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VOLUME 13, NUMBER 13 PHYSICAL REVIEW LETTERS 28 SEPTEMBER 1964 SPLITTING OF SPIN-UNITARY SPIN SUPERMULTIPLETS Mirza A. Baqi Beg and Virendra Singh* Institute for Advanced Study, Princeton, New Jersey (Received 26 August 1964) 1. Recent work of Giirsey, Pais, and Radi- cati 1 "" 3 appears to indicate that the ideas pro- pounded by Wigner 4 27 years ago may, with ap- propriate generalization to accommodate strange- ness, find spectacular fulfillment in the domain of particle physics. In the Sakata model one im- mediately gets SU(6) in place of Wigner y s SU(4); in the eightfold way a similar picture is easily constructed with the help of quarks. 5 In a rela- tivistic theory the full invariance group, of course, is not SU(6); however, the classification of particle states with respect to SU(6) still seems permissible. 6 The qualitative success of SU(6) classifications, in spite of the marked lack of degeneracy in the super multiple ts, prompts one to ask: What is the nature of the phenomenological interaction responsible for the breakdown? Unless one can pin down the transformation properties of this interaction the symmetry will be of little practi- cal use. In this connection it should be noted that mass formulas were written down in references 1 and 2 on the basis of physical intuition. It is not clear, a priori, whether these formulas can be derived by starting with any number of SU(6) tensor operators. The purpose of this note is to report some re- sults that have emerged in a systematic study of the problems mentioned above. We consider all the mass formulas that can be derived by con- sidering tensor operators transforming accord- ing to real representations of dimensionality less than 1000, which can contribute to the me- son, baryon, and low-lying resonance spectra. These representations and their SU(3)®SU(2) content are 7 35 = U, 3)©(8,3)©(8,1.), (1) 189 = (j., l.)©(8, J.)©(27, JL)©2(8,3)©(10, 3) ©(10*,3)©(1.,5)©(8,5), (2) 405 = (1 > 1)©(8,1)©(27 > 1_)©2(8 > 3)©(10,3) ©(10*,3)©(27,3)©U,5) ©(8, 5)©(27, 5). (3) 418 The tensors we consider will, of course, 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. Incidentally, the 3^ representation has already been considered by Kuo and Yao. 8 The choice turns out to be rather inadequate since the spin degeneracy is not lifted at all and for baryons the isospin degeneracy is not lifted either. Before we write down the mass formulas it is necessary to establish the requisite notation. 2. The first problem is to set up a scheme for labeling SU(6) states. Mathematically, such a scheme is afforded by the reduction chain SU(6)DU(1)®SU(5)DU(1)®U(1)®SU(4)« (4) We are unable, however, to find any physical meaning for the quantum numbers that emerge in this chain. We begin therefore by considering * ne physical chain (P chain) SU(6)DSU(2)®SU(3)DSU(2)®U(1)®SU(2), (5) which fails to furnish us with enough labels. We therefore supplement this chain with an unphys- ical chain (U chain) SU(6) 3 U (1) ® SU(2) ®SU(4) D U(l) ®SU(2) ®SU(2) ®SU(2). (6) The subgroups in one chain do not generally com- mute with those in another and appropriate "re- coupling" transformations are needed. 3. We denote by Ap a , a,p= 1,2, • • •, 6, the 35 infinitesimal generators of SU(6) satisfying the canonical commutation rules. From an inspec- tion of the adjoint representation one can pick out the generators of the commuting SU(3) and SU(2), respectively. These are SU(3): A 3 +A ; + 3 i + 3 z,j = l,2,3; (7) SU( 2)j : J + =5>/ . i = l 1 = 1

Splitting of Spin-Unitary Spin Supermultiplets

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Page 1: Splitting of Spin-Unitary Spin Supermultiplets

VOLUME 13, NUMBER 13 P H Y S I C A L R E V I E W L E T T E R S 28 SEPTEMBER 1964

SPLITTING OF SPIN-UNITARY SPIN SUPERMULTIPLETS

Mi rza A. Baqi Beg

and

Vi rendra Singh* Institute for Advanced Study, Princeton, New Jersey

(Received 26 August 1964)

