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SU(3) isoscalar factors
~
Nuclear Physics B2 (1967) 697-712. North-Holland Publ. Comp., A m s t e r d a m
SU(3) ISOSCALAR FACTORS* C. - K . C H E W Physics Department. McGill University. Montreal. Canada and R. T . S H A R P ** Department of Physics. University qf lllinois. Urbana. Illinois Received 24 April 1967
Abstract: The general non-orthonormalized SU(3) i s o s e a l a r factor is e x p r e s s e d as a six-fold sum involving a t r i p l y - s t r e t c h e d 15-j coefficient which contains a quadruple sum. The s y m m e t r y p r o p e r t i e s of the s y m m e t r i c i s o s e a l a r factor and its relation to the a s y m m e t r i c i s o s c a l a r factor a r e discussed. Two o r t h o n o r m a l i z a tion s c h e m e s a r e discussed which diagonalize respectively a metric o p e r a t o r and a mixed C a s i m i r o p e r a t o r .
1. I N T R O D U C T I O N S i n c e t h e d i s c o v e r y of SU(3) s y m m e t r i e s in n u c l e a r a n d p a r t i c l e p h y s i c s t h e r e h a s b e e n m u c h i n t e r e s t in t h e g r o u p a n d in i t s C l e b s c h - G o r d a n c o e f f i c i e n t s . A u t h o r s h a v e p r e s e n t e d t a b l e s of t h e m o s t c o m m o n l y u s e d i s o s c a l a r f a c t o r s [1, 2] a n d h a v e g i v e n f o r m u l a s of v a r y i n g d e g r e e of c o m p l e x i t y and generality [3-11]. Hou [9] f o l l o w s t h e v a n d e r W a e r d e n a p p r o a c h , c o n s t r u c t i n g a s c a l a r w h i c h i s a p o l y n o m i a l in t h e v a r i a b l e s of t h r e e " o n e - p a r t i c l e " s p a c e s ; P o n z a n o [10] e x p a n d s s u c h a s c a l a r in p r o d u c t s of one p a r t i c l e s t a t e s and d e t e r m i n e s t h e g e n e r a l i s o s c a l a r f a c t o r f r o m t h e c o e f f i c i e n t s . He a v o i d s t h e tracelessness conditions by generating the general one-particle state from two i n d e p e n d e n t q u a r k (1, 0) b a s e s ; he e v a l u a t e s t h e g e n e r a l n o n o r t h o n o r m a l i z e d i s o s c a l a r f a c t o r in t e r m s of s p e c i a l 2 1 - j s y m b o l s of SU(2). H i s f o r m u l a h a s 17 s u m s . S h a r p a n d von B a e y e r [11] (we w i l l r e f e r to t h i s p a p e r a s I) d e r i v e a formula for the general isoscalar factor with fifteen summations, synthes i z i n g it out of s i m p l e r s p e c i a l i s o s c a l a r f a c t o r s . T h e i r o n e - p a r t i c l e s t a t e s a r e g e n e r a t e d f r o m one q u a r k a n d one a n t i q u a r k (0, 1) b a s i s ; t h e t r a c e l e s s n e s s c o n d i t i o n i s i m p o s e d b y d i s c a r d i n g a s u b s p a c e of " u n w a n t e d " s t a t e s . * This work was supported by the National R e s e a r c h Council of Canada and the Dep a r t m e n t of P h y s i c s , University of Illinois. ** On leave f r o m MeGill University, Montreal, Canada.
698
C. - K . C H E W a n d R . T . S H A R P
K l e i m a [12] has a l s o d e s c r i b e d q u a r k - a n t i q u a r k b a s i s p o l y n o m i a l s . In this p a p e r we follow P o n z a n o in c o n s t r u c i n g a s c a l a r p o l y n o m i a l in t h r e e o n e - p a r t i c l e s p a c e s , but we u s e the q u a r k - a n t i q u a r k v a r i a b l e s of I. The i s o s c a l a r f a c t o r s a r e e x t r a c t e d by p r o j e c t i n g on p r o d u c t s of o n e - p a r t i c l e s t a t e s ; this is m o r e d i r e c t than the synthetic method of I and y i e l d s a s i m p l e r g e n e r a l f o r m u l a , with ten s u m s . The t r a c e l e s s n e s s condition is not a c o m plication, for the o n e - p a r t i c l e s t a t e s on which the s c a l a r is p r o j e c t e d contain no unwanted s t a t e s . I s o s p i n m a n i p u l a t i o n s play an i m p o r t a n t r o l e in the calculation. An o r t h o g o n a l i z a t i o n s c h e m e b a s e d on m i x e d C a s i m i r o p e r a t o r s , d e ; s c r i b e d e a r l i e r by us [13], is shown to apply with little modification to the present approach.
2. SYMMETRIC AND ASYMMETRIC ISOSCALAR FACTORS Much of the implicit form. in our p r e s e n t scalar factors, the f o r m e r .
content of this s e c t i o n m a y be found e l s e w h e r e , at l e a s t in We thought it worthwhile to collect the r e s u l t s in one p l a c e notation. We define s y m m e t r i c and a s y m m e t r i c SU(3) i s o r e l a t e t h e m to each other, and exhibit the s y m m e t r i e s of
We use the o n e - p a r t i c l e s t a t e s [P T q } defined by eq. (3.1) of I; they a r e p o l y n o m i a l s of d e g r e e p ; q in s i x independent v a r i a b l e s ~ 77~ ;4" 77* {*, and s a t i s f y the t r a c e l e s s n e s s condition by v i r t u e of t h e i r orthogonality to the unwanted s t a t e s of the s a m e d e g r e e N G ( ~ *
+ 7777* + ~ . ) G
I P-G n TqM- G ) ; h i s
the n u m b e r of r o w s above that containing the h e a v i e s t s t a t e in the hexagonal weight d i a g r a m and is r e l a t e d to h y p e r e h a r g e Y by n = Y + ~ ( q - p ) . The p h a s e convention is that of de Swart [1] and of H a r v e y and Elliott [4], d e s c r i b e d c o r r e c t l y in I but i n c o r r e c t l y a t t r i b u t e d to B i e d e n h a r n t h e r e ; the m a t r i x e l e m e n t s of R± (the -K:~ of de Swart) a r e negative; in the B i e d e n h a r n convention, u s e d by Ponzano [10] the r e d u c e d m a t r i x e l e m e n t s of R e a r e $
negative. Our s t a t e I P,~ T q~) d i f f e r s f r o m the c o r r e s p o n d i n g s t a t e of P o n z a n o 2V~
by a p h a s e (-1)½(P+q+n)-T. We define a s c a l a r p o l y n o m i a l in t h r e e s e t s of v a r i a b l e s Su = _~ p BUl2 ~u21 ~u13 ~u31 ~u23 ~u32 (:s u 12 --21 --13 --31 --23 --32 - '
(2. la)
S u = N P B u 1 2 ~ u 2 1 R u l 3 R u 3 1 ~ u 2 3 ~ u32 ~ * s u 12 --21 --13 --31 --23 --32 v ,
(2.1b)
or
where
SU(3) ISOSCALAR FACTORS
699
~1 ~2 ~3 C=
~?1 772 7?3
,
(2.3)
~1 ~2 ~3 and C* is C with the replacements ~ --~ ~* etc. The operator P is an instruction to project out unwanted states; this operation is achieved automatically by projecting S u later on one-particle states. N u normalizes S u to unity; it is evaluated in appendix B. Our Su, eqs. (2.1a) and (2.1b) correspond to Ponjano's invariant I given by his eqs. (38) and (39) respectively. From the degree of S u in starred and unstarred variables one finds Pl =u12 +u13 + s ,
ql =u21 +u31 ,
P2 =u21 +u23 + s ,
q2 =u12 +u32 '
P3 =u31 +u32 + s ,
q3 =u13 +u23 '
Pl = u12 + u13 '
ql =u21 +~31 + s ,
P2 = u21 + u23 ,
q2 =u12 +u32 + s ,
s = ~1 ~. (Pi - q i ) z (2.4a)
Or
s = ~.
(qi-Pi) ,
(2.4b) q3 =u13 +u23 + s ' P3 = u31 + u32 ' for Su given by eq. (2.1a) or (2.1b). It is apparent that increasing u12 , u23 , u31 and decreasing u21 , u32, u13 by the same integer leaves the p and q unchanged; two Su whose u are thus related are called degenerate; the number of S u in a degenerate set is determined by the restriction that the u are non-negative integers; besides the p and q any one of the six u can be used to specify S u uniquely. We expand S u in one particle states ~11
Su =
P2
q2)
P3
q3
nln2M1M 2 T 1T2 T3
× (Pl
ql.
P2
q2.
n 1 T1M 1 ' n2 T2 M2 '
P3
q3.
u)
n3 T3 M 3 '
(2.5) ,
defining thereby an SU(3) Wigner coefficient P2 q2 -P3 q3. u ) =
( P l Tq l1'. n1
P2 q2. P3 q3. u~] \( MT 11 T 2 T 3
n 2 T 2' n 3 T 3'
)
M2 M 3 -
(2.6) '
700
C. - K . C H E W a n d R. T . S H A R P
which f a c t o r s by v i r t u e of the W i g n e r - E c k a r t t h e o r e m f o r isospin into a s y m m e t r i c i s o s c a l a r f a c t o r and an SU(2) Wigner coefficient, shown in the second line of eq. (2.6). The M s a t i s f y ZiMi = 0 and the n satisfy Zin i =
"-:i(qi - 1,i)
: ±s.
