P H Y S I C A L R E V I E W V O L U M E 1 4 2 , N U M B E R 4 F E B R U A R Y 1 9 6 6

Mass Formulas for the 189 Multiplet of SU(fi)* CHIA-HWA CHAN AND NGUYEN-HUU XUONG

University of California, San Diego, La Jolla, California (Received 24 September 1965)

Mass relations for the 189 multiplet of SU(6) are discussed in detail. They (the masses of the 25 different isomultiplets belonging to 189) are given in terms of six mass-splitting parameters. If the newly discovered mesons are assumed to belong to 189, as soon as the determination of spin and parity of six of them is made, one can use their masses as input and predict all the masses of the remaining particles belonging to 189.

THE original static SU(6) theory1 has been used successfully in classifying the low-lying baryons

into a 56 multiplet and the low-lying mesons into a singlet and a 35 multiplet. The experimental values of the masses of these particles agree well with the SU(6) mass formula proposed by Beg and Singh.2 Other baryon-meson resonances can be classified into the 70 multiplet,3 and Beg and Singh4 have given the mass relations among them. Recently, many new meson resonances have been discovered.5 As these particles decay mostly into two members of the 35, they can be classified into

(35®35)sym= 103501890405.

Particles with spin 2, like the /° and A2, must belong either to the 189 or to the 405. In this article we would like to give in detail the mass relations of the 189.6

We note that

189<g>189=l+2(35)+175+3(189)+280+280*+405 +490+490*+ 2 (896+896*)+2 (3675)

+3969+5670+5670*+6720.

Following Beg and Singh, we consider all the mass formulas that can be derived by considering tensor operators transforming according to real representa­tions of dimensionality less than 1000. The tensors we consider will be singlets under SU(2); under SU(3) we shall take the ones that either are singlet or transform like the I=Y=0 member of an octet. We shall also neglect the contribution from the octet part of the 189 as is done in the work of Beg and Singh. The mass for­

mula for the 189 we derive is7

*This work supported in part by the U. S. Atomic Energy Commission.

^ i i r s ey and Radicati, Phys. Rev. Letters 13, 173 (1964); B. Sakita, Phys. Rev. 136, В1756 (1964).

2 M. A. B. Beg and V. Singh, Phys. Rev. Letters 13; 418 (1964). 3 A. Pais, Phys. Rev. Letters 13, 175 (1964). 4 M. A. B. Beg and V. Singh, Phys. Rev. Letters 13, 509 (1964). 6 See, for example: Lyndon M. Hardy, Sul Urk Chung, Orin I.

Dahl, Richard I. Hess, Janos Kirz, and Donald H. Miller, Phys. Rev. Letters 14, 401 (1965); Donald H. Miller, Sul Urk Chung, Orin I. Dahl, Richard I. Hess, Lyndon M. Hardy, Janos Kirz, and Werner Koellner, ibid. 14, 1074 (1965); V. E. Barnes, В. В. Culwick, P. Guidoni, G. R. Kalbfleisch, G. W. London, R. B. Palmer, D. Radojicic, D. С Rahm, R. R. Rau, C. R. Richardson, N. P. Samios, J. R. Smith, B. Goz, N. Horowitz, T. Kikuchi, J. Leitner^and R. Wolfe, ibid. 15, 322 (1965).

6 There is no real reason to prefer 189 to 405; it can only be decided by experiment. We choose it because it has a smaller dimension.

M2= +W(/+l)+4»4[C1w-2S(.S ,+ l ) - iF 1 ] + 2 « , [ ^ ( ^ + l ) - 5 ( 5 + l ) + / ( / + l ) - i P ] +^m£2j(j+ 1)+22C2<

3) - (j(j+1))2

-iO'+W- •(СУ8>)2], (1)

where wi, • • -m6 are parameters, C2(3) and C2

(4) are the quadratic Casimir operators of SU(S) and St/(4), N and S are the spins of the non-strangeness-bearing and strangeness-bearing quarks, and / , F, and / have the usual meaning of isotopic spin, hypercharge, and spin.

