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Parity determination in particle physics
Volume 15, number4
PHYSICS
LETTERS
trinsically broken M(12) invariance [6] of strong interactions.
4. F. Gffrsey, M. Nauenberg and T.D. l,ee, Phyvl. Rev.
135 (1964) B467. 5, P . G . O . F r e u n d and B.W.Lee, Phys. I~ev. t,etter~ 13 (1964) 592; O.W.Greenberg, ibid 13 (1964) 59~. 6. K. Bardakei, J.M. Cornwall, P. G, O. Freur~d ~a¢l B. W. Lee, Phys. Rev. Le~ers 14 (1965) 48. See also subsequent work on the intrinsic symmetry-breaking mechanism by R. Delboargo, A. Salam, J. Stxadhee, Proc.Rov.Soe. (to be published), and B. Sakita and K. C. Wall, Phys. Ray. Lvt'~er ~ (~o be published). See also M.B~g and A. Pais, Phys. Ray. Letters 14 (1965} 267.
T h e a u t h o r w o u l d like to t h a n k P r o f e s s o r R. O p p e n h e i m e r f o r h i s h o s p i t a l i t y at t h e I n s t i t u t e f o r A d v a n c e d Study. Refe~'enees 1. M.Gell-Mann, Physics Letters 8 (1964) 214. 2. G.Zweig, CERN report (1964). 3. M.Gell-Man~, Physics 1 (1964} 63.
PARITY
DETERMINATION
P. L. C 8 O N K A ,
15 April 1!#~5
IN
PARTICLE
M. J. M O R A V C S I K
PHYSICS
*
and M. D. S C A D R O N
Lawrence Radiation Laboratory, University of California, Liver~nore, California Received I March 1965
T h e p u r p o s e of t h i s note i s to e s t a b l i s h s o m e v e r y g e n e r a l t h e o r e m s a b o u t t h e type of e x p e r i m e n t s w h i c h c a n be u s e d f o r t h e d e t e r m i n a t i o n of t h e p a r i t y of one of t h e p a r t i c l e s in a r e a c t i o n (henceforth callea briefly parity experiments), It s h o u l d be s t a t e d at the o u t s e t t h a t p a r i t y e × p e r i m e n t s a r e of t h r e e k i n d s : 1. T h o s e b a s e d only on r o t a t i o n a n d r e f l e c t i o n invarianae. 2. T h o s e i n v o l v i n g a l s o s o m e a s s u m p t i o n s about the angular momentum states which cont r i b u t e s i g n i f i c a n t l y to t h e p r o c e s s , but w h i c h do n o t involve a s s u m p t i o n s a b o u t t h e d y n a m i c s of t h e r e a c t i o n (e.g. t h e v a r i o u s t h r e s h o l d theorems). 3. T h o s e built on a n e x p l i c i t d y n a m i c a l m o d e l of t h e i n t e r a c t i o n (e.g, d i s p e r s i o n r e l a t i o n s ) . T h e r e s u l t s of t h i s note a r e r e l e v a n t only to the e x p e r i m e n t s of t h e f i r s t kind, w h i c h a r e , h o w e v e r , the most attractive since they are the most general and hence, theoretically, the most conclusive. Let us consider the particle reaction A 1 + A2 -- B1 + B2
it is i m p o s s i b l e to d e t e r m i n e the p a r i t y of B 2 (or
A2). Proof, L e t u s denote t h e m a t r i x e l e m e n t o~ eq. (1) by Me1 d e p e n d i n g on w h e t h e r the i32 (or A2) p a r i t y is p o s i t i v e o r n e g a t i v e . Let u~3 fur'thermore denote the matrix element of the re~ctlon O+ +A 2 - , B 1 + B2
(2)
by M~2 depending again on whether the B2 (or A2) parity is positive or nagative. Here O ~ denoteu a pariticle with zero spin and positive internal parity. Furthermore, we denote the matrix element of the reaction O+ +A I - , O + +O :~
(3)
by M ~ depending on whether the second particle on the right-hand side has positive or neg;itlve parity. Then we can write ** MI and
(I)
and the spins of the particles are arbitrary. Let us assume that the parity of all particles in eq. (I) is known e x c e p t t h a t of B 2 (but t h i s p a r t i c l e could j u s t a s well be A2) Theorem 1. In an e x p e r i m e n t in w h i c h A 1 is a b o s o n and no s p i n i n f o r m a t i o n is o b t a i n e d a b o u t A1,
* Work done under the auspices of the U.g.Atomie Energy Comissien. ** The details of the formalism which decomposes general M matrix into the outer product~ of simpler particle reaction matrices ,Nil be discussed in a forthcoming paper (11.
