Fortschritte der Physik / Progress of Physics: Band 14, Heft 12 1966 [Reprint 2021 ed.] 9783112500422, 9783112500415

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FORTSCHRITTE DER PHYSIK HERAUSGEGEB EIN IM AUFTRAGE DER PHYSIKALISCHEN GESELLSCHAFT IN DER DEUTSCHEN DEMOKRATISCHEN REPUBLIK VON F. KASCHLUHN, A. LÖSCHE, R. RITSCHL UND R. ROMPE

B A N D 14 • H E F T 12 • 1966

A K A D E M I E

- VERLAG-

•

B E R L I N

I N

HALT

V . S . B A R A S H E N K O V : Dispersion Analysis of Elastic Scattering of High Energy Particles 741 A. A. MAKAROV, NGUYEN VAN H I E U , P. W I N T E R N I T Z : Antiproton Annihilation into Two Mesons and Higher Symmetries 771

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Fortschritte der Physik 14, 7 4 1 - 7 6 9 (1966)

Dispersion Analysis of Elastic Scattering of High Energy Particles Joint Institute for Nuclear Research, Laboratory of Theoretical Physics, Dubna V . S . BAEASHENKOV

I. Introduction I t is well known that the situation in strong interaction physics is at present very difficult. Except for the results obtained on the basis of formal symmetry rules and of a number of semiphenomenological and very often inconsistent models, strong interactions remain still a "thing in itself". Even those effects as elastic scattering of nucleons or single pion production, which seem to be relatively simple, turn out to be inaccessible for accurate calculations. As a matter of fact, we do not even know what is the main reason for our failures: either we are not able to solve complicated systems of field equations or, perhaps, the ideas underlying our concept of the nature of strong interactions are imperfect. Of special importance, from this point of view, is an experimental check of the results and conclusion which are a consequence of the basic principles of the theory and not associated with any approximate methods of calculations. This is the reason for great interest which has been recently paid to a dispersion analysis of small-angle elastic scattering of high-energy particles. A prominent feature of dispersion relations is that they establish on the basis of the most general postulates of quantum field theory a relationship between the experimentelly observable physical quantities: total cross sections for particle and antiparticle interaction with a target and the real part of the amplitude for elastic scattering of particles at zero angle. A violation of dispersion relations, therefore, will point out to an invalidity of some general principles of the theory and in the first turn — the causality principle, which plays the most important role in the derivation of dispersion relations. When comparing dispersion relations with experiment we do not, of course, verify the causality principle in its general philosophical meaning. The question is to clear up the space-time limits of applicability of the concrete formulation of the causality principle which is used in modern quantum physics. In its general form the causality principle is essentially that each phenomenon in nature gives rise necessarily to another phenomenon — its consequence, and vice versa: any phenomenon was caused by some other phenomenon of nature — its cause1). Now we have no reasons to doubt the validity of this principle. However, A detailed dicussion of different philosophical formulations of causality can be found e.g. in the monographies [1, 2]. 54

Zeitschrift „Fortschritte der Physik", Heft 12

742

V . S . BARASHENKOV

in the field theory a more limited formulation of this principle is used, according t o which it is assumed t h a t in any whatever small space-time region the velocity of signal propagation does not exceed t h a t of light; it means t h a t any event occuring in the physical system can effect the evolution of this system only in the future and cannot at the moments of time preceding the given phenomenon (see e.g., monography [3]), the requirement of such a "microscopic causality" restricts the theory essentially, but makes it possible to obtain a number of import a n t experimentally verifiable results, in particular to ground the derivation of dispersion relations. I t is worth while noting t h a t the formulation of the causality principle used in the field theory is essentially an extrapolation of the corresponding macroscopic formulation to the region of very small space-time scales and it is not at all clear beforehand t h a t such an extrapolation is justified. A variation of the formulation of microscopic causality, if required by experiment, will lead to great changes in our concepts of the nature of phenomena in the subatomic world. II. Zero-Angle Elastic Scattering Amplitude The differential cross section for elastic scattering of a particle with energy T is expressed in terms of the imaginary and real parts of the amplitude 2 ) a(T,B) = \D{T,8)

+ iI{T,e)\*=

\D{T,e)\*+

|I(T,0)|2

(1)

I n the case of scattering at the angle 0 = 0 the imaginary part of the amplitude can be expressed, b y using the optical theorem, in terms of the coresponding total cross section a t (T): I(T) = I{T,0)

= ^-

B

± =

0

(9>

The first of these identities can be satisfied by putting

E*jE

(In doing so we have taken into account that A± = B± = 0 , see expression (8))«). Since at all the values of x ^ 1 the integral J > 0, then the difference A D (T) also remains positive over the whole region T > 30 GeV [92]. In particular, for T*jT < 1 and % = 0.5

~

^ T ~ 0.26 ^ ( G e V ) 10" 1 3 cm

(24)

) More exactly: the numerical coefficients in (22) are calculated for T* = 20 GeV, however at T*jT < 1 the shift of T* by 1 GeV nonessential^ affects the quantity A D(T),

6

762

V. S. Babashenkov

(laboratory system) and, consequently, oc.iT)

-

a+{T)

=

±

1

tan

+

1 fM

O-ex(T)

0

0.29 j/?1 (GeV)'

27.

1 2 (4ti)2

I

MT

_ yi-2x Ma

2

1 + tan2

(1 —

x)jt

(25)

+

0

(?)

Ma 2

~ 0.31 mb ~~ 32 7i2

(26)

(in c.m.s.). It should be stressed that irrespective of any theory, at the point where D-(T) = D+(T) the charge exchange cross section a e x ( T ) must decrease up to its minimum value ffex

(T)

m l n

=

1

( A ) V

+

[ T ) ~ a- [T)}\

(27)

The presently available experimental data (see Fig. 6) over the whole energy region T > 1.3 GeV are about as twice as large than 3 0 G e V and 0-{T) for T > 1 9 G e V were approximated by expression

F i g . 12. E n e r g y dependence of| t h e r a t i o oi t h e real a n d i m a g i n a r y p a r t s of t h e a m p l i t u d e of elastic p — p a n d p — p scattering a t a n angle 6 = 0. T h e values of a + ( T ) a t T < 0 . 7 G e V are calculated b y t h e SODING m e t h o d [7], b u t for a n o t h e r choice of t h e asymptotics of t h e cross sections a ± (see t h e t e x t ) . The curve