- Email: [email protected]
Spin test for a fermion (J > 12 ) decaying into a spin-12 fermion (ΛO) plus a spin-1 photon
~
Nuclear Physics B25 (1971) 519-524. North-Holland Publishing Company
SPIN TEST FOR A FERMION ( j > ~) DECAYING INTO A SPIN-I/z FERMION (A °) PLUS A SPIN-1 PHOTON S. KRASZNOVSZKY
Central Research Institute for Physics, Budapest Received 16 September 1970 Abstract: The decay of a fermion of arbitrary spin into an unstable spin-½ fermion (A°) plus a photon is treated with density-matrix techniques and the helicity formalism of Jacob and Wick. A complete determination of the multipole parameters of the parent fermion is possible only in terms of the particle - and polarization angular distributions of the decay hyperon (A°).
1. I N T R O D U C T I O N R e c e n t l y a p e a k h a s b e e n o b s e r v e d in the A% e f f e c t i v e - m a s s d i s t r i b u tion b e t w e e n 1300 and 1400 MeV (refs. [1-3]), which m a y be e x p l a i n e d by a A°~(1350) r e s o n a n c e d e c a y i n g into h ° and V (ref. [3]). T h e a i m of this p a p e r is to give a m e t h o d f o r the d e t e r m i n a t i o n of the spin of the A°~(1350).
2. F I N A L D E N S I T Y M A T R I X In g e n e r a l the final d e n s i t y m a t r i x p(A ° +y) is known if the initial d e n s i t y m a t r i x p(h°~(1350)) and the d e c a y m a t r i x cy~ a r e known [4], p(A°+'/) = ~p(h°y(1350))c~ + ,
(1)
where
{c~]g+) ij =c)E ~i . T h e spin state of a s p i n - J A°v(1350) at p r o d u c t i o n is d e s c r i b e d by a set of ( 2 J + 1) 2 c o m p l e x m u l t i p o l e p a r a m e t e r s t L M (ref. [5]) with L ~< 2J, w h e r e M = - L , . . . , L and L =0, 1 , 2 , . . . , 2J. It is c o n v e n i e n t to expand p(A°~(1350)) a s a s e r i e s of ( 2 J + 1) - d i m e n s i o n a l m a t r i c e s TLM ( r e f s . [6, 7]) p(A°(1350)) i , M
= ( 2 J + l ) -1
2J ~
L ~
L=OM=-L
(2L+I)tLM* TLM i"i''
w h e r e T L M a r e a c o m p l e t e o r t h o n o r m e l set of ( 2 J + 1) 2 c o m p l e x m a t r i c e s that m a y be c o n s t r u c t e d f r o m the o p e r a t o r s of the c o m p o n e n t s of the a n -
(2)
520
S. KRASZNOVSZKY
g u l a r m o m e n t u m J in a m a n n e r s i m i l a r to that in which the s p e r i c a l h a r m o n i c s Y L M a r e c o n s t r u c t e d f r o m the c o o r d i n a t e s x, y and z (ref. [7]). F o l l o w i n g the t r e a t m e n t of B y e r s and F e n s t e r [6] 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 m a y be s u b s t i t u t e d f o r the m a t r i x e l e m e n t s of the TLM:
TLM[M,, M, = { J , L , M ' , M [ J,M"}
M" = M ' + M .
with
T h e p r o p e r t i e s of the m u l t i p o l e p a r a m e t e r s e r t i e s of the d e n s i t y m a t r i x :
(3)
follow f r o m the g e n e r a l p r o p -
IL_ M = (-)Mt*LM ,
too = 1 .
(4)
F o r a p a r i t y - c o n s e r v i n g p r o d u c t i o n p r o c e s s (for A°y(1350)) when the p r o d u c t i o n n o r m a l ~ is c h o s e n a s the p o l a r a x i s [8],
tLM = 0
for
M odd.