The s y m m e t r i e s of the i s o s c a l a r f a c t o r a r e a p p a r e n t f r o m eqs. (2.6) and (2.1). Interchanging two of the t h r e e s p a c e s m u l t i p l i e s S u by (-1) s and a c cording to the s y m m e t r y p r o p e r t y of the SU(2) Wigner coefficient m u l t i p l i e s the i s o s c a l a r f a c t o r by (-1)s+T1 +T2+T3. Another s y m m e t r y d e a l s with r e v e r s i n g " m a g n e t i c " quantum n u m b e r s n and M. It can be shown that
P
q)*
nTM
:
-
q
P \
T-M)
(-
1)½(p-q-n)-M
,
(2.7)
w h e r e the a s t e r i s k m e a n s substitute ~ ..... ~* etc. Putting a s t a r on (1; 2; 3 IS) in (2.6) does not change its value, s i n c e the v a r i a b l e s a r e all d u m m i e s . But it induces the r e p l a c e m e n t s p'--" q, n ...... n, M - , -M, uij ..... uji and i n t r o duces a factor
(_l)'-'i[~(Pi - qi - hi) - Mi] : (_1) 2s : 1. By the s y m m e t r y of the SU(2) Wigner coefficient the i s o s c a l a r f a c t o r changes by a p h a s e (-1)T,+T:+T3 o n r e v e r s i n g n , interchanging p with q and uij with uji. We need c o n s i d e r only eq. (2.1a) f r o m now on, since this t r a n s f o r m a t i o n int e r c h a n g e s it with (2.1b). We have s e a r c h e d u n s u c c e s s f u l l y f o r s y m m e t r i e s of the g e n e r a l SU(3) i s o s c a l a r f a c t o r analogous to the Regge s y m m e t r i e s of the SU(2) Wigner c o e f f i Cient; however by e m b e d d i n g SU(3) in SU(4) in d i f f e r e n t w a y s one could find identities s i m i l a r to the g e n e r a l i z e d Regge identities s a t i s f i e d by SU(2) Wign e r c o e f f i c i e n t s [14]. The a s y m m e t r i c i s o s c a l a r f a c t o r
p l q l . p2q2 n l T ' n2T 2
p3q3 u -n3T 3 /
is defined by
Pl ql P2 q2 q3P3 u -n3T 3 -M3}12 = n l M 1 nl T1M1) n2 T 2 M 2) T 1 T2
×~Plql P2q2
q3P3u} ~ TIT 2
\ n i T l n2T 2 -n3T 3 Here U
q3P3 _M3112 - n3 T3
T3
\ M I M 2 -M31"
(2.8)
SU(3) ISOSCALAR FACTORS
701
is the product state given by eq. (5.2) of I; the heaviest state (n 3 = 0, -M 3 = T3 = ½(P3 +q3)) is given by eq. (4.4) of I whose phase is fixed by the convention (P:
ql
. P2 ~ q2
[q3
P3
0
½(q3+P3)
½(Pl+ql) ' -s ~(p2+q2-s)
u) ) 0
the last factor in eq. (2.8) is an SU(2) Clebsch-Gordan coefficient. It is not hard to show that o
q nTM n T M
nTM
P
1 ½(p-q-n)-M
nTM n
is a n o r m a l i z e d invariant; D = ½(P+l)(q+l)(p+q+2) is the dimension of the representation (p,q). It follows that within a phase factor Su
(_l)q3+u21
D~½
~ P3 q3 n3T3M3 n3 T 3 1143)3
q3P3 u
-n 3 T 3 -M3)12 (-1) ~(p3+q3-n3)-M3 (2.9) The phase is fixed by comparing in eq. (2.1) and (2.9) the coefficients of say the state P3
q3
,
0 ½(P3+q3)-½(P3+q3))
= (p3 ! q3 ! )-½(-~3)P3~ q3 .
Substitution of eq. (2.9) in (2.6) yields
Plql P2q2 q3P3 u, {nlT1; n2T2 -n3T 3 )
1
= I (/53,+ 1)(q3 + 1)(P3 + q3 + 2) i ~ 2(2T 3 + 1)
A
1 Plql P2q2"P3q3" u), × (_l)~(P3-q3-n3)+u21+T2-T1 (nlT1; n2T 2' n3T 3' the
Incidentally the N of eq. (4.4)of I, called N u of eq. (2.1) by
N3u
in eq. (2.11), is related to 1
N3u = (P3 ! q3 ! D3)aNu •
(2.11)
3. DERIVATION OF GENERAL ISOSCALAR FACTOR In this section we extract the general i s o s c a l a r factor by projecting Su on products of one-particle states. Isospin manipulations play an important role. Taking the s c a l a r product of eq. (2.5) with
702
C. -K.CHEW and R.T.SHARP
P2
q2~ P3
q3
TI T2T3
M1M2 g i v e s an e x p r e s s i o n for the i s o s c a l a r f a c t o r
T1T2T3~,Pl
ql P2
q2 P3
q3
. P l q l , P2q2 P3q3 u)= ~ (\M1M2M3/~/\nlT1MI' " n2T2M2; n3T3M3[Su) nlT " n2T2; n3T3; MIM2
(3.1) We now e x a m i n e the isospin s t r u c t u r e of both f a c t o r s in the s c a l a r p r o d u c t on the right side of eq. (3.1). F i r s t the left f a c t o r . By eq. (3.1) of I we can w r i t e
I p q ) = ~ N'~-n'(-~*)n-n' I T' n T n' (-n')'.(n-n'): ' T* 3/
(3.2)
where 1 1 1 1 N' = ~ {-~(p+q-n)+ T+l I)'.V _{½(p+q+n)+ T+l }'~ {½(p+q+n)T}:T {½(p+q_-n)T}'. ~
(p+q+l)' {½(p+q-n)+T+n'+l}'.{½(p+q-n)-T+n'}'
I_
1
( )J33.:
In eq. (3.2) T ' is the i s o s p i n s t a t e T ' = ½(p + n ' ) f o r m e d f r o m u n s t a r r e d ~ v a r i a b l e s and T* = ½(q -n+n') is the isospin s t a t e f o r m e d f r o m s t a r r e d v a r iables; the two i s o s p i n s T'T* a r e c o m b i n e d to give a s t a t e I T ) . The isospin f a c t o r s on the left of the s c a l a r p r o d u c t in eq. (3.1) can thus 5e w r i t t e n
M1M 2
I
M1 /
M3/
M2 . ?
T
M2
T
1
= (-1)2(Tl+T2+T3)[(2Tl+l)(2T2+l)(2T3+l)] 5 (3.4) where
(~a~7b - 77a~b)a+b-c (~ bT?c _ 77b~c)b+c -a (~c~?a _ 77c~ a)C+a-b (abc) =
1
(3.5)
[(a+b-c)' (b÷c-a)' (c+a-b)' (a+b+c+l)' ]-~ is the n o r m a l i z e d van d e r W a e r d e n i n v a r i a n t which is the SU(2) analog of the s c a l a r Su in SU(3). (abc) can be expanded [15].
M1M2 ma
mc
a mb mc
The s c a l a r p r o d u c t on the right of eq. (3.4) is to be taken only with r e s p e c t to the v a r i a b l e s f o r T1T2T 3 which a p p e a r in both f a c t o r s ; the vari-
703
SU(3) ISOSCALAR FACTORS ables for ~1~1~2~2~3~3 w h i c h a p p e a r o n l y on t h e r i g h t a r e f r e e . s u b s t i t u t i o n (3.4) eq. (3.1) r e a d s !
=
AIA'2A3
5
(1 1(1 1
!
T
With the
T
1(2 2 ( 2 2
g3
nln2n 3
(3n)
× [(-~[)" (~1 -~[)' (-"~)' (~. -~[)" (-'3)' (~3 -~)' ]-1, t
9
( 2) T ' +~ n i - n ' ( 2 T i + 1)e1_N ,i . H e r e I S F u h a s b e e n w r i t t e n f o r b r e v i t y w i t h A i' = to r e p r e s e n t t h e i s o s c a l a r f a c t o r on t h e l e f t of eq. (3.1). N o w e x p a n d S u s i m i l a r l y in p o w e r s of the ( v a r i a b l e s . T h e r e s u l t c a n b e written Su = N u
, 2 , , B u ( T 3 T 1 T 2 ) ( T I T 2 T 3 ) ( T 2 T 3 T 1) nln2n 3 T 1T 2 T 3 T
T
T
T
T'
"¢
X (1 n l ~lr*nl-nl (2 n2 ~2"*n2-n2 (3 n3 g3'~*n3-n3 '
(3.8)
where (-1)
i~z
~i -~iJul2!
u21, u13! u31! u23: u32, " s'.
' U 3 2 - ~ ) . a' . ' B u = 5..(U12_5).e,.(U21_e),.~:(U13_O)!K:(U31_K),.~:(U23_~), " P.( ,
,
fi!y:
1
× [ ( T 3 + T ~ I - T ~ ) ! ( T 3" + T 2~'- T 1 )*! (T~I+T "2 - T 3 ") i ( T 3" + T 2,, + T I. + I ) " , ]2-
× [ ( T I + T 2 - T 3 ). ( T I + T 3 - T 2 ) . ( T 2 + T 3 - T 1 ) . ( T I + T 3 + T 2 + l ) ] z 1 *
v.
"
T
"
m
*
T
× [(T2+T3-T , * ,,1 )., ( T 2, ~ T ~1 - T 3•) . , ( T 3 + T 1 - T 2 ) - ( T 2 + T 1 + T 3 + l ) . ] T1T2T 3 are independent summation T1 = T
variables
and T~T
- T1 , tt
???
*
T?
~ + T *1 ,
0 = T~" - T 2 + T 3
5 = TI-I
~=T2-T
K = T~ - T 2"~ + T *1 ,
~t = T
(3.10)
*
y = T 1 + T~ - T 3
fi = T 3 + T 1 - T
3 +1 2 ,
T
= T3 - T3
2 ,
a = T 2 +T 3 -T 1 ,
(3.9)
-T 1 + T 3 ,
/x = T 3 - T 1 + T 2 ,
704 It
C.K. CHEW and R . T . S H A R P t~
T i , Ti
a r e s t a t e s in the s a m e u n s t a r r e d i v a r i a b l e s (i = 1 , 2 , 3 ) . W h e n eq. (3.8) i s s u b s t i t u t e d in eq. (3.7) t h e f i n a l r e s u l t i s
ISF u = N u
~ A'A' a' B n,ln,2n ~ 1 2~3 u T~T~T~
(3.11)
× ((T~ T i T1)(T~T'2T2)(T~TT3 T3)
(T 1 T 2 T 3 ) ( T "1T~T~)(T~
T~TI")(T~T ~T~')).
T h e e x p r e s s i o n (3.11) c o n t a i n s s i x e x p l i c i t s u m m a t i o n s . T h e i s o s p i n f a c t o r i s a t r i p l y s t r e t c h e d 1 5 - j s y m b o l . It i s e v a l u a t e d in a p p e n d i x A in t e r m s of a q u a d r u p l e s u m . H e n c e t h e r e s u l t (3.11) c o n t a i n s t e n s u m s in a l l . T h e n o r m a l i z a t i o n c o n s t a n t N u i s e v a l u a t e d in a p p e n d i x B; it i n v o l v e s a seven-fold sum. In a p p e n d i x C we s h o w t h a t a s c h e m e f o r o r t h o g o n a l i z i n g d e g e n e r a t e SU(3) i s o s c a l a r f a c t o r s b a s e d on m i x e d C a s i m i r o p e r a t o r s and d i s c u s s e d e a r l i e r b y us [13] a p p l i e s r e a d i l y to t h e p r e s e n t s c h e m e . T h e o r t h o g o n a l i z a t i o n d o e s not s p o i l t h e s y m m e t r y p r o p e r t i e s . One of us (C. - K . C) i s p r e p a r i n g a p r o g r a m f o r c o m p u t a t i o n of i s o s c a l a r f a c t o r s , o r t h o n o r m a l i z e d a c c o r d i n g to t h e s c h e m e d e s c r i b e d in a p p e n d i x C. O u r t h a n k s a r e d u e to H. C. y o n B a e y e r f o r h e l p f u l d i s c u s s i o n s .
