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Electromagnetic form factors in the bootstrap theory
~
Nuclear Physics B1 (1967) 301-308. North-Holland Publ. Comp., Amsterdam
ELECTROMAGNETIC FORM FACTORS IN THE B O O T S T R A P T H E O R Y M. NOGA
Laboratory of Physics. Komensky University, Bratislava. Czechoslovakia Received 30 January 1967
Abstract: It is shown that form factors calculated within the framework of the DashenFrautschi off-the-mass-shell bootstrap method have the same analytical properties as those prescribed by the S-matrix theory, i.e. only unitary cuts are present. The discontinuity calculated along these cuts satisfies the unitarity condition only when certain relations among various coupling constants are fulfilled. Using these relations, the ratios among the isovector magnetic moment of the nucleon, magnetic dipole of the A --*N+ ~ transition and isovector magnetic moment of the A-resonance are calculated. The results are the same as those followingfrom on the mass-shell bootstrap method and from the Lie group methods of the strong-coupling model.
1. INTRODUCTION Two y e a r s ago D a s h e n and F r a u t s c h i p r o p o s e d an o n - t h e - m a s s - s h e l l , Sm a t r i x m e t h o d f o r the c o m p u t a t i o n of s m a l l p e r t u r b a t i o n s to p r o c e s s e s i n v o l v i n g s t r o n g l y i n t e r a c t i n g p a r t i c l e s . T h e i r m e t h o d is b a s e d on the a n a l y t i c i t y , s e l f - c o n s i s t e n c y and b o o t s t r a p d y n a m i c s within the f r a m e w o r k of the N/D method. U s i n g t h i s m e t h o d it is p o s s i b l e to c a l c u l a t e the s c a t t e r i n g a m p l i t u d e 5T which c o u p l e s the c h a n n e l s which w e r e d e c o u p l e d b e f o r e i n t r o ducing the p e r t u r b a t i o n . P a r t i c l e s a r e d e s c r i b e d in the D a s h e n - F r a u t s c h i m e t h o d ( h e r e a f t e r c a l l e d DF) a s p o l e s of the s c a t t e r i n g a m p l i t u d e ST, and r e s i d u e s in t h e s e p o l e s a r e r e l a t e d to the weak o r e l e c t r o m a g n e t i c p r o p e r t i e s of the h a d r o n s i n c l u d i n g in t h i s way the e l e c t r o m a g n e t i c m a s s s p l i t t i n g , m a g n e t i c m o m e n t , weak d e c a y etc. [1]. R e c e n t l y , the s a m e a u t h o r s have e x t e n d e d t h e i r o r i g i n a l o n - t h e - m a s s s h e l l p r o c e s s e s [2]. This e x t e n s i o n of t h e i r m e t h o d to o f f - t h e - m a s s - s h e l l p r o c e s s e s e n a b l e s the c a l c u l a t i o n of the e l e c t r o m a g n e t i c and weak f o r m f a c t o r s of the h a d r o n s . Since the c a l c u l a t i o n s m a d e in the p r e s e n t p a p e r a r e b a s e d on the DF m e t h o d , we s h a l l b r i e f l y outline i t s b a s i c i d e a s . L e t us c o n s i d e r , f o r the s a k e of s i m p l i c i t y , the t w o - c h a n n e l p r o b l e m . T h e s e two c h a n n e l s a r e a s s u m e d to be c o u p l e d only by the e l e c t r o m a g n e t i c o r weak i n t e r a c t i o n s . The a m p l i t u d e 5T c o u p l i n g the a f o r e - m e n t i o n e d two c h a n n e l s is c o n s i d e r e d a s an a m p l i t u d e with one e x t e r n a l p a r t i c l e b e i n g off the m a s s s h e l l . The r e s i d u e
~)2
M. NOGA
[ 6T in a p o l e c o r r e s p o n d i n g to a p a r t i c l e i n the d i r e c t c h a n n e l i s t h e n e l a t e d in a s i m p l e way to the e l e c t r o m a g n e t i c o r weak f o r m f a c t o r . T h e y n a m i c a l e q u a t i o n f o r D F f o r m f a c t o r i s a r e l a t i o n e n a b l i n g the c a l c u l a t i o n [ the r e s i d u e i n the s c a t t e r i n g a m p l i t u d e 5T with f o u r e x t e r n a l p a r t i c l e s , ae p a r t i c l e of w h i c h i s of the m a s s s h e l l . T h i s p r o c e d u r e is t h e r e f o r e e r y d i f f e r e n t f r o m the u s u a l S - m a t r i x one, w h e r e the f o r m f a c t o r s a r e c a l u l a t e d f r o m the b e g i n n i n g a s q u a n t i t i e s r e l a t e d to a m p l i t u d e s with t h r e e e x ~rnal p a r t i c l e s . T h e S - m a t r i x m e t h o d is b a s e d on the a n a l y t i c i t y p r o p e r t i e s of the f o r m t c t o r (as a f u n c t i o n of the m a s s of the v i r t u a l p a r t i c l e ) a n d on the u n i t a r i t y o n d i t i o n f o r the f o r m f a c t o r . S i n c e the s t a r t i n g p o i n t of the D F m e t h o d d i f ~rs r a d i c a l l y f r o m that of the S - m a t r i x m e t h o d it i s not a p r i o r i o b v i o u s lat f o r m f a c t o r s c a l c u l a t e d f r o m the D F m e t h o d h a v e the a n a l y t i c a l p r o p e r e s a s s u m e d i n S - m a t r i x t h e o r y , a n d if it is t r u e , t h e r e a r i s e s , of c o u r s e , le s e c o n d q u e s t i o n w h e t h e r D F f o r m f a c t o r s f u l f i l l the u n i t a r i t y c o n d i t i o n . T h e m a i n p u r p o s e of the p r e s e n t p a p e r is to show t h a t the f o r m e r q u e s o n c a n b e a n s w e r e d in a n a f f i r m a t i v e way. P r o v i d e d t h a t c e r t a i n r e l a t i o n s mong the v a r i o u s c o u p l i n g c o n s t a n t s a r e f u l f i l l e d , the a n s w e r to the l a t t e r u e s t i o n i s the s a m e . In o u r c o n s i d e r a t i o n we r e s t r i c t o u r s e l v e s to the c a l c u l a t i o n of the e l e c : o m a g n e t i c f o r m f a c t o r of the n u c l e o n with one n u c l e o n off the m a s s s h e l l . o r the s a k e of s i m p l i c i t y the c a l c u l a t i o n s a r e p e r f o r m e d in the s t a t i c m o d e l , h e r e the whole p r o b l e m is m o r e t r a n s p a r e n t a n d c a n b e s i m p l y s o l v e d .
