MIRACULOUS HELICITY RULES, BASES, AND AMPLITUDES FOR VECTOR BOSONS

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MIRACULOUS HELICITY RULES, BASES, AND AMPLITUDES FOR VECTOR BOSONS

R. Brown

To cite this version:

R. Brown. MIRACULOUS HELICITY RULES, BASES, AND AMPLITUDES FOR VECTOR BOSONS. Journal de Physique Colloques, 1985, 46 (C2), pp.C2-3-C2-11. �10.1051/jphyscol:1985201�.

�jpa-00224510�

JOURNAL DE PHYSIQUE

Colloque C2, supplément au n°2, Tome 46, février 1985 page C2-3

MIRACULOUS HELICITY RULES, BASES, AND AMPLITUDES FOR VECTOR BOSONS

R.W. Brown

Physics Department, Case Western Reserve University, Cleveland, OH 44106, U.S.A.

Résumé - Nous décrivons les développements récents de la théorie de perturbation qui résultent de calculs faits sur des réactions avec des bosons de gauge.

Abstract - We describe what we have learned in recent years about perturbation theory as a result of doing calculations for processes with gauge bosons.

I - INTRODUCTION

During the past half-dozen years many of us have been doing higher-order Born ampli- tude calculations for quark-gluon processes and for weak-boson production. Among the motives are the interest in multi-jet phenomena in the former case and tests for trilinear couplings in both cases.

Bonuses have been earned from these labors. Simple results corresponding to unfore- seen symmetries have been found, despite complicated steps, that tell us that we may very well profit by a reformulation of the vector-boson helicity bases and the asso- ciated amplitude decomposition with which we start.

The simple results are seen in both analytical work and numerical computations. We describe the analytical discoveries in terms of three miracles, in the discussion to follow, and we can relate the numerical and analytical results in a unified ap- proach. This approach and certain new conclusions are due to a collaboration in- volving J. Donohue, M. Karlsson, G.T. Kleppe and myself.

II - HELICITY RULES

A) Helicity conservation

Every child in Marseilles knows that a massless spin-Jj particle preserves its he- licity throughout any series of vector/axial-vector (V/A) couplings. With mass, helicity flip is possible but the high-energy limit is smooth since there is no re- duction in the number of spin states.

Massless spin - 1 particles also partake of helicity conservation in a process such as a Compton reaction. This is manifest, for both QED and QCD, in the forms of the Compton amplitude to be discussed in due course. That gauge bosons with mass fol- low suit at high energy is expected at first glance because renormalizable theories are advertised to make a smooth transition from the 3 spin states to the 2 massless spin states.

One may wonder, however, about helicity conservation for vector bosons with mass since unitarity constraints do not generally require that the longitudinal-helicity cross sections vanish relative to the positive-helicity (right-handed = RH) and negative-helicity (left-handed = LH) rates. We have in mind both non-renormaliza- ble theories and composite vector particles. In fact, angular momentum conservation appears to force one vector particle to be in a longitudinal state in the forward Compton Born amplitude for V/A couplings. Also, the high-energy limits may well depend on which helicity basis (s,t,u channel, etc.) is utilized for nonzero mass.

The naive expectations are realized, however, and we will see presently how elegant- ly the non-Abelian theories follow the Abelian theories that in turn follow the massless conservation rules.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1985201

JOURNAL DE PHYSIQUE

B) H e l i c i t y c o r r e l a t i o n

What i s t h e r e l a t i o n of t h e vector-boson h e l i c i t y t o t h e e m i t t i n g l a b s o r b i n g D i r a c fermion? Years ago, McVoy and Dyson /l/ c a l c u l a t e d t h a t e l e c t r o n s " t r a n s f e r r e d "

t h e i r h e l i c i t y t o h a r d photons i n b r e m s s t r a h l u n g . We s e e a n a n a l o g / 2 / i n t h e e e -+ W d i f f e r e n t i a l c r o s s s e c t i o n s f o r a g i v e n e l e c t r o n handedness ( F i g . 1 ) . A LH y i s found i n t h e LH-e- hemisphere and a RH y i s found i n t h e RH-e+ hemisphere. We d e f i n e t h e hemisphere dominance of h e l i c i t y - t r a n s f e r o r h a n d e d n e s s - t r a n s f e r a s s p j n p f f , an e x p e c t e d o c c u r r e n c e when a p r o p a g a t o r p o l e i s p r e s e n t a t t h e phase s p a c e boundary. [The p o l e a l s o e x p l a i n s t h e h e l i c i t y conundrum r e f e r e n c e d e a r l i e r : The angular-momentum z e r o s , 0 ( 8 ) , from h e l i c i t y non-conservation (122) i n t h e forward and backward d i r e c t i o n s a r e dominated by t - c h a n n e l and U-channel p o l e s , 0 ( 8 - ~ ) , r e s p e c t i v e l y . ] S p i n o f f i s q u a n t i t a t i v e l y u s e f u l , even when t h e p o l e i n f i n i t y i s avoided by, s a y , an a n g u l a r c u t o f f B C . The e r r o r i n c h a r a c t e r i z i n g t h e p o l a r i z a t i o n i s r o u g h l y O(Oc).

