SP (q') .- e.RB(q')} + ’2' (20)q/z(2h.)/2‘ C q’-0 x Mm!A (lawman [F5,:}()-\o((p>> . (6) In section X we show that the surface-Born term in (6) is zero in the soft-pion limit as long as we use the deriva- H I tive coupling for the strong §Bn vertex. Then the nonBorn soft-pion amplitude on the LHS of (6) is proportional to the ETC term which is essentially the amplitude for the de- 1 sired two—body radiative weak decay. ; In section XI the amplitude RN for a+B+w°+y is related to MN, the amplitude for the nonradiative weak de- cay a+B+n°, in the soft—pion limit via Low's soft-photon theorem. 1 Since the parameters for the amplitude MN have already been determined, through this chain of low-energy theorems we are able to express the invariant amplitudes, A' and B', in terms of known parameters and predict the values of several experimentally accessible quantities such as decay rates, branching ratios, and asymmetry parameters. As usual the starting point in deriving the soft- pdxni theorem (6) is the LSZ reduction formula for the ma- trix element of the Hamiltonian: 94x . l ')h‘(3‘3 (x) #{(o))lo<(p)> (1) without the photon in the final state because in what fol— lows the line from which the photon is emitted can be the final leg as well as the initial one, and also from inter— mediate states. To avoid carrying along the nonessential normali- zation factors in (l) we introduce the integrals I A [qx 5X 1 a (due eun> , iq€x (2) A 4. 5X J .=. [d x e e(—X.)\°“P3> . (4) fl Depending on the position of the photon line, the integral (4) can be split in two parts corresponding to the first two diagrams and the last two in Figure 6: Li) (Z) (3) (4') Figure 6. s-channel one—baryon intermediate state in radi- ative decays. 71 e{(q+P ',,-*P) X 1",=d§\m9000318))31:"\s ( h w)x (5) + + — 1:, ér-dee 9.qu em).)l2(lo«p>>. Integration by parts with respect to xo transforms 6(xo) into a delta function, 6(xo). Then the integration in x can f1 1. J a- “ '11:. be readily performed to give a delta function in 3—momenta. Integration with respect to p implicit in the summation % can now be carried out to yield {I ‘—‘ 1210 ZWm»3”)s1)+9)><3(p+q))ma 1‘2 ) st q; + EFW) -— E S’M) ( s , , 6) 1" = 11211)"): <41”) 3‘. tWWI+k1>lduaq>>> 34 SP)“ 9', + EPW) +19, - E$(P'+q’+k) A In the soft-pion limit, q'+0, the denominator of I34 does not vanish so that this term makes no contribution to the surface-Born term. Hence we have to deal with 122 only. The strong BB" vertex can be written as ()5:\8(.3 _——-.__‘.__"""(xx ,, <(3M P>s1szFN up) SU(PHK,;$) (7) with z a a. In; (K) = K , C A p: p8 (8) whereas the weak-electromagnetic vertex take the form 72 ($(pn)\j{|o<(p)) = ._l_q _i_‘ _L__. 3(P.)[(cvisv) __..._.‘___ (mu/31213.14 N, N, 43—h.— M... 313 + ““ M¢>+1¢+ “‘ M) ——-1——-1w3> <9) 2M, 2M, {WK'Ms corresponding to (l) and (2) in Figure 6. By inserting (7) a and (9) into (6a) and summing over spins we get X in) 'Y( n+M ,St‘ :. V ‘4 s P" s) [(KAC‘YsKAV)—L—— e . r" " I 256114)lq.’+E,.