INTERNATIONAL CENTRE FOR THEORETICAL PHYSICSstreaming.ictp.it/preprints/P/69/111.pdffor the decay amplitudes. These will then be compared with the presently available experimental

IC/ 69/111

INTERNATIONAL ATOMIC ENERGY AGENCY

INTERNATIONAL CENTRE FOR THEORETICALPHYSICS

EFFECTIVE LAGRANGIA.N

FOR NON-LEPTONIC HYPERON DECAYS

WITH SU(3) 8 SU(3) SYMMETRY BREAKING

K A H M E D

G. M U R T A Z A

and

A . M . H A R U N - A R R A S H I D

1969

MIRAMARE - TRIESTE

IC/69/111ERRATA

INTERNATIONAL ATOMIC ENERGY AGENCY

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

EFFECTIVE LAGRANGIAN

FOR NON-LEFTONIC HYPERON DECAYS

WITH SU(3) SI SU(3) SYMMETRY BREAKING *

Page 2:

K. A H M E D

G. M U R T A Z A

and

A.M. H A R U N-A R R A S H I D

E R R A T A

Para. 2 lines 2 and 3* interchange "s-wave" and "p-wave".

Page 6: Replace the third equation by the following:

Page 7: Replace page by new page 7 attached.

* To be published in "Physical Review".

Page 8: Line 3: Replace "by means" by "by no means".

Page 8: Replace the second equation by the following:

where w© take

Page 9: Replace the first equation by the following:

- g' (i + e to .

Page 11: Table I. In the column "Total (Present work)" 5th line,

replace "0. 09" by "1.09" .

- 2 -

u • • • " • - • • • ^ • ' • ^ ' V ^ ; *

• ^ ^"' " ^ ^A+N'1

.̂ _r t.Vg,:'".:;,:..^ «q A / •••••. ffj S* Jft/^.!Er.A.

V7' ~K~- "*"""•.•

1 ' '•'• "?,}•.'"

Ic/69/lll

I n t e r n a t i o n a l Atomic Energy Agency

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

EFFECTIVE LAGRANGIAN FOR NON-LEPTONIC HYPERON DECAYS

WITH SU(3) ® SU(3) SYMMETRY BREAKING *

K. AHMED, G. MURTA^A and A.M. HARUN-AR RASHID

University of Islamabad, Rawalpindi, Pakistan,

and

International Centre for Theoretical Physics, Trieste, Italy.

ABSTRACT

The badronic decays of hyperons are discussed in the context

of broken c h i r a l symmetry, the breaking being introduced according

to the p r e s c r i p t i o n of Gell-Mann, Oakes and Renner. The c o r r e c t i o n s

due t o symmetry breaking l ead t o con t r i bu t i ons t o the p-wave

amplitudes g iv ing b e t t e r agreement with the experimental da ta where-

as the s—wave amplitudes remain almost unchanged.

MIRAMARE - TRIESTE

September 1969

* To be submitted for publication.

Non-leptonic hyperon deoays have "been disoussed from the point

of view of chiral invariant effective Lagrangians by Lee ' as well as2)"by Scheohter ' who have shown that the method reproduces essentially

the current algebra results* These current algebra results obtained

especially by Sugawara 'and by Suzuki ' can be summarized by saying

that the parity-violating hyperon decays are described very well while

the predicted values of the parity-conserving decay amplitudes do not

agree at all with experiments. In fact, Brown and Sommeifield^' have

demonstrated that their current algebra results as well as those of

Hara, ffambu and Sohechter •' yield p-wave amplitudes half as

small as experimental results.

7")In view of this large discrepancy, Kumar and Pati ' have

proposed a model for the hyperon decays in which the p-wave amplitudes

remain essentially unchanged while the s-wave amplitudes are significantly

altered leading to better agreement with experiments* The model is

designed to obtain corrections due to the mass-splitting within the

baryon octet and i t is found that these corrections, which were neglected

by Brown and Sommerfield,contribute significantly to the parity-conserving

deoays.

The success ofAKumar and Pati calculation encourages one to

believe that a similar situation will also obtain in the context of the

effective Lagrangian theory, provided symmetry breaking is introduced

in a systematic way. The construction of the chiral SU(3) x SU(3)

invariant Lagrangian has been described by many authors and we shall

follow here closely the method given by Zumino . We shall then

introduce the SU(3) x SU(3) symmetry breaking of the type proposed by

Gell-Mann, Oakes and Renner ' using for this purpose the elegant method

of Yoshida V This gives us a systematic way of introducing symmetry-

breaking effects which can be exhibited in a transparent manner, unlike

the. model-dependent calculation of Kumar and Pati. We shall first of

all write down the relevant Lagrangians and then give the expressions

for the decay amplitudes. These will then be compared with the presently

available experimental data as well as the predictions of th'e model \>y

Kumar and Pati.