APPEND~
A
W e w i s h to e v a l u a t e t h e i s o s c a l a r e x p r e s s i o n
Y = YT3T3
(A. 1) ?
$
?
~
t
~
~
~
tt
$
tn
~
$
v11
=- ((T1T1T1)(T2T2T2)(T3T3T3)[(T1T2T3)(T1T2T 3 )(T2T3T 1 )(T3T1T 2 )) , ' " "1 are T1T1T
w h i c h a p p e a r s in o u r f i n a l f o r m u l a (3.11).
in t h e s a m e i s o s p i n
variables ~i~/i and satisfy T~ +T~ = TI; similarly for 2 and 3. We follow closely the methods used by Bandzaitis, Karosiene and Jucys [16] to evaluate the X-coefficient to which our Y is formally similar [17]:
Fabc~ X t degh f ~ =
<(abc)(def)(ghj)l(adg)(beh)(cfJ))"
T h i n k of e x p r e s s i o n (A.1) a s t h e s c a l a r p r o d u c t of
((T3T3T3) I (T2T3 T 1 )(T1T2T 3) ) ,
(A.2)
with tT
$
tit
tt
$
tlt
~
~
,
<(T1T2T 3 )(T3TI T2 )t (TI TI T1)(T2T2T2) > •
(h.3)
In e x p r e s s i o n s (A.2) and (A.3) s c a l a r p r o d u c t s a r e to b e t a k e n w i t h r e s p e c t to v a r i a b l e s w h i c h a p p e a r in b o t h f a c t o r s ; o t h e r v a r i a b l e s a r e f r e e . A l s o t h e scalar product
SU(3) ISOSCALAR FACTORS
705
r~ rl for example, is to be i n t e r p r e t e d a s
D1
., ) M1
with T~'
DI=( =
i
.,(
T i?1
T ,1
M 1 M"1 MI
)
(T,I - M'I)I (T'1 +M'I)'
]½
,, ~ , i (T~ - M[): (T~ + M~)I (T~ - M 1 )[ (T~ + M1 ).J
(A.4)
The state (A.3) has isospins T'~T~T1T 2 added to give T~. The states (A.2) with f r e e T3T ~ are a complete set of such states with orthonormality e x p r e s s e d by ( T 3 T~ IT3 T~ ) = 6 T3 T35 T~ T~ / (2 T 3 + 1)(2 T~ + 1) in an obvious notation. Hence we may write
<(T1T2T3 )( T3T1T2 )1 (T~T'I T1) (T~.T'2T2) >
= E ((T~T;T3)[(T~T;T~I')(T1T2T3)) r3r;
(2T 3 + 1)(2T~ +
1)YT3T~ ,
(A.5)
where the coefficient YT^T*,, is just the expression (A.1) to be evaluated. Expand both sides of ~q.~(A.5) substituting f r o m eq. (3.6) for all the threeisospin invariants and equate coefficients of
Mr
M;. M1 M2 M~
The r e s u l t is
r~ r;. rj
r; r~ <2
rl* rl rl
r*2 rl r2
M'~ ) (M~ M'I M 1) ( M~. ) D1D2D3 • IiM~ (M~ M~. M~ ) (M~ M~ M'2 M2 T* T~ T 3
T3T~
MT3
T" T~ T"I
T1 T2 T3 "
M~ M2 r e g a r d M3M3M1M 1 as free
In eq. (A.6) we may by the f r e e ones. Now multiply by
(A.6)
and
M3M~M2 as
determined
706
C.-K. CHEW and R.ToSHARP __
"
m
T"
(M; M~ M 1 with M;M 3 (and
~T!
and sum over M I M I p r o p e r t y of the Wigner coefficients a
b
T 1 T2 T3 1 M2 M3 T
hence M3) fixed, using the orthogonality
c
a
b
~, 2~+i
Y~/a The r e s u l t is T*3 T~ T 3 -1
Y
(M~ M'3 M 3
T"2 T~ T "1
M~M~M~M1
(MT~ T~ T 3 x / T 3 T I T2, ×
~,,,,,,\~,,,,}{ M~.I~, 3 '~'3
i
T l T2 T3
M~
T 1 T~ T 1
T 2 T~. T 2
M,2,,)~M~ M I M 1)(
) D1D2D3 .
M~M~M 2
(A.7) The right side of eq. (A.7) cannot depend on M3M~ which a r e still f r e e , !
~r
T
*
*
W
,
~
'T
o
T!
.
so we may s e t M 3 = T3, M 3 = - T 3. T h e n a l s o M 3-- T 3 ; t h e M 3 s u m c o n t a m s one t e r m only. T h u s
r~ T~
T3
Y : ( - T~ T'3 T ; - T 3' )
( T3"
x'T3
rl
×
T1* -M1-MI
r2
2
-1
T ~
T3 rl *
~ ('~* " " -T~ M "1 ) MIM'IM; 13-1v11 T2"
MI+M1-T3 ' "
r3
T"1
T2*
T ~"
""1 .... 1 MI-MI-T 3 T'~
r~
rl r l
( r; r~ r2 ., , ,. , . ) DID2D3 • ~M1-MI-r3 r~-r~+M1-M~+M~ T3-r3-MI
(A.8)
The last t h r e e Wigner coefficients in the summation contain one sum each, so we have a sixfold sum. The M1M 1 sums can be effected. I n s e r t the values of the s u m l e s s Wigner coefficients and substitute
SU(3) ISOSCALAR FACTORS
T~I
T'I T1
+
(-M1-M i M I M1 )=(-1)M1
707
,_TI
2T1
I(T~+T'I-TI)~ (T'I+TI-T*I)~ (T~-MI-MI)! (TI+MI)! (TI+TI+T~+I)! 1½ ×
.
t_
.
.
.
.
.
.
(T -M )' (T' +M' )' (T' -M' )' (T*+M +M' )' (T +T*-T' )' 1
1"
×~
(-
1
1"
1
1"
1
1
1"
1
1
I
1"
i)% ( 2 T ~ - x ) ! (T~+TT1-MI-%)~
(A.9)
x (T~-M1-M'I-X): (T~+T'I-TI-X):x~(TI+T'I+T~+I-x)~ Eq. (A.9) i s o b t a i n e d f r o m eq. (A.3) of r e f . [17] with ,
,
'
a = T1 - M1 - M1 ,
b = T
d:
e : T 1 + M 1 - T~ + M 1 ,
T 1 - T~ + M~ ,
+ T - T1 ,
c = T
- M1 ,
f = 0,
a n d with a R e g g e p e r m u t a t i o n a p p l i e d ,to ~r o w s a n d c o l u m n s ; an e x a c t l y a n? a - iT! l o g o u s s u b s t i t u t i o n i s m a d e f o r t h e (T2T2T2) W i g n e r c o e f f i c i e n t . T h e M1M 1 s u m s c a n t h e n b e e f f e c t e d with t h e h e l p of t h e w e l l - k n o w n f o r m u l a
X
1 (a+b+c+d)! (a+x)! (b+x)~ (c-x)! (d-x)~ = (a-¢c)~ (a+d)' (b+c)~ (b+d)! "
The result is *
Y = ( - 1) T2+ T 3 - T~'+ T ~ - 2 T ~ - T{+ T~
~
T
(T 3 +T 3 - T 3).
[
~(T2+T 3 -T~)~
×
~
(2~+T'2'-T3)! (T~+T'I-T1)'. (T'I+T1-T~)! T?T ?? ~ * T! ~. ? * TVT Vt T (T 2 +T3-T1)!(TI+T3-T2 ). ( T I + T 2 + T 3 + l ) .
(TI+TI+TI+I). (T2+T2-T2). (T2+T2-T2). }½ t 1 2 + z 3 - 1 1 ~. [ 3 1 + T 3 - T 2 ) . [TI+T1-T1). J ?
*
T
I
?
(-1)
xyM 1
?
!
*
(T;+T2+T~+I):
. 2 + T 3* + l ) . ' (T .1. .+. T
×
*
* ' '(T~'+ (T2+T2-T2).
x+y+TI+M 1
(T2+T2-
T2-Yj.
]-~
*" T 2" - T3)A
,
(2 T2-Y)'.
( T 2 + T 1- T 3 - y ) .
!
*
Tv
*
V
1
(T3+T3+T3 + 1)I ( T I +T2- T'3")!I~ (TI+T3 _T2). (TI+T2+T3+I). i
C. -K.CHEW
708
(2T!-x)!
X
y! (Tb+T2+Tz+l-y)!x!
and R.T.SHARP
(Ti-Th-Ti+Ti+T;+TT-x-y)!
(TT+Ti-Tl-x)!
(Tl+Ti+TT+l-x)!
(Tg-T{+Ti-x)!
x 1 I-(T2-Th+T$+M1)!
If T; = Tt = 0, Y reduces
APPENDIX
(A.lO)
(Tl-Ml)! to a singly stretched
9-j,
or X coefficient.
B
We wish to calculate the metric matrix (SU,,1s,> which arises when the unnormalized SU = N;lSU are used as base vectors in the degenerate space which they span. The no;ma,liza$ion constant NU which appears in eqs. (2.1)(3.11) is given by N, = (S,[ S,) -2 in terms of its diagonal elements. Because of (2.9) we may write SU = (P3! %!)3 (-1)
fi3+“21
&
43
0
%P3+43) -&3+43)
>
;3u+!D3-1)
X more terms
,
where 43
-1
+3u = N3U
0
@. 1)
u
P3
$(P3+43) $(P3+43) >12
u13 U23~*“31~*z’32 = (-1) u21 PBu12Du21 (7@2 -7?2sl)s 12 21 771 q2 1 2
=N;;
X
c T1T2n l”1
I$ll2l,~~~,,C::~
p1q1
p2q2
493
i nlT1’
n2T2
0 $(P3+43) >
I:;;::;)
u (B.2)
.