This formula8 is in fact rather similar to the general mass formula suggested by Beg and Singh [Eq. (31) of Ref. 2] except that here we have an additional term—the last term in Eq. (1). This term comes from the second symmetric 189 multiplet in the reduction of the direct product 189® 189. (In fact, in the reduction of 189(8)189, 189 occurs thrice, but the third 189 is antisymmetric and therefore does not contribute to the meson mass-breaking.)

Our procedure closely follows that in Ref. 4. We first decompose 189 into multiplets belonging to $U(3)®SU(2) (P chain):

189= (1+8+27, l )+(8i+82+10+10, 3 ) + ( l + 8 , 5).

Our notation for particle states which are eigenstates of operators in the P chain is9

(M)=-xy, (8,1) = i£ ' , тр, rjp7 Kf,

(27,1) = S, T, Up, V, Wp, Zp, Up, T, S, (8i,3) = iT/~ , pp, ы, Kp*~ , ( 8 2 , 3 b i £ / W , ^ X / ' ; (2)

(10,3) = L, (RdP,Mp,R,

(iO,3) = N,Mp,(R2)p,L,

(1,5) = / p , (8,5) = й,г,Ор, к.

7 This formula is only good for the 189 multiplet, with the assumptions we made in the text.

8 The term involving Y vanishes here because we have both particles and antiparticles belonging to the same representation.

9 For a more precise definition of the names of the particles, see Table I.

142 1132

142 MASS F O R M U L A S FOR 189 M U L T I P L E T OF 517(6) 1133

There are two octets with / = 1; we choose them to be orthogonal to each other with 8i and 82 to be normal and abnormal in the sense defined by Feynman, Gell-Mann, and Zweig.10 The decomposition of the 189 into SU(4:)®SU(2)S®U(1)Y (U chain) is as follows:

189= (6,1)F=2+ (4,2)F==1+ (202,2)F==1+ (20i,l)r_0

+ (15,1) F==0+ (15,3) F==0+ (202,2) r _ x + (4,2)F=_1+(6,l)r=-2,

where the first number in the brackets refers to the dimension of 5/7(4) and the second one to (25+1). Our notation for particle states which are eigenstates of operators in the U chain is

TABLE I. Quantum numbers of meson resonances in the 189-dimensional representation of SU(6).

(2,0;1,0,0) = S,N,

(3) V1* 2 > 2> 2> 2 / ~ ~ u u, J^u ,

(0 ,0 :2 ,0 ,0 )=7 ,Z U , (Ri)u,fu, (0,0;1, 1,0) = ячД (2?,)u, a>', (0, 1:1, 1, 0) = riJ9 Wu, Pu, w, pu, Ou, r.

Here the number in parentheses are (Y,S;Qi,Q2}Qz), where Qi, Q2, and Q% are the Wigner numbers. The anti-particle states are not given here. They can be easily obtained from the first three 5/7(4) submultiplets in Eq. (3) by changing F to —Y and Qz to — Q3. Re-coupling formulas, relating states (with same F, / , and J) in the two chains are as follows:

[м л f f - | i]fM k*i = - * - * *\\K*

Tf*f 1. 2 2 J?*t I ^ v J w L 3 3 JJ J v-tV J _

V* VI Vi V(V12) V(2/15) -V(9/20), LV(5/12) -V(8/15) V(V20)J

*'1 = fV* ~Vi)fK') , VVlu), Vf

#1") fVI - V * 0 R>\ = V* VI -V* P' \Vi VI VI

К lz J rx k lz

[u):[ Vi

lp?-]. M (4)

-vr 0 0 Vi

R2 p'

lp .

