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The analytic continuation of nucleon form factors
Nuclear P h y s m s B3 (1967) 95-105. North-Holland Publ. Comp., A m s t e r d a m
THE A N A L Y T I C C O N T I N U A T I O N OF N U C L E O N FORM FACTORS J. E. B O W C O C K , W. N. C O T T I N G H A M and J. G. W I L L I A M S
Department of Malhematical Physics. Universzty of Birmingham Received 7 August 1967
A b s t r a c t . A new reversion formula for the S t m l t j e ' s t r a n s f o r m is d e r i v e d and used for the analytic continuation of nucleon form factors. F o r given experimental a c c u r a c y the extent to which the s p e c t r a l functions may be determined is analysed and with p r e s e n t data it is c l e a r that only a very pooor resolution may be obtained. An e s h m a t e of the expemmental a c c u r a c y r e q u i r e d to r e s o l v e the rho contributmn is given. 1. I N T R O D U C T I O N O v e r t h e l a s t few y e a r s t h e r e h a v e b e e n s e v e r a l a t t e m p t s to e x t r a p o l a t e the 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 of t h e n u c l e o n s to t h e u n p h y s i c a l r e g i o n [1-8 I. A s a f u n c t i o n of m o m e n t u m - t r a n s f e r s q u a r e d , t, the f o r m f a c t o r s G(I) a r e a n a l y t i c in t h e c o m p l e x t - p l a n e e x c e p t f o r a r i g h t - h a n d b r a n c h c u t r u n n i n g f r o m s o m e p o s i t i v e t h r e s h o l d v a l u e 1o to i n f i n i t y . T h u s w e m a y w r i t e a Cauchy integral oO
G(0=~to t -t
'
o r a s u i t a b l y s u b t r a c t e d f o r m ; g(l) i s I m G(t) f o r t >I lo. G(t) i s d i r e c t l y a c c e s s i b l e to e x p e r i m e n t f o r _,o < l < 0 and h a s b e e n m e a s u r e d [9-16] o v e r a w i d e r a n g e of v a l u e s of t. On t h e o t h e r h a n d the s p e c t r a l f u n c h o n g ( 0 m a y o n l y b e c a l c u l a t e d f r o m s o m e t h e o r y of t h e i n t e r a c t i o n of t h e e l e c t r o m a g n e t i c f i e l d w i t h m e s o n s , a n d m e s o n - b a r y o n i n t e r a c t i o n s . T h e s i m p l e s t a s s u m p t i o n i s t h a t i t m a y be r e p r e s e n t e d a s the s u m of t h r e e d e l t a f u n c t i o n s p l a c e d a t the k n o w n p o s i t i o n s of t h e w - , qS-, p - r e s o n a n c e s , but a n y r e a l i s t i c t h e o r y m u s t a l s o t a k e i n t o a c c o u n t the f i n i t e w i d t h of the r h o a n d t h e c o n t i n u u m c o n t r i b u t i o n . A n u m b e r of d i f f e r e n t s p e c t r a l f u n c t i o n s h a v e b e e n s u g g e s t e d a l l of w h i c h g i v e a r e a s o n a b l e f i t to the p h y s i c a l f o r m f a c t o r d a t a [14, 17-21]. I t i s t h e r e f o r e of i n t e r e s t to s e e how w e l l p r e s e n t e x p e r i m e n t s c a n d i s hnguish between these spectral functions. Further, since it is known that an e x a c t k n o w l e d g e of G(t) s p e c i f i e s a u n i q u e s p e c t r a l f u n c t i o n i t i s d e s i r a b l e to find out w h a t e x p e r i m e n t a l a c c u r a c y i s r e q u i r e d to g i v e a s p e c i f i e d r e s o l u t i o n of t h e s p e c t r a l f u n c t i o n .
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J. BOWCOCK et al.
This may be done by the p r o c e s s of analytic continuation f r o m the p h y s ical r e g i o n onto the b r a n c h cut a c r o s s which the discontinuity is simply 2ig(t). Several a t t e m p t s [1-8] in this d i r e c t i o n have p r e v i o u s l y been made but these a r e l a r g e l y u n s a t i s f a c t o r y a s we shall a r g u e below. In what follows we d i s c u s s the g e n e r a l p r o b l e m of analytic continuation and the p r e v i o u s work in this field. We then d e r i v e an i n v e r s i o n f o r m u l a for eq. (1) and apply it to the nucleon f o r m - f a c t o r data. We also p r e s e n t a d i s cussion of the relation between e x p e r i m e n t a l a c c u r a c y in the physical r e g i o n and the r e s o l u t i o n which m a y be achieved on the s p e c t r a l function.
2. REMARKS ON ANALYTIC CONTINUATION In any attempt to analytically continue f r o m the finite set of e x p e r i m e n t a l points with e r r o r b a r s it m u s t be a s s u m e d that the f o r m f a c t o r is a ' s m o o t h ' function which p a s s e s through these points without rapid oscillations. This is entirely r e a s o n a b l e and could if n e c e s s a r y be f o r m u l a t e d m a t h e m a t i c a l l y in t e r m s of r e s t r i c t i o n s on the magnitude of the higher d e r i v a t i v e s of the f o r m factor. P r e v i o u s work by L e v i n g e r and c o - w o r k e r s [1-4] has been c o n c e r n e d with extrapolating the n u c l e a r f o r m f a c t o r s by m e a n s of a c o n f o r m a l mapping, 1
b - (1 - t / l o ) ~ b + (1
-
t/to)~
which takes the cut t - p l a n e onto the inside of the unit c i r c l e in the zT-plane. The physical region is along the r e a l axis -1 < ~ < ( b - 1)/(b+ 1) and b >~ 1 but o t h e r w i s e a r b i t r a r y . G(t(v)) may then be written as oo
anT?n
n=0 and the s p e c t r a l function is given by oo
g(~) = 1 [G(ei~) _ G(e-i~)] = ~ a n s i n n ~ , n=l
(2)
where ~ = exp i~ c o r r e s p o n d s to the boundary of the unit circle. The coefficients a n could be d e t e r m i n e d f r o m an exact knowledge of G0?). B e c a u s e of the inadequacy of the e x p e r i m e n t a l data the s e r i e s has to be truncated a f t e r five or six t e r m s , and the a n d e t e r m i n e d by a l e a s t s q u a r e s fit. The g(~) function is then obtained by substitution into eq. (2). Thus in effect, the a n a lytic continuation is made by the a s s u m p t i o n of a specific analytic form. However, the question a r i s e s a s to what is the relation between the s p e c t r a l function obtained in this way and the exact s p e c t r a l function i.e. between N co gNL~) = ~ an sin n~ n=l
and
g(~) = ~ a n sin n~ . n=l
NUCLEON FORM F A C T O R S
97
To the extent that the f i r s t few coefficients of the t r i g o n o m e t r i c a l s e r i e s as calculated by Levinger a r e equal to the F o u r i e r coefficients
an
1 7[ =~f g(4') sin
n4' d4'
-TT
we have, r e m e m b e r i n g t h a t g ( 4 ) is an odd function of 4, 1 1 N gff,4) =f "n FL~-~+-~~ -~
1 (sinn4 s i n n 4 ' + c o s n 4 c o s n 4 ' ) _ j g ( 4 ' ) d 4 '
n=l
7[
=f _~
sin [(N+ ½)(4 - 4')] =(~,~ d~' 2sin½(~-4') ....
(3)
The function KN(~ - 4') = sin [(N+ ½)(4 - 4')]/2 sin½(4 - 4') is shown in fig. 1 for N = 5, one of the values chosen by Levinger. The rho m a s s c o r r e s p o n d s to a value ~ = 104 ° and a width of 120 MeV c o r r e s p o n d s to the range 98 ° < ~ ~< 107 o. Since the width of the main peak of K5(4 - 4') is s e v e r a l t i m e s bigger than this i n t e r v a l , it is c e r t a i n l y not p o s s i b l e to r e s o l v e the rho in the s p e c t r a l function. This a g r e e s with the conclusions of Levinger. The function K 5 ( 4 - 4') also has the u n d e s i r a b l e p r o p e r t y that it is not positive defininite so that the calculated s p e c t r a l function is not simply a smoothed out v e r s i o n of the s p e c t r a l function but something which is the s p e c t r a l function weighted with an o s c i l l a t o r y factor. A s i m i l a r a n a l y s i s could probably be made on any method of analytic continuation which involves fitting the f o r m f a c t o r s by a t r u n c a t e d infinite s e r i e s of given functions each p o s s e s s i n g the c o r r e c t analytic s t r u c t u r e . However, in addition to the fact that the r e l a t i o n between the s p e c t r a l function thus c a l culated and the true s p e c t r a l function is not in g e n e r a l a simple one, the c a l culated s p e c t r a l function depends on the f o r m of the analytic functions used. ( C o m p a r e , for example the smoothed s p e c t r a l functions of Levinger and Wang with those of Orman. ) We shall d e r i v e below a f o r m a l i n v e r s i o n f o r m u l a for eq. (1) into which n u m e r i c a l v a l u e s of the f o r m f a c t o r may be i n s e r t e d so that no p a r t i c u l a r analytic f o r m need to a s s u m e d . The method is m a t h e m a t i c a l l y t r a n s p a r e n t so that we shall be able to investigate the reliability of the continuation and obtain a smoothed s p e c t r a l function which is m o r e c l o s e l y r e l a t e d to the true one.
gN(4)
3. THE INVERSION FORMULA Eq. (1) (for t < 0) is simply a Stielt,jes t r a n s f o r m and thereby p o s s e s e s a s t a n d a r d i n v e r s i o n in the f o r m of a double Laplace t r a n s f o r m i n v e r s i o n [22]. However, since the i n v e r s e Laplace t r a n s f o r m involves a contour integration in the complex plane this would r e q u i r e an analytic continuation of b e f o r e it can be p e r f o r m e d . We should like to have an i n v e r s i o n f o r mula of the type
G([)
98
J.E.BOWCOCK
et a l .
