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General classification of conspiracy relations
~
Nuclear Physics B2 (1967) 657-668. North-Holland Publ. Comp., Amsterdam
GENERAL CLASSIFICATION OF CONSPIRACY RELATIONS H. H(JGAASEN and Ph. SALIN CERN.
Geneva
Received 29 June 1967
Abstract: We present a study of constraints on helicity amplitudes when the external particles have a r b i t r a r y spins. The reaction 7~N--' ON is studied in detail and we find two types of "conspiracy relations". The first one occurs at t - 0 when the m a s s e s of 7r and p are different; it is different from the Gribov-Volkov type as it vanishes when the m a s s e s of ~ and p are equal. The Gribov-Volkov type of relations is very different from the preceding one and occurs when the m a s s e s of and p are the same. We show how all these relations can be found without explicit reference to the invariant amplitudes and argue that generally all conspiracy r e l a tions can be obtained directly from the Trueman-Wick crossing relations between helicity amplitudes.
1. I N T R O D U C T I O N Due to the g r e a t u t i l i t y of the R e g g e pole m o d e l , t h e r e h a s f o r a t i m e b e e n a g r e a t i n t e r e s t in the r e g g e i z i n g and a n a l y t i c a l c o n t i n u a t i o n of h e l i c i t y a m p l i t u d e s c o r r e s p o n d i n g to e x t e r n a l p a r t i c l e s with a r b i t r a r y s p i n s . In a now c l a s s i c a l p a p e r , W a n g [1] h a s g i v e n a s e t of p r e s c r i p t i o n s f o r c o n s t r u c t i n g h e l i c i t y a m p l i t u d e s f r e e of k i n e m a t i c a l s i n g u l a r i t i e s . T h e r e h a s , howe v e r , b e e n s o m e u n e a s i n e s s a b o u t how to t a c k l e p r o b l e m s l e a d i n g to the s o c a l l e d c o n s p i r a c y r e l a t i o n s . T h e s e a r e l i n e a r c o m b i n a t i o n s of h e l i c i t y a m p l i t u d e s t h a t m u s t v a n i s h at c e r t a i n v a l u e s of the k i n e m a t i c a l v a r i a b l e s of the p r o c e s s s t u d i e d . T h i s v a n i s h i n g i s n e c e s s a r y to a v o i d c o n s t r u c t i n g r e g g e i z e d a m p l i t u d e s that v i o l a t e M a n d e l s t a m a n a l y t i c i t y . We d e c i d e d to t r e a t a c o n c r e t e , s u f f i c i e n t l y c o m p l i c a t e d a n d s i m p l e p r o b l e m in d e t a i l to get a n u n d e r s t a n d i n g of the d i f f e r e n t m e c h a n i s m s that c a n l e a d one to c o n s t r u c t m o d e l s with w r o n g a n a l y t i c a l p r o p e r t i e s . T h e e x a m p l e t h a t we c h o o s e i s 7rN --' pN: i n a d d i t i o n to r e a s o n a b l e high s p i n s , it a l s o c o n t a i n s t h r e e u n e q u a l m a s s e s . S t i l l it i s s i m p l e r to h a n d l e and we b e l i e v e t h a t what we h a v e l e a r n t s h o u l d m a k e w o r k e a s i e r f o r t h o s e who want to r e g g e i z e high s p i n r e a c t i o n s . T h e p l a n of the p a p e r is t h e n to s t a r t f r o m t h e s p e c i f i c e x a m p l e ~N ~ pN that we t r e a t i n the m o s t p e d e s t r i a n m a n n e r by c a l c u l a t i n g all h e l i c i t y a m p l i t u d e s i n t e r m s of t h e M a n d e l s t a m i n v a r i a n t f u n c t i o n s ( s e c t . 2). We t h e n f i n d , ( s e c t . 3), two c l a s s e s of c o n s t r a i n t s d e p e n d e n t on w h e t h e r o r not the ~ and the p m a s s e s a r e e q u a l . Next, we show how t h e s e c o n s t r a i n t s a r e o b t a i n e d f r o m the c r o s s i n g r e -
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H. H()GAASEN and Ph. SALIN
l a t i o n s of T r u e m a n a n d W i c k ( s e c t . 4). In s e c t . 5 we t h e n s h o w how one c a n , from the crossing matrix, tackle the general spin problem.
2. G E N E R A L I T I E S F O R 7;N -~ p N S C A T T E R I N G A M P L I T U D E 2.1. Kinematics in lhe l-channel T h e g e n e r a l s t r u c t u r e of t h e s c a t t e r i n g a m p l i t u d e f o r t h e p r o c e s s nN , pN i s w e l l known, b e c a u s e of i t s a n a l o g y with p h o t o p r o d u c t i o n o r e l e c t r o p r o d u c t i o n [2]. N e v e r t h e l e s s , o u r r e s u l t s d e p e n d s o c r i t i c a l l y upon t h e k i n e m a t i c s , t h a t to m a k e o u r d e r i v a t i o n a s c l e a r a s p o s s i b l e , we s h a l l d e f i n e e v e r y t h i n g we n e e d . In o r d e r to m a k e the t r a n s p o s i t i o n to p h o t o p r o d u c t i o n t r i v i a l , we t a k e t h e k i n e m a t i c s of t h e p r o c e s s p N - ~ nN, w h i c h a c c o u n t s f o r t h e s a m e t h i n g in w h a t we i n t e n d to do. W e s h a l l c a l l k, q, P l a n d P2, r e s p e c t i v e l y t h e f o u r - m o m e n t a of p, n and n u c l e o n s , a n d m, g. and M t h e i r m a s s e s :
T h e n , in t h e c e n t r e - o f - m a s s k : ( k , ko) ,
of t h e / - c h a n n e l , we w r i t e t h e f o u r - v e c t o r s
q = (k,-u~),
Pl = (p' -E),
as
P2 = ( p ' E) ,
where ~/(t- (re+U)2)(/-
(m-~)2)
l + m 2 _ ~2 ,
iki
:
2~/t
2(t
p--½Jt-4M2, k'p = kpcosOt,
k0 =
E:A;l,
u- s ( u - s) it c o s Or= 4kp = ~ 4 i l _ ( m + ~ ) 2 ) ( t _ ( m _ l ~ . ) 2 ) s = - ( k + p l )2
,
u =
- ( k - p2) 2
A l l t h e s e e x p r e s s i o n s a r e a s s u m e d to h o l d f o r l ) 4 M 2, w i t h p o s i t i v e d e t e r m i n a t i o n f o r t h e s q u a r e r o o t s . In t h e f o l l o w i n g we s h a l l p e r f o r m an a n a l y t i c c o n t i n u a t i o n to n e g a t i v e t f o l l o w i n g a p a t h in t h e u p p e r half p l a n e of t h e c o m plex t-plane. 2.2• Invariant amplitudes W e e x p a n d t h e s c a t t e r i n g a m p l i t u d e on t h e e i g h t f o l l o w i n g i n v a r i a n t s
CONSPIRACY RELATIONS
659
T = u ( P 2 ) ( / ~1"= S i B i ) u ( p 1)
,
S 1 = iY5P 1 •
S 5 : V 5 ( v . k ) ( p 1 " ¢) ,
$2 = iY5P2 "c
S 6 = V5(v'k)(P2.
S 3 = iY5k.e
,
S 7 = V5(7. k)(k" E),
S4 = y 5 7 . c
,
S 8 = i y 5 ( ~ . k ) ( y . ~) ,
¢) ,
(2)
and, a s shown by Ball [2], the f u n c t i o n s B i a r e f r e e of k i n e m a t i c a l s i n g u l a r i t i e s , and will be p o s t u l a t e d to s a t i s f y the M a n d e l s t a m r e p r e s e n t a t i o n . 2.3. H e l i c i l y a m p l i t u d e s As u s u a l we define h e l i c i t y a m p l i t u d e s f i n the t - c h a n n e l f r e e of k i n e m a t i c a l s i n g u l a r i t i e s in s by f c d a b = [½(1 + c o s Ot)] -2-1 t~+Pl [½(1 - c o s
ot)l-½1 x-~l f c d a b '
(3)
with ~ = a - b and t~ = c - d, and h e l i c i t y a m p l i t u d e s with d e f i n i t e p a r i t y f ± by
fcdab = fcdab ± ~c-dab"
(4)
A f t e r s o m e a l g e b r a i c m a n i p u l a t i o n s , one o b t a i n s the h e l i c i t y a m p l i t u d e s in t e r m s of the i n v a r i a n t f u n c t i o n s B i. The r e s u l t i s . . . . fo1 =f½½10-
2 ~ ip M JR(B1 + B 2 ) - M k o ( B S + B 6 ) + k o s 8 1 '
-+ f+o1 =f½½10
2 ~/2 ik E B 8 = - ~ '
fil
-= f ½-½10 : -
-+ fll
-+ =f½-½10--
2(2ip ~
IS4+
(k.p)(B5+
B6) ]
2 ~/2 ik M [ p 2 ( B 5 + B 6 ) + MBs] ' ko
foo =f½½oo - - M ~
--
2ik
co~ Otfo1 ÷ ~
[(k'P)B8
- E 2 ( B 1 - B2) + M ( B 4 + k o E ( B 5 - B6))I ,
fl0-+ = f ½½ -00+ These expressions
= - Mrn4ip [ k o B 4 + k o ( k . p ) ( B 5 + B 6 ) + k 2 E ( B 5 _ B 6 ) ] .
(5)
c o n t a i n all i n f o r m a t i o n of i n t e r e s t c o n c e r n i n g the k i n e
660
H. H~)GAASEN and Ph. SALIN
m a t i c a l s i n g u l a r i t i e s and c o n s t r a i n t s a m o n g h e l i c i t y a m p l i t u d e s . now f o c u s o u r a t t e n t i o n on this.