cos 8 cos 0

F i g . 1 - Photon p o l a r i z a t i o n f o r _{Fig. 2 - W} + p o l a r i z a t i o n f o r ua + w + ~

e h e i+ ~W from /2/. The dashed from 121. 8 = c.m. a n g l e between W+

curve c o r r e s p o n d s t o a nonrenormal- and U.

i z a b l e coupling.

S p i n o f f a p p l i e s a s w e l l t o t h e Born p r o d u c t i o n of p a i r s of v e c t o r bosons w i t h mass i n t h e a n n i h i l a t i o n of D i r a c f e r m i o n s ,

flit -f W' . ^{( 1 )}

o r i n i t s c r o s s e d v e r s i o n (Compton). While peaking i s c e r t a i n l y s e e n i n t h e e a r l i e r c a l c u l a t i o n s of Gaemers and Gounaris / 3 / and of Hellmund and R a n f t / 4 / , t h e s e

a u t h o r s employed a r e c t a n g u l a r p o l a r i z a t i o n b a s i s , s o we t u r n t o B i l c h a k S&. / 2 / who d i r e c t l y a d d r e s s e d S-channel h e l i c i t y t r a n s f e r i n e e + w,Zy,WW,ZZ and

q e ' ⁺ W,ZY ,Wg,Zg,WW,WZ,ZZ.

For ua ⁺ W + y ( 1 2 1 and F i g . 2 ) t h e r e i s 80% (20%) W+ s p i n o f f from a ( U ) i n agreement w i t h t h e s q u a r e d quark charge r a t i o . T h i s i s t r u e even n e a r t h r e s h o l d . By c o n t r a s t t h e l o n g i t u d i n a l c o n t r i b u t i o n dominates, as t h e energy i s i n c r e a s e d , f o r nonrenor- m a l i z a b l e c o u p l i n g s ( e . g . , t h e ~ = 3 c u r v e s ) .

F i g u r e s 3 and 4 show r e s u l t s f o r p a i r p r o d u c t i o n where b o t h v e c t o r s have mass and where s p i n o f f a p p l i e s only away from t h r e s h o l d , w i t h two-pole enhancement f o r e e ⁺ ZZ and one-pole enhancement f o r e e ⁺ WW. The s c a l e i n t h e f i g u r e s i s some-

times t o o c o a r s e t o show t h e s t r u c t u r e a t forward and backward a n g l e s t h a t r e s u l t s from t h e k i n d s of a n g u l a r momentum c o n s t r a i n t s d i s c u s s e d p r e v i o u s l y .

Fig. 3 - Z0 p o l a r i z a t i o n f o r e f H e h + ZOZO from /2/.

COS 8

Fig. 4 - W+ p o l a r i z a t i o n f o r e+e- ⁺ h-

from 121. 0 = c.m. a n g l e between W+ and e-.

The r e l e v a n c e of s p i n o f f i s n o t obvious s i n c e t h e l o n g i t u d i n a l h e l i c i t y s t a t e per- m i t s forward/backward a n g u l a r momentum c o n s e r v a t i o n and s i n c e a t r i l i n e a r - b o s o n

c o u p l i n g c o n t r i b u t e s t o Wy and WW p r o d u c t i o n . Indeed s p i n o f f i s s p o i l e d f o r non- gauge W c o u p l i n g s , a s s e e n i n t h e f i g u r e s . W e e x p e c t t h a t t h e l o n g i t u d i n a l h e l i c i - t y s t a t e c o m p l i c a t e s m a t t e r s f o r n o n r e n o r m a l i z a b l e i n t e r a c t i o n s , even when t h e n e c e s s a r y u n i t a r i t y c o r r e c t i o n s a r e made, o r f o r t h e p r o d u c t i o n of composite v e c t o r bosons. I n g e n e r a l , r e n o r m a l i z a b l e gauge t h e o r i e s a r e r e q u i r e d f o r s p i n o f f t o apply t o Born a m p l i t u d e s .

