CLASSIFYING BARYONS

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CLASSIFYING BARYONS

D. Faiman

To cite this version:

D. Faiman. CLASSIFYING BARYONS. Journal de Physique Colloques, 1973, 34 (C1), pp.C1-167-

C1-170. �10.1051/jphyscol:1973118�. �jpa-00215198�

CLASSIFYING BARYONS

CLASSIFYING BARYONS D. FAIMAN+

CWN, Geneva, S w i t z e r l a n d

1.- AN UNSYMMETRY. - For a number o f y e a r s now t h e c o l l i n e a r symmetry ~ ~ ( - 6 concocted by L i p k i n and 1 ~ Meshkov [ I 1 t o overcome t h e non-covariance of SU(6) -

h a s enjoyed an undeniable measure o f s u c c e s s i n i n t e r - r e l a t i n g t h e v a r i o u s observed hadronic decay r a t e s . T h i s i s most remarkable s i n c e t h e lowest o r d e r t h r e e p o i n t v e r t e x i s t h e only p r o c e s s t h a t c a n f u l f i l l t h i s c r i t e r i o n of c o l l i n e a r i t y and t h e r e i s c e r t a i n l y no r e a s o n t o b e l i e v e t h a t such v e r t i c e s dominate s t r o n g i n t e r a c t i o n s . True, t h e observed spectrum of hadrons seems t o f a l l i n t o SU(6)W m u l t i p l e t s i n a most remarkable manner [z], b u t t h i s i n i t s e l f i s no r e a s o n t o e x p e c t t h e symmetry t o apply a t v e r t i c e s (c. f . t h e O(4) symmetry of t h e hydrogen spectrum b u t n o t of i t s s c a t t e r i n g s t a t e s [?,I).

S u ( 6 ) used i n t h e above manner has had W i t s f a i - l u r e s too. Perhaps t h e most s p e c t a c u l a r being i t s p r e d i c t i o n t h a t t h e B-meson should n o t decay i n t o

v i a t h e h e l i c i t y 5 1 s t a t e s - t h e dominantly obser- ved p o l a r i z a t i o n [ 4 ]

Rosner L.51 s p o t t e d t h a t t h e problem h e r e seems t o be t h e bZ

p r o p e r t y of SU(6Iw 8 O(2)Lz which h a s t h e e f f e c t on t h o s e decay p r o c e s s e s t h a t c a n proceed v i a two f i n a l s t a t e o r b i t a l a n g u l a r momenta 8 (e.g. t h e s-wave and d- wave amplitudes o f B +a) o f imposing a r e l a t i o n -

s h i p on t h e amplitudes of d i f f e r e n t 8 . ^{I t} i s t h i s r e l a t i o n s h i p t h a t i n t h e c a s e of B

l e a d s t o t h e wrong p o l a r i z a t i o n . I f t h e f u l l symmetry could be broken by t h e i n t r o d u c t i o n of a A L ~ = + ^{1 ampli-}

t u d e i t might s t i l l be p o s s i b l e t o r e t a i n t h e appea- r e n c e of SU(6IW symmetry among a l l decays of t h e same 8 b u t t o have t h e l i n k between d i f f e r e n t

broken. Such a n "e-broken" symmetry does indeed appear t o be r e a l i z e d by n a t u r e [5,6,7 1.

But look what we have done. We s t a r t e d w i t h a n ad hoc symmetry having no t h e o r e t i c a l f o u n d a t i o n : we

broke i t i n a h i g h l y a r b i t r a r y manner. Now we a s s e r t t h a t n a t u r e seems t o r e s p e c t t h e r e s u l t i n g unsymmetry.

+

a d d r e s s a f t e r t h i s Conference, Weizmann I n s t i t u t e , Rehovot, I s r a e l .

