9
PHYSICAL REVIEW D VOLUME 13, NUMBER 7 1 APRIL 1976 Spin-unitary-spin splitting of SU(8) supermultiplets* C. A. Nelson Department of Physics, State University of New York at Binghamton, Binghamton, New York 13901 (Received 8 September 1975) The surprising narrowness of the J or ((3.1) is interpreted as indication of a_pure cF state, and hence as evidence for the SU(8)+SU(6) X SU(2)sc X U(l),c symmetry-breaking chain (S, = charmed-quark-spin generators, Y , = hypercharm generator) instead of an approximate SU(8)+SU(4) X SU(2), chain (S = quark spin generators) which would imply strong mixings. Decompositions under both chains of the s-wave qq meson states of the 64 = L+ 9 of SU(8) and of the 3 q baryon states of the three-particle symmetric 120 representation are given. The most general mass-splitting operators with breaking in the Y and Y, directions for these two multiplets are derived, which commute with the Casimir operators of the SU(6) X SU(2), X U(1), chain, which contain only one- and two-body operators, and which are invariant under rotations. Two independent mass relations follow for mesons containing charmed quarks; six, for baryons containing charmed quarks. In an appendix, for reference relative to previous SU(6)-symmetric quark-model mass analyses, the reduced numerical coefficients as determined by the meson 3 of SU(6) are listed. I. INTRODUCTION Our purpose is to discuss here two topics from the standpoint of the charmed symmetric quark model: (i) The ground state 64 = 1 + 63 (meson) and 120 --- (baryon) representations of SU(8) together w i t h their decompositions under the subgroups SU(6) x SU(2), x U(l) , and SU(4) x SU(2), where S stands fir spin. The SU(2) ,c subgroup acts on the c quark's spin, and Y, is the hypercharm operator (see Sec. 111) with eigenvalues -a, -$, -$, and $ for, respectively, the 6, X, A, and c type quarks. (ii) The mass operator for these states in the SU(6) X SU(2),, x U(l)ycchain which is derived by extending the one- and two-body force analysis of the SU(6)-symmetric quark model1 which previ- ously gave the successful mass formula^"^ for the baryons, e.g., the Giirsey-Radicati formula4 for the 56 of SU(6) theory2" Electromagnetic ef- fects willbe ign~red.~ The motivation, of course, is the recent dis- covery7 of narrow resonances J or $(3.1), and $'(3.7), which can be interpreted as charmed8 quark-antiquark objects, JPC = I--, IG= Om, with N= 0 and 2 harmonic-oscillator quanta excited, respectively. The N = 2 state is either a radial or an orbital excitation, It is important to recog- nize that present difficulties with the charm in- terpretation (the rise of R to 5-3 i0.6 at 7.8 GeV, the absence of narrow peaks in missing-mass plots, the absence of increased kaon to pion pro- duction ratio, etc.), principally involve phenom- ena in eZ- hadrons above the transition region at about 3.6-4.1 GeV. Hence, these difficulties may, in fact, not exist if the transition region is due to excitation of first the charm and then the color degrees of freedom at about 3.9 GeV, which would be a natural o c c u r r e n ~ e ~ - ~ ~ in the Han-Nambu ver- sion of the three-quartet model. In this case, the details of the discussion in this article apply to the Su(3)"-color singlet states. On the other hand, this type of mass analysis remains relevant, though not in detail, even if additional heavy quarks1'-l4 are found to be necessary, because this analysis preserves successful ~ ~(61 results and accepts the heavy-quark explanation, based on the Okubo- Zweig-Iizuka rule, of the narrowness of the new particles. A single charmed quark is certainly the simplest of such heavy-quark models. Lastly, we emphasize the basic contrast between (a) the present spectra and mass analyses in which the cF purity of the + and $' is given greatest im- portance, and (b) various previous analyses15 in which broken SU(4) is treated in analogy with bro- ken SU(3) so as to derive mass relations and mass mixing angles and to predict specific mass values from existing data, but where # -cc+ ~(p$+ nZ) + 6hX results with E, 6 +0. We first discuss the mesons in Secs. 11 and III, and then the baryons in Sec. IV. 11. SPIN-UNITARY-SPIN SPLITTTNG OF THE MESONfi SUPERMULTIPLET OF SU(8) We will make use of the well-known fact16 that the breaking of an approximate symmetry group can be simply expressed in terms of a "chain" of successively smaller subgroups which are valid to an increasingly better approximation. The prime example is the chain SU(6) -SU(3) x SU(2)sq, where S , stands for the spin of the noncharmed

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Page 1: Spin—unitary-spin splitting of SU(8) supermultiplets

P H Y S I C A L R E V I E W D V O L U M E 1 3 , N U M B E R 7 1 A P R I L 1 9 7 6

Spin-unitary-spin splitting of SU(8) supermultiplets*

C. A. Nelson Department of Physics, State University of New York at Binghamton, Binghamton, New York 13901

(Received 8 September 1975)

