the pomeranchuk theorem and its modifications

ISSN 0015 - 8208

Fortschritte der Physik 28, 237-258 (1980)

The Pomeranchuk Theorem and Its Modifications

JAN FISCIIER and RUDOLF ~ L L Y

Institute of Physics, Czechoslovak A d m y of Sciences, Prague, CSSRI)

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 2. The Pomeranchuk-Okun hypothesis . . . . . . . . . . . . . . . . . . . . . . . 238 3. The original Pomeranchuk theorem . . . . . . . . . . . . . . . . . . . . . . . 239 4. Basic properties of the scattering amplitude . . . . . . . . . . . . . . . . . . . 241 5. Mathematical refinements and extensions on rising cross sections . . . . . . . . . . 242 6. Present status of the Pomeranchuk relation based on dispersion relations only . . . . 244 7. Use of unitarity, analyticity in t and of isospin invariance . . . . . . . . . . . . . . 247 8. Rates of the vanishing of the total cross-section difference . . . . . . . . . . . . . 250 9. Differential cross sections. . . . . . . . . . . . . . . . . . . . . . . . . . . . 252

10. Concludingremarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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. It is more than 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 Ponieranchuk theorem [ I ] 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 labo- ratory 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 froni every new accelerator have been confirming it and it is widely believed that the asymptotic equality of a+(E) and a-(E) should be a consequence of some very general physical principles. It has not been proved, however, that this asymptotic equality rigorously foIIows from local field theory. Pomeranchuk's proof rests on the dispersion relation and some additional assumptions, which never have been removed completely froni the proof. Nevertheless, a continuous effort of a number of authors has resulted in

l) Na Slovance 2, CS-180 40 Prague

16 Zeitschrift ,Fortschritte der Physik", Heft 5

238 JAN FISCI~ER and RUDOLF L%LY

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 Poineranchuk 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 i t in the next section.

2. The Pomeranchuk-Okun Hypothesis

The Pomeranchuk-Okun hypothesis [2, 31 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) < (ineZastic) M (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) Fp +- pp 2) Fn -+ Fn 3) Fp + 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 f and 9, respectively, we obtain

(elastic scattering) (elastic scattering) (charge exchange).

1) (1/4) I f + 912 2) If12 3) (V4) If - 91a

apart from a normalization factor. Then, using (2.1), we have

If - 91 < If1 . This implies that the differential cross sections of processes 1) and 2) are equal; further, since this equality holds in the forward direction too, the optical theorem gives

atot(Pp) - otot(F4

provided that the real parts are negligible. Analogous relations have been derived for other processes like pion-nucleon or nucleon- nucleon scattering [3]. The generalization of Pomeranchuk-Okun’s results to a general isotopic multiplet is straightforward [4]. The hypothesis has been confirmed with the increasing energy of accelerators during the past two decades. It is in keeping with the intuitive idea that the target particle is less and less able to distinguish between the different projectiles belonging to one isotopic multiplet if the speed of the projectpila becomes larger and larger.

The Pomeranchuk Theorem 839

This rule, however, does not apply to projectiles froni different isotopic niultiplets like pions and kaons, pions and protons and so on. Indeed, the target proton or nucleus is able to distinguish between such particles in the sense that the corresponding cross sections are significantly different from each other even a t the highest available energies and, apparently, tend to different asymptotic values. A satisfactory quantitative explanation of this fact is given in the quark model [5 ] . It is natural to ask now whether analogous asymptotic relations hold between particle- particle and antiparticle-particle scattering, no matter whether the particle and the antiparticle belong to the same isomultiplet (as in the case of x+ and rc-) or not (as in the case of the proton and the antiproton for instance). This extension of the original idea was performed by POMERAXCHUK [ l ] and by VAN HOVE [34] and LOGUNOV et al. [35] for total cross sections and differential cross sections respectively. We discuss them here in section 3 and section 9 respectively.

3. The Original Pomeranchuk Theorem

POMERANCHUK [ I ] used the following assumptions : I ) The forward scattering amplitudes P+(E) and F-(E) fulfil the dispersion relations

where O(E) involves only terms which stay bounded by const. E as E --+ m. 2) The high-energy limits of o+(E) and c ( E ) exist and are finite:

liin oh(E) = o+

3) ( F , ( E ) / E ( are bounded by finite constants Cf :

E+m (3.2)

(3.3)

We stress that assumptions (3.2) and (3.3) seemed quite natural at that time. Dis- cussing 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 Conipton wave length, only those partial waves contribute to the amplitudes F* (E) for which 1 < C fz, which implies (3.3). Suppose the limit d o of Aa(E) = u+(E) - c ( E ) is not zero:

limida(E) = d o $. 0. h r n

(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

f do 4n2

Re F*(E) N - E l 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) showsthat Re F*(E) is asyiiiptoti- cally a factor In E greater than In1 F,(E) unless 50 = 0. So, the Pomeranchuk theorein

l6*

240 JAN FISCIIER and RUDOLF &LY

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

lim Ao(E) = 0 . E-w

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 althoughit 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 (3.0) as to the “Pomeranchuk relation”. This distinction allows us to discern two aspects in the contribution made by Pomeran- chuk in [ I ] . One of them is the theorem and its proof, of course. The other aspect is, however, the very suggestion that i t is the vanishing of Ao(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

Fig. 1. At Pomeranchuk’s time, experimental data on total crow sections up to 0.7 GeV only were available. They hardly suggest the high-energy vanishing of Au(E). Taken from Ref. [66).

the Pomeranchuk relation is even more interesting than the original shape of the theorem, for some of its assumptions became out of date. Therefore many authors have been modifying the assumptions, but they have always tried to preserve the physical content of the Pomeranchuk relation. It turned out necessary, for instance, to abandon the assumptions (3.2) and (3.3) becaual they had lost their experimental plausibility after the discovery of the rise of totss cross sections in 1972 [ti]. On the other hand, there was already a firmer though lees restrictive relation a t that time to replace these phenomenological assumptions : the FROISSART bound [?‘I, which had been proved by n ’ i r l R ~ ~ ~ [8] to be a rigorous consequenc- of local field theory, unitarity and the assumption that the lightest hadron has a none zero mass. Similarly, also the dispersion relation (3.1) was just an assumption twenty years ago for processes like the proton-proton or kaon-proton scattering. A weaker form of analyticity was proved for them later by BROS, EPSTEIN and GLASER [9]. They showed that these scattering amplitudes are analytic a t least in the upper half of the complex energy plane except a semicircular disc of unknown, finite radius around the origin. This “re- duced” analyticity property, nevertheless, is able to produce asymptotic theorems of