(5)
In g e n e r a l an e l e m e n t of the d e c a y m a t r i x m a y be w r i t t e n [5]
~a~fl, M =
~ A 4~
D J* " 0,0) ~a~fl M, ~-~fl(¢~' "
T h e A~a~fi a r e the h e l i c i t y a m p l i t u d e s ,
(6)
and ;ta, ;tfi a r e h e l i c i t i e s f o r the
d e c a y p r o d u c t s of the p a r e n t r e s o n a n c e [9]. If p a r i t y is c o n s e r v e d in the d e c a y p r o c e s s [9]:
A_~c _~fl = 77~a 77t3(-) J-Sa-Sfi A ~a~ fi •
(7)
H e r e S a , S fi and ~ a , ~fi a r e the s p i n s and i n t r i n s i c p a r i t i e s of a and fl, J, ~? a r e the spin and the p a r i t y of the p a r e n t r e s o n a n c e , r e s p e c t i v e l y , and j* DM, ha_hfi(~, 0,0) s t a n d s f o r the m a t r i x e l e m e n t of the r o t a t i o n o p e r a t o r [7], w h e r e 0 and ~b define the d i r e c t i o n of the r e l a t i v e m o m e n t u m in the c . m . of the two (a and fi) p a r t i c l e s . U s i n g the r e l a t i o n s [7] J*
D M,, ~ = (-)
M"-;~
J
D_M,,_; ~ , L
=
L and the s y m m e t r y a n d o r t h o g o n a l i t y r e l a t i o n s of the 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 [7], p c a n be e x p r e s s e d a s p ?) a~ ~o A ~ ^Of +l ~
= (_)J+X'-2;t 4~1 A ol~fi A ~ x J
2 J + I ) I / 2 L ,~M
x(J,J,~,-~'IL,~-A
(2L+1)1/2
?~LMJJM, ~_k,(¢),0,0) ,
SPIN TEST FOR FERMION DECAY
521
where =
?t0~
•
Obviously eq. (8) is t r u e even if the p a r e n t boson r e s o n a n c e decays into two bosons.
3. DECAY DISTRIBUTIONS We d e r i v e equations containing only the p a r a m e t e r s of the A ° (because the e l e c t r o n - p o s i t r o n p a i r is not a p e r f e c t a n a l y z e r for the p o l a r i z a t i o n of the photon). Taking (9) we obtain
0~x~(a°) = (_) J + ~ f i - 2 h a + ~ × ~ (2L+I)I/2(J,J,X L,M
14n (2J+ 1)1/2
~kflA~c~ht3A*~'~hfi
-Xfl,-X +X~IL, X -a /tLMZJM, X _X,(~,O,O). (10)
H e r e we r e m a r k that in the special case, when the ;~fi= 0 and ~ = - ~, - ½, and ~ our eq. (10) is the same as B u t t o n - S h a f e r ' s [10]. One can easily get the angular distribution, the angular distribution of the longitudinal p o l a r i z a t i o n s and the angular distribution of the t r a n s v e r s e p o l a r i z a t i o n s for the A° hyperon using the following e x p r e s s i o n s : (i) The angular distribution
I(O, ~)) = T r
(p(A°)) .
