IL NUOVO CIMENTO VOL. 43 A, N. 3 1 Febbraio 1978

Mass Formula for Charmed.Hadron Multiplets (*).

DAo Vo~G D u c

Institute of Physics - Hanoi, Vietnam

(rieevuto 1'11 Ottobre 1977)

Summary. - - A general mass formula is derived for all observed SU 4 hadron multiplets. The formula involves, besides hypercharge and charm, the second-order Casimir operator of the subgroup SU3, as well as tha t of some other subgroup named ~'Ua. The classification of hadrons according to the decomposition SU4D~'~ 3 is also given.

1 . - I n t r o d u c t i o n .

The d i scovery of t he ne w fami ly of par t ic les assoc ia ted to t he new q u a n t u m n u m b e r cal led c h a r m has m o t i v a t e d a g r ea t in te res t in t he s t u d y of t he SU4

s y m m e t r y p roposed earlier b y m a n y a u t ho r s ('~). I n this no te we will der ive t he mass f o r m u l a for c h a r m e d - h a d r o n mul t ip le t s

in t he SU4 s y m m e t r y scheme. There h a v e been severa l papers d e v o t e d t o t h i s p rob lem, in which mass f o r m u l a e h a v e b e e n g iven for severa l mul t ip le t s of SU4 (see, for example , (7-1~) a n d references there in) . The n o t e w o r t h y fac t is t ha t , a l t h o u g h the m e t h o d used here is a r a t h e r s t r a i g h t f o r w a r d genera l iza-

(*) To speed up publication, the author of this paper has agreed to not receive the proofs for correction. (1) D. AMATI, It. BACRY, J. I~U]TTS and J. PRENTKI: NUOVO Cimento, 34, 1732 (1964). (3) J. D. BJORKEN and S. L. GLASGOW: Phys. Lett., 11, 255 (1964). (a) Y. HARA: Phys. Rev., 134, B 701 (1964). (4) Z. MAKI and Y. OHNuKr: Prog. Theor. Phys., 32, 144 (1964). (5) P. TARJA~NE and V. L. T~,PLITZ: Phys. Rev. Lett., 11, 447 (1963). (s) S. L. GLASGOW, J. ILLIOPOULOS and L. ~¢[AIANI: Phys. Rev. D, 2, 1285 (1970). (7) M. K. GAILLARD, B. W. LE~ and J. L. ROSNER: Rev. Mod. Phys,, 47, 277 (1975). (s) S. 0KU]~O, V. S. MATHUR and S. BORC~ARD2: Phys. Bey. Lett., 34, 236 (1975). (9) S. O~uBo: Phys. Rev. D, 11, 3261 (1975). (lo) A. J. MACFARLANE: Journ. Phys. G, 1, 601 (1975). (11) A. W. SMITH: Journ. Phys. G, 1, 907 (1975).

365

366 PAP VONG Due

t ion of the Okubo ' s m e t h o d (12) for the SU3 scheme, the procedure of finding generul expression proves to be in rea l i ty ve ry complicated, the difficulty be ing to eva lua te the SU4 m a t r i x e lements of some t e rms enter ing the s y m m e t r y - b reak ing in terac t ion t t ami l ton ian . Fo r this reason one often handles specific cases sepura te ly nnd, to our knowledge, a general unified mass formul~ h a s

not ye t been found. Our der iva t ion of the mass fo rmula is based on using some subalgebra we

n a m e S~'U3. Tha t ~llows us to get r id of eva lua t ing the m a t r i x e lements men- t ioned above, and the final fo rmula involves, ins tead of them, the SU3 and SU~ second Casimir operutors. The conten t of the p~per is a r ranged as follows. Sect ion 2 is devo ted to the der iva t ion of the mass formula . Due to the presence of the S"U3 Casimir opera tor , in order to app ly the ob ta ined mass formula to concrete cases, the remain ing p r o b l e m is to classify the hadrons of the SU4 mult ip le t s according to the decomposi t ion SU4 ~ SU3. This is done in sect. 3, where the mass sum rules for the mul t ip le t s of 1+ and ~+ baryons , as well as

those of 0 - and 1- mesons are ex t rac ted .

