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Electroproduction of the P33 resonance
~
Nuclear Physics B28 (1971) 573-593. North-Holland Publishing Company
ELECTROPRODUCTION OF
THE P33 RESONANCE
R. L. CRAWFORD
Department of Natural Philosophy, University of Glasgow, Scotland Received 11 June 1970 (Revised manuscript received 23 December 1970) Abstract: The fixed t-dispersion relations for pion electroproduction are solved for energies below and in the region of the first resonance by making detailed use of the coupled multipole equations and Watson's theorem. The P33 multipoles, GE, n and F~r are evaluated by fitting to experimental data.
i. INTRODUCTION
Until r e c e n t l y , the e x p e r i m e n t a l d a t a f o r the e l e c t r o p r o d u c t i o n of pions in the r e g i o n of the N*(1236) r e s o n a n c e have c o n s i s t e d a l m o s t e n t i r e l y of m e a s u r e m e n t s of the t o t a l c r o s s s e c t i o n s , ~T and ~S, f r o m i n e l a s t i c e l e c t r o n - p r o t o n s c a t t e r i n g . Since t h e s e give i n f o r m a t i o n about only the g r o s s f e a t u r e s of the r e a c t i o n , d i s p e r s i o n c a l c u l a t i o n s f o r e l e c t r o p r o d u c t i o n [1,2] have tended to be c o n s i d e r a b l y l e s s d e t a i l e d than c o r r e s p o n d i n g s t u d i e s of p h o t o p r o d u c t i o n [3-5] w h e r e the e x p e r i m e n t a l r e s u l t s a r e m o r e demanding. It has been s u f f i c i e n t to r e t a i n only the P33 m u l t i p o l e s in the r e s c a t t e r i n g and to have only the Born t e r m s in the f o r c e t e r m s f o r t h e s e m u l t i p o l e s . In c o m p a r i s o n , an a d e q u a t e d e s c r i p t i o n of the p h o t o p r o d u c t i o n d a t a r e q u i r e s the e f f e c t s of the r e s c a t t e r i n g in o t h e r final s t a t e s and it is n e c e s s a r y to take a c c o u n t of the coupling between the m u l t i p o l e s . H o w e v e r , the i n f o r m a t i o n which i s now a v a i l a b l e f r o m c o i n c i d e n c e e x p e r i m e n t s [6-8], with the p r o m i s e of m o r e to c o m e , has a l t e r e d this c o n s i d e r a b l y and the e x p e r i m e n t a l s i t u a t i o n is m o v i n g c l o s e r to the one found in p h o t o p r o d u c t i o n . It should be noted that the s i m p l e e l e c t r o p r o d u c t i o n m o d e l s d e s c r i b e d above do not give a good fit to the d a t a f o r 7r+ p h o t o p r o duction n e a r the f o r w a r d d i r e c t i o n o r f o r 7r° p h o t o p r o d u c t i o n n e a r the b a c k w a r d d i r e c t i o n . Since t h e s e c o r r e s p o n d to the m o s t l i k e l y r e g i o n s f o r c o i n c i d e n c e m e a s u r e m e n t s of e l e c t r o p r o d u c t i o n , the t h e o r y ought to be r e f i n e d if the new e l e c t r o p r o d u c t i o n d a t a a r e to be u s e d p r o p e r l y . A move in this d i r e c t i o n has been m a d e by von Gehlen [9] who has given s o l u t i o n s f o r the i s o v e c t o r m u l t i p o l e s but without any d i r e c t c o n f r o n t a t i o n with data. In this p a p e r , the coupled i n t e g r a l e q u a t i o n s f o r the e l e c t r o p r o d u c t i o n m u l t i p o l e s a r e a p p l i e d in d e t a i l to fit to the new d a t a and to u s e it to e v a l u a t e the c h a r g e f o r m f a c t o r s of the pion and neutron. The equations a r e s o l v e d n u m e r i c a l l y and i n c l u d e all the n o n - n e g l i g i b l e r e s c a t t e r i n g which can
574
R.L. CRAWFORD
be calculated f r o m W a t s o n ' s t h e o r e m . All the couplings between the m u l t i pole equations a r e retained. It is shown in sects. 2 and 3 that this detail can be easily i n c o r p o r a t e d by choosing to w o r k with the Pault amplitudes. Sect. 3 also gives the details of the methods which a r e used to solve for the individual multipoles. In the next section, a p a r t i a l check on the methods employed is obtained by calculating the photoproduction amplitudes. Finally, in sect. 5, the details and r e s u l t s of the e l e c t r o p r o d u c t i o n calculation a r e p r e s e n t e d . The notation and conventions of Zagury [1] a r e used throughout this paper.
2. THE INTEGRAL EQUATIONS FOR THE M U L T I P O L E AND PAULI AMPLITUDES W a t s o n ' s t h e o r e m [10], using unitarity and t i m e r e v e r s a l i n v a r i a n c e , links the complex p h a s e s of the multipoles with the p h a s e shifts for elastic pion-nucleon s c a t t e r i n g . Although the r e l a t i o n s a r e exact only below the threshold for the production of two pions, it is p o s s i b l e to u s e the t h e o r e m at significantly higher e n e r g i e s provided the i n e l a s t i c i t i e s of the p h a s e shifts r e m a i n small. The i n f o r m a t i o n which is thus obtained is conveniently introduced into a fixed t - d i s p e r s i o n relation by p r o j e c t i n g to the coupled int e g r a l equations for the multipoles. This a p p r o a c h has a l r e a d y p r o v e d to be s u c c e s s f u l in photoproduction [3-5]. The techniques which a r e r e q u i r e d f o r the calculation of the multipole equations f o r e l e c t r o p r o d u c t i o n a r e s i m i l a r and have a l r e a d y been d e s c r i b e d [4,9]. It is usual to e x p r e s s the multipole equations in t e r m s of a set of a m p l i tudes, Ha(W, k 2) with a = 1,2, etc., which a r e r e l a t e d to the m u l t i p o l e s by s i m p l e k i n e m a t i c f a c t o r s that explicitly i m p o s e the known b e h a v i o u r of the m u l t i p o l e s at s m a l l values of k and q (ref. [11]). T h e s e f a c t o r s a p p e a r quite n a t u r a l l y during the p r o j e c t i o n and help to m i n i m i s e the coupling of the m u l t i p o l e s to t h e m s e l v e s . They will be defined l a t e r as r e q u i r e d . The multipole equations have the f o r m ReHa(W, X2 )
=Ha,i(W,)t2)+lP ~
dW'
Im Ha(W', k 2)
W'-W
(1)
m+Lt
The inhomogeneous t e r m ,
Ha, I,
Ha' I(W' k2) = Ha' B(W' k2)
1 m~+ ~ + ~-
can be written as
dW,~Kafi(W,W,,~.2)ImHr(W,,X2), r
(2)
w h e r e Ha, B is the Born t e r m and Kar t a r e n o n - s i n g u l a r k e r n e l s which give the coupling between the multipoles. In g e n e r a l , the s - and u - c h a n n e l cuts link all multipoles of the s a m e isospin type and the u - c h a n n e l cut also c o u p l e s the i s o v e c t o r m u l t i p o l e s with isospin indices of 1 and 3. At ~t2 = 0, in the photoproduction limit, the s c a l a r m u l t i p o l e s uncouple f r o m the t r a n s v e r s e multipoles and f r o m the production amplitude. Since the multipole equations c o m e f r o m fixed t - d i s p e r s i o n r e l a t i o n s , they r e q u i r e that the p a r t i a l wave expansions of the i m a g i n a r y p a r t s of the
ELECTROPRODUCTION OF THE P33 RESONANCE
575
i n v a r i a n t amplitudes c o n v e r g e at unphysical values of cos 0. As a r e s u l t , it is not possible to calculate the k e r n e l s in eq. (2), and hence Ha, I, for l a r g e values of W. F o r photoproduction, the limit is f o r m a l l y at 1340 MeV (ref. [12]) but it can be shown [13] that the weakness of the t - c h a n n e l s t r u c t u r e p e r m i t s the use of the multipole equations up to the second r e s o n a n c e r e gion. It has been argued that this limit is also valid in e l e c t r o p r o d u c t i o n for ~t2 > -200 fm -2 (ref. [9]). A detailed study of e l e c t r o p r o d u c t i o n in the region of the P33 r e s o n a n c e is considerably m o r e difficult than for photoproduction. Due to the s c a l a r multipoles, the n u m b e r of equations which have to be solved is a l m o s t doubled and t h e r e is a c o r r e s p o n d i n g i n c r e a s e in the complexity of the couplings between the multipoles. Also, the calculation has to be repeated for s e v e r a l values of X2 and it cannot be a s s u m e d , as in photoproduction, that it is always possible to i g n o r e all but a s m a l l n u m b e r of the couplings. T h e r e f o r e , even if the k e r n e l s in eq. (2) a r e obtained n u m e r i c a l l y , the r e m a i n d e r of the calculation is c u m b e r s o m e and p o s e s s e v e r a l technical difficulties. This is p a r t i c u l a r l y true since the P33 multipoles cannot be a c c u r a t e l y predicted and have to be evaluated by fitting to e x p e r i m e n t a l data. The calculation b e c o m e s significantly s i m p l e r by choosing to use the int e g r a l equations for the amplitudes, 5ri, in the Pauli decomposition of the T - m a t r i x . These equations a r e obtained d i r e c t l y f r o m the d i s p e r s i o n r e l a t-ions and it is r e l a t i v e l y simple to p r o j e c t n u m e r i c a l l y to the multipoles to use W a t s o n ' s t h e o r e m . In effect this a p p r o a c h can be used to solve s i m u l taneously all the r e q u i r e d multipole equations with a l a r g e n u m b e r of r e s c a t t e r i n g contributions and with the automatic inclusion of all the couplings. The p r o j e c t i o n of the equations for the Pauli amplitudes f r o m the d i s p e r s i o n r e l a t i o n s is s t r a i g h t f o r w a r d but lengthy [14]. T h e r e a r e , however, two points which r e q u i r e care. It is n e c e s s a r y to i m p o s e a subtraction in the d i s p e r s i o n relation for A 5 to e n s u r e that the c o n s t r a i n t s r e q u i r e d by gauge i n v a r i a n c e a r e satisfied. Secondly, the s c a l a r amplitudes must have additional subtraction t e r m s for W equal to infinity to give the c o r r e c t highenergy behaviour. This is i m p o r t a n t even for calculations at the 1'33 r e s o nance. The equations a r e conveniently given in t e r m s of Fi(W,z , ~t2) defined by 5rl(W , z, k 2) =
8--~4(E1 + m)(E 2 + rn) (W - m) FI(W , z,
k2),
5r2(W, z, ~.2) = ~ -1- ~ 4(E1 - m ) ( E 2
-m) (W+m)F2(W,z,k2),
5r3(W , z, ~.2) = ~ 1
,/(E 1 _ m)(E 2
-m) (E2+m)(W+m)F3(W,z,k2),
~:4(W,z,k2 ) = ~1
4(El+m)(E2+m )
(E 2
_m)(W_m)F4(W,z,~t2) ,
576
R.L. CRAWFORD
~:7(W,z,X2) = 8~1 ~/.(El+m)(E2 _m) (E 1 _m)F7(W,z,k2 ) , 5rs(W,z,X2 ) = ~ 1
~/(El_m)(E2+m)
m)F8(W,z,k2)
.
(3)
dW' ~/kij(W',W,z',k2)Im~(W',z',k2) ,
(4)
(El+
They have the f o r m
dW' imFi(W,,z,,X2) ReFi(W,z,X2) = Fi,B(W,z,k2 ) + l p ? m+~ W' -W +y1 ? rn+~ where
Fi, B
J
a r e the Born t e r m s ,
kij
a r e n o n - s i n g u l a r k e r n e l s and
z' =k ~ (k~q~ - koq o +kqz).
(5)
The p r i m e s indicate that the energy v a r i a b l e s have values c o r r e s p o n d i n g to W'.