(11)
(ii) Expectation value of a angular m o m e n t u m o p e r a t o r : (A} = T r (p(A °) A ) . T r (p(A°)) 1 hfl=+landeq. Bearing in mind that ~ =+3,
(12) (7) we get
px_~(A°)= ~
[[A½112(-)Ln(L1o)+[A½_ ll2n(3L~]t*LMYLM(0,(0),
°)1, (A
[IA½_ll 2(_)L nLo(3)+!AI1
2z
L,M
:
,
L,M ~-~I(A°)
Ol
=
~ [E(A½1A*_I(-) L+I+A L,M
½1)]
(3) . L* nLltLMC~MI(¢P'
0,0)
'
(13)
522
S. KRASZNOVSZKY
and
o_~½
=
T, , p~_~
where n(1) = (_) J-½ ff2J+ 1 (j, j, I, - ½]LO} Lo 47T ' n (3) : (-) Lo
J-~ ~r2j+ 1 (J, J, ~ - ~/L0) T '
riLl = (-)J-½ YLM(O, (p) :
4~ L* . f~ 2+ -1 L DMo(~ , O, O)
L* v/-2L + 1 L* Q)MI(qS,0,0) = ~ DMl(qb, 0, 0) , j 3 j_~ = ~?/(1,~p(-) -~ = 7(-) Eqs. (11) and (12) and the r e c u r s i o n relations for the Clebsch-Gordan coefficients [7, 10] yield
I(O,(~) = G G 2 L~ven M= - L
12+
-1
L(L+ 1)liA~_ll2
1 - (J+ ~) (g- ½)
1
x n (1) Lo I I M Y L M ( O , 4)) _
^
IPAoA :
2J
L
G
G
Lod d M= - L
(14a)
213J(J+I)-•-L(L+I)iA½_112 L
(J+ ~)(J- ~)
- ]A½1[ 21
×n (1) Lo t I M Y L M ( O , 4))
I P A ° x : 2Rep12 21: 2Re L,~ M E(AI~I AT~-l" (-)L+I +A ½-1 A i l ) { L ( L + I
t
[l+(-1)L+l]}[L(L+l)]-2[(J+a2)(J-~)]-~n(1)t
L*
* (DMI(~b, 0,0) Lo L M
(14b)
) _ (j+ ½)2 (14c)
SPIN TEST FOR FERMION DECAY
IPAo;=-2Imp,
I =-2Im
~-~
E
L,M
523
6(A½1A*}_I(-)L+I+A½_IA*~I){L(L+I)
_(J+})2[I+(_I)L+I]}[L(L+I)]-2[(j+~)(j_½)]
n L o ~ L M ~ , M l t q J , s , 0 ) . (14d)
H e r e A, } , ~ a r e unit v e c t o r s pointing in the d i r e c t i o n s of A°, (fix ~ × n × A r e s p e c t i v e l y and PAO is the p o l a r i z a t i o n v e c t o r of A°.
4. SPIN DETERMINATION The p u r p o s e of this p a r t is to give s o m e useful e x p r e s s i o n s for the m u l tipole p a r a m e t e r s , which can be u s e d in a " m o m e n t u m a n a l y s i s " [11]. F u r t h e r , s e v e r a l inequalities will be given which have a grea.t value in p r a c t i c a l d e c i s i o n s , concerning the higher a l l o w e d - s p i n quantum n u m b e r s . Calculating the r a t i o of the weighted a v e r a g e s we obtain e x p r e s s i o n s which a r e independent of tLM (provided that tLM¢ 0) f o r all the allowed v a l u e s of L and M, and in the c a s e of any given J , one can d e r i v e the following useful relation:
fIP AoA I~LM(O, 4~)d~ •^) CbMl(q5 , 0,0) dl2 f / ( p AO~_ Zphoy L
E
[3J(J+I)-I-L(L+I)IA~_ll2
Lodd, M L.
IA}I}21
(J+ ~ ) } J - ½). . . . .
Lodd, M
o
., L(L+I):2(J+½) 2 c(A ~l d i - l +d~ -l A½1) ~/(L(L+ I) (J+ ~2)(J- ~)
(15)
Eq. (15) follows f r o m the orthogonality p r o p e r t i e s of the functions and O ~/1((~, 0,0)
YLM(O, ~)
f YLM(O, ~) YL,M,(O, c~)dr2 : 5LL, 5MM, , L' f e d ~/1 (qS,0,0) CbM,l(qS,0,0) di2 - 2 L4~+ 1
5LL'SMM''
The r e a d e r can find in r e f s . [11,12] s o m e useful f o r m u l a e for the s t a t i s t i c a l t r e a t m e n t of weighted a v e r a g e s . The l e f t - h a n d side of eq. (15) can be d e t e r m i n e d f r o m the e x p e r i m e n t a l data [11]. F r o m eq. (15), which r e l a t e s longitudinal to t r a n s v e r s e p o l a r i z a t i o n s , 1 2 , ]A_1 11 2 and one can * d e t e r m m " e t h e p a r a m e t e r s [A_ll 2 ;~e(AXlA~1 +A~-IA~1)B e c a u s e of the n o r m a h z a t i o n of I(O, dp), we have 2 22 2one m o r e equation f o r the p a r a m e t e r s . Hence e v e r y multipole p a r a m e t e r can be d e t e r m i n e d f r o m the decay d i s t r i b u t i o n s of eq. (14). Knowing the multipole p a r a m e t e r s for any J the spin of h°7 s y s t e m can be obtained [11,13].