The (D3-1) other terms in eq. (B.l) refer to a summation over n3T3M3. Since all D3 terms contribute equally we have
(i,, 1itu) =
(-l)ui1-u21
p3! q3! D3< $3u, j 4,,,
s
(B.3)
SU(3) ISOSCALAR
FACTORS
709
We have only to calculate (~h3u , ]~3u ) Taking the scalar product of eq. (B. 2) with
Plql
M1
½(P3 q3)
P2q2 \ / T1T2
lniTIM1) n2T2M2/\MIM2 \
½(P3+q3) ) '
gives an expression for the a s y m m e t r i c i s o s c a l a r factor
Plql. P2q2 [q3 P3 < n 1T 1' n2 T2 ½(P3+q3) )
( T1 T2 = N3u ~M1 \M1M2
½(P3+q3)\
Plql
P2q2 n2T2M2
(p3+q3)/(nlT1MI' [
~3u>"
(B.4)
Using eq. (3.2) we can transform the left factor of the scalar product on the right side of eq. (B.4): / TIT2
½(P3+q3) \ n plql P2q2 L ½(P3+q3) / 1TIM1; n2T2M2 ) 1 2
M1
(B.5)
N,1N~2~lni~nl-n'l~2nT2~n2-n'2 T'1T~ n,ln,2
(-nl)I (nl-n~)! (-n~)! (n2-n~)I v
T~. T~.
T1
T2
½(P3+q3)
,
The isospin state on the right is formed by adding TIT 1 to give T1, then T2T 2 to give T2, and finally TIT 2 to give ½(P3+q3) (with M = ~(p3+q3)). Similarly expanding J/3u we get *
~3u = (-1)U21p
F~lnl ~1,n 1-n'1~2n2 ~2,n 2 -n'2
l/U
t
v
T 2 ~ 13+u32+nl-nl-n2) \ 1 /U v v ) T~. T 1 ~t 23+u31+n2-n2-nl) ' ½(P3+q3)
where
(B.6) T
?
u12! u21! s! (-1) -nl-n2+n2
F =
(n2-n~.)! (u12-n2+n2)!' (nl_n~): (u21-nl+nl)~' (nl_n2_nl):, , ,
v
× [(u13+nl-n2-n1)I u32'. (u12-n2+n2)!,
t
, (u12+u32+u13+s÷n 1 ,
(n2_nl_n2)!' , v
v
+1).]
1
(u13 +u32+nl-nl-n2+ l ). ×I
(u23+n~.-~l-n2)! u31: (u21-n1+n~): (u21+n23+u31+s+n'2+l )[~ ½ ' ' ' (u23 +u31+n2-n2-n l +1).
(B.7)
710
C. -K. CHEW and R. T.SHARP
When eqs. (B.5) and (B.6) a r e i n s e r t e d in the s c a l a r p r o d u c t in eq. (B.4) t h e r e a r i s e s the i s o s p i n s c a l a r p r o d u c t
T1 T1
T1
T'1 T 2 5(u13+u32+n1-nl-n2) ~
T'2 T 2
T2
T'2 T~
= GX
~(u23+u31+n2-n2-nl)
½(P3+q3)
½(P3+q3 )
,
(B.8)
/
where *
*
I
!
T 2- T 2 + T 1 -~(u23 +u 3 l + n 2 - n 2 - n l ) G : (-1)
(B.9) t
~
v
1
X [(2Tl+l)(2T2+l)(u13+u32+nl-nl-n 2 + 1) (u23+u31+n2-n2-n 1 + 1) ]~
,
and X is the X - c o e f f i c i e n t I
X =X
*
1
T
T
t
1
!
T
T'1 T 2 ~(u13+u32+nl-nl-n 2) T~ T 2 ~(u23+u31+n2-n2-n1)
LT1 T2
,
(B.10)
½(P3+q3 )
which is s t r e t c h e d in its t h i r d c o l u m n and t h e r e f o r e c o n t a i n s a double s u m
[16]. H e n c e we find
I
Plql . P2q2 I q3 P3 u \
u21
n 1 T 1 ' n 2 T 2 ] 0 ½(P3+q3)/ = (-1)
,
,
N3u ~ , N1N2FGX, nln 2
(B.11)
in t e r m s of which the m e t r i c m a t r i x is g i v e n b y ^ ,, >= /Plql P2q2 I q3 P3 u' \ ( @3u, IJ/3u ~ \ " nlTIT2 n l T l ' n 2 T 2 I 0 ½ 0 3 + q 3 ) /
x
< Pl ql . P2q2 I q3 n 1 T 1' n2T 2 0
P3
u )
½(P3+q3 )
(--uN3u')-I N3
(B.12)
a triple sum over a product of two factors each containing a quadruple sum. The matrix" A different orthogonalization scheme is discussed in appendix C. APPENDIX
C
Following a suggestion of O'Raifeartaigh, Macfarlane and Rao [18], the present authors [13] evaluated the matrix of the mixed Casimir operator
SU(3) ISOSCALAR FACTORS
711
(c.1)
G12 = ½ T r (A1A1A 2 - A2A2A1)
in the d e g e n e r a t e s u b s p a c e of the ~ 3 u of eq. (B.2). H e r e A is the m a t r i x whose e l e m e n t s a r e g e n e r a t o r s of SU(3) t r a n s f o r m a t i o n s ; G12 , a H e r m i t i a n o p e r a t o r which c o m m u t e s with all the other o p e r a t o r s of i n t e r e s t , is s u i t able f o r lifting the d e g e n e r a c y . Diagonalizing its m a t r i x y i e l d s o r t h o g o n a l eigenstates
(c.2)
:
w h e r e eg is a p h a s e f a c t o r to be assigned. In the context of the p r e s e n t p a p e r G12 should be r e p l a c e d by G123 = -~(G12 +G23 +G31 ) ,
(C.3)
to o p e r a t e on the s u b s p a c e of d e g e n e r a t e S u of eq. (2.1). Since S u is an SU(3) s c a l a r we m a y s u b s t i t u t e A 3 = -(A 1 +A2), whereupon we get G123 = G 1 2 + I ,
(C.4)
w h e r e I is a multiple of the identity in the d e g e n e r a t e s p a c e and m a y be d i s regarded here. In the f o r m (C.4), G123 is an SU(3) s c a l a r in the s p a c e of 1 and 2 and we need r e t a i n only a single t e r m in the expansion (2.9) of the o p e r a n d s t a t e Su, say the t e r m
(_l)U21~3u = N31u(_l)U21 t q3 0
!)3
u
)
½(P3+q3) ½(P3+q3 )
(we d r o p its coefficient). Thus the old s c h e m e is equivalent to the new with the r e p l a c e m e n t ~ 3u ~ (-1) u21 Su ; by eq. (C.2) the o r t h o g o n a l i z e d s c a l a r s are
sg=
gu(-1)u21Cug.
u12 It r e m a i n s to a s s i g n an o v e r a l l p h a s e to Sg. This is conveniently done by r e q u i r i n g that for e a c h g the (Pug with g r e a t e s t magnitude have the p h a s e (-1) u2~l. This choice does not affect any of the s y m m e t r i e s of the i s o s c a l a r f a c t o r d i s c u s s e d in s e c t . 2, and can be g e n e r a l i z e d to c o v e r the c a s e in which the g r e a t e s t m a g n i t u d e is a s s u m e d by m o r e than one (Pug; b e c a u s e of the a n t i s y m m e t r y of G123 in s t a r r e d and u n s t a r r e d v a r i a b l e s the e i g e n v a l u e g is r e v e r s e d in sign u n d e r r e v e r s a l of m a g n e t i c quantum n u m b e r s . The r e l a t i v e p h a s e of s y m m e t r i c and a s y m m e t r i c i s o s c a l a r f a c t o r s b e c o m e s m o r e c o m p l i c a t e d . When the u label in eq. (2.10) is r e p l a c e d by g the p h a s e f a c t o r (-1) u21 on the right is r e p l a c e d by ~, which is the sign of
(.1)e(P3+P2-Pl-q3+q21- - ql - s ) ~ u
( Pl ql ~°ug\ ~
P2 q2 P3 q3 u~'] 1 ; , ; 0 ~ ( p l + q l ) z-s ~(p2+q2-s) 0 ~(p3+q3) ;
(c.6)
712
C. -K. CHEW and R. T. SHARP
T h e c o e f f i c i e n t of e a c h (Pug i n the e x p r e s s i o n (C.6) h a s p h a s e ( - 1 ) u 2 1 ; the o v e r a l l s i g n c a n only be d e t e r m i n e d a p o s t e r i o r i i n e a c h c a s e of i n t e r e s t .
REFERENCES [1] J. J. De Swart, Rev. Mod. Phys. 35 {1963} 916, [2] P. McNamee, S.J. and F. Chilton, Rev. Mod. Phsy. 36 (1964) 1005. [3] L. C. Biedenharn, J. Math. Phys. 4 {1963} 436; G. E. Baird and L. C. Biedenharn, J. Math. Phys. 4 {1963} 1449; 5 (1964} 1723; 5 {1964) 1730. [4] J. P. Elliott and M. Harvey, Proc. Roy. Soc. {London} A272 {1963} 557. [5] M. Moshinsky, Rev. Mod. Phys. 34 (1962) 813. [6] J. K. Kuriyan, D. Lurie and &. J. Macfarlane, J. Math. Phys. 6 {1965) 772. [7] N. Mukunda and L.K. Pandit, J. Math. Phys. 6 (1965} 746. [8] K.T. Hecht, Nucl. Phys.62 {1965} 1. [9] Hou Pei-Yu, Scientia Sinica {1965} 367. [10] G. Ponzano, Nuovo Cimento 41A (1966) 142; Istituto di Fisica, Torino, Italy preprint. [11] R. T. Sharp and H. yon Baeyer, J. Math. Phys. 7 {1966) 1105. [12] D. Kleima, Nuc].Phys. 70 {1965} 577. [13] C. -K. Chew and R. T. Sharp, Can. J. Phys. 44 (1966) 2789. [14] R. T. Sharp, Nuovo Cimento 47A (1967) 860. [15] V. Bargmann, Rev. Mod. Phys. 34 (1962) 829. [16] A. Bandzaitis, A. Karosiene and £. Jucys, Liet. Fiz. Rin. 4 {1964} 457. [17] R. T. Sharp, Nucl. Phys., to be submitted. [18] L. O'Raiffeartaigh, ~. J. Macfarlane and P. Rao, talk given by O'Raifeartaigh at Toronto Conference on Particle Symmetries, October 1965.