3. SOLUTION OF THE INTEGRAL EQUATIONS The equations for the F - a m p l i t u d e s , and hence the multipoles, are solved by a simple i t e r a t i v e method. Suitable s t a r t i n g f o r m s for the i m a g i n a r y p a r t s of the multipoles a r e used to evaluate I m F i in the i n t e g r a l s in eq. (4) and ReF i a r e calculated for nodes in W and z. N u m e r i c a l integration o v e r z gives the c o r r e s p o n d i n g real p a r t s of the multipoles and these can be used with W a t s o n ' s t h e o r e m to give a new set of i m a g i n a r y p a r t s of the m u l t i poles for input to the next cycle. Alternatively, for multipoles which do not have sufficiently stable solutions for this method, it is easy to calculate Ha, I by using eq. (1) to c o m p a r e the input i m a g i n a r y p a r t with the resulting real part. The new i m a g i n a r y p a r t is obtained by solving the single m u l t i pole equation. The complete p r o c e s s is repeated until, after a few c y c l e s , the i m a g i n a r y p a r t s of the multipoles settle to suitably stable f o r m s . The final r e s u l t s , for c o m p a r i s o n with e x p e r i m e n t , contain Im Sri f r o m all the multipoles with non-negligible i m a g i n a r y p a r t s and Re ~ i a r e polynomials in z whose o r d e r is one less than the n u m b e r of nodes at which Re Fi(W , z , k 2) a r e calculated. The m e a s u r e d phase shifts for pion-nucleon s c a t t e r i n g [15] d e t e r m i n e the multipoles which must be included with the P33 p a r t i a l waves in the r e scattering. The l o w - e n e r g y p a r t s of the S-waves have been shown to be i m portant in photoproduction and a r e t h e r e f o r e d e s i r a b l e here. The possible effects of the r e s c a t t e r i n g at the P l l and D13 r e s o n a n c e s in the second r e sonance region r e q u i r e the inclusion in s o m e f o r m of the i m a g i n a r y p a r t s of these amplitudes. Finally, the P31 final state gives a s m a l l r e s c a t t e r i n g
ELECTROPRODUCTION OF THE P33 RESONANCE
577
Table 1 The multipoles which are included in the rescattering contributions to the dispersion integrals. Final pion-nucleon state SII
$31
Isovector multipoles S(1) E(1)
0+ 0+
Isoscalar multipoles S(0) E (0)
0+ 0+
S(3) E(3)
0+ 0+
1)33
.q(3) E(3) M(3) ~1+ 1+ 1+
D13
.q(1) E(1) M(1) -2- 2- 2-
S~0_)E(0) M(0) 2- 2-
c o r r e c t i o n . A l t o g e t h e r , this r e q u i r e s the solution of the i m a g i n a r y p a r t s of the multipoles listed in table 1. F o r the real p a r t s , it is n e c e s s a r y to calculate up to and including D-waves. In p r a c t i c e , for n u m e r i c a l c o n s i d e r a tions, Re F i a r e calculated at five nodes in z and t h e r e f o r e effectively contain up to G-waves. Since W a t s o n ' s t h e o r e m cannot be used at high e n e r g i e s , a cut-off is introduced into the i m a g i n a r y p a r t of each multipole at the energy where the elasticity of the c o r r e s p o n d i n g phase shift falls below 0.8. The good b e h a v ious of the solution at this point is e n s u r e d by r e p r e s e n t i n g the r e s c a t t e r i n g in the multipole at higher e n e r g i e s by a tail which is joined smoothly to the solution at the cut-off and which vanishes quickly as W goes to infinity. It is found that the r e s u l t s below and in the region of the P33 r e s o n a n c e a r e insensitive to the details of these tails. Due to this cut-off, the calculation contains only p a r t of the contributions f r o m the P l l and D13 r e s o n a n c e s . However, it is found in photoproduction that the inclusion of the P l l and D13 multipoles in this f o r m gives useful c o r r e c t i o n s and that the c o n s e q u e n c e s of omitting the full r e s o n a n c e c o n t r i bution a r e not s e r i o u s for W less than 1290 MeV. T h e r e is, of c o u r s e , the f r e e d o m to add bumps to the tails to r e p r e s e n t the r e m a i n d e r of these eff e c t s but, at p r e s e n t , the available e x p e r i m e n t a l data is not good enough even f o r a rough e s t i m a t e of their size. In p a r t i c u l a r , for the D13 m u l t i poles, it is n e c e s s a r y to m e a s u r e a total of six unknown and potentially r e sonant t e r m s . Except for the P33 multipoles, the h i g h - e n e r g y tails in the d i s p e r s i o n i n t e g r a l s imply that the complex p h a s e s of the multipoles go to zero as W goes to infinity. As a r e s u l t , the multipole equations have unique solutions [16] and the simple i t e r a t i v e method, using W a t s o n ' s t h e o r e m to give the i m a g i n a r y p a r t f r o m the real p a r t , is s a t i s f a c t o r y . In g e n e r a l , all these
578
R.L. CRAWFORD
m u l t i p o l e s c o n v e r g e r a p i d l y to s o l u t i o n s w h i c h a r e s t a b l e to b e t t e r than 1%. F o r the P33 m u l t i p o l e s , the c o m p l e x p h a s e is c l o s e to n at the c u t - o f f and it is well known that this c a u s e s d i f f i c u l t i e s in any a t t e m p t to get p a r a m e t e r - f r e e s o l u t i o n s f o r t h e s e a m p l i t u d e s . The p r o b l e m is c o m p o u n d e d by the inability of the p r o j e c t e d m u l t i p o l e e q u a t i o n s to give Ha, I at l a r g e v a l u e s of Wo It has to be a s s u m e d that both Ha, I and the c o m p l e x p h a s e , qSa(W), a r e known only f r o m the t h r e s h o l d at Wo to s o m e finite e n e r g y , W L. To r e f l e c t this, the P33 m u l t i p o l e s a r e s o l v e d by a d a p t i n g the m e t h o d of E n g e l s and S c h m i d t [17]. This is a m o d i f i e d O m n e s - M u s k h e l i s h v i l l i m e t h o d which a l l o w s a c l e a n s e p a r a t i o n of the known and unknown c o n t r i b u t i o n s to the solution. W a t s o n ' s t h e o r e m g i v e s the D - f u n c t i o n e x a c t l y
Da
(6)
Wo F o r Wo < W < WL, the solution to eq. (1) is g i v e n by ga(k 2)
Na(W' X2) = Ha(W' ~'2) Da(W) 1
=
Ha, I(W' x2) ReDa(W) + WL - W
WL
+ 7r(WL - W ) p f
(W' - WL )Ha, I(W', k 2) Im Da(W') dW'
Wo
1
?
7r(WL - W) WL
W v - W
(W' - WL) Da(W') Im Ha(W' , X2) dW'
W' - W
(7)
where ga(k 2) = lim WHa, I(W, ;t 2) W ~ oo
_1 ? dW' ImHa(W',k2). -Tr W o
(8)
(9)
Eq. (7) is e x a c t and it can be shown that the s o l u t i o n s which it g i v e s s a t i s f y lim --W'~Ha(W,X 2) = constant. W ~ oo
(10)
The f a c t o r of (W' - W L ) in the i n t e g r a l s in (7) p r e v e n t s any d i f f i c u l t i e s in a n u m e r i c a l c a l c u l a t i o n f r o m Da(W) which b e h a v e s like (W- WL)-~ba(WL)/lr c l o s e to W L. The s e c o n d and l a s t t e r m s in eq. (7) a r e the unknown p a r t s of the s o l u tion. It is a s s u m e d that t h e r e a r e no l a r g e r e s c a t t e r i n g c o n t r i b u t i o n s f r o m a s e c o n d P33 r e s o n a n c e a b o v e W L and Im Ha in the s e c o n d i n t e g r a l is a p -
ELECTROPRODUCTION
OF
THE
P33 RESONANCE
579
Table 2 The validity of the solutions for the P33 multipoles which are given by eq. (7) with ga as a free p a r a m e t e r is checked by comparing the values given by the solutions for the left-hand side of eq. (1), ReHa, and the right-hand side, H~. W (pion mass units)
gm =o Re Hm
gm =g'
Re H m - Hm
Re Hm
gm =2g'
Re Hm-H'm
Re Hm
Re Hm -H'm
7.720
0.102x10-1 0.153x10-5
0.137x10-1 0.096x10-5
0.172xi0- I
0.017xi0-5
8.098
0.820xi0-2 0.262x10-5
0.138x10- I 0.110x10-5
0.196x10-1
0.039x10-5
8.350
0.623x10-2 0.237x10-5
0.133x10-1 0.093x10-5
0.204x10-1
0.042x10-5
8.601
0.263x10-2 0.258x10-5
0.835x10-2 0.127x10-5
0.141x10-1
0.042x10-5
8.854
0.0
0.0
0.0
0.005x10 -5
9.106
0.326x10-3 0.296x10-5 -0.308x10-2 0.114x10-5 -0.648x10-2 -0.032x10-5
9.358
0.115x10-2 0.352x10-5 -0.287×10-2 0.149x10-5 -0.689x10-2 -0.028x10-5
9.735
0.190x10-2 0.488x10-5 -0.185x10-2 0.242x10-5 -0.561x10-2 -0.016x10-5
0.270x10 -5
0.i03x10 -5
10.114
0.232x10-2 0.872x10-5 -0.114x10-2 0.447x10-5 -0.460x10-2
0.018x10-5
10.492
0.259x10-2 2.960x10-5 -0.645x10-3 1.671x10-5 -0.388x10-2
0.388x10-5
3) 2 The three solutions , are for M('I+ at k = 0 and use the same H~ , I_ between W° and W LThe one with gG = g corresponds to the solutions for M~3+) in ref. [4].
p r o x i m a t e d by a t a i l w h i c h v a n i s h e s r a p i d l y a s W g o e s to i n f i n i t y . S i n c e ~ba(WL) i s c l o s e to ~, this i s n o t a c r i t i c a l p a r t of the c a l c u l a t i o n and a s i m p l e i t e r a t i v e m e t h o d c a n be u s e d to j o i n the t a i l s m o o t h l y to the s o l u t i o n f o r I m H a b e l o w WL. H o w e v e r , ga, the o t h e r u n k n o w n t e r m , d e t e r m i n e s the s i z e of the s o l u t i o n to the e x t e n t t h a t the m a i n e f f e c t of Ha, I i s to c o n t r o l o n l y the s h a p e of the m u l t i p o l e a c r o s s the r e s o n a n c e . S i n c e the p r o j e c t e d m u l t i p o l e e q u a t i o n s c a n n o t g i v e ga f r o m eq. (8), the P33 m u l t i p o l e s c a n n o t b e p r e d i c t e d and m u s t be o b t a i n e d f r o m e x p e r i m e n t a l d a t a by t r e a t i n g ga a s a free parameter. B e c a u s e of the k n o w n t h r e s h o l d b e h a v i o u r , eq. (7) i s u s e d to give s o l u tions for Hm(W'k2) =
1 1/-El + m k-qV ~
M(13)(W' ~'2) '
1 1 / / E - 1 + rn
E~3+)(W,;~2)
He(W'~'2) = kq VE2+m Hs(W, ~2 )
W ] E/~I + m
=
--.- ~ = . k..q
, ~2
+ ,,,
(3)
sl+(w,x
2 .
)
(Ii)
580
R.L. CRAWFORD
Table 2 shows examples of the validity of the v a r i o u s solutions which a r e given by eq. (7) for the s a m e Ha, I by v a r y i n g ga" The r e s u l t s a r e for M~.3) at k2 = 0 but a r e typical of all three multipoles at all values of ~t2. If, l+ as an additional check, eq. (9) is used to r e c a l c u l a t e ga from the solution for ImHa, it is found that the original value is r e p r o d u c e d to about 0.5%. These c h a r a c t e r i s t i c s appear to be independent of the r e s c a t t e r i n g above WL since they also apply if bumps a r e added to the tail for Im H a to r e p r e sent additional 1)33 r e s o n a n c e s . Clearly this technique gives v e r y s a t i s f a c t o r y solutions for the calculable l o w - e n e r g y p a r t s of the equations for the 1)33 multipoles. It shows that the projected multipole equations on their own fail completely to define unique solutions for these amplitudes. In p a r t i c u l a r , it d e m o n s t r a t e s that the v a r i ational approach can have no predictive power since, for example, all three solutions in table 2 would be v e r y acceptable end points for this type of calculation. The sensitivity of the solutions to ga is not s u r p r i s i n g since any attempt to impose a specific a s y m p t o t i c f o r m on the solution as in eq. (10) m u s t make the final r e s u l t c r i t i c a l l y dependent on the a s y m p t o t i c f o r m of Ha, I. It is t h e r e f o r e not possible to use A d l e r ' s c r i t e r i o n [2] of choosing the amplitude which d e c r e a s e s f a s t e s t as W goes to infinity in o r d e r to get a unique solution. To elaborate on this, the usual assumption that (~a(W) goes to ~ as W goes to infinity m e a n s that eq. (10) is sufficient to define a unique solution. The contribution f r o m the c i r c l e at infinity r e q u i r e d for the conventional O m n e s - M u s k h e l i s h v i l l i method [2] is then given by ga.
