IL NUOV0 CIMENT0 VOL. X X X I I I , N. 4 16 Agosto 1964

Note on High-Energy Proton-Proton Scattering.

H. A. BETttE (*)

C E R N - Geneva

(ricevuto il 6 Aprile 1964)

S u m m a r y . - - I t has been suggested that the elastic scattering of high- energy protons can be considered as the sum of a statistical contribution depending on the total energy s, and a diffraction term depending on the four-momentum transfer, fit). Detailed examination of the Cornell- Brookhaven measurements shows that this is not possible, even under very general assumptions about ](t), but that the <, diffraction )~ term, for any given t, decreases strongly with increasing energy s.

C o c c o ~ I et al. (1) have measu red a t B r o o k h a v e n the sca t te r ing of p ro tons

b y p ro tons up to ve ry h igh m o m e n t u m transfer , - - t = 24 (GeV/c) 2. OREA~ (2)

has po in ted ou t t h a t the cross-sect ion a t fa i r ly large angles (~ 30 ~ can be

represen ted r a the r well b y a un iversa l func t ion of the t ransverse m o m e n t u m

of the sca t te red p ro ton , P . , t hus

d~ (1) dw - - = 10 -~5-4~ exp [-- pz/0.151]

where da/dw is in cm2/sr, a nd p~ in GeV/c. The diff ract ion peak a t smal l

angles can be represen ted b y

da (ka~~ ~ exp [9t] (0 < 20 ~ (2) d-~ -~ \ 4 ~ ] '

(') On sabbatical leave from Physics Department, Cornell University, Ithaca, N. Y. (1) G. COCCONI, V. T. COCCONI, A. D. Kaisn, J. OREAR, R. RUBINSTEIh T, B. D.

SCARL, W. F. BAKER, E. W. J~NKrNS and A. L. READ: Phys. Rev. Lett., l l , 499 (1963); 12, 132 ( 1 9 6 4 ) .

(2) j . ()a~g~: Phys. Rev. Lett., 12, 112 (1964).

1168 ~. A. BETHE

w h e r e - - t = 2 p ~ ( 1 - - c o s O ) is measured in tGeV/c) 2. F r o m 65 ~ to 90 ~ the

cross-section does not change much wi th angle bu t depends strongly on energy:

Cocco~-i (3) finds t h a t this can be wri t ten as

da - - -~ 10 -'24.6~ exp V- 3.38w] , (3) d o

where w is the centre-of-mass energy in GeV. FAST, HAGEDORN and JO~ES (4) have developed a s tat is t ical theory in which they assume the init ial fo rmat ion

of a compound s ta te which then can dis integrate into var ious final s tates

according to phase-space volumes available. They calculate the f ract ional p robabi l i ty of dis integrat ing into two protons in thei r ground s ta te and find

th is to be:

(4) F~, : exp [-- 3.30(w -- 1.88)].

This agrees ve ry well wi th the w-dependence of (3) which m a y be t aken to be

an a rgumen t for the s ta t is t ical theory. The cross section for fo rmat ion of

the compound s ta te is then

da/dw 10_27. ~ 0.5 m b . (4a) ar F~I

Hagedorn , in an unpubl ished paper , has shown tha t this result is theoret ical ly

not unreasonable f rom the s tandpoin t of the s ta t is t ical model. I f tlfis s tat is t ical in terpre ta t ion of the large angle scat ter ing is accepted,

i t is t emp t ing to in terpre t the scat ter ing a t smaller angles as a sum of a

s tat is t ical and a diffraction contr ibut ion, and to assume t h a t the la t ter (apar t

f rom exchange effects) depends just on t. In fact , 5~I~A~I (5) has made a

specific assumpt ion on the t-dependence, viz., t ha t the diffraction pa r t of the scat tered ampl i tude is the sum of two exponentials in t which interfere con-

s t ruct ively, and t ha t to the diffraction cross-section there is added a stat ist ical

cont r ibut ion depending only on s, thus

da (5) d~-~9 - - (A e x p [~t] 4- B e x p [fit]) 2 ~- C exp [-- yw].