Since the basic tool in all calculations of non-leptonic hyperon

deoays is s t i l l the time-honoured pole model of Feldman, Matthews andAbdus

Salam ' , let us first write down the strong meson-baryon interaction

- 2 -

wbioh we need for the strong vertex part. A simple SU(3) x SU(3)

obiral invariant Lagrangian describing the interaction ofApseudoscalar

meson octet with the baryon ootet is the following ' :

inv. " I _ z ^n

where

and

From the first term it follows that the normalized pseudoscalar fields

are given by

5 1 tP - 7 5

and hence the relevant meson-baryon interaction Lagrangian comes out

to be

L. = a Tr j B y -y_ (b. 9 PB + b o B3 P ) iinv. [ v '5 1 fi 2 JU J.

together with a contact interaction term which, however,does not con-

tribute in the hyperon decays. The PCAC condition is written in this

is the usual pion decay oonstant known to be about 100 MeV,

construction as d, A » (m,,/2a) ^ so that a » l/2 f-r where f_

As regards the non-leptonic Lagrangian, we shall follow the12)method of Lee ' who has shown that if a) the weak interaction

Lagrangian is of the current X current type and b) if the currents

are of the form proposed by Cabibbo ' , then CP invariance xsf the

theory determines that the non-leptonic weak interaction Lagrangian

must transform like Gell-Mann's A,. Furthermore, in the spirit of14rZumino's as well as Weinberg's ' methods of oonstruction of chiral

invariant effeotive Lagrangians, we shall adopt here the point of view

that we have only derivative oouplings in the theory. This then gives

us the following non-leptonic Lagrangian:

- 3 -

. P l V

+ "2

-~

^ .B[X 6 , P] yd Br

, P]

B—• B[\Q, P]

There are other terms involving derivatives of meson fields, some of

whioh vanish in the SU(3) limit, and some, are ruled out "by current

algebra . We shall not consider them here.

We must now introduce the symme try-"breaking term. Recently,o\

Gell-Mann, Oakes and Renner ' have proposed a definite way of "breaking

chiral SU(3) x STJ(3) symmetry in current algebra and Macfarlane and

Weisz •*' as well as Toshida ' have shown how to construct a parallel

theory in terms of chiral Lagrangians. Following Gell-Mam et al . ,

we write the symmetry-breaking Hamiltonian as

H - c vQ

whioh transforms as (3»3*) + (3*,3) representation of chiral 3U(3) x SU(3)

and we obtain the v 's from the following expression given by Yoshida:

88

where XQ and X. (i = 1,...,8) are bilinear functions of the baryon

fields transforming as singlet and ootet under SU(i)

v-4-

and Q* ({/ - 1,...,8) are the 18 x 18 generators of (3,3*) + (3*,3):

-D*.^ 0 ,/

d. .. g.

For further details of this method of construction we refer to Yoshida's

paper* For our purposes we simply pick up terms corresponding to the

prooess B1 •*• B + P so that

LSB

where tc is a scale factor and

-0 ? (-i) fijk B.

m0 ̂ C"i} fjJU ̂ V 3 B

B ]

The complete Lagrangian is then given by

inv* NL SB

We are now in a position to write down the deoay amplitudes* In

standard notation, we hare the p-wave amplitudes

A-M k-r)'

V5

A-N

""" —'t ' J\J

s2

£.-A

« t

/ /

6 •6- 6 -

and the s-wave amplitudes

A-N

A+N

S+"£

12

m -¥k ~

A-N

-7 -

The strong coupling F/D ratio is denoted "by f so that ab. - g./2£It

and aCt^ + bg) o (l-fjg^/f^. , To facilitate comparison with thecalculations of Kumar and Pati, although i t is "by means essential forthe analysis, we further introduce d' + f' - g'//^2 and f'/(<*' + f' ) = iNotice that, since we have used pseudoveotor coupling, our Born terms B''incorporate the AM/2M mass corrections. However, because the ratiosof the sums of baryon masses are not SU(3) symmetrio, the use of SU(3)for the pseudoveotor oouplings leads to a f i t different from that ofKumar and Pati . Since we are interested to show the olose correspondenceof the present work with the model of these authors, we rewrite ouramplitudes using Goldberger-Treiman relation

B Pi j k

and absorb some overall factors in a re-definition of the weak couplingconstant g1, We thus get, e .g. ,

f A-N-g' (l-2f« ) y T J*,

where we take

g' - -6 1 10" 6 MeV, f« - 6

and

f . 0.34 •

We must emphasize that this redefinition of the coupling constants isdone for the purpose only of comparison with the calculations of Kumarand Pati and is in no way essential for the model here considered. Toreduqe the number of parameters further, we assume that d" and f" areproportional to strong interaction d and f , respectively,and,introducing as before g" and f", we identify g" with f^ / J~2 ofKumar and Pati* Thus we are able to rewrite our amplitudes in the form,e.g..

- 8 -

MO - e 1A+8

A-B

~ 2 ^k/l

>(*+p) 1A+N

+ « f k m k2g (1-f) A-N

A+X - k/ i+^

. g' (1 + 2f eto.

where we u s e f, m, • 1 . 4 x 10 MeV. As r e g a r d s t h e p a r a m e t e r s 0( and

(b f we u s e Y o s h i d a ' s e s t i m a t e from Gell-Mann-Okubo mass ' f o rmu la

m0 o

with the value of c » -1.26.