We s h a l l
3. CONSTRAINTSAMONGHELICITY AMPLITUDES F r o m eq. (5) one now s e e s that for I going to z e r o , one g e t s two " c o n s p i r a c y r e l a t i o n s " a m o n g the a m p l i t u d e s f , r e q u i r i n g only that the B i a r e finite at this point [3]. This m a y be shown by i n v e r t i n g the p r e c e d i n g r e l a t i o n s , o r , m o r e s i m p l y , by looking at the l i m i t when t --. 0 + of the h e l i c i t y a m p l i t u d e s . One o b t a i n s -. ~ ( m 2 _ ~. 2 ) lim f01 ~ [M(B5+B6) - B 8 1 + O ( ~ t ) ' t ~0 + lim
~-+ , ~ ( m 2 _ ~2) fll =-i [M(B5+B6)-BsI+O(Jt ) .
t -~0 +
Z/
Then
lim I/f01 + f l l ] = O(4t) ,
(6)
t '0 +
and by a s i m i l a r p r o c e d u r e ,--[-
l i m [iflO t-~0 +
2/0ol = o(#t).
(7)
-+
The two r e m a i n i n g a m p l i t u d e s f 0 1 and f l l go to a c o n s t a n t . One h a s now a s t r a n g e s i t u a t i o n : if we put the two b o s o n m a s s e s equal and go to the l i m i t t = 0, then the two r e l a t i o n s v a n i s h and one o b t a i n s a t h i r d " c o n s p i r a c y r e l a t i o n " w h i c h h a s n o t h i n g to do with the two p r e c e d i n g ones. That r e l a t i o n s (6) and (7) v a n i s h is m o s t e a s i l y s e e n in the following way. We define h e l i c i t y a m p l i t u d e s Fij f r e e of k i n e m a t i c a l s i n g u l a r i t i e s by F01 =
1 '
Y00 = ~t f o o ,
Fll
-
r,l k
'
FIO = ~ f lO '
then r e l a t i o n (6) b e c o m e s
m 2 _ ix 2 [F01
2M
Fll]--+ 0 ,
(8.a)
and r e l a t i o n (7)
i M ( m ~ _ ~2)
El0+ F00 i -~ 0(" 2 - 9 )
F o r m = p, this g i v e s no r e l a t i o n .
(8.b)
CONSPIRACY RELATIONS
661
D i r e c t l y f r o m e x p r e s s i o n s (5) one o b t a i n s f. 0 1 . -~. 0 , .
fll
.... 2~/2 [B 4 + ~ - 2s ( B 5 + B 6 ) ] ,
--+ f01 ~ 0,
fll
fl0
f00
--I-
--* 2~-2 ~[M(B5+B6) - B8] , 2
"~ 0 ,
....
i
~~: l=-
B 8 + MB4 I ' ..
(9)
w h e r e ~ = m and Z = 2 M 2 + 2 p 2. It is c l e a r that now the t h r e e h e l i c i t y flip a m p l i t u d e s vanish. But the t h r e e r e m a i n i n g n o n - f l i p a m p l i t u d e s a r e not independent. In fact, they satisfy: ~il
~4 -~ / ~2 s -f+l l
+ ~ / 2 / 0 0 ] --' 0 ,
(10)
and this r e l a t i o n h a s n o t h i n g to do with (6) and (7). The r e a s o n why t h e s e two s e t s of c o n s t r a i n t s a r e so d i f f e r e n t , and why t h e r e c a n n o t be a c o n t i n u a t i o n f r o m one s e t to the o t h e r one is s i m p l y due to the f a c t that
lim
2 l i m (k o - t + m 2 - m 2 . )
m l - ~ m 2 t--~O
2v~
¢lim lim ko, t-~O ml-~m 2
the l e f - h a n d s i d e is infinite, the r i g h t - h a n d side is z e r o . To c o m p l e t e this d i s c u s s i o n , let u s n o t i c e that: (i) when all four m a s s e s a r e equal, then the c o n s t r a i n t e q u a t i o n (10) s u r v i v e s f o r t ~ 0; (ii) when all f o u r m a s s e s are different, a s f o r i n s t a n c e in nN--, K*E o r in n - p -~ pOrt (if one t a k e s into a c c o u n t the e l e c t r o m a g n e t i c m a s s d i f f e r e n c e of p and n) then the helicity amplitudes become all independent f o r t = O. There is no longer any constraint of types (6), (7) or (10). In this c a s e one is of c o u r s e giving up the SU 3 and even the i s o s p i n i n v a r i a n c e of the s c a t t e r ing, w h i c h would i m p l y f u r t h e r r e l a t i o n s a m o n g s c a t t e r i n g a m p l i t u d e s . In o r d e r to p r o v e this l a s t point, which l o o k s at f i r s t s i g h t r a t h e r a s t o n i s h i n g , we should w r i t e down e x p l i c i t l y the r e l a t i o n s b e t w e e n h e l i c i t y a m p l i t u d e s and i n v a r i a n t f u n c t i o n s in the f o u r u n e q u a l m a s s c a s e . This we s h a l l not do, b e c a u s e in s e c t . 5 we s h a l l p r o v e it in a m o r e d i r e c t and m o r e g e n e r a l way. But in a v e r y intuitive w a y , let us n o t i c e that the f a c t o r E w h i c h r e p r e s e n t s the n u c l e o n e n e r g y is z e r o when t -~ 0, b e c a u s e E = ½~t, but tends to infinity when the two f e r m i o n s h a v e d i f f e r e n t m a s s e s , a s E 1 = ( ( t + m 2 - m 2 ) / ( 2 ( t ) ) , (even if the m a s s d i f f e r e n c e is 0.1 eV!). We now t r y to get s o m e i n s i g h t into the v a r i o u s t y p e s of c o n s t r a i n t s , and we f i r s t s h o w the c o n n e c t i o n b e t w e e n o u r c o n s t r a i n t s and the G r i b o v - V o l k o v [4] type of c o n s t r a i n t s .
662
H. HOGAASEN and Ph. SALIN
4. C O N S T R A I N T S IN T H E FORWARD D I R E C T I O N The f i r s t " c o n s p i r a c y r e l a t i o n " h i s t o r i c a l l y found i s the w e l l - k n o w n r e l a t i o n of G r i b o v a n d Volkov. T h i s r e l a t i o n w a s found i n NN s c a t t e r i n g , that i s , i n the four e q u a l m a s s c a s e , for t = 0 c o r r e s p o n d s to f o r w a r d s c a t t e r i n g i n the s - c h a n n e l . In f a c t 2t c o s 0s = 1 + s - 4 M
2
G r i b o v a n d Volkov found t h e i r " c o n s p i r a c y r e l a t i o n " by i n v e r t i n g the r e l a t i o n b e t w e e n h e l i c i t y a m p l i t u d e s i n the t - c h a n n e l a n d i n v a r i a n t f u n c t i o n s , r e q u i r i n g the l a t t e r to be r e g u l a r a r o u n d t = 0, a n d they gave a v e r y n a t u r a l e x p l a n a t i o n of t h e i r r e l a t i o n , by n o t i c i n g that for c o s 0s = +1 a m o n g the five i n d e p e n d e n t a m p I i t u d e s i n the s - c h a n n e l , only t h r e e s u r v i v e , n a m e l y t h o s e with h e l i c i t y n o n - f l i p . T h e n , b e c a u s e the n u m b e r of i n d e p e n d e n t a m p l i t u d e s s h o u l d be the s a m e at that p o i n t i n a l l t h r e e c h a n n e l s , one s h o u l d get two r e l a t i o n s b e t w e e n h e l i c i t y a m p l i t u d e s i n the l - c h a n n e l . One of t h e s e r e l a t i o n s i s the c o n s p i r a c y r e i a t i o n , the o t h e r one i s the v a n i s h i n g of one a m p i i t u d e . All t h i s h a s b e e n e x p l i c i t l y d e m o n s t r a t e d by A b e r s a n d T e p l i t z [5], u s i n g the c r o s s i n g r e l a t i o n of T r u e m a n a n d Wick [6]:
f~#-~ = ~; 9'I[ X'~'
'/a'"
(11)
B e c a u s e of f o r m u l a (3), w h i c h i s a c o n s e q u e n c e of t o t a l a n g u l a r m o m e n t u m conservation, f~(cos 0 s = +1) = 0 if ;t ¢ ~. I n the c a s e of NN s c a t t e r i n g , t h e r e a r e two h e l i c i t y flip a m p l i t u d e s f s 1 1 1 and f s _1 z!. T h e v a n i s h i n g of f s ~ ! ! l e a d s to the GV r e l a t i o n 222--2
--2-2-22
2--2-22
- J+'+__ - J'+_+_ + z _t_ + •
o
S
: o .
