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Crossing matrix for processes involving massless particles
[7[-~-.B.~ Nuclear Physics B l l (1969) 186-194. North-Holland Publ. Comp., Amsterdam
CROSSING MATRIX FOR PROCESSES INVOLVING MASSLESS PARTICLES J. R. FOX P h y s i c s Department, Imperial College, London SW7 * Received 28 February 1969
Abstract: Using the covariant helicity formalism, we derive the crossing matrix for processes involving m a s s l e s s particles and find, as usual, that the helicity of a m a s s l e s s particle changes from one channel to the other if and only if it is ingoing in one and outgoing in the other. We show that, for consistency in the definitions of the cosines of the crossing and c e n t r e - o f - m a s s scattering angles, the functions "/[x-(mi+m))2][x-(mi-mj)2 ] for m i = 0 must be taken to be analytic.
1. I N T R O D U C T I O N We u s e the f e l i c i t y ( c o v a r i a n t h e l i c i t y ) f o r m a l i s m of F e l d m a n a n d M a t t h e w s [1] to d e r i v e the c r o s s i n g m a t r i x f o r h e i i c i t y a m p l i t u d e s in p r o c e s s e s i n v o l v i n g m a s s l e s s p a r t i c l e s . F o l l o w i n g c l o s e l y the t r e a t m e n t of the m a s s i v e c a s e by F e l d m a n a n d K i n g [2], we find that the c e n t r e - o f - m a s s ( c . m . ) h e l i c i t i e s of the m a s s l e s s p a r t i c l e i n t h e d i r e c t a n d c r o s s c h a n n e l s a r e opp o s i t e ff a n d only if t h e p a r t i c l e is i n g o i n g i n one c h a n n e l a n d o u t g o i n g i n the o t h e r . T h i s r e s u l t h a s p r e v i o u s l y b e e n o b t a i n e d by o t h e r m e t h o d s [ 3 , 4 ] . We show t h a t the f u n c t i o n A(X, i , j ) , d e f i n e d in s e c t . 2 m u s t f o r c o n s i s t e n c y b e t a k e n to b e a n a l y t i c f o r m i = O, when it a p p e a r s i n e x p r e s s i o n s f o r c o s i n e s of c r o s s i n g a n d c . m . s c a t t e r i n g a n g l e s .
2. T H E F E L I C I T Y O P E R A T O R
W e shall consider the crossing matrix between s and t channel helicity amplitudes in the process defined by s: A + B ~ C + D , t: D + B ~ C + A . , w h e r e f o r t h e m o m e n t the p a r t i c l e s m a y h a v e a n y m a s s e s . * The research reported in this document has been sponsored in part by the Air Force Office of Scientific Research (OAR) through the European Office of Aerospace Research, United States Air Force.
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187
The P a u l i - L u b a n s k i f o u r - v e c t o r o p e r a t o r Wp(i) for p a r t i c l e i is defined by
W~(i) = - ½ ¢ ~ vp(~J vP(i) P (~(i) ,
(1)
w h e r e for each i, jvP(i) and P~(i) obey the commutation r e l a t i o n s of a P o i n c a r d algebra, P~(i) is the f o u r - m o m e n t u m o p e r a t o r for p a r t i c l e i, and Jvp(i) its rotation and boost o p e r a t o r s , which we r e w r i t e in the f o r m
J = (J23,J31,J12) , K =- (Jo1,Jo2,J03) .
(2)
As we shall always be c o n c e r n e d with e i g e n s t a t e s of the m o m e n t u m ope r a t o r s P~(i), we r e p l a c e the l a t t e r by t h e i r eigenvalues P ~ ( 0 in what follows. The s - c h a n n e l felicity o p e r a t o r s for p a r t i c l e s A and B a r e defined [1] by 2 2 FS(A) = A(s, A, B) WP (A)pp(B) = -A WP (A)[p~(B) + p~(A)] ,
(3)
FS(B) = A(S,2A, B) Wp(B)p~(A) = 2 ~ Wp(B)[pP(A) +p#(B)] .
(4)
Using eqs. (I) and (2) we may write eq. (3) as 2
FS(A) = A(S,A, B) [J(A). (p°(B)p(A) - p°(A)p(B)) - K(A)" (p(B) x p(A))] ,
(5)
and similarly for eq. (4): FS(B)
2
=
A(S,A, B) [J(B) • (p°(A)p(B) - p ° ( B ) p ( A ) ) - K(B)" (p(A) x p(B))] ,
(6)
where p(i) is the space p a r t of PU(i) and 1 2sin 2B - 2mAre 2 2B . A 2 ( s , A , B ) = s 2 + m A4 + m 4B - 2sin..We may r e w r i t e eqs. (5) and (6) by introducing r i g h t - h a n d e d sets of unit t h r e e - v e c t o r s m , n and ~ for each of p a r t i c l e s A and B. H e r e re(A) is the unit v e c t o r in the d i r e c t i o n of p(B) x p(A); n(A) is the unit v e c t o r in the d i r e c t i o n of p(A) x re(A) and s i m i l a r l y for re(B) and n(B). In t e r m s of t h e s e v e c t o r s , F s (A) - A(s,2A, B) [J(A)" ~(A)(Ip(A) IPo(B) - Po(A)p(B) "~(A)
- J ( A ) . n(A)Po(A ) Ip(B) I - K(A) . r e ( A ) ] p ( A ) l i p ( B ) I ] ,
FS(B) . . . .
(A ~-~ B) .
F u r t h e r , the t - c h a n n e l f e l i c i t i e s of K and B a r e defined by
(7)
(8)
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J.R. FOX
2 Ft(k') - A(t, A, Ci W~z(A)PP(C) ' 2
Ft~Bj'' =-A(t,B,D)
Wp~BJI~*")' " '~' "
(9)
(10)
Now, since the felicity o p e r a t o r is L o r e n t z invariant, it m a y be calculated in any convenient f r a m e . In the s - c h a n n e l e.m. f r a m e , p(A) = - p ( B ) , so that
FS(A)
2
-
A(s, A, B) W0(A)(p°(A)+pO(B))
-
2~ A(s,A, B) J(A)'p(A)
= J(A)./~(A) ,
(11)
and FS(B)
=
J(B)" p ( B )
,
(12)
which are respectively the s-channel c.m. helicities of particles A and B. Similarly, calculating Ft(A.), and Ft(B) in the t-channel c.m. frame, we find that they are respectively the t-channel c.m. helicities of particles ~. and B.
3. MASSLESS PARTICLES L e t us now take p a r t i c l e B to be m a s s l e s s . Its e i g e n s t a t e s [5, 6] a r e IP(B), XB> w h e r e the helicity kB is fixed, and is the eigenvalue of the h e l i c ity o p e r a t o r A(B), the g e n e r a t o r of r o t a t i o n s in the little group E(2) of p(B). The two t r a n s l a t i o n g e n e r a t o r s a r e ~m and ~n, which have z e r o e i g e n v a l u e s in the o n e - d i m e n s i o n a l r e p r e s e n t a t i o n s of E(2) to which m a s s l e s s p a r t i c l e s belong. In t e r m s of the v e c t o r s re(B) and n(B) introduced e a r l i e r , ~m = J ( B ) ' m ( B ) -
K(B)'n(B) ,
~'n = J ( B ) . n ( B ) + K ( B ) " r e ( B ) .