Ks(~-O
/ Fig. 1. The wexghtmg function KS(~ - ~') . 0
g(l') = f -
K(t',l)G(l) dl ,
(4)
o0
into w h i c h we m a y p l a c e d i r e c t l y t h e e x p e r i m e n t a l v a l u e s of G(l). A l t h o u g h o u r f i n a l a n s w e r c a n n o t be w r i t t e n in s u c h a s i m p l e f o r m it i s p o s s i b l e to g e t q u i t e c l o s e to it. Let us f i r s t change v a r i a b l e s
2t' - lo//0 = e o s h ~ ,
2t - I o / l o = - e o s h 0 ,
and d e f i n e
c,(0) = c / o , g'(~) =g(t') =-g(t')
f o r ¢5 > 0 for ~ 5 ( 0 .
We t h e n h a v e oo
1
f G'(O) = ~ j o
g'(~5)_sin_hdadd~ eosh~ + eoshO
'
G'(O) c a n in p r i n c i p l e be d e t e r m i n e d by e x p e r i m e n t f o r a l l r e a l O. N o t i n g t h a t G'(O) i s an e v e n f u n c t i o n of 0 we m a y t a k e the F o u r i e r t r a n s f o r m of t h i s e q u a t i o n to g i v e * • See footnote on next page.
NUCLEON FORM FACTORS
99
¢(3
G(p) = f
G'(O) e -ipO dO
-o0
=
2 f o
G'(O) dO
cosp0 co
=2f
co
g,((p)sinh (p d~b f
o
o
c o s p 0 dO c o s h ~b + c o s h 0
oo
= 2
f o
g ' @ ) s i n p $ dq~ _ - i~ (p) s i n h 7rp s i n h ~rp
(5)
where oo
~(p) = f
g'($) e -ip(p ddp .
_o0
T h u s we h a v e @(p) = s i n h p u G(p) .
(6)
T h i s i s a s i m p l e r e l a t i o n t e l l i n g u s t h e t r a n s f o r m of t h e s p e c t r a l f u n c t i o n in t e r m s of t h e t r a n s f o r m of t h e f o r m f a c t o r . W e m a y a l s o w r i t e t h e i n v e r s i o n of t h i s e q u a t i o n a s
2S
g'(4~) =~-
o
sinpq5 s i n h p n d p f o
c o s p 0 G'(O) d O .
(7)
Eq. (7) i s t h e f o r m u l a w e w e r e s e e k i n g . It i s n o t q u i t e of t h e f o r m of eq. (4) s i n c e i f w e t r y to i n t e g r a t e i n i t i a l l y o v e r p w e g e t a d i v e r g e n c e d u e to t h e s i n h (pTr) f a c t o r . T h i s i s a r e f l e c t i o n of t h e s i n g u l a r n a t u r e of t h e i n v e r s e of t h e S t i e l t j e s t r a n s f o r m . H o w e v e r , we m a y f e e d n u m e r i c a l d a t a f o r G'(O) i n t o this expression regardless of an specific analytic form. Further, since it is b a s e d on t h e s t a n d a r d t h e o r y o f F o u r i e r t r a n s f o r m s i t i s m o r e u s e f u l f o r analytic investigation. L e t u s c o n s i d e r w h a t w i l l h a p p e n w h e n we t r y t o a p p l y eq. (7). In o r d e r t h a t the s e c o n d i n t e g r a t i o n o v e r p m a y c o n v e r g e , G(p) m u s t d e c r e a s e a t l e a s t a s f a s t a s exp(-pTr). T h i s i s g u a r a n t e e d f o r t h e F o u r i e r t r a n s f o r m of t h e e x a c t G'(O) by the a s s u m p t i o n s a b o u t i t s a n a l y t i c s t r u c t u r e ( s e e eq. (1)). It i s u n l i k e l y t h a t a n y i n t e r p o l a t i o n of t h e e x p e r i m e n t a l p o i n t s w i l l p o s s e s s t h i s p r o p e r t y , a s w e s h a l l s e e in t h e n e x t s e c t i o n . It i s a l s o c l e a r t h a t a s p b e c o m e s l a r g e c o s p 0 w i l l o s c i l l a t e s e v e r a l t i m e s in t h e r e g i o n w h e r e G'(O) i s s i z e a b l e . T h e e x p e r i m e n t a l u n c e r t a i n t y in G ' ( 0 ) w i l l c a u s e t h e v a l u e s of * Footnote from preceding page. In o r d e r to justify our change of o r d e r of integration some r e s t m c t i v e assumptions about g(/) may be n e c e s s a r y . We shall not consider this point m detail because, as we shall see m sect. 4, our final analvsm avoids this step.
i00
J . E. B O W C O C K et a l .
G(p) to b e c o m e u n r e l i a b l e . Both t h e s e p o i n t s f o r c e u s to i n c l u d e s o m e kind of c u t - o f f in the p - i n t e g r a t i o n . L e t u s call the c u t - o f f f u n c t i o n c ( p ) and i n t r o d u c e it s i m p l y a s m u l t i p l y ing both s i d e s of eq. (6).
i~(p)~(p) = ~(p)d(p) ,
(8)
where /~(p)
=
sinh pTr C(p) .
We will only c o n s i d e r f u n c t i o n s ~(p) w h i c h d e c r e a s e s u f f i c i e n t l y r a p i d l y a s IPl ~ oo that/~(p) a l s o tends to z e r o . T h u s ~'(p) m u s t d e c r e a s e at l e a s t a s f a s t a s exp(-Tr]p]) f o r l a r g e IPl. T h i s h a s the c o n s e q u e n c e that oo
c(o) = ~1 f
U(p) eipO dp
-o0
is an a n a l y t i c f u n c t i o n of 0 in the s t r i p [ I m 0 t < ~r. A l s o , f o r r e a l 0, 0(3
D(O) = ~1 f
c(p) s i n h p ~ eipO dp
-OO
= ½[C(O -iv} - C(O+iTr)] . If we now i n v e r t eq. (8) u s i n g the c o n v o l u t i o n t h e o r e m we find ¢o
co
g,(~,)c(~ - 4,,1 d~' =71 f G'(O')D((p- 0')dO'
f -o0
-00 O0
= 2i- i f
G'(0 ') [C(q~- O' - i 7 r } - C ( 4 ) - 0 ' + i ~ - } ] d 0 '
--~O
= g's(~}.
