Regge pole method and one-meson approximation

Nuclear Not

to

Physics

44 (1963) 116-122;

be reproduced

REGGE POLE

by

photoprint

METHOD

D. S. CHERNAVSKY,

AND

Institute,

Received

without

Publishing

written

ONE-MESON

I. M. DREMIN,

P. N. Lebedev Physical

@ North-Holland

o microfilm

of Sciences,

5 November

1962

Co., Amsterdam from

the

publisher

APPROXIMATION

E. L. FEINBERG

Academy

permission

and I. I. ROYZEN Moscow,

USSR

Abstract: All the main results of the Regge pole method concerning the asymptotic behaviour of cross sections are also given by the one-particle approximation, if we additionally specify some quantities. The one-particle approximation, moreover, gives some basic characteristics of inelastic processes.

1. Introduction The elegance of the relativistic Regge pole method (RPM), recently developed for strong interactions on some general principles lS3), as well as its particular results, which have been verified experimentally to some extent (for elastic scattering and promising. However, at Elab M 10 GeV “)), make this method very attractive three main problems arise in connection with it: (a) Does it describe all possible high-energy processes or does an essential contribution come asymptotically not only from the extreme right pole but also from other singularites (cuts, etc.) ensuring the non-vanishing of elastic scattering? This point may be clarified by comparing some predictions of the RPM with experimental data, for instance along the lines of ref. ‘). The answer is not yet quite definite experimentally. (b) The method is restricted to elastic processes; as to inelastic collisions it makes only such predictions as may be drawn from the optical theorem and from the twoparticle unitarity in the t-channel; it is not clear at all how the method can be used for obtaining other characteristics of inelastic events. (c) It is often stated that the asymptotic characteristics of high energy interaction events, predicted by the RPM and first of all the unlimited increase of the effective impact parameter (“long range forces”) - cannot be represented by any model. However, this statement may well be questioned. On the other hand, there exists an effective method of treating both inelastic and elastic collisions, based on a special class of diagrams: the one-particle approximation (OPA). If we could establish a correspondence between the two methods, we would have an additional tool for treating inelastic phenomena and obtain some suggestions concerning a possible generalization of the RPM, etc. This correspondence has in fact been obtained to a considerable extent in ref. “). Similarly, the correspondence between both methods and the “strip approximation” method is also essential. 116

REGGE POLE METHOD

1 17

Let us stress that by the OPA we do not mean the one-particle exchange (OPE) method of refs. 7- 9), which practically coincides with the pole approximation 1o, 11), but the method developed in refs. 12, 13). It coincides with the OPE at not very high energies (E~ab < I0 GeV), but differs from the OPE at higher energies with respect to the assumption about vertex parts (i.e. the interaction cross section of virtual pions: see below). We believe that the following statements are valid: a) It is proved 6) that the RPM and OPA are based on a special kind of neglect of many-meson exchanges in the asymptotic region. Their fundamental assumptions (concerning the asymptotic behaviour of the S-matrix): the single pole contribution in the RPM; neglect of all diagrams besides those of one-meson exchange in the OPA, are essentially equivalent in the sense that if one of these assumptions is wrong (and therefore the corresponding method fails), then the other is also wrong. If, on the other hand, one of these methods holds only for some part of the interactions, then the other can not describe all interactions. Both methods include arbitrary elements: the position of the pole l(t) and its residue r(t) in the RPM, the virtual meson cross section (vertex part) a(s, k 2) in the OPA, which are determined from some additional postulates or from experiment. They are closely connected with each other. b) All the main results of the RPM concerning the asymptotic behaviour of cross sections (the increase of effective impact parameter, the logarithmic decrease of the elastic cross section, relations connecting total cross sections for various particles, etc.) are also given by the OPA, if we in addition specify some quantities in the OPA. After this specification, the OPA description of the elastic processes becomes equivalent to the RPM. Moreover, the OPA gives some basic characteristics of the inelastic processes responsible for the elastic scattering. Therefore, the OPA theory of high energy interactions may be considered as a "microscopic" picture, underlying the RPM "phenomenological" description, and by checking experimentally the OPA predictions, we check the RPM predictions.