524
s. KRASZNOVSZKY WehaveA½_ 1=0,
1
1
if J=5, b e c a u s e / ~ ( A ° ) = - f i ( y ) a n d I x A O ± ~ y ] ~
S i n c e t h e e x p e r i m e n t s s h o w c o n s i d e r a b l e a n i s o t r o p y in t h e a n g u l a r d i s t r i b u t i o n of A°~(1350) ~ A ° + y d e c a y one c a n c o n c l u d e t h a t t h e s p i n of t h e A%(1350) i s h i g h e r t h a n ½. We w a n t to e m p h a s i z e t h a t t h i s m e t h o d g i v e s a p o s s i b i l i t y to d e t e r m i n e t h e s p i n of t h e A ° ~ 1 3 5 0 ) , s u p p o s i n g t h e m u l t i p o l e p a r a m e t e r s of t h e i n i t i a l s t a t e (A°~(1350)) c a n n o t b e n e g l e c t e d (tLM ~0) a n d g (the A°~(1350) spin) > ½, F u r t h e r e x p e r i m e n t a l c h e c k s c a n b e m a d e on t h e d a t a , u s i n g s o m e i n e q u a l i t i e s w h i c h a r e i m p o s e d on t h e m u l t i p o l e p a r a m e t e r s by t h e p o s i t i v i t y p r o p e r t y of t h e d e n s i t y m a t r i x [14]. U s i n g t h e r e l a t i o n d e r i v e d f r o m t h e h e r m i t i c i t y of t h e d e n s i t y m a t r i x p ( r e f . [15])
tLM(n) = (-)MtL_M(n) , o n e h a s [15]
S n = ( 2 J + l ) -1 ~ L=O
(2L+l)(tLo)2+2
~ ltLM(½n)] 2 M=I
,
for n even,
Sn= (2J+ l ) - i
2J r L .~ (2L+ I) LtLo(P)tLo(q)+ Z R e (-)MI LM(P) I L_M( q) 1 , L=O M=I
f o r n o d d , w h e r e tLM(n) = f(J, tLM) a n d n=p+ q (p, q a r e i n t e g e r s ) a n d t h e i n e q u a l i t i e s c o n s i s t i n g of S n - s ( r e f . [14]) a r e d e p e n d e n t on J.
REFERENCES
[1] Van Yun-Tchan et al., preprint, JINR P-1615 (1964); P r o c . XII. Int. ConZ. on high energy physics, Dubna, Vol. 1 (1964) 615. [2] E . G . B u b e l e v e t al., Phys. L e t t e r s 24B (1967) 246. [3] G . B o z d k i e t al., Phys. L e t t e r s 28B (1968) 360. [4] U.Fano, Rev. Mod. Phys. 29 (1957) 74. [5] J . D . J a c k s o n , in High energy physics, 1965 (Gordon and Breach, Science P u b l i s h e r s , New York, 1966). [6] N . B y e r s a n d S . F e n s t e r , Phys. Rev. L e t t e r s 11 (1963) 52. [7] M. E. Rose, E l e m e n t a r y theory of angular momentum (Wiley, New York, 1957). [8] R . H . C a p p s , Phys. Rev. 122 (1961) 929. [9] M. Jacob in G. F. Chew and M. Jacob, Strong interaction physics (Benjamin, New York, 1964). [10] J.Button-Shafer, Phys. Rev. 139B (1965) ~07. [11] J.Button-Shafer, Phys. Rev. 134B (1964) 1372. [12] N. B y e r s , CERN 67-20. [13] P . E . S c h l e n e t al., Phys. Rev. L e t t e r s 11 (1963) 167. [14] P . M i n n a e r t , Phys. Rev. 151 (1966) 1306. [15] S . K r a s z n o v s z k y et al., Report (Budapest) of the Central Res. Inst. for Phys. 18 (1970) 3.