(9}
T h e r e is an a l t e r n a t i v e way of d e r i v i n g eq. (9) which s h o w s c l e a r l y the n a t u r e of o u r a n a l y t i c continuation. This is by c o n s i d e r i n g the c h a n g e of v a r i a b l e - 2 t + t o = t o c o s h 0 a s a m a p p i n g of the s t r i p {Im 0 [ < 7r into the cut t - p l a n e . Since c o s h 0 is an even function of 0, +0 m a p into the s a m e point. The p h y s i c a l r e g i o n t < 0 then t r a n s f o r m s into the line I m 0 = 0. The u p p e r side of the cut in the t - p l a n e g o e s onto the l i n e s I m 0 = +Tr, Re 0 > 0 and I m 0 = - % Re 0 > 0. The l o w e r side of the cut g o e s onto the l i n e s I m 0 = -~, Re 0 > 0 and I m 0 = +Tr, Re 0 < 0. We t h e r e f o r e have oo
f
g'((p')C(q5 - qb') dq)'
--OO
+oo+#r
+oo-i7r
=f G'(O')C(ep-O'-iTr}dO'-f C'(O')C(O-O'+i~) dO'. -~+ iTr -~o-i~
NUCLEON FORM FACTORS
101
Eq. (9) i s t h e n o b t a i n e d b y d i s p l a c i n g t h e l i n e i n t e g r a l s f r o m t h e u p p e r a n d l o w e r e d g e s of t h e s t r i p onto the r e a l a x i s . It c a n a l s o b e s e e n t h a t w i t h a s u i t a b l e c h o i c e of t h e f u n c t i o n C , eq. (9) i s t r u e e v e n i f s e v e r a l s u b t r a c t i o n s a r e n e e d e d in t h e d i s p e r s i o n r e l a t i o n eq. (1). ( A l s o , t h i s r e m o v e s t h e n e c e s s i t y of i n v e s t i g a t i n g t h e v a l i d i t y of t h e c h a n g e of o r d e r of i n t e g r a t i o n in eq. (5).) F r o m eq. (9) w e s e e t h a t t h e s p e c t r a l f u n c t i o n i s d e t e r m i n e d o n l y to t h e e x t e n t of i t s i n t e g r a l w e i g h t e d w i t h a s m o o t h i n g f u n c t i o n C(~ - @'). To d e t e r m i n e g'(@) e x a c t l y w o u l d m e a n c h o o s i n g C(@ - @') to b e e q u a l to 5(@ - ~ ' ) . T h i s w o u l d l e a d b a c k d i r e c t l y to eq. (7) w h i c h w e h a v e a r g u e d a b o v e c a n n o t b e u s e d f o r t h e h i g h f r e q u e n c y c o m p o n e n t s . T h e s m o o t h i n g f u n c t i o n C(@ - @') w i l l p o s s e s s a w i d t h w h i c h w i l l b e i n v e r s e l y p r o p o r t i o n a l to t h e f r e q u e n c y a t w h i c h t h e c u t - o f f ~ ( p ) b e c o m e s e f f e c t i v e . T h u s the e x p e r i m e n t a l a c c u r a c y on G ' ( 0 ) w i l l d e t e r m i n e s o m e f r e q u e n c y to w h i c h ~ ( p ) m a y b e w e l l d e t e r m i n e d a n d t h i s in t u r n w i l l d e t e r m i n e the m i n i m u m w i d t h of C(@ - ¢b') a n d h e n c e how w e l l we m a y r e s o l v e t h e s p e c t r a l f u n c t i o n . One m a j o r a d v a n t a g e of eq. (9) i s t h a t the s m o o t h i n g f u n c t i o n m a y b e c h o s e n to b e a p o s i t i v e d e f i n i t e f u n c t i o n w i t h a s i m p l e p e a k a t ~ ' = ~, e . g . (1/a~Tr) e x p {-[(@ - @ ' ) / a ] 2 } , s o t h a t t h e i n t e r p r e t a t i o n of t h e s m o o t h e d s p e c t r a l f u n c t i o n i s q u i t e c l e a r . A f u r t h e r f e a t u r e i s t h a t i t e n a b l e s u s to e s t i m a t e the e x p e r i m e n t a l a c c u r a c y n e e d e d f o r a s p e c i f i e d f i n e n e s s of r e s o l u t i o n of t h e s p e c t r a l f u n c t i o n . B o t h t h e s e p o i n t s w i l l be a m p l i f i e d in t h e f o l lowing section.
4. A P P L I C A T I O N TO N U C L E O N F O R M F A C T O R D A T A W e s h a l l a p p l y o u r e x t r a p o l a t i o n p r o c e d u r e to t h e p r o t o n - m a g n e t i c f o r m f a c t o r , GmP , w h i c h i s t h e b e s t k n o w n e x p e r i m e n t a l l y . T h e e x p e r i m e n t a l i n f o r m a t i o n o n G m P i s s h o w n i n fig. 2 t o g e t h e r w i t h t h r e e t y p i c a l c u r v e s w h i c h w e h a v e d r a w n t h r o u g h t h e d a t a b a r s . T h e s e a r e r e p r e s e n t a t i v e of a l l the c u r v e s t h a t w e h a v e t r i e d . A t t h e p r e s e n t t i m e no e x p e r i m e n t a l p o i n t s e x i s t a b o v e 0 = 6.0. T h e e x p e r i m e n t a l i n d i c a t i o n i s t h a t G'(0) d e c r e a s e s a t l e a s t e x p o n e n t i a l l y a s a f u n c t i o n of 0 ( i n v e r s e l y p r o p o r t i o n a l to t). W e h a v e e x t r a p o l a t e d G'(p) to z e r o f o r 0 > 6.5. T h e s e n s i t i v i t y of t h e c a l c u l a t e d ~(p) to t h i s e x t r a p o l a t i o n p r o c e d u r e i s n e g l i g i b l e i n c o m p a r i s o n w i t h the o t h e r u n certainties. T h e ~ ( p ) c o r r e s p o n d i n g to e a c h of t h e s e c u r v e s w a s e v a l u a t e d n u m e r i c a l ly a n d m u l t i p l i e d by s i n h (pTr) to g i v e zg(p). T h e s e a r e p l o t t e d in fig. 3 a n d a s w e e x p e c t e d t h e r e i s c l o s e a g r e e m e n t b e t w e e n t h e c a r v e s up to s o m e v a l u e of p ( ~ 1.7) a b o v e w h i c h t h e c u r v e s d i f f e r c o n s i d e r a b l y . T h i s i s n o t o n l y t r u e f o r t h e t h r e e c u r v e s p l o t t e d b u t a l s o f o r a l l t h e c u r v e s w h i c h we t r i e d . T h i s s e e m s to b e t h e n a t u r a l l i m i t , d e t e r m i n e d by t h e m c c u r a c y of t h e e x p e r i m e n t a l d a t a , up to w h i c h w e m a y c a l c u l a t e t h e F o u r i e r t r a n s f o r m of t h e s p e c t r a l f u n c t i o n . If t h e d a t a w e r e g i v e n to b e t t e r a c c u r a c y t h i s w o u l d f u r t h e r r e s t r i c t t h e c u r v e s w e c o u l d d r a w f o r G'(0) a n d p u s h t h i s l i m i t to h i g h e r v a l u e s of p. W e m a y e s t i m a t e t h e r e s o l u t i o n o b t a i n a b l e on g'(@) f r o m the r e l a t i o n A p ~ ~ 1 w h i c h f o r Ap = 1.7 g i v e s Ad~ ~ 0.6. T h i s g i v e s u s t h e s m a l l e s t
102
J . E . BOWCOCK et al.
G'(e)
0
I
I
I
1
2
3
6
I
I
4
5
F i g . 2. T h r e e typical c u r v e s d r a w n t h r o u g h the e x p e r i m e n t a l points of the p r o t o n m a g n e t m f o r m f a c t o r . The e x p e r i m e n t a l points have b e e n taken f r o m r e f s . [9-16].
I /
20 ¸
(~(p)
I
/ /
I0
/ /
o
,~
~ \
~
"
J ~l; /
--10
F i g . 3. The values of g(p) c o r r e s p o n d i n g to the c u r v e s of fig. 2.
NUCLEON FORM FACTORS
103
width of the smoothing function C(@ - @'). It c o r r e s p o n d s to about t h r e e t i m e s the rho width at the position of the rho, s m a l l e r than this n e a r the t h r e s h o l d and i n c r e a s e s for l a r g e r v a l u e s of t. The fact that the r e s o l u t i o n which may be achieved v a r i e s with the position on the b r a n c h cut is simply a c o n s e q u e n c e of the n o n - l i n e a r i t y of cosh ~ i.e. cosh(~b + Aq~) - cosh(q~ - A~b) i n c r e a s e s with ~bfor a fixed A@. It is thus evident that any a t t e m p t to r e s o l v e a rho contribution to the s p e c t r a l function with a width of 120 MeV cannot succeed. This is in a g r e e m e n t with L e v i n g e r ' s findings. The b e s t we can do is to choose a s m o o t h ing function C(~b - ~'), with a width d e t e r m i n e d by the value of p w h e r e the c u r v e s f o r ~ ( p ) begin to d i v e r g e , and evaluate a smoothed s p e c t r a l function g~(~b) given by eq. (9). Let us choose C(~b- q>') = ~ e x p { -
-
'
whence
g;(q~) : ~
1 ~ G'(O')exp{ -~o
7r2-(
sin{ 27r(~b-0 a2
')} dO'
.
I n t e g r a t i n g n u m e r i c a l l y we get f r o m the t h r e e chosen c u r v e s of G'(O) the t h r e e smoothed s p e c t r a l functions shown in fig. 4. These a r e quite c l o s e to each other and have the g e n e r a l f e a t u r e of a b r o a d positive contribution spanning the m a s s region w h e r e the p, 00 and q~ r e s o n a n c e s would contribute.
oil 510 0
0
I
t
2OO I
(m~)
F i g . 4. The s m o o t h e d s p e c t r a l f u n c t i o n s f o r the p r o t o n m a g n e t m f o r m - t a c t o r of f~g. 2. A G a u s s m n s m o o t h i n g f u n c t m n h a s b e e n u s e d with c~ equal to 1.6.
104
J . E . BOWCOCK et al.
