Symmetry Schemes for the Elementary Particles

7/30/2019 Symmetry Schemes for the Elementary Particles

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C h a p t e r 1 3S y m m e t r y s c h e m e s for t h ee l e m e n t a r y p a r t i c l e s

L e p t o n s a n d h a d r o n sThe s t a r t i ng po in t o f a ll sym m et ry schemes fo r t he e l emen ta ry pa r t i c l e s is t heobse rva t ion tha t t he re appea r t o be fou r fundamen ta l i n t e rac t ions be tweenthese par t ic les . Th ese are , in decreasing ord er of s t rength :( i) th e s t ron g in ter ac t ion , f i rs t d i scussed in the context of the b inding ofthe nucleons in the nucleus;

( i i ) the e lec t romagnet ic in terac t ion;( iii) the w eak in terac t io n (which , for exam ple , is responsib le for be ta dec ay);( iv) the gravi ta t ional in terac t ion .

(Recen t deve lopmen t s sugges t t ha t t hese in t e rac t ions m ay no t be d i s t inc t , bu tm ay be man i fes t a t ions o f a s ing le fund am en ta l i n t e rac t ion . )In te rm s of these four in terac t ions i t is possib le to d iv ide the observ edpart ic les in to two m ajo r ca tegories , the " lep tons" (and "a nt i lep tons") whichn e v e r exper i ence s t rong in te rac t ions , and the "had rons" (and "an t ihad rons" )wh ich , a t l east i n some c i rcumstances , i n t e rac t t h roug h the s t rong in t e rac t ion .In add i ti on the re a re t he " in t e rmed ia t e" pa r t i c l e s t ha t a re t he ca r r i e rs o f t hein t e rac tions (o f wh ich the pho ton , W + and Z ~ have ac tua l ly been obse rveda t t he time o f wr i t i ng ) . Th e ca t egory o f h a d r o n s can be fu r the r d iv ided in totwo c lasses , tho se w hose in t r ins ic sp in j i s an in teger (= 0 , 1 , 2 , . . . ) be ing

1 3cal led "mesons" and the o thers ( for which j = 2 , 2 , ' " ") be ing referred to as"ba ryons" . T he " l ep ton num ber" and "ba ryo n number" may then be de f inedfor a l l the present ly observed par t ic les by

1, i f the part ic le is a lepton,L - - 1 , i f the par t ic le i s an an t i lep ton ,

0 , for any o th er typ e of par t ic le ,255


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256 GROUP THEOR Y IN PHYSIC Sa n d 1,B = - 1 ,

0,i f the par t ic le is a baryon,i f t he pa r t i c l e i s an an t iba ry on ,for any o ther type of par t ic le .

2 T h e g l o b a l i n t e r n a l s y m m e t r y g r o u p S U ( 2 )and isotopic spin

T he objec t of th i s sec t ion is to in t rod uce the concept of i so topic sp in a ndpresent the basic ideas in such a way tha t they are eas i ly genera l izable too the r i n t e rna l symmet r i e s .

Cons ide r f ir st t he case o f t he p ro ton (p ) and the neu t ron (n ). The i r re s tmasses mp a n d mn a r e a l mo s t identical ( m p C 2 - 938.3 MeV, mnc 2 = 939.6MeV) , and the i r i n t e rac t ions wi th each o the r ( t ha t i s p -p , p -n and n -n ) a reindepende n t o f how they a re pa i red (p rov ided tha t t hey a re a lways coup ledin to the same s t a t e o f t o t a l sp in and pa r i t y ) . I t is a s t hou gh the re is on lyone par t ic le , the "nucleon" (N), which m ight ex is t in e i ther of two s ta tes , onecor respond ing to t he p ro ton and the o the r t o t he neu t ron , t hese two s t a t e sbeing d is t inguished only by an e lec t rom agn et ic f ie ld . Th is is a s imi lar s i tua t io nt o t h a t o f a n a t o m i n a s ta t e w i t h o rb i t a l a n g u l a r mo m e n t u m l s u b j e c t e d t oa smal l mag net ic f ie ld H. As noted in Ch ap ter 10, Sect ion 6 , i f a l l the effec tsof the e lec t rons ' sp ins are neglec ted ( inc luding degeneracies caused by the m )the n the ene rgy e igenvalue o f a s t a t e w i th an gu la r m om en tu m I is (2l + 1 ) -fo lddege nera te in the absence of the f ie ld , bu t sp l i ts in to ( 2 /+ 1) d i fferent va luesw hen the f ie ld is appl ied . N atu ra l ly one does not rega rd thes e as be ing (21 + 1)d i ffe ren t a toms , b u t r a the r t he y a re t hou gh t o f a s (2 /+ 1 ) d i ffe ren t s t a t e s o ft h e s a me a t o m.The co r respondence be tween these two s i t ua t ions depends on the connec-t ion between energy and ma ss in the specia l theo ry of re la t iv i ty . I t l eads tothe proposal tha t the nucleon N should be ass igned an " i so topic sp in" I wi thvalue 89 ( th is va lue being chosen so th a t 21 + 1 - 2 , so tha t i t can e x is t in21 + 1 (=2) d i ffe ren t s t a te s , one co r respond ing to t he p ro ton and one to t h eneu t ron . Fu r the r , i t is sugges t ed tha t i n t he absence o f e l ec t romagn e t i c i n t e r -ac t ions ( tha t is , i n a un ive rse w i th no e l ec t rom agne t i c i n t e rac t ions ) t he p ro to nand the ne u t ron wou ld be iden t ica l , and each o f t he i r i n t e rac t ions , wh ich a real l "strong", would also be ident ical .Developing th is fur ther , one can in t roduce three se l f-ad jo in t l inear opera-to rs I1 , I2 and Z3 tha t sa t i s fy the commuta t ion re l a t i ons