L 0 JM Wi -VI

V fl[7] • iJ loj ,

Particles

Л.П

x; Kp Ku

1 TTp

7T« Vp Vu

s T Up Uu V Wp Wu

ZiU

Kv*~ Ku*~ pp Pu CO Kp*' Ku*' pp

f Pu

f CO

L (RI)P (Ri)u N Mp Mu Шр №)« fp fu k r Op Ou

Y

0 0 1 1 0 0 0 0 2 1 1 1 0 0 0 0 0 1 1 0 0 0 1 1 0 0 0 1 0 0 2 1 1 0 0 0 0 1 0 0 0

/ 0 0 i 1 2 1 1 0 0 1 * i 1 2 2 1 1 0 0 i 1

1 1 0 \ i 1 1 0 3 2 1 1 0 i 1 1 1 0 0 i 1 0 0

N

0

"i 0

1 0 1 2

1 2 0

1

0

1 2

1 1 1 2

0 1 1 2

1 1 3 2

1

2 3 2 1

1

S

0

1 2

0

1 0 1 2

1 2 0

1

0

"i 1 1 1 2

1 0 1 2

0 0 1 2

0

0 * 1

1

/ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

2 2 2 2 2 2

C2(3>

0

6

6

6

16 16 16

16 16

16

6

6

6 6

6

6 12 12

12 12

12

0

6 6 6

C 2( 4 )

0

39/4

8

8 5

39/4

15/4 12

8

12

15/4

8 8

39/4

8 8

39/4

12 5

39/4

8

12 39/4

8

8

S = wi+ 8W2+ 2m4-|- 2ШБ+ 8жб, Г= wi+ 8W2+4w4+ 7шб+ 8шб, F=mi+8w2+6w4+12w6+8w6, Z=wi+6w2+w3+4m4+7w6+8w6,

We have tabulated in Table I the quantum numbers associated with the states listed above. In the table, particles with negative Y are omitted, because they are just antiparticles of particles with positive F. From Table I and Eqs. (1) and (4) we can calculate all the masses in terms of rnv'-niQ. For particles with the same / , F, and / , the physical states are neither eigen­states of the P chain nor of the U chain. To get these masses, we start with the U chain as the basis and diagonalize the mass operator. We obtain the following 25 equations for the 25 different isomultiplets belonging to 189, where we have used the particle symbol to denote the square of its mass.

со'=mi+ Зш2+mz+4^4+4w6+ 7 We, CO = Ш1+ЗЖ2+Шз+ 2^4+ 7^6 , k=wi+3 Ш2+3w3+4^4+ 7 ть+Згпв, r=Wi+3W2+v3w3+2W4+4W5+3W6,

R. P. Feynman, M. Gell-Mann, and G. Zweig, Phys. Rev. Letters 13, 678 (1964).

1134 С. Н. C H A N A N D N . H. XUONG 142

N= mi+6m2+ms+ 2w4+ 2m^+ SMQ , (К^в+(и)в=2т1+11т%+5ш^+2ть+16т%, {K')R(U)R— (ш1+4т2+4ш4+Мб+8шб)(^1+7ш2+^4+^5+8шб)--4ш227 (7г/)я+ (W)R= 2mi+11^2+6^4+8^6+ 16m6, W)R (W)R= (Wi+ 5^2+4^4+ 4^5+ 8^б) (ш1+6Ш2+2^4+4Ш5+8Шб) — 6W22 , (#i) д+ (p) R=2mi+ 9m2+ 2^з+ 8^4+12m5+15ш6, (RI)R(P)R= [mi+ 4^2+Жз+ 6^4+ 8w5+ (22/3) w6][wi+5ш2+^3+2w4+4w5+ (23/3)?%]—2(^2+im6)2, №0д+ (p') i?= 2^i+9f#2+2m3+ 6w4+8w6+ 15w6 , (R2)R(P')R= [ W I + 4 W 2 + W 3 + 2 M 4 + (22/3)ш6][м1+5ш2+Мз+4ш4+8ш5+ (23/3)w6]—2(w2+|m6)2, (/)д+(0)д=2^1+3ш2+6шз+8ш4+12шб+^б, (J)B(P)B= (mi+m2+Smz+2m4l— \т%) (^i+2w2+Зш3+6ш4+ 12жб+ |ш6)—2[m2+ (5/3)ш6]2 , (Xf)R+(v,)R+(Z)R=3ml+llm2+Sm,+ 16m,f (Х%(ч%+ (ri%(Z)B+ (Z)R(X%= [ > I + 5 W , + (20/3)мб][^1+4ш2+2ш4+ (14/3)m6]

