ON A G E N E R A L I N E Q U A L I T Y B E T W E E N T H E

S L O P E O F D I F F R A C T I O N P E A K A N D T O T A L

C R O S S S E C T I O N

G. G. A r u s h a n o v a n d I . I . P i r m a t o v UDC 539.101

We consider the general inequality

where

> ~ (1) 16~el a

d ( d ~ / (2)

represen ts the slope of the diffraction cone, a t is the total c ross section of interaction, O-e [ is the total c ross section of elast ic scat ter ing, and 5 is the ra t io of the rea l par t of the amplitude of forward s ca t t e r - ing to its imaginary par t . As we shall see f rom a compar ison with the experiment , inequality (1), which must be sat isf ied for sufficiently large values of energy, is close to equality.

The resul t of type (1) can be obtained in different models . In the s imples top t ica l mode, when a -~ R 2 /4 , o- t = ~rR 2, where R is the radius of the region of interaction, we have

1 -~ ~ t /4~ a. ( 3 )

The right hand side of (1) differs f rom the right hand side of (3) by the factor at(1 + 52)/4ael, which is close to unity. It is well known that the differential c ross section of e last ic sca t te r ing inside the diffraction cone is descr ibed by the express ion

ct__!a _~ [ct:~ e~t, (4) dt \ dt ]t=o

where according to the optical theorem

- ~ t=o = d (l + ~,)/16 ~, (5)

while outside the diffraction cone the experimental data lie considerably above the curve given by (4). Therefore , integrat ing (4) over t we obtain resu l t (1). Because of the sharp decrease of the differential c ross section with the growth of the t ransmit ted pulse smal l values of ItJ give the dominant contribution to the elast ic sca t te r ing c ross section. Hence inequality (1) obtained in this way must be close to equality. Finally, a formula of type (3) was derived and analyzed recent ly by A. B. Kaidalov [1] within the f r ame- work of the theory of complex angular moments taking Mandel' s tam branch points into considerat ion.

The p r imary objective of the present note is to s t r e ss the general nature of inequality (1) and com- pare it with the exist ing experimental data. In fact, the following res t r ic t ion on the slope of the diffrac- tion peak at high energies was obtained in [2] f rom completely general considerat ions:

a ~ ~:-(l + ~'-') aoz ' -

where a0R and aoi are in ter re la ted through the equation

aoR -- ~ aos = 2:et/:t. (7)

V. L Lenin Tashkent State Universi ty . Trans la ted f rom Izvest iya Vysshikh Uehebnykh Zavedenii Fizika, No. 11, pp. 107-109, November, 1972. Original ar t ic le submitted July 20, 1971.

C 1974 Consultants Bureau, a division of Plenum Publishing Corporation, 227 g;est 17th Street, New York, N. Y. i0011. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission of the publisher. ,~ copy of this article is available from the publisher for $15.00.

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Solving this equation for a0I and substituting into (6) we find the extremum /~ ~_ r (minimum) of the right hand side of (6) with respec t to a0R and immedi- 0.9 ately a r r ive at the resul t (1). Similarly, f rom [3] we have the inequality t 15 2 for the nth derivative

o,9 at" ~ -dr t=o t=o ,.

g5 t g 4 e gO ~ 6ev /C A mer i t of inequalities (1) and (8) is the possibil i ty of their direct ve r i - fication by experiment , since both par ts of these inequalities contain

Fig. 1 qualities that can be measured in experiment . Making use of the ob- vious inequality a t > ae l , f rom (1) we have

a > ~t (1 + ~2)/16~ > ~t/16~, (9)

which is Mart in ' s resul t [4] for the slope of the square of the imaginary par t of the amplitude. We com- bine this inequality with the res t r ic t ions on the total interact ion c ross section f rom below. Using quite general assumptions J in and Martin [5, 6] and Sugawara [7] have shown that

~t ~> C/E, (1 O)

where E is the energy of the incident par t ic le in the labora tory sys tem of re fe rence . On the basis of the works of Khuri and Kinoshita [11, 12] Vernov [8-10] proved a considerably s t ronger inequality

~t ~ cE 2~-I-', (11)

provided that

~, I ~ ctg ~ , 0 ~ 1/2; (12)

where e is a positive number as smal l as desired. It follows f rom (11) that in the case of asymptot ical ly purely imaginary amplitude (c~ = 1/2) a t may decrease only more weakly than any power of the energy (roughly speaking, it can decrease only as a logari thmic power) [8-10]. F r o m (9) we see that a s imi la r s ta tement is valid also for the slope of the diffraction peak.

The right hand side of inequality (1) is represen ted in Fig. 1 for the case of ~ + p- interact ion, for which it has been measured as a function of the momentum PL of the incident pion in the laboratory sys tem of coordinates . The experimental data a re taken f rom [13-15]. It is evident that within the experimentaI e r r o r s (less than 10%) it is close to unity f rom below even for smal l energies . A noteworthy feature of the right hand side of (1) is its constancy inspi te of the noticeable s t ruc ture in the energy dependence (at low energies) of the individual factors in the right hand side. The right hand side of (1) for pp-sca t te r ing and other p rocesses has a s imi l a r nature . The best measured quantity is at; the measurements of ~el and a are not very accura te . If one of these is measured more accura te ly than the other (or the other is, in general , not measured) , then for this other quantity a lower bound can be obtained f rom (1), which will be hopefully close to the rea l value.

The authors thank Prof . S. A. Azimov for d iscuss ions .

LITERATURE CITED

i. A.B. Kaidalov, Paper Presented at the Session of the Department of Nuclear Physics of the Acad-

emy of Sciences of the USSR [in Russian], Moscow (1971). 2. G.G. Arushanov, Izv. Vuz. SSSR, Fizika, No. 12, 86 (1967). 3. S.A. Azimov and G. G. Arushanov, Izv. Vuz SSSR, Fizika, No. 4, 103 (1970).

4. A. Martin, Phys. Rev., 129, 1432 (1963). 5. Y.S. Jin and A. Martin, Phys. Rev., 135B, 1369 (1964). 6. Y.S. Jin and A. Martin, Phys. Rev., 135B, 1375 (1964). 7. M. Sugawara, Phys. Rev. Letters, 14, 336 (1965). 8. Yu. S. Vernov, Zh. Eksp. Teoret. Fiz., 50, 672 (1966). 9. Yu. S. Vernov, Zh. F, ksp. Teoret. Fiz., 5_33, 191 (1967).

I0. Yu. S. Vernov, Yade. Fiz., i0, 176 (1969). ii. N.N. Khuri and T. Kinoshita, Phys. Rev., 137B, 720 (1965). 12. N.N. Khuri and T. Kinoshita, Phys. Rev., 140B, 706 (1965). 13. V . S . Barashenkov, Cross Section of Interaction of Elementary Par t ic les [in Russian], Nauka,

Moscow (1966).

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14. V . S . Barashenkov, Fortschrit te d. Phys . , 14, 741 (1966). 15. T. Lasinski, R. Levi Setti, and E. Predazzi, Phys. Rev. , 179, 1426 (1969}.

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