1. Recent work of Giirsey, P a i s , and Rad i -cati1""3 appears to indicate that the ideas p r o ­pounded by Wigner4 27 y e a r s ago may, with a p ­p ropr i a t e general iza t ion to accommodate s t r a n g e ­n e s s , find spec tacu la r fulfillment in the domain of pa r t i c l e phys ics . In the Sakata model one i m ­mediately gets SU(6) in place of Wigner ys SU(4); in the eightfold way a s imi l a r p ic ture is easi ly const ructed with the help of qua rks . 5 In a r e l a -t iv is t ic theory the full invar iance group, of cou r se , is not SU(6); however, the classif icat ion of pa r t i c l e s t a t e s with r e spec t to SU(6) s t i l l s e e m s p e r m i s s i b l e . 6

The quali tat ive s u c c e s s of SU(6) c lass i f ica t ions , in spi te of the marked lack of degeneracy in the super mult iple t s , p rompt s one to ask: What is the na ture of the phenomenological in teract ion respons ib le for the breakdown? Unless one can pin down the t ransformat ion p rope r t i e s of this in teract ion the s y m m e t r y will be of li t t le p r a c t i ­cal u se . In this connection it should be noted that m a s s formulas we re wri t ten down in r e f e r ences 1 and 2 on the bas i s of physical intuition. It is not c l ea r , a p r i o r i , whether these formulas can be der ived by s ta r t ing with any number of SU(6) tensor o p e r a t o r s .

The purpose of this note is to r e p o r t some r e ­sul ts that have emerged in a sys temat ic study of the p rob lems mentioned above. We consider al l the m a s s formulas that can be der ived by con­s ider ing t ensor o p e r a t o r s t ransforming a c c o r d ­ing to r e a l r ep re sen ta t ions of dimensionali ty l e s s than 1000, which can contribute to the m e ­son, baryon, and low-lying resonance spec t r a . These r ep re sen ta t i ons and thei r SU(3)®SU(2) content a r e 7

35 = U, 3)©(8,3)©(8,1.) , (1)

189 = (j., l.)©(8, J.)©(27, JL)©2(8,3)©(10, 3)

©(10*,3)©(1. ,5)©(8,5), (2)

405 = (1>1)©(8,1)©(27>1_)©2(8>3)©(10,3)

©(10* ,3)©(27 ,3)©U,5)

©(8, 5)©(27, 5). (3)

418

The t e n s o r s we consider will , of cou r se , be s inglets under SU(2); under SU(3) we shal l take the ones that e i ther a r e singlet or t r ans fo rm like the I=Y = 0 m e m b e r of an octet .

Incidentally, the 3 ^ rep resen ta t ion has a l ready been considered by Kuo and Yao.8 The choice tu rns out to be r a t h e r inadequate s ince the spin degeneracy is not lifted at all and for baryons the isospin degeneracy is not lifted e i ther .

Before we wr i t e down the m a s s formulas it is n ece s sa ry to es tab l i sh the requ is i t e notation.

2. The f i r s t p roblem is to se t up a scheme for labeling SU(6) s t a t e s . Mathematical ly , such a scheme is afforded by the reduct ion chain

SU(6)DU(1)®SU(5)DU(1)®U(1)®SU(4)« (4)

We a r e unable, however , to find any physical meaning for the quantum numbers that emerge in this chain. We begin there fore by consider ing * n e physical chain (P chain)

SU(6)DSU(2)®SU(3)DSU(2)®U(1)®SU(2), (5)

which fails to furnish us with enough labels . We therefore supplement this chain with an unphys-ical chain (U chain)

SU(6) 3 U (1) ® SU(2) ®SU(4)

D U(l) ®SU(2) ®SU(2) ®SU(2). (6)

The subgroups in one chain do not general ly c o m ­mute with those in another and appropr ia te " r e -coupling" t rans format ions a r e needed.

