Charm changing weak decays 1/2+ → 3/2+ + 0−/γ in SU(4) and SU(8)w

Pram~na, Vol. 11, No. 3, September 1978, pp. 333-351, ~) printed in India

Charm changing weak decays 3/2 + + 0 - / ' / i n SU(4) and SU(8)w

RAMESH C VERMA and M P K H A N N A Department of Physics, Panjab University, Chandigarh 160 014 MS received 9 December 1977; revised 17 April 1978

Abstract. Weak decay modes (1/2 + ~ 3]2 + + 0-]y) of charmed baryons are studied.

Relations among the various decay amplitudes are derived in isospin, SU(3), SU(4) and SU(8)w symmetries. Sextet dominance in SUO) forbids B(3) --~ D(10) + P(3*) decays. 20" dominance in SU(4) specifies all the decays in terms of fi- decays. Weak decays of ~a *++ and El- are also discussed. SU(8)w symmetry predicts a (~') = 0, which is consistent with the experimental value.

Keywords. Weak nonleptonic decays; charmed baryons; SU(4) and SU(8)w.

1. Introduction

Weak nonleptonic decays of the type B(1/2 +) -+ B(1/2+)+P(O -) have been discussed in higher symmetry frameworks (Iwasaki 1975; Altarelli et al 1975; Gupta 1976a;

Verma and Khanna 1977 a, b; Karino 1977). In this paper we wish to discuss the weak mesonic and radiative decays: B(1/2 +) --> B(3/2+)+P(O-)/~. The charmed quark being the heaviest among the four quarks, the decays would be allowed in the charm chang- ing mode. In the next section, we comment on the possible masses of the charmed baryons. We note that B(6) multiplet can decay to B(3*) via strong and/or electro- magnetic interactions and so their weak decays are not expected to be interesting.

We study the weak decays of B(3*) and B(3) in the channels:

B(3*)~D(IO)+P(9);

B(3) ~ D(6)+P(9) and B(3)-~ D(10)+P(3*) allowed energetically. Weak Hamiltonian is described in §3. A C : A S = - - I mode is Cabibbo-enhanced. From among the Cabibbo suppressed modes

AC---1, A S = 0 and AC---AS=--I

the AC---1, AS-:--0 mode may be enhanced due to the SU(4) 15-admixture.

(Verma and K.hanna 1977c). So the decays corresponding to A C--- A S : - - 1 are not discussed. In § 4 several decay amplitude relations are obtained in the SU(2), SU(3) and SU(4) symmetries assuming the GIM model of weak interaction. H AC=AS

w

obeys A I = l selection rule. The A C = - - I , A S = 0 Hamiltonian contains both the A I - - 1 / 2 and AI----3/2 pieces but we assume A I : l / 2 enhancement. In SU(3) we obtain sum rules with and without sextet dominance. SU(4) symmetry relates the decays in different channels of B(3) and B(3*) multiplets. CP invariance further 333

334 Ramesh C Verma and M P Khanna

Table 1. MasseS of charmed baryon isomultiplets in SU(4)

1/2 + Multiplets Mass GeV 3/2 + Mulfiplets Mass GeV

f ~'x 2.50 (input) [" ~'x* 2.50 (input)

B(6) ~t 2-69 D(6) l ~t* 2.65

f~l 2"88 f~x* 2"80

~ ~, 3"75 ~ ~," 3"77

B(3) D(3)

~a 4"00 ~a* 3"92

A'l 2"26 (input) D(I) fls* 5.04

B(3*) ~x' 2"49

relates these decays with those of O,-. /)(3/2 +) ~ B(1/2 +) q- P(0-) decays of ~a *++

( C : 3 ) singlet are also discussed. In §5 we study all these processes in spin symmetry considerations. Adjoint representation admixture to weak deca~ys is considered in § 6. Summary and conclusion are given in the last section.

2. Mass spectrum of the charmed baryons

Various mass relations have been obtained in quark models and in the frameworks of SU(4) and SU(8) (Hendry and Lichtenberg 1975; Lichtenberg 1975; Franklin 1975; Verma and Khanna 1977d). Experimentally a state has been observed (Cazzoli

et al 1975; Knapp et al 1976) at 2"26 GeV decaying to A and 3¢r. Another state

decaying to the first one and a pion has also been observed at 2.5 GeV. Taking the first state to be A1 '+ and second state to be (i) 271"° (3/2 +) or (ii) Z'1° (1/2 +) following mass spectrum can be obtained (Gupta 1976b).

We see that B(6) can decay to B(3*) and pseudo-scalar meson/photon through strong/electromagnetic interactions (Kobayashi et al 1972; Gupta 1976c). The remaining charmed multiplets B(3*) and B(3) may decay through weak interaction via B(3*)~/9(10) -kP(9), B(3) -~ D(6) + P ( 9 ) and B(3)-~ D(10) q-P(3*) channels.

These decay modes would be allowed in mass spectrum of charmed baryons as predicted by de Rujula et al 1975.

In the Jr=3/2+ charmed isobars, f~a *++ seems to be stable against the strong electromagnetic decays D(3/2 +) -~ B(1/2 +) -q-P(0-)/y. Therefore the weak decays of

~ a *++ would be interesting and we shall discuss them too.