2. - Der iva t ion o f the m a s s formula .

In ana logy wi th the SU3 theory , i t is commonly assumed t h a t the SU~ mass split opera tor has the s t ruc ture

(2.1) A M --~ Ms ~- MI~

wi th Mo t rans fo rming as the a - th componen t of the adjoint represen ta t ion

of SU4, n a m e l y

(2.2) [F~, Mb] = ifa~,.M~, a, b, c = 1 ,2 , .. . , 15

where F~ denote the SU4 generators , ]~b~ being SU, s t ruc ture constants . The operutors of isospin I , charge Q, hype rcha rge Y ~nd cha rm C are expresse4 in t e rms of the generutors F~ th rough the re la t ions (*)

I ~ = F ~ , a = 1 , 2 , 3 ,

t 2 1

2 ~ 1 (2.3) r = ~ F ~ - - FI~ + ~ ( n - - g ) ,

3 1 (n - - ~)

(12) S. 0KVBO: Prog. Theor..Phys., 27, 949 (1962). (*) We employ the model in which the quarks belong to the 4-dimensional first funda- mental representation of SU 4. Moreover, the fourth quark has Q ~ an-, /7 = ½, C = 1~ and hence the generalized Gell-Manu-Nishijima formula is to be Q = I n + ½ ( r + c).

MASS F O R M U L A F O R C } ~ [ A R M E D - H A D R O N MULTIPL}~TS 367

where n and ~ are the numbers of quarks and an t iquarks of which the hadrons of the mul t ip le t are made up.

In general , there are th ree opera tors of such k ind of M~ sat isfying (2.2) which m a y be wr i t ten as

(2.4) F~ , d.b~FbF¢ , d~o~d,~dFbF~Fa

(sum over repea ted indices), where the tensors d~b~ are constants appear ing in the an t i commuta to r s of the SU4 Gel l -Mann's matr ices 2,,

(2.5)

The (( vec tor )) operators (2.4) can be ob ta ined b y fo rma l differentiat ion of the second-, third- and four th-order SU4 Casimir operators , respec t ive ly :

~C~ ~C.~ ,~ d ~C4 .~ d~b~d~dFbF~F~ (~.6) ~F---~ Fo, ~ oJ'oF~, ~---

I n the f r amework of co loured-quark theo ry all observed hadrons are colour singlets and made up of three quaxks or a pa i r of qua rk and au t iquark . Thus, b y assuming tha t the hypotheses abou t the perfec t conf inement of all coloured s tates is correct, we have to deal only wi th the representa t ions which appea r in the p roduc t of th ree f u n d a m e n t a l 4-dimensional qua rk representa t ions or of qua rk and an t iqua rk representa t ions . All of t h e m belong to the t y p e of so-called representa t ions wi th degeneracy (*) (in Okubo ' s t e rminology (~)), and, as was shown in (~3), for these represen ta t ions the th i rd iorder (( vec tor )~ oper- a tor d~b~d~c~Fb_F~Pa can be expressed in t e rms of the first- and second-order o n e s -/~a and d,bcFbF~.

Tak ing all s t a ted above into account , for the observed physica l hadron mult iplets , we can use the SU4 mass split opera to r in the following fo rm:

(2.7) M = a ~ bFs + cDs -{- dFls -[- eD~5,

where a, b, c, d, e are some constants , Da is an a b b r e v i a t e d no ta t ion of d~b~FbFc,

(2.s) Db ~ dsocFbFc , D15 ~ ~ l S b c F b F c •

(*) That is a representation such that the Young tableau associated to it has rows with equal number of boxes. (13) S. OKUBO: Journ. Math. Phys., 16, 528 (1975).