(13)
F o r p(B) p h y s i c a l (real, with Po(B) > 0), A(B) is given by J ( B ) . p(B), and its m a t r i x e l e m e n t s , f o r all p(B), a r e hkB ' pB(B) ---- = •B 5)~B, UB '
(14)
w h e r e we have o m i t t e d the m o m e n t u m delta function. Now FS(B) a c t s in the s p a c e of t w o - p a r t i c l e s t a t e s of A and B, but if P(A) is r e p l a c e d by its eigenvalue p(A) in FS(B), we m a y think of it as a s i n g l e - p a r t i c l e o p e r a t o r in the s p a c e of s t a t e s of B. With this in mind, we
CROSSING MATRIX
189
s e e f r o m eq. (8) that FS(B) is a l i n e a r c o m b i n a t i o n of J ( B ) - ~ ( B ) and ~n (since Po(B) = Ip(B) I), and conclude that its e i g e n s t a t e s a r e just the helicity e i g e n s t a t e s of B, at l e a s t f o r p(B) p h y s i c a l . We find s s - rn 2 F k B ' #B(B) - ( p ( s ) , kBI FS(B)I p(B),/~B> - A(s, A, B) AB 6kB, #B
(15)
4
f o r p(B) p h y s i c a l , so if s > m 2 , F ~ B , ~B(B) = AkB, ~B(B). We m a y now fix p(B) and continue in p(A) so that s - m 2 b e c o m e s negative. In this c a s e , AA, #(B) is c l e a r l y unaffected, but F ~ , / l ( B ) m a y change sign u n l e s s A(s, A, B) is t a k e n to continue a s s - m 2. With this choice, F ~ ,,(B) and h A ,,(B) a r e identical f o r all s, and p(B) p h y s i c a l . Now AA,u(B) is by defini[ion i n v a r iant u n d e r any c o m p l e x L o r e n t z t r a n s f o r m a t i o n (which m a y take p(B) to an u n p h y s i c a l value), and F ~ ,,(B) is i n v a r i a n t by c o n s t r u c t i o n , so that F ~ #(B) and AA, #(B) a r e i d e n t i c a l ' ~ r all p(B) and s, and the e i g e n s t a t e s of the two o p e r a t o r s a r e identical (in our r e p r e s e n t a t i o n ) . As FsA, p(B) and AA, p(B) a r e identical in this r e p r e s e n t a t i o n we m a y find an explicit f o r m f o r A(B) f r o m eq. (8): hA, p(B)
=
FSA~(B) , P°(B)2 p ( A ) . p(B))
- ~n(..){P(B),~B)=(P(B),AB] J(B).p(B) {p(B),gB> •
Therefore operator.
Po(B )
"
J . P/Po is the c o r r e c t f o r m for the m a s s l e s s p a r t i c l e helicity
4. THE CROSSINGMATRIX We now c o n s i d e r the b e h a v i o u r of p a r t i c l e B under c r o s s i n g . We c a l c u l a t e F S ( B ) and F t ( B ) in the s a m e f r a m e , so that the m o m e n t a p(B) a p p e a r ing in the two e x p r e s s i o n s a r e the s a m e . A convenient choice is an s - c h a n nel c.m. f r a m e such that the s c a t t e r i n g is in the y z - p l a n e and pp(B) = = (p, 0 , 0 , p ) . * We could have made the other choice. The felicity and helicity would then have 2 and the felicity amplitudes and cross'ng , matrix been of opposite sign for s < m~, would be different from, though-simply related to, the helicity amplitudes and crossing matrix. Our impending calculation of the helicity crossing matrix would be altered in detail, but not in its result.
190
J . R . FOX F r o m eq. (12) we find F S ( B ) = J3(B) .
(16)
F r o m eq. (10), e x p a n d e d in the m a n n e r of eqs. (5) and (6) we find 2 F t ( B ) - A(/, B, D) [J3(B)(D°(D) - P3(D))P - ~ n ( B ) P 2 ( 5 ) P ] in the r e p r e s e n t a t i o n , Therefore
= J3(B) '
(17)
with o u r c h o i c e of sign of A(I, B, D) f o r t < rn 2 . FS(B) = Ft(S) ,
(16)
and the e i g e n s t a t e s a r e r e l a t e d , up to a p h a s e by ]p(B), ~s) = [P(B),~tt = ~ts) ,
(19)
w h e r e ~ts and ~tt a r e the s - and t - c h a n n e l c . m . h e l i c i t i e s of p a r t i c l e B (for a m a s s l e s s p a r t i c l e , the h e l i c i t y is the s a m e in any f r a m e , and so is the s a m e a s the c . m . helicity). F r o m eq. (19) we s e e that the c r o s s i n g r e l a t i o n b e t w e e n the s - and tc h a n n e l h e l i c i t y a m p l i t u d e s , in the spin s p a c e of p a r t i c l e B, is, a p a r t f r o m a possible phase r s
..;.k B
(s,t,u)
(s,t,u) ,
= r t
(20)
..;.)t B
s i n c e the g e n e r a l - f r a m e t - c h a n n e l h e l i c i t y a m p l i t u d e is the s a m e as the c . m . f r a m e h e l i c i t y a m p l i t u d e as f a r a s the m a s s l e s s p a r t i c l e label is c o n cerned. Now let p a r t i c l e A, r a t h e r than B, be m a s s l e s s and let us c a l c u l a t e FS(A) and F t ( ~ , ) in an s - c h a n n e l c . m . f r a m e , with the s c a t t e r i n g in the y z plane and p p ( A ) = -p/z(~,) = (q, 0, 0, q). F r o m eq. (11), FS(A) : J3(A) .
(21)
F r o m eq. (9), e x p a n d e d in the m a n n e r of eqs. (5) and (6) 2
F t ( A ' ) = A(t, A,C-) [ - J 3 ( ~ ' ) ( P ° ( C ) - P 3 ( C ) ) q + ~ n ( ~ ' ) q P 2 ( I ) ) ]
= J3(~') '
(22)
in o u r r e p r e s e n t a t i o n . T h e spin o p e r a t o r s J ( A ) and d(~,) of p a r t i c l e and a n t i p a r t i c l e a r e r e l a t e d by J(A) = _dW(~)
.
(23)
H e n c e J3(A) = -J3(~,), a s the r e p r e s e n t a t i o n is o n e - d i m e n s i o n a l (and in any c a s e , d i a g o n a l f o r J3)- T h e r e f o r e FS(A) = - F t ( ~ - ) ,
(24)
and the e i g e n s t a t e s a r e r e l a t e d up to a p h a s e by IP(A), ~ts) = ]P(A) = -p(A), h t = - h s ) .
(25)
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191
T h e r e f o r e t h e s - c h a n n e l c . m . h e l i c i t y of A i s o p p o s i t e to t h e t - c h a n n e l c . m . h e l i c i t y of ~, a n d t h e c r o s s i n g r e l a t i o n , in t h e s p i n s p a c e of p a r t i c l e A i s , up t o a p h a s e
Ts
(s, t, u) = T t
• . ; h A.
(s, t, u) .
(26)
.-hA;..
