Fortschritte der Physik / Progress of Physics: Band 28, Heft 5 [Reprint 2021 ed.] 9783112571309, 9783112571293

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

H E F T 5 • 1980 • B A N D 28

A K A D E M I E - V E R L A G EVP 1 0 , - M 31728

•

B E R L I N

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Zeitschrift „Fortschritte der Physik" Herausgeber: Prof. Dr. Frank KaschJuhn, Prof. Dr. Artur LSsche, Prof. Dr. Rudolf Ritsohl, Prof. Dr. Robert Rompe, im Auftrag der Physikalischen Gesellschaft der Deutsohen Demokratischen Republik. Verlag: Akademie-Verlag, DDR - 1080 Berlin, Leipziger StraBe 3 - 4 ; Fernruf: 22 36221 und 22 36 229; Telex-Nr. 111420; Bank: Staatsbank der DDR, Berlin, Konto-Nr. 6836-26-20712. Chefredakteur: Dr. Lutz Rothkiroh. Anschrift der Redaktion: Sektion Physik der Humboldt-Universität zu Berlin, DDR - 1040 Berlin, Hessische Straße 2. Veröffentlicht unter der Lizenznummer 1324 des Presseamtes beim Vorsitzenden des Ministerrates der Deutschen Demokratischen Republik. Gesamtherstellung: VEB Druckhaus „Maxim Gorki", DDR - 7400 Altenburg, Corl-von-OssieUky-Straße 30/31. Erscheinungsweise: Die Zeitschrift „Fortschritte der Physik" erscheint monatlich. Die 12 Hefte eines Jahres bilden einen Band. Bezugspreis je Band 180,— M zuzüglich Versandspesen (Preis für die DDR: 120,— M). Preis je Heft 15,— M (Preis filr die DDR : 1 0 , - M). Bestellnummer dieses Heftes: 1027/28/5. (c) 1980 by Akademie-Verlag Berlin. Printed in the German Democratio Republic. AN (EDV) 57618

ISSN 0015 - 8208 Fortschritte der Physik 28, 237-258 (1980)

The Pomeranchuk Theorem and Its Modifications J A N FISCHEK a n d R U D O L F SÂLY

Institute of Physics, Czechoslovak Academy of Sciences, Prague,

CSSR1)

Contents 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

Introduction The Pomeranchuk-Okun hypothesis The original Pomeranchuk theorem Basic properties of the scattering amplitude Mathematical refinements and extensions on rising cross sections Present status of the Pomeranchuk relation based on dispersion relations only Use of unitarity, analyticity in t and of isospin invariance Rates of the vanishing of the total cross-section difference Differential cross sections Concluding remarks References

. . . .

237 238 239 241 242 244 247 250 252 255 257

1. Introduction The concept of antiparticle raises the question whether there is a deeper connection between the scattering of a particle on a target and the scattering of the corresponding antiparticle on the same target. I t is more t h a n twenty years ago that this problem became topical in connection with the discovery of the antiproton and, also, because of an intense development of various applications of dispersion relations, in which the direct F+ and the crossing F_ scattering amplitude figure together, in one relation. A general connection, which would be valid at all energies, is not known. If, however, the primary energy of the collision is sufficiently high, we can use the Pomeranchuk theorem [1] stating that a+(E) and a_(E) should approach each other as E tends to infinity. Here, a+(E) and a_(E) is the total cross section of the particle-particle and antiparticle-particle collision, respectively, and E is the energy of the projectile in the laboratory frame. Being almost a standard part of current textbooks in high-energy physics, yet, the Pomeranchuk theorem is still of rather a puzzling nature. Experimental data from every new accelerator have been confirming it and it is widely believed t h a t the asymptotic equality of a+(E) and a-(E) should be a consequence of some very general physical principles. I t has not been proved, however, that this asymptotic equality rigorously follows from local field theory. Pomeranchuk's proof rests on the dispersion relation and some additional assumptions, which never have been removed completely from the proof. Nevertheless, a continuous effort of a number of authors has resulted in Na Slovance 2, CS-180 40 Prague 16

Zeitschrift „Fortschritte der Physik", Heft 5

Jan Fischer and Rudolf Saly

238

the weaking, removal or justification of some of the original assumptions within the framework of local axiomatic field theory. In the present paper we give a review of the various modifications and improvements of the Pomeranchuk theorem and also of related statements, for which a common name "Pomeranchuk-type theorems" is often used. Historically, the Pomeranchuk theorem was motivated by the Pomeranchuk-Okun hypothesis. We briefly recall it in the next section.

2. The Pomeranchuk-Okun Hypothesis The Pomeranchuk-Okun hypothesis [2, 3] amounts to the assumption that charge exchange reactions are negligible compared both to the elastic and the inelastic cross sections at sufficiently high energies. Symbolically, {charge exchange)

(inelastic) on (elastic).