similar kind as the dispersion relation itself. Details will be discussed later in Sect. 5 and 6. Finally, let us point out that (3.6) is not the only form to express the intuitive idea that a+(E) and a-(E) approach each other at very high energies. Indeed, there are a number of other possible ways of expressing i t ; for instance, to consider the limit of A u(E) on a sequence of discrete energies tending to infinity, to take the continuous or a discrete limit of the ratio o+(E)/a-(E), to use various integral conditions, etc. These are possibilities which also can be used to describe the physical content of the Pomeranchuk relation. Some interesting possibilities will be discussed in Sections 6 and 7. After these remarks one could be under the impression that it is only the relation (3.6) which has some physical value, whereas the assumptions and the derivation of the Po- meranchuk theorem are of historical interest only. However, Pomeranchuk's assumption on the constancy of the total cross section has had far-reaching consequences because its interpretation in terms of Regge theory has led to the concept of the Pomeron and to postulating its dominant role in the high-energy limit. In the next section we give a survey of the properties of the scattering amplitude which follow from local field theory and are usually adopted as the basis of the S-matrix approach.

4. Basic Properties of the Scattering Amplitude

Let us consider the two reactions

(+) a + b + a + b

(-) d + b + d + b

where a and b are hadrons with masses M , and M , respectively, the bar denoting the corresponding antiparticle. The scattering amplitnde is a function of the kinematic in- variants s, t and u. All the theorems connected with the Pomeranchuk relation rest on the amplitudes F , (E, t ) for (+) and (-) being the boundary values of a function which for fixed t , t 5 0, it analytic in the upper half of the complex energy plane except a finite region. This also means that the amplitude can be analytically continued from the direct to the crossed channel. We shall use, instead of s, the variable E = (s - u)/(2Mb), which for the case of forward scattering (t = 0 ) coincides with the energy of particle a in the rest frame of particle b. Throughout the paper, E is real and positive. Complex values of the energy variable are denoted by z. The scattering amplitudes F+(E, t) and F-(E, t) have the following general properties (see [ lo] for details): (1) F* (E, t ) for A' > T and t 5 0 are the boundary valuesof the functions F&, t ) of coin- plex z, which are analytic in z (fort fixed) for z in the domain D, = (2: Im z > 0, jz/ > r ) where r is a nonnegative constant. Thus,

F& (E, t ) = lim Fi (E + i ~ , t ) . &>O+

(2) F+(z, t ) and FJz , t ) are connected by the relation

F-(z, t ) = F+*(-z*, t )

for every z E Dr, where the asterisk means the complex conjugate.

242 JAN FISCHER and RUDOLF &LY

(3) There exists an integer N such that lF*(z, t)l < I z / ~ in some neighbourhood of t = 0 for all sufficiently high z, where N is independent of t and of the direction of z in Dr. These three properties constitute the common basis from which asymptotic theorems for elastic scattering are usually derived. Besides them, for deriving a number of results, so~nc of the following general properties of the scattering amplitude are used: (4) PA (E , t ) are analytic in t in the Lehmann-Martin ellipse. (5) The scattering amplitude satisfies the unitarity condition, which has the following consequences : a) the optical theorem

In1 F,(E) = (p/41c) oi(S), for E > Ma (4- 1)

where p = (E2 - M,)l/Z and F,(E) = F+(E, t = 0) b) t,he partial wave amplitudes at (E) (we omit the subscripts + and -) satisfy the inequalities

Im a@) 2 jal(E)/?. (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 substi- tuted by the Froissart-Martin bound, which follows from them: (7) There exist real constants C and Eo such that

[ F , ( E ) ( 5 C E h 2 E , for every E > E,.

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 Yomeranchuk 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 iinitarity) but not possessing the property that do(E) tends to zero. This development, nevertheless, has led to considerable pro- gress in the rigour of formulations and an economical selection of assumptions. WEINBERG [4 ] was the first tothrow donbt of the assumptions (3.2) and (3.3). Actually, on one hand, the existence of the asymptotic limits of the cross sections o+(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, o+(E) behating like C + sin (In E ) ) . Beside this, Froissart’s work [7] has strongly suggested that the highest partial wave that gives a contribution to the amplitude may rise as fl In E (cf. Yonieranchuk’s supposition that ZmaE N VE). So, neither the limits of ui (E) , if any, have to be finite, nor F i ( E ) / E be bounded by constants. Thus the removal of (3.2) and (3.3) in an adequate theoretical treatment seems imperative. Deriving a theorem from the properties of the Herglotz functions, Weinberg succeeded in relaxing (3.2), replacing it by the weaker assumption that do($?) does not change its sign indefinitely: There exists E, such that, for every E > E,, da(E) is of one sign. (6.1)


He then showed that the integral W

converges provided that (3.3) is fulfilled. Besides this, Weinberg obtained a few results for rising crow sections, thereby extending the validity of the Pomeranchuk relation. Let 11s quote here the following one: if F+(E) = O(E Inm E) and o&(E) - C* Inm E , then C+ = C-for m 5 1. (Note that the premises mean the boundedness of the ratios Re P i (E)/Im P $ ( E ) , i.e. the amplitudes cannot be purely real asymptotically). Hence, though da(E) may not tend to zero, the ratio c+(E)/c-(E) does tend to one. And in the fact that this ratio of cross sections tends to one we have another possibility (besides (3.6)) of expressing the intuitive idea that o+(E) and c (E) approach one another. Also the assumption (3.1) was examined, since it involves the following presuppositions: the analyticity of the amplitudes F* (z) in the complex z plane except for two cuts and poles on the real axis, the crossing symmetry F+(z) = F-*(-z*), and the vanishing of the contribution from the infinite circle of the Cauchy contour integral we started with. The latter assumption is connected with the number of subtractions, and it was just this assumption which was questioned and given effort to avoid it or to justify. To this end, SUGAWARA and KANAZAWA [ 113 formulated a theorem (stating, roughly speaking, that the behaviour of F k ( z ) / z is essentially the same everywhere in the complex z-plane provided that this function is analytic in the cut plane, polynomially bounded and that some additional assumptions are satisfied) and, using the crossing symmetry together with (3.2) and (3.3), they derived the Pomeranchuk relation. This idea has been used by MEIMAN [12] in a more general and more elegant manner. His method is based on the following two Phragmh-Lindelof theorems [ 121. (L4) If g(z) is analytic in D, and is bounded on the real axis by a constant H, Jg(E)I < M , then either Ig(z)l < M at all points in D,, or g(z) increases exponentially. (B) Let g(z) be analytic in D , and let t+ (or t-) be the limiting value of g(E) as E -+ fm (or -00) along the real axis. Then either t+ = t- or g(z) cannot be bounded. We now apply these theorems to the function