3. We denote by Apa, a,p= 1,2, • • • , 6, the 35 infinitesimal gene ra to r s of SU(6) satisfying the canonical commutat ion r u l e s . F r o m an inspec­tion of the adjoint r ep resen ta t ion one can pick out the gene ra to r s of the commuting SU(3) and SU(2), respec t ive ly . These a r e

SU(3): A3 +A ; + 3

i + 3 z, j = l , 2 , 3 ; (7)

SU(2)j: J + = 5 > / . i = l

1 = 1

Page 2: Splitting of Spin-Unitary Spin Supermultiplets

VOLUME 13, NUMBER 13 P H Y S I C A L R E V I E W L E T T E R S 28 SEPTEMBER 1964

The subscript J implies ordinary spin. The iso-spin and hyper charge operators in SU(3) are, of course,

SU(2)7: I^Af+Af,

I_=A2l+A5*,

I^^-Af+AS-Af); (9)

U(l): y = - ( A 3s + V ) . (1Q)

Equations (7)-(10) complete the identification of operators in the P chain. For the U chain one has

SU(4): A? + {6?<AJ+A*), t , i = l , 2 , 4 , 5 ; (11)

U(l): Y; (12)

SU(2)S: S+=A3e,

S_ ~AQ ,

S 8 = | ( A j 8 - V ) . (13)

We use the subscript S to make explicit the fact that in the defining representation S is the spin of the strangeness-bearing quark. The commuting subgroups of SU(4) are SU(2)j and

SU(2)^: N^Af+Af,

N„=A41+A5\

^ ^ ( V - V + V - V ) . d4) N is the spin of quarks with no strangeness. Note the relationship between the two chains and the identity

J = N + S. (15)

We shall need only the quadratic Casimir oper­ators of the various groups. For SU(6) our defini­tion is

c2(8)=$i>-V=-ss V'V}-(i6)

A = l A,M = 1 Simi lar ly , C2

<3), C2<4), and C2

{2)(L) = L(L + 1) (L = I,J,S,N).

4. We proceed to the construction of tensor operators. For T(35) a general construction is immediately available from Ginibre's theorem9

and an analogous construction can be worked out for T(189) and T(405). We shall not quote these

general constructions since we are interested only in representations a such that OL*<8CI con­tains 189 and 405 no more than once.

For T<189) [T(405)], we first extract the anti­symmetric (symmetric) part of 606 = 1J3©21. to obtain the basis tensors of the 15- (21-) dimen­sional representation and at the same time iden­tify the quantum numbers associated with each component. With this information in hand we can write down the five I = J= Y = 0 states that oc­cur in the reducible representation 15*015 = 1®35®189 (21*021 = 1©35£>4Q5). The extrac­tion of orthogonal linear combinations with p re ­scribed transformation properties is then a straightforward task. A knowledge of the basis tensor leads immediately to the corresponding tensor operator.

All of these tensor operators can be expressed in terms of the Casimir operators of subgroups in either the P or the U chains. The mass opera­tors10 can then be read off from these expres­sions.

5. We indicate by M(nym> the mass operator

containing a symmetry-breaking term transform­ing like an SU(6) tensor of multiplicity n with an SU(3) component of multiplicity m and I=Y = 0, and singlet under SU(2)j.

The five "irreducible" mass formulas are

^(35) ( 8 ,=«i + V ^ i [ 2 5 ( S + l ) - C 2( 4 ,

+ i n (17)

M(i89>(1> -a2 + b2[2J(J+l)^C2iz)l (18)

M ( 1 8 9 )( 8 ) = «3 + 6 3{f2J(J r+l)- .C 2

( 3 ) ] + 3[2/(/+l)

- |I*-2J\r(tf+l) + 2S(S+l)]

- | [ 2S(S + l ) -C 2( 4 )

+ i y 2 ] } , (19)

Mim)il> = a4 + b4[2J(J+l) + C2<*>], (20)

M ( 4 0 5 )( 8 ) = a5 + 65{[2J(J+1)+C2

< 3 )] + (21/8)[2S(S+1)

-C2< 4 ) + iY2] + 3[2/( /+ l )

- | Y 2 + 2N(N+ 1)-2S(S+ 1)]}, (21)

where the coefficients depend only on the Casimir operators of SU(6)0

If the symmetry-breaking term in the actual mass operator contains contributions from all the five tensors listed above, the mass operator10

is

M = a + bC2i3)+cJ(J+l) + dY

+ e[2S(S+l)-C2(4)+\Y2]

+f[N(N+l)-S(S+l)]

+*[/( /+1H[Y2 l (22)

419

Page 3: Splitting of Spin-Unitary Spin Supermultiplets

VOLUME 13, NUMBER 13 PHYSICAL R E V I E W LETTERS 28 SEPTEMBER 1964

We proceed to examine the consequences of Eq. (22) for the 56- and 35-dimensional repre­sentations.