We wish to point out t ha t Minami 's assumpt ion is not tenable in view

of the exper imenta l data , and t h a t in fact the cross-section cannot be wri t ten

in the more general form

da (6) d(,~ =/~(t) + g(s).

(3) G. COCCONI: to be published in Nuovo Cimo~to. if) G. FAST and R. HAGEDORN: NUOVO Cimettto, 27, 208 (1963); G. FAST, R. HAGE-

DORN a~nd L. W../ONES: ~Vttovo Ci~t~ento, 27, 856 (1963). (5) S. MINA~H: Phys. Rev. Lett., 12, 200 (1964).

NOTE ON H I G H - E N E R G Y PROTON-PROTON SCATTERING 1 1 6 9

Before comparing this expression with experimental data, we want to show tha t there is no appreciable contr ibution from exchange. I f / ( t ) is the ampli tude for direct scattering with four -momentum change t, then ](u) will be the ampli tude for exchange scattering. The maximum possible interference con- t r ibu t ion to da/d(,J is then

(7) In t = 2 I](t)](u)].

Now to give the hypothesis (6) the most favourable t rea tment , we shall assume tha t ]2(t) is, for moderate scat ter ing angles (say 30 ~ 50~ as close as possible

to expression (1) which is known to represent the da*a very well for 0 > 30 ~

Now, for moderate 0,

(s) p • p sin0 ~ 2p sin �89 = ~ / ~ ,

so tha t the best approximat ion for ](t) is (in this region)

(9) /(t) = 10 -'2"73 exp [-- 3.3(-- t) �89 .

5~ow we assume tha t (9) is also valid at sufficiently large argument to obtain the exchange contr ibution ](u). We therefore set, using (7),

(10) In t -- 2.10 -25"47 exp [-- 3.3[(-- t) �89 Jr (-- u)�89 ----

---- 10 -25'~7 exp [-- 6.6p(sin �89 + cos �89 .

Now for small 0, say 0 < 45 ~ In t is obviously negligible compared with p(t). But for 0 ~ 45 ~ we have sin �89 ~- cos �89 ~ 1.305 and therefore

(11) In t ~ 10 -25"~7 exp [-- 8.64p] ~ 10 -25"~7 exp [-- 4.2w]

using Cocconi's approximation w = 2.1p which is valid on the average. I t is seen tha t (11) is negligible compared with (3). This means tha t if the direct diffraction ampli tude depends only on t (and not also on s), then the inter- ference term is always negligible compared to the statistical t e rm at large

angles (tO ~ 45~ Thus we can use t6) wi thout an added interference term. The experimental da ta (i) are known to show, for any given t, a strong

decrease with increasing s. To fit these as well as possible by a formula of

type (6), we should use the largest possible g(s) which is just Cocconi's (, stati-

stical ~) expression (3), derived from 90 ~ scattering. In Table I we have used

the data for - - t = 6.0 and 10.0, the two t values for which the largest number

of observations (4) exist. We give the observed da/d~o in cm~-/sr, the statistical contr ibution g(s) calculated from (3), and the difference. This difference should

7 4 - I1 Nuovo CDnento.

1 1 7 0 H . A . BETIIE

be c o n s t a n t /~(t), if (6) were val id , i t is obvious t h a t this is no t the case. F o r

- - t = 6.0, even af ter s u b t r a c t i n g the (~statist ical ~ c o n t r i b u t i o n , the cross-

sect ion decreases b y a fac tor of 4 as w increases. Indeed , the cor rec t ion g(s)

TABLE 1. -- Cross-sections.

- - t

(GeV/eV

6.0 6.0 6.0 6.0

9.9 10.0 10.0 10.0

P0

GeV/c

l l .1 15.7 21.7 31.5

11.4 14.2 20.9 28.7

Oena

degree

68.3 55.4 46.2 37.7

90.0 78.5 62.1 52.0

GeV/e

4.77 5.60 6.53 7.83

4.82 5.34 6.41 7.47

d~/(t~

obs.