Nuclear Physics B25 (1971) 519-524. North-Holland Publishing Company
SPIN TEST FOR A FERMION ( j > ~) DECAYING INTO A SPIN-I/z FERMION (A °) PLUS A SPIN-1 PHOTON S. KRASZNOVSZKY
Central Research Institute for Physics, Budapest Received 16 September 1970 Abstract: The decay of a fermion of arbitrary spin into an unstable spin-½ fermion (A°) plus a photon is treated with density-matrix techniques and the helicity formalism of Jacob and Wick. A complete determination of the multipole parameters of the parent fermion is possible only in terms of the particle - and polarization angular distributions of the decay hyperon (A°).
1. I N T R O D U C T I O N R e c e n t l y a p e a k h a s b e e n o b s e r v e d in the A% e f f e c t i v e - m a s s d i s t r i b u tion b e t w e e n 1300 and 1400 MeV (refs. [1-3]), which m a y be e x p l a i n e d by a A°~(1350) r e s o n a n c e d e c a y i n g into h ° and V (ref. [3]). T h e a i m of this p a p e r is to give a m e t h o d f o r the d e t e r m i n a t i o n of the spin of the A°~(1350).
2. F I N A L D E N S I T Y M A T R I X In g e n e r a l the final d e n s i t y m a t r i x p(A ° +y) is known if the initial d e n s i t y m a t r i x p(h°~(1350)) and the d e c a y m a t r i x cy~ a r e known [4], p(A°+'/) = ~p(h°y(1350))c~ + ,
(1)
where
{c~]g+) ij =c)E ~i . T h e spin state of a s p i n - J A°v(1350) at p r o d u c t i o n is d e s c r i b e d by a set of ( 2 J + 1) 2 c o m p l e x m u l t i p o l e p a r a m e t e r s t L M (ref. [5]) with L ~< 2J, w h e r e M = - L , . . . , L and L =0, 1 , 2 , . . . , 2J. It is c o n v e n i e n t to expand p(A°~(1350)) a s a s e r i e s of ( 2 J + 1) - d i m e n s i o n a l m a t r i c e s TLM ( r e f s . [6, 7]) p(A°(1350)) i , M
= ( 2 J + l ) -1
2J ~
L ~
L=OM=-L
(2L+I)tLM* TLM i"i''
w h e r e T L M a r e a c o m p l e t e o r t h o n o r m e l set of ( 2 J + 1) 2 c o m p l e x m a t r i c e s that m a y be c o n s t r u c t e d f r o m the o p e r a t o r s of the c o m p o n e n t s of the a n -
(2)
520
S. KRASZNOVSZKY
g u l a r m o m e n t u m J in a m a n n e r s i m i l a r to that in which the s p e r i c a l h a r m o n i c s Y L M a r e c o n s t r u c t e d f r o m the c o o r d i n a t e s x, y and z (ref. [7]). F o l l o w i n g the t r e a t m e n t of B y e r s and F e n s t e r [6] 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 m a y be s u b s t i t u t e d f o r the m a t r i x e l e m e n t s of the TLM:
TLM[M,, M, = { J , L , M ' , M [ J,M"}
M" = M ' + M .
with
T h e p r o p e r t i e s of the m u l t i p o l e p a r a m e t e r s e r t i e s of the d e n s i t y m a t r i x :
(3)
follow f r o m the g e n e r a l p r o p -
IL_ M = (-)Mt*LM ,
too = 1 .
(4)
F o r a p a r i t y - c o n s e r v i n g p r o d u c t i o n p r o c e s s (for A°y(1350)) when the p r o d u c t i o n n o r m a l ~ is c h o s e n a s the p o l a r a x i s [8],
tLM = 0
for
M odd.
