New dispersion sum rules for inelastic e - p scattering

Volum e 3 l B . n u m b e r 7

NEW

PHYSICS

DISPERSION

SUM

RULES

LETTERS

FOR

30 M a r c h 1970

INELASTIC

e-p

SCATTERING

H. L E U T W Y L E R *

CERN. Geneva. Switzerland and J . S T E R N **

Inslitut de Physique Nucl~aire. Di~,ision de Physique ThSorique, Orsay, France R e c e i v e d 23 F e b r u a r y 1 9 7 0

C a u s a l i t y c o m b i n e d with s c a l i n g i m p l i e s a s e t of s u m r u l e s f o r the e l e c t r o m a g n e t i c s t r u c t u r e f u n c t i o n s .

We propose a local interpolation between the Regge and Bjorken limits to convert these relations into finite energy sum rules relating the deep inelastic and resonance regions.

In the p r e s e n t p a p e r we e x p l o i t two p r o p e r t i e s of the e l e c t r o m a g n e t i c s t r u c t u r e f u n c t i o n s

12~j'dxexp(iqx){pj [j~t(:,'), = {qlzq v-xpvq2} 1

+~

Jv(0)]i P}c =

vfet(v,av_b)

VI+

(ppqv+pv qp)vm -pppuq 2 -gtiv

u2rn2} V2

which determine the inelastic e-p cross sections: c a u s a l i t y , w h i c h r e q u i r e s t h e c o m m u t a t o r to v a n i s h f o r . v 2 0: t h e f a c t t h a t V1 a n d V2 a p p e a r to o b e y the scaling laws Vl . ~ ( 2 v m 2 ~ )- 1 Vl

(~)

V 2 ~ ( 2 v 2 m ~ ) -1 F 2 (~)

(1)

in the B j o r k e n l i m i t v - ~ o : (_q2).2mv=~. We s h o w t h a t t h e s e p r o p e r t i e s i m p l y a s e t of s u m r u l e s [1~ f o r V1 and V2. S i n c e t h e F o u r i e r t r a n s f o r m s Vi(.£'), ( i = 1 , 2), of t h e s t r u c t u r e f u n c t i o n s v a n i s h f o r .x.2< 0, t h e c o r r e s p o n d i n g r e t a r d e d q u a n t i t i e s Vir e t (x') m a y be e x p r e s s e d a s 0 (n.r) Vi(x) by u s i n g a l i g h t l i k e v e c t o r n p r a t h e r t h a n a t i m e l i k e one. T h e v i r t u e of t h e c h o i c e npnP =0 l i e s in t h e f a c t t h a t t h e i n t e g r a t i o n in the corresponding dispersion relation 1 -dE vret (#) : .,,. J ~--~e Vi (k +En ) (2) r u n s a l o n g a s t r a i g h t l i n e in t h e ( v . q 2 ) p l a n e a n d t h e l i m i t E ~ = oo c o i n c i d e s w i t h t h e B j o r k e n l i m i t . * On leave of absence from Institut ftir T h e o r e t i s c h e Physik. Universit~t Bern. ** L a b o r a t o i r e associ~ au CNRS. 458

T h e s c a l i n g l a w s (1) t h e r e f o r e g u a r a n t e e the c o n v e r g e n c e of t h e i n t e g r a l a n d t h e no s u b t r a c t i o n a s s u m p t i o n i s j u s t i f i e d . In t e r m s of the i n v a r i a n t s v, q2 the d i s p e r s i o n r e l a t i o n m a y be r e w r i t t e n

1 ( d r ' Vi(v,,av,_b) a v'-v-i6

-2~i

(3)

where a and b are arbitrary parameters varying in the r a n g e $ - ~ < a < ~ , b > / ] a 2. W i t h t h i s r e p r e s e n t a t i o n o n e i m m e d i a t e l y v e r i f i e s t h a t Vret m u s t s c a l e l i k e (2vm2)-lF_lret(~) i n t h e B j o r k e n limit, where

1_1_( d~;

Flet(~)r = 2 ~ i a ~ - ~

-ic

Fl(~')

(4)

~'

