IL NUOVO CI~IENTO

VOL. X X X V I I , N. 1

1o Maggie 1965

A Theory of Elementary Particles ('). C. LOVELkCE

Physics Department, Imperial College - London (ricevuto il 30 Ottobre 1964)

-We explore what happens if the four leptons are allowed to form a quartet, instead of being chopped up to make a triplet. If we also require S U3 invariance for the baryons, we are led very naturally *o the third-rank group B 3. The third q u a n t u m number, L = leptocharge, is nonzero only for leptons among known particles. It also cancels fractional charge for SU3 triplets. Leptons and antileptons form an 8-fold massless representation. When B 3 is broken down to S U3, only the rouen is allowed a mass. Preferred assignments for baryons, mesons and vector mesons are (21}, (27} and (21}, respectively. Yukawa coupling is then pure D, and there are no unwanted selection rules. (21) and (27} each contain an SUa octet and singlet of normal particles (L=0), together with triplets or sextuplets of ~,leptobars ,~. The latter are strongly interacting particles with the q u a n t u m numbers of a baryon (or meson) + 2 leptons. If the symmetry is broken by an L = 3 boson (M÷+), this explains the muon mass, and makes the leptobars much heavier than normal baryons, while preserving SU3. M++ would decay strongly into 2~'-. We give tables of the struetm'e of B 3 representations up to (448}, ~nd techniques which should be useful in other big groups.

Summary.

1.

-

Introduction.

T h e success of SU~ has led t o n u m e r o u s a t t e m p t s to i m p r o v e it. E s p e cially, s e v e r a l a u t h o r s h a v e t r i e d to i n c l u d e t h e l e p t o n s . U s u a l l y t h e s e schemes

(*) The research reported in this document has been sponsored in part by the Air Force Office of Scientific Research OAR through the European Office Aerospace Research U. S. Air Force. 15 - II Nuovo Cimento.

226

C. LOVELACE

have p u t the leptons into an SUa triplet (~). The existence of two neutrinos then forces ~ mixing of the space-time a n d internal symmetries. Such m i x i n g is no doubt an elegant idea in principle, but it m a y not be true. The present paper explores alternatives. I n Sect. 2 we search for a purely internal s y m m e t r y group, which will combine baryons and leptons. We conclude t h a t B~ is b y far the neatest solution. This is the seven-dimensional rotation group, which has been used in physics before ("-7), but a p p a r e n t l y not with the present assignments and consequences. Section 3 contains m a t h e m a t i c a l properties of this group and its representations. We follow D ~ L x ¢), especially in using the simple roots, ~nd covariant a~id c o n t r a v a r i a n t co-ordinates for the weights. This has considerable practical advantages. A m o n g other things, it gets rid almost entirely of the messy ~/2 and %/3 factors in the generators. We give the structure (weights, multiplicites, S U~ content) of all the representations up to {448}. These were calculated by a v e r y simple method, which we have not seen given before. I t works b y considering how the representations split up into muitiplets under various subgroups, and then constructing the bigger ones from direct products. I t could p r o b a b l y be used for a n y group, and extended to calculate the Clebseh-Gordan coefficients. We also give information on the generators, and a seven-dimensional tensor analysis for constructing the invariants. Section 4 is concerned with the physical application. The three q u a n t u m

(1) C. Iso: Nuoro Ci~etdo, 25, 456 (1962); R. GATTO: NUOVO Cimento, 27, 313 (1963); 28, 567 (1963); S. A. BLL'D~IAN:Xtwro Cime~do, 27, 751 (1963); Proc. Easter~v Them'. Phys. Co~/. (1963), p. 303; Y. NE'E)t.aN: NUOVO Cime~to, 27, 922 (1963); (~. M.~Rx: AcI~ Phys. A~stri~cc,, 17, 231 (1964); T. TOYODA: NUOVO Cime~to, 32, 1721 (1964); V. OUPTA: Phys. Ret'., 135, 783 (1964); E. M. L~p_~L~xov: Xucl. Phys., 53, 350 (1964); A. S.~LA3~ and J. C. W~aD: Electro~ag,~etic a~l(t ll*ea~: I~ter(~ctions (London preprint, 1964). (~) J. TIo~xo: Nuoro Cime~to, 6, 69, 255 (1957); A. I%Is: Jour~. M~th. Phys., 3, 1135 (1962). (3) G. FETXBER(; and F. Gt;_~sEY: Pl, ys. Rer., 114, 1153 (1959). (4) R. ]~. BEHRENI)S: _N~UOVO CimeMo, 11, 424 (1959); R. E. BEHR~NDS and A. 8IRLLX: Phys. Rer., 121, 324 (1961). (5) D. C. PEASLEE: Pl, ys. Rec., 117, 873 (1960). (6} j. ~i. SOURIAU: Com pt. t:eml., 250, 2807 (1960); 251, 1612 (1960); a. M. SovnL~u and D. KASTLEi/: Proe. Aiz" Cots/., vol. 1 (1961), p. 169. (;) G. LouPL~s, M. S~u,~v£ and J. C. TnOT~X: ComTt. Re~d., 256, 2311 (1963). (s) E. B. DYNKIN: _-Ira.. Moth. ,s'oc. Tra~sl:~tio~s, 6, I i I , 245 (1957); especially the Appendix of the latter, and introduction of the former. Dynkin's approach has previously been applied to physics by P. A. ROWLATT:London Thesis (1962) (unpublished), and A. S.~L:X~: P,'oc. I A E A Trieste Sem i~ar (1962), p. 731. The identification of the quantum numbers is discussed by M. IKEDA: Progr. Theor. Phys., 30, 915 (1963).

A TtlEORY

OF E L E M E N T A R Y

PARTICLES

227

n u m b e r s of t h e B3 group are the h y p e r c h a r g e , charge and ~,leptocharge ~. The l a t t e r is zero for all the k n o w n strongly i n t e r a c t i n g particles, - - ~ for the ~z-, We show how - - o S1 for the e-, u' and ~, a n d positive for t h e i r antiparticles. all k n o w n particles can be fitted into the group, a n d h o w this explains such a p p a r e n t l y unrelated puzzles as the lepton masses, the 39 n a t u r e of Y u k a w a coupling, a n d the absence of triplets. We also show how the s y m m e t r y m u s t be broken, reducing it to SU3. The most interesting predictions are the existence of a quite new class of strongly i n t e r a c t i n g particles, which we call leptobars, a n d of a M++ boson decaying strongly into 2~x+. Section 5 is a discussion. The theory is a d m i t t e d l y speculative, in fact it s t a r t e d life as a toy. However, it has t u r n e d out so n e a t l y (relative to its competitors), t h a t I a m now half-convinced.

2. -

S e a r c h for a g r o u p .

Leprous' interactions, unlike b a r y o n s ' , are so simple t h a t we need no group theory to tell us w h a t they are. Nevertheless, classifying leptons is worthwhile, since it tells us how to combine t h e m w i t h b a r y o n s in a unified t h e o r y of e l e m e n t a r y particles. I n this Section, we recount the a r g u m e n t s which led to the B3 scheme. T h e y are i n t e n d e d to be suggestive, r a t h e r t h a n compulsiye, a n d t h e r e are various loopholes t h r o u g h which more c o m p l i c a t e d theories could be reached. We s t a r t e d b y c o n t e m p l a t i n g the way in which weak interactions violate S U3(9). T h e y do so by m a k i n g the c o v a r i a n t , i n v a r i a n t . We w a n t e d to find a s y m m e t r y which would m a k e this true for leptons also. Therefore the leprous m u s t all belong to the same representation, a n d t h e i r observed vector currents m u s t be conserved b y the gauge. The t h e o r y will be nearer if the choice of q u a n t u m n u m b e r s which diagonalizes the leptons, diagonalizes t h e i r k n o w n currents also. Otherwise, we would expect an a r b i t r a r y m i x i n g angle, like the Cabibbo angle (% which the lepton currents do not seem to possess. Now comes the big divide. E i t h e r the two neutrinos are altogether different, or else t h e y are particles and a n t i p a r t i c l e in some sense. I n the latter case, the k n o w n leprous f o r m a triplet, and S['3 is the obvious choice for the group. We h a v e two objections to such theories. Firstly t h e y already exist (~). Secondly, the t w o - n e u t r i n o e x p e r i m e n t requires m i x i n g of the internal symm e t r y with L o r e n t z invariance, if the letpons are to f o r m a triplet. This (9) 31. GELL-3IASS and 3I. I,£~-Y: Nuor.) Cime~do, 16, 705 (1961); Z. 3h~;I, ~[. NAKA(;-A'~VA~n(l S. SAI,7.ATA:Progr. Theor. Phys., 28, 8"/0 (196:2); Proc. CERN Conj. (1962), p. 663; R. E. BzrtnZ.XDS and A. SIRLIX: Phys. Rer. Lett., 8, 221 (1962); N . CABIBBO: Phys. Rev. Lett., 10, 531 (1963); N. Br~EME, B. ttELLESE-', and M. Roos: Phys. Lett., l l , 344 (1964).

228

C. LOVELACE

sounds like a good point, b u t in practice it a p p e a r s r a t h e r forced. Also, if we e x a m i n e it closely, we find t h a t the lepton current does in fact contain a m i x i n g angle, a n d an additional s y m m e t r y - - R - i n v a r i a n c e - - i s needed to determ i n e it. I n the present work, we consider the possibility t h a t the two neutrinos are really different, a n d t h a t the ( 1 + Yb) factor is not a p r o p e r t y of the leptons themselves, b u t of t h e i r interactions, just as it is for the baryons. This m e a n s at least four leprous. 5;ew leptons would in general be easy to observe (1,), so we shall assume t h a t there are none. We m u s t therefore look for c o m p a c t semi-simple (u) Lie groups w i t h l-fold representations. The following are the only semi-simple groups with this p r o p e r t y : a) A1. This is the ordinary r o t a t i o n group, whose spin-~ r e p r e s e n t a t i o n has 4 components. We reject it because there is no internal structure. I t is clear e x p e r i m e n t a l l y t h a t the leptons are a r r a n g e d 2 × 2 , b y the charge a n d muon number.

b) A~®A~. r o t a t i o n group, served currents c o m b i n a t i o n s of expressions will the v e c t o r p a r t

(2.1)

This is the spin-½ r e p r e s e n t a t i o n of the four-dimensional a n d has been suggested b y several people (~). The six conwill be d e t e r m i n e d b y the D i r a c m a t r i c e s a~, or b y linear t h e m , analogous to the I + a n d I _ of 03. Some of these bilinear contain just two terms, a n d can therefore be identified with of the observed lepton currents

J

~y~,(1 + ys)v + ~7~(1 q- 75)v',

I ~7z(1+ 75)e + Vy,(1 + 75)/z, (lo) A scheme with six leptons, the two new ones being heavy, has been suggested by E. M. LIP)tANOV: 2urm Eksp. Teor. Fiz., 43, 893 (1962); 46, 1917 (1964). See however, V. COOK, D. K ~ r E , L. T. KERTH, P. G. J[UnPHY, W. A. WENZEL and T. F. ZIeF: Phys. t~ev., 123, 655 (1961). Three neutrinos have been suggested by K. Hill)A: ~TUOVO Cin~ento, 27, 1439 (1963). (11) For the relation between conserved vector currents and the adjoint representation of semi-simple Lie groups, see R. UTI~,_~I~x: Phys. Rev., i0i, 1597 (1956); M. E. MAYER: ~UOVO CimeMo, 11, 760 (1959); S. L. GLASHO~" and M. GELL-MANN: A~n. Phys., 15, 437 (1961); P. A. ROWLATT: London Thesis (1961) (unpublished); V. I. OGIEVETSKII and I. V. POLUBARINOV: .t~t~. Pl~ys., 25, 358 (1963): 2urm Eksp. Teor. 2¢i.z., 45, 966 (1963). (1~) L. Dn BROGLIE, D. BOH~, P. HILLIOX, F. HALBWACItS, T. TAKABAYASI and J. P. VIGInn: Proc. Aix Con]., vol. 1 (1961), p. 503; P)~ys. Bey., 129, 438, 451 (1963); F. HALBWACHS: Compt. Rend., 255, 2724 (1962); Xuoro CimeMo, 28, 695 (1963); ]). HILLION: Cahiers Phys., 16, 381 (1962); T. TAKABAYASI: NUOVO Cime~to, 27, 504 (1963); W. KROLIKOWSKI: Acta Phys. Polon., 22, Suppl., 51 (1962); G. FEIXB>;RG and F. Gi)RSEY: Phys. Rev., 128, 378 (1962); S. N~tK~&~URAand S. SATO: Progr. Tbeor. Phys., 28, 323 (1962); 29, 325 (1963).

A THEORY OF ELEMENTARY PARTICLES

229

w i t h o u t introducing a n y a r b i t r a r y m i x i n g angle. The objection to this scheme is t h a t it cannot be combined with A2(---- S U3) for the baryons, w i t h o u t adding new leptons. The b a r y o n a n d lepton groups share one q u a n t u m n u m b e r - - c h a r g e - - , so we can use the t h i r d r a n k group A, ® A~ to include t h e m both. The smallest r e p r e s e n t a t i o n of this g r o u p suitable for leptons is {6}, which is the p r o d u c t of an A2 t r i p l e t w i t h an A~ doublet. I t would therefore require two new leptons (lo), either b o t h charged or b o t h uncharged. I t has the a d v a n t a g e of needing no new baryons. However, there m u s t be three m o r e v e c t o r mesons (all u n i t a r y singlets) to complete the adjoint representation. The disconnectedness m a k e s A, ® A~ a r a t h e r u n i n t e r e s t i n g group, solving no problems (e.g., fractional charges, lepton masses), a n d m a k i n g few new predictions. c) Aa. This is the basic ~t-fold r e p r e s e n t a t i o n of SU,, analogous to the SUa triplets (~3). The 15 conserved currents are given b y the 4 × 4 traceless matrices. The diagonal ones are no use, because t h e y are uncharged, while the off-diagonal ones only contain a single t e r m each. To get the observed lepton currents, we h a v e to introduce a m i x i n g angle, a n d d e t e r m i n e it b y R-invariance. Also, three different q u a n t u m n u m b e r s for only four particles seem excessive. d) B~. This is the spin -1 r e p r e s e n t a t i o n of the 5-dimensional r o t a t i o n group. I t has b e e n suggested b y HA~ARI and b y Kt)H.NELT and UlC]3A~" (~4). The conserved currents were given b y BEHRENDS et al. (15), eq. (3.42) (they used it for baryons). We see i m m e d i a t e l y t h a t some of t h e m are sums of two terms, j u s t like {2.1). L e t us t h e n consider it further. The conserved currents m u s t belong to the 10fold adjoint r e p r e s e n t a t i o n (11). This corresponds to the a n t i s y m m e t r i c tensor in five dimensions. I t s structure, in the plane of the two q u a n t u m n u m bers, is shown in Fig. 1. The two elements in the middle will be uncharged, while those a t the cor-

Fig. 1 . - Structure of the lO-fold representation of B 2.