CA L C U L A TION To c a l c u l a t e the N ~ N + y 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 we s h a l l s t a r t :om D a s h e n - F r a u t s c h i e q u a t i o n f o r f o r m f a c t o r s [1, 2]. T h i s e q u a t i o n u p lied to the p r o c e s s N' + y - - N + ~ d e t e r m i n e s t h e r e s i d u e in the d i r e c t ~ a n n e l n u c l e o n p o l e of the N' + y ~ N + ~ p h o t o p r o d u c t i o n a m p l i t u d e 5 T in h i c h the i n i t i a l n u c l e o n N' is off the m a s s s h e l l with m a s s e q u a l to M + v V/is the n u c l e o n m a s s a n d v is a f i x e d p a r a m e t e r ) . In o u r c a s e , the a m p l i 1+ lde 6 T c o u p l e s two c h a n n e l s ~N a n d yN. L e t u s t a k e c h a n n e l one to b e J = ~ , 1 2 , yN. T h e p h o t o n c a n b e c o n s i d e r e d a s ~, ~N a n d c h a n n e l two to b e J = -*+ p a r t i c l e with i s o s p i n e i t h e r z e r o o r one t h u s e n a b l i n g to u s e the i s o s p i n ) r m a l i s m . In t h i s c a s e the D F e q u a t i o n f o r N ~ N + y f o r m f a c t o r is of the }rm -
F(v) = -
1 f
1 f D(w) I m 6 T ( w , v )
dw
(1)
D'(0) ~ L
h e r e F(v) i s i s o s c a l a r o r i s o v e c t o r m a g n e t i c m o m e n t f o r m f a c t o r of t h e u c l e o n with one n u c l e o n off the m a s s s h e l l , the q u a n t i t y f l 1 i s the r e d u c e d i o n - n u c l e o n c o u p l i n g c o n s t a n t [( f l ) 2 = 3 f 2 = 0.24], the f u n c t i o n D is the e n o m i n a t o r f u n c t i o n of the u s u a l N / D m e t h o d f o r the P l l a m p l i t u d e of ~N c a t t e r i n g . D(w) i s a s s u m e d to b e known. T h e d y n a m i c a l v a r i a b l e w i s the ion e n e r g y i n the p h o t o p r o d u c t i o n a m p l i t u d e ~ T(w, v) a n d t h e i n t e g r a t i o n i s
ELECTROMAGNETIC FORM FACTORS
303
p e r f o r m e d a l o n g all l e f t - h a n d c u t s due to f o r c e s of the a m p l i t u d e 5T(¢0, ~). To c a l c u l a t e F(~) we have to d e t e r m i n e Im 5T(~o, ~). A c c o r d i n g to the K r o l l R u d e r m a n t h e o r e m [3] w h i c h s a y s that at t h r e s h o l d the s i n g l e pion p h o t o p r o d u c t i o n a m p l i t u d e is given with a c c u r a c y of the o r d e r (pion m a s s ) / ( n u c l e o n m a s s ) by the s u m of the B o r n t e r m s , the f o r c e s in eq. (1) will be a p p r o x i m a t e d well at low e n e r g y , if we r e s t r i c t o u r s e l v e s to the e x c h a n g e of n u c l e o n and A - r e s o n a n c e , n e g l e c t i n g the s m a l l c o n t r i b u t i o n f r o m pion e x c h a n g e a s is c u s t o m a r y in the s t a t i c m o d e l . If we c o n s i d e r the i s o s p i n of the p h o t o n equal to one, the B o r n t e r m s of 5T a s s o c i a t e d with n u c l e o n and e x c h a n g e a r e of the f o r m 1 /Xvfl 16 /~*f~ b(w, r,) =-~ w - [, + Y co + w33 - ,
'
(2)
w h e r e /~v is the i s o v e c t o r m a g n e t i c m o m e n t of the nucleon, p* is the m a g n e t i c dipole of the A ~ N + y t r a n s i t i o n and f31 is the r e d u c e d ~ + N ~ A c o u p l i n g c o n s t a n t . T h e i n t e g r a l (1) can be e a s i l y p e r f o r m e d u s i n g the r e l a tion I m 6T(w, v) = I m b(~0, ~). A f t e r a s i m p l e c a l c u l a t i o n we can w r i t e f o r the i s o v e c t o r m a g n e t i c m o m e n t N ~ N + y f o r m f a c t o r (denoted by Fv(~)) the f o l l o w i n g e x p r e s s i o n 1 .
Fv(')
Pv .
~D,---7-~
16
D(v) .
/1"f31 D ( ' - w 3 3 )
.
,
+ 9 f~D,(0 )
(u)
u - w33
The d e n o m i n a t o r f u n c t i o n D(w) is the a n a l y t i c f u n c t i o n with u n i t a r y cut s t a r t ing at the point co = m (m is the pion m a s s ) and r u n n i n g to infinity. On this cut the u n i t a r i t y c o n d i t i o n is valid: Im D(w) = - p ( w ) g ( ~ v ) ,
(4)
w h e r e p(w) is a k i n e m a t i c a l f a c t o r [¢02 - m2]~ and N(w) is the n o m i n a t o r f u n c t i o n of the N / D m e t h o d . We s e e that the i s o v e c t o r m a g n e t i c m o m e n t f o r m f a c t o r Fv(V ) is the a n a l y t i c f u n c t i o n in the c o m p l e x u - p l a n e with cuts s t a r t i n g at the p o i n t s r, = m and v = m + w 3 3 . It should be noted that the b o o t s t r a p f o r m f a c t o r (3) h a s no p o l e s at v = 0 and at ~ = o)33 b e c a u s e the r e s i d u e s in t h e s e p s e u d o p o l e s a r e equal to z e r o . (D(0) = 0). T h i s r e s u l t is in c o m p l e t e a g r e e m e n t with the a n a l y t i c i t y p r o p e r t i e s p r e s c r i b e d by S - m a t r i x t h e o r y . F u r t h e r , we can e x p e c t t h a t the a f o r e - m e n t i o n e d b r a n c h p o i n t s a r e c o n n e c t e d with t h r e s h . olds of s c a t t e r i n g a m p l i t u d e s e n t e r i n g the u n i t a r i t y condition f o r the f o r m f a c t o r (fig. 1). The b r a n c h point of the f i r s t t e r m on the r i g h t - h a n d side of eq. (3) is due to the i n t e r m e d i a t e u + N ~ V + N s c a t t e r i n g p r o c e s s in the P l l w a v e (fig. la), while the s e c o n d t e r m of the s a m e e q u a t i o n c o n t a i n s the + A ~ V + N s c a t t e r i n g a m p l i t u d e (also in the P l l w a v e (fig. lb)). T h e s e two t e r m s a r e d e s i g n a t e d by FN(v) and FvA(V-o)33 ) r e s p e c t i v e l y . The s u p e r s c r i p t s a r e c o n n e c t e d with the b a r y o n s o c c u r r i n g in the i n t e r m e d i a t e s t a t e s , while the a r g u m e n t s ~ and v - ~ 3 3 a r e p i o n e n e r g i e s in the c o r r e s p o n d i n g p i o n - b a r y o n i n t e r m e d i a t e s t a t e s . Since we w o r k in the l o w e s t o r d e r of p e r t u r b a t i o n t h e o r y ( f o r c e s a r e t a k e n f r o m B o r n t e r m s ) the u n i t a r i t y condition
304
M. NOGA
f o r t h e F N ( v ) f o r m f a c t o r i s of t h e f o l l o w i n g f o r m Im FN(v)
= c N f ~ p(v)BN(v) ,
(5)
w h e r e C N i s a n o r m a l i z a t i o n c o n s t a n t , BN(v) i s t h e s u m of the B o r n t e r m s (the d i r e c t c h a n n e l n u c l e o n p o l e , n u c l e o n a n d A - r e s o n a n c e e x c h a n g e - p o l e s ) of t h e ~ + N - - ' 7 + N p h o t o p r o d u c t i o n a m p l i t u d e in P l l w a v e . T h e e x p r e s s i o n f o r t h e BN(u) i s g i v e n b y :
v
9
v
9 v+w33
It s h o u l d b e n o t e d t h a t in o u r c a s e ( s t a t i c m o d e l ) t h e s p i n i s c o n s e r v e d j u s t l i k e t h e i s o s p i n a n d we r e m i n d t h a t t h e p h o t o n i s c o n s i d e r e d a s a p a r t i c l e w i t h i s o s p i n e q u a l to one. T h e v a r i a b l e v p l a y s h e r e t h e r o l e of t h e p i o n e n e r g y a n d t h e B o r n a m p l i t u d e (6) i s on t h e m a s s s h e l l w h e r e a s t h e a m p l i t u d e 5T(c~,v) i s not. Now we i n v e s t i g a t e w h e t h e r t h e u n i t a r i t y c o n d i t i o n ( d i s c o n t i n u i t y c a l c u l a t e d a l o n g the cut) f o l l o w i n g f r o m t h e e x p r e s s i o n (3) i s c o m p a t i b l e w i t h t h e u n i t a r i t y c o n d i t i o n (5) w h i c h i s v a l i d f o r S - m a t r i x p e r t u r b a t i o n t h e o r y . U s i n g r e l a t i o n (4) we c a n w r i t e
(a}
(b) Fig. 1. D i a g r a m m a t i c r e p r e s e n t a t i o n of the electromagnetic form factor calculated within the D a s h e n - F r a u t s c h i method. The solid lines a r e nucleons, the double lines a r e the A - r e s o n a n c e s , the dashed lines a r e pions, and the wiggly lines r e p r e s e n t photons.
