Derivative amplitudes in photoproduction

~

Nuclear Physics B28 (1971) 125-140. North-Holland Publishing Company

DERIVATIVE AMPLITUDES IN PHOTOPRODUCTION H. J. W. M U L L E R Sektion F h y s i k d e r Universit~tt Mi2nchen *

Received 7 January 1971 Abstract: Partial-wave decompositions of amplitudes for reactions of non-spinless particles generally involve derivatives of Legendre functions. In an approach suggested by Fubini and Rebbi the invariants are constructed in such a way that their amplitudes possess simple scalar-like partial-wave decompositions and are therefore particularly suitable for reggeization and applications of current algebra. Here we explore the usefulness of the derivative approach by application to a non-trivial example-pion photoproduetion. The results are of interest both as an application of the method and as a compact treatment of photoproduction amplitudes.

I. INTRODUCTION We c o n s i d e r the t-channel k i n e m a t i c s of the pion photoproduction a m p l i tude although the i n t r o d u c t o r y r e m a r k s p e r m i t a r b i t r a r y generalization. The horizontal double line shown in fig. 1 divides the amplitude into two p i e c e s , one piece containing the kinematical v a r i a b l e s of the initial state, the other those of the final state - both, of c o u r s e , being r e l a t e d by e n e r g y m o m e n t u m conservation. The r e q u i r e m e n t s of rotational or L o r e n t z i n v a r iance, p a r i t y c o n s e r v a t i o n etc. of the o v e r a l l t r a n s i t i o n amplitude a r e gene r a l l y taken into account by writing down all possible invariants which m a y be f o r m e d f r o m the given v e c t o r s , s p i n o r s and t e n s o r s . The differential a p p r o a c h of Fubini et al. I f , 2] and Rebbi [3] p r e s c r i b e s in addition that the invariants should contain no coupling between v a r i a b l e s belonging to the inc o m i n g s y s t e m and v a r i a b l e s belonging to the outgoing s y s t e m . Then, c l e a r l y , in o r d e r to get the complete n u m b e r of invariants, some new v e c t o r s which have not p r e v i o u s l y been used in this context m u s t be introduced. T h e s e a r e taken to be the d e r i v a t i v e s with r e s p e c t to the relative e n e r g y m o m e n t u m v e c t o r s of the initial and final states - hence the n a m e s ' d e r i v a tive i n v a r i a n t s ' and ' d e r i v a t i v e a m p l i t u d e s ' . In o r d e r to u n d e r s t a n d the simple s c a l a r - t y p e p a r t i a l - w a v e expansions of the d e r i v a t i v e amplitudes, we r e c a l l f i r s t the behaviour of the t r a n s i t i o n amplitude under r o t a t i o n s in the c.m. s y s t e m . Let R~ be a rotation ( c h a r acterized~ by E u l e r angles a) of the k i n e m a t i c a l v a r i a b l e s of the initial state, R~ a rotation of the k i n e m a t i c a l v a r i a b l e s of the final state. Overall * Lehrstuhl Professor Dr. F. Bopp.

126

H . J . W . MTjLLER

rotational invariance then r e q u i r e s that the amplitude be invariant under the product G ad = R ai Rfa which r o t a t e s both initial and final v a r i a b l e s by the s a m e angle a. Now an a r b i t r a r y rotation -

=

,

may be written as Ga/3

=

R~(Rifl)-I

fi fi [RiR f] ,

or

so that an a r b i t r a r y ' m i x e d ' rotation of a rotationally invariant s y s t e m is equivalent to a c o r r e s p o n d i n g rotation of the initial v a r i a b l e s only or of the final v a r i a b l e s only. Although the t r a n s i t i o n amplitude is (by requirement) invariant under G aa or G fl~, it is not so in general under rotations of the kinematical v a r i a b l e s of the ingoing (or outgoing) state only. But if the inv a r i a n t s a r e now chosen in such a way that they a r e invariant under r o t a tions of the initial (or final) kinematical v a r i a b l e s s e p a r a t e l y , then the initial and final s t a t e s a r e s e p a r a t e l y r e p r e s e n t e d as rotational invariants and the only quantity which changes under the group of t r a n s f o r m a t i o n s G aB is the s c a l a r p r o d u c t of the c.m. m o m e n t u m of the initial state and that of the final state, i.e. that quantity which couples the v a r i a b l e s of one to the other Then - just as in s c a l a r - s c a l a r s c a t t e r i n g - the accompanying amplitudes p o s s e s s ' s c a l a r ' p a r t i a l - w a v e expansions in the total angular m o m e n t u m , i.e. expansions of the f o r m (2J+ 1 ) C j P j ( c o s