At h i g h e r m a s s e s the s m o o t h e d s p e c t r a l f u n c t i o n s b e c o m e n e g a t i v e a n d s i n c e o u r w e i g h t i n g f u n c t i o n i s p o s i t i v e t h i s i m p l i e s t h a t the t r u e s p e c t r a l f u n c t i o n becomes negative. T h e s e c u r v e s c o r r e s p o n d to a = 1.6, a r e s o l u t i o n of a b o u t s e v e n t i m e s the r h o width. It i s n o t p o s s i b l e to a c h i e v e the o p h m u m r e s o l u t i o n of t h r e e t i m e s the r h o width (a =0.6) with a G a u s s i a n c u t - o f f . T h i s i s b e c a u s e G(p) sinhTrp b e c o m e s so g r e a t for p > 1.7 ( s e e fig. 3) that the C-aussian d o e s n o t c u t off f a s t enough. S e v e r a l o t h e r p o s i t i v e - d e f i n i t e s m o o t h i n g f u n c t i o n s h a v e led u s to a s i m i l a r r e s o l u t i o n . It m i g h t be p o s s i b l e to find a s m o o t h i n g f u n c t i o n w h i c h would i m p r o v e t h i s r e s o l u t i o n s o m e w h a t b u t n o t of c o u r s e b e y o n d the l i m i t d i s c u s s e d . S i n c e we a r e so f a r f r o m r e s o l v i n g the a n t i c i p a t e d s t r u c t u r e of the s p e c t r a l f u n c t i o n we have n o t p u r s u e d t h i s p o i n t f u r t h e r . The s m o o t h e d s p e c t r a l f u n c t i o n s for the p r o t o n e l e c t r i c f o r m - f a c t o r a n d the n e u t r o n m a g n e t i c f o r m - f a c t o r a r e v e r y s i m i l a r to that of the p r o t o n m a g netic form factor.
5. R E L A T I O N B E T W E E N E X P E R I M E N T A L A C C U R A C Y AND THE R E S O L U T I O N O F THE S P E C T R A L F U N C T I O N We h a v e s e e n i n the l a s t s e c t i o n t h a t the a c c u r a c y of the p r e s e n t e x p e r i m e n t a l d a t a only a l l o w s u s to d e d u c e a v e r y s m o o t h e d out v e r s i o n of the s p e c t r a l f u n c t i o n . It i s of i n t e r e s t to e s t i m a t e what i m p r o v e m e n t s would be n e c e s s a r y to r e s o l v e a r e s o n a n c e of g i v e n width w h i c h we m i g h t a s s u m e to b e p r e s e n t i n the s p e c t r a l f u n c t i o n . In p a r t i c u l a r if we looked a t the m a g n e t i c i s o v e c t o r f o r m - f a c t o r w h e r e only one r e s o n a n c e , the r h o m e s o n , i s b e l i e v e d to c o n t m b u t e to the s p e c t r a l f u n c t i o n we c a n e s t i m a t e the a c c u r a c y r e q u i r e d to r e s o l v e s u c h a r e s o n a n t c o n t r i b u t i o n . It should be p o i n t e d out h o w e v e r , t h a t the p o s i t i o n a n d width of the r h o c o n t r i b u t i o n to the s p e c t r a l f u n c t i o n a r e not n e c e s s a r i l y t h o s e of the f r e e r h o m e s o n [24, 25]. H o w e v e r , l e t u s for s i m p l i c i t y a s s u m e they a r e , i.e. Ep = 750 MeV, Fp = 120 MeV. I n o r d e r to b e a b l e to s e e the s t r u c t u r e of s u c h a r e s o n a n c e by o u r m e t h o d of a n a l y t i c c o n t i n u a t i o n the s m o o t h i n g f u n c t i o n C(q5 - ~ ' ) i n eq. (9) m u s t h a v e a w i d t h of the o r d e r of the r h o - m e s o n width. In t e r m s of ~ , the p a r a m e t e r s of 1 the r h o m e s o n quoted in the p r e c e d i n g p a r a g r a p h a r e (~p = 3 . 2 , ~Fp = A~ =0.2. If we s i m p l y m a k e u s e of Ap. A~ ~ 1 th~s g i v e s Ap ~ 5. T h u s we m u s t be a b l e to c a l c u l a t e G(p) a c c u r a t e l y for a l l 0 ~ p ~ 5 if we w i s h to r e s o l v e the r h o c o n tmbution. In t e r m s of the e x p e r i m e n t a l a c c u r a c y r e q u i r e d f o r G'(O) we m a y n o t e that a L o r e n t z i a n s h a p e d r e s o n a n c e m the s p e c t r a l f u n c t i o n g i v e s r i s e to a g(P) = C e - ~ P . s m ap w h e r e ~ i s the width of the r e s o n a n c e , C i s a c o n s t a n t w h i c h for a r h o - d o m i n a n t s p e c t r a l f u n c t i o n i s [4] of o r d e r 10 a n d a i s the p o s i t i o n of the r e s o n a n c e . T h u s w h e n p ~ 5, ~(p) = 1 a n d s i n h (p~) ~ 106 . H e n c e ~,(p)
=
i g ( p ) / m n h pg ~ 10 -~ .
We m u s t t h e r e f o r e m e a s u r e the f o r m f a c t o r to a p p r o x i m a t e l y 1 p a r t i n
NUCLEON FORM FACTORS
105
106 if w e w i s h e d to e x t r a p o l a t e to the p o s i t i o n of the r h o m e s o n a n d d i s t i n g u i s h a p e a k with the w i d t h of the r h o . T h i s v e r y high a c c u r a c y r e q u i r e d i l l u s t r a t e s the s e n s i t i v e n a t u r e of a n a l y t i c c o n t i n u a t i o n a n d i s a l s o a r e f l e c t i o n of the f a c t the r e g i o n of the r h o m e s o n i s q u i t e f a r f r o m the p h y s i c a l region. If, h o w e v e r , we a r e i n t e r e s t e d i n d e t e r m i n i n g w h e t h e r o r n o t t h e r e i s a s i z e a b l e a m o u n t of s p e c t r a l f u n c t i o n i s a g i v e n m a s s i n t e r v a l the r e q u i r e m e n t s w i l l n o t b e so s t r i n g e n t . F o r e x a m p l e i t i s of i n t e r e s t to d i s c o v e r how l a r g e the s p e c t r a l f u n c t i o n i s i n the r e g i o n 4 < t < 16 w h e r e t h e r e a r e no r e s o n a n c e s . I n t h i s c a s e we m a y c h o o s e a b r o a d e r s m o o t h i n g f u n c t i o n a n d the c o r r e s p o n d i n g a n a l y s i s g i v e s a r e q u i r e d e x p e r i m e n t a l a c c u r a c y of ~ 1 % which is not u n r e a s o n a b l e . F r o m the p o i n t of v i e w of b e i n g a b l e to d i s t i n g u i s h b e t w e e n d i f f e r e n t s p e c t r a l f u n c t i o n s o r to d e d u c e the s p e c t r a l f u n c t i o n f o r the f o r m f a c t o r s i t i s t h e r e f o r e i m p o r t a n t that one should have v e r y a c c u r a t e e x p e r i m e n t s at some m o m e n t u m t r a n s f e r s . It i s m o r e v a l u a b l e to h a v e o n e e x p e r i m e n t a l p o i n t of high a c c u r a c y t h a n s e v e r a l p o i n t s of low a c c u r a c y . F o r the n e u t r o n f o r m fac. t o r s , w h i c h a r e d e d u c e d f r o m e l e c t r o n d e u t e r o n s c a t t e r i n g , the c o r r e c t i o n s m u s t a l s o b e k n o w n to the s a m e a c c u r a c y . O n e of u s (J. G. W. ) would l i k e to t h a n k the S c i e n t i f i c R e s e a r c h C o u n c i l f o r a m a i n t e n a n c e g r a n t d u r i n g the p e r i o d t h i s w o r k w a s u n d e r t a k e n .
REFERENCES [1] [21 [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]
J . S . L e v l n g e r a n d R . F . P e l e r l s . Phys. Rev. 134 (1964) B1314. J . S . L e v m g e r a n d C . P . W a n g , Phys.Rev. 136 (1964} B733. J . S . L e v i n g e r and C . P . W a n g . Phys.Rev. 138 {1965) B1207. J . S . L e v m g e r , preprlnt. Jan. 1967. G. L.Kane and R . A . Z d a m s . Phys. Rev. 151 (1966) 1239. F. Chllton and F. J. Uhrhane. Bull. Am. Phys. Soc. 11 (1966) 396, and preprints. B . O r m a n , Phys. Rev. 138 (1965) B1308. B . O r m a n , Phys. Rev. 145 (1966) 1140. K.W. Chen et al., Phys. Rev. 141 (1966) 1267. R.Hofstadter et al.. Phys.Rev. 142 (1966) 922. D.Benaksas et al., Phys. Rev. 141 (1966) 1308. W. Bartel et al., Phys.Rev. Letters 17 (1966) 608. P.Stem et al., Phys. Rev. Letters 16 (1966) 592. E.B.Hughes et al., Phys. Rev. 139 (1965) B458. E.B.Hughes et al., Phys. Rev. 146 (1966) 973. J . R . D u n n l n g e t al., Phys. Rev. 141 ~1966) 1286. L . H . C h a n et al., Phys.Rev. 141 (1966) 1298. R.Wllson, Springer T r a c t s m Modern Physms, 39 (1965). T . M a s s a m and A. Zichmhl Nuovo Clmento 43 (1966) 1137. S. Furulchl and K. Watanabe, PI og. Theor. Phys. 35 (1966) 174. N.G.Antomou and J. E, Bowcock, Phys. Rev.. to be pubhshed. Wldder. The Laplace T r a n s f o r m a t i o n (Princeton Umverslty Press) Tables of integral t r a n s f o r m s (Bateman Manuscmpt Project) Vol. 1, p. 36. J.S. Ball and D. Y. Wong, Phys. Rev. 130 (1963) 2112. M.W. Klrson. Phys. Rev. 132 (1963) 1249.