-E.tr’+ca’>l 45-1444. -3 “33 K . £8.13 1 - x1¢+ 2M, 4) ($43 2M; ”WSW“ (SKAvaP) (10) where we have written 8“" for the part of 8“" corresponding I to I" in (3). The other part J" of 8“" has associated with it the graphs of the Figure 7. Of these graphs, the last two do not contribute to the surface term in the soft-pion li- mit just as corresponding graphs did not to I". Repeating the procedure followed in the calculation of I" we obtain er q; 8:" == - WV) [(¢ + i 11¢) ______L__. (c KA- xsv KA)+(<;KA- xvaA) ‘ Z MP 19w. - MP 1 (4 + P! 4143)] {fn+M,)fi’Y5 (HP) 14. -K - M, 2 M, 2 E,(p-q’)[q;+ E,(p—q’)-Ed(|>)] (ll) X 73 -. ._._._ Figure 7. u-channel one—baryon intermediate state in radi- ative decays. We can immediately see from (10) and (ll) that the surface term vanishes in the soft-pion limit when M6#Mé and MY#MG. However, if at least one of these mass inequali- ties does not hold, an ambiguity arises which can be can- celed by a corresponding ambiguity in the Born term. The net result is a well-defined limit whose value depends on the type of interaction (derivative or nonderivative) adopted for the strong vertex. Consider M6=MB and MY=Ma, in which case expres- sions (10) and (11) can be put in a more convenient form. In fact, note that in the limit q'+0 we can make the fol- 74 lowing replacements: P“ L P, ’ Fn L4, P 3 7— Esw’wu') [613+ Em —E.'+qr>1 .4 2w , ZEN-qr)ng+E,tr-qr>-Ed

]-—+ 2m , so that in the limit q'+0, (10) and (11) reduce to X __ I . EPq/xS: g IMF) 94 ISL/pip?) [(KAC “ i5 KAV)“"'“-"“" (¢ +13" K¢) ZP‘C' Pck-M,‘ 2’ at +(¢+ P“ JD/~¢)—-—-—1--—-—-(KC-'1(K\/)]u(), (12a) 2M“ #H'K—MP A 5 A P I F) — “P i equ z—UW)[(¢+_———Ké)__.___(cK -7 vK) t‘ J 2MP P’Ufi-MP A s A + (c Ky. 35v KA) ‘ (¢ . £114” 9:" MA“: mp) . (12b) P‘wK' Md 2M4 2 P-Q’ Let us now turn our attention to the Born term. There are altogether six Feynman diagrams of which four present singularities of the nature discussed in the case of the surface term. These singular diagrams are almost identical to those contributing to the surface term, the difference being the appearance of off-mass-shell propa— gators in the Feynman diagrams instead of on-mass-shell intermediate states. The four diagrams of interest are given in Figure 8. The amplitudes corresponding to these diagrams are __ _w__.— jwt‘4.|‘u"'nl , L} 75 P' h , P p’+h / thh/ Vim, ‘ X, \P a] $ P"qHY /’q Y //Cl' P'+q ’+b. / Pal, I, a P a Figure 8. Born diagrams for the radiative decays. €~Ri= my) 94’15 ____1__[(KC_XSKV)———i——(¢+P—5— 44¢) P'+i4'-Ms PvK-Ma 2” Ké) ‘ (Kc —- XSKVH 1MP) . (13a) + (¢ + 2M5 P’*K+fl”Mg e. R‘: -‘a(y)[(¢+ 29:4 M) Fm ikcx— 15vK)+(cK XvK) P F x— a “'14 ‘fi van. (13> p—g-K—M (¢+ 2M Y H] A" 54% W F b In the soft-pion limit and for M6=M8 and My=Ma, the above 76 amplitudes can be put in the following form: B e.R1="”"5"’*MP)[). (7) From (1), (7), and (IX.3) we immediately obtain the expres- sion for the invariant amplitudes, 81 (8) Ey:= ___l___.(’_tfl_ + fk")v . Mq+ M!3 2MP Expression (8) is good for the radiative weak decay of a charged particle. For the decay of neutral hyperons, how- ever, (8) is not complete because diagrams in which the purely electromagnetic transition 2°»A can occur also con— tributes. These diagrams are of the general form shown in Figure 11(a) with the electromagnetic vertex of the Figure 11(b). (a) (b) Figure 11. (a) Diagrams for the radiative decay of a neu- tral hyperon. (b) Electromagnetic 20A vertex. In case of neutral hyperons, therefore, in addition to (8) there are terms arising from interactions of the form 82 6‘? i—tfi—Ofivh,+ (9) 2M1 where uTEu(XOA) is the transition anomalous magnetic mo- ment, and the average MTE(MZO+MA)/2 has been taken as the transition mass. If all four types of diagrams occur in a radiative weak decay the amplitudes will be given by ’d=_.l__(_fl£.