Pram~na, Vol. 12, No. 1, January 1979, pp. 57-75, t~) printed in India.
Weak nonleptonic decays of 3/2 + isobars in SU(3)
K U S U M S H A R M A * and R A M E S H C V E R M A Department of Physics, Panjab University, Chandigarh 160 014
*Permanent Address: DAV College for Women, Karnal 132 001, MS received 5 April 1978; revised 29 September 1978
Abstract. Weak transitions of decuplet isobars are expanded in terms of eigen-ampli- tudes of the direct channel in the framework of SU(3). Starting with the most general weak Hamiltonian and assuming intermediate states to be non-exotic, we obtain AI=½ rule for D,- decays. Invoking of the CP invariance forbids aI1 the pv weak processes D(10)-~D(10)+ P(8). Decays of the charmed multiplets are also dis- cussed in these dynamical considerations. We obtain triplet dominance of charm changing weak Hamiltonian for ~3 *++ decays.
Keywords. Non leptonic decays; 3/2 + isobars; weak transitions.
1. Introduction
Structure o f the weak Hamiltonian for hadronic decays is not clear, though many models o f weak interaction have been proposed (Glashow et al 1970; Bran¢o et al 1975; Fritzsch and Minkowski 1976; Abe et al 1977). Structure o f the weak Hamil- tonian for the non-leptonic decays o f 1/2 + hyperons has been analysed in a dynamical consideration (Taha 1968; Bajaj et al 1974). P h e n o m e n o n o f octet dominance was obtained with a simple dynamical assumption corresponding to the intermediate states. In this paper, we study the structure o f the weak Hamiltonian for the weak non-leptonic decays o f 3/2 + charmed and uncharmed isobars in the SU(3) symmetry framework using similar dynamical assumption.
Inclusion o f charm quark predicts the existence o f ten [/)(6) D(3), D(I)]3/2 + charmed states (Amati et al 1964) ih addition to the uncharmed decuplet isobars.
Recently observed mass values (Cazzoli et al 1975; K n a p p et al 1976; Goldhaber et al 1976) o f few charmed states indicate ( G u p t a 1976; de Rujula et al 1977; Verma and K h a n n a 1977a) that ~ a *++ (D(1)) may be the only possible purely weakly decaying.
We discuss the weak non-leptonic decays o f ~ 3 *++ and ~ - in the modes 3/2 + -+ 3/2+/1/2++0 -. Starting with the most general Hamiltonian and assuming that only the eigen amplitudes corresponding to the non-exotic states can contribute to the transitions, we obtain A I = l / 2 rule for pv as well as p c decays o f ~ - in the modes ~ - --> ,~/~* + ~. CP invariant SU(3) weak Hamiltonian further forbids all the pv decays D(IO)-~D(IO)+P(8), but it gives no constraints for the decays o f charmed baryons. F o r ~ a *++ decays, dynamical assumption alone leads to triplet dominance o f the charm-changing weak Hamiltonian. The Cabibbo enhanced decays (/~C----/~S---1) o f ~ 3 *++ are forbidden in pv as well as p c modes. We include weak decays o f D(3) and D(6) multiplets for the sake o f completeness.
57
58 Kusum Sharma and Ramesh C Verma
Simple relations are obtained among the decay amplitudes o f these charmed isobars.
An outline o f the method and the weak Hamiltonian are described in § 2 whereas
§ 3 gives the details for pv and pc decays.
2. Preliminaries
We assume current @ current form o f the weak interaction. G I M weak current transforms like
J = ~n cos 0 + ~A sin 0 -- "~n sin O + ~h cos 0, (la) component o f 15 representation o f SU(4), where 0 is the Cabibbo angle. (Dirac operators have been omitted). It has the following SU(3) decomposition
j s = pn cos 0 q- p~ sin 0, j3 = _ ~'n sin 8 q- ~A cos 0.
Then weak H a m i l t o n i a n / / w ' ~ ~J, jt~. has the following components:
Hw ,~., (-~8, 8 ) -q- ( 3 , 3*)Ac=0) -q- ( 3 , 8 ) A c = _ 1 + ( 8 , 3 * ) A C = + I . (2a) Thus charm conserving weak Hamiltonian belongs to the representations present in the direct products
8®8 = i~2(8)~I0~I0.~2_.7 3®3* = 1 ~ 8 .
Tensor structure T(SII:) o f SU(3) weak Hamiltonian can be written as:
HAc=O W e~a TS 11, l/n~ -1/2) + T~ °, s/2, -1/n) -t- T x°* {15, 1/2~ i l l S ) --1- T sv 11~ 1/$p - l / n )
- - ~/5 r z7 {I~ 31n, -11n) (2b)
whereas charm changing ( A C = - I) weak Hamiltonian lies in the direct product:
_8@3 = 3 ~ ) 6 . 0 ) 15.