(13) As for leptons is considered or implicit in Y. KATAYAMA, K. •ATUMOTO, S. TANAKA and E. YA~IADA: Progr. Theor. Phys., 28, 675 (1962); Z. MARl: Progr. Teor. Phys., 31, 331, 333 (1964); K. )IATUMOTOand S. TANAKA: Progr. Theor. Phys., 31, 723 (1964); B. J. BJORKEN and S. L. GLASHOW: Phys..bert., l i , 255 (1964). (14) H. HARARI: XUOVO Cin~ento, 29, 1068 (1963); H. Ki3HNELT and P. URBAN: Proposal o] a -Yew Symm.etry Group /or the Leprous, Graz preprint (Aug. 1964). (is) R. E. BEHRENDS, J. DREITLEIN, C. FRONSDAL and B. W. LEE: Rev. Mod. Phys., 34, 1 (1962).

230

c. LOVELACE

n e r s w i l l o n l y c o n t a i n one t e r m , w h e n e x p : e s s e d i n t h e p r o d u c t {4} G {4}. W e m u s t t h e r e f o r e i d e n t i f y t h e o b s e r v e d c u r r e n t s (2.11 w i t h a p a i r of elem e n t s of {10} in t h e m i d d l e of o p p o s i t e sides. I t is j u s t a m a t t e r of n a m e s , w h i c h p a i r we t a k e . F i g u r e 2 shows t h e s t r u c t u r e of t h e 4 - f o l d r e p r e s e n t a t i o n of B2. W e m u s t so p l a c e t h e l e p t o n s in it, t h a t t h e c u r r e n t s (2.1) c o m e o u t in t h e r i g h t p l a c e in t h e 10-fold r e p r e s e n t a t i o n . T h e r e a r e t w o i u e q u i v a l e n t w a y s of d o i n g t h i s . One is o b v i o u s l y u n Fig. 2. - Assignment of lep n a t u r a l , since i t p l a c e s l e p t o n s w i t h t h e s a m e tons and antileptons to 4-fold c h a r g e in o p p o s i t e c o r n e r s of t h e s q u a r e . T h e representation of B e . o t h e r one is s h o w n in F i g . 2. W e h a v e also s h o w n t h e a s s i g n m e n t for a n t i l e p t o n s . Tile p h a s e s a r e d e t e r m i n e d so as t o g i v e t h e c o r r e c t i n v a r i a n t a n d c u r r e n t s . F i g u r e 3a s h o w s t h e D y n k i n d i a g r a m (s) of t h e L i e a l g e b r a B~. T h e r e a r e t w o s i m p l e r o o t s , one l o n g e r t h a n t h e o t h e r . T h e y c o r r e s p o n d t o v e c t o r s a l o n g t h e ho:q0==0 0 0 0 0==0 z o n t a l a x i s , a n d a l o n g t h e l e f t d i a g o n a l in a) b) c) F i g . 2. W e see t h a t o n l y t h e l a t t e r c a n be Fig. 3. Dynkin diagrams for identified with the isospin raising operator a) B 2, b) A2_,S'U a, ~) B a. 1+. T h i s m e a n s t h a t we m a k e (v', e - ) i n t o a n i s o d o u b l e t , a n d F - , v i n t o s i n g l e t s (16). O n e of t h e t w o B., q u a n t u m h u m bers will then be the charge.

(101 Besides ref. (1,10,12"141 other authors who have assigned quantum numbers to leptons are I. KAWAKAMI: Progr. Theor. Phys., 19, 459 (1958); I. S~&AV~:IIRA:Nucl. Phys., 10, 6 (1959); 11, 569 (1959); W. KROLIKOWSKI: Ntl('l. Phys., 11, 687 (1959); 17, 421 (19601; 23, 53 (1961); 33, 261 (1962); Phys. Rev., 126, 1195 (1962); Bull. Acad. Polon. Sci., 10, 595 (1962); Xuoc9 Cimet~to, 24, 52, 969 (1962); E. 3I. LIP)IANOV: 2ur~. Eksp. Teor..Fiz., 37, 10St (19591; 44, 1396 (19631; J. LUKIERSKI: B~dl. Acad Polot~. Sci., 8, 553 (1960); N. C.&BIBBO and R. (;ATTO: Phys. Rev. Lett., 5, l l 4 (1960); K. H. Tzou: Cahiers Phys., 17, 48 (1963) and eight earlier papers; S. R. DE GROOT: Phys. Lett., 2, 55 (19621; D. HORN: Phys. Lett., 2, 303 (1962); C. RYAN: Phys. Lett., 1, 135 (1962); Xucl. Phys., 36, 464 (1962); H. YAMAMOTO: Prog~'. ~l']teor. Phys., 28, 881 (19621 ; D. HORN- a n d Y . NE'EMAx:Physiea, 28, 1023(1962); S. SASAKI, H. SE(~AWA and H. YAMAMOTO: Progr. Theor. Phy.~., 30, 214 (1963); YA. P. TERLETSKII: ~url!. Eksp. Teor. Fiz., 44, 1583 (19631; P. HILLION: Act'~ Phys. Polo~., 24, 31 (19631; P. K. KABIR: XUOCO Cimento, 28, 165 (1963); R. L. IN(;RAHA.~ and M. A. MELVIN: XUOVO Cimento, 29, 1034 (19631; M. TAKETANI and T. TATI: Progr. Tbeor. Phys., 30, 134 (1963); R. E. MARSIIAK, C. RYAN, T. K. RADHA and K. RAMAN: Phys. gev. Lett., 11, 396 (19631; Xuovo Cime~tto, 32, 408 (196~t): R. V. S_~IIRXOV: Zto'th Eksp. Teor. Fiz., 46, 1637 (1964); H. KXTSUM1)VI: Progr. Theot'. Phys., 31, 642 (1964); M. A. MARKOV: Xucl. Phys., 55, 130 (1964). \Ve have (lnly listed schemes with two neutrinos.

A THEORY

OF E L E M E N T A R Y

PARTICLES

231

For baryons, symmetry under the group A2(= SU3) is now fairly well established. This suggests looking for a larger ~o'roup, which will contain both the baryon group A~ and the lepton group B2. As we have just seen, the two symmetries are not completely independent. They share one quantum n u m b e r - - c h a r g e - - a n d one simple root--isospin. In B2 this is the longer of the two roots, corresponding to the white circle in the Dynkin diagram (Fig. 3a). We therefore identify it with one of the circles in the Dynkin diagram representing A2 (Fig. 3b). The result is Fig. 3c. Consulting DYNKIN (s) shows we have arrived at the third-rank group B3. B3 requires no new leptons, since it has an 8-fold representation, consisting of two B~ quartets, which is just right for leptons and antileptons.

3. - Representations of B3.

I n this Section we give some mathematical properties of the Lie algebra Ba and i t s representations. The corresponding Lie group is O:--the sevendimensional rotation group. We follow the notation of DY~'KIN" (s). Figure 3c shows the Dynkin diagram for the algebra Ba. The three circles correspond to the three simple roots, which we denote, from left to right, by a , D, y. They are vectors in a 3-dimensional space. If we normalize 72= 1, then their scalar products are

/3

(3.1)

g,~=(St, v)=

-

-

2

7

-

-

--].

(#,

v =

~,/~

or

7)"

¢t, ~, y define an oblique co-ordinate system in this 3-dimensional space. To each element of a representation, there corresponds a point x in this space, known as its weight. The hypercharge Y, charge Q, and teptoeharge L, are defined as the covariant co-ordinates of x relative to this oblique co-ordinate system, i.e. (3.2)

x= Ya+Qg÷ Ly.

We shall refer to them collectively as the charges. An irreducible representation of B3 is uniquely defined by assigning nonnegative integers (1, m, n) to the three circles of the Dynkin diagram. The

232

c. LOVELACE

Pi

[i

H

i,

i~

~h

~

I

•

II

r

I

~ ~

I b

li

b

! -x-

I

P

[

B

[] 2

°~

v

294

330

(0, O, 4)

(2, 1, o)

(0, 1, 2)

189

(1, O, 2)

(4, O, O)

.

.

.

@

.

.

.

.

15" 6 3*

10" 8

3

6 3* 3*

1 O*

8 8

24 15" 15" 6 6 3* 3*

27 lO 10" 8 8 8 1

1

8

6*

15"

10

3

3

6*

15

15

6*

15

15"

24*

6*

3

15'

15

3

27

1

24

10

3*

3*

6" 3

g 6 /

15

15"

10

I

lO*

.

27

.

8

10

8

1 O*

8 1

10

1

8

3*

6

15"

3*

6

3*

m

1~

35

35'

35 r

1

10

14

14

8l

I4

35

35 ~

10

i:

M

5

14

10

14

35

35'

5

5

3O

35'

35

35'

5

lO

35 ~

30

55

.

35'

I

10

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.

- - I

.

35

lO

1

14

.

.

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.

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----

.

.

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.

.

.

.

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.

.

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23~

C. LOVELACE

dimension of this representation is given by (117)

2 5

3

')

t

3

The numbers l, m, n/2 are the contravariant co-ordinates x ~ = (x, ~t) of the g r e a t e s t weight of the representation (18). Its covariant c o - o r d i n a t e s - - t h e charges Y, Q, L - - a r e obtained from them by

(3.4)

( Y, Q, L) =

2

2

m

2

3

n/2

.

H e r e the m a t r i x is g,,, the inverse of (3.1). Note the n/2 (~s). Knowing the greatest weight, the other weights of the representation can be obtained from it, by successively subtracting the three simple roots, according to the rules of the layer algorithm (s). In general, the n u m b e r of weights is less t h a n the dimension of the representation, showing t h a t some of them must occur several times. We shall discuss shortly how to calculate the multiplicity of each weight. Knowing the weights and their multiplicities, we can classify the corresponding particles according to the leptocharge L. This will show how the representation splits into SU3 multiplets, invariant under the A2 subgroup. I n the left half of Tables I and II, we have shown this structure for the various representations of B.~. The rows give the B3 representations, and the columns correspond to values of the leptocharo'e. The entries in the table list the S U3 multiplets with this value of L, contained in the representation. The n o t a t i o n for the SU3multiplets is t h a t of BEIIRENDS et al. (~5). We have only shown L > 0 , since L - + - - L takes the SU3 multiplets into their conjugates. F r o m this table, we can easily reconstruct the weights and nlultiplicities of the B3 representations, using the well-known weights and multiplicities of the elements within each S U3 multiplct (~9), together with the displacement of the origin in the SU3 plane. This latter depends only on the leptoeharge, (17) This can be calculated trom Dynkin's eq. (0.139) [ref. (s), p. 355]. (t8) In general, the numbers on the Dynkin diagram are 2(x, [.t)/(~, ~), where x is the greatest weight. (tg) E. Wm_~ER: Phys. Rec., 51, 106 (1937); P. TARJANNE: ~4nm .lead. Nci. I:e~nicae, A 6, no. 105 (1962); J. P. ANTOINE: AIoL ,Not'. Nci. Bruxelles I, 77, 150 (1963).

A THEORY

OF E1.E_~IENTARY

235

PARTICLES

TABLE II. -- ,qtrueture o/ the odd represe~d~tio~,s o/ B 3. t!

(l, m, n)

Decomposition under _t~.

i Dimension

and under B2

r.~,r

1 b

(o, o, ~)

8

(1, 0, 1)

(o, 1, 1)

(0, 0, 3)

(2,0, l)

(3, 0, 1)

3

1

4 i

6•

112

112'

168'

448

3 3

8 1

3*

16 4

15 6* 3 3

8 8

6 3*

20 16 4

16

15 6* 3

10 8

20 16

20

15 6* 6* 3 3

10" 8 8

40 16 4

16 4

4

24* 15 15 6* 6 ~' 3 3

27 10* 10" 8 8 1

80 40 16 4

40 16 4

16 4

1

6 3*

3

1

1

15" 3*

6*

15" 15" 15'* 3*

24* 6*

1

10"

a n d is g i v e n b y t h e f o r m u l a e

(3.5)

~

~ r

1

~L ,

Q = Q' + ~L 3

•

H e r e (Y, Q, L) a r e t h e c h a r g e s of t h e w e i g h t , a n d ( Y ' , Q') a r e t h e h y p e r c h a r g e a n d c h a r g e i t w o u l d p o s s e s s , j u d g i n g f r o m i t s p o s i t i o n w i t h i n t h e SU3 n m l t i p l e t a l o n e , if t h e l a t t e r w e r e c e n t r e d a t t h e o r i g i n . T h e t e r m ~L 2 cancels t h e f r a c t i o n a l c h a r g e of t h e S U, t r i p l e t s . T h u s , s u p p o s e we w a n t t o k n o w t h e m u l t i p l i c i t y of t h e p o i n t w i t h (Y, Q, L ) = (2, 3, 3) i n t h e r e p r e s e n t a t i o n {294}. W e see f r o m T a b l e I t h a t t h i s r e p r e s e n t a t i o n c o n t a i n s t h r e e S U ~ m u l t i p l e t s w i t h L = 3, n a m e l y 10, 8, 1.

!

]

236

c. LOVELACE

F r o m (3.5), we m u s t look for the m u l t i p l i c i t y of the p o i n t with (X', Q ' ) = ( 1 , 1) in each of these SUn multiplets, a n d sum. This gives 1 + 1 + 0 - - 2. Similarly, b y classifying the weights according to their h y p e r c h a r g e , we can split the r e p r e s e n t a t i o n into multiplets i n v a r i a n t u n d e r the subgroup B2. These are shown in the right half of Tables I a n d I f . We h a v e only given t h e m for :Y > 0, since t h e y are identical for ± ~'. The f o r m u l a corresponding to (3.5) for the d i s p l a c e m e n t of the origin in the B.~ plane is (3.6)

Q:Q'+Y,

L:L'+TY.