4. P H O T O P R O D U C T I O N
As in m o s t e l e c t r o p r o d u c t i o n calculations, the photoproduction a m p l i tudes a r e obtained by solving the e l e c t r o p r o d u c t i o n equations at ~t2 = 0. Since photoproduction has been studied s u c c e s s f u l l y and in detail using the multipole equations, the principal i n t e r e s t , in this c a s e , is to get a test of the p a r t i c u l a r methods that a r e used and to examine the consequences of omitting p a r t of the r e s c a t t e r i n g in the second r e s o n a n c e region. It is t h e r e f o r e sufficient to s u m m a r i s e the i m p o r t a n t r e s u l t s and to display a few typical c u r v e s (figs. 1 and 2). The two 1)33 multipoles w e r e evaluated by fitting to the ~o data [18]. In general, the solutions for the multipoles, the c r o s s sections, p o l a r i sations and a s y m m e t r y r a t i o s a g r e e well with those of B e r e n d s , Donnachie and W e a v e r [4]. The fits to experimental data for ~+ production n e a r the f o r w a r d direction and for 7r° production n e a r the backward direction a r e s a t i s f a c t o r y and give a hopeful p r o g n o s i s for the ability of this approach to d e s c r i b e e l e c t r o p r o d u c t i o n data f r o m coincidence experiments. It is found that the effects of the 1)11 r e s o n a n c e a r e probably s m a l l for proton t a r g e t s due to cancellation between the i s o s c a l a r and i s o v e c t o r amplitudes. This, however, will not r e m a i n true at all values of ~t2. It is possible that the lack of knowledge about this r e s o n a n c e in photoproduction [19] and in e l e c troproduction may be one of the main limitations for an e l e c t r o p r o d u c t i o n calculation. The effect of the D13 r e s o n a n c e can be e s t i m a t e d by adding ap-
ELECTROPRODUCTION OF THE P33 RESONANCE
581
d__~ 2O
IO
0
i 30
i 60
I 90
i 120
I 150
180
0 Fig. l(a). Differential cross sections for pion photoproduction at fixed energy ~p -* y+n at k L = 340 MeV. p r o p r i a t e B r e i t - W i g n e r t e r m s to the h i g h - e n e r g y t a i l s of the m u l t i p o l e s . The c o r r e c t i o n s a r e m o s t m a r k e d f o r ~+ p r o d u c t i o n a t s m a l l v a l u e s of O b u t a r e n o t i m p o r t a n t f o r W l e s s t h a n a b o u t 1290 MeV.
5. E L E C T R O P R O D U C T ~ N E l e c t r o p r o d u c t i o n e x p e r i m e n t s a r e a p o t e n t i a l s o u r c e of i n f o r m a t i o n a b o u t the p i o n and n u c l e o n e l e c t r o m a g n e t i c f o r m f a c t o r s b u t i t i s not p o s s i b l e at p r e s e n t to m e a s u r e a l l five t e r m s s i m u l t a n e o u s l y due to the l i m i t a t i o n s of p r e s e n t d a t a and b e c a u s e it i s n e c e s s a r y a t the s a m e t i m e to e v a l u a t e the P33 m u l t i p o l e s . F o r t u n a t e l y , the p r o t o n f o r m f a c t o r s a r e w e l l
582
R.L. CRAWFORD
30
20
"LL I0
o
I
I
I
I
I
30
6o
9o
12o
L5o
fSo
U
Fig. l(b). Differential cross sections for pion photoproduetion at fixed energy 7p ~ ~Op at k L = 340 MeV. known from elastic e l e c t r o n - p r o t o n s c a t t e r i n g and it is only n e c e s s a r y to c o n s i d e r the m e a s u r e m e n t of the r e m a i n d e r , GM, n, GE, n and F~. All t h r e e couple strongly to the ~+ production amplitude but, since they i n t e r f e r e badly, p r e s e n t experimental data does not allow their separation. GM n and F~ also contribute strongly to the inhomogeneous amplitude for M(13+) a~d hence, by virtue of the r e s c a t t e r i n g in the P33 final state, they have a considerable effect on the lr o production amplitude in spite of being absent f r o m the o Born t e r m s . However, as a l r e a d y d e s c r i b e d , they control the shape but not the size of the r e s o n a n c e in M(3~ 1' " This gives a very weak m e a s u r e ment which is susceptible to s m a l l e r r o r s in the n o n - r e s o n a n t multipoles and to the o m i s s i o n of h i g h - e n e r g y r e s c a t t e r i n g . T h e r e f o r e it is u n r e a l i s t i c to include m o r e than two f o r m f a c t o r s as f r e e p a r a m e t e r s in the calculation. The ones which are chosen a r e the l e a s t well known, that is GE, n and F~, and it is a s s u m e d that GE - , and satisfy the nucleon dipole f o r m and scaling law [20]. ' ~ G M ' p GM'n These two f o r m f a c t o r s and the three p a r a m e t e r s for the P33 multipoles are fitted to the e x p e r i m e n t a l data of M i s t r e t t a et al. [7]. The e x p e r i m e n t gives m e a s u r e m e n t s of the u+ and o differential c r o s s sections for proton t a r g e t s o v e r a range of values for W, X2, 0 and ~?, the angle between the electron s c a t t e r i n g plane and the hadron production plane. A notable f e a ture of the data is the c l e a r detection of the i n t e r f e r e n c e between the t r a n s -
E L E C T R O P R O D U C T I O N OF THE P33 RESONANCE
583
do" dn
t t
15
I0
I
I
200
i
I
300
I
400
kL (MeV) Fig. 2(a). D i f f e r e n t i a l c r o s s se~)Aons f o r pion photoproduction at fixed angle ~ p - - * ~+n at '0 = 30 °. doI
dn 15
I0
5
I
200
I'
300
I
I
400
I
EL(MeV )
F i g . 2(b). 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 pion photoproduction at fixed angle yp -~ ~Op at 0 = 150 °.
584
R.L. CRAWFORD
i 0 -'~
io,,~.
0 -I ~
12~I~-
,
~3~o
W
(M eV)
-2 -3
F i g . 3(a). M u l t i p o l e s o l u t i o n s at ),2 = ~1.2 f m - 2 . I s o s c a l a r . T h e s y s t e m of u n i t s i s o b t a i n e d b y a s s u m i n g t h a t c = /~ = g = 1.
xlO-a
,=
,
El.
"
13Mi" c:)o
L
W (MeV)
-!
F i g . 3(b). M u l t i p o l e s o l u t i o n s a t k 2 = - 1 . 2 f m - 2 . I s o v e c t o r f o r T = ½. T h e s y s t e m of u n i t s i s o b t a i n e d b y a s s u m i n g t h a t c = /~ = gt = 1.
ELECTROPRODUCTION OF THE P33 RESONANCE
585
x I 0 -a
2
I
El+ 0
-2 ~
~
~J
~
~ " " " 1 ='~" W
Ee*
Fig. 3(c). Multipole solutions at k2 = -1.2 fm -2. Isovector for T = 3. The system of units is obtained by assuming that c = ~ = ~ = 1. v e r s e and s c a l a r a m p l i t u d e s . T h i s g i v e s a c o n t r i b u t i o n to the d i f f e r e n t i a l c r o s s s e c t i o n w h i c h h a s t h e f o r m [7] ~/½e(~+ 1) s i n 0
cosT? S(W,
~, X2),
w h e r e • d e s c r i b e s the r a t i o of s c a l a r to t r a n s v e r s e p o l a r i s a t i o n f o r t h e v i r t u a l p h o t o n . S i n c e t h i s i s the o n l y t e r m w h i c h c o n t r i b u t e s l i k e cosT? to 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 , i t p r o v i d e s a s u i t a b l e l e v e r f o r s e p a r a t i n g the t r a n s v e r s e a n d s c a l a r p a r t s of t h e P33 r e s o n a n c e if t h e r e i s d a t a o v e r a s u f f i c i e n t l y l a r g e r a n g e of 7. T h e f i v e f r e e p a r a m e t e r s a r e d e t e r m i n e d f r o m a l e a s t - s q u a r e s fit. S i n c e a l a r g e p a r t of t h e p r o d u c t i o n a m p l i t u d e i s o b t a i n e d in a p a r a m e t e r f r e e m a n n e r , t h e n o r m a l i s a t i o n e r r o r s of t h e d a t a a r e i n c l u d e d q u a d r a t i c a l l y w i t h the s t a t i s t i c a l e r r o r s in the t o t a l e r r o r s w h i c h a r e u s e d to c a l c u l a t e X2. The P33 m u l t i p o l e s a r e c a l c u l a t e d f r o m t h e ~o c r o s s s e c t i o n s w h e r e t h e y c o n t r i b u t e m o s t s t r o n g l y a n d , by s i m i l a r r e a s o n i n g , GE, n and F ~ a r e o b t a i n e d f r o m the y+ d a t a . T h e two s e t s of p a r a m e t e r s a r e t r e a t e d a l t e r n a t e l y d u r i n g t h e m i n i m i s a t i o n and a t e a c h s t a g e the c u r r e n t v a l u e s of o n e s e t a r e u s e d in t h e f i t t i n g of t h e o t h e r . T h e d a t a g i v e f o u r s e p a r a t e c a l c u l a t i o n s ; f o r k 2 c l o s e to - 1 . 2 fm - 2 , - 3 . 4 f m - 2 , - 6 . 3 f m -2 and - 1 0 . 2 f m -2 r e s p e c t i v e l y . It i s n o t p o s s i b l e to u s e t h e ~+ d a t a a t - 1 5 . 2 8 f m -2 s i n c e t h e r e a r e no c o r r e s p o n d i n g ~o r e s u l t s . S i n c e t h e d a t a d e p e n d on f o u r v a r i a b l e s , W, O, k 2 and ~?, i t i s n o t e a s y in a f e w ~ d i a g r a m s (figs. 5 and 6) to g i v e an a c c u r a t e i m p r e s s i o n of t h e f i t w h i c h i s o b t a i n e d . T h e r e f o r e , a d d i t i o n a l d e t a i l s of t h e X2 f o r e a c h e n e r g y a r e
R.L. CRAWFORD
586
.io-' lOOxM~_ S I.
'HOO
'
12OO
'
W (MeV)
,~5o
-I
-2
J
-3
F i g . 4(a). M u l t i p o l e s o l u t i o n s at ~.2 = - 6 . 3 f m - 2 . I s o s c a l a r . T h e s y s t e m of u n i t s i s o b t a i n e d b y a s s u m i n g t h a t c = /i = ~ = 1.
j~z
IO~M.l~. 0 _ ~
S.+
.
~eV)
M+.
-I F i g . 4(b). M u l t i p o l e s o l u t i o n s at X2 = - 6 . 3 f m - 2 . I s o v e c t o r f o r T = ½. T h e s y s t e m of u n i t s i s o b t a i n e d b y a s s u m i n g t h a t c = /~ = ~ = 1.
E L E C T R O P R O D U C T I O N OF THE P33 RESONANCE
587
3 IO
I
o
_
,,,oo
,
,,2oo -
-I
~
\ d ~ "
, - -
~ '~
w
Eo+
Fig. 4(e). Multipole solutions at X2 = -6.3 fm -2. Isoveetor for T = ~. The system of units is obtained by assuming that c = h-= p. = 1. given in table 3. In g e n e r a l , the r e s u l t s a r e v e r y s a t i s f a c t o r y e x c e p t at -10.2 fm -2 w h e r e it is not p o s s i b l e to get a s t a b l e fit to the n + data. H o w e v e r , at this value of k2, it is p o s s i b l e to u s e data at only one value of W s i n c e the r e m a i n d e r is at an e n e r g y w h i c h is s u f f i c i e n t l y high f o r the c a l c u l a t i o n to be u n r e l i a b l e . It h a s to be e x c l u d e d s i n c e it r e p r e s e n t s a l a r g e p r o p o r t i o n of the total d a t a and can t h e r e f o r e d i s t o r t the final r e s u l t s badly if it is used. A l s o , at this k 2, the e x p e r i m e n t is c l o s e to its l i m i t s . T h e r e is t h e r e f o r e no r e a s o n to b e l i e v e that this f a i l u r e has m u c h s i g n i f i c a n c e . M~,3) is of c o u r s e the d o m i n a n t f e a t u r e of the r e a c t i o n and is t h e r e f o r e the n~+st a c c u r a t e l y m e a s u r e d of the P33 m u l t i p o l e s . S t a t i s t i c a l l y , it is d e t e r m i n e d to b e t t e r than one p e r c e n t and the 10% u n c e r t a i n t y in the n o r m a l i s a t i o n of the d a t a g i v e s a s y s t e m a t i c e r r o r that is t y p i c a l l y 6% o r 7g~ The s h a p e of the 7NN* f o r m f a c t o r , e s s e n t i a l l y that of f~/I~.3)/k, is not v e r y c l e a r x+ Its slope a p p e a r s to ly s e e n f r o m the s m a l l r a n g e of k 2 c o v e r e d by the data. be ::?P;
t hbld a ~ tdhutehtto ° ~ : h e : l l l : : : .