C o m p a r i s o n of t h i s s e a l i n g law f o r V r e t w i t h t h e l i m i t v ~ o in eq. (3) t h e n l e a d s to t h e d e s i r e d sum rule:

f dvVl(V,-2m~v-b) = ~m 2 -

~

~,

(5)

oo

w h e r e i t i s u n d e r s t o o d t h a t t h e i n t e g r a l on t h e l e f t - h a n d s i d e i s c a r r i e d out in a s y m m e t r i c f a s h i o n s u c h t h a t t h e a s y m p t o t i c t a i l s of V 1 do not c o n t r i b u t e . In t h i s s u m r u l e t h e q u a n t i t i e s a n d b a r e i n d e p e n d e n t p a r a m e t e r s v a r y i n g i n the r a n g e - ~o <~ <~o; b > / m 2 ~ 2. T h e s a m e a n a l y s i s m a y of c o u r s e b e a p p l i e d to t h e f u n c t i o n V 2 w h i c h v a n i s h e s o n e p o w e r f a s t e r in t h e B j o r k e n l i m i t . T h i s f u n c t i o n t h e r e f o r e , in a d d i t i o n to t h e s u m r u l e (5) w h i c h in t h e present case reads In t he l i m i t i n g c a s e b - ~a 2 t he s t r a i g h t l i n e al~ - b

becomes tangent to the kinematical boundary, q2 = u2,

Volume 3lB. n u m b e r 7

PHYSICS

LETTERS

30 March 1970

oO

)~dvV2(v,-2m~v-b)

= 0

satisfies the further sum rule

f d w V2(v,-2m~v-b) -o0

=

(7) -1

which may be derived in an analogous fashion. Note that these sum rules are satisfied ident i c a l l y by f u n c t i o n s V 1 a n d V2 t h a t a r e e x a c t l y scaling invariant. The sum rules therefore test the deviations from exact scaling. In t h e r e m a i n d e r of t h i s p a p e r we d i s c u s s t h e p o s s i b i l i t y of e v a l u a t i n g t h e s e s u m r u l e s in t e r m s of t h e e x p e r i m e n t a l d a t a [2] f o r e - p s c a t t e r i n g . W e c o n c e n t r a t e on t h e two s u m r u l e s f o r V2 f o r which the small angle data provide rather accur a t e i n f o r m a t i o n . N o t e t h a t V 2 i s r e l a t e d to t h e s t a n d a r d f u n c t i o n [2] W 2 by V2 = ( - q 2 ) - l w 2. Since the scaling law is valid both for timelike a n d s p a c e l i k e q2 t h e a s y m p t o t i c t a i l s of t h e i n t e g r a l s in (6) a n d (7) m a y b e e x p r e s s e d in t e r m s of t h e s c a l i n g f u n c t i o n F 2. H o w e v e r , e v e n if we choose the parameters } and b such that the s t r a i g h t l i n e q2 = - 2 m ~ v -b c u t s t h e a x i s q 2 = 0 a t a l a r g e v a l u e of v, t h e s c a l i n g l a w c a n n o t be u s e d to e s t i m a t e t h e f u n c t i o n V2 in t h e e n t i r e u n p h y s i c a l r_egion q 2 > 0 , s i n c e it e x h i b i t s a s i n g u l a r i t y at 6,2 = 0, w h e r e a s t h e f u n c t i o n V2 i s w e l l b e h a v e d a n d i s r e l a t e d to t h e p h o t o a b s o r p t i o n cross-section. What is therefore needed is a reliable interpolation in the strip -A
V2 = l p f ~ s

~(S)F {s-q2 s_q2 2 \ ~ )

~(q2) : Fa(0 )PFds e(s) 0 s-q2

(6)

(8)

w h e r e ~(s) i s a s p e c t r a l f u n c t i o n n o r m a l i z e d by fdscr(s) = 1. T h i s m o d e l h a s t h e f o l l o w i n g p r o p e r ties: i) it s a t i s f i e s t h e s c a l i n g law in t h e B j o r k e n l i m i t . If t h e s p e c t r a l f u n c t i o n d o e s n o t e x t e n d to l a r g e v a l u e s of s t h e s c a l i n g l a w w i l l b e v a l i d d o w n to r a t h e r s m a l l v a l u e s of q2 a s o b s e r v e d e x p e r i m e n t a l l y . On t h e o t h e r h a n d the s i n g u l a r i t y at q2 = 0 i s s m e a r e d out; ii) f o r f i x e d q2 t h e m o d e l e x h i b i t s t h e R e g g e b e h a v i o u r V2 ~ v- lf~(q2) c o r r e s p o n d i n g to t h e Pomeron contribution with a residue given by* * Note that the P o m e r o n contribution is related to the scaling function F2(~) only insofar as fi(q2) ~(_q2)-1F2(O ) for q2 ~oo. The behaviour of /9 at s m a l l values of q2 has nothing to do with the scaling function. The value of fl(0) deduced from the a s y m p totic photoabsorption c r o s s - s e c t i o n d e t e r m i n e s the integTal s~ 1 :k]dss-l(y(s). One finds a s m a l l value s O ~ 0 . 3 GeV+Z~in a g r e e m e n t with i) [3].