Thus the B~ m u l t i p l e t s t r u c t u r e could also be used to calculate the multiplicities. Before describing how these tables were obtained, we w a n t to c o n s t r u c t a m u l t i p l i c a t i o n t a b l e for the reduction of the direct p r o d u c t of two represent a t i o n s into its irreducible components. There are a n u m b e r of m e t h o d s we can use for this (2o): a) The relation t o the seven-dimensional r o t a t i o n group. T h e even representations of T a b l e I corl:espond to irreducible tensors (integer spin) in 07. The t h i r d column of Table I shows t h e i r s y m m e t r y t y p e , r e p r e s e n t e d b y Y o u n g t a b l e a u x . I t is understood, of course, t h a t all possible c o n t r a c t e d tensors h a v e been s u b t r a c t e d off. I t is t h e n easy to see (21) w h a t t y p e s of irreducible tensor can be f o r m e d f r o m an outer product. (Note t h a t in O~ the t o t a l l y a n t i s y m m e t r i c tensors of r a n k 3 a n d 4 are equivalent.) b) There are general t h e o r e m s (s) which tell us the first t w o t e r m s in the expansion of a direct product. c) The p r o d u c t of an even r e p r e s e n t a t i o n w i t h an odd r e p r e s e n t a t i o n can only give rise to odd representations, and so on. This is because the h y p e r charge a n d leptocharge are always integers in the former, a n d half-integers in the latter. The even a n d odd r e p r e s e n t a t i o n s correspond to integer a n d half-integer spin in t h e g r o u p O 7. d) The p r o d u c t of the dimensions on the l e f t - h a n d side m u s t equal the sum of those on the r i g h t - h a n d side. e) The direct p r o d u c t of three r e p r e s e n t a t i o n s is associative. ]) The a d j o i n t r e p r e s e n t a t i o n (21}, because it corresponds to the generators of the group, m u s t ~lways occur in the direct p r o d u c t of two identical (20) A complete formula for the reduction, using characters, has been given by R. BRAUER: Co~l)t. Rend., 20¢, 1784 (1937), but it seems too complicated to be useful for third-rank groups. (21) See, for example, H. WEYL: The Theory o] Groups at~d Quantu n~ Mechanics, (New York), p. 358 et seq.

A THEORY

OF E L E M E N T A R Y

237

PARTICLES

r e p r e s e n t a t i o n s , except for the i n v a r i a n t r e p r e s e n t a t i o n {1}. (In groups like A~, (~identical )) m u s t be replaced b y (( conjugate ~)). B y these m e a n s we were able to construct a m u l t i p l i c a t i o n table, Table I I I , for the irreducible representations. TABLE I I I . Product

Decompgsition o] )roducts of irreducible representations o] B s. Decomposition

7@

7

27 + 21+

7@

8

48 +

1

8

Product

Decomposition

8~27

168'+ 48

8~35

112'+112+ 48+

8

7@ 21

105 + 35+

7

8~48

189 + 1 0 5 + 35+ 27+ 21+ 7

7Q 27

77 + 1 0 5 +

7

8 @ 77

448 + 168'

7@ 35

1 8 9 + 35+ 21

21@21

1 6 8 + 1 8 9 + 35+ 27+ 21+

7@ 48

168'+ 112+ 48+ 8

21@27

330 + 1 8 9 + 27+ 21

7~

1 8 2 + 3 3 0 + 27

21~35

378 + 1 8 9 + 1 0 5 + 35+ 21+ 7

330 + 1 6 8 + 1 8 9 + 2 7 + 2 1

27~27

182 + 3 3 0 + 1 6 8 + 27-- 21+ 1

27@35

616 + 1 8 9 + 1 0 5 + 35

35C35

294 x - 3 7 8 + 1 6 8 + 1 8 9 + 1 0 5 + 3 5 + +27+21+7+1

77

7~105 8G

7

48 +

8

8~"

8

3 5 + 21+

8~

21

112 + 48+

7~: 1

8

1

I

_Now we shall describe how Tables I a n d I I were constructed. First, the weights of the r e p r e s e n t a t i o n {7}, {8}, {21} were o b t a i n e d f r o m the layer a l g o r i t h m (8). I t t u r n e d out t h a t {7} and {8} had as m a n y weights as elements, so t h a t all their multiplicities m u s t be ]. The multiplicities in the adjoint r e p r e s e n t a t i o n {21} ara given by the relation to the group generators (see below). This shows t h a t [0, 0, 0) has m u l t i p l i c i t y 3 (the r a n k of the group), :and e v e r y other weight has multiplicity l. K n o w i n g the weights a n d multiplicities of these representations, we can infer their m u l t i p l c t s t r u c t u r e under the A~ a n d Bn subgroups. This is done, e.g., b y considering a p a r t i c u l a r layer w i t h fixed L, finding the largest A2 weight in it, s u b t r a c t i n g off the An multiplet it generates, finding the largest An weight in the remainder, a n d so on. This is not difficult for such small representations. To calculate the An a n d B2 s t r u c t u r e of the higher representations, we now use our table of direct p r o d u c t s (Table I I I ) . We select an e n t r y in which we

238

c. LOVELACE

know the nmltiplet structure of b o t h factors on the left-hand side, and all b u t one of the terms on the right-hand side. Such entries exist for all representations bigger t h a n {21}. N e x t we calculate the multiplet structure of the reducible direct p r o d u c t representation. This is easily done, using the fact t h a t L and Y are additive, and the known tables for the reduction of direct products in the subgroups A2 and Ba (~5). F o r example, {7} ® {7} can be written in terms of A~ multiplets as

(3.7)

([3*; 1] Jr [ 1 ; 0 ] .-f-[3 ; --1]) @ ([a* ;1] -k [1; 0] . [3 ; - - 1 ] ) .

Here the first n u m b e r in square brackets gives the A2 nmltiplet, and the second n u m b e r is L. We now multiply each pair of terms and add. The known multiplication table for A.,, and the a d d i t i v i t y of /5 gives, for example, [3"; 1] ~,[3"; 1] = [6*; 2] . [ 3 ;

2].

Adding all such products, gives us the multiplet structure of the reducible direct p r o d u c t representation Us). We now subtract the multiplet structures of the known terms on the right-hand side, to obtain the multiplet structure of the representation we are looking for. Thus, for example, {27} is obtained b y subtracting {21} and {1} from {7} .@ {7}. F o r the higher representations, we can simplify the calculations considerably, by doing some algebra with the multiplets b e f m e multiplying' them. F o r example, we can write

(3.s)

{112'} = {8} Q ({35} - - -t_±O r')~'~l = {8} C ([1; 3] _ [6; 1] + [6*; - - 1 J .

[1; - - 3 ] ) ,

which is much easier to evaluate. The calculations using the A2 and B2 sub.o'roups will, of course, be independent of each other. A ('heck is thus provi(~ed, since the multiplicities inferred from the A~ and B~ decompositions outzht to agree. We have vmified in every case t h a t this was so, and have also checked t h a t the wei~'ht dia~oiams were identical with those deduced by the layer altzorithm (s). I t appears to us t h a t this m e t h o d of calculating multiplicities could be e x t e n d e d to any group, and t h a t it is very much easier t h a n previous methods such as the point-set algebra of BEnREXDS et al. (,5). It could probably be refined to calculate the Clebsch-Go:dan coefficients, in terms of the known Clebsch-Gordan coefficients of some subgroup. (2~) A convenient method in practice, is to arrange the multiplication in columns according to L, like t.he multiplication of two large numbers.

A THEORY

OF ELEMENTARY

PARTICLES

239

:Next we discuss the generators of the group B3. As mentioned before, there are three s i m p l e roots a, ~, y, with scalar products (3.]). We shall denote t h e m collectively b y small Greek letters such as ~t, ,. There are altogether 18 roots in the Lie algebra B3. We shall denote roots which m a y or m a y not be simple, b y lower case bold f~ce Latin letters such as i , j . All 18 roots are linear combinations (with integer coefficients) of the three simple ones

f~

They are identical with the 18 non zero weights of the adjoint representation (21}. Thus all the roots are

(3.10) ~+2y,

a+~+2y,

a÷2~+2y,

and their negatives. Knowing the roots, we can write down the generators of the group and their c o m m u t a t i o n relations. There are three c o m m u t i n g generators H~, which are the operators whose eigenvalues are the three q u a n t u m numbers. Thus

H ! T[', Q, L) ----Tf ] Y, Q, L) ,

(3.1a)

H~I]',Q,L) = Q r I ~,(2, L~, H v l Y , Q, L~ = L I Y , Q, L ) .

These correspond to the element i0, 0, 0} of the adjoint representation, which has multiplicity 3. We stroll also find it convenient to use certain linear combinations of t h e m (3.12)

H ~' :

~ gUSH ,

where g~" is given by (3.1). The 18 remaining generators E ~ correspond to the 18 roots i, a given in (3.10). The c o m m u t a t i o n relations for these 21 generators

are

(8,15)

(3.13)

[ H , H ] = 0,

(3.14)

[ H , E ~] = a'it E ~

(3.15)

[ Ei, E-~] ---- Z a~ HI' ,

(3.16)

[ E i, E ~] : ~V~E i-i ,

ix--j,

240

c. LOVELACE

where a~ is defined b y (3.9) a n d (3.10), and H ~ b y (3.11). 2V~ vanishes unless i ~ - j is a root. The nonvanishing N " all satisfy (3.17)

(N~J) ~ = 1

(this is due to the w a y we n o r m a l i z e d the roots a n d generators) a n d also (s.,5) N i i = _ _ y j i = __ ~ - i , - j _= .h~-¢,-i, (3.18)

T h e i r phases are p a r t l y a r b i t r a r y . A consistent choice is shown in Table I V (.~3). The roots ¢t, [3, ¢ t + [~ also belong to the subgroup A2. The corresponding g e n e r a t o r s are related to those used b y BEHRENDS et al. (~5) for SU3 b y (3.19)

E ~ = ~/~E~,

E ~ = v~E,,

E ~+~ = v / ~ E ~ .

I n each r e p r e s e n t a t i o n {d}, the generators will be d × d matrices. We now ~how h o w to construct these. The H , will be diagonal matrices, d e t e r m i n e d b y t h e i r relation to the q u a n t u m numbers, (3.11). E ~ will be the adjoint of E ~. Once we k n o w the m a t r i c e s for the simple generators E , corresponding to the three simple roots, we can construct all the others f r o m the c o m m u t a t i o n relations (16). N o w for each simple root, we can define operators J ± = ~/2/(Ix, [t)E ±~ , (3.20)

Jo = H~I(Ix, bt) , w h i c h satisfy the well-known c o m m u t a t i o n relations for three-dimensions a n g u l a r m o m e n t u m operators. This follows f r o m {3.13), (3.14), (3.15). (Ix, Ix) is g i v e n b y (3.1). We use this fact to calculate their m a t r i x elements. Suppose we h a v e a weight x belonging to some representation, such t h a t x ÷ i t , ..., x -[--¥Ix a r e weights, b u t x + ( % ' + l ) I x is not, a n d such t h a t x - - I x , ..., x - - M I x are weights, but x - - ( M + I ) I x is not. I t is clear t h a t , in t e r m s of the angular m o m e n t a defined b y (3.20), x will have q u a n t u m numbers

(3.21)

j=½(M+Y),

m

~(M--X).

(23) For a third-rank group, (3.18) are not the only consistency conditions on the N ij. The easiest way of obtaining consistent phases for the structure constants, is to calculate them in a particular representation.

241

A THEORY OF ELEMENTARY PARTICLES TABLE IV. - The structure constants N~.

i

i a

~ i-+Z

fl+r

~+fl+? fl+2y a+fl+2y u+2fl+2y

1

-1 I - - - - 7 - --1

--1

--1 1

1

--1

fl+y

~+t~

1

+r

fl~-2y ~+fl

1

1

--1

I

--1

--1 +2y

1

u ÷ 2fi + 2y 1

1

1 --I

--I I

--1

1

--1

--1

1

--I

--~--,,

1

--1

--1

1

--a--fl --y

-~-2r

--1

--~--fl --27

--1

--1

--:e--2fl--2y

--1

1

--1

--1

--1

--1 1

--1

--1

1

B y t h e u s u a l a r g u m e n t s of a n g u l a r m o m e n t u m t h e o r y (.2~), t h e c o m m u t a t i o n r e l a t i o n s t h e n d e t e r m i n e t h e m a t r i x e l e m e n t s of t h e oloeratols (3.20), a p a r t from some phase factors. Since we have ~ rank-3 group, there will be no c o n t r a d i c t i o n if w e i n s i s t on t h e C o n d o n - S h o r t l e y p h a s e c o n v e n t i o n (54) f o r (~) See, for example (Princeton 1957). 16

-

A. ED~IOND8: Angular Momentum in Quantum Mechanics

I1 N u o v o Cimentoo

242

c. LOVEL~CE

each of the three simple roots. We can now use the sta~tdar4 matrix elements for angular momenta (e~), together with (3.20) and (3.21), to give us the matrix of the simple generators

{

(3.22)

E~. Ix> = ~ ( t , , p.)~V(M -t- 1)I2 Ix + ~>, E - , I*> = V ( ~ , t~)M(~ ~ ÷ 1)12 Ix - - t*>,

where 2V, M are the number of elements above and below Ix> in the submultiplet generated by E ~ . Together with the commutation relations (16), this determines all the generators, apart from some complications with multiple weights. The generators for {8} are shown in Table V. TABLE V. - Generators in the 8-/old (lepton) representation.

.E~'

= J½, 0, ½><--L 0, ½/+ ]L 0, -- ½><--1, 0 , - - ~ ] ,

E~

= 1 ½ , 1, ~ \ / '

E~

-

l

V

O, ½1 -]- 1-- ½, 0 , - - ½> < - - ½,

1,--~1

~ { 1 ½ , 1 , ~~><~,],½/+11~,o, 1~><~,' o , - ½ 1 +

+ I-- ½, o, ½><-- ½, o,-- ½1+ l-- L - - L - - ½><-- ½,- 1,---~!}, Egt-}-fl

= [½, 0 , - - ½ > < - - ½ ,

Ee+~'

- ~/~-{1½,1,½><--½,-1,-~l-

-- 1,--½[ -I1~,

1, } > { - - ½ ,

o, ½1 ,

1

-1½, ~, ~><½, o, ~ ' - I Ea+~+~ '

L o, ~ > < - ½ , -

~,

-- 1

~.},

1 V ~ {l}, 0 , - - ½> <-- ½, -- 1 , - - ~ t - - I ~ , 1, ½> <-- }, 0 , - - ~ I -}~-1½, 1,~><--½, 0,½]--i½, 0 , ½ > < - - ½ , - - 1 , - - ½ ! } ,

~> <~, o,

I-!-~,

o,

½>

~, - 1,

1,

E~,+e+=r =--1½,0, ½><--~,-- L - - } I + I}, t,~><--~, 0,--~1,

E~'+=e+=y=]½, 1, ½><--½,--1,--~1+ /½, 1,}><

½,--1,--½[.