2.- CONSTITUENT QUARKS FOR CURRENT QUARKS. - ^Such

grubby phenomenology seems f a r removed from t h e uto- p i a n world of c u r r e n t a l g e b r a . T h i s l a t t e r abounds with well-defined, p h y s i c a l l y measurable o p e r a t o r s such a s t h e a x i a l v e c t o r and v e c t o r c h a r g e s which mediate weak and e l e c t r o m a g n e t i c t r a n s i t i o n s . I t i s of c o u r s e t r u e t h a t t h e c u r r e n t d e n s i t i e s whose space i n t e g r a l s t h e s e c h a r g e s a r e , c a n be e x p r e s s e d i n terms of f i c t i c i o u s quark f i e l d s :

and i t i s p e r f e c t l y t r u e t h a t one c a n e n l a r g e t h e a l z e b r a t o U(12) by i n c l u d i n g a l l s i x t e e n D i r a c c o v a r i a n t s r :

Moreover i t

t r u e t h a t i n t h e i n f i n i t e momentum frame 35 o u t of t h e s e 144 o p e r a t o r s - t h o s e having t h e s o - c a l l e d "good" l i g h t - c o n e p r o p e r t i e s - ^do

g e n e r a t e a n SU(6)W a l g e b r a o f c u r r e n t s

:

b u t t h i s SU(6)W i s a completely d i f f e r e n t symmetry from t h e group which i s u s e d t o c l a s s i f y hadrons.

Indeed i t s g e n e r a t o r s do n o t even commute w i t h t h e f r e e quark Hamiltonian

The q u e s t i o n t h u s a r i s e s a s t o whether t h e s e two

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1973118

C1-168 D. FAIMAN

SU(6) w groups a r e r e a l l y d i f f e r e n t , s a y i n t h e way t h a t SU(2)isospin i s d i f f e r e n t from SU(2)paull spin O r , i f n o t , i s t h e r e a u n i t a r y t r a n s f o r m a t i o n t h a t t a k e s one from t h e w e l l - d e f i n e d SU(6)W of c u r r e n t s t o t h e symmetry group of t h e hadron spectrum - t h e SU(6)W o f c o n s t i t u e n t s . Melosh [91 d i s c o v e r e d t h a t i f one d e f i n e s a n o p e r a t o r

V z exp

i Y)

t h e n one can c o n s t r u c t 35 q u a n t i t i e s Wa :

5 6 , L = 0 f o r n u c l e o n s . The most remarkable property

-

o f V however i s t h a t a l t h o u g h v I ~ ) i s i n f i n i t e l y c o m p l i c a t e d , V-I Fi5 V i s i n f i n i t e l y s i m p l e

I t h a s p r e c i s e l y two p i e c e s : one l o o k i n g l i k e a p i o n i n t h a t i t t r a n s f o r m s under t h e S U ( ~ ) ~ o f c o n s t i - t u e n t q u a r k s a s t h e 1 3 5 , LZ

01 r e p r e s e n t a t i o n ;

-

t h e o t h e r p i e c e l o o k i n g l i k e a n A1 -meson, i . e . t h e

135, LZ = 2 1 1 r e p r e s e n t a t i o n of . Under t h e weaker c h i r a l SU(3)

SU(3) t h e s e two terms t r a n s - form r e s p e c t i v e l y a s l ( 8 , l ) o , ~ ] -

, S ) ~ , O ~ and

!(3,?)1 ,-I 1 - 1(T,3)-1 , I 1 .

I t i s of c o u r s e i m p o r t a n t n o t t o l o s e s i g h t o f t h e W: = V

V-I , f a c t t h a t t h e s e s t a t e m e n t s a r e e x a c t only i n t h e which d o i n d e e d commute w i t h Hfree , and which there- f r e e - q u a r k model. I t was however s u g g e s t e d by Melosh f o r e q u a l i f y a s t h e g e n e r a t o r s of a spectrum symmetry. that t h e

defined here f o r t h e f r e e quark s i t u a t i o n , might n o t be t o o d i f - 3.- SOME NOT-SO-GRUBBY PHENOMENOLOGY. - C o n s i d e r f e r e n t from t h o s e of t h e t r u e t r a n s f o n n a t i o n t h a t t h e commonly e n c o u n t e r e d element of the

Would a p p l y t o t h e " r e a l " world of i n t e r a c t i n g quarks.

v e c t o r c h a r g e between two h a d r o n s t a t e s a t i n f i n i t e I f t h i s i s t h e c a s e t h e n one h a s v i a t h e PCAC hypo-

monentum t h e s i s a t h e o r e t i c a l m o t i v a t i o n f o r t h a t e x t r a