2. One-Particle Approximation Let us recall, first of all, the general structure and the results of the one-meson approximation 13). It is based on the assumption that the matrix element of an inelastic collision producing at least two groups of particles according to fig. 1 ( k 2 =

Fig. 1.

118

D.S. CHERNAVSKYet aL

k 2 - k 2, # and m are the masses of meson and nucleon, and # = h = c = 1) m a y be represented in the form

M = A~(k 2, (Pl + k) 2) D(k2)Ab( k2, (P2 -- k)2),

O)

where the vertex amplitudes Aa and A b depend, besides, on the kind of colliding particles a and b and on the m o m e n t a o f the particles produced, q~, qj(i = 1 , . . . , n;j = 1 , . . . , m) and O(k 2) is 'the propagator for a pion of m o m e n t u m k. Neglecting the interference of the two groups of particles generated (which is suggested by the sharpness of their angular distribution 13)), we may integrate the differential cross sectien corresponding to eq. (1) over momenta. The total cross section thus obtained (for simplicity, we take the case s >> m 2, m 2, k 2 13)) can be represented as 3 O'ab -- 8 d s 2 f

2

s2 a..(k , sx)O2(k2)ab.(k 2, s2)dsl ds2 dk 2,

(2)

where Sl = - ( P l + k ) 2 and s 2 = - - ( p 2 - - k ) 2 are the squares of total energies of the systems a + zc and b + 7r in their respective centre-of-mass systems. By aa~(k 2, sl) and ab~(k 2, $2) we designate functions (proportional to the integrals over the secondary particle m o m e n t a of ]Aa(k 2, sl)l 2 and IAb(k 2, s2)] z) which give, for a real meson, i.e., for k 2 = - 1, the total a + rr and b + n interaction cross sections. They may be treated as a continuation of the real process cross sections aatb),(-1, si) to the region of virtual particles, i.e., to the integration interval k 2 > 0 7,8). Besides the assumption expressed by eq. (1) (one-meson exchange), the method implies a definite choice of the functions aa~(k 2, Sl) and ab~(k 2, s2). The simplest possibility is to substitute for them the real meson cross sections O'a(b)n(--l, Si) (which corresponds to the pole approximation 7-11). It is clear a priori that for large k 2 such a procedure is unwarranted. Though the integration interval o f k 2 is bounded by conservation laws, it covers very large k 2. Furthermore, it was shown 11) that if we calculate, on such a basis, the contribution of a limited (but fixed) interval of small k 2, by assuming

ffa(b)n(k 2, Si) -~ O'a(b)n(-- 1, s~) = aa(b)~ -- const for k 2 "( c, aa(b).(k 2, s~) = 0 for /? > c,

C ~ 1,

even this contribution increases logarithmically with s, and the constancy of trab (which, as the experiment shows, seems to be a necessary requirement for any theory) cannot be secured. In ref. 13) it was shown that the constancy of aab requires a non-factorial dependence o f ffan(k 2, $1) and of trb~(k 2, S2) on k 2 and si, viz. they should decrease, when k 2 increases, so that the effective values of k 2 be the smaller, the larger sv This dependence may be obtained more accurately if the analytical properties of O'a(b)g(k 2, Si) in the k 2 plane are used.

REGGE POLE METHOD

119

The analytical properties o f tra(b)z(k 2, Si) can be investigated 14) if it is expressed through the imaginary part of the corresponding zero angle scattering amplitude when the "mass" of the particle is x / - k 2 and is variable. The results are obtained via the Jost-Lehman-Dyson representation. It turns out that the analyticity region depends on si; t h e r e f o r e O'a(b)~(k 2, Si) cannot be split into two factors, one depending o n k 2 the other on sv Moreover, it is shown that aa(b)~(k2, si) should be positive for real k 2 and should not have singularities for k 2 > - 9 when s~ ~ 00. All these properties essentially restrict the class of admissible functions. If we also require that gab does not depend on energy for s -~ 00, it appears 1s) that aatb)~(k 2, st) cannot, owing to the above conditions, be the ratio of two polynomials in k 2. Therefore, if we put

st) =

1. s,)F(k

s,).