The surprising narrowness of the J or ((3.1) is interpreted as indication of a_pure cF state, and hence as evidence for the SU(8)+SU(6) X SU(2)sc X U(l),c symmetry-breaking chain (S, = charmed-quark-spin generators, Y, = hypercharm generator) instead of an approximate SU(8)+SU(4) X SU(2), chain (S = quark spin generators) which would imply strong mixings. Decompositions under both chains of the s-wave qq meson states of the 64 = L+ 9 of SU(8) and of the 3 q baryon states of the three-particle symmetric 120 representation are given. The most general mass-splitting operators with breaking in the Y and Y, directions for these two multiplets are derived, which commute with the Casimir operators of the SU(6) X SU(2), X U(1), chain, which contain only one- and two-body operators, and which are invariant under rotations. Two independent mass relations follow for mesons containing charmed quarks; six, for baryons containing charmed quarks. In an appendix, for reference relative to previous SU(6)-symmetric quark-model mass analyses, the reduced numerical coefficients as determined by the meson 3 of SU(6) are listed.

I. INTRODUCTION

Our purpose i s to discuss here two topics from the standpoint of the charmed symmetric quark model:

(i) The ground state 64 = 1 + 63 (meson) and 120 - - - (baryon) representations of SU(8) together w i t h their decompositions under the subgroups SU(6) x SU(2), x U(l) ,, and SU(4) x SU(2), where S stands fir spin. The SU(2) ,c subgroup acts on the c quark's spin, and Y , i s the hypercharm operator (see Sec. 111) with eigenvalues -a, -$, -$, and $ for, respectively, the 6, X, A, and c type quarks.

(ii) The mass operator for these states in the SU(6) X SU(2),, x U(l)yc chain which i s derived by extending the one- and two-body force analysis of the SU(6)-symmetric quark model1 which previ- ously gave the successful mass formula^"^ for the baryons, e.g., the Giirsey-Radicati formula4 for the 56 of SU(6) theory2" Electromagnetic ef- fects willbe i g n ~ r e d . ~

The motivation, of course, i s the recent dis- covery7 of narrow resonances J o r $(3.1), and $'(3.7), which can be interpreted a s charmed8 quark-antiquark objects, JPC = I--, IG= Om, with N = 0 and 2 harmonic-oscillator quanta excited, respectively. The N = 2 state i s either a radial or an orbital excitation, It i s important to recog- nize that present difficulties with the charm in- terpretation (the rise of R to 5-3 i0 .6 at 7.8 GeV, the absence of narrow peaks in missing-mass plots, the absence of increased kaon to pion pro- duction ratio, etc.), principally involve phenom- ena in eZ- hadrons above the transition region a t about 3.6-4.1 GeV. Hence, these difficulties may, in fact, not exist if the transition region i s due to

excitation of f irst the charm and then the color degrees of freedom at about 3.9 GeV, which would be a natural o c c u r r e n ~ e ~ - ~ ~ in the Han-Nambu ver- sion of the three-quartet model. In this case, the details of the discussion in this article apply to the Su(3)"-color singlet states. On the other hand, this type of mass analysis remains relevant, though not in detail, even if additional heavy quarks1'-l4 a r e found to be necessary, because this analysis preserves successful ~ ~ ( 6 1 results and accepts the heavy-quark explanation, based on the Okubo- Zweig-Iizuka rule, of the narrowness of the new particles. A single charmed quark is certainly the simplest of such heavy-quark models.

Lastly, we emphasize the basic contrast between (a) the present spectra and mass analyses in which the cF purity of the + and $' i s given greatest im- portance, and (b) various previous analyses15 in which broken SU(4) i s treated in analogy with bro- ken SU(3) so as to derive mass relations and mass mixing angles and to predict specific mass values from existing data, but where # -cc+ ~ ( p $ + nZ) + 6hX results with E, 6 +0.

We first discuss the mesons in Secs. 11 and III, and then the baryons in Sec. IV.

11. SPIN-UNITARY-SPIN SPLITTTNG OF THE MESONfi SUPERMULTIPLET OF SU(8)

We will make use of the well-known fact16 that the breaking of an approximate symmetry group can be simply expressed in terms of a "chain" of successively smaller subgroups which a r e valid to an increasingly better approximation. The prime example is the chain SU(6) -SU(3) x SU(2)sq, where S, stands for the spin of the noncharmed

Page 2: Spin—unitary-spin splitting of SU(8) supermultiplets

2140 C . A . NELSON

quarks, with SU(3) - SU(2), x U(l), in SU(6) theory. Here, for instance, the hypercharge operator, which breaks the SU(3) symmetry, is conserved at the level of the smaller SU(2), x U(l), subgroup. The eigenvalues of commuting se ts of generators in the chain provide quantum numbers with which to label, in practice uniquely, the states in the irreducible representations of the initial approxi- mate symmetry group. Often two o r more chains a re relevant physically, and then superposition effects occur such that the physical resonances are eigenstates of neither chain. In the preceding example, there i s also the chain SU(6) - SU(4),,, x SU(2)SA X U(1) Y with SU(4),,, - SU(2), x SU(2)S3T,6, where S,,, stands for the spin of the 32- and 6- type quarks, whose existence i s announced by the "mixing" of the I= Y = 0 pairs of s-wave meson states, the 4 - w and 9-11!.