ELECTROMAGNETIC FORM FACTORS
1 try
N
ImFv(V) =
9 D'(0)
305
N(v) p(v) -~
(7)
for v > m. We d e t e r m i n e the n o m i n a t o r function N(v) of the ~N s c a t t e r i n g in the P l l p a r t i a l wave amplitude by the f o r c e s due to nucleon and A - r e s o n a n c e exchange. Using c r o s s i n g m a t r i c e s for spin and isospin in the static model [4] we get 1 (f:)2D(-•)
N(v)
-
9
v +•
16 ( f # ) 2 D ( - w 3 3 ) +
- -
9
v+w33
(8)
'
w h e r e we have slightly displaced the nucleon exchange pole f r o m v = 0 to make it different f r o m the d i r e c t - c h a n n e l nucleon pole. Taking the limit • ~ 0 and using the conditions D(0) = 0 and v > m we find out that only the second t e r m of eq. (8) contributes to relation (7). If we take the r e s u l t of the static b o o t s t r a p model [4],
(fll) 2 = 2(f#) 2 ,
(9)
we can r e w r i t e eq. (7) in the following f o r m N rv (.)
8 ~tvf~ .
= c N/ pc.) [- 9
- - +
84-2 Uv f l ] 9 - - v+w33 '
(10)
where C N - 1 D(-w33) 9 ¢o33D'(0 ) " As can be easily seen, the f o r m of the relation (10) is analogous to the unit a r i t y condition given by eq. (5). Now, we a r e in a position to d i s c u s s the b a s i c idea of the p r e s e n t paper. F o r m f a c t o r s calcultated by the DF method fulfill the unitarity condition only when the discontinuity given by eq. (10) is identical to that given by the unitarity condition eq. (5). This identity r e q u i r e s the e x p r e s s i o n in the s q u a r e b r a c k e t s of eq. (10) to be identical with the Born a p p r o x i m a t i o n given by eq. (6). This condition leads to the following relation ~v = ~r2 U* ,
ill)
which is c o n s i s t e n t with the experiment. This r a t i o is the s a m e as that obtained by D a s h e n and F r a u t s c h i [2] in t h e i r static b o o t s t r a p model with l i n e a r a p p r o x i m a t i o n of the D-function. V a r i o u s a p p r o x i m a t i o n s of D - f u n c t i o n s used in t h e DF method have been c r i t i c i z e d [5]. Our a p p r o a c h shows the p o s s i bility of avoiding at l e a s t p a r t i a l l y , this difficulty. The point is this: Our r e sult eq. i l l ) is e s s e n t i a l l y independent of the detailed b e h a v i o u r of the Dfunction and is only the consequence of the unitarity condition for the f o r m factor. If we r e d u c e the o f f - t h e - m a s s - s h e l l s e l f - c o n s i s t e n c y condition given by eq. (3), taking the limit v -~ 0, to an o n - t h e - m a s s - s h e l l one, we get the r e lation
306
M. NOGA 1
16 ~* f l
btv=~-~v--
~-
f~
D(_w33 ) 00330'(0 )
(12)
In o r d e r to fulfill the r e l a t i o n s (9), (11) and (12) at the s a m e t i m e , we m u s t a l l o w the d e n o m i n a t o r f u n c t i o n D(0)) to obtain the following r e l a t i o n D(-0)33) = -0)33 D'(0) .
(13)
T h e l a s t r e l a t i o n is u s e f u l f o r the c a l c u l a t i o n of the ± - r e s o n a n c e m a s s and is t r i v i a l l y fulfilled in the l i n e a r a p p r o x i m a t i o n of the D - f u n c t i o n a s it h a s b e e n u s e d by D a s h e n and F r a u t s c h i [2]. In a s i m i l a r way we c a n c a l c u l a t e the d i s c o n t i n u i t y of the rv~(V - 0)33) a l o n g the cut s t a r t i n g at v = m + 0)33, which is g e n e r a t e d by the ~ + A ~ N + y s c a t t e r i n g (fig. lb). T h i s d i s c o n t i n u i t y h a s to b e i d e n t i c a l with that g i v e n by the following u n i t a r i t y c o n d i t i o n Im F vA ( v - 0)33 ) =
CAf3p(v - 0)33)BA(v 0)33 ) ,
(14)
w h e r e C A is a n o r m a l i z a t i o n c o n s t a n t , f 3 is the r e d u c e d A + ~ --- N c o u p l i n g c o n s t a n t , p(w) is a known k i n e m a t i c a l f a c t o r , the v a r i a b l e v - 0)33 is the p i o n e n e r g y in the 7r + A i n t e r m e d i a t e s t a t e and BA(v - w33) r e p r e s e n t s the s u m of the B o r n t e r m s (nucleon d i r e c t - c h a n n e l p o l e , n u c l e o n and A - r e s o n a n c e e x c h a n g e - p o l e s ) in the P l l w a v e of the ~ + A ~ y + N p h o t o p r o d u c t i o n p r o c e s s . T h e e x p r e s s i o n of the B A(v - 0)33) is g i v e n by:
BA(v- 0)33 ) = - - -v
+ 9 v - 0)33
+ 9
v
(i5)
'
w h e r e ~* is the i s o v e c t o r m a g n e t i c m o m e n t of the A - r e s o n a n c e . It should be n o t e d t h a t in the c a l c u l a t i o n of the BA( V- 0)33) the r e l a t i o n b e t w e e n the m a g n e t i c d i p o l e s of the A --- N + y and N ~ A + y t r a n s i t i o n s , (A ~ N + y ) / (N--" A+ ~) = ½ haS b e e n u s e d . T h i s r e l a t i o n f o l l o w s f r o m s i m p l e k i n e m a t i c a l c o n s i d e r a t i o n s . In the s a m e way, f l and f 3 a r e k i n e m a t i c a l l y r e l a t e d b y the r e l a t i o n f l 3 = 2f31. Now l e t us ca'~culate the d i s c o n t i n u i t y of the F @ ( v - 0)33) following f r o m eq. (3). U s i n g r a t i o s of coupling c o n s t a n t s d e r i v e d a b o v e we c a n w r i t e
I m FA(V-v ¢°33) =
CAf31P(V-W33)I- f~ll2vv - - ~
8
f~u*
9 v-0)33
+ 190 U v f -v~ j
'
(16)
where CA =
8 D(-0)33) 9 w33D'(0)
"
S i m i l a r l y as in the a b o v e c a s e , the D F f o r m f a c t o r FvA(V- co33) o b e y s the unit a r i t y condition, if we r e q u i r e the d i s c o n t i n u i t y of the l~v~(V - w33) g i v e n by r e l a t i o n (16) to be i d e n t i c a l with the d i s c o n t i n u i t y g i v e n by the u n i t a r i t y c o n dition (14). It m e a n s t h a t the e x p r e s s i o n b e i n g in the s q u a r e b r a c k e t s of eq.