3. Weak decay Hamiltonian

In a current ® current model, SU(4) weak Hamiltonian Hw transforms like 15® I5.

Charm changing weak decays

335 Since/-/w behaves like [3, Jr]+, SU(4) weak Hamiltonian is reduced to:

H w N 1 5 : (3+3*) + (8+1)

I ~ C l = l

A C = O + 2 0 " : ( 6 + 6 * ) + 8

IAcI

^{= 1 C = O}

+ 84: (3+3"+15+15") + (27+8+1)

(i)

IACl=l A c = 0

where SU(3) decomposition is given in brackets. GIM model forbids Hw xs to con- tribute. However

AC=-4-AS

decay modes do not occur in 15 representation of SU(4) and A C = - - 1, A S = 0 mode may get enhanced through possible 15-admixture (Branco

et al

1976; Bajaj and Kapoor 1977; Shin-Mura 1976). We discuss A C

= A S = - - I and A C = - - I , A S : 0 decays only.

3.1. Isospin selection rule

A C = A S = - - I decays satisfy A I = I selection rule while A C ~ - - I , A S = 0 decays contain both the A I = l / 2 and 3/2 pieces. We obtain relations among the decay amplitudes assuming A I : 1/2 enhancement.

3.2.

SU(3) framework

In SU(3) we first obtain amplitude relations by assuming sextet dominance which follows from 20" dominance in SU(4). Here we notice that

B(3)~D(IO)+P(3*)

decays are forbidden totally since the direct product (10*× 3 × 3) does not contain sextet representation. Similarly all the decays involving 7' are also forbidden.

These decays can arise through 15 component of 84. Sum rules are also obtained for the SU(3) weak Hamiltonian H'~ *+15.

3.3. SU(4)

framework

SU(4) symmetry relates all the three channels B(3*) -+ D(10) + P(9), B(3) ~ D(6) + P(9) and B(3) ~ D(10) + P(3*). 20" dominant weak Hamiltonian has the following

parts:

H~°"-:al/2 ~abmn Dcep B[m, pale tt[c,d][a'b]

(2a)

d //la, b] - (2b)

+ a~[2 ~apmn Dcep BIm'n]e P b

[c,d]

336 Ramesh C Verma and M P Khanna Table 2. ( A C = A S = - - 1) decay amplitudes 2.1 B(3*) ~ D(10) + e(9)

Decay H w t°' ' H w sz

At A t '+ --* W*°K + ad3v:2 b~]~/2 + b d ~ 2

A t 2",*°*r + all6 btl2 -- b'sl6 + bd2

At ~*+*r ° at/6 bt/2 -- b's/6 + bu/2

As ~'*+*1 ax/2V'3 bi/2"v/3 +bt'/6"V'3 -- b s / 2 ~ 3

As ~*+*l' 0 b~/2v/6 -- bt'/3V'6 + bd~/6 + bd2x/6

As A++K - -- at/~/6 3bdv'6

A~, A + & "° -- at/3 V'2 bdv~2 As W(+ --. ~*%+ 0 -- b ' t / 3 v ' 2

/Is Z'*+X '° 0 b s ' / 3 v ' 2

Alo ~x '° -+ FZ-K+ atl~/6 3bdv'6 dxt ~ * - , r ÷ a z / 3 ~ 2 b s / ~ 2

At~ W*% ° atl6 -- ba'/6 + bs/2

Ats ~*oq al/2x/3 -- bt/~/3 + bt'/6x/3 -- bd2~/3

At, ~*°~1' 0 b t / 2 ~ 6 -- b(/3V'6 + bs/V6 + b,/2~/6

Axs ~ * + K - -- ax/3V'2 b,l.v/2 + bd~/2

Axe ..~*°K° - at~6 bs/2 - ba'/6 + bd2

bs' =: bs - b,

2.2 B(3) ~ D(6) + e(9)

Decay H w to'' H w s*

Air ~S + -'* fix*°K + ax/v'3

A . ~-t*% + a , / V ' 6 - at~v/6

Axe ~ t * + # at/2V3

Ate ~t*+*l atl2

Art ~1"+*? ' 0

A , ~x*++K - -- axl.v/3 Ats ,.v't*+KO -- at/v'6 + at/v'6

bd~/3

bdV6 + bd~/6

-- ba/2~/3 + be/2V'3

bs/6 -- bd6

bt/2~/2 - bd3~/2 + 5b./6v~2 + bd3x/2

bd ~/ 3

bd~/6 + bd~/6

As, ~++ --* Wz*+*r + -- atlv'6 bl/~/6 + ba/v'6

,'Its ~ l * + + K "° ai/vt3 bz/V3 Ast ~ t + -* ~ x * % + -- at/v:3 bl/~/3

Am .~z*+K ° at]v/6 bl/~/6 + bt/~/6

2.3 B(3) - . D(10) + P(3*)

Decay Hwt°" HwS,

Ats ~ t +'-* ~*°F+ 0 b=/~/3

ABe ~*°D+ 0 b=l~/6 + b41~/6

Aao •*+ D ° 0 btl'v: 3

Atx .~t ++ --* ~*+ D+ 0 bd ~ 3

Am ~ t + -'* .W*°D + 0 bd'v/3

Charm changing weak decays

337

CP

invariance expresses the decay amplitudes in terms of those of ~ - . We also consider the weak Hamiltonian

Hw ~°" + ~,

where Hw s4 has the following pieces:

H'ws4 = - - b : t / 2

Ceamn Depc B[m'n]p Pbd Hla:b)d) -- b2/4 ¢apmn Decd B~ m'n] PPe n(c,(a,b)d)

"-~ b3/4 ¢apmn Decd B elm' n] PPb H(c,(a, d)t5) Decd B[m'n] PPb H(a'b)

bJ4 Ceamn p (c, d)

~ b5/2 ¢apmn Depc B~ m'n] pd H(a,b)

e (c, d)"

⁽³⁾

In the tables 2 to 5 we have given contributions to the weak decays, from different components of SU(4) weak Hamiltonian.