(2.1)

Note that this has nothing to do with dispersion relations. The reasoning was based on the assumption of the isotopic invariance of strong interaction, which had been confirmed by experiment. To illustrate this by an example, consider the following three processes 1) p p - > p p

(elastic scattering)

2) pn - > pn

(elastic scattering)

3) pp

(charge exchange).

fin

Since the nucleon-antinucleon system has two isotopic states with the total isospin T = 1 and T = 0, we may express the differential cross sections of the reactions 1) to 3) in terms of amplitudes which correspond to these two states. Denoting them by / and g, respectively, we obtain 1) (1/4) 1/ + g\*

2) l/l 2 3) (1/4) |/ - g\2 apart from a normalization factor. Then, using (2.1), we have \f-g\
oo 3)

\ F

±

( E ) j E \

are bounded by finite constants \ F

±

(3.1)

P'

oo. (3.2)

:

C±

( E ) / E \ ^ C

±

.

(3.3)

We stress that assumptions (3.2) and (3.3) seemed quite natural at that time. Discussing the latter condition, Pomeranchuk argued that, because of finite range of interaction in the sense that the interaction takes place within the range of the Compton wave length, only those partial waves contribute to the amplitudes F± (E) for which l < C \ j E , which implies (3.3). Suppose the limit Aa of Aa(E) = a+(E) — aJE) is not zero: lim

A a ( E ) =

4= 0 .

Aa

(3.4)

Taking into account (3.2) and (3.4) we can easily calculate the integral on the r.h.s of (3.1) and, for large enough energies, the leading term gives R e F ± ( E )

±Aa

-E

\ n E .

(3.5)

Now, the essential feature of the proof is a contradiction between the assumption (3.3) and the requirement of analyticity (3.1). Indeed, (3.5) shows that R e F±(E) is asymptotically a factor In E greater than Im F± (E) unless Aa = 0. So, the Pomeranchuk theorem 16*

240

JAN FISCHER a n d RUDOLF SALY

reads as follows: under the assumptions (3.1), (3.2) and (3.3), the total-cross-section difference Ao(E) tends to vanish at infinity: lim

Aa(E)

=

0.

(3.6)

¿?—00

A few remarks are in order. First of all we must take care in using the word "theorem". A theorem consists of assumptions and of a statement following from them. In current physical language, however, the statement itself is often called theorem. So, (3.6) is mostly called Pomeranchuk theorem although it is only a relation which, according to the Pomeranchuk theorem, holds if the conditions (3.1), (3.2) and (3.3) are satisfied. In order to avoid confusion, we shall systematically refer to (3.6) as to the "Pomeranchuk relation". This distinction allows us to discern two aspects in the contribution made by Pomeranchuk in [J]. One of them is the theorem and its proof, of course. The other aspect is, however, the very suggestion that it is the vanishing of Aa(E) which is worth proving. This was not evident from experimental data at Pomeranchuk's time (see Fig. 1). The subsequent development in high-energy physics has shown that the physical content of

GeV Fig. 1. At Pomeranchuk's time, experimental data on total cross sections up to 0.7 GeV only were available. They hardly suggest the high-energy vanishing of Aa(E). Taken from Eef. [ Ma

(4.1)

where p = {E2 - M^ and F±(E) = F±{E, t = 0) b) the partial wave amplitudes at(E) (we omit the subscripts -f- and —) satisfy the in equalities Im ai(E) ^ \a,(E)\K (4.2) (6) The interaction is invariant under rotations in the isotopic space. In the proof of the property (4) the empirical fact it used that the lightest hadron has a nonvanishing rest mass. In some cases, (4) and (5) are not used explicitly but are substituted by the Froissart-Martin bound, which follows from them: (7) There exist real constants C and E0 such that |J±(.E)| ^ CE In2 E,

for every

E >

E0.

5. Mathematical Refinements and Extensions on Rising Cross Sections If we now look into the papers following on, we observe that all the assumptions made by Pomeranchuk have come in for critical examination and were shown not to be warrantable in general. And yet, in view of the possible physical significance of the Pomernachuk relation, there has been a constant effort to make it a strict consequence of first principles. This effort has not been fully successful. Examples of functions were found satisfying first principles (including unitarity) but not possessing the property that Aa{E) tends to zero. This development, nevertheless, has led to considerable progress in the rigour of formulations and an economical selection of assumptions. W E I N B E R G [ 4 ] was the first to throw doubt of the assumptions (3.2) and ( 3 . 3 ) . Actually, on one hand, the existence of the asymptotic limits of the cross sections a+{E) looks physically plausible but has not so far been justified mathematically in any way. Therefore it amounts to an ad hoc assumption (consider, for example,