ELECTROMAGNETIC FORM FACTORS
307
(16) is identical with the Born a p p r o x i m a t i o n BA(u - w33 ) given by eq. (15). This a r g u m e n t leads to the ratio Pv = ~v ,
(17)
which is the s a m e as that obtained by Singh [6] using the L i e - g r o u p s t r u c t u r e f o r s t r o n g coupling model. It is a pity that we have no e x p e r i m e n t a l data on this quantity and thus we cannot c o m p a r e our p r e d i c t i o n with the experiment. We see that the u n i t a r i t y condition f o r DF f o r m f a c t o r r e q u i r e s c e r t a i n r e l a t i o n s between magnetic m o m e n t s of the nucleon and A - r e s o n a n c e to be fulfilled. In addition to those, t h e r e is a relation (13), which r e l a t e s the m a s s of the A - r e s o n a n c e to that of the nucleon. The DF f o r m f a c t o r can be r e p r e s e n t e d d i a g r a m m a t i c a l l y as shown in fig. 1. The a m p l i t u d e s f o r the p r o c e s s e s ~ + N ~ 7 + N and n + A - ~ + N, when they e n t e r the unitarity condition (eqs. (10) and (11)), a r e multiplied by weight f a c t o r s C N and CA r e s p e c t i v e l y . It is i n t e r e s t i n g to note that in the p r e s e n t a p p r o a c h weight f a c t o r s have definite values, while in an a p p r o a c h b a s e d on the u n i t a r i t y condition and the O m n e s - M u s k e l i s h v i l i equation within the f r a m e w o r k of the N/D method they cannot be d e t e r m i n e d . It is intuitively c l e a r that the weight f a c t o r s C N and C A d e t e r m i n e the weights of the v a r i o u s channels in the s t r u c t u r e of the c o r r e s p o n d i n g hadron.
3. CONCLUSION We can conclude that t h e r e is a quantitative evidence that e l e c t r o m a g n e t i c and weak f o r m f a c t o r s calculated by the m e a n s of the D a s h e n - F r a u t s c h i method have analytic p r o p e r t i e s , which a r e the s a m e as those p r e s c r i b e d by S - m a t r i x t h e o r y , i.e. only u n i t a r y cuts a r e p r e s e n t . The c a l c u l a t e d d i s continuity along t h e s e cuts is compatible with the unitarity condition, p r o vided that c e r t a i n r e l a t i o n s among m a s s e s and coupling constants a r e fulfilled. The a n a l y s i s of the discontinuity of the b o o t s t r a p f o r m f a c t o r s allows us to r e p r e s e n t t h e s e f o r m f a c t o r s d i a g r a m a t i c a l l y (fig. 1). T h e s e d i a g r a m s a r e the s a m e as t h o s e used in the S - m a t r i x p e r t u r b a t i o n method, but t h e r e is an i m p o r t a n t d i f f e r e n c e : The unitarity condition of the b o o t s t r a p f o r m f a c t o r s r e q u i r e s that the coupling constants o c c u r r i n g in v a r i o u s v e r t i c e s of the d i a g r a m a r e not independent in c o n t r a d i s t i n c t i o n to the usual S - m a t r i x p e r t u r b a t i o n method. In the p r e s e n t method we have obtained the r a t i o s among i s o v e c t o r m a g netic m o m e n t of the nucleon ~v, magnetic dipole of the A ~ 7 + N t r a n s i t i o n g* and i s o v e c t o r magnetic m o m e n t of the A - r e s o n a n c e p~ without any a r t i ficial a s s u m p t i o n like the l i n e a r a p p r o x i m a t i o n of the D - f u n c t i o n used in the p r e v i o u s on the m a s s shell b o o t s t r a p method [2] or infinite coupling constants in the s t r o n g - c o u p l i n g model [6]. The r e s u l t s of the p r e s e n t s i m p l e calculations show the c o n s i s t e n c y of the o f f - t h e - m a s s - s h e l l b o o t s t r a p d y n a m i c s p r o p o s e d by Dashen and F r a u t s c h i with s o m e p r i n c i p l e s of the S - m a t r i x theory. M o r e o v e r , we believe that the D a s h e n - F r a u t s c h i method m a y play a m o r e e s s e n t i a l role in the question of f o r m f a c t o r s , since the f o r m e r can be completed with the u n i t a r i t y condi-
308
M. NOGA
t i o n t h u s e n a b l i n g t o o b t a i n t h e n e w p h y s i c a l i n f o r m a t i o n s . It h a s b e e n p r o v e d t h a t t h e a n a l y s i s of s t r o n g f o r m f a c t o r s c a l c u l a t e d w i t h i n t h i s f r a m e w o r k p r e d i c t s t h e s e t s of h a d r o n s [7] w h i c h c o r r e s p o n d to i r r e d u c i b l e r e p r e s e n t a t i o n s of t h e L i e - g r o u p d e s c r i b i n g t h e s t r o n g - c o u p l i n g m o d e l [6, 8 ] . It i s s u r p r i s i n g t h a t t h e r a t i o s of t h e c o u p l i n g c o n s t a n t s a r e t h e s a m e in b o t h a p proaches, but mass formulae differ essentially. T h e a u t h o r w o u l d l i k e to t h a n k D r . M. Petr~i~ f o r d i s c u s s i o n . He i s a l s o g r a t e f u l t o D r . J. P i ~ g t f o r c r i t i c a l r e m a r k s to c e r t a i n d i f f i c u l t i e s e n c o u n t e r e d in a n e a r l i e r v e r s i o n of t h i s p a p e r .
REFERENCES [1] [2] [3] [4] [5] [6] [7] [8]
R. F. Dashen and S. C. F r a u t s c h i , Phys. Rev. 137 (1965) B1318. R. F. Dashen and S. C. F r a u t s c h i , Phys. Rev. 143 (1966} 1171. N.M. Kroll and Yl. A. Ruderman, Phys. Rev. 93 {1954} 233. G . F . C h e w , P h y s . R e v . L e t t e r s 9 (1962) 233; E . A b e r s , L. Balazs and Y . H a r a , Phys. Rev. 136 {1964) B1382. G.Shaw and D.Wong, Phys.Rev. 147 {1966) 1028. V.Singh, P h y s . R e v . 144 (1966} 1275. M. No~a, Nucl. Phys. B1 (1967} 85. V. Singh and B. M. Udgaonkar, Phys. Rev. 149 (1966} 1164.