353
Volume 15, ~rnbtrr 4
PHYSICS L E T T E R S
Any experimental observable L I for reaction (I) is of the general form
nI
--
Tr (M~ ~MlS f)
46)
where Sl and Sf denote the sets of spirt o p e r a t o r s measured in the initialand final states, respectively. F o r an experiment which yielde no spin information on A 1, t h e r e f o r e , we get, substituting eqs.
(4) and (S)anto eq. (s)
2 Tr L +I = [a+[ .
(~?siM~s f) +
+ ++++, r r ( ~ * - - ) Tr
(M?~IM~5)+
O ++A 2~O
(7)
+ +++++Tr <~;*~;) Tr ( M ? S i ~ ¥ L;o ++" ++Tr (M;*M3) Tr
(@StM2S f) +
+ ~'+-, Zr (M;%) Tr
(~?siM'+s f) +
c+> +
~+
.
.
+ +b" +2 r r (Me*Me) T r (M2*SiM2Sf).
,t~e Tr (~,;*.,;) = Tr (~;*M;) + 0. and Tr (g'M;>
>0. both ~1'+ ~ve
the form
LImA ~ T r ( M ~ S i M ~ i ) + B++Tr (M2*SiM2Sf) , (0) where the A's and B's are positive. In the absence of dyrmmical information the A's and B'e are unknown, and hence t h e r e Is no way of distinguishing between L+I and L'I+ Theorem 2. In a parity experiment some spin J~formatlon must be acquired about all bosone participating in the reaction. Proof. This Is a simple c o r o l l a r y of T h e o r e m 1, C n c e any boson about which we did not acquire sptn tnformation~ could be A1 of T h e o r e m 1, and according to it the experiment could not be a parity experiment. It should be r e m a r k e d that in p r a c t i c e a "parity experiment" might consist of a set of measuremeats. In such a case the theorem requires only that we should obtain spin information about each boson somewhe~'e In the s e t and not in each mea~ surement. Arty spin-0 p a r t i c l e should be cons~dered aJ~ automatically revealing spin Informattma a b o ~ ltsc~lf. Consider next the e a s e when A 1 in reaction (I) Is ;t fermton, 8 ~ c e f e r m i o n s a p p e a r in p a i r s , 354
++B 2
(10)
and
and
• r ~.,;',,;)~0.
t h e r e m u s t be at l e a s t one o t h e r f e r m i o n in (1). We have then the following result: Theorem 3. In an experiment in which no spin information is obtained about at l e a s t two of the f e r m i o n s it is i m p o s s i b l e to d e t e r m i n e the parity of B 2 (or A2). Proof. The proof is a l m o s t identical with that of T h e o r e m s 1 and 2, except that now we b r e a k up r e a c t i o n 41) in a slightly different way. Instead of r e a c t i o n s (2) and (3), we will now c o n s i d e r
(MS*M3! T r (M~*SfM2Sf) +
+ ~+*b ~ rr (~;.M;) Tr
15 April 1965
A 1 +O--' B 1 + O ±
(11)
(ifA 1 and B 1 are the two fermions). W e can then use eqs. (4)-(9), except for the substitution M 2 ~ M I0 a n d M 3 - ~ M 11" W h e r e M10 and M l l r e f e r to equations (10) and 411) respectively. A similar decomposition can be made ff the two fermlons are beth in the initialor both in the final state, and the proof foUows in a s i m i l a r way. Theorem 4. In an experiment in which spin information is obtained for all fermlons, but this information on two of the fermions is equivalent to coupling them together into an unpolarized boson, the parity of B m (or A n) cannot be determined. Proof. This is an immediate corollary of Theorem 1. Theorems I and 2 have told us that ff a boson is unpoiarized in reaction (I), no parity determinatlon can be made. W e will now consider the parity experiments that we can carry out ff the bosons are polarized. Theorem 5. If A 1 is a linearly polarized photon in reaction (1), any parity experiment in that reaction can be duplicated by replacing A 1 by a spin-zero bosom Proof. Let us consider the reaction O+A 2~