E q s . (20) a n d (26) s h o w t h a t t h e c . m . h e l i c i t y of the m a s s l e s s p a r t i c l e c h a n g e s if a n d only if t h e p a r t i c l e i s c r o s s e d in g o i n g f r o m one c h a n n e l t o t h e o t h e r . T h i s s t a t e m e n t i s i n d e p e n d e n t of t h e p a r t i c l e m o m e n t u m o r t h e v a l u e s of s, t, a n d u, b e c a u s e t h e h e l i c i t i e s a r e . In o t h e r w o r d s , t h e c r o s s ing a n g l e i s e i t h e r 0 o r ~, a n d is c o n s t a n t in t h e e n t i r e i s , t, u) p l a n e . T h e e x t e n s i o n of t h i s a n a l y s i s to p r o c e s s e s i n v o l v i n g m o r e t h a n one m a s s l e s s p a r t i c l e i s t r i v i a l , a s t h e r e a s o n i n g a p p l i e s to e a c h p a r t i c l e i n d e p e n d e n t l y . T h e c r o s s i n g - m a t r i x in t h e p r o d u c t s p a c e of t h e ( r e m a i n i n g ) m a s s i v e p a r t i c l e s h a s b e e n g i v e n by T r u e m a n a n d W i c k [7].
5. D E T E R M I N A T I O N O F A(X, i,j), (m i = O) W e w o u l d l i k e to c o m m e n t on t h e d e t e r m i n a t i o n of A(X, i,j) b e l o w t h e p s e u d o - t h r e s h o l d x = (m i - mj)2 to b e u s e d in t h e e x p r e s s i o n s f o r t h e c e n t r e o f - m a s s s c a t t e r i n g a n g l e s a n d t h e c r o s s i n g a n g l e s of t h e m a s s i v e p a r t i c l e s in t h e r e a c t i o n . L e a d e r ' s [8] e x p r e s s i o n s f o r t h e c o s i n e s of t h e c r o s s i n g angles may be used for the massive particles, with the appropriate masses s e t e q u a l to z e r o , b e c a u s e t h e m a s s l e s s n e s s of o t h e r p a r t i c l e s m a k e s no e s s e n t i a l d i f f e r e n c e to t h e r e a s o n i n g u s e d t o o b t a i n t h e s e e x p r e s s i o n s . H o w e v e r , t h e r e i s a n a m b i g u i t y in t h e i n t e r p r e t a t i o n of A(x, i,j). In e a r l i e r s e c t i o n s of t h i s p a p e r we u s e d t h e a n a l y t i c d e t e r m i n a t i o n of A(X, i,j) when m i o r rnj w a s z e r o , b u t t h i s c h o i c e w a s not e s s e n t i a l to o u r r e s u l t ; it m e r e l y e n a b l e d u s to a r r i v e at t h i s r e s u l t m o r e e a s i l y . C o n s e q u e n t l y , we m u s t d e c i d e a l l o v e r a g a i n how we a r e to i n t e r p r e t A in t h e cosine expressions. We s h o w t h a t , w h e n a m a s s l e s s p a r t i c l e i s p r e s e n t , we a r e f o r c e d b y c o n s i s t e n c y c o n s i d e r a t i o n s to c h o o s e A(X, i,j), (m i = 0) to b e a n a l y t i c in t h e x - p l a n e . W i t h t h i s d e t e r m i n a t i o n of A, L e a d e r ' s e x p r e s s i o n s f o r t h e c r o s s ing angles are valid for the massless particles as well. T h e b o u n d a r y of t h e p h y s i c a l r e g i o n h a s s i x a s y m p t o t i c p o r t i o n s , on e a c h of w h i c h (with t h e p o s s i b l e e x c e p t i o n of t h e l i n e s s = 0, t = 0, u = 0) t h e c r o s s i n g a n d c . m . s c a t t e r i n g a n g l e s a r e c o n s t a n t a n d s e p a r a t e l y e q u a l to 0 o r ~. W e l a b e l t h e s e p o r t i o n s s+, s - , t+, t - , u+, u - , w h e r e s+ i s t h a t p o r t i o n of t h e b o u n d a r y of t h e s - p h y s i c a l r e g i o n on w h i c h c o s 0s = +1, s ~ ~, etc. N e a r s u c h a p o r t i o n , to w i t h i n a p h a s e , s
t
TXCXD;XA)~B(S, t, u) = T)kC~;,,
A~)AB'~''(S, t, U) + ~,, ,
(27)
w h e r e t h e s e t {;~'} i s d e t e r m i n e d in t e r m s of t h e s e t {),} b y t h e c r o s s i n g a n g l e s , a n d ~ g o e s to z e r o on t h e b o u n d a r y , a n d a t l e a s t a s f a s t a s
192
J.R. FOX
Tt{A'}(s, t, u) if t h i s a m p l i t u d e v a n i s h e s t h e r e . B e c a u s e B is a l i n e a r c o m b i n a t i o n of i n d e p e n d e n t t - c h a n n e l h e l i c i t y a m p l i t u d e s , ~ a n d Tt{~, '} c a n n o t in g e n e r a l ' c o n s p i r e ' to m a k e the r i g h t - h a n d s i d e of eq. (27) v a n i s h f a s t e r t h a n Tt{a '} on the b o u n d a r y . F u r t h e r m o r e , n e a r t h i s p o r t i o n of t h e b o u n d a r y [9], t u)
7),C~,D; XA)~B(S, ,
,'x
( s i n 0 s ) [;~C-;~D:FxA+~'B]
(28)
a c c o r d i n g a s c o s 6 s = ± 1, a n d
r b k; xi h(s,
(sin ot)
lab- xk;xb±xh I,
(29)
a c c o r d i n g a s cos 0 t = ± 1. so that in v i e w of o u r p r e v i o u s r e m a r k s , the exp o n e n t s in the r i g h t - h a n d s i d e s of eqs. (28) a n d (29) m u s t b e equal. T h i s i m p l i e s a n i n t i m a t e c o n n e c t i o n b e t w e e n the c r o s s i n g a n d s c a t t e r i n g a n g l e s . L e t u s take p a r t i c l e A a s m a s s l e s s , and c o n s i d e r the c r o s s i n g f r o m s to t - c h a n n e l . Let u s o r d e r the m a s s e s of the p a r t i c l e s : m D > m B ", m C ~ m A = 0. T h e n , with the d e f i n i t i o n s ( c o n s i s t e n t with o u r p r e v i o u s d e f i n i t i o n s of s a n d l c h a n n e l s )
2
2
2
s(t - u) + m B ( m D - mC) cosO s-
A(s, A, B)&(s, C, D)
t(s cos
2
-
2
'
2
u) + r n c ( m D - roB)
0t =
A(t, B, D) A(t, A, C ) 2
2
2
u ( s - t) + m D ( m C - roB) 0u =
cos
A(u, B, C) A(u, A, D)
,
(30)
w h i c h a r e u n a m b i g u o u s a b o v e the r e l e v a n t c h a n n e l t h r e s h o l d s , a l o n g * S+
s --, +oo ,
I ~0-
U
S-
S ~+oo ,
t --~-oo
u~O+
t+
s ~0-
,
t 4+00
?~ ---~ - oo
t-
S
,
t
u~0-
U+
s 40+
,
t ---~-oo
U
U-
s ~-oo
,
t --0+
U ~ +
~-oo
~+oo
- - - ~ - oo ,
~q.
,
, oo ,
c°
•
(31)
* We refer the reader to fig. 1 of ref. [10]. With our ordering of particle m a s s e s , Kibble's region II is our s-physical region, his region I is our t-physical region, and his region III is our u-physical region.