[z z2] = iz ,}(13.1)

That i s , more br ief ly ,3

r= l(13.2)


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E L E M E N T A R Y P A R T IC L E S Y M M E T R Y S CH E M E S 257for p , q = 1 , 2 , 3 . The se are ident ica l to the c om m uta t ion re la t ions in Eq ua t ion(10 .9 ) . Indeed one can wr i t e , by ana logy wi th Equa t ion (10 .7) ,

Z p = - i O ( a p ) , (13.3)(for p = 1, 2 , 3), whe re al , a2 and a3 are basis elem ents of the real Lie alg eb rasu (2 ) . The ana logy may be ex tended so tha t /1 , 2 :2 and Z3 may be rega rdedas be ing ope ra to rs co r respond ing to t he mea su rem en t o f t he "compon en t s" o fi so topic sp in in three mutual ly perpendicular d i rec t ions in an " i so topic sp inspace" . In t roduc ing the l i nea r ope ra to r Z2 by

z 2 _ ( Z l ) 2 + ( z 2 ) 2 + ( z 3 ) 2 ( 1 3 . 4 )(by analogy wi th E qu at io n (10 .16)), i t is c lear th a t a ll the pr ope rt ies of theope ra to rs A1, A2, A3 and A 2 considered in C ha pte r 10, Sect ion 3 , a pplyequal ly to the ope ra tor s 2"1, I2 , I 3 and Z 2 . In par t icu lar , the op era tor Z 2 has

1 3eigenvalues of the form I ( I + 1), where I takes one of the values 0, ~, 1 , ~, . . . .This quant i ty I i s then regarded as the " i so topic sp in" , and the poss ib le va luesof i t s "c om pon ent in the th i rd d i rec t ion in iso topic sp in space" assoc ia ted w i ththe op era tor 2:3 are g iven by the e ig env alue /3 of 2 :3 , which a ssum e any of the(2 I + 1) v a lu e s I , I - 1 , . . . , - I . T h e s i mu l ta n e o u s e ig e n ve c to r o f / 2 a n d / 3wi th e igenvalues I ( I + 1 ) a n d / 3 ma y b e d e n o t e d (b y a n a l o g y w i t h E qu a t i o n s(10.23) an d (10.24)) as r so th at

~ ( 2 , , I }= /3r 9 (13.5)

Indeed , for any e lement a of the su(2) Lie a lgeb ra span ned by the b asis ele-men t s a l , a2 and a3 o f Eq ua t ion (13.3) ,

I(I)(a)r = E D ' ( a ) I i I 3 r 'I ~ = - I (13.6)

where D / is t he i r reduc ib l e rep resen ta t ion o f su (2 ) i n t roduc ed in Cha p te r 10,Sect ion 3.I t may a l so be a ssumed tha t a l l t hese i so top ic sp in ope ra to rs commutewi th a l l t he ope ra to rs co r respond ing to space - t ime t rans fo rmat ions , so tha t

the s t a t e vec to r o f each had ron is t he d i rec t p rodu c t o f a func t ion o f space-t ime an d one o f t he vec to rs ~ / I . Each va lue o f /3 co r responds to a pa r t ic l e , t hese t of (2I + 1) par t ic les asso cia ted wi th a par t icu la r va lue I be ing sa id to forman " i so topic m ul t ip le t" . I t is impl ied from E qu at io n (13.6) th a t th e vectorsr fo rm the bas is o f t he (2 I + 1 ) -d imens ional i r reduc ib l e rep resen ta t ion D I

1of su(2) . In the case of the nucleons, the pro ton is ass igned the va lu e/ 3 =1a n d t h e n e u t r o n t h e v a l u e / 3 = - 5 "

These considera t ions imply tha t a l l the par t ic les in an i so topic mul t ip le tmus t have the same in t r in s i c sp in and pa r i t y , a s we l l a s t he same ba ryonn u mb e r ( a n d o t h e r qu a n t u m n u mb e r s , s u c h a s s t r a n g e n e s s a n d c h a rm) .