B I+B~
(12)
whose matrix element will be denoted by M~I2. Then we can write + + 6- m M12 M +1 = 6. I M12
(13)
M; = e. i M;2 + e . m M12
(14)
and w h e r e e Is the photon polarization v e c t o r , and I and m a r e the unit v e c t o r and unit pseudovector, respectively, which span the momentum space p e r p e n d i c u l a r to the photon momentum. If the photon is l i n e a r l y polarized in the i o r m directions, only the f i r s t o r second t e r m s of
Volume 15, n u m b e r 4
PHYSICS
these expressions will survive and hence any parity information must c o m e from MI2. T h e o r e m 6. If A 1 is a circularly polarized photon in reaction (I), the parity experiments in that reaction will not be duplications of "simpler" experiments in the sense of T h e o r e m 5. Proof. The proof of T h e o r e m 5 breaks down for circularly polarized photons, since these have no definite parity (being linear combinations of e - I a n d e . m ) . T h e o r e m 7. T h e o r e m s 5 a n d 6 a l s o hold f o r s p i n - o n e besons. P r o o f . T h e e x p r e s s i o n s (13) and (14) c a n be m a d e to hold f o r s p i n - o n e bosoms by a d d i n g t h e terms e. nM+12 and e .n M-12 , respectively. Here n is the second vector spanning the mom e n t u m s p a c e . T h e s e e x t r a t e r m s , h o w e v e r , do not a f f e c t t h e r e s t of t h e p r o o f of T h e o r e m 5 o r 6. An e x a m p l e f o r T h e o r e m s 5 a n d 6 c a n be g i v e n in s p i n - z e r o b o s o n p h o t o p r o d u c t i o n [2]. T h e r e a r e no p a r i t y e x p e r i m e n t s t h e r e u s i n g u n p o l a r i z e d p h o t o n s only, t h u s i l l u s t r a t i n g T h e o r e m 1. F u r t h e r m o r e , all p a r i t y e x p e r i m e n t s u s i n g l i n e a r l y p o l a r i z e d p h o t o n s c a n be d u p l i c a t e d in t h e s p i n zero buson - nucleon reaction*. The experiments utilizing circularly polarized photons, however, are genuinely "new" experiments. It is p o s s i b l e , of c o u r s e , that in a p r a c t i c a l s e n s e it i s e a s i e r to c a r r y o u t t h e p h o t o n - i n d u c e d reactions than the spin-zero boson induced ones. An i l l u s t r a t i o n of T h e o r e m 3 is g i v e n in ref. 3, w h e r e it i s s h o w n t h a t r e l a t i v e p a r i t i e s of p a r t i c l e s c a n n e v e r be d e t e r m i n e d in an e x p e r i m e n t * Since we do not k~ow of a complete treatment of spin-J~ - spln-0 reactions in the literature, we list the results here. The notation is as follows: SP denotes the same intrinsicparities on the two sides of the reaction, while OP denotes opposite intrinsic parities. By ~r we denote the spin--~ particle spin, I, m and n denote the three orthogonal unit vectors spanning the momentum space, of which I and n are true vectors and m a pseudoveetor. M denotes the reaction matrix. We then have M = a 1 + a2~.m
(SP)
M = bla'l
(OP)
and + b2a'.
which lead to the experimental observables given in table 1. From this table we can see that the parity experiments here, namely Pin = ~ A m , Tln=thosenl ~ 7" , Tl! = ~ T , , , I s = ± 10Tram, etc. are exaetly lib'ted in re'~. 1, eqs. (t)-~C9, for photoproductlon with linearly polarized photons. On the other hand. eqs. (vit)-(xitt) of ref. 1, using circularly polarized photons, have no analogues in table I.
