Non-leptonic weak decays of charmed baryons

Non-leptonic weak decays of charmed baryons

V GUPTA

Tata Institute of Fundamental Research~ Bombay 400005 MS received 26 March 1976

Abstract. It is shown that the ~Ch = AS decays of a baryon sextuplet, triplet and singlet of SU (3), into meson and baryon, can provide simple tests of lhe iscspin and SU (3) transformation prop~:ties of the ACT, = AS non-teptonic interaclion in the Glashow Iliopoulos-Maiani scheme.

Keywords. Charmed baryons; non-leptonic decays; sum rules.

1. Introduction

The extremely narrow width of the new vector mesons at 3" 1 GeV (Aubert et al

1974, Augustin ^{et al} 1974, Bacci ^{et al} 1974) and 3"7 GeV (Abrams ^{et al} 1974) is at present understood on the basis of a larger symmetry group, than SU(3), under- lying the strong interactions of hadrons. A popular choice for the larger sym- metry group is SU(4) in which the new quantum number Ch, is called ' charm' (Bjorken and Glashow 1964; De Rujula and Glashow 1975). A direct test of the underlying SU(4)would be relations between stronger interaction quantities like masses (Borehardt et al 1975; Gaillard e t a l 1975) and coupling constants (Gupta 1976 a). However, at least some of the low-lying charmed hadrons (not yet found) would be stable with respect to strong decays and will manifest them- selves through their weak interaction decays. To study their weak decays one uses the weak interaction proposed by Glashow ^{et al} (1970) and this will be referred to as the GIM-scheme for short.

The lowest charmed mesons with charm Ch = -k 1 are expected to belong to the 3* representation of SU(3) and denoted by P (3*) for J r = 0-, etc. [We will refer to SU(3) and SU(4) representations by their dimensionality Ns and N4 and where necessary to avoid confusion denote them by N3 (Pl, P~) or N4 (Pl, Ps, ps) in the highest weight notation. We use the notation for multiplets, states, ere, used earlier in Gupta 1976 a]. The low-lying charmed baryons are expected to belong to two inequivalent twenty dimensional representations of SU(4) denoted by D(20) and B(20') for 0rv = (3/2)+ and (1/2) + respectively. The SU(3) multi- plets contained in B(20') are B(8), B(6), B(3*) and B(3) with Ch = 0, 1, 1 and 2 respectively, while D(20) contains D(10), D(6) and D(3) and D(1) with Ch = 0, 1, 2 and 3. B(8) and D(10) are the usual (1[2) + octet and (3/2) ÷ decuplet of baryons. The lowest lying baryons are expected to belong to B (3"), B (6) or D (6) as they carry the least amount of charm, namely C~ ---- -b 1.

277

278 V Gupta

However, it is not clear which of these would be stable with respect to strong interactions and electromagnetic decays. In fact, at present, in the absence of a mass scale (or the discovery of charmed hadrons) it is a moot point as to which of the charmed baryon SU(3) multiplets (D(N~) or B(N3)) will decay only by weak interactions.

The non-leptonic weak decays of B(3*) and P(3*), using the GIM-scheme, have already been discussed (Altarelli et al 1975 a and 1975 b; Gupta 1976 b; Kingsley et al 1975; Einhorn and Quigg 1975) and one finds simple relations, between decay amplitudes satisfying ACh---- A S rule (S = strangeness), which can provide a test of the GIM-scheme. However, the relations for B(3*) decays may become useless because depending on the mass breaking interaction the lowest lying baryons may belong to B(6) or D(6) (Note the higher of the two will decay pre- dominantly into the other electromagnetically). Further depending on spacing between the various SU(3) multiplets and the mass of Ch = -4- 1 mesons the lower one of B(3) or D(3) and even D(1) may be stable with respect to both strong and electromagnetic interactions and have only weak decays. Consequently in this note we discuss the following AC~ = A S decays.

B(6) ~ B(8) -1- P(9) (1)

B(3) ~ B(8) q- P(3*) (2 a)

B(3*) + P(9) (2 b)

-+ B(6) ÷ P(9) (2.c)

as well as D (1) decays at the isospin and SU(3) level and show that they can provide simple tests of the isospin and SU(3) properties of the full AC~ = A S weak inter- act.ion in the GIM-scheme. Since our analysis is at the SU(3) level it covers all the possible ACh= A S decays involving D(6), D(3), B(3) and B(6) and a meson (0% 1-, etc.) nonet with obvious appropriate replacements. The analysis presented here together with earlier work (Altarelli et al 1976 ; Gupta 1976 b ; Kingsley et al.