C ROSSING M A T R I X
193
T h i s i n f o r m a t i o n e n a b l e s u s to d e t e r m i n e e a s i l y t h e s i g n of t h e c o s i n e s on a n y of t h e s i x p o r t i o n s of t h e b o u n d a r y . F r o m L e a d e r [8], we h a v e t h e e x p r e s s i o n s f o r t h e c o s i n e s of t h e m a s sive particle crossing angles:
(s + m2-)(tl~ cos
XB
-
-
-
+
roB2 - mr,.u 2 ) - 2 m B2 ( m C2 + m B2
2
roD)
A ( s , A, B) A(t, B, D)
2 2 ( s + m 2 - roD)(/+ m c ) COS XC =
-
2
2
2
2
2 m c .Cm( + m B - roD)
A(s, C, D)A(t, A, C)
(s+ m2D - m2)(t+ m2D - m 2B ) + 2 m D2 ( m C 2 + m 2B - m D) 2 c o s XD = -
A(S, C , D ) ~ ( t , B , D )
, (32)
N o w we c a n c h o o s e p a r t i c u l a r d e t e r m i n a t i o n s f o r t h e A a n d s e e if t h e y a r e c o n s i s t e n t . We f i n d c o s 6 s a n d c o s 0t on e a c h of t h e s i x p o r t i o n s of t h e b o u n d a r y . T h e e q u a l i t y of t h e e x p o n e n t s in e q s . (28) a n d (29) w i t h t h e a p p r o p r i a t e s i g n s g i v e s u s {~'} in t e r m s of {)~} a n d t h e r e f o r e c o s XB, c o s XC a n d c o s XD ( b e c a u s e c o s ×A - - 1). W e c a n t h e n c o m p a r e t h e s e v a l u e s of t h e c o s i n e s of t h e c r o s s i n g a n g l e s w i t h t h e v a l u e s o b t a i n e d f r o m eq. (32). T h e r e s u l t s a r e p r e s e n t e d in t a b l e 1, w h i c h a s s u m e s the c o n v e n t i o n a l d e t e r m i n a t i o n of A(x, ~ j ) f o r mi, mj ¢ 0, t h a t i s , t h e one w h i c h * g i v e s A> 0forx<(m i - m j ) . T h e two d e t e r m i n a t i o n s of A ( x , i , j ) , (m i=O), a r e p r e s e n t e d t o g e t h e r ; w h e n t h e r e i s m o r e t h a n one e n t r y , the u p p e r c o r r e s p o n d s t o A = ] x - m ~ l , j t h e l o w e r to A = x - m 2. (The e n t r i e s a r e a l l + l o r -1; we g i v e o n l y t h e s i g n s . ) It m a y b e s e e n in t a b l e 1 t h a t t h e l o w e r e n t r i e s , but not t h e u p p e r , a r e consistent between lines 4-6 and lines 7-9. This means that, for consiste n c y , A(x, i,j), (m i = 0), m u s t b e t a k e n a s x - m 2, t h e a n a l y t i c f o r m of A in d this case. H a d w e u s e d t h e o t h e r ( a n a l y t i c ) d e t e r m i n a t i o n of A(x, i,j), (mi, mj ¢ 0), our conclusion about the m i = 0 case would have been unaltered. This is a r e f l e c t i o n of t h e f a c t t h a t t h e r e a r e two w a y s of c o n t i n u i n g , in m o m e n t u m , m a s s i v e p a r t i c l e c . m . h e l i c i t y s t a t e s [3]. T h e c o n c l u s i o n s a r e u n a l t e r e d when t h e a n a l y s i s i s e x t e n d e d to t h e r e m a i n i n g , f i n i t e p o r t i o n s of t h e p h y s i c a l r e g i o n b o u n d a r y (on e a c h of w h i c h i t i s s t i l l t r u e t h a t t h e c o s i n e s a r e c o n s t a n t a n d s e p a r a t e l y e q u a l to ± 1). W i t h t h e a b o v e d e t e r m i n a t i o n s of t h e f u n c t i o n s A, a l l t h e c r o s s i n g a n g l e c o s i n e s a r e c o r r e c t l y g i v e n b y L e a d e r ' s e x p r e s s i o n s ; t h a t f o r c o s XA r e d u c i n g to - 1 i d e n t i c a l l y . * Here, x is taken to v a r y along paths which connect the upper side of the r i g h t hand cut of the S-matrix in the x-plane with the lower side of the left-hand cut. Thus if A(x, i,)), (m., m- ¢ 0) is g r e a t e r than zero f o r x below pseudothreshold, it is analytic in the x - pzl a n ]e cut from _oo to (m i- ~mj)2 and (m i ~ mj) 2 to +~ (rather than from pseudothreshold to threshold).
194
J . R . FOX
Table 1 Values of the c o s i n e s of the c r o s s i n g angles on the d i f f e r e n t portions of the boundary.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
t-
u+
u-
=~
~
+
±
~:
+
-
T
cos
01
T
~-
+
-
_
_
.
.
.
.
.
.
.
cosxB
and
see
t+
0s
f r o m eqs. (28) (29),
s-
c o s
cosxA .
s+
.
.
.
.
.
.
.
.
.
.
.
.
.
i
.
.
.
.
.
.
.
.
.
.
.
.
.
.
i
-
~
~:
+
cosxc
-
+
+
+
+
~:
text.
cos XD
-k
:~
+
:F
+
+
f r o m eq. (32)
cos XC
cosx
cos
B
XD
_
_
+
:F
~:
:~
:~
:r
_
_
~
~:
-
~
-
+
+
+
A n a l o g o u s c o n c l u s i o n s a r e o b t a i n e d f r o m a c o n s i d e r a t i o n of s-u a n d crossing, and when more than one massless particle is present.
6.
t-u
CONCLUSION
We have derived the usual massless particle crossing matrix using the felicity (covariant helicity) formalism. We find, incidentally, that the helicity o p e r a t o r f o r a m a s s l e s s p a r t i c l e i s J . P/Po, d e f i n e d f o r a l l pp. We have shown that, for consistency between the crossing matrix and t h e b e h a v i o u r of h e l i c i t y a m p l i t u d e s in t h e t w o c h a n n e l s n e a r t h e b o u n d a r y of t h e p h y s i c a l r e g i o n , t h e f u n c t i o n s a(x, i,j), (m i = 0), a p p e a r i n g in t h e e x p r e s s i o n s f o r t h e c o s i n e s of t h e c r o s s i n g a n d c . m . s c a t t e r i n g a n g l e s , m u s t h a v e t h e i r a n a l y t i c d e t e r m i n a t i o n x - m y f o r a l l x. T h e a u t h o r w o u l d l i k e t o t h a n k P r o f e s s o r s G. F e l d m a n a n d P . T . M a t t h e w s f o r h e l p f u l d i s c u s s i o n s a n d a c r i t i c a l r e a d i n g of t h e m a n u s c r i p t . H e a c k n o w l e d g e s t h e s u p p o r t of a C o m m o n w e a l t h S c h o l a r s h i p .
REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
G. Feldman and P . T . Matthews, Phys. Rev. 168 (1968) 1587. G. F e l d m a n and M. King, Nuovo Cimento 60 (1969) 86. A.H. M u e l l e r and T. L. T r u e m a n , Phys. Rev. 160 (1967) 1306. J . P . Ader, M . C a p d e v i l l e and H . N a v e l e t , N u o v o C i m e n t o 56 (1968) 315. S. Weinberg, Phys. Rev. 134 (1964) B882. S. Schwcber, An introduction to r e l a t i v i s t i c quantum field t h e o r y (Row, P e t e r s o n and Co., Evanston, Illinois, 1961) Ch. 2. T . L . T r u e m a n a n d G . C . W i c k , Ann. of Phys. 26 (1964) 322. E. L e a d e r , Phys. Rev. 166 (1968) 1599. M. J a c o b and G. C. Wick, Ann. of Phys. 7 (1959) 404. T . W . B . K i b b l e , Phys. Rev. 117 (1960) 1159.