39 m f/2 190 MeV

The numerical results for the amplitudes are given in Talale 1.

The olose correspondence of the present work with the model of Kumar

and Pati is clearly exhibited.

In this work we have not attempted to obtain an exact numerical

agreement with experimental data since, in view of the number of parameters

of the theory, such an agreement would not be very significant. Our

purpose in this note has been to demonstrate that the effective Lagrangian

theory incorporating Gell-Mann/symmetry breaking is well adapted to

reproduce the current algebra results of Kumar and Pati.

ACKNOWLEDGMENTS

The authors would like to thank Professors Abdus Salam and

P. Budini as well as the International Atomic Energy Agency for

hospitality at the International Centre for Theoretical Physics,

Trieste. ' One author (AMHR) i s grateful to the Ford Foundation

for making possible his Assooiateship at the I.C.T.P,

- 9 -

REFERENCES

1) B.ff. Lee, Pbys. Rev. 1^0, 1359 (1968).

2) J. Schechter, Phys. Rev. 1^, 1829 (1968).

3) H. Sugawara, Phys. Rev. Letters 1£, 87O and'997 (1965).

4) M. Suzuki, Phys. Rev. Letters 1J5_, 986 (1965)5

0. Murtaza and P . J , O'Donnell, Can. J . Phys. 4£, 2375 (1967).

5) L.S. Brown and CM. Sommerfield,Phys. Rev. Letters 16, 751 (1966).

6) Y. Hara, T, Nambu and J , Soheohter, Phys. Rev. Letters 16_, 380(1966).

7) A. Kumar and J.C. P a t i , Phys. Rev. Letters 18, 1230 (1967).

8) B, Zuraino, "Symmetry Principles at High Energy", ed. Perlmutterand Kursunoglu, (Benjamin 1968),

9) M. Gell-Hann, R. Oakes and B. Renner, Phys. Rev.,12^, 2195 (1968).

10) K. Yoshida, "Broken chiral SU(3) @ SU(3) in an effective

Lagrangian model11, University of Durham preprint , 1969.

11) G, Peldman, P.T. Matthews and Abdus Salam, Phys. Rev. 121., 302

(1961).

12) See, e.g.jB.W, Lee, Proceedings of the Argonne InternationalConference on Weak Interact ions , Argonne National Laboratory,Report No. AHL-7130, 1965, p.421j see also Ref.l

13) N. CabiVbo, Phys. Rev. Letters 10, 531 (1963).

14) S. Weinterg, Phys. Rev. Letters 18, 188 (1967)} 166, 1568 (1968).

15) A.J. Maofarlane and P.H. Weisz, DAMPT, Cambridge, preprint 69/20,

1969.

-10-

p(A °) x 10s

P(S-J x !0«

P{ZZ) x 106

P(Zt) x 106

[P(A.)+2P(3I)] x

s(A?) x io6

S ( c l ) x 106

SCI I) x 106

S ( l t ) i IO6

[S(A-)+ 2SCSI)] xx LV/FSCLQ)]"1

- 0

0

- 0

0

BORN TERM

Kumar & Pati

4.9-3.8

-0.7+2.4

-2.6+2.4

7.0-5.0

BORN +SE

Kumar & Pati

.12+0.03+0- 0.01

.08+0.01-0- -0.01

.07-0.02+0=. -0.01

.05-0.05-0=^-0,02

. 1

. 1

.08

.02

Present work

5.34-3.41

-0.71+2.19

-2.43+2.16

8.05+4.59

} TERMS

Present work

0.013+0.002-0.- -0.099

O.OO69-O.0005-0.016 » - 0 .

-0.007-0.0019-0.0066 = -0

-0.0072-0.0013-0.075 = - 0 .

114

01

.15

08

TABLE 1

SYMMETRY-BREAKINGTERM

Kumar

0,

- 0 ,

- 0 ,

1,

&Pati

,88

22

05

4 .

Presentwork*)

-0.36

-0.42

+0.34

-0.08:

NON-POLE TERM

Kumar

-o .

0.

- 0 .

c

& Pati

27

49

57

>

Presentwork

-O.32:

O.56

-0.66

0

Kumar

1 .(+0.08

1.(+0.02

- 0 .(-0.04

3 .

1 .

Kumar

- 0 .

0 .

- 0 .

- 0 .

1.

TOTAL

& Pati * * J

98- 2.1)

48- 1.46)

25- -0.3)

4

1

(4

TOTAL

&Pati

26

48

58

02

02

Present work

1

1

0

3

0

.57

.06

.07

.37

.09

Present work

- 0

0

- 0

-0

1

.42

.55

.59

.08

.1

2

1

- 0

4

EXPERIMENT

.267

.611

.119

.143

-(0.33

0,

<o.

0 .

405

406

004

dt

i

*

±

*

±

±

0

0

0,

0,

0,

0.

0 .

0 .

.071

.141

,013

,076

004)

007

007)

009

D E C A Y A M P L I T U D E S+ ) For K =100.

**) ETC contribution is given in the bracket.***) Filthuth, CERN 69-7.