1975; Einhorn and Quigg 1975) will complete the discussion of the AC~ = /kS decays of all the low lying charmed baryons one may expect to find in the future.

In section 2 we briefly discuss mass relations and the ordering of the various baryon SU(3) multiplets in broken SU(4) and SU(3) symmetry and substantiate the above remarks about the various possibilities. In section 3 the non-leptonic AC~ = /kS weak interaction in the GIM-scheme together with its transforma- tion properties is presented. In section 4 the relations for the decays of B(6) given in (1) are given, while in section 5 we give the results for the decays of B(3) listed above. The decays of D(I) are briefly remarked upon in section 5 d. A summary together with concluding remarks are given in the last section.

2. Comment on ordering of baryon masses in broken SU(4) and SU(3)

In discussing mass relations one assumes that the SU(4) symmetry breaking inter- action transforms like the fifteenth component of a SU(4) 15-plet (i.e., like T44 in tensor notation) plus the eighth compouent of SU(4) 15-plet (i.e., like T3 a) which is also the eighth component of the SU (3) octet in the 15-plet. The T33 term gives the usual breaking of SU(3). For a meaningful discussion it is necessary to

define the SU(3) and SU(4) breaking parameters aa, b3, c~ and aa, b, and c4 in terms of which the masses of the various isospin multiplets in B(20') and D(20)

are given. These are defined by the mass terms

m o = mo ° D ~r° D~i~ ÷ c~ D '~a D~a -q- c4 D '~a D ~ , (3) and

rn B : mo B/~jk B,~ + a3 B i~3

Bi~3

~3~ Btz~

-~" a4 B i~4 Bi~4 + b4/]~a, Bt4~ ' (4)

for the B(20') and D(20) which are described by the tensors Bl~ and D,~ respec- tively (Gupta 1976 a). In this definition mo ° : mN* = 1 "232 GeV and me ~ = mN

= 0' 94 GeV and do not represent the average masses of the D(20) and B(20').

This definition of the parameters a~, etc. is convenient for discussing the ordering cf the masses as most of the parameters turn out to be positive. From the known masses one has c3 = 0"45 GeV, a3 : 0" 19 GeV and b3 = 0"23 GeV.

Our discussion will be based on first order breaking and linear mass formula for the baryons and we also neglect the small mixing between B(3*) and B(6) due to SU(3) breaking (Borchardt et al 1975, Gaillard et al 1975). Denote the masses of the SU(3) multiplet B(N~) and D(N3) by m" (N~) and m ° (N3) respec- tively. Since one has to have mD(6) > m r) (10) so that ca > 0 and one has the unique ordering

mD(1) ~- mD(3) > m°(6)

(5)

for D(20) in which mass increases with charm. In case of B(20') since mB(3 *)

> mB(8) one has ba > 0 however a4 could be negative but its magnitude is limited by 4 ] a 4 [ < ba because one expects m~(6) > roB(8). This gives the unique ordering.

m~(3) )- mS(3 *) > mS(6).

For aa > 0 threz cases arise depending on the relative m:~gnit.ldes of a4 and ba.

One finds the unique orderings.

m~(6) ~ roB(3) > mS(3*), m~(3) ~ mS(6) > mS(3*), m~(3) > mB(3 *) ~> roB(6),

if a~ ~ 2b4; (6a)

if a4 < 2b4, b4 < 2a4; (6 b)

if b ~ 2aa. (6 c)

In the literature (Borchardt et al 1975 ; Gaillard et al 1975) a further assump- tion is made that the two terms transforming like T~ a and T44 in the symmetry breaking interaction belong to the same representation. This would imply a~/az = ba/b~ = yB. Though c4/c3 = yO need not be equal to yS, in general, it is further assumed that y'~ = yD = y; so that knowledge of the only one para- meter fixes all the masses in B(20') and D(20). Since one expects SU(4) to be much more badly broken than SU(3) one expects y to be much larger unity. Note since b~--- 1 "2 as therefore for any y the ordering of the masses of the SU(3) multiplets is given by eq. (6 b). With a linear mass formula we find