0),

J where O is the angle between the c.m. m o m e n t a of initial and final states. Of c o u r s e , it is wellknown that p a r t i a l - w a v e analysis of c u s t o m a r y a m p l i tudes for p a r t i c l e s with spin leads to d e r i v a t i v e s of L e g e n d r e polynomials so these d e r i v a t i v e s must come out again somehow - in fact they do but a r e now contained in the derivative invariants. Thus it is c l e a r that the amplitudes t h e m s e l v e s when d e c o m p o s e d into p a r t i a l waves p o s s e s s only simple L e g e n d r e polynomials. -

2. THE t-CHANNEL DERIVATIVE AMPLITUDES We f i r s t collect essential kinematical definitions. The f o u r - m o m e n t a of the ingoing lr (spin s 1) and 7 (spin s 2) a r e denoted by P l , P2 - those of the outgoing N , N by q l , q 2 r e s p e c t i v e l y . We use the m e t r i c goo = -1, g i i = +1 f o r i = 1, 2, 3. Defining

DERIVATIVE

AMPLITUDES

1

A =pl+P2 =ql+q2,

127

1

P=-~(Pl-P2) ,

Q =~(ql-q2) ,

= -A

2

we have in t e r m s of c.m. momenta p , q of initial and final states P.Q=p.q

=-v,

P l = P = -P2 ,

ql = q = - q 2 ,

Q.A=0, P.A

i 2 , = --~r/'/~. 7T

~

~

"

It is advantageous to collect here also the following relations Iql = ½(t- 4m21½ , 1,

12

P l ' P 2 = ~[ ~ - t ) ,

JPl = (t- m ~ ) / 2 C / , El=E2

1

= ~mm2 + q 2 = ½ t ~ ,

H2 =-(m + E1)(m + E2 ) - q2 = 2m(m +St ½) K 2=- ( m + E 1 ) ( m + E 2 ) + 09=

q2

1= H(m+½t ~) ,

(p21 +i¢~ 2) !2 s - m 2 + 2 1 p [(m 2 + q2)½ = - 2 p . q .

In the spirit of Fubini's method we now define three mutually orthogonal v e c t o r s constructed from the relative f o u r - m o m e n t u m of the initial channel; these v e c t o r s are PU = (0,p) ,

(vector)

0 _ _½iAO(o,p × ~p) a , (axialvector) LPI~ = ie t2vp(r AUpp OP(~ • V~P.ff T p u = el~vp ,~ ~" .Lp = -½AO(o, p X L p ) .

(vector)

With the help of the polarization vector ~t~ of the ingoing photon we may then construct the following derivative invariants for the initial state I p a = a(e .P) ,

(scalar)

Ipb : b(¢ . Lp) ,

(pseudoscalar)

I p c = c(e " Tp) ,

(scalar)

128

H.J.W. MULLER

w h e r e a, b, c a r e suitable n o r m a l i z a t i o n f a c t o r s . S i m i l a r l y we c o n s t r u c t d e r i v a t i v e i n v a r i a n t s for the nucleon-antinucleon s y s t e m in the outgoing state. T h e s e a r e

I Q a = a(~75 u) ,

(scalar)

IQ~ = fi(Sy ~u) Q~ ,

(pseudoscalar)

IQ7 = 7(~7 u ) L ~ ,

(scalar)

IQ8 ' = ~(vT~u) T ~ ,

(pseudoscalar)

w h e r e again a, fl, 7, 5 a r e suitable n o r m a l i z a t i o n f a c t o r s . One can show that t h e s e i n v a r i a n t s f o r m a c o m p l e t e o r t h o n o r m a l i z e d set. Since p a r i t y is c o n s e r v e d in strong and e l e c t r o m a g n e t i c i n t e r a c t i o n s , the t r a n s i t i o n amplitude m u s t be a p u r e s c a l a r having positive parity. F u r t h e r , since we a r e only i n t e r e s t e d in a f i r s t - o r d e r coupling of the e l e c t r o m a g n e t i c field, the p o l a r i z a t i o n v e c t o r m a y a p p e a r only once in any i n v a r iant. We obtain t h e r e f o r e six p a r i t y - c o n s e r v i n g invariants. T h i s n u m b e r is f u r t h e r r e d u c e d to four by taking into account the s u b s i d i a r y condition of gauge i n v a r i a n c e as well as the L o r e n t z condition which r e q u i r e the invariant I p a to vanish for p h y s i c a l , i.e. t r a n s v e r s e l y p o l a r i z e d photons (for s i m p l i c i t y we c o n s i d e r v i r t u a l photons f i r s t ) , The total t r a n s i t i o n a m p l i tude for total i n t e g e r spin m a y t h e r e f o r e be written ( f l M l i ) : E. . I v i l p j ~ i f l v ,

t),

w h e r e i = q, fl, 7, 5 and j = a, b, c subject to the above c o n s t r a i n t s . The inv a r i a n c e of Ipb , I p c under the gauge t r a n s f o r m a t i o n is r e a d i l y s e e n to be g u a r a n t e e d since if e/l is r e p l a c e d by P p t h e s e i n v a r i a n t s obviously vanish (a concise s t a t e m e n t of gauge invariance). The d e r i v a t i v e a m p l i t u d e s ~ij now p o s s e s s s i m p l e s c a l a r p a r t i a l - w a v e expansions, i.e. ~ 1"" : ~ (2J+ 1)(DJijPr(c°su J

e) ,

f o r if we write
R( a)

ei a. J

j2 = _

a2 R(~) a=O

we have (f 1J2M IJ)