+ [^i+4w2+2w4+ (14/3)w6][wi+2w2+6w4+ (14/3)ш6]+[ш1+2ш2+6ш4+ (14/3)w6] X[wi+Sw2+ (20/3)w6]~ (40/9)m6

2- 10(ш2+|ж6)2-[2ш2+ (10/3)w6]2, (X')B(fi%(Z)B= [wi+Sw2+ (20/3)м6][ш1+4ш2+2м4+ (14/3)w6][wi+2w2+6w4+ (14/3)*»«]

- [ 2 w 2 + (10/3)м6]2[ш1+5ш2+ (20/3)m6]~- C(40/9)w62][mi+4w2+2m4+ (14/3)w«]

- 1 0 (w2+ f w 6 ) 2 Oi+ 2w2+6w4+ (14/3) w6] - (40/3) w6 (w2+ fwe) [2w2+ (10/3) w 6 ] , (Af)*+ (Х*)й+ (iT*,)^=3w1+12w2+3^3+9w4+9w&+22w6

(Mb(i?*b+ (£*)д(£*')*+ (**0* (Mb = [_тг\- (13/3)^2+Шз+4ш4+7шб+ (67/9)шб][^1+ (13/3)м2+^з+^4+Ш5+ (67/9)we]

+ [wi+ (13/3)^2+^3+^4+^5+ ( 6 7 / 9 ) W 6 ] [ M I + (10/3)w2+w3+4?#4+M6+ (64/9)we] + [W+- (10/3)^2+^3+4^4+^5+ (64/9)w6][wi+ (13/3)ш2+ш3+4^4+7шб+ (67/9)МВ]

- (24/81) (Зж2+Мб)2, (ло* (**')*(£*)*

= [V»r+ (13/3)т2+шз+4ж4+7шб+ (67/9)^б]С^1+ (13/3)м 2+^з+^4+^5+ (67/9)we] X [tni+ (10/3)w2+W3+4w4+ ШьЛ- (64/9)w6]

- (4/81)(3w2+w6)2C6wi+22w2+6w3+21w4+12w5+ (130/3)w6]+ (32/729) (3w2+w6)3. It is worthwhile to note that the particle states <a and a/ do not mix with each other. The former has positive G parity and therefore can not decay into р+ж; its main decay mode will be J?+iT*. The latter has negative G parity and therefore can decay into р+ж. Also, the particles Ri, R%, p and p (with / = 1, F = 0 and / = 1) break into two groups with (p)R and (RI)R having negative G parity and (p% and (J^2)B having positive G parity.11

At present, the spin and parity of the newly discovered mesons are not well known. However, as soon as the determination of spin and parity of six new mesons is made, one can use their masses as input into Eq. (5) and predict all the masses of the remaining particles belonging to 189.

We would like to thank Professor Freeman J. Dyson and Professor M. A. B. Beg for many interesting discussions. 11 We note that (Ri)p and (i?2)P do not have a definite G-parity? but the real physical states (RI)E and (R2)R Cso a r e (ROu a n ( l № ) « ] ,

being a mixture of (Ri)p, (R2)P, (pf)Pi and (P)P, do have a definite G-parity opposite to each other. See also B. W. Lee, S. Okubo, and J. Schecter (unpublished).