I L NUOVO CIMENTO VOL. XXXV, N. 4 16 Febbraio 1965

Nonlocal Form Factors tor High-Energy Proton-Proton Scattering at Large Momentum Transfer.

V. WATAGHIN

Is t i tu to di Fis ie~ Teorica de l l 'U~ ive r s i t d - Torino Is t i tu to Nazionale di Fis ica Nueleore - Sezione di Torino

H . R . RE1SS (*)

U.S. Nav.~l Ordn~n~ce L~boratgry . Si lver Spri~q, Md.

(ricevuto il 3 Agosto 1964)

S u m m a r y . - - A nonlocal field-theoretical model in which a well defined choice of form factors is made, is applied to high energy proton-proton elastic scattering. Only the exchange oi strongly interacting mesons and this only in the lowest order of perturbat ion theory, is taken into account. The form factors are functions of the total c.m. energy squared s and momentum transfer squared t, and their s and t dependence is almost, but not completely, fixed by the requirements of relativistic invariance and macroscopic causality, b.t high s and t vMues they probably reduce considerably the contribution of higher-order diagrams. A comparison with the existing experimental da ta is made. All the quali tat ive features of the da ta are reproduced while the quant i ta t ive fit is good only in a l imited range of values of t.

1. - I n t r o d u c t i o n .

R e c e n t e x p e r i m e n t a l r e su l t s for h i g h - e n e r g y p - p e l a s t i c s c a t t e r i n g a t l a r g e

m o m e n t u m t r a n s f e r (1) h a v e shown some new a n d i n t e r e s t i n g f ea tu r e s . A l t h o u g h

t h e d a t a a r e n o t v e r y n u m e r o u s a n d n o t v e r y p rec i s e , e s p e c i a l l y a t t h e h i g h e s t

ene rg i e s a n d m o m e n t u m t r a n s f e r s , one can p e r h a p s s u m m a r i z e t h e n e w f e a t u r e s

in t h e fo l lowing w a y :

(') At present a visiting Professor at the Is t i tu to di Fisica dell 'Universit~ di Torino. (1) W. F. BAKER et al.: Phys . Rev. Lett., 12, 132 (1964); G. Coccoy~ et al.: Phys .

Rev. Lett., l l , 499 (1963).

1154 v. ~VATAGHIN and H. R. RE~SS

a) At fixed energy the differential cross-section a t first drops rap id ly as the square of the m o m e n t u m t rans fe r t increases in absolute value

and then levels out near the m a x i m u m value of - - t .

b) For large fixed values of - - t the cross-section is ve ry sensi t ive to the

value of the square of the to ta l center-of-mass energy s, and in general

is smal]er the larger is s. (This effect is somet imes called (( shrinkage ~).)

SERBER (2) has t r ied to fit the p-p elastic scat ter ing da ta b y int roducing

empir ica l ly a fo rm-fac tor funct ion of t, bu t his resul ts do not exhibi t the

s-dependence of the cross-section shown by the exper imenta l da ta . I t seems therefore worthwhile to app ly to this process a s imple form of nonlocal field theory recent ly re formula ted (3), in which a well defined choice of form-fac tors

is made. These fo rm factors are Loren tz - invar ian t and preserve macroscopic causal i ty.

I t is the purpose of this pape r to repor t on the appl icat ion of this me thod

and to discuss the implicat ions of the results obtained. I t is found tha t , a l though a quan t i t a t i ve predic t ion of cross-sections cannot be obta ined for

all values of m o m e n t u m t rans fe r and of beam energy, nevertheless the theory

yields the correct orders of magni tude and gives the ma in qual i ta t ive features of the exper imenta l results. I n view of the very simple na ture of the nonlocal i ty t h a t is pos tu la ted in the calculations, these results seem to be encouraging

and to lend suppor t to the nonlocal assumpt ions of the theory.

2. - C a l c u l a t i o n s and resul ts .

Our calctflations are based on a nonlocal modificat ion (a) of pe r tu rba t ion

a) b)

Fig. 1. - Feynman diagrams for lowest- order nonlocal p-p elastic scattering via. a strongly coupled meson: a) direct

diagram; b) exchange diagram.

theory in the F e y n m a n formulat ion. We l imit ourselves to the considerat ion

of the lowest-order F e y n m a n diagrams cont r ibut ing to p-p elastic scat ter ing indicated in Fig. 1 a) and 1 b). The only contr ibut ions which are t aken into

account correspond to the exchange of s t rongly in te rac t ing mesons. For in- s tance, the cont r ibut ion due to the ex- change of a pho ton is neglected because

of its weaker coupling.