3 6 8 D A 0 V O N G D U G

B y using the explicit values of dsb~ and d15~ (see, for example , ref. (~4)), we have

1 3

(e .9 ) D~ = ; - ~ Z ~ - - - V ~ a ~ l

8

( s n 0 ) D~5 - - ~ Z F ~ - - - ~ / D a=1

1 ~ 2 2

Le t us t r ans fo rm these expressions into the following more convenient forms including the second-order SU, and SU3 Casimir opera tors :

(s.11)

( s n s )

D. - 2 ~ / ~ f : ~ f z + 2 ~ a o~F~ +

1 15 2 2 s

Now, inser t ing (2.11) and (2.12) into (2.7) and t ak ing into account the identi- fications (2.3), we obta in

(2.]3) M : ~ - t - f l Y ~ - T C - [ - 8 K - - - ~ C 2 ~-

q- a - - K + I ( I ~- I ) - - -~ Y~--- -~ - - g Y C - - ( F ~ 3 - ~ F~4) ,

where ~, fl, 7, (~, ~t are some new constants , K s tands for the second-order SU3 Casimir opera tor (*)

8

(2.14) K ~- ~ F~ ; a = l

its explicit expression has been given, for example , in ref. (15). Thus, we have K--~ 4/3, 10/3, 3, 6, ... for 3-plet, 6-plet, 8-plet, 10-plet, ... representat ions.

However , work ing wi th fo rmula (2.13), we encounter the difficulty of eva lua t ing the m a t r i x e lements of the opera tors F[3 and F~4 in the closed form. To avoid this difficulty, we proceed as follows. First , f rom the commuta to r s

(14) H. H.~YASHI, I. ISHIWATA, S. IWAO, M. SHAKO and S. TAKESHITA: Ann. o] Phys., 101, 394 (1976). (*) We use here the notation K instead of the conventionally adopted C2, in order not to confuse with charm. (15) I~. C. BI]~I)ENtIAt~N: Journ. Math. Phys., 4, 436 (1963).

MASS FORMULA FOR CHARMED-MADRON MULTIPL:ETS 369

of the SU4 generators

(2.~5) [Fa, FO] : i f a b c F ~ , a, b~ v = 1 , 2 , . . . , 15 ,

with the explicit values of the s t ructure constants f.b~, we can see t h a t the

generators

(2.~6) P~-F~, Po-=F,o, ~ o - F , , , ~v,_=F,~, /~---~(F~+2V2~,~)

form another SU3 subalgebra which hereaf te r we shall refer to a s A ~ U 3 . Wi t h this in view we t rans form the t e rm -- (F[3 -~-F~4) as follows:

(2.17) 15 8

- (F~ + Fh) - - ~ F~ + ~ (~v~ + _p~) _ a = a a = l

Inser t ing (2.17) into (2.13) and taking into account (2.3), we obta in t h e final

formula for the mass operator (*):

(.o.18) M o: ~- t i l t -~- ~¢ -~ (~[K 1 ¢ 2 1 -~- ) , [ ½ K - ~ / ~ - - 1 v ~ -aC2]

where /~ stands for the second-order S~3 Casimir opera tor

8

(2.19) /£ - - ~ / ~ . a ~ a

To apply formula (2.18) to each concrete hadron of a given SU4 multi- plet, we have to know its value o f /~ , or, all the same, which representa t ion

of the SUs subgroup i t belongs to.

N 3. - S Ua classification. Mass s u m rules.

Note tha t the operator - - 2 V ~ F s + Fa~ commutes with all the gener- ators (2.16) of the S~'Ua subalgebra, and therefore the q u a n t u m number asso- ciated with it takes the same value for all t he members of a given irreducible representa t ion of the S~3 subgroup. On the other hand, it follows f rom (2.3) t h a t

(3.1) 1 (n--~).

(*) The constants ~, fi . . . . in (2.18) need not to be the same as those in (2.13).

24 - I1 Nuovo Cimento A .