P--6

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and

V Gupta

enD(l) > roB(3) ~> too(3) ~ mZ3(6) ~ m°(6) ~- roB(3*), 3 . 4 ~< y < 11 "8

(7 a)

m°O) > roB(3) > ,nD(3~ ~-- roB(6) > mB(3 *) ~> mD(6), y ~ 11-8. (7 b) Using the P(3*) masses ( ~ 2 " 1 7 and 2"22 GeV) predicted by Borchardt e t a l (1975) one finds that for 3 . 4 < y < 13 D ( 3 ) and for 11"8 < y < 13 D ( 6 ) would be stable with respect to strong and electromagnetic decays. It is very interesting to note t h a t in the above range 3.4 < y < 13 even D(1) with Ch = 3 (analogue in SU(4) of the t2- in SU(3)) turns out to be stable with respect to decay by strong interactions*. To illustrate these remarks the predicted masses for the isospin multiplets D(N3, I), etc. in D(20) and B (20') for y = 10 a n d 12 are. given in table 1.

T o sum up, it is clear that even in the above cne p a r a m e t e r model for masses and more so with more parameters (a4, b a, etc.), there is a case for studying the weak decays of a charmed baryon sextuplet, triplet and singlet of SU(3), in addition to those of B(3*) wl~ich have already been considered.

3. The non-leptonic ACh =- A S interaction

The hadronic weak current in the G I M - s c ~ e m e (suppressing space-time structure) is given by

Jh = cos O J.,3 ~- sin O J31 -~ COS 0,]3 4 - - sin OJ, 4, (8) Table 1. Predicted masses (in GeV) of ch~.rn-.ed brryen isospin, I, mrhiplels, in the one pe.rameter model, in the B(20') and D(20). Tl-e masses of D ( N , I ) and B(N3, I) :~xe denoted by mD(N~, 1) a.nd mB(Nz, 1) respectively.

y 10 12 y 10 12

mB(6, 1 ) 3.44 3- 94 roD(6, 1 ) 2" 73 3" 03

roB(6, ½) 3- 63 4' 13 m~(6, ½) 2" 88 3.18

roB(6, 0) 3.82 4" 32 mz~(6, 0) 3-03 3' 33

m~(3*, -~-) 2"89 3"24 lnDt3, ~) 4"23 4-83 roB(3*, 0) 2" 69 3" 04 roD(3, 0) 4' 38 4" 98 roB(3, ½) 4" 80 5" 56 roD(l, 0) 5" 73 6' 63

mS(3, 0) 5" 05 5" 81 . . . .

* Note wid~ a h-~rger y -- 20, the value determined for the vector mesons, (Borch~xdt et al 1975) and a linear mass formula all ihe charmed b~ayons in B(20 ~) and D(20) will decay by S'.roLg inle~actions, in which ce.se tt~.e coupling cor.st~.nt sum rules for (DBP)-couplings (Gvpta 1976 c.) would provide a rich testing ground of the underlying SU(4) symmet, y. In such a case only tee members of the P(3*) will decay by weak interactions and provide a test of Hcs.

281 where 0 is the Cabibbo angle and the indices indicate the transformation properties under SU(4), etc. Jh belongs to the fifteen dimensional representation of SU(4).

d~ and J31 are the usual AS = 0 and AS = 1 terms with ~_Ch = 0. The terms j 4 and j 4 satisfy the selection rules A C h = AS--- 1 and ± C h = 1, z~S----0 respectively, where S is strangeness. Tire non-leptonic interaction is given by the anti-commutator {Jh, J~,*} so that the leading charm changing term (since 0 is small) is

H c s = cos ~ O {J91, ,]4 z} --{- h.c. (9)

and satisfies the AC~, --- AS : + 1 rule. Further, Hcs satisfies a pure [ A1 I : 1 isospin selection rule. To see its SU(3) and SU(4) transfo;mation propelties, we write

Hcs = 1t- + H+, (10)

H± = ½ cos ~ 0 [(j1, j43} ± {j23, j41}] + h.c. ( l l ) Clearly both H+ and H_ satis~, a pure [ ~1 I = 1 rule. It is easy to see that (i) H_transforms as (6 + 6*) under SU(3) and as the self-conjugate twenty dimensional representation (0, 2, 0), denoted by 2_0", under SU(4)" (ii)H+ transforms as (15 + 15") under SU(3) and belongs to the 84 dimensional :epresentations (2, 0, 2) of SU (4). The fifteen dimensional representation of SU(3) which enters in H+ is (2,1) and its conjugate (1, 2) and will be denoted by simply 15 and 15".