(5)
In g e n e r a l an e l e m e n t of the d e c a y m a t r i x m a y be w r i t t e n [5]
~a~fl, M =
~ A 4~
D J* " 0,0) ~a~fl M, ~-~fl(¢~' "
T h e A~a~fi a r e the h e l i c i t y a m p l i t u d e s ,
(6)
and ;ta, ;tfi a r e h e l i c i t i e s f o r the
d e c a y p r o d u c t s of the p a r e n t r e s o n a n c e [9]. If p a r i t y is c o n s e r v e d in the d e c a y p r o c e s s [9]:
A_~c _~fl = 77~a 77t3(-) J-Sa-Sfi A ~a~ fi •
(7)
H e r e S a , S fi and ~ a , ~fi a r e the s p i n s and i n t r i n s i c p a r i t i e s of a and fl, J, ~? a r e the spin and the p a r i t y of the p a r e n t r e s o n a n c e , r e s p e c t i v e l y , and j* DM, ha_hfi(~, 0,0) s t a n d s f o r the m a t r i x e l e m e n t of the r o t a t i o n o p e r a t o r [7], w h e r e 0 and ~b define the d i r e c t i o n of the r e l a t i v e m o m e n t u m in the c . m . of the two (a and fi) p a r t i c l e s . U s i n g the r e l a t i o n s [7] J*
D M,, ~ = (-)
M"-;~
J
D_M,,_; ~ , L
=
L and the s y m m e t r y a n d o r t h o g o n a l i t y r e l a t i o n s of the 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 [7], p c a n be e x p r e s s e d a s p ?) a~ ~o A ~ ^Of +l ~
= (_)J+X'-2;t 4~1 A ol~fi A ~ x J
2 J + I ) I / 2 L ,~M
x(J,J,~,-~'IL,~-A
(2L+1)1/2
?~LMJJM, ~_k,(¢),0,0) ,
SPIN TEST FOR FERMION DECAY
521
where =
?t0~
•
Obviously eq. (8) is t r u e even if the p a r e n t boson r e s o n a n c e decays into two bosons.
3. DECAY DISTRIBUTIONS We d e r i v e equations containing only the p a r a m e t e r s of the A ° (because the e l e c t r o n - p o s i t r o n p a i r is not a p e r f e c t a n a l y z e r for the p o l a r i z a t i o n of the photon). Taking (9) we obtain
0~x~(a°) = (_) J + ~ f i - 2 h a + ~ × ~ (2L+I)I/2(J,J,X L,M
14n (2J+ 1)1/2
~kflA~c~ht3A*~'~hfi
-Xfl,-X +X~IL, X -a /tLMZJM, X _X,(~,O,O). (10)
H e r e we r e m a r k that in the special case, when the ;~fi= 0 and ~ = - ~, - ½, and ~ our eq. (10) is the same as B u t t o n - S h a f e r ' s [10]. One can easily get the angular distribution, the angular distribution of the longitudinal p o l a r i z a t i o n s and the angular distribution of the t r a n s v e r s e p o l a r i z a t i o n s for the A° hyperon using the following e x p r e s s i o n s : (i) The angular distribution
I(O, ~)) = T r
(p(A°)) .