E-i = (£i)+, H~ ca~ be written 4o~wa immediately from (3.11).

Knowing the generators, we can easily calculate the Clebsch-Gordan coefficients for the reduction of direct product representations, by separating off the representation generated from the largest weight, as described by BEItRENDS et al. (16). In fact, to do this, we need only know the three simple generators E -u. It is convenient to start by constructing the weight diagrams (s) for each of the representations contained in the direct product. This tells us

A THEORY

OF E L E M E N T A R Y

PARTICLES

243

i m m e d i a t e l y which of the E -~ we should a p p l y to a given weight to get the n e x t one. I n the case of multiple weights there will be a n a m b i g u i t y . This is resolved b y defining the degenerate weights according to the w a y t h e y separate into mlfltiplets u n d e r some subgroup. This m e a n s t h a t we define one weight to be t h a t a t t a i n e d b y a p a r t i c u l a r p a t h in the weight diagram, a n d the others to be orthogonal to it. The Clebsch-Gord~n coefficients for the reduction of {8} × {8} can be calculated in a m a t t e r of minutes. L a s t l y we w a n t to m a k e explicit the relation to the seven-dimensional r o t a t i o n group, since this is also useful in calculating Clebseh-Gordan coefficients. {7} is the v e c t o r r e p r e s e n t a t i o n of this group. We n a m e the elements J Y, Q, L> as follows V~ - [ 1 , 1, 1>,

= l o, o, ~5, (3.23)

V4 = ]0, o, 05,

E=

lo, o , - 1 5 ,

V6=]O ,

--1,--15,

G ----]--1,

--1~ --I).

(3.22) a n d (3.1) t h e n give for the simple generators

(3.24) E : V, = V: ,

~ ~; = V, ,

with all their other m a t r i x elements vanishing. To a v o i d confusion, we will use L a t i n capitals for the subscripts corresponding to this seven-dimensional space. Table I I I shows t h a t we can construct an i n v a r i a n t , q u a d r a t i c in the vector Yz. Obviously, we m u s t choose pairs for which each q u a n t u m n u m b e r adds up to zero, and t h e n p u t t h e m t o g e t h e r so as to c o m m u t e w i t h the three simple generators (3.24). The c o m m u t a t i o n relations (3.16) will t h e n ensure invariance under the full group. This gives for the i n v a r i a n t

(3.25) So we see we ~re in an oblique co-ordinate s y s t e m in the 7-dimensional space,

244

e. LOVELAC:E

in which the metric tensor is f

--1 1 --1

(3.26)

1

G~ ~ Gx~ = --1 1

--1 I t is preferable to keep to this oblique co-ordinate system, r a t h e r t h a n transform to Cartesian co-ordinates, in order to preserve the simple relation to the q u a n t u m numbers Y, Q, L, given b y (23). The representation {21} formed out of {7} ® (7}, will now be given b y (3.27)

A ~ --

1

V5

(L~ l ~ - - T~v l ~ ) ,

while {27} will be (3.28)

Tx~V

1 (u~, v~ + v~, v~,) v 5 ~/~ - T

:--

AB

u, v,] .

The identification of the various elements A ~ , T ~ with the weights ]Y, Q, L) is easily obtained f r o m (3.23). There is a one-to-one correspondence, except for the triply degenerate weights 10, 0, 0), which lie along the right-hand diagonals of the tensors. Higher-rank tensors can be constructed in a similar manner, using the metric tensor (3.26), and the totally a n t i s y m m e t r i c tensor with seven indices whose nonvanishing elements ~re ~=1 according to the sign of the p e r m u t a t i o n . The latter is invariant (2~), despite the oblique co-ordinates, because det (G~.v) = 1. Finally, we state a simple rule for decomposing 07 tensors into U3 tensors: W r i t e 1~ 2, 3 as lower indices, 7, 6, 5 as the corresponding upper indices (15), a n d omit 4. The leptocharge is the n u m b e r of upper indices minus the n u m b e r of lower ones. F o r example, {21} decomposes (3.29)

A~z~ ~ A u" + A ~'+ A u + At, + A

=

=[3;2]+[3";1]+[8+1;

0]+[3;--1]+[3";--2],

(25) See any book on general relativity, for example V. FOCK: The Theory o/ Space-

Time and Gravitation (London, 1959), p. ll6.

A TH]~OI~Y OF ELEMENTAR~r PARTICLES

245

w h e r e A ~', A a r e a n t i s y m m e t r i c in 3 d i m e n s i o n s . T h i s m a k e s i t e a s y t o c a l c u l a t e t h e i s o s c a l a r f a c t o r s r e l a t i u g t h e C t e b s c h - G o r d a n coefficients of B3 a n d S U~.

4. - P h y s i c a l

interpretation.

4"1. Lepton8. - W e see f r o m T a b l e I I t h a t t h e 8 - f o l d r e p r e s e n t a t i o n of B3 s p l i t s i n t o t w o 4 - f o l d r e p r e s e n t a t i o n s of B2, w i t h o p p o s i t e h y p e r c h a r g e . I f w e w a n t t o p r e s e r v e B 2 - s y m m e t r y for t h e l e p t o n s , a n d h a v e n o n e w ones, t h e n we m u s t p u t l e p t o n s a n d a n t i l e p t o n s i n t o {8}. H o w e v e r , p u t t i n g f e r m i o n s a n d a u t i f e r m i o n s i n t o t h e s a m e r e p r e s e n t a t i o n , r e q u i r e s s o m e c a r e . To see t h e c o r r e c t p r o c e d u r e , we w i l l c o n s i d e r first a s i m p l i f i e d m o d e l . S u p p o s e we w i s h t o m a k e e+ a n d e - i n t o a n i s o d o u b l e t . T h i s m e a n s we m u s t h a v e u 3 - p a r a m e t e r c o n t i n u o u s g r o u p of t r a n s f o r m a t i o n s c o n n e c t i n g p a r t i c l e a n d a n t i p a r t i c l e , i n a d d i t i o n t o t h e L o r e n t z g r o u p . O b v i o u s l y , in g e n e r a l , w e c a n n o t h o p e f o r t h i s . H o w e v e r , if t h e p a r t i c l e s h a v e zero mass~ s u c h a g r o u p e x i s t s - - i t is t h e P a u l i t r a n s f o r m a t i o n s (2~) w h i c h a r e i s o m o r p h i c t o 03. T h u s , we c a n o n l y p u t f e r m i o n a n d a n t i f e r m i o n i n t o a n i s o d o u b l e t if t h e y h a v e zero m a s s , a n d we m u s t m a k e t h e i d e n t i f i c a t i o n in s u c h a w a y t h a t t h e i s o s p i n g r o u p is t h e P a u l i g r o u p . W e t h e r e f o r e define t h e i s o d o u b l e t as

(4.1)

~+ ½[= ~ e°,

~ - ½/= e,

where e°=--~C~* is t h e c h a r g e - c o n j u g a t e s p i n o r (~7). I t is e a s y t o see t h a t t h e i s o s p i n r o t a t i o n s I~ a n 4 I w i l l t h e n g e n e r a t e t h e P a u l i t r a n s f o r m a t i o u s . T h e g e n e r a t o r I is (27) 1

(4.2) = : ½(e°+~,~,~e ° -

1

1

e + e ) : ~-- "½(~C-~474 Ce*

-

-

e+e): =-

= : ½{~e* - - e+e)" = - - : e+e : = - - : ~ e : , (2~) W. PAULI: .Yuovo Cimento, 6, 204 (1957); D. ]~. PURSEY: Nuovo Cimento, 6, 266 (1957); G. ~ D E R S : Nuovo Cimeuto, 7, 171 (1958}; B. TOUSCH]~](: Nuovo Cim.euto, 8, 181 (1958); F. Gi2RSEY: Nuovo Cimento, 7, 411 (1958); S. A. WOUTHUYSE.~': Proc. K. Ned. Akad. Wetensch., B 61, 54 (1958); J. GEHENIAU: Bull. Acad. Roy. Belgique C1. Sci., 44, 418 (1958); 46, 826 (1960); R. N.aTAF: Journ. Phys. Radiu.m, 20, 470 (1959); J. LUKIERSKI: Bull. Acad. Polon. Sci., 7, 577 (1959); D. LURId: Physica, 25, 1139 {1959); Z. TOKUOKA: Progr. Theor. Phys., 21, 471 (1959); B. TOUSCHEK: .Vuovo Cimento, 13, 394 (1959); E. CORINALDESI and L. A. RADICATI: NUOVO Ci~ento, 13, 667 (1959); J. M. JAuC~: Nuovo Cin~ento, 16, 1068 (1960); H. P. Di2RR: Zeits. Natur]orsch., 16a, 327 (1961); C. R. HAGEN: XUOVO Cimento, 29, 306 (1963). (~7) Our spinor notation is t h a t of P. RO)IAN: Theory o] Elementary Particles (Amsterdam, 1960).

246

C. LOVELACE

in view of the normal product. This is just the charge, which confirms our definition of isospin. Similarly, we find for the other g e n e r a t o r , I+ = :g747~e* :,

I _ - :~747~e :.

]~eplacing y4 by 7~ gives ut the three conserved currents. Now we consider the situation for the B3 group. The B~ subgroup, determined b y the roots ~ and y, can transform any one of the leptons into any other, and will thus ensure equal masses. T h e generators E ~, E -~ and H ~ will form a n additional 3-parameter group transforming v into ¢'. Plainly this can only exist for zero masses, and must be the product of the Pauli group with the transformation v-->v'. Using the B~ identifications of Fig. 2, we are therefore led to the following lepton assignment

<½,1, ~1=

r0~°,

<-~,-~,-~!=

~,,

(½, 1,

75e,',

( - ½ , - 1 , -½1=

e,

½[=

(4.3)

<½,0, ½ [ = - 7 J ' ,

<-~,

o,-½1=-¢,

<½,0,-~1=

<--½,

o,

yS,

½1=

~.

The conserved lepton currents can be calculated from the generators given in Table V, and are shown in Table VI. E ,~+r and E -~-r are the vector parts of the weak-interaction current, H~ is the electromagnetic current, and //~ is the lepton-number current. The others are unfamiliar. According to Table I I I , the product {8} ® {8) decomposes into {35} ~+{21} ~-{7} + {1}. I t turns out t h a t the vector currents transforming like {35} and {1} involve particle and antiparticle in such a way t h a t they vanish when the normal prohuct is taken. The {21} are the conserved ones, we have just given. The {7} vector currents are nonvanishiug, b u t not conserved. For example,

(4.4)

I (7} ; 0, 1, 1> = V'2[/~7/-- eYT].

There are two sorts of anomalous lepton interactions which might occur under the present theory, an4 would be compatible with experiment if they were sufficiently weaker t h a n the usual weak interactions. Firstly, t h e y could involve others of the 21 conserved currents (Table VI). Secondly, the axial lepton current might not be pure I (2t} ; 0, 1, 1> but might contain an a d m i x t u r e of [{7) ; 0, 1, 1>, just as the axial b a r y o n currents are p a r t l y D. According to (4.4), this would m a k e one or b o t h neutrinos helicity mixtures. Now we consider what happens to the leptons when the B~ s y m m e t r y is broken down to SU~. According to (~.3) and Table II, the m u o n will form a

A THEORY

247

OF E L E M E ~ ' T A R Y P A R T I C L E S

TA]~LE VI. -- The 21 cow,served lepton currents. i

E-i

E~

i

l

+ t3

- - v- ¢ 7~Tav C --vT#ysV ¢C -

- - 2~yuv'

- - 2~' yue

~),#},~e~ - ~7~7~ ~

~7,uV~v--~y~y~e

1

fi + 27 ~+~

1

- - 2#Tuv

+2~

:~ + 2fl + 2y

-

-

- - 2~7~#

fi7~75 r'c + ~' 7#751~c

eYuYs/~e + fiYu75e c

#~y~yae 4- e~Yu751~

H~

2 [fiyMt ÷ ~y~e]

H7

3fiyul, ÷ ~Tz~e + v y t ~ v - O~z,v

u n i t a r y s i n g l e t , a n d c a n t h e r e f o r e a c q u i r e a m a s s - t e r m , e, - - v ' a n d 75v ~ w i l l f o r m a u n i t a r y t r i p l e t . T h e t r a n s f o r m a t i o n l e a d i n g f r o m v' t o v w i l l t h e r e f o r e b e a P a u l i t r a n s f o r m a t i o n , a n d r e v e r s e t h e s i g n of t h e m a s s t e r m . T h u s t h e m a s s w i l l b e t h e n e g a t i v e of t h e e a n d ~' m a s s e s . W e i n t e r p r e t t h i s as m e a n i n g t h a t a l l t h r e e m a s s e s v a n i s h . So, w h e n B3 is b r o k e n d o w n t o S b ~ , o n l y t h e m u o n is a l l o w e d t o a c q u i r e a m a s s . I f we a p p l y t h e n o r m a l SU3 m a s s - s p l i t t i n g f o r m u l a ( 2 s ) t o t h e l e p t o n s , w e f i n d i t p r o d u c e s a v'-~ m a s s - s p l i t t i n g . E x p e r i m e n t a l l y t h i s m u s t b e q u i t e (~s) S. 0KUBO: Progr. Theor. Phys., 27, 949 (1962); J. GI~IBRE: Journ. Math. Phys., 4, 720 (1963); Nuovo Cim.ento, 30, 406 (1963); M. A. RxSliI~) and I. I. YA~A~'AKA: Phys. Rev., 131, 2797 (1963); H. GOLDBEm~ and Y. LEHnEn-ILAMED: Journ. Math. Phys., 4, 501 (1963); B. DIU: Nuovo Cimento, 28, 466 (1963); R. E. CUTKOSKY and ]>. TARJANNE: Phys. Rev., 132, 1354 (1963); P. HILLIOY: Compt. Rend., 257, 1225 (1963); J. J. DE SWART: Rev. Mod. Phys., 35, 916 (1963); R. H. CAPES: Phys. Rev., 134, 649 (1964); S. P. RosE~-: Jour~. Math. Phys., 5, 289 (1964); S. COL>:_~IA~', S. L. GLASHOW and D. J. KLEIT~[AN: Phys. Rer., 135, 779 (1964).