4.0 �9 10 -a2 1.29" lO -32 6.0 �9 1 0 -38

3.5 �9 10 -a3

2.2 �9 10 -32 5.1 �9 10 -33

4.8 �9 10 TM

1.47.10 -3~

g(s)

calc.

2.5" 10 -a2 1.6" 10 -aa 6.8" 10 -35 0.9" lO -a6

2.2" 10 -a~ 3.9" 10 -aa 1.0" 1 0 T M

2.8" 10 -~6

Diff.

1.5 �9 10 -32 1.13.10 -32 5.9 �9 l0 -33 3.5 �9 l0 -33

0 1.2 �9 lO -3a 3.8 �9 1 0 -3~

1.4 �9 10 -3a

is negl ig ib le already for the second l ine, Po -~ 15.7, so t h a t /2(t) should be ve ry

nea r ly equa l to the d i rec t ly observed cross-section. T h a t the l a t t e r decreases

s t rong ly w i th inc reas ing energy is ev iden t (1). The decrease is fur ou ts ide the

e x p e r i m e n t a l error of a b o u t 25 9'0. F o r - - t = 10.0, the first l ine is zero, s imply

because this l ine refers to 90 ~ sca t t e r ing a nd was therefore used to deduce

the s t a t i s t i ca l g(s). To o b t a i n a smooth beha v i ou r of the difference, less should

have been s u b t r a c t e d ; i.e., only p a r t of the 90 ~ cross-sect ion should be a t t r i -

b u t e d to the s ta t i s t i ca l process. This, of course, would m a k e the energy-

dependence of the (~ difference )~ even s t ronger . I f the first l ine of - - t - : 10.0

is d is regarded the o ther l ines st i l l show s t rong energy-dependence , a nd a t

least for the las t two ent r ies (Po ~ 20.9 and 28.7) this conc lus ion is well out-

side the e x p e r i m e n t a l error. We thus conc lude t h a t (6) is n o t val id .

I f we e x a m i n e 5 [ inami ' s own ana lys i s which is on ly a t one energy

( p o : 16.7 GeV/c) we f ind t h a t his fit is s lso n o t ve ry good. His theore t ica l

curve (Case I I ) is a b o u t a fac tor 1.8 too low a l o u n d 600 - - 70 ~ a n d nea r ly b y

the same fac tor too high a r o u n d 40 ~ bo th outs ide e x p e r i m e n t a l error. I f the

40 ~ reg ion had been used for a b e t t e r a d j u s t m e n t of the pa ramete r s , the

d i sc repancy a r o u n d 60 ~ 70 ~ would have been even larger, a t least a fac tor

of 2. We have m a d e the same obse rva t i on a t o ther energies: if the para-

meters i n fit) are ad jus ted to fit smal l a n d m e d i u m angle da ta , a n d g(s) to

fit t he 90 ~ da ta , t h e n the theory gives too low cross-sections a r o u n d 60 ~

This is because the theore t ica l f(t) falls too fast w i th 0 so t h a t essent ia l ly on ly

g(s) is lef t a t 60~ b u t the observed cross-sect ion a t 60 ~ is s u b s t a n t i a l l y

NOTE ON H I G I I - E N E R G Y PROTON-PROTON S C A T T E R I N G 1171

larger t han t ha t at 90 ~ f rom which g(s) is d e d u c e d - - a b o u t a factor of

6 according to the accumula ted da ta curve of ref. (3), Fig. 2.

We mus t therefore accept the fact t ha t the cross-section is not just a

funct ion of t alone bu t depends also on s, even if a s tat is t ical contr ibut ion has first been subtracted. Equa t ion (1) is a possible form, not ing t ha t

t o~�89 (12) p• = p sin 0 = (-- t) �89 1 -~ s -- -4m-]

in which 4m"- m a y be neglected. The formula m~y of course be modified to

be linear for smM1 t (which were not used in deriving (1)), e.g., by subst i tu t ing

(13) p• --~ ( q _ - t)~(l + t/s) ~ ,

where q can still be chosen to improve the fit. However , the complete curve cannot be fi t ted by any choice of t, since the coefficient in (1), 10 -25"4v, is too

small to fit the forward cross-section, and the second exponent ia l in (1) e~n

only decrease the cross-section.