4 . D e c a y a m p l i t u d e r e l a t i o n s

4.1. A C : A S : -- 1

decay mode

A C : A S decays of 1/2 + baryons are denoted by set of amplitudes A's in tables 2.

A 1 to A16 describe the decays of B(3*) multiplet. A17 to A27 and A2s to Aa2 represents the decays of C - - 2 multiplet in the channels: B ( 3 ) ~ D(6) q- P(9) and B ( 3 ) ~ D(10)

+ P(3*) respectively.

(a) lsospin selection rule

tt AC=

w AS Satisfies a A I = 1 selection rule which gives Aa : As

(V'3) ,47 : A6 ,411 - - ( ~ / 2 ) A12 = ,48 .415 - - (V'2) .416 - - .4~

AlS -- (~/2)

AI9 : A2a

A22 - - ( V 2 ) A23 = - - A25 ABe ^-- (~/2) A29= ^-- ,43i.

(4a) (4b) (4c) (4d)

(4e) (40 (4g)

338 Ramesh C Verma and M P Khanna (b) SU(3) framework

In addition to the relations (4) of isospin, sextet dominance gives:

0 . : ./15 : A 8 ~--/19 : A14 : / / 2 1 : "42s = "/129 = A3o = A81 ~--- A s 2 ('Sa)

( V 2 ) a l l = a~o (5b)

( v ' 3 ) a l = ( C 6 ) an = (V2) A4 = - - a s = a l s = - - (~/3) a ~ ( 5 0

A l v = 2ax9 : (2/C3) A~o = -- Ann (Sd)

- - ( % / 2 ) A u = A ~ = - - A26 = ('V/2) A27. ( 5 e )

Notice that the decays B(3) ---> D(10)+P(3*) are forbidden.

Inclusion of H , ) 5 maintains (4) and (5b), other relations are modified to:

Ax = (V2) An -- As (6a)

(3.V'2) A, -- (a/6) An -- 2As -- (23/3) As (6b) A 5 --- A1, - : 1/6 [(.v/6) An + As + (.v'3) As)] (6c)

A 9 = - - A s (6d)

6Azs = -- (4~/3) A2 + (V'6) As -- (2V'6) A s + (3~/6) A n (6e) ( ~ 6 ) Als = (V'6) As + A6 - - (V'3) Axx (6f)

A17 = 2Ax9 -I- (V'2) A~t -- An6 (6g)

( ' ~ / 2 ) A18 - - AX7 = A2e (6h)

(v'3) Ano = Ax9 -- A2~ + (V'2) A u -- A~ (6i)

(2V'2) A23 = 2A22 + An5 -- A u (6j)

(V'2) An, -- A ~ ⁼ (V'2) Aa4 -- An6 (6k)

B(3) ---> D(10+P(3*) may arise through Hw xb and obey the relations

Az~ = Ant (61)

Aso = Ann (6m)

('V:2) ans -- A3o = aax. (6n)

Charm changing weak decays 339 (c) SU(4) framework

In SU(4), the different channels get related. 20" dominance, in addition to (4) and (5) gives:

(~/3) Aa ^{= A 1 9} (7)

while the total SU(4) weak Hamiltonian (20"+84) satisfies (4), (Sb) and (6). In addi- tion we get:

(V'2) Au - - A ~ = - - (V'6) As (8a)

(3'V/2) ,419 = - - ,46 "1- (3V'3) As q- (2X/3) All (8b) (3/'V ~2) -432 = ~`46 - - ('V z3) `411 ^(8c)

x/3-~ Aso = ( V 2 ) ,43 - ,4s - ,411.

(Ca)

Relations obtained till now are valid for both the parity conserving (pc) and the parity violating (pv) modes. In SU(4) symmetry, CP invariance relates these decays (1/2 +

~ 3 / 2 + + 0 -) with (3/2+--> 1/2+-t-0 -) decays. 20" dominance completely specify these decays in terms of Q - decay amplitudes through the relations:

A3 ---- -- (1/2V'6) Q~r cot 0 (93)

A ~ = ^-- (l/x/6) ^{Q -} cot 0 (9b)

for pe mode.

Parity violating (d-wave) decays are expected to have very small amplitudes due to large centrifugal barrier. However, pv decays are also related in similar manner as (9), except that now minus sign is to be replaced by plus. Inclusion of 84 does not lead to any useful relations.

4.2. A C = - - I , A S : 0 decay mode

These decays of 1/2 + baryons are denoted by B's in tables 3.1 to 3.3. B 1 to B2a describes the weak decays of B(3*) multiplet. B~ to B43 and B~3 to B51 represent the decay amplitudes in the channels B(3)--> D(6)q-P(9) and B(3)~D(10)-~P(3*) respectively.

