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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
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
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)
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