The v a n i s h i n g of f ! ! ! 1_l e a d s to n o t h i n g new. 22--2 All t h i s i s w e l l ~ n o w n , but we h a v e r e c a l l e d it i n o r d e r to give s o m e i n s i g h t i n t o the f o l l o w i n g . The c o n c l u s i o n of t h i s d i s c u s s i o n i s that the c o n s t r a i n t s b e t w e e n h e l i c i t y a m p l i t u d e s f t i n the t - c h a n n e l , t h r o u g h the TW c r o s s i n g r e l a t i o n , due to the v a n i s h i n g of the h e l i c i t y flip a m p l i t u d e s f s for c o s 0 s = +1, s e e m to be a v e r y g e n e r a l p r o p e r t y w h i c h s h o u l d be m e t i n a l l s c a t t e r i n g p r o c e s s e s a + b ~ c + d . In the e q u a l m a s s c a s e c o s 0s = + l c o r r e s p o n d s to t = 0. But i n the u n e q u a l m a s s c a s e c o s es = +1 l e a d s to a r e l a t i o n b e t w e e n s a n d t, s = s ( t ) . T h u s i n the u n e q u a l - m a s s c a s e the h e l i c i t y a m p l i t u d e s of the t - c h a n n e l s h o u l d s a t i s f y a n u m b e r of r e l a t i o n s a l o n g a c u r v e s = s(/). We s h a l l show t h a t t h e s e r e l a tions are satisfied in a trivial manner. We r e t u r n to o u r 7rN -+ pN p r o b l e m , a n d n o t e t h a t t = 0 d o e s n o t c o r r e s p o n d to f o r w a r d d i r e c t i o n i n the s - c h a n n e l . But n e v e r t h e l e s s we found a c o n s t r a i n t a t t = 0, a n d it i s a p p a r e n t i n eq. (5) t h a t t h e r e i s no o t h e r s i n g u l a r p o i n t e x c e p t for /9 = 0 o r k = 0. The a s s u m e d c o n s t r a i n t for cos 0s = +1 s e e m s to be r a t h e r h i d d e n i n o u r c a s e .
CONSPIRACY
The answer is that there is no constraint cos OS = +l. In the TN ----)pN problem the s-channel
663
RELATIONS
in the unequal-mass
we have six independent helicity
case _foor
amplitudes
in
For cos OS= +1 only f$+ and f_+_$ are non-zero. Thus we expect four relations between helicity amplitudes in the f-channel. In fact, there are four relations, but they are completely trivial, because in that case cos OS = +l corresponds to cos 0~ = +l, and due to the factor (l+ cos Q) +/x+P*/ (1 _cos ,&+ IX-PI in
flp, f;/JCOSHt =*l) = 0 Explicitly,
the following f&
=
helicity
fUl0 9
if
h*p#O.
amplitudes
vanish:
fi-1 =f+$10
,
and
ffo =fpoo
f
for cos et = -1,
for cos Bt = +l . There is no “conspiracy relation” of the Gribov-Volkov type. On the other hand, if we make the two boson masses equal, m = p, then cos OS= +l corresponds to t = 0, and as cos 8t = -(2(s -M2 - /~2)/4pM) f rtl, Application of the TW crossing relaone does not find trivial zeros in fllL. tion leads to the “conspiracy relation” of eq. (10). For example
fS dcba = f:+,go 1
=H
cosxc [sin+(xd+xb)f& - cos &-J+xb)f~l]
+
i[sin+(Xd
+
sinxc _ 2~ [-fO(J sin%xd+xb) +fio cm k$-J+xb)] .
- Xb)f&
+ cos
%Xd -
Xb)f;I] (12)
H. HOGAASEN and Ph. SALIN
664
F o r cos 0s =+1, then ×c =×b = ×d = ½ ~ a n d f Slll^ - ~ ~o =0" This gives
f:l
-½
(13)
foo = o.
If one u s e s
-÷ f;1 = ½ cos0/ flll
~- ~f 1 l
~ - 2s and COS 0 t (t:O) : 4MA '
then eq. (13) l e a d s to eq. (10) --
~ - 2s -+
- 4M~ fll
+ 4"2 f 0 0 ] --" 0 .
Let us now show how we can obtain the ! = 0 conditions (6) and (7) on the f t , when m ¢ ~, f r o m the c r o s s i n g r e l a t i o n s , without making r e f e r e n c e to t h e ' i n v a r i a n t a m p l i t u d e s {Bi}. Evidently, at l = 0, cos 0s ¢ i l , and we have no z e r o s on the left-hand side of the c r o s s i n g r e l a t i o n (115. However, the left-hand side is r e g u l a r when l is continued f r o m the s m a l l negative value of ! (when cos 0s = +15 to t = 0. Inspection of the c r o s s i n g m a t r i x c~X~
,
[TW eq. (42) and (43)] shows, h o w e v e r , that s o m e of~the e l e m e n t s of this a r e singular. By looking up which combinations of f ~ , ~ , a r e multiplied by these singular f a c t o r s , and r e q u e s t i n g their p r o d u c t t o b e r e g u l a r , we r e c o v e r eqs. (6) and (75. We shall r e t u r n to this in the next section. We have worked out explicitly, on a specific example, n a m e l y the nN ~ pN p r o c e s s , the v a r i o u s m e c h a n i s m s of c o n s t r a i n t s a m o n g helicity amplitudes. We now t r y to g e n e r a l i z e this to the g e n e r a l c a s e for any m a s s e s and any spins.
5. THE GENERAL P R O B L E M OF CONSTRAINTS AMONG HELICITY AMPLITUDES In the g e n e r a l spin c a s e it is v e r y difficult to c o n s t r u c t invariant functions with no k i n e m a t i c a l s i n g u l a r i t i e s . On the other hand we know how to obtain helicity amplitudes f r e e of k i n e m a t i c a l s i n g u l a r i t i e s [1]. T h e r e f o r e , we shall f r o m now on work only with helicity amplitudes. In o r d e r to exhibit and c h a r a c t e r i z e the v a r i o u s kinds of c o n s t r a i n t s amohg helicity a m p l i t u d e s , we shall need only the two following p r o p e r t i e s . (i) Each helicity amplitude m a y be written fP
: [ ½ ( l + c o s g p ) l ½]~t+"l[½(1-cos0p)1½i~t-~tf~/~
p=s,
,
(141
uor t,
w h e r e ~ t z is f r e e of k i n e m a t i c a l s i n g u l a r i t i e s in t. (ii) The helicity amplitudes f s in the s - c h a n n e l and f t in the t-channel a r e r e l a t e d through the TW c r o s s i n g r e l a t i o n
CONSPIRACY RELATIONS
s
:
fcdab
~
665
dsa . . . .
A'b'c'D'
Sb, ,~Sc, ,,Sd , ,~t A ' a t × a l a b ' b ~×blac 'c ~Xc l a D 'd~ Xd)]c ' A ' D ' b ' "
(15)
5.1. The forward direcliou couslrainls It i s w e l l k n o w n t h a t t h e b o u n d a r y of t h e p h y s i c a l d o m a i n f o r c o s 02 = +1, c o s 2 0 u = +1 o r c o s 2 01 = +1, i s g i v e n i n a l l t h r e e c h a n n e l s by t h e s a m e c u r v e i n t h e i s , l, tl) p l a n e , n a m e l y ~(s,l)
= 0,
where
(s,t) = stu - s(m 2 -m2~',-2djv_a-m2) -t(m2 -m2)(m2b 2 2 m2m2)(m2+
- (roared-
b
If 0s i s t h e c . m . a n g l e f o r t h e plies cos 0 t = ±i, when m a ¢ mc, There is an ambiguity in sign in s i m p l i c i t y to b e c o s 0! = + l w h e n In this kinematical situation f~
~
c - m2)
2 2 2 md-m c-rob).
(16)
p r o c e s s a + b ~ c + d, t h e n c o s 0s = + i i m 0 t b e i n g t h e c . m . a n g l e of d + b --' c + -~. t h e d e f i n i t i o n of c o s 0t. W e c h o o s e i t f o r c o s 0 s = +1.
=0
f~/,~, =0
if
~ - ~
if X'-~'¢
¢0, 0.
B u t t h e r e a r e e x a c t l y t h e s a m e n u m b e r of h e l i c i t y a m p l i t u d e s i n e a c h c h a n nel satisfying these relations because ~t - p =~t a - ~ t b - ( ~ t c - ~ t d ) = h d - ~ t b - ( ~ t c - ) t a )
=~t,_p,
W e a r e t h e n l e d to t h e f o l l o w i n g s t a t e m e n t : If 0 s i s t h e c . m . a n g l e of t h e p r o c e s s a + b - ~ c + d, t h e n c o s 0 s = +1 i m p l i e s n o c o n s t r a i n t s a m o n g h e l i c i t y a m p l i t u d e s i n t h e / - c h a n n e l a + b -~ c ~- ~t, w h e n m e ¢ m a. If m c = m a a n d m d = m b , t h e n t h e T r u e m a n - W i c k c r o s s i n g r e l a t i o n l e a d s to r e l a t i o n s a m o n g h e l i c i t y a m p l i t u d e s of t h e G r i b o v - V o l k o v type. 5.2. The t = 0 constraints (i) S i n c e a l l k i n e m a t i c a l s i n g u l a r i t i e s (1 + c o s Os) f a c t o r s , a n d a s c o s 0 s =[2st+s 2 - s ~ m 2 + ( m 2
i n t of f ~ p
are contained in the
m2~,~2 2 b,~,,Oc - rod)~ eJab eJcd ,
with
~ 2 b = [s - ( m a - mb)2][s - (ma+ mbl 2] , ~ 2 d _- [~ - (m c - md)21[~ - ( m c , mdl 2] , i t i s c l e a r t h a t f ~ t ~ h a s n o p o l e f o r ! = 0.
666
H. HDGAASEN and Ph. SALIN
(ii) On the o t h e r hand, one h a s , for e x a m p l e , for the a n g l e s e n t e r i n g the crossing matrix 1
sin ×a = 2 m a [@(s, t ) ] ~ / C i a b ~ a c with
~ac
= [t - ( m a + m c ) 2 J [ t - (m a - m c ) 2] .
If m a = rnc, then s i n X a and c o s x a have a pole f o r t = 0, and d J p ( x a ) h a s a pole of o r d e r n, with
n =~(!~+ Izl + i ~ - ~ I) + ½(J-sup(l'~l, i~i)) = ' ~ J , because
dJ/x(Xa) where
= Yj[½(l
+ cos Xa)]½ i A+M I [½(I - cos Xa)] ½]~-p ip[X-~ J - M ], [x+~ I' ( c o s
is a Jacobi p o l y n o m i a l
o r d e r J - M , and M : sup(iX i, I
Xa) ,
I).