Nuclear Physics B2 (1967) 697-712. North-Holland Publ. Comp., A m s t e r d a m
SU(3) ISOSCALAR FACTORS* C. - K . C H E W Physics Department. McGill University. Montreal. Canada and R. T . S H A R P ** Department of Physics. University qf lllinois. Urbana. Illinois Received 24 April 1967
Abstract: The general non-orthonormalized SU(3) i s o s e a l a r factor is e x p r e s s e d as a six-fold sum involving a t r i p l y - s t r e t c h e d 15-j coefficient which contains a quadruple sum. The s y m m e t r y p r o p e r t i e s of the s y m m e t r i c i s o s e a l a r factor and its relation to the a s y m m e t r i c i s o s c a l a r factor a r e discussed. Two o r t h o n o r m a l i z a tion s c h e m e s a r e discussed which diagonalize respectively a metric o p e r a t o r and a mixed C a s i m i r o p e r a t o r .
1. I N T R O D U C T I O N S i n c e t h e d i s c o v e r y of SU(3) s y m m e t r i e s in n u c l e a r a n d p a r t i c l e p h y s i c s t h e r e h a s b e e n m u c h i n t e r e s t in t h e g r o u p a n d in i t s C l e b s c h - G o r d a n c o e f f i c i e n t s . A u t h o r s h a v e p r e s e n t e d t a b l e s of t h e m o s t c o m m o n l y u s e d i s o s c a l a r f a c t o r s [1, 2] a n d h a v e g i v e n f o r m u l a s of v a r y i n g d e g r e e of c o m p l e x i t y and generality [3-11]. Hou [9] f o l l o w s t h e v a n d e r W a e r d e n a p p r o a c h , c o n s t r u c t i n g a s c a l a r w h i c h i s a p o l y n o m i a l in t h e v a r i a b l e s of t h r e e " o n e - p a r t i c l e " s p a c e s ; P o n z a n o [10] e x p a n d s s u c h a s c a l a r in p r o d u c t s of one p a r t i c l e s t a t e s and d e t e r m i n e s t h e g e n e r a l i s o s c a l a r f a c t o r f r o m t h e c o e f f i c i e n t s . He a v o i d s t h e tracelessness conditions by generating the general one-particle state from two i n d e p e n d e n t q u a r k (1, 0) b a s e s ; he e v a l u a t e s t h e g e n e r a l n o n o r t h o n o r m a l i z e d i s o s c a l a r f a c t o r in t e r m s of s p e c i a l 2 1 - j s y m b o l s of SU(2). H i s f o r m u l a h a s 17 s u m s . S h a r p a n d von B a e y e r [11] (we w i l l r e f e r to t h i s p a p e r a s I) d e r i v e a formula for the general isoscalar factor with fifteen summations, synthes i z i n g it out of s i m p l e r s p e c i a l i s o s c a l a r f a c t o r s . T h e i r o n e - p a r t i c l e s t a t e s a r e g e n e r a t e d f r o m one q u a r k a n d one a n t i q u a r k (0, 1) b a s i s ; t h e t r a c e l e s s n e s s c o n d i t i o n i s i m p o s e d b y d i s c a r d i n g a s u b s p a c e of " u n w a n t e d " s t a t e s . * This work was supported by the National R e s e a r c h Council of Canada and the Dep a r t m e n t of P h y s i c s , University of Illinois. ** On leave f r o m MeGill University, Montreal, Canada.
698
C. - K . C H E W a n d R . T . S H A R P
K l e i m a [12] has a l s o d e s c r i b e d q u a r k - a n t i q u a r k b a s i s p o l y n o m i a l s . In this p a p e r we follow P o n z a n o in c o n s t r u c i n g a s c a l a r p o l y n o m i a l in t h r e e o n e - p a r t i c l e s p a c e s , but we u s e the q u a r k - a n t i q u a r k v a r i a b l e s of I. The i s o s c a l a r f a c t o r s a r e e x t r a c t e d by p r o j e c t i n g on p r o d u c t s of o n e - p a r t i c l e s t a t e s ; this is m o r e d i r e c t than the synthetic method of I and y i e l d s a s i m p l e r g e n e r a l f o r m u l a , with ten s u m s . The t r a c e l e s s n e s s condition is not a c o m plication, for the o n e - p a r t i c l e s t a t e s on which the s c a l a r is p r o j e c t e d contain no unwanted s t a t e s . I s o s p i n m a n i p u l a t i o n s play an i m p o r t a n t r o l e in the calculation. An o r t h o g o n a l i z a t i o n s c h e m e b a s e d on m i x e d C a s i m i r o p e r a t o r s , d e ; s c r i b e d e a r l i e r by us [13], is shown to apply with little modification to the present approach.
2. SYMMETRIC AND ASYMMETRIC ISOSCALAR FACTORS Much of the implicit form. in our p r e s e n t scalar factors, the f o r m e r .
content of this s e c t i o n m a y be found e l s e w h e r e , at l e a s t in We thought it worthwhile to collect the r e s u l t s in one p l a c e notation. We define s y m m e t r i c and a s y m m e t r i c SU(3) i s o r e l a t e t h e m to each other, and exhibit the s y m m e t r i e s of
We use the o n e - p a r t i c l e s t a t e s [P T q } defined by eq. (3.1) of I; they a r e p o l y n o m i a l s of d e g r e e p ; q in s i x independent v a r i a b l e s ~ 77~ ;4" 77* {*, and s a t i s f y the t r a c e l e s s n e s s condition by v i r t u e of t h e i r orthogonality to the unwanted s t a t e s of the s a m e d e g r e e N G ( ~ *
+ 7777* + ~ . ) G
I P-G n TqM- G ) ; h i s
the n u m b e r of r o w s above that containing the h e a v i e s t s t a t e in the hexagonal weight d i a g r a m and is r e l a t e d to h y p e r e h a r g e Y by n = Y + ~ ( q - p ) . The p h a s e convention is that of de Swart [1] and of H a r v e y and Elliott [4], d e s c r i b e d c o r r e c t l y in I but i n c o r r e c t l y a t t r i b u t e d to B i e d e n h a r n t h e r e ; the m a t r i x e l e m e n t s of R± (the -K:~ of de Swart) a r e negative; in the B i e d e n h a r n convention, u s e d by Ponzano [10] the r e d u c e d m a t r i x e l e m e n t s of R e a r e $
negative. Our s t a t e I P,~ T q~) d i f f e r s f r o m the c o r r e s p o n d i n g s t a t e of P o n z a n o 2V~
by a p h a s e (-1)½(P+q+n)-T. We define a s c a l a r p o l y n o m i a l in t h r e e s e t s of v a r i a b l e s Su = _~ p BUl2 ~u21 ~u13 ~u31 ~u23 ~u32 (:s u 12 --21 --13 --31 --23 --32 - '
(2. la)
S u = N P B u 1 2 ~ u 2 1 R u l 3 R u 3 1 ~ u 2 3 ~ u32 ~ * s u 12 --21 --13 --31 --23 --32 v ,
(2.1b)
or
where
SU(3) ISOSCALAR FACTORS
699
~1 ~2 ~3 C=
~?1 772 7?3
,
(2.3)
~1 ~2 ~3 and C* is C with the replacements ~ --~ ~* etc. The operator P is an instruction to project out unwanted states; this operation is achieved automatically by projecting S u later on one-particle states. N u normalizes S u to unity; it is evaluated in appendix B. Our Su, eqs. (2.1a) and (2.1b) correspond to Ponjano's invariant I given by his eqs. (38) and (39) respectively. From the degree of S u in starred and unstarred variables one finds Pl =u12 +u13 + s ,
ql =u21 +u31 ,
P2 =u21 +u23 + s ,
q2 =u12 +u32 '
P3 =u31 +u32 + s ,
q3 =u13 +u23 '
Pl = u12 + u13 '
ql =u21 +~31 + s ,
P2 = u21 + u23 ,
q2 =u12 +u32 + s ,
s = ~1 ~. (Pi - q i ) z (2.4a)
Or
s = ~.
(qi-Pi) ,
(2.4b) q3 =u13 +u23 + s ' P3 = u31 + u32 ' for Su given by eq. (2.1a) or (2.1b). It is apparent that increasing u12 , u23 , u31 and decreasing u21 , u32, u13 by the same integer leaves the p and q unchanged; two Su whose u are thus related are called degenerate; the number of S u in a degenerate set is determined by the restriction that the u are non-negative integers; besides the p and q any one of the six u can be used to specify S u uniquely. We expand S u in one particle states ~11
Su =
P2
q2)
P3
q3
nln2M1M 2 T 1T2 T3
× (Pl
ql.
P2
q2.
n 1 T1M 1 ' n2 T2 M2 '
P3
q3.
u)
n3 T3 M 3 '
(2.5) ,
defining thereby an SU(3) Wigner coefficient P2 q2 -P3 q3. u ) =
( P l Tq l1'. n1
P2 q2. P3 q3. u~] \( MT 11 T 2 T 3
n 2 T 2' n 3 T 3'
)
M2 M 3 -
(2.6) '
700
C. - K . C H E W a n d R. T . S H A R P
which f a c t o r s by v i r t u e of the W i g n e r - E c k a r t t h e o r e m f o r isospin into a s y m m e t r i c i s o s c a l a r f a c t o r and an SU(2) Wigner coefficient, shown in the second line of eq. (2.6). The M s a t i s f y ZiMi = 0 and the n satisfy Zin i =
"-:i(qi - 1,i)
: ±s.