which is supposed to be analytic in D , and polynomially bounded (see Sec. 4, properties (1) and (3)). If we make use of the additional Pomeranchuk's assumptions (3.2) and (3.3), i.e., the boundedness of g(E), we immediately obtain lim g(E) = lim g(E) . And, on

account of crossing symmetry and the optical theorem, this obviously gives c+ = c-. However, Meiman in fact used a generalized version of the theorem (B) and on such grounds he showed that one did not lose everything when the assumption (3.2) was dropped. Namely we can still claiiii that there exists an infinite sequence of energies En, En -+ 00 as n + 00, on which Ae(E) tends to zero. This is actually a very general result since it follows from the rigorous properties (l), (2) and (3) and the only additional assuinption (3.3). Historically, analyticity and crossing in D , was proved by BROS, EPSTEIN and GLASER in 1965. This fact (note that Meiman's method can, as we have seen, be applied to D , instead of the cut plane) served the starting point of the work of MLIRTIN [13]. His approach is based on the following idea. First we transform D,. through Z = 9. The analyticity domain becomes the whole 2-plane except for a finite disc of radius r2 and one cut along

I&+- 00 E++W

244 JAN FISCRER and RUDOLF &LY

the positive real axis. Then we make two assumptions: we suppose the crossing-odd amplitude F(E) = F+(E) - F-(E) to be such that

(5.3) F(E) = o(E In E )

along the real axis, and we assume F ( E ) to be polynomially bounded. The Phragmh- Lindelof theorem then allows (5.3) to be extended to all directions in the 2-plane. so, using the Cauchy theorem and crossing symmetry, we obtain a rigorous once-subtracted dispersion relation for F(Z) /p . Then we follow Pomeranchuk’s way, assuming the existence of the limit lim Aa(E), and arguing that its nonvanishing value would contra-

dict (5.3). We see, however, that (5.3) is far more general than (3.3). As seen from the work of Meiman and Martin, the use of the Phragmh-Lindelof theorems is very powerful and is noted for a particular simplicity. There are only two ad hoc assumptions inMartin’s approachleft: the constraint (5.3) and the existence of lim Ao(E). The latter can be avoided by introducing an average, for instance [14]

E a O

.E-=c

TRUONG and LAM [ l a ] show that ( A U ) ~ tends to vanish with E 4 00 if (5.3) holds, which makesdo(E) vanishing in the mean. This approach allows us to dispense with theassump- tion that the limit lim A a(E) exists.

E+W

6. Present Status of the Pomeranchuk Relation Based on Dispersion Relation only

It turns out that also the assumption (5.3) can still be weakened. Indeed, using the Froissart-Martin bound and the dispersion relation one can replace (5.3) by

Re P(E) = o(E In E ) . (6.1)

The result has been obtained by GRUNBERG and TRUONG [I51 and by MEIMAN [16]. The method of Grundberg and Truong starts with the inverse dispersion relation,

dE’ Re F(E’) - bE‘ E’ &7’2-E2

0

where b is a constant. Integrating Eq. (6.2) between 0 and E we get

d E E 2 E’2

d l r Im F Q ’ ) - = -- 7G ’ sL 1-1 E - E Re F(E’) - + b 4 2 . 0 s 0

(6.3)

(Although we have assumed the existence of the dispersion relation, this technique is also applicable to functions which are analytic in D, only, for example, by using the variable E = E f r21E). Now, using the assumption (6.1) we can estimate the r.h.s. of (6.3). Dividing (6.3) by In E we get

lim ( A O ) ~ = 0 (6.4) E-00

(where the definition (5.4) and the optical theorem have been used).

The Pomeranchnk Theorem 245

Let us formulate this result as a theorem. Theorem 1.- Assume thatF(z) possesses the properties (I), (2), (3), (5a) and (7). Then (6.1) implies (6.4). It is interesting to compare this theorem with a result obtained later by Meiman. Mei- man’s method consists of representing the reaction amplitude in the form of a complex potential of a simple layer, and of introducing a certain new remarkable concept of generalized limit, Lim Aa(E). The latter is such that a well-defined set of points of zero

asymptotic density may be omitted (exact definition and further discussion of this concept is given in Refs. [16, 171). We again formulate Meiman’s result as a theorem. Theorem 2: Assume that F ( z ) possesses the properties (l), (2), (3), (5a) and (7). If (6.1) is satisfied and Aa(E) does not change sign above some energy, then the generalized limit Lim da(E) exists and is equal to zero,

E+o2

E+m

Lim A @ ) = 0. E +m

Theorems 1 and 2 overlap to a great extent, but not fully. At present, they represent two main results regarding the problem of the Pomeranchuk relation in the case when no further information besides (1), (2), (3), (5a) and (7) is available. Thus far, we have only considered the oscillations of Aa(E) and not those of the real part of P(E). Both Grunberg and Truong and Xleiman restrict the oscillations of Re F ( E ) by the condition (6.1). The subsequent Theorem 3 shows what happens if this condition is replaced by an integral condition on the phase [17]. Theorem 3: Let P(z) satisfy (1)) (2) , (3), (58) and (7) and be continuous in the closure of D, except, possibly, the point z = 00. Let Aa(E) not change sign above some energy E,. If

m

then the integral W defined by (5.2) converges:

IWI < 00. (6.7)

Here, E(E) = Im F(E)/Re P(E) and the phase p(x) is defined as follows:

y ( x ) = arctan x - n for x > 0

q ( x ) = arctan x for x < 0

p(0) = -742.

The COMeCtiOn of this theorem with Meiman’s convergence (6.5) is given by the following lemma [17]: Lemma If W converges and d o ( E ) does not change sign above some energy, then Lim do(E) exists and equals zero.

To make condition (6.6) easier to read, let us give two typical situations in which it is automatically satisfied: a ) the signs of Aa(E) and Re P(E) are asymptotically equal, b) [ (E) 5 -n(l + e)/ln E for some E > 0 and all E > E,,.

J3+.ca

246 JAN FISCHER and RUDOLF ~ B L Y

These samples indicate that asymptotic oscillations of Re F ( E ) are allowed to a large extent. However, the main advantage of (6.6) is that it is an integral condition. Theorem 3 extends the class of functions claimed by Theorem 2 to satisfy (6.5) by a nonempty set. One element of this set is

W 2a,,b,, In b, In ( - i z ) +,; [i - ia,,(z - b,,)] [l - ia,(z + b,)] F ( z ) = z

(where a,, = lo3 exp n3 and b, = exp n2, n = I , 2 , . . .), which satisfies the conditions of Theorem3 but violates (6.1). On the other hand, there is, of course, a notable set of functions obeying (6.1) but violating (6.6) ; for instance,

~ ' ( z ) = z(1n (--iz))y y E (0, 1 ) .