(11)
(ii) Expectation value of a angular m o m e n t u m o p e r a t o r : (A} = T r (p(A °) A ) . T r (p(A°)) 1 hfl=+landeq. Bearing in mind that ~ =+3,
(12) (7) we get
px_~(A°)= ~
[[A½112(-)Ln(L1o)+[A½_ ll2n(3L~]t*LMYLM(0,(0),
°)1, (A
[IA½_ll 2(_)L nLo(3)+!AI1
2z
L,M
:
,
L,M ~-~I(A°)
Ol
=
~ [E(A½1A*_I(-) L+I+A L,M
½1)]
(3) . L* nLltLMC~MI(¢P'
0,0)
'
(13)
522
S. KRASZNOVSZKY
and
o_~½
=
T, , p~_~
where n(1) = (_) J-½ ff2J+ 1 (j, j, I, - ½]LO} Lo 47T ' n (3) : (-) Lo
J-~ ~r2j+ 1 (J, J, ~ - ~/L0) T '
riLl = (-)J-½ YLM(O, (p) :
4~ L* . f~ 2+ -1 L DMo(~ , O, O)
L* v/-2L + 1 L* Q)MI(qS,0,0) = ~ DMl(qb, 0, 0) , j 3 j_~ = ~?/(1,~p(-) -~ = 7(-) Eqs. (11) and (12) and the r e c u r s i o n relations for the Clebsch-Gordan coefficients [7, 10] yield
I(O,(~) = G G 2 L~ven M= - L
12+
-1
L(L+ 1)liA~_ll2
1 - (J+ ~) (g- ½)
1
x n (1) Lo I I M Y L M ( O , 4)) _
^
IPAoA :
2J
L
G
G
Lod d M= - L
(14a)
213J(J+I)-•-L(L+I)iA½_112 L
(J+ ~)(J- ~)
- ]A½1[ 21
×n (1) Lo t I M Y L M ( O , 4))
I P A ° x : 2Rep12 21: 2Re L,~ M E(AI~I AT~-l" (-)L+I +A ½-1 A i l ) { L ( L + I
t
[l+(-1)L+l]}[L(L+l)]-2[(J+a2)(J-~)]-~n(1)t
L*
* (DMI(~b, 0,0) Lo L M
(14b)
) _ (j+ ½)2 (14c)
SPIN TEST FOR FERMION DECAY
IPAo;=-2Imp,
I =-2Im
~-~
E
L,M
523
6(A½1A*}_I(-)L+I+A½_IA*~I){L(L+I)
_(J+})2[I+(_I)L+I]}[L(L+I)]-2[(j+~)(j_½)]
n L o ~ L M ~ , M l t q J , s , 0 ) . (14d)
H e r e A, } , ~ a r e unit v e c t o r s pointing in the d i r e c t i o n s of A°, (fix ~ × n × A r e s p e c t i v e l y and PAO is the p o l a r i z a t i o n v e c t o r of A°.
4. SPIN DETERMINATION The p u r p o s e of this p a r t is to give s o m e useful e x p r e s s i o n s for the m u l tipole p a r a m e t e r s , which can be u s e d in a " m o m e n t u m a n a l y s i s " [11]. F u r t h e r , s e v e r a l inequalities will be given which have a grea.t value in p r a c t i c a l d e c i s i o n s , concerning the higher a l l o w e d - s p i n quantum n u m b e r s . Calculating the r a t i o of the weighted a v e r a g e s we obtain e x p r e s s i o n s which a r e independent of tLM (provided that tLM¢ 0) f o r all the allowed v a l u e s of L and M, and in the c a s e of any given J , one can d e r i v e the following useful relation:
fIP AoA I~LM(O, 4~)d~ •^) CbMl(q5 , 0,0) dl2 f / ( p AO~_ Zphoy L
E
[3J(J+I)-I-L(L+I)IA~_ll2
Lodd, M L.
IA}I}21
(J+ ~ ) } J - ½). . . . .
Lodd, M
o
., L(L+I):2(J+½) 2 c(A ~l d i - l +d~ -l A½1) ~/(L(L+ I) (J+ ~2)(J- ~)
(15)
Eq. (15) follows f r o m the orthogonality p r o p e r t i e s of the functions and O ~/1((~, 0,0)
YLM(O, ~)
f YLM(O, ~) YL,M,(O, c~)dr2 : 5LL, 5MM, , L' f e d ~/1 (qS,0,0) CbM,l(qS,0,0) di2 - 2 L4~+ 1
5LL'SMM''
The r e a d e r can find in r e f s . [11,12] s o m e useful f o r m u l a e for the s t a t i s t i c a l t r e a t m e n t of weighted a v e r a g e s . The l e f t - h a n d side of eq. (15) can be d e t e r m i n e d f r o m the e x p e r i m e n t a l data [11]. F r o m eq. (15), which r e l a t e s longitudinal to t r a n s v e r s e p o l a r i z a t i o n s , 1 2 , ]A_1 11 2 and one can * d e t e r m m " e t h e p a r a m e t e r s [A_ll 2 ;~e(AXlA~1 +A~-IA~1)B e c a u s e of the n o r m a h z a t i o n of I(O, dp), we have 2 22 2one m o r e equation f o r the p a r a m e t e r s . Hence e v e r y multipole p a r a m e t e r can be d e t e r m i n e d f r o m the decay d i s t r i b u t i o n s of eq. (14). Knowing the multipole p a r a m e t e r s for any J the spin of h°7 s y s t e m can be obtained [11,13].