LETTERS
~SApril 1965
Table l Experimental observables for the reaction: spin-~ spin-0 ~ s p t n ~ + spin-0. For notation, see foo~mte on this page. Observables
TrM*M<: 1 -= IoP1
~3P
0
OP
0
?OPrn
Re e~a2
] I m blb 2
loV~
o
,,
T r M* ~ I M ~ IOA1
IOAm IoA n
0
0
Re ala2
-~ !zn b~b2
0
0
Tr M*~lMCra =- 10Tlm
0
0
lO Tml
e
0
10rln
4 I~
,q,,~
} ~o ,,i,,, ~
loTmn
0
loTtm a
0
0
~1~iI2-~]~212
~l~,12-~le,~l 2
torii
0
l o r ~ ,tl,hlz+~,t%l 2 ql¢,,t zo }If,~l~ 'ornn ~[,,,t2-~1,,21 e -~1~:112+},1t,~12 involving four fermions only, in which no spin irdormation is o~4ained on two of them. With the g r o w i n g n u m b e r of " e l e m e n t a r y " p a r t i c l e s p a r i t y e x p e r i m e n t s will contiv~e to gain in i m p o r t a n c e , and s o m e of t h e m a r e }icing p l a n n e d at the p r e s e n t t i m e . We believe that o u r p r e s e n t r e s u l t s a r e a l r e a d y of s u f f i c i e n t i n t e r e s t to affect the s e n s i b l e c h o i c e of s u c h e ) ~ e r t m e n t s , although t h e y a r e only a f i r s t s t e p in the s y s t e m a t i z a t i o n of the k i n e m a t i c and spin s t r u c t u r e of p a r t i c l e r e a c t i o n s . T h e o r e m s applying to m o r e c o m p l e x s y s t e m s and involving o t h e r p r o p e r t i e u of the M - m a t r i x will he g i v e n at a l a t e r t i m e . W e w i s h to thank It. P. Stapp for v e r y hedpfut c o m m e n t s on the f i r s t d r a f t of this paper.
References I. P.L.Csonka, M.J. Moravcsik and M.D.Seadrtm, to be published. 2. M.J.Moravseik, Phys.Rev,125 0962) 1088. 3. P.L.Csonka, Rev. Mod, Phys. (January ]965).
355
PHYSICS
LETTERS
trinsically broken M(12) invariance [6] of strong interactions.
4. F. Gffrsey, M. Nauenberg and T.D. l,ee, Phyvl. Rev.
135 (1964) B467. 5, P . G . O . F r e u n d and B.W.Lee, Phys. I~ev. t,etter~ 13 (1964) 592; O.W.Greenberg, ibid 13 (1964) 59~. 6. K. Bardakei, J.M. Cornwall, P. G, O. Freur~d ~a¢l B. W. Lee, Phys. Rev. Le~ers 14 (1965) 48. See also subsequent work on the intrinsic symmetry-breaking mechanism by R. Delboargo, A. Salam, J. Stxadhee, Proc.Rov.Soe. (to be published), and B. Sakita and K. C. Wall, Phys. Ray. Lvt'~er ~ (~o be published). See also M.B~g and A. Pais, Phys. Ray. Letters 14 (1965} 267.
T h e a u t h o r w o u l d like to t h a n k P r o f e s s o r R. O p p e n h e i m e r f o r h i s h o s p i t a l i t y at t h e I n s t i t u t e f o r A d v a n c e d Study. Refe~'enees 1. M.Gell-Mann, Physics Letters 8 (1964) 214. 2. G.Zweig, CERN report (1964). 3. M.Gell-Man~, Physics 1 (1964} 63.