Charm changing decays can occur through three modes depending upon the change in strangeness. The tensor structure o f various components is as follows:
H A c = - I "~ ~ c-1, 1, 1~ rT6* q- T 15 c-1, 1, 1)] cot 0 q- [T 6. ~1, o, o~ -~- T 15 a, 1, o~] tan 0
+ iT30, 1/2, l/n) + r6*(0, 1/,, l/n) + Z(lo 5 1/2, 1/2, -- 5v/2 T{ 15, 3/n, 1/211" (20)
Weak nonleptonic decays o f 3]2 t isobars in SU(3) 59 The hyperon decay process A->B+P can be understood as:
S + A - > m - > B + P ,
where the spurion s has the same tensor structure as the weak Hamiltonian so as to conserve all the quantum numbers in the above reaction. The transition amplitude are expressed in terms o f reduced amptitudes (Gourdin 1967) by using SU(3) Clebsch Gordan coefficients corresponding to the projection o f the initial and the final states on the eigen states [rn). We use the Biedenharn's (1963) conventions for the isoscalar factors. A p p e n d i x B gives the isoscalar factors for the direct product 2 7 ® 10 as calculated from Pandit and M u k u n d a (1965). For other products, the isoscalar factors are taken from Haacke et al (1976).
2.1. D(10)-> D(10)+P(8) decays
Reduced amplitudes are defined as:
a m = ( l O o , i 8 e [ m )
(m[811%>
b m = ( 10 D, [ 82, '[ m ) (ra [ 27 I 10D)
c. = <1%,18, Im> <m t'0110s>
d. = <10s, { 8, [ m> <m { 10" { I%>. (3)
In all, we have twelve reduced matrix elements corresponding to a 8, ale, aav, ads, b s, bxo, b2v, ba~, e2T, ,as, d, and d2v. Assuming the intermediate states to be non-exotic, we get the following constraints:
a f t , 35 = b27 35 = •7, 35 = d27 = 0 . (4)
2.2. D(10)-'> B(8) + P($)decays
Here, CP invariance gives no constraints. We define:
a'. = <sBl 8, I m> <~ l 8'l I%>
b'm = (8B I 8 , l m) <m [ 27 { 10D) C'm = (8 B / 8 , ]rn) (m ] 1 0 1 1 0 0 )
d'm ffi <8 B 1 8 , I m ) <m { 10" { 10s>. (5) The consequences of CP-invariance are derived by writing the cross channel :
D'+~ --> D + ~ transition amplitudes and then applying the conditions:
A (D --> S' if) = -- A (D' --> DP) for pv amplitudcs and B (D --> D'p) = B (D' o D~) for pc amplitudes.
60 Kusum Sharma and Ramesh C Verma
t r a I i t r p t
We have 14 parameters i.e. a 27, a 10, sD, a SF, b 27, b so, b 10,, b sD, b'sF c'ao., c'a7 d'sz~, d's~ and d'27. The dynamical assumption for the intermediate states imposes the following conditions
a'a7 = 0 ; b' 27, a 0 * : 0 ; c' a 7 , 1 0 * - - 0 ; d'a7 = 0 . -- (6) The number of independent parameters is reduced to eight.
ill++
2.3. Weak decays of ~Z 3
* + +
~ a can decay through the following channels:
0(1)-~ 0(3) / B(3) + P(8)
D(1) -~ D(6) / B(6) / B(3*) + P(3*).
We denote the reduced amplitudes as:
(i) D(1) ~ D(3) / B(3) + P(8)
a 3 . = ( 3 1 8 l m ) ( m l 3 l l ) m=3, n=I/2or3/2.
~ , = (3118 I n ) (m[6* [ 1) m=6*.
~5 = (3[8im) ( n i l 5 [1) m=15, (7)
(ii) D ( 1 ) ~ D(6) / B(6) + P(3*)
bg=(613*irn)(rn[311 )
m = 3 ,b], = (613* Ira) <m 16" I I)
no intermediate state,b~s=(6[3*lrn)(m
1511) m=lS, (8)(iii) D(1) ~//(3*) + P(3*)
c a = ( 3 * 1 3 * l m ) ( m t 3 l l ) m = 3 , c . , = ( 3 1 3 , lm)<ml6,[1 ) m=6*,
els
= (3*
13" Ira) (rnI ~511)
no intermediate state,(9)
where n denotes the spin of the baryon in the final state. Assumption of the inter- mediate states leads toN n
a6,.15~--0, b15=0; c 6 , = 0 . (10)
Weak nonleptonic decays of 3/2 + isobars in SU(3) 2.4. Decays of D(3) multiplet
D(3) isobars decay through the channels n(3) --> D(6)/B(6)+P(8),
--> D(IO)/B(8)+P(3*), B(3*)+P(8).