Nuclear Physics B1 (1967) 301-308. North-Holland Publ. Comp., Amsterdam
ELECTROMAGNETIC FORM FACTORS IN THE B O O T S T R A P T H E O R Y M. NOGA
Laboratory of Physics. Komensky University, Bratislava. Czechoslovakia Received 30 January 1967
Abstract: It is shown that form factors calculated within the framework of the DashenFrautschi off-the-mass-shell bootstrap method have the same analytical properties as those prescribed by the S-matrix theory, i.e. only unitary cuts are present. The discontinuity calculated along these cuts satisfies the unitarity condition only when certain relations among various coupling constants are fulfilled. Using these relations, the ratios among the isovector magnetic moment of the nucleon, magnetic dipole of the A --*N+ ~ transition and isovector magnetic moment of the A-resonance are calculated. The results are the same as those followingfrom on the mass-shell bootstrap method and from the Lie group methods of the strong-coupling model.
1. INTRODUCTION Two y e a r s ago D a s h e n and F r a u t s c h i p r o p o s e d an o n - t h e - m a s s - s h e l l , Sm a t r i x m e t h o d f o r the c o m p u t a t i o n of s m a l l p e r t u r b a t i o n s to p r o c e s s e s i n v o l v i n g s t r o n g l y i n t e r a c t i n g p a r t i c l e s . T h e i r m e t h o d is b a s e d on the a n a l y t i c i t y , s e l f - c o n s i s t e n c y and b o o t s t r a p d y n a m i c s within the f r a m e w o r k of the N/D method. U s i n g t h i s m e t h o d it is p o s s i b l e to c a l c u l a t e the s c a t t e r i n g a m p l i t u d e 5T which c o u p l e s the c h a n n e l s which w e r e d e c o u p l e d b e f o r e i n t r o ducing the p e r t u r b a t i o n . P a r t i c l e s a r e d e s c r i b e d in the D a s h e n - F r a u t s c h i m e t h o d ( h e r e a f t e r c a l l e d DF) a s p o l e s of the s c a t t e r i n g a m p l i t u d e ST, and r e s i d u e s in t h e s e p o l e s a r e r e l a t e d to the weak o r e l e c t r o m a g n e t i c p r o p e r t i e s of the h a d r o n s i n c l u d i n g in t h i s way the e l e c t r o m a g n e t i c m a s s s p l i t t i n g , m a g n e t i c m o m e n t , weak d e c a y etc. [1]. R e c e n t l y , the s a m e a u t h o r s have e x t e n d e d t h e i r o r i g i n a l o n - t h e - m a s s s h e l l p r o c e s s e s [2]. This e x t e n s i o n of t h e i r m e t h o d to o f f - t h e - m a s s - s h e l l p r o c e s s e s e n a b l e s the c a l c u l a t i o n of the e l e c t r o m a g n e t i c and weak f o r m f a c t o r s of the h a d r o n s . Since the c a l c u l a t i o n s m a d e in the p r e s e n t p a p e r a r e b a s e d on the DF m e t h o d , we s h a l l b r i e f l y outline i t s b a s i c i d e a s . L e t us c o n s i d e r , f o r the s a k e of s i m p l i c i t y , the t w o - c h a n n e l p r o b l e m . T h e s e two c h a n n e l s a r e a s s u m e d to be c o u p l e d only by the e l e c t r o m a g n e t i c o r weak i n t e r a c t i o n s . The a m p l i t u d e 5T c o u p l i n g the a f o r e - m e n t i o n e d two c h a n n e l s is c o n s i d e r e d a s an a m p l i t u d e with one e x t e r n a l p a r t i c l e b e i n g off the m a s s s h e l l . The r e s i d u e
~)2
M. NOGA
[ 6T in a p o l e c o r r e s p o n d i n g to a p a r t i c l e i n the d i r e c t c h a n n e l i s t h e n e l a t e d in a s i m p l e way to the e l e c t r o m a g n e t i c o r weak f o r m f a c t o r . T h e y n a m i c a l e q u a t i o n f o r D F f o r m f a c t o r i s a r e l a t i o n e n a b l i n g the c a l c u l a t i o n [ the r e s i d u e i n the s c a t t e r i n g a m p l i t u d e 5T with f o u r e x t e r n a l p a r t i c l e s , ae p a r t i c l e of w h i c h i s of the m a s s s h e l l . T h i s p r o c e d u r e is t h e r e f o r e e r y d i f f e r e n t f r o m the u s u a l S - m a t r i x one, w h e r e the f o r m f a c t o r s a r e c a l u l a t e d f r o m the b e g i n n i n g a s q u a n t i t i e s r e l a t e d to a m p l i t u d e s with t h r e e e x ~rnal p a r t i c l e s . T h e S - m a t r i x m e t h o d is b a s e d on the a n a l y t i c i t y p r o p e r t i e s of the f o r m t c t o r (as a f u n c t i o n of the m a s s of the v i r t u a l p a r t i c l e ) a n d on the u n i t a r i t y o n d i t i o n f o r the f o r m f a c t o r . S i n c e the s t a r t i n g p o i n t of the D F m e t h o d d i f ~rs r a d i c a l l y f r o m that of the S - m a t r i x m e t h o d it i s not a p r i o r i o b v i o u s lat f o r m f a c t o r s c a l c u l a t e d f r o m the D F m e t h o d h a v e the a n a l y t i c a l p r o p e r e s a s s u m e d i n S - m a t r i x t h e o r y , a n d if it is t r u e , t h e r e a r i s e s , of c o u r s e , le s e c o n d q u e s t i o n w h e t h e r D F f o r m f a c t o r s f u l f i l l the u n i t a r i t y c o n d i t i o n . T h e m a i n p u r p o s e of the p r e s e n t p a p e r is to show t h a t the f o r m e r q u e s o n c a n b e a n s w e r e d in a n a f f i r m a t i v e way. P r o v i d e d t h a t c e r t a i n r e l a t i o n s mong the v a r i o u s c o u p l i n g c o n s t a n t s a r e f u l f i l l e d , the a n s w e r to the l a t t e r u e s t i o n i s the s a m e . In o u r c o n s i d e r a t i o n we r e s t r i c t o u r s e l v e s to the c a l c u l a t i o n of the e l e c : o m a g n e t i c f o r m f a c t o r of the n u c l e o n with one n u c l e o n off the m a s s s h e l l . o r the s a k e of s i m p l i c i t y the c a l c u l a t i o n s a r e p e r f o r m e d in the s t a t i c m o d e l , h e r e the whole p r o b l e m is m o r e t r a n s p a r e n t a n d c a n b e s i m p l y s o l v e d .