( B ~ F ~ ~ ~ A > . ^{AL = 2} 1 a m p l i t u d e o f Rosner, d i s c u s s e d above. T h a t i s t o s a y , t h e Melosh-transformed a x i a l c h a r g e and I t i s w e l l known

[ A ) i s a n u c l e o n s t a t e the phenomenoiogical 3po model [ 5 , 6 1 b o t h y i e l d t h e n i n t h i s r e p r e s e n t a t i o n i t h a s t o be a f a r more

two i n d e p e n d e n t S U ( E ~ ) ~ a m p l i t u d e s . A p p l i c a t i o n of c o m p l i c a t e d o b j e c t t h a n t h e 56, L = 0 o f i t s cons-

t h e s e i d e a s t o p i o n i c t r a n s i t i o n s among t h e b a r y o n s t i t u e n t quark c l a s s i f i c a t i o n . Indeed t h e s t a t e pro-

( w i t h i n t h e c o n t e x t o f t h e Melosh formalism) h a s duced by t h e o p e r a t i o n

been v i g o r o u s l y pursued by t h e a u t h o r s o f r e f s . [ l o ,

~ 1 2 ) 11,121 and [131 . (Refs. 1 4 , 1 5 and 16 examine e l e c - c a n e a s i l y be shown [ 9 I t o c o n s i s t of an i n f i n i t e t r o m a g n e t i c t r a n s i t i o n s where i t i s found t h a t t h e m i x t u r e of SU(6)W

0 ( 2 ) L - r e p r e s e n t a t i o n s i f 1;) r e l e v a n t Melosh-transformed t r a n s i t i o n o p e r a t o r h a s

L

i t s e l f s t a r t s o u t a s pure 56, L = 0 . N e v e r t h e l e s s f o u r i n d e p e n d e n t terms).

.U

V I A ) i s a bona-fide

S U ( 6 ) ~ , c u r r e n t s s t a t e ( f r o m Thus w i t h renewed e n t h u s i a s m f o r t h a t unsymmetry t h e d e f i n i t i o n of V) and h e n c e t h e above m a t r i x e l e - "&broken SU(~),,," let u s t u r n o u r a t t e n t i o n t o t h e

rnent can be r e - w r i t t e n new Berkeley-SLAC

x A phase s h i f t s o l u t i o n

(S~V-' ^F~~ v (i) , t h a t P r o f e s s o r R o s e n f e l d [I?] h a s p r e s e n t e d a t t h i s where 12) and Iz) now r e p r e s e n t t h e c o n s t i t u e n t Conference. Our f i g u r e i s s i m i l a r t o one which appea- q u a r k c l a s s i f i c a t i o n of t h e hadyons i n q u e s t i o n , e . g . r e d i n r e f e r e n c e b u t modified s o t h a t t h e

T a b l e I P a r t i a l wave

21

('in l o u t

Associated r e s o n a n c e

Assignment

S u ( 6 ) w , L (SU(3),SU(2))

" a n t i - ~ ~ ( 6 ) ~ " s o l u t i o n

C l a s s of p r e d i c t i o n

P P l l DS13 DD13 SD31 DS33 DD33 DD15 FP15 FF15 PP33 SDll DS13 P P l l FF35 PP31 Fm

N(1470) ~ ( 1 5 2 0 ) A(1650) A(1670)- ~ ( 1 6 7 0 ) ~ ( 1 6 9 0 ) ~(l690)~(1700)~(1730)N(1750)~(1~)4(1~0)~(1960)

-

70 ^{7 0} z! ^{7 0} - ⁵⁶ - 56 702_070l35656

L-0 L Z l

L - 1 L Z l L Z 2 L = o L = l L = l L = o L = 2 L = 2 L : 2

( 8 , 2 ) ( 8 , 2 ; ( 1 0 , 2 ) (10,Z) (8.4) ( 8 , 2 ) ( 1 0 , 4 ) ( 8 , 4 ) ( 8 , 4 ) ( 8 , 2 ) ( 1 O p 4 ) ( 1 o , 4 ) ( 1 q 4 )