I t i s s a t i s f y i n g t h a t s p i n r e s u l t s f o r o t h e r vector-boson r e a c t i o n s can b e d e s c r i b - ed s i m i l a r l y . The weak bosons s p i n o f f t h e n e u t r i n o i n t h e c l a s s i c r e a c t i o n f o r W

C2-6 JOURNAL DE PHYSIQUE

s e a r c h e s , Vz + &wz', and o f f t h e i n i t i a l c h a r g e d l e p t o n i n &z + & z z l b u t n o t i n

& Z + vWz1/5/. A c r o s s - c h a n n e l l e p t o n exchange i s p r e s e n t i n t h e f i r s t two Compton s u b a m p l i t u d e s b u t n o t i n t h e t h i r d . The i n t e g r a t i o n o v e r t h e v i r t u a l photon, from t h e e l e c t r o m a g n e t i c r e c o i l o f f t h e n u c l e u s z , d o e s b r i n g o u t d i f f e r e n c e s i n d e t a i l - e d comparisons, s u c h a s a d e c r e a s e i n s p i n o f f a s t h e &z l a b o r a t o r y e n e r g y i s i n - c r e a s e d .

No p o l e i s p r e s e n t i n t h e i m p o r t a n t c h a r g e asymmetry / 6 / i n pp ⁺ WX, which i s t h e tendency of t h e decay l e p t o n ( a n t i l e p t o n ) t o f o l l o w t h e p (p). (The e f f e c t s of p o l a r i z e d p r o t o n beams on t h e asymmetry a r e d e t a i l e d i n a c o n t r i b u t i o n t o t h i s con- f e r e n c e by P. C h i a p e t t a and J. S o f f e r . ) But.we s t i l l may s a y t h a t t h e W+ s p i n s o f f

U, when i t i s found i n t h e p r o t o n h e m i s p h e r e , backward b i a s i n g t h e e+, and s o f o r t h . 111 - HELICITY BASES

A) CALKUL M i r a c l e

The s i m p l e c o n s e r v a t i o n and s p i n o f f r u l e s s u g g e s t t h a t we s e a r c h f o r v e c t o r h e l i c i t y b a s e s t h a t most d i r e c t l y e x h i b i t t h e r u l e s a n a l y t i c a l l y . The CALKUL b a s i s / 7 / (Cambridge, L e i d e n , g a t h o l i e k e g n i v . Leuven) i s j u s t what we n e e d b o t h t o s i m p l i f y m a s s l e s s c a l c u l a t i o n s and t o c o n n e c t w i t h s p i n o f f . We d e f i n e a b a s i s f o r a n G- going RH(+) o r LH(-) photon w i t h r e s p e c t t o two s p i n o r momenta p , p l :

4 .

N = ( p - p ' p - q p ' - q ) With i d e n t i t i e s s u c h a s

we b e n e f i t from t h e d i s a p p e a r a n c e of many t e r m s due t o c h i r a l i t y mismatch, m a s s l e s s - n e s s , and c u r r e n t c o n s e r v a t i o n , f o r p h o t o n a t t a c h m e n t s t o a v e r t e x A ( p l , p ) = y ,Ylfi5

.

t f e t e r m s t h a t r e m a i n s i m p l i f y s i g n i f i c a n t l y .

B e s i d e s t h e m i r a c u l o u s s i m p l i f i c a t i o n of h i g h - o r d e r Born a m p l i t u d e s , s u c h a s f o r e e + y w , we a l s o a c h i e v e a q u i c k g r a s p of s p i n o f f . The c h i r a l m a s s l e s s f e r m i o n w i t h fourmomentum p h a s t h e p o l a r i z a t i o n l i m i t s p a f. p p . Thus t h e photon a t - t a c h e d t o e i t h e r s i d e , y i e l d i n g a Compton s u b p r o c e s s , h a s i t s p o l a r i z a t i o n d e t e r - mined by p , p l and t h e

*

B) CALKUL f o r mass

I f o n l y one v e c t o r boson h a s mass i n ( 1 ) o r i n i t s c r o s s e d (Compton) v e r s i o n , t h e n t h e CALKUL b a s i s i s r e a d i l y g e n e r a l i z e d t o r e p r e s e n t t h a t boson. P u t t i n g t h e mass- l e s s n e s s of t h e o t h e r boson t o good u s e and a d d i n g a l o n g i t u d i n a l component, we c a n make an e a s y a n a l y t i c a l d e m o n s t r a t i o n of l o n g i t u d i n a l s u p p r e s s i o n , and of s p i n o f f w i t h t h e a d a p t e d S-channel b a s i s / 2 / . An i m p o r t a n t s t e p i s t h e i m p l e m e n t a t i o n of r a d i a t i o n symmetry, a p o i n t t o which we w i l l r e t u r n .