THE A N A L Y T I C C O N T I N U A T I O N OF N U C L E O N FORM FACTORS J. E. B O W C O C K , W. N. C O T T I N G H A M and J. G. W I L L I A M S
Department of Malhematical Physics. Universzty of Birmingham Received 7 August 1967
A b s t r a c t . A new reversion formula for the S t m l t j e ' s t r a n s f o r m is d e r i v e d and used for the analytic continuation of nucleon form factors. F o r given experimental a c c u r a c y the extent to which the s p e c t r a l functions may be determined is analysed and with p r e s e n t data it is c l e a r that only a very pooor resolution may be obtained. An e s h m a t e of the expemmental a c c u r a c y r e q u i r e d to r e s o l v e the rho contributmn is given. 1. I N T R O D U C T I O N O v e r t h e l a s t few y e a r s t h e r e h a v e b e e n s e v e r a l a t t e m p t s to e x t r a p o l a t e the 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 of t h e n u c l e o n s to t h e u n p h y s i c a l r e g i o n [1-8 I. A s a f u n c t i o n of m o m e n t u m - t r a n s f e r s q u a r e d , t, the f o r m f a c t o r s G(I) a r e a n a l y t i c in t h e c o m p l e x t - p l a n e e x c e p t f o r a r i g h t - h a n d b r a n c h c u t r u n n i n g f r o m s o m e p o s i t i v e t h r e s h o l d v a l u e 1o to i n f i n i t y . T h u s w e m a y w r i t e a Cauchy integral oO
G(0=~to t -t
'
o r a s u i t a b l y s u b t r a c t e d f o r m ; g(l) i s I m G(t) f o r t >I lo. G(t) i s d i r e c t l y a c c e s s i b l e to e x p e r i m e n t f o r _,o < l < 0 and h a s b e e n m e a s u r e d [9-16] o v e r a w i d e r a n g e of v a l u e s of t. On t h e o t h e r h a n d the s p e c t r a l f u n c h o n g ( 0 m a y o n l y b e c a l c u l a t e d f r o m s o m e t h e o r y of t h e i n t e r a c t i o n of t h e e l e c t r o m a g n e t i c f i e l d w i t h m e s o n s , a n d m e s o n - b a r y o n i n t e r a c t i o n s . T h e s i m p l e s t a s s u m p t i o n i s t h a t i t m a y be r e p r e s e n t e d a s the s u m of t h r e e d e l t a f u n c t i o n s p l a c e d a t the k n o w n p o s i t i o n s of t h e w - , qS-, p - r e s o n a n c e s , but a n y r e a l i s t i c t h e o r y m u s t a l s o t a k e i n t o a c c o u n t the f i n i t e w i d t h of the r h o a n d t h e c o n t i n u u m c o n t r i b u t i o n . A n u m b e r of d i f f e r e n t s p e c t r a l f u n c t i o n s h a v e b e e n s u g g e s t e d a l l of w h i c h g i v e a r e a s o n a b l e f i t to the p h y s i c a l f o r m f a c t o r d a t a [14, 17-21]. I t i s t h e r e f o r e of i n t e r e s t to s e e how w e l l p r e s e n t e x p e r i m e n t s c a n d i s hnguish between these spectral functions. Further, since it is known that an e x a c t k n o w l e d g e of G(t) s p e c i f i e s a u n i q u e s p e c t r a l f u n c t i o n i t i s d e s i r a b l e to find out w h a t e x p e r i m e n t a l a c c u r a c y i s r e q u i r e d to g i v e a s p e c i f i e d r e s o l u t i o n of t h e s p e c t r a l f u n c t i o n .
96
J. BOWCOCK et al.
This may be done by the p r o c e s s of analytic continuation f r o m the p h y s ical r e g i o n onto the b r a n c h cut a c r o s s which the discontinuity is simply 2ig(t). Several a t t e m p t s [1-8] in this d i r e c t i o n have p r e v i o u s l y been made but these a r e l a r g e l y u n s a t i s f a c t o r y a s we shall a r g u e below. In what follows we d i s c u s s the g e n e r a l p r o b l e m of analytic continuation and the p r e v i o u s work in this field. We then d e r i v e an i n v e r s i o n f o r m u l a for eq. (1) and apply it to the nucleon f o r m - f a c t o r data. We also p r e s e n t a d i s cussion of the relation between e x p e r i m e n t a l a c c u r a c y in the physical r e g i o n and the r e s o l u t i o n which m a y be achieved on the s p e c t r a l function.
2. REMARKS ON ANALYTIC CONTINUATION In any attempt to analytically continue f r o m the finite set of e x p e r i m e n t a l points with e r r o r b a r s it m u s t be a s s u m e d that the f o r m f a c t o r is a ' s m o o t h ' function which p a s s e s through these points without rapid oscillations. This is entirely r e a s o n a b l e and could if n e c e s s a r y be f o r m u l a t e d m a t h e m a t i c a l l y in t e r m s of r e s t r i c t i o n s on the magnitude of the higher d e r i v a t i v e s of the f o r m factor. P r e v i o u s work by L e v i n g e r and c o - w o r k e r s [1-4] has been c o n c e r n e d with extrapolating the n u c l e a r f o r m f a c t o r s by m e a n s of a c o n f o r m a l mapping, 1
b - (1 - t / l o ) ~ b + (1
-
t/to)~
which takes the cut t - p l a n e onto the inside of the unit c i r c l e in the zT-plane. The physical region is along the r e a l axis -1 < ~ < ( b - 1)/(b+ 1) and b >~ 1 but o t h e r w i s e a r b i t r a r y . G(t(v)) may then be written as oo
anT?n
n=0 and the s p e c t r a l function is given by oo
g(~) = 1 [G(ei~) _ G(e-i~)] = ~ a n s i n n ~ , n=l
(2)
where ~ = exp i~ c o r r e s p o n d s to the boundary of the unit circle. The coefficients a n could be d e t e r m i n e d f r o m an exact knowledge of G0?). B e c a u s e of the inadequacy of the e x p e r i m e n t a l data the s e r i e s has to be truncated a f t e r five or six t e r m s , and the a n d e t e r m i n e d by a l e a s t s q u a r e s fit. The g(~) function is then obtained by substitution into eq. (2). Thus in effect, the a n a lytic continuation is made by the a s s u m p t i o n of a specific analytic form. However, the question a r i s e s a s to what is the relation between the s p e c t r a l function obtained in this way and the exact s p e c t r a l function i.e. between N co gNL~) = ~ an sin n~ n=l
and
g(~) = ~ a n sin n~ . n=l
NUCLEON FORM F A C T O R S
97
To the extent that the f i r s t few coefficients of the t r i g o n o m e t r i c a l s e r i e s as calculated by Levinger a r e equal to the F o u r i e r coefficients
an
1 7[ =~f g(4') sin
n4' d4'
-TT
we have, r e m e m b e r i n g t h a t g ( 4 ) is an odd function of 4, 1 1 N gff,4) =f "n FL~-~+-~~ -~
1 (sinn4 s i n n 4 ' + c o s n 4 c o s n 4 ' ) _ j g ( 4 ' ) d 4 '
n=l
7[
=f _~
sin [(N+ ½)(4 - 4')] =(~,~ d~' 2sin½(~-4') ....
(3)
The function KN(~ - 4') = sin [(N+ ½)(4 - 4')]/2 sin½(4 - 4') is shown in fig. 1 for N = 5, one of the values chosen by Levinger. The rho m a s s c o r r e s p o n d s to a value ~ = 104 ° and a width of 120 MeV c o r r e s p o n d s to the range 98 ° < ~ ~< 107 o. Since the width of the main peak of K5(4 - 4') is s e v e r a l t i m e s bigger than this i n t e r v a l , it is c e r t a i n l y not p o s s i b l e to r e s o l v e the rho in the s p e c t r a l function. This a g r e e s with the conclusions of Levinger. The function K 5 ( 4 - 4') also has the u n d e s i r a b l e p r o p e r t y that it is not positive defininite so that the calculated s p e c t r a l function is not simply a smoothed out v e r s i o n of the s p e c t r a l function but something which is the s p e c t r a l function weighted with an o s c i l l a t o r y factor. A s i m i l a r a n a l y s i s could probably be made on any method of analytic continuation which involves fitting the f o r m f a c t o r s by a t r u n c a t e d infinite s e r i e s of given functions each p o s s e s s i n g the c o r r e c t analytic s t r u c t u r e . However, in addition to the fact that the r e l a t i o n between the s p e c t r a l function thus c a l culated and the true s p e c t r a l function is not in g e n e r a l a simple one, the c a l culated s p e c t r a l function depends on the f o r m of the analytic functions used. ( C o m p a r e , for example the smoothed s p e c t r a l functions of Levinger and Wang with those of Orman. ) We shall d e r i v e below a f o r m a l i n v e r s i o n f o r m u l a for eq. (1) into which n u m e r i c a l v a l u e s of the f o r m f a c t o r may be i n s e r t e d so that no p a r t i c u l a r analytic f o r m need to a s s u m e d . The method is m a t h e m a t i c a l l y t r a n s p a r e n t so that we shall be able to investigate the reliability of the continuation and obtain a smoothed s p e c t r a l function which is m o r e c l o s e l y r e l a t e d to the true one.
gN(4)
3. THE INVERSION FORMULA Eq. (1) (for t < 0) is simply a Stielt,jes t r a n s f o r m and thereby p o s s e s e s a s t a n d a r d i n v e r s i o n in the f o r m of a double Laplace t r a n s f o r m i n v e r s i o n [22]. However, since the i n v e r s e Laplace t r a n s f o r m involves a contour integration in the complex plane this would r e q u i r e an analytic continuation of b e f o r e it can be p e r f o r m e d . We should like to have an i n v e r s i o n f o r mula of the type
G([)
98
J.E.BOWCOCK
et a l .