This gives do(E) = (--7cy/2) (In E)y--l, ( ( E ) = --ny/(2 In E ) and, hence, a violation of (6.6). Theorems 1, 2 and 3 give sufficient conditions for various high-energy limits to exist and be zero. Still, none of these conditions is sufficient for the original Pomeranchuk relation (3.6), because the existence of the limit lim A @ ) has not been assumed. (It is worth

mentioning in this connection that the existence of this limit represents a rather strong assumption.) On the other hand, once the exstence of this limit is additionally assumed, life becomes much easier : firstly, the theorems quoted yield conditions of a just asymptotic vanishing of Ao(E) and, secondly, the relation (3.6) itself follows under compara- tively very weak conditions. The former statement is demonstrated in Sec. 8, whereas the latter feature is seen from the following [I71 Theorem 4: Let F ( z ) be continuous in Dr (except a t z = co) and satisfy (J), ( 2 ) , (3), (5a) and ( 7 ) . If lim ( ( E ) In E (finite or infinite) exists and liesoutsidetheinterval (-n/2, -nc>

(see Big. 2 ) , then lini do(E) = 0, provided that this limit exists2).

Reversing this statement, we infer that if lim do(E) =/= 0 then ( ( E ) is asymptotically

confined between -n/(2 In E ) and -n/ln E at least on some infinite sequence of energies.

E+m

E-tm

E--.oO

E-XC

Fig. 2. The assumption that li i i i du(E) + 0 implies that t ( B ) should, a t least on some infinite sequence of energies, fall

between the two curves shown in the figure. Experimental data [24 ] point towards a very different behaviour, giving € ( E ) nearly coiistant with the value around 1. Taken from Ref. [17].

E+m

2, If the limit lim dcr(E) does not exist it follows that a t least lim inf \da(E)I = 0. E+m E-tm


This is an answer to the question [%I “How to live with a violation of the Pomeranchuk relation”. The consequences of the opposite assumption, lini /la(E) = 0, have been recently dis-

The Theorem 3 yields also a bound on Ao(E). Indeed, reversing, the theorem we obtain the following result: Theorem 5 : Let P(z) satisfy (I) , (2), (3), (5a) and (7) and be continuous in nr except for z = 03. Then either (6.7) holds or

cussed also in Ref. [MI. E - a

IIm F(En)I 5 4 1 + E ) IRe p(En)I/(ln En) (6.8)

where {En) is an infinite sequence, En 3 00 as n + 00.

This result means that the total cross-section difference Ao(E,) is bounded either by (6.7) or by (6.8). Therefore, this bound can be written, for inst,ance, in the following form

This is a generalization of a result obtained by LOMSADZE [20] and by MNATSAKAT I OVA

and 1-ERNOV [ZY]. Comparing (6.9) with the bound of ROY and SINGH [HI,

~ i m 5 const., E - , ~ 1nE -

(6.10)

we see that (6.10) is improved in all cases unless Re P(E) saturates theFroissart-Martin boiind. Another generalization of (6.10) in terms of the mean values was obtained by GRUNBERG and TRUONG [ I G ] .

7. The Use of Unitarity, Analyticity in t and of Isospin Invariance

Froai the mathematical point of view, Theorems I and 2 are theorems about the connection between the behaviour of the real and the imaginary part of an analytic function in the neighhourhood of a boundary point. (Note that this connection is of local nature, since only conditions in the neighbourhood of the infinitely removed point are iniportant). Besides analyticity in energy, there have been assumed crossing symmetry, the Frois- sart-Martin bound and the temperadness. These assumptions are regarded as well esta- blished (see 8ec. 4), but they are insufficient to prove the Pomeranchuk relation. An important role is played by the assrirnption (6.1), which, although plausible, has no deeper justification. Making use of analyticity of the amplitude in t and of unitarity, EDEN [21] was able to manage without (6.1) and derive an alternative version of the Poineranchuk relation, viz.

o+(E) lim - = 1 E+w

(7.1)

assiiniing, on the other hand, that the total cross sections uc (E) behave like C , (In E)”, ,111 > 0, aspiptotically. His method has been refined in a number of papers [22, 23, 14, I;] in which, eventually, only an unbounded rise of (u+)~ or (u- )~ is assumed. This additional assnniption seems natural a t least from the point of view of recent experiments. This remarkable result is often given a paradoxical form stating that “the Pomeranchuk relation is rigorous when cross-sections rise but not when they are bounded”.

248 JAN FISCHER and RUDOLF ~ I L Y

Relation (7.1) implies that if o+(E) rises unboundedly then c ( E ) also rises unboundedly. Using rather general assumptions, VOLKOV, LOGUNOV and MESTVIRISHVILI [23] have also proved that if o+(E) tends to a finite, nonvanishing limit then also c ( E ) must tend to a finite, nonvanishing limit and, similarly, if a+(E) tends to zero then also o-(E) must tend to zero. To give an idea how analyticity in t and unitarity can be taken into account we will follow the argument given by KINOSHITA [22]. A forward scattering amplitude (say F+(E) or F-(E)) may be expanded as

where ul(E) is the I-th partial wave amplitude. The unitarity condition can be expressed by the inequalities

Im q ( E ) Z [Im al(E)]% + [Re a@)]%, I = 0, 1,2, . . . (7.3)

Now, note that in the expansion (7.2) only the first L,,, partial waves contribute to the scattering amplitude, where L,,, = C fE In E under the assumption of analyticity in t . Contributions from partial waves with 1 > L,,, can be shown to be smaller than E-" for sufficiently large E . Making uae of unitarity (Eq. (7.3)) and the Schwartz inequality, we obtain

Lmu 5

5 C (21 + 1) Im a1

(22 + 1) (Im al)l/a l=O

l / Z L X ila E (21 + 1) [:: f L o I

5 LmaxPm F+(E)I"'

and an analogous inequality for F-(E). Thus, for large enough energies we have

IRe P(E)/EI 5 Re IF+(E)/BI + IRe p-(E)/fll

(7.4) 1 5 - In E ( [ U + ( E ) ] ~ / ~ + [u-(E)]"~)