d T ~ : : : l u t m o n sE~:+r) itis inn°twellfigs.3 deter-and 4
have no g r e a t s i g n i f i c a n c e and i n d i c a t e m e r e l y that t h e r e is no n e e d to have a l a r g e E ! 3) and that it is p r o b a b l y negative. The s ~ + l a r - t r a n s v e r s e i n t e r f e r e n c e in the c r o s s s e c t i o n s g i v e s a c l e a r m e a s u r e m e n t of S(13+). The b e s t r e s u l t is at 6.3,fm "2 w h e r e the 307 u s a b l e d a t a p o i n t s b e t w e e n 1132 and 1284 MeV ~ i v e a s t a t i s t i c a l e r r o r w h i c h is l e s s than 50/0. The v a l u e s o b t a i n e d f o r S~.)/M~, 3) at r e s o n a n c e a r e r e s p e c * k 21+ tively -0.147, - 0 . 2 5 4 , -0.244 and -0.065~o~ equal to -1.2 fro- 2 , -3.4 f m - 2 , -6.3 f m -2 and -10.2 fm -2. The l a s t r e s u l t a p p e a r s to be a n o m a l o u s and m a y be r e f l e c t i n g the l a c k of d a t a at -10.2 f m - 2 . The s y s t e m a t i c u n c e r t a i n t y in S(3) . i ~. . .a. i I i i c u l t to e s t i m a t e in a r e a l i s t i c way s i n c e it is s e n s i t i v e to any e r I+ . f o r m the ~ dependence of the data. O b v i o u s l y , o t h e r i n d e p e n d e n t e x p e r i m e n t a l r e s u l t s a r e r e q u i r e d . The final r e s u l t a l s o d e p e n d s on the a c c u r a c y
R. L. CRAWFORD
588
Table 3 Details of the fits at different values of W. (a) 7r° production. W(MeV)
k L (MeV)
1197 1223
294 328
1226 1270
332 391
X2(fm -2)
Total X2
Number of data points
-1.22 -1.19
41.7 28.0
37 32
totals
69.7
69
-3.34 -3.27
50.5 65.8
55 58
totals
116.3
113
1132 1166 1200 1228 1259 1284
214 255 298 334 376 409
-6.54 -6.47 -6.41 -6.35 -6.29 -6.24
20.2 44.4 35.4 77.1 49.5 71.2
26 35 60 59 66 61
totals
297.8
307
1238
343
-10.37
39.5
26
(b) ~'+ production W (MeV)
k L (MeV)
1170 1197 1223
260 294 328
1185 1226 1270
279 332 391
1198 1281
296 407
~.2(fm-2)
Total X2
Number of data points
-1.24 -1.22 -1.19
26.4 16.7 16.3
21 19 17
totals
59.4
57
-3.40 -3.34 -3.27
19.4 24.9 51.5
18 22 26
totals
95.8
66
-6.41 -6.29
35.8 40.6
28 31
totals
76.4
59
of the s o l u t i o n s f o r SO+ and S 1_ and t h e s e can only b e i m p r o v e d when t h e r e i s a b e t t e r u n d e r s t a n d i n g of the s t r u c t u r e of the s e c o n d r e s o n a n c e r e g i o n . H o w e v e r , t h e s c a l a r - t r a n s v e r s e i n t e r f e r e n c e i s m a i n l y b e t w e e n M~.3Yand 1+ the s c a l a r m u l t i p o l e s and, a s h a s b e e n p o i n t e d o u t [7], a l l c o n t r i b u t i o n s , e x c e p t f r o m SI3) , c h a n g e sign at the r e s o n a n c e . A s a r e s u l t , t h e e n ere-v d e p e n d e n c e of t h e * i n t e r f e r e n c e p r o v i d e s a good t e s t of the s o l u t i o n f o r S! 3~. It can be s e e n f r o m t h e ×2 at d i f f e r e n t v a l u e s of W, e s p e c i a l l y f o r 42 e~+al to - 6 . 3 f m - 2 , t h a t a s a t i s f a c t o r y f it i s o b t a i n e d f o r a l a r g e r a n g e of e n e r g i e s and t h a t the i n t e r f e r e n c e i s w e l l d e s c r i b e d . In the f i t to t h e 7r+ d a t a , t h e r e i s c o n s i d e r a b l e i n t e r a c t i o n b e t w e e n t h e
E L E C T R O P R O D U C T T O N O F T H E P33 R E S O N A N C E
589
4O do" 30
20
J
IO
I
0
I
I
I
I
90
180
r~ F i g . 5(a). 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 ~.o e l e c t r o p r o d u c t i o n at f i x e d E, W, 0 and X2. W = 1198 M e V , k2 = - 6 . 4 1 f m - 2 a n d e = 125 ° .
dodn~ 30
20
I0
0
~
180
rl F i g . 5(5). 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 ~or ~.o e l e c t r o p r o d u c t i o n at f i x e d E, W, 0 and X2. W = 1260 M e V , ~t = - 6 . 2 9 f m - 2 and 0 = 1 3 5 ° .
590
R.L. CRAWFORD
dodfl= 30
20
JO
I
I
I 90
0
I
I 18o
rl F i g . 6(a). 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 ~ o r ?r+ e l e c t r o p r o d u c t i o n at f i x e d E, W, 0 a n d ) 2 . W = 1200 M e V , ~t~ -- - 6 . 4 7 f m - 2 a n d 8 = 25 ° .
3o
20
t
IO
o
9o
18o
+ F i g . 6(b). 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 ff e l e c t r o p r o d u c t i o n a t f i x e d E W, 0 a n d A2. W = 1259 MeV, A Z = - 6 . 2 9 f r o - 2 a n d e = 25 ° .
591
ELECTROPRODUCTION OF THE P33 RESONANCE Table 4 The g a p a r a m e t e r s for M (3) E (3) and S (3) 1+ '
k2 (fro-2)
gm x 102
1+
1+ "
ge x 103
gs x 102
-1.2
0.279±0.006
-0.347±0.092
-0.649±0.312
-3.4
0.228±0.003
-0.113±0.020
-1.401±0.084
-6.3
0.172±0.001
-0.048±0.013
-0.782±0.037
-10.2
0.158±0.004
-0.013±0.012
-0.215±0.082
Only the statistical errors are shown. I.O
0-8
O'6
t
O'4 O-2 I
i
I
i
i
2
4
6
8
iO
i
-~z ( f m - ~ )
Fig. 7. The form factor for the ~NN* coupling: 0)/k(WR, 0)] with WR=1236MeV.
[M~3)+(WR,X2)/k(WR,X2)I/[M~3)+(WR,
The solid curve is the nucleon dipole. values of F~ and GE, n. It is t h e r e f o r e c l e a r l y unwise to ignore GE, n and to set it to z e r o wheh m e a s u r i n g F~ f r o m e l e c t r o p r o d u c t i o n data. S i m i l a r l y , the uncertainty in GM, n limits the a c c u r a c y with which F~ can be found. The r e s u l t s for F~ (fig. 8) a r e not v e r y i n f o r m a t i v e but they appear to favour the nucleon dipole shape r a t h e r than the shape given by simple rho dominance. The r e s u l t s for GE, n (fig. 9) a r e encouraging. They show that it is s m a l l but n o n - z e r o and positive in the region 0 > k2 > -5 fm -2 and a r e in good a g r e e m e n t with the r e s u l t s f r o m e l e c t r o n - d e u t e r o n scattering. In conclusion, it can be claimed that d i s p e r s i o n theory has been shown to give an adequate d e s c r i p t i o n of p r e s e n t e x p e r i m e n t a l data at s m a l l values of k2 and that it is a p r a c t i c a l method for e x t r a c t i n g useful information f r o m the r e s u l t s of coincidence m e a s u r e m e n t s of electroproduction. It is to be hoped that additional suitable data will a p p e a r and that they will c o v e r both the
592
R . L . CRAWFORD
o-s
f
o
1 2
I 4
I 6
-k= (fm -2 )
Fig. 8. The pion form factor. Curves (A) and (B) a r e r e s p e c t i v e l y the predictions from simple rho dominance and the nucleon dipole form.
•E,n 0.06 o-o/~
0.02
t
i
U 1
i
I
2
3
L,
I
I
5
6
- X=(fro -=)
Fig. 9. The neutron charge form factor. The open c i r c l e s are the r e s u l t s of this calculation and the solid c i r c l e s a r e from electron-deuteron scattering [20]. f i r s t and s e c o n d r e s o n a n c e r e g i o n s . O b v i o u s l y , f r o m t h e r e s u l t s a b o v e , i m p r o v e d d a t a a r e r e q u i r e d to g i v e a b e t t e r d e t e r m i n a t i o n of F ~ . T h e d a t a f o r t h e s e c o n d r e s o n a n c e r e g i o n w i l l a l l o w a b e t t e r e s t i m a t e of t h e b a c k g r o u n d to the P33 r e s o n a n c e and w i l l t h e r e f o r e p e r m i t m o r e a c c u r a t e m e a s u r e m e n t s f r o m the l o w - e n e r g y d a t a . In p a r t i c u l a r , the v a l u e w h i c h i s o b t a i n e d f o r F~ i s s e n s i t i v e to t h e s u b t r a c t i o n c o r r e c t i o n s f o r t h e s c a l a r a m p l i t u d e s a n d , a s a r e s u l t , i t m a y b e s e n s i t i v e to h i g h - e n e r g y r e s c a t t e r i n g c o n t r i b u t i o n s . H o w e v e r , the i n t e r p r e t a t i o n of the s e c o n d r e s o n a n c e r e g i o n w i l l n o t b e e a s y d u e to t h e l a r g e n u m b e r of p o t e n t i a l l y r e s o n a n t m u l t i p o l e s w h i c h h a v e to b e c o n s i d e r e d . T h e a u t h o r w i s h e s to t h a n k P r o f e s s o r J. C. Gunn and P r o f e s s o r R. G. M o o r h o u s e f o r h e l p and e n c o u r a g e m e n t t h r o u g h o u t t h i s w o r k and P r o f e s s o r A. D o n n a c h i e f o r m a n y u s e f u l d i s c u s s i o n s and f o r s u p p l y i n g t h e C E R N p h a s e shifts.
ELECTROPRODUCTION OF THE P33 RESONANCE
593
R E FE R E N C E S [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
[19] [20]
N. Zagury, Phys. Rev. 145 (1967) 1112. S . L . A d l e r , Ann. of Phys. 50 (1968) 189. A. Donnachie and G.Shaw, Ann. of Phys. 37 (1966) 333. F . A . Berends, A. Donnachie and D . L . W e a v e r , Nucl. Phys. B4 (1967) 1, 55 and 103. D. Schwela, H. Rollnik, R.Weizel and W. Korth, Z. Phys. 202 (1967) 452; D. Schwela and R.Weizel, Z. Phys. 221 (1969) 71; D. Schwela, Z. Phys. 221 (1969) 158. C.W. Akerlof et al., Phys. Rev. 163 (1967) 1482. C.Mistretta et al., Phys. Rev. 184 (1969) 1487. K. Baba et al., Nuovo Cimento 59A (1969) 53. G.v. Gehlen, Nucl. Phys. B9 (1969) 17; B20 (1970) 102. K.M.Watson, Phys. Rev. 95 (1954) 228. H . F . J o n e s , Nuovo Cimento 40A (1965) 1018. J . S . B a l l , Phys. Rev. 124 (1961) 2014. G. Shaw, Nuovo Cimento 44A (1966) 1276. R. L. Crawford, Gauge invariant dispersion relations for pion electroproduetion, Glasgow University preprint, September 1970; submitted to Nucl. Phys. B. A. Donnachie, R. G. Kirsopp and C. Lovelace, Phys. Letters 26B (1968) 161. G. Mennesier, Nuovo Cimento 46A (1966) 459. J. Engels and W. Schmidt, Phys. Rev. 169 (1968) 1296. J. T. Beale, S.D. Ecklund and R . L . W a l k e r , California Institute of Technology report CTSL-42, 1966; G. F i s c h e r et al., Bonn contribution to the XIVth Int. Conf. on high-energy physics, Vienna, 1968; M. Croissiaux et al., Phys. Rev. 164 (1967) 1623; R.Morand et al., Phys. Rev. 180 (1969) 1299. R. G. Moorhouse and W. A. Rankin, Nucl. Phys. B23 (1970) 181. E. Lohrmann, Proc. of the Lund Int. Conf. on elementary particles, 1969, p. 11.