J(J+ 1)(f]M]J)
(2.1)

DERIVATIVE AMPLITUDES

129

and so

i,j

IQi l p j j 2 ~ij(v, t) = ~,j I Q i l p j J ( J + 1)¢ij(v, t) , i

i m p l y i n g (2.1). We note h e r e the weliknown r e l a t i o n s j2

=

.

a

8cosO

(sin 2 O)

~cosa

'

j 2 P j ( c o s O) : J(J+ 1 ) P j ( c o s ~) . In o r d e r to c o m p l e t e the s p e c i f i c a t i o n of the a m p l i t u d e s we have to s p e c ify the n o r m a l i z a t i o n f a c t o r s c o n t a i n e d in the i n v a r i a n t s . T h i s m a y be done by r e q u i r i n g the i n v a r i a n t s to s a t i s f y s o m e o r t h o n o r m a l i t y r e l a t i o n s

i,pj-x z]'

spins

i,oj-x

z

spins

z z]

In the case of the m a s s l e s s photon the m o s t general f o r m of the s u m

over

s p i n s is given by I/

spins

12

e Cp Ev = g;lV_ • P2 P2 2

T=

1.

P2

In all s u b s e q u e n t c a l c u l a t i o n s the s e c o n d t e r m on the r i g h t - h a n d side does not c o n t r i b u t e ; we t h e r e f o r e take f o r m a l l y T = 0. One then finds 1

*

1

2

c = 8/(Xt) ~ = 8 / t ~ ( t - mlr) ,

b = ½t~ ,

w h e r e X = [ t - (m~ + m y ) 2 ] [ t - ( m ~ - my) 2] t o g e t h e r with K a = 1, K b = g c = j 2 , ( o p e r a t o r ) . T h e s u m m a t i o n o v e r s p i n s of the outgoing n u c l e o n - a n t i n u c l e o n s y s t e m is c a r r i e d out in the s a m e m a n n e r as in the c a l c u l a t i o n of f e r m i o n f e r m i o n c r o s s s e c t i o n s . F o r the s a k e of c l a r i t y we quote the r e l e v a n t f o r mula: (~(ql)O1 v(q2))(v(q2)O 2 u(ql)) = T r ( O 1 A_(q2)O 2 A+(ql)) , spins with p r o j e c t i o n o p e r a t o r s given by h+(q) = ~

1

(m ± i T " q ) •

One then finds: !

a = rn(2/t) 2 , l y = 22m/t ,

= ( 2 / t - 4m2 )~* , 5 = 2m (2/t3(t - 4m 2)) ~1 ,

130

H.J.W. MULLER

t o g e t h e r with Ky = K 5 : j 2 .

K a = l : Kfi ,

An i m p o r t a n t and useful consequence of t h e s e o r t h o n o r m a l i t y r e l a t i o n s is the i n v e r s i o n f o r m u l a (first derived by Rebbi)

(2.2) spins with R e b b i ' s notation ~iJ= K.K.,~.. z

J

~! = K . ' b . . .

Ij'

J

z

(2.3)

~j

It is now p o s s i b l e to p r o v e using the H a l l - W i g h t m a n t h e o r e m [5] that the inv a r i a n t d e r i v a t i v e a m p l i t u d e s have 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 in the m o m e n t u m t r a n s f e r v a r i a b l e s and that t h e i r only 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 a r e those explicitly introduced by the n o r m a l i z a t i o n f a c t o r s . T h i s then c o m p l e t e s the specification of t h e / - c h a n n e l photoproduction amplitudes. F o r the s - c h a n n e l the a n a l y s i s is s i m i l a r , though slightly m o r e c o m p l i cated since it has half i n t e g e r total spin.