(2) R. SERBER: Rev. Mod. Phys., 36, 649 (1964); Phys. Rev. Lett., 10, 357 (1963). (a) G. W.~TAC,~IN: Ann. Inst. Henri.Poincar~, 1, 47 (1964); Nuovo Cimento, 25,

1383 (1962); 30, 483 (1963).

NONLOCAL FORM FACTORS :ETC. 1 1 5 5

Another contr ibut ion t ha t is neglected in our calculat ion is t h a t of dif-

f rac t ion scat ter ing since the m a t r i x elements corresponding to 1 a) and 1 b)

are real in our model. Diffract ion sca t ter ing is i m p o r t a n t near the forward

direction, i.e. for small m o m e n t u m transfers.

We consider the exchange of the s t rongly coupled f~, ¢o, and ,~ mesons as

giving the re%in cont r ibu t ion to p-p elastic scat ter ing (~). Besides the con- t r ibut ions of d iagram 1 a) and 1 b) for each of the mesons exchanged, we have

t aken into account both the cont r ibut ion of the in ter ference be tween the

direct and exchange diagrams and also t h a t ar is ing f rom the p-re, (o-r:, 9-(o interference.

The modificat ion of the F e y n m ~ n p e r t u r b a t i o n me thod in t roduced by the

nonlocal i ty of the in te rac t ion consists of the uppearance of form-factors mul-

t ip ly ing the local ma t r ix elements. According to the nonlocal theory (3), which

we are applying, a fo rm fac tor G(t) is associated with the m a t r i x e lement

corresponding to the direct d iagram la) , and a fo rm fac tor G(u) is associated wi th the m a t r i x e lement corresponding to the exchange d iagram 1 b). u and t

are re la ted in our case b y the equat ion:

u = 4 M 2 - - s - - t ,

where M is the mass of the proton. For the case of p-p elastic scat ter ing the prescr ip t ion given in ref. (3)

yields the following expression for G(t):

G(t) = (1--¼l~t)-4[1 ÷ l~(M2--¼t)]-2(1 --/2t)-2(1 + l~M2) 2 .

With these form-factors it is a m a t t e r 0f s t ra igh t forward appl icat ion of the F e y n m a n rules to find the following expression for the differential elastic

sca t te r ing cross-section:

de 1 Y

dD 4s

y = G2(t ) { (g~x~ 2 16

4~ 4p td t ~ m ~ ÷G(t ) G(u ) [ [4~]

2M 2 4M 4 • 8M 4 ÷ ( s - 2 M 2 ) ~ - - t t ~ ( s - -4M2) 2 + ~-d

\ ~ ! ( t - m2) ~

16 (t- ~)(u- ~)

- - - ÷

÷

(~) We have not taken into account the contribution of other known strongly interacting mesons (~, ~, f .... ) because little is known about their interaction xvith nuc}.eons. However we do not expect that the contribution of the ~ and ~ mesons will change substantially our fit unless their coupling constants are unusually large.

1156 v. WATAGttIN and g. R. REI~S

+ \~Y~] ( t - m : ) ( u - m 2) :t~ 4~ &:~-/2-)~- ,n~) + ( t - / ) ( u - m:)

g~ovgp~v t u ' ' 4 + G~(u)

/~ 4:~ 4:~ ~ + u - ~ I ( u - ~)~

. [ ( s _ 2 M 2 ) . , 4 - u ( s + ½ u ) j ~ / g ~ v uS M ~ ,, 4.~ ( u - - m~) 2 + 1 6 t t'~ 4Jz 47e U - - m "2

4-

m a n d u a r e r e s p e c t i v e l y t h e ~ a n d p m e s o n r e s t masses . I l l t h e a b o v e e x p r e s -

s ion t h e r e a p p e a r as p a r a m e t e r s a f u n d a m e n t a l l e n g t h a n d t h e m e s o n - n u c l e o n

coup l ing c o n s t a n t s . T h e l a t t e r a r e t a k e n f r o m t h e l i t e r a t u r e t o be :

2

g~-° = 15; g~ov _ g~.v 4~ 4~ 4~

- - 2 .