In the SU(8) theory there a re two analogous re- duction chains, the " s u ( ~ ) chain"

where the SU(6) subgroup is that discussed above; it acts on the 6-, X-, and h-type quarks. The other chain i s

The SU(4) subgroup here acts on the 6- , 37-, A-, and c-type quarks; it does not involve spin and i s not to be confused with the SU(4),,@ subgroup of SU(6) theory. It has the further reduction

where SU(3) i s the usual group for the 6-, 37-, and A-type quarks. This second chain we will call the "SU(4) chain."

The physical resonances, a s we noted, need not be eigenstates of either chain so we will, first, consider the meson eigenstates in each chain sep- arately. The ground state of an s-wave, Fermi quark-antiquark pair with negative parity and spin J = S = O , 1 i s the reducible 64 of SU(8) -

In the "Su(6) chain, " the direct sum

with the notation [dim SU(6), dim ~ ~ ( 2 ) , ~ ] 2 , wheren, is the totalnumber of charmed quarks plus charmed antiquarks. Mesons associated with the first two rep- resentations contain no charmed quarks and a re the familiar ones from the SU(6) theory. Continu- ing the chain, there next a r e the further reduc- tions, SU(6) - SU(3) X SU(2)sq with the notation

(dim sU(3), dim SU(2)sq)

and similarly for the other SU(6) subchain SU(6) - SU(4),,, x SU(2)sx x U(1), with the notation (dim SU(4),,,, ~ ~ ( 2 ) s ~ ) : ~

The Su(6) - SU(3) X SU(2)sa subchain yields, for the n%+O states, after recoupling the spins by E = z q + So

[6,2*];'= (3,2)1'+ (3,2)0',

with the notation (dim S U ( ~ ) , dim SU(~),*) JP, J = S for the 64 representation. The (3*, 2*)lm consists of the i z s p i n singlet F*+= (Xc)' and a doublet D*+ = (%c)+ and D*O= PC)'. The (3*, 2*)0- consists of a singlet F+= (Xc)' and a doublet D+= @c)+ and Do =(PC)'. The [6, 2*]i1 contains their antiparticles. The [I , 3 i s the JP = 1- isospin singlet @: = (Fc)', and the [I, 11; is the 0- singlet 17: = (FC)'~

On the other hand, for the "SU(4) chain" under SU(8) - SU(4) x SU(2), these SU(8) representations decompose into

with the notation {dim SU(4), dim S U ( ~ ) ~ } . Then, under SU(4) - SU(3) x U(l),c the SU(4) representa- tions decompose into

1 = I;, - 15=1:+8:+3:+3:-l, -

with the notation dim S L T ( ~ ) ~ , The corresponding wave functions can be easily written down; we on- ly note that in this chain the eigenstates a re super- positions of w, - @, - $,, and of 9, - 17, - 17,.

For several reasons, we will assume that the SU(6) X SU(2)sc x U(l),c subgroup of SU(8) and the chain associated with it a re of major importance for the breaking of SU(8) for mesons and baryons.

Page 3: Spin—unitary-spin splitting of SU(8) supermultiplets

13 - S P I N - U N I T A R Y - S P I N S P L I T T I N G O F S U ( 8 ) S U P E R M U L T I P L E T S 2141

Firs t and foremost, the striking narrowness of the Q(3.1) and the q'(3.7) suggests that they a re pure cF states due to some new dynamical invari- ance principle, for example, nc i s exactly con- served in the strong interactions responsible for the mass spectra, In particular, the J / $ will be identified with the [I, 3 x irreducible represen- tation of the SU(6) X SU(2)sc x U(l) yc subgroup of SU(8). Note that decay modes such a s Q- 5n can go, for instance, via unitarity corrections rather than from mixings of the quark content of the 11, and 11,'. This17 has been pointed out in the context of Okubo-Zweig-Iizuka-rule suppressions, e.g., 4 - KZ and K x - 377 both have connected duality diagrams so @ - 39, with a hairpin diagram, can go via unitarity correlations which a re difficult to distinguish from $ being other than an eigenstate of the SU(6) - S U ( ~ ) X , px SU(~) , , x U(1), chain. How- ever, in the case of the @ resonance, the mass spec- trum (see Appendix B) indicates that the latter $ - k); + E(@P + XZ), c +O, indeed occurs. Second, the lowest mesons can be identified in the 64 of SU(8) with the 1 and 35 of SU(6), and this 1 +% can be - - identifiedwiththe [I + 35,19 representation of the SU(6) x SU(2)sn x U(l),p subgroup of SU(8). Also, a s discussed i"n See. fi, the lowest baryons can be similarly identified in the 120 of SU(8) with the 56 of SU(6), and this 56 can bedentified with the - [56,1g irreducible representation of the SU(6) X SU(2)sc x U(l)yc subgroup of SU(8).