The SU(4) symmetry of the leading terms at short distances (Gaillard and Lee 1974; Altarelli and Maiani 1974) suggests that the SU(4) representation 20"

m~

(i.e., H_) is enhanced relative to the 84 i.e., H+. The consequences of 20"-domi- nance have already been considered for the AC~ = AS decays of B(3*) -+ B(8) + P (8) aild P (3*) ~ P (8) + P (8) (Gaillard et a11975; Altarelli et al 1975 a and 1975b). In particular Altarelli etal 1975a find that 20"'-aommance' in the SU(4) limit, gives an extra relation for the S-wave decays of the hyperons, namely S ( ~ - ) ~ 2S(A_°), which is violated by about 50%, which is not too bad. Further it has been shown that the use of the full Hcs at the isospin and SU(3) level only gives simple relations for the B(3*) and P(3*) decays which would provide tests of the presence of both the pieces H_ and H+ in Hcs (Gupta 1976 b). Consequently we consider the AC~ = A S decays of B(6) and B(3) and D(I) d,ae to /-/_s only at the isospin and SU(3) level and present ampli- tude relations which arise from

(i) (ii)

The I A / l = 1 selection rule obeyed by the full Hcs

Tile hypothesis that H_ is enhanced relative to H+ at the SU(3) level i.e., Hcs transforms as (6 + 6*) at the SU(3) level. This follows from 20"-dominance but is weaker than that since one doe5 not use SU(4), which would have, for example, related the B(6) and B(3) decay ampli- tudes and which is not the case for the H--enhancementhypothesis at the SU(3) level.

282 V Gupta

(iii) The full Hcs at the SU(3) level. In this case unlike for the H--enhance- ment hypothesis in (ii), owing to the large number of parameters at the SU(3) level, most of the relations between decay amplitudes are not simple. Consequently we give only relations involving at most 4 or 5 decay amplitudes since they would be easier to check experimentally.

An advantage of the relations obtained from Hcs at the isosopin and SU(3) level is that they are ~,alid for both the parity violating and the parity conserving

amplitudes.

4. B(6) -+ B(8) + P(9) dec~ys

For the charmed baryons we will use the notation BZ(N3, I) where N3 denotes its SU(3) representation, I its isospin and its charge Q will indicate its Is-value. The 30 decay amplitudes Ba . . . . Bn0 which satisfy AC~ = A S are defined in table 2.

The physical states in the SU(3) nonet

= sin OvPl ~ cos OvP8,

~7' = cos OpP1 + sin Ove8, where P1 is a pure SU(3) singlet which

P(9) are given by (mixing angle 0p) (12 a) (12 b) mixes with P8 the eighth component of

Table 2. The 30 A C~ = A S decay amplitudes for B(6)-+B(8) + P(9). The eight primed amplitudes B ( , etc. are also defined as the amplitude relatior.s are more compactly given in terms of them.

B 1 = A ( B ~'+ (6, 1) --> ~r+ I + ) B~o = A ( B ° (6, ½) ---> K - l +)

B2 = A ( B ~ (6, l) -+ KOp) B2~ = A(B°(6, ½)-> ~+ ~-)

B~ = A ( B + (6, 1)-~, K + ~o) B2~ = A ( B ° (6, ½)-> K~ I °)

