D. Amati, S. Fubini, and A. Stanghellini

Transcription

1 560 Session H 1 LIST OF REFERENCES 1. T. Regge, Nuovo Cim. 14, 951 (1959). 2. V.N. Gribov, JETP, 41, 1962 (1961); 41, 667 (1961). 3. M. Jacob, G. C. Wick, Ann. of Phys, 7, 404 (1959). 4. M. E. Rose "Elementary theory of angular momentum", New York A. J. Macfarlane, Rev. Mod. Phys. 34, 41 (1962). 6. G. C. Wick, Ann. of Phys., 18, 65 (1962). 7. R. E. Cutkosky, Journ. of Math. Phys., 1, 429 (1960). 8. J. S. Ball, W. R. Fraser, M. Nauenberg. preprint, April V.N. Gribov, I. Ya. Pomerancbuk, JETP, 42, 1141 (1962). 10. M. Gell-Mann, Phys. Rev. Lett. 8, 263 (1962). 11. V.N. Gribov, B. L. loflfe, A. P. Rudik, I. Ya. Pomeranchuk, preprint, May T H E O R Y O F H I G H - E N E R G Y S C A T T E R I N G A N D M U L T I P L E P R O D U C T I O N D. Amati, S. Fubini, and A. Stanghellini CERN, Genève (Invited paper presented by D. Amati) We wish to report about the different consequences of a model for high-energy interactions 1} in the investigation of which Bertocchi, Ceolin, Duimio and Tonin collaborated with us. This model has been suggested to us by the structure of the strip approximation to the Mandelstam representation 2 ) and can be simply understood as a generalization to very high energy of the peripheral model. The basic idea is that the main contributions to multiple production are given by a combination of a large number of low-energy processes. The graphs we are considering are shown in Fig. 1. Each bubble represents a low-energy two-body process. The number of multiperipheral graphs does, of course, increase with increasing energy. We wish to show that the sum of all multiperipheral effects exhibits, in the high-energy limit, particularly simple features, both for elastic scattering and for multiple production. It is clear that the knowledge of the production amplitude allows to compute not only the production cross-section, but also the imaginary part of the elastic scattering amplitude through the unitarity relation Fig. 1 In fig. 2 are shown the multiperipheral diagrams that give the elastic amplitude. They represent the shadow scattering of the multiple production as given by the multiperipheral model.

2 High energy physics (Theoretical) 561 Fig. 2 Let us consider the sum of all multiperipheral effects giving rise to A. This sum can be performed by making use of a recurrence relation which allows to compute the (n+1) contribution, once the n,h is known. This recurrence relation is visualized in Fig. 3. cross-section, whereas it will be shown that the forward off-mass shell amplitude A{s, u, u, 0) leads to predictions concerning the average asymptotic properties of high energy multiple production. In the asymptotic limit (high multiplicities), the integral equation reduces considerably. First, the term A 0 can be dropped and the kernel turns out to depend only on the ratio s'/s, so that the equation is invariant under the transformation s-+cs, s'->cs'. This allows us to factorize the s dependence of the amplitude in the simple form Fig. 3 Eq. (2) shows that A n+i can be computed only when A n is known not only on the mass shell, but in correspondence to all space-like values of the fourmomenta q, q', The recurrence formula does indeed allow to calculate all multiperipheral contributions in terms of the low-energy one A 0. This procedure can be summarized by means of the integral equation where The problem is then reduced to the solution of a homogeneous integral equation for (p(uv, t), whose solution determines both the exponent a(t) and the eigenfunction q>(uv, t). Both eigenvalues and eigenfunctions have a physical meaning: the eigenvalue determines <x(t) and gives therefore the well-known shrinking of the diffraction peak, whereas the eigenfunction as already pointed out is connected with the average properties of multiple production. The detailed analysis of the integral equation 3 ' 4 ) allowed us to obtain approximate solutions for a and (p. It was found indeed A form of the scattering amplitude analogous to Eq. (4) has been obtained by many people by adapting to high-energy scattering the results of Regge in potential theory. This analogy can be understood by considering that our multiperipheral graphs, observed in the crossed channel, are the relativistic analogue of the different iterations of the potential model used by Regge. The predictions obtained by means of the model can be divided into two categories : THE INTEGRAL EQUATION The knowledge of the solution A(s, u, v, t) of the integral equation is the fundamental problem of our work. Indeed, the on-mass-shell amplitude A(s 9 ft 2, - /i 2, i) leads to the elastic diffraction (a) many general trends of the high-energy collisions do only depend on the transformation property of the integral equation, which is a consequence only of the topology of the multiperipheral graphs. (b) the specific numerical answers (like, for example, the value of the total cross-sections) do depend, of course, on the choice of A 0 and on the manner in