Ks(~-O
/ Fig. 1. The wexghtmg function KS(~ - ~') . 0
g(l') = f -
K(t',l)G(l) dl ,
(4)
o0
into w h i c h we m a y p l a c e d i r e c t l y t h e e x p e r i m e n t a l v a l u e s of G(l). A l t h o u g h o u r f i n a l a n s w e r c a n n o t be w r i t t e n in s u c h a s i m p l e f o r m it i s p o s s i b l e to g e t q u i t e c l o s e to it. Let us f i r s t change v a r i a b l e s
2t' - lo//0 = e o s h ~ ,
2t - I o / l o = - e o s h 0 ,
and d e f i n e
c,(0) = c / o , g'(~) =g(t') =-g(t')
f o r ¢5 > 0 for ~ 5 ( 0 .
We t h e n h a v e oo
1
f G'(O) = ~ j o
g'(~5)_sin_hdadd~ eosh~ + eoshO
'
G'(O) c a n in p r i n c i p l e be d e t e r m i n e d by e x p e r i m e n t f o r a l l r e a l O. N o t i n g t h a t G'(O) i s an e v e n f u n c t i o n of 0 we m a y t a k e the F o u r i e r t r a n s f o r m of t h i s e q u a t i o n to g i v e * • See footnote on next page.
NUCLEON FORM FACTORS
99
¢(3
G(p) = f
G'(O) e -ipO dO
-o0
=
2 f o
G'(O) dO
cosp0 co
=2f
co
g,((p)sinh (p d~b f
o
o
c o s p 0 dO c o s h ~b + c o s h 0
oo
= 2
f o
g ' @ ) s i n p $ dq~ _ - i~ (p) s i n h 7rp s i n h ~rp
(5)
where oo
~(p) = f
g'($) e -ip(p ddp .
_o0
T h u s we h a v e @(p) = s i n h p u G(p) .
(6)
T h i s i s a s i m p l e r e l a t i o n t e l l i n g u s t h e t r a n s f o r m of t h e s p e c t r a l f u n c t i o n in t e r m s of t h e t r a n s f o r m of t h e f o r m f a c t o r . W e m a y a l s o w r i t e t h e i n v e r s i o n of t h i s e q u a t i o n a s
2S
g'(4~) =~-
o
sinpq5 s i n h p n d p f o
c o s p 0 G'(O) d O .
(7)
Eq. (7) i s t h e f o r m u l a w e w e r e s e e k i n g . It i s n o t q u i t e of t h e f o r m of eq. (4) s i n c e i f w e t r y to i n t e g r a t e i n i t i a l l y o v e r p w e g e t a d i v e r g e n c e d u e to t h e s i n h (pTr) f a c t o r . T h i s i s a r e f l e c t i o n of t h e s i n g u l a r n a t u r e of t h e i n v e r s e of t h e S t i e l t j e s t r a n s f o r m . H o w e v e r , we m a y f e e d n u m e r i c a l d a t a f o r G'(O) i n t o this expression regardless of an specific analytic form. Further, since it is b a s e d on t h e s t a n d a r d t h e o r y o f F o u r i e r t r a n s f o r m s i t i s m o r e u s e f u l f o r analytic investigation. L e t u s c o n s i d e r w h a t w i l l h a p p e n w h e n we t r y t o a p p l y eq. (7). In o r d e r t h a t the s e c o n d i n t e g r a t i o n o v e r p m a y c o n v e r g e , G(p) m u s t d e c r e a s e a t l e a s t a s f a s t a s exp(-pTr). T h i s i s g u a r a n t e e d f o r t h e F o u r i e r t r a n s f o r m of t h e e x a c t G'(O) by the a s s u m p t i o n s a b o u t i t s a n a l y t i c s t r u c t u r e ( s e e eq. (1)). It i s u n l i k e l y t h a t a n y i n t e r p o l a t i o n of t h e e x p e r i m e n t a l p o i n t s w i l l p o s s e s s t h i s p r o p e r t y , a s w e s h a l l s e e in t h e n e x t s e c t i o n . It i s a l s o c l e a r t h a t a s p b e c o m e s l a r g e c o s p 0 w i l l o s c i l l a t e s e v e r a l t i m e s in t h e r e g i o n w h e r e G'(O) i s s i z e a b l e . T h e e x p e r i m e n t a l u n c e r t a i n t y in G ' ( 0 ) w i l l c a u s e t h e v a l u e s of * Footnote from preceding page. In o r d e r to justify our change of o r d e r of integration some r e s t m c t i v e assumptions about g(/) may be n e c e s s a r y . We shall not consider this point m detail because, as we shall see m sect. 4, our final analvsm avoids this step.
i00
J . E. B O W C O C K et a l .
G(p) to b e c o m e u n r e l i a b l e . Both t h e s e p o i n t s f o r c e u s to i n c l u d e s o m e kind of c u t - o f f in the p - i n t e g r a t i o n . L e t u s call the c u t - o f f f u n c t i o n c ( p ) and i n t r o d u c e it s i m p l y a s m u l t i p l y ing both s i d e s of eq. (6).
i~(p)~(p) = ~(p)d(p) ,
(8)
where /~(p)
=
sinh pTr C(p) .
We will only c o n s i d e r f u n c t i o n s ~(p) w h i c h d e c r e a s e s u f f i c i e n t l y r a p i d l y a s IPl ~ oo that/~(p) a l s o tends to z e r o . T h u s ~'(p) m u s t d e c r e a s e at l e a s t a s f a s t a s exp(-Tr]p]) f o r l a r g e IPl. T h i s h a s the c o n s e q u e n c e that oo
c(o) = ~1 f
U(p) eipO dp
-o0
is an a n a l y t i c f u n c t i o n of 0 in the s t r i p [ I m 0 t < ~r. A l s o , f o r r e a l 0, 0(3
D(O) = ~1 f
c(p) s i n h p ~ eipO dp
-OO
= ½[C(O -iv} - C(O+iTr)] . If we now i n v e r t eq. (8) u s i n g the c o n v o l u t i o n t h e o r e m we find ¢o
co
g,(~,)c(~ - 4,,1 d~' =71 f G'(O')D((p- 0')dO'
f -o0
-00 O0
= 2i- i f
G'(0 ') [C(q~- O' - i 7 r } - C ( 4 ) - 0 ' + i ~ - } ] d 0 '
--~O
= g's(~}.
