IL NUOVO C I M E N T 0

VOL. IX, N. 2

16 Luglio 1958

Elastic Scattering and Intrinsic Structure ol Elementary Particles. D . I. BLOHINCEV, V. S. ]~ARA~ENKOV a n d V. G. GRI~IN The J o i n t I~.stitut+ /or Nuclear R,esearc]~ - Dubtta

(ricevuto il 21 Marzo 1958)

Summary. - - Experimcut~tl (L~t~ (') on elastic sc~dt.ei'ing of ,~-mesons ,)n protons with energy E ~ 1.3 GeV have been analysed. I t is sh(nw~ th;tt from the ml;~lysis of t;heir a.n/ular (lis{ribution it is I)ossible lo determine the root-mc~m-sqm~re ra.(lius and to ~'(,J: (he (l~ta ~ll)out fh(; disiribut~io,~ of m:,.tler inside the n,w~]eon. The rooi-mean-squ~re <(pion r~(lius ,> is

fon.nd to be equal to ~¢/
1.-

Introduction.

P a r t i c l e s t r u c t u r e is d e t e r m i n e d b y s t u d y i n g t h e e l a s t i c s c a t t e r i n g of s o m e r~ys b y this p a r t i c l e . W h e n o b s e r v i n g s u c h s c a t t e r i n g i n p a r t i c l e e n s e m b l e s we g e t t h e <~m e a n optic,~l i m a g e ~) of t h e p a r t i c l e f r o m w h i c h one m a y o b t a i n t h e s p a c e - t i m e p i c t u r e of t h e i n t r i n s i c s t r u c t u r e of t h e p a r t i c l e w i t h r e s p e c t t o t h e s e l e c t e d r a y s . W e s h o u l d use s h o r t e r w a v e s , if we i n t e n d t o o b t a i n m o r e detailed information about particle structure. I n Sect. 4 we s h a l l d i s c u s s t h e p r i n c i p l e l i m i t s for tile c o n s t r u c t i o n of s u c h o p t i c a l i m a g e of t h e

particle

a n d t h e l i m i t s of a p p l i c a b i l i t y of t h e space

s t r u c t u r e of e l e m e n t a r y p a r t i c l e s . T h e e x p e r i m e n t s o n t h e e l a s t i c s c a t t e r i n g of f'~st e l e c t r o n s o n a n u c l e o n carr i e d o u t b y H o f s t a d t e r ' s g r o u p w h i c h a l l o w e d t o d e t e r m i n e t h e f o r m - f a c t o r of (1) M. CIIRETIEN, I. LEITNER, N. P. SAMIOS, M. SCI1WARTZand J. ~TEINB]ERGER: P h y s . Rev., 108, 383 (1957).

250

D. I. BLOIIINCEV, V. S. B A R A S E N K O V

and

V. G. GRI~IN

t h e electric charge a n d the n u c l e o n ' s m a g n e t i c m o m e n t (2.3) are p r e s e n t l y t h e o n l y e x a m p l e of m e a s u r e m e n t s of t h e s t r u c t u r e of e l e m e n t a r y particles. H o w e v e r , t h e s t u d y of the elastic s c a t t e r i n g of fast particles of o t h e r kinds also gives t h e possibility for o b t a i n i n g i n f o r m a t i o n a b o u t t h e n u c l e o n ' s a n d n u c l e u s ' s t r u c t u r e . This i n f o r m a t i o n c o n t r i b u t e s to the d a t a o b t a i n e d f r o m t h e electron s c a t t e r i n g e x p e r i m e n t s which, s t r i c t l y speaking, in their t u r n give inform a t i o n only a b o u t t h e d i s t r i b u t i o n of the electric charges a n d c u r r e n t s inside t h e particles u n d e r i n v e s t i g a t i o n , i.e., a b o u t the (, electric ~) particle structure. F u r t h e r , a detailed analysis of 1.3 GeV ~ - m e s o n s c a t t e r i n g on p r o t o n is g i v e n (1). This analysis, as it will be shown, m a k e s it possible to get inform a t i o n b o t h a b o u t t h e ~(n u c l e a r )) s t r u c t u r e of a n u c l e o n a n d a b o u t its (( nuclear ~) or (( pion ~) radius.