-_ifz_)c +4 FTC,___L___rf_: P MVMP 2M{3 2M“ P“ Mx‘Ms 2M, 3“ MVMP 2M1 F” f‘ P‘ (10) Bid: I J— "|' Pat V + I T V + 1 HT V P Md+MP 2MP 2 Md) Pd Mu+ M5 2M1 5“ M1+MP ZMT Pk Once the amplitudes are known the decay rate __L 3 fl d’p’ 9:5. (21x34 S‘(p’+'a—p) (2 my) (A’ + X;B’)J5.¢ MM2 (11) E Z P 0 can be evaluated (Appendix C) to yield the result 2 z 3 1“: g: (M) (IM‘HB’H . (12) an M, Measurement of the decay rate alone does not always discriminate among several models, in the same sense that cross-section data sometimes accommodate conflicting models to describe the same scattering process. In such cases polarization measurements are needed to test the mod- els. In a decay process the additional measurable quantity 4? .A V ‘wbah. d - ‘ll- ’ 1 83 is the asymmetry parameter defined as 1* , a: 2R8 Z°+Y Figure 12. Diagrams 87 '2‘ \— Z" / I, / 11'0 I A C" I—I A n- C; for radiative weak ['1 decay modes. \ \ [G x “O 88 For the numerical computation the experimentally determined (total) magnetic moments of the proton and of the neutron43 are used: “Meta”: 2.793 (AN ) (l) Hn(t0{a‘) =“.9‘3 H“ , where is the nuclear magneton. The total magnetic mo— u N ments of the other baryons are taken to be the SU(3) val- ues44 with appropriate "mass corrections":45 Pd = PM SU(3) value) x .13.?— . (2) 4 These "mass-corrected" values come closer to the existing experimental values for DA and p£+. The numerical results are displayed in Table 6. The only experimental numbers available at present24 are those for the decay Z++p+yt +_, —3 0.52 T‘U'. P”) =(2..6_to.3)x10 , a=.-1.03:’O.42 - (3) r‘(z* -—9 P+fl‘) I The theoretical branching ratio and asymmetry parameter are 2.82x10'3 and —0.78 respectively. The agreement is excel— lent, although further reduction of the experimental errors is highly desired. Note that the relatively large violation of unitary symmetry found in nonradiative decays gives rise to the relatively large magnitude of B' which is the symme- try breaking amplitude for the radiative decay. Therefore 89 the measured asymmetry parameter of about -1.0 strongly supports breaking of unitary symmetry in both types of de- cays, which is a result quite different from that of pre— vious models. Table 6. Theoretical radiative weak decay amplitudes of hyperons, decay rates, and asymmetry parameters. Decay Amplitude Decay rate Asymmetry (105 m;%sec-%GeV-l) (107 sec-1) parameter A' B' r a A+n+y —0.147 0.927 0.795 -0.309 Z++p+y 0.887 -0.429 2.328 -0.783 Z°+n+y -1.538 -0.304 6.009 +0.380 EO+A+Y 0.483 -0.512 0.653 -0.998 EO+ZO+Y 1.574 0.226 0.846 +0.281 E-+Z—+y 0.0015 -0.062 0.0013 -0.049 In concluding this last section we briefly men— tion what remains to be done. We have seen in section VIII that our overall fit to the experimental data is very good, with the exception of a discrepancy in the s-wave A(2:). We argued that this discrepancy can be removed without dis- turbing the good fit by incorporating the contribution of 90 the 