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LETT]~I~E AL NUOV0 CIM]~NTO VOL. 15, N. 16 17 Apr i l e 1976
Strong Decays of Baryons in SU2~ Symmetry (*).
S. IwAo
Departme~t o] Physics, College o] Liberal Arts, K a n a z a w a University - Kanazawa
( r icevuto il 9 F e b b r a i o 1976)
I n two p rev ious p a p e r s (~) we i n v e s t i g a t c d t he h a d r o n s in t h e f r a m c w o r k of b r o k e n SU2z~ s y m m e t r y . T he new f u n d a m e n t a l c o n s t i t u e n t s of had rons , (~ fall ~) (]), (( r ise ~) (r), ]i(i ~ 1 . . . . . _~-- 4) a n d r i ( i -- 1 . . . . , N - - 4(5) for 22v*= even(odd) ) are i n t r o d u c e d in ad- d i t i on to u, d, s, c, b a n d t (2). Especia l ly , t he r e l a t i on b e t w e e n masses of p.s. mesons a n d qua rks a n d hence t he mass fo rmu lae for t h e f o r m e r were g iven, b y e x t e n d i n g t h e ideas of G~LL-MAz~N ct al. (GMOR) (a) a n d of GAILLARD et al. (a). The b a r y o n mass f o r m u l a e are d iscussed a long t he same l ine a n d t he n u m e r i c a l e s t i m a t e s of 1-+ a n d ,~+ b o t t o m e d b a r y o n s are specif ical ly g iven (~).
I n t h i s p a p e r we w a n t to discuss t he s t rong decays of b a r y o n s w i t h J P -~ �89 and 3+ in to b a r y o n s a n d k n o w n p.s. mesons of t h e S U 3 oc te t m e m b e r .
Before g e t t i n g in to t h i s p r o b l e m it wou ld be i m p o r t a n t to get a r o u g h idea on t he masses of ~ : - b a r y o n s in b r o k e n S U 4 and S U 5 s y m m e t r y . F o r t h i s pu rpose we shal l choose J~0(1535), A(1670), E(1750) a n d E(1820) as t he fu l l m e m b e r s of t he lowes t �89 oc te t in SU3 s y m m e t r y . E x p e r i m e n t a l i n f o r m a t i o n on these r e sonances is poor a n d n e i t h e r b r a n c h i n g ra t ios no r t he masses are e s t ab l i shed def in i t ive ly . I f we use t h e n u m b e r s of t he b r a c k e t in t h e first t h r ee s ta tes , we ge t 1845 a n d 1833 GeV for t h e ~ (1820) - s t a t e in t h e l inea r a n d q u a d r a t i c G e l l - M a n n - O k u b o mass fo rmula , respec t ive ly . Thus , we shal l use t e n t a t i v e l y four n u m b e r s in the b a r y o n symbo l s in t h i s paper .
L e t us deno te t he c h a r m e d b ' t ryons w i t h .~- b y a d d i n g a p r ime , ', to t he p rev ious ly def ined n o t a t i o n s ( " ) . Then , wc f ind
(1) C ~ - - ~ ( 1 5 3 5 ) = S ' - - �88 + X(1750)) =
- : T ' - - X(1820) x ( Z ( 1 7 5 0 ) - - ~ ( 1 5 3 5 ) ) ,
(2) Co- -JV(1535) = A ' - - ~ - ( 3 Z ( 1 7 5 0 ) + A(1670)) = x ( A ( 1 6 7 0 ) - - ~ : ( 1 5 3 5 ) )
(*) W o r k is s u p p o r t e d ii1 p a r t b y t h e 1975 Scient i f ic Resct~rch F u n d of M i n i s t r y of E d u c a t i o n . (1) S. IWAO: Part ic le phys ics i~ broke~e Slf2,v s ymme t ry , H P I C K - 0 2 1 ; Baryon masses i'a broken SU2.v symme t ry , HPICI / : -022 , p r c p r i n t s ( J a n u a r y 1976). (3) I t . FRITZSCH, M. GELL-MANN a n d P. ~[INKOVSKI: P h y s . L e t t . , 5 9 B , 256 (1975); Y. ACIIISIAN, K . KOLLER a n d T. F. ~VALsH: P h y s . Le t t . , 59 B, 261 (1975); H . HARARI: HGW ma~y quarks?, A t a l k g i v e n a t R I F P ( N o v e m b e r 1975). (a) M. GELL-)/IANN, I t . J . OAKES a n 4 B. RENNER: P h y s . Bev . , 175, 2195 (1968). (4) M. K . GAILLAn]), B. W. LEE a l l4 J . L. ]:~OSNER: I~CV. Mod. P h y s . , 47, 277 (1975).