248

c. LOVELACE

small (29) if not zero. W e therefore conclude t h a t the i n t e r a c t i o n which breaks SU3 does not involve leptons. E l e c t r o m a g n e t i s m will split off e f r o m ( - - v ' , $5v~), allowing the f o r m e r to acquire a mass. The two neutrinos will still be connected b y a P a u l i t r a n s f o r m a t i o n a n d therefore massless, as long as the U-spin s y m m e t r y , g e n e r a t e d b y E ~, is not violated. This m a y v e r y well be an e x a c t s y m m e t r y for leptons. Thus Ba s y m m e t r y gives a v e r y n e a t quMitative e x p l a n a t i o n of the lepton masses.

4"2. Vector mesons. - The vector mesons should obviously be placed in the adjoint r e p r e s e n t a t i o n (21}, since this will couple t h e m to conserved currents (~). We see f r o m Table I t h a t this contains an SU3 octet and singlet with leptocharge 0, plus four triplets w i t h nonzero leptocharge. We shall call strongly i n t e r a c t i n g particles with L ~ 0 (~leptobars )~. To m a k e the assignment of the k n o w n v e c t o r mesons, we use the oblique co-ordinate s y s t e m in the seven-dimensional space, defined b y (3.23). (21~ will be a 7-dimensional a n t i s y m m e t r i c tensor, As~, given b y (3.29). (3.23) a n d (3.29) enable us to find the q u a n t u m n u m b e r s :Y, Q, L of each element A ~ s . Because of the oblique co-ordinate system, there will be a one-to-one correspondence, except for the element along the r i g h t - h a n d diagonal As. s-.~, which will all h a v e ( Y, Q, L) -----(0, 0, 0). There are only t h r e e of these, since the tensor is a n t i s y m m e t r i e . We relate t h e m to k n o w n v e c t o r mesons, b y using the way in which the l a t t e r f o r m m u l t i p l e t s u n d e r the S U3 and isospin subgroups. Thus p0 is defined to be E-~[O, 1, 0), because E -~ is the isospin lowering operator and 10, 1, 0) is p+. ~0 is defined to be t h a t p a r t of E-~[1, 0, 0) orthogonal to p0, a n d ;~ to be t h a t p a r t of E - r I 0 , 0, 1) orthogonal to b o t h po a n d % k will be the u n i t a r y singlet. The ordinary v e c t o r mesons (with L = 0) will all be in the u p p e r r i g h t (or lower left) q u a r t e r of the a n t i s y m m e t r i e tensor. This corner of the tensor is shown in Table V I I to give the identification of the k n o w n v e c t o r mesons, and t h e i r q u a n t u m n u m b e r s . A calculation shows t h a t the conserved currents to which po, ~ a n d k are

(29) W. H. BARKAS, W. BIRNBAUM and F. M. S_~11Tlt: Phys. Rev., 101, 778 (1956); L. FRIEDMAN: Phys. Rev. Lett., 1, 101 (1958); J. BAHCALL and R. B. CURTIS: _Yttovo

Citr~ento, 21, 422 (1961); A. S. GOLDHABER: Phys. Rev., 130, 760 (1963); K. N.~GY: Acta Phys. Hu~tgar., 16, 69 (1963); 17, 157, 163 {1964); M. NAKAGAWA, H. OKONO(;I, S. SAKATA and A. TOYODA: Progr. Theor. Phys., 30, 727 (1963); S. SNIEGOCKI: NUOVO Ciraento, 27, 536 (1963); P. DENNERY and A. PRIMAKOr'F: Phys. Lett., 6, 67 (1963); A. N. KAMAL: Nuovo Cimento, 33, 1108 (1964);T. Lu, K. S. YANG and L. F. Lo: Acta Phys. Sinica, 20, 19 (1964); O. A. ZAIMIDOROGA,•. ~. KULYUKIN, R. M. SULYAEV, ]. V. FALOMKIN, A. I. FILIPPOV, V. M. TSUPO-SITNIKOV and Yu. A. SCIIERBAKOV: 2ur~. E,ksp. Teor. Fiz., 46, 1240 (1964).

A

THEORY

ELEMENTARY

OF

c o u p l e d , ~re r e l a t e d t o t h e c u r r e n t s

PARTICLES

249

of t h e t h r e e c h a r g e s b y

O o ~ - - l J, + JQ--½J~ , 1

(4.5)

[3J -

1 2 ,~V,-~JL • TABLE V I I . - The <~normal ~>( L = 0 ) p a r t

o] the antisymmetric te~tsor {21}, with (Y, Q, L) wlues, and assignments oj vector mesons, bargo~ts a~td antibargo~s. X

3I 6

(1, o, o)

(1, 1,0)

(0, O, O)

K*O

K*+

V3

~g Zo

(o, l, o)

5/ x + ~

(--], o, o)

(o, o, o) ;,

00

q+

~3

m

K*O

X'o

X~

y

A

M

X-

(o, ~, o)

(0, O, O) ~o°

~.

9~ --9-

2: °

y

- ~/-~ + ~/~

A

~-~

--X-

__

~'+

(-- t , - - 1 , o) --- ~

*+

--.5-

250

C. LOVELACE

T h u s ~ is n o t c o u p l e d t o t w o pions (since t h e y h a v e Y = L ~ - 0 ) , while ~ is o n l y c o u p l e d to l e p t o b a r s . 5Tow w h a t a b o u t t h e ¢o? This c a n n o t be d i r e c t l y identified w i t h ~, since t h e l a t t e r is n o t s t r o n g l y c o u p l e d to a n y k n o w n particle. T h e r e are t w o possibilities: e i t h e r t h e co is a B3 singlet c a r r y i n g b a r y o n c o n s e r v a t i o n , or else t h e ~ a n d k are s t r o n g l y m i x e d b y t h e s y m m e t r y - b r e a k i n g force, t o give t h e o b s e r v e d ~ a n d co. I n t h e l a t t e r case, t h e ~ a n d (o should be coupled to i d e n t i c a l c u r r e n t s , as far as k n o w n particles are concerned. W e t h i n k this s e c o n d case less likely~ since it conflicts w i t h t h e s y m m e t r y - b r e a k i n g m e c h a n i s m discussed below. 4"3. Baryons. - F o r t h e b a r y o n s , we n e e d a r e p r e s e n t a t i o n c o n t a i n i n g a n L = 0 octet, a n d as little else as possible. T a b l e I shows t h a t {21}, {27}, {35} a n d {77} are the possibilities. E a c h c o n t a i n s a u n i t a r y singlet, besides t h e octet, b u t no o t h e r (~o r d i n a r y ~) particles w i t h L = 0. T h e y all c o n t a i n v a r y i n g n u m b e r s of l e p t o b a r s , w i t h L # 0. W e shall discuss these later. T h e u n i t a r y singlet m i g h t be identified w i t h t h e Y*(1405), if this t u r n s out t o h a v e the r i g h t p a r i t y (3o).

(30) As in Schwinger's theory [ref. (54)], this would require ~+ for the Y* (1405). The ½- assignment is much preferred by theorists since it would give a simpler model of low energy I~.~" scattering. See, R. H. DALITZ and S. F. TuA~-: Phys. Rev. l~ett., 2, 425 {1959); Ann. Phys., 8, 100 (1959); 10, 307 (1960); R. H. DALITZ: Proc. Ai): Co~/., vol. 2 (1961), p. 151; Proc. 1962 High-Et~ergy Phys. Con/. (Geneva, 1963), p. 391; Ann. Rev..Yucl. Sci., 13, 339 (1963); S. F. TUAN: Phys. Lett., 2, 62 (1962); S. MINAMI: NHovo Cim.ento, 14, 767 (1959); R. ]~. SCHULT and R. H. CAPPS: Phys. Rer., 122, 1659 (1961); Nuovo Cimeato, 23, 416 (1962); M. M. ISLAM: NUC~. Phys., 29, 635 (1962); R. C. ARNOLD and J. J. SAKURAI: Phys. Rev., 128, 2808 (1962); T. AKIBA and R. H. CAPES: Phys. Rev. Lett., 8, 457 (1962); T. AKIBA: Progr. Theor. Phys., 29, 439 (1963); G. L. SHAW and M. H. Ross: Phys. t~ev., 126, 814 (1962); E. PA(~IOLA: XUOVO Cimento, 25, 169 (1962); Y. F u J I I : P]iys. Rev., 131, 2681 (1963); Progr. Theor. Phys., 29, 574 (1963); R. CilhNI): XHOVO Cim,e~to, 29, 967 (1963); 31, 827 (1964); 32, 1348 (1964); 33, 148 (1964); S. X. BISWAS and S. K. Bos]~: Phy S. flev., 135, 1045 (1964). However, it does not seem possible to exclude ½+at present, because, except for MINA~I, nobody has taken this possibility seriously enough to calculate its consequences for K interactions. The Y~ experiments do not at present give any evidence on the parity, except perhaps for the sharp upper slope. See, ~f. H. ALSTON, L. W. ALVAREZ, P. EBERHARD, ~I. L. GOOD, U . GRAZIANO, n . K. TICHO and S. G. WOJClCKI: Phys. Bec. Lett., 6, 698 (1961); P. BAST~]~N, 5I. FERRO-]mZZI and A. H. ROSENFELD: Phys. Rev. Lett., 6, 702 (196l); A. FRISK and A. G. E}~sPo~-o: Phys. Lett., 3, 27 (1962); A. FRISK: Ark. Fys., 24, 221 (1963); R. H. MARCH, A. R. ERWlN and W. D. WALKER: Phys. Lett., 3, 99 (1962); G. ALEXANDER, G. R. KALBFLEISCII, D. H. MILLER and G. A. S~tlTH: Phys. Rev. Lett., 8, 447 (1962); D. COLLEY, ~N. GEL'FAND, U. NAUENB]ERI;,0. STEINBERG~R, S. WOLF, H. R. BRUGGER,P. R. KRAM]~R and R. J. PLANO: Phys. Rev., 128, 1930 (1962); A. BARBARO-GALTIERI, F. M. SMITH

and J. W. PATRICK: Phys. Lett., 5, 63 (1963); R. K. ADAIR: Phys. Left., 6, 86 (1963);

A THEORY

OF E L E M E N T A R Y

PARTICLES

251

{21} is the same a n t i s y m m e t r i c tensor representation t h a t we used for the v e c t o r mesons. The identification of the ordinary b a r y o n s in it is shown in Table V I I . We have called the u n i t a r y singlet y. The phases have been chosen to agree with the SU~ ones used by BEItlgENDS et al. (~). GELL-~[ANN (at) defines the E - with a minus sign. The a n t i b a r y o n s will belong to another {21}. Their assignments are shown in Table V I I , and can be deduced from the invariant

(~.6)

~_, ~ G ~

~A~

,

LEM~I

where G s~ is given in (3.26). {27} is the s y m m e t r i c traceless second-rank tensor, which can be cons t r u c t e d from 2 vectors b y (3.28). I t has more leptobars t h a n {21}. The assignments for the ordinary b a r y o n s will differ from {21} only for the u n i t a r y singlet y. This occurs in the m a n n e r shown for the meson singlet h in Table V I I I . {35} is the totally a n t i s y m m e t r i c third rank tensor

(4.7)

B~

1 =-~_ [ I~A~-V3

~ A~-

~A~].

The assignments for the k n o w n baryons in this representation can be obtained f r o m Table V I I , b y replacing A ~ b y B , ~ r . {77} is the totally s y m m e t r i c t h i r d - r a n k tensor Sz~.v. The assignments for the k n o w n baryons in it are obtained from those in {27}, by replacing T~.v b y S~.~. E a c h of those representations can be subjected to the t r a n s f o r m a t i o n

(4.8)

X~vR.." --> X ~ ' ' "

~

~ G~aG'~BG R~ .., XaBc... , ABC...

N. KWAK:and J. ScgxEPs: Xuovo Cimento, 28, 1200 (1963); B. BHow_~tI~:, 1). C. J.~ix, P. C. ~i.~THUR and N. T. V. L~KSH:~[I:X~oco Cime~do, 31, 465 (1964); A. Z. M. I S ~ L , S. LOKAXATHAN~nd Y. PRAK,~SH: Nuoro Cimetdo, 33, 25 (1964). I am grateful to Prof. R. H. CAPPS for a discussion on this point. (3~) G. FEINBER(~ and R. E. BEHRESDS: Phys. Rev., 115, 745 (1959); S. COL~_~I.XN and S. L. GLASHOW:Phys. Rev. Lett., 6, 423 (1961); J. J. S.~KURXI: Phys. Rev. Lett 7, 426 (1961); M. GELL-MA:;.X:CM. Tech. Report, CTSL-20 (1961), Appendix (unpublished); P. BREITENLOHNEI¢:XUOVO Cime~to, 26, 231 (1962); H. GOLDBERG and Y. Nn'E)~N: Nuovo Cimento, 27, 1 (1963); S. OKUBO and R. E. MARSHAK: NUOVO Cim.ettto, 28, 56 (1963); J. J. DE SWART:Rev. Mod. Phys., 35, 916 (1963); A. W. MAI~TIN and K. C. WALI: Phys. Rev., 130, 2455 (1963); Y. HA~A and Y. MIY.~MOTO: Progr. Thegr. Phys., 29, 466 (1963);H. SUG.XWARA: Progr. Theor. Phys., 31, 213 (1964); W. WAD_t: Phys. Rev., 133, 439 (1964); K. KhW.~RABAYASHI:Phys. l~ev., 134, 877 (1964); K. KIKKAWA: Progr. Theor. Phys., 31, 328 (1964).