(9)

iii) it satisfies the sum rules identically. We now assume that the model (8) provides a good phenomenological representation of the actual function V2 in the entire high energy region, with a function if(s) that contributes significantly only for small values of s. (This assumption may be tested at least partially by fitting the small q2 high u data. ) In this case the difference 172-V2 vanishes as soon as we reach the high energy region and we therefore get a finite energy sum rule version of eq. (6):

f dvV2(v,-2m(v-b) =

v-

0

s

s+° ~+(s)

77 (i0)

where

~(s) = +[~+(s+b) 2m%] A similar relation corresponding to (7) is easily obtained. Finally, if we choose the parameter b large enough such that s can be neglected as compared to b, then the sum rules simplify to v+

j d v V2(v , -2m~ v-b ) P

j'~+d ~ v-

1 -~- F 2 ( ~ )

= ~-~+d~ [

v2(~,-2m~-~) -_

1 [11 2rn[~+

7?

d,r/F2(z/)

(11)

+ }d~

F2(T/) 1

(12) w h e r e ~+ = ~+(0). In t h i s l a r g e b l i m i t t h e s h a p e of t h e f u n c t i o n ~(s) i s i r r e l e v a n t . T h e d u a l i t y c h a r a c t e r of t h e s e s u m r u l e s i s e v i d e n t : in p r i n c i p l e t h e y a l l o w to d e t e r m i n e t h e a s y m p t o t i c s c a l i n g f u n c t i o n s i n t e r m s of t h e low energy data. The published results which unfort u n a t e l y r e p r e s e n t only a s m a l l f r a c t i o n of t h e d a t a do not p e r m i t u s to c a r r y out a n e x t e n s i v e t e s t of t h e s e s u m r u l e s . W e h a v e p e r f o r m e d o n l y a r o u g h e v a l u a t i o n f o r o n e p a r t i c u l a r c h o i c e of t h e p a r a m e t e r s ~ a n d b: we h a v e c h o s e n t h e s t r a i g h t l i n e to c o r r e s p o n d to t h e 10 GeV, 6 ° d a t a , so t h a t t h e s e d a t a c o u l d b e u s e d in t h e r e g i o n v ' 0 . In t h e r e g i o n v, 0 w e h a v e performed a n i n t e r p o l a t i o n b e t w e e n d a t a t a k e n a t d i f f e r ent energies and angles. The cut-offs were c h o s e n a t W = 2 GeV. In t h i s way we o b t a i n e d t h e v a l u e 0.107 ± 0 . 0 1 0 G e V -2 f o r t h e l e f t - h a n d s i d e of (11). T h i s v a l u e a g r e e s r a t h e r w e l l w i t h t h e r i g h t - h a n d s i d e of the s u m r u l e w h i c h w a s e v a l u a t e d a t 0.097 + 0 . 0 0 6 GeV - 2 . F o r t h e s e c o n d sum rule the contribution from the resonance r e g i o n a n d t h e e l a s t i c p e a k a m o u n t s to 0.72 ± 0.02 G e V -1 a s c o m p a r e d to 0 . 6 3 ± 0.04 G e V -1 f o r t h e r i g h t - h a n d s i d e .

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References 1. A m o r e d e t a i l e d a c c o u n t of t h i s w o r k w i l l be published elsewhere. 2. F o r a r e v i e w s e e R . T a y l o r . P r o c . 4th I n t e r n . Symp. on E l e c t r o n and photon i n t e r a c t i o n s at high e n e r g i e s . L i v e r p o o l (1969).

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3. H . D . I . A b a r b a n e l . M . L . G o l d b e r g e r a n d S . B . T r e i m a n P h y s . R e v. L e t t e r s 22 (1969) 500 and R . B r a n d t , P h y s . R e v. L e t t e r s 22 (1969) 1149.