B e c a u s e ) ~ p should not h a v e this p o l e in t - n , it is n e c e s s a r y that s o m e c o m b i n a t i o n s of h e l i c i t y a m p l i t u d e s f / go to z e r o a s t n L e t us w o r k out e x p l i c i t l y this for ~N -* pN. We e x p r e s s the f -S1 ~ l ~10 h e l i c i t y a m p l i t u d e in t e r m s of h e l i c i t y a m p l i t u d e s in t h e / - c h a n n e l . This h a s b e e n given in (12). C o s Xc and sin Xc a r e finite when t -~ 0, but Xd and Xb a r e not. c o s Xb = [ ( s + M 2 - m 2 ) t - 2M2(p 2 - m 2 ) ] / C i a b ~ b d
,
c o s Xd = [-(s + M 2 - tz2)t - 2M2(~ 2 - r n 2 ) ] / C i c d ~ b d '
sin ×b -
2M (4~ ~ a b -~bd
,
sin
J a b = 2 ~/s q s '
Xd
:
2M ( ~ ~bdCicd
'
(17)
Cicd = 2 ~/sk s ,
~ b d = 2 ~ i - P l = ~ t vrt - 4M 2 A f t e r s o m e a l g e b r a i c m a n i p u l a t i o n s , it m a y be shown that t 2 A ( s ) + t B ( s ) + 8M4(~ 2 - rn2) 2 c o s (Xd+X b ) 4 s k s q s ( t - 4M2)t and c o s 0( b - × d ) = t 2 A ' ( s ) + t B ' ( s ) 4 s k s q s C l - 4M2)t
(18)
Now, when t -~ O, c o s ( ~ + Xb) is going to infinity like 1 / t and c o s (Xb- Xd) b e h a v e s a s a f u n c t i o n of s. F r o m this we d e d u c e that c o s ½ix d + Xb) and
CONSPIR ACY RE LATIONS sin ½(×d + ×b) a r e going to infinity like 1/~/l,
667
with
cos½(Xd+×b) ~ i sin½0(d+Xb) . F r o m eq. (12), then, it is n e c e s s a r y that --÷
---
f l l ÷ if01 .-÷
iflo
- 2f00 ~ ~/.
(19)
T h e s e r e l a t i o n s a r e e x a c t l y the s a m e a s t h o s e d e d u c e d in eqs. (6) and (7), f r o m the i n v a r i a n t a m p l i t u d e s [the f a c t o r two in (19) is i n t r o d u c e d when one g o e s f r o m the f to the f ] . One m a y c o n v i n c e o n e s e l f that the two r e l a t i o n s (19) a r e r e a l l y i n d e p e n d e n t , by l o o k i n g at o t h e r r e l a t i o n s with t h e f s, w h e r e the s a m e c o m b i n a t i o n s of h e l i c i t y a m p l i t u d e s a l w a y s o c c u r , but with d i f f e r ent m u l t i p l i c a t i v e f a c t o r s . It is r e m a r k a b l e to n o t i c e that if the four m a s s e s a r e d i f f e r e n t , then c o s ×b and c o s × d a r e no l o n g e r s i n g u l a r f o r t = 0, so that t h e r e is no c o n s t r a i n t at this point and we m a k e the g e n e r a l s t a t e m e n t : When the four m a s s e s of a s c a t t e r i n g p r o c e s s a + b -* c + d a r e all d i f f e r e n t , t h e r e is no c o n s t r a i n t a m o n g h e l i c i t y a m p l i t u d e s for t = 0. T h e r e a r e c o n s t r a i n t s at this point only when t h r e e k i n d s of d i f f e r e n t m a s s e s o c c u r with m a ¢ m c and m b = rod, and t h e s e c o n s t r a i n t s have n o t h i n g to do with the f o r w a r d type of c o n s t r a i n t s .
6. CONCLUSIONS It r e s u l t s f r o m the p r e c e d i n g d i s c u s s i o n t h a t t h e r e a r e two t y p e s of c o n s t r a i n t s a m o n g h e l i c i t y a m p l i t u d e s in the t - c h a n n e l : (i) the f i r s t type of c o n s t r a i n t s o c c u r s only in r e a c t i o n s a + b -~ c + d with p a i r w i s e equal e x t e r n a l m a s s e s (m a = m c , m-o = m d )' in the s - c h a n n e l f o r w a r d d i r e c t i o n (or b a c k w a r d d i r e c t i o n ) , and is due to the v a n i s h i n g of h e l l city flip a m p l i t u d e s f s; (ii) The s e c o n d type of c o n s t r a i n t s o c c u r s at t h o s e p o i n t s w h e r e the r o t a tion m a t r i c e s a r e infinite, i . e . , when It
-
(rob+ md)2][t - (m b - r o d )2] = 0 .
T h e y a r e u s u a l l y known a s " t h r e s h o l d c o n d i t i o n s " , s e e for i n s t a n c e J o n e s [7], but o c c u r a t t = 0 o n l y w h e n m b = m d , a n d ~ cm.. We s u m m a r i z e the v a r i o u s p o s s i b i l i t i e s in table 1. ]In all c a s e s t h e s e two t y p e s of c o n s t r a i n t s c a n be c o m p u t e d d i r e c t l y f r o m the T r u e m a n - W i c k c r o s s ing r e l a t i o n , without r e f e r e n c e to i n v a r i a n t a m p l i t u d e s . T h e r e f o r e , one m a y a v o i d the t r e m e n d o u s a l g e b r a i c m a c h i n e r y of going f r o m the i n v a r i a n t a m p l i t u d e s to the h e l i c i t y a m p l i t u d e s and then i n v e r t i n g the o b t a i n e d r e l a t i o n s in o r d e r to get the c o n s t r a i n t s . It is not n e c e s s a r y e i t h e r to go f r o m the h e l i c i t y f r a m e to the u s u a l p a r t i a l wave f r a m e to o b tain the t h r e s h o l d c o n d i t i o n s . T h i s l a s t p r o c e d u r e i n v o l v e s r a t h e r c o m p l i -
668
H. HOGAASEN and Ph. SXLIN Table 1 Masses
rna = m c
mb - m d
Type of constraint for t -- 0 =L1 type I
Number of constraint points
ql -= 0
3
m b e rn d
t - 0
typeII
q! 0: Pt 0
4
m a c rn c rnb - rnd
t - 0
typelI
qt :0;
Pt-O
4
qt=O;
Pt=:O
4
mbern d
cOSOs
Other constraints
none
c a t e d C l e b s c h - G o r d a n c o e f f i c i e n t c a l c u l a t i o n s a n d f u r t h e r a s s u m p t i o n s on the t h r e s h o l d b e h a v i o u r of p a r t i a l wave a m p l i t u d e s . In o u r o p i n i o n , t h i s i s a n o n - n e g l i g i b l e p r o g r e s s . N e v e r t h e l e s s , i t s e e m s t h a t the g e n e r a l f o r m a l i m s d e v e l o p e d by W a n g [1] i n o r d e r to e x t r a c t the k i n e m a t i c a l s i n g u l a r i t i e s , l o s e s s o m e of i t s g e n e r a l i n t e r e s t b e c a u s e one i s f o r c e d to e v a l u a t e e x p l i c i t l y the c r o s s i n g r e l a t i o n s , w h i c h , a t the s a m e t i m e , g i v e s both the k i n e m a t i c a l s i n g u l a r i t i e s of h e l i c i t y a m p l i t u d e s a n d the c o n s t r a i n t s a m o n g them.
Addendum: A f t e r t h i s p a p e r w a s w r i t t e n , we h a v e b e e n i n f o r m e d t h a t the whole p r o b l e m of k i n e m a t i c a l s i n g u l a r i t i e s , c r o s s i n g s a n d c o n s t r a i n t s a m o n g h e l i c i t y a m p l i t u d e s , h a s b e e n r e v i e w e d a n d s o l v e d by G. C o h e n T a n n o u d j i , A. M o r e l a n d H. N a v e l e t f r o m S a c l a y . A m o n g o t h e r p o w e r f u l r e s u l t s , t h e s e a u t h o r s h a v e r e a c h e d the s a m e c o n c l u s i o n a s we h a v e w i t h r e g a r d to the c o n s t r a i n t p r o b l e m , u s i n g t r a n s v e r s i t y a m p l i t u d e s . T h e y found a l s o t h a t s o m e p h a s e s b e t w e e n m a t r i x e l e m e n t s of the c r o s s i n g m a t r i x of T r u ~ m a n a n d W i c k h a v e to be c h a n g e d . T h i s would m e a n t h a t the . It /~' m a t r l x g ~ x t ~ of eq. (11) s h o u l d n o t be e x a c t l y the T W m a t r i x . We a r e m u c h i n d e b t e d to Dr. C o h e n - T a n n o u d j i for h a v i n g i n f o r m e d u s of h i s r e s u l t s p r i o r to p u b l i c a t i o n , a s w e l l a s for v e r y i l l u m i n a t i n g d i s c u s s i o n s .
REFERENCES [1] L. L. C.Wang, Phys. Rev. 142 {1966} 1187. [2] J. S. Ball, Phys. Rev. 124 {1961) 2014; N. Dombey, Nuovo Cimento 31 {1964} 1025. [3] M. B. Halpern, Princeton preprint {1967}; B. Diu and M. Le Bellac, Orsay preprint 11967). [4] V. N. Gribov and D. V. Volkov, Soviet Phys. - J E T P 17 {1963} 720. [5] E. Abets and V. Teplitz, Los )mgeles p r e p r i n t {1967}. [6] T. L. Trueman and G.C.Wick, Ann. Phys. 26 (1964) 322. [71 H. F. Jones, Imperial College London p r e p r i n t {1966).
