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In fact one can give a more general version of the above remark [43]: The transformation (5.6) may be replaced by a general transformation which preserves normalization and orthogonality

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and the assumption made that all nonzero a are of the same order of magnitude. It introduces "democracy" among all physical states and treats the ground state of the incident particle on the same footing as the excited states. This makes good sense if one believes (following Good and Walker [23]) that in the high energy limit the ground and excited states of the incident system (e.g., a pion) are considered to be approximately degenerate.

The formula which gives all diffractive elastic and production amplitudes is

(f| T(A) | i) =

Σan*an [ db exp (i▲·b) {1— exp [— 1⁄2o‚AT (b)]},

ik 2π เ


where the "total cross sections" σ are free parameters. In order to obtain (5.8), e.g., we have to make all σ approximately equal to σ, (they cannot be exactly equal to σ, because we would have no production!). When we do that, we have approximately (due to (5.9)) the elastic amplitude of the form (5.8), and the production amplitudes become approximately (again due to (5.9); for an explicit example see (5.7)),

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Since o≈≈o, we obtain the result obtained in experiment [S4].

Let us make two comments to close this "strong coupling" description of diffractive production which, in contrast to the "one step" description, is not a perturbative approach:

(i) In the "strong coupling" model there is an internal relation between elastic and production processes, whereas in the "one step" picture there is none. To put it differently, a definite relation between attenuations in the entrance and exit channels exists in this model.

(ii) The "strong coupling" description implies the following general prediction: In all coherent diffractive production processes where there are strongly interacting particles in the entrance and exit channels, and diffractive production processes are much weaker than elastic scattering amplitudes, one should see comparable attenuations in the entrance and all-exit channels. Before indicating other possibilities of interpretation of the low absorption effect, let us consider an example of a two component system (such as was described before, e.g., by eq (5.6)) penetrating a piece of nuclear matter.

The following remark about penetration through a sequence of thin slabs of nuclear matter is in order here.

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Assume that p (the density) does not depend on the transverse coordinate b (each slab extends to infinity in the b plane). The profiles of the elastic scattering amplitudes for one slab are

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Let us consider the attenuation of the incident wave, eikz: After A slabs we have

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The exponentials give the attenuation. This is the attenuation in the case of strong coupling between | 1) and 2). Then, let us take the case where | 1) scatters only elastically (hence we forbid the intermediate states | 2)). Then the attenuation of the incident wave is

(ii) eik z (1 − (1 | T | 1 )) ^ = e ik z[1 − 1⁄2((Ï | г | î )+(2 | г | 2 )) ]a

=eikz exp[-14 (i+σ2) p2].

When σ =σ, the attenuation in case (i) is the same as in case (ii), but in the case oσ the situation changes. Take, e.g., o=340, o=140. Then the attenuation factor in (i) is

(i) 11⁄2 (e-(3/8)σT+e−(1/8)σT).

Instead, in case (ii) it is

(ii) e-(1/4) T.

Hence, for large enough T, the attenuation in (ii) is stronger than the attenuation in (i). A very important conclusion follows from this observation: by introducing strong coupling between the initial state of the incident particle and some other states one may reduce considerably the absorption of the initial state in nuclear matter.

One can put it differently: the effect of allowing intermediate states to occur during the multiple scattering process may be an increased penetrability (or decreased absorption). Hence the so-called inelastic shadowing effect increases the penetrability of a specific component.

We have considered only the 2×2 case but one can consider much more involved systems (which contain more than two components). Such a scheme was developed by L. Van Hove ([44], see also [41]).

There is, therefore, still another possibility of explaining low absorption of the 37 (5π) systems produced coherently on nuclei: to consider it a "one step" process but to assume that the produced object, 3π (5π), is a superposition of several strongly coupled channels. This coupling may reduce the absorption of a 3 system as we have seen on the example quoted above. Graphically such process would look as follows:

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This is, at the moment, the most commonly considered model of coherent diffractive production. In order to have a satisfactory solution, one would have to know precisely the superposition of states which form the produced 3′′ (5) system and their interactions with the nucleons of the target. In other words, one has to specify the structure of the produced object. No one has, so far, proposed a model detailed enough and convincing enough. We shall give a few more details after making the following remark.


One may also look at the problem of the penetration of a many component system through nuclear matter as a problem of diagonalization of the "profile-matrix." Let us again assume (for simplicity sake) that just two states are operative. Then the single scattering profile matrix is

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Hence the following transformation of the physical states diagonalizes the interaction

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Note that | Ĩ) and | 2) are the “eigenstates” of the scattering; they scatter only elastically.

Note also that even if the transition between 1→2 (or 2→1) is very weak we have a superposition of these physical states propagating through the nuclear matter, a superposition in which both these states are equally important.

Now let us go back and close our discussion of the coherent diffractive production by pions of 3 and 57 systems on nuclei by formulating more completely the "one step" description which allows for reduction of absorption for the produced composite systems (L. Van Hove [44], C. Rogers and C. Wilkin [41], A. Białas and K. Zalewski [45]). It is a direct generalization of the multiple scattering we have given above for the standard "one step" amplitude.

We want to keep the "one step" production model, hence we assume the incident particle ground state | 1) to be "well separated" from a set of excited states | m) from which the system | 2) emerges. On the mass scale

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One can perform the sum S by diagonalizing (m' | T | m). In an abstract form:

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The statesm) are physical states. The states | ã) are their linear combinations. (E.g., in the process of K1→K, regeneration they are degenerate and identical with | Ko) and | Řo)). In general they do not have a well defined energy. The states | a) are decoupled, that is to say only diagonal elements (aa) are different from zero. Thus

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The attenuation of the outgoing beam is small when the λa's are small. So, in this approach the problem is to construct a physically plausible matrix (m' | I | m) which produces the correct eigenvalues a. Hence one has to go deeply into the structure of the produced object. There is no commonly accepted model of such a structure but there are many examples ([41], [44], [45], [S4]) which show that one may obtain the desired low absorption in many ways.

6. Shadowing Effects in Inelastic Electron-Nucleus Scattering

Experiments performed to see shadowing effects in inelastic electron-nucleus scattering failed to show it-see H. Kendall in ref. [S5]. Inelastic electron scattering cross sections were measured at: 0=6°, the incident electron energies were 4.5, 7, 10, 13.5, 16, 19.5 GeV and the energy losses, v, were 0.1 Gev <<17 GeV. The targets were Be, Cu, and Au nuclei. They plotted

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All data points were consistent with S = 1 for all momentum transfers and energy losses (although the errors were quite large).

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