524
s. KRASZNOVSZKY WehaveA½_ 1=0,
1
1
if J=5, b e c a u s e / ~ ( A ° ) = - f i ( y ) a n d I x A O ± ~ y ] ~
S i n c e t h e e x p e r i m e n t s s h o w c o n s i d e r a b l e a n i s o t r o p y in t h e a n g u l a r d i s t r i b u t i o n of A°~(1350) ~ A ° + y d e c a y one c a n c o n c l u d e t h a t t h e s p i n of t h e A%(1350) i s h i g h e r t h a n ½. We w a n t to e m p h a s i z e t h a t t h i s m e t h o d g i v e s a p o s s i b i l i t y to d e t e r m i n e t h e s p i n of t h e A ° ~ 1 3 5 0 ) , s u p p o s i n g t h e m u l t i p o l e p a r a m e t e r s of t h e i n i t i a l s t a t e (A°~(1350)) c a n n o t b e n e g l e c t e d (tLM ~0) a n d g (the A°~(1350) spin) > ½, F u r t h e r e x p e r i m e n t a l c h e c k s c a n b e m a d e on t h e d a t a , u s i n g s o m e i n e q u a l i t i e s w h i c h a r e i m p o s e d on t h e m u l t i p o l e p a r a m e t e r s by t h e p o s i t i v i t y p r o p e r t y of t h e d e n s i t y m a t r i x [14]. U s i n g t h e r e l a t i o n d e r i v e d f r o m t h e h e r m i t i c i t y of t h e d e n s i t y m a t r i x p ( r e f . [15])
tLM(n) = (-)MtL_M(n) , o n e h a s [15]
S n = ( 2 J + l ) -1 ~ L=O
(2L+l)(tLo)2+2
~ ltLM(½n)] 2 M=I
,
for n even,
Sn= (2J+ l ) - i
2J r L .~ (2L+ I) LtLo(P)tLo(q)+ Z R e (-)MI LM(P) I L_M( q) 1 , L=O M=I
f o r n o d d , w h e r e tLM(n) = f(J, tLM) a n d n=p+ q (p, q a r e i n t e g e r s ) a n d t h e i n e q u a l i t i e s c o n s i s t i n g of S n - s ( r e f . [14]) a r e d e p e n d e n t on J.
REFERENCES
[1] Van Yun-Tchan et al., preprint, JINR P-1615 (1964); P r o c . XII. Int. ConZ. on high energy physics, Dubna, Vol. 1 (1964) 615. [2] E . G . B u b e l e v e t al., Phys. L e t t e r s 24B (1967) 246. [3] G . B o z d k i e t al., Phys. L e t t e r s 28B (1968) 360. [4] U.Fano, Rev. Mod. Phys. 29 (1957) 74. [5] J . D . J a c k s o n , in High energy physics, 1965 (Gordon and Breach, Science P u b l i s h e r s , New York, 1966). [6] N . B y e r s a n d S . F e n s t e r , Phys. Rev. L e t t e r s 11 (1963) 52. [7] M. E. Rose, E l e m e n t a r y theory of angular momentum (Wiley, New York, 1957). [8] R . H . C a p p s , Phys. Rev. 122 (1961) 929. [9] M. Jacob in G. F. Chew and M. Jacob, Strong interaction physics (Benjamin, New York, 1964). [10] J.Button-Shafer, Phys. Rev. 139B (1965) ~07. [11] J.Button-Shafer, Phys. Rev. 134B (1964) 1372. [12] N. B y e r s , CERN 67-20. [13] P . E . S c h l e n e t al., Phys. Rev. L e t t e r s 11 (1963) 167. [14] P . M i n n a e r t , Phys. Rev. 151 (1966) 1306. [15] S . K r a s z n o v s z k y et al., Report (Budapest) of the Central Res. Inst. for Phys. 18 (1970) 3.