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258 G R O U P T H E O R Y I N P H Y S I C Si so top ic mu l t ip le t B Y I /3 Q pa r t i c le

- 1 - 1 7 r-, p -Ir, p 0 0 1 0 0 7r~ ~

1 1 7r+,p +_ ! 0 K ~ K *~1 2K , K * 0 1 ~ ! 1 K + K *+2

~ , r 0 0 0 0 0 ~ ~ 1 6 2 1 7 6 1 7 6_ 1 0 nN 1 1 89 2

! 1 p2_ 3 --1 A -2_ ! 0 A ~A 1 1 3 2! 1 A +23 2 A ++2

A 1 0 0 0 0 A ~- 1 - 1 E -

E 1 0 1 0 0 E ~1 1 E +

_ ! - I = -9 . 1 - I 5 ! 0 =o

2

1 - 2 0 0 - 1 g t -Tab le 13.1 : I so top ic sp in , hype rcha rg e and ba ryo n num ber a s s ignm ents o fs o m e o f t h e m o s t i m p o r t a n t h a d ro n s .

I t i s a s s u m e d t h a t al l hadrons can be c lassi f ied wi thin th is scheme. His-to r ica l ly , the ea r l i e s t pa r t i c le s to be incorpora ted in th i s s cheme a f te r thenuc leons we re the th ree p ions ~ r+, 1r~ and ~ r - , which we re a s s igned by to ani so top ic mul t ip le t w i th I = 1 , the va lue s o f /3 be ing 1 , 0 and -1 re spec t ive ly .For bo th the nuc leons and the p ions the e lec t r i c cha rge Q e of the pa r t i c le i sg iven by

1Q = / 3 + ~ B , (1 3.7 )w h e r e B , t h e b a r y o n n u m b e r i n t r o d u c e d i n t h e p r e v i o u s s e c t i o n , h a s v a l u e 1for the nuc leons and 0 fo r the p ions . In f ac t E qu a t io n (13 .7 ) ho lds on ly fo r a lln o n - s t r a n g e a n d u n - c h a r m e d h a d r o n s , t h e g e n e r a l i z a t i o n f o r s t r a n g e h a d r o n sbe ing g iven la te r in E qu a t io n (13 .9 ) . A l i st o f i so top ic sp in a s s ignmen ts fors o m e o f t h e m o s t i m p o r t a n t h a d r o n s i s c o n t a i n e d i n T a b le 1 3.1 .


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E L E M E N T A R Y P A R T I C L E S Y M M E T R Y S C H E M E S 259The essen t ia l a s sumpt ion under ly ing the above ana lys i s i s tha t the SU(2)group correspon ding to th e Lie a lgebra su(2) is the invar iance group of thes t rong in te rac t ion H ami l ton ian . Th is impl ie s tha t th i s Ham i l ton ian and the

cor respond ing T-mat r ix a re i r reduc ib le t enso r ope ra to rs t r ans fo rming as theone-dimensional ident i ty i r reducib le representa t ion . This enables predic t ionsto be m ade of ra t ios of cross-sec t ions and s imilar dyn am ical qu ant i t ies us ingthe Wigner-Eckar t Theorem and the Clebsch-Gordan coeff ic ients for su(2) .(See, for example, Chapter 18, Section 2, of Cornwell (1984) for an introduc-tory deta i led analys is) .

3 T h e g l o b a l i n t e r na l s y m m e t r y g r o u p S U ( 3 )a n d s t r a n g e n e s s

The p resen t accoun t o f the su (3 ) sy m m etry scheme fo r hadron s is in tended toin trodu ce i ts m ost s ignif icant fea tures and to emph asize the ro le of the group-theore t ica l and Lie-a lgebra ic arguments developed in ear l ie r chapters . Therehave been m an y long and de ta i led reviews of the su(3) schem e, and to thesethe reader is referred for more specif ic inform at ion on ce r ta in topics . The fol-lowing list gives a selection of these: B ehre nds et a l . (1962), Behrends (1968),Seres te tsk i i (1965) , Carru thers (1966) , Charap et e l . (1967), de Franceschiand Maiani (1965), de Swart (1963, 1965), Dyson (1966), Emmerson (1972),Lo ndon (1964), G at t o (1964), G el l -M ann and Ne 'e m an (1964), Gou rdin(1967), Kokkedee (1969), Lichtenberg (1978), Mathews (1967), Ne'eman(1965) , O 'Raifear ta igh (1968) and Smorodinsky (1965) .