4P

where the constant C in L,,, has been set equal to 1/(4p m. Note that this reasoning is similar to that which led Pomeranchuk to relation (3.3), the main difference being that the property \al(E)ll 5 1 is replaced by (7.3) and the condition L,,, = C @ is replaced by LmaX = C @ In E . And now, combining this result together with the dispersion relations we get the ROY- SINGH bound r18,151

where (En) is an infinite sequence, E,, --f bo, n + 00. The bound (7.5) was originally proved by ROY and SINGH [18] for monotonic amplitudes and then generalized by GRUNBERG and TRUONG [15] for any high-energy behaviour. We sketch the proof of (7.5) under the assumption of a smooth behaviour of the amplitude using our Theorem 5

The Pomeranchuk Theorem 2 49

because the complete proof is too long as to be reproduced here. If the integral W diver- ges we can use (6.8) and, with regard to (7.4), the relation (7.5) follows. On the other hand, the convergence of W is not possible if (7.5) does not hold. Indeed, assume that Ida(E)l >= C [ ( U + ( E ) ) ~ / ~ + (a-(E))u2] above some energy. Then also Ido(E)I L C(a+(E) + o-(E))~/* 2 C [ A o ( E ) ~ ~ / ~ and the integral W cannot converge, Using the bound (7.5), Grunberg and Truong were able to prove the Pomeranchuk re- ation for rising cross sections in a very general form. Indeed, assuming that

lim (a, + aJE = 00 E-PW

and dividing (7.5) by (a+ + aJEn, we find

In this way, we obtain [I51 Theorem 7: Let P(z) fulfil (l), (2), (3), ( 4 ) and (5) . Then (7.5) holds and thus (7.6) implies (7.7). Thus, the unbounded rise (7.6) is the only additional assumption required for (7.7) to be valid. Note that eq. (7.7) implies that there exists an infinite sequence of energies { E n ] , En --f 00 with n --f 00, on which

and, in particular, if dcr(E)/(o+(E) + o-(E)) has a limit, this limit is zero. Thus, (7.7) represents another generalization of Eden’s result (7.1). It the preceding we have seen that an essential tool in proving the Pomeranchuk relation is the dispersion relation and the bound (7.5). It is obvious that the asymptotic equality of the total cross sections u=+,(E) and a,-,(E) for the scattering of pions on protons is a special case of the Pomeranchuk relation. However, since the pions belong to one isotopic multiplet, the asymptotic equality can also be derived from the Pomeran- chuk-Okun hypothesis. Combining the basic steps of these two approaches, ROY and SINUH [I81 were able to obtain a new, nontrivial relation. Three points are crucial in their procedure. Firstly, they derived the following unitarity bound

Further, they used the isotopic spin invariance :

and, thirdly, they used the dispersion relation. Combining all this gives [I81

(7.10)

(7.11)

provided that these limits exist. Now, if the exchange process has vanishing cross section it follows from (7.11) that Aa(E) -+ 0 as E + 00. And this is actually an improvement of the Pomeranchuk-Okun

250 JAN FISUEER and RUDOLF S h y

hypothesis, since no assumption on the real parts has been made. Transition to other reactions, e.g. Kn, is straightforward. Although all the cross sections in (7.11) can be determined experimentally and the constant on the right-hand side is known, it is not trivial to perform an experimental test of (7.11), however high the available energy may be. The reason is that (7.11) is a relation between two high-energy limits and that, besides, also equality between them is admitted. Usually, an asymptotic relation means that, for all energies above a certain value Eo (which may be unknown but definitely exists and is finite), the relation should be valid. One could, therefore, be liable to ignore the lim signs in (7.11) and consider

the corresponding inequality a t finite energies. But is it not to be forgotten that (7.11) admits even the possibility that the opposite inequality

L-)

(7.12)

holds for every E, while the limits are equal according to (7.11). This is particularly important because of the fact that a naive extrapolation of data according to the existing fits, [25]

,y-wnon(E) - E-1.15 , do(E) - E-093’ (7.13)

would suggest that (7.12) does hold a t all energies E above 200 GeV. But even such a behaviour would not contradict the bound (7.1 1). It is not difficult to derive a finite-energy counterpart of (7.11):

where (En) is an infinite sequence of energies tending to infinity with n --f cw. (This relation immediately follows from (6.9), combined with (7.9) and (7.10).) Certainly, it is compatible with (7.13) and coincides with (7.11) in the high-energy limit. Another bound following from (7.9) and (7.10), namely,

l G + p E ) - (7.15)

represents a more serious problem, because it does contradict the parametrization (7.13). This fact argues either against the parametrization (7.13) or against (7.9) or (7.10).

8. Rates of the Vanishing of the Total Cross-Section Difference

Once conditions for the vanishing of the total cross section difference have been establi- shed, nothing can prevent us from inquiring after the rate of this vanishing. I n the early sixties the question was posed whether, starting from Pomeranchuk’s assumptions, the integral W can be proved to be convergent (see (5.2)) and consequently, Ab(E) tends to zero with E + 00 faster then l/ln I3 in the mean. We have already seen in Sec. 5 that WEINBERG [4] was able to prove the convergence of W under assumptions which were even weaker than those made by Yomeranchuk. Experimental data on Ao(E) gathered till now strongly suggest that a rather fast asymptotic vanishing of do@‘) takes place [25]. Both the hadron-hadron total cross section measurements and K,O regeneration experiments point towards a negative-


power like decrease of da(E), Aa(E) M const. E-l

where A is a constant close to 1/2. This confirms that the search for sufficient conditions of a given vanishing rate is not a purely academical question but has deeper physicaI background. Let us remark that with (8.1) the integral W would converge. It has turned out during the years that the assumption needed for W to converge can be considerably weakened. In 1974, GRURBERG and TRUONG [I51 proved the convergence requiring only that

and that Aa(E) be of a definite sign above some E,. This result immediately follows from relation (6.3) by splitting the integral on the r.h.s. and by using (8.2)

IRe P(E)/EI S C , E > E, (8-2)

dE' E' + E dE' In ~ -

E'2 n cf l E - E l E" IRe F(E')( - + - E - E

EO

The first integral on the r.h.s. has a compact support and behaves like 1/E as E + w. The second integral converges and can be estimated by nC/2, thus

and if Aa(E) does not change sign indefinitely, then the integral W converges. And still, not even the condition (8.2) is the only one to ensure the convergence of W . This has been shown in Refs. [17, 261, where it was replaced by (6.6), which is satisfied if, for example, t ( E ) = Im F(E)/Re F ( E ) > 0 or if Q E ) < -n/ln E , asymptotically (see Theorem 3 of the present paper). There have been derived a number of sufficient conditions for the convergence of the integral W in literature. We wish to make a general remark that any of these conditions, if the existence of the limit lim Ao(E) is additionally assumed, converts into a sufficient

condition of a fast vanishing of Aa(E), viz. Ida(E)I < C/ln E. This rule is easily extended to the integrals