27 representation. This addition, however, implies an undesirable increase in the number of free parameters, un- less a way is found to determine some of these parameters independently of the experimental decay amplitudes. For in- stance, the parameters Yv and y' are essentially residue functions evaluated at t=0 so that they might be determined from the high energy total cross-section data. Recall that the total cross-section is related to the imaginary part of the forward scattering amplitude through the optical theorem. LIST OF REFERENCES LI ST OF REFERENCES l. T.D. Lee and C.N. Yang, Phys. Rev. 104, 254(1956). 2. E. Fermi, Z. Physik 88, 161(1934); E.C.G. Sudarshan and R.E. Marshak, Prod. Padua-Venice Conf. on Mesons and Newly Discovered Particles, 1957; Phys. Rev. 199, 1860(1958); J.J. Sakurai, Nuovo Cimento l, 647(1958). 3. R.P. Feynman and M. Gell-Mann, Phys. Rev. 109, 193 (1958). 4. See Ref.3; S. Gerstein and Y. Zeldovich, Soviet Phys. JETP(English Transl.) 2, 576(1956). 5. Y. Nambu, Phys. Rev. Lett. 5, 380(1960); M. Gell-Mann and M. Levy, Nuovo Cimento 16, 705(1960). 6. M. GellsMann, Phys. Rev. 125, 1667(1962); Physics 1, 63(1964). 7. W.I. Weisberger, Phys. Rev. Lett. 11, 1047(1965); S.L. 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The new data, however, do not differ significantly from the old ones, so do not change appreciably the values of our searched parame- ters. 40. R.E. Behrends, Phys. Rev. 111, 1691(1958). 41. F.E. Low, Phys. Rev. 110, 974(1958); S.L. Adler and Y. Dothan, Phys. Rev. 151, 1267(1966). 42. Y. Hara, Phys. Rev. Lett. 11, 378(1964). 43. A.H. Rosenfeld et al., Rev. Mod. Phys. 11(Supplement), 94 Apri1(1970). 44. S. Coleman and S.L. Glashow, Phys. Rev. Lett. 6, 423 (1961).? ' 45. M.A.B. Beg and A. Pais, Phys. Rev. 137, B1514(1965). APPENDICES APPENDIX A. Notation and Conventions We describe here the notation and conventions used in the text. The metric and gamma matrices are those defined by Bjorken and Drell. The coordinates t,x,y,z are denoted by the con- travariant four-vector x“, XH=(XO,X',XL,X3)=(t)x))’12):(t1x)o (1) The covariant four—vector xu is XH=(X0,X,,XL,X3)=(t,—X,-Y,'Z)=grvxu (2) with the metric tensor guv=g11v given by 1 ' <3 9‘ ° (3) The four-momentum of a particle of mass m is de- fined by V P“=(E.F.»P1»P.)=UP‘+"“)"P) ‘4) so that (summing over repeated indices) PsFrFr‘r-m o (5) 95 96 The scalar product of two four-momenta is ' = F ___._ _ q = E E “51C! . Pq F qr Foqo “3 P ‘74 (6) Four-vector will always be set in lightface, three-vectors in double line. For the four-dimensional gradient we use the ab- breviations 3 s 2. 3r25.2__ 3 3x" Dxr (7) For example, the four-divergence of Bu=(B°,B.) is BVBV‘ =3 £0 + i c 21: Ex The four-dimensional Laplacian, denoted by [32, is defined by clearanzze-Zi-liezim". <8) axrax, 7t‘ 2x‘ W Let P11 denote the four-momentum operator. If F(x) is any field operator. its commutator with Pu is [PKFm] = Aamx) . (9) This gives the translation formula (Rx -LP.x F(x) = C FUD) 6 - (10) The gamma matrices in the Dirac equation satisfy the anticommutation relations [7",1’L e mwvxr = zgr‘“ . (11, 97 Useful combinations are or“ e izxwhmr‘) = 11.1821’1 , (12) 15 a 1' 7°X‘1‘X’ 21“ » ‘13) *5 5 “Pr 1.. 1°90 + tapas 1°?