569
5 7 0 s. :wAo
a n d
(3) X : , , ~ a - - ~ ( 1 5 3 5 ) : X:~ - - X(1750) : x(E(1820) --~4z(1535))
as a gene ra l i z a t i on of t h e G L R . S imi l a r r e l a t ions m a y s imply be o b t a i n e d for �89 in b r o k e n S U 5 b y i n t r o d u c i n g a p a r a m e t e r y in place of x. M a k i n g use of x = 20.53 a n d y = 34.06 (5), we f ind t h e resu l t s t a b u l a t e d in t ab l e I. The n u m b e r s in t he b r a c k e t
TABLE I. -- Mass spectra o] charmed and bottomed baryons with j e = �89 and �89 in units o] GeV.
N o t a t i o n s L i n e a r Q u a d r a t i c mass f o r m u l a mass f o r m u l a
1 + 1 + �89 �89 ~ �89 �89
C 1 C~ 6.156 5.95 3.465 4.11
S S" 6.352 6.10 3.523 4.17
T T ' 6.535 6.23 3.586 4.22
A A ' 4.801 4.50 2.972 3.45
C O C o 4.566 4.31 2.887 3.35
Xr X~.~a 8.723 7.39 4.295 4.69
X ~ X ~ 8.977 7.60 4.358 4.76
B 1 B~ 9.596 8.86 4.398 5.13
/~s B~ 9.792 9.01 4.444 5.19
B T B~ 9.975 9.14 4.494 5.23
B A B~a 7.192 6.33 3.707 4.21
B 0 B o 6.957 6.13 3.640 4.13
Xbu.o 4 X~u.b a 13.86 11.2 5.480 5.91
Xb~ X ~ 14.11 11.5 5.530 5.97
Br B~s,~as 14.81 13.3 5.519 6.40
BcuA.cd A B~uA.cd A 10.58 8.90 4.549 5.10
B~ s B'~s 15.01 13.4 5.556 6.44
B cs A B ~ A 10.82 9.10 4.604 5.16
Bc~ B~ 17.38 14.7 6.075 6.79
Bb~ B~c 19.07 (23.47) 15.7 6.415 (6.994) 7.03
in t he las t l ine of t he t a b l e come f rom a n o t h e r poss ible choice of c o m b i n a t i o n s of i n p u t . These r e su l t s m a y con t a i n some a m b i g u i t y due to t h e i n d i r e c t ca l cu la t ions i n v o l v e d . W e hope t h a t t he mass di f ferences m a y no t be l a rge ly v i o l a t e d f rom t h e t r ue va lues .
I t is i n t e r e s t i n g to no t ice t h a t al l ~ - - b a r y o n s a p p e a r a t lower masses t h a n t h e cot-
(5) I-I. HAYASHI, T. ISHIWAT~., S. IWAO, ~I. SHAKO a n d S. TAKESHITA: SUj , SUe and strong interactions, HPICK:-O20, p r e p r i n t ( J a n u a r y 1976). A m o r e d e t a i l e d r e p o r t i s i~ p r e p a r a t i o n .
STRONG DECAYS OF BARYONS IN SU2N SYMMETRY 571
1 § responding va lues for ~ -states in the l inear fit, whi le only the quadra t i c fit suppor ts ]+
the s tab i l i ty of ~ -baryons for s t rong decays both in S U 4 and in S U 5. This t endency m a y no t be changed even if we proceed to h igher symmetr ies . Our resul t m a y be jo ined wi th the observa t ion of mass re la t ions be tween �89 and ~+ states in SU~ by BORCHARDT et al. (6) and wil l give an addi t iona l cr i ter ion to single out the correct use of the mass formula.
We shall consider S U 3 octet p .s . -meson associated s t rong decays of new baryons belonging to the S U a subclassifications 6 + 3 and 3 of �89 and 6 and 3 of ~+ in SU~N(2~ ~ = 4, 5 . . . . ). As we po in ted out before (1) 6 and 3 for �89 s ta tes should be t rea ted separa te ly ( the convent ion (7) was used so as to d iscr imina te i t f rom the o ther sex te t wi th �89 appear ing in the SUa subclassification (1)).
There arise three kinds of al lowed Yukawa t rans i t ions :
(4) 6 - + 6 + 8 ,
(5) 3 -+ 6 + 8
and
(6) 3 - + 3 + 8
a m o n g baryons belonging to i r reducible representa t ions 6, 3 and 3. The mul t ip l ica t ions of these three i r reducible representa t ions y ie ld
(7)
(8)
and
(9)
8 • 2 1 5 1 + 4 .8 + 2 .10 + 2 .10 + 3-27 + 35 + 35 + 64 ,
8 • 2 1 5 1 + 3 . 8 + 1 0 + 2 . 1 0 + 2 . 2 7 + 3 5
3 • 2 1 5 1 + 3 , 8 + 1 0 + 1-0+ 27 ,
where the numbers m and n of m . n on the r.h.s, of eqs. (7)-(9) represent the mul t i - p l ic i ty m of the i r reducible representa t ion n. These relat ions te l l us t h a t all the pro- cesses under considerat ion m a y be described by a single coupl ing cons tant for the respec- t ive process. Looking th rough table I , once more, one sees t h a t exc luding several cases m a n y s ta tes admi t energe t ica l ly the pion, kaon and e ta-meson associated decays.