2. - P h a s e - s h i f t

analysis.

As is k n o w n , t h e differential cross-section of the elastically s c a t t e r e d p a r t icles m a y be p r e s e n t e d in the l o r m (*):

(1)

da~

)~2 l~0

H e r e u s u a l n o t a t i o n s are used, in particualr, flz = exp [-t-2i~z] , where +7, is t h e c o m p l e x s c a t t e r e d w a v e phase. O n t h e basis of t h e o r e t i c a l considerations (~) a n d fl'om t h e direct c o m p a rison of cnlculation w i t h e x p e r i m e n t a l d a t a (5+) it follows, t h a t in sufficiently h i g h e n e r g y regions E ~ E* (E* ~ I GeV for ,-:--mesons a n d 5 GeV for nucleons) t h e real p a r t of the p h a s e m a y be p u t equal to zero w i t h sufficient

(2) E. ]~. CHAMBERS and R. HOFSTADTER: C E R X ,~'ymposium, 7, 295 (1956); R. HOFSTADTER" Rev. Mod. Phys., 28, 214 (1956). (3) D. R. YEN~IE, M. M. L~v¥ and D. G. RAVENflALL: leer. Mod. Phys., 29, 144 (1957). (*) For simplicity we do not take into account spin dependence of tile interaction and neglect the <er. Teor. Fiz., 33, 1051 (1957). (9) V. G. GRZ~IN, I. S. SAITOV a n d [. V. ~JUVILO: ~/u. Eksper. Teor. Fiz. (in print).

ELASTIC

SCATTERING

AND

INTRINSIC

STRUCTURE

O1° E L E M E N T A R Y

a c c u r a c y . I n t h i s c a s e t h e q u a n t i t y fi~ w i l l b e r e a l , p h a s e s h i f t a n a l y s i s is c o n s i d e r a b l y s i m p l i f i e d (~). T h e v a l u e s of t h e f u n c t i o n

(2)

PARTICLES

251

due to this fact the

I ( l ) = 2 I m ~ t = - - In fl,

~ r e g i v e n in F i g . 1 in t h e f o r m of a h y s t o g r a m . T o c a l c u l a t e t h e v a l u e s b e t w e e n t h e e x t r e m e e x p e r i m e n t a l v a l u e s of t h e d i f f e r e n t i a l c r o s s - s e c t i o n of t h e d i f f r a c t i o n s c a t tering" daa(0)/dK2 , t h e c u r v e s w i t h t h e l a r g e s t and the smallest curvatures were plotted 5 f r o m (~) (ef. F i g . 3). T h e e l a s t i c s c a t t e r i n g crosss e c t i o n a t z e r o a n g l e in t h i s c a s e w a s n o r m a l i z e d t o t h e t o t a l c r o s s - s e c t i o n a~ = G , d - a i , ~= = {33.2 ~ 3 ) m b according to the optical theor e m (lo). I n a ( . c o r d a n e e w i t h so p l o t t e d c u r v e s t w o l h y s t o g r a m s a r e g i v e n in F i g . 1. S o l i d c u r v e s a r e d r a w n t h r o u g h t i m r c ( ' t a n g u h ~ r c e n t r e s of Fig. 1. The histogram of tile the hystogr~mls. values of the function I(l) = T h e c r o s s - s e c t i o n of t i m n o n - d i f f r a c t i o n 2 Im 7h calculated for the exe l a s t i c s c a t t e r i n g da,,,JdL) -- da,/dK2 - - d a , / d . Q treme experimental values of at the energy E ] . 3 G e V in ( r : - - p ) - c o l ] i s i o n the differential diffraction scattering cross-section. Solid curves is o n l y s o m e p e r c e n t of da,~/d~ in t h e a n g l e are drawn through the centres r e g i o n 0 ~ 40 ° a n d w i t h g o o d a c c u r a c y o n e of the histogram rectangles. m'~y a s s u m e t h a t t h e f u n c t i o n s I(1) i n F i g . 1 concern only the pure diffraction scattering. T l l e e r r o r m i g h t a p p e a r d u e t o t l l e b i g a n g l e s r e g i o n , w h e r e da,,JdK2 )> d a d / d Q . I - I o w e v e r , t h e a p p r o x i m a t i o n da~(O)/dt'2 in t h i s r e g i o n of r a p i d l y f a l l i n g f u n c t i o n e x c l u d e s t h e i s o t r o p i e n o n - d i f f r a c t i o n s c a t t e r i n g (*) (see also Sect. 3). I n a c c o r d a n c e w i t h t h e v a l u e s I(1) (see F i g . 1) t h e c r o s s - s e c t i o n w e r e calculated: 10