CROSSING MATRIX FOR PROCESSES INVOLVING MASSLESS PARTICLES J. R. FOX P h y s i c s Department, Imperial College, London SW7 * Received 28 February 1969
Abstract: Using the covariant helicity formalism, we derive the crossing matrix for processes involving m a s s l e s s particles and find, as usual, that the helicity of a m a s s l e s s particle changes from one channel to the other if and only if it is ingoing in one and outgoing in the other. We show that, for consistency in the definitions of the cosines of the crossing and c e n t r e - o f - m a s s scattering angles, the functions "/[x-(mi+m))2][x-(mi-mj)2 ] for m i = 0 must be taken to be analytic.
1. I N T R O D U C T I O N We u s e the f e l i c i t y ( c o v a r i a n t h e l i c i t y ) f o r m a l i s m of F e l d m a n a n d M a t t h e w s [1] to d e r i v e the c r o s s i n g m a t r i x f o r h e i i c i t y a m p l i t u d e s in p r o c e s s e s i n v o l v i n g m a s s l e s s p a r t i c l e s . F o l l o w i n g c l o s e l y the t r e a t m e n t of the m a s s i v e c a s e by F e l d m a n a n d K i n g [2], we find that the c e n t r e - o f - m a s s ( c . m . ) h e l i c i t i e s of the m a s s l e s s p a r t i c l e i n t h e d i r e c t a n d c r o s s c h a n n e l s a r e opp o s i t e ff a n d only if t h e p a r t i c l e is i n g o i n g i n one c h a n n e l a n d o u t g o i n g i n the o t h e r . T h i s r e s u l t h a s p r e v i o u s l y b e e n o b t a i n e d by o t h e r m e t h o d s [ 3 , 4 ] . We show t h a t the f u n c t i o n A(X, i , j ) , d e f i n e d in s e c t . 2 m u s t f o r c o n s i s t e n c y b e t a k e n to b e a n a l y t i c f o r m i = O, when it a p p e a r s i n e x p r e s s i o n s f o r c o s i n e s of c r o s s i n g a n d c . m . s c a t t e r i n g a n g l e s .
2. T H E F E L I C I T Y O P E R A T O R
W e shall consider the crossing matrix between s and t channel helicity amplitudes in the process defined by s: A + B ~ C + D , t: D + B ~ C + A . , w h e r e f o r t h e m o m e n t the p a r t i c l e s m a y h a v e a n y m a s s e s . * The research reported in this document has been sponsored in part by the Air Force Office of Scientific Research (OAR) through the European Office of Aerospace Research, United States Air Force.
CROSSING MATRIX
187
The P a u l i - L u b a n s k i f o u r - v e c t o r o p e r a t o r Wp(i) for p a r t i c l e i is defined by
W~(i) = - ½ ¢ ~ vp(~J vP(i) P (~(i) ,
(1)
w h e r e for each i, jvP(i) and P~(i) obey the commutation r e l a t i o n s of a P o i n c a r d algebra, P~(i) is the f o u r - m o m e n t u m o p e r a t o r for p a r t i c l e i, and Jvp(i) its rotation and boost o p e r a t o r s , which we r e w r i t e in the f o r m
J = (J23,J31,J12) , K =- (Jo1,Jo2,J03) .
(2)
As we shall always be c o n c e r n e d with e i g e n s t a t e s of the m o m e n t u m ope r a t o r s P~(i), we r e p l a c e the l a t t e r by t h e i r eigenvalues P ~ ( 0 in what follows. The s - c h a n n e l felicity o p e r a t o r s for p a r t i c l e s A and B a r e defined [1] by 2 2 FS(A) = A(s, A, B) WP (A)pp(B) = -A WP (A)[p~(B) + p~(A)] ,
(3)
FS(B) = A(S,2A, B) Wp(B)p~(A) = 2 ~ Wp(B)[pP(A) +p#(B)] .
(4)
Using eqs. (I) and (2) we may write eq. (3) as 2
FS(A) = A(S,A, B) [J(A). (p°(B)p(A) - p°(A)p(B)) - K(A)" (p(B) x p(A))] ,
(5)
and similarly for eq. (4): FS(B)
2
=
A(S,A, B) [J(B) • (p°(A)p(B) - p ° ( B ) p ( A ) ) - K(B)" (p(A) x p(B))] ,
(6)
where p(i) is the space p a r t of PU(i) and 1 2sin 2B - 2mAre 2 2B . A 2 ( s , A , B ) = s 2 + m A4 + m 4B - 2sin..We may r e w r i t e eqs. (5) and (6) by introducing r i g h t - h a n d e d sets of unit t h r e e - v e c t o r s m , n and ~ for each of p a r t i c l e s A and B. H e r e re(A) is the unit v e c t o r in the d i r e c t i o n of p(B) x p(A); n(A) is the unit v e c t o r in the d i r e c t i o n of p(A) x re(A) and s i m i l a r l y for re(B) and n(B). In t e r m s of t h e s e v e c t o r s , F s (A) - A(s,2A, B) [J(A)" ~(A)(Ip(A) IPo(B) - Po(A)p(B) "~(A)
- J ( A ) . n(A)Po(A ) Ip(B) I - K(A) . r e ( A ) ] p ( A ) l i p ( B ) I ] ,
FS(B) . . . .
(A ~-~ B) .
F u r t h e r , the t - c h a n n e l f e l i c i t i e s of K and B a r e defined by
(7)
(8)
188
J.R. FOX
2 Ft(k') - A(t, A, Ci W~z(A)PP(C) ' 2
Ft~Bj'' =-A(t,B,D)
Wp~BJI~*")' " '~' "
(9)
(10)
Now, since the felicity o p e r a t o r is L o r e n t z invariant, it m a y be calculated in any convenient f r a m e . In the s - c h a n n e l e.m. f r a m e , p(A) = - p ( B ) , so that
FS(A)
2
-
A(s, A, B) W0(A)(p°(A)+pO(B))
-
2~ A(s,A, B) J(A)'p(A)
= J(A)./~(A) ,
(11)
and FS(B)
=
J(B)" p ( B )
,
(12)
which are respectively the s-channel c.m. helicities of particles A and B. Similarly, calculating Ft(A.), and Ft(B) in the t-channel c.m. frame, we find that they are respectively the t-channel c.m. helicities of particles ~. and B.
3. MASSLESS PARTICLES L e t us now take p a r t i c l e B to be m a s s l e s s . Its e i g e n s t a t e s [5, 6] a r e IP(B), XB> w h e r e the helicity kB is fixed, and is the eigenvalue of the h e l i c ity o p e r a t o r A(B), the g e n e r a t o r of r o t a t i o n s in the little group E(2) of p(B). The two t r a n s l a t i o n g e n e r a t o r s a r e ~m and ~n, which have z e r o e i g e n v a l u e s in the o n e - d i m e n s i o n a l r e p r e s e n t a t i o n s of E(2) to which m a s s l e s s p a r t i c l e s belong. In t e r m s of the v e c t o r s re(B) and n(B) introduced e a r l i e r , ~m = J ( B ) ' m ( B ) -
K(B)'n(B) ,
~'n = J ( B ) . n ( B ) + K ( B ) " r e ( B ) .