(9}
T h e r e is an a l t e r n a t i v e way of d e r i v i n g eq. (9) which s h o w s c l e a r l y the n a t u r e of o u r a n a l y t i c continuation. This is by c o n s i d e r i n g the c h a n g e of v a r i a b l e - 2 t + t o = t o c o s h 0 a s a m a p p i n g of the s t r i p {Im 0 [ < 7r into the cut t - p l a n e . Since c o s h 0 is an even function of 0, +0 m a p into the s a m e point. The p h y s i c a l r e g i o n t < 0 then t r a n s f o r m s into the line I m 0 = 0. The u p p e r side of the cut in the t - p l a n e g o e s onto the l i n e s I m 0 = +Tr, Re 0 > 0 and I m 0 = - % Re 0 > 0. The l o w e r side of the cut g o e s onto the l i n e s I m 0 = -~, Re 0 > 0 and I m 0 = +Tr, Re 0 < 0. We t h e r e f o r e have oo
f
g'((p')C(q5 - qb') dq)'
--OO
+oo+#r
+oo-i7r
=f G'(O')C(ep-O'-iTr}dO'-f C'(O')C(O-O'+i~) dO'. -~+ iTr -~o-i~
NUCLEON FORM FACTORS
101
Eq. (9) i s t h e n o b t a i n e d b y d i s p l a c i n g t h e l i n e i n t e g r a l s f r o m t h e u p p e r a n d l o w e r e d g e s of t h e s t r i p onto the r e a l a x i s . It c a n a l s o b e s e e n t h a t w i t h a s u i t a b l e c h o i c e of t h e f u n c t i o n C , eq. (9) i s t r u e e v e n i f s e v e r a l s u b t r a c t i o n s a r e n e e d e d in t h e d i s p e r s i o n r e l a t i o n eq. (1). ( A l s o , t h i s r e m o v e s t h e n e c e s s i t y of i n v e s t i g a t i n g t h e v a l i d i t y of t h e c h a n g e of o r d e r of i n t e g r a t i o n in eq. (5).) F r o m eq. (9) w e s e e t h a t t h e s p e c t r a l f u n c t i o n i s d e t e r m i n e d o n l y to t h e e x t e n t of i t s i n t e g r a l w e i g h t e d w i t h a s m o o t h i n g f u n c t i o n C(~ - @'). To d e t e r m i n e g'(@) e x a c t l y w o u l d m e a n c h o o s i n g C(@ - @') to b e e q u a l to 5(@ - ~ ' ) . T h i s w o u l d l e a d b a c k d i r e c t l y to eq. (7) w h i c h w e h a v e a r g u e d a b o v e c a n n o t b e u s e d f o r t h e h i g h f r e q u e n c y c o m p o n e n t s . T h e s m o o t h i n g f u n c t i o n C(@ - @') w i l l p o s s e s s a w i d t h w h i c h w i l l b e i n v e r s e l y p r o p o r t i o n a l to t h e f r e q u e n c y a t w h i c h t h e c u t - o f f ~ ( p ) b e c o m e s e f f e c t i v e . T h u s the e x p e r i m e n t a l a c c u r a c y on G ' ( 0 ) w i l l d e t e r m i n e s o m e f r e q u e n c y to w h i c h ~ ( p ) m a y b e w e l l d e t e r m i n e d a n d t h i s in t u r n w i l l d e t e r m i n e the m i n i m u m w i d t h of C(@ - ¢b') a n d h e n c e how w e l l we m a y r e s o l v e t h e s p e c t r a l f u n c t i o n . One m a j o r a d v a n t a g e of eq. (9) i s t h a t the s m o o t h i n g f u n c t i o n m a y b e c h o s e n to b e a p o s i t i v e d e f i n i t e f u n c t i o n w i t h a s i m p l e p e a k a t ~ ' = ~, e . g . (1/a~Tr) e x p {-[(@ - @ ' ) / a ] 2 } , s o t h a t t h e i n t e r p r e t a t i o n of t h e s m o o t h e d s p e c t r a l f u n c t i o n i s q u i t e c l e a r . A f u r t h e r f e a t u r e i s t h a t i t e n a b l e s u s to e s t i m a t e the e x p e r i m e n t a l a c c u r a c y n e e d e d f o r a s p e c i f i e d f i n e n e s s of r e s o l u t i o n of t h e s p e c t r a l f u n c t i o n . B o t h t h e s e p o i n t s w i l l be a m p l i f i e d in t h e f o l lowing section.
4. A P P L I C A T I O N TO N U C L E O N F O R M F A C T O R D A T A W e s h a l l a p p l y o u r e x t r a p o l a t i o n p r o c e d u r e to t h e p r o t o n - m a g n e t i c f o r m f a c t o r , GmP , w h i c h i s t h e b e s t k n o w n e x p e r i m e n t a l l y . T h e e x p e r i m e n t a l i n f o r m a t i o n o n G m P i s s h o w n i n fig. 2 t o g e t h e r w i t h t h r e e t y p i c a l c u r v e s w h i c h w e h a v e d r a w n t h r o u g h t h e d a t a b a r s . T h e s e a r e r e p r e s e n t a t i v e of a l l the c u r v e s t h a t w e h a v e t r i e d . A t t h e p r e s e n t t i m e no e x p e r i m e n t a l p o i n t s e x i s t a b o v e 0 = 6.0. T h e e x p e r i m e n t a l i n d i c a t i o n i s t h a t G'(0) d e c r e a s e s a t l e a s t e x p o n e n t i a l l y a s a f u n c t i o n of 0 ( i n v e r s e l y p r o p o r t i o n a l to t). W e h a v e e x t r a p o l a t e d G'(p) to z e r o f o r 0 > 6.5. T h e s e n s i t i v i t y of t h e c a l c u l a t e d ~(p) to t h i s e x t r a p o l a t i o n p r o c e d u r e i s n e g l i g i b l e i n c o m p a r i s o n w i t h the o t h e r u n certainties. T h e ~ ( p ) c o r r e s p o n d i n g to e a c h of t h e s e c u r v e s w a s e v a l u a t e d n u m e r i c a l ly a n d m u l t i p l i e d by s i n h (pTr) to g i v e zg(p). T h e s e a r e p l o t t e d in fig. 3 a n d a s w e e x p e c t e d t h e r e i s c l o s e a g r e e m e n t b e t w e e n t h e c a r v e s up to s o m e v a l u e of p ( ~ 1.7) a b o v e w h i c h t h e c u r v e s d i f f e r c o n s i d e r a b l y . T h i s i s n o t o n l y t r u e f o r t h e t h r e e c u r v e s p l o t t e d b u t a l s o f o r a l l t h e c u r v e s w h i c h we t r i e d . T h i s s e e m s to b e t h e n a t u r a l l i m i t , d e t e r m i n e d by t h e m c c u r a c y of t h e e x p e r i m e n t a l d a t a , up to w h i c h w e m a y c a l c u l a t e t h e F o u r i e r t r a n s f o r m of t h e s p e c t r a l f u n c t i o n . If t h e d a t a w e r e g i v e n to b e t t e r a c c u r a c y t h i s w o u l d f u r t h e r r e s t r i c t t h e c u r v e s w e c o u l d d r a w f o r G'(0) a n d p u s h t h i s l i m i t to h i g h e r v a l u e s of p. W e m a y e s t i m a t e t h e r e s o l u t i o n o b t a i n a b l e on g'(@) f r o m the r e l a t i o n A p ~ ~ 1 w h i c h f o r Ap = 1.7 g i v e s Ad~ ~ 0.6. T h i s g i v e s u s t h e s m a l l e s t
102
J . E . BOWCOCK et al.
G'(e)
0
I
I
I
1
2
3
6
I
I
4
5
F i g . 2. T h r e e typical c u r v e s d r a w n t h r o u g h the e x p e r i m e n t a l points of the p r o t o n m a g n e t m f o r m f a c t o r . The e x p e r i m e n t a l points have b e e n taken f r o m r e f s . [9-16].
I /
20 ¸
(~(p)
I
/ /
I0
/ /
o
,~
~ \
~
"
J ~l; /
--10
F i g . 3. The values of g(p) c o r r e s p o n d i n g to the c u r v e s of fig. 2.
NUCLEON FORM FACTORS
103
width of the smoothing function C(@ - @'). It c o r r e s p o n d s to about t h r e e t i m e s the rho width at the position of the rho, s m a l l e r than this n e a r the t h r e s h o l d and i n c r e a s e s for l a r g e r v a l u e s of t. The fact that the r e s o l u t i o n which may be achieved v a r i e s with the position on the b r a n c h cut is simply a c o n s e q u e n c e of the n o n - l i n e a r i t y of cosh ~ i.e. cosh(~b + Aq~) - cosh(q~ - A~b) i n c r e a s e s with ~bfor a fixed A@. It is thus evident that any a t t e m p t to r e s o l v e a rho contribution to the s p e c t r a l function with a width of 120 MeV cannot succeed. This is in a g r e e m e n t with L e v i n g e r ' s findings. The b e s t we can do is to choose a s m o o t h ing function C(~b - ~'), with a width d e t e r m i n e d by the value of p w h e r e the c u r v e s f o r ~ ( p ) begin to d i v e r g e , and evaluate a smoothed s p e c t r a l function g~(~b) given by eq. (9). Let us choose C(~b- q>') = ~ e x p { -
-
'
whence
g;(q~) : ~
1 ~ G'(O')exp{ -~o
7r2-(
sin{ 27r(~b-0 a2
')} dO'
.
I n t e g r a t i n g n u m e r i c a l l y we get f r o m the t h r e e chosen c u r v e s of G'(O) the t h r e e smoothed s p e c t r a l functions shown in fig. 4. These a r e quite c l o s e to each other and have the g e n e r a l f e a t u r e of a b r o a d positive contribution spanning the m a s s region w h e r e the p, 00 and q~ r e s o n a n c e s would contribute.
oil 510 0
0
I
t
2OO I
(m~)
F i g . 4. The s m o o t h e d s p e c t r a l f u n c t i o n s f o r the p r o t o n m a g n e t m f o r m - t a c t o r of f~g. 2. A G a u s s m n s m o o t h i n g f u n c t m n h a s b e e n u s e d with c~ equal to 1.6.
104
J . E . BOWCOCK et al.