CA L C U L A TION To c a l c u l a t e the N ~ N + y 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 we s h a l l s t a r t :om D a s h e n - F r a u t s c h i e q u a t i o n f o r f o r m f a c t o r s [1, 2]. T h i s e q u a t i o n u p lied to the p r o c e s s N' + y - - N + ~ d e t e r m i n e s t h e r e s i d u e in the d i r e c t ~ a n n e l n u c l e o n p o l e of the N' + y ~ N + ~ p h o t o p r o d u c t i o n a m p l i t u d e 5 T in h i c h the i n i t i a l n u c l e o n N' is off the m a s s s h e l l with m a s s e q u a l to M + v V/is the n u c l e o n m a s s a n d v is a f i x e d p a r a m e t e r ) . In o u r c a s e , the a m p l i 1+ lde 6 T c o u p l e s two c h a n n e l s ~N a n d yN. L e t u s t a k e c h a n n e l one to b e J = ~ , 1 2 , yN. T h e p h o t o n c a n b e c o n s i d e r e d a s ~, ~N a n d c h a n n e l two to b e J = -*+ p a r t i c l e with i s o s p i n e i t h e r z e r o o r one t h u s e n a b l i n g to u s e the i s o s p i n ) r m a l i s m . In t h i s c a s e the D F e q u a t i o n f o r N ~ N + y f o r m f a c t o r is of the }rm -
F(v) = -
1 f
1 f D(w) I m 6 T ( w , v )
dw
(1)
D'(0) ~ L
h e r e F(v) i s i s o s c a l a r o r i s o v e c t o r m a g n e t i c m o m e n t f o r m f a c t o r of t h e u c l e o n with one n u c l e o n off the m a s s s h e l l , the q u a n t i t y f l 1 i s the r e d u c e d i o n - n u c l e o n c o u p l i n g c o n s t a n t [( f l ) 2 = 3 f 2 = 0.24], the f u n c t i o n D is the e n o m i n a t o r f u n c t i o n of the u s u a l N / D m e t h o d f o r the P l l a m p l i t u d e of ~N c a t t e r i n g . D(w) i s a s s u m e d to b e known. T h e d y n a m i c a l v a r i a b l e w i s the ion e n e r g y i n the p h o t o p r o d u c t i o n a m p l i t u d e ~ T(w, v) a n d t h e i n t e g r a t i o n i s
ELECTROMAGNETIC FORM FACTORS
303
p e r f o r m e d a l o n g all l e f t - h a n d c u t s due to f o r c e s of the a m p l i t u d e 5T(¢0, ~). To c a l c u l a t e F(~) we have to d e t e r m i n e Im 5T(~o, ~). A c c o r d i n g to the K r o l l R u d e r m a n t h e o r e m [3] w h i c h s a y s that at t h r e s h o l d the s i n g l e pion p h o t o p r o d u c t i o n a m p l i t u d e is given with a c c u r a c y of the o r d e r (pion m a s s ) / ( n u c l e o n m a s s ) by the s u m of the B o r n t e r m s , the f o r c e s in eq. (1) will be a p p r o x i m a t e d well at low e n e r g y , if we r e s t r i c t o u r s e l v e s to the e x c h a n g e of n u c l e o n and A - r e s o n a n c e , n e g l e c t i n g the s m a l l c o n t r i b u t i o n f r o m pion e x c h a n g e a s is c u s t o m a r y in the s t a t i c m o d e l . If we c o n s i d e r the i s o s p i n of the p h o t o n equal to one, the B o r n t e r m s of 5T a s s o c i a t e d with n u c l e o n and e x c h a n g e a r e of the f o r m 1 /Xvfl 16 /~*f~ b(w, r,) =-~ w - [, + Y co + w33 - ,
'
(2)
w h e r e /~v is the i s o v e c t o r m a g n e t i c m o m e n t of the nucleon, p* is the m a g n e t i c dipole of the A ~ N + y t r a n s i t i o n and f31 is the r e d u c e d ~ + N ~ A c o u p l i n g c o n s t a n t . T h e i n t e g r a l (1) can be e a s i l y p e r f o r m e d u s i n g the r e l a tion I m 6T(w, v) = I m b(~0, ~). A f t e r a s i m p l e c a l c u l a t i o n we can w r i t e f o r the i s o v e c t o r m a g n e t i c m o m e n t N ~ N + y f o r m f a c t o r (denoted by Fv(~)) the f o l l o w i n g e x p r e s s i o n 1 .
Fv(')
Pv .
~D,---7-~
16
D(v) .
/1"f31 D ( ' - w 3 3 )
.
,
+ 9 f~D,(0 )
(u)
u - w33
The d e n o m i n a t o r f u n c t i o n D(w) is the a n a l y t i c f u n c t i o n with u n i t a r y cut s t a r t ing at the point co = m (m is the pion m a s s ) and r u n n i n g to infinity. On this cut the u n i t a r i t y c o n d i t i o n is valid: Im D(w) = - p ( w ) g ( ~ v ) ,
(4)
w h e r e p(w) is a k i n e m a t i c a l f a c t o r [¢02 - m2]~ and N(w) is the n o m i n a t o r f u n c t i o n of the N / D m e t h o d . We s e e that the i s o v e c t o r m a g n e t i c m o m e n t f o r m f a c t o r Fv(V ) is the a n a l y t i c f u n c t i o n in the c o m p l e x u - p l a n e with cuts s t a r t i n g at the p o i n t s r, = m and v = m + w 3 3 . It should be noted that the b o o t s t r a p f o r m f a c t o r (3) h a s no p o l e s at v = 0 and at ~ = o)33 b e c a u s e the r e s i d u e s in t h e s e p s e u d o p o l e s a r e equal to z e r o . (D(0) = 0). T h i s r e s u l t is in c o m p l e t e a g r e e m e n t with the a n a l y t i c i t y p r o p e r t i e s p r e s c r i b e d by S - m a t r i x t h e o r y . F u r t h e r , we can e x p e c t t h a t the a f o r e - m e n t i o n e d b r a n c h p o i n t s a r e c o n n e c t e d with t h r e s h . olds of s c a t t e r i n g a m p l i t u d e s e n t e r i n g the u n i t a r i t y condition f o r the f o r m f a c t o r (fig. 1). The b r a n c h point of the f i r s t t e r m on the r i g h t - h a n d side of eq. (3) is due to the i n t e r m e d i a t e u + N ~ V + N s c a t t e r i n g p r o c e s s in the P l l w a v e (fig. la), while the s e c o n d t e r m of the s a m e e q u a t i o n c o n t a i n s the + A ~ V + N s c a t t e r i n g a m p l i t u d e (also in the P l l w a v e (fig. lb)). T h e s e two t e r m s a r e d e s i g n a t e d by FN(v) and FvA(V-o)33 ) r e s p e c t i v e l y . The s u p e r s c r i p t s a r e c o n n e c t e d with the b a r y o n s o c c u r r i n g in the i n t e r m e d i a t e s t a t e s , while the a r g u m e n t s ~ and v - ~ 3 3 a r e p i o n e n e r g i e s in the c o r r e s p o n d i n g p i o n - b a r y o n i n t e r m e d i a t e s t a t e s . Since we w o r k in the l o w e s t o r d e r of p e r t u r b a t i o n t h e o r y ( f o r c e s a r e t a k e n f r o m B o r n t e r m s ) the u n i t a r i t y condition
304
M. NOGA
f o r t h e F N ( v ) f o r m f a c t o r i s of t h e f o l l o w i n g f o r m Im FN(v)
= c N f ~ p(v)BN(v) ,
(5)
w h e r e C N i s a n o r m a l i z a t i o n c o n s t a n t , BN(v) i s t h e s u m of the B o r n t e r m s (the d i r e c t c h a n n e l n u c l e o n p o l e , n u c l e o n a n d A - r e s o n a n c e e x c h a n g e - p o l e s ) of t h e ~ + N - - ' 7 + N p h o t o p r o d u c t i o n a m p l i t u d e in P l l w a v e . T h e e x p r e s s i o n f o r t h e BN(u) i s g i v e n b y :
v
9
v
9 v+w33
It s h o u l d b e n o t e d t h a t in o u r c a s e ( s t a t i c m o d e l ) t h e s p i n i s c o n s e r v e d j u s t l i k e t h e i s o s p i n a n d we r e m i n d t h a t t h e p h o t o n i s c o n s i d e r e d a s a p a r t i c l e w i t h i s o s p i n e q u a l to one. T h e v a r i a b l e v p l a y s h e r e t h e r o l e of t h e p i o n e n e r g y a n d t h e B o r n a m p l i t u d e (6) i s on t h e m a s s s h e l l w h e r e a s t h e a m p l i t u d e 5T(c~,v) i s not. Now we i n v e s t i g a t e w h e t h e r t h e u n i t a r i t y c o n d i t i o n ( d i s c o n t i n u i t y c a l c u l a t e d a l o n g the cut) f o l l o w i n g f r o m t h e e x p r e s s i o n (3) i s c o m p a t i b l e w i t h t h e u n i t a r i t y c o n d i t i o n (5) w h i c h i s v a l i d f o r S - m a t r i x p e r t u r b a t i o n t h e o r y . U s i n g r e l a t i o n (4) we c a n w r i t e
(a}
(b) Fig. 1. D i a g r a m m a t i c r e p r e s e n t a t i o n of the electromagnetic form factor calculated within the D a s h e n - F r a u t s c h i method. The solid lines a r e nucleons, the double lines a r e the A - r e s o n a n c e s , the dashed lines a r e pions, and the wiggly lines r e p r e s e n t photons.
