L:ETTER:E AL NUOVO CIMENTO VOL. 16, N. 9 26 Giugno 1976

Isoplet Mass Splitting Determined by the Difference in Action

of the u- and d-Quarks (*).

C. S. K A ~ A ~

Physics Department Concordia University - Montreal, Qudbec, Canada (**)

(ricevuto il 26 Aprile 1976)

DOTHAN and NF,'~,MAN (~) likened the emergence of spectrum generating algebras in particle physics to the strong-coupling theories of the nineteen forties, or their more recent equivalent, the nucleon-isobar reciprocal bootstrap and its generalized sequels. In such a model the radial meson field variable is limited to a fixed (~ radius ~. Since the space of meson operators is a tensor representation of S U 3 built from the direct product of ~( quark ~ and (~ antiquark ~ operators, it makes sense to replace the meson variable by a corresponding quark variable. The corresponding limitation of the radial quark field variable is analogous to the MIT bag model (2) or to the concept of infra- red slavery (3).

If M~ is a meson field operator, then

(1) Mir = BidB~j, i, j = 1, 2, 3 ,

where B~r transforms according to the triplet representation of SU3 and B4~ according to the antitriplet representation (for the moment the index 4 is an arbitrary addition). Now suppose that A~r i , j = 1, 2, 3 are the generators of S U 3, then

(2) [Aej, Ak~] = ~imAk~-- ~k~Ai~, i, j , k, m = 1, 2, 3,

(3) ~ [a o, Ak~] = 0, i, j = 1, 2, 3 , k = l

(4) [A~j, B,k] = a,,Baj, r k = 1, 2, 3 ,

(5) [Aii , B j = - - dk~B~4, i, j , k = 1, 2, 3 .

(*) Work supported in pa r t by the Nat ional Research Council of Canada. ( . . ) Concordia Univers i ty de Qu6bec, E lementary Par t ic le Physics Group, CUQ/EPP-I1/76. (1) Y. D o ~ and Y. I~~ AEC research and deveZapmen~ report, CALT-68-41 (~965). (l) R. C. JA~FE and J. KISKIS: Phys. Rev. 1), to be publ ished (March 1976). (*) A. DE ROJUL~, H. GEORGI and S. L. GLASHOW: Phys. Fev. D, 12~ 147 (1975).

276

ISOPLET MASS SPLITTING D E T E R M I N E D BY THE D I F F E R E N C E ETC. 277

In analogy with the Kuriyan-Sudarshan-Chew-Low scheme (a,5), the dynamical algebra of B's and A's (eq. (2)-(5)) is closed by the condition

(6) [B4i , Bj4 ] = O(~r Aj~) , i, j = 1, 2, 3 ,

so that with

O : q - 1

0 : - - 1

0 = 0

the dynamical algebra is the Lie algebra of S U 4 ,

it is that of SU~, 3,

it is that of T T ~ S U 3 .

The group T : | 3 is ruled out because by its use one cannot predict transitions between elementary particles.

Suppose that A~., i , j = 1,2, generate the isospin subgroup of SUa. Then B3~ can be identified with the s-quark, B~4 with the u-quark and B24 with the d-quark. Also the SU4 and SU~,~ representations can be constructed by means of a Gel ' land basis (6)

(7) ~(,~) = m] 3 m., m=3 } .

)t~12D~ll ?~122 / /

All the parameters mij are integers of any size subject to certain conditions. In the first-order t ime-independent perturbation theory, the mass of each baryon is given by

(8) i , j=l

The action of the s-quark breaks the S U 3 symmetry. In the case of the dynamical group SU1, ~ (v), one obtains

(9)

and

(10)

M(&') + M(E) = 2M(A)

_~(A) + 2M(E) = 3;g(Z) .

These sum rules are satisfied within a 5% error; moreover the combination of these two sum rules yields the Gel1-Mann-Okubo mass formula. In the case of the dynamical group SU4 (s) the GM0 formula is also obtained. The action of the u- and d-quarks break the SU2 symmetry. For this calculation only the first three rows of the Gclf'and pat tern (eq. (7)) are needed. For an octet representation, m13= m23+ 1 = m33 + 2.

(4) J . KURIY~r a n d E . C. G. SUD~kRSI3[2klff: Phy s . Rev . , 162, 1650 (1967). (~) C. S. IEALMAN: Can. Journ . P h y s . , 50, 481 (1972). (a) I . M . GEL'FA~CD a n d M. I . GRAEV: Amer . Math . Soc. Transl . , SeE 2, 64, 116 (1967). (7) C. S. KAI~IA~r Can. Journ . P h y s . , 51, 1573 (1973). (s) C . S . KAD~A~r Particles and Nucle i , 2, 185 (1971).

278

The last two rows are as follows:

m,3 m~a-- 11 n-->- m,3 /

~o..+ ( mla m13-- 2 ) ,

\ re ,a-- 1 ]

~__+ ( m*3-1 m'3-- 21, \ m13-- 1 ]

p-+ \ m13-- 1 l

Z+_+ ( ml~ m'3-- 2) ,

\ m13--2 /

~o.._> (m13-- 1 m,,--2). \ mla - - 2 J

Using eq. (8), one then obtains

(11) M(n) =

(12) .]/(p) =

(13) M(~..-) ----

(14) M(Z ~ =

(15) 2d(:~ +) =

(16) M(A) =

(17) M(Z-) =

( is) M(~ ~ =

Hence

(19)

(2o)

C. S. KALM'AN

mia ~ t a - - 2

Z - ~ t ' 'm~la /

h"+( mla-1 m n - l ) ,

\ m 1 3 - 1

Me(5') + 3C~ + 4Va,

2~o(~) + 4c~ + 3v~,

Me(Z) + (Cu + ~3)(5.5 + v/~/2(2:olA>), ~o(:~) + SC~ + 303,

Me(A) + (C~ + 0~)(4.5 + V~/2<AI-ro>),

.~o(~) --{- 4C 2 -~ 8C3,

2~o(~) + 8c2 + 4c~.

~(~-) - - M(E o) = 4[M(n) - - ~ ( p ) ] .

Equat ion 19 is satisfied by the current experimental values and eq. (20) is satisfied within 10% of M(E- ) - -M(~~ Note also tha t the combination of these two sum rules yields the Coleman-Glashow mass formula.