3. CORRELATION OF DERIVATIVE AMPLITUDES TO BALL AND CGLN AMPLITUDES It is an i n t e r e s t i n g and i m p o r t a n t question to ask, how the d e r i v a t i v e a m p l i t u d e s a r e r e l a t e d to the f a m i l i a r Ball [6] and CGLN [7, 8] a m p l i t u d e s . T h i s is the p r o b l e m we e x a m i n e in this section. A suitable s e t of t - c h a n n e l p a r i t y - c o n s e r v i n g rotation i n v a r i a n t s and t h e i r r e s p e c t i v e a m p l i t u d e s Gi, i = 1, 2, 3,4 have been defined by Ball [6]. In t e r m s of the k i n e m a t i c a l v a r i a b l e s of fig. 1 the t r a n s i t i o n amplitude is then w r i t t e n as

= x 2 G x I , where q.c

a. q x e

Iqt

/ql

a. p x e

G2+i ~

G4+i

(a.p)(q.p

q2lp I

×v)

G3 ,

the Pauli s p i n o r s × being r e l a t e d to D i r a c s p i n o r s by

(0) u(p)

[2m(E++m)]~

'

[2m(Z+m)]½

X2 '

where

x1,×2 (+.1) or (i), c o r r e s p o n d i n g to nucleon and antinucleon spin up or down r e s p e c t i v e l y . F o r

DERIVATIVE A M P L I T U D E S

'~,P, ¢"

131

~ "c, P~,%

Fig. 1. t - c h a n n e l k i n e m a t i c s f o r pton p h o t o p r o d u e t i o n .

subsequent calculations it is advantageous to r e - e x p r e s s G in the following form

G = ~q . 6 G1 +i vq ~ ( p . o ) ( 6 . p x q ) [ G 2 + z G 4 ]

-i lpl v

( 6 . q ) ( o . q x p ) [ G 2 z +G4] v

- i IP_]v ( q . o ) ( 6 . p X q ) [ G 2 z + G 4 + q~-p-~G3] ,

(3.1)

where

z =p'q/Ipllq] =cosO,

v = p 2 q 2 - ( p ' q ) 2.

In d e r i v i n g (3.1) the following r e l a t i o n s were used which we quote b e c a u s e of t h e i r r e p e a t e d u s e f u l n e s s in l a t e r calculations:

(tr.a)(o.b) = a . b + i t r . ( a x b ) aX (bXC) =

,

( a . c ) b - ( a . b ) c,

v(a. p x 6) = ( 6. p x q){(p, q)(p. a)- p2( q . a ) } - p2( 6.q)( o. q x p) , v ( a . q x e) = ( e . p x q ) { q 2 ( p . a ) - ( p . q ) ( q . o ) } - ( p . q ) ( 6.q)(a. q x p ) . We o b s e r v e that the i n v a r i a n t s a r e all s c a l a r s as they should be f o r the tchannel; gauge i n v a r i a n c e is also taken c a r e of. Our f i r s t step is to e s t a b lish the r e l a t i o n between the amplitudes G and the d e r i v a t i v e amplitudes ~ij" We t h e r e f o r e r e - e x p r e s s the D i r a c s p i n o r s contained in the IQi in t e r m s of Pauli s p i n o r s and finally equate coefficients of the four i n v a r i a n t s

132

H.J.W.

MULLER

in (3.1) to the Gi. It is convenient to redefine the 6 ' s in t e r m s of g ' s :

~ac = -Wgac,

4'[3b = wgfib,

'~yc : 2iwgyc,

~Sb = 4iwgsb,

(3.2)

where 1/w = 4 f 2 m 2 ( m +½t½) 2 . It m a y be useful to r e m a r k that s o m e c a r e is n e c e s s a r y in dealing with the d e r i v a t i v e s ~/~P, ~/~Q contained in the i n v a r i a n t s , since e.g. 0P

o ~-v + 2Pl ~

a(p )

H o w e v e r , it is one of the g r e a t a d v a n t a g e s of the differential method that the unpleasant d e r i v a t i v e s with r e s p e c t to the e x t e r n a l m a s s e s m 2, m 2 a r e c o m p l e t e l y avoided in that the d e r i v a t i v e s a r e c o n t r a c t e d only with r e s p e c t to v e c t o r s which a r e orthogonal to (in this case) Pl, P2, i.e. 4, P (since P l P 2 = ½ A ± p ) - a p r o p e r t y known as t h e i r t r a n s v e r s a l i t y condition. "The r e l a t i o n s linking the G ' s and g ' s a r e found to be 2G1 = gac '

=

+Z

C +

C ~

- 2 G 4 = gTc+ZgSb+gSb , - 2 ( G3 + G4 ) = g~ b '

(3.3)

w h e r e the p r i m e s denote differentiation wi*~h r e s p e c t to z. The r e l a t i o n s (3.3) a r e effectively f i r s t - o r d e r d i f f e r e n t i a l equations in the g ' s . T h e i r s o lutions will be useful below. We r e c a l l f i r s t that the d i f f e r e n t i a l a m p l i t u d e s have s i m p l e s c a l a r p a r t i a l - w a v e expansions; hence also the g ' s by (3.2).