PARITY
DETERMINATION
P. L. C 8 O N K A ,
15 April 1!#~5
IN
PARTICLE
M. J. M O R A V C S I K
PHYSICS
*
and M. D. S C A D R O N
Lawrence Radiation Laboratory, University of California, Liver~nore, California Received I March 1965
T h e p u r p o s e of t h i s note i s to e s t a b l i s h s o m e v e r y g e n e r a l t h e o r e m s a b o u t t h e type of e x p e r i m e n t s w h i c h c a n be u s e d f o r t h e d e t e r m i n a t i o n of t h e p a r i t y of one of t h e p a r t i c l e s in a r e a c t i o n (henceforth callea briefly parity experiments), It s h o u l d be s t a t e d at the o u t s e t t h a t p a r i t y e × p e r i m e n t s a r e of t h r e e k i n d s : 1. T h o s e b a s e d only on r o t a t i o n a n d r e f l e c t i o n invarianae. 2. T h o s e i n v o l v i n g a l s o s o m e a s s u m p t i o n s about the angular momentum states which cont r i b u t e s i g n i f i c a n t l y to t h e p r o c e s s , but w h i c h do n o t involve a s s u m p t i o n s a b o u t t h e d y n a m i c s of t h e r e a c t i o n (e.g. t h e v a r i o u s t h r e s h o l d theorems). 3. T h o s e built on a n e x p l i c i t d y n a m i c a l m o d e l of t h e i n t e r a c t i o n (e.g, d i s p e r s i o n r e l a t i o n s ) . T h e r e s u l t s of t h i s note a r e r e l e v a n t only to the e x p e r i m e n t s of t h e f i r s t kind, w h i c h a r e , h o w e v e r , the most attractive since they are the most general and hence, theoretically, the most conclusive. Let us consider the particle reaction A 1 + A2 -- B1 + B2
it is i m p o s s i b l e to d e t e r m i n e the p a r i t y of B 2 (or
A2). Proof, L e t u s denote t h e m a t r i x e l e m e n t o~ eq. (1) by Me1 d e p e n d i n g on w h e t h e r the i32 (or A2) p a r i t y is p o s i t i v e o r n e g a t i v e . Let u~3 fur'thermore denote the matrix element of the re~ctlon O+ +A 2 - , B 1 + B2
(2)
by M~2 depending again on whether the B2 (or A2) parity is positive or nagative. Here O ~ denoteu a pariticle with zero spin and positive internal parity. Furthermore, we denote the matrix element of the reaction O+ +A I - , O + +O :~
(3)
by M ~ depending on whether the second particle on the right-hand side has positive or neg;itlve parity. Then we can write ** MI and
(I)
and the spins of the particles are arbitrary. Let us assume that the parity of all particles in eq. (I) is known e x c e p t t h a t of B 2 (but t h i s p a r t i c l e could j u s t a s well be A2) Theorem 1. In an e x p e r i m e n t in w h i c h A 1 is a b o s o n and no s p i n i n f o r m a t i o n is o b t a i n e d a b o u t A1,
* Work done under the auspices of the U.g.Atomie Energy Comissien. ** The details of the formalism which decomposes general M matrix into the outer product~ of simpler particle reaction matrices ,Nil be discussed in a forthcoming paper (11.
353
Volume 15, ~rnbtrr 4
PHYSICS L E T T E R S
Any experimental observable L I for reaction (I) is of the general form
nI
--
Tr (M~ ~MlS f)
46)
where Sl and Sf denote the sets of spirt o p e r a t o r s measured in the initialand final states, respectively. F o r an experiment which yielde no spin information on A 1, t h e r e f o r e , we get, substituting eqs.
(4) and (S)anto eq. (s)
2 Tr L +I = [a+[ .
(~?siM~s f) +
+ ++++, r r ( ~ * - - ) Tr
(M?~IM~5)+
O ++A 2~O
(7)
+ +++++Tr <~;*~;) Tr ( M ? S i ~ ¥ L;o ++" ++Tr (M;*M3) Tr
(@StM2S f) +
+ ~'+-, Zr (M;%) Tr
(~?siM'+s f) +
c+> +
~+
.
.
+ +b" +2 r r (Me*Me) T r (M2*SiM2Sf).