(13)
F o r p(B) p h y s i c a l (real, with Po(B) > 0), A(B) is given by J ( B ) . p(B), and its m a t r i x e l e m e n t s , f o r all p(B), a r e hkB ' pB(B) ----
(14)
w h e r e we have o m i t t e d the m o m e n t u m delta function. Now FS(B) a c t s in the s p a c e of t w o - p a r t i c l e s t a t e s of A and B, but if P(A) is r e p l a c e d by its eigenvalue p(A) in FS(B), we m a y think of it as a s i n g l e - p a r t i c l e o p e r a t o r in the s p a c e of s t a t e s of B. With this in mind, we
CROSSING MATRIX
189
s e e f r o m eq. (8) that FS(B) is a l i n e a r c o m b i n a t i o n of J ( B ) - ~ ( B ) and ~n (since Po(B) = Ip(B) I), and conclude that its e i g e n s t a t e s a r e just the helicity e i g e n s t a t e s of B, at l e a s t f o r p(B) p h y s i c a l . We find s s - rn 2 F k B ' #B(B) - ( p ( s ) , kBI FS(B)I p(B),/~B> - A(s, A, B) AB 6kB, #B
(15)
4
f o r p(B) p h y s i c a l , so if s > m 2 , F ~ B , ~B(B) = AkB, ~B(B). We m a y now fix p(B) and continue in p(A) so that s - m 2 b e c o m e s negative. In this c a s e , AA, #(B) is c l e a r l y unaffected, but F ~ , / l ( B ) m a y change sign u n l e s s A(s, A, B) is t a k e n to continue a s s - m 2. With this choice, F ~ ,,(B) and h A ,,(B) a r e identical f o r all s, and p(B) p h y s i c a l . Now AA,u(B) is by defini[ion i n v a r iant u n d e r any c o m p l e x L o r e n t z t r a n s f o r m a t i o n (which m a y take p(B) to an u n p h y s i c a l value), and F ~ ,,(B) is i n v a r i a n t by c o n s t r u c t i o n , so that F ~ #(B) and AA, #(B) a r e i d e n t i c a l ' ~ r all p(B) and s, and the e i g e n s t a t e s of the two o p e r a t o r s a r e identical (in our r e p r e s e n t a t i o n ) . As FsA, p(B) and AA, p(B) a r e identical in this r e p r e s e n t a t i o n we m a y find an explicit f o r m f o r A(B) f r o m eq. (8): hA, p(B)
=
FSA~(B) , P°(B)2 p ( A ) . p(B))
- ~n(..){P(B),~B)=(P(B),AB] J(B).p(B) {p(B),gB> •
Therefore operator.
Po(B )
"
J . P/Po is the c o r r e c t f o r m for the m a s s l e s s p a r t i c l e helicity
4. THE CROSSINGMATRIX We now c o n s i d e r the b e h a v i o u r of p a r t i c l e B under c r o s s i n g . We c a l c u l a t e F S ( B ) and F t ( B ) in the s a m e f r a m e , so that the m o m e n t a p(B) a p p e a r ing in the two e x p r e s s i o n s a r e the s a m e . A convenient choice is an s - c h a n nel c.m. f r a m e such that the s c a t t e r i n g is in the y z - p l a n e and pp(B) = = (p, 0 , 0 , p ) . * We could have made the other choice. The felicity and helicity would then have 2 and the felicity amplitudes and cross'ng , matrix been of opposite sign for s < m~, would be different from, though-simply related to, the helicity amplitudes and crossing matrix. Our impending calculation of the helicity crossing matrix would be altered in detail, but not in its result.
190
J . R . FOX F r o m eq. (12) we find F S ( B ) = J3(B) .
(16)
F r o m eq. (10), e x p a n d e d in the m a n n e r of eqs. (5) and (6) we find 2 F t ( B ) - A(/, B, D) [J3(B)(D°(D) - P3(D))P - ~ n ( B ) P 2 ( 5 ) P ] in the r e p r e s e n t a t i o n , Therefore
= J3(B) '
(17)
with o u r c h o i c e of sign of A(I, B, D) f o r t < rn 2 . FS(B) = Ft(S) ,
(16)
and the e i g e n s t a t e s a r e r e l a t e d , up to a p h a s e by ]p(B), ~s) = [P(B),~tt = ~ts) ,
(19)
w h e r e ~ts and ~tt a r e the s - and t - c h a n n e l c . m . h e l i c i t i e s of p a r t i c l e B (for a m a s s l e s s p a r t i c l e , the h e l i c i t y is the s a m e in any f r a m e , and so is the s a m e a s the c . m . helicity). F r o m eq. (19) we s e e that the c r o s s i n g r e l a t i o n b e t w e e n the s - and tc h a n n e l h e l i c i t y a m p l i t u d e s , in the spin s p a c e of p a r t i c l e B, is, a p a r t f r o m a possible phase r s
..;.k B
(s,t,u)
(s,t,u) ,
= r t
(20)
..;.)t B
s i n c e the g e n e r a l - f r a m e t - c h a n n e l h e l i c i t y a m p l i t u d e is the s a m e as the c . m . f r a m e h e l i c i t y a m p l i t u d e as f a r a s the m a s s l e s s p a r t i c l e label is c o n cerned. Now let p a r t i c l e A, r a t h e r than B, be m a s s l e s s and let us c a l c u l a t e FS(A) and F t ( ~ , ) in an s - c h a n n e l c . m . f r a m e , with the s c a t t e r i n g in the y z plane and p p ( A ) = -p/z(~,) = (q, 0, 0, q). F r o m eq. (11), FS(A) : J3(A) .
(21)
F r o m eq. (9), e x p a n d e d in the m a n n e r of eqs. (5) and (6) 2
F t ( A ' ) = A(t, A,C-) [ - J 3 ( ~ ' ) ( P ° ( C ) - P 3 ( C ) ) q + ~ n ( ~ ' ) q P 2 ( I ) ) ]
= J3(~') '
(22)
in o u r r e p r e s e n t a t i o n . T h e spin o p e r a t o r s J ( A ) and d(~,) of p a r t i c l e and a n t i p a r t i c l e a r e r e l a t e d by J(A) = _dW(~)
.
(23)
H e n c e J3(A) = -J3(~,), a s the r e p r e s e n t a t i o n is o n e - d i m e n s i o n a l (and in any c a s e , d i a g o n a l f o r J3)- T h e r e f o r e FS(A) = - F t ( ~ - ) ,
(24)
and the e i g e n s t a t e s a r e r e l a t e d up to a p h a s e by IP(A), ~ts) = ]P(A) = -p(A), h t = - h s ) .
(25)
CROSSING MATRIX
191
T h e r e f o r e t h e s - c h a n n e l c . m . h e l i c i t y of A i s o p p o s i t e to t h e t - c h a n n e l c . m . h e l i c i t y of ~, a n d t h e c r o s s i n g r e l a t i o n , in t h e s p i n s p a c e of p a r t i c l e A i s , up t o a p h a s e
Ts
(s, t, u) = T t
• . ; h A.
(s, t, u) .
(26)
.-hA;..
E q s . (20) a n d (26) s h o w t h a t t h e c . m . h e l i c i t y of the m a s s l e s s p a r t i c l e c h a n g e s if a n d only if t h e p a r t i c l e i s c r o s s e d in g o i n g f r o m one c h a n n e l t o t h e o t h e r . T h i s s t a t e m e n t i s i n d e p e n d e n t of t h e p a r t i c l e m o m e n t u m o r t h e v a l u e s of s, t, a n d u, b e c a u s e t h e h e l i c i t i e s a r e . In o t h e r w o r d s , t h e c r o s s ing a n g l e i s e i t h e r 0 o r ~, a n d is c o n s t a n t in t h e e n t i r e i s , t, u) p l a n e . T h e e x t e n s i o n of t h i s a n a l y s i s to p r o c e s s e s i n v o l v i n g m o r e t h a n one m a s s l e s s p a r t i c l e i s t r i v i a l , a s t h e r e a s o n i n g a p p l i e s to e a c h p a r t i c l e i n d e p e n d e n t l y . T h e c r o s s i n g - m a t r i x in t h e p r o d u c t s p a c e of t h e ( r e m a i n i n g ) m a s s i v e p a r t i c l e s h a s b e e n g i v e n by T r u e m a n a n d W i c k [7].