6. In the 56-dimensional representation there exist the following identities:

2J(J+l)-C2< 3 ) = -f, (23)

2S(S+l)-C2(4> + ! Y 2 = - 8 y - 1 5 / 2 , (24)

/(/+ 1)- ±J2-N(N+ 1) + S(S+ 1) = - F + \. (25)

Equation (22) therefore collapses into

M = M0+M1J(J+l)+M2Y + Ms[l(I + l)-\Y2], (26)

a result conjectured by Giirsey and Radicati.1

Mass relationships based on this formula are satisfied to great accuracy. One now has an ex­planation for the empirically known fact that the mass formula for the bar yon octet can be used with the same coefficients for the resonance de-cuplet in broken SU(3).

7. For the 35-dimensional representation, Eqc (22) must be used with care since there is no analog of the identity (23) and hence the mass operator is not a priori diagonal in either the P or the U chains. Only the u) and <p states, how­ever, are affected.

Let CL>U,<^U b e eigenstates of operators in the U chain, and a>p, <pp of those of the P chain. They are related through the equations

W u = ( i ) 1 / 2W p + ( l ^ V p , (27)

^=- ( l ) 1 / 2 w p+(3> I / 2 V (28)

Since the bulk of the mass operator is diagonal in the U chain, it is convenient to start with w^j and cp\j as the basis and subsequently carry out the diagonalization of the mass matrix. The new eigenvectors are the physical a> and <p.

By a straightforward evaluation of the quantum numbers that occur in Eq. (22) (see Table I), we can write down the squares of meson masses1 1

in terms of a,b,c,d,e,f and obtain sum rules by elimination. For pseudoscalar mesons one r e ­covers the usual sum rule [meson label = (meson mass)2]

4K-IT = 3T]. (29)

No other sum rules are possible since our origi­nal formula was much too general.

If we drop the contribution of M(189)(8), we get

the constraint f=g. One extra sum rule is now obtained, to wit,

w<p = ±(7t+K*-K)(3K*-p +K-ir)

-$(4ff *-p)(&fi:*-p + -n-K-2u-2(p). (30)

420

Table I. Quantum numbers of mesons and baryons. Center dots mean "not an eigenstate."

Particle

7T

P u>u o;p

V K # * K K*

<?U <p-p N N*

2 A

Yt* E w*

a

i

l

I 0 0

0 1/2

1/2

1/2

1/2 0

0 1/2 3/2

1

0 1

1/2

1/2 0

N

0

1 1

0 1/2

1/2

1/2 1/2

0

1/2 3/2

1

0 1

1/2 1/2

0

S

0

0 0

0 1/2

1/2 1/2

1/2 1

0

0

1/2

1/2 1/2

1 1

3/2

J

0 1 1

1

0 0

1 0 1

1

1 1/2 3/2

1/2

1/2 3/2

1/2 3/2

3/2

r <3> C 2

6 6

. . . 6

6 6

6

6 6

0 6

12 6

6 12

6 12

12

C 2

8

8 8

0 15/4 15/4 15/4 15/4

0

63/4 63/4

9

9 9

15/4 15/4

0

With the present mass values Eq. (30) appears to be obeyed quite well. We are thus led to con­jecture that the 189-octet contribution is indeed absent. It is important to state, however, that no further contributions can be dropped without running into serious contradiction with physical reality.

8. The mass operator,10 we are led to propose, is therefore

M = a + bC2i3) + cJ(J+l) + dY

+ 4 2 S ( s + i ) - c 2( 4 ) + -1y2]

+/[/(/+ l)-iY» 4 N(N+ 1)-S(S+ 1)]. (31)

Applications of this formula to the 70-dimen­sional representation will be the subject of a forthcoming communication.