For A C = / ~ S = - - I mode, we define the reduced amplitudes as follows:
(i) D(3) -~ D(6)/B(6)+P(8)
A." = (618 Ira) (m [ 6* [3) =3", 15" n = ½ or ~, B,," = (61 81 m) (m 11513) m = 6, 15", 24, where n denotes the spin of the final state baryon.
(ii) D ( 3 ) ~ D(10) + e(3*)
C. = (1013* I m) ( m l 6* [ 3) no intermediate state D•-=<lOl3*lm)(m[15[3 ) m = 6, 24.
(iii) D (3) --> B (8) + P (3*)
Em ~- (8 ] 3 * [ m ) ( m l 6 * I 3) m = 3 " , 15", F,. = (81 3*Ira) (m 11513) m = 6 , 15".
(iv) D (3) --, B (3*) + P (8)
G.--- (3"[81 m) ( m l 6 * l 3) m = 3 * , 15",
H m = (3*181rn) ( m 1151 3 ) m = 6 , 15".
The dynamical assumption leads to the following constraints A~5* = B~6., ~4 = 0,
D ~ = E~5, = F15, = O1~, = / ~ 1 ~ , = H I ~ , = 0.
2.5. Weak decays of D (6) multiplet
/)(6) isobars decay through the following channels:
D (6) -* D (10)/B (8) q- P (8).
61
(11)
(12a)
0 2 b )
(13)
(14)
62 Kusum Sharma and Ramesh C Verma
For A C = A S = --1 mode, the reduced amplitudes are defined as follows.
(i) D (6) -~ D (10) + P (8)
g m = ~ l O l S I m ) ( m [ 6 * [ 6 ) m = 8 , 27,
hm = (10181m) (ml15[
6> m = 8 , 10, 27. (15)(ii) D (6) --> B (8) + P (8)
Jm = < 8 1 8 [ m )
<m16"16>
m = 8 , 2 7 ,km --- (8 18 1 m> (m
11516> m=8, 10, 10",
27. (16) The assumption of intermediate states imposes the conditionsga7 =h27 = 0 , .~v =kt0,,27 = 0 . (I7)
3. Decay amplitudes
In this section we give the results obtained for various decay modes of charmed and uncharmed isobars. Expressions for all the possible decay amplitudes of charmed multiplets are given in appendix A.
3.1. D (10)-), D (I0) + P (8)
Most of these decays are not observed as the parent particle decays strongly. But, in principle, these can occur and we consider them to obtain conditions from CP-invariance. The relations obtained among the reduced matrix elements are
2a27 = -- (9 C6) a 8 - - (3 V'30) al0, 2a~ = -- (~70) a 8 + (V'14) at0,
1/2 ('V'70) b35 = (V'3/6 V'7) b27 = -- (~10/15 V'7) blo --- bs, (18a) for pv decays,
and an5 = -- V'14/5 a s -- (V'14/2) at0 + 2V'7/~/~ a27
bas := (4~/7)/(5V'10) b s + 4/5 bx0 + (6~/3)/(5V'10) b27 (18b) for pc decays.
This does not lead to any useful relation since the number o f parameters is large, while the dynamical assumption alone reduces the ~ - decay amplitudes to the following:
W e a k nonleptonic decays o f 3/2 + isobars in SU(3) 63 -- (~*°zr-/~-) = -- (1/V'15) as -- (I/4vt3) al0 -- (1/3~/10) bs
- - (1/3X/7)
blo
+ (1V'30) ds,( - ~ * ~ ° / ~ - ) = -- (1/V'30) a s -- (1/4V'6) at0 -- (1/6V'5) bs
-- (1/3x/14) b~o + (I/2v'15) d s (19)
and further it gives null contribution to the matrix element
(10, 8 I T27 ~1,~:~,-1/~ 110 (-2, 0)).
Thus A I = l / 2 rule is obtained for both the pv as well as pc modes. Actually in pv mode, none of the decays D(10)-> D(10)+P(8) are allowed to occur in nature under dynamical assumption and CP invariance. The result (F,*Tr [ ~ - ) = 0 in pv mode has been obtained in SU(4) (Verma and Khanna 1977b) but with 20" dominance. It is to be noted that in SU(3) tensor analysis, 2__7 piece of weak Hamiltonian under CP invariance gives null contribution to ~--->F.*~r decays in pv mode. In SU(4) current algebra framework, (~**r I ~ - ) are forbidden in pc mode too (Khanna 1976). In our analysis for the pc mode 27 contribution vanishes for ~-->.~**r decays under the dynamical assumption thus leading to octet dominance for ~ - decays. We would like to point out that CP invariance along with dynamical assumption does not lead to octet dominance for pc decays o f other resonances in D (10).