ELECTROMAGNETIC FORM FACTORS
1 try
N
ImFv(V) =
9 D'(0)
305
N(v) p(v) -~
(7)
for v > m. We d e t e r m i n e the n o m i n a t o r function N(v) of the ~N s c a t t e r i n g in the P l l p a r t i a l wave amplitude by the f o r c e s due to nucleon and A - r e s o n a n c e exchange. Using c r o s s i n g m a t r i c e s for spin and isospin in the static model [4] we get 1 (f:)2D(-•)
N(v)
-
9
v +•
16 ( f # ) 2 D ( - w 3 3 ) +
- -
9
v+w33
(8)
'
w h e r e we have slightly displaced the nucleon exchange pole f r o m v = 0 to make it different f r o m the d i r e c t - c h a n n e l nucleon pole. Taking the limit • ~ 0 and using the conditions D(0) = 0 and v > m we find out that only the second t e r m of eq. (8) contributes to relation (7). If we take the r e s u l t of the static b o o t s t r a p model [4],
(fll) 2 = 2(f#) 2 ,
(9)
we can r e w r i t e eq. (7) in the following f o r m N rv (.)
8 ~tvf~ .
= c N/ pc.) [- 9
- - +
84-2 Uv f l ] 9 - - v+w33 '
(10)
where C N - 1 D(-w33) 9 ¢o33D'(0 ) " As can be easily seen, the f o r m of the relation (10) is analogous to the unit a r i t y condition given by eq. (5). Now, we a r e in a position to d i s c u s s the b a s i c idea of the p r e s e n t paper. F o r m f a c t o r s calcultated by the DF method fulfill the unitarity condition only when the discontinuity given by eq. (10) is identical to that given by the unitarity condition eq. (5). This identity r e q u i r e s the e x p r e s s i o n in the s q u a r e b r a c k e t s of eq. (10) to be identical with the Born a p p r o x i m a t i o n given by eq. (6). This condition leads to the following relation ~v = ~r2 U* ,
ill)
which is c o n s i s t e n t with the experiment. This r a t i o is the s a m e as that obtained by D a s h e n and F r a u t s c h i [2] in t h e i r static b o o t s t r a p model with l i n e a r a p p r o x i m a t i o n of the D-function. V a r i o u s a p p r o x i m a t i o n s of D - f u n c t i o n s used in t h e DF method have been c r i t i c i z e d [5]. Our a p p r o a c h shows the p o s s i bility of avoiding at l e a s t p a r t i a l l y , this difficulty. The point is this: Our r e sult eq. i l l ) is e s s e n t i a l l y independent of the detailed b e h a v i o u r of the Dfunction and is only the consequence of the unitarity condition for the f o r m factor. If we r e d u c e the o f f - t h e - m a s s - s h e l l s e l f - c o n s i s t e n c y condition given by eq. (3), taking the limit v -~ 0, to an o n - t h e - m a s s - s h e l l one, we get the r e lation
306
M. NOGA 1
16 ~* f l
btv=~-~v--
~-
f~
D(_w33 ) 00330'(0 )
(12)
In o r d e r to fulfill the r e l a t i o n s (9), (11) and (12) at the s a m e t i m e , we m u s t a l l o w the d e n o m i n a t o r f u n c t i o n D(0)) to obtain the following r e l a t i o n D(-0)33) = -0)33 D'(0) .
(13)
T h e l a s t r e l a t i o n is u s e f u l f o r the c a l c u l a t i o n of the ± - r e s o n a n c e m a s s and is t r i v i a l l y fulfilled in the l i n e a r a p p r o x i m a t i o n of the D - f u n c t i o n a s it h a s b e e n u s e d by D a s h e n and F r a u t s c h i [2]. In a s i m i l a r way we c a n c a l c u l a t e the d i s c o n t i n u i t y of the rv~(V - 0)33) a l o n g the cut s t a r t i n g at v = m + 0)33, which is g e n e r a t e d by the ~ + A ~ N + y s c a t t e r i n g (fig. lb). T h i s d i s c o n t i n u i t y h a s to b e i d e n t i c a l with that g i v e n by the following u n i t a r i t y c o n d i t i o n Im F vA ( v - 0)33 ) =
CAf3p(v - 0)33)BA(v 0)33 ) ,
(14)
w h e r e C A is a n o r m a l i z a t i o n c o n s t a n t , f 3 is the r e d u c e d A + ~ --- N c o u p l i n g c o n s t a n t , p(w) is a known k i n e m a t i c a l f a c t o r , the v a r i a b l e v - 0)33 is the p i o n e n e r g y in the 7r + A i n t e r m e d i a t e s t a t e and BA(v - w33) r e p r e s e n t s the s u m of the B o r n t e r m s (nucleon d i r e c t - c h a n n e l p o l e , n u c l e o n and A - r e s o n a n c e e x c h a n g e - p o l e s ) in the P l l w a v e of the ~ + A ~ y + N p h o t o p r o d u c t i o n p r o c e s s . T h e e x p r e s s i o n of the B A(v - 0)33) is g i v e n by:
BA(v- 0)33 ) = - - -v
+ 9 v - 0)33
+ 9
v
(i5)
'
w h e r e ~* is the i s o v e c t o r m a g n e t i c m o m e n t of the A - r e s o n a n c e . It should be n o t e d t h a t in the c a l c u l a t i o n of the BA( V- 0)33) the r e l a t i o n b e t w e e n the m a g n e t i c d i p o l e s of the A --- N + y and N ~ A + y t r a n s i t i o n s , (A ~ N + y ) / (N--" A+ ~) = ½ haS b e e n u s e d . T h i s r e l a t i o n f o l l o w s f r o m s i m p l e k i n e m a t i c a l c o n s i d e r a t i o n s . In the s a m e way, f l and f 3 a r e k i n e m a t i c a l l y r e l a t e d b y the r e l a t i o n f l 3 = 2f31. Now l e t us ca'~culate the d i s c o n t i n u i t y of the F @ ( v - 0)33) following f r o m eq. (3). U s i n g r a t i o s of coupling c o n s t a n t s d e r i v e d a b o v e we c a n w r i t e
I m FA(V-v ¢°33) =
CAf31P(V-W33)I- f~ll2vv - - ~
8
f~u*
9 v-0)33
+ 190 U v f -v~ j
'
(16)
where CA =
8 D(-0)33) 9 w33D'(0)
"
S i m i l a r l y as in the a b o v e c a s e , the D F f o r m f a c t o r FvA(V- co33) o b e y s the unit a r i t y condition, if we r e q u i r e the d i s c o n t i n u i t y of the l~v~(V - w33) g i v e n by r e l a t i o n (16) to be i d e n t i c a l with the d i s c o n t i n u i t y g i v e n by the u n i t a r i t y c o n dition (14). It m e a n s t h a t the e x p r e s s i o n b e i n g in the s q u a r e b r a c k e t s of eq.