The concep t o f the s t rangeness q ua n tu m num ber was deve loped ou t o f the"associa ted product ion " hypo thesis of Pais (1952) to expla in the observat iontha t ce r ta in hadrons a re c rea ted by s t rong in te rac t ions , bu t decay th roughthe weak in te rac t ion (Ge l l -Mann 1953 , Nakano a nd Nish i j ima 1953 , Nish i-j ima 1954 , Ge l l -Mann and P a is 1955). The p roposa l was tha t eve ry hadronpossesses a " s t rangeness qu an t um num ber" S , wh ich is a ssum ed to be an in te-ge r , and tha t p roduc t ion o r decay takes p lace th rough the s t rong in te rac t ioni f and on ly if the qu an t i ty AS , de f ined by

A S -- {sum of in i t ia l va lues of S} - {sum of f inal va lues of S},is zero , tha t is , i f and only if s t rang enes s is addi t ive ly conserved. Th e gener-a l iza t ion of E qu at io n (13 .7) is g iven by the "Gel l -M ann-N ishi j ima formula"

Q = / 3 + ( 1 / 2 ) B + (1 / 2) S , (13.9)(which is consis tent wi th Equat ion (13 .7) , as nucleons and p ions are ass ignedthe value S - 0). This form ula indica tes th a t i t i s m ore convenient to workwith the "hypercharge" Y def ined by

y = B + S, ( 3.10)


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260 G R O U P T H E O R Y I N P H Y S I C Si n t e rms o f wh ich Eq ua t ion (13 .9 ) becom es

Q = I3 + ( 1/ 2) Y . (13.11)Assuming tha t B i s conserved , the se lec t ion ru le for s t rong in terac t ions i s tha tthey ac t i f and only i f

A Y = 0 . (13 .12)Tab le 13 .1 g ives t he a ss ignmen t o f hype rcha rge for some o f t he m os t im por t an thad rons .

I t is na tura l to ass um e th a t the p oss ib le va lues of Y a re e igenvalues ofa se l f-ad jo in t l inear ope ra to r Y. As a ll the pa r t ic les in an i so topic mu l t i -p l e t a re a ssum ed to have the same va lue o f Y , and as Y is a ssum ed to bes imul t aneous ly measu rab le w i th /3 , i t i s necessa ry tha t

[y , 2:p] = 0 (13.1 3)for p = 1, 2 , 3 , imp lying th at

[Y ,2 2] = 0 (13.14)as we l l . Moreover , Y i s a ssumed to be unchanged by space - t ime t rans fo rma-t ions.

As Y i s an in teger for a l l observed par t ic les , i t i s reasonable to assumetha t iY i s the basis e lem ent of a rea l Lie a lgebra th a t is i som orphic to a u(1)rea l Lie a lgeb ra ( the corres pond ing basis e lem ent of u(1) be in g [ i] ). ) (As theun i t a ry i r reduc ib le rep re sen ta t ions o f t he co r respond ing L ie g roup U(1 ) a rea l l one-d imensional and are g iven by

Fu(1)([ei~]) = [ei~],where p = 0 , and where x i s rea l , i t fol lows th a t the corre spon dingi r reduc ib l e rep resen ta t ions o f u (1 ) a re such tha t

= [ i p ] .

Th en the e igenvalues of Y take the va lues p - 0 ,-1-1, =h2, . . . . ) Co nseq uen t lythe set cons ist ing of iY, iZ1, iZ2 and iZ3 forms th e basis of a u(1) @ su(2) rea lL ie a lgeb ra ( the commuta t ion re l a t i ons be ing Equa t ions (13 .1 ) and (13 .13 ) ) .However , th i s a lone does not imply any corre la t ion between the e igenvalueso f Y and 23. To ob ta in th i s it is necessa ry to make the fu r the r a ssum pt ionthat th i s u(1) G su(2) Lie a lgebra i s the proper subalgebra of a la rger rea l Liea lgebra .

The na tu ra l cand ida t es t o cons ide r a re t he rank-2 compac t semi -s imp lerea l Lie a lgebras , because a l l the i r re levant propert ies are known. Being com-pact , a l l the f in i te-d imensional re pres enta t io ns of the i r asso cia ted L ie groupsa re equ iva l en t t o un i t a ry rep resen ta t ions , wh ich the i so top ic sp in a rgumen t sof the previous sec t ion suggest to be a desi rab le fea ture . A rank-2 a lgeb ra isa p p ro p r i a t e b e c a u s e i t c a n a c c o mmo d a t e t w o mu t u a l l y c o mmu t i n g o p e ra t o r s


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E L E M E N T A R Y P A R T I C L E S Y M M E T R Y S C H E M E S 26 1(co r respond ing to y and 23 ) in i ts Car t a n suba lgeb ra . Th e non-s imple can -d id ate su(2) @ su(2) can be e l imina ted because i t wo uld leave the va lues ofY and i f3 unre la ted , so the choice i s narrowed to the rank-2 compact s implerea l L ie algeb ras . The ana lys is o f Ch ap te r 11 shows tha t t he re a re on ly th reenon- i somorph ic a lgeb ras w i th the requ i red p roper t i e s , namely su (3 ) ( t he com-pac t rea l form of A2), so(5) (which i s the com pac t rea l form of B2 an d C2,as these are i somorphic) , and th e com pact rea l form of G2. I t is now c leartha t t he scheme based on su (3 ) ag rees we l l w i th exper imen ta l obse rva t ion ,and tha t t h i s i s no t t he case fo r t he schemes based on the o the r a lgeb ras .Consequent ly the present account wi l l be confined so le ly to the su(3) scheme.Even wi th su (3 ) se l ec t ed a s be ing the appropr i a t e a lgeb ra , t he re s t i l l r ema insthe qu est ion of the prec ise re la t ionship of y and 2"3 to th e basis e lem ents ofthe C ar t an suba lgeb ra o f A2 . Th i s i s equ iva l en t t o t he p rob lem o f a ss ign ingpart ic les to mul t ip le t s , which was reso lved by Gel l -Mann (1961, 1962) andNe 'eman (1961) , and which wi l l be d iscussed short ly .