Writing J and using the r e l a t i o n

(2j÷ ll[z pS÷ %1 -- JeS÷ 1 ÷(J÷ 1)P _ 1' we obtain f r o m (3.3)

G1:2 J

DERIVATIVE

AMPLITUDES

133

G 2 = ½ ~j [(2J + 1)g~J)P~i(z)+g(J)yc {JP"J+l~(z)+ (J + 1)P~_ l(Z)}] ,

(z) +gSb {JPJ+I (z) +

G4 = -½ ~ [(2J+ i) J i

(J)

"-

,

G3 +G4 = - ~ (J+~)gflb P j ( z ) . J

(3.4)

These are the partial-wave expansions derived by Ball [6, 9] by a different method. In fact we m a y use his expressions for relating the g(J) to the Tm a t r i x elements. In Ball's notation +

g~c =

(J)

'

j

=

'

(J) g6b

+

'

where

+ Tj(+, ", 1) ± Tj(-, +, 1)

%= j(j+l)[2lpllq]]½

+ Tj(+, +, 1) + Tj(-, -, I)

'

j(j+l)[2]pllql]½

(3.5)

'

and Tj(+, +, 1) are the T-matrix elements for transitions initiated by a phoz and an antinucleon of ton of helicity +1 producing a nucleon of helicity +-i z with total angular momentum J, the first argument referring to helicity +-I the nucleon, the second to the antinucleon. Thus the differential amplitudes are effectively the amplitudes having these T-matrix elements as their

partial waves. E x p r e s s e d m o r e p r e c i s e l y ~ , fl~ are the amplitudes r e p r e senting a t r a n s i t i o n by magnetic radiation to a triplet final state of angular momentum J and parity (- 1)J , a~ is an amplitude r e p r e s e n t i n g a magnetic t r a n s i t i o n to a triplet final state of parity (-1) J + l , and ~ an amplitude r e p r e s e n t i n g an electric transition to a singlet final state of parity (-1) J + l . (For the calculations of these parities see for instance Childers and Holladay [131.) Our next step is to relate the differential amplitudes to the L o r e n t z - i n v a r i a n t amplitudes derived by Chew et al. [7]. F o r this purpose we consider the f i r s t - o r d e r differential equations (3.3) - and of these only the coupled equations in g y c , gSb require discussion. Defining the o p e r a t o r s d d P-

d

dzdz'

d

q = d z Z dz '

the equations may be r e w r i t t e n 2G2 = P g 6 b + Q g y c ,

-2G4 = P g v c +Qg6b •

Using j2 =

and the operator relations

~d. (z 2 - 1) d

dz

'

134

H.J.W. MULLER (p+Q)-I+(p_

Q)-I = _2(j2)-1

,

[(p +Q)- 1 _ (p_ Q)- 1]pQ- I = 2(j2)- 1 Qp-1 = z

\dz

'

one finds

~12g6b

eJ' g v c = G4 +z G 2 +

G2+zG4+(~)

-1 G 4 .

(3.6)

T h e L o r e n t z - i n v a r i a n t C G L N a m p l i t u d e is in o u r k i n e m a t i c a l n o t a t i o n given

by 4

= c ( ~ ( q 2 ) O U ( q l ) )

0 = ~ AiMi , i=l

,

w h e r e c is a s u i t a b l e n o r m a l i z a t i o n f a c t o r and M 1 : -i~5(~. e)(~.p2) ,

M2 = 2 i ~ 5 ( - Q ' e P l ' P 2

+Q'p2pl

" e) ,

" ~) '

M3 = 7 5 ( - 7 " e P l ' P 2 + ' / ' p 2 p l b

M 4 = M 4 + 2 r a M 1 = 2~5(~. E Q . p 2 - 7 . p 2

Q. c) .

We u s e the i n v e r s i o n f o r m u l a (2.2) to c a l c u l a t e the ~ ' s in t e r m s of the A ' s and h e n c e the A ' s in t e r m s of G ' s . By d i r e c t a p p l i c a t i o n of the i n v e r s i o n f o r m u l a (i.e. p e r f o r m i n g the spin s u m m a t i o n s ) one finds i

q~aC = _c 4i 2z

[(

T~

1 ~ .Q)A 2 + ~(T .P)A1] ,

(t- m2)(t-4m2) ½ ,d,-1 ~ac = c

5 ~ 2~ rn t~

&/3b= c i

_3 22 i

2'

t-~(t - 4m )~ m (t-m2)

& flb : - c - 2~- m

[~zz)

[A1 + tA2] '

,~T ('P" Q)[A1-2mA4]

'

d-1 (~z)

[A1- 2mA4] '

5

gP~/C = c

~ t2

Lp. LQ[AI - ~

A4] - c ~ T mD

. T A3 ,

DERIVATIVE AMPLITUDES 1

(j2)-l~c

(j2)-1 ®sb

1

= c

~Sb= c

= c

135

[A 1 - ~ A 4 ] - c

1 t~

z+

2~5 i T S . T~ t 2 ~ [AI-~A4]-c t2 ( t - 4 m )-~

i ( t - 4 m 2 ) 71 ( t - m 2 ) ' " Lp'LQA3 2-~ t rn

i(t-

t A4 ] _ c

2~ t-~

A3,

2~ m

L(

-1

*+

P"" - ~

i(t- 4m )~(t- m2)

2~ m

'

A3 .