Thus, in our derivation of the mass-splitting operators for the s-wave meson 64 = 1 + 63 and - - - baryon 120 representations of SU(8), we will as- sume t h F t h e operator (i) commutes with the Casimir operators of the SU(6), SU(2)sc, and U(l),- subgroups of this chain, and i s invariant unde; rotations. We want the derivation to be a direct extension of SU(6) analyses in the symme- tr ic quark model used to rederive1 the Giirsey- Radicati result, used to study the first excited baryon multiplet, the (70, I-), in the notation (dim SU(6), LP), with N the number of orbital or radial quanta excited in harmonic-oscillator shells, and used3 to treat uniformly all of the bar- yon multiplets with hr=O, 1, o r 2 harmonic-oscil- lator excitation quanta. Hence, we will assume that the mass-splitting operator (ii) contains only one- and two-body operators,'* and that it (iii) transforms like a linear combination of three types of terms which transform, respectively, a s a sin- glet, a s the hypercharge operator under SU(3), and as the hypercharm operator under SU(4). For one-body operators, this transformation assump- tion is equivalent to mass splitting between the nonstrange and strange quarks, and to an indepen- dent mass splitting between the noncharmed and charmed quarks.

111. MESON MASS OPERATOR AND INDEPENDENT MASS FORMULAS

We use a formalism in t e rms of the generators of SU(8) to derive the ground state SU(6) X SU(2)sc X U(l) meson and baryon mass-splitting opera-

yo tors. The 63 generators of S U ( ~ ) , IiS, with M = 1, 2, 3 , 4 o r 6 , X , X , c f o r S U ( 4 ) a n d ~ = l , 2 o r + , + f o r SU(2),, a re constructed from Fermi creation and annihilation operators for s-wave quarks and anti- quarks in the charmed symmetric quark model in Appendix A. The Su(8) commutation relations, which can be easily computed from Eq. (Al), a re

[I%, I;; ;: ] = &ifr, I;;"' - b;LS' INS M' r" (3

The charm operator C with eigenvalue 1 (-1) for a c quark (antiquark) and 0 for 6, X, X quarks and their antiquarks is not a linear combination of generators of SU(8) so we introduce the operator

with B the baryon number operator. This relation i s the analog of Y = S+ B = SJkf + g:f which relates the the hypercharge SU(3) generator and strangeness operator for the 6, 37, and X quarks. The genera- tors of SU(6) and i ts subgroups will be denoted by script letters to distinguish them from SU(8) gen- erators. Since the c quark has B = i, S = 0, the phenomenological extension to include the c quark is ~ = g i f + g : f =B+s-+c.

In Sec. 11, we discussed the relevant reduction chains which occur in Su(8). Generatorslg for the subgroups in these chains a re tabulated in Table I.

TABLE I. Generators of SU(8 ) subgroups.

Subgroup Generators

Page 4: Spin—unitary-spin splitting of SU(8) supermultiplets

LSON 13 - The tensor operators in the mass formula will be

expressed in terms of the Casimir operators for the various subgroups. Casimir operators needed a re

and those for the several SU(2) subgroups describ- ing the spins of particular se ts of quarks. All these Casimir operators can be expressed a s bi- linear terms in the SU(8) generators of the form [x, yl,.

We can now derivez0 the mass-splitting operator for the s-wave meson 64 = 1 + 63. Our assumptions require the mass operator be a quadratic polyno- mial in the generators of SU(8), commute with

(S,), and Y;, be invariant under rota- Cis), Ctz) tions, and transform like a linear combination of three types of terms which, respectively, trans- form a s a singlet, a s Y under SU(3), and a s Y, under SU(4), For mesons, the mass operator must be invariant under charge conjugation. We group the 63 generators of SU(8) into seven types: I:', I:, I: and I:, I;:, I%, I;?, I: and 1:;. Modulo pieces to make them traceless, I:' a r e the genera- tors of SU(3); I: is the generator of U(l),,; I: and i ts adjoint, I:, a re generators of SU(4) not con- tained in the SU(3) and U(l),, subalgebras, etc.

The most general term linear in the generators and a scalar under rotations must be a linear com- bination of the generators of SU(4). Generators I$ and I: a re clearly admissible. We next consid- e r the linear combination of the remaining genera- tors O = a:Iz+ b: I:. Using the identity

and Eq. (I),

Since symmetrized expressions which have differ- ent numbers of different types of generators a re linearly independent, the conditions that this com- mutator vanish a re

a:=bg=O (vq). (6)

Thus n, and n, a re the only admissible one-body terms which satisfy the transformation require- ment and which a re invariant under charge conju- gation,

For bilinear terms in the generators of SU(8),

invariance under rotations implies that there a re two classes of terms that can be considered sep- arately: those constructed from I:, I;, I: and I:, and those constructed from I::, I::, IsS, 1:: and I::. We write the terms quadratic in the genera- tors in the form [x, Y]+ so that they are linearly independent of the terms linear in the generators. The most general term in I:', c, and I,C and i t s ad- joint i s a linear combination of the six expressions of the form [x, Y]+. Of these, only [I:, I,"], when commuted with $[I:, I:], yields combinations of the form

so these can be considered separately. The van- ishing of

implies that

asq,p= 0 ( ~ 4 , q', PI. The adjoint [I:,, $1, i s also eliminated. The same argument with I: -I: eliminates terms of the form [I:, I;]+ and i t s adjoint. Similarly, the terms [c, I;]+ and their adjoint a re excluded. Under com- mutation with

r C S Irn 2[Icr, CSl+, only

[I:, 1: I+ leads to combinations of

and i t s adjoint; so, this term can be considered separately. The vanishing of

implies that

a q q ~ = O (V 4, 4')

This then leaves a s admissible terms

[I:', $' I+, [I:, I:]+, and [If, 13, Linear combinations of these lead to

which transforms a s a singlet,

which transforms a s Y, and

which transforms a s Y , and a singlet. Here the numerical diagonal matrices y: =diag(5, 5, -%) and c$=diag(-$, -4, -+, a) have been used.