Th e b asic phi losophy of the su(3) schem e is th a t Y and i f3 are me m bers ofthe C ar t an suba lgeb ra of A2 , and the i r e igenvalues Y an d /3 a re de t e rm inedby the weights of the i r reducib le repre senta t io ns of A2. Th e se t of had ronsco r respon d ing to a pa r t i cu l a r i r reduc ib le rep resen ta t ion is sa id t o fo rm a "uni -t a ry mul t ip l e t " and the had rons invo lved a re a ssumed to be iden t i ca l apa r tf rom the i r va lues o f Y , / 3 and I , so t ha t t hey a ll have the sam e sp in, pa r -i t y and ba ryo n number . Moreover, i t is a ssum ed th a t i n an idea l un iversethe re is on ly one type o f i n t e rac tion , t he s t rong in t e rac t ion , and t ha t a ll t hepa r t i c l e s in a un i t a ry m ul t ip le t have exac t ly t he sam e mass . A t t h i s po in tthere i s a problem, because i t wi l l become apparent tha t in the rea l worldthe pa r t i c l e s i n a un i t a ry mul t ip l e t have masses t ha t a re on ly ve ry rough lyequal . Th e s i tua t ion is qu an t i ta t ive ly qui te d ifferent f rom that in the iso-topic sp in scheme, where the masses wi th in an i so topic mul t ip le t d i ffer by a tmo s t a few pe r cen t, and where the d i ffe rence can be a t t r i b u ted to t he w eakere l ec t romag ne t i c i n t e rac tion . I t is c l ea r t ha t t he cons ide rab le m ass -sp l i tt i ngsbe tween i so top ic mu l t ip l e t s i n a un i t a ry mul t ip l e t canno t be a t t r i bu ted to t hee l ec t romagne t i c i n t e rac t ion , so tha t i t i s necessa ry to make the a ssumpt iontha t t he re a re two types o f s t rong in t e rac tion . Th e w eaker ve rsion, wh ich wil lbe ca l l ed the "med ium-s t rong in t e rac t ion" , i s a ssumed to be re spons ib l e fo rthese m ass-sp l i t t ings . Th e s t ron ger vers ion wi ll s t il l be referred to as " the"s t rong in t e rac t ionThe fi rst priori ty is to establ ish the relat ionship of Y, ffl , i f2 and if3 to thebas is e l em en t s h~ 1, h ~ , ea l , e - ~ l , e~2 , e -~2 , e~1+~2 and e_ (~ l+~2) o f t heWeyl canon ica l basi s o f A2 . T he req u i remen t s a re t ha t :

(i) Z1, 2"2, 2"3 sat isfy the co m m uta t io n relat io ns in Eq ua t ion s (13.1);(ii) Y sa t i s f i e s t he commuta t ion re l a t i ons in Equa t ion (13 .13 ) ; and

(iii ) i f any par t ic le in a un i tar y m ul t ip le t ha s in tegra l e lec t r ic charge ( th a tis , i f Q is an integer), then a ll t he pa r t i c l e s i n t he mul t ip l e t mus t havein tegra l e lec t r ic charge .


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262 G R O U P T H E O R Y I N P H Y S I C S

_ 2 0

23

f ~

Figure 13.1: Values of/3 and Y for the irreducible representation{3}.

These requirements lead to the assignments"Y = l(I)(Ha ~) + 2(I)(Ha2) = 2(I)(ha~) + 4(I)(ha2) = 2(I)(H2),2"1 - 89 89 = v / - ~{ (I ) ( e~ , ) - ( I ) (e_~)} ,Z2 = - 89 + 89 = - i x / ~ { ( I ) ( e ~ ) + (I )( e_ ~) },I 3 = ~ 1 0 ( H ~ ) = 3( I) (h~ ) = vf3(I)(H1)

(13.15)(where HI a n d / / 2 a re the o r tho-normal basis e lements of the Ca r tan sub-algebra of A2 of Exa mple II of Ch apt er 11, Sect ion 6) . (The detai led a rgum entthat leads to Equations (13.15) may be found, for example, in Chapter 18,Section 3, of Cornwell (1984)).Th e irreducible rep resentat ion s of A2 were invest igated in detail in Ch apte r12, Section 4. For a weight