3.2. D ( 1 0 ) ~ B(8) + P(8) decays
Here, we discuss the ~ - decays only. We obtain the following amplitudes after retaining only those amplitudes which correspond to physical states.
- - ( ~ ° * r - I ~ - ) = (V/3/5) a's o + (l/v/15) a'ar + (1/2V6) a'xo + (1/5~/2) b'sn + (1/3v'lO) b'sr + (2/3v'14) b'xo -- (3/5a/6) d'8 o - - (1/V'30) d's F,
( ~-*r° I ~ - ) = (X/6/10) dad + (I/'v/30) a'sr + (1/4V'3) a'xo + (1/10) b'so + (1/6X/5) b'sr + (1/3C7) b'xo -- (3/I0~/3) d'ao -- (1/2X/15) d'ar,
( A K - I ,0,-) = (1/5V/2) a'so + (l/x/10) a'sF -- (1/4) a'xo
+ (1]lOx/3) b'so + (1]2~/15) b'sr -- (l/x/21) b'10 (20) - - (1/10) d'8o -- (1/2,x/5) d'sF.
64 Kusum Sharma and Ramesh C Verma
Here also dynamical assumption alone gives null contribution to the matrix element (8, 8 ] T 27 (1, $12,-1/2) 10 ( - - 2, 0)~. Thus leading to A I = l / 2 rule i.e.
~-_ = - - v ' 2 ~ o .
3.3. ~ + + decays
Using conditions (10) we obtain triplet dominance for SU(3) charm changing weak Hamiltonian. For A C = - - I , A S = 0 , we obtain the following relations:
(i) D ( 1 ) ~ D(3)/B(3) --}- P(8)
< ~ ' * + - , , - ~ , , + l f ~ , I ~ + * + + > = - , / 2 e = * + + / ~ + = ° l a , , - . , . , + + >
= -- V'6 '/=*++ / =++ , - = -= ~lg~3 > - - - v ' 3 ( f ~ + / f / g K + l £ ~ *++ *++ ' ) (21a) (ii) D(1)-~ D(6)/B(6) + P(3*)
X/2 <~-;+ / ~.~ F + I a;++> --- -- X/2 (2:; + / 27~- O + I a ; + + >
* + ÷ + + ~+ +
= ( ~ r /2~, DO l a , > (21b) (iii) D(1) --, B(3*) + P(3*)
<z'~ F+I a;++> = -- <a'~ D ÷ I a T ÷ > • (21c)
Notice that A C = -4- A S decays are not allowed.
3.4. Decays of Dr3) multiplet
Conditions (141 yield the following relations among the decay amplitudes:
(i) O (3) ~ D (6) / B (6) + P (8)
. . = / ~ . + ~+ ~Olg~:* >
0 = < a l o / a o , ~ + l a = > . - , / - ,
- - ( ~ * + / ~ ' + ~ ' * + + \
- - \ - x • " , ~r+ I -2 ," = (Z,~,++ / XI+ ~o [ _==*++\/, (22a)
Weak nonleptonic decays of 3/2 + isobars in SU(3)
, + ,t,+
v'2 *+ + - o ,++
(ii) D(3)---> D(10) + P(3*)
o = < s *o D+I [2~+> = @:*+ D+ I ~++>,
~ > = ~ , , .
<~,OF+ I *+ < z , + D o l [ * + > = ~/2 <2:*OD+[~*+\
(iii) D(3) ~ B(8) + P(3*)
0 = < ~ o 0 +] n , > = < z + *+ D + =*++\ I-~ , , X/2 <Z '° D + l .~*+> -- <2 "÷ O ° [ ~*+),
- 2 <~OF+ I *+ ~, ) = v ' 6 ( A O + l *+
(iv) 0(3) --* B(3*) + P(8)
* + +
o = <...'~ ~o I n*+> = <-";'r+ I
~ >,
X/2 < a ' ~ ¢ I [~÷> -=- <~.'%r + I [*+),
V'6 < ~'~ ~1 [ ~*+> = <~,O,r + ] .~[+> _ 2 <a'~ X '~ [ ~*+>,
65 (22b) (22¢)
/ ~ * o -/~o ~+ I *+ *+
(22d) (233) (23b)
(243) (24b) (24c)
(253) (25b) (25c) Notice that weak decays of ~:+ and E~ ++ are totally forbidden. Cabibbo enhanced component (6*) of the weak Hamiltonian does not contribute to D(3)->.D(IO)+P(3*) decays.