6 ^Q

^Q

^Q

6 6 6 ⁰

6 6 6 ^Q ? ^Q ? 6 3

3 2 2 1 1 1 1 2 2 3 2 2 3 2 3 1

CLASSIFYING BARYONS C1-169

new e x p e r i m e n t a l s o l u t i o n c a n b e compared w i t h e x p e r i m e n t . T h e r e a r e now s i x t e e n XN

a m p l i t u d e s t h a t show some s i g n o f r e s o n a n c e . T h e s e a r e d e p i c t e d b y c r o s s e s when t h e s i g n o f t h e a m p l i t u d e i s c l e a r , a n d by q u e s t i o n m a r k s when b a c k g r o u n d e f f e c t s p r o - d u c e a n y a m b i g u i t y i n t h e i n t e r p r e t a t i o n . T h e t h e o - r e t i c a l p r e d i c t i o n s a r e i n d i c a t e d b y s i n g l e a r r o w s o r d o u b l e a r r o w s a c c o r d i n g as t h e i n c o m i n g a n d o u t - g o i n g o r b i t a l a n g u l a r momenta a r e t h e same o r d i f f e - r e n t r e s p e c t i v e l y . The d i s t i n c t i o n h e r e i s t h a t s i n - g l e a r r o w p r o c e s s e s a r e n o t c a p a b l e o f d i s t i n g u i s h i n g a s t o which o f t h e two Melosh a m p l i t u d e s d o m i n a t e s . T h e d o u b l e a r r o w p r o c e s s e s , o n t h e o t h e r h a n d , a r e s e n s i t i v e t o t h i s d i s t i n c t i o n a n d f o r c o n v e n i e n c e t h e s e h a v e a l l b e e n set i n t h e ' ' a n t i - ~ ~ ( 6 ) ~ ' ' p o s i - t i o n i . e . t h a t p e r t a i n i n g t o AL = 2 1 dominance.

A g a i n as i n r e f e r e n c e [ 1 8 ] we a t t a c h t h r e e d e g r e e s o f i m p o r t a n c e t o o u r p r e d i c t i o n s . C l a s s

p r e d i c t i o n s are unambiguous i n t h a t t h e p a r t t c l e a s s i g n m e n t i s u n i q u e . C l a s s 2 p r e d i c t i o n s c o u l d i n p r i n c i p l e b e a l t e r e d by c o n f i g u r a t i o n m i x i n g , b u t t h e e f f e c t o f s u c h m i x i n g i s c o n s i d e r e d t o be r e a s o n a b l y w e l l un- d e r s t o o d a n d h a s b e e n t a k e n i n t o a c c o u n t f o r t h e s e p r e d i c t i o n s [ 7 , 1 9 ] . I n a l l s u c h c a s e s i t t u r n s o u t t h a t m i x i n g a l t e r s o n l y t h e m a g n i t u d e s b u t n o t t h e p h a s e s f o r t h e a s s i g n m e n t s i n d i c a t e d . L a s t l y , c l a s s 3 p r e d i c t i o n s r e p r e s e n t e d u c a t e d g u e s s e s a t c l a s s i f i - c a t i o n . M i x i n g c o u l d c e r t a i n l y c h a n g e t h e s e l a t t e r p r e d i c t i o n s , b u t we h a v e made t h e s i m p l e s t p o s s i b l e m u l t i p l e t a s s i g n m e n t s b a s e d on t h e o b s e r v e d

'id p h a s e s . They a r e t h u s n o t s o much p r e d i c t i o n s as i n t e r p r e t a t i o n s .

A p a r t Tcom t h e f a c t t h a t t h e r e a r e now n o d i s a - g r e e m e n t s b e t w e e n t h e o r y a n d e x p e r i m e n t ( c . f . r e f . [ 1 8 1 w h e r e t h i s was n o t s o ) t h e f o l l o w i n g s i x p o i n t s a r e o f s p e c i a l i n t e r e s t .

1. T h e 70, L'- 1 - s t a t e s a r e d o m i n a t e d by t h e AL = 2 1 t r a n s i t i o n a m p l i t u d e .

z

2. B a s e d o n l y o n t h e FP15 wave i t a p p e a r s t h a t t h e 56, ^L'

m u l t i p l e t i s d o m i n a t e d by t h e AL = ⁰ a m p l i t u d e .

z

3.