A- - #1 0 ~ 1 -~ - ~2 OL 2 , (13.16)the associated eigenv alues/3 and Y of the operators :/'3 and y are given by1}/3 = #I- ~#2, (13.17)Y = #2.The argument is s imply that , by Equations (13.15) above,

/3 = A(3h~, ) = 3 { # 1 ( o ~ 1 , O L l > - ~ - # 2 ( o ~ 1 , o l2 > } ~ - - # 1 - ( 1 / 2 ) # 2 ,and

Y = A(2hal +4 ha 2)= #1{2(O~1,O~1> "~- 4 ( a l , a2> } + #2{2(C~1, a2 ) + 4 (a 2, a2 )} = #2"


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E L E M E N T A R Y P A R T I C L E S Y M M E T R Y S C H E M E S 263

Y

I I

ol

Figure 13 .2 : Values of /3 and Y for the i r reducib le representa t ion{8}.

The re su l t i ng pa i r s o f e igenva lues /3 and Y fo r t he i r reduc ib le rep rese n ta t ions{3}, {8}, {6} and {10} can be rea d off Figu res 12.1, 12.3, 12.4 and 12.6, andare d isp layed in Figures 13.1 , 13.2, 13 .3 and 13 .4 . For the repr esen ta t ion s{3*}, {6*} and {10"} the va lues of / 3 and Y are the negat ives of those of {3},{6} and { 10} respect ive ly . By Eq ua t ion (13.11) the corre spon ding values ofthe e lec t r ic charge Q e are given by

Q - P l . (13.18 )The we igh t o f mu l t ip l ic i t y 2 o f t he i r reduc ib le re p resen ta t ion {8} m ay bethough t o f a s be ing assoc i a t ed wi th two e igenvec to rs , one co r respond ing tothe e igenvalues I = 0 , /3 = 0 , Y = 0 , and the o ther to I = 1 , /3 = 0 , Y = 0 .Th e bes t -e s t ab l i shed non- t r iv i a l un i t a ry mul t ip l e t s a re i nd ica t ed in F igu res13.5, 13.6, 13.7 an d 13.8. In each case the figure on the rig ht ha nd sidei s t he quan t i t y m c 2, quo ted in MeV, where m i s t he ave rage re s t mass o fthe co r resp ond ing i sotop ic mu l t ip le t . (The m em bers o f an i so top ic m u l t ip l e tnecessar i ly l ie in the sam e horizonta l l ine in each of these f igures .) To each ofthe b a ryo n mu l t ip le t s { 8} and { 10} the re co r respond an t iba ryon s t rans fo rm ingas {8} and {10"} respect ive ly ({8} being ident ica l to i t s complex conjugate) .A t t he t ime t ha t t h i s scheme was p roposed a ll t he pa r t i c l e s o f t he ba ryo ndecup le t had a l ready been obse rved , excep t fo r t he ~ - . Th e subseq uen td i scovery o f t h i s pa r t ic l e w i th p rec i se ly the p red ic t ed q ua n tu m num bers (and ares t mass a s p red ic t ed by the Ge l l -Mann-Okubo mass fo rmula ) was a t r i umphfor the theory . In addi t ion to the hadro ns l i s ted in the f igures , there are a


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_1. .- I 2 i0 2 I

I

43

r 13

Figure 13.3: Values of /3 and Y for the i rreducible representat ion{6}.

- ~ - I 29 , ,

i_ _30 2 I z

- I 9

- 2

, , . __

Figure 13.4 : Values of /3 and Y for the irreducible represen tation{ 10}.


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n( u d d )

PI 9( u u d )

( d s s )

9 3 9

~ o(u s s )

I - I o ( u d s ) ~ - I I- I - ~ 0 ~ I I 1 9 3( d d s ) A~ (u d s ) (u u s ) 1 11 5

- I 1 3 1 8

I3

Figu re 13 .5 : Th e baryon octe t {S} wi th j = 89 a n d p a r i t y + . (T h e qu a rkcon ten t s a re in pa ren theses . The f igu res on the r igh t han d s ide give m c 2 (inMev ), w here m is the average m ass of the i so topic mul t ip le t to i t s le f t. )

A - A 09 @( d d d ) ( u d d ) 9 1 2 3 2(u u d ) (u u u )

3 ~ . - _ ! ~ . o I ~ - + ~ 1 3 8 5( d d s ) ( u d s ) ( u u s ) Z 5

,..~o- I 9 1 5 3 0(u s s )

" 2( s s s )

( d s s )

1 6 7 2

Figure 13 .6 : The ba ryon decup le t {10} wi th j = 3 and pa r i t y + . (The qua rkcon ten t s a re in pa ren theses . Th e f igu res on the r igh t ha nd s ide g ive m c 2 (inM ev), w here m is the average ma ss of the i so topic mu l t ip le t to i t s le f t. )


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- I ~(~

K o( d ~ )

I2,_

K w

( s S )

I

7tO~ 0

K . l -( u ~ )

( u ~ , d d )I

(uS,dd,s~,)

4 9 6

l ~ r + 13 7: ~ - I 3( u d ) 5 4 9

Ko- I 9 4 9 6( s d )

Fi g u re 1 3 . 7 : T h e me s o n o c t et {8 } w i t h j = 0 a n d p a r i t y - . (T h e qu a rkcontents are in parenthes es . The f igures on the r ight han d s ide g ive m c 2 (inMev), whe re m is the average mass o f the i so topic m ul t ip le t to i t s le f t. )num ber of s ingle ts (be longing to the i r reducib le rep rese nta t io n {1}) .