5. D E T E R M I N A T I O N O F A(X, i,j), (m i = O) W e w o u l d l i k e to c o m m e n t on t h e d e t e r m i n a t i o n of A(X, i,j) b e l o w t h e p s e u d o - t h r e s h o l d x = (m i - mj)2 to b e u s e d in t h e e x p r e s s i o n s f o r t h e c e n t r e o f - m a s s s c a t t e r i n g a n g l e s a n d t h e c r o s s i n g a n g l e s of t h e m a s s i v e p a r t i c l e s in t h e r e a c t i o n . L e a d e r ' s [8] e x p r e s s i o n s f o r t h e c o s i n e s of t h e c r o s s i n g angles may be used for the massive particles, with the appropriate masses s e t e q u a l to z e r o , b e c a u s e t h e m a s s l e s s n e s s of o t h e r p a r t i c l e s m a k e s no e s s e n t i a l d i f f e r e n c e to t h e r e a s o n i n g u s e d t o o b t a i n t h e s e e x p r e s s i o n s . H o w e v e r , t h e r e i s a n a m b i g u i t y in t h e i n t e r p r e t a t i o n of A(x, i,j). In e a r l i e r s e c t i o n s of t h i s p a p e r we u s e d t h e a n a l y t i c d e t e r m i n a t i o n of A(X, i,j) when m i o r rnj w a s z e r o , b u t t h i s c h o i c e w a s not e s s e n t i a l to o u r r e s u l t ; it m e r e l y e n a b l e d u s to a r r i v e at t h i s r e s u l t m o r e e a s i l y . C o n s e q u e n t l y , we m u s t d e c i d e a l l o v e r a g a i n how we a r e to i n t e r p r e t A in t h e cosine expressions. We s h o w t h a t , w h e n a m a s s l e s s p a r t i c l e i s p r e s e n t , we a r e f o r c e d b y c o n s i s t e n c y c o n s i d e r a t i o n s to c h o o s e A(X, i,j), (m i = 0) to b e a n a l y t i c in t h e x - p l a n e . W i t h t h i s d e t e r m i n a t i o n of A, L e a d e r ' s e x p r e s s i o n s f o r t h e c r o s s ing angles are valid for the massless particles as well. T h e b o u n d a r y of t h e p h y s i c a l r e g i o n h a s s i x a s y m p t o t i c p o r t i o n s , on e a c h of w h i c h (with t h e p o s s i b l e e x c e p t i o n of t h e l i n e s s = 0, t = 0, u = 0) t h e c r o s s i n g a n d c . m . s c a t t e r i n g a n g l e s a r e c o n s t a n t a n d s e p a r a t e l y e q u a l to 0 o r ~. W e l a b e l t h e s e p o r t i o n s s+, s - , t+, t - , u+, u - , w h e r e s+ i s t h a t p o r t i o n of t h e b o u n d a r y of t h e s - p h y s i c a l r e g i o n on w h i c h c o s 0s = +1, s ~ ~, etc. N e a r s u c h a p o r t i o n , to w i t h i n a p h a s e , s
t
TXCXD;XA)~B(S, t, u) = T)kC~;,,
A~)AB'~''(S, t, U) + ~,, ,
(27)
w h e r e t h e s e t {;~'} i s d e t e r m i n e d in t e r m s of t h e s e t {),} b y t h e c r o s s i n g a n g l e s , a n d ~ g o e s to z e r o on t h e b o u n d a r y , a n d a t l e a s t a s f a s t a s
192
J.R. FOX
Tt{A'}(s, t, u) if t h i s a m p l i t u d e v a n i s h e s t h e r e . B e c a u s e B is a l i n e a r c o m b i n a t i o n of i n d e p e n d e n t t - c h a n n e l h e l i c i t y a m p l i t u d e s , ~ a n d Tt{~, '} c a n n o t in g e n e r a l ' c o n s p i r e ' to m a k e the r i g h t - h a n d s i d e of eq. (27) v a n i s h f a s t e r t h a n Tt{a '} on the b o u n d a r y . F u r t h e r m o r e , n e a r t h i s p o r t i o n of t h e b o u n d a r y [9], t u)
7),C~,D; XA)~B(S, ,
,'x
( s i n 0 s ) [;~C-;~D:FxA+~'B]
(28)
a c c o r d i n g a s c o s 6 s = ± 1, a n d
r b k; xi h(s,
(sin ot)
lab- xk;xb±xh I,
(29)
a c c o r d i n g a s cos 0 t = ± 1. so that in v i e w of o u r p r e v i o u s r e m a r k s , the exp o n e n t s in the r i g h t - h a n d s i d e s of eqs. (28) a n d (29) m u s t b e equal. T h i s i m p l i e s a n i n t i m a t e c o n n e c t i o n b e t w e e n the c r o s s i n g a n d s c a t t e r i n g a n g l e s . L e t u s take p a r t i c l e A a s m a s s l e s s , and c o n s i d e r the c r o s s i n g f r o m s to t - c h a n n e l . Let u s o r d e r the m a s s e s of the p a r t i c l e s : m D > m B ", m C ~ m A = 0. T h e n , with the d e f i n i t i o n s ( c o n s i s t e n t with o u r p r e v i o u s d e f i n i t i o n s of s a n d l c h a n n e l s )
2
2
2
s(t - u) + m B ( m D - mC) cosO s-
A(s, A, B)&(s, C, D)
t(s cos
2
-
2
'
2
u) + r n c ( m D - roB)
0t =
A(t, B, D) A(t, A, C ) 2
2
2
u ( s - t) + m D ( m C - roB) 0u =
cos
A(u, B, C) A(u, A, D)
,
(30)
w h i c h a r e u n a m b i g u o u s a b o v e the r e l e v a n t c h a n n e l t h r e s h o l d s , a l o n g * S+
s --, +oo ,
I ~0-
U
S-
S ~+oo ,
t --~-oo
u~O+
t+
s ~0-
,
t 4+00
?~ ---~ - oo
t-
S
,
t
u~0-
U+
s 40+
,
t ---~-oo
U
U-
s ~-oo
,
t --0+
U ~ +
~-oo
~+oo
- - - ~ - oo ,
~q.
,
, oo ,
c°
•
(31)
* We refer the reader to fig. 1 of ref. [10]. With our ordering of particle m a s s e s , Kibble's region II is our s-physical region, his region I is our t-physical region, and his region III is our u-physical region.