Nuclear Physics B2 (1967) 657-668. North-Holland Publ. Comp., Amsterdam
GENERAL CLASSIFICATION OF CONSPIRACY RELATIONS H. H(JGAASEN and Ph. SALIN CERN.
Geneva
Received 29 June 1967
Abstract: We present a study of constraints on helicity amplitudes when the external particles have a r b i t r a r y spins. The reaction 7~N--' ON is studied in detail and we find two types of "conspiracy relations". The first one occurs at t - 0 when the m a s s e s of 7r and p are different; it is different from the Gribov-Volkov type as it vanishes when the m a s s e s of ~ and p are equal. The Gribov-Volkov type of relations is very different from the preceding one and occurs when the m a s s e s of and p are the same. We show how all these relations can be found without explicit reference to the invariant amplitudes and argue that generally all conspiracy r e l a tions can be obtained directly from the Trueman-Wick crossing relations between helicity amplitudes.
1. I N T R O D U C T I O N Due to the g r e a t u t i l i t y of the R e g g e pole m o d e l , t h e r e h a s f o r a t i m e b e e n a g r e a t i n t e r e s t in the r e g g e i z i n g and a n a l y t i c a l c o n t i n u a t i o n of h e l i c i t y a m p l i t u d e s c o r r e s p o n d i n g to e x t e r n a l p a r t i c l e s with a r b i t r a r y s p i n s . In a now c l a s s i c a l p a p e r , W a n g [1] h a s g i v e n a s e t of p r e s c r i p t i o n s f o r c o n s t r u c t i n g h e l i c i t y a m p l i t u d e s f r e e of k i n e m a t i c a l s i n g u l a r i t i e s . T h e r e h a s , howe v e r , b e e n s o m e u n e a s i n e s s a b o u t how to t a c k l e p r o b l e m s l e a d i n g to the s o c a l l e d c o n s p i r a c y r e l a t i o n s . T h e s e a r e l i n e a r c o m b i n a t i o n s of h e l i c i t y a m p l i t u d e s t h a t m u s t v a n i s h at c e r t a i n v a l u e s of the k i n e m a t i c a l v a r i a b l e s of the p r o c e s s s t u d i e d . T h i s v a n i s h i n g i s n e c e s s a r y to a v o i d c o n s t r u c t i n g r e g g e i z e d a m p l i t u d e s that v i o l a t e M a n d e l s t a m a n a l y t i c i t y . We d e c i d e d to t r e a t a c o n c r e t e , s u f f i c i e n t l y c o m p l i c a t e d a n d s i m p l e p r o b l e m in d e t a i l to get a n u n d e r s t a n d i n g of the d i f f e r e n t m e c h a n i s m s that c a n l e a d one to c o n s t r u c t m o d e l s with w r o n g a n a l y t i c a l p r o p e r t i e s . T h e e x a m p l e t h a t we c h o o s e i s 7rN --' pN: i n a d d i t i o n to r e a s o n a b l e high s p i n s , it a l s o c o n t a i n s t h r e e u n e q u a l m a s s e s . S t i l l it i s s i m p l e r to h a n d l e and we b e l i e v e t h a t what we h a v e l e a r n t s h o u l d m a k e w o r k e a s i e r f o r t h o s e who want to r e g g e i z e high s p i n r e a c t i o n s . T h e p l a n of the p a p e r is t h e n to s t a r t f r o m t h e s p e c i f i c e x a m p l e ~N ~ pN that we t r e a t i n the m o s t p e d e s t r i a n m a n n e r by c a l c u l a t i n g all h e l i c i t y a m p l i t u d e s i n t e r m s of t h e M a n d e l s t a m i n v a r i a n t f u n c t i o n s ( s e c t . 2). We t h e n f i n d , ( s e c t . 3), two c l a s s e s of c o n s t r a i n t s d e p e n d e n t on w h e t h e r o r not the ~ and the p m a s s e s a r e e q u a l . Next, we show how t h e s e c o n s t r a i n t s a r e o b t a i n e d f r o m the c r o s s i n g r e -
658
H. H()GAASEN and Ph. SALIN
l a t i o n s of T r u e m a n a n d W i c k ( s e c t . 4). In s e c t . 5 we t h e n s h o w how one c a n , from the crossing matrix, tackle the general spin problem.
2. G E N E R A L I T I E S F O R 7;N -~ p N S C A T T E R I N G A M P L I T U D E 2.1. Kinematics in lhe l-channel T h e g e n e r a l s t r u c t u r e of t h e s c a t t e r i n g a m p l i t u d e f o r t h e p r o c e s s nN , pN i s w e l l known, b e c a u s e of i t s a n a l o g y with p h o t o p r o d u c t i o n o r e l e c t r o p r o d u c t i o n [2]. N e v e r t h e l e s s , o u r r e s u l t s d e p e n d s o c r i t i c a l l y upon t h e k i n e m a t i c s , t h a t to m a k e o u r d e r i v a t i o n a s c l e a r a s p o s s i b l e , we s h a l l d e f i n e e v e r y t h i n g we n e e d . In o r d e r to m a k e the t r a n s p o s i t i o n to p h o t o p r o d u c t i o n t r i v i a l , we t a k e t h e k i n e m a t i c s of t h e p r o c e s s p N - ~ nN, w h i c h a c c o u n t s f o r t h e s a m e t h i n g in w h a t we i n t e n d to do. W e s h a l l c a l l k, q, P l a n d P2, r e s p e c t i v e l y t h e f o u r - m o m e n t a of p, n and n u c l e o n s , a n d m, g. and M t h e i r m a s s e s :
T h e n , in t h e c e n t r e - o f - m a s s k : ( k , ko) ,
of t h e / - c h a n n e l , we w r i t e t h e f o u r - v e c t o r s
q = (k,-u~),
Pl = (p' -E),
as
P2 = ( p ' E) ,
where ~/(t- (re+U)2)(/-
(m-~)2)
l + m 2 _ ~2 ,
iki
:
2~/t
2(t
p--½Jt-4M2, k'p = kpcosOt,
k0 =
E:A;l,
u- s ( u - s) it c o s Or= 4kp = ~ 4 i l _ ( m + ~ ) 2 ) ( t _ ( m _ l ~ . ) 2 ) s = - ( k + p l )2
,
u =
- ( k - p2) 2
A l l t h e s e e x p r e s s i o n s a r e a s s u m e d to h o l d f o r l ) 4 M 2, w i t h p o s i t i v e d e t e r m i n a t i o n f o r t h e s q u a r e r o o t s . In t h e f o l l o w i n g we s h a l l p e r f o r m an a n a l y t i c c o n t i n u a t i o n to n e g a t i v e t f o l l o w i n g a p a t h in t h e u p p e r half p l a n e of t h e c o m plex t-plane. 2.2• Invariant amplitudes W e e x p a n d t h e s c a t t e r i n g a m p l i t u d e on t h e e i g h t f o l l o w i n g i n v a r i a n t s
CONSPIRACY RELATIONS
659
T = u ( P 2 ) ( / ~1"= S i B i ) u ( p 1)
,
S 1 = iY5P 1 •
S 5 : V 5 ( v . k ) ( p 1 " ¢) ,
$2 = iY5P2 "c
S 6 = V5(v'k)(P2.
S 3 = iY5k.e
,
S 7 = V5(7. k)(k" E),
S4 = y 5 7 . c
,
S 8 = i y 5 ( ~ . k ) ( y . ~) ,
¢) ,
(2)
and, a s shown by Ball [2], the f u n c t i o n s B i a r e f r e e of k i n e m a t i c a l s i n g u l a r i t i e s , and will be p o s t u l a t e d to s a t i s f y the M a n d e l s t a m r e p r e s e n t a t i o n . 2.3. H e l i c i l y a m p l i t u d e s As u s u a l we define h e l i c i t y a m p l i t u d e s f i n the t - c h a n n e l f r e e of k i n e m a t i c a l s i n g u l a r i t i e s in s by f c d a b = [½(1 + c o s Ot)] -2-1 t~+Pl [½(1 - c o s
ot)l-½1 x-~l f c d a b '
(3)
with ~ = a - b and t~ = c - d, and h e l i c i t y a m p l i t u d e s with d e f i n i t e p a r i t y f ± by
fcdab = fcdab ± ~c-dab"
(4)
A f t e r s o m e a l g e b r a i c m a n i p u l a t i o n s , one o b t a i n s the h e l i c i t y a m p l i t u d e s in t e r m s of the i n v a r i a n t f u n c t i o n s B i. The r e s u l t i s . . . . fo1 =f½½10-
2 ~ ip M JR(B1 + B 2 ) - M k o ( B S + B 6 ) + k o s 8 1 '
-+ f+o1 =f½½10
2 ~/2 ik E B 8 = - ~ '
fil
-= f ½-½10 : -
-+ fll
-+ =f½-½10--
2(2ip ~
IS4+
(k.p)(B5+
B6) ]
2 ~/2 ik M [ p 2 ( B 5 + B 6 ) + MBs] ' ko
foo =f½½oo - - M ~
--
2ik
co~ Otfo1 ÷ ~
[(k'P)B8
- E 2 ( B 1 - B2) + M ( B 4 + k o E ( B 5 - B6))I ,
fl0-+ = f ½½ -00+ These expressions
= - Mrn4ip [ k o B 4 + k o ( k . p ) ( B 5 + B 6 ) + k 2 E ( B 5 _ B 6 ) ] .
(5)
c o n t a i n all i n f o r m a t i o n of i n t e r e s t c o n c e r n i n g the k i n e
660
H. H~)GAASEN and Ph. SALIN
m a t i c a l s i n g u l a r i t i e s and c o n s t r a i n t s a m o n g h e l i c i t y a m p l i t u d e s . now f o c u s o u r a t t e n t i o n on this.