At h i g h e r m a s s e s the s m o o t h e d s p e c t r a l f u n c t i o n s b e c o m e n e g a t i v e a n d s i n c e o u r w e i g h t i n g f u n c t i o n i s p o s i t i v e t h i s i m p l i e s t h a t the t r u e s p e c t r a l f u n c t i o n becomes negative. T h e s e c u r v e s c o r r e s p o n d to a = 1.6, a r e s o l u t i o n of a b o u t s e v e n t i m e s the r h o width. It i s n o t p o s s i b l e to a c h i e v e the o p h m u m r e s o l u t i o n of t h r e e t i m e s the r h o width (a =0.6) with a G a u s s i a n c u t - o f f . T h i s i s b e c a u s e G(p) sinhTrp b e c o m e s so g r e a t for p > 1.7 ( s e e fig. 3) that the C-aussian d o e s n o t c u t off f a s t enough. S e v e r a l o t h e r p o s i t i v e - d e f i n i t e s m o o t h i n g f u n c t i o n s h a v e led u s to a s i m i l a r r e s o l u t i o n . It m i g h t be p o s s i b l e to find a s m o o t h i n g f u n c t i o n w h i c h would i m p r o v e t h i s r e s o l u t i o n s o m e w h a t b u t n o t of c o u r s e b e y o n d the l i m i t d i s c u s s e d . S i n c e we a r e so f a r f r o m r e s o l v i n g the a n t i c i p a t e d s t r u c t u r e of the s p e c t r a l f u n c t i o n we have n o t p u r s u e d t h i s p o i n t f u r t h e r . The s m o o t h e d s p e c t r a l f u n c t i o n s for the p r o t o n e l e c t r i c f o r m - f a c t o r a n d the n e u t r o n m a g n e t i c f o r m - f a c t o r a r e v e r y s i m i l a r to that of the p r o t o n m a g netic form factor.
5. R E L A T I O N B E T W E E N E X P E R I M E N T A L A C C U R A C Y AND THE R E S O L U T I O N O F THE S P E C T R A L F U N C T I O N We h a v e s e e n i n the l a s t s e c t i o n t h a t the a c c u r a c y of the p r e s e n t e x p e r i m e n t a l d a t a only a l l o w s u s to d e d u c e a v e r y s m o o t h e d out v e r s i o n of the s p e c t r a l f u n c t i o n . It i s of i n t e r e s t to e s t i m a t e what i m p r o v e m e n t s would be n e c e s s a r y to r e s o l v e a r e s o n a n c e of g i v e n width w h i c h we m i g h t a s s u m e to b e p r e s e n t i n the s p e c t r a l f u n c t i o n . In p a r t i c u l a r if we looked a t the m a g n e t i c i s o v e c t o r f o r m - f a c t o r w h e r e only one r e s o n a n c e , the r h o m e s o n , i s b e l i e v e d to c o n t m b u t e to the s p e c t r a l f u n c t i o n we c a n e s t i m a t e the a c c u r a c y r e q u i r e d to r e s o l v e s u c h a r e s o n a n t c o n t r i b u t i o n . It should be p o i n t e d out h o w e v e r , t h a t the p o s i t i o n a n d width of the r h o c o n t r i b u t i o n to the s p e c t r a l f u n c t i o n a r e not n e c e s s a r i l y t h o s e of the f r e e r h o m e s o n [24, 25]. H o w e v e r , l e t u s for s i m p l i c i t y a s s u m e they a r e , i.e. Ep = 750 MeV, Fp = 120 MeV. I n o r d e r to b e a b l e to s e e the s t r u c t u r e of s u c h a r e s o n a n c e by o u r m e t h o d of a n a l y t i c c o n t i n u a t i o n the s m o o t h i n g f u n c t i o n C(q5 - ~ ' ) i n eq. (9) m u s t h a v e a w i d t h of the o r d e r of the r h o - m e s o n width. In t e r m s of ~ , the p a r a m e t e r s of 1 the r h o m e s o n quoted in the p r e c e d i n g p a r a g r a p h a r e (~p = 3 . 2 , ~Fp = A~ =0.2. If we s i m p l y m a k e u s e of Ap. A~ ~ 1 th~s g i v e s Ap ~ 5. T h u s we m u s t be a b l e to c a l c u l a t e G(p) a c c u r a t e l y for a l l 0 ~ p ~ 5 if we w i s h to r e s o l v e the r h o c o n tmbution. In t e r m s of the e x p e r i m e n t a l a c c u r a c y r e q u i r e d f o r G'(O) we m a y n o t e that a L o r e n t z i a n s h a p e d r e s o n a n c e m the s p e c t r a l f u n c t i o n g i v e s r i s e to a g(P) = C e - ~ P . s m ap w h e r e ~ i s the width of the r e s o n a n c e , C i s a c o n s t a n t w h i c h for a r h o - d o m i n a n t s p e c t r a l f u n c t i o n i s [4] of o r d e r 10 a n d a i s the p o s i t i o n of the r e s o n a n c e . T h u s w h e n p ~ 5, ~(p) = 1 a n d s i n h (p~) ~ 106 . H e n c e ~,(p)
=
i g ( p ) / m n h pg ~ 10 -~ .
We m u s t t h e r e f o r e m e a s u r e the f o r m f a c t o r to a p p r o x i m a t e l y 1 p a r t i n
NUCLEON FORM FACTORS
105
106 if w e w i s h e d to e x t r a p o l a t e to the p o s i t i o n of the r h o m e s o n a n d d i s t i n g u i s h a p e a k with the w i d t h of the r h o . T h i s v e r y high a c c u r a c y r e q u i r e d i l l u s t r a t e s the s e n s i t i v e n a t u r e of a n a l y t i c c o n t i n u a t i o n a n d i s a l s o a r e f l e c t i o n of the f a c t the r e g i o n of the r h o m e s o n i s q u i t e f a r f r o m the p h y s i c a l region. If, h o w e v e r , we a r e i n t e r e s t e d i n d e t e r m i n i n g w h e t h e r o r n o t t h e r e i s a s i z e a b l e a m o u n t of s p e c t r a l f u n c t i o n i s a g i v e n m a s s i n t e r v a l the r e q u i r e m e n t s w i l l n o t b e so s t r i n g e n t . F o r e x a m p l e i t i s of i n t e r e s t to d i s c o v e r how l a r g e the s p e c t r a l f u n c t i o n i s i n the r e g i o n 4 < t < 16 w h e r e t h e r e a r e no r e s o n a n c e s . I n t h i s c a s e we m a y c h o o s e a b r o a d e r s m o o t h i n g f u n c t i o n a n d the c o r r e s p o n d i n g a n a l y s i s g i v e s a r e q u i r e d e x p e r i m e n t a l a c c u r a c y of ~ 1 % which is not u n r e a s o n a b l e . F r o m the p o i n t of v i e w of b e i n g a b l e to d i s t i n g u i s h b e t w e e n d i f f e r e n t s p e c t r a l f u n c t i o n s o r to d e d u c e the s p e c t r a l f u n c t i o n f o r the f o r m f a c t o r s i t i s t h e r e f o r e i m p o r t a n t that one should have v e r y a c c u r a t e e x p e r i m e n t s at some m o m e n t u m t r a n s f e r s . It i s m o r e v a l u a b l e to h a v e o n e e x p e r i m e n t a l p o i n t of high a c c u r a c y t h a n s e v e r a l p o i n t s of low a c c u r a c y . F o r the n e u t r o n f o r m fac. t o r s , w h i c h a r e d e d u c e d f r o m e l e c t r o n d e u t e r o n s c a t t e r i n g , the c o r r e c t i o n s m u s t a l s o b e k n o w n to the s a m e a c c u r a c y . O n e of u s (J. G. W. ) would l i k e to t h a n k the S c i e n t i f i c R e s e a r c h C o u n c i l f o r a m a i n t e n a n c e g r a n t d u r i n g the p e r i o d t h i s w o r k w a s u n d e r t a k e n .
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J . S . L e v l n g e r a n d R . F . P e l e r l s . Phys. Rev. 134 (1964) B1314. J . S . L e v m g e r a n d C . P . W a n g , Phys.Rev. 136 (1964} B733. J . S . L e v i n g e r and C . P . W a n g . Phys.Rev. 138 {1965) B1207. J . S . L e v m g e r , preprlnt. Jan. 1967. G. L.Kane and R . A . Z d a m s . Phys. Rev. 151 (1966) 1239. F. Chllton and F. J. Uhrhane. Bull. Am. Phys. Soc. 11 (1966) 396, and preprints. B . O r m a n , Phys. Rev. 138 (1965) B1308. B . O r m a n , Phys. Rev. 145 (1966) 1140. K.W. Chen et al., Phys. Rev. 141 (1966) 1267. R.Hofstadter et al.. Phys.Rev. 142 (1966) 922. D.Benaksas et al., Phys. Rev. 141 (1966) 1308. W. Bartel et al., Phys.Rev. Letters 17 (1966) 608. P.Stem et al., Phys. Rev. Letters 16 (1966) 592. E.B.Hughes et al., Phys. Rev. 139 (1965) B458. E.B.Hughes et al., Phys. Rev. 146 (1966) 973. J . R . D u n n l n g e t al., Phys. Rev. 141 ~1966) 1286. L . H . C h a n et al., Phys.Rev. 141 (1966) 1298. R.Wllson, Springer T r a c t s m Modern Physms, 39 (1965). T . M a s s a m and A. Zichmhl Nuovo Clmento 43 (1966) 1137. S. Furulchl and K. Watanabe, PI og. Theor. Phys. 35 (1966) 174. N.G.Antomou and J. E, Bowcock, Phys. Rev.. to be pubhshed. Wldder. The Laplace T r a n s f o r m a t i o n (Princeton Umverslty Press) Tables of integral t r a n s f o r m s (Bateman Manuscmpt Project) Vol. 1, p. 36. J.S. Ball and D. Y. Wong, Phys. Rev. 130 (1963) 2112. M.W. Klrson. Phys. Rev. 132 (1963) 1249.