If one uses Eq. (31) to define the meson central mass and Eq. (26) to define the baryon central mass, one obtains -610 MeV and ~970 MeV, r e ­spectively. Equation (9) of reference 3 now gives £*ps2/47r~ 13, a gratifying result.

We are deeply indebted to Professor A. Pais for his interest and enthusiastic encouragement. Many conversations with Professor F. J. Dyson are gratefully acknowledged. One of us (V.S.) wishes to thank Professor J. R. Oppenheimer for hospitality at the Institute for Advanced Study.

*On leave from (and address after 1 September 1964)

Page 4: Splitting of Spin-Unitary Spin Supermultiplets

VOLUME 13, NUMBER 13 P H Y S I C A L R E V I E W L E T T E R S 28 SEPTEMBER 1964

Tata Institute for Fundamental Research, Bombay, India.

1 F . Glirsey and L. Radieati, Phys. Rev. Letters 13, 173 (1964).

2A. Pais, Phys. Rev. Letters 13, 175 (1964). 3 F . Giirsey, A. Pais, and L. Radieati, Phys. Rev.

Letters 23, 299 (1964). 4E. Wigner, Phys. Rev. 5>1, 105 (1937). 5M. Gell-Mann, Phys. Le t te r s^ , 214 (1964). 6 F . Giirsey, A. Pais, and L. Radieati, private com­

munication. 7We are aware that representations are not, in gen­

eral, determined by their dimensionality. However,

FINAL-STATE INTERACTIONS IN THE DE­CAY 7?°-37r. Frank S. Crawford, J r . , Ronald A. Grossman, L . J .L loyd , LeRoy R. Price, and Earle C. Fowler [Phys. Rev. Letters jU, 564 (1963)].

Our cutoff criterion "remove events with m(e+e"~) <100 MeV" was designed to eliminate 77° Dalitz-pair background events u* +p-~ir+ +p +e+ + e~ +X° from the desired sample of events 7r +/>-~7r +/> + 7r +TT~+X°. Unfortunately, it also removes 7r+7r"" pairs with a small laboratory opening angle. We were aware of this bias but wrongly estimated that the effect on the spec­trum was negligible. The cutoff turns out to be unnecessary, and we have now rescued by ioni­zation criteria the cut-off events. Our sample of 97 decays is thus increased to 109. Also, we now constrain the mass of the j\ to 548 MeV.

there is no ambiguity in the representations considered in this paper.

8 T. K. Kuo and T. Yao, preceding Letter [Phys. Rev. Letters 13,415(1964)].

9 J . Ginibre, J . Math. Phys. 4, 720 (1963). 10Note that these mass operators commute with all

the Casimir operators of SU(6) and thus cannot r e ­produce the off-diagonal elements of the symmetry-breaking interaction. The dependence on state labels, in a given SU(6) representation, is, however, correct­ly reproduced (Wigner-Eckart theorem).

uWe have followed the canonical practice of using masses for fermions and (masses)2 for bosons.

Our corrected (vs published) number of events in the seven 12-MeV intervals of TT° kinetic en­ergy from 0 to 84 MeV are 10 (was 10), 29 (23), 22 (29), 20 (17), 12 (12), 11 (5), and 5 (1). Our corrected (vs published) values of fitted param­eters are, for the linear matrix-element theory, a = 0.45±0.07 (was 0.71 ±0.09), with x2 = 4.2 (was 6.1), which predicts R = (000)/(+-0) = 1.63 ±0.03 (was 1.50±0.04). For the Brown and Singer theory, we find m a = 392±9 (was 381 ± 5), r a = 8 8 ± 1 5 (was 48±8), and x2 = 2.4 (was 2.7), which predict R = 1.28±0.07 (was 1.02±0.07). The experimental value of R according to the compilation of Rosenfeld et al.1 is R = 1.16 ±0.15.

*A. H. Rosenfeld, A. Barbaro-Galtieri, W. H. Bar-kas, P. L. Bastien, J. Kirz, and M. Roos, to be pub­lished.

E R R A T U M

421