P a s s a r i n o /8/ h a s employed t h e D i r a c s p i n o r f o r m a l i s m of C a f f o and Remiddi / 9 / t o o b t a i n a g e n e r a l r e p r e s e n t a t i o n f o r v e c t o r p o l a r i z a t i o n b a s e s . The i d e a i s t o u s e

Bargmann-Wigner formulas t o w r i t e a wave f u n c t i o n f o r a v e c t o r boson with mass m, Ell = 1 1 ; ^y"u ,

4m symm

i n terms of s p i n o r s d e f i n e d according t o / g / . The use of t h e fermion c o o r d i n a t e s pre,sent i n a s c a t t e r i n g problem i s c l e a r l y suggested by ( 7 ) t h e l i m i t of which i s fundamentally r e l a t e d t o t h e CALKUL formula. Simpler c a l c u l a t i o n s ensue, even when no p a r t i c l e i s massless.

I V - HELICITY AMPLITUDES A) Donohue Miracle

The u l t i m a t e s t e p t h a t should be taken t o make m a n i f e s t the h e l i c i t y r u l e s i s t o de- compose t h e Feynman Born amplitudes i n t o t h e a p p r o p r i a t e h e l i c i t y b a s i s . I f t h e r e were h e l i c i t y s t a t e s t h a t d i d n o t c o n t r i b u t e a t a l l , t o take a s i m p l e example, t h e i r omission would be c l e a r a t t h e o u t s e t . The q u e s t i o n i s , a r e t h e r e o t h e r b a s e s f o r v e c t o r bosons with mass, b e s i d e s t h e S-channel ones under d i s c u s s i o n , where t h e amplitudes a r e even s i m p l e r ?

Consider a s a b u i l d i n g block t h e Abelian ( n o t r i l i n e a r coupling) Compton r e a c t i o n ,

vf + V ' £ ' (8)

with massless Dirac fermions and both v e c t o r masses nonzero. A c r o s s i n g argument connects our c o n c l u s i o n s t o ( l ) . Donohue 1101 found t h e s u r p r i s i n g r e s u l t t h a t the Born amplitude f o r (8) h a s a 2x2, r a t h e r t h a n 3x3, form,

f o r RH f , f '

.

By t h e removal of p r e j u d i c e a g a i n s t non-orthogonal b a s e s , we f i n d o u r s e l v e s with only two terms, one connecting s-channel h e l i c i t i e s and a n o t h e r U-channel h e l i c i t i e s i n diagonal form and w i t h one s p i n dimension missing. The non-orthogonality i m p l i e s t h a t h e l i c i t y i s n o t conserved b u t i f , f o r example, V i s i n I+lU> then V ' must be i n l+ls> and s o on. P a i r p r o d u c t i o n ( 1 ) l i k e w i s e has a d i a g o n a l form o b t a i n e d by c r o s s i n g ( S + t ) . The V , V ' bosons may be v i r t u a l w i t h h e l i c i t i e s d e f i n e d a s i n deep i n e l a s t i c e l e c t r o n s c a t t e r i n g f o r t h e s p a c e l i k e c a s e .

The s e c r e t of Donohue's r e p r e s e n t a t i o n i s r e l a t e d t o t h e s u c c e s s of CALKUL - ^{we can}

show t h a t t h e Donohue b a s i s reduces t o t h e CALKUL b a s i s o b t a i n e d by dropping t h e

4 y g term i n ( 6 ) . The d i f f e r e n c e s between the h e l i c i t y frames d i s a p p e a r i n t h e massless l i m i t . We have a g a u g e - i n v a r i a n t decomposition i n t o two r a d i a t i o n p o l e s whose r e s i d u e s a r e pure h e l i c i t y p r o j e c t i o n o p e r a t o r s with t h e channel and s e n s e determined by t h e p a r e n t s p i n o r . The s p i n o f f observed f o r e e -t ZZ i s thereby de- r i v e d , n o t i n g t h a t t h e forward and backward s c a t t e r i n g l i m i t s of+.the orthogonal S-channel decomposition of Fig. 3 c o i n c i d e with t h e r e s p e c t i v e t , u s i n g u l a r i t i e s i n t h e c r o s s e d form of ( 9 ) . (We o b t a i n S-channel l o n g i t u d i n a l c o n t r i b u t i o n s from + l h e l i c i t y s t a t e s i n t h e t , u channels.) A s a n o t h e r example, a n a l y s i s of t h e

s i n g u l a r i t i e s i n ( 9 ) shows t h a t V' s p i n n i n g o f f f i s t o be expected f o r ( 8 ) . B ) Mikaelian Miracle