(3)

a t = a~,, + a~ ;

a~, = ~ a , f l ) ; l=o

w h i c h a r e in g o o d a g r e e m e n t

-- ( 7.7--

at :

aft) , 1-O

with the experimental values:

a:,, = ( 2 4 . 6 . 2 9 . 0 ) a,

a~ =

mb ;

7.9) rob;

(32.3--36.9) mb;

a~:" - - 26 m b , a .... d =

(7.5 _L ~ 1.2) m b ,

a~*" = (33.2 d: 3) m b .

(*) Tile non-diffraction cross-section a.,~ is ~ 15% of a~,, due to great angles. (lO} L. I. LAP]DUS: 2U. Eksper. Teor. Fiz., 31, 1099 (1956).

252

D . I . BLOHINCEV, V. S. BARA~ENKOV &lld V. G. (;ItI,~IN

The calculated values

/l~.(l) = a~n(l)/a~

and

,4~(1) = a fl)/a,

(in per cent}

are given in Fig. 2. I t is seen from this figure t h a t with l > 6-7 partial cross-sections rapidly decrease with the increase of 1. The angular distribution of elastically scattered particles reconstructed ia consistence with (1) b y the first ten values are in good agreement with the initiul curves. The insignificant contribution of terms with gre~t values of l, which were not taken into account, is due to the fact t h a t pion nucleon innteraetio is short-range one. 25 ,.~ ln

20

10

-el

3

z.

5

6

?

3. - Q u a s i - c l a s s i c a l

B

9

10-

Fig. 2. - Relative contributions of the partial ~bsorption cross-sections A~,~(1)~ ain(/)/ff t and of the p~rtial diffraction cross-sections Aa(/): ad(l)/a, (in per cent). The indices ~, + ~> and <~--~> distinguish respectively the curves drawn for the eases of the differential diffraction scattering cross-section with the largest and the smallest curvatures. (See Fig. 3).

a p p r o x i m a t i o n and proton structure.

A t high energies of tile scattering particles, when the wave length ~ becomes considerably smaller in comparison with the dimensions of the scattering system and the relative change of the absoi~ption coefficient in nuclear m a t t e r in the interval 2 is A K / K < 4 1, the qu~si-classical a p p r o x i m a t i o n is applicable with good accuracy. I n our case Z = 0.28-10 13 cm and some times less t h a n the nucleon dimensions. Using the values I(1) according to the well-known for.~ulas (t~) and assuming t h a t the nucleon is purely ~bsorbing and r ~ Z ~ / i ( l ~ - ] ) ~ l , the cross-sections a~, ~ (25.5 ~ 1 . 5 ) rob; ad--~ (7.4 ~ : 0 . 1 ) m b were calculated. Tile angular distribution of elastically scattered particles is represented, in Fig. 3, b y the dotted lines calculated in accordance with (t~). Solid curves designate the extreme values of experimental angular distribution from (~) with the largest and the smallest curvatures. The good agreement of the caleulated magnitudes with tile corresponding ones calculated in the previous section and with their experimental values m a y be considered as one of the justifications of the further application of

(11) S. FERNBACH, R. SEIB.BEt~and T. B. TAYLOR: Phys.

Rev., 75, 1352 (1949).

