Mass formula for charmed-hadron multiplets

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 multiplet, 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 generators (2.16) of the S~'Ua subalgebra, and therefore the q u a n t u m number associated 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 mixtures 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 independent 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~.

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Mass formula for charmed-hadron multiplets