W e h a v e chosen t h e v a l u e l - ~ - - 2 . 5 M ~ in our c a l c u l a t i on , b u t as d i s cus sed

be low i t s v a l u e is n o t r e a l l y a r b i t r a r y .

W i t h t h e s e v a l u e s of t h e p a r a m e t e r s we o b t a i n e d t h e cu rves shown i a

F ig . 2. These cu rves r e p r e s e n t t h e d i f f e r en t i a l c ross - sec t ion n o r m a l i z e d b y

d i v i d i n g i t b y t h e d i f f e r e n t i a l c ross - sec t ion a t ! = 0. The l a t t e r was c a l c u l a t e d

w i t h t h e a i d of t h e o p t i c a l t h e o r e m :

= ~ - ~ 1 ,

w h e r e a,o L is t a k e n f r o m e x p e r i m e n t as b e i n g a p p r o x i m a t e l y c o n s t a n t a n d

e q u a l to 40 m b a t t h e s e energ ies .

F i g u r e 2a shows t h e c a l c u l a t e d d i f f e r en t i a l c ros s - sec t ion n o r m a l i z e d w i t h

r e s p e c t to t h e f o r w a r d s c a t t e r i n g c ro s s - s ec t i on p l o t t e d as a f u n c t i o n of t h e

i n v a r i a n t f o u r - m o m e n t u m t r a n s f e r s q u a r e d - - t (5). This is t h e s ame w a y of

r e p r e s e n t i n g t h e d a t a t h a t was e m p l o y e d in t h e e x p e r i m e n t a l work . T h e

e x p e r i m e n t a l p o i n t s a r e t a k e n f r o m ref. ( ') .

I t can be seen t h a t our c a l c u l a t i o n g ives co r r ec t q u a n t i t a t i v e r e su l t s o n l y

for a l i m i t e d r a n g e of va lue s of t. The a c t u a l va lues of t for which t h i s

a g r e e m e n t is a c h i e v e d a r e n o t v e r y s ign i f i can t , s ince b y c h a n g i n g t h e v a l u e

of t h e sole u n d e t e r m i n e d p a r a m e t e r (l 2} c o n t a i n e d in t he t h e o r y , t h e a r e a of

a g r e e m e n t c a n b e s h i f t e d c o r r e s p o n d i n g l y . F i g u r e 2 r e p r e s e n t s r e su l t s o b t a i n e d

(~) The differential cross-section in the forward direction is calculated as in ref. (1), i.e., from the optical theorem with a constant energy-independent value employed for the to ta l p-p cross-section. Our calculated value of the differential cross-section for the forward direction has the same s-dependence. This explains why the curves in Fig. 2a all converge to the same nonunit value of X at t = 0 .

NONLOCAL FORM FACTORS ETC. 1157

: 1 11 GeVlc

10-6L 18.9

! 18.1 ° I I . I , I . 2~ ~ l k h l O I 1 A ~ 13

31.5 18.2 "~.~ ,9.6 ~8.6 20 GeV/c 2 o.~ °~'o~ ~ o 0 I n 0 2 1 , 9

10-; ~ 2~.7 ~ ~ / . 3CL7

10-,21

10 - ~ . . . . . 0 4 8 12 16

- - t a)

o 2 t ' 8 i

I 31.8 30.9 [

C

20 24 28

I % \ ~ - 6 7 o

i o 6 5 " p ~ 2 3 . 8 10! 70° P:19"6

/ 68 ° p = 2 6 . 6 ~

10 2 3~0 40 50 60 S

b)

70

Fig. 3. - E l a s t i c differential cross-section (normalized as expla ined in the text ) as a funct ion of squared fou r -momen tum trans- fer t, a); and squared center-of-mass to ta l energy s for 0o.m" equal to 67 ° b). c) Angular d is t r ibut ion obta ined by p lo t t ing (log10 X)/P½ ~s ~ funct ion of cos 0. The expe r imen ta l

points are taken from ref. 0).

0.2r

!'/ - 0 . 2

- 0 . 4 ~

- 0 . 8 - V .