Nuclear Physics B28 (1971) 573-593. North-Holland Publishing Company
ELECTROPRODUCTION OF
THE P33 RESONANCE
R. L. CRAWFORD
Department of Natural Philosophy, University of Glasgow, Scotland Received 11 June 1970 (Revised manuscript received 23 December 1970) Abstract: The fixed t-dispersion relations for pion electroproduction are solved for energies below and in the region of the first resonance by making detailed use of the coupled multipole equations and Watson's theorem. The P33 multipoles, GE, n and F~r are evaluated by fitting to experimental data.
i. INTRODUCTION
Until r e c e n t l y , the e x p e r i m e n t a l d a t a f o r the e l e c t r o p r o d u c t i o n of pions in the r e g i o n of the N*(1236) r e s o n a n c e have c o n s i s t e d a l m o s t e n t i r e l y of m e a s u r e m e n t s of the t o t a l c r o s s s e c t i o n s , ~T and ~S, f r o m i n e l a s t i c e l e c t r o n - p r o t o n s c a t t e r i n g . Since t h e s e give i n f o r m a t i o n about only the g r o s s f e a t u r e s of the r e a c t i o n , d i s p e r s i o n c a l c u l a t i o n s f o r e l e c t r o p r o d u c t i o n [1,2] have tended to be c o n s i d e r a b l y l e s s d e t a i l e d than c o r r e s p o n d i n g s t u d i e s of p h o t o p r o d u c t i o n [3-5] w h e r e the e x p e r i m e n t a l r e s u l t s a r e m o r e demanding. It has been s u f f i c i e n t to r e t a i n only the P33 m u l t i p o l e s in the r e s c a t t e r i n g and to have only the Born t e r m s in the f o r c e t e r m s f o r t h e s e m u l t i p o l e s . In c o m p a r i s o n , an a d e q u a t e d e s c r i p t i o n of the p h o t o p r o d u c t i o n d a t a r e q u i r e s the e f f e c t s of the r e s c a t t e r i n g in o t h e r final s t a t e s and it is n e c e s s a r y to take a c c o u n t of the coupling between the m u l t i p o l e s . H o w e v e r , the i n f o r m a t i o n which i s now a v a i l a b l e f r o m c o i n c i d e n c e e x p e r i m e n t s [6-8], with the p r o m i s e of m o r e to c o m e , has a l t e r e d this c o n s i d e r a b l y and the e x p e r i m e n t a l s i t u a t i o n is m o v i n g c l o s e r to the one found in p h o t o p r o d u c t i o n . It should be noted that the s i m p l e e l e c t r o p r o d u c t i o n m o d e l s d e s c r i b e d above do not give a good fit to the d a t a f o r 7r+ p h o t o p r o duction n e a r the f o r w a r d d i r e c t i o n o r f o r 7r° p h o t o p r o d u c t i o n n e a r the b a c k w a r d d i r e c t i o n . Since t h e s e c o r r e s p o n d to the m o s t l i k e l y r e g i o n s f o r c o i n c i d e n c e m e a s u r e m e n t s of e l e c t r o p r o d u c t i o n , the t h e o r y ought to be r e f i n e d if the new e l e c t r o p r o d u c t i o n d a t a a r e to be u s e d p r o p e r l y . A move in this d i r e c t i o n has been m a d e by von Gehlen [9] who has given s o l u t i o n s f o r the i s o v e c t o r m u l t i p o l e s but without any d i r e c t c o n f r o n t a t i o n with data. In this p a p e r , the coupled i n t e g r a l e q u a t i o n s f o r the e l e c t r o p r o d u c t i o n m u l t i p o l e s a r e a p p l i e d in d e t a i l to fit to the new d a t a and to u s e it to e v a l u a t e the c h a r g e f o r m f a c t o r s of the pion and neutron. The equations a r e s o l v e d n u m e r i c a l l y and i n c l u d e all the n o n - n e g l i g i b l e r e s c a t t e r i n g which can
574
R.L. CRAWFORD
be calculated f r o m W a t s o n ' s t h e o r e m . All the couplings between the m u l t i pole equations a r e retained. It is shown in sects. 2 and 3 that this detail can be easily i n c o r p o r a t e d by choosing to w o r k with the Pault amplitudes. Sect. 3 also gives the details of the methods which a r e used to solve for the individual multipoles. In the next section, a p a r t i a l check on the methods employed is obtained by calculating the photoproduction amplitudes. Finally, in sect. 5, the details and r e s u l t s of the e l e c t r o p r o d u c t i o n calculation a r e p r e s e n t e d . The notation and conventions of Zagury [1] a r e used throughout this paper.
2. THE INTEGRAL EQUATIONS FOR THE M U L T I P O L E AND PAULI AMPLITUDES W a t s o n ' s t h e o r e m [10], using unitarity and t i m e r e v e r s a l i n v a r i a n c e , links the complex p h a s e s of the multipoles with the p h a s e shifts for elastic pion-nucleon s c a t t e r i n g . Although the r e l a t i o n s a r e exact only below the threshold for the production of two pions, it is p o s s i b l e to u s e the t h e o r e m at significantly higher e n e r g i e s provided the i n e l a s t i c i t i e s of the p h a s e shifts r e m a i n small. The i n f o r m a t i o n which is thus obtained is conveniently introduced into a fixed t - d i s p e r s i o n relation by p r o j e c t i n g to the coupled int e g r a l equations for the multipoles. This a p p r o a c h has a l r e a d y p r o v e d to be s u c c e s s f u l in photoproduction [3-5]. The techniques which a r e r e q u i r e d f o r the calculation of the multipole equations f o r e l e c t r o p r o d u c t i o n a r e s i m i l a r and have a l r e a d y been d e s c r i b e d [4,9]. It is usual to e x p r e s s the multipole equations in t e r m s of a set of a m p l i tudes, Ha(W, k 2) with a = 1,2, etc., which a r e r e l a t e d to the m u l t i p o l e s by s i m p l e k i n e m a t i c f a c t o r s that explicitly i m p o s e the known b e h a v i o u r of the m u l t i p o l e s at s m a l l values of k and q (ref. [11]). T h e s e f a c t o r s a p p e a r quite n a t u r a l l y during the p r o j e c t i o n and help to m i n i m i s e the coupling of the m u l t i p o l e s to t h e m s e l v e s . They will be defined l a t e r as r e q u i r e d . The multipole equations have the f o r m ReHa(W, X2 )
=Ha,i(W,)t2)+lP ~
dW'
Im Ha(W', k 2)
W'-W
(1)
m+Lt
The inhomogeneous t e r m ,
Ha, I,
Ha' I(W' k2) = Ha' B(W' k2)
1 m~+ ~ + ~-
can be written as
dW,~Kafi(W,W,,~.2)ImHr(W,,X2), r
(2)
w h e r e Ha, B is the Born t e r m and Kar t a r e n o n - s i n g u l a r k e r n e l s which give the coupling between the multipoles. In g e n e r a l , the s - and u - c h a n n e l cuts link all multipoles of the s a m e isospin type and the u - c h a n n e l cut also c o u p l e s the i s o v e c t o r m u l t i p o l e s with isospin indices of 1 and 3. At ~t2 = 0, in the photoproduction limit, the s c a l a r m u l t i p o l e s uncouple f r o m the t r a n s v e r s e multipoles and f r o m the production amplitude. Since the multipole equations c o m e f r o m fixed t - d i s p e r s i o n r e l a t i o n s , they r e q u i r e that the p a r t i a l wave expansions of the i m a g i n a r y p a r t s of the
ELECTROPRODUCTION OF THE P33 RESONANCE
575
i n v a r i a n t amplitudes c o n v e r g e at unphysical values of cos 0. As a r e s u l t , it is not possible to calculate the k e r n e l s in eq. (2), and hence Ha, I, for l a r g e values of W. F o r photoproduction, the limit is f o r m a l l y at 1340 MeV (ref. [12]) but it can be shown [13] that the weakness of the t - c h a n n e l s t r u c t u r e p e r m i t s the use of the multipole equations up to the second r e s o n a n c e r e gion. It has been argued that this limit is also valid in e l e c t r o p r o d u c t i o n for ~t2 > -200 fm -2 (ref. [9]). A detailed study of e l e c t r o p r o d u c t i o n in the region of the P33 r e s o n a n c e is considerably m o r e difficult than for photoproduction. Due to the s c a l a r multipoles, the n u m b e r of equations which have to be solved is a l m o s t doubled and t h e r e is a c o r r e s p o n d i n g i n c r e a s e in the complexity of the couplings between the multipoles. Also, the calculation has to be repeated for s e v e r a l values of X2 and it cannot be a s s u m e d , as in photoproduction, that it is always possible to i g n o r e all but a s m a l l n u m b e r of the couplings. T h e r e f o r e , even if the k e r n e l s in eq. (2) a r e obtained n u m e r i c a l l y , the r e m a i n d e r of the calculation is c u m b e r s o m e and p o s e s s e v e r a l technical difficulties. This is p a r t i c u l a r l y true since the P33 multipoles cannot be a c c u r a t e l y predicted and have to be evaluated by fitting to e x p e r i m e n t a l data. The calculation b e c o m e s significantly s i m p l e r by choosing to use the int e g r a l equations for the amplitudes, 5ri, in the Pauli decomposition of the T - m a t r i x . These equations a r e obtained d i r e c t l y f r o m the d i s p e r s i o n r e l a t-ions and it is r e l a t i v e l y simple to p r o j e c t n u m e r i c a l l y to the multipoles to use W a t s o n ' s t h e o r e m . In effect this a p p r o a c h can be used to solve s i m u l taneously all the r e q u i r e d multipole equations with a l a r g e n u m b e r of r e s c a t t e r i n g contributions and with the automatic inclusion of all the couplings. The p r o j e c t i o n of the equations for the Pauli amplitudes f r o m the d i s p e r s i o n r e l a t i o n s is s t r a i g h t f o r w a r d but lengthy [14]. T h e r e a r e , however, two points which r e q u i r e care. It is n e c e s s a r y to i m p o s e a subtraction in the d i s p e r s i o n relation for A 5 to e n s u r e that the c o n s t r a i n t s r e q u i r e d by gauge i n v a r i a n c e a r e satisfied. Secondly, the s c a l a r amplitudes must have additional subtraction t e r m s for W equal to infinity to give the c o r r e c t highenergy behaviour. This is i m p o r t a n t even for calculations at the 1'33 r e s o nance. The equations a r e conveniently given in t e r m s of Fi(W,z , ~t2) defined by 5rl(W , z, k 2) =
8--~4(E1 + m)(E 2 + rn) (W - m) FI(W , z,
k2),
5r2(W, z, ~.2) = ~ -1- ~ 4(E1 - m ) ( E 2
-m) (W+m)F2(W,z,k2),
5r3(W , z, ~.2) = ~ 1
,/(E 1 _ m)(E 2
-m) (E2+m)(W+m)F3(W,z,k2),
~:4(W,z,k2 ) = ~1
4(El+m)(E2+m )
(E 2
_m)(W_m)F4(W,z,~t2) ,
576
R.L. CRAWFORD
~:7(W,z,X2) = 8~1 ~/.(El+m)(E2 _m) (E 1 _m)F7(W,z,k2 ) , 5rs(W,z,X2 ) = ~ 1
~/(El_m)(E2+m)
m)F8(W,z,k2)
.
(3)
dW' ~/kij(W',W,z',k2)Im~(W',z',k2) ,
(4)
(El+
They have the f o r m
dW' imFi(W,,z,,X2) ReFi(W,z,X2) = Fi,B(W,z,k2 ) + l p ? m+~ W' -W +y1 ? rn+~ where
Fi, B
J
a r e the Born t e r m s ,
kij
a r e n o n - s i n g u l a r k e r n e l s and
z' =k ~ (k~q~ - koq o +kqz).
(5)
The p r i m e s indicate that the energy v a r i a b l e s have values c o r r e s p o n d i n g to W'.