,t~e Tr (~,;*.,;) = Tr (~;*M;) + 0. and Tr (g'M;>
>0. both ~1'+ ~ve
the form
LImA ~ T r ( M ~ S i M ~ i ) + B++Tr (M2*SiM2Sf) , (0) where the A's and B's are positive. In the absence of dyrmmical information the A's and B'e are unknown, and hence t h e r e Is no way of distinguishing between L+I and L'I+ Theorem 2. In a parity experiment some spin J~formatlon must be acquired about all bosone participating in the reaction. Proof. This Is a simple c o r o l l a r y of T h e o r e m 1, C n c e any boson about which we did not acquire sptn tnformation~ could be A1 of T h e o r e m 1, and according to it the experiment could not be a parity experiment. It should be r e m a r k e d that in p r a c t i c e a "parity experiment" might consist of a set of measuremeats. In such a case the theorem requires only that we should obtain spin information about each boson somewhe~'e In the s e t and not in each mea~ surement. Arty spin-0 p a r t i c l e should be cons~dered aJ~ automatically revealing spin Informattma a b o ~ ltsc~lf. Consider next the e a s e when A 1 in reaction (I) Is ;t fermton, 8 ~ c e f e r m i o n s a p p e a r in p a i r s , 354
++B 2
(10)
and
and
• r ~.,;',,;)~0.
t h e r e m u s t be at l e a s t one o t h e r f e r m i o n in (1). We have then the following result: Theorem 3. In an experiment in which no spin information is obtained about at l e a s t two of the f e r m i o n s it is i m p o s s i b l e to d e t e r m i n e the parity of B 2 (or A2). Proof. The proof is a l m o s t identical with that of T h e o r e m s 1 and 2, except that now we b r e a k up r e a c t i o n 41) in a slightly different way. Instead of r e a c t i o n s (2) and (3), we will now c o n s i d e r
(MS*M3! T r (M~*SfM2Sf) +
+ ~+*b ~ rr (~;.M;) Tr
15 April 1965
A 1 +O--' B 1 + O ±
(11)
(ifA 1 and B 1 are the two fermions). W e can then use eqs. (4)-(9), except for the substitution M 2 ~ M I0 a n d M 3 - ~ M 11" W h e r e M10 and M l l r e f e r to equations (10) and 411) respectively. A similar decomposition can be made ff the two fermlons are beth in the initialor both in the final state, and the proof foUows in a s i m i l a r way. Theorem 4. In an experiment in which spin information is obtained for all fermlons, but this information on two of the fermions is equivalent to coupling them together into an unpolarized boson, the parity of B m (or A n) cannot be determined. Proof. This is an immediate corollary of Theorem 1. Theorems I and 2 have told us that ff a boson is unpoiarized in reaction (I), no parity determinatlon can be made. W e will now consider the parity experiments that we can carry out ff the bosons are polarized. Theorem 5. If A 1 is a linearly polarized photon in reaction (1), any parity experiment in that reaction can be duplicated by replacing A 1 by a spin-zero bosom Proof. Let us consider the reaction O+A 2~
B I+B~
(12)
whose matrix element will be denoted by M~I2. Then we can write + + 6- m M12 M +1 = 6. I M12
(13)
M; = e. i M;2 + e . m M12
(14)
and w h e r e e Is the photon polarization v e c t o r , and I and m a r e the unit v e c t o r and unit pseudovector, respectively, which span the momentum space p e r p e n d i c u l a r to the photon momentum. If the photon is l i n e a r l y polarized in the i o r m directions, only the f i r s t o r second t e r m s of
Volume 15, n u m b e r 4
PHYSICS
these expressions will survive and hence any parity information must c o m e from MI2. T h e o r e m 6. If A 1 is a circularly polarized photon in reaction (I), the parity experiments in that reaction will not be duplications of "simpler" experiments in the sense of T h e o r e m 5. Proof. The proof of T h e o r e m 5 breaks down for circularly polarized photons, since these have no definite parity (being linear combinations of e - I a n d e . m ) . T h e o r e m 7. T h e o r e m s 5 a n d 6 a l s o hold f o r s p i n - o n e besons. P r o o f . T h e e x p r e s s i o n s (13) and (14) c