We c o n s i d e r n e x t a non-Abelian Compton r e a c t i o n ( 8 ) where a t r i l i n e a r coupling e x i s t s f o r t h e two v e c t o r bosons a t l e a s t one of which i s massless. An example i s t h e photoproduction of a weak boson o f f quarks o r e l e c t r o n s ,

y f -+ Wf' (10)

C2-8 JOURNAL DE PHYSIQUE

Sometime a g o M i k a e l i a n /l11 found t h a t t h e u n p o l a r i z e d d i f f e r e n t i a l c r o s s s e c t i o n f o r ( 1 0 ) c o u l d b e w r i t t e n i n f a c t o r i z e d form w i t h p e r f e c t l y s q u a r e d c o e f f i c i e n t , b u t o n l y f o r t h e gauge t h e o r y v a l u e of t h e W m a g n e t i c moment. T h i s was e v e n t u a l l y b e a t i f i e d by t h e a m p l i t u d e a n a l y s i s (Born Again) of Goebel, H a l z e n and ~ e v e i l l e / l 2 / . It i s h e l p f u l t o d i g r e s s t o d i s c u s s t h e symmetry u n d e r l y i n g t h i s f a c t o r i z a t i o n . The c r o s s e d r e a c t i o n t o ( 1 0 ) ,

h a s b e e n s u g g e s t e d a s a p r o b e of t h e W c o u p l i n g s and h a s r e l a t e d f a c t o r i z a t i o n 1131.

M i k a e l i a n , Samuel and Sahdev 1141 showed t h a t t h e f a c t o r c o u l d produce z e r o s a t c e r - t a i n c.m. a n g l e s i n t h e d i f f e r e n t i a l c r o s s s e c t i o n s f o r ( 1 1 ) . [The z e r o s can a l s o b e s e e n i n t h e c u r v e s o f 1 1 3 1 and i n F i g . 2.1 I t remained, however, f o r Goebel e t a l . t o r e c o g n i z e and d e v e l o p t h e i m p l i e d u n i v e r s a l f a c t o r i z a t i o n of e a c h 4-body

h e l i c i t y a m p l i t u d e .

Brodsky, Kowalsky and Brownsky have d i s c o v e r e d t h a t f a c t o r i z a t i o n i s a s p e c i a l c a s e of a g e n e r a l r a d i a t i o n symmetry, which can b e c o n n e c t e d back t o c l a s s i c a l i n t e r - f e r e n c e e f f e c t s and which l e a d s t o s p i n - i n d e p e n d e n t z e r o s ( o f t e n u n p h y s i c a l ) i n a l l gauge-theory Born a m p l i t u d e s ( s p i n s 2 1 ) f o r gauge boson e m i s s i o n / a b s o r p t i o n 1151.

( I t i s a m u s i n g t h a t we had t o w a i t f o r W c a l c u l a t i o n s t o s t i m u l a t e t h e d i s c o v e r y of a symmetry t h a t i s p r e s e n t i n a J a c k s o n problem!) There i s a l s o a complementary theorem 115,161 f o r s p i n - d e p e n d e n t symmetry and a s s o c i a t e d s p i n - d e p e n d e n t z e r o s . G e n e r a l i z a t i o n s t o a n a l l - o r d e r s e x t e r n a l plane-wave d e c o u p l i n g theorem 1171 and t o g a u g i n o e m i s s i o n i n supersymmetry 1181 now e x i s t .

R e t u r n i n g t o t h e h e l i c i t y a n a l y s i s , we use what we have l e a r n e d from 1 1 2 1 t o w r i t e t h e non-Abelian 4-body a m p l i t u d e i n t h e s i m p l e form

Z i s t h e s p i n - i n d e p e n d e n t c o e f f i c i e n t i n which t h e r a d i a t i o n z e r o r e s i d e s and MA i s an A b e l i a n Cornpton a m p l i t u d e . V a r i o u s forms a r e p o s s i b l e by v i r t u e of t h e r a d i a t i o n symmetry 1151. Namely, t h e g e n e r a l Born (n+l)-body t r e e r a d i a t i o n a m p l i t u d e