We s h a l l
3. CONSTRAINTSAMONGHELICITY AMPLITUDES F r o m eq. (5) one now s e e s that for I going to z e r o , one g e t s two " c o n s p i r a c y r e l a t i o n s " a m o n g the a m p l i t u d e s f , r e q u i r i n g only that the B i a r e finite at this point [3]. This m a y be shown by i n v e r t i n g the p r e c e d i n g r e l a t i o n s , o r , m o r e s i m p l y , by looking at the l i m i t when t --. 0 + of the h e l i c i t y a m p l i t u d e s . One o b t a i n s -. ~ ( m 2 _ ~. 2 ) lim f01 ~ [M(B5+B6) - B 8 1 + O ( ~ t ) ' t ~0 + lim
~-+ , ~ ( m 2 _ ~2) fll =-i [M(B5+B6)-BsI+O(Jt ) .
t -~0 +
Z/
Then
lim I/f01 + f l l ] = O(4t) ,
(6)
t '0 +
and by a s i m i l a r p r o c e d u r e ,--[-
l i m [iflO t-~0 +
2/0ol = o(#t).
(7)
-+
The two r e m a i n i n g a m p l i t u d e s f 0 1 and f l l go to a c o n s t a n t . One h a s now a s t r a n g e s i t u a t i o n : if we put the two b o s o n m a s s e s equal and go to the l i m i t t = 0, then the two r e l a t i o n s v a n i s h and one o b t a i n s a t h i r d " c o n s p i r a c y r e l a t i o n " w h i c h h a s n o t h i n g to do with the two p r e c e d i n g ones. That r e l a t i o n s (6) and (7) v a n i s h is m o s t e a s i l y s e e n in the following way. We define h e l i c i t y a m p l i t u d e s Fij f r e e of k i n e m a t i c a l s i n g u l a r i t i e s by F01 =
1 '
Y00 = ~t f o o ,
Fll
-
r,l k
'
FIO = ~ f lO '
then r e l a t i o n (6) b e c o m e s
m 2 _ ix 2 [F01
2M
Fll]--+ 0 ,
(8.a)
and r e l a t i o n (7)
i M ( m ~ _ ~2)
El0+ F00 i -~ 0(" 2 - 9 )
F o r m = p, this g i v e s no r e l a t i o n .
(8.b)
CONSPIRACY RELATIONS
661
D i r e c t l y f r o m e x p r e s s i o n s (5) one o b t a i n s f. 0 1 . -~. 0 , .
fll
.... 2~/2 [B 4 + ~ - 2s ( B 5 + B 6 ) ] ,
--+ f01 ~ 0,
fll
fl0
f00
--I-
--* 2~-2 ~[M(B5+B6) - B8] , 2
"~ 0 ,
....
i
~~: l=-
B 8 + MB4 I ' ..
(9)
w h e r e ~ = m and Z = 2 M 2 + 2 p 2. It is c l e a r that now the t h r e e h e l i c i t y flip a m p l i t u d e s vanish. But the t h r e e r e m a i n i n g n o n - f l i p a m p l i t u d e s a r e not independent. In fact, they satisfy: ~il
~4 -~ / ~2 s -f+l l
+ ~ / 2 / 0 0 ] --' 0 ,
(10)
and this r e l a t i o n h a s n o t h i n g to do with (6) and (7). The r e a s o n why t h e s e two s e t s of c o n s t r a i n t s a r e so d i f f e r e n t , and why t h e r e c a n n o t be a c o n t i n u a t i o n f r o m one s e t to the o t h e r one is s i m p l y due to the f a c t that
lim
2 l i m (k o - t + m 2 - m 2 . )
m l - ~ m 2 t--~O
2v~
¢lim lim ko, t-~O ml-~m 2
the l e f - h a n d s i d e is infinite, the r i g h t - h a n d side is z e r o . To c o m p l e t e this d i s c u s s i o n , let u s n o t i c e that: (i) when all four m a s s e s a r e equal, then the c o n s t r a i n t e q u a t i o n (10) s u r v i v e s f o r t ~ 0; (ii) when all f o u r m a s s e s are different, a s f o r i n s t a n c e in nN--, K*E o r in n - p -~ pOrt (if one t a k e s into a c c o u n t the e l e c t r o m a g n e t i c m a s s d i f f e r e n c e of p and n) then the helicity amplitudes become all independent f o r t = O. There is no longer any constraint of types (6), (7) or (10). In this c a s e one is of c o u r s e giving up the SU 3 and even the i s o s p i n i n v a r i a n c e of the s c a t t e r ing, w h i c h would i m p l y f u r t h e r r e l a t i o n s a m o n g s c a t t e r i n g a m p l i t u d e s . In o r d e r to p r o v e this l a s t point, which l o o k s at f i r s t s i g h t r a t h e r a s t o n i s h i n g , we should w r i t e down e x p l i c i t l y the r e l a t i o n s b e t w e e n h e l i c i t y a m p l i t u d e s and i n v a r i a n t f u n c t i o n s in the f o u r u n e q u a l m a s s c a s e . This we s h a l l not do, b e c a u s e in s e c t . 5 we s h a l l p r o v e it in a m o r e d i r e c t and m o r e g e n e r a l way. But in a v e r y intuitive w a y , let us n o t i c e that the f a c t o r E w h i c h r e p r e s e n t s the n u c l e o n e n e r g y is z e r o when t -~ 0, b e c a u s e E = ½~t, but tends to infinity when the two f e r m i o n s h a v e d i f f e r e n t m a s s e s , a s E 1 = ( ( t + m 2 - m 2 ) / ( 2 ( t ) ) , (even if the m a s s d i f f e r e n c e is 0.1 eV!). We now t r y to get s o m e i n s i g h t into the v a r i o u s t y p e s of c o n s t r a i n t s , and we f i r s t s h o w the c o n n e c t i o n b e t w e e n o u r c o n s t r a i n t s and the G r i b o v - V o l k o v [4] type of c o n s t r a i n t s .
662
H. HOGAASEN and Ph. SALIN
4. C O N S T R A I N T S IN T H E FORWARD D I R E C T I O N The f i r s t " c o n s p i r a c y r e l a t i o n " h i s t o r i c a l l y found i s the w e l l - k n o w n r e l a t i o n of G r i b o v a n d Volkov. T h i s r e l a t i o n w a s found i n NN s c a t t e r i n g , that i s , i n the four e q u a l m a s s c a s e , for t = 0 c o r r e s p o n d s to f o r w a r d s c a t t e r i n g i n the s - c h a n n e l . In f a c t 2t c o s 0s = 1 + s - 4 M
2
G r i b o v a n d Volkov found t h e i r " c o n s p i r a c y r e l a t i o n " by i n v e r t i n g the r e l a t i o n b e t w e e n h e l i c i t y a m p l i t u d e s i n the t - c h a n n e l a n d i n v a r i a n t f u n c t i o n s , r e q u i r i n g the l a t t e r to be r e g u l a r a r o u n d t = 0, a n d they gave a v e r y n a t u r a l e x p l a n a t i o n of t h e i r r e l a t i o n , by n o t i c i n g that for c o s 0s = +1 a m o n g the five i n d e p e n d e n t a m p I i t u d e s i n the s - c h a n n e l , only t h r e e s u r v i v e , n a m e l y t h o s e with h e l i c i t y n o n - f l i p . T h e n , b e c a u s e the n u m b e r of i n d e p e n d e n t a m p l i t u d e s s h o u l d be the s a m e at that p o i n t i n a l l t h r e e c h a n n e l s , one s h o u l d get two r e l a t i o n s b e t w e e n h e l i c i t y a m p l i t u d e s i n the l - c h a n n e l . One of t h e s e r e l a t i o n s i s the c o n s p i r a c y r e i a t i o n , the o t h e r one i s the v a n i s h i n g of one a m p i i t u d e . All t h i s h a s b e e n e x p l i c i t l y d e m o n s t r a t e d by A b e r s a n d T e p l i t z [5], u s i n g the c r o s s i n g r e l a t i o n of T r u e m a n a n d Wick [6]:
f~#-~ = ~; 9'I[ X'~'
'/a'"
(11)
B e c a u s e of f o r m u l a (3), w h i c h i s a c o n s e q u e n c e of t o t a l a n g u l a r m o m e n t u m conservation, f~(cos 0 s = +1) = 0 if ;t ¢ ~. I n the c a s e of NN s c a t t e r i n g , t h e r e a r e two h e l i c i t y flip a m p l i t u d e s f s 1 1 1 and f s _1 z!. T h e v a n i s h i n g of f s ~ ! ! l e a d s to the GV r e l a t i o n 222--2
--2-2-22
2--2-22
- J+'+__ - J'+_+_ + z _t_ + •
o
S
: o .