E+m

m 00

J da(E) Ea(ln E)@ dE and j" Re P ( E ) EY-l(ln E)d dE (8.3) L EQ

in which A o(E) or, also, Re F(E) are weighted by some power of E . The convergence properties of these two integrals can be used to characterize the asymptotic behaviour of F(E): the highest values of N , j3 and y , 6 for which they are finite indicate how fast A o(E) and Re P(E)/E, respectively, tend to zero with E --f 00. Constraints to be im- posed on the phase of F(E) in order to ensure the convergence of (8.3) for given OL, p and y, 6 have been found [ 271. We give two examples here to illustrate the result: 1. For the convergence of the integrals (8.3) with OL E ( - 1 , O ) it is sufficient to require

t ( E ) B (1 + E ) cot (a 1 4 2 )

2. If ( ( E ) 2 -n(l + c)/ln E , then the integrals 03 03

J do@) (In E)-1-6 dE and J Re F(E) E-' dE EO EO

converge.

252 JAN FISCHER and RUDOLF &LY

Another way of examining the vanishing rate of Au(E) and ReF(E) is to consider asymptotic relations for their derivatives, Relevant theorems have been derived in Ref. [59]. As an example, let us quote here the following relation:

d arctan E(E) + - (In Aa(E))

E-tm d l n E

which follows from the general assumptions 1,2,3,5a and 7, provided that the amplitude is continuous up to the real axis, certain high-energy limits exist and Aa(E) does not vanish too fast. Formula (8.4) relates the first derivative of the total cross section difference to the phase at asymptotic energy. If applied to the existing high-energy fits (8.1), it leads to the relation

lim arctan E(E) = 4 2 .

Experimental data on K: regeneration above 10 GeV are in good agreement with this relation, giving E(E) close to 1. Another relation is

E+CC

(8.5) M E ) lim Re F(E)/E = lim

E-xc E-XC tan (2 d 1n d a m ) 2 d l n E

with a slightly narrower domain of validity. For details we refer the reader to Ref. [59], where also other analogous relations are derived and problems connected with practical applications are discussed. The results that have been quoted in the present section are correlations between Re F(E) and Aa(E) or between E(E) and Aa(E). Analogous correlations have been investigated for the crossing even amplitude (see Refs. [15,26-32,591) but they can simply be trans- ferred to the crossing-odd case by replacing the symmetric amplitude by iF(E). For the crossing-odd case we refer the reader to Refs. [12, 15, 17, 26, 32, 33 and 591.

9. Differential Cross Sections

The Pomeranchuk-Okun hypothesis (see Section 2) suggests that the differential cross sections of projectiles scattered by the same target are asymptotically equal provided that the projectiles belong to one isomultiplet. Why not then try to extend this result to particle-particle and antiparticle-particle scattering? This extension was first performed by VAN HOVE [34]. Assuming that the amplitude for direct reaction behaves like Ef+(t) he provedf+(t) = -f-(t). The result was generalized by LOGUNOV et al. [35] to amplitudes of the type E”(l)(ln E)sct) (In In with a, B and y real. Both the approaches discussed that problem in models, but later on one began to realize what was the relevant feature of thismodel: it was the fact that the phase of the reaction amplitude has a finite limit a t infinite energy. This problem has been analyzed by number of authors [13, 16, 36- 451 since. Without entering into details of these gradual generalizations we only sketch the method of MEIMAN [16]. We introduce the following auxiliary function

h(z, t ) = iE In (F+(z, t) /F-(z, t ) ) . (9.1)

If for given t the ratio of the amplitudes has no zeros in the upper half plane, Im z > 0, then the function h(z, t ) is analytic in this half plane (due to analyticity of F*(z, t ) ) .

The Pomeranchnk Theorem 253

On the Imaginary axis h(z , t ) is real and therefore

h(-z*) = h*(Z) On the real axis we have

h(E, t ) = E(Arg F-(E, t ) - Arg F+(E, t ) ) + iE In &(E, t ) (9.3)

= !F+(E, t)/F-(E, t ) I 2 - (9-4)

where da+(E, t ) do-(E, t )

at I dt Q2(E, t ) =

Since F+(z, t ) are polynomially bounded, h(z, t ) does not grow faster than Iz In 21.

These properties obviously indicate that, in deriving the asymptotic value of Q2(E, t ) , h(E, t ) plays the same role as F(E) has played for do(E) . Thus, we can immediately apply all the results of Sec. 6 to this case. Theorem 1 implies that if the difference of the phases does not grow too fast,

then Arg F+(E, t ) - Arg F-(E, t ) = o(ln E ) ,

&(E, t ) += 1

(9.5)

(9.6)

in the mean. Application of the other results is straightforward, too. An interesting contribution to these problenis has been made by CORNILLE and MARTIN [42, 431. We shall mention some of their results here. We have seen in Section 7 that if the technical assumption (6.1) is replaced by unitarity and analyticity in t one can derive (7.1), which is an alternative to the Poineranchuk relation. There is a similar situation also in the case of differential cross sections. The role of the real part Re F(E) /E is played here by the phase difference Arg F+(E, t ) - Arg- F-(E, t ) which, in analogy to relation (6.1), is assumed to obey (9.5). This is also a technical assumption, which hardly could be checked experinlentally [ 4 4 ] . It was shown by Cornille and Martin, however, that it can be replaced by a more plausible one, which can be checked a posteriori by looking a t measurements of differential cross sections. To motivate their reasoning [ 42, 441 consider first the case t = 0. By virtue of unitarity Im F+(E, t = 0) 2 0 and, therefore, the phases of the forward amplitudes stay bounded between 0 and n. The condition (9.5) is then satisfied and (9.6) follows [I.?]. The main point of Cornille's and Martin's work is to generalize this to t $; 0. To do this they establish the following bound on the phase of the elastic scattering amplitudes away from the forward direction :

The bound follows from the positivity of the Legendre expansion coefficients of the absorptive part of the amplitude. The result allows the asymptotic equality of differential cross sections for ab -+ ab and rib + 6b to be proveds) for any momentum transfer inside the diffraction peak, no matter whether the peak shrinks or not. Also, the bound on the phases allows control of the number of zeros of the amplitudes in the coniplex plane. For further details concerning this result we refer the reader to Refs. [ 4 2 ] and [44] ; for other problems related to differential cross sections see also Refs. [I61 and [ 4 5 ] .