( p a r t i c l e s w i t h c h a r g e s Qi, momenta p i ; p h o t o n momentum q ) c a n b e w r i t t e n

The e x p r e s s i o n ( 1 3 ) i s i n v a r i a n t under

Whatever, ( 1 2 ) i s what we want. With t h e t r i l i n e a r c o u p l i n g t r a n s f o r m e d away ( r e a l l y , n o l o n g e r e x p l i c i t ) , we may i m m e d i a t e l y a p p l y t h e Donohue form ( 9 ) t o MA and h e n c e , &A. The Donohue - G o t t l i e b r e s u l t 1 1 0 1 s u f f i c e s , a c t u a l l y , s i n c e one v e c t o r is m a s s l e s s . The f i n a l f o r m u l a a c h i e v e d makes m a n i f e s t t h e r a d i a t i o n symme-

t r y and t h e s p i n symmetry ( h e l i c i t y r u l e s ) , c a l c u l a t i o n s w i t h which a r e p a r t i c u l a r l y s i m p l e . C o n t r a s t s h o u l d b e made w i t h t h e p r o c e d u r e of s q u a r i n g and s p i n summing where terms p r o l i f e r a t e and p o l a r i z a t i o n i n f o r m a t i o n i s l o s t . We may combine t h e

t e c h n i q u e of B j o r k e n and Chen 1191 f o r p r o j e c t i n g o u t a r b i t r a r y p o l a r i z a t i o n s w i t h o u r approach.

C. More p a r t i c l e s

A Born amplitude f o r t h e r a d i a t i o n of ( a t l e a s t ) one massless gauge boson by Dirac p a r t i c l e s may be r e g a r d e d a s the sumof graphs g e n e r a t e d by a t t a c h i n g t h e gauge boson i n a l l p o s s i b l e ways t o a s e t of s o u r c e g r a p h s . T h e s e t of graphs may be r e a r r a n g e d a s a gauge-invariant v e r t e x expansion where t h e attachments a r e c l u s t e r e d a t each v e r t e x 1151. An i n t e r n a l l i n e attachment l e a d s t o two r a d i a t i o n f a c t o r s , one f o r each a s s o c i a t e d v e r t e x . Then f o r each 3-vertex we have g e n e r a t e d a Compton s e t of subgraphs t o which we can apply a h e l i c i t y form, u t i l i z i n g t h e f a c t o r i z a t i o n (12) i n the e v e n t t h a t t h e r e is a non-Abelian t r i l i n e a r a t t a c h n e n t involved. Of c o u r s e , the h e l i c i t y channels a r e now defined r e l a t i v e t o t h e i n d i v i d u a l l e g s , e x t e r n a l and i n t e r n a l , b u t t h i s i s expected from s p i n o f f .

I n g e n e r a l , ( 1 2 ) i s n o t r e a l i z e d f o r n-body Born r a d i a t i o n a m p l i t u d e s , n 2 5. We wish t o emphasize t h a t the symmetry under (15) and (16) reduces (13) t o (12) only f o r n=3 and t h a t r a d i a t i o n zeros a r e the g e n e r a l r e s u l t , n o t f a c t o r i z a t i o n . We make n o t e , however, of t h e 5-body f a c t o r i z a t i o n discovered by CALKUL / 7 / i n t h e Born amplitude f o r f f + f ' f ' y , a r e s u l t t h a t r e s t s on the fermions being massless.

Fermion h e l i c i t y c o n s e r v a t i o n i s presumably c r u c i a l , s i n c e s c a l a r v e r t i c e s s p o i l the f a c t o r i z a t i o n . This a d d i t i o n a l symmetry may be r e l a t e d t o a combination of r a - d i a t i o n symmetry, which only reduces t h e amplitude t o a two-dimensional z e r o , and t h e h e l i c i t y f o r m f o r t h e Compton subgraphs (one v e c t o r i s t h e exchanged v i r t u a l photon).

D. More masses

I n t h e circumstance where both v e c t o r bosons have mass, r e a l o r v i r t u a l , t h e non- Abelian Compton f a c t o r i z a t i o n (12) i s no l o n g e r p o s s i b l e . MNA r e v e r t s t o 3x3 form,

though i t i s s t i l l d i a g o n a l i n terms of t h e t h r e e independent p o l a r i z a t i o n v e c t o r s i n t r o d u c e d i n /10/. The t h i r d d i a g o n a l m a t r i x element vanishes r e l a t i v e t o t h e two terms t h a t a r e analogous t o those i n ( 9 1 , a s energy i s i n c r e a s e d . A t high e n e r g i e s , non-Abelian Compton Born amplitudes go o v e r t o Abelian forms t h a t i n t u r n a r e ex- p r e s s e d more c l e a r l y w i t h a non-orthogonal b a s i s .