3 562 Session H 1 which A 0 is continued off the mass shell. As is generally known, it has not yet been possible to find a completely satisfactory way of performing such a continuation, especially in the case of higher waves. We have therefore concentrated our attention on the general model independent predictions, which we shall try to summarize now. If we wish to obtain the average spectra of secondaries, i.e., which is the number dn s (k) of secondaries with 4 momentum k, what must be done is to compute how many secondaries in any multiperipherism can have such a momentum and then sum over all multiperipherisms. The procedure is indeed simple and can be visualized in Eq. (6) and Fig. 4: 1. Elastic amplitude The high-energy behaviour of the scattering amplitude T(s, t) is given by where the first sign stands for amplitudes symmetric under crossing (s<r^s) as, for instance, absolute elastic scattering, while the second sign stands for antisymmetric amplitudes under the crossing. The exponent for the charge exchange amplitude is always smaller than the one for the purely elastic one. Eq. (5) turns out to be independent of the scattering particles, apart from the value of C(f). The C{t) can be factorized in such a manner that the relation between different amplitudes (dominated by the same pole) is the following: where x, y, z and w represent any kind of particles. 2. Inelastic scattering The average properties of multiple production are also easy to obtain. To find an average value, for instance, we must perform the sum over all multiperipheral processes with suitable weighting factors. The multiplicity, for instance, shall be given by due to the fact that the n ih peripherism gives just n+l final states. The expression for <N> is indeed extremely simple to obtain: Fig. 4 The actual evaluation leads to where M 2 represents the mass, k r the transverse momentum, and E lab the lab. energy of the secondary in question. Eq. (7) has the feature that the spectrum of mass and transverse momentum is independent of both the incident and secondary energies (as suggested by experiments). The energy spectrum (for not too high or too low energies) is given by The transverse spectrum is strongly peaked in the forward direction, and its actual shape F(k?) is model dependent. Another result is that both the inelasticity (energy released by the incident nucléon) and the branching ratios between different secondaries are energy independent. 3. Relation between asymptotic properties and bound states All our equations, established for t^o, can be continued to positive t. At t = 4/i 2, a develops an imaginary part. Eq. (7) shows already that, for the

4 High energy physics (Theoretical) 563 value t = t B for which a(t B ) is an integer, Re T develops a pole. This means that the particles exchange a bound state (or resonance if t B >4fi 2 ) of mass t B and of spin oc(t B ). This relation between asymptotic properties of scattering and bound states can be understood without making appeal to polology. It is possible to show, in fact, that our fundamental integral equation for a integer, written for t>0, coincides with the Bethe-Salpeter ladder equation for a bound state of angular momentum a. 4. Cuts in the angular momentum variable The weakest point of the whole approach is that unitarity in the s channel is not taken into account. A similar difficulty appears in all approaches based on the Regge theory, since in the potential model s means only momentum transfer, and therefore unitarity in this channel has no meaning. Now, Froissart has shown that unitarity in the s channel, together with the Mandelstam representation, do not allow cross-sections increasing more rapidly than (log ^) 2. So this means that Eq. (4) is only acceptable for a(0^1, for t^o. Now both our equation and the potential model do not give any such limitation for a(t). In order to have an idea of the effect of unitarity, we have tried to take into account the corrections due to elastic unitarity corresponding to effects of multiple scattering or, in other words, corresponding to the shadow scattering of the elastic scattering given by (5). These new terms show a new interesting feature; they give rise to continuous distributions of powers (cuts in the angular momentum') : The position of the upper limit of integration for t = 0 is simply given by: So the dominance of the pole on the cut depends on whether a(0) is larger or smaller than 1. For a(0)>l the cut dominates on the pole, so the whole theory breaks down, as predicted by the Froissart theorem. In the a(0) = 1 case, which seems to be suggested by experiment, the cut and the pole coincide. We wish to emphasize that we have not proved the existence of cuts in the angular momentum, but have just given arguments in favour of their existence. It is, however, interesting to note that these cuts enter in the theory in a non-trivial manner, and that they are a possible mechanism by which unitarity leads to the Froissart limitation in the cross-section. In conclusion, we have seen that in the framework of our work, many characteristics of high-energy physics show simple regular features. One of such features concerns the Regge pole behaviour of elastic scattering. We want to stress, however, that many other characteristics as, for instance, multiplicities and spectra of production processes, show analogous regularities which in our opinion have exactly the same physical basis. A very interesting confirmation concerning of the results relativistic elastic scattering has been obtained independently by B. Lee and R. Sawyer 5). These authors have succeeded in extending to the Bethe- Salpeter equation the methods of Regge and Blankenbecler-Goldberger for Yukawa potential scattering and have proved that the amplitude is meromorphic in the complex angular momentum half-plane Re />. An expression for a(t), in the case of weak coupling, is obtained, and coincides in that limit with the one obtained in reference 4). LIST OF REFERENCES 1. Amati, D., Fubini, S. and Stanghellini, A., Nuovo Cimento (in press), Physics Letters 1, 29 (1962) and CERN report TH 264(1962). 2. Amati, D., Fubini, S., Stanghellini, A. and Tonin, M., Nuovo Cimento, 22, 569 (1961). 3. Ceolin, C, Duimio, F., Fubini, S. and Stroffolini, R., Nuovo Cimento, (in press). 4. Bertocchi, L., Fubini, S. and Tonin, M., Nuovo Cimento 25, 626 (1962). 5. Lee, B. and Sawyer, R. (preprint).

