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The previous case dealt with an elementary object scattering from a composite object. We already saw in the examples of scattering of relativistic particles from external electromagnetic fields that "internal structure" (in these cases the internal spin quantum numbers + anomalous magnetic moment) breaks down the "ansatz" of additivity of phase shifts. We can also have a look at this problem from the point of view of a Glauber-like description of scattering of two composite objects. The formulae given below are interesting also because they may be used to analyze high energy nucleus-nucleus collisions (which is not an academic problem because there are experimental projects under way).
The geometry of the process is shown in figure 3.
For the sake of simplicity let us take the wave functions of (a) and (b) in the form of products of single particle wave functions. Let us assume also that all particles have the same single particle wave functions. The ground state wave functions are:
What kind of formulae would we have if the "ansatz" of additivity of phase shifts of the composite system (b) colliding with nucleons of (a) were valid? Let us look at the profile of the jth nucleon:
which is the profile for elastic scattering of (b) from the jth nucleon of (a) and a two-dimensional
p (s) = ƒ ** dz 4o* (r) ço (r).
This is different from (3.2). What is the difference? First let us note that (3.2) is a sum rule. For instance, we can extract from (3.2) the following contribution of the second order
Hence the formula (3.2) sums over all intermediate excited states. For instance, the above contribution gives:
ALL POSSIBLE EXCITED STATES
Suppose we reject the intermediate excited states and take only the ground state as a possible intermediate state (this is the way to eliminate all channels but one). Then each y; can be averaged over r1(b):
(to make it identical to (3.3) we should substitute s,a)→→s;(«)). Hence we get a formula which follows from additivity of phase shifts.
It would seem, therefore, that indeed "compositeness" of the incident particle is decisive in destroying or satisfying additivity. The other "moral" is that if we know the structure of the composite body (b) we may still use a generalized Glauber model with additivity of all possible phase shifts of the pairs of components of (a) and (b).
Let us consider some limiting cases of eq (3.2) (compare ref. ). Let the radii of the two composite objects be R. and Rь. The calculations of ref.  show that the smaller is R, the nearer we are to the additivity of (b)-nucleon phase shifts. But that means that this additivity improves with increase of the binding of (b). Of course for R-0 the additivity becomes exact. One can see this explicitly by replacing for (b), pb) (s)~(2) (s) (then (b) is a point-like object). When R→0 we in fact remove all the intermediate excited states already mentioned: In this case
Having written down the formula (3.2) this is a good place to discuss it a little further. As we have already said, it would be very interesting to test formulae of the type (3.2) against some experimental data. There is, however, very little data in existence to analyze. To the best of my knowledge, only deuteron-deuteron scattering data are available, but reliable calculations are very difficult because of the high spins involved. Nevertheless, there exist some calculations [S1] and there seems to be reasonable agreement between theory and experiment. But we shall talk about comparison with experiment at other occasions.
Some special cases of formula (3.2) were also employed to describe hadron-hadron scattering in the high energy limit. For example, the limit when A and B become very large was considered  (compare also ):
where the p's were defined before and we assume that all y's are the same. One gets this formula trivially from
as AB→∞. But this formula has no intermediate excited states, neither of (b) nor of (a). So, the Chou & Yang  limit A, B→ looses all excited state contributions and becomes (3.4). Equation (3.4) gives the well-known "droplet model"  elastic scattering amplitude of two composite objects whose hadronic matter distributions are given by pa) (s(a)) and p(b) (s(b)).
If one assumes that y(b) is a very narrow function of b (hence the components of the two hadrons are very small) we can write
[ [ d3sp) (8) p) (b+s)
d2g(a)ď2s(b)p(a) (s(a)) y (b−s(b)+s(a))p(b) (s(b)) = k
where K is a free parameter.
(2x) 2 √ dq exp (−iq•b) F (a) (9) Fo (9),
If we accept that the densities of hadronic matter are the same as charge densities, F(a), F(b) are the charge form factors of the colliding hadrons. This formula was used successfully to:
(i) reproduce the proton charge form factors from elastic scattering hadron-hadron cross sections.
(ii) predict diffractive structure (e.g., diffractive minima) of the high energy hadron-hadron collisions.
The very recent measurements of p-p elastic collisions confirm the existence of such a structure (CERN-Serpukhov experiment).
In the form given above, the droplet model is very crude and I do not want to go beyond this qualitative description. One should perhaps mention at this point that the amplitude (3.4) contains the geometric shape of the colliding objects (e.g., their transverse density distributions). If these geometric characteristics do not depend on energy, one gets the total cross section (from the optical theorem) which is energy independent. So, it seems to be difficult to reconcile this model with the recent evidence for the increase of the total hadron-proton cross sections at very high energies (compare the data e.g., analyzed in ref. ).
Selection of formulae taken from a standard partial wave expansion [S6]
Let us first consider the "elementary" collisions (whose scattering amplitude is determined by the profile (b)). As the wave passes a scatterer it gets modified by a factor 1-y(b). Hence, the probability that the particle gets removed from the incident beam is 1— | 1—y (b) |2 = 2 Rey (b)(b) 2 (at the impact parameter b). Notice that here we use the same expression as
in the following paragraphs: we identify 1–7 with y, and 1— | ʼn |2 = 1− | 1—y |2. Hence,
ƒ ď2b[2 Re y (b) — | √ (b) |3].
As oldby (b) 12, (see Remark below) we have
Consider the case when one "elementary" particle scatters from a "composite" nucleus. In this case the profile is
because, due to the same arguments as before, 1-1-(r) 2 gives the probability (at the impact parameter b) of losing the incident particle from the elastic channel. It is convenient however to