The s y m m e t r i e s of the i s o s c a l a r f a c t o r a r e a p p a r e n t f r o m eqs. (2.6) and (2.1). Interchanging two of the t h r e e s p a c e s m u l t i p l i e s S u by (-1) s and a c cording to the s y m m e t r y p r o p e r t y of the SU(2) Wigner coefficient m u l t i p l i e s the i s o s c a l a r f a c t o r by (-1)s+T1 +T2+T3. Another s y m m e t r y d e a l s with r e v e r s i n g " m a g n e t i c " quantum n u m b e r s n and M. It can be shown that
P
q)*
nTM
:
-
q
P \
T-M)
(-
1)½(p-q-n)-M
,
(2.7)
w h e r e the a s t e r i s k m e a n s substitute ~ ..... ~* etc. Putting a s t a r on (1; 2; 3 IS) in (2.6) does not change its value, s i n c e the v a r i a b l e s a r e all d u m m i e s . But it induces the r e p l a c e m e n t s p'--" q, n ...... n, M - , -M, uij ..... uji and i n t r o duces a factor
(_l)'-'i[~(Pi - qi - hi) - Mi] : (_1) 2s : 1. By the s y m m e t r y of the SU(2) Wigner coefficient the i s o s c a l a r f a c t o r changes by a p h a s e (-1)T,+T:+T3 o n r e v e r s i n g n , interchanging p with q and uij with uji. We need c o n s i d e r only eq. (2.1a) f r o m now on, since this t r a n s f o r m a t i o n int e r c h a n g e s it with (2.1b). We have s e a r c h e d u n s u c c e s s f u l l y f o r s y m m e t r i e s of the g e n e r a l SU(3) i s o s c a l a r f a c t o r analogous to the Regge s y m m e t r i e s of the SU(2) Wigner c o e f f i Cient; however by e m b e d d i n g SU(3) in SU(4) in d i f f e r e n t w a y s one could find identities s i m i l a r to the g e n e r a l i z e d Regge identities s a t i s f i e d by SU(2) Wign e r c o e f f i c i e n t s [14]. The a s y m m e t r i c i s o s c a l a r f a c t o r
p l q l . p2q2 n l T ' n2T 2
p3q3 u -n3T 3 /
is defined by
Pl ql P2 q2 q3P3 u -n3T 3 -M3}12 = n l M 1 nl T1M1) n2 T 2 M 2) T 1 T2
×~Plql P2q2
q3P3u} ~ TIT 2
\ n i T l n2T 2 -n3T 3 Here U
q3P3 _M3112 - n3 T3
T3
\ M I M 2 -M31"
(2.8)
SU(3) ISOSCALAR FACTORS
701
is the product state given by eq. (5.2) of I; the heaviest state (n 3 = 0, -M 3 = T3 = ½(P3 +q3)) is given by eq. (4.4) of I whose phase is fixed by the convention (P:
ql
. P2 ~ q2
[q3
P3
0
½(q3+P3)
½(Pl+ql) ' -s ~(p2+q2-s)
u) ) 0
the last factor in eq. (2.8) is an SU(2) Clebsch-Gordan coefficient. It is not hard to show that o
q nTM n T M
nTM
P
1 ½(p-q-n)-M
nTM n
is a n o r m a l i z e d invariant; D = ½(P+l)(q+l)(p+q+2) is the dimension of the representation (p,q). It follows that within a phase factor Su
(_l)q3+u21
D~½
~ P3 q3 n3T3M3 n3 T 3 1143)3
q3P3 u
-n 3 T 3 -M3)12 (-1) ~(p3+q3-n3)-M3 (2.9) The phase is fixed by comparing in eq. (2.1) and (2.9) the coefficients of say the state P3
q3
,
0 ½(P3+q3)-½(P3+q3))
= (p3 ! q3 ! )-½(-~3)P3~ q3 .
Substitution of eq. (2.9) in (2.6) yields
Plql P2q2 q3P3 u, {nlT1; n2T2 -n3T 3 )
1
= I (/53,+ 1)(q3 + 1)(P3 + q3 + 2) i ~ 2(2T 3 + 1)
A
1 Plql P2q2"P3q3" u), × (_l)~(P3-q3-n3)+u21+T2-T1 (nlT1; n2T 2' n3T 3' the
Incidentally the N of eq. (4.4)of I, called N u of eq. (2.1) by
N3u
in eq. (2.11), is related to 1
N3u = (P3 ! q3 ! D3)aNu •
(2.11)
3. DERIVATION OF GENERAL ISOSCALAR FACTOR In this section we extract the general i s o s c a l a r factor by projecting Su on products of one-particle states. Isospin manipulations play an important role. Taking the s c a l a r product of eq. (2.5) with
702
C. -K.CHEW and R.T.SHARP
P2
q2~ P3
q3
TI T2T3
M1M2 g i v e s an e x p r e s s i o n for the i s o s c a l a r f a c t o r
T1T2T3~,Pl
ql P2
q2 P3
q3
. P l q l , P2q2 P3q3 u)= ~ (\M1M2M3/~/\nlT1MI' " n2T2M2; n3T3M3[Su) nlT " n2T2; n3T3; MIM2
(3.1) We now e x a m i n e the isospin s t r u c t u r e of both f a c t o r s in the s c a l a r p r o d u c t on the right side of eq. (3.1). F i r s t the left f a c t o r . By eq. (3.1) of I we can w r i t e
I p q ) = ~ N'~-n'(-~*)n-n' I T' n T n' (-n')'.(n-n'): ' T* 3/
(3.2)
where 1 1 1 1 N' = ~ {-~(p+q-n)+ T+l I)'.V _{½(p+q+n)+ T+l }'~ {½(p+q+n)T}:T {½(p+q_-n)T}'. ~
(p+q+l)' {½(p+q-n)+T+n'+l}'.{½(p+q-n)-T+n'}'
I_
1
( )J33.:
In eq. (3.2) T ' is the i s o s p i n s t a t e T ' = ½(p + n ' ) f o r m e d f r o m u n s t a r r e d ~ v a r i a b l e s and T* = ½(q -n+n') is the isospin s t a t e f o r m e d f r o m s t a r r e d v a r iables; the two i s o s p i n s T'T* a r e c o m b i n e d to give a s t a t e I T ) . The isospin f a c t o r s on the left of the s c a l a r p r o d u c t in eq. (3.1) can thus 5e w r i t t e n
M1M 2
I
M1 /
M3/
M2 . ?
T
M2
T
1
= (-1)2(Tl+T2+T3)[(2Tl+l)(2T2+l)(2T3+l)] 5 (3.4) where
(~a~7b - 77a~b)a+b-c (~ bT?c _ 77b~c)b+c -a (~c~?a _ 77c~ a)C+a-b (abc) =
1
(3.5)
[(a+b-c)' (b÷c-a)' (c+a-b)' (a+b+c+l)' ]-~ is the n o r m a l i z e d van d e r W a e r d e n i n v a r i a n t which is the SU(2) analog of the s c a l a r Su in SU(3). (abc) can be expanded [15].
M1M2 ma
mc
a mb mc
The s c a l a r p r o d u c t on the right of eq. (3.4) is to be taken only with r e s p e c t to the v a r i a b l e s f o r T1T2T 3 which a p p e a r in both f a c t o r s ; the vari-
703
SU(3) ISOSCALAR FACTORS ables for ~1~1~2~2~3~3 w h i c h a p p e a r o n l y on t h e r i g h t a r e f r e e . s u b s t i t u t i o n (3.4) eq. (3.1) r e a d s !
=
AIA'2A3
5
(1 1(1 1
!
T
With the
T
1(2 2 ( 2 2
g3
nln2n 3
(3n)
× [(-~[)" (~1 -~[)' (-"~)' (~. -~[)" (-'3)' (~3 -~)' ]-1, t
9
( 2) T ' +~ n i - n ' ( 2 T i + 1)e1_N ,i . H e r e I S F u h a s b e e n w r i t t e n f o r b r e v i t y w i t h A i' = to r e p r e s e n t t h e i s o s c a l a r f a c t o r on t h e l e f t of eq. (3.1). N o w e x p a n d S u s i m i l a r l y in p o w e r s of the ( v a r i a b l e s . T h e r e s u l t c a n b e written Su = N u
, 2 , , B u ( T 3 T 1 T 2 ) ( T I T 2 T 3 ) ( T 2 T 3 T 1) nln2n 3 T 1T 2 T 3 T
T
T
T
T'
"¢
X (1 n l ~lr*nl-nl (2 n2 ~2"*n2-n2 (3 n3 g3'~*n3-n3 '
(3.8)
where (-1)
i~z
~i -~iJul2!
u21, u13! u31! u23: u32, " s'.
' U 3 2 - ~ ) . a' . ' B u = 5..(U12_5).e,.(U21_e),.~:(U13_O)!K:(U31_K),.~:(U23_~), " P.( ,
,
fi!y:
1
× [ ( T 3 + T ~ I - T ~ ) ! ( T 3" + T 2~'- T 1 )*! (T~I+T "2 - T 3 ") i ( T 3" + T 2,, + T I. + I ) " , ]2-
× [ ( T I + T 2 - T 3 ). ( T I + T 3 - T 2 ) . ( T 2 + T 3 - T 1 ) . ( T I + T 3 + T 2 + l ) ] z 1 *
v.
"
T
"
m
*
T
× [(T2+T3-T , * ,,1 )., ( T 2, ~ T ~1 - T 3•) . , ( T 3 + T 1 - T 2 ) - ( T 2 + T 1 + T 3 + l ) . ] T1T2T 3 are independent summation T1 = T
variables
and T~T
- T1 , tt
???
*
T?
~ + T *1 ,
0 = T~" - T 2 + T 3
5 = TI-I
~=T2-T
K = T~ - T 2"~ + T *1 ,
~t = T
(3.10)
*
y = T 1 + T~ - T 3
fi = T 3 + T 1 - T
3 +1 2 ,
T
= T3 - T3
2 ,
a = T 2 +T 3 -T 1 ,
(3.9)
-T 1 + T 3 ,
/x = T 3 - T 1 + T 2 ,
704 It
C.K. CHEW and R . T . S H A R P t~
T i , Ti
a r e s t a t e s in the s a m e u n s t a r r e d i v a r i a b l e s (i = 1 , 2 , 3 ) . W h e n eq. (3.8) i s s u b s t i t u t e d in eq. (3.7) t h e f i n a l r e s u l t i s
ISF u = N u
~ A'A' a' B n,ln,2n ~ 1 2~3 u T~T~T~
(3.11)
× ((T~ T i T1)(T~T'2T2)(T~TT3 T3)
(T 1 T 2 T 3 ) ( T "1T~T~)(T~
T~TI")(T~T ~T~')).
T h e e x p r e s s i o n (3.11) c o n t a i n s s i x e x p l i c i t s u m m a t i o n s . T h e i s o s p i n f a c t o r i s a t r i p l y s t r e t c h e d 1 5 - j s y m b o l . It i s e v a l u a t e d in a p p e n d i x A in t e r m s of a q u a d r u p l e s u m . H e n c e t h e r e s u l t (3.11) c o n t a i n s t e n s u m s in a l l . T h e n o r m a l i z a t i o n c o n s t a n t N u i s e v a l u a t e d in a p p e n d i x B; it i n v o l v e s a seven-fold sum. In a p p e n d i x C we s h o w t h a t a s c h e m e f o r o r t h o g o n a l i z i n g d e g e n e r a t e SU(3) i s o s c a l a r f a c t o r s b a s e d on m i x e d C a s i m i r o p e r a t o r s and d i s c u s s e d e a r l i e r b y us [13] a p p l i e s r e a d i l y to t h e p r e s e n t s c h e m e . T h e o r t h o g o n a l i z a t i o n d o e s not s p o i l t h e s y m m e t r y p r o p e r t i e s . One of us (C. - K . C) i s p r e p a r i n g a p r o g r a m f o r c o m p u t a t i o n of i s o s c a l a r f a c t o r s , o r t h o n o r m a l i z e d a c c o r d i n g to t h e s c h e m e d e s c r i b e d in a p p e n d i x C. O u r t h a n k s a r e d u e to H. C. y o n B a e y e r f o r h e l p f u l d i s c u s s i o n s .