T h e r e l a t i o n b e t w e e n t h e A ' s a n d G ' s now f o l l o w s f r o m (3.2) and (3.3). U s i n g t h e r e l a t i o n s (2.3) a n d (3.6) a n d c h o o s i n g t h e n o r m a l i z a t i o n f a c t o r c a s tom c

=

-

3

1

27 t~ we o b t a i n B a l l ' s r e l a t i o n s

(t- m 2 ) ( t - 4m21½ G1 =

[AI+tA2] '

16~t

(t- m 2 ) ( t -

4m2)½ 1 A3 , 16~ t~

G2 =

1

( t - m2~)(t~ - 2m) 63 = -

16yt

[A1 + t ~ A 4 ] '

(tG4 = "

m~) 167rt [ 2 m A l - t A 4 ]

"

(3.7)

F o r t h e s a k e of c o m p l e t e n e s s we a d d s o m e o p e r a t o r r e l a t i o n s w h i c h a r e u s e f u l in d e r i v i n g t h e a b o v e e q u a t i o n s :

L p . LQ = ¼td 2 ,

d

Lp.Q

-1

= LQ.P=O.

Of c o u r s e w e c o u l d a l s o h a v e w o r k e d w i t h a n o n - g a u g e - i n v a r i a n t s e t of i n v a r i a n t s a s B a l l [6] d o e s a n d t h e n i m p o s e t h e g a u g e c o n d i t i o n a t t h e v e r y e n d . H o w e v e r , it i s o b v i o u s f r o m t h e a b o v e how t h e c a l c u l a t i o n s w o u l d p r o c e e d in t h a t c a s e , s o t h e r e w a s no p o i n t in g o i n g t h r o u g h m o r e c o m p l i c a t e d

136

H.J.W.

MULLER

calculations. N e v e r t h e l e s s it m a y be e s s e n t i a l - although this is not always done - to work in t e r m s of a m a s s i v e v i r t u a l photon which p e r m i t s a l s o longitudinal radiation. Ball and J a c o b [11] e m p h a s i z e this point by saying that one should not d i s s o c i a t e photoproduction f r o m pion production induced by v i r t u a l photons, and Dombey [12] even c o n s i d e r s it e s s e n t i a l f o r the r e g g e i z a t i o n of the photoproduction a m p l i t u d e s . He s t a r t s f r o m the o b s e r vation that a wellknown s e l e c t i o n rule f o r b i d s r a d i a t i v e t r a n s i t i o n s (induced by t r a n s v e r s e radiation) between two s t a t e s with total a n g u l a r m o m e n t u m z e r o - and p r e c i s e l y this would rule out pion exchange (into the singlet NN state) in c h a r g e d pion photoproduction. However, working in t e r m s of f o r m a l l y g a u g e - i n v a r i a n t a m p l i t u d e s and a m a s s i v e photon he can show that the longitudinal t r a n s i t i o n to the singlet final state contains the pion pole, thus p e r m i t t i n g r e g g e i z a t i o n without violation of a s e l e c t i o n rule. We m a y i n c o r p o r a t e t h e s e r e s u l t s a p o s t e r i o r i by r e l a x i n g the condition e • P = 0, which we i m p o s e d b e f o r e , and working with a m a s s i v e photon. We then have a n o t h e r i n v a r i a n t I p a of the k i n e m a t i c a l v a r i a b l e s which combined with the f i n a l - s t a t e i n v a r i a n t s leads to the additional d i f f e r e n t i a l amplitude invariants

IQv~Ipa ,

IQTI.pa ,

and hence to two additional differential a m p l i t u d e s • a, ~ - A c c o r d i n g l y the r~" • t h r e e - d i m e n s i o n a l p o l a r i z a t i o n v e c t o r e has to be r e p l a c e d by a f o u r - d i m e n s i o n a l v e c t o r containing a longitudinal component, and a m p l i t u d e s A5, A6, GS, G 6 have to be defined. The r e l a t i o n s (3.3) a r e then a m e n d e d by (setting rn? = photon m a s s ) -2m2yG5 =2m27zgow' - g a a '