252

C. LOV~LACE

where G ~ is the m e t r i c tensor (3.26). F o r {27} a n d (77} this is the well-known R - t r a n s f o r m a t i o n of SUs(3~) while for {21} a n d {35} it is the R - t r a n s f o r m a t i o n w i t h a - - 1 change of phase. T h o u g h there are four possible a s s i g n m e n t s for b a r y o n s , {21} is to be preferred, not only for economy, b u t also in connection w i t h the s y m m e t r y b r e a k i n g (see below). 4"4. Mesons and Y u k a w a coupling. - The possible assignments for mesons are m o r e r e s t r i c t e d t h a n those for baryons. I n S U3 the p r o d u c t [ 8 ] ® [ 8 ] contains [8] twice. This degeneracy results in the existence of two sorts of Y u k a w a coupling: D which is even under t h e R - t r a n s f o r m a t i o n , a n d F which is odd (31). E x p e r i m e n t a l l y , D coupling m u s t be m u c h stronger (33). H o w e v e r , f r o m Table I I I we see t h a t in B3 the p r o d u c t s {21} ® {21}, {27} ® {27}, etc. involve each r e p r e s e n t a t i o n only once. Therefore, the D - F degeneracy is r e m o v e d b y the a d d i t i o n a l B3 t r a n s f o r m a t i o n s , a n d Y u k a w a coupling will either be p u r e D or p u r e F , depending on which repr e s e n t a t i o n we use for the mesons. Obviously, we should select our m e s o n a s s i g n m e n t so as to m a k e it p u r e D. Using the metric tensor (3.26) and the t o t a l l y a n t i s y m m e t r i c tensor w i t h 7 indices, we can easily construct the i n v a r i a n t Y u k a w a couplings for the various b a r y o n a n m e s o n assignments. I f the mesons were (77}, we could not h a v e a n y Y u k a w a coupling ut all, so this is excluded. Also, a ¥ u k a w a coupling to {35} mesons is only possible if the b a r y o n s are either {21} or {35} (see T~ble I I I ) . The beh~viour of each coupling u n d e r the R - t r a n s f o r m a t i o n (4.8) can be investigated. W e find t h a t , if the mesons are {27} or {35}, the Y u k a w a coupling will be R-even a n d therefore pure D, while if t h e y are (21}, the coupling will be R - o d d a n d therefore pure 2~. This can also be checked b y direct calculation, using the assignments of Tables V I I a n d V I I L Therefore, the observed d o m i n a n c e of D coupling shows t h a t the mesons m u s t either be {27} or {35}. T h e y can only be {35} if the b a r y o n s are either {21} or {35}, so there are six possible choices in all. The {27} a s s i g n m e n t is to be preferred, b o t h for economy, a n d in connection w i t h the s y m m e t r y - b r e a k i n g . This is the s y m m e t r i c traceless tensor (3.28). The a r r a n g e m e n t of the k n o w n mesons in it, is shown in T a b l e V I I I . h is the u n i t a r y singlet, which m i g h t be identified w i t h the recently discovered "rd~r: res-

(as) R. H. DALITZ: Proc. CERN Con]. (1962), p. 391; Phys. Lett., 5, 53 (1963); J. J. DE SWART and C. K. IDDI~C,S: Phys. Rev., 130, 319 (1963); J. J. DE SWART: Phys. Lett., 5, 58 (1963); S. IJ. GLASHOW and A. H. ]~OSENFELD: Phys. Rev. Lett., 10, 192 (1963); Y. HARA: Phys. Rev., 133, 1565 (1964); M. HOSODA: Progr. Theor. Phys., 31, 126 (1964); V. SINGH: Nuovo Cimento, 33, 763 (1964); R. F. DAS~EN': Phys. Lett., l i , 89 (1964). See also ref. (al).

A THEORY OF ]~LE.~IENTARY PARTICLES

253

TABLE VIII. - The (~normal ,) ( L = 0 ) part o] the symmetric tensor {27}, with (Y, Q, L)

values, and assignments o] mesons. N M 6

(i, o, o)

(1, 1, o)

:~o

zo

2h

~'~

(o, o, o) r/+~V ~

(-- 1, o, o)

(o, o, o)

(o, 1, o)

(o, o, o)

I

K°

K+

yl +

/

7~/~+~:~

/7o

( - - 1 , - - 1,0)

( o , - - 1 , o)

2h

(o, o, o) 4 ~/3h 7

o n u n c e (33) if t h e q u u n t u m n u m b e r s t u r n o u t r i g h t .

T h e a s s i g n m e n t of t h e

n o r m a l m e s o n s i n (35) is o b t a i n e d b y c h a n g i n g T ~ v to B 4 ~ ,

and replacing

t h e coefficients of h b y t h o s e of y i n T ~ b l e V I I . So far as n o r m a l p a r t i c l e s (L = 0) are c o n c e r n e d , these six p o s s i b l e ussignm e n t s for b a r y o n s a n d m e s o n s differ o n l y i n t h e c o u p l i n g c o n s t a n t s of t h e

(s3) G. R. KALBFLEISCH, ~L. W. ALVAREZ, A. BARBARO-GALTIERI, O. I. DAHL, P. EBERHARD, W. E. HUMPHREY, J. S. LINDSEY, D. W. MERRILL, J. J. ~IURRAY, A. RITTENBERG, R. R. Ross, J. B. SHAFER, F. T. SHIVELY,D. M. SIEGEL, G. A. S_MITK .and R. D. TRIPr: Phys. Rev. Lett., 12, 527 (1964); M. GOLDBERG, M. GUNDZIK, S. LICIITMAN, J. ]~EITNER, ~'~[. PRIMER, P. L. CONNOLLY, E. L. HART, K. W. LAX, G. LONDON-, N. P. SAMIOS and S. S. YAMA.~IOTO: Phys. Rev. Lett., 12, 546 (1964); 13, 249 (1964); G. R. KALBFLEISCft, 0. I. DAHL and A. RITTENBERG: Phys. Rev. Lett., 13, 349 (1964); P. M. DAUBER, W. E. SLATER,L. T. SMITH, D. H. STORK and H. K. TICHO: Phys. Rev. Lett., 13, 449 (1964). Theoretical papers on this resonance are S. R. CIIOUDHURY and ],. K. PANDE: NUOVO Cimento, 33, 1736 (1964); L. M. BRowx and H. FAIER: Phys. Rev. Lett., 13, 73 (1964); G. BECCHI and G. MORPURGO: Phys. Bee. Lett., 13, 110 (1964); H. BACRY, J. NUYTS and L. VAN HOVE: CERN preprint (Sept. 1964).

25~:

e. LOVELACE

u n i t a r y singlets. These will, of course, be reflected in their e x p e r i m e n t a l widths. T a b l e I X shows the y a n d h coupling constants in the various assigmnents. We h a v e expressed t h e m in t e r m s of the r a t i o to the A a n d 7:° couplings. I n each t e r m of the D Y u k a w a coupling (a~) i n v o l v i n g a pion~ A m u s t be replaced b y the c o m b i n a t i o n of A and y given in the t h i r d column of Table I X .

'FABLE

IX.

-

I"ukawa coupli~gs o] the ~tnitary sin.qlets, ]or various baryon and meso~t assignments.

!Baryons Mesons! Coupled to =

Coupled to K

Coupled to ".q

4\

{21}

{27}

A- v~y

{27}

{27}

A----y

A+

{27}

A -- X+2y

A + 2 ~."2y

AA -- ~ '2 (Ay + yA)

{77}

{eT}

2 V'2 A--~--y

4V2 ,J+ ~- y

AA+--~

{21}

{35}

{35}

~ph/pp= °

A 4 - 2 \ 2y

2Vff 7

4xff , y ~

f l A - - ~ 2+ly ÷ ~/2~A

~A-

1

2%{ (~y+~A) 7

(Ay+~A)

1

\2 1

A + -V- g y

A--%/2y

.IA

,~:;?~ 1 (3y + 9A)

--

~

"21

4\g '21

....

8\79 _

8 ~.'iJ

7 '2

~/6 "21

2 x6

-2

I n each t e r m involving a K-meson, A is to be replaced b y the c o m b i n a t i o n shown in the f o u r t h column, while in the t e r m s involving ~, AA becomes the expression in the fifth column. Since h is a u n i t a r y singlet, it m u s t be coupled to the SUa i n v a r i a n t . Thus we need only give its coupling to one b a r y o n t o determine t h e m all. Column six of Table I X shows the r a t i o of the eouplino/s (~ph)/(Pp~°). There are two inequivalent forms for the meson-meson coupling, assuminff {27} mesons. T h e y are

(4.9)

(Z

G~LGJIXT~ , T . . ~2

KLMN

and

(4.10)

~ KZM2IPQRS

[~¢KL#'~MN[~CPQI'CRSITI "717 [17 ~T ~ ~ ~r ~LM.LNp~I_QR

rJFSK *

A T H E O R Y OF E L E M E N T A R Y P A R T I C L E S

255

As far as the pion-pion interaction is concerned, they are equiwtlent, t a k i n g crossing s y m m e t r y into account. The main difference between t h e m is t h a t (4.10) allows a direct h~r:r: coupling, wh~re~s (L9) does not. I n view of the large experimental width of the ~7:rc resonance (33), it therefore looks as if (4.10) is dominant. F r o m the coupling constants of Table I X , we can infer the widths of the Y* and h for various assignments, assuming t h a t these arc the u n i t a r y singlets. Thus, for example, {27} baryons would give a narrower Y* t h a n would [21}. However, the coupling constants are all of the same order of magnitude, so t h a t the calculation of the widths must be done carefully, t a k i n g finalstate interactions into account. We therefore postpone it to a subsequent paper. 4"5. Resonances. I n SU~, the ~ b a r y o n - m e s o n resonances are assigned to a 10-fold representation. Looking at Table I, we see t h a t the smallest B3 representation containing an SU~ deeuplet is (189}. This is not quite as b a d us it sounds, since m o s t of these 189 particles would have L ¢ 0 a n d therefore have u n o b s e r v a b l y large masses, as we shall argue later. ~Nevertheless there remains, with L = 0, a 10", a 1 and three 8's, besides the 10 we want. We m i g h t hope t h a t vanishing Clebsch-Gordan coefficients would stop some of t h e m decaying into n o r m a l particles. However, the rule at the end of Sect. 3 can be used to show this is not so. E x p e r i m e n t a l l y , there is no evidence for ~3+ particles a p a r t from the usual decuplet, now t h a t the Y~* (1660) p a r i t y has been reversed (ad). F o u r of these particles would be ~:p P~s resonances. Experimentally, it is very unlikely t h a t such particles exist with mass < 1700 MeV (35). W i t h higher mass, t h e y would be difficult to see in ,-:p, because of the low spin, b u t should show up in the ~ p - > Y K channels, according to the Clebsch-Gordan coefficients (ae). F o u r such undiscovered particles seems too much. This is the only bad feature of the B3 scheme that, we have y e t found. The answer probably lies in the fact t h a t the B3 mass splitting must be m u c h greater t h a n the SU3. I t is known from dynamical calculations (37), _

3 +

(34) D. BERLEY et al.: Prec. Dubna Cott]. (1964), to be published. (3s) p. AUVIL and C. LOVELACE: NU,OVO Cime~do, 33, 473 (1964). (36) A. 1~. EDMONDS: Proc. Roy. Soc., A268, 567 (1962); 5[. A. RASHID: Nttova Cime~to, 26, 118 (1962); Y. DOTHAN ~nd H. ttAnAaI: Israel AEC report IA-777 (1962, unpublished); P. TARJAN'NE: Anl~. Acid. Sci. Fel~icl~e, A 6, no. 105 (1962); J. J. DE SWART: _~eY. Mod. Phys., 35, 916 (1963); V. V. VANA¢~ASand A. P. YUTSIS: Litov. l¢iz. Sbornik, 2, 199 (1962). (a:) The best discussion of the ~'* mechanism is by A. DONNACHIEand J. HAMILTON: Phys. Rev., 133, 1053 (1964). Most of the cut-off required by other authors is to compensate for their neglect of low-energy ~n S-wave exchange. It has been shown in numerous papers by the Hamilton and Shirkov groups that the =A" partial-wave dispersion relations are incompatile with experiment unless such a 7:= interaction is introduced.

256

c. LOVELACE

t h a t the ~3 + resonances are produced b y long-range forces, n a m e l y the exchange of single b a r y o n s a n d of low-energy meson pairs. Such forces will obviously be far more sensitive to mass differences t h a n the short-range forces would be. l~ecent calculations (38) h a v e indicated t h a t the m a s s - f o r m u l a for the decuplet can be derived on purely d y n a m i c a l grounds, a s s u m i n g the masses of the b a r y o n s a n d mesons. I t s success m u s t therefore be p a r t l y a c c i d e n t a l - - t h e mass-splitting h a p p e n s to leave a b a r y o n - e x c h a n g e pole sufficiently close to each ~3 + resonance. I n general, we h a v e no r i g h t to expect weakly b o u n d multiplets to be p r e s e r v e d b y b a d l y b r o k e n s y m m e t r i e s , especially when, as in B3, the m a s s - s p l i t t i n g m u s t be m a n y times larger t h a n either the binding energy or the range of the force producing it. 4"6. Leptobars. - As we h a v e noticed, the theory predicts a considerable n u m b e r of strongly i n t e r a c t i n g particles w i t h Z ¢ 0, which we h a v e called ((leptobars ~), because t h e y h a v e the q u a n t u m n u m b e r s of an ordinary h a d r o n + 2 leptons. T h e r e %ill be b a r y o n , m e s o n a n d v e c t o r - m e s o n leptobars. T h e y are the missing SU3 triplets, for which so m a n y tears h a v e recently b e e n shed (39.4o). T h e y will h a v e integral charge, however, because of our e x t r a q u a n t u m n u m b e r leptocharge. This diplaces the S U3 multiplets b y an a m o u n t depending on the triality, eq. (3.5), as several authors h a v e suggested m i g h t h a p p e n (4o). Since leptobars h a v e not y e t been observed, p r e s u m a b l y t h e y m u s t h a v e masses large enough to forbid pair p r o d u c t i o n a t accelerator energies, i.e. ~ 3 GeV. We shall discuss in the n e x t Subsection how the s y m m e t r y b r e a k i n g i n t e r a c t i o n can be a r r a n g e d to give this. ~Now some r e m a r k s on l e p t o b a r decays. We apologize t h a t t h e y are not more q u a n t i t a t i v e , b u t the t h e o r y of conventional nonleptonic decays is not y e t sufficiently developed to predict w i t h a n y c e r t a i n t y w h a t would h a p p e n to a h y p o t h e t i c a l h e a v y particle. The S U3-breaking i n t e r a c t i o n will induce a

(3s) R. E. CUTKOSKY: A~t~l. Phys., 23, 415 (1963); M. HOSODA: Progr, Yheor. Phys., 30, 400 (1963); 31, 126 (1964); A. W. MARTIN and K. C. WALI: ]Phys. Rev., 130, 2455 (1963); Nuovo Cimento, 31, 1324 (1964); E. GOTSMAN: NUOVO Cimvnto. 32, 628 (1964); J. C. P.~Tz: Pl~ys. Rev., 134, 387 (1964); G. L. KANE: Phys. Rev., 135, 843 (1964); S. R. CHOUDHURY and L. K. PANDE: Phys. Rev., 135, 1027 (1964); K. C. WALI and R. L. WAR.~OCK: Phys. Rev. (to be published). (ag) M. GELL-MANN: Phys. Lett., 8, 214 (1964); G. ZWEIG: CERN preprint (1964); see also ref. (sa). (4o) L. C. BIEDENHARN and E. C. FOWLER: XUOVO Cimento, 33, 1329 (1964); C. R. HAGEN and A. J. MACFARLANE: Phys. Rev., 135, 342 (1964); A. SIMOn1 and B. VITALE: ~'UOVO Cimento, 33, 1199 (1964); F. ENGLERT and R. BROUT: Phys. Rev. Lett., 12, 682 (1964); C. NOACK: Zeits. Phys., 179, 1 (t964}; S. OKUBO, C. RYAN ~nd R. E. MARSHAK: Rochester preprint UR-875-37 (1964).