\ \ - - 1.6' z~.7o \ \ ' \

2~.: \ \ ' \ L ~8- ~0:o \ \ , r

20 2 °" 9 °kk<~L�~, ~6 4 1 - " - X ~ £ X , ~ ° ~ , 8 o o i

\,~,.8 \ ,?0~i -2.2- % "- I

• . f

- i 1 - 1 , 0 - 0 . 5 - 0 . 6 - O . q

c)

-5.2 %

1158 v. WATAGHIN and H. R. REISS

with 1 -= = 2 . 5 M 2. With l -S= 3 M ~, good agreement is obta ined at max imum

m o m e n t u m t ransfe r for all energies. Two impor tan t points mus t be made here. The first is t h a t l ~ is not t ru ly

an a rb i t r a ry parameter . I f the ideas embodied in the nonlocal form factors employed here really have a correct physical basis, t hen l 2 should be the same for all interactions. Since l i t t le da ta are present ly available a t sufficiently high m om e n tum transfers (in the limit of low energies and small m o m e n t u m transfers the nonlocal theory results should coincide with the local theory ones) to t ru ly tes t these form factors for a va r ie ty of interact ions, we simply use the available p-p scat ter ing da ta as a means of inferr ing the likely magni tude of l ~.

A second impor t an t point is that , a l though the quan t i t a t ive agreement obta ined is limited, most of the principal qual i ta t ive features of the experi- ments are reproduced quite well. Fo r example, as can be seen in Fig. 2b, t he var ia t ion of the cross-section with s (the squared to ta l center-of-mass energy) is predic ted proper ly for fixed center-of-mass scat ter ing angle (~). The cal- culated curve was chosen at an angle which gives good quant i ta t ive agreement , bu t even at o ther angles t h e t r end of the theoret ical results is exac t ly tha t given by exper iment . This correct qual i ta t ive dependence on s is also seen in Fig. 2a, where the curves for different labora tory momenta occur in the proper relat ive orientation, and exhibi t the shrinkage with increased energy tha t was pointed out in ref. (1). This contrasts with the empirical calculations of SER]3ER (~) who obtains a single curve for all energies in a plot such as Fig. 2a.

There are several possible causes of the failure to obtain a quan t i t a t ive fit a t small and in termedia te momen tum transfers. F i r s t ly the funct ional form of the space par t of the form factors is not suggested by theoret ical arguments and m a y therefore be different f rom the one chosen. Secondly as a l ready remarked the ma t r ix element in our model being real we get no con- t r ibu t ion to the cross-section f rom the diffraction scat ter ing which is impor t an t at small values of t. Thirdly the quant i ta t ive disagreement m a y be due also to the l imitat ions of per tu rba t ion theory. Fu r the r tests of the theory will

perhaps clarify some of these points. Since we are dealing with large coupling constants, the ex ten t to which

our calculation reproduces the exper imenta l da ta seems to us to be quite encouraging. Presumably , we could significantly improve the predictions of the theory by making refinements in the nonlocal model employed.

(6) This is one way of representing the shrinkage effect. In Fig. 2a one can observe another form of the shrinkage effect shown as the s-dependence of the cross-section at fixed t. The fact that our calculation seems to reproduce much better the first type of shrinkage effect is because at fixed angle the s-dependence of the form-factors is much stronger.

NONLOCAL FORM FACTORS ~TC. 1159

The q u a l i t a t i v e a g r e e m e n t o b t a i n e d u s i n g on ly l o w e s t - o r d e r p e r t u r b a t i o n

t h e o r y of s t r o n g i n t e r a c t i o n s sugges t s t h e p o s s i b i l i t y t h a t t h e cut -off p r o v i d e d

b y t h e f o r m f a c t o r m a y b e suf f ic ien t ly s t r o n g t o l e a d t o c o n v e r g e n c e or semi -

c o n v e r g e n c e in p e r t u r b a t i o n t h e o r y .

R I A S S U N T O

Si appliea al processo di diffusione elastica protone-protone ad elevate energie un modello non locale di teoria dei campi. Si tiene conto solo dello scambio di mesoni che interagiscono fortemente e questo solo nell 'ordine pitt basso della teoria delle pertur- bazioni. I fa t tor i di forma sono funzioni delt 'energia totale al quadrato nel s.c.m, s e del momento trasferito al quadrato t, e la loro dipendenza da s e t ~ fissata quasi, ma non completamente, dalle condizioni di invarianza relativist ica e causalit.~ macrosco- pica. Per grandi valori di s e t essi probabilmente riducono in modo considerevole il contributo dei diagrammi di ordine superiore. Si confrontano i r lsul tat i con i dat i speri- mentali. L 'accordo quali tat ivo ~ buono mentrc quello quant i ta t ivo ~ soddisfacente solo in un l imitato campo di valori di t.