V - SPIN DEPENDENT ZEROS

Besides t h e spin-independent zeros t h a t a r i s e i n charge-dependent phase-space re- g i o n s , corresponding t o e q u a l Qi/pi-q [ s e e (15)1, we have spin-dependent z e r o s t h a t a r i s e i n charge-independent ways, corresponding t o e q u a l ~ ~ 1 p i . q [ s e e ( 1 6 ) l and r e s i d i n g , f o r example, i n MA r a t h e r than i n Z f o r ( 1 2 ) . Thls i s t h e complementary r a d i a t i o n theorem f o r gauge t h e o r i e s t h a t we have a l r e a d y i n t r o d u c e d .

An i n t e r e s t i n g i n s t a n c e of a spin- dependent zero i s the complete v a n i s h i n g of t h e Born amplitude f o r (11) whenever t h e photon i s p o l a r i z e d t r a n s v e r s e t o t h e s c a t t e r - i n g plane and t h e W i s l o n g i t u d i n a l /20,4,16/. Nongauge c o u p l i n g s o r photons with mass g i v e a nonzero answer. This and o t h e r examples s u g g e s t ways t o measure se- l e c t e d h e l i c i t y c o n t r i b u t i o n s o r t o t e s t gauge t h e o r i e s .

We might ask how t h e s e zeros a r e r e l a t e d t o t h e Donohue form, which i t s e l f p r e d i c t s z e r o s i n h e l i c i t y amplitudes. Regarding t h e form a s a l i n e a r f u n c t i o n a l D - E f o r photon p o l a r i z a t i o n v e c t o r E and a given W p o l a r i z a t i o n i n ( l l ) , t h e r e i s a d i r e c - t i o n ^E, t h a t y i e l d s a n u l l f u n c t i o n a l , ⁼ 0 . When E, i s p h y s i c a l we o b t a i n a spin-dependent r a d i a t i o n zero. This i s n o t , however, t h e z e r o on t h e diagonal of t h e m a t r i x obtained from t h e non-orthogonal bases.

V I - OUTLOOK

We have n o t y e t progressed t o the p o i n t where a r e c i p e i s a v a i l a b l e f o r t h e ready s i m p l i f i c a t i o n of a r b i t r a r y t r e e gauge-theory amplitudes with a r b i t r a r y masses.

C2-I 0 JOURNAL DE PHYSIQUE

Closed-loops, a b o u t which we w i l l comment s h o r t l y , p r e s e n t f u r t h e r problems.

N e v e r t h e l e s s t h e view of v e c t o r - b o s o n a m p l i t u d e s a s a s e r i e s of a t t a c h m e n t s , gene- r a t i n g a n e x p a n s i o n of p o l e t e r m s w i t h s p e c i f i c h e l i c i t i e s , s h o u l d c o n t i n u e t o b e u s e f u l , p e r h a p s i n c o n c e r t w i t h a f o r m a l i s m s u c h a s t h a t o f P a s s a r i n o /8/. We have made p r o g r e s s on t h e high-energy l i m i t o f an a r b i t r a r y Born a m p l i t u d e , where t h e masses a r e n e g l e c t e d . I n t h a t l i m i t we c a n make t h e r a d i a t i o n v e r t e x e x p a n s i o n , e l i m i n a t e c e r t a i n t r i l i n e a r c o u p l i n g s by r a d i a t i o n s y m e t r y , and t h e n decompose i n t o CALKUL/Donohue p o l e s . The CALKUL d e c o m p o s i t i o n i s e s p e c i a l l y s i m p l e , u s i n g i d e n t i t i e s l i k e ( 6 1 , f o r t h e A b e l i a n Compton s u b g r a p h s . Any s h a r p v a l u e of h e l i c i t y found i n a p a r t i c u l a r phase-space r e g i o n i s i n v a r i a b l y c o r r e l a t e d w i t h a p o l e enhancement ( s p i n o f f ) .