3) To see directly how the bound (9.7) works i t is useful t o write explicitly the following bound

which follows immediately from (9.7) and the analogue [I71 of (6.9) written down in terms of In F+/F- and (Arg F+ - Arg F-).

17 Zeitsehrift ,Fortachritte der Physik", Heft 5

Tab

le 1

~ ~

Ass

umpt

ions

St

atem

ent

Form

ula

Ref

eren

ce

num

ber

num

ber

gene

ral

addi

tiona

l in

this

re

view

E

1,2, 3,5a, 7

1,2,3,4, 5a, 7

-1, 2, 394, 5

1329 3,495, 6

1) 2,

39495

1,29394, 5

1, 2, 3, 5

a, 7

1,2,3,5a, 7

1, 2,3,5a, 7

1, 2, 3, 5a) 7

1, 2, 3,5a, 7

1, 2, 3,5a, 7

1, 2,3, 5&) 7

~(

z)

co

ntin

uous

in 4

,

both

the

lim

its e

xist

lirn (a+ +

u-)E

=

00

lim a_@)

= 0

0

F(E

) sm

ooth

R

e F

(E) =

o(E

In E

)

Re

P(E

) = o

(E In

E)

fixe

d si

gn o

f A

a(E

) lir

n E

(E) I

n E

a (-n, -42

)

F(z

) con

tinuo

us in

6,

~(

z)

co

ntin

uous

in 4

, fi

xed

sign

of

Aa(

E)

E--

mo

E+

W

&oo

m

Re

F(E

) = O

(E)

fixe

d si

gn o

f do

(E) 4

4

5(E

) r (1

+ 4

co

t2

F(z

) con

tinuo

us in

4,

F(z

) con

tinuo

us in

b,

the

limits

exi

st

I Re

F(E

) E-l

dE =

00

W

lim (

a+(E

)/a-

(E))

=

1

E-t

w

lirn

(A&

=

0

E4

m

lim

da(E

) = 0

E-

lirn

inf

IAo(

E)I =

0

8-02

W I A

o(E

) E-*

dE

con

verg

es

W 1 da

(E) E

-l dE

con

verg

es

1 da

(E) Ea dE

con

verg

es fo

r a E

(--

1,O)

m

arct

an E

(E) +

- do

(E)]

= 0

d

In E

c171


10. Concluding Remarks

We select here the following topics out of the numerous problems related to the Poine- ranchuk theorem :

1. The Pomeranchuk relation has not been derived from local field theory. What has been proved rigorously for the difference (or ratio) of the total cross sections? (Sec- tion 5, 6) 2. Although Nature does not seem to favour such a possibility, a violation of the Po- meranchuk relation is theoretically possible. What would follow froni the violation of (3.6)? (Section 6) 3. Rigorous theory does not tell the rate by which da(E) should tend to zero. What are the sufficient conditions for a given, fast, vanishing of Aa(E)? (Section 8) 4. Mathematically, the existence of the limit lim da(E) represents a rather strong

assumption. Can we introduce another definition of the high-energy limit so that its existence would follow from some plausible assumptions? (Section 6, 7)

Some of the answers to these four problems are collected in Table 1.

E+w

Fig. 3. Experimental data for the n*p, Kfp and p i p total eross sections known till 1978. The picture strongly suegests that the corresponding particle-target and antiparticle-target total cross sections approach each other nlth 111- creasing energy. Taken from Ref. [671.

17*

256 JAN FIscIiER and RUDOLF 8dLY

It should be noted, however, that a number of interesting contributions to problems related to the Pomeranchuk theorem have not been discussed here. There is, for instance, a notable series of papers motivated by the seeming violation of the Pomeranchuk relation (3.6) after early Serpukhov measurements [as]. These authors [47-581 have investigated in detail the consequences of assuming the violation of (3.6) and, thereby, have contributed to the problem quoted here as No. 2. Other results related to the Pomeranchuk theorem, which are not discussed in the present review, can be found in the excellent reviews of EDEN'S [60], KINOSHITA'S [22], LOGUNOV'S et al. [33, 621, MARTIN'S [56] and ROY'S [GI] . Let us comment briefly in conclusion on the confrontation of the Pomeranchuk relation and asymptotic theorems in general with experimental data. Of course, such a compa- rison is possible only if we assume that the existing data determine the asymptotic behaviour. This is, naturally, an additional assumption which although plausible is not

'y 0

0 .

' t t 0'4 0,2

10 20 40 700 200 400 Pbb TGeV/cl

Fig. 4. The differences of total cross sections f o r d , K* and pf interactions with protons seem to tend to zero with a negative power of energy. Taken from Ref. [67].

evident a priori. It is, therefore, desirable to have a finite-energy analogue of the Pomeranchuk relation or to draw some finite-energy consequences from the Pomeran- chuk relation itself. Some results in this direction have been obtained by several authors [19, 63, 641. It is possible that the smooth picture of high-energy elastic scattering is only approxi- mative and new structures will be found by detailed measurements or after passing to higher energies. In any case, the Pomeranchuk theorem has had a great scientific value both experimentally and theoretically, stimulating further theoretical research, serving as a guide to model building and collecting facts to have in mind before constructing them. Fig. 1 shows that the idea of the asymptotic equality of o+(E) and o-(E) was far from trivial in 1957. Pigs. 3 and 4 indicate how beautifully this idea has been supported experimentally during the subsequent twenty years.

It is a pleasure for us to thank A. Martin for helpful correspondence and M. N. Mnataakkanova, Yu. S. Vernov and I. VrkoE for useful discussions.


References

[ I ] I. YA. POMERANCEUK, Zh. eksper. teor. Fiz. 34, 725 (1958). [ Z ] I. YA. POMERANCHUK, Zh. eksper. teor. Fiz. SO, 423 (1956). [3] L. B. OKUN, I. YA. POMERANCHUK, Zh. eksper. teor. Fiz. 30, 424 (1956). [a] S. WEINBERG, Phys. Rev. 124, 2049 (1961). [5] J. J. J. KOKKEDEE, The Quark Model (Benjamin, New York, 1969). [6] 8. AMALDI et al., Phys. Letters 43 B, 231 (1973). [7] If. FROISSART, Phys. Rev. 123, 1053 (1961). [8] A. MARTIN, Phys. Rev. 129, 1432 (1963); Nuovo Ciniento 42, 930 (1966). [9] J. BROS, H. EPSTEIN, V. GLASER, Nuovo Cimento 81, 1265 (1965); Comrn. Math. P h p . 1,