ELASTIC

SCATTERING AND INTRINSIC

STRUCTUR~

O1," E L E M E N T A R Y

PAICTICLES

253

the quasi-classi(;a] ~pproxima.tion. Using this :rpproxim:~tion, from the integral e q u a t i o n d e t e r m i n i n g t h e ima,g'inary p a r t of t h e pha.se

o , Ci0" ( 1 ~ 2 6 C ¢ ~ 2

y~

( h e r e L = l , , , ~ ) w e e~m e a l ( . u l a t e t h e p i o n t~bsol'ption c o e f f i c i e n t in nueh, ons us a, fun('ti(m of the distan(,e r from the mwleon c e n t r e u s i n g t h e k n o w n v a h w s of I(1). F o r t h i s p u r p o s e we r e w r i t e e q u a t i o n (4) in the form :

•

\

O6

, 10"

(~)

z(e)

20 ~

30 ~

~0 °

50"

6~°

70 °

80 ~ 90 ~

,)dr , Fizz. 3. Solid curves show lhc exl r e m e e.Xl)crimenlal values(,f the diffen,nlial (tiffraciion sea.tterin~ crosss('cti(m (xxith the larg(,st or the s/hal. lest ( q l F V I I I l l F O S ) . l ) ~ t s h i ' d (qll'v(h'; s h o w the l~nKula.r distribution of the dif-

where

Q(r, 0) : /

)

r/~/r"--

/

(22

0

for

r>

~2,

for

r ~ cj.

fra,(~t, i o l l

F o r t h e mmlerie~fl s o h d i o n i t is s u i t a b l e t o p r e s e n t (5) in t h e f o r m :

(7)

s t ' I H t e l ' i l l ~ l " W]li(~}l ~t(i|~d, (~ll](ql-

l;~ted ll(!(tOl'dill~ i0 l.lll~ optical model formul:~c (~). Two era'yes correspond to the two curves for I(1) (see Fig. 1).

[j ~ f kiPi~ , i:j

= 1

where

)-L;

. -

Ki

K(rJ;

1)~ - Q(r~; ~

n

.L = J[t(~,~)];

~ = (}--[)L/'a b e i n g t i l e m e a n p o i n t of t h e j - i n t e r w ~ l . T h i s line~rr e q m ~ t i o n s y s t e m h~s ({ trbm~'ul:~r f o r m )> in v i r t u e of (6) a n d t i l e s o l u t i o n m~y be e~,sily f o u n d b y s u c c e s s i v e s u b s t i t u t i o n s . T h e f u n c t i o n

K(r) t h u s e ~ h ' u l ~ t e d is r e p r e s e n t e d in F i g . 4.

This function determines the ~ p i o n s t r u c t u r e ~) of ~ p r o t o n ~ v e r a ~ e d o v e r t h e spa~ce interv~fl A r ~ 2. F o r t h e r o o t - m e a n - s q m ~ r e (~p i o n r a d i u s ~ of ~ p r o t o n L

(s)

( r 2>

L

4K(r) 0

,

r 2 K ( r ) dr, 0

~54

D.I.

BLOHINCEV, V. $. BARA~ENKOV

and

V. G. GRISIN

the following value was obtained <9)

= (0.82

o.o6).1o-,

cm,

which coincides with the ~ electromagnetic radius ,~ of a p r o t o n obtained from H o f s t a d t e r ' s group experiments (2.3). As is seen from Fig. 4, the absorption coefficient essentially increases in the central region of the proton. However, the values ~0~ c m in this region are n o t quite precisely determined, as they are dependent on the approximation of da~(O)/d[2 in the large angle interval. This inaccuracy in the separation of the Y I diffraction scattering decreases rapidly with the energy increase, as the fraction of the I non-diffraction elastic processes becomes nep ~1013Cm 0 I | ! | i i F"--"-~ ' gligible. 01 03 0 5 0 2 0.9 11 13 15 17 So with E : 5 GeV, a,,,= 0.06%a~,,; with Fig. 4. - The absorption coefficient E : 7 GeV, a,~.,: 0.014 °/oa~,' (calculation acK = K(r) as a function of the discording to the statistical theory (~2)). tance from the nucleon centre. I f in the peripheral regions of the proton Two curves correspond to the two extreme experimental values of the r:-mesons are mainly present one m a y assume that angular distribution of the diffraction scattering (with the largest (10) K ( r ) : K . o~(r), or the smalles curvatures). where K is the energy dependent coefficient of the meson absorption b y tile peripheral ~-meson field, and ~(r) is the mean density of the ::-meson cloud near the point r. Within the accuracy of the experimental data the analytic form of o(r) m a y be ~pproximated b y different curves of the type described in (~). I n the central regions of the nucleon K0") is very likely to be determined b y the other kinds of particles {nucleons, hypcrons, K-mesons) and the formula (10) is not applicable.