2m (G 2+G6) = g ~ a ,

where

~cea = Wg~a ,

~va = Wgva •

Then again

gaa = ~ (2J+ 1) ~'aa -(J) P j ( z ) , J etc. w h e r e g(~Ja) r e p r e s e n t s a longitudinal t r a n s i t i o n to a t r i p l e t final s t a t e of p a r i t y (- 1) J + l and g(-J) f f a a longitudinal t r a n s i t i o n to a singlet final state. It is c l e a r then that g(-J) ~ a m a y contain the pion pole. 4. CORRELATION OF DERIVATIVE AMPLITUDES WITH H E L I C I T Y AMPLITUDES We can a l s o e a s i l y e s t a b l i s h the connection between the d e r i v a t i v e a m plitudes - or any of the o t h e r a m p l i t u d e s - with the J a c o b - W i c k helicity

DERIVATIVE AMPLITUDES amplitudes. In t e r m s of the T - m a t r i x element defined by

i

m54(q2+ql-P2"Pl)

Sfi - (2~)2

•

(4E1E 2 P20) ~

V(q2)T(q l,q2,P l,P2)u(q l) ,

the latter are given by (for helicity +1 of the photon) 1 ++1 J d01(0) , f1!(0)22 _ [Ipllq1 1]½ ~(J+~)Tj j 1 ~ i +- id Jl(0) ' f~_½(O) : [lpllql] ½ J (J +-~)Tj 1 ~ (j+½)r~+ld~ l(e),

fl_½½(O) -[ipl]ql] ½ J f~_~_½(o) - [l~']lq1 I]~ ~(j+½)Tj-ldJl(O), : with

dJl(0) d

2 =

!

LJ(J+I).]

i(0) - S(J+ I)

and by (3.5) T+±I j :

[½J(:+~)Ipll#1]½(g~J~g~¢')

T±:FI j = J(J+ 1)[½1p]lq I]½ (g~J) vg(J)'yc) '

But by (3.4) -(G 3 + G4) • G 1 = ~J (J + ½)(g~J) ~ ~c~(J)~P"J '

G2+G4 =~-~Ij+I_w~(J)~:~(J)~tzp" ' pj) " j ~ 2J~Syc ~6b ~ J+PJ~ Hence 1

f~.~2~.½(O) = 2-~ sin e [-(G3 +G4) :FG1] , 1

fl½:~½(O) : -2 -~ (cos 0 + 11 [G 2 + G4] •

137

138

H.J.W. MfJLLER

Next we define the following m a t r i x f ' f~S~N

~

X2 f '

X1 ,

(4.1)

w h e r e X2, ×1 a r e the Pauli s p i n o r s given above, and

On the o t h e r hand we know that the t r a n s i t i o n m a t r i x e l e m e n t is given by ( f i M l i ) = ×2 G × 1 ,

G a matrix.

T.he connection between the G i ' s and the helicity a m p l i t u d e s m u s t t h e r e f o r e be such that we m a y w r i t e f ' = G1 el@ . where ~ is a p h a s e and G 1 is G as defined e a r l i e r but with photon helicity +1. This, of c o u r s e , m e a n s that we want to identify the z - c o m p o n e n t of the nucleon spin with its helicity. T h i s m a y be done for the NN s y s t e m of the t channel p r o v i d e d the z - a x i s is chosen to lie in the d i r e c t i o n of the final nucleon m o m e n t u m . In o r d e r to obtain the explicit equivalence f o r photon helicity +1, we choose k-= P2 to lie in the x - z plane as shown in fig. 2. We a l s o choose the p o l a r i z a t i o n v e c t o r -~ ~

1 = _2-½ ( e x

cosO+

(6 -1 would be + 2 - ½ ( e x

ez

sin~ + i e y )

c o s 0 + e z sin0 - i e y ) )

, ,

iy -Z

--X

X

/

% Z

f

Z

F i g . 2. Choice of k = P 2 in the

xz

plane.

/

×

DERIVATIVE AMPLITUDES P2

Ip2l-cos0

139 ql

ez-sinO

Iqll- %,

eX,

0, of c o u r s e , is the s c a t t e r i n g angle - i.e. the angle b e t w e e n P2 = - P and q 1 = q in the c.m. s y s t e m . I n s e r t i n g t h e s e v a l u e s for p , q and ~ in G we find

(Gl+G3+G4)sin6

+(G2+G4)(l+cosO)

2~ I~.(G2_ G 4 ) ( l _ cos 0)

-(G1-G3-G4)sine

eiq5 f'_

1

in a g r e e m e n t with o u r e x p a n s i o n s above if we choose +e iq5 = 1. The c u s t o m a r y p a r i t y - c o n s e r v i n g c o m b i n a t i o n s of t-channel helicity a m p l i t u d e s a r e then defined as (cf. Ball and J a c o b [11])

F1, 2 = ~

([I!22

.....