APPEND~
A
W e w i s h to e v a l u a t e t h e i s o s c a l a r e x p r e s s i o n
Y = YT3T3
(A. 1) ?
$
?
~
t
~
~
~
tt
$
tn
~
$
v11
=- ((T1T1T1)(T2T2T2)(T3T3T3)[(T1T2T3)(T1T2T 3 )(T2T3T 1 )(T3T1T 2 )) , ' " "1 are T1T1T
w h i c h a p p e a r s in o u r f i n a l f o r m u l a (3.11).
in t h e s a m e i s o s p i n
variables ~i~/i and satisfy T~ +T~ = TI; similarly for 2 and 3. We follow closely the methods used by Bandzaitis, Karosiene and Jucys [16] to evaluate the X-coefficient to which our Y is formally similar [17]:
Fabc~ X t degh f ~ =
<(abc)(def)(ghj)l(adg)(beh)(cfJ))"
T h i n k of e x p r e s s i o n (A.1) a s t h e s c a l a r p r o d u c t of
((T3T3T3) I (T2T3 T 1 )(T1T2T 3) ) ,
(A.2)
with tT
$
tit
tt
$
tlt
~
~
,
<(T1T2T 3 )(T3TI T2 )t (TI TI T1)(T2T2T2) > •
(h.3)
In e x p r e s s i o n s (A.2) and (A.3) s c a l a r p r o d u c t s a r e to b e t a k e n w i t h r e s p e c t to v a r i a b l e s w h i c h a p p e a r in b o t h f a c t o r s ; o t h e r v a r i a b l e s a r e f r e e . A l s o t h e scalar product
SU(3) ISOSCALAR FACTORS
705
r~ rl for example, is to be i n t e r p r e t e d a s
D1
., ) M1
with T~'
DI=( =
i
.,(
T i?1
T ,1
M 1 M"1 MI
)
(T,I - M'I)I (T'1 +M'I)'
]½
,, ~ , i (T~ - M[): (T~ + M~)I (T~ - M 1 )[ (T~ + M1 ).J
(A.4)
The state (A.3) has isospins T'~T~T1T 2 added to give T~. The states (A.2) with f r e e T3T ~ are a complete set of such states with orthonormality e x p r e s s e d by ( T 3 T~ IT3 T~ ) = 6 T3 T35 T~ T~ / (2 T 3 + 1)(2 T~ + 1) in an obvious notation. Hence we may write
<(T1T2T3 )( T3T1T2 )1 (T~T'I T1) (T~.T'2T2) >
= E ((T~T;T3)[(T~T;T~I')(T1T2T3)) r3r;
(2T 3 + 1)(2T~ +
1)YT3T~ ,
(A.5)
where the coefficient YT^T*,, is just the expression (A.1) to be evaluated. Expand both sides of ~q.~(A.5) substituting f r o m eq. (3.6) for all the threeisospin invariants and equate coefficients of
Mr
M;. M1 M2 M~
The r e s u l t is
r~ r;. rj
r; r~ <2
rl* rl rl
r*2 rl r2
M'~ ) (M~ M'I M 1) ( M~. ) D1D2D3 • IiM~ (M~ M~. M~ ) (M~ M~ M'2 M2 T* T~ T 3
T3T~
MT3
T" T~ T"I
T1 T2 T3 "
M~ M2 r e g a r d M3M3M1M 1 as free
In eq. (A.6) we may by the f r e e ones. Now multiply by
(A.6)
and
M3M~M2 as
determined
706
C.-K. CHEW and R.ToSHARP __
"
m
T"
(M; M~ M 1 with M;M 3 (and
~T!
and sum over M I M I p r o p e r t y of the Wigner coefficients a
b
T 1 T2 T3 1 M2 M3 T
hence M3) fixed, using the orthogonality
c
a
b
~, 2~+i
Y~/a The r e s u l t is T*3 T~ T 3 -1
Y
(M~ M'3 M 3
T"2 T~ T "1
M~M~M~M1
(MT~ T~ T 3 x / T 3 T I T2, ×
~,,,,,,\~,,,,}{ M~.I~, 3 '~'3
i
T l T2 T3
M~
T 1 T~ T 1
T 2 T~. T 2
M,2,,)~M~ M I M 1)(
) D1D2D3 .
M~M~M 2
(A.7) The right side of eq. (A.7) cannot depend on M3M~ which a r e still f r e e , !
~r
T
*
*
W
,
~
'T
o
T!
.
so we may s e t M 3 = T3, M 3 = - T 3. T h e n a l s o M 3-- T 3 ; t h e M 3 s u m c o n t a m s one t e r m only. T h u s
r~ T~
T3
Y : ( - T~ T'3 T ; - T 3' )
( T3"
x'T3
rl
×
T1* -M1-MI
r2
2
-1
T ~
T3 rl *
~ ('~* " " -T~ M "1 ) MIM'IM; 13-1v11 T2"
MI+M1-T3 ' "
r3
T"1
T2*
T ~"
""1 .... 1 MI-MI-T 3 T'~
r~
rl r l
( r; r~ r2 ., , ,. , . ) DID2D3 • ~M1-MI-r3 r~-r~+M1-M~+M~ T3-r3-MI
(A.8)
The last t h r e e Wigner coefficients in the summation contain one sum each, so we have a sixfold sum. The M1M 1 sums can be effected. I n s e r t the values of the s u m l e s s Wigner coefficients and substitute
SU(3) ISOSCALAR FACTORS
T~I
T'I T1
+
(-M1-M i M I M1 )=(-1)M1
707
,_TI
2T1
I(T~+T'I-TI)~ (T'I+TI-T*I)~ (T~-MI-MI)! (TI+MI)! (TI+TI+T~+I)! 1½ ×
.
t_
.
.
.
.
.
.
(T -M )' (T' +M' )' (T' -M' )' (T*+M +M' )' (T +T*-T' )' 1
1"
×~
(-
1
1"
1
1"
1
1
1"
1
1
I
1"
i)% ( 2 T ~ - x ) ! (T~+TT1-MI-%)~
(A.9)
x (T~-M1-M'I-X): (T~+T'I-TI-X):x~(TI+T'I+T~+I-x)~ Eq. (A.9) i s o b t a i n e d f r o m eq. (A.3) of r e f . [17] with ,
,
'
a = T1 - M1 - M1 ,
b = T
d:
e : T 1 + M 1 - T~ + M 1 ,
T 1 - T~ + M~ ,
+ T - T1 ,
c = T
- M1 ,
f = 0,
a n d with a R e g g e p e r m u t a t i o n a p p l i e d ,to ~r o w s a n d c o l u m n s ; an e x a c t l y a n? a - iT! l o g o u s s u b s t i t u t i o n i s m a d e f o r t h e (T2T2T2) W i g n e r c o e f f i c i e n t . T h e M1M 1 s u m s c a n t h e n b e e f f e c t e d with t h e h e l p of t h e w e l l - k n o w n f o r m u l a
X
1 (a+b+c+d)! (a+x)! (b+x)~ (c-x)! (d-x)~ = (a-¢c)~ (a+d)' (b+c)~ (b+d)! "
The result is *
Y = ( - 1) T2+ T 3 - T~'+ T ~ - 2 T ~ - T{+ T~
~
T
(T 3 +T 3 - T 3).
[
~(T2+T 3 -T~)~
×
~
(2~+T'2'-T3)! (T~+T'I-T1)'. (T'I+T1-T~)! T?T ?? ~ * T! ~. ? * TVT Vt T (T 2 +T3-T1)!(TI+T3-T2 ). ( T I + T 2 + T 3 + l ) .
(TI+TI+TI+I). (T2+T2-T2). (T2+T2-T2). }½ t 1 2 + z 3 - 1 1 ~. [ 3 1 + T 3 - T 2 ) . [TI+T1-T1). J ?
*
T
I
?
(-1)
xyM 1
?
!
*
(T;+T2+T~+I):
. 2 + T 3* + l ) . ' (T .1. .+. T
×
*
* ' '(T~'+ (T2+T2-T2).
x+y+TI+M 1
(T2+T2-
T2-Yj.
]-~
*" T 2" - T3)A
,
(2 T2-Y)'.
( T 2 + T 1- T 3 - y ) .
!
*
Tv
*
V
1
(T3+T3+T3 + 1)I ( T I +T2- T'3")!I~ (TI+T3 _T2). (TI+T2+T3+I). i
C. -K.CHEW
708
(2T!-x)!
X
y! (Tb+T2+Tz+l-y)!x!
and R.T.SHARP
(Ti-Th-Ti+Ti+T;+TT-x-y)!
(TT+Ti-Tl-x)!
(Tl+Ti+TT+l-x)!
(Tg-T{+Ti-x)!
x 1 I-(T2-Th+T$+M1)!
If T; = Tt = 0, Y reduces
APPENDIX
(A.lO)
(Tl-Ml)! to a singly stretched
9-j,
or X coefficient.
B
We wish to calculate the metric matrix (SU,,1s,> which arises when the unnormalized SU = N;lSU are used as base vectors in the degenerate space which they span. The no;ma,liza$ion constant NU which appears in eqs. (2.1)(3.11) is given by N, = (S,[ S,) -2 in terms of its diagonal elements. Because of (2.9) we may write SU = (P3! %!)3 (-1)
fi3+“21
&
43
0
%P3+43) -&3+43)
>
;3u+!D3-1)
X more terms
,
where 43
-1
+3u = N3U
0
@. 1)
u
P3
$(P3+43) $(P3+43) >12
u13 U23~*“31~*z’32 = (-1) u21 PBu12Du21 (7@2 -7?2sl)s 12 21 771 q2 1 2
=N;;
X
c T1T2n l”1
I$ll2l,~~~,,C::~
p1q1
p2q2
493
i nlT1’
n2T2
0 $(P3+43) >
I:;;::;)
u (B.2)
.