B 4 = A ( B + (6, 1) -~ ~r+I°) B n = A ( ~ ° (6, ½) --> K~ A ) B5 = A ( B + (6, 1)--> 7r + /\ ) B2a = A ( B ° (6, ½)--~ ~r ° 8 °) B6 = A ( B + (6, 1) --~ ~ I + ) B2u = A ( B ° (6, ½) -> ~ S °) B 7 = A ( B + (6, 1) ---> 7r ° I +) B26 = A ( B ° (6, 0) -* ~ ~o) Bs = A(B°( 6, 1)---> rr ° A ) B~7 = -4(B + (6, 1)---> "q' I +) B~ = A(B°(6, 1)--> ~r ° l °) B2s = A(B° (6, 1)--> r/' A) Bto = A(/~ (6, I)--> ~ A) B.,g = A(B°(6, 1 ) ~ ~ ' l °) B l l = A ( B ° (6, 1) -+ ~ l °) Bzo = A ( B ° (6, 1) --~ ~' ~o) Bxz = A(B° (6, 1 ) ~ ~r- I + ) B ( = A ( B + (6, 1)--+ P s i +) Bx3 = A ( B ) (6, 1) --> 7r+ 1 - ) B'lo = A ( B ° (6, 1) ~ Ps.A) B14 = A ( B ~ (6, 1) ~ K ° ~o) B'11 = A ( B ° (6, 1) -+ P s l ° ) Bls = A(B ° (6, 1)---> K~ n) B'z6 = A I B ° t 6, ½ ) ~ P~ E °) B16 = A ( B ~ (6, 1) ---> K + ~ - ) B'z7 = A(B+(6, l) ---> P~ Z+) B~7 = A ( B ° (6, 1) ---> K - p - ) B'2s = A ( B ° (6, 1) --+ P1 A ) BI~ = A ( B + (6, ½) ---> K~ l + ) B'I~ = A ( B ° (6, 1) --~ Px l ° ) B ~ = A ( B + (6, ½) --~ ~r+ ~o) B'~o = A ( B ° (6, 1) ~ P~ ~o)

Non-leptonic weak decal,s of charmed baryons 283 the octet. It is convenient to define the eight primed amplitudes Be, etc., also given in table 2, which can be expressed in terms of the corresponding unprimed ones using equations (12a) and (12b).

The decay amplitudes B~ will depend on eight parameters (or unknown ampli- tudes) which can be counted by considering B ( 6 ) + H + ~ B ( 8 ) + P(9) and treating H+ as spurious. The 6* part of H_ having Ch = - - 1 contributes giving

- -_3

three amplitudes g27 = (27-+ 27 (BP)), gs(8-+ 8s(BP)) and g~(8-+ 8a (BP)), where 8s(BP) means the symmetric and 8A(BP) the antisymmetric octet formed out of B(8) and P(8), etc. Similarly the 15 part of H+ only contributes and gives rise to five amplitudes namely, g~r = (27 ~ 27(BP)), g'lo = (10 -+ 10 (BP)), g ~ = (10" -+ 10' (BP)) and g's,A = (8 ~ 8s,A (BP)). One obtains eight

I A l l = 1 sum rules namely

B5 = Bs, B6 = B11; B27 = B~9; (12)

B1 = B4 + B7 = 2B9 + Bt~ + B13; (13 a)

B4 ~ B 7 = Bja --B13 ; (13 b)

Bls ---- B2o + (~/2) Bz2, (14 a)

B19 = B21 + (~/2) B24; (14 b)

which are true for the full Hcs. On the H_-enhancement hypothesis only g27, gs and ga are non-zero and one obtains 19 SU(3) sum rules. These are

- - ( V 2 ) B~ = B4 + (V3) B~; (15 a)

--(V'2~B3 = B7 + (x/3) Bs, B6' = Bs; (15 b)

BI~ = ~B14, B13 : --B~5, B16 = B17; (16 a)

- - = B . + ( V 3 ) B8 = + 0 6 b)

- - ( x / 6 ) B;5 = B~, --B~.~ + 2Bzo (17 a)

- - ( 1 / 6 ) B29 = Bxs + 2B,1 - - B z o (17 b)

Bxs = B~9, Bn = Bzo, B~ = B2,, B1 = B~6 = - - ( X / 2 ) B l s (18) B ~ = - - (V3) B;s = --Bio = (V2) Bs - - 2 (V6) B~.. (19) For the full Hes though there are 14 SU(3) sum rules only two are relatively simple, namely

- - ( ~ + B~.,) = 2 ( ~ 8 + ~ , ) = 20 (B~, + B~7). (20) These can provide a test of the specific SU(3) transformation property of the ful Hcs.