3. SOLUTION OF THE INTEGRAL EQUATIONS The equations for the F - a m p l i t u d e s , and hence the multipoles, are solved by a simple i t e r a t i v e method. Suitable s t a r t i n g f o r m s for the i m a g i n a r y p a r t s of the multipoles a r e used to evaluate I m F i in the i n t e g r a l s in eq. (4) and ReF i a r e calculated for nodes in W and z. N u m e r i c a l integration o v e r z gives the c o r r e s p o n d i n g real p a r t s of the multipoles and these can be used with W a t s o n ' s t h e o r e m to give a new set of i m a g i n a r y p a r t s of the m u l t i poles for input to the next cycle. Alternatively, for multipoles which do not have sufficiently stable solutions for this method, it is easy to calculate Ha, I by using eq. (1) to c o m p a r e the input i m a g i n a r y p a r t with the resulting real part. The new i m a g i n a r y p a r t is obtained by solving the single m u l t i pole equation. The complete p r o c e s s is repeated until, after a few c y c l e s , the i m a g i n a r y p a r t s of the multipoles settle to suitably stable f o r m s . The final r e s u l t s , for c o m p a r i s o n with e x p e r i m e n t , contain Im Sri f r o m all the multipoles with non-negligible i m a g i n a r y p a r t s and Re ~ i a r e polynomials in z whose o r d e r is one less than the n u m b e r of nodes at which Re Fi(W , z , k 2) a r e calculated. The m e a s u r e d phase shifts for pion-nucleon s c a t t e r i n g [15] d e t e r m i n e the multipoles which must be included with the P33 p a r t i a l waves in the r e scattering. The l o w - e n e r g y p a r t s of the S-waves have been shown to be i m portant in photoproduction and a r e t h e r e f o r e d e s i r a b l e here. The possible effects of the r e s c a t t e r i n g at the P l l and D13 r e s o n a n c e s in the second r e sonance region r e q u i r e the inclusion in s o m e f o r m of the i m a g i n a r y p a r t s of these amplitudes. Finally, the P31 final state gives a s m a l l r e s c a t t e r i n g
ELECTROPRODUCTION OF THE P33 RESONANCE
577
Table 1 The multipoles which are included in the rescattering contributions to the dispersion integrals. Final pion-nucleon state SII
$31
Isovector multipoles S(1) E(1)
0+ 0+
Isoscalar multipoles S(0) E (0)
0+ 0+
S(3) E(3)
0+ 0+
1)33
.q(3) E(3) M(3) ~1+ 1+ 1+
D13
.q(1) E(1) M(1) -2- 2- 2-
S~0_)E(0) M(0) 2- 2-
c o r r e c t i o n . A l t o g e t h e r , this r e q u i r e s the solution of the i m a g i n a r y p a r t s of the multipoles listed in table 1. F o r the real p a r t s , it is n e c e s s a r y to calculate up to and including D-waves. In p r a c t i c e , for n u m e r i c a l c o n s i d e r a tions, Re F i a r e calculated at five nodes in z and t h e r e f o r e effectively contain up to G-waves. Since W a t s o n ' s t h e o r e m cannot be used at high e n e r g i e s , a cut-off is introduced into the i m a g i n a r y p a r t of each multipole at the energy where the elasticity of the c o r r e s p o n d i n g phase shift falls below 0.8. The good b e h a v ious of the solution at this point is e n s u r e d by r e p r e s e n t i n g the r e s c a t t e r i n g in the multipole at higher e n e r g i e s by a tail which is joined smoothly to the solution at the cut-off and which vanishes quickly as W goes to infinity. It is found that the r e s u l t s below and in the region of the P33 r e s o n a n c e a r e insensitive to the details of these tails. Due to this cut-off, the calculation contains only p a r t of the contributions f r o m the P l l and D13 r e s o n a n c e s . However, it is found in photoproduction that the inclusion of the P l l and D13 multipoles in this f o r m gives useful c o r r e c t i o n s and that the c o n s e q u e n c e s of omitting the full r e s o n a n c e c o n t r i bution a r e not s e r i o u s for W less than 1290 MeV. T h e r e is, of c o u r s e , the f r e e d o m to add bumps to the tails to r e p r e s e n t the r e m a i n d e r of these eff e c t s but, at p r e s e n t , the available e x p e r i m e n t a l data is not good enough even f o r a rough e s t i m a t e of their size. In p a r t i c u l a r , for the D13 m u l t i poles, it is n e c e s s a r y to m e a s u r e a total of six unknown and potentially r e sonant t e r m s . Except for the P33 multipoles, the h i g h - e n e r g y tails in the d i s p e r s i o n i n t e g r a l s imply that the complex p h a s e s of the multipoles go to zero as W goes to infinity. As a r e s u l t , the multipole equations have unique solutions [16] and the simple i t e r a t i v e method, using W a t s o n ' s t h e o r e m to give the i m a g i n a r y p a r t f r o m the real p a r t , is s a t i s f a c t o r y . In g e n e r a l , all these
578
R.L. CRAWFORD
m u l t i p o l e s c o n v e r g e r a p i d l y to s o l u t i o n s w h i c h a r e s t a b l e to b e t t e r than 1%. F o r the P33 m u l t i p o l e s , the c o m p l e x p h a s e is c l o s e to n at the c u t - o f f and it is well known that this c a u s e s d i f f i c u l t i e s in any a t t e m p t to get p a r a m e t e r - f r e e s o l u t i o n s f o r t h e s e a m p l i t u d e s . The p r o b l e m is c o m p o u n d e d by the inability of the p r o j e c t e d m u l t i p o l e e q u a t i o n s to give Ha, I at l a r g e v a l u e s of Wo It has to be a s s u m e d that both Ha, I and the c o m p l e x p h a s e , qSa(W), a r e known only f r o m the t h r e s h o l d at Wo to s o m e finite e n e r g y , W L. To r e f l e c t this, the P33 m u l t i p o l e s a r e s o l v e d by a d a p t i n g the m e t h o d of E n g e l s and S c h m i d t [17]. This is a m o d i f i e d O m n e s - M u s k h e l i s h v i l l i m e t h o d which a l l o w s a c l e a n s e p a r a t i o n of the known and unknown c o n t r i b u t i o n s to the solution. W a t s o n ' s t h e o r e m g i v e s the D - f u n c t i o n e x a c t l y
Da
(6)
Wo F o r Wo < W < WL, the solution to eq. (1) is g i v e n by ga(k 2)
Na(W' X2) = Ha(W' ~'2) Da(W) 1
=
Ha, I(W' x2) ReDa(W) + WL - W
WL
+ 7r(WL - W ) p f
(W' - WL )Ha, I(W', k 2) Im Da(W') dW'
Wo
1
?
7r(WL - W) WL
W v - W
(W' - WL) Da(W') Im Ha(W' , X2) dW'
W' - W
(7)
where ga(k 2) = lim WHa, I(W, ;t 2) W ~ oo
_1 ? dW' ImHa(W',k2). -Tr W o
(8)
(9)
Eq. (7) is e x a c t and it can be shown that the s o l u t i o n s which it g i v e s s a t i s f y lim --W'~Ha(W,X 2) = constant. W ~ oo
(10)
The f a c t o r of (W' - W L ) in the i n t e g r a l s in (7) p r e v e n t s any d i f f i c u l t i e s in a n u m e r i c a l c a l c u l a t i o n f r o m Da(W) which b e h a v e s like (W- WL)-~ba(WL)/lr c l o s e to W L. The s e c o n d and l a s t t e r m s in eq. (7) a r e the unknown p a r t s of the s o l u tion. It is a s s u m e d that t h e r e a r e no l a r g e r e s c a t t e r i n g c o n t r i b u t i o n s f r o m a s e c o n d P33 r e s o n a n c e a b o v e W L and Im Ha in the s e c o n d i n t e g r a l is a p -
ELECTROPRODUCTION
OF
THE
P33 RESONANCE
579
Table 2 The validity of the solutions for the P33 multipoles which are given by eq. (7) with ga as a free p a r a m e t e r is checked by comparing the values given by the solutions for the left-hand side of eq. (1), ReHa, and the right-hand side, H~. W (pion mass units)
gm =o Re Hm
gm =g'
Re H m - Hm
Re Hm
gm =2g'
Re Hm-H'm
Re Hm
Re Hm -H'm
7.720
0.102x10-1 0.153x10-5
0.137x10-1 0.096x10-5
0.172xi0- I
0.017xi0-5
8.098
0.820xi0-2 0.262x10-5
0.138x10- I 0.110x10-5
0.196x10-1
0.039x10-5
8.350
0.623x10-2 0.237x10-5
0.133x10-1 0.093x10-5
0.204x10-1
0.042x10-5
8.601
0.263x10-2 0.258x10-5
0.835x10-2 0.127x10-5
0.141x10-1
0.042x10-5
8.854
0.0
0.0
0.0
0.005x10 -5
9.106
0.326x10-3 0.296x10-5 -0.308x10-2 0.114x10-5 -0.648x10-2 -0.032x10-5
9.358
0.115x10-2 0.352x10-5 -0.287×10-2 0.149x10-5 -0.689x10-2 -0.028x10-5
9.735
0.190x10-2 0.488x10-5 -0.185x10-2 0.242x10-5 -0.561x10-2 -0.016x10-5
0.270x10 -5
0.i03x10 -5
10.114
0.232x10-2 0.872x10-5 -0.114x10-2 0.447x10-5 -0.460x10-2
0.018x10-5
10.492
0.259x10-2 2.960x10-5 -0.645x10-3 1.671x10-5 -0.388x10-2
0.388x10-5
3) 2 The three solutions , are for M('I+ at k = 0 and use the same H~ , I_ between W° and W LThe one with gG = g corresponds to the solutions for M~3+) in ref. [4].
p r o x i m a t e d by a t a i l w h i c h v a n i s h e s r a p i d l y a s W g o e s to i n f i n i t y . S i n c e ~ba(WL) i s c l o s e to ~, this i s n o t a c r i t i c a l p a r t of the c a l c u l a t i o n and a s i m p l e i t e r a t i v e m e t h o d c a n be u s e d to j o i n the t a i l s m o o t h l y to the s o l u t i o n f o r I m H a b e l o w WL. H o w e v e r , ga, the o t h e r u n k n o w n t e r m , d e t e r m i n e s the s i z e of the s o l u t i o n to the e x t e n t t h a t the m a i n e f f e c t of Ha, I i s to c o n t r o l o n l y the s h a p e of the m u l t i p o l e a c r o s s the r e s o n a n c e . S i n c e the p r o j e c t e d m u l t i p o l e e q u a t i o n s c a n n o t g i v e ga f r o m eq. (8), the P33 m u l t i p o l e s c a n n o t b e p r e d i c t e d and m u s t be o b t a i n e d f r o m e x p e r i m e n t a l d a t a by t r e a t i n g ga a s a free parameter. B e c a u s e of the k n o w n t h r e s h o l d b e h a v i o u r , eq. (7) i s u s e d to give s o l u tions for Hm(W'k2) =
1 1/-El + m k-qV ~
M(13)(W' ~'2) '
1 1 / / E - 1 + rn
E~3+)(W,;~2)
He(W'~'2) = kq VE2+m Hs(W, ~2 )
W ] E/~I + m
=
--.- ~ = . k..q
, ~2
+ ,,,
(3)
sl+(w,x
2 .
)
(Ii)
580
R.L. CRAWFORD
Table 2 shows examples of the validity of the v a r i o u s solutions which a r e given by eq. (7) for the s a m e Ha, I by v a r y i n g ga" The r e s u l t s a r e for M~.3) at k2 = 0 but a r e typical of all three multipoles at all values of ~t2. If, l+ as an additional check, eq. (9) is used to r e c a l c u l a t e ga from the solution for ImHa, it is found that the original value is r e p r o d u c e d to about 0.5%. These c h a r a c t e r i s t i c s appear to be independent of the r e s c a t t e r i n g above WL since they also apply if bumps a r e added to the tail for Im H a to r e p r e sent additional 1)33 r e s o n a n c e s . Clearly this technique gives v e r y s a t i s f a c t o r y solutions for the calculable l o w - e n e r g y p a r t s of the equations for the 1)33 multipoles. It shows that the projected multipole equations on their own fail completely to define unique solutions for these amplitudes. In p a r t i c u l a r , it d e m o n s t r a t e s that the v a r i ational approach can have no predictive power since, for example, all three solutions in table 2 would be v e r y acceptable end points for this type of calculation. The sensitivity of the solutions to ga is not s u r p r i s i n g since any attempt to impose a specific a s y m p t o t i c f o r m on the solution as in eq. (10) m u s t make the final r e s u l t c r i t i c a l l y dependent on the a s y m p t o t i c f o r m of Ha, I. It is t h e r e f o r e not possible to use A d l e r ' s c r i t e r i o n [2] of choosing the amplitude which d e c r e a s e s f a s t e s t as W goes to infinity in o r d e r to get a unique solution. To elaborate on this, the usual assumption that (~a(W) goes to ~ as W goes to infinity m e a n s that eq. (10) is sufficient to define a unique solution. The contribution f r o m the c i r c l e at infinity r e q u i r e d for the conventional O m n e s - M u s k h e l i s h v i l l i method [2] is then given by ga.