(3)

(with F ( - 1, si) = 1, of course), F will be of the form of exp to(k 2, si) (or of the sum of such exponents) where tO can be expanded in powers of k 2 (for small k2). Keeping only the two first terms of this series, we find that for small k 2 and large In s i all the above requirements are satisfied if the function F behaves asymptotically as F(k 2, sl) ~ In st e x p ( - y k 2 In s,),

(Ik21 << 1),

(4)

(7 > 0 is a constant). All the requirements can also be satisfiedbysomeother functions, for instance, i f F ~ In~ s~ e x p ( - ~k 2 Ina si), where ct and fl are connected by the relation 3fl = 2e + 1. However, it is exactly in the case ofeq. (4) that we come to the asymptotic behaviour predicted by the Regge pole method (see below). Thus, for increasing s (and hence si) the effective values of k 2 tend to k 2 = 0. Herefrom it can easily be found ~3) that for increasing s the effective values of the transversal component k ; decrease as well: k2eff -- (In s ) - ' .

(5)

Therefore the effective impact parameter increases as x/ln s. Of course, we still put tr~tb~(--1, Sl) = a~(b)~ = const.

(6)

3. Elastic Scattering The elastic scattering a + b --* a + b due to the above inelastic processes is given by a two-meson graph (fig. 2). The square of the momentum transfer t =

Fig. 2.

D.S. CHERNAVSKYet aL

120

- (k I + k2) 2, for increasing s, should also tend to zero, as It] ~ (In s ) - 1: in fact, according to eq. (4) aatb),(k 2, Si) is not small only i f k 2 < 1/~ In si, and since it is an integral o f [Aa(b)~(k ~, s,, qj)[ 2, the function A~(b). cannot be large either beyond this region. Let us now assume Aatb). to be proportional to ~/F, i.e., Aa(b)~(k 2, s~, qj) = ~/F(k 2, s~) Aa(b), ( - 1, S~, qj). Then the unitarity condition leads (for - t << l, - t In s ~> 1) to the relation 13) I m Aab(S, t) ~ S e x p ( - k T t In S)a..ab, ,.

(7)

The elastic process described by this f o r m u l a has the same properties as those given by the Regge pole method.

4. Inelastic One-Pion Exchange Let us now turn to the relations between the total cross sections for different kinds o f particles. These relations m a y be obtained f r o m the one-meson approximation under the additional assumption that the function F(k 2, si) is universal, i.e., that it does not depend on the kind o f particles a and b. This universal character corresponds to that o f the pole trajectory in the R P M , which is quite natural. In fact, the analytical properties o f 0"a(b),~(k 2, Si) investigated in ref. 14) are practically independent of the nature o f strongly interacting particles. Moreover, the asymptotic cross section aab(S ~ ~ ) should also be constant for any real particles. Therefore, the analysis of ref. 13) holds for any strongly interacting particles a and b, which leads to a universal F(k z, si). In this case, substituting eqs. (3) and (6) into eq. (2) and taking the constant factors aa~ and ab~ out o f the integrand, we obtain O'ab = O'anO'bn J ,

(8)

where J is a constant which does not depend on the kind o f particles a and b (their masses enter the lower integration limits, but their effect vanishes asymptotically). F o r a = b = rc we find J = 1/a,, (this relation defines y through a,,). N o w we come to the well-known relations for cross s e c t i o n s w h i c h were obtained by the Regge method 15). Putting, for instance, (1) a = b = N, (2) a = b = K and (3) a = N, b = K, we have, for nucleons (N) and K-mesons, GNN~nn ~ ~2n,

(8a)

UKK O'nn .~- 0"2n,

(8b) (8c)

O'KNO'nn ~ O'nNO'nK,

f r o m which it follows t that =

(8d)

t Similar results may be obtained in the many-peripheral model which is a particular case of the OPA. We learnt this from a preprint of D. Amati, S. Fubini and A. Stanghellini (1962), which we saw after this paper had been written.