3 7 0 DAO v o ~ G D v c

Hence , t he h y p e r c h a r g e Y m a y serve t o sepa ra te t he m e m b e r s of different

SUa submul t ip le t s . As t o t he m e m b e r s be long ing to t he same SUa submul t ip le t , t h e y differ f r o m each o the r b y the i r isospin a n d cha rm, because

(3.2) { "~1,2,3 ~ /~al,2:3 = IIj',a ,

~ - ~ ~-~ r - y c + (n-~).

These r e m a r k s al low us t o t r e a t t he classif icat ion acco rd ing t o t h e de-

compos i t i on SUa ~ SUa in a r a t h e r s imple w a y , w h e n e v e r t he SU3 c o n t e n t of a g iven SUa mul t ip l e t is known . L e t us cons ider some concre te cases.

a) The 20-plet o] ½ + baryons. This mu l t i p l e t is descr ibed b y t he th i rd -

r a n k t enso r ~otn~ sa t i s fy ing t h e condi t ions

(3.3) YJ[~l~ : - - lP~,~, Wt~j~ -]- W ~ -+- ~ a ~

I t s SU3 c o n t e n t is

2 0 : 8 ~ - 6 ~ 3 + 3

wi th t h e fo l lowing a s s i g n m e n t :

[ oc t e t w i th C : 0:

X = I , I : !" 2" : Y = O , I = 1 :

Y = O , I = 0 :

~Y = - - 1, I : ½ :

6-ple t w i th C = 1 :

~ Y = I , I = 1 :

( 3 . 4 ) I 7 = 0 , I~--- 1.

:Y : - - I , I = 0 :

3-ple t w i t h C : 1:

I t = l , I = 0 :

17= O, i = 1 :

3-ple t w i th C ---- 2 :

Y = 1, I : ½ :

Y : O , I = 0 :

= 0 .

t~ ++ = C2~W~.j~, R + = - ( V ' ~ l l + V'fl41~),

R ° = - - " V ~ E ~ 4 J ~ ,

Sq- = '~/)[1413 -]-- ~[3411, S0 : ~/)[2413 -~ ~)[2,~]2, T O : 'V/~~r~4~3 ;

A + = V ' ~ ' ~ l m ,

B + = V'g~ram, B ° = C'3y-':a214 ;

U ++ : v/2~vr1414, U + = V ~ ( , , l , ,

V + : C 2 ~ ¢ 3 , j 4 .

MASS FORMULA FOR CHARMED-HADRON MULTIPLETS 371

With the above-quoted remarks in mind we now see that almost all these particles are definite states according to the decomposition BU, 3 Kg3, except for the particles S and B which have identical quantum numbers I , Y and 0, and therefore enter the Kg3 submultiplets by definite mixtures. These mix- tures can also be easily found from the transformation laws of the particle wave functions. The result is

octet with Y = 1:

C = O , I = r 2 : P,

C = 1 7 I = 1 : R f f , R+, RO,

C = l , I = O : A+,

C = 2 , I = l : 2 U+f, u+; 6-plet with Y = 0 :

C=O, 1 2 1 : z+, zO,z-, c = 1, I = 4 : 4(S+ + GB+), &(So + d 5 B 0 ) 7

C = 2 , I=O: v+; - 3-plet with Y = 0 :

C = 0 7 I = O : A ,

C = 1, I = 4: *(- d38+ + B+) , & ( - d 3 B 0 + BO) ;

3-plet with Y = - 1:

G=O, I= l : T O 7- 2 - ! 7 - 9

C = l , I = O : TO.

Now by taking the matrix elements of the mass operator (2.18) between various particle states, we can express their masses in terms of the constants a , j3, y , 6, A as follows:

- N = a + / 3 + 3 6 + L 3 L 1 ,

Z = a + 3 6 + 1 & 1 ,

A = a + 3 6 + 9 1 ,

S = c ~ - / 3 + 3 6 + 2 1 ,

R = ~ + / 3 + ~ + 3 6 + . ~ ~ ~ 1 ,

B = ~ + y + 3 6 + ~ , i l ,

T = a - P + y + 3 6 + 2 1 ,

A = a + B + y + 6 + 3 A ,

B = a + y + 6 + W ,

u = a + B + 2 y + % A , v = ~ + 2 y + ~ # A .