The v a n i s h i n g of f ! ! ! 1_l e a d s to n o t h i n g new. 22--2 All t h i s i s w e l l ~ n o w n , but we h a v e r e c a l l e d it i n o r d e r to give s o m e i n s i g h t i n t o the f o l l o w i n g . The c o n c l u s i o n of t h i s d i s c u s s i o n i s that the c o n s t r a i n t s b e t w e e n h e l i c i t y a m p l i t u d e s f t i n the t - c h a n n e l , t h r o u g h the TW c r o s s i n g r e l a t i o n , due to the v a n i s h i n g of the h e l i c i t y flip a m p l i t u d e s f s for c o s 0 s = +1, s e e m to be a v e r y g e n e r a l p r o p e r t y w h i c h s h o u l d be m e t i n a l l s c a t t e r i n g p r o c e s s e s a + b ~ c + d . In the e q u a l m a s s c a s e c o s 0s = + l c o r r e s p o n d s to t = 0. But i n the u n e q u a l m a s s c a s e c o s es = +1 l e a d s to a r e l a t i o n b e t w e e n s a n d t, s = s ( t ) . T h u s i n the u n e q u a l - m a s s c a s e the h e l i c i t y a m p l i t u d e s of the t - c h a n n e l s h o u l d s a t i s f y a n u m b e r of r e l a t i o n s a l o n g a c u r v e s = s(/). We s h a l l show t h a t t h e s e r e l a tions are satisfied in a trivial manner. We r e t u r n to o u r 7rN -+ pN p r o b l e m , a n d n o t e t h a t t = 0 d o e s n o t c o r r e s p o n d to f o r w a r d d i r e c t i o n i n the s - c h a n n e l . But n e v e r t h e l e s s we found a c o n s t r a i n t a t t = 0, a n d it i s a p p a r e n t i n eq. (5) t h a t t h e r e i s no o t h e r s i n g u l a r p o i n t e x c e p t for /9 = 0 o r k = 0. The a s s u m e d c o n s t r a i n t for cos 0s = +1 s e e m s to be r a t h e r h i d d e n i n o u r c a s e .
CONSPIRACY
The answer is that there is no constraint cos OS = +l. In the TN ----)pN problem the s-channel
663
RELATIONS
in the unequal-mass
we have six independent helicity
case _foor
amplitudes
in
For cos OS= +1 only f$+ and f_+_$ are non-zero. Thus we expect four relations between helicity amplitudes in the f-channel. In fact, there are four relations, but they are completely trivial, because in that case cos OS = +l corresponds to cos 0~ = +l, and due to the factor (l+ cos Q) +/x+P*/ (1 _cos ,&+ IX-PI in
flp, f;/JCOSHt =*l) = 0 Explicitly,
the following f&
=
helicity
fUl0 9
if
h*p#O.
amplitudes
vanish:
fi-1 =f+$10
,
and
ffo =fpoo
f
for cos et = -1,
for cos Bt = +l . There is no “conspiracy relation” of the Gribov-Volkov type. On the other hand, if we make the two boson masses equal, m = p, then cos OS= +l corresponds to t = 0, and as cos 8t = -(2(s -M2 - /~2)/4pM) f rtl, Application of the TW crossing relaone does not find trivial zeros in fllL. tion leads to the “conspiracy relation” of eq. (10). For example
fS dcba = f:+,go 1
=H
cosxc [sin+(xd+xb)f& - cos &-J+xb)f~l]
+
i[sin+(Xd
+
sinxc _ 2~ [-fO(J sin%xd+xb) +fio cm k$-J+xb)] .
- Xb)f&
+ cos
%Xd -
Xb)f;I] (12)
H. HOGAASEN and Ph. SALIN
664
F o r cos 0s =+1, then ×c =×b = ×d = ½ ~ a n d f Slll^ - ~ ~o =0" This gives
f:l
-½
(13)
foo = o.
If one u s e s
-÷ f;1 = ½ cos0/ flll
~- ~f 1 l
~ - 2s and COS 0 t (t:O) : 4MA '
then eq. (13) l e a d s to eq. (10) --
~ - 2s -+
- 4M~ fll
+ 4"2 f 0 0 ] --" 0 .
Let us now show how we can obtain the ! = 0 conditions (6) and (7) on the f t , when m ¢ ~, f r o m the c r o s s i n g r e l a t i o n s , without making r e f e r e n c e to t h e ' i n v a r i a n t a m p l i t u d e s {Bi}. Evidently, at l = 0, cos 0s ¢ i l , and we have no z e r o s on the left-hand side of the c r o s s i n g r e l a t i o n (115. However, the left-hand side is r e g u l a r when l is continued f r o m the s m a l l negative value of ! (when cos 0s = +15 to t = 0. Inspection of the c r o s s i n g m a t r i x c~X~
,
[TW eq. (42) and (43)] shows, h o w e v e r , that s o m e of~the e l e m e n t s of this a r e singular. By looking up which combinations of f ~ , ~ , a r e multiplied by these singular f a c t o r s , and r e q u e s t i n g their p r o d u c t t o b e r e g u l a r , we r e c o v e r eqs. (6) and (75. We shall r e t u r n to this in the next section. We have worked out explicitly, on a specific example, n a m e l y the nN ~ pN p r o c e s s , the v a r i o u s m e c h a n i s m s of c o n s t r a i n t s a m o n g helicity amplitudes. We now t r y to g e n e r a l i z e this to the g e n e r a l c a s e for any m a s s e s and any spins.
5. THE GENERAL P R O B L E M OF CONSTRAINTS AMONG HELICITY AMPLITUDES In the g e n e r a l spin c a s e it is v e r y difficult to c o n s t r u c t invariant functions with no k i n e m a t i c a l s i n g u l a r i t i e s . On the other hand we know how to obtain helicity amplitudes f r e e of k i n e m a t i c a l s i n g u l a r i t i e s [1]. T h e r e f o r e , we shall f r o m now on work only with helicity amplitudes. In o r d e r to exhibit and c h a r a c t e r i z e the v a r i o u s kinds of c o n s t r a i n t s amohg helicity a m p l i t u d e s , we shall need only the two following p r o p e r t i e s . (i) Each helicity amplitude m a y be written fP
: [ ½ ( l + c o s g p ) l ½]~t+"l[½(1-cos0p)1½i~t-~tf~/~
p=s,
,
(141
uor t,
w h e r e ~ t z is f r e e of k i n e m a t i c a l s i n g u l a r i t i e s in t. (ii) The helicity amplitudes f s in the s - c h a n n e l and f t in the t-channel a r e r e l a t e d through the TW c r o s s i n g r e l a t i o n
CONSPIRACY RELATIONS
s
:
fcdab
~
665
dsa . . . .
A'b'c'D'
Sb, ,~Sc, ,,Sd , ,~t A ' a t × a l a b ' b ~×blac 'c ~Xc l a D 'd~ Xd)]c ' A ' D ' b ' "
(15)
5.1. The forward direcliou couslrainls It i s w e l l k n o w n t h a t t h e b o u n d a r y of t h e p h y s i c a l d o m a i n f o r c o s 02 = +1, c o s 2 0 u = +1 o r c o s 2 01 = +1, i s g i v e n i n a l l t h r e e c h a n n e l s by t h e s a m e c u r v e i n t h e i s , l, tl) p l a n e , n a m e l y ~(s,l)
= 0,
where
(s,t) = stu - s(m 2 -m2~',-2djv_a-m2) -t(m2 -m2)(m2b 2 2 m2m2)(m2+
- (roared-
b
If 0s i s t h e c . m . a n g l e f o r t h e plies cos 0 t = ±i, when m a ¢ mc, There is an ambiguity in sign in s i m p l i c i t y to b e c o s 0! = + l w h e n In this kinematical situation f~
~
c - m2)
2 2 2 md-m c-rob).
(16)
p r o c e s s a + b ~ c + d, t h e n c o s 0s = + i i m 0 t b e i n g t h e c . m . a n g l e of d + b --' c + -~. t h e d e f i n i t i o n of c o s 0t. W e c h o o s e i t f o r c o s 0 s = +1.
=0
f~/,~, =0
if
~ - ~
if X'-~'¢
¢0, 0.
B u t t h e r e a r e e x a c t l y t h e s a m e n u m b e r of h e l i c i t y a m p l i t u d e s i n e a c h c h a n nel satisfying these relations because ~t - p =~t a - ~ t b - ( ~ t c - ~ t d ) = h d - ~ t b - ( ~ t c - ) t a )
=~t,_p,
W e a r e t h e n l e d to t h e f o l l o w i n g s t a t e m e n t : If 0 s i s t h e c . m . a n g l e of t h e p r o c e s s a + b - ~ c + d, t h e n c o s 0 s = +1 i m p l i e s n o c o n s t r a i n t s a m o n g h e l i c i t y a m p l i t u d e s i n t h e / - c h a n n e l a + b -~ c ~- ~t, w h e n m e ¢ m a. If m c = m a a n d m d = m b , t h e n t h e T r u e m a n - W i c k c r o s s i n g r e l a t i o n l e a d s to r e l a t i o n s a m o n g h e l i c i t y a m p l i t u d e s of t h e G r i b o v - V o l k o v type. 5.2. The t = 0 constraints (i) S i n c e a l l k i n e m a t i c a l s i n g u l a r i t i e s (1 + c o s Os) f a c t o r s , a n d a s c o s 0 s =[2st+s 2 - s ~ m 2 + ( m 2
i n t of f ~ p
are contained in the
m2~,~2 2 b,~,,Oc - rod)~ eJab eJcd ,
with
~ 2 b = [s - ( m a - mb)2][s - (ma+ mbl 2] , ~ 2 d _- [~ - (m c - md)21[~ - ( m c , mdl 2] , i t i s c l e a r t h a t f ~ t ~ h a s n o p o l e f o r ! = 0.
666
H. HDGAASEN and Ph. SALIN
(ii) On the o t h e r hand, one h a s , for e x a m p l e , for the a n g l e s e n t e r i n g the crossing matrix 1
sin ×a = 2 m a [@(s, t ) ] ~ / C i a b ~ a c with
~ac
= [t - ( m a + m c ) 2 J [ t - (m a - m c ) 2] .
If m a = rnc, then s i n X a and c o s x a have a pole f o r t = 0, and d J p ( x a ) h a s a pole of o r d e r n, with
n =~(!~+ Izl + i ~ - ~ I) + ½(J-sup(l'~l, i~i)) = ' ~ J , because
dJ/x(Xa) where
= Yj[½(l
+ cos Xa)]½ i A+M I [½(I - cos Xa)] ½]~-p ip[X-~ J - M ], [x+~ I' ( c o s
is a Jacobi p o l y n o m i a l
o r d e r J - M , and M : sup(iX i, I
Xa) ,
I).