A THEORY OF :EL~EMENTARY PARTICLES

257

mass splitting among the leptobars, and the ordinary weak b a r y o n currents will t h e n make the heavier leptobars decay into the lighter ones. The latter will be stable, unless a AL ~ 0 b a r y o n current also exists. There seems to be no reason why it should not, especially since the weak lepton current is ] A L I = 1. I f it does, leptobars will decay into ordinary particles. Their high mass will make much more phase-space available. We can therefore expect f r e q u e n t nonleptonic decays into ~ large n u m b e r of pious. I f these occurred in cosmic rays, t h e y would not easily be distinguishable from strong interactions in the emulsion. F u r t h e r m o r e , we would expect the large phase-space to shorten the lifetime. I f the leptobars were sufficiently h e a v y and the [AL [= 1 coupling were sufficiently strong, the track would not be observable at all. The p r o d u c t i o n and nonleptonic decay of a leptobar pair, would t h e n be indistinguishable from a strong-interaction two-fireball event, except b y the sharpness of the fireball mass. One other point seems worth mentioning. I f leptobars exist and decay t h r o u g h a [AL [---- 1 current, then t h e y should be produced in stars b y neutrino interactions. If the star were sufficiently cold and dense, its core could consist mostly of leptobars, made stable by the Pauli principle, just like the h y p e r o n stars of AMBARTSUMYAN and SAAKYA>" (~). Since leptobars must be m u c h heavier t h a n normal baryons, such stars could easily achieve very highdensities. This m a y be of interest in connection with quasi-stellar objects.

4"7. Symmetry-breaking. - A v e r y a t t r a c t i v e hypothesis is t h a t Ba is broken b y the existence of an L = 3 singlet boson, which we call ~++ (it will be doubly charged b y (3.5), the antiparticle 3[-- will have L------3). This has the following pleasant consequences: i) The only lepton with a large mass will be the rouen. This is because #+#+ is the only L = 3 bilinear q u a n t i t y which we can form from the leptons. The muon should therefore have an anomalous interaction:

(4.11)

~

+ ~F.~/.

This will not affect the SU s y m m e t r y for the letpons, so t h a t e, v, v' will still have zero mass, as we argued above. ii) If the baryons and mesons belong to (21} and (27} respectively~ t h e n the strong interactions which break Ba will never involve ordinary baryons o1" mesons, but only leptobars. This is because (21} and (27} do not contain (al) V. V. AMBARTSUMYAN o~nd G- S. SAAKYAN" AS~'O~. Zu'r'~., 37, 193 (1960); 3 8 , 785, 1016 ( 1 9 6 1 ) ; G. S. S.a.aKH~'A-~~ a n d Y r . \ ' . ~AI~TANYAN: Astron. Zur~., 4t 193 ( 1 9 6 4 ) ; Nuovo Cim.e~do, 30, 82 (1963). 17 - II Nuovo Cimento.

258

C. LOVELACE

a n y L = 3 p a r t i c l e s , so t h a t t h e b i l i n e a r L = 3 e x p r e s s i o n s f o r m e d f r o m t h e m , w i l l n o t i n v o l v e L = 0 p a r t i c l e s a t all. T h i s e x p l a i n s w h y t h e l e p t o b a r s s h o u l d have a much larger mass than the normal strongly interacting particles. iii) S U3 is n o t a f f e c t e d , b e c a u s e t h e ~ + + is a u n i t a r y s i n g l e t . w i t h L = 3 h a v e zero t r i a l i t y (~o), j u s t as d o t h o s e w i t h L - - 0 .

Particles

iv) L w i l l r e m a i n a g o o d q u a n t u m n u m b e r for s t r o n g i n t e r a c t i o n s . T h i s w i l l e n s u r e l e p t o n c o n s e r v a t i o n . (The v a l i d i t y of l e p t o n c o n s e r v a t i o n for w e a k i n t e r a c t i o n s d e p e n d s on t h e a b s e n c e of A Y ~ 0 l e p t o n c u r r e n t s . ) lqow we c o n s i d e r t h e l i m i t s on t h e m a s s of t h e ~[, a n d t h e ~I coupling" c o n s t a n t , g i v e n b y t h e p r e s e n t e x p e r i m e n t a l a b s e n c e of a n o m a l o u s m u o n i n t e r a c t i o n s (4~). A s t r o n g l i m i t c o m e s f r o m t h e l a c k of r e n o m a l i z a t i o n effects i n r~-~t d e c a y , as w a s p o i n t e d o u t b y KOBZAgEV a n d OKU.~- (a3). E x p e r i m e n t a l l y , t h e ~tL,' a n d r:e, c o u p l i n g c o n s t a n t s a r e e q u a l t o w i t h i n 7 O//o(a4). T h i s i m p l i e s for t h e w a v e - f u n c t i o n r e n o r m a l i z a t i o n of t h e m u o n , d u e t o i t s a n o m a l o u s interaction, (4.12)

Z~-1 - - 1 < 0.07 .

We can write the muon mass and wave-function renormalization constant in t e r m s of i t s L e h m a n n s p e c t r a l f u n c t i o n s (45):

(4.13)

105.144 5 I e V = tt - - rn~ = Z2 |

[tta4(u -~) q- xa2(x'2)] d x 2 ,

.3

(M+/~)8

(4.14)

(Z~) -1 = 1 ~- |

(r~(z -°) d z 2

~J

(M+/~)*

(4~) Muon experiments have recently been reviewed by G. FEINBERG and L . M. LEDERMAN: ~lttl~. Rev. Nucl. Sci., 13, 431 (1963); A. ZICHICtlI: Suppl. Nuovo Cimet~to, 1, l l (1963); N. RAMSAY: Proc. Dubna Con]. (196t), to be published. (4a) I. Yu. KOBSAREV and I,. B. 0KUX: 2urn. Eksp. Teor. Fiz., 4i, 1205 (1961); Nucl. Phys., 35, 311 (1962). See also, T. D. LEE a n d D . N. YAN(~: Phys. Rev., 108, 1611 (1957); M. TAKETANI, Y. KATAYAMA, P. L. FERREIRA, G. W. BUND and P. R. D}~ PAULA :a SILVA: Progr. Theor. Phys., 21, 799 (1959); G. SHI and G. MARX: Acta Phys. Hu~gar., 15, 251 (1963); H. FISSNER, Z. KJELLMAN and A. STAUDE: =Yuovo Cit~ento, 32, 782 (1964). (44) E. DI CAPUA, R. GARLAND, L. PONDROM and A. STRELZOFF: Phys. Rer., 133, 1333 (1964). (45) G. K-KLL~\~: Helv. Phys. Act_q, 25, 417 (1952); tl. [,E1).~IANN: Nuovo Cimento, 11, 342 (1954).

A TIIEORY

OF E L E M E N T A R Y

PARTICLES

259

with

(4.15)

o< [(~(:~")l< ~(~ ~) .

Here M ~ n d / , are the masses of these particles. Plainly, to get a large selfmass without a large wave-function renormalization, we need (~2 near its upper limit. Therefore we take

(4.16)

(~(u~) = a2(~.2) = g.,(~(y2__ [ M - - tel ~) .

This is optimistic as far as the ratio of the two spectral functions is concerned, but pessimistic a b o u t their mass distribution. (t.12) and (4.13) then give (4.17)

M ~ 1395 MeV,

(4.18)

g~ ~ 0.07.

We would expect the 2~ ÷fi intermediate state alone to contribute to the spectral functions something of the order of [~/4~, where [ is the ~ [ coupling const~nt. Thus (4.19)

/2 g 4~g~ < 0.88 .

(4.19) is valid in second order p e r t u r b a t i o n theory, provided the cut-off is greater th~n 1.65 M (43). We want ] 2 to be near its upper limit, firstly because on the basis of the bt/e mass ratio, we would expect 1'2~1, a n d secondly because the larger the coupling constant, the more chance t h a t p e r t u r b a t i o n theory will be invalidated by the existence of a ~5I resonance. We need this in order to explain the ratio of the two spectral functions. I n second-order p e r t u r b a t i o n theory, a 2 ~ 0 a n d it is impossible to obtain a large m u o n mass without a large wave-function renormalization also (,a.~6). The contribution of an anomalous interaction to the m u o n magnetic mom e n t has been calculated by a n u m b e r of authors (~). The effect depends on the spin of the new particle, but is always limited by

(4.20)

(ds) The most complete calculations are by R. 8V~;AXO: Progr. Theor..Phys., 28, 508 (1962). Others are in V. B. BEI~ESTETSKII,(). N. KROKIIINand A. K. KHLEBXIKOV: ~urn. ]~ksp. Theor. Fiz., 30, 788 (1956); W. S. ('OWLANI):Xucl. Phys., 8, 397 (1958); H. PIETSCHMANN:Acta Phys. Austriaca, 13, 315 (1960); B. DE TOLLIS: _Yttovo Cimento, 16, 203 (1960); B. JOUVET and L. GOLDZAHL:-¥tlOVO CimeMo, 18, 702 (1960).

260

c. LOVELACE

Their calculations assume t h a t the 5[ is uncharged. I n our case, it is doubly charged, a n d this introduces a n o t h e r t h i r d - o r d e r d i a g r a m for the ~-~y v e r t e x . However, this will h a v e two internal 5~ lines, so its c o n t r i b u t i o n will be O(M -4) a n d therefore negligible. E x p e r i m e n t M l y (47), s u b t r a c t i n g the e l e e t r o d y n a m i c contribution, (~.2~)

5g = ( - - 3 ~ 5 ) . J o - ~ .

T a k i n g j-" at its u p p e r limit in (19), t h e n gives (4.22)

M > 10.7 G e V .

The m u o n f o r m factor is close to 1, up to m o m e n t u m transfers t = 4 (GeV) -~(ds). Conceivable i n t e r m e d i a t e states are MM pairs, l e p t o b a r - a n t i l e p t o b a r pairs, a n d the vector mesons p, ?, k. M~I pairs will h a v e v e r y large masses b y (4.22). L e p t o b a r s can only enter t h r o u g h their coupling to the 5I. This requires t h a t we h a v e either a b a r y o n - a n t i b a r y o n , or at least six meson leptobars in the i n t e r m e d i a t e state. This restriction arises because 5[++ is a u n i t a r y singlet, a n d the meson r e p r e s e n t a t i o n {27} contMns [6*; 2] instead of [3; 2], m a k i n g it impossible to f o r m a [1; 3] bilinearly. Therefore, the i n t e r m e d i a t e state will a g a i n be v e r y massive. The coupling of ~ to the vector mesons can also only t a k e place t h r o u g h )1. Because 5[ has (Y, Q, L ) = (1, 2, 3) b y (3.5), it will only i n t e r a c t w i t h k of the three v e c t o r mesons, b y (4.5). J u d g i n g f r o m ¥0 a n d h, we m a y expect X to h a v e a mass similar to the octet vector mesons, s a y 1 GeV. The lowest-order g r a p h for the ~-~k v e r t e x will contain two ~[ i n t e r n a l lines. We h a v e to e v a l u a t e it a t the k mass, which is small c o m p a r e d w i t h the IV[ mass, b y (22). We therefore can ge~ a rough idea of its m a g n i t u d e b y counting the powers of ~1 in the integrand. This gives g~z ~ ]2/M~, which is e x t r e m e l y small. So the X is not likely to m a k e an appreciable c o n t r i b u t i o n t o the m u o n form-factor. The limits (4.19) a n d (4.22) therefore seem to be sufficient to explain all the experiments. U n f o r t u n a t e l y , (4.22) shows t h a t we would need an accelerator of at least 264 GeV to produce 5f pMrs. Nevertheless it is amusing to consider w h a t t h e i r e x p e r i m e n t M effects would be. 5[ would decay stro~gly into 2~+. The production of an ~ p a i r would a p p e a r as four m u o n s energing f r o m a strong i n t e r a c t i o n vertex. The e x p e r i m e n t s on p p - - > ~ would therefore not h a v e

(~7) G. CHARPAK, F. J. M. FA'XlJEY, R. l,. GARWIN, T. :~IULLER, J. C. SE×S and A. ZICHICHI: Phys. Lett., 1, 16 (1962). (48) R. COOL, l~. ELLSWORTH, L. LEDERMAN, A. 3[ASCI1KE, A. •ELISSINOS, M. TANNENBAUXI, J. TINLOT and T. YA_~I,~XOUCHI: Proc. Dublin ('o~/. (1964), to be published.

~, THEORY OF ELEMENTARY PARTICLES

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seen it, since t h e y looked specifically for muon pairs (49). I n other experiments the muons would appear as four tracks leaving the chamber. Scmmers would not notice ~nything peculiar a b o u t such an event, and it would be fed to the computer as if normal. The kinematics program, h a v i n g been told that m u o n s do not inter~ct strongly, would do its best to turn t h e m into b a d l y m e a s u ; e d pions. I t would usually succeed, since the muons would come out fast. I f not, the event would be rejected as unfittable. We conclude t h a t the most obvious effect of M pair p r o d u c t i o n at accelerator ene:gies, would probably be an increased t u r n o v e r of measuring staff. Once we suspect the existence of such a particle, and have a sufficiently big accelerator, detecting it will be easy. The simplest experiment (50) would use a p r o t o n benin of the highest energy and intensity. The target would be surrounded b y lead shielding, as closely as possible, so as to catch pions before t h e y decay. B e y o n d the shielding would be spark chambers, triggered only b y events with at least four eha;'ged particles. With well-designed shielding, b a c k g r o u n d could p r o b a b l y be eliminated almost completely, and the 5[ detected in a m a t t e r of minutes. This would make a very nice direct experim e n t for detecting a n y strong interaction of the m n o n . 3~-pair production will always be small compared to pion p r o d u c t i o n because of its high q u a n t u m numbers, just as strange-particle p r o d u c t i o n is. Thus most muons in cosmic rays will still come from pion decay, and the existence of M will have little effect on the m u o n spectrum. I t is t e m p t i n g to relate M's to the evidence for collimated ((beams)) of muons in extensive air showers of p r i m a r y energy > 10 ~5 eV (5~). These m u o n beams are very difficult to u n d e r s t a n d , if the muons come f r o m pion decay.