[IO] A. MARTIN, Scattering Theory: Unitary, Analyticity and Crossing (Springer, New York, 1969). [11] M. SUQAWARA, A. KANAZAWA, Phys. Rev. 132, 1895 (1961). [I21 IV. N. MEIMAN, Zh. eksper. teor. Fiz. 43, 2277 (1962). [I31 A. MARTIN, Nuovo Cimento 39, 704 (1965). [ l a ] T. N. TRUONG, W. S. LAM, Phys. Rev. D 6, 2875 (1972). [I51 G. GRUNBERQ, T. N. TRUONQ, Phys. Rev. D 9,2874 (1974); Phys. Rev. Lette s 31,63 (1966). [I61 N. N. MEIMAN, Zh. eksper. teor. Fiz. 68, 791 (1976). [I71 J. FISCHER, R. &LY, I. VRKOC, Phys. Rev. D 18, 4271 (1978). [I81 S. M. ROY, V. SINGH, Phys. Letters 32 B, 60 (1970). [I91 M. N. MNATSARANOVA, Yu. S. VERNOV, Teor. Mat. Fiz. 32, 153 (1978). [20] Yu. M. LOMSADZE, Preprint ITP-75-126P, Kiev, 1975. [2I] R. J. EDEN, Phys. Rev. Letters 16, 39 (1966), High Energy Collisions of Elementary Particles

[ZZ] T. KINOSHITA, in: Perspectives in Modern Physics, ed. R. E. Marshak (Wiley, New York, 1966),

[23] G. G . VOLKOV, A. A. LOGUNOV, M. A. NESTVIRISHVILI, Teor. Mat. Fiz. 4, 196 (1970). [24] Budapest-Dubna-Praha etc. Collaboration, Nuclear Phys. B 116, 249 (1976). [25] G. GIACOMELLI, Phys. Rep. 23 C, 125 (1976). [26] J. FISCHER, P. KoLLR, I. V R K O ~ , Phys. Rev. D 13, 133 (1976). [Z7] N. N. KHURI, T. KINOSHITA, Phys. Rev. 137 B, 720 (1965); Phys. Rev. 140 B, 706 (1965). [28] J. S. JIM, S. W. MCDOWELL, Phys. Rev. 138B, 1279 (1965). 1291 R. Wrr, Acta Phys. Polon. 29, 771 (1966). [30] Yu. S. VERNOV, Yadern Fiz. 10, 176 (1969). [31] H. CORNILLE, Nuovo Cimento 66 A, 217 (1970). [32] S. Yu. MEDVEDEV, Yadern Fiz. 19, 930 (1974); ibid. 18, 926 (1973). [33] A. A. LOQUNOV, M. A. MESTVIRISHVILI, 0. A. KHRUSTALEV, Teor. Mat. Fiz. 9, 153 (1971). [34] L. VAN HOVE, Phys. Letters 6, 252 (1963); ibid. 7, 76 (1963); Rev. Mod. Phys. 38, 655 (1965). [35] A. A. LOQUNOV, N. VAN HIEU, I. T. TODOROV 0. A. KHRUSTALEV, Phys. Letters 7,69 (1963). [36] N. N. MEIMAN, Zh. eksper. teor. Fiz. 46, 1039 (1964). [37] J. L. GERVAIS, F. J. YNDURAIN, Phys. Rev. 167, 1289 (1968); ibid. 169, 1187 (1968). [38] A. BIALAS, J. SICIAK, R. WIT, Acta Phys. Polon. 32, 651 (1967). [39] Yu. S. VERNOV, Teor. Mat. Fiz. 4, 3 (1970). [40] Yu. S. MEDVEDEV, M. Yu. LOMSADZE, Nuclear Phys. B 4S, 283 (1972). [al l H. CORNILLE, A. MARTIN, Nuclear Phys. B 48, 104 (1972). [42] H. CORNILLE, A. MARTIN, Phys. Letters 40 B, 671 (1972). [43] A. MARTIN, in: Properties of the Fundamental Interactions, ed. A. ZICEICHI, Bologna 1973,

[44] H. CORNILLE, A. MARTIN, Nuclear Phys. B 49, 413 (1972). [45] G. GRUNBERG, T. N. TRUONQ, T. N. PHAM, Phys. Rev. D 10, 3829 (1974). [46] J. V. ALLABY et al., Phys. Letters 30 B, 500 (1969). [47] A. A. ANSELM et al., Yadern Fiz. 11, 896 (1970). [48] G. AUBERSON, T. KINOSHITA, A. MARTIN, Phys. Rev. D 3, 3185 (1971). [49] R. C. CASELA, Phys. Rev. Letters 24, 1463 (1970); jbid. 26 54 (1971). [5O] H. CORNILLE Nuovo Cimento Lettere 4, 267 (1970).

240 (1966).

(Cambridge Univ. Press,.1967).

p. 211.

p. 221.

258 JAN FISCHER and RUDOLF S ~ L Y

[ S l ] R. J. EDEN, G. KAISER, Nuclear Phys. B 28, 253 (1971). [52] J. FINKELSTEIN, Phys. Rev. Letters 24, 172 (1970). [53] J. FINKELSTEIN, S. M. ROY, Phys. Letkers 34B, 322 (1971). [54] V. N. GRIBOB, I. Yn. KOBSAREV, V. D. MUR, L. B. OKUN, V. S. POPOV, Phys. Letters 32 B,

129 (1970); Yadern Fiz. 12, 1271 (1970). [55] T. KINOSHITA, Phys. Rev. D 2, 2346 (1970). [56] A. MARTIN, Lectures given at the McGill University, Summer School, 1972. [57] S. M. ROY, Phys. Letters 34 B, 407 (1971). [58] R. WIT, Nuovo Cimento 69 A, 312 (1970). [59] J. FISCHER, P. KOL& Phys. Letters 64 B, 45 (1976); Phys. Rev. D 17, 2168 (1978). [SO] R. J. EDEN, Rev. Mod. Phys. 43, 15 (1971). [61] S. hf. ROY, Phys. Rep. C 6, 128 (1972). [62] 8. A. LOQUNOV, M. A. MESTVIRISHVILI, V. A. PETROV, Ann. Phys. (N. Y.) 114, 46 (1978). [63] Yu. S. VERNOV, Teor. Mat. Fiz. 21, 195 (1974). [64] H. CORNILLE, R. E. HENDRIOK, Phys. Rev. D 10, 3805 (1974). [65] E. NAGY, Acta Phys. Hungarica 43, 237 (1977). [66] B. CORK et al., Phys. Rev. 107, 248 (1957). [67] A. S. CARROLL et al., Phys. Letters 80 B, 319 (1979).

the pomeranchuk theorem and its modifications

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