4. - A possible e x p e r i m e n t a l criterion for t h e e x i s t e n c e of an e l e m e n t a r y l e n g t h .

During the last years the idea t h a t there m a y really exist a limit of ~pplicability of the conventional space-time description of the particle structure connected with the existence of a certain (, elementary length )) has been fre(13) V. S. BARA~ENKOVand V. M. MALTSEV: Acta Phys. Pol. (in print).

ELASTIC SCATTERING

AND INTRINSIC

STRUCTURE

OF ELEMENTARY

PARTICLES

25~

q u e e r l y suggested. This idea was expressed in different versions of the t h e o r y -of (( n o n - l o c a l fields ~) or (~non-local interaction ~) (see, e.g., (~)). Such theoretical schemes lead to form-factors which w e a k e n the interaction for the s h o r t waves. I n tMs w a y one m i g h t hope to eliminate the divergencies from the m o d e r n q u a n t u m t h e o r y due to v e r y s h o r t wavelengths. H o w e v e r , this p o i n t m ~ y be the weakest one in the (( non-local ,~ theories (*). G. V. WATAGHIN ~nd E. FERMI were the first to notice in their statistical t h e o r y of multiple p r o d u c t i o n (15) t h a t at high energies the interaction becomes not a weak~ b u t , on the c o n t r a r y , ~ strong one. The calculations show t h a t the weak (in the sense of the generally accepted classification) i n t e r a c t i o n of F e r m i t y p e (v, e, ~) also becomes strong at high energies (cross-section > ;~2) (~6-1s). N o w we should like to p u t a question: u n d e r w h a t conditions from a p u r e l y empirical s t a n d p o i n t would it be possible to speak a b o u t the nonlo('ality? E v i d e n t l y , these conditions should occur when it would become impossible to use the el;~sti(; scattering of p~,rticles as ~ me{ms of s t u d y i n g their str|l('ture. Thus, the m a t t e r depends Ul)On t h e a s y m p t o t i c behaviour of the ('ross-sections at high energies. If with ~ -> 0 all the elastic s(.:ttterings for a certain internal region R will tend to the diffraction s c a t t e r i n g on a (( black sphere ~) of radius R, the elastic s c a t t e r i n g will cerise to give i n f o r m a t i o n a b o u t the intrinsic structure of this region nnd the m a x i m u m information will be limited by the d a t a ~bout the outer dimensions of the (( sphere ,. The scattering cross-section which is due to the process in this region will be equal to ~R 2 (>>z)~2) a n d equals the corresponding inelastic scattering. I n this ease, instead of the description of the space-time structure, the problem a b o u t possible w a y s of particle t r a n s f o r m a t i o n will become i m p o r t a n t . The m a g n i t u d e of R f r o m tMs point of view is the same length scale which d e t e r m i n e s the real non-locality, i.e., the limit of the applicability of spacet i m e description of the particle structure. I t can be seen f r o m the analysis of the pion scattering on protons t h a t the central region of the nucleon tends to a p p e a r as (i black ~).