372 DAO VONG DUC

From (3.6) the following mass sum rules are obtained:

(3.7) N+c"=&L'+3A) (well-known 8 U, result) ,

Relations (3.8) and (3.9) have been first derived in the paper of Gaillard, Lee and Rosner (7). Note that relations (3.11) and (3.12) hold only for pure 8 and B states, which in reality may be mixed together because of the presence of symmetry breaking. Nevertheless, by summing them we get

This relation holds also for mixed 8 and B, since the mass sum 8 + B is in- dependent of the mixing parameter.

b ) The 20-plet of ++ baryons. This multiplet is described by the com- pletely symmetric third-rank tensor yrjk. Its S U , content is

with the following assignment :

10-plet with C = 0:

Y = 1, I = : N*++ = ylll, N*+ = dgyllz7 N*O = f l Y 1 2 z 7

AT*- = y222,

P r o 7 I Z 1 : 2*+ =Gy113)L'*O =&y123, L'*-=1/3ys237

Y = - 1, I = + : S*O = d3y133 , E*- = dzy233 7

Y=-2, I= 0: - =y3,,;

6-plet with C = 1 :

singlet with C = 3 :

Y = l , I= 0: X++ = y444

M A S S F O R M U L A F O R C H A R M ~ D - I I A ] 0 R O N M U L T I P L ] ~ T S

The SU3 a r r angemen t for this case is t r anspa ren t . We have

(3.15)

10-plet wi th [Y--~ 1:

C ~--O, I ~ - 8 : N*++, N*+, N* o, N * -

C 1, I ~ - 1 : R *++, R *+, R *°

C ~ 2 , I ~ ~-" U *++, U *+ 2 "

C--~3, I - ~ 0 : X ++;

6-plet wi th Y ~- 0 :

C-----0, I ~ l : Z *+, X *°, 27*-,

C 1, I - ~ ½ : S*+, S *°

C = 2 , I ~ 0 : V*+;

3-plet wi th Y ~ - - 1 :

C 0, I - ~ 1. ~.o ~ . - ~- - ~ . ~ , ~ ,

C~--1 , I - ~ O: T * ° ;

singlet wi th Y ---- - - 2 :

C = O , I = 0 : .(2-.

3 7 3

Now proceeding as in the previous case, we get the following mass sum rules (s):

(3.16) N * - - ~ * ~ - - ~ ' * - - ~ * ~ S* - - .(2 ---- R * - - S*--~ S * - - T*--~ U * - - V*,

(3.17) ) 7 * - R* -~ R * - - U* ~ U * - X ++ .

The origin of these equidis tance rules is t ha t , for the comple te ly s y m m e t r i c representa t ion 20 considered, one has

1 K ~ C ( C - - 9 ) - [ - 6 ,

1 10 _ ,7= 5 r ( Y + 7) + T '

(3.18)

and therefore the mass formula (2.18) tu rns out to become l inear wi th respec t

bo th to Y and C:

(3.19) M .= a -t- b Y -~- eC.

3 7 4 DA0 voN~ ~)vc

This is in full a g r e e m e n t w i t h t h e fac t t h a t for th is r ep re sen t a t i on (as a special case of degene ra t e r ep resen ta t ions ) b o t h t he second- a n d th i rd -o rde r (( vec to r ))

operators dabcF~F~ a n d d~b,d, odEoF~Fa can be expressed in t e r m s of F~ (~a).

e) The 15-plet o] O- mesons. This m u l t i p l e t is descr ibed b y t he t raceless

t e n s o r ~ . I t s SUa c o n t e n t is

1 5 = S + 3 + 3 + 1

w i t h t h e fo l lowing a s s i g n m e n t :

(3.20)

o c t e t w i th C = 0:

Y= I, I = ½: K+= ~ , K ° = ~ ,

i ] ~ = o , I=1:~+ = ~ , ~ o = ~ ( ~ _ ~ ) ,

1 2 8 = O, Z = O : n - - ~ ( ~ + q 4 - - ~ ) ,

3-ple t w i t h C = 1 :

Y=I , I = 0 : ~+=9~,

I 7 = 0~ T - - 1. D + - - ~2 D O = ( p ~ ,

3-plet w i th C = - - 1:

Y O, r _ _ 1. ~ o _ , = ~ - - ~ . ~ --%, D- = ~ ,

Y = - - I , I = 0 : F - = q~ ;

single* w i t h C = 0:

2 4

T h e mass f o r m u l a (2.18) t h e n gives (with fl = 7 = 0)

(3.21)

K ~ ~ + 3~ -4- 2~ ,

= ~ + 3~-t- ~ ,

D = a + ~ + ½ a 2 , - ~ . ,/o = ~ A - s

MASS ~ORMULA FOR CHAI~M]~D-HADI~ON MULTIPL]~TS 375

F r o m (3.21) we obta in

(3.22)

(3.23)

(3.24)

K--~=F--D,

z~ -~- ~ ~- ~ = ~ ( K @ D) .

(well-known SU3 resul t ) ,

Here the part icle symbols s t and for the masses or quadra t ic masses depending on e i ther the l inear or quadra t ic mass formnl~ adopted , l~elation (3.23) has been first der ived in ref. ('), i t is in nice ag reemen t wi th expe r imen t for the quadra t ic mass formula . Note t h a t , besides t he 15-plet of 0- mesons, there m a y exist ano the r SU~ singlet n a m e d ~' and the ~-~o-~' mix ing is also possible. !qevertheless, if one ignores the exis tence of t he ~' , t hen re la t ion (3.24) is independent of the ~-~o mix ing p a r a m e t e r and predic ts (for the quadra t ic

mass formula) a m~ of abou t 2.3 G e ¥ .

d) The 15-plet o] 1 - mesons. This mul t ip le t is t r e a t ed quite s imilar ly

to the one of pseudoscalar mesons. I n ana logy wi th (3.23) and (3.24) we h a v e

(3.25)

(3.26)

K * - - 0 = F * - - D * ,

q + ~ + ~ = ~ ( K * + D*) .

l~elation (3.25) is also satisfied well b y exper iment . As to re la t ion (3.26), i t holds only for pure q0 and ~, because the q0-~?-¢o mix ing effect is in fac t considerable.

• R I A S S U N T O (*)

8i deriva una formula generale di massa per tutti i multipletti osservati di adroni S U4. La formula eoinvolge assieme all'ipere~riea e al charm l'operatore di seconclo ordine di Casimir del sottogruppo SUz cosi come quello di qualche altro sottogruppo denomi- nate ~Uz. Si dg anche la classificazione degli adroni seguendo la decomposizione

(*) T,'aduzione a cura della Redazione.

Maecoua- ~popMy~la ~l~a O~lapoBaHHblX a~lpOUHblX MyJIbTHHJIeTOB.

Pe3mMe (*). - - B b I B O ~ I H T C ~ [ o6~a~ ~bopMyna ~ Bcex Ha6nio~aeMsiX S U 4 a~pOHH],IX MyIIbTHYIYIeTOB. ~ T a ~opMy~Ia BKJIIO~aeT, nOMHMO rHnep3apa~a ~ mapMa, oneparop Ka3nMgpa Broporo n o p ~ a ~aa no~rpynrmI S Uz, a Taxme onepaTop Ka3aMHpa ~zi~ ~pyroi~ no~rpynnbL ~a3bmaeMoft ~ 3 . YIp~mo~arcz xaacca~pH~auH~ a~poHon B COOT- BeTCTBHH C pa3no~eHHeM SU 4 D ~ z .

(*) Ilepese3eno peDaKque~.