F 3 , 4 = 1 + cos e f - - -

e 1 - c o s ~ fl½½ .

T h e s e a m p l i t u d e s have well defined p a r i t y .

5. ASYMPTOTIC

BEHAVIOUR

OF

INVARIANT

AMPLITUDES

Next we wish to e x a m i n e the a s y m p t o t i c b e h a v i o u r of the i n v a r i a n t (CGLN) a m p l i t u d e s . F o r this p u r p o s e we solve eqs. (3.7) for the A i and obtain 16~t

A1 = (t- m2)(2m+t½) (G3+G4)

16= [ Cl

'

1

l

A2- (t-m 2) (t-4m2)½ +--2m+t~' (G3+G4) ' 1

A3 A4

=

=

(t-

16~ t~ m2)(t- 4m2)½ G2 '

(t-

m2)(2m +t½) [t~

16~

1

G4 - 2m G3]

.

Substituting the r e l a t i o n s (3.3) we obtain the i n v a r i a n t CGLN a m p l i t u d e s in t e r m s of d i f f e r e n t i a l a m p l i t u d e s : 8/rt

y

A1 = (t- m2)(2m +t-~) g~b '

140

H . g . W . MULLER A2 =

8~

1 ,

i

(t- m 2) [(t- 4m2) -~ gac- (2m +t~) - lg'flb]

,

1

A3 =

"44 =
8ut~

8ff

[-(2,,

±

+g;b ) +2, g b ] .

It i s now an e a s y m a t t e r to d e r i v e , f o r a g i v e n p r o c e s s , t h e l a r g e z b e h a v i o u r of t h e C G L N a m p l i t u d e s a t f i x e d t. It f o l l o w s f r o m t h e s c a l a r p a r t i a l w a v e e x p a n s i o n s of t h e d e r i v a t i v e a m p l i t u d e s t h a t a s p i n J s t a t e e x c h a n g e d in the t - c h a n n e l p r o d u c e s a c o n t r i b u t i o n to t h e d e r i v a t i v e a m p l i t u d e w h i c h f o r l a r g e z b e h a v e s a s s o m e f u n c t i o n of t m u l t i p l i e d b y z J. But f r o m t h e a b o v e we s e e t h a t t h e C G L N a m p l i t u d e s a r e c o m b i n a t i o n s of t h e d e r i v a t i v e a m p l i t u d e s a n d d e r i v a t i v e s of t h e l a t t e r - s o t h e b e h a v i o u r f o r l a r g e z of a s p i n J c o n t r i b u t i o n to a C G L N a m p l i t u d e A i i s s o m e f u n c t i o n of t m u l t i p l i e d b y z J - r i , w h e r e r i i s t h e m i n i m u m o r d e r of d e r i v a t i v e s of t h e d e r i v a t i v e a m p l i t u d e s a p p e a r i n g in A i ( c o n s i d e r i n g z g " a s e q u i v a l e n t to a f i r s t d e r i v a t i v e ) . R e g g e i z a t i o n t h e n i m p l i e s the r e p l a c e m e n t J ~ a(t) w h e r e a(t) i s t h e c o r r e s p o n d i n g R e g g e t r a j e c t o r y , a n d we o b t a i n t h e r e s u l t t h a t a C G L N a m plitude A i behaves for large z and fixed t as

A i ~ K(t)za(t)'ri.

RE FE RE NCES

[1] S. Fubini, talk presented at the High-energy physics Conf. in Heidelberg, 1967, Proceedings p. 644. [2] V. de Alfaro, S. Fubini, C. Rosetti and C. F u r l a n , Ann. of Phys. 44 (1967) 165. [3] C.Rebbi, Ann. of Phys. 49 (1968) 106. [4] See for instance, S. S. Schweber, Introduction to r e l a t i v i s t i c quantum field theory (Harper and Row, New York, 1961). [5] D. Hall and A. S. Wightman, Mat. Fys. Medd. Dan. Vid. Selsk. 31 (1957) No. 5. [6] J. S. Ball, UCRL-8858 (1959); UCRL-9172 (1960); Phys. Rev. 124 (1961) 2014. [7] G . F . C h e w , M. L. Goldberger, F. E. Low and Y. Nambu, Phys. Rev. 106 (1957) 1345. [8] G. F. Chew, AECU-4664 (1957) 116. [9] See also R.W. Childers and W. C. Holladay, Phys. Rev. 132 (1963) 1809. [10] M. Jacob and G. Wick, Ann. of Phys. 7 (1959) 409. [11] J. S. Ball and M. Jacob, Nuovo Cimento 54A (1968) 620. [12] N. Dombey, Nuovo Cimento 31 (1964) 1025.