5 564 Session H 1 DISCUSSION MATTHEWS: This group of Italian theoreticians was in the business before Regge became popular and I think that they can be congratulated on the ingenious way in which they got on to the band wagon... Any questions? OEHME: I would like to add a remark concerning possible branch point trajectories in the complex /-plane. A possible branch-point trajectory in the /-plane corresponds to a branchpoint in the energy (s) which depends upon /. One may approach the problem by trying to continue the inelastic unitary condition into the complex /-plane. As a simple example, take a 3-particle intermediate state where two particles are correlated; take the angular momentum of the correlated pair in its own cm. system to be equal to /, i.e., the total angular momentum. As one continues from large values of Re / to smaller ones, a single particle pole s = m 2 (I) comes out of the elastic threshold (4 /i 2 ) and simultaneously a branch-point s = (w(/)+//) 2 comes out of the three particle threshold (s = 9/i 2 ). This does not prove the existence of moving branch-points, because one may also leave the angular momentum of the correlated pair quantized. The remaining kinetic angular momentum is then continuous, and one has a situation which is very similar to a two-channel problem in the potential scattering: one treats the correlated pair essentially as an elementary particle and hence one finds only the branch-points corresponding to integer angular momenta. I do not know at present what is the correct continuation. MANDELSTAM: DO you propose using this equation for actually getting numbers out, or rather just for finding the general properties of high-energy scattering? AMATI: In the framework of the theory, the importance of the cuts with respect to the poles depends very much on the value of a (0), For us this appears very reasonable because we expect that the cuts could ensure consistently the Froissart limit a^l. In this sense it is clear that a (0) = 1 is a limiting point. 1 would say that for the moment we would not give any particular attention to the actual numbers obtained by the iterative method up to the second iteration for what regards diffraction, but just only as a qualitative indication. MANDELSTAM: May I make another remark regarding these cuts in the angular momentum plane? If you actually look and see what happens in the unphysical region, for positive rather than negative t, that is if you start with something going up like t a with a>\ and iterate the diagrams, the output has a worse asymptotic behaviour than the input. If we then repeat this procedure and try to calculate the double-spectral function by the standard method we essentially get a double spectral function with an infinite number of subtractions. Chew, Frautschi and I came to the conclusion that probably it indicated an inconsistency in the equations, and one should be careful before deducing consequences from them. AMATI: I realize this and I have discussed it with Chew. I believe that what happens is that we are looking at a different region; you are looking to the spectral function while we are looking at the actual function for a negative value of /. I believe that your and our iteration procedures are not each other's continuation. In fact, the simple continuation of our " pole " and " branch-point " lines show that for positive t the first of them always dominates the second one. Low: There is a further difficulty associated with the cuts Amati mentioned, if they are not cancelled by inelastic diffraction scattering. That is, as soon as t becomes infinitesimally log (log $} large (like, in the asymptotic limit) they become logs larger than the assumed form s a (0 and so invalidate it. The cancellation is, therefore, very important if one wishes to be able to determine a (t) for finite t, such as t 50/i 2. In case of a failure of the cancellation, one could only determine a'(0). AMATI: Yes, we have realized it. We will not believe in the actual contribution of the " cuts " to diffraction when this is the situation. Now we are investigating (for the moment we have no definite answers) what can be the self-consistent method to bring cuts and poles into the game and playing with them together from the beginning. This is a rather complicated problem and what we have seen is that one characteristic that can come out from there is that the pole trajectory would be renormalized by the cut one, and perhaps in such a way so as to overcome the cut. COCCONI: I would like to make two remarks, not theoretical but experimental. One is that, it was stated by Amati that the spectra of the secondaries should go as 1/2T. Actually the evidence is that they go down exponentially. The second remark is that if one looks at the extreme high-energy cosmic ray evidence, one has the impression of a strong coherence of particles as if they come out from two single centres, two fireballs, and not through many small centres. AMATI: Regarding the first question: remember that our secondaries are low energy systems and not pions, therefore still you must do a sort of convolution in order to obtain the spectra of pions and this would change somewhat the behaviour of the curve. Besides, I said that we do not expect the 1 JE law to be valid over the whole spectrum: it should break down when E approaches a reasonable fraction of the primary energy (let us say x / 1 0 ). In regard to the second one, I think that the interpretation of cosmic ray events is very controversial in this sense. I have just seen some results of Hasegawa which will be presented in this conference: they have done a fit of all the cosmic ray events obtained up to now and their point of view is that, in some cases, you need at least 6 fireballs. I would like to wait for more experimental data. By the way, I would say that cosmic rays can still give very interesting information.