(a) lsospin selection rule

Assuming A I = I / 2 dominance, we obtain:

- - (V'2) B1 : B3 (10a)

( ¢ 6 )

~

= ( V 3 ) B, = - ( V 2 ) B7 (10b)

Table 3. (AC=-I, AS=0) decayampftudes 3.1. B(3*)-+D(10)+P(9) Decay Hw =¢" Hw s" Hw x~ B~ A~ '+ ~ 2~*°K + B~ ~*+K ° Ba A °~+ B, A+~ ° B~ A+q Be A+~/" B, A++,, - Bs ~'+ "-> ~*°K+ Bo E*% + B~o ~,,+~.o Bn ~'*+~/ Bx= 27*+*¢ Bxs A + +K - Bx, A +K'° B~ ~x '° -'> ~*-K + B~e ~*°K° B~ ~'*-~r + B~s E *%0 B~. ~,o.q B~e Z'*°~ ' B,~ E*+,, - B,,,, A+K - B=~ A°K ° a~]6 ba/2 + bs'16 + bs/2 -ad3~/2 b/3C2 + bd~/2 ad3~/2 bdC2 -- b/13~/2 + bd~/2 ad3 b,/2- bs'/6 0 bd2~/3 + b//6~/3 + bd~/3 0 b=/2~/6 -- b//3~/6 + bJ~/6+bd2~/6 --ad.v/ 6 3bJ ~ 6 ax/3V2 b=/~/2 -- b='13~/2 + b6[~/2 ad6 b=/2 + ba'/6 + bd2 ad6 b=/2 + bd2 ad2v'3 b=/2v'3-b,'/3C3- bd2C3 0 bd2~/6 - bs'13~/6 + be/~/6 + bd2~/6 -axle~6 3bJ ~/ 6 -ad3~/2 --b'a/3~/2 + bd~/2 2ad3v'2 2b,1~/2 --ad3v'2 bd~/2 + b,'/3~2 + b~/2 2ad3~/2 2bd~/2 at/2v~2 --b=/2~/2 -- bs'/6V'2 + bJ2~2 ad2~6 -bd2~/6 - b/16~/6 + b=12~/6 0 b=/2~/3 -- b,'/3~/3 + bd~/3 + bd2~3 ad3~/2 bd~/2 + b~[~/2 --ad3~/2 bd~/2 + bs/~/2 -ad3~2 b,/~/2 - b,'/3~/2 + b~/~/2 Cd6 - c,13 -Cd3~/2 + 2Cd3V2 Ca13~/2 - 2Cd3~2 Cd3 - 2Cd3 0 0 --CJ~6 + 2C,[~6 --C,13~/2 + 2Cd3~2 - Csl6 + Cd3 -C,/6 + C,13 -C,/2~/3 + C,/~/3 0 Ca/~/6- 2Cd~/6 Cs/3~/2 - 2C,/3~2 o --Ca/3V'2 + 2Cd3v'2 0 C816~/2- C,/3~/2 -Cd2~/6 + Cd~/6 0 -Cd3~/2 + 2Cd3~/2 C813~/2 -- 2Cd3~/2 Cd3~/2 -- 2Cd3~/2

C3

3.2. B(3)~D(6) P(9) i Decay Hw to" Bs, $~s+ -~ ~t*oK + axle6 + atl~6 B~, .=.x* +K ° --at/~/6 Bl6 ~vIt*O~r+ at/ ~/ 3 -- ad ~/ 3 B~v ~vt*+~l~ ax/~3 -- a~2~/3 Bu ~ x* + *l ad 2 B ~ e ,~,1" + ,I " 0 Bee ~x*++~r - --atl~3 Bat ~l ++ "~ ~t *+K+ at/~/6 Bat ~t*+,r+ -- a~/~/6 Baa ~'x*++,re --at/~6 Ba, ~t*++*l ail ~ 2 B~s ~'x*++q' 0 Bs~ fZ~ + "-" ~x *°K+ --axlV3 + a,l~/3 Bs~ ~x*%+ --ax/ ~/ 6 -- ad ~ 6 Bat 5~x *+~e --at/2~/3 -- at/2~/3 Bs~ ~t*+*/ --at~2 + at/2 B,o ~t*+,( 0 B~t ~'x*++K- atl~/3 B~ ~t*+K,° at/~/6 Hw 94 --bxl~/6 + bd~/6 -bdV6 + bJ~/6 bd~/3 + bd~/3 -bd2V3 - bd2~/3 bd2 + bs/6 + bd3 b,12~/2 -- b3/3V2 + 5bd6~2 + bd3~/2 bi/ ~ 3 -bd's/6 - bd~6 bl/~6 + b,l~/6 --bd~6 bdv'2 0 --bd~/3 -- bd.s/3 --bt!~/6 -- bd~/6 --btlXvt3 - bd2~/3 --bt/6 + bal3 + bd6 bs/2x/2 -- bd3v/2 + 5bt6V2 + bd3x/2 --bd~/3 b,/~6 - b,~6