C ROSSING M A T R I X
193
T h i s i n f o r m a t i o n e n a b l e s u s to d e t e r m i n e e a s i l y t h e s i g n of t h e c o s i n e s on a n y of t h e s i x p o r t i o n s of t h e b o u n d a r y . F r o m L e a d e r [8], we h a v e t h e e x p r e s s i o n s f o r t h e c o s i n e s of t h e m a s sive particle crossing angles:
(s + m2-)(tl~ cos
XB
-
-
-
+
roB2 - mr,.u 2 ) - 2 m B2 ( m C2 + m B2
2
roD)
A ( s , A, B) A(t, B, D)
2 2 ( s + m 2 - roD)(/+ m c ) COS XC =
-
2
2
2
2
2 m c .Cm( + m B - roD)
A(s, C, D)A(t, A, C)
(s+ m2D - m2)(t+ m2D - m 2B ) + 2 m D2 ( m C 2 + m 2B - m D) 2 c o s XD = -
A(S, C , D ) ~ ( t , B , D )
, (32)
N o w we c a n c h o o s e p a r t i c u l a r d e t e r m i n a t i o n s f o r t h e A a n d s e e if t h e y a r e c o n s i s t e n t . We f i n d c o s 6 s a n d c o s 0t on e a c h of t h e s i x p o r t i o n s of t h e b o u n d a r y . T h e e q u a l i t y of t h e e x p o n e n t s in e q s . (28) a n d (29) w i t h t h e a p p r o p r i a t e s i g n s g i v e s u s {~'} in t e r m s of {)~} a n d t h e r e f o r e c o s XB, c o s XC a n d c o s XD ( b e c a u s e c o s ×A - - 1). W e c a n t h e n c o m p a r e t h e s e v a l u e s of t h e c o s i n e s of t h e c r o s s i n g a n g l e s w i t h t h e v a l u e s o b t a i n e d f r o m eq. (32). T h e r e s u l t s a r e p r e s e n t e d in t a b l e 1, w h i c h a s s u m e s the c o n v e n t i o n a l d e t e r m i n a t i o n of A(x, ~ j ) f o r mi, mj ¢ 0, t h a t i s , t h e one w h i c h * g i v e s A> 0forx<(m i - m j ) . T h e two d e t e r m i n a t i o n s of A ( x , i , j ) , (m i=O), a r e p r e s e n t e d t o g e t h e r ; w h e n t h e r e i s m o r e t h a n one e n t r y , the u p p e r c o r r e s p o n d s t o A = ] x - m ~ l , j t h e l o w e r to A = x - m 2. (The e n t r i e s a r e a l l + l o r -1; we g i v e o n l y t h e s i g n s . ) It m a y b e s e e n in t a b l e 1 t h a t t h e l o w e r e n t r i e s , but not t h e u p p e r , a r e consistent between lines 4-6 and lines 7-9. This means that, for consiste n c y , A(x, i,j), (m i = 0), m u s t b e t a k e n a s x - m 2, t h e a n a l y t i c f o r m of A in d this case. H a d w e u s e d t h e o t h e r ( a n a l y t i c ) d e t e r m i n a t i o n of A(x, i,j), (mi, mj ¢ 0), our conclusion about the m i = 0 case would have been unaltered. This is a r e f l e c t i o n of t h e f a c t t h a t t h e r e a r e two w a y s of c o n t i n u i n g , in m o m e n t u m , m a s s i v e p a r t i c l e c . m . h e l i c i t y s t a t e s [3]. T h e c o n c l u s i o n s a r e u n a l t e r e d when t h e a n a l y s i s i s e x t e n d e d to t h e r e m a i n i n g , f i n i t e p o r t i o n s of t h e p h y s i c a l r e g i o n b o u n d a r y (on e a c h of w h i c h i t i s s t i l l t r u e t h a t t h e c o s i n e s a r e c o n s t a n t a n d s e p a r a t e l y e q u a l to ± 1). W i t h t h e a b o v e d e t e r m i n a t i o n s of t h e f u n c t i o n s A, a l l t h e c r o s s i n g a n g l e c o s i n e s a r e c o r r e c t l y g i v e n b y L e a d e r ' s e x p r e s s i o n s ; t h a t f o r c o s XA r e d u c i n g to - 1 i d e n t i c a l l y . * Here, x is taken to v a r y along paths which connect the upper side of the r i g h t hand cut of the S-matrix in the x-plane with the lower side of the left-hand cut. Thus if A(x, i,)), (m., m- ¢ 0) is g r e a t e r than zero f o r x below pseudothreshold, it is analytic in the x - pzl a n ]e cut from _oo to (m i- ~mj)2 and (m i ~ mj) 2 to +~ (rather than from pseudothreshold to threshold).
194
J . R . FOX
Table 1 Values of the c o s i n e s of the c r o s s i n g angles on the d i f f e r e n t portions of the boundary.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
t-
u+
u-
=~
~
+
±
~:
+
-
T
cos
01
T
~-
+
-
_
_
.
.
.
.
.
.
.
cosxB
and
see
t+
0s
f r o m eqs. (28) (29),
s-
c o s
cosxA .
s+
.
.
.
.
.
.
.
.
.
.
.
.
.
i
.
.
.
.
.
.
.
.
.
.
.
.
.
.
i
-
~
~:
+
cosxc
-
+
+
+
+
~:
text.
cos XD
-k
:~
+
:F
+
+
f r o m eq. (32)
cos XC
cosx
cos
B
XD
_
_
+
:F
~:
:~
:~
:r
_
_
~
~:
-
~
-
+
+
+
A n a l o g o u s c o n c l u s i o n s a r e o b t a i n e d f r o m a c o n s i d e r a t i o n of s-u a n d crossing, and when more than one massless particle is present.
6.
t-u
CONCLUSION
We have derived the usual massless particle crossing matrix using the felicity (covariant helicity) formalism. We find, incidentally, that the helicity o p e r a t o r f o r a m a s s l e s s p a r t i c l e i s J . P/Po, d e f i n e d f o r a l l pp. We have shown that, for consistency between the crossing matrix and t h e b e h a v i o u r of h e l i c i t y a m p l i t u d e s in t h e t w o c h a n n e l s n e a r t h e b o u n d a r y of t h e p h y s i c a l r e g i o n , t h e f u n c t i o n s a(x, i,j), (m i = 0), a p p e a r i n g in t h e e x p r e s s i o n s f o r t h e c o s i n e s of t h e c r o s s i n g a n d c . m . s c a t t e r i n g a n g l e s , m u s t h a v e t h e i r a n a l y t i c d e t e r m i n a t i o n x - m y f o r a l l x. T h e a u t h o r w o u l d l i k e t o t h a n k P r o f e s s o r s G. F e l d m a n a n d P . T . M a t t h e w s f o r h e l p f u l d i s c u s s i o n s a n d a c r i t i c a l r e a d i n g of t h e m a n u s c r i p t . H e a c k n o w l e d g e s t h e s u p p o r t of a C o m m o n w e a l t h S c h o l a r s h i p .
REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
G. Feldman and P . T . Matthews, Phys. Rev. 168 (1968) 1587. G. F e l d m a n and M. King, Nuovo Cimento 60 (1969) 86. A.H. M u e l l e r and T. L. T r u e m a n , Phys. Rev. 160 (1967) 1306. J . P . Ader, M . C a p d e v i l l e and H . N a v e l e t , N u o v o C i m e n t o 56 (1968) 315. S. Weinberg, Phys. Rev. 134 (1964) B882. S. Schwcber, An introduction to r e l a t i v i s t i c quantum field t h e o r y (Row, P e t e r s o n and Co., Evanston, Illinois, 1961) Ch. 2. T . L . T r u e m a n a n d G . C . W i c k , Ann. of Phys. 26 (1964) 322. E. L e a d e r , Phys. Rev. 166 (1968) 1599. M. J a c o b and G. C. Wick, Ann. of Phys. 7 (1959) 404. T . W . B . K i b b l e , Phys. Rev. 117 (1960) 1159.