3.5. Decays of D(6) multiplet
Dynamical assumption leads to the following relations among decay amplitudes (i) D (6) -~ D (10) + P (8)
o = < n - . + la~O> = <~,o ~ O l a ~ > = <s *°¢+ 1 s~÷>
= <Z,,+ ~.o [ =.I+> = <Z,+ w+ 1:27~*+> = <At+ ~e.[ 2;;++) (26a)
<Z,,÷ zr- 12;~o > ~_ _ <~,o KO l ~ > , (26b) P - - 5
O~ ~,. A .~ A A A <. I e... II I I A 4- A V %/ ~ + ~ ~ v ~ ~'o* v v
II + :I I~ ,il, ".~ N v II v /x + ~ D tn ~* + V 4- v ~ I i,,~ II,
V ~ ~ v 4- + N ~
+ v V I~ II [~] v <.. __ A v ~ ~ + "~- "~ v ~ v ~ v <~. v II ,,X- I 4- v
J A i i'~ A .]{- -i- i ~TI I + v ~] ~ i.J ,11. + = V v ~ o ~ --, .,~ ~ "
Weak nonleptonic decays o f 3/2 + isobars in SU(3) -- ( p K- [27.o) + (n T ° l Z* °) = -- x/2 ( p -~o ] Z , + )
_ ( o K o] 2770} + ( ~ - K+ I 277°} -~ -- V'2 (~,o K + [Z,,+}
( s - 127 .o> - (270 1 27,0> = (270 1
67 (27h) (27i) (27j)
Notice that weak decays of D. *°, =,~'+ and ~*++ _~ are forbidden.
4. Conclusions
In this paper we have discussed the weak nonleptonic decays of 3/2 + isobars in SU(3) symmetry framework. Without assuming the symmetric nature of current ® current weak interaction and including all components (8, 10, I0", 27) of the weak Hamil- tonian we obtain A I = l / 2 selection rule for ~ - decays in both the pv and pc decay modes under dynamical assumption. Using CP invariance further, all the parity violating amplitudes become zero for D(10) -+ D(10)+P(8) processes. For the ~ a *++
decays, dynamical assumption alone forbids Cabibbo enhanced sextet component to contribute and weak Hamiltonian is predicted to be triplet dominant which allows only the/~ C = - 1, A S : 0 decay processes. For D(3) and D(6) multiplets, our dyna-
, ~ + + *0 it+ , + +
mical assumption forbids all the weak decays of ~ + -9. ~1 , -=x and 271 . We obtain simple relations among the decay amplitudes from the most general Hamil- tonian. Our relations are valid for both pv and pc modes. The dynamical assump- tion used to derive these results is physically understandable (Bajaj et al 1974), since
ql++
quarks appear in qqq and ~q combinations only. Experimental search for ~ s decay modes may further justify this assumption. It is interesting to carry out these consi- derations to SU (4), where spurion may belong to 15, 20" and 84 representations.
For 20" spurion, there is no physical intermediate state for the processes.
D(20) + S-+ m ~ D(20)/B(20') + P(15)
15 spurion contributes only to A C : 0 , & S = - - I and A C = - - I , A S = 0 processes.
84 spurion may vanish for pv decays similar to 27 spurion in SU(3). This work will be taken up elsewhere.
Acknowledgements
We are thankful to Prof. M P Khanna for interesting discussions. One of us (RCV) gratefully acknowledges the financial support given by CSIR, New Delhi.
68 Kusum Sharma and Ramesh C Yerma
Appendix A
The complete transition amplitudes in terms of all the reduced matrix elements for various possible decays are listed below:
f~*++ Decays
(i) A C = A S = - - I deeaymode(× cot O)
(~:++ / ~++, ~0 / a;+÷) =_(1/vT0) <~ + (l/Vi~) ~ .
<a~+l z~ D ~ i a; ++) = - O l ~ / i ~ ) hi6
(ii) AC---I, A S = - k l decay mode ( x tan 0)
(~;+ / s; K+ I fi;~> =-(1/v'3-6) a~ +(llV'i-2) ~,
(=.*++/~++
r ° [ [~;÷+)=--(I/V'~) ~6-b (1/VF2) a;,
( Z * + / Z ; F +I a*++>
= - - ( I / V ' ~ ) b~s( A'~ F+ i a*") =(I/v'~) eo,
(iii) A C = - - I , A S = 0 decay mode
( a *÷ / a~ r ÷ I g~++)---0/2 v'io) a~5+(l/2V-2)a~,--(l12~6 ) 4.=*÷/~ ~'+ I £~*+÷~=(3/2 "VI-O) <6--(1/2 V'-6)~,+(1/2 ~/2)a:
<~++ / ~+ ,~° I aI++> = (v'~/4) ~ , + (1/4 ~/3~ a ; , - (1/4) al
( ~ * + + / ¢[++ -2 , -~ ~[ f~*++~ -- (3/4V'1-5)<5--(I/4) a~,-- (1/4V'3) a~ ~__
( Z~ ÷ I Z~ 1)+ I t~ *++) =
(7/6 v'5) b]6-- (1/2 'V3) b;
<~*+÷/z ++~ Dol a*~>=Cs/3 v55) bi~+O/v'~ b;
Weak nonleptonic decays of 3/2 + isobars in SU(3)
( ~ + / n ~ F+ I a *++) = - - (1/2 V'5) bi~ + 0/2 v'~) b;
( A ' ~ O + l a *++) =(l/~/l'2) ee, --(1/V~12")ea
< ~ '; F + [ a *++) ----(1/~/~2) %, +(l/V'12") ca
Hereafter, we omit the reduced amplitudes corresponding to the exotic states.