C,1~/ 6 -CdV6 + GI~/6 GI~/3 Cd2vt3 + Cs!2~/3 - Cd2q3 Cd6 -- Cd6 + Cd6 G/2V2 + G/6~/2 - G/6~/2 + 2G/3~/2 -CdV3 + Cdx/3 cdv'6 - c81~'6 + cd,v'6 Cd#6 - cd,v'6 + Cdv'6 cd,v'6 - cs/~6 + c,1~6 Cd3,v'2 - cd3~/2 + Cd3ff2 Cd2 + cd6 - cd6 + 2Cd3 c,1~/ 3 c,1~ 6 C,/2V'3 c,16 + c,13 - Cd3 c,12v'2 + CJ6~2- Cd6V'2 + 2Cd3,v'2 -C31,v"3 + CdV3 -c, IV6 + C,I~/6

t~ e~ t~ 4~

342 R a m e s h C V e r m a a n d M P K h a n n a 3.3 B(3) --. D(10) + P(3*)

D e c a y H w ~°'' HwS 4 HwX6

B4s .~a + ---> Z * ° F + 0 b2/~/6 -- b d ~ / 6 -- C d V 6 B44 A°D + 0 b d v ' 3 + b d ~ 3 -- C d x / 3

B46 A + D ° 0 b ~ / v ' 3 - - C d v ' 3

B4 6 ~ Z + + -"> .Z'*+F + 0 -- b,l~/3 -- C d ~ / 3

B47 A+D + 0 b d ~ / 3 -- C4/v'3

Bas A + + D ° 0 0 - - C4

B , , ~ + ~ ~ * ° F + 0 - - b a / x / 3 - - b d x / 3 - C d ~ / 3

B,o ~P*°D+ 0 -- b s / V 6 + b d ~ 6 - - C d V 6

Bst ~ ' * + D ° 0 - - b ~ / V 3 ^-- C d V 3

o = s . = B e

~ = Blo = (v'2) (B17 - B~s)

Bs = Bls + B16

2 B l s -+- B2s = ax7

(v'2) t% = Br,

B26 - - 2B27 = ~30

( V2)

B~ = &4

(V2) ~2o = &5

&2 = B~3 a~7 - - ( v ~2) B~s

(.V2) &~ = B~

B,4 = 8,~ = a4, = 0/x/3) B~

(v'2) Bso = & v

Ooc) Ood) (1o0 (lof) (lOg) (1Oh)

00i) OOj) (lOk) (100 OOm) OOn) (I0o) OOv) OOq) OOr)

Oos)

(lo0

Charm changing weak decays

(b) SU(3) frame work

In SU(3), sextet dominance o f weak Hamiltonian, in addition to (10), yields:

343

0 --- Bz~ -~ B~o (1 l a )

0 = / ~ 9 : : / ~ 3 5 = &o =/~43 = B . = / ~ =/~4e = B47 = / ~ 8

B4~ : B6o - B s l ( l i b )

( I / x / 2 ) B a = B 9 : ( I / v ' 3 ) B n - - (2/V'6)B19 --- (I/x/2)Bx4 : _ (1/X/6)B~a - : - - (1/X/2)B22 : - - (1/X/2)B2a

& = 0 / 2 ) B 4 = - (1/~/2)Bxo = ( 1 / V 2 ) B ~ ( V 2 ) B 1 = B 8

B17 ~--- B15 ( ~ / 3 ) Bx9 = Bls

Ba~ --- - - ( V 3 ) Baa B38 ---- - - B26

B37

--= - / ~

B41 ~ - _ B3o

(v'2)B38 = B37 ( V 2 ) B ~ = A3o

- (2/v:6) B~8 = B ~ 3 = B ~

Note that B(3)--> D(10)q-P(3*) are forbidden. A C : - 1, Cabibbo suppressed and are related to ~ C : A S : -- 1 as:

B4 ---- -- 2A2 tan0 B32 ~ - - A ~ tan0 Bao = 2 Ax~ tan0.

( l l c ) (11d) (1 le) ( 1 1 0 (11g) (1111) (l li) ( l l j ) ( I l k )

(llI)

(11m) ( l l n ) (1 l o ) (1 l p) ( l l q ) 0 1 0 A S = 0 decays are

(12a) (12b)

02c)

344 Ramesh C Verma and M P Khanna

I n the presence o f Hw 15 c o m p o n e n t in SU(3), relations ( l l f ) to ( l l m ) are valid.

O t h e r relations are modified t o :

(V'2)Bx q- B~ = - - (1/2)B a -k (1/x/3)B~ -b (3/~/2)B9 (13a)

( 2 x / 2 ) B 4 = 2 B a - - ( 2 / v ' a ) B 7 (13b)

B e = (1/X/2)B2o = B12 (13c)

(2V'2)B~o = Ba + ('V'2)B9 (13d)

B a -~ B14 : (3/2)B a + (1]X/3)Bv - - (1/X/2)B 9 (13e)

B2 -t- B14 = (2/~/3)B7 (13f)

( ~ / 2 ) B 1 9 - - B u = - - ( 1 / ~ 6 ) B 3 + (1/3X/2)B 7 - (1/V'3)Ba-b(3/2X/2)BIv (13g) B2a - - B14 = B l e - - B2 = (1/2)B3 q- ( 1 / V ' 2 ) B o - - (1/2)B1~ (13h) V 3 (B~a - - Bz~) = (V3)Bx4 - - Bx3 = ~/6(Blo - - Bg) (13i)