One po in t t ha t i s immed ia t e ly apparen t f rom Figu re 13 .1 i s t ha t fo r t hei rreducib le rep rese nta t ion {3} the va lues of Q are 2 1 and 1 i .e . t h e y, - 5 - 5a r e n o t i n t e g e r s . This is ac tual ly a specia l case of the ge nera l res u l t th a tthe e igenvalues Q for the uni tary mul t ip le t be longing to the i r reducib le rep-r e s e n t a t i o n r ( { n ~ , n 2 } ) a r e i n t eg e r s i f a n d o n ly i f (n l - n 2 ) /3 i s a n i n t e g e r .(The a rgu m en t i s t ha t , by Eq ua t ions (12 .9 ) and (12.10 ) , eve ry we igh t ~ inr({n ~, n2}) is of t he fo rm

2 2A -- n lA1 + n2A2 - q l a l - q2c~2 = ~ -~ k = l {~ -~ j = l n j ( A - t ) k J - qk }c ~ k ,so tha t , f rom Eq ua t ions (13 .16 ) and (13.18) ,

2Q = Z n j ( A - 1 ) l J - q l - - ( 2 / 3 ) n l + ( 1 / 3 ) n 2 - q l - - - ( 1 / 3 ) ( n l - n 2 ) + n l - q l .

j--1As n l , n2, ql , a nd q2 are al l integers, this e xpre ssion is an in tege r i f an d o nlyif ( n l - n 2 ) / 3 is a n in t eg e r .)

The mos t f ru i t fu l p roposa l fo r dea l ing wi th th i s obse rva t ion was made byOel l -Mann (1964) and Zweig (1964) , and i s tha t the par t ic les correspondingto the i r reducib le representa t ions {3} and {3*} do exis t , and are the basiccon st i tuents of a ll the observed hadrons . Ge l l -Ma nn (1964) ca l led the par t ic leso f t he {3} "quarks" , so tha t t hose o f t he {3*} become "an t iqua rks" . The

1 wh i l e t he an t i qu a rkss s u mp t i o n is t h a t t h e q u a rk s h av e b a ry o n n u m b e r B =


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p -- I(dS)

K .O( d ~ )

_ !2

K ~-(sS)

K ~ I . 8 9 2( u { )

p O ( u O , d d )i P * 7 7 00 , ~ 2 I( u u , d d ,s ~ ) ( u d ) 7 8 3 Z3

~ , o- I =_ 8 9 2( s d )

Figure 13 .8" The meson oc t e t {8} wi th j = 1 and pa r i t y - . (The qua rkcon tents are in parentheses . Th e f igures on the r ight han d s ide g ive m c 2 (inMe v), w here m is the av erage mass of the i so topic m ul t ip le t to i t s lef t .)

1 Th e th ree qu a rks a re now usua l ly ca l led the u , d ando r r e s p o n d t o B = - 5 "s quarks (u corresponding to i so topic sp in "up" , d to i so topic sp in "down",and s t o non-ze ro s t rangeness ) , and the a ssoc i a t ed an t i qua rks a re deno ted byfi, d and $. The pro pert ies of the q uark s are sum m ariz ed in Table 13.2.

In t he s imp les t mode l t he m e s o n s are m ade of q~ pai rs ( i.e. qu ark a ndan t iq ua rk pai rs) . As (10 .38) shows th a t D 1/ 2 | 1 /2 ~ D 1 ~ two par t ic les1 combine to p roduce compos i t e s w i th sp in 1 and sp in 0 .i th in t r ins ic sp inM ore ove r, as note d in (1 2.18), for A2 {3} | {3*} ..~ {8} @ {1}, so th a t th eqc7 pai rs t ra nsfo rm as the {8} and the {1}. Th is expla ins ver y neat ly th eobse rva t ion tha t t he re ex i s t su (3 ) meson oc t e t s and s ing le t s w i th bo th sp in 1and sp in 0 .