5. B(3) and D(1) deca~s

There are three types of decays of B(3), at the SU(3) level, as given in (2 a), (2 b) and (2 c). The 24 decay amplitudes, C,, for the various decays are defined in table 3. In addition the four primed amplitudes C~o, etc., are also defined in table 3 for convenience. We give the sum rules for the three types of decays under their subheads. The decays of D(1) are given in section 5 d.

284 V G u p t a

Table 3. The B(3) dezay amplitudes satisfyir, g L\Ct, = z ~ S selection rtde

B ( 3 ) ~ B ( 8 5 + P(3"5 B ( 3 ) ~ B ( 6 ) + P ( %

C1 = A ( B ++ (3, ½) ~ D + S+) C~ = A ( B ~÷ (3, _4-) ~ B +~ (6.1)~ -a)

C2 ~- A ( B ÷ (3, ½) ~ D o E +) C~ = A ( B ++ (3, ½) ~B+(6, ½) ~r+)

C.~ = A ( B + (3, ½) ~ D + 2 '~) C ~ = A~B + t3, ½) ~ B + + f f , I)K-)

C 4 = A ( B + (3, ½) ~ D + A) C ~ = A ( B + (3, ½) ~ B+(6, 1)h t--)

C~ = A ( B + (3, ½ ) ~ F + S °) C,8 = A ( B + (3, 3 ) ~ B°(6, ½)~r+)

C~ = A ( B + (3, O) ~ D + S °) C1~ = A ( B + (3, 3) ~ B~(6, ½) ~r°)

B (3) ~ B (3*) + P ~9) C~o = ,4(n+ (3, 3) ~ B+(6, ½5~)

C7 = A (B ++ (3, 3) ~ B + (3", 3) 7r÷) C2x = A ( B + (3, ½) --~ B°(6, 0)K+) C s = A (B ~ (3, ½) ~ B ° (3", ½) ,+) C2~ = A ( B + (3, 0) ~ B+(6, 3)~ %) C9 = A (B + (3, 3) ~ B + (3 *, k) ~o) C2a : A ( B + (3, 0) --~ B°(6, O)~r+)

C~o --- A (B + (3, ½) ~ B + (3', 3) ~) C2a - A ( B * (3, 3) ~ B+(6, 3)~/') C u --- A ~.B + (3, ½) -+ B + 1.3 *, 0) K °) C':0= ^{A ( B +} (3, ½) ~ B~(6, ½)es) Ct2 = A (B ~ (3, 0) ~ B+ (3% 3) g~-) C'.4= A(B+ (3, 3) -~ B+(6, 3)P~) Q = ,4 (B~ (3, 3) ~ O + (3 *, 35 ~'5

Q o = .4 (B+ (3, ~) ~ B+ (3", 3) Ps) C'~3= .4 (B + (3,½) ~ B + (3", 3) Ca) 5 a. B(3) -~ B(3*) + P(9) d e c a y s

T h e seven decay a m p l i t u d e s C7 to C1~ are given in t e r m s o f f o u r p a r a m e t e r s , two o f which arise f r o m the 6* p a r t o f H_ a n d t w o f r o m the 15 p a r t o f H+. T h e r e is only one I A I [ = 1 s u m rule

= + ( V 2 ) co. (21)

W i t h the H_ - e n h a n c e m e n t h y p o t h e s i s one expects four SU(3) s u m rules:

C7 = - - C 1 2 , Cs = C , . C~ = - - ( V 3 ) C ; o (22)

Cs = ¼ C7 ÷ ( a / 3 ) C~a. (23)

F o r t h e full H c s one expects two SU(3) s u m rules, h o w e v e r each involves six a m p l i t u d e s a n d consequently we o m i t t h e m as they would b e h a r d to verify.

5 b. B(3) - + B(8) + P(3*) d e c a y s

I n this case t h e r e are only six decay a m p l i t u d e s 6'1 to C~ as the b a r y o n s d o n o t f o r m a n o n e t unlike the mesons. T h e s u m rules in this can b e o b t a i n e d f r o m (21) and (22) b y r e p l a c i n g C7, C8, C12 by C 1, Ci . . . C6. T h e r e is no s u m rule c o r - r e s p o n d i n g to ( 2 3 ) a s there is no a m p l i t u d e c o r r e s p o n d i n g to C ~ in this caso.