The (D3-1) other terms in eq. (B.l) refer to a summation over n3T3M3. Since all D3 terms contribute equally we have
(i,, 1itu) =
(-l)ui1-u21
p3! q3! D3< $3u, j 4,,,
s
(B.3)
SU(3) ISOSCALAR
FACTORS
709
We have only to calculate (~h3u , ]~3u ) Taking the scalar product of eq. (B. 2) with
Plql
M1
½(P3 q3)
P2q2 \ / T1T2
lniTIM1) n2T2M2/\MIM2 \
½(P3+q3) ) '
gives an expression for the a s y m m e t r i c i s o s c a l a r factor
Plql. P2q2 [q3 P3 < n 1T 1' n2 T2 ½(P3+q3) )
( T1 T2 = N3u ~M1 \M1M2
½(P3+q3)\
Plql
P2q2 n2T2M2
(p3+q3)/(nlT1MI' [
~3u>"
(B.4)
Using eq. (3.2) we can transform the left factor of the scalar product on the right side of eq. (B.4): / TIT2
½(P3+q3) \ n plql P2q2 L ½(P3+q3) / 1TIM1; n2T2M2 ) 1 2
M1
(B.5)
N,1N~2~lni~nl-n'l~2nT2~n2-n'2 T'1T~ n,ln,2
(-nl)I (nl-n~)! (-n~)! (n2-n~)I v
T~. T~.
T1
T2
½(P3+q3)
,
The isospin state on the right is formed by adding TIT 1 to give T1, then T2T 2 to give T2, and finally TIT 2 to give ½(P3+q3) (with M = ~(p3+q3)). Similarly expanding J/3u we get *
~3u = (-1)U21p
F~lnl ~1,n 1-n'1~2n2 ~2,n 2 -n'2
l/U
t
v
T 2 ~ 13+u32+nl-nl-n2) \ 1 /U v v ) T~. T 1 ~t 23+u31+n2-n2-nl) ' ½(P3+q3)
where
(B.6) T
?
u12! u21! s! (-1) -nl-n2+n2
F =
(n2-n~.)! (u12-n2+n2)!' (nl_n~): (u21-nl+nl)~' (nl_n2_nl):, , ,
v
× [(u13+nl-n2-n1)I u32'. (u12-n2+n2)!,
t
, (u12+u32+u13+s÷n 1 ,
(n2_nl_n2)!' , v
v
+1).]
1
(u13 +u32+nl-nl-n2+ l ). ×I
(u23+n~.-~l-n2)! u31: (u21-n1+n~): (u21+n23+u31+s+n'2+l )[~ ½ ' ' ' (u23 +u31+n2-n2-n l +1).
(B.7)
710
C. -K. CHEW and R. T.SHARP
When eqs. (B.5) and (B.6) a r e i n s e r t e d in the s c a l a r p r o d u c t in eq. (B.4) t h e r e a r i s e s the i s o s p i n s c a l a r p r o d u c t
T1 T1
T1
T'1 T 2 5(u13+u32+n1-nl-n2) ~
T'2 T 2
T2
T'2 T~
= GX
~(u23+u31+n2-n2-nl)
½(P3+q3)
½(P3+q3 )
,
(B.8)
/
where *
*
I
!
T 2- T 2 + T 1 -~(u23 +u 3 l + n 2 - n 2 - n l ) G : (-1)
(B.9) t
~
v
1
X [(2Tl+l)(2T2+l)(u13+u32+nl-nl-n 2 + 1) (u23+u31+n2-n2-n 1 + 1) ]~
,
and X is the X - c o e f f i c i e n t I
X =X
*
1
T
T
t
1
!
T
T'1 T 2 ~(u13+u32+nl-nl-n 2) T~ T 2 ~(u23+u31+n2-n2-n1)
LT1 T2
,
(B.10)
½(P3+q3 )
which is s t r e t c h e d in its t h i r d c o l u m n and t h e r e f o r e c o n t a i n s a double s u m
[16]. H e n c e we find
I
Plql . P2q2 I q3 P3 u \
u21
n 1 T 1 ' n 2 T 2 ] 0 ½(P3+q3)/ = (-1)
,
,
N3u ~ , N1N2FGX, nln 2
(B.11)
in t e r m s of which the m e t r i c m a t r i x is g i v e n b y ^ ,, >= /Plql P2q2 I q3 P3 u' \ ( @3u, IJ/3u ~ \ " nlTIT2 n l T l ' n 2 T 2 I 0 ½ 0 3 + q 3 ) /
x
< Pl ql . P2q2 I q3 n 1 T 1' n2T 2 0
P3
u )
½(P3+q3 )
(--uN3u')-I N3
(B.12)
a triple sum over a product of two factors each containing a quadruple sum. The matrix
C
Following a suggestion of O'Raifeartaigh, Macfarlane and Rao [18], the present authors [13] evaluated the matrix of the mixed Casimir operator
SU(3) ISOSCALAR FACTORS
711
(c.1)
G12 = ½ T r (A1A1A 2 - A2A2A1)
in the d e g e n e r a t e s u b s p a c e of the ~ 3 u of eq. (B.2). H e r e A is the m a t r i x whose e l e m e n t s a r e g e n e r a t o r s of SU(3) t r a n s f o r m a t i o n s ; G12 , a H e r m i t i a n o p e r a t o r which c o m m u t e s with all the other o p e r a t o r s of i n t e r e s t , is s u i t able f o r lifting the d e g e n e r a c y . Diagonalizing its m a t r i x y i e l d s o r t h o g o n a l eigenstates
(c.2)
:
w h e r e eg is a p h a s e f a c t o r to be assigned. In the context of the p r e s e n t p a p e r G12 should be r e p l a c e d by G123 = -~(G12 +G23 +G31 ) ,
(C.3)
to o p e r a t e on the s u b s p a c e of d e g e n e r a t e S u of eq. (2.1). Since S u is an SU(3) s c a l a r we m a y s u b s t i t u t e A 3 = -(A 1 +A2), whereupon we get G123 = G 1 2 + I ,
(C.4)
w h e r e I is a multiple of the identity in the d e g e n e r a t e s p a c e and m a y be d i s regarded here. In the f o r m (C.4), G123 is an SU(3) s c a l a r in the s p a c e of 1 and 2 and we need r e t a i n only a single t e r m in the expansion (2.9) of the o p e r a n d s t a t e Su, say the t e r m
(_l)U21~3u = N31u(_l)U21 t q3 0
!)3
u
)
½(P3+q3) ½(P3+q3 )
(we d r o p its coefficient). Thus the old s c h e m e is equivalent to the new with the r e p l a c e m e n t ~ 3u ~ (-1) u21 Su ; by eq. (C.2) the o r t h o g o n a l i z e d s c a l a r s are
sg=
gu(-1)u21Cug.
u12 It r e m a i n s to a s s i g n an o v e r a l l p h a s e to Sg. This is conveniently done by r e q u i r i n g that for e a c h g the (Pug with g r e a t e s t magnitude have the p h a s e (-1) u2~l. This choice does not affect any of the s y m m e t r i e s of the i s o s c a l a r f a c t o r d i s c u s s e d in s e c t . 2, and can be g e n e r a l i z e d to c o v e r the c a s e in which the g r e a t e s t m a g n i t u d e is a s s u m e d by m o r e than one (Pug; b e c a u s e of the a n t i s y m m e t r y of G123 in s t a r r e d and u n s t a r r e d v a r i a b l e s the e i g e n v a l u e g is r e v e r s e d in sign u n d e r r e v e r s a l of m a g n e t i c quantum n u m b e r s . The r e l a t i v e p h a s e of s y m m e t r i c and a s y m m e t r i c i s o s c a l a r f a c t o r s b e c o m e s m o r e c o m p l i c a t e d . When the u label in eq. (2.10) is r e p l a c e d by g the p h a s e f a c t o r (-1) u21 on the right is r e p l a c e d by ~, which is the sign of
(.1)e(P3+P2-Pl-q3+q21- - ql - s ) ~ u
( Pl ql ~°ug\ ~
P2 q2 P3 q3 u~'] 1 ; , ; 0 ~ ( p l + q l ) z-s ~(p2+q2-s) 0 ~(p3+q3) ;
(c.6)
712
C. -K. CHEW and R. T. SHARP
T h e c o e f f i c i e n t of e a c h (Pug i n the e x p r e s s i o n (C.6) h a s p h a s e ( - 1 ) u 2 1 ; the o v e r a l l s i g n c a n only be d e t e r m i n e d a p o s t e r i o r i i n e a c h c a s e of i n t e r e s t .
REFERENCES [1] J. J. De Swart, Rev. Mod. Phys. 35 {1963} 916, [2] P. McNamee, S.J. and F. Chilton, Rev. Mod. Phsy. 36 (1964) 1005. [3] L. C. Biedenharn, J. Math. Phys. 4 {1963} 436; G. E. Baird and L. C. Biedenharn, J. Math. Phys. 4 {1963} 1449; 5 (1964} 1723; 5 {1964) 1730. [4] J. P. Elliott and M. Harvey, Proc. Roy. Soc. {London} A272 {1963} 557. [5] M. Moshinsky, Rev. Mod. Phys. 34 (1962) 813. [6] J. K. Kuriyan, D. Lurie and &. J. Macfarlane, J. Math. Phys. 6 {1965) 772. [7] N. Mukunda and L.K. Pandit, J. Math. Phys. 6 (1965} 746. [8] K.T. Hecht, Nucl. Phys.62 {1965} 1. [9] Hou Pei-Yu, Scientia Sinica {1965} 367. [10] G. Ponzano, Nuovo Cimento 41A (1966) 142; Istituto di Fisica, Torino, Italy preprint. [11] R. T. Sharp and H. yon Baeyer, J. Math. Phys. 7 {1966) 1105. [12] D. Kleima, Nuc].Phys. 70 {1965} 577. [13] C. -K. Chew and R. T. Sharp, Can. J. Phys. 44 (1966) 2789. [14] R. T. Sharp, Nuovo Cimento 47A (1967) 860. [15] V. Bargmann, Rev. Mod. Phys. 34 (1962) 829. [16] A. Bandzaitis, A. Karosiene and £. Jucys, Liet. Fiz. Rin. 4 {1964} 457. [17] R. T. Sharp, Nucl. Phys., to be submitted. [18] L. O'Raiffeartaigh, ~. J. Macfarlane and P. Rao, talk given by O'Raifeartaigh at Toronto Conference on Particle Symmetries, October 1965.