REGGE POLE METHOD

121

The universality of F opens new possibilities within the one-meson approximation: after determining (or estimating) F from some inelastic process (from N N collisions, for instance) we can apply it to other processes (e.g. to K N collisions). 5. O t h e r I n e l a s t i c P r o c e s s e s

Let us consider other possible inelastic processes besides the one-pion exchange. (a) To begin with, such processes may result from the exchange of a single particle other than a pion, e.g. K-meson. Calculations perfectly similar to those for the pion exchange a ~ ) lead to the total cross section due to the one-K-meson exchange ,~(K) Vab differing from "ab~(~)by a factor (D(k 2) = D~(k 2) and D t ( k 2) are the n and K-meson propagators): ab

~---

O'aK GbK

If we assume that DK(O)/D~(O) ~ m K a (as is the case if the non-pole terms of the propagators are small for k 2 ~ 0 ) , and O'a(b) ~ ~ aa(b)K, then Uab _(K) ~ 10-2 ~ ,,(~) b , i.e., the K-meson exchange contribution may be neglected. (b) According to the current point of view, a one-pion inelastic interaction leads to a two-pion elastic one which is treated in the Regge method as being due to the exchange of a single "vacuum particle". It seems natural to investigate inelastic processes, due to the exchange of a "vacuum particle". This was in fact done in refs. ~6) and ~7). These processes should be treated as diffractional inelastic or quasi elastic processes. Hypothetically, it may be anticipated that the cross section for such an inelastic process decreases with energy. In general, the character of such inelastic processes essentially differs from that of inelastic processes due to one-pion exchange. (c) Of considerable importance is the problem of the relative part played by multimeson exchanges. An example is given by the Fermi-Landau hydrodynamical multiple production. As is shown in ref. 5), this process does not have the properties of a one-particle exchange scheme, and hence elastic scattering due to hydrodynamical inelastic scattering cannot be involved in the RPM, according to the present point of view. In ref. 5) it is shown that experimental data even for very high energies of 10 3 to 105 GeV seem to point to multiple pion-exchange inelastic processes. If this is supported by further experiments, it will be clear that the RPM does not hold for all observed interactions. 6. C o n c l u s i o n

Thus, the RPM and OPA are different aspects of the same phenomenon. This correspondence of the two methods enables us, first of all, to specify the quantities entering the OPA (a(k 2, sl), for instance) and thus to make the predictions of the OPA concerning inelastic processes more definite; and secondly, to use the information on inelastic processes to check the RPM. To some extent this was done in ref. 5). However, in our opinion, this approach is by no means exhausted.

122

D.S. CHERNAVSKY et al.

References 1) V. N. Gribov, JETP 42 (1962) 1260; 41 (1961) 1962 2) V. N. Gribov and I. Ya. Pomeranchuk, preprint of the Institute for Theoretical and Experimental Physics (ITEP) No. 43 (1962) 3) G. F. Chew, preprint UCRL No. 10058 (1962) 4) Yu. D. Bayukov, N. G. Birger, G. A. Leksin and D. A. Suchkov, preprint ITEP No. 63 (1962) 5) E, L. Feinberg and D. S. Chernavsky, preprint of the Lebedev Physical Inst. Acad. of Sci. (t962) 6) I. I. Royzen and D. S. Chernavsky, preprint of the Lebedev Physical Inst., Acad of Sci., also JETP (to be published) 7) I. M. Dremin and D. S. Chernavsky, JETP 38 (1960) 229 8") F. Salzman and G. Salzman, Phys. Rev. 121 (1961) 1541 9) E. Ferrari and F. Selleri, Phys. Rev. Lett. 7 (1961) 387 10) G. F. Chew and F. E. Low, Phys. Rev. 113 (1959) 1640 11) V. B. Berestetsky and I. Ya. Pomeranchuk, JETP 39 (1960) 1078 12) I. M. Dremin and D. S. Chernavsky, JETP 40 (1961) 1333 13) I. M. Dremin and D. S. Chernavsky, JETP 43 (1962) 551 14) I. M. Dremin, JETP 41 (I961) 821 15) V. N. Gribov and I. Ya. Pomeranchuk, JETP 42 (1962) 1141 16) V. N. Gribov and I. Ya. Pomeranchuk, Phys. Rev. Lett. 8 (1962) 343; G. Domokos, JETP 42 (1962) 538; M. Gell-Mann, Phys. Rev. Lett. 8 (1962) 263 16) V. N. Gribov, B. L. Ioffe, I.Ya. Pomeranchuk and A. P. Rudick, JETP 42 (1962) 1419 17) K. A. Ter-Martirosyan, preprint ITEP (1962)