°—%S.fi> 1 (14) 76 _=. “Mr .5 yr; ._. x°a_ + 7‘9, .722. .5’1115) a 2X” at 3x DY 72 Let 1 and o (a=l,2,3) denote the unit and the Pauli spin matrices: 1 O O 1 O “L. 1 O 1 l O 1 O 0 -\ We adopt the representation a. 7;a._ (00) 601(105 X‘-(O') . ‘ a 1 I ‘ I (17) *0 O o -1 10 The positive-energy spinor u(p,s) and its adjoint u(p,s)su+(p,s)y° normalized H(p,s)u(p,s)=1, satisfy Dirac equation (:5 -M)uu(P,S)= 32‘ M . (19) 5 2M APPENDIX B. SU(3) Invariant Vertices SU(3) invariant interactions among various octets can be constructed with the help of the F and Dk matrices k defined by (Wm = 41...... , (Duh... = dam <1) where the value of the structure constants are displayed below: klm fklm klm dklm 123 1 118 1/J3 147 1/2 146 1/2 156 -1/2 157 1/2 246 1/2 228 1/73 257 1/2 247 -1/2 345 1/2 256 1/2 367 -1/2 338 1/J§ 458 V572 344 1/2 678 «J372 355 1/2 366 -1/2 377 -1/2 448 -1/(273) 558 -1/(2J§) 668 -1/(2J§) 778 -1/(2J3) 888 -1/(2J§) 98 —v‘I 1 in"; um 99 fklm 1s antisymmetric and d change of any two indices. klm is symmetric under inter- The SU(3) invariant Yukawa coupling, for example, is written fl :: quNN ‘6‘ L 15 [a (Fh)(m+ ("°‘)-— E( (+1 ZMO> lZ°> = 55W) (3) - = 1(5‘-'B,_)|o) 12> E l M) = 3810) “ a°> ‘;->=-.\1.(B4—-1BS)IO) Similar relations hold for the pseudoscalar meson states 1(5/870 «.2: e. l 7 > and the operators Pk' Using the structure constants given . a above we can determine K _ a a BézzgnNN[a(F )86+(l—a)(D ’86] for any particular vertex. APPENDIX c. Decay Rate The decay rate of the radiative weak decay, a(p)+8(p')+v(k), is given by T“: .L .L. _M_P- d' ’ .815. (210" SN ’+h- )2 \MV' (1) 2' (217W EF P ZR. P P spin where M=11(p')(A’+x.8’)K¢u(p> . (2) The Hermitian conjugate of M is M’{ = mp) M (A‘— 1.8" ) NP) (3) so that ZIM12=Tr\wL(AI+‘/SB')K¢ 8+ M4 ¢K(AI*-rsa’*)1 . (4) Wm 2MP 2MP Inserting the identity K¢(F+Md)=(V+Md)k¢—2(R.P)¢ (5) in (4) we obtain two pieces for the trace: one containing the factor kilk and the other the factor ilk. Because 100 101 ¢¢=€Z=4 and Kfiskzr-O the first piece is zero. Thus the trace (4) reduces to Z IMP: 2“” Tr[(P+Mp)(A’+x.8')MA’*—x.e”>] SW" 4 M“ MP . 3112.312? «(Ammo . (a) Mmr. Now, the energyémomentum conservation p=p'+k gives Z. 2 kp’: Mai—W3 There is also a factor 2 coming from the two states of po- larization of the photon. Then expression (6) becomes 2. .3. Z M)": M UA’I‘HB’I‘) . (7) 5pn NLJWP Substituting (7) in the decay rate (1) we get 1. Z T‘: i 3— (Mi‘MP) (IMZHB’F') d’P’ d’k 4 z--——— .._. .. S( 4&- . (8) 2(211) 4 1 EF zh. F P) Integration in k yields 1 (MS-Méyummwug d’?’ 8888.42,) . (9) _‘ P= i 2' (21172. Md El, 2k0 Write d3p'=pzdpd9 where 0=|P'|. Also let HP) 5 PQHZVP. = (92+ Mg)” + {3 ”M4 . (10) 102 If 00 is the value of p at which f(p) is zero, then o . M (if) = P + 1 “ (11) P'eo d? (9,Z+M;)‘/l ’ (p,’-+ Mpy‘ The argument in the 6-function of (9) can be changed accord— ing to 2 V $(Pg+h.—P.)= 3‘9"” e “’5“ Md; 8(9-9.) (12) (41.) M. 4? (=9. so that 3 z z z & d: SQPhko—Pflsdcfl S €dP|__ (P°+MP) “91%) E4 ° (FWD/‘29 .4 2, 2. -_-. 2W 1’: e. 11 ME. (13) 94‘ )4: where we have replaced 2 z P ..._ M“ —-MP 2M4 Therefore, inserting (13) into (9) we finally obtain M1 M‘ 3 .. T‘=..L( «7 g) ()MHBI) . (14) 8Tr ’44 This is the expression for the decay rate used in the text.