(II) D ( 3 ) ~ D(6)/B(6) + P(8) ( aT ° / a °~+ I a *+) = 0
<~*+/~7. ~ 1 a*+) = o
< ~'+ / ~ I,+ I ~*++) •o
I , - , )?*++
/ 2:7-+ ~o I ~++) = o
(s~ '+ / ~7. ,,° i -~*+) =(114q2) ,,11, -(3/10v'6) sil (-~'+/-=7 ,71 ~*+)---(3/4v '6) ,4.", +(l/lO,v'2) ( ~ o / ~o ,,'+1 ~*+5
= 1/4 A;, --(3/10x/3)(~*+_~ ~# --~s:+ Tt-e [ ~ * + ) = - - ( 1 / 4 ) A~,--(3/10%/3) B~6 --(27~ ++ / 27~ + K - I ~ + ) =(1/2~/2) A~,+(3/5C6) B~.
(IID D (3) ~ D (10) -4- P (3*)
( il*°D+ I Qt*+) (27*+D+ ] .~*++ )
< 2:*0 ~ 1 ~*+ )
<Z*+ ~ 1 ~ *+ ) (~*OF+ i~; + ) (iv) D (3)-~ ~ (S) + P (3*)
< ~oD+ t n *÷)
= 0 ----0
= 1/5 De
---- (2/5V'2) D e= (2/5V~2) D 6.
- - 0
69
70 Kusum Sharma and Ramesh C Verma ( 2 ' + D + t/-2~*++ )
( A D ' [ ' ~ { + )
(Z+DO [.~*+ >
(.~oF+
1~ *+ -2 >(V) D (3) + B (3*) + P (8)
<H', +x-z I n , + >
~ , * + +
<~÷ ,,+l-, >
<AT m'=*+>
i h"dg<H; +:IH~ ,+>
~ + + , +
(-'1 ' 7 ] ~ , )
( v l ) D ( 6 ) - ~ D O o ) + P (8)
= 0
---- ( 3 / 4 x / 6 ) E a , - - (1/2~/10) F e - - - - (1[4X/2) E a , - - (3[2V'30) F e --- (3[4X/3) E s , - (1/2X/5) F e
= (3/4V'3) E a , + (1/2X/5) Fa
= 0
= 0
= (3[4x/3) Ga, - - (1[2x/5) Ha,
= ( 3 / 4 v ' 6 ) Gs, + (1/2x/10) H e - - - - ( 1 / 4 x / 2 ) G* + (3/2~/30) H a
= (3[4x/3) G s , + (1[2x/5) H e.
o = <a-,~+ I n * ° ) = <z,o ~:o ! n , o > = <~,o,,+ i ~ + >
-_ ( z , ~ o + i.~ + > = (~:,+ ,~+ [ z~ +÷ > = ( A++ ~ 1 -x~*++ )
_ (H,O ~,o =~,o) 1
- < z , + x - I H ~,o>
<H,o,71 H,*o>
<z',o kol HI'o>
<n-K+[ ~I'o >
( z,o ,,+ 1 z~ '÷ )
= (ga/5) + (2/15x/2) h a - - ( l ] 3 x / 1 0 ) hlo
= - - ( I / 5 x / 2 ) ga - - (1/15) h a + (1/6x/5) hlo
= (1[5) ga + (2/15x/2) h a + ( 2 / 3 x / I 0 ) hxo
= ( 3 / 5 x / 6 ) g a + (115x/3)h a + (1[2x/15)hlo --- - - (1/5~22) ga - - (1[15) h a - - (1/3 ~/5) hlo
= - - (3/5.v/3) gs - - (2/5x/6) h a + ( l / V 3 0 ) hxo
= 015,V'2) gs - - : 1 / 1 5 ) hs - - 0 / 3 V 5 ) h~o
Weak nonleptonic decays of 3/2 + isobars in S U ( 3 ) 71
(Vlt)
_ <2:,+¢o i s ~ ,÷ >
<s*+'Jl 2:1 *+ >
- < A++K- I ,q*+>
< A÷K o I Z~'+ >
<n*OK+ I S~">
-- < A+K- I Z*°>
<Z*+¢- i Z~'°>
_ < 2:,o,,o I ,~,o>
< 2:*% t 2:*°>
< z"-~÷ I z*o>
<_~*- K+ 12:~o>