For b a r y o n s t he s imp les t a ssum pt ion is t ha t each ba ryon cons i st s o f t h reequ arks (and so each an t iba ry on cons i s ts o f t h ree an t i qua rks ) . As th ree pa r t i -1 3cles wi th in t r ins ic sp in ~ couple to produce a composi te wi th in t r ins ic sp in

qu a rk B I /3 Y S Q1/3 1/2 1/2 1/3 0 2//31 //3 1 / 2 - 1 / 2 1 /3 0 - 1 / 31//3 0 0 - 2 / 3 - 1 - 1 / 3

Tab le 13.2" Q ua n tu m num bers o f t he q ua rks u , d and s .


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268 G R O U P T H E O R Y I N P H Y S I C So r 8 9 because , b y Equat ion (10 .38) ) ,D 1/2 | D 1/2 | D 1/2 ( D 1 G D ~ | D 1/2 ~ (D ~ @ D 1/2) 9 (D O D 1/2)

(D3/2 @ D 1/2) @ D 1/2,an d, as wa s not ed in (12.18), for A2 {3} | {3} | {3} ~ {10} @ 2{8} @ {1}, thisprov ides a s imple exp lana t ion o f the ex i s tence o f baryon o c te t s o f sp in 89a n d3bary on d ecuple t s o f sp in ~ .The quark con ten t s sugges ted by the cons idera t ions a re ind ica ted in F ig-ures 13.5, 13.6, 13.7 and 13.8.W he n the u n i t a ry sp in pa r t s o f the s t a t e vec to r s fo r the b aryons a re in -ves t igated along the l ines indicated above for mesons , one very s igni f icantfea tu re em erges. I t can shown tha t the t r ip le p roduc t s o f {3} basi s vec to r sthat form bas is vectors for the {10} are s y m m e t r i c wi th r espec t to the in te r -chan ge of indices . Also, as D 3/2 correspo nds to the highe s t weight ap pe ar in gin D 1/2 | D 1/2 | D 1/2 th e intr in sic spin p ar t of the s ta te ve ctors for the 3' 2sp in compos i t es a re s y m m e t r i c produc t s o f the sp in par t s o f the cons t i tuen t s .As the gen era l ized Pau l i Exc lus ion Pr inc ip le s t a t es tha t f e rmion s t a t e vec to r sm u s t b e a n t i s y m m e t r i c with respect to in terchanges such as these, i t fo l lowstha t , i f t he on ly d i s t ingu i sh ing l abe l s fo r the quarks a re those a l r eady in t ro -3duced , then the o rb i t a l par t o f the th ree-quark wave func t ions fo r the sp in -~d ecu p l e t b a r y o n s m u s t b e a n t i s y m m e t r i c . W hile th is .is no t impo ssible, i t iscon t ra ry to exper ience wi th g ro und s t a t e conf igura tions in o ther systems . Thed i l em m a can b e av o i d ed b y m ak i n g t h e f u r t h e r a s s u m p t i o n t h a t e ach o f t h eth ree quarks u , d and s comes in three var ie t i es tha t a re d i s t ingu i shed by afur th er featu re , which is cal led "colour" . ( I t wi ll be app recia ted th at th is ispure ly a m at t e r o f t e rmino logy , and th a t i t has no th ing to do wi th "colour" inthe norm al s ense o f the word . ) I f each o f the th ree q uarks o f a sp in -3 decup le thas a d i f fe ren t co lour, then the in te rna l sym m et ry par t o f the s t a t e vec to r isno longer symmet r i c , and so the p rob lem wi th the o rb i t a l par t does no t a r i s e .Th i s idea fo rms the bas is o f the "SU(3) co lour symm et ry scheme" and thenceo f "q u a n t u m ch r o m o d y n am i cs " . I n th i s sch em e t h e s t ro n g i n t e rac t i o n tak esp lace th rough the exchange of 8 "g luons" , which be long to the i r r educ ib lerepre sen ta t ion {8} of the SU(3) co lour g roup .

Thi s in t roduc t ion wi l l be conc luded by no t ing tha t i t has p roved veryf ru i tfu l to ex tend th e above co ns idera t ions in var ious d i r ec t ions . Th e m os ts t r a igh t fo rward genera l i za t ion , f rom su(3) to su (4 ) , p roduces a s cheme wi tht h e ad d i t i o n a l q u an t u m n u m b er " ch a r m " . T h e m o r e s o p h i s ti c a ted s u g g es ti o ntha t symmet ry b reak ing i s " spon taneous" in o r ig in g ives r i s e to p rob lemswithin "global" schemes (Golds tone 1961, Golds tone et al. 1962). However,as was shown by Higgs (1964a,b , 1966) , when incorporated in a gauge theory(Yang a nd Mil ls 1954 , an d Shaw 1955) these di f ficul ties not only disap pea rbu t p erm i t m ass genera t ion o f the in te rm edia te par t ic l es , thereb y a l lowingthe cons t ruc t ion o f a un i f i ed theory o f weak and e lec t romagnet i c in te rac t ions(c.f . Sa lam 1980, W einberg 1980, and Glashow 1980), based on a u(1) G su(2)a lgebra .

Symmetry Schemes for the Elementary Particles

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