a n be m a d e to hold f o r s p i n - o n e bosoms by a d d i n g t h e terms e. nM+12 and e .n M-12 , respectively. Here n is the second vector spanning the mom e n t u m s p a c e . T h e s e e x t r a t e r m s , h o w e v e r , do not a f f e c t t h e r e s t of t h e p r o o f of T h e o r e m 5 o r 6. An e x a m p l e f o r T h e o r e m s 5 a n d 6 c a n be g i v e n in s p i n - z e r o b o s o n p h o t o p r o d u c t i o n [2]. T h e r e a r e no p a r i t y e x p e r i m e n t s t h e r e u s i n g u n p o l a r i z e d p h o t o n s only, t h u s i l l u s t r a t i n g T h e o r e m 1. F u r t h e r m o r e , all p a r i t y e x p e r i m e n t s u s i n g l i n e a r l y p o l a r i z e d p h o t o n s c a n be d u p l i c a t e d in t h e s p i n zero buson - nucleon reaction*. The experiments utilizing circularly polarized photons, however, are genuinely "new" experiments. It is p o s s i b l e , of c o u r s e , that in a p r a c t i c a l s e n s e it i s e a s i e r to c a r r y o u t t h e p h o t o n - i n d u c e d reactions than the spin-zero boson induced ones. An i l l u s t r a t i o n of T h e o r e m 3 is g i v e n in ref. 3, w h e r e it i s s h o w n t h a t r e l a t i v e p a r i t i e s of p a r t i c l e s c a n n e v e r be d e t e r m i n e d in an e x p e r i m e n t * Since we do not k~ow of a complete treatment of spin-J~ - spln-0 reactions in the literature, we list the results here. The notation is as follows: SP denotes the same intrinsicparities on the two sides of the reaction, while OP denotes opposite intrinsic parities. By ~r we denote the spin--~ particle spin, I, m and n denote the three orthogonal unit vectors spanning the momentum space, of which I and n are true vectors and m a pseudoveetor. M denotes the reaction matrix. We then have M = a 1 + a2~.m
(SP)
M = bla'l
(OP)
and + b2a'.
which lead to the experimental observables given in table 1. From this table we can see that the parity experiments here, namely Pin = ~ A m , Tln=thosenl ~ 7" , Tl! = ~ T , , , I s = ± 10Tram, etc. are exaetly lib'ted in re'~. 1, eqs. (t)-~C9, for photoproductlon with linearly polarized photons. On the other hand. eqs. (vit)-(xitt) of ref. 1, using circularly polarized photons, have no analogues in table I.
LETTERS
~SApril 1965
Table l Experimental observables for the reaction: spin-~ spin-0 ~ s p t n ~ + spin-0. For notation, see foo~mte on this page. Observables
TrM*M<: 1 -= IoP1
~3P
0
OP
0
?OPrn
Re e~a2
] I m blb 2
loV~
o
,,
T r M* ~ I M ~ IOA1
IOAm IoA n
0
0
Re ala2
-~ !zn b~b2
0
0
Tr M*~lMCra =- 10Tlm
0
0
lO Tml
e
0
10rln
4 I~
,q,,~
} ~o ,,i,,, ~
loTmn
0
loTtm a
0
0
~1~iI2-~]~212
~l~,12-~le,~l 2
torii
0
l o r ~ ,tl,hlz+~,t%l 2 ql¢,,t zo }If,~l~ 'ornn ~[,,,t2-~1,,21 e -~1~:112+},1t,~12 involving four fermions only, in which no spin irdormation is o~4ained on two of them. With the g r o w i n g n u m b e r of " e l e m e n t a r y " p a r t i c l e s p a r i t y e x p e r i m e n t s will contiv~e to gain in i m p o r t a n c e , and s o m e of t h e m a r e }icing p l a n n e d at the p r e s e n t t i m e . We believe that o u r p r e s e n t r e s u l t s a r e a l r e a d y of s u f f i c i e n t i n t e r e s t to affect the s e n s i b l e c h o i c e of s u c h e ) ~ e r t m e n t s , although t h e y a r e only a f i r s t s t e p in the s y s t e m a t i z a t i o n of the k i n e m a t i c and spin s t r u c t u r e of p a r t i c l e r e a c t i o n s . T h e o r e m s applying to m o r e c o m p l e x s y s t e m s and involving o t h e r p r o p e r t i e u of the M - m a t r i x will he g i v e n at a l a t e r t i m e . W e w i s h to thank It. P. Stapp for v e r y hedpfut c o m m e n t s on the f i r s t d r a f t of this paper.
References I. P.L.Csonka, M.J. Moravcsik and M.D.Seadrtm, to be published. 2. M.J.Moravseik, Phys.Rev,125 0962) 1088. 3. P.L.Csonka, Rev. Mod, Phys. (January ]965).
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