4. P H O T O P R O D U C T I O N
As in m o s t e l e c t r o p r o d u c t i o n calculations, the photoproduction a m p l i tudes a r e obtained by solving the e l e c t r o p r o d u c t i o n equations at ~t2 = 0. Since photoproduction has been studied s u c c e s s f u l l y and in detail using the multipole equations, the principal i n t e r e s t , in this c a s e , is to get a test of the p a r t i c u l a r methods that a r e used and to examine the consequences of omitting p a r t of the r e s c a t t e r i n g in the second r e s o n a n c e region. It is t h e r e f o r e sufficient to s u m m a r i s e the i m p o r t a n t r e s u l t s and to display a few typical c u r v e s (figs. 1 and 2). The two 1)33 multipoles w e r e evaluated by fitting to the ~o data [18]. In general, the solutions for the multipoles, the c r o s s sections, p o l a r i sations and a s y m m e t r y r a t i o s a g r e e well with those of B e r e n d s , Donnachie and W e a v e r [4]. The fits to experimental data for ~+ production n e a r the f o r w a r d direction and for 7r° production n e a r the backward direction a r e s a t i s f a c t o r y and give a hopeful p r o g n o s i s for the ability of this approach to d e s c r i b e e l e c t r o p r o d u c t i o n data f r o m coincidence experiments. It is found that the effects of the 1)11 r e s o n a n c e a r e probably s m a l l for proton t a r g e t s due to cancellation between the i s o s c a l a r and i s o v e c t o r amplitudes. This, however, will not r e m a i n true at all values of ~t2. It is possible that the lack of knowledge about this r e s o n a n c e in photoproduction [19] and in e l e c troproduction may be one of the main limitations for an e l e c t r o p r o d u c t i o n calculation. The effect of the D13 r e s o n a n c e can be e s t i m a t e d by adding ap-
ELECTROPRODUCTION OF THE P33 RESONANCE
581
d__~ 2O
IO
0
i 30
i 60
I 90
i 120
I 150
180
0 Fig. l(a). Differential cross sections for pion photoproduction at fixed energy ~p -* y+n at k L = 340 MeV. p r o p r i a t e B r e i t - W i g n e r t e r m s to the h i g h - e n e r g y t a i l s of the m u l t i p o l e s . The c o r r e c t i o n s a r e m o s t m a r k e d f o r ~+ p r o d u c t i o n a t s m a l l v a l u e s of O b u t a r e n o t i m p o r t a n t f o r W l e s s t h a n a b o u t 1290 MeV.
5. E L E C T R O P R O D U C T ~ N E l e c t r o p r o d u c t i o n e x p e r i m e n t s a r e a p o t e n t i a l s o u r c e of i n f o r m a t i o n a b o u t the p i o n and n u c l e o n e l e c t r o m a g n e t i c f o r m f a c t o r s b u t i t i s not p o s s i b l e at p r e s e n t to m e a s u r e a l l five t e r m s s i m u l t a n e o u s l y due to the l i m i t a t i o n s of p r e s e n t d a t a and b e c a u s e it i s n e c e s s a r y a t the s a m e t i m e to e v a l u a t e the P33 m u l t i p o l e s . F o r t u n a t e l y , the p r o t o n f o r m f a c t o r s a r e w e l l
582
R.L. CRAWFORD
30
20
"LL I0
o
I
I
I
I
I
30
6o
9o
12o
L5o
fSo
U
Fig. l(b). Differential cross sections for pion photoproduetion at fixed energy 7p ~ ~Op at k L = 340 MeV. known from elastic e l e c t r o n - p r o t o n s c a t t e r i n g and it is only n e c e s s a r y to c o n s i d e r the m e a s u r e m e n t of the r e m a i n d e r , GM, n, GE, n and F~. All t h r e e couple strongly to the ~+ production amplitude but, since they i n t e r f e r e badly, p r e s e n t experimental data does not allow their separation. GM n and F~ also contribute strongly to the inhomogeneous amplitude for M(13+) a~d hence, by virtue of the r e s c a t t e r i n g in the P33 final state, they have a considerable effect on the lr o production amplitude in spite of being absent f r o m the o Born t e r m s . However, as a l r e a d y d e s c r i b e d , they control the shape but not the size of the r e s o n a n c e in M(3~ 1' " This gives a very weak m e a s u r e ment which is susceptible to s m a l l e r r o r s in the n o n - r e s o n a n t multipoles and to the o m i s s i o n of h i g h - e n e r g y r e s c a t t e r i n g . T h e r e f o r e it is u n r e a l i s t i c to include m o r e than two f o r m f a c t o r s as f r e e p a r a m e t e r s in the calculation. The ones which are chosen a r e the l e a s t well known, that is GE, n and F~, and it is a s s u m e d that GE - , and satisfy the nucleon dipole f o r m and scaling law [20]. ' ~ G M ' p GM'n These two f o r m f a c t o r s and the three p a r a m e t e r s for the P33 multipoles are fitted to the e x p e r i m e n t a l data of M i s t r e t t a et al. [7]. The e x p e r i m e n t gives m e a s u r e m e n t s of the u+ and o differential c r o s s sections for proton t a r g e t s o v e r a range of values for W, X2, 0 and ~?, the angle between the electron s c a t t e r i n g plane and the hadron production plane. A notable f e a ture of the data is the c l e a r detection of the i n t e r f e r e n c e between the t r a n s -
E L E C T R O P R O D U C T I O N OF THE P33 RESONANCE
583
do" dn
t t
15
I0
I
I
200
i
I
300
I
400
kL (MeV) Fig. 2(a). D i f f e r e n t i a l c r o s s se~)Aons f o r pion photoproduction at fixed angle ~ p - - * ~+n at '0 = 30 °. doI
dn 15
I0
5
I
200
I'
300
I
I
400
I
EL(MeV )
F i g . 2(b). 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 pion photoproduction at fixed angle yp -~ ~Op at 0 = 150 °.
584
R.L. CRAWFORD
i 0 -'~
io,,~.
0 -I ~
12~I~-
,
~3~o
W
(M eV)
-2 -3
F i g . 3(a). M u l t i p o l e s o l u t i o n s at ),2 = ~1.2 f m - 2 . I s o s c a l a r . T h e s y s t e m of u n i t s i s o b t a i n e d b y a s s u m i n g t h a t c = /~ = g = 1.
xlO-a
,=
,
El.
"
13Mi" c:)o
L
W (MeV)
-!
F i g . 3(b). M u l t i p o l e s o l u t i o n s a t k 2 = - 1 . 2 f m - 2 . I s o v e c t o r f o r T = ½. T h e s y s t e m of u n i t s i s o b t a i n e d b y a s s u m i n g t h a t c = /~ = gt = 1.
ELECTROPRODUCTION OF THE P33 RESONANCE
585
x I 0 -a
2
I
El+ 0
-2 ~
~
~J
~
~ " " " 1 ='~" W
Ee*
Fig. 3(c). Multipole solutions at k2 = -1.2 fm -2. Isovector for T = 3. The system of units is obtained by assuming that c = ~ = ~ = 1. v e r s e and s c a l a r a m p l i t u d e s . T h i s g i v e s a c o n t r i b u t i o n to the d i f f e r e n t i a l c r o s s s e c t i o n w h i c h h a s t h e f o r m [7] ~/½e(~+ 1) s i n 0
cosT? S(W,
~, X2),
w h e r e • d e s c r i b e s the r a t i o of s c a l a r to t r a n s v e r s e p o l a r i s a t i o n f o r t h e v i r t u a l p h o t o n . S i n c e t h i s i s the o n l y t e r m w h i c h c o n t r i b u t e s l i k e cosT? to 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 , i t p r o v i d e s a s u i t a b l e l e v e r f o r s e p a r a t i n g the t r a n s v e r s e a n d s c a l a r p a r t s of t h e P33 r e s o n a n c e if t h e r e i s d a t a o v e r a s u f f i c i e n t l y l a r g e r a n g e of 7. T h e f i v e f r e e p a r a m e t e r s a r e d e t e r m i n e d f r o m a l e a s t - s q u a r e s fit. S i n c e a l a r g e p a r t of t h e p r o d u c t i o n a m p l i t u d e i s o b t a i n e d in a p a r a m e t e r f r e e m a n n e r , t h e n o r m a l i s a t i o n e r r o r s of t h e d a t a a r e i n c l u d e d q u a d r a t i c a l l y w i t h the s t a t i s t i c a l e r r o r s in the t o t a l e r r o r s w h i c h a r e u s e d to c a l c u l a t e X2. The P33 m u l t i p o l e s a r e c a l c u l a t e d f r o m t h e ~o c r o s s s e c t i o n s w h e r e t h e y c o n t r i b u t e m o s t s t r o n g l y a n d , by s i m i l a r r e a s o n i n g , GE, n and F ~ a r e o b t a i n e d f r o m the y+ d a t a . T h e two s e t s of p a r a m e t e r s a r e t r e a t e d a l t e r n a t e l y d u r i n g t h e m i n i m i s a t i o n and a t e a c h s t a g e the c u r r e n t v a l u e s of o n e s e t a r e u s e d in t h e f i t t i n g of t h e o t h e r . T h e d a t a g i v e f o u r s e p a r a t e c a l c u l a t i o n s ; f o r k 2 c l o s e to - 1 . 2 fm - 2 , - 3 . 4 f m - 2 , - 6 . 3 f m -2 and - 1 0 . 2 f m -2 r e s p e c t i v e l y . It i s n o t p o s s i b l e to u s e t h e ~+ d a t a a t - 1 5 . 2 8 f m -2 s i n c e t h e r e a r e no c o r r e s p o n d i n g ~o r e s u l t s . S i n c e t h e d a t a d e p e n d on f o u r v a r i a b l e s , W, O, k 2 and ~?, i t i s n o t e a s y in a f e w ~ d i a g r a m s (figs. 5 and 6) to g i v e an a c c u r a t e i m p r e s s i o n of t h e f i t w h i c h i s o b t a i n e d . T h e r e f o r e , a d d i t i o n a l d e t a i l s of t h e X2 f o r e a c h e n e r g y a r e
R.L. CRAWFORD
586
.io-' lOOxM~_ S I.
'HOO
'
12OO
'
W (MeV)
,~5o
-I
-2
J
-3
F i g . 4(a). M u l t i p o l e s o l u t i o n s at ~.2 = - 6 . 3 f m - 2 . I s o s c a l a r . T h e s y s t e m of u n i t s i s o b t a i n e d b y a s s u m i n g t h a t c = /i = ~ = 1.
j~z
IO~M.l~. 0 _ ~
S.+
.
~eV)
M+.
-I F i g . 4(b). M u l t i p o l e s o l u t i o n s at X2 = - 6 . 3 f m - 2 . I s o v e c t o r f o r T = ½. T h e s y s t e m of u n i t s i s o b t a i n e d b y a s s u m i n g t h a t c = /~ = ~ = 1.
E L E C T R O P R O D U C T I O N OF THE P33 RESONANCE
587
3 IO
I
o
_
,,,oo
,
,,2oo -
-I
~
\ d ~ "
, - -
~ '~
w
Eo+
Fig. 4(e). Multipole solutions at X2 = -6.3 fm -2. Isoveetor for T = ~. The system of units is obtained by assuming that c = h-= p. = 1. given in table 3. In g e n e r a l , the r e s u l t s a r e v e r y s a t i s f a c t o r y e x c e p t at -10.2 fm -2 w h e r e it is not p o s s i b l e to get a s t a b l e fit to the n + data. H o w e v e r , at this value of k2, it is p o s s i b l e to u s e data at only one value of W s i n c e the r e m a i n d e r is at an e n e r g y w h i c h is s u f f i c i e n t l y high f o r the c a l c u l a t i o n to be u n r e l i a b l e . It h a s to be e x c l u d e d s i n c e it r e p r e s e n t s a l a r g e p r o p o r t i o n of the total d a t a and can t h e r e f o r e d i s t o r t the final r e s u l t s badly if it is used. A l s o , at this k 2, the e x p e r i m e n t is c l o s e to its l i m i t s . T h e r e is t h e r e f o r e no r e a s o n to b e l i e v e that this f a i l u r e has m u c h s i g n i f i c a n c e . M~,3) is of c o u r s e the d o m i n a n t f e a t u r e of the r e a c t i o n and is t h e r e f o r e the n~+st a c c u r a t e l y m e a s u r e d of the P33 m u l t i p o l e s . S t a t i s t i c a l l y , it is d e t e r m i n e d to b e t t e r than one p e r c e n t and the 10% u n c e r t a i n t y in the n o r m a l i s a t i o n of the d a t a g i v e s a s y s t e m a t i c e r r o r that is t y p i c a l l y 6% o r 7g~ The s h a p e of the 7NN* f o r m f a c t o r , e s s e n t i a l l y that of f~/I~.3)/k, is not v e r y c l e a r x+ Its slope a p p e a r s to ly s e e n f r o m the s m a l l r a n g e of k 2 c o v e r e d by the data. be ::?P;
t hbld a ~ tdhutehtto ° ~ : h e : l l l : : : .