Page 5: Spin—unitary-spin splitting of SU(8) supermultiplets

13 - S P I N - U N I T A R Y - S P I N S P L I T T I N G O F S U ( 8 ) S U P E R M U L T I P L E T S 2143

Next, we discuss the second class of bilinear terms: those formed from generators transform- ing a s vectors under rotations. These terms must also commute with the Casimir operators CjE', CF)(Sc) and with Y: SO the candidates a r e the rotation-vector analogs of the surviving first- class terms, plus those constructed with (S,); and (5);. These analogs a r e

[I,', ,fi'T ,, , *, I+, [IZZ::]+, and I::],.

Since the admissible terms linear in the genera- tors a re I: and I:, the only candidates constructed from (Sc)S, and (5); a re

I(%);, (sc):l+, [(s);, @):I+ [PC);, I::]+, and

[(a;, Ig"',.

Commutation with Y t , CZ(SC), and Ci" shows that all these terms enter. The additional linear com- binations of bilinear terms, satisfying the trans- formation requirement, a r e

Finally, we collect the admissible terms and obtain the mass formula for the s-wave meson 64 = 1 + 63. The terms, together with their form - - - in terms of quantum numbers, a r e tabulated in Table 11. These give the following twelve-param- eter mass formula:

M=m,+ m,n,+ rn,C; ')+rn,2~(~+ l ) + r n , ~ ~ ) + m , [ I ( ~ + 1) - + Y ~ ] + ~ , [ ~ s ~ ( s ~ + 1) - c$'(x, 6 ) + + y 2 ]

+ m.,[2~~,,.(~,,, + 1) - 2S,(S,+ 1) - *2S,(S,+ l ) ] + men,+ m,Y,2+ m,,2Sc(Sc+ 1)+ rnl12Sq(Sq + 1). (9)

This mass formula predicts the following inde- triplets, a re equal; and from the observed mass pendent equalities: splittings of the states in the meson 36 of SU(6),

the magnitude of this mass d i f fe rence E* - D * = F - D ; (lo)

F - D = 76 MeV linear mass formula

the strange-nonstrange mass differences for the pseudoscalar and vector SU(3) triplets, and anti-

(=0.074 GeV2 quadratic mass formula). (11)

TABLE 11. Independent t e r m s contributing to mass-splitting opera tor .

F o r m in t e r m s of quantum numbers In form of genera tors

2 Linear T e r m s :

A

"c

9 Bilinear T e r m s :

~ $ 6 )

2S(S+ 1)

c( 3)

I ( I + l ) - + ~ z - + c ( ~ 3 )

s, ( S , + 1) - Sc(Sc + l ) - +s(s+ 1)

Page 6: Spin—unitary-spin splitting of SU(8) supermultiplets

2144 C . A . NELSON

The numerical reduced coefficients a s determined by the 36 a r e given in Appendix B. The quadratic formulafor a D mass of 2 GeV implies an almost degenerate F mass of 2.02 GeV. Equation (lo), a s well a s Eq. ( l l ) , is an SU(8) prediction, since the four states a re in the [6,2*]y1 multiplet of SU(6) X SU(2) ,c x U(1) ,, which involves recoupling the charmed and noncharmed quark spins.

Note that in Eq. (9) the SU(6)-breaking terms enter with the same coefficients for each SU(6) multiplet. This i s the same a s for the coefficients of the SU(3)-breaking terms in the Giirsey-Radi- cati formula and in the SU(8) baryon mass formula obtained below, Eq. (13). Here Eq. (9) mixes the 1 and 63 of SU(8) only a s a consequence of the - standard 1 and 35 mixing of ql and q, in the SU(6) theory, Relativeto the SU(6) - SU(4),,, x SU(2),, chain, the 77 - 71' mixing i s due to the Ci6' and Ci3' terms; the breaking of ideal @-w mixing i s due to the Cp ' term, Leo, it is solely responsible for @

not being a pure AX state. The choice M , = M , re- quires, in addition to c?' being absent, the ab- sence of [1(I+ 1) - $y2]. From the viewpoint of an SU(3) singlet-octet system, ~z, mixes the singlet and octet whereas [1(1+ 1) - $Y '1 only breaks the octet; however, the rn, and nz, terms also mix the states and break the octet. If systematic use of SU(6) i s used to classify the terms, as in the SU(6) irreducible tensor approach, the m,, m,, and m, terms arise16921 from SU(6) tensors 35T:;1, 35T;;i, and 35q;j. Irreducible tensor operators a re labeled m ~ $ ~ ~ ~ ~ ~ 7 d ' n ' S U ( 2 ) ~ , where n? specifies the SU(6) state of q and/or 5. ~ x p e r i e n c e ~ ? ~ with baryon levels and past confusions22 over inade- quate meson-mass operators indicate that such operators with the larger SU(6) representations should not be excluded, but should be retained a s has been done here, This means that only mixing angles can be predicted for the s-wave mesons of the 36 of SU(6); however, these predictions alone aresignificant for decay tests.