2B21 = B 3 + (2/V'3)B, + (X,/2)B8 - - B~7 (13j)

B4o---B2a (13k)

- - BaT = 2(B25 -k Bs2) (131)

(v'2)B,3 - - B4~ = B , 5 = - - B s ~ (13m)

- - B46 = B4~ =B4a - - B45 (13n)

B4a = 0 ( 1 3 o )

B6o = - - B4a ( l a p )

B4, = - - B , , (13c0

A C = - - 1 , / k S = 0 decays a r e r e l a t e d to A C = A S = - - 1 as:

B7 -= - - A s tart0 (14a)

B17 = - - 2A11 tan0 (14b)

( V ' 2 ) B 7 - - B 3 = 2A 8 t a n 0 (14c)

B4~ = Aa0 tan0 (14d)

B47 = A31 t a n 0 (14e)

Charm changing weak decays 345 (c) SU(4) symmetry

20" dominance maintains relations (10), (11) and (12) and gives

Bso = - ( V 3 ) B,. (15)

Weak Hamiltonian (20"+84) obeys relations ( l l f t o llm), (13) and (14). In addition it gives:

(V'3) B~ = (1/V'2)B z -t- B9 -- (1/V'2)B17 (16a) B~6 - ( V 2 ) (B32 + B25) = (3/Ve)B~7 - ( V 2 ) B7 (16b)

Bs0 = -- (I/V'6)B17 q- (2V'2/3)B7 (16c)

(V'2) B2s -- Ba0 = (3/V'6)Ba -- (V'3) Ba. (16d) Relations (I0) to (16) are valid for both the pv and pc decays. CP invariance with 20" dominance relates these decay with ~ - decays as:

B, = ± ( I / V 6 ) [/,- (17a)

K

(V'6) Ban = 4- [~- (17b)

where -a t- and -- sign stand for pc and pv modes respectively.

4.3.

B(I/2+)-~

D(3/2 +) + y

decays

We have discussed the charm changing weak radiative decays of the type B(1 [2 +)--~

B(1/2+)+y elsewhere (Verma et al 1977). The decays B(I/2 +) --> D(3/2+)+y can be calculated from the weak Hamiltonian (2) but in this case Pb ° should be replaced by photon transforming as 15 ~ 1. It is to be noted that singlet component of photon gives null contribution here. Moreover, the second term (2b) of weak Hamiltonian does not contribute to these decays. We obtain the following relation for Cabibbo- enhanced mode ( A C----AS).

n (A1 '+ ~ Z '*+ ~,) : M (_~'° a ~ ~,0 7) =: (1/~/3) M (~2 + --,'- ,~1 *÷ ~,). (18) Relation is valid for both the pc and pv modes, since Hamiltonian (2) is independent of the behaviour of Hamiltonian under charge conjugation. It ofily relates these decay with D(3/2~)~B(l[2+)q-~ ,, out of which ~--->-~-q-~, is observable. But the decay ~ - ~ ~-~, obtains contribution only from (2b) part of weak Hamiltonian.

4.4. ~ a *++ decays

Most of the charmed isobars may decay through strong and or electromagnetic interactions. Only ~ a *++ ( C = 3 ) singlet is expected to decay via weak interaction P.--8

346 Ramesh C Verma and M P Khanna Table 4. ~**++ decays ( A C -- A S ---- -- 1 mode)

Decay product Hw s°'" Hw u

Ass $~t+D + 0 --(~/2)b~

As, [~x'+D + -- 2/'e'6 at 0

Ass ~s++K ° a, bx

Ass ~e+*r + as bx

Table 5. t~, *++ decays ( A C ---- -- 1, A S --- 0 mode)

Decay product H w e°" H w s' Hw x5

B.e ~x+F+ o - ( 4 2 ) b~ Cdv'2

Bss Zt+D + 0 (C2) bs G / C 2

B61 ~'x++D ° 0 0 - C5

Be5 ~,'+F + - - 2a, I V 6 0 - - C d V ' 6

Boo At'+D + -- 2at/V'6 0 Cd~/6

B67 l'~*+K + - - as bx C1

Be, ~,+,r+ aj - - bl Cl

Bs, ~,÷+~ adx/2 bx/x/2 Cd~/2

Boo .'~++q -- 2adv'6 -- 3bt/~/6 Cx/~/6

Bet .~a++q ' 0 0 1/(2~/2)(G -- 3C, -- 3C,)

t h r o u g h t h e c h a n n e l / 3 ( 1 ) - o - B ( 6 ) / B ( 3 * ) + P ( 3 * ) a n d B ( 3 ) + P ( 9 ) . Sextet d o m i n a n c e f o r b i d s D(1)-o. B ( 6 ) + P ( 3 * ) p r o c e s s e s . V a r i o u s a m p l i t u d e s a r e g i v e n in t a b l e s 4 a n d 5.