(13) D. I. BLOIIINCEV: Usp. Fiz. Nauk, 61, 137 (1957); V. S. BARASENKOV: Nuovo Cimento, 5, 1469 (1957). (*) This circumstance was also noted by M. A. MARKOV(14). (la) M. A. MARKOV: Usp. Fiz. Nauk, 51, 317 (1953). (15) C~. WATAGIIIN: Symposium sobre raios cosmicos, Rio de Janeiro (August 4-8, 1941); E. FEICMI: Progl'. Theov. Phys,, 5, 570 (1949). (16) D. I. BLOIIINCEV: Usp. Fiz. Nauk, 62, 381 (1957). (17) I. E. TAMM: Zu. ~)ksper. Teor. Fiz., 32, 178 (1957). (is) V. S. BARA~ENKOV:Proceeditegs o/ the Co~]erence in Padua-Venice (1957); Nucl. Phys. (in print).

256

D.I.

BLOHINCEV, V. S. BARASENKOV a n d v. G. GRISIN

F r o m the point of view given here the further s t u d y of the energetic dependence of elastic pion scattering m a y be of principal importance. I t can be said t h a t the dimension of the non-locality radius R must not be a universal length So; it m a y depend upon the kind of the interaction. The m i n i m u m scale of the space-time description R determined b y the dimensions of the (~black sphere )~ m a y be introduced into the t h e o r y in a relativistic in variant way. Indeed, the scattered wave phase ~]~.is an invariant. W e intend to consider it as a function of two inw~riants (19): D = F,,F" ~ ,,q),,

(ll)

a,nd

F = PI,P' + (P"q)~'P q),,cp ,,

Here ~D is the four-dimensional e n e r g y - m o m e n t u m vector of tile whole system whereas P;, is the same for tile relative motion of the incident particle and the particle of the scatterer, and finally

Here e,,,p is the fully a n t i s y m m e t r i c a l unit tensor of fourth rank, Ms.~ is the m o m e n t u m a n t i s y m m e t r i c a l tensor. Using these invariants the ((black sphere)) m a y be determined as follows: (13)

~7~~ 0

if

DIE > R ~

and

,@ = + ic~,

if

D/F < Rz ,

for F ~ Po, where Po is the great value of the m o m e n t u m at which the opacity occurs. The q u a n t i t y D/F is an operator, therefore, the inequalities (13) are determined for its eigenvalues. I t can be easily seen t h a t in tile center of mass system ( ~ = 0)

D/F--

M ~

h~l(1 + 1 )

V~o -

p~,

where M is the three-dimensional angular m o m e n t u m , while Po is the threedimensional m o m e n t u m of the relative motion. DIE determines the collision p a r a m e t e r in a relativistic-invariant way. :Note in conclusion t h a t in p e r t u r b a t i o n theory there are k n o w n ((propagation functions ~) which lead to diver~'cncies in the region of great frequencies. These functions are constructed with the help of tile plane waves which were used as ,~ zero approximation. Meanwhile at gre~t frequencies (19) Yu M. ~IROKOV: ~U. Eksper. Teor. Fiz.. 21. 748 (1951); M. A. 5[ARKOV:Dokl. Akad. Nauk SSSR, 101, 449 (1955L

ELASTIC SCATTERING AND INTRINSIC STRUCTURE OF ELEMENTARY PARTICLES

257

o f t h e field in t h e presence of particles the plane w~ve will be quite a bad ~ p p r o x i m ~ t i o n due, to t h e diffraction scattering. I n s t e a d of the plane wave o n e o u g h t to t~ke a series expansion a c c o u n t i n g for the shal~p change of t h e w a v e field at g r e a t relative m o m e n t a of the interacting particles. This gives rise to the reh~tivistic i n v a r i a n t cut-off form-factors. However, these f o r m - f a c t o r s are n o t due to weakening of the interaction at high frequencies, as it is a s s u m e d in the c o n v e n t i o n a l non local theories but, on the e o n t r ~ r y , to its strengthening. As a whole, phenomenologically, this factor t a k e s into a c c o u n t the intensive ineb~stic processes of a n y origin which ocv u r at high energies.

R 1 A S S l T N T t ~ (*) .~i stm~ ~malizz~ti d~ti sperim~,ntali ,*ullo so,tittering eht.~tic~, di me.~oili ~ ,~ll p r o | t m i di E 1.3 GeV. ,
(*} T r a d u z i o n c

a cura della llcdazionc.