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") >
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 = 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 +'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° X.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;) > ,
iq€x (2)
A 4. 5X
J .=. [d x e e(—X.)
1‘2 )
st q; + EFW) -— E S’M) (
s , , 6)
1" = 11211)"): <41”) 3‘. tWWI+k1> -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)[)>6(xo)=
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
\ (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)u.)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))malduaq>>>
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——-