A behaviour , for large t, as

(14)

ruther than as

(15)

da/d(o -~ exp [ - - ~(-- t)] �89

exp [~t]

is reasonable. As KI~OSmTA (6) h,~s shown, a formula like (14) ~nould be expected if the scat ter ing ,~mplitude for lurge reM s is sufficiently bounde4

in the complex plane of cos0 (i.c., of t).

Since the cross-section now depends on s for a given t, we m a y obta in a

Regge- type representat ion. Fo r - - t (.< s, (13) and (1) give

(16) in (da/d(,))e ~ = A(t) + 3.3(-- t).~'-l.

I n the Regge theory

(17) In (da /d(o)~ = fi(t) -v [:~(t) - - 1] In s .

The behavieur is not identical bu t only a l imited range of s was used in the

experiments , and only the dependence on s mat ters . Equa t ing the derivat ives

(~) T. KINOSHITA: Phys. Rev. Lett., 12, 257 (1964).

1172 u. Ao BETttE

of (16) and (17) with respect to s, and using the definitions for s and t, we ge t

3 3 (18) ~(t) ~-- 1 - - ~'-)- (1 -- cos 0 ) ( - t ) � 89

I n Table I , we have used da t a a r o u n d 50 ~ for - - t = 6, and a r o u n d 60 ~ for

- - t = 10. In se r t i ng these values

~ = - - 0 . 5 f o r - - t = 6 ,

z r for - - t - - 1 0 .

:Not m u c h mean i ng can be g iven to these results, especially because ~ for the

leading t r a j ec to ry m a y no t be able to become < - - 1 .

We have shown t h a t the a s sumpt ion of a s ta t i s t ica l con t r ibu t ion to the

cross-sect ion does no t s ignif icant ly s implify the r emainder of the cross-section,

in t h a t this r emainder still depends on s as well as t. The ques t ion then is

whe the r there is a ny evidence for the s ta t is t ica l par t . The d(r/deJ is abou t

c o n s t a n t f r o m 70 ~ to 90 ~ (ref. (3), Fig. 2 ) which means t h a t the 90 ~ cross-

sect ion is abou t three t imes h igher t h a n expec ted front the smoo th func t ion (1).

P a r t of this m a y still be due to some cons t ruc t ive in terference effect bu t

p r o b a b l y no t all. Thus there is some, r a the r slight, evidence for ano the r

p h e n o m e n o n near 90 ~ . I t would be in te res t ing to see whe the r this is real ly

s tat is t ical . I f so, then exci ted s tates of the nucleon should be fo rmed wi th

a p robab i l i ty p ropor t iona l to the i r s ta t i s t ica l weight . I t should be possible

to observe the energy spec t rum of the nucleons emi t t ed a t a given, large

angle 0l~b; this should indicate the fo rma t ion of exci ted s tates in the recoil

nucleon. The cont inuous b a c k g r o u n d in the energy spec t rum, aris ing f rom the

d i s in tegra t ion of exci ted s ta tes e m i t t e d in the direct ion ~l~b' should no t be too

d is turb ing , at least for the first 1 GeV down f rom tim energy of the elast ical ly

sca t t e red nucleon.

R I A S S U N T O (*)

Si ~ suggerito che lo scattering elastico di protoni di alta energia pub essere consi- derato come somma di un contributo statistico dipendente dall'energia totale s e di un termine di diffrazione dipendente dal quadrimomento trasferito ](t). Una dettagliata analisi delle misure di Cornell-Brookhaven mostra chc questo non ~ possibile, anche entro ipotesi generali circa ](t), ms che il ternfine (< diffrattivo )>, per un dato t, decresce fortemen~e per s crescenti.

(*) T r a d u z i o n e a vura del la Redaz ione .