L e t us begin wi th �89 decays. Firs t ly , f rom a new tabu la t ion of isoscalar factors for S U 3 by RABL et al. (s), we find the fol lowing superposi t ions of �89 charmed
1+ baryons in t e rms of ~ and 0- s ta tes :
6 - + 6 + 8:
(10) rc~> = I c l ~ > - ~/~6 Ic1~> + ]~ ISK>,
(11) Is') = 2 gg f l R ~ + '2 ~/l~ [s=) + 2 ~/~ lS'J) + V ~ ITK)
(s) S. BORCI-IARDT, V. S. MATtIUR and S. OKUBO: Phys. Rev. Left., 34, 38 (1975); Phys. Rev. D, 11, 2572 (1975). (~) D. AMATI, H. BACRY, J. NUYTS and J. PRENTKI: .Yllovo Cimento, 34, 1732 (1964). (s) V. ]=~)~BL, G. CASIPBELL jr. arid K. C. ~V.XLI: SU4 Clebsch-Gordan eoe]/ieienls, C00-3533-50, SU4206-50, prcprint (March 1975).
572
and
S . I W A O
02) IT%= V~ISK>+ V~IT~>. Similar expressions may be obtained for B~, B's and B~ by appropria te sub-
st i tut ions.
3 - > 6 + 8:
(13)
and
(14)
and s imilar ly for B~' and B~ in a proper account. F ina l ly
3 - - > 3 + 8 :
(15)
and
(16) Ix~,> = ~ [xo.K> 1
Similar formulae apply to X~, and X~,: A generalization of eqs. (10)-(16) to the baryons belonging to SU2N ( 2 N = 6, 7 . . . . ) is straightforward.
Now we decompose the states appearing on the r.h.s, of eqs. (10)-(16) into the charge states for a given I~ (SU, C.G. coefficients will be used). We then apply the so-cMled reduction technique (of dispersion theory) to p.s. mesons. At tha t stage we assume the well-known PCAC relation. We get a matr ix element of axial vector current
1 + densi ty between �89 and : -baryon states, e.g., for eq. (10) (8):
(17)
(18)
and
(19)
! oz~5 < r : ig~(t)ud 17~e l + ....
t 5 <C~i~-~[S } = igK(t)gol 7~,Us + ...
! S <C1]~:~1C1> = ig~(t) uo', 7~ue~ "k . . . .
I t is known tha t the PCAC is not good as far as the eta-meson is concerned. We as- sume it formally, then we find from eqs. (17)-(19) the following approximate relations:
1 (20) g. (0)(m~;- m~) ~ ~ go,~,=,
1 (21) gK(0)(m0;-- ms) ~ ~]~ gd, sK
STRONG D]~CAYS OF BA_-~YONS IN S U2N SY~IM]~TICY 573
and
(22) 1
Here, we discriminate particle and mass symbols explicitly. From these relations we find
(23) gO~SK ~ (me i - - ms) ]r:
and
go'~o,,~ h:
Similar quantities may easily be obtained for all remaining amplitudes. These rela- tions will be checked as soon as the experimental data become available.
1+ For transitions of ~+-states into ~ - and 0--states, we find their amplitudes by replacing the symbols on the 1.h.s. of eqs. (10)-(12), (15) and (16) by C~, S*, T*, Xc* and Xc*, respectively. We do have not a proper application of PCAC in these cases. However, we can find a rough estimate of the decay branching probabilities from the amplitudes thus obtained. A further generalization to SU2N(2N= 5, 6 . . . . ) may be obvious from our preceding discussions (*).
We hope that the generalization discussed in this paper will be checked experimentally and we can proceed to the dynamics of the constituents (flavors) inside the hadron box with confidence.
Recently, B. W. LEE (Possibility o] measuring .~.parity, FERMILAB-76/14-THY/EXP (January 1976)) has pointed out that usually people assign ~(1820) as an SUa octet member with J ~ = 8- in his proposal for determining ~T: and AK relative parity. If this is the case we should replace ~(1820) in eqs. (1) and (3) by 1845 and 1833 MeV obtained from the GMO mass formula for linear and quadratic fit, respectively. The numbers in table I will be modified correspondingly. However, the main conclusion obtained in this paper may not be changed.
(*) For new sexte ts , s inglets , etc. which h a v e not a p p e a r e d in SU,, b u t a re i nvo lved i~ the subelas- siflcation of SUn of SU6,~,... ( t he whole job wil l be comple te4 up to SU, s y m m e t r y w i th o u t going up to h igher ones (1)), we need add i t iona l C.G. coefficients i~ o rder to comple te th i s k ind of job.