- < ~ , o K o l z ~ o >
D (6) o a (8) + P (8)
= - - (1/5X/2) gs + (1/15) h a -4- (1/3X/5) hxo, - - (3/5X/6) gs - - (1/5X/3) ha,
= (3/5X/3) gs - - ( 2 / 5 x / 6 ) hs -4- ( l / x / 3 0 ) hs
= - - (1/5) gs + (2/15X/2) ha - - ( I / 3 X / 1 0 ) h,o,
= - - (1/5) gs + (2/15X/2) h s + ( 2 / 3 v ' 1 0 ) h~o,
= - - ( 2 / 5 v ' 2 ) gs - - (2/15) h s + (1/3X/5) h~o,
= (2/5X/2) gs + (2/15) h s - - ( 1 / 3 x / 5 ) h~o,
= - - (4/15) hs - - (1/3X/5) h~o,
= - - ( l / 5 x / 2 ) g s + (l/5)hs,
= (3/5X/6) gs + ( 1 / 5 x / 3 ) h s,
= (2/5X/2) gs - - (2/15) hs + (1/3X/5) h~o
= - - ( 2 / 5 x / 2 ) gs + (2/15hs) - - (1/3 x/S) hlo,
= - - ( 4 / 1 5 ) hs - - ( 1 / 3 V 5 ) h,o
0 = <z+¢+ I ~.*++
__ <~o ~ i a , o > ,
( ~ o K J [ 21.+>
_ <2:o,,, 12:,+)
< A ,r+ I E *+ >
- < 2:+ ¢o I ,v~ '÷)
<~'+ '71
2:1 ) **= - - (3/10)is, + ( l / 2 x / 5 ) j s j + (1/5 v'2)ks, - - (l/3x/10)ks~ + ( 1 / 3 v ' 5 ) k l o
= - - (I/X/10)jsj + (1/3x/5)ks~ -4- ( 1 / 3 x / 1 0 ) k : o
= (3/5x/6)jsl - - (1/5 x/3)ks~ - ( 1 / v / 3 0 ) k l o
= ( 1 / x / l O ) j s ~ - ( l / 3 x / 5 ) k s , - - (1/3x/lO)klo
= (3/5X/6)j s - - (1/5x/3)ks~ + (1/x/30)k,o
72 Kusum Sharma and Ramesh C Verma
< p ~,o I z;'+>
_ <uo Ko ) z~o>
< ~ - K + ~'~'o)
<,~o ~ z~o>
._ <zo ,71 z~o>
<A~ ] 2:,o>
< z+ "-I z*o>
_ <pr- I Z~,o>
"~'o I ,V;o>
- < w ,~ I ~ , °>
- <~-,,o I-~*°>
< ~o ,71 ~'0>
- <z+ K- I ut">
- <zo ~ l u *°>
= _ (3/lO)jsl -- (l/2x/5)js~ + ( 1 / 5 x / 2 ) k s , + ( 1 / 3 V l O ) k s , - - (1/3V'5)klo
= (l/5.V/2)jsx - - (1/v'lO)ja, - - (1/5)ksx - - ( l / 3 v t 5 ) k s , - - ( 1 / 3 v ' l O ) k l o
= (2/5 X/2)js~ d- ( 2 / 3 ~ 5 ) k s , - - ( 1 / 3 x / l O ) k z o
= _ (1/5~,/2)/s~ + (1/V'10)/a2 - - (1/5)k8, - - (1/3V'5)ks, - - ( 1 / 3 x / l O ) klo
= _ ( I / S V 2 ) j s l - (l/5)kat ,
= - - (3/5V'6)ja, + (1/5x/3)k8, + ( 1 / ~ 3 0 ) k ~ o . --- - - (3/Sv~6)jal + (1/5V'3)kal - - ( I / v ' 3 0 ) kx o,
= ( l / 5 v ' 2 ) / s x + (1/5)ksx
= ( 1 / 5 ~ / 2 ) ] s . - - ( I / V ' l O ) j a , -t- ( 1 / 5 ) k s , + (1/3 x/5)ka, -4- ( l /3 x / l O)kxo
--- (2/5x/2)ja~ - - ( 2 / 3 x / 5 ) k s , + (1/3x/lO)kxo.