IV. MASS OPERATOR AND INDEPENDENT MASS FORMULAS FOR THE BARYON 120

SUPERMULTIPLET

For completeness, we first discuss the " s u ( ~ ) chain7' reduction of the totally symmetric three- particle representation of SU(8), the 120, in which we place the baryons. As in the symmetric quark model, to be consistent with the spin and statistics theorem, we assume that there exists an SU(3)"- color degree of freedom and that the states in the 120 a r e in the totally antisymmetric three-particle 7

representation of SU(3)"-color, the singlet. The

direct sum

120 ={20,, 4)+{20,, 2) -

in the SU(8) - SU(4) X SU(2), chain with the notation {dim SU(4), dim SU(~),), where a permutation- symmetry subscript, s = symmetric and m =mixed, suffices to distinguish the two twenty-plet Young diagrams, Under SU(4) - SU(3) x U(l)yc,

20, = 8,+ 6, + 3: + 3,,

20,=100+61+3,+ I,,

with the notation dim SU(3),,. Note that n,= C, i.e., n, has the charm eigenvalue, for states containing no antiqilarks. While the SU(3) decuplet and octet a r e obtained by this reduction, their physical re- lation a s submultiplets of the 56 of SU(6) i s not made manifest by the SU(4) reduction chain. Hence, we return to the SU(6) chain.

The relevant reductions of the 120 under the SU(8) - Su(6) x Su(2) ,, xU(1) ., c h z a r e

120= [56, I],+ [21,2],+ [6,3],+ [I , 41, - with the notation, as before, of [dim SU(6), dim SU(~),,],, and then under Su(6) - SU(3) x SU(2)Sq

56=(10,4)+ (8,2) -

6 = (3,2), -

1 = (1, 11, -.

with the notation (dim SU(6), dim SU(2),q). In or- der to obtain the physical states with n,PO, the spin of the charmed-and_ no_ncharmed quarks must be recoupled, i.e., S = S,+ S,. This yields

[21, 11, = (6, 3)$*+ (6, 3)$++ (3*, I)$+,

[I, 41, = (1, I)$+,

with the notation (dim SU(3), dim SU(~) ,~)J ' , S= J for the 120 representation. It is a straightforward e x e r c i s e 0 tabulate the wave functions for the thirteen charmed states, and from their composi- tion in terms of 6-, 37-, A-, and c-type quarks to read off their respective I, B, Y, and C quantum numbers. The (6,3) consists of an isospin singlet (AXc),, a doublet (37Xc); and (6Xc)i, and a triplet (XXc)', (XBc)',, and (66~)" . The (3*, 1) consists of a doublet (Xhc): and (@kc)*,, and a singlet (37Pc)+,. The (3,2) consists of a singlet ( k c ) + and a doublet (Xcc)* and (8cc)*+. The S and ii sub- scripts denote the permutation symmetry of the two-particle combination of 6, 37, X quarks.

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13 - S P I N - U N I T A R Y - S P I N S P L I T T I N G O F S U ( 8 ) S U P E R M U L T I P L E T S 2145

For the 120 representation of SU(8) there exists Thus, by the derivation of Sec. 111, the most gen- the i d e n t i t y era1 SU(6) X SU(2),c x U(l),c mass formula for the

120 which satisfies our conditions has eleven pa- ~ S , ( S , + ~ ) = C + + C ~ .

- (12) rameters and is

On the 56 of SU(6) only the first four terms, the Giirsey-Radicati formula, a re independent. Here, a s in SU(6) theory, for baryons the two-body dominance assumption has led to a significant simplification.

For the 56 of SU(6) the Gursey-Radicati formula yields four independent sum rules. For the thirteen ad- ditional states in the - 120 of SU(8), Eq. (13) predicts the following six new independent equalities:

= -188 MeV, (19)

where the subscripts denote (dim SU(3), ing thirteen new baryon levels, we predicted six dim SU(~),~)J ' . Equalities (14) and (15) specify new mass relations. In an appendix, fo r refer- equal spacing for both of the two SU(3) sextets, ence relative to previous SU(6) symmetric-quark- Eq. (16) specifies that this spacing i s also com- model mass analyses, we gave the reduced nu- mon, Eq. (17) specifies a common spacing for the merical coefficients a s determined by the meson two SU(3) triplets, Eq. (18) specifies the same - 3 6 of SU(6). separation between iostopic spin multiplets in the J~ = and ++ levels for the triplets a s for the sex- ACKNOWLEDGMENTS

tets, and Eq. (19) specifies a relation between the The author wishes to thank the Joint Awards splitting of the antitriplet and those of the other Counc i l /~n ive r s i t~ Awards Committee for a State charm levels and the nucleon octet. University of New York Research Foundation

Equations (14) and (15) a r e SU(3) equal-spacing Award. He also thanks SLAC for its fine hospi- statements. Both Eq. (16) and (17) are SU(8) re-

tality during the completion of this work. sults because recoupling of 8, and $ is involved, and clearly Eqs. (18) and (19)-are Su(8) results.