ELECTROMAGNETIC FORM FACTORS
307
(16) is identical with the Born a p p r o x i m a t i o n BA(u - w33 ) given by eq. (15). This a r g u m e n t leads to the ratio Pv = ~v ,
(17)
which is the s a m e as that obtained by Singh [6] using the L i e - g r o u p s t r u c t u r e f o r s t r o n g coupling model. It is a pity that we have no e x p e r i m e n t a l data on this quantity and thus we cannot c o m p a r e our p r e d i c t i o n with the experiment. We see that the u n i t a r i t y condition f o r DF f o r m f a c t o r r e q u i r e s c e r t a i n r e l a t i o n s between magnetic m o m e n t s of the nucleon and A - r e s o n a n c e to be fulfilled. In addition to those, t h e r e is a relation (13), which r e l a t e s the m a s s of the A - r e s o n a n c e to that of the nucleon. The DF f o r m f a c t o r can be r e p r e s e n t e d d i a g r a m m a t i c a l l y as shown in fig. 1. The a m p l i t u d e s f o r the p r o c e s s e s ~ + N ~ 7 + N and n + A - ~ + N, when they e n t e r the unitarity condition (eqs. (10) and (11)), a r e multiplied by weight f a c t o r s C N and CA r e s p e c t i v e l y . It is i n t e r e s t i n g to note that in the p r e s e n t a p p r o a c h weight f a c t o r s have definite values, while in an a p p r o a c h b a s e d on the u n i t a r i t y condition and the O m n e s - M u s k e l i s h v i l i equation within the f r a m e w o r k of the N/D method they cannot be d e t e r m i n e d . It is intuitively c l e a r that the weight f a c t o r s C N and C A d e t e r m i n e the weights of the v a r i o u s channels in the s t r u c t u r e of the c o r r e s p o n d i n g hadron.
3. CONCLUSION We can conclude that t h e r e is a quantitative evidence that e l e c t r o m a g n e t i c and weak f o r m f a c t o r s calculated by the m e a n s of the D a s h e n - F r a u t s c h i method have analytic p r o p e r t i e s , which a r e the s a m e as those p r e s c r i b e d by S - m a t r i x t h e o r y , i.e. only u n i t a r y cuts a r e p r e s e n t . The c a l c u l a t e d d i s continuity along t h e s e cuts is compatible with the unitarity condition, p r o vided that c e r t a i n r e l a t i o n s among m a s s e s and coupling constants a r e fulfilled. The a n a l y s i s of the discontinuity of the b o o t s t r a p f o r m f a c t o r s allows us to r e p r e s e n t t h e s e f o r m f a c t o r s d i a g r a m a t i c a l l y (fig. 1). T h e s e d i a g r a m s a r e the s a m e as t h o s e used in the S - m a t r i x p e r t u r b a t i o n method, but t h e r e is an i m p o r t a n t d i f f e r e n c e : The unitarity condition of the b o o t s t r a p f o r m f a c t o r s r e q u i r e s that the coupling constants o c c u r r i n g in v a r i o u s v e r t i c e s of the d i a g r a m a r e not independent in c o n t r a d i s t i n c t i o n to the usual S - m a t r i x p e r t u r b a t i o n method. In the p r e s e n t method we have obtained the r a t i o s among i s o v e c t o r m a g netic m o m e n t of the nucleon ~v, magnetic dipole of the A ~ 7 + N t r a n s i t i o n g* and i s o v e c t o r magnetic m o m e n t of the A - r e s o n a n c e p~ without any a r t i ficial a s s u m p t i o n like the l i n e a r a p p r o x i m a t i o n of the D - f u n c t i o n used in the p r e v i o u s on the m a s s shell b o o t s t r a p method [2] or infinite coupling constants in the s t r o n g - c o u p l i n g model [6]. The r e s u l t s of the p r e s e n t s i m p l e calculations show the c o n s i s t e n c y of the o f f - t h e - m a s s - s h e l l b o o t s t r a p d y n a m i c s p r o p o s e d by Dashen and F r a u t s c h i with s o m e p r i n c i p l e s of the S - m a t r i x theory. M o r e o v e r , we believe that the D a s h e n - F r a u t s c h i method m a y play a m o r e e s s e n t i a l role in the question of f o r m f a c t o r s , since the f o r m e r can be completed with the u n i t a r i t y condi-
308
M. NOGA
t i o n t h u s e n a b l i n g t o o b t a i n t h e n e w p h y s i c a l i n f o r m a t i o n s . It h a s b e e n p r o v e d t h a t t h e a n a l y s i s of s t r o n g f o r m f a c t o r s c a l c u l a t e d w i t h i n t h i s f r a m e w o r k p r e d i c t s t h e s e t s of h a d r o n s [7] w h i c h c o r r e s p o n d to i r r e d u c i b l e r e p r e s e n t a t i o n s of t h e L i e - g r o u p d e s c r i b i n g t h e s t r o n g - c o u p l i n g m o d e l [6, 8 ] . It i s s u r p r i s i n g t h a t t h e r a t i o s of t h e c o u p l i n g c o n s t a n t s a r e t h e s a m e in b o t h a p proaches, but mass formulae differ essentially. T h e a u t h o r w o u l d l i k e to t h a n k D r . M. Petr~i~ f o r d i s c u s s i o n . He i s a l s o g r a t e f u l t o D r . J. P i ~ g t f o r c r i t i c a l r e m a r k s to c e r t a i n d i f f i c u l t i e s e n c o u n t e r e d in a n e a r l i e r v e r s i o n of t h i s p a p e r .
REFERENCES [1] [2] [3] [4] [5] [6] [7] [8]
R. F. Dashen and S. C. F r a u t s c h i , Phys. Rev. 137 (1965) B1318. R. F. Dashen and S. C. F r a u t s c h i , Phys. Rev. 143 (1966} 1171. N.M. Kroll and Yl. A. Ruderman, Phys. Rev. 93 {1954} 233. G . F . C h e w , P h y s . R e v . L e t t e r s 9 (1962) 233; E . A b e r s , L. Balazs and Y . H a r a , Phys. Rev. 136 {1964) B1382. G.Shaw and D.Wong, Phys.Rev. 147 {1966) 1028. V.Singh, P h y s . R e v . 144 (1966} 1275. M. No~a, Nucl. Phys. B1 (1967} 85. V. Singh and B. M. Udgaonkar, Phys. Rev. 149 (1966} 1164.