= (1/Sx/2)jst + ( 1 / v ' l O ) j s , - - ( 1 / 5 ) k a x + ( 1 / 3 x / 5 ) k s , + (1/3 x/lO)klo
= (3/lO)js~ + ( t / 2 x / 5 ) A , - (1/5x/2)ks,--(1/3"v'lO)ks, + (1/3"v'5)kxo
= - - ( 3 / l O x / 2 ) j s , - - (1/2x/lO)ja, + (l/lO)ks, q- ( I / 6 x / 5 ) k a , - - (1/3 v ' l O)k~
= (3/10Vt6)]8~ - - (3/2 V'3)]8, - - (1/lOv'3)ka~
-4- (1/2 v'lS)ks, "d- ( 1 / ~ / 3 0 ) k l o ---- (3/-lO)ja, - - (1/2 V'5)ja, - - ( l / 5 v ' 2 ) k a x
+ (1/3~/10)k8, - - (1/3,~/5)klo
-- - - (3/lOx/2)ja, d- ( l / 2 x / l O ) j s , d- (1/lO)ksx - - (1/6.~/5)ksa -4- (l[3~/lO)kxo
Weak nonleptonic decays of 3/2+ isobars in SU(3) 73
= (3/IOv'6)js~ ÷ ( 3 / 2 x / 3 0 ) j s , - - (1/lOx/3)ks~
-- (1/2x/15)ks,- (1/x/30)klo.
Appendix B
Following are give the isoscalar factors for the direct product 27 ® 10 used in the paper.
Y=2, I = 2
(Y~, I x) : (Y~, 12) 81 64 35 1 2x/2 x/5 (1, 3/2) : (1, 3/2) 2x/5 x/15 2V'3
1 - - 1 1 (1, 1/2) : (1, 3/2)
x/2 x/3 x/6 Y--2, I=1
(Y1, Ix) : (Y2, I,) 64 27 35*
(1, 3/2) : (1, 3/2) 2x/2 5 x/5 3V'7 x/42 3x/2 (1, 1/2) : (1, 3/2) 5 1 --V'5
3X/7 V'21 3 Y=2, I=0
(Y,, 10 : (r~, 1~) lO* 35*
- t v'2 (2,1) : ( 0 , 1 )
V'3 "v/3 (1, 3/2) : (1, 3/2) X/___2 __1
• V'3 ~/3
Y = I , 1=5/2 (Y~, I~) : (r~, ld
(1, 3/2) : (0, 1)
81 64 35
~/21 2 --V'5
2V10 vq5 2v'~
Y=I, 1=3/2 (Y1, I1) : (r~, I~)
(1, 3/2~ : (0, 1) (1, 1/2) : (0, l)
81 64 35 35* 27 I0
X/3 2X/14 V'5 x/5 --53/2
~/70 3x/15 3x/6 3x/3 0 3V'21 V'15 --~/5 --x/5 - - V 5 x/3 --4 2V7 3v'21 6V3 3v'6 X / 1 4 3V'21
74 Kusum Sharma and Ramesh C Verma Y = I , 1=1/2
(Iq, I0 : (I'~, I2) (1, 3/2) : (0, 1) (1~ 1/2) : (0, 1)
64 35* 27 10" 8
21/5 7 1/3 1 --21/2
31/21 61/3 21/7 31/3 31/3 10 --1/5 --.v'3 1/5 --1/2 31/21 31/3 1/35 31/3 31/15 Y=O, I = 0
(I"1, Ix) : (Y~, Is) 64 27 8 (1, 1/2) : (--1, 1/2) 1/10 --4 1 1/21 1/35 1/15 Y=O, I = 2
(Y1, I1) : (Y2, I2) 81 (1, 3/2) : (--1, 1/2) 3
64 35 35* 27
4 1/2 -- 1/5 1 -- V'5 21/10 1/105 21/6 2V'3 21/7
Y=O, I----1 ( I,'1,/1) : ( Y2, 18) 81 64
1 81/2
(1, 3/2) : (--1, 1/2) (1, 1/2) : (--1,1/2)
35 35* 27 10 10" 8
v'5 11 --1 --41/5 --1/2 --4 21/14 91/7 181/2 18 21/21 9V'7 9 9
1/5 1/10 --7 --1/5 1 2 1/10 --7
1/14 91/7 91/2 9 1/105 91/7 9 93/5
Y = - - I , I=3/2
. . . . . . . , , , - T i | 1 . . . . .
(rx, Ix) : (Ya, I~) 81 64 35 35" 27 10"
1 2 - - ~ 5 1/5 --~/15 --1 (1, 3/2) : (--2, O)
4 1/21 41/3 2V'6 21/14 V~6 Y - - - 1 , t = 1/2
(Iq, fi) : (I'~, I~) (1, 1/2) : (--2,0)
81 64 35 27 10 8
1 1/2 --1 - - V 6 2 1
V;;7 1/El V'---'3 1/35 1/21 V15
Weak nonleptonic decays of 3/2 + isobars in SU(3) 75 References
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