(49) ~[. CONVERSI, T. MASSA_~LTm 3IULLER and A. ZICH~CnI: Proc. Dubt~a Con]. (1964), to be published; A. BARBARO-GALTIERIand R. D. TRIPP: Proc. Dubna Con]. (1964), to be published; and Berkeley preprint U('Rl,-l1428 Rev. (1964). (~o) This was suggested to me by Dr. I. SRILLICORN. (.~1) This question is reviewed by S. I. NIKOLSKII: Uspekhi ~_'iz. Xauk, 78, 365 (1962). Subsequent papers are I. L. ROZENTAL: 2Urtt. Eksp. Teor. Fiz., 36, 943 (1959); S. N. \~ERNOV, B. A. KHRENOV a n d G. B. KHRISTIANSEN: ~ur~. Eksp. Teor. Fiz., 37, 1252 (1959); S. N. VERNOV, [. P. IVAX~:YEO, G. V. KULIKOV and G. B. KHRJSTIA~SE.~: 2urn. Eksp. Teor. Fiz., 39, 509 (1960); S. N. VER~OV, LI DON KEVA, B. A. KHRENOV and G. B. KnRISTIANSE~-: 2ur~i. Eksp. Teor. J,'iz., 42, 758 (1962); Jour~l. Phys. Soc. Japalt, 17, Suppl. A-III, 213 (1962); S. N. VERXOV, G. B. KHRISTIANSEN, I. F. BELYAEVA, V. A. DMITRIEV, G. ~'. KULIKOV, YU. A. '.-N~ECHIN, V. I. SOLOV]~VAand B. A. KHRF,NOV: Izr. Akad. Xauk SSSR, 26, 651 (1962); S. MIYAK~, K, H1NOTANI, T. KANEKO and N. ITO: Jour~. Phys. Soe. Jetpan, 18, 464 (1963); H. W. HUNTER and P. T. TRENT: Proc. Phys. Soc., 79, 487 (1962); J. HIBX~R, R. FIRKOWSKI, J. GAWIN and A. ZAWADZKI:Acta P]~ys. Polo~., 25, 101 (1964). For contrary evidence, see Yu. N. VAVlLOV, G. I. PUI;ACIIOVAand V. M. FEDOROV: ~ur}~. F,lcsp. Teor. Fiz., 44, 487 (1963).

262

c. LOVELACE

To explain t h e m , we would need not only the existence of 5['s, b u t also a m e c h a n i s m for their production. This would h a v e to be a quasi-diffraction process, so the ~ ' s would come out in the forward direction, a n d w i t h an energy large c o m p a r e d to their mass, thus lining up the decay muons. The absence to their mass, thus lining up the decay muons. The absence of (( m u o n b e a m s ~> at lower energies could mean, not t h a t l~['s were absent, b u t m e r e l y t h a t they were not produced in this way. F r o m the analogy with SUn, we would expect the B3 mass-splitting operator to t r a n s f o r m like a sum of L = 0 u n i t a r y singlets t a k e n f r o m various representations (~8). U n f o r t u n a t e l y , Table I shows t h a t e v e r y even represent a t i o n possesses such a u n i t a r y singlet, so there will be a large n u m b e r of t e r m s in the mass formula, a n d we cannot predict the l e p t o b a r masses. 4"8. Miscellaneous. - The vector r e p r e s e n t a t i o n (7}, which we h a v e not y e t filled, is the B3 quark, and could p l a y the same role in composite models (39). H o w e v e r , it has integral charge, a n d its ordinary L----0 p a r t would be just a u n i t a r y singlet, so nothing spectacular would be seen. There h a v e been a n u m b e r of previous models for e l e m e n t a r y particles based on the seven-dimensional r o t a t i o n group. TIOMNO (e), FEI2~BERG and Gt~RSEY (3), BEIIt~E~NDS (4), I)EASLEE (5), SOURIAU and KASTLEI~ (6) and L o g t'IAs et al. (7) all suggested {8} for the b a r y o n s a n d {7} for the mesons. The closest to the present scheme was PEASLEE (5), who also p u t leprous and antileptons into (8}, as we h a v e done. However, as PEASLEE noted, h a v i n g leptons a n d b a r y o n s in similar representations, m a k e s lepton conservation v e r y difficult to u n d e r s t a n d . I n our theory, leprous a n d hadrons are quite distinct, since the f o r m e r h a v e half-integer spin and the l a t t e r h a v e integer spin in the seven-dimensional isospace. Also, none of these older theories satisfied S U a i n v a r i a n c e for the ba:'yons (52) (despite the 8), a n d their Y u k a w a couplings a p p e a r i n c o m p a t i b l e w i t h e x p e r i m e n t .

5.

-

Discussion.

B y working in larger a n d larger groups, e x p e r i m e n t a l i s t s h a v e been able to produce resonances faster t h a n the theorists could predict t h e m . R e c e n t l y the theorists h a v e discovered a counterbalancing a c t i v i t y . A3, B3, C3, Dd, (A2) ~ a n d other horrors are all now on record as the basis of physics (53). The difference between us a n d all these authors is t h a t we get s o m e t h i n g b a c k for our m o n e y - - t h e e x t r a q u a n t u m n u m b e r is used to include leptons. (55) D. R. SPEISER and J. TARSKI: Journ. Math. Phys., 4, 588 (1963). (53) For the full gallery, see A. SALA)[: Proc. Dub~ta Co~/. (1964), to be published

A

THEORY

OF

]~L]~MENTARY

PARTICLES

263

An objection, sure to be made, is t h a t our theory does not fulfil the fashionable ideal of uniting space-time with internal s y m m e t r y . However, fashionable ideals are not always correct, and it would be dangerous to say t h a t alternatives were not even w o r t h considering. Of these purely internal b a r y o n + + l e p t o n theories, the present scheme is the neatest we could find. Let us list some questions answered b y the theory, i) W h y are leptons light? Because leptons a n d antileptons are in the same B3 representation, ~nd this forces t h e m to h a v e zero muss in the B3 limit, it) W h y is the m u o n the heaviest of t h e m ? Because, when B3 is broken down into SU3, only the m u o n is allowed a m~ss. iii) W h a t gives the m u o n its mass? I n t e r a c t i o n w i t h the missing triplets, p r o b a b l y b y m e a n s of the M ++ boson, iv) W h y are triplets not observed? Because t h e y are m a d e h e a v y by the strong B3-breaking interactions, which are f o r b i d d e n to n o r m a l baryons, v) W h y is the charge always an integer? Because the t h i r d B3 q u a n t u m n u m b e r displaces SU3 m u l t i p l e t s according to their t r i a l i t y (4o). vi) W h y are ¥ u k a w a couplings p r e d o m i n a n t l y D? Because the mesons are in a 27-fold, not a 21-fold, repr e s e n t a t i o n of B3. I t also goes as far towards explaining the weak interactions of leptons, as 2U3 does for the b a r y o n s (9). The predictions for k n o w n strongly i n t e r a c t i n g particles differ only m a r ginally f r o m SU3, a m t wilt ~berefore be difficult to test. E a c h octet is accomp a n i e d b y a singlet, as in Schwinger's W3 t h e o r y (~*). There are e x p e r i m e n t a l candidates for these, t h o u g h the q u a n t u m n u m b e r s are not certain, especially the Y* p a r i t y (3o). We can predict their coupling constants (Table I X ) , a n d therefore their widths. However, W3 has the v e r y serious d i s a d v a n t a g e of forbiddino' p-->2:: ~nd K * - ~ K + = (55). This is not so in our t h e o r y - - t h e r e are no u n w a n t e d selection rules. I n s t e a d the theory can be m a d e to forbid /~ Y u k a w a coupling, which is e x p e r i m e n t a l l y small (32). However, we 40 share the other d r a w b a c k of ~ - - u n w a n t e d ~3 + resonances. ~ndeed, we h a v e even more of t h e m . We t h i n k the answer is t h a t there is a l i m i t to how far a badly

(~) J. SCHWINGER: Phys. Rec. Lett., 12, 237 (1964); Phys. Rec., 135, 816 (1964); Phys. Rev. Lett., 13, 355 (1964); P. SIN(~ER: Phys. Rev. Lett., 12, 524 (1964); R. E. CUTKOSKY: Phy8. Rev. Lett., 12, 530 (1964); I. S. GERSTEIN and K. T. MAIthNTHAPFA: Phys. Rev. Lett., 12, 570 (1964); A. PAIS: Phys. Rev. Lett., 12, 632 (1964); F. ENGLERT and R. BROUT: Phys. Rev. Lett., 12, 682 (1964); E. JOHNSON, R. F. SAWYER and K. C. WALI: Phys. Rev. Lett., 13, 141 (1964); E. JOHNSON and R. F. S~wYER: Phys. Lett., 9, 212 (1964); J. J. SAKURAI: Phys. Lett., 10, 132 (1964); S. L. GLASHOW and D. J. KLEITMAN: Phys. Lett., 11, 84 (1964); W. ALLES: Phys. Lett., lt, 86, 176 (1964); J. J. BREH~ and K. C. WALI: Phys. Lett, 11, 355 (1964); M. SUZUKI: Phys. Rev., to be published. (~s) j . j. SAKURAI: Proc. Dubna Con]. (1964), to be published; C. A. L~vI~so~, H. J. LIPKIN and S. MESHKOV: ~VUOVOCiraento, 32, 1376 (1964).

264

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b r o k e n s y m m e t r y can be believed, when applied to weakly b o u n d composite particles (56). To some extent, this difficulty is already present in S(~, since there are e x p e r i m e n t a l resonances which s t u b b o r n l y refuse to lie down amono" the rest (e.g. the ×(725), a n d p e r h a p s the incomplete ~-- m u l t i p l e t f o r m e d by the ~ ' * (1525) a n d ¥ * (1660)). The really novel prediction is the existence of new t y p e s of particle. The (( leptobars )) are the missing SU~ t:'iple~s. T h e y have the q u a n t u m n u m b e r s of a b a r y o n (or m e s o n ) q - 2 leptons, integer charge, a n d big masses. I f the mass were greater t h a n 3 GeV, t h e y would not be seen in accelerator experiments, since t h e y h a v e to be produced in pairs. I n cosmic ~'ays, the decays m i g h t be difficult to distinguish f r o m strong interactions. The other new object is the 5I++. This is a doubly cha:'ged boson, decaying s~rongly into 2~+. I t s existence breaks B3 down to S U~: 5Iuon electrodyn:~.mics requires its mass to be ) 1 0 . 7 GeV, so t h a t a 264 GeV accelerato:' would be needed to p a i r - p r o d u c e it. H o w e v e r , even if it could be produced, it would not be seen unless specially looked for. Finally some philosophical points. I t m a y be argued t h a t a n y theory which predicts the existence of large n u m b e r s of undiscovered particles contradicts the principle of simplicity. Our answer would be that, historically, physicists h a v e always erred on the side of economy (e.g. the S a k a t a model, and the original Dirac electron theory, which identified the positive electron with the p:'oton). The philosophy of discovery should be as m u c h a m a t t e r for experim e n t a l test as physics itself. Now the opposite question? Assuming B3 is right, how final is it? I n ordinary isospin theory, some multiplets such as the nucleon have to be displaced f r o m the origin, to m a k e the charge an integer. This suggested an e x t r a q u a n t u m n u m b e r - - h y p e r c h a r g e - - t o describe the displacement. I n SU~, :'epresentations with non-zero triality (4o) likewise need shifting to keep the charo'e an integer. Again a n e x t r a q u a n t u m n u m b e r is plausible, and this has a p p a r ently been the m a i n m o t i v e behind the recent vogue for rank-3 groups (sa). Similarly, in the lepton s y m m e t r y B:, Fig. 1 shows a displacement f r o m the origin in the charge. H o w e v e r , in the group B3 there is never a n y displacement. All particles h a v e integer charge, and all observed particles are given their observed charge b y the group, w i t h o u t f u r t h e r additions. This is some indication t h a t Ba m a y be the m a x i m a l s y m m e t r y of the e l e m e n t a r y particles, a n d not m e r e l y a n o t h e r stage in the eternal ascension to higher a n d higher r a n k groups. The fact t h a t all k n o w n particles now find a n a t u r a l place in the scheme is also encouraging in this respect. So, t h o u g h the n u m b e r of

(ss) Of course this was said in the much maligned paper of R. J. OAKES and C. N. YANG: Phys. Rev. Lett., 11, 174 (1963), though they seem to have chosen their ground unfortunately.

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e l e m e n t a r y p a r t i c l e s still t o be d i s c o v e r e d is c o n s i d e r u b l e (if we are r i g h t ) , i t is n o t i n f i n i t e .

I a m g r u t e f u l for d i s c u s s i o n s to P r o f . A. SAL.~I, P r o f . P. T. -~[ATTHEWS~ Dr. J . TARSKI~ Dr. J . 5[. CgAR.~P, Dr. T. KIBBLE, Dr. S. GOLDSACK a n d Dr. I. SKILLICOm~~.

R1ABSUNT()

(*)

Si e~amina quello ehe ~ccade se si fa in modo che i quattro leptoni formino u n quar~etto, invece di separarli per formare un tripletto. Be si esige anche l'invarianza dei barioni rispetto alla SU3, si ~ portati in modo molto naturale al gruppo di terzo tango B a. I1 terzo numero quantico, L--eariea leptonica, /~ diverso da zero solo per i leptoni compresi fra le p~rticelle note. Esso anmflla anehe la cariea frazionale per i tripletti della SU3. I Ieptoni e gli antileptoni formeno un ottuplice rappresentazione priva di massa. Quando la B 3 ~ ridotta a SU3, solo il muone pub avere una mass~. Le assegnazione preferite per i barioni, i m e s o n i e i mesoni vettoriali sono, rispettivamente, {21}, {27} e {21}. L'aeeoppiamento di Yukava, allora, ~ puro D, e non ei sono regole di selezione indesiderabili. La {21 } e la {27} eonCengono eiascuna un ottetto e un singoletto di partieelle normali ( L = 0 ) in S U a, assieme a tripletti o sestupletti di ((leptobari ~. Queste ultime sono particelle con interazione forte che hanno i humeri quantiei di u n barione (o mesone) + 2 leptoni. Be la simmetria ~ infranta da un bosone con L 3 (M++), si spiega la massa de] muone, ed i leptobari diventano molto pift pesanti dei normali barioni, pur conservando la SUa. M ~+ dovrebbe decadere /ortemet~te in 2p.*. Si d'~nno te tabelle della struttura della rappresentazione B.~ .,ino a {448}, e teeniche utili in altri grandi gruppi.

(*) T r a d u z i o n e a c u r a delia R e d a z i o n e .