Again these decays do not p r o v i d e a simple a m p l i t u d e r e l a t i o n t o test the SU(3) t r a n s f o r m a t i o n p r o p e r t i e s o f the full H c s .

5 c. B(3) -+ B(6) + P(9) d e c a y s

T h e 11 decay a m p l i t u d e s C~4 t o Cz~ in tiffs case are given in t e r m s o f five p a r a m e t e r s a t the SU(3) level. T w o o f these c o m e f r o m the 6* p a r t o f H_ a n d three f r o m

the 15 part of H+. One obtains two I ~ l l =- 1 sum rules, namely c . = c,~ + (V2) G~

G~ = q . + (V2) G~.

(24 a) (24 b) As tests of the H_ -enhancement hypothesis one obtains the following seven SU(3) sum rules :

(716 = C~ = 2C~9 = (2/v/3) C20, C',

C14 : C23 , C15 : C22 , C17 = C18 ,

Note C', is no longer zero for the full Hcs.

= 0 (25 a)

(25 b) For the full Hcs the only simpl.

checkable relation which emerges, out of four SU(3) sum rules, is

C~n ÷ C2~ : Ca9 --k (v/3) C'0. (26)

Thus the decays of baryon triplet can p'ovide a test of the SU(3) transformation properties of the full Hcs.

5 d. D(I) decays

It was noted in section 2 that the triply charmed, doubly charged, I = 0, S = 0e SU(3) singlet state D +÷ (1, 0) or D(1) may turn out to have only weak decays, Its ~C1, = .~S decay amplitudes are

D1 = A(D ++ (I, 0) ---> B~+(3, ½) K°), D2 = A(D ++ (I, 0) --+ B+(3, 0) 7r +) (27 a) D3 = A(D~+ (1, 0) - B~(3*, ½) D+), D , = A(D+* (l, 0) - B+(6, ½) ~ ) .

(27 b) Note our remarks can be applied to decays with B(3) and B(6) replaced by D(3) and D(6), if eae:g.~tically allowed. The only interesting relations in this case are obtained fo: H_ dominance hypothesis and these are

D1 = D.~; D, = 0. (28)

6. Summmy and concluding remarks

On the basis of first order breaking of SU (4) and a linear mass formula, with one parameter, we have argued that one may expect stable (except for weak interaction decays) charmed (1/2) + and (3/2) + baryons belonging to the representations 1, 3", 3 and 6 of SU(3). Since these would decay only by weak interactions, we have studied their non-leptonic decays which satisfy the ACh ---- A S rule since these are the dominant non-leptonic decays in the GIM-scheme. In particular we have analysed the /~Ch = L\S decays of a baryon, 3, 6 and 1 into baryon and meson, from the point of view of providing tests of the isospin and SU(3) transfor- mation p;operties of the ± G , = A S interaction, Hcs, in the GIM-scheme. We have shown that a number of simple relations can be obtained which, in future can provide tests of the fall Hcs. As pointed earlier, our analysis is at tile SU(2) and SU(3) level and as such is valid for the type of decays shown in (1) and (2)

286 V Gupta

for all possible decays involving D(3), D(6), B(3), B(6), B ( 8 ) a n d a meson nonet be it 0-, 1-, etc., (e.g., D ( 3 ) - + B(3*) ÷ V (9)) with a p p r o p r i a t e replace- ments. F u r t h e r m o r e , in each case, all the amplitude relations o b t a i n e d are (i) valid for the parity violating a n d parity conserving amplitudes; (ii) independent of whether the current JR is pure (V-A) or not a n d (iii) independent of the a d d i - t i o n of a (V + A) piece t r a n s f o r m i n g like J24 (De Rujula et al 1975) as this does not affect the ACh = ~ S non-leptonic decays.

The fascinating possibility (on the basis of mass formula in section 2) that D (1) with Ch = 3 (analogue of g2- in SU (3)) may turn out to be stable, with respect to strong a n d electromagnetic decays, has been r e m a r k e d u p o n a n d a brief discussion of its weak decays has been included.

Acknowledgement

I a m grateful to J Pasupathy for c o m m e n t s a n d suggestions.

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