An a r g u m e n t p r o p o s e d i n r e f e r e n c e [ l g ] t o e x p l a i n t h e n o n - o b s e r v a t i o n o f t h e l T 3 5 wave i n

A ( l 8 9 0 )

Ax r e m a i n s v a l i d s o l o n g as t h e 70, L' x

m u l t i p l e t a l s o c h o o s e s AL z =

. N o t i c e

t h a t p o i n t s 1 , 2 a n d 3 are a l l i n a c c o r d w i t h t h e p r e d i c t i o n o f B u c c e l l a e t al. [ 2 1 1 who p r o p o s e a m o d e l , i n many w a y s e q u i v a l e n t t o t h e scheme o f Melosh, t h a t r e q u i r e s a l l e v e n L m u l t i p l e t s t o d e c a y t o t h e g r o u n d s t a t e v i a AL z =

t r a n s i t i o n s , and a l l odd L m u l t i p l e t s t o c h o o s e ALZ = 1 .

4. T h e R o p e r r e s o n a n c e p h a s e c l e a r l y f a v o u r s t h e 5 6 , L P

a s s i g n m e n t o v e r t h e c o m p e t i n g 70,

- P + P +

L = ^{0 and 70, L} = ² . B o t h o f t h e l a t t e r ' s P P l 1 s t a t e s would h a v e t h e o p p o s i t e p h a s e t o t h a t o b s e r y x l .

5. T h e s e c o n d P I 1 r e s o n a n c e ~ ~ ( 1 7 5 0 , 1 / 2 + ) l o o k s a s i f i t h a s t h e o p p o s i t e p h a s e t o t h e Roper.

I f t h i s i s t h e c a s e t h e n i t c a n not b e a t h i r d 56,

P + P +

L r 0 . Of t h e r e m a i n i n g p o s s i b i l i t i e s , 7 0 , L =O i s f a r more l i k e l y t h a n 70, L P = ²⁺ s i n c e t h e l a t - t e r i s e x p e c t e d t o b e f a r t o o w e a k l y c o u p l e d t o t h e

N a n d t h e x A c h a n n e l s t o

v i s i b l e [ 2 0 1 .

6 . T h e l o n g - s o u g h t A X p a r t n e r t o t h e R o p e r N*

a p p e a r s t o h a v e t u r n e d up. H e r e a g a i n t h e p h a s e o f t h e PP33 wave f i t s t h e 56, L' = ^O+ a s s i g n m e n t b u t n o t t h a t o f t h e c o m p e t i n g 56, L' =

m u l t i p l e t .

4.- CONCLUSIONS. - A t t h e p r e s e n t s t a g e o f i t s d e v e l o p m e n t i t i s c l e a r t h a t t h e Melosh t r a n s f o r m a - t i o n a s a p p l i e d t o b a r y o n s p e c t r o s c o p y g o e s a con- s i d e r a b l e way t o w a r d s p r o v i d i n g a t h e o r e t i c a l foun- d a t i o n f o r some a l r e a d y s u c c e s s f u l q u a r k model a n d symmetry i d e a s . lYhethcr

w i l l go a l l t h e way, and/

o r w h e t h e r i t w i l l f i n d u s e f u l a p p l i c a t i o n w h e r e t h e more s i m p l e minded a p p r o a c h e s f a i l i s f o r t h e f u t u r e t o d e c i d e .

5.- ACKNOWLECGEMENTS. - T h e f o r m a l i s m i n s e c t i o n

i s a h i g h l y i n a d e q u a t e summary o f tile e x c e l l e n t d i s - c u s s i o n i n r e f e r e n c e 1 3 . I am i n d e b t e d t o Tony Hey, J o n R o s n e r and J a c q u e s Weycrs f o r numerous d i s c u s - s i o n s a n d t o A r t R o s e n f e l d f o r p r e p u b l i c a t i o n d e t a i l s o f t h e new B e r k e l e y - SLAC

p h a s e s h i f t s s o l u t i o n .

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