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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.