d T ~ : : : l u t m o n sE~:+r) itis inn°twellfigs.3 deter-and 4
have no g r e a t s i g n i f i c a n c e and i n d i c a t e m e r e l y that t h e r e is no n e e d to have a l a r g e E ! 3) and that it is p r o b a b l y negative. The s ~ + l a r - t r a n s v e r s e i n t e r f e r e n c e in the c r o s s s e c t i o n s g i v e s a c l e a r m e a s u r e m e n t of S(13+). The b e s t r e s u l t is at 6.3,fm "2 w h e r e the 307 u s a b l e d a t a p o i n t s b e t w e e n 1132 and 1284 MeV ~ i v e a s t a t i s t i c a l e r r o r w h i c h is l e s s than 50/0. The v a l u e s o b t a i n e d f o r S~.)/M~, 3) at r e s o n a n c e a r e r e s p e c * k 21+ tively -0.147, - 0 . 2 5 4 , -0.244 and -0.065~o~ equal to -1.2 fro- 2 , -3.4 f m - 2 , -6.3 f m -2 and -10.2 fm -2. The l a s t r e s u l t a p p e a r s to be a n o m a l o u s and m a y be r e f l e c t i n g the l a c k of d a t a at -10.2 f m - 2 . The s y s t e m a t i c u n c e r t a i n t y in S(3) . i ~. . .a. i I i i c u l t to e s t i m a t e in a r e a l i s t i c way s i n c e it is s e n s i t i v e to any e r I+ . f o r m the ~ dependence of the data. O b v i o u s l y , o t h e r i n d e p e n d e n t e x p e r i m e n t a l r e s u l t s a r e r e q u i r e d . The final r e s u l t a l s o d e p e n d s on the a c c u r a c y
R. L. CRAWFORD
588
Table 3 Details of the fits at different values of W. (a) 7r° production. W(MeV)
k L (MeV)
1197 1223
294 328
1226 1270
332 391
X2(fm -2)
Total X2
Number of data points
-1.22 -1.19
41.7 28.0
37 32
totals
69.7
69
-3.34 -3.27
50.5 65.8
55 58
totals
116.3
113
1132 1166 1200 1228 1259 1284
214 255 298 334 376 409
-6.54 -6.47 -6.41 -6.35 -6.29 -6.24
20.2 44.4 35.4 77.1 49.5 71.2
26 35 60 59 66 61
totals
297.8
307
1238
343
-10.37
39.5
26
(b) ~'+ production W (MeV)
k L (MeV)
1170 1197 1223
260 294 328
1185 1226 1270
279 332 391
1198 1281
296 407
~.2(fm-2)
Total X2
Number of data points
-1.24 -1.22 -1.19
26.4 16.7 16.3
21 19 17
totals
59.4
57
-3.40 -3.34 -3.27
19.4 24.9 51.5
18 22 26
totals
95.8
66
-6.41 -6.29
35.8 40.6
28 31
totals
76.4
59
of the s o l u t i o n s f o r SO+ and S 1_ and t h e s e can only b e i m p r o v e d when t h e r e i s a b e t t e r u n d e r s t a n d i n g of the s t r u c t u r e of the s e c o n d r e s o n a n c e r e g i o n . H o w e v e r , t h e s c a l a r - t r a n s v e r s e i n t e r f e r e n c e i s m a i n l y b e t w e e n M~.3Yand 1+ the s c a l a r m u l t i p o l e s and, a s h a s b e e n p o i n t e d o u t [7], a l l c o n t r i b u t i o n s , e x c e p t f r o m SI3) , c h a n g e sign at the r e s o n a n c e . A s a r e s u l t , t h e e n ere-v d e p e n d e n c e of t h e * i n t e r f e r e n c e p r o v i d e s a good t e s t of the s o l u t i o n f o r S! 3~. It can be s e e n f r o m t h e ×2 at d i f f e r e n t v a l u e s of W, e s p e c i a l l y f o r 42 e~+al to - 6 . 3 f m - 2 , t h a t a s a t i s f a c t o r y f it i s o b t a i n e d f o r a l a r g e r a n g e of e n e r g i e s and t h a t the i n t e r f e r e n c e i s w e l l d e s c r i b e d . In the f i t to t h e 7r+ d a t a , t h e r e i s c o n s i d e r a b l e i n t e r a c t i o n b e t w e e n t h e
E L E C T R O P R O D U C T T O N O F T H E P33 R E S O N A N C E
589
4O do" 30
20
J
IO
I
0
I
I
I
I
90
180
r~ F i g . 5(a). 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 ~.o e l e c t r o p r o d u c t i o n at f i x e d E, W, 0 and X2. W = 1198 M e V , k2 = - 6 . 4 1 f m - 2 a n d e = 125 ° .
dodn~ 30
20
I0
0
~
180
rl F i g . 5(5). 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 ~or ~.o e l e c t r o p r o d u c t i o n at f i x e d E, W, 0 and X2. W = 1260 M e V , ~t = - 6 . 2 9 f m - 2 and 0 = 1 3 5 ° .
590
R.L. CRAWFORD
dodfl= 30
20
JO
I
I
I 90
0
I
I 18o
rl F i g . 6(a). 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 ~ o r ?r+ e l e c t r o p r o d u c t i o n at f i x e d E, W, 0 a n d ) 2 . W = 1200 M e V , ~t~ -- - 6 . 4 7 f m - 2 a n d 8 = 25 ° .
3o
20
t
IO
o
9o
18o
+ F i g . 6(b). 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 ff e l e c t r o p r o d u c t i o n a t f i x e d E W, 0 a n d A2. W = 1259 MeV, A Z = - 6 . 2 9 f r o - 2 a n d e = 25 ° .
591
ELECTROPRODUCTION OF THE P33 RESONANCE Table 4 The g a p a r a m e t e r s for M (3) E (3) and S (3) 1+ '
k2 (fro-2)
gm x 102
1+
1+ "
ge x 103
gs x 102
-1.2
0.279±0.006
-0.347±0.092
-0.649±0.312
-3.4
0.228±0.003
-0.113±0.020
-1.401±0.084
-6.3
0.172±0.001
-0.048±0.013
-0.782±0.037
-10.2
0.158±0.004
-0.013±0.012
-0.215±0.082
Only the statistical errors are shown. I.O
0-8
O'6
t
O'4 O-2 I
i
I
i
i
2
4
6
8
iO
i
-~z ( f m - ~ )
Fig. 7. The form factor for the ~NN* coupling: 0)/k(WR, 0)] with WR=1236MeV.
[M~3)+(WR,X2)/k(WR,X2)I/[M~3)+(WR,
The solid curve is the nucleon dipole. values of F~ and GE, n. It is t h e r e f o r e c l e a r l y unwise to ignore GE, n and to set it to z e r o wheh m e a s u r i n g F~ f r o m e l e c t r o p r o d u c t i o n data. S i m i l a r l y , the uncertainty in GM, n limits the a c c u r a c y with which F~ can be found. The r e s u l t s for F~ (fig. 8) a r e not v e r y i n f o r m a t i v e but they appear to favour the nucleon dipole shape r a t h e r than the shape given by simple rho dominance. The r e s u l t s for GE, n (fig. 9) a r e encouraging. They show that it is s m a l l but n o n - z e r o and positive in the region 0 > k2 > -5 fm -2 and a r e in good a g r e e m e n t with the r e s u l t s f r o m e l e c t r o n - d e u t e r o n scattering. In conclusion, it can be claimed that d i s p e r s i o n theory has been shown to give an adequate d e s c r i p t i o n of p r e s e n t e x p e r i m e n t a l data at s m a l l values of k2 and that it is a p r a c t i c a l method for e x t r a c t i n g useful information f r o m the r e s u l t s of coincidence m e a s u r e m e n t s of electroproduction. It is to be hoped that additional suitable data will a p p e a r and that they will c o v e r both the
592
R . L . CRAWFORD
o-s
f
o
1 2
I 4
I 6
-k= (fm -2 )
Fig. 8. The pion form factor. Curves (A) and (B) a r e r e s p e c t i v e l y the predictions from simple rho dominance and the nucleon dipole form.
•E,n 0.06 o-o/~
0.02
t
i
U 1
i
I
2
3
L,
I
I
5
6
- X=(fro -=)
Fig. 9. The neutron charge form factor. The open c i r c l e s are the r e s u l t s of this calculation and the solid c i r c l e s a r e from electron-deuteron scattering [20]. f i r s t and s e c o n d r e s o n a n c e r e g i o n s . O b v i o u s l y , f r o m t h e r e s u l t s a b o v e , i m p r o v e d d a t a a r e r e q u i r e d to g i v e a b e t t e r d e t e r m i n a t i o n of F ~ . T h e d a t a f o r t h e s e c o n d r e s o n a n c e r e g i o n w i l l a l l o w a b e t t e r e s t i m a t e of t h e b a c k g r o u n d to the P33 r e s o n a n c e and w i l l t h e r e f o r e p e r m i t m o r e a c c u r a t e m e a s u r e m e n t s f r o m the l o w - e n e r g y d a t a . In p a r t i c u l a r , the v a l u e w h i c h i s o b t a i n e d f o r F~ i s s e n s i t i v e to t h e s u b t r a c t i o n c o r r e c t i o n s f o r t h e s c a l a r a m p l i t u d e s a n d , a s a r e s u l t , i t m a y b e s e n s i t i v e to h i g h - e n e r g y r e s c a t t e r i n g c o n t r i b u t i o n s . H o w e v e r , the i n t e r p r e t a t i o n of the s e c o n d r e s o n a n c e r e g i o n w i l l n o t b e e a s y d u e to t h e l a r g e n u m b e r of p o t e n t i a l l y r e s o n a n t m u l t i p o l e s w h i c h h a v e to b e c o n s i d e r e d . T h e a u t h o r w i s h e s to t h a n k P r o f e s s o r J. C. Gunn and P r o f e s s o r R. G. M o o r h o u s e f o r h e l p and e n c o u r a g e m e n t t h r o u g h o u t t h i s w o r k and P r o f e s s o r A. D o n n a c h i e f o r m a n y u s e f u l d i s c u s s i o n s and f o r s u p p l y i n g t h e C E R N p h a s e shifts.
ELECTROPRODUCTION OF THE P33 RESONANCE
593
R E FE R E N C E S [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
[19] [20]
N. Zagury, Phys. Rev. 145 (1967) 1112. S . L . A d l e r , Ann. of Phys. 50 (1968) 189. A. Donnachie and G.Shaw, Ann. of Phys. 37 (1966) 333. F . A . Berends, A. Donnachie and D . L . W e a v e r , Nucl. Phys. B4 (1967) 1, 55 and 103. D. Schwela, H. Rollnik, R.Weizel and W. Korth, Z. Phys. 202 (1967) 452; D. Schwela and R.Weizel, Z. Phys. 221 (1969) 71; D. Schwela, Z. Phys. 221 (1969) 158. C.W. Akerlof et al., Phys. Rev. 163 (1967) 1482. C.Mistretta et al., Phys. Rev. 184 (1969) 1487. K. Baba et al., Nuovo Cimento 59A (1969) 53. G.v. Gehlen, Nucl. Phys. B9 (1969) 17; B20 (1970) 102. K.M.Watson, Phys. Rev. 95 (1954) 228. H . F . J o n e s , Nuovo Cimento 40A (1965) 1018. J . S . B a l l , Phys. Rev. 124 (1961) 2014. G. Shaw, Nuovo Cimento 44A (1966) 1276. R. L. Crawford, Gauge invariant dispersion relations for pion electroproduetion, Glasgow University preprint, September 1970; submitted to Nucl. Phys. B. A. Donnachie, R. G. Kirsopp and C. Lovelace, Phys. Letters 26B (1968) 161. G. Mennesier, Nuovo Cimento 46A (1966) 459. J. Engels and W. Schmidt, Phys. Rev. 169 (1968) 1296. J. T. Beale, S.D. Ecklund and R . L . W a l k e r , California Institute of Technology report CTSL-42, 1966; G. F i s c h e r et al., Bonn contribution to the XIVth Int. Conf. on high-energy physics, Vienna, 1968; M. Croissiaux et al., Phys. Rev. 164 (1967) 1623; R.Morand et al., Phys. Rev. 180 (1969) 1299. R. G. Moorhouse and W. A. Rankin, Nucl. Phys. B23 (1970) 181. E. Lohrmann, Proc. of the Lund Int. Conf. on elementary particles, 1969, p. 11.