APPENDIX A: SECOND-QUANTIZED FORMALISM

V. SUMMARY

We again emphasize, from the point of view of future $ spectroscopy, that the mass relations de- rived in this article preserve cc purity of the J/$(3.1) resonance. We studied the charmed sym- metric quark model for mesons and baryons using approximate SU(6) x SU(2),c x U(l),, symmetry with breaking in the Y and Yc directions in order to re- solve mass degeneracies among resonances in the same submultiplets. To reduce the number of pos- sible mass formulas for baryons, we assumed that one- and two-body contributions to the mass-split- ting operator dominate. For the six meson levels containing charmed quarks, we predicted two new independent mass relations. For the correspond-

We introduce a set of Fermi creation and anni- hilation operators1 for the s-wave quarks and anti- quarks, a$:, a?, b:?, and b$i where M = l , 2 ,3 , 4 for SU(4), Y= 1,2 for SU(2), and q" = l , 2 , 3 for SU(3)" color. The spin and SU(4) indices for the antiquark operators can be grouped together since complex-conjugate representations in SU(2) a re unitarily equivalent to the original ones. The gen- erators of Su(8) constructed in terms of these a re

with

Page 8: Spin—unitary-spin splitting of SU(8) supermultiplets

NELSON 13 -

From these expressions, Yc(q) = *(-iV,+ 3N3 and F c ( a = -$(-Nt+ 3N,). These lead in the s-wave meson mass-splitting operator, for a system composed of a fixed number of quarks and anti- quarks, to a single charge-conjugation invariant term,

This i s the number operator for the total number of charmed quarks and antiquarks.

From the other linearly independent terms in the Su(6) X SU(2),c x U(l),c meson and baryon mass-splitting operators in the text, this second quantized formalism can also be used to extract specific dynamical parameters characterizing single quarks and the two-body interquark forces. Such an explicit interpretation in the three-quar- tet model of the forces responsible for the ob- served hadronic mass splittings brings these mass operators in closer contact with more ba- sic quantum-field-theory approaches to quark dy- namics, for example, gauge fields on a lattice and the bag model.

APPENDIX B: NUMERICAL COEFFICIENTS AS DETERMINED BY M E S O N S OF SU(6)

On the s-wave meson 36 multiplet of SU(6) the terms in the meson m a s ~ f o r m u l a derived in the text, Eq. (9), reduce to the first eight terms. For

each term a normalization factor 32, = (Tk,, - Thin + I)-' is introduced so a s to treat them in a comparable manner. It i s in order of the terms in Eq. (9), 1, 3, &, $, $, i , &, and i. The reduced coefficients, M i = ??zi /3l i , for the linear (quadratic) mass formula a s determined from the experimental dataz3 a re 1.000, -0.489, -1.853, 0.594, 1.800, -0.411, 0.593, and 0.530 (0.975, -0.353, -2.364, 0.697, 1.967, -0.710, -0.176, and 0.054) in units of GeV (Gev2). The last term does not contribute significantly to the quadratic mass formula. Otherwise, for a simultaneous treatment of JP= 0- and 1- states i t does not seem possible to reduce the number of terms a pvioyi, for example, by abstracting rules from the nearness of mesons to eigenstates of the SU(6) - SU(4),,, x SU(2),, chain. Note that for both the linear and quadratic formulas, the two terms with largest reduced coefficients are Ci6' and CA3' which are the operators responsible for q - q' &d 4 - w mixing of the associated eigenstates of the SU(6) - SU(4), x SU(2),;1 chain.

The ideal mixing angle is

to be compared with the empirical mixing angles 8, = 37'27' (linear), 39'59' (quadratic) and 0, = -24" (linear), -10°33' (quadratic) a s determined from sinz8, = [@ - * ( 4 ~ * - p ) ] / ( @ - w ) , etc.

*Work supported i n p a r t by t h e U. S. Energy Research and Development Administration and in par t by a State University of New Yorlc Research Foundation Award.

'0 . W. Greenberg, Univ. of Maryland Report No. 680, 1967 (unpublished).

2 0 . W. Greenberg and M. Resnikoff, Phys. Rev. 163, 1844 (1967); 172, 1850 (E) (1968); D. R. Divgi and 0 . W. Greenberg, Phys. Rev. 2, 2024 (1968); D. R. Divgi, Phys. Rev. 178, 2487 (1969).

3 ~ . H. Dalitz, Univ. of Oxford r e p o r t presented a t Triangle Meeting on Low-Energy Hadron Physics, 1973, at Smolenice, Czechoslovakia (unpublished); R. Horgan and R. H. Dalitz, Nucl. Phys. E, 135 (1973); 3, 546 (E) (1974); R. Horgan, ibid. m, 514 (1974).