(a) A C = A S = - - I decays

H e r e A I = I s e l e c t i o n r u l e gives n o r e l a t i o n . A3a = 0

A~s = A86

(b) A C = - - I , A S = 0 decays A I = 1/2 selection r u l e g i v e s :

- ( ~ / : ) B ~ = B . Be6 = a6s (v'2) Bs~ = B~

S e x t e t d o m i n a n c e f u r t h e r g i v e s :

0 = Be2 = B ~ = B54 = Be1

- B~7 = B6~ = ( C 2 ) ~s~ = - v 5 ) - ~ s ~

B ~ ~--- - - .436 t a n O Bee ---- - - Aa4 t a n 0

S e x t e t d o m i n a n c e l e a d s t o :

(19a) (19b)

(2Oa) (20b) (20o)

(21a) (21b) (21c) (210

Charm changing weak decays

H . 6.+1s maintains (20b) and (21d). In addition it yields:

o = B ~ = B 6 t Bs2 = / ) s 6

(2V6) Beo = -- (5V2) B, e + B68 B ~ = As8 tan 0

( V 2 ) Bs9 = - - Aa tan 06

SU(4) with 20~ dominance, in addition to (19), (20) and (21) gives:

A~ --- ~ cot 0 A64 ---- ~lr cot 0.

Relations (19) to (23) are valid for both the pv and pc modes.

347

(22a) (22b) (22c) (22d) (22e)

(23a) (23b)

5. SU(8)w Considerations

Since ordinary SU(8) is valid only in the static limit, we employ SU(8)w symmetry (Horn et al 1965; Carter et al 1965; Lipkin and Meshkov 1965) to discuss these decays. Here GIM weak Hamiltonian transforms like 720 and 1232 representations.

Weak Hamiltonian is described in detail in Verma and Khanna (1977a and b). In SU(4) × SU(2)w structure it can immediately be seen that pc Hamiltonian transforms as(20 ~, 1) and (84,1) components of 720 and 1232 representations and pv Hamiltonian as (20", 3) and (84, 3) components.

If 720 dominance is assumed (as an extension of 20" dominance in SU(4)), all the pv decays of charmed and uncharmed baryons are forbidden. In the pc mode B(3*) ~ D(10)+P(9); B(3) --> D(10)+P(3*) and B(1/2 +) -~ D(3/2+)+~ and ~ - -+ A K - modes are not allowed. But ~---> AK- is an observed decay hence 1232 part of weak Hamiltonian must be included as in the case of nonleptonic decays 1/2 +--> 1/2 + + 0 - (Verma and Khanna 1977a). Now if 20" dominance is assumed at the SU(4) sublevel of SU(8)w all the results obtained in SU(4) are reproduced. PV decays B

(1/2+)

~ D(3/2+)+P(O -) still remain forbidden since 1232 does not contain (20~,3) component. Total GIM weak Hamiltonian (including 84 at SU(4) level) yieMs (4), (5b), (6), (8), ( l l f ) to (llm), (13), (16) and (22). In addition we obtain:

(i) pc decays

Ae = - - (V'3)All Bt7 = -- (2[x/3)/~

( V 2 ) B ~ s - - B~o = V'2[(V2)B3o - - B3d 0 = ,483 = Bs~ = B ~ = B54

( ~ a ) (24b)

(~tc) (24d)

348 Ramesh C Verma and M P Khanna (ii) pv decays

A e = (~/3)A n

Bx¢ ---- (2/X/3)B~ = Bs -- (v:2)B9

0 = / ~ 3 = / h 4 = B~ = / h 8 = B~I.

(24¢) (240

(24g)

(24h)

We notice that 12- decays and ~ * + + --> B(3) + P(9) and B(3*) + P(3*) are forbidden in pv mode, f2a*++--> B(6)+P(3*) decays vanish in pc mode. Thus assymmetry parameters for ~ - modes ( ~ _ - , ~0- and flk-) vanish.

6. Adjoint representation admixture to weak decays

We have considered elsewhere (Verma and Khanna 1977c) adjoint representation admixture to weak Hamiltonian in SU(4) and SU(8) for B(1/2 +) --> B(1/2 +) -k P(0-) decays. This admixture may be expected to arise from the large symmetry breaking even in the case of GIM model (Shin-Mura 1976 Igarashi and Shin Mura 1977) 15-admixture in SU(4) has also been proposed in the models of weak interactions (Branco et al 1976; Bajaj and Kapoor 1977).

In SU(4), 15-weak Hamiltonian can be written as

-1-½Cz "afmn Dabe Bib m'n] pC 11:

-']-½C4, "abmn Dace B[f m'n] peb Hfc. ₍₂₅₎ Hw 15 does not contain A C --- 4- A S decays and s o A C = ^-- 1, A S = 0 mode gets enhanced. Hw ^TM obeys the AI--1/2 selection rule, hence the relations (I0) and (20) are maintained. At the SU(3) level weak Hamiltonian transforms like triplet com- ponent of HwlL Neglecting 15 component at the SU(3) level, Hw s+8* obeys (1 l a), (llc), (lid) and (llf). In addition we get:

885 = ( x/2) B~o = ( v'2) B~o

('~/6)B2s = 2Bat -- Bsa

~sI--B32----Bse--B~

(26a) (26b)

(26c)

(26d) (26¢)

Charm changtng weak decays 349 The B(3) --> D(10) + P(3*) and ~ a *++ -~ B(6) + P(3*), forbidden by sextet dominance, now arise through triplet component and obey:

(1/2)B~ = B44 = B45 ---- B4e = B47 = (l/1/3)B4s = B4~

(1/2) Bs0 = B~I (27a)

B~3 =/~5~ (27b)

3Bsv -- Bss = (21/6)Beo. (27c)

In SU(4) symmetry Hw 15+ zo", in addition to SU(3) results gives:

B~ -- (1/2) BaT = 1/6 (B 8 + 1/2 B,) (28a)

B57 = f L - . ( 2 8 b )

All these relations (26) to (28) are valid for pv and pe modes. Most general Hamil- tonian Hw 15+z°"+~ does not lead to any useful relations:

In SU(8)w symmetry, 15-admixture is extended to 63 admixture (Verma and Khanna 19770). Neglecting 84 component at the SU(4) sublevel, SU(8) weak Hamiltonian (63-t-720+ 1232) gives (lla), (llc), (lid), (llf), (26), (27) and (28).