The d e s i d e r a t a a l s o i n c l u d e t h e n e e d t o c o n s i d e r t h e h e l i c i t y a n a l y s i s of Compton s c a t t e r i n g o f f o t h e r v e c t o r p a r t i c l e s . K i m and T s a i have shown t h a t t h e h e l i c i t y of a W i s c o n s e r v e d i n t h e s c a t t e r i n g from a n e l e c t r o m a g n e t i c f i e l d b u t o n l y a t s m a l l a n g l e s /21/. More i n f o r m a t i o n comes from a c o n t r i b u t i o n t o t h i s c o n f e r e n c e by V.A. Koval'chuk a n d I . V . S t o l e t n i i . We i n t e r p r e t t h e i r d e n s i t y - m a t r i x computa- t i o n f o r y e -t W a s i n d i c a t i n g t h a t t h e W h e l i c i t y matches t h a t of t h e y i n t h e dominant f o r w a r d peak. T h i s ,is s p i n o f f a s s o c i a t e d w i t h a t - c h a n n e l p o l e . But f u r t h e r work on t h e g e n e r a l V1V2 + V3V4 Born h e l i c i t y t r e a t m e n t a n a l o g o u s t o Donohue's r e s e a r c h s h o u l d b e done.

For c o m p l e t e n e s s , we n o t i c e t h a t s p i n - z e r o p a r t i c l e s U a r e e a s i l y i n c l u d e d and f i t n i c e l y i n t o t h e g e n e r a l p i c t u r e . F o r example, t h e p o l e s i n t h e VU + VO a m p l i t u d e and i t s c r o s s e d c h a n n e l s do n o t l e a d t o s h a r p v e c t o r h e l i c i t i e s - t h e c a l c u l a t i o n s u s i n g ( 2 ) - ( 5 ) a r e s i m p l e h e r e - u n l e s s i t i s V s p i n n i n g o f f V. We c a n n o t s p i n o f f a s c a l a r p a r t i c l e .

Open and i n t e r e s t i n g q u e s t i o n s r e v o l v e a r o u n d c o n n e c t i o n s t o o t h e r c o n t r i b u t i o n s t o t h i s symposium. I t r e m a i n s t o b e s e e n i f v e c t o r h e l i c i t y b a s e s e f f e c t i v e i n Born a m p l i t u d e s a r e u s e f u l i n c a l c u l a t i o n s s u c h a s t h a t by M. Anselmino and P.

K r o l l on p h o t o p r o d u c t i o n of v e c t o r mesons o r i n t h e program t o t e s t dynamical models by G.R. G o l d s t e i n and M . J . Moravcsik. I n a d d i t i o n , i t may b e p o s s i b l e t o combine t h e Weyl r e p r e s e n t a t i o n i n v o l v i n g two-component s p i n o r s , where F i e r z i d e n t i t i e s b r i n g s u b s t a n t i a l s t r e a m l i n i n g t o t h e g i v e n problem ( s e e , e . g . , / 2 2 / ) , w i t h what we have t o s a y .

I t may b e p o s s i b l e t o s i m p l i f y c l o s e d - l o o p c a l c u l a t i o n s b e c a u s e of t h e symmetries we have d i s c u s s e d . F o r i n s t a n c e , t h e r a d i a t i o n symmetry under ( 1 5 ) and ( 1 6 ) can b e e x t e n d e d t o c l o s e d l o o p s p r o v i d e d t h a t t h e i n t e r n a l c h a r g e s a n d c u r r e n t s a r e s h i f t e d a s w e l l . P e r h a p s a n o f f - s h e l l v e r s i o n of t h i s would l e a d t o c a n c e l l a t i o n s i n h i g h e r - o r d e r QED r a d i a t i v e c o r r e c t i o n s .

L e t us n o t l o s e s i g h t , f i n a l l y , of a c e n t r a l a p p l i c a t i o n of o u r work. The i d e n t i - f i c a t i o n of t h e p o s s i b l e p o l e s and t h e i r c o n t r o l of h e l i c i t y y i e l d s a q u a n t i t a - t i v e l y u s e f u l e s t i m a t e of v e c t o r - b o s o n decay d i s t r i b u t i o n s . The d e t a i l s of t h e demise of weak b o s o n s a r e a major c o n c e r n a t t h e p r e s e n t time.

ACKNOWLEDGMENT

I am most g r a t e f u l t o P r o f e s s o r J. S o f f e r , t o t h e symposium committee, t o my c o l l a - b o r a t o r s (C. B i l c h a k , J. Donohue, M. K a r l s s o n , G. Kleppe and J. S t r o u g h a i r ) , and t o t h e N a t i o n a l S c i e n c e F o u n d a t i o n (USA) f o r making t h i s p r o g r e s s r e p o r t p o s s i b l e . I t h a n k K. Kowalski f o r h e l p f u l r e s e a r c h d i s c u s s i o n s and c r i t i c i s m .

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