B e c a u s e ) ~ p should not h a v e this p o l e in t - n , it is n e c e s s a r y that s o m e c o m b i n a t i o n s of h e l i c i t y a m p l i t u d e s f / go to z e r o a s t n L e t us w o r k out e x p l i c i t l y this for ~N -* pN. We e x p r e s s the f -S1 ~ l ~10 h e l i c i t y a m p l i t u d e in t e r m s of h e l i c i t y a m p l i t u d e s in t h e / - c h a n n e l . This h a s b e e n given in (12). C o s Xc and sin Xc a r e finite when t -~ 0, but Xd and Xb a r e not. c o s Xb = [ ( s + M 2 - m 2 ) t - 2M2(p 2 - m 2 ) ] / C i a b ~ b d
,
c o s Xd = [-(s + M 2 - tz2)t - 2M2(~ 2 - r n 2 ) ] / C i c d ~ b d '
sin ×b -
2M (4~ ~ a b -~bd
,
sin
J a b = 2 ~/s q s '
Xd
:
2M ( ~ ~bdCicd
'
(17)
Cicd = 2 ~/sk s ,
~ b d = 2 ~ i - P l = ~ t vrt - 4M 2 A f t e r s o m e a l g e b r a i c m a n i p u l a t i o n s , it m a y be shown that t 2 A ( s ) + t B ( s ) + 8M4(~ 2 - rn2) 2 c o s (Xd+X b ) 4 s k s q s ( t - 4M2)t and c o s 0( b - × d ) = t 2 A ' ( s ) + t B ' ( s ) 4 s k s q s C l - 4M2)t
(18)
Now, when t -~ O, c o s ( ~ + Xb) is going to infinity like 1 / t and c o s (Xb- Xd) b e h a v e s a s a f u n c t i o n of s. F r o m this we d e d u c e that c o s ½ix d + Xb) and
CONSPIR ACY RE LATIONS sin ½(×d + ×b) a r e going to infinity like 1/~/l,
667
with
cos½(Xd+×b) ~ i sin½0(d+Xb) . F r o m eq. (12), then, it is n e c e s s a r y that --÷
---
f l l ÷ if01 .-÷
iflo
- 2f00 ~ ~/.
(19)
T h e s e r e l a t i o n s a r e e x a c t l y the s a m e a s t h o s e d e d u c e d in eqs. (6) and (7), f r o m the i n v a r i a n t a m p l i t u d e s [the f a c t o r two in (19) is i n t r o d u c e d when one g o e s f r o m the f to the f ] . One m a y c o n v i n c e o n e s e l f that the two r e l a t i o n s (19) a r e r e a l l y i n d e p e n d e n t , by l o o k i n g at o t h e r r e l a t i o n s with t h e f s, w h e r e the s a m e c o m b i n a t i o n s of h e l i c i t y a m p l i t u d e s a l w a y s o c c u r , but with d i f f e r ent m u l t i p l i c a t i v e f a c t o r s . It is r e m a r k a b l e to n o t i c e that if the four m a s s e s a r e d i f f e r e n t , then c o s ×b and c o s × d a r e no l o n g e r s i n g u l a r f o r t = 0, so that t h e r e is no c o n s t r a i n t at this point and we m a k e the g e n e r a l s t a t e m e n t : When the four m a s s e s of a s c a t t e r i n g p r o c e s s a + b -* c + d a r e all d i f f e r e n t , t h e r e is no c o n s t r a i n t a m o n g h e l i c i t y a m p l i t u d e s for t = 0. T h e r e a r e c o n s t r a i n t s at this point only when t h r e e k i n d s of d i f f e r e n t m a s s e s o c c u r with m a ¢ m c and m b = rod, and t h e s e c o n s t r a i n t s have n o t h i n g to do with the f o r w a r d type of c o n s t r a i n t s .
6. CONCLUSIONS It r e s u l t s f r o m the p r e c e d i n g d i s c u s s i o n t h a t t h e r e a r e two t y p e s of c o n s t r a i n t s a m o n g h e l i c i t y a m p l i t u d e s in the t - c h a n n e l : (i) the f i r s t type of c o n s t r a i n t s o c c u r s only in r e a c t i o n s a + b -~ c + d with p a i r w i s e equal e x t e r n a l m a s s e s (m a = m c , m-o = m d )' in the s - c h a n n e l f o r w a r d d i r e c t i o n (or b a c k w a r d d i r e c t i o n ) , and is due to the v a n i s h i n g of h e l l city flip a m p l i t u d e s f s; (ii) The s e c o n d type of c o n s t r a i n t s o c c u r s at t h o s e p o i n t s w h e r e the r o t a tion m a t r i c e s a r e infinite, i . e . , when It
-
(rob+ md)2][t - (m b - r o d )2] = 0 .
T h e y a r e u s u a l l y known a s " t h r e s h o l d c o n d i t i o n s " , s e e for i n s t a n c e J o n e s [7], but o c c u r a t t = 0 o n l y w h e n m b = m d , a n d ~ cm.. We s u m m a r i z e the v a r i o u s p o s s i b i l i t i e s in table 1. ]In all c a s e s t h e s e two t y p e s of c o n s t r a i n t s c a n be c o m p u t e d d i r e c t l y f r o m the T r u e m a n - W i c k c r o s s ing r e l a t i o n , without r e f e r e n c e to i n v a r i a n t a m p l i t u d e s . T h e r e f o r e , one m a y a v o i d the t r e m e n d o u s a l g e b r a i c m a c h i n e r y of going f r o m the i n v a r i a n t a m p l i t u d e s to the h e l i c i t y a m p l i t u d e s and then i n v e r t i n g the o b t a i n e d r e l a t i o n s in o r d e r to get the c o n s t r a i n t s . It is not n e c e s s a r y e i t h e r to go f r o m the h e l i c i t y f r a m e to the u s u a l p a r t i a l wave f r a m e to o b tain the t h r e s h o l d c o n d i t i o n s . T h i s l a s t p r o c e d u r e i n v o l v e s r a t h e r c o m p l i -
668
H. HOGAASEN and Ph. SXLIN Table 1 Masses
rna = m c
mb - m d
Type of constraint for t -- 0 =L1 type I
Number of constraint points
ql -= 0
3
m b e rn d
t - 0
typeII
q! 0: Pt 0
4
m a c rn c rnb - rnd
t - 0
typelI
qt :0;
Pt-O
4
qt=O;
Pt=:O
4
mbern d
cOSOs
Other constraints
none
c a t e d C l e b s c h - G o r d a n c o e f f i c i e n t c a l c u l a t i o n s a n d f u r t h e r a s s u m p t i o n s on the t h r e s h o l d b e h a v i o u r of p a r t i a l wave a m p l i t u d e s . In o u r o p i n i o n , t h i s i s a n o n - n e g l i g i b l e p r o g r e s s . N e v e r t h e l e s s , i t s e e m s t h a t the g e n e r a l f o r m a l i m s d e v e l o p e d by W a n g [1] i n o r d e r to e x t r a c t the k i n e m a t i c a l s i n g u l a r i t i e s , l o s e s s o m e of i t s g e n e r a l i n t e r e s t b e c a u s e one i s f o r c e d to e v a l u a t e e x p l i c i t l y the c r o s s i n g r e l a t i o n s , w h i c h , a t the s a m e t i m e , g i v e s both the k i n e m a t i c a l s i n g u l a r i t i e s of h e l i c i t y a m p l i t u d e s a n d the c o n s t r a i n t s a m o n g them.
Addendum: A f t e r t h i s p a p e r w a s w r i t t e n , we h a v e b e e n i n f o r m e d t h a t the whole p r o b l e m of k i n e m a t i c a l s i n g u l a r i t i e s , c r o s s i n g s a n d c o n s t r a i n t s a m o n g h e l i c i t y a m p l i t u d e s , h a s b e e n r e v i e w e d a n d s o l v e d by G. C o h e n T a n n o u d j i , A. M o r e l a n d H. N a v e l e t f r o m S a c l a y . A m o n g o t h e r p o w e r f u l r e s u l t s , t h e s e a u t h o r s h a v e r e a c h e d the s a m e c o n c l u s i o n a s we h a v e w i t h r e g a r d to the c o n s t r a i n t p r o b l e m , u s i n g t r a n s v e r s i t y a m p l i t u d e s . T h e y found a l s o t h a t s o m e p h a s e s b e t w e e n m a t r i x e l e m e n t s of the c r o s s i n g m a t r i x of T r u ~ m a n a n d W i c k h a v e to be c h a n g e d . T h i s would m e a n t h a t the . It /~' m a t r l x g ~ x t ~ of eq. (11) s h o u l d n o t be e x a c t l y the T W m a t r i x . We a r e m u c h i n d e b t e d to Dr. C o h e n - T a n n o u d j i for h a v i n g i n f o r m e d u s of h i s r e s u l t s p r i o r to p u b l i c a t i o n , a s w e l l a s for v e r y i l l u m i n a t i n g d i s c u s s i o n s .
REFERENCES [1] L. L. C.Wang, Phys. Rev. 142 {1966} 1187. [2] J. S. Ball, Phys. Rev. 124 {1961) 2014; N. Dombey, Nuovo Cimento 31 {1964} 1025. [3] M. B. Halpern, Princeton preprint {1967}; B. Diu and M. Le Bellac, Orsay preprint 11967). [4] V. N. Gribov and D. V. Volkov, Soviet Phys. - J E T P 17 {1963} 720. [5] E. Abets and V. Teplitz, Los )mgeles p r e p r i n t {1967}. [6] T. L. Trueman and G.C.Wick, Ann. Phys. 26 (1964) 322. [71 H. F. Jones, Imperial College London p r e p r i n t {1966).