4 ~ . Giirsey and L. A. Radicati, Phys. Rev. Lett . 13, 173 (1964); B. Sakita, Phys. Rev. 136, B1756 (1964).

'see, f o r example, A. Pa is , Rev. Mod. Phys. 38, 215 (1966); F. J . Dyson, Symmetry Groups in Nuclear and Particle Physics Benjamin, Reading, Mass. , 1966).

'Mass relat ions based on approximate SU(8) symmetry should be no be t te r than those based on SU(6) because of the l a r g e r m a s s splittings. With the l inear Giirsey- Radicati formula, t h e r e i s a l imit to the accuracy with which a l l the m a s s e s in the 56 can b e f i t ; i t i s to get within 11 MeV of a l l of them.

7 ~ . J. Aubert e t a1 ., Phys. Rev. Lett. 33, 1404 (1974);

J.-E. Augustin e t a1 ., M . 33, 1406 (1974); C. Bacci e t a1 ., ibid. 3, 1408 (1974); G. S. Abrams et a1 ., ibid. 33, 1453 (1974).

8 ~ - ~ . Bjorken and S. L . Glashow, Phys. Lett. 11, 255 (1964); S. L. Glashow, J. Iliopoulos, and L. Maiani, Phys. Rev. D 2, 1285 (1970).

'c. A. Nelson, State University of New York a t Bingham- ton Technical Report No. 3/24/75, 1975 (unpublished); Lett. Nuovo Cimento 14, 369 (1.975).

'OJ. C. Pati and A . Salam, Univ. of Maryland Technical Report No. 75-056, 1975 (unpublished).

' i ~ . Machacek, Y. Tomozawa, and S. K. Yun, Phys. Rev. D 12, 2925 (1975).

' 2 ~ . M , Barnett , Phys. Rev. Lett. 34, 41 (1975). ' 3 ~ . Suzuki , Berkeley Report, 197 5 (unpublished). '%IH. H a r a r i , Phys. Lett. E, 265 (1975); SLAC Report

No. SLAC-PUB-1589, 1975 (unpublished). l5Many authors have car r ied out such broken-SU (4)

analyses; s e e , fo r example, M. K. Gail lard, B. W. Lee , and J. L. Rosner, Rev. Mod. Phys. 9, 277 (1975); S. Borchardt , V. S. Mathur, and S. Okubo, Phys. Rev. Lett. 34, 38 (1975); 2, 236 (1975). F o r work on SU(8) u s i n g Z ( 4 ) x SU (2), chain s e e , and re fe rences therein, S. Iwao, Ann. Phys. (N.Y .) 2, 1 (1965) ; J . W. Moffat, Phys. Rev. 140, B1681 (1965); Phys. Rev. D 12, 288 (1975); S. Okubo, ibid. 11, 3261 (1975); S. E . E l iezer and B. R . Holstein, ibid. 11, 3344 (1975); D. B.

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13 - S P I N - U N I T A R Y - S P I N S P L I T T I N G O F S U ( 8 ) S U P E R M U L T I P L E T S 2147

Lichtenberg, Nuovo Cimento z, 563 (1975); A. W. Hendry and D. B. Lichtenberg, Phys. Rev. D 12, 2756 (1975). As s t ressed in the text, we have tr ied to follow the physics in the choice of our input assump- tions, for instance, the J ($) narrowness suggests the SU(6)x SU(2)sc x U ( ~ ) ~ ~ chain [the possibility of mixing with the SU(4) chain i s discussed in the text and i s treated, e.g., in Okubo and in Eliezer and Holstein] and previous successful SU(6) treatments of baryons sup- port dynamical one- and two-body operator dominance [in SU(8), some of the possible three-body operators have been treated by Okubo (see this reference)].

'%IM. A. B. Beg and V . Singh, Phys. Rev. Lett. s, 418 (1964); 2, 509 (1964).

"0. W. Greenberg, Univ. of Maryland Technical Report No. 75-064, 1975 (presented a t Orbis Scientiae 11, Coral Gables, Florida, 1975) (unpublished).

'*P. Federman, H. R . Rubenstein, and I. Talmi, Phys.

Lett. 22, 208 (1966). See also A. W . Hendry, Nuovo Cimento 48, 780 (1967).

" ~ o t e that for a particular spin group, say SU(2)sc, the familiar spin generators in the spherical basis a r e given by SC,= (u,)tlg<, and S $ = $ ( u 3 ) ~ 1 ~ : .

' ' ~ h i s method of derivation was f i rs t used in 0 . W. Greenberg and C. A. Nelson, Phys. Rev. 179, 1354 (1969).

"L. Licht, U. S. Naval Ordinance Laboratory Report, 1968 (unpublished).

"J. Schwinger, Phys. Rev. 9, B816 (1964); H. Harari and M. A. Rashid, Phys. Rev. 143, 1354 (1966); R. H. Dalitz, in Proceedings of an Informal Meeting on Ex- perimental Meson Spectroscopy, Philadelphia, 1968, edited by C. Baltay and A. H. Rosenfeld (Benjamin, New York, 1968), p. 497.

2 3 ~ a r t i c l e Data Group, Phys. Lett. e, 1 (1974).