In addition it yields:

(i) pc decays

B4s=B~---3 [ ( 1 / 1 / 2 )

B3--Bg]

= ( 1 / 3 / 2 ) ( B ~ + ( 1 / 2 )

B3~--21/2 1~)

(29a)

1/6 (Bs8 -- Bs0 = 2 ~ _ - (29b)

1/3 (B s + (31/2) B.) ---- 2 ~ s - . (290)

(ii) pv decays

Decays of the type B(3*)-~ D(10)+P(9) and B ( 3 ) ~ D(10)+P(3*) are forbidden and can arise only through 84 component of 1232 Hamiltonian. B(3)-~ D(6)+P(9) decays are expressed in terms of one parameter obeying:

0 = f ~ F = B ~ = B3o ---- B41 = B4z (30a)

- f i _ - - - ( 1 / 2 ) a ~ --- B ~ = 2B~7 = ( 2 v ' 3 ) B ~ = ( 4 / 1 / 6 ) B ~ ,

= (1/2) B31 = (1/2) B82 --- (1/2) Bz3 = ( 1 / 6 ) / ] 3 4 ^{- ~} (21/3) B35

= Bze = (1/2) B37 = 2B33 = (21/3) B39 --- (21/6) B,o. (30b) It may be noted that most general SU(8)w Hamiltonian (ineluding 84 at the SU(4) level) forbids ~ - ~ AK- decay in pv mode. The parity violating ~ - ---> ,~rr decays earl occur only through the 63 admixture. Hence SU(8)~, symmetry alone predicts

350 Ramesh C Verma and M P Khanna

asymmetry parameter a ( ~ - - ~ AK-) to be zero. It is to be compared with the experimental value (Kocher and Wernhard 1974).

K- +0.36

a ( ~ - - - > A )=0"66_0.30 from 15events.

7. Summary and conclusions

In this paper we have discussed weak decay modes of J P = I / 2 + baryons other than the modes B(1/2+)~B(1/2+)q-P(O-)/y. Mass spectrum of charmed baryons suggests that the charm changing decays B(1/2+)--rD(3/2+)q-P(O-)/y are allowed energetically.

The charm changing decays of 1/2 + baryons to 3•2 + isobars are interesting to study as they involve a large number of decay products which can be observed experimentally.

Only parity conserving decays (p-wave) will be important, because parity violating (d-wave) decays are expected to be suppressed due to centrifugal barrier. Assuming GIM model of weak Hamiltonian we obtain several relations among the amplitudes of B(3*)--> D(10)+P(9); B ( 3 ) ~ D(6)+P(9) and B ( 3 ) ~ D(10)-{-P(3*) decays in the SU(2), SU(3) and SU(4) frameworks. Full H aC= AS obey A I = I selection rule,

W

H A C = - I , AS=0 contains both a I = l / 2 and 3/2 pieces and A I = l / 2 dominance is

w

assumed. At SU(3) level, sextet dominance forbids B(3)-~ D(10)q-P(3*) decays.

SU(4) symmetry relates different decay modes of B(3) and B(3*) multiplets. CP invariance with 20" dominance completely specify all these decays in terms of ~ - decay amplitudes. We have considered the 15-admixture to weak Hamiltonian.

This enhances the A C = - - I , A S = 0 decays and satisfy a l = I / 2 rule. We have also studied these charm changing processes in SU(8)w considerations. We observe that in the GIM model 720 dominance forbids all the pv decays, hence both the 720 and 1232 representations should be included in the weak Hamiltonian. Since the experimental data regarding the charmed baryons and their decays are meagre we are at present unable to decide about the structure of weak Hamiltonian.

Note added in Proof

Recently in a CERN experiment, (Gaillard 1978) the assymetry parameter for

~ - -~ A K - decay is observed to be a ( ~ x - ) ] = 0.06-1-0.14, which is consistent with zero, the value predicted by SU(8)w symmetry.

Acknowledgement

One of us (RCV) acknowledges the financial support given by CSIR.

List of Symbols SU(4) Tensors

Ba[ m, n] 20' multiplet of 1/2 + baryons Da~c 20 multiplet of 3/2 + baryons Pb a 15 multiplet of 0- mesons

H b ^~l

[a, b 1 I'/io ~r]

H(a,b) ( c,d)

~abmn

flit-

Charm changing weak decays

15 component of weak Hamiltonian 20" component of weak Hamiltonian 84 component of weak Hamiltonian Levi-Civita symbol for SU(4)

decay amplitude for the